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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 THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in T H E FACULTY OF GRADUATE STUDIES (Department of Electrical Engineering) We accept this thesis as conforming to the required standard THE 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 Mov/ 2.5-, H g g DE-6 (2/88) Abstract Two improvements to a comprehensive analytic model which describes the steady-state current in a metal-insulator-semiconductor tunnel junction are re-ported. The first modification replaces the conventional two-band representa-tion 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 im-provement 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-insulator-semiconductor (MIS) tunnel junctions is demonstrated by simulating the d.c. and a.c. performance of three major types of transistor with tunnel oxide emit-ters, 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 well-ii 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 PET 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 xi 1 Introduction 1 1.1 General Background 1 1.2 Objective 10 1.3 Thesis Outline • • • 1 2 2 Basic Theory of Thin 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 3 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 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 5 Modeling the M I S E T Device 86 5.1 Model Formulation 88 5.2 DC Characteristics 94 5.3 Small-Signal Analysis . 99 5.4 Summary . . . 112 6 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 7 Conclusion 135 Bibliography 138 Appendix A : Computer Program for Modeling T E T R A N s 145 Appendix B : Computer Program for Modeling MISETs 156 Appendix C : Computer Program for Modeling Unannealed P N P P E T s 1 6 5 Appendix D : Computer Program for Modeling Annealed P N P P E T s 1 7 5 v 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 5.1 Model parameter values for the simulation of the MISET 95 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 PET 7 2.1 Energy band diagrams of MIM junctions, (a) Symmetrical struc-ture, (b) Non-symmetrical structure. 16 2.2 Energy band diagrams for MIS junctions in which (a) Epm < ECQ and (b) EFm > Eco 20 2.3 Dispersion curves for the one-band and two-band representations of the oxide 24 2.4 Different types of tunnel barrier, (a) Rectangular, (b) Trape-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 degen-erate MIS tunnel diode 38 vii 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 repre-sentations 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 inte-grand 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 3.6 Percentage errors, w.r.t. numerical integration results, of J m _ c 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 67 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]. $e independent of energy as in [22] 72 viii 4.4 Prediction of TETRAN characteristics: same parameters as for Fig. 4.3, except for the use of a one-band representation of the oxide. 9C given by one term of (3.4) with \e = 0.8eV 74 4.5 Prediction of TETRAN characteristics: same parameters as for Fig. 4.4, except for the use of three terms in (3.4) for 6e, and Xe = 1.1^ 75 4.6 Prediction of TETRAN characteristics: same parameters as for Fig. 4.5, except for the inclusion of the surface-state tunneling effect, and \ t = 0.85cV. The dashed lines represent experimental curves from (4] 77 4.7 Common-emitter small-signal hybrid-?r equivalent circuit for the TETRAN 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 1 7cm- 3 96 5.3 Common-emitter characteristic of the MISET with PB = 5 x 10 1 6cm- s 97 5.4 Common-emitter characteristic of the MISET with PB = 2.5 x 1016cm~8. The dashed lines represent experimental curves from [5] 98 5.5 Dependence of current gain on collector current of the MISET. . 103 5.6 Dependence of transconductance and emitter resistance on col-lector current of the MISET. 104 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 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 dop-ing density of the MISET I l l 6.1 Structure of the pnp polysilicon emitter transistor [70] 116 6.2 Energy band diagram of the pnp oxidized PET without post-deposition annealing 120 ix 6.3 Energy band diagram of the pnp oxidized PET with post-deposition annealing 122 6.4 Gummel plots of the unannealed pnp PET from computer simu-lation (dashed line) and published experimental data (solid line) [70] 125 6.5 Common emitter characteristics of the annealed pnp PET from computer simulation (dashed line) and published experimental data (solid line) [70] 129 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 supervi-sion 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 resis-tance devices such as the MIS-switch (2). This device has a metal-tunnel oxide-n-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 in-1 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 nor-mally 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 TETRAN (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 SOURCE METALLIC EMITTER ULTRA-THIN INSULATOR N-SUBSTRATE COLLECTOR /N Ic CE Figure 1.1: Structure of a TETRAN 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 approach-ing 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 ex-tensions 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 BASE o EMITTER p 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 junc-tion 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. Un-fortunately, 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 ttlown state, 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 SiOt insulator. Recent work [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 PET [9,15]. The TETRAN 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 hetero-junction transistors is the removal of the necessity to trade-off the transconduc-tance (gm) against the base resistance (Rb) to obtain a high maximum oscillation frequency (fmax)- This property still needs to be demonstrated in the MISET. The npn PET 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 experimen-tal 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 energy-dependent 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. 1.3 Thesis Outline Chapter One introduces the background to, and current work on, MIS and SIS devices. The necessity of developing an improved tunneling model is empha-sized. The first two sections of Chapter Two summarize the mathematical for-mulation 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 sur-veys 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. The 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. The in-appropriateness 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. The trade-off between the base conductance and cut-off frequency is studied in order to optimize the device for high maximum oscillation frequency. Chapter Six describes the modeling of pnp polysilicon emitter transistors, and treats both annealed and unannealed devices. The fact that these transis-tors 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 * 1 eM-lVxD - Xa exp(--yv/xT)] (2-1) 15 Figure 2.1: Energy band diagrams of MIM junctions, (a) Symmetrical struc-ture, (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)) where m* is the effective transverse mass of electrons in tne metal, es is defined in Fig. 2.1(b) and 8e{e%) is the tunneling probability, which is energy dependent, and can be expanded using the Taylor's series 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(eK) represents the difference between the number of electrons per second per unit area, having (x-directed) energy in the range (2.2) m[Me,)J = -{h + cxet + fa\ + •••) (2.3) 17 Ex to Es + dEx, incident on opposite sides of the barrier. Jo = F Jo = kT In 1 + exp.(5g£*) 1 + e x p ( ^ ± ^ ) J " l + exp(ft)  > 1 + e X p ( 2 # ) (2.4) Truncating the expression for 0e, i.e. (2.3), to its first two terms and putting this into (2.2), gives after much mathematical treatment (and use of (2-4)), Jn = 94*?t»Texpi-biH^,*. LrrAl - exp{-ciqV)] - Ry - R2} cths ' sin(7rciA;T) (2.5) where R2 = f Jo .eifcT-1 •\dx l l + x 1 + xexpiqV/kT)1 In the case of a metal, f is usually very large and R\ and Rj are negligible. Jn 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) s exp(-60 e x p ( - ) - e x p ( - ^ - ) (2.6) " H / i s l-ClkT 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)) 4irmtek2T,/l  n = q h1— ' T / Epno — Ee0 . T . Epm — Ee, kT (2.7) where m* is the electron transverse effective mass in the semiconductor. The tunneling probability 9e is assumed to be constant. Except for large tunnel barrier heights (x«) and small oxide thickness (d), and the case when the Fermi-levels are well below the conduction band edge, this expression can cause very large errors. For EFno - Eeo < 0 and EFm - Eeo < 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] Q<t>b J n = A T ^ e e x P ( - ^ ) exp(f— ) - l qV (2.8) kT' I rKnkT' where A* is the effective Richardson constant (= 47rgm*fc l//i 3) and ^ is the Schottky barrier height. The ideality factor n (usually ~ 1 - 1.5) is defined as 19 E F M //////////// EFT ( a ) (b) Figure 2.2: Energy band diagram? for MIS junctions in which (a) Epm < Ecc and (b) EFm > Eco. 20 —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: Jn = A*T\(KV + Kt) e x p [ -7ete)* ] - K3 exp[-<7e(*;)*]} (2.9) where 1 1 + K2 = K3 = 2(jg F m-i? C 0)te)i 3*27/H 2{EFm - Ejfc)* 3)fcar87e and 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 9C. The tunneling probability, using the WKB approximation, 21 is given by (25] 0 = exp[-2 [*'\klx\dx] (2.10) where xt, xm are essentially the positions of the oxide interfaces with the semi-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 for-bidden gap of the oxide is required. The complex electron wavevector fcj in the oxide is usually taken to obey the Franz dispersion relationship [26] kj ~ 2 m ; ( £ - EH) + 2mU{Evi - E) ( ' ' 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 ^ ( * - *,)*(! - - (1 - ^ ) ^ ] - « (213) where is the oxide bandgap {—Ed - Evi) In the past, a two-band model (or Franz-single mass model) has generally been accepted. It assumes that m*d and are equal, and renders (2.13) to 22 [22,27] t , = a f i i t ( , - _ f t ( , i ( 1 _ * L i * , i ,2.14) However, recent calculations of the band structure for Si02 in the a-quartz form indicate that mj,- ~ 0.5me (m, is the electron rest mass) and m*,- ~ (5 — 10)me [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?&{E_Eji ( 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 prob-ability 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 = ~ (2.16) where Ex is the energy associated with the electron momentum in the 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{Et) = exp[~(2m^{Eei - Ez)i] (2.17) for a trapezoidal barrier, $(Ea) = exp{~(2m^H^)\(Ed - Em)\ - - - EM)\\) (2.18) for a triangular barrier, ${Ea) = « p [ - ^ ( 2 m ; ) * ( 5 A . ) K " (2-19) Once 0(EX) is known, J„ in (2.7) can be determined by setting 9e = 9(Eeo). 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~^ - Eeo (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 (cn,cp) and thermal emission (c„,ep) rates for electrons and holes per active trap at the semiconductor surface are given by CP C n EP where n„ p, are the electron and hole concentrations at the surface, on and ap are the thermal capture cross sections for electrons and holes, riy and pi are defined as Neexp(^t^A) and Nvexp(Et^u) respectively, and vtn is the carrier thermal velocity. Neglecting trap photoemission [35], the various currents that 27 — = vthannt (2.20) *en — = vthoppt (2.21) Tep £ Un vthonni (2.22) 1 VthOpPi (2.23) Figure 2.5: Surface-state tunneling in a MIS system 28 enter and leave the trap can be written as Jet = -qNt[cn(l - ft) - enft] (2.24) Jvt = qNt[cpft - e„(l - ft)} (2.25) Jnu'= qNtvm{ft - fm) (2.26) where Nt is the density of states at trap energy level Etr, ft is the occupancy probability at Eti fm is the occupancy of Et in equilibrium with the metal, and um (=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 Jet + Jvt + Jmt = 0, the surface state occupancy ft has a value between fm and / , and is given by ft = Ttf' + T ' f m (2.27) U + r. where and T. = Vth\vn{n, + n,) + (7p(p, + Pi)] (2.28) /, = nton + pxap  {n, + ni)on + {p, + Pi)0p 1 s — s — if p. n. l + e x p ( ^ » ) 1 + e x pkg8*.) i f * kT 29 Putting (2.27) into (2.26), the surface state tunneling current via a single level trap is given by Jmt = qNt(^-^) (2.30) T, + Tt For rt » rt, e.g. in the case of oxide thickness < 15A, (2.27) simplifies to ft « fm. 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 rt » r,, e.g. in the case of oxide thickness > 25A, (2.27) reduces to ft « 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 rt. We call this the oxide tunneling controlled case. The tunneling rate can be defined as [33,36] Vm = I/* = Vmo{Etr)0{Etr) (2.31) where Vmo (Etr) is the attempt-to-escape frequency at the trap energy level Etr, and 0(Etr) is the tunneling probability at Etr. A variety of experimental meth-ods have yielded values of Vmo in the range from 10s to 1013 Hz. Recently, Jain 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 1015 Hz at midgap to 1010 Hz at the silicon conduction band edge. 30 //////////// t t Figure 2.6: Energy band diagram of a metal-St'CVpSt tunnel diode. The surface state density distribution Nt(E) within the semiconductor bandgap and the carrier capture cross section areas on, op can be estimated by small signal capacitance and conductance techniques [37,38]. Consider a metal-Si02-pSi tunnel diode as shown in Fig. 2.6, the applied bias VB is varied such that the hole Fermi level Epp can scan across the whole Si bandgap. With each biasing point, the a.c. capacitance Cm and conductance Gm 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 > 20A, the recombination currents are much larger than the tunneling current Jmt under the application of an a.c. signal. The circuit 31 + vB > Vox > Ips C0x Cgc R]3 O—i 11 1 (I , — w \ — o I 1 -r-Cs (a) R* c, ox C>ctt.) R B — | | 1 — W v—O C.(> f) £ , (0 , 1 J | — W V — - W v (b) -j I 1 1 I i WV—O o_ 1| o ( c ) (d) Figure 2.7: Small-signal equivalent circuits of a MIS diode. Steps (a)-(d) rep-resent 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 (CM(V'«,u;),Gm(^,,a;)) curves [37] are shown in Fig. 2.8. Once Cm and Gm are known, Cd and Gd can be deduced from Fig. 2.7(c) and (d) providing the bulk resistance RB is given. The surface state capacitance C, is obtained by subtracting the calculated space charge capacitance Cte from Cd at equilibrium (or very low) frequency weq: C.ty.) = C R F(V'„u; e,)-C. e(0,) (2.32) Then, the interface state density, Nt = Ct/q, is found from C, and its energy position in the Si bandgap. From the conductance, the distribution Nt(E) can 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 (ap or an) can also be found. 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) EV EC (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 1011 to 10 1 3cm" JeV _ 1 , and are greatly affected by metal diffusion through the oxide film. The upper limit of Nt can possibly be due to a limited solubility of metals in the oxide film. The surface states are usually found to be acceptor-like. The typical values of <rn, ap are found to be 1013cmJ, although a spread of several orders of magnitude is possible [36,37]. 2.3 The Determination of Barrier Height There are several methods to determine the tunnel barrier energies. Dressendor-fer 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 il-luminated 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. The fai, 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, fai is 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 Jn)/#V]-vs-V curves from this experiment are shown in Fig. 2.10. Abrupt changes in the slope of Jn versus V should be observed at V= xe/q and 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 2 on degenerate p-type Si, and from 0.64 eV at 14 A to 1.27 eV for 29.3 A of Si0 2 on degenerate n-type Si. The Al-SiOj barrier height (xm) is determined to be about 0.61 eV. In the case of non-degenerate semiconductor, Card et al. [23,41] have de-duced 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(Jn)]-vs-V curve. Therefore, the unknown 9e is determined, and x« is found from the relation 0e ~ exp(—\\l*d). Kumar and Dahlke [42] have proposed another curve fitting technique. 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 ) qVm<X (b) qVm > 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. 4 0 Jn = A^ec\Jx{-q^)-U-q^)\' (2.35) where The measured forward and reverse currents, and the surface Fermi po-tentials Vt and Vm obtained from the measured surface potential V',(V), are used to calculate the barrier function r/(Vm) = ln(A*rJ0,.). The values of U as functions of V ^ 1 and V m are shown in Fig. 2.12. The slopes \si\ = x'^/v 7^ of U{V~l) at large V m and s2 = d/{4Xl/2) of U{Vm) at small Vm, and the difference of their intercepts with the ordinate s3 = xxl*d are also indicated. By finding 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 Jn = J . [ exp(^ ) - 1] (2.36) J, is the extrapolated saturation current and is related to the metal-semiconductor barrier height fa by J . = A-r'* eexp(-A;) (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 Kumar and Dahlke [42]. 42 the experimental data. The Richardson plot \n(Jt/T2)-vs-l/nT results in a straight line, and the values of fa and A*0e are obtained fron the slope and the y-intercept. Once 0e is known, the tunneling barrier height can be found. Realizing that the previous studies of barrier height have been confined to the tunneling of electrons, Ng and Card [24] proposed a procedure to deter-mine the tunneling barrier of holes for Au-Si02-hSi tunnel junctions with oxide 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 inten-sity. In the steady state, the photocurrent JPN is equal to the summation of the short-circuit hole tunnel current J,C and the back diffusion current J<j- One can formulate an explicit expression for the hole tunneling probability \h as «p(-xi"rf) = , D p { ± ^ ! l ! £ , e x p [ ( ^ ) ( ^ - 1 ) ]} - (2.38) The magnitudes of JPH and JTE 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 is replaced by a one-band 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 i0 2 46 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. There-fore 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 — Et — Xe (3.1) where E^ is the oxide band gap, Et is the semiconductor band gap and Xe is 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 sym-bolic band diagram of Fig. 3.2(b) which we propose. The evanescent states in 47 OXIDE CONDUCTION BAND ^ >kt / < COMPLEX CONDUCTION BAND •—' N / \ ' \ V _ _ ~ - / \ / o e ~ r \-l SEMICONDUCTOR I ^ J FORBIDDEN GAP ^ \ COMPLEX VALENCE BAND—is ^ - -« OXIDE VALENCE BAND (b) Figure 3.1: Band structures of the SiOi barrier assumed in (a) the two-band model and (b) the one-band model. 48 Figure 3.2: Energy band diagrams for (a) two-band and (b) one-band represen-tations 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 0n are similar to the set given by (2.17)-(2.19), with Ex, Ed and replaced by Ex, E'^ and E^. 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 (xe ~3.2 eV, 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 New Power Series for Tunnel ing 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 z . This means the expressions can be adapted 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 dif-ferent operating regimes, and is not restricted to particular relative positions of E F m , Eeo and E f n . Thirdly, the series are fast converging and calculate, within 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 tunnel-ing current can be written as (see Fig. 3.3) 51 Jem — 1 + e x p ( ^ ± % ^ * ) 1 1 + exp{&±&f^) | 9edEx (3.3) The electron tunneling probability 6e(Ex) can be expanded using the Taylor series with respect to the conduction band edge 9e(Ex) = ee(Eeo) + 9\{Eeo){Ex-Eco) + 9-^^{Ex-EeoY + ••• + nl (Ex - Eeo)n (3.4) where d"9c dE? as Substituting (3.4) into (3.3), the electron tunnel current can be written 4nqm'e Jem — (3.5) where JE.0 [JO T n=0,l,l dEt [Ex - Eeo)'dEx and Imj is the same with EFm in place of Epn. 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. _ f°° f fE« H(Eeo)  inJ~Jo [JEeo j\ 1 + Substituting n = E'~E-, we get (Et-Eeoy e x p ( g ^ y r " ) dEt - f l / 0 ^ ( ^ ^ ( ^ 1 + m p ( ; ; d L ^ , dEt Since the normalized barrier height E t i k T E " is usually very large (~40 for a 1 eV barrier), we could write °°.rfEFn- Eeo-Et = (kTy+le>(Eeo) /o°°[^( CO ' (3.6) Substituting /„, and Imj back into (3.5), we obtain Jem = — ^ E \kT)ne:{Eeo) fc3 n=0,l,2 ^ F n ~ ECL kT EFW. — EEO, kT (3.7) where 0{{Eeo) is defined in (3.4). Similarly, the hole tunneling current is given by (see also Fig. 3.2) 4nqmlk2T2 ^ / . ^ n - r w j f i x f T - EFp^ T ,EV0 - EFm^ J»m -~" 2 - \ k l I *v\t,*o) |'n+lV £ p ) - ?n+H ^ ) n=0,l,J kT (3.8) where e:(EV0) = drev 54 D Notice that the first terms of the power series expressions (3.7-3.8) are exactly equivalent to the Jem and Jvm expressions obtained in [22]. The impor-tance of the additional terms that have been derived can be readily seen in the MIS system in Fig. 2.2(b) where Epm < Eeo. Suppose electrons are tunneling from the metal to the semiconductor, and assume that Eeo — Efm ^> 0, the magnitude of this component of the electron current can be written as (from 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 Et, and the curve is exponentially increasing with Et (see (2.17)-(2.19)). If we approx-imate this curve (# 3) using a Taylor series expansion ».bout Ex = Eeo as in (3.4), then 9\ is the first term, 93 and <?s are the summations of the first two and three terms respectively. This implies 9\ is constant, 0 2 is linear and 03 is quadratic, etc. In the case where Epmi < Eeo (see curve # l), the integration of the product of fi and 9\ in the interval [Eeo,Eci\ gives a good estimate of Jm_, c , because of the exponential drop of fi near Ez — Eco. This is why the assignment of a constant tunneling probability is satisfactory in this case. As the Fermi level rises above Eeo to a position #FmJ> see curve # 2, the integral of the product of / a and 9 in the interval [Eeot Epmi\ contributes much to J m - e . (3.3)) (3.9) 55 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 Jem series (3.7) should be kept. To investigate how fast the series expression for J m _ e (3.7) with Eco — 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 com-binations 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 rectan-gular 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 nor-mal operation range where Epm — Eeo < 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 (Acnr2) 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 Jm-,e as computed using Simmons' expression and various numbers of terms in the series expression, (d = 16A, x« = 1-leV.) 59 % error 20 I -80 i Simmons' eq. (kT) 3 terms 2 terms 1 term 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« = MeV. 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 — Eeo — 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 Em — Eeo, therefore the truncated series 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 Formulat ion The revised model for the MIS tunnel junction utilizes a one-band representa-tion of the oxide and allows for the energy dependence of the tunneling prob-ability factors. These two improvements to the original model are included in 62 the new model by replacing the expressions for Jcm and Jom ((12a) and (13) in [22]) by (3.7) and (3.8). The expressions for the electron and hole tunneling probabilities at the band edges (0cm and 0vm 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 appli-cations. 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 (emit-ter 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 inversion-channel 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 T E T R A N . 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 deple-tion 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. Mod-est 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 two-band model for Si02 based on the Franz dispersion relation has been replaced 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 JCM and hole tunnel current JVM are given in (3.7) and (3.8), and the electron and hole barriers (xe and XH) are related by (3.2). In the model, x« is taken as an adjustable fitting parameter. Any solution for the tunnel current JEM and JVM must be consistent with the stipulations of Kirchoff's laws. Regarding voltages, summation of the potential drops shown in Fig. 4.2 yields: ^ + ^  + - + ^ - — - ^ = 0 (4.1) q q where VCB is the collector-emitter voltage and if)j is the potential drop across the oxide given by 0/ = —Q. (4.2) where €/ is the oxide permittivity and Q„ the total charge in the semiconductor, is derived in Appendix B of [49] as: Qt = sgn(^y/2l^t[NJl/2(EFnkTEeo) - n 0 + Nvfl/2( E^^)-Po + N D ^ (4.3) 68 Figure 4.2: Energy band diagram for the T E T R A N . 69 where n0 and p0 are the equilibrium concentrations of electrons and holes re-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. = Jvm + Jd (4.4) where Jj is the current due to diffusion of holes from the source into the quasi-neutral 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 Jvm 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,, VCE, \h and Xe ( s e e (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 Jem can be com-70 puted, as well as the terminal currents: JB — Jem + Jvm Jc = Jem ~~ Ji (4.6) (4.7) A computer program has been written to evalute the steady-state charac-teristics 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 character-istics were generated for a device structure resembling that used in the experi-mental work reported in (4). The physical parameters of the device were: oxide thickness = 16A, collector doping density = 7 x 1014cm~s and collector thick-ness = 10/im. The device exhibits a current gain of about 120 at a collector current density of around 10sAcm~2. To simulate the experimental I-V curve in [4], the tunnel model in [22] was first used, i.e. the two-band model with con-stant 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 TETRAN characteristics: d = 16A, other parameters as in Table I of [22]. 6e independent of energy as in [22]. 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 0.5me 0.2m, 0.66m, II. 9c, 3.9c0 4 8 0 c m 2 V - V l 8/w 1.45 x 101 0cm- 3 2.8 x 101 0cm-8 1.04 x 10 l 9cm- 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 Jem and J v m , 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. 0e given by one term of (3.4) with Xe = O.BeV. 74 (X 10 3AC»"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 0C, and x« = l.leV. 75 gain AJe/Av/, is 100 at a collector current density of around 2 x 10 3 Acm~ 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 30% 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 Lp in the expression 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 Jd over Jem. Turning attention to the low gain region at low source current, it is sug-gested that the neglect of surface-state tunneling is responsible for the ex-perimental 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 J-V 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 on, ap ~ 5 x 10"1 8cm J, the surface state density is Nt ~ 5 x 1012cm~' and the tunneling time constant is r m ~ 1.2 x 10~8s. All these parameters lie within 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 TETRAN 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 104Acm~2. This effect was attributed to the generation of surface states at the oxide-semiconductor interface. The results of the proposed model suggest an alter-native 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-1, which is about the breakdown field strength of thin oxides [48]. The saturation voltage VCg is observed to be about IV in the simulation, and is much less than the value of about 5V measured in the exper-imental results. This discrepancy is most probably due to the emitter contact 78 resistance. The results of this work on the TETRAN device have been published [49]. The value for electron tunnel barrier height (xe) that fits the T E T R A N 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: dJc Transconductance gm = dVSE Common-emitter input capacitance C„ Collector-source capacitance C,e = Input resistance rte = Output resistance r„ = dQ. dVSE dQ, CE dVSE dJs dVCE dJc Js The unity current gain cut-off frequency can be expressed as: 9m h = 2*C.e\l + 2Cte/C.e\1!> (4.8) 79 Figure 4.7: Common-emitter small-signal hybrid-7r equivalent circuit for the T E T R A N device. 80 0m (Sm-») C.e Cte (Fm"8) (Orn8) (Orn8) fr (GHz) VCB = 5V X t = LOeV J, = Z2Acm~1 2.7 x 10* 1.8 x IO - 1 2.0 x 10"T 2.0 x IO - 7 1.9 x IO"4 2.3 VCE = 5V X, = l.OeK J, = 8i4cm~J 0.9 x 10* 2.1 x IO"1 5.0 x 10~T 6.5 x IO"7 1.8 x IO - 4 0.7 VCE = 2V Xe = LOeV J, = 32>1 cm - 1 1.8 x 10* 1.6 x IO - 8 2.0 x 10~7 3.0 x IO"7 1.9 x 10~4 1.8 VCJE = 5V X« = 1 K V J, = 32j4cm-8 1.5 x 10* 1.7 x IO"8 1.8 x IO"7 2.0 x 10~7 3.3 x 10~4 1.4 Table 4.2: Simulation results for the small-signal hybrid-* parameters of the T E T R A N . 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 semi-conductor in response to small changes in either J c , V$E or VCE- The pertur-bations had a magnitude of 1% of the operative steady-state values. The results for a range of operating conditions and two different values of electron barrier height, Xe, are listed in Table 4.2. With respect to the conditions used to obtain the first row of the Table 4.2, note that a reduction in either J$ or VSE, or an increase in x«5 leads to a reduction in /r , principally 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 «600 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 gm for the BICFET should exceed that for the T E T R A N . The input capacitance Ctt usually exceeds Cte, certainly in the case of the T E T R A N (see Table 4.2), 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 gm and C0t will cause fx in the case of the T E T R A N to be inferior to that of the 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 gm we neglect the contribution of the diffusion 82 current to Jc, see (4.7) and seek an expression for Jem 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: Jem = JR'exp[-7e(Xm - ^//2) 1 / J] (4.10) Jem w Jc as noted above, and 0/, the potential drop across the insulator, is the main contributor to VSE, thus: „ dJe _ dJem qle Taking Xm = l.leV, d = 16A, VCE=5 V and Js = 3.2 x 106 Acm" 2 (i.e. as per row 4 of Table 4.2) we find from the model that Jc = 3.8 x 107 A m - 1 and 0/=O.68 V. Substituting these values into (4.11) gives: gm = 3.3JC = 1.1 x 10 8n- lm _ 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 €/ = 3.9c0 we have C„ = 2.2 x l O - ' F m - 1 . Using these approximate values of gm and Cte in (4.9) yields /r=0.8 GHz. This figure is in 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 care-fully studied by numerical simulations. Computer analysis indicates that the T E T R A N is a modest gain (~ 100), low current (Jc up to lOMcm - 2 ) and 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 ex-tent by optimizing the oxide parameters, without introducing serious reliability problems. A possible way of improving the performance of the TETRAN is by chang-ing 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 (Jcm), and therefore increase the 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 un-derneath 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 maximum d.c. gains of 600, 10000, and 25000 with base implant doses 5x10", 10 1 2, and 5x 1 0 n c m - 2 respectively. Theoretical work has not been reported to explain the extremely high gain achieved by these transistors. Also no theoretical (or experimen-tal, 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 character-istics 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. The 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 . Fro 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 fmaX, the trade-off between the base conductance and / r is also studied. 87 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)t. Once <f> and 0, are known, all the potentials and 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. = - S gn(^)(2tr ( . ) " ! [ JV c 7 1 / I ( - 1 ^i) -n s exp(i^) + " v 7 V , ( ^ ) - P B + ^ V (5.1) where qa3 = Eah -q<}>- q<f>0 - q$, Assuming the absence of fixed charge or surface-state charge at the semi-conductor/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 ipt. As a matter of fact, 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 Jem = A\T2 ±(kTY9i\7j+l{^) - Tj+1(^)) (5.3) and Jvm = A\T* t(kTy9i[Ti+l(^p-)-?i+1(=^)\ (5.4) 90 respectively, where = Xm + - Xe 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*T and A*H are defined as in [22]. Generally, series expressions with three terms will be sufficient to approx-imate 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 VCB as ne qPebn0b Leb {coth(^)[exp A - 1] - csch(— )^[exp( -qVcB kT ) 1]} (5.5) J, ne qDeb^ob {csch(g)[exp(|£) - 1] vv' - c o t h ( - ± ) [ e x p ( --qVcB kT 1]} (5.6) where VCB = V C E + 0. + 4>0 - E t h / q + V/ + Xm/q - Xe/q 91 not is the equilibrium electron density, Dtb is the electron diffusion constant and La is the electron diffusion length in the neutral base. W'b is the effective base 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 Dei is related to fieb by the Einstein relation kT Deb = — ( 5 . 8 ) 9 The mobility of electrons fieb is dependent of the base doping density Pg and is given by the empirical relation (valid in the range of PB ~ 1018 — 10 J 0cm_ 3) [57] 1 2 5 2 x 103 . . . , x * » = 8 8 + 1 + 6.984 x 10-" x PB(cm->) C m V ° ^ The electron diffusion length Ltb is given by Leh = yjDeirlh (5.10) The electron lifetime TEB also depends upon the base doping density PB, and is normally related experimentally by [55] T ' b = 1 + (PB/5 x 1016em-3) ( 5 1 1 ) 92 In the base-collector depletion region, the generation current is expressed as Jt = —^-\exp(~2~kf~) ~ li (5-12) where lf,e is the depletion width at the base-collector junction and rt is the generation lifetime of the carriers. The hole current due to diffusion from the collector to the base is formu-lated as Jd = "9\/^ P°c(eXP(~lr£) " 11 (5ll3) where Dhc is the hole diffusion constant, is the hole lifetime and poc is the equilibrium hole concentation at the collector. To obtain <j> and rp, when the voltage applied across the collector and emitter (VCE) and the base current (JB) are given, two non-linear equations need to be solved: Jne-Jem =0 (5.14) JB + Jt + Jd - Jvm ~Jrb = 0 (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: JE = Jem + Jvm (5.16) 93 Jc = Jne + Jt + Jd (5.17) A computer program has been written to evaluate the steady-state char-acteristics 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 101Jcm"J), the device h?s a moderate current gain (/?) of about 600. For a base dose of 101 ,cm" J, 0 increases from about 3000 at low voltages to about 10000 near the punchthrough voltage which is in excess of 25V. For a base dose of 5 x 10ncm" J, /? increases from above 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 e is assumed equal to X m ) , the oxide thickness (d), the base doping density {PB), the base width (W») 94 T temperature 300A: silicon bandgap at base region l.UV m ; electron effective mass in St'Oj 0.5m, ml electron transverse mass in Si 0.2m, < hole transverse mass in St 0.66m, «» permittivity of Si 11.9ce «/ permittivity of Si'Oj 3.9e, U carrier generation lifetime at base-collector depletion region 0.2ns *kc hole recombination lifetime in collector 0.2ns Dkc hole diffusion constant in collector 12cm8*-1 nt intrinsic carrier concentration 1.45 x 10I0em-8 Nc conduction band density of states 2.8 x 10wcrrr 8 Nv valence band density of states . 1.04 x 10"cm-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" 8 « and three different values of P^, the collector 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 1017, 5 x 1016 and 2.5 x 10 1 6cm - 8, which are equivalent to the base implant doses of 5 x 1018, 1018 and 5 x 10 n cm - 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 PB = 2.5 x 10 1 7 cm - 8 (Fig. 5.2), the current gain /? increase from 500 at low VCE (~1V) to about 750 at high VCE (~30V) when Jc is low (~ 8 x 10~ 3Acm" 2). For PB = 5 x 10 , 6 c m - 8 (Fig. 5.3), 0 increases from 4300 at VCB ~ IV when Jc ~ 1.6 x 10 _ 8 Acm- 8 , to 95 COLLECTOR-EMITTER BIAS ( V ) Figure 5.2: Common-emitter characteristic of the M I S E T with PB = 2.5 x 10 1 7cm- 9. 96 Jc 97 Figure 5.4: Common-emitter characteristic of the MISET with PB = 2.5 x 1016cm~8. The dashed lines represent experimental curves from (5). 98 about 7000 at VCE ~ 20K which is near to the base punchthrough voltage. In the case of PB = 2.5 x 101 6cm-3 (Fig. 5.4), 0 increases from 10600 at VCE ~ IV to 25000 at VCE ~ 3V near base punchthrough, when JC ~ 3.2 x 10~4Acm~2. 5.3 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 (5.18) dJc <7m Vats dVBE For high performance transistor design, the emitter series resistance Re needs 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 (5.19) R t = ^ dJB The base-emitter capacitance C\,t is also a critical parameter determining 99 the speed of transistor operation. It is defined as dQB C 6 e = (5.20) VuB dVBE QB 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 Qmin in the neutral base region, i.e. QB = Q. + Qmin (5.21) 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 dQ. Cie » (5.23) VOB dVsE The unity current gain cutoff frequency can be expressed as In the MISET device, the base-collector capacitance Cbe is mainly a depletion layer capacitance, and is usually small compared to Cbe. Thus, fr can be 100 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 VBE-In 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, W9 = 0.2/im, PB = 2.5 x 1016cm~3 and rob = 4 x 10_8s. At VCE - 2V, a base current {JB) of 32Acm~2 gives rise to an collector current (Jc) of 6.6 x 10 3Acm - 2 . By changing JB slightly, i.e. perturbing the steady-state system, we obtain d<j>/dVBE ~ 0.08, dQg/dVBE ~ 1.3fiFcm~3 and gm ~ 2 x 10 4Scm - 2 . The base storage capacity Cji/f is estimated by the formula ^ , / / = ^ ; ( ^ ) n o t e x p ( ^ ) (5.26) where n„> is the equilibrium electron concentration in the neutral base. This equation gives a value of 1.2^tFcm-2 for C ^ / / . Even though the magnitude of Cji/f is comparable to dQtjdVBEi the small ratio d<f>/dVBE justifies the use of (5.23) for finding C\,t. Also C\,c is found to be about 0.018^Fcm -2, which is small compared to C\,t, thus justifing the approximation of fa in (5.25): 2xl0 4 5cm" 2  h * 2* x 1.3 x l O - ' F c m - 2 = 2 A G H * 101 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 pop-ulation of holes. Under these conditions, it is possible to estimate fr by analyt-ical 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 103j4cm -2 into (4.11) gives gm = 4.8V - 1 x Jc « 3.2 x l O ^ ^ c m - 2 For d = 18A and tj = 3.9e0, (4.12) gives Ctt = 1.9/iFcrrr2. Using (5.25), the 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, gm, Rg and /r on the collector current Jc, we plot all thede parameters against Jc in Figs. 5.5-5.7. All the parameters have been calculated numerically, since the previous analytical expressions for gm and Clt are only valid in the regime of 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 104Acm~7. The plot resolves the large discrepancy 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 lAcm - 2 ) , while in (65] Jc was several orders of 102 Figure 5.5: Dependence of current gain on collector current of the MISET. 103 gm He • I I J 1— IO" 1 1 10 102 IO3 COLLECTOR CURRENT DENSITY ( A c m - 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 gm rises while Rt falls as, Jc increases. It is interesting to note that Re can be higher than 104fiflcm2 at low collector 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 K r M c m - 2 is about 35jiftcm', that is an order of magnitude higher than typical values for the polysilicon emitter transistor (see [66]). Re degrades gm, 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 Re, and this is precisely the reason for reducing the interfacial oxide thickness as much as possible in high speed PET 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 Jc, to about 2.4GHz 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 Jc-In conventional high-speed BJTs, there is a trade-off between the base 106 conductance and / r , and the base doping density must be carefully chosen 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] where Rb is the base sheet resistance and C\,e is the base-collector capacitance 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 fmax 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 (5.27) 107 / is the length of the contact. pb is the resistivity of the base region, and the 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 fmac are plotted against base doping density PB for two values of instrinsic base lifetime TOH (see (5.11)). The device parameters are: \t — O.BeV, d = 18A, Wb = 0.2/nm, AE = h x / = 5 x Sfim2 (emitter area), and the transistor is operating with JB = IGAcm-2 and VCE = 2V. Referring to the case of the larger base lifetime (rgb = 5 x 10"7s as used in the SEDAN III program [55]), 0 and fT stay more or less constant up to PB ~ 7.5 x 10 1 7cm - 3. /maj,. rises rapidly up to this point because the base resistance decreases as PB increases. Beyond Pg ~ 7.5 x 10 1 7cm _ s, fmax tends to increase slowly because fa starts to fall. Even at the high base doping density of 5 x 1018cm~3, fmax has still not reached its maximum value because 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 rob = 5 x 10 -8s, which might be a typical value in actual devices due to excessive base recombination, / r drops rapidly beyond the doping density of Pg = 2 x 1017cm~3. When it reaches PB = 7.5 x 1017cm"3, the increase in base conductance cannot compensate for 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* o I H 1 0.5 H io17 io18 BASE DOPING DENSITY (cm-3) 10 1 9 Figure 5.9: Dependence of cut-off frequency on the base doping density of the MISET. 110 ' ' *->PB 10" 1018 10»9 BASE DOPING DENSITY (cm"3) Figure 5.10: Dependence of maximum oscillation frequency on the base doping density of the MISET. I l l 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 VBE, but low current gain and high fx at large bias. The reason for this is that the emitter resistance Re decreases as the bias increases; this is an inherent property of MIS tunnel junction emitters. The cut-off frequency of the MISET is found to be about 2GHz at high collector current, close to the fT predicted for the TETRAN. 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 TETRAN 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 fmat 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 / m a * - 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 low-power applications. If pnp PETs can be optimized to exhibit values of fmax 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 PET re-ported 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 tem-perature. 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 Gtof 1 - 2 x 10 1 4scm - 4, combined with an emit-ter resistance Rt of about 26/xncm2. If the device is annealed at 900°C for 30 minutes after the polysilicon film deposition, a monocrystalline emitter ap-proximately 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 Rt (~ 1 — 2^tf2cmJ). Both types of device can exhibit current gain of up to about 300. The reasons why pnp PETs can exhibit good gain and low emitter resis-tance 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 back-injected 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 success-fully 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]. The high frequency performance of these devices is assessed by computing some important a.c. parameters. 6.1 M o d e l Formulat ion In the unannealed device, there is no mono-emitter region underneath the tun-nel oxide. The 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 P E T 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 unan-nealed 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 con-sidered. The minority carrier diffusion in polysilicon film can be described by exactly the same equation used for diffusion in mono-silicon. Taking into con-sideration 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-)\Nc exp( — '-)- npol] (6.1) where npoi is the equilibrium concentration of minority carriers, Wpol is the polysilicon layer thickness, Dpo\ and Lpoi are the effective diffusion constant 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 Eeo, the energy above which electron tunneling occurs, is in line with the poly-Si conduction band edge, while Evo, below which hole tunneling occurs, is coin-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 Eco and Evo respectively. The parameters ax, a 2 , Pi and which determine the tunneling currents (see (5.3) and (5.4)), are defined as the differences between the quasi-Fermi levels with respect to Eeo or Evo. The effective electron (or hole) tunneling barrier height is the energy 119 Figure 6.2: Energy band diagram of the pnp oxidized PET without post-deposition annealing. 120 difference between Eeo (or Evo) and the middle point of the oxide conduction (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 JnX and the tunneling currents Jtn and Jtp are treated in exactly the same way as for the unannealed devices. J r m is the recombination current in the mono-emitter region, and Jn2 and Jnm 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 — VCB and Wl by fa and the mono-emitter depth, respectively. Jrb is the recombination current, while Jpt and Jpe 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 Jt at the base/collector junction is 121 Figure 6.3: Energy band diagram of the pnp oxidized PET with post-deposition annealing. 122 given by (5.12), and the recombination current J r at the base/mono-emitter junction is also given by (5.12), but with — VCB and lbc replaced by <f> and the 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 = Jtp ~ Jrm — Jr (6.2) Jnm = J B + J t + J d - J r - Jrh (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, fa and <f> are determined, the terminal currents can be com-puted by JE = Jtn + Jtp (6.6) Jc = Jpc + J f + Jd (6.7) 123 Computer programs that are used to obtain the steady-state character-istics of unannealed and annealed pnp PETs are listed in Appendix C and D respectively. 6.2 D C Characterist ics 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 base-emitter bias (VBE)- Some of the physical parameters for this simulation are: electron barrier height (xe) = 0.5eV, oxide thickness (d) = 10A, £1 = O.leV, base width (M j^) = 0.3/im, base doping density (Pp) = 101Tcm~3 and intrinsic base lifetime (r„&) = 5 x 10~8s. The value of ft used here corresponds to a polysilicon doping density of 1020cm~3, i.e. about the solid solubility limit for boron in silicon. The value of x« u s ed gave the best fit of predicted and 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 (Npoi ~ 2 x 1019cm~3), the minority carrier mobility and lifetime are estimated to be about 8 c m 2 V - 1 a - 1 and 2 x 10"10s respectively [12]. Therefore, in the simulation performed here Dpoi and Lpoi are taken to be 0.2cm2s_1 and 0.06/xm, 124 BASE-EMITTER BIAS ( V ) Figure 6.4: Gummel plots of the unannealed pnp PET from computer simula-tion (dashed line) and published experimental data (solid line) [70]. 125 respectively. The d.c. characteristics are not affected significantly by changing the val-ues of Dpoi or LPoi by one or two orders of magnitude. This implies that the quantum mechanical reflection of electrons at the oxide barrier is the dominant mechanism for preventing back-injection, consequently minority carrier diffu-sion 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 experimen-tal 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 polysil-icon 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 Gt and the emitter resistance 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, JB can be related to Ge via qn] fqVEB* ( E I O X JB = exP(-Jtr~) (6-8) This means Ge can be easily computed by determining the base saturation cur-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 Gt 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 1014Scm~*, which is in the same order as the experimental result of 1 — 2 x 10 1 4acm - 4. An expression for the emitter resistance RT has already been given in (5.19). For a high collector current level (~ 103Acra -2), RE is computed to be about 15/ifkm J, which is close to the experimental value of 2GnUcm2. The annealing step can affect the integrity of the oxide and reduce the tunnel oxide thickness. Also during annealing the dopants from the heavily-doped 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 1 9cm - 3 and 2.3 x 1017cm~3 respectively. These values are based on SUPREM simulation results of the processing sequence used in [70]. With an electron barrier height (xe) of 0.5eV, an interfacial oxide thickness (d) of 7 A is required to give a low Re of 2.7fiUcm2. The simulations indicate that this reduction of oxide thickness (from the value of 10A used for the unannealed de-vice) is essential to bring Re down to the value of 2/xflcm2 found in experimental 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, Ge is computed to be 1.2 x 10 1 4scm - 4, which is in the experimentally observed range of 1 — 2 x 1 0 1 4 « c m - 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 10~3Acm~3, which is in good agreement with the experimental value of 250 at the same current level. 128 COLLECTOR-EMITTER BIAS, VCE ( V ) Figure 6.5: Common emitter characteristics of the annealed pnp PET 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 a figure of 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 gm and 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 gm, but different expressions for Cj e are needed. For the unannealed device, Cj e is computed in exactly the same way as for the MISET. 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\,t can be approximated by (5.23). In the annealed device, Cbe is the equivalent of two series capacitances Cox and Caff, i.e. I l l . , where Cox(= ej/d) is the oxide capacitance and Cdi/f is the base minority carrier storage capacitance. An expression for C^y/ is given in (5.26). This can be used in this case with n o e replaced by the base equilibrium hole concentration p0\,. 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 55 tt s tt a: tt u< tt o i g H 55 W OS H /r 10 1 0 IO9 • 108 ANNEALED DEVICE UNANNEALED DEVICE IO"2 IO"1 1 10 BASE CURRENT DENSITY (Xcm" 2) 102 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 JB = 10"sAcm~2 to 2.1GHz at JB = 35Acm - 2 , while for the annealed device fx remains constant at about 16GHz regardless of the current level. The dependence of fx on JB for the unannealed device is similar to that of the MISET, where fx can be significantly increased by an increase in JB-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 gm also rises rapidly. On the other hand the value of Cbe, which approaches the oxide capacitance Cos, only rises slowly. This explains the increase of fT with JB shown in Fig. 6.6 for the unannealed device. In the annealed device, Cbe mainly depends on Cajf due to the large Cot. Since both gm 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 _ ^ 1 1 _ 1 22. 2n Gdiff 2n rB 2n 2Db where rB is the base transit time, Wb is the base width and Db is the diffusion 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, Cbt is essentially dorminated by C * / / rather than Cox, and gm is controlled by 132 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,// < Cox). 6.4 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 unan-nealed 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 prac-tical device, then a fx value of 8GHz should be considered feasible. The highest value reported so far for an experimental pnp PET device is 1.6GHz [75]. Cur-rently, 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 emit-ters. 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. char-acteristics 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 den-sities. 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, simu-lated 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 probabili-ties of electrons and holes, which were not considered in previous analytic mod-els, is shown to be crucial to accurate device modeling, especially at high current levels. New power series expressions for the tunneling currents have been for-mulated 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 MES-FETs [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 pa-rameters 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. By considering diffusion or thermionic emission currents, rather than tunnel currents, it may be possible to adapt the present model to analyse BICFETs. 137 Bibliography [1] D. L. Pulfrey, "MIS solar cells: A review", IEEE Trans. Electron Devices, vol. 25, 1308-1317, 1978. [2] T. Yamamoto and M. Morimoto, "Thin MIS-structure Si negative-resistance diode", Appl. Phys. Lett., vol. 20, 269-270, 1972. [3] J. G. Simmons and G. W. Taylor, "Concepts of gain at an oxide-semiconductor interface and their application to the T E T R A N - A tunnel emitter transistor - and to the MIS switching device", Solid-State Elec-tronics, vol. 29, 287-303, 1986. (4] M. K. Moravvej-Farshi and M . A. Green, "Operational silicon bipolar inversion-channel field effect transistor (BICFET)", IEEE Electron De-vices Letter, vol. 7, 513-515, 1986. (5] M. A. Green and R. B. 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Card, "Potential barriers to electron tunneling in ultra-thin films of Si0 2", Solid-State Communications, vol. 14, 1011-1014, 1974. V. Kumar and W. E. Dahlke, "Characteristics of Cr-Si02-nSi tunnel diodes", Solid-State Electronics, vol. 20, 143-152, 1977. S. Ashok, J. M. Borrego and R. J. Gutmann, "Electrical characteristics of GaAs MIS Schottky diodes", Solid-State electronics, vol. 22, 621-631, 1979. M . A. Green and J. Shewchun, "Current multiplication in metal-insulator-semiconductor (MIS) tunnel diodes", Solid-State Electronics, vol. 17, 349-365, 1974. F. Campabadal, X. Aymerich-Humet and F. Serra-Mestres, "Characteri-zation of the switching in tunnel MISS devices", Solid-State Electronics, vol. 29, 381-385, 1986. H. C. Card and K. K. Ng, "Tunneling in ultra-thin SiOj layers on sili-con: Comments on dispersion relations for electrons and holes", Solid-State Communications, vol. 31, 877-879, 1979. K. M. Chu and D. L. Pulfrey, "An improved analytic model for the metal-insulator-semiconductor tunnel junction", IEEE Trans. Electron Devices, vol. 35, 1656-1663, 1988. K. Yamabe and Taniguchi, "Time-dependent dielectric breakdown of thin thermally grow Si0 2 films", IEEE Journal of Solid-State Circuits, vol. 20, 343-348, 1985. K. M. Chu and D. L. Pulfrey, "An analysis of the dc and small-signal ac performance of the tunnel emitter transistor (TETRAN)", IEEE Trans. Electron Devices, vol. 35, 188-194, 1988. S. Y. Yung and D. E. Burk, "A temperature-dependent study of the in-terfacial resistances in polysilicon-emitter contacts", IEEE Trans. Electron Devices, vol. 35, 1494-1500, 1988. 141 G. Baccarani, B. Ricco and G. Spadini, "Transport properties of polycrys-talline silicon films", J. Appl. Phys., vol. 49, 5565-5570, 1978. J. G. Fossum and F. A. Lindholm, "Theory of grain-boundary and intra-grain recombination current in polysilicon p-n junction solar cells", IEEE Trans. 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[69] A. B. Glaser and G. E. Subak-Sharpe, Integrated circuit engineering, Addison-Wesley: Reading, MA, 51-52, 1977. [70] C. M. Maritan and N. G. Tarr, "Polysilicon emitter pnp transistors", sub-mitted to IEEE Trans. Electron Devices, 1988. [71] J. A. Hutchby, "High performance pnp AlGaAs/GaAs heterojunction bipo-lar transistors: A theoretical analysis", IEEE Electron Device Letters, vol. 7, 108-111, 1986. [72] D. A. Sunderland and P. D. Dapkus, "Optimizing np.i and pnp heterojunc-tion bipolar transistors for speed", IEEE Trans. Electron Devices, vol. 34, 367-377, 1987. [73] P. A. Potyraj, D. L. Chen, M. K. Hatalis and D. W. Greve, "Interfacial oxide, grain size, hydrogen passivation effects on polysilicon emitter tran-sistors", IEEE Trans. Electron Devices, vol. 35, 1334-1343, 1988. [74] G. R. Wolstenholme, D. C. Browne, P. Ashburn and P. T. Landsberg, "An investigation of the transition from polysilicon emitter to SIS emitter behaviour", submitted to IEEE Trans. Electron Devices, 1988. [75] Y. Kobayashi, Y. Yamamoto and T. Sakai, "A new bipolar transistor struc-ture 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 metal-GaAs 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 THIS PROGRAM IS WRITTEN TO GENERATE THE STEADY-STATE 1 C CHARACTERISTICS OF THE TETRANS 2 C 3 C 4 C 5 C 8 C 7 C 8 C C MOOIFIED IN LINES 198. 208-7. 43, 82-4. 201. 204. 271 S C 304-5. 307-9 (USE F0102) 7 C BANDGAP OF SI02 UNSPECIFIED! (LINE 225) 73 C ONE-BAND MODEL (LINES 88. 91-7. 175. 178-181. 225. 338-9, AREA 1) 78 C USE FIRS-T THREE TERMS OF POWER SERIES FOR HOLE CURRENT ONLY 79 C (LINES 338-49, 497) 82 C USE EXACT INTEGRAL (IMSL PROQM) FOR ELECTRON CURRENT 85 C (LINES 180-2, 8. 10. 14, AREAS, 195-6, 285) 88 C MODIFY LINES 188. 195-8, 209-7 91 C MODIFY LINES 501. 508 94 C C 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 C 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) C COMMON /AREA1/ CBAR,CHISC,CHISV,CJCM, • CJVM.CPSII.COS.CTNLCM.CTNLVM.EOAP. f JOO.JORG.JUPC.NC.ND.NNO.NV.O.PHIO.PNO. • PSISIO.VTHERM.OFIX COMMON /AREA2/ NT.RH0.N1,P1.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 COMMON /AREA5/ CFO1,FDFLA,FDFLC,FDFLE.FDFLF,FDFLO 5 COMMON /AREAB/ BRC COMMON /SE$$OM/ A(20.22),B(20).Y(22.21) C 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./ C C C READ IN DATA DESCRIBING DEVICE C REAO(KR.S) (LABEL(I).I=1.15> 5 FORMAT(15A4) Listing of MIS9.S at 22:12:01 on OCT 10. 1988 for CC1d*KCHU on 0 Page 2 40 READ(KR.FREE) ITMAX1,ITMAX2,ERR1.ERRS.OELY 41 READ(KR.FREE) CNTRL1 42 C 43 READ(DATFL.FREE) ND.CHISC.PHIM.MISTAP.D.KI.NCONV.AE.AH.OFIX 44 READ(OATFL.FREE) JOD.JORG.JUPC 45 DO 100 1=1.100 46 READ!DATFL,FREE) RHOII).NT!I).SIGMAN!I).SIGMAP!I).TAUM!I).IADNR!I) 47 IF(NTU) .LE. O.ODO) GOTO 200 48 100 CONTINUE 49 200 NTRAP=I-1 50 c 51 READ(FOFL.FREE) (FDFLF!I),FOFLA(I),FDFLC(I).FOFLE(I). 52 # FDFLG(I).I»1,81> 53 C 54 C ECHO PRINT 55 c 58 WRITEUP. 10) (LABEL(I).I=1.15) 57 10 FORMAT!'120X,15A4///) 58 WRITEILP.15) ND.CHISC.PHIM.MISTAR.D.KI.NCONV 59 15 FORMATCIX,'ND=',D10.3,2X,'CHIS=",F5.3.2X.'PHIM='.F5.3.2X. 80 f'MISTAR*' .F5.3.2X, '0=' .010. 3.2X. 'HI"*1 .F5.2.2X. 61 #'NCONV='. 7 5/) 62 WRITE<LP.20) JOD.JORG.JUPC.AE.AH.OFIX 63 20 FORMATCIX.'JOD='.010.3.2X.'JORG='.010.3.2X.'JUPC*'.012.5.5X, 84 f'AE=',010.3,2X,'AH='.010.3.2X,'QFIX='.D11.4.2X//) 85 IF(NTRAP .EO. 0) GOTO 400 86 WRITEUP.25) 67 25 FORMAT(2X,11:',3X,'ETA:',5X,'NT:',7X.'SIGMAN:',SX.'SIQMAP:1. 68 #5X.'TAUM:•,5X.'IADNR:'/) 89 DO 300 I=1.NTRAP 70 300 WRITE(LP,30) I.RHO(I),NT(I),SIGMAN(I).SIGMAP!I).TAUM!I).IADNR!I) 71 30 FORMAT!IX.13.2X.F5.3.2X.D10.3.2X.3(010.3.2X).12) 72 WRITE!LP.35) 73 35 FORMAT!///) 74 C 75 C 78 C NORMALIZE POTENTIALS TO KBOLTZ'T/Q AND COMPUTE CONSTANTS 77 C 78 400 VTHERM=KBOLTZ*T/Q 79 CTUNL=2.0D0'DSQRT(2.0D0,ME)/HBAR"0S0RT(0) 80 CFD1=PI*PI/6.0D0 81 EPSIS=KS*EPSIO 82 EPSII=KI*EPSIO 83 NC=NI*DEXPL(EGAP/(2.000*VTHERM))*DSQRT(NCONV) 84 NV=NI'DEXPL(EGAP/(2.ODO * V THERM))/OSORT(NCONV) 85 EGAPI=EGAPI/VTHERM 88 EGAP=EGAP/VTHERM 87 CHISC=CHISC/VTHERM 88 CHISV=EGAP*CHISC 89 PHIM=PHIM/VTHERM 90 C 91 C 92 C 93 C TNLCM=-C TUNL * D'DSOR T(MISTAR•V THERM) 94 c 95 C 98 c 97 CTNLVM=-CTUNL *D*DSORT(MISTAR*VTHERM) Dat ing of MIS9.S at 23:12:01 on OCT 10. 198B for CC1d=KfHU on Q Page 3 98 C 99 NN0=ND 100 PNO=NI'NI/ND 101 PHI0=0L0G(NC/N0> 102 PSISI0=0LOG(NO/PNO) 103 CJVMaAH'T'T 104 CJCM=AE*T«T 105 CBAR=PHIO*CHISC-PHIM 106 CPSII=0/EPSII/VTHERM 107 CQS=DS0RT(2.0D0*KB0LTZ'T*EPSIS) 108 IFINTRAP .EQ. 0) GOTO 800 109 00 500 I=1,NTRAP 110 RHO(I)=RHO(I)/VTHERM 111 CN(I)=SIGMAN(I)'VELTH 112 CP(I)=SIGMAP<I>*VELTH 113 NMI)=NC'DEXPL(RHO(I)-EGAP) 114 500 P1U)=NVDEXPL(-RH0(I)) 115 C 116 C 117 C REAO IN VOLTAGE V AND A STARTING ESTIMATE FOR PHI 118 C 119 800 REAO(KR,FREE.END=1400) V.PHI 120 V=V/VTHERM 121 PHI=PHI/VTHERM A. 122 C -»* 123 U=V-PHl 124 C GIVEN PHI. COMPUTE A STARTING ESTIMATE FOR PSIS 125 CALL FPSIS 128 C 127 C CALL SSM TO FINO A SOLUTION FOR THE COUPLED POTENTIAL 128 C AND HOLE CURRENT CONTINUITY EOUATIONS 129 C 130 NTIMES=0 131 X(1)=U 132 X(2)=PSIS 133 00 700 1=1.3 134 Y(1.I)=U 135 Y<2.I)=PSIS 138 700 YU.I)=Y(I.I)*DELY 1-37 NEWY=.TRUE. 138 CALL SSM(X.F.2.0.ERR1.ITMAX2.C0MPF.NEWY.NENA,NEWB.IFAIL.ft8O0> 139 800 U1=X(1) 140 PSIS1=X(2) 141 NE»»Y=. FALSE. 142 C CALL SSM AGAIN TO OBTAIN AN ESTIMATE OF THE ERROR IN THE SOLUTION 143 CALL SSM(X.F,2.0.ERR2.ITMAX2.C0MPF,NEMY.NEWA.NEWB.IFAIL,*.90O> 144 C 145 C PREPARE FOR OUTPUT OF RESULTS 148 C 147 900 U=X(1) 148 PSIS=X(2) 149 PHIER=DABS(U-U1)'VTHERM 150 PSISER=DABS(PSIS-PSIS1)'VTHERM 151 PHI=V-U 152 U0=U*VTHERM 153 C 154 CALL FOS 155 C L i s t i n g of MIS9.S at 22:12:01 on OCT 10. 1988 for CC1d=KCHU on G Page 158 JVT=0.0D0 157 JCT=0.0D0 158 JMT=0.000 159 0SS=0.000 180 IF(NTRAP .EQ. 0) GOTO 1300 161 CVALEN=PHIO-EGAP*PSIS 182 00 1200 I=1.NTRAP 163 ZETA=CVALEN*RH0(I)*V 184 FM=1.0DO/<1.0D0*DEXPL(ZETA) 165 F TNUM=NSURF * CN1I) *P1(I)*CP(I)*FM/TAUM(I) 166 FT0NM=(NSURF*N1<IM'CN(I)*(PSURF*P1(I)*CP(I)•1.0OO/TAUM(11 167 FT=FTNUM/FTONM 168 IFUAONR(I) .EQ. 1) GOTO 1000 189 QSS=OSS-FT,0*NT(I) 170 GOTO 1100 171 1000 QSS=QSS+(1.0D0-FT)*Q*NT(I) 172 1100 JVTsJVTtO'NTlD'CPdJMPSURF'FT-PKDMI.ODO-FT)) 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) 175 1300 PSIsCPSIMQS+QSS^OFIX) 178 C 177 ETAMC=-(PHIO*PSIS«V) 178 C 179 C 180 C 180.3 BRC = CHISC • PSII/2.0D0 •-* 160.4 THCM e OEXPL(CTNLCM'OSQRT(BRCn 5 180.6 LOWER(1) = 0.00 181 UPPER(1) = 1.202 181.1 L0WER(2) = 0.00 181.2 UPPER*2) = BRC 181.3 DOUI = OMLINIFUNCM,LOWER.UPPER.2,20000.0.DO.1.0-3.IER) 181.4 JCM = CJCM * DOUI 181.5 C 182 C 183 C 184 CALL FJVM 185 C 186 CALL FJPN 187 C 188 JTOT=JCM*JVM*JMT 188.5 " JCOLL= JCM • JPN • JUPC 188.7 VSE = (ETAV-ETAMV) • VTHERM 189 C 190 UO=U'VTHERM 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 196 40 FORMAT(IX.' V=',F8.4,3X,'EMITTER J='.014.7.3X.'COLLECTOR J='. 198.5 f 014.7.3X.'SOURCE J=',D14.7.3X.'IFAIL='.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) PSURF.NSURF.ETAC.ETAV,ETAMC,ETAMV 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 MIS9.S at 22:12:01 on OCT 10. 1988 for CC1d=KCHU on 0 Page 5 203 WRITE!LP.5S) JCM.JVM.JCT.JVT.JMT.JPN 204 55 FORMAT!1X.'JCM=',D14.7.2X.'JVM=',D11.4.?X.'JCT='.011.4.2X, 205 #'JVT='.D11.4.2X.1JMT='.D11.4.2X.'JPN='.011.4/) 206 WRITEUP.80) THCM, THVM, OS, VSE 207 60 FORMAT!IX,'THCM=',D11.4,2X,'THVM=',D11.4,2X,'QS=',D14.7.2X. 207.5 • 'SOURCE-EMITTER V=',014.7////) 208 C 209 GOTO 600 210 C 211 C 212 1400 STOP 213 END 214 BLOCK DATA 215 C 218 IMPLICIT REAL*8 (A-Z) 217 INTEGER CNTRL1.ITMAX1.LP.NTIMES 218 C 219 COMMON /AREA*/ CBAR.CHISC.CHISV,CJCM. 220 # CJVM.CPSII.COS.CTNLCM.CTNLVM.EGAP. 221 # JOO.JORG.JUPC.NC,NO.NNO.NV.Q.PHIO,PNO. 222 • PSIS10,VTHERM,OFIX 223 COMMON /AREA4/ CNTRL1.ITMAX1.LP.NTIMES 224 C 225 DATA EGAP/1.079862456D0/.0/1.80210-19/.LP/8/ 228 C _ 227 END £ 228 SUBROUTINE COMPF(X.F) O 229 C 230 C THIS ROUTINE COMPUTES THE RESIDUE OF THE POTENTIAL AND 231 C HOLE CURRENT CONTINUITY EQUATIONS 232 C 233 IMPLICIT REAL*8 <A-Z) 234 INTEGER CNTRL1.1,IADNR.ITMAX1.LP.NTIMES,NTRAP 235 C 236 DIMENSION X!2),F(2),TAUM!100),RHO(100), 237 # NT!100).CN!100).CP!100).N1(100).P1(100).IADNR! 100) 238 C 239 COMMON /AREA 1/ CBAR,CHISC.CHISV,CJCM, 240 f CJVM.CPSII.COS.CTNLCM.CTNLVM.EGAP, 241 # JOO.JORG.JUPC.NC.ND.NNO.NV.Q.PHIO.PNO. 242 I PSISIO.VTHERM.QFIX 243 COMMON /AREA2/ NT,RH0.N1.PI.CN,CP,TAUM.IADNR.NTRAP 244 COMMON /AREA3/ ETAC.ETAMC.ETAMV.ETAV.JVM.JPN.NSURF.PHI,PSII.PSIS. 245 f PSURF.OS,THVM.THCM.U.V 248 COMMON /AREA4/ CNTRL1.ITMAX1.LP.NTIMES 247 C 248 NTIMES=NTIMES*1 249 U=X!1) 250 PHI=V-U 251 PSIS=X!2) 252 C 253 CALL FQS 254 C 255 QSS=0.0D0 256 JVT=0.0D0 257 IF!NTRAP .EO. 0) GOTO 300 258 CVALEN=PHIO-EGAP*PSIS 259 DO 200 1=1.NTRAP L i s t i n g of MIS9.S at 22:12:01 on OCT 10. 1988 for CCId-KCHU on 0 Page 8 260 ZETA=CVALEN*RHO(I)*V 281 FM=1.0DO/(1.0DO»DEXPL(ZETA)) 282 FTNUM=NSURF*CN(I)*P 1 (I)*CP(I)*FM/TAUM<I) 28 3 FTDNM=(NSURF*N1(I))•CN(I)•(PSURF *P1(I•)* CP(I)•1.ODO/TAUM( I) 284 FT=FTNUM/FTDNM 285 IF(IADNRU) .EO. 1) GOTO 100 268 QSS=QSS-FT'0'NT(I) 267 GOTO 200 268 100 QSS=QSS*(1.000-FT)*Q'NT(I) 269 200 JVT=JVT»a'NT<I)'CP<I>'(PSURF'FT-P1<I)'<1.000-FT)) 270 C 271 300 PSII=CPSII'(QS+OSS*QFIX) 272 F<1)=CBAR*V+PSIS*PSII 273 C 274 CALL FJVM 275 CALL FJPN 276 F(2)=JVM-JPN-JVT 277 C 278 IF(CNTRL1 .EQ. 0) GOTO 400 279 UO=U'VTHERM 280 PSISO=PSIS'VTHERM 281 WRITE(LP.5> NTIMES,UO.PSISO.F(1),F(2) 282 5 FORMAT(1X,I3.2X,D23.18.2X.D14.7.2X,2(D14.7.2X>) ^ 283 C 5* 284 400 RETURN O 285 END 285.05 C 285.1 C 285.15 DOUBLE PRECISION FUNCTION FUNCM(N.X) 285.2 IMPLICIT REAL'S (A-Z) 285.25 INTEGER N 285.3 DIMENSION XIN) 285.35 COMMON /AREA 1/ CBAR. CHISC. CHISV. CJCM. 285.4 I CJVM. CPSII. COS, CTNLCM. CTNLVM, EGAP. 285.45 t JOD, JORG. JUPC, NC. ND, NNO, NV, 0, PHIO. PNO. 285.5 # PSISIO. VTHERM. QFIX 285.55 COMMON /AREA3/ ETAC, ETAMC. ETAMV. ETAV. JVM, JPN, NSURF, PHI, 285.6 # PSII. PSIS. PSURF. OS. THVM. THCM, U. V 285.65 COMMON /AREA6/ BRC 285.7 ETTO = X(1) • X(2) - ETAC 285.75 FSO = O.DO 285.8 IF (ETTO .GE. 150.00) GO TO 100 285.81 FSO = 1.D0/M.D0 • DEXPL(ETTO)) 285.82 100 CONTINUE 285.83 ETTM = X(1) • X(2) - ETAMC 285.84 FM = O.DO 285.85 IF (ETTM .GE. 150.DO) GO TO 200 285.86 FM = 1.D0/(1.D0 • OEXPL(ETTM)) 285.87 200 CONTINUE 285.88 FUNCM = (FSO - FM) • OEXPL(CTNLCM'DSQRT(BRC-X ( 2))) 285.89 RETURN 285.9 END 285.91 C 285 92 C 286 SUBROUTINE FQS 287 C 288 C THIS ROUTINE COMPUTES THE CHARGE OS STORED ON THE SEMICONDUCTOR 289 C L i s t ing 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 C 10O 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 .GE. O.ODO) GOTO 100 WRITEUP.5) ARGMNT 5 FORMAT(IX.'WARNING: SQUARE OF SURFACE FIELD IS NEGATIVE' ,5X,011 .4 ) ARGMNTsO.000 OS=COS'DSORT(ARGMNT) IF(PSIS . 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 /AREA3/ 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)) Page 7 ( Listing of MIS9.S at 22:12:01 on OCT 10. 1988 for CC1d-KCHU on Q Page 8 341 C 342 C 343 C 344 C 345 C 346 C 347 C 348 C 349 JVM= JVMO • JVMt * JVM2 350 RETURN 351 END 352 SUBROUTINE FJPN 353 C 354 C THIS ROUTINE COMPUTES THE MINORITY CARRIER HOLE CURRENT FLOWING 355 C INTO THE SEMICONDUCTOR 358 C 357 IMPLICIT REAL'S (A-Z) 358 C 359 COMMON /AREA1/ CBAR.CHISC.CHISV.CJCM. 360 f CJVM.CPSI.COS.CTNLCM.CTNLVM.EGAP. 361 # JOD.JORG.JUPC.NC.NO.NNO.NV.Q.PHIO.PNO. 362 # PSISIO.VTHERM.OFIX 363 COMMON /AREA3/ ETAC.ETAMC.ETAMV.ETAV.JVM,JPN.NSURF.PHI,PSI.PSIS. 364 # PSURF.OS.THVM.THCM.U.V & 385 C K9 366 PSISI=PSISIO-PHI 387 IFtPSIS GT. PSISI) PSISCfl=PSISI 388 IFIPSIS .LT. PSISI) PSISCR=PSIS 369 IFCPSISCR LT. O.ODO) PSISCR=O.0D0 370 C 371 JRG=JORG'DSQRT(PSISCR/PSISIO)'(DEXPL(PHI/2.ODO)-1.ODO) 372 C 373 JD=J0D'(DEXPL(PHI)-1.0D0) 374 C 375 JPN=JRG*J0-JUPC 376 RETURN 377 END 378 SUBROUTINE FPSIS 379 C 380 C THIS ROUTINE COMPUTES PSIS GIVEN PHI. ASSUMING NO SURFACE STATES 381 C 382 IMPLICIT REAL'S (A-Z) 383 INTEGER CNTRL1.I.ITMAX1.LP.NTIMES 384 C 385 COMMON /AREA1/ CBAR.CHISC.CHISV.CJCM. 386 # CJVM.CPSI.COS.CTNLCM.CTNLVM.EGAP, 387 # JOO,JORG.JUPC.NC.NO.NNO.NV.O.PHIO.PNO. 388 # PSISIO.VTHERM.OFIX 389 COMMON /AREA3/ ETAC.ETAMC.ETAMV,ETAV,JVM,JPN,NSURF,PHI.PSI.PSIS. 390 • PSURF.OS.THVM.THCM.U.V 391 COMMON /AREA4/ CNTRL1.ITMAX1.LP.NTIMES 392 C 393 CBARV=-(CBAR*V) 394 IF(CBARV .GT. O.ODO ) GOTO 100 395 IF(CBARV .LT. O.ODO) GOTO 200 396 PSIS=0.000 397 RETURN 398 100 PSISLO=0.000 L i s t i n g of MISS.S at 22:12:01 on OCT 10, 1988 for CC1d=KCHU on 0 Page 9 399 PSISHI=CBARV 400 GOTO 300 401 200 PSISLO=CBARV 402 PSISHI=0.0D0 403 C 404 300 DO 500 I=1.ITMAX1 405 PSIS=(PSISLO*PSISHI)/2.0DO 408 CALL FOS 407 PSII=CPSII'OS 408 F1=PSIS*PSII-CBARV 409 IF(F1 .EQ. O.ODO) RETURN 410 IFJF1 .GT. O.ODO) GOTO 400 411 PSISLO=PSIS 412 GOTO 500 413 400 PSISHI=PSIS 414 500 CONTINUE 415 C 418 RETURN 417 END 418 DOUBLE PRECISION FUNCTION FDI(ETA) 419 C 420 C FERMI-OIRAC INTEGRAL OF ORDER ONE 421 C 422 IMPLICIT REAL'S (A-Z) 423 INTEGER INDEX ^ 424 C Cn 425 DIMENSION FOFLA(81),F0FLC(81),FOFLE(81).FDFLF(811,F0FLG(81) 420 C 427 COMMON /AREA5/ CF01,FOFLA.FDFLC.FDFLE.FDFLF.FOFLG 428 C 429 IF(ETA .LT. -4.ODO) GOTO 100 430 IF(ETA .GT. 4.ODO) GOTO 200 431 X=(ETA*4.ODOI/O.100+0.500 432 IN0EX=X*1 433 ETA0=DFLOAT(INDEX-41)'0.1D0 434 0ELETA=ETA-ETA0 435 DXPETA=DEXPL(ETAO) 438 FD1=FDFLF(INDEX)+DELETA*(DLOG(1.ODO*DXPETA)• 437 # 0ELETA/2.000/(1.000*1.ODO/DXPETA)) 438 RETURN 439 C 440 100 F01=DEXPL(ETA) 441 RETURN 442 C 443 200 F01=-DEXPL(-ETA)*ETA'ETA/2.000*CFD1 444 RETURN 445 END 448 DOUBLE PRECISION FUNCTION FD102(ETA) 447 C 448 C FERMI-DIRAC INTEGRAL OF ORDER ONE-HALF 449 C 450 IMPLICIT REAL'8 (A-Z) 451 INTEGER INDEX 452 C 453 DIMENSION F0FLA(81) .F0FLC(81) ,F0FLE(81) ,FDFLF(«1) .F0FL0(8)> 454 C 455 COMMON /AREA5/ CFO1.FOFLA.FOFLC.FOFLE.FOFLF.FOFLG 456 C L i s t i n g of MIS9.S at 22:12:01 on OCT 10, 198B f o r CC1d=KCHU on Q 457 IF(ETA .LT. -4.ODO) GOTO 100 458 IF(ETA .GT. 4.000) GOTO 200 459 X=(ETA*4.0DO)/0.100*0.500 480 INDEX=X*1 461 ETAO=DFLOAT(INDEX-41)'0.1D0 482 DELETAsETA-ETAO 463 FD102=FDFLE(INDEX)+DELETA*(FDFLC(INDEX)+DELETA/2.ODO'FDFLA(INDEX)) 484 RETURN 465 C 466 100 F0102=0EXPL(ETA) 487 RETURN 468 C 469 200 FD102=-DEXPL(-ETA)*ETA'ETA/2.0D0*CFD1 470 RETURN 471 END 472 DOUBLE PRECISION FUNCTION F0302(ETA) 473 C 474 C FERMI-DIRAC INTEGRAL OF ORDER THREE-HALVES 475 C 476 IMPLICIT REAL'S (A-Z) 477 INTEGER INDEX 478 C 479 DIMENSION FDFLA(81).FDFLC(81).FDFLE(81),FOFLF(81),FDFLG(81) 480 C 481 COMMON /AREA5/ CFD1.FDFLA.FDFLC.FDFLE.FDFLF,FDFLG 482 C 483 IF(ETA .LT. -4.000) GOTO 100 484 IF(ETA .GT. 4.000) GOTO 200 485 X=(ETA»4.0D0)/0.100*0.500 486 INDEX=X*1 487 ETA0=DFL0AT(INDEX-41)'0.1D0 488 0ELETA=ETA-ETA0 489 FD302=FDFLG(INDEX)*DELETA*(FDFLE(INDEX)*0ELETA/2.000'FDFLC( INDEX)) 490 RETURN 491 C 492 100 FD302=DEXPL(ETA) 493 RETURN 494 C 495 200 FD302=-0EXPL(-ETA)*ETA'ETA/2.00O*CFD1 496 RETURN 497 END 497. 01 DOUBLE PRECISION FUNCTION FDIKX) 497. 02 IMPLICIT REAL*8 (A-Z) 497. .03 DATA PI/3.14592B54DO/ 497. .04 Y = X 497. 05 IF(Y .LE. O.DO) GOTO 50 497. 08 Y = -V 497. 07 50 Z = 1.00'DEXPHY) -. 250052'OEXPL (2.'V) * . 111747'DEXPL (3.' V) 497. 08 • - 084557'DEXPL(4.'Y) •.040754'DEXPL(5.*Y) 497. 09 # -.020532'DEXPL(6.*Y) •.005108'DEXPL(7.'Y) 497. 1 IF(X .LE. O.DO) GOTO 100 497. 11 Z = -Z • (X"2)/2.DO • (PI"2)/8.D0 497. 12 100 CONTINUE 497. 13 F0I1 = Z 497. 14 RETURN 497. 15 END 497. 16 C 497. 17 C -—--»>- r f .n fwifw Listing of Mlsg.S at 22:12:01 on OCT 10. 19S8 for CC1d=KCHU on 0 Page 11 497 . 18 DOUBLE PRECISION FUNCTION FDI2(K) 497 . 19 IMPLICIT REAL'S (A-Z) 497 .2 . DATA PI/3.14592654D0/ 49* .21 Y B X 497. .22 IF(Y .LE. 0.00) GOTO 50 497 .23 Y = -Y 497 .24 50 Z = I.DO'DEXPL(Y) -.1Z504B'DEXPL(2.'Y> •.037842'DEXPL(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 CONTINUE 497 .3 F0I2 = Z 497 .31 RETURN 497 32 END 497. 33 C 497 .34 C 497 35 DOUBLE PRECISION FUNCTION FDI3U) 497. .36 IMPLICIT REAL'S (A-Z) 497. .37 DATA PI/3.14592654DO/ 497. 38 Y = X 497. 39 IF(Y .LE. 0.00) GOTO 50 497. .4 Y = -Y 497. .41 50 Z = I.OO'DEXPL(Y) -.082592'DEXPL(2.*Y) •.013661'DEXPL(3 497 .42 • -.009796*DEXPL(4.*Y) •.012978*0EXPL(5.*Y) 497 .43 f -.O10859'DEXPL(6.*Y) •.003448'DEXPL(7.*Y) 497. .44 IF1X .LE. 0.00) GOTO 100 497. .45 Z - -Z • (X"4)/24.D0 • ((PI'X)"2)/12.D0 • 7.'(PI"4)/ 497. .48 * 360.00 497. .47 100 CONTINUE 497 .48 FDI3 = Z 497. .49 RETURN 497. .5 END 496 DOUBLE PRECISION FUNCTION DEXPL(X) 499 C 500 IMPLICIT REAL'S (A-Z) 501 C 501. 5 IF(X .GE. 150.ODO) GOTO 200 502 IF(X .LT. -150.ODO) GOTO 100 503 DEXPL=DEXP(X) 504 RETURN 505 100 DEXPL=O.ODO 506 RETURN 506. 3 200 0EXPL=OEXP(150.000) 506. 8 RETURN 507 ENO 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 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 10 15 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) 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 JD. JB. OFIX, JRNB. JR. JG COMMON /AREA1/ XCHtM. XCHIE, XEGB. XPHIO. JDO • KT. XCHIH. JNO. JRO. JGO. XVBI. XVCE, JNE 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/ NC a 2.8025 NV B 1.04D25 READ(5.5> ( L A B E L ( I ) . I B 1 , 1 5 ) FORMAT!15A4) READI5.FREE) ITMAX1, ITMAX2, ERR1. ERR2. DELY WRITE(6.10) (LABEL(1).I=1.15) FORMAT!'1'.20X, 15A4///) READ!3,FREE) T. EGB, CHIM, CHIE FORMAT(2X, 'To', F6.2.2X, •EOBB',F4.2.2X, ,CHIM»',F4.2.2X. f 'CHIEB'.F4.2/) Paga 1 Listing of MISETS.S at 22:05:34 on OCT 10. 1988 for CCId-KCHU on Q I'aga 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. 'WBASE1*'.D9.3.2X, 40.6 I 'DI = '.D9.3.2X. ' LEB=',09.3/i 46 C 47 WRITE<8.10) (LABEL!I).I=1.15) 48 10 FORMAT! ' 1' ,20X, 15A4//7) 49 WRITE(6,15) T, EGB. CHIM. CHIE 50 15 F0RMAT(2X.'T='. F8.2.2X. 'EGB=',F4.2,2X. 'CHIM=',F4.2.2X. 51 • *CHIE=".F4.2/) 62 WRITE<8,20) MCDME. MVDME, MIDME 53 20 FORMAT!2X,*MCDME=',F4.2,2X. 'MVOMEo'.F4.2.2X. 'MIDMEo-,F4.2/> 54 WRITE(6.25> NCOLL. PB. WBASE, DI, LEB 55 25 FORMAT!2X,*NCOLL=',09.3,2X, •PB=*,09.3.2X. 'WBASE=',09.3,2X, 56 * 'Dl=-.09.3,2X. 'LEB='.09.3/) 57 WRITE(6.30) DB. OP. TAUCP. TAUR, TAUG 58 30 FORMAT!2X,'08=',09.3,2X, '0P=',D9.3.2X, 'TAUCP=*.09.3,2X. 59 * 'TAUR='.09.3,2X, 'TAUG=',09.3/) 60 WRITE(6.35) NEPSII. NEPSIS, OFIX 81 35 FORMAT!2X,"NEPSI1=',F5.2,2X; 'NEPSIS=',F5.2,2X. *OFIX=',010.4/) 61 5 COS = 0SQRT!2*KT* EPSIO • NEPSIS) 62 WRITE(8.40) JB 63 40 FORMAT!2X,'JB=',010.4///) 64 C 65 C 68 C 66 5 JNO = (Q * OB * NI * NI / PB) / LEB 67 KT = KBOLTZ * T 68 VTHERM = KT / 0 89 XEGB = EGB / VTHERM 69 5 DELWBO a DSORT!2*NEPSIS*EPSI0*(NCOLL/tPB*(NCOLL*PB)))'VTHERM/Q) 70 XCHIM = CHIM / VTHERM 71 XCHIE * CHIE / VTHERM 72 XCHIH e XCHIE • XEGB 73 MC = MCDME * ME 73 3 WRITE(8,50) VTHERM 73 6 SO FORMAT(2X.'VTHERM='.09.4/) 74 MV = MVDME * ME 74 5 60 FORMAT(2X,'NB=',09.4,2X, 1XPHIOe".09.4.2X. 'CPSII=',D9.4/) 75 MI = MIDME ' ME 75 5 65 FORMAT(2X,'GAMMA=',09.4.2X. •ASTOC=',D9.4,2X. •ASTOV=",D9.4/) 76 C 78 3 70 F0RMAT!2X.'JDO='.010.5,2X. 'JNO=',010.5,2X, 'JRO=',010.5.2X. 76 6 # 1JGO='.D10.5.2X, •OELWBOs'.010.5/) 77 C 77 5 75 FORMAT!2X.-XVB^* .D9.4///) 78 C 79 NB = NI • NI / PB L i s t i n g of MISETS.S at 22:05:34 on OCT 10. 1986 for CC1d=KCHU on 0 Page 3 80 XPHIO = DL0G(NV/PB) 81 CPSII a DI / (NEPSII'EPSIO) / VTHERM 82 COS = DSQRT(2*KT* EPSIO * NEPSIS) 82. 1 GO TO 90 82. 2 80 READ!5.FREE.END=340) VCE 82. 5 PHIO = PHI 82. 7 PSISO a PSIS 83 GAMMA a 4 * PI * DI * 0SQRT(2*MI) / H 84 ASTQC = (4'PI/H) ' (Q/H) * (MC/H) • ( K V 2 ) 85 ASTOV = (4*PI/H) ' (Q/H) * (MV/H) • (KT*'2) 86 JDO = 0 • DSQRT(DP/TAUCP) * Nl • Nl / NCOLL 87 C JNO s 0 * DB * Nl * Nl / PB 88 JNO = (0 * DB • Nl * Nl / PB) / LEB 89 JRO = Q * Nl • DSQRT(2 ,NEPSIS ,EPSI0*VTHERM/(Q 4PB)) / TAUR 90 JGO = Q • Nl • 0SQRT(2'NEPSIS ,EPSI0M(NCOLL*PB)/(NCOLL*PB)) 91 # 'VTHERM/Q) / TAUG 92 DELWBO = DSQRm'NEPSIS'EPSIOMNCOLL/lPBMNCOLL+PBMl'VTHERM/Q) 93 XVBI = XEGB • XPHIO • OLOG(NC/NCOLL) 94 C 95 C 96 C 97 WRITE(6.50) VTHERM 98 50 FORMAT!2X,'VTHERM=',09.4/) 99 WRITE(8,eO) NB. XPHIO, CPSII 100 60 FORMAT(2X,'NB='.D9.4,2X, 1XPHIO=',09.4,2X. *CPSII='.09.4/) 101 WRITE16.65) GAMMA, ASTOC. ASTOV 102 65 FORMAT(2X,'GAMMA=',09.4,2X, 'ASTQC=".09.4.2X. 1ASTQV='.09.4/) 103 WRITE(6.70) JDO. JNO, JRO. JGO. DELWBO 104 70 F0RMAT(2X.•JDO='.D10.5.2X. 'JN0=\D10.5.2X. 'JROs',D10.5,2X. 104. .5 VBE = (XEGB -XPHIO -X(2) -XCHIM -XPSII *XCHIE) • VTHERM 104 .6 JE = JTN • JTP 104 .7 C JC = JN • JG • JO 104 .8 JC = JNE - JRNB • JG • JD 104 .86 CBC = 0SQRT(Q*NEPSIS'EPSIO*NCOLL'PB / (2*(NCOLL+PB)• 104 .92 • (VTHERM'XVBI • VCE - VBE))) 105 • •JGO='.D10.5.2X. 'DELWBO=•.D10.5/) 106 WRITE(6.75) XVBI 108 .5 300 FORMAT!2X,'VCEa1.F5.2.2X, 'VBEB'.D11.5.2X. 106 .6 f 1JE='.D11.5.2X. 'JC=',D11.5.2X. 106 .7 # 'PHIOa'.F6.3.2X, 'PSISO='.F6.3.2X, 1lFAIL='.15/) 107 75 FORMAT(2X,'XVBI='.09.4///) 107 .3 310 FORMAT!2X.,PSI1='.011.5.2X, 'PHIa1.011.5.2X. 'PHIERa'.D11.5.2X. 107 .6 • 'PSISs*.011.5.2X. 'PSISER=*.D11.5.2X. 'DELWB=',011.5/) 108 C 108 3 320 FORMAT(2X,'QS=",011.5,2X, 'JTN=',D11.5.2X. •JTP='.011.5.2X. 108 6 • •JNE='.D11.5.2X. 'JRs'.D11.5.2X, 1JG=',011.5,2X, 108. 8 # 'JDa'.D11.5/) 108. 83 WRITE(6,330) JRNB. CBC 108. 86 330 FORMAT!2X,"JRNB='.011.5.2X. 'CBCa'.011.5////) 108. 9 GOTO 80 109 C 110 C 111 C 112 READ!5.FREE) VCE. PHIO. PSISO 113 GO TO 90 114 80 REAO!5.FREE.END=340) VCE 115 PHIO = PHI 116 PSISO a PSIS L i s t i n g of MISETS.S at 22:05:34 on OCT 10. 1988 for CC1d=KCHU on Q 118.5 INTEGER FREE 117 90 XVCE = VCE / VTHERM 118 Y l l . l l = PHIO / VTHERM 119 VI2.1) = PSISO / VTHERM 120 C 121 00 220 1=2.3 121.2 COMMON /AREA4/ WBASE. DELWB. DELWBO, LEB 121.5 DATA FREE/'*'/ 122 DO 210 J=1.2 123 210 Y(J,I) * Y U . 1) 123.3 C WflITE(8,60) X(1), X(2) T23.8 CO FORMAT(2X,'X(1)=',D11.5.2X, 1X(2)=',D11.5/) 124 220 Y d , I ) = Y d . I ) • DELY 124.5 C WRITE(8,70) 124.7 CO FORMAT(2X.'STEP FOS*) 125 C 126 NEWY • .TRUE. 128.3 C WRITE(8.80) 126.6 CO FORMAT(2X.'STEP FJTN') 127 CALL SSM(X,F.2.0.ERR1.ITMAX1,COMF.NEWY,NEWA,NEWB.IFAIL.&240) 127.3 C WRITE(6.90) 127.6 CO FORMAT(2X, 'STEP FJTP') 127.65 XVCB = XVCE-XEGB*XPHI0*X(2)*XCHIM*XPSII-XCHIE 127.7 C SRXVCB = DSORTlXVBI+XVCE-XEGB+XPHIO*X(2)*XCHIM*XPSII-XCHIE) 127.75 SRXVCB = DSORT(XVBI*XVCB) 127.8 DELWB = DELWBO • SRXVCB 127.81 WBEFF = WBASE - DELWB 127.82 ITANHY a 1/ OTANH(WBEFF/LEB) 127.83 ISINHY a 1/ OSINH(WBEFF/LEB) 127.84 EXPX1 =DEXPL(X(D) - 1 127.85 EXPXCB = DEXPL(-XVCB) - 1 127.86 JRNB = JNO * tITANHY - ISINHY) * (EXPX1 • EXPXCB) 127.87 JNE = JNO * (ITANHY • EXPX1 - ISINHY • EXPXCB) 128 C 129 240 XIPHI = X( 1) 130 XIPSIS = X(2) 130.5 C JD = -JDO • (DEXPL(XEGB-XPHI0-X(2)-XCHIM-XPSII*XCHIE-XVCE)-1) 130.7 JD a -JDO * EXPXCB 131 NEWY = FALSE. 132 CALL SSM(X,F.2.0.ERR2.ITMAX2,COMF,NEWY,NEWA,NEWB,IFAIL.4260) 133 C 133.03 F<1) = (JNE - JTN) • 1.0-4 133.06 F(2) = JB • JG • JD - JTP - JRNB 133.1 C WRITE(6,100) X(1>. X(2). F( 1 ) . F(2) 133.2 COO FORMAT(2X.,X(1)=,.D11.5.2X. 1X(2)=',D11.5,2X. 133.3 C f 'F(1)=',011.5.2X, 'F(2)a',D11.5) 133.4 C WRITE(6.110) JTN, JTP, JN. JR, JG 133.5 C10 FORMAT(2X,'JTN=',D11.5,2X, •JTPa',011.5.2X. •JN=',D11.5,2X. 133.6 C • 'JRa 1,011.5,2X. 'JGa\D11.5> 133.7 C WRITE(6.120) OS. XPSII 133.8 C20 FORMAT{2X.'QS='.011.5.2X. •XPSII='.Dl1.5/) 134 280 PHI = X(1) * VTHERM 135 PSIS = X(2) • VTHERM 136 PSII a XPSII * VTHERM 137 PHIER = DABS(X(1)-XIPHI) • VTHERM 138 PSISER a DABS(X(2)-XIPSIS) • VTHERM 139 VBE a (XEGB -XPHIO -X(2) -XCHIM -XPSII *XCHIE) • VTHERM 140 JE a JTN • JTP L i s t i n g of MISET5.S at 22:05:34 on OCT 10. 1988 for CC1d=KCHU on 0 141 C JC = JN • JG • JD 142 JC a JNE - JRNB • JG • JD 143 CBC = DSORT(Q*NEPSIS*EPSIO*NCOLL*PB / (2*(NCOLL+PB)• 144 9 (VTHERM*XVBI • VCE - VBE))) 145 c 148 WRITE(6,300) VCE. VBE. JE. JC, PHIO. °SISO. IFAIL 14/ 300 FORMAT!2X.'VCE*'.F5.2.2X. 'VSE*'.011.5.2X. 148 f 'JE='.011.5.2X. 'JC=',011.5.2X. 149 « 'PHIO*1.F6.3.2X. 'PSISO='.F3.3.2X, 'IFAIL=' ,15/) 150 WRITE(8.310) PSII, PHI. PHIER. PSIS. PSISER. DELISTS 151 310 F0RMAT(2X.'PSII='.D11.5.2X. 'PHI = ',D11.5,2X. 'PHIER*' .D11.5.2X. 152 f 'PSIS=-'.011.5.2X. 'PSISER='.011.5,2X. 'DELWB=',011.5/) 153 WRITE(6.320> O , JTN. JTP JNE. JR. JG. JD 154 320 F0RMAT(2X.'0S='.011.5.2X. 'JTN='.Dl1.5.2X. 'JTP='.D11 I.5.2X. 155 # 'JNEa'.011.5.2X. ' JR='.011. 5. 2X. 'JG='.OM. 5.2X. 158 • 'JO-*'.011.5/) 157 WRITEJ6,330) JRN8. CBC 158 330 FORMAT<2X.'JRNB=',011.5.2X. 'CBC*',011.5////) 159 GOTO 80 180 340 STOP 181 END 182 C 183 C 184 C 185 C 168 SUBROUTINE COMF(X.F) 167 IMPLICIT REAL'S (A-Z) 168 INTEGER FREE 169 DIMENSION X(2), F( 2 ) . FREE(I) 170 COMMON /AREA 1/ XCHIM. XCHIE, XEGB, XPHIO, JOO. JO. JB, QFIX. 171 f KT. XCHIH. JNO. JRO, JGO. XVBI. XVCE, JNE, JRNB. JR. JG 172 COMMON /AREA2/ NB. PB. NC. NV, COS. CPSII. OS 173 COMMON /AREA3/ JTN. JTP. GAMMA. ASTOC. ASTOV. XPSII 174 COMMON /AREA4/ WBASE. DELWB, OELWBO. LEB 175 DATA FREE/'*'/ 176 C 177 C 178 C WRITE(6,60) X(1). X(2) 179 CO FORMAT(2X,'X(1)='.D11.5.2X, 'X(2)=',D11.5/) 180 CALL FOS(X) 181 c WRITE(6.70) 182 CO FORMAT(2X, STEP FOS') 183 XPSII = -CPSII * (QS+OFIX) 184 CALL FJTN(X) 185 c WRITE(6.80) 188 CO FORMAT(2X,'STEP FJTN') 187 CALL FJTP(X) 188 c WRITE(6,90) 189 CO FORMAT(2X,'STEP FJTP') 190 XVCB = XVCE-XEGB*XPHI0*X(2)*XCHIM*XPSII-XCHIE 191 C SRXVCB = DSQRT(XVBI*XVCE-XEGB*XPHI0*X(2)*XCHIM*XPSII- XCHIE) 192 SRXVCB = DSORT(XVBI*XVCB) 193 DELWB = DELWBO * SRXVCB 194 WBEFF = WBASE - OELWB 195 ITANHY = 1/ DTANH(WBEFF/LEB) 196 ISINHV = 1/ OSINH(WBEFF/LEB) 197 EXPX1 = OEXPL(X(1)) - 1 198 EXPXCB = DEXPL(-XVCB) - 1 L i s t i n g of MISETS.S at 22:05:34 on OCT 10, 1988 for CC1d=KCHU on 0 Paga 8 199 JRNB = JNO * (ITANHY - ISINHY) • !EXPX1 • EXPXCB) 200 JNE = JNO * (ITANHY • EXPX1 - ISINHY • EXPXCB) 201 C JN = JNO * DEXPL(XM) / (WBASE-DELWB) 202 C JR = JRO * 0S0RT(DABS(X(2))) • (DEXPL(X(1)/2)- 1) 203 JQ = JGO * SRXVCB 204 C JD = -JDO * (DEXPL(XEGB-XPHI0-X(2)-XCHIM-XPSII*XCHIE-XVCE)-1) 205 JD = -JDO * EXPXCB 206 C 207 C F(1) x (JN • JR - JTN) • 1.0-4 208 C F(2) • JB • JG • JD - JR - JTP 209 F(1) = (JNE - JTN) * 1.D-4 210 F(2) * JB • JG • JD - JTP - JRNB 211 C WRITEI6.100) X(1). X(2), F(1), F(2) 212 COO F0RMAT(2X.'X!1)*'.011.5.2X. 'X(2)=',D11.5,2X. 213 C f 'F(1)='.D11.5.2X. 'F(2)s-,D11.5) 214 C WRITE(8.110) JTN. JTP, JN, JR, JQ 215 CIO FORMAT(2X,'JTN=',D11.5,2X, •JTP=',D11.5.2X. 'JN»",D11.5,2X, 216 C # 'JR='.011.5,2X, 'JG='.D11.5) 217 C WRITE(6,120) OS. XPSII 218 C20 FORMAT(2X,'QS=',011.5.2X, •XPSII"",D11.5/) 219 RETURN 220 END 221 C 222 C 223 C 224 C 225 SUBROUTINE FQS(X) <5> 228 IMPLICIT REAL*8 (A-Z) H* 227 DIMENSION X(2) 228 229 COMMON /AREA 1/ XCHIM. XCHIE. XEGB. XPHIO. JDO. JO. JB. QFIX. 230 * KT. XCHIH. JNO. JRO. JGO. XVBI. XVCE, JNE. JRNB. JR. JG 231 COMMON /AREA2/ NB. PB. NC. NV. COS. CPSII. OS 232 C 233 NSURF = NC ' F0102(X(2)*XPHIO*X(1)-XEGB) 234' PSURF = NV • FD102( - XPHIO - X ( 2)) 235 ARGMNT = NSURF - NB*DEXPL(X(1)  • PSURF - PB • PB*X(2) 236 IF (ARGMNT .GE. O.ODO) GOTO 100 237 WRITE(6.5) ARGMNT 238 5 FORMAT!IX.'WARNING: SOURE OF SURFACE FIELD IS NEGATIVE'. 239 # 5X, D11.4) 240 ARGMNT = 0.000 241 100 OS = -COS * DSORT(ARGMNT) 242 IF (X(2) .LT. O.ODO) OS = -OS 243 RETURN 244 END 245 C 246 C 247 C 248 C 249 SUBROUTINE FJTN(X) 250 IMPLICIT REAL'8 (A-Z) 251 . DIMENSION X(2). LOWER!2), UPPER!2) 252 COMMON /AREA 1/ XCHIM. XCHIE. XEGB. XPHIO. JOO. JD. JB. QFIX, 253 • KT. XCHIH. JNO. JRO. JGO. XVBI, XVCE, JNE. JRNB. JR. JG 254 COMMON /AREA3/ JTN. JTP, GAMMA. ASTQC, ASTOV. XPSII 255 COMMON /AREA5/ XBRC. XNALP1, XNALP2 256 C L i s t i n g of MISET5.S at 22:05:34 on OCT 10. 1988 for CCldsKCHU on G Fags 7 257 XNALP1 = XCHIE - XCHIM - XPSII 258 XNALP2 = X(2) • XPHIO • X(1) - XEGB 259 XBRC = XCHIE - XPSII/2 260 LOWER!1) = O.DO 281 UPPER!1) =1.202 262 L0WER(2) = 0.00 263 UPPER!2) = XBRC 264 DOUI = DMLIN(FUNCM, LOWER. UPPER. 2. 20000. O.DO.-1.0-5. IER) 265 JTN = ASTOC • DOUI 288 RETURN 267 END 288 C 269 C 270 C 271 C 272 DOUBLE PRECISION FUNCTION FUNCM(N.XE) 273 IMPLICIT REAL'S <A-Z) 274 INTEGER N 275 DIMENSION XE(N) 278 COMMON /AREAS/ XBRC, XNALP1. XNALP2 277 XNETT1 = XE(1> *XE(2) -XNALP1 278 FN1 = O.DO 279 IF (XNETT1 .GE. 150.00) GO TO 100 280 FN1 = 1.00/(1.DO *0EXPL(XNETT1)) 281 100 CONTINUE 282 XNETT2 = XE(1) *XE(2) -XNALP2 283 FN2 = O.DO 284 IF (XNETT2 .GE. 150.00) GO TO 200 285 FN2 = 1.D0/I1.D0 •DEXPHXNETT2)) 288 200 CONTINUE 287 FUNCM = (FN1 -FN2) 'DEXPL(-GAMMA'(XBRC-XE(2))'KT) 288 RETURN 289 END 290 C 291 C 292 C 293 C 294 SUBROUTINE FJTP(X) 295 IMPLICIT REAL*8 (A-Z) 296 DIMENSION XI2) 297 298 COMMON /AREA1/ XCHIM. XCHIE. XEGB, XPHIO. JDO. JD. JB. QFIX. 299 • KT, XCHIH. JNO. JRO, JGO. XVBI, XVCE. JNE, JRNB, JR. JG 300 COMMON /AREAS/ JTN, JTP, GAMMA. ASTOC. ASTOV. XPSII 301 C 302 XNBET1 = XCHIM • XPSII - XCHIE - XEGB 303 XNBET2 = -X(2) - XPHIO 304 BRV = (XCHIH + XPSII/2) ' KT 305 THETVO•= DEXPLI-GAMMA'DSORT(BRV)) 306 THETV1 = GAMMA ' THETV0/(2'DS0RT(BRV)) 307 THETV2 = GAMMA ' (GAMMA/BRV • 1/(DSQRT(BRV))•*3) • THETVO/4 308 JTP = ASTOV ' (THETVO ' (FD1(XNBET2)-FOHXNBET1)) 309 • • (KT'THETVI) * (FD2(XNBET2)-FD2(XNBET1)) 310 # • l(KT"2)'THETV2) ' (FD3( XNBET2) -FD3( XNBET 1).)) 311 RETURN 312 END 313 C 314 C 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) 318 IMPLICIT REAL'S (A-Z) 319 IF(X .LT. -150.ODO) GOTO 100 320 DEXPL*DEXP(X) 321 RETURN 322 100 DEXPL=0.000 323 RETURN 324 END 325 C 326 C 327 DOUBLE PRECISION FUNCTION FD1(X) 328 IMPLICIT REAL'S (A-Z) 329 DATA PI/3.14159285400/ 330 Y = X 331 IF(Y .LE. O.DO) GOTO 50 332 Y B -Y 333 50 Z = 1.OO'DEXPL(Y) -.250052'DEXPL(2.*V) • . 1 11747'0EXPL(3.*Y) 334 * - .064557'DEXPL(4.'Y) • .040754'DEXPH5. *Y) 335 f -.020532*0EXPL(8.'V> •.005108'DEXPL(7.'Y) 336 IF(X .LE. O.DO) GOTO 100 337 Z = -Z • (X"2)/2.DO • (PI"2>/8.00 338 100 CONTINUE 339 FDI = Z 340 RETURN 341 END 342 C 343 C 344 DOUBLE PRECISION FUNCTION FD2(X) 345 IMPLICIT REAL'S (A-Z) 346 DATA PI/3.14159265400/ 347 Y = X 348 IF(V .LE. 0.00) GOTO 50 349 Y = -Y 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) 352 • -.007486*0EXPL(8.'Y) • .002133'OEXPH7.*V) 353 IF(X LE. O.DO) GOTO 100 354 Z = Z • (X"3)/8.DO • (PI"2)'X/8.DO 355 100 CONTINUE 358 FD2 = Z 357 RETURN 358 END 359 C 360 C 361 DOUBLE PRECISION FUNCTION FD3(X) 382 IMPLICIT REAL*8 (A-Z) 363 DATA PI/3. 14159265400/ 364 Y = X 365 IF(Y .LE. O.DO) GOTO 50 366 Y = -Y 367 50 Z= I.DO'DEXPL(Y) - 082592*DEXPL(2.*Y) •.013881'DEXPL(3.*Y) 368 f -.009798*DEXPL(4.*Y) •.012976*DEXPL(5.*Y) 389 • -.010659*DEXPL(6.*Y) •.003446*DEXPL(7.*Y) 370 IF(X .LE. O.DO) GOTO 100 371 Z = -Z • (X**4)/24.DO • ((PI *X) * *2)/12 .DO • 7MPI"4)/ 372 f 380.00 Listing of MISET5.S at 22:05:34 on OCT 10. 1986 for CC1d=KCHU on Q Page 9 373 100 CONTINUE 374 FD3 = Z 375 RETURN 378 ENO 377 C 378 C 379 C 380 C 381 DOUBLE PRECISION FUNCTION F0102(X> 382 IMPLICIT REAL'S (A-Z) 383 DATA PI/3.14159265400/ 384 IF (X .LE. O.DO) GOTO 100 385 IF (X .LE. 2.DO) GOTO 200 386 IF (X .LE. 4.00) GOTO 300 387 GOTO 400 388 100 FD102 = DEXPL(X) -0.353568'DEXPL(2'X) *0.192439*DEXPL<3'X) 389 • -0.122973*DEXPL(4*X) *6.077134*DEXPL(5*X) 390 * -0.03B228*DEXPL(6,X) *0.008348*0EXPL(7*X) 391 GOTO 500 392 200 FD102 = 0.765147 • X'(0.604911 • XMO. 189885 • 393 # X*(0.020307 • XM-0.004380 • X*(-0.000388 • 394 # X*0.000133))))) 395 GOTO 500 396 300 FD102 = 0.777114 • XMO.581307 • XMO.206132 • 397 # XM0.017680 • XM-O.006549 • XMO.000784 -398 # X'O.000036))))) 399 GOTO 500 400 400 XN2 = 1/(X"2) 401 F0102 = (X'*1.5) • (0.752253 • XN2M0.928195 • 402 # XN2M0.880839 • XN2M25.7829 • XN2M-553.636 • 403 # XN2M3531.43 - XN2* 3254.65)))))) 404 500 RETURN 405 END Listing of PISETP.S at 21:56:53 on OCT 10. 1988 for CC1iJ=KCHU on G 1 C 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 REAL'S (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) 15 C 16 COMMON /AREA 1/ XCHIM. XCHIE, XEGB. XPHIO. JDO. JD. JB. OFIX. 17 • KT, XCHIH. JNO. JRO, JGO. XVBI. XVCE. JNE. JRNB. JR. JG. 18 » XPEGB. XZETA 19 COMMON /AREA2/ NB. PB. NC, NV. COS. CPSII, OS 20 COMMON /AREA3/ JPOL, JTN, JTP, GAMMA. ASTOC. ASTOV, XPSII 21 COMMON /AREA4/ WBASE. DELWB. DELWBO. LEB 22 COMMON /AREA5/ OPOL. LPOL 23 COMMON /SESJOM/ AI20.22). B(20). VI22.21) 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 CONDUCTION 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. MIOME 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 Page 2 SB C ECHO DEVICE PARAMETERS TO OUTPUT FILE 8 60 C 61 C 62 WRITE(6,10) (LABEL(I),I*1.1S) 63 10 FORMAT* • V ,20X. 15A4///J 64 WRITE(6.15) T, EGB, PEGB. CHIM. CHIE 65 15 FORMAT!2X,'T=', F6.2.2X, 'EGB*',F4.2,2X, •PE08*',F4.2.2X. 86 # 'CHIM=',F4.2,2X, 'CHIE*',F4.2/) 87 WRITE(6,20) MCDME. MVDME, MIOME 88 20 FORMAT! 2X , ' MCDME*' ,F4 . 2, 2X , 'MVDME*' .F4.2.2X, 'MIDME=- ,F4.2/) 89 WRITE(8,25) NCOLL. PB. WBASE, 01. LEB 70 25 FORMAT!2X,'NCOLL*',D9.3.2X, 'PB='.D9.3.2X. 'WBASE*',09.3,2X, 71 • *DI='.D9.3.2X, 'LEB*',D9.3/) 72 WRITE!6,30) DB. OP. TAUCP, TAUR. TAUG 73 30 FORMAT!2X,'DB='.09.3,2X, 'DP*'.09.3.2X, 'TAUCP*'.09.3.2X, 74 f 'TAUR*'.09.3,2X. 'TAUG*',D9.3/) 75 WRITE<6.35) NEPSII, NEPSIS, QFIX 78 35 FORMAT!2X,'NEPSI1='.F5.2.2X, 'NEPSIS=',F5.2.2X. •OFIX*',D10.4/) 77 WRITE!6.37) DPOL, LPOL. ZETA 78 37 FORMAT!2X,'DPOL='.09.3.2X,'LPOL*',D9.3.2X,'ZETA*'.F4.2/) 79 WRITE«B,40) JB 80 40 FORMAT!2X,'JB=',D10.4///) 81 C 82 C 83 C NORMALIZE DEVICE PARAMETERS TO UNITS OF KT 84 C ALL VARIABLES STARTING WITH AN X ARE IN KT'S 85 C 86 C 87 KT = KBOLTZ • T 88 VTHERM = KT / 0 89 XZETA = ZETA / VTHERM 90 XEGB * EGB / VTHERM 91 XPEGB = PEGB / VTHERM 92 XCHIM = CHIM / VTHERM 93 XCHIE = CHIE / VTHERM 94 XCHIH = XCHIE - XEGB 95 C 98 MC = MCDME ' ME 97 MV * MVDME ' ME 98 MI * MIDME ' ME 99 C 100 C 101 c COMPUTE ALL CONSTANTS 102 c 103 c 104 NB = NI * NI / PB 105 XPHIO = OLOG!NV/PB) 108 CPSII = 01 / (NEPSII-EPSIO) / VTHERM 107 COS = 0S0RT(2*KT' EPSIO : NEPSIS) 108 GAMMA = 4 • PI * 01 • DS0RT(2'MI) / H 109 ASTOC = (4*PI/H) * (Q/H) • (MC/H) • !KT"2) 110 ASTOV = (4-PI/H) * (Q/H) • (MV/H) • !KT**2) 111 JDO = Q * DSORT!DP/TAUCP) * NI * NI / NCOLL 112 c JNO = 0 * DB * NI • NI / PB 113 JNO = (0 * DB * NI * NI / PB) / LEB 114 JRO = 0 ' NI ' DSQRT(2*NEPSIS,EPSI0,VTHERM/(0,PB)) / TAUR 115 JGO = 0 • NI • DS0RT(2'NEPSIS-EPSI0,((NC0LL*PB)/(NC0LL*PB)) 118 # *VTHERM/0) / TAUG L i s t i n g of PISETP.S at 21:58:53 on OCT 10. 1988 for CC1d=KCHU on 0 Page 3 117 DELWBO a 0S0RT(2-NEPSIS*EPSI0*(NC0LL/(fBMNC0LL*PB)))*VTHERM/0) 118 XVBI = XEGB - XPHIO - OL00(NC/NCOLL) 119 C 120 C 121 C WRITE ALL CONSTANTS TO OUTPUT FILE 8 122 C 123 C 124 WRITE(6.50) VTHERM 125 50 FORMAT(2X,'VTHERM='.D9.4/) 126 WRITE(6.60) NB, XPHIO, CPSII 127 60 FORMAT(2X,'NB=',09.4,2X, 'XPHIO=",D9.4.2X, 'CPSIIa•,09.4/) 128 WRITE(6.65) GAMMA. ASTOC, ASTOV 129 65 FORMA T(2X,'GAMMA=',09.4,2X, 1ASTOC='.09.4.2X. 'ASTOV='.09.4/) 130 WRITE(8,70) JOO, JNO, JRO. JGO. OELWBO 131 70 F0RMAT(2X,'J00='.D10.5.2X, 'JNO='.010.5.2X. 'JRO='.010.5.2X. 132 « •JG0=1.010.5.2X, 'OELWBO='.010.5/) 133 WRITE(8.75) XVBI 134 75 FORMAT(2X.'XVBI='.09.4///) 135 C 136 C 137 C READ INITIAL VALUES 138 C ' FOR THE VARIABLES TO BE USED IN THE ITERATION 139 c FROM INPUT FILE 5 140 c 141 c 142 80 READ(5,FREE,END=340) VCE, PHIO, PSISO. PHI10 143 XVCE = VCE / VTHERM 144 C 145 C 146 C SET UP INITIAL CONDITIONS 147 C FOR NUMERICAL SUBROUTINE PACKAGE SSM 148 c 149 c 150 Y( 1. 1) = PHIO / VTHERM 151 Y(2.1) = PSISO / VTHERM 152 Y(3.1) = PH110 / VTHERM 153 c 154 DO 220 1=2.4 155 DO 210 J=1.3 156 210 Y(J,I) = YIJ.1) 157 220 Y d . I ) = Y d . I ) • OELY 158 C 159 C 180 C RUN SUBROUTINE SSM 161 C 162 C 163 NEWY = TRUE. 164 CALL SSM(X.F.3.0.ERR1,ITMAX1.COMF.NEWY.NEWA,NEWB,IFAIL.&240) 165 c 166 240 XIPHI = X(1) 167 XIPSIS = X(2) 168 XIPHI1 = X(3) 169 NEWY = FALSE. 170 CALL SSMCX.F,3,0.ERR2.ITMAX2,COMF.NEWY.NEWA,NEWB,IFAIL.S280) 171 C 172 C 173 C CONVERT RETURNED VALUES TO VOLTS 174 C d a t i n g of PISETP.S at 21:56:53 on OCT 10. 1888 for CC1d*KCHU on Q Page 4 175 C 176 260 PHI = X(1) * VTHERM 177 PSIS = X(2) * VTHERM 178 PHI1 = X(3) ' VTHERM 179 PSII = XPSII * VTHERM 180 C 161 C 182 C CALCULATE DIFFERENCE BETWEEN FIRST AND SECOND CALL TO SSM 183 C 184 C 185 PHIER = DABS(X(D-XIPHI) ' VTHERM 186 PSISER = DABS(X(2)-XIPSIS) * VTHERM 187 PHI1ER = DABS(X(3)-XIPHI1) * VTHERM 188 C 189 C 190 C COMPUTE FINAL VOLTAGES AND CURRENT DENSITIES TO BE OUTPUT 191 C 192 C 193 VBE = (XEGB -XPHIO -X(2) -XCHIM -XPSII *XCHIE -XZETA) 194 f • VTHERM 195 JE = JTN • JTP 196 C JC = JN • JG • JO 197 JC = JNE - JRNB • JG • JD 198 CBC = DSORKQ'NEPSIS'EPSIO'NCOLL'PB / (2*(NCOLL*PB)• 199 # (VTHERM*XVBI • VCE - VBE))) 200 C 201 c 202 c WRITE VOLTAGES. CURRENT DENSITIES AND VARIABLES 203 c DETERMINED FOR THE FINAL SOLUTION 204 c TO OUTPUT FILE 6 205 c 206 c 207 WRITE(6.300) VCE, VBE. JE. JC. PHIO. PSISO. PHI10. IFAIL 208 300 F0RMAT(2X,'VCE='.F5.2.2X, 1VBE=".014.8.2X, 209 # 'JE=',011.5,2X. "JC=".D14.8,10X, "PHIO=",F8.3.2X, 210 # "PSISO=".F6.3.2X. •PHI10='.F6.3.2X. 'IFAIL*'.15/) 211 WRITE(8.310) PSII. PHI. PHIER, PSIS. PSISER. 212 # PHI1, PHI1ER. DELWB 213 310 FORMAT(2X,•PSII=".011.5.IX. "PHIs".014.8,IX. "PHIER*'.011.5.2X, 214 f "PSISs".011.5.2X, "PSISERs".D11.5.2X, 215 # "PHIls".D11.5.2X. "PHI1ER='.011.5.2X. "DELWBB",D11.5/) 218 WRITE(6,320) OS. JTN, JTP, JNE. JG. JD, JPOL 217 320 F0RMATI2X,'0S='.D14.8.2X, "JTN=",D11.5.2X, "JTP=",D11.5.2X, 218 # "JNE=".D11.'5.3X, "JG=" .011.5.2X, 219 # "JD=".D11.5.2X. 'JPOL=".011.5/) 220 WRITE(6.330) JRNB, CBC 221 330 F0RMAT(2X."JRNB=\011.5.2X. 'CBC=",011.5////) 222 GOTO 80 223 340 STOP 224 END 225 C 226 C 227 C MAIN SUBROUTINE USED TO CALCULATE 228 C CURRENT OENSITIES USED IN CONTINUITY EQUATIONS 229 C 230 C 231 SUBROUTINE COMF(X.F) 232 IMPLICIT REAL'S (A-2) L i s t i n g of PISETP.S at 21:56:53 on OCT 10, 1988 for CCM=KCHU on 0 PiiQ« 5 233 INTEGER FREE 234 OIMENSION X(3). F<3). FREE(I) 235 COMMON /AREA1/ XCHIM, XCHIE. XEGB, XPHIO. JDO. JD. JB. OFIX, 236 * KT. XCHIH, JNO. JRO. JGO, XVBI. XV:E. JNE. JRNB, JR. JG. 237 » XPEGB, XZETA 238 COMMON /AREA?/ NB. PB, NC, NV, COS. CPSII, OS 239 COMMON /AREA3/ JPOL, JTN, JTP, GAMMA, ASTOC. ASTOV, XPSII 240 COMMON /AREA4/ WBASE, DELWB, DELWBO, LEB 241 COMMON /AREA5/ DPOL. LPOL 242 DATA FREE/'"/ 243 C 244 C 245 C WRITE(6,60) X(1). X(2). X(3) 246 CO FORMAT(2X,'X(1)=',011.S.2X, 'X(2)='.011.5.2X. 'X<3)='.011.5/> 247 C 248 C 249 C COMPUTE OS:CHARGE STORED AT OXIDE-SEMICONDUCTOR INTERFACE 250 C AND PSII:VOLTAGE ACROSS THE OXIDE 251 C 252 c 253 CALL FOS(X) 254 c WRITE(6.70) 255 CO FORMAT(2X, STEP FQS') 256 XPSII = -CPSII * !QS*QFIX) 257 c 258 c 259 c COMPUTE JPOL .'MINORITY CARRIER CURRENT DENSITY IN POLYSILICON 260 c 261 c 262 CALL FJPOL(X) 263 c WRITE(6.75) 264 C5 FORMAT<2X. STEP FJPOL') 265 c 266 c 267 c COMPUTE JTN:MAJORITY CARRIER TUNNELING CURRENT DENSITY 288 c 269 c 270 CALL FJTN(X) 271 c WRITE(8.80) 272 CO FORMAT!2X.'STEP FJTN') 273 C 274 c 275 c COMPUTE JTP:MINORITY CARRIER TUNNELING CURRENT DENSITY 278 c 277 c 278 CALL FJTP(X) 279 c NRITE(6.90) 280 CO F0RMAT!2X.'STEP FJTP') 281 c 282 c 283 c COMPUTE VCB:COLLECTOR-BASE VOLTAGE 284 c 285 c 286 XVCB = XVCE-XEGB*XPHI0*X!2)*XCHIM*XPSII-XCHIE»XZETA 287 c 288 c 289 c COMPUTE DELWB:BASE WIDTH-MODULATION 290 c M a t i n g of PISETP.S at 21:56:53 on OCT 10. 1986 for CCId»KCHU on 0 Pago 6 O 291 C 292 C SRXVCB a DSQRT(XVBI*XVCE-XEGB*XPHI0*X(2)*XCHIM*XPSII-XCHIE) 293 SRXVCB = DSQRT(XVBI*XVCB) 294 DELWB = DELWBO * SRXVCB 295 C 296 C 297 C COMPUTE JRNB:RECOMBINATION CURRENT DENSITY IN THE BASE 298 C 299 C 300 WBEFF = WBASE - DELWB 301 ITANHY = 1/ DTANH(WBEFF/LEB) 302 ISINHY a 1/ DSINH(WBEFF/LEB) 303 EXPX1 a DEXPL(X(1)) - 1 304 EXPXCB a DEXPL(-XVCB) - 1 305 JRNB a JNO • (ITANHY - ISINHY) * (EXPX1 • EXPXCB) 308 C 307 c 308 c COMPUTE JNE:MAJORITV CARRIER EMITTER CURRENT DENSITY 309 C 310 C 311 JNE a JNO • (ITANHY * EXPX1 - ISINHY • EXPXCB) 312 C 313 C JN a JNO ' DEXPLIX(D) / (WBASE-DELWB) 314 C JR = JRO • DS0RT(DABS(X(2))) • (OEXPL(X(1)/2)-1) 315 C 316 C 317 c COMPUTE JG:GENERATION CURRENT DENSITY IN THE CB JUNCTION 318 C 319 C 320 JG a JQO * SRXVCB 321 C 322 c 323 c COMPUTE JDDIFFUSION CURRENT DENSITY IN THE CB JUNCTION 324 c 325 c 328 c JD a -JDO ' (DEXPL(XEGB-XPHI0-X(2)-XCHIM-XPSII*XCHIE-XVCE)-1) 327 JO a -JDO • EXPXCB 328 c 329 c 330 c COMPUTE ERROR FUNCTIONS 331 c USING CURRENT DENSITY CONTINUITY EQUATIONS 332 c FOR NUMERICAL PACKAGE SSM 333 c 334 c 335 F(1) a (JNE - JTN) • 1.D-4 336 F(2) a JB • JG • JD - JTP - JRNB 337 F(3) a JPOL - JTP 338 c 339 c WRITEI6.100) X(1). X(2), X(3). F( 1 ) . F ( 2 ) . F(3) 340 coo FORMAT(2X,'X(1)s',D11.5,2X, 'X(2)=-.Dl1.5.2X. 'X(3)='.011.5,2X, 341 c # 'Ft1)a-.011.5.2X. ^(2)='.D11.5.2X. 'F(3)=-,D11.5) 342 c WRITE(8,110) JTN, JTP, JG, JPOL 343 C10 FORMAT(2X,'JTN='.D11.5.2X, 'JTPa".D11.5.2X, 344 C # 'JGa'.011.5.2X. •JPOLa'.011.5/) 345 C WRITE(6.120) OS, XPSII 348 C20 FORMAT(2X.•QS='.011.5.2X, "XPSIIa".011.5/) 347 RETURN 348 END Listing of PISETP.S at 21:58:53 on OCT 10. 1988 for CC1d=KCHU on 0 Page 7 349 C 350 C 351 C SUBROUTINE TO CALCULATE CHARGE STORED AT OXIDE -SEMI INTERFACE 352 C 353 C 354 SUBROUTINE FOSIX) 355 IMPLICIT REAL'S (A-Z) 356 DIMENSION X(3) 357 358 COMMON /AREA 1/ XCHIM. XCHIE, XEGB, XPHIO. JOO. JO. JB. OFIX. 359 • KT. XCHIH. JNO. JRO. JGO. XVBI. XVCE. JNE. JRNB. JR. JG. 380 » XPEGB, XZETA 361 COMMON /AREA2/ NB. PB. NC. NV. COS. CPSII. OS 362 C 363 NSURF = NC * F0102(X(2)*XPHIO*X(1)-XEGB) 364 PSURF = NV ' FD102(-XPHI0-X(2)) 365 ARGMNT = NSURF - NB*DEXPL(X(1)) • PSURF - PB • PB*X(2) 366 IF (ARGMNT .GE. 0.000) GOTO 100 367 WRITE(6.5) ARGMNT 368 5 FORMAT!IX.'WARNING: SOURE OF SURFACE FIELD IS 1 NEGATIVE*. 389 i 5X. D11.4) 370 ARGMNT = 0.000 371 100 OS = -COS * DSORT(ARGMNT) 372 IF (X(2) .LT. O.ODO) OS = -OS 373 RETURN 374 END 375 c 376 c 377 c SUBROUTINE TO CALCULATE 378 c MINORITY CARRIER CURRENT DENSITY IN THE POLYSILICON 379 c 380 c 381 SUBROUTINE FJPOL(X) 382 IMPLICIT REAL'8(A-Z) 383 DIMENSION X(3) 384 COMMON /AREA 1/ XCHIM. XCHIE. XEGB, XPHIO. JOO. JO. JB. OFIX, 385 # KT. XCHIH. JNO. JRO. JGO. XVBI. XVCE. JNE. JRNB. JR. JG. 386 # XPEGB, XZETA 387 COMMON /AREA2/ NB, PB, NC. NV. COS. CPSII. OS 388 COMMON /AREA3/ JPOL. JTN. JTP, GAMMA, ASTOC. ASTOV. XPSII 389 COMMON /AREA5/ DPOL. LPOL 390 c 391 XBETA1 = XPEGB - X(3) - XZETA 392 PJ = NV ' FD102(-XBETA1) 393 JPOL = 1.60218D-19 ' DPOL ' PJ / LPOL 394 RETURN 395 END 396 c 397 c 398 c SUBROUTINE TO CALCULATE 399 c MAJORITY CARRIER TUNNELING CURRENT DENSITY 400 c USING A THREE TERM SERIES APPROXIMATION 401 c 402 c 403 SUBROUTINE FJTN(X) 404 IMPLICIT REAL'S (A-Z) 405 DIMENSION X(3) 408 COMMON /AREA 1/ XCHIM. XCHIE. XEGB, XPHIO, JDO. JD. JB, QFIX, L i s t i n g of PISETP.S at 21:58:53 on OCT 10. 1988 for CC1<J=KCHU on Q Paga 6 407 # KT, XCHIH. JNO. JRO, JGO, XVBI. XVCE. JNE. JRNB, JR. JG. 408 # XPEGB, XZETA 409 COMMON /AREA3/ JPOL. JTN. JTP, GAMMA, ASTOC, ASTOV, XPSII 410 C 411 IF (XPSII .LT. 0) GOTO 10 412 XNALP1 = - XPSII - XZETA 413 XNALP2 = X(2) • XPHIO • X(1) - XEGB 414 BRC = (XCHIE - XPSII/2) * KT 415 GOTO 20 416 10 XNALP1 = -XZETA 417 XNALP2 = XPHIO • X(2) • X(1) - XEGB • XPSII 418 BRC = (XCHIE • XPSII/2) * KT 419 C 420 C 421 20 THETCO = DEXPH-GAMMA*DSORT(BRC)) 422 THETC1 a GAMMA * THETC0/(2*DSORT(BRC)) 423 THETC2 a GAMMA * (GAMMA/BRC • 1/(DSORT(BRC))* * 3) * THETCO/4 424 JTN a ASTOC * (THETCO * (FD1(XNALP1)-FD1(XNALP2)) 425 f • (KT*THETC1) * (F02(XNALP1)-F02(XNALP2)) 426 # • ((KT*'2)'THETC2) * (FD3(XNALP1)-FD31XNALP2))) 427 RETURN 428 END 429 C 430 C 431 C SUBROUTINE TO CALCULATE 432 C MINORITY CARRIER TUNNELING CURRENT DENSITY 433 C USING A THREE TERM SERIES APPROXIMATION 434 C 435 C 436 SUBROUTINE FJTP(X) 437 IMPLICIT REAL'S (A-Z) 438 DIMENSION X(3) 439 440 COMMON /AREA1/ XCHIM. XCHIE. XEGB, XPHIO. JDO. JD. JB. OFIX. 441 # KT. XCHIH, JNO. JRO. JGO. XVBI, XVCE. JNE, JRNB, JR. JG. 442 * XPEGB. XZETA 443 COMMON /AREAS/ JPOL, JTN. JTP. GAMMA. ASTOC. ASTOV, XPSII 444 c 445 IF (XPSII .LT. 0) GOTO 10 448 XNBET1 a XZETA • X(3) - XPEGB 447 XNBET2 = -XPEGB - XPSII • XEGB - X(2) - XPHIO 448 BRV a (XCHIH • XPSII/2) * KT 449 GOTO 20 450 10 XNBET1 = -XEGB • XPSII • XZETA • X(3) 451 XNBET2 = -XPHIO - X(2) 452 BRV = (XCHIH - XPSII/2) • KT 453 c 454 c 455 20 THETVO a DEXPL(-GAMMA*OSQRT(BRV)) 456 THETV1 a GAMMA • THETVO/<2*DSQRT(BRV)) 457 THETV2 a GAMMA • (GAMMA/BRV • 1/(DSORT(BRV))* * 3) * THETVO/4 458 JTP = ASTOV * (THETVO * (F01(XNBET2)-F01(XNBET1)) 459 # • (KT'THETVI) • (F02(XNBET2)-FD2(XNBET1)) 460 # • <(KT**2)*THETV2) * (FD3(XNBET2)-FD3(XNBET1))) 461 RETURN 482 END 463 c 464 c -if—»-»-»-- f - t - m m | - ... ...... Listing of PISETP.S a t 21:58:53 on OCT 10. 1988 for CC1d=KCHU on 0 Page 485 C EXPONENTIAL FUNCTION 486 C 467 C 468 DOUBLE PRECISION FUNCTION OEXPL(X) 469 IMPLICIT REAL * 8 (A-Z) 470 IF(X .LT. -150.000) GOTO 100 471 OEXPL=OEXP(X) 472 RETURN 473 100 DEXPL=O.ODO 474 RETURN 475 END 476 C 477 C 478 C FERMI-DIRAC FUNCTION OF ORDER 1 479 C USING A SERIES APPROXIMATION 480 C 481 C 482 DOUBLE PRECISION FUNCTION FDKX) 483 IMPLICIT REAL'S (A-Z) 484 DATA PI/3. 14159285400/ 485 Y = X 488 IF(V .LE. O.DO) GOTO 50 487 Y -e -Y 488 50 Z = 1.OO'DEXPL(Y) -.250052'DEXPL(2.'Y) •.111747'DEXPL(3.*Y) 489 • -.084557'0EXPL(4.*Y) •.040754*DEXPL(5.'Y) 490 • -.020532'0EXPL(6.*Y) •.005108'DEXPL(7.*Y) t— 491 IF(X .LE. O.DO) GOTO 100 rj 492 Z = -Z • (X"2)/2.D0 • (PI"2)/6.00 w 493 100 CONTINUE 494 FD1 = Z 495 RETURN 498 END 497 C 498 C 499 C FERMI-DIRAC FUNCTION OF ORDER 2 500 C USING A SERIES APPROXIMATION 501 C 502 C 503 DOUBLE PRECISION FUNCTION FD2(X) 504 IMPLICIT REAL'S (A-Z) 505 OATA PI/3.141592654D0/ 506 Y = X 507 IF(Y .LE. O.DO) GOTO 50 508 Y = -Y 509 50 Z= I.DO'DEXPL(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. O.DO) GOTO 100 513 Z = Z • (X"3)/8.00 • (PI"2)'X/6.D0 514 100 CONTINUE 515 FD2 = Z 516 RETURN 517 END 518 C 519 C 520 C FERMI-DIRAC FUNCTION OF OROER 3 521 C USING A SERIES APPROXIMATION 522 C 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 REAL'S (A-Z) 526 DATA PI/3.141592654DO/ 527 V = X 528 IF(V .LE. O.DO) GOTO 50 529 Y = -Y 530 50 Z = 1 .DO'OEXPL(Y) -.062592'0EXPL(2.'Y! •.013661'DEXPL(3.'Y) 531 • -.009796'DEXPL(4.'Y) •.012976'0CXPL(5.*Y) 532 • -.010659'DEXPL(6.'Y) •.003448*DKXPL(7.'Y) 533 IF(X .LE. O.DO) GOTO 100 534 Z = -Z • (X"4)/24.00 • < <PI'X)"2J/12.00 • 7.MPI"4)/ 535 • 380.DO 536 100 CONTINUE 537 FD3 * Z 538 RETURN 539 END 540 C 541 C 542 C FERMI-DIRAC FUNCTION OF ORDER 1/2 543 C USING A SERIES APPROXIMATION 544 C 545 C 548 DOUBLE PRECISION FUNCTION FD102IX) 547 IMPLICIT REAL*8 (A-Z) 548 DATA PI/3.141592854D0/ 549 IF (X .LE. O.DO) GOTO 100 550 IF (X .LE. 2.DO) GOTO 200 551 IF (X .LE. 4.00) GOTO 300 552 GOTO 400 553 100 FD102 = OEXPL(X) -0.35356B'DEXPL(2'X) *0.192439'OEXPLI3'X) 554 * -0.122973'0EXPL(4'X) *0.077134'DEXPL(5'X) 555 # -0.036228'0EXPL(6'X) *0.008348'DEXPL(7'X) 558 GOTO 500 557 200 F0102 = 0.785147 • X'(0.804911 • X'(0.189885 • 558 • X'(0.020307 • XM-0.004380 • XM-0.000366 • 559 # X'O.000133))))) 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 -563 # X'O.000036))))) 584 GOTO 500 585 400 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 M a t i n g of PIMSP.S at 21:52:10 on OCT 10. 1988 for CC1d«KCHU on 0 Page 1 Cn 1 C 2 C 3 C 4 C 1 5 C 6 C 7 C e c 1 9 c 10 c 11 12 13 1 14 1 15 1 16 c 17 1 18 • 19 • 20 1 21 1 22 1 23 1 24 25 c 26 1 27 • 28 f 29 f 30 c 31 c 32 c 1 33 c 34 c 35 1 36 1 37 c 38 c 39 c 1 40 c 4 1 c 42 1 43 5 1 44 1 45 c 46 c 47 c 1 48 c 49 c 50 1 51 1 52 1 53 1 54 1 55 1 56 1 57 c 58 c THIS PROGRAM IS WRITTEN TO GENERATE THE STEADY-STATE CHARACTERISTICS OF THE ANNEALED PNP PETS. "d •c rt C3 a DEFINE ALL VARIABLES X IMPLICIT REAL'S (A-Z) O INTEGER FREE. I. J , IFA1L. ITMAX1, ITMAX2 LOGICAL NEWY, NEWA. NEWB (*} EXTERNAL COMF O DIMENSION FREE(1), LABEL!15). X(4). F<4) g COMMON /AREA 1/ XCHIM, XCHIE, XEGB, XPHIO, JDO. JD. JB. OFIX, ^ KT, XCHIH. JNO. JRO. JGO. XVBI, XVCE. JNE. JRNB. JR. JG. £• XPEGB, XZETA. XZETA2, XV8IE. JPO. JRM. JPE. JP2 rt COMMON /AREA2/ NE, NB. PB. NC. NV. COS. CPSII ^ COMMON /AREA3/ JPOL. JTN. JTP. GAMMA. ASTOC. ASTOV M COMMON /AREA4/ WBASE. DELWB. DELWBO. LEB. LHE. LE. DE - l COMMON /AREA5/ OPOL. LPOL O COMMON /SEJJOM/ A(20,22). B(20). YJ22.21) <2 DATA FREE/'"/. NEWY/.TRUE./. NEWA/.FALSE./. NEWB/.FALSE./. 3 NI/1.45D16/. KB0LTZ/1.3B086D-23/. 0/1.6021BD-19/. " 3 « EPSIO/6.654180-12/. M > PI/3.141592654D0/. ME/0.910950-30/. H/6.62817D-34/ O o NV = 2.8025 ' S t NC = 1 .04025 B > s a « JT Cu REVERSE CONDUCTION AND VALENCE FOR PNP DEVICE READ ITERATION SPECIFICATIONS FROM INPUT FILE 5 READ(5.5) (LABEL(I).I=1.15) FORMAT!15A4) READ!5.FREE) ITMAX1. ITMAX2, ERR1. ERR2, DELY READ DEVICE PARAMETERS FROM INPUT FILE 3 READ!3.FREE) T. EGB. CHIM. CHIE READ!3.FREE) MCDME, MVDME. MIDME k . READ!3.FREE) NE. NCOLL. PB, WBASE. DI, LEB. LHE H READ!3.FREE) LE. DE. DB. DP. TAUCP. TAUR. TAUG H READ!3.FREE) NEPSII. NFPS1S, OFIX » READ!3.FREE) OPOL. LPOL. ZETA, PEGB READ!3.FREE) JB L i s t i n g of PIMSP.S at 21:52:10 on OCT 10. 1088 for CCIdHCCHU on 0 Page 2 59 C ECHO DEVICE PARAMETERS TO OUTPUT FILE " 80 C 61 C 82 WRITE(8.10> (LABEL(I).I=1. 15) 63 10 FORMAT!'1',20X, 15A4///) 64 WRITE<8.15) T. EGB. PEGB, CHIM, CHIE 65 15 FORMAT(2X.'7='. F6.2.2X. 'EGB='.F4.2.2X. 'PEGB*',F4.2, 2X, 68 # 'CHIM*'.F4.2.2X. 'CHIE*'.F4.2/) 87 WRITE(8,20) MCDME, MVDME. MIDME 68 20 FORMAT!2X,'MCOME=',F4.2,2X, 'MVOME=•,F4.2.2X. 'MIDME*' .F4.2/I 89 WRITE(8.25) NE. NCOLL. PB, DI. LEB. LHE. WBASE 70 25 FORMAT(2X,' NE*',09.3,2X, 'NCOLL*',D9.3,2X, 'PB='.D9.3, 2X. 71 # '0I=',09.3.2X. 'LEB=',09.3.2X, 'LHE=',D9.3,2X, 72 • 'WBASE*',D9.3/) 73 WRITE(e.30) LE. DE. DB. DP. TAUCP. TAUR. TAUG 74 30 FORMAT!2X,'LE*',09.3,2X, 'DE*'.09.3.2X. 'DB=',D9.3,2X, 'DP*'. 75 • D9.3.2X, 'TAUCP=',09.3,2X, 'TAUR*',09.3,2X, 'TAUG*'.D9.3/) 76 WRITE(6,35) NEPSII. NEPSIS, QFIX 77 35 F0RMAT(2X.'NEPSII='.F5.2.2X. 'NEPSIS*'.F5.2.2X. 'QFIX* '.010.4/) 78 WRITE(6,37) DPOL, LPOL. ZETA 79 37 FORMAT!2X, DPOL*'.09.3.2X,'LPOL*'.D9.3.2X,'ZETA*'.F4.2/) 80 WRITE<6.40) JB 81 40 FORMAT(2X.'JB*'.D10.4///) 82 C 83 C 84 C NORMALIZE DEVICE PARAMETERS TO UNITS OF KT 85 C ALL VARIABLES STARTING WITH AN X ARE IN KT'S 86 c 87 c 88 KT = KBOLTZ • T 89 VTHERM = KT / 0 90 XZETA = ZETA / VTHERM 91 XEGB = EGB / VTHERM 92 XPEGB = PEGB / VTHERM 93 XCHIM = CHIM / VTHERM 94 XCHIE = CHIE / VTHERM 95 XCHIH = XCHIE - XEGB 98 c 97 MC = MCDME * ME 98 MV = MVOME • ME 99 MI * MIDME * ME 100 c 101 c 102 c COMPUTE ALL CONSTANTS 103 c 104 c 105 NB = NI * NI / PB 106 XPHIO = DLOG!NV/PB) 107 XZETA2 = OLOG(NC/NE) 108 CPSII = 01 / (NEPSII'EPSIO) / VTHERM 109 COS = DSORT!2-KT* EPSIO * NEPSIS) 110 GAMMA = 4 * PI ' DI * DSQRT(2'MI) / H 111 ASTOC = <4'PI/H) * (Q/H) • (MC/H) • (KT"2) 112 ASTOV = (4'PI/H) * (Q/H) • (MV/H) • (KT«*2) 113 JOO = Q • DSQRT(DP/TAUCP) * NI * NI / NCOLL 114 c JNO = 0 • OB • NI • NI / PB 115 JNO = (0 * DB • NI * NI / PB) / LEB 118 JPO = 0 • OE * NI • NI / LHE / NE L i s t i n g of PIMSP.S st 21:52:10 on OCT tO. 1988 for CC1d=KCHU on 0 Page 3 117 JRO = 0 • NI * DS0RT(2,NEPSIS,EPSI0M(Nf•PB)/(NE*PB>) 118 • 'VTHERM/0) / TAUR 119 JOO s fl 1 NI * DSQRm'NEPSIS'EPSIOMINrOLUPBl/lNCOLl'PB)) 120 • 'VTHERM/Q) / TAUO 121 OELWBO * OSQRT (2*NEPSIS*EPSIO*(NCOLL/(PC*(NCOLL*PBI • 1 * VTHERM/O) 122 XVBI = XEOB - XPHIO - OLOG(NC/NCOLL) 123 XVBIE = XEGB - XPHIO - XZETA2 124 C 125 C 126 C WRITE ALL CONSTANTS TO OUTPUT FILE 6 127 C 128 C 129 WRITE<6,50) VTHERM. XZETA2 130 50 FORMAT(2X.'VTHERM*'.09.4.2X, 'XZETA2*'.011.5/) 131 WRITE(8,80> NB. XPHIO. CPSII 132 60 FORMAT!2X,"NB='.09.4,2X, 'XPHIO*',D9.4.2X. 'CPSII*',D9.4/) 133 WRITE(6,85) GAMMA, ASTOC, ASTOV 134 65 FORMAT(2X,'GAMMA*'.D9.4,2X, 'ASTOC*'.09.4.2X, 1ASTOV*',09.4/) 135 WRITE(6.70) JDO. JNO. JPO, JRO. JGO. OELWBO 136 70 FORMAT(2X,'JOO*',D10.5,2X. 'JNO*',D10.5,2X, 'JPO*'.010.5.2X. 137 f 'JRO*',D10.5,2X. 'JGO*'.D10.5.2X, 'DELWBO*'.010.5/) 138 WRITE(6.75) XVBI. XVBIE 139 75 FORMAT(2X.'XVBI*'.09.4.2X. 'XVBIE*',09.4///) 140 C 141 C 142 c READ INITIAL VALUES 143 c FOR THE VARIABLES TO BE USED IN FIRST ITERATION 144 c FROM INPUT FILE 5 145 c 146 c 147 80 REAO(5.FREE.ENO=340) VCE. PHIO, PH110, PHI20. PSIIO 146 XVCE = VCE / VTHERM 149 C 150 C 151 C SET UP INITIAL CONDITIONS 152 C FOR NUMERICAL SUBROUTINE PACKAGE SSM 153 c 154 c 155 Y(1.1) = PHIO / VTHERM 158 Y(2. 1) = PH110 / VTHERM 157 Y<3.1) = PHI20 / VTHERM 158 Y(4. 1) = PSIIO / VTHERM 159 c 160 00 220 1=2.5 161 DO 210 J=1,4 182 210 Y(J.I) = Y(J.I) 163 220 Y(I.I) = Y(I.I) • DELY 164 C 185 C 166 C RUN SUBROUTINE SSM 187 C 188 C 189 NEWY = .TRUE. 170 CALL SSM(X.F.4.0.ERR1.ITMAX1.COMF.NEWY.NEWA.NEWB.IFAIL.«240) 171 C 172 240 XIPHI * X(1) 173 XIPHI1 = X(2) 174 XIPHI2 = X(3) L i s t i n g of PIMSP.S at 21:52:10 on OCT 10, 19B8 f o r CCId-KCHU on 0 Page 4 175 XIPSII = X(4) — i u . . 176 NEWY = .FALSE. 177 CALL SSM(X.F.4,0,ERR2,ITMAX2.COMF.NEWY,NEWA.NEWB,IFA1L,&260) 178 C 179 C 180 C CONVERT RETURNED VALUES TO VOLTS 181 C 182 C 183 260 PHI = XM) 4 VTHERM 184 PHI1 = X(2) * VTHERM 185 PHI2 o X(3) * VTHERM 166 PSII a X<4) * VTHERM 187 C 168 C 189 C COMPUTE BAND BENDING AT EB JUNCTION 190 C 191 C 192 PSIS = (XEGB - XPHIO - XM) - XZETA2) • VTHERM' 193 C 194 C 195 c CALCULATE DIFFERENCE BETWEEN FIRST AND SECONO DALL TO SSM 196 c 197 c 198 PHIER a DABS(X(D-XIPHI) ' VTHERM 199 PHI1ER = DABS(X(2)-XIPHI1) * VTHERM 200 PHI2ER = 0ABS(X(3)-XIPHI2) • VTHERM 201 PSIIER « DABS(X(4)-XIPSII) * VTHERM 202 c 203 c 204 c COMPUTE FINAL VOLTAGES AND CURRENT DENSITIES TO BE OUTPUT 205 c 206 c 207 VBE = (XM) • XZETA2 - X(4) -XZETA) •' VTHERM 208 JE = JTN • JTP 209 JC - JNE - JRNB • JG • JD 210 CBC = DSQRKO'NEPSIS'EPSIO'NCOLL'PB / (2*(NCOLL*PB)• 211 • (VTHERM'XVBI • VCE - VBE))) 212 c 213 c 214 c WRITE VOLTAGES. CURRENT DENSITIES AND VARIABLES 215 c DETERMINED FOR THE FINAL SOLUTION 216 c TO OUTPUT FILE 8 217 c 218 c 219 WRITE(8.300) VCE. VBE. JE. JC 220 300 FORMAT(2X,'VCE=1,F5.2.2X, 'VBE=',D14.8.2X. 'JE='.011.5.4X. 221 • 'JCa1.014.8/) 222 WRITEI8.305) PHIO. PHI10. PHI20, PSIIO. IFAIL 223 305 FORMAT (2X , •PHIO*' .F6.3.2X, "PHUOa1 .F8.3.2X. 224 # 'PHI20='.F6.3.2X. •PSI10=',F6.3.2X. •IFAIL='.15/) 225 WRITE(6.310) PSIS. PHI. PHI1. PHI2. PSII 226 310 F0RMAT(2X,'PSIS='.011.5.2X. 'PHI = '.014.8.7X. •PH11 = '.011.5.2X. 227 * ,PHI2=1.D11.5.2X, 'PSIIs'.011.5) 228 WRITE(6.315) PHIER, PHIIER. PHI2ER, PSIIER. DELWB 229 315 FORMAT(2X,'PHIER='.D11.5.2X. 'PHI1ER*'.011.5.2X. 230 # PHI2ER='.011.5.2X. 'PSIIER=',D11.5,2X, DELWB='.Dl1.5/) 231 WRITE(8,320) JTN. JTP. JPE. JNE. JR. JG. JD. JPOL 232 320 FORMAT(2X,'JTN=',D11.5.2X, 'JTP=-.011.5.2X. •JPE=*.011.5.2X. Listing of PIMSP.S at 21:52:10 on OCT 10, 1988 for CC1d=KCHU on 0 Page 5 233 # 'JNE=',011.5.3X. 'JR='.011.5.38X, 'JG»'.011.5.2X. 234 « 'JD='.011.6.2X. •JPOL"',011.5/) 235 WRITE(8,330) JRNB. JRM, JP2. CBC 238 330 FORMAT( 2X . ' JRNB=' ,011 .5,2X , ' JRM=>' ,011 . 5.2X , 237 • •JP2='.011.5.2X. 'CBC**.011.5////) 238 GOTO 80 239 340 STOP 240 END 241 C 242 C 243 C MAIN SUBROUTINE USED TO CALCULATE 244 C CURRENT DENSITIES USED IN CONTINUITY EQUATIONS 245 C 248 C 247 SUBROUTINE COMF(X.F) 248 IMPLICIT REAL'S (A-2) 249 INTEGER FREE 250 DIMENSION X(4), F(4). FREEH) 251 COMMON /AREA1/ XCHIM, XCHIE. XEGB, XPHIO, JOO. JD. JB. OFIX. 252 • KT. XCHIH. JNO. JRO. JGO. XVBI. XVCE. JNE, JRNB. JR. JG. 253 * XPEGB. XZETA, XZETA2, XVBIE. JPO, JRM, JPE. JP2 254 COMMON /AREA2/ NE. NB. PB. NC. NV, COS. CPSII 255 COMMON /AREA3/ JPOL. JTN. JTP, GAMMA. ASTOC. ASTOV 258 COMMON /AREA4/ WBASE. DELWB. DELWBO. LEB. LHE. LE. DE 257 COMMON /AREAS/ OPOL. LPOL 258 DATA FREE/"/ 259 C 260 C 281 C COMPUTE JPOL: MINORITY CARRIER CURRENT DENSITY IN POLYSILICON 262 C 263 C 264 CALL FJPOL(X) 265 C 266 C 267 C COMPUTE JTN: MAJORITY CARRIER TUNNELING CURRENT DENSITY 268 C 289 c 270 CALL FJTN(X) 271 c 272 c 273 c COMPUTE JTP: MINORITY CARRIER TUNNELING CURRENT DENSITY 274 c 275 c 278 CALL FJTP(X) 277 c 278 c 279 c COMPUTE VCB: COLLECTOR-BASE VOLTAGE 280 c 281 c 282 XVCB = XVCE-X(1)-XZETA2*X(4)*X2ETA 283 c 284 c 285 c COMPUTE OELWB: BASE WIDTH MODULATION 286 c 287 c 288 SRXVCB = OSQRT(XVBI*XVCB) 289 OELWB = DELWBO ' SRXVCB 290 c i t l n g of PIMSP' S at 21:52:10 on OCT 10. 1988 for CC1d=KCHU on 0 291 WBEFF = WBASE - DELWB 292 ITANHV a 1/ DTANH(WBEFF/LEB) 293 ISINHY a 1/ DSINHIWBEFF/LEB) 294 ITANHU = 1/ DTANHUE/LHE) 295 ISINHU s 1/ OSINH(LE/LHE) 296 EXPX1 = DEXPL(X(1)J - 1 297 EXPX1R a DEXPL(X( 1)/2) -1 296 EXPX3 a 0 E X P L ( X ( 3 ) 1 - 1 299 EXPXCB = DEXPL(-XVCB) - 1 300 C 301 C 302 C COMPUTE JRNB: RECOMBINATION CURRENT DENSITY IN THE BASE 303 C 304 C 305 JRNB a JNO * (ITANHV - ISINHY) * (EXPX1 • EXPXCB) 306 C 307 c 309 c COMPUTE JNE: MAJORITY CARRIER EMITTER CURRENT DENSITY 309 c 310 c 311 JNE a JNO * (ITANHY • EXPX1 - ISINHY • EXPXCB) 312 c 313 c 314 c COMPUTE JPE: MINORITY CARRIER DIFFUSION CURRENT DENSITY AT 315 c THE MONOCRYSTALLINE EMITTER DEPLETION EDGE 316 c 317 c 318 JPE = JPO • (ITANHU • EXPX1 - ISINHU * EXPX3) 319 c 320 c 321 c COMPUTE JP2: MINORITY CARRIER EMITTER CURRENT DENSITY 322 c 323 c 324 JP2 = JPO * (ISINHU * EXPX1 - ITANHU * EXPX3) 325 c 326 c 327 c COMPUTE JRM: RECOMBINATION CURRENT DENSITY IN THE EMITTER 328 c 329 c 330 JRM a JPO * ((ITANHU - ISINHU) • (EXPX1 • EXPX3)) 331 c JN a JNO * DEXPL(X(D) / (WBASE-OELWB) 332 c 333 c 334 c COMPUTE JR: RECOMBINATION CURRENT DENSITY IN THE EB JUNCTION 335 c 336 c 337 JR a JRO • EXPX1R • DSORT (XVBIE - X ( U ) 338 c 339 c 340 c COMPUTE JG: GENERATION CURRENT DENSITY IN THE CB JUNCTION 341 c 342 c 343 JG a -JGO * EXPXCB * SRXVCB 344 c 345 c 346 c COMPUTE JO: OIFFUSION CURRENT DENSITY IN THE CB JUNCTION 347 c 348 c L i s t i n g of PIMSP.S at 21:52:10 on OCT 10. 1988 for CC1d=KCHU on G Page 7 349 C JD = -JDO • (DEXPL(XEGB-XPHI0-X(2)-XCHIM-XPSI*XCHIE-XVCE)-350 JD = -JOO * EXPXCB 351 C 352 C COMPUTE ERROR FUNCTIONS 353 C USING CURRENT DENSITY CONTINUITY EQUATIONS 354 C FOR NUMERICAL PACKAGE SSM 355 C 356 c 357 F(1) e (JNE - JTN -JRM - JR) * 1.0-4 358 F(2) = JPE - JB - JD - JG • JRNB • JR 359 F<3> = JPOL -JTP 360 F(4) = JP2 - JTP 361 c 382 RETURN 363 END 364 c 365 c 366 c 387 c 366 c SUBROUTINE FOS(X) 389 c IMPLICIT REAL*8 (A-Z) 370 c DIMENSION X(3) 371 c 372 c COMMON /AREA 1/ XCHIM. XCHIE, XEGB, XPHIO. JDO. JD. JB. OFIX 373 c f KT. XCHIH, JNO. JRO. JGO. XVBI. XVCE. JNE. JRNB. JR, JG 374 c « XPEGB. XZETA 375 c COMMON /AREA2/ NB. PB. NC. NV. COS. CPSII. OS 376 c 377 c NSURF a NC • FD102(X(2)*XPHI0*X(D-XEGB) 378 c PSURF = NV * FD102I-XPHI0-X(2)) 379 c ARGMNT = NSURF - NB*OEXPL(X(1)  • PSURF - PB • PB*X(2) 380 c IF (ARGMNT .GE. O.ODO) GOTO 100 381 c WRITEI6.5) ARGMNT 382 c FORMAT(IX,'WARNING: SOURE OF SURFACE FIELD IS NEGATIVE'. 383 c • 5X. D11.4) 384 c ARGMNT a 0.000 385 coo OS = -COS * DSORT(ARGMNT) 388 C IF (X(2) .LT. 0.000) OS = -QS 387 C RETURN 388 C END 389 C 390 c 391 c SUBROUTINE TO CALCULATE 392 c MINORITY CARRIER CURRENT DENSITY IN THE POLYSILICON 393 c 394 c 395 SUBROUTINE FJPOL(X) 398 IMPLICIT REAL'8(A-Z) 397 DIMENSION X(4) 398 COMMON /AREA 1/ XCHIM. XCHIE. XEGB, XPHIO, JDO. JO. JB. QFIX, 399 # KT, XCHIH, JNO, JRO. JGO. XVBI. XVCE. JNE, JRNB, JR. JG, 400 • XPEGB, XZETA, XZETA2, XVBIE. JPO. JRM, JPE. JP2 401 COMMON /AREA2/ NE. NB, PB. NC. NV. COS. CPSII 402 COMMON /AREA3/ JPOL. JTN. JTP. GAMMA, ASTQC. ASTOV 403 COMMON /AREA5/ OPOL, LPOL 404 c 405 XBETA1 = XPEGB - X(2) - XZETA 408 PJ = NV • FD102(-XBETA1I L i s t i n g of PIMSP.S at 21:52:10 on OCT 10. 1988 for CC1d-KCHU on Q Page 8 407 JPOL = 1.602180-19 • OPOL * PJ / LPOL 408 RETURN 409 END 410 C 411 C 412 C SUBROUTINE TO CALCULATE 413 C MAJORITY CARRIER TUNNELING CURRENT DENSITY 414 C USING A THREE TERM SERIES APPROXIMATION 415 C 418 C 417 SUBROUTINE FJTN(X) 418 IMPLICIT REAL'S (A-Z) 419 DIMENSION X(4) 420 COMMON /AREA1/ XCHIM. XCHIE. XEGB. XPHIO. JDO, JO, JB, OFIX. 421 f KT, XCHIH, JNO, JRO. JGO, XVBI, XVCE. JNE, JRNB. JR, JG. 422 « XPEGB. XZETA. XZETA2. XVBIE. JPO. JRM, JPE, JP2 423 COMMON /AREA3/ JPOL. JTN. JTP. GAMMA. ASTOC. ASTOV 424 C 425 IF (X(4) .LT. 0) GOTO 10 426 XNALP1 = -XPEGB - X(4) - XZETA • XEGB 427 XNALP2 = -XZETA2 428 BRC = (XCHIE - X(4)/2) ' KT 429 GOTO 20 430 10 XNALP1 = -XZETA 431 XNALP2 * -XEGB • X(4) • XPEGB • XZETA2 432 BRC = (XCHIE • X(4)/2) • KT 433 C 434 C 435 20 THETCO = DEXPL(-GAMMA'DSORT(BRC)) 436 THETC1 * GAMMA ' THETC0/(2*DSQRT(BRC)) 437 THETC2 = GAMMA * (GAMMA/BRC • 1/(0SQRT(BRC) > "3> • THETCO/4 438 JTN = ASTOC ' (THETCO ' (F01 (XNALP1)-FD1{XNALP2)) 439 # • (KVTHETC1) ' (FD2(XNALP1)-FD2(XNALP2)) 440 » * l(KT"2)'THETC2) ' (F03(XNALP1) -FD3IXNALP2) ) ) 441 RETURN 442 END 443 C 444 C 445 C SUBROUTINE TO CALCULATE 446 C MINORITY CARRIER TUNNELING CURRENT DENSITY 447 C • USING A THREE TERM SERIES APPROXIMATION 448 C 449 C 450 SUBROUTINE FJTP(X) 451 IMPLICIT REAL'8 (A-Z) 452 DIMENSION X(3) 453 454 COMMON /AREA1/ XCHIM. XCHIE. XEGB, XPHIO. JOO. JD. JB. OFIX. 455 1 KT, XCHIH. JNO. JRO. JGO. XVBI. XVCE. JNE. JRNB. JR. JG. 456 * XPEGB. XZETA. XZETA2. XVBIE, JPO. JRM. JPE, JP2 457 COMMON /AREA3/ JPOL, JTN, JTP, GAMMA. ASTOC. ASTOV 458 C 459 IF (X(4) .LT. 0) GOTO 10 460 XNBET1 = -XPEGB • XZETA • X(2) 461 XNBET2 = -XPEGB - X(4) • XZETA2 • X(3) 462 BRV = (XCHIH • X(4)/2) ' KT 463 GOTO 20 484 10 XNBET1 = -XEGB • X(4) • XZETA • X(2) • • P V T . . , . n  L i s t i n g of PIMSP.S at 21:52:10 on OCT 10. 1080 for CC1d-KCHU on 0 Page 0 465 XN8ET2 = -XEGB • XZETA2 • X(3) 488 BRV = (XCHIH - X(4)/2) • KT 467 C 488 C 469 20 THETVO s DE XPL(-GAMMA * DSORT(BRV)) 470 THETV1 a GAMMA • THETVO/(2'DSQRT(BRV)) 471 THETV2 a GAMMA • (GAMMA/BRV • 1/( DSORT'.BRV)) • *3) • THETVO/4 472 JTP = ASTOV • (THETVO * (FD1(XNBET2)-FU1(XNBET1)) 473 • • (KVTHETV1) * (FD2(XNBET2) F02(XNBET 1)) 474 t * <(KT"2)'THETV2) * (FD3(XNBET2)-FD3IXNBET1))) 475 RETURN 476 END 477 C 478 C 479 c EXPONENTIAL FUNCTION 480 c 481 C 482 DOUBLE PRECISION FUNCTION OEXPL(X) 483 IMPLICIT REAL*8 (A-Z) 484 IF(X .LT. -150.000) GOTO 100 485 DEXPL=OEXP(X) 486 RETURN 487 100 DEXPL=0.000 488 RETURN 489 END 490 C 491 C 492 C FERMI-OIRAC FUNCTION OF ORDER 1 493 C USING A SERIES APPROXIMATION 494 c 495 c 498 DOUBLE PRECISION FUNCTION FOKX) 497 IMPLICIT REAL'S (A-Z) 498 DATA PI/3.14159265400/ 499 Y = X 500 IF(Y .LE. O.DO) GOTO 50 501 Y = -Y 502 50 Z = I.DO'OEXPL(Y) -.250052'DEXPL(2.*Y) •.111747*DEXPL(3.*Y) 503 • • 064557 ,0EXPL(4.'Y) • 040754«DEXPL(5.*Y) 504 # • .020532'DEXPHB. *Y) • .005108*DEXPL(7. • Y) 505 IF(X .LE. 0.00) GOTO 100 506 Z = -Z • (X*'2)/2.D0 • (PI"2)/8.00 507 100 CONTINUE 508 F01 3 Z 509 RETURN 510 END 511 c 512 c 513 c FERMI-OIRAC FUNCTION OF ORDER 2 514 c USING A SERIES APPROXIMATION 515 c 516 c 517 DOUBLE PRECISION FUNCTION FD2(X) 518 IMPLICIT REAL'S (A-Z) 519 DATA PI/3.14159265400/ 520 Y = X 521 IF(Y LE. O.DO) GOTO 50 522 Y = - Y L i s t i n g of PIMSP.S at 21:32:10 on OCT 10, 1088 for CC1d=KCHU on 0 Pag* 10 523 50 Z " 1.D0*DEXPL(Y> -.125046*DEXPL(2."Y) •.037642*DEXPL(3. •Y) 524 « -.018183'0EXPL(4.*Y) • .012484*DEXPL(5.*Y) 525 « -.007486*DEXPL(6.*Y) • .002133*DEXPL<7.*Y) 528 IF(X .LE. O.DO) GOTO 100 527 Z = Z • <X"3)/6.00 • (PI- 42)'X/6.00 528 100 CONTINUE 529 FD2 = Z 530 RETURN 531 END 532 C 533 C 534 C FERMI*DIRAC FUNCTION OF ORDER 3 635 c USING A SERIES APPROXIMATION 536 c 537 c 538 DOUBLE PRECISION FUNCTION F03(X) 539 IMPLICIT REAL*8 (A-Z) 540 DATA PI/3.14159285400/ 541 Y => X 542 IF(Y .LE. O.DO) GOTO 50 543 Y = -Y 544 50 Z a I.DO'DEXPL(Y) -.082592*0EXPL(2. ,Y) •.013661'OEXPLt3. •Y) 545 # -.00979B*DEXPL(4.'Y) •.012978*DEXPL(5.*Y) 548 » -.010859"DEXPL(8.*Y) •.003446'DEXPL(7.*Y) 547 IF(X .LE. O.DO) GOTO 100 548 Z = -Z • (X"4)/24.D0 • ((PI*X)* ,2)/12.D0 • 7.MPI # ,4)/ 549 » 360.00 550 100 CONTINUE 551 FD3 = Z 552 RETURN 553 END 554 C 555 C 558 C FERMI-DIRAC FUNCTION OF ORDER 1/2 557 c USING A SERIES APPROXIMATION 558 c 559 c 560 DOUBLE PRECISION FUNCTION F0102(X) 561 IMPLICIT REAL'S (A-Z) 562 DATA PI/3.141592854D0/ 563 IF (X .LE. O.DO) GOTO 100 564 IF (X .LE. 2.00) GOTO 200 565 IF (X .LE. 4.DO) GOTO 300 586 GOTO 400 567 100 FD102 = OEXPL(X) -0.353588*DEXPL(2*X) *0.192439*DEXPL(3* X) 588 # -0.122973*DEXPL(4*X) *0.077134'DEXPL(5'X) 569 # -0.036228 ,DEXPL(8*X) *0.008346'OEXPL(7*X) 570 GOTO 500 571 200 FD102 = 0.765147 • XMO.804911 • XMO. 189885 • 572 • XMO.020307 • XM-0.004380 • XM-0.000366 • 573 # X'O.000133))))) 574 GOTO 500 575 300 FD102 = 0.777114 • XMO.581307 • XMO.208132 • 576 • XMO.017680 • XM-0.006549 • XMO.000784 -577 # X-0.000036))))) 578 GOTO 500 579 400 XN2 a 1/(X"2) 580 FD102 = (X*'1.5) * (0.752253 • XN2M0.928195 • Ut)t1n(j of PIMSP.S at 21:52:10 on OCT 10. 1988 for CC1d=KCHU on G p«9 e 1 1 561 • XN2M0.880839 • XN2M25.7829 • XN2M-553.638 • 582 * XN2M3531 43 - XN2' 3254 . 8 5 ) ) ) ) ) ) 583 500 RETURN 584 END 00 PUBLICATIONS Chu, K.M. and D.L. Pulfrey, "Design procedures for differential cascode voltage switch circuits", 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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