924 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-26, NO. 6 , J U N S i 979

Exact Equivalent Circuit Model for Steady-State Characterization of Semiconductor Devices

with Multiple-Energy-Level Recombination Centers

PHILIP CHING HO CHAN AND CHIH-TANG SAH, FELLOW, IEEE

Abstract-One-dimensional steady-state characteristics of a two- terminal semiconductor device is modeled by an exact transmission-line circuit model. This is generalized to include Shockley-Read-Hall (SRH) recombination centers with an arbitrary number of energy levels. I t is then applied to a study of the double-acceptor zinc centers in silicon p-n junction diodes. The exact internal characteristics of the diodes; such as the spatial variation of carrier densities, electric field, net charge density, recombination rate, and current densities: are obtained numerically from the circuit model and studied at different injection levels. The zinc population and the relative importance of recombina- tion at each of the two zinc energy levels is interpreted by the &rh- Shockley theory for multiple-energy-level SRH centers.

T I. INTRODUCTION

HIS PAPER is organized as follows. A historical introduc- tion is given in this section which traces the deve1opmt:n.t

of the various numerical techniques for obtaining the ex;xt electrical characteristics of semiconductor devices. Section I1 presents the theory and development of the new steady-stiite circuit model for a semiconductor or a semiconductor device with a multiple-energy-level recombination center. It also in- cludes a discussion of the procedure used to solve the error circuit model from which the final accurate solutions of l.he internal and terminal characteristics of a device can be ob- tained. Section I11 gives an illustration of the application of the circuit model to the double-acceptor zinc centers in silicon p-n junctions.

One-dimensional characteristics of semiconductor devices are described by a set of coupled nonlinear partial differexial equations [I J , [2], known as the Shockley equations [ 3 ] , 143

Manuscript received November 7, 1978; revised January 23, 1S79. This work was supported by the Air Force Office of Scientific Reseuch under Grants AFOSR-76-2911 and AFOSR-78-3714, and by the Rome Air Development Center under Contract RADC-F-19628-7740138 with the U.S. Department of the Air Force. P. C. H. Chan was ilso supported by IBM Postdoctoral Fellowship. This work is based in part on a Ph.D. dissertation submitted to the Department of Electrical Engineering, University of Illinois, Urbana.

The authors are with the Department of Electrical Engineering, University of Illinois, Urbana, IL 61801.

IIEEE standard notation is used. XA = XA +x, where X A is the static component and x, is the small-signal component.

Equations (1) and (2) are the electron and hole continuity equations. Equations (3) and (4) are the electron and hole current equations. Equation ( 5 ) is the Poisson's equation, and (6) is the rate equation at the Shockley-Read-Hall (SRH) re- combination centers p ] , [ 6 ] . In (1)-(6), we have assumed a single-energy-level acceptor-like SRH recombination center.

In general, it is not possible to find a solution to the set of equations in closed analytical form, even in the absence of re- combination centers. To simplify the mathematics, the device is usually divided into regions. In each region the dominating effects are linearized. In order to obtain analytical results, impurity concentrations, carrier lifetimes, and mobilities are assumed constant in each region. The overall performance of the device is found by joining and adding solutions of each region using appropriate boundary conditions [ l ] , [ 7 ] - [ 9 ] . For practical devices, impurity concentrations, mobilities, carrier lifetimes are all spatially dependent. Analytical solu- tions are difficult if not impossible to obtain.

A significant advance was made by Gummel [ lo] , who proposed a scheme for numerical integration of the one- dimensional steady-state Shockley equations. Boundary condi- tions were applied only at points representing external contacts. Spatial variation of impurity doping profile and the dependence of mobility on dopant impurity concentration were included. The recombination terms in (1) and (2) were modeled by a single-level SRH center with spatially constant concentration of recombination centers. The steady-state recombination rate is given by

Gwyn et al. [ 1 1 1 expressed the Shockley equations in the difference form. The solution is obtained by recursively sub-

CHAN AND SAH: EQUIVALENT CIRCUIT MODEL FOR SEMICONDUCTOR DEVICES 925

stituting the Poisson’s equation into the current equations and then substituting again into the continuity equations to elimi- nate the field variable. The two continuity equations were then solved simultaneously. Scharfetter and Gummel [12] applied the same model to Read diode. DeMari has studied the Gummel method in great detail [ 131. He subsequently extended it to transient problems [14]. The procedure for the steady-state solution was outlined by DeMari [ 131. Basically, the iterative procedure involves integration of continuity and current equations, and solution of Poisson’s equation by the finite difference method. Convergence is good for low and moderate recombination throughout the device. However, the procedure diverges for high recombination [ 151 - [ 181. Choo [15] attempted to improve the convergence by introducing internal loop and damping factor without much success.

Sah [4], [ 191-[21] has linearized the Shockley equations. He synthesized the resulting difference equations into equiva- lent distributed circuits. Linearization technique was also used by Gokhale [22]. The first comprehensive numerical work using linearization technique was given by Graham and Hauser [23]. With this method, they found the convergence to be quadratic if the initial trial solution is close to the true solution.

Extensions of the Gummel technique to two-dimensional problems are also found in the literature [24]-[26]. Methods using finite-element techniques were also investigated [27] - [29]. However, the finite-element technique has not been shown to be superior to the commonly used finite-difference technique for one-dimensional application. The finite-element technique is more attractive for two- or three-dimensional problems with nonrectangular geometry.

In this work, we shall show how Sah’s equivalent circuit model can be modified to expedite computation. This ap- proach not only preserves all the advantages of linearization, but also provides more insight to the physics of the problem.

11. THE THEORY OF EQUIVALENT CIRCUIT MODEL

A . Formulation of the Model If Boltzmann statistics are assumed, (1)-(6) can be expressed

in terms of u,, uN, up, and U T , where UT is the electrostatic PO-

tential, and uN, up, and U T are the quasi-Fermi potentials of the electrons, holes, and recombination centers, respectively. Diffusivities D,, D, are replaced by mobilities using Einstein relations

Dn = E*n (kT/q) (sa 1 and

D, = Ilp(kT/q). (8b)

The current equations (3) and (4) are combined with the continuity equations. The result is a set of four second-order coupled nonlinear partial differential equations with uN, up, UT, and VI as dependent variables, and x, t as independent variables. The equations are

T = an at rCP- rCN

ET is the energy level of the center. ItT and p T are the con- centration of occupied and empty centers. Equations (sa) to (9d) are expanded as follows:

Here VA is the static component and u, is the small-signal component of the dependent variables. The spatial and time derivatives are replaced by finite differences. The expanded equations may be separated into two sets. One set contains only the static components. The other set contains only the small-signal components. A distributed circuit was synthe- sized based upon each set of the equations. We shall call the distributed circuit synthesized from the set of static equations the dc steady-state or static equivalent circuit, while the dis- tributed circuit synthesized from the set of small-signal equa- tions is the small-signal equivalent circuit. The small-signal equivalent circuit for a single-level SRH center was first given by Sah (see [21], fig. 5.l(a)]) and it is shown in Fig. 1. The circuit elements are defined in Table I. The general large- signal equivalent circuit was also given by Sah [4].

The small-signal equivalent circuit contains only linear ele- ments. The values of these elements depend on the static po- tentials or static carrier densities which represent the dc operating point of the small-signal problem. The small-signal equivalent circuit also shows that the four potentials un , u p , ui, and ut are not independent. Given any three potentials, the fourth one can be computed from the equivalent circuit.

B. The Steady-State Model Green and Shewchum [30] have shown that by using

Newton-Raphson iteration technique for solving nonlinear equations, similar to the method used by Gokhale [22], the small-signal equivalent circuit of Sah [ 191 -[2 1 ] can be used to obtain dc and transient solutions. The model was partially described [30], [35]. A complete description is given here.

At steady state, the time derivatives vanish. Equations (sa) to (9d) can be expressed symbolically as:

FN(vL)=FN(vJ, vL, vj, &)=o (1 l a )

Fp(&) = Fp(& vfv, vj, VjT) = 0 (1 1b)

PI(&) = F&, Vfv, Vj, Vb) = 0 (1 IC)

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-26, NO. 6 , JUNE 1979

Fig. 1 . \Small-signal equivalent circuit for singleenergy-level SRH centers. Only one section of the circuit is shown.

TABLE I DEFINITION OF CIRCIJIT ELEMENTS IN FIG. 1

U = qV,/kT

EN = qVNlkT

Up = qVp/kT

N = n e x p ( U - U N ) i

P = n . e x p ( U p - U )

ck = E

Cn = qzN/kT

C = q 2 P / k T

G . = q J N / k T n l

Gpi = qJp/kT

P, = l/(q2nN)

P p = l / ( q Y p P )

N~ = N ( e + c ! i ) / ( e + e + c N+,- P ) T T p n n p n P

RCN = c [WTT- (N.+nl)NTl

RCP = c [ ( P + ? l ) N T - ~ L N T T l

Ct = q'N P /kTNTT T T

Gnt = q2cnNPT/kT

G = q'c PNT/kT Pt

G n t i = q2R P /kTNTT C N T

G . = q z R N /kTN ?fl CP T TT

FT(VL) = F*(V$, VL, vj, Vh) = 0. ( 1 1 4

v' - +

Here j is the index for the j th iteration and

M - (1 2) M = P, N , I , or T and m = p , n , i, or t. The upper case sub- scripts denote steady-state potentials and the lower case sub- scripts denote error potentials. Using (12), (11) is expandzd in terms of u&' by Taylor series. Retaining only first-order terms, then

F ~ ( ~ ~ l ) t f ~ ( v ~ l , u ~ l ) ~ o

&(Vx;I , UL') = [dFN(V&')/dV&'] u p .

where

Thus

fN( vx;' , UL' ) = -FN( V&') (13a)

f p ( V&' , u p ) = -Fp( vL' ) (13b)

fr( v p , Ul;l' ) = -F1( v p ) (13c)

f T ( V & l , u k ' ) = -F*(V&'). ( 1 3 4

The terms on the left-hand side of (13a)-(13d) are the same as the small-signal equations used to obtain the small-signal equivalent circuit if V&' is replaced by V, and u k ' by u,. The error equivalent circuit derived from (13) has the same topology as the small-signal equivalent circuit given in Fig. 1. The terms on the right-hand side of (13) depend only on the trial solutions V&' (M = N , P, I , T). They can be modeled as additional current sources to be called error sources or "correc- tors." They represent a measure of deviation of the trial solution from the true solution. The complete equivalent circuit for dc analysis is shown in Fig. 2. The expressions for, error sources are given in Table 11. The presence of capaci- tances in the dc equivalent circuit in Fig. 2 requires some explanation. The capacitances come from charge conservation or Poisson's equation. They are only used to compute vi, the electrostatic potential, which cannot be obtained otherwise. For a given set of VL1, the error sources on the right-hand side of (13) can be evaluated. uL1 are then determined by solving the transmission-line equivalent whose section between x and x + Ax is shown in Fig. 2 . A new and corrected set of V&' is then obtained from (12). The new V L obtained serves as the trial solutions for the next iteration. The details of the iteration scheme will be given in the next section. The process i s repeated until Vhl becomes sufficiently close to the true solutions VL or uL' (error) becomes sufficiently small. After convergence, internal device parameters such as carrier densi- ties, lifetimes, currents, and recombination rates may be calcu- lated from the converged VL.

CHAN AND SAH: EQUIVALENT CIRCUIT MODEL FOR SEMICONDUCTOR DEVICES 927

A k-1 P.

k+1 F 1

k+l k-1 n n

A X k ______)

Fig, 2. Error equivalent circuit of single-energy-level SRH centers. Only one section of the circuit is shown.

TABLE I1 EXPRESSIONS FOR ERROR SOURCES IN FIG. 2

If' = q (RCp k - REN)bxk

k = node number index

It must be pointed out that the error sources drop out when convergence is reached. The error equivalent circuit reduces to Sah's small-signal equivalent circuit [21] given here in Fig. 1. Hence, the same circuit can be used for small-signal analysis. The small-signal admittance is the ratio of the current flowing into the distributed circuit divided by the input voltage. The small-signal conductance and capacitance are precisely defined and are easily computed.

C. Computational Procedures The iteration procedure for the dc analysis is given in Fig. 3.

The dopant impurity concentration profile and the spatial

lnput impurity and N profile, etc. Set v = o T T

+ Calculate ti , equilibrium Fermi level

x = 0 uskng charge neutrality conditions at x = x an$ diffusion potential at

i Use piecewise linear approximation for the electrostatic potential as the initial trial solution - I A

r * 7

circuit in Figure 2.2 using the most Compute the circuit elements of the

recent set of v j - l +

Calculate X, Y, and A matrices

I

Use converged V j from

age as initial trial previous a p p l d volt- . solution

i I

Fig. 3. Iteration procedures for dc analysis.

variation of the recombination center densities are inputs. Calculation begins at zero bias. The electrostatic potential is chosen to be zero at x = xL. The equilibrium Fermi potential is calculated at x = xL using the charge neutrality condition. The diffusion potential is calculated at x = 0 using the equilib- rium Fermi potential and charge neutrality.

A piecewise-linear approximation is used for the initial values for VI, the electrostatic potential. For example, V' is taken to be constant in the quasineutral regions and linear in the space-

C 2 8 IEEE 'XANSACTIONS ON ELECTRON DEVICES, VOL. ED-26, NO. 6, JUNE 1979

charge region. Due to the self-correcting nature of the sclleme, this proved to be a good initial trial solution. At equilibrium, the quasi-Fermi potentials coincide with the Fermi potential. The number of sections (or steps or nodes or grid points) of the distributed circuit in Fig. 2 is then selected. The circuit elements for each section are calculated using the set of as- sumed values of the trial potentials.

By applying Kirchhoff's current law and charge conservittion, four linear algebraic equations can be written, one for each potential. However, only three of the four equations are independent. The quasi-Fermi potential of the recombination center can be expressed in terms of the other three poter..tials, u p , u, , and ui, using the rate equation. Relating the potentials between the adjacent section, we write

[U]k+l = [X]k[U]k + [Y]k[u]k-i + [Alk,

f o r k = 2 , 3 ; . * , n (1 4)

where

[ u p =[

[X] and [Y] are 3 by 3 matrices containing the circuit ele- ments of Fig. 2. [A] is a 3 by 1 matrix containing the error sources. We shall assume perfect contacts on both ends 0:' the device. The boundary condition is shown in Fig. 4. Fcr an n-section model, the boundary condition is uFti = u t c ' = upt1 = 0 at x = XL and u; = u; = uf = 0 at x = 0 as indicatemd in Fig. 4. The interior 3(n - 1) potentials are still unknown. Equation (14) provides 3(n - 1) equations needed to solve for the 3(n - 1) unknown potentials. Since there are couplings only between the adjacent nodes, the system of equations formed from (14) is a very sparse one. The system may be solved by standard Gauss elimination [31], [32] or L-U fac- torization [33] techniques.

The set of trial solutions is corrected by the error potentials obtained from the solution to the system of linearized equa- tions. The set of improved potentials is used to compute the circuit elements and error sources in Fig. 2 for the next itera- tion. This procedure is repeated until the error potentials are sufficiently small. The dc potentials have then converged to the true potentials. The set of converged potentials at this applied voltage is used as the trial solution for the next applied voltage except at k = 1. At k = 1, the three dc potentials are changed by AV, the increase in applied voltage, accordin[; to

v$i = vk,i-l M +(-l)"V&k,i &j,l (15)

where

M = N , P , I

2 = 1, for n+-p devices

1 = 2, for p+-n devices.

Here, the first superscript of VM, k, is the node number index and the second superscript, j , is the iteration number index. The second term of (15) is nonzero if and only if k = 1 and

Fig. 4. Boundary conditions for computing the error potentials. Each of the n-sections of the p+-n junction has the form shown in Fig. 2.

j = 1, i.e.,

Vdl = V$ + (- 1)'AV

where Vho are the converged solution at node k = 1 from the previous dc bias, Vh' is the trial potential at k = 1 at the new applied voltage, and V2' are the converged solutions from the previous applied voltage. The increase in applied voltage is introduced only at the first node during the first iteration. In subsequent iterations, this change in potentials is propagated down the transmission line of Fig. 2. The same iteration scheme of (12) is used to improve the dc potentials until UL, the error potentials throughout the transmission line, are less than the preset value. The dc potentials have then converged to the true potentials at the new applied voltage. The process is repeated until the desired bias voltage is reached.

D. Multiple Energy Level SRH Centers

In previous sections, we have limited our discussion to models with single-energy-level SRH centers. Sah and Shockley [34] have treated the recombination kinetics of multiple- energy-level SRH centers. The general large-signal problem is a very complex one. In addition to (1) and ( 9 , we now have one rate equation for each of the charge states of the SRH center. Using the notation of [20], the rate equations are

- [ TCN (s + i) - YCP (s + ;)I 9

f o r s # - r a n d s + t , b u t - r < s < t (16)

-- aN-r - yCp (-r + -$ - rcN (-r + i), for s = -r (17) at

and

s denotes the initial charge state of the center, t is the total number of electrons the centers can capture, and r is the total

CHAN AND SAH: EQUIVALENT CIRCUIT MODEL FOR SEMICONDUCTOR DEVICES 929

Fig. 5. The general large-signal equivalent circuit for arbitrary number of energy-level SRH centers. Only 3 energy levels ( E l l 2 , E312, E512) are shown in order to include all the features of the circuit. Only one section of the circuit is shown.

number of electrons the centers can release. The limiting charge states are s = -r and s = t. The rate equations can only have processes making the transitions -r * -r t 1, -r t 1 * -r t 2 , * , t - 1 * t. Further complications result from the fact that the recombination terms in continuity equations must include recombinations at all energy levels, i.e.,

and

t -1

s =-r

Since YCN and rcp are time dependent, they are not equal, in general. Furthermore, density of each charged state of the SRH centers must be included in the Poisson’s equation. For the complete problem, (16)-( 18) must be solved simulta- neously with the modified continuity equations and Poisson’s equation. At steady state, the time derivatives in (16)-(18) vanish and we have the familiar relation

R ~ ~ ( s t 6) = R ~ ~ ( s t 3) = R ~ ~ ( S t +),

for -r < s < t - 1. (21)

We shall see that the multiple energy level SRH centers can be handled by the equivalent circuit approach in an elegant way.

Sah [20] has treated the small-signal equivalent circuit for

multiple-energy-level SRH centers. The general large-signal equivalent circuit is the same as the small-signal circuit with the addition of error sources. Maes and Sah [35] have used the two-level model to study gold in silicon p-n junctions. The general large-signal equivalent circuit is given in Fig. 5. Only three energy levels are included in order to show all the fea- tures of the circuit. The error source expressions for the general large-signal transient analysis can be obtained by ex- tending the one-level treatment given by Green and Shewchun [30]. They will not be given here. Instead, we shall concen- trate on steady-state problems.

The steady-state equivalent circuit of t t r - 1 energy level SRH centers is given in Fig. 6. The potentials of the t t r - 1 energy levels of the recombination center are represented by t t r - 1 nodes. We note that the coupling capacitances be- tween the recombination center nodes are removed. This can be established through the same reasoning as in Section 11-B. The Poisson’s equation expressed in terms of the capacitances is

a2 vi ax CK 7 + c, (v, - Vi) t c p ( v p - u j )

t-1

t Cs+1/2 (~s+ , / , - v i ) = o S=-Y

where u p , v,, vi, and us+l12 are the error potentials. The Poisson’s equation does not contain the coupling capacitances.

The error sources at vi, u,, and u p can be derived from the

IEEE 7 RANSACTIONS O N ELECTRON DEVICES, VOL. ED-26, NO. 6 , JUNE 1979

Poisson's equation and equations. They are

I

. . . J n k+ 1

L - ~

Fig. 6 . The large-signal steady-state equivalent circuit for arbitrary number of energy-level SRH centers. Only one section of the circuit is shown.

the electron and hole continuity and

vp- Axk-' + \ f o r s f - r a n d s f t - 1 , b u t - r < s < t - 1. (29)

S =-r

+ 4 R&v(St ;)AXk s=t-1

(24) S = - Y

where

R C P ~ ti> = Cps-lpNst1 - epsNs l(25)

R C N ( ~ + 3 ) = c , ~ N N ~ - enstlNs+l a ((26)

Equation (25) is the net hole capture rate for the charge state transition s + 1 2 s and (26) is the net electron capture rate for s 2 s t 1, The error sources at the recombination center nodes can be derived from the rate equations of aach charge state of the recombination center. The error sources are

I ~ r + l / 2 = 4 [ R : ~ ( - Y + 3 ) - R k C p ( - r t 3 ) ] A ~ k , f o r s = - r l(27)

f o r x = t - 1 It-1/2 = 4 [R:p(t - f) - R&(t - 4)] Axk , k

(.28)

For a double-acceptor SRH center, it has three charge states s = 0, 1 , 2 and the two quasi-trap potential nodes are denoted by v112 and ~312. The error sources at each of these nodes are

4 2 = 4 [ R $ N ( + > - R5P'p(4)lAXk, at u1/2 (30)

'4[Rk?P(3> - Rk?NV(~>IAXk, at u 3 / 2 (31) The multiple-energy-level center is only slightly more com-

plex than the single-level center. In calculating the error sources contained in the X and A matrices in (14), more terms must be included. However, the bulk of the computations remains unchanged. We are dealing with the same system of linear equations with the same sparsity. There is no significant increase in either storage or computing time. The same is also true for the large-signal transient problem.

111. APPLICATION OF THE MODEL TO THE STUDY OF

SILICON p-n JUNCTION DIODE DOPED WITH ZINC A. Zinc as a Recombination Center in Silicon

Zinc exhibits two acceptor levels in silicon [37]-[39], Electrical properties of zinc centers can be characterized using the multiple-level SRH statistics of Sah and Shockley [34]. The definitions of the eight rate coefficients shown in Fig. 7 are

epo total hole emission coefficient of neutral zinc centers, epl total hole emission coefficient of singly ionized zinc

e,, total electron emission coefficient of singly ionized zinc centers,

centers,

CHAN AND SAH: EQUIVALENT CIRCUIT MODEL FOR SEMICONDUCTOR DEVICES 931

Fig. 7. Shockley-Read-Hall (SRH) coefficients for zinc in silicon.

total electron emission coefficient of doubly ionized zinc centers, total hole capture coefficient of singly ionized zinc centers, total hole capture coefficient of doubly ionized zinc centers, total electron capture coefficient of neutral zinc centers, total electron capture coefficient of singly ionized zinc centers.

All the emission coefficients are expressed in s-l and all the capture coefficients are expressed in cm3/s. Ellz is the energy level associated with the transition between neutral and singly charged zinc. E31z is the energy level associated with the tran- sition between singly charged and the doubly charged zinc center. The total rate coefficients are the sum of thermal, optical, and Auger-impact rates [4], i.e.,

ep =e$ t e; t eP,p t eFn.

Here we shall assume that the thermal rates dominate. Therefore, we can write

ep E e;.

For convenience, we will drop the superscript t . Hence, e: is the hole thermal emission rate coefficient at neutral zinc centers.

The eight SRH coefficients were obtained from extrapolating the experimental data of Herman and Sah [39] together with the detail balance relations

en2 =ni2cnlcpzlepl

en1 =ni2cnoCp1lepO. The values at 297 K are given in Table 111. The basis for ob- taining the values of these recombination parameters is de- scribed in the following paper [41].

B. Numerical Results The device to be used in this section is a p+-n junction

diode. It has the dopant impurity profile

NI(x) = 3 X l O I 9 exp ( - ~ ~ 1 2 . 3 X lo-*)- 5.7 X

We shall assume the concentration of zinc to be constant in this device and equal to 1014 ~ m - ~ . The effect of spatial variation on Nzn will be considered later. All the calculations to be discussed are performed at 297 K. The device length is 8 mils or 203.2 pm.

To start the calculation, a piecewise-linear approximation for the electrostatic potential was used. Convergence to the final solution was obtained in sixteen iterations. The initial trial solution was obtained from first-order theory for abrupt junction using depletion approximation. The trial solution

I I I 1 0 -

- ?

I I I 1 I I 0 2 4 6 8 10

X (microns) Fig. 8. Initial trial solution and converged solution of electrostatic

potential at zero bias.

X (microns)

Fig. 9. Population of each zinc charge state versus position at equilibrium.

TABLE I11 EXPERIMENTAL SRH COEFFICIENTS AT 297 K FOR ZINC IN SILICON

enl = 8.8x10-' sec-'

sec-l e = 2 . zX1o7 PO

nO

P I

c = ZxlO-' cm3/sec

c = 9x10-8 cm3fsec

en2 = 1.1~10~ sec-I

e = 2.9~101 sec-'

c = 5x10-" crnsfsec

= 6 . 0 ~ 1 0 - ~ cm3/sec

PI

nl

P2 -

and the converged solution for the device at zero bias are illus- trated in Fig. 8. Only the first 10 pm of the device is plotted in Fig. 8.

Other electrical properties of the device can be computed from the normalized potentials. In Fig. 9, the equilibrium concentrations of each charge state of zinc impurity is plotted as function of position. In the p+ region, almost all the zinc atoms are almost in the natural state. In then region, zinc atoms are all double-negatively charged. In the space-charge region, zinc atoms are mostly single-negatively charged. The dominance of each charge state can be reasoned from the Sah-Shockley [34] statistics for multiple-energy-level SRH centers.

932 IEEE TI:ANSACTIONS ON ELECTRON DEVICES, VOL. ED-26, NO. 6, JUNE 1979

The steady-state ratio of populations Ns+l to Ns is give81 by

(32)

For zinc, there are three charge states: s = 0, 1 , 2 which ccrre- spond to Ngn, N&, and NZ;, respectively. From (32), the ratios for the three charge states are

and

Using (33) and (34), the population of each charge state can be expressed in terms of a, p, and the total zinc concentration N Z n .

( 3 5 )

(36)

(37)

The eight SRH coefficients in (33) and (34) at room tempera- ture are tabulated in Table 111. We shall examine (35)-(37) in detail.

Case I: p" Neutral Region: Since P >> N , hole captures are the dominant processes, i.e.,

cp2P >> cnlN or epl or en2

and

cp P >> cnoN or epo or e,,

01 and are both small compared to unity. Hence, from (35)-(37) we have

N$!n ANzn

and

Ngn >>Nzn or N i l .

Case 11: IZ Neutral Region: Here electron capture by singly ionized zinc (c, N ) and hole emission from neutral zinc (epo ) are the dominant processes, i.e.,

cnlN>> cpzp, epl, e p ~ and

epo >> c p ~ p , en,, en1

Hence a and 0 are both large compared to unity, and a0 >> 8 %

or 0. From (35)-(37), we have

N i ; i NZn

and

O t l

I , I 1 I 0 50 100 I50 200

X (microns)

Fig. 10. Electrostatic U and quasi-Fermi potentials Up and UN, as function of position at 0.10-V forward bias.

Case III: Space-Charge Region: In the space-charge region both N and P are small. In this case, the hole emission from the neutral zinc center ( e p o ) dominates. Hence 0 >> a or 1. From (35)-(37), we have

NZn A Nzn

and

N&, >> NZi or hr&,

The quasi-Fermi potentials and the electrostatic potential for small forward bias is shown in Fig. 10. The majority carrier quasi-Fermi potentials are essentially constant in both the space-charge region and the quasineutral region. The minority hole quasi-Fermi potential in the n-type bulk decreases linearly. The steady-state recombination rate is given by

Rss = Rss@ + 3) t - 1 (38)

S=-r

where

(39)

For zinc centers in silicon, we have

Rss =Rss (+)+Rss ($ ) (40)

where

The steady-state current is obtained by integrating the total steady-state recombination rate.

J O (43)

N i ; >> Ngn or N i n . Fig. 11 shows the steady-state electron and hole current as

CHAN AND SAH: EQUIVALENT CIRCUIT MODEL FOR SEMICONDUCTOR DEVICES 933

I

0.3 X IO-' 0 \ a - < 0 2 x lo-' z

3

o I X

0

I I I I

X (microns)

Fig. 11. Electron and hole current as function of position at 0.10-V forward bias.

i J

8 10

X (microns)

Fig. 12. Electric field as function of position at several bias voltages.

'OI6 1 I R I

v = 0.2 vo l t v = 0.0 Volt

v = 0 4 Volt

V = 0 6 Volt V ' 0.8 Volt

- 0 2 4 6 8 10

X (microns)

Fig. 13. Net charge density as function of position at several bias voltages.

function of position at 0.1 V. They are obtained from

(44)

assuming JN (0) = 0 and

J P ( X ) = Jss - J N W . (45)

Fig. 11 shows that both recombination current in space- charge region and diffusion current are important at this bias voltage. In Fig. 12, the electric field is plotted as function of

electric field is reduced to less than 20 percent of the peak value at zero bias. Fig. 13 shows the net charge region width is about 1 pm on the lowly doped n side, and 0.5 pm on the heavily doped p side. The width of the space-charge region decreases with increasing applied voltage. At 0.6 V, the p+ peak moves past the metallurgical junction. This is due to the large number of holes injected from the p side.

In Fig. 14, the total steadystate recombination rate is plotted as function of position at several bias voltages. At low forward bias, Fig. 14(a), recombination current in the space- charge region dominates, as is evident from the large peak in the space-charge region. However, diffusion current is not negligible due to the large number of recombination centers in the bulk. At bias voltage of 0.3 to 0.4 V, Fig. 14(b), the peak disappears and diffusion current dominates.

It is interesting to examine how each component of (40) con- tributes to the total steady-state recombination rate. Fig. 15 shows Rss($-) and Rss($) as function of position at four dif- ferent applied voltages. Rss(z) dominates at low forward bias. However, as the applied voltage increases, Rss(4) increases much faster than Rss(3). At 0.5 V and above,Rss(i) domi- nates. A plot of zinc population as function of position at low forward bias shows essentially the same features as in the equilibrium case. The picture changes considerably at moder- ate forward bias. The zinc population at four different bias voltages is shown in Fig. 16. The change in zinc population can be accounted for by referring back to (33) through (37).

At 0.4 V, low-level injection conditions still prevail in the n region. However, hole capture becomes comparable to elec- tron capture due to the increasing number of injected minority holes, i.e., c n l N A c p Z P >> ep l or e n 2 . As a result, Q: from (33) is close to unity. But 0 is still large due to the large value of epo , i.e., epo >> cpl P A cnoN >> enl . In short, we have 0 >> Q: L 1. Hence from (36) we have NZn "Nzn in the n region. The zinc population under these conditions is shown in Fig. 16(a) and (b).

At 0.6 V, the injected minority hole carrier concentration is sufficiently high so that cpl P >> epo >> cnoN or enl , and C U , ~ >> I . Hence from (35)-(37), we have Ngn GNz, . This shift in zinc population is responsible for the relative domi- nance of Rss(4) and&(:) as is evident from (41) and (42). At forward bias above 0.5 V, Rss(S) is limited by the absence of both Nzn and N i , in the base.

Fig. 17 shows the carrier concentration at 0.6 and 0.8 V. At 0.8 V, the minority- and majority-carrier concentrations are comparable. Further evidence of high-level injection can be observed from the potentials against position plot in Fig. 18. We see that the minority-carrier quasi-Fermi potential and the electrostatic potential in the n neutral region are no longer constant.

IV. CONCLUSION

The one-dimensional steady-state equivalent circuit model for semiconductor devices was generalized to include SRH centers of arbitrary number of energy levels. The model was subsequently applied to study zinc-doped p-n junction diodes.

position at several voltages. At 0.6 V, the peak value of the The electrical characteristics of the diodes were studied in

934. E E E TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-26, NO. 6, JUNE 1979

Fig. 14. Total steady-state recombination rate as function of position at (a) 0.2-V, (b) 0.4-V, (c) 0.6-V, and ('d) 0.8-V forward bias.

2 4 6 8 IC

Fig. 15. Rs& and R,y& as functions of posiiion at (a) 0.2-V, (b) 0.3-V, (c) 0.4-V, and (d) 0.5-V forward bias.

CHAN AND SAH: EQUIVALENT CIRCUIT MODEL FOR SEMICONDUCTOR DEVICES 935

X (microns) X (microns) (b)

X (microns) ( c ) X (microns) ( d 1

Fig. 16. Zinc population as function of position at (a) 0.4-V, (b) 0.5-V, (c) 0.6-V, and (d) 0.7-V forward bias.

I I I I , I I

N = I I x ~ ~ ' ~ c m - ~ P ~ 6 0 x I O " ~ m - ~

N = P = I 2 x IONG cm-'

- lof5 - vs - a i

a 2

10'0 -

1 0 5 ~ . I -

2 4 6 8 10 2 4 6 8 10

X (microns) (a 1 X (microns) [ I ) )

lo5

Fig. 17. Carrier densities as function of position at (a) 0.6-V and (b) 0.8-V forward bias.

I I I I detail at different injection levels. The zinc population and the relative importance of recombination at each zinc level was explained in terms of Sah-Shockley theory for multiple- energy-level SRH centers.

The model is applicable to any SRH centers with multiple- energy level. It can be used to obtain the characteristics of any two-terminal one-dimensional semiconductor devices at any applied voltage.

U

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936 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-26, NO. 6, JUNE 1979

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