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arXiv:1307.4008

v2

[cond-mat.mes

-hall]7Sep2013

Magneto-optical conductivity of graphene on polar substrates

Benedikt Scharf,1 Vasili Perebeinos,2 Jaroslav Fabian,3 and Igor Zutic1

1Department of Physics, University at Buffalo, State University of New York, Buffalo, NY 14260, USA2IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA

3Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany(Dated: September 10, 2013)

We theoretically study the effect of polar substrates on the magneto-optical conductivity of dopedmonolayer graphene, where we particularly focus on the role played by surface polar phonons (SPPs).Our calculations suggest that polaronic shifts of the intra- and interband absorption peaks can besignificantly larger for substrates with strong electron-SPP coupling than those in graphene onnon-polar substrates, where only intrinsic graphene optical phonons with much higher energiescontribute. Electron-phonon scattering and phonon-assisted transitions are, moreover, found toresult in a loss of spectral weight at the absorption peaks. The strength of these processes isstrongly temperature-dependent and with increasing temperatures the magneto-optical conductivitybecomes increasingly affected by polar substrates, most noticeably in polar substrates with smallSPP energies such as HfO2. The inclusion of a Landau level-dependent scattering rate to accountfor Coulomb impurity scattering does not alter this qualitative picture, but can play an importantrole in determining the lineshape of the absorption peaks, especially at low temperatures, whereimpurity scattering dominates.

PACS numbers: 81.05.ue,78.67.Wj,63.22.Rc,72.10.Di

Keywords: graphene, magnetic field, phonons, substrate, optical conductivity, magneto-optical conductivity,

spintronics

I. INTRODUCTION

One of the main reasons for the tremendous in-terest shown in graphene during the past decade isthat its excellent transport and optical properties13

make it an attractive candidate for possible applicationsin nanoscale electronics and optoelectronics.46 Recentbreakthroughs714 in graphene indicate that it may alsobe particularly suitable for spintronics.1519

In addition to its possible technological applications,graphene is also highly appealing for fundamental re-search: Its low-energy excitations can be described bya 2D Dirac-like Hamiltonian of massless fermions withan effective speed of light vF 108 cm/s, which essen-tially allows one to study quantum electrodynamics in(2+1) dimensions by studying the electronic propertiesof graphene.1 The Dirac-like behavior of electrons nearthe K and K points in graphene is also reflected in theirresponse to a magnetic field. If a perpendicular magneticfield is applied to the graphene plane, the linear disper-sion in the vicinity of those points evolves into a dis-crete spectrum of non-equidistant Landau levels (LLs),which gives rise to an unconventional, half-integer quan-

tum Hall effect.20,21

Likewise, graphene subject to a magnetic field exhibitspeculiar optical properties, conveniently described by theso-called magneto-optical conductivity, the optical con-ductivity in the presence of a magnetic field: In contrastto conventional two-dimensional electron gases, whichonly exhibit one pronounced optical absorption peak cen-tered around the cyclotron resonance frequency, a se-quence of distinct optical absorption peaks can be exper-imentally observed in monolayer graphene2230 due to its

non-equidistant LL spectrum.31,32 Those peaks broadlycorrespond to optical transitions between different LLsn, which must satisfy Paulis principle [see Figs. 1 (a)and (b)].

The interaction between electrons and phonons modi-fies this simple single-particle picture in the following way[see Fig. 1 (c)]: (i) Electrons moving through the crystallattice generate a polarization field, which results in theemergence of polarons, quasiparticles describing an elec-tron and its accompanying polarization field. The forma-

tion of polarons modifies the individual LLs giving rise todressed (and even split) LLs, which in turn leads to shiftsof the absorption peaks observed in the magneto-opticalconductivity. (ii) During optical transitions phonons canbe absorbed or emitted, which results in the appearanceof phonon sidepeaks in the absorption spectrum. Botheffects have been discussed in impressive detail in Ref. 33for the case of graphene. In addition, the formation ofnew electron-phonon bound statesboth at zero34 andfinite35 magnetic fieldfurther refines this picture.

If graphene is situated on a polar substrate, elec-trons couple not only to intrinsic graphene phonons,but also to surface polar phonons (SPPs), that is, sur-face phonons of polar substrates which interact withthe electrons in graphene via the electric fields thosephonons generate [see Fig. 1 (d)]. Those phonons canhave a profound impact on transport or optical proper-ties of graphene: Surface polar phonons have, for exam-ple, been argued to be responsible for current saturationin graphene3641 and to affect the carrier mobility,4244

which can also be influenced by the emergence of interfa-cial plasmon-phonon modes due to the coupling betweenSPPs and graphene plasmons,45,46 the spin-relaxation,47

the optical absorption,48 the renormalization of the Fermi

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FIG. 2: (Color online) (a) Feynman diagram correspondingto the lowest-order self-energy. Solid and wiggly lines denoteelectron and phonon propagators, respectively. Equation (6)is obtained by averaging over the electron-phonon couplingmatrix elements. (b) Schematic view of the frequency depen-

dence of the phonon spectral function 2F() for a typicalpolar substrate.

interaction is described by the self-energy. For an elec-tron with spin quantum number s the retarded electronicself-energy due to the electron-phonon coupling caninthe lowest order [see Fig. 2 (a)]be approximated by

Rph,s () =

0

d2F()

dNs( + )

nBE () + nFD () + i0+ +

nBE () + nFD ()

+ + i0+

(6)following Migdals approach and averaging over theelectron-phonon coupling matrix elements at the Fermisurface.33,5357 Here 2F() denotes the Eliashbergelectron-phonon spectral function normalized to the elec-tronic density of states (DOS) per spin at the chemicalpotential = (T) and Ns() the electronic DOS for spins, while nFD/BE() = 1/[exp() 1], with = 1/(kBT),the temperature T, and the Boltzmann constant kB,are the Fermi-Dirac and Bose-Einstein distribution func-tions, respectively.79

The spectral function 2F() contains the averaged(and spin-independent) coupling of electrons in grapheneto the optical and SPPs. In the following, we will assumethat the phonon frequencies and the electron-phononcoupling are not significantly affected by the magneticfield and that the averaged coupling can be reasonablywell described by its zero-field value. Then the dominantelectron-optical phonon coupling is that to longitudinal-optical (LO) and transverse-optical (TO) phonons at the point and to the TO phonon at the K point.5863

The dispersion of LO and TO phonons near the pointcan be approximated by the constant energy 197meV, that of TO phonons near the K and K points

by K 157 meV. Moreover, there are typically twosurface optical (SO) phonons in polar substrates thatinteract with the electrons in graphene and whose dis-persion can again be approximated by substrate-specific,constant frequencies SO1 and SO2 summarized in Ta-ble I.

The resulting phonon spectral function in our modelthen reads as

2F() =

A( ) , (7)

which consists of four contributions [see Fig. 2 (b)]: thecombined contribution from both phonon modes witha total coupling5863 A = D

2A/(2Mc), the contribu-

tion from the KTO phonon mode with a coupling twice aslarge as that of each phonon at the point,5860 AK =D2

A/(2McK), and the contribution from the = SO1

and = SO2 modes described by the coupling39,43

A = (e2/2)20 dF

2

[q()] e2q()z0 (1 + cos ) /q()

with q() ||2 2cos /(vF). Here Mc is the car-bon mass, D

11.2 eV/A the strength of the electron-

phonon coupling,38 A = 33a2/2 the area of thegraphene unit cell, a 1.42 A the distance between twocarbon atoms, z0 3.5 A the van der Waals distance be-tween the graphene sheet and the substrate, and F2

(q)

the Frohlich coupling given by48,6466

F2SO1/2

(q) =SO1/2

2

1

i/ + (q) 1

0/i + (q)

(8)

with the optical, intermediate, and static permittivi-ties , i, and 0 of the substrate as well as thestatic, low temperature dielectric function (q) = 1 +2e2g (q, = 0) /(q), where is the background di-

electric constant and g (q, ) the polarization functionof graphene. We find that, for the relatively high dop-ing and the magnetic fields considered in this work, theresults are not noticeably affected by whether the polar-ization function calculated for zero67,68 or finite69 mag-netic field is used. In our model the dielectric mediumabove the graphene plane is air, and for simplicity weuse the average = (1+ 0)/2 for the background dielec-tric constant. The dielectric function (q) accounts forthe screening of the Coulomb interaction in the graphenesheet above the polar substrate.

Moreover, the spin-polarized DOS for the spectrumgiven in Eq. (1) is needed to calculate the self-energy (6),and we also include the effect of scattering (other thanscattering with optical or SPPs, such as scattering atcharged impurities) on a phenomenological level by in-troducing the constant scattering rate . The self-energyobtained in this way is then inserted into Kubo formulasfor the magneto-optical conductivities xx() = yy()and xy() = yx(), and we refer to the Appendix Afor further details on this procedure. Here we are pri-marily interested in the absorption, that is, essentiallyin Re[xx()]. In the following, we will thus computethe real part of xx() as well as the imaginary part

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Al2O3a h-BNb HfO2

c SiCd SiO2e

0 12.53 5.09 22.0 9.7 3.90

i 7.27 4.575 6.58 - 3.36

3.20 4.10 5.03 6.5 2.40

SO1 [meV] 56.1 101.7 21.6 116.0 58.9

SO2 [meV] 110.1 195.7 54.2 - 156.4

F2SO1 [meV] 0.420 0.258 0.304 0.735 0.237

F2

SO2 [meV] 2.053 0.520 0.293 - 1.612

TABLE I: Optical, intermediate, and static permittivities aswell as frequencies and the strengths of the bare Frohlich cou-plings [(q) set to 1 in Eq. (8)] for the SPP scattering on thesubstrates Al2O3, hexagonal BN, HfO2 SiC, and SiO2 (takenfrom Ref. 48).a Refs. 45,65, b Refs. 38,45,70, c Refs. 38,45,65, d Refs. 38,71,e Ref. 38, which uses averages of values from Refs. 39,43,65.

0

5

10

15

Re[

x

x()]/

0

no phonons

nonpolar

BN

SiO2SiCHfO

2

Al2O

3

0 25 50 75h_

[meV]

0

5

10

15

20

25

Re[

+()]/

0

-15

-10

-5

0

Im[

x

y()]/

0

0 50 10025 75h_

[meV]

0

3

4

6

Re[

-()]/

0

0 200 400 600

h_

[meV]

0

0.5

1

Re[

xx

()]/

0

T=300 K=0.2 eVB=10 T=5 meV

(a) (b)

(c) (d)

(e)

phonon-assistedtransitions

n=3 n=4B=0

FIG. 3: (Color online) Calculated frequency dependence ofthe magneto-optical conductivity of graphene on several dif-ferent substrates at room temperature: (a) Re [xx()], (b)Im [xy()], (c) Re[+()], (d) Re [()]. For comparison,Re [xx()] for zero magnetic field is presented in (e).

of xy() following the procedure outlined in the Ap-pendix A. Those calculations are performed for severaldifferent substrates with the corresponding parameterssummarized in Table I and the results discussed in thefollowing section.

III. RESULTS

A. General behavior

To give a general impression of the effect the electron-phonon interaction has on the magneto-optical conduc-tivity and of the quantitative differences between sub-strates, Figs. 3 (a) and (b) show the conductivitiesRe [xx()] and Im[xy()] as fractions of the univer-sal ac conductivity 0 = e2/(4) at room temperature,B = 10 T, a fixed chemical potential = 0.2 eV,and = 5 meV for graphene on several different sub-

-4

-2

0

2

4

6

8

Im[

xx

()]/

0

nonpolar

SiO2

HfO2

Al2O

3

0 25 50 75h_

[meV]

-10

-5

0

5

10

Im[

+

()]/

0

-8

-6

-4

-2

0

2

4

Re[

xy

()]/

0

0 50 10025 75h_

[meV]

0

3

4

6

8

Im[

-

()]/

0

T=300 K=0.2 eVB=10 T=5 meV

(a) (b)

(c)

(d)

FIG. 4: (Color online) Calculated frequency dependence ofthe magneto-optical conductivity of graphene on several dif-ferent substrates at room temperature: (a) Im [xx()], (b)Re [xy()], (c) Im [+()], (d) Im [()].

strates: Al2O3, hexagonal BN, HfO2, SiC, SiO2, and anon-polar substrate (where only the intrinsic graphene

optical phonons affect the conductivity). For compari-son, Re[xx()] in the absence of any phonons is alsopresented in Fig. 3 (a). Moreover, the real parts of() = xx() ixy(), describing the absorption ofright- and left-handed circularly polarized light, respec-tively, are presented in Figs. 3 (c) and (d).

One can clearly see a pronounced absorption peak inFigs. 3 (a)-(c), which corresponds to intraband transi-tions, the main contribution to which arises from n =3 4 transitions. The interaction between electrons andphonons leads to a modification of the LLs (see Sec. I),which is reflected in a shift of the position of the intra-band absorption peak to lower energies compared to the

case where no phonons are considered. The magnitude ofthis polaronic shift is smallest for non-polar substrates,followed by that of the polar substrates BN and SiO2.For substrates with a strong electron-SPP coupling suchas Al2O3 or with low SPP frequencies such as HfO2 themagnitude of the shift is larger. In the case of HfO2one can furthermore also discern several sidepeaks corre-sponding to phonon-assisted transitions [see Fig. 3 (c)].In addition to the absorption presented in Fig. 3, the cor-responding refractive components of the magneto-opticalconductivities, which are also of experimental interest,are shown in Fig. 4.

B. Temperature dependence

The pronounced difference at T = 300 K describedabove between HfO2 and other substrates is a conse-quence of the strong temperature dependence introducedby the SPPs. This is illustrated in Figs. 5 and 6, whichcompare the optical absorption Re [xx()] of SiO2 withthat of HfO2 at different temperatures. As can be seenin Fig. 5, which focuses on the intraband absorption, theintraband absorption peak (here: n = 3 4) becomes

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5

0

5

10

15

e

xx

)0

0 200 400

h_

[meV]

0

0.5

1

I()/I0

0 20 40 60 80 100h_

[meV]

0

5

10

15

e

xx

)0 T=1 K

T=50 K

T=100 KT=200 KT=300 K 0 200 400

h_

[meV]

0

0.5

1

I()/I0

SiO2

(d)

(c)

HfO2

(a)

(b)

=0.2 eVB=10 T=5 meV

FIG. 5: (Color online) Calculated frequency dependence ofRe [xx()] and of the corresponding spectral weight I() forgraphene on (a,c) SiO2 and (b,d) HfO2 substrates at differenttemperatures T.

0.1

1

1020

Re[

xx

()]/

0

0 100 200 300 400 500h_

[meV]

0.1

1

10

Re[

xx

()]/

0 T=1 KT=100 KT=200 KT=300 K

SiO2

HfO2

(a)

(b)

=0.2 eVB=10 T=5 meV

n=3 n=4

n=-2 n=3

n=-5/6 n=6/5

n=-4/5 n=5/4

n=0 n=1

n=-1 n=2

n=-3 n=4phonon-assistedtransitions

n=3 n=4

phonon-assistedtransitions

FIG. 6: (Color online) Calculated frequency dependence ofRe [xx()] for graphene on (a) SiO2 and (b) HfO2 substratesat different temperatures T. The different intra- and inter-band transitions are labeled in the case of SiO2.

broader and loses spectral weight with increasing tem-perature.

There are two reasons for this loss of spectral weight,which happens for both SiO2 and HfO2: (i) As tempera-ture increases, LLs close below the Fermi level are ther-mally depopulated, while LLs above the Fermi level arepopulated. Hence, new transitions (satisfying the selec-tion rules) are possible into the now no longer fully oc-cupied LLs slightly below or from the no longer com-pletely empty LLs slightly above . In the case of in-traband transitions the new transitions are close to themain absorption peak, which leads to a broadening of theintraband absorption peak. (ii) With increasing temper-ature the number of phonons and thus the probability ofelectron-phonon scattering (leading also to a broadeningof the absorption peaks) as well as of phonon-assistedtransitions increases.

0

0.25

0.5

0.75

1

N(h_)/N

0[eV]

B=10 TB=5 TB=1 TB=0

10

5

-Im[

ph,

R

()][meV]

15

-1 -0.5 0 0.5 1h_

[eV]

-20

-10

0

10

Re[

ph,

R

()][meV]

20

(a)

(b)

(c)

FIG. 7: (Color online) (a) Energy dependence of the spin-unpolarized DOS N() for several different magnetic fields,where N0 = 2/(

2v2F). At high energies where the spacingbetween LLs becomes comparable or smaller than the life-time broadening, N() converges to the DOS of zero mag-netic field. (b) Imaginary and (c) real parts of the self-energyRph, () at T = 0 and = 0.2 eV if only intrinsic grapheneoptical phonons at the point are taken into account.

Comparing SiO2 and HfO2, one can discern that themain mechanism for the behavior of SiO2 (up to roomtemperature) is (i), while HfO2 is also strongly affectedby (ii) for temperatures above approximately 200 K. Fig-ures 5 (c) and (d) show the spectral weight

I() =

0

dRe [xx()] (9)

normalized to the non-interacting spectral weight I0 =2||0/ for of SiO2 and HfO2, respectively. They illus-trate that the spectral weight lost at the main peak ismainly redistributed in the region between the intrabandpeak and the first interband peaks.

The mechanisms (i) and (ii) detailed above for the in-traband transitions also apply to the interband transi-tions: In the case of SiO2 [Fig. 6 (a)] one can clearly ob-serve the emergence of new pronounced interband tran-sitions due to the thermal depopulation of LLs belowthe Fermi level (n = 0 1, 1 2, 2 3) at highertemperatures, while phonon-assisted transitions also leadto an enhanced midgap absorption, that is, the absorp-tion in the gap between intra- and interband transitions.For graphene on a HfO2 substrate [Fig. 6 (b)] one cannoteven observe pronounced interband transition peaks any-more at higher temperatures due to increased electron-phonon scattering. Likewise, the midgap absorption inHfO2 rapidly increases from approximately 20% at lowtemperatures to about 40% at room temperature due toadditional phonon-assisted transitions, whereas the in-crease of the midgap absorption from about 12 13% to15 16% is much more modest in SiO2.

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6

25

0

-Im[

ph,

R

()][meV]

50

T=1 KT=300 K

-100

-50

0

50

Re[

ph,

R

()][meV]

-500 -250 0 250h_

[meV]

100

50

-Im[

ph,

R

()][meV] 0

-500 -250 0 250 500h_

[meV]

-200

-100

0

100

Re[

ph,

R

()][meV]

SiO2

(d)

(b)

(c)

(a)

HfO2

SiO2

HfO2

=0.2 eVB=10 T=5 meV

FIG. 8: (Color online) Calculated energy dependence of theself-energy Rph, () for graphene on (a,b) SiO2 and (c,d)HfO2 substrates at T = 1 K and T = 300 K.

C. Temperature dependence of the self-energy

While mechanism (i) is a consequence of Fermi statis-

tics and described by the factor nFD () nFD ()in Eqs. (A4) and (A5), the effect of phonons, that is,mechanism (ii), is described by the self-energy (6). Itis this self-energy due to phonons, R

ph,s(), from which

the qualitative difference between SiO2 and HfO2 at roomtemperature arises.

Since the self-energy is calculated from the spin-polarized DOS, which in turn can be expressed by thespin-unpolarized DOS N(), Fig. 7 (a) shows N() atseveral different magnetic fields for the readers orienta-tion. The imaginary part of R

ph,s () describing electron-phonon scattering is essentially given by contributionsfrom the DOS centered around +sgBB/2 andmultiplied by the thermal factor nBE(

) + nFD(

) for each individual phonon branch . At T = 0these thermal factors are just Heaviside step functionsas nBE() vanishes, resulting in a gap of width 2centered around = 0 for each phonon branch. Thisis illustrated in Fig. 7 (b), where the contribution from = to Im

Rph, ()

at T = 0 and = 0.2 eV is

presented, with the corresponding real part displayed inFig. 7 (c). Since Zeeman splitting for g = 2 at B = 10 Tis small compared to the LL broadening, the spin-degreeof freedom does not affect the self-energy noticeably andthe self-energies of spin-up and spin-down electrons arealmost identical.

The total self-energy due to phonons is then given bythe sum of the individual contributions from each branch and shown in Fig. 8 for spin-up electrons at B = 10T, = 0.2 eV, and T = 1 K as well T = 300 K. Atfinite temperatures the discontinuities at = dis-appear and are thermally broadened with increasing tem-perature. Moreover, there is now also a finite contribu-tion from nBE() which grows with increasing temper-ature. For phonons with relatively high frequencies ,such as in Figs. 8 (a) and (b), this contribution results inthe emergence of peaks inside the region [, ], but

-0.2

0

0.2

0.4

Re[

xxd

if(

)]/

0

0 20 40 60 80 100h_

[meV]

-0.2

0

0.2

0.4

Re[

xxd

if (

)]/

0

T=1 KT=50 KT=100 KT=300 K

SiO2

HfO2

(a)

(b)

=0.2 eVB=10 T=5 meV

(n=2 n=3)

(n=3 n=4)

(n=2 n=3)

(n=3 n=4)

FIG. 9: (Color online) Calculated frequency dependence ofthe spin-polarized magneto-optical conductivity Re

difxx ()

for graphene on (a) SiO2 and (b) HfO2 substrates at differenttemperatures T.

does not significantly affect the high-frequency behav-

ior at room temperature dominated by nFD(||) 1as nBE() 1. In HfO2 the situation is quite dif-ferent at room temperature, which is entirely due tothe contribution from the SO1 phonon: Because of thelow frequency SO1 = 21.6 meV, which corresponds tonBE(SO1) 0.77, the self-energy at high frequencies isstrongly enhanced compared to its low-temperature val-ues [see Figs. 8 (c) and (d)]. As a consequence of the sig-nificantly enhanced imaginary part of the self-energy thetransition peaks in the optical conductivity of grapheneon HfO2 are strongly broadened, while the increased realpart of the self-energy results in a growing polaronic shiftto lower energies [compare Fig. 5 (b)].

The behavior of the self-energy of SiO2 and HfO2 de-scribed above reflects the fact that at room temperaturethere are many more SO1 phonons available for scatter-ing with electrons or phonon-assisted optical transitionsin graphene on HfO2 than there are in graphene on SiO2.In this way, the stronger effect of polar substrates withlow SPP frequencies, such as HfO2, compared to polarsubstrates with higher SPP frequencies or non-polar sub-strates can be understood intuitively. Finally, we notethat, if the temperature was increased further, grapheneon SiO2 or other polar substrates would eventually alsoshow a behavior similar to that of HfO2.

D. Spin-polarized absorption

As mentioned above, the spin-degree of freedom doesnot play a crucial role for magneto-optical absorptionexperiments because Zeeman splitting is small at experi-mentally relevant magnetic fields, a fact not only reflectedin xx(), but also in the self-energy seen above. Nev-ertheless, there can be some spin effectsalbeit small,namely at a level of 1% of the total absorptionasdemonstrated by Fig. 9. There the difference between

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7

0

4

8

12

16

20

e

xx

0

0

no phonons

SiO2

HfO2

0.05 0.1 0.15 0.21/B [1/T]

0

1

2

3

Re[

xx

(0

)]/

0

(a)

(b)

=5 meVT=1 K=0.2 eV

h_

0=20 meV

h_

0=500 meV

FIG. 10: (Color online) Calculated magnetic field dependenceof Re [xx(0)] for graphene on SiO2 and HfO2 substrates at(a) 0 = 20 meV and (b) 0 = 500 meV. For comparison,Re [xx(0)] in the absence of any phonons is also presented.

the (intraband) absorption by spin-up and by spin-downelectrons is shown, that is, Re [difxx ()] = Re

xx()

Rexx()

, where /xx () is given by Eq. (A4) if the

spin-degree of freedom is fixed to s = / and the sum-mation over s is omitted. The substrates are again cho-sen to be SiO2 and HfO2 with = 0.2 eV, B = 10 T, = 5 meV, and several different temperatures, that is,the same choice of parameters as in Figs. 3-6.

As can be seen in Fig. 9, there is a slight differencebetween the amount of absorption by spin-up and spin-down electrons in the intraband transition peak seen inFigs. 3-6. This is primarily a consequence of the factthat the n = 3 LL for spin-up is extremely close to theFermi level = 0.2 eV and (due to disorder) slightlydepopulated. Thus, even at low temperatures there is aslightly enhanced probability of transitions n = 2 3for spin-up electrons compared to spin-down electrons.Consequently, the probability of transitions n = 3 4at low temperatures is slightly higher for spin-down elec-trons than for spin-up electrons. The most pronouncedpeaks in Fig. 9 can be attributed to this imbalance be-tween optical transitions. In addition to this essentiallysingle-particle effect there is also a substructure arisingfrom the electron-phonon interaction as can particularlybe seen in Fig. 9 (b). At higher temperatures the spin-polarized conductivity difxx () quickly decreases as kBTexceeds the value of the Zeeman splitting.

E. Dependence on the magnetic field and chemical

potential

Having investigated the dependence of the absorptionon temperature, we now briefly discuss how changing themagnetic field or chemical potential affects the magneto-optical conductivity. Figure 10 depicts Re [xx(0)] at0 = 20 meV and 0 = 500 meV as a function of

0.1

1

1020

Re[

xx

()]/

0

B=5 TB=10 T

B=20 T

0 100 200 300 400 500h_

[eV]

0.1

1

10

Re[

xx(

)]/

0

(a)

(b)

SiO2

HfO2

=5 meVT=100 K=0.2 eV

FIG. 11: (Color online) Calculated frequency dependence ofRe [xx()] for graphene on (a) SiO2 and (b) HfO2 substrateswith different magnetic fields B.

the inverse magnetic field for graphene on SiO2 and on

HfO2 with = 0.2 eV, T = 1 K, and = 5 meV.For comparison, we also show the absorption obtainedfrom the single-particle picture, that is, in the absenceof any phonons. As before, the qualitative behavior canbe understood from the single-particle picture: With in-creasing B the spacing between LLs increases and dif-ferent intraband and interband transitionsgoverned bythe selection rules and the position of the Fermi levelwith respect to the LLsbecome possible. This leads tothe behavior depicted in Fig. 10 (a) for intraband tran-sitions monitored at low frequencies such as 0 = 20meV and the oscillatory behavior of interband transitionsmonitored at frequencies above 2|| and illustrated inFig. 10 (b). One can also clearly observe a polaronic shiftdue to the interaction between electrons and phonons forboth intra- and interband transitions. Moreover, with de-creasing magnetic field the amplitudes of the interbandtransition oscillations around 0 in Fig. 10 (b) decreaseas the ratio between the LL spacing and the broadening is diminished.

The statements given above are corroborated inFig. 11, which displays the frequency dependence ofRe [xx()] for graphene on SiO2 and HfO2 substratesat T = 100 K, = 0.2 eV, = 5 meV, and differ-ent magnetic fields. The major trends that can be seenhere are that with increasing magnetic field the intra-band transition peak moves to higher energies, while theamplitudes of interband transition peaks and of phonon-assisted peaks increase. A somewhat opposite behaviorcan be observed in Fig. 12, which shows the dependenceof Re [xx()] on the chemical potential at T = 100 K,B = 10 T, = 5 meV. As the chemical potential in-creases, the intraband transition peak moves to lower en-ergies and the amplitudes of the oscillations for interbandtransitions decrease. Furthermore, the onset of interbandtransitions in the vicinity of 2|| can also be clearly fol-lowed in Fig. 12.

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0.1

1

10

30

Re[

xx

()]/

0

=200 meV=300 meV

=400 meV

0 200 400 600 800h_

[eV]

0.1

1

10

Re[

xx(

)]/

0

(a)

(b)

SiO2

HfO2

=5 meVT=100 KB=10 T

FIG. 12: (Color online) Calculated frequency dependence ofRe [xx()] for graphene on (a) SiO2 and (b) HfO2 substratesat different chemical potentials .

Both the behavior of the intraband transition peakwith increasing magnetic field and with increasing chem-ical potential can be qualitatively explained in the single-particle picture as originating from the LL spectrum inthe vicinity of the Fermi level: With increasing magneticfield the LL spacing increases giving rise to a higher en-ergy of the intraband transition. On the other hand, fora fixed magnetic field the LL spacing near the Fermi leveldecreases if the absolute value of the chemical potentialis increased and thus situated in a denser region of theLL spectrum. Consequently, the energy of the intrabandtransition is decreased. Likewise, the decrease of the am-plitudes of the interband peaks with increasing chemicalpotential , decreasing magnetic fields B, or high pho-ton energies can be interpreted as arising from optical

transitions between increasingly denser regions of the LLspectrum, where the energy difference between differenttransitions is small compared to the broadening due toscattering.

While the main trends in Figs. 11 and 12 can thusbe readily understood from the single-particle picture,the electron-phonon coupling has still a profound im-pact and is needed to explain features such as polaronicshifts, the emergence of phonon sidebands, and enhancedbroadening of the transition peaks, particularly strikingat high photon energies or higher temperatures, dueto electron-phonon scattering.

F. Scattering at Coulomb impurities

So far, our model has accounted for scattering otherthan electron-optical phonon/SPP scattering by the ad-dition of a constant scattering rate in the total self-energy A2. However, in a more realistic model this scat-tering rate would also be dependent on the energy andLL of the state considered. To model this behavior atleast partially, we assume that scattering is dominated

0

50

100

150

Re[

xx

()]/

0 suspended

SiO2

HfO2

Al2O

3

0 20 40 60 80 100h_

[eV]

0

5

10

15

Re[

xx

()]/

0

(a)

(b)

=0.2 eVB=10 T

ni=51011

cm

-2

T=1 K

T=300 K

phonon-assistedtransition

n=3 n=4

FIG. 13: (Color online) Calculated frequency dependence ofRe [xx()] for graphene on several different substrates at (a)T = 1 K and (b) T = 300 K in the presence of Coulombimpurity scattering.

by scattering at charged impurities and replace by/[2(k = s(n))], where for simplicity we use the trans-

port scattering time (k) due to Coulomb impurities foran electron with energy k as calculated in Ref. 72 forB = 0 and evaluate it at the energy corresponding tothe LL considered. Since Coulomb impurity scatteringalso depends on the dielectric environment, the scatter-ing rate /[2(k = s(n))] is also affected by the choiceof the substrate.

This can be seen in Fig. 13, which shows Re[ xx()]at T = 1 K and T = 300 K for suspended graphene( = 1) and for graphene on several different substrateswith an impurity concentration of ni = 5 1011 cm2, = 0.2 eV, and B = 10 T. At low temperatures [seeFig. 13 (a)] the broadening of the intraband transition

peaks is mainly determined by the Coulomb impurityscattering and markedly different for different substrates.By comparing Fig. 13 (a) with the low-temperature be-havior in Fig. 5, one can also conclude that the actualintraband peak positions themselves are not noticeablyaffected by the inclusion of a LL-dependent scatteringrate. If the temperature is increased, electron-phononscattering becomes the dominant scattering mechanismand thus the quantitative differences between the re-sults obtained for constant and LL-dependent scatteringrates are no longer as pronounced as at low tempera-tures as can be deduced from a comparison of Figs. 3 (a)and 13 (b).

G. Comparison with experiments

As mentioned in Sec. I, several distinct absorptionpeaks can be seen in experiments.2229 The positions ofthose peaks, however, are not exactly where the single-particle picture predicts they are, and the experimentaldata is usually fitted to the single-particle picture usinga renormalized Fermi velocity. This procedure resultsin different values of the fitted Fermi velocity in different

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experiments and in the case of Refs. 23,28 even for differ-ent LL transitions. The experimental values for the fittedFermi velocity range from around c 1.03 106 m/s toc 1.12 106 m/s. Those values are higher than thebare Fermi velocity, indicating that the transition peaksare shifted to higher energies.

This shift to higher energies is attributed to theelectron-electron interaction, whichin contrast to a

standard two-dimensional electron gas73is argued toplay an important role in the LL spectroscopy ofgraphene.23,26,74,75 However, while electron-electron in-teraction predicts the correct sign of the shift (that is,to higher energies), it also predicts a very strong correc-tion of approximately 30%, not seen in experiments (seeabove).

As described in the previous sections as well as inRef. 33, the electron-phonon interaction also leads toa shift of the transition peaks, albeit to lower energies.Thus, it stands to reason that the strong effect due to theelectron-electron interaction is somewhat compensatedby the electron-SPP coupling. Most magneto-optical ex-

periments up until now have been conducted on SiO2or SiC substrates, where our calculations suggest a shiftto lower energies corresponding to a correction of ap-proximately 15 20% compared to the position of thebare transition peak at low temperatures, = 0.2 eV,and B = 10 T [see Fig. 13 (a)]. Experiments on sub-strates with strong electron-phonon coupling, such asAl2O3, or with low SPP frequencies, such as HfO2, couldfurther clarify the role played by SPPs. Our calcula-tions suggest that for such substrates the effect of theelectron-phonon coupling reducing the electron-electron-interaction-induced shift is even more pronounced andmight possibly even completely compensate for it.

To give a crude estimate of the combined impact ofthe electron-phonon and electron-electron interaction, wehave calculated the optical absorption at T = 1 K, =0.2 eV, and B = 18 T using the procedure outlined inSec. II, where we have replaced the Fermi velocity vF 106 m/s by a higher value of 1.30 106 m/s to reflectthe effect of the Coulomb interaction. In this way wefind that the positions of the intraband transition peaksfor graphene on SiO2 and SiC substrates are shifted tohigher energies compared to the bare case (that is, in theabsence of phonons and with a Fermi velocity ofvF 106m/s) and obtain fitted Fermi velocities of c 1.07 106 m/s for SiC and of c 1.11 106 m/s for SiO2.While these values are significantly closer to experimental

data, we stress that these values are only simple estimatesand that a more detailed theory is required to treat therenormalization of the Fermi velocity consistently.7678

Due to the small size of the Zeeman splitting for g = 2compared to the electron-phonon or electron-impurityscattering, there is only a weak spin dependence of themagneto-optical conductivity. However, substantiallylarger g-factors have been reported for hydrogenatedgraphene,12 making an enhancement of the spin effectsbriefly discussed in Sec. III D appear more feasible, al-

though in that case there is also enhanced scattering dueto the presence of hydrogen adatoms, somewhat compen-sating for the effect of an increased g-factor.

IV. CONCLUSIONS

In this work we have investigated how the magneto-optical conductivity of (doped) monolayer graphene is af-fected by the substrate the graphene layer is placed upon.Here our particular focus has been on the effects of SPPs.Our calculations suggest that polaronic shifts of the intra-and interband absorption peaks can be significantly en-hanced for substrates with strong electron-SPP couplingas compared to those for non-polar substrates, where onlyintrinsic graphene optical phonons contribute. Moreover,electron-phonon scattering and phonon-assisted transi-tions result in a broadening and loss of spectral weightat the transition peaks. The strength of these processesis strongly temperature-dependent and with high tem-peratures the magneto-optical conductivity becomes in-

creasingly affected by polar substrates. This is especiallytrue for polar substrates with small SPP energies such asHfO2, where many phonons are available for scatteringand phonon-assisted transitions and most of the spectralweight has been transferred away from the main absorp-tion peaks already at room temperature.

Furthermore, we have also briefly studied the effectof LL-dependent scattering rates modeling Coulomb im-purity scattering. While the qualitative picture of theimpact of optical and SPPs on the magneto-optical con-ductivity, outlined above, remains unaffected by the in-clusion of a LL-dependent broadening, it can play a pro-found role in determining the lineshape of the absorption

peaks, especially at low temperatures, where impurityscattering dominates.

Acknowledgments

We gratefully acknowledge John Cerne from the Uni-versity at Buffalo for stimulating discussions. This workwas supported by U.S. ONR, NSF-NRI NEB 2020, andSRC. J.F. acknowledges support from DFG Grant No.GRK 1570.

Appendix A: Total self-energy and Kubo formulasfor the magneto-optical conductivity

In order to calculate the self-energy (6) due to phonons,the spin-polarized DOS is needed. Using a constantGaussian broadening80 of width for each state to de-scribe scattering other than scattering with optical orSPPs, such as scattering at charged impurities, on a phe-nomenological level and employing Poissons summationformula, we can write the spin-polarized DOS for the

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spectrum given in Eq. (1) as Ns() = N(sgBB/2)/2,where

N( = 0) = 2||2v2

F

1+

2

m=1 exp

2ml2B

2v2F

2 cos

ml2B

2

2v2F

,

N( = 0) =4

2v2F

2

2v2

F/(22l2B)

tanh[2v2F/(22l2B)]

.

(A1)Thus, Eqs. (6)-(8) and (A1) allow us to calculate theelectronic self-energy due to phonons. The real part ofEq. (6) is related to the polaron formation, while its imag-inary part describes the scattering between electrons andphonons.

Finally, the effect of non-phonon-related scattering isalso included in the electronic self-energy by adding theconstant scattering rate . Then the total self-energy canbe obtained as

Rs () = i +

A2

dN( + sgBB/2)

nBE () + nFD () + i0+ +

nBE () + nFD ()

+ + i0+,

(A2)which can in turn be used to extract the electronicGreens function and the corresponding spectral function

An,s() = 2 Im

1

+ i0+ [s(n) + Rs ()] /

.

(A3)

The electronic spectral function (A3) can then be usedto calculate the magneto-optical conductivities as81

xx () =i0v2F

43l2B

s,n=0

ddnFD () nFD ()

+ + i0+

sn,n+1(, ) + sn+1,n(, )(A4)

and

xy () =0v

2F

43l2B

s,n=0

ddnFD () nFD ()

+ + i0+ sn,n+1(, ) sn+1,n(, ) .

(A5)In the derivation of the Kubo formulas (A4) and (A5)vertex corrections have been ignored. Moreover, both for-mulas include the universal ac conductivity 0 = e

2/(4)as well as the auxiliary function

sm,n(, ) = [Am,s() + Am,s()] [An,s() + An,s()] ,

(A6)

which reflects the fact that (in the dipole approximation)only spin-conserving optical transitions from a given LL|n| to LLs |n 1| are permitted. Here we are primar-ily interested in the absorption, that is, essentially inRe [xx()]. In the following, we will thus use Eqs. (A2)-

(A6) to calculate the real part ofxx() = yy() as wellas the imaginary part of xy() = yx(), where the integration can be preformed using the Dirac- func-tion arising from the denominator, while the remainingintegral is computed numerically.82 The imaginary partof xx() and the real part of xy(), determining therefractive index in the graphene plane, can then be de-termined using the Kramers-Kronig relations.

1 A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109(2009).

2 N. M. R. Peres, Rev. Mod. Phys. 82, 2673 (2010).3 S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Rev.

Mod. Phys. 83, 407 (2011).4 A. K. Geim, Science 324, 1530 (2009).5 P. Avouris, Nano Lett. 10, 4285 (2010).6 F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Nature

Phot. 4, 611 (2010).7 N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and

B. J. van Wees, Nature 448, 571 (2007).

8 S. Cho, Y.-F. Chen, and M. S. Fuhrer, Appl. Phys. Lett.91, 123105 (2007).

9 M. Nishioka and A. M. Goldman, Appl. Phys. Lett. 90,252505 (2007).

10 W. Han, W. H. Wang, K. Pi, K. M. McCreary, W. Bao,Y. Li, F. Miao, C. N. Lau, and R. K. Kawakami, Phys.Rev. Lett. 102, 137205 (2009).

11 W. Han, K. Pi, K. M. McCreary, Y. Li, J. J. I. Wong,A. G. Swartz, and R. K. Kawakami, Phys. Rev. Lett. 105,167202 (2010).

12 K. M. McCreary, A. G. Swartz, W. Han, J. Fabian, andR. K. Kawakami, Phys. Rev. Lett. 109, 186604 (2012).

7/29/2019 1307.4008v2.pdf

11/12

11

13 M. Gmitra, D. Kochan, and J. Fabian, Phys. Rev. Lett.110, 246602 (2013).

14 I. Neumann, M. V. Costache, G. Bridoux, J. F. Sierra, andS. O. Valenzuela, Appl. Phys. Lett. (in press) (2013).

15 I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.76, 323 (2004).

16 J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, andI. Zutic, Acta Phys. Slov. 57, 565 (2007).

17 C. Jozsa and B. J. van Wees, in Handbook of Spin Trans-

port and Magnetism, edited by E. Y. Tsymbal and I. Zutic(CRC Press, New York, 2011).

18 H. Dery, H. Wu, B. Ciftcioglu, M. Huang, Y. Song,R. Kawakami, J. Shi, I. Krivorotov, I. Zutic, and L. J.Sham, IEEE Trans. Electron. Dev 59, 259 (2012).

19 P. Seneor, B. Dlubak, M. B. Martin, A. Anane, H. Jaffres,and A. Fert, MRS Bull. 37, 1245 (2012).

20 V. P. Gusynin and S. G. Sharapov, Phys. Rev. Lett. 95,146801 (2005).

21 N. M. R. Peres, F. Guinea, and A. H. Castro Neto, Phys.Rev. B 73, 125411 (2006).

22 M. L. Sadowski, G. Martinez, M. Potemski, C. Berger, andW. A. de Heer, Phys. Rev. Lett. 97, 266405 (2006).

23 Z. Jiang, E. A. Henriksen, L. C. Tung, Y.-J. Wang, M. E.Schwartz, M. Y. Han, P. Kim, and H. L. Stormer, Phys.Rev. Lett. 98, 197403 (2007).

24 M. L. Sadowski, G. Martinez, M. Potemski, C. Berger, andW. A. de Heer, Solid State Commun. 143, 123 (2007).

25 R. S. Deacon, K.-C. Chuang, R. J. Nicholas, K. S.Novoselov, and A. K. Geim, Phys. Rev. B 76, 081406(2007).

26 E. A. Henriksen, P. Cadden-Zimansky, Z. Jiang, Z. Q.Li, L.-C. Tung, M. E. Schwartz, M. Takita, Y.-J. Wang,P. Kim, and H. L. Stormer, Phys. Rev. Lett. 104, 067404(2010).

27 M. Orlita and M. Potemski, Semicond. Sci. Technol. 25,063001 (2010).

28 I. Crassee, J. Levallois, D. van der Marel, A. L. Walter,T. Seyller, and A. B. Kuzmenko, Phys. Rev. B 84, 035103

(2011).29 M. Orlita, C. Faugeras, R. Grill, A. Wysmolek,W. Strupinski, C. Berger, W. A. de Heer, G. Martinez,and M. Potemski, Phys. Rev. Lett. 107, 216603 (2011).

30 C. Ellis, A. Stier, D. George, J. Tischler, E. Glaser,R. Myers-Ward, J. Tedesco, J. C.R. Eddy, D. Gaskill,A. Markelz, et al., Multi-component response in multilayergraphene revealed through terahertz and infrared magneto-spectroscopy, 37th International Conference on Infrared,Millimeter, and Terahertz Waves (IRMMW-THz), pp.1-3(2012).

31 V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, Phys.Rev. Lett. 98, 157402 (2007).

32 M. Koshino and T. Ando, Phys. Rev. B 77, 115313 (2008).33 A. Pound, J. P. Carbotte, and E. J. Nicol, Phys. Rev. B

85, 125422 (2012).34 S. M. Badalyan and F. M. Peeters, Phys. Rev. B 85,

205453 (2012).35 J. Zhu, S. M. Badalyan, and F. M. Peeters, Phys. Rev.

Lett. 109, 256602 (2012).36 I. Meric, M. Y. Han, A. F. Young, B. Ozyilmaz, P. Kim,

and K. L. Shepard, Nature Nanotech. 3, 654 (2008).37 M. Freitag, M. Steiner, Y. Martin, V. Perebeinos, Z. Chen,

J. C. Tsang, and P. Avouris, Nano Lett. 9, 1883 (2009).38 V. Perebeinos and P. Avouris, Phys. Rev. B 81, 195442

(2010).

39 A. Konar, T. Fang, and D. Jena, Phys. Rev. B 82, 115452(2010).

40 X. Li, E. A. Barry, J. M. Zavada, M. B. Nardelli, and K. W.Kim, Appl. Phys. Lett. 97, 232105 (2010).

41 A. M. DaSilva, K. Zou, J. K. Jain, and J. Zhu, Phys. Rev.Lett. 104, 236601 (2010).

42 J. H. Chen, C. Jang, S. Xiao, M. Ishigami, and M. S.Fuhrer, Nature Nanotech. 3, 206 (2008).

43 S. Fratini and F. Guinea, Phys. Rev. B 77, 195415 (2008).44

S. H. Zhang, W. Xu, S. M. Badalyan, and F. M. Peeters,Phys. Rev. B 87, 075443 (2013).

45 Z.-Y. Ong and M. V. Fischetti, Phys. Rev. B 86, 165422(2012).

46 Z.-Y. Ong and M. V. Fischetti, Phys. Rev. B 88, 045405(2013).

47 C. Ertler, S. Konschuh, M. Gmitra, and J. Fabian, Phys.Rev. B 80, 041405 (2009).

48 B. Scharf, V. Perebeinos, J. Fabian, and P. Avouris, Phys.Rev. B 87, 035414 (2013).

49 E. H. Hwang and S. Das Sarma, Phys. Rev. B 87, 115432(2013).

50 H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li,F. Guinea, P. Avouris, and F. Xia, Nature Phot. 7, 394(2013).

51 J. W. McClure, Phys. Rev. 104, 666 (1956).52 G. D. Mahan, Many-Particle Physics (Kluwer/Plenum,

New York, 2000).53 A. B. Migdal, Sov. Phys. JETP 7, 996 (1958).54 P. Allen and B. Mitrovic, in Solid State Physics, edited by

H. Ehrenreich, F. Seitz, and D. Turnbull, vol. 37 (TurnbullAcademic, New York, 1982).

55 F. Dogan and F. Marsiglio, Phys. Rev. B 68, 165102(2003).

56 E. J. Nicol and J. P. Carbotte, Phys. Rev. B 80, 081415(2009).

57 A. Pound, J. P. Carbotte, and E. J. Nicol, Phys. Rev. B84, 085125 (2011).

58 S. Piscanec, M. Lazzeri, F. Mauri, A. C. Ferrari, and

J. Robertson, Phys. Rev. Lett. 93, 185503 (2004).59 M. Lazzeri, S. Piscanec, F. Mauri, A. C. Ferrari, andJ. Robertson, Phys. Rev. Lett. 95, 236802 (2005).

60 V. Perebeinos, J. Tersoff, and P. Avouris, Phys. Rev. Lett.94, 086802 (2005).

61 T. Ando, J. Phys. Soc. Jpn. 75, 124701 (2006).62 M. Lazzeri, C. Attaccalite, L. Wirtz, and F. Mauri, Phys.

Rev. B 78, 081406 (2008).63 K. M. Borysenko, J. T. Mullen, E. A. Barry, S. Paul, Y. G.

Semenov, J. M. Zavada, M. B. Nardelli, and K. W. Kim,Phys. Rev. B 81, 121412 (2010).

64 S. Q. Wang and G. D. Mahan, Phys. Rev. B 6, 4517 (1972).65 M. V. Fischetti, D. A. Neumayer, and E. A. Cartier, J.

Appl. Phys. 90, 4587 (2001).66 A. S. Price, S. M. Hornett, A. V. Shytov, E. Hendry, and

D. W. Horsell, Phys. Rev. B 85, 161411 (2012).67 B. Wunsch, T. Stauber, F. Sols, and F. Guinea, New J.

Phys. 8, 318 (2006).68 E. H. Hwang and S. Das Sarma, Phys. Rev. B 75, 205418

(2007).69 P. K. Pyatkovskiy and V. P. Gusynin, Phys. Rev. B 83,

075422 (2011).70 R. Geick, C. H. Perry, and G. Rupprecht, Phys. Rev. 146,

543 (1966).71 G. Harris, Properties of Silicon Carbide (INSPEC, Insti-

tution of Electrical Engineers, London, UK, 1995).

7/29/2019 1307.4008v2.pdf

12/12

12

72 E. H. Hwang and S. Das Sarma, Phys. Rev. B 79, 165404(2009).

73 W. Kohn, Phys. Rev. 123, 1242 (1961).74 A. Iyengar, J. Wang, H. A. Fertig, and L. Brey, Phys. Rev.

B 75, 125430 (2007).75 Y. A. Bychkov and G. Martinez, Phys. Rev. B 77, 125417

(2008).76 C.-H. Park, F. Giustino, M. L. Cohen, and S. G. Louie,

Phys. Rev. Lett. 99, 086804 (2007).77

C.-H. Park, F. Giustino, C. D. Spataru, M. L. Cohen, andS. G. Louie, Nano Letters 9, 4234 (2009).

78 D. C. Elias, R. V. Gorbachev, A. S. Mayorov, S. V. Mo-rozov, A. A. Zhukov, P. Blake, L. A. Ponomarenko, I. V.Grigorieva, K. S. Novoselov, F. Guinea, et al., Nat. Phys.7, 701 (2011).

79 Both Eq. (6) and averaging the phonon spectral function

and electron-phonon coupling matrix elements over theFermi surface constitute very good approximations to cal-culate the electronic self-energy if the electron-phonon cou-pling or the typical phonon energies compared to the Fermienergy are small (see Refs. 54,55).

80 For convenience, we use Gaussian instead of Lorentzianbroadening to calculate the DOS here. However, we findthat the results presented in this manuscript are not sig-nificantly affected by using Lorentzian broadening.

81

Those formulas are straight-forward extensions of the for-mulas found in V. P. Gusynin, S. G. Sharapov, and J. P.Carbotte, J. Phys.: Condens. Matter 19, 026222 (2007)and in Ref. 33 to account for the spin-degree of freedom.

82 Our numerical integrations over have been conducted ongrids with () = 0.1 meV.