Magnetophonon Resonance in Monolayer Graphene

The conductivity describing magnetophonon resonances is calculated in monolayer graphene, with the Fermi level located near the Dirac point. Intervalley scattering due to zone-edge phonons gives dominant contribution to the conductivity compared to intravalley scattering due to zone-center optical phonons mainly because of lower frequency. Resonances are classified into three types, i.e., principal, symmetric, and asymmetric transitions. The magnetophonon oscillations due to the principal and symmetric transitions are periodic in inverse magnetic field, while those due to the asymmetric transitions are not precisely periodic. The amplitude of the oscillation is shown to be weakly dependent on magnetic field.


Introduction
2][3][4] MPR provides detailed information on carrier effective mass and phonon frequency at higher temperatures, typically between liquid nitrogen and room temperature.Recently, ambipolar field-effect transistors of an atomically thin graphene were successfully fabricated. 5Since then, graphene has been a subject of considerable theoretical and experimental study [6][7][8][9] and graphene-based electronic devices are considered as a promising candidate for future integrated circuits.In view of these developments, it is essential to obtain a deeper understanding of the role of electronphonon interaction at higher temperatures.In this paper, we theoretically investigate magnetophonon resonances in monolayer graphene considering zone-edge 10,11 and zone-center 10,[12][13][14][15][16] optical phonons.
For conventional group IV elements or group III-V compound semiconductors, such as Si and GaAs, magnetophonon resonances occur periodic in inverse magnetic field because the resonance takes place when the optical-phonon energy equals an integer multiple of the separation between nearest-neighbor Landau levels.For graphene, the Landau-level separation is not constant.We shall first examine the condition of magnetophonon resonances in graphene.We then calculate the magnetoconductivity in the ohmic regime using the Kubo formula.We take into account both intervalley scattering by zone-edge optical phonons and intravalley scattering by zone-center optical phonons.
The paper is organized as follows: In § 2, the resonance condition of graphene MPR is considered.The conductivity due to the magnetophonon effect is calculated in § 3. Some examples of numerical results are presented and discussed in § 4. A summary is given in § 5. Harmonic analysis of oscillatory components is discussed in Appendix.

Resonance Condition
For conventional semiconductors, energy levels in strong magnetic field B are quantized into a set of Landau levels equally separated by cyclotron energy ω c .2][3][4] The MPR condition of conventional semiconductors is, therefore, given by Since the cyclotron frequency is given by ω c = eB/m * , the conductivity maxima appear periodically in inverse magnetic field with the fundamental field of B 0 = m * ω 0 /e.Here, m * is the electron effective mass.
For monolayer graphene, the electron energy in magnetic field B perpendicular to a graphene sheet is given by where )γ 0 a is the band parameter, γ 0 is the transfer integral between π orbitals of the nearest-neighbor carbon atoms, a is the lattice constant, and sgn(n) is the sign function defined by Since the level separation between successive Landau levels is not constant, MPR does not occur simultaneously for all the Landau levels in contrast to conventional semiconductors.We first consider the resonance condition in graphene before calculating the MPR conductivity.
As will be shown latter, MPR in graphene produces only weak structures on magnetoconductivity because of the higher optical-phonon energies and small occupation number of optical phonons.Moreover, Shubnikov-de Haas oscillations occur at relatively higher temperatures in graphene. 17,18 herefore, we consider the case that the Fermi level lies in the Landau level at zero energy (n = 0) in the following analysis, because otherwise Shubnikovde Haas oscillations could obscure any possible features of MPR in actual experiments.
Figure 1 shows a schematic diagram of electron occupation considered.At higher magnetic field ω B > kT and lower electron density n e < (1/2)g s g v /(2πl 2 ) with g s = 2 and g v = 2 being the spin and valley degeneracy, respectively, the occupation factor of each Landau level, f n , becomes where kT is the thermal energy, n e is the electron density, and 0 < f 0 < 1.
As will be shown below, transitions by absorbing and emitting phonons with large momentum giving large change in the center coordinates of the cyclotron motion contribute to MPR and there is essentially no selection rule for transitions between initial and final Landaulevels.For the occupation factor given by eq. ( 4), we separate MPR transitions into the following three types: (1) The principal transitions are between n = 0 and n = 1, 2, 3, . ... Similar transitions also occur between n = 0 and n = −1, −2, −3, . . .(Fig. 1(a)).
The resonance condition of the principal transitions is given by Since ω B = γ √ 2eB/ , the condition can be written as with fundamental magnetic field B 0 being given by The oscillations are periodic in inverse magnetic field as in conventional MPR.The zone-edge optical-phonon energy is ω K = 162 meV, which gives the fundamental magnetic field B K = 20.7 T for γ 0 = 3.03 eV and a = 0.246 nm.For the zone-center optical phonon, the phonon energy is ω Γ = 196 meV and the fundamental magnetic field is B Γ = 30.3T. When B > B 0 /4, only principal transitions occur.When B ≤ B 0 /4, symmetric transitions contribute to MPR with condition This can be written as The oscillations are also periodic in inverse magnetic field with the fundamental magnetic field of B 0 /4.For further reducing magnetic field, asymmetric transitions finally contribute to MPR.Since the resonance condition for the transition between Landau levels −m and n is given by asymmetric transitions occur when 8. These resonances do not occur periodically in inverse magnetic field.It is worth mentioning that other transitions such as ε n+N − ε n = ω 0 with n ≥ 1 contribute to the current when excited Landau levels are occupied by electrons.

Electronic states
0][21][22][23][24][25] The effective Hamiltonian for an electron is given by with π = −i∇ + (e/ )A.In the Landau gauge, A = (0, Bx), the eigenstates can be specified by a set of quantum numbers λ ≡ {v, n, X}, where v is a valley index (v = K or K ′ ), n is a Landau level index, and X is a center coordinate.The wave functions, F λ (r), are written as (13)   with where L is the linear dimension of the system and Hn(x) is the Hermite polynomial.Here, we follow the notation as in ref. 14.

Electron-phonon interaction
We consider the system described by the Hamiltonian H e is the Hamiltonian for electrons in magnetic field where c † λ and c λ are the creation and destruction operators, respectively.H p is the Hamiltonian for optical phonons where q = (q x , q y ) is the phonon wave vector, µ denotes the phonon mode, and b † qµ and b qµ are the creation and destruction operators, respectively.We take into account four optical-phonon modes: zone-edge modes K and K ′ , longitudinal zone-center mode l, and transverse zonecenter mode t (µ = K, K ′ , l, t).We neglect q dependence of the optical-phonon frequencies because of the small dispersion both at zone-edge and zone-center and have ω µ = ω Γ for µ = t and l and ω µ = ω K for µ = K and K ′ .
The electron-phonon interaction for zone-center modes can be written as 12,13 with where N is the number of unit cells and M is the mass of a carbon atom.Further, we have defined where ϕ(q) = tan −1 (q y /q x ) and b is the bond length.
Note that e µ (−q) = e µ (q) * .The interaction for zone-edge modes is written as 11 where ω = e 2πi/3 , β K is the same as β Γ in the nearestneighbor tight-binding model, but can be different in actual systems, and The total interaction potential becomes The interaction Hamiltonian becomes with where for µ = t and l and with

Conductivity
Using the Kubo formula, 26,27 MPR conductivity σ xx is given by where π(ω + i0) is the retarded current-current correlation function For MPR, the current is carried by hopping processes with the aid of emission or absorption of optical phonons. 26,28,29 Tus, the current is given by that carried by the center coordinate, i.e., j x = (−e) Ẋ = (−e)(1/eB)[∂U ep (r)/∂y].In the second quantized form we have ) and we have Here, and β = 1/kT .][32][33] The MPR conductivity given by eq. ( 38) diverges whenever a resonance condition is satisfied.Effects of higher order terms or other scattering mechanisms will remove the divergences in σ xx .In the present study, we phenomenologically include those effects by replacing the delta function in eq. ( 38) with the Lorentzian function We then assume that the level width Γ is determined by short-range scatterers and is given by with the dimensionless parameter W characterizing the scattering strength, which is the same as that defined previously [34][35][36][37] for scatterers with potential range smaller than the electron wavelength (W = A −1 in refs.

38-40)
Here, n i the scatterer density and u is the strength.From eq. ( 38), we see that the MPR conductivity can be given by a sum of the conductivity due to the zoneedge modes, σ K xx , and that due to the zone-center modes, σ Γ xx .For the electron occupation given by eq. ( 4), σ K xx is written as with Here, K nm = max(n, 1) + max(m, 1), N K = n B ( ω K ), and λ K is the dimensionless coupling parameter defined by 11 where max(n, m) = n for n ≥ m and m for n < m.For zone-center modes, we have with Here, G nm = n + m, N Γ = n B ( ω Γ ), and λ Γ is obtained from λ K given by eq. ( 44) by replacing ω K with ω Γ and β K with β Γ . 13r the nth Landau level to be clearly resolved as resonance, the level width should be less than the distance to the nearest neighbor Landau levels.The condition is written as For Γ given by eq. ( 40), this condition is reduced to n < N * with N * = (W −1 − 2) 2 W/8.When W −1 = 40, we have N * = 4.5, suggesting clear oscillations due to the principal transitions for B > B K /N * ≈ 0.2B K and those due to the symmetric transitions for B > B K /4N * ≈ 0.06B K .However, when W −1 = 10, we have N * = 0.8 and oscillations will not be well resolved.Note that many kinds of experimental techniques have been developed to detect such a weak oscillatory structure of the MPR conductivity, such as two cascaded RC networks 41 and magnetic-field modulation technique. 42ccording to the Einstein relation, the conductivity is given by σ = e 2 D * D(ε), where D * is the diffusion constant and D(ε) is the density of states.In the absence of a magnetic field, we have D * ∼ v 2 F τ , where v F = γ/ and τ is the relaxation time.Because τ −1 ∝ (2π/ )n i u 2 D(ε), we immediately have σ ∼ (e 2 /π 2 )W −1 . 38,43 his clearly indicates that the conductivity is independent of electron concentration n e for constant scattering strength W . Experimentally, however, the conductivity increases almost linearly with n e for sufficiently large n e , 17,44 showing that the effective scattering strength in actual graphene on SiO 2 substrate varies considerably with the electron energy, i.e., W ∝ n −1 e .Plausible scatterers giving rise to such strong n e dependence are likely to be charged impurities. 43,45 ecent measurement of τ /τ 0 ∼ 2 is also consistent with charged-impurity scattering, although the claim otherwise, 46 where τ 0 is a conventional relaxation time and τ is a transport relaxation time giving the conductivity.For such scatterers, the broadening of Landau levels decreases with |n|, roughly Γ n ∝ (|n| + 1) −1/2 , and the above condition of observation of well-resolved MPR is likely to be relaxed, because Γ ∼ (Γ n + Γ m )/2 for transition between Landau levels n and m.

Results and Discussion
Figure 2 shows magnetic field dependence of MPR conductivity σ K xx due to the intervalley scattering by zoneedge phonons.We replace the upper limit of summation in eq. ( 42) with N max = 10 in this calculation.Clear oscillatory structures can be seen for small W (or smaller level width Γ) and become weaker with increasing W . Weak oscillations remain to be visible even for W −1 = 10, corresponding to the scattering strength near the Dirac point in typical samples. 17,35  show the relative contributions of the principal, symmetric, and asymmetric transitions, we decompose σ K xx into each component in Fig. 3.When B > B K /4 (or B K /B < 4), only the principal transition occurs.8][49] The MPR conduction takes place because of jumps of the center coordinates of a cyclotron orbit caused by opticalphonon scattering.The conductivity is proportional to the product of the diffusion constant D * and the density of states D = ∂n e /∂µ at around the Fermi level, i.e. σ xx ∼ e 2 D * D, where n e is the electron concentration and µ is the chemical potential.
Consider, for example, the principal transition due to optical-phonon absorption between n = 0 and n = ±N for N ∼ B K /B.The diffusion constant D * is given by where ∆X is the hopping distance of each jump and is given by ∆X ∼ (N + 1) 1/2 l.The optical-phonon scattering rate τ −1 ep is written as where ρ is the final density of states for the opticalphonon scattering.MPR arises from the oscillation in the scattering rate.At a resonance magnetic field, ρ is given by ρ ≈ (2πl 2 ) −1 (πΓ) −1 , and we have At high temperatures when the thermal energy is larger than the level broadening, D is given by We thus obtain Note that this equation agrees with eq. ( 42) for the principal transitions between n = 0 and n = ±N with phonon absorption processes.From eq. ( 52), we see that the magnetic-field dependence of σ K xx comes from term (N + 1)/(πl 2 Γ).For Γ ∝ √ B, we have This equation explains the field dependence of the principal contribution of σ K xx shown in Fig. 3.There is a background conductivity in all of principal, symmetric, and antisymmetric components.This partly arises due to the assumption of Lorentzian function (39), giving large low-and high-energy tails in the final-state density of states.In the region of weak magnetic fields, the background increases with a higher power in B −1 than that given in eq. ( 53).This mainly arises due to contributions of many transitions among Landau levels overlapping with each other because broadening exceeds Landau-level separations.In such weak magnetic fields, however, scattering by acoustic-phonon branches should also be considered. 50o far we have been considering the numerical results of the intervalley scattering by the zone-edge phonons.Qualitatively, the essential features are not different between zone-edge phonon MPR and zone-center phonon MPR.We present in Fig. 4 magnetic field dependence of the total conductivity σ xx = σ K xx + σ Γ xx at T = 300 K. We set β K = β Γ = 2, although actual values can be slightly larger. 15,33 e find that the intervalley scattering gives dominant contribution to the conductivity compared to the intravalley scattering.This is mainly due to the smaller phonon energy for the zone-edge mode, i.e., ω K = 162 meV in contrast to ω Γ = 196 meV.The difference between σ K xx and σ Γ xx becomes larger when temperature is lowered.

Summary
In this paper, the conductivity due to magnetophonon resonances has been calculated in monolayer graphene in the case that the Fermi level lies in the n = 0 Landau level at zero energy or the Dirac point.Resonances are classified into three types transitions, principal, symmetric, and asymmetric.The principal and symmetric oscillations are periodic in inverse magnetic field, while the asymmetric oscillations are not precisely periodic in inverse magnetic field.In high magnetic field, the amplitude of the MPR conductivity due to principal resonances shows weak magnetic field dependence.In weak magnetic fields, asymmetric transitions become more dominant.The intervalley scattering due to zoneedge phonons gives dominant contribution to the conductivity compared to the intravalley scattering due to zone-center phonons, mainly because of lower phonon frequency.

Acknowledgement
This work was supported in part by Global COE Programs at Osaka University "Center for Electronic Devices Innovation" and at Tokyo Tech "Nanoscience and Quantum Physics," by Grants-in-Aid for Scientific Research, and by Grant-in-Aid for Scientific Research on Priority Area "Carbon Nanotube Nanoelectronics" from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

Appendix
For analyzing experimental results, it may be convenient to extract oscillatory components from the MPR conductivity.Here, we present results of a harmonic analysis. 28,29 r the zone-edge modes, the oscillatory part due to the principal transitions is given by

Fig. 1 .
Fig. 1. (Color online) Schematic diagram of electron occupation and the electron transition processes.

Fig. 2 .Fig. 3 .
Figure2shows magnetic field dependence of MPR conductivity σ K xx due to the intervalley scattering by zoneedge phonons.We replace the upper limit of summation in eq.(42) with N max = 10 in this calculation.Clear oscillatory structures can be seen for small W (or smaller level width Γ) and become weaker with increasing W . Weak oscillations remain to be visible even for W −1 = 10, corresponding to the scattering strength near the Dirac point in typical samples.17,35To show the relative contributions of the principal, symmetric, and asymmetric transitions, we decompose σ K xx into each component in Fig.3.When B > B K /4 (or B K /B < 4), only the principal transition occurs.Resonances due to symmetric transitions appear at a B K /B = 4 and 8 and those due to asymmetric transitions appear atB K /B = (1 + √ 2) 2 ≈ 4.8, (1 + √ 3) 2 ≈ 7.5, • • • .Asymmetric transitions become more dominant with

Fig. 4 .
Fig. 4. (Color online) Total conductivity σxx = σ K xx + σ Γ xx as a function of magnetic field at T = 300 K (solid line).Dotted line shows σ K xx due to the intervalley scattering and dashed line σ Γ xx