Time and field dependence of excess carrier concentration in semiconductors

InJared Printed

Phys. Vol. 26, No. 4, pp. 201-208, in Great Britain

1986

0020-089 l/86 $3.00 + 0.00

Pergamon Journals Ltd

TIME AND FIELD DEPENDENCE OF EXCESS CARRIER CONCENTRATION IN SEMICONDUCTORS S. Microelectronics

Research

Center,

E. SCHACHAM

Department Technology,

of Electrical Engineering, Haifa 32000, Israel

Technion-Israel

Institute

of

(Received 22 January 1986) Abstract-The time and space distribution of excess carriers in semiconductor devices is derived through the solution of the ambipolar continuity equation. Both build-up of excess carriers due to external excitation and their decay following its termination are presented. A detailed solution for a I-D photoconductive device with various boundary conditions is obtained. The solution is expanded to the 3-D distribution, where an analytical solution is given for the diffusion region of photodiodes. The procedure for solving the 3-D problem in photoconductor devices is outlined. The derived equations are used to demonstrate the advantages of the “covered electrodes” structure and in order to obtain an accurate value for the ambipolar mobility as measured in a four-probe device.

1.

INTRODUCTION

The behavior of excess electrons and holes in a semiconductor material is governed by generation, recombination, diffusion and drift processes. The motion of both carriers can be formulated through two continuity equations, one for each type of carrier. These equations can be combined the 3-D distribution into one ambipolar equation, eliminating the internal field term. (‘I Frequently, can be simplified to a 1-D problem, Such is the case with most photoconductive detectors where the excess carriers are swept across by the applied electric field. The entire device or a part of it is exposed to radiation. The output is detected as a change in voltage or current across the input electrodes or across additional contacts, such as in the three-port integrating detector known as SPRITE.“’ The mathematical basis for solving the ambipolar equation was laid by Shockley.‘3’ A detailed analysis of the photoconductive process was given by Rittner. (4)It includes the expressions for the distribution of excess carriers under steady-state condition, and the build-up of the total amount of carriers following a uniform illumination of the entire device. For the case of a smaller optical window (“dark electrodes”), the steady-state solution is presented under the large-field approximation. Other particular solutions are given for problems such as the decay of total excess carriers following 6 -function following illumination of the entire device,‘5’ or excess carrier distribution are treated as 1-D illumination using the Laplace transform. ~5.‘)All these solutions and applications problem. In order to be able to interpret measured results and to facilitate the design of new devices, a solution of the ambipolar continuity equation is presented in this paper. It includes both the time and space dependence of the excess carrier distribution in the semiconductor as a function of the generating excitation. The 1-D expansion is presented in its final compact form for either ohmic or electrically reflecting contacts. Both build-up of excess carriers and their decay following the termination of illumination are analyzed. The solution is extended to the 3-D problem, with the exact expression depending on the recombination velocities at the various surfaces. The application of the solution to the design of covered electrode photoconductive detectors and to the interpretation of mobility measurements in a four-port device is demonstrated.

2.

DECAY

OF

EXCESS

CARRIERS

The dependence of the excess carrier distribution on time ambipolar continuity equation. If the equilibrium majority-carrier 201

IN

1-D

and space is derived from the density is large compared to the

S. E.

202

SCHACHAM

trap density, the equation in n-type material is a$ -=I: at

^ -;+g+DV2~-p~.V@,

where p(r, t) is the excess hole concentration, g is the external generation rate and t is the excess carrier lifetime; /Aand D are the ambipolar mobility and diffusion constant, which, for extrinsic material and for low injection levels, are identical to those of the minority carriers. Due to the quasineutrality, the concentration of excess electrons ti(r, r), is almost identical to that of excess holes. In this section we analyze the decay of excess carriers following turning off of the external excitation at t = 0, i.e. g(r >, 0) = 0. Following the procedure outlined by Shockley,‘3) the solution of the ambipolar equation can be obtained through separation of variables. The recombination term I_i/t can be eliminated by multiplying by the exp( - t/r) factor, reducing the equation to 1 d2X ,uE dX __---=_-_ X dx2 DX d.x

1 dT DT dt

= K,

(2)

where fi(x, t) = X(x). T(t)rexp(t/r) and K is the constant of the separation. The solution for the time-dependent part, T, is of the form T = exp(KDt)

(3)

and, since the concentration decays with time, the constant of the separation must be negative, i.e. K = -a2. For a bar of length x0 and for ideal ohmic contacts, i.e. with infinite surface recombination velocity, the boundary conditions are P(x =0)=/5(x The solution for the space-dependent

=x0)=0.

(4)

part of the equation is then given by

X=C,exp

$

sin::

(

1

(5)

which, when inserted in equation (2), implies az _ p2E2 ( n2n2 4D2 3’ Therefore,

one gets J!(x, f) =

exp

( t E!$f-!.C$)z, -;

+

[C~exp(-~)sin~~)].

(6)

In order to calculate the Fourier coefficients C,,, one has to know the distribution of excess carriers at t = 0: $(x, 0) = exp g (

f C, sin y. ) “= I

(7)

The profile of excess carrier distribution at t = 0 depends on the preceding excitation. If, for example, a steady state prevailed prior to the cessation of excitation, rf(x, 0) can be derived from the ambipoiar equation (I) with @/at = 0. The steady-state distribution pss,(x) depends on the form of the excitation and usually a straightforward solution can be obtained.‘4.5,8)Equation (7) is a Fourier series, however its terms are not orthogonal. The orthogonality is achieved by transferring the exponential factor, then one obtains

Thus, the time and space dependence of the excess carriers following the termination excitation can be determined analytically according to the initial distribution.

of the

Excess carrier concentration

203

As can be seen from equation (6), the decay of the carriers is not a function of their lifetime r only, but also of their drift and diffusion. One can define a decay coefficient r, by

where L = (0~)“~ is the diffusion length and u’ = @r is the drift length. Using this notation, equation (6) is reduced to fi(x,t)=exp

Wx f (ZL’),_,

C p [ flex (+n(Y:)].

(10)

When the difksion length is comparable to the length of the device, the main contribution to the sum is from the first term. Thus the effective lifetime is r,. For example, if at I = 0 a d-function pulse was applied at x = a, i.e. 8(x, 0) = P,6(x -a), the excess carrier distribution following it is given by

0.24

g .;; L

0.20

E 8

O.f6

ti 8 .k 8 s

D.f2

0.08

i?

‘._,

w

0.04

0 .UQ 0

90

20

30

40

50

60

* (pm)

irl

0.08

0.08

:

,u

2

ii

ii

m

0.1

‘C

I.

A ,,... ..-”

I 8 lz

..‘. . . ..____. 0.04

Fig. I. Excess carrier decay for a “covered electrode” photoconductor. The electric field sweeps to the right; gr = I. (a) Decay following illumination to steady-state. Exposed region between 20 and 40 pm of a ~O.LMIlong sample. (b) As in (a); exposed region between It? and 30gm. (c) As in (b); i~~urn~natjo~ terminated after 4. IO-‘s. (d) A 20 ym long device exposed completely till steady state and following decay.

S.E. SCHACHAM

204

Another practical example is the case of “dark electrodes ” .(4)In this device the optically active area is only a part of the semiconductor bar. Thus, it is possible to obtain a high resolution, due to the small optical window, and to reduce the destrucrtive effect of the ohmic contacts. This overlap structure results in a significant enhancement of voltage responsivity and D* for a given resolution, as was demonstrated by Kinch et al. c9)for HgCdTe photoconductive detectors. In order to derive the expression for the decay of excess carriers following the steady-state distribution, let us assume the optically active area is between the points x! and x2 on the device. The segments [0, s,] and [x2, x,,] are covered. The steady-state distribution is expressed by three different functions, one for each section (as demonstrated by Rittner (4)for the case of a large field). After switching off the external excitation g, the distribution during the decay is given by a single function throughout the device:

exp(s)[

-$sin(y)

- zcosr$)]

+exp(-~)[~sin~~)+~cos(1~)] w

x

*

[($+~<~][($)2+(~~+1]

This series is used to derive part of the curves of Fig. 1, as explained in the next section. Using the functions produced above, it is possible to derive the expression of the integral of the excess carriers over any section of the bar. The change in current, due to the external generation, in a device to which a constant voltage is applied, or under constant current conditions, is proportional to this integral. Integrating equation (6) on an arbitrary section of the device [xi, x2], one obtains “2 @n(t)

=

d(x,

t>

s XI

dx = 5 H;exp( n=l

- t/z,>,

(13)

where

H = c n

-exp(~)[~sin~~)-~cos~~)] (141

” (~~+~)

and C, is given by equation (8). When integrating over the entire device, i.e. xi = 0 and x2 = ~0, then

(15) For this particular case solutions were derived for uniform illumination(4.*} and for the case of sweep-out.@’

Excess carrier concentration

3. BUILD-UP

OF

EXCESS

205

CARRIERS

In order to derive the distribution of excess carriers during their build-up as a result of the external generation g, it is necessary to solve the inhomogeneous continuity equation, equation (1). The solution of the inhomogeneous equation is thus equal to the difference between the steady-state distribution j&(.x) [as obtained from equation (1) with a$/at = 0] and the solution of the homogeneous equation describing the decay. The second part was solved in the previous section. Combining the two processes it is possible to determine the decay of excess carriers following excitation for any period of time. Assuming the excitation was turned on at t = 0 and off at t = &,, using equations (6) and (g), and the orthogonality of the C,‘s, one obtains

j?(x,r>r,)=exp

(w’x)nI, 5 C C,[ 1 -ew (-$]exp(

-y)sinF,

(16)

where C,, are the Fourier coefficients for the steady-state initial distribution. Figure 1 demonstrates the effect of dark electrodes on the build-up and decay of excess carriers in an n-type device with an ambipolar mobility of 600 cm’/V*s; a diffusion constant of 4 cm’js and an excess carrier lifetime of 10m6s. The diffusion length is therefore 20pm. An electric field of 10 V/cm is applied from left to right so that the drift length is 60 pm. In Fig. la, a 60 pm long device has an optical window of 20pm located at its center. The decay of the excess carriers following steady state is demonstrated. Due to the effect of the ohmic contacts combined with the drift and diffusion, the maximum concentration is 17%, as demonstrated in Fig. 1b. The same con~guration is used for Fig. lc, however here the initial distribution is not a steady state but rather the excitation was terminated after 4. lo-‘s. Although the duration of the external generation was considerably smaller than the lifetime, the maximum concentration obtained is >80% of that at steady state. The reason is the shorter effective lifetime with t, = 0.23 *10e6 s and r2 = 0.13. 10e6 s. Finally, in Fig. Id a 20 pm device is examined. The entire device is exposed to radiation and the decay following steady state is demonstrated. The advantage of covered electrodes for the same optical window is obvious. The integral of excess carriers is six times smaller in Fig. Id as compared to Fig. lb. Also, the effective lifetime is considerably shorter, as can be seen from the figure, with r1 = 0.76. lo-‘s and r> = 0.23. lo-‘s. 4.

BLOCKING

CONTACTS

If instead of ohmic contacts the device is terminated by electrically reflecting or blocking contacts, i.e. with zero recombination velocity, the boundary conditions of equation (4) are replaced by the condition of zero derivatives of excess carriers at x = 0 and x = x0. In this case all the sine terms, in all previous expansions, have to be replaced by cosines. In the genera1 case, for arbitrary surface recombination velocities, both sine and cosine terms must be used in the series. The appropriate boundary conditions should be applied to determine the coefficients of their arguments. It should be noted that unlike other cases treated in the literature where, due to symmetry, a rather simple transcendental equation can be derived for these coefficients, in our case even when the excitation and the surface recombination velocities are symmetrical, the presence of the electric field destroys the symmetry and complicates the expansion.

5. 3-D EXPANSION The decay of excess carriers following the termination of the external excitation can be extended into 3-D. Once again, the initial distribution $(x, y, z, 0) must be known. The simplest configuration to solve is when there is symmetry about the y- and z-axes. This implies that the electric field is in the x-direction. Secondly, the initial distribution, and, therefore, the external excitation that generates it must be symmetrical about the y- and z-axes. Such a symmetry is unlikely to be found in photoconductive devices, however it is applicable to the diffusion region of photodiodes which

S. E. SCHACHAM

206

are illuminated along the x-axis. If the photodiode is short-circuited, the boundary conditions given by equation (4) apply for this device as well. This is due to the ohmic contact on the back end and to the Boltzmann condition on the edge of the space-charge region in the front end. The boundary conditions in the y- and z-directions are given by (17) and ,

where y0 and z0 are the dimensions in the y- and z-directions. Due to this symmetry the solution of the y- and z-components in the separation of variables is given by cosine terms, i.e. cos /3y ‘cos yz. The constants p and y are deduced from the boundary conditions (17) leading to the transcendental equations BD, tan fly,/2 = s?

and

yD,, tan

yZO/2 = &.

(18)

Each one of these equations has one root for 0 < /3,,y0 < rt or 0 < yOzo< 7-tand one root for every section for which (2m - 1)~ < bmy, < (2m + 1)~ or (21 - 1)~ < yIzo < (21+ 1)n. Inserting these terms in the separation of the ambipolar equation in 3-D, shows that the constants must satisfy the relationship K=

Therefore, B(x,Y,

the 3-D solution

-($yp?l-yi.

(19)

is given by

(5 1E $ E

z, f> = exp

n-lm-01-O

x{C.,,exp[-~-(St,+y1)Dr]sin(~)cos(8,y)cos(g,j)j,

(20)

where W,Y, cnm, = XO(P,Y~+ sin P,Y~)(Y,z~ + sin YP,)

sss x0

X

0

YO/2

:();2

mYo/2

-

@(x,y,z,O)exp(

-g)sin(G)cos(/Ly)cos(y,z)dxdydz

(21)

zo/2

due to the orthogonality of the coefficients. The same procedure as described in Section 3 for obtaining the build-up and the decay after a short excitation, can be applied to the 3-D problem as well. For the specific cases of either ohmic or electrically reflecting contacts in the y- and z-directions, leading to either pm = (2m + l)rc/yo or &, = 2mn/y, and similarly for y,, an analytical expression for d(x, y, z, t) can be obtained using equation (20). In most cases in photoconductors the symmetrical distribution discussed above does not apply. Moreover, the electric field is not constant but is a function of the coordinates. However, if the gradient of the electric field is not too large, i.e.
Excess carrier

concentration

207

neglect the change in the electric field due to the presence of excess carriers. The derived field is then applied in the Fourier expansion to find the distribution. A numerical solution 2-D problem is given by Kolodny and Kidron.“” 6. APPLICATION

TO

MOBILITY

MEASUREMENT

Finally, let us demonstrate the application of the I-D ambipolar mobility. According to the technique described from the travel time of excess carriers between two probes. the entire device of length x0. The change in voltage drop x2, spaced xIZ apart, is given by A&=+$)

electric for the

solution to the direct measurement of in Ref. (11) the mobility is determined A constant voltage is maintained across AI’,, between the two probes at x, and

(22)

{ -AP,;+APo~-$&[~(x,,-~(x,)]} x0

Here n and p are the electron and hole concentrations, respectively, and b is the mobility ratio. APO is the excess carrier concentration integrated over the entire device, while AP,, is the integral ZAR and RAZ, respectively, while over x,*. The first two terms are due to the drift, representing the third term is a result of the diffusion. The sum of the drift terms leads to a negative voltage drop whenever most of the excess carriers are in between the output electrodes. Thus, generating excess carriers outside the probes results in a positive voltage change. As the perturbation is swept across, the voltage reverses its sign, and finally as the excess carriers come out on the other side, the voltage switches a second time and becomes positive again. Thus a trace like the one shown in Fig. 2 is obtained. To the first approximation the ambipolar mobility is derived from ,u = x,,lEAt, where At is the time between the two zeros of the AV,* trace. Using the solution to the continuity equation, we can fit a theoretical curve to the measured trace, thus deriving an accurate value for the ambipolar mobility. The values of fi (x,) and J?(x2) can be obtained using equations (8) and (16), while the integral terms are derived from equations (13) and (14). Figure 2 shows a trace of A V,2 obtained at 234 K of the device used in Ref. (11). The Hall-bar is a p-type Hg, _,Cd,Te with x = 0.29 and NA = 10’6cm~3. The best fit is obtained with the ambipolar mobility p‘, = 1.9. IO4 cm*/V.s. This result is substantially larger than the result of 1.65. IO4 given in Ref. (11). There, pLEwas calculated according to the first-order approximation. However, at this high temperature, the diffusion term proportional to kT.@ cannot be neglected. It causes the difference between the first approximation, based on drift only, and the accurate computation including all terms. It is interesting to note that the corrected value of pL,when inserted in Fig. 3 of Ref. (1 I), results in the extension of the linear dependence of the mobility on the logarithm of the temperature to the higher temperature range.

-61 0

I

I

I

I

I

I

1

2

3

4

5

6

Time ( IO-‘s Fig. 2. Voltage

difference

between

two inner probes

1

on a semiconductor

device following

illumination.

208

S. E. SCHACHAM

In addition to deriving the mobility, the fitting process emphasizes the strong dependence of the theoretical curve on the excess carrier lifetime. While the amplitude of the first peak is very sensitive to the length of the exciting pulse and to the location of application, the ratio between the amplitudes of the negative part and the second positive part depends almost entirely on the lifetime. Thus the excess carrier lifetime can also be obtained from the fit. The best fit for the curve of Fig. 2 was derived using a lifetime of 200 ns, while increasing the lifetime by 20% results in a comparable increase in the second peak. Figure 2 shows the curve computed with a lifetime of 240 ns. The larger amplitude of the second peak is obvious. 7. CONCLUSION The solution to the continuity equation can be expanded into a Fourier series. The expansion is presented for the 1-D problem, while the exact presentation in 3-D depends on the surface recombination in the other directions as well. Both cases of build-up of excess carriers, when the external generation is applied, and their decay following its termination can be obtained directly from these expansions. These solutions are applicable to the diffusion region of a short-circuited photodiode as well as to photoconductors. Ackno~kedgement-The

author

wishes to thank

Professor

I. Kidron

for his enlightening

comments

and encouragement.

REFERENCES I. W. van Roosbroeck. Phys. Rec. 91, 282 (1953). 2. C. T. Elliott, D. Day and D. J. Wilson, Infrared Phys. 22, 31 (1982). 3. W. Shockley, Electrons and Holes in Semiconductors. Van Nostrand, New York (1950). 4. E. S. Rittner, in Photoconducticity Conference, Atlantic City, N.J., 1954 (Edited by R. C. Breckenridge), New York (1956). 5. B. K. Ridley, Proc. phys. Sot. 75, 157 (1960). 6. G. Duggan and D. E. Lacklison, J. appl. Phys. 53, 3088 (1982). 7. Y. Shacham-Diamand and I. Kidron, J. appl. Phys. 56, 1104 (1984). 8. R. L. Williams, Irzfrared Phys. 8, 337 (1968). 9. M. A. Kinch, S. B. Borrello, B. H. Breazeale and A. Simmons, Infrared Phys. 17, 137 (1977). IO. A. Kolodny and I. Kidron. Infrared Phys. 22, 9 (1982). 11. S. E. Schacham and E. Finkman, J. appl. Phys. 57, 1161 (1985).

p. 215. Wiley,