Faraday effect in graphene enclosed in an optical cavity and the equation of motion method for the study of magneto-optical transport in solids

2.1 MB·English

by  A. Ferreira

Faraday effect in graphene enclosed in an optical cavity and the equation of motion
method for the study of magneto-optical transport in solids
Aires Ferreira1,2, J. Viana-Gomes1, Yu. V. Bludov1, Vitor M. Pereira2, N. M. R. Peres1,2, A. H. Castro Neto2,3
1 Department of Physics and Center of Physics,
University of Minho, P-4710-057, Braga, Portugal
2 Graphene Research Centre and Department of Physics,
National University of Singapore, 2 Science Drive 3, Singapore 117542 and
3 Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA
We show that by enclosing graphene in an optical cavity, giant Faraday rotations in the infrared
regime are generated and measurable Faraday rotation angles in the visible range become possible.
2 Explicit expressions for the Hall steps of the Faraday rotation angle are given for relevant regimes.
1 In the context of this problem we develop an equation of motion (EOM) method for calculation of
0 themagneto-opticalpropertiesofmetalsandsemiconductors. Itisshownthatproperlyregularized
2 EOM solutions are fully equivalent to the Kubo formula.

n
a PACSnumbers: 8.20.Ls,78.67.Wj,72.80.Vp,81.05.ue
J

6
I. INTRODUCTION electronic density; and (ii) the quantum Hall regime, of

] strong fields and/or a low electronic density.
ll For interpretation of the optical Faraday rotation, in
a Electromagnetic radiation emitted by far stellar ob-
the semi-classical regime, the Drude theory of metals
h jects travels for long periods of time through very di-
suffices.3 In the case of graphene, it is possible to change
- luted concentrations of interstellar gases, traversing re-
s itselectronicdensityeitherbyuseofagateorbythead-
e gions where week magnetic fields exist. In this circum-
sorption of molecules.5,6 At high doping, graphene is in
m stance,thepolarizationoftheelectricfieldrotatesdueto
thesemiclassicalregimeandBoltzmanntransporttheory
t. its interaction with the gases immersed in the magnetic can be used to compute the Hall conductivity.7
a field. Due to the enormous traveling distances through
Intheabsenceofdisorderandotherrelaxationmecha-
m suchinterstellarregions,thedegreeofrotationofthepo-
nisms (such as electron-phonon scattering), the conduc-
larization can be important. This magnetic rotational
- tivityofgraphene(atzeromagneticfield)wouldbeexclu-
d effect turns out to be a problem in astrophysics, since it
sivelydeterminedbyinterbandtransitions. Inthelimitof
n modifies, in an unpredictable way, the polarization state
o no disorder, the optical conductivity of doped graphene,
of the emitted radiation, introducing additional difficul-
c in the infrared region of the spectrum and at zero mag-
ties in the interpretation of astronomical observations.
[ netic field, is given by8–15
In the electrodynamics of metals and insulators the ef-

2 fect of polarization rotation induced by a magnetic field σ =σ n ((cid:126)ω−2E ), (1)
v xx g F F
wasfirstdiscussedbyFaraday1 and, onEarth, hasmany
8 where σ = πe2/(2h) is the so-called ac universal con-
different applications. g
8 ductivity of graphene.8,16–18
1 In magneto-optics, the effect coined optical Faraday
When a magnetic field is applied perpendicularly to
3 rotation1 refers to the rotation of the plane of polariza-
graphene’ssurface,thesystemdevelopsafiniteHallcon-
9. tion of light when it transverses either a dielectric2 or a ductivity. In the quantum regime, it was shown that the
0 metal,3 in the presence of a static magnetic field applied Faradayrotationangleθ issolelydeterminedbythefine
F
1 alongthedirectionofpropagationoftheelectromagnetic
structureconstantα,andpresentsastep-likestructureas
1 wave. In addition to the rotation of the plane of polar- theFermienergycrossesdifferentLandaulevels(LLs).19
:
v ization, the polarization itself acquires a certain degree TheestimatedFaradayrotationsteps’heightinthiscase
Xi of ellipticity. In dielectrics, the effect can be explained is of the order of θF ∼ 0.4◦,19 a magnitude that can be
using a model of harmonic oscillators coupled to light.2 resolved experimentally.20 In the context of topological
r In metals, the effect has its roots in the Hall effect.4
a insulators, similar quantization rules in certain thin-film
For a two-dimensional (2D) metal, such as graphene, geometries have been derived in Refs. 21 and 22. We
in the Hall regime, the conductivity becomes a tensor note in passing that, when the external magnetic field
σˆ, with finite (nonzero) values for both diagonal and is absent, a dynamic Hall effect can still be induced by
off-diagonal components. In magneto-optics, the com- usingcircularlypolarizedlightimpingingongrapheneat
ponents of the tensor depend both on the frequency of a finite angle with the normal to the graphene surface.23
theimpingingelectromagneticwaveandonthecyclotron On the theoretical side, the magneto-optical trans-
frequency of the electrons, due to the magnetic field per- port properties of graphene have been investigated with
pendiculartotheplaneofthemetal. Theresponseofthe the Green’s function method8,10, and by means of
electrons to the external magnetic field has two regimes: numerical implementations of the Kubo formula, us-
(i) the semiclassical limit, of low fields and/or a high ing exact diagonalization19 and Chebyshev polynomial
2
expansions.24Theseapproachescomewithprosandcons: the scale of interest. The goal is to reach cyclotronic
numericalstudiesallowtheexplorationofgeneralscenar- energies, usually O(10) meV in fields of 1-10 T, where
ios, whereas Green’s functions allows one to obtain ana- strongopticalresponsestakeplace. Theso-calledoptical
lytic results, but many times at the expense of lengthy quantum Hall conductivity of 2D electron gases shows a
calculations. robustplateauxastheFermienergyisswept,althoughno
Motivatedbytheneedforanalyticalflexibleanalytical quantization rule for the plateaux’s height exists.32 Due
tools, the equation of motion (EOM) method employed to its peculiar band structure, graphene has been pre-
in Ref. 25 is generalized to include the effect of a mag- dicted to display a characteristic optical quantum Hall
netic field. As shown later, starting from a small set of effect which should be detectable via Faraday rotation
EOMs, an adequate treatment permits the derivation of measurements.19Inthesemiclassicalregime,ontheother
responsefunctionswithcorrectanalyticalproperties(i.e., hand, the Faraday rotation of graphene was reported to
satisfying Kramers-Kronig causality relations). be O(1) degrees in fields of a few tesla,33 a surprisingly
The present paper is divided into two main parts. In high value for a one-atom-thick electronic system.
Sec.IIwepresenttheEOMmethodforcalculationofthe
magneto-opticaltransportinmetalsandsemiconductors;
to be concrete, the method is described in the context
of the properties of graphene. In Sec. III we describe
in detail the Faraday effect in graphene and propose an
experimental setup that is able to enhance the Faraday
effect up to the visible range. Section III relies heavily
on the results derived in Sec. II. Some technical details
are given in the Appendixes.
We have chosen to organize the subjects according
to the following interests of different readers: a reader
FIG. 1: Lattice structure and Brillouin zone of monolayer
having a primary interest in the Faraday effect, and fa-
graphene. Left: Hexagonallatticeofgraphene,withthenext
miliar with the details on the magneto-optical proper-
nearest neighbor, δ , and the primitive, a , vectors depicted.
ties of graphene, should be able to read Sec. III with a i i√
The area of the primitive cell is A = 3 3a2/2 (cid:39) 5.1 Å2,
bird’s-eye reading of Sec. II. A reader interested in the c 0
and a (cid:39)1.4 Å. Right: Brillouin zone of graphene, with the
0
Faraday effect in graphene but not well acquainted with Dirac points K and K(cid:48) indicated. Close to these points, the
its magneto-optical properties may want to go through dispersion of graphene is conical and the density of states is
Sec. II first. Finally, reading Sec. II alone may appeal to proportional to the absolute value of the energy.
readers interested in applying the EOM method to an-
otherproblemofinterestbearingnorelationtographene.
A. Graphene
II. EQUATION OF MOTION METHOD FOR
CALCULATION OF THE MAGNETO-OPTICAL The starting point of the present analysis is the low-
CONDUCTIVITY
energy continuum description of single-layer graphene;
havingtwo(carbon)atomsperunitcellandsixfoldsym-
Here,wedeveloptheEOMapproachtothecalculation metry, its elementary excitations obey a 2D Dirac equa-
ofthemagneto-opticalpropertiesofasemiconductor. To tion with linear electronic dispersion.34 This section is
beconcrete,themethodispresentedinthecontextofthe meanttofixthenotation. TheBrillouinzoneofgraphene
optical response of graphene. has six corners, and among these, only two are inequiv-
Electrons constrained to two dimensions are responsi- alent, the so-called K and K(cid:48) Dirac points (see Fig. 1).
ble for a variety of quantum manifestations, a striking Atthesepoints,thevalenceandconductionbandstouch,
example being the integer quantum Hall effect (IQHE). withalinearelectronicspectrumuptoenergiesof∼2eV.
Measured in semiconductor 2D electron gases more than We assume, in what follows, that the two Dirac points
30 years ago26 and in the early days of graphene, in can be treated independently, and introduce the val-
both monolayer5,27 and bilayer samples28 (very recently ley degeneracy index, g = 2, when pertinent. This
v
also in trilayer graphene29), the static quantum Hall ef- consideration is justified for typical experimental condi-
fect is a hallmark of elementary excitations in electronic tions (i.e., low concentrations of scattering centers, finite
systems.30 temperatures, etc.) and provides an accurate descrip-
Its dynamical analog—the ac quantum Hall effect— tionofgraphene’selectronictransportpropertiesatfinite
canprovideadditionalinformationaboutchargecarriers, densities.16,35
suchastheopeningofgapsinthespectrum.31Recentad- In accordance, we resort to the 2×2 Dirac Hamilto-
vances in time-domain spectroscopy in the Thz regime20 nian of graphene, describing the physics of elementary
have paved the way to measurement of dynamical op- excitations within the K valley, H = v σ ·p, where
K F
tical conductivities at impinging field energies closer to v (cid:39) 106m/s is the Fermi velocity, σ = (σ ,σ ) [with
F x y
3
σ (i = x,y) denoting Pauli matrices], and p is the mo- B. Theoretical methods
i
mentum of the low-energy excitation (measured relative
to the K point).34 H has eigenvalues given by
K
In the context of electronic systems, the EOM was
E =±(cid:126)v |k|, (2) extensively used in calculations of light polarization in
F
semiconductor laser theory.37 Recently, it has been used
[with k=(kx,ky) denoting a 2D wave vector], and (nor- to study excitons in graphene in zero field.25
malized) wave functions given by
The EOM approach avoids the calculation of current-
(cid:18) (cid:19)
1 1 currentcorrelators(i.e.,Kuboformula),and,hence,pro-
ψ (r)= √ eik·r, (3)
λ,k 2A λeiθk videsashortcuttodeterminationoftheresponseofelec-
tronic systems to external perturbations. As shown in
where A is the area of the graphene sample, λ =
detailinAppendixC,withanappropriateregularization
+1(−1) for electron(hole)-like excitation, and θ =
k procedure, the EOM solutions become fully equivalent
arctan(k /k ).
y x to the Kubo formula, and hence provide an accurate de-
Theelectromagneticfieldcanbeincorporatedviamin-
scription of transport in the linear response regime. An-
imal coupling, p → p + eA , where −e < 0 is the
g other advantage of the present approach is that it allows
electron charge, and the vector potential A relates to
g for the calculation of non-linear corrections to the con-
theelectromagneticfieldaccordingtotheusualrelations,
ductivity.
B=∇×A and E=−∂A /∂t.
g g
Here, the vector potential contains the information At the heart of the EOM approach to calculation of
about the impinging electromagnetic radiation, and pos- themagneto-opticalconductivityistheHeisenbergequa-
sible external static magnetic fields. Assuming light lin- tion for the electronic current density, J(t), in the pres-
earlypolarizedalongthexaxis,theradiationtermreads ence of an external electromagnetic field, i.e., dJ/dt =
A = [A (r)e−iωt +c.c.]e , where ω stands for the fre- (i/(cid:126))[H,J],withH beingthetotalHamiltonian,Eq.(4).
0 x
quency of the radiation field and A (r) describes its po- Havingsolvedforthecurrentdensityofthesysteminthe
0
sition dependence. For clarity of exposition, we separate presenceofexternalperturbation,infirstorderintheex-
the light-matter interaction term from the free Hamilto- ternal field A, the optical conductivity follows from the
nian, constitutive electromagnetic relation
H =H +ev σ·A, (4)
0 F
J˜(ω)
where H0 ≡ HK +evFσ ·AB, with AB describing the σij(ω)=gsgv× E˜i(ω), (7)
static magnetic field. j
A typical experimental scenario corresponds to a con-
stant magnetic field B > 0 applied in the transverse di-
where O˜(ω) relates to the average O(t) [O = J ,E ] ac-
rectionwithrespecttothegrapheneplane. Insuchcase, i j
cording to O(t) = O˜(ω)e−iωt + c.c., with appropriate
LLs develop and the eigenenergies of charge carriers be-
come quantized according to36 regularization implicit (Appendix C; Sec. IID). Having
graphene in the Dirac cone approximation in mind, the
E =sign(n)(cid:126)vF(cid:112)2|n| ,n=0,±1,±2,..., (5) latter equation contains the relevant degeneracies. The
n lB spin contribution as a degeneracy factor, gs, should be
valid for typical magnetic fields ((cid:46)15 T) when the Zee-
with lB = (cid:112)(cid:126)/(eB) denoting the magnetic length. man effect does not manifest.
Choosing the gauge A = (0,Bx,0) results in the fol-
B
The first step is to project the Heisenberg EOM
lowing set of Landau eigenfunctions,
for the current onto the space of unperturbed single-
(cid:18) (cid:19)
C φ (x) particlestates: weintroducethefieldoperatorΨ (r,t)=
ψn,ky(r)= √Ln isign|n(n|−)1φ|n|(x) eikyy, (6) (cid:80)αcˆα,σ(t)ψα(r) (and the respective Hermitiσan con-
jugate), where cˆ (cˆ† ) is the annihilation (cre-
whereφ (x)=e−ξ(x)2/2H (ξ(x))/(cid:112)n!2n√πl , H (x)is ation) operator oαb,σeyiαn,gσ fermionic anticommutation
n n B n
theHermitepolynomialofdegreen≥0,φ−1(x)=0,and rules: {cˆα,σ,cˆ†α(cid:48),σ(cid:48)} = δαα(cid:48)δσ,σ(cid:48) and {cˆα,σ,cˆα(cid:48),σ(cid:48)} =
ξ(x) stands for the dimensionless center of the Landau {cˆ† ,cˆ† }=0. Thesymbolα=(λ,k,...)specifiesthe
orbit,ξ(x)=l k +x/l . Here,Listhelineardimension α,σ α(cid:48),σ(cid:48)
B y B single-particle state of the electron (or hole) and σ =±1
ofthesystemintheydirectionandC isanormalization
n is the spin variable. The kets |α,σ(cid:105) ≡ cˆ† |0(cid:105) represent
constantthatdistinguishesthezero-energylevelfromthe α,σ
√ eigenstatesofH ,and,therefore,thepositionrepresenta-
remaining levels, C = 1 for n = 0 and C = 1/ 2 for 0
n n tion, (cid:104)r|α,σ(cid:105)≡ψ (r), equals Eq. (3) at zero magnetic
|n|≥1. α,σ
field or Eq. (6) in the presence of a transverse uniform
Havingreviewedthebasicsofthegraphene’selectronic
magnetic field.
low-energy theory, in what follows we present the EOM
approach to the study of magneto-optical transport. Thesecond-quantizedformofthefullHamiltonianand
4
the current density operator is given by
(cid:90)
Hˆ(t)=(cid:88) drΨ†(r,t)HΨ (r,t), (8)
σ σ
σ
(cid:90)
Jˆ(t)=(cid:88) drΨ†(r,t)j Ψ (r,t), (9)
i σ i σ
σ
respectively, where
ev
j =− Fσ, (10)
A
is the current density of graphene in the continuum
FIG. 2: Allowed interband transitions (vertical arrows) in
description.16,35 We omit the spin dependence of the op-
graphene;aphotonofenergy(cid:126)ω producesanexcitationfrom
erators hereafter for clarity of exposition. the lower to the upper Dirac, as long as (cid:126)ω >2µ. The tran-
We now define the generic operator,
sitions conserve k and hence are said to be “vertical.” For
Pˆ (t)≡cˆ†(t)cˆ (t), (11) (cid:126)ω ≤ 2µ, Pauli blocking forbids any (interband) transition.
αβ α β In practice, due to disorder (impurities, etc.), the interband
conductivity can be non-zero even for (cid:126)ω≤2µ.
whose EOM reads
dPˆ (t)= i (cid:88)h (cid:104)Pˆ (t), Pˆ (t)(cid:105) , (12)
dt αβ (cid:126) γδ γδ αβ With this notation, the current density along the x di-
γ,δ
rection reads,
Hwhamerielthoγnδia=n [(cid:104)Eγq|H.ˆ(|4δ)(cid:105)].arSeotlvhienmg afotrrixPˆαelβe(mt)engtivseosfdtihreecftullyl Jx(t)=−eAvF (cid:88) jλx,λ(cid:48),k,k(cid:104)cˆ†λ,k(t)cˆλ(cid:48),k(t)(cid:105). (15)
the current density according to, λ,λ(cid:48),k
Jˆ(t)=(cid:88)(cid:104)α|j |β(cid:105)Pˆ (t), (13) Thenon-nullmatrixelementsinEq.(14)contributingto
i i αβ the conductivity correspond to transitions between dif-
α,β
ferent bands conserving the momentum k. These tran-
and hence the (yet non-regular) optical conductivity via sitions are said to be “vertical,” and, in addition, since
Eq. (7). The regularization is the final step of the they connect states in different bands, they are refereed
EOM approach needed for obtaining a fully-consistent to as being interband-like (see Fig. 2).
conductivity (in particular, obeying Kramers-Kronig Taking the dipole approximation, A(r) → A0, the
relations).38 The respective technical procedure is given Hamiltonian [Eq. (8)] reads
in Appendix C.
In the following section, we solve Eq. (12) explicitly in Hˆ = (cid:88)E (k)cˆ† cˆ +ev A (cid:0)e−iωt+c.c.(cid:1)×
λ λ,k λ,k F 0
the linear response regime (i.e., first order in the electric λ,k
field) for any pair of quantum states α,β, in the absence ×(cid:88)jx cˆ† cˆ +(c←→v). (16)
ofamagneticfield. Thecaseoffinite(nonzero)magnetic c,v,k,k c,k,σ v,k,σ
k
field intensity is left for Sec. IID.
In the latter equation, E (k) ≡ λ(cid:126)v k , and the sub-
λ F
scripts c (v) denote electrons (holes).
C. Graphene in a zero magnetic field As described above, we need to compute the time evo-
lution of the operator Pˆ (t)=cˆ† (t)cˆ (t). Straight-
v,c,k v,k c,k
ThepurposeofthissectionistoshowtheEOMmethod forward algebra yields
at work in the context of a simple problem, which allows
us to derive well-known results. In the absence of mag- dPˆ = i (cid:110)[E (k)−E (k)]Pˆ
neticfields,themacroscopicelectroniccurrentfollowsthe dt v,c,k (cid:126) v c v,c,k
applied optical field, and thus only the longitudinal con- +ev A (cid:0)e−iωt+c.c.(cid:1)jx [nˆ (t)−nˆ (t)](cid:9) ,
F 0 c,v,k,k c v
ductivity is nonzero. From symmetry considerations, we (17)
also have σ (ω) = σ (ω). According to the statement
xx yy
Eq. (13), the relevant set of EOMs to be solved is de- where we have defined the occupation operator for elec-
termined by the non-zero matrix elements of the current trons (holes) as nˆ (t) ≡ cˆ† (t)cˆ (t). A similar
c(v) c(v),k c(v),k
density.
equation holds for Pˆ which can be obtained by in-
Defining (cid:104)k,λ|j |k(cid:48),λ(cid:48)(cid:105) = −(ev /A)jx and us- c,v,k,σ
x F λ,λ(cid:48),k,k(cid:48) terchanging c←→v.
ing the wave functions Eq. (3), we easily find
To proceed, we take the average of Eq. (17) with re-
jλx,λ(cid:48),k,k(cid:48) = δk2,k(cid:48) (cid:0)λ(cid:48)eiθk +λe−iθk(cid:1) . (14) smpaetcet(cid:104)tnˆoct(ht)e−unnˆpve(rtt)u(cid:105)0rb(cid:39)ed(cid:104)nˆHca−mnˆilvt(cid:105)o0n.iaBno,tHh0p,raoncdedauprpesroaxrie-
5
consistent with an expansion of Jˆ (t) up to first order in not included in the Hamiltonian Eq. (16). This contri-
x
the parameter A . The solution of the above differential bution gives the Drude response and reads39
0
equation reads
2e2 Γ
(cid:104)Pˆv,c,k(t)(cid:105)0 =P˜v,c,k(ω)e−iωt+P˜v,c,k(−ω)eiωt, (18) Reσxinxtra(ω)= h |µ|(cid:126)2ω2+Γ2 . (24)
with, Interestingly enough, the latter result can be derived
from a full quantum mechanical calculation by consider-
(cid:104)nˆ (cid:105) −(cid:104)nˆ (cid:105)
P˜ (ω) = ev A jx c 0 v 0 (1,9) ing a finite magnetic field intensity and taking the limit
v,c,k F 0 c,v,k,kEc(k)−Ev(k)−(cid:126)ω−iΓ B → 0 in the end.10 This is because a magnetic field
open gaps in the spectrum of a clean system, allowing
and we have introduced an imaginary energy Γ by hand,
for intraband transitions (see Sec. IID). A semiclassical
so to account for disorder phenomenologically. The re-
calculation also leads to an equivalent result (Sec. IIE).
mainingtermP˜ (ω)canbeobtainedfromthelatter
v,c,k,σ
expression by making ω → −ω and Γ → −Γ . From
Eq. (15), the oscillator strength of the current density
D. Optical conductivity of graphene in a magnetic
along the x direction J˜x(ω) is seen to be given by field
J˜ (ω)=−evF (cid:88)[jx P˜ (ω)+jx P˜ (ω)].
x A v,c,k,k v,c,k c,v,k,k c,v,k In what follows, we show that the EOM method can
k be employed to study the magneto-optical response of
(20)
graphene along the same lines as in Sec. IIC. The pres-
The longitudinal optical conductivity, σ , follows from
xx ence of a transverse magnetic field in the Hamiltonian
Eq. (7),
develops LLs, and hence we must start from the eigen-
states given in Eq. (6). The latter defines the field oper-
σinter(ω)=g g e2vF2 (cid:90) d2k(cid:0)sin2θ (cid:1)× ator, Ψ(r,t)=(cid:80)n,kycˆn,ky(t)ψn,ky(r) (together with the
xx v s iω 4π2 k respective Hermitian conjugate); the index n labels the
degenerate LL with energy given by Eq. (5). The field
n [E (k)]−n [E (k)]
× F v F c +(c↔v). operator can be written as
E (k)−E (k)−(cid:126)ω−iΓ
c v
(21) 1 (cid:88) (cid:18) φ (x) (cid:19)
Ψ(r,t) = √ |n|−1 eikyycˆ
where nF(E)=1/[e(E−µ)/kBT +1] stands for the Fermi- 2Ln(cid:54)=0,ky isign(n)φ|n|(x) n,ky
Diracdistribution(µisthechemicalpotential). Inderiv- 1 (cid:88)(cid:18) 0 (cid:19)
ing this expression, we have used the relation E˜x(ω) = +√L φ0(x) eikyycˆ0,ky. (25)
iωA0. Taking the clean limit Γ → 0 and considering ky
ω >0 and T =0, one obtains the well- known result
Thispeculiarspinorialstructure,withasinglelevelbeing
πe2 highlighted, is on the basis of non-standard features in
Re σxinxter(ω) = 2h θ((cid:126)ω−2|µ|) . (22) the magneto-optical conductivity of graphene.8,11,31,40
The latter result is the T → 0 limit of Eq. (1). For
photonenergieshigherthan2µ(seeFig.2),theinterband 1. The longitudinal conductivity
conductivity is essentially frequency independent (up to
energies of ∼2 eV) and equals According to Eq. (7), the calculation of the longitu-
dinal conductivity requires computation of the average
πe2
σ = , (23) value of the current density operator along the x direc-
g 2h tion,
whichisnothingotherthantheuniversalconductivityof (cid:88) (cid:88)
J (t)= (cid:104)n,k |j |n(cid:48),k(cid:48)(cid:105)(cid:104)cˆ† (t)cˆ (t)(cid:105). (26)
graphenementionedinSec.I.Forµ=0,andcontraryto x y x y n,ky n(cid:48),ky(cid:48)
ordinarysemiconductors,thereisnofrequencythreshold n,n(cid:48)ky,ky(cid:48)
for interband transitions: according to Eq. (22), some
Using the LL wavefunctions [Eq. (6)], we easily find the
interband transitions will always be available for a suffi-
non-zero matrix elements to be,
cientlyhighphotonfrequency. Asaconsequence,Drude’s
descriptionwillnotsufficeforageneraldescriptionofthe ev
optical response of graphene. (cid:104)0,ky|jx|±1,ky(cid:48)(cid:105)=− √2FAδky,ky(cid:48) , (27)
Inadditiontotheinterbandtransitionsdiscussedhere,
thereisanintrabandcontributioningraphenewhichcan (cid:104)n,ky|jx|n(cid:48),ky(cid:48)(cid:105)=− e2vAFi(cid:2)sign(n(cid:48))δ|n|−1,|n(cid:48)|
be appreciable for µ (cid:54)= 0. This contribution comes from (cid:3)
−sign(n)δ δ , (28)
nonvertical processes (e.g., via collisions with phonons), |n|,|n(cid:48)|−1 ky,ky(cid:48)
6
where in the last line n,n(cid:48) (cid:54)= 0. These statements show thetimeevolutionofoperatorsinvolvingthezero-energy
that the optical transitions conserve ky and occur be- state (c†1a0, v1†a0, and Hermitian conjugates), while the
tween levels with indexes n and n(cid:48) satisfying |n|−|n(cid:48)|= other set refers to higher energy LLs. Take, for instance,
±1. the operator Pˆ(1) belonging to the latter set; as in the
n
Two sets of transitions are thus allowed: intraband case of zero magnetic field (Sec. IIC), the commutator
transitions, occurring within the same band, and, as [H,Pˆ(1)] gives rise to (i) occupation number operators
in the absence of a magnetic field, transitions connect- n
(v† v andc†c ), and(ii)afreeevolutionterm, that
ing LLs in the valence and conduction bands, which n+1 n+1 n n
are interband-like. Transitions involving the zero-energy is, the operator Pˆn(1) itself. In addition, intraband terms
state n = 0 can be considered either intraband- or with |n|−|n(cid:48)|=±2 show up, namely, c†c , v† v
n n+2 n−1 n+1
interband-like, since the zero-energy state is shared be- anda†v δ . Thesetermsdonotoriginaterealintraband
0 2 n,1
tween electrons and holes. For the sake of simplicity in transitions, since the respective current density matrix
definingthesetofEOMs, throughout, weclassifytransi- elements are null.
tions involving the zero-energy state as being interband.
We are now in the position to write the prototype
Inordertoclearlydistinguishamongthepossibletypes
EOMs governing the interaction of Landau quasiparti-
of transitions, we define
cles with an external oscillating electric field,

cn for n>0 (cid:126) dPˆ(1) =(cid:2)E −E (cid:3)Pˆ(1)− iev A(t)×
cˆn,ky ≡ v|n| for n<0 , (29) i dt n n −(n+1) n 2 F
a0 for n=0 ×[vn†+1vn+1−c†ncn], (34)
(cid:126) d 1
with the Hermitian conjugates following identical redefi- Pˆ =E Pˆ + √ ev A(t)[a†a −c†c ], (35)
i dt c 1 c 2 F 0 0 1 1
nitions. Note that with these definitions the subscript n
in the operators take only positive integer values.
where we have omitted the time dependence of the op-
a. Interband transitions—Using the field operator in
erators and defined Pˆ = c(v)†a . The remaining op-
the presence of a magnetic field [Eq. (25)], and keeping c(v) 1 0
trackofjusttheinterbandtermsforthemoment,thefull erators obey similar equations. [The EOM for Pˆn(2) is
Hamiltonian takes the form obtained making Pˆ(1) → Pˆ(2)and interchanging n with
n n
n+1ontheright-handsideofEq.(34). AsforPˆ ,welet
Hˆ =(cid:88)(cid:2)Enc†ncn+E−nvn†vn(cid:3) Pˆ →Pˆ , E →E , and c (c†)→v (v†) in Eq.v(35).]
c v 1 −1 1 1 1 1
n≥1 To solve the above set of differential equations to first
+evF√A(t)(cid:104)c†a +v†a +h.c.(cid:105) orderinA0,weproceedasinSec.IIC.Takingtheaverage
2 1 0 1 0 value(cid:104)...(cid:105)0 ofeachEOMwithrespecttotheunperturbed
−evF2A(t)i(cid:88)(cid:104)Pˆn(1)+Pˆn(2)−h.c.(cid:105) , (30) wHraimttielntoansia(cid:104)nO,(Ht)0(cid:105),0th=eOs˜o(lωu)teio−niωftor+eOa˜c(h−oωp)eeriaωtto,rwOhecraentbhee
n≥1 oscillator strengths read
where A(t)≡A0(e−iωt+c.c.), and i (cid:104)v† v (cid:105) −(cid:104)c†c (cid:105)
P˜(1)(ω)=− ev A n+1 n+1 0 n n 0 , (36)
Pˆ(1) =c†v , (31) n 2 F 0E−(n+1)−En−(cid:126)ω−iΓ
n n n+1
Pˆn(2) =c†n+1vn. (32) P˜c(ω)= √12evFA0(cid:104)a−†0Ea01(cid:105)−0−(cid:126)ω(cid:104)c−†1ci1Γ(cid:105)0 , (37)
(Also, for clarity, we have omitted k under all the sum-
y
mation signs.) The first line in Eq. (30) describes mass- and where, as in Sec. IIC, we have added a imaginary
lessDiracfermionsinatransversemagneticfieldandthe energy Γ to account for level broadening. The solutions
remaining lines contain the electronic transitions among for P˜(2)(ω) and P˜ (ω) can be obtained from the latter
n v
different LLs induced by the external electric field. expressions as described below Eq. (35).
The interband current density along the x direction Combining these results and Eq. (33), we easily find
can be recast into the form

J˜ (ω) = 1 ev (cid:88)(cid:88)(cid:104)iP˜1(ω)+iP˜2(ω)(cid:105)
Jˆx(t)=−√12AevF (cid:16)c†1a0+v1†a0+h.c.(cid:17) x 2A√ (cid:104)F ky n≥1 n(cid:105) n (cid:111)
+ 1 ev (cid:88)(cid:16)iPˆ(1)+iPˆ(2)+h.c.(cid:17) . (33) − 2 P˜c(ω)+P˜v(ω) +”c.c. term” ,(38)
2A F n n
n≥1
where the summation over k has been restored. This
y
(cid:80)
From the form of the current we see that there are two summation yields the degeneracy of the LLs ky =
basic sets of EOMs to be solved: the first set refers to A/(2πl2). The last term in the above equation (i.e., the
B
7
b. Intraband transitions—The intraband interaction
Hamiltonian reads
Hˆintra = iev A(t)(cid:88)(cid:2)v†v −c†c −h.c.(cid:3) , (40)
int 2 F n n+1 n n+1
n≥1
and the zero-energy operators (a and a†) are ab-
0 0
sent given our classification of intraband transitions [see
Eq. (29) and the following text]. The calculation follows
identical steps to the interband conductivity and, hence,
is not repeated. The final expression for the (regular)
intraband diagonal conductivity reads,
FIG. 3: Schematic of electronic transitions contributing to e2v2(cid:126) (cid:88) (cid:88)Nc (cid:20) 1
σintra(ω)= F i α ×
σxx(ω) of doped graphene in a magnetic field. In this exam- xx 2πl2 E −E
ple, E ≥ E , and thus the last occupied LL, n = N ≥ 1, B α=±1 n=1 n+1 n
F 1 F
belongs to the conduction band. Two types of transitions n [E ]−n [E ]
takeplace: (i)interbandtransitions,connectingLLsfromthe × E F−En+1−α(F(cid:126)ω+niΓ)
lower cone (valence band) with LLs in the upper cone (con- n+1 n
duction band), and (ii) intraband transitions within the up- +(En →−En∧En+1 →−En+1)] . (41)
per cone. Intraband transitions are limited to adjacent LLs:
NF →NF+1. Thefigureshowsthefollowinginterbandtran- The full longitudinal conductivity σxx(ω) is given by
sitions: (a) the pair −N → N +1 and −N −1 → N , addingitsinterbandandintrabandcounterparts,thatis,
F F F F
whose energy difference is E +E (the lowest interband Eqs. (39) and (41), respectively; straightforward algebra
N+1 N
energy;notethattransitions−NF−1→NF areforbiddenbe- yields
causen=N isoccupied);and(b)thepair−N −1→N +2
F F F
aEnd −+NEF −2(→theNsFec+on1d.lTowheesrteisnpteecrbtiavnedeneneregrgyyddiffiffeerreenncceei)s, σ (ω)= e2 (cid:88)Nc Λxnxm nF(En)−nF(Em), (42)
N+1 N+2 xx h iE (cid:126)ω+E +iΓ
andinthiscasebothtransitionstakeplace. Transitionswith nm nm
n(cid:54)=m=−Nc
higher energy differences are not represented.
with E = E −E , and where we have defined the
nm n m
longitudinal matrix elements
c.c. term) is obtained taking the complex conjugate and
making ω →−ω of all the previous terms. (cid:126)2v2
Λxx = F(1+δ +δ )δ . (43)
The final expression for the longitudinal (interband) nm l2 m,0 n,0 |m|−|n|,±1
B
conductivity is derived in two steps: (i) dividing the
Eq. (38) by E˜ (ω) [Eq. (7)], and (ii) undertaking ap- Equation (42) is the main result of the present sec-
x
propriate regularization to remove the divergent factor tion. It coincides with Eq. (7) in Ref. 41 obtained via
1/ω, a Green’s function calculation in the bubble approxima-
tion and, also, with a Kubo formula calculation within
theDiracconeapproximation(seeAppendixC).Wenote
e2v2(cid:126) (cid:88)Nc (cid:88) in passing that, on top of the interband and intraband
σinter(ω)= F i (1+δ ) α
xx 2πl2 n,0 contributionsdiscussedhere, thereisacorrectionarising
B n=0 α=±1 fromphonon-electroncoupling. Atlowtemperaturesand
(cid:20) 1 nF[E−(n+1)]−nF[En] zero field, this correction is expected to be small.39 At a
× ×
E −E E −E −α((cid:126)ω+iΓ) high magnetic field, though, a recent calculation shows
−(n+1) n −(n+1) n
that phonon energy peaks split the LLs nearby,42 which
+(n↔n+1)] . (39)
can lead to a measurable signature in magneto-optical
The above expression is analytic in the upper-half plane experiments.
andfiniteatω =0,thusobeyingKramers-Kronigcausal-
ity relations. (We refer to Appendix C for the derivation
and physical grounds of the regularization procedure.) 2. The general properties of σxx(ω)
Note that, as usual when dealing with low-energy theo-
ries, a cutoff energy E of the order of the bandwidth In what follows, we overview the main features of
cut
must be considered for consistency; we take n ≤ N , graphene’slongitudinalmagneto-opticalconductivity,an
c
with N = int[(E /E )2], where int[...] denotes the in- essential step to understanding the Faraday rotation in
c cut 1
tegerpart. N variesroughlyas104B−1 withB inteslas. graphene (Sec. III).
c
Within the physical relevant range for E , these sum- a. Low electronic density—Atalowelectronicdensity,
cut
mationsconvergequiterapidly;thefiguresinthepresent more precisely, for |E | < E , no intraband transitions
F 1
work have E ≈t(cid:39)2.7 eV. can take place. Because the LL energy scale in graphene
cut
8
Let us first consider the limiting case when the energy
gap ∆ is larger than the level broadening, ∆ (cid:38) Γ.
NF NF
The latter typically happens at high magnetic fields and
not too high Fermi energies; in this limit, the real part
of Eq. (44) displays a maximum at ω (cid:39) ∆ω , with an
NF
intensity falling off as B/∆ω ,
NF
(cid:18) 2eBv2 (cid:19)
Reσintra(∆ω )(cid:39) F ×σ . (46)
xx NF πΓ∆ω g
NF
The intraband magneto-peak, Eq. (46), is the lowest fre-
quency peak in the absorption spectrum of graphene
with E > E ; its magnitude increases with increas-
F 1
ing Fermi energy and/or magnetic field intensity. An
example of an intraband absorption line occurring at
ω (cid:39) ∆ω is shown in Fig. 5. In that case, the param-
FIG.4: Longitudinalmagneto-opticalconductivityasafunc- NF
eters correspond to ∆ = 22.6 meV and Γ = 6.8 meV,
tion of the photon energy for a field of 7 T, zero chemical and hence ∆ (cid:38) ΓN.F Some points are worth men-
potential, T = 17 K, and Γ = 6.8 meV (∼ 79 K). The hori- NF
tion: (i) the intraband contribution to the conductivity
zontaldashed-dot(black)linemarksthegraphene’suniversal
[Eq. (44)] dominates at low photon frequencies; and (ii)
ac-conductivity background [Eq. (1)].
thecurveforReσ (ω)showsthattheremainingabsorp-
xx
tion peaks are found in the higher frequency part of the
isrelativelyhigh(e.g.,E (cid:39)36meVforafieldof1T),the spectrum, above the threshold for interband transitions,
1
magneto-opticalconductivityisfullydrivenbyinterband (cid:126)ω ≥E +E . (Note that, at a low magnetic field
NF NF+1
transitions even close to room temperature. and/or high Fermi energy, the level spacing between ad-
Figure4showsaplotofEq.(42)forzeroFermienergy jacent LLs is so reduced that E (cid:39) E (cid:39) E , and
NF NF+1 F
and a magnetic field of 7 T: a sequence of absorption thus one recovers the condition found earlier, namely,
peaks, correspondingto the maximum of the realpart of (cid:126)ω > 2E .) Such interband peaks cause Shubnikov–de
F
eachterminEq.(39),(cid:126)ω (cid:39)E , E −E , E −E ,etc., Haas oscillations despite the finite electronic density.
1 2 −1 3 −2
isclearlyobserved[seeEq.(49)andtextthereafter]. The For a general relation between the broadening and the
conductivity never vanishes, even though the concentra- energy gap ∆ , the maximum for the intraband peak
NF
tion of carriers is low (EF → 0), a genuine signature of occurs at
graphene’s LL structure.5
(cid:114)
The contributions from different interband transitions (cid:113)
ωintra =Re 2∆ω ∆ω2 +Γ2/(cid:126)2−∆ω2 −Γ2/(cid:126)2.
[Eq. (39)] partially overlap at a high frequency, with the peak NF NF NF
effect that the real part of σ (ω) displays the so-called (47)
xx √
Shubnikov–de Haas oscillations around the universal ac When ∆ω ≤ Γ/( 3(cid:126)) (typically the case for a very
NF
opticalconductivityofgraphene,σg (theimaginarypart, high Fermi energy and/or low magnetic field), the intra-
in turn, oscillates around 0).8–15 The semiclassical con- band conductivity is maximal at null frequency, with an
ductivity is null, on the other hand, thus failing to de- intensity given by Eq. (46) multiplied by a factor of 2.
scribe the magneto-transport in neutral graphene. Theregime∆ (cid:46)Γisillustratedinthebottompanel
b. High electronic density—Awayfromchargeneutral- in Fig. 12. TwNoFmagnetic fields are considered, at a
ity, moreprecisely, for|EF|>E1, thepictureismorein- fixed Fermi energy, EF = 0.3 eV, with Re σxx(ω) be-
volved; intraband transitions can now occur, while some ing represented by the solid lines. When B = 7T (left-
interband transitions will be blocked. We take T = 0 hand panel), although a considerable number of levels
a(snidm,ilwaritchoonuctlulsoisosnsofhogledneforralhitoyl,esa)s;sudmireectthinastpEecFtio>n o0f acorerdoinccguptoiedEq(.N(F47=),9c)o,rornesephoansds∆tNoFa(cid:39)m1a.4xΓim, uwmhicohf,tahce-
Eq. (41) shows that a single type of intraband transition longitudinal conductivity at ω (cid:39) ∆ω . This is in-
isallowed,whosecontributiontotheopticalconductivity deedconfirmedbythenumericalcalculatNiFonshownthere.
reads Decreasing the magnetic field down to B =3 T (right-
σxinxtra(ω)= eh2∆2ωi(cid:126)NvFF2lB2 ((cid:126)ω+i(cid:126)Γω)2+−i(cid:126)Γ2∆ωN2F . (44) hvbaaerrnideosfpaoascncleBu−l)p1,ie∼rded√leuBvcee)ls,sw∆tohNiNFch(irn=ectau2lrl2n.thiAnactsreatahsceeosnLstLehqeeunneenurcmgey-,
F
In the above formula, ∆ (cid:39) 0.67Γ, and the maximum of the intraband peak
NF
is seen to be shifted to zero frequency, again in accor-
∆ ≡(cid:126)∆ω =E −E , (45)
NF NF NF+1 NF dance with Eq. (47).
denotes the intraband gap, with N being the index for Given the intrinsic large cyclotron gap of graphene,
F
the last occupied LL. E , the intraband contribution [Eq. (44)] controls the
1
9
photon energy, whereas the interband peaks appear at
energies (cid:126)ω (cid:38)2E =0.4 eV.
F
Wefinallyremarkthat,aslongasnottoolowmagnetic
fields are considered (B (cid:46) 0.1 T), the above considera-
tions are valid even close to room temperature (e.g., for
B =1T,thefirstLLcorrespondstoathermalenergyof
420 K).
3. The Hall conductivity
TheHallopticalconductivityofgraphene,σ (ω),fol-
xy
lows directly from Eq. (7); choosing i = y, j = x, we
obtain
J˜(ω)
σ (ω)=−g g × y , (50)
FIG.5: Thelongitudinalconductivityasfunctionofthepho- xy s v E˜ (ω)
x
ton energy for E = 0.2 eV. Other parameters as in Fig. 4.
F
where we have invoked graphene’s sixfold crystallo-
The solid horizontal (black) line shows graphene’s universal
ac-conductivity background [Eq. (1)]. graphic symmetry to write σxy(ω)=−σyx(ω). The cen-
tral quantity to be computed this time is the average
value of the current density operator along the y direc-
magneto-optical response of this material in the mi- tion; using Eqs. (6) and (25), we get
crowaveregionuptoterahertz(THz)frequenciesinsam- (cid:88)
J (t)=−ev (cid:104)n,k |j |n(cid:48),k (cid:105)(cid:104)cˆ† (t)cˆ (t)(cid:105).
ples with a finite electronic density (EF >E1). y F y y y n,ky n(cid:48),ky(cid:48)
The interband contribution, on the other hand, is im- n,n(cid:48)
(51)
portant both in samples with a low electronic density,
The non-zero matrix elements read
E < E , where it determines the full magneto-optical
F 1
ev
response (discarding the effect of phonons as discussed (cid:104)0,k |j |±1,k (cid:105)=−i√ F , (52)
y y y
above),andinsampleswitharbitrarycarriersconcentra- 2A
tions, for photon energies above the threshold for inter- (cid:104)n,k |j |n(cid:48),k (cid:105)=− evF (cid:2)sign(n(cid:48))δ
band transitions, (cid:126)ω = E +E (typically within y y y 2A |n|−1,|n(cid:48)|
NF NF+1 (cid:3)
the near-infrared region). +sign(n)δ , (53)
|n|,|n(cid:48)|−1
The positions of each interband peak can be obtained
(plus respective complex conjugates) where, in the last
from Eq. (47), with ∆ω replaced by
NF line, n,n(cid:48) (cid:54)= 0. Omitting the summation over k , the
y
∆Ω =(E +E )/(cid:126), (48) total current density reads
n n+1 n
i (cid:16) (cid:17)
with the constraint n ≥ NF. At finite electronic densi- Jˆy(t)= √2AevF c†1a0+v1†a0−h.c.
ties, N ≥ 1, typically one has (cid:126)∆Ω (cid:38) Γ, and thus we
arrive aFt the following useful approxinmation − 1 ev (cid:88)(cid:16)Pˆ(1)−Pˆ(2)+h.c.(cid:17)
2A F n n
n≥1
ωpinetaekr(n) (cid:39)∆Ωn ,n≥NF . (49) − 1 ev (cid:88)(cid:0)c†c −v†v +h.c.(cid:1). (54)
2A F n n+1 n n+1
For not too√small fields, B (cid:38) 0.1 T, the cyclotron gap n≥1
E (cid:39) 36× B meV·T−1/2 is larger than the LL broad-
1 The EOMs resemble those derived for the longitudinal
ening, and thus, in practice, the latter statement can be
conductivity [Eqs. (34)-(35)], the reason being that the
generalized to include the case of N =0.
F currentmatrixelementsinthexandy directionsarethe
Forgeneralparameters,theintensityofeachinterband
same except for phase factors [compare Eqs. (27) and
peak is no longer given by a simple expression, because
(28) with Eqs. (52) and (53)]. The final formula (after
manyinterbandtransitionscancontributetothespectral
regularization) yields,
weight close to each of the resonances ω (cid:39) ∆Ω . As
n
aaroreusnudlta, caosnωstavnatrvieasl,uethoefarbeaolutpσagrt. Eofxaσmxxp(lωes)aorsecsihlloawtens σxreyg(ω)= eh2 (cid:88)Nc iΛExnym nF(cid:126)(ωE+n)E−nF+(EiΓm), (55)
nm nm
in Fig. 4 for EF = 0 and in Fig. 5 for EF = 0.2 eV. In n(cid:54)=m=−Nc
the first case, we have N = 0 and therefore all the
F with matrix elements Λxy related to Λxx [Eq. (42)] ac-
observed peaks are interband-like. The second case has mn nm
cording to,
N = 4 and therefore one intraband peak is observed,
F
corresponding to transitions n = 4 → n = 5, at low Λxy =iΛxx (δ −δ ). (56)
nm nm |m|,|n|−1 |m|−1,|n|
10
the interband gap, (cid:126)∆Ω . The only exception is in-
NF
deed the interband transition −N → N +1 because,
F F
contrary to interband transitions involving larger energy
differences, it cannot be canceled by the other member
of the pair, n = −N −1∧m = N , since the latter is
F F
forbiddenviaPauliblockade;aschematicpictureisgiven
in Fig. 6.
The extremum points of the real part of the Hall con-
ductivity occurs at zero frequency, ω =0, and
ωintra (cid:39)∆ω ±Γ/(cid:126), (59)
± NF
ωinter (cid:39)∆Ω ±Γ/(cid:126), (60)
± NF
FIG. 6: Schematic of electronic transitions contributing to wherewehaveconsideredΓ/(cid:126)(cid:46)∆ω [seeEq.(46)and
the Hall conductivity of doped graphene in a magnetic field. text therein] and made use of Γ/(cid:126)(cid:28)NF∆Ω . The latter
Contrarytothelongitudinalconductivity(Fig.3),symmetry NF
consideration is true in virtually all situations except for
impliesthatonlyinterbandtransitionsinvolvingthesmallest
graphene at a low electronic density and small magnetic
energy difference, (cid:126)∆Ω = E +E , contribute to
NF NF NF+1 fieldB. Withinthesameaccuracy,theHallconductivity
σ . The remaining interband transitions (∆Ω , with n >
xy n
at ω =0 reads
N ) come in pairs whose contribution to the Hall current
F
mutually cancel as explained in the text: an example of a
pair of interband transitions that cancel is shown in zig-zag (cid:18)1−δ 1+δ (cid:19)(cid:18)4eBv2 (cid:19)
arrows. Note: The schematic picture is strictly adequate for Reσ (0)(cid:39)− NF,0 + NF,0 F ×σ ,
N ≥1;thecaseofN =0admitsasingletypeofelectronic xy ∆ω2 ∆Ω2 (cid:126)π g
F F NF NF
transition, namely, n=0→n=1. (61)
whereas at the point ωintra it is given by
±
btLaaiiksneeewddiosuensitσnhxgexG(EωrO)e,eMnthmfueentcrhetoisoudnltscocfiaonlrcciudtlheasetiwoHnitash.l4lt1hcoenrdeusuclttivoitby- Reσxy(ω±intra)(cid:39)F∆±ωNF (cid:18)1∆−ωδNNFF,0(cid:19)(cid:18)eBπΓvF2 (cid:19)×σg,
(62)
Symmetry considerations imply that only two terms
and for ωinter it reads
contribute in general for the zero-temperature Hall con- ±
ductivity, and hence the formula Eq. (55) can be consid- (cid:18)1+δ (cid:19)(cid:18)eBv2 (cid:19)
erably simplified. The first term is the intraband contri- Reσ (ωinter)(cid:39)F± NF,0 F ×σ ,
bution and reads, xy ± ∆ΩNF ∆ΩNF πΓ g
(63)
σxinytra(ω)= eh22(cid:126)l22vF2 ((cid:126)ω+1iΓ−)2δ−NF(cid:126),02∆ω2 , (57) wtehnesriteywoef hthaveeHdaelfilnpeedakFsω±de=pe±nd(cid:126)eωn/c(e(cid:126)ωon±thΓe).mTahgneeitnic-
B NF field intensity B is the same as for t√he longitudinal (in-
and the second is interband-like, connecting electronic traband) peaks [Eq. (46)], i.e., as ∼ B. Also, similarly
states with n=−NF and n=NF +1, and reads, to σxx(ω), in doped graphene with NF > 1, the inter-
band peak is very low compared with the intraband Hall
σinter(ω)= e22(cid:126)2vF2 1+δNF,0 . (58) peak for ∆ωNF (cid:28) ∆ΩNF. We, finally, remark that the
xy h l2 ((cid:126)ω+iΓ)2−(cid:126)2∆Ω2 anomalyassociatedwiththezeroenergyLLispresentin
B NF all the latter expressions via the factor 1+δ .
NF,0
A single interband transition play a role in setting the Figure 7 shows the Hall conductivity of graphene at
Hall conductivity, even for zero Fermi energy. This is a high magnetic field (B = 7 T) for N = 0 (top) and
F
at odds with the situation for σ (ω), where many non N = 4 (bottom), corresponding to neutral and highly
xx F
equivalentinterbandtransitionscontributetotheoptical doped graphene samples, respectively. The main charac-
spectral weight. To understand this peculiar feature of teristics of Reσ (ω) can be explained using Eqs. (61)-
xy
σ (ω), let us consider the second lowest interband res- (63). In particular, for doped graphene, the spectral
xy
onant energy, namely, ∆E = E −E : there weight concentrates around two well-separated parts of
2 NF+2 −NF−1
are two distinct sorts of interband transitions n → m the spectrum: (i) an intraband-dominated region (n =
involving such energy difference, namely, the pair n = 4 → n = 5 ), at low photon energies, with a maxi-
1
−N −2∧m =N +1andn =−N −1∧m =N +2, mum (minimum) intensity occurring at (cid:126)ω (cid:39)∆ +Γ(cid:39)
F 1 F 2 F 2 F + 4
whose Hall matrix elements read, Λxy = iΛxx and 30 meV ((cid:126)ω (cid:39) ∆ − Γ (cid:39) 16 meV ) [intensity equal
Λxy = −iΛxx , respectively. Whnen1ms1ubstitunt1imng1 into to (cid:39)10e2/h−((cid:39)−204e2/h), in accordance with Eq. (62)],
n2m2 n2m2
Eq. (55), these contributions cancel each other at T =0 and(ii)aninterband-dominatedregionn=−4→n=5,
because Λxx =Λxx . The same argument applies to at high photon energies, with a maximum (minimum)
all transitino2nms2involvni1nmg1an energy difference larger than intensity occurring at (cid:126)Ω (cid:39) (cid:126)∆Ω + Γ (cid:39) 413 meV
+ 4