The coupled electromagnetic field effects on quantum transport in an electrically modulated graphene

Physics Letters A 380 (2016) 509–515

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Physics Letters A www.elsevier.com/locate/pla

The coupled electromagnetic field effects on quantum transport in an electrically modulated graphene Ning Ma a,b,∗ , Shengli Zhang b,∗∗ , Daqing Liu c , Wanzhou Zhang a a b c

Department of Physics, MOE Key Laboratory of Advanced Transducers and Intelligent Control System, Taiyuan University of Technology, Taiyuan 030024, China Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, China School of Mathematics and Physics, Changzhou University, Changzhou, 213164, China

a r t i c l e

i n f o

Article history: Received 5 July 2015 Received in revised form 3 November 2015 Accepted 9 November 2015 Available online 14 November 2015 Communicated by R. Wu Keywords: Landau level Weiss oscillation Conductivity

a b s t r a c t Recent experiments have revealed that the periodic graphene ripples can cause a periodic potential. Motivated by this, we have explored the electromagnetic field effects on the transport of graphene subjected to an electric periodic potential. Comparing with recently obtained results for magnetically modulated graphene, we also find an abrupt disappearance of oscillations in the magnetoconductivity at γe = 1, and in a strong magnetic field, the Shubnikov–de Haas oscillations on it are superimposed on the Weiss oscillations. However, these oscillations with the electric and magnetic modulations have a π phase shift. Our results should be valuable for electronic applications of graphene. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Graphene (or a graphite monolayer) has drawn much attention in the field of the physics of electronic systems with reduced dimensionality [1–4], owing to its promising applications in nanoelectronics because of the exotic chiral features [5–9] in electronic structure. To date, graphene has already led to some of the most startling discoveries in condensed matter physics, which are especially found to be tied to the remarkable ‘relativistic-like’ spectrum of electrons and holes. One of those that have been testified in experiment is the abnormality of the two-dimensional (2D) quantum Hall effect [10–12]. Starting from these, Gusynin and Sharapov theoretically study the Shubnikov–de Haas (SdH) oscillations and Hall plateaus in the zero frequency (dc) collisional and Hall conductivities [13,14]. These magnetic oscillations appear due to the interplay of the quantum Landau levels (LLs) with the Fermi energy, and serve as a powerful technique to investigate the Fermi surface and the spectrum of electron excitations. Nevertheless, they do not refer to the dc diffusive conductivity as well with the expected Weiss oscillations on it. This type of oscillation was found with an artificially created periodic potential in the submicrometer range,

*

Corresponding author at: Department of Physics, Taiyuan University of Technology, Taiyuan 030024, China. Corresponding author. E-mail addresses: [email protected] (N. Ma), [email protected] (S. Zhang).

**

http://dx.doi.org/10.1016/j.physleta.2015.11.007 0375-9601/© 2015 Elsevier B.V. All rights reserved.

earlier in a conventional 2D electron gas (2DEG) [15–17], and now in graphene [18–20]. These researches show that the electric and magnetic periodic potentials have modulated their electronic structure in unique ways leading to fascinating physics and possible applications. Subsequently, Ma et al. have studied the electrostatic field effects on the Weiss oscillations in the dc diffusive conductivity in the presence of a spatially modulated magnetic field [21] on the basis of these works [15–20] and the recent reports [22–29]. Moreover, we note that the Coulomb interaction is also an important problem in graphene since it is not screened due to the 2D character of the system. To date, several studies have already been devoted to this issue [30–35]. Especially, some recent works have reported the oscillatory phenomena on the thermodynamic quantities (like magnetization and specific heat) in graphene with the Coulomb interaction [36–38]. Essentially, these oscillations originate from the interplay of the quantum LLs with the Fermi energy, which is meanwhile responsible for the SdH oscillations in present work. Their results show that the Coulomb interaction causes an increase in the energy and spacing of LLs, and in the period of oscillations in magnetization. And, it leads to a decrease of the specific heat, as well as to an increase and shifting of the oscillations, suggesting that it favors a decrease of the chemical potential. Motivated by this, we will focus on the related effects of Coulomb interaction on the quantum transport of graphene in our another work. In present work, we mainly complement the current studies mentioned above to clarify the coupled electromagnetic (EM) field

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N. Ma et al. / Physics Letters A 380 (2016) 509–515

effects on the low temperature magnetotransport in graphene subjected to a weak 1D electric periodic potential. The additional electric modulation introduces a length scale, period of modulation, in the system giving rise to interesting physical effects on the transport response. In practice, the effects of periodic potential on electron transport in 2D electron systems have been the subject of continued interest, where the electrical modulation can be carried out by two interfering laser beams [15], or depositing an array of parallel metallic strips on the surface [16], or controlled adatom deposition [39], or interaction with a substrate [40], and/or the periodic ripples in suspended graphene [41,42]. Our findings show that the Weiss oscillations in the dc conductivity are much more robust with respect to the electric field. Specifically, the value of conductivity and the amplitude of Weiss oscillations on it are both shown to increase as E increases with a given B, but the opposite behavior for the SdH oscillations. More interestingly, both oscillations abruptly vanish when the value of electric to magnetic field ratio approaches a threshold value γe = υ E B = 1, which is atF tributed to the anomalous EM coupling effects on the spectrum of graphene. Here we just briefly touch upon these issues, and below detailedly elucidate them with involving the well-known zero mode (a Landau level at zero energy). This paper is organized as follows. In section 2, a brief introduction is given to the 2D model for graphene. The energy eigenvalues and eigenstates are obtained in the presence of crossed uniform electric and magnetic fields as well with a 1D weak periodic electric modulation potential. Furthermore, we analytically derive the dc diffusion conductivity for the case of quasielastic scattering in the Kubo formalism, and describe the EM coupling effects on it in detail. In section 3, the asymptotic expression is derived for the exact solution of the dc conductivity. In the last section, we present brief summary and conclusions.

subbands within an energy range equal to the potential drop across the ribbon. Consequently, it breaks the even-k symmetry of the energy for electrons (or holes) (i.e. E e/h,k y = E e/h,−k y ), but merely reserves the odd one between them with E e,k y = − E h,−k y . Also, the energies of electrons and holes become degenerate at n = 0 with E e,k y = E h,k y = h¯ υ F γe k y . Furthermore, the EM coupling is shown to cause a collapse of the energy spectrum at γe = 1 from Eq. (2). In fact, the transition at γe = 1, according to Bohr–Sommerfeld quantization condition, can be linked to the classical dynamics of a massless particle,   characterized by closed orbits at γe  < 1 and open trajectories     at γe  > 1 [21]. Specifically, for γe  < 1 (i.e. the strong magnetic field), one can get  rid  of the electric field, and then obtain closed orbits. But for γe  > 1 (i.e. the small magnetic field), one can retrieve the electric regime, and get rid of the magnetic field rather than the electric one, thereby causing the magnetic breakdown due to the absence of closed orbits. On the other hand, the transition occurs when the Lorentz force  is overwhelmed by the electrostatic potential. That means, for γe  < 1 the spectrum con  sists of discrete quantized energy bands, while for γe  ≥ 1 the spectrum becomes (quasi-) continuous, which is beyond the scope of the present work. Especially, according to the related reports [22,43,44], the contraction and collapse of LLs spectrum can be both measured at fields attainable in laboratories. For example, at B ∼ 1 T the spacing between the 0th and 1st LLs is ∼ 29 meV for E ∼ 5 × 105 V m−1 , 20% reduction in the spacing is expected. And the collapse of the LLs should be also observable by applying E ∼ 1 × 106 V m−1 . In fact, we emphasize here the action of EM coupling parameter γe (i.e. the value of electric to magnetic field ratio), what means that these above phenomena could be realized at a smaller electric field, as the magnetic field synchronously decreases. The eigenfunctions of n,k y are given by

2. Electrical conductivity with periodic electric modulation

n,k y = 



exp ik y y

We consider the carriers moving in the x– y plane subjected to a perpendicular magnetic field [B = (0, 0, B )] and an in-plane electrostatic potential U (x) = U 1 + U 2 , which comprises a uniform electric field potential [U 1 (x) = e E x] and a weak electric modulation potential [U 2 (x) = V 0 cos ( K x)]. The parameter V 0 is the modulation amplitude, and K = 2π /a with a for the modulation period. For the present case, the 2D massless particle Hamiltonian reads

H = υ F α · π + IU (x) .

(1)

In Eq. (1), the Fermi velocity υ F is 1.0 × 106 m/s, α is the vector of Pauli matrices, I is the identity matrix, and π = p+eA is the canonical momentum with A for the vector potential following Landau gauge A = (0, Bx, 0). Along the same lines [21,27], we firstly obtain the eigenvalues of H without U 2 (x)



n,k y = sgn(n)ω g |n| + h¯ γe k y .

(2)

In Eq. (2), integer n denotes the LLs for electrons (n ≥ 0) and holes (n ≤ 0), and k y is a good quantum number corresponding to the translation symmetry along the y axis. For concision, we define

ωg =

√

2h¯ ωc with



ωc = υ F 1 − γe2

3/4

/c for the cyclotron fre quency corrected by the electric field, c = h¯ /e B for the magnetic length, and γe = E / ( B υ F ) for the  EM coupling parameter (yielding to Lorentz covariant demand γe  < 1). Eq. (2) reveals that the lowlying LLs spacing scales as the EM contract factor of (1 − γe2 )3/4 that is absent in a conventional 2DEG. This then leads to a mixing of LLs as γe increases, which is supported by the numerical results (Ref. [24]). As reported, the concurrent action of the electric field dragging force and the Lorentz force, mixes the electron and hole



2L y c∗

−i n−1 (ξ )

n (ξ )



χ± ( E , B ) ,

(3)

in which the orbital function is 2 e −ξ /2

n (ξ ) = 

√ H n (ξ )

2n n!

(4)

π

with the normalized Hermitian polynomials H n (ξ ), and the spinor function is

χ+T = (γe m , 0, 0, i m n ), χ−T = (0, γe m , − i m n , 0) with



m = γe2 + n2 

−1/2

n = 1 − 1 − γe2

(5)

,

1/2

(6)

. 

1/4

Here we define ξ = (x + xc ) /c∗ with c∗ = lc / 1 − γe2 and xc = √ c2 k y − sgn(n) 2 |n|c∗ γe in graphene, distinguishing from xccon = c2 k y − υ F2 γe2 m/e E in a conventional 2DEG. The imposition of peri  odic boundary conditions x, y + L y = (x, y ) over some suitably large length L y leads to the allowed values k y = 2π / L y ( = 0, ±1, ±2, . . .). The localized character of the bound-state eigenfunctions n (x) ensures that the electronic motion is bounded in the x direction, so that no runaway electrons develop under these conditions. From Eq. (3), we conclude that the EM field squeezes the oscillator states and mixes the electron and hole wave functions. Also the location of the Gaussian shifts as a function of the index n, indicating effective spatial separation between the electrons and holes, resembling the case in [24].

N. Ma et al. / Physics Letters A 380 (2016) 509–515

As mentioned above, the electric modulation is approximated by the first Fourier component of the periodic potential U 2 (x) = V 0 cos ( K x), that is a good approximation, in particular, for the electrostatically induced 1D periodic potential. However, carrying the potential U 2 (x), one can hardly obtain the exact eigenvalues of Eq. (1) in closed analytical form. Considering the practical V 0 is sufficiently small in experiment, we can evaluate the correction to the energy by first-order perturbation theory with the unperturbed wave functions n,k y (r ) in Eq. (3):

∞ n,k y =

L y dx

−∞

dy n∗,k y (r ) H (x) n,k y (r ) .

(7)

0

diff d col σμν (ω) = σμν (ω) + σμν (ω), accounts for both diffudiff col sive σμν (ω) and collisional σμν (ω) contributions whereas the Hall nd contribution is obtained from nondiagonal conductivity σμν (ω).

conductivity

Our focus, in this work, will be the calculation of the dc diffusive contribution to the conductivity, but without involving the collisional and Hall ones, which are the key issues in our next work. In a magnetic field, the main contribution to Weiss oscillations mainly comes from the scattering induced migration of the Larmor circle center. This is so-called diffusive (band) contribution to the conductivity, which is actually the extended state contribution and arises due to the finite drift velocity acquired by the charge carriers in the presence of the electric field and the electric mod-

σ ydiff y

e −x cos (bx) H n (x) H n+2m (x) dx

Lx L y

2

0



= 2n−1/2

π 2

2 n! (−1)m b2m e −b /4 L n2m

and for Laguerre polynomials

L n0 (x)



b2 2

ζ

provided that the scattering is elastic or quasielastic. Here we define β as β = k 1T with k B for the Boltzmann constant, f E ζ as





B

, [b > 0]

(8)

= Ln (x), one can obtain (9)

υ yζ ≡ υ y

in which

V 0 −u /2 (10) e [L n−1 (u ) + L n (u )] 2   with u = K 2 c2 / 2 1 − γe2 . By virtue of the n, B , one can es   τ  timate the nth LL width as W = 2 nz, B . As seen, the Landau

√

n,k y = sgn(n)ω g n + h¯ υ F γe k y + n, B cos ( K xc ) .

(11)

In Eq. (11), the first term arises from the Landau energy, the second one from the electric field, and the last one from the electric modulation. Here we remark that, in stark contrast to the results in [18], the three terms are closely related to the electric field, i.e., the first and last terms are both modified by it, and the second one is completely determined by it. That means that the energy corrections from the electric field and the electric modulation, make the degenerate LLs k y dependent, expand them into bands, what finally leads to nonzero velocity and the dc diffusive conductivity (see below). But in Ref. [18] the energy correction merely arises due to the electric modulation. In other words, for null electric modulation, the zeroth LL is different from the rest since its energy is zero and thus the electrons in this level do not contribute to the conductivity. On the contrary, the electrons in the zeroth LL here actually contribute to the conductivity because of the nonzero k y -dependence term from the electric field, i.e., the second term in Eq. (11). Following the Kubo formula, we explore the conductivity correction from the electrostatic potentials. In the linear response regime, the conductivity tensor has been evaluated as: σμν (ω) = d nd d σμν (ω) + σμν (ω), μ, ν = x, y , z, σμν (ω) stems from the diagonal







d ρ d of the density operator ρ . In any basis J μ = Tr ρ d J μ =

d nd σμν E ν , where J μ is the current density, and σμν (ω) comes from   d nd the nondiagonal part of ρ ρ = ρ + ρ . In general, the diagonal

=

1 ∂ h¯ ∂ k y

n,k y

V 0 K c2 e −u /2

sin ( K xc ) [ L n−1 (u ) 2h¯ + Ln (u )] + υ F γe .

=−

n, B =

bandwidth oscillates as a function of index n due to the oscillatory functions of L n (u ) with respect to n. The electric field can modulate the perturbation energy n,k y and Landau bandwidth W through u since the EM term γe takes part in the u, which is not reported in the previous works [18–20]. From Eqs. (2) and (9), we can finally write the eigenvalues of the total Hamiltonian (1)



the Fermi–Dirac distribution function, τ E ζ as the electron relaxation time, and ζ as the energy quantum number. The component of velocity in the y direction follows the expression: n,k y

n,k y = n, B cos ( K xc ) ,

part

σ ydiff y , we have [18,19,21,46–48]: 2         ζ 2 βe = f Eζ 1 − f Eζ τ Eζ υy , (12)

ulation. For dc conductivity

Using the formulas [45] for Hermite polynomials

∞

511

σ has no conζ = 0 since υx = 0. However,

Note, the diagonal component of the conductivity diff xx

(13) diff xx

tribution to the modulation, i.e. σ it should be stressed here that there exists a nonzero collisional col (hopping) contribution (σxx ) originating from the localized state contribution and the migration of the cyclotron orbit due to scattering by charge impurities. This accordingly gives rise to a drift current in x direction (i.e. the direction of electric field). Without the collision, the drift current in x is null since the electron moves merely normal to E and B under the Lorentz force in the presence of crossed uniform electric and magnetic fields, thereby causing the current in y. These points will be discussed detailly in our another work. The first term in Eq. (13), originates from the electric modulation and is modified by the electric field, but the second one is independent of the electric modulation, and solely generated by the EM field. Substituting (13) into (12), and summating over ζ



=

ζ

Ly

L x /c2

dk y

2π

∞  ,

(14)

n =0

0

we obtain the dc diffusion conductivity:

σ ydiff y =



V 0e

2

h¯

τ β ,

(15)

with the dimensionless conductivity that has a

nE, B

= a + b ,

γe -dependence (16)

in which



a =

∞ 1 − γe2 u 

=

2π V 02

g ( En)

[g ( E n ) + 1]2 n =0

∞ 1 − γe2 ue −u 

8π

n =0

n2, B

g (En ) [g ( E n ) + 1]2

× [Ln−1 (u ) + Ln (u )]2 ,

(17)

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N. Ma et al. / Physics Letters A 380 (2016) 509–515

lessly symmetric electron–hole spectrum of the conductivity with a single peak at the charge neutrality point (CNP). The CNP peak confirms the zero-energy states in graphene [49], suggesting the π -Berry phase difference of the SdH oscillations to a conventional 2DEG. And because massless Dirac fermions in a magnetic field always exhibit such  a zero mode, the Hall conductance is given by

Fig. 1. (Color online.) The dimensionless conductivity vs Fermi energy for different electric fields Es, E 1 = 0 V m−1 , E 2 = 1500 V m−1 , and E 3 = 3000 V m−1 in graphene. The velocity is taken to be 1 × 106 m/s, B = 1 T, T = 6 K, V 0 = 0.5 meV, and a = 350 nm.

and

b =

1 2π

c2



h¯ υ F γe V0

2  ∞ n =0

g (En ) [g ( E n ) + 1]2

(18)

.

In Eq. (17), the term of a is caused by the electric modulation and then revised by the electric field, but the other term of b in Eq. (18) completely arises from the electric field, i.e., if E = 0, then b = 0. In addition, the exponential function is defined as

√



3/4

g ( E n ) = exp [β ( E − E F )] with E F = υ F h¯ 2π ne 1 − γe2 for the Fermi energy and ne for the electron density. Apparently, Eq. (16) reduces to the conductivity of Eq. (15) at E = 0 in Ref. [18]. Clearly, the dimensionless conductivity is the key role of the dc conducdiff

tivity σ y y in Eq. (15), and the n, B existing in Eq. (17) demonstrates that the HB oscillations are responsible for the Weiss and SdH oscillations in the conductivity (see below). In Fig. 1, we graphically plot the dimensionless conductivity

nE, B as a function of the Fermi energy E F for different electric

fields Es, E 1 = 0 V m−1 , E 2 = 1500 V m−1 , and E 3 = 3000 V m−1 in graphene. The related parameters are given by: a = 350 nm, υ F = 1 × 106 m/s, B = 1 T, V 0 = 0.5 meV, and T = 6 K. Fig. 1 reflects the periodicity of the Weiss and SdH oscillations in the conductivity. As seen, for low Fermi energy (smaller n), the SdH oscillations are shown to be superimposed on the Weiss oscillations. In the regime of high Fermi energy (larger n), one can see the quenching of both oscillations, and thereinto the Weiss oscillation is the faster one. For null electric field, there exists a fault-

σxy = v + 12 4e2 /h (v is an integer) [13,14]. Additionally, the

symmetry implies the same occupancy for electrons and holes in the LLs. With the electric field applied, we find that, from Eqs. (17) and (18), besides the contribution from the electric modulation, there is an extra contribution to the longitudinal conductivity from the electric field. Such a contribution leads to the lifting of the oscillating center in the conductivity, in company with the amplitude increasing as E increases. It should be stressed here that the CNP peak of zero mode conductivity does not split for the electrons and holes, and indeed indicates two points: (i ) The two Dirac cones are not gapped and also remain the inversion symmetry; (ii) The electric field and the electric modulation both preserve the symmetric distribution of electrons and holes in the zeroth LL, which is revealed from their identical amplitudes. Inspection of Fig. 2 illustrates the magnetic response of the dimensionless conductivity for zero and finite electric field at T = 6 K. The dimensionless magnetic field is introduced here as B ∗ / B with B ∗ = h¯ /ea2 . As seen, the SdH oscillations, in the region of high magnetic field, are superimposed on the Weiss oscillations similar to the Fig. 1 above. That means, the results of Eq. (16) can also exhibit SdH-type oscillations, though the calculations only include the extra contribution to the dc conductivity resulting from the electric modulation with involving the electric field, without including the pure SdH oscillations. Here we emphasize that the amplitude and phase of both oscillations are both modulated by the electric field, but this phenomenon is absent in Ref. [18]. Panel (a) presents that the electric field lifts the oscillating center with increasing B ∗ / B, i.e., the value of conductivity increases in company with the periodic oscillation, while B ∗ / B rises. Panel (b) demonstrates the same electric field effect and meanwhile describes that the dimensionless conductivity is magnetic nonuniform with respect to B. Furthermore, the amplitude of Weiss oscillations is shown to augment as E increases, but the opposite behaviors for the SdH oscillations. More interestingly, both oscillations abruptly vanish at the threshold point γe = 1 (i.e. E = υ F B), according to Eq. (15). These phenomena, not known in a conventional 2DEG, should be attributed to the EM field effects on Landau spectrum in graphene [21–25]. Furthermore, the minima of the longitudinal conductivity do not approach zero, implying

Fig. 2. (Color online.) The dimensionless conductivity of electron and hole states with respect to (a) B ∗ / B and (b) B for different electric fields Es, E 1 = 0 V m−1 , E 2 = 900 V m−1 , and E 3 = 1500 V m−1 in graphene. We use B ∗ = h¯ /ea2 with a = 350 nm, υ F = 1 × 106 m/s, T = 6 K, V 0 = 0.5 meV, and ne = 3 × 1015 m−2 .

N. Ma et al. / Physics Letters A 380 (2016) 509–515

513

Fig. 3. (Color online.) The dimensionless conductivity versus inverse magnetic field: curve 1 (electric modulation) and curve 2 (magnetic modulation), Dirac electrons; curve 3 (electric modulation), electrons with parabolic energy spectrum at E = 700 V m−1 in graphene. We use B ∗ = h¯ /ea2 with a = 350 nm, V 0 = 0.5 meV, υ F = 1 × 106 m/s, and ne = 3 × 1015 m−2 .

the additional contribution to σ y y from the electric field. In other words, once again this demonstrates that, besides the contribution from the electric modulation, there exists another one from the electric field. Moreover, one can see that finite temperature can cause the Weiss oscillations damping fast in the region of small magnetic field, which resembles the results in [18,19,21]. From these panels, we address that the Weiss oscillations are much more robust for the low temperature, strong electric and magnetic fields. It is interesting to compare the case of a standard 2D electron system with a parabolic energy dependence, which could be obtained as follows:

nE,,Bs =

∞  n =0

+

g ( En ) [g ( E n ) + 1] 1

2π c2



h¯ E BV0

2

{

u s e −u s 2π

3. Asymptotic expressions To better appreciate the given exact solutions, we seek an asymptotic expression for the dimensionless conductivity (17) using the successful approach [18,19,21,46–48], which is applicable for the case that many LLs are filled. Using the asymptotic expression of the Laguerre polynomials:

 √  1 exp (−u /2) L n (u ) →  √ cos 2 nu − π /4 , π nu

[ Ln (u )]2

2 }

figure indicates that the latter case is much more sensitive to the electric field comparing with the former one under the same conditions.

(20)

and the continuum limit under the electric field

(19)

with u s = K 2 c2 /2. And E ns = (n + 1/2) h¯ ωcs − h¯ k y υ F γe − υ F2 γe2 m/2 with ωcs = e B /mc should be used for the parabolic electron spectrum case. It should be stressed here that we calculated here only the extra contribution to the conductivity resulting from the electrical 1D modulation and as such the pure SdH oscillations are not included in our results. Nevertheless, this extra contribution to the conductivity can exhibit SdH-type oscillations and it is interesting to highlight their character. For this purpose, in Fig. 3 we show these oscillations for Dirac electrons and standard ones at an arbitrary E = 700 V m−1 . We see a clear phase shift between these oscillations in such two cases (especially in a high magnetic field) in agreement with the known behavior of SdH oscillations caused by the presence of half-filled zero energy LL for Dirac electrons. Furthermore, curve 1 presents that the Weiss oscillations are much more robust as compared with the system of electrons with the standard parabolic energy spectrum (curve 3). The physical reasons for these differences are in different Fermi velocities of Dirac and standard electrons. As shown in curves 1 and 2, Fig. 3 also compares with the dimensionless conductivity with the electric modulation potential and the spatially modulate magnetic field for Dirac electrons (see Ref. [21]) in graphene. The two lines present some common features between them, for example, in a high magnetic field, the SdH oscillations are superimposed on the Weiss oscillations for finite electric field, and the predicted novel transition at γe = 1, etc. However, the main difference is that these oscillations in such two cases are actually out of phase with π phase shift, whereas they are in phase with the conventional 2DEG [47]. In addition, this

n→

1



lc E

υF Q

2

2  2 ∞  ∞ lc , → EdE

υF Q

n =0



with u = 2π 2 /b and Q = h¯ 1 − γe2 the following integral:



 u 1 − γe2  lc

a =

2π

√ 2 2

× cos







3/4

2 ∞ dE

υF Q

n

0 2

1/2 u /n cos

(21)

0

, we transform Eq. (17) into

E g (E) [g ( E ) + 1]2

 √

2 un −

π 4

.

(22)

For low temperature β −1 E F , we rewrite the above integral by replacing E by E F + sβ −1 in the sine term in the integrand part and by E F in the non-integrand part,



  2u 1 − γe2  lc

a =

2π 2 β

∞ × −∞

π 4

+

υF Q

dse s

(e s

+ 1) 2

cos2 (

2π alc s), υ F Q blc∗ β



 cos2

2π ph¯ lc Q blc∗

π Q lc∗



lc ph¯

− (23)

√



3/4

where p = υE F ha¯ = k F a = 2π ne a 1 − γe2 is the dimensionless F Fermi momentum of the electron. Furthermore, the above expression can be revised as

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N. Ma et al. / Physics Letters A 380 (2016) 509–515

Fig. 4. (Color online.) The graphene conductivity as a function of the inverse magnetic field: exact solution (solid curve); asymptotic expression (dashed curve) at (a) E = 600 V/m, (b) E = 900 V/m, (c) E = 1200 V/m, and (d) E = 1500 V/m. We use B ∗ = h¯ /ea2 with a = 350 nm, V 0 = 0.5 meV, υ F = 1 × 106 m/s, ne = 3 × 1015 m−2 , and T = 6 K.



a =

  2u 1 − γe2  lc 8π 2 β

∞ ×

4

dse s cosh (s/2)

+

2π alc s

υ F Q blc∗ β

 2

cos

υF Q 2

−∞

π



cos2 (

π Q lc∗



the amplitude of SdH oscillations. In a word, the electrostatic potentials could largely modulate the amplitude and phase of both oscillations as shown in Figs. 2 and 4.

lc ph¯

2π ph¯ lc Q blc∗

−

4. Conclusions

(24)

).

Employing the identity [45]

∞ dx −∞

cos ax 2

cosh β x

aπ

=

2β 2 sinh (aπ /2β)

we finally obtain

√

a =

2ulc∗

 cos2

π Q lc∗

,



{1 − R T +    lc ph¯ 1 }. 2R T cos2 2π − ∗ 4π 2 β h¯ υ F

(25)

lc ph¯

Q blc

(26)

8

Here we introduced the temperature reduction factor

RT =

4π 2 alc k B T / sinh 4πυ



υ F Q blc∗

2 al F

ckB T Q blc∗



.

(27)

In Eq. (27), the temperature reduction factor R T depends on not only the temperature, but also the EM field. A closer analytical examination of Eq. (26) further presents that the oscillatory part of the conductivity follows both Weiss and SdH frequencies. As exhibited in Fig. 4, for finite electric field both oscillations in the conductivity for the exact solution (solid curve) agree well with their asymptotic expression (dashed curve) in the region of small magnetic field. For strong magnetic field, the SdH oscillations become superimposed on top of the Weiss oscillations, leading to a larger deviation of the asymptotic expression to the exact solution. Furthermore, one can see the electric field could enhance the amplitude of Weiss oscillations in the conductivity, but weaken

In summary, we have theoretically investigated the dc diffusive conductivity σ y y with periodic electric modulation in the presence of crossed uniform electric and magnetic fields in graphene. We analytically derive the dc diffusion conductivity for the case of quasielastic scattering in the Kubo formalism, and compare these results to recently obtained results for magnetically modulated graphene, as well as those for an electrically modulated conventional 2DEG system. We find that in the electrically modulated graphene system considered in this work, Weiss oscillations in σ y y have a higher amplitude than that in the conventional 2DEG, and there is a clear phase shift between these oscillations in such two cases in agreement with the known behavior of SdH oscillations caused by the presence of half-filled zero energy LL for Dirac electrons. Furthermore, there are some common features between the electric modulation potential and the spatially modulate magnetic field for Dirac electrons, for example, in a high magnetic field, the SdH oscillations are superimposed on the Weiss oscillations for finite electric field, and both oscillations are shown to be largely modulated by the electric field. As E increases, the value of conductivity and the amplitude of Weiss oscillations on it both increase with the increase of E, and both oscillations are shown to abruptly vanish at γe = 1, which is attributed to the coupled EM field induced anomalous effect on the Landau level. The main difference is that these oscillations in such two cases are actually out of phase with π phase shift, whereas they are in phase with the conventional 2DEG. Our results are a good starting point and guide for further experimental works. Acknowledgements This work is financially supported by the National Natural Science Foundation of China (Grants Nos. 11074196 and

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