Combined temperature- and magnetic field-induced optical responses of phosphorene

Chemical Physics 524 (2019) 113–117

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Combined temperature- and magnetic field-induced optical responses of phosphorene Doan Quoc Khoaa,b, Tran Cong Phongc, Vo Thanh Lamd, Bui Dinh Hoie,

T

⁎

a

Division of Computational Physics, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Viet Nam c The Vietnam National Institute of Educational Sciences, 101 Tran Hung Dao Str., Ha Noi City, Viet Nam d Faculty of Natural Sciences Pedagogy, Sai Gon University, 273 An Duong Vuong Str., District 5, Ho Chi Minh City, Viet Nam e Center for Theoretical and Computational Physics and Department of Physics, University of Education, Hue University, Hue City, Viet Nam b

ARTICLE INFO

ABSTRACT

Keywords: Phosphorene Optical conductivity Linear response theory

The paper presents a theoretical investigation of the temperature and an out-of-plane Zeeman magnetic field effects on the interband optical conductivity of phosphorene. The electronic states are calculated with the tightbinding Hamiltonian and to compute the optical conductivity, the Kubo formalism is employed. The main result is that the multiple bands due to the perpendicular Zeeman splitting of the energy levels lead to the multiple resonance structures of the optical conductivity. It is found that the strong anisotropic interband optical conductivity of pristine phosphorene keeps the anisotropic feature when the temperature is increased. Importantly, our results show that the optical properties can be externally tuned via a magnetic field. In particular, when the magnetic field is applied, the unique anisotropic property breaks down and there is a little to no sign for temperature- and orientation-dependent transitions. The scientific soundness of findings is useful for practical aspects of phosphorene in spintronic.

1. Introduction Novel properties of gapless graphene as the first two-dimensional (2D) material made it remarkable for logic applications in industry [1–6]. However, due to the absence of the band gap in this material, many other layered materials with inherent non-zero band gap have been synthesized [7–10]. Depending on their composition and stacking, these materials have very different properties. Differently from the flat hexagonal lattice of graphene stemming from the -hybridization of carbon atomic orbitals, sp3 -hybridization Xmakes phosphorene a puckered honeycomb structure with a direct finite band gap of 1.52 eV [11–18]. Interestingly, phosphorene owns an on/off ratio and carrier mobility of about 105 and 103 cm2/Vs, respectively [19]. This property converts phosphorene to a promising candidate for developing flexible electronics and optoelectronics. Phosphorene experimentally can be exfoliated using different methods [19]. For instance, layer-dependent photoluminescence in two to five layers phosphorene with the thickness from 1.3 nm for bilayer phosphorene to 3.0 nm for five layers phosphorene are synthesized using mechanical exfoliation method [20]. The remarkable property in this material refers to the high anisotropic feature between the Fermi

⁎

velocity and the effective mass of carriers along X- and Y-directions of the first Brillouin zone (FBZ) [13]. This unique property introduces unique physical properties as well, which has not been widely seen in other electronic 2D materials. In view of improvement, a vast amount of methods and approaches have put forth to tailor the anisotropic electronic, optical, thermal, and magnetic properties of phosphorene such as mechanical forces [21–24], doping [25–27] and applying electric/magnetic field [28–31]. Although numerous theoretical works on electronic and mechanical perturbation effects in phosphorene have been done [32–35], there come a small number of theoretical concepts trying to describe magnetic perturbation effects [36,37]. In a theoretical work [38], people have tried to address magnetic quantization effects in phosphorene by combined magnetic and electric fields. Or, the Landau levels have been investigated in multilayer phosphorene to find the role of a perpendicular magnetic field in magneto-electronic and -optical transport properties [39,40]. To the best of our knowledge, few studies have been reported analytically to date on the optical conductivity of phosphorene in the presence of combined effects of temperature and magnetic field. The wide variety of real applications of phosphorene in the spintronic community and of their defects calls for an enormous exploratory

Corresponding author. E-mail addresses: [email protected] (D.Q. Khoa), [email protected] (B.D. Hoi).

https://doi.org/10.1016/j.chemphys.2019.05.012 Received 15 April 2019; Received in revised form 10 May 2019; Accepted 12 May 2019 Available online 13 May 2019 0301-0104/ © 2019 Elsevier B.V. All rights reserved.

Chemical Physics 524 (2019) 113–117

D.Q. Khoa, et al.

Fig. 1. Side view of phosphorene with a unit cell consisting of four phosphorus atoms, as shown with a blue dashed rectangle. The Zeeman magnetic field is applied to the system perpendicularly, i.e. B = Bz ez . x (y) axis refers to the armchair (zigzag) direction.

Fig. 2. The direction-dependent electronic band structure of phosphorene along X and (b) Y directions in the absence and presence of Zeeman the (a) magnetic field. We set the Fermi energy level to zero (the black horizontal dashed lines). The multiple bands due to the Zeeman spin-splitting field is evident.

effort, which is still in its early stage. In this context, the optical responses of phosphorene in the presence of temperature and magnetic field effects are essential tools to understand the physical origin of its behavior in spintronic and to exploit it for designing innovative devices. The focus will be set on the interband optical transitions in phosphorene, which have not already been extensively studied in the literature. The study will base on the atomistic (tight-binding) description of the material and on the linear response theory. Such an approach allows the extraction of optical responses using the Kubo formula. The remainder of this paper is divided into four sections. Section 2 gives a brief overview of the geometric structure and the theoretical framework. Section 3 deals with the optical conductivity. The impacts of temperature and magnetic field on the interband optical transitions in phosphorene are reported in Section 4 and finally, the conclusions are presented in Section 5.

for a = 4.42936 Å (the length of the unit cell into the armchair direction) and b = 3.27 Å (the length of the unit cell into the zigzag direction) [13,41,42]. By diagonalizing the Hamiltonian in Eq. (2), the magnetic field (spin)-dependent electronic band structure can easily be calculated as:

Es

k ,±

In this section, we start with the atomic structure of monolayer phosphorene in order to propose the tight-binding Hamiltonian model for two sublayers with four atoms per sublayer (see Fig. 1). The effective Hamiltonian of phosphorene in the absence of Zeeman perpendicular magnetic field can be taken from Refs. [13,14,41], however; it is easy to extend it when the magnetic field is present to †

tij f i, s fj, s + gµ B Bz

i, j , s

†

i

f i, fi, + H. c. ,

(1)

where the summation runs over up to fifth nearest neighbors and the summation over s = ± 1 refers to the spin degree of freedom. The op†

erators f i, s and fj, s are introduced to create and annihilate an electron at i-th and j-th atomic site with spin s, respectively. The hopping integral energies tij between atomic sites i and j are given by t1 = 1.220 eV, t2 = + 3.665 eV, t3 = 0.205 eV, t4 = 0.105 eV, and t5 = 0.055 eV [13,41], as presented in Fig. 1. The g factor is the Lande factor, µ B is the Bohr’s magneton and Bz is the perpendicular magnetic field strength. As can be seen from the unit cell of phosphorene, for both sublayers, one can consider the same atoms, aiming at decreasing the dimension of the Hamiltonian. In the low-energy limit for one pz -like orbital, after the Fourier transformation, the Hamiltonian matrix reads [41,42]

h 11

Hk =

k h 12 k h 13 k

0

h 12 h 13 k h 11 k

0 h 13 k

k

0 h 11 k h 12 k

k h 12 k h 11 k

k

(3)

k

In this section, we intend to use the Kubo formalism [43–45] to compute the optical responses of the monolayer phosphorene to an applied optical field E ( , t ) in the presence of the perturbation Zeeman magnetic field. The total Hamiltonian including the interacting term in the presence of the optical field is given by H = H 0 + Hint where H 0 k

k

is the non-interacting Hamiltonian and Hint = J ·A . The vector potential A can be calculated using the relation A =

, (2)

k

t 0

E ( , t ) dt . On the

other hand, the current density J along the Cartesian direction in the interacting term is related to the components of the respective electric field E within the linear response of the optical conductivity as J = E where the Greek indices and may take the values of the

where the momenta k = (k x , k y ) belong to the FBZ of phosphorene, h 13 = 2t1 h 11 = 4t4 cos(k x a/2)cos(k y b /2), h 12 = gµ B Bz , and k

k

3. Interband optical conductivity

0 h 13

h 13 h 13 + sh 12.

k

The spin- and orientation-dependent electronic band structure of monolayer phosphorene are presented in Fig. 2 in order to see how the Zeeman magnetic field affects the carrier dynamics. The high anisoY is expected from the anisotropic X and tropic bands along carrier Fermi velocities and effective masses of phosphorene [13], as shown in Fig. 2(a) and (b). Our model results in the band gap of about Eg = 1.52 eV, in good agreement with Refs. [13,14,41,42]. In the absence of magnetic field, the spin degeneracy is evident (the blue curves) and once the Zeeman field is turned on, the spin-splitting emerges (the orange and black curves). To ensure that the Zeeman field is strong enough to split the bands, we set gµ B Bz = Eg /2 and gµ B Bz = Eg . As usual, the spin-up state is shifted down in energy by gµ B Bz , while the spin-down state is shifted up by the same amount, leading to a gapless phase for spin-up states and a gapped phase for spin-down ones at gµ B Bz = Eg /2 . However, while a gapped phase appears for spin-down states at gµ B Bz = Eg , a Dirac-like cone takes place along both directions Y for spin-up ones. The main purpose of this work X and from is focused on the dependency of interband optical transitions between these split bands on the temperature of the system.

2. Theoretical framework

H=

= h 11 ±

flavors x and y. After converting the wave vector from k to k + (e/ ) A in the presence of optical field, the -component of J reads

k

e ikx a1x cos(k y b /2) + t2 e ikx a2x + 2t3 e ikx a3x cos(k y b/2) + t5 e ikx a5x . The distance between the intra- and inter-planar nearest-neighbor atoms projected to the x direction are given by the parameters a1x = 1.41763 Å, a2x = 2.16400 Å, a3x = 3.01227 Å, a4x = 2.21468 Å, and a5x = 3.63258 Å

J =

e

†

fk, f k, k,

114

k

+i

e

†

fk, f k, k,

k

,

(4)

Chemical Physics 524 (2019) 113–117

D.Q. Khoa, et al.

Fig. 3. Interband optical conductivity along the armchair edge {(a), (b)} and zigzag edge {(c), (d)} when the temperature is increased. Panels {(a), (c)} are the real parts of IOC and panels {(b), (d)} refer to the imaginary parts for both directions. The height of absorption peaks and also the scattering peaks decreases with T.

where { , } are connected to the velocity of carriers in phosk k phorene, given by [13,14,41,42]

4. Discussion We begin with some general remarks. In what follows, all physical constant g , µ B , kB, c, me , , and e set to unity for simplicity. Also, all IOC curves normalized to the universal value of 0 = e 2/ have two real and imaginary parts. In order to see the temperature effects on the IOC of phosphorene in the presence of magnetic field, we assume constant Zeeman splitting fields gµ B Bz = Eg /2 and gµ B Bz = Eg because we want to achieve some information from the changes made in the band structure shown in Fig. 2. It should be noted that for the case of undoped phosphorene at zero temperature there are two sets for four types of interband transitions between E , E , E , and E bands. The first set includes four

x k

= + 2t1 a1x sin(k x a1x + (k x a3x

k

k

) + t2 a2x sin(k x a2x

k

) + 2t3 a3x cos(k y b /2)sin

) + 2t4 asin(k x a/2)cos(k y b/2) + t5 a5x sin(k x a5x +

k

), (5a)

y k

= + bt1sin(k y b/2)cos(k x a1x +

k

) + bt3sin(k y b /2)cos(k x a3x +

k

)+2 (5b)

t4 bcos(k x a/2)sin(k y b/2),

the {E

k,

2t1 a1x cos(k y b /2)cos(k x a1x + cos(k y b/2)cos(k x a3x

y k

k

)

k

) + t2 a2x cos(k x a2x

t5 a5x cos(k x a5x +

= + bt1sin(k y b /2)sin(k x a1x +

k

)

k

k

) + 2t3 a3x (5c)

),

bt3sin(k y b /2)sin(k x a3x

k

k

k

).

1

dte i

0

t

k,

k

[J (t ), J (0)] .

(6)

In the present work, we focus only on the interband optical transitions because we assume that the incident optical field is strong enough and will be adsorbed by carriers, leading to the transition from the valence band to the conduction band. Of course, the intraband ones can also be considered if one considers disorder or scattering of the host electrons in the THz region (Drude-like conductivity), however; this is out of the scope of the present paper. Since the Hall conductivities are zero, i.e. xy ( ) = yx ( ) = 0 due to the structural symmetry of phosphorene, the longitudinal interband optical conductivity (IOC) along the armchair and/or zigzag directions are calculated via inter

( )

0

=

2i k

FBZ

( k)

n FD, s k ,+ Es k ,+

2·

n FD, s k, +Es k,

n FD, s +

+i

k ,+ Es k ,+

n FD, s k,

Es

k,

+i

. (7)

where 0 = e 2/ and = 10 meV are the universal value for the optical conductivity and the finite damping between the valence and conducµ)/ kB T ] is the tion bands, respectively. n FD, s = 1/1 + exp [(E s k ,±

k ,+

k ,+

k ,+

k,

k ,+

k ,+

k,

k ,+

and smallest (zero) gaps, respectively. As for gµ B Bz = Eg , the first and second types of transitions are the same but for the frequencies equal to 2Eg , while the two other ones correspond to the largest (3Eg ) and smallest (Eg ) gaps, respectively (see Fig. 2). After addressing the possible interband transitions, we express the general treatments for the optical conductivity representing the main features of transition at low and high frequencies. We start all calculations for the pristine phosphorene ( gµ B Bz = 0 eV) when the temperature is increased. Fig. 3 shows the behaviors of IOC for both directions as a function of optical frequency at different temperatures. First of all, we see that the band gap of the system does not change with temperature because the peaks take place at the same points of the optical energy axis and only their height decreases. It can be seen that the IOC of pristine phosphorene along the armchair direction has a 6Eg transiEg and a weak maximum at strong minimum at tion (see Fig. 3(a) and (b)), while it has only a strong maximum at 5Eg transition along the zigzag edge (see Fig. 3(c) and (d)). These results are in excellent agreement with Ref. [42]. Interestingly, temperature affects all transitions along both directions significantly and leads to a zero conductivity at high enough temperatures. This means that the IOC decreases with the temperature if the band gap remains unchanged. Based on the Kramers-Kroning rule, the variation of imaginary parts of IOCs with T should not take place at different optical energies compared to the real parts, as confirmed in Fig. 3(b) and (d). One interesting phenomenon from the behavior of imaginary parts with temperature is that the peaks are disappearing in both directions, which means that the maximum profound optical field-induced scattering of electrons occurs. This was expected because the temperature is the macroscopic version of the kinetic energy of electrons and the scattering rate of carriers also strongly depends on their kinetic energy. From this point, increasing temperature is somehow increasing the inherent scattering rate between electrons themselves and eventually

Eventually, the normalized optical conductivity with respect to the planar area is calculated via

( )=

k,

k ,+

(because of the spin degeneracy) } independent of the direction when

k,

ilog[ h 13/ h 13 ].

=

k ,+

equally because of the same splitting gap (equal to Eg ) between them (see Fig. 2). However, the two other ones occur between the bands E E E and E corresponding to the largest (2Eg )

(5d) where

k,

gµ B Bz = 0 eV (referring to the blue curves in Fig. 2), whereas the second set of interband transitions refer to the non-zero magnetic fields. In the latter, at gµ B Bz = Eg /2 , the first and second types of transitions E E occur between the energy bands E and E

x k

=

k,

same transitions ,E } {E , E

k ,±

Fermi–Dirac distribution function at the chemical potential µ . As usual, T is the absolute temperature and kB is the Boltzmann constant.

115

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D.Q. Khoa, et al.

value of the conductivity with respect to the temperature. It points that inter ( )/ 0 has finite amounts up to 15 K at all optical energy reRe xx Eg and remains gions, whereas approaches zero after about 5 K for > Eg . The results above tell us that the abfinite up to 20 K for sorption of the frequency-dependent optical field depends strongly on the temperature, in good agreement with Ref. [46]. In the case of scattering mechanism of electrons when interacting with the optical field, panels (b) and (f) offer an increasing trend in both Zeeman field strengths, however; the intensity of scattering at gµ B Bz = Eg /2 is more than the one at gµ B Bz = Eg . In the very low-temperature region of the conductivity at gµ B Bz = Eg /2 , a dip takes place at T 1 K which the scattering process changes its conduct before and after that. We traced this effect back to the backscattering of electrons appearing when the Zeeman field is high enough (see Fig. 2). Similar to the armchair direction, we have discussed the optical responses of the zigzag direction as a function of temperature with respect to its alteration on the strengths of the low and high Zeeman fields, i.e. gµ B Bz = Eg /2 (panels (c) and (d)) and gµ B Bz = Eg (panels (g) inter ( )/ 0 the intensity and (h)). Using the real and imaginary parts of yy of absorbed and scattered optical field of phosphorene can be obtained. inter ( )/ 0 for both Zeeman Unlike the armchair edge, the real part of yy field strengths becomes regular as the optical frequency is increased, inter ( )/ 0 increases with . However, still in the y-direction the i.e. Re yy absorption process in the intermediate frequencies has a critical value at T 1 K with two different behaviors before and after that, while this was not the case in panel (a). These discrepancies are a straight consequence of inherent anisotropy of charge carriers in phosphorene. inter ( )/ 0 is not changed much for Nevertheless, the imaginary part of yy different Zeeman fields and behave almost similarly with constant values at very small and large Zeeman fields, as shown in panels (d) and (h). Also, they show an increasing trend of .

Fig. 4. Real {(a), (e)} ({(c), (g)}) and imaginary {(b), (f)} ({(d), (f)}) part of the armchair (zigzag) component of the optical conductivity as a function of temperature for two different Zeeman field strengths, namely gµ B Bz = Eg /2 and gµ B Bz = Eg corresponding to the {(a)-(d)} and {(e)-(h)} panels, respectively.

5. Conclusions

between electrons and optical field. Now we turn to the combined effects of temperature and magnetic field on the interband optical transitions in phosphorene. In the following, we investigate the effect of temperature on the optical conductivity of phosphorene when the Zeeman field hits the surface of phosphorene with two different strengths gµ B Bz = Eg /2 and gµ B Bz = Eg , corresponding to the {(a)-(d)} and {(e)-(h)} panels of Fig. 4, respectively, at different values of optical frequency , namely < Eg , Eg , and > Eg . This is revealed by the phase shift E k , ±/ kB T of the Fermi–Dirac distribution function. In a nutshell, the {(a)-(d)} and {(e)-(h)} panels have little to no change when gµ B Bz = Eg /2 and gµ B Bz = Eg because the panels show exactly the same behaviors but with some peaks occurring at fairly low temperatures. The temperature T which varies from 0 K to 20 K is the right range of non-zero conductivity in our model because both components vanish for T > 20 K. As indicated in Fig. 4 the shape of curves for different optical frequencies is roughly the same for imaginary parts (panels (b), (d), (f), and (h)), while they take a slight change in the real parts (panels (a), (c), (e), and (g)) at very low temperatures. This change is around the critical point T = 1 K. The main point of temperature effect can be understood from the kinetic energy of host carriers, affecting the scattering rate of carriers when interacting with the optical light. Let us focus on each panel. We first peruse the armchair edge. In panel (a), two reverse procedures emerge before and after T = 1 K at inter < Eg . Re xx ( )/ 0 increases from 0 K to low-frequency region, i.e. 1 K and decreases after 1 K slightly. However, this is not the case for Eg and there is a smooth decreasing treatment with temperature. These arise from the different selection rules of transition between bands at gµ B Bz = Eg /2 when the optical frequency is increased. On the other hand, it provides the peak at all frequencies when the Zeeman splitting field is increased ( gµ B Bz = Eg ) as presented in panel (e). The remarkable difference between panel (a) and (e) can be linked to the

The temperature effects on the interband optical transitions in phosphorene are studied based on a tight-binding Hamiltonian model in the absence and presence of Zeeman magnetic field. By using the linear response theory, and by explicitly including the optical field, the longitudinal optical conductivities along the armchair and zigzag edges are calculated. Crucially, while the paper reports that the real (imaginary) part of optical transitions decreases (increases) with temperature independent of the optical frequency, the key finding refers to a little to no alteration of temperature- and orientation-dependent optical transitions at weak and strong Zeeman fields. The theoretical findings here provide support that can potentially the opto-spintronic engineering of phosphorene be realized in experiment. Acknowledgement This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.01-2017.361. References [1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Science 306 (2004) 666. [2] Y. Wang, D.R. Andersen, J. Phys.: Condens. Matter 28 (2016) 475301. [3] M. Khandelwala, A. Kumar, J. Mater. Chem. 3 (2015) 22975. [4] X. Yu, B. Wang, D. Gong, Z. Xu, B. Lu, Adv. Mater. 29 (4) (2017) 1604118. [5] M. Yarmohammadi, M. Zareyan, Chin. Phys. B 25 (2016) 068105. [6] M. Yarmohammadi, Physics Letters A 380 (2016) 4062. [7] N. Takagi, C.L. Lin, K. Kawahara, E. Minamitani, N. Tsukahara, M. Kawai, R. Arafune, Prog. Surf. Sci. 90 (2015) 1. [8] M. Derivaz, D. Dentel, R. Stephan, M. Hanf, A. Mehdaoui, P. Sonnet, C. Pirri, Nano Lett. 15 (2015) 2510. [9] F.F. Zhu, W.J. Chen, Y. Xu, C.L. Gao, D.D. Guan, C.H. Liu, D. Qian, S.C. Zhang, J.F. Jia, Nat. Mater. 14 (2015) 1020. [10] N. Alem, R. Erni, C. Kisielowski, M.D. Rossell, W. Gannett, A. Zettl, Phys. Rev. B 80

116

Chemical Physics 524 (2019) 113–117

D.Q. Khoa, et al. (2009) 155425. [11] H. Liu, A.T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tomnek, P.D. Ye, ACS Nano 8 (2014) 4033. [12] A. Castellanosgomez, L. Vicarelli, E. Prada, J.O. Island, K.L. Narasimhaacharya, S.I. Blanter, D.J. Groenendijk, M. Buscema, G.A. Steele, J.V. Alvarez, 2D Mater. 1 (2014) 025001. [13] A.N. Rudenko, M.I. Katsnelson, Phys. Rev. B 89 (2014) 201408(R). [14] M. Ezawa, New J. Phys. 16 (2014) 115004. [15] Y. Du, C. Ouyang, S. Shi, M. Lei, J. Appl. Phys. 107 (2010) 093718. [16] J. Qiao, X. Kong, Z.-X. Hu, F. Yang, W. Ji, Nature Commun. 5 (2014) 4475. [17] A.S. Rodin, A. Carvalho, A.H. Castro Neto, Phys. Rev. Lett. 112 (2014) 176801. [18] P.T.T. Le, M. Davoudiniya, K. Mirabbaszadeh, B.D. Hoi, M. Yarmohammadi, Physica E: Low-dimensional Syst. Nanostruct. 106 (2019) 250. [19] L. Li, Y. Yu, G.J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X.H. Chen, Y. Zhang, Nature Nanotechnol. 9 (2014) 372. [20] J. Dai, X.C. Zeng, J. Phys. Chem. Lett. 5 (2014) 1289. [21] X. Han, H. Morgan Stewart, S.A. Shevlin, C.R.A. Catlow, Z.X. Guo, Nano Lett. 14 (2014) 4607. [22] X. Peng, Q. Wei, A. Copple, Phys. Rev. B 90 (2014) 085402. [23] T. Zhang, J.-H. Lin, Y.-M. Yu, X.-R. Chen, W.-M. Liu, Sci. Rep. 5 (2015) 13927. [24] N.D. Hien, M. Davoudiniya, K. Mirabbaszadeh, Le T.T. Phuong, M. Yarmohammadi, Chem. Phys. 522 (2019) 249. [25] D.H. Bui, M. Yarmohammadi, Phys. Lett. A 382 (2018) 1885. [26] P. Rastogi, S. Kumar, S. Bhowmick, A. Agarwal, Y.S. Chauhan, IETE J. Res. 63 (2016) 205.

[27] P.T.T. Le, K. Mirabbaszadeh, M. Davoudiniya, M. Yarmohammadi, Phys. Chem. Chem. Phys. 20 (2018) 25044. [28] R. Fei, L. Yang, Nano Lett. 14 (2014) 2884. [29] H.D. Bui, Le T.T. Phuong, M. Yarmohammadi, EPL 124 (2018) 27001. [30] Q. Liu, X. Zhang, L. Abdalla, A. Fazzio, A. Zunger, Nano Lett. 15 (2015) 1222. [31] H.D. Bui, M. Yarmohammadi, Solid State Commun. 280 (2018) 39. [32] H.D. Bui, M. Yarmohammadi, Physica E: Low-dimensional Syst. Nanostruct. 103 (2018) 76. [33] P.T.T. Le, M. Yarmohammadi, Physica E: Low-dimensional Syst. Nanostruct. 107 (2019) 11. [34] P.T.T. Le, M. Davoudiniya, K. Mirabbaszadeh, B.D. Hoi, M. Yarmohammadi, Physica E: Low-dimensional Syst. Nanostruct. 106 (2019) 250. [35] H.D. Bui, M. Yarmohammadi, Superlattices Microstruct. 122 (2018) 453. [36] H.D. Bui, M. Yarmohammadi, J. Magn. Magn. Mater. 465 (2018) 646. [37] P.T.T. Le, M. Yarmohammadi, Chem. Phys. 519 (2019) 1. [38] J.-Y. Wu, S.-C. Chen, G. Gumbs, M.-F. Lin, Phys. Rev. B 95 (2017) 115411. [39] J.M. Pereira Jr., M.I. Katsnelson, Phys. Rev. B 92 (2015) 075437. [40] X.Y. Zhou, W.K. Lou, F. Zhai, K. Chang, Phys. Rev. B 92 (2015) 165405. [41] S. Yuan, A.N. Rudenko, M.I. Katsnelson, Phys. Rev. B 91 (2015) 115436. [42] C.H. Yang, J.Y. Zhang, G.X. Wang, C. Zhang, Phys. Rev. B 97 (2018) 245408. [43] R. Kubo, J. Phys. Soc. Jpn. 12 (1957) 570. [44] G. Mahan, Many Particle Physics, Plenum Press, New York, 1993. [45] G. Grosso, G.P. Parravicini, Solid State Physics, 2nd edn., Academic Press, New York, 2014. [46] A. Iurov, G. Gumbs, D. Huang, Phys. Rev. B 98 (2018) 07541.

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