SUPPLEMENTAL MATERIAL
Electron drift-mobility measurements in polycrystalline CuIn1-xGaxSe2solar cells
S. A. Dinca,1 E. A. Schiff,1 W. N. Shafarman,2 B. Egaas,3 R. Noufi,3 and D. L. Young3
1Department of Physics, Syracuse University, Syracuse, New York 13244-1130, USA
2Institute of Energy Conversion, University of Delaware, Newark, Delaware 19716, USA
3National Renewable Energy Laboratory, Golden, Colorado 80401, USA
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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 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 forthe 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]
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) techniquein whicha semiconductor material is illuminated near an electrode interface with a pulse of strongly absorbed light. Following the illumination, a sheet of photocarriersis created, ideally at position x= 0 and time t = 0. The analysis for this situation is well-known[19]; 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 themobile photocarriersunder the influence of anelectric field Egives riseto a transient photocurrentI(t) in the externalcircuit. Thephotocurrent densityj(t) in the sample is given by the sum of the conduction anddisplacement current densities[20]
, (B1)
where 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 densityis equal to the space-average conduction current density
.(B2)
This is a well-known result.20[21]-[22]The photocurrent density j(t) (see eq.(B2)) is related to transient photocurrents I(t) measured in the external circuit through the expression:
, (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 photocurrentI(t)is the sum of the electron and hole transient photocurrentsIe(t) and Ih(t):
.(B4)
The electrontransient photocurrent[23]Ie(t)given by eq. (B3) is with related to the total electroncarrier density n(x,t) by the following formula:
,(B5)
wheree is the elementary charge, μeis the electron drift mobility and dis the thickness of the active region of the CIGS film as measured by capacitance. We assume a uniform external field
.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 density[24]reflects both the charge densityof free carriers and also the trappedphotocarriers.Assuming that the surviving charge at time tis , where τe is the deep trapping lifetime, we can write the following equationfor :
(B6)
whereis the carrier densitycreated by impulse illumination,H(x) is the Heavisidefunction and represents the position of the mobile carrierat time t.
Substituting Eq. (B6)into Eq. (B5) and solving the integral, the resulting transient photocurrent is:
(B7)
where is the electron transit-time and is the total injected photocharge at t = 0.
The transient photocharge Q(t) is obtained by integrating the transient photocurrent I(t):
. (B8)
where Qe(t) and Qh(t) are the electron and hole transient photocharge, respectively. The equation for Qe(t) is:
(B9)
Ananalogous equation applies for Qh(t).Evaluating Eq. (B9) at ,and making an allowance for a uniform internal electric field Ebi = V0/d,gives
,(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 times (τ→∞) each term in Eq. (B10) is only half of the total collected charge 0.5Q0 andthe asymptotic charge is.
Appendix C: Transient Photocharge Measurements with Voltage-Dependent 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 timestTare measured for photocarriers that are photogenerated at time t=0 and then drift across a layer in an electric field. The drift mobility µis estimated according to the expression:
,(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 photochargeQ(t)/Q0 at 293 K measured with two pulsed laser wavelengths, 690 and 1050 nm; Q0 is the total photocharge. 690 nm illuminationis absorbed near the CdS/CIGS interface, and this transient photocharge is dominated by hole drift. For 690 nmwe identify the time at which half of the ultimate photocharge Q0 has been collected as the hole risetime[26]; 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 analysis. The 50% of the photocharge attributable to holes is collected relatively promptly, and
the electron transit time teis identified with collection of half of the remaining photocharge.
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 voltagefor the two illuminations: 690 nm and 1050 nm; this means that both photocarries (electron and hole) were able to traverse the depletion layer without being trapped. We identified these “plateau”photocharges as the total photocharge Q0
absorbed by the sample.
In Figure C2 (b)and (d) we have graphed the reciprocal of the hole and electron transit timeobtained from the photocharge transients. The hole mobilities were obtained from fitting lines using the following equation ((2b) from ref. 19):
(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:
,(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 inand .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 builtin potentials VBI for these cells.We speculated that the difference between VBI and V0reflects a rapid drop of the built-in electric potential near the CIGS/CdS interface.
1
1
[*]CIS – CuInSe2, CIGS – CuIn1-xGaxSe2, CGS – CuGaSe2
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