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First-principles study of the elastic, electronic and optical properties of ε-GaSe layered semiconductor
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First-principles study of the elastic, electronic and optical properties of ε-GaSe layered semiconductor Shun-Ru Zhang a,n, Shi-Fu Zhu b, Bei-Jun Zhao b, Lin-Hua Xie c, Kei-Hui Song a a
Department of Physics and Information Engineering, Huaihua University, Huaihua 418008,China Department of Materials Science, Sichuan University, Chengdu 610064, China c Institute of Solid State Physics & School of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610066, China b
art ic l e i nf o
a b s t r a c t
Article history: Received 27 February 2013 Received in revised form 1 December 2013 Accepted 9 December 2013
The elastic, electronic and optical properties of ε-GaSe layered semiconductor have been studied from the first principles with the local density approximation (LDA). The optimized structure of GaSe has been found to be in good agreement with the experimental results. The mechanical stability of ε-GaSe is confirmed by calculations of the elastic constants. The calculated band structure shows that the crystal has a direct band gap of 0.816 eV. Furthermore, the linear photon-energy-dependent dielectric functions and some optical constants such as refractive index, extinction, reflectivity and absorption coefficients have been calculated. & 2013 Published by Elsevier B.V.
Keywords: GaSe Density functional theory Elastic constants Electronic structure Optical properties
1. Introduction In recent years, III–VI layered semiconductors have attracted a lot of interests because they have remarkable nonlinear optical properties and are promising for optical switching devices [1–4]. GaSe is a member of the family of the III–VI type semiconductor crystals, crystallizing in a layered structure in which each layer consists of four atoms stacked along the c-axis with a repeating unit of Se–Ga–Ga–Se. These layered compounds are characterized by highly anisotropic bonding forces [5]. In GaSe, intralayerbonding forces are the ionic covalent type and interlayer ones are the van der Waals type, which causes a weak interlayer interaction between layers. Therefore, GaSe can easily be cleaved along these layers. Several polytypes have been reported, which differ in the stacking sequence of the basic layer units [2]. Bridgman grown samples are commonly found to follow the hexagonal structure with D13h symmetry [6]. Many experiments have been carried out to grow and characterize GaSe crystal [6–10], the properties of GaSe at ambient conditions are reasonably well know [11], various important applications of GaSe in the areas of broadband mid-infrared electromagnetic waves, terahertz (THz) generator/detectors and radiation detectors were reported [12,13]. Recently, the phase transition from the ε-GaSe to the NaCl-type structure motif is
n
Corresponding author. Tel.: þ 86 745 2851011. E-mail address: [email protected] (S.-R. Zhang).
observed near 21 GPa [14,15]. To improve the optical, thermal and mechanical properties, further experiments were reported about GaSe doped with various substitutional impurities (In, Er, S, and Te) [16–21]. In the aspect of theory investigations, the first-principles calculations of the bulk GaSe band structure and elastic properties have been reported in Refs. [22–24], while electronic properties of two dimensional GaSe crystals were investigated theoretically and experimentally in Ref. [25], Jahangirli [26] studied of the phonon spectrum of ε-GaSe, Rak et al. [27] investigated the electronic structure and mechanical properties for pure and doped GaSe. Despite large number of experimental studies on this material, it is seen that there are no theoretical reports of the optical properties ε-GaSe. The present study focuses on the elastic, electronic and optical properties of ε-GaSe crystal using density functional theory with the local density approximation (LDA). The article is organized as follows: in Section 2, we briefly describe the computational method used in the present work. Results and discussion will be presented in Section 3. A summary of the work will be given in Section 4. 2. Calculation details The crystal lattice parameters and atom positions of ε-GaSe have been optimized using density functional theory (DFT) based on a plane-wave pseudo-potential technique implemented in the CASTEP package [28]. The plane-wave energy cutoff and the
0921-4526/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.physb.2013.12.014
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Brillouin zone sampling were fixed at 330 eV and 16 16 3 Monkhorst and Pack [29] special k-point meshes, respectively. In this work, the interactions between the valence electrons and the ion cores have been modeled by the ultrasoft pseudo-potential [30]. The valence electronic configurations assumed for the atoms in GaSe are as Ga: 3d104s24p1 and Se: 4s24p1. The electronic exchange-correlation energy is evaluated with the local-density approximation (LDA) [31], using Perdew–Zunger parametrization [32] of Ceperley–Adler electron-gas data [33]. The Broyden– Fletcher–Goldfarb–Shanno (BFGS) algorithm [34] was used to minimize the total energy and internal forces. The tolerances for geometry optimization were set as followings: the difference in total energy being within 1 106 eV/atom, the maximum ionic Hellmann–Feynman force being within 0.01 eV/Å, the maximum stress being within 0.02 GPa and the maximum ionic displacement being within 1 104 Å.
3. Results and discussions 3.1. Volume optimization The ε-GaSe crystal is hexagonal and belongs to the space group P-6m2 (No. 187). The investigated GaSe primitive cell contains four formula units and two layers, the crystallographic position of eight atoms in the unit cell are following: For Ga(1) in position (0, 0, 70.075), for Ga(2) in position (2/3, 1/3, 0.575), for Se(1) in position (2/3, 1/3, 0.150), and for Se(2) in position in position (1/3, 2/3, 0.650) and (1/3, 2/3, 0.650). Calculations were carried out using the lattice constant a ¼b¼3.743 Å and c ¼15.919 Å for GaSe [35]. The lattice parameters a, c, the Ga–Ga and Ga–Se distances of ε-GaSe obtained from the structure optimization are presented in Table 1 along with other calculations and experimental data reported in the literature. Considering that the zero-point vibration and thermal effects are not taken into account, the calculated lattice parameters agree quite well with the experimental ones. Our calculations underestimate the equilibrium lattice parameter a(c) with the maximal error of 0.85–1.17% (1.56–1.69%) compared with respect to experimental values. This is largely sufficient to allow the further study of elastic, electronic and optical properties. 3.2. Elastic properties The elastic constants are related to the stress tensor and the strain tensor by Hooke's law and they are calculated as the second derivatives of the internal energy with respect to the strain tensor. The elastic constants Cijkl can be expressed as [38–40] ∂sij ðxÞ ð1Þ C ijkl ¼ ∂ekl X
Not only the elastic tensor specifies the response of a material to applied stresses, but also it gives criteria about the actual stability of the structure. The elastic stiffness tensor of hexagonal compounds has five independent components because of the symmetry properties of the D13h space group, namely c11, c33, c44, c12 and c13 in Young's notation., We adopted a 21 21 5 Monkhorst– Pack special k-point meshes to ensure the reliability of the calculated results. The elastic compliance tensor elements of ε-GaSe are displayed in Table 2. Elastic stiffness tensor components must satisfy certain relations known as the Born stability criteria [44] which for the hexagonal ε-GaSe lattice requires that c12 4 0, c33 40, c66 ¼(1/2) (c11 c12)40, c44 4 0, ðc11 þ c12 Þðc233 2c213 Þ4 0. The values reported in Table 2 satisfy all these constraints, implying that ε-GaSe crystals are mechanically stable. We display experimental and theoretical calculated values for ε-GaSe in the same table. Except for c44 ( E68% overestimated), the agreement between the calculated and experimental values is very good. The origin of this discrepancy between the results of experimental work and our calculated value of c44 may be illustrated by the calculated distance between the atomic layers. The calculated distance along the c axis is obviously compressed in comparison to the experiment under the LDA, it means that the calculated distance between the atomic layers are shorten. The shorter interlayer distance, the stronger interaction between the atomic layers. According to the definition [38–40], the phenomenon about the overestimated theoretical value of c44 is reasonable. In order to check the internal consistency of our elastic constants, we can compare the bulk modulus reported in Table 2 with an equivalent combination of cij. The Voigt (V) [45] and Reuss (R) [46] limits for the bulk modulus (B) and shear modulus (G) are [47] BV ¼ ð1=9Þð2c11 þ 2c12 þ c33 þ 4c13 Þ
BR ¼
ðc11 þ c12 Þc33 2c213 c11 þ c12 þ 2c33 4c13
This work Exp. Theor. a
Ref. [35]. Ref. [36]. c Ref. [37]. d Ref. [8]. b
a, b(Å)
c(Å)
ð4Þ
1 2 1 1 2 6 6 þ þ þ GR ¼ 15 4 c11 c33 c12 c13 c44 c11 c12
ð5Þ
In the Voigt–Reuss–Hill (VRH) approximation [48], the B and G of the polycrystalline material are approximated as the arithmetic Table 2 Elastic constants cij of hexagonal ε-GaSe, calculated from them: bulk (BVRH) and shear modulus (GVRH) in the Voigt–Reuss–Hill approximation, and the Young modulus. All values are given in gigapascal.
This work Theor. Exp.
3.711 3.743a 3.755b 3.715c
15.67 15.919a 15.94b 15.77c
dGa–Ga(Å)
dGa–Se(Å)
2.405 2.388a 2.520d 2.41c
2.437 2.469a 2.473d
ð3Þ
GV ¼ ð1=30Þð7c11 5c12 4c13 þ 2c33 þ 12c44 Þ
where ekl, sij, X, x are respectively Eulerian strain tensor, applied stress tensor, the coordinates before and after deformation.
Table 1 Crystal structure data of hexagonal ε-GaSe.
ð2Þ
This work Exp.
c11
c12
c13
c33
c44
103.56 100.9a 100.8 b 97.83 c 105.0d 103.3e
27.35 27a 26.9 b 40.25 c 32.4d 28.9e
10.91 9.7a 12.6 b 25.67 c 12.6d 12.0e
28.86 33.9a 32.8 b 35.8 c 35.1d 34.1e
17.50 8.3a 10.6 b 9.11 c 10.4d 9.0e
Yx 94.0
Yy 94.0
Yz 27.04
GVRH 24.42
BVRH 30.78 28.7d 28.57f
a
Ref.[37]. Ref [23]. c Ref [24]. d Ref. [41]. e Ref. [42]. f Ref. [43]. b
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mean of the Voigt and Reuss limits:
3
Table 3 Mulliken charge (e), bond type, bond population, and band gap of ε-GaSe.
1 BVRH ¼ ðBV þ BR Þ; 2
ð6Þ
1 GVRH ¼ ðGV þ GR Þ: 2
ð7Þ
The calculated bulk and shear modulus in the Voigt–Reuss–Hill approximation are also listed in Table 2 along with the Young modulus. According to Pugh's criterion [49], a material is brittle if the B/G ratio is less or equal than 1.75. Our results show that the B/G ratio is 1.26 for ε-GaSe, which means the crystal is brittle. Such behavior of this material can be connected indirectly with the elastic and Young modulus anisotropy. There are three independent elastic shear constants for hexagonal crystals, thus, three shear-type anisotropic factors can be defined as [50]: α1 ¼ 2c44 =ðc11 c12 Þ ¼ 0:46;
ð8Þ
α2 ¼ c66 =c44 ¼ 2:18;
ð9Þ
Species
S
p
D
Total
Charge
Bonds
Population
Ga(1) Ga(2) Se(1) Se(2)
1.11 1.09 1.90 1.83
1.71 1.72 4.32 4.32 This work 0.816
10 10 0 0
12.82 12.81 6.22 6.15 Theor. 0.267a
0.18 0.19 0.22 0.15
Ga(1)–Ga(1) Ga(2)–Ga(2) Ga(1)–Se(2) Ga(2)–Se(1) Exp. 2.12a
1.07 0.14 3.65 3.94
Eg(eV) a
Ref. [51].
and α3 ¼
ð1=6Þðc11 þc12 þ 2c33 4c13 Þ ¼ 1:38 c44
ð10Þ
These three independent elastic shear constants largely deviate from unity, showing the strong elastic anisotropy of the hexagonal crystal. Furthermore, the Young modulus perpendicular to the c axis is almost four times as that parallel to the c axis. 3.3. Electronic properties The investigation of electronic band structure of hexagonal ε-GaSe is very useful as it helps us to understand the electronic and optical properties of the material better. At this point, what we first need is to describe our calculated electronic structure. The energy-band structure calculated using the LDA for hexagonal ε-GaSe is shown in Fig. 1. It shows that the top of the valence bands and the bottom of the conduction bands lie at the Γ-point, indicating that ε-GaSe is a direct band gap semiconductor. The most prominent features of the calculated band gap of ε-GaSe are listed in Table 3. The obtained direct gap of 0.816 eV for ε-GaSe, agrees well with the other theoretical result of Plucinski et al. [51]. However, the band gap of this material is underestimated in the LDA, when compared with the experimental data (see Table 3 for comparison) and this is an intrinsic feature of the DFT-LDA.
Fig. 1. Calculated band structure for hexagonal ε-GaSe single crystal.
Fig. 2. Partial density of states (PDOS) for hexagonal ε-GaSe single crystal.
From the partial density of states (PDOS) we are able to identify the angular momentum characteristics of the various structures. We present the s, p, and d projected DOS in the Ga and Se atom of ε-GaSe in Fig. 2. The lower valence bands from 15.75 to 12.06 eV for ε-GaSe are derived mostly from the cation Ga 3d states and Se 4s states and it is noted that the 3d state of cation Ga is highly localized. From 8.85 to 0 eV, the upper valence bands, we find the anions Se 4s and 4p orbitals hybridized with the cation Ga 3s and 3p orbitals and from 0.81 to 21.68 eV, the bands are derived also from a combination of cation Ga 3s and 3p states and anion Se 4s and 4p states. In the high energy zone, the s and p projected DOS of Ga (or Se) have some difference because of the difference in chemical environment. The Mulliken overlap population is one of the more useful and widely used indicators of the strength of a bond. For the bond overlap population, it indicates the degree of electron cloud overlap between two bonding atoms. The highest and lowest values imply that the chemical bond exhibits strong covalency and ionicity, respectively. Positive and negative values indicate bonding and antibonding states. A value for the overlap population close to zero indicates that there is no significant interaction between the electronic populations of the two atoms. The calculated Mulliken charge populations are given in Table 3. From Table 3, one can see that charge transfer from Ga(1) and Ga(2) to Se(1) and Se(2) is about 0.37e, the charge population of Ga(1)–Ga (1) is positive indicating the chemical bond bonding states while the left chemical bonds are anti-bonding states. The electron distribution locates between the Ga(1)–Ga(1), indicating that the bonding behavior of the Ga(1)–Ga(1) is a combination of strong covalent and weak ionic character. Noting that charge populations of the Ga–Se bonds are large, it means that the Ga–Se bonds are strong covalence bonds. By the analysis above, one can see that the strong bonds mean the good mechanical property along the c axis
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in the layer and the low hardness of ε-GaSe crystal are mainly due to the interlayer interactions. 3.4. Optical properties Although there are many experimental works on the optical properties of ε-GaSe crystal [21,52–55], the theoretical investigations are inadequate still. It has been earlier found that the calculated optical properties for AgGaS2, AgGaSe2 [56] and CuGaS2 [57] are in excellent agreement with the experimental findings, and we have used the same theory to predict the linear optical properties of hexagonal ε-GaSe in the present work. By using the scissors operator, we have corrected our band gap values with 2.12 eV for ε-GaSe. The optical functions calculated by neglecting all lattice vibrational effects and pertaining only the electronic transitions are shown in Figs. 3–7. The optical response was calculated in the photon energy range of 0–15 eV from the calculated band structure. The optical properties of the material are determined by the dielectric function ɛ(ω) given by ɛ(ω)¼ ɛ1(ω)þ iɛ2(ω) The imaginary part of ɛ(ω), ɛ2(ω) depends on the joint density of states and the momentum matrix elements. The real part, ɛ1(ω), was obtained from ɛ2(ω) by the Kramers–Kronig relations. For the hexagonal structure, the dielectric functions are resolved into two components ɛxy(ω) which is the average of the spectra for polarizations along the x and y directions (E ? c-axis) and ɛz(ω) corresponding to the z-direction(E//c-axis). The calculated real and imaginary parts of the frequencydependent linear dielectric function are shown in Fig. 3. Along the crystallographic c-direction, the imaginary part of the dielectric constant, ɛ2z(ω), shows several main peaks located at 2.0, 5.0, 7.5 and 11.0 eV, while perpendicular to the c-direction, peaks are located at 6.0 eV. These peaks correspond to the optical transitions
Fig. 5. The extinction coefficient of ε-GaSe crystal.
Fig. 6. The absorption coefficient of hexagonal ε-GaSe crystal.
Fig. 3. The real and imaginary parts of optical dielectric function of ε-GaSe cryst.
Fig. 7. The reflectivity spectrum of hexagonal ε-GaSe crystal.
Fig. 4. The refractive index of ε-GaSe crystal.
from the valence band to the conduction band. From the curves of ɛ2z(ω), it is seen that 0–2.12 eV photon-energy range is characterized by high transparency, no absorption and a small reflectivity which explains the origin of the peak structure in the reflectivity and absorption coefficient spectra. Fig. 3 also shows the real parts of the frequency-dependent linear dielectric functions. The static dielectric constants ɛ0 of ε-GaSe is calculated as 7.63 along
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c-direction and 7.64 perpendicular to c-axis. We find that the calculated dielectric function results including the locations of main peaks and the general shape of the curves are in accord with experimental results [52–54]. The refractive index and the extinction coefficient are displayed in Figs. 4 and 5, respectively. The static refractive index nxy(0) and nz(0) are found to be 2.764 and 2.763, respectively, the anisotropy is quite small and negative for ε-GaSe in the region 0–0.21 eV. As can be seen, the refractive index n(ω) slowly increases with energy in the infrared region. The local maxima of the extinction coefficient k(ω) correspond to the zero of ε1(ω), we have also resolved into in parallel and perpendicular components in order to estimate the degree of anisotropy of this constant, and found that the extinction coefficient is lower when the light propagation parallels along the c-direction. According to Fig. 6, the average absorption edge is located at 2.0 eV for ε-GaSe, which is smaller than the experimental values (2.12 eV for ε-GaSe) [51]. This is mainly due to the underestimated band gap. The absorption coefficients decrease rapidly in the low energy region, which is the typical characteristic of semiconductors and insulators. In Fig. 7, the calculated optical reflectivity R(ω) is displayed. The calculated reflectivity increases slowly from 22% to 29.2% with the photon energy in the in-frare area for ε-GaSe. From the absorption and reflectivity spectra, we can conclude that ε-GaSe is transmitting for frequencies o 2.0 eV. 4. Conclusions In the present work, the elastic, electronic and linear optical properties of ε-GaSe crystal have been investigated in detail by using the density functional methods under the LDA. It is found that the results of implemented structural optimization by the LDA and the experiments agree with each other well, the fundamental gap of ε-GaSe crystal is direct at the Γ point of the BZ. The calculated band gap by LDA is 0.816 eV and is smaller than the experimental result of 2.12 eV. According to the elastic constants data, the structure of ε-GaSe is mechanically stable. The calculated Young modulus reveals that ε-GaSe compound is a highly anisotropic material and can be destroyed easily along the c axis. Furthermore, Mulliken overlap population analysis shows that the layered of the crystal is brittle because of the weak interaction. The calculated optical properties indicate that ε-GaSe are transmitting for frequencies o 2.0 eV, and in the transmitting region the absorption and reflectivity coefficients are low, so the compound should be an excellent far-IR transparent crystal material, which has been proved by experiments and theory. Acknowledgments This work is supported by National Natural Science Foundation of China under Grant No. 11174100. References [1] Ch. Ferrer-Roca, J. Bouvier, A. Segura, M..V. Andrés, V. Muñoz, J. Appl. Phys. 85 (1999) 3780. [2] A. Gouskov, J. Camassel, L. Gouskov, Prog. Cryst. Growth Charact. 5 (1982) 323. [3] K.C. Mandal, S.H. Kang, M. Choi, Proc. MRS 969 (2007) 111.
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First-principles study of the elastic, electronic and optical properties of ε-GaSe layered semiconductor Shun-Ru Zhang a,n, Shi-Fu Zhu b, Bei-Jun Zhao b, Lin-Hua Xie c, Kei-Hui Song a a
Department of Physics and Information Engineering, Huaihua University, Huaihua 418008,China Department of Materials Science, Sichuan University, Chengdu 610064, China c Institute of Solid State Physics & School of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610066, China b
art ic l e i nf o
a b s t r a c t
Article history: Received 27 February 2013 Received in revised form 1 December 2013 Accepted 9 December 2013
The elastic, electronic and optical properties of ε-GaSe layered semiconductor have been studied from the first principles with the local density approximation (LDA). The optimized structure of GaSe has been found to be in good agreement with the experimental results. The mechanical stability of ε-GaSe is confirmed by calculations of the elastic constants. The calculated band structure shows that the crystal has a direct band gap of 0.816 eV. Furthermore, the linear photon-energy-dependent dielectric functions and some optical constants such as refractive index, extinction, reflectivity and absorption coefficients have been calculated. & 2013 Published by Elsevier B.V.
Keywords: GaSe Density functional theory Elastic constants Electronic structure Optical properties
1. Introduction In recent years, III–VI layered semiconductors have attracted a lot of interests because they have remarkable nonlinear optical properties and are promising for optical switching devices [1–4]. GaSe is a member of the family of the III–VI type semiconductor crystals, crystallizing in a layered structure in which each layer consists of four atoms stacked along the c-axis with a repeating unit of Se–Ga–Ga–Se. These layered compounds are characterized by highly anisotropic bonding forces [5]. In GaSe, intralayerbonding forces are the ionic covalent type and interlayer ones are the van der Waals type, which causes a weak interlayer interaction between layers. Therefore, GaSe can easily be cleaved along these layers. Several polytypes have been reported, which differ in the stacking sequence of the basic layer units [2]. Bridgman grown samples are commonly found to follow the hexagonal structure with D13h symmetry [6]. Many experiments have been carried out to grow and characterize GaSe crystal [6–10], the properties of GaSe at ambient conditions are reasonably well know [11], various important applications of GaSe in the areas of broadband mid-infrared electromagnetic waves, terahertz (THz) generator/detectors and radiation detectors were reported [12,13]. Recently, the phase transition from the ε-GaSe to the NaCl-type structure motif is
n
Corresponding author. Tel.: þ 86 745 2851011. E-mail address: [email protected] (S.-R. Zhang).
observed near 21 GPa [14,15]. To improve the optical, thermal and mechanical properties, further experiments were reported about GaSe doped with various substitutional impurities (In, Er, S, and Te) [16–21]. In the aspect of theory investigations, the first-principles calculations of the bulk GaSe band structure and elastic properties have been reported in Refs. [22–24], while electronic properties of two dimensional GaSe crystals were investigated theoretically and experimentally in Ref. [25], Jahangirli [26] studied of the phonon spectrum of ε-GaSe, Rak et al. [27] investigated the electronic structure and mechanical properties for pure and doped GaSe. Despite large number of experimental studies on this material, it is seen that there are no theoretical reports of the optical properties ε-GaSe. The present study focuses on the elastic, electronic and optical properties of ε-GaSe crystal using density functional theory with the local density approximation (LDA). The article is organized as follows: in Section 2, we briefly describe the computational method used in the present work. Results and discussion will be presented in Section 3. A summary of the work will be given in Section 4. 2. Calculation details The crystal lattice parameters and atom positions of ε-GaSe have been optimized using density functional theory (DFT) based on a plane-wave pseudo-potential technique implemented in the CASTEP package [28]. The plane-wave energy cutoff and the
0921-4526/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.physb.2013.12.014
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Brillouin zone sampling were fixed at 330 eV and 16 16 3 Monkhorst and Pack [29] special k-point meshes, respectively. In this work, the interactions between the valence electrons and the ion cores have been modeled by the ultrasoft pseudo-potential [30]. The valence electronic configurations assumed for the atoms in GaSe are as Ga: 3d104s24p1 and Se: 4s24p1. The electronic exchange-correlation energy is evaluated with the local-density approximation (LDA) [31], using Perdew–Zunger parametrization [32] of Ceperley–Adler electron-gas data [33]. The Broyden– Fletcher–Goldfarb–Shanno (BFGS) algorithm [34] was used to minimize the total energy and internal forces. The tolerances for geometry optimization were set as followings: the difference in total energy being within 1 106 eV/atom, the maximum ionic Hellmann–Feynman force being within 0.01 eV/Å, the maximum stress being within 0.02 GPa and the maximum ionic displacement being within 1 104 Å.
3. Results and discussions 3.1. Volume optimization The ε-GaSe crystal is hexagonal and belongs to the space group P-6m2 (No. 187). The investigated GaSe primitive cell contains four formula units and two layers, the crystallographic position of eight atoms in the unit cell are following: For Ga(1) in position (0, 0, 70.075), for Ga(2) in position (2/3, 1/3, 0.575), for Se(1) in position (2/3, 1/3, 0.150), and for Se(2) in position in position (1/3, 2/3, 0.650) and (1/3, 2/3, 0.650). Calculations were carried out using the lattice constant a ¼b¼3.743 Å and c ¼15.919 Å for GaSe [35]. The lattice parameters a, c, the Ga–Ga and Ga–Se distances of ε-GaSe obtained from the structure optimization are presented in Table 1 along with other calculations and experimental data reported in the literature. Considering that the zero-point vibration and thermal effects are not taken into account, the calculated lattice parameters agree quite well with the experimental ones. Our calculations underestimate the equilibrium lattice parameter a(c) with the maximal error of 0.85–1.17% (1.56–1.69%) compared with respect to experimental values. This is largely sufficient to allow the further study of elastic, electronic and optical properties. 3.2. Elastic properties The elastic constants are related to the stress tensor and the strain tensor by Hooke's law and they are calculated as the second derivatives of the internal energy with respect to the strain tensor. The elastic constants Cijkl can be expressed as [38–40] ∂sij ðxÞ ð1Þ C ijkl ¼ ∂ekl X
Not only the elastic tensor specifies the response of a material to applied stresses, but also it gives criteria about the actual stability of the structure. The elastic stiffness tensor of hexagonal compounds has five independent components because of the symmetry properties of the D13h space group, namely c11, c33, c44, c12 and c13 in Young's notation., We adopted a 21 21 5 Monkhorst– Pack special k-point meshes to ensure the reliability of the calculated results. The elastic compliance tensor elements of ε-GaSe are displayed in Table 2. Elastic stiffness tensor components must satisfy certain relations known as the Born stability criteria [44] which for the hexagonal ε-GaSe lattice requires that c12 4 0, c33 40, c66 ¼(1/2) (c11 c12)40, c44 4 0, ðc11 þ c12 Þðc233 2c213 Þ4 0. The values reported in Table 2 satisfy all these constraints, implying that ε-GaSe crystals are mechanically stable. We display experimental and theoretical calculated values for ε-GaSe in the same table. Except for c44 ( E68% overestimated), the agreement between the calculated and experimental values is very good. The origin of this discrepancy between the results of experimental work and our calculated value of c44 may be illustrated by the calculated distance between the atomic layers. The calculated distance along the c axis is obviously compressed in comparison to the experiment under the LDA, it means that the calculated distance between the atomic layers are shorten. The shorter interlayer distance, the stronger interaction between the atomic layers. According to the definition [38–40], the phenomenon about the overestimated theoretical value of c44 is reasonable. In order to check the internal consistency of our elastic constants, we can compare the bulk modulus reported in Table 2 with an equivalent combination of cij. The Voigt (V) [45] and Reuss (R) [46] limits for the bulk modulus (B) and shear modulus (G) are [47] BV ¼ ð1=9Þð2c11 þ 2c12 þ c33 þ 4c13 Þ
BR ¼
ðc11 þ c12 Þc33 2c213 c11 þ c12 þ 2c33 4c13
This work Exp. Theor. a
Ref. [35]. Ref. [36]. c Ref. [37]. d Ref. [8]. b
a, b(Å)
c(Å)
ð4Þ
1 2 1 1 2 6 6 þ þ þ GR ¼ 15 4 c11 c33 c12 c13 c44 c11 c12
ð5Þ
In the Voigt–Reuss–Hill (VRH) approximation [48], the B and G of the polycrystalline material are approximated as the arithmetic Table 2 Elastic constants cij of hexagonal ε-GaSe, calculated from them: bulk (BVRH) and shear modulus (GVRH) in the Voigt–Reuss–Hill approximation, and the Young modulus. All values are given in gigapascal.
This work Theor. Exp.
3.711 3.743a 3.755b 3.715c
15.67 15.919a 15.94b 15.77c
dGa–Ga(Å)
dGa–Se(Å)
2.405 2.388a 2.520d 2.41c
2.437 2.469a 2.473d
ð3Þ
GV ¼ ð1=30Þð7c11 5c12 4c13 þ 2c33 þ 12c44 Þ
where ekl, sij, X, x are respectively Eulerian strain tensor, applied stress tensor, the coordinates before and after deformation.
Table 1 Crystal structure data of hexagonal ε-GaSe.
ð2Þ
This work Exp.
c11
c12
c13
c33
c44
103.56 100.9a 100.8 b 97.83 c 105.0d 103.3e
27.35 27a 26.9 b 40.25 c 32.4d 28.9e
10.91 9.7a 12.6 b 25.67 c 12.6d 12.0e
28.86 33.9a 32.8 b 35.8 c 35.1d 34.1e
17.50 8.3a 10.6 b 9.11 c 10.4d 9.0e
Yx 94.0
Yy 94.0
Yz 27.04
GVRH 24.42
BVRH 30.78 28.7d 28.57f
a
Ref.[37]. Ref [23]. c Ref [24]. d Ref. [41]. e Ref. [42]. f Ref. [43]. b
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mean of the Voigt and Reuss limits:
3
Table 3 Mulliken charge (e), bond type, bond population, and band gap of ε-GaSe.
1 BVRH ¼ ðBV þ BR Þ; 2
ð6Þ
1 GVRH ¼ ðGV þ GR Þ: 2
ð7Þ
The calculated bulk and shear modulus in the Voigt–Reuss–Hill approximation are also listed in Table 2 along with the Young modulus. According to Pugh's criterion [49], a material is brittle if the B/G ratio is less or equal than 1.75. Our results show that the B/G ratio is 1.26 for ε-GaSe, which means the crystal is brittle. Such behavior of this material can be connected indirectly with the elastic and Young modulus anisotropy. There are three independent elastic shear constants for hexagonal crystals, thus, three shear-type anisotropic factors can be defined as [50]: α1 ¼ 2c44 =ðc11 c12 Þ ¼ 0:46;
ð8Þ
α2 ¼ c66 =c44 ¼ 2:18;
ð9Þ
Species
S
p
D
Total
Charge
Bonds
Population
Ga(1) Ga(2) Se(1) Se(2)
1.11 1.09 1.90 1.83
1.71 1.72 4.32 4.32 This work 0.816
10 10 0 0
12.82 12.81 6.22 6.15 Theor. 0.267a
0.18 0.19 0.22 0.15
Ga(1)–Ga(1) Ga(2)–Ga(2) Ga(1)–Se(2) Ga(2)–Se(1) Exp. 2.12a
1.07 0.14 3.65 3.94
Eg(eV) a
Ref. [51].
and α3 ¼
ð1=6Þðc11 þc12 þ 2c33 4c13 Þ ¼ 1:38 c44
ð10Þ
These three independent elastic shear constants largely deviate from unity, showing the strong elastic anisotropy of the hexagonal crystal. Furthermore, the Young modulus perpendicular to the c axis is almost four times as that parallel to the c axis. 3.3. Electronic properties The investigation of electronic band structure of hexagonal ε-GaSe is very useful as it helps us to understand the electronic and optical properties of the material better. At this point, what we first need is to describe our calculated electronic structure. The energy-band structure calculated using the LDA for hexagonal ε-GaSe is shown in Fig. 1. It shows that the top of the valence bands and the bottom of the conduction bands lie at the Γ-point, indicating that ε-GaSe is a direct band gap semiconductor. The most prominent features of the calculated band gap of ε-GaSe are listed in Table 3. The obtained direct gap of 0.816 eV for ε-GaSe, agrees well with the other theoretical result of Plucinski et al. [51]. However, the band gap of this material is underestimated in the LDA, when compared with the experimental data (see Table 3 for comparison) and this is an intrinsic feature of the DFT-LDA.
Fig. 1. Calculated band structure for hexagonal ε-GaSe single crystal.
Fig. 2. Partial density of states (PDOS) for hexagonal ε-GaSe single crystal.
From the partial density of states (PDOS) we are able to identify the angular momentum characteristics of the various structures. We present the s, p, and d projected DOS in the Ga and Se atom of ε-GaSe in Fig. 2. The lower valence bands from 15.75 to 12.06 eV for ε-GaSe are derived mostly from the cation Ga 3d states and Se 4s states and it is noted that the 3d state of cation Ga is highly localized. From 8.85 to 0 eV, the upper valence bands, we find the anions Se 4s and 4p orbitals hybridized with the cation Ga 3s and 3p orbitals and from 0.81 to 21.68 eV, the bands are derived also from a combination of cation Ga 3s and 3p states and anion Se 4s and 4p states. In the high energy zone, the s and p projected DOS of Ga (or Se) have some difference because of the difference in chemical environment. The Mulliken overlap population is one of the more useful and widely used indicators of the strength of a bond. For the bond overlap population, it indicates the degree of electron cloud overlap between two bonding atoms. The highest and lowest values imply that the chemical bond exhibits strong covalency and ionicity, respectively. Positive and negative values indicate bonding and antibonding states. A value for the overlap population close to zero indicates that there is no significant interaction between the electronic populations of the two atoms. The calculated Mulliken charge populations are given in Table 3. From Table 3, one can see that charge transfer from Ga(1) and Ga(2) to Se(1) and Se(2) is about 0.37e, the charge population of Ga(1)–Ga (1) is positive indicating the chemical bond bonding states while the left chemical bonds are anti-bonding states. The electron distribution locates between the Ga(1)–Ga(1), indicating that the bonding behavior of the Ga(1)–Ga(1) is a combination of strong covalent and weak ionic character. Noting that charge populations of the Ga–Se bonds are large, it means that the Ga–Se bonds are strong covalence bonds. By the analysis above, one can see that the strong bonds mean the good mechanical property along the c axis
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in the layer and the low hardness of ε-GaSe crystal are mainly due to the interlayer interactions. 3.4. Optical properties Although there are many experimental works on the optical properties of ε-GaSe crystal [21,52–55], the theoretical investigations are inadequate still. It has been earlier found that the calculated optical properties for AgGaS2, AgGaSe2 [56] and CuGaS2 [57] are in excellent agreement with the experimental findings, and we have used the same theory to predict the linear optical properties of hexagonal ε-GaSe in the present work. By using the scissors operator, we have corrected our band gap values with 2.12 eV for ε-GaSe. The optical functions calculated by neglecting all lattice vibrational effects and pertaining only the electronic transitions are shown in Figs. 3–7. The optical response was calculated in the photon energy range of 0–15 eV from the calculated band structure. The optical properties of the material are determined by the dielectric function ɛ(ω) given by ɛ(ω)¼ ɛ1(ω)þ iɛ2(ω) The imaginary part of ɛ(ω), ɛ2(ω) depends on the joint density of states and the momentum matrix elements. The real part, ɛ1(ω), was obtained from ɛ2(ω) by the Kramers–Kronig relations. For the hexagonal structure, the dielectric functions are resolved into two components ɛxy(ω) which is the average of the spectra for polarizations along the x and y directions (E ? c-axis) and ɛz(ω) corresponding to the z-direction(E//c-axis). The calculated real and imaginary parts of the frequencydependent linear dielectric function are shown in Fig. 3. Along the crystallographic c-direction, the imaginary part of the dielectric constant, ɛ2z(ω), shows several main peaks located at 2.0, 5.0, 7.5 and 11.0 eV, while perpendicular to the c-direction, peaks are located at 6.0 eV. These peaks correspond to the optical transitions
Fig. 5. The extinction coefficient of ε-GaSe crystal.
Fig. 6. The absorption coefficient of hexagonal ε-GaSe crystal.
Fig. 3. The real and imaginary parts of optical dielectric function of ε-GaSe cryst.
Fig. 7. The reflectivity spectrum of hexagonal ε-GaSe crystal.
Fig. 4. The refractive index of ε-GaSe crystal.
from the valence band to the conduction band. From the curves of ɛ2z(ω), it is seen that 0–2.12 eV photon-energy range is characterized by high transparency, no absorption and a small reflectivity which explains the origin of the peak structure in the reflectivity and absorption coefficient spectra. Fig. 3 also shows the real parts of the frequency-dependent linear dielectric functions. The static dielectric constants ɛ0 of ε-GaSe is calculated as 7.63 along
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c-direction and 7.64 perpendicular to c-axis. We find that the calculated dielectric function results including the locations of main peaks and the general shape of the curves are in accord with experimental results [52–54]. The refractive index and the extinction coefficient are displayed in Figs. 4 and 5, respectively. The static refractive index nxy(0) and nz(0) are found to be 2.764 and 2.763, respectively, the anisotropy is quite small and negative for ε-GaSe in the region 0–0.21 eV. As can be seen, the refractive index n(ω) slowly increases with energy in the infrared region. The local maxima of the extinction coefficient k(ω) correspond to the zero of ε1(ω), we have also resolved into in parallel and perpendicular components in order to estimate the degree of anisotropy of this constant, and found that the extinction coefficient is lower when the light propagation parallels along the c-direction. According to Fig. 6, the average absorption edge is located at 2.0 eV for ε-GaSe, which is smaller than the experimental values (2.12 eV for ε-GaSe) [51]. This is mainly due to the underestimated band gap. The absorption coefficients decrease rapidly in the low energy region, which is the typical characteristic of semiconductors and insulators. In Fig. 7, the calculated optical reflectivity R(ω) is displayed. The calculated reflectivity increases slowly from 22% to 29.2% with the photon energy in the in-frare area for ε-GaSe. From the absorption and reflectivity spectra, we can conclude that ε-GaSe is transmitting for frequencies o 2.0 eV. 4. Conclusions In the present work, the elastic, electronic and linear optical properties of ε-GaSe crystal have been investigated in detail by using the density functional methods under the LDA. It is found that the results of implemented structural optimization by the LDA and the experiments agree with each other well, the fundamental gap of ε-GaSe crystal is direct at the Γ point of the BZ. The calculated band gap by LDA is 0.816 eV and is smaller than the experimental result of 2.12 eV. According to the elastic constants data, the structure of ε-GaSe is mechanically stable. The calculated Young modulus reveals that ε-GaSe compound is a highly anisotropic material and can be destroyed easily along the c axis. Furthermore, Mulliken overlap population analysis shows that the layered of the crystal is brittle because of the weak interaction. The calculated optical properties indicate that ε-GaSe are transmitting for frequencies o 2.0 eV, and in the transmitting region the absorption and reflectivity coefficients are low, so the compound should be an excellent far-IR transparent crystal material, which has been proved by experiments and theory. Acknowledgments This work is supported by National Natural Science Foundation of China under Grant No. 11174100. References [1] Ch. Ferrer-Roca, J. Bouvier, A. Segura, M..V. Andrés, V. Muñoz, J. Appl. Phys. 85 (1999) 3780. [2] A. Gouskov, J. Camassel, L. Gouskov, Prog. Cryst. Growth Charact. 5 (1982) 323. [3] K.C. Mandal, S.H. Kang, M. Choi, Proc. MRS 969 (2007) 111.
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Please cite this article as: S.-R. Zhang, et al., Physica B (2013), http://dx.doi.org/10.1016/j.physb.2013.12.014i