SUPPLEMENTAL MATERIAL
Electron drift-mobility measurements in polycrystalline CuIn1-xGaxSe2 solar
cells
S. A. Dinca,1 E. A. Schiff,1 W. N. Shafarman,2 B. Egaas,3 R. Noufi,3 and D. L. Young3
1
Department of Physics, Syracuse University, Syracuse, New York 13244-1130, USA
2
Institute of Energy Conversion, University of Delaware, Newark, Delaware 19716, USA
3
National Renewable Energy Laboratory, Golden, Colorado 80401, USA
1
APPENDIX A: ELECTRON MOBILITIES in CuIn1-xGaxSe2 MATERIALS
Figures A1 illustrates Hall-effect measurements of electron mobilities in CIGS as well as
the present drift-mobility measurements. The Hall measurements are done in n-type CIGS; the
Cu(In,Ga)Se2 alloys
s
r
q
p
o
n
m
l
k
j
i
h
g
f
e
d
c
b
a
0.01
0.1
1
10
100
1000
2
Electron mobility (cm /Vs)
Figure A1. Room temperature electron mobilities reported for CuIn1-xGaxSe2 (CIGS). Each
symbol represents a particular sample; the letters indicates the reference for each
measurement according to the key in Table A.I. Line s shows the present drift mobilities from
the present work (). All other mobilities are Hall mobilities. Solid and open symbols indicate
mobility measurements on thin film polycrystalline and single crystal samples, respectively.
2
drift-mobility measurement is in thin-film, p-type CIGS. Table A1 has the references for these
different measurements.
Table A.I. Key to the letter codes (a–s) used in Fig. A1 to identify the experimental references
for the room temperature electron Hall mobility measurements.
Specimen*
Code[reference]
CIS (single crystal)
a[1], b[2], c[3], d[4], e[5], f[6], g[7], h[8], i[9]
CIGS (single crystal)
j[10], r[11]
CGS (single crystal)
k[12]
CIS (thin film)
l[13], m[14], n[15], o[16], p[17], q[18]
CIGS (polycrystalline films)
r[11], s[present work]
---------------------------------------------------*
CIS – CuInSe2, CIGS – CuIn1-xGaxSe2, CGS – CuGaSe2
3
APPENDIX B: HECHT ANALYSIS: TRANSIENT PHOTOCHARGE MEASUREMENTS
UNDER UNIFORM ILLUMINATION
Experimentally, charge transport properties (electron and hole drift mobilities) are
typically investigated using the time-of-flight (TOF) technique in which a semiconductor
material is illuminated near an electrode interface with a pulse of strongly absorbed light.
Following the illumination, a sheet of photocarriers is created, ideally at position x = 0 and time t
= 0. The analysis for this situation is well-known19; in this section we present the extension to
weakly absorbed illumination that generates carriers uniformly throughout the volume of the
specimen.
We start with some generalities. The motion of the mobile photocarriers under the
influence of an electric field E gives rise to a transient photocurrent I(t) in the external circuit.
The photocurrent density j(t) in the sample is given by the sum of the conduction and
displacement current densities20
j t   jc x, t    r  0
E x, t 
,
t
(B1)
where  r  0 is the dielectric constant. This can be simplified if the voltage, V across the sample is
constant. Integrating the eq. (B1) over the sample thickness d, the photocurrent density is equal
to the space-average conduction current density
j (t ) 
1
d
d
0 jc x, t dx .
(B2)
This is a well-known result.20-22 The photocurrent density j(t) (see eq.(B2)) is related to transient
photocurrents I(t) measured in the external circuit through the expression:
4
I t   jt A ,
(B3)
where A is the cross-sectional area of the specimen.
When a pulse of weakly absorbed light is incident on one of the cell electrodes, electron
and hole photocarriers are uniformly photogenerated in the volume of the sample. Under this
condition, the measured photocurrent I(t) is the sum of the electron and hole transient
photocurrents Ie(t) and Ih(t):
I (t )  I e (t )  I h (t ) .
(B4)
The electron transient photocurrent23 Ie(t) given by eq. (B3) is with related to the total electron
carrier density n(x,t) by the following formula:
I e (t ) 
A d
enx, t  e E dx ,
d 0
(B5)
where e is the elementary charge, μe is the electron drift mobility and d is the thickness of the
active region of the CIGS film as measured by capacitance. We assume a uniform external field
E  V d . An analogous equation applies for Ih(t).
In the process of the drift we assume that some of carriers are captured and immobilized
by traps. As a result, the total charge density24 reflects both the charge density of free carriers
and also the trapped photocarriers. Assuming that the surviving charge at time t is exp(  t  e ) ,
where τe is the deep trapping lifetime, we can write the following equation for n( x, t ) :
n exp  t /  e H x   e Et , for t  d  e E
nx, t    o
0,
for t  d  e E

(B6)
where n 0 is the carrier density created by impulse illumination, H(x) is the Heaviside function
and  e E t  x represents the position of the mobile carrier at time t.
5
Substituting Eq. (B6) into Eq. (B5) and solving the integral, the resulting transient
photocurrent is:
 Q0

t 
 exp  t  e 1  , t  t e
I e t    t e
 te 
0,
t  te

(B7)
where t e  d  e E is the electron transit-time and Q0  eno dA is the total injected photocharge
at t = 0.
The transient photocharge Q(t) is obtained by integrating the transient photocurrent I(t):
Qt    I (t ' )dt '  Qe t   Qh t  .
t
(B8)
0
where Qe(t) and Qh(t) are the electron and hole transient photocharge, respectively. The equation
for Qe(t) is:
 e  e
 t  t   e  
 1 , t  t e
Q0 1   exp    
 te  te
  e  t e

Qe t   
 Q  e 1   e 1  exp   t e  ,
t  te
   
 0 t e  t e 

e







(B9)
An analogous equation applies for Qh(t). Evaluating Eq. (B9) at t  te , and making an allowance
for a uniform internal electric field Ebi = V0/d, gives

 
QV   e V0  V    e V0  V  
d2



1

1

exp


2
  V  V   
Q0
d2
d

e 0


 


 h V0  V  
d2
1 

 h V0  V  
d2

 
d2
 
1  exp  




V

V

h
0

 
,
(B10)
which is the extended Hecht equation 25 for the photocharge collection as a function of the
applied voltage measured with uniformly absorbed excitation. Note that, for long deep-trapping
6
times (τ→∞) each term in Eq. (B10) is only half of the total collected charge 0.5Q0 and the
asymptotic charge is Q  Q0 .
7
APPENDIX C: TRANSIENT PHOTOCHARGE MEASUREMENTS WITH VOLTAGEDEPENDENT DEPLETION LAYER
The measurements of transit times presented in the body of this paper were done with a
sample and temperature for which the electric field was nearly uniform across a depletion region,
and which showed little capacitance variation with voltage. This was atypical; nearly all mobility
estimates were done using samples with depletion widths that increased substantially with
increasing reverse bias, and in this section we show the associated analysis.
In the typical time-of-flight (TOF) technique transit times tT are measured for
photocarriers that are photogenerated at time t=0 and then drift across a layer in an electric field.
The drift mobility µ is estimated according to the expression:
  L EtT  ,
(C1)
where L is the average displacement of the carriers at the transit time and E is the electric field.
Figure C1 shows the normalized photocharge Q(t)/Q0 at 293 K measured with two pulsed laser
wavelengths, 690 and 1050 nm; Q0 is the total photocharge. 690 nm illumination is absorbed
near the CdS/CIGS interface, and this transient photocharge is dominated by hole drift. For 690
nm we identify the time at which half of the ultimate photocharge Q0 has been collected as the
hole risetime26; we corrected for the optical pulsewidth and the RC time constant to convert this
to a transit time.
To obtain sensitivity to electron motion, we used 1050 nm, which is absorbed uniformly
throughout the depletion width. Both electron and hole photocarriers contribute equally to the
ultimate photocharge. We defined the electron risetime as the time required for 75% photocharge
collection, as illustrated in Fig. C1. This is reasonable as long as the hole drift mobility is
significantly larger than the electron drift-mobility, which proves to be self-consistent with our
8
analysis. The 50% of the photocharge attributable to holes is collected relatively promptly, and
the electron transit time te is identified with collection of half of the remaining photocharge.
Photocharge Q/Q0
1.0
tR,e
0.8
3Q0/4
0.6
x0.5
0.4
Q0/2
0.2
1050 nm
690 nm
tR,h
0.0
0.0
0.1
0.2
Time (s)
1.5
2.0
Figure C1: Normalized photocharge transients Q(t)/Q0 measured using two optical
wavelengths, 690 nm and 1050 nm in a NREL cell at 293 K with a bias voltage of -0.3 V (690
nm) and 0 V (1050 nm). The total photocharge Q0 is defined as the total photocharge
collected at longer times and larger bias voltages. The intersection of the transients with the
horizontal lines at Q0/2 (690 nm), and 3Q0/4 (1050 nm) were used to determine the hole and
electron risetimes t R , h and t R, e (690 nm: Q0 = 5.53x10-12 C; 1050 nm: Q0 = 1.44x10-11 C).
Mobilities were obtained by fitting the voltage-dependent photocharge transients. Figure
C2 (a) and (c) illustrates the voltage-dependence of the photocharge at 293 K for two specimens:
NREL-1 and IEC-1. Note that an IEC-1 cell was used at 150 K for the data presented in the body
of the paper. The open and solid symbols indicate the photocharge Q measured at 4 μs with the
two wavelengths; photocharge collection was complete by this time. For these CIGS cells, the
photocharge is fairly independent of the voltage for the two illuminations: 690 nm and 1050 nm;
this means that both photocarries (electron and hole) were able to traverse the depletion layer
9
without being trapped. We identified these “plateau” photocharges as the total photocharge Q0
absorbed by the sample.
2
2
0.0
IEC-1
20
15
15
10
10
5
690 nm
1050 nm
(a)
(b)
5
(c)
(d)
0
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
Photocharge Q (pC)
-0.5
NREL-1
0
0.5
ln(2)d /2th (cm /s)
-1.0
2
0.0
2
-0.5
ln(2)d /4te (cm /s)
Photocharge Q (pC)
-1.0
20
0.0
-1.0
-0.5
0.0
Bias Voltage (V)
-1.0
-0.5
0.0
Bias Voltage (V)
Figure C2: Panels (a) and (c) indicate the photocharges measured at 1 µs for the NREL-1 and
IEC-1 samples. The symbols in panels (b) & (d) are the photocharge transit-time
measurements for varying bias voltages at 293 K. For 690 nm wavelength these are the Q0/2
transit times; for 1050 nm these are the 3Q0/4 transit times. The lines in these panels are
fittings that yield the hole drift mobility μh and the electron drift mobility μe estimates,
respectively. The voltage intercepts are related to the built-in potential VBI (see text). (NREL1: μh = 0.38 cm2/Vs and μe = 0.05 cm2/Vs; IEC-1 μh = 0.09cm2/Vs and μe = 0.02 cm2/Vs)
10
In Figure C2 (b) and (d) we have graphed the reciprocal of the hole and electron transit
time obtained from the photocharge transients. The hole mobilities were obtained from fitting
lines using the following equation ((2b) from ref. 19):
 h  ln 2d 2 V  2t h V0  V 
(C2)
where d is the voltage-dependent depletion width (inferred from capacitance measurements) and
the offset potential V0 is a fitting parameter to account for the built-in field’s effects. In Fig.
C2(d), the hole transit-times under reverse bias were too short to be measured accurately with the
diode laser setup, and we show the data only through -0.2 V. It is interesting that the straight-line
fit to these data misses the reverse bias points systematically; this might reflect a hole mobility
that increases with depth in the material. We discussed these issues at greater length in ref. 19.
The electron transit-times, the major concern in this paper, were longer and more readily
measured. The electron mobility was fitted to:
 e  ln 2d 2 V  4t h V0  V  ,
(C3)
which accommodates the uniform generation of the electrons.
The statistical error bars on each point in Fig. C2 (b) and (c) were determined by making
several risetime measurements at a given voltage and propagating the risetime error into the error
in d 2 2t h and d 2 2t e . The large errors for more negative bias voltages occur because the
photocharge risetime approaches the shortest value permitted by the laser pulsewidth and the RC
time constant.
As was discussed in the body of the paper, the offset potentials V0 inferred from this
fitting (about 0.24 V for IEC-1 and around 0.30V for NREL-1) are smaller than the built-in
potentials VBI for these cells. We speculated that the difference between VBI and V0 reflects a
rapid drop of the built-in electric potential near the CIGS/CdS interface.
11
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