Advanced Characterization Techniques for Thin Film Solar Cells (RAU:SOLARCELLS CHARACT. O-BK) ||...

8Steady-State Photocarrier Grating MethodRudolf Br€uggemann

8.1Introduction

The excess-carrier properties of a photoexcited semiconductor are important indi-cators of its quality with respect to applications in optoelectronic devices or forstudying the recombination physics. In contrast to the majority-carrier properties,which can be determined rather straightforwardly by stationary photocurrent mea-surements, the minority-carrier properties can only be revealed by more sophisti-cated methods. In this respect the steady-state photocarrier grating (SSPG) methodhas had an enormous impact since it was suggested by Ritter, Zeldov, and Weiser,named RZW hereafter, in 1986 [1].

The SSPGmethod is based on the carrier diffusion under the presence of a spatialsinusoidal modulation in the photogeneration rate G, which induces a so-calledphotocarrier grating. From photocurrent measurements at different grating periodsL, the ambipolar diffusion lengthL can be determined by an analysis that assumesambipolar transport and charge neutrality.

The proposal by RZW, following papers on the analysis [2, 3] and the simple setuptriggered a rapid widespread application [4–8]. In parallel, critical andmore in-depthaccounts on the underlying theory were given to put the technique on a firm groundor to describe its limits when applied to semiconductors with traps. Ritter et al. [9],Balberg et al. [10, 11], Li [12], and Shah et al. [13] analyzed the transport equations, alsowith respect to the �lifetime� or �relaxation time� regimes. In the lifetime regime, thecarrier lifetimes are longer than the dielectric relaxation time.

The previous publications were later criticized byHattori et al. [14] who performeda second-order perturbation approach and pointed out deficiencies of other earlieranalyses. Nevertheless, Hattori et al. showed that under the conditions in the�lifetime regime� the analysis and evaluation of the SSPGmethod are correct. Theseauthors also suggested a correction method to avoid incorrect values of L.

A novel aspect was introduced by Abel and Bauer [15] who numerically solved thetransport and Poisson equations and compared the solutions with a generalizedtheory which enables to study the SSPG results by the variation of L and/or electric

Advanced Characterization Techniques for Thin Film Solar Cells,Edited by Daniel Abou-Ras, Thomas Kirchartz and Uwe Rau.� 2011 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2011 by Wiley-VCH Verlag GmbH & Co. KGaA.

j177

field E. These authors derive their expressions in terms of the mobility-lifetimeproduct ðmtÞmin of the minority carriers, which can be determined from the SSPGmethod and related to L.

More recently, Schmidt and Longeaud [16] developed a generalized derivation ofthe solution of the SSPG equations at low E. They identified the shortcomings in theprevious derivations which differ from the numerical solution. Their approach alsoallows the SSPGexperiment to beused for the density-of-states (DOS) determination.

An important aspect is the above-mentioned association of L with the ðmtÞmin

product of the minority carriers. Together with the majority-carrier mobility-lifetimeproduct ðmtÞmaj at the same G from the photoconductivity sph via sph ¼ eGðmtÞmaj,where e equals 1.6� 10�19C, the excess-carrier properties canbe related to each otherwith the aim to consistently describe them in relation to models for recombination,the role of the Fermi level Ef, and the relevant DOS in the band gap [17].

The above brief and not complete description of SSPG-related aspects indicatesthat both the theoretical understanding and the variety of applications have evolved inthe last 20 years. While some treatments point to modifications that may be neededfor the correct determination of L and the corresponding ðmtÞmin, the originalformulation and analysis by RZW is still often used as deviations are consideredto be small.

8.2Basic Analysis of SSPG and Photocurrent Response

8.2.1Optical Model

Figure 8.1a sketches the arrangement of the SSPG experiment. Two coherent planewaves, which originate from one laser beam that has been split into the beams L1 and

L1

L’1

L2

L’2

’

’ ’

air

semiconductor

(a)

L1

L’1

L2

L’2

’’’

air

semiconductor

(b)x

z

Figure 8.1 Schematics of the interference experiment with two plane waves L1 and L2 withwavelength l. The coordinate system is sketched.

178j 8 Steady-State Photocarrier Grating Method

L2withwavelength l, impinge symmetrically onto a semiconductorwhere they sufferrefraction according to Snell�s law. The angle between the two beams changes from q

in air to q0 in the semiconductor with refractive index ns. The wavelength changesfrom l to l0 where l0 ¼ l nair/ns. These relations hold accordingly when thesurrounding medium is not air but glass. The plane wave of L1 can be describedby the electric field E1

! ð~rÞ ¼ E10!

exp ik1 �~r�ivtð Þ in air and by E01

!~rð Þ ¼

E010

!exp ik01

! �~r�ivt� �

in the semiconductor, where the variables have their typicalmeanings. A similar expression holds for E2

!~rð Þ of L2.

In Figure 8.1b, L1 is perpendicular to the semiconductor surface so that norefraction occurs.

The local photon flux W in the semiconductor is related to the square of the localelectric field by

W ~rð Þ / E01

!~rð Þþ E0

2!

~rð Þ�� 2 ¼ E0

10!

exp ik01! �~r�ivt

� �þ E0

20!

exp ik02! �~r�ivt

� �� 2ð8:1Þ

where ~k�� is given by (2pn/l) andwe assume low absorption so that there is negligible

decay of the photon flux. It is then found from Eq. (8.1) that in the z-direction thevalue ofW is constant. In the x-direction the influence of the refractive-index changecancels. The sinusoidal modulation in the x-direction, sketched in Figure 8.2, whichforms a so-called grating, has a grating period L given by

L ¼ l=½2sinðq=2Þ� ð8:2Þ

Assuming E10! ¼ E0

10!

and for a generation rateG�aW, independent in z, leads toa variation of G in x according to

G ~rð Þ ¼ GðxÞ ¼ ðG1 þG2Þþ 2c0ðG1G2Þ0:5 cosð2px=LÞ ð8:3Þ

where G1¼aW1, G2¼aW2. The additional parameter c0 has been introduced hereby RZWas a grating-quality factor in order to account for a nonideal grating becauseof optical scattering, nonideal coherence, ormechanical vibrations. The amplitude ofthemodulation 2c0 G1G2ð Þ0:5 is typically larger thanG2 so that regions exist in whichG(x) with the two beams is less than G1 only.

L1 L2

air

semiconductor

x

z

Figure 8.2 Schematic of the interference fringes between coplanar electrodes. The parallel planesof constant photon flux are perpendicular to the air–semiconductor interface.

8.2 Basic Analysis of SSPG and Photocurrent Response j179

For the more explicit G profile within the semiconductor with the absorptioncoefficient a one obtains [18]

Gðx; zÞ ¼ GðxÞ ¼ ðG1 þG2Þexpð�azÞ 1þ 2c0ðG1G2Þ0:5G1 þG2

cosð2px=LÞ" #

ð8:4ÞAn account of the influence of the polarization-dependent reflection on G was

given by Nicholson [19].In the asymmetric case of Figure 8.1b, in which the stationary beam hits the

sample at right angle, the analysis yields that the fringes are no longer perpendicularto the surface but that they are tilted. This may cause problems by blurring of therelevant photocarrier grating relative to the electric field that probes its distribution.

8.2.2Semiconductor Equations

Poisson�s equation consists of the local total charge r to which the free carriers in thebands, trapped charge, and/or charged defects in addition to any charge fromimpurities or dopants contribute. In terms of E, it takes the form

1edEdx

¼ ree0

ð8:5Þ

with the dielectric constants e of the semiconductor and e0. For the special case of ahomogeneous semiconductor with coplanar electrodes as in Figure 8.2, Poisson�sequation reduces to r¼ 0 and E is given by the ratio of the applied voltage and theelectrode distance.

In addition, with homogeneous free-carrier distributions the current densitiesare determined by the electric field so that the current equation reduces to

j ¼ jn þ jp ¼ enmn þ epmp� �

E ¼ sn þ sp� �

E ð8:6Þ

with the free electron (hole) density n (p) and with the conductivities of the electrons(holes) sn (sp). The index denotes the respective carrier type of the electrical currentdensity j and the extended-state mobility m.

The continuity equations with div jn¼ div jp¼ 0 become

G ¼ R ð8:7Þwith the recombination rate R.

The special case under almost homogeneousG is used for the determination of themobility-lifetime (mt)maj product of the majority carriers. The measured photocur-rent is assumed to be determined by one carrier type, say electrons. It is then given bythe difference of the total to the dark current density. The photoconductivityis determined from Ohm�s law jph¼ sphE and it is determined by the excesselectron density nph.


The recombination lifetimes of the free electrons and holes are defined by [20]

R ¼ ntot�nthtn

¼ nphtn

¼ ptot�pthtp

¼ pphtp

ð8:8Þ

where ntot (ptot) is the total electron (hole) density under illumination and nth and pthare the thermal equilibrium values. With Eq. (8.7) and the substitution of nph itfollows that

ðmtÞn ¼ mntn ¼ spheG

� �ð8:9Þ

which gives experimental access to themajority-carrier ðmtÞmaj product. Correspond-ing equations hold if holes are the majority carriers.

8.2.3Diffusion Length: Ritter–Zeldov–Weiser Analysis

RZWdevised the �SSPG technique for diffusion length measurement in photocon-ductive insulators.� Their starting point of analysis are the transport equations ofSection 19.2 under the assumption of ambipolar transport, so that only one carriertype is analyzed, and charge neutrality. They reduce to

1edd x

�eDd nðxÞd x

� �¼ GðxÞ�RðxÞ ð8:10Þ

where D is an effective diffusion coefficient and where n(x) is the sum of thecorresponding homogenous excess density n0 and the modulated density n(x)¼n0 þ Dn(x). Splitting off the homogeneous terms results in a differential equationfor Dn(x) according to

Dd2 DnðxÞ

d x2¼ gðxÞ�DnðxÞ

tðnÞ ð8:11Þ

with gðxÞ ¼ 2c0ðG1G2Þ0:5cosð2px=LÞ and the carrier lifetime t. The ansatzDnðxÞ ¼ Dn0cosð2px=LÞ in phase with g(x) results in

Dn0 ¼ 2c0ffiffiffiffiffiffiffiffiffiffiffiG1G2

p

1þ 2pLL

� �2 t ð8:12Þ

where the ambipolar diffusion length L is introduced by

L ¼ffiffiffiffiffiffiDt

p: ð8:13Þ

It is necessary to correctly account for a possibly G-dependent t in Eq. (8.12).Experimentally, the relation between photoconductivity or excess-carrier density andthe generation rate is usually given by

n / Gc ð8:14Þ


with the power-law exponent c. Onemay expand Eq. (8.14) to eliminate t in Eq. (8.12)so that with t� cn0= G1 þG2ð Þ one determines

Dn0 ¼ 2c0ffiffiffiffiffiffiffiffiffiffiffiG1G2

p

1þ 2pLL

� �2

cn0G1 þG2

: ð8:15Þ

The local conductivity which is proportional to n(x)¼ n0 (1 þ Dn(x)/n0) may nowbe written as

sðxÞ ¼ sðL1 þ L2Þ½1þA cosð2px=LÞ� ð8:16Þ

where n0 is related to sðL1 þ L2Þ. The photocarrier grating amplitude A is given by

A ¼ 2cc0ffiffiffiffiffiffiffiffiffiffiffiG1G2

pG1 þG2

cg ð8:17Þ

in which the last term

cg ¼1

1þ 2pLL

� �2 ð8:18Þ

with L and L describes the influence of the grating.So far, the diffusion length L is related to the local conductivitys(x).What is needed

for the electrical measurements is a relation between L and themeasured current. Ina next phenomenological step, RZW thus assumed that themagnitude of the averagecurrent density jcoh in the direction of the electric field under coherent conditions inthe presence of the grating can be modeled by a series resistor model of localresistivities or inverse conductivities, given by

jcohðL; L1 þ L2Þ=E ¼ savðLÞ ¼ LÐL0

dxsðxÞ

¼ sðL1 þ L2Þffiffiffiffiffiffiffiffiffiffiffiffi1�A2

pð8:19Þ

A problem with this approach is that, as j must be constant, E(x) must oscillate ifs(x) oscillates, in contradiction to the assumption of no space charge and a constantE.

RZW also suggested that a lock-in technique may be used to determine theparameter b that, as will be shown below, is related to the diffusion length and isdefined by RZW by

bðLÞ ¼ UcohðLÞUincðLÞ ð8:20Þ

at a given grating period. Here Ucoh(Uinc) denotes the lock-in amplifier (LIA) signalunder coherent (incoherent) beam conditions. The LIA detects the difference in thecurrents with respect to the current density j1 under the bias level of the beam L1. Thepolarization change of 90� of L1 together with the unchanged chopped L2 results in acomparison of jcohðL; L1 þ L2Þ, under coherent condition, with jincðL; L1 þ L2Þ,


under incoherent condition. In terms of current densities the parameter b is thusgiven by

bðLÞ ¼ Ucoh

Uinc¼ jcohðL; L1 þ L2Þ�j1ðL; L1Þ

jincðL; L1 þ L2Þ�j1ðL; L1Þ ð8:21Þ

The link between the experimentally determined b and the diffusion length L isadjusted by an approximation of the current densities with the assumed and alsotypically observed power-law dependence of the photocurrent density jph/Gc

and the condition that G2/G1� 1. The G terms can then be eliminated with, forexample, jincðL; L1 þ L2Þ�jincðL; L1Þ / ðG1 þG2Þc�ðG1Þc and jcohðL; L1 þ L2Þ fromEq. (8.19) so that b(L) reads

bðLÞ ¼ð1þG2=G1Þc

ffiffiffiffiffiffiffiffiffiffiffiffi1�A2

p�1

h i½ð1þG2=G1Þc�1� �

ð1þ cG2=G1Þ 1�2ðcc0cgÞ2G2=G1

� ��1

cG2=G1

� 1�2cðc0cgÞ2 ð8:22Þ

where the quadratic terms inG2/G1 have been neglected in the last step. Substitutingcg from Eq. (8.18) yields

bðLÞ ¼ 1� 2Z

1þ 2pLL

� �2" #2 ð8:23Þ

with Z ¼ cc20.

8.2.3.1 Evaluation SchemesEquation (8.23) is the central equation of the SSPG analysis: the data sets of themeasured b values at different positionsL can be analyzed with the fit function b(L)and the two fit parameters: the diffusion length L and a fit parameterZ. While L has aphysical meaning, the parameter Z is just a fit parameter. However, it can beevaluated for a self-consistency check of the analysis and the measured b data. Fromthe Z value and the separate experimental determination of c and cd, the gratingquality factor c0 can be calculated. It should be �close to 1.�

Balberg et al. [7] suggested rearranging the data according to a linear form as

1=L2 ¼ ½Z1=2=ð2pLÞ2�½2=ð1�bÞ�1=2�ð2pLÞ�2 ð8:24Þwith the ordinate values 1=L2 and the abscissa values ½2=ð1�bÞ�1=2. The diffusionlength L can be read from the extrapolation to the abscissa or from the fit parametersfor the straight line.

In our laboratory it has been customary to plot the values related to themeasured bas ð1�bÞ�1=2 on the ordinate scale by

ð1�bÞ�1=2 ¼ ½ð2ZÞ�1=2ð1=LÞ2�ð2p=LÞ2 þð2ZÞ�1=2 ð8:25ÞThis kind of linear plot was also used by Nicholson [19].


The advantage of any linear plot is that a disagreement from the theoreticalexpectation may more easily be identified by deviations from the linear behavior.Ambipolarity is not obeyed by charge separation under high electric fields [21], whichcan also be identified from a concave instead of a linear shape of the graph according toEq. (8.24). It is then still possible to evaluate the ambipolar diffusion length fromportions of the data set. Any super- or sublinear behavior depends, of course, on the plotspecification with respect to abscissa and ordinate values. Balberg [22] analyzed themeasurement data for the identification of nonambipolarity under low-field conditions.

For identifying theeffect of surface recombination, the rearrangement according to

½2=ð1�bÞ��1=2 ¼ ½L2=Z1=2�ð2p=LÞ2 þðZÞ�1=2n o

g2s ðLÞ ð8:26Þ

was suggested [18], where gsðLÞ is a function which is 1 for negligible surfacerecombination and which decreases with increasing L. In a linear plot, gs(L) thustakes effect at large ordinate values so that L can still be determined from the b data atshorter values ofL. In the case of surface recombination there is a sublinear increasein the plot.

8.2.4More Detailed Analyses

Analyses of the SSPG method are available which give a more detailed account byincluding the effects of the dark conductivity, of traps,E at low andhigh values and theaspect of space charge. They also relate, in more detail, the measured L with theminority-carrier (mt) product.

8.2.4.1 Influence of the Dark ConductivityIn a following paper [3], RZWalso took into account the effect of the underlying darkconductivity sd. The introduction of sd in Eq. (8.16) yields a slightly modified versionof Eq. (8.23).

With the dark-conductivity coefficient cd, given by

cd ¼sph

sph þ sd; ð8:27Þ

Equation (8.23) reads

bðLÞ ¼ 1� 2ccdc20

½1þð2pL=LÞ2�2 ¼ 1� 2Z

½1þð2pL=LÞ2�2 ð8:28Þ

where now Z ¼ ccdc20. Formally, the fit functions (8.23) and (8.28) with the two fit

parameters L and Z are the same.

8.2.4.2 Influence of TrapsIn the original RZWapproach, assumptions have beenmade. Especially with respectto semiconductors with band tails, it will be necessary to introduce the contributionsof the trapping states and of the trapped charge. This is often accomplished by the


introduction of effective or drift mobilities in which the reduction in mobilitythrough the trapping and emission processes is expressed. We denote the referenceto the total charge by a capital letter of the index so that the drift mobilities read

mN ¼ mnnN

mP ¼ mppP

ð8:29Þ

with the total density N and P. A ratio qn

N ¼ nþ ntnN

¼ qnð8:30Þ

can be defined with the trapped-electron density nt and with a corresponding relationfor the holes.

Poisson�s equation then takes the form

1edEdx

¼ P�Nee0

ð8:31Þ

where the right-hand side is zero under charge neutrality with N¼P.The effective diffusion coefficient DN may be introduced via the condition

kTemn

dndx

¼ DNdNdx

ð8:32Þ

which yields [3]

DN ¼ kTemn qn þN

dqndN

� �ð8:33Þ

where k is the Boltzmann constant and T is the temperature.The ambipolar diffusion equation with DN¼DP and a common lifetime tR reads

DNPd2 DNdx2

�DNtR

þDGðxÞ ¼ 0 ð8:34Þ

which is similar to Eq. (8.10). The ambipolar diffusion coefficient DNP is a rathercomplicated construct (see Eq. B6 of Ritter et al. [3]). For Boltzmann statistics

DNP ¼ 2 kTe

mNmPmN þmP

. It is noted that the parameters tR ¼ tn=qn ¼ tp=qp and DNP

contain the common response of the total electron and hole contributions and thuspertain to the ensemble of free and trapped carriers.

Upon inspection of the diffusion equation (Eq. (8.34)), the diffusion length L isdefined by

L ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiDNPtR

q: ð8:35Þ

The same b(L,L) function as in Eq. (8.28) results from the analysis.


An effective lifetime tq is defined by Balberg [10] who obtains

L ¼ ffiffiffiffiffiffiffiffiffiffiffiffi2Dqtq

p ð8:36Þ

with the effective diffusion coefficient Dq. It can be shown that DNPtR ¼ 2Dqtq.

8.2.4.3 Minority-Carrier and Majority-Carrier Mobility-Lifetime ProductsTo relate the diffusion length L with the individual mobilities and the individuallifetimes and their products of Eqs. (8.8) and (8.9), we rewriteDNP and tR according to

L ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiDNPtR

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kTe

mntnmptp

mntn þ mptp

sð8:37Þ

We can then associate either the electrons or the holes with the majority andminority carriers. Writing (mt)min¼mmintmin and (mt)maj¼mmajtmaj with the free-carriermobilities mmin and mmaj and the free-carrier recombination lifetimes tmaj andtmin embraces the respective mobilities and lifetimes. The corresponding equationfor L is

L ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kTe

ðmtÞminðmtÞmaj

ðmtÞmin þðmtÞmaj

sð8:38Þ

For the case that (mt)min� (mt)maj, Eq. (8.38) takes the form

ðmtÞmin � e2kT

L2 ð8:39Þ

which allows the separate determination of (mt)min. As a rule of thumb, a room-temperature value of L¼ 200 nm corresponds to (mt)min¼ 8� 10�9 cm2V�1.

The factor of 2 stems from the common contribution of the full electron and holedensities in an intrinsic-type semiconductor, with n� n0, p� p0, to the recombi-nation rate. This relation is also deduced independently without SSPG backgroundfor the surface-photovoltage experiment or collection lengths in solar cells [23]. If theabove inequality is notmet, the respective two unknowns (mt)min and (mt)maj can onlybe determined with the additional photoconductivity measurement and

sph ¼ eG ðmtÞmin þðmtÞmaj

h ið8:40Þ

Equations (8.38) and (8.40) with the two unknowns allow the (mt)min and (mt)maj

determination but no straightforward correlation whether the electrons or the holesare the majority/minority carriers.

In an analysis based on solutions for the free-carrier densities, Shah et al. [24]deduce the relation

L ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCkTe

ðmtÞminðmtÞmaj

ðmtÞmin þðmtÞmaj

sð8:41Þ


which is similar to Eq. (8.38). Here, C is a sample-dependent factor between 1 and 2which depends on the photogeneration-rate dependence of the excess-carrier den-sities. For n-type a-Si:H samples, Shah et al. [25] estimate C� 1 so that a discrepancyof a factor of 2 exists in relationwith Eq. (8.39). ForG-independent lifetimes,C can beshown to be equal to 2.

The factor 2 of Eq. (8.39) also appears in the treatment by Schmidt and Long-eaud [16] who suggest to fit Eq. (8.28) and apply Eq. (8.39) for an apparent ðmtÞappmin

which can then be corrected.Abel et al. [15] derive the relation

bðLÞ ¼ 1�2c

1þ 2pL

2kTeðmtÞp

mntreld

mtð Þn

� �1=2 !2

1þ 2pkTe

2ðmtÞpþmntreld

h i� �1=2L

!2

þ 2pkTe

ðmtÞpmntreld

h i1=2� �1=2L

!4" #2

ð8:42Þsimplified for the case that electrons are the majority carriers and that the E-relateddrift terms can be neglected. If the term mnt

reld with the effective dielectric relaxation

time treld can be neglected, because treld ¼ ðn=NÞtd � td, Eq. (8.42) reduces to theRZW relation (Eq. (8.23)) combined with Eq. (8.39) and ðmtÞmin ¼ ðmtÞp, includingthe factor of 2.

Hattori et al. [14] suggest using additional information from frequency-dependentlifetime measurements for a correction of L from the RZW analysis.

8.3Experimental Setup

Figure 8.3 sketches two of the possible arrangements for themeasurements of thebparameter as a function of the grating periodL. The two beams hit the sample S inthe gap between the electrodes. In Figure 8.3a the sample ismoved on a linear stage

P1P2

M2

M1

BS

H

C

L

S

(a)

P1 P2M2

M1

BS

H

CL

S

(b)

Figure 8.3 Schematics of experimental setups. The double arrows indicate that the sample ismoved in (a) and an optical table is moved in (b)1) in order to change the angle between the twobeams.

1) The set-up of b) has been devised by C. Longeaud and R. Br€uggemann at LGEP, Paris and at theUniversity of Oldenburg.

8.3 Experimental Setup j187

to change the angle between the two beams, split by the beam splitter BS. Themirrors M1 and M2 are rotated according to the sample position. The half-waveretardation plate H is positioned in the strong beam, alternatively placed betweenBS andM1, in order to rotate the polarization. If H was in the weak beam, choppedby the chopper C, an intensity change by H-rotation would directly enter into themeasured signal. In the strong beam, any intensity changes from the rotation of Hwould only change the light bias slightly. The optional polarization filters define thepolarization and can also be used for a variation ofW by rotation of the polarizer P2,keeping P1 fixed.

The linear polarization is typically changed from 0� to þ 90� or �90�. A changefrom �45� to þ 45� can also be used in order to reduce the polarization-dependentreflection variations of L1. A change of reflection upon rotating the polarization of L1is usually not taken further into consideration as it only slightly changes the bias level.

Figure 8.3b is different through themovement of an optical table onwhichmost ofthe optical elements are arranged. The sample S isfixedwhich gives a different optionfor the cryostat, compared with Figure 8.3a. The advantage here is that the length ofthe light path is almost the same for every L.

Arrangements with one perpendicular beam can also be found in the literature.Niehus and Schwarz [26] achieve long L by introducing an additional beam splitter.One beam is transmitted and the second beam is reflected on the beam splittersurface close to the other�s transmission position which defines small angles q.

A glass prismwas employed by Nowak [27] while a compact goniometer-like setupis used at Utrecht University where the sample is rotated with no linear movementinvolved [28].

ForT-dependentmeasurements, a suitable cryostat is incorporated into the setups.One should check that the grating quality factor is not reduced by the influence of thecryostat window or additional vacuum-pump-related vibrations. For very thin sam-ples, a cold finger with a hole may be helpful as a beam dump for the transmittedlight.

The photocurrent is typically measured by an LIA. The photocurrents undercoherent and incoherent illumination conditions for Eq. (8.21) could also bemeasured in the steady state, for example, by an electrometer but usually the LIAmeasurements will result in lower experimental errors.

Figure 8.4 sketches a strategy for the alignment of the two beams if their diametersare smaller than the electrode length. After adjusting L1 and L2 independentlyhorizontally for maximum photocurrent signal, the vertical position must beoptimized for overlap. A slight mismatch in the vertical direction may be difficultto assess by inspection with the eye. However, better alignment can be achieved byadjustment of the vertical position of L2, as indicated by the double arrow inFigure 8.4a, and by monitoring the photocurrent or the LIA signal under coherentillumination. Best overlap is achieved when the LIA signal shows a minimum formaximum interference, as sketched in Figure 8.4b if b is positive.

For negative b, the phase shift of 180� must be taken into consideration.Depending on the experimenter�s and the LIA options, the negative values of thereal part or the amplitude can be maximized in this case.


One can always make a consistency check for the fitting results for L and Z. Aspointed out in the literature [7, 18], the determined value of thefit parameterZ shouldbe reasonable when comparing its value with Z ¼ ccdc

20 of Eq. (8.28) with the

experimentally determined c and cd and a grating quality factor c0 close to 1. Duringthemeasurements, a check of an approximately constant jinc(L) or LIA-signalUinc(L)at the different grating periods is helpful. For the different measurement positions,the measured b data should decrease with increasing L. It may also be helpfulinitially, after having set-up the experiment, to compare the b values measured withtwo different setups in order to check the data [29, 30].

8.4Data Analysis

The experimental results comprise a set of values of the parameter b, measured ata number of grating periods L. The (L,b) data can be evaluated according to oneof the schemes from Eqs. (8.23) to (8.28). A comparison of the evaluation methodsis discussed in [31]. Figure 8.5 illustrates graphical representations for long andshort L. Through the large value for L¼ 200 nm and Z¼ 0.8, the b values inFigure 8.5a cover a wide range and do not even reach the limit 1–2 Z¼� 0.6 forlong L. The b values with Z¼ 0.1 cover only a small range between 0.8 and 1; forL¼ 200 and 20 nm. This makes a precise L-determination difficult. There is almostno variation inb forL¼ 20nm, even forZ¼ 0.8.Ritter et al. [3] consider thatL� 20nmcan just be determined as a lower limit. It must be possible to measure at leastsome variation in b at the shortest experimental L to determine L. It is noted thatresults for determining b for L< 500nm are difficult to achieve experimentally.

Figure 8.5b with the linear plots shows that the two cases for Z¼ 0.1 can be moreeasily distinguished compared to Figure 8.5a but it has been pointed out [31] that any

(a)

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Figure 8.4 Schematic of the adjustment of L1 and L2 between the electrodes, (a). L1 is fixed. Theverticalmovement of the lightly shaded L2 results in the typical signal at the LIA, sketched in (b). Theposition of minimal signal results in optimized overlap, shown here for theL range with positive b.

8.4 Data Analysis j189

small variation in the large b values would lead to a large error because of the (1�b)term in the denominator of the linear plots.

The two full lines for L¼ 20 nm and the two dashed lines for L¼ 200 nmextrapolate to the same ordinate value, respectively, which represents L.Figure 8.5b has thus very steep straight lines for L¼ 20 nm because the ordinateintercept is a large negative value.

Figure 8.6 illustrates the effect ofmeasurement errors in the determination of b. Itis more likely that errors in the beam adjustments yield b values that are rather toolarge due to incomplete overlap or nonideal grating. The artificial data points with thefull circles in Figure 8.6 represent the ideal b(L) variations in Figure 8.6a and b withL¼ 140 nm. The open symbols show the shift by increasing b by 15%.

The dashed curve in Figure 8.6a shows the fit to the open symbols and is onlyslightly different compared to the full linewhich is the fit of the full symbols. Becausea constraint in a (b,L) fit is given by b(L¼ 0)¼ 1 according to Eq. (8.28), the fitting

121086420

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Figure 8.5 Analytical b–L plot (a) and linear plot (b) for L¼ 200 and 20 nm and Z¼ 0.1 and 0.8.The b values are almost constant for L¼ 20 nm (a). The almost vertical lines of the data pointscorrespond to L¼ 20 nm (b).

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Figure 8.6 The symbols represent artificialmeasurement data (full symbols are ideal). A 15% erroris imposed on b for L> 500 nm (upward shift, L¼ 140 nm, Z¼ 0.8, open symbols). The dashedcurves show the fits in (a) and (b).


procedure according to Eq. (8.28) is found to be less affected by too large a value of bat short L. The apparent L of 144 nm is close to the original value of 140 nm as is Zwith Z¼ 0.78 instead of 0.8.

The nominally straight line in Figure 8.6b shows some curvature in the range ofshortL, that is, large ordinate values. Restricting the linearfit in the range of abscissavalues<4 results in a diffusion length L¼ 169 nm, which is quite a large discrepancyof 29 nm. Only if one restricts the fitting to a more linear region with abscissa values<2, an apparent value of L< 150 nm is achieved.

Figure 8.7 shows typical experimental data of a device quality and ahigh-defect a-Si:H sample in the b(L) plot (Figure 8.7a) and in a linear plot (Figure 8.7b). There is verygood agreement between the two evaluation schemes with L¼ 154 nm (and a goodvalue of c0¼ 0.95 from Z and c) and L¼ 41.5 nm.

For the evaluation of the experimental data, the following analysis may beinstructive.With saym values of (b,L), one can estimate the error in L by performingseveral fits with variations of (m� 1) or (m� 2) data points to determine the variationin L values. Single measurement data should be skipped, if too large a disagreementbetween the two evaluation schemes is observed. Especially for large values of b, anyof the linear plots with the term (1� b) in the denominator suffers from errorenhancement and should be handled with caution.

It is noted at the end of this section that the sample alignment in the SSPGexperiment is quite robust against small variations in the sample orientation.Experimental tests in our laboratory have shown that such rotations of the sampleor the sample holder by a few degrees, do not lead to a significant variation in themeasured b values. Unintentional misalignment of the sample is thus not severe.Any larger misalignment is easy to spot by the eye.

Substantial intentional rotation of the sample leads to a decrease of the effect of themodulation of the photocarrier grating. Balberg et al. [7] rotated the sample by 90� sothat the photocurrent is parallel to the interference fringes. Meeting the require-ments of the photocurrentmeasurement under incoherent conditions in this way, nohalf-wave plate is needed.

2.82.42.01.61.20.80.40.0

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Figure 8.7 Experimental data in different representations in (a) and (b) for two undoped a-Si:Hsamples. There are data points with both negative and positive values of b.

8.4 Data Analysis j191

8.5Results

This chapter presents and reviews results on a number of thin-film semiconductors.The limited and partially personal account that is given for each semiconductor isrestricted to the photoconductive properties in relation to the SSPGmethod, that is,the minority-carrier properties in the steady state. As these are linked with themajority-carrier properties, the latter will also be discussed in comparison whenappropriate. Typically, Eq. (8.39) is used to convert the experimental SSPG-derived Linto (mt)p.

8.5.1Hydrogenated Amorphous Silicon

Hydrogenated amorphous silicon (a-Si:H) has been in the focus of SSPGapplicationssince the early work by Ritter et al. [3, 32] The main issues in a-Si:H-related researchwith SSPG deal with the identification of deficiencies of the method, the physics ofrecombination and its relation to the DOS, the effect of doping on the excess-carrierproperties, and the role of the Ef position in general.

8.5.1.1 Temperature and Generation Rate DependenceThemonotonous decrease of (mt)p with decreasing T [33] in Figure 8.8 is a signaturein a-Si:H. To demonstrate at least one example of numerical-modeling results of thephotoresponse and the minority-carrier properties in particular, Figure 8.8b shows

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Figure 8.8 T dependence of (mt)p of an undoped a-Si:H sample at different G, after 33 (a).Numerical-simulation results with a free-hole mobility of 1 cm2 V�1 s�1 which reproduce thedecrease of (mt)p with decreasing T and increasing G, after [34] (b).


the simulated (mt)p that can roughly reproduce the experimental findings ofFigure 8.8a [34].

At any T, (mt)p drops with increasing G. There has been some controversy in theliterature about the consequences of the G dependence with respect to DOS modelsin the band gap of a-Si:H. Balberg and Lubianiker [35] suggested that one correlateddangling bond (DB) level is not sufficient to explain the experimental data while theNeuchatel group [36] pointed out that a proper balancing of charge in one correlatedDB and in the band-tail states can also explain the experimental findings.

8.5.1.2 Surface RecombinationThe application of Eq. (8.26) is illustrated in Figure 8.9 with data, measured with aHeNe laser and an Ar laser [18]. From the b versus L plot in Figure 8.9a theidentification of deviations is not so easy, given that there is always an experimentalerror. A reduced Lwill be determined if all the circle-data points are taken for the fit.

In Figure 8.9b, there is a curvature in the data for the Ar illumination especially incontrast to the linear behavior of the squares. It is noted that the b data at the longerLcoincide from the two measurements. This means, as noted by Haridim et al., that Lcan be determined in the usual way for longerL. For the HeNe measurements withthe larger absorption depth all data points are suitable.

Haridim et al. also determined the two surface recombination velocities at the air/a-Si:H and the a-Si:H/substrate by illuminating from the air and substrate sides andby fitting the experimental data with Eq. (8.26) with suitable gs(L).

8.5.1.3 Electric-Field InfluenceIn the early publications on SSPG there has been concern with respect to the electric-field limitations in two ways: Emodulation due to space charge and phase shift in the

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Figure 8.9 Digitized data from Haridimet al. [18] in ab versusL plot (a) and in the linearrepresentation (b). The b values weredetermined under HeNe and under Arillumination. A force fitting of the data results in

a 40 nm different L of the latter. Note that atsmall abscissa and ordinate values in (b) thedata points of the two kinds of experiments falltogether.

8.5 Results j193

modulation of n(x) and p(x) because E pulls these apart. Accounts on the �high�-Eonset, the value at which such a deviation from the ideal behavior according toEq. (8.28) occurs, have been summarized by Schmidt and Longeaud [16]. They pointout that depending on the theoretical approach, different E values for the low-fieldlimit have been determined. Experimentally, one may study the b dependence on Eand take a constant value of b as an indicator of the low-field limit. At higher fieldvalues, a drift length has been determined from the field dependence [3, 21].

The analysis by Abel et al. [15], which includes both the effect of space charge andexternal electric field, has been applied to study the b dependence onL and E. Goodagreement could be achieved between the experiment and thefit function (Eq. (8.42))for b(L,E) [37]. Accounts on the effective dielectric relaxation time and space chargecould thus be given.

8.5.1.4 Fermi-Level PositionThe aim of SSPG measurements with a-Si:H films was also to add complimentaryinformation from the minority-carrier side to the majority-carrier propertiesfrom sph.

Shortly after the SSPG proposal, the properties of boron-doped films were studiedby Yang et al. [6]. Low-concentration boron doping was shown to lead to an increasein L. With higher doping level, L dropped drastically. These results were interpretedas a change in the minority-carrier type. For the higher doping levels and the shift ofEf toward the valence-band edge, the electrons become the minority carriers and theincreasing defect density upon doping then leads to the reduction in L. Animportant observation was a benefit in L at small shifts of Ef toward mid-gapwhich was thought to be beneficial for the net i-layer carrier collection in a-Si:Hbased pin solar cells.

In undoped a-Si:H with Ec�Ef in the range of 0.65–0.75 eV, where Ec�Ef iscalculated from Ec�Ef¼ kT ln(s0/sd), with the prefactor s0¼ 150–200V�1 cm�1,the (mt)maj values are typically a factor of 50–100 higher than the (mt)min [17]. Thereis thus quite a variation in Ef for undoped a-Si:H samples and this variationexpresses itself in a variation in the mobility-lifetime products [38]. Typically,(mt)maj increases with decreasing Ec�Ef [39] accompanied by a decrease in (mt)min.This anticorrelated behavior is also seen when the electrons are the minoritycarriers in p-type a-Si:H, where (mt)min decreases with increasing Ec�Ef> 0.8eV [36, 40, 41, 46].

In a field-effect configuration, the Fermi level in the conduction channel can bechanged by the gate voltage. This effect was exploited by Balberg and Lubianiker [36]and also Schwarz et al. [42] to study the Ef -dependent variation of sph and Lwithoutbeing hampered by a doping-dependent increase in the DB density.

8.5.1.5 Defects and Light-Induced DegradationThe time-dependent decrease of the photocurrent upon illumination is a signature ofmany a-Si:H films. Any increase in the defect density should also have an influenceon the minority-carrier properties so that SSPG may reveal the underlying recom-bination physics. It is known in the literature thatEf shifts towardmid-gap upon light-


soaking [43]. In view of the Fermi level related discussion above, it is illustrative topoint out the possibility that L may increase upon light-soaking, as illustrated inFigure 8.10 [44].Here, light-soakingwith the concomitant increase inEc�Ef leads toan increase in L while new defects are being created. Defects become less negativeand thus less attractive for hole capture, so that the minority-carrier propertiesimprove.

It is usually reported that Ldoes not change in thefirst hours of light-soaking –withdifferent interpretations.Wang and Schwarz [45] reported that for a-Si:H filmswith alowDBdensity, themain recombination channel is via band-tail states. An increase intheDBdensity upon illumination thus does not lead to a drop in L in the early stage ofdegradation.

The interpretation by the Neuchatel group [21] was different in that they assumedthat the main recombination channel is via DB. Because of conversion of the newlycreated DB into neutral DB they argued that these become then ineffective as themain capture process of holes is capture by negatively charged DB.

Sakata et al. [46] argued that errors have been made in the determination of L byWang and Schwarz [45] because the measured L by SSPG should be corrected. Theymaintain that there is a drop in the true diffusion length also in the initial stages ofdegradation.

With respect to the influence of the (mt) products on Ef and the reported change ofEf upon light-soaking, it appears necessary to always monitor the dark current, too.This has been done byMorgado [47] who determined (mt)p, (mt)n, and the Ef variationupon light-soaking and also annealing.

8.5.1.6 Thin-Film Characterization and Deposition MethodsSSPG has also been applied in order to characterize the samples that were depositedby newly developed deposition methods, for example, hot-wire chemical vapordeposition (HWCVD). For example, Mahan et al. [48], Unold et al. [49], Lubianikeret al. [50], and Feenstra [51] presented SSPGresults with L in the range of 150–170 nmfor typical device quality a-Si:H. These values were correlated with depositionparameters and, for example, with the hydrogen content of the films. Under specific

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Figure 8.10 Increase of L upon light-soaking of a-Si:H, after [44].

8.5 Results j195

deposition conditions, the so-called polymorphous silicon can be prepared. It hasbeen shown to exhibit favorable minority-carrier properties [52], which enhances thered response in solar cells.

8.5.2Hydrogenated Amorphous Silicon Alloys

One of the first reports on the application of SSPG was given by Bauer et al. [4] onhydrogenated amorphous silicon germanium (a-Si1�xGex:H) alloys. Typically, bothsph as a signature of themajority carriers and L as a signature of theminority carriersdecrease with increasing x but they are affected in a different way by the concomitantshift of Ef toward mid-gap.

Abel et al. [53] report on a decrease in (mt)p with increasing x in the range of x up toabout 0.43. The values of (mt)p stay>10�9 cm2V�1, that is, L of about 100 nm. F€olschet al. [54] report slightly larger values in this x-range. These authors report a strongdecrease in L for x larger than 0.5 at which the combination of increasing sd anddecreasing sph also makes L very difficult to measure.

The data by Gueunier et al. [55] on polymorphous silicon germanium alloys (pm-Si1�xGex:H) range between 100 nm and 130 nm, that is, (mt)p between about 10�9

and 3� 10�9 cm2V�1, for x 0.35 at which the optical gap E04 (defined as the energywhere the absorption coefficient a (E04)¼ 104 cm�1) was 1.5 eV. Light-soaking wasshown to only affect the majority-carrier (mt)n product. An L-value of 100 nm wasreported by Bhaduri et al. [56] in the range x¼ 0.4–0.5.

Yang et al. [57] determined L for hydrogenated amorphous silicon carbide films(a-Si1�xCx:H) of different band gaps and different feed stock gases. These authorsfind a correlation between the increase ofL andbetter solar cell properties. Theymakeit clear though that, as mentioned above, the sole value of L may not be the onlyindicator of the defect density of a sample. They deduce from the complementarysd- and sph-results that Ef -related occupation of defects may have lead to highervalues of L of some of their a-Si1�xCx:H samples.

Mohring et al. [33] reported on the T-dependent andG-dependent (mt) products ofmajority andminority carriers in optimized a-Si1�xCx:H with an optical band gap upto 1.95 eV. For the alloys, there is a monotonous decrease of (mt)p with decreasing T,similar to the findings for a-Si:H. At any T, (mt)p decreases with increasing alloycontent which has been related by Mohring et al. to the previous observation ofincreased DB density with increasing x. For a-Si1�xCx:H with a band gap of 1.94 eV,the room-temperature values for (mt)p were (1.2)� 10�9 cm2V�1 and with about afactor of 10 higher for (mt)n.

8.5.3Hydrogenated Microcrystalline Silicon

The application of SSPG to hydrogenated microcrystalline silicon (mc-Si:H) is ratherstraightforward on the one hand as, for example, sample geometries and relevantabsorption coefficients are not significantly different. On the other hand, there are


some obstacles involved because of the typically higher sd and some possible effect ofsurface roughness, which reduces c0.

Goerlitzer et al. [58] and Droz et al. [59] find smaller Z in mc-Si:H because of thesmall ratio sph./sd. With respect to optical scattering, Zwas shown to increase afterpolishing the rough surface of mc-Si:H samples [32].

The Tdependence of (mt)p of mc-Si:H was studied by Br€uggemann and Kunz [60],and also by Balberg [61]. For the analysis, they also performed numerical simulationsin order to correlate the experimental and simulation results.Otherwork is devoted torelating Lwith the deep defect density [29, 62–64] and the crystalline volume fractionand the deposition technique [29, 65].

There are indications that transport is inhomogeneous in mc-Si:H with respect tothe transport path parallel or perpendicular to the film-growth direction [66].

8.5.4Hydrogenated Microcrystalline Germanium

Badran et al. [67] reported on L in microcrystalline germanium (mc-Ge:H). Because Lwas quite short, the conclusion was that mc-Ge:H appears suitable for diodeapplications as a sensor, as was demonstrated previously, under reverse bias butnot as a solar cell.

8.5.5Other Thin-Film Semiconductors

Thin-film chalcopyrite semiconductors were characterized soon after the SSPGmethod was proposed. Balberg et al. [5] determined the majority and minority-carrier properties of CuGaSe2 thin films.Here, electrons are theminority carriers forwhich the authors determined L of 115 nm. The reliability of the data was increasedby a self-consistency check by comparing the fit parametersZ from SSPGand c fromsph. From the intensity dependence of (mt)p it was concluded that there is a sharp dropin the DOS close to the band edges – in contrast to a-Si:H.

Menner et al. [68] report SSPG results on CuGaSe2 thin films and determined thehighest L for a Cu/Ga ratio at 0.95, that is, close to the stochiometric point. SSPGmeasurements in theCu-rich regionwere impossible because of the highsd.Menneret al. also aimed at revealing inhomogeneity in the growth process by application of alaser with a short absorption depth and by illumination from either the air/film andthe substrate/film side. They concluded that a better crystal structure exists close tothe substrate.

BelevichandMakovetskii [69]giveanaccountof L measurements inpolycrystallinep-type CuInSe2. From their analysis they determine G-dependent minority-electrondiffusion lengths between 360 and 420 nm, decreasingwith increasingG. They relatethe values of L to the dimension of the crystallites in the polycrystalline film.

Generally, it is noted that one should not be too optimistic with respect to theapplication of SSPGto theCu(In,Ga)Se2 system as itmay be hampered by the largesdin Cu(In,Ga)Se2, for which Zweigart et al. [70] determined L for some chemical

8.5 Results j197

compositions, only. Because the dark-current activation energy is also low, a decreaseof T will only help if sph does not decrease too much as well.

8.6Density-of-States Determination

Schmidt and Longeaud [16] developed an analysis of SSPG with emphasis on theDOS determination at the quasi-Fermi energy of the majority carrier. For the energyscale, the quasi-Fermi energy can be shifted by a variation of TorG, so that a scan inenergy can be performed. The experimentally determined DOS values are given by acombination of photocurrent and SSPG measurements. The application of themethod was demonstrated on a-Si:H [16] and mc-Si:H [71].

8.7Summary

We have given an account of the SSPG method with some emphasis on the setup,on the necessary and complementary measurement steps, and on consistencychecks with the aim to obtain a more complete picture on the excess-carrierproperties. The evaluation of the experimental data on the basis of the RZWapproach provides the basis of determining L when the experiment is performedunder the necessary condition of small modulation depth at low E, short enoughdielectric relaxation time and sufficiently high photocurrent values with distinctlydifferent values between (mt)min and (mt)maj. A simple relationship between L and(mt)min can then be applied.

Reference has beenmade to alternative and additional analyses by Ritter et al. [3],Balberg et al. [10, 11], Li [12], Hattori et al. [14], Abel et al. [15], Schmidt andLongeaud [16], and shah et al. [24] which cover possible shortcomings, limitations,and additional aspects. Some of the relevant literature on the applications of SSPGon thin-film amorphous and microcrystalline semiconductors and on chalcopyriteand other semiconductors was sketched. The examples illustrate the potential ofthe method for the characterization of the photoelectronic properties of thin-filmsemiconductors.

References

1 Ritter, D., Zeldov, E., andWeiser, K. (1986)Steady-state photocarrier gratingtechnique for diffusion lengthmeasurement in photoconductiveinsulators. Appl. Phys. Lett., 49, 791.

2 Ritter, D. andWeiser, K. (1986) Ambipolardrift-length measurement in amorphous

hydrogenated silicon using the steady-state photocarrier grating technique.Phys.Rev. B, 34, 9031.

3 Ritter, D.,Weiser, K., andZeldov, E. (1987)Steady-state photocarrier gratingtechnique for diffusion-lengthmeasurement in semiconductors – theory


and experimental results for amorphous-silicon and semiinsulating GaAs.J. Appl. Phys., 62, 4563.

4 Bauer, G.H., Nebel, C.E., and Mohring,H.-D. (1988) Diffusion lengths in a-SiGe:H and a-SiC:H alloys from optical gratingtechnique. Mater. Res. Soc. Symp. Proc.,118, 679.

5 Balberg, I., Albin, D., and Noufi, R.(1989) Mobility-lifetime products inCuGaSe2. Appl. Phys. Lett., 54, 1244.

6 Yang, L., Catalano, A., Arya, R.R., andBalberg, I. (1990) Effect of low-levelboron doping and its implication to thenature of gap states in hydrogenatedamorphous-silicon. Appl. Phys. Lett.,57, 908.

7 Balberg, I., Delahoy, A.E., and Weakliem,H.A. (1988) Self-consistency and self-sufficiency of the photocarrier gratingtechnique. Appl. Phys. Lett., 53, 992.

8 Liu, J.Z., Li, X., Roca i Cabarrocas, P.,Conde, J.P., Maruyama, A., Park, H., andWagner, S. (1990) Ambipolardiffusion length in a-Si-H(F) and a-Si,Ge-H,F measured with the steady-statephotocarrier grating technique.Conference Record 21st IEEEPhotovoltaic Specialists Conference,p. 1606.

9 Ritter, D., Zeldov, E., and Weiser, K.(1988) Ambipolar transport inamorphous-semiconductors in thelifetime and relaxation-time regimesinvestigated by the steady-statephotocarrier grating technique. Phys. Rev.B, 38, 8296.

10 Balberg, I. (1990) The theory of thephotoconductance under the presence of asmall photocarrier grating. J. Appl. Phys.,67, 6329.

11 Balberg, I. (1991) Theory of the smallphotocarrier grating under the applicationof an electric-field. Phys. Rev. B, 44, 1628.

12 Li, Y.M. (1990) Phototransport under thepresence of a small steady-statephotocarrier grating. Phys. Rev. B, 42,9025.

13 Hubin, J., Sauvain, E., and Shah, A.V.(1989) Characteristic lengths for steady-state transport in illuminated, intrinsica-Si:H. IEEE Trans. Electron. Devices, 36,2789.

14 Hattori, K., Okamoto,H., andHamakawa,Y. (1992) Theory of the steady-state-photocarrier-grating technique forobtaining accurate diffusion-lengthmeasurements in amorphous-silicon.Phys. Rev. B, 45, 1126.

15 Abel, C.-D., Bauer, G.H., and Bloss, W.H.(1995) Generalized theory for analyticalsimulation of the steady-statephotocarrier grating technique. Philos.Mag. B, 72, 551.

16 Schmidt, J.A. and Longeaud, C. (2005)Analysis of the steady-state photocarriergrating method for the determination ofthe density of states in semiconductors.Phys. Rev. B, 71, 125208.

17 Br€uggemann, R. (1998) Steady-statephotoconductivity in a-Si:H and itsalloys, in Properties of Amorphous Siliconand its Alloys (ed. T. Searle), INSPEC,London, p. 217.

18 Haridim, M., Weiser, K., and Mell, H.(1993) Use of the steady-statephotocarrier-grating technique for thestudy of the surface recombinationvelocity of photocarriers and thehomogeneity of hydrogenatedamorphous-silicon films.Philos. Mag. B, 67, 171.

19 Nicholson, J.P. (2000) Fresnel correctionsto measurements of ambipolar diffusionlength. J. Appl. Phys., 88, 4693.

20 Blakemore, J.S. (1987) SemiconductorStatistics, Dover, New York.

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