On the theory of Nernst–Ettingshausen oscillations in monolayer and
bilayer graphene
Z.Z. Alisultanov a,b,c,∗a Amirkhanov Institute of Physics of Dagestan Science Centre, Russian Academy of Sciences, Makhachkala, Russiab Prokhorov General Physics Institute, Russian Academy of Sciences, Moscow, Russiac Dagestan State University, Makhachkala, Russia
a r t i c l e i n f o a b s t r a c t
Article history:Received 21 May 2014Accepted 7 June 2014Available online 11 June 2014Communicated by V.M. Agranovich
The Landau levels in graphene in crossed magnetic and electric fields are dependent on the electric field. However, this effect is not taken into account in many theoretical studies of graphene in crossed fields. In particular, it is not considered in the Nernst–Ettingshausen effect, in which the regime of crossed fields is realized. In this Letter, we considered the Nernst–Ettingshausen effect in monolayer and bilayer graphene, taking into account the dependence of Landau levels on the electric field. We showed that the magnitude and period of the Nernst coefficient oscillations depend on the electric field. This fact is important for the fundamental theory of Nernst–Ettingshausen effect in graphene and gives the new possibility for control of this effect using an applied electric field. The latter is very interesting for practical applications.
Graphene is the 2D carbon allotrope, in which atoms are ar-ranged in hexagonal structure with carbon–carbon distance 1.42 Å. Graphene has the unique physical properties. By virtue of this graphene can become a promising material for modern electronics. The crystal symmetry and valence of carbon leads to Dirac energy spectrum for electronic excitations [1]. Most of the unique proper-ties of graphene are associated with such spectrum [2].
Unusual behavior of graphene in a magnetic field is one of the major consequences of the Dirac spectrum [1–4]. Landau levels (LL) in graphene are non-equidistant. Moreover, there is the zero level. Distances between LL are large. For example, a distance between first two levels is more than 1000 K at magnetic field 10 T. This fact leads to giant magneto-optical [5,6] and thermo-magnetic [7]effects and unusual quantum Hall effect [1,8,9]. The latter can be observed even at room temperature.
In [10,11], LL in graphene in crossed magnetic and electric fields was studied. In the presence of longitudinal electric field, LL depend on the latter. Such dependence is related with the non-quadratic dispersion law. Indeed, yet Lifshitz and Kaganov [12,13]showed that in case of the non-parabolic dispersion law, the res-onance frequency in crossed magnetic and electric fields depends on the applied electric field. Actually, a cyclotron mass doesn’t de-pend on the energy only for the parabolic dispersion law. In recent paper [14], the magnetization oscillations in graphene in crossed
* Correspondence to: Amirkhanov Institute of Physics of Dagestan Science Centre, Russian Academy of Sciences, Makhachkala, Russia.
fields was studied. It was shown that an oscillation form is depen-dent on the electric field.
In this Letter, we will investigate the Nernst–Ettingshausen (NE) oscillations in monolayer (MG) and bilayer graphene (BG). The NE effect is thermal analog of the Hall effect [15,16]. The effect con-sists in the induction of a temperature gradient ∇T = (∇xT , 0, 0) in a sample in a magnetic field H = (0, 0, H) and in an electric field E = (0, E, 0). Quantitatively, the effect is characterized by the NE coefficient (NC): ν = E/(−∇xT )H . The NE effect in graphene was studied in detail in [7,17]. However, in these papers, the depen-dence of LL on the electric field was not considered, although the regime of crossed fields was realized in the NE effect. In present paper, we study the NE effect considering this dependence.
We will use the quasi-classical approach based on the quanti-zation conditions of Lifshitz–Onsager type [12,13], like Lifshitz and Kaganov. This condition for the two-dimensional system can be written as (we assume that the plane of electronic system is XY and the magnetic field is directed along the Z axis)
A(ε∗) = 2π h̄eH
c(n + γσ ), (1)
where A(ε∗) is an area enclosed by the electron trajectory for the energy ε∗(p) = ε(p) − υ0 p in momentum space, υ0 = c[E H ]/H2
is the average drift velocity of the electron perpendicularly E Hplane, E and H are the electric and magnetic fields, respectively, γσ = γ + 1/2(m(μ)/m)σ with σ = ±1 and the electron–cyclotronmass m(ε) = (2π)−1dA/dε, γ = 1/2 for non-relativistic gas and γ = 0 for graphene, m is the electron mass, μ is the chemical
2330 Z.Z. Alisultanov / Physics Letters A 378 (2014) 2329–2331
potential. In this work, we ignore the Zeeman splitting of the Lan-dau levels, i.e. we assume that m(μ)/m = 0. In our case E⊥H . Then υ0 = cEe y/H and ε∗(p) ≡ ε(p) − υ0 p y .
The energy spectrum of MG in the vicinity of Dirac point has form ε(p) = νbυF |p|, υF ≈ 108 cm/s, νb = ±1 is the band index. Then, it is easy to show that curve ε∗ = const is the ellipse with following parameters: a = ε/
√υ2
F − υ20 , b = ευF /(υ2
F − υ20 ). Then
A(ε) = πab = πε2υF /(υ2F − υ2
0 )3/2. For energy, we obtain
εn,p y = sgn(n)(1 − β2)3/4√
2h̄υF l−1H
√|n| + υ0 p y, (2)
where β = υ0/υF . The expression (2) completely coincides with the expression obtained by solving the Dirac equation. The en-ergy spectrum of BG in the vicinity of Dirac point can be written as [4]: ε(p) = νb(
√υ2
F p2 + t2⊥/4 − νsbt⊥/2), where t⊥ is the effec-tive hopping energy between two layers, νsb = ±1 is the subband index. We assume that t⊥ = 0.4 eV. In crossed fields, we obtain: A = π(ε2 + εt⊥ + β2t2⊥/4)/(υ2
F (1 − β2)3/2).Similarly, for a parabolic spectrum ε(p) = p2/2m we have
A(ε) = 2πmε and
εn,p y = sgn(n)h̄ωc(n + 1/2) + υ0 p y,
i.e. LL don’t change in crossed fields.To investigate the NE oscillations we will use results of papers
[7,17]. For NC we have
ν = νterm + νmag, (3)
where νterm is the thermal contribution to the NC [7]
νterm = σxx
e2nc
dμ
dT, (4)
and νmag is the magnetization contribution to the NC [7,16]
νmag = cρyy
H
dM
dT. (5)
Here σxx is the diagonal component of the conductivity tensor, nis the concentration of carriers, ρyy is the diagonal component of the resistivity tensor, μ is the chemical potential, T is the temper-ature, M is the magnetization. Moreover, we will use relation with thermodynamic potential Ω [17]
dμ
dT= ∂2Ω
∂T ∂μ
(∂2Ω
∂μ2
)−1
T,
dM
dT= 1
LxL y
∂2Ω
∂T ∂ H. (6)
Eqs. (4)–(6) reveal the essential physics of NE oscillations in the quantizing magnetic fields.
The thermodynamic potential of 2D system in crossed fields can be written as
Ω = −kB TL y
π h̄
p y max∫0
dp y
∑n
ln
(1 + exp
μ − εn,p y
kB T
), (7)
where L y is the linear system size along Y axis. Value p y max is de-termined from the condition of the degeneracy of LL. Using result of [10], we obtain
0 < x = c
eH p y < Lx.
Then p y max = eH Lx/c. Using the Poisson summation formula, we obtain
Ω = −kB T L y
π h̄
{ ∞∫a
eH Lx/c∫0
dp y ln
(1 + exp
μ − εn − υ0 p y
kB T
)dn
+ 2 Re∞∑
k=1
∞∫a
eH Lx/c∫0
dp y ln
(1 + exp
μ − εn − υ0 p y
kB T
)
× ei2πkndn
}
= Ω1 − 2kB T L y
π h̄
∞∑k=1
Ik, (8)
where −1 < a < 0. We will assume that εn=a = 0. After integrating by parts, and then over p y
Ik = Re
{1
i2πkυ0
∞∫0
dεnei2πk( c2π h̄eH A(εn)−γ )
× lnexp μ−εn
kB T + 1
exp μ−εn−eE LxkB T + 1
}, (9)
where we use the Lifshitz–Onsager quantization conditions (1). The energies ε ∼ μ are important for the magnetic oscillations. Therefore, we expand the function A(ε) in the vicinity of μ, i.e. A(ε) ≈ A(μ) + 2πm(μ)(ε − μ). Then, integrating and taking into account that μ kB T , we obtain (we consider the case μ eE Lx)
Ω̃ ≈ kB T L yωc
2π2υ0
∞∑k=1
1
k2
sin kAh̄ωcm − sin k(A−2πmeE Lx)
h̄ωcm
sinh 2π2kkB Th̄ωc
, (10)
where ωc = eH/mc. In (10) we put γ = 0. Analogically, we can obtain the Ω1
Ω1 = kB T L y
2π2h̄2eE
∞∫0
ln1 + exp μ−ε−eE Lx
kB T
1 + exp μ−εkB T
A(ε)dε. (11)
Next, we can expand functions (10) and (11) in powers eE Lx/μ. This gives
Ω̃ = mE(μ)ω2c S
2π3
∞∑k=1
ψ(xk)
k2e− 2πkΓ
h̄ωc cos
(k
AE(μ)c
h̄eH
), (12)
Ω1 = − S
2π2h̄2
∞∫0
AE(ε)dε
exp ε−μkB T + 1
, (13)
where xk = k2π2kB T /h̄ωc , ψ(xk) = xk/ sinh xk . In (12) we add the term with Γ , which appears when Dingle LL broadening is consid-ered. Finally, we obtain
dμ̃
kBdT=
∑∞k=1 ψ(xk)L(xk)e− 2πkΓ
h̄ωc sin(k AE (μ)ch̄eH )
1 + 2∑∞
k=1 ψ(xk)e− 2πkΓh̄ωc cos(k AE (μ)c
h̄eH )
, (14)
where L(x) = coth x − 1/x is the Langevin function.Thus, using (3)–(6), (14) we can investigate the NE oscillations
in graphene taking into account the dependence of LL on the elec-tric field. We will consider only the oscillating part of NC. From (6)we obtain
ν̃ =[
σxx
e2nc+ cnρyy
H2
]dμ̃
dT= κ(H)
dμ̃
kBdT. (15)
Finally, in this paper we consider a case, when ωcτ ≤ 1, as in [17]. At ωcτ ≤ 1 the Shubnikov–de Haas oscillations are small,
Z.Z. Alisultanov / Physics Letters A 378 (2014) 2329–2331 2331
Fig. 1. Normalized Nernst–Ettingshausen (NE) oscillation as function of the inverse magnetic field (a) and carriers concentration (b) for monolayer graphene.
Fig. 2. Normalized Nernst–Ettingshausen (NE) oscillation as function of the inverse magnetic field (a) and carriers concentration (b) for bilayer graphene.
i.e. at this condition we can neglect the conductivity oscillations. Then we can write ν̃ = κdμ̃/dT and consider only oscillations of the function ν̃/κ = dμ̃/dT . These oscillations in MG and BG are shown in Figs. 1 and 2. As follows from figures, the period and amplitude of oscillations depend on the electric field. Note, such dependence is not observed in materials with parabolic dispersion law. Therefore, we need take into account this dependence in the-oretical studies of NE oscillations.
Influence of longitudinal electric field on the LL gives a unique opportunity to control the magnetism in graphene by electric field. Lifshitz and Kaganov [12] proposed the use of the effect for the study of isoenergy surfaces in semiconductors. Indeed, the disper-sion law of semiconductor is non-parabolic due to the existence of the band gap. Therefore, the influence of electric field on the LL can be observed in semiconductors. In work [18], the optical ab-sorption in semiconductors in crossed magnetic and electric fields were investigated. In the work, cases E < H and E > H were con-sidered in detail. Due to linear dispersion law, the influence of the electric field on the LL in graphene is much larger than same effect in traditional semiconductors and other non-relativistic materials.
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