The first principle study: Electronic and optical properties in Bi2Se3

Computational Condensed Matter 4 (2015) 59e63

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The first principle study: Electronic and optical properties in Bi2Se3 Geoffrey Tse a, *, Dapeng Yu a, b a b

State Key Laboratory for Mesoscopic Physics, and Electron Microscopy Laboratory, School of Physics, Peking University, Beijing 100871, PR China Collaborative Innovation Center of Quantum Matter, Beijing 100871, PR China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 11 August 2015 Received in revised form 6 September 2015 Accepted 7 September 2015 Available online 10 September 2015

The crystal structure of Bi2Se3 using ab initio calculations have been investigated using density functional theory (DFT). According to the atomic positions provided by the previous literature, we were able to construct a lattice structure using visualization software Material Studio. This structure is found in rhombohedral, of the space group D53d with R3m (#166) and lattice parameter of a ¼ b ¼ 4.143 Å, c ¼ 28.836 Å, and the bond angle of a ¼ b ¼ 90 , g ¼ 120 , while treating the exchange-correlation potential with the general gradient Approximations (GGA) method. The structural calculations were performed to investigate, the electronic, charge density, optical, and phonon properties. To conclude, the partial density of state plots indicates the material properties near to the fermi level determined by the p state orbitals. © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Keywords: Bi2Se3 GGA Electronic structure Charge density Partial density of state Optical properties

1. Introduction Bi2Se3 is a thermoelectrical material, can be widely used for cooling application at room temperature. Thermoelectric, the conversion of heat to electrical energy, can be applied on applications of thermoelectric cooling devices and refrigerators. This effect is based on the Peltier, Seeback and Thompson effect [1]. The advantage of solid-state cooling devices is that they are user friendly and there are no moving parts. The material efficiency and its performance is governed by the figure of merit ZT, which is ZT: S2sT/k. Where S is the Seebeck coefficient, s is the electrical conductivity, and T is the absolute temperature of the material provided, and k is the thermal conductivity. The performance of ZT can be improved with high value of thermal conductivity and low value of electrical conductivity. The Seeback coefficient is intrinsic, and cannot be adjusted via physical or chemical process. Phonon vibration could have an impact on increasing the temperature and could affect the performance of the ZT. The thermoconductivity corresponds to the phonon transportation of the thermoelectrical materials. Previous study has shown, phonon blocking could enhance the ZT by means of, material fabrication in form of superlattice [2] or nanowire [3]. In addition, the

* Corresponding author. E-mail address: [email protected] (G. Tse).

thermoconductivity can be reduced by using phonon electron glass crystals [4e7]. Despite the costs of fabrication, the ZT quantity can be further improved with the use of Bi2Te3/Sb2Te3 superlattice [2] or Bi2Te3 nanowire [3]. Recent literature have shown, BiAlO3 (perovskite) and HfTe5 (transition-metal pentatalluride) are thermoelectric materials, which can used as a cooling devices above and below the ambient temperature respectively [9]. “More previous studies have involved the experimental work in quantum transport and photothermoelectric effect of topological insulator Bi2Se3 [10e14]”. Similar computational modelling works for other materials have also been investigated [15e19]. In this work, the ab. initio DFT calculations will be used in exploring the electronic structure, charge density, partial density of state and optical properties. To the best of knowledge from the author, there are no theoretical and experimental data on charge density, PDOS and optical properties. 2. Computational methods We used DFT calculations in predicting material properties. Electronic properties can be explored in performing geometry optimization, single point energy and bandstructure calculations. Partial density of states (PDOS) are also investigated. According to our previous study, the energy-cutoff of 550 eV with k-point mesh of 4  4  1 was chosen, with the brillouin sampling of 4 points.

http://dx.doi.org/10.1016/j.cocom.2015.09.001 2352-2143/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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G. Tse, D. Yu / Computational Condensed Matter 4 (2015) 59e63

There are 12 empty bands the k point separation for bandstructure calculation is 0.015 Å1. Here, the Monkhorst-Pack [20] special point's integration scheme within the Brillouin-zone (BZ) was determined, based on our experience. For the exchangecorrelation-interaction, the description was provided by the Perdew-Berke-Erzndof (PBE) [21] of the Generalized Gradient Approximation (GGA). The norm-conserving [22e25] is preferred other than ultrasoft potential [26] when calculating the optical properties. In providing the description of the valanceecore interaction, such potential are employed with the Martin-Troullier scheme [27], where the electron orbital, 4s2 4p4 for Selenide atom (atomic no. 34), and the 6s2 6p3 5 d10 of Bismuth atom (atomic no. 83) was chosen. The C in Z plane, A along XZ plane criteria is needed in calculating optical material properties. The total energy SCF tolerance was set to a value of 5  107 eV/atom, with the maximum force of 0.01 eV/Å, maximum stress of 0.02 GPa, and displacement of 5  104 Å. The performance and the reliability of CASTEP (Cambridge-Sequential-Total-Energy-Package) under Material Studio [28], within periodic boundary conditions, in comparing with SIESTA [29], have been testified by the previous literature [30]. Spin-orbit interaction effect must included, since the elements used in this work involves atomic number being greater than value of 50 [31]. This simulation would require a total of four processors to perform overall. For material properties, we have investigated the optical quantity such as electron losses function L(u), the reflectivity R(u), reflective index n(u) and absorption a(u), using the real ε1(u) and the imaginary part ε2(u) of the dielectric tensor, calculated with density functional perturbation theory (DFPT). This is shown in equations (1)e(6), where term u represents the frequency.

2 ε1 ðuÞ ¼ 1 þ p p

ε2 ðuÞ ¼

aðuÞ ¼

Z∞ 0

u0 ε2 ðu0 Þ u0 2  u2

du0

   2e2 p  c b  rjjvk d Ekc  Ekv  E jk j u Uε0

1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ε21 ðuÞ þ ε22 ðuÞ  ε1 ðuÞ

(1)

(2)

(3)

 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ε ðuÞ þ jε ðuÞ  12   1 2 RðuÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   ε1 ðuÞ þ jε2 ðuÞ þ 1

(4)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1 ε21 ðuÞ þ ε22 ðuÞ þ ε21 ðuÞ nðuÞ ¼ pffiffiffi 2

(5)

LðuÞ ¼

sðuÞ ¼

ε2 ðuÞ ε1 ðuÞ þ ε2 ðuÞ 1 ðuε2 ðuÞÞ 4p

positions of Selenide (0.0000, 0.0000, 0.0000), (0.0000, 0.0000, 0.2117) and Bismuth (0.0000,0.0000,0.4008) are shown. The geometry optimization was then performed. We reported, at this point, the electron system would be able to converge after 30 steps of convergence, when the material was treated as an insulator. Determining the DFT lattice constant will always be the first steps, in avoiding certain sources of error when determining other quantities. In the case of DFT, experimental lattice constants cannot be applied. The over binding issues in DFT will result in the underestimating of the self-interaction between the charges, the theoretical value will be smaller than the experimental, one in what the calculations performed will be away from the equilibrium for the atomic configurations [35]. The lattice structure we used to perform the simulation run was a superlattice cell, consisting of 3 primitive cells, with the total of 21 Se atoms and 12 Bi atoms. It was shown in Fig. 1. The bulk DFT lattice length a ¼ b, and c after the relaxation is found to be 4.113 and 28.5659 Å respectively, bond angle of a ¼ 90.148, b ¼ 89.852 , g ¼ 119.854 . The total energy calculated is observed to be 13436.21 eV, the lattice volume is found to be 419.12 Å. The total energy plot against the volume of the specified 3D periodic systems was indicated in Fig. 2. The minimum point found on such plot will provide adequate information, that the DFT calculation performed have been geometry optimized. Fig. 3 explicitly demonstrates, the high symmetry points applied on the lattice structure within brillouin zone. Here, the local point G(0.000, 0.000, 0.000), among the high symmetry points F(0.000,0.500,0.000), Z(0.000,0.500,0.000), and other symmetry points Q(0.000,0.500,0.500) and B(0.500,0.000,0.000) within the zone boundary was given, to generate the k-points in the desired kpaths (The direction is given as G / F / Q / Z / G / B). Fig. 4 shows the electronic bandstructure of Bi2Se3. According to this figure, the material is found to be indirect. The energy bandgap

(6)

(7)

Here, equation (1) use the formalism of Ehrenreich and Cohen [32]. Equation (2) can be derived from equation (1) with KramerseKronig relation. Equations (3)e(6) can be found in previous study [33]. Equation (7) can be referred to the recent literature [34]. 3. Results and discussion The structural, density of states (DOS), optical and phonon properties of Bi2Se3 have been studied in this work. The Wyckoff

Fig. 1. Bi2Se3 bulk with the total of 33 atoms. The Bismuth and Selenide atom positions are indicated.

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61

-13435.0 Total Energy

Total Energy (eV)

-13435.2 -13435.4 -13435.6 -13435.8 -13436.0 -13436.2 360

380

400

420

440

460

480

Lattice volume (Å) Fig. 2. Total energy plot against lattice volume of Bi2Se3.

from direct G/G point is found to be 0.53 eV. According to experimental value, the value is 0.2e0.3 eV. At this point, we acknowledge that, the previous literature [36,37], was able to report a theoretical value of 0.3 and 0.32 eV respectively. However, the lattice constant used was extracted directly from the experimental data, and then perform the structural relaxation on atoms by fixing the cell length and bond angle, but the cell optimization procedure is not carried out [35]. On the other hand, our previous literature have mentioned, only a single 5-atoms cell is used, instead of a conventional cell [36]. Looking at our bandstructure diagram in Fig. 3 comparing to the reference [36], there is a direct G / G transition. In other word, there is no anti-crossing feature, the inversion between conduction and valance band, found at the G point. Thus, the calculation carried out is without spin-orbitcoupling (SOC). Furthermore, it is a direct bandgap semiconductor. The difference between our theoretical value and the previous study [35] is by 0.19 eV. Here, the p orbitals of Bi and Se atoms mainly contribute to the states near to the fermi surface, rather than s orbitals. The following describes how the crystal field splits between the p orbitals. There

Fig. 4. A diagram illustrating bandstructure with the variation of electronic bands.

are three states that makes up the p orbitals for Bi atoms and two states for Se atoms respectively. At this point, the Pz orbital splits from Px and Py orbitals, which the Px and Py orbitals remain degenerate. Nearest to the fermi level, the Pz level is determined to þ be the two of the five hybridized states P1þ z and P2z . This is where the þ sign indicates the parity of the corresponding states. It is the chemical bonding that hybridizes the states on Bi and Se atoms, in what raises all the Bi states and reduces the Se states. Fig. 5 shows partial density of states (PDOS) of the Bi2Se3. The states 11 eV below the fermi level corresponds to the s orbital states from Bi and Se atoms. The states 1 eV near to the fermi level are responsible for the p orbitals from Se atoms. The states at around 1 eV are relating to the valance from Bi 2- and Se 3þatoms. Our theoretical data includes the 5 d10 orbital states, which improves over the existing model, in comparing to our previous studies [36,37]. The induction of an incident light is caused by the electron band-to-band transition from valence band to the conduction band. The power dissipation of electromagnetic wave, which is the total amount of photon energy absorbed in the material, is corresponding to the imaginary part of the dielectric tensor ε2(u) shown

120

Bi

6s2 6p3 5d10

Se

4s2 4p4

Partial Density of States (states/eV)

80 40 0 16 12 8 4

0 120

Bi2Se3

Total

80 40 0 -25

Fig. 3. A diagram showing the high symmetry points created within the brillouin zone (BZ).

-20

-15 -10 -5 Energy (eV)

0

5

Fig. 5. A diagram illustrating the variation of partial density of states for Bismuth, and Selenide.

62

G. Tse, D. Yu / Computational Condensed Matter 4 (2015) 59e63 250000 30

Absorption

Re Im

25

200000 150000

15 10

100000

5 0

50000

Absorption (cm )

Dielectric function

20

-5 0

-10

6 n k

Refractivity

1.0

5

0.6

3

0.4

2

0.2

1

0.0

0

10

Re Im

8

Loss function

10

Conductivity (fs )

6

8

4 6

2 0

4

-2

2

Loss Function

Reflectivity

4

Refractive Index

0.8

-4 0

-6 0

5

10

15

20

25

Frequency (eV)

0

5

10

15

20

25

30

Frequency (eV)

Fig. 6. A diagram illustrating the optical properties at bulk, with scissors operator ¼ 0 eV, instrumental smearing of 0.5 eV, polarized, the polarization in direction of (1.0, 0.0, 0.0).

in Equation (2). Fig. 6 represents the real ε1(u) and the imaginary ε2(u) parts of the dielectric function, which depends on the frequency, for the Bi2Se3. The ionic and the electronic part of a nonpolar material at high frequency, is indicated in the static dielectric permittivity tensor ε(0). The dielectric constant ε(∞) at high frequency is 15.06. In Fig. 6, the imaginary part of the dielectric tensor ε2(u) is indicating its first peak at 2.72 eV. with the first edge is at 1.069 eV, which is associated to the fundamental bandgap Eg. This peak is related to the transition of the interband between the valence band maximum (VBM) and the conduction band minimum (CBM). Fig. 6 indicates the refractive index n(u) and the extinction coefficient k(u). The static refractive index n(0) is found to be 3.88. The refractivity spectrum R(u) is displayed in Fig. 6. The first edge value of this quantity is at 92.9% of the ultra-violet (UV) and the magnitude starts to rise to attain the maximum at 9.11 eV, which is outside the UV region. The absorption coefficient a(u) is found in Fig. 6. The first absorption value is 1.069 eV, and the strong absorption coefficient is at 4.60 eV. This means, there are low electron losses and strong absorption of the crystal at low energy. This leads to the fact that, the material is partially transparent in visible region. The conductivity s(u) is shown in Fig. 6. The first edge is reported at the value of 1.07 (fs)1at energy of 1.53 eV. The value of 0.17 (fs)1 is related to the fundamental bandgap, and the maximum conductivity is 10.0 (fs)1.

In describing the loss of energy on a fast electron, the prominent peaks found in the electron energy loss function (EELS) spectra L(u) is responsible to a collective oscillation of the electrons in valance band, which is known as the plasma resonance. This is shown in Fig. 6. Such peak is located at the plasma frequency up of 10.5 eV. At this frequency, the real part of dielectric tensor is going through zero. The electron loss is found to be small and strongly reflected in the UV region. Fig. 7 indicates the charge density contour plot of Bi2Se3. Here, the strong bonding for the interlayer in bulk, Se3 and Bi, Se1 and Bi, Se and Bi atoms for instance, are known to be covalent. Likewise, the electrostatic force of attraction between Se1 and Se, Se1 and Se1, Se3 and Se atoms are resulting from the Van de Waal force, which are found to be weak. It is observed that, the quintuple layer 1 appears to have the inverse symmetry of quintuple layer 2. They are the main key to determine the electronic material properties. Also, the interlayer coupling is important for near-gap electronic structure of the layered compound, in bulk. 4. Conclusion The electronic structure calculations of Bi2Se3 have been investigated using first principle methods, using PBE scheme of the GGA method within the DFT framework. It is determined,

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Acknowledgement

C Bi Se Se3

Se

Bi Se Bi

Bi Se1

quintuple layer 2

References

Se1 Bi Bi Se Bi

quintuple layer 1

Se3 Se Se Bi Se

O

The author would like to thank Dr. J. Pal for proof-reading this manuscript. This work was funded by the National Science Foundation of China (NSFC) project number 11234001, under the recipient of D. Yu. Finally, the author would also like to thank Peking University (PKU) for the Postdoctoral Research Fellowship. This work made use of the computing facilities at International Centre of Quantum Materials (ICQM), supported by PKU.

A

Fig. 7. The contour plot of charge density in showing the lattice direction of A and C of Bi2Se3.

according to both the bandstructure and the PDOS plots, the porbital near to the fermi level dominates, rather than s-orbital. Our PDOS results are reported in good comparison with both theoretical and experimental data in previous studies. Another thermoelectric material Bi2Te3 are shown to obtain similar electrical anomaly and also the electronic structure. The Seeback coefficient and the electrical resistivity of Bi2Se3 can be improved by substitution of other materials such as Sb, Ba. Such quantities are not enhanced in significant when being replaced by Bi2Te3. So far, there are no results available for PDOS, charge density and optical properties. We believe our predicted data will be useful to the experimentalist in the future. Author contributions G. Tse performed the material search, computational simulations, analysis of the data and writing the manuscript. D. Yu for providing useful discussion, and the financial support on the research work. Conflict of interest The authors declare no competing financial interests in this work.

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