591 IFEE TK4NSACTIONS OK ELECTRON DEVICES, VOL. ED-17, NO. 8, AUGUST 1970

ACKNOWLEDGMEKT [3] H. S. Carslaw and J . C. Jaeger, Conduction of Heat in Sol ids , 2nd ed. Oxford: Oxford University Press, 1962. ~h~ authors ,,.auld like to t h a n k D. DeLi:itt and [1] J: J . Sparkes, “Voltage feedback and thermal resistance in junc-

Dr. A. Reisman for suggesting these problems for [5] E. S. Scklig, “Low-temperature operation of Ge picosecond logic tlon transistors,!‘ Proc . IRE, vol. 46, pp. 1305-1306, June 1958.

investigation. circuits, IEEE J . Solid-State Circuits, vol. SC-3, pp. 271-276, [6] F. H . Dill, A. S. Farber, and H. N . Yu, “Picosecond integrated

REFERENCES circuits in germanium and silicon,” IEEE J . Solid-State Circuits,

September 1968.

[ I ] R. LV. Keyes, “Physical problems and limits i n computer logic,” [7] J . -4. &;ruthers, T. H. Geballe, H. h l . Rosenberg, and J. Y l , IEEE Spectruna, pp. 36-45, May 1969. Ziman, The thermal conductivity of germanium and silicon

[ 2 ] R. Holm, Electrical Contacts Handbook. Berlin: Springer-Verlag, 1958,

between 2 and 300°K,” Proc . Roy. Soc., vol. A238, pp. 502-514, January 1957.

vol. SC-3, pp. 160-162, June 1968.

Numerical Solutions for a One-Dimensional Silicon n-p-n Transistor

B. V. GOKHALE

Abstract-This paper describes a technique of obtaining numeri- cal solutions of the basic carrier transport equations for a semicon- ductor and the results of some calculations pertaining to a silicon n-p-n transistor.

The calculations include dc characteristics in direct and inverse operation, saturation parameters, and small-signal ac common emitter h-parameters. Both Boltzmann and Fermi statistics have been used, and the depenience of carrier mobilities on electric field has been taken into account.

0 ISTRODUCTION

IVING to rapid technological advances in the field of semiconductor devices, the need for detailed and accurate analysis of device opera-

tion has become very urgent. Current analytical theory is applicable to a number of limiting situations which are treated under special simplifying approximations appropriate to each case, but the extension of the theory to cover more general cases has not proved feasible. For this reason, many authors, notably Sparkes [ l ] , have emphasized the need for a reappraisal of the standard theory and have pointed out the suitability of numerical solutions to overcome mathematical complexity.

Gummel [ 2 ] achieved a notable breakthrough r\.ith his technique of numerical integration of the one- dimensional carrier transport equations. His xvork n-as further elucidated and mathematically generalized by de LIari [S I , [4] and was used by Ghosh [3] in develop- ing a very general transistor model. But the usefulness of numerical solutions in device design theory does not

hlanuscript received January 13, 1970; revised March 13, 1970. The author is with the I B l I -Components Division, East Fishkill

Facility, Hopewell Junction, N.Y. 12533.

seem to be sufficiently exploited. The present paper describes some calculations of an exploratory nature, made a-ith the object of interrelating electrical proper- ties of transistors \\-ith their physical parameters such as the doping profile, carrier mobilities, and lifetimes. These calculations may be of interest to device de- signers.

THE BASIC EQUATIONS The basic one-dimensional equations x\\-hich govern

transport of carriers in a semiconductor comprise the continuity equations for electrons and holes:

and Poisson’s equation for the electrostatic potential $:

These must be supplemented by equations which define the electron and hole currents j,, j p in terms of the carrier densities n, p and the respective quasi-Fermi potentials &, &,:

GOKHALE: NUMERICAL SOLUTIONS FOR ONE-DIMEKSIONAL TRANSISTOR 595

and the Boltzmann relations between the potentials and carrier densities:

12 = n . e P / k T ( + h ) p = n i e q / k T ( $ p - # ) , (6)

The ra te of recombination R(n, e) is assumed to follow the Shockley-Read-Hall steady-state recombination law corresponding to uniformly distributed recombina- tion centers with a single energy level in the center of the bandgap :

The lifetime parameters T,, 7, are taken to be constants independent of x. Any possible contribution of the re- combination centers to the charge density is neglected.’

The base current enters the base of the transistor at right angles to the x axis. It cannot therefore be included in a one-dimensional analysis in any natural may. We can take i t into account by assuming the existence of hole sources distributed in the base of the transistor. The function G ( x ) represents the positive charge gen- erated per second per unit volume at the point x. There are no obvious clues to guide the choice of the function G ( x ) , but in the present calculation, for simplicity, it is taken to be a nonzero constant over a suitably chosen region in the base and is zero everywhere else. The density of the base current Jb is given by

J b = JBAsEG(T)IIX. (8)

The quant i ty Ja is, in this sense, the ratio of the base current to the area of the emitter junction.

The portion of the base over which G ( x ) is nonzero defines, in a certain sense, the “electrical base”. In the calculations reported here the electrical base included the region + < + m i n + O . l volt (see Fig. 3).

The functions N D ( ~ ) and N A ( x ) represent the densi- ties of n- and p-type impurities. All impurities are as- sumed to be fully ionized.

The mobilities of minority as well as majority carriers are assumed to obey an empirical law proposed by Caughey and Thomas [ 6 ] :

The same authors suggest [6] that the field dependence of mobilities may be taken into account by multiplica- tive factors:

1 The omission of recombination center density from (3) and the use of (7) in ac as well as dc calculations implies a vanishingly small density of recombination centers possessing infinitely large capture

1 dl 4- (E/8000)2 ’ rn =

1 rp = 1 + I E I /1.95 x 104’

(1 2 )

E being the electric field in volts per centimeter.

dence of mobilities are Three simple alternatives in regard to the field depen-

1) r,=rp=l, 2) F,, rpl given by (1 1) and (12), 3) F,, I?, given by (11) and (12) but with the modifi-

cation that the argument E is replaced by the gradient of the appropriate quasi-Fermi potential.

Alternative 1 ignores the field dependence altogether. Almost all the published work which takes field de- pendence into account follows 2 . This procedure is open to objection because it implies a decrease of mobilities in the vicinity of a forward-biased emitter junction where a strong electric field may exist a t low current densities. Carriers crossing the emitter junction move against the electric field; hence they cannot gain energy from the field and cannot become hot.

Alternative 3 seems physically the most plausible one; i t has been used in almost all the calculations reported here. h’ith this alternative, the division of current into drift and diffusion components is not possible except when the quasi-Fermi potential gradient is small. However, this does not seem to contradict any basic principle.

Equations (l), (2), and (3) are to be solved subject to the following boundary conditions:

a t the emitter contact: 4, = 4, = 0, (13)

at the collector contact: +,, = & = VCE. (14)

The values of at the boundaries are determined from the Boltzmann relations (6) and the neutrality condi- tion :

N D - AT^ + p - n = 0. (1 5)

THE TECHNIQUE OF SOLUTION The equations (l), (2) , (3) are solved by a finite

difference method. The values of the functions 4,, q5p,

and + are to be found a t a finite number of fixed, dis- crete, and not necessarily equidistant points, known as the grid points, in the range of interest. With suitable difference approximations for derivatives, n.e obtain a system of algebraic equations of the form

F k ( . $ l , E ’ , . * ; U , b , C, . . .) = 0. (1 6)

The arguments tT stand for the values of the potentials a t the grid points, and a , b , c, . . for external param- eters such as the bias voltage V/CE, the base current

cross sections. ~~ density J b , the minority carrier lifetimes 7,, T,, the

doping profile, etc. There are three equations of the form ( 1 6 ) which must be satisfied a t each grid point.

Suppose that an approximate steady-state solution of' the equations is available for certain values of the external parameters al , b l , c1, * . To obtain a more exact solution we assume that corrections A&, A&, . must be added to the potentials. To the first order, then, we have

Equation (17) is a system of linear equations which can be solved by inversion of the so-called Jacobian matrix (dFk/dtr). The method of approximating (I), ( 2 ) , and ( 3 ) by finite difference expressions leads to a Jacobian matrix of a peculiarly simple structure known as a band matrix. The only nonzero elements are those lying on the principal diagonal or on any one of four adjacent upper or lower co-diagonals. The inversion of such a matrix is accomplished with the help of a sub- routine based on a modified Gaussian algorithm.

On solving (17) the values of the potentials are up- dated by adding the corrections At,., and this process is repeated as often as necessary until the successive cor- rections decrease in magnitude and are all smaller than a specified small number (usually a millionth part of k T / q ) . The culmination of this process in a satisfactory solution is possible only when the starting approximate solution is sufficiently close to the exact solution. This technique of solving (16) is known as the Newton- Raphson technique.

When an exact solution corresponding to the values

lye can proceed to find a new solution for different values u2, bz, cq, * . . of the parameters. We only need to solve (17) with the current values of $ 1 , (2, . . and new values az, bs, cz, - . . substituted in place of al , b l , c1, . . . A convergent process results when the changes (a2 -a l ) , (bz - b l ) , * are sufficiently small. In prac- tice, convergence is usually possible if , on substituting az, bz, CZ, * * . for all b l , c1, . . , none of the corrections A& arising from (17) is large in comparison with kT/q . The external parameters should therefore be changed in sufficiently small steps.

This requirement of small steps can be automated. When az, bz, cz, . . are given, it is possible to find, if necessary, intermediate values ai, bi , ci, - of the external parameters for which a convergent process is possible. The step from (a l , b l , c1, . . . ) to (az, b2, c2 , . e e ) is thus subdivided into a number of substeps to facilitate convergence. T o determine suitable inter- mediate values ai, bi, c i , ' . . we have, from (17),

a1, bl , C l , * . ' of the external parameters is available,

TEFF TR 4 Y E 4CTTOXS gY ELFCTRON DEVICES. AUGI:ST 1970

Equation (18) is again a system of linear equations, \vith a Jacobian matrix of the same structure, and calculated in the same way as in (17). The corrections A& found by solving (18) are linear functions of the changes (u i -u l ) , ( b i - b l ) , . We now specify the largest permissible value, denoted by CP, that any correction a t a n y grid point may have. If any actual correction exceeds CP, the increments ( a i - a l ) ( b i - b l ) , . * are proportionately decreased to make the largest correction equal to CP. The corrections At,. are then used to update the potentials and we return to (17), with the values ai, bi , ci, . . - substituted for ai , b l , C I , * * .

In a small-signal ac calculation, the equations to be solved are, again, those following from (18). The parameter increments (a2 - Q ) , (bz - b l ) , . . . are now infinitesimals and have the time factors e k t . The solu- tions At, are the complex amplitudes of the ac potentials rather than corrections to their dc values, and there is no iteration and no requirement of convergence.

The calculations described in the following section always started with the equilibrium solution; that is, the one corresponding to J b = O and VCE = O . Approxi- mate values of the potentials in the equilibrium situa- tion were determined from standard theory based on the division of the transistor into quasi-neutral and space-charge regions; and these approximate values were refined by the matrix-inversion technique just described. The parameters Jb and VCE were then given the desired values and the calculation proceeded via (18) and (17).

The technique of division of a problem into substeps effectively eliminates all danger of divergence but does not guarantee convergence. Occasionally i t happens that the corrections remain approximately constant in magnitude in successive iterations, or go up and down in a periodic fashion. The reasons for this behavior are not clear. As a practical matter, a convergent solution can be usually found in such cases by slightly perturb- ing one of the external parameters, such as J b , T ~ , T ~ , etc.

RESULTS OF SOME RECENT CALCULATIONS

The following calculations were performed in an attempt to develop a computer program incorporating the technique just discussed and capable of answering a wide variety of questions regarding the operation of a transistor.

The doping profiles used in the calculations were generated by means of mathematical formulas of sufficient flexibility to incorporate the salient features of diffused transistors. A typical profile, and the elec- tron and hole mobilities as functions of x , are shown in Fig. 1.

The sheet resistance of the base for transverse cur- * (bi - bl) + * * * = 0. rent flow was calculated from the equilibrium solution

GOKHALE: NUMERICAL. SOLUTIONS FOR ONE-DIMENSIONAL TRANSISTOR

700 1 600

x ( p )

Fig. 1. Doping profile and mobilities.

and found to be 33.8 kQ/n, a value that is rather high but not exorbitant. Alinority carrier lifetimes were as- sumed to be r p = 1 ns, r , = 10 ns.

Figs. 2 and 3 show detailed solutions for a very few of the calculations. For the rest of the calculations, only the terminal voltages and currents will be reported.

The Gummel Plot The J,- VBE characteristics for V/CS = 0, the so-called

Gummel plots, were calculated for both direct and in- verse operation. The two characteristics coincided to a very satisfactory degree.

The standard analytical theory [7] of these char- acteristics permits the calculation of the average elec- tron diffusivity in terms of P,, the total number of holes in the base under equilibrium conditions (usually called “the integrated base doping”). Using the value of P,, given by the equilibrium solution, turns out to be 11.6 cm2/s. On the other hand, 5, may be calcu- lated directly from its definition:

I Po, D =----, (19)

The value in this case is 15.2 cm2/s, indicating a rea- sonably close agreement between the analytical and numerical techniques.

J , - VCE Characteristics J , - VCE characteristics were calculated both for

direct and inverse operation. I t was found that the saturation region as well as the active region was amenable to calculation. The so-called cutoff region, although not investigated in any detail, seems to offer no special difficulty.

A calculation of the saturation voltage V C E ~ ~ ~ was made in the following way. A value of Jb was specified. By a slight modification of the computer program, the

597

4.5 <

1.6

1.2

0.8

- > * 0.4

0

- 0.4 0

r

t

(X )P

Fig. 2. Net charge density versus X.

REGION OF NONZERO G(x)

-4

\ , I , .12 .36 .60 .E4 1.08 1.32 1.56 1.80

X ( P )

Fig. 3 . The electrostatic potential II. versus X.

value of VCE, instead of being fixed, n a s continually adjusted until a specified collector current density was achieved. This procedure was repeated for successively higher values of Jb. I t may be seen from Fig. 4 that the saturation voltages of about 4 and 9 mV correspond to J , = 1000 and 3000 A/cm2.

The total number of holes, P, was also calculated for each value of Jb. The plot of P versus Jb was approxi- mately a straight line in the saturation region. For J , = 1000 and 3000 A/cm2, the saturation time constant T,, given by q.AP/AJb, was found to be 3.9 and 4.2 ns, respectively.

30

25 '

10 .

I 0 100 200 300 400 500

Jb A/cm2

Fig. 4. VCB versus Jb at constant J,.

A c Calculations

Small-signal ac calculations lvere made a t a number of bias points, a t several frequencies up to 10 GHz, to obtain common emitter h-parameters, defined by

GVBE = hie.GJb + h r e . G V c ~ ,

and

6Jc = hr,'GJb -I- h0,.6VcE. (20)

At a particular bias point the values of any one of the parameters a t different frequencies may be plotted in the complex plane. The low-frequency portions of all such plots gave smooth semicircles with centers on the real axis. The frequency a t which the real part of h was equal to the abscissa of the center was very nearly the same for all four parameters and was equal to fa , the 3-dB cutoff frequency for the common emitter current gain. [See Figs. 5-10, The bias conditions labeled 1, 2 , and 3 signify 1) L'CE = 1 volt, Jc=3938 A/cm2; 2 ) VCE = 0.3 volt; J C = 3465 A/cm2; and 3) VCE= 0.03 volt, JC = 2288 A/cm2.]

At low frequencies the input impedance h;, can be accurately represented by a parallel RC circuit. The short-circuit, forn-ard-current gain hre can be expressed in the form

The gain bandwidth product f T is given by f r = P . f o . At frequencies of 1 GHz and above, the real part of hfe be- comes negative, although i t is impossible to show this in Fig. 6 because of the smallness of the absolute mag- nitude of hr,. The negative phase of hfe is larger than the high-frequency limit of 90' predicted by (21); and the

-100

-500

-1000

2 -1500

bo0 MHz

-2000 c

- 2500

-3000 0 i SO0 1000 1500 2000 2500 3000 3500 4000

R E A L S ( ~ A I - ~ ~ ~ )

Fig. 5 . Input impedance hi,, bias conditions 1, 2, and 3.

f I

250 I 0 50 100 150 200 250 300 350 400 450

, -1-

REALS

Fig. 6 . Short-circuit forward current gain hjd, bias conditions 1, 2, and 3.

"excess phase," seems to increase rapidly as saturation is approached. A t the frequency of 10 GHz, this excess phase amounts to 13.5O for the V C E = 1-volt curve, 17.4' for the V~~=0.3-vol t curve , and 85.8' for the V C E = 0.03-volt curve.

Figs. 7-10 show the output admittance hoe and the reverse transfer voltage ratio hre. The low-frequency behavior of these parameters is simple. The magnitude of hoe increases with increasing collector current density ( J c ) and with decreasing collector bias ( V C E ) . At low collector biases the imaginary part of hoe becomes nega- tive, so that the reactance is inductive rather than capacitive. The semicircles corresponding to h,, shrink with increasing Jc , and the absolute magnitudes de- crease with increasing V C E . The sign of the imaginary part is, again, different a t low collector bias from that in the active region.

Emitter and Collector Capacitances p-n junction capacitances may be calculated in tu.0

different ways. 1) The stored-charge method, in which capacitance

is defined by C = d Q / d V , where V is the terminal voltage and Q is the total charge of any one kind of carrier (electrons or holes) integrated over the whole profile. The derivative is determined by a dc calculation or a zero-frequency ac calculation.

GOKHALE: NUMERICAL SOLUTIONS FOR ONE-DIMENSIONAL TRANSISTOR 599

16 ,

-2 -

REALS

- 4 ' I I 86 88 90 92 94 96 98 100

Kmho/cm2

Fig. 7 . Output admittance h,,, bias condition 3.

3.0 I

05 1.0 1.5 2.0 2 5 3 0 3.5 4.0 4.5 5.0 55 Kmho/cm2

Fig. 8. Output admittance h,,, bias conditions 1 and 2.

The ac admittance method, in which a small-signal ac calculation gives the value of the complex admittance Y. When the frequency dependence of Y is analyzed, an equivalent circuit in terms of resistances and a capacitance is usually possible. In the simplest case a parallel RC circuit, in which C= l/u Im( Y ) , is adequate. For a single p-n junc- tion under reverse or lorn forward bias, the two methods give identical results. However, under large forward bias the results are different; as a matter of fact, the ac admittance of a diode in heavy injection can be inductive.

The two ways of calculating capacitance may be readily generalized for a transistor structure. In the stored-charge method, emitter and collector capaci- tances are

.06 I I

IO GHr

-1

-.04 - ' O 2 1 - I -.06 I

S O .52 .54 .56 .58 .60 .62 .64 .66 REALS

Fig. 9. Reverse voltage transfer ratio h,,, bias condition 3.

f, .02 -

10

0 0 .01 .02 .03 .04 .OS

.o 1

0 0 .01 .02 .03 .04 .OS

REALS

Fig, 10. Reverse voltage transfer ratio h,,, bias conditions 1 and 2.

C E = - , c, = ( X ) . ( 2 2 ) ~ V B C vBE

I n the ac admittance method the appropriate admit- tances to be considered are YEBS, YCBS, which are, re- spectively, admittances measured between emitter and base terminals with the base ac short-circuited to the collector, and between collector and base terminals with the latter ac short-circuited to the emitter. These admit- tances may be easily calculated from the common emitter h-parameters:

YCBS = h i e . h o e - hm* h f e

hie

- (23)

Capacitance values obtained from an analysis of the frequency dependence of these admittances are shown alongside the stored-charge capacitance in Table I .

I t is seen from Table I that the values of CE calculated by the two methods are substantially different at all bias points. The difference is not quite so striking in the

600 IEEE TP,.4,VS,4CTTONS ON ELECTRON DEVICES, AUGUST 1970

TABLE I COMPARISON OF CAPACITAXCES CALCULATED BY DIFFEREST METHODS

(rn = 10 ns, rP = 1 ns) .__ - __

Bias Conditions I CB (pF/mi12) 1 I C, (pF/mi12)

0.03 0.03

0.3

0.03 1

0.03 0.3 1 0.3 1 0.3 1

396 1 3.19 1 z:;; 4.28 2.05 2 . 2 2 1 1.93

4.93 90

800 3 . 5 7 4.63 4.63 I 0.490 89 7 3.59 4.64 4.64 0.310

1106 5.30 9 .91 ! 9.87 10.76 2138 8.41 21.03 21.13 19.58 3459 7.48 12.24 3929 7.63

12.23 0.549

9532 12.24

14.44 12.24 0.326

28 .00 28 .oo 0.728 10 880 14.57 27.85 27.85 0.382 18 840 24.32 21 590

53.35 24.90

53.37 1.18 52.73 52.71 0.472

2 .oo 0.499 5.44

0.310 13.21 25.16 0.586

0.802 0.334

0.381 1.35 0,466

YBC

2 .oo 0.499 5.43

0.313 13.06 25.14 0.586 0.334 0.802 0.381

0 . 4 6 i 1.35

yBC

Fig. 11. The common emitter equivalent circuit. YBE = (1 -hra)/hc.; Y~~=h, , /hi . ; Y~~=(h i , . h , , -h / , .~ , , -h~ , ) /h i , ; and A=(hf.+h,,)/hi..

case of C,. At VCE = 1 volt the two values of C, are in good agreement. At VCE =0.3 volt the agreement is poorer, and is quite bad at VCE = 0.03 volt.

The stored-charge capacitance thus equals the ac ad- mittance capacitance when a junction is reverse biased but differs considerably when forward biased.

Lindmayer and Wrigley [8] present persuasive ar- guments that such disagreement is only to be expected. The stored-charge capacitance is proportional to the increase in the stored charge of only one kind of carrier, while the ac admittance-capacitance is proportional to the net charge (due to carriers of both kinds) entering one terminal. The two are conceptually different, al- though related to each other. The relationship, however, is by no means a simple one.

A common emitter T-equivalent circuit and its rela- tionship to the h-parameters is shown in Fig. 11. It turns out that the capacitances associated with the admit- tances YBE and YBC are very nearly equal to the stored- charge capacitances a t all bias points2 These capaci- tances are also listed in Table I.

The gain-bandwidth product fT is often calculated as

This formula follows from the circuit of Fig. 11 if the capacitances associated with YBE and YBc are taken to

similar result (private communication). In an independent investigation, Dr. G. Hachtel has obtained a

be the stored-charge capacitances. In fact, the values of fT calculated from (25) agreed within about 5 percent with those deduced from the frequency dependence of hf,.

Dependence of hfe on the Lifetimes and Other Parameters Some calculations were made with a different value of

T~ (50 ps). The results indicate that a low lifetime in the emitter region does not materially affect fT, although p and fp are quite different.

In order to study the effect of the electric field in the base region, the doping profile 2 of Fig. 1 2 was used. This profile is identical with the one in Fig. 1 except in the base. The total base doping and the peak value of the base dope are the same for the two. The results are shown in Table 11.

To evaluate the effect of different doping levels in the neutral emitter region, the profiles A , B , C of Fig. 13 were studied. These profiles are identical with the one of Fig. 1 (which is redrawn in Fig. 13 for comparison) ex- cept in the neutral emitter regions. The results are shown in Table 111.

Table I11 indicates the possible effect of some factors, such as strong degeneracy of the electron gas and par- tial ionization of impurities, which have been neglected so far and which tend to reduce the effective majority carrier density in the emitter region. This situation is simulated by the use of a low Co doping profile. The results suggest that the use of Fermi statistics and the partial ionization of impurities are quite important in the prediction of 0.

GOKHALE: NUMERICAL SOLUTIONS FOR OSE-DIMENSIONAL TRAXSISTOR 601

i

1 I .70 74 .78 .82 .86 .90 .94 .98 1.02 1.06

X

Fig. 12. Doping profiles differing in the base only.

TABLE I1 COMPARISON OF DOPING PROFILES 1 AND 2 OF FIG. 12

( V c ~ = l volt, Jb=25 A/cm2, r,=lO ns, ~ ~ = 1 ns)

Profile ~ vER ~ , ( ~ / c m z ) j @ 1 jT ( G H ~ ) fs ( N I H ~ ) ~ - 1

1 i 0 .833 I 10 938 2 0.822 7434

x ( c )

Fig. 13. Doping profiles with different dopes in the neutral emitter region.

TABLE 111 COMPARISON OF DOPING PROFILES A , B , AKD C OF FIG. 13

( VC/CE = 1 volt, 7, = 10 ns, T~ = 1 ns)

1 23.0 0.829 10 004 460 A

10.9

B 1 t"9: 1 0.831 1 9997 ! iii ~ 10.4 0.830 10 001 419 I 10.8

C 41.2 0.833 9992 1 9 . 3

TABLE IV

(AP=P-P,,; V c ~ = l volt; J,=lO 000A/cm*) COMPARISON OF DIFFERENT VALUES OF Tn, 7 p

\I 0.31611s 850 528 326 p = 195 10 ns

10 ns 3.1611s 1 ns

Tn

f~ = 10.4 GHz 10.2 10.2 10.3 AP = 1,6472X 1012crn-2 1.6583X1012 1.6578 X 10I2 1.6535 X loL2

31 .6 ns

1 ,6586 X 10l2 1.6572 X 10I2 1.6478 X lox2 1.6538 X 10l2 10.2 10.2 10.4 ~ 10.3 1390 9 03 312 540

T.L\BLE V TABLE V I COMPARISOK OF DIFFEREKT, BASE SHEET RESISTAXES ;\LTERKATIVE FORMCLATIONS OF FIELD DEPEXDESCE OF MOBILITIES

( V c ~ = l volt; J,=6000 AJ'cm2; 7*=10 ns; ~ ~ = 1 ns) ( v C E = l volt; Ja=25 .A/cmZ;r,=lO ns; ~ # = 1 ns)

I Base Sheet Resistance W!O) P f~ (GHz)

- -

35.3 1 309

5 . 6 9 . 3 ~ 145

9 . 4 19.6 260 8 . 5 12 .9 194

4 . 5 109

6 . 9

7 I

Partial ionization of impurities was incorporated into the computer program by replacing N D , N A in (3) by N D + , NA- given by

N D 1YD+ = -- - 1 + ~ ~ ( E C - E D ) I ~ T ~ ~ / ~ T ( * - - ~ ~ ) - - E G I ~ ~ T ( 2 6 )

and

N A LITA- =

1 + 2e(E~-Ev)/kTeq/kT(Op-~)--EG/2kT (2’i)

where EQ is the energy bandgap. For simplicity both the ionization energies (Ec - E D ) and (EA -Ev) were taken to be 0.04 volt.

Fermi statistics were used by changing (6) to

and

Table VI. ’The last t ~ a alternatives for r,, P P , sur- prisingly, give \ w y similar results.

A computer program for the numerical integration of the basic equations has been shown to be a useful tool for the analysis of semiconductor device operation. Device parameters calculated in this way are consistent in the sense that they are derived from the same mate- rial parameters, using the same technique of calculation. This feature is likely to be important in device design.

ACKNOWLEDGMEXT The author lvishes to thank S. Schmidt, who wrote

the subroutine for the inversion of the band matrix; 11. B. Vora, G. Hachtel, 11. l lock, and E. Sxvatt; and P. H. Bardell, A. Rideout, J . H. Bleher, C. S. Chang, and G. Steinberg for a careful reading of the manuscript and for many helpful suggestions.