Structures, optical properties, and electrical transport processes of SnO$_2$ films with oxygen deficiencies

0.66 MB·English

by  Yu-Chen Ji

Structures, optical properties, and electrical transport processes of SnO films with
2
oxygen deficiencies
Yu-Chen Ji,1 Hua-Xing Zhang,1 Xing-Hua Zhang,2 and Zhi-Qing Li1,∗
1Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology,
Department of Physics, Tianjin University, Tianjin 300072, China
2Hebei University of Technology, School of Material Science and and Engineering, Tianjin 300130, China.
(Dated: February 6, 2013)
3 The structures, optical and electrical transport properties of SnO2 films, fabricated by rf sput-
tering method at different oxygen partial pressures, were systematically investigated. It has been
1
0 found thatpreferred growth orientation ofSnO2 film is strongly related totheoxygen partial pres-
2 sure during deposition, which provides an effective way to tune the surface texture of SnO2 film.
All films reveal relatively high transparency in the visible range, and both the transmittance and

n optical band gap increase with increasing oxygen partial pressure. The temperature dependence
a of resisitivities was measured from 380 K down to liquid helium temperatures. At temperature
J above∼80K,besidesthenearest-neighbor-hoppingprocess, thermalactivationprocessesrelatedto

1 two donor levels (∼30 and ∼100 meV below the conduction band minimum) of oxygen vacancies
3 are responsible for the charge transport properties. Below ∼80 K, Mott variable-range-hopping
conduction process governs the charge transport properties at higher temperatures, while Efros-
] Shklovskii variable-range-hopping conduction process dominates the transport properties at lower
i
c temperatures. Distinct crossover from Mott type to Efros-Shklovskii type variable-range-hopping
-s conduction process at several to a few tenskelvin are observed for all SnO2 films.
l
r PACSnumbers: 73.61.Le,81.15.Cd,73.50.Bk
t
m
.
t I. INTRODUCTION the temperature dependence of resistivities from liquid
a
helium temperatures to 380 K for seven polycrystalline
m
SnO films deposited by rf sputtering method at differ-
- In recent years, great interest has been focused on 2
d SnO duetoitsimportanceinmanytechnicalfields,such ent oxygen partial pressures. (It is known that the car-
n as ga2s sensors,1 solar cells,2 and optoelectronic devices.3 rierconcentrationsinSnO2 filmsareseriouslyinfluenced
o by the oxygen contents.) The electrical transport pro-
The nominally undoped(unintentionaldoped) SnO is a
c 2 cess was found to be dominated by thermal activation
[ transparentmetal-oxidesemiconductor which is natively
n-type and usually has a resistivity in the range of 1- andvariable-range-hopping(VRH)processesathighand
1 100 Ω cm.4 The optical transparency of SnO thin films lowtemperatures,respectively. Particularly,a transition
v 2 from Mott type to Efros-Shklovskii(ES) type VRH con-
can be up to 97% in the visible range (for films of thick-
5 duction process was observed for all the samples. The
7 ness 0.1-1 µm) with a direct optical band gap of 3.6-4.2
variationsin surfacemorphologiesandopticalproperties
0 eV.5,6 The conductivity of SnO2 film can be greatly en- withoxygenpartialpressureforthe films werealsostud-
1 hanced by doping small amount of fluorine, antimony,
ied.
. tantalumandotherdopants.7–9Currently,fluorinedoped
2
0 SnO2 (FTO) film is one of the widely used transparent
3 conducting oxide (TCO) materials. In addition, the p-
1 type conductivity, which is crucial in fabricating trans- II. EXPERIMENTAL METHOD
: parent p-n junction, was recently observed in Li, In, Al,
v
Xi and N doped SnO2,3 which further stimulates the stud- The SnO2 thin films were deposited on quartz glass
ies on SnO2 materials. The influences of the fabricating by standard rf sputtering method. A SnO2 target (with
ar methodandfabricatingconditiononthestructures,elec- diameter of 60 mm and 5 mm thick), synthesized by the
tricaltransportandoptical propertiesof SnO2 films had traditional ceramic process, was used as the sputtering
been extensively investigated.7 The electrical transport source. SnO powders with 99.9% purity were ground
2
properties of different forms of SnO2 such as nanobelt, by ball milling for 12 h before being compressed into a
nanowire,andthinfilmshadalsobeenstudied.10–12How- disk. The diskwascalcinedat1673Kinairfor12hand
ever, the origins of electrical conductivity as well as the then furnace cooled. The base pressure of the chamber
chargetransportmechanismsofSnO2havenotbeenfully was pumped to 1.0×10−4 Pa and then a mixture gas of
understood, which seriously limits optimizing the prop- Ar and O (both were 99.999% purity) was introduced
2
erties of this material. Thus detailed measurements of asthesputteringgas. Oxygenpartialpressure(O )was
pp
thetemperaturebehaviorofresistivitiesoverawidetem- controlled (0, 0.5%, 1%, 1.5%, 5%, 10%, and 20%) by
perature range and then extracting the chargetransport adjusting the flux ratio of Ar to O and the pressure of
2
mechanisms in SnO2 films are necessary and nontrivial. chamberwasmaintainedat1Pa. The substratetemper-
In the present paper, we systematically investigated ature and the sputtering power were kept at 973 K and
2
150 W, respectively. (112) peaks increased sharply. The different preferred
Film thickness (∼1 µm) was determined by a sur- growth orientations would lead to different surface tex-
face profiler (Dektak, 6 m). Crystal structures of the ture of the films, which is important in improving the
films were measured in a powder x-ray diffractometer efficiency of solar cells.2 Our results indicate that alter-
(D/MAX-2500, Rigaku) with Cu Kα radiation. The ing the oxygenpartialpressure in the sputtering process
surface morphology of SnO thin films was character- is an effective way to tune the texture of SnO films.
2 2
ized by a scanning electron microscope (SEM, S-4800,
Hitachi). The resistivity was measured by the stan-
dard four-probe technique. A Keithley 236 unit and a
Keithley 2182Anano-voltmeterwere used as the current
source and voltmeter, respectively. The temperature en-
vironmentwasprovidedbyaphysicalpropertymeasure-
ment system (PPMS-6000, Quantum Design). Optical
absorptionandtransmittancespectraweremeasuredina
UV-VIS-NIRscanningspectrophotometer(UV-3101PC,
SHIMADZU).
(110) (101) (111) (211) (112)
7
6
)
arb. units (101) (200) (002) (310) 45
nsity ( (200) 3 FpoIGsit.e2d.un(cdoelroroxoynglienne)paSrEtiMal pimreasgseusreosfoSf:nO(a2) t0h,i(nb)fil1m%s,d(ce)-
e 1.5%, (d) 5%.
nt
I

(002) 2
1
Figure2showstheSEMimagesforfourrepresentative
20 40 60 80 films. The variation in the texture of the films can also
2 (deg.) beseeninthe morphologiesofthe films. FilmsNos. 1to
3 show square based pyramid-like structures. Film No.
FIG. 1. (color online) XRD patterns of SnO2 thin films de-
4 shows a gravel-like morphology. Films Nos. 5 to 7 ex-
positedunderdifferentoxygenpartialpressures(0,0.5%,1%,
hibittriangularpyramidalshapegrains. Accordingtothe
1.5%, 5%, 10%, and 20%).
morphology simulation results by Smith,13 the variation
insurfacemorphologyisconsistentwiththechangeinthe
preferred growth orientation observed in XRD measure-
ments. ThemeangrainsizeDwasobtainedbysurveying
III. RESULTS AND DISCUSSION ∼60 grains in each film and was listed in Table I. The
grainsize shows a decrease trend with increasing oxygen
partial pressure, which may be related to the variation
Figure 1 shows the XRD patterns of the SnO thin
2
in the preferred orientations of the films.
films deposited at different oxygen partial pressures.
Samples were numbered from 1 to 7 according to the Figure 3 shows the optical transmittance (T ) versus
R
oxygen partial pressures (0, 0.5%, 1%, 1.5%, 5%, 10%, wavelength for four representative SnO films, as indi-
2
and 20%) during deposition. Comparing to the stan- cated. HerethetransmittanceT istheintensityratioof
R
dard PDF card (No. 41-1445), one can know that the transmitted light to incident light, i.e., T =I/I . From
R 0
filmshavetetragonalrutile-typestructureandthereisno Fig. 3, we can see that all films exhibit relatively high
secondary phase. The lattice constants are a≈4.742 ˚A, transparencies in the visible range (390-780 nm). At a
c≈3.189˚A, and insensitive to the oxygencontents,while certain wave range of visible spectra, the transparency
thepreferredgrowthorientationvarieswiththevariation of the film enhances with increasingoxygenpartialpres-
in oxygen partial pressure. The strongest peak is from sure. In addition, the absorption edge slightly shifts to
the diffraction of (002) plane for sample No. 1, while it shorter wavelength side with increasing oxygen content.
becomes(200)planeforsampleNos. 2and3. Forsample Sincethe relationbetweentransmittanceandabsorption
No. 4, the diffraction corresponding to the (101) plane coefficient(α) is T =exp(−αd) (where dis the thickness
R
is the strongest peak. With further increasing oxygen of the film), one can readily obtain the absorption coef-
contents,the relativeintensity of(101)peak is obviously ficient versus photon energy hν from the data in Fig. 3.
reduced and the relative intensity of (111), (211), and Theoretically,therelationshipbetweenabsorptioncoeffi-
3
cientandthe incidentphotonenergycanbe writtenas14
4 7
(αhν)1/n =A(hν−E ), (1) 10
g
4
where A is a constant and Eg is the band gap. Using 103 3
n=1/2 for direct band gap transition, the values of E
g )
were obtained by extrapolating the linear portion of the cm102 2
plotto(αhν)2=0. TheinsetofFig.3showstheobtained
(
E value versus oxygen partial pressure. Clearly, the 1
g 10
opticalbandgapofthefilmincreaseswithoxygenpartial
pressurewhichisinaccordancewiththeblue shiftofthe 0
10
absorptionedge. The oxygenvacancies are often treated
as the main reason for the opaque of SnO2 thin films.15 -1
10
Thus, the enhancement in the transmittance and optical
0.0 0.1 0.2 0.3 0.4 0.5
bandgapofthefilmscanbeattributedtothereductionof -1
1/T (K )
oxygenvacancieswithincreasingoxygenpartialpressure.
Figure 4 shows the resistivities as functions of recip- FIG. 4. (color online) Variation in the logarithm of resistiv-
rocal temperature from 380 K down to liquid helium ity with reciprocal temperature for four SnO2 thin films, as
temperatures for four representative films, as indicated. indicated.
Inspection of this figure indicates that the resistivity in-
creaseswithdecreasingtemperatureovertheentiremea-
sured temperature range which shows a typicalsemicon-
TABLEI.Valuesofrelevantparametersforsevenoxygende-
ductorbehaviorintheelectricaltransportproperty. Ata
ficient SnO2 polycrystalline films. ρi (i=1, 2, 3) and Ei are
certaintemperature,the filmdepositedunder loweroxy-
definedin Eq. (2). D is themean grain size of thefilms.
genpartialpressurepossessessmallerresistivity,e.g.,the
resistivityofsampleNo. 7isabout1000timeslargerthan
that of sample No. 1 at 10 K. Our observation supports Film Opp D ρ1 E1 ρ2 E2 ρ3 E3
No. (%) (nm) (Ω cm) (meV) (Ω cm) (meV) (Ω cm) (meV)
Kılı¸c and Zunger’s prediction that oxygen vacancies are
1 0 141 0.067 105 0.297 30.8 1.01 3.82
the main electron suppliers in SnO2.5 2 0.5 144 0.055 91 0.228 32.6 1.38 4.12
Weanalyzethecarriertransportprocessesinhightem- 3 1 132 0.064 114 0.316 34.5 3.06 4.45
perature regions. In Fig. 5, we replotted the resistivities 4 1.5 98.3 0.078 102 0.621 35.7 5.59 5.60
asfunctions ofreciprocaltemperaturefrom∼80to∼380 5 5 78.1 0.127 108 2.92 31.3 16.9 8.29
K for the four representative SnO thin films. The high 6 10 70.3 0.097 119 1.42 36.4 18.6 6.39
2
temperatureconductionprocessinn-typesemiconductor 7 20 75.4 0.154 114 1.77 37.0 17.7 7.81
usually comes from the electrons hopping from the shal-
lowdonorlevelstotheconductionbandwhichcanbeex-
pressedbytheArrheniusfunctionρ(T)−1=ρ−1e−Ea/kBT.
0
However, no straight lines can be found in Fig. 5. Ac- level (∼30 meV below the conduction band minimum),
cording to Samson and Fonstad,16 the oxygen vacancies the other is an intermediately deep donor level (∼100
inSnO canformtwodonorlevels,oneisashallowdonor meVbelowtheconductionbandminimum). Inaddition,
2
the electrons in the impurity band can also take part in
theconductingprocessbyhoppingfromanoccupiedsite
tothe nearestempty siteviaphononassistance[whichis
100
callednearest-neighbor-hopping(NNH)process]atinter-
%) 80 mediate temperature range.17 (Note that the NNH con-
e ( ductionalsohasanArrhenusform.) Thus,wetentatively
c
n 60 4.08 fit our ρ(T) data by
a
anmitt 40 E (eV)g44..0004 ρ(T)−1 =ρ−1e−E1/kBT +ρ−1e−E2/kBT +ρ−1e−E3/kBT,
Tr 3.96 7 1 2 3 (2)
20 5
3.920 5 10 15 20 4 where ρ1, ρ2, and ρ3 are prefactors independent of tem-
0 Opp (%) 2 perature, E1 and E2 are the activation energies associ-
ated with the two donor levels, and E is the activation
3
300 400 500 600 700 800
energyofthe NNHprocess. The theoreticalpredications
Wavelength (nm) ofEq.(2)arealsoshowninFig.5(thesolidcurves),from
FIG.3. (coloronline)Roomtemperatureopticaltransmission which one can see that the measured ρ(T) data can be
spectraoffourSnO2thinfilms,asindicated. Theinsetshows well described by Eq. (2). [Precisely, samples Nos. 1 to
thevaluesofopticalbandgapversusoxygenpartialpressure. 4canbe fittedby Eq.(2) from60to 380K,andsamples
Nos. 5 to7canbe fittedfrom90to 380K.]The relevant
4
parameters obtained in the fitting processes are listed in gap can no longer be ignored, the temperature depen-
Table I. Inspection of Table I indicates that the values dence of resistivity would become21,22
of E and E are ∼100±20 meV and ∼30±7 meV, re-
1 2
spectively,which areconsistentwith the activationener- ρ(T)=ρ e(TES/T)1/2, (5)
ES
gies of the two donor levels of oxygenvacancies.16 While
the value of E3 varies from 3.82 to 8.29 meV, which is where ρES is a temperature independent prefactor and
in good accordance with the activation energy of NNH TES is a characteristic temperature. The characteristic
conduction.4,10,18 temperature TES is given by21,22
The empty sites among the nearest neighbors will de-
β e2
crease dramatically with decreasing temperature. At T = 1 , (6)
ES
sufficient low temperatures, the NNH process would ǫξkB
be quenched and the electrons will hop to empty sites where β is a constant with a value of ≃2.8, ǫ is the
that are not nearest to them but have smaller energy 1
dielectric constant of the material. In ES VRH model,
discrepancies.20 In this case, there are usually two forms
the width of the Coulomb gap is22
of hopping. At relatively high temperatures, Mott sug-
gested the VRH conduction in the form of19–21 e3 N(E )
F
∆ = . (7)
ρ(T)=ρ e(TM/T)1/4, (3) CG pǫ3/2
M
for three dimensions, where ρ is a temperature inde-
M
pendent prefactor and TM is a characteristic tempera- (a) 4 (b)
ture. The Mott VRH conduction theory is based on a
constant density of states (DOS) near the Fermi level.
3
The characteristic temperature can be written as21
3
w
18
T = , (4)
M k N(E )ξ3
B F 2
2
where N(E ) is the DOS without long range coulomb
F
interactions and ξ is the relevant electron localization
length. When the temperature is further decreased, the 20 40 60 20 40 60
Coulombinteractionsbetweenthe chargecarriersshould T (K) T (K)
be considered. In this situation, ES found that the elec-
FIG.6. (coloronline)Log-logplotofw=−∂lnρ/∂lnT versus
tronic DOS inthe vicinity ofE is no longera constant,
F T for samples (a) No. 5 and (b) No. 7. The solid lines are
andasoftgap(Coulombgap)willforminthe DOSnear
thelinear fitsto thedata.
the Fermi energy. When the influence of the Coulomb
To check whether the Mott and ES types VRH pro-
cesses exist in our samples, we plotted w against T
7
on double logarithmic scales, where w is defined as
4 w(T)=−∂lnρ/∂lnT.23 Using the general form of VRH
1 conduction ρ(T)=ρ exp(T /T)x, one can obtain the ex-
10 0 0
3 pression logw≃log(xTx)−xlogT. Thus when the hop-
m) 0
ping mechanisms dominate the carrier transport pro-
c
2 cesses, one can get the magnitudes of exponent x from
(
the slopes of the linear parts in logw versus logT data.
0
10 Figure6showswasafunctionofT indouble-logarithmic
scales from ∼10 to 70 K for two representative films
(Nos. 5 and 7). Clearly, there are two distinct linear
parts in each curve, which suggests the existence of two
different hopping mechanisms in our films at low tem-
0.005 0.010 0.015
peratures. The values of x obtained from each plot are
-1
1/T (K ) ∼0.56and∼0.25atlowertemperatureregionandhigher
FIG. 5. (color online) Logarithm of resistivity as a function temperatureregion,respectively,whichsuggeststhatthe
of reciprocal temperature for four SnO2 thin films from ∼80 crossoverfromESVRH(x∼0.56)toMottVRH(x∼0.25)
to ∼380 K, as indicated. The symbols are the experimental processoccurswithincreasingtemperature. Forsamples
data and the solid curvesare least-squares fits to Eq.(2). Nos. 5and7,boththecrossovertemperaturesarearound
40 K.
5
TABLE II. Values of relevant parameters for seven oxygen deficient SnO2 polycrystalline films studied in this work. ρM, TM
′
and ρES, TES are defined in Eq. (3) and Eq. (5), respectively. Tcross and Tcross are the crossover temperatures from Mott to
ES VRH conduction which were obtained from theoretical calculation and observation of the data plot, respectively. ∆ is
CG
thevalue of theCoulomb gap. Whop,Mott and Whop,ES are theaverage hopping energies of electrons.
′
Film No. ρM TM ρES TES Tcross Tcross ∆CG Whop,Mott Whop,ES
(Ω cm) (K) (Ω cm) (K) (K) (K) (meV) (meV) (meV)
1 0.151 2780 1.86 31.6 5.76 4 0.262 1.57×10−4T3/4 2.43×10−4T1/2
2 0.302 2159 2.95 26.2 5.07 6 0.223 1.47×10−4T3/4 2.21×10−4T1/2
3 0.282 7172 5.65 49.6 5.49 8 0.320 1.98×10−4T3/4 3.04×10−4T1/2
4 0.361 13053 8.84 81.2 8.07 8 0.496 2.30×10−4T3/4 3.88×10−4T1/2
5 0.060 155608 2.55 688 48.7 40 3.55 4.28×10−4T3/4 1.13×10−3T1/2
6 0.039 187461 2.04 746 47.5 40 3.65 4.49×10−4T3/4 1.18×10−3T1/2
7 0.043 195370 2.27 771 48.6 40 3.75 4.53×10−4T3/4 1.20×10−3T1/2
The experimental ρ(T) data at higher and lower tem-
67 T (K) 2.4 25 11 T (K) 2.8
perature regions are least-squares fitted to Eq. (3) and
4 7 (a) 4 7 (b)
Eq.(5), respectively,andthe resultsareshownin Fig. 7. 10 10
4
For samples Nos. 1 to 4, the experimental data can be 3 4
well described by Eq. (3) in temperature range of ∼10 m)10 3 103 3
iKngtote∼m4p0erKat.uFreorrasanmgepilses∼N4o0s.K5toto∼77,0tKhe. Ocourtrseisdpeonthde- ( c1012 2 102 2
10
above temperature regions, the ρ(T) data deviates from 1
the predication of Eq.(3). The deviation parts are quite 100 10
coincide with the predicationofEq.(5)below ∼10K for
0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7
samples Nos. 1 to 4 and ∼30 K for samples Nos. 5 to 1/4 -1/4 1/2 -1/2
(1/T) (K ) (1/T) (K )
7. Thus, the crossoverfromthe Mott to ESVRH occurs
near ∼10 K for samples Nos. 1 to 4 and ∼40 K for sam- FIG. 7. (color online) (a) Logarithm of resistivity as a func-
ples Nos. 5 to 7. The fitting parameters ρ , T , ρ , tionofT−1/4forfourSnO2thinfilms,asindicated. Thesym-
M M ES bolsaretheexperimentaldataandthestraightsolidlinesare
and T are listed in Table II.
ES least-squares fits to Eq. (3), (b) Logarithm of resistivity as a
hoIpnpitnhgeeMneortgtieVsRofHelaencdtroEnSsVarReHgivtehneobryie1s9,,2t1he average functionof T−1/2 forfourSnO2 thinfilms, asindicated. The
symbolsaretheexperimentaldataandthestraightsolidlines
are least-squares fitsto Eq. (5).
1/4
1 T
M
W = k T (8)
hop,Mott B
4 (cid:18) T (cid:19)
and Eqs. (4), (6), and (7), the width of the Coulomb gap
1/2 can be rewritten as
1 T
ES
W = k T , (9)
hop,ES 2 B (cid:18) T (cid:19) T3 1/2
∆ =0.9k ES . (11)
CG B
(cid:18)T (cid:19)
respectively. The hopping energy in Eq. (8) should be M
equal to that in Eq. (9) at T , thus the exact value of
cross Thus the values of ∆ are obtained and also listed in
T can be written as CG
cross Table II, from which one can see that ∆ increases
CG
16T2 with increasing oxygen partial pressure. The average
Tcross = T ES. (10) hopping energies Whop,Mott and Whop,ES are calculated
M
using Eq. (8) and Eq. (9), respectively. The criterion
The values of the crossover temperature T together of W >2∆ in the Mott VRH theory24 requires
cross hop,Mott CG
with that estimated from logw-logT plots (written as a temperature higher than ∼5 K, ∼7 K, and ∼40 K for
′
T ) are listed in Table II. Since the transition from samplesNos. 1to3,No. 4,andNos. 5to7,respectively.
cross
the MottVRHconductionbehaviorto the ESVRHcon- Taking account of the crossover temperatures listed in
ductionbehaviorisasmoothprocess,ratherthanasharp Table II, we can see that the criterion of the Mott VRH
′
crossover,theslightdiscrepancybetweenT andT conductionisfullysatisfiedforallsamples. However,the
cross cross
is physically reasonable. requirement of W <∆ in the ES VRH theory is
hop,ES CG
We further analyze the applicability of the Mott and hardlysatisfiedforallsamples,sinceitrequirestempera-
ES VRH conduction laws to our films. Combining turestobe lowerthan∼2Kand∼10KforsamplesNos.
6
1to4andNos. 5to7,respectively. Thediscrepancybe- temperaturerange80K.T.380K,theNNHconduction
tweenexperimentalresultsandtheoreticalpredictionsin processtogether with two thermalactivationconduction
the W was also observed in other systems,17,24,25 processes,onefromtheshallowdonorlevelsandtheother
hop,ES
which still needs further investigation. from the intermediately deep donor levels, are the main
charge transport mechanisms. Below ∼80 K, the Mott
type and EStype VRH sequentially governsthe conduc-
IV. CONCLUSION tion processes with decreasing temperature. Our work
establishesaquitecompletepictureoftheoverallelectri-
Insummary,wesystematicallyinvestigatedthecharge cal conduction mechanisms in the SnO2 polycrystalline
transport processes of SnO thin films, which were de- films from liquid-helium temperatures up to 380 K.
2
posited by rf sputtering method at different oxygen par-
tial pressures, from 380 K down to liquid helium tem-
peratures. The influences of oxygen partial pressure on
ACKNOWLEDGMENTS
the structures and optical properties were also studied.
Thepreferredcrystalgrowthdirectionaswellasthesur-
face morphology of the films can be tuned by varying We would like to thank Z. M. Lu for his contribution
oxygen partial pressure in the deposition process. All in SEM measurements. This work was supported by the
films reveal relatively high transparency in the visible NSF of China through grant no. 11174216, Research
range,and both the transmittance and optical band gap FundfortheDoctoralProgramofHigherEducation,and
increase with increasing oxygen partial pressure. In the Tianjin City NSF through grant no. 10JCYBJC02400.
∗ Author to whom correspondence should be addressed. 14 J. I. Pankove, Optical processes in semiconductors
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