PHYSICAL REVIEW B 92, 195431 (2015)

Two- scattering in in the quantum Hall regime

A. M. Alexeev,1 R. R. Hartmann,2 and M. E. Portnoi1,3,* 1School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom 2De La Salle University-Manila, 2401 Taft Avenue, 1004 Manila, Philippines 3International Institute of Physics, Universidade Federal do Rio Grande do Norte, Natal, RN, Brazil (Received 28 April 2015; revised manuscript received 31 October 2015; published 30 November 2015)

One of the most distinctive features of graphene is its huge inter-Landau-level splitting in experimentally attainable magnetic fields which results in the room- quantum . In this paper we calculate the longitudinal conductivity induced by two-phonon scattering in graphene in a quantizing magnetic field at elevated . It is concluded that the purely phonon-induced scattering, negligible for conventional heterostructures under quantum Hall conditions, becomes comparable to the disorder-induced contribution to the dissipative conductivity of graphene in the quantum Hall regime.

DOI: 10.1103/PhysRevB.92.195431 PACS number(s): 73.43.Cd, 73.22.Pr

I. INTRODUCTION the Hall resistance. In conventional semiconductor systems the quantum Hall effect is observed at helium temperatures The quantum Hall effect, discovered in 1980 by Klaus only, and the value of σ is governed by scattering on von Klitzing et al. [1], allows one to determine the quantum xx disorder. In the QHR, σ depends exponentially on the energy resistance standard in terms of the electron charge and Plank’s xx separation between the and the nearest LL. The lon- constant with spectacular accuracy. However, the level of gitudinal conductivity prefactor has a theoretically predicted precision necessary for applications (a few parts per 2 2 billion) for conventional quasi-two-dimensional semiconduc- value of e /hin the presence of short-range disorder and 2e /h tor systems requires the use of ultralow temperatures and high in the presence of long-range disorder [16,17]. It was shown magnetic fields [2,3]. The discovery of graphene [4] in 2004 that the contribution from phonon scattering is negligibly led to a revival of interest in quantum Hall effect physics [5,6]. small. The exceptions are exotic cases such as magnetoroton The energy of Landau levels (LLs) in graphene is given by dissociation, in which the whole effect arises from scattering on [18], or an enhancement of phonon-induced √ E =±(v /l ) 2N, (1) scattering near the intersection of two LLs corresponding N F B to different size- subbands [19–21]. However, √ 6 as we show in this paper, the phonon-scattering mechanism where vF = 1 × 10 m/s is the Fermi velocity, lB = /eB is the magnetic length, B is the magnitude of the applied dominates in the high-temperature QHR in graphene, since at =  = × 4 magnetic field, N is the LL number, and the “±” signs T>TlB ( s/lB )/kB (where s 2 10 m/s is the sound correspond to the electrons and holes, respectively [7–9]. For velocity in graphene [22,23]) the energy of an acoustic phonon B = 10 T the separation between the zero and the first LL in with a wave vector comparable to the inverse magnetic length graphene is E01/kB ≈ 1300 K (in comparison, E/kB ≈ is much smaller than the temperature; therefore, the number 200K in conventional quasi-two-dimensional semiconductor of such phonons increases drastically. systems). Therefore, the quantum Hall effect in graphene In this paper we restrict our consideration to electron can be observed even at room temperatures in experimentally scattering induced by the interaction with intrinsic in-plane attainable magnetic fields. Indeed, the room-temperature quan- phonons only, neglecting the effects of disorder as well tum Hall effect in graphene was first observed in 2007 [10]; as electron-electron interactions. Our calculations result in however, extremely high magnetic fields (up to 45 T) were the lower estimate of the longitudinal conductivity σxx in needed. graphene in the QHR, as we do not study electron interactions It has been understood since the 1930s that the longitu- with various other types of phonons such as out-of-plane dinal conductivity of metals in a quantizing magnetic field flexural phonons, which are present in suspended samples increases with increasing electron scattering [11]. To develop [24,25], or bulk acoustic phonons in a substrate, which a graphene-based quantum Hall standard of resistance [12–15], interact with electrons in graphene on polar substrates such which would work at elevated temperatures and in moderate as boron nitride [26] or carbide via piezoelectric magnetic fields, it is essential to examine the contributions coupling [27,28]. from different scattering processes to the longitudinal conduc- tivity σxx in the quantum Hall regime (QHR), T  E01/kB, as σxx provides the major correction to the quantized value of II. THEORETICAL FORMALISM: TWO-PHONON SCATTERING WITHIN ONE LANDAU LEVEL The of a charged carrier (electron or hole) in graphene subjected to a magnetic field B normal to the *[email protected] graphene plane is given in the Landau gauge, A = (0,Bx,0),

1098-0121/2015/92(19)/195431(6) 195431-1 ©2015 American Physical Society A. M. ALEXEEV, R. R. HARTMANN, AND M. E. PORTNOI PHYSICAL REVIEW B 92, 195431 (2015) by the following expression [7–9]: conductivity σxx at the Nth LL due to two-phonon scattering 2 using the generalized form of the Einstein relation [21,36,37]: CN (x − x0) N,k = √ exp − y N 2 2 2 N! πlB 2lB e D σxx = νN (1 − νN ) , (3) − 2πl2 k T √ HN [(x x0)/lB ] exp(iky y) B B × , (2) 4 ± 2NH − [(x − x )/l ] lB  2 N 1 0 B Ly = →  − D Wky ky (ky ky ) , (4) √ 2  = = = 2 ky where C0 1 and CN=0 1/ 2, x0 lB ky is the guiding center coordinate, ky is the electron wave vector y component, −1 where νN ={exp [(EN − EF)/kBT ] + 1} is the LL filling HN are the Hermite polynomials, Lx and Ly are the graphene factor, EN is the LL energy defined by Eq. (1), EF is the Fermi sample dimensions, the “±” sign corresponds to electrons and  energy, k and k are the y components of the electron wave holes, and the and valley indices are omitted. Here the z y y vector before and after scattering, D is the diffusion coefficient, axis has been chosen in the direction of the applied magnetic and W →  is the probability of scattering. field and the x axis and y axis are in the graphene plane. ky ky Two-phonon scattering in graphene in the QHR is possible The wave function given by Eq. (2) is defined for one Dirac through two different virtual intermediate states: a phonon point only and its two components correspond to graphene’s + with wave vector q is first emitted or a phonon with wave two sublattices. In what follows all results will be obtained − vector q is first absorbed. Transitions changing the electron for electrons only. Calculations for the nonequivalent Dirac LL number in the intermediate states are strongly suppressed point and holes can be easily repeated in a similar fashion. due to small values of corresponding matrix elements and the From here on we will omit the “±” sign in order to simplify presence of large denominators in the expression for W →  notation. We account for the valley and spin degeneracy by ky ky (see Appendix). Therefore, in what follows we consider only multiplying the final result for σ by the factor of 4 at the xx transitions with no change in the electron LL number. Then, final stage of calculations. the probability of two-phonon scattering is given by Fermi’s Electron scattering on phonons leads to a change in the golden rule with the matrix element ky component of the electron wave vector, which results    in a change to the electron guiding center coordinate x0. | ˆ + | | ˆ − | − + ky Vq ky ky Vq ky  = − + + Electron transitions between LLs due to one-phonon scat- Mky ,k (q ,q ) nq (nq 1) y  −  q tering are suppressed due to the large energy gaps be- ky tween LLs in graphene in the QHR. Indeed, the number    k |Vˆq− |k k |Vˆq+ |ky of phonons with the energy required for such transitions − y y y , (5)  −  + is nq exp ( EN1N2 /kBT )(EN1N2 is the energy gap q between two different LLs in graphene), which is very small =  − −1 in the QHR. Inter-LL scattering on acoustic phonons has where nq [exp ( q /kBT ) 1] is the phonon occupation = ˆ an additional exponentially strong suppression in the matrix number, q sq is the acoustic phonon frequency, and Vq is element of transition which is markedly different from the case the electron-phonon coupling operator. of conventional semiconductor systems as discussed in the Operators describing electron scattering on intrinsic longi- Appendix. It is also evident that in the QHR only scattering on tudinal (LA) and transverse (TA) acoustic phonons in graphene acoustic phonons can provide a noticeable contribution to the are [38,39] longitudinal conductivity σxx. The optical phonon energy in ig g ei2ϕ graphene corresponds to the temperature range 1800–2300 K Vˆ LA = U q exp (iqr) d h , (6) q q −g e−i2ϕ ig [29,30] which leads to very small optical phonon occupation h d numbers at room temperatures and below. In this work, we i2ϕ ˆ TA = 0 ighe Vq Uq q exp (iqr) −i2ϕ , (7) are interested in phonon-induced equilibrium longitudinal ighe 0 conductivity, whereas magnetophonon resonance associated −1/2 1/2 with optical phonons has been studied in Refs. [31–33]. where Uq = (LxLy ) (/2ρq ) , ρ is the graphene two- One-phonon scattering within the same LL is ineffective dimensional mass density, r is the position vector in the because of the small width of the LLs, which can be arguably graphene plane, and ϕ is the angle between the phonon wave achieved in pristine graphene samples. Interestingly, even in vector q and the x axis. The diagonal matrix elements in heavily disordered samples the broadening of LLs was found to Eqs. (6) and (7) describe electron coupling to the phonon- shrink in several particular cases [34,35]. For our calculations created deformation potential, and the off-diagonal matrix we assume an infinitely narrow band of delocalized states in elements originate from the phonon-induced bond-length the middle of each LL with a vanishing in modulations, which effect hopping amplitudes between two between LLs. In this limit one-phonon scattering within one LL neighboring sites. The corresponding coupling constants were is forbidden by energy and momentum conservation. The next estimated as gd ≈ 20–30 eV and gh ≈ 1.5–3.0eV[22,23,38– order process to consider is two-phonon scattering through 42]. Note that the deformation potential couples electrons with virtual states with no change in the electron initial and final LA phonons only. Furthermore, since the second component of LL numbers but with the change in its in-plane momentum (or the electron wave function defined by Eq. (2) vanishes for the the guiding center coordinate). We calculate the longitudinal zero LL, electrons in this LL do not interact with TA phonons.

195431-2 TWO-PHONON SCATTERING IN GRAPHENE IN THE . . . PHYSICAL REVIEW B 92, 195431 (2015)

Substituting the electron-phonon scattering operators given by Eqs. (6) and (7) into Eq. (5) yields ⎛ ⎞ μ,γ − μ,γ + μ,γ + μ,γ − M  (q )M   (q ) M  (q )M   (q ) μ,γ − + − + ky ,ky ky ,ky ky ,ky ky ,ky = 2 − − + + + ⎝ − ⎠ Mk ,k (q ,q ) Gμ,γ Uq q nq Uq q (nq 1) , (8) y y  −  +  q  q ky ky where μ ={LA,TA}, γ ={d,h} are the indices introduced to separate the contributions to the longitudinal conductivity σxx from LA and TA phonons and from the two different scattering mechanisms discussed above, Gμ,γ is the generalized electron-phonon μ,γ coupling constant with the values GLA,d = gd , GLA,h = gh, GTA,d = 0, GTA,h = gh, and the matrix elements M  (q)are ky ,ky given by LA,d † 2 0 0 M  (q) = i   (r)exp(±iqr)I (r)dr = iC exp(±β − α) L (α) + L (α) δ  ± , (9) ky ,k N,k N,ky N N N−1 ky ,ky qy y y LA,h † M  (q) =   (r)exp(±iqr)LAN,k (r)dr ky ,ky N,ky y

2 −1/2  1 =− ± − ± −  ± CN N lB [qx cos 2ϕ (ky ky )sin2ϕ]exp( β α)LN−1(α)δky ,ky qy , (10) TA,h † M  (q) =   (r)exp(±iqr)TA N,k (r)dr ky ,ky N,ky y

2 −1/2  1 =− ∓ − ± −  ± CN N lB [qx sin 2ϕ (ky ky) cos 2ϕ]exp( β α)LN−1(α)δky ,ky qy . (11) = 2  − 2 + 2 = 2  + = ± In Eqs. (9)–(11), α (lB /4)[(ky ky ) qx ], β i(lB qx /2)(ky ky ), N 0, the “ ” sign refers to emitted and absorbed 0 1 × phonons, LN and LN are the Laguerre polynomials, I is the 2 2 identity matrix, and LA and TA are given by 0 ei2ϕ 0 iei2ϕ  = , = . (12) LA −e−i2ϕ 0 TA ie−i2ϕ 0 0 = 1 = −1/2 Equation (9) is also valid for the zero LL when substituting LN− 0 and LN− 0. Note the N factor in Eqs. (10) and 1 1  (11) which suppresses the off-diagonal contribution to σxx in higher LLs. Summation over all possible values of ky in Eq. (8) results in the following expressions for two-phonon scattering matrix elements: LA,d − + 2 8 4 2 0 2 2 0 2 2 4 M  (q ,q ) = C (g U ) [q/(s)] L l q /2 + L l q /2 ky ,k N d q N B N−1 B y 2 2 2 2 + − − + × exp(−l q )sin l (q q − q q )/2 δ  + −− + , (13) B B y x y x ky ,ky qy qy LA,h − + 2 8 4 2 −2 1 2 2 4 2 2 4 + + + + 2 M  (q ,q ) = C (g U ) [q/(s)] N L l q /2 exp −l q l (q sin 2ϕ + q cos 2ϕ ) ky ,k N h q N−1 B B B x y y − − − − 2 2 2 + − − + × (q sin 2ϕ + q cos 2ϕ ) sin l (q q − q q )/2 δ  + −− + , (14) x y B y x y x ky ,ky qy qy TA,h − + 2 8 4 2 −2 1 2 2 4 2 2 4 + + + + 2 M  (q ,q ) = C (g U ) [q/(s)] N L l q /2 exp − l q l (q cos 2ϕ − q sin 2ϕ ) ky ,k N h q N−1 B B B x y y − − − − 2 2 2 + − − + × (q cos 2ϕ − q sin 2ϕ ) sin l (q q − q q )/2 δ  + −− + . (15) x y B y x y x ky ,ky qy qy  →  Substituting the calculated probability of the two-phonon scattering Wky ky into Eq. (4) and performing summation over ky as well as integration over all possible values of q+ and q− yield the following result for the longitudinal conductivity: = LA,d + LA,h + TA,h − σxx σ˜xx σ˜xx σ˜xx νN (1 νN ), where ∞ LA,d = 2 8 4 2 4 + 4 − 2 2 − 2 2 0 2 2 + 0 2 2 4 σ˜xx (e /h) CN /2π gd lB /ρ s TlB /T ηq (ηq 1)q exp lB q 1 J0 lB q LN lB q /2 LN−1 lB q /2 dq, 0 (16) ∞ LA/TA,h = 2 −2 8 4 2 4 + 8 − 2 2 − 2 2 1 2 2 4 σ˜xx (e /h)N CN /2π ghlB /ρ s (TlB /T) ηq (ηq 1)q exp lB q 1 J0 lB q LN−1 lB q /2 dq. 0 (17) μ,γ Here J0 is the Bessel function of the first kind. The expressions for σxx were multiplied by a factor of 4 to account for the LA,d LA/TA,h valley and spin degeneracy. From Eqs. (16) and (17) it is evident thatσ ˜xx σ˜xx due to the relatively small value of gh −2 compared to gd [22,23,38–42] and the small N factor contained in Eq. (17).  There are two distinctive temperature limits. In the low-temperature limit, T TlB , the main value of the integrals in Eqs. (16) and (17)isformedbyq  kB T/s and qlB can be considered as a small parameter. Expanding the integrand into power series μ,γ in qlB and taking into account only the lowest power term yields the following expression for σxx in the low-temperature

195431-3 A. M. ALEXEEV, R. R. HARTMANN, AND M. E. PORTNOI PHYSICAL REVIEW B 92, 195431 (2015) limit: LA,d  2 3 7 4 2 22 6 8 σ˜xx (e /h)(2 π /15) gd / lB ρ s (T/TlB ) , (18) LA/TA,h  2 N−1 2 11 4 2 22 6 12 σ˜xx (e /h) A0 /N (5528π /1365) gh/ lB ρ s (T/TlB ) . (19) N−1 1 4 = 4N−4 N−1 j  LA/TA,h = In Eq. (19) the coefficients A0 are defined by [LN−1(x/2)] j=0 Aj x , where N 1. For the zeroth LL,σ ˜xx 0. LA,d Note thatσ ˜xx in Eq. (18) does not depend on the LL number. The temperature dependencies given by Eqs. (18) and (19) are different from the case of conventional semiconductor heterostructures for both two-phonon scattering [21,36,37] and phonon-assisted hopping conductivity [16,17]. In Eq. (18), which corresponds to the deformation potential scattering mechanism, the lower power in the temperature dependence of mobility compared to that obtained in Refs. [21,36,37] is due to phonons in graphene being two dimensional. As was mentioned above, the contribution to longitudinal conductivity at higher LLs given by Eq. (19) is a distinctive feature of graphene without analogy in semiconductor systems.

In the more interesting high-temperature limit, T>TlB , the main value of the integrals in Eqs. (16) and (17)isformedby q  1/lB , and sq/kBT can be considered as a small parameter. By expanding the integrand into power series in sq/kBT and μ,γ taking into account only the lowest power term we obtain the following result for σxx in the high-temperature limit: LA,d = 2 8 d 2 4 2 22 6 σ˜xx (e /h) CN N /2 π gd /lB ρ s T/TlB , (20) where 4N 3 2j + 3 2j + 5 d = BN  j + 1 − F ; ;1;−1 , (21) N j 2 2 1 4 4 j=0 and LA/TA,h = 2 −2 h 6 4 2 22 6 σ˜xx (e /h)N N−1/(2 π) gh/ lB ρ s T/TlB , (22) where 4N−4 + + h N−1 7 2j 7 2j 9  − = A  j + 1 − F ; ;1;−1 . (23) N 1 j 2 2 1 4 4 j=0

In Eqs. (21)–(23),  is the gamma function and 2F1 is the hypergeometric function. In Eq. (22), N = 0, since in the zeroth LL σ˜ LA/TA,h = 0. The coefficients AN−1 are the same as in Eq. (19) and the coefficients BN are defined by [L0 (x/2) + L0 (x/2)]4 = xx j j N N−1 4N N j j=0 Bj x .

III. DISCUSSION AND CONCLUSION scattering. To compare our results with the analysis provided in Ref. [37] it is necessary to express them in terms of the In Fig. 1 we plotσ ˜ = σ˜ LA,d + σ˜ LA,h + σ˜ TA,h obtained by xx xx xx xx diffusion coefficient numerical integration of Eqs. (16) and (17) from 0 to 300 K for 2 B = 10 T (T ≈ 22 K). It can be seen from Fig. 1(a) that for 2πl σxx T lB D = σ˜ B k T = sl . T T ,˜σ linearly increases with temperature in line with xx 2 B 2 B lB xx e e /h TlB Eqs. (20)–(22). From Fig. 1(b) one can see the change of the In Ref. [37] the diffusion coefficient for T>Tl in the σ˜xx temperature dependence from the high power to the linear B lowest (second) order of perturbation theory is written as D = law occurring around T = Tl . Substituting the numerical B (αT/T )2sl . Thus, there is a simple connection between the values of all the constants into Eqs. (20) and (22) results lB B in the following simplified expression forσ ˜ at elevated dimensionless electron-phonon interaction constant α and our xx 2 temperatures: dimensionless constantsσ ˜N /(e /h). Namely, 1/2 1/2 ≈ 1/2 σ˜N B σ˜xx σ˜N (T/300 K)(B/10 T) , (24) α = 0.27 . 4e2/h 10 T whereσ ˜N has the following values for the six lowest LLs:σ ˜0 = 2 2 2 2 0.65e /h,˜σ1 = 0.06e /h,˜σ2 = 0.20e /h,˜σ3 = 0.19e /h, According to Ref. [37] higher orders of perturbation theory 2 2 = σ˜4 = 0.15e /h,˜σ5 = 0.14e /h. Here we used the unscreened can be neglected for T

195431-4 TWO-PHONON SCATTERING IN GRAPHENE IN THE . . . PHYSICAL REVIEW B 92, 195431 (2015)

FIG. 1. (Color online) (a) The longitudinal conductivity prefactorσ ˜xx as a function of temperature for B = 10T. (b) Expanded view in the 0–30 K temperature range. well exceeds 300 K and the lowest order perturbation analysis where is fully valid for ambient conditions. Ni ,Nf − + M  (q ,q ) In conclusion, we obtained the value of the longitudinal ky ,ky conductivity in graphene in the quantum Hall regime due    Nf ,k |Vˆq+ |k ,Nm Nm,k |Vˆq− |ky ,Ni to two-phonon scattering at elevated temperatures which is = y y y − +  −  Ei Em q comparable to the disorder-induced longitudinal magneto- Nm,ky conductivity in conventional semiconductor heterostructures    Nf ,k |Vˆq− |k ,Nm Nmk |Vˆq+ |ky,Ni [16,17,43]. The predicted distinctive temperature and mag- + y y y , (A1) − −  + netic field dependence of the phonon scattering contribution  Ei Em q Nm,ky to the preexponential factor in σxx given by Eq. (24) can be easily separated from the temperature- and field-independent and contribution caused by disorder when analyzing experimen- = − −  + +  − tal data. This should allow the parameters of electron- (Ei ,Ef ) δ(Ei Ef q q ). phonon interaction in graphene to be extracted with enhanced Here Ei , Em, and Ef are the energies of the electron LLs in the accuracy. initial, intermediate, and final states. Clearly, the transitions changing the electron LL number in the intermediate states are suppressed compared to the transitions conserving the LL ACKNOWLEDGMENTS number due to the presence of large denominators in Eq. (A1). We thank Charles Downing for a critical reading of the Notably, single-phonon scattering on acoustic phonons with manuscript. A.M.A. is grateful to Daniil Alexeev for fruitful a change in the LL number is also suppressed because of the discussions. This work was supported by the EU H2020 very small value of the corresponding matrix element RISE project CoExAN (Grant No. H2020-644076), EU FP7 N |Vˆ LA/TA |N ∼exp −l2 q2/4 . Initial Training Network NOTEDEV (Grant No. FP7-607521), f q i B √ √ and FP7 IRSES projects CANTOR (Grant No. FP7-612285), 2vF Due to energy conservation, q = ( Nf − Ni ), which QOCaN (Grant No. FP7-316432), and InterNoM (Grant No. slB results in the following estimate: FP7-612624). R.R.H. acknowledges financial support from URCO (Grant No. 15 F U/S 1TAY13-1TAY14) and Travel 2 LA/TA vF 2 Grant by the British Council Newton Fund. Nf |Vˆ |Ni ∼exp − ( Nf − Ni ) . (A2) q 2s2 Interestingly, unlike the case of a conventional quasi-two- APPENDIX: ELECTRON-PHONON SCATTERING dimensional semiconductor system [44,45], the exponential INVOLVING DIFFERENT LANDAU LEVELS factor in Eq. (A2) does not depend on the magnetic field. In- stead, it has a nontrivial dependence on the difference between The probability of two-phonon scattering through a virtual the LL numbers of the two involved levels and becomes nonva- intermediate state is calculated using Fermi’s golden rule as nishing for very high adjacent LLs. However, these high levels given below: are not relevant for the high-temperature quantum Hall effect, which is the subject of our interest; whereas, for Ni = 0 and 2π LA/TA 2 2 2 Ni ,Nf − + 2 Nf = 1, Nf |Vˆq |Ni ∼exp(−v /2s ) ≈ exp(−50 /2) Wk →k = M  (q ,q ) nq− (nq+ + 1)(Ei ,Ef ), F y y  ky ,ky leading to a complete suppression of inter-LL transitions.

195431-5 A. M. ALEXEEV, R. R. HARTMANN, AND M. E. PORTNOI PHYSICAL REVIEW B 92, 195431 (2015)

[1] K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, [23] P. G. Klemens and D. F. Pedraza, Carbon 32, 735 (1994). 494 (1980). [24] E. Mariani and F. von Oppen, Phys. Rev. Lett. 100, 076801 [2] B. Jeckelmann, B. Jeanneret, and D. Inglis, Phys. Rev. B 55, (2008). 13124 (1997). [25] E. V. Castro, H. Ochoa, M. I. Katsnelson, R. V. Gorbachev, [3] B. Jeckelmann and B. Jeanneret, Rep. Prog. Phys. 64, 1603 D. C. Elias, K. S. Novoselov, A. K. Geim, and F. Guinea, Phys. (2001). Rev. Lett. 105, 266601 (2010). [4] K. S. Novoselov, A. K. Geim, S. V.Morozov, D. Jiang, Y.Zhang, [26] L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, Taniguchi, K. Watanabe, L. M. Campos, D. A. Muller, J. Guo, 666 (2004). P. Kim, J. Hone, K. L. Shepard, and C. R. Dean, Science 342, [5] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. 614 (2013). Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, [27] R. Geick and C. H. Perry, Phys. Rev. 146, 543 (1966). Nature (London) 438, 197 (2005). [28] H.-c. Hwang and J. H. Henkel, Phys. Rev. B 17, 4100 (1978). [6] Y.Zhang,Y.-W.Tan,H.L.Stormer,andP.Kim,Nature (London) [29] M. Lazzeri, S. Piscanec, F. Mauri, A. C. Ferrari, and J. 438, 201 (2005). Robertson, Phys.Rev.B73, 155426 (2006). [7] J. W. McClure, Phys. Rev. 104, 666 (1956). [30] H. Suzuura and T. Ando, J. Phys. Soc. Jpn. 77, 044703 (2008). [8] A. K. Geim and K. S. Novoselov, Nat. Mater. 6, 183 (2007). [31] M. O. Goerbig, J.-N. Fuchs, K. Kechedzhi, and V.I. Falko, Phys. [9] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, Rev. Lett. 99, 087402 (2007). and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009). [32] N. Mori and T. Ando, J. Phys. Soc. Jpn. 80, 044706 (2011). [10] K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. [33] Y. Kim, J. M. Poumirol, A. Lombardo, N. G. Kalugin, T. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and Georgiou, Y. J. Kim, K. S. Novoselov, A. C. Ferrari, J. Kono, A. K. Geim, Science 315, 1379 (2007). O. Kashuba, V. I. Falko, and D. Smirnov, Phys. Rev. Lett. 110, [11] V. S. Titeika, Ann. Phys. Leipzig 22, 129 (1935). 227402 (2013). [12] A. J. M. Giesbers, G. Rietveld, E. Houtzager, U. Zeitler, R. Yang, [34] A. J. M. Giesbers, U. Zeitler, M. I. Katsnelson, L. A. K. S. Novoselov, A. K. Geim, and J. C. Maan, Appl. Phys. Lett. Ponomarenko, T. M. Mohiuddin, and J. C. Maan, Phys. Rev. 93, 222109 (2008). Lett. 99, 206803 (2007). [13] A. Tzalenchuk, S. Lara-Avila, A. Kalaboukhov, S. Paolillo, M. [35] A. L. C. Pereira, C. H. Lewenkopf, and E. R. Mucciolo, Phys. Syvaj¨ arvi,¨ R. Yakimova, O. Kazakova, T. J. B. M. Janssen, V. Rev. B 84, 165406 (2011). Falko, and S. Kubatkin, Nat. Nanotechnol. 5, 186 (2010). [36] Yu. A. Bychkov, S. V. Iordanskii, and G. M. Eliashberg, Pis’ma [14] W. Poirier and F. Schopfer, Nat. Nanotechnol. 5, 171 (2010). Zh. Eksp. Teor. Fiz. 34, 496 (1981) [JETP Lett. 34, 473 (1981)]. [15] J. A. Alexander-Webber, A. M. R. Baker, T. J. B. M. Janssen, [37] A. P. Dmitriev and V. Y. Kachorovskii, Phys. Rev. B 52, 5743 A. Tzalenchuk, S. Lara-Avila, S. Kubatkin, R. Yakimova, B. A. (1995). Piot, D. K. Maude, and R. J. Nicholas, Phys. Rev. Lett. 111, [38] H. Suzuura and T. Ando, Phys. Rev. B 65, 235412 (2002). 096601 (2013). [39] E. Mariani and F. von Oppen, Phys.Rev.B82, 195403 (2010). [16] D. G. Polyakov and B. I. Shklovskii, Phys. Rev. Lett. 73, 1150 [40] G. Pennington and N. Goldsman, Phys. Rev. B 68, 045426 (1994). (2003). [17] D. G. Polyakov and B. I. Shklovskii, Phys. Rev. Lett. 74, 150 [41] J. H. Chen, C. Jang, S. Xiao, M. Ishigami, and M. S. Fuhrer, (1995). Nat. Nanotechnol. 3, 206 (2008). [18] V. M. Apalkov and M. E. Portnoi, Phys.Rev.B66, 121303 [42]K.I.Bolotin,K.J.Sikes,J.Hone,H.L.Stormer,andP.Kim, (2002). Phys. Rev. Lett. 101, 096802 (2008). [19] V. M. Apalkov and M. E. Portnoi, Phys.Rev.B65, 125310 [43] R. G. Clark, J. R. Mallett, A. Usher, A. M. Suckling, R. J. (2002). Nicholas, S. R. Haynes, Y. Journaux, J. J. Harris, and C. T. [20] V. M. Apalkov and M. E. Portnoi, Physica E 15, 202 (2002). Foxon, Surf. Sci. 196, 219 (1988). [21] V. N. Golovach and M. E. Portnoi, Phys. Rev. B 74, 085321 [44] M. Sh. Erukhimov, Fiz. Tekh. Poluprovodn. 3, 194 (1969) (2006). [Sov. Phys. Semicond. 3, 162 (1969)]. [22] S. Ono and K. Sugihara, J. Phys. Soc. Jpn. 21, 861 (1966); [45] V. V. Korneev, Fiz. Tverd. Tela (Leningrad) 19, 357 (1977) K. Sugihara, Phys.Rev.B28, 2157 (1983). [Sov. Phys. State 19, 205 (1977)].

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