Hall conductivity of silicene and germanene

C O M P U T A T I O N & T H E O R Y

Low-energy bands, optical properties, and spin/valley- Hall conductivity of silicene and germanene

Pham Thi Huong^1,2, Do Muoi³, Huynh V. Phuc⁴, Chuong V. Nguyen⁵, Le T. Hoa^6,7,*, Bui D. Hoi⁸, and Nguyen N. Hieu^6,7,*

1Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam

2Faculty of Environment & Labour Safety, Ton Duc Thang University, Ho Chi Minh City, Viet Nam

3Department of Physics and Engineering Physics, University of Science-VNU.HCM, Ho Chi Minh City, Viet Nam

4Division of Theoretical Physics, Dong Thap University, Cao Lanh 870000, Viet Nam

5Department of Materials Science and Engineering, Le Quy Don Technical University, Ha Noi, Viet Nam

6Institute of Research and Development, Duy Tan University, Da Nang 550000, Viet Nam

7Faculty of Natural Sciences, Duy Tan University, Da Nang 550000, Viet Nam

8Department of Physics, University of Education, Hue University, Hue, Viet Nam

Received:1 April 2020 Accepted:5 July 2020

Ó

Springer Science+Business Media, LLC, part of Springer Nature 2020

ABSTRACT

In this work, we study systematically low-energy bands, optical absorbance and spin/valley-Hall conductivity of silicene and germanene in the presence of a perpendicular electric field. Our analytical calculations indicate that both sil- icene and germanene are semiconductors with a tiny energy gap and we can control their energy gap by the perpendicular electric field. Our calculations also demonstrate that the low-frequency optical absorbance of silicene is much greater than that of germanene and the external electric field plays an important role in determining the optical absorption peaks. When the Fermi level is in the forbidden band, the Hall conductivity is quantized, while spin/valley-Hall conductivities of both silicene and germanene depend strongly on the Fermi energy when the Fermi level is in the conduction band. Analytical results for spin/valley-Hall conductivities of silicene and germanene are presented in detail in this work.

Introduction

Since the successful exfoliation of graphene mechanically [1], a revolution in search for new two- dimensional (2D) layered materials has taken place shortly thereafter. An excellent consequence of this is

that many new 2D layered materials have been dis- covered with promising applications in spintronics and optoelectronics, silicene and germanene are two of them [2, 3]. Silicene [4] and germanene [5], monolayers of silicon and germanium atoms, respectively, have been successfully synthesized

Address correspondence toE-mail: lethihoa8@duytan.edu.vn; hieunn@duytan.edu.vn https://doi.org/10.1007/s10853-020-05044-0

J Mater Sci

Computation & theory

experimentally. Unlike graphene, one atom thickness 2D nanomaterial with planar structure, silicene and germanene have low-buckling structures. Besides, silicene and especially germanene have quite large spin-orbit gap [6,7]. The spin-orbit gap of germanene has been predicted up to 43 meV, much larger than that of silicene (3.9 meV) [7]. It is known that in the semimetal form with gapless, graphene has been shown to not suitable to apply in electronic devi- ces [8]. The possession of strong intrinsic spin-orbit interaction, which can open band gap in their elec- tronic band structures, silicene and germanene have solved the energy gap problem encountering in graphene.

Low-energy bands of silicene and germanene have been investigated by different approaches [7, 9–11].

Based on density functional theory (DFT) calcula- tions, Ni and co-workers have demonstrated that band gap sizes of both silicene and germanene can be driven (increased) by external vertical electric field [12]. Matthes and co-workers have indicated that the spin-orbit effect plays a key role in the optical absorbance of silicene and germanene at low-fre- quency regions [11]. In another work [13], based on analytical approaches, Matthes and co-workers also showed that the optical absorbance of silicene and germanene can be defined via the Sommerfeld fine- structure constant. Due to their strong spin-orbit coupling (SOC), the electronic characteristics of sil- icene and especially germanene have many interest- ing features [14]. Recently, valley- and spin- dependent quantum Hall conductivity in bilayer sil- icene has been investigated using dynamic Kubo formula [15]. Previously, Hall conductivities of sil- icene and germanene have been considered by dif- ferent methods [9, 10, 16, 17]. The quantum Hall effect has been experimentally observed and mea- sured in graphene [18, 19] and is predicted to be possible in silicene and germanene soon [6,20].

In this study, we systematically consider the low- energy bands, optical absorbance, and spin/valley- Hall conductivity of silicene and germanene using the analytical approaches of low-energy Hamiltonian and Kubo formalism. We focus on the influence of the external electric field on low-energy electron states and optical absorbance of silicene and germanene.

The spin- and valley dependence of Hall conductiv- ities in silicene and germanene has been calculated analytically in detail.

Low-energy electron states

The low-energy effective Hamiltonian of silicene and germanene in a perpendicular electric field can be written as [21]

Hg ¼hvFðgk_xr_xþkyr_yÞ gsDSOr_zD_zr_z; ð1Þ where vF is the Fermi velocity, g¼ 1 is for K/K’

valley, s¼ 1 are for spin-up and spin-down of electrons, respectively, r_x;r_y;r_z are the Pauli matri- ces, DSO is the intrinsic spin-orbit coupling (SOC) strength,D_z¼edE_z, withE_z being the perpendicular applied electric field and, 2d is the vertical distance between the two sublattices. Here, we neglected the Rashba SOC effect due to its small magnitude [21–23].

Using the Pauli matrices, the Hamiltonian (1) can be expressed as the following

Hg ¼ D_gs hv_Fðgk_xik_yÞ hvFðgk_xþikyÞ D_gs

; ð2Þ

where D_gs¼D_zgsDSO. By diagonalization of the Hamiltonian (2) of detjHgEgsðkÞj ¼0, we receive the low-energy dispersion relations in the form:

EgsðkÞ ¼ke_gs¼k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D²_gsþh²v²_Fk² q

; ð3Þ

where e_gs¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D²_gsþh²v²_Fk² q

and k¼ 1 is the band index. For numerical calculations, we use available parameters for silicene and germanene reported in Refs. [7, 21]. The lattice constants a of silicene and germanene are, respectively, 3.86 A˚ and 4.02 A˚ [7].

The Fermi velocityv_Fand the intrinsic SOC strength DSO of silicene (germanene) are 5:4210⁵ m/s (5:2410⁵ m/s) and 3.90 meV (43.00 meV), respec- tively [7]. The vertical distance between two sublat- tices 2d in silicene and germanene is 0.46 A˚ and 0.66 A˚ , respectively [21].

Low-energy bands of silicene and germanene as a function of the dimensionless parameter K ¼ hv_Fk=DSO are shown in Fig.1. Our calculations demonstrate that the band gap of silicene atD_z¼0 is only 7.80 meV, quite smaller than that of germanene (86 meV). However, compared with the transition metal dichalcogenides, the band gaps of both silicene and germanene are very small [24]. Our obtained results for the band gap of silicene is consistent with the previous study [25]. The band gaps of both sil- icene and germanene can be driven by the electric field D_z. When the electric field energy D_z equals to

the intrinsic SOC strengthDSO, the band gaps of both silicene and germanene are equal to zero as shown in Fig.1. Dependence of band gaps of silicene and germanene on electric field (here we use parameter D_z=DSO) are shown in Fig. 2. The graph depicts the band gap dependence on theD_z=DSOof monolayers at the K point (or K’ point) in the case of spin-up and spin-down that are symmetrical with each other through a vertical line passing through the zero points. Compared to silicene, the band gap of ger- manene varies drastically with the electric field.

Wave function of the system is the eigenfunction of the Hamiltonian (1). This is two-component spinor, which can be written as

w_kkgsðrÞ ¼ 1 ffiffiffiS p e^ikr /

v ; ð4Þ

where S¼L_xL_y is the volume of the sample and functions/andvsatisfy the conditionj/j²þ jvj² ¼1.

Solve the eigenvalue–eigenfunction equation Hgw_kkgs¼E_kkgsw_kkgs, we obtain

/¼ 1 ffiffiffi2

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þkcosh

p ;

v¼ k ffiffiffi2

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1kcosh p

:

ð5Þ

Finally, the wave function becomes

w_kkgsðrÞ ¼ 1 ffiffiffiffiffiffi p2S

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þkcosh p

k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1kcosh p e^iguk

!

e^ikr: ð6Þ

The density of states (DOS) can be written as DðEÞ ¼X

k;s;k

dðEE_kkgsÞ: ð7Þ

In Eq. (7), the sum over k can turn into a form P

k! ½L=ð2pÞ²R

kd²k. Then, we obtain the DOS as follows:

DðEÞ ¼ S ð2pÞ²

X

k;s

Z

k

dðEE_kkgsÞd²k

¼ S 2p

1 2

X

s

Z

k

dðk²Þh

dðE jE_kkgsjÞ þdðEþ jE_kkgsjÞi

¼ S 2p

1 2

X

s

2jEj Z

k

dðk²ÞdðjE_kkgsj²E²Þ

¼ S 2pjEjX

s

Z

k

dðk²ÞdðE²h²v²_Fk²D²_gsÞ:

ð8Þ By using the Heaviside step functionHðxÞ, Eq. (8) can be rewritten in the form:

DðEÞ ¼ S 2p

1 h²v²_FjEjh

ðHjEj jDzgDSOjÞ þHðjEj jDzþgDSOjÞi

:

ð9Þ

The calculated DOS of silicene and germanene is shown in Fig. 3. We can realize that the DOS of sil- icene and germanene has the same profile. We see that the DOS of both silicene and germanene increa- ses linearly with a vertical jump. This vertical jump is consistent with the Heaviside step function HðxÞ as expressed in Eq. (9). The tendency of changes in energy due to the effect of the external electric field as shown in Fig.3 is consistent with the previous cal- culations by Vargiamidis and co-workers [26].

(a)

(b)

Figure 1 Low-energy bands of silicene (a) and germanene (b) as a function of the dimensionless parameter K ¼hvFk=DSO at different value of electric field energyD_z.

Ehrenreich–Cohen dielectric function and low-frequency optical absorbance

We first calculate the velocity elements which is needed for the calculation of the Hall conductivity tensor. Hereafter, we use the symbolsa¼ ðk;kÞand a⁰¼ ðk⁰;k⁰Þfor simplicity. The velocity elements are

v^gs_x;aa0 ¼ hw_kkgsjv^xjw_k⁰_k⁰_gsi;

v^gs_y;aa0 ¼ hw_kkgsjv^yjw_k⁰_k⁰_gsi; ð10Þ where

v^x ¼1 h

oH^g

okx

¼gvFr_x; v^_y ¼1

h oH^g

ok_y ¼gv_Fr_y:

ð11Þ

Using the wave function in Eq. (6), we have

v^gs_x;aa0 ¼ hw_kkgsjv^_rjw_k⁰_k⁰_gsi

¼ 1 2Sd_kk⁰h

gv_Fk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1kcoshÞð1þk⁰coshÞ q

e^igukþ þgv_Fk⁰

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1þkcoshÞð1k⁰coshÞ q

e^iguk

¼gv_F 2Sd_kk⁰h

k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1kd_kÞð1þk⁰d_kÞ q

e^iguk þk⁰

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1þkd_kÞð1k⁰d_kÞ q

e^iguki

¼gvF

2Sd_kk⁰_þ;

ð12Þ and

v^gs_x;aa⁰ ¼iv_F

2Sd_kk⁰: ð13Þ

Here, we set d_k¼coshand

(a) (b) (c)

Figure 2 (a) Band gap of silicene at the K point as a function of parameterD_z=DSO. A comparison of band gaps of silicene and germanene at the K point in the cases of spin-up (b) and spin-down (c).

(a) (b) (c)

Figure 3 Density of states of silicene (a) and of silicene and germanene atD_z=DSO¼0:5(b) andD_z=DSO¼1:0(c).

_þ¼h k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1kd_kÞð1þk⁰d_kÞ q

e^iguk þk⁰

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1þkd_kÞð1k⁰d_kÞ q

e^iguki

; ¼h

k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1kd_kÞð1þk⁰d_kÞ q

e^iguk k⁰

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1þkd_kÞð1k⁰d_kÞ q

e^iguki :

ð14Þ

The components of the transfer momentum matrices can be written as

P^gs_x;aa0¼mv^gs_x;aa⁰¼mgvF

2S d_kk⁰_þ; P^gs_y;aa0¼mv^gs_y;aa0¼imvF

2S d_kk⁰;

ð15Þ

wheremis the electron mass. Hence, jPðkÞj² ¼1

2

jP^gs_x;aa0j²þ jP^gs_y;aa0j²

¼m²v²_F

2S² ð1kk⁰D²_gs e²_gsÞ:

ð16Þ

It is well-known that we can infer the optical char- acteristics of layered nanomaterials from parts of in- plane dielectric functionðxÞ. In the case of undoped 2D layered nanomaterials, the dielectric function can be obtained by using the Ehrenreich–Cohen formula via the optical transfer matric M_cvðkÞ for the filled valence band (v) and completely unfilled conduction band (c) as follows [13]:

ðxÞ ¼1þ8p LS

X

c;v

X

k

jM_cvðkÞj² X

b¼1

1

e_cðkÞ e_vðkÞ þbðhxþiCÞ;

ð17Þ

whereLrefers to the lattice constant,Sstands for the area of the sheet, and M_cvðkÞ is the optical transfer matrix which can be given by

M_cvðkÞ ¼ e m ffiffiffiffiffiffiffiffiffiffiffiffi

4pe₀h

p hw_ckjP_cvðkÞjw_vki

e_cðkÞ e_vðkÞ : ð18Þ Using the equalities

1

xiC¼ x

x²þC²þi C x²þC²; 1

xþiC¼ x

x²þC²i C x²þC²;

ð19Þ

the sum in Eq. (17) is rewritten as

X

b¼1

1

e_cðkÞ e_vðkÞ þbðhxþiCÞ¼

¼ 1

e_cðkÞ e_vðkÞ þ hxþiCþ 1

e_cðkÞ e_vðkÞ hxiC;

¼ e_ce_vþhx

ðece_vþhxÞ²þC²þ e_ce_vhx ðece_vhxÞ²þC² þi

C

ðece_vþhxÞ²þC² C

ðece_vþhxÞ²þC²

: ð20Þ Substituting Eqs. (18) and (20) into Eq. (17), we obtain

ðxÞ ¼1þ8p LS

e² m²4pe0h

X

k

jP_cvðkÞj² ðe_ce_vÞ² e_ce_vþhx

ðece_vþhxÞ²þC²þ e_ce_vhx ðece_vhxÞ²þC² þi

C

ðece_vþhxÞ²þC² C

ðece_vþhxÞ²þC²

: ð21Þ Using the transformation P

k! ½L=ð2pÞ²R

kd²k, the sum over kin Eq. (21) can be rewritten in the form

X

k

jPcvðkÞj² ðece_vÞ²

e_ce_vþhx

ðece_vþhxÞ²þC²þ e_ce_vhx ðece_vhxÞ²þC² þi

C

ðe_ce_vþhxÞ²þC² C

ðe_ce_vþhxÞ²þC²

!

! S ð2pÞ²

Z

k

d²kjP_cvðkÞj² ðe_ce_vÞ²

e_ce_vþhx ðe_ce_vþhxÞ²þC² þ e_ce_vhx

ðe_ce_vhxÞ²þC² þi

C

ðece_vþhxÞ²þC² C

ðece_vþhxÞ²þC²

¼ m² 8h²Si

1 4Dgs

ðhxÞ² 1

hx þ m²

16ph²S Z

k

dðecevÞ

1kk⁰ 4D²_gs ðece_vÞ²

1 ecev

e_ce_vþhx

ðe_ce_vþhxÞ²þC²þ e_ce_vhx ðe_ce_vhxÞ²þC²

: ð22Þ Finally, we obtain the Ehrenreich–Cohen dielectric function as follows

ðxÞ ¼1þ e² LS²₀h³p

Z

dðece_vÞfðece_vÞ þi e²

4LS²₀h³

1þ4D_etas h²x²

1

hx;

ð23Þ

where

fðece_vÞ ¼

1kk⁰ 4D²_gs ðecevÞ²

1 e_ce_v

e_ce_vþhx ðecevþhxÞ²þC² þ ecevhx

ðece_vhxÞ²þC²

:

ð24Þ The optical absorbance for the interband transitions can be expressed by

AðxÞ ¼8p²x cS

X

c;v

X

k

jM_cvðkÞj²dðe_ce_vhxÞ: ð25Þ By using Eq. (16) and Eq. (18), we can rewrite Eq. (25) as

AðxÞ ¼8p²x cS

X

k

e²v²_F 8p0hS²

1 ðece_vÞ²

1kk⁰4D²_gs h²x²

dðe_ce_vhxÞ

¼p²x cS

e²v²_F p₀hS²

S ð2pÞ²

Z

k

d²k 1 ðece_vÞ²

1kk⁰4D²_gs h²x²

dðece_vhxÞ:

ð26Þ Finally, we obtain the optical absorbance as

AðxÞ ¼ e² 8c₀h³S²

1kk⁰4D²_gs h²x²

: ð27Þ

In Fig.4 we show our calculations for absorbance of silicene and germanene at different values of the electric field energyD_z. We can see that atD_z¼DSO, the absorbance (in units of a=4 with a being Som- merfeld’s fine-structure constant) is equal to 1 [AðxÞj_Dz¼DSO ¼1] and it does not depend on the incoming photon energy. For silicene as shown in Fig.4a, in the case ofD_z\DSO, after not responding to the incident light in the small energy region (about a few meV), the absorbance increases gradually and reaches the value of AðxÞj_D

z¼DSO ¼1 in the high energy region. In the case of D_z[DSO, as soon as responding to the incident light, the absorbance of silicene suddenly reaches maximum value then gradually decreases and approaches the value of AðxÞj_D

z¼DSO ¼1 at the higher energy domain. Also, the closer the value of D_z is to DSO, the higher the peak. Figure4d shows the absorbance of silicene at D_z=DSO 1. For germanene, the vertically jumping of the absorbance has occurred for both casesD_z\DSO

and D_z[DSO as shown in Fig.4b. The vertical increase then decreasing to AðxÞj_Dz¼DSO ¼1 of absor- bance as mentioned above only occurs in the case of D_z[DSO. Meanwhile, for D_z\DSO, after increasing vertically, the absorbance of germanene continues to increase slightly to approach the value of AðxÞj_Dz¼DSO ¼1. The absorption peak of silicene is much greater than that of germanene as shown in Fig.4c; the peak formation in the optical absorbance of silicene and germanene in our calculations is consistent with the obtained results by first-principles simulations [11].

Spin- and valley-dependence of quantum Hall conductivity in silicene

and germanene

The linear response conductivity tensor is given by Kubo formula [27,28]

r^nd_yxðg;sÞ ¼ihe² S

X

f6¼f⁰

ðfff_f⁰Þv_x;ff⁰v_y;ff⁰ ðEfE⁰_fÞðEfE⁰_fþhxþiCÞ:

ð28Þ Here, jfi ¼ jk;k_x;k_y;siand jf⁰i ¼ jk⁰;k⁰_x;k⁰_y;s⁰i.

The off-diagonal component (x!0;C!0) of the DC Hall conductivity tensor is given by

r^nd_yxðg;sÞ ¼ihe² S

X

f6¼f⁰

ðfff_f⁰Þv_x;ff⁰v_y;ff⁰

4e²_gs : ð29Þ

We only consider the case of interband transition with k¼k⁰, all other cases give zero. In the case of interband transition, product of the velocity elements in Eq. (12) and Eq. (13) is written as

v^gs_x;aa0v^gs_y;aa0 ¼igv²_F

S² d; ðdd_k¼coshÞ: ð30Þ Substituting Eq. (30) into Eq. (29), we have

r^nd_yxðg;sÞ ¼ihe²gv²_F S³ D_gsX

k

ðf_kf_kÞ

4e³_gs ; ð31Þ

where D_gs¼D_zgsDSO ande_gs¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D²_gsþh²v²_Fk² q

. Hence, we obtain the spin-dependence rspin

yx ðg;sÞ and valley-dependence rvalley

yx ðg;sÞ Hall conductiv- ities as the followings

rspin

yx ðg;sÞ ¼X

g

r^nd_yxðg;s¼ þ1Þ r^nd_yxðg;s¼ 1Þ

;

rvalley

yx ðg;sÞ ¼X

g

r^nd_yxðg¼ þ1;s¼ 1Þ r^nd_yxðg¼ 1;s¼ 1Þ : ð32Þ We consider separately two possible cases of the Fermi level position: Fermi level is in the forbidden band and the conduction band.

When the Fermi level is in the forbidden band, at zero temperature, we havef_k¼1 andf_þk¼0. Using the transformation of sum overkinto the integral as above-used, Eq. (31) becomes

r^nd_yxðg;sÞ ¼ihe²gv²_F S³ D_gs S

ð2pÞ² Z

k

d²k 1

ðh²v²_Fk²þD²_gsÞ³⁼²

¼g e² 8phS²

D_gs jD_gsj:

ð33Þ We known that the spin current is formulated by J_s¼ ðh=2eÞðJ"J#Þ, then we need multiply the spin- Hall conductivityrspin

yx ðg;sÞbyh=2pand the valley- Hall conductivityrvalley

yx ðg;sÞby 1/2e[28,29]. From Eq. (33) we can see that when the Fermi level is in the gap, the Hall conductivity is quantized.

In the case the Fermi level is in the conduction band, the sum overkin Eq. (31) should be calculated for both occupied states in the valence band and in the electron band which locate below the Fermi level [26]. Then, we can rewrite Eq. (31) in the form:

r^nd_yxðg;sÞ ¼ihe²gv²_F

4S³ D_gs X

k

1 e³_gsX

k;kF

1 e³_gs

: ð34Þ

Finally, the spin-Hall rspin

yx and valley-Hall rvalley

yx

conductivities can be given by

rspin

yx ¼ e²

2hS²

D_zDSO

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðD_zDSOÞ²þh²v²_Fk²_F q

D_zþDSO

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðDzþDSOÞ²þh²v²_Fk²_F q

;

ð35Þ

rvalley

yx ¼ e²

2hS²

D_zDSO

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðD_zDSOÞ²þh²v²_Fk²_F q

þ D_zþDSO

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðD_zþDSOÞ²þh²v²_Fk²_F q

:

ð36Þ

The spin-Hall rspin

yx and valley-Hall rvalley

yx con-

ductivities of silicene and germanene as a function of a D_z=DSO at different values of a dimensionless

(a) (b) (c)

(d) (e)

Figure 4 (color online) Absorbance of silicene (a) and germanene (b) at difference values of electric field energy. (c) Absorbance of silicene and germanene atDz=DSO¼0:5andDz=DSO¼1:7for a

comparison. Absorbance of silicene (d) and germanene (e) at Dz=DSOis around 1.0.

parameter K_F¼hvFkF=DSO in the case of the Fermi level in the conduction band are presented in Figs.5 and6, respectively.

As shown in Fig.5a, b, we can see that, atK_F¼0 (corresponding to the Fermi energy equals to zero), the graphs describing the dependence of the spin- Hall conductivities of silicene and germanene on the D_z=DSO are the same profile and value described by a ladder shape. The spin-Hall conductivity rspin

yx (in

units of e=8p) of silicene and germanene is equal to 2:0 when energy of the electric field varies from 0 to D_z¼DSO. WhenD_z[DSO, the spin-Hall conductivity rspin

yx jumps immediately to zero. This type of ladder shape is no longer available when KF[0 for both silicene and germanene. For silicene, the spin-Hall conductivity rspin

yx depends strongly on the param- eter K_F¼hv_Fk_F=DSO while the graph showing the dependence of rspin

yx for germanene on D_z=DSO at different values of K_F is insignificant as shown in

Fig.5b. In Fig. 5c, we present a comparison of spin- Hall conductivities of silicene and germanene as a function of D_z=DSO the at K_F¼3 to show this difference.

As shown in Fig.6, we can see that, atD_z¼0, the valley-Hall conductivity rvalley

yx of silicene and ger- manene is equal to zero and this zero value is inde- pendent of the parameter KF. This is different from the spin-Hall conductivity. However, whenD_z[DSO, the valley-Hall conductivity rvalley

yx depends

strongly on KF, especially in the case of silicene. The ladder graph form (at K_F¼0) will be broken when K_F[0. The larger the value ofK_F, the more straight the graph is. The biggerK_Fis, the greater the value of the electric field energy to the valley-Hall conduc- tivity (in units of e=8p) being asymptotic to the maximum value of 2.0 is. Similar to the case of the spin-Hall conductivity, the effect ofKFon the valley- Hall conductivity of silicene is more pronounced than that of germanene.

(a) (b) (c)

Figure 5 Electric field dependence of the spin-Hall conductivity rspin

yx of silicene (a) and germanene (b) at different values of dimensionless parameter KF¼hvFkF=DSO. (c) Electric field

dependence of the spin-Hall conductivity of silicene and germanene atKF¼3.

(a) (b) (c)

Figure 6 Electric field dependence of the spin-Hall conductivityrvalley

yx of silicene (a) and germanene (b) at different value ofK.cSpin- Hall conductivity of silicene and germanene as a function ofD_z=DSOatKF¼3.

Conclusion

In conclusion, we performed systematically the ana- lytical calculations for low-energy bands, optical absorbance and spin/valley-Hall conductivity of sil- icene and germanene when a perpendicular electric field was introduced. The low-energy bands of sil- icene and germanene have been calculated based on the low-energy effective Hamiltonian including the SOC effect. The detailed analytical calculations for the Ehrenreich–Cohen dielectric function and low- frequency optical absorbance of silicene and ger- manene have been performed and our estimations for optical absorbance are consistent with the previous first-principles calculations. Our analytical calcula- tions indicated that the spin/valley-Hall conductivi- ties of silicene depend strongly on the Fermi level, especially in the case of the Fermi level in the con- duction band.

Acknowledgements

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 103.01-2017.309.

Compliance with ethical standards

Conflict of interest The authors declare that they have no conflict of interest.

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