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Theory of surface photovoltage in a semiconductor with deep impurities
Solid-State Electronics Vol. 36, No. I, pp. 989-999, 1993 Printed in Great Britain. All rights reserved
THEORY
Copyright
OF SURFACE
SEMICONDUCTOR
PHOTOVOLTAGE
WITH
0038-I lOI/ $6.00 + 0.00 0 1993 Pergamon Press Ltd
IN A
DEEP IMPURITIES
S. C. CHOO, L. S. TAN and H. H. SEE Centre
for Optoelectronics, Electrical Engineering Kent Ridge Crescent, Singapore
Department, National University 051 I, Republic of Singapore
of Singapore,
(Received 21 August 1992; in revised form 21 January 1993) Abstract-A recent theory of the surface photovoltage is extended to a semiconductor with deep impurities, whose concentration NT $ O.l)N,I, where N, is the net concentration of shallow impurities. Numerical solutions, which have been obtained for both n-type and p-type Si with gold as an example of a deep impurity, are used to guide the development of the theory. By approximating the gold acceptor and donor levels as two independent levels, expressions are derived for the relationships between the surface photovoltage and the splitting of the quasi-Fermi potentials vsc in the surface space charge region, and between vsc and the photon flux density in terms of recombination in the space charge region and at surface states, as well as carrier diffusion in the bulk. From these expressions, a complete theory is built up which is capable of predicting the photon flux density required to yield a specified photovoltage for a given wavelength of light. The theory is shown to agree well with the numerical solutions. In particular, it explains the unexpectedly large surface photovoltage observed from the numerical solutions for n-type gold-doped Si with NT = O.llN, I. As an application of the theory, it is shown that Goodman’s surface photovoltage method will yield the appropriate minority carrier diffusion lengths in the bulk regions of n-type and-p-type gold-doped Si ma%ial.
NOTATION
Y
1.64 x 10m9 cm’js, capture coefficient for electrons on gold acceptor centres[3] 6.30 x 10m8 cm’js, capture coefficient for electrons on gold donor centres[3] 1.44 x lo-’ cm’is, capture coefficient for holes on gold acceptor centre$3] 5.60 x lo-* cm’is. capture coefficient for holes on gold donor ce&s[3i electron, hole diffusion coefficient energy level of the gold acceptor state, at 0.02 eV above El [2] energy level of the gold donor state, at 0.20 eV below E, [2] Fermi level of intrinsic semiconductor energy level of fast surface states electron, hole current density
“$ WLi X OL 6 en’ Pp P
Subscript o m S
(~v,/~141)“2 electron, hole diffusion length [~V,Iq(lN,I - &)I” net shallow impurity concentration net shallow acceptor, donor concentration concentration of gold atoms total electron, hole concentration electron, hole concentration when Fermi level is at E, (r,p,), (r;‘n,Wl electron, hole concentration when Fermi level is at E2
(r2P2), @;‘n2)Pl intrinsic electron, at 4,
electron concentration hole concentration when Fermi level is
excess electron, hole concentration electron charge ($1 ic., ), (cp2/c.2) surface recombination velocity for electrons, holes thermal voltage, normalisation parameter for potential ( = 0.025875 V)
normalised electrical potential normalised surface photovoltage ( = Iv, - viol width of space charge region ( = W,, x IOL [‘h VtIq(l4 I - NT)I”~ spatial coordinate, measured from surface absorption coefficient of light permittivity of semiconductor electron, hole mobility incident photon flux density electric potential electron, hole quasi-Fermi potential
equilibrium (usually) value at position where recombination-rate mum occurs in space charge region surface
HI)
maxi-
Superscript + , - ,0
charged
states of gold atom 1.
INTRODIJCIION
In a recent paper[l], detailed numerical solutions have been presented for the surface photovoltaic effect in a semiconductor. It has been shown that with the numerical solutions as a guide, an analytical theory of the steady-state surface photovoltage can be derived, which takes into account bulk carrier diffusion,
space-charge
recombination
and
recombi-
at surface states. However, in that paper, the treatment assumes the presence of a set of ShockleyRead-Hall (SRH) recombination centres in the semiconductor, whose concentration is so small as to have no direct influence on the space charge. In the present paper, we extend the treatment of the surface photovoltage to include the effects of a set of multiple-level nation
989
990
S. c.
CHOO
deep impurities, which are not only effective recombination centres but whose concentration is sufficiently large that their presence cannot be ignored in the solution of Poisson’s equation. For this treatment, we choose gold in silicon as an example of a deep impurity, because its recombination properties are well known[2,3] and because it is commonly used as a lifetime controller. As in the previous paper[l], we shall rely on the numerical solutions to guide us in the development of the theory. 2. MATHEMATICAL
AND PHYSICAL
MODEL
Gold can exist in silicon in one of three charge states: a neutral state, a positively charged donor state and a negatively charged acceptor state. It has therefore two deep levels: an acceptor level lying in the middle of the band gap, which is associated with the neutral and negatively charged states; and a donor level lying in the lower half of the band gap, associated with the neutral and the positively charge states. Allowing for the presence of gold in the semiconductor, the system of equations used in our numerical solutions to describe carrier transport is as follows: J. = - qp,nd$/dx
+ qD, dnldx,
(1)
Jp = - qp&rd$ ldx - qDP dpldx,
(2)
dJ,,ldx d2$/dx2=
= - dJ,/dx
= q(U, - G),
(3)
-(q/c)(p-n+N,--NN;+N;),
(4)
where N; and NG are the concentrations of gold in the negatively charged and positively charged state, respectively, and are given, in terms of the trapping parameters (see “Notation” for values of the parameters[2,3]), by:
and
In eqn (3), G is the photogeneration hole pairs and is given by[l]:
rate of electron-
G(x) = or@exp( - ax), while Ur is the total electron-hole and is given as follows:
(7)
recombination
u,=u,+u,,
rate
(8)
where U, and U2 are the recombination rates at the acceptor and donor levels, respectively, and are given by the following expressions[2]:
U, = (N; + NO,)
pn -nf
cg’(n +n,)+c;‘(p
+p,)’
(9)
et ai.
pn
uz=(NI:+N;) cp2’(n
In addition,
+
-nf
n,)+
c;l(p +p2)'
we have the auxiliary
(lo)
equations:
At the illuminated front surface (x = 0), the electric field is fixed at a specified value, corresponding to a constant surface charge density, while at the back surface, the specimen is assumed to be terminated in an ohmic contact. Surface recombination is allowed for by assuming a set of Shockley-Read-Hall type of surface states, so that at x = 0, we have: J.=
-J,=q(pn-nf)lb;‘(~+p,,) +s;‘(n
+ n,,)].
(12)
The system of eqns (1) to (4), subject to the above boundary conditions, is solved numerically using the method described by Shieh[4], to yield the surface potential $, for specified values of photon flux @ and absorption coefficient Q (corresponding to a particular wavelength of light used). The surface photovoltage A$, is then obtained from A$, = I$, - tisO], where tisO is the equilibrium value of $,. 3. PRELIMINARY
STUDIES BASED ON NUMERICAL SOLUTIONS
In order to understand the roles played by the gold acceptor level and donor level in determining the surface photovoltage, it is necessary to consider both n-type and p-type silicon. In n-type silicon, the acceptor level is expected to be the dominant recombination level, at least in the neutral bulk material, while in p-type silicon, it is the donor level that should be dominant. For such a situation to occur, however, the concentration of the gold atoms must not be so large that the equilibrium Fermi level in the n-type or p-type bulk material is significantly altered by their presence. Thus, although we have obtained the numerical solutions for cases where NT = IN, 1, we shall concentrate the discussions in the present paper on cases where Nr = O.l]N,]. As we have found from our preliminary investigations, at such a concentration, the gold atoms, while not affecting the Fermi level in the neutral bulk region significantly, nevertheless has a major effect on the space charge in the surface space charge region. In the first phase of our work, we obtained numerical solutions for the surface photovoltage as a function of the photon flux for p-type. and n-type silicon at an absorption coefficient a = 918.1 cm-‘, with IN,] = 1 x 10” cmm3 and Nr ranging from 1 x 10” cme3 to 1 x 1014cm-3. The surface recombination velocities s, and sp are both set equal to lOOcm/s. Typical results for both n-type and p-type. silicon are shown in Fig. 1 at an equilibrium surface potential of lOV, and Fig. 2 at 17.5 V, . The two surface potentials are chosen because the smaller value corresponds to a surface space charge region which is in depletion
991
Surface photovoltage with deep impurities
Type NT --
-*
.,11
1”
0
0.2
0.4
(cd)
1x1o’4
---
P
1x1o’4
._.-w
n
1x10’”
““‘.“’
P
1x1o’3
0.6
0.8
1
1.2
“SP
relationship between vsc and the normalised surface photovoltage vsP. Figures 3(a) and 4(a) show the numerical solutions of the photon flux density @ plotted against vsc at the surface (x = 0)t for the cases shown in Figs 1 and 2, respectively. It can be seen from the figures that the Q - vsc relationships for both types of material are essentially similar. For comparison, we also show in Fig. 3(a) the results obtained by replacing the gold centres in the n-type Si where NT = 1 x lOI cmm3, with a set of SRH centres located at the same energy as the gold acceptor level and with the same capture parameters, but whose concentration is so small as to have no influence on the space charge anywhere in the semiconductor.# The purpose of this replacement is to replicate, as closely as possible, the recombination behaviour of the gold centres, whilst eliminating their effect on the space charge. It can be seen from the figure that the curve due to the “equivalent” SRH centres follows closely the curve due to the gold centres, demonstrating that the gold acceptor level is indeed the dominant recombination level in n-type Si
and that there is no anomaly in the relationship between the photon flux and vsc for the case with gold. Next we turn our attention to the numerical solutions of vsPvs vsc in Figs 3(b) and 4(b), corresponding to the cases shown in Figs 3(a) and 4(a). From these figures it is immediately clear that there is a and the larger value to one in weak inversion[ I]. Also, difference in the vsP- vsc relationships between the following Ref. [l], the value of 918.1 cm-’ has been used for a as it corresponds to the case of medium CL. n-type gold-doped material with NT = 1 x lOI cmm3 and those for the p-type material with the same gold Unless otherwise indicated, all the results presented in
Fig. 1. Numerical solutions of photon flux density @plotted against normalised surface photovoltage ysP for n-type and p-type Si with IN, 1= 1 x 10” cm-j and various gold concentrations, at ludl = 10. a: =918.1 cm-‘, and s.=s= 100 cm/s.
this paper have been obtained for this value of a, as well as for s, = sP = 100 cm/s. We observe from the figures that the behaviour of the surface photovoltage in n-type material is distinctly different from that in the p-type material. In the latter case, with an increase in Nr, the magnitude of the surface photovoltage at a given photon flux decreases, as one would expect intuitively from a reduction in the carrier lifetime. However, in the case of n-type material at vd = - 10, the normalised surface photovoltages up to 0.7, instead of decreasing as NT increases from 1 x 1Ol3 to 1 x 1014cm-3, show an unexpectedly large increase. To understand the cause of the “anomalous” behaviour in the n-type material, we made use of the approach taken in Ref. [l], by breaking the problem into two parts. In the first part, we studied the relationship between the photon flux and the splitting of the quasi-Fermi levels vsc[ E (& - 4,)/V,] in the space charge region, and in the second part, the
10
‘O
0
Vhroughout this paper whenever the numerical solutions of “SC are mentioned, these refer to the value of vsc evaluated at x = 0. $For these SRH centres, the SRH lifetime parameters rpo and r, are set equal to (cp,NT)-’ and (c,, NT)-‘, respectively. At the same time, the presence of NT and NT is removed from Poisson’s equation, eqn (4).
0.2
0.4
0.6
0.8
1
1.2
“SP
Fig. 2. Numerical solutions of photon flux density @plotted against normalised surface photovoltage ysP for n-type and p-type Si with lN, l = 1 x lOI cme3 and various gold concentrations, at lurol= 17.5. a = 918.1 cm-‘, and s, =sP = 100 cm/s.
s. c.
992
CHOCI
et al.
independent rather than two interacting levels[S], at least as far as the contribution of Nf and NT to the space charge is concerned. Therefore, instead of using eqns (5) and (6), we approximate them as follows:
NT
NT z
(13)
*+r
0
2
4
6
_-.
p
.___
n
1x10’4 1x1o’3
“““”
P
1x10’3
8
10
(14) 12
vsc
(4 1.0
IYSOI = 0.8
/I
10
Type NT (cm-3) ---.
p
1x1014 1x10L4
0.6 t .---
n
1x1o’3
-”
/
/I
The validity of the above expressions has been checked by comparing the calculated values for NT’ and N; with those obtained numerically. We found that in most of the surface space charge region, these expressions proved a good approximation to the numerical solutions. Where deviations do occur, the contribution to the space charge of the term in question (whether it is N,f or N;) is negligible anyway, and the errors incurred by the approximations do not matter. We next assume that the quasi-Fermi potentials are constant for electrons and holes in the space charge region. Thus, we can write: n = nb e’;
Vsc Fig. 3. Numerical solutions for (a) the photon flux density @ vs the splitting of the quasi-Fermi potentials Q.. and (b) the normalised surface photovdltage vsp vs vsc, for n-type
p =pb e ‘,
(15)
10I3
IV#-Jl=17.5
and p-type gold-doped Si with various gold concentrations and for n-type Si with “equivalent” SRH centres, at Ic’rOI= 10. IN,1 = I x 10”cm ‘. For the SRH centres, the SRH lifetime parameters rro and ~~~ are set equal to (co, NT)-’ and (c,,, NT)-‘, respectively.
concentration and for the n-type material with the equivalent SRH centres. This difference is particularly marked in the case of the n-type gold-doped material at v,,, = - 10, where an almost-linear relationship exists between vsP and vsc for values of vsc ranging from 2 to 7, suggesting that it is this relationship which is the cause of the anomalous behaviour of the surface photovoltage, and that the key to the development of a theory of the surface photovoltage for the material with gold impurities lies in our understanding the relationship between vsP and vsc. In the treatment that follows, we shall show that by making suitable approximations it is possible to achieve this understanding, and with this as the base, an analytical theory of the surface photovoltage can be derived. 4. THEORY
4.1. Relationship
,../
lop
O
Although N: and NT are related by the totality condition, NT = Nf + Nt + NT, the two energy levels of gold are quite far apart. As an approximation, it thus seems reasonable to treat them as two
,
,
’
2
(a)
,
_-.
p
.-em
n
1x1o’4 1x10”
, P
4
1X10” I 5
6
4
5
6
“““”
Vi, 1.4
IVs,l= 17.5
1.2 1.0
a
>
0.8
0.6 0.4 0.2 0
OF SURFACE PHOTOVOLTAGE
between vsc and vsP
./ ,!: r.: z’ .$
0
(b)
Fig. 4. Numerical
1
2
3 VSC
solutions for (a) the photon flux density 8 vs the splitting of the quasi-Fermi potentials vs,-, and (b) the normalised surface photovoltage vsp vs vsc. for n-type and p-type gold-doped Si with various gold concentrations at IV,1 = 17.5. IN,1 = I x lOI5 cme3.
Surface photovoltage with deep impurities where nb and Pb are, respectively, the electron and hole concentrations immediately outside the space charge region. Poisson’s equation, together with eqns (13) and (14) for NT’ and N; , respectively, then becomes: c?‘, d*v --T=pbe-‘-n,e’+N, 4 dx
NT
+r
1
I
p?+pbe-’
+
nf +n,e'
Letting: c,s-’
nb n, +n:’
c* s
Pb P2+P:’
nb
1 nb a,=---; r2Pb
993
potential under photoexcitation, and F. is the equilibrium value of F at the surface.t In general, since and nbPb = n f exp(vSc ) (nb - nw) = (pb - pw) at the edge of the space charge region, we can find nb and pb for a given vsc. Then, using eqns (19) and (25) we can determine v, (or, equivalently, vsp) for a given vsc, or vice versa. However, we can achieve some simplification by assuming that low-level injection conditions prevail in the neutral bulk region and specialising the solutions to n-type and p-type material. For n-type material then: N,= Nn;
nb%q,,,z
ND- NT;
pb z pboeyx. (26)
For p-type material; (17)
N,= -NA;
Pb=Pt.o=(NA-Nr)
and with the boundary conditions that in the neutral bulk region, (dv/dx) = 0 and v = 0, we get: dvldx = r F(v)/L,
(18)
+m;
NTP~ nbz%e
yK.
(27)
Equation (19), together with eqns (25)-(27), provides the basis for understanding the difference in the
where F(v)=*
l)+k(eVI
$(e-‘-
I
l)T v aj+c/’
Both in eqns (18) and (19X the minus and plus signs refer to n-type and p-type material, respectively;&(v) is given by:
(
f;(v) = 21/ tan-’ 8,
2c, exp[( - l)j- ‘v] + 1 Bj -tan-r’
2c. + 1 bj
4ajcf > 1;
(20)
> '
-f;(v)=Bj Yi ( 1n2c,exp[(-l)j-1v]+ 2cj exp[( - l)j- Iv] + 1 +Bj - pi
_ ln
2cj+
1 -8j
2cj + 1 + /Ij>’ A(V) =
Yj[(2Cj
+
4ajcj < 1;
(21)
I)-’
-(Zc,exp[(-l)j-‘v]+l)-‘1,
4ajcf=
exp[(- I)‘-‘v]+exp[(aj + c,- ’ + 1
l)‘-‘2v]
) +xw]}-“2.
(19) /
relationships between vsp and vsc for n-type and p-type material shown in Figs 3(b) and 4(b). To explain this difference in the relationships, we begin by noting that for n-type material, since p2 = 3.18 x 10” cme3, v5z - 10 and pb[ =pwexp(v,,-)] varies from 1.6 x 106cmb3 to 6.5 x 108cm-3 corresponding to vsc = 2 to 7, we can ignore the term associated with the donor level (j = 2) in the summation in eqn (19). The f, (v,) term is due mainly to the presence of the n: term in eqn (5). However, nf = 5.67 x 10” cme3 and, as a simplification, may be regarded as small compared to n over the range of vsc being considered. Therefore, they, (v,) term is also neglected. For the same reason and because 1 $ a, $ exp(2v,), we find that: a,+c,‘e”s+e*“s a,+c,l+*
1; (22)
N pboeyX =alNrlT.
(28)
where n:--n
YI=
I
n:+
y2=P:-P2 P: +P2 ’
sj=Jm.
(23) (24)
To find the relationship between vsr and vsc, we use the condition that the surface field is a constant, unaltered by the photoexcitation. Thus: F(v,) = F,,
(25)
where F(v,) is the value of F when v = v,, the surface tExact expressions for calculating the Fermi level and the equilibrium surface field can be found in Ref. [6]. SSE 36,7--E
The net result is that, from eqns (19), (25) and (26), we get:
5~&eWe-vs-“M-y 2
ND
ND
’
(29) We note in passing that the above result can also be obtained, but in a more direct manner, by ignoring n : and p r in eqn ( 13), and approximating NT in eqn (16) to N; z NT/[ 1 + r, (p/n)]. Now, over the range of vsc = 2 to 7; the first term on the RHS of eqn (29), representing the effect of the
s. c. CHoo et al.
994
minority carrier term. Therefore.
holes, is small compared to the vsc eqn (29) simplifies to the following: v, = gvs, - h,
(30)
where (31) Equations (30) and (31) thus give, to a first order, a simple linear relationship between vsr and vsc. In the case of p-type material, on the other hand, we can neglect the contribution due to NT. Moreover, we note that for all cases of interest, n G nf and p $pT, so that we can further approximate N: in It is easy to show eqn (14) to N: z N,/(l +p,/p). then that, to a good approximation: az+(.;‘e In where have:
“+em?‘Y +,/;(v b)-lnbtem”’ /FF. -
a>+ (‘2+ 1 h = pzipb;
and
since
h +e-“‘ -zh. hi-1
(32)
I 9 b g exp( - v,), we (33)
Thus, from eqns (19), (25) and (27), the following approximate result is obtained for p-type material:
The above equation shows that the presence of the deep impurities simply introduces an additional constant term to the r.h.s. of this equation. For NA = 1 x 1On cmm3 and N, = 1 x lOI cm-‘, this term has a numerical value which is about one-third of the value of the other constant term, which represents the effect of the majority carriers. The relationship between rs and rse for p-type material is therefore essentially what one would expect in a semiconductor without the gold impurities and may be treated in the same manner as described in Ref. [I]. A comparison of eqns (29) and (34) shows that they are nearly analogous except for the additional term in eqn (29). This term arises from the lI2(N,IN,)vsc variation of N; with ysc for n-type material. To see this, we recall that in the space charge region of the n-type material NY in eqn (13) can be further approximated to Nr/[l + r,(p/n)]. Now, as vsc increases, the minority carrier concentration in the space charge region increases exponentially by the factor exp(vs,), while the majority carrier concentration is relatively unchanged. As a result N; varies with vsc and this gives rise to the extra term in eqn (29). For p-type material, it is NT’, rather than N; , which needs to be taken into account in considering the effect of the gold impurities on the space charge. In this case, as we have noted earlier, Ni in eqn (14) can be further approximated to NT/( 1 + p,/p). Since p, the majority carrier concentration in p-type material, is relatively unaffected by changes in vsc, NT’ will be constant and this explains the absence of a term similar to l/2(Nr/Nn)vsc in eqn (34).
The significance of the I/Z(N,/N,)v,, term in eqn (29) for n-type material is that, as vsc increases, this term tends to produce an increase in the surface electric field. However, the boundary condition at .Y = 0 requires that the surface field be maintained constant. Therefore, to satisfy this boundary condition, va, which is a negative quantity, must decrease in magnitude, and this results m a larger vsP than would be the case if the gold impurities were not present. This explains the difference in the behaviour of the vsP - vsc relationship between the n-type and the p-type material. 3.1. I. Comparison herween cIp/7muitnute theor? und numericd solutions. To assess the adequacy ol the approximations used in eqns (13) and (14) to the values of vs,, determine the rsP--vsc relationship, calculated from eqn (19) as a function of \I~(. are compared with the numerical solutions presented earlier for n- and p-type gold-doped material, with N, = I x 1O”cm -‘. This comparison is shown in Fig. 5, and as can be seen from the figure. the approximate values of Ye,, calculated from eqn ( 19) agree very well with the numerical solutions. Apart from providing a simple, yet accurate means of calculating vsr, eqn (19) is useful in allowing us to gain an insight into the roles played by the various charge concentrations in determining the rIyp-18~~relationship. To illustrate this for the n-type material with NT = I x 10’4cm ’ at \a,”= -. IO, we show in Fig. 6 the plot for the complete solution for car. as given by eqn (19>-this is represented by the solid circles in the figure; and compare it with the straightline plot (dotted curve A) given by eqns (30) and (3 I ), 1.4
-
Numerical
0.8 2 0.6
0
2
4
6
8
10
12
14
“SC
Fig. 5. Comparison of numerical solutions for the normalised surface photovoltage ySP vs ysc with the values calculated from eqn (19) for n-type and p-type gold-doped Si with IN,/ = 1 x 10’5cm-3
and NT=
I x 1014cm~‘.
photovoltage with deep impurities
Surface 1.2
,
I 0
and w
Eqn(19)
. . . Eqn (30) 1.0
.----. - -
JUT =
2
-
Eqn (29)
,’
+ Donor Term
0.6
0.4
0.2
0
4
Eqn(30)+DonorTerm
0.8
z
995
2
4
6
8
10
12
14
%c Fig. 6. Plots of normalised surface n-type No for Si with NT = 1 x lOI cmm3, showing linear eqns (30) and (31) and the effects terms neglected in deriving
photovoltage vsp vs vsc = 1 x 10’Scm-3 and relationship given by of including the other these equations.
as well as other plots (short-dashed curve B and long-dashed curve C) resulting from the inclusion of various terms neglected in deriving these equations. The straight-line plot is seen to account for the greater part of vsr over the range of vsc from about 5 to 7. From about 7 to 9, most of the discrepancy between the straight-line plot and the complete solution for vsP can be removed by including the effect of the donor term (curve B), and from 9 onwards, the minority-carrier hole term, which increases exponentially with increases in vs,-, becomes important (curve C). Finally, in order to yield the solid circles at the lower end of vsc, the presence of n: and pf in the expression for NT in eqn (13), which was neglected in arriving at eqn (29), now needs to be taken into account.
u, =
dx.
U2,)
pn -n; $,(n
+ n,) + %o,CP+ P,) ’
(38)
j=l,2
(39)
where (cbiNr)-‘;
TV, = (c,Nr)-‘.
(40)
p = p,* eaJ4 ; n = n* e-o/4 I
(41)
7 po, =
We then let:
where xi =x -X0,, and p:, n,?, a, and X, are constants whose significance will be apparent later. Substituting for p and n in eqn (39) and dropping the subscript j for convenience, we get, after some algebraic manipulation: =
2n,sWv,,/2) 6
To derive the relationship between the photon flux and vsc, we proceed as in Ref. [I] by starting with the following current-component equations which are derived from the current continuity equations: For n-type material:
+
The calculation of the minority carrier current at the edge of the space charge region (x = IV) and the surface recombination current follows the procedure described in Ref. [I]. The analytic expressions for these currents are given in Appendix 1. 4.2.1. Recombination current. In calculating the recombination current in the space charge region, we again follow the same approach as given in Ref. [l]. However, because of the large capture asymmetry r, of the acceptor level, we find that in p-type material CJ, may not exhibit the characteristic maximum in the space charge region, as assumed in Ref. [l]. The previous treatment must therefore be extended to take this into account. Moreover, in order to obtain a closed-form expression for the recombination-rate integral, we need to assume that the effective concentrations of the recombination centres, (NT + NT) for the gold acceptor level and (N!: + N: ) for the gold donor level, are constant throughout the space charge region, and for simplicity, we set both these effective concentrations equal to NT. We shall examine later the error incurred by this assumption. With the assumption of Nr being the effective concentration of recombination centres for both levels, eqns (9) and (10) for U, and U, can be expressed in the familiar SRH form. Thus
u
4.2. Relationship between photon Jrux and vsc
(U,
f0
1 e(B+(IXI) + e-(8+oXl + 2k t
(42)
where k
=
e-Y~~12COSh
E,---E 2 4K
(43) JJIV-J,(O)=&-J”,, and for p-type
(35)
material:
J”(O)-J,(IV=Jo-Ju,,
(36)
Ej being the energy level of the recombination centres concerned. By integrating U from x = 0 to x = W, we obtain the following expression for the recombination current J, at a given recombination level:
where J
=
u
Jo = q@(l - e-lw),
(37)
2qnisWvscPlf -
(
u hc
)
3
(44
s. C. CHOO et al.
996
1.0
where ,”
‘m ‘E 0.8 s ::
1 > k*;
(45)
k2 > 1;
(46)
,
-tan-‘*
z 0.6
>
_,nz”+k
-,,@=
Z.+k+v’
>
fr,(vsc)-(i~+k)~‘-(z,+k)~‘,
k’=
I,
,
I
p-type YsI)= 10, vsc = 5.14 -
Numerical
~~~~~~~~ Analytical
A
(47)
with Q, and z,, as constants to be determined. We now distinguish between two cases: Case 1, where U has a maximum and Case 2, where Ii has no maximum. The condition for I/ to have a maximum is that for n-type material pS > pm, and for p-type material n, > n,, where:
Case I (U with maximum) For this case, a = E,,,/V,. where E, is the field at the point .Y = X,,, where the maximum U occurs[l]; p*=p,,,, n*=n,, X,=X,,,; and - = exp( - IalX,); -0
zh = 2, exp(lal W).
(49)
3.5 ,
I “-‘YF
0
0.2
0.4
(b)
0.6
0.8
1
1.2
1.4
x Olm)
Fig. 8. Comparison of analytical and numerical solutions for recombination rates U, and U, for p-type gold-doped Si with vSO= 10. NA = 1 x IOi5cm-’ and NT = I x 10’4cm-‘. at (a) vsc = 5.14 and (b) vsc = 8.39.
ys,,= -10, ysc = 6.93 -
Case 2 (V without maximum) In this case, a = Es/V,, where E, is the surface field; p *_-p,, n*=n,, X,=0; and
Numerical
~~~~.~.. Analytical
I
5”OPS 2, = _ -- ._ ; zh = z, exp()al W).
0.6
(a)
0.8
1
1.2
1.4
1
1.2
1.4
x Ocm) 10 n-type
:
_”
d2
8
“$J =
t 2
6
Numerical
~~~~~~~~ Analytical
0
(b)
-10, Ysc = 8 88
0.2
0.4
0.6
0.8
x ocm)
Fig. 7. Comparison of analytical and numerical solutions for recombination rates U, and U, for n-type gold-doped Si with vSO= - 10, ND = 1 x 10’5cm-’ and NT = 1 x lOI cmm3, at (a) vsc = 6.93 and (b) vsc = 8.88.
(50)
To complete our treatment of the recombination current, it is necessary to explain how E,,, and E, are determined. E, is the equilibrium surface field, which is a known quantity. E, and X, are determined by applying the Schottky-barrier approximation to eqn (29) for n-type material, and to eqn (34) for p-type material. The required expressions for these two parameters are given in the Appendix. 4.2.2. Comparison between analytical and numerical solutions. We begin by comparing the numerical solutions for U, and U, with the values calculated from the analytical expression for U. The results for n-type material with vSo= - 10 are shown in Fig. 7 for two values of vsc corresponding to two different photon fluxes; similar results for p-type material are shown in Fig. 8. Recombination rates in n-type material As can be seen from Fig. 7, the acceptor level is the dominant recombination level for n-type material, although the donor level does increase in importance somewhat as vsc increases. The behaviour of CJ, is seen to be of the classical Sah-Noyce-Shockley
Surface photovoltage with deep impurities type[7], with its characteristic maximum and rapid decay on both sides of the maximum. The analytical solutions for U,, as calculated from eqns (42) and (43), are in good agreement with the numerical solutions, and this is due to the fact that the effective concentration of recombination centres (NT + N;) or equivalently, (Nr - NT+), is well approximated by N-r, except near the surface where N+ is significant. However, when this occurs, CJ, has decreased to such a small value that the error incurred is not of great consequence. A point worth noting is that on the right-hand of the maximum, all the U, -curves decay towards a bulk value of ApPb/rpO, . The behaviour of U,, on the other hand, is different. At the smaller value of vsc, it is nearly flat near the surface. Although a maximum can be observed at the larger value of vsc, the maximum is much broader than in the case of U, These differences from U, arise because the donor level lies in the lower half of the bandgap, which makes the p,-term large in comparison to all the other terms in the denominator of eqn (39). This dominance of the p,-term causes U, to saturate near the surface. In the case of the larger vsc, the increase in p results in a slight fall-off in U, as the surface is approached. The analytical solutions for U,, obtained by using the same procedure as for U, , agree less well with the numerical solutions than in the case of U,, because the effective concentration of recombination centres, which is now given by (Nr - N;), decreases rapidly as x increases beyond 0.1 pm. We note that where the
1o-4
t : _
1o-5
-
Numerical 0
Analytical
r
Fig. 9. Comparison of analytical and numerical solutions for the space charge recombination current JUTI plotted against vsc, for n-type and p-type gold-doped Si with ludl = 10, IN,1= 1 x lOI crnm3 and NT = 1 x 10’4cm-3.
997
maximum of U, occurs, p = n/r,, and therefore at this point, since N; z Nr/[l + r, (p/n)] (see discussion in Section 4.1) N; has already increased to half of NT. Another mechanism for U, to decay as it approaches the bulk is the increase in n in the denominator of eqn (lo), and this is the mechanism which causes U, to decrease in the analytical expression in eqn (39). However, this mechanism does not produce as rapid a decrease in U, as the increase of NT does. Recombination
rates in p-type
material
For p-type material, the large capture asymmetry r, of the acceptor level can make n, < n, [see eqn (48)]. As a consequence, U, may not have a maximum, and this is borne out by the numerical solutions shown in Fig. 8(a) for vsc = 5.14. However, at the larger value of vsc, as shown in Fig. 8(b), a maximum has started to appear. The analytical solutions are again obtained from eqns (42) and (43) and, as can be seen from the figures, they provide a very good fit to the numerical solutions. Where the donor centres are concerned, there is again a region where the effective concentration of the recombination centres,(Nr - NT), shows a significant decrease from the assumed value of Nr, but this region is now located near the surface. This accounts for the discrepancy between the analytical and numerical solutions near the surface in Fig. 8(b). To some extent the error to the left of the broad maximum of the U2-curves is compensated by the error to the right. As with the U,-curves for n-type material, all the CJz-curves fall towards a bulk value which is now given by An,/r,,,. Having examined the behaviour of CJ, and U, and how the analytical solutions for these recombination rates compare with the numerical solutions, we now compare the solutions for the recombination currents. Figure 9 shows the numerical solutions of the total recombination current Jcr in the space charge region and the corresponding analytical solutions, as a function of vsc. The analytical solutions are obtained from eqn (44) with the necessary accompanying equations. Also, to take account of the long tails of the U, -curves in n-type material and of the U,-curves in p-type material, we add a correction term q Apb W/T@, to the recombination current in n-type maternal and q An,, W/T,,, in the case of p -type material. Only the cases for Iv&]= 10 are presented, as over the range of vsc concerned (approximately between 0 and 5) the curves for ]vr,,]= 17.5 are practically indistinguishable from the curve shown for the n-type material. The agreement between the analytical and numerical solutions is found to be quite satisfactory throughout the range of vsc considered. The error in the analytical solutions for the donor level does not have a serious effect because of the smaller contribution this level makes to the total recombination current, as compared to the acceptor level.
s. c. CHoo
998
et al.
errors in the analytical value Apb and hence in the analytical computation of the diffusion current component Jn {see eqn (30) of Ref. [I]}. These errors are particularly large when the surface recombination current J, is the dominant current component on the r.h.s. of eqn (5 I) at small values of vsP for vSO= - 10. Fortunately, under such conditions, because J,, is not the dominant current component and is in fact much smaller than J, and Jvr, the errors in Apb translate into errors of only about 10% in the photon flux. On the other hand, in the case of v,” = - 17.5, although the errors in ApPbare not as large, the contribution of Jr, to the r.h.s. of eqn (51) is greater, and therefore the errors in the predicted values of the photon flux are also about 10%.
:J , , , 0
0.2
( 0.4
(
yiyZ+
, 0.6
0.8
1
6. APPLICATION
Fig. 10. Comparison of numerical solutions and results of the present theory for the photon flux plotted against vsp for n-type and p-type Si with NT = 1 x 10’4cmm’ and IN,1 = 1 x 10’5cmm’ at /LJ,~I= 10 and 17.5. OF COMPLETE
THEORY
As shown in Ref. [I], the complete pression can be written in the following l-
q@
[
1
-ev-aW) EL,,”+ 1
I
current form:
=Jo+Ju,+Js.
LENGTH
1
VSP
5. VALIDITY
TO CARRIER DIFFUSION MEASUREMENTS
ex-
(51)
where (52) and where Ju, is given by eqn (44) with the relevant accompanying equations. J, is the surface recombination current and is given by eqn (12) or (42) in Ref. [l]. In eqns (51) and (52) the first subscript refers to n-type material and the second subscript to p-type material, while N, in eqn (52) denotes the equilibrium majority carrier concentration in the bulk, which is given by eqn (26) and (27) for n-type and p-type material, respectively. The complete theory is then obtained by using eqns (19) and (25) to determine vsc for a given vsP. Using the above theory, we have computed the photon flux as a function of vsP and compared the results with the numerical solutions for the cases shown in Figs I and 2 for Nr = 1 x lOI cm- ‘. The results of the present theory are shown as solid circles in Fig. IO, and as can be seen, the agreement between these results and the numerical solutions is very satisfactory in the case of p-type material. In the case of n-type material, the agreement is quite good with errors of f&10% throughout for v,,, = - 17.5 and errors of the same order for vSO= - 10 at values of vsP smaller than 0.5. The main source of these errors is the quasi-equilibrium assumption, which leads to
One application of the theory that has been developed is to carrier diffusion length measurements based on Goodman’s method[8]. The validity of this method has been established in Ref. [I] for a material without deep impurities. On the basis of the present theory, it is clear that the method will continue to be valid for gold-doped material. To demonstrate this, we refer to eqns (19) and (51). Equation (19) shows that as long as quasi-equilibrium conditions apply in the surface space charge region, fixing vsP in Goodman’s method is the same as holding vsc. constant. Then, according to eqn (51) a plot of photon flux @ against r ’ should be a straight line intercepting the LX ’ axis at r ’ = L,,,ll the minority carrier diffusion length. The solutions of the present theory for the Goodman plots, together with the corresponding numerical solutions, are shown in Fig. II for a p-type material with v,,, = 10 and vSP= 0.1 and for an n-type material with v,” = - 10 and vsP = 0.5.
*.‘-I5
-*,, -
-
1
Numerical
Present Tixory
0
l/a
km)
Fig. Il. Plots of photon flux density @ vs 2-l for p-type gold-doped Si with Y,~= 10 and vsp = 0.1, and for n-type gold-doped Si with vsO= - 10 and vsp = 0.5. NT = 1 x lOI cm-j and IN,1 = 1 x IO”cm~’ in both cases.
Surface
photovoltage
A larger value of vsr is used for the n-type material in order to give roughly the same value of vsc as in the case of the p-type material. The intercept values given by both these two sets of solutions are found to yield the correct minority carrier diffusion lengths, with the numerical solutions giving values which agree with the expected bulk values of the diffusion length to within about 1%. In the case of the p-type material, the present theory provides agreement which is as good as the numerical solutions, while in the case of the n-type material, the agreement is to within about 6%. In the latter case, the discrepancy between the present theory and the numerical solutions for the photon flux at the larger values of l/a is due to errors arising from the quasi-equilibrium assumption, which we have noted earlier. 7. CONCLUSIONS
An approximate theory of the surface photovoltage has been developed for a Si specimen doped with gold as the deep impurity, whose concentration NT < O.l]N,]. The theory gives results which are in good agreement with the numerical solutions, and provides an insight into the physical processes that control the surface photovoltage. In particular, it explains the unexpectedly large photovoltage that is observed from the numerical solutions for n-type material with NT = 0.11N, 1. The theory is not completely analytical, since it is not possible in general to derive an explicit relationship between the surface photovoltage vsP and the splitting of the quasi-Fermi potentials vsc in the surface space charge region. However, the equation governing the relationship between these two parameters can be solved using any simple iterative technique. As an application of the theory, it has been shown that Goodman’s method of minority carrier diffusion
with deep impurities
999
length measurement, whose validity was established in Ref. [I] for a material without deep impurities, will continue to work well for gold-doped Si material, yielding the minority carrier diffusion length in the bulk material. REFERENCES 1. S. C. Choo, L. S. Tan and K. B. Quek, Solid-St. Electron. 35, 269 (1992). 2. C.-T. Sah, Proc. IEEE 55, 672 (1967). 3. C. C. Abbas. IEEE Trans. Electron Devices ED-31. 1428 (1984). (1985). 4. T.-J. Shieh, Ph.D. thesis, Univ. of Cincinnati 5. S. C. Choo, Phys. Rev. B 1, 687 (1970). 6. L. Forbes and C.-T. Sah, IEEE Trans. Electron Devices ED-16, 1036 (1969). 7. C.-T. Sah, R. N. Noyce and W. Shockley, Proc. IRE45, 1228 (1957). 8. A. M. Goodman, J. appl. Phys. 32, 2550 (1961). APPENDIX
Calculation of E,,, and A’,,, For n-type material, applying proximation to eqn (29) yields: E,(L/V,)=F,,,=
the Schottky-barrier
-JD.
ap-
(Al)
where v,=lnz;
P=~-~~sc+ln~.).
(A2)
X,/L
= F, + J-P.
(A3)
for p-type
material,
and
Similarly,
E,(L/V,)
we get
= F,,, = J2vm - 2Q,
(A4)
where vm =lnE;
Q=!$f-iLInP?,
(A5) A
PM
and Xm/L=J2v,-2e-F,.
(‘46)
THEORY
Copyright
OF SURFACE
SEMICONDUCTOR
PHOTOVOLTAGE
WITH
0038-I lOI/ $6.00 + 0.00 0 1993 Pergamon Press Ltd
IN A
DEEP IMPURITIES
S. C. CHOO, L. S. TAN and H. H. SEE Centre
for Optoelectronics, Electrical Engineering Kent Ridge Crescent, Singapore
Department, National University 051 I, Republic of Singapore
of Singapore,
(Received 21 August 1992; in revised form 21 January 1993) Abstract-A recent theory of the surface photovoltage is extended to a semiconductor with deep impurities, whose concentration NT $ O.l)N,I, where N, is the net concentration of shallow impurities. Numerical solutions, which have been obtained for both n-type and p-type Si with gold as an example of a deep impurity, are used to guide the development of the theory. By approximating the gold acceptor and donor levels as two independent levels, expressions are derived for the relationships between the surface photovoltage and the splitting of the quasi-Fermi potentials vsc in the surface space charge region, and between vsc and the photon flux density in terms of recombination in the space charge region and at surface states, as well as carrier diffusion in the bulk. From these expressions, a complete theory is built up which is capable of predicting the photon flux density required to yield a specified photovoltage for a given wavelength of light. The theory is shown to agree well with the numerical solutions. In particular, it explains the unexpectedly large surface photovoltage observed from the numerical solutions for n-type gold-doped Si with NT = O.llN, I. As an application of the theory, it is shown that Goodman’s surface photovoltage method will yield the appropriate minority carrier diffusion lengths in the bulk regions of n-type and-p-type gold-doped Si ma%ial.
NOTATION
Y
1.64 x 10m9 cm’js, capture coefficient for electrons on gold acceptor centres[3] 6.30 x 10m8 cm’js, capture coefficient for electrons on gold donor centres[3] 1.44 x lo-’ cm’is, capture coefficient for holes on gold acceptor centre$3] 5.60 x lo-* cm’is. capture coefficient for holes on gold donor ce&s[3i electron, hole diffusion coefficient energy level of the gold acceptor state, at 0.02 eV above El [2] energy level of the gold donor state, at 0.20 eV below E, [2] Fermi level of intrinsic semiconductor energy level of fast surface states electron, hole current density
“$ WLi X OL 6 en’ Pp P
Subscript o m S
(~v,/~141)“2 electron, hole diffusion length [~V,Iq(lN,I - &)I” net shallow impurity concentration net shallow acceptor, donor concentration concentration of gold atoms total electron, hole concentration electron, hole concentration when Fermi level is at E, (r,p,), (r;‘n,Wl electron, hole concentration when Fermi level is at E2
(r2P2), @;‘n2)Pl intrinsic electron, at 4,
electron concentration hole concentration when Fermi level is
excess electron, hole concentration electron charge ($1 ic., ), (cp2/c.2) surface recombination velocity for electrons, holes thermal voltage, normalisation parameter for potential ( = 0.025875 V)
normalised electrical potential normalised surface photovoltage ( = Iv, - viol width of space charge region ( = W,, x IOL [‘h VtIq(l4 I - NT)I”~ spatial coordinate, measured from surface absorption coefficient of light permittivity of semiconductor electron, hole mobility incident photon flux density electric potential electron, hole quasi-Fermi potential
equilibrium (usually) value at position where recombination-rate mum occurs in space charge region surface
HI)
maxi-
Superscript + , - ,0
charged
states of gold atom 1.
INTRODIJCIION
In a recent paper[l], detailed numerical solutions have been presented for the surface photovoltaic effect in a semiconductor. It has been shown that with the numerical solutions as a guide, an analytical theory of the steady-state surface photovoltage can be derived, which takes into account bulk carrier diffusion,
space-charge
recombination
and
recombi-
at surface states. However, in that paper, the treatment assumes the presence of a set of ShockleyRead-Hall (SRH) recombination centres in the semiconductor, whose concentration is so small as to have no direct influence on the space charge. In the present paper, we extend the treatment of the surface photovoltage to include the effects of a set of multiple-level nation
989
990
S. c.
CHOO
deep impurities, which are not only effective recombination centres but whose concentration is sufficiently large that their presence cannot be ignored in the solution of Poisson’s equation. For this treatment, we choose gold in silicon as an example of a deep impurity, because its recombination properties are well known[2,3] and because it is commonly used as a lifetime controller. As in the previous paper[l], we shall rely on the numerical solutions to guide us in the development of the theory. 2. MATHEMATICAL
AND PHYSICAL
MODEL
Gold can exist in silicon in one of three charge states: a neutral state, a positively charged donor state and a negatively charged acceptor state. It has therefore two deep levels: an acceptor level lying in the middle of the band gap, which is associated with the neutral and negatively charged states; and a donor level lying in the lower half of the band gap, associated with the neutral and the positively charge states. Allowing for the presence of gold in the semiconductor, the system of equations used in our numerical solutions to describe carrier transport is as follows: J. = - qp,nd$/dx
+ qD, dnldx,
(1)
Jp = - qp&rd$ ldx - qDP dpldx,
(2)
dJ,,ldx d2$/dx2=
= - dJ,/dx
= q(U, - G),
(3)
-(q/c)(p-n+N,--NN;+N;),
(4)
where N; and NG are the concentrations of gold in the negatively charged and positively charged state, respectively, and are given, in terms of the trapping parameters (see “Notation” for values of the parameters[2,3]), by:
and
In eqn (3), G is the photogeneration hole pairs and is given by[l]:
rate of electron-
G(x) = or@exp( - ax), while Ur is the total electron-hole and is given as follows:
(7)
recombination
u,=u,+u,,
rate
(8)
where U, and U2 are the recombination rates at the acceptor and donor levels, respectively, and are given by the following expressions[2]:
U, = (N; + NO,)
pn -nf
cg’(n +n,)+c;‘(p
+p,)’
(9)
et ai.
pn
uz=(NI:+N;) cp2’(n
In addition,
+
-nf
n,)+
c;l(p +p2)'
we have the auxiliary
(lo)
equations:
At the illuminated front surface (x = 0), the electric field is fixed at a specified value, corresponding to a constant surface charge density, while at the back surface, the specimen is assumed to be terminated in an ohmic contact. Surface recombination is allowed for by assuming a set of Shockley-Read-Hall type of surface states, so that at x = 0, we have: J.=
-J,=q(pn-nf)lb;‘(~+p,,) +s;‘(n
+ n,,)].
(12)
The system of eqns (1) to (4), subject to the above boundary conditions, is solved numerically using the method described by Shieh[4], to yield the surface potential $, for specified values of photon flux @ and absorption coefficient Q (corresponding to a particular wavelength of light used). The surface photovoltage A$, is then obtained from A$, = I$, - tisO], where tisO is the equilibrium value of $,. 3. PRELIMINARY
STUDIES BASED ON NUMERICAL SOLUTIONS
In order to understand the roles played by the gold acceptor level and donor level in determining the surface photovoltage, it is necessary to consider both n-type and p-type silicon. In n-type silicon, the acceptor level is expected to be the dominant recombination level, at least in the neutral bulk material, while in p-type silicon, it is the donor level that should be dominant. For such a situation to occur, however, the concentration of the gold atoms must not be so large that the equilibrium Fermi level in the n-type or p-type bulk material is significantly altered by their presence. Thus, although we have obtained the numerical solutions for cases where NT = IN, 1, we shall concentrate the discussions in the present paper on cases where Nr = O.l]N,]. As we have found from our preliminary investigations, at such a concentration, the gold atoms, while not affecting the Fermi level in the neutral bulk region significantly, nevertheless has a major effect on the space charge in the surface space charge region. In the first phase of our work, we obtained numerical solutions for the surface photovoltage as a function of the photon flux for p-type. and n-type silicon at an absorption coefficient a = 918.1 cm-‘, with IN,] = 1 x 10” cmm3 and Nr ranging from 1 x 10” cme3 to 1 x 1014cm-3. The surface recombination velocities s, and sp are both set equal to lOOcm/s. Typical results for both n-type and p-type. silicon are shown in Fig. 1 at an equilibrium surface potential of lOV, and Fig. 2 at 17.5 V, . The two surface potentials are chosen because the smaller value corresponds to a surface space charge region which is in depletion
991
Surface photovoltage with deep impurities
Type NT --
-*
.,11
1”
0
0.2
0.4
(cd)
1x1o’4
---
P
1x1o’4
._.-w
n
1x10’”
““‘.“’
P
1x1o’3
0.6
0.8
1
1.2
“SP
relationship between vsc and the normalised surface photovoltage vsP. Figures 3(a) and 4(a) show the numerical solutions of the photon flux density @ plotted against vsc at the surface (x = 0)t for the cases shown in Figs 1 and 2, respectively. It can be seen from the figures that the Q - vsc relationships for both types of material are essentially similar. For comparison, we also show in Fig. 3(a) the results obtained by replacing the gold centres in the n-type Si where NT = 1 x lOI cmm3, with a set of SRH centres located at the same energy as the gold acceptor level and with the same capture parameters, but whose concentration is so small as to have no influence on the space charge anywhere in the semiconductor.# The purpose of this replacement is to replicate, as closely as possible, the recombination behaviour of the gold centres, whilst eliminating their effect on the space charge. It can be seen from the figure that the curve due to the “equivalent” SRH centres follows closely the curve due to the gold centres, demonstrating that the gold acceptor level is indeed the dominant recombination level in n-type Si
and that there is no anomaly in the relationship between the photon flux and vsc for the case with gold. Next we turn our attention to the numerical solutions of vsPvs vsc in Figs 3(b) and 4(b), corresponding to the cases shown in Figs 3(a) and 4(a). From these figures it is immediately clear that there is a and the larger value to one in weak inversion[ I]. Also, difference in the vsP- vsc relationships between the following Ref. [l], the value of 918.1 cm-’ has been used for a as it corresponds to the case of medium CL. n-type gold-doped material with NT = 1 x lOI cmm3 and those for the p-type material with the same gold Unless otherwise indicated, all the results presented in
Fig. 1. Numerical solutions of photon flux density @plotted against normalised surface photovoltage ysP for n-type and p-type Si with IN, 1= 1 x 10” cm-j and various gold concentrations, at ludl = 10. a: =918.1 cm-‘, and s.=s= 100 cm/s.
this paper have been obtained for this value of a, as well as for s, = sP = 100 cm/s. We observe from the figures that the behaviour of the surface photovoltage in n-type material is distinctly different from that in the p-type material. In the latter case, with an increase in Nr, the magnitude of the surface photovoltage at a given photon flux decreases, as one would expect intuitively from a reduction in the carrier lifetime. However, in the case of n-type material at vd = - 10, the normalised surface photovoltages up to 0.7, instead of decreasing as NT increases from 1 x 1Ol3 to 1 x 1014cm-3, show an unexpectedly large increase. To understand the cause of the “anomalous” behaviour in the n-type material, we made use of the approach taken in Ref. [l], by breaking the problem into two parts. In the first part, we studied the relationship between the photon flux and the splitting of the quasi-Fermi levels vsc[ E (& - 4,)/V,] in the space charge region, and in the second part, the
10
‘O
0
Vhroughout this paper whenever the numerical solutions of “SC are mentioned, these refer to the value of vsc evaluated at x = 0. $For these SRH centres, the SRH lifetime parameters rpo and r, are set equal to (cp,NT)-’ and (c,, NT)-‘, respectively. At the same time, the presence of NT and NT is removed from Poisson’s equation, eqn (4).
0.2
0.4
0.6
0.8
1
1.2
“SP
Fig. 2. Numerical solutions of photon flux density @plotted against normalised surface photovoltage ysP for n-type and p-type Si with lN, l = 1 x lOI cme3 and various gold concentrations, at lurol= 17.5. a = 918.1 cm-‘, and s, =sP = 100 cm/s.
s. c.
992
CHOCI
et al.
independent rather than two interacting levels[S], at least as far as the contribution of Nf and NT to the space charge is concerned. Therefore, instead of using eqns (5) and (6), we approximate them as follows:
NT
NT z
(13)
*+r
0
2
4
6
_-.
p
.___
n
1x10’4 1x1o’3
“““”
P
1x10’3
8
10
(14) 12
vsc
(4 1.0
IYSOI = 0.8
/I
10
Type NT (cm-3) ---.
p
1x1014 1x10L4
0.6 t .---
n
1x1o’3
-”
/
/I
The validity of the above expressions has been checked by comparing the calculated values for NT’ and N; with those obtained numerically. We found that in most of the surface space charge region, these expressions proved a good approximation to the numerical solutions. Where deviations do occur, the contribution to the space charge of the term in question (whether it is N,f or N;) is negligible anyway, and the errors incurred by the approximations do not matter. We next assume that the quasi-Fermi potentials are constant for electrons and holes in the space charge region. Thus, we can write: n = nb e’;
Vsc Fig. 3. Numerical solutions for (a) the photon flux density @ vs the splitting of the quasi-Fermi potentials Q.. and (b) the normalised surface photovdltage vsp vs vsc, for n-type
p =pb e ‘,
(15)
10I3
IV#-Jl=17.5
and p-type gold-doped Si with various gold concentrations and for n-type Si with “equivalent” SRH centres, at Ic’rOI= 10. IN,1 = I x 10”cm ‘. For the SRH centres, the SRH lifetime parameters rro and ~~~ are set equal to (co, NT)-’ and (c,,, NT)-‘, respectively.
concentration and for the n-type material with the equivalent SRH centres. This difference is particularly marked in the case of the n-type gold-doped material at v,,, = - 10, where an almost-linear relationship exists between vsP and vsc for values of vsc ranging from 2 to 7, suggesting that it is this relationship which is the cause of the anomalous behaviour of the surface photovoltage, and that the key to the development of a theory of the surface photovoltage for the material with gold impurities lies in our understanding the relationship between vsP and vsc. In the treatment that follows, we shall show that by making suitable approximations it is possible to achieve this understanding, and with this as the base, an analytical theory of the surface photovoltage can be derived. 4. THEORY
4.1. Relationship
,../
lop
O
Although N: and NT are related by the totality condition, NT = Nf + Nt + NT, the two energy levels of gold are quite far apart. As an approximation, it thus seems reasonable to treat them as two
,
,
’
2
(a)
,
_-.
p
.-em
n
1x1o’4 1x10”
, P
4
1X10” I 5
6
4
5
6
“““”
Vi, 1.4
IVs,l= 17.5
1.2 1.0
a
>
0.8
0.6 0.4 0.2 0
OF SURFACE PHOTOVOLTAGE
between vsc and vsP
./ ,!: r.: z’ .$
0
(b)
Fig. 4. Numerical
1
2
3 VSC
solutions for (a) the photon flux density 8 vs the splitting of the quasi-Fermi potentials vs,-, and (b) the normalised surface photovoltage vsp vs vsc. for n-type and p-type gold-doped Si with various gold concentrations at IV,1 = 17.5. IN,1 = I x lOI5 cme3.
Surface photovoltage with deep impurities where nb and Pb are, respectively, the electron and hole concentrations immediately outside the space charge region. Poisson’s equation, together with eqns (13) and (14) for NT’ and N; , respectively, then becomes: c?‘, d*v --T=pbe-‘-n,e’+N, 4 dx
NT
+r
1
I
p?+pbe-’
+
nf +n,e'
Letting: c,s-’
nb n, +n:’
c* s
Pb P2+P:’
nb
1 nb a,=---; r2Pb
993
potential under photoexcitation, and F. is the equilibrium value of F at the surface.t In general, since and nbPb = n f exp(vSc ) (nb - nw) = (pb - pw) at the edge of the space charge region, we can find nb and pb for a given vsc. Then, using eqns (19) and (25) we can determine v, (or, equivalently, vsp) for a given vsc, or vice versa. However, we can achieve some simplification by assuming that low-level injection conditions prevail in the neutral bulk region and specialising the solutions to n-type and p-type material. For n-type material then: N,= Nn;
nb%q,,,z
ND- NT;
pb z pboeyx. (26)
For p-type material; (17)
N,= -NA;
Pb=Pt.o=(NA-Nr)
and with the boundary conditions that in the neutral bulk region, (dv/dx) = 0 and v = 0, we get: dvldx = r F(v)/L,
(18)
+m;
NTP~ nbz%e
yK.
(27)
Equation (19), together with eqns (25)-(27), provides the basis for understanding the difference in the
where F(v)=*
l)+k(eVI
$(e-‘-
I
l)T v aj+c/’
Both in eqns (18) and (19X the minus and plus signs refer to n-type and p-type material, respectively;&(v) is given by:
(
f;(v) = 21/ tan-’ 8,
2c, exp[( - l)j- ‘v] + 1 Bj -tan-r’
2c. + 1 bj
4ajcf > 1;
(20)
> '
-f;(v)=Bj Yi ( 1n2c,exp[(-l)j-1v]+ 2cj exp[( - l)j- Iv] + 1 +Bj - pi
_ ln
2cj+
1 -8j
2cj + 1 + /Ij>’ A(V) =
Yj[(2Cj
+
4ajcj < 1;
(21)
I)-’
-(Zc,exp[(-l)j-‘v]+l)-‘1,
4ajcf=
exp[(- I)‘-‘v]+exp[(aj + c,- ’ + 1
l)‘-‘2v]
) +xw]}-“2.
(19) /
relationships between vsp and vsc for n-type and p-type material shown in Figs 3(b) and 4(b). To explain this difference in the relationships, we begin by noting that for n-type material, since p2 = 3.18 x 10” cme3, v5z - 10 and pb[ =pwexp(v,,-)] varies from 1.6 x 106cmb3 to 6.5 x 108cm-3 corresponding to vsc = 2 to 7, we can ignore the term associated with the donor level (j = 2) in the summation in eqn (19). The f, (v,) term is due mainly to the presence of the n: term in eqn (5). However, nf = 5.67 x 10” cme3 and, as a simplification, may be regarded as small compared to n over the range of vsc being considered. Therefore, they, (v,) term is also neglected. For the same reason and because 1 $ a, $ exp(2v,), we find that: a,+c,‘e”s+e*“s a,+c,l+*
1; (22)
N pboeyX =alNrlT.
(28)
where n:--n
YI=
I
n:+
y2=P:-P2 P: +P2 ’
sj=Jm.
(23) (24)
To find the relationship between vsr and vsc, we use the condition that the surface field is a constant, unaltered by the photoexcitation. Thus: F(v,) = F,,
(25)
where F(v,) is the value of F when v = v,, the surface tExact expressions for calculating the Fermi level and the equilibrium surface field can be found in Ref. [6]. SSE 36,7--E
The net result is that, from eqns (19), (25) and (26), we get:
5~&eWe-vs-“M-y 2
ND
ND
’
(29) We note in passing that the above result can also be obtained, but in a more direct manner, by ignoring n : and p r in eqn ( 13), and approximating NT in eqn (16) to N; z NT/[ 1 + r, (p/n)]. Now, over the range of vsc = 2 to 7; the first term on the RHS of eqn (29), representing the effect of the
s. c. CHoo et al.
994
minority carrier term. Therefore.
holes, is small compared to the vsc eqn (29) simplifies to the following: v, = gvs, - h,
(30)
where (31) Equations (30) and (31) thus give, to a first order, a simple linear relationship between vsr and vsc. In the case of p-type material, on the other hand, we can neglect the contribution due to NT. Moreover, we note that for all cases of interest, n G nf and p $pT, so that we can further approximate N: in It is easy to show eqn (14) to N: z N,/(l +p,/p). then that, to a good approximation: az+(.;‘e In where have:
“+em?‘Y +,/;(v b)-lnbtem”’ /FF. -
a>+ (‘2+ 1 h = pzipb;
and
since
h +e-“‘ -zh. hi-1
(32)
I 9 b g exp( - v,), we (33)
Thus, from eqns (19), (25) and (27), the following approximate result is obtained for p-type material:
The above equation shows that the presence of the deep impurities simply introduces an additional constant term to the r.h.s. of this equation. For NA = 1 x 1On cmm3 and N, = 1 x lOI cm-‘, this term has a numerical value which is about one-third of the value of the other constant term, which represents the effect of the majority carriers. The relationship between rs and rse for p-type material is therefore essentially what one would expect in a semiconductor without the gold impurities and may be treated in the same manner as described in Ref. [I]. A comparison of eqns (29) and (34) shows that they are nearly analogous except for the additional term in eqn (29). This term arises from the lI2(N,IN,)vsc variation of N; with ysc for n-type material. To see this, we recall that in the space charge region of the n-type material NY in eqn (13) can be further approximated to Nr/[l + r,(p/n)]. Now, as vsc increases, the minority carrier concentration in the space charge region increases exponentially by the factor exp(vs,), while the majority carrier concentration is relatively unchanged. As a result N; varies with vsc and this gives rise to the extra term in eqn (29). For p-type material, it is NT’, rather than N; , which needs to be taken into account in considering the effect of the gold impurities on the space charge. In this case, as we have noted earlier, Ni in eqn (14) can be further approximated to NT/( 1 + p,/p). Since p, the majority carrier concentration in p-type material, is relatively unaffected by changes in vsc, NT’ will be constant and this explains the absence of a term similar to l/2(Nr/Nn)vsc in eqn (34).
The significance of the I/Z(N,/N,)v,, term in eqn (29) for n-type material is that, as vsc increases, this term tends to produce an increase in the surface electric field. However, the boundary condition at .Y = 0 requires that the surface field be maintained constant. Therefore, to satisfy this boundary condition, va, which is a negative quantity, must decrease in magnitude, and this results m a larger vsP than would be the case if the gold impurities were not present. This explains the difference in the behaviour of the vsP - vsc relationship between the n-type and the p-type material. 3.1. I. Comparison herween cIp/7muitnute theor? und numericd solutions. To assess the adequacy ol the approximations used in eqns (13) and (14) to the values of vs,, determine the rsP--vsc relationship, calculated from eqn (19) as a function of \I~(. are compared with the numerical solutions presented earlier for n- and p-type gold-doped material, with N, = I x 1O”cm -‘. This comparison is shown in Fig. 5, and as can be seen from the figure. the approximate values of Ye,, calculated from eqn ( 19) agree very well with the numerical solutions. Apart from providing a simple, yet accurate means of calculating vsr, eqn (19) is useful in allowing us to gain an insight into the roles played by the various charge concentrations in determining the rIyp-18~~relationship. To illustrate this for the n-type material with NT = I x 10’4cm ’ at \a,”= -. IO, we show in Fig. 6 the plot for the complete solution for car. as given by eqn (19>-this is represented by the solid circles in the figure; and compare it with the straightline plot (dotted curve A) given by eqns (30) and (3 I ), 1.4
-
Numerical
0.8 2 0.6
0
2
4
6
8
10
12
14
“SC
Fig. 5. Comparison of numerical solutions for the normalised surface photovoltage ySP vs ysc with the values calculated from eqn (19) for n-type and p-type gold-doped Si with IN,/ = 1 x 10’5cm-3
and NT=
I x 1014cm~‘.
photovoltage with deep impurities
Surface 1.2
,
I 0
and w
Eqn(19)
. . . Eqn (30) 1.0
.----. - -
JUT =
2
-
Eqn (29)
,’
+ Donor Term
0.6
0.4
0.2
0
4
Eqn(30)+DonorTerm
0.8
z
995
2
4
6
8
10
12
14
%c Fig. 6. Plots of normalised surface n-type No for Si with NT = 1 x lOI cmm3, showing linear eqns (30) and (31) and the effects terms neglected in deriving
photovoltage vsp vs vsc = 1 x 10’Scm-3 and relationship given by of including the other these equations.
as well as other plots (short-dashed curve B and long-dashed curve C) resulting from the inclusion of various terms neglected in deriving these equations. The straight-line plot is seen to account for the greater part of vsr over the range of vsc from about 5 to 7. From about 7 to 9, most of the discrepancy between the straight-line plot and the complete solution for vsP can be removed by including the effect of the donor term (curve B), and from 9 onwards, the minority-carrier hole term, which increases exponentially with increases in vs,-, becomes important (curve C). Finally, in order to yield the solid circles at the lower end of vsc, the presence of n: and pf in the expression for NT in eqn (13), which was neglected in arriving at eqn (29), now needs to be taken into account.
u, =
dx.
U2,)
pn -n; $,(n
+ n,) + %o,CP+ P,) ’
(38)
j=l,2
(39)
where (cbiNr)-‘;
TV, = (c,Nr)-‘.
(40)
p = p,* eaJ4 ; n = n* e-o/4 I
(41)
7 po, =
We then let:
where xi =x -X0,, and p:, n,?, a, and X, are constants whose significance will be apparent later. Substituting for p and n in eqn (39) and dropping the subscript j for convenience, we get, after some algebraic manipulation: =
2n,sWv,,/2) 6
To derive the relationship between the photon flux and vsc, we proceed as in Ref. [I] by starting with the following current-component equations which are derived from the current continuity equations: For n-type material:
+
The calculation of the minority carrier current at the edge of the space charge region (x = IV) and the surface recombination current follows the procedure described in Ref. [I]. The analytic expressions for these currents are given in Appendix 1. 4.2.1. Recombination current. In calculating the recombination current in the space charge region, we again follow the same approach as given in Ref. [l]. However, because of the large capture asymmetry r, of the acceptor level, we find that in p-type material CJ, may not exhibit the characteristic maximum in the space charge region, as assumed in Ref. [l]. The previous treatment must therefore be extended to take this into account. Moreover, in order to obtain a closed-form expression for the recombination-rate integral, we need to assume that the effective concentrations of the recombination centres, (NT + NT) for the gold acceptor level and (N!: + N: ) for the gold donor level, are constant throughout the space charge region, and for simplicity, we set both these effective concentrations equal to NT. We shall examine later the error incurred by this assumption. With the assumption of Nr being the effective concentration of recombination centres for both levels, eqns (9) and (10) for U, and U, can be expressed in the familiar SRH form. Thus
u
4.2. Relationship between photon Jrux and vsc
(U,
f0
1 e(B+(IXI) + e-(8+oXl + 2k t
(42)
where k
=
e-Y~~12COSh
E,---E 2 4K
(43) JJIV-J,(O)=&-J”,, and for p-type
(35)
material:
J”(O)-J,(IV=Jo-Ju,,
(36)
Ej being the energy level of the recombination centres concerned. By integrating U from x = 0 to x = W, we obtain the following expression for the recombination current J, at a given recombination level:
where J
=
u
Jo = q@(l - e-lw),
(37)
2qnisWvscPlf -
(
u hc
)
3
(44
s. C. CHOO et al.
996
1.0
where ,”
‘m ‘E 0.8 s ::
1 > k*;
(45)
k2 > 1;
(46)
,
-tan-‘*
z 0.6
>
_,nz”+k
-,,@=
Z.+k+v’
>
fr,(vsc)-(i~+k)~‘-(z,+k)~‘,
k’=
I,
,
I
p-type YsI)= 10, vsc = 5.14 -
Numerical
~~~~~~~~ Analytical
A
(47)
with Q, and z,, as constants to be determined. We now distinguish between two cases: Case 1, where U has a maximum and Case 2, where Ii has no maximum. The condition for I/ to have a maximum is that for n-type material pS > pm, and for p-type material n, > n,, where:
Case I (U with maximum) For this case, a = E,,,/V,. where E, is the field at the point .Y = X,,, where the maximum U occurs[l]; p*=p,,,, n*=n,, X,=X,,,; and - = exp( - IalX,); -0
zh = 2, exp(lal W).
(49)
3.5 ,
I “-‘YF
0
0.2
0.4
(b)
0.6
0.8
1
1.2
1.4
x Olm)
Fig. 8. Comparison of analytical and numerical solutions for recombination rates U, and U, for p-type gold-doped Si with vSO= 10. NA = 1 x IOi5cm-’ and NT = I x 10’4cm-‘. at (a) vsc = 5.14 and (b) vsc = 8.39.
ys,,= -10, ysc = 6.93 -
Case 2 (V without maximum) In this case, a = Es/V,, where E, is the surface field; p *_-p,, n*=n,, X,=0; and
Numerical
~~~~.~.. Analytical
I
5”OPS 2, = _ -- ._ ; zh = z, exp()al W).
0.6
(a)
0.8
1
1.2
1.4
1
1.2
1.4
x Ocm) 10 n-type
:
_”
d2
8
“$J =
t 2
6
Numerical
~~~~~~~~ Analytical
0
(b)
-10, Ysc = 8 88
0.2
0.4
0.6
0.8
x ocm)
Fig. 7. Comparison of analytical and numerical solutions for recombination rates U, and U, for n-type gold-doped Si with vSO= - 10, ND = 1 x 10’5cm-’ and NT = 1 x lOI cmm3, at (a) vsc = 6.93 and (b) vsc = 8.88.
(50)
To complete our treatment of the recombination current, it is necessary to explain how E,,, and E, are determined. E, is the equilibrium surface field, which is a known quantity. E, and X, are determined by applying the Schottky-barrier approximation to eqn (29) for n-type material, and to eqn (34) for p-type material. The required expressions for these two parameters are given in the Appendix. 4.2.2. Comparison between analytical and numerical solutions. We begin by comparing the numerical solutions for U, and U, with the values calculated from the analytical expression for U. The results for n-type material with vSo= - 10 are shown in Fig. 7 for two values of vsc corresponding to two different photon fluxes; similar results for p-type material are shown in Fig. 8. Recombination rates in n-type material As can be seen from Fig. 7, the acceptor level is the dominant recombination level for n-type material, although the donor level does increase in importance somewhat as vsc increases. The behaviour of CJ, is seen to be of the classical Sah-Noyce-Shockley
Surface photovoltage with deep impurities type[7], with its characteristic maximum and rapid decay on both sides of the maximum. The analytical solutions for U,, as calculated from eqns (42) and (43), are in good agreement with the numerical solutions, and this is due to the fact that the effective concentration of recombination centres (NT + N;) or equivalently, (Nr - NT+), is well approximated by N-r, except near the surface where N+ is significant. However, when this occurs, CJ, has decreased to such a small value that the error incurred is not of great consequence. A point worth noting is that on the right-hand of the maximum, all the U, -curves decay towards a bulk value of ApPb/rpO, . The behaviour of U,, on the other hand, is different. At the smaller value of vsc, it is nearly flat near the surface. Although a maximum can be observed at the larger value of vsc, the maximum is much broader than in the case of U, These differences from U, arise because the donor level lies in the lower half of the bandgap, which makes the p,-term large in comparison to all the other terms in the denominator of eqn (39). This dominance of the p,-term causes U, to saturate near the surface. In the case of the larger vsc, the increase in p results in a slight fall-off in U, as the surface is approached. The analytical solutions for U,, obtained by using the same procedure as for U, , agree less well with the numerical solutions than in the case of U,, because the effective concentration of recombination centres, which is now given by (Nr - N;), decreases rapidly as x increases beyond 0.1 pm. We note that where the
1o-4
t : _
1o-5
-
Numerical 0
Analytical
r
Fig. 9. Comparison of analytical and numerical solutions for the space charge recombination current JUTI plotted against vsc, for n-type and p-type gold-doped Si with ludl = 10, IN,1= 1 x lOI crnm3 and NT = 1 x 10’4cm-3.
997
maximum of U, occurs, p = n/r,, and therefore at this point, since N; z Nr/[l + r, (p/n)] (see discussion in Section 4.1) N; has already increased to half of NT. Another mechanism for U, to decay as it approaches the bulk is the increase in n in the denominator of eqn (lo), and this is the mechanism which causes U, to decrease in the analytical expression in eqn (39). However, this mechanism does not produce as rapid a decrease in U, as the increase of NT does. Recombination
rates in p-type
material
For p-type material, the large capture asymmetry r, of the acceptor level can make n, < n, [see eqn (48)]. As a consequence, U, may not have a maximum, and this is borne out by the numerical solutions shown in Fig. 8(a) for vsc = 5.14. However, at the larger value of vsc, as shown in Fig. 8(b), a maximum has started to appear. The analytical solutions are again obtained from eqns (42) and (43) and, as can be seen from the figures, they provide a very good fit to the numerical solutions. Where the donor centres are concerned, there is again a region where the effective concentration of the recombination centres,(Nr - NT), shows a significant decrease from the assumed value of Nr, but this region is now located near the surface. This accounts for the discrepancy between the analytical and numerical solutions near the surface in Fig. 8(b). To some extent the error to the left of the broad maximum of the U2-curves is compensated by the error to the right. As with the U,-curves for n-type material, all the CJz-curves fall towards a bulk value which is now given by An,/r,,,. Having examined the behaviour of CJ, and U, and how the analytical solutions for these recombination rates compare with the numerical solutions, we now compare the solutions for the recombination currents. Figure 9 shows the numerical solutions of the total recombination current Jcr in the space charge region and the corresponding analytical solutions, as a function of vsc. The analytical solutions are obtained from eqn (44) with the necessary accompanying equations. Also, to take account of the long tails of the U, -curves in n-type material and of the U,-curves in p-type material, we add a correction term q Apb W/T@, to the recombination current in n-type maternal and q An,, W/T,,, in the case of p -type material. Only the cases for Iv&]= 10 are presented, as over the range of vsc concerned (approximately between 0 and 5) the curves for ]vr,,]= 17.5 are practically indistinguishable from the curve shown for the n-type material. The agreement between the analytical and numerical solutions is found to be quite satisfactory throughout the range of vsc considered. The error in the analytical solutions for the donor level does not have a serious effect because of the smaller contribution this level makes to the total recombination current, as compared to the acceptor level.
s. c. CHoo
998
et al.
errors in the analytical value Apb and hence in the analytical computation of the diffusion current component Jn {see eqn (30) of Ref. [I]}. These errors are particularly large when the surface recombination current J, is the dominant current component on the r.h.s. of eqn (5 I) at small values of vsP for vSO= - 10. Fortunately, under such conditions, because J,, is not the dominant current component and is in fact much smaller than J, and Jvr, the errors in Apb translate into errors of only about 10% in the photon flux. On the other hand, in the case of v,” = - 17.5, although the errors in ApPbare not as large, the contribution of Jr, to the r.h.s. of eqn (51) is greater, and therefore the errors in the predicted values of the photon flux are also about 10%.
:J , , , 0
0.2
( 0.4
(
yiyZ+
, 0.6
0.8
1
6. APPLICATION
Fig. 10. Comparison of numerical solutions and results of the present theory for the photon flux plotted against vsp for n-type and p-type Si with NT = 1 x 10’4cmm’ and IN,1 = 1 x 10’5cmm’ at /LJ,~I= 10 and 17.5. OF COMPLETE
THEORY
As shown in Ref. [I], the complete pression can be written in the following l-
q@
[
1
-ev-aW) EL,,”+ 1
I
current form:
=Jo+Ju,+Js.
LENGTH
1
VSP
5. VALIDITY
TO CARRIER DIFFUSION MEASUREMENTS
ex-
(51)
where (52) and where Ju, is given by eqn (44) with the relevant accompanying equations. J, is the surface recombination current and is given by eqn (12) or (42) in Ref. [l]. In eqns (51) and (52) the first subscript refers to n-type material and the second subscript to p-type material, while N, in eqn (52) denotes the equilibrium majority carrier concentration in the bulk, which is given by eqn (26) and (27) for n-type and p-type material, respectively. The complete theory is then obtained by using eqns (19) and (25) to determine vsc for a given vsP. Using the above theory, we have computed the photon flux as a function of vsP and compared the results with the numerical solutions for the cases shown in Figs I and 2 for Nr = 1 x lOI cm- ‘. The results of the present theory are shown as solid circles in Fig. IO, and as can be seen, the agreement between these results and the numerical solutions is very satisfactory in the case of p-type material. In the case of n-type material, the agreement is quite good with errors of f&10% throughout for v,,, = - 17.5 and errors of the same order for vSO= - 10 at values of vsP smaller than 0.5. The main source of these errors is the quasi-equilibrium assumption, which leads to
One application of the theory that has been developed is to carrier diffusion length measurements based on Goodman’s method[8]. The validity of this method has been established in Ref. [I] for a material without deep impurities. On the basis of the present theory, it is clear that the method will continue to be valid for gold-doped material. To demonstrate this, we refer to eqns (19) and (51). Equation (19) shows that as long as quasi-equilibrium conditions apply in the surface space charge region, fixing vsP in Goodman’s method is the same as holding vsc. constant. Then, according to eqn (51) a plot of photon flux @ against r ’ should be a straight line intercepting the LX ’ axis at r ’ = L,,,ll the minority carrier diffusion length. The solutions of the present theory for the Goodman plots, together with the corresponding numerical solutions, are shown in Fig. II for a p-type material with v,,, = 10 and vSP= 0.1 and for an n-type material with v,” = - 10 and vsP = 0.5.
*.‘-I5
-*,, -
-
1
Numerical
Present Tixory
0
l/a
km)
Fig. Il. Plots of photon flux density @ vs 2-l for p-type gold-doped Si with Y,~= 10 and vsp = 0.1, and for n-type gold-doped Si with vsO= - 10 and vsp = 0.5. NT = 1 x lOI cm-j and IN,1 = 1 x IO”cm~’ in both cases.
Surface
photovoltage
A larger value of vsr is used for the n-type material in order to give roughly the same value of vsc as in the case of the p-type material. The intercept values given by both these two sets of solutions are found to yield the correct minority carrier diffusion lengths, with the numerical solutions giving values which agree with the expected bulk values of the diffusion length to within about 1%. In the case of the p-type material, the present theory provides agreement which is as good as the numerical solutions, while in the case of the n-type material, the agreement is to within about 6%. In the latter case, the discrepancy between the present theory and the numerical solutions for the photon flux at the larger values of l/a is due to errors arising from the quasi-equilibrium assumption, which we have noted earlier. 7. CONCLUSIONS
An approximate theory of the surface photovoltage has been developed for a Si specimen doped with gold as the deep impurity, whose concentration NT < O.l]N,]. The theory gives results which are in good agreement with the numerical solutions, and provides an insight into the physical processes that control the surface photovoltage. In particular, it explains the unexpectedly large photovoltage that is observed from the numerical solutions for n-type material with NT = 0.11N, 1. The theory is not completely analytical, since it is not possible in general to derive an explicit relationship between the surface photovoltage vsP and the splitting of the quasi-Fermi potentials vsc in the surface space charge region. However, the equation governing the relationship between these two parameters can be solved using any simple iterative technique. As an application of the theory, it has been shown that Goodman’s method of minority carrier diffusion
with deep impurities
999
length measurement, whose validity was established in Ref. [I] for a material without deep impurities, will continue to work well for gold-doped Si material, yielding the minority carrier diffusion length in the bulk material. REFERENCES 1. S. C. Choo, L. S. Tan and K. B. Quek, Solid-St. Electron. 35, 269 (1992). 2. C.-T. Sah, Proc. IEEE 55, 672 (1967). 3. C. C. Abbas. IEEE Trans. Electron Devices ED-31. 1428 (1984). (1985). 4. T.-J. Shieh, Ph.D. thesis, Univ. of Cincinnati 5. S. C. Choo, Phys. Rev. B 1, 687 (1970). 6. L. Forbes and C.-T. Sah, IEEE Trans. Electron Devices ED-16, 1036 (1969). 7. C.-T. Sah, R. N. Noyce and W. Shockley, Proc. IRE45, 1228 (1957). 8. A. M. Goodman, J. appl. Phys. 32, 2550 (1961). APPENDIX
Calculation of E,,, and A’,,, For n-type material, applying proximation to eqn (29) yields: E,(L/V,)=F,,,=
the Schottky-barrier
-JD.
ap-
(Al)
where v,=lnz;
P=~-~~sc+ln~.).
(A2)
X,/L
= F, + J-P.
(A3)
for p-type
material,
and
Similarly,
E,(L/V,)
we get
= F,,, = J2vm - 2Q,
(A4)
where vm =lnE;
Q=!$f-iLInP?,
(A5) A
PM
and Xm/L=J2v,-2e-F,.
(‘46)