Microwave Semiconductor Devices
Theory, Technology, and Performance
0101
2006
978-3-8169-7202-0
978-3-8169-2202-5
expert verlag
Johann-F. Luy
The book starts with the basics of semiconductor physics required for the understanding of high frequency semiconductor devices. The active two-terminal devices are well suited to study the occurence of drift and diffusion mechanisms, heat conduction problems, loss mechanisms, millimeter wave semiconductor technology and packaging aspects. Impedance matching and application examples conclude the section on Gunn and transit time devices. The physics of Schottky contacts and their application to design and properties of Schottky diodes is discussed in the second focal point of this part. Detector and mixer circuits with Schottky diodes play an important role up to THz frequencies. Recently investigated two terminal devices employing heterostructures is considered. The second part is entirely devoted to three terminal devices. The bipolar transistor and its basic properties and limits with respect to high frequency operation are discussed. Questions of technology as well as modelling are adressed. The heterojunction bipolar transistor then can be understood as a natural bypass of these limits. The extraction of characteristic transistor parameters at high frequencies is of importance for the performance evaluation of the different transistor types and therefore treated in more detail. Field effect transistors are introduced and the metal semiconductor field effect transistor is discussed in detail. The family of heterostructure field effect transistors and especially the high electron mobility transistor is explained. Application examples and links to latest research activities are given. Aspects of multiplier and switching devices conclude this second part.
<?page no="0"?> Microwave Semiconductor Devices JOHANN-F. LUY <?page no="2"?> Johann-F. Luy Microwave Semiconductor Devices <?page no="3"?> Bibliografische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http: / / dnb.dnb.de abrufbar. © 2006 · expert verlag GmbH Dischingerweg 5 · D-72070 Tübingen Das Werk einschließlich aller seiner Teile ist urheberrechtlich geschützt. Jede Verwertung außerhalb der engen Grenzen des Urheberrechtsgesetzes ist ohne Zustimmung des Verlages unzulässig und strafbar. Das gilt insbesondere für Vervielfältigungen, Übersetzungen, Mikroverfilmungen und die Einspeicherung und Verarbeitung in elektronischen Systemen. Alle Informationen in diesem Buch wurden mit großer Sorgfalt erstellt. Fehler können dennoch nicht völlig ausgeschlossen werden. Weder Verlag noch Autoren oder Herausgeber übernehmen deshalb eine Gewährleistung für die Korrektheit des Inhaltes und haften nicht für fehlerhafte Angaben und deren Folgen. Internet: www.expertverlag.de eMail: info@verlag.expert CPI books GmbH, Leck ISBN 978-3-8169-2202-5 (Print) ISBN 978-3-8169-7202-0 (ePDF) <?page no="4"?> Preface The wireless communication technologies are the first huge mass market for the application of high speed, microwave semiconductor devices. In the hand-helds and in the base stations for the cellular communication standards GSM and UMTS there is a requirement for devices for the generation of microwave power, for switching and for detection and mixing of signals at microwave frequencies. These mobile communication systems work at frequencies around 2 GHz and there is still research ongoing on the improvement of device characteristics as noise performance or efficiency and realizations in cost effective material systems as silicon. Applications at frequencies between 2 and 10 GHz are close to market: wireless local area networks at 2.4 GHz and around 5 GHz, new satellite navigation systems, and satellite radio services. Also several military systems operate at X-band frequencies around 10 GHz. At higher frequencies we enface applications like satellite TV (around 12 GHz), future wireless local area network standards (WLANs at 17 GHz), point to point and point to multi point links (29, 38 and 60 GHz), last mile video transmission systems (at 42 GHz) and new mass markets caused by automotive applications: radar sensors at 24, 60 or 76/ 79 GHz - even at operation frequencies beyond 100 GHz. All these applications require basically the same functionalities as required for the mobile communication frequencies in order to realize transmitters and receivers. The properties of the microwave semiconductor devices are decisive for the overall system performance. Sensitivity and conversion loss of a detector or mixer depend directly on the device properties as device junction capacitance and resistance. If new high frequency architectures for receivers shall be developed (e.g. digital receivers) the basic devices will be the same and the knowledge of their physics and corresponding performance will remain important to specify new architecture details. The phase noise of a microwave oscillator influences the signal to noise ratio of the transceiver and therefore the maximum possible range of a radar system. At millimeter wave frequencies the properties of actual devices do not only depend on the underlying semiconductor physics but they depend also on the technological details of the device fabrication and the packaging. It is essential for system designers as well as for device engineers to dispose of basic knowledge of the physics, properties, technologies and system implications of microwave and millimeter wave devices based on silicon and compound semiconductor devices. This book tries to serve these needs and is based on a lecture developed for master course students of high frequency technology. The book starts with the basics of semiconductor physics required for the understanding of high frequency semiconductor devices. Special attention is given to the differences in the coarse of the energy bandgaps in the different semiconductor materials and the resulting differences in the physical properties. The active two-terminal devices are well suited to study the occurence of drift and diffusion mechanisms, heat conduction problems, loss mechanisms, millimeter wave semiconductor technology and packaging aspects. Impedance matching and application examples conclude the section on Gunn and transit time devices. The physics of Schottky contacts and their application to design and properties of Schottky diodes is discussed in the second focal point of this part. Detector and mixer circuits with Schottky diodes play an important role up to THz frequencies. Recently investigated two terminal devices employing heterostructures will be considered. The second part will entirely be devoted to three terminal devices. The bipolar transistor and its basic properties and limits with respect to high frequency operation are discussed. Questions of technology as well as modelling are <?page no="5"?> Preface adressed. The heterojunction bipolar transistor then can be understood as a natural bypass of these limits. The extraction of characteristic transistor parameters at high frequencies is of importance for the performance evaluation of the different transistor types and therefore treated in more detail. Field effect transistors are introduced and the metal semiconductor field effect transistor is discussed in detail. The family of heterostructure field effect transistors and especially the high electron mobility transistor are explained. Always application examples and links to latest research activities are given. In a final chapter new device concepts are introduced. Basic knowledge of electronic devices is helpful to the reader, however not undispensable. At the end of the chapters some typical exercises and questions are given which shall help the reader to reconsider the material. The author hopes that this textbook is helpful to students and also to practicing engineers in the field of high frequency systems, who recognize, that a basic understanding of the device physics is necessary to improve the quality and reliability as well as the performance of future microwave systems. The author is indebted to M.S. Jung Han Choi, who did not only include the tutorials but also proofread the material and helped a lot to improve the manuscript. My thanks are also due to my son Theo who helped me to organize the material and my great acknowledgement is directed towards Sabine Kaiser and Kathrin Ruoff, who gave helpful support in the preparation of the manuscript. J.-F. Luy, Ulm October 2004 <?page no="6"?> Content 1. Basics .................................................................................................... 1 1.1. Basics of carrier dynamics ...................................................................... 1 1.2. Energy band diagrams ............................................................................ 3 1.3. Temperature dependence....................................................................... 7 1.4. Drift velocity ............................................................................................ 9 2. Transferred Electron Effect: The Gunn device................................. 11 2.1. Basics ................................................................................................... 11 2.2. Operation modes and design rules ....................................................... 16 2.3. Thermal properties of the Gunn element ............................................. 19 2.4. Techniques to reduce thermal resistances............................................ 23 2.4.1. Spreading of the active area ................................................................. 24 2.4.2. Use of heatsink with high thermal conductivity...................................... 25 2.5. Technology ........................................................................................... 26 2.6. Conclusions .......................................................................................... 30 2.7. Outlook ................................................................................................. 31 3. Transit time diodes ............................................................................. 33 3.1. Basics of the pn-junction ....................................................................... 33 3.2. Transit Time Effect................................................................................ 35 3.3. Injection mechanisms ........................................................................... 40 3.3.1. Impact ionization ................................................................................... 40 3.3.2. Thermionic emission ............................................................................. 48 3.3.3. Tunneling with avalanche injection ....................................................... 51 3.4. Loss mechanisms ................................................................................. 52 3.4.1. Space charge effects ............................................................................ 52 3.4.2. The Skin effect ...................................................................................... 54 3.5. The IMPATT diode ................................................................................ 58 3.5.1. Design constraints: current density limits .............................................. 65 3.5.2. Design constraints: matching considerations ........................................ 66 3.5.3. Pulsed mode ......................................................................................... 68 3.6. Integrated transit time devices .............................................................. 70 3.7. Self oscillating mixer operation ............................................................. 74 3.8. Discussion and Conclusions ................................................................. 76 <?page no="7"?> Content 4. Schottky diodes .................................................................................. 79 4.1. Schottky contact modeling .................................................................... 79 4.2. Basic structures of Schottky Barrier diodes for high frequency applications ........................................................................................... 84 4.3. DC Analysis of Schottky diodes ............................................................ 85 4.4. The Schottky diode detector ................................................................. 87 5. Heterojunctions and heterostructure diodes ................................... 93 5.1. Formation: Types of heterostructures ................................................... 93 5.2. Modulation doping................................................................................. 94 5.3. Strain and stress: Example Si/ SiGe ...................................................... 95 5.4. Semi-classical Heterojunction diodes ................................................... 98 5.4.1. Hot electron injection in Gunn devices.................................................. 98 5.4.2. Heterostructure IMPATT ....................................................................... 99 5.5. Resonant Tunneling ........................................................................... 100 5.5.1. Resonant tunneling diode operation and III-V realizations.................. 102 5.5.2. Silicon/ Germanium resonant tunneling structure ................................ 104 5.5.3. A full wave rectifier with a resonant tunneling diode ........................... 105 5.5.4. The quantum well injection transit time diode (QWITT) ...................... 106 5.5.5. Esaki tunneling diode with SiGe quantum well.................................... 108 5.5.6. Further tunneling diode applications ................................................... 108 6. The Bipolar Junction Transistor...................................................... 110 6.1. Basics of the Bipolar Transistor .......................................................... 111 6.2. The Ebers-Moll equivalent circuit ........................................................ 112 6.3. Current-voltage relations..................................................................... 113 6.4. RF modeling........................................................................................ 116 6.5. Frequency limits.................................................................................. 117 6.6. Technology ......................................................................................... 118 6.7. Power limits......................................................................................... 120 7. The Heterojunction Bipolar Transistor............................................ 123 7.1. Basics of HBTs ................................................................................... 123 7.2. Types of HBTs .................................................................................... 124 7.3. Experimental devices.......................................................................... 128 7.4. High frequency characterization.......................................................... 131 7.4.1. Current gain cut-off frequency............................................................. 132 7.4.2. Maximum oscillation frequency ........................................................... 133 7.5. Conclusions ........................................................................................ 136 <?page no="8"?> Content 8. The Field Effect Transistor............................................................... 139 8.1. Basics of MESFETs ............................................................................ 140 8.2. Principles of MESFET operation ......................................................... 141 8.3. Technology ......................................................................................... 142 8.4. Principles of MOSFET operation......................................................... 144 8.5. Equivalent circuit and cut-off frequency .............................................. 145 8.6. Conclusions and outlook ..................................................................... 148 9. The High Electron Mobility Transistor ............................................ 151 9.1. Basic HFET theory.............................................................................. 153 9.2. Types of HFETs .................................................................................. 154 9.3. SiGe HFETs........................................................................................ 156 9.4. Technological aspects ........................................................................ 158 9.5. High power FETs ................................................................................ 158 9.5.1. Concept of GaN HEMTs ..................................................................... 159 9.5.2. Device optimization ............................................................................. 162 9.5.3. Conclusions ........................................................................................ 164 10. Future Devices .................................................................................. 166 10.1. Heterostructure varactor diodes.......................................................... 166 10.2. PIN diodes and MEMS switches ......................................................... 169 10.3. The Resonance Phase Transistor....................................................... 174 List of Abbreviations ....................................................................................... 178 Index.................................................................................................................. 179 <?page no="9"?> Glossary Acceptor: An atom which is likely to take on one or more electrons when placed in a crystal Bandgap: The range of energies between existing energy bands where no energy levels exist Compensation: The process of adding donors and acceptors to a crystal Conduction band: Lowest empty or partially filled band in a semiconductor Conductivity: The ratio of the current density to the applied electric field Crystal: A solid which consists of atoms placed in a periodic arrangement Density of states: The density of electronic states per unit energy and per unit volume Diffusion: Motion of carriers caused by thermal energy Donor: An atom which is likely to give off one or more electrons when placed in a crystal Drift: Motion of carriers caused by an electric field Electron: Particle with spin 1/ 2 and carrying a single negative charge (1.6 x 10 -19 Coulomb) Energy band: A collection of closely spaced energy levels Energy level: The energy which an electron can have Fermi energy: The average energy per particle when adding particles to a distribution but without changing the entropy or the volume. Chemists refer to this quantity as being the electro-chemical potential Hole: Particle associated with an empty electron level in an almost filled band Impurity: A foreign atom in a crystal Intrinsic semiconductor: A semiconductor free of defects or impurities Intrinsic carrier density: The density of electrons and holes in an intrinsic semiconductor Ionization: The process of adding or removing an electron to/ from an atom thereby creating a charged atom (ion) Mobility: The ratio of the carrier velocity to the applied electric field Resistivity: The ratio of the applied voltage to the current Saturation velocity: Maximum velocity which can be obtained in a specific semiconductor State: Solution to the Schroedinger equation defined by a unique set of quantum numbers Thermal equilibrium: A system is in thermal equilibrium if every ongoing process is exactly balanced by its inverse. Valence band: Highest filled or almost filled band in a semiconductor Valence electrons: Electrons in the outer shell of an atom <?page no="10"?> 1. Basics 1 1. Basics 1.1 Basics of carrier dynamics In a semiconductor crystal the electrons move in a three-dimensional lattice. The wavefunction describes the motion of the electrons and is a solution of the Schrödinger equation. The squared modulus of the wavefunction gives the probability of the location. A general representation of a plane wave function is z) k y k x k ωt i ( z y x ( exp with the same amplitude everywhere, the angular frequency ω , and the wave vector k [1.1]. There is a definite relation between ω and k which can be visualized in the ω k diagram. Fig. 1: General appearance of a ω k diagram. The energy of the electron is ω W (with Planck´s constant) and the momentum is k p . The velocity with which the electron moves is called the group velocity ) / , / , / ( z y x g dk d dk d dk d v and the phase velocity is given by ) ω/ k , ω/ k , k (ω v z y x p / A general solution of the wavefunction will also include the structural periodicity P(r) of a crystal z)) k y k x k ωt i ( P(r)* z y x ( exp as indicated in Fig. 1: According to the Bloch´s theorem the ω-k relation is periodic in k if the structure is periodic. The periodicity is the inverse of that in space. If the crystal has a period a z then the crystal has the periodicity 2π/ a z and we do not have to worry about k values outside the periodicity interval. So it is sufficient to use only k values in a primitive cell of the reciprocal lattice. This cell is called the Brillouin zone or the Brillouin cell. -/ a z / a z 0 k <?page no="11"?> 1. Basics 2 Fig. 2: Brillouin zone for cubic semiconductors [1.2] The most important symmetry points and symmetry lines within the Brillouin zone are indicated in Fig. 2 and Fig. 3. The points (and lines) inside the Brillouin zone are denoted with Greek letters. The points on the surface of the Brillouin zone are denoted with roman letters. The center of the so-called Wigner-Seitz cell is always denoted by a Γ . )) (Σ axes 110 along edge (zone ) 0 , 4 / 3 , 4 / 3 ( / 2 : )) (Δ axes 100 along edge (zone ) 1 , 0 , 0 ( / 2 : )) (Λ axes 111 along edge (zone ) 2 / 1 , 2 / 1 , 2 / 1 ( / 2 : center) (zone ) 0 , 0 , 0 ( / 2 : a K a X a L a Fig. 3: Brillouin symmetry points. , , denote constant energy surfaces. To describe the behaviour of electrons and holes within a crystal lattice basically quantum mechanics is necessary. Newton´s law is not applicable directly. However, as an useful approximation electrons can be treated as classical particles if the particle mass is replaced by the carrier effective mass. We introduce mass effective hole mass effective electron * * n n m m and can determine the force F on an electron or hole dt v d m E q F g n * and dt v d m E q F g p . <?page no="12"?> 1. Basics 3 Using the formula for kinetic energy and the relationship for the electron momentum, k v m p g , we get 2 2 2 2 1 2 1 k v m W g and from this ) , ( k W relationship we can write 1 2 2 2 * k d W d m We can now describe the carrier dynamics with the following set of equations: The group velocity is a function of y ,k x k and z k and can be converted to the energy. dt dk dt dp F i i gi k W k v 1 with z y x i , , . The acceleration of the carriers under an applied field is given by ) dt dk k k W dt dk k k W dt dk k W ( k W dt d dt dv z z x y y x x x i gi 2 2 2 2 1 1 and using k p we obtain from Newton´s law ) F k k W F k k W F k W ( dt dv z z x y y x x x gi 2 2 2 2 2 1 which yields then for the acceleration of the carriers and which finally is completed by the effective mass approximations 1.2 Energy band diagrams For any semiconductor there are forbidden energy regions in which allowed states cannot exist. The highest energy band which is completely filled with electrons at zero temperature is called valence band. The conduction band is directly above the valence band and is empty at zero temperature. The bandgap is the minimum width along the forbidden band. The energy levels in the most important semiconductors are drawn as a function of the wave vector in Fig. 4. The conduction band minima in silicon are at the X-points. In GaAs the conduction band minimum is at the zone center. In both materials the valence band maximum is at the zone center but there are more valence bands with different effective masses. At temperatures above zero the electrons can hop from the valence band to the conduction band leaving holes in the valence band. F m dt dv g * 1 j i k k W m ij 2 * ] 1 [ <?page no="13"?> 1. Basics 4 Fig. 4: Energy levels as a function of the wave vector A representation of the conduction band minimum versus the position is corresponding to an upside-down picture of the electric potential. The conduction band gives the minimum energy an electron can have at every position. The slope is a measure of the electric field. If we consider now an intrinsic undoped semiconductor, the periodic boundary conditions for the wave function together with the allowable k values (integer numbers m, n, p) yield the possible number of states filled with two electrons in a volume element. 1 z y x z y x Δk Δk Δk π) ( L L L ΔN(k) 3 2 Assuming a parabolic effective mass approximation for the case of an isotropic effective mass * c m we find the number of states in an energy interval and the density of states is obtained dividing by the volume and ΔW 1 L X , L Y and L Z represent the dimensions in the three coordinate directions Ψ(x) L) Ψ(x z z y y x x L p k L n k L m k 2 , 2 , 2 ΔW W m π L L L ΔN(W) c z y x 2 / 1 2 / 3 2 * 2 2 4 <?page no="14"?> 1. Basics 5 with the factor 2 because of the spin of the electrons. The larger the effective mass the more available states per interval are occupied. The Fermi function F(w) gives the probability that an electron occupies a state: The total number of electrons in the conduction band per unit volume is kT W W F N dW g(W)*F(W n C F C W C 2 / 1 ) with the effective conduction band density of states 2 3 2 * 2 2 π kT m * N n C and the Fermi integral 0 if exp or exp 1 2 0 2 1 x x dy x y y (x) F then , exp kT W W N n C F C which is the density of conduction electrons. A very similar calculation for holes (replacing F(W) by 1-F(W) and integration from - to W V ) results: , 2 3 * 2 2 exp π kT m with N kT W W N p p V F v V which is the density of valence band holes. energy : Fermi , W JK * . k F 1 23 10 380662 1 kT W W F(W) kT if W-W F F exp kT W W W F F / exp 1 1 ) ( 2 1 2 3 2 * 2 2 4 1 2 W m π g(W) c <?page no="15"?> 1. Basics 6 In doped semiconductors extra atoms are added to the crystals. If W D is the donor energy the probability, that the extra atoms are ionized at room temperature, is given by D D C D D N kT W W N N e In a semiconductor we can formulate the charge neutrality condition. Wherever there are free carriers we must have 0 0 ) N N p n q( ρ ρ D A For non-degenerate materials and fully ionized impurities this leads to a simple solution: 2 2 2 2 i A D A D n N N N N n 2 2 2 2 i D A D A n N N N N p If we consider now non-compensated materials we obtain the approximation: D i D N n p N n 2 The Fermi energy level is determined by the number of available carriers, where the number of electrons and holes are equal: kT W W N kT W W N F v V C F C exp exp yielding c v V C F N N kT W W W ln 2 1 2 1 And we obtain for the intrinsic carrier density kT g W V N C N kT c W v W V N C N i n np exp exp 2 <?page no="16"?> 1. Basics 7 Fig. 5: N-type doped semiconductor; Energy levels in the k-space and as function of the density of states [1.1] Fig. 6: P-type doped semiconductor; Energy levels in the k-space and as function of the density of states [1.1] 1.3 Temperature dependence Practically all parameters depend on the temperature: The energy gap and the intrinsic carrier densities, the effective masses and the mobilities. Three different temperature regions can be distinguished: Region I: Freeze out - The temperature is so low that most of the impurities are not ionized. The intrinsic carriers are frozen out, too. The material is like an insulator. 0 p n Region II: Normal - The temperature is large enough to excite some intrinsic carriers and all of the impurity atoms A D N N n Region III: Intrinsic - The temperature is so high that intrinsic carriers take over i n p n Fig. 7 illustrates the influence of the temperature on the intrinsic carrier density. <?page no="17"?> 1. Basics 8 0 5 10 15 1000/ T (1000/ K) Intrinsic density (cm-3) Fig. 7: The intrinsic carrier density as a function of the inverse temperature for GaAs ( ), silicon ( ), germanium ( ). The temperature dependence of the intrinsic carrier density is calculated from kT )/ (W e N N n g V C i 2 . The temperature dependence of the energy bandgap is shown in Fig. 8 for Ge, Si and GaAs. Fig. 8: Energy bandgap as a function of the temperature 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 0 200 400 600 800 1000 1200 Temperature (K) Energy Bandgap (eV) GaAs Si Ge 1,E +17 1,E +13 1,E +09 1,E +05 1,E +01 1,E -03 1,E -07 1,E -11 <?page no="18"?> 1. Basics 9 1.4 Drift velocity In a gallium-arsenide semiconductor, empty electron conduction bands exist that are at a higher energy level than the conduction bands occupied by most of the electrons. Any electrons that do occupy the higher conduction band essentially have no mobility. If an electric field of sufficient intensity is applied to the semiconductor electrons, they will move from the low-energy conduction band to the high-energy conduction band and become essentially immobile. The immobile electrons no longer contribute to the current flow and the applied voltage progressively increases the rate at which the electrons move from the low band to the high band. As the v-E curve shows, the maximum velocity rate is reached and begins to decrease even though the applied voltage continues to increase. The point at which the velocity on the curve begins to decrease is called the threshold. This point is the beginning of the negative-resistance region. Negative resistance is caused by electrons moving to the higher conduction band and becoming immobile. Fig. 9: v-E characteristics. Data for undoped material (solid lines) and doped 10 17 cm -3 (dotted lines) [1.1] For the occurence of a negative differential mobility we can state three preconditions 1. At least two valleys in the conduction band required 2. The energy difference between the main and the satellite valley has to be less than the bandgap difference (avoiding impact ionization) 3. There must be higher energetic satellite valleys with reduced curvature (increased eff. mass) compared to main valley References [1.1] Theo G. Van de Roer, Microwave Electronic Devices, Chapman & Hall, 1994 [1.2] S.M.Sze, Physics of Semiconductor Devices, John Wiley & Sons, 1981 <?page no="20"?> 2. Transferred Electron Effect: The Gunn device 11 2. Transferred Electron Effect: The Gunn device 2.1 Basics Fig. 10: Energy band diagrams for two materials showing negative differential mobility The properties to be discussed in this chapter had been proposed in 1961 for a semiconductor to display the Transferred Electron Effect. The oscillations observed then experimentally by Gunn in 1963 can be fully explained using this effect [2.1]. When a sample is subjected to a strong electric field (above a certain threshold valve of several thousand volts per centimeter) the electrons gain enough energy to be transferred to the higher conduction band valley. Here, since the mobility is much lower, their velocity decreases. So the average velocity of all the carriers also decreases. Eventually, with the field high enough, the majority of electrons are transferred to the higher valley and the velocity starts increasing with field again. In between this transition, however, there is a region where the drift velocity (and hence the current density) actually decreases with an increase in electric field. This region is said to have Negative Differential Resistance (NDR). If the electrons in the upper valley enter a region where the electric field is lower than the threshold field, they will drop back to increase the average current density. This is the effect which causes the microwave current oscillations observed by Gunn. Since it takes a finite amount of time for the electrons to lose or gain energy from an electric field, there is a material specific upper limit on the operating frequency of Gunn Devices. This intervalley relaxation time is in the InP material approximately 50 % lower than in GaAs (Tab. 1). <?page no="21"?> 2. Transferred Electron Effect: The Gunn device 12 Properties GaAs InP Bandgap [eV] E(-L) [eV] Threshold field [kV/ cm] Max.neg.diff.mob.[cm 2 / Vs] Intervalley relax. Time [ps] Max. op. Frequency [GHz] 1.42 0.31 3.2 -2400 1.5 ~100 1.35 0.69 10.5 -2000 0.75 ~200 Tab. 1: Properties of GaAs and InP In order to use the negative differential mobility region in III-V semiconductors for Gunn oscillations the preconditions for the semiconductor materials can now be completed and summarized as follows: 1. The material has to have at least two ‘valleys’ in the conduction bands 2. The minima are far enough apart to ensure that, initially the majority of electrons are in the lowest valley. 3. This energy difference between the minima is smaller than the energy band Gap (W g ). This avoids significant impact ionisation. 4. The time taken for an electron to go from one conduction band valley to another is much less than one period of the microwave output frequency. 5. The upper valley has a much higher effective mass and density of states compared to the main valley. This also implies that the mobility of the carriers is much lower in the upper valley. The principal structure of a Gunn device is shown in Fig. 11: Fig. 11: Scheme of a transferred electron device + n + n + n <?page no="22"?> 2. Transferred Electron Effect: The Gunn device 13 Fig. 12: Formation of a high field domain in GaAs. 12a shows the electric field profile and 12b shows the charge density distribution [2.2] A phenomenological understanding of the Gunn effect is possible with a „Gedanken“- Experiment and the help of Fig.12: In a GaAs probe an electric field almost approaching threshold E t is applied. The Electrons will drift to the anode at the right side. By a local thermal fluctuation or doping inhomogenity the electric field exceeds E t locally. The electrons will become more slowly and an accumulation occurs (at x=x 1 ) compared to the equilibrium. A dipol domain is formed. The domain forces the current to be reduced as there is a constant voltage at the „element“. The domain travels to the anode the current reaches its original value. A new domain starts, travels and will be resolved at the cathode. A pulsed current for microwave generation is obtained. In order to study the theory of the dipole layers in more detail and to calculate the Gunn device parameters which can be measured it will be necessary to know the domain velocity v D . The electric field is below threshold on the left and right sides of the domain. The field rises inside the domain to a peak value E m =E high . From an analysis of Poisson´s equation and the current continuity equation Butcher has shown, that the domain travels at the same velocity v 0 as the electrons in the uniform region outside the domain: D v v 0 A dynamic characteristic velocity-electric field curve as shown in Fig. 13 relates v 0 to the peak field E high . The value of E high can be determined from Butcher´s equal area rule which states, that the two shaded regions in Fig. 13 have equal areas. <?page no="23"?> 2. Transferred Electron Effect: The Gunn device 14 Fig. 13: Illustration of Butcher’s equal area rule. Dynamic velocity - electric field characteristic (dashed) and static velocity electric field characteristic (solid) The equal-area rule states then: high low E E D dE v E v 0 From that representation we can determine e.g. the high field value E high from a graphical representation of a velocity-field characteristic. A further simplification of the calculations is possible using the triangular domain assumption (Fig. 14). Fig. 14: Space charge density and field distribution for the triangular domain assumption By the help of the triangular domain assumption the domain parameters E high , b, E low can now be calculated: b E E d E V b n ε e E E low hifh low o low high 2 n-n o x E high b Field distribution Space charge density E low <?page no="24"?> 2. Transferred Electron Effect: The Gunn device 15 Here we have assumed that the total width of the active layer is d and the width of the domain is calculated from low high D E E qN b with D N n 0 . The equal-area rule can now be used to calculate the operation voltage of a certain Gunn element. This is illustrated using an example: The length of an active region shall be µm L 10 the electron mobility is µ n =6000 cm 2 / Vs, the doping of the active region is N D =2*10 15 cm -3 and the device area is A=3*10 -4 cm 2 . The DC current is given with I=1.152 Ampere and then the domain velocity is calculated to be If we take now a v - E curve for the material under consideration at the specified doping level and operation temperature we obtain now from the v - E curve: and from Butcher’s rule and a graphical representation of the v-E characteristic we obtain Consequently the domain voltage is calculated to be and the total voltage across the device is then V/ cm * E l 3 10 2 . kV/ cm . E h 5 10 . V . D V L l E V 12 2 s cm eAN I v D / 10 * 2 . 1 7 0 V E E eN V l h D D 12 . 0 2 2 <?page no="25"?> 2. Transferred Electron Effect: The Gunn device 16 2.2 Operation modes and design rules The three most important operation modes of Gunn elements are the accumulation layer mode, the transit time mode and the quenched-domain or dipol-layer mode (Fig. 15). x (m) x (m) x (m) (V/ cm) Fig. 15: Most common operation modes of Gunn elements [2.3] The accumulation layer mode is observed in TEDs (Transferred Electron Devices) with subcritical N D *L product (N D *L <10 12 cm -2 ; N D represents the doping of the active layer and L is the length of the active layer; see also next pages). Here the voltage is always above threshold value V T . This mode provides comparatively low efficiency due to non ideal waveforms. The transit time mode occurs, if the N D *L product is greater than 10 12 cm -2 . The space charge perturbations increase exponentially in space and time to form domains or dipole layers. The domain formation usually takes place at the cathode because the largest doping fluctuation and space charge perturbation exists there. The cyclic formation and subsequent disappearance of the domains at the anode yield the experimentally observed Gunn oscillations. The conversion efficiency is up to about 10 % for GaAs. The Quenched domain mode is characterized by a reduced bias voltage which causes a width reduction of the domain up to zero and therefore a suppression of domains: When the bias voltage is reduced below a value V s , the domain is quenched. In a resonant circuit the TED can operate at frequencies higher than the transit time frequency if the domain is quenched before reaching the anode. Multiple domains are formed (c.f. Fig. 15) if the N D *L product is > 3*10 12 cm -2 . This mode provides the highest possible efficiency (up to 13 % for GaAs) but is however much more difficult to adjust than the predominantly used transit time mode. <?page no="26"?> 2. Transferred Electron Effect: The Gunn device 17 In the transit time mode it is straight forward to calculate the operation frequency f from cm/ s * . f*L L v T f s t 7 10 2 1 1 or with the transit time T t and the saturation velocity v s with parameters valid for GaAs. The corresponding dependence of the operation frequency on the length of the active region is shown in Fig. 16. Fig. 16: Oscillation frequency [GHz] versus active region length [µm] for GaAs- Gunn elements operating in transit time regime In order to derive a condition for the doping level design in Gunn elements we have to deal with the semiconductor transport equations. The continuity equation in n-doped GaAs is x J q t n 1 The Poisson equation is n N ε q x E D and we have the convection current qnv(E) J and the conservation of current t x,t E ε x,t J J tot Introducing the Poisson equation and current convection in the conservation equation yields <?page no="27"?> 2. Transferred Electron Effect: The Gunn device 18 v(E) x E q ε N q t x,t E ε t J D tot and with the small signal approximation n = N D + n 1 we obtain 1 1 1 1 1 n E v ε q N x n v x E E v N x n v x v N x n v x nv t n D D D The differential equation 1 1 1 n E v ε q N x n v t n D will now be considered in a range, where D μ E v is constant and therefore 1 1 1 n μ ε q N x n v t n D D can be re-written. Now we introduce D D D μ ε q N τ 1 which is the dielectric relaxation time. The general solution of the differential equation is D t )e vt, (x n (x,t) n 0 1 1 This equation describes the carrier density or space charge wave which can grow exponentially (Fig. 17). A strong increase and a traveling domain formation during the transit time ( L s v t T 1 ) occurs if 1 D D D μ ε q ν L N τ t or 2 12 10 cm μ q εν L N D D In other words, the condition N D *L>10 12 cm -2 guarantees that the relaxation time is comparable to the transit time, in order not to damp the space charge waves ! <?page no="28"?> 2. Transferred Electron Effect: The Gunn device 19 n + n n + Fig. 17: Visualization of a growing inhomogenity The growth of the domain in Fig. 17 is possible if N D *L > 10 12 cm -2 (in GaAs). If the length of the active region is chosen to meet a certain operation frequency, the doping of the active region has to exceed the border line as shown in Fig. 18. Fig. 18: Length and doping of the active region 2.3 Thermal properties of the Gunn element Limited conversion efficiencies of active millimeter wave devices poses severe restrictions to the design of the devices and the corresponding heat sinking techniques. Especially the high DC current in Gunn elements causes a high DC power disspation in a Gunn diode. Length of the active region [µm] L> 10 12 / N D No Dipol Domains Doping of active region [10 16 cm -3 ] <?page no="29"?> 2. Transferred Electron Effect: The Gunn device 20 The condition in GaAs for which the charge domain will grow is given as assuming all donors are fully ionized (n 0 =N D ). Therefore the carrier concentration at threshold must be if we assume that L=10 µm. If we assume for the threshold field in GaAs now 3.5 kV/ cm then the velocity follows 2*10 7 cm/ s. The current in the diode can then be calculated from (A is the diode area). The dissipated power in the diode is then given by where E t is the threshold field and L is the length of the diode. The power per unit volume is then or P=3.36W for a diode area of 3*10 -4 cm 2 . Fig. 19: General scheme and possible mounting of a GaAs Gunn device Typically, active millimeter wave devices are mounted on a heat sink. Fig. 19 shows a possible configuration for a mesa-like device. The junction temperature increase T and the dissipated heat power P w are related via the thermal resistance : W P T e.g. copper heatsink Active region Substrate (remaining) upside up config. Contact regions Soldering or bonding Bond wire 2 12 10 cm *L N D 3 15 12 0 10 10 cm L n L AE v qn IV P t d 0 A v qn I d 0 3 3 7 / 2 . 11 / 10 * 12 . 1 * cm MW cm W L A P <?page no="30"?> 2. Transferred Electron Effect: The Gunn device 21 If we neglect radiation from the surfaces, we assume one dimensional heat flow and constant thermal conductivity we can derive a simple expression for the thermal resistance n i i i HS d r 1 2 1 with r HS HS 1 with r: device radius i : thermal conductance in layer i or heat sink (HS) d i : thickness of layer i r HS HS 1 represents a thermal spreading resistance and is inversely proportional to the periphery of the device. A focal plane with radius r would have this thermal resistance. For a more general analysis in cylinder symmetric coordinates the nonlinear heat conduction equation has to be solved with suited boundary conditions. Fig. 20: Model structure to analyze the nonlinear heat conduction equation. Heat is generated in a focal plane of radius r 1 at a height z=0 which is located on top of a cylinder with height H. The cylinder with radius r 2 has a thermal conductivity 1 and is placed on a material with thermal conductivity 2 0 1 z T κ z r T κr r r <?page no="31"?> 2. Transferred Electron Effect: The Gunn device 22 For the case of temperature dependent thermal conductivities this equation can be rearranged: 0 1 2 2 2 2 2 2 z T z T T r T r T r r T r T r If we look at the temperature dependence of semiconductor materials like silicon or GaAs (Fig. 21) it becomes obvious, that the temperature dependence of the thermal conductivities may play an important role in integrated circuits, which employ semiconductor materials as an interface medium between the active device and the heat sink. The thermal conductivity of the semiconductor materials silicon and GaAs decreases with increasing temperature. At room temperature the thermal conductivity of silicon is approximately three times higher than the thermal conductivity of GaAs. Fig. 21: Thermal conductivity of silicon and GaAs as a function of the temperature A solution of the nonlinear heat conduction equation can be based on the boundary conditions that heat radiation from the surface is neglected ( 2 0 r r for r Τ ) and that there exists a semi-infinite heat sink ( z for Τ 0 ). Employing a Kirchhoff- Transformation the non-linear heat conduction equation can be transformed into a Laplace equation for a heat-potential and finally a back-transformation yields the temperature distribution [2.4]. <?page no="32"?> 2. Transferred Electron Effect: The Gunn device 23 Fig. 22: Thermal resistance as a function of the device diameter [GaAs and silicon; different input powers] In the case of temperature dependent thermal conductivities the thermal resistance is dependent on the supplied heat flux density: the thermal resistance will have an exponential dependence on the supplied heat flux density caused by the decreasing thermal conductivity with increasing heat flux density - the thermal resistance is dependent on the input power (Fig. 22) ! Fig. 22 shows the results of a calculation, where a 100 m thick semiconductor cylinder is mounted on a copper heat sink. If we include the temperature dependence of the thermal conductivities we recognize that the thermal resistance depends on the input power. It can be seen that in small devices the self-heating effect is more pronounced, i.e. for small input powers the thermal resistance is approximately inversely proportional to the junction diameter whereas at large input powers the thermal resistance approaches a quadratic dependence. 2.4. Techniques to reduce thermal resistances From the preceding calculations it is evident that there exist mainly two possibilities two reduce the thermal resistance: An increase of the heat generating area and an increase in the thermal conductivities can reduce thermal resistances. <?page no="33"?> 2. Transferred Electron Effect: The Gunn device 24 2.4.1. Spreading of the active area In most cases the device area is fixed by impedance matching constraints. A reduction of thermal resistances is possible with non-concentrated structures: The solid circular mesa diode is not thermally optimized. Fig. 23 shows the improvement in thermal resistance which can be obtained dependent on the ratio of the inner radius to the outer cylinder radius. Fig. 23: Improvement of the thermal resistance by the use of a ring structure instead of a solid circular structure of the same area. Thermal resistances ratio of a circular to a ring structure as a function of the ratio of the inner radius to the cylinder thickness The calculations shown in Fig. 23 are based on the expressions a R R m K R r r dr aR a mesa Θ ring Θ 2 2 1 2 2 0 2 sin 1 m d Κ(m) (complete elliptic integral of the first hind) 2 4 R r rR m aR a πκ mesa Θ 2 1 2 <?page no="34"?> 2. Transferred Electron Effect: The Gunn device 25 Obviously, a ring structure causes a significant reduction of the thermal resistance. If the inner radius exceeds the cylinder thickness by more than a factor five the thermal resistance is reduced by a factor of two. 2.4.2 Use of heatsink with high thermal conductivity Improvement of the thermal resistance is possible using diamond as a heatsink material. The thermal conductivity of diamond at room temperature is and at 500 K ) 10 500 1 K W/ (cm K) ( Fig. 24: Thermal resistance of a focal plane on a diamond heatsink as a function of the height of the diamond. Parameter: radius of the diamond. (Dashed lines valid for assumption of constant thermal conductivity) Because of this high thermal conductivity diamond is used as a heat sink material for active mm-wave power devices. It can be seen that the thermal resistance shows a minima at small cylinder (diamond heat sink) radii r 2 (Fig. 24). With increasing radius r 2 the minima becomes weaker and is not more visible for radii > 300 µm. This minima is dependent on the expression for the temperature dependence of the thermal conductivity: For a constant thermal conductivity the rule H=1/ 3 r 2 seems appropriate. At a higher conductivity the minima occurs for H=1/ 2 r 2 . If r 2 >400 µm the calculated thermal resistance-values are very close together a significant improvement in thermal resistance by further increasing the diamond dimensions is not possible. For the design of diamond heatsinks we obtain the rule 1/ 3 r 2 <H<1/ 2 r 2 . The cross section of a diamond heat sink equipped package is shown in Fig. 25. [K/ W] 50 100 150 250 200 350 300 11 19 15 7 23 H [m] K(500K) 100m 250m 100m 400m K(300K) 100m 800m K) W/ (cm K) ( 20 300 1 <?page no="35"?> 2. Transferred Electron Effect: The Gunn device 26 Fig. 25: Cross section of Gunn device (mesa diode) mounted on diamond heat sink in mm-wave package. 2.5 Technology A typical fabrication process of a discrete Gunn device starts with the growth or deposition of the active epitaxial layer on a highly n + -doped GaAs substrate.The epitaxial layer is then covered with a multi layer metallization system Au/ Ge-Ni, Ag, Au. The last Au layer is selectively electroplated in order to guarantee sufficient stability. Then the substrate is mechanically (by polishing) and/ or chemically thinned to a rest thickness typically below 10 m. The remaining substrate is now also covered with evaporated metallization (Fig. 26A). Au/ G e-Ni Substrate 3m Epitaxial Layer Au/ G e-Ni Ag 500nm Au 5m Forming of Au Top- Contact by Electroplating Mesa Definition Plasma Etch Removal of the Au/ G e-Ni Contact layer (d) (c) (b) (a) Fig. 26A : A beamlead process (I) The top contact is then defined in a photolithography step and the active device area realized by dry chemical etching. Wet chemical etching of the Au/ Ge-Ni layer provides individual Gunn devices ready for mounting in a package. The mounting process itself turns-out to be one of the most critical process steps. Due to the small device areas required at millimeter wave frequencies the bonding of a top contact is very difficult and can reduce the overall yield drastically. It is therefore very desirable to introduce a beamlead process as illustrated in Fig. 26B. <?page no="36"?> 2. Transferred Electron Effect: The Gunn device 27 Spin-On of Photoresist And Litography Evaporation of Gold (200nm) Lithography and Electroplating of Cross-Strips Spin-On of Photoresist (4m) (e) (h) (g) (f) Fig. 26B: Beamlead process (II) Fig. 27 shows a top view on integrated beamleads. The mesa top contact can be clearly identified. Fig. 27: Gunn device with integrated beamlead. Top view. <?page no="37"?> 2. Transferred Electron Effect: The Gunn device 28 The bonding problems become even more evident if distributed structures like ring geometries are used (Fig. 28). These devices then are mounted in a microwave or millimeter wave package as sketched in Fig. 25. The packaged device can now be understood as a device with a negative real part of the impedance and an imaginary part which is transformed by the inductance of the bond wires and by the packaging capacitance into a packaged device impedance which now has to be matched to a resonator impedance. Fig. 28: A ring structure The reproducibility of the process can be improved significantly if not only the leads but also the package is fabricated in an integrated process sequence. Fig. 29: Integrated Gunn device Fig. 29 shows the concept where the semi-insulating GaAs substrate material is used as packaging material. Fig. 30 shows a schematic overview on the process sequence which is employed to realize a pretuned Gunn module. <?page no="38"?> 2. Transferred Electron Effect: The Gunn device 29 Semi-Insulating GaAs Contact and Etch Stop Layer Oxide Gunn Oxide Contact and Etch Stop Layer Gunn Au/ Ge/ Ni Contact and Etch Stop Layer Oxide Au Isolation Oxide Au/ Ge/ Ni Gunn Contact and Etch Stop Layer Gunn Au/ Ge/ Ni Contact and Etch Stop Layer Au Isolation Metallization “Pretuned Gunn Module“ Oxide Fig. 30: Process sequence for an integrated mm-wave Gunn module Fig. 31a shows a SEM picture of a GaAs Gunn and Fig. 31b shows a packaged millimeter wave Gunn device. Fig. 31a: Scanning Electron Micrography Fig. 31b: Packaged Gunn • Semi insulating GaAs substrate • Definition of an oxide window • Forming a highly doped buried layer • Lift off process: resist deposition, • mask,exposure, developer • metal deposition, „lift off“ • Active area etching • Deposition of „cold“ isolation (Si 3 N 4 ) • Opening of contact window • Evaporation of heat sink metallization • Electroplating • Back side via hole process • (> 20 technological steps) 70µm diameter • Deposition of active n-doped • layer (MBE, MOCVD) <?page no="39"?> 2. Transferred Electron Effect: The Gunn device 30 Performance examples in terms of output power versus frequency of mm-wave Gunn devices are given in Fig. 32. Fig. 32: Performance of mm-wave Gunn devices as published for GaAs and InP Gunn devices in continuous wave (cw) operation. Numbers next to the symbols denote the dc-to-RF conversion efficiencies in percent [2.5] 2.6 Conclusions A negative differential mobility region is used in Gunn elements for the generation of microwave oscillations. The three basic requirements for this kind of transferred electron mechanism, called k-space transfer are: 1. The semiconductor must provide two valleys in the conduction band, 2. The energy difference between the valleys is less than the band gap difference 3. The higher energetic valley provides a reduced curvature and therefore an increased effective mass or a reduced mobility. These preconditions are fulfilled by GaAs and InP (and GaN). The intervalley relaxation time of InP is approximately 50 % of the intervalley relaxation time in GaAs. This makes the InP devices operational in a fundamental mode up to higher frequencies than GaAs devices. Design restrictions are due to transit time constraints and thermal limitations. Packaging aspects deal with parastics and yield. Beam lead technology is a first step to improve the yield, pretuned modules also improve the reproducibility. Typical output powers of up to 50 mW at 100 GHz are obtained with Gunn elements in CW operation. <?page no="40"?> 2. Transferred Electron Effect: The Gunn device 31 2.7 Outlook The real-space transfer is a new concept for an electron transfer between two different semiconductor materials of varying mobilities. Fig. 33: Real space transfer The real-space transfer may occur in heterostructure semiconductor systems. The example shows a GaAs layer sandwiched between two AlGaAs heterostructure layers. The electrons move between the two heterolayers. Within the GaAs layer the electrons move under the action of an applied electric field. The electric field is applied parallel to the heterolayers. Under the influence of the electric field the electrons are heated and the average electron energy increases until the carriers reach an energy which is greater than the heterojunction discontinuity E c . The electrons will then be scattered and their momentum will be redirected perpendicular to the heterojunction: They will be transferred from the GaAs layer into the AlGaAs layer. This is called the real-space transfer mechanism. If the composition of the AlGaAs layer is chosen in a way, that the mobility in the AlGaAs layer is lower than in the GaAs layer, conditions exist like those in bulk GaAs to cause a negative differential resistance (NDR). A possible device realization is shown in Fig. 34. Fig. 34: The NERFET [2.3] AlGaAs AlGaAs GaAs E c Electric field electrons electrons electrons electrons Principle Gate N-AlGaAs Graded AlGaAs Layer GaAs - Channel GaAs substrate Source Drain <?page no="41"?> 2. Transferred Electron Effect: The Gunn device 32 Problems In order to design 77 GHz oscillator circuits with Gunn elements, InP and GaAs materials are considered [v sat,GaAs =1.2*10 7 cm/ s, v sat,InP =2.4*10 7 cm/ s]. Give the proper layer thickness and possible doping for GaAs and InP Gunn elements, respectively, if they operate in the fundamental mode. If we assume that the device diameter is 30 µm, what is the thermal resistance of a pure copper heat sink [ Cu =3.84 W/ cmK] ? Discuss the principles of high field domain in Gunn elements and how to calculate an effective transit velocity of the domain Enumerate k-space transfer conditions and compare with the real-space transfer mechanism References [2.1] Comprehensive overview and original contributions are found in: B.G.Bosch, R.W.H.Engelmann; Gunn-effect Electronics; Pitman Publishing, 1975 [2.2] A. Schlachetzki, Halbleiterbauelemente der Hochfrequenztechnik, Teubner Studienskripten, 1984 [2.3] S.M.Sze, High-Speed Semiconductor Devices, J. Wiley & Sons, 1988 [2.4] J.-F.Luy, P. Russer, Silicon Based Millimeter Wave Devices, Springer 1994 [2.5] S.M.Sze, Modern Semiconductor Device Physics, J. Wiley & Sons, 1998 <?page no="42"?> 3. Transit time diodes 33 3. Transit time diodes 3.1 Basics of the pn-junction If we bring a p-type and a n-type material together, we can expect that the electrons move from the n-type region to the p-type region and that the holes move from the p-type region to the n-type part. Away from this transition region the junction will behave like bulk material and the rule of Fermi level alignment can be applied. In the depletion region or space charge region there are no free carriers, but there is an electric field which causes a potential energy difference. Fig. 35: Energy band diagram, position of p-n junction and coarse of potential The application of Poisson´s equation (actually the first of four Maxwell equations) to a p-n junction ε ρ E can be re-written for the x-direction only a d N N n p ε q dx dE and yields in the p-region ) ( 0 p a Χ x Ν ε q - Ε(x) and in the n-region ) ( 0 -x Χ Ν ε q - Ε(x) n d where the charge densities in the p - and n - regions are defined in Fig. 36. E C p type n type - - - - + + + + W Fermi qV 0 potential W V n p <?page no="43"?> 3. Transit time diodes 34 Fig. 36: Charge density and electric field across a p-n junction in thermal equilibrium We calculate the voltage across the junction from C x Χ x Ν ε q Ε(x)dx V(x) p a 0 2 2 where we define zero of V(x) at the p side and obtain for the coarse of the voltage within the p-side 2 2 2 0 0 2 p p a Χ x Χ x Ν ε q V(x) and within the n-side 0 2 0 ) 2 ( p n d Φ x x Χ Ν ε q V(x) we can obtain the p-side voltage at the junction from 2 2 0 0 p a p Χ Ν ε q Φ which also is called the buildt-in voltage at the p-side. Similarily we obtain the buildtin potential at the n-side from 2 0 0 2 n d n n Χ Ν ε q Φ In order to calculate the depletion region width, we have additionally 2 0 0 ln ) / ( i D A n p g n p bi n N N q kT / q Φ Φ Ε Φ Φ Φ with the neutrality condition for the charge densities N eAX N qAX Q Q a p d n n p 0 0 and we obtain for the depletion layer width on the p-side ) / 1 ( 1 2 0 d a a bi p N N N q X <?page no="44"?> 3. Transit time diodes 35 and on the n-side ) / 1 ( 1 2 0 a d d bi n N N N q X which gives for the total depletion layer width ) N N ( q ε X X W a d bi n p 1 1 2 0 0 In order to have a feeling for the order of quantities we provide a short numerical example: With 3 16 10 cm Ν d and 3 18 10 cm Ν a we obtain μm . Χ Χ W n 33 0 0 0 and Angstroms . Χ 3 8 0 3.2 Transit Time Effect Active millimeter wave devices may be distinguished regarding their injection mechanisms and the corresponding transit time effect. There are electron transfer elements which employ k-space transfer or the real space transfer mechanisms (cf. Chapter 2). The transit time devices differ regarding the phase angle between the injected carriers and the voltage. Gunn elements GaAs AlGaAs/ GaAs RST Heterostructure TUNNETT Tunnelgeneration READ Diode Avalanche SiGe Gunn ? Si/ SiGe RST Heterostructure HJ IMPATT Avalanche + Tunneling BARITT (Schottky) BARITT (PD) Barrier-Inj. (TA,AA) BARITT Double drift concepts Electron transfer elements real space In-phase injection Injection Angle π / 2 K-Space Transit time devices Fig. 37: Family of active millimeter wave devices <?page no="45"?> 3. Transit time diodes 36 In order to gain a general understanding of the influence of the injection mechanisms on the complex device impedance an analysis of a general transit time device is performed. Fig. 38: Model structure of a general transit-time device with an n-doped drift region We make some simplifying assumptions: The model is used to study only carrier transport by drift within 0 < x < d. The positive polarity of the applied potential is at the injector and the negative potential at the collector. A general analysis of this structure is possible using the following assumptions: only one kind of carriers is considered: holes in this case the analysis may be performed one-dimensional diffusion may be neglected and the transit time is small compared to the carrier lifetime there is no generation or recombination of carriers in the drift region Then we can write the basic semiconductor equations equation density current Particle equation Continuity equation Poisson 1 0 qvp J x J q t p p N ε ε q x E D r J, p, v, E are now assumed to follow t j t j e p p p , e J J J 1 0 1 0 which is the small signal assumption. If only AC contributions are considered and higher order terms are neglected we obtain 0 1 1 0 1 1 1 1 0 1 1 p p v p v q J x J q jω p ε ε q x E r Injector Contact n-type semiconductor + - 0 d x <?page no="46"?> 3. Transit time diodes 37 with the abbreviations E: electric Field E 1 : time dependent (AC) electric Field q: electron charge (1.6*10 -19 cm -3 ) 0 : absolute permittivity (8.85416*10 -12 F/ m) r : relative permittivity (12 for silicon) p: hole density p 1 : time dependent (AC) hole density J 1 : time dependent (AC) current density v,v 0 ,v 1 : drift velocities Assuming constant carrier mobility the second order differential equation is obtained dx dE v jω ε v μe p dx E d 1 0 0 0 2 1 2 The general solution of this wave equation is given by 0 0 0 1 v ω j ε v μe p k b kx ae (x) E where the complex propagation constant k has been introduced. We have now to apply suited boundary conditions and assume, that the total current is comprised from the particle and displacement current (x) E ε jω J J r t 1 0 1 By comparison with the differential equation and the general solution the quantities b and a can be determined r r t ε jω ) ( J a ε jω J b 0 1 0 0 The AC electric field is now obtained from 1 0 1 0 1 kx e J ) ( J ε jω J (x) E t r t and the drift region impedance of the semiconductor structure is obtained by integration over the drift region kd e kd J ) ( J ωC j J dx d (x) E Z t t D 1 1 0 1 0 1 1 where we have used the normalized drift region capacitance d r ε ε C 0 and define the complex injection ratio ) ( 0 1 j Α(ω)e J ) ( J t <?page no="47"?> 3. Transit time diodes 38 The impedance level is inversely proportional to the capacitance and device area. The complex injection ratio with an amplitude A() and an angle () denominating the phase shift between the injection current density and the total current density obviously can determine the behaviour of the device: This expression describes the injection mechanism and causes the differences in the impedances of IMPATT , BARITT or TUNNETT diodes. To study the importance of the injection ratio it is convenient to investigate the real and imaginary part of the drift region impedance : θ φ φ A θ θωC Z φ) (θ φ θωC A Z D D sin sin 1 Im cos cos Re The propagation constant has been restricted to the case where the applied field is high enough to punch through the region, and carriers are drifting at a saturated velocity. s v d ω Θ is the Transit angle s v ω j k holds for drift with saturated velocity A general representation of the real part of a drift region impedance as a function of the transit angle and the phase injection angle is shown in Fig. 39. The results have been normalized using A/ C=1 Fig. 39: 3D representation of the real part of the drift region impedance as a function of the transit angle and the phase injection angle <?page no="48"?> 3. Transit time diodes 39 It can be seen, that the real part may become negative for values of the injection phase angle > 0. The optimum transit angle (for maximum amount of the real part) depends on the injection phase angle and decreases with increasing phase angle simultaneously the amount of the negative resistance increases. Fig. 40 shows a corresponding plot of the imaginary part. Fig. 40: 3D representation of the imaginary part of a drift region impedance as a function of the transit angle and the phase injection angle The imaginary part is always capacitive. From the previous discussion, it is apparent that a mechanism in which the carriers are injected at a large phase angle without a decrease in amplitude may be favourable. Furthermore, the injection mechanism should ensure that the carriers are injected with saturated velocity. Another approach would be to let =0, ,...and to make A() < 0 which occurs in the case of an IMPATT diode as will be shown in the next section. We have seen, that the transit time diodes can be distinguished considering the injection mechanisms: Carrier injection from a high field region where impact ionization occurs (IMPATT diode), or thermionic emission across a potential barrier (BARITT diode), or tunnel injection (TUNNETT and QWITT diode) or diodes with intermixed drift and injection mechanisms (MISAWA type). <?page no="49"?> 3. Transit time diodes 40 3.3. Injection mechanisms The discussion of the injection ratio ) ( 1 ) 0 j t e A(ω J ) ( J has shown that a mechanism in which the carriers are injected at a large phase angle without a decrease in amplitude is favourable and to let A() < 0 which occurs in the case of an IMPATT diode! The mechanism of impact ionization which will be discussed in detail in section 3.3.1 can be described in three phases: Fig. 41: Phases of the impact ionization process 1. Electrons from conduction band gain energy at least W g 2. Each electron generates an electron hole pair by impact ionization (one electron from valence band into conduction band! ) 3. New electron-hole pair contributes to transport and/ or to impact ionization 3.3.1 Impact ionization One of the key parameters in this device is the ionization rate, i.e. the number of electron-hole pairs which are generated by a carrier per unit distance. For the device analysis and design it is necessary to know the dependence on the "driving force" the energy or the electric field in a local model. The ionization rates may be calculated from the band structure using Monte Carlo simulations. These calculations are in good agreement with measurements (Fig. 42). An analytical representation is obtained by the semi-empirical equation 2 , ) ( exp , , , qE F F W qE (E) α p n p n p n i i i p n with the high field ionization threshold energies for electrons and for holes eV . W eV; . W p n i i 0 5 6 3 , W V W C W g + - - - - - >W g Heating - - Pair generation Transport + + + + + + + + - - >W g Transport Pair generation Heating <?page no="50"?> 3. Transit time diodes 41 the threshold fields compensating for Coulomb scattering eV/ cm . F eV/ cm, . F p n i i 6 6 10 091 3 10 954 1 and for optical phonon scattering for electrons and holes, respectively. eV/ cm . F eV/ cm, . F p n r r 5 5 10 110 1 10 069 1 These values are obtained by a fitting to the measurements from silicon material. The temperature dependence of the ionization rates can be introduced using elements of the "Baraff-Theory“ the mean free path for optical phonon scattering. kT W λ λ p p n p n 2 tanh , 0 , with W p = 63 meV denoting the optical phonon energy. Fig. 42: Ionization rates as function of the electric field for different semiconductors (adapted from [3.1]) <?page no="51"?> 3. Transit time diodes 42 elec. 300K holes 300K elec. 500K holes 500K 0 200000 400000 600000 800000 1E+006 E(V/ cm) 1E+006 100000 10000 1000 100 10 1 alpha(1/ cm) Fig. 43: Calculation of the ionization rates for silicon at different temperatures If we assume a homogeneous field between two scattering events, the threshold field due to optical phonon scattering can be determined from kT W λ W λ W F p p p n p r p n p n 2 tanh , , 0 , with nm . λ n 6 7 0 and nm . λ p 5 5 0 . The ionization rates at two different temperatures for silicon as a function of the electric field are shown in Fig. 43. Now, imagine a semiconductor structure where the electric field is high enough to ensure (partly) impact ionization. How can we observe ionization and avalance multiplication in DC characteristics of devices? <?page no="52"?> 3. Transit time diodes 43 This behaviour the breakdown mechanism can be very well observed in avalanche diodes. Fig. 44 shows the I-V Characteristics of a high power IMPATT diode made from silicon material. It can be seen in the logarithmic scale that the diode in forward direction performs according the expected exponential dependence nKT V I I S exp with the ideality factor n, the temperature T, the Voltage V and Boltzmann´s constant k. A sharp breakdown in reverse direction is detected. Fig. 44: DC charcteristics of a silicon IMPATT diode; forward bias(left) and reverse bias (right) Fig. 45: DC charcteristics of a GaAs IMPATT diode; forward bias (left) and reverse bias (right) The Fig. 45 shows the I-V characteristics for a GaAs device. There are low saturation current densities (in the range of 5*10 -7 A/ cm 2 ). For the analysis of the p-n junction it is important to look at the forward characteristic at high currents. <?page no="53"?> 3. Transit time diodes 44 This characteristic shows, that there exists a high DC series resistance Rs in the order of 5 Ohms. This has to be accounted for in the I-V charcteristic. nKT R I V I I S S exp . Impact ionization Drift region How is carrier generation by impact ionization used to generate a dynamic negative resistance? Let us consider a simple p + nn + structure. Fig. 46: Electric field profile in a single drift Impact Avalanche Transit Time structure (left) and voltage, injected current and terminal current as a function of the phase (right) There will be a region of width l a , where the field is high enough to cause impact ionization. We can perform a “Gedankenexperiment” assuming we have a breakdown at a certain threshold voltage and a superimposed RF voltage (Fig. 46, topright). The number of carriers generated by impact ionization increases as long as the voltage is above threshold! This leads to the inductive phase shift between injected current and applied voltage (Fig. 46). To describe the dynamic behaviour quantitavely we apply the continuity equations G x J q t n n 1 G x J q t p p 1 with the current densities p p n n v p q J v n q J and the simplifications p n p n v v α α p) av(n pv α nv α G p p n n <?page no="54"?> 3. Transit time diodes 45 which yields the differential equation C n p C J v α ) J (J x v t J 2 with n p C J J J which is also known as the avalanche equation. This equation can now be integrated over the length of the avalanche region with the quasi stationary approximation: J c (t,x)=J c (t) is independent from space. 0 0 2 a a l C l n p C a adx J v J J v dt dJ l . To evaluate the different current components within the avalanche region we introduce thermally generated saturation current components according to Fig. 47: ps ns s J J J and we obtain for the avalanche equation s l C a C J adx J l v dt dJ a 0 1 2 Fig. 47: Current components in an avalanche region Now, we assume that within the high field region (avalanche width) the conduction current is independent from the space and that we neglect the influence of the saturation currents. Then we come up with an approximation for the ionization integral a l a l a adx 0 and J s =0 and obtain c a c i J l dt dJ τ ) 1 ( which is also known as the Read equation. The intrinsic response time is v l τ a i 2 . We see, that the factor 2 in the expression for the intrinsic response time is a consequence of our assumptions on the current components in an avalanche region. Higher order solutions of the Read equation show that v l τ c i 3 is more appropriate at millimeter wave frequencies [3.2] J c =J n +J p -l a 0 x J n J p J ps J ns <?page no="55"?> 3. Transit time diodes 46 For an analytical large signal representation we use a Taylor series representation of the ionization rate ωt (E)mE ' α (E) α α(E(t)) cos 0 .The electric field is spatially constant but cosinusoidal time varying. (E) ' α represents the first derivative of the ionization rate with respect to the electric field. Second and higher order terms are neglected. Using harmonic expansions for the solution of the Read equation we obtain ωt (z) I (z) I J (z) I J ωt z e J(t) sin 2 1 sin 0 1 0 0 0 (I 0 , I 1 are modified Bessel functions). We introduce the parameter z ω J εmE ω ωτ mE α'l z a i a 0 2 and the small signal avalanche resonance frequency i a a ετ J α'l ω 0 and the modulation depth m. The total current is obtained switching to a complex notation and adding the displacement current to the frequency dependent component of J(t) (z) zI (z) I ω ω z J J a t 0 1 2 2 0 2 The injection ratio is now jφ e Α(ω) (z)ω I (z)ω zI J ) ( J a t 2 1 2 0 1 2 1 1 0 which shows that the negative impedance of the IMPATT diode is caused by the frequency dependence of A because there is no phase shift between the injection current density and the total current density (=) ! The impedance of the Impact Avalanche Transit Time diode can now be written as (z) zI (z) I ω ω l l θ θ ω A ωC Z θ θ ωC ω A Z a d a D D d D 0 1 2 2 2 1 1 sin ) ( 1 1 Im cos 1 1 ) ( Re The real part contains the amplitude of the injection ratio, the device capacitance C d and the drift region design with the transit angle . Fig. 48 shows the effect of increasing the modulation index the amount of the real part decreases. The pole <?page no="56"?> 3. Transit time diodes 47 caused by the avalanche resonance is also dependent on the modulation index and is shifted towards lower frequencies with increased modulation. This behaviour can be described by the introduction of the large signal avalanche frequency (z) zI (z) I ω ω a a L 0 1 2 which is shifted to lower frequencies with increased modulation z. Fig. 48: Real part of the impedance of an IMPATT diode as a function of the frequency and the modulation index. Contour lines every Ω .1 0 . Parameters for the calculation: Drift velocity cm/ s * . 7 10 65 0 ; drift region width μm .25 0 , V/ cm * E 5 10 5 . Current density 2 10kA/ cm . Real part normalized to an area of 2 6 10 7 cm * . Fig. 49: Real part of mm-wave IMPATT diode as a function of the frequency and the current density <?page no="57"?> 3. Transit time diodes 48 The typical behaviour of IMPATT diodes is represented in Fig. 49: The real part of the impedance is positive below the avalanche frequency and negative above the avalanche frequency. Therefore power generation at high frequencies becomes possible. Fig. 50: Imaginary part of an IMPATT diode as a function of the frequency and the modulation index The imaginary part is always capacitive above the avalanche resonance frequency. The dependence on the modulation index is weaker than for the real part. 3.3.2 Thermionic emission + p-region n-region p-region E V=V DR V=V FB V >V>0 DR x 0 d Fig. 51: Electric field in a p-n-p BARITT diode at different bias conditions We consider a p-n-p structure as shown in Fig. 51. <?page no="58"?> 3. Transit time diodes 49 If the applied voltage is equal to the reach through voltage U DR holes are injected across the potential barrier. Minority carriers are injected at the forward biased p-n junction. The holes injected at x 0 drift through the n-region and are collected at x=d. This thermionic emission leads to an injection current density which is in phase with the driving RF voltage: σE ) ( J 0 1 following Ohm´s law. The total current density is obtained adding the displacement current E jω σE J t Comparing with the expression for the injection current we obtain σ ω arctg ω φ ω σ σ ω A ) ( ) ( 2 2 2 and for the impedance of a Barrier Injection Transit Time diode we have θ σ ωC θ C ω σ θωC σ Z i i D sin cos 1 Re 2 2 2 2 θ θ σ ωC σ ωC C ω σ σ θ θωC Z i i i D sin cos 1 Im 2 2 2 2 Still unknown is the injection conductivity which can be calculated using the Schottky approximation. The flat band voltage follows from Poisson´s equation 2 0 2 1 d N ε ε q V D r FB . The operation regime of a BARITT diode is between the reach through voltage V DR and the flat band voltage V FB . The reach through voltage is V) (V V V V bi FB FB DR 4 where D bi V V V is the diffusion voltage at the junction and is about 0.9V for a Pt-Schottky contact on p-type silicon. The barrier caused by the applied voltage is FB FB pot V V V V 4 2 and the injected hole current is ) exp( t pot FB p V V J J with the thermal voltage q kT V t <?page no="59"?> 3. Transit time diodes 50 Fig. 52: Impedance contour lines of a BARITT diode with µm . D 3 2 , 3 15 10 8 cm * N Parameter is the frequency in GHz The pre-factor J FB is dependent on the kind of the forward biased junction: n p or n - M . The conductivity is then t FB FB p p V V V V J dV dJ σ 2 and the voltage for maximum conductivity is obtained from 0 dV dσ which gives us t FB FB V V V V 2 and 2 t pot V V The maximum amount of the real part is achieved at an operation frequency of 30 GHz at a current density of 900 A/ cm 2 . At a typical diode diameter of 60 µm for Ku band devices this corresponds to a current of 25 mA and a real part of - 1.7 . The DC characteristic of a BARITT diode is symmetrical over a large range of voltages. No avalanche breakdown is observed (Fig. 53). <?page no="60"?> 3. Transit time diodes 51 Fig. 53 IV characteristics of a pnp BARITT diode Fig. 54 shows a photo of a linear IV characteristic from a BARITT diode designed for V-band operation. From the width of the active region (0.75 µm) and the intended doping level (n-type, 2.5*10 16 cm -3 ) a flat band voltage of 10.6 V with a buildt in voltage of 0.9 V and a reach through voltage of 4.5 V is expected. Fig. 54: DC characterization of a V-Band BARITT diode. Origin marked by bright dot. 2V/ div. in x-direction and 1mA/ div. in y-direction. 3.3.3. Tunneling with avalanche injection In case of a time dependent injection current contribution which is related to tunneling (J r (t)) this has to be accounted for in the Read equation S T l C C i J t J adx J dt dJ τ a 0 1 0V <?page no="61"?> 3. Transit time diodes 52 A general solution of this equation is obtained from * dt * t ** dt l ** t α τ e t * ) * (t J J τ C t * dt l * t α τ e (t) J a i T s i a i C 0 1 1 0 1 0 1 1 The evaluation of this equation depends on the integral dt α(t)l a ] 1 [ and it can be shown that a solution is possible, if i T a τ γ l (E) α 1 0 The parameter T depends on the tunneling generation rate [3.3]. The conclusion is, that tunneling reduces the stationary ionization integral and that tunneling causes a negative phase shift compared to pure avalanche injection. 3.4 Loss mechanisms In transit time devices we may have loss mechanisms originating from effects in the active part of the device and loss mechanisms which can be attributed to parasitic effects. As typical representatives of these effects we discuss space charge effects and skin-effect induced losses. 3.4.1. Space charge effects Within the space charge of a transit time device we have a drift current caused by a carrier charge density =J/ v s when the electrons travel at saturated velocity across the depletion region. From the Poisson equation we obtain a local field deterioration s v ε x J x E ) ( We now assume, that all carriers are generated within the avalanche region l a . From the disturbance in voltage caused by the carriers in the drift region I d we obtain ) ε ε (ε ) A I (J εv l J v J dx x E b ΔU s r s d l s x l d d 2 2 0 0 and the space charge resistance is d l d sc Aε l dx x E I R 0 s 2 v 2 ) ( 1 This electronic resistance R SC has a quadratic dependence on l d. <?page no="62"?> 3. Transit time diodes 53 Fig. 55: Current density as a function of breakdown voltage for a p + nn + diode. “x” represents measured values (Length of n region 0.4 m; n doping 2*10 17 cm -3 ). This resistance predicts a linear I-V characteristic after the on-set of avalanche multiplication. This is confirmed for low current densities. If the current density is increased, we observe significant deviations from this approximation (Fig. 55). An improved approximation becomes possible if we introduce a depletion layer width, which is dependent on the current density a dl d l (J) l l with s D a s M dl εv J N ε q l εv J E (J) l and the maximum electric field E M . If we calculate the maximum electric field in silicon from [2.3] V/ cm N * . E eff M 16 10 5 10 log 3 1 1 10 5 4 and consider the effective doping approximation s D eff ev J N N we obtain a very good agreement with the results at high current densities. <?page no="63"?> 3. Transit time diodes 54 The relevance of the effective doping approximation may be illustrated by an example. If we assume C * . cm/ s; q * ; v kA/ cm J s 19 7 2 10 6 1 10 1 100 we obtain 3 16 12 5 10 6 10 6 1 10 1 cm * * . * qv J q ρ s which is in the same order of magnitude as the doping levels in millimeter wave transit time devices. The calculation of the I-V characteristic of an IMPATT diode depends further on the drift region design. We have to distinguish the situations where l dl >l n with n eff M dl l qN εE l for a p + nn + device from the situation l dl <l n where we simply can calculate the voltage from eff M B qN εE V 2 2 These two different situations are distinguished by the punch through factor n dl l l D If the “punch through” is larger than one, i.e. if l dl >l n we calculate the voltage from ε N ql l E V eff n n M B 2 2 3.4.2 The Skin effect Two terminal RF devices have very often a cylindrical geometry. The advantage is, that parasitics can be kept small. However, substrate, buffer or contact regions may contribute to losses. Therefore lossy regions have to be removed. In order to investigate the loss mechanisms in cylinder-symmetric geometries, we consider a straight metallic wire with circular cross section and constant conductivity . In case of an applied DC current we will obtain a uniform current density across the cross section. This is not more the case if we apply an AC or RF current. Now current displacement takes place from the inner to the outer sides. This effect is increasingly important with increasing frequency. Using copper and a frequency of 100 MHz we have current conduction only in a surface layer of approximately 1/ 100 mm! Therefore it is called skin effect. A phenomenological explanation is as follows: If the current density J increases with the time, then the magnetic field will also increase. The magnetic field induces an electric filed which is opposite to the electric field in the inner part of the cylinder the total field and therefore the current density will therefore decrease in the inner part of the structure. <?page no="64"?> 3. Transit time diodes 55 Generally, the electromagnetic field can be obtained from the Maxwell equations 0 0 divH t H -μ rot E divE J rot H The divergence of E and H are chosen to be zero, because we want to know the solution in the inner part of the device and there are no free electric or induced magnetic charges. As the operating frequency of the devices is increased into the millimeter wave range, the current will be confined to flow within a skin depth „Delta“ of the surface of the substrate. To calculate the skin effect induced losses more exactly we have the Maxwell equations in cylinder coordinates r z z r z r r r z z E E H iω H γ σ σE j rH r γ σ σE j 0 μ 1 Harmonic time dependence is already separated with the complex conductivity γ i γ σ ω ε iε γ r Im Re 0 where sigma is the real conductivity. We simplify this assuming no relevant displacement current. σ γ The displacement current can be neglected in highly conductive material up to frequencies of 100 GHz ! Fig. 56: Perspective picture of the model structure The boundary conditions required to solve the differential equations are illustrated in Fig. 56. The current input density takes place with constant current density 2 0 0 πa I (r) j <?page no="65"?> 3. Transit time diodes 56 The current output is assumed to take place on an infinitesimal ring with radius a 0 ) a δ(r πa I (r) j 0 0 0 1 2 with the DIRAC function (r-a 0 ) {e.g. appendix} We now get from Maxwell´s equations )H p r r ( z r r 0 1 1 2 2 2 2 with 2 2 1 p δ i p Here the penetration depth is 2 2 / 1 0 / δ depth skin σ ωμ δ s p With the help of Fig. 56 we deduce: At z=0 the current density is given j z (r,z=0)=j 0 (r) and at z=L the current density is given j z (r,z=L)=j 1 (r). There is no current flow from the cylinder jacket j r (r=a,z)=0. Lines of current flux are commonly given by dr: dz=j r : j z and we can derive an implicit representation of the lines of flux C s e B e A x) (ζ xJ (pa) J (pax) J x π I rH y k s y k s s s s 1 1 1 0 2 with x=r/ a and y=z/ a. A s and B s are to be determined from the boundary conditions and s represent the zeros of the Bessel function of first order J 1 ( s )=0 and k s2 = s2 -(pa) 2 . Fig. 57: A plot of lines at which the current amounts to 36.9% (=1/ e) (Cylinder with overall radius a and current extraction radius a 0 ; Parameter cylinder length L in µm; z: Vertical coordinate; r: radial coordinate) Typical dimensions and quantities for a 100 GHz two terminal device are <?page no="66"?> 3. Transit time diodes 57 diameter [m] 30 Substrate height [m] 2-20 skin_substrate [m] 5 contact thickness [m] 0.1 skin_contact [m] 0.236 (gold) In order to have a more quantitive measure of the skin effect we introduce the definition of an effective ohmic resistance via Joule’s heat power 2 σE dV N from which we define the effective ohmic resistance 2 0 I N R eff From the preceeding discussion and the calculation of the current distribution in cylindric semiconductor material solving Maxwell’s equations we evaluate an effective value of the skin effect resistance 1 2 tanh 2 1 1 Re 1 2 2 , 1 4 2 s r L k L k r k j δ r σπr L R s s s s p eff DC Skin L : Substrate thickness, r: device radius, j 1,s : zeros of the Bessel function J 1 , 2 2 , p s j s δ r j j k Fig. 58: Effective ohmic resistance as a function of the cylinder length L. Parameter: cylinder radius r <?page no="67"?> 3. Transit time diodes 58 We see from Fig. 58, that the effective ohmic resistance can reach values which are comparable to usual device impedance values in the millimeter-wave range.If we look at Fig. 59, which shows the effective ohmic resistance normalized with respect to the DC resistance, we see that there are especially unsuited geometries. Fig. 59: Normalized effective resistance as a function of the normalized cylinder length for different values of p / a The figure shows the effective normalized ohmic resistance and hence the heat power produced in the semiconductor cylinder dependent on the normalized cylinder length at different penetration depths. The effective ohmic resistance, related to the DC resistance shows at first a quadratic increase with increasing length. At very long cylinders we get a limit which is length independent and the effective resistance increases proportionally to the length like the DC resistance. The skin effect is responsible for a reduction of the effective area and hence for an increase of resistance dependent on the penetration depth. At medium cylinder lengths and penetration depths which are small enough, this effect is accompanied by an extension of the current path, because the current displacement in the bottom and top region now makes itself noticeable in comparison to the DC case. This provides a maximum of the normalized effective resistance in the region of medium cylinder lengths for small penetration depths and large cylinder radius, respectively. We identify regions with p / a>0,2 and L/ a<2/ 3 as the limiting cases for the doubling of the substrate resistance relative to the DC resistance. This can be applied for a 100 GHz device fabricated on high conductivity silicon and results in a substrate thickness requirement of less than 9m. 3.5 The IMPATT diode In an early work the use of transit-time effects was discussed as a mechanism to generate a frequency dependent negative resistance which may be used for the generation of oscillations [3.4]. Later these ideas were implemented in semiconductor devices [3.5]. W.T.Read recognized the suitability of the impact ionization and avalanche multiplication mechanisms for the injection of hot carriers, and proposed the IMPATT diode (Fig. 60) [3.6]. <?page no="68"?> 3. Transit time diodes 59 Fig. 60: The original READ concept Fig. 61: Single and double drift IMPATT diodes and BARITT diode <?page no="69"?> 3. Transit time diodes 60 The READ concept is very often modified to be applied to flat profile devices as is shown in Fig. 61. Flat profile means a constant doping in the active region. That is n or p in single drift devices and means n and p in double drift devices. In double drift devices electrons and holes contribute to power generation. Fig. 62: IMPATT doping profiles In Fig. 62 the most important IMPATT structures are summarized. In the structures with doping spikes a confinement of the avalanche region widths is aimed at. For a design optimization of sophisticated double drift structures at millimeter waves it is necessary to analyze the large signal behaviour of the respective behaviour without the restrictive assumptions made with the analytical large signal model. The analysis of the coupled semiconductor equations for this purpose is done using numerical schemes: Continuity Generation Current density Poisson n) p q(N δx δE ε p p n n v p α v n α G G δx δJn q δt δn 1 δx δn qD n qv J n n n δx δp qD p qv J p p p G δx δJp q δt δp 1 <?page no="70"?> 3. Transit time diodes 61 Usually these equations are solved applying a constant current density and a DC voltage with a superimposed RF voltage swing Fig. 63: Large signal simulation of E-field distribution in DD device: Electric field in [10 5 V/ cm] as function of the x-coordinate within the device. Fig. 63 shows the electric field within an optimized flat profile double drift diode at four subsequent moments during one period of the RF cycle. It can be seen that inspite of the large field swing there are no undepleted parts of the semiconductor. Fig. 64: E field distribution in DLHL device (Large signal simulation) The DLHL - Design of doping spikes is very sensitive with respect to the absolute levels of doping in the spikes and with respect to the width of these spikes. Fig. 65 shows a SIMS (Secondary Ion Mass Spectroscopy) measurement of the doping profile of a V-Band (50 - 75 GHz) DLHL strcuture. <?page no="71"?> 3. Transit time diodes 62 t Fig. 65: Silicon DLHL IMPATT on n+ substrate, phosphorus doped n-type doping: Sb (Antimony) and p-type doping: Ga (Gallium) The mesa structure and the individual layers and thicknesses of a 100 GHz Quasi Read double drift diode are shown in Fig. 66. Fig. 66: Mesa structure of a Quasi Read Double Drift Region Diode <?page no="72"?> 3. Transit time diodes 63 Fig. 67: Scanning electron micrograph of an IMPATT mesa In the SEM picture the transition from the active regions to the substrate rest can be well recognized due to the different etch behaviour of the layers 2 . Breakdown voltage is 12.6 volts Fig. 68: I-V characteristic of DD IMPATT A suited package of millimeterwave IMPATT diodes consists of a carrier with high thermal conductivity, e.g. copper, a surrounding quartz ring and a contact ribbon. A cross section of a packaged device is shown in Fig. 69. 2 The I-V characteristic typically reveals a sharp breakdown in reverse direction (Fig. 68, linear representation). 0V <?page no="73"?> 3. Transit time diodes 64 Fig. 69: Cross section of a package with a stud to obtain low capacitance and short contact ribbons to obtain low inductance For optimum heat removal diamond heat sinks are used (Fig. 70). High power diodes benefit from the thermal conductivity of diamond Cu κ W/ cmK IIa Dia κ 4 20 Fig. 70: Cross section of the mounting of high power silicon IMPATT diodes with diamond heatsinks. Dimensions in µm if not otherwise stated. Stub Diode Contact ribbon Quartz ring Quartz ring <?page no="74"?> 3. Transit time diodes 65 Fig. 71: SEM of thermo-compression mounted diode in quartz ring package Crossed ribbons seem to be the best compromise between the low inductance requirements and the handling properties (Fig. 71). The packaged diode is then introduced in a wave guide resonator, contacted with a bias pin with low pass filter sections (“choke”) and tuned with a short. The resonator may be tapered from standard waveguide height to ease impedance matching (Fig. 72). Fig. 72: Cross section of a waveguide oscillator circuit 3.5.1 Design constraints a) Current density limits The design of transit-time devices is strongly affected by the device area. Thermal limitations are of special importance for mm-wave IMPATT diodes. The input power P in =VJA is limited by the maximum allowed junction temperature rise, T P in . If we use for simplicity <?page no="75"?> 3. Transit time diodes 66 A 1 which denominates the spreading resistance of the heat sink, we obtain the thermal current density limit A V T J th . If ionization has to be avoided in the drift region, a space charge limit can be derived from Poisson’s equation and with Q/ E we obtain Q=2 J/ or E J sc 2 1 . For IMPATT diodes the avalanche resonance limit provides a further constraint on the maximum current density. The current density must be below J av to ensure READ type operation. s av v J 3 2 3.5.2 Matching considerations The power delivered to the load is with describing the matching losses. The power generated by the diode is with the intrinsic conversion efficiency i . Fig. 73: Oscillator equivalent circuit In Fig. 74 experimental data from W-band (75-110 GHz) IMPATT diodes are represented. It can be seen, that with Quasi Read doping profiles higher output powers at lower current densities are obtained than with flat profile devices. D s L R R v 1 VJA R A J P i D t D 2 2 2 1 -jX -jR jX R D D L L R S L D L v P P <?page no="76"?> 3. Transit time diodes 67 Fig. 74: RF power (cw mode) of silicon IMPATT diodes. W-band frequencies. The conversion efficiency i depends on the current density as does the negative resistance. It also depends on the capacitance and device area, respectively. The measured data (Fig. 75) contain matching losses, which play an increasing role if the device capacitance increases, i.e. if the device impedance reduces. Fig. 75: Efficiency of QRDD IMPATT diodes <?page no="77"?> 3. Transit time diodes 68 3.5.3 Pulsed mode Considering the small signal impedance of IMPATT diodes it is found from Read´s equation and using the small signal approximations with 1 0 ´ 0 ´ 1 E J l J j i a and 1 a l at breakdown, that the total current density is given by ) 1 1 1 ( 1 2 0 ´ 0 ´ 1 1 1 0 ´ 0 ´ 1 1 1 J l E j E j E J l j E j J J i a i a tot A pole of the current density is reached if the avalanche frequency a reaches the operation frequency: This means, that the total current is zero if the avalanche frequency equals the operation frequency. Therefore we expect at very high current densities which may be reached in short pulse operation at first a decrease of the output power and finally a quenching of oscillations. The experimental observations deviate from these findings. In Fig. 76 we observe a moderate increase of the pulsed RF output power at current densities up to 100 KA/ cm 2 . The oscillation frequency simultaneously increases from 92 GHz to 104 GHz. If the current density is further increased - which shifts the avalanche frequency beyond the operation frequency - a sudden increase of the RF power and a constant oscillation frequency of 96 GHz is observed. Fig. 76: IMPATT operation below the avalanche frequency.Avalanche resonance frequency is marked; measurement results are given: circles represent power values crosses mark frequencies a l a a c i c l dx l J dt dJ 0 ) 1 ( 1 t j C e J J J 1 0 0 ´ 0 2 J l i a a with <?page no="78"?> 3. Transit time diodes 69 Fig. 77 shows current and RF pulse of this high power operation mode. At 90 GHz operation frequency values of more than 40 W output power with pulse lengths between 10 and 100 nsec and a duty cycle of 1% are possible. Using diode diameters of 120 m the currents reach peak values of 20A. Fig. 77: DC current and RF pulse from DD-MISAWA type IMPATT diode Fig. 78: SIMS profile of a DD diode for pulsed operation at 94 GHz The SIMS profile analysis of a high power pulsed IMPATT diode is shown in Fig. 78. The n-side and p-side of the double drift diode are comparatively high doped for W-band operation (3*10 17 cm -3 and 2*10 17 cm -3 ) and are not yet depleted at punch through. However, if we consider the influence of the injected carriers at the employed high current densities, we expect a significant broadening of the depleted regions and of the avalanche region. Large signal simulations show, that the device in fact at current densities above 100 KA/ cm 2 has avalanche multiplication across the whole device and operates as a pin - or MISAWA type diode. RF pulse 42 W 20 A Current pulse (10A/ div.) <?page no="79"?> 3. Transit time diodes 70 Assuming a flat E-field we can explain the pin/ MISAWA type of operation: Fig. 79: Dynamic description of pin diode behaviour At a time step "a" a small excess of electrons and holes is assumed in the center of the diode. These carriers will drift to the electrodes with increasing concentration due to avalanche multiplication ("b"). The large amount of carriers causes a decrease of the electric field in the center of the diode and consequently a minima during the voltage cycle. At the same time the current is maximum due to the maximum of the carrier density. Due to the decreased field in the center of the diode there is now a lack of carriers ("c") which will drift in the opposite direction. At time step "d" the electric field is increased in the center of the diode due to the carriers which have reached the electrodes. This means voltage is maximum and current is minimum. To close the cycle, the field maximum in the center of the diode will lead to the situation of time step "a". The occurence of a negative resistance is obvious from the phase shift between voltage and current. 3.6 Integrated transit time devices To realize planar integrated transit time devices a semi-insulating or high resistivity substrate (HR) is used [e.g. 10000 cm material as indicated in Fig. 80]. Fig. 80: Scheme/ cross section of coplanar silicon IMPATT with an epitaxial grown etch stop and contact layer(design for W-band operation) <?page no="80"?> 3. Transit time diodes 71 The pnn + double drift layers are grown by Si- MBE on a highly p-doped etch stop and contact layer. To achieve a small series resistance, large p contacts are defined in a photoresist process and formed by an isotropic etch (HNO 3 - HF). A second photoresist process is adjusted to define the top contact of the diodes. In an anisotropic etch (KOH) the mesalike diode is formed and the junction area is controlled. The etchant has a self-stopping behaviour with respect to the p + -B/ Ge layer. The resist serves as passivation of the diodes. Titanium and gold are evaporated and the thickness of the gold layer is increased up to 3 µm by electroplating. Fig. 81: Scheme of a molecular beam epitaxy machine In Fig. 81 a scheme of a Silicon molecular beam epitaxy (MBE) equipment is represented. The ultra high vacuum (UHV) chamber is loaded through a gate valve from a wafer storage chamber, where up to 20 wafers can be stored under low vacuum conditions. The wafers are fixed in substrate holders. Under UHV conditions the wafer is then exposed to a molecular silicon beam which can be generated by electron gun controlled evaporation. Doping is done by evaporation from effusion cells: Ga, B for p-type doping and Sb, As, P for n-type doping. Important for the dopant incorporation is the substrate heating. Typical values are in the range between 450 and 750 o C (Typical values for CVD (chemical vapour deposition) are 1000-1200 o C). There are several measurement and analysis instruments to monitor the layer growth: Quadrupole Mass Analysis (QMA) and Reflection High Energy Electron Diffraction (RHEED). <?page no="81"?> 3. Transit time diodes 72 Fig. 82: SEM micrograph of a planar IMPATT diode The SEM micrograph of a quasi-planar diode (it is not really planar as the layer sequence is still vertical) clearly resembles the different layers from the device concept (Fig. 82). The resonator structure may then be formed by a final photoresist and etch step. P + -Diffusion MBE Mesa-Definition Self-Aligned Process SiO 2 SiO 2 SiO 2 SiO 2 Fig. 83: Scheme of coplanar BL(buried layer)-IMPATT technology Fig. 83 shows a different technology where buried layer (BL) substrates are used. Highly doped (> 10 20 cm -3 ) buried layers are realized on HR substrates by multiple diffusion of Arsenic (a). SiO 2 covers the wafer where no BLs are located. Then the active layers are grown by MBE. N + contact a b c d P + contact Substrate <?page no="82"?> 3. Transit time diodes 73 Monocrystalline growth is obtained within the opened BL windows whereas the polycrystalline material on the SiO 2 is removed later on in an etch step (b). A gold metallization is used as an etch mask for the definition of the active device area. A thick mushroom - like gold contact (c) eases the realization of a self - aligned contact to the p + - BL in order to reduce contact resistances as far as possible (d). An airbridge technology can be applied to reduce parasitic capacitances and reduce variations in bond wire parameters. A sputter deposition of silicon nitride (“cold” process at T= 600K) provides a diode passivation (Fig. 84a). Then the isolated diode structure is covered with a thick photo resist (3...5 m) and a photo litography process is applied. The structured photoresist is used as a mask to open the nitride layer (Fig. 84b). The opened windows are then covered with an evaporated metallization layer: the thin plating base. Again a lithography process is applied with a thick resist and the contact windows are opened (c). The gold layers are electroplated to a thickness of several m. Typical values of gold layer thickness are 4 m. The remaining photo resist is removed using solvents and ultrasonic: The thin Ti/ Au layer is removed where it was covered by photo resist. This is a lift-off process. Fig. 84: Airbridge process The IMPATT diode can now be integrated in a resonant structure. In order to match the IMPATT diodes impedance the impedances of diode and resonator have to be conjugate complex. a b c d <?page no="83"?> 3. Transit time diodes 74 In order to care for the reduced real part of the diode under large signal operation it is a good approach to assume that the real part of the resonator part is three times lower than the real part of the small signal diode impedance. A symmetric dipole is a suited structure for low impedance levels at high frequencies (Fig. 85). With this configuration optimized for 76 GHz a radiated CW output power of up to 10 mW can be obtained (Fig. 86). Chip size: 3000 m x 2000 m Fig. 85: Symmetric Dipole Fig. 86: Oscillation frequency f and radiated power P versus bias current I 3.7 Self oscillating mixer operation Fig. 87: Fuze block diagram Transit time diodes may be operated in a self-oscillating mixer (SOM) mode. The SOM mode was first introduced in proximity sensors [3.8] (Fig. 87). As the current - voltage characteristic of transit time devices can show a strong dependence on the load, a direct detection of a Doppler frequency is possible. To understand the influence of the SOM sensitivity on a radar system the round trip loss (RTL) is introduced: RTL= P 0 - MDS and the Minimum Detectable Signal is given by MDS = P N -G with the conversion gain G and the low frequency noise power P N . All quantities are given in logarithmic units. It can be shown that the most important contributions to the conversion gain origin from the effects of RF recification, AM modulation and power matching [3.7]. If we calculate the maximum range of a radar system from the radar equation S R l L N S G MDS P R p 5 / 3 2 0 2 0 4 10 ) / ( ) 4 ( where we assume an antenna gain G of 30dB, a RTL= 100 dB, a wavelength of 0 =3.75 mm corresponding to a frequency of 80 GHz, a radar cross section of <?page no="84"?> 3. Transit time diodes 75 =1 m 2 , a S/ N= 10 dB, an antenna efficiency L S =1.55 dB ( 70%) and an attenuation of l p = 0.02 dB/ m corresponding to heavy rain we obtain a maximum radar range of R= 42 m. In order to measure MDS the set-up in Fig. 88 can be used. The radiated power from the SOM is fed into a waveguide measurement set-up, the power is measured, fed through an attenuator and a tunable phase shifter to a switchable short. The short is controlled by a square wave modulator which leads to a change in the reflection coefficient between “1” and “0”. In other words, the signal is reflected back to the SOM with the same carrier frequency but with sidebands representing an AM. In order to adjust known phase relations between the side bands; the phase shifter is required. The SOM is connected to a frequency selective voltmeter and the “Doppler” voltages can be measured at selected offset-frequencies in certain bandwidths. The attenuation is increased up to the value where the downconverted signal vanishes in the noise floor. SOM absorber short Fig. 88: MDS measurement set-up A very simple method to determine RTL from DC measurements is possible, if we assume that the conversion gain is only due to RF rectification. Then the RTL is approximately 2 0 , / 2 n DC HF V V RTL with the voltage shift 2 ' ' ' / 2 ) ( 4 mE l V a DC HF ‘, ‘‘ : first and second derivative of ionization rate with respect to the electric field l a : avalanche region length m : modulation depth E : electric field and the mean square of the low frequency noise voltage IU n,0 I 2 in an IMPATT diode. The low frequency noise voltage of the SOM can be measured at the operating current. The voltage shift can be determined from the measured voltage at the operating current under oscillation conditions and from the measured voltage with suppressed oscillations. In Fig. 89 measured RTL is shown as a function of the space charge resistance of different integrated SOMs with double drift diodes. It can be seen that the RTL variies between 105 and 135 dB. Additionally it is found that for optimum SOM operation the diodes should be designed with a punch through factor >1; i.e. the drift regions should be shortened compared to a maximum efficiency design. <?page no="85"?> 3. Transit time diodes 76 Fig. 89: RTL of 61 GHz Self Oscillating Mixers employing planar integrated IMPATT diodes as a function of the space charge resistance 3.8 Discussion and Conclusions Transit Time devices can deliver high output powers at microwave and millimeter wave frequencies. The level of the output power and the operation frequency depend on the impedance matching between resonator and device. The diode doping profile, the device area, parasitics and the resonator impedance influence the matching. The noise behaviour of the oscillator depends on the quality factor of the resonator as well as on the device properties. As some important device parameters like ionization rates, barrier heights, tunneling rates, drift velocities are temperature dependent so are the oscillator properties. Therefore frequency stability of two terminal oscillators is an issue. Fig. 90 shows the variation of operation voltage, oscillation frequency and radiated power as a function of the temperature for a 76 GHz monolithically integrated IMPATT oscillator. A frequency variation of 0.7% over the temperature range from -10 o C up to 85 o C may be too much for applications and may violate regulatory issues. Fig. 90: Temperature dependence of the oscillation frequency change f´ osc ,operation voltage change U´ 0 and the radiated power P rad of a 76 GHz oscillator 76 GHz <?page no="86"?> 3. Transit time diodes 77 Possible approaches to improve the frequency stability include frequency stabilization with high quality factor resonators as a passive concept. Active stabilization is done using frequency dividers and set-up of a PLL either tuning the bias conditions of the device or a varactor. A third possibility is to synchronize the two terminal oscillator with a stable oscillator which may operate at a lower frequency f sync . This subharmonic synchronization at n f f sync 0 (with n= 1,2,3...) is an interesting concept as it shifts the problem to generate a stable signal to lower frequencies. The drawback is, that also the signal to noise ratio S 0 / N 0 of the signal is reduced corresponding to the number n n N S N S sync sync log * 20 / 0 0 The thermal properties do not only affect the noise properties but also the lifetime. Fig. 91 shows the Median Time Between Failure (MTBF) related to the junction temperature. It can be recognized that reducing the junction temperature can greatly improve the lifetime. Fig. 91: MTBF and temperature (after Sze) Distributed structures (e.g. ring geometries), improved heat sinking and active cooling can improve the heat removal whereas doping profile optimization can increase the conversion efficiency and reduce the required input power. Most published results ! <?page no="87"?> 3. Transit time diodes 78 Problems A single drift IMPATT diode made from silicon material has a 0.5 µm wide 2*10 16 cm -3 Antimony doped active layer between a p + contact (10 19 cm -3 ) and a n + contact (2*10 19 cm -3 ). o Determine the maximum electrical field if U=0 o Draw the electric field distribution if U=20V and calculate the maximum electric field at the p-n junction o When the current density is 20 kA/ cm 2 , find the effective doping concentration in the intrinsic layer. Assume the electron saturation velocity is 10 7 cm/ s. o Using the effective doping approximation, find the maximum electric field o Draw the electric field distribution again o Calculate the depletion layer length using the space charge approximation method. o Determine the punch-through voltage In IMPATT diodes, operation near avalanche frequency may lead to increased output power, which cannot be explained by conventional READ theory. Discuss this phenomena. In IMPATT devices, what is the optimum transit angle for maximum efficiency ? In BARITT diodes of pnp type, of which kind are the injected carriers ? References [3.1] A. Möschwitzer, K. Lunze, Halbleiterelektronik, 1984, VEB Verlag Berlin [3.2] L.H.Holway, S.L.G.Chu, „Theory and Measurement of back bias voltage in IMPATT diodes”, IEEE Trans. MTT-31, 916-922 (1983) [3.3] J.F.Luy, R.Kühne, “Tunneling Assisted IMPATT Operation” IEEE Transactions on Electron Devices, vol. 36., 1989, pp. 589-595 [3.4] J. Müller; Elektronenschwingungen im Hochvakuum, Hochfrequenztrechnik u. Elektroakustik, 41, 1933, 156-167 [3.5] W. Shockley; Negative Resistance arising from transit time in semiconductor diodes; Bell Syst. Tech. J., 33; 1954; 799-826 [3.6] W.T.Read; A proposed high-frequency, negative resistance diode; Bell Syst.Tech. J., 37, 1958; 404-446 [3.7] M.Claassen, “Self-Mixing Oscillators”, in ´Silicon-Based Millimeter-Wave Devices, ed. by. J.-F.Luy and P.Russer; 1994; Springer Verlag, Heidelberg [3.8] M.Skolnik, „Radar Handbook“; McGrwal-Hill, 1990 <?page no="88"?> 4. Schottky diodes 79 4. Schottky diodes Schottky contacts can be considered like pn contacts where one part of the junction (p or n) is replaced by a metal (Fig. 92). The space charge is only in the semiconductor part. Because there is almost no diffusion capacitance, Schottky diodes can be very fast. Therefore they are suited as switch or mixer diodes. Fig. 92: Metal - semiconductor junction 4.1 Schottky contact modeling To discuss the basic properties of Schottky diodes it is helpful to consider the bandgap physics of the metal-semiconductor transition before and after intimate contact of the materials according to the Schottky-Mott model [4.1]. Fig. 93: a) Metal and n-type semiconductor before contact b) Metal and n-type semiconductor after intimate contact c) Metal and p-type semiconductor after intimate contact In Fig. 93 the following abbreviations are introduced: : energy difference between fermi energy and conduction band edge, W FM , W F : Fermi levels within the metal and the semiconductor, <?page no="89"?> 4. Schottky diodes 80 p b n b g W ε qN Ψ(z) z D 2 2 ). / ln( D C N N kT 2 w 2 z) ( ε eN Ψ(z) D w max D eN E w: space charge region width, V d0 : band bending voltage at zero bias; W g : bandgap difference W C -W V . From the work function M of the metal and the electron affinity S of the semiconductor the barrier for electrons bn can be determined It can be seen that the band-gap energy is given by However, in reality the barrier height does not increase linearily with the work function of the metal. This finding is usually ascribed to interface states. So we have to look a bit more in detail of the Schottky contact. In the space charge region of the Schottky diode we can determine the electrostatic potential (z) from the Poisson equation with the doping level N D in the semiconductor region of the Schottky diode. In this equation we have neglected free carriers within the space-charge region. The solution for this “depletion approximation” is a parabolic solution for (z) . The electric field E(z) increases linearly from the edge of the space charge region up to the maximum. The space charge region width w is related to the bending or diffusion potential V D with and And we can derive for the total charge In order to describe deviations from the Schottky type model, Bardeen developed a model which considers interface states. The semiconductor surface without metal has already a certain density of surface states within the forbidden gap. Even without being in contact with the metal the bands of the semiconductor can therefore be bent upwards (Fig. 94). D D D V qN qN 2 w sc Q S M n b D D V qN 2 w qV b qV D <?page no="90"?> 4. Schottky diodes 81 F d W qV 0 0 F W Fig. 94: Metal and n-type semiconductor with surface states (a) before contact and (b) after contact with the semiconductor An interfacial layer of width and a charge neutrality level 0 of the interface states are introduced. The total charge in the surface states is where D it represents the density of surface states per unit area and unit energy and V d0 the band bending voltage for zero bias conditions. The charge neutrality condition expresses the counterbalance between surface state charge and space charge! From the charge neutrality condition and the equation for the total charge in the surface states we obtain the pinning of the Fermi level by a large number of surface states D it . It is required that This can only be obtained if corresponds to the energy location of the neutrality level In the Bardeen limit, the Schottky Barrier height is independent of the chemical nature of the metal and depends only on the neutrality level 0 of the surface states on the semiconductor: 0 ) ( 0 0 g d W qV ) ( 0 0 g d it SS W qV qD Q 0 g n b W 0 sc Q ss Q <?page no="91"?> 4. Schottky diodes 82 . 0 M b d d . 3 2 c c M n b . 1 M b d d We have obtained from the Schottky-Mott model the result, that the Schottky barrier height varies linearly with the work function of the metal according to Whereas in the Bardeen model the Schottky Barrier height becomes independent of the metal Often both models do not predict experimental observations. Experimental data may be approximated by Experimental data for the barrier heights versus the work function of the metal are shown in Fig. 95 with n-type silicon. Fig. 95: Experimentally determined barrier heights as function of the metal work function on n-type silicon It can be seen, that the Schottky-Barrier height varies with the work function of the metal. The experimental data can be fitted from The band diagram of a Schottky contact is used to illustrate the different contributions to the carrier transport. Fig. 96 shows the band diagram for forward bias. 175 . 0 2 M b d d c S eV c 12 . 0 3 <?page no="92"?> 4. Schottky diodes 83 Fig. 96: Band diagram of a Schottky contact under forward bias. Electrons as majority carriers From Fig. 96 different carrier transport mechanisms can be distinguished: Pure thermionic emission (a) Thermionic field emission (b 1 ) Pure field emission (b 2 ) The Quasi Fermi level for thermionic emission is indicated by the dashed line and the Quasi Fermi level for diffusion is given by the dotted line. Correspondingly the transport processes for electrons as majority carriers are a ) Emission of electrons over the top of the barrier b 1 ) Thermally assisted tunneling b 2 ) Tunneling at the bottom of the conduction band in the quasi neutral region Fig. 97: Band diagram of Schottky contact involving minority carriers <?page no="93"?> 4. Schottky diodes 84 If additionally minority carriers are considered, we have the situation shown in Fig. 97 with Quasi Fermi levels n and p split across the quasi neutral region of the sample. The additional transport mechanisms are c ) Recombination via deep trap levels d 1 ) Recombination in quasi neutral region d 2 ) Recombination at the back contact The total diode current is the sum of the contributions of the individual current paths. The transport processes a to d are connected in parallel. Usual models for these transport processes describe electron transport only in one dimension and assume that Schottky barrier is spatially uniform. 4.2 Basic structures of Schottky Barrier diodes for high frequency applications The choice of the substrate for the realization of a Schottky diode depends on the intended use of the diode. For a discrete device a low resistivity, highly doped substrate may be used in order to reduce series resistances. For integrated, planar devices a high resistivity or semi-insulating substrate is used in order to avoid dielectric losses within the carrier material of the integrated circuit (Fig. 98). Fig. 98: Structure for a discrete device (left) and integrated device (right) The highly doped cathode may be realized by diffusion. Fig. 99 shows a practical example. Fig. 99: Basic layout of an integrated Schottky diode <?page no="94"?> 4. Schottky diodes 85 1 0 kT qV e J J kT b e T A J 2 * * 0 The layer sequence suited to mm-wave Schottky diodes consists of a 100-200 nm thick epitaxial layer with a doping concentration between 1*10 16 cm -3 and 2*10 16 cm -3 grown on highly n-doped buried layers. Both p-type and n-type Schottky-barrier diodes can be grown. Fig. 100 shows a coplanar layout with an anode finger providing low parasitic capacitances. Fig. 100: Coplanar layout of integrated Schottky diode 4.3 DC Analysis of Schottky diodes The I-V characteristic of a Schottky diode is described by the thermionic field emission theory with the voltage V, the Boltzmann constant k and the temperature T. The saturation current density is given by with the effective Richardson constant A ** which is A ** =112 A/ (K*cm 2 ) for n-type silicon and A ** =32 A/ (K*cm 2 ) for p-type silicon. It is obvious, that the saturation current is dependent on the barrier height. If it is intended to reduce the barrier height and to increase J 0 a thin highly doped layer (-doping) may be introduced between metal and active layer in order to make the semiconductor at its surface more like a metal. Fig. 101 shows the layer sequence of a p-type barrier lowered Schottky detector. Fig. 101: Layer sequence of p-type mm-wave Schottky diode with delta-doping in order to reduce the barrier height <?page no="95"?> 4. Schottky diodes 86 ) ln( ) / ln( * * 2 0 kT A T J b The ultimate goal in reducing the barrier height may be to obtain zero bias operation capability. This leads to a lowering of the threshold voltage (Fig. 102). Fig. 102: linear I-V of silicon p-type Schottky diode. Zero marked by a bright dot The non-ideal behaviour of Schottky diodes is usually described by the introduction of an ideality factor n, so that the current-voltage relation for voltages V>3kT/ e reads as with the zero-bias Schottky barrier b0 . The dependence of the ideality factor n on the voltage dependence of the current Schottky barrier bj is given by It is straightforward, that the Schottky barrier height can be determined from I-V measurements. If the equation for the diode current is written in the form Which corresponds to a straight line fit in semilogarithmic representation, then the saturation current is obtained from the y-axis intercept and the ideality from the slope e/ (nkT). If the diode area A and the effective Richardson constant A ** are known, the barrier height can be obtained from If the effective Richardson constant is not known, the barrier height can be obtained from an activation energy plot (Richardson plot) of The slope of the Richardson plot yields the barrier height and from the y-axis intercept one obtains the effective Richardson constant. 1 exp 0 nkT qV AJ I nkT qV kT T A J b exp exp 0 2 * * n dV d q j b 1 1 1 kT T A J b exp 2 * * 0 <?page no="96"?> 4. Schottky diodes 87 2 2 2 sc 2 ) ( C 1 A N q kT qV V D kT b i qV 2 2 2 A εN q dV (V) dC D sc j R 0 0 The space charge in the depletion region is dependent on the voltage. It can be shown, that the space charge capacitance Csc of a Schottky diode of area A can be represented by with denominating the energy difference between the Fermi energy and the conduction band energy. This formula shows that a plot of the measured capacitance (1/ C 2 ) versus the voltage yields a straight line, with an x-axis intercept at V=Vi. The Schottky barrier height can be obtained from The slope of the straight line yields the doping concentration in the semiconductor 4.4 The Schottky diode detector The operation of a Schottky diode detector is described using the equivalent circuit of Fig. 103 Fig. 103: Detection Circuit [4.2] For the low frequency quantities the relation holds between the junction resistance the current sensitivity and the voltage sensitivity In the detection circuit the RF source is represented as a voltage source with impedance Z 0 which is capacitively coupled to the diode. dI dV i R 1 det 0 P I . 1 det 0 P V <?page no="97"?> 4. Schottky diodes 88 . 1 ) ( 2 0 0 0 V V g V V R I V I J J J J D ) ( / 1 0 S J I I q nkT dV dI R ) cos( 1 0 t V V V J J . ) )( 2 1 ( 2 1 0 0 S DC J DC R R gV I V V 0 2 1 ) ( 2 1 0 0 J gV V I DC S J S V R gV V I R V 2 1 0 0 0 2 1 ) ( The DC bias voltage V DC is fed through a resistor and an inductor to the diode, so that a Taylor series expansion around the bias voltage V 0 taking the first three terms yields For the Schottky diode we had derived and considering the detection circuit it follows the voltage at the diode and the current through the diode is obtained as This formula illustrates the importance of the diodes non-linearity: a DC current contribution is proportional to the second derivative of the current with respect to the voltage (g)! This DC current contribution can influence the bias point depending on an external load resistance as the DC Voltage is now obtained from according to the DC equivalent circuit (Fig. 104) Fig. 104: DC equivalent circuit of the detector It is now straightforward to distinguish two possible extreme cases: a) There is no bias voltage and the total current is zero b) If there is a DC short circuit with R DC =0 the voltage can be calculated from 2 0 2 2 2 ) ( 2 ) ( nkT I I q dV I d g S )] 2 cos( 1 [ 2 1 ) cos( 1 2 1 1 0 t gV t V R I I J J J D <?page no="98"?> 4. Schottky diodes 89 2 2 2 2 1 1 ) / ( 1 / 1 ) ( 2 S J J S J S J J S J J S J R R R R C R R R R R C R R V P and the detected current is which may be further simplified with In order to perform a sensitivity analysis it is helpful to introduce a diode equivalent circuit (Fig. 105) Fig. 105: Detection diode equivalent circuit The voltage divison network causes a voltage across the junction which holds for a fundamental harmonic approach. The dissipated power is and the DC current sensitivity () 1 det 0 P I can be calculated from the DC analysis with R J >>R S which corresponds to 19.3 [A/ W] at room temperature (n=1, T=300K). The voltage sensitivity becomes The RF analysis yields (also neglecting R S with respect to R J ) which corresponds to a low pass characteristic. It is obvious, that for a good highfrequency performance a low junction capacitance and a low series resistance should be obtained. 2 ! 0 0 det 2 1 ) ( ) ( J DC gV V I V I I 1 1 V R R C j R R R V S J J J R J J nkT q gR J 2 0 S J J R R C 2 2 0 1 0 ) ( ) ( 0 0 DC V I V I ) ( 2 1 0 0 S <?page no="99"?> 4. Schottky diodes 90 In order to reduce the junction resistance, the temperature may be reduced, and a forward bias may be applied (increase of I 0 ) and the saturation current may be increased by reducing the barrier height (zero or low-barrier diodes). Schottky diode detectors for millimeter-wave operation are realized in waveguides employing whisker contacts or fin-line technology (Fig. 106). Fig. 106: 94 GHz Schottky diode detector in waveguide-finline technology Another frequently used figure of merit for the characterization of microwave detectors is the tangential sensitivity (TSS). It is defined as the input power in dBm which is required to change the DC voltage output by an amount which is equal to the noise fluctuations [4.1]. Fig. 107 shows TSS measurements dependent on the bias conditions. The tangential sensitivity improves from -35 dBm at zero bias up to -50 dBm at 0.2 V bias voltage. Fig. 107: Tangential sensitivity and DC current as a function of the DC voltage for a W-band Schottky diode detector The voltage sensitivity of a detector circuit which is integrated in an antenna circuit is normalized with respect to the effective antenna area (Fig. 108). With silicon Schottky diodes integrated in antenna structures on high resistivity silicon substrate (SIMMWIC technology) a sensitivity of 83 mV/ mW/ cm 2 is obtained which can be increased up to 150 mV/ mW/ cm 2 with bias. <?page no="100"?> 4. Schottky diodes 91 Fig. 108: Detector voltage as a function of the RF power density In a mixer operation mode the conversion loss characterizes the high frequency properties of the device and the circuit. Fig. 109 shows a flip-chip mounted double series Schottky diode configuration in a front-end for a 76 GHz Intelligent Cruise Control Radar System. The conversion loss is typically below 10 dB. Fig. 109: Flip-chip mounted Schottky diodes in a 76 GHz mixer circuit Problems Consider a chrome-silicon metal-semiconductor junction with N D =10 17 cm -3 . [ M =4.5 eV, =4.05 eV, N C =2.82*10 19 cm -3 , N V =1.83*10 19 cm -3 ] o Calculate the barrier height and the built-in potential o Repeat for a p-type semiconductor with the same doping density o Calculate the depletion layer width, the electric field in the silicon at the metal-semiconductor interface, the potential across the semiconductor and the capacitance per unit area for an applied voltage of -5 V. o Calculate the forward current under forward bias of 0.5 V <?page no="101"?> 4. Schottky diodes 92 References [4.1] J.H.Werner, “Schottky Diodes”, in ´Silicon-Based Millimeter-Wave Devices, ed. by. J.-F.Luy and P.Russer; 1994; Springer Verlag, Heidelberg [4.2] Theo G. Van de Roer, Microwave Electronic Devices, Chapman & Hall, 1994 <?page no="102"?> 5. Heterojunctions and heterostructure diodes 93 5. Heterojunctions and heterostructure diodes This chapter deals with the formation of heterostructures, with the concept of modulation doping, the properties and problems caused by strain and stress and the corresponding material properties. The discussion of two-terminal heterostructure diodes based on transit time effects and quantum mechanical tunneling concludes the chapter. 5.1 Formation: Types of heterostructures In all practical relevant casers, the energy gaps of constituent semiconductors are different. Therefore at least one of the bands, the conduction or the valence band are discontinuous at the heterointerface. Fig. 110: Types of heterostructure In type I heterostructures, the sum of the conduction band and valence band edge discontinuities is equal to the energy gap difference The type II heterostructure is arranged in a way, that the discontinuities have different signs. Type III heterostructures are formed in a way, that the top of the valence band material lies above the conduction band minimum of the other material. As Ga Al x x 1 GaAs C W V W C W V W C W V W As In Al 52 . 0 48 . 0 InP C W V W C W V W C W V W GaSb InAs C W V W C W V W C E V W As Ga Al x x 1 GaAs C V C V V As In Al 52 . 0 48 . 0 InP C V C V C V GaSb InAs C V C V C E V Type I Type II Type III v c g W W W <?page no="103"?> 5. Heterojunctions and heterostructure diodes 94 Type I heterostructure is formed e.g. by the material system Al x Ga 1-x As/ GaAs. This is a lattice matched material system. If the materials are not lattice matched, the mismatch can be accomodated through strain or by the formation of misfit dislocations. For the construction of energy band diagrams it is helpful to remember, that 1. At equilibrium the Fermi level is flat everywhere, 2. Far from the junction the properties are bulk-like Further, we remember, that , the workfunction, is describing the energy level e*, which is required to shift an electron from the Fermi level W Fermi to the vacuum level. In order to remove an electron from the material, this amount of energy is required. is the electron affinity. The energy required to shift an electron from the conduction band edge to the vacuum level is e*. Fig. 111: Valence band, Fermi level, conduction band and vacuum level in GaAs 5.2 Modulation doping The concept, to separate free carriers spatially from the doping atoms was proposed by Dingle and co-workers from the AT&T Bell Labs in 1978. If the carriers can be separated from the doping atoms a reduced impurity scattering can be expected and the carrier concentration may be increased without affecting the mobility ! Fig. 112: Layer structure (top), energy band diagram of AlGaAs and GaAs as separated materials (middle) and coarse of the conduction band energy of AlGaAs and GaAs in intimate contact (bottom). N-type AlGaAs Intrinsic GaAs Layer structure for a modulationdoped heterostructure AlGaAs GaAs Energy band diagrams of AlGaAs and GaAs when not in contact and in equil. E f E f E f GaAs AlGaAs Energy band diagrams of AlGaAs and GaAs when in contact and in equil. Subbands N-type AlGaAs Intrinsic GaAs Layer structure for a modulationdoped heterostructure AlGaAs GaAs Energy band diagrams of AlGaAs and GaAs when not in contact and in equil. E f E f E f GaAs AlGaAs Energy band diagrams of AlGaAs and GaAs when in contact and in equil. Subbands <?page no="104"?> 5. Heterojunctions and heterostructure diodes 95 The modulation doping scheme is lllustrated in Fig. 112. The material with the larger bandgap (AlGaAs) is doped n-type whereas the other material is left undoped. As the Fermi level must align throughout the structure and the AlGaAs material is n-doped, the Fermi level lies closer to the conduction band edge than in GaAs (Fig. 112 middle). When the materials come into contact, electrons must be transferred from AlGaAs into GaAs to align the Fermi levels ! Therefore the electron concentration within GaAs is significantly increased without adding ionized donor impurities ! The result in the energy band diagram is, that a band bending occurs: The sharp bending of the conduction band edge and the presence of the conduction band discontinuity forms a potential well within the GaAs layer. (Fig. 112 bottom). If the spatial dimensions are small and the band bending is strong, then potential wells with spatial quantization effects occur. This spatial quantization produces so-called subbands. 5.3 Strain and stress: Example Si/ SiGe The lattice constant of Ge is 4.2% larger than that of Si: The material systems are not matched. Therefore if a bulk Ge layer is placed on a bulk Si layer in an attempt to form a single crystal, every 24 th Si atom at the interface would not be able to form a bond with a Ge atom. Figure 113 demonstrates the bulk lattice constant of Si and Si 1-x Ge x crystals. Fig. 113: Bulk Si and Si 1-x Ge x not in contact Provided only a thin layer of Si 1-x Ge x is grown pseudomorphically (lattice matched) to bulk Si then the layer is strained and the symmetry changes from cubic to tetragonal (Fig. 114). Fig. 114: Lattice matched, pseudomorphic growth <?page no="105"?> 5. Heterojunctions and heterostructure diodes 96 For thin layers of Si 1-x Ge x grown on bulk Si there exists a maximum thickness called the critical thickness above which it costs too much energy to strain additional layers of material into coherence with the substrate. Defects in the system appear to relieve the strain, in this case misfit dislocations (Fig. 115). Fig. 115: Formation of misfit dislocations Strained layers well above the equilibrium critical thickness may be grown epitaxially to form metastable strained layers but these layers may subsequently relax forming defects if the layers are thermally processed. It is straightforward that strain and stress have impacts on the physics of the band alignments of Si/ SiGe heterostructures Fig. 116: Influence of strain on Si/ SiGe band alignments If Si 1-x Ge x layers with sub-critical thicknesses are grown on silicon substrate we obtain band alignment of type I (left side of Fig. 116). This valence band discontinuity can be advantageously used in the SiGe heterojunction bipolar transistor (Ch. 8). It can be considered as a further added value that in this material system there is no discontinuity in the conduction band. The situation changes, if strained silicon layers are grown on Si 1-x Ge x substrates. These Si 1-x Ge x substrates may be formed by thick (means beyond the critical thickness) Si 1-x Ge x buffer layers on top of which thin silicon layers lead then to a type II band alignment situation (right part of Fig. 116) which finds application in Si 1-x Ge x heterojunction field effect transistors (Ch. 10). Fig. 117 further illustrates the influence of the buffer on Si/ SiGe band alignments. SiGe Si SiGe E C E V Buffer E C E V Strained Si on SiGe SiGe Si SiGe E C E V Buffer E C E V Strained Si on SiGe SiGe Si SiGe E C E V Buffer E C E V Strained Si on SiGe Si SiGe Si W =0 C W V Su bst rat e W C W V Strained SiGe on Si Si SiGe Si V Su bst rat e C V Strained SiGe on Si Si SiGe Si V Su bst rat e C V Strained SiGe on Si <?page no="106"?> 5. Heterojunctions and heterostructure diodes 97 Fig. 117: The influence of the buffer on Si/ SiGe band alignments The type I situation is obtained if SiGe is under compressive strain (left side of Fig. 117) and the type II situation is obtained if silicon is under tensile strain (right side in Fig. 117). GaAs and AlGaAs have almost similar lattice constants but different bandgaps. The III-V semiconductors also provide high mobilities and are therefore well suited for the formation of heterostructures. Silicon and Germanium have different lattice constants and different bandgaps, this material systems enables the realization of alloys - and the mobilities are lower compared to the III-V materials. Fig. 118: Bandgaps and lattice constants of some semiconductors In case of alloy formation the bandgap depends on the alloy composition, that is the Al mole fraction in AlGaAs and the Ge content in SiGe. The details of the band structure may be influenced as the , L and X points may shift different with varying alloy composition. This is illustrated in Fig. 119 for an Al Ga 1- As material. <?page no="107"?> 5. Heterojunctions and heterostructure diodes 98 Fig. 119: Al Ga 1- As bandgap dependence on Al fraction for , L and X point 5.4 Semi-classical Heterojunction diodes In this section the use of heterojunctions in microwave diodes is described and discussed in order to improve the performance of the known device concepts from Gunn and IMPATT diodes 5.4.1 Hot electron injection in Gunn devices The Gunn device is a typical high field device. The domain build-up mechanism requires to drive the device in the negative differential mobility region. This means, the carriers have to gain enough energy from the electric field in order to reach this condition. The injection of already “hot” carriers, which have already high energy, will reduce device losses and hence increase the efficiency. This can be achieved by the introduction of an injection barrier, which is basically possible by heterojunctions or by appropiate doping sequences as known from a camel or planar doped barrier injector (Fig. 120) Fig. 120: Hot electron injection in Gunn elements <?page no="108"?> 5. Heterojunctions and heterostructure diodes 99 5.4.2. Heterostructure IMPATT From the discussion of IMPATT diode theory it is known, that the breakdown condition is given by 1 * a l and that the length of the avalanche region l a directly influences the conversion efficiency. The introduction of Si 1-x Ge x in the high field region of a silicon IMPATT structure in order to reduce the bandgap can therefore be helpful to increase the ionization rate at a given field strength in comparison to pure silicon (Fig. 121). Fig. 121: Layer sequence, layer thicknesses and composition for a 100 GHz double drift Si/ Si 1-x Ge x heterojunction IMPATT diode The increase of the ionization rate is accompanied by an increase of the number of carriers which are injected by interband tunneling. This becomes evident by the inspection of the I-V reverse characteristic of the device (Fig. 122) Fig. 122: I-V of the device from Fig. 121 in reverse irection at 300 and at 77 K It is evident, that at low voltages the temperature coefficient is negative which is typical for tunneling. Therefore this device is called heterojunction Mixed Tunneling Avalanche Transit Time (MITATT) diode. As the reduction of bandgap in lattice mismatched material systems is strain limited by the critical thickness an additional degree of freedom may be obtained if a superlattice (SLS) is inserted in the high field region of an IMPATT diode: A change of the semiconductor type in every mono layer. The formation of so-called minibands across the superlattice leads to virtual energetic state caused by the overlap of the wavefunctions. Such a superlattice sequence provides larger critical thickness for the same bandgap as a standard alloy. Fig. 123 shows a possible band diagram of a Super Lattice Avalanche Transit Time diode. <?page no="109"?> 5. Heterojunctions and heterostructure diodes 100 Fig. 123: Band diagram of a Super Lattice Avalanche Transit Time device 5.5 Resonant Tunneling The resonant tunneling devices are considered to be based on a quantum mechanical tunneling mechanism: If a particle is confined by a potential V(r) on a scale comparable to it´s de Broglie wavelength the particles momentum k is quantized. The continuos energy spectrum m k k W 2 2 2 is broken up into energy subbands k E n which are approximated by 2 2 2 2 2 w n mL n W [ w L denotes the width of the potential well] The particle has a finite probability of being in the classically forbidden region, where its energy E is lower than the local value of the potential. Consider now two semiconductor materials with different bandgaps. The material with the larger bandgap forms the two barriers and is sandwiched between the material with the smaller bandgap, in this case. If the potential well, which is formed between the barriers, is small enough (typical values are below 5 nm) , only discrete energy levels are allowed in the well: quantized states (Fig. 124) Typical material systems which are used are AlAs/ GaAs/ AlAs AlAs/ InGaAs/ AlAs AlSb/ InAs/ AlSb Si/ SiGe/ Si If we now apply a voltage to this double barrier diode the energy diagram is modified. We can identify a voltage where the position of the Fermi level equals the position of the first discrete state in the quantum well. q E V / 2 0 <?page no="110"?> 5. Heterojunctions and heterostructure diodes 101 Fig. 124: Conduction band (W L ) in a double barrier structure without bias (left) and with bias in the first resonance (right) In a double barrier quantum well diode the conditions for resonant tunneling can then be stated as follows: 1. The energy of the incoming electrons is equal to the energy of a discrete state 2. The de Broglie amplitude of the waves in the quantum well builds up due to multiple scattering. Then the waves are leaking in both directions, and the reflection is reduced and the transmission increased. This is similar to the Fabry Perrot effect which describes the generation of standing waves between two conducting planes. As the response time constants are below 0.3 ps, the frequency limit of resonant tunneling is in the THz range. Fig. 125: Double barrier structure and transmission coefficient versus incident energy The transmission coefficient as shown in Fig. 125 can be calculated from where T 1 and T 2 represent the transmission coefficients of the two barriers at the energy W=W i [5.1]. The lifetime width of the resonant state is given by In cid en t En er gy Transmission Coefficient Incident Energy 2 2 2 2 2 1 2 1 ) ( 4 ) ( i W W T T T T E T / <?page no="111"?> 5. Heterojunctions and heterostructure diodes 102 ( is the intrinsic time constant) and can be expressed in a quasi-classically approximation 5.5.1 Resonant tunneling diode operation and III-V realizations In Fig. 126 the basic structure of a AlGaAs/ GaAs double barrier resonant tunneling diode is shown. The barriers are formed by the semiconductor material with the larger bandgap (AlGaAs). The well and the cladding layers beneath the barriers are formed by GaAs. Fig. 126: Layer sequence (top) and conduction band diagram of a resonant tunneling AlGaAs/ GaAs diode ) ( 2 1 T T W i AlGaAs GaAs n + n + AlGaAs GaAs n + n + F E C E F E C E <?page no="112"?> 5. Heterojunctions and heterostructure diodes 103 For the operation of the device it is essential to understand the influence of the bias voltage on the band diagram Fig. 127: Conduction band diagram of a resonant tunneling diode in thermal equilibrium without bias (top), small bias voltage (middle) and large bias voltage (bottom) If a bias voltage is applied across a RTD, the energy level diagram changes as shown in the upper figures of Fig. 127. Because of the bias voltage, the energy level on the left side becomes higher. Now some electrons in the left can tunnel into the well. After that, they can continue to tunnel out to the right. Current from left to right increases due to tunneling. Now the bias voltage is increased even more. The energy level on the left continues to increase. Though electrons in the well region can tunnel out to the right side by inelastic tunneling, no electrons can tunnel from left to the well region now. As a result, current beginns to decrease, and we get the NDR (negative differential resistance) region in the I-V curve. Fig. 128: I-V curve of a resonant tunneling diode <?page no="113"?> 5. Heterojunctions and heterostructure diodes 104 5.5.2 Silicon/ Germanium resonant tunneling structure Besides the lattice mismatch in the silicon/ germanium material system, it is also necessary to consider the strain induced splitting of heavy and light holes and the pecularities of the energy band diagrams. Without using a buffer the preferred structure on silicon substrate uses hole tunneling. Fig. 129 illustrates the situation. Fig. 129: Valence band diagram in a DBRT with hh lh splitting It is important to consider, that due to the small energy offsets it is essential to employ cladding layers in order to thermalize the carriers before they reach the barriers. These cladding layers lead to an additional voltage drop which has to be accounted for in the analysis of measurement results (Fig. 130). Fig. 130: Schematic valence band diagram in a SiGe RT with bias A layer sequence for a hole resonant tunneling diode is shown in Fig. 131. The germanium content in the cladding layers and the well is limited to 25 % to cope with the critical layer thickness. Fig. 131: Design of a hole resonant tunneling diode in SiGe <?page no="114"?> 5. Heterojunctions and heterostructure diodes 105 The resonances can be observed as kinks in the I-V characteristic (Fig. 132). Fig. 132: DC measurements of Hole SiGe RT structure at 77 K . Photo from a curve tracer, linear scale It is observed from mesa type devices, that there is a strong dependence of the I-V characteristic on the device diameter. It can be seen, that at very large diameters, leakage currents can be larger than the tunneling contributions so that we don´t see anymore the resonances. Fig. 133: DC measurements of hole SiGe RT structure at 77 K from devices with different diameters in [µm]. Linear scale. 5.5.3 A full wave rectifier with a resonant tunneling diode The specific I-V characteristic of a tunneling diode can be used to form a full wave microwave rectifier. The diode is biased into the first peak of the resonance. The microwave voltage is then full wave rectified due to the symmetric behavior of the diode in the vicinity of the peak voltage. Fig. 134 illustrates the concept. <?page no="115"?> 5. Heterojunctions and heterostructure diodes 106 Fig. 134: Current voltage characteristic of a tunneling diode (solid line) and a Schottky diode (dashed) and voltage response of the tunneling diode (top left) and the Schottky diode on a modulating RF signal A significant increase in detector sensitivity is observed if the RTD is biased into the first peak compared to lower and higher bias voltages. The maximum sensitivity of an integrated 94 GHz resonant tunneling full wave detector is obtained at a bias voltage of 0.47 V. Fig. 135: Detector voltage of a resonant tunneling full wave rectifier as a function of the microwave frequency 5.5.4 The quantum well injection transit time diode (QWITT) The low response time and the negative differential resistance region make the resonant tunnelling diode ideally suited as a device in oscillator applications. The highest frequencies up to now are 712 GHz with RTDs in the AlSb/ InAs/ AlAs material system. The output power at this frequency reached 0.2 µW. These rather low values are the consequence of the very limited modulation voltage in RTDs due to the neighbouring resonances. The absolute level of the voltage can be much larger if the resonant tunnelling mechanism is employed as an injector and combined with a drift region (Fig. 136). <?page no="116"?> 5. Heterojunctions and heterostructure diodes 107 Fig. 136: Concept of a quantum well injection transit time diode illustrated by means of a conduction band diagram Currents and voltages in a QWITT diode are very similar to that of a BARITT or pure TUNNETT diode. Fig. 137: Modulating RF voltage (top), injected current (middle) and current flux to be measured at the connectors of a QWITT diode (bottom) The discussion of RTDs in the silicon/ germanium material system showed, that it is easier to form a hole RTD than an electron RTD. For the construction of a SiGe QWITT therefore a hole injector is used an the I-V curve shown in Fig. 138. The resonant kinks can be seen and the asymmetry of the characteristic caused by the drift layer. <?page no="117"?> 5. Heterojunctions and heterostructure diodes 108 Fig. 138: I-V of a p-SiGe-QWITT at 77 K 5.5.5 Esaki tunneling diode with SiGe quantum well The valence band offset in the SiGe material system and the small bandgap in germanium can also be used to improve the characteristics of an interband tunneling device. The introduction of a SiGe quantum well (QW) at the p-n junction leads to an enhanced interband tunneling current and results in an increased peak to valley ratio compared to a homojunction interband tunneling diode. Fig. 139 illustrates the concept. Fig. 139: Schematic band structure of an interband tunneling device with a SiGe QW 5.5.6 Further tunneling diode applications Analog microwave oscillators and detectors as well as frequency converters are possible applications of tunneling diodes [5.2]. Digital applications in memories, and logic elements providing multivalued logic are expected to be the most interesting future applications of tunneling diodes. Energy SiGe QW V B C B eU n-Si p -Si Energy SiGe QW V B C B eU n-Si p -Si <?page no="118"?> 5. Heterojunctions and heterostructure diodes 109 References [5.1] K.F. Brennan, A.S. Brown “Theory of Modern Semiconductor Devices”, J. Wiley, 2002 [5.2] A. Cidronali, V. Nair, G. Collodi, J. Lewis, M. Camprini, G.Manes, H. Goronkin: “MMIC Application of Heterostructure Interband Tunneling Diodes” IEEE Transaction on Microwave Theory and Techniques special issue on ”RF and Microwave Tutorials” vol. 51, pp. 1351-1367, April 2003. <?page no="119"?> 6. The Bipolar Junction Transistor 110 6. The Bipolar Junction Transistor In 1947, Bardeen and Brattain invented the Ge point contact transistor. A photo of this historic device is shown in Fig. 140. They wanted to realize a Field-Effect Transistor, but ended up with a Bipolar Transistor. Shockley then developed the bipolar junction transistor. Fig. 140: Photo of the historical Ge point contact transistor The Nobel Prize was granted "for their researches on semiconductors and their discovery of the transistor effect" to William Bradford Shockley (1910 - 1989) Semiconductor Laboratory of Beckman Instruments, Inc. Mountain View, CA, USA John Bardeen (1908-1991) University of Illinois Urbana, IL, USA and Walter Houser Brattain (1902-1987) Bell Telephone Laboratories, Murray Hill, NJ, USA Fig. 141: The Inventors of the transistor: William Shockley (seated), John Bardeen (left), Walter Brattain [1947/ 1948] <?page no="120"?> 6. The Bipolar Junction Transistor 111 6.1 Basics of the Bipolar Transistor To discuss the basic properties of a bipolar junction transistor the energy-band of a npn layer sequence is drawn (Fig. 142) Fig. 142: npn device with two p-n junctions If we inspect the band diagram, we can state the following: 1. The electrons in the highly n-doped emitter are at the most negative potential 2. The first p-n junction is biased so electrons flow into the base. It will happen that some electrons recombine in the base and that some electrons will make it across the base without recombining. 3. The second p-n junction is reverse biased and has no barrier for electrons flowing from p-type to n-type material. Those electrons that make it to the collector will immediately enter the collector to continue as current. The pnp transistor is very similar except that the current is due to holes. Let us now care for the current flow in a Bipolar Junction Transistor (BJT). The current in the collector is proportional to the emitter current reduced by electrons absorbed by the base where is the fraction of electrons which are able to get through the base to the collector. I C0 is the reverse current due to the normal base-collector junction, but this is very small. An ideal junction would have 1 . Real transistors have 99 . 0 95 . 0 . is increased for thin bases or lightly doped bases. If the base doping is equal to the emitter doping ) ( E B N N , then half the emitter current would be due to base majority carriers, and would not exceed 0.5. The schematic symbols of bipolar transistors are shown in Fig. 143 Fig. 143: Schematic symbols of npn and pnp transistors 0 C E C I I I <?page no="121"?> 6. The Bipolar Junction Transistor 112 Biasing of npn and pnp transistors is illustrated in Fig. 144 Fig. 144: Transistor biasing From Fig. 143 and Fig. 144 it can be seen, that the transistor can be viewed as two diodes and in fact can be checked with an ohm meter. The forward current I BE is like in a diode, with an exponential dependence of the current on the voltage over any orders of magnitude. Typical currents I C are 1 mA at 0.6 V in Silicon and 0.25 V in Germanium. V CB is limited by reverse breakdown like in a diode: typically in the range 10-20 V. The transistor limits V Cemax and P max =I C V CE . However, the collector current I C is not like in a diode ! If Kirchoff´s law is used at the junction, then it follows C B E I I I and if the effect of carriers through the base is included: The relation between the base current and the emitter current is : B E I I 1 1 and the relation between base current and collector current is B B C I I I 1 6.2 The Ebers-Moll equivalent circuit The analysis of an equivalent cicuit will yield further insight in the transistor operation. Fig. 145 shows the Ebers-Moll equivalent circuit. Fig. 145: Ebers-Moll equivalent circuit E B CO E B E I I I I I I <?page no="122"?> 6. The Bipolar Junction Transistor 113 The current through the Base-Emitter Diode (I EN ) can be written as 1 T BE V V ES EN e I I and the current through the Base-Collector Diode (I CI ) 1 T BC V V CS CI e I I These currents cause transfer currents: The current through the base-emitter diodes causes a collector transfer current 1 T BE V V CES CT e I I and due to symmetry reasons the current through the base-collector diode causes an emitter transfer current 1 T BC V V ECS ET e I I where transfer saturation currents are given by and with By summation of the currents at emitter, collector and base contact the follwing relations are obtained which are known as the Ebers-Moll equations. 6.3 Current-voltage relations The emitter efficiency for npn transistors is defined as the ratio of the electron current injected from the emitter to the base over the total emitter current: EN n n I I The base transport factor is defined as the ratio of the electron current arriving at the collector over the electron current injected from the emitter to the base ES N CES I A I CS I ECS I A I S ECS CES I I I C E B I I I 1 1 T BC T BE U U CS U U CES C e I e I I 1 1 T BC T BE U U ECS U U ES E e I e I I <?page no="123"?> 6. The Bipolar Junction Transistor 114 The current gain factor is defined as These general relations can now be applied to the three possible circuit configurations of the Bipolar Junction Transistor (BJT) as shown in Fig. 146. Fig. 146: Basic BJT circuit configurations: Common base (left), common emitter (center), common collector (right) As an example, the current-voltage relation for the common emitter configuration is given. The voltages are CE BE BC V V V and the currents are with and which yields T CE T CE T CE T CE V V I ES CS N V V ECS N V V CS V V ES CS N B C e A I I A e I A e I e I I A I I 1 1 1 1 This is the exact expression for the collector current in common emitter configuration. N N EN CT N I I A * n CT n I I 1 1 T CE BE T BE V V V CS V V CES C e I e I I 1 1 T CE BE T BE V V V ECS V V ES E e I e I I C B E I I I ES CES N I I A CS ECS I I I A <?page no="124"?> 6. The Bipolar Junction Transistor 115 If then the collector current is given by and the current gain is A measurement circuit for common emitter configuration is shown in Fig. 147. Fig. 147: Measurement circuit for common emitter configuration The voltage-current curves of a BJT can be displayed in three different graphs: 1. The input characteristic shows the dependence of the base current on the base-emitter voltage and looks like a typical diode graph. The different curves depend on U CE biasing. 2. The output characteristic shows the dependence of the collector current on the collector-emitter voltage with the base current as a parameter. This is a current source graph. 3. The transfer characteristic shows the dependence of the collector current on the base current. Fig. 148 shows all three graphs in one diagram: The output characteristic in the first quadrant, the transfer characteristic in the second quadrant and the input characteristic in the third quadrant. The DC load line is marked by V CE =V L and I C =I K with (c.f. Fig. 147). Indicated by dashed lines is, how a small change in input voltage U BE causes a small change in base current, resulting in a large collector current change and a large output voltage change V CE . This is the basic amplification mechanism of the BJT. It can be recognized, that the phase will be changed by 180 o and that the transistor will enter the dark marked saturation regime for large input signals. B ECS B CS T CE I I I I V V N N B C A A I I 1 * N N B C N A A I I B 1 L L K R V I <?page no="125"?> 6. The Bipolar Junction Transistor 116 Fig. 148: Output, transfer and input characteristic of a BJT. Indicated is also the load line and the amplification of a small sinusoidal input voltage resulting in a larger output voltage swing. 6.4 RF modeling An exact modeling of the BJT under large signal conditions at microwave frequencies requires an RF large signal model (Fig. 149). Voltage dependent capacitances of base-emitter and base-collector junctions have to be considered. Fig. 149: An RF large signal model of a BJT For a lot of applications it is however sufficient to work with a small signal approach and to approximate the non-linear characteristics in certain sections by straight lines. This is illustrated using the exponential current-voltage characteristic of a baseemitter diode. At first a diode current-voltage characteristic is linearised in a certain operation point (Fig. 150). <?page no="126"?> 6. The Bipolar Junction Transistor 117 Fig. 150: Linearization of an exponential diode current voltage characteristic The small signal resistance is obtained from which is The small signal resistance decreases with increasing current. The equivalent circuit from Fig. 149 can now be simplified if the operation regime is restricted to the normal active mode (V CE >0 and V BE <0). The -equivalent circuit is obtained for common emitter operation. Fig. 151: -equivalent circuit 6.5 Frequency limits The current gain cut off frequency is defined as the frequency where the current gain becomes unity. The current gain cut-off frequency of a BJT can be calculated using the -equivalent circuit (Fig.151). The input current is c be e be be be B C j V C j V r V I * * and the current at the shorted output is It is assumed that = T which yields and for the current gain cut off frequency it is obtained BE B be dV dI r 1 EN T n be I V b r ) 1 ( c be be be C C j V r V b I * * c e T be c e T be c T be B C C C r b C C r C r b I I * 1 1 2 2 2 2 <?page no="127"?> 6. The Bipolar Junction Transistor 118 c e be T T C C b r f * * 2 1 2 If it is considered, that the collector capacitance is caused by the space charge capacitance of the reverse biased base-collector junction and the emitter capacitance is composed by the diffusion capacitance and the space charge capacitance of the forward biased base-emitter junction then we obtain for the current gain cut off frequency ) ( 2 1 2 1 C E B sc be se be de be T C b r C b r C b r f with the base transit time B , the emitter charging time E and the collector charging time C . Power gain can still be obtained beyond the current gain cut off frequency. In order to determine the power gain cut off frequency, the -equivalent circuit has to be extended by a base resistance r b in order to account for the consumed power in the input circuit (Fig. 152). Fig. 152: Equivalent circuit for the determination of the power gain cut off frequency Starting from the condition max max f P f P out in after lengthy calculations the well known expression c b T C r f f 8 max is obtained. The power gain cut off frequency is dependent on the current gain cut off frequency and is also dependent on the base resistance and the collector capacitance. 6.6 Technology As an example for a fabrication process of bipolar transistors a process sequence for the fabrication of planar silicon bipolar junction transistors is given (Fig.153 and Fig. 154). There are diffused or implanted doped regions in the planar transistor. The collector is grown by epitaxy. There are direct semiconductor-metal transitions and no self-alignment method is employed in this example. sc c C C se de e C C C <?page no="128"?> 6. The Bipolar Junction Transistor 119 The process starts with the oxidation of a high resistivity silicon wafer. Diffusion windows are opened and buried n + subcollectors are formed by implantation of arsenic or antimony. An n - -doped layer is grown by epitaxy to form the collector layer and subsequently oxidized. Selective boron diffusion isolates the pn-junctions. Fig. 153: Fabrication process of silicon bipolar transistors up to the boron diffusion for pn-isolation Diffusion of boron forms the base layer, subsequent oxidation, emitter window opening and diffusion of donors form the emitter. After an oxidation step and another contact window opening the metallization layers providing the ohmic contacts are evaporated and after a final photo resist step the contact layers are etched. <?page no="129"?> 6. The Bipolar Junction Transistor 120 Fig. 154: Fabrication process of silicon bipolar transistors from the isolation diffusion up to the final metallization Emitters and extrinsic bases diffused from polysilicon layers can significantly improve the conventional planar technology. [6.2] The separation between the emitter contact and the base contact can be self-aligned using a side-wall spacer with very precise control possibility of the spacer. The emitter polysilicon can extend over a considerably wider region than the area of the contact, reducing its series resistance. The base-collector current junction area can be kept small. In total, significant reductions in device area are made over non self-aligned devices. 6.7 Power limits Temperature increase in the BJT is the essential power limitation of bipolar transistor operation. The thermal resistance of power transistors is therefore a major concern and adequate heat sinks have to be provided. To handle a large amount of power the emitter stripe width and the base thickness shall be optimized. At high currents electromigration of the metal contacts into the semiconductor can occur. This is a limitation for the maximum collector current. At high collector voltages the onset of avalanche breakdown occurs, which leads to increased power loss and eventually to thermal destroy. Microwave transistors may be very sensitive with respect to negative base-emitter voltage, as the highly doped diode causes a small breakdown voltage. <?page no="130"?> 6. The Bipolar Junction Transistor 121 Problems In the Ebers-Moll model, the following reciprocity argument is valid for symmetrical and non-symmetrical transistors [6.3]. Prove it where N is the ratio of collected current to injected current in the normal mode I is the ratio of collected current to injected current in the inverted mode I ES is the emitter saturation current I CS is the collector saturation current Assume, the transit time for electrons across the base of an n-p-n transistor is 100 ps, and the electrons cross the 1 µm depletion region of the collector junction at their scattering limited velocity. The emitter-base charging time is 30 ps and the collector capacitance and resistance are 0.1 pF and 5 , respectively. Find the cutoff frequency. Answer to the following questions after reading the following paragraph It is necessary to use lightly doped material for the base region and heavily doped material for the emitter to maintain a high value of (emitter injection efficiency), (current transfer ratio), and (base-to-collector current transfer factor) in homojunction Si n-p-n transistors. o Explain, why a high value of , , and can be expected in the above situation o Discuss the advantages of lightly doped base and heavily doped emitter transistors in high-speed operation view point. CS I ES N I I * * <?page no="131"?> 6. The Bipolar Junction Transistor 122 References [6.1] S.M.Sze, Physics of Semiconductor Devices, J. Wiley, 1981 [6.2] S.M.Sze, Modern Semiconductor Device Physics, J. Wiley, 1998 [6.3] Ian E. Getreu, Modeling the Bipolar Transistor, Elsevier Scientific Publishing Company, 1978 <?page no="132"?> 7. The Heterojunction Bipolar Transistor 123 7. The Heterojunction Bipolar Transistor From the discussion of the bipolar junction transistor it is concluded, that the collector current shall be larger than the base current in order to obtain a current gain which is larger than one. The current gain BJT can be expressed by b b p e e n B C BJT L N D L N D J J where D n is the diffusion constant of the electrons in the base, D p is the hole diffusion constant, N e is the emitter doping, N b is the base doping, L e is the emitter thickness and W b is the base thickness. The diffusion constants are in the same order of magnitude as are the emitter and base width, which means that b e N N in order to guarantee transistor operation with current gain. This requires low base doping which leads to high base resistance and correspondingly negative influence on the high frequency performance. An increase of the base area does not help as this leads to increased C bc which influences f max negatively. This is the reason, that conventional BJTs can only be used uo to several GHz. The usual limits are in the range of N emax =10 20 cm -3 and N b =10 18 cm -3 which yields ß=100 . 7.1 Basics of HBTs The limits of Bipolar Junction Transistors can be circumvented in the Heterojunction Bipolar Transistor (HBT). The key feature of the HBT is the difference in barrier height for electrons and holes in the emitter-base junction. This is obtained by a base-emitter heterojunction where the band gap of the emitter semiconductor W g(e) is larger than the band gap in the base W g(b) (Fig. 155). This controls the injection of majority carriers from the emitter into the base versus the back injection of base majority carriers into the emitter. Fig. 155: Banddiagrams of BJT (left) and HBT (right) The HBT has a number of advantages over the BJT: A high current gain can be achieved in a HBT which is virtually independent of the emitter and the base doping. <?page no="133"?> 124 A high base doping can be used which reduces the base resistance and contact resistance. A lower doping in the emitter and collector regions reduces the emitterbase and collector-base capacitances. A graded base and a built-in electric field can be realized without compromise on base doping. Overall in a HBT compared to a BJT there are greater design limits using band gap engineering regarding breakdown voltages and velocity control in the collector-base depletion region as well as the use of ballistic injection mechanisms. Under applied voltage, the heterojunction behaves similar like the homojunction, with the exception that, the intrinsic carrier densities are different on both sides: density 0 0 n p at the emitter- If we consider, that the hole base junction T BE g T BE V V kT W D C V V V D i n e e N N N e N n p 1 2 1 0 0 and the electron density 0 0 p n at the base-emitter junction is given by T BE g V V kT W A C V p e e N N N n 2 0 0 and the hole current is given by e n p p L p qD J ) 0 ( and the electron current is given by b p n n L n qD J ) 0 ( then the current gain as the ratio of electron to hole current at the hetero-interface follows from It is obvious from this relation, that the base doping N A may be increased beyond the emitter doping N D if a bandgap difference compensates. A bandgap difference of e.g. 200 meV improves the ratio of p n J J / by a factor of 2200 at room temperature. 7.2 Types of HBTs A typical representative of the single heterojunction bipolar transistor family is shown in Fig. 156. kT W W A b p D e n n p b p e n p n g g e N L D N L D p n L D L D J J / 2 1 * ) 0 ( ) 0 ( * kT W C V i g e N N n / 2 <?page no="134"?> 7. The Heterojunction Bipolar Transistor 125 Fig. 156: Layer sequence, doping, composition and bandgap of a GaAs-GaAlAs HBT The bandgap of the emitter material is larger than the bandgap of the base material. The construction of the band-diagram of a single heterojunction GaAlAs/ GaAs bipolar transistor is illustrated in Fig. 157. The GaAlAs layer is n-doped and forms the wide gap emitter, the GaAs layer is p-doped and forms the base. Fig. 157: Band-diagram of emitter-base contact: Without contact (top), and ideal contact, without bias (bottom) From Fig. 118 (bandgaps versus lattice constants) it became evident, that GaAs/ GaAlAs is a very suited material combination for the fabrication of heterostructures, as the materials obey similar lattice constants and different bandgaps and show high electron mobilities. The table shows several III-V material combinations for HBTs Tab. Single heterojunction transistors with wide gap emitters. The base material is the same as the collector material. <?page no="135"?> 126 Silicon based HBTs may also employ wide band-gap emitter technology, e.g. semiinsulating polycrystalline silicon, amorphous silicon or microcrystalline silicon. The high emitter resistance associated with the wide-gap material degrades device performance. Another approach is to form a double heterojunction bipolar transistor in the material system silicon/ germanium. In case of pseudomorphic growth of a silicon-germanium alloy on silicon substrate (cf. Fig. 114) the band alignment is shown in Fig. 116 (left side) and yields an ideal situation for the construction of a npn HBT. The strain in pseudomorphic material systems influences the bandgap and the band alignment properties: If a substrate with a lattice constant a 0 is used and a strained layer with a lattice constant a parallel to the surface and a perpendicular to the surface is grown, then the pseudomorphic growth conditions require a = a 0 The design of a Si/ SiGe HBT requires that the thickness of the SiGe layer remains below the critical layer thickness. In order to gain benefit from the HBT concept, the base doping is increased beyond the values which are common in BJTs - base doping in HBTs can even exceed the emitter doping, which is called doping inversion. In case of very high base doping using boron as a dopant, special care has to be taken to avoid outdiffusion of boron into the n-doped collector and emitter regions. For this reason, undoped interface layers i EB and i BC are necessary to tolerate an outdiffusion of boron towards the emitter or collector (Fig. 158) Fig. 158: Vertical structiure of a high speed Si/ SiGe HBT The i BC - layer has a significant influence on the high frequency properties of the transistor due to the Kirk-effect occurring under high current conditions: The drifting carriers cause a change in the local electric field which itself causes a widening of the neutral base (which is free of field) to the collector. By this extension the base transit time b is increased and the cut-off frequencies are reduced. n+, 200 nm n, 100 nm i , 3 nm EB p+, 20 nm i , 10 nm n, 500 nm n+ buried layer Si Ge Emitter Basis Kollektor BC 1-x x <?page no="136"?> 7. The Heterojunction Bipolar Transistor 127 Fig. 159: High current effects in bipolar transistors. Collector current increases from 1 to 4. a) shows the electric field and b) the potential versus the location inside the device and c) shows the dependence of the emitter-collector transit time on the inverse collector current At small collector currents there is an abrupt field increase at the base/ I CB transition (Situation 1 in Fig. 159). If the current is increased, then drifting electrons will reduce the electric field at the base/ I CB transition towards 0 (Situation 2 in Fig. 159). The base push-out (Kirk effect) has reached the full intrinsic layer (Situation 3 in Fig. 159). The additional barrier which is generated at the heterointerface base/ I CB generates a barrier with further increased collector current (a) and (b) in Fig. 159) as the electrons E-Feld Kollektor 1 2 3 4 i Basis BC a) Potential 1 2 3 4 b) z z W +i B BC W B 1 2 f T 1/ Kollektorstrom 1 2 3 4 c) <?page no="137"?> 128 are not compensated by holes anymore. Therefore the transit frequency of the transistor is reduced rapidly (Situation 4c in Fig. 159). Usually the transit frequency values given in the literature are the maximum attainable values close to the operation point 3 in Fig. 159. There the width of the neutral base, which is free of electric field, is given by BC B i L L where B L is equal to the width of the p-doped region. Therefore the expression for the calculation of the transit time frequency under this operation conditions is modified to r s C n CB B p E EC T v L D i L D L f 2 2 ) ( 1 2 2 1 2 2 where r represents remaining time constants. 7.3 Experimental devices The growth of the submicron doping profiles for Si-Ge HBTs can be done by MBE or UHV/ CVD. Fig. 160 shows a doping profile of an UHV/ CVD grown layer sequence. The poly-silicon emitter is arsenic doped (As), the base is boron doped (B) and shows a trapezoidal Ge-profile (Ge), the collector is phosphor doped (P) and has an arsenic doped subcollector. Fig. 160: SIMS doping profile and Ge profile of a UHV/ CVD grown SiGeHBT The technological realization of a high frequency test transistor is described using Fig. 161. In a high resistivity (silicon) wafer a highly n + doped buried layer is formed. Then the n-doped collector is grown, the p + - SiGe - layer for the base, the n - layer for the emitter and the n + layer for the emitter contact are formed. As SiGe shows a significantly different behaviour with respect to wet chemical etching as pure silicon layers, a self-stopping etch procedure can be applied to etch the emitter mesa. This situation is shown in the upper part of Fig. 161. <?page no="138"?> 7. The Heterojunction Bipolar Transistor 129 Fig. 161: Double mesa process A small underetching of the emitter contact is desirable, as this mushroom-like situation can now be exploited to facilitate a self-aligned base metallization, as indicated in the middle of Fig. 161 and shown in the SEM in Fig. 162. Fig. 162: Multi layer resist process in order to fabricate the mushroom (left side) and etched emitter fingers (right side) Collector etching is straightforward. Finally, trenches define the collector area and isolate the device (cf. Fig. 163) emitter emitter emitter base base collector collector n + n - PtAu PtAu TiPtAu self-aligned air bridge <?page no="139"?> 130 Fig. 163: Trench etching The described double mesa fabrication process is easy and suited for quick device characterization. For highly integrated circuits process technologies which guarantee a high yield are employed. The schematic cross section of a UHV/ CVD grown SiGe HBT which was fabricated in a planar process is shown in Fig. 164. Fig. 164: Schematic cross section of a UHV/ CVD SiGe HBT The cross section of a f T =40 GHz AlGaAs HBT is shown in Fig. 165 Fig. 165: Cross section of an AlGaAs HBT [7.1] A planar configuration of a HBT realized on a InP substrate is shown in Fig. 166 <?page no="140"?> 7. The Heterojunction Bipolar Transistor 131 Fig. 166: HBT on semi-insulating InP substrate [7.2] 7.4 High frequency characterization High frequency devices are characterized by scattering measurements (S-parameter measurements). As the HBT is a three-terminal-device (emitter, base, collector) it is operated in a two-port configuration. Both "common emitter" (see Fig. 165) and "common base" configurations are used. Fig. 165: Measurement principle of a S-parameter setup with a HBT which is operated in the common emitter configuration The device is contacted on wafer by coplanar probe tips (Fig. 166). After the transistor is set to a certain DC bias point (V BE , I B , V CE , I C ) a high frequency signal is fed to the device under test (DUT) by the probe tips, where the frequency is swept between 45MHz to 50 GHz (in some measurement systems up to 110 GHz). To ensure the so called "small signal" operation mode, the applied HF-signal has to be small in comparison to the applied DC bias (V HF <<Vce, V BE ). At the input terminal, part of the power is scattered back and part of the power is transmitted to the output of the device, and at the output terminal part of the power is scattered back and part of the power is transmitted to the input. The ratio between reflected and incoming power at the input and at the output is given by the quantities s 11 and s 22 , respectively. As the network analyzer, which is used for characterization, is capable to measure both magnitude and phase of the incoming signal, the measured quantities are complex numbers. S 21 describes the normalized power, which is transmitted from the input to the output. In the same way s 12 describes the (undesired) reaction of the output on the input. <?page no="141"?> 132 C E BC BC BE C B sat C n B p E E C B E T R R C C C qI T k v x D L D L f 2 2 2 2 1 2 1 2 1 2 * 21 12 22 11 21 0 21 * ) 1 )( 1 ( 2 s s s s s I I h CE V B C / * 1 * * 1 : 0 21 i i I I h CE V B C Fig. 166: Typical layout suitable for coplanar probing, showing both ground-signaground (GSG) and ground-signal (GS) probe configurations 7.4.1 Current gain cut-off frequency It is now necessary, to extract from the S-parameter measurements the current gain and the cut-off frequencies. The current gain in common emitter operation is given by or with (the DC current gain in common Emitter) and (the angular frequency). The current gain h21 has therefore at frequencies below the "roll-off"-frequency 1 (or -3dB cutoff frequency for the a.c. current gain) the DC value . Above the gain h 21 2 decreases as 1 2 and therefore looks like a stright line with slope - 20dB/ dec. in a double log scale. The frequency at which this extrapolation passes unity (0dB) is defined as the common emitter transit frequency f T . If the device was measured in the common base configuration, the scattering parameters first have to be transformed in common emitter configuration mathematically to determine f T . As fT describes the speed of the HBT, it is composed of the individual times the carriers need to pass base ( ) B and collector ( ) c and the time which is needed to charge emitter-base (CBE) and base-collector (CBC) junctions, where RE and RC are the parasitic emitter and collector resistances: <?page no="142"?> 7. The Heterojunction Bipolar Transistor 133 Therefore, a high f T requires a very narrow base width w B , a narrow collector depletion region x C and the operation at a high collector current level I C . Fig. 167 shows measured data from SiGe and GaAs HBTs. Fig. 167: Ft as a function of collector current density for a GaAs HBT and a SiGe base bipolar transistor. Experimental data from 1997. The comparison in Fig. 167 uses data from Hitachi´s 130 GHz f T SiGe HBT technology and from the InGaP/ GaAs HBT from Oka and Hirata - both data sets are reported at the IEDM 1997. The GaAs HBT has higher f t than the SiGe base transistor. The larger energy gap of GaAs means, that the GaAs HBT is less susceptible to the base-collector junction avalanche effect. Therefore, a GaAs HBT can be designed to operate at much higher collectorcurrent densities than a SiGe base transistor and still meets the breakdown voltage requirements. That is, GaAs HBTs can be scaled down to smaller dimensions than Si-base or SiGe-base transistors. Compared to GaAs HBTs, the real advantage of SiGe Base bipolar transistors is their compatibility with silicon VLSI processes. 7.4.2 Maximum oscillation frequency Another important figure of merit of a microwave transistor is its maximum oscillation frequency. If the maximum oscillation frequency is reached, the power amplification drops to unity (0dB). The maximum oscillation frequency is dependent on the base resistance R B and the base collector capacitance C BC according to How is this quantity extracted from S-Parameter measurements ? It is important to note that there exist two definitions to extract this quantity from S-parameter measurements: The maximum available gain (MAG) is the power amplification, which can be achieved without external feedback if the device is perfectly conjugately matched at the input and at the output terminal simultaneously. This can only be the case if the transistor is unconditionally stable. To determine this, we have to calculate the stability factor (Rolletfactor) : BC B T C R f f 8 max <?page no="143"?> 134 k s s s s s s s s 1 2 11 2 22 2 11 22 12 21 2 12 21 * * * * For k<1 the transistor is potentially unstable (or conditional stable), that is, applying a certain combination of passive load and source impedance can induce oscillation (without external feedback). If k>1, the device is unconditionally stable, that is, in the absence of external feedback, a combination of passive load or source impedance will not cause oscillation . For the case of unconditional stability (k>1) the MAG is defined by In the region "k<1" the maximum stable gain (MSG) is defined as the power amplification which is obtained, if the device is forced to be stable by terminating it with lossy impedances In Fig.168 one can clearly see the kink in the MSG/ MAG curve at the point at which k exceeds unity. MAG also reaches a -20dB/ decade slope and from this the maximum oscillation frequency fmax can be extrapolated. Another method assumes, that the undesired reaction of the output on the input caused by parasitic capacitances and inductions of the outer transistor can be eliminated by an appropriate lossless feedback network around the transistor, at least for the frequeny the circuit is designed for . If under this condition the transistor is matched with conjugate complex impedances, the power amplification we now obtain is the maximum unilateral gain (U or MUG, Mason's invariant gain) Or in terms of y or h -parameters U is higher compared to the MSG/ MAG value and U approaches a slope of -20 dB/ decade. U is a useful quantity, if the device can be made unilateral by embedding it into a lossless feedback network which compensates for the parasitic effects of the device. Comparing fmax values from different publications, it is therefore important to note, wheather these quantities have been obtained both from an extrapolation of MAG or U, respectively, otherwise they are not to be compared. The existence of two different definitions for extrapolating the maximum oscillation frequency of a transistor requires to treat the quantity RB*CBC in the expression for the maximum frequency of oscillation as an effective base-collector time constant, which takes different values, depending on weather fmax is extrapolated from MAG or from U. This time constant is made up from different combinations of the 12 21 12 21 2 12 21 / Re / * 1 / * 2 1 s s s s k s s U ) Im( * ) Im( ) Re( * ) Re( * 4 1 ) Re( * ) Re( ) Re( * ) Re( * 4 1 21 12 22 11 2 12 21 21 12 22 11 2 12 21 h h h h h h y y y y y y U 12 21 s s MSG 1 * 2 12 21 k k s s MAG <?page no="144"?> 7. The Heterojunction Bipolar Transistor 135 0.05 0.1 0.5 1 5 10 50 100 frequency [GHz] 0 5 10 15 20 25 30 35 40 |h 21 | 2 , MSG/ MAG, U [dB] 0 1 2 stability factor k MSG MSG MAG MAG UU |h 21 | 2 |h 21 | 2 kk f T f T f max (MAG) f max (U) f T = 36 GHz f T = 36 GHz f max = 83 GHz (MAG) f max = 83 GHz (MAG) f max = 111 GHz (U) f max = 111 GHz (U) components of the extrinsic and intrinsic base resistances and base-collector capacities. eff BC B C R is higher for MAG in comparison to U, as in the first case the contribution of the delay associated with the product of the intrinsic base resistance and the extrinsic base collector capacity is accounted for, which is neglected if fmax is extrapolated from U. Fig. 168: On wafer measurements of a SiGe HBT F max from U is higher than from MSG/ MAG: U has to be used if k is below unity over the whole measurement range, which is possible for very high speed devices. <?page no="145"?> 136 Fig. 169: Measured SiGe HBT record performances [7.3] The cut-off frequencies of the SiGe HBTs were contiously increased in the last years. Fig. 169 shows record performance data from 2002. On-wafer measurements up to 110 GHz were performed ant pad-parasitics were de-embedded. The measurements yield an f T in excess of 200 GHz for a current density range of 830- 1650kA/ cm 2 . A maximum power gain cut-off frequency of more than 270 GHz is determined from the 20 dB/ dec extrapolation of Masons´s unilateral power gain (U) at 40 GHz. The peak f max amounts to 285 GHz. An extrapolation of f max from the maximum available gain in the 50 - 80 GHz frequency range results in a peak f max of 194 GHz. 7.5 Conclusions The bandgap difference in the valence band between silicon and germanium (as an example for a HBT concept) helps to suppress the back injection of holes from the base to the emitter by an additional barrier and leads to an increase in the current gain of a heterojunction bipolar transistor compared to a homojunction device. This current gain increase is exponentially dependent on the band gap difference. The heterojunction induced current gain increase can be traded against an increase in the base doping in order to reduce the base resistance and to increase the cut-off frequencies. Ultra-thin base layers become possible still maintaining a low base resistance. Heavy base doping also helps to reduce RC effects: The heterostructure transistor design decouples the material composition and the control of carrier movement. <?page no="146"?> 7. The Heterojunction Bipolar Transistor 137 Problems The fundamental reason for the better high frequency performance in HBT´s than in homojunction BJT´s lies in the ability to simultaneously achieve high , low base resistance (r b ) and small transit time (high f T ). Please explain the underlying device physics in detail. Draw a schematic graph which shows the dependence of the HBT´s current gain cut-off frequency on the base thickness and the dependence of the HBT´s maximum oscillation frequency on the base thickness o (The absolute values are not important ! Assume suited parameters for your calculation and explain your derivations and the result) The Early effect in bipolar transistors describes the dependence of the collector current on the collector-emitter voltage under active forward operation of the transistor: If the collector-emitter voltage is increased, then the reverse voltage at the base-collector junction is increased and the collector depletion region will extend into the base - the effective basewidth is reduced and the collector current increases. What is the impact of the heterojunction on the Early effect ? Employing double mesa structure, the following Si/ SiGe HBT device is fabricated. Vertical dimensions of each layer is illustrated in the right figure as shown below. Please answer to the following questions. o What makes the HBT superior at microwave frequencies to the BJT? o Draw an exact energy band diagram between A and A’. (Assume that Wg(Si 0.8 Ge 0.2 )=1.01 eV, and Wg(Si)=1.12 eV. ΔW C =0.022 eV , ΔW V =0.088 eV ) o Assume that 1 B B p P D n L N D L N D , calculate the DC current gain β. o Why is the i BC layer(10nm) included in the above structure and how does this layer affect to the frequency performance of the device? o Discuss the mesa and the airbridge structures. The fabricated Si/ SiGe structure is measured using the network analyser. Si 0.8 Ge 0.2 A <?page no="147"?> 138 The measured S-parameters at 800 MHz are as follows. Magnitude Angle ( Degree ) S11 0.65 -95º S12 0.035 40º S21 5 115º S22 0.8 -35º o For the purpose of the coplanar probing, we are willing to use Ground- Signal-Ground (GSG) RF probes for the S-parameter measurement. Please draw the probe configuration which is connected with the HBT. Indicate the HBT terminal at the drawing. ( Assume, the common emitter configuration ) o Calculate the Rollet factor, k, and evaluate the stability of this HBT at the measured frequency. o Find out the maximum stable gain(MSG) or the maximum available gain(MAG). References [7.1]U. Scaper, P. Zwicknagl, "Physical Scaling Rules for AlGaAs/ GaAs power HBTs Based on a Small Signal Equivalent Circuit," IEEE Trans. Microwave Theory Tech. Vol. 46, No. 7, pp. 1006-1009, 1998. [7.2] M. Hafizi, "New Submicron HBT IC Technology Demonstrates Ultra-Fast, Low-Power Integrated Circuits," IEEE Trans. Electron Devices, Vol. 45, No. 9, pp. 1862-1864, 1998. [7.3] B.Jagannathan et al. IEEE-EDL, vol. 23, no.5, May 2002, pp. 258-260 <?page no="148"?> 8. The Field Effect Transistor 139 8. The Field Effect Transistor The Field Effect Transistor (FET) was proposed about twenty years before the BJT by Julius Lilienfeld in 1926 as a metal-semiconductor field-effect transistor (MESFET). The junction field-effect transistor (JFET) was proposed by Shockley in 1952 and first built by Dacey and Ross in 1953. The Metal-Oxyde-Semiconductor field-effect transistor (MOSFET) was invented in 1960 and is commercially available since 1964.0 Field effect devices use the electric field of a gate or grid to modulate the number of charges (i.e. the electron current) moving from the source to the drain. The FET can have very small parasitic capacitances as all contacts are on the surface. The cutoff-frequencies are mainly determined by the transit time of the electrons under the gate: very high speeds are possible and the structure is favourable for microwave frequencies. With the MESFET there is a technologically simple structure known, which is realized since 1969. The principle is indicated in Fig. 170. Fig. 170: Basic configurations of JFET (left) and MESFET (right) Besides the FETs with depletion regions (JFET and MESFET) there exists the family of MISFET (Metal Insulator Semiconductor FET) devices which in the case an oxide is used as insulating layer are called Metal Oxide Semiconductor FETs (MOSFET). MOSFETs can be realized with a channel which is cut-off without gate voltage. These are enhancement type devices. In case the channel is open without gate voltage, the devices are called depletion type devices (Fig. 171) Fig. 171: FETs with isolated gate: Enhancement and depletion type <?page no="149"?> 140 8.1 Basics of MESFETs The basic scheme of a GaAs-MESFET is shown in Fig. 172 [8.1]. The semi-insulating substrate keeps parasitic capacitances from the contact pads low. Additional advantages of GaAs are the high carrier velocities and high electron mobilities, as source and drain parasitic resistances provide low field regions where the mobility dominates. Fig. 172: Cross section of a MESFET In order to understand the MESFET operation, we look at first into the currentvoltage characteristics of the different metal contacts. The source and drain contacts reveal a symmetrical and linear behaviour at low voltages. These contacts are ohmic and they provide the current flow in and out of the active layer. At higher voltages the current starts to saturate and the source-drain current as a function of the sourcedrain voltage enters the saturated region. If a conducting channel is built-in, then the device is called a normally ON device. The Schottky gate in a MESFET is a rectifying contact like a diode - it´s not an ohmic contact. Reverse bias creates a depletion region below the gate. There are no carriers in the depletion region. The carriers are swept away by the electric field. Consequently, the number of carriers in the channel is reduced and the resistance is increased. The configuration shown in Fig. 172 is a typical GaAs configuration as GaAs cannot be oxidized thermally like silicon, which has a native oxide. Thus, the gate in the MESFET is a Schottky gate. Gate width S G D Gate length Depl. region n - GaAs Semi-insulating GaAs <?page no="150"?> 8. The Field Effect Transistor 141 8.2 Principles of MESFET operation From a cross section through the gate a band diagram in a MESFET is drawn (Fig. 173) [8.3]. Fig. 173: Band diagram in a MESFET below the gate (left side) and carrier density distribution The following abbreviations are used : permittivity : mobility of electrons w: width of channel a: height of channel L ch : length of channel b: thickness of channel and the voltage drop between source and gate is V GS = V bi - V depl with V depl representing the voltage drop over the depletion region and the value of V depl at V GS = 0 is called the built-in voltage. From the Poisson equation the voltage drop over the depletion region is obtained and the built-in voltage follows from the band diagram with B : Schottky Barrier potential F : Fermi level at source contact The Fermi level can be calculated from known parameters D C F N N e kT ln with the effective conduction band density of states (cf. chapter 1) N C and the channel doping N D . F B bi V 2 2 b qN V D depl <?page no="151"?> 142 The channel resistance follows from w b a N q L R D ch ch This describes the linear part of the I D - V GS characteristic. The current density along the channel is given by the ohmic law equation. If a constant mobility is assumed across the channel and if E Z represents the longitudinal electric field in the channel, then the drain current is given by It follows with the narrow channel approximation (ab) After the integration across the channel we obtain The drain current saturates, when the channel thickness becomes zero and for the saturated drain current we obtain which is valid in the saturation regime. The FET is a voltage contolled device, and it is common to define the transconductance as figure of merit of this device 8.3 Technology A fabrication sequence for a GaAs MESFET is described with the help of Fig. 174. Semi-insulating GaAs material is used as a substrate. SiO 2 is deposited and a resist is patterned for the mask for ion implanting Si + ions which are used to dope the channel. For the source and the drain, the patterned resist is again the mask. The high doping of the n + implantation damages the GaAs surface. Thus, an annealing process is required to repair the GaAs crystal. But GaAs cannot be exposed to high temperatures. One problem is, that As outdiffuses from the wafer. Thus, a cap layer is deposited on the surface. SiO 2 may be used for that purpose. For the preparation of the metallization, a Si 3 N 4 film is deposited and patterned. The metallization in this example has 3 layers. First, a thin layer of Ge with a thickness of approximately 0.1 µm is deposited. A thin Ni layer serves as a barrier metal. The Ni Barrier is needed as the Au can diffuse into the GaAs and can decompose the GaAs. A thicker Au layer provides metallization for contact and bond wires. For the definition of the metallization areas a lift-off process is used. The excess metal is “lifted” when the resist is removed. The Si 3 N 4 film is etched for the gate contact. Again using a lift-off process the Ti-Pd-Au film is patterned. The Ti is deposited first and forms the Schottky contact. Au and Al cannot be used in direct contact with GaAs as they react z D D E b a w N q I ) ( dz dV V V V b a w I ch ch T GS D 2 ] 2 1 [ 2 DS DS T GS ch D V V V V aL w I 2 2 T GS ch DSat V V aL w I ) ( T GS GS DSat m V V aL w dV dI g <?page no="152"?> 8. The Field Effect Transistor 143 with it. The gate is deposited partially into the GaAs surface. This improves the threshold voltage control under the gate, while allowing the rest of the channel to be thick to minimize parasitic resistance. Fig. 174: Process steps for the fabrication of a GaAs MESFET The use of a second gate may be advanategous in order to control the amplification. Such a dual gate MESFET (Fig. 175) may be used in heterodyne mixers, where the second gate can be used as an amplitude modulation input port. In contrast to passive mixers, it is possible to achieve conversion gain with an active mi xer. Fig. 175: Layout of a dual-gate MESFET Power MESFETs are operated at high drain current denities. Plated heatsinks and interdigital source and drain fingers are used for high current density operation. Essentially, a number of devices is connected in parallel and air bridges may connect all sources. <?page no="153"?> 144 8.4 Principles of MOSFET operation In a MOSFET it is important to consider the charges which are influenced by an applied voltage below the channel. Assuming the source, the drain and the substrate are connected to the same potential 0V (Fig. 176, left side) this leads to an accumulation of holes beneath the gate insulator. No current can flow between the source and the drain since the material in between has no free electrons. When a moderate, positive voltage is applied at the gate, the majority holes in the substrate will be pushed away from the region immediately beneath the gate, as in the middle of the figure. A depletion region is formed. The transistor is now in the depletion operation region. Still no current can flow between the source and the drain. Now, when V GS is increased even further, electrons in the n-type drain and source regions will be attracted to the insulator-semiconductor interface region. A n-type material in the shape of the channel has been created, a channel through which electrons can go from the source to the drain, which is illustrated in the most right part of Fig. 176. The transistor can now conduct current in the channel; the transistor is said to be in the inversion operation region. Fig. 176: Operation regions of MOSFET devices Complementary MOS (CMOS) provides the basis for modern high speed silicon based electronics. In a simple and reliable technology pand n-type FETs are realized on the same substrate (Fig. 177). Due to the high yield and simultaneously high integration density more than 10 8 MOSFETs can be integrated per chip. The possible miniaturization leads to good RF and switching performance. Fig. 177: Cross section of complementary field effect transistors CMOS operation is now demonstrated well in the GHz-range. The limits regarding the velocity and integration density are not yet reached. The interconnect problem in multilayer CMOS circuits is a limitation regarding the integration density. Basic limitations regarding the velocity are coming from the electron mobility through the channel. Typical electron mobilities are between 300 and 600 cm 2 / Vs and the <?page no="154"?> 8. The Field Effect Transistor 145 maximum drift velocity is 8*10 6 cm/ s. Alternatives may be to use semiconductor structures with higher mobilities and higher maximum drift velocities as GaAs or InP. In silicon material the electron mobility can be increased if the silicon layer is under tensile strain (Fig. 178) 3 . Due to the lattice constant mismatch, silicon experiences a symmetrical lattice deformation. The biaxial strain contributes twofold to the silicon band structure: a hydrostatic stress, which causes a forbidden band-gap energy decrease in the X direction and an uniaxial stress, which produces an additional splitting of the degenerate levels at the Γ point (cf. Fig. 4). Lower energy is required for the electrons to enter the conduction band. Fig. 178: Strained layer Si-MOSFET The electron mobility in a tensile strained silicon layer on Si 0.55 Ge 0.45 buffer can exceed 2000 cm 2 / Vs at room temperature [8.4]. 8.5 Equivalent circuit and cut-off frequency In order to derive a physically related small signal equivalent circuit, in a MESFET structure (Fig. 179) the most relevant capacitances and resistances are included, the channel contains a current source. In the intrinsic FET, the elements C GD + C GS are the total gate to channel capacitance, the resistances R GS and R DS under the gate show the effects of the channel resistance. The extrinsic and parasitic elements include the source resistance R S , the drain resistance R D , the gate resistance R G and the substrate capacitance C DS . 3 Due to the lattice constant mismatch, silicon experiences a symmetrical lattice deformation. The biaxial strain contributes twofold to the silicon band structure: a hydrostatic stress, which causes a forbidden band-gap energy decrease in the X derection and an uniaxial stress, which produces an additional splitting of the degenerate levels at the point (cf. Fig. 4) Lower energy is required for the electrons to enter the conduction band. strained Si channel relaxed SiGe buffer Si substrate G D S strained Si channel relaxed SiGe buffer Si substrate G D S <?page no="155"?> 146 Fig. 179: Physically related small signal equivalent circuit elements The simplified intrinsic small signal equivalent circuit shows the influence of the transconductance: The influence of the gate-siurce voltage on the drain current is modeled with a current generator g m V gs in the intrinsic FET. Fig. 180: Simplified small signal circuit of intrinsic FET At high frequencies the currents through the gate capacitances are not any longer neglectable and allow a cut-off frequency definition which is similar to the currentgain cut-off frequency of Bipolar Transistors. The small signal equivalent circuit (Fig. 180) is used to determine the cut-off frequency [8.2]. The gate current and the drain current are obtained from CHANNEL CHANNEL D S S G C gs C gd V ds =0 g m V gs V gs i d i g D S S G C gs C gd V ds =0 g m V gs V gs i d i g <?page no="156"?> 8. The Field Effect Transistor 147 if the output is shorted. The second term in the expression for the drain current is neglected and the amount of the ratio of the drain current to the gate current is set to be one for the transit frequency: The transit frequency follows then from In order to gain more insight into the physical meaning of this equation the transconductance is inserted and it is obtained It can be seen, that an increase of the mobility will increase the transit frequency and that the cut-off frequency is proportional to the square of the gate length. This is the main reason for the enormous speed enhancement of MOS based circuits with reduced gate lengths. The maximum oscillation frequency is obtained from a similar procedure from with the gate resistance R G . To increase the maximum oscillation frequency, the transit frequency may be increased, the series resistance of the control electrode reduced and the feedback capacitance from the output to the input reduced. Finally, a comparison between FET and BJT is shown in table 3. gd gs gs m d gd gs gs g C j V V g i C j C j V i ) ( 1 gd gs T m g d C C g i i ) ( 2 2 gd gs m T T C C g f ) ( / 2 ) ( 2 ) ( 2 2 T GS i T GS i m gd gs m T V V L a wL C V V aL w C g C C g f G gd T R C f f 4 max <?page no="157"?> 148 Tab.3: Comparison of FET and BJT 8.6 Conclusions and outlook FETs are succesfully used in small signal amplifiers with low noise voltages, in oscillators, in power amplifiers and mixers at microwave frequency. In order to further improve the operation speed of FETs the reduction of the gate length is an appropriate means. A reduction of the gate length from 450 nm to 40 nm (Fig. 181) requires technologies to compensate for an increased gate resistance e.g. by mushroom gates and requires the transition from optical lithography to e-beam lithography. Fig. 181: Scanning electron micrographs of FETs with different gate lengths In case of short channels (below 0.5 µm) the electric field in MESFETs can become very high. The average electric field in the channel will be in the saturation part of the v-E characteristic. It may be assumed that the electron velocity in the whole channel is at its saturation value v s . For very small gate lengths the transit frequency increases linearly with reduced gate length according to FET BJT voltage current parallel to surface perpendicular to surface [small lateral dimensions, [small vertical dimensions advanced lithography, >> epitaxy] influence of surface] rarely temperature exponential temperature dependent dependence (Majority carriers) (Minority carriers) Control current transport temperature behaviour 450 nm 70 nm 40 nm 450 nm 70 nm 40 nm <?page no="158"?> 8. The Field Effect Transistor 149 where instead of the mobility µ the saturation velocity v s is of importance. The cutoff frequency is now inversely proportional to the gate length. Another important short channel effect in high frequency FETs is the velocity overshoot effect. The maximum dynamic velocity can be much larger than the maximum static velocity. Another possibility regarding a FET speed enhancement is to use materials with higher mobilities, e.g. compound semiconductors or to use strained layer silicon technology. Next generation CPUs are expected to be build on a strained layer process on 12-inch silicon substrates. Problems The figure below represents an InP MESFET structure. If the following parameters are given, solve the following questions. 1500 , 10 7 . 5 , 10 * 2 , 150 , 5 . 0 , 8 . 0 sec, / 4600 , 5 . 12 3 17 3 17 2 T cm N cm N m W m L V V cm C D B n r where, B means the schottky barrier height. 1. Calculate the depletion length due to the built-in potential at the gate 2. Find the pinch-off voltage 3. Calculate the saturation current ( Idss), and the transconductance when Vgs=0 V. 4. Estimate the Cgs when Vgs=0V 5. Using your answers to the subtask 3, 4, find the approximate unit current gain frequency, neglecting the capacitance between gate and drain L v f s T 2 Gate Source Drain L W T <?page no="159"?> 150 References [8.1]S.M.Sze, “High-Speed Semiconductor Devices”, 1990, J.Wiley & Sons [8.2] C.A.Liechti, “Microwave Field-Effect Transistors - 1976”, IEEE-MTT, vol. 24, 1976, pp. 279-300 [8.3] T.G.van de Roer, “Microwave Electronic Devices”, 1994, Chapman&Hall [8.4] U.König, “SiGe/ Si Heterostructure Devices - Status, Problems and Prospects” <?page no="160"?> 9. The High Electron Mobility Transistor 151 9. The High Electron Mobility Transistor One of the disadvantages of a MESFET is, that the mobility is degraded at high channel doping. It is the most important advantage of a High Electron Mobility Transistor (HEMT) 4 , that the channel doping is decoupled from the disadvantageous effect of Coulomb scattering. As was seen in the discussion of the AlGaAs/ GaAs HBT, the electrons tend to move from the AlGaAs to the GaAs and to form a conducting channel near the interface. Now the electrons are separated from the donors and the electrons have the mobility of undoped material although their density can be very high. With a spacer layer of undoped AlGaAs the Coulomb field of the ionized donors in the AlGaAs is shielded and ionized impurity scattering is further reduced. The band diagram of a HFET is shown in Fig. 182 Fig. 182: Band diagram of a HFET The AlGaAs layer is n-doped and the GaAs layer is p-doped. Both layers are separated by an undoped spacer layer. Electrons are supplied from the AlGaAs layer into the GaAs section, which forces the conduction band to fall below the Fermi level. An electron inversion layer is generated within a small potential well. The energy levels in this potential well are quantized, i.e., the electrons are confined in the zdirection. Two distinct advantages of HEMTs become obvious: 1. The spacer moves ionized impurities away from the 2-DEG, reducing scattering. Typical spacers are between 1 and 2 nm for MODFETs. 2. The electrons are confined in the z-direction by a triangular well of very small dimensions, thus the allowed electron energies are quantized. 4 There are other acronyms like MODFET (Modulation Doped Field Effect Transistor), HFET (Heterojunction Field Effect Transistor) and TEGFET (Two-dimensional electron gas FET) Undoped spacer 1 - 2 nm Electron inversion layer Metal N + - AlGaAs p - - GaAs Undoped spacer 1 - 2 nm Electron inversion layer Metal N + - AlGaAs p - - GaAs <?page no="161"?> 152 Less scattering by lattice vibrations (phonons) is a consequence especially at lower temperatures. The concept of modulation doping is illustrated with the help of Fig. 183. Fig. 183: The concept of modulation doping We can build a quantum well, in which the carriers prefer to stay, and provide these carriers by using “donor” layers which are heavily doped, but do not contribute to large amounts of scattering. An important quantity for the HEMT operation is the sheet carrier density in the channel. Obviously the sheet carrier density depends on the operation conditions as is illustrated in Fig. 184. Fig. 184: Conduction band (upper part), electric field and sheet carrier density (lowest part) in a HEMT for different gate voltages. <?page no="162"?> 9. The High Electron Mobility Transistor 153 9.1 Basic HFET theory Following the representation of Fig. 181, we obtain for the junction electric field with the sheet carrier density n s . The electric field in the AlGaAs layer - if the undoped spacer is neglected - is given by if the doping in the AlGaAs is given by N D and the thickness of the AlGaAs layer is given by d Al . The position x min of the potential minimum - which means, that the electric field id zero - can be determined from The depth of the potential well below the Fermi level is given by where V min is given in terms of the gate voltage and V 1 and V 2 are the areas of triangular field regions in the AlGaAs layer: and Now we obtain for the depth of the well And we recognize, that the well depth increases with increasing gate voltage, and the well is filled with electrons. The well depth reduces with increased sheet density. We call this the first finding from Poisson considerations. In order to consider the quantum mechanical nature of the effects in a HEMT we determine the position of the subbands in a triangular potential well from Where i denotes the number of the subband. The subband levels move up when the field increases and the well becomes narrower. When only the first subband level is below the Fermi level, the sheet density is given by s j qn E ) ( Al D j d x qN E E D s D j Al N n qN E x d min 2 min V V W C w 1 min V V V G B 2 2 min 1 2 2 D s Al D D N n d qN x qN V D s Al D N qn x d qN V 2 ) ( 2 2 2 min 2 Al s G Al D B C w d qn V d qN W 2 2 3 / 2 3 1 * 2 4 3 2 3 2 i qE m W j i <?page no="163"?> 154 There is a monotonic increase of n s with the depth of the potential well below the Fermi level! The second finding from quantum mechanical considerations is that the sheet density increases with the potential depth. The crossing of these two curves gives the actual values of n s and w . The drain current in an ideal HEMT is calculated from if saturated velocity is assumed. The sheet carrier density is proportional to the drain current. The influence of the gate voltage on the junction field is given by and we obtain for the transconductance with the gate capacitance The transconductance of an ideal HEMT is independent on the gate voltage. In a real HEMT however, the channel width is not constant. An effective width would have to consider the spatial spread of electrons. Furthermore, there exist parasitic FET/ MESFET channels. The AlGaAs potential minimum always will contain some electrons. 9.2 Types of HFETs In order to increase the drain current, which can be carried by the transistor, the sheet carrier density shall be increased. The problem in AlGaAs/ GaAs devices are so called DX centers which have an energy level near the middle of the forbidden gap and which are formed by the silicon atoms which are used as donors. The result are low-frequency relaxation phenomena, noise and photoconductivity. Therefore the Aluminium content is limited to approcimately 25 % which means that the conduction kT W kT m n w c s 1 2 * exp 1 ln 1 ) ( and ) ( w s s w n n sat s ch D v n qw I E d j Al g V ch sat G Al sat ch g D m L v C d v w V I g Al ch ch g d w L C <?page no="164"?> 9. The High Electron Mobility Transistor 155 band discontinuity W C stays below 0.23 V. The AlGaAs HEMT is therefore a high transconductance but a low current device. In order to keep the silicon donors away from the Aluminium atoms several approaches have been developed. Fig. 185: Two types of spacers: Modulation doped superlattice (a) and delta doped quantum well (b) In Fig. 185 a it is shown, how the AlGaAs is replaced by a superlattice where only GaAs is doped. Minibands are formed. From outside it looks loke a doped AlGaAs layer. An alternative approach is shown in Fig. 185 b: A thin (10 nm) layer of GaAs is put into the AlGaAs layer close to the channel. The GaAs has a very high and thin doping spike, a delta doping. Other schemes to increase the sheet density of HFETs are given in Fig. 186. Fig. 186: Double sided channel (left side) and multiple quantum well (right side) <?page no="165"?> 156 Employing two channels in a HFET can increase the channel conductivity by a factor of 2 (Fig. 186 left). The next step would be to use multiple quantum wells (Fig. 186 right) with very thin barriers in order to ensure, that the electrons can tunnel through the barriers avoiding the formation of independent channels. 9.3 SiGe HFETs In order to obtain a conduction band offset in the silicon/ germanium material system, a strained silicon layers grown on Si 1-x Ge x is required. These Si 1-x Ge x layers may be formed by Si 1-x Ge x buffer layers on top of which thin silicon layers lead then to a type II band alignment situation (right part of Fig. 116). The two dimensional electron gas channel is then formed by the silicon layer. Fig. 187 shows the layer sequence and the conduction band structure of a n-HFET in Si/ SiGe with a double sided channel together with its conduction band diagram. Fig. 187: Double sided channel (left side) and multiple quantum well (right side) In Fig. 188 the structure is visualized as a 3-D drawing in order to explain the position of the contacts. Electrons can travel from source to drain, if cariers are in the channel, i.e. if th GS V V and if a potential differenhce between source and drain exists 0 DS V QW E F E C Si 2 DEG - Si Si S i G e S i G e Substrate buffer doping 2-dimensional electron gas channel doping Cap spacer <?page no="166"?> 9. The High Electron Mobility Transistor 157 Fig. 188: The concept of a HEMT (n-channel MODFET) in SiGe The transconductance of optimized n-SiGe HFETS can now exceed 400 mS/ mm and shows regions of high linearity proving a nearly ideal HFET behaviour (Fig. 188) Fig. 189: Transconductance of a n-SiGe HFET as a function of the gate bias and drain current as a function of the gate bias. The ellipse marks the region of constant transconductance. Cap Si Cap Spacer SiGe Doping layer SiGe Spacer SiGe channel Si (2DEG) Doping layer SiGe Spacer SiGe Constant buffer SiGe graded buffer Si..SiGe Source Drain Gate contactimplantation contactimplantation L G W G V GS V DS 350 250 150 50 0 Transconductance g m [mS/ mm] -0.6 -0 4 -0.2 0.0 0.2 0.4 0.6 Gate Bias V GS [V] Drain Current [mA/ mm] 350 250 150 50 0 <?page no="167"?> 158 9.4 Technological aspects For the realization of millimeter-wave FETs, it is essential to fabricate submicron gates. An important role in the technology plays the generation of photoresist masks. Some typical steps for the realization of a mask from a photoresist are shown in Fig. 190. Fig. 190: Typical steps for the generation of a mask from photoresist This process sequence can use UV light or e-beams to expose the photo resist. Fig. 191 shows an e-beam written mushroom-gate. Fig. 191: E-beam written Schottky - gate 9.5 High power FETs Substrate Adhesion layer Photo resist Mask and alignment Exposure • UV - light • e-beam Exposed region Developing Remaining resist rests O 2 Plasma dip Photo resist mask Substrate Adhesion layer Photo resist Mask and alignment Exposure • UV - light • e-beam Exposed region Developing Remaining resist rests O 2 Plasma dip Photo resist mask <?page no="168"?> 9. The High Electron Mobility Transistor 159 Traditionally GaAs is used for the realization of solid state power HEMTs. High output power requires high current and/ or high voltage. Due due to the material constants the maximum power density in GaAs is limited to about 1 Watt/ mm. To compare electrical microwave power performance limits, the Johnson figure of merit can be used: 2 / max sat br T C v E f X P with P max representing the output power, f T is the transit frequency and X C is the capacitive input reactance at the frequency of the P max measurement. The Johnson figure of merit shows, that the Watts per unit input capacitance for a given f T are limited by the product of breakdown field strength (E br ) and saturated velocity (v sat ) of the material. It describes, the ease with which technologies can deliver a specific output power at a given frequency. Table 2 contains materials for power transistors and shows come important power transistor parameters including the Johnson figure of merit. Table 4: Comparison of materials for power transistors The limitations of existing technologies become evident: The output power versus frequency is limited (e.g. 1 W/ mm at 10 GHz with GaAs); the supply voltages are limited (typically 10 V at X-band) which requires high currents leading to losses and reliability problems. The high temperature stability is poor as well as the robustness against electromagnetic pulses. 9.5.1 Concept of GaN HEMTs Wide-band gap Gallium-Nitride based semiconductors were used for the realization of blue diode lasers and demonstration of FETs with record power density of more than 10 W/ mm at X-band. The large bandgap is associated with a large breakdown field and a large breakdown voltage. High temperature stability, EMP hardness and the potential high reliability are other reasons for the use of GaN for RF power applications. Studies of electron transport in GaN suggest a velocity - field characteristic with a region of negative differential mobility, high threshold and critical fields, and reduced energy-relaxation times. These properties make GaN also suited <?page no="169"?> 160 to realize negative resistance diodes for high power generation of mm-waves as an alternative to three-terminal devices. In Fig. 192 the basic concept for the realization of AlGaN/ GaN HEMTs is shown. The Two-Dimensional Electron Gas is formed within the GaN layer. Fig. 192: HEMT concept in the AlGaN/ GaN material system The layer stack of a millimeter wave GaN HEMT is given in Fig. 192 [10.2]. Because single crystal GaN is not readily available, a suited substrate material has to be choosen. Sapphire substrates are available as low cost substrates, however provide a poor thermal conductivity and a poor lattice mismatch with GaN. Low cost silicon substrates provide a poor lattice and thermal expansion match. Insulating SiC substrates provide a superb thermal conductivity and a better lattice match but are very expensive. The HFET layers consist of a 20 nm AlN buffer layer, followed by a series of layers grown by ammonia MBE (molecular beam epitaxy). These consist of a 1 µm insulating carbon-doped GaN layer, a 0.2 µm undoped GaN channel layer and an AlGaN layer of 25 nm total thickness, consisting of a 5 nm spacer, a 15 nm donor layer (nominally doped with Si to 1x1019 cm-3), and a 5 nm cap layer. The 2-DEG formation in this device is due to supply layer doping. In the literature, there are also designs reported without a doped supply layer, where the 2-DEG formation is due to spontaneous polarization or piezo-doping. Fig. 193: Vertical structure of a GaN/ AlGaN HEMT The T-gate process uses a three-layer resist stack, patterned by e-beam lithography. The gate length was 0.13 µm (Fig. 194). Source-to-drain and source-to- GaN AlGaN W F W L E 2DEG y + + + + + + 1µm UID GaN 15nm Si: AlGaN 5nm UID AlGaN 5nm UID AlGaN Saphir/ SiC 1µm UID GaN 15nm Si: AlGaN 5nm UID AlGaN 5nm UID AlGaN Saphir/ SiC <?page no="170"?> 9. The High Electron Mobility Transistor 161 gate spacings are 2.1 and 0.7 μ m, respectively. Contact resistances for these devices are 1 Ω -mm. Fig. 194: Scanning electron micrograph image of a T-Gate structure The maximum drain current was > 1.25 A/ mm, while the peak transconductance is 250 mS/ mm (Fig. 195, left). The current is still pinched off completely at -5 V gate bias. Fig. 195 (left): Transconductance and Drain current measured as a function of the gate voltage from the device structure given in Fig. 193. Fig. 195 (right): Short circuit current gain (H 21 , circles) and unilateral power gain (squares) versus frequency for the same device at a source drain voltage of 10 V and a gate bias of -2.5 V. The extrapolation of the measured current gain and power gain as a function of the frequency yields values of f T and f max of 103 GHz and 170 GHz, respectively. 200 nm 0 50 100 150 200 250 300 0 0.2 0.4 0.6 0.8 1 1.2 -6 -4 -2 0 2 4 Transconductance (mS/ mm) Drain current (A/ mm) Gate Voltage V g (V) Frequency (GHz) 10 100 Gain (dB) 0 10 20 30 40 H 21 UPG V ds =10 V V g =-2.5 V Frequency (GHz) 10 100 Gain (dB) 0 10 20 30 40 H 21 UPG V ds =10 V V g =-2.5 V Frequency (GHz) 10 100 Gain (dB) 0 10 20 30 40 Frequency (GHz) 10 100 Gain (dB) 0 10 20 30 40 H 21 UPG V ds =10 V V g =-2.5 V Frequency (GHz) 10 100 Gain (dB) 0 10 20 30 40 H 21 UPG V ds =10 V V g =-2.5 V Frequency (GHz) 10 100 Gain (dB) 0 10 20 30 40 H 21 UPG V ds =10 V V g =-2.5 V <?page no="171"?> 162 9.5.2 Device optimization A significant problem in AlGaN/ GaN HEMTs is the RF current collapse occurring at higher current densities due to surface effects. SiN x surface passivation has been reported to reduce the RF current collapse phenomenon with corresponding improvement of the RF output power density and power-added efficiency. Fig. 196 (left): GaN HEMT I-V characteristic before SiN x passivation. Fig. 196 (right): GaN HEMT I-V characteristic after SiN x passivation. Pulsed I-V measurements before and after passivation are shown in Fig. 196 [9.3]. Without passivation a significant current collapse and reduced RF power is obtained. It can be seen, that passivation helps a lot: The current collapse is reduced and the RF power is improved by a factor of 3 to 4. The beneficial effect of surface passivation on the RF performance of AlGaN/ GaN HEMTs is due to a decrease in the surface trap density and in the surface negative fixed charge density. To improve the device reliability is a big issue. The material quality has to be improved, i.e. the dislocation density and the defect density has to be reduced. External stress which may be caused by packaging and environemental influences which relate to surface stability and passivation are of importance for the device reliability.The contact quality and stability are of special importance for high field, high current and high channel temperature operation. The theoretical investigation of a field effect transistor with a field-modulating plate (FP) using a two-dimensional ensemble Monte Carlo simulation revealed, that the introduction of FP is effective in canceling the influence of surface traps under forward bias conditions and in reducing the electric field intensity at the drain side of the gate edge under pinch-off bias conditions [9.4]. A field-plate effectively reduces the peak electric-field strength on the drain side of the gate edge, thereby increasing the breakdown voltage (Fig. 197). The lower field due to the field-plate significantly reduces tunneling from the gate into surface traps; field-plate gates successfully eliminate high-voltage leakage currents under pulsed operation. <?page no="172"?> 9. The High Electron Mobility Transistor 163 Fig. 197: Implementation of a field-plate in a GaN The GaN power density exceeds 30 W/ mm and 55 % PAE with field plates up to Xband frequencies. The tremendous progress in GaN microwave power performance achieved in the last years is indicated in Fig. 198. Fig. 198: GaN microwave power device progress <?page no="173"?> 164 9.5.3 Conclusions Present day GaN HEMTs exhibit outstanding performance. They deliver 10 times more power than GaAs pHEMTs at X-band. A reliability of more than 700 hours CW at 30 V has been obtained. GaN is considered to be the enabling technology for a next generation of phased arrays with increased range and sensitivity, increased bandwidth and an improved cost to weight ratio. There are challenges remaining regarding the device uniformity, the efficiency and the high-voltage reliability, the thermal management and the costs. Problems Using a Si/ SiGe technology, a Si/ SiGe HEMT structure is fabricated according to Fig. 188. o What is the role of ‘Graded buffer Si..SiGe’ layer in the above structure? o Under the source and the drain contacts, there are deep contact implantation layers. Discuss their contributions to the current transport mechanism of the device with respect to DC and RF performances. o Explain the properties of the Si/ SiGe HEMT technologies compared to the III-V HEMT devices. o Discuss in detail the current transport mechanism of the HEMT device using the above the Si/ SiGe HEMT structure. o What are the motivations of the so-called mushroom type gate structure in the figure? o Draw a simplified small-signal equivalent circuit of the above structure. o Draw a energy band diagram between A and A’. <?page no="174"?> 9. The High Electron Mobility Transistor 165 References [9.1] S.M.Sze, “High-Speed Semiconductor Devices”, 1990, J.Wiley & Sons [9.2] J.A. Bardwell, Y.Liu, H.Tang, J.B. Webb, S.J.Rolfe, J.Lapointe, “AlGaN/ GaN HFET devices grown by ammonia-MBE with high f T and f MAX ”, Electronics Letters, 2003 [9.3] J.Würfl, N.Chaturvedi, P.Heymann, H.Klockenhoff, A.Liero, R.Lossy, M.Mai, “High Power GaN-Devices for L-Band and X-Band Applications: Technology and Performance”, IEEE-MTT-S Workshop on GaN Devices, June 2004 [9.4] Y. Hori, M.Kuzuhara, Y.Ando, M.Mizuta, “Analysis of electric field distribution in GaAs metal-semiconductor field effect transistor with a field-modulating plate”, J. Appl. Phys., vol. 87, no. 7, Apr. 2000, pp. 3483-3487 <?page no="175"?> 166 10. Future Devices The properties of the microwave devices together with the advantages and disadvantages of microwave devices were discussed in the preceeding chapters. In this chapter two further microwave two terminal devices are described together with recently developed alternative approaches for the tuning of resonant circuits and for the switching and routing of microwave signals. Finally the approach of the resonant phase transistor is considered as a potential scheme for the generation of microwave signals up to the THz frequency range. 10.1 Heterostructure varactor diodes Varactor diodes are used frequently in voltage controlled oscillators or in multipliers.The structure of a simple p-n varactor diode is shown in Fig. 199. The space charge layer extends from - w to + w. The Poisson equation for the potential reads and the boundary conditions are fixed by and the doping distribution is assumed to obey Fig. 199: VARiable reactor (Varactor) From the integration of the Poisson equation the potential distribution is obtained from where and the electric field is obtained from 0 w x p n 0 w x p n A D N N e 2 0 ) ( 0 ) 0 ( V V w n A D x B N N N * ) 2 )( 1 ( ) 1 ( ) ( ) ( 1 2 0 n n y y eBw y V V y n n w x y / ) 2 )( 1 ( ) ) 2 ( 1 ( 1 * ) ( ) ( 1 1 0 n n y n eBw w V V y E n n <?page no="176"?> 10. Future Devices 167 From the space charge boundary The space charge width is obtained: The maximum electric field at the pn-junction is then given by In case of the varactor, we are interested in the dependence of the capacitance on the voltage. The capacitance C per unit area follows from with the device area A. With the expression for E max we obtain which gives with If we now consider a resonant circuit with a resonance frequency and require a linear dependence of the resonance frequency on the tuning voltage, then we can conclude that which requires a hyperabrupt doping profile as scetched in Fig. 200 Fig. 200: Hyperabrupt doping profile (dotted line) and appximation in real devices (solid line) If the sheet resistance of a varactor diode is given by, 0 )) ( 1 ( w x y E ) 2 / ( 1 0 ) )( 2 ( n eB V V n w ) 2 / ( 1 1 0 max ) 2 ( ) ( ) 1 ( ) 2 ( )) 0 ( 0 ( n n n V V eB n n x y E E dV dE dV dQ A C max 1 ) 2 / ( 1 0 1 ) )( 2 ( ) ( n n V V n eB C ) ( ~ 0 ) 2 / ( 1 0 V V V V C n 2 / 1 ) ( LC r 0 V V r 1 2 / 2 / 3 n 2 / 3 x Doping level Distance 2 / 3 x Doping level Distance A d R b max ) 2 / ( 1 n <?page no="177"?> 168 with d representing the length of the undepleted regions and σ the conductivity, the cut off frequency is determined from and finally the frequency dependent quality factor Q of the device is obtained from Due to the p-n junction, the classical varactor diode exhibits a non-symmetric I-V and C-V characteristic. If a device would provide a symmetric capacitance-voltage characteristic, in a multiplier operation odd harmonics could then be suppressed, permitting sideband generation at frequencies of 2f p +2f in " where f p is the pump frequency and f in is the frequency of the carrier signal. A Heterostructure Barrier Varactor (HBV) may exhibit a symmetric capacitance-voltage characteristic. Fig. 201: Capacitance voltage characteristic of a heterostructure barrier varactor [10.2] The heterostructure barrier varactor is a symmetric varactor consisting of a high band gap semiconductor (barrier), surrounded by moderately doped modulation layers of a semiconductor with a lower band gap. The barrier prevents electron transport through the structure. When an external signal is applied to the HBV, carriers are accumulated at one side and depleted at the other side of the barrier, causing a symmetric, voltage dependent capacitance.The HBV can achieve high power handling capability by epitaxial stacking of the barriers. This results in higher breakdown voltages and allows the generation of higher levels of sideband power. HBVs are normally fabricated from III-V heterostructures, principally GaAs/ AlGaAs on GaAs and InGaAs/ InAlAs on InP. GaAs based HBVs exhibit quite high breakdown voltages but suffer from a comparatively low electron mobility and excessive leakage, which lowers the conversion efficiency significantly [10.1]. Stateof-the-art HBVs are fabricated using In 0.53 Ga 0.47 As/ In 0.52 Al 0.48 As on InP [10.2]. This system offers higher electron velocities and the electron potential barrier is higher compared to GaAs based systems, which means lower leakage currents and, thus, higher conversion efficiencies of frequency multipliers. Fig. 200 shows a semiconductor layer sequence corresponding to 2 HBV diodes in series. The symmetric capacitance-voltage characteristic of this device together with a relatively high modulation ratio and power handling capability make this device well suited for subharmonic parametric sideband generation. With HBV Sideband C R f b c 2 1 f f f Q c 2 <?page no="178"?> 10. Future Devices 169 generators (SBGs) a conversion loss of 10-15 dB is achieved with output frequencies of 200 GHz. Fig. 202: HBV epitaxial layer characteristics [10.2] 10.2 PIN diodes and MEMS switches The basic structure of the PIN diode is illustrated in Fig. 203. It consists of the lightly doped high resistivity („intrinsic“) layer contacted on one side by the high conductivity p layer and on the other side by the high conductivity n layer. Fig. 203: Structure of a PIN diode In the forward bias state (positive voltage on the p layer) holes are injected into the intrinsic i-layer from the p-layer, and electrons are injected from the n-layer. The injection of holes and electrons in the i-layer results in storage of electric charge in the i-layer of quantity Q. The presence of freely mobile charge in the i-layer results in the phenomenon known as conductivity-modulation. In other words, the formerly high resistance i-layer, becomes a high conductance (low resistance) layer due to the ability of injected mobile holes and electrons to conduct current. The amount of charge Q in the i-layer is proportional to the concentrations of holes and electrons injected by the forward current I F . Neglecting diffusion currents, the current in the i-layer is given by p + n + intrinsic + p + n + intrinsic p + n + intrinsic + - GLE AE p n e I p n F <?page no="179"?> 170 Q I dt dQ R Q I dt dQ F 1 ) ( S S T R T F S e I e I T Q with the mobilities and carrier densities for electrons and holes, µ n ,µ p and n, p, respectively. A is the device area, E the electric field, L the length of the active region and G is the device conductance. In switching applications the switching time is an important parameter. The governing equation for the mobile charges in the i-layer is with the carrier lifetime and the total charge In steady state the mobile charge in the i-layer is constant The removal of mobile charge is described by The charge is either removed by withdrawing it caused by the reverse current I R or by carrier recombination (Q/ ). The charge decreases steadily until the i-layer is totally depleted of mobile charge at a time which we shall call switching time T S . The reverse operation equation is solved by integrating from t=0 to t=T S and using that Q at t=0 is equal to I F .: The i-layer is depleted of carriers at t=T S , Q(T S )=0 so that the result is It can be seen that the ratio of forward to reverse current influences the switching time as does the carrier lifetime. When the injecting current is stopped by rapidly reversing the bias voltage, the injected carriers take time to leave the i-layer. They leave by several mechanisms 1) Some of them decay within the i-layer with a semiconductor bulk lifetime b 2) Some of them diffuse to the surface and decay at the surface with surface lifetime s 3) Some of them are withdrawn by the driver reverse current which flows until the i-layer is depleted of mobile charge. Since the carrier decay mechanisms are independent and competing mechanisms the resultant carrier lifetime is given by The PIN diode can yield very small switching times due to small cartrier lifetimes and optimized current modulator design. However, the PIN diode causes an insertion loss, which is due to the intrinsic resistance of the PIN diode under forwared bias state ) 1 ln( R F S I I T ) ( p n e Q Q I dt dQ F : 0 4 2 exp tan 2 sinh 4 0 1 0 L L L L eI kT R i i F I S b 1 1 1 <?page no="180"?> 10. Future Devices 171 and simplified for in the case of mm-wave diodes with The total resistance of the diode consists of the intrinsic layer resistance R I in series with an ohmic contact resistance R C. R C is inversely proportional to the area. The Pin diode presents a constant junction depletion capacitance to an RF signal at a reverse bias. This capacitance is given by The insertion loss of semiconductor switches may be a severe problem in phased array, where several combinations of switches are connected close to the antenna. Also the power consumption is a draw back. Micro-Electro-Mechanical Systems (MEMS) switches are advantageous with this respect. Their operation is similar to a classical relay. Additionally they provide a high linearity, cause no intermodulations, the driver circuits may be simple and the concept of a capacitive MEMS switch shown in Fig. 204 is compatible to a monolithic integration on low-cost substrates Fig. 204: Capacitive MEMS switch, embedded in a coplanar line. Not a „physical“ semiconductor device but on semiconductor substrate. 0 2L L i F p n p n i I I µ µ µ µ L R 1 ) )( ( 4 2 region i in the lifetime carrier effective : holes and electrons of mobility : µ constant diffusion ) / 1 / ( 2 : D length diffusion D : L current DC bias forward : I charge electron : e re temperatu diode : T constant s Boltzmann´ : K region intrinsic the of Thickness : L p n, 0 0 0 0 F i p n n µ µ D i L A C <?page no="181"?> 172 The capacitive coupling switches, mostly realised in an airbridge configuration as illustrated in Fig. 201 [10.3] have a thin dielectric film and an air gap between the two metallic contact surfaces. The air gap is electromechanically adjusted to achieve a capacitance change between the “up” and “down” state. The capacitance ratio of the downstate value to the upstate value is a key parameter for such a device; a high capacitance ratio is always desirable. Because of the capacitive coupling nature, in most cases these switches are not suitable for low-frequency applications. However, the contact lifetime is typically a smaller issue compared with that in a metal contacting switch. A DC voltage of arbitrary polarity will pull down the membrane from the open (on) state to the off state. In order to describe the typical hysteresis occurring during the switching cycle of MEMS switches, the simple model depicted in Fig. 205 is used. Fig. 205: MEMS model with a spring with spring constant k, a capacitance with width g (which is the height of the bridge) and an applied voltage. The electrostatic force is given by 2 2 0 . . 2 bias stat el V g Ww F and the area of the effective capacitance is wW A with the width of the bridge w and the width of the central conductor W. The spring constant is given by 3 3 * * 16 L t w Y k with Young´s modul Y=80*10 10 N/ m 2 in the case of gold as membrane metal and with the thickness t of the bridge. L is the length of the bridge. The spring force follows from g g k F spring 0 * with the relaxe height g 0 of the bridge. The electrostatic force and the spring force are functions of the bridge height and yield a diagram with several intersections as shown in the example of Fig. 206. <?page no="182"?> 10. Future Devices 173 Fig. 206 left side: Operation regions of capacitive MEMS switches. Dashed line show the electrostatic force as function of the bridge height, the solid line shows the spring force. Fig. 206 right side: Switching hysteresis. Bridge height as a function of the applied voltage, starting from the open position (g=4 µm) Characterisation of the RF MEMS capacitive switches consists of S-parameter measurements in the offand on-state. The measured S-parameters depend on the width, the length and the material of the membrane. The RF measurement results of a serial capacitive airbridge switch with a width of 120 µm and a length of 400 µm are shown in Fig. 207. The membrane consists of 0.76 µm sputtered Au and the height above the electrode is 2.8 µm. Insertion loss lower than 0.32 dB up to 40 GHz in the off-state (0 V) and isolation down to 13 dB @ 40 GHz in the on-state (28 V) are measured. The return loss in off-state is better -22 dB up to 40 GHz. Fig. 207: RF measurement result of a serial capacitive airbridge switch [10.4] There exists a wide variety of different MEMS switches: Capacitive membrane switches for phase shifting, extremely broadband switches for the selection of <?page no="183"?> 174 different front-ends and antennas, ohmic switches for signal routing and more. Fig. 208 shows the scheme of a toggle switch which was designed for improved power handling capability [10.4]. It is an ohmic contact switch realised by an movable cantilever in the signal line of an coplanar transmission line environment. Thereby, the signal line can be connected by electrostatic actuation. Metal to metal or capacitive coupling is possible. The advantage is the extremely high isolation and the high bandwidth. Fig. 208: Schematic concept of a so called "Toggle Switch" [10.4] 10.3 The Resonance Phase Transistor Fabrication of oscillator circuits with transistors is using commonly the amplification character of these active elements in a way - by means of embedding them into an appropriate circuit network - that there is a reference plane that is characterized by a negative resistance. This conventional procedure of power generation is limited to frequencies below the maximum frequency of oscillation. Although, by continuously decreasing structural dimensions, cut-off frequencies have been raised impressively in the past, there are limits existing, both by technological and physical reasons, that make it questionable if oscillators can be fabricated at frequencies well above several 100 GHz. A new way to overcome this limitations is opened by the principle of resonance phase amplification that was first discussed in 1998 [10.5]. In this scheme, by increasing the collector current phase angle beyond a value of π, power generation is accomplished by creating a negative resistance reference plane at the output of the two port device. Up to now two concepts of resonance phase amplification are distinguished, namely coherent base transport quantum-well injection Perspectives of power generation at frequencies close to 100 GHz using coherent base transport were discussed first by Greenberg and Luryi in 1993 [10.6]. In order to obtain coherent transport, the carriers have on off <?page no="184"?> 10. Future Devices 175 1. to travel through the base in a ballistic manner, i.e., without any scattering or loss of energy 2. to enter the base mono-energetic and with a homogeneous velocity Coherent like transport is essential in order to achieve a high value of the total transport factor in the device. This is possible to realise by a step or linearly graded base with high enough band energy difference over the base layer. The phase shift itself is obtained by a delayed injection into a drift region. Besides the transport factor, the optimum choice of the injection phase and the collector transit θ is of importance. The phase angle condition to get optimum negative resistance values reads [10.7] 2 / 3 2 / Basically, this condition may be fulfilled either by choosing sufficiently lage values of and θ. However, the collector transport factor becomes zero at 2 / and thus, the task is to attain injection angles θ significantly in excess of π/ 2. Fabrication of first RPTs is very similar to a mm-wave HBT technology [10.8]. Active operation at ~ 40 GHz is reached with very low current densities. Fig. 206 shows the measured current gain of a RPT with a 120 nm thick SiGe base layer extracted from on-wafer S-parameter measurements without deembedding. The maximum transit frequency of this transistor is f T,max =5.6 GHz with I B =5 mA; at 40 GHz the current gain is again increased to 6.5 dB as a result of the resonance phase effect. Fig. 209: Measured current gain h 21 with I B =1mA and V CE as parameter [10.9] <?page no="185"?> 176 References [10.1] J. Stake, L. Dillner, S. H. Jones, C. M. Mann, J. Thornton, J. R. Jones, W. L. Bishop and E. L.Kollberg, “Effects of Self-Heating on Planar Heterostructure Barrier Varactor Diodes”, IEEE Transactions on Electron Devices, vol. 45, pp. 2298-2303, 1998. [10.2] Haiyong Xu, Jeffrey L. Hesler, Yiwei Duan, Thomas W. Crowe, Robert M. Weikle, II, “A Heterostructure Barrier Varactor Sideband Generator”, IEEE- MTT-S, 2003, pp. 2031-2034 [10.3] Goldsmith C, Lin T-H, Powers B, Wu W-R and Norvell B.: "Micromechanical membrane switches for microwave applications", Tech. Digest, IEEE Microwave Theory and Techniques Symp., 1995 pp. 91-94 [10.4] B.Schauwecker, J.-F. Luy, K.M.Strohm, T.Mack, „Microwave electrostatic micro-machined devices for on-board applications”, Final Report for the European Space Agency, ESTEC / Contract No. 14547/ 00/ NL/ CK, 2003 [10.5] H. Jorke, J. Weller, J.-F. Luy “Resonance Phase Amplification” in “Future Trends in Microelectronics: Off the Beaten Path” Editors S. Luryi and J. Zaslavsky, 1998 [10.6] A.A. Grinberg and S. Luryi “Coherent Transistor”, IEEE Trans. on Electron Devices 40, 1512 (1993) [10.7] H.Jorke, M.Schäfer, J.-F. Luy, „Resonance Phase Transistor - Concepts and Perspectives“, Silicon Monolithic Integrated Circuits in RF Systems (SiRF), 2001, pp. 149 - 156 [10.8] E.Kasper, J.Eberhardt, H.Jorke, J.-F.Luy, H.Kibbel, M.W.Dashiell, O.G.Schmidt, M.Stoffel, „SiGe resonance phase transistor: active transistor operation beyond the transit frequency f T ”, Solid State Electronics 48 , 2004, pp. 837 - 840 [10.9] R.Wanner, G. Olbrich, H. Jorke, J.-F. Luy, S. Heim, E. Kasper, P. Russer, „Experimental Verification of the Resonance Phase Transistor Concept“, IEEE-MTT-S, 2004, pp. 991-994 <?page no="186"?> 177 List of Abbreviations: A area α ionization rate C capacitance E electric field ε permittivity (ε o * ε r ) f frequency f D Doppler frequency G antenna gain G c conversion gain Reduced Planck’s constant, s J h 34 10 * 05457 , 1 2 I current J current density K Boltzmann constant, K J / 10 * 38066 , 1 23 thermal conductance length m mass MDS Minimum Detectable Signal µ mobility N doping levels ΔN number of states n i intrinsic carrier density n electron density p hole density P power P N low frequency noise power q electron charge, 1.6*10 -19 C r radius charge density R resistance RTL Round Trip Loss σ conductivity σ R Radar cross section Θ thermal resistance time constant(s) T temperature V voltage v velocities W energy levels angular frequency Z impedance <?page no="187"?> Index accumulation 160 accumulation layer mode 19 air bridges 159 alloy comoposition 111 anisotropic etch 81 avalanche equation 50 avalanche frequency 51, 53, 76, 78 back injection 152 Baraff-Theory 46 Bardeen model 93 BARITT 56 barrier heights 93, 100, 104 base transport factor 129 beamlead process 32 Bipolar Junction Transistor BJT 126 breakdown 48, 113, 180 Brillouin zone 2 buffer 111 build-in potential, voltage, field 39, 157, 140 bulk lifetime 188 buried layer 82, 97 Butcher's rule 15, 17, 18 carrier lifetime 187 cladding layers 118 CMOS 160 coherent base transport 192 complex injection ratio 43 conduction band 6 conductivity 63 conductivity-modulation 187 conservation of current 20 continuity equation 20, 41, 49, 70 convection current 20 conversion efficiency 76 conversion loss 105 coplanar probing 148 Coulomb scattering 46 critical layer thickness 142 current density 70 current gain 127, 139, 148 current gain factor 129 current sensitivity 101, 103 cut-off frequency 134, 162, 163 de Broglie wavelength 114 delta doped quantum well 171 density of states 7 density of valence band holes 8 depletion 160 depletion approximation 91 diamond 30 dielectric relaxation time 21, 23 diffusion capacitance 90 dirac function 67 discontinuities 107 domain 15, 18 Doppler frequency 85 double mesa process 145 double sided channel 171 drift region impedance 42 DX center 170 Early effect, voltage 132 e-beam 174, 177 effective conduction band density of states 7 effective doping approximation 61 effective mass 2, 3, 4, 7, 14 effective ohmic resistance 65 effective Richardson constant 98 electron affinity 91, 94 emitter efficiency 128 energy band diagrams 13 energy bandgap 11 Fermi energy 9 Fermi function 7 Fermi integral 7 Fermi level 38, 108, 115 field plate 179, 181 flat band voltage 56 Flat profile 69 Freeze out 10 full wave microwave rectifier 121 Gallium-Nitride 175 GaN HEMT 176 Generation 70 graded base 140 group velocity 1, 3, 5 Gunn 13 ff HEMT 167 ff Heterojunction Bipolar Transitor HBT 139 ff <?page no="188"?> heterostructure 107 Heterostructure Barrier Varactor HBV 185 hyperabrupt doping profile 184 IMPATT diode 68 injection conductivity 56 injection ratio 51 interband tunnelling 124 intrinsic carrier density 9, 10 intrinsic response time 50 inversion 160 ionization rate 45 JFET 155 Johnson figure 175 Kirk effect 132, 143 k-space transfer 35, 40 Laplace equation 27 lift-off process 83, 158 majority carriers 94 Mason's invariant gain 150 matching losses 77 maximum available gain 150 maximum oscillation frequency 163 maximum stable gain 150 maximum unilateral gain 150 Maxwell equation 63 Median Time Between Failure 88 MEMS 189 MESFET 155 ff minimum detectable signal 85 minority carriers 94 MISAWA type diode 80 MISFET 155 misfit dislocations 110 mismatch 108 modulation doping 109, 167, 171 modulation index 52 molecular beam epitaxy 81, 176 MOSFET 155 ff multiple quantum well 171 negative differential mobility 12 neutral base 144 ohmic contact 99 optical phonon scattering 46 outdiffusion 142 package 33 particle current density equation 41 particle velocity 5 passivation 178 penetration depth 64 phase velocity 1 photoresist masks 174 PIN diode 186 planar doped barrier 112 Poisson 70 Poisson equation 20, 38, 41, 60, 70, 75, 91, 157, 183 potential well 169 power amplification 149 power gain cut off frequency 135 pulse operation 78 punch through factor 62 quality factor 185 quantized states 114 quantum well 168, 192 quasi fermi level 94 quasi read double drift diode 72 quasi stationary approximation 50 quenched domain mode 19 QWITT 122 radar equation 85 reach through voltage 57 Read equation 50, 54 real-space transfer 36, 40 recombining 126 rectification 85 resonance phase 192 resonant tunnelling 114 resonator 75, 82 response time 115 ribbons 75 Richardson plot 100 ring structure 29 round trip loss 85 satellite valley 12 scattering measurements 147 Schottky contact, diodes 90 ff self-aligned devices 137 self-oscillating mixer 85 sheet carrier density 168 sheet resistance 185 sideband generators 186 SIMS 71 skin depth 64 skin effect 62 skin effect resistance 65 small signal resistance 133 space charge capacitance 101 <?page no="189"?> space charge resistance 60, 87 stationary ionization integral 59 strain 109, 142 strained layer 162, 172 stress 109 subbands 169 subharmonic synchronization 88 superlattice 113 surface lifetime 188 surface states 92 surface traps 179 switching time 187 tangential sensitivity 104 temperature dependence 87 thermal conductivities 28 thermal resistance 25, 29, 30 thermal spreading resistance 26 thermal voltage 57 thermionic emission 56, 98 transconductance 158, 170, 173, 177 transfer current 128 transit angle 43 transit frequency 164 transit time effect 40 transit time mode 19 transmission coefficient 115 trenches 145 tunneling 113 tunneling generation rate 59 two dimensional electron gas 167 Ultra High Vacuum / Chemical Vapour Deposition 144 valence band 6 varactor 183 vector operators 67 voltage sensitivity 101, 103 wave equation 42 wavefunction 1 wide band-gap emitter 142 Wigner-Seitz 2 work function 91, 96 zero bias operation 100 δ -doping 98 π -equivalent circuit 133
