Optical properties and electronic band structure of AgGaTe_2 chalcopyrite semiconductor

Optical properties and electronic band structureof AgGaTe2 chalcopyrite semiconductor

Shinya Arai, Shunji Ozaki, and Sadao Adachi*Graduate School of Engineering, Gunma University, Kiryu-shi, Gunma 376-8515, Japan

*Corresponding author: [email protected]‑u.ac.jp

Received 23 October 2009; revised 18 December 2009; accepted 18 December 2009;posted 11 January 2010 (Doc. ID 118925); published 2 February 2010

The optical properties of AgGaTe2 chalcopyrite semiconductor are studied by optical absorption, spectro-scopic ellipsometry (SE), and thermoreflectance (TR) spectroscopy. Optical absorption spectra suggestthat AgGaTe2 is a direct-gap semiconductor having a bandgap of ∼1:2 eV at T ¼ 300K. The pseudodi-electric-function spectra of AgGaTe2 are determined by SE in the range between E ¼ 1:2 and 5:2 eV forboth states of polarization. These spectra reveal distinct structures at energies of the critical points in theBrillouin zone. The TR spectra are also measured in the E ¼ 1:0–5:3 eV ranges at T ¼ 20K–300K. Thespin–orbit and crystal-field splitting parameters of AgGaTe2 are determined to be Δso ¼ 0:70 eV andΔcr ¼ −0:23 eV, respectively. © 2010 Optical Society of America

OCIS codes: 160.4760, 300.6470.

1. Introduction

A family of the ternary semiconducting compoundsAIBIIICVI

2 have been widely investigated because oftheir potential applications to photovoltaic solarcells, light-emitting diodes, and nonlinear optical de-vices [1]. Most of these compounds have a chalcopy-rite structure in the space group of D12

2d − I�42d [2].Silver gallium ditelluride (AgGaTe2) is one in thisfamily and has a nearly ideal chalcopyrite structure(c=a ¼ 1:90 and u ¼ 0:25) [3]. Although the materialhas been the subject of many research efforts, manyfundamental properties are not sufficiently evalu-ated or are even unknown [1,2].Little is known about the optical properties of

AgGaTe2 [4–13]. Tell et al. [4] measured the electro-reflectance (ER) spectra of a AgGaTe2 single crystalat 77K and 300K. They reported that the materialhas a direct energy gap at 1:32 eVð300KÞ, which wasprobably the lowest energy gap. Raman scatteringand far-infrared optical spectra of a AgGaTe2 singlecrystal were measured by Julien et al. [7]. Ohmeret al. [8,9] determined the refractive-index dispersionat wavelengths of 3–5 μm and absorption spectra for

light polarizations E⊥c and E∥c. The optical absorp-tion spectra were also reported by several authors[6,11,12]; however, there is a wide variation in thereported bandgaps determined from the optical ab-sorption measurements in the literature, with valuesranging from 0.99 to 1:32 eV. The temperature-dependent photoluminescence spectra of AgGaTe2were also measured by Bodnar et al. [10] at T ¼10K–220K and by Krustok et al. [13] at T ¼10K–100K.

In this work, we investigate the electronic energy-band structure of AgGaTe2 single crystals grown bythe conventional Bridgman method. The experimen-tal techniques used here are optical absorption, spec-troscopic ellipsometry (SE), and thermoreflectance(TR) spectroscopy. We also calculate the electronicenergy-band structure of AgGaTe2, using the empiri-cal pseudopotential method (EPM, see Appendix A)and considering the spin–orbit interaction, and com-pare it with our experimental results.

SE is an excellent technique for investigating theoptical response of semiconductors [14,15]. To ourknowledge, however, no SE measurement has beenperformed on AgGaTe2 to date. Our measured SEspectra will be analyzed on the basis of a simplifiedmodel of the interband transitions, namely, themodel dielectric function (MDF) [16].

0003-6935/10/050829-09$15.00/0© 2010 Optical Society of America

10 February 2010 / Vol. 49, No. 5 / APPLIED OPTICS 829

Modulation spectroscopies, such as TR and ER, arepopular means of analyzing critical-point (CP) ener-gies in semiconductors [17]. All these techniqueshave, as a common feature, the application of a smallperturbation, such as a change in the lattice constantor an external field, that destroys some inversionsymmetry of the crystal structure. The resultantTR and ER spectra reflect the first and third deriva-tives of the linear dielectric function, respectively[18]. Although Tell et al. [4] reported the ER spectrafor AgGaTe2 measured at 77K and 300K, no de-tailed modulation spectroscopy study has been per-formed on AgGaTe2 to date. We will measure TRspectra for both E⊥c and E∥c at T ¼ 20K–300Kand analyze these spectra using the standard criti-cal-point (SCP) expression [19]. It is evident that adetailed knowledge of the optical response is of greatimportance for many device applications and is alsoof scientific interest.

2. Experiment

The AgGaTe2 single crystals used in this study weregrown by the conventional Bridgman method. Aquartz ampoule graphitized by acetone was filledwith a charge of a Ag : Ga : Te ¼ 1∶1∶2 mixturein atomic ratio and was then sealed off under10−6 Torr. Ag, Ga, and Te were 99.999% (Ag) or99.9999% pure (Ga and Te). The ampoule was slowlyheated to 850 °C, melted, and then equilibrated for atleast three days. The crystals were grown by slowlylowering the ampoule to achieve its cooling rateof ∼1:5 °C=h.The AgGaTe2 single crystals had a size of a few

tens of cubic millimeters. They did not contain spur-ious phases, like Ag2Te and Ga2Te3. The x-ray dif-fraction trace showed that the angles and relativestrengths of the diffraction peaks were in good agree-ment with those reported by Hahn et al. [3] (a ¼ 6:3Å and c ¼ 12:0 Å). The hot-probe measurements in-dicated that the electrical conductivity of the crystalswas of the p type. The ingots were cut to have surfaceorientation parallel to the c axis. Before optical mea-surements, the samples were mechanically polishedwith 0:05 μm Al2O3 powder, followed by chemicallypolishing with a solution of Br2 in methanol, andthen were finally rinsed with deionized water.The optical absorption measurements were carried

out using a JASCO CT-25C spectrometer, a 50Whalogen lamp, and a liquid-nitrogen-cooled Ge photo-diode. The thickness of the samples used for thesemeasurements was about 80 μm. The optical absorp-tion spectra were recorded at T ¼ 15K–300K.The SE equipment was of the polarizer-sample-

rotating-analyzer type (DVA-36VW-A, Mizojiri Opti-cal, Co, Ltd.). A 150W xenon lamp was used as thelight source. The SE data were measured over thephoton-energy range of 1:2–5:2 eV at room tempera-ture. The angle of incidence and the polarizer azi-muth were set at 70° and 30°, respectively.Ellipsometry yields the complex ratio ρ between

Fresnel’s reflection coefficients rp and rs [14,15]:

ρ ¼ rsrp

¼ tanΨeiΔ; ð1Þ

whereΔandΨare theellipsometric variables.Ateachphoton energy E ¼ ℏω, the four optical parameters(ε1⊥, ε2⊥, ε1∥, ε2∥) can be determined from two pairsof SE measurements ½ρmeasðδ ¼ 0° and δ ¼ 90°Þ� byminimizing the error function Gðε1⊥;ε2⊥;ε1∥;ε2∥Þ[20,21]:

Gðε1⊥; ε2⊥; ε1jj; ε2jjÞ ¼X

δ¼0°;90°

jfReðρmeasδ Þ − Reðρcalδ Þg2

þ fImðρmeasδ Þ − Imðρcalδ Þg2j;

ð2Þ

whereρmeasðδ ¼ 0° and δ ¼ 90°Þare the reflection coef-ficients measured when the c axis is parallel to boththe interface and the plane of incidence (δ ¼ 0°) oris parallel to the interface and perpendicular to theplane of incidence (δ ¼ 90°). ρcal is the calculated re-flection coefficient ratio.

The experimental setup for TR measurement wasessentially the same as that described in the litera-ture [22]. The heater consisted of a Au film (∼20nm)deposited on the sapphire substrate by evaporation.The AgGaTe2 sample was mounted on this heaterusing silicon grease. Temperature modulation wasachieved by passing a pulsed electric current at∼7Hz (square pulses with a 30% duty cycle) throughthe deposited Au-film heater. The probe light from a150W xenon lamp or a 50W halogen lamp wasirradiated near normal to the sample surface. TRspectra were then measured in the 1:0–5:3 eVphoton-energy range using a grating spectrometerand a thermoelectrically cooled photomultiplier tubeor a liquid-nitrogen-cooled Ge photodiode. The TRmeasurements were carried out at T ¼ 20K–300K.

3. Results and Discussion

A. Optical Absorption Spectra

One of the most essential semiconductor parametersis the bandgap energy. To determine the fundamen-tal absorption edge of AgGaTe2, we measured theoptical absorption spectra of this material. Theabsorption coefficient α was calculated from themeasured transmittance T using the expression

T ¼ ð1 − RÞ2e−αx1 − R2e−2αx

; ð3Þ

where R is the reflectivity. We used a value of 0.27for R. This value was obtained from the presentSE measurements (see Fig. 4).

Figures 1(a) and 1(b) show the squares of the ab-sorption coefficient, α2, as a function of photon energyfor AgGaTe2 at T ¼ 15K, 200K, and 300K for E⊥c[Fig. 1(a)] and E∥c [Fig. 1(b)]. At α ≥ 300 cm−1, thedependence on photon energy E can be written as

830 APPLIED OPTICS / Vol. 49, No. 5 / 10 February 2010

αðEÞ ¼ AðE − E0Þ1=2: ð4Þ

This dependence is expected for direct optical transi-tions. The plots in Figs. 1(a) and 1(b) give the band-gap energies of E0 ∼ 1:261 (15K), ∼1:230 (200K),and ∼1:192 eV (300K) for E⊥c; and ∼1:238 (15K),∼1:205 (200K), and ∼1:164 eV (300K) for E∥c.We show in Fig. 1(c) the E0 versus T data for

AgGaTe2 obtained from the optical absorption mea-surements at T ¼ 15K–300K. The E0 gaps for bothE⊥c and E∥c have nearly constant values at tempera-tures up to T ∼ 80K, and then show a gradualdecrease with increasing T. Traditionally, temper-ature variation of the bandgap energy E0ðTÞ is ex-pressed in terms of Varshni’s formula [23]. Pässler[24,25] proposed an analytical expression that takesinto account the bandgap shrinkage effect in accor-dance with general equations and parameter rela-tionships governing the electron–phonon interactionmechanism:

EgðTÞ ¼ Egð0Þ −αpΘp

2

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

�2TΘp

�p

p

s− 1

�; ð5Þ

where αp plays the role of a T → ∞ limiting value ofthe bandgap shrinkage coefficient −dEgðTÞ=dT,Θp isapproximately equal to the average phonon tempera-ture, and the power exponent p is closely related tothe overall shape of the electron–phonon spectralfunction in the given material. Pässler’s expressionis more palatable than Varshni’s equation [23] fromthe theoretical point of view.The solid curves in Fig. 1(c) represent the fitted

results of the experimental data to Eq. (5). The fit-determined parameters are listed in Table 1. Differ-entiating Eq. (5) with respect to T, we obtain

dEgðTÞdT

¼ −αp�2TΘp

�p−1

�1þ

�2TΘp

�p� ð1−pÞ=p

: ð6Þ

The temperature coefficients dEgðTÞ=dT at T ∼

300K for AgGaTe2 obtained from Eq. (6) are dE0ðTÞ=

Fig. 1. Square of the absorption coefficient α2 as a function ofphoton energy for AgGaTe2 at T ¼ 15K, 200 K, and 300K for(a) E⊥c and (b) E∥c. (c) Temperature variation of the fundamentalabsorption edge (E0) for AgGaTe2. The solid curves represent thecalculated results using Eq. (5). The fit-determined parametervalues are listed in Table 1.

Table 1. Values of Eg ð0Þ, αp , Θp , and p in Eq. (5) for AgGaTe2a

CP Egð0Þ ðeVÞ αp ð10−4 eV=KÞ Θp (K) p dEg=dT ð10−4 eV=KÞE0

b 1.262 3.8 260 3.0 −3:6E0

c 1.238 3.8 220 3.5 −3:7E0A 1.36 2.5 190 4.8 −2:5E0B 1.19 2.6 190 5.0 −2:6E0C 2.00 2.4 180 4.8 −2:4E1 2.20 2.3 170 4.8 −2:3E2 2.67 2.4 180 4.8 −2:4E3 3.11 2.4 180 4.8 −2:4E4 3.34 2.3 170 4.8 −2:3E5 3.64 2.3 170 4.8 −2:3E6 3.78 2.3 170 4.8 −2:3E7 4.01 2.3 170 4.8 −2:3E8 4.40 2.3 170 4.8 −2:3E9 4.50 2.3 170 4.8 −2:3E10 4.80 2.3 170 4.8 −2:3E11 5.20 2.3 170 4.8 −2:3

aTemperature coefficients calculated from Eq. (6) are shown in the last column (T ¼ 300K).bFrom optical absorption ðE⊥cÞ.cFrom optical absorption ðE∥cÞ.


dT ∼ −3:6 × 10−4 eV=K (E⊥c) and dE0ðTÞ=dT ∼

−3:7 × 10−4 eV=K (E∥c). These values are also listedin Table 1.

B. Spectroscopic Ellipsometry Spectra

Figure 2 shows the εðEÞ spectra of AgGaTe2 for bothE⊥c and E∥cmeasured by SE. The experimental εðEÞspectra reveal several CP’s (open circles). The calcu-lated MDF results are shown by the solid curves.They are obtained from the sum of Eqs. (B1) and(B5) presented in Appendix B. The fit-determinedMDF parameters are listed in Table 2. It is notedthat the experimental ε1ðEÞ values are usually some-what larger than those calculated from our model. Toimprove the fit, therefore, we considered an addi-tional term, ε1∞, to ε1 (see Table 2). This term isassumed to be constant and may arise from otherhigher-lying gaps and core exciton transitions.The weak structure observed at E∼ 1:2 eV is due

to transitions at the fundamental absorption edge(E0). There is an accumulation of CP’s in the2–5 eV spectral region. This accumulation consistsof the E1 − E11 CP’s. We can also find a clear differ-ence in the polarization dependence of εðEÞ in the3–4:5 eV spectral region. The relatively strong peakobserved at ∼2:5 eVðE2Þ is mainly due to the two-dimensional (2D) excitonic and one-electron transi-tions. Excellent agreement is achieved between themeasured εðEÞ spectra and our model calculationover the entire range of photon energies.Optical spectra, such as the complex refractive in-

dex n�ðEÞ ¼ nðEÞ þ ikðEÞ, absorption coefficientαðEÞ, and normal-incidence reflectivity RðEÞ, canbe easily obtained from the present study in the formof practical functions, since they are directly relatedto the complex dielectric function εðEÞ ¼ ε1ðEÞþ

iε2ðEÞ. The real refractive index nðEÞ and extinctioncoefficient kðEÞ can now be given by [16]

nðEÞ ¼�½ε1ðEÞ2 þ ε2ðEÞ2� 1

2 þ ε1ðEÞ2

�12

; ð7aÞ

kðEÞ ¼�½ε1ðEÞ2 þ ε2ðEÞ2� 1

2 − ε1ðEÞ2

�12

: ð7bÞ

Fig. 2. Complex dielectric-function spectra εðEÞ ¼ ε1ðEÞ þ iε2ðEÞof AgGaTe2 for (a) E⊥c and (b) E∥c. The open circles are deter-mined by SE at T ¼ 300K. The solid curves are obtained fromthe sum of Eqs. (B1) and (B5), and ε1∞; see also Table 2.

Table 2. Critical-Point Parameters used in the Calculation of Optical Constants for AgGaTe2 at T ¼ 300K

Parameter E⊥c E∥c Parameter E⊥c E∥c

E0A (eV) 1.31 E6 (eV) 3.73 3.73E0B (eV) 1.14 1.14 C6 0.91 0.92E0C (eV) 1.95 1.95 γ6 0.27 0.27A0A (eV1:5) 0.64 E7 (eV) 3.96 3.96A0B (eV1:5) 0.14 1.00 C7 0.15 0.20A0C (eV1:5) 0.50 0.27 γ7 0.28 0.28Γ (meV) 30 30 E8 (eV) 4.35E1 (eV) 2.15 2.15 C8 1.17C1 0.14 0.35 γ8 0.23γ1 0.24 0.26 E9 (eV) 4.45E2 (eV) 2.59 2.59 C9 1.03C2 1.41 1.44 γ9 0.23γ2 0.30 0.26 E10 (eV) 4.75 4.75E3 (eV) 3.06 3.06 C10 0.46 0.24C3 1.13 0.84 γ10 0.23 0.23γ3 0.28 0.29 E11 (eV) 5.15 5.15E4 (eV) 3.29 C11 0.90 0.72C4 0.10 γ11 0.23 0.23γ4 0.28 E12 (eV) 6.30 6.30E5 (eV) 3.59 3.59 C12 1.30 1.30C5 0.10 0.38 γ12 0.30 0.30γ5 0.28 0.28 ε1∞ 1.46 1.53


The numerically calculated nðEÞ and kðEÞ spectra ofAgGaTe2 by introducing the MDF-derived εðEÞ va-lues into Eq. (7) are shown in Fig. 3 (solid curves).The open circles show the experimental data takenfrom SE. The open triangles also show the experi-mental data determined by Ohmer et al. [8] usingthe polarization interference method. The strongestpeaks seen at E∼ 2:6 eV in n and k are due to transi-tions at the E2 CP.The absorption coefficient αðEÞ and normal-

incidence reflectivity RðEÞ can be written as [16]

αðEÞ ¼ 4πλ kðEÞ; ð8Þ

RðEÞ ¼ ½nðEÞ − 1� 2 þ kðEÞ2½nðEÞ þ 1� 2 þ kðEÞ 2 ; ð9Þ

where λ is the light wavelength in the vacuum. Thecalculated spectral dependence of αðEÞ and RðEÞ forAgGaTe2 is shown in Fig. 4 (solid curves). The opencircles represent the experimental α and R data de-termined by SE. Like n and k, α and R reveal theweak, but distinct, CP structures in their opticalspectra.

C. Thermoreflectance Spectra

Modulation spectroscopies, such as TR and ER, mea-sure the change of the complex dielectric functionεðEÞ ¼ ε1ðEÞ þ iε2ðEÞ [26]:

ΔRR

¼ 1R∂R∂ε1

Δε1 þ1R∂R∂ε2

Δε2 ¼ αΔε1 þ βΔε2; ð10Þ

where the fractional coefficients α and β are functionsof photon energy and their sign and relative magni-tude determine the result of the analysis in the dif-ferent spectral region. In TR, the modulation oftemperature results in a change Δε1;2 of the crystal,which is induced by a shift of the bandgap energy Egand by the change of the broadening parameter Γ[22]:

Δε1;2 ¼ ∂ε1;2∂T

ΔT ¼ ∂ε1;2∂Eg

dEg

dTΔT þ ∂ε1;2

∂ΓdΓdT

ΔT: ð11Þ

Figures 5(a) and 5(b) show the TR spectra forAgGaTe2 obtained at temperatures between T ¼20K and 300K for E⊥c and E∥c, respectively. Thedominant structures seen at ∼1:2 eV are due to theE0αðα ¼ A;B;CÞ transitions. The CP structures arealso observed at ∼2:2 (E1), ∼2:7 (E2), ∼3:1 (E3),∼3:8 (E6), ∼4:4 (E8), ∼4:9 (E9), and ∼5:2 eV (E11).All these structures come from transitions at the Γ,N,P, orSpoint in theBrillouin zone (seeAppendixA).

We analyzed our measured TR spectra using theSCP model [19]. The SCP model gives

εðEÞ ¼ C − AeiθðE − Eg þ iΓÞ n; ð12Þ

where each CP is defined by the amplitude A, CPenergy Eg, broadening energy Γ, and phase angle θ.The exponent n has the value of −1=2 for one-dimensional CP, 0 [logarithmic, i.e., lnðE − EgþiΓÞ] for 2D CP, and 1=2 for three-dimensional (3D)CP. Discrete excitons can be represented by n ¼ −1.

In the present analysis, we take into account onlythe bandgap shift term ∂ε1;2=∂Eg in Eq. (11). This isbecause the contribution from the bandgap shift ismuch larger than that from the broadening term.

Fig. 3. Numerically calculated spectral dependence of the com-plex refractive index n�ðEÞ ¼ nðEÞ þ ikðEÞ for AgGaTe2 (solidcurves). The open circles represent the experimental data takenfrom SE. The open triangles also show the experimental data ob-tained using the polarization interference method [8].

Fig. 4. Numerically calculated spectral dependence of the absorp-tion coefficient αðEÞ and normal-incidence reflectivity RðEÞforAgGaTe2 (solid curves). The open circles represent the experimen-tal data determined by SE.


The α and β values used here were calculated fromthe experimental SE data and were assumed to showno strong temperature variation. The correspondingfirst-derivative spectrum of the dielectric functionwith respect to the bandgap energy Eg can be writtenas

dεðEÞdEg

¼�AeiθðE − Eg þ iΓÞ−1; n ¼ 0nAeiθðE − Eg þ iΓÞn−1; n ≠ 0

: ð13Þ

The open circles in Fig. 6 show the experimentalTR spectra for AgGaTe2 measured at T ¼ 20K.The solid curves represent the SCP fit results withn ¼ −1. We can see that the SCP model with n ¼−1 agrees well with the experimental ΔR=R spectra.The fit-determined phase angles varied from θ ¼ 9°to 350°. Note that the discrete-exciton line shapewith θ ≠ 0° corresponds to the Fano profile, i.e., theline shape that results from the interaction of the dis-crete exciton with a continuous background [27].The temperature-modulated CP parameters

ΔEg ¼ ðdEg=dTÞΔT determined from TR analysisare about −1 × 10−4 eV=K ðE0 − E11 CP’s). These va-lues are obtained fromminimizing the mean-squaresdeviation between the experimental and calculatedΔR=R spectra. The temperature coefficient dEg=dTcan, in principle, be either positive or negative, butour experimental results suggest that it is negativeand of the order of ∼10−4 eV=K. Therefore, we can es-timate the temperature modulation achieved in thepresent work to be ΔT ∼ 1K. It should be noted that

the positive temperature coefficient for the E0 gap atT below ∼100K was observed in chalcopyrite semi-conductors AgGaSe2 [28] and AgInSe2 [29,30]. Wecannot, however, find any such positive temperaturecoefficient in AgGaTe2. This may be because of thedifferent hybridization coefficients of the valence-band Se or Te p electrons with the 4d electrons of Ag.

We show in Fig. 7 the temperature variationsof the E0αðα ¼ A;B;CÞ and E1 CP energies inAgGaTe2 determined from the SCP analysis. The so-lid curves represent the calculated results usingEq. (5). The fit-determined Pässler’s parametersfor the En CPs (n ¼ 0 − 11), together with their tem-perature coefficients dEnðTÞ=dT at T ¼ 300K, calcu-lated from Eq. (6), are summarized in Table 1. It is

Fig. 5. TR spectra for AgGaTe2 measured at T ¼ 20K–300K.

Fig. 6. TR spectra for AgGaTe2 measured at T ¼ 20K. The solidcurves represent the fits using Eq. (13), with n ¼ −1, to the experi-mental data.

Fig. 7. Temperature variations of the E0αðα ¼ A;B;CÞ and E1

CP energies in AgGaTe2 determined from the SCP analysis.The fit-determined parameter values are summarized in Table 1.


understood from Fig. 7 that Eq. (5) can successfullyexplain the experimental data over the entire tem-perature range of T ¼ 20K–300K.Finally, we consider the quasi-cubic model pro-

posed by Thomas and Hopfield [31,32]. Under thespin–orbit interaction, represented by thematrix ele-mentΔso, and that of the crystal field, represented byΔcr, the energy differences of the split-off valencebands are given by

EBA ¼ E0B − E0A

¼ Δso þΔcr

2−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Δso þΔcr

2

�2−23ΔsoΔcr

s;

ð14aÞ

ECA ¼ E0C − E0A

¼ Δso þΔcr

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�Δso þΔcr

2

�2−23ΔsoΔcr:

s

ð14bÞ

In Fig. 8 we schematically show the conduction-and valence-band structures at the Γ point ofAgGaTe2. The Γ6 conduction band is predominantlys-like in character. The essential difference betweenthe potential that an electron experiences in an idealchalcopyrite lattice and that in a zinc-blende latticeis the relatively small difference in crystal field dueto a site beyond the next-nearest neighbors. Thecrystal-field perturbation (Δcr), without spin–orbitcoupling (Δso), causes a splitting of the p-like Γ15 va-lence band into Γ5 and Γ4 states (single-group nota-tion). The combined effect of both spin–orbit andcrystal-field perturbations in chalcopyrite latticesplits the p-like Γ15 valence band into Γ6ðAÞ,Γ7ðBÞ, and Γ7ðCÞ states, while the wave functionfor each band may be written as a linear combinationof px, py, and pz and spin function, provided the inter-action of the Γ6 and Γ7 levels with the conduction

Fig. 8. Relationships between the valence-band splitting in thezinc-blende and chalcopyrite lattices and between the irreduciblerepresentations of the bands at the Γ point. The selection rules ofthe optical dipole transitions are also shown in (c) and (d).

Fig. 9. Electronic energy-band structure of AgGaTe2 calculatedby EPM. The main interband optical transitions are indicatedby vertical arrows.

Table 3. Experimentally Determined Critical Point Energies at T ¼ 0K in AgGaTe2, with Those Obtained from the Empirical PseudopotentialMethoda

CP E⊥c E∥c

Expt. (eV) Calc. (eV) Transition Expt. (eV) Calc. (eV) Transition

E0A 1.36 1.2 Γð29; 30Þ → Γð33;34ÞE0B 1.19 1.1 Γð31; 32Þ → Γð33;34Þ 1.19 1.1 Γð31;32Þ → Γð33;34ÞE0C 2.00 1.7 Γð27; 28Þ → Γð33;34Þ 2.00 1.7 Γð27;28Þ → Γð33;34ÞE1 2.20 2.5 Γð31; 32Þ → Γð35;36Þ 2.20 2.5 Γð31;32Þ → Γð35;36ÞE2 2.67 2.9 Nð31; 32Þ → Nð33; 34Þ 2.67 2.9 Nð31;32Þ → Nð33;34ÞE3 3.11 3.0 Nð31; 32Þ → Nð35; 36Þ 3.11 3.0 Nð31;32Þ → Nð35;36ÞE4 3.34 3.1 Pð31; 32Þ → Pð33;34ÞE5 3.64 3.3 Pð31; 32Þ → Pð37;38Þ 3.64 3.3 Pð31; 32Þ → Nð37; 38ÞE6 3.78 3.7 Nð31; 32Þ → Nð37; 38Þ 3.78 3.7 Nð31;32Þ → Nð37;38ÞE7 4.01 4.2 Nð27; 28Þ → Nð39; 40Þ 4.01 4.2 Nð27;28Þ → Nð39;40ÞE8 4.40 4.4 Sð29;30Þ → Sð37;38ÞE9 4.50 4.6 Sð31; 32Þ → Sð39;40ÞE10 4.80 4.7 Pð27; 28Þ → Pð39;40Þ 4.80 4.7 Pð27;28Þ → Pð39;40ÞE11 5.20 5.1 Nð31; 32Þ → Nð41; 42Þ 5.20 4.9 Nð31;32Þ → Nð41;42Þ

aThe valence bands are labeled 1–32, while the conduction bands are from 33. (Note that only the valence bands labeled 9–32 fall in theenergy range between −5 and þ6 eV, see Fig. 9.) Transitions using these notations are given in parentheses.


band is neglected. This result is known as the quasi-cubic model [31,32].The lowest direct bandgaps determined from the

SCP analysis are E0A ¼ 1:36 eV, E0B ¼ 1:19 eV, andE0C ¼ 2:00 eV at T ¼ 20K. Introducing these band-gap energies into Eq. (14), we obtain Δso ¼ 0:70 eVand Δcr ¼ −0:23 eV. These values largely differ fromthose obtained by Sermage et al. [5], who used reflec-tance spectroscopy (Δso ∼ 0:85 eV andΔcr ∼ −0:16 eVat T ¼ 300K).Using the values of Δso ¼ 0:70 eV and

Δcr ¼ −0:23 eV, we obtain the squared p-matrix ele-ments P2 ¼ j < cjpjv > j2 between the p-like valenceand the s-like conduction band [21]:

P20A⊥ : P2

0B⊥ : P20C⊥ : P2

0Ajj : P20Bjj : P

20Cjj

¼ 1∶0:21∶0:79∶0∶1:58∶0:42 ð15Þ

The strength parameters A0αðα ¼ A;B;CÞ in Eq. (B1)are constrained to satisfy the above strength ratio inthe present MDF analysis (see Table 2).

4. Conclusions

We studied the optical properties of the chalcopyritesemiconductor AgGaTe2 by means of optical absorp-tion, SE, and TR spectroscopy. AgGaTe2 single crys-tals were grown from stoichiometric melts by theconventional Bridgman method. The optical absorp-tion spectra suggested that the material has a directenergy gap at ∼1:2 eV at 300K. The εðEÞ spectra ofAgGaTe2 were determined using SE in the photon-energy range between 1.2 and 5:2 eV. The measuredεðEÞ spectra revealed distinct structures at the EnCPs (n ¼ 0–11). The temperature-dependent CP en-ergies were determined by performing SCP analysis.These CPs were assigned to specific points in theBrillouin zone by the aid of a band-structure calcula-tion using EPM. The spin–orbit and crystal-fieldparameters of AgGaTe2 were also determined tobe Δso ¼ 0:70 eV and Δcr ¼ −0:23 eV.

Appendix A: Electronic Energy-Band Structure

The electronic energy-band structure of AgGaTe2has been studied theoretically by several authors[33–38]. We have calculated the electronic energy-band structure of AgGaTe2 using the EPM. The pseu-dopotential form factors for Ag, Ga, and Te are takenfrom [39]. The spin–orbit interaction is taken intoconsideration in the calculation. Figure 9 showsthe result of this calculation. Because no experimen-tal information about the spin–orbit splitting ofAgGaTe2 is available, we used the splitting para-meter of ZnTe [40] as that of this chalcopyrite semi-conductor. The calculated energy-band structure ofAgGaTe2 is similar in overall appearance to thoseof Cu-based AIBIIICVI

2 chalcopyrite semiconductorsobtained by the first-principles calculation [41].TheE0A,E0B, andE0C transitions are dominated at

the Γ point. The E1 peak may be due to transitionsfrom the highest valence to the second lowest conduc-

tion band at the Γ point. The E2 and E3 peaks arisefrom transitions at the N point. The E4 structureseen only for E⊥c originates from transitions nearthe P point. TheE5 structure may also originate fromtransitions near the P point. TheE6 and E7 peaks aremainly caused by transitions at the N point. The E8and E9 structures, which are observed only for E∥cand E⊥c, respectively, are principally arising fromtransitions at the S point. The E10 transitions maybe dominated at the P point, while the highest CPE11 observed in the present study may be due to tran-sitions near the N point. The εðEÞ spectrum is alsocalculated from our obtained energy-band structureof AgGaTe2. Table 3 summarizes the main contribu-tions to εðEÞ of the interband transitions in the spe-cific parts of the Brillouin zone of AgGaTe2.

Appendix B: Model Dielectric Function

The E0 gaps in chalcopyrite semiconductors may beof the 3D M0 CP. Assuming the valence and conduc-tion bands are parabolic and using the Kramers–Kronig transformation, we obtain the contributionof these gaps to εðEÞ [16]:

εðEÞ ¼X

α¼A;B;C

A0αE−1:50α f ðx0αÞ; ðB1Þ

with

A0α ¼43

�32μ0α

�1:5

P20α; ðB2Þ

f ðx0αÞ ¼ x−20α ½2 − ð1þ x0αÞ1=2 þ ð1 − x0αÞ1=2�; ðB3Þ

x0α ¼ ðEþ iΓÞ=E0α: ðB4Þ

In Eqs. (B1)–(B4), μ0α is the combined density-of-states mass, P2

0α is the squared momentum-matrixelement, Γ is the broadening energy of the E0αgap, and αð¼ A;B;CÞ stands for the Γ6ðAÞ → Γ6,Γ7ðBÞ → Γ6, and Γ7ðCÞ → Γ6 transitions.

The fundamental optical spectra of AgGaTe2 re-veal several CPs at energies higher than the E0gap. These CPs are labeled Eiði ≥ 1Þ and may corre-spond to transitions at points Γ, N, T, etc. in theBrillouin zone (see Fig. 9 and Table 3). Here we sim-ply assume the contributions to εðEÞ from these CPsas damped harmonic oscillators (DHOs) [16]:

εðEÞ ¼ Cn

ð1 − x2nÞ − ixnγn; ðB5Þ

withxn ¼ E=En; ðB6Þ

where Cn and γn are, respectively, the strength andnondimensional broadening parameters of the nthDHO. Note that the DHO is a different representa-


tion of a 2D M1 CP, both with and without the exis-tence of the excitonic interaction [42].

Theauthors thankT.MiyazakiandT.Nakamura fortheir experimental support and fruitful discussion.

References1. S. Siebentritt and U. Rau, Wide-Gap Chalcopyrites (Springer,

2006).2. O. Madelung, Semiconductors: Data Handbook (Springer,

2004).3. H. Hahn, G. Frank, W. Klingler, A. D. Meyer, and G. Störger,

“Über einige ternäre chalkogenide mit chalkopyritstruktur,”Z. Anorg. Allg. Chem. 271, 153–170 (1953).

4. B. Tell, J. L. Shay, and H. M. Kasper, “Some properties ofAgAlTe2, AgGaTe2, and AgInTe2,” Phys. Rev. B 9, 5203–5208 (1974).

5. B. Sermage, F. Barthe-Lefin, and A. C. Papadopoulo-Scherle, “Variation avec la température de la bande interditedu champ cristallin et du couplage spin-orbite en centre dezone de AgGaSe2 et AgGaTe2,” J. Phys. Colloq. 36, C3-137(1975).

6. N. N. Konstantinova and Y. V. Rud’, “Optical properties ofAgGaTe2 single crystals,” Sov. Phys. Semicond. 23, 1101–1104(1989).

7. C. Julien, I. Ivanov, A. Khelfa, F. Alapini, and M. Guittard,“Characterization of the ternary compounds AgGaTe2 andAgGa5Te8,” J. Mater. Sci. 31, 3315–3319 (1996).

8. M. C. Ohmer, J. T. Goldstein, D. E. Zelmon, A. W. Saxler, S. M.Hegde, J. D. Wolf, P. G. Schunemann, and T. M. Pollak, “Infra-red properties of AgGaTe2, a nonlinear optical chalcopyritesemiconductor,” J. Appl. Phys. 86, 94–99 (1999).

9. P. G. Schunemann, S. D. Setzler, T. M. Pollak, M. C. Ohmer, J.T. Goldstein, and D. E. Zelmon, “Crystal growth and proper-ties of AgGaTe2,” J. Cryst. Growth 211, 242–246 (2000).

10. I. V. Bodnar, V. F. Gremenok, R.W.Martin, andM.V. Yakushev,“Photoluminescence spectra of the AgGaTe2 single crystalsdoped with hydrogen,” Opt. Spectrosc. 88, 377–379 (2000).

11. U. N. Roy, B. Mekonen, O. O. Adetunji, K. Chattopahhyay, F.Kochari, Y. Cui, A. Burger, and J. T. Goldstein, “Compositionalvariations and phase stability during horizontal Bridgmangrowth of AgGaTe2 crystals,” J. Cryst. Growth 241, 135–140(2002).

12. R. Kumar and R. K. Bedi, “Characterization of thermallyevaporated AgGaTe2 films grown on KCl substrates,”J. Mater. Sci. 40, 455–459 (2005).

13. J. Krustok, A. Jagomägi, M. Grossberg, J. Raudoja, andM. Danilson, “Photoluminescence properties of polycrystallineAgGaTe2,” Sol. EnergyMater. Sol. Cells 90, 1973–1982 (2006).

14. H. G. Tompkins and E. A. Irene, Handbook of Ellipsometry(William Andrew, 2005).

15. H. Fujiwara,Spectroscopic Ellipsometry: Principles and Appli-cations (Wiley, 2007).

16. S. Adachi, Optical Properties of Crystalline and AmorphousSemiconductors: Materials and Fundamental Principles(Kluwer Academic, 1999).

17. R. K. Willardson and A. C. Beer, Semiconductors and Semime-tals (Academic, 1972), Vol. 9.

18. D. E. Aspnes, “Direct verification of the third-derivativenature of electroreflectance spectra,” Phys. Rev. Lett. 28,168–171 (1972).

19. P. Lautenschlager, M. Garriga, L. Viña, and M. Cardona,“Temperature dependence of the dielectric function and inter-band critical points in silicon,” Phys. Rev. B 36, 4821–4830(1987).

20. S. Logothetidis, M. Cardona, P. Lautenschlager, andM. Garriga, “Temperature dependence of the dielectric func-

tion and interband critical points of CdSe,” Phys. Rev. B 34,2458–2468 (1986).

21. T. Kawashima, S. Adachi, H. Miyake, and K. Sugiyama,“Optical constants of CuGaSe2 and CuInSe2,” J. Appl. Phys.84, 5202–5209 (1998).

22. E. Matatagui, A. G. Thompson, and M. Cardona, “Thermore-flectance in semiconductors,” Phys. Rev. 176, 950–960 (1968).

23. Y. P. Varshni, “Temperature dependence of the energy gap insemiconductors,” Physica 34, 149–154 (1967).

24. R. Pässler, “Basic model relations for temperature dependen-cies of fundamental energy gaps in semiconductors,” Phys.Stat. Solidi B 200, 155–172 (1997).

25. R. Pässler, “Parameter sets due to fittings of the temperaturedependencies of fundamental bandgaps in semiconductors,”Phys. Stat. Solidi B 216, 975–1007 (1999).

26. B. O. Seraphin and N. Bottka, “Band-structure analysis fromelectro-reflectance studies,” Phys. Rev. 145, 628–636 (1966).

27. P.Lautenschlager,M.Garriga,S.Logothetidis,andM.Cardona,“Interband critical points of GaAs and their temperaturedependence,” Phys. Rev. B 35, 9174–9189 (1987).

28. S. Ozaki, M. Sasaki, and S. Adachi, “Positive temperaturevariation of the bandgap energy in AgGaSe2,” Phys. Stat.Solidi A 203, 2648–2652 (2006).

29. S. Ozaki and S. Adachi, “Optical absorption and photolumi-nescence in the ternary chalcopyrite semiconductorAgInSe2,” J. Appl. Phys. 100, 113526 (2006).

30. S. Ozaki and S. Adachi, “Temperature dependence of thelowest-direct-bandgap energy in the ternary chalcopyritesemiconductor AgInSe2,” J. Mater. Sci. Mater. Electron. 18,S25–S28 (2007).

31. D. G. Thomas and J. J. Hopfield, “Exciton spectrum of cad-mium sulfide,” Phys. Rev. 116, 573–582 (1959).

32. J. J. Hopfield, “Fine structure in the optical absorption edgeof anisotropic crystals,” J. Phys. Chem. Solids 15, 97–107(1960).

33. A. S. Poplavnoi, Y. I. Polygalov, and A. M. Ratner, “Energyband structure of the compounds AgGaS2, AgGaSe2, andAgGaTe2,” Sov. Phys. J. 17, 1495–1499 (1974).

34. R. Asokamani, R. M. Amirthakumari, R. Rita, and C. Ravi,“Electronic structure calculations and physical properties ofABX2 (A ¼ Cu;Ag; B ¼ Ga; In; X ¼ S;Se;Te) ternary chalco-pyrite systems,” Phys. Stat. Solidi B 213, 349–363 (1999).

35. S. N. Rashkeev and W. R. L. Lambrecht, “Second-harmonicgeneration of I − III − VI2 chalcopyrite semiconductors:Effects of chemical substitutions,” Phys. Rev. B 63, 165212(2001).

36. A. H. Reshak, “Linear, nonlinear optical properties and bire-fringence of AgGaX2 (ðX ¼ S;Se;TeÞ compounds,” Phys. B369, 243–253 (2005).

37. L. Bai, Z. Lin, Z. Wang, and C. Chen, “Mechanism of linear andnonlinear optical effects of chalcopyrite AgGaX2 (X ¼ S, Se,and Te) crystals,” J. Chem. Phys. 120, 8772–8778 (2004).

38. C. Mitra and W. R. L. Lambrecht, “Band-gap bowing inAgGaðSe1−xTexÞ2anditseffectonthesecond-orderresponsecoef-ficient and refractive indices,” Phys. Rev. B 76, 205206 (2007).

39. M. L. Cohen and V. Heine, in Solid State Physics, H.Ehrenreich, F. Seitz, and D. Turnbull, eds. (Academic, 1970),Vol. 24.

40. J. P. Walter, M. L. Cohen, Y. Petroff, and M. Balkanski,“Calculated and measured reflectivity of ZnTe and ZnSe,”Phys. Rev. B 1, 2661–2667 (1970).

41. J. E. Jaffe and A. Zunger, “Electronic structure of the ternarychalcopyrite semiconductors CuAlS2, CuGaS2, CuInS2,CuAlSe2, CuGaSe2, and CuInSe2,” Phys. Rev. B 28, 5822–5847(1983).

42. S. Adachi, “Excitonic effects in the optical spectrum of GaAs,”Phys. Rev. B 41, 1003–1013 (1990).


Documents

Optical properties and electronic band structure of AgGaTe_2 chalcopyrite semiconductor