Quantum magnetooscillations in the ac conductivity of disordered graphene

1.6 MB·English

by  U. Briskot

Quantum magnetooscillations in the ac conductivity of disordered graphene
U. Briskot,1,2 I. A. Dmitriev,1,2,3 and A. D. Mirlin1,2,4
1Institut fu¨r Nanotechnologie, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany
2Institut fu¨r Theorie der Kondensierten Materie and Center for Functional Nanostructures,
Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
3Ioffe Physical Technical Institute, 194021 St. Petersburg, Russia
4Petersburg Nuclear Physics Institute, 188300 St. Petersburg, Russia
(Dated: January 31, 2013)
The dynamic conductivity σ(ω) of graphenein the presence of diagonal white noise disorder and
quantizing magnetic field B is calculated. We obtain analytic expressions for σ(ω) in various para-
3
metric regimes ranging from the quasiclassical Drude limit corresponding to strongly overlapping
1
Landau levels (LLs) to the extreme quantum limit where the conductivity is determined by the
0
optical selection rules of the clean graphene. The nonequidistant LL spectrum of graphene renders
2
its transport characteristics quantitatively different from conventional 2D electron systems with

n parabolic spectrum. Since the magnetooscillations in the semiclassical density of states are anhar-
a monic and are described by a quasi-continuum of cyclotron frequencies, both the ac Shubnikov-de
J Haasoscillationsandthequantumcorrectionstoσ(ω)thatsurvivetohighertemperaturesmanifest

0 aslowbeatingontopoffastoscillationswiththelocalenergy-dependentcyclotronfrequency. Both
3 types of quantum oscillations possess nodes whose index scales as ω2. In the quantum regime of
separated LLs, we study both the cyclotron resonance transitions, which have a rich spectrum due

] to the nonequidistant spectrum of LLs, and disorder-induced transitions which violate the clean
l
l selectionrulesofgraphene. Weidentifythestrongestdisorder-inducedtransitionsinrecentmagne-
a
totransmissionexperiments. Wealsocomparethetemperature-andchemicalpotential-dependence
h
of σ(ω) in various frequency ranges from the dc limit allowing intra-LL transition only to the uni-
-
s versalhigh-frequencylimitwheretheLandauquantizationprovidesasmallB-dependentcorrection
e to the universal value of the interband conductivity σ=e2/4(cid:126) of the clean graphene.
m
. PACSnumbers: 72.80.Vp,73.43.Qt,78.67.Wj
t
a
m
I. INTRODUCTION spectrum makes it highly selective in the frequency do-
-
d main.
n Since its discovery in 2004,1 the two dimensional carbon While experimental data on quantum magnetooscil-
o allotrope – graphene – is attracting outstanding interest lations in graphene is limited (although growing),14–17
c
[ inthecondensedmattercommunity. Ithasbeenverified therearecomprehensivestudiesofrelatedphenomenafor
thatcarriersingrapheneshowalineardispersionrelation semiconductor 2DES where the effects both near and far
1 withFermivelocityv ≈106m/sandaregovernedbythe from equilibrium have been extensively studied.18,19 In
v masslessDiracequati0on.2,3 Duetothecarriers’Diracna- thelinearresponseregime,quantummagnetooscillations
6
ture, graphene shows remarkable properties. The clean in the ac conductivity were theoretically predicted20,21
4
2 densityofstates(DOS)islinearinenergyandvanishesat and observed22 already in the seventies. In Ref. 23
7 theDiracpoint,ν (ε)=|ε|/2πv2((cid:126)=1). Theabsenceof the theory was generalized to high mobility 2DES with
0 0
. scales at the Dirac point gives rise to a universal dc con- smoothdisorderpotential;thefindingsarealsoconfirmed
1
0 ductivity of the order of the conductance quantum. The in recent experiment.24
3 universal high-frequency conductivity σ = e2/4(cid:126) yields This work is (i) motivated by experimental and tech-
1 the constant absorption coefficient of 2.3% which makes nical advances in the graphene research and (ii) gen-
: graphene attractive for broadband optical applications.4 eralizes the theory on quantum magnetooscillations in
v
i The nontrivial topology of graphene leads to the char- the conventional 2DES mentioned above. We study
X acteristic half-integer quantum Hall (QH) effect5 and to quantum oscillations in Landau quantized graphene in
r the nonequidistant Landau level (LL) spectrum includ- the presence of disorder. Previous theoretical works
a
ing the unusual zeroth LL at ε = 0.6 Due to the large on the magnetoconductivity in graphene accounted for
(cid:112)
cyclotron energy ω = v 2eB/c compared to conven- the disorder in terms of a phenomenological broaden-
c 0
tional 2D electronic systems (2DES) the QH effect can ing in the single-particle spectrum25–27 or focused on
be observed up to room temperature.7 Recent rapid de- the dc magnetoconductivity.28,29 The optical conductiv-
velopments demonstrate great potential of graphene in ity has also been studied for vacancies within the T-
optoelectronics.8–12 From this perspective, the applica- matrix approximation.30 Here we perform a systematic
tion of a quantizing magnetic field creates a suitable en- calculation of the dynamic magnetoconductivity within
vironment for applications in e.g. laser physics.13 The the self-consistent Born approximation (SCBA).
quantizationintroducesatunableenergyscaleintheoth- The paper is organized as follows. In Sec. II we out-
erwise scale-free graphene, while the nonequidistant LL line the model of the SCBA in graphene and present re-
2
sults on the spectrum of disordered graphene in a mag- Within the SCBA,28,31 the self-energy is given by
netic field, in the semiclassical as well as in the quantum
regime. Further details on the calculation of the density Σˆ =(cid:90) d(cid:126)q W ((cid:126)q)e+iq(cid:126)·(cid:126)rˆGˆ e−iq(cid:126)·(cid:126)rˆ. (7)
ofstatescanbefoundinAppendixA.SectionIIIpresents νµ (2π)2 νησµ ησ
theformalismforthecalculationofthedynamicconduc-
tivity. The following Secs. IV and V are devoted to the Weakdisorderischaracterizedbythedisordercorrelator
dynamic conductivity in the regimes of strongly over- W((cid:126)q) which in the case of graphene is generally a fourth
lapping and well-separated LLs respectively. In Sec. VI rank tensor in valley and sublattice space. Here we con-
we calculate the high-frequency conductivity of a mod- sidershort-rangeimpuritieswhichdoscatterbetweenthe
erately disordered graphene. The summary of results is sublatticesbutdonotproduceanyintervalleyscattering.
presented in Sec. VII. Details on the calculation regard- In a given valley, this gives diagonal white noise disorder
ing Secs. IV and V can be found in Appendix B. with the correlator
W ((cid:126)q)=2πv2αδ ⊗δ , (8)
νησµ 0 νη σµ
II. LANDAU LEVEL SPECTRUM WITHIN
characterized by a single parameter α (cid:28) 1. For the di-
SCBA
agonal disorder, the self-energy is independent of the LL
indexandbecomesdiagonalinthesublatticespace; still,
In the following we assume that disorder does not mix
it carries an asymmetry between the sublattices a and b,
the two valleys in graphene and therefore calculate all
quantities per spin and valley. Correspondingly, to get Σˆ =diag(Σ ,Σ ). (9)
a b
thefullconductivityofgrapheneoneshouldmultiplythe
results for conductivity below by the degeneracy factor Theasymmetryisduetothefactthat–inagivenvalley
of 4. In the presence of a constant magnetic field in z- ofthecleangraphene–thewavefunctionofthezerothLL
directiontheelectronsinasinglevalleyofcleangraphene residesinonesublatticeonly. Inwhatfollows, wechoose
are described by the 2D Dirac equation thevalleysuchthatthewavefunctionofthecleanzeroth
LL resides in the sublattice b. The SCBA equation (7)
(cid:104) (cid:105)
Hˆ =v (cid:126)σ· p(cid:126)ˆ−eA(cid:126)((cid:126)rˆ) , A(cid:126)((cid:126)r)=−yB(cid:126)xˆ. (1) for disorder with the correlator (8) acquires the form
0
Here we have chosen the Landau gauge for the vector Σ = αωc2 (cid:88)+∞ ε−Σb(a) . (10)
potential A(cid:126). The vectors (cid:126)r = (x,y)T, (cid:126)xˆ = (1,0)T, and a(b) 2 (ε−Σ )(ε−Σ )−ω2n
a b c
n=1(0)
(cid:126)σ =(σ ,σ ) denote the Pauli matrices. The positions of
x y
the Landau levels (LLs) in clean graphene are given by Away from the zeroth LL the difference between the two
self-energy components is negligible, Σ (cid:39) Σ , [see dis-
(cid:112) a b
En =sign(n)ωc |n|, n∈Z. (2) cussion under Eq. (16) and Fig. 1] yielding
The optical selection rules of the clean graphene allow αω2 (cid:88)+∞ (ε−Σ)
transitions from the LL n to m if Σ (cid:39)Σ= c . (11)
a,b 2 (ε−Σ)2−ω2n
n=0 c
|n|−|m|=±1, (3)
Apart from specifics of the zeroth LL due to its pro-
therefore enabling both intra- [sign(n) = sign(m)] and nouncedsublatticeasymmetry,onecanconsidertwolim-
interband [sign(n)(cid:54)=sign(m)] transitions. Since the LLs iting cases we address separately below: (i) clean, or
movecloserathigherenergy,itisconvenienttointroduce quantumlimit,whenthedisorder-broadenedLLsremain
a local cyclotron frequency well separated and (ii) dirty, or classical limit when LLs
strongly overlap almost restoring the linear slope of the
ω =E −E ; (4) DOS at B = 0. Unlike conventional 2D systems with
n |n|+1 |n|
parabolic spectrum, in graphene the two cases (i) and
that is, the distance between neighboring LLs. In high (ii)frequentlycoexist: LLs,wellseparatedneartheDirac
LLs, n (cid:29) 1 (ε (cid:29) ω ), it approaches the quasiclassical point, start to overlap at higher energies where the local
c
cyclotron frequency of a massless particle cyclotronfrequency(5)stronglyreducescomparedtoωc.
ω2
ω (cid:39)ωloc = c . (5)
n c 2|ε| A. Separated Landau levels
The disorder is included into the self-energy Σˆ which We start with the limit of well separated LLs. In this
enters the impurity averaged electronic Green’s function case, the main contribution to the self-energy at energy
ε comes from the states in the nearest LL of the clean
Gˆ =(ε−Hˆ −Σˆ)−1. (6) graphenetowhichweassigntheintegernumberN closest
3
to ε2/ω2. LLs with index n (cid:54)= N contribute to logarith- B. Overlapping Landau levels
c
mic energy renormalization as detailed below.
ForN (cid:54)=0,thesublatticeasymmetrycanbeneglected, In view of the condition α (cid:28) 1 of weak disorder, the
and the solution to Eq. (11) for the retarded self-energy regime of strongly overlapping LLs is realized at high
reads [the advanced self-energy ΣA =(ΣR)∗] energies ε>ε (cid:39)ω /√α(cid:29)ω . Indeed, the LL number
ov c c
ε˜+E i(cid:113) Nov where LLs start to overlap is given by the condition
ε−ΣR = 2 N+2 Γ2N −(ε˜−EN)2, N (cid:54)=0, (12) ΓN (cid:39)ωN which gives [see Eqs. (2), (4), and (13)]
where the width of the Nth LL N (cid:39)α−1 (cid:29)1. (19)
ov
√
Γ = αω /Z(E ), N (cid:54)=0, (13)
N c N Accordingly, we assume Σ (cid:39) Σ and use the Poisson
a,b
summation formula to rewrite Eq. (11) into a rapidly
and the condition of applicability is Γ (cid:28) ω . Apart
N N
convergent sum in the Fourier space [see Eq. (A1)]. The
fromtheusualrenormalizationofenergybythefactorof
2 inside the LLs,21,31 in graphene the energy gets addi- small parameter that controls such an expansion is the
Dingle or coherence factor
tionally renormalized according to
ε˜=ε/Z(EN). (14) λ=e−2απ2ε˜2/ωc2 , (20)
Here the renormalization constant is
which is the analog of the Dingle factor δ = e−π/ωcτq in
2DES with parabolic spectrum and describes the smear-
Z(ε)=1−αln(∆ /|ε|), (15)
c
ing of quantum oscillations in the disordered system. To
and ∆c is the high energy cut-off of the order of the zeroth order in λ, we obtain a broadening of the single-
band width. Within the SCBA, the additional logarith- particle states at B =0,
mic correction describes the influence of states in dis-
tant LLs [n (cid:54)= N in Eq. (10)], similar to renormaliza- ΣR(cid:12)(cid:12)λ→0 =−i/2τq(ε˜). (21)
tion group (RG) corrections.5 For the type of disorder
we are investigating here, the SCBA corrections with- with the energy-dependent quantum scattering time
out magnetic field are known to be in quantitative ac- given by
cordance with RG calculations.5 Apart from the addi-
Z(ε)
tional logarithmic renormalization and from the non-
τ (ε˜)= . (22)
equidistant spectrum E of clean LLs, Eq. (12) repro- q απ|ε˜|
N
ducesthewell-knownsemicircularlawobtainedbyAndo
for2DESwithparabolicspectrum.21,31Fromtherequire- The renormalization constant Z(ε) which defines the
mentαln(∆ /|ε|)(cid:28)1,weobtainthattheresultsarejus- renormalized energy ε˜ = ε/Z(ε) is given by Eq. (15).
c
tified for energies above the exponentially small energy To first order in λ(cid:28)1, the self-energy acquires the form
scale ∆ e−1/α.
c i (cid:104) (cid:105)
In the vicinity of the zeroth LL, N = 0, one needs ε−ΣR =ε˜+ 2λei2πε˜|ε˜|/ωc2 +1 . (23)
2τ (ε˜)
to take into account the explicit sublattice asymmetry. q
Equation (10) yields the self-energies5
ε˜ i(cid:113) C. Density of states
ε−ΣR = + Γ2−ε˜2, (16)
b 2 2 0
ε−ΣR = 1+Z(E1)ε˜+i1−Z(E1)(cid:113)Γ2−ε˜2. (17) TheDOSperspinandvalleyν(ε)=−ImtrGR((cid:126)r,(cid:126)r,ε)/π
a 2 2 0 in the case of white noise disorder is given by
The width of the zeroth LL in both sublattices is
1
√ √ ν(ε)=− ImtrΣR. (24)
Γ = 2αω /Z(E )= 2Γ ; (18) 2π2v2α
0 c 1 1 0
√
the factor 2 comes from different degeneracy of the ze- The DOS in the sublattice b obtained numerically from
roth LL – the total number of states N = L2/2πl2 is the SCBA equation (10) is plotted in Fig. 1 as the thin
tot B
the same in all LLs, but in the zeroth LL all these states line. SeparatedLLsshowasemicircleDOS,seeEqs.(12)
reside in one sublattice. We observe that only a small and (16). The dashed line for the DOS in the sublattice
amount of the spectral weight ∝ 1−Z(E ) ∝ α (cid:28) 1 a shows that (i) in the zeroth LL the DOS transferred
1
is scattered into the afore empty sublattice a. At the into the sublattice a is small and vanishes if disorder is
same time, the strong renormalization of energy by the turnedoff,seeEq.(17)and(ii)athigherenergytheDOS
factorof2relatedtothehighdegeneracyofcleanLLsin (self-energy) in both sublattices is approximately equal.
Eqs. (12) and (16) is absent in Eq. (17). Only the log- Finally, the thick line, calculated according to Eqs. (23)
arithmic renormalization due to distant LLs with n (cid:54)= 0 and(24),illustratesthelimitofstronglyoverlappingLLs
remains. corresponding to large ε(cid:29)ε (which gives λ(cid:28)1).
ov
4
to massive electrons with mass, energy, and cyclotron
energy given by
m =ε˜∗/v2, (28)
eff 0
ε =ε˜/2−ω(eff)/2, (29)
eff c
ω(eff) =eB/m c, (30)
c eff
such that the local velocity in the parabolic band
v =(cid:112)2ε /m coincides with v . The term −ω(eff)/2
eff eff eff 0 c
accounts for the shift of LLs due to vacuum fluctuations
intheparabolicbandwhichisabsentduetothenontriv-
ial Berry’s phase in graphene, see also Eq. (33) below.
The effective cyclotron frequency ω(eff) coincides with
c
the renormalized local cyclotron frequency in graphene,
FIG.1. TheDOSindisorderedgraphene. Thethinlineisthe ω˜loc(ε˜∗) = ω2/2|ε˜∗| = Z(ε∗)ωloc(ε∗). If one addition-
c c c
DOSinthesublatticebobtainednumericallyfromtheSCBA
ally introduces the renormalized τ˜ (ε˜) = τ (ε)/Z(ε) =
equation (10) for α = 0.01. For the first two levels we show q q
1/πα|ε˜|, equation (26) transforms into conventional ex-
the DOS in the sublattice a as the dashed line. Already in
pression for parabolic band, specifically,
the second LL there is no noticeable difference between the
a and b sublattice. The thick curve shows our result for the ε˜∗ m
DOSaccordingtoEqs.(23)and(24). Itisingoodagreement ν˜ = → eff, (31)
0 2πv2 2π
with the numerical results for high energies. 0
λ=e−2απ2ε˜2/ωc2 → e−π/ω˜c(eff)τ˜q(ε˜), (32)
2πε˜|ε˜| 2πε
D. Semiclassical regime → eff +π. (33)
ωc2 ωc(eff)
Here we address the semiclassical regime of high LLs
in graphene, N (cid:29) 1, additionally assuming that these
√ III. DYNAMIC CONDUCTIVITY
levels strongly overlap, ε (cid:29) ε (cid:39) ω / α. The aim
ov c
is to provide a link to the the results in the corre-
Below we calculate the dynamic conductivity of disor-
spondentparabolicbandwitheffectiveenergy-dependent
dered graphene in the presence of a quantizing magnetic
mass such that the local characteristics of graphene and
field. We use the Kubo formula for the real part of the
conventional2DESwithsuchparabolicspectrumathigh
diagonal conductivity
LLs are identical.
In the limit λ→0, Eqs. (21) and (24) yield the B =0 (cid:90) dεf −f
renormalization of the clean DOS ν by disorder σ(ω)= ε ε+ω K(ε,ε+ω), (34)
0 4π ω
ε ε˜
ν = → ν˜ ≡Z(ε)ν(ε˜)| = . (25) to calculate the magnetoconductivity in the linear re-
0 2πv2 0 λ→0 2πv2
0 0 sponse. Here fε is the equilibrium Fermi-Dirac distribu-
tion function, and the conductivity kernel is given by
According to Eq. (23), the correction ∝λ reads
(cid:104)(cid:16) (cid:17) (cid:16) (cid:17) (cid:105)
K(ε ,ε )=−tr GˆR−GˆA ˆj GˆR −GˆA ˆj .
2πε˜|ε˜| 1 2 ε ε1 x ε2 ε2 x
ν˜ ≡ν˜(ε˜)−ν˜ =2ν˜ λ cos , (26)
osc 0 0 ω2 (35)
c
From the Hamiltonian (1), the current density operator
TheenergyrenormalizationinducedbytheLandauquan-
tization shows similar (but phase-shifted by π/2) oscilla- (cid:126)ˆj = e ∂Hˆ = ev0 (cid:126)σ, (36)
tions L ∂(cid:126)k L
λ 2πε˜|ε˜| where L is the length of the system. Due to the linear
∆ReΣ(ε˜)= sin . (27)
τ (ε˜) ω2 spectrum the current operator does not depend on the
q c
magnetic field. Since the two valleys are decoupled, we
Apart from (i) the large-scale logarithmic renormaliza- calculate the conductivity per spin and valley.
tion (25) which is specific for the linear spectrum of The effect of the disorder averaging in Eq. (35) is
graphene and (ii) a phase shift related to a nontriv- twofold: (i) the bare Green’s function is replaced by the
ial Berry’s phase of graphene the local characteristics impurity averaged Green’s function (6), (ii) the summa-
of graphene and 2DES with parabolic spectrum at high tion of the diagrams 11(c) leads to vertex corrections
LLs are identical as expected. Namely, quasiparticles in to the current operator (36). In 2DES with parabolic
graphene at energy ε˜close to ε˜∗ (cid:29)ω behave equivalent spectrumandwhitenoisedisorder,thevertexcorrections
c
5
are absent. By contrast, in graphene the vertex correc- It follows from Eq. (23) that to zeroth order in λ only
tions are present for the diagonal white noise disorder as theRA-sectorofEq.(40)contributestotheconductivity
well. TheyoriginatefromthenontrivialBerry’sphaseof for the intraband processes. For ω = |ε −ε | (cid:28) ε the
1 2 F
Dirac fermions. We present details on the calculation of kernel (39) has a Drude form with the energy-dependent
the conductivity including vertex corrections in App. B. broadening (22)andlocalcyclotronfrequency (5). Ifthe
As expected, we find that in the quasiclassical regime temperatureT (cid:28)ε , theseparameterscanbeevaluated
F
of strongly overlapping LLs the vertex corrections give on-shell. The conductivity σ = σ + σ in this
D D,+ D,−
rise to the replacement of τ by the transport scattering regime reads
q
time τ =2τ in the Drude part of the conductivity, see
tr q
Eq. (41) below, while the quantum time appears only in σ (ω)= 1 D/τtr . (41)
the quantum corrections related to the Landau quanti- D,± 4π(ω±ωloc)2+τ−2
c tr
zation. In particular, τ enters the Dingle factor λ, see
q
Eq. (20). Due to the vertex corrections the transport scattering
time τ = 2τ replaces the quantum scattering time
tr q
as explained in App. B. The Drude weight D for Dirac
IV. OVERLAPPING LANDAU LEVELS fermions is
e2|ε |
In this subsection we consider the case of highly doped D = F , (42)
2
graphene with the Fermi energy ε (cid:29) ω ,ω. Therefore,
F c
onlyintrabandprocessesarepossibleandonecanneglect which produces the standard Drude weight D = e2ε
eff
the sublattice asymmetry in the self-energy. It follows
in the correspondent parabolic band, see Eq. (29). The
that the conductivity in this regime should not change
Drude part of the conductivity in graphene was also ob-
inthepresenceofintervalleyscattering. Usingtheconti-
tained in Refs. 32 and 33. Without magnetic field the
nuity equation, the conductivity kernel (35) is expressed Drudeformhasbeenconfirmedexperimentally34 though
in terms of density-density correlators Π. Their general
deviations (possibly due to interactions) were also ob-
form is given in Eq. (B14), which simplifies to
served.
On top of the semiclassical Drude conductivity quan-
ΠRR(RA) =ωc2 (cid:88)∞ (cid:20)GR (ε )+GR (ε )(cid:21) tum oscillations are superimposed. There are two major
ε1,ε2 2π n+1,− 1 n+1,+ 1 and competing damping mechanisms present. Finite T
n=0 (37)
(cid:20) (cid:21) leads to the thermal damping of the Shubnikov-de Haas
× GR(A)(ε )+GR(A)(ε ) (SdH) oscillations in the ac- and dc-response. Scattering
n,− 2 n,+ 2
off disorder also smears quantum oscillations and this is
captured by the coherence factor λ. The latter mech-
if the sublattice asymmetry is absent. Here we intro-
anism is dominant for T below the Dingle temperature
duced the chiral Green’s functions
T =1/2πτ .35
D q
GR(A)(ε˜)= 1 . (38) The leading order quantum corrections which are
n,± ε˜−ImΣR(A)∓|E | strongly damped by finite temperature describe SdH os-
n
cillations in the dynamic and dc conductivity. The cor-
In the following we write ε for ε˜for brevity. In terms of responding contribution to the kernel K = K +K ,
+ −
the Π-correlators, the kernel (35) acquires the form Eq. (35), reads
K(ε ,ε )=e2Re(cid:104)(cid:16)(cid:2)ΠRA (cid:3)−1−απ(cid:17)−1 K(1)(ε ,ε )=4πσ
1 2 ε1,ε2 ± 1 2 D,±
−(cid:16)(cid:2)ΠRR (cid:3)−1−απ(cid:17)−1+{ε ↔ε }(cid:105). (39) ×(cid:26) 2a2± (cid:20)ν˜osc(ε2) + ν˜osc(ε1)(cid:21)
ε1,ε2 1 2 a2±+1 ν˜0(ε2) ν˜0(ε1) (43)
(cid:20) (cid:21)(cid:27)
a ∆ReΣ(ε ) ∆ReΣ(ε )
As previously, we use the Poisson formula to rewrite the + ± 2 − 1 ,
sumsoccurringinEq.(37)asrapidlyconvergentsumsin a2±+ω2/ε21 1/τq(ε2) 1/τq(ε1)
the Fourier space. With the self-energies for overlapping
LLs from Eq. (23), we obtain where we used the abbreviation a± =τtr(ω±ωcloc).
ThefirstterminEq.(43)isproducedbythequantum
2(cid:0)ε −ΣR(cid:1)(cid:0)ε −ΣR(A)(cid:1) correction to the DOS, Eq. (26), while the second term
ΠRε1R,ε(2RA) = (cid:0)ε −ΣR1 (cid:1)2−ε1(cid:0)ε −2 ΣR(εA2)(cid:1)2−ω2 accounts for the energy renormalization, Eq. (27). The
1 ε1 2 ε2 c (40) firsttermisdominantawayfromthecyclotronresonance,
(cid:26) (cid:27)
see Fig. 2. In this case a (cid:29) 1, and the kernel (43)
× sign(ε )τ ΣR − sign(ε )τ ΣR(A) . ±
1 q,ε1 ε1 2 q,ε2 ε2 acquires the form
(cid:20) (cid:21)
Weorganizethefollowinganalysisinordersofλ. Details ν˜ (ε ) ν˜ (ε )
K(1)(ε ,ε )(cid:39)8πσ osc 2 + osc 1 . (44)
of the calculation are presented in Appendix B. ± 1 2 D,± ν˜ (ε ) ν˜ (ε )
0 2 0 1
6
6 1
4
2
0 0
2
4
6
1
0 1 2 3 4 5 6 7 8 9 10 11 12
0 1 2 3 4 5 6
FIG. 2. The prefactors from Eq. (43): (a) 2a2/(a2 +1), (b) FIG. 3. The SdH oscillations in the dynamic conductivity
− −
a /(a2 +ω2/ε2), (c) 2a2/(a2 +1), (d) a /(a2 +ω2/ε2), (solid line) and the envelope function showing slow modu-
− − F + + + + F
where a = τ (ω ± ωloc). All curves are calculated for lation due to nonequidistant spectrum of LLs (dashed line)
± tr c
εF/ωc =6andα=0.01. Awayfromthecyclotronresonance, calculated according to Eq. (45) for εF/ωc =6.
a (cid:29) 1, the subleading effects due to the energy renormal-
±
ization (thin lines) can be ignored.
Detuning of the Fermi energy from the center
of the LL shifts the frequency nodes such that
This form of the kernel is expected from a golden ε2 +ω2 =ω2(n/2+1/4), n ∈ N holds. Note that al-
F c
rule consideration where the conductivity (in the clas- though we assumed ω (cid:28)εF the frequencies in Eq. (46)
sically strong field, a (cid:29) 1) is determined by the are well within the range of our approximation for not
±
product of initial and final DOS ν(ε)ν(ε + ω). To too large k in view of εF (cid:29) ωc. For the values of ω
first order in the coherence factor λ, this yields in Eq. (46) the DOS oscillations acquire a relative phase
ν(ε)ν(ε+ω)−ν2 (cid:39)ν [ν (ε+ω)+ν (ε)]∝λ. shift of approximately (2k + 1)π between ε and ε + ω
0 0 osc osc
Forsmall T,ω (cid:28)ε we evaluatethe smoothfunctions due to the ε-variation of the period which leads to the
F
σ and λ on-shell but keep the energy dependence in destructive interference.
D,±
the rapidly oscillating parts while calculating the con- At T >TD, the temperature smearing dominates over
ductivity (34). The asymptotics of the resulting Fresnel the quantum mechanical smearing. However, there are
integrals provides a reasonably simple expression for the additionalquantumoscillations∝λ2 thatsurvivehigher
correction to the Drude conductivity away from the cy- temperaturesandbecomeexponentiallylargerthanSdH
clotron resonance at T =0, oscillationsatT (cid:29)TD. Theyarewellknownforsystems
with a parabolic spectrum21,23 and originate from the
σ(±1)(ω)=4σD,±λcos(cid:18)2π(ε2Fω2+ω2)(cid:19)2sπiωn/ω2ωπcloωlcoc . (45) tweirtmhsmineatnheenRerAg-yseεc1t+orεi2nbEuqt.d(o40o)scwilhlaictehwdoithnottheosecnilelragtye
c c differenceω ≡|ε −ε |. Suchω-oscillationsareinsensitive
1 2
Equation (45) describes SdH oscillations in the dynamic to the position of the chemical potential with respect to
conductivity illustrated in Fig. 3. LLs. Therefore, averaging over the temperature window
Thefastharmonicoscillationswiththelocalcyclotron accordingtoEq.(34)doesnotleadtoadditionaldamping
frequency ω(loc) are similar to those known for systems incontrasttoSdHoscillations. Therelevantcontribution
c
at T (cid:29)T is
with a parabolic spectrum except for the absence of the D
vacuum shift due to the Berry’s phase of Dirac fermions (cid:18) a2 −1 2πω
and hence a shift in the zero of energy to εF/2−ωc(loc). σ±(q)(ω) = 2σD,±λ2 a2±+1 cos ωloc
± c
However, the magnetooscillations in Fig. 3 further (47)
show a slow modulation on the scale of the cyclotron + 2a± sin2πω(cid:19) sin2πωωc22 .
frequency ωc at the Dirac point. This beating in the a2 +1 ωloc 2πω2/ω2
± c c
quantum oscillations is due to the difference of the cy-
clotron frequencies at the initial and final state of the Away from the cyclotron resonance, a± (cid:29) 1, the quan-
optical transition. Remarkably, thequantum oscillations tum correction (47) acquires a simpler form,
shownodesstemmingfromdestructiveinterferenceofthe
densityofstatesintheinitialandfinalstate. IftheFermi σ(q)(ω) (cid:39) 2σ λ2 sin2πωωc22 cos2πω . (48)
energy ε is situated in the center of a LL, the node oc- ± D,± 2πω2/ω2 ωloc
F c c
curs at
ω =√2k+1 ωc , k ∈N. (46) tedTihneFciogn.d4u.cAtipviatrytafrtohmigthhteeimntpeegreartucyreclTotr(cid:29)onTrDesiosnpalnocte-
2
7
in either sublattice a or b for K (cid:54)=0, and
1 (cid:112)
ν (ε)= Re 2Γ2−(ε)2, (50)
K=0 l2π2Γ2
B
residing in sublattice b for K = 0. The total DOS, in-
1
cluding both valleys, spin components and sublattices is
givenby8ν forK (cid:54)=0andby4ν forK =0. The
K(cid:54)=0 K=0
width of the individual resonances is then 2(Γ +Γ ),
N K
determined by the width of the DOS in L and L .
K M
In the following we distinguish the cyclotron resonance
1 1 1 1 peaks, ||K|−|M||=1, from the disorder-induced peaks,
7 6 5 4 ||K|−|M|| (cid:54)= 1. The latter vanishes if the disorder is
switchedoff,whiletheformersurviveastheyrespectthe
clean selection rules in graphene.
FIG. 4. Magnetoconductivity σ (ω)+σ(q)(ω) of graphene
D,± ± Using ω (cid:29) Γ for the Π-correlators (40), we cast the
at high temperatures [normalized to the Drude value (41)], c
conductivity (34) in the form
calculated according to (48) for ε /ω = 6. Vertical lines
F c
mark the position of integer harmonics of the cyclotron reso-
nance. σ(ω)=σ0 (cid:88) P(K,M) ωΓc2cKΓcM F˜KM(ω,µ,T), (51)
K M
K,M
harmonics ω =nωcloc, n∈N, that have an analog in sys- where σ0 =e2/4π2 and cK =1+δK,0, see App. B and C
tems with a parabolic spectrum, we encounter again an for details. The coefficients
additionalmodulationofthequantumoscillationsdueto
the presence of multiple cyclotron frequencies. Since the P(K,M)=1, ||K|−|M||=1, (52)
temperature-stable quantum corrections ∝λ2 are insen-
sitive to the position of the Fermi energy with respect to for the cyclotron resonance transitions; for the disorder-
LLs, the positions of the nodes, ω/ω = (cid:112)n/2, n ∈ N, induced transitions not involving the zeroth LL (||K|−
c
arethereforesolelydeterminedbytheprobingfrequency |M||=(cid:54) 1, K (cid:54)=0 and M (cid:54)=0),
ω.
Γ2(E2 +E2 )
P(K,M)= M K +{M ↔K}; (53)
2(E2 −E2 +ω2)2
K M c
V. SEPARATED LANDAU LEVELS finally, for the disorder-induced transitions involving the
zeroth LL (||K|−|M||=(cid:54) 1, K =0 or M =0)
WenowaddressthequantumregimeofwellresolvedLLs,
Γ2(E2 +ω2)
whichinvolvesLLswithnumbers|n|<Nov,seeEq.(19). P(K,M)= K c +{M ↔K}. (54)
In the following, LK denotes the K-th LL. 2(EM2 −EK2 +ωc2)2
Our results for the dynamic conductivity at T =0 are
The function F˜ in Eq. (51) describes the shape of
illustrated in Fig. 5. As long as LLs are separated, the KM
the peaks in the conductivity; its general form is given
conductivity is the sum of contributions from individual
in App. C, see Eq. (C5). In the case of the disorder-
transitions between L and L . Despite LLs are sepa-
K M inducedtransitions||K|−|M||=(cid:54) 1,thevertexcorrections
rated, the corresponding peaks in σ(ω) overlap. Indeed,
are negligible: the conductivity kernel can be calculated
some transitions are degenerate (for instance L → L
0 1 usingthebarepolarizationbubbleanddependsonenergy
and L → L in Fig. 5). In other cases, the excita-
−1 0 only via the product ν (ε)ν (ε+ω). Correspondingly,
tion energies for different transitions may become close K M
for ||K| − |M|| =(cid:54) 1 the function F˜ in Eq. (51) is
toeachotherduetothenonequidistantspectrumofLLs. KM
reduced to the bare F (cid:39)F˜ , given by
From the Kubo formula (34) we find that to leading KM KM
order in Γ/ω the conductivity kernel is proportional to
c l4π4Γ Γ
the density of the initial and final states. In this section, F (ω,µ,T)= B K M
KM c c
weneglecttherenormalizationoftheenergydescribedby K M (55)
(cid:90) f −f
Z(ε), Eq. (15), assuming that the contribution of states × dε ε ε+ω ν (ε)ν (ε+ω),
with energies |ε| < ∆ e−1/α is negligible. We thus put ω K M
c
Z(ε) = 1 in Eqs. (12)-(18), which gives ImΣR = 0 in
a see Eq. (C1). At high T (cid:29)Γ and K (cid:54)=M, the distribu-
Eq. (17). Using Eq. (24), we obtain the partial DOS
tion function can be considered smooth on the scale Γ;
Eq. (55) reduces to
1 (cid:112) √
ν (ε)= Re Γ2−(ε−E )2, Γ= αω ,
K(cid:54)=0 l2π2Γ2 K c 4Γ
B (49) FKM(ω,µ,T)= 3ωK (fK −fM)FKM(δω/Γ), (56)
8
thecaseofdisorder-inducedtransitions,theconductivity
12 (a)
vanishes in the clean limit Γ→0.
10
1.0
8
0.8
6
4 0.6
2 0.4
0 0.2
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
(b) 0.0 0.5 1.0 1.5 2.0 2.5 3.0
6
FIG. 6. Solid line: The function F , Eq. (C3), describing
KM
the shape of the disorder-induced conductivity peaks (||K|−
4 |M|| =(cid:54) 1) in the regime T,ωc (cid:29) Γ, see Eqs. (56) and (51).
Dashedline: ThefunctionF˜ ,Eq.(C6),replacingF in
KM KM
the case of the cyclotron resonance (||K|−|M|| = 1). The
strength and the width of the cyclotron resonance is reduced
2
duetothevertexcorrectionsascomparedtoF thatwould
KM
resultfromthecalculationwiththebarepolarizationbubble.
Here we assume K,M (cid:54)=0, in which case F and F˜ do
0 KM KM
not depend on K and M.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
7 (c)
Forthecyclotronresonance||N|−|M||=1,thevertex
6 corrections are important leading to a more complicated
ε-dependence of the kernel; the corresponding functions
5 F˜ and F˜ entering Eq. (51) are given in App. C. The
behaviorofthefunctionsF andF˜ isillustratedinFig.6.
4
The figure shows that the vertex corrections reduce the
3 strength and width of the cyclotron resonances.
We now discuss several limiting cases which describe
2
the rich pattern of resonances in Fig. 5.
1
0
A. Intra-Landau level transitions
0.0 0.5 1.0 1.5 2.0 2.5 3.0
We start with the case ω (cid:46) Γ enabling only intra-LL
transitions. This includes the dc limit ω → 0 and two
FIG. 5. The dynamic conductivity according to Eq. (51) for
distinct temperature regimes: a) T (cid:28) Γ and b) T (cid:29) Γ.
different chemical potentials and T =0. The cyclotron reso-
The contribution of the intra-LL transitions L → L
nancesareindicatedbythesolidarrowsandtheparticipating 0 0
and L →L is illustrated in Fig. 5.
LLs,whereL denotesthen-thLL.Thedashedarrowsmark 1 1
n
the disorder-induced transitions. The spectrum is calculated a) When the temperature is the smallest scale, T (cid:28)
for a disorder strength α=0.01. Γ, only the level LN, determined by the position of the
chemical potential, contributes. The conductivity (51)
becomes (σ =e2/4π2)
0
where we denote f = f(E ) and δω =ω−E +E .
K K M K
The function FKM is given by Eq. (C3), see also Fig. 6. σ =σ cNωc2 F (ω,µ,T). (58)
For T (cid:29)Γ and K =M (intra-LL transitions), 0 ω2 NN
N
F (ω,µ,T)= ΓK FKK(δω/ΓK) , (57) In the dc limit ω →0, Eq. (58) acquires the form28
KK 3T cosh2[(E −E )/2T]
K N c ω2 (cid:18) (µ−E )2(cid:19)
σ =σ N c 1− N . (59)
whereN istheLLclosesttothechemicalpotentialµ. In 0 ω2 Γ2
N N
9
NotethatF isclosetounityforsmallfrequencies, hence
the dependence on the disorder strength drops out for
µ = E . In this regime, the dependence σ ∝ τ−1 ∝
N tr
α characteristic for classically strong magnetic fields,
ω τ (cid:29)1,isexactlycompensatedbytheincreasedDOS
N tr
inside the LL: the average of ν(ε)2 over L is propor-
N
tional to ω2/Γ2 ∝α−1.
c
b) T (cid:29)Γ, hence T (cid:29)ω: With the help of the high T
0.1 0.2
expression (57), the conductivity (51) reads
σ =σ (cid:88)∞ cnωc2ΓnFnn(ω/Γn) . (60)
0 3ω2T cosh2[(E −E )/2T]
n=−∞ n n N 0 1 2 3 4 5 6
The summation limits are sent to infinity since the con-
tributionoflargeenergieswhereLLsoverlapisexponen-
FIG. 7. The temperature dependence of the conductivity for
tially small.
T (cid:29)Γ >ω andµ=ω . Theinsetshowsthesmalltemper-
In Eq. (60) the zeroth LL is special since its width N c
√ aturebehavioraccordingtoEqs.(61)and(62)(dashedlines),
is bigger by a factor of 2 and its oscillator strength is
describing the 1/T decrease followed by a saturation into a
enhanced by a factor of two. However, its contribution T-independentregime. Thethicklineiscalculatedaccording
is significant only for N = 0 and T (cid:46) ωc. Note that to Eq. (60). The shaded area indicates the regime T (cid:46) Γ,
for all other levels (n (cid:54)= 0) the function F (ω/Γ ) does where the conductivity saturates at a µ-dependent value, see
nn n
not depend on the LL index n. We obtain three high T Eq. (58).
regimes:
b.1) For Γ(cid:28)T (cid:28)ω only L contributes, while the
N N
contributionfromthelevelsfartherawayfromthechem-
ical potential is exponentially suppressed,
c ω2Γ
σ =σ N c N F (ω/Γ ). (61)
0 3ω T NN N
N
Theconductivityσ ∝T−1 isproportionaltotheslopeof
the Fermi function in L , f (cid:39)1/2−(ε−E )/4T. Note
N N
thatthewidthofthepeakinEq.(61)forN =0isbigger
√ √
by a factor 2 in view of Γ = 2Γ.
0
b.2) ω (cid:28)T (cid:28)µ<ε : As µ (cid:29) T, the influence of
N ov
zeroth LL can be neglected. Furthermore, at T (cid:29) ωN 0.1 0.2 0.3 0.4 0.5 0.6
the sum in Eq. (60) can be converted into an integral,
which gives
FIG. 8. The temperature dependence of the conductivity for
64Γ|µ|3
σ =σ0 3ω4 FNN(ω/Γ). (62) ωtrib<utΓionanfdro|mµ|t<heΓz0ero(cid:28)thTL.LT[Ehqe.d(a6s1h)e]danlidneasllaorethtehrelecvoenls-
c
[Eq. (63)]. The thick line is calculated according to Eq. (60).
The conductivity in this regime is T-independent. The shaded area indicates the regime T (cid:46)Γ, where the con-
b.3) For ω ,µ(cid:28)T <ε we obtain ductivity saturates at a µ-dependent value, see Eq. (58).
N ov
96ζ(3)ΓT3
σ =σ F (ω/Γ), (63)
0 ω4 NN T-independent regime (62) is not present since the cor-
c
responding conditions cannot be met. At small T the
where ζ(z) is the Riemann ζ-function. We observe that contribution from L dominates. It decreases due to
0
the temperature takes the role of the chemical potential thermal smearing within the zeroth LL. With increas-
in Eq. (62). ing T the influence of L decreases and the other LLs
0
The dc-limit ω → 0 of Eqs. (61), (62) and (63) is ob- take over, which leads to an enhancement of the conduc-
tained using F (0) = 1, and hence shows the same tivity due to thermal activation of higher energy states.
NN
dependence on µ and T. In Figs. 7 and 8 the shaded areas indicate the crossover
The overall T-dependence of the dc conductivity totheregime(58), wheretheconductivitysaturatesata
(ω →0) for T (cid:29) Γ is shown in Fig. 7 (for µ (cid:39) ω ) µ-dependent value. The unusual T3 and µ3 dependence
c
and in Fig. 8 (for µ (cid:39) Γ). In both cases it shows a originates from the interplay between the energy depen-
nonmonotonous temperature dependence. In Fig. 8 the dence of the transition rates (53) and (54) and the level
10
spacing(5). ItisthereforespecialforDiracfermionsand b) ||K|−|M|| (cid:54)= 1: According to Eqs. (51) and (53),
hence graphene. the partial contribution of the disorder-induced transi-
Inthedclimitω →0,Eqs.(63)and(62)reproducethe tions L →L in the case K,M (cid:54)=0 is
K M
resultsofRef.29,wheretheregimeT (cid:29)ω wasstudied
N
for the dc conductivity. In addition, we find the 1/T 2(|K|+|M|)[1+(|K|−|M|)2]
σ =σ F (ω,µ,T).
behavior, Eq. (61), in the low-T range of Γ(cid:28)T (cid:28)ω . KM 0 [(|K|−|M|)2−1]2 KM
N
(67)
For M =0 or K =0,
B. Inter-Landau level transitions
c c Γ2
σ =σ K M F (ω,µ,T)
Figure 5 demonstrates that the nonequidistant LL spec- KM 0 2ΓKΓM KM
trum of graphene leads to a rich spectrum of resonances. (K+1)[M −(K+1)]2+(M +1)[K−(M +1)]2
× .
Their strength depends strongly on the chemical po- [(M −K)2−1]2
tential. In what follows we discuss separately a) the (68)
cyclotron resonance transitions and b) disorder-induced
transitions. where the function F is given by Eq. (C2).
KM
To make the result more transparent, we rewrite
Eq. (67) for a particular set L → L (M > 0) of
−M M
mirror transitions and for T (cid:29)Γ,
10
4ω2Γ
8 σ(ω)=σ c (f −f )F (δω/Γ). (69)
0 3ω2 ω −M M MM
M
6
The function F is illustrated in Fig. (6). Since ω (cid:39)
MM
2E , it holds
4 M
ω2Γ ωΓ
2 c = . (70)
ω2 ω ω2
M c
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 We see that the strength of the disorder-induced transi-
√
tion is enhanced with increasing frequency ω ∝ M.
Our results are illustrated in Figs. 5 and 9. The
FIG. 9. The dynamic conductivity, Eq. (51), for µ = 0 and
strongest response corresponds to the cyclotron reso-
three different temperatures. The spectrum is calculated for
nances. Indeed, Eqs. (64) - (66) show that the cyclotron
a disorder strength α=0.01.
peaks are a factor ω2/Γ2 stronger than the disorder-
c
induced peaks (67) - (69). Additionally, we observe that
a) We start with the cyclotron resonances, ||K| −
theinterbandcyclotronresonanceissuppressedbyafac-
|M|| = 1. Apart from the transition L → L also the
K M tor ω /ω < 1 in comparison to the intraband cyclotron
transition L → L needs to be taken into account, c
−M −K resonances, for which ω /ω ≥1.
since it has the same transition energy. c
Among the disorder-induced transitions, the intra-LL
a.1) For T (cid:28)Γ, Eq. (51) reduces to
andthemirrortransitionsL →L arethestrongest.
−M M
The latter become more pronounced with increasing fre-
c c σ ω2
σ(ω)= K M 0 c [F˜ (ω,µ,T)+F˜ (ω,µ,T)]. quency. The presence of the L → L and the L → L
Γ Γ KM −M,−K 1 3 1 2
K M peaks in Figs. 5(b) and (c) indicates that the chemical
(64)
potential lies in the first LL. Furthermore, the intensity
a.2) For T (cid:29)Γ, Eq. (51) yields
ratio of the L → L and the L → L peaks provides
1 2 0 1
σ(ω)=σ0cK3cMΓω4ωc2(fK−fM+f−M−f−K)F˜KM(cid:18)δΓω(cid:19). icnofmorpmaraitsioonnoofnthtehecaolccucluaptaedtiosnpeocftrLa1i.nFFiogr.i5ntsotatnhceem, tehae-
surements reported in Ref. 15 shows that in this par-
(65)
ticular experiment the chemical potential was lying in
Heref areintroducedbelowEq.(56),andthefunctions
K L ; since the L → L resonance was stronger than the
F˜ and F˜ are given by Eqs. (C5) and (C6). For M = 0, 1 0 1
L → L resonance, we conclude that L was less than
K =−1 the occupation of L drops out, 1 2 1
0 half-filled. No measurements so far have reported the
8ω2 disorder-induced transitions. However, a closer inspec-
σ(ω)=σ0 3Γωc (f−1−f1)F˜−1,0(δω/Γ). (66) tion of Fig. 1(a) and (b) from Ref. 17 reveals a peak at
roughly140meVwhichshouldbeattributedthedisorder-
In Eqs. (65) and (66), µ and T enter via f only. induced transition L → L according to its position
K,M −1 1
The corresponding expression in brackets takes values with respect to the cyclotron resonances L → L and
0 1
between zero and one. L →L .
1 2