Surface voltage and surface photovoltage - Dieter Schroder ...

Surface voltage and surface photovoltage
R
Φ(λ)
s 1 s 2
p-Type
τ, D, L
0 d x
(a)
R
Φ(λ)
s 1 s 2
p-Type
τ, D, L
∆n(W)
0 W d x
(b)
Figure A1. Homogeneous p-type sample geometry with optical
excitation: (a) neutral, (b) with surface-induced space-charge region.
photon flux density , wavelength λ and absorption coefficient
α is incident on one side of this wafer. We assume that carriers
generated by absorbed photons diffuse in the x-direction and
that the wafer has infinite extent in the y–z-plane, allowing the
neglect of edge effects. The steady state, small-signal excess
minority carrier density n(x) is obtained from a solution of
the one-dimensional continuity equation
D d2 n(x)
dx 2
− n(x)
τ
subject to the boundary conditions
+ G(x) = 0 (A1)
x = 0: dn(x) n(0)
= s 1
dx x=0 D
and
x = d: dn(x) n(d)
=−s 2
dx x=d D . (A2)
The electric field in the semiconductor sample is low and
is neglected for simplicity. The generation rate is given by
G(x, λ) = (λ)α(λ)[1 − R(λ)]exp(−α(λ)x).
(A3)
An implicit assumption in this expression is that each
absorbed photon generates one ehp. This is usually a good
assumption for semiconductors.
The solution to equation (A1) using (A2) and (A3) is [62]
{[ ( )
(1 − R)ατ
d − x
n(x) = K
(α 2 L 2 1 sinh
− 1)
L
( ) [ ( d − x
x
+K 2 cosh +e −αd K 3 sinh
L
L)
( )]][( x s1 s 2 L
+K 4 cosh
L D
+ D ) ( d
sinh
L L)
( d −1 }
+(s 1 + s 2 ) cosh − e
L)] −αx (A4)
where
K 1 = s 1s 2 L
D + s 2αL K 2 = s 1 + αD
K 3 = s 1s 2 L
D − s 1αL K 4 = s 2 − αD.
For SPV measurements, a surface charge induced spacecharge
region of width W exists as shown in figure A1(b). The
light generates excess carriers and it is well known from pn
junction theory (law of the junction) that the excess carrier
density at the scr edge is related to the resulting surface
voltage V S through the relation [25]
n(W) = n 0 (exp(qV S /kT) − 1) ≈ n 0 qV S /kT
(A5)
where n 0 is the equilibrium minority carrier density. The
approximation in equation (A5) holds for V S ≪ kT/q.
To determine the surface voltage, we need an expression
for n(W). Forx = W, equation (A4) becomes
{[ ( ) ( )
d − W
d − W
n(W) = A K 1 sinh + K 2 cosh
L
L
( ) ( )]]
W W
+e
[K −αd 3 sinh + K 4 cosh
L L
[(
s1 s 2 L
×
D
+ D ) ( ( d d −1
sinh + (s 1 + s 2 ) cosh
L L)
L)]
}
−e −αW (A6)
with A = (1 − R)ατ/(α 2 L 2 − 1). Let us now examine the
various assumptions that are made for the simplified equation
usually used to analyse SPV data. The space-charge region
width W is usually much smaller than the wafer thickness d,
hence d −W ≈ d. Assumption (i): d −W ≫ L, i.e., the wafer
is much thicker than the minority carrier diffusion length. This
allows equation (A6) to be written as
{[ [ ( ) ( )]
W W
n(W) = A (K 1 + K 2 ) cosh − sinh
L L
( ) ( )]/ ]
W W
+e
[K −αd 3 sinh + K 4 cosh cosh(d/L)
L L
[
s1 s 2 L
×
D + D ] −1 }
L + s 1 + s 2 − e −αW . (A7)
For W ≪ L and α(d − W) ≈ αd ≫ 1, we have
{( [
(1 − R)ατ
n(W) = (K
(α 2 L 2 1 + K 2 ) 1 − W ])
− 1)
L
(
s1 s 2 L
×
D + D ) −1 }
L + s 1 + s 2 − e −αW . (A8)
The assumption αW ≪ 1 leads to
{[(
(1 − R)αL
n(W) = s
(α 2 L 2 2 + D ) (
(αL − 1) 1+ s )]
1W
− 1) L
D
[
× s 2 s 1 + D ( )] D −1 }
L L + s 1 + s 2
{[(
(1 − R)αL
= 1+ D )(
1+ s )]
1W
(αL +1) s 2 L D
[
× s 1 + D (
1+ D
L s 2 L + s )] −1 }
1
. (A9)
s 2
If we assume the back contact to have very high surface
recombination, i.e., s 2 →∞, equation (A9) becomes
n(W) =
(1 − R)αL
(αL +1)
( 1+s1 W/D
s 1 + D/L
The term s 1 W/D is usually ≪1, leading to
(1 − R) αL
n(W) =
(s 1 + D/L) (1+αL)
which is the desired result.
)
. (A10)
(A11)
R29