minority-carrier transport in nonuniformly doped silicon-an analytical approach

2 10 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 37, NO. 1 , JANUARY 1990

Minority-Carrier Transport in Nonuniformly Doped Silicon-An Analytical Approach

LEENDERT A. VERHOEF AND WIM C. SINKE

Abstract-In this paper, we derive a general analytical solution for minority-carrier transport in a nonuniformly doped quasi-neutral silicon region. It is shown that in the case of an exponential doping-density profile N ( x ) = FeGx and in the case of a general power-law doping- density profile N ( x ) = A ( x + B)', a closed-form solution for the minority-carrier concentration is obtained for doping densities up to 10'' ~ m - ~ . In the analysis, the experimentally observed dependencies of minority-carrier lifetime, minority-carrier mobility, and band-gap narrowing on doping density are taken into account. Contrary to earlier analytical solutions, our solution is free of integrals of minority-carrier transport parameters over the semiconductor region under study.

Three important bipolar device configurations in which a nonuni- form doping density plays a role are analyzed with the analytical solution. The first is the drift-field (DF) solar cell, for which we calculate a factor-of-20 reduction in the dark saturation current compared with a uniformly doped solar cell. Second, the effective back-surface recombination velocity of a highllow junction back-surface field (BSF) cell is shown to decrease with increasing BSF region thickness. Third, the influence of surface recombination velocity on the minority-carrier concentration profile in a heavily doped emitter is reduced when a strong power law ( C large) doping profile in an n-p junction device is employed.

I. INTRODUCTION INORITY -carrier transport in semiconductor de- M vices is often analyzed under the assumption of po-

sition-independent doping density and transport parameters in the quasi-neutral regions. In that case, the only driving force for carrier transport is the gradient in the minority-camer concentration, i.e., current flows by diffusion only. The minority-carrier continuity and current equations in these quasi-neutral regions can then be solved analytically, and device performance can be calculated exactly using the appropriate boundary conditions for the device under study. In many configurations, however, the assumption of position-independent parameters cannot be made, and quasi-electric fields due to gradients in doping density and band-gap narrowing, as well as gradients in minority-carrier lifetime and mobility, are present. Either an inclusion of these position-dependent transport parameters in the analytical solution or a numerical analysis is needed. Numerical approaches are widely used, and many

Manuscript received August 28, 1988. This work was supported by the Netherlands Organization for the Advancement of Research (Nederlandse Organisatie voor Wetenschoppelijk Onderzoek (NWO)) and the Nether- lands Technology Foundation (Stichting Technische Wetenschappen (STW)). The review of this paper was arranged by Associate Editor S . J . Fonash.

The authors are with the FOM Institute for Atomic and Molecular Phys- ics, Amsterdam, The Netherlands.

IEEE Log Number 8930910.

software packets are available for the simulation of devices that contain doping gradients. The plug-in variables for these programs are the doping-density profile and fits to the empirical dependences on doping density of band- gap narrowing and minority-carrier lifetime and mobility. To obtain physical insight into semiconductor device performance, analytical solutions to the transport equations are preferred.

Earlier, the so-called drift-field (DF) solar cell, which employs a base region with a position-dependent doping density, as proposed by Wolf [ 11, has been analyzed analytically. Several authors have solved the minority-carrier transport by assuming uniform mobility [2] or a linear dependence of lifetime and mobility on position [3], [4]. This is not an appropriate description since the dependence on doping density leads to nonlinear lifetime and mobility profiles in all practical cases. Lindholm and Chen [5] arrived at an analytical solution in the case of a DF silicon solar cell taking into account a power-law dependence of lifetimes on doping density, but they neglected the influence of the mobility gradient. Band-gap narrowing was neglected because they only considered moder- ately doped ( c 5 x 10l8 ~ m - ~ ) regions, where the effects are still rather small [6].

A general solution for minority-carrier transport, taking into account all these dependencies, was obtained by Park et al. [7], but it is not in a closed form. Del Alamo and Swanson [8] have investigated nonuniformly high-doped regions by assuming quasi-transparency of emitters. In both of these papers, the solutions are valid for general doping profiles but still contain double integrations of minority-carrier transport parameters over the emitter region. Furthermore, in Park's case, at least two iterations have to be performed to obtain an accurate result [7]. Our approach is different; we limit ourselves to certain doping profiles, but the result is a closed-form expression for minority-carrier density, which is free of integrations and iterations. It is based on experimentally observed dependencies of minority-carrier lifetime and mobility and band- gap narrowing on doping density. Although these parameters are, strictly speaking, not independent [6], they can be used, provided they are consistent. The solution is obtained for two general doping density profiles: an exponential and a power-law dependence on position. Two doping-density regimes will he discussed (one in which the effect of band-gap narrowing is negligible). For the

0018-9383/90/0100-0210$01 .OO O 1990 IEEE

VERHOEF AND SINKE: MINORITY-CARRIER TRANSPORT IN NONUNIFORMLY DOPED SILICON 21 1

first regime, we illustrate the relative importance of gradients in the other transport parameters. The second regime deals with doping densities (over approximately 10'' ~ m - ~ ) , wherre the effective doping density nearly saturates due to the band-gap narrowing.

To illustrate the application of the solutions thus obtained, we evaluate the influence of graded regions in three sections of a silicon solar cell. Two examples deal with the no-narrowing approximation, and one example deals with a heavily doped emitter, including band-gap narrowing.

First, we demonstrate the relative influence of the effective electric field associated with a doping gradient on the dark saturation current of the base in a DF silicon solar cell. The merit of such a device is the increased radia- tion resistance due to a reduced dependence of efficiency on minority-carrier lifetime, which makes it suitable for space applications [ 11.

Second, at the back surface of a modem back-surface field (BSF) silicon solar cell, a high/low junction is present. In earlier analyses, the effective back-surface recombination velocity, which is used to characterize such a high/low junction region, is derived by treating it as a step junction [9]. However, the change in transport parameters is not abrupt: an interfacial region exists where gradients in carrier lifetime, mobility, and doping density are non- zero. Here, we will give an analytical solution for the effective back-surface recombination velocity as a function of the spatial extension of the BSF region.

Third, the solution that includes band-gap narrowing can be used to analyze minority-carrier transport in heavily doped emitter regions. In these regions, the doping density is strongly depth dependent. Use of the analytical solution yields insight into the dependence on doping profile of the ratio of surface- to bulk-emitter recombination. This ratio is a function of various emitter parameters and can now be stuided analytically. We calculate the minority-carrier concentration profile in such a heavily doped emitter.

11. GENERAL SOLUTION FOR MINORITY -CARRIER CONCENTRATION

In this section, we solve the minority-carrier continuity and current equations for a nonuniformly doped n-type silicon region. The analysis for p-type silicon is equiva- lent. The assumptions made here are that the transport parameters are a function of depth ( x ) only, so we can treat transport in one dimension. Furthermore, we assume quasi-neutrality and low-injection conditions, so we treat minority carriers only. The device is in steady state. The set of equations to be solved is

7 ( N ) = K T k

p ( N ) = M N P m

AEG = efV,ln (E). Here, J is the hole-current density, which is equal to the sum of the diffusion-current density Jdf l and the drift-current density Jdrifr. p is the hole mobility, T the minority- hole recombination lifetime, D the hole diffusivity, p the hole density, and N the doping density. AEG is the apparent band-gap narrowing [6]. All these parameters are x dependent. E is the electric field. Furthermore, e is the electron charge, V, = kB T / e the thermal voltage in which kB is Boltzmann's constant, and K, k , M , in, No, andfare (positive) constants.

Equations (1) and (2) are the well-known one-carrier current and continuity equations. Equation (3) describes the electric field associated with the gradient in the doping density and the gradient in the apparent band-gap narrowing [lo]. The effective band-gap narrowing contains all heavy doping effects, such as rigid band shift, band tail- ing, fermi-dirac statistics, and degeneracy.

Equation (4) is a fit to experimental data for silicon (see [ l l ] for a compilation of data) with k = 1 for lOI5 < N < 5 x 10l8 ~ m - ~ , where K = 1.3 x 1OI2 ~ m - ~ / s and k = 2 for 5 X 10l8 < N < lo2' cmP3, where K is a band- to-band Auger recombination coefficient equal to 5.5 x lo3' cmP6 s . Equation (4) is also supported by theory [ 121, [13]. The data together with the fits are shown in Fig. 1 . The doping ranges used for the fits are somewhat flexible; the fit with k = 1 still holds for doping densities up to 1019 ~ m - ~ , and the fit with k = 2 still holds reasonably down to 10l8 ~ m - ~ . Equation (4) also holds for the electron recombination lifetime in p-type silicon, with K = 1.0 X 1031 cmP6 s for k = 2.

Equation (5) is a somewhat simplified form of the dependence of p on N , which was empirically determined by Burk and De la Torre [ 171. It fits the experimental data on hole mobility [17]-[20] and on electron mobility [21], [22] versus doping density reasonably well, as can be seen from Fig. 2, and is no more than 20-percent off for holes for any doping density between 1017 and 1019 ~ m - ~ . For electrons, the fit is even better. From the fits, we find in

= 0.38 and M = 1.4 x lo9 cmo.86 V-' s-l for holes and in = 0.42 and M = 1 . 1 x 10" cm0.74 V-' s-' for electrons. However, stronger or weaker dependences can be analyzed as well since we will solve the (1) through (6) for general K , M, k , and m.

Equation (6) is an empirical fit to apparent band-gap narrowing data [ l l ] , [16] (see Fig. 3 for a compilation of

dP dx

data together with the fit according to (6)). The original fit treats the narrowing in two ranges of doping density: for N < No = 7 x 1017 ~ m - ~ , f = 0, and hence, AEG = 0; for N > No: fV, = 18.7 mV. However, due to large error bars in the data, reasonable fits are obtained for val-

d l n N 1 dAE, ( 3 ) uesfbetween 0.75 and 1.0, with values forNo between 7 X 1017 and 2 X 10l8 ~ m - ~ . The special casef = 1 implies

J ( x ) J d f i 4- Jdrifr = - eD - 4- epEp

(2) - + - = o P edx T

E = -V,- + - - dx e dx

212 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 37, NO. I , JANUARY 1990

10-8

n 10-0-

w lo-'-

0 v1 Y

E! 9 10-8 W

3 0 x 10-9-

-

1

I - . - .

-

-

OBeck and Conradt. Ref. 14

0 Dziewor and Schmid, Ref. 15

&\ - a'? PP -

ASSnirhun and Swanson, Ref. 16 A [ del Alamo et. al., Ref. 9 1

10-101 ' ' ' ' ' ' - 1 1 I I I I I I I I , , I

10" 10'0 10" O b . d : l I , , , , I I I , , , I 10'8 10'0 10"

DONOR DENSITY (cm-3) DONOR DENSITY ( c ~ n - ~ )

Fig. 1 . Data for minority-hole lifetime versus doping density after a com- pliation from [6] with fits according to (4). The solid line is a fit according to r 0: N - k with k = 1; the dashed line is a fit with k = 2.

10' , , , , , , , , , , , , , , , , , , , , , , I , , , , , , ,

0

0

-+-

--e--

\ *.

ELECTRONS

HOLES

10' loia 10" lo18 40'9 1020

DOPING DENSITY (CM- ) Fig. 2. Mobility versus doping density for holes in n-type silicon and for

electrons in p-type silicon. In the doping-density range from 10'' to over lOI9 crK3, a power-law dependence of mobility on doping density is apparent.

that no electrical field is present in the region where N > No; the effective doping density N,$ = N exp ( -( AEG/kB T ) ) saturates at No for N > No. Equation (6) implies that No has to be outside the doping range under consideration; otherwise, a discontinuity in E will appear at N = No.

As is seen from the above, all of the quantities, including mobility, lifetime, band-gap narrowing, and electric field, depend on doping density. Combining (1) through (6) yields an equation for the hole concentration p, where the only remaining x-dependent parameter (apart from p) is the doping density N

- (y)'- (1 -f)7 d2 In ") p = 0 (7)

Fig. 3. Apparent band-gap narrowing versus doping density. The fit is according to a simple logarithmic dependence.

in which we have used the Einstein relation D = VTp. In the simple case of p , 7, and AEG independent of N , that is m = k = f = 0, the third term in (7) is just p/D7 (provided d2 In N/dx2 = 0) and the prefactor d In N/dx of the first derivative dp/dx is equal to E / V,. In the general case described by (7), the third term depends on all three parameters p, 7, and AEG , whereas the second term only depends on mobility and band-gap narrowing. The prefactor of the second term, representing an effective electric field, is reduced by a fraction ( f + m).

Diffusion-doped and implanted regions usually give an error function or Gaussian profile in doping density, but for the analysis here, we consider two resembling doping profiles: a power law and an exponential. These two are given in (8) and (9).

Npow(x) = A(x + B)C ( 8 )

N,,,(x) = FeGx (9)

where A, B, C, F, and G are positive constants. By adjusting these, a whole range of practical doping profiles can be fitted. Fig. 4 shows a typical profile obtained by ion implantation and subsequent annealing, together with fits according to (8) and (9).

The solution of (7) with respect to (8) and with respect to (9) has been carried out in the Appendix. This yields the following solution for the minority-hole concentration in case of a power-law dependence (8):

VERHOEF AND SINKE: MINORITY-CARRIER TRANSPORT IN NONUNIFORMLY DOPED SILICON 213

t t

1 i

101'b.o ' ' 'o? ' ' '012' ' ' '013' ' ' '014 ' ' 0.4 POSITION (pm)

Fig. 4. Doping density profile obtained by ion implantation followed by thermal annealing together with fits according to a power law ( N oc ( -x + B)B, solid line) and an exponential ( N a CGX, dashed line) dependence.

where

v = ( 1 + 2 ( 1 -.f + m) c + ( 1 - f + rn)' c2p2

2cY

1 + (1 - f + m ) C abs ( (rn + k ) C + 2

2

In case of an exponential doping density profile (9), the hole concentration is given by

p e x p ( x ) C 3 P l , e ~ p ( ~ ) + C 4 P 2 , e x p ( X )

+ c4Ku ((m + k)GLexp(x) )) (11)

for m + k * e 0, where z1 = (1 - f + r n ) / ( r n + k ) and

I,( y ) and K,( y ) are the modified Bessel functions of the first and second kind, respectively [ 2 8 ] . The constants CI and C2 or C3 and C, are to be determined from the boundary value problem for the specific device under analysis. Equations (10) and (1 1) are exact solutions for the minority-carrier concentration in a silicon semiconductor re-

TABLE I ANALYTICAL SOLUTIONS FOR MINORITY-CARRIER CONCENTRATION IN

SEE (3) TO (6). ONLY (x) Is GIVEN FOR THE BESSEL FUNCTION DIFFERENT DOPING RANGES I N CASE OF A POWER LAW DOPING PROFILE;

SOLUTIONS. THE SECOND SOLUTION p2,pow ( X ) IS FOUND BY REPLACING I,, WITH K,

doping density range transport parameters solutions

homogeneous 0 0 0 e a

1-c ~ 1 0 1 7 0 0 0 ( x + B ) T I v r G ) ab(!$)

\ ,

10'7<N<7X10'7 0.38 1 0

1-(1-f-m)C

1-(1-f-m)C

7x1017 < N < 5x1018 0.38 I 0.75 ( X + B ) ~ I ,

5x1018 < N < l@o 0.38 2 0.75 (x+B)-I,

TABLE I1 ANALYTICAL SOLUTIONS FOR MINORITY-CARRIER CONCENTRATION IN

VARIOUS DOPING RANGES IN CASE OF AN EXPONENTIAL DOPING PROFILE. ONLY P , . ~ ~ ~ ( X ) Is GIVEN FOR THE BESSEL FUNCTION SOLUTIONS. THE

SECOND SOLUTIONp2,,,p (1) IS FOUND BY REPLACING 1, WITH K,

doping density range transport parameters solutions

d-

m k f D1 e.&) V

homogeneous 0 0 0 ea

<io17 0 0 0 eFe.i$+Yy 10L7<N<7X10'7 0.38 1.0 0 11 (&) 1.0

( xp x,) 0.46 7x1017 < N < 5X10t8 0.38 1.0 0.75 e0~065GxI0.qg & 5x1018 < N < 1020 0.38 2.0 0.75 e0'065Gx10.26 (a) 0.26

gion. Earlier treatments resulted in equations that still contain double integrations over the semiconductor region [6 ] , [ 7 ] , and to obtain an exact result, two iterations have to be performed [ 7 ] .

These equations become more transparent by examin- ing the prefactors and the arguments of the Bessel functions. In the power-law case (lo), the argument contains Lp,,(x) = [ V , c l ( x ) ~ ( x ) ] ~ / ~ which is the position- dependent diffusion len th. The prefactor is equal to (x + B ) / [ N ( x ) p ( x ) ] ' f i . In the exponential case ( l l ) , the argument is also inversely proportional to the local diffusion length Lexp(x) = [ v T p ( x ) 7 ( x ) ] 1 / 2 but has no explicit position dependence. The prefactor is again inversely proportional to [ N ( x ) p (:vtx)]'/* but also not ex-

214 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 37, NO. I . JANUARY 1990

plicitly dependent on position. For comparison, in a uniformly doped base, the solutions are exponentials with argument x/L. The various parameters incorporated in (10) and (1 1) come from the empirical fits to data according to (4) through (6). As discussed above, the parameters m, k , andfhave different values in different doping density regimes; they are summarized in Tables I and I1 together with the analytical solutions (10) and (11) for power-law doping profiles and exponential doping profiles, respectively. In addition, the casse of uniform doping is shown, which gives simple exponentials for p1 (x) and p 2 (x). One exception has to be made in the case of an exponential doping profile when m = k = 0; (11) is no longer valid, but a simple solution is obtained, as given in Table 11.

111. EFFECT OF THE VARIOUS PARAMETERS ON

MINORITY-CARRIER CONCENTRATION To evaluate the effect of band-gap narrowing on the so-

lutions (10) and (1 l ) , we defined a silicon region 0.5-pm wide and a doping density of 7 x 10" cmP3 at x = W = 0.5 pm and lo2' cmP3 at x = 0 pm. Then, we calculate pl(x) (see Fig. 5(a)) andp2(x) (see Fig. 5(b)) withf = 0 andf = 0.75 for a power-law profile with C = 5.0. We find that p 2 ( x ) is very sensitive to f , whereas p1 (x) is completely independent off. In Fig. 5(c), the hole concentration p (x) is calculated with and without band-gap narrowing. The hole-concentration gradient at x = W is larger in the case off = 0 than in the case off = 0.75. This implies that carriers (either injected at x = W or gen- erated somewhere in the semiconductor) can more easily reach the x = 0 interface in the latter case; from carriers injected at x = W , only a fraction of 0.01 can diffuse to x = 0.1 pm forf = 0, whereas a fraction of 0.15 reaches the same depth in the case off = 0.75. In Section IV we expand more on carrier injection.

In Fig. 6(a), we have plotted five different doping profiles in a semiconductor region of width W = 0.5 pm. Profile I represents a step profile with a uniform doping density of 1019 ~ m - ~ . Cases 11, 111, and IV are power-law doping profiles with coefficients C = 4.0, 1.0, and 0.2. The constants A and B are taken such that for all these profiles N ( 0 ) = l O I 7 cmP3 and N ( W ) = 1019 cmP3. These values also hold for V , which is the exponential profile. In Fig. 6(b) and (c), we have plotted the two contributions p1 (x) and p 2 ( x ) to p ( x ) for all these profiles, taking W = 50 pm and assuming m = 0.38, k = 1, and f = 0. Here, p1 (x) represents the term of (lo), and (11) belonging to the Bessel function of the first-kind I , , and p 2 ( x ) represents the term of (10) and (11) belonging to the Bessel function of the second-kind K,. In the homogeneous case I , these are simple functions and eXlL. As in the case of the dependence on f, p l ( x ) is almost independent of C , whereas p 2 (x) is very sensitive to the power exponent C .

In the limiting case of the power-law dependence with C -+ 0, the profile approaches the uniform case I (Fig.

10-3b.0 " ' 0.1 I " " 0.2 I " " 0.3 I " " 0.4 I " ' " 0.5 POSITION (pm)

c /

0

(C) Fig. 5. Effect of band-gap narrowing AEG on the solutions for minority-

hole concentration contributions (a) p , ( x ) and (b) p 2 ( x ) and (c) on the total minority-hole concentration p ( x ) under the boundary conditions p ( 0 ) = 0 a n d p ( W ) = 1. W = 0.5 pm, N ( W ) = 7 x 10" cm-3 and N ( 0 ) = 10'' cm-3 (see text). A power-law doping profile (8) is used with C = 5. The solid curves are calculated neglecting band-gap narrowing ( f = 0) in (6); the dashed curves include band-gap narrowing ( f = 0.75).

10, the profile approaches the exponential curve V (Fig. 6(a)), and the p ( x ) contributions approach those of the

6(a)), andp,(x) approaches the uniform case. When C > exponential case (Fig. 6(b) and (c)).


1 0.6

L , \

- - - _II ,C=O.Z

- - - - I I I , C = l . O

__ I V ; C = 4 0

POSITION (pm) (C)

Fig. 6 . (a) Doping density profiles. Profile I is a step profile corresponding to uniform doping; profiles 11, 111, and I V correspond to power-law dependences (7) with C = 0.2, 1.0, and 4.0, respectively, and V i s an exponential profile. For all profiles, N ( 0 ) = I O l 9 cm-' and N ( W ) = 10'' cm-3. (b), (c): Contributions to minority-hole concentration p ( x ) = C,p,(x) + C2p2(x ) for these five profiles normalized with respect to x = W. The increasing terms pI (x ) belong to eXlL or the first Bessel function. The decaying termpz ( x ) belongs to or the second Bessel function.

IV. INFLUENCE OF DOPING-DENSITY PROFILE ON SILICON SOLAR CELL PERFORMANCE

We now evaluate three device configurations: two in which the traditionally assumed independence of lifetime and mobility on doping density becomes invalid but where band-gap narrowing is negligible and one configuration in which band-gap narrowing is of importance. First, in a DF solar cell, the doping density increases from a low value (typically 1017 cmP3) at the p-n junction to a high value (typically 5 X 10'' ~ m - ~ ) at the back metal contact. This is a range in which, according to Figs. 1 and 2, the lifetime and mobility depends on doping density by a power law, whereas in good approximation, band-gap narrowing can be neglected (Fig. 3). We have calculated the dark saturation current density.

Second, at the low/high region of a p+-n-n+ BSF silicon solar cell, a spatially extending interfacial region in- fluences the minority-hole reflection properties. In this configuration, we take only mobility and lifetime gradients into account. When the low/high region is fab- ricated by implantation followed by a drive-in anneal, the doping profile can very well be described by the exponential profile as used for derivation of the analytical solution (1 1). We determine the effective back-surface recombination velocity as a function of the BSF-thickness.

Third, we consider the transport in a heavily doped emitter region. Here, we take into account the dependences of all transport parameters and study the influence of the power-law coefficient C on the hole-concentration profile.

A. DF Silicon Solar Cells

A drift field p+-n(x) silicon solar cell [ l ] consists of a heavily doped p-type emitter at the front and an n-type base of thickness W. We assume that the doping density at x = 0 (edge-space-charge region at the base side) is N ( 0 ) and increases exponentially with x to N ( W ) at x = W. We solve ( 1 1 ) subject to the following boundary conditions:

where the intrinsic carrier concentration ni = 1.2 x 10" cmP3 at T = 300 K. Equation (12) implies camer injection over the junction under applied voltage V >> V T , and (13) characterizes the back surface at x = W by an effective back-surface recombination velocity S.

Insertion of ( 1 1 ) in (12) and (1 3) yields the coefficients C, and C, as a function of K, M , k, m, S, W, F, and G:

216 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 31, NO. I . JANUARY 1990

with

P e x p ( 0 ) The relative importance of the gradient in mobility and lifetime is studied by calculating Jo with fixed values of W = 50 pm, N ( 0 ) = 1017 cmP3, N ( W ) = 5 X 10l8

c4 = ( 15) Ku(bexp) - OL'Zu(bexp)

, s = S / G D ( W ) and bexp as where zw = bexp defined in the Appendix. Here, we have used the two rules (A10) and (Al l ) as given in the Appendix for the derivative of a Bessel function.

Furthermore, we can use p (x) to find the hole current density ( 1 ) at x = 0:

e ( m + k ) G W / 2

e p ( x ) E p ( x ) - e D ( x ) - d p ( x ) ) . (16) dx x = o

Insertion of (14) and (15) in (1 1 ) and of ( 1 1 ) in (16) yields

J ( 0 ) = (17) where

m + k + bexp 4

Zu-l(bexp) + Iu+l(bexp) Zu(bexp) - PsKu(bexp)

Ku- 1 ( b e x p ) + Ku+ 1 ( bap) Ku(bexp) - O F ' Z u ( b e x p )

-

In Fig. 7 , we give Jo as a function of base thickness with parameters, the recombination velocity at the back surface S, and the doping density N ( W ) at the back. For the calculation, we use N ( 0 ) = 1017 cm-3 and values of m = 0.38, k = 1 with corresponding M , K . We neglect band-gap narrowing, i.e., we let f = 0. The reason for this is twofold. First, for doping densities < 5 X 10l8 cmP3, the effect of AEG is small, and second, we concen- trate on the relative influence of mobility and lifetime gradients only, which is similar when band-gap narrowing is accounted for.

Several observations can be made here. First, the dark- saturation current density is considerably lower in the case of large N( W ) . The current of a uniformly doped base ( N ( W ) = 1017 ~ r n - ~ ) is a factor of 20 larger when compared with a strong DF case ( N ( W ) = 5 X 10l8 ~ m - ~ ) . For larger uniform doping densities (e.g., 5 x 10l8 crnp3), Jo is even larger. Second, because minority carriers are effectively shielded from the back surface by the effective electric field E in the latter case, the influence of the surface recombination velocity S is smaller when a DF is present. These two observations are consistent with earlier calculations in which the gradient in mobility was neglected [5].

~ m - ~ , and S = lo4 cm/s for various values of m ( = 0 or 0.38), k( = 0 or l ) , M , and K . In the case of k = 0, we have used uniform lifetimes K of 28 and 0.56 ps, which is associated with the doping densities N ( 0) and N ( W ) , respectively. For m = 0, we have done the same for the mobility M . The results are shown in Table 111.

First, the realistic values for m = 0.38, k = 1 give a current density of 0.401 X A/cm2, which is in agreement with Fig. 7. Second, using the realistic depen- dency for mobility ( m = 0.38 and M = 1.4 X lo9 cm0.86/V - s ) but a hypothetical uniformity for the lifetime ( k = 0) results in a strong dependence of current density on the lifetime T = K . A large lifetime gives a small Jo , and a small lifetime causes a large Jo. Third, in the case of k = 1 with a hypothetical uniform mobility ( m = 0), a large mobility of 485 cm2/V s results in a higher Jo then does a small mobility M = 110 cm2/V s. Apparently, the diffusion current in (1) grows faster with mobility than does the drift current, which is of opposite sign. Fourth, when both m and k are zero, i.e. , only a gradient in doping density is present, the solution (11) becomes invalid but (7) is readily solved. We have calculated Jo with this solution (see also Table 11). Finally it is clear from Table I11 that the influence of the minority- carrier lifetime on the saturation-current density is larger than that of the mobility, which is caused by 7 changing almost two orders of magnitude from front to back, whereas p only changes by a factor of 5. In addition, according to our calculations, a small mobility is beneficial, whereas a small lifetime is detrimental.

B. BSF Region and the Effect on the Back-Sugace Recombination Velocity

We now consider the high/low junction at the back of the base of a p+-n-n+ silicon solar cell. Traditionally, this junction is treated as a step function [7]. The step in carrier lifetime, mobility, and conduction band then causes repulsion of minority carriers from this highly doped region. The repelling quality is usually characterized by an effective back-surface recombination velocity Se,, which in the case of an n-type base is defined by

Here, x = Wb is the edge of the quasi-neutral base. When the n-n+ interfacial region has a spatial extension W, the gradient in lifetime and the electric field associated


\ 4

* rn z w

l o - l P -

n

E 10-13:

5 w p: p:

io-'' g 2 1 0 - ' J o '

,,r I I I I I I I , 3 I I I 9 8 8 I 9 I j \ \ _ - - S= 10' CM/S

. - s=104 CM/S . \

- \ \ ----; - - -__ --_ -- -- -- -- '.

\ '-.-__ N ( w ) = ~ o ' ~ \ \

\

---- _ _ _ - - - -_ -- -- -- 1

5x1 0''

I

- - - - - - - - _ _ - - - - _ _ _ - - - .. 'L. -

I

: -

10 20 30 40 50 " ' I " " I " ' ' I " " ' " "

with the gradient in mobility will enforce the repulsion of minority carriers from the heavily doped region.

We consider the following configuration: A homoge- neously doped n-type base is bordered at x = 0 by an n+ (x) region in which the doping density increases with position by an exponential (9). At the end of this region of thickness W , an interface is present, and it is characterized by a surface recombination velocity S. This may be due either to a metal contact or to a heavily doped silicon region. In Fig. 8, this configuration is shown with a metal/silicon interface at W.

Next, we derive S for minority holes in the base region. Current continuity at x = Wgives the boundary con- dition for the solution of this problem:

eff

(19) Se, can now be determined by using the definition (1 8) at x = 0:

(20) The hole concentration in the interfacial region pexp (x) is again given by (1 1).

The solution of ( 1 1 ) with respect to (19) through (20) yields both the minority-hole profile in the interfacial region and the effective recombination velocity Se, as a function of the interfacial region thickness W.

TABLE I11 DARK SATURATION CURRENT DENSITY Jo IN A/cM' FOR VARIOUS

COMBINATIONS OF MOBILITY AND LIFETIME GRADIENTS. AN EXPONENTIAL

N ( W ) = 5 X 10'' CM-', S = io4 CM /S AND W = 50 p m ARE USED DOPING PROFILE N (x) = F exp (Gxvt) WITH N ( 0 ) = 10'' CM-3 AND

T T (N(O))= T (N(x))= T (NO)=

L4 28 us 2s - 0.56 LLS 0.56 us

k(N(0)) = 485 cm% 0.2 15 0.486 4.42

s(N(x)) = 485 - 1 in cm% 0.202 0.403 4.22

UNOVI) = 11Ocmz/s 0.140 0.333 3.22

with Ps as defined under (15). Sefis plotted in Fig. 9. The values used for the calculation are N ( 0) = lOI7 cmP3 and N ( W ) = 5 X 1OI8 cm-3 and for the back-surface recombination velocity, a typical value for a silicon/metal interface S = lo6 cm/s. Furthermore, we put f = 0. Sef decreases with increasing BSF region thickness W . For small values of W, Se, approaches the velocity S = lo6 cm/s of the metal/silicon interface (which is out of scale in Fig. 9).

C. The Emitter in an n+ (x)-p Junction Device Another important application is the use of the analyt-

ical solution (10) or (1 1) for determination of the emitter contribution Jo, e to the total dark-saturation current JO in an n+-p junction device and the influence of the emitter donor-density profile N ( x) on the minority-hole concentration in the emitter region. Earlier attempts have been made to solve this problem [29], [30], but these involve numerical calculation of integrals with integrants contain- ing diffusivity and equilibrium minority-camer density. Here, we will derive an equation for Jo,e and show some hole-concentration profiles for various doping profiles.

We take the simple configuration of Fig. 10. The position x = 0 is the boundary of the depletion region on the emitter side. We assume that the doping-density profile is given by (8), with N ( 0 ) = 7 X lOI7 cmP3 and N ( W e ) = lo2' ~ m - ~ , where We is the emitter thickness. The boundary conditions are now given by

218 IEEE rRANSACTIONS ON ELECTRON DEVICES, VOL. 37, NO. I , JANUARY 1990

Solving this set of equations yields determined by (16) and (17) as a function of We and emitter-surface-recombination velocity S . The emitter current consists of the following two terms:

J0,e = JO,e,drifl + JO,e,dSq. (24) with

entire emitter region for C > 4. The effect of various profiles on the emitter saturation current density is still under investigation.

V. DISCUSSION AND CONCLUSIONS In this paper, we have given for the first time an itera-

tion-free and integral-free full analytical solution for minority-carrier transport in a silicon semiconductor region

N ( 0 ) B

a, U are given below (10) in Section I1 and b = bpow is defined in the Appendix. 0, contains the boundary con- dition at x = Wand is given by

(m + 1 - f ) C + 1

m + 1 - f ) C + 1 P, =

Here, it is zw = (b / c r ) (W, + B)* and s = ( S / D ( We))( We + B ) . Thus, insertion of the emitter- doping profile parameters A , B , and C in (24) through (27) yields an exact expression for Jo, e .

In Fig. 11, we have plotted the minority-hole concentration as a function of depth for three values of C ( = 0.3, 1 .O, and 4.0) as well as for the step-junction case already shown in Fig. 6(a). Note here that the emitter surface is at x = 0, and the SCR interface is at 0.5 pm. For the mobility and the lifetime dependencies, we use m = 0.38 and k = 2, and band-gap narrowing is taken into account usingf = 0.75. Fig. 1 shows that a single value of k = 2 yields a reasonable fit over the doping range considered here. For the power-law profiles, which are seen in Fig. 1 l(b), (c), and (d), we have calculated p (x) using S = lo4 cm/s and S = lo6 cm/s.

The influence of S on the profiles is apparent. In the case of a step profile and the step-like profile ( C = 0.3), this influence is strong. The reason for this is that in a large part of the emitter, no electric field exists ( d N / d x = 0 together with (3) and (6) yields E = 0), so holes are not repelled from the emitter surface. In the case of C = 4.0, there is a smaller influence of the emitter-surface- recombination velocity on the camer profile; the electric field effectively shields the holes from the surface. Fron the definition of the electric field (3) in combination with the expression for doping profile (8) and (6 ) , it can be shown that the electric field is almost constant over the

where the doping density is position dependent. For the case of an exponential as well as for the case of a power- law doping profile, the solution is given in terms of modified Bessel functions. The assumptions made in the derivation are low-injection conditions, the Einstein relation holds, quasi-neutrality , and steady-state conditions. Fur- thermore, we have used empirical relationships between minority-carrier mobility and lifetime and the doping density and between apparent band-gap narrowing and doping density.

Four doping density regimes with different sets of m, k , and f (the coefficients of the empirical relationships) can be distinguished (see Tables I and 11). When the semiconductor device encompasses several of these doping regimes, one can divide the region under study into seg- ments corresponding with these different doping regimes. In each segment, the solution for p1 (x) and p 2 (x) can then be determined using (10) or (1 1). The hole concentration in the entire semiconductor region is readily calculated by imposing the appropriate boundary conditions (current-density and carrier-concentration continuity at all points) on the set of solutions.

In this study, we have assumed power-law and exponential-doping profiles, whereas in most practical cases, the profiles resemble error functions or Gaussian curves. These practical profiles can be fitted reasonably well by adjusting the coefficients A , B , and C or F and G. In case

VERHOEF AND SINKE: MINORITY-CARRIER TRANSPORT IN NONUNIFORMLY DOPED SILICON 2 19

ME&

Fig. 8. Configuration of a BSF region in a solar cell. The doping density increases from the base doping density N ( 0 ) at the n-n+ interface to N( W ) at the silicon nf/metal interface, where S > IO6 cm/s.

uration current density of this device is larger than (and opposite to) that of the mobility. A small mobility results in a decrease in the current density, whereas a small lifetime gives an increase in current density. This shows that a low mobility can be beneficial for device performance. The BSF region analysis reveals that the repelling action of the high/low junction is enhanced if the thickness of the graded region is increased. In device technology, the highllow junction is used to fabricate ohmic contacts. The given solution can be used to optimize these contacts. Fi- nally, the emitter region of a silicon bipolar junction device (such as a solar cell) was analyzed. This structure is widely used in technology, but an analytical solution for carrier transport has not been derived so far. The given calculation is suited to be an onset for further exact analysis of the influence of the emitter-doping profile on total device performance.

APPENDIX SOLUTION OF (6)

First, we solve (7) using the power-law doping density profile (8). Insertion of (8) in (7) yields

(. + B ) 2 p ( x ) = 0. (A1

' - 0 1 2 3 L 5 mC2 + c BSF REGION THICKNESS (prn)

+ ( 1 - f ) Fig. 9. Effective back-surface recombination velocity Sec as a function of

the thickness W(see Fig. 8) of the back-surface field region. Here, N ( 0 ) = IO1' ~ m - ~ , N( W ) = IOt9 ~ m - ~ , and S = IO6 cm/s are used. Here, a prime denotes a derivative with respect to the ar-

gument. We now substitute y = (x + B ) and g ( y ) = p ( x ) to find

METAL GRID

g"(y) + (1 - f - m ) C - - Y

POSITION

Fig. 10. Configuration of an n+ ( x ) emitter in a solar cell. The doping density is low at the space-charge region interface ( x = 0) and high at the surface ( x = We).

the profile cannot be fitted by a single exponential or power-law curve, it can be divided into two or more regions where such fits can be made. The solutions in the different regions then have to be matched using carrier and current continuity to yield the full analytical solution.

We have briefly discussed three typical device configurations in which our analytical solution enables exact analysis of the device performance. The DF solar cell ex- hibits a lower saturation current than does a standard ho- mogeneously doped solar cell. As an example, the relative importance of gradients in the three parameters (doping density, lifetimes, and mobility) was calculated for this cell type. The importance of lifetime on the sat-

Multiplying (A2) by y 2 and introducing b& -Am + k - /vTKM yields

C ( m + k ) + 2 Y 2 g " ( Y ) + (1 - f - m ) c Y g ' ( Y ) - [b&Y

+ ( 1 - f ) ( m C 2 + c ) ] g ( y ) = 0. ('43) The solution for (A3) involves the modified Bessel

functions of the first- and second-kind Z, and K , [28] (see p. 440)

(1 - ( 1 - f - m ) C ) / 2 d Y ) = Y

with U = ( 1 + 2 ( 1 - f + m ) C + ( 1 - f + m ) 2 C 2 ) 1 / 2 / 2 a anda = (( m + k)C + 2 ) / 2 . Resubstitution of y = x + B and rewriting the argument of the Bessel function in terms of a local diffusion length yields (10). Second, the exponential-doping profile (9) is inserted into

220 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 37, NO. I , JANUARY 1990

loL1 0 21

10" 5 U M

10'O 5

X 10-50.0 0.1 0.2 0.3 0.4 0.5 0.6

POSITION ( p m )

POSITION ( p m )

lo2' 0 21

loeo 3

1o17Vw

loz1 0 21

10" 5

(b) ( 4 Fig. 11. Minority-hole concentration p ( x ) normalized with respect to

p (0.5) (solid lines and dashed lines) and doping-density profile (dotted lines) as function of position. The surface is at x = 0, and the SCR interface is at 0.5 pm. The four cases shown are (a) a step junction emitter, (b) a step-like power profile C = 0.3, (c) a linear profile C = 1.0, (d) and a strong power profile C = 4.0. The two curves in (b) to (d) correspond to S = lo4 cm/s (solid lines) and S = lo6 cm/s (dashed lines).

(7). This gives

p"(x) + ( 1 - f - m)Gp'(x) with U = (( 1 - a ) 2 + 4c)'l2/2 -- a h ( ( 1 - f + m ) / ( m + k ) ) > 0. Resubstitution of z in (A7) yields the result ( l l ) , which holds for m + k # 0.

The two modified Bessel functions are given in [31], p. p(') = O' 661:

m

s = o s!(s + U ) ! (A91 (A5)

Substitution of z = e ( m f k ) G x / 2 and g(z) = p ( x ) yields I&) = c

(A10) T I&) - K , ( x ) = - 2 sin ( U T ) .

G2zg ' (z ) Two useful rules that concern the derivative of these functions are

(A1 1 ) d l , ( x ) L d - 4 + 4 l + l ( ~ )

( 1 - f - m ) ( m + k ) + 2

-- - Dividing by ( ( m + k ) G/2)2 and introducing a = 1 + dx 2 22" 1 - f - m ) / ( m + k ) ) , c = 4m( 1 - f > / ( m + k ) 2 and

(A7)

bexp = 4Fmfk/G2(m + k)2VTKMgives = - K U - l ( x ) + K"+l(x). (A12) dx 2

z 2 g " ( z ) + azg'(z) - (b,',z2 + C)g(z) = 0. The solution for (A7) also involves the modified Bessel functions:

ACKNOWLEDGMENT

The work described in this paper is part of the research program of the Foundation for Fundamental Research on

(A8) Matter (Stichting voor Fundamental Onderzoek (FOM)).

-(U - f - m ) / ( m + k ) ) g(z) = z ( C 3 z z 1 ( b e x ~ z ) + C 4 K u ( b e x ~ z ) )

. . . . . . .

22 1 VERHOEF AND SINKE: MINORITY-CARRIER TRANSPORT IN NONUNIFORMLY DOPED SILICON

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* Leendert A. Verhoef was born in Sittard, The Netherlands, on August 21, 1961. He received the B.S. and M.S. degrees in experimental physics from the State University of Utrecht, Utrecht, The Netherlands, in 1982 and 1986, respectively.

He spent one year at the Department of Elec- trical Engineering of the University of Florida, Gainesville, where he worked on the characterization of high-efficiency solar cells. Currently, he is with the FOM Institute for Atomic and Molec- ular Physics in Amsterdam, The Netherlands. His

main research interests are device modeling and poly-crystalline solar cell processing and characterization.

* Wim C. Sinke was born in Vlissingen, The Neth- erlands, on November 21, 1955. He studied experimental physics at the State University of Utrecht, Utrecht, The Netherlands, from which he obtained the M.S. degree in 1981.

From 1982 to 1985, he worked at the FOM In- stitute for Atomic and Molecular Physics in Am- sterdam, The Netherlands, on ion implantation and laser annealing of silicon. For this work, he obtained the Ph.D. degree from the State University of Utrecht in 1985. In 1986, he joined the Hitachi

Central Research Laboratory in Tokyo, Japan, to study structural properties of amorphous silicon. He returned to the FOM Institute in 1987, where he is now leading a group that is involved in research on silicon solar cells, ion implantation, and laser processing of semiconductors.

minority-carrier transport in nonuniformly doped silicon-an analytical approach

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