Transient photoconductivity analysis near a grain boundary of polycrystalline semiconductors

~Solid ~Prlnted

State C o ~ u n i c a t i o n s , in Great Brltaln.

Vol.64,No.6

TRANSIENT

pp.851-854

PHDTOCONDUCTIVITY

BOUNDARY

1987.

ANALYSIS

OF P O L Y C R Y S T A L L I N E

0038-1098/87 $3.00 + .00 ©|987 Pergamon Journals Ltd.

NEAR

A GRAIN

SEMICONDUCTORS

C. A. Dimitriadis D e p a r t m e n t of Physics, Solid STate Physics Section, U n i v e r s i t y of T h e s s a l o n l k i , T h e s s a l o n l k l 540 06, Greece (Received: January , 1987, by C . W . M c C o m b i e ) The effect of grain b o u n d a r y r e c o m b i n a t i o n on the p h o t o c o n d u c T a n c e transient r e s p o n s e , after e x c i t a t i o n with a delta function light pulse near a grain b o u n d a r y , is studied here. An a n a l y t i c a l e x p r e s s i o n is derived for This decay which can be r e a d i l y computed n u m e r i c a l l y . The photocond u c t a n c e Turns out to be a n o n - e x p o n e n t i a l function of Time even at large decay Times. A p r a c t i c a l method, based on such n o n - e x p o n e n t i a l decay curves o b t a i n e d wlth the light beam p o s i t i o n e d away or near a grain b o u n d a r y , is d e v e l o p e d %o deduce accurate value of the bulk lifetime and the grain b o u n d a r y r e c o m b i n a t i o n velocity.

Various e x p e r i m e n t a l Techniques have been used to obtain i n f o r m a t i o n about the r e c o m b i n a t i o n process at grain b o u n d a r i e s of p o l y c r y s t a l l i n e s e m i c o n d u c t o r s . (a) I-V, C-V and deep level t r a n s i e n t s p e c t r o s c o p y (DLTS) t e c h n i q u e s which require large c o n t a c t areas, thus g i v i n g only a v e r a g e d r e s u l t s l-s. (b) E l e c t r o n - b e a m - i n d u c e d current (EBIC) or l i g h t - b e a m - l n d u c e d - c u r r e n t (LBIC) t e c h n i q u e s which require the f a b r i c a t i o n of a p-n j u n c t i o n or s e m i t r a n s p a r e n t S c h o t t k y barrier for the c o l l e c t i o n of m i n o r i t y c a r r i e r s 6-10. These m e t h o d s have the advantage that the e x c i t a t i o n by e l e c t r o n or light beam allow local p r o b i n g of small area of the grain b o u n d a ries. H o w e v e r , the high precessing T e m p e r a t u r e may change The grain b o u n d a r y c h a r a c t e r i s t i c s , which is u n d e s i r a b l e II. (c) P h o t o c o n d u c T i v i t y m e a s u r e m e n t s which, also, allows local probing of small area of the grain boundaries, but require v e r y simple n o n - d e s t r u c t i v e sample prep a r a t i o n I~,13.

\

X$

IBEAM T t

~X

h

""GRAIN BOUNDARY

Z

R e c e n t l y a s t e a d y - s t a t e p h o t o c o n d u c t i v i t y experimental t e c h n i q u e , using a focused laser beam to g e n e r a t e e x c e s s carriers in a b i a s e d sample, was used to obtain information c o n c e r n i n g the m a g n i t u d e of the r e c o m b i n a t i o n v e l o c i t y at the grain b o u n d a r y i n t e r f a c e s I~. In this m e t h o d the fitting of the e x p e r i m e n t a l data with the theory r e q u i r e s a k n o w l e d g e of the hulk m i n o r i t y carrier l i f e t i m e (T) and %he d i f f u s i o n coefficient (D). Here, we present a t r a n s i e n t - s t a t e p h o t o c o n d u c t i v i t y method which enables us to determine quite e a s i l y both hulk lifetime and grain b o u n d a r y r e c o m b i n a t i o n velocity if the optical a b s o r p t i o n c o e f f i c i e n t and the d i f f u s i o n c o e f f i c i e n t of m i n o r i t y carriers are known.

Fig.

i: Geometry

tance, The geometry rier mobility.

of the problem.

of the sample and the car-

In the analysis the following assumptions are made: (i) The sample is of semi-infinite T h i c k n e s s , i.e. b o u n d e d only by the top surface at z =0. (2) The grain b o u n d a r y can be c o n s i d e r e d as a surface with r e c o m b i n a t i o n velocity s. (3) The d u r a t i o n of ~he pulsed light is much smaller than the lifetime (T). (4) The injection level is low. (5) The excess m i n o r i t y carrier p o p u l a t i o n decay is not i n f l u e n c e d by the effect of surface r e c o m b i n a t i o n . This can be a c c o m b ~ s h e d either by p o l i s h i n g The surface so thai the surface r e c o m b i n a t i o n velocity To be zero or by using e x c i t i n g light of small absorption c o e f f i c i e n t so That the g e n e r a t i o n To take place deep in the sample. (6) The e l e c t r o n - h o l e pair generation rate can be w r i t t e n

The g e o m e t r y of the method is shown in Fig. 1. The n - t y p e sample is illuminated by a finely focused m o n o c h r o m a t i c pulsed light beam, incident n o r m a l l y to The surface, near a grain boundary. Excess c a r r i e r are created at (Xs,Ys,Zs). by the f o c u s e d b e a m which s u b s e q u e n t l y d i f f u s e and drift away. If an e l e c t r i c field along the xaxis is e s t a b l i s h e d by a constant bias current, Then a p h o t o c o n d u c t i v e signal (AV) r e s u l t s the m a g n i t u d e of which is p r o p o r t i o n a l to the total m i n o r i t y - c a r r i e r c o n t e n t I~, i.e. AV = Bfffdp(x,y,z)dxdydz

--

SURFACE

(i)

g(z,T)

where the c o n s t a n t B d e p e n d s on the e l e c t r i c field a l o n g the x-axis, %he sample bulk resis-

= N(l-R)me-~Z6(t)

(2)

where T is the Time, N is the number of photons carried by a simple pulse just out-

851

TRANSIENT PHOTOCONDUCTIVITY

852

side the s a m p l e s u r f a c e , R is the s u r f a c e r e f l e c t i v i t y and 6(t) is the Dirac d e l t a function. In o r d e r to c a l c u l a t e the total p o p u l a t i o n of the m i n o r i t y c a r r i e r s A P t o t ( t ) i n d u c e d in the sample by a pulse, we will first c a l c u l a t e t h e i r d e n s i t y A p ( x , y , z , t ) due to a p o i n t s o u r c e Z at d e p t h h (Fig. i) and then i n t e g r a t e over all the p o i n t s a l o n g the beam and over the entire s a m p l e volume. The d e n s i t y of the m i n o r i t y excess c a r r i e r s due to an i n s t a n t a n e o u s p o i n t source of u n i t y s t r e n g t h has been c a l c u l a t e d by Van R o o s b r o e c k Is. U n d e r the c o n d i t i o n s stated above~ by comb i n i n g Eqn. (22) of Ref. 15 with Eqn. (2) for the g e n e r a t i o n rate, we obtain the f o l l o w i n g exp r e s s i o n for Ap: (Z-H)2+y 2 N(I-R)A(41U)-3/2 Ap = L3

e -AH

e

~U

=2J6(

I -H/2/'U

= 2/~-U

,

and

Vol. 64, No. 6

ANALYSIS

using

e_b 2

e I H/2/'U

db +

_b 2 db):

(7) the p r o p e r t i e s

for

the

i n t e g r a t i o n 16

b2-ac I e - ( a t 2 + 2 b t + C ) d t = ~--[ e2 a o

f e-aXerf(bx)dx o we o b t a i n

6Ptot(t)

the

=

= ~

final ~(I-R)

a

ea2/4b 2

e r f C ( ~ ab)

erfc(

) ,

(B)

(9)

result e-U[l+eA2Uerfc(A¢~)]×

x X ~X +S2U X x [err(2_~U) + e s erf c (2 + u + S ¢r~)] ( i0 )

x[e

(X-Xs)2 ~U +e

(X+Xs)2 S(X+Xs)+S2U ~U -2/~'USe ×

Also,substituting

X+X x e r f c ( 2~

+ SvrU)]

,

A : eL,

U : t/T,

Z = z/L,

H : h/L

X : x/L, and

APtot(0)

(3)

w h e r e e r f c ( x ) is the c o m p l e m e n t a r y error function and A, U, X, Y, Z, H and S are the d i m e n s i o n l e s s q u a n t i t i e s n o r m a l i z e d with the bulk c a r r i e r l i f e t i m e • and d i f f u s i o n l e n g t h L = (DT)I/2:

S : s/(L/~)

(Z_H)2 x~

e

4U

(i0)

2N(I-R) = L----~

(II)

A P t o t (t) AV : A P t o t ( 0 )

re-

:

Y : y/L

2N(I-R)A(4~U)-3/2e-AHe L3

:

in Eqn.

The p h o t o c o n d u c t i v e signal normalized with spect to its v a l u e at t=0 is given by

.

(4)

e

The total c o n t e n t of e x c e s s m i n o r i t y c a r r i e r s can now be o b t a i n e d b y i n t e g r a t i n g Eqn. (3) w i t h r e s p e c t to H f r o m zero to i n f i n i t y and then over the e n t i r e s a m p l e v o l u m e , i.e.

APtot(t)

t=U:0

-U ×

x

When from

-U

2 [ l+eA

Uerfe(A/U) ]x

X SXs+S2U X s [erfc(2---~)+e erfc(2-~U

+ S~)].(12)

the p u l s e d light beam is p o s i t i o n e d away the g r a i n b o u n d a r y , Eqn. (12) b e c o m e s

y2 dZ

f

o

e

~U

dY

AV = ~e -U [ l + e A 2 U e r f c ( A/6)]

x

.

(13)

o

(X-Xs)2 4U

×U (e

+e

In o r d e r to c a l c u l a t e the bulk l i f e t i m e T f r o m the d e c a y of the p h t o c o n d u c t i v e signal, the par a m e t e r s t and • are s e p a r a t e d by p u t t i n g %:a2Dt. Then A 2 : ~ 2 % T , U : % / A 2 and s u b s t i t u t i n g for U in Eqn. (13) we o b t a i n

(X+Xs)2 4U )dX -

o

-2/~-U" Se

SX +S2U s x

X+X x I e SX e r f c ( ~ o

AV - e - t / A 2 ----~[l+e%erfc(/%)]

+ S~)dX]

(5)

Since

®_y_2 f e O

~U dY

(X-Xs)2 oo f (e o

4U

~6)

: 2/~-U

(X+Xs)2 + e

4U

) dX:

.

(14)

Thus, when a and D are known, AV can be m e a s u r e d at a g i v e n v a l u e of % and E~n. (14) can be s o l v e d for A to y i e l d T as T=A / s 2 D . F i g u r e 2 s h o w s the p a r a m e t e r A as a f u n c t i o n of % for v a r i o u s v a l u e s of AV as o b t a i n e d from E q n . ( 1 4 ) . C l e a r l y , the s l o p e of the c u r v e s i n c r e a s e s w~th i n c r e a s i n g % and b e c o m e s i n f i n i t e at a c e r t a i n v a l u e of %. T h i s v a l u e of % b e c o m e s s m a l l e r for l a r g e r v a l u e s of the p h o t o c o n d u c t i v e s i g n a l AV and~ t h e r e f o r e , the a c c u r a c y b e c o m e s p r o g r e s s i v e l y h i g h e r for s m a l l e r v a l u e s of AV. W h e n the p u l s e d light b e a m is p o s i t i o n e d n e a r the g r a i n b o u n d a r y , then for S=0 Eqn. (12) red u c e s to Eqn. (13). For S~O, the p h o t o c o n d u c -

TRANSIENT PHOTOCONDUCTIVITY ANALYSIS

Vol. 64, No. 6

853

.soZ

JV 0.95

0.9

0.8

0.7

0.6

0.5

0.4

IO

J~

10-I

10-1

.

10-2 ~o-Z

~-3

-jO-Z

i

10-

1

i

i

10

10Z ~=a2Dt

Fig.

2: N o r m a l i z e d a b s o r p t i o n c o e f f i c i e n t A v e r s u s the n o r m a l i z e d time % for var i o u s values of the p h o t o c o n d u c t a n c e d e c a y ratio AV from eqn. (14).

10-3

10-4

I 1

I 2

I 3

I 4

I 5

I 6

U tire signal d e c a y for large v a l u e s of U a p p r o a c h e s a s y m p t o t i c a l l y to

-U e Av ~ U ~

'

Fig.

(15)

i.e. even at large d e c a y t i m e s the p h o t o c o n d u c tance decay is still not e x p o n e n t i a l . F r o m Eqn. (15) it f o l l o w s that the plot of ( A V F ~ ) v s U a p p r o a c h e s an e x p o n e n t i a l for large d e c a y times. An e x a m p l e of a £ n ( A V v ~ ) v s U plot, as o b t a i n e d from Eqn. (12), for X_:0.5 A = 0 . 1 and two values of S is shown in ~ig. ~. A m a x i m u m of the p h o t o e o n d u c t a n t e decay c u r v e a p p e a r s at a pos i t i o n U m w h i c h d e p e n d s on the value of S . F o r S=0 the m a x i m u m lies at U m = 0 . 5 and for lar.ge v a l u e s of S the m a x i m u m is at Um:0. F i g u r e 4 shows a plot of the p o s i t i o n of the m a x i m u m U m as a f u n c t i o n of S for two v a l u e s of A. F r o m Fig. 4 and the e x p e r i m e n t a l value of U m it is p o s s i b l e to e s t i m a t e the grain b o u n d a r y r e c o m b i n a t i o n v e l o c i t y if the a b s o r p t i o n c o e f f i c i e n t of the e x c i t i n g light and the diffusion c o e f f i cient of m i n o r i t y c a r r i e r s are known. In c o n c l u s i o n , a n a l y t i c a l e x p r e s s i o n for the p h o t o c o n d u c t a n c e d e c a y is o b t a i n e d when the m o n o c h r o m a t i c p u l s e d light b e a m is p o s i t i o n e d at d i f f e r e n t d i s t a n c e s f r o m a grain boundary. It turns out that the p h o t o c o n d u c t a n c e d e c a y is not e x p o n e n t i a l even at large decay times. A method, b a s e d on such n o n - e x p o n e n t i a l d e c a y curves, is d e v e l o p e d to d e t e r m i n e the bulk l i f e t i m e and the grain b o u n d a r y r e c o m b i n a t i o n velocity.

3: P l o t of the n o r m a l i z e d p h O t o c o n d u e t a n c e AV versus the n o r m a l i z e d time U for n o r m a l i z e d a b s o r p t i o n c o e f f i c i e n t A=0.1, d i s t a n c e b e t w e e n grain b o u n d a r y light beam Xs=0.5 and two v a l u e s of the grain b o u n d a r y r e c o m b i n a t i o n velocity S.

102

101

lO 0

.I

I •1

I -2

I -3

I% .4

% I .5

I

.6 Um

Fig. Ackgowled~ements The a u t h o r w i s h to thank the Greek M i n i s t r y of R e s e a r c h and T e c h n o l o g y for f i n a n c i a l support.

4: Plot of the m a x i m u m U m of the decay of the s e m i l o g a r l t h m i c A V v ~ curve as a f u n c t i o n of the n o r m a l i z e d grain houndary r e c o m b i n a t i o n ve.locity for two values of the n o r m a l i z e d a b s o r p t i o n coeff i c i e n t A.

854

TRANSIENT PHOTOCONDUCTIVITY ANALYSIS

Vol. 64, No. 6

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