Physica B 406 (2011) 3299–3302

Contents lists available at ScienceDirect

Physica B

0921-45

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/physb

Theoretical studies on the structural, electronic, and optical propertiesof Cu2CdGeSe4

Dan Li a, Furi Ling b, Zhenye Zhu a, Xinghong Zhang a,n

a Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, PR Chinab Wuhan National Laboratory for Optoelectronics, College of Optoelectronic Science and Engineering, Huazhong University of Science and Technology,

Wuhan, Hubei 430074, PR China

a r t i c l e i n f o

Article history:

Received 28 February 2011

Received in revised form

20 May 2011

Accepted 20 May 2011Available online 27 May 2011

Keywords:

Stannite

Density functional theory

Generalized gradient approximation

26/$ - see front matter & 2011 Elsevier B.V. A

016/j.physb.2011.05.044

esponding author. Tel./fax: þ86 755 2603375

ail address: lidanhanadusino@21cn.com (X. Z

a b s t r a c t

We investigate the electronic structure for Cu2CdGeSe4 in stannite structure with the first-principles

method. This crystal is the direct band gap compound. In addition, the dielectric function, absorption

coefficient, reflectivity, and energy-loss function are studied using the density functional theory within

the generalized gradient approximation. We discuss the optical transitions between the valence bands

and the conduction bands in the spectra of the imaginary part of the dielectric function at length.

We also find a very high absorption coefficient and wide absorption spectrum for this material. The

prominent structures in the spectra of reflectivity and energy-loss function are discussed in detail.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

Recently, the quaternary compounds I2–II–IV–VI4 (I¼Cu, Ag;II¼Zn, Cd, Hg; IV¼Si, Ge, Sn; VI¼S, Se) have drawn wide interestfor their potential applications in thin-film solar cells due to theirp-type conductivity and high optical absorption (41�104 cm�1)[1–11]. They are synthesized by replacing half of the III atoms internary I–III–VI2 (I¼Cu, Ag; III¼Al, Ga, In; VI¼S, Se) with II atoms andthe other half with IV atoms through the cation cross-substitu-tion [12,13]. Some groups reported that quaternary I2–II–IV–VI4compounds have four typical types of crystal structures [14],namely kesterite (KS), stannite (ST), wurtzite–kesterite (WKS), andwurtzite–stannite (WST) structures. Among these quaternary alloysCu2CdGeSe4 has strong optical anisotropy and the energy gap value isequal to 1.29 eV [15] or 1.20 eV [16]. Two polymorphous modifica-tions were reported in Refs. [17,18]: an orthorhombic high-tempera-ture modification with the wurtzite–stannite structure (space groupPmn21, a¼0.80968(9), b¼0.68929(6), and c¼0.66264(6) nm), and atetragonal low-temperature modification with the stannite structure(space group I42m, a¼0.57482(2) and c¼1.10533(3) nm [17]). Andwith the calculations of the total energy the ST structure ofCu2CdGeSe4 is more stable than the others, hence this phase is theground-state structure. Its phase stability and electronic structurehave been studied with ab initio calculations [14,19,20]. But toour knowledge there is no systemic report about electronic and

ll rights reserved.

9.

hang).

optical properties of Cu2CdGeSe4 using the first-principles method.In addition, increasing number of elements and structural freedommake their physical and chemical properties more flexiblerelative to binary or ternary compound. With such inducement wewill study the structural, electronic, and optical properties forCu2CdGeSe4 in stannite phase with the first-principles methods indetail.

2. Method of calculation

The calculations are performed based on the density functionaltheory (DFT) using ultrasoft pseudopotential and plane-wavepseudopotential method in the CASTEP (CAmbridge Serial TotalEnergy) package [21,22]. The exchange-correlation energy of theelectrons is described in the generalized gradient approximation(GGA) in the scheme of Perdew et al. [23]. During whole calculationwe only consider the valence electrons; the electronic config-urations of constituent elements are Cu(3d104s1), Cd(4d105s2),Ge(4s24p2), and Se(4s24p4), respectively.

The quaternary compound Cu2CdGeSe4 is optimized for theprimitive cells with a specific 8�8�8 k-point grid accordingto the Monkhorst–Pack scheme [24]. The tolerance in the self-consistent field (SCF) calculation is 1.0�10�6 eV/atom. Weemploy 350 eV as the plane-wave cutoff energy. For the calcula-tions of the electronic and optical properties 8�8�8 and16�16�16 k-point grids are used for numerical integrationsover the Brillouin zone (BZ).

0

2

4

6

8

10

12

14

16

18

20

TDO

S (e

lect

rons

/eV

)

D. Li et al. / Physica B 406 (2011) 3299–33023300

3. Result and discussion

3.1. Structural parameters

We use the initial crystallographic parameters analyzed by X-raydiffraction (XRD) as a beginning for geometry optimization. Thecalculated equilibrium lattice constants (a and c) and u-parameters(ux(Se) and uz(Se)) for Cu2CdGeSe4 are compared with the previousdata in Table 1. The parameter ux(Se) is the x-coordinate of Se, anduz(Se) is the z-coordinate of Se in the unit cell. Our calculated latticeparameters a and c are in agreement with the available experimentaland theoretical results [18,20,25–27]. And the two theoreticalu-parameters have satisfactory agreement with their experimentalvalues [17]. It can be seen that the optimized structure is reasonable.Therefore the following calculations about the electronic and opticalproperties for this compound are very credible.

-16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10

Energy (eV)

Fig. 2. Total density of states for Cu2CdGeSe4.

16

02468

Cu 4s Cu 3d

3.2. Electronic structure

We calculate the energy band structure of Cu2CdGeSe4 andpresent it in Fig. 1(a). A similar electronic structure of stannite-typeCu2CdSnSe4 is shown in Fig. 2(b) for reference. The valence bandmaximums (VBMs) and conduction band minimums (CBMs) of bothcrystals occur at G point, so they are direct gap semiconductorsG–G). The calculated band gap of Cu2CdGeSe4 (0.017011 eV) isbigger than that of Cu2CdSnSe4 (0.000484 eV). This result hassatisfactory agreement with the previous experimental report [16].

Table 1Equilibrium lattice constants a, c, and u-parameters ux(Se) and uz(Se) of stannite-

type Cu2CdGeSe4.

a (A) c (A) ux (Se) Uz (Se)

Cu2CdGeSe4

This work 5.75216 10.9454 0.26143 0.13911

Expt. 5.7482(2)a 11.0533(3)a 0.2705(2)a 0.1337(1)a

5.657b, 5.65c 10.988b, 10.96c

5.7489d, 5.753e 11.055d, 11.057e

Other calc. 5.500f 11.0605f

a Ref. [22].b Ref. [18].c Ref. [19].d Ref. [20].e Ref. [21].f Ref. [13].

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

Γ

Ene

rgy

(eV)

Z Γ X P N

0.017011 eV

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

0.000484 eV

ΓΓZ X P N

Fig. 1. Band structures of (a) Cu2CdGeSe4 and (b) Cu2CdSnSe4.

-16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 100.00.40.81.21.62.0 Se 4s Se 4p

PD

OS

(ele

ctro

ns/e

V)

Energy (eV)

0.00.40.81.21.62.0

Ge 4s Ge 4p

048

12 Cd 5s Cd 4d

Fig. 3. Partial density of states for Cu2CdGeSe4.

Both the band gaps are underestimated compared with the experi-mental values of 1.20 and 0.96 eV [16]. Therefore, we cannot discussthe absolute band gaps simulated with the GGA functional. In aprevious study Maeda and Wada [28] reported that the screenedexchange-LDA (sX-LDA) calculation greatly improved the band gapof chalcopyrite-type CuInSe2 (0.96 eV). With such functional wesuccessfully obtained a band gap of Cu2CdGeSe4 as 1.12 eV, which isbigger than that of CuInSe2 but in good agreement with experi-mental value of 1.2 eV.

Figs. 2 and 3 show total density of the states (TDOS) andpartial density of states (PDOS) for Cu2CdGeSe4. The VBM ismainly dominated by Cu 3d and Se 4p states, while the CBM isdominated by Se 4p and Ge 4s states. Moreover, Se 4s and 4pstates are clearly located between �15 and �13 eV, and between�6 and 0 eV in the valence bands (VBs), respectively. The VBsat about �9 eV are mainly influenced by Cd 4d states, whichpresent a sharp peak because of its strong localization character.A obvious hybridization between Se 4p and Cu 3d states isobserved in the energy range from �4 eV to the Fermi level.The Ge 4s states dominate a narrow band located 8–10 eV below

0 2 4 6 8 10 12 14 16 18 20 22 240

5

10

15

20

L

Energy (eV)

0.0

0.2

0.4

0.6

0.8

1.0

R

05

1015202530

Fig. 5. Optical constants for Cu2CdGeSe4: (a) absorption coefficient I(o),

(b) reflectivity R(o), and (c) energy-loss spectrum L(o).

D. Li et al. / Physica B 406 (2011) 3299–3302 3301

the Fermi level. While the lower conduction bands (CBs) (o2 eV)are dominated by the hybridization of Ge 4s and Se 4p states, andthe higher CBs (Z2 eV) mainly consist of Cd 5s, Ge 4s, and 4pstates. Apparently, this crystal has some covalent features.

3.3. Optical properties

The dielectric function e(o) of Cu2CdGeSe4 is calculated basedon the electronic structure. The imaginary part e2(o) of thedielectric function e(o) can be obtained from the momentummatrix elements between the occupied and unoccupied electronicstates, which is given as follows [29]:

e2ðoÞ ¼2e2pOe0

Xk,v,c

9/cck9u dr9cv

kS92dðEck�Ev

k�EÞ, ð1Þ

where the integral is over the first Brillouin zone (BZ), o is thelight frequency, e is the electronic charge, and cc

k and cvk are the

conduction and valence band wave functions at k, respectively.According to the Kramers–Kronig transformation, the real parte1(o) of the dielectric function e(o) can be obtained from theimaginary part e2(o) [30]

e1ðoÞ ¼ 1þ2

pP

Z 10

o0e2ðo0Þo02�o2

do0 ð2Þ

where P implies the principal value of the integral.The Cu2CdGeSe4 crystal has strong optical anisotropy, hence

we calculate the average imaginary part e2(o) of the dielectricfunction e(o) by e2ðoÞ ¼ ðex

2ðoÞþey2ðoÞþe

z2ðoÞÞ=3, where ex

2ðoÞ(¼ey

2ðoÞ) is the average of the spectra for polarization along thex and y directions and ez

2ðoÞ corresponding to the z direction.The average real part e1(o) is obtained from Eq. (2). We show theaverage imaginary part and the real part of the dielectric functionspectra of Cu2CdGeSe4 in Fig. 4 from 0 to 25 eV. There are fourprominent peaks in e2(o) spectrum for this compound. Peaks A(1.51 eV) and B (3.78 eV) are mainly ascribed to the transitionsfrom Cu 3d and Se 4p states in the VBs to Ge 4s states in the CBs.Peaks C and D are situated at around 4.97 and 5.98 eV, which maycorrespond to the transitions from Ge 4p, Se 4p, and Cd 5s statesin the VB to the band-edge of the CBs. From the e1(o) spectrum inFig. 4 we also can easily obtain the static dielectric constant forCu2CdGeSe4, corresponding to the value of 14.32.

We also calculate the absorption coefficient I(o), reflectivity R(o),and energy-loss function L(o) and show them in Fig. 5(a)–(c).

0 2 4 6 8 10 12 14 16 18 20 22 24-4

0

4

8

12

16

Die

lect

ric F

unct

ion

Energy (eV)

ε1ε2

A

BC

D

Fig. 4. Imaginary part e2(o) and real part e1(o) of the dielectric function for

Cu2CdGeSe4.

The optical absorption coefficient is one of the most crucial evalua-tion criterions in application of the photoelectric materials. We

calculate its spectrum by IðoÞ ¼ffiffiffi2p

o½ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie1ðoÞ2þe2ðoÞ2

q�e1ðoÞ�1=2for

Cu2CdGeSe4. From Fig. 5(a) we observe a very high absorptioncoefficient (4104 cm�1) and a wide absorption spectrum(0–21 eV) for this material, hence it could be a good substitutefor CIGS as absorber layer in thin-film solar cells. In Fig. 5(b), thereis a prominent peak located at 11.84 eV for Cu2CdGeSe4. To ourknowledge the energy-loss function is usually used to describethe energy-loss of a fast electron traversing in the material [31].Its main peak is always defined as the bulk plasma frequency,which occurs e2(o)o1 and e1(o) reaches the zero point [32]. Thesharp structure is located at 16.10 eV for Cu2CdGeSe4, corre-sponding to a rapid decrease of reflectance. This process isassociated with the transitions from the Se s mixed with Ge sstates to the CBs. To our knowledge there have not been anydetailed theoretical results on the electronic and optical proper-ties of Cu2CdGeSe4 in stannite phase, including the dielectricfunction, absorption coefficient, reflectivity, and energy-loss func-tion. Therefore, we hope that our results could serve as areference for future study.

4. Conclusions

In conclusion we report a systematic study of the structural,electronic, and optical properties of Cu2CdGeSe4 in stannite phaseusing the DFT with GGA. Our calculated lattice constants are ingood agreement with the previous data. We discuss the energyband structure, TDOS, and PDOS for this material in detail. It is adirect gap semiconductor and the band gap is bigger than that ofCu2CdSnSe4. The VBM is made up of Cu 3d and Se 4p orbits, andthe CBM is an antibonding of Ge 4s and Se 4p orbits. In additionwe simulate many optical spectra including the dielectric func-tion, absorption coefficient, reflectivity, and energy-loss function.The electron transition in e2(o) spectra is studied at length. Thehigh absorption of Cu2CdGeSe4 is very important for applicationin absorber material of thin-film solar cells. We also discuss theprominent peaks of R(o) and L(o) spectra.

D. Li et al. / Physica B 406 (2011) 3299–33023302

Acknowledgments

This project was supported by the National Natural ScienceFoundation of China (no. 10902029).

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