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Modeling of tunnel oxide transistors Chu, Kan Man 1988

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MODELING OF TUNNEL OXIDE TRANSISTORS By Kan Man Chu B.Eng.(Hons), McGill University, 1984 M.A.Sc., University of British Columbia, 1986 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF PHILOSOPHY in T H E FACULTY OF G R A D U A T E STUDIES (Department of Electrical Engineering)  We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH COLUMBIA Nov 1988 © Kan Man Chu, 1988  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department  or by his or her representatives.  It is understood that copying or  publication of this thesis for financial gain shall not be allowed without my written permission.  Department The University of British Columbia Vancouver, Canada Date  DE-6 (2/88)  Mov/  2.5-,  Hgg  Abstract  Two improvements to a comprehensive analytic model which describes the steady-state current in a metal-insulator-semiconductor tunnel junction are reported. The first modification replaces the conventional two-band representation of the thin oxide band structure with a one-band model. In this approach the electron barrier height for tunneling is always less than the hole barrier height by an amount equal to the semiconductor band gap. The second improvement enables the energy dependence of the electron and hole tunneling probability factors to be taken into account. This is accomplished by expressing the tunneling probabilities as short-term series expressions. The capability of the model to accurately predict the electrical characteristics of metal-insulatorsemiconductor (MIS) tunnel junctions is demonstrated by simulating the d.c. and a.c. performance of three major types of transistor with tunnel oxide emitters, namely the tunnel emitter transistor (TETRAN), MIS-emitter transistor (MISET) and polysilicon emitter transistor (PET). Experimental data for the d.c. characteristics of all these devices are available and are found to be wellii  described by the predictions of the models. No experimental data for the limits of high frequency operation of the T E T R A N , MISET and pnp P E T have yet been reported. The models presented here suggest what those limits can be expected to be.  iii  Contents Abstract  Ii  List of Tables  vi  List off Figures  vii  Acknowledgement 1  2  3  xi  Introduction  1  1.1  General Background  1  1.2  Objective  1.3  Thesis Outline  10 •••  1 2  Basic Theory of T h i n Oxide Tunnel Junctions  15  2.1  Existing Models of Band-To-Band Direct Tunneling .  15  2.2  Surface-State Tunneling  27  2.3  The Determination of Barrier Height  35  The Formulation of an Improved Tunnel Junction Model  46  3.1  The One-Band Model  46  3.2  The New Power Series for Tunneling Current  51  3.3  The Revised Model Formulation  62  iv  4  5  6  7  Modeling the T E T R A N Device  64  4.1  Model Formulation  66  4.2  DC Characteristics  71  4.3  Small-Signal Analysis  79  4.4  Summary  84  Modeling the M I S E T Device  86  5.1  Model Formulation  88  5.2  DC Characteristics  94  5.3  Small-Signal Analysis  5.4  Summary  .  99  . . .  112  Modeling the P N P Poly silicon Emitter Transistor  114  6.1  Model Formulation  .118  6.2  DC Characteristics  124  6.3  Small-Signal Analysis  130  6.4  Summary  133  Conclusion  135  Bibliography  138  Appendix A : Computer Program for Modeling T E T R A N s  145  Appendix B : Computer Program for Modeling M I S E T s  156  Appendix C : Computer Program for Modeling Unannealed PNP PETs  1  6  5  Appendix D : Computer Program for Modeling Annealed PNP PETs  1  v  7  5  List of Tables 4.1  Model parameter values for simulation of the T E T R A N . . . . . .  73  4.2  Simulation results for the small-signal hybrid-7r parameters of the T E T R A N  81  Model parameter values for the simulation of the MISET  95  5.1  vi  List of Figures 1.1  Structure of a T E T R A N  3  1.2  Structure of a MISET  5  1.3  Structure of a npn P E T  7  2.1  Energy band diagrams of MIM junctions, (a) Symmetrical structure, (b) Non-symmetrical structure.  16  2.2  Energy band diagrams for MIS junctions in which (a) Ep and (b) E > E  20  Fm  2.3  m  < E  CQ  co  Dispersion curves for the one-band and two-band representations of the oxide  24  2.4  Different types of tunnel barrier, zoidal, (c) Triangular.  26  2.5  Surface-state tunneling in a MIS system  28  2.6  Energy band diagram of a metal-St'CVpSt tunnel diode  31  2.7  Small-signal equivalent circuits of a MIS diode. Steps (a)-(d) represent different stages of simplication  32  2.8  Typical capacitance (a) and conductance (b) curves of a MIS tunnel diode [37].  34  2.9  Typical result of photoemission measurements on a MIS diode [40]  36  2.10 The current characteristic (a) and its derivative (b) for a degenerate MIS tunnel diode  38  vii  (a) Rectangular,  (b) Trape-  2.11 Energy band diagrams of a MIS diode at (a) reverse and (b) forward bias, from [42]. Note that the field reversal in (a) is caused by a large negative charge density at the interface  40  2.12 Curve fitting technique proposed by Kumar and Dahlke [42].  42  . .  2.13 Energy band diagram of a short-circuit MIS tunnel diode under optical illumination  44  2.14 The J-V characteristics of a MIS tunnel diode under dark and illuminated conditions  45  3.1  Band structures of the 5t'Oj barrier assumed in (a) the two-band model and (b) the one-band model.  48  3.2  Energy band diagrams for (a) two-band and (b) one-band representations of the oxide  49  3.3  Complete energy band diagram of a MIS diode with tunneling currents indicated  53  3.4  Graphical representation of the factors appearing in the integrand of (3.9)  56  3.5  Comparison of exact numerical integration of (3.9) for the case of rectangular barriers (solid lines) and trapezoidal barriers (dashed lines)  58  Percentage errors, w.r.t. numerical integration results, of J _ as computed using Simmons' expression and vanous numbers of terms in the series expression, (d = 16A, x« = l.leV.) . . . . . .  59  3.7  As for Fig. 3.6 but with d = 16A, x« = 2.2cK  60  3.8  As for Fig. 3.6 but with d = 12A, x« = l.leV. .  61  4.1  Effect of source current on the potential distribution and charge flow in the MIS junction, (a) I, = 0. (b) I, > 0  4.2  Energy band diagram for the T E T R A N  69  4.3  Prediction of T E T R A N characteristics: d = 16A, other model parameters as in Table I of [22]. $ independent of energy as in [22]  72  3.6  m  c  67  e  viii  4.4  Prediction of T E T R A N characteristics: same parameters as for Fig. 4.3, except for the use of a one-band representation of the oxide. 9 given by one term of (3.4) with \e = 0.8eV  74  Prediction of T E T R A N characteristics: same parameters as for Fig. 4.4, except for the use of three terms in (3.4) for 6 , and Xe = 1.1^  75  Prediction of T E T R A N characteristics: same parameters as for Fig. 4.5, except for the inclusion of the surface-state tunneling effect, and \ = 0.85cV. The dashed lines represent experimental curves from (4]  77  4.7  Common-emitter small-signal hybrid-?r equivalent circuit for the T E T R A N device  80  5.1  Energy band diagram of a MISET  89  5.2  Common-emitter characteristic of the MISET with PB = 2.5 X 10 cm-  96  Common-emitter characteristic of the MISET with PB = 5 x 10 cm-  97  Common-emitter characteristic of the MISET with PB = 2.5 x 10 cm~ . The dashed lines represent experimental curves from [5]  98  C  4.5  e  4.6  t  17  5.3  16  5.4  16  3  s  8  5.5  Dependence of current gain on collector current of the MISET. . 103  5.6  Dependence of transconductance and emitter resistance on collector current of the MISET.  5.7  Dependence of cut-off frequency on collector current of the MISET. 105  5.8  Dependence of current gain on the base doping density of the MISET 109  104  5.9 Dependence of cut-off frequency on the base doping density of the MISET 110 5.10 Dependence of maximum oscillation frequency on the base doping density of the MISET  Ill  6.1  Structure of the pnp polysilicon emitter transistor [70]  6.2  Energy band diagram of the pnp oxidized P E T without postdeposition annealing 120 ix  116  6.3  Energy band diagram of the pnp oxidized P E T with post-deposition annealing 122  6.4  Gummel plots of the unannealed pnp P E T from computer simulation (dashed line) and published experimental data (solid line) [70]  125  Common emitter characteristics of the annealed pnp PET from computer simulation (dashed line) and published experimental data (solid line) [70]  129  6.5  6.6  Dependence of the unity current gain cut-off frequency on the base current for both the annealed and unannealed pnp PETs. . 131  x  Acknowledgement  The author wishes to thank Professor David L. Pulfrey for his patient supervision and his valuable contributions to this research. I am also grateful to Alan Laser for his efforts in developing and using numerical programs for modeling polysilicon emitter transistors in the course of a summer research project under the author's supervision. Special thanks go to my parents and brothers for their encouragement throughout this work. Financial assistance from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.  xi  Chapter 1 Introduction 1.1  General Background  In the recent years, the metal-insulator-semiconductor (MIS)  tunnel junction  has played an important role in the development of new device structures. One very promising application of MIS tunneling devices is in solar cells. It has been shown that this structure gives a higher open-circuit photovoltage than that of the Schottky barrier solar cell, while maintaining the attractive features that make the Schottky junction potentially more suited to very large area, terrestrial, solar cell applications [1], than the p-n junction. Another application of tunnel oxide structures is found in negative resistance devices such as the MIS-switch (2). This device has a metal-tunnel oxiden-p structure. The p-n junction is biased in the forward direction. In the high +  impedance state there is a wide depletion layer at the oxide-semiconductor in1  terface, and the current in the device is low. In the low impedance state there is a significant inversion layer charge formed by holes at the oxide-semiconductor interface, leading to a high field strength in the oxide, which in turn enhances the electron current tunneling from the metal into the conduction band of the semiconductor. This device has potential as a static memory element since the two distinct states allow it to perform the memory storage function which normally requires a bistable circuit. However, the switching times are calculated as being too long for high speed usage [3]. Tunnel oxide structures have also found application in the design of bipolar current amplifiers. Recently three such devices, namely, the tunnel emitter transistor (TETRAN), MIS-emitter transistor (MISET) and polysilicon emitter transistor (PET) have been reported. The structure of a T E T R A N (3] is shown in Fig. 1.1.  Without an excess supply of minority earners a deep depletion  layer develops at the oxide-semiconductor interface because the holes generated thermally in this layer can tunnel through the oxide. Most of the voltage is dropped across the depletion layer while the field strength in the oxide is low, and there is a small flow of electrons tunneling from the metal into the semiconductor conduction band. Increasing the supply of holes by injection through the p diffusion region results in the formation of a hole inversion layer +  at the interface, which in turn increases the voltage drop across the oxide and the electron current tunneling from the metal into the semiconductor. Modest 2  METALLIC EMITTER ULTRA-THIN INSULATOR  SOURCE  N-SUBSTRATE COLLECTOR /N  I  c  CE  Figure 1.1: Structure of a T E T R A N  gains of around 120 have been measured in operational devices. Of greater interest is the prediction of an intrinsic cut-off frequency as high as 600 GHz  The structure of the MISET is shown in Fig. 1.2.  The electron to hole  injection ratio of the MIS junction forming the emitter can be quite high, if the metal has a low work function, as for example in the case for aluminium. The electrons, injected by the metal emitter, are collected with the aid of the reversed biased p-n junction forming the collector. Green and Godfrey [5] have recently reported a MISET device with common emitter current gain approaching 25000, which is extremely high for a bipolar device. Another advantage of this device is the simple structure, thus it has small lateral and vertical extensions as well as fewer preparation steps than conventional BJTs. However, other aspects of this device, Buch as the degradation of the ultra-thin tunnel oxide (~20A) [6] and the high frequency performance need to be studied further before the practicality of this device can be assessed. The replacement of the metal emitter in the MISET by a polysilicon emitter can result in a more stable structure, because now the thin tunnel oxide is sandwiched between silicon layers. This arrangement is likely to lead to less degradation and create lower surface states than when contacted with metal on one side. A typical polysilicon emitter transistor structure is shown  4  EMITTER p  BASE o THICK OXIDE P-DIFFUSION  TUNNEL 0XIOE  N-EPITAXIAL N-SUBSTRATE  COLLECTOR  Figure 1.2: Structure of a MISET  5  in Fig. 1.3. The thickness of the tunnel oxide can greatly affect the electrical characteristics of the devices [7]. There are two type of device: one with an intentional, chemically-grown interface oxide (~20-30A) [8], and one with a "clean" interface [9,10] at which there is an unavoidable native oxide (~5A). The first type of device has a higher current gain and a lower (or even negative) temperature coefficient for the current gain than the second type [11]. This behavior can be roughly predicted by using tunneling theory [8] for the first device and minority carrier transport theory [10,12] in polycrystalline silicon for the second device. Eltoukhy and Roulston [13] later proposed a unified theory which could apply to any polysilicon transistor. For the high frequency performance, the second type of device would seem to be superior to the first type. The use of an intentionally grown oxide significantly degrades the high frequency performance capability by increasing the emitter resistance by an order of magnitude [14]. The polysilicon layer can be deposited after the monocrystalline emitter is formed by conventional ion implantation techniques [10], or the polysilicon can itself be used as a source for the emitter diffusion [8,9]. For the device with a "clean" interface, the high temperature polysilicon anneal (e.g. 1000°C/30 min.) degrades the integrity of the native oxide layer and destroys its blocking action [14], thus increasing the base current and reducing the gain of the device. The best alternative for avoiding this problem is the use of in-situ doped polysilicon. 6  POLY-EMITTER  N+-MONO-EMITTER N-COLLECTOR  Figure 1.3: Structure of a npn P E T  7  Rowlandson and Tarr [15] report a "true" polysilicon emitter transistor, formed by depositing in-situ doped polysilicon at a temperature (627°(7) which is low enough to prevent diffusion into the substrate, and by not subjecting the device to any additional annealing. Therefore, in this structure the emitter-base junction coincides with the interface between polycrystalline and monocrystalline material. A common emitter current gain as high as 10000 has been reported for this device. The major application of MIS and semiconductor-insulator-semiconductor (SIS) junctions has already been mentioned above. However, there are some other interesting applications. Moravvej-Farshi and Green [16] have replaced the normally implanted source and drain regions of NMOS transistors by SIS diodes to produce devices with near-zero junction depth, which is a critical parameter in determining the onset of short-channel effects in MOSFETs. Unfortunately, a self-aligned structure has not yet been fabricated. Fossum and Barker [17] have attempted to use a MIS junction to serve as a charge packet threshold detector. A charge-coupled device (CCD) input structure was designed to inject a metered packet of holes into the MIS junction. The MIS junction may switch to a "high" current state or stay at a low state, tt  n  depending on the amount of the charge injected. However, the slow switching speed (10-100 ms) of this device may prevent it from being very useful.  8  Because of the interest in the MIS tunnel junction, several models to aid in device analysis and design have been formulated by diffferent researchers. The first general model was that of Green and Shewchun [44]. It has been shown that the results of this comprehensive numerical model can be duplicated for a wide variety of operating conditions by a much simpler analytical model [22].  This latter model has been incorporated in the popular semiconductor  device analysis program SEDAN III [55]. One of the simplifying features of the analytic model in [22] is that the electron and hole tunnel currents can be derived assuming constant tunneling probability factors. This approximation is valid so long as the semiconductor surface is not strongly inverted and the metal Fermi level lies within the range of energies defined by the semiconductor band gap at the semiconductor surface. These conditions may not hold in some tunnel junction applications. Another feature of the original analytical model [22] is its utilization of a two-band representation of the thin SiO insulator. Recent work t  [30], which compares experimental data on electron and hole barrier heights [24] with theoretical predictions, suggests that a one-band model of the oxide is more appropriate for analysis of tunneling phenomena in MIS junctions. However, until the work described in this thesis, this representation of the oxide had not been incorporated in models to explain the characteristics of MIS or SIS devices.  9  1.2  Objective  Many interesting applications of MIS and SIS tunnel junctions incorporating thin oxide layers on silicon have been proposed and, in some case, demonstrated. The particular area that is presently attracting much attention is the application of MIS and SIS junctions in the design of small-sized, easily-fabricated and high speed silicon bipolar transistors. Of special interest is the modeling and characterization of the T E T R A N [4], MISET [5] and P E T [9,15]. The T E T R A N and its gallium arsenide counterpart, the inversion-base bipolar transistor (IBT) [64], represent a completely new type of device. Unlike the conventional bipolar transistor there is no doped neutral base region. The base is actually formed by a very thin inversion layer of mobile holes induced by the emitter; consequently, all of the problems associated with the diffusion of minority electrons in a neutral base region are eliminated. Furthermore, the device is expected to operate at extremely high frequencies because the base transit time is essentially zero. This is on account of the very high field in the induced base which assists the passage of electrons to the collector. The MISET can exhibit very high common emitter current gain; values up to 25000 have already been reported [5]. This is because the MIS contact behaves as a heterojunction, which causes the tunneling of the conduction-  10  band electrons between the metal and silicon to be greatly favored over that of valence-band holes. In this respect the silicon MISET is similar to the GaAs heterojunction emitter transistor [56]. Another interesting property of heterojunction transistors is the removal of the necessity to trade-off the transconductance (g ) against the base resistance (Rb) to obtain a high maximum oscillation m  frequency (fmax)- This property still needs to be demonstrated in the MISET. The npn P E T can be optimized to obtain high d.c. gain (~10000) [15] or high cutoff frequency (~16 GHz) [58], depending on the processing treatment of the emitter-base contact. The gain-speed trade-off for npn PETs with different tunnel oxide thicknesses, base-emitter junction depths, base and emitter doping concentrations has been studied [59] by numerical simulations and experimental measurements. Recently, Maritan and Tarr [70] have fabricated some pnp PETs and shown that these devices can exhibit reasonable values of gain and emitter series resistance. This is an extraordinary result as one would expect that the pnp device performance would be very poor, due to the fact that the probability of holes tunneling through the thin oxide is considerably less than that of electrons. For the design, understanding and development of MIS and SIS tunnel junction devices, an accurate model of the structure is needed. The primary goal of this work is to refine and improve the analytical MIS model of [22]  11  by representing the oxide via a one-band model, and by allowing for energydependent tunneling probability factors for electrons and holes. The new model is used to predict the d.c. characteristics of TETRANs, MISETs and PETs, and to make comparisons with the published experimental results for these devices. Some important a.c. model parameters are extracted in order to evalute the high frequency performance of these devices. The interesting gain-speed trade-off in MISETs and the surprisingly good performance of pnp PETs are investigated by computer simulations.  Thesis Outline  1.3  Chapter One introduces the background to, and current work on, MIS and SIS devices. The necessity of developing an improved tunneling model is emphasized. The first two sections of Chapter Two summarize the mathematical formulation of two major tunneling mechanisms in MIS tunnel junctions, namely band-to-band direct tunneling and surface-state tunneling. A key parameter in modeling tunnel junctions is the tunneling barrier height. The last section surveys the different experimental methods of determining the tunneling barrier height. Chapter Three details the improvement of an earlier analytical model of 12  the MIS tunnel junction. T h e improvements include: (1) replacement of the conventional two-band formulation of the oxide band structure with a one-band representation, (2) making allowance for the energy dependence of the tunneling probabilities for electrons and holes, (3) extending the previous model so that it can deal with three terminal transistor structures, rather than only two-terminal diode structures. Chapter Four is devoted to the use of the new model to predict both the d.c. and the small-signal a.c. performance of the T E T R A N . T h e predicted characteristics are compared with the published experimental data. T h e inappropriateness of using the previous model to fit the experimental data is demonstrated. In Chapter Five, the d.c. and high frequency performance of M I S E T s is evaluated and compared to the previous experimental results. T h e tradeoff between the base conductance and cut-off frequency is studied in order to optimize the device for high m a x i m u m oscillation frequency. Chapter Six describes the modeling of pnp polysilicon emitter transistors, and treats both annealed and unannealed devices. The fact that these transistors exhibit reasonable gain and low values of emitter series resistance is shown to be consistent with the new model proposed. The potential for high frequency operation of both annealed and unannealed devices is assessed. 13  Conclusions are drawn and suggestions for further work are made in the last chapter.  14  Chapter 2 Basic Theory of Thin Oxide Tunnel Junctions 2.1  Existing Models of Band-To-Band Direct Tunneling  In metal-insulator-metal (MIM), MIS or SIS systems, electrons and holes can tunnel through the thin insulator by three different mechanisms, namely, direct band tunneling, surface state tunneling and trap-assisted tunneling. We will consider the direct band tunneling in detail, since it is generally the dominant mechanism. For a symmetrical MIM structure (see Fig. 2.1(a)) at 0 K, the electron tunnel current through the insulator is [18, p.553]  J n  = ^ i b i I * M-lVxD  - Xa  1e  15  exp(--y xT)] / v  (2-1)  Figure 2.1: Energy band diagrams of MIM junctions, (a) Symmetrical structure, (b) Non-symmetrical structure.  16  where Xi  7  h  and m!, is the effective mass of electrons in the conduction band of the insulator. For large applied voltages, V>/ > Xm/9> (2.1) can be simplified to the well-known Fowler-Nordheim equation. Extending the MIM theory to non-symmetrical structures and non-zero temperatures, Stratton [19] has derived an equation for the electron tunnel current (see Fig. 2.1(b)) (2.2) where m* is the effective transverse mass of electrons in tne metal, e is defined s  in Fig. 2.1(b) and 8 {e ) is the tunneling probability, which is energy dependent, e  %  and can be expanded using the Taylor's series  m[Me,)J =  -{h + c e + fa\ + •••) x  t  (2.3)  where coefficients 6 i , c i , /i, • • • are dependent on the shape of the tunnel barrier. The first two terms are enough if c\ < 1/kT. The supply function S(e ) represents the difference between the number K  of electrons per second per unit area, having (x-directed) energy in the range 17  E to E + dE , incident on opposite sides of the barrier. x  s  x  Jo  =F =  exp.(5g£*)  1 +  1+ e x p ( ^ ± ^ ) J  Jo " l + exp(ft) kT In > 1 +  e  X  (2.4)  p(2#)  Truncating the expression for 0 , i.e. (2.3), to its first two terms and e  putting this into (2.2), gives after much mathematical treatment (and use of (2-4)), J = 94*?t»Texpi-biH^,*. n  ch  Al - exp{-c V)] - R - R } iq  ' sin(7rciA;T)  s  t  Lrr  y  2  (2.5)  where  R  2  = f  Jo  .eifcT-1 l  l + x  1+  •\dx xexpiqV/kT) 1  In the case of a metal, f is usually very large and R\ and Rj are negligible. J  n  can therefore be approximated by the first term of (2.5). Although this  approximation is often used in the case of heavily doped semiconductors [20] (where f is only several kT), or even in non-degenerate semiconductors [21] (where f < 0), the error caused by this approximation may be significant. For electron tunneling at non-degenerate semiconductor surfaces [13] (where f < 0), we could use Maxwell-Boltzmann statistics rather than Fermi-Dirac 18  statistics in deriving the supply function in (2.4). From (2.2) we obtain 47rm;(ikr) "  /i  H  exp(-60  s  l- kT  s  exp(-)-exp(-^-)  (2.6)  Cl  The same form of expression is also used to describe minority carrier tunneling between semiconductor surfaces and metals [8,20], where Maxwell-Boltzmann statistics can apply. To model the tunneling current in a MIS system without borrowing from the MIM theory, Tarr et al. [22] derived an expression for the electron tunnel current (see Fig. 2.2(a)) 4irm k T h— t e  n  =  q  2  , /l  1  T  / Epno — E . e0  T  '  . Epm — E , e  kT  (2.7)  where m* is the electron transverse effective mass in the semiconductor. The tunneling probability 9 is assumed to be constant. e  Except for large tunnel  barrier heights (x«) and small oxide thickness (d), and the case when the Fermilevels are well below the conduction band edge, this expression can cause very large errors. For E  Fno  -E  eo  < 0 and E  Fm  -E  eo  < 0, the Fermi-Dirac integral  in (2.7) can be replaced by the exponential function. After a few mathematical steps, the electron tunnel current can be approximated by [23,24] J  n  qV Q<t>b = A T ^ e x ( - ^ ) exp(f— ) - l kT' I nkT' e  (2.8)  P  rK  where A* is the effective Richardson constant (= 47rgm*fc //i ) and ^ is the l  3  Schottky barrier height. The ideality factor n (usually ~ 1 - 1.5) is defined as 19  EFT E  F  ////////////  M  (a)  (b) Figure 2.2: Energy band diagram? for MIS junctions in which (a) Ep and (b) E > E .  m  Fm  co  20  <  E  cc  —V/ A V'*? where AV»« is the change in semiconductor surface potential due to the bias. In a MIS system where the metal Fermi-level rises above the conduction band edge (Fig. 2.2(b)), Simmons and Taylor [3] have suggested an approximate expression for the electron tunnel current which takes the (parabolic) energy dependence of the tunneling probability into account. This expression is in the following form: J = A*T\(K n  V  + K ) e x p [ - 7 e t e ) * ] - K exp[-<7 (*;)*]} t  3  e  (2.9)  where  1 1+ K  2  K and  3  = =  2(jg -i? )te)i Fm  C0  3* 7/H 2  2{E  Ejfc)* 3)fc r 7  Fm  a  8  e  is the effective mass of electrons in the conduction band of the oxide. The expressions presented so far allow the electron tunnel current to be  calculated if the tunneling probalility $ is known. However, there is controversy over the tunneling band structure appropriate to thin silicon oxides, and on the different forms of 9 . The tunneling probability, using the WKB approximation, C  21  is given by (25] 0 = exp[-2 [*'\k \dx]  (2.10)  lx  where x , x are essentially the positions of the oxide interfaces with the semit  m  conductor and metal respectively, and kjx is the x-component of the complex electron wavevector in the oxide. In order to evaluate (2.10), knowledge of the band structure in the forbidden gap of the oxide is required. The complex electron wavevectorfcjin the oxide is usually taken to obey the Franz dispersion relationship [26]  kj ~ 2 m ; ( £ - E ) H  +  2mU{E - E) vi  (  ' '  where m^, mV are the effective masses associated with the conduction and valence bands in the oxide respectively, and Ed, E„i are the energies of the band edges in the oxide. Once the transverse wavevector kt is known, the x-component of wavevector in the oxide is given by kix = yjk) - k}  (2.12)  Note that (2.11) can be rewritten as  *, = £ 2 j ^ ( * where  - *,)*(! -  - (1 - ^ ) ^ ] - «  (213)  is the oxide bandgap {—Ed - E ) vi  In the past, a two-band model (or Franz-single mass model) has generally been accepted. It assumes that m*d and 22  are equal, and renders (2.13) to  [22,27] t, afiit(,-_ =  f t (  ,i  ( 1  _*Li*,i  ,2.14)  However, recent calculations of the band structure for Si0 in the a-quartz form 2  indicate that mj,- ~ 0.5m (m, is the electron rest mass) and m*,- ~ (5 — 10)m e  e  [28). Also, Weinberg [29] observes that the conduction band and the top of the valence band originate from very different electronic orbitals, as indicated in Fig. 1 of [28]. It is possible that the conduction band should be connected (in the complex domain) to one of the deeper lying valence,bands, rather than to the top of the valence band. This implies that the appropriate value of E^ in (2.13) could be as high as 18 eV. A small ratio of m^/m^ or a large E^ both reduce (2.13) to k i  = Q?& _ ji {E  E  ( 2 1 5 )  This is the so-called parabolic or one-band model. O'Neill [30] has shown that the one-band model gives better agreement between estimates of tunnel probability and some experimental results than the two-band model. Eq. (2.15) also suggests that the effective barrier height for holes is higher than that for electrons and that the energy difference is equivalent to one silicon band gap. Fig. 2.3 compares the dispersion curves of the one-band and two-band models. Note that kj represents the damping factor of electron waves in the oxide. The smaller kj is, the higher will be the tunneling probability.  23  24  Combining (2.15) and (2.12), we obtain  kix = where E  ~  (2.16)  is the energy associated with the electron momentum in the x-  x  direction. Putting (2.16) into (2.10), the tunneling probability can be found. Since Ed can be a function of x, the shape of the oxide potential barrier can affect 0. Although many forms of barrier, such as parabolic or triangular shape [19], have been considered, it is commonly assumed that the barrier is either rectangular or trapezoidal (see Fig. 2.4). From (2.10), it can be shown that: for a rectangular barrier, 6{E ) = exp[~(2m^{E t  - E )i]  ei  (2.17)  z  for a trapezoidal barrier, $(E ) = exp{~(2m^H^)\(Ed  - E )\ -  a  m  -  -E  M  )\\)  (2.18)  for a triangular barrier, ${E ) = a  «p[-^(2m;)*( A.)K "  (2-19)  5  Once 0(E ) is known, J„ in (2.7) can be determined by setting 9 = 9(E ). X  e  eo  Moreover, the coefficients of the Taylor's series in (2.3) can be fixed. Another factor affecting the tunneling probability $ is the effective electron barrier height xT = ~E~^ - E  eo  (see Fig. 2.2). The values for xT (or Xe) reported  25  E  0->  (a)  Figure 2.4: Different types of tunnel barrier, (a) Rectangular, (b) Trapezoidal, (c) Triangular.  26  in the literature vary widely, e.g. from 0.25 to 3.3 eV for a 25 A thick oxide [40]. The barrier height also appears to depend upon oxide thickness, being smaller for thinner oxides [32]. This barrier lowering effect is not fully understood at present and may be due to a number of factors, including image forces, surface effects, fixed oxide charge, and the presence of amorphous silicon oxide.  2.2  Surface-State Tunneling  In a MIS system electrons can tunnel through the thin oxide to surface states lying within the forbidden gap at the semiconductor-oxide interface, and then communicate with the bulk by recombination processes [33,34] as shown in Fig. 2.5. The capture (c ,c ) and thermal emission (c„,e ) rates for electrons n  p  p  and holes per active trap at the semiconductor surface are given by —  =  v an  =  v op  th  n  (2.20)  t  *en C  —  P  th  p  (2.21)  t  ep  £  T  Cn  E  v o ni  (2.22)  VthOpPi  (2.23)  th  Un  n  1  P  where n „ p, are the electron and hole concentrations at the surface, o and a n  p  are the thermal capture cross sections for electrons and holes, riy and pi are defined as N exp(^ ^ ) e  t  A  and N exp( ^ ) v  Et  u  respectively, and v is the carrier tn  thermal velocity. Neglecting trap photoemission [35], the various currents that 27  Figure 2.5: Surface-state tunneling in a MIS system  28  enter and leave the trap can be written as Jet =  -qNt[c (l - f ) - e f ]  (2.24)  Jvt =  qN [c f - e„(l - ft)}  (2.25)  Jnu'=  qN v {f - f )  (2.26)  n  t  p  t  t  n  t  t  m  t  m  where Nt is the density of states at trap energy level E , ft is the occupancy tr  probability at E u  m  f  ti  m  is the occupancy of E in equilibrium with the metal, and t  ( = l / r « ) is the tunneling rate. The occupancy of the interface states is determined by a competition  between tunneling transitions to the metal and the capture of carriers from the conduction and valence bands of the semiconductor surface. Under the steady state condition J  et  between f  m  + J  vt  + Jmt = 0, the surface state occupancy f has a value t  and / , and is given by f = t  Ttf  '  + ' T  (2.27)  f m  U + r.  where T. = Vth\v {n, + n,) + (7 (p, + Pi)] n  p  and /,  no  =  t  + pa  n  x  p  {n, + ni)o + {p, + Pi)0 n  1  s — s —  l+exp(^») 1+exp  kg8*.) kT 29  p  if p.  i  f  n.  *  (2.28)  Putting (2.27) into (2.26), the surface state tunneling current via a single level trap is given by J t = qN (^-^) m  (2.30)  t  T, + Tt  For r » r , e.g. in the case of oxide thickness < 15A, (2.27) simplifies to f « f . t  t  t  m  Interface states and metal are in equilibrium, the metal Fermi level is pinned to the energy level of the surface states, and Jmt is controlled by the interface recombination time r . We call this the interface recombination controlled case. #  For r » r,, e.g. in the case of oxide thickness > 25A, (2.27) reduces to f « t  t  the majority carrier quasi-Fermi level at the semiconductor surface is pinned to the states, and Jmt is controlled by the time constant for tunneling through the oxide r . We call this the oxide tunneling controlled case. t  The tunneling rate can be defined as [33,36] Vm = I / * = Vmo{Etr)0{E )  (2.31)  tr  where Vmo (E ) is the attempt-to-escape frequency at the trap energy level E , tr  tr  and 0(E ) is the tunneling probability at E . A variety of experimental methtr  tr  ods have yielded values of Vmo in the range from 10 to 10 Hz. Recently, Jain s  13  and Dahlke [36], by measuring the photo- and dark capacitance transients in Cr-SiOj-nSi MIS tunnel diodes and assuming a two-band model, found that decreases from about 10  15  Hz at midgap to 10 Hz at the silicon conduction 10  band edge. 30  t  //////////// t  Figure 2.6: Energy band diagram of a metal-St'CVpSt tunnel diode.  The surface state density distribution N (E) t  within the semiconductor  bandgap and the carrier capture cross section areas o , o can be estimated n  p  by small signal capacitance and conductance techniques [37,38]. Consider a metal-Si0 -pSi tunnel diode as shown in Fig. 2.6, the applied bias VB is varied 2  such that the hole Fermi level Ep can scan across the whole Si bandgap. With p  each biasing point, the a.c. capacitance C  m  and conductance G  m  of the diode  are measured at different frequencies. From the capacitance and conductance dispersion curve, useful surface state parameters can be extracted. The small-signal equivalent circuit for a MIS diode is shown in Fig. 2.7(a). For oxide thickness > 2 0 A , the recombination currents are much larger than the tunneling current J  mt  under the application of an a.c. signal. The circuit  31  >V Cx 0  O—i  11 1  ox  I  +  v  >  Ips  B  c,ox  Cgc  R]3  (I  ,—w\—o  1  -r-C  1  >ctt.)  —| | C.(> ) f  1  s  R*  R  B  1—Wv—O  £,(0,  J |—WV—  -Wv  (b)  (a)  -j I  C  1 I  i  WV—O  o_  1|  o  (d)  (c)  Figure 2.7: Small-signal equivalent circuits of a MIS diode. Steps (a)-(d) represent different stages of simplication.  32  in Fig. 2.7(a) can be reduced to (b), and subsequently to (c) and (d). Some typical measured capacitance and conductance (C (V'«,u;),G (^,,a;)) curves M  m  [37] are shown in Fig. 2.8. Once C and G are known, C and G can be deduced from Fig. 2.7(c) m  m  d  d  and (d) providing the bulk resistance RB is given. The surface state capacitance C, is obtained by subtracting the calculated space charge capacitance C from te  Cd at equilibrium (or very low) frequency w : eq  C.ty.) = C (V'„u; ,)-C. (0,) e  RF  (2.32)  e  Then, the interface state density, Nt = C /q, is found from C, and its energy t  position in the Si bandgap. From the conductance, the distribution N (E) can t  be independently determined by setting  ty = ^(Gd(&,w)M««  (2-33)  The interface recombination constant r, can be determined by the fact that the maxima occurs at wr, = 1. From (2.28), the majority carrier capture area (a  p  or a ) can also be found. n  The surface state distribution depends on the type of metal used and the processing steps. For example, magnesium exhibits one peak in the center of the bandgap, while gold exhibits a large peak near the conduction band and a small peak near the valence band [39]. An annealing step also tends to increase 33  MVB)  —  E  E  V  C  (b) Figure 2.8: Typical capacitance (a) and conductance (b) curves of a MIS tunnel diode [37].  34  the surface state density. This arises most probably from thermally stimulated migration of the metal into the oxide. The typical values of Nt range from 10  11  to 10 cm" eV , and are greatly affected by metal diffusion through the oxide 13  J  _1  film. The upper limit of N can possibly be due to a limited solubility of metals t  in the oxide film. The surface states are usually found to be acceptor-like. The typical values of <r , a are found to be 10 cm , although a spread of several n  13  p  J  orders of magnitude is possible [36,37].  2.3  The Determination of Barrier Height  There are several methods to determine the tunnel barrier energies. Dressendorfer and Barker [40] have measured the Si-SiOj and Al-SiOj electron barrier heights on oxides of thickness ~50 A by photoemission measurements. Since the current across the oxide consists of a background tunneling current and a much smaller photoemissive current, a slow light-on/light-off cycle is needed to resolve these two components. The currents through the device under illuminated and dark conditions are time averaged and subtracted to find the photoemissive component. The typical relation between the square root of the photoyield and photon energy is shown in Fig. 2.9. There are three distinct sections of the curves: the portion from <J>TI to 4>T2 caused by photoemission from the conduction band of the silicon; the linear  35  (Photoyield)  1/2  >  TI  ^12  Photon Energy (eV)  Figure 2.9: Typical result of photoemission measurements on a MIS diode [40].  36  section from <f>T2 onwards; and the steepest section at higher energies. These latter two portions of the curves are caused by indirect and direct absorption processes from the silicon valence band. Thefai,which is the barrier between the Si valence band and SiC>2 conduction band, is obtained to be about 3.6 eV for a slightly forward biased MIS junction with degenerate n-substrate and oxide thickness about 56 A. This implies the average electron barrier xT on the Si side (see Fig. 2.2 for definition) is about 2.5 eV, assuming a Si bandgap of 1.1 eV. For the same device under reverse bias,faiis found to be about 2.6 eV. This value can be understood as the average electron barrier Ym on the metal side. The difference from Xm(~3.2 eV) and Xe(~3.1 eV) measured with much thicker oxide (>300 A) suggests that a barrier lowering correction of about 0.5 eV must be included. This technique cannot be u^ed to determine the barrier heights of thin oxide <40 A, because of the difficulty of resolving the very small photoemissive current from the large background tunneling current. Kasprzak et al. [32] have deduced the energy barriers of ultrathin (10-30 A) SiOj layers between aluminium and degenerate silicon. The MIM tunneling theory was used to interpret the result (see also Fig. 2.1(b)). The J„-vs-V and [d(ln J )/#V]-vs-V curves from this experiment are shown in Fig. 2.10. n  Abrupt changes in the slope of J versus V should be observed at V = x /q and n  e  37  38  I  —Xm/? due to the potential barrier changes from a trapezoid to a triangle (see Fig. 2.4(b) and (c)). The mean barrier height  <f>  0  at V=0 can also be obtained  from the low-voltage approximated relation ln{R/d) = Cd{m*Jl)^  (2.34)  where R is the resistance of the tunnel diode at V=0, C is a constant, and m* is the electron effective mass in the oxide. The Si-SiOj barrier height (x«) has been found to increase from 0.42 eV at 10 A to 0.65 eV for 25.5 A of Si0 on degenerate p-type Si, and from 0.64 eV 2  at 14 A to 1.27 eV for 29.3 A of Si0 on degenerate n-type Si. The Al-SiOj 2  barrier height (xm) is determined to be about 0.61 eV. In the case of non-degenerate semiconductor, Card et al. [23,41] have deduced the Si-Si02 barrier height by fitting the experimental tunnel current at forward bias with the I-V characteristics predicted by MIS theory (see Fig. 2.2(a)). Refering to (2.8), <f\>  c a r  * be measured by the extrapolation to  the V-axis of the (linear) C - V plot, and n can be obtaind from the slope of _ ,  the [ln(J )]-vs-V curve. Therefore, the unknown 9 is determined, and x« is n  e  found from the relation 0 ~ exp(—\\l*d). e  Kumar and Dahlke [42] have proposed another curvefittingtechnique. For the MIS system (x« = Xm = Xi large surface state density) shown in Fig. 2.11, the tunnel current can be expressed as 39  (a)  qV <X  ( b ) qV  m  m  >  X  Figure 2.11: Energy band diagrams of a MIS diode at (a) reverse and (b) forward bias, from [42]. Note that the field reversal in (a) is caused by a large negative charge density at the interface.  40  J = n  A^e \J {- ^)-U- ^)\' c  q  x  (2.35)  q  where  The measured forward and reverse currents, and the surface Fermi potentials V and V obtained from the measured surface potential V',(V), are t  m  used to calculate the barrier function r/(Vm) = ln(A*r 0,.). The values of U as J  functions of V ^ and V are shown in Fig. 2.12. The slopes \si\ = x'^/v ^ of 1  7  m  U{V~ ) at large V and s = d/{4 l  m  ) of U{V ) at small V , and the difference  l/2 X  2  m  m  of their intercepts with the ordinate s = x l*d are also indicated. By finding x  3  the values of Si, sj and S3, and the intercepts of the slope lines, the parameters X, d and A*T* can be obtained simultaneously. A Richardson plot provides an alternate means of determining the barrier height. Ashok et al. [43] have analyzed the I-V characteristics of Au-nGaAs MIS diodes, and the forward current is found to obey the relation J = J . [ e x p ( ^ ) - 1] n  (2.36)  J, is the extrapolated saturation current and is related to the metal-semiconductor barrier heightfaby  J . = A-r'* exp(-A;) e  (2.37)  The inclusion of the ideality factor in (2.37) has been found necessary to explain 41  Figure 2.12: Curve fitting technique proposed by K u m a r and Dahlke [42].  42  the experimental data. The Richardson plot \n(J /T )-vs-l/nT t  2  results in a  straight line, and the values of fa and A*0 are obtained fron the slope and the e  y-intercept. Once 0 is known, the tunneling barrier height can be found. e  Realizing that the previous studies of barrier height have been confined to the tunneling of electrons, Ng and Card [24] proposed a procedure to determine the tunneling barrier of holes for Au-Si0 -hSi tunnel junctions with oxide 2  thickness in the range of 20-30 A. The energy band diagram of the short-circuit junction under optical illumination is shown in Fig. 2.13. For an oxide thickness > 20A, a suppression of the short-circuit photocurrent is observed and the hole concentration at the semiconductor surface increases with illumination intensity. In the steady state, the photocurrent J  PN  is equal to the summation of the  short-circuit hole tunnel current J, and the back diffusion current J<j- One can C  formulate an explicit expression for the hole tunneling probability \h as  «p(-xi"rf)  = ,  D  p  {  ±^!l! ,exp[(^)(^  The magnitudes of J H and J P  £  TE  - 1)]}-  (2.38)  can be obtained from the J-V characteristics of  the diode under dark and illuminated conditions as shown in Fig. 2.14.  43  Figure 2.13: Energy band diagram of a short-circuit MIS tunnel diode under optical illumination.  44  Figure 2.14: The J-V characteristics of a MIS tunnel diode under dark and illuminated conditions.  45  Chapter 3 The Formulation of an Improved Tunnel Junction Model  The analytic model formulated by Tarr et al. [22] is commonly used to calculate the steady-state current in a metal-insulator-semiconductor junction. In this chapter two improvements to this model are proposed. Firstly, the two-band model for S1O2 based on the Franz dispersion relation i s replaced by a oneband model. Secondly, the restriction of the tunneling probability factor to a constant value is removed.  3.1  The One-Band Model  The motivation for switching to a one-band model is the success achieved by O'Neill [30] in using it to explain the asymmetry in electron and hole tunnel currents in thin SiOj layers. In the two-band model, the band structure of 5 i 0  46  2  barrier is assumed to have the shape shown in Fig. 3.1(a), where the complex conduction and valence bands extend into the tunneling energy range. Therefore the evanescent states in the oxide are derived from both the conduction and valence bands. Recent experimental data on the band structure of 510  2  in the a-quartz  form suggests that the one-band model is more appropriate, and that only the complex conduction band can extend into the energy band gap at reasonably small values of ki (Fig. 3.1(b)). This constitutes the central postulate of the one-band model, namely: that the evanescent states in the oxide are derived only from the conduction band edge [30]. The switching from the two-band model to the one-band model affects the way which tunnel barriers are represented in the energy band diagram. The corresponding barrier height for holes, \h, depends on the model used to represent the oxide (see Fig. 3.2). For the two-band model we have Xh = Egi — E — Xe  (3.1)  t  where E^ is the oxide band gap, E is the semiconductor band gap and Xe is t  the silicon electron barrier height. Therefore, the electron and hole tunneling probabilities depend on two adjustable parameters, i.e. E^ and x e  The situation is different in the one-band model represented by the symbolic band diagram of Fig. 3.2(b) which we propose. The evanescent states in  47  OXIDE CONDUCTION BAND  ^  /  /  /  <  \  COMPLEX CONDUCTION BAND •—'  V __  ~- /  ol  e~  SEMICONDUCTOR I J FORBIDDEN GAP ^  ^  \  N  '  r  COMPLEX VALENCE BAND—is ^  >k  \  \  \-  - -«  OXIDE VALENCE BAND  (b)  Figure 3.1: B a n d structures of the SiOi barrier assumed in (a) the two-band model and (b) the one-band model.  48  t  Figure 3.2: Energy band diagrams for (a) two-band and (b) one-band representations of the oxide.  49  the oxide are derived from the oxide conduction band only, and the electron and hole barriers are related by: XH = Xe + E,  (3.2)  The electron and hole tunneling probabilities depend on only one adjustable parameter, i.e. the silicon electron barrier height x«- Note that in Fig. 3.2(b), the lower portion of oxide energy band does not represent the insulator valence band. It is, instead, a reflected version of the top part of the energy band configured in such a way as to give the correct barrier shape for holes. Due to the symmetry, the expressions for hole tunneling probability 0 are similar to n  the set given by (2.17)-(2.19), with E , Ed and x  replaced by E , E'^ and E^. x  As mentioned in Section 2.1, the values for x« reported in the literature vary widely, and the barrier lowering effect is not fully-understood at present. In the tunnel junction model of the past [3,22,44], it has been assumed that the unlowered bulk oxide values for electron and hole barrier height (x ~3.2 eV, e  Xn ~4.2 eV) and the independence of Xe and Xfc with tunnel oxide thickness are valid. As pointed out by [45,46], the predictions of tunneling current based on these assumptions disagree with the experimental results.  Therefore, to  describe the conduction mechanism in a tunnel junction we adopt the one-band model with only one adjustable parameter, namely Xe> whose value may be much smaller than the bulk oxide value of 3.2 eV. The barrier height Xe depends 50  on the oxide thickness and surface condition, and can only be determined by parameter fitting theoretical results to experimental data.  3.2  T h e N e w P o w e r Series for T u n n e l i n g C u r rent  In Section 2.1, the inaccuracy of using (2.5)-(2.9) to approximate the tunneling currents in MIS junction was discussed. We have developed new power series expressions for J  e m  and J  v m  which have several advantages over the previous  analytic expressions ((2.7), for example). Firstly, the new expression accounts for the energy dependence of the tunneling probability 0, and can be evaluated given the relation between 0 and E . This means the expressions can be adapted z  into both one-band and two-band models and used with any shape of tunneling barrier. Secondly, this single expression can accurately predict current in different operating regimes, and is not restricted to particular relative positions of E F m , Eeo  and E f . Thirdly, the series are fast converging and calculate, within n  the constraints of the WKB approximation, to any desired accuracy depending on the number of terms employed. Using Harrison's independent electron approach [25], the electron tunneling current can be written as (see Fig. 3.3)  51  Jem  —  1+ exp(^±%^*) 1  | 9 dE e  exp{&±&f^)  1+  (3.3)  x  The electron tunneling probability 6 (E ) can be expanded using the Taylor e  x  series with respect to the conduction band edge 9 (E ) = e  x  e (E ) + 9\{E ){E -E ) + e  eo  eo  x  -^^{E -E Y  9  co  + ••• +  nl  (E - E ) x  eo  x  eo  (3.4)  n  where d"9 dE?  c  Substituting (3.4) into (3.3), the electron tunnel current can be written as 4nqm' Jem —  e  (3.5)  n=0,l,l  where dE  t  JE.  0  [JO  and I j is the same with E m  [E x  T Fm  E )'dE eo  x  in place of Ep . n  Note that /„,• can be further simplified by changing the order of integration 52  Figure 3.3: Complete energy band diagram of a MIS diode with tunneling currents indicated.  53  of the double integral. (E -E y  _ f°° f f « H(E ) E  inJ  [J  ~Jo  j\  Eeo  Substituting n = '~ -, E  t  eo  eo  dE  1+exp( ^y ") g  r  t  we get  E  - f l/ ^(^^(^ 0  Since the normalized barrier height  E t i  k T  E  ;; L^, d  1+mp(  dE  t  " is usually very large (~40 for a 1 eV  barrier), we could write =  °°. E (kTy+ e>(E ) / °°[^( rf  l  eo  E -E  Fn  eo  o  t  CO '  (3.6)  Substituting /„, and I j back into (3.5), we obtain m  Jem = —  ^  ^Fn ~ E  E \kT) e:{E ) n  fc  3  eo  EFW. — E ,  CL  EO  kT  n=0,l,2  kT (3.7)  where 0{{E ) is defined in (3.4). eo  Similarly, the hole tunneling current is given by (see also Fig. 3.2) 4nqmlk T 2  J»  m  -~"  2  ^  /.^n-rwjfi  2- \ n=0,l,J  k  l  xf  T  I *v\ *o) |'n+lV t,  -  E ^ Fp  £p  T  ,E  V0  ) - ?n+H  - E ^ Fm  ^ kT  )  (3.8)  where e:(E ) = V0  dre  v  54  D  Notice that the first terms of the power series expressions (3.7-3.8) are exactly equivalent to the J  em  and J  expressions obtained in [22]. The impor-  vm  tance of the additional terms that have been derived can be readily seen in the MIS system in Fig. 2.2(b) where Epm < E . Suppose electrons are tunneling eo  from the metal to the semiconductor, and assume that E  eo  — Ef  m  ^> 0, the  magnitude of this component of the electron current can be written as (from (3.3)) (3.9) The current magnitude can intuitively be understood as the integration of the product of two functions within an interval. The situation can be represented graphically in Fig. 3.4. The tunneling probability 0 is plotted against E , and t  the curve is exponentially increasing with E (see (2.17)-(2.19)). If we approxt  imate this curve (# 3) using a Taylor series expansion ».bout E = E x  eo  as in  (3.4), then 9\ is the first term, 9 and <?s are the summations of the first two 3  and three terms respectively. This implies 9\ is constant, 0 is linear and 03 is 2  quadratic, etc. In the case where Ep i < E m  eo  (see curve # l), the integration  of the product of fi and 9\ in the interval [E ,Eci\ gives a good estimate of eo  J _, , because of the exponential drop of fi near E — E . This is why the m  c  z  co  assignment of a constant tunneling probability is satisfactory in this case. As the Fermi level rises above E  eo  to a position  of the product of / and 9 in the interval [E a  55  eot  #FmJ>  see curve # 2, the integral  Ep \ contributes much to J - . mi  m  e  decrease here  Figure 3.4: Graphical representation of the factors appearing in the integrand  of (3.9).  56  Since /j is flat in this interval, the fact that 0 is approximated by 0\ can cause large error. We can use 02,0s, or even higher order curves depending on the accuracy desired. Of course, the more the curve / shifts to the right, the more terms of the Taylor series will be required, thus more terms of the J  em  series  (3.7) should be kept. To investigate how fast the series expression for J _ (3.7) with E — m  e  co  Epn ^ 0 converges, and how accurate the series expressions with one, two, and three terms are compared with the result from numerical integration (3.3) and the expression derived by Simmons (2.9), we present Figs. 3.5 to 3.8. Fig. 3.5 shows the tunnel current from exact numerical integration for six different combinations of oxide thickness, barrier height and shape. We can see that the currents calculated assuming rectangular and trapezoidal barrier only differ by a few %. Therefore we conclude that the shape of the barrier, whether rectangular or trapezoidal, does not affect the tunneling current much. Figs. 3.6-3.8 show the relative error of the power series expressions and Simmons' expression with respect to the exact numerical solution in the case of a rectangular barrier. Generally, the convergence rate of the series is satisfactory, only three terms are needed to achieve the accuracy obtained by Simmons' expression within the normal operation range where Ep — E m  eo  < 25kT. Note that Simmons' expression  is not as good as the series expression in the way that it always underestimates the current at low bias and overestimates the current at high bias, while the  57  Jpi-*c  (Acnr) 2  Figure 3.5: Comparison of exact numerical integration of (3.9) for the case of rectangular barriers (solid lines) and trapezoidal barriers (dashed lines).  58  % error  Figure 3.6: Percentage errors, w.r.t. numerical integration results, of J -, as computed using Simmons' expression and various numbers of terms in the series expression, (d = 16A, x« = 1-leV.) m  59  e  % error  20  I  Simmons' eq. (kT) 3 terms 2 terms 1 term  -80  i  Figure 3.7: As for Fig. 3.6 but with d - 16A, Xt = 2.26^.  60  % error  20  I  Figure 3.8: As for Fig. 3.6 but with d = 12A, x« = M e V .  61  series expression can predict quite accurately the current at low bias, and by adding more terms the current at high bias can be predicted at any desired accuracy. The insufficiency of Tarr's expression (3.5) also deserves our attention. For example, in Fig. 3.6 when Epm — E  eo  — 25kT, the actual current by numerical  calculation is about 5 times larger than the prediction from the first term of the series, but only 1.5 times larger than the prediction from the sum of the first three terms. The improvement by adding more terms is pronounced. By comparing Figs. 3.6-3.8, we notice that the series expression in general predicts the actual current more accurately with increasing barrier height and decreasing oxide thickness. This can intuitively be explained by referring to Fig. 3.4. Increasing the barrier height or decreasing the oxide thickness can reduce the slope of the curve # 3 near E — E , therefore the truncated series m  eo  expression can approximate the exponential curve # 3 better and give rise to a more accurate result.  3.3  T h e Revised M o d e l Formulation  The revised model for the MIS tunnel junction utilizes a one-band representation of the oxide and allows for the energy dependence of the tunneling probability factors. These two improvements to the original model are included in 62  the new model by replacing the expressions for J  cm  and J  om  ((12a) and (13) in  [22]) by (3.7) and (3.8). The expressions for the electron and hole tunneling probabilities at the band edges (0  cm  and 0  vm  as derived in Appendix II of [22]) are replaced in  the new formulation by the energy-dependent probabilities given in (2.17-2.19). Selection of a rectangular, trapezoidal or triangular barrier is now possible. The tunnel probabilities in (2.17-2.19) are incorporated in the tunnel current expressions via the series expansions of (3.4). All the other equations in [22] are preserved, except for the replacement of J3/2 in the expression for semiconductor charge density ((25) in [22]) by J i /  2  [49]. This Fermi-Dirac integral, and those required in the evaluation of the tunnel currents, are computed by short-series approximations [60].  63  Chapter 4 Modeling the T E T R A N Device  Historically device designers have sought relentlessly to increase the speed of transistor operation. Initially the bipolar transistor was the superior high speed device while the MOS transistor was more useful for low-speed/low-power applications. With continued scaling however, the MOS transistor has the potential to outperform bipolar transistors for very high speed Kxid very high density circuits. This is because in MOS devices, capacitances are more amenable to scaling and the shrinking of vertical dimensions in the MOSFET (junction depth and oxide thickness) is more easily accomplished than in the BJT (emitter depth and base width). Unfortunately, the endless thrust to ever-decreasing device geometries leads to problems in both junction and MOS transistors. A very serious problem encountered in either device is the punchthrough effect. In the bipolar transistor, punchthrough occurs when the collector merges with the emitter, and in the MOS transistor when the drain and source depletion 64  regions begin to merge. Recently, Taylor and Simmons [61,62] have proposed the bipolar inversionchannel field-effect transistor (BICFET) which operates on the principles of inversion in heterojunctions. This is a new bipolar transistor concept which has combined the virtues of the bipolar and MOS concepts. The inversion channel replaces the base in the conventional bipolar (or heterojunction bipolar) device, and all of the problems associated with the neutral base region, which include scattering in a very heavily doped layer and the storage of minority carriers, are eliminated in the BICFET. One of the claims for this device is a value for fr of around 10000 GHz [62]. Following the announcement of the BICFET, a different structure called the T E T R A N (see Fig. 1.1) which operates on the same principle, was reported [3,4]. Note that in [4] the device is referred to as a BICFET. This could be confusing because in the BICFET described in [61,62] the region of the emitter is a thick, wide band gap semiconductor rather than a thin tunnel oxide, and the current transport through this region is deemed to be by diffusion or thermionic emission rather than by tunneling. The BICFET of [4] is, in our terminology, a TETRAN. The operating principle of the T E T R A N is quite simple. Application of a reverse bias voltage to the metal emitter leads to depletion of the underlying 65  semiconductor, see Fig. 4.1.  Holes can be injected into the depletion region  from the p contact, the source, so inverting the surface region of the semicon+  ductor underneath the emitter. This increase in hole concentration leads to a redistribution of the voltage drops across the emitter insulator and the depletion region, see Fig. 4.1(b). The increased field in the insulator paves the way for an increase in the electron tunnel current. There will be some tunneling of holes from the semiconductor to the metal but, if this current is less than the enhancement in electron current, the structure will exhibit current gain. Modest gains of around 120 have been measured in operational devices [4]. The d.c. characteristics of this device will be analysed in Section 4.2. Moravvej-Farshi and Green [4] also suggest that, in the T E T R A N , the transconductance is lower and the input capacitance is higher than in a BICFET, and they hint that the intrinsic cut-off frequency for the T E T R A N is about 600 GHz. This is still a sensational figure, one that has prompted the work in Section 4.3. This, to the present author's knowledge, is the first detailed analysis of the high frequency capabilities of the T E T R A N .  4.1  Model Formulation  The basis for the model of the T E T R A N is the improved analytical model for the MIS tunnel junction discussed in Chapter 3. This model has been modified to  66  Figure 4.1: Effect of source current on the potential distribution and charge flow in the MIS junction, (a) /, = 0. (b) / , > 0.  67  accommodate a third electrode (the source) and to improve the characterization of the tunneling process. The latter improvements are twofold: firstly, the twoband model for Si0  based on the Franz dispersion relation has been replaced  2  by a one-band model; secondly the restriction of the tunneling probability factor to a contant value [22] has been removed. The detailed energy band diagram of the device is illustrated in Fig. 4.2. The expressions for electron tunnel current J  CM  and hole tunnel current J  are given in (3.7) and (3.8), and the electron  VM  and hole barriers (x and XH) are related by (3.2). In the model, x« is taken e  as an adjustable fitting parameter. and J  VM  Any solution for the tunnel current  J  EM  must be consistent with the stipulations of Kirchoff's laws. Regarding  voltages, summation of the potential drops shown in Fig. 4.2 yields:  ^ +^ +- +^ - — q  q  ^ =0  (4.1)  where VCB is the collector-emitter voltage and if)j is the potential drop across the oxide given by  (4.2)  0/ = —Q.  where €/ is the oxide permittivity and Q„ the total charge in the semiconductor, is derived in Appendix B of [49] as: Q  t  =  sgn(^y/2l^ [NJ ( t  l/2  EFn  kT  )  - n  Eeo  + N f ( ^^)E  v l/2  Po  68  0  +  N  D  ^  (4.3)  Figure 4.2: Energy band diagram for the T E T R A N .  69  where n and p are the equilibrium concentrations of electrons and holes re0  0  spectively. Considering now the need for continuity of charged particle flows across the interface we have, for the case of holes, neglecting recombination/generation in the depletion region [3], impact ionization effects [17], surface state tunneling [36] and trap-assisted tunneling [63]: J. = J  vm  + Jd  (4.4)  where Jj is the current due to diffusion of holes from the source into the quasineutral region of the collector (see Fig. 4.2), and is given by  Equations (4.1) and (4.4) are two non-linear equations which need to be solved simultaneously. By substituting equation (3.8) for J  vm  and (4.5) for Jj  into (4.4), and equations (4.2) and (4.3) into (4.1), the equations (4.1) and (4.4) can be written in terms of two independent variables tp, and <j>. Thus, for given values of J , , V , \h and Xe ( CE  see  (3-2)) solutions for 0, and <f> can be obtained.  A standard iterative technique based on a generalized secant method was used for this purpose. Once ty, and <j> are known, the electron tunnel current J  em  70  can be com-  puted, as well as the terminal currents: JB  Jc  Jem +  —  (4.6)  Jvm  (4.7)  Jem ~~ Ji  =  A computer program has been written to evalute the steady-state characteristics of T E T R A N devices using numerical methods to solve the equations of the above model. This program is listed in Appendix A.  4.2  D C Characteristics  To test the capabilities of the T E T R A N model, d.c. current-voltage characteristics were generated for a device structure resembling that used in the experimental work reported in (4). The physical parameters of the device were: oxide thickness = 16A, collector doping density = 7 x 10 cm~ and collector thick14  s  ness = 10/im. The device exhibits a current gain of about 120 at a collector current density of around 10 Acm~ . To simulate the experimental I-V curve s  2  in [4], the tunnel model in [22] was first used, i.e. the two-band model with constant tunneling probability. The J-V characteristics are shown in Fig. 4.3, and it is clear there is a strong disagreement with the experimental results shown in Fig. 2 of [4]. The d.c. gain predicted is only about 5 and the predicted collector current is an order of magnitude below that of the experimental current.  71  Figure 4.3: Prediction of T E T R A N characteristics: d = 16A, other parameters as in Table I of [22]. 6 independent of energy as in [22]. e  72  T  temperature silicon bandgap electron effective mass in oxide electron transverse mass in Si hole transverse mass in Si permittivity of silicon permittivity of SxO% hole mobility hole lifetime intrinsic carrier concentration conduction band density of states valence band density of states  300X I. 12*7 0.5m 0.2m, 0.66m, II. 9c, 3.9c 480cm V-V e  0  2  l  8/w 1.45 x 10 cm2.8 x 10 cm1.04 x 10 cm10  10  3  8  l9  3  Table 4.1: Model parameter values for simulation of the T E T R A N .  On replacing the two-band model in [22] by the one-band model with a lowered barrier height of 0.8 eV, but still using the constant tunnel probability approximation in the calculation, the J-V characteristics shown in Fig. 4.4 are predicted. The parameters used in the simulation are listed in Table 4.1. The collector current level has been raised an order of magnitude higher than the level in Fig. 4.3. However, the gain declines very rapidly as the source current increases. This is due to the fact that the Jem and J„m expressions based on constant tunnel probability deviate increasingly from the actual current as the junction is more reverse biased. This shortcoming of the model can be corrected by using the series expressions for J  em  and J  vm  , in which case the characteristics  in Fig. 4.5 are obtained, assuming x« = Xm = M eV. The collector current level is about the same as that of the actual device of [4], and the predicted current 73  Figure 4.4: Prediction of T E T R A N characteristics: same parameters as for Fig. 4.3, except for the use of a one-band representation of the oxide. 0 given by one term of (3.4) with Xe = O.BeV. e  74  (X 10 AC»" ) 3  2  Figure 4.5: Prediction of T E T R A N characteristics: same parameters as Fig. 4.4, except for the use of three terms in (3.4) for 0 , and x« = l.leV. C  75  gain AJe/Av/, is 100 at a collector current density of around 2 x 10 Acm~ . 3  3  This is in close agreement with the value of 120 measured by Moravvej-Farshi and Green [4] at similar collector current densities. As would be expected, the predicted current gain is sensitive to the value chosen for \ . e  For example,  reducing Xe to l.OeV leads to about a 3 0 % increase in gain. A unique feature of the T E T R A N I-V characteristic is the reversal in polarity of the collector curent at low collector-emitter voltages.  The effect  is clearly visible in Fig. 4.5. The value of VQE at which the current reversal occurs is called the cut-in voltage. The predicted value of 0.4V is close to the measured value of 0.6V [4]. No doubt even better agreement could be achieved by adjusting the value of the hole lifetime used to compute L in the expression p  for Jd (4.5), or by including in the model effects due to other currents, e.g., recombination/ generation in the depletion region. In the present model the collector current at voltages below the cut-in voltage is due to the dominance of J over d  J . em  Turning attention to the low gain region at low source current, it is suggested that the neglect of surface-state tunneling is responsible for the experimental data not being modeled very satisfactorily. By incorporating into the T E T R A N model the surface-state tunneling model of Section 2.2, the JV curve shown in Fig. 4.6 is predicted. It is assumed that all the surface-  76  Figure 4.6: Prediction of T E T R A N characteristics: same parameters as for Fig. 4.5, except for the inclusion of the surface-state tunneling effect, and X« = 0.85eV. The dashed lines represent experimental curves from [4].  77  states are acceptor-like. The electron and hole capture cross sections are o , n  a  p  ~ 5 x 10" cm , the surface state density is N ~ 5 x 10 cm~' and the 18  J  t  12  tunneling time constant is r ~ 1.2 x 10~ s. All these parameters lie within 8  m  the range of values obtained from previous experimental work [36,37]. When surface-state tunneling is included, there is an additional leakage charge flow from the hole inversion region in the semiconductor surface to the metal, thus reducing the gain of the T E T R A N device. Therefore a lower barrier height of 0.85 eV is needed to fit the experimental curve in Fig. 2 of [4]. The predicted J-V curve agrees closely with the experimental result. The cut-in voltage is found to be 0.5 V, and the maximum gain is about 115. Another interesting correlation of predicted and experimental data is the effect of the passage of high collector currents. In [4] irreversible reduction of current gain was observed in devices for which Jc was tt~ken above 10 Acm~ . 4  2  This effect was attributed to the generation of surface states at the oxidesemiconductor interface. The results of the proposed model suggest an alternative mechanism. The calculations indicate that at these current levels the voltage drop across the oxide is about 0.55V. This corresponds to an oxide field strength of 3.5 x lOPVcm , which is about the breakdown field strength of -1  thin oxides [48]. The saturation voltage V g is observed to be about IV in the C  simulation, and is much less than the value of about 5V measured in the experimental results. This discrepancy is most probably due to the emitter contact 78  resistance. The results of this work on the T E T R A N device have been published [49]. The value for electron tunnel barrier height (x ) that fits the T E T R A N e  d.c. characteristics best is found to be around leV, which is close the to the values 0.7 and 0.9eV which have been used by others [30,50] in the analysis of devices with thin tunnel oxides.  4.3  Small-Signal Analysis  The small-signal behaviour of the T E T R A N device can be described using the common-emitter hybrid-* model which is shown in Fig. 4.7. The parameters are defined as follows: Transconductance g  m  =  dJ  c  dV  SE  dQ.  Common-emitter input capacitance C„ Collector-source capacitance C, =  dV  SE  dQ,  e  Input resistance r = te  Output resistance r„ =  CE  dV  SE  dJ  s  dV dJ  CE  c  Js  The unity current gain cut-off frequency can be expressed as:  h=  9m  2*C. \l + 2C /C. \ !> te  e  79  e  1  (4.8)  Figure 4.7: Common-emitter small-signal hybrid-7r equivalent circuit for the T E T R A N device.  80  0m  V  = 5V = LOeV J, = Z2Acm~ V = 5V X, = l.OeK J, = 8i4cm~ V = 2V Xe = LOeV J, = 32>1 cm CB  X t  C (Fm" )  C.e  fr (GHz)  te  (Sm-») 2.7 x 10*  1.8 x I O  0.9 x 10*  2.1 x IO"  1.8 x 10*  1.6 x I O  1.5 x 10*  1.7 x IO"  -1  (Orn )  (Orn )  8  8  8  2.0 x 10"  T  2.0 x I O  1.9 x IO"  -7  4  2.3  -4  0.7  1  CE  1  5.0 x 10~  6.5 x IO"  7  1.8 x I O  -8  2.0 x 10~  3.0 x IO"  7  1.9 x 10~  1.8  3.3 x 10~  1.4  T  J  CE  7  4  -1  VCJE =  5V  X« = 1 K V J, = 32j4cm  8  1.8 x IO"  7  2.0 x 10~  7  4  -8  Table 4.2: Simulation results for the small-signal hybrid-* parameters of the TETRAN.  The hybrid-JT parameters listed above were computed from the d.c. model (neglecting surface-state tunneling) described in Section 4.2 by examining the changes in terminal currents, terminal voltages and stored charge in the semiconductor in response to small changes in either J , V$E or V E- The perturc  C  bations had a magnitude of 1% of the operative steady-state values. The results for a range of operating conditions and two different values of electron barrier height, Xe, are listed in Table 4.2.  With respect to the  conditions used to obtain the first row of the Table 4.2, note that a reduction in either J$ or V E, or an increase in x« leads to a reduction in / r , principally S  5  81  via a decrease in transconductance. However, the changes are not great and fx remains in the neighbourhood of 1-2 GHz. This is a far cry from the value of fx « 6 0 0 GHz previously suggested [4] as being appropriate for a T E T R A N of the type under discussion. Comparing the structure of the BICFET and T E T R A N , it is clear that the thin insulator employed in the T E T R A N will cause it to have a larger input capacitance than the BICFET. Also, the effective electron barrier height Xt is likely to be much larger for a semiconductor-semiconductor interface. Thus g  m  for the BICFET should exceed that for the T E T R A N . The input capacitance C  tt  usually exceeds C , certainly in the case of the T E T R A N (see Table 4.2), te  and, therefore (4.8) can be reduced to:  h  =2%7.  <'> 4 9  It is quite clear from this equation that the aforementioned differences in g  m  and C t will cause fx in the case of the T E T R A N to be inferior to that of the 0  BICFET. The results presented in this section indicate the large extent of this difference in high frequency capability. Confirmation of the estimate of fx resulting from the numerical analysis can be obtained by carrying-out an approximate analytical evaluation of fx via (4.9). This is demonstrated below. To obtain an estimate for g we neglect the contribution of the diffusion m  82  current to Jc, see (4.7) and seek an expression for J  em  which can be readily  differentiated. Such an equation appears as equation (A.32) in [3j. Taking the dominant term in this equation and ignoring the voltage dependence of pre-exponential factors we have, in the notation of the present work: J  = R'exp[-7e(Xm - ^ / / 2 )  em  Jem  w  J  1/J  ]  (4.10)  Jc as noted above, and 0/, the potential drop across the insulator, is  the main contributor to VSE, thus: „  dJ  e  _ dJ  q  em  le  Taking m = l.leV, d = 16A, V =5 V and J = 3.2 x 10 A c m " (i.e. as X  CE  6  s  2  per row 4 of Table 4.2) we find from the model that Jc = 3.8 x 10 A m 7  - 1  and  0/=O.68 V. Substituting these values into (4.11) gives: g = 3.3J = 1.1 x 10 n- m m  8  C  l  _a  As regards the input capacitance, this can be taken as being approximately equal to the capacitance of the ultra-thin emitter oxide, i.e.: (4.12) For d = 16A and €/ = 3.9c we have C „ = 2.2 x l O - ' F m . - 1  0  Using these  approximate values of g and C in (4.9) yields / =0.8 GHz. This figure is in m  te  r  good agreement with the results of the numerical analysis given in Table 4.2. 83  4.4  Summary  Both the d.c. and a.c. performance of the T E T R A N device have been carefully studied by numerical simulations. Computer analysis indicates that the T E T R A N is a modest gain (~ 100), low current (J  c  up to l O M c m ) and -2  reasonably high frequency (~ 2GHz) device. The performance of the device depends largely on the tunnel oxide parameters.  Theoretically, reducing the  thickness or barrier height of the tunnel oxide improves the gain and frequency response of the device. However, practically, a stable oxide with a thickness of 16 A in an MIS junction is just about the thinnest that one can obtain. Further reduction of oxide thickness may affect the reliability of the device. Therefore, the performance of the T E T R A N is not expected to improve to a large extent by optimizing the oxide parameters, without introducing serious reliability problems. A possible way of improving the performance of the T E T R A N is by changing the emitter material from metal to heavily n-doped polysilicon. Since the Fermi-level in heavily n-doped polysilicon can occur above the conduction band edge, this will increase the emitter current (J ), and therefore increase the cm  gain, transconductance and cut-off frequency. Also the SIS junction is likely to have less surface-state traps than the MIS junction. This will increase further the gain of the device, because the leakage current from base to emitter due  84  to surface-state tunneling will be minimized. With these potential merits the polysilicon T E T R A N should be further investigated.  85  Chapter 5 Modeling the M I S E T Device  MIS emitter transistors are potentially very useful devices. They are simple to fabricate and have potential for high current gain [5]. They can be viewed as having operating principles similar to heterojunction bipolar transistors. The structure of the MIS emitter transistor is shown in Fig. 1.2. The p-type base region is defined by implanting boron into an n-type epitaxial substrate which forms the collector. The heterojunction emitter is formed by growing a thin thermal oxide  (~20A) on the wafer surface  and capping this oxide with a layer  of low work function metal such as Mg, or Al. The low work function metal electrostatically induces a thin layer of electrons along the silicon surface underneath it. This causes the tunneling of conduction-band electrons between the metal and silicon to be greatly favored over that of valence-band holes. The first MIS emitter transistor structure was proposed and fabricated by Kisaki [65]. A current gain of 120 and a unity gain cut-off frequency of about 1GHz 86  were measured. Gains have increased dramatically in recent years. Green and Godfrey [5] have fabricated some operational devices with m a x i m u m d.c. gains of 600, 10000, and 25000 with base implant doses 5 x 1 0 " , 10 , and 5 x 1 0 c m 12  n  2  respectively. Theoretical work has not been reported to explain the extremely high gain achieved by these transistors. Also no theoretical (or experimental, for that matter) investigation of the high frequency performance of these transistors has been carried out. T h e work described in this chapter seeks to provide a theoretical analysis of the d.c. and high frequency properties of the M I S E T using the improved tunnel junction model developed in Chapter 3. The theoretical I-V characteristics of the transistors are then compared with the d.c. experimental results in [5], and a general agreement is sought by adjusting the tunnel barrier height. T h e next step is to consider the effect of small variation ; of voltages and cur-  rents, thus generating a small-signal hybrid-* model of the M I S E T . F r o m the parameters of the a.c. model, it is possible to estimate the unity gain cut-off frequency using a similar approach to that taken in [49]. In order to optimize the transistor for high  fma , X  the trade-off between the base conductance and /  is also studied.  87  r  5.1  Model Formulation  The operation of the MISET is similar to that of the conventional npn BJT in the sense that the supply of holes from the base lead into the p-neutral base region biases the emitter-base potential and controls the flow of electrons from the emitter to the collector. The only difference is that the transport of carriers across the emitter-base junction is through tunneling, rather than diffusion. Since the hole tunneling barrier is always higher by an amount equal to one silicon energy band gap than the electron tunneling barrier (as proposed by the one-band model), the back-injected hole current is greatly reduced and the emitter efficiency is improved. This is analogous to the effect achieved in AlGaAs/GaAs heterojunction devices, where the injected holes are discouraged from diffusing to the emitter by the energy barrier resulting from the larger band gap of the emitter [56]. The energy band diagram of the MISET is shown in Fig. 5.1.  All the  quantities indicated in the diagram are positive. The two independent variables of the system are <f> and \j) . Once <f> and 0, are known, all the potentials and t  current components in the system can be calculated. The total charge stored in the semiconductor space-charge region is given by [49]  88  Figure 5.1: Energy band diagram of a MISET.  89  Q.  =  - gn(^)(2tr .)" [JV 7 S  !  (  + "v7  c  V  1/I  (- ^i)-n exp(i^) 1  s  ,(^)-P  B  +  (5.1)  ^V  where qa  =  3  E  ah  -q<}>- q<f> - q$, 0  Assuming the absence of fixed charge or surface-state charge at the semiconductor/oxide interface, the potential drop V*/ across the oxide is related to the stored charge Q, by 1>i = -Q,  (5.2)  Notice that rf>i is a function of the two variables <j> and ip . As a matter of fact, t  all the current components in the system are dependent on the variables <j>, t/)  t  or tpj. Let's examine each current component in the system. The electron and hole currents are given by the series expressions  J = A\T ±(kTY9i\7 {^) 2  em  j+l  - T (^)) j+1  (5.3)  and  J  vm  = A\T* t(kTy9i[T (^p-)-? (=^)\ i+l  90  i+1  (5.4)  respectively, where =  - Xe  Xm +  qPi = Egb - q<*i The derivatives of the tunneling probabilities 9{ and 9{ are properly defined in Section 3.2. The effective Richardson constants A* and A* are defined as in T  H  [22]. Generally, series expressions with three terms will be sufficient to approximate the tunneling currents. However, an exact integration may be needed to accurately compute the electron current under high bias because of the low electron barrier [47,49]. The electron diffusion currents at the depletion edges of the neutral base near the emitter and collector are represented by Jne and Jne respectively. They can be given in terms of <j> and V B as C  {coth(^)[exp A - 1]  qPebn b 0  ne  Leb  - csch(—^)[exp(  kT  ) 1]}  (5.5)  1]}  (5.6)  csch(g)[exp(|£) - 1]  qDeb^ob J,  -qVcB  ne  {  vv'  -qVcB  - c o t h ( - ± ) [ e x p ( - kT where V B C  =  V  C  E  +  0.  +  4> 0  E  t  91  h  / q  +  V/  +  Xm/q  -  Xe/q  not is the equilibrium electron density, D is the electron diffusion constant and tb  La is the electron diffusion length in the neutral base. W' is the effective base b  width, which is voltage dependent due to the basewidth modulation effect. The base recombination current is obtained as  Jrb  — =  Jnt~  Jnc  —f (coth(—2-) - csch(-*-)] • *->tb LJA L,th {(exp(^) - 1] +  [exp(n^) "  1]}  (5-7)  The diffusion constant D is related to fi by the Einstein relation ei  eb  D  eb  =  kT  —  9  (  5  .  8  )  The mobility of electrons fi b is dependent of the base doping density Pg and e  is given by the empirical relation (valid in the range of PB ~ 10 — 10 cm ) 18  J0  _3  [57] 5 2 x 10 + 1 + 6.984 x 10-" x P (cm->) 12  *» =  8  8  B  The electron diffusion length L  tb  .  3  C  m  V  . °  .  ,  x  ^  is given by Leh = yjD r ei  (5.10)  lh  The electron lifetime T also depends upon the base doping density P , and is EB  B  normally related experimentally by [55]  T  '  b =  1 + (P /5 x 10 em- ) B  92  16  3  (  5  1  1  )  In the base-collector depletion region, the generation current is expressed as J = —^-\ (~2~kf~) ~ i exp  t  (- )  l  5  12  where lf, is the depletion width at the base-collector junction and r is the e  t  generation lifetime of the carriers. The hole current due to diffusion from the collector to the base is formulated as  Jd=  " \/^ ° ~lr " 9  P c(eXP(  where Dhc is the hole diffusion constant,  £)  11  (5ll3)  is the hole lifetime and p  oc  is the  equilibrium hole concentation at the collector. To obtain <j> and rp, when the voltage applied across the collector and emitter (V E) and the base current (JB) are given, two non-linear equations C  need to be solved: Jne-Jem  =0  J + J + Jd - Jvm ~Jrb = 0 B  t  (5.14) (5.15)  These equations represent the conditions for the continuity of electron and hole flows in the transistor, and can be solved using a standard iterative technique based on a generalized secant method. Once <f> and rp, are known, all the current components can be computed and the terminal currents can be obtained as: J  E  = Jem + Jvm  93  (5.16)  (5.17)  Jc = Jne + J + Jd t  A computer program has been written to evaluate the steady-state characteristics of MISET devices using numerical methods to solve the equations of the above model. The program is listed in Appendix B.  5.2  D C Characteristics  To test the validity of the model, a set of J-V characteristic curves are generated and compared with experimental results from similar structures reported in [5]. Devices with three different base doping densities were constructed in [5], and each exhibited a different collector characteristic. The experimental J-V curves for three different values of the base implant dose are shown in Fig. 2 of [5]. For the highest base dose (5 x 10 cm" ), the device h?s a moderate current 1J  J  gain (/?) of about 600. For a base dose of 10 cm" , 0 increases from about 1,  J  3000 at low voltages to about 10000 near the punchthrough voltage which is in excess of 25V. For a base dose of 5 x 10 cm" , /? increases from above n  J  10000 at low voltages to nearly 25000 near the punchthrough voltage of 4V. To simulate these experimental curves, the physical parameters of the device listed in Table 5.1 are used. The d.c. characteristics are especially sensitive to five parameters, namely the electron tunnel barrier height (x is assumed equal to e  X m ) , the oxide thickness (d), the base doping density {PB), the base width (W») 94  temperature silicon bandgap at base region m ; electron effective mass in St'Oj ml electron transverse mass in Si hole transverse mass in St permittivity of Si «» permittivity of Si'Oj «/ carrier generation lifetime at base-collector depletion region U *kc hole recombination lifetime in collector Dkc hole diffusion constant in collector nt intrinsic carrier concentration conduction band density of states N N valence band density of states . T  <  c  v  300A: l.UV 0.5m, 0.2m, 0.66m, 11.9c 3.9e, 0.2ns 0.2ns 12cm *1.45 x 10 em2.8 x 10 crrr 1.04 x 10"cme  8  1  I0  w  8  Table 5.1: Model parameter values for the simulation of the MISET.  and the base intrinsic lifetime (r^ defined in (5.11)). With Xe = 0.8eV, d=18A, rVj=0.2/xm, T<& = 4 x 10" « and three different values of P^, the collector 8  characteristics shown in Figs. 5.2-5.4 are obtained.  The values for the base  doping density used in the simulation are deliberately set to 2.5 x 10 , 5 x 10 17  16  and 2.5 x 10 cm , which are equivalent to the base implant doses of 5 x 10 , 16  10  18  -8  and 5 x 1 0 c m n  18  -8  in the experimental structures with a base width equal to  0.2nm. The simulated curves in Figs. 5.2-5.4 are found to agree closely with the experimental curves in Fig. 2(a)-(c) of [5j. In the case of P = 2.5 x 1 0 c m 17  B  (Fig. 5.2), the current gain /? increase from 500 at low V  CE  at high V  CE  (~30V) when J  c  -8  (~1V) to about 750  is low (~ 8 x 10~ Acm" ). For P = 5 x 1 0 c m 3  (Fig. 5.3), 0 increases from 4300 at V  CB  2  ~ IV when J  95  ,6  B  c  8  ~ 1.6 x 1 0 A c m - , to _8  8  8  8  COLLECTOR-EMITTER BIAS  Figure 5.2: Common-emitter P = 2.5 x 10 cm- . B  17  characteristic  9  96  (V)  of the M I S E T  with  Jc  97  Figure 5.4: Common-emitter characteristic of the MISET with P = 2.5 x 10 cm~ . The dashed lines represent experimental curves from (5). B  16  8  98  about 7000 at VCE ~ 20K which is near to the base punchthrough voltage. In the case of P  = 2.5 x 10 cm- (Fig. 5.4), 0 increases from 10600 at V 16  B  3  ~ IV  CE  to 25000 at V E ~ 3V near base punchthrough, when J ~ 3.2 x 10~ Acm~ . C  5.3  C  4  2  Small-Signal Analysis  Using the same approach as described in Section 4.3 for the T E T R A N , the d.c. model can be used to examine the small-signal parameters which affect directly the high frequency performance of the MISET device. The transconductance is defined as dJ dV  (5.18)  c  <7m  BE  Vats  For high performance transistor design, the emitter series resistance R needs e  to be minimized because it can degrade the transconductance. In the area of high speed polysilicon emitter transistors, a lot of effort has been expended in attempting to design an emitter contact with minimal series resistance [66]. In the MISET, the emitter series resistance is mainly due to the thin tunnel oxide, and can be expressed as R  t  = ^ dJ  B  (5.19)  The base-emitter capacitance C\, is also a critical parameter determining t  99  the speed of transistor operation. It is defined as C Q  B  6e  =  dQ  (5.20)  B  dV  BE  VuB  is the total charge stored in the base region, including the stored charge Q,  in the space-charge region and the storage charge of minority carriers Q in in m  the neutral base region, i.e. Q  (5.21)  = Q. + Qmin  B  By differentiating (5.21) w.r.t. VBE, we obtain  dVBE  dV^BE  dV^BE  where C<j,// is the base diffusion capacitance and is defined as dQ^n/difi. It can be shown that although the magnitude of Ca// may become comparable to dQt/dVsE, the ratio d^fdVBE  is usually small, so the second term of the right  side of (5.22) can be neglected (see the numerical example later). Therefore, the base-emitter capacitance can be approximated as C  ie  »  dQ.  dVsE  (5.23) VOB  The unity current gain cutoff frequency can be expressed as  In the MISET device, the base-collector capacitance C  be  is mainly a depletion  layer capacitance, and is usually small compared to C . be  100  Thus, fr can be  approximated as !  T  K  J ^  (5.25)  All the above small-signal parameters can be computed numerically from the d.c. model by examining the changes in Jc> 0/ or Q, in response to small changes in JB or VBEIn order to give an estimate of the high frequency cut-off of the MISET, we consider a particular structure with Xe = 0.8cV, d = 18A, W P  = 0.2/im,  = 2.5 x 10 cm~ and r = 4 x 10 s. At V E - 2V, a base current {J ) of 16  B  9  3  _8  C  ob  B  32Acm~ gives rise to an collector current (Jc) of 6.6 x 10 Acm . By changing 2  3  -2  JB slightly, i.e. perturbing the steady-state system, we obtain d<j>/dVBE ~ 0.08, dQg/dVBE ~ 1.3fiFcm~ and g ~ 2 x 10 Scm . 3  4  The base storage capacity  -2  m  Cji/f is estimated by the formula ^,// = ^;(^)n  o t  exp(^)  (5.26)  where n„> is the equilibrium electron concentration in the neutral base. This equation gives a value of 1.2^tFcm for C ^ / / . Even though the magnitude of -2  Cji/f is comparable to dQ jdVBEi the small ratio d<f>/dV E justifies the use of t  B  (5.23) for finding C\, . Also C\, is found to be about 0.018^Fcm , which is t  -2  c  small compared to C\, , thus justifing the approximation of fa in (5.25): t  2xl0 5cm" 4  h  2  * 2* x 1.3 x l O - ' F c m 101  2  =  2  A  G  H  *  Note that the above fr value is obtained under conditions of high base-emitter bias, where the emitter metal Fermi-level rises well above the semiconductor conduction band and the semiconductor surface is degenerated by a large population of holes. Under these conditions, it is possible to estimate fr by analytical expressions, such as (4.11) and (4.12). Substituting Xm = O.SeV, d = 18A, V>/ = 0.67K (found from the simulation), and Jc = 6.6 x 10 j4cm 3  -2  into (4.11)  gives g = 4.8V m  -1  x J « 3.2 x l O ^ ^ c m c  For d = 18A and tj = 3.9e , (4.12) gives C 0  tt  - 2  = 1.9/iFcrrr . Using (5.25), the 2  unity current gain cutoff frequency fr is estimated to be about 2.7GHz, which closely agrees with the value obtained from numerical analysis. To investigate the dependence of the small-signal parameters such as 0, g , Rg and / r on the collector current Jc, we plot all thede parameters against m  Jc in Figs. 5.5-5.7. All the parameters have been calculated numerically, since the previous analytical expressions for g and C are only valid in the regime of m  lt  high Jc, and are not satisfactory for low Jc- In Fig. 5.5, the gain decreases quite rapidly as the collector current increases, from 0 ~ 20000 near Jc = l A c m  - 2  to P ~ 120 near Jc = 0.7 x 10 Acm~ . The plot resolves the large discrepancy 4  7  between two different reports on the current gain of MISETs: 0 ~ 20000 in [5] and 0 ~ 120 in [65]. In the former case the measurements of 0 were made at very low Jc (of the order of l A c m ) , while in (65] Jc was several orders of -2  102  Figure 5.5: Dependence of current gain on collector current of the MISET.  103  gm  He  •  IO"  1  I  I  J  1  10  10  COLLECTOR CURRENT DENSITY  1— 2  IO  3  (Acm ) - 2  Figure 5.6: Dependence of transconductance and emitter resistance on collector current of the MISET.  104  Figure 5.7: Dependence of cut-off frequency on collector current of the MISET.  105  magnitude higher. In Fig. 5.6, we observe that g  m  rises while R falls as, Jc increases. It t  is interesting to note that R can be higher than 10 fiflcm at low collector 4  e  2  current, which is in general agreement with the extremely high values of emitter series resistance measured in [67]. Experimentally, Moravrej-Farshi [67] has also observed that increasing the collector current level by an order of magnitude or more results in a 2 to 3 fold drop in the measured series resistance, which is essentially the predicted tendency in Fig. 5.6. The predicted Rg at Jc = 7 x KrMcm  -2  is about 35jiftcm', that is an order of magnitude higher than  typical values for the polysilicon emitter transistor (see [66]). R degrades g , e  m  which explains why the fa for MISETs (~ 2GHz) is relatively low compared to PETs (~ 15(7ffz). Of course by reducing the oxide thickness we can decrease R , and this is precisely the reason for reducing the interfacial oxide thickness e  as much as possible in high speed P E T design [66]. Fig. 5.7 shows that fa of the device increases monotonically with Jc. The / r increases from about 10MHz when 0 ~ 20000 at low J , to about 2.4GHz c  when 0 ~ 120 at high Jc- The prediction is in good agreement with the experimental result in [65], where an operational device exhibited a gain of 120 and a fa of about 1GHz at high JcIn conventional high-speed BJTs, there is a trade-off between the base  106  conductance and / , and the base doping density must be carefully chosen r  to optimize the maximum oscillation frequency fmaz- This is the frequency at which the forward power gain of a transistor becomes unity, and can be expressed as [68]  (5.27) where Rb is the base sheet resistance and C\, is the base-collector capacitance e  of the device. Increasing the base doping density will increase the transit time of minority carriers in the base and therefore decrease fj.  Also, Rb will be  reduced due to the lower base resistivity. On the other hand, decreasing the base doping density will increase both the fr and Rb. Thus there is an optimal base doping density at which fmaz is highest. To determine whether a similar trade-off exists in MISETs, the effect of the base doping density PB on /?, fr and f  max  for a particular device at  certain base current was considered. Neglecting the parasitic resistance and capacitance, Rb and Cbc are essentially the base-spreading resistance and the base-collector depletion capacitance respectively. For a rectangular base layer with two contacts at two opposite sides, which is the structure assumed in this work, the base-spreading resistance can be calculated as [69]  where Wb is the base width, h is the distance between the two contacts and  107  / is the length of the contact. p is the resistivity of the base region, and the b  dependence of its value on base doping density is shown in [18, p.32]. In Figs. 5.8-5.10, three small-signal parameters 0,  and f  mac  are plotted  against base doping density PB for two values of instrinsic base lifetime T  OH  (5.11)).  (see  The device parameters are: \ — O.BeV, d = 18A, W = 0.2/nm, t  b  AE = h x / = 5 x Sfim (emitter area), and the transistor is operating with 2  J  B  = IGAcm- and V 2  CE  = 2V.  Referring to the case of the larger base lifetime (r = 5 x 10" s as used 7  gb  in the SEDAN III program [55]), 0 and f  stay more or less constant up to  T  P  B  ~ 7.5 x 10 cm . 17  -3  /maj,. rises rapidly up to this point because the base  resistance decreases as P increases. Beyond Pg ~ 7.5 x 10 cm , f 17  B  _s  max  tends  to increase slowly because fa starts to fall. Even at the high base doping density of 5 x 10 cm~ , fmax has still not reached its maximum value because 18  3  is  dropping more rapidly than /r- This suggests that the base regions of MISETs, like GaAs heterojunction transistors, can be doped as heavily as wished in order to improve high frequency performance. In the case of the short base lifetime r  ob  = 5 x 10 s, which might be a -8  typical value in actual devices due to excessive base recombination, / r drops rapidly beyond the doping density of Pg = 2 x 10 cm~ . 17  3  When it reaches  PB = 7.5 x 10 cm" , the increase in base conductance cannot compensate for 17  3  108  Figure 5.8: Dependence of current gain on the base doping density of the MISET.  109  ST  /is &5 1.5 >*  o  g  or W PS la*  oI  H  1  0.5  H  io  io  17  10  18  19  BASE DOPING DENSITY (cm ) -3  Figure 5.9: Dependence of cut-off frequency on the base doping density of the MISET.  110  '  '  *->P  B  10" 10 BASE DOPING DENSITY (cm" ) 18  10»9  3  Figure 5.10: Dependence of maximum oscillation frequency on the base doping density of the MISET.  Ill  the drop in / r , which is more rapid than that in the case of long base lifetime. Therefore fmaz begins to decrease after this point. The result suggests that for a transistor with short base lifetime, there is only a limited advantage to be gained regarding improving the high frequency performance by increasing the base doping density. This is because base recombination generally reduces the gain, transconductance and cut-off frequency of the MISET device. With a short base lifetime, this effect becomes important. The rise of base sheet conductance cannot compensate for the drop of fx as the base doping density increases, leading to degradation of fmaz-  5.4  Summary  The MISET device displays the interesting behaviour of having high current gain and low fx at small bias V , BE  but low current gain and high fx at large  bias. The reason for this is that the emitter resistance R decreases as the bias e  increases; this is an inherent property of MIS tunnel junction emitters. The cutoff frequency of the MISET is found to be about 2GHz at high collector current, close to the f  T  predicted for the T E T R A N . This is by no means coincidental,  since at high bias the relative positions of the Fermi-levels and band edges near the oxide interfaces for both the T E T R A N and MISET are essentially identical. The fx of the MISET is not likely to be improved by reducing the oxide  112  thickness or the barrier height, since an oxide thickness of 18A is about the reliability limit for a MIS junction. However, the f  mat  of the MISET device can  be optimized by adjusting the base doping density. The simulations indicate that, for reasonable values of base minority carrier lifetime, fmax continues to increase as the base layer doping density increases to the practical limit.  113  Chapter 6 Modeling the P N P Polysilicon Emitter Transistor  High performance bipolar integrated circuits have significantly benefited from the advent of polysilicon emitter contact technology. Because the polysilicon contact reduces the back-injected emitter current, the emitter efficiency of this kind of device is improved. The resultant increase in current gain can then be traded-off with higher base doping (lower base sheet resistance). The outcome is an increase in switching speed. Polysilicon emitter transistors (PETs) that have been studied extensively in recent years are mainly of the npn type. The best devices have exhibited a high cut-off frequency of 16GHz [58]. Recently, Maritan and Tarr [70] have fabricated pnp PETs with different surface treatments, and demonstrated that they can exhibit reasonable gain and acceptably low values of emitter series  114  resistance. These results are extremely important, because they imply that the cut-off frequency of pnp devices might be very high. Although pnp devices will exhibit larger base transit times than npn devices due to the lower hole mobility, they have smaller base sheet resistance compared with npn devices. This should be advantageous in terms of /  ma  * - The resistivity of n-type (phosphorus doped)  silicon is generally two to three times lower than that of p-type (boron doped) silicon, see [18, p.31]. As long as fa for pnp PETs is not 2-3 times less than that for equivalent npn devices, the  values for these two transistor types  will be comparable to each other. In the area of GaAs heterojunction bipolar transistors (HBTs), theoretical analysis and computer simulations [71,72] indicate that the values of / „ „ , for pnp and npn structures are very close. This feature may open up the possibility of complementary npn/pnp design approaches for circuits such as amplifiers and A / D converters, and provide a solution to the long-standing problem of developing very high speed complementary circuits in III/V materials for lowpower applications. If pnp PETs can be optimized to exhibit values of f  max  close enough to npn PETs, then silicon bipolar technology might enjoy the advantages of complementary circuit design which have already been forseen in GaAs HBTs. In this section, we are concerned mainly with pnp devices with a deliberately-  115  p COLLECTOR (SUBSTRATE)  Figure 6.1: Structure of the pnp polysilicon emitter transistor [70].  grown interfacial oxide to which the improved tunneling model developed in Chapter 3 can be applied. Two types of device, one with and the other without post polysilicon-deposition anneals, are simulated by the model and compared to the experimental results in [70]. The device structure of the pnp P E T reported in [70] is illustrated in Fig. 6.1.  The intrinsic base is formed by ion  implantation of phosphorus, and has a depth of 0.3/im after annealing. A thin layer of oxide (10 — 20A) is chemically grown in the emitter window, then a boron-doped amorphous Si film is deposited and later recrystallized at low temperature. An emitter polysilicon film of about O.l^m thick is obtained. This device, without any post polysilicon-deposition annealing, gives an effective  116  emitter Gummel number G of t  1 - 2 x 10 scm , combined with an emit14  ter resistance R of about 26/xncm . t  2  -4  If the device is annealed at 900°C for  30 minutes after the polysilicon film deposition, a monocrystalline emitter approximately 0.15/Ltm deep forms underneath the oxide due to dopant diffusion from the polysilicon into the base region. The annealed device gives the same G, as the unannealed device, but much lower R (~ 1 — 2^tf2cm ). Both types t  J  of device can exhibit current gain of up to about 300. The reasons why pnp PETs can exhibit good gain and low emitter resistance are not obvious. The one-band model [30], which implies the tunneling probability of holes in silicon MIS structures is inevitably smaller than that for electrons, suggests that the ratio of hole emitter current to electron backinjected base current should be small, leading to inferior current gain. Also because of the larger hole tunnel barrier height Xh in pn^. devices compared to electron barrier height x« in npn devices, the emitter resistance of pnp devices is expected to be large. All these reasons suggests that pnp device performance should be very poor. In this chapter it is shown that the one-band model, which has successfully predicted the characteristics of TETRANs and MISETs, is capable also of predicting the characteristics of pnp oxidized PETs (PETs with chemically grown emitter oxides). The simulated d.c characteristics of both unannealed  117  and annealed devices agree well with the experimental d.c. data in [70]. T h e high frequency performance of these devices is assessed by computing some important a.c. parameters.  6.1  M o d e l Formulation  In the unannealed device, there is no mono-emitter region underneath the tunnel oxide. T h e energy band diagram is simply as shown in Fig. 6.2. It can be seen that the charge flows and potential drops across the unannealed  PET  are similar to those existing in the M I S E T , except that here we have a pnp rather than a npn structure. The similarity is obvious if the band diagram of the M I S E T (Fig. 5.1) is viewed upside down. Therefore, the method of solution used in the M I S E T is directly applicable to solving the the case of the unannealed P E T . However, two modifications need to be made in order to simulate the device correctly, as the emitter material is polysilicon rather than metal. Firstly, the effect of minority carrier conduction in poly-Si must be considered. The minority carrier diffusion in polysilicon film can be described by exactly the same equation used for diffusion in mono-silicon. Taking into consideration the boundary conditions for minority carrier concentrations at the metal/poly-Si and oxide/poly-Si interfaces, the electron diffusion current at the  118  poly-Si/ oxide junction can be written as Jm = -jr^coth{-f-)\N  c  exp(  —  '-)- n ]  (6.1)  pol  where n i is the equilibrium concentration of minority carriers, W po  pol  is the  polysilicon layer thickness, D \ and L i are the effective diffusion constant po  po  and diffusion length in the p-doped polysilicon, respectively. In addition to the two non-linear equations (see (5.14) and (5.15)) describing the continuity of electron and hole flows at the mono-Si/ oxide junction, another equation Jni = Jtn which represents the electron (minority carrier) continuity at the poly-Si/ oxide junction (see Fig. 6.2), must be solved simultaneously. Secondly, the forbidden energy range for tunneling is no longer coincident with the mono-Si band gap as in the case of the MISET, but is determined by the range of overlap of the poly-Si and mono-Si band gaps. In Fig. 6.2 E , eo  the energy above which electron tunneling occurs, is in line with the poly-Si conduction band edge, while E , below which hole tunneling occurs, is coinvo  cident with the mono-Si valence band edge. If the potential drop across the oxide V/ is reversed, then the mono-Si conduction band edge and the poly-Si valence band edge will become E  co  a , Pi and 2  and E  vo  respectively. The parameters a , x  which determine the tunneling currents (see (5.3) and (5.4)), are  defined as the differences between the quasi-Fermi levels with respect to E  eo  or E . The effective electron (or hole) tunneling barrier height is the energy vo  119  Figure 6.2: Energy band diagram of the pnp oxidized P E T without post-deposition annealing.  120  difference between E  eo  (or E ) and the middle point of the oxide conduction vo  (or valence) band edge. In the annealed device a layer of p-doped monocrystalline silicon about 0.15/xm thick exists underneath the thin oxide. The energy band diagram shown in Fig. 6.3 is seen to be quite different from that of the unannealed device. As a result, a different solution procedure is required. The independent variables chosen to solve the equations describing the system are 0j, fa, fa and <j>. These are defined in Fig. 6.3. All the current components and potential drops indicated in the band diagram can be expressed in terms of these four variables. The minority carrier diffusion current in the polysilicon J  nX  and the tunneling currents J  same way as for the unannealed devices. J mono-emitter region, and J  n2  and J  nm  and J  tn  r m  tp  are treated in exactly the  is the recombination current in the  are the electron diffusion currents across  the oxide interface and depletion edge of the mono-emitter respectively. Their magnitudes are given by (5.5)-(5.7), after replacing — V B and Wl byfaand C  the mono-emitter depth, respectively. J b is the recombination current, while r  J  pt  and J  pe  are the recombination current and hole diffusion currents at the  depletion edges of the neutral base. These currents can be related to <j> and VCB by (5.5)-(5.7), except all the minority carrier parameters are required to change to those of holes. The generation current J at the base/collector junction is t  121  Figure 6.3: Energy band diagram of the pnp oxidized P E T with post-deposition annealing.  122  given by (5.12), and the recombination current J at the base/mono-emitter r  junction is also given by (5.12), but with — V  and lb replaced by <f> and the  CB  c  base/emitter depletion width, respectively.  is the electron diffusion current  from the collector to the base, and is given by (5.13), after changing the minority carrier parameters to those of electrons. Finally, there are four non-linear equations to be solved. They are:  Jpe  Jnm  =  =  J  B  Jtp  +  J  ~  Jrm  +  t  J  —  d  -  (6.2)  Jr  J  r  -  Jh r  (6.3)  Jn2  =  Jtn  (6.4)  Jnl  =  Jtn  (6.5)  The first equation is concerned with the continuity of electron current at the base depletion edge near the emitter. The other three take care of the continuity of hole flows at the mono-emitter depletion edge and the two silicon/ oxide interfaces, respectively. After rfci,fa,faand <f> are determined, the terminal currents can be computed by  JE  =  Jtn  +  Jtp  Jc  =  Jpc  +  J  123  f  (6.6) +  Jd  (6.7)  Computer programs that are used to obtain the steady-state characteristics of unannealed and annealed pnp PETs are listed in Appendix C and D respectively.  6.2  D C Characteristics  The first simulation carried out was for the case of an unannealed device. In Fig. 6.4 the collector and base currents (Jc & JB) are plotted against the baseemitter bias (VBE)-  Some of the physical parameters for this simulation are:  electron barrier height (x ) = 0.5eV, oxide thickness (d) = 10A, £1 = O.leV, e  base width (M^j) = 0.3/im, base doping density (Pp) = 10 cm~ and intrinsic 1T  base lifetime (r„&) = 5 x 10~ s. 8  3  The value of ft used here corresponds to  a polysilicon doping density of 10 cm~ , i.e. about the solid solubility limit 20  3  for boron in silicon. The value of x« e d gave the best fit of predicted and us  experimental data. Similar values have been used by others in modeling SIS tunnel structures [59].  Since no experimental data for the minority carrier  mobility and lifetime in p-doped polysilicon is available, we use here the data for minority carrier transport in n-doped polysilicon. In heavily n-doped polysilicon (N i ~ 2 x 10 cm~ ), the minority carrier mobility and lifetime are estimated po  19  3  to be about 8 c m V a 2  -1  -1  and 2 x 10" s respectively [12]. 10  Therefore, in the  simulation performed here D i and L i are taken to be 0.2cm s po  po  124  2  _1  and 0.06/xm,  BASE-EMITTER BIAS  (V)  Figure 6.4: Gummel plots of the unannealed pnp P E T from computer simulation (dashed line) and published experimental data (solid line) [70].  125  respectively. The d.c. characteristics are not affected significantly by changing the values of D i or L oi by one or two orders of magnitude. This implies that the po  P  quantum mechanical reflection of electrons at the oxide barrier is the dominant mechanism for preventing back-injection, consequently minority carrier diffusion in the polysilicon is relatively unimportant. It is somewhat similar to the case of the npn polysilicon transistor, where the d.c. characteristics are greatly changed by altering the interfacial oxide thickness, but not nearly so much by changing the grain size of the polysilicon or by using hydrogen treatments for the passivation to the poly-emitter [73]. Fig. 6.4 shows that the simulated curves closely agree with the experimental curves reported in [70], except for the feature that Jg is underestimated in the high forward bias (VBE > 0.8V) regime. This discrepancy could probably be removed by including a surface-state tunneling current component in the model. "Kinks" in JB-VBE curves are often observed in practical npn polysilicon emitter transistors, for example in [73,74]. Simulations show that these "kinks", or departures from linearity, can be smoothed-out by increasing the surface-state density [74]. The emitter Gummel number G and the emitter resistance t  can be  easily obtained from the simulation. For a conventional pnp transistor in which  126  the base curent is dominated by the back injection of electrons into the emitter, J  B  can be related to G via e  JB =  qn]  qV B*  f  ex  E  ( E I O X  P(-Jtr~)  (-) 6 8  This means G can be easily computed by determining the base saturation cure  rent from the y-intercept of the Gummel plot. Although in polysilicon emitter transistors JB is given by the tunneling current expression (3.8), rather than by such a simple expression as (6.8), the Gummel number concept is still very useful for evaluating the transistor's d.c. performance. The JB curve in the Gummel plot deviates from linearity at high emitter-base bias, therefore G  t  can only be computed by extrapolating the linear portion of the curve in the low bias region. G« is found to be about 6.1 x 10 Scm~*, which is in the same 14  order as the experimental result of 1 — 2 x 10 acm . An expression for the 14  -4  emitter resistance R has already been given in (5.19). For a high collector T  current level (~ 10 Acra ), R is computed to be about 15/ifkm , which is 3  -2  J  E  close to the experimental value of 2GnUcm . 2  The annealing step can affect the integrity of the oxide and reduce the tunnel oxide thickness. Also during annealing the dopants from the heavilydoped polysilicon diffuse across the tunnel oxide to the mono-silicon region, pushing the emitter-base junction away from the oxide interface. Therefore, a mono-emitter is formed underneath the tunnel oxide. In the annealed device 127  the mono-emitter region is 0.15/zm deep, while the base width is reduced from the original O.Zfim to 0.15/zm. The mono-emitter and base doping densities are taken to be 10 cm 19  -3  and 2.3 x 10 cm~ respectively. These values are based 17  3  on SUPREM simulation results of the processing sequence used in [70]. With an electron barrier height (x ) of 0.5eV, an interfacial oxide thickness (d) of 7 A e  is required to give a low Re of 2.7fiUcm . The simulations indicate that this 2  reduction of oxide thickness (from the value of 10A used for the unannealed device) is essential to bring R down to the value of 2/xflcm found in experimental 2  e  devices. The result suggests that the interfacial oxide becomes thinner after the high temperature annealing, thus reducing the emitter resistance by almost an order of magnitude. From the simulated Gummel plot, G is computed to e  be 1.2 x 10 scm , which is in the experimentally observed range of 1 — 2 x 14  10 «cm . 14  -4  -4  The simulated common emitter characteristics for the annealed  device are shown in Fig. 6.5, plotted along with the experimental data from [70]. The current gain is 300 at 0.5 x 10~ Acm~ , which is in good agreement 3  3  with the experimental value of 250 at the same current level.  128  COLLECTOR-EMITTER B I A S ,  V  CE  (V)  Figure 6.5: Common emitter characteristics of the annealed pnp P E T from computer simulation (dashed line) and published experimental data (solid line) (70).  129  6.3  Small-Signal Analysis  In this section the unity current gain cut-off frequency fx is used as afigureof merit to compare the a.c. performance of annealed and unannealed devices. By perturbing the steady-state parameters by 1% in the d.c. model, small signal parameters such as g  and  m  can be determined and fx can be computed by  (5.25). In both types of device (annealed and unannealed) , the same formula (5.18) is used to obtain g , but different expressions for Cj are needed. For the m  e  unannealed device, C j is computed in exactly the same way as for the MISET. e  The contribution of minority carrier storage capacitance (second term on the right side of (5.22)) is always small compared to charge storage in the depletion region underneath the oxide, therefore C\, can be approximated by (5.23). t  In the annealed device, C is the equivalent of two series capacitances C be  ox  and Caff, i.e.  I  l  l  .  ,  where C (= ej/d) is the oxide capacitance and Cdi/f is the base minority carrier ox  storage capacitance. An expression for C^y/ is given in (5.26). This can be used in this case with n  oe  replaced by the base equilibrium hole concentration p \,. 0  The unity current gain cut-off frequencies of both types of device are plotted in Fig. 6.6. For the unannealed device, fx increases quite rapidly from  130  /r ANNEALED  10  DEVICE  10  55 tt s tt a: tt u< tt o i  UNANNEALED  DEVICE  IO • 9  g H  55 W OS  10  8  H  IO"  2  IO"  1  1  BASE CURRENT DENSITY  10  10  2  (Xcm" ) 2  Figure 6.6: Dependence of the unity current gain cut-off frequency on the base current for both the annealed and unannealed pnp PETs.  131  0.12GHz at J  B  = 10" Acm~ to 2.1GHz at J s  2  B  = 35Acm , while for the -2  annealed device fx remains constant at about 16GHz regardless of the current level. The dependence of fx on J  B  for the unannealed device is similar to that  of the MISET, where fx can be significantly increased by an increase in J B  This is because in unannealed devices, where there is no mono-emitter region, the device performance is mainly determined by the tunneling mechanism in the emitter tunnel oxide. Since the resistance of the tunnel oxide decreases rapidly as the device becomes more forward-biased, the transconductance g  m  also rises rapidly. On the other hand the value of C , which approaches the be  oxide capacitance C , only rises slowly. This explains the increase of f with os  J  T  shown in Fig. 6.6 for the unannealed device.  B  In the annealed device, C mainly depends on Cajf due to the large C . be  Since both g  m  ot  and C<a// increase as the bias curent increases, a more or less  constant fx results. In the case of conventional BJTs, neglecting the emitter and collector transit times, fx can be written as  /  =  i _ 2n  where r  B  ^ Gdiff  1  2n  _  1  r  B  1  22.  2n  2D  b  is the base transit time, W is the base width and D is the diffusion b  b  constant of minority carrier in base. Note that fx is constant and only depends upon the base width and base doping density. Considering the annealed PETs, C  bt  is essentially dorminated by C * / / rather than C , and g ox  132  m  is controlled by  the bias at the mono-emitter/base junction, therefore a constant fa is to be expected, as in the case of conventional BJTs. Generally speaking, the high frequency performance of the unannealed device is much worse than that of the annealed device. At high forward bias, ft of the annealed device is still several times higher than that of the unannealed device, due to the lower emitter resistance (thinner tunnel oxide) and smaller base/emitter capacitance (Cd,// <  6.4  C ). ox  Summary  The d.c. performance of both the annealed and unannealed pnp devices is good, in as much as reasonable current gains are possible. This is an unexpected result in view of the low hole tunnel probability predicted by the one-band model. For a tunnel oxide thickness of around 10A, pnp devices exhibit current gains close to 300, which are much less than that of npn devices (~  2000) with the same  thickness of oxide [7j. Simulations show that as the oxide thickness increases the current gain of the pnp PETs decreases. The current gain can even drop below unity if the oxide thickness is larger than 20A. This trend is contrary to what is observed in npn devices. It is further evidence of the fact that the low tunnel probability of holes makes tunneling the current limiting mechanism in pnp PETs.  133  From the high frequency performance study of the annealed and unannealed devices, it is predicted that fx for the annealed devices will be higher than that for unannealed devices. This result is mainly due to the lower emitter resistance and emitter-base capacitance of the annealed devices. The computed fx (based on a calculation which includes only the base transit time) for pnp annealed devices, is about 16GHz, which is extremely good. Even if the base transit time only comprises half of the emitter-to-collector delay time in a practical device, then a fx value of 8GHz should be considered feasible. The highest value reported so far for an experimental pnp P E T device is 1.6GHz [75]. Currently, a state of the art npn polysilicon emitter transistor can have a fx of 16GHz. If a pnp device with fx approaching 8GHz can be made, than the complementary circuit design approach, which implies low power consumption and high speed, will be feasible in Si technology.  134  Chapter 7 Conclusion The new formulation of the MIS tunnel junction model proposed in this thesis is used to simulate several silicon bipolar structures with tunnel oxide emitters. The new formulation represents the tunnel oxide band structure via a one-band model, and also allows for the energy dependence of the electron and hole tunneling probabilities. For the T E T R A N device, which has d.c. characteristics that cannot be adequately predicted by an earlier analytic model, the new model generates d.c. curves which agree closely with the experimental data. The unity current gain cut-off frequency is estimated to be about 2GHz, which is far more reasonable than the estimate of 600GHz recently suggested by others. In the modeling of MISETs, the new formulation also correctly predicts the experimental d.c. characteristics of devices with different base doping densities. The simulated current gains, emitter series resistances and unity current gain cut-off frequencies are in good agreement with the experimental data at  135  selected collector current levels. For pnp polysilicon emitter transistors, simulated results based on the new model fit the experimental d.c. curves very well. Contrary to intuitive expectations, the simulations suggest that these devices can exhibit moderate current gains and cut-off frequency values. The predicted results are in good agreement with experimental data reported in the literature. The major contribution of the work is a demonstration that the one-band model, which has never been employed previously to characterize transistors with tunneling emitters, leads to an accurate description of carrier transport through thin silicon oxides. The energy dependence of the tunneling probabilities of electrons and holes, which were not considered in previous analytic models, is shown to be crucial to accurate device modeling, especially at high current levels. New power series expressions for the tunneling currents have been formulated in order to accommodate the energy dependent t jnneling probabilities in a computationally economic manner. The resulting new model should be easy to incorporate into a more general device simulation program, such as the SEDAN program. There is now much interest in GaAs devices and it appears that the model developed here may be of use in analyzing some advanced structures utilizing this semiconductor material. The enhancement of gate barrier height in MESFETs [76,77] may be due to the presence of a thin insulating layer. If so, the  136  gate MIS structure could be investigated by modifying the program to use parameters relevant to GaAs. Another interesting device is the GaAs BICFET [78].  This device is similar to the T E T R A N , but has a wide bandgap semi-  conductor in place of a thin insulator. 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Electron Devices, 1988. [75] Y. Kobayashi, Y. Yamamoto and T. Sakai, "A new bipolar transistor structure for very high speed VLSI", Dig. of Tech. papers, IEEE Symposium on VLSI Technology, 40-41, 1985. [76] M . T . Schmidt, D. V . Podlesnik, H. L. Evans, C. F. Yu, E. S. Yang and R. M . Osgood, Jr., "The effect of a thin ultraviolet grown oxide on metalGaAs contacts", J. Vac. Sci. Technol., vol. A6, 1446-1450, 1988.  143  [77] Q. Y. Ma, M. T . Schmidt, X. Wu, H. L. Evans and E. S. Yang, "Effect of Schottky barrier height on EL2 measurement by deep-level transient spectroscopy", J. Appl. Phys., vol. 64, 2469-2472, 1988. [78] G. W. Taylor, M. S. Lebby, A. Izabelle, B. Tell, K. Brown-Goebeler, T . Y. Chang and J. G. Simmons, "Demonstration of a p-channel GaAs/AlGaAs BICFET", IEEE Electron Device Letters, vol. 9, 84-86, 1988.  144  of MIS9.S at 22:12:01 on OCT 10. 1988 for CC1d=KCHU on 0 C C C 1 C 2 C 3 C 4 C 5 C 8 C 7 C 8 C C S C 7 C 73 C 78 C 79 C 82 C 85 C 88 C 91 C 94 C C  C  C  5 C  C C C C  THIS PROGRAM IS WRITTEN TO GENERATE THE STEADY-STATE CHARACTERISTICS OF THE TETRANS  MOOIFIED IN LINES 198. 208-7. 43, 82-4. 201. 204. 271 304-5. 307-9 (USE F0102) BANDGAP OF SI02 UNSPECIFIED! (LINE 225) ONE-BAND MODEL (LINES 88. 91-7. 175. 178-181. 225. 338-9, AREA 1) USE FIRS-T THREE TERMS OF POWER SERIES FOR HOLE CURRENT ONLY (LINES 338-49, 497) USE EXACT INTEGRAL (IMSL PROQM) FOR ELECTRON CURRENT (LINES 180-2, 8. 10. 14, AREAS, 195-6, 285) MODIFY LINES 188. 195-8, 209-7 MODIFY LINES 501. 508 IMPLICIT REAL'S (A-2) INTEGER CNTRL1.DATFL,FDFL.FREE,I,IADNR,IFAIL,ITMAX1,ITMAX2.KR. f LABEL.LP.MINO.NTIMES.NTRAP. IER LOGICAL NEWY,NEWA,NEWS EXTERNAL COMPF, FUNCM DIMENSION NT(100),RHO(100),N1(100),P1(100),SIGMAN(100), • SIGMAP(100),CN(100).CP(100),IADNR(100),TAUM(100) DIMENSION ACCEST(2),X(2).F(2I, UPPER(2). L0WER(2> DIMENSION FREE(1),LABEL!15) DIMENSION FDFLA(81),FDFLC(81),FDFLE(81).FDFLF(81).FDFLGI81) COMMON • f • COMMON COMMON f COMMON COMMON COMMON COMMON  /AREA1/ CBAR,CHISC,CHISV,CJCM, CJVM.CPSII.COS.CTNLCM.CTNLVM.EOAP. JOO.JORG.JUPC.NC.ND.NNO.NV.O.PHIO.PNO. PSISIO.VTHERM.OFIX /AREA2/ NT.RH0.N1,P1.CN.CP.TAUM.IADNR.NTRAP /AREA3/ ETAC.ETAMC.ETAMV.ETAV.JVM.JPN.NSURF.PHI.PSII.PSIS. PSURF.OS.THVM.THCM.U.V /AREA4/ CNTRL1.ITMAX1.LP.NTIMES /AREA5/ CFO1,FDFLA,FDFLC,FDFLE.FDFLF,FDFLO /AREAB/ BRC /SE$$OM/ A(20.22),B(20).Y(22.21)  DATA EPSIO/8.850-12/.HBAR/1.0540-34/.KBOLTZ/1.380540-23/. • KS/11.7D0/.ME/9.110-31/.Nl/1.45016/.PI/3.14592654DO/. • T/300.ODO/.VELTH/1.005/ DATA FDFL/^/.0ATFL/3/.KR/5/.FREE/•••/ DAT* NEWY/.TRUE./.NEWA/.FALSE./.NEWB/.FALSE./ READ IN DATA DESCRIBING DEVICE REAO(KR.S) (LABEL(I).I=1.15> 5 FORMAT(15A4)  Listing of MIS9.S at 22:12:01 on OCT 10. 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 58 57 58 59 80 61 62 63 84 85 86 67 68 89 70 71 72 73 74 75 78 77 78 79 80 81 82 83 84 85 88 87 88 89 90 91 92 93 94 95 98 97  1988 for CC1d*KCHU on 0  Page  READ(KR.FREE) ITMAX1,ITMAX2,ERR1.ERRS.OELY READ(KR.FREE) CNTRL1  C  READ(DATFL.FREE) ND.CHISC.PHIM.MISTAP.D.KI.NCONV.AE.AH.OFIX READ(OATFL.FREE) JOD.JORG.JUPC DO 100 1=1.100 READ!DATFL,FREE) RHOII).NT!I).SIGMAN!I).SIGMAP!I).TAUM!I).IADNR!I) IF(NTU) .LE. O.ODO) GOTO 200 CONTINUE NTRAP=I-1  100 200 c #  READ(FOFL.FREE)  (FDFLF!I),FOFLA(I),FDFLC(I).FOFLE(I). FDFLG(I).I»1,81>  C C ECHO PRINT c WRITEUP. 10) (LABEL(I).I=1.15) 10 FORMAT!'120X,15A4///) WRITEILP.15) ND.CHISC.PHIM.MISTAR.D.KI.NCONV 15 FORMATCIX,'ND=',D10.3,2X,'CHIS=",F5.3.2X.'PHIM='.F5.3.2X. f'MISTAR*' .F5.3.2X, '0=' .010. 3.2X. 'HI"* .F5.2.2X. #'NCONV='.F7.5/) WRITE<LP.20) JOD.JORG.JUPC.AE.AH.OFIX 20 FORMATCIX.'JOD='.010.3.2X.'JORG='.010.3.2X.'JUPC*'.012.5.5X, 1  f'AE=',010.3,2X,'AH='.010.3.2X,'QFIX='.D11.4.2X//) IF(NTRAP .EO. 0) GOTO 400 WRITEUP.25) 25 FORMAT(2X, 1:',3X,'ETA:',5X,'NT:',7X.'SIGMAN:',SX.'SIQMAP: . #5X.'TAUM:•,5X.'IADNR:'/) DO 300 I=1.NTRAP 300 WRITE(LP,30) I.RHO(I),NT(I),SIGMAN(I).SIGMAP!I).TAUM!I).IADNR!I) 30 FORMAT!IX.13.2X.F5.3.2X.D10.3.2X.3(010.3.2X).12) WRITE!LP.35) 35 FORMAT!///) C C C NORMALIZE POTENTIALS TO KBOLTZ'T/Q AND COMPUTE CONSTANTS C 400 VTHERM=KBOLTZ*T/Q CTUNL=2.0D0'DSQRT(2.0D0ME)/HBAR"0S0RT(0) CFD1=PI*PI/6.0D0 EPSIS=KS*EPSIO EPSII=KI*EPSIO NC=NI*DEXPL(EGAP/(2.000*VTHERM))*DSQRT(NCONV) NV=NI'DEXPL(EGAP/(2.ODO * V THERM))/OSORT(NCONV) EGAPI=EGAPI/VTHERM EGAP=EGAP/VTHERM CHISC=CHISC/VTHERM CHISV=EGAP*CHISC PHIM=PHIM/VTHERM C C C C TNLCM=-C TUNL * D'DSOR T(MISTAR•V THERM) c C c CTNLVM=-CTUNL *D*DSORT(MISTAR*VTHERM) 1  1  ,  2  D a t i n g of  A. -»*  98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 128 127 128 129 130 131 132 133 134 135 138 1-37 138 139 140 141 142 143 144 145 148 147 148 149 150 151 152 153 154 155  MIS9.S at C  23:12:01 on OCT  10. 198B for  NN0=ND PNO=NI'NI/ND PHI0=0L0G(NC/N0> PSISI0=0LOG(NO/PNO) CJVMaAH'T'T CJCM=AE*T«T CBAR=PHIO*CHISC-PHIM CPSII=0/EPSII/VTHERM CQS=DS0RT(2.0D0*KB0LTZ'T*EPSIS) IFINTRAP .EQ. 0) GOTO 800 00 500 I=1,NTRAP RHO(I)=RHO(I)/VTHERM CN(I)=SIGMAN(I)'VELTH CP(I)=SIGMAP<I>*VELTH NMI)=NC'DEXPL(RHO(I)-EGAP) P1U)=NVDEXPL(-RH0(I))  CC1d=KfHU on Q  500 C C C REAO IN VOLTAGE V AND A STARTING ESTIMATE FOR PHI C 800 REAO(KR,FREE.END=1400) V.PHI V=V/VTHERM PHI=PHI/VTHERM C U=V-PHl C GIVEN PHI. COMPUTE A STARTING ESTIMATE FOR PSIS CALL FPSIS C C CALL SSM TO FINO A SOLUTION FOR THE COUPLED POTENTIAL C AND HOLE CURRENT CONTINUITY EOUATIONS C NTIMES=0 X(1)=U X(2)=PSIS 00 700 1=1.3 Y(1.I)=U Y<2.I)=PSIS 700 YU.I)=Y(I.I)*DELY NEWY=.TRUE. CALL SSM(X.F.2.0.ERR1.ITMAX2.C0MPF.NEWY.NENA,NEWB.IFAIL.ft8O0> 800 U1=X(1) PSIS1=X(2) NE»»Y=. FALSE. C CALL SSM AGAIN TO OBTAIN AN ESTIMATE OF THE ERROR IN THE SOLUTION CALL SSM(X.F,2.0.ERR2.ITMAX2.C0MPF,NEMY.NEWA.NEWB.IFAIL,*.90O> C C PREPARE FOR OUTPUT OF RESULTS C 900 U=X(1) PSIS=X(2) PHIER=DABS(U-U1)'VTHERM PSISER=DABS(PSIS-PSIS1)'VTHERM PHI=V-U U0=U*VTHERM C CALL FOS C  Page  3  158 MIS9.SJVaTt=022: .0D102:01 on OCT 10. 1988 f o r CC1d=KCHU on G 157 JCT=0.0D0 158 JMT=0.000 1 5 9 0 S S=T0R.0A 0P0 .EQ. 0) GOTO 1300 1 8 0 I F ( N 161 CVALEN=PHIO-EGAP*PSIS 1 8 2 00VAI=L1EN .NT*R RH A0P(I)*V 163 Z0E0TA1=2C 184 FM=1.0DO/<1.0D0*DEXPL(ZETA) 1 6 5 M==N SUSRU FR*FC N11<I)IM *P1(I)*CP(I)*FM/TAUM(I) 166 FFT0TN NU M (N *N C ' N(I)*(PSURF*P1(I))*CP(I)•1.0OO/TAUM(11 167 FT=FTNUMF/TONM 1 OSN I)0*N .EQ. 16 88 9 QIFSU S=AO S-R F(T T(I)1) GOTO 1000 170 GOTO 1100 1 7 1 1 0 0 0 QSS=QSS+ .0D T)*Q*NT(I) FT-PKDMI.ODO-FT)) 172 1100 JVTsJVTtO' N(1Tl D0'C-FPdJMPSURF' 173 JCT=JCT*Q*NT(I)«CN(I)*(NSURF•I 1.OOO-FT)-N1(I)*FT) 174 1200 JMT=JMT*Q*NT<I)•(FT-FM)/TAUM(I) 1 78 5 C1300 PSIIsCPSIIMQS+QSS^OFIX) 17 177 ETAMC=-(PHIO*PSIS«V) 1 7 8 C 1 C 17 890 C 180. 3 B RC = CHISC • PSII/2. 0D0 160.4 THCM e OEXPL(CTNLCMO ' SQRT(BRCn LO 1180. 81 6 U PW PEE RR(1(1)) == 10..20002 181. L0P W R*(22)) = =B 0.R0C0 181.1 2 U PEER 181. 3 D O UI ==CO MLI*NIFDU NICM,LOWER.UPPER.2,20000.0.DO.1.0-3.IER) 181. 4 J C M J C M O U 181. C 1825 C 1 843 C 18 CALL FJVM 1 8 5 C 186 CALL FJPN 187 C 1 885 " JTJC OO T=LJC M*JV M*JM T JPN • JUPC 188. L = J C M • 188.7 VSE = (ETAV-ETAMV) • VTHERM 189 C 190 UO=UV ' THERM 191 VO=VVTHERM 192 PHIO=PHI*VTHERM 193 PSISO=PSIS * V THERM 194 PSIO=PSI*VTHERM 195 WRITEUP.40) VO. JTOT. JCOLL. JUPC , IFAIL . IER 1 9 6 4 0 F M7A.3TX.(IX .' V=',F8.4,3X,J=' 'EMITTER 014.7.3X.'C OLLECTOR J='. 198.5 f OR014. 'SOURCE ,D14.7.3X.'J='I.FAIL=' .14.3X.' IER='.14/) 197 WRITEUP.45) UO.PHIO.PHIER.PSISO.PSISER.PSIIO 198 45 FORMAT!IX.'U='.D14.7.2X.'PHI='.D14.7.2X,'(PHIER=",D10.3,')',2X, 199 ••PSIS='.D14.7,2X,'(PSISER=',D10.3.')'.2X.'PSII=.D14.7/) 200 WRITE(LP.50) PSURFN . SURFE . TACE . TAVE , TAMCE , TAMV 201 50 FORMAT!IX,'PSURF=',D14.7,2X.'NSURF='.D14.7.2X,'ETAC='.F13.6.2X. 202 #'ETAV='.F10.6.2X,'ETAMC='.F10.6.2X.'ETAMV='.F10.8/)  L i s t i n g of  ,  •-*  5  ,  Page  L i s t i n g o f MIS9.S a t 22:12:01 on OCT 10. 1988 f o r CC1d=KCHU on 0  _ £ O  203 204 205 206 207 207.5 208 209 210 211 212 213 214 215 218 217 218 219 220 221 222 223 224 225 228 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 248 247 248 249 250 251 252 253 254 255 256 257 258 259  WRITE!LP.5S) JCM.JVM.JCT.JVT.JMT.JPN FORMAT!1X.'JCM=',D14.7.2X.'JVM=',D11.4.?X.'JCT='.011.4.2X, #'JVT='.D11.4.2X. JMT='.D11.4.2X.'JPN='.011.4/) WRITEUP.80) THCM, THVM, OS, VSE 60 FORMAT!IX,'THCM=',D11.4,2X,'THVM=',D11.4,2X,'QS=',D14.7.2X. • 'SOURCE-EMITTER V=',014.7////)  55  1  C  GOTO 600 C C 1400 STOP END BLOCK DATA C IMPLICIT REAL*8 (A-Z) INTEGER CNTRL1.ITMAX1.LP.NTIMES C COMMON /AREA*/ CBAR.CHISC.CHISV,CJCM. # CJVM.CPSII.COS.CTNLCM.CTNLVM.EGAP. # JOO.JORG.JUPC.NC,NO.NNO.NV.Q.PHIO,PNO. • PSIS10,VTHERM,OFIX COMMON /AREA4/ CNTRL1.ITMAX1.LP.NTIMES C DATA EGAP/1.079862456D0/.0/1.80210-19/.LP/8/ C END SUBROUTINE COMPF(X.F) C C THIS ROUTINE COMPUTES THE RESIDUE OF THE POTENTIAL AND C HOLE CURRENT CONTINUITY EQUATIONS C IMPLICIT REAL*8 <A-Z) INTEGER CNTRL1.1,IADNR.ITMAX1.LP.NTIMES,NTRAP C DIMENSION X!2),F(2),TAUM!100),RHO(100), # NT!100).CN!100).CP!100).N1(100).P1(100).IADNR! 100) C COMMON /AREA 1/ CBAR,CHISC.CHISV,CJCM, f CJVM.CPSII.COS.CTNLCM.CTNLVM.EGAP, # JOO.JORG.JUPC.NC.ND.NNO.NV.Q.PHIO.PNO. I PSISIO.VTHERM.QFIX COMMON /AREA2/ NT,RH0.N1.PI.CN,CP,TAUM.IADNR.NTRAP COMMON /AREA3/ ETAC.ETAMC.ETAMV.ETAV.JVM.JPN.NSURF.PHI,PSII.PSIS. f PSURF.OS,THVM.THCM.U.V COMMON /AREA4/ CNTRL1.ITMAX1.LP.NTIMES C NTIMES=NTIMES*1 U=X!1) PHI=V-U PSIS=X!2) C CALL FQS C QSS=0.0D0 JVT=0.0D0 IF!NTRAP .EO. 0) GOTO 300 CVALEN=PHIO-EGAP*PSIS DO 200 1=1.NTRAP  Page  5  L i s t i n g o f MIS9.S at 22:12:01 on OCT 10. 1988 f o r CCId-KCHU on 0  ^ 5* O  260 281 282 28 3 284 285 268 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 285.05 285.1 285.15 285.2 285.25 285.3 285.35 285.4 285.45 285.5 285.55 285.6 285.65 285.7 285.75 285.8 285.81 285.82 285.83 285.84 285.85 285.86 285.87 285.88 285.89 285.9 285.91 285 92 286 287 288 289  100 200 C 300 C  PSII=CPSII'(QS+OSS*QFIX) F<1)=CBAR*V+PSIS*PSII CALL FJVM CALL FJPN F(2)=JVM-JPN-JVT  C  C 400 C C  100  200  C C  ZETA=CVALEN*RHO(I)*V FM=1.0DO/(1.0DO»DEXPL(ZETA)) FTNUM=NSURF*CN(I)*P 1 (I)*CP(I)*FM/TAUM<I) FTDNM=(NSURF*N1(I))•CN(I)•(PSURF * P 1 ( I • ) * CP(I)•1.ODO/TAUM( I) FT=FTNUM/FTDNM I F ( I A D N R U ) .EO. 1) GOTO 100 QSS=QSS-FT'0'NT(I) GOTO 200 QSS=QSS*(1.000-FT)*Q'NT(I) JVT=JVT»a'NT<I)'CP<I>'(PSURF'FT-P1<I)'<1.000-FT))  IF(CNTRL1 .EQ. 0) GOTO 400 UO=U'VTHERM PSISO=PSIS'VTHERM WRITE(LP.5> NTIMES,UO.PSISO.F(1),F(2) 5 FORMAT(1X,I3.2X,D23.18.2X.D14.7.2X,2(D14.7.2X>) RETURN END DOUBLE PRECISION FUNCTION FUNCM(N.X) IMPLICIT REAL'S (A-Z) INTEGER N DIMENSION XIN) COMMON /AREA 1/ CBAR. CHISC. CHISV. CJCM. I CJVM. CPSII. COS, CTNLCM. CTNLVM, EGAP. t JOD, JORG. JUPC, NC. ND, NNO, NV, 0, PHIO. PNO. # PSISIO. VTHERM. QFIX COMMON /AREA3/ ETAC, ETAMC. ETAMV. ETAV. JVM, JPN, NSURF, PHI, # P S I I . PSIS. PSURF. OS. THVM. THCM, U. V COMMON /AREA6/ BRC ETTO = X(1) • X(2) - ETAC FSO = O.DO IF (ETTO .GE. 150.00) GO TO 100 FSO = 1.D0/M.D0 • DEXPL(ETTO)) CONTINUE ETTM = X(1) • X(2) - ETAMC FM = O.DO I F (ETTM .GE. 150.DO) GO TO 200 FM = 1.D0/(1.D0 • OEXPL(ETTM)) CONTINUE FUNCM = (FSO - FM) • OEXPL(CTNLCM'DSQRT(BRC-X ( 2 ) ) ) RETURN END SUBROUTINE FQS  C C C  THIS ROUTINE COMPUTES THE CHARGE OS STORED ON THE SEMICONDUCTOR  Page  8  Page  L i s t i n g of MIS9.S at 22:12:01 on OCT 10. 1988 for CC1d*KCHU on Q 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 328 327 328 329 330 331 332 333 334 335 336 337 338 338. 1 338.2 339 339.05 339. 1 339.15 339.2 339.25 340  IMPLICIT REAL*8 ( A - Z ) INTEGER CNTRL1,ITMAX1.LP.NTIMES COMMON /AREA1/ CBAR.CHISC.CHISV.CJCM. CJVM.CPSII.COS.CTNLCM.CTNLVM.EGAP, JOD.JORG,JUPC.NC.ND.NNO.NV.Q.PHIO.PNO. PSISIO.VTHERM,QFIX COMMON /AREA3/ ETAC.ETAMC.ETAMV.ETAV.JVM,JPN.NSURF.PHI,PSII,PSIS. # PSURF.OS.THVM.THCM.U.V COMMON /AREA4/ CNTRL1.ITMAX1.LP.NTIMES # # »  PXN=PNO*OEXPL(PHI) NXN=NNO*PXN-PNO ETAC=-(PSIS*PHIOI NSURF=NC*FD102(ETAC) NNSURF=NC*FD302(ETAC) ETAV=-(EGAP-PHI0-PSIS-PHI) PSURF=NV*FD102(ETAV) PPSURF=NV*FD302(ETAV) ARGMNTaNSURF-NXN*PSURF-PXN*N0*PSIS IF(ARGMNT . G E . O.ODO) GOTO 100  C 10O  W R I T E U P . 5 ) ARGMNT 5 FORMAT(IX.'WARNING: SQUARE OF SURFACE FIELD IS ARGMNTsO.000  NEGATIVE',5X,011.4)  OS=COS'DSORT(ARGMNT) I F ( P S I S . L T . O.ODO) 0S=-QS RETURN END SUBROUTINE FJVM THIS ROUTINE COMPUTES THE CURRENT FLOW JVM BETWEEN THE VALENCE BAND ANO THE METAL IMPLICIT REAL'S ( A - Z ) COMMON /AREA 1/ CBAR,CHISC.CHISV,CJCM, CJVM.CPSII.COS.CTNLCM.CTNLVM.EGAP. JOD.JORG.JUPC.NC.ND.NNO.NV.Q.PHIO.PNO. PSISIO.VTHERM.QFIX COMMON / A R E A 3 / ETAC.ETAMC.ETAMV.ETAV.JVM.JPN.NSURF.PHI.PSII.PSIS. # PSURF.OS.THVM,THCM.U.V  « # #  ETAMV=-(EGAP-PHIO-V-PSIS)  BRV = CHISV - P S I I / 2 . D 0 BRVSR = DSORT(BRV) THVM=DEXPL(CTNLVM* BRVSR) THVMDt = -CTNLVM * THVM /(2.D0*BRVSR) THVM02 = -CTNLVM*(-CTNLVM/BRV • 1.00/(BRVSR**3))*THVM/4.DO JVMO = CJVM * THVM *(FDI1(ETAMV)-FDI1(ETAV)) JVM1 = CJVM *THVMD1*(F0I2(ETAMV)-FDI2IETAV)) JVM2 = CJVM *THVMD2*(FDI3(ETAMV)-FDI3(ETAV))  (  7  Listing 341 ofCMIS9.S at 22:12:01 on OCT 10. 1988 for CC1d-KCHU on Q 342 C 343 C 344 C 345 C 346 C 347 C 348 C 349 JVM= V JMO • JVMt * JVM2 350 RETURN 351 END 352 SUBROUTINE FJPN 353 C 354 C THIS ROUTINE COMPUTES THE MINORITY CARRIER HOLE CURRENT FLOWN IG 355 C INTO THE SEMC IONDUCTOR 358 C 357 IMPLICIT REALS' (A-Z) 3 5 8 C 3 5 90 COMMON /AREA1/ CBC A RIC .CO HSISC CT.C H ISV .CC JC M .VME 3 6 f C J V M . P S . . N L C M . T N L .H GA P.P.NO. 3 6 1 # J O D . J O R G . J U P C . N C . N O . N N O . N V . Q . P I O 3 2 #COMMON /AREA3/ PSISIO V .ATC H.EETRAM O .CF.EIXTAMV.ETAV.JVM,JPN.NSURF.PHI,PSII.PSIS. 336 6 3 E T M 6845 C # PSURFO . ST . HVMT . HCMU .V . & 3 K9 3 676 PIFtPSIS SISI=PSISIO -P. H I PSISCfl=PSISI 3 8 G T PSISI) 388 IFIPSIS .LT. PSISI) PSISCR=PSIS 369 IFCPSISCR LT. OO . DO) PSISCR=O0.D0 3 7 0 C 371 JRG=JORGD ' SQRT(PSISCR/PSISIO)('DEXPL(PHI/2.ODO)-1.ODO) 372 C 373 JD=J0D'(DEXPL(PHI)-1.0D0) 374 C 375 JPN=JRG*J0-JUPC 3 7 6 R ED TURN 377 EN 3 79 8 C SUBROUTINE FPSIS 3 7 3 81 0 CC THIS ROUTINE COMPUTES PSIS GIVEN PHI. ASSUMN I G NO SURFACE STATES 3 8 382 IMPLICIT REALS' (A-Z) 3 8 3 I NTEGER CNTRL1.I.ITMAX1.LP.NTIMES 384 C 385 COMMON /AREA1/ CBAR.CHISC.CHISV.CJCM. 386 # CJVMC . PSIC . OSC . TNLCMC . TNLVME . GAP, 387 # JOO,JORG.JUPC.NC.NO.NNO.NV.O.PHIO.PNO. 388 # PSISIOV . THERMO . FIX 389 COMMON /AREA3/ ETAC.ETAMC.ETAMV,ETAV,JVM,JPN,NSURF,PHI.PSII.PSIS. •COMMON /AREAP RFN O . TSR T .L H1V .AH .V . IMES 339910 4/SUC I.TM MT XC 1L .M PN .U T 339923 C CBARV=-(CBAR*V) 394 IF(CBARV .GT. OO . DO ) GOTO 100 395 IF(CBARV .LT. OO . DO) GOTO 200 3 9 6 P S I S = 0 . 0 0 0 397 RETURN 398 100 PSISLO=0.000  Page 8  L i s t i n g of MISS.S at  ^ Cn  399 400 401 402 403 404 405 408 407 408 409 410 411 412 413 414 415 418 417 418 419 420 421 422 423 424 425 420 427 428 429 430 431 432 433 434 435 438 437 438 439 440 441 442 443 444 445 448 447 448 449 450 451 452 453 454 455 456  200 C 300  400 500 C  C C C C C C  C 100 C 200  C C C C C C  22:12:01 on OCT 10,  1988  for CC1d=KCHU on 0  PSISHI=CBARV GOTO 300 PSISLO=CBARV PSISHI=0.0D0 DO 500 I=1.ITMAX1 PSIS=(PSISLO*PSISHI)/2.0DO CALL FOS PSII=CPSII'OS F1=PSIS*PSII-CBARV IF(F1 .EQ. O.ODO) RETURN IFJF1 .GT. O.ODO) GOTO 400 PSISLO=PSIS GOTO 500 PSISHI=PSIS CONTINUE RETURN END DOUBLE PRECISION FUNCTION FDI(ETA) FERMI-OIRAC INTEGRAL OF ORDER ONE IMPLICIT REAL'S (A-Z) INTEGER INDEX DIMENSION FOFLA(81),F0FLC(81),FOFLE(81).FDFLF(811,F0FLG(81) COMMON /AREA5/ CF01,FOFLA.FDFLC.FDFLE.FDFLF.FOFLG IF(ETA .LT. -4.ODO) GOTO 100 IF(ETA .GT. 4.ODO) GOTO 200 X=(ETA*4.ODOI/O.100+0.500 IN0EX=X*1 ETA0=DFLOAT(INDEX-41)'0.1D0 0ELETA=ETA-ETA0 DXPETA=DEXPL(ETAO) FD1=FDFLF(INDEX)+DELETA*(DLOG(1.ODO*DXPETA)• # 0ELETA/2.000/(1.000*1.ODO/DXPETA)) RETURN F01=DEXPL(ETA) RETURN F01=-DEXPL(-ETA)*ETA'ETA/2.000*CFD1 RETURN END DOUBLE PRECISION FUNCTION FD102(ETA) FERMI-DIRAC INTEGRAL OF ORDER ONE-HALF IMPLICIT REAL'8 INTEGER INDEX  (A-Z)  DIMENSION F 0 F L A ( 8 1 ) . F 0 F L C ( 8 1 ) , F 0 F L E ( 8 1 ) , F D F L F ( « 1 ) . F 0 F L 0 ( 8 ) > COMMON /AREA5/ CFO1.FOFLA.FOFLC.FOFLE.FOFLF.FOFLG  Page  9  L i s t i n g o f MIS9.S a t 22:12:01 on OCT 10, 198B f o r CC1d=KCHU on Q 457 458 459 480 461 482 463 484 465 466 487 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 497. 01 497. 02 497..03 497..04 497. 05 497. 08 497. 07 497. 08 497. 09 497. 1 497. 11 497. 12 497. 13 497. 14 497. 15 497. 16 497. 17  C 100 C 200  C C C  FERMI-DIRAC  C C  C 100  50  100  C C  F0102=0EXPL(ETA) RETURN FD102=-DEXPL(-ETA)*ETA'ETA/2.0D0*CFD1 RETURN END DOUBLE PRECISION FUNCTION F0302(ETA)  C  C 200  IF(ETA .LT. -4.ODO) GOTO 100 IF(ETA .GT. 4.000) GOTO 200 X=(ETA*4.0DO)/0.100*0.500 INDEX=X*1 ETAO=DFLOAT(INDEX-41)'0.1D0 DELETAsETA-ETAO FD102=FDFLE(INDEX)+DELETA*(FDFLC(INDEX)+DELETA/2.ODO'FDFLA(INDEX)) RETURN  INTEGRAL OF ORDER THREE-HALVES  IMPLICIT REAL'S (A-Z) INTEGER INDEX DIMENSION FDFLA(81).FDFLC(81).FDFLE(81),FOFLF(81),FDFLG(81) COMMON /AREA5/ CFD1.FDFLA.FDFLC.FDFLE.FDFLF,FDFLG IF(ETA .LT. -4.000) GOTO 100 IF(ETA .GT. 4.000) GOTO 200 X=(ETA»4.0D0)/0.100*0.500 INDEX=X*1 ETA0=DFL0AT(INDEX-41)'0.1D0 0ELETA=ETA-ETA0 FD302=FDFLG(INDEX)*DELETA*(FDFLE(INDEX)*0ELETA/2.000'FDFLC( INDEX)) RETURN FD302=DEXPL(ETA) RETURN FD302=-0EXPL(-ETA)*ETA'ETA/2.00O*CFD1 RETURN END DOUBLE PRECISION FUNCTION FDIKX) IMPLICIT REAL*8 (A-Z) DATA PI/3.14592B54DO/ Y = X IF(Y .LE. O.DO) GOTO 50 Y = -V Z = 1.00'DEXPHY) -. 250052'OEXPL (2.'V) * . 111747'DEXPL (3.' V) • - 084557'DEXPL(4.'Y) •.040754'DEXPL(5.*Y) # -.020532'DEXPL(6.*Y) •.005108'DEXPL(7.'Y) IF(X .LE. O.DO) GOTO 100 Z = -Z • (X"2)/2.DO • (PI"2)/8.D0 CONTINUE F0I1 = Z RETURN END  -—--»>-  r f .nfwifw  Listing of Mlsg.S at 22:12:01 on OCT 10. 19S8 for CC1d=KCHU on 0 497. 18 DOUBLE PRECISION FUNCTION FDI2(K) 497. 19 IMPLICIT REALS' (A-Z) 449* 97 .2 . D ATAXPI/3.14592654D0/ . 2 1 Y 497..22 IF(Y .LE. 0.00) GOTO 50 497 .23 Y = -Y 497.24 50 Z = I.DOD ' EXPL(Y) -.1Z504B'DEXPL(2.'Y> •.037842D ' EXPL(3 497 .25 • - 018183'DEXPL(4.*Y) • .012484'DEXPL(5.'Y) 497.28 # -.007486*0EXPL(8.*Y) • .002133'DEXPL(7.'Y) 497.27 IF(X LE. 0.00) GOTO 100 497 28 Z = Z • <X"3)/6.00 • <PI"2)'X/6.D0 497 .29 100 CONTN I UE 449977 .3 F0I2 .31 RETURN= Z 497 32 END 497. 33 C 497 .34 C 497 35 DOUBLE PRECISION FUNCTION FDI3U) 497. . 3 6 IM ITI/3.1R4E5A (A-Z) 497..37 D ATPALICP 92L6S' 54D O/ 497. 3 8 Y = X 497. 39 IF(Y= .LE. 0.00) GOTO 50 497. . 4 Y Y 497..41 50 Z = I.OOD ' EXPL(Y) -.082592'DEXPL(2.*Y) •.013661D ' EXPL(3 497 .42 • -.009796*DEXPL(4.*Y) •.012978*0EXPL(5.*Y) 497 ..4443 fIF1X .LE. -.O10859' EXPL(6. 497. 0.00)DG OTO 1*0Y) 0 •.003448'DEXPL(7.*Y) 497. ..4485 Z - -Z36•0.00(X"4)/24.D0 • ((PI'X)"2)/12.D0 • 7.'(PI"4)/ 497. * 497..47 100 CONTN I UE 4 9 7 . 4 8 F D I 3 497..49 RETURN= Z 497..5 END 496 DOUBLE PRECISION FUNCTION DEXPL(X) 459090 C IMPLICIT REALS' (A-Z) 501 C 501.5 IF(X .GE. 150O . DO) GOTO 200 550023 IF(X .LT. 1 5 0 DEXPL=DEXP(X).ODO) GOTO 100 550045 100 D REETX UPRLN =OO . DO 506 RETURN 506. 3 200 0EXPL=OEXP(150.000) 506.8 RETURN 507 ENO B  Page 11  L i s t i n g of MISETS.S at 22:05:34 on OCT 10. 1988 for CC1d=KCHU on 0 1 1 1 1 1 1 1 1 1 2 3  THIS PROGRAM IS WRITTEN TO GENERATE THE STEADY-STATE CHARACTERISTICS OF THE MISETS  MODIFY 132-133. 85-69, 78. 108-108. 127-130 NOTE THE DABS IN 129 ADD JDO TO AREA 1, ALSO ADO JRNB AND CHANGE JN TO JNE IN AREA1 AOO AREA4, INCLUDE GAIN FEEDBACK BV BASEWIDTH MODULATION NOTE LEB IN AREA4 MODIFY 31. 40. 66. 104. 108. 133 MODIFY 82 (GET A CONTINUOUS CURVE) USE EXACT INTEGRAL FOR ELECTRON TUNNEL CURRENT (17. 170. 174-180. 188)  4 9  8 7 7.2 7.4 7.9 7.8 8 9 9.5 10 11 12 13 15 IB 17 17.5 17.7 18 19 20 21 22 23 24 25 25.3 25.8 28 27 28 29 30 31 32 33 34 35 38 37 37.5 37.7 38 38.5 38.7  MOOIFY 82 (BET A CONTINUOUS CURVE) USE EXACT INTEGRAL FOR ELECTRON TUNNEL CURRENT (17. 170. EXTERNAL FUNCM 174-180. 188) IMPLICIT REAL'S (A-Z) INTEGER FREE. I . J . IFAIL. ITMAX1. ITMAX2 LOGICAL NEWY. NEWA. NEWB EXTERNAL COMF. FUNCM DIMENSION FREE(I). LABEL!15), X(2). F(2) COMMON /AREA4/ WBASE. OELWB. DELWBO. LEB COMMON /AREA5/ XBRC. XNALP1, XNALP2 COMMON /AREA1/ XCHtM. XCHIE, XEGB. XPHIO. JDO JD. JB. OFIX, • KT. XCHIH. JNO. JRO. JGO. XVBI. XVCE, JNE JRNB. JR. JG COMMON /AREAS/ NB. PB. NC, NV, COS. CPSII. OS COMMON /AREA3/ JTN. JTP, GAMMA, ASTOC, ASTOV, XPSII COMMON /AREA4/ WBASE. DELWB, DELWBO. LEB COMMON /AREA5/ XBRC, XNALP1, XNALP2 COMMON /SESSOM/ A(20,22). 8(20). YI22.21) NC = 2.8D25 NV•« 1.04025 DATA FREE/'"/. NEWY/.TRUE./. NEWA/.FALSE./, NEWB/.FALSE./. • NI/1.45018/. KBOLTZ/1.38066D-23/. 0/1.802180-19/. • EPSIO/8.85418D-12/. • PI/3.141592654D0/, ME/0.91095D-30/. H/6.82B17D-34/  10 15  NC a 2 . 8 0 2 5 NV B 1.04D25 READ(5.5> ( L A B E L ( I ) . I B 1 , 1 5 ) FORMAT!15A4) R E A D I 5 . F R E E ) ITMAX1, ITMAX2, E R R 1 . ERR2. DELY WRITE(6.10) (LABEL(1).I=1.15) FORMAT!'1'.20X, 15A4///) READ!3,FREE) T. EGB, CHIM, C H I E FORMAT(2X,'To', F 6 . 2 . 2 X , •EOBB',F4.2.2X, C H I M » ' , F 4 . 2 . 2 X . f 'CHIEB'.F4.2/) ,  Paga  1  Listing of MISETS.S at 22:05:34 on OCT 10. 1988 for CCId-KCHU on Q 39 READ'3.FREE) MCDME. MVDME. MIDME 39.3 20 FORMAT!2X,'MCDME=',F4 . 2. 2X , MVDME"',K4.2.2X. 'MI0ME=',F4.2/) 40 READ!3.FREE) NCOLL. PB. WBASE, 01. LEB 40.3 25 FORMAT!2X.'NCOLL='.09.3.2X, 'PB=',D9. t.2X. 'WBASE *'.D9.3.2X, 40.6 I 'DI = '.D9.3.2X. ' LEB=',09.3/i 1  46  47 48 49 50 51 62 53 54 55 56 57 58 59 60 81 61 62 63 64 65 68 66 67 68 89 69 70 71 72 73 73 73 74 74 75 75 76 78 76 77 77 78 79  C 10 15 20 25 30 35 5 40 C C C 5  5  3 6  SO  5  60  5  65 C 70  3 6 5  WRITE<8.10) (LABEL!I).I=1.15) FORMAT! ' 1' ,20X, 15A4//7) WRITE(6,15) T, EGB. CHIM. CHIE F0RMAT(2X.'T='. F8.2.2X. 'EGB=',F4.2,2X. 'CHIM=',F4.2.2X. • *CHIE=".F4.2/) WRITE<8,20) MCDME. MVDME, MIDME FORMAT!2X,*MCDME=',F4.2,2X. 'MVOMEo'.F4.2.2X. 'MIDMEo-,F4.2/> WRITE(6.25> NCOLL. PB. WBASE, DI, LEB FORMAT!2X,*NCOLL=',09.3,2X, •PB=*,09.3.2X. 'WBASE=',09.3,2X, * 'Dl=-.09.3,2X. 'LEB='.09.3/) WRITE(6.30) DB. OP. TAUCP. TAUR, TAUG FORMAT!2X,'08=',09.3,2X, '0P=',D9.3.2X, 'TAUCP=*.09.3,2X. * 'TAUR='.09.3,2X, 'TAUG=',09.3/) WRITE(6.35) NEPSII. NEPSIS, OFIX FORMAT!2X,"NEPSI1=',F5.2,2X; 'NEPSIS=',F5.2,2X. *OFIX=',010.4/) COS = 0SQRT!2*KT* EPSIO • NEPSIS) WRITE(8.40) JB FORMAT!2X,'JB=',010.4///)  JNO = (Q * OB * NI * NI / PB) / LEB KT = KBOLTZ * T VTHERM = KT / 0 XEGB = EGB / VTHERM DELWBO a DSORT!2*NEPSIS*EPSI0*(NCOLL/tPB*(NCOLL*PB)))'VTHERM/Q) XCHIM = CHIM / VTHERM XCHIE * CHIE / VTHERM XCHIH e XCHIE • XEGB MC = MCDME * ME WRITE(8,50) VTHERM FORMAT(2X.'VTHERM='.09.4/) MV = MVDME * ME FORMAT(2X,'NB=',09.4,2X, XPHIOe".09.4.2X. 'CPSII=',D9.4/) MI = MIDME ' ME FORMAT(2X,'GAMMA=',09.4.2X. •ASTOC=',D9.4,2X. •ASTOV=",D9.4/) 1  F0RMAT!2X.'JDO='.010.5,2X. 'JNO=',010.5,2X, 'JRO=',010.5.2X. # JGO='.D10.5.2X, •OELWBOs'.010.5/) 1  C 75 C  FORMAT!2X.-XVB^* .D9.4///) NB = NI • NI / PB  I'aga  L i s t i n g o f MISETS.S a t 22:05:34 on OCT 10. 1986 f o r CC1d=KCHU on 0 80 81 82 82. 1 82. 2 82. 5 82. 7 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 104..5 104 .6 104 .7 104 .8 104 .86 104 .92 105 106 108 .5 106 .6 106 .7 107 107 .3 107 .6 108 108 3 108 6 108. 8 108. 83 108. 86 108. 9 109 110 111 112 113 114 115 116  80  C  XPHIO = DL0G(NV/PB) CPSII a DI / (NEPSII'EPSIO) / VTHERM COS = DSQRT(2*KT* EPSIO * NEPSIS) GO TO 90 READ!5.FREE.END=340) VCE PHIO = PHI PSISO a PSIS GAMMA a 4 * PI * DI * 0SQRT(2*MI) / H ASTQC = (4'PI/H) ' (Q/H) * (MC/H) • ( K V 2 ) ASTOV = (4*PI/H) ' (Q/H) * (MV/H) • (KT*'2) JDO = 0 • DSQRT(DP/TAUCP) * N l • N l / NCOLL JNO s 0 * DB * N l * N l / PB JNO = (0 * DB • N l * N l / PB) / LEB JRO = Q * N l • DSQRT(2 NEPSIS EPSI0*VTHERM/(Q PB)) / TAUR JGO = Q • N l • 0SQRT(2'NEPSIS EPSI0M(NCOLL*PB)/(NCOLL*PB)) # 'VTHERM/Q) / TAUG DELWBO = DSQRm'NEPSIS'EPSIOMNCOLL/lPBMNCOLL+PBMl'VTHERM/Q) XVBI = XEGB • XPHIO • OLOG(NC/NCOLL) ,  ,  4  ,  C C C  50 60 65 70 C  300  WRITE(6.50) VTHERM FORMAT!2X,'VTHERM=',09.4/) WRITE(8,eO) NB. XPHIO, CPSII FORMAT(2X,'NB='.D9.4,2X, XPHIO=',09.4,2X. *CPSII='.09.4/) WRITE16.65) GAMMA, ASTOC. ASTOV FORMAT(2X,'GAMMA=',09.4,2X, 'ASTQC=".09.4.2X. ASTQV='.09.4/) WRITE(6.70) JDO. JNO, JRO. JGO. DELWBO F0RMAT(2X.•JDO='.D10.5.2X. 'JN0=\D10.5.2X. 'JROs',D10.5,2X. VBE = (XEGB -XPHIO -X(2) -XCHIM -XPSII *XCHIE) • VTHERM J E = JTN • J T P JC = JN • JG • JO JC = JNE - JRNB • JG • JD CBC = 0SQRT(Q*NEPSIS'EPSIO*NCOLL'PB / (2*(NCOLL+PB)• • (VTHERM'XVBI • VCE - VBE))) • •JGO='.D10.5.2X. 'DELWBO=•.D10.5/) WRITE(6.75) XVBI FORMAT!2X,'VCEa .F5.2.2X, ' V B E B ' . D 1 1 . 5 . 2 X . 1  1  1  # 'PHIOa'.F6.3.2X,'JC=',D11.5.2X. 'PSISO='.F6.3.2X, l F A I L = ' . 1 5 / ) f JE='.D11.5.2X. FORMAT(2X,'XVBI='.09.4///) FORMAT!2X. PSI1='.011.5.2X, 'PHIa .011.5.2X. 'PHIERa'.D11.5.2X. • 'PSISs*.011.5.2X. 'PSISER=*.D11.5.2X. 'DELWB=',011.5/) 1  1  75 310  C 320  ,  1  FORMAT(2X,'QS=",011.5,2X, 'JTN=',D11.5.2X. •JTP='.011.5.2X. • •JNE='.D11.5.2X. 'JRs'.D11.5.2X, JG=',011.5,2X, # 'JDa'.D11.5/) WRITE(6,330) JRNB. CBC FORMAT!2X,"JRNB='.011.5.2X. 'CBCa'.011.5////) GOTO 80 1  330 C C C 80  READ!5.FREE) VCE. PHIO. PSISO GO TO 90 REAO!5.FREE.END=340) VCE PHIO = PHI PSISO a PSIS  Page  3  L i s t i n g o f MISETS.S a t 22:05:34 on OCT 10. 1988 f o r CC1d=KCHU on Q 118.5 117 118 119 120 121 121.2 121.5 122 123 123.3 T23.8 124 124.5 124.7 125 126 128.3 126.6 127 127.3 127.6 127.65 127.7 127.75 127.8 127.81 127.82 127.83 127.84 127.85 127.86 127.87 128 129 130 130.5 130.7 131 132 133 133.03 133.06 133.1 133.2 133.3 133.4 133.5 133.6 133.7 133.8 134 135 136 137 138 139 140  90  INTEGER FREE XVCE = VCE / VTHERM Y l l . l l = PHIO / VTHERM VI2.1) = PSISO / VTHERM  C  210 C CO 220 C CO C C CO C CO C  C 240 C  C C COO C C C10 C C C20 280  00 220 1=2.3 COMMON /AREA4/ WBASE. DELWB. DELWBO, LEB DATA FREE/'*'/ DO 210 J=1.2 Y ( J , I ) * Y U . 1) WflITE(8,60) X ( 1 ) , X(2) FORMAT(2X,'X(1)=',D11.5.2X, X(2)=',D11.5/) Y d , I ) = Y d . I ) • DELY WRITE(8,70) FORMAT(2X.'STEP FOS*) 1  NEWY • .TRUE. WRITE(8.80) FORMAT(2X.'STEP FJTN') CALL SSM(X,F.2.0.ERR1.ITMAX1,COMF.NEWY,NEWA,NEWB.IFAIL.&240) WRITE(6.90) FORMAT(2X, 'STEP FJTP') XVCB = XVCE-XEGB*XPHI0*X(2)*XCHIM*XPSII-XCHIE SRXVCB = DSORTlXVBI+XVCE-XEGB+XPHIO*X(2)*XCHIM*XPSII-XCHIE) SRXVCB = DSORT(XVBI*XVCB) DELWB = DELWBO • SRXVCB WBEFF = WBASE - DELWB ITANHY a 1/ OTANH(WBEFF/LEB) ISINHY a 1/ OSINH(WBEFF/LEB) EXPX1 = D E X P L ( X ( D ) - 1 EXPXCB = DEXPL(-XVCB) - 1 JRNB = JNO * tITANHY - ISINHY) * (EXPX1 • EXPXCB) JNE = JNO * (ITANHY • EXPX1 - ISINHY • EXPXCB) XIPHI = X( 1) XIPSIS = X(2) JD = -JDO • (DEXPL(XEGB-XPHI0-X(2)-XCHIM-XPSII*XCHIE-XVCE)-1) JD a -JDO * EXPXCB NEWY = FALSE. CALL SSM(X,F.2.0.ERR2.ITMAX2,COMF,NEWY,NEWA,NEWB,IFAIL.4260) F<1) = (JNE - JTN) • 1.0-4 F ( 2 ) = JB • JG • JD - JTP - JRNB WRITE(6,100) X(1>. X ( 2 ) . F ( 1 ) . F ( 2 ) FORMAT(2X. X(1)= .D11.5.2X. X(2)=',D11.5,2X. f 'F(1)=',011.5.2X, 'F(2)a',D11.5) WRITE(6.110) JTN, JTP, JN. JR, JG FORMAT(2X,'JTN=',D11.5,2X, •JTPa',011.5.2X. •JN=',D11.5,2X. • 'JRa ,011.5,2X. 'JGa\D11.5> WRITE(6.120) OS. XPSII FORMAT{2X.'QS='.011.5.2X. •XPSII='.Dl1.5/) PHI = X(1) * VTHERM PSIS = X(2) • VTHERM PSII a XPSII * VTHERM PHIER = DABS(X(1)-XIPHI) • VTHERM PSISER a DABS(X(2)-XIPSIS) • VTHERM VBE a (XEGB -XPHIO -X(2) -XCHIM -XPSII *XCHIE) • VTHERM J E a JTN • JTP ,  ,  1  1  L i s t i n g o f MISET5.S a t 22:05:34 on OCT 10. 1988 f o r CC1d=KCHU on 0 141 142 143 144 145 148 14/ 148 149 150 151 152 153 154 155 158 157 158 159 180 181 182 183 184 185 168 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 188 187 188 189 190 191 192 193 194 195 196 197 198  C  c  JC = JN • JG • JD JC a JNE - JRNB • JG • JD CBC = DSORT(Q*NEPSIS*EPSIO*NCOLL*PB / (2*(NCOLL+PB)• 9 (VTHERM*XVBI • VCE - VBE))) WRITE(6,300) VCE. VBE. J E . JC, PHIO. °SISO. IFAIL FORMAT!2X.'VCE*'.F5.2.2X. 'VSE*'.011.5.2X.  300 « f 310 320  330 340 C C C C  C C C CO  c  CO  c  CO  c  CO C  'PHIO* .F6.3.2X. 'JC=',011.5.2X. 'PSISO='.F3.3.2X, 'IFAIL=' ,15/) 'JE='.011.5.2X. WRITE(8.310) P S I I , PHI. PHIER. PSIS. PSISER. DELISTS F0RMAT(2X.'PSII='.D11.5.2X. 'PHI = ',D11.5,2X. 'PHIER*' .D11.5.2X. 1  OS, JTN. J T P . JNE. JR. JG. JD f WRITE(6.320> 'PSIS=-'.011.5.2X. 'PSISER='.011.5,2X. 'DELWB=',011.5/) F0RMAT(2X.'0S='.011.5.2X. 'JTN='.Dl1.5.2X. 'JTP='.D11I.5.2X. 5.2X. # 'JNEa'.011.5.2X. ' JR='.011. 5. 2X. 'JG='.OM. • 'JO-*'.011.5/) WRITEJ6,330) JRN8. CBC FORMAT<2X.'JRNB=',011.5.2X. 'CBC*',011.5////) GOTO 80 STOP END  SUBROUTINE COMF(X.F) IMPLICIT REAL'S (A-Z) INTEGER FREE DIMENSION X ( 2 ) , F ( 2 ) . F R E E ( I ) COMMON /AREA 1/ XCHIM. XCHIE, XEGB, XPHIO, JOO. JO. JB, QFIX. JR. JG NC. NV,XVBI. COS. CXVCE, P S I I . JNE, OS f COMMON KT./AREA2/ XCHIH. NB. JNO.PB. JRO, JGO. JRNB. COMMON /AREA3/ JTN. J T P . GAMMA. ASTOC. ASTOV. XPSII COMMON /AREA4/ WBASE. DELWB, OELWBO. LEB DATA FREE/'*'/ WRITE(6,60) X ( 1 ) . X(2) FORMAT(2X,'X(1)='.D11.5.2X, 'X(2)=',D11.5/) CALL FOS(X) WRITE(6.70) FORMAT(2X, STEP FOS') XPSII = -CPSII * (QS+OFIX) CALL FJTN(X) WRITE(6.80) FORMAT(2X,'STEP FJTN') CALL F J T P ( X ) WRITE(6,90) FORMAT(2X,'STEP FJTP') XVCB = XVCE-XEGB*XPHI0*X(2)*XCHIM*XPSII-XCHIE SRXVCB = DSQRT(XVBI*XVCE-XEGB*XPHI0*X(2)*XCHIM*XPSII- XCHIE) SRXVCB = DSORT(XVBI*XVCB) DELWB = DELWBO * SRXVCB WBEFF = WBASE - OELWB ITANHY = 1/ DTANH(WBEFF/LEB) ISINHV = 1/ OSINH(WBEFF/LEB) EXPX1 = OEXPL(X(1)) - 1 EXPXCB = DEXPL(-XVCB) - 1  199 200 2 20 01 2 2 0 2043 2 06 5 2 0 2 0 7 2 0 8 2 0 9 2 1 0 2 1 1 212 213 214 215 216 2 21178 219 220 221 222 2 243 22 225 2 27 8 22 2 2 8 2 2 9 2 0 23 31 2 3 2 32 3 234' 2 36 5 23 237 23 39 8 2 240 241 2 24 42 3 244 2 46 5 24 247 248 249 250 251 . 252 253 254 2 56 5 25  MISETSJR.SNBa t=22:JN 05: TY 10, 98H 8Yf)o• rC C!E 1dX =K 0PXCB) O34* on(ITAO NC H - IS1IN PC XH 1U•onEX JNE = JNO * (ITANHY • EXPX1 - ISINHY • EXPXCB) C J L(X ELWB) C JN R == JRJN OO **0D S0ERXTP(D AM BS)(X/(2)()W ) B•ASE-D (DEXPL(X(1)/2)1) J Q = J G O * S R X V C B C JD = -JDO * (DEXPL(XEGB-XPHI0-X(2)-XCHIM-XPSII*XCHIE-XVCE)-1) JD = -JDO * EXPXCB CC F(1) x (JN • JG JR•- JT N)- • 1.0J-4 C F(1) F(2) • JB • J D J R TP = ( J N E J T N ) * 1 . D 4 F(2) *EJB •0) JG • X(2), JD - JF(1), TP - JR NB C W R I T I 6 . 1 0 X(1). F(2) COO F0RMAT(2X.'X!1)*'.011.5.2X. 'X(2)=',D11.5,2X. C f 'F(1)='.D11.5.2X. 'F(2)s-,D11.5) C WRITE(8.110) JTN. JTP, JN, JR, JQ CIO FORMAT(2X,'JTN=',D11.5,2X, •JTP=',D11.5.2X. 'JN»",D11.5,2X, C # 'JR='.011.5,2X, 'JG='.D11.5) C W R I T E ( 6 , 1 2 0) 'Q OS=' S.,011. XPSII C20 FORMAT(2X, 5.2X, •XPSII"",D11.5/) RETURN END C C C C SUBROUTINE FQS(X) IM IC ITNRE AL*8 (A-Z) D IMPELN SIO X(2)  L i s t i n g of  <H*5>  OMMON X/ACRH EA.1JN /O XC.H IM. JX HIEXVBI. . XEGB . XEP,HJNE. IO. JDO . JO. JB. QFIX. *C JRO. GC COMMKT. ON /AREIH A2/ NB. PB. N CO. . N V. CX OVSC . CPSII. JR ON SB. JR. JG C N UR RFF = =N NV C '•F0F 1D 021(X O*X 1)-X GB) PSSU 02(2()*X -P XPH HIIO - (X (E2)) A TMN =T N.SG UE. RF O -O L(X(1 10 ))0 • PSURF - PB • PB*X(2) IFRGM A (N RG .N DB O*)DEGX OP TO WRITE(6.5) ARGMNT 5 #FORMAT!I5X, XW '. AR NIN4G): SOURE OF SURFACE FIELD IS NEGATIVE.' D11. ARGMNT = 0.000 100 OS = -COS * DSORTA ( RGMNT) IF (X(2) .LT. O O . D O ) OS = -OS RETURN END C C C C SUBROUTINE FJTN(X) IMPLICIT REAL8' (A-Z) DIMENSION X(2). LOWER!2), UPPER!2) COMMON /AREA 1/ XCHIM. XCHIE. XEGB. XPHIO. JOO. JD. JB. QFIX, • KT. XCHIH. JNO. JRO. JGO. XVBI, XVCE, JNE. JRNB. JR. JG COMMON /AREA3/ JTN. JTP, GAMMA. ASTQC, ASTOV. XPSII COMMON /AREA5/ XBRC. XNALP1, XNALP2 C  Paga  8  L i s t i n g o f MISET5.S at 22:05:34 on OCT 10. 1988 f o r CCldsKCHU on G 257 258 259 260 281 262 263 264 265 288 267 288 269 270 271 272 273 274 275 278 277 278 279 280 281 282 283 284 285 288 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314  XNALP1 = XCHIE - XCHIM - XPSII XNALP2 = X(2) • XPHIO • X(1) - XEGB XBRC = XCHIE - XPSII/2 LOWER!1) = O.DO UPPER!1) = 1 . 2 0 2 L0WER(2) = 0.00 UPPER!2) = XBRC DOUI = DMLIN(FUNCM, LOWER. UPPER. 2. 20000. O.DO.-1.0-5. IER) JTN = ASTOC • DOUI RETURN END C C C C  DOUBLE PRECISION FUNCTION FUNCM(N.XE) IMPLICIT REAL'S <A-Z) INTEGER N DIMENSION XE(N) COMMON /AREAS/ XBRC, XNALP1. XNALP2 XNETT1 = XE(1> *XE(2) -XNALP1 FN1 = O.DO IF (XNETT1 .GE. 150.00) GO TO 100 FN1 = 1.00/(1.DO *0EXPL(XNETT1)) CONTINUE XNETT2 = XE(1) *XE(2) -XNALP2 FN2 = O.DO IF (XNETT2 .GE. 150.00) GO TO 200 FN2 = 1.D0/I1.D0 •DEXPHXNETT2)) CONTINUE FUNCM = (FN1 -FN2) 'DEXPL(-GAMMA'(XBRC-XE(2))'KT) RETURN END  100  200  C C C C  SUBROUTINE F J T P ( X ) IMPLICIT REAL*8 (A-Z) DIMENSION XI2) •  C  C C  COMMON /AREA1/ XCHIM. XCHIE. XEGB, XPHIO. JDO. JD. JB. QFIX. KT, XCHIH. JNO. JRO, JGO. XVBI, XVCE. JNE, JRNB, JR. JG COMMON /AREAS/ JTN, JTP, GAMMA. ASTOC. ASTOV. XPSII  XNBET1 = XCHIM • XPSII - XCHIE - XEGB XNBET2 = -X(2) - XPHIO BRV = (XCHIH + XPSII/2) ' KT THETVO•= DEXPLI-GAMMA'DSORT(BRV)) THETV1 = GAMMA ' THETV0/(2'DS0RT(BRV)) THETV2 = GAMMA ' (GAMMA/BRV • 1/(DSQRT(BRV))•*3) • THETVO/4 JTP = ASTOV ' (THETVO ' (FD1(XNBET2)-FOHXNBET1)) • • (KT'THETVI) * (FD2(XNBET2)-FD2(XNBET1)) # • l ( K T " 2 ) ' T H E T V 2 ) ' (FD3( XNBET2) -FD3( XNBET 1).)) RETURN END  Fags  7  Listing of MISET5.S at 22:05:34 on OCT 10. 1968 for CCH=KCHU on Q 315 C 316 C 317 DOUBLE PRECISION FUNCTION OEXPL(X) 3 1 8 IMPLIC.LT. IT RE A L.O S' DO (A-Z) 3 1 9 IF(X 1 5 0 ) GOTO 100 3 2 0 D E X P L * D E X P ( X ) 3 2 1 R ETEUXRP NL=0.000 3 2 2 1 0 0 D 323 RETURN 324 END 325 C 326 C 327 DOUBLE PRECISION FUNCTION FD1(X) 328 IMPLICIT REALS' (A-Z) 3 2 9 D ATA PI/3.14159285400/ 3 3 0 Y = X 3 3 1 IF(Y-Y .LE. OD . O) GOTO 50 3 3 2 Y 333 50 Z = 1.OOD ' EXPL(Y) -.250052'DEXPL(2.*V) • . 1 11747'0EXPL(3.*Y) 3 3 4 * . 0 64557' DEXPL(4. 'Y ) •. •0.05108' 040754'D DEXPL(7. EXPH5.'Y)*Y) 3 3 5 f -. 0 20532*0EXPL(8. ' V > 336 IF(X .LE. OD . O) GOTO 100 3 3 7 Z = -Z • (X"2)/2.DO • (PI"2>/8.00 3 3 8 1 0 0 C O N T N I U E 3 3 9 FTD IRN =Z 3 4 0 R E U 34 42 1 C END 3 343 C 34 44 D OU BLIC EITPRREECA ISIL OS'N(A-Z) FUNCTION FD2(X) 3 5 IM PL 3 47 6 D PI/3.14159265400/ 34 YATA =X 3 IF(V= .LE. 34 48 9 Y -Y 0.00) GOTO 50 350 50 Z= I.DO'DEXPL(Y) -.125046'DEXPL(2.*Y) •.037642*DEXPL< 3.*Y) 351 f -.018183*DEXPL(4.*Y) • .012484'0EXPL(5.*Y) 3 5 2 • -. 007486*0EXPL(8. 353 IF(X LE. OD . O) GOTO 1'Y00) • .002133'OEXPH7.*V) 35 4 Z =N ZE • (X"3)/8.DO • (PI"2)'X/8.DO 3 5 5 1 0 0 C O N T I U 3 578 FT DU2RN =Z 35 RE 35 59 8 C END 3 360 C 3 D OU BLIC EITPRREECAISL IO*8N(A-Z) FUNCTION FD3(X) 36 81 2 IM PL 3 64 3 D PI/3. 14159265400/ 36 YATA =X 3 66 5 IF(Y= .LE. . O) GOTO 50 36 Y -Y OD 367 50 Z= I.DOD ' EXPL(Y) - 082592*DEXPL(2.*Y) •.013881'DEXPL(3.*Y) 3 6 8 f -. 0 09798*DEXPL(4. *Y) Y) •. 0012976*DEXPL(5. **Y) 3 8 9 • -. 0 10659*DEXPL(6. * •. 03446*DEXPL(7. Y) 370 IF(X .LE. OD . O) GOTO 100 371 Z = -Z • (X**4)/24.DO • ((PI *X) * *2)/12 D . O • 7MPI"4)/ 372 f 380.00 B  L i s t i n g of MISET5.S at 22:05:34 on OCT 10. 1986 for CC1d=KCHU on Q 373 374 375 378 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405  100  C C C C  100  CONTINUE FD3 = Z RETURN ENO  DOUBLE PRECISION FUNCTION F0102(X> IMPLICIT REAL'S (A-Z) DATA PI/3.14159265400/ IF (X .LE. O.DO) GOTO 100 IF (X .LE. 2.DO) GOTO 200 IF (X .LE. 4.00) GOTO 300 GOTO 400 FD102 = DEXPL(X) -0.353568'DEXPL(2'X) *0.192439*DEXPL<3'X) • -0.122973*DEXPL(4*X) *6.077134*DEXPL(5*X) * -0.03B228*DEXPL(6 X) *0.008348*0EXPL(7*X) GOTO 500 FD102 = 0.765147 • X'(0.604911 • XMO. 189885 • # X*(0.020307 • XM-0.004380 • X*(-0.000388 • # X*0.000133))))) GOTO 500 FD102 = 0.777114 • XMO.581307 • XMO.206132 • # XM0.017680 • XM-O.006549 • XMO.000784 # X'O.000036))))) GOTO 500 XN2 = 1/(X"2) F0102 = (X'*1.5) • (0.752253 • XN2M0.928195 • # XN2M0.880839 • XN2M25.7829 • XN2M-553.636 • # XN2M3531.43 - XN2* 3254.65)))))) RETURN END ,  200  300  400  500  Page  9  Listing 1 ofCPISETP.S at 21:56:53 on OCT 10. 1988 for CC1J=iKCHU on G 2 C 3 C THIS PROGRAM IS WRITTEN TO GENERATE THF. STEADY -STATE 4 C CHARACTERISTICS OF THE UNANNEALED PNP I'ETS. 5 C 6 C 7 C DEFINE ALL VARIABLES 8 C 9 C 10 IMPLICIT REALS' (A-Z) 11 INTEGER FREE. I. J. IFAIL. ITMAX1, ITMAX2 12 LOGICAL NEWY, NEWA. NEWB 13 EXTERNAL COMF 14 DIMENSION FREEM), LABEL! 15), X(3), F(3) 1 5 C 1 6 COMMKT, ON X/ACRH EA 1/JNXO CH IM. XJG CO H.IEXVBI. , XEGB .VCXEP.H IO. JJR DN OB. .JD. JB.JG.OFIX. 1178 • I H . . JRO, X JNE. JR. » XPEGB. XZETA 1 9 CO OM MM MO ON N //A EA2/ JPOL, NB. PB. NC,JTP,NVG .AM CM OA S.. A CPSII, 20 C AR REA3/ JTN, STOC. O ASSTOV, XPSII 21 COMMON /AREA4/ WBASE. DELWB. DELWBO. LEB 2 2 C OM MM MO ON N //A AM 5// O POL. LPOLB(20). VI22.21) 23 CO SERSEJO AI20.22). 24 C 25 DATA FREE/'*•/. NEWY/.TRUE./. NEWA/.FALSE./. NEWB/.FALSE./, 26 f NI/1.45D16/. KBOLTZ/1.380660-23/. 0/1.602180-19/. 27 • EPSI0/8.85418D-12/. 28 # PI/3. 14159285400/. ME/0.910950 - 30/. H/6.626170-34/ 29 C 30 C 31 C REVERSE CONDUCTO I N AND VALENCE 32 C TO MODEL PNP TRANSISTOR 33 C 34 C 35 NV = 2.8D25 38 NC = 1.04D25 37 C 38 C 39 C READ ITERATION SPECIFICATIONS FROM INPUT FILE 5 40 C 41 C 42 READ(5.5) (LABEL(I).1=1.15) 43 5 FORMAT(15A4) 44 READ(5.FREE) ITMAX1. ITMAX2, ERR1. ERR2, DELY 45 C 46 C 47 C REAO OEVICE PARAMETERS FROM INPUT FILE 3 46 C 49 C 50 READ(3,FREE) T, EGB, CHIM. CHIE 51 READ(3.FREE) MCDME, MVDME. MO IME 52 READ!3,FREE) NCOLL. PB. WBASE. DI, LEB 53 REAOI3.FREE) DB. OP. TAUCP. TAUR. TAUQ 54 READ(3.FREE) NEPSII. NEPSIS. OFIX 55 READ(3.FREE) DPOL. LPOL. ZETA. PEGB 56 READ)3.FREE) JB 57 C 58 C  Listing of PISETP.S at 21:56:53 on OCT 10, 1988 for CC1d=KCHU on 0 SB 60 61 62 63 64 65 86 87 88 89 70 71 72 73 74 75 78 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 98 97 98 99 100 101 102 103 104 105 108 107 108 109 110 111 112 113 114 115 118  C C C 10 15 20 25 30 35 37 40 C C C C C C  C C C c c c  c  Page  ECHO DEVICE PARAMETERS TO OUTPUT FILE 8 WRITE(6,10) (LABEL(I),I*1.1S) FORMAT* • V ,20X. 15A4///J WRITE(6.15) T, EGB, PEGB. CHIM. CHIE FORMAT!2X,'T=', F6.2.2X, 'EGB*',F4.2,2X, •PE08*',F4.2.2X. # 'CHIM=',F4.2,2X, 'CHIE*',F4.2/) WRITE(6,20) MCDME. MVDME, MIOME FORMAT! 2X , ' MCDME*' ,F4 . 2, 2X , 'MVDME*' .F4.2.2X, 'MIDME= ,F4.2/) WRITE(8,25) NCOLL. PB. WBASE, 01. LEB FORMAT!2X,'NCOLL*',D9.3.2X, 'PB='.D9.3.2X. 'WBASE*',09.3,2X, • *DI='.D9.3.2X, 'LEB*',D9.3/) WRITE!6,30) DB. OP. TAUCP, TAUR. TAUG FORMAT!2X,'DB='.09.3,2X, 'DP*'.09.3.2X, 'TAUCP*'.09.3.2X, -  f WRITE<6.35)'TAUR*'.09.3,2X. 'TAUG*',D9.3/) NEPSII, NEPSIS, QFIX FORMAT!2X,'NEPSI1='.F5.2.2X, 'NEPSIS=',F5.2.2X. •OFIX*',D10.4/) WRITE!6.37) DPOL, LPOL. ZETA FORMAT!2X,'DPOL='.09.3.2X,'LPOL*',D9.3.2X,'ZETA*'.F4.2/) WRITE«B,40) JB FORMAT!2X,'JB=',D10.4///) NORMALIZE DEVICE PARAMETERS TO UNITS OF KT ALL VARIABLES STARTING WITH AN X ARE IN KT'S KT = KBOLTZ • T VTHERM = KT / 0 XZETA = ZETA / VTHERM XEGB * EGB / VTHERM XPEGB = PEGB / VTHERM XCHIM = CHIM / VTHERM XCHIE = CHIE / VTHERM XCHIH = XCHIE - XEGB MC = MCDME ' ME MV * MVDME ' ME MI * MIDME ' ME COMPUTE ALL CONSTANTS NB = NI * NI / PB XPHIO = OLOG!NV/PB) CPSII = 01 / (NEPSII-EPSIO) / VTHERM COS = 0S0RT(2*KT' EPSIO : NEPSIS) GAMMA = 4 • PI * 01 • DS0RT(2'MI) / H ASTOC = (4*PI/H) * (Q/H) • (MC/H) • !KT"2) ASTOV = (4-PI/H) * (Q/H) • (MV/H) • !KT**2) JDO = Q * DSORT!DP/TAUCP) * NI * NI / NCOLL JNO = 0 * DB * NI • NI / PB JNO = (0 * DB * NI * NI / PB) / LEB JRO = 0 ' NI ' DSQRT(2*NEPSISEPSI0VTHERM/(0PB)) / TAUR JGO = 0 • NI • DS0RT(2'NEPSIS-EPSI0((NC0LL*PB)/(NC0LL*PB)) # *VTHERM/0) / TAUG ,  ,  ,  ,  2  L i s t i n g of PISETP.S at 21:58:53 on OCT 10. 1988 for CC1d=KCHU on 0 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 180 161 162 163 164 165 166 167 168 169 170 171 172 173 174  DELWBO a 0S0RT(2-NEPSIS*EPSI0*(NC0LL/(fBMNC0LL*PB)))*VTHERM/0) XVBI = XEGB - XPHIO - OL00(NC/NCOLL) C C C C C  WRITE ALL CONSTANTS TO OUTPUT F I L E 8 WRITE(6.50) VTHERM FORMAT(2X,'VTHERM='.D9.4/) WRITE(6.60) NB, XPHIO, CPSII FORMAT(2X,'NB=',09.4,2X, 'XPHIO=",D9.4.2X, 'CPSIIa•,09.4/) WRITE(6.65) GAMMA. ASTOC, ASTOV FORMA T(2X,'GAMMA=',09.4,2X, ASTOC='.09.4.2X. 'ASTOV='.09.4/) WRITE(8,70) JOO, JNO, JRO. JGO. OELWBO F0RMAT(2X,'J00='.D10.5.2X, 'JNO='.010.5.2X. 'JRO='.010.5.2X. « •JG0= .010.5.2X, 'OELWBO='.010.5/) WRITE(8.75) XVBI FORMAT(2X.'XVBI='.09.4///)  50 60 65  1  70  1  75 C C C C  c c c  80 C C C C  c c c  210 220 C C C C C  c  240  C C C C  '  READ INITIAL VALUES FOR THE VARIABLES TO BE USED IN THE ITERATION FROM INPUT F I L E 5 READ(5,FREE,END=340) VCE, PHIO, PSISO. PHI10 XVCE = VCE / VTHERM SET UP INITIAL CONDITIONS FOR NUMERICAL SUBROUTINE PACKAGE SSM Y( 1. 1) = PHIO / VTHERM Y(2.1) = PSISO / VTHERM Y(3.1) = PH110 / VTHERM DO 220 1=2.4 DO 210 J=1.3 Y ( J , I ) = YIJ.1) Y d . I ) = Y d . I ) • OELY RUN SUBROUTINE SSM NEWY = TRUE. CALL SSM(X.F.3.0.ERR1,ITMAX1.COMF.NEWY.NEWA,NEWB,IFAIL.&240) XIPHI = X ( 1 ) XIPSIS = X(2) XIPHI1 = X(3) NEWY = FALSE. CALL SSMCX.F,3,0.ERR2.ITMAX2,COMF.NEWY.NEWA,NEWB,IFAIL.S280) CONVERT RETURNED VALUES TO VOLTS  Page  3  d a t i n g of  117765 117778 179 180 116812 183 184 185 118876 188 118990 191 119923 194 195 119976 198 1 20909 220012 203 220045 220076 208 209 210 211 221132 221145 218 217 218 219 220 221 222223 222245 226 227 228 229 230 231 232  PISETP.S a t 21:56:53 on OCT 10. 1888 f o r CC1d*KCHU on Q C 260 PHI = X(1) * VTHERM P SIS VTTH HEERRM M PH I1 == X(2) X(3) *' V PSII = XPSII * VTHERM C C C CALCULATE DIFFERENCE BETWEEN FIRST AND SECOND CALL TO SSM C C PHIER = DABS(X(D-XIPHI) ' VTHERM P SISI1E PH ER R= = DABS(X(2)-XIPSIS) DABS(X(3)-XIPHI1) ** V VTTH HEER RM M C C C COMPUTE FINAL VOLTAGES AND CURRENT DENSITIES TO BE OUTPUT C C VBE = (XEGB -XPHIO -X(2) -XCHIM -XPSII *XCHIE -XZETA) f • VTHERM JE = JTN • JTP C JJC == JNJNE• -JG •B JO C J R N •N JDOLLP CBC = DSORKQN ' E• PSISJE 'GPSIO 'C ' B / (2*(NCOLL*PB)• # ( V T H E R M * X V B I • V C E VBE))) cC c WRITE VOLTAGES. CURRENT DENSITIES AND VARIABLES c DETERMINED FOR THE FINAL SOLUTION c TO OUTPUT FILE 6 c c WRITE(6.300) VCE, VBE. JE. JC. PHIO. PSISO. PHI10. IFAIL 300 F0RMAT(2X,'VCE='.F5.2.2X, VBE=".014.8.2X, # 'JE=',011.5,2X. "JC=".D14.8,10X, "PHIO=",F8.3.2X, # "PSISO=".F6.3.2X. •PHI10='.F6.3.2X. 'IFAIL*'.15/) WRITE(8.310) PSII. PHI. PHIER, PSIS. PSISER. I1E ELWB 310 #FORMAT(2X,PHI1, •PSII=".P0H 11. 5.R IX.. D "PHIs".014.8,IX. "PHIER*'.011.5.2X, f# "PSISs".011.5.2X, "PSISERs". D 11. 5 . 2 X, "PHIls".D11.5.2X. "PHI1ER='.011.5.2X. "DELWBB",D11.5/) WRITE(6,320) OS. JTN, JTP, JNE. JG. JD, JPOL 320 F0RMATI2X,'0S='.D14.8.2X, "JTN=",D11.5.2X, "JTP=",D11.5.2X, # "JNE=".D11.'5.3X, "JG=" .011.5.2X, # "JD=".D11.5.2X. 'JPOL=".011.5/) WRITE(6.330) JRNB, CBC 330 F0RMAT(2X."JRNB=\011.5.2X. 'CBC=",011.5////) TO 340 G STO O P 80 C END C C MAN I SUBROUTINE USED TO CALCULATE C CURRENT OENSITIES USED IN CONTINUITY EQUATO I NS C C SUBROUTINE COMF(X.F) IMPLICIT REALS' (A-2) 1  Page  4  L i s t i n g of  233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 288 269 270 271 272 273 274 275 278 277 278 279 280 281 282 283 284 285 286 287 288 289 290  PISETP.S  C C C CO C C C C C c c CO c c c c c c C5 c c c c c c CO C c c c c c CO c c c c c c c c c  at  21:56:53  on  OCT 10, 1988  for  CCM=KCHU on 0  INTEGER FREE OIMENSION X(3). F<3). FREE(I) COMMON /AREA1/ XCHIM, XCHIE. XEGB, XPHIO. JDO. JD. JB. OFIX, * KT. XCHIH, JNO. JRO. JGO, XVBI. XV:E. JNE. JRNB, JR. JG. » XPEGB, XZETA COMMON /AREA?/ NB. PB, NC, NV, COS. CPSII, OS COMMON /AREA3/ JPOL, JTN, JTP, GAMMA, ASTOC. ASTOV, XPSII COMMON /AREA4/ WBASE, DELWB, DELWBO, LEB COMMON /AREA5/ DPOL. LPOL DATA FREE/'"/ WRITE(6,60) X(1). X(2). X(3) FORMAT(2X,'X(1)=',011.S.2X, 'X(2)='.011.5.2X. 'X<3)='.011.5/> COMPUTE OS:CHARGE STORED AT OXIDE-SEMICONDUCTOR INTERFACE AND PSII:VOLTAGE ACROSS THE OXIDE CALL FOS(X) WRITE(6.70) FORMAT(2X, STEP FQS') XPSII = -CPSII * !QS*QFIX) COMPUTE JPOL .'MINORITY CARRIER CURRENT DENSITY IN POLYSILICON CALL FJPOL(X) WRITE(6.75) FORMAT<2X. STEP FJPOL') COMPUTE JTN:MAJORITY CARRIER TUNNELING CURRENT DENSITY CALL FJTN(X) WRITE(8.80) FORMAT!2X.'STEP FJTN') COMPUTE JTP:MINORITY CARRIER TUNNELING CURRENT DENSITY CALL FJTP(X) NRITE(6.90) F0RMAT!2X.'STEP FJTP') COMPUTE VCB:COLLECTOR-BASE VOLTAGE XVCB = XVCE-XEGB*XPHI0*X!2)*XCHIM*XPSII-XCHIE»XZETA COMPUTE DELWB:BASE WIDTH-MODULATION  PiiQ«  5  M a t i n g o f PISETP.S a t 21:56:53 on OCT  O  291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 308 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 328 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 348 347 348  C C C C C C C  10. 1986 f o r CCId»KCHU on 0  SRXVCB a DSQRT(XVBI*XVCE-XEGB*XPHI0*X(2)*XCHIM*XPSII-XCHIE) SRXVCB = DSQRT(XVBI*XVCB) DELWB = DELWBO * SRXVCB COMPUTE JRNB:RECOMBINATION CURRENT DENSITY IN THE BASE WBEFF = WBASE - DELWB ITANHY = 1/ DTANH(WBEFF/LEB) ISINHY a 1/ DSINH(WBEFF/LEB) EXPX1 a DEXPL(X(1)) - 1 EXPXCB a DEXPL(-XVCB) - 1 JRNB a JNO • (ITANHY - ISINHY) * (EXPX1 • EXPXCB)  C  c c C C  C C C C C  c C C  COMPUTE JNE:MAJORITV  CARRIER EMITTER CURRENT DENSITY  JNE a JNO • (ITANHY * EXPX1  - ISINHY • EXPXCB)  JN a JNO ' D E X P L I X ( D ) / (WBASE-DELWB) JR = JRO • DS0RT(DABS(X(2))) • (OEXPL(X(1)/2)-1) COMPUTE JG:GENERATION  CURRENT DENSITY IN THE CB JUNCTION  JG a JQO * SRXVCB  C  c c c c c c c c c c c c c c coo c c  C10 C C C20  COMPUTE J D D I F F U S I O N CURRENT DENSITY IN THE CB JUNCTION JD a -JDO ' (DEXPL(XEGB-XPHI0-X(2)-XCHIM-XPSII*XCHIE-XVCE)-1) JO a -JDO • EXPXCB COMPUTE ERROR FUNCTIONS USING CURRENT DENSITY CONTINUITY EQUATIONS FOR NUMERICAL PACKAGE SSM F ( 1 ) a (JNE - JTN) • 1.D-4 F ( 2 ) a JB • JG • JD - JTP - JRNB F ( 3 ) a JPOL - JTP WRITEI6.100) X ( 1 ) . X ( 2 ) , X ( 3 ) . F ( 1 ) . F ( 2 ) . F ( 3 ) FORMAT(2X,'X(1)s',D11.5,2X, 'X(2)=-.Dl1.5.2X. 'X(3)='.011.5,2X, # 'Ft1)a-.011.5.2X. ^(2)='.D11.5.2X. 'F(3)=-,D11.5) WRITE(8,110) JTN, JTP, JG, JPOL FORMAT(2X,'JTN='.D11.5.2X, 'JTPa".D11.5.2X, # 'JGa'.011.5.2X. •JPOLa'.011.5/) WRITE(6.120) OS, XPSII FORMAT(2X.•QS='.011.5.2X, "XPSIIa".011.5/) RETURN END  Pago  6  Listing of PISETP.S at 21:58:53 on OCT 10. 1988 for CC1d=KCHU on 0 349 350 351 352 353 354 355 356 357 358 359 380 361 362 363 364 365 366 367 368 389 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 408  C C C C C  SUBROUTINE TO CALCULATE CHARGE STORED AT OXIDE-SEMI INTERFACE  SUBROUTINE FOSIX) IMPLICIT REAL'S (A-Z) DIMENSION X(3) COMMON /AREA 1/ XCHIM. XCHIE, XEGB, XPHIO. JOO. JO. JB. OFIX. • KT. XCHIH. JNO. JRO. JGO. XVBI. XVCE. JNE. JRNB. JR. JG. » XPEGB, XZETA COMMON /AREA2/ NB. PB. NC. NV. COS. CPSII. OS C NSURF = NC * F0102(X(2)*XPHIO*X(1)-XEGB) PSURF = NV ' FD102(-XPHI0-X(2)) ARGMNT = NSURF - NB*DEXPL(X(1)) • PSURF - PB •PB*X(2) IF (ARGMNT .GE. 0.000) GOTO 100 WRITE(6.5) ARGMNT 5 FORMAT!IX.'WARNING: SOURE OF SURFACE FIELD IS NEGATIVE*. 1 i 5X. D11.4) ARGMNT = 0.000 100 OS = -COS * DSORT(ARGMNT) IF (X(2) .LT. O.ODO) OS = -OS RETURN END c c c SUBROUTINE TO CALCULATE c MINORITY CARRIER CURRENT DENSITY IN THE POLYSILICON c c SUBROUTINE FJPOL(X) IMPLICIT REAL'8(A-Z) DIMENSION X(3) COMMON /AREA 1/ XCHIM. XCHIE. XEGB, XPHIO. JOO. JO. JB. OFIX, # KT. XCHIH. JNO. JRO. JGO. XVBI. XVCE. JNE. JRNB. JR. JG. # XPEGB, XZETA COMMON /AREA2/ NB, PB, NC. NV. COS. CPSII. OS COMMON /AREA3/ JPOL. JTN. JTP, GAMMA, ASTOC. ASTOV. XPSII COMMON /AREA5/ DPOL. LPOL c XBETA1 = XPEGB - X(3) - XZETA PJ = NV ' FD102(-XBETA1) JPOL = 1.60218D-19 ' DPOL ' PJ / LPOL RETURN END c c c SUBROUTINE TO CALCULATE c MAJORITY CARRIER TUNNELING CURRENT DENSITY c USING A THREE TERM SERIES APPROXIMATION c c SUBROUTINE FJTN(X) IMPLICIT REAL'S (A-Z) DIMENSION X(3) COMMON /AREA 1/ XCHIM. XCHIE. XEGB, XPHIO, JDO. JD. JB, QFIX,  Page  7  Listing 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 448 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 482 463 464  Paga  o f PISETP.S a t 21:58:53 on OCT 10. 1988 f o r CC1<J=KCHU on Q # KT, XCHIH. JNO. JRO, JGO, XVBI. XVCE. JNE. JRNB, JR. # XPEGB, XZETA COMMON /AREA3/ JPOL. JTN. JTP, GAMMA, ASTOC, ASTOV, XPSII C  10 C C 20  JG.  IF  (XPSII .LT. 0) GOTO 10 XNALP1 = - XPSII - XZETA XNALP2 = X(2) • XPHIO • X(1) - XEGB BRC = (XCHIE - XPSII/2) * KT GOTO 20 XNALP1 = -XZETA XNALP2 = XPHIO • X(2) • X(1) - XEGB • XPSII BRC = (XCHIE • XPSII/2) * KT THETCO = DEXPH-GAMMA*DSORT(BRC)) THETC1 a GAMMA * THETC0/(2*DSORT(BRC)) THETC2 a GAMMA * (GAMMA/BRC • 1/(DSORT(BRC))* * 3) * THETCO/4 JTN a ASTOC * (THETCO * (FD1(XNALP1)-FD1(XNALP2)) • (KT*THETC1) ((KT*'2)'THETC2) * (FD3(XNALP1)-FD31XNALP2))) f# • * (F02(XNALP1)-F02(XNALP2)) RETURN END  C C C C C C C  c  SUBROUTINE TO CALCULATE MINORITY CARRIER TUNNELING CURRENT DENSITY USING A THREE TERM SERIES APPROXIMATION SUBROUTINE F J T P ( X ) IMPLICIT REAL'S (A-Z) DIMENSION X(3) COMMON /AREA1/ XCHIM. XCHIE. XEGB, XPHIO. JDO. JD. JB. OFIX. # KT. XCHIH, JNO. JRO. JGO. XVBI, XVCE. JNE, JRNB, JR. JG. * XPEGB. XZETA COMMON /AREAS/ JPOL, JTN. J T P . GAMMA. ASTOC. ASTOV, XPSII IF  (XPSII  .LT. 0) GOTO 10  XNBET1 = a XZETA XPEGB XNBET2 -XPEGB •-X(3) XPSII- • XEGB - X(2) - XPHIO 10  c c  20  c c  GOTOBRV 20 a (XCHIH • XPSII/2) * KT XNBET1 = -XEGB • XPSII • XZETA • X(3) XNBET2 = -XPHIO - X(2) BRV = (XCHIH - XPSII/2) • KT  THETVO a DEXPL(-GAMMA*OSQRT(BRV)) THETV1 a GAMMA • THETVO/<2*DSQRT(BRV)) THETV2 a GAMMA • (GAMMA/BRV • 1/(DSORT(BRV))* * 3) * THETVO/4 JTP = ASTOV * (THETVO * (F01(XNBET2)-F01(XNBET1)) # # RETURN END  • (KT'THETVI) • (F02(XNBET2)-FD2(XNBET1)) • <(KT**2)*THETV2) * (FD3(XNBET2)-FD3(XNBET1)))  6  -if—»-»-»-- f - t - m m |  - ...  ......  Listing PISETP.EX SPOatNEN 21: on OCT 10. 1988 for CC1d=KCHU on 0 485 of C TIA5L8:5F3 UNC TION 486 C 446678 C DOUBLE PRECISION FUNCTION OEXPL(X) 446790 IMPLIC EAL0*00) 8 (A-Z) IF(X .LITT. R-150. GOTO 100 447712 O E X P L = O E X P ( X ) RETURN 447743 100 R DEETX =OO . DO UPRLN 475 END 476 C 447778 CC FERMI-DIRAC FUNCTION OF ORDER 1 479 C USING A SERIES APPROXIMATION 480 C 444888123 C D OU BLIC EITPRREECA ISIL OS'N(A-Z) FUNCTION FDKX) I M P L 484 DATA PI/3. 14159285400/ 448885 Y = .LE. X OD IF(V . O) GOTO 50 448878 50 Y -e -Y D ' EXP0L (Y) -.250052' 'Y) •.111747' 489 •Z = 1.-.O0O 84557' EXPL(4. *Y) D•.EXPL(2. 040754*DEXPL(5. 'Y)DEXPL(3.*Y) 449910 •IF(X .LE. -.020532' 0)EXPL(6. t— OD .O GOTO *1Y) 00 •.005108'DEXPL(7.*Y) rj 492 Z = -Z • (X"2)/2.D0 • (PI"2)/6.00 449943 100 CFOD N1TN IU EZ = 449985 ED TURN ER N 497 C 498 C FU RD 540909 C C FUESR INM GI-DIR AACSER IEN SCTIO AN PPROO XFIMAO TIO NER 2 550021 C C 503 DOUBLE PRECISION FUNCTION FD2(X) 504 IMPLICIT REALS' (A-Z) 550065 O PI/3.141592654D0/ YATA =X 550078 IF(Y= .LE. . O) GOTO 50 Y -Y OD 509 50 Z= I.DOD ' EXPL(Y) -.125046*DEXPL(2.*Y) •.037642*DEXPL(3.'Y) 510 • -,018183'DEXPL(4.'Y) • .012484'DEXPL(5.*Y) 511 f -.007488'DEXPH8.*Y) • .002133'DEXPL(7.'Y) 512 IF(X .LE. OD . O) GOTO 100 513 Z = Z • (X"3)/8.00 • (PI"2)'X/6.D0 514 100 CONTN I UE 551165 F D 2 RETURN= Z 517 END 518 C 519 C 520 C FERMI-DIRAC FUNCTION OF OROER 3 521 C USING A SERIES APPROXIMATION 522 C w  Page  Listing of PISETP.S at 21:56:53 on OCT 10. 1966 for CC<d=KCHU on Q 523 C 524 DOUBLE PRECISION FUNCTION F03(X) 525 IMPLICIT REALS' (A-Z) 526 DATA PI/3.141592654DO/ 5 V X OD 52 27 8 IF(V= .LE. . O) GOTO 50 5 2 9 Y = Y 530 50 Z = 1 .DOO ' EXPL(Y) -.062592'0EXPL(2.'Y! •.013661'DEXPL(3.'Y) 5 3 1 • -. 0 09796' 12976'0CXPL(5.*Y) 532 • -.010659'D DEXPL(4. EXPL(6.''Y Y)) •. •.0003448*DKXPL(7. 'Y) 5 3 3 IF(X .LE. O D . O ) G O T O 1 0 0 534 Z = -Z • (X"4)/24.00 • < <PI'X)"2J/12.00 • 7.MPI"4)/ 5 3 5 • 3E80D .O 5 3 6 1 0 0 C O N T N I U 537 FD3 * Z 538 RETURN 539 END 5 41 0 C 54 C 5 4 2 C FUESR M I-DIR ACSER FU N CTIO N O FIMAO R D ER 1/2 5 4 3 C I N G A I E S A P P R O X T I O N 544 C 545 C 548 DOUBLE PRECISION FUNCTION FD102IX) 547 IMPLICIT REAL*8 (A-Z) 548 DATA PI/3.141592854D0/ 549 IF (X .LE. OD . O) GOTO 100 5 5 0 IF ( X .LE. 2 . D O) GG TO 302000 551 IF (X .LE. 4.00) OO TO 5 OT1O 00 OEXPL(X) -0.35356B'DEXPL(2'X) *0.192439'OEXPLI3'X) 55 52 3 100 G FD 02 4= 5 5 4 *# -0. 1 22973' 0 EXPL(4' X )) *0. *0.0008348' 77134'D EXPL(5' X )) 5 5 5 -0. 0 36228' 0 EXPL(6' X D EXPL(7' X 558 GOTO 500 557 200 F0102 = 0.785147 • X'(0.804911 • X'(0.189885 • 5 5 8 • X'O (0..0 000133))))) 20307 • XM-0.004380 • XM-0.000366 • 5 5 9 # X' 580 GOTO 500 561 300 F0102 = 0.777114 • X'(0.581307 • X'(0.208132 • 582 f X'(0.017880 • XM-0.006549 * X'(0.000784 5 6 3 # X' 584 GOTO 500O.000036))))) 585 400 XN2 = 1/(X"2) 586 FD102 = (X"1.5> * (0.752253 • XN2' (0.928195 • 567 # XN2M0.680839 • XN2M25.7829 • XN2'(-553.636 • 568 # XN2'(3531.43 -XN2'3254.65)))))) 569 500 RETURN 570 END  Page  M a t i n g o f PIMSP.S at 21:52:10 on OCT 10. 1988 f o r CC1d«KCHU on 0 1 2 3 4 5 6 7  e  3 Cn  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  C C C C C C C  c c c  c  c  c c c c c c c c c c 5  c c c c c  THIS PROGRAM I S WRITTEN TO GENERATE THE STEADY-STATE CHARACTERISTICS OF THE ANNEALED PNP PETS. 1  •c  rt C3  a  DEFINE ALL VARIABLES 1  X  O  IMPLICIT REAL'S (A-Z) INTEGER FREE. I . J , IFA1L. ITMAX1, ITMAX2 LOGICAL NEWY, NEWA. NEWB 1 EXTERNAL COMF 1 DIMENSION F R E E ( 1 ) , LABEL!15). X ( 4 ) . F<4)  (*} O g  1  COMMON /AREA 1/ XCHIM, XCHIE, XEGB, XPHIO, JDO. JD. J B . OFIX, KT, XCHIH. JNO. JRO. JGO. XVBI, XVCE. JNE. JRNB. JR. JG. XPEGB, XZETA. XZETA2, XV8IE. JPO. JRM. JPE. JP2 1COMMON /AREA2/ NE, NB. PB. NC. NV. COS. CPSII 1COMMON /AREA3/ JPOL. JTN. JTP. GAMMA. ASTOC. ASTOV 1COMMON /AREA4/ WBASE. DELWB. DELWBO. LEB. LHE. LE. DE 1COMMON /AREA5/ OPOL. LPOL COMMON /SEJJOM/ A(20,22). B ( 2 0 ) . YJ22.21)  ^ £• rt ^ M - l O <2  •1 •  3  DATA F R E E / ' " / . NEWY/.TRUE./. NEWA/.FALSE./. NEWB/.FALSE./. NI/1.45D16/. KB0LTZ/1.3B086D-23/. 0/1.6021BD-19/. « EPSIO/6.654180-12/. PI/3.141592654D0/. ME/0.910950-30/. H/6.62817D-34/  •1  " > O  M  f f  REVERSE CONDUCTION AND VALENCE FOR PNP DEVICE 1  1  NV = 2.8025 NC 1 = 1 .04025  o '  S  t B  >  READ ITERATION SPECIFICATIONS FROM INPUT F I L E 5 1  s a «  1 1 1  READ(5.5) (LABEL(I).I=1.15) FORMAT!15A4) READ!5.FREE) ITMAX1. ITMAX2, ERR1. ERR2, DELY  JT  Cu  READ DEVICE PARAMETERS FROM INPUT FILE 3  1  READ!3.FREE) 1 READ!3.FREE) 1 READ!3.FREE) 1 READ!3.FREE) 1 READ!3.FREE) 1 READ!3.FREE) READ!3.FREE) 1  1 c c  "d  T. EGB. CHIM. CHIE MCDME, MVDME. MIDME NE. NCOLL. PB, WBASE. D I , LEB. LHE LE. DE. DB. DP. TAUCP. TAUR. TAUG NEPSII. NFPS1S, OFIX OPOL. LPOL. ZETA, PEGB JB  k  . H H »  1  L i s t i n g o f PIMSP.S at 21:52:10 on OCT 10. 1088 f o r CCIdHCCHU on 0 59 80 61 82 63 64 65 68 87 68 89 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 98 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 118  C C C 10 15 20 25  30 35 37 40 C C C C  c c  c c c c c c  c  ECHO DEVICE PARAMETERS  TO OUTPUT F I L E "  WRITE(8.10> ( L A B E L ( I ) . I = 1 . 15) FORMAT!'1',20X, 15A4///) WRITE<8.15) T. EGB. PEGB, CHIM, CHIE FORMAT(2X.'7='. F6.2.2X. 'EGB='.F4.2.2X. 'PEGB*',F4.2, 2X, # 'CHIM*'.F4.2.2X. 'CHIE*'.F4.2/) WRITE(8,20) MCDME, MVDME. MIDME FORMAT!2X,'MCOME=',F4.2,2X, 'MVOME=•,F4.2.2X. 'MIDME*' .F4.2/I WRITE(8.25) NE. NCOLL. PB, DI. LEB. LHE. WBASE FORMAT(2X,' NE*',09.3,2X, 'NCOLL*',D9.3,2X, 'PB='.D9.3, 2X. # '0I=',09.3.2X. 'LEB=',09.3.2X, 'LHE=',D9.3,2X, • 'WBASE*',D9.3/) WRITE(e.30) L E . DE. DB. DP. TAUCP. TAUR. TAUG FORMAT!2X,'LE*',09.3,2X, 'DE*'.09.3.2X. 'DB=',D9.3,2X, 'DP*'. • D9.3.2X, 'TAUCP=',09.3,2X, 'TAUR*',09.3,2X, 'TAUG*'.D9.3/) WRITE(6,35) NEPSII. NEPSIS, QFIX F0RMAT(2X.'NEPSII='.F5.2.2X. 'NEPSIS*'.F5.2.2X. 'QFIX* '.010.4/) WRITE(6,37) DPOL, LPOL. ZETA FORMAT!2X, DPOL*'.09.3.2X,'LPOL*'.D9.3.2X,'ZETA*'.F4.2/) WRITE<6.40) JB FORMAT(2X.'JB*'.D10.4///) NORMALIZE DEVICE PARAMETERS TO UNITS OF KT ALL VARIABLES STARTING WITH AN X ARE IN KT'S KT = KBOLTZ • T VTHERM = KT / 0 XZETA = ZETA / VTHERM XEGB = EGB / VTHERM XPEGB = PEGB / VTHERM XCHIM = CHIM / VTHERM XCHIE = CHIE / VTHERM XCHIH = XCHIE - XEGB MC = MCDME * ME MV = MVOME • ME MI * MIDME * ME COMPUTE ALL CONSTANTS NB = NI * NI / PB XPHIO = DLOG!NV/PB) XZETA2 = OLOG(NC/NE) CPSII = 01 / (NEPSII'EPSIO) / VTHERM COS = DSORT!2-KT* EPSIO * NEPSIS) GAMMA = 4 * PI ' DI * DSQRT(2'MI) / H ASTOC = <4'PI/H) * (Q/H) • (MC/H) • ( K T " 2 ) ASTOV = (4'PI/H) * (Q/H) • (MV/H) • (KT«*2) JOO = Q • DSQRT(DP/TAUCP) * NI * NI / NCOLL JNO = 0 • OB • NI • NI / PB JNO = (0 * DB • NI * NI / PB) / LEB JPO = 0 • OE * NI • NI / LHE / NE  Page  2  L i s t i n g o f PIMSP.S s t 21:52:10 on OCT  117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 146 149 150 151 152 153 154 155 158 157 158 159 160 161 182 163 164 185 166 187 188 189 170 171 172 173 174  tO. 1988 f o r CC1d=KCHU on 0  JRO = 0 • NI * DS0RT(2NEPSISEPSI0M(Nf•PB)/(NE*PB>) • 'VTHERM/0) / TAUR JOO s fl NI * DSQRm'NEPSIS'EPSIOMINrOLUPBl/lNCOLl'PB)) • 'VTHERM/Q) / TAUO OELWBO * OSQRT (2*NEPSIS*EPSIO*(NCOLL/(PC*(NCOLL*PBI • 1 * VTHERM/O) XVBI = XEOB - XPHIO - OLOG(NC/NCOLL) XVBIE = XEGB - XPHIO - XZETA2 ,  ,  1  C C C C C 50 60 65 70 75 C C c c c c c 80 C C C C c c c 210 220 C C C C C C 240  WRITE ALL CONSTANTS TO OUTPUT FILE 6 WRITE<6,50) VTHERM. XZETA2 FORMAT(2X.'VTHERM*'.09.4.2X, 'XZETA2*'.011.5/) WRITE(8,80> NB. XPHIO. CPSII FORMAT!2X,"NB='.09.4,2X, 'XPHIO*',D9.4.2X. 'CPSII*',D9.4/) WRITE(6,85) GAMMA, ASTOC, ASTOV FORMAT(2X,'GAMMA*'.D9.4,2X, 'ASTOC*'.09.4.2X, ASTOV*',09.4/) WRITE(6.70) JDO. JNO. JPO, JRO. JGO. OELWBO FORMAT(2X,'JOO*',D10.5,2X. 'JNO*',D10.5,2X, 'JPO*'.010.5.2X. f WRITE(6.75)'JRO*',D10.5,2X. 'JGO*'.D10.5.2X, 'DELWBO*'.010.5/) XVBI. XVBIE FORMAT(2X.'XVBI*'.09.4.2X. 'XVBIE*',09.4///) 1  READ INITIAL VALUES FOR THE VARIABLES TO BE USED IN FIRST ITERATION FROM INPUT FILE 5 REAO(5.FREE.ENO=340) VCE. PHIO, PH110, PHI20. PSIIO XVCE = VCE / VTHERM SET UP INITIAL CONDITIONS FOR NUMERICAL SUBROUTINE PACKAGE SSM Y(1.1) Y(2. 1) Y<3.1) Y(4. 1)  = = = =  PHIO / VTHERM PH110 / VTHERM PHI20 / VTHERM PSIIO / VTHERM  00 220 1=2.5 DO 210 J=1,4 Y(J.I) = Y(J.I) Y(I.I) = Y(I.I) • DELY RUN SUBROUTINE SSM NEWY = .TRUE. CALL SSM(X.F.4.0.ERR1.ITMAX1.COMF.NEWY.NEWA.NEWB.IFAIL.«240) XIPHI * X(1) XIPHI1 = X(2) XIPHI2 = X(3)  Page  3  L i s t i n g of  175 176 177 178 179 180 181 182 183 184 185 166 187 168 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232  PIMSP.S a t 21:52:10 on OCT 10, 19B8 f o r CCId-KCHU on 0 XIPSII = X(4) — iu.. NEWY = .FALSE. CALL SSM(X.F.4,0,ERR2,ITMAX2.COMF.NEWY,NEWA.NEWB,IFA1L,&260) C C C CONVERT RETURNED VALUES TO VOLTS C C 260 PHI = XM) VTHERM PHI1 = X(2) * VTHERM PHI2 o X(3) * VTHERM PSII a X<4) * VTHERM C C C COMPUTE BAND BENDING AT EB JUNCTION C C PSIS = (XEGB - XPHIO - XM) - XZETA2) • VTHERM' C C c CALCULATE DIFFERENCE BETWEEN FIRST AND SECONO DALL TO SSM 4  c c  c c c c c  c c c c c c c  PHIER a DABS(X(D-XIPHI) ' VTHERM PHI1ER = DABS(X(2)-XIPHI1) * VTHERM PHI2ER = 0ABS(X(3)-XIPHI2) • VTHERM PSIIER « DABS(X(4)-XIPSII) * VTHERM COMPUTE FINAL VOLTAGES AND CURRENT DENSITIES TO BE OUTPUT VBE = (XM) • XZETA2 - X(4) -XZETA) •' VTHERM JE = JTN • JTP JC - JNE - JRNB • JG • JD CBC = DSQRKO'NEPSIS'EPSIO'NCOLL'PB / (2*(NCOLL*PB)• • (VTHERM'XVBI • VCE - VBE))) WRITE VOLTAGES. CURRENT DENSITIES AND VARIABLES DETERMINED FOR THE FINAL SOLUTION TO OUTPUT FILE 8  WRITE(8.300) VCE. VBE. JE. JC 300 FORMAT(2X,'VCE=,F5.2.2X, 'VBE=',D14.8.2X. 'JE='.011.5.4X. • 'JCa.014.8/) WRITEI8.305) PHIO. PHI10. PHI20, PSIIO. IFAIL 305 FORMAT (2X , •PHIO*' .F6.3.2X, "PHUOa .F8.3.2X. # 'PHI20='.F6.3.2X. •PSI10=',F6.3.2X. •IFAIL='.15/) WRITE(6.310) PSIS. PHI. PHI1. PHI2. PSII 310 F0RMAT(2X,'PSIS='.011.5.2X. 'PHI = '.014.8.7X. •PH11 = '.011.5.2X. * PHI2=.D11.5.2X, 'PSIIs'.011.5) WRITE(6.315) PHIER, PHIIER. PHI2ER, PSIIER. DELWB 315 FORMAT(2X,'PHIER='.D11.5.2X. 'PHI1ER*'.011.5.2X. # PHI2ER='.011.5.2X. 'PSIIER=',D11.5,2X, DELWB='.Dl1.5/) WRITE(8,320) JTN. JTP. JPE. JNE. JR. JG. JD. JPOL 320 FORMAT(2X,'JTN=',D11.5.2X, 'JTP=-.011.5.2X. •JPE=*.011.5.2X. 1  1  1  ,  1  Page  4  Listing of PIMSP.S at 21:52:10 on OCT 10, 1988 for CC1d=KCHU on 0 233 234 235 238 237 238 239 240 241 242 243 244 245 248 247 248 249 250 251 252 253 254 255 258 257 258 259 260 281 262 263 264 265 266 267 268 289 270 271 272 273 274 275 278 277 278 279 280 281 282 283 284 285 286 287 288 289 290  # 'JNE=',011.5.3X. 'JR='.011.5.38X, 'JG»'.011.5.2X. « 'JD='.011.6.2X. •JPOL"',011.5/) WRITE(8,330) JRNB. JRM, JP2. CBC 330 FORMAT( 2X . ' JRNB=' ,011 .5,2X , ' JRM=>' ,011 . 5.2X , • •JP2='.011.5.2X. 'CBC**.011.5////) GOTO 80 340 STOP END C C C MAIN SUBROUTINE USED TO CALCULATE C CURRENT DENSITIES USED IN CONTINUITY EQUATIONS C C SUBROUTINE COMF(X.F) IMPLICIT REAL'S (A-2) INTEGER FREE DIMENSION X(4), F(4). FREEH) COMMON /AREA1/ XCHIM, XCHIE. XEGB, XPHIO, JOO. JD. JB. OFIX. • KT. XCHIH. JNO. JRO. JGO. XVBI. XVCE. JNE, JRNB. JR. JG. * XPEGB. XZETA, XZETA2, XVBIE. JPO, JRM, JPE. JP2 COMMON /AREA2/ NE. NB. PB. NC. NV, COS. CPSII COMMON /AREA3/ JPOL. JTN. JTP, GAMMA. ASTOC. ASTOV COMMON /AREA4/ WBASE. DELWB. DELWBO. LEB. LHE. LE. DE COMMON /AREAS/ OPOL. LPOL DATA F R E E / " / C C C COMPUTE JPOL: MINORITY CARRIER CURRENT DENSITY IN POLYSILICON C C CALL FJPOL(X) C C C COMPUTE JTN: MAJORITY CARRIER TUNNELING CURRENT DENSITY C c CALL FJTN(X) c c c COMPUTE JTP: MINORITY CARRIER TUNNELING CURRENT DENSITY c c CALL FJTP(X) c c c COMPUTE VCB: COLLECTOR-BASE VOLTAGE c c XVCB = XVCE-X(1)-XZETA2*X(4)*X2ETA c c c COMPUTE OELWB: BASE WIDTH MODULATION c c SRXVCB = OSQRT(XVBI*XVCB) OELWB = DELWBO ' SRXVCB c  Page  5  i t l n g o f PIMSP' S a t 21:52:10 on OCT 291 292 293 294 295 296 297 296 299 300 301 302 303 304 305 306 307 309 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348  10. 1988 f o r CC1d=KCHU on 0  WBEFF = WBASE - DELWB ITANHV a 1/ DTANH(WBEFF/LEB) ISINHY a 1/ DSINHIWBEFF/LEB) ITANHU = 1/ DTANHUE/LHE) ISINHU = s DEXPL(X(1)J 1/ OSINH(LE/LHE) EXPX1 - 1 EXPX1R a DEXPL(X( 1)/2) -1 EXPX3 a = 0DEXPL(-XVCB) E X P L ( X ( 3 ) 1 - 1- 1 EXPXCB C C C C C  COMPUTE JRNB: RECOMBINATION  C  JRNB a JNO * (ITANHV - ISINHY) * (EXPX1 • EXPXCB)  c c c c  COMPUTE JNE: MAJORITY  CURRENT DENSITY IN THE BASE  CARRIER EMITTER CURRENT DENSITY  JNE a JNO * (ITANHY • EXPX1 - ISINHY • EXPXCB)  c c c c c c  COMPUTE J P E : MINORITY CARRIER DIFFUSION CURRENT DENSITY AT THE MONOCRYSTALLINE EMITTER DEPLETION EDGE  c c c c c  COMPUTE J P 2 : MINORITY CARRIER EMITTER CURRENT DENSITY  c c c c c  COMPUTE JRM: RECOMBINATION  c c c c c c c c c c c c c c c c  JPE = JPO • (ITANHU • EXPX1 - ISINHU * EXPX3)  JP2 = JPO * (ISINHU * EXPX1 - ITANHU * EXPX3) CURRENT DENSITY IN THE EMITTER  JRM a JPO * ((ITANHU - ISINHU) • (EXPX1 • EXPX3)) JN a JNO * D E X P L ( X ( D ) / (WBASE-OELWB) COMPUTE JR: RECOMBINATION  JR a JRO • EXPX1R  CURRENT DENSITY IN THE EB JUNCTION  • DSORT (XVBIE - X ( U )  COMPUTE JG: GENERATION  CURRENT DENSITY IN THE CB JUNCTION  JG a -JGO * EXPXCB * SRXVCB COMPUTE JO: OIFFUSION CURRENT DENSITY IN THE CB JUNCTION  L i s t i n g o f PIMSP.S a t 21:52:10 on OCT  349 350 3 521 3 5 353 354 355 3 35576 3 598 35 360 3 621 38 363 364 365 366 3 38676 389 3 37 701 372 373 3 75 4 37 3 776 3 7 37798 3 380 381 382 3 84 3 38 385 388 387 3 898 38 390 391 392 393 3 95 4 39 3 39978 398 399 400 401 402 403 404 405 408  C C C C C cC c c c c c c c c c c c c c c c c c c c c c c coo  C C C cC c c c c  c  10. 1988 f o r CC1d=KCHU on G  JD = -JDO • (DEXPL(XEGB-XPHI0-X(2)-XCHIM-XPSII*XCHIE-XVCE)JD = -JOO * EXPXCB C MG PUTECUE OR FU TIO UO SIN RR RR EN TD EN NC SIT YNSCONTINUITY EQUATO I NS FOR NUMERC IAL PACKAGE SSM F(1) e (JNE - JTN -JRM - JR) * 1.0-4 F(2) B - JD - JG • JRNB • JR F<3> ==JPJPEOL- J-JTP F(4) = JP2 - JTP RETURN END SUBROUTINE FOS(X) IMPLICIT REAL*8 (A-Z) DIMENSION X(3) COMMON /AREA 1/ XCHIM. XCHIE, XEGB, XPHIO. JDO. JD. JB. OFIX f KT. XCHIH, JNO. JRO. JGO. XVBI. XVCE. JNE. JRNB. JR, JG «COMMO XN PEG/A B.REA2/ XZETANB. PB. NC. NV. COS. CPSII. OS N SUR RFF = a NC • FD102(X(2)*XPHI0*X(D-XEGB) P ASRU GMN T =NV NSU*RFFD102I-XPHI0-X(2)) - NB*OEXPL(X(1) • PSURF - PB • PB*X(2) IF A (RGMNT .GE. OO . DO) GOTO 100 WRITEI6.5) ARGMNT FORMAT(IXW ', ARNING: SOURE OF SURFACE FIELD IS NEGATIVE.' •ARGMNT a5X.0.0D11. 00 4) OS = -COS * DSORT(ARGMNT) IF (X(2) .LT. 0.000) OS = -QS RETURN END SUBROUTINE TO CALCULATE MINORITY CARRIER CURRENT DENSITY IN THE POLYSILICON SUBROUTINE FJPOL(X) IM IC ITNREAL' D IMPELN SIO X(4)8(A-Z) COMMON /AREA 1/ XCHIM. XCHIE. XEGB, XPHIO, JDO. JO. JB. QFIX, # KT, XCHIH, JNO, JRO. JGO. XVBI. XVCE. JNE, JRNB, JR. JG, • XPEGB, XZETA, XZETA2, XVBIE. JPO. JRM, JPE. JP2 COMMON /AREA2/ NE. NB, PB. NC. NV. COS. CPSII COMMON /AREA3/ JPOL. JTN. JTP. GAMMA, ASTQC. ASTOV COMMON /AREA5/ OPOL, LPOL XBETA1 = XPEGB - X(2) - XZETA PJ = NV • FD102(-XBETA1I  Page  7  L i s t i n g o f PIMSP.S a t 21:52:10 on OCT 10. 1988 f o r CC1d-KCHU on Q 407 408 409 410 411 412 413 414 415 418 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 484  Page  JPOL = 1.602180-19 • OPOL * P J / LPOL RETURN END C C C C C C C  SUBROUTINE TO CALCULATE MAJORITY CARRIER TUNNELING CURRENT DENSITY USING A THREE TERM SERIES APPROXIMATION SUBROUTINE FJTN(X) IMPLICIT REAL'S (A-Z) DIMENSION X(4) COMMON /AREA1/ XCHIM. XCHIE. XEGB. XPHIO. JDO, JO, JB, OFIX. « f  C  THETCO = DEXPL(-GAMMA'DSORT(BRC)) THETC1 * GAMMA ' THETC0/(2*DSQRT(BRC)) THETC2 = GAMMA * (GAMMA/BRC • 1/(0SQRT(BRC) > " 3 > • THETCO/4 JTN = ASTOC ' (THETCO ' (F01 (XNALP1)-FD1{XNALP2)) # • ( K V T H E T C 1 ) ' (FD2(XNALP1)-FD2(XNALP2)) » * l ( K T " 2 ) ' T H E T C 2 ) ' (F03(XNALP1) -FD3IXNALP2) ) ) RETURN END  C C C C C • C C  SUBROUTINE TO CALCULATE MINORITY CARRIER TUNNELING CURRENT DENSITY USING A THREE TERM SERIES APPROXIMATION SUBROUTINE F J T P ( X ) IMPLICIT REAL'8 (A-Z) DIMENSION X(3) 1 *  COMMON /AREA1/ XCHIM. XCHIE. XEGB, XPHIO. JOO. JD. JB. OFIX. KT, XCHIH. JNO. JRO. JGO. XVBI. XVCE. JNE. JRNB. JR. JG. XPEGB. XZETA. XZETA2. XVBIE, JPO. JRM. JPE, JP2 COMMON /AREA3/ JPOL, JTN, JTP, GAMMA. ASTOC. ASTOV  C IF  10  JG.  IF ( X ( 4 ) .LT. 0) GOTO 10 XNALP1 = -XPEGB - X(4) - XZETA • XEGB XNALP2 = -XZETA2 BRC = (XCHIE - X(4)/2) ' KT GOTO 20 XNALP1 = -XZETA XNALP2 * -XEGB • X(4) • XPEGB • XZETA2 BRC = (XCHIE • X ( 4 ) / 2 ) • KT  10 C C 20  XPEGB. XZETA. XZETA2. XVBIE. JPO. JRM,JNE, JPE,JRNB. JP2 JR, KT, XCHIH, JNO, JRO. JGO, XVBI, XVCE. COMMON /AREA3/ JPOL. JTN. JTP. GAMMA. ASTOC. ASTOV  ( X ( 4 ) .LT. 0) GOTO 10 XNBET1 = -XPEGB • XZETA • X(2) XNBET2 = -XPEGB - X(4) • XZETA2 • X(3) BRV = (XCHIH • X ( 4 ) / 2 ) ' KT GOTO 20 XNBET1 = -XEGB • X(4) • XZETA • X(2)  8  • •PVT..  , .  n  L i s t i n g o f PIMSP.S a t 21:52:10 on OCT 10. 1080 f o r CC1d-KCHU on 0 465 488 467 488 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 498 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522  XN8ET2 = -XEGB • XZETA2 • X ( 3 ) BRV = (XCHIH - X(4)/2) • KT C C 20  THETVO s DE XPL(-GAMMA * DSORT(BRV)) THETV1 a GAMMA • THETVO/(2'DSQRT(BRV)) THETV2 a GAMMA • (GAMMA/BRV • 1/( DSORT'.BRV)) • *3) • THETVO/4 JTP = ASTOV • (THETVO * (FD1(XNBET2)-FU1(XNBET1)) • • ( K V T H E T V 1 ) * (FD2(XNBET2) F02(XNBET 1)) t  * <(KT"2)'THETV2) * (FD3(XNBET2)-FD3IXNBET1)))  RETURN END  C C  c c C  100 C C C C  c c  50  EXPONENTIAL  FUNCTION  DOUBLE PRECISION FUNCTION OEXPL(X) IMPLICIT REAL*8 (A-Z) IF(X .LT. -150.000) GOTO 100 DEXPL=OEXP(X) RETURN DEXPL=0.000 RETURN END FERMI-OIRAC FUNCTION OF ORDER 1 USING A SERIES APPROXIMATION DOUBLE PRECISION FUNCTION F O K X ) IMPLICIT REAL'S (A-Z) DATA PI/3.14159265400/ Y = X I F ( Y .LE. O.DO) GOTO 50 Y = -Y Z = I.DO'OEXPL(Y) -.250052'DEXPL(2.*Y) •.111747*DEXPL(3.*Y) • • 064557 0EXPL(4.'Y) • 040754«DEXPL(5.*Y) # • .020532'DEXPHB. *Y) • .005108*DEXPL(7. • Y) IF(X .LE. 0.00) GOTO 100 Z = -Z • (X*'2)/2.D0 • ( P I " 2 ) / 8 . 0 0 CONTINUE ,  100  c c c c c c  3  RETURN F01 END  Z  FERMI-OIRAC FUNCTION OF ORDER 2 USING A SERIES APPROXIMATION DOUBLE PRECISION FUNCTION FD2(X) IMPLICIT REAL'S (A-Z) DATA PI/3.14159265400/ Y =X IF(Y L E . O.DO) GOTO 50 Y = -Y  Page  0  L i s t i n g o f PIMSP.S a t 21:32:10 on OCT 523 524 525 528 527 528 529 530 531 532 533 534 635 536 537 538 539 540 541 542 543 544 545 548 547 548 549 550 551 552 553 554 555 558 557 558 559 560 561 562 563 564 565 586 567 588 569 570 571 572 573 574 575 576 577 578 579 580  50  10, 1088 f o r CC1d=KCHU on 0  Pag*  Z " 1.D0*DEXPL(Y> -.125046*DEXPL(2."Y) •.037642*DEXPL(3. •Y) « -.018183'0EXPL(4.*Y) • .012484*DEXPL(5.*Y) « -.007486*DEXPL(6.*Y) • .002133*DEXPL<7.*Y) IF(X .LE. O.DO) GOTO 100 Z = Z • <X"3)/6.00 • ( P I - 2 ) ' X / 6 . 0 0 CONTINUE FD2 = Z RETURN END 4  100  C C C  c c  c  50  FERMI*DIRAC FUNCTION OF ORDER 3 USING A SERIES APPROXIMATION DOUBLE PRECISION FUNCTION F03(X) IMPLICIT REAL*8 (A-Z) DATA PI/3.14159285400/ Y => X I F ( Y .LE. O.DO) GOTO 50 Y = -Y Z a I.DO'DEXPL(Y) -.082592*0EXPL(2. Y) •.013661'OEXPLt3. •Y) # -.00979B*DEXPL(4.'Y) •.012978*DEXPL(5.*Y) » -.010859"DEXPL(8.*Y) •.003446'DEXPL(7.*Y) IF(X .LE. O.DO) GOTO 100 Z = -Z • ( X " 4 ) / 2 4 . D 0 • ( ( P I * X ) * 2 ) / 1 2 . D 0 • 7 . M P I 4 ) / » 360.00 CONTINUE FD3 = Z RETURN END ,  ,  100  C C C  c c c  100  FERMI-DIRAC FUNCTION OF ORDER USING A SERIES APPROXIMATION  300  400  1/2  DOUBLE PRECISION FUNCTION F0102(X) IMPLICIT REAL'S (A-Z) DATA PI/3.141592854D0/ IF (X .LE. O.DO) GOTO 100 IF (X .LE. 2.00) GOTO 200 IF (X .LE. 4.DO) GOTO 300 GOTO 400 FD102 = OEXPL(X) -0.353588*DEXPL(2*X) *0.192439*DEXPL(3* X) # -0.122973*DEXPL(4*X) *0.077134'DEXPL(5'X) # -0.036228 DEXPL(8*X) *0.008346'OEXPL(7*X) GOTO 500 FD102 = 0.765147 • XMO.804911 • XMO. 189885 • • XMO.020307 • XM-0.004380 • XM-0.000366 • # X'O.000133))))) GOTO 500 FD102 = 0.777114 • XMO.581307 • XMO.208132 • • XMO.017680 • XM-0.006549 • XMO.000784 # X-0.000036))))) GOTO 500 XN2 a 1 / ( X " 2 ) FD102 = (X*'1.5) * (0.752253 • XN2M0.928195 • ,  200  # ,  10  Ut)t1n(j o f PIMSP.S a t 21:52:10 on OCT 10. 1988 f o r CC1d=KCHU on G 561 582 583 584  00  • * 500  XN2M0.880839 • XN2M25.7829 • XN2M-553.638 • XN2M3531 43 - XN2' 3254 . 8 5 ) ) ) ) ) ) RETURN END  p  «9  e  1 1  PUBLICATIONS Chu, K.M. and D.L. Pulfrey, "Design procedures for differential cascode voltage switch c i r c u i t s " , IEEE J . Solid-State Circuits, SC-21, 1082-1087, 1986. Chu, K.M. and D.L. Pulfrey, "A comparison of CMOS circuit techniques: differential cascode voltage switch logic versus conventional logic", IEEE J . Solid-State Circuits, SC-22, 528-532, 1987. Chu, K.M. and D.L. Pulfrey, "An analysis of the DC and small-signal AC performance of the tunnel emitter transistor (TETRAN)", IEEE Trans. Electron Devices, ED-35, 188-194, 1988. Chu, K.M. and D.L. Pulfrey, "An improved analytic model of the MIS tunnel junction", IEEE Trans. Electron Devices, ED-35, 1656-1663, 1988.  

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