Drude weight, plasmon dispersion, and pseudospin response in doped graphene sheets

2.8 MB·English

by  Marco Polini

Drude weight, plasmon dispersion, and pseudospin response in doped graphene sheets
Marco Polini,1,∗ A.H. MacDonald,2 and G. Vignale3
1NEST-CNR-INFM and Scuola Normale Superiore, I-56126 Pisa, Italy
2Department of Physics, University of Texas at Austin, Austin, Texas 78712, USA
3Department of Physics and Astronomy, University of Missouri, Columbia, Missouri 65211, USA
Plasmonsinordinaryelectronliquidsarecollectiveexcitationswhoselong-wavelengthlimitisrigid
center-of-mass motion with a dispersion relation that is, as a consequence of Galileian invariance,
unrenormalized by many-body effects. The long-wavelength plasmon frequency is related by the
f-sum rule to the integral of the conductivity over the electron-liquid’s Drude peak, implying that
transportpropertiesalsotendnottohaveimportantelectron-electroninteractionrenormalizations.
InthisarticlewedemonstratethattheplasmonfrequencyandDrudeweightoftheelectronliquidin
9 adopedgraphenesheet,whichisdescribedbyamasslessDiracHamiltonianandnotinvariantunder
0 ordinary Galileian boosts, are strongly renormalized even in the long-wavelength limit. This effect
0 is not captured by the Random Phase Approximation (RPA), commonly used to describe electron
2 fluids. It is due primarily to non-local inter-band exchange interactions, which, as we show, reduce
both the plasmon frequency and the Drude weight relative to the RPA value. Our predictions can
n
be checked using inelastic light scattering or infrared spectroscopy.
a
J

8
INTRODUCTION iscorrectlycaptured bythecelebratedRPA[2, 3, 6], but
2
also by rigorous arguments[7] in which the selection of

] The first theory of classical collective electron density a particular center-of-mass position breaks the system’s
el oscillations in ionized gases by Tonks and Langmuir[1] Galileaninvarianceandplasmonexcitationsplaytherole
- of Goldstone bosons. In two-dimensional (2D) systems
r in the 1920’s helped initiate the field of plasma physics. (cid:112)
st The theory of collective electron density oscillations in Vq =2πe2/q so that ωpl(q →0)= 2πne2q/m.
t. metals, quantum in this case because of higher electron Electrons in a solid, unlike electrons in a plasma or
a densities, was developed by Bohm and Pines[2, 3] in electronswithajellium model[6]background,experience
m
the 1950’s and stands as a similarly pioneering contribu- a periodic external potential created by the ions which
- tion to many-electron physics. Bohm and Pines coined breaks translational invariance and hence also Galilean
d
n the term plasmon to describe quantized density oscil- invariance. Solid state effects can lead in general to a
o lations. Today plasmonics is a very active subfield of renormalization of the plasmon frequency, or even to the
c optoelectronics[4, 5], whose aim is to exploit plasmon absence of sharp plasmonic excitations. In semiconduc-
[
properties in order to compress infrared electromagnetic tors and semimetals, however, electron waves can be de-

1 waves to the nanometer scale of modern electronic de- scribedatsuper-atomiclengthscalesusingk·ptheory[8],
v vices. This wide importance of plasmons across different which is based on an expansion of the crystal’s Bloch
8
fields of basic and applied physics follows from the ubiq- Hamiltonianaroundbandextrema. Inthesimplestcase,
2
uity of charged particles and from the strength of the for example for the conduction band of common cubic
5
4 long-range Coulomb interaction. semiconductors, this device leads us back to a Galileian-
. invariant parabolic band continuum model with isolated
1 The physical origin of plasmons is very simple. When
electron energy E (p) = p2/(2m ). The crystal back-
0 electrons in a plasma move to screen a charge inhomo- c b
9 geneity, they tend to overshoot the mark. They are ground for electron waves appears only via the replace-
0 mentofthebareelectronmassbyaneffectivebandmass
thenpulledbacktowardthechargedisturbanceandover-
: m . It is this type of k·p Galilean invariant interact-
v shootagain,settingupaweaklydampedoscillation. The b
Xi restoring force responsible for the oscillation is the av- ing electron model, valid for many semiconductor and
semiconductor heterojunction systems, which has been
erage self-consistent field created by all the electrons.
r of greatest interest in solids. The absence of electron-
a Because of the long-range nature of the Coulomb in-
electron interaction corrections to plasmon frequencies
teraction, the frequency of oscillations ω (q) tends to
pl
at very long wavelengths in these systems has been am-
be high and is given in the long wavelength limit by
ω2(q → 0) = nq2V /m where n is the electron density, ply demonstrated experimentally by means of inelastic
pl q light scattering[9, 10].
m is the bare electron mass in vacuum, and V is the
q
Fourier transform of the Coulomb interaction. This sim- The situation turns out to be quite different in
ple explicit plasmon energy expression is exact because graphene – a monolayer of carbon atoms tightly packed
long-wavelengthplasmonsinvolverigidmotionoftheen- in a 2D honeycomb lattice[11, 12, 13], which has engen-
tireplasmawhichisindependentofthecomplexexchange dered a great deal of interest because of the new physics
andcorrelationeffectsthatdress[6]themotionofanindi- it exhibits and because of its potential as a new mate-
vidualelectron. Theexactplasmonfrequencyexpression rial for electronic technology. The agent responsible for
2
many of the interesting electronic properties of graphene
sheets is the bipartite nature of its honeycomb lattice.
The two inequivalent sites in the unit cell of this lattice
are analogous to the two spin orientations of a spin-1/2
particlealongthe+zˆand−zˆdirections(thezˆaxisbeing
perpendicular to the graphene plane). This observation
opens the way to an elegant description of electrons in
grapheneasparticlesendowedwithapseudospindegree-
of-freedom[11, 12, 13] (in addition to the regular spin
degree-of-freedomwhichplaysapassiverolehere). When
k·p theory is applied to graphene it leads to a new type
of electron fluid model, one with separate Dirac-Weyl
Hamiltonians for electron waves centered in momentum
spaceononeoftwohoneycomblatticeBrillouin-zonecor-
ners: Hˆ =vσ·p. Herev isthebareelectronvelocity, p
D
isthek·pmomentum, andσ isthepseudospinoperator
constructedwithtwoPaulimatrices{σi,i=x,y},which
act on the sublattice pseudospin degree-of-freedom. It
follows that the energy eigenstates for a given p have
pseudospins oriented either parallel (upper band) or an- FIG. 1: Breakdown of Galileian invariance in
tiparallel (lower band) to p. Physically, the orientation graphene. Panel 1a) shows the occupied electronic states
of the pseudospin determines the relative amplitude and in the upper band of graphene in the ground state. Notice
thateverystateischaracterizedbyavalueofmomentum(the
the relative phase of electron waves on the two distinct
origin of the arrow) and a pseudospin orientation (the direc-
graphene sublattices.
tion of the arrow). Panel 1b) shows the occupied states after
The feature of graphene that is ultimately responsible
a Galilean boost. An observer riding along with the boost
for the large many-body effects on the plasmon disper- wouldclearlyseethattheorientationofthepseudospins,rel-
sionandtheDrudeweightisbroken Galilean invariance. ative to the center of the occupied region has changed. It
What happens is that the oriented pseudospins provide lookslikethepseudospinsaresubjectedtoa“pseudomagnetic
an “ether” against which a global boost of the momenta field”thatcausesthemtotilttowardsthe+xˆ direction. The
appearance of this pseudomagnetic field is the signature of
becomes detectable. This is explained in detail in the
broken Galilean invariance. In contrast, in a Galilean invari-
caption of Fig. 1.
ant system [Panels 1c) and 1d)] the energy eigenstates are
From Fig. 1 we can also see why the plasmon fre-
characterized by momentum only: an observer riding along
quency in graphene is so strongly affected by exchange with the boost would not see any change in the character of
and correlation. In a plasmon mode the region of oc- the occupied states.
cupied states (Fermi circle) oscillates back and forth in
momentum space under the action of the self-induced
electrostatic field. In graphene however, this oscillatory above. Electron-electroninteractionsingraphenearede-
motion is inevitably coupled with an oscillatory motion scribed by the usual non-relativistic Coulomb Hamilto-
of the pseudospins. Since exchange interactions depend nianHˆ ,whichiscontrolledbythe2DFouriertransform
C
ontherelativeorientationofpseudospinstheycontribute of the Coulomb interaction, V =2πe2/((cid:15)q) with (cid:15) an ef-
q
to plasmon kinetic energy and renormalize the plasmon fective average dielectric constant.
frequency even at leading order in q. Electron carriers with density n can be induced in
Inwhatfollowswepresentamany-bodytheoryofthis graphene by purely electrostatic means, creating a cir-
subtle pseudospin coupling effect and discuss the main cular 2D Fermi surface in the conduction band with
√
implications of our findings for theories of charge trans- a Fermi radius k , which is proportional to n[14].
F
portandcollectiveexcitationsindopedgraphenesheets. The model described by Hˆ = Hˆ +Hˆ requires an ul-
D C
traviolet wavevector cut-off, k , which should be as-
max
signed a value corresponding to the wavevector range
GRAPHENE DIRAC MODEL over which Hˆ describes graphene’s π bands. This cor-
D
responds to taking k ∼ 1/a where a ∼ 1.42 ˚A is
max 0 0
Graphene’s honeycomb lattice has two-atoms per unit the carbon-carbon distance. This model is useful when
cellanditsπ-valencebandandπ∗-conductionbandtouch k is much larger than k . In this low-energy descrip-
max F
at two inequivalent points, K and K(cid:48), in the honey- tion,themany-bodypropertiesofdopedgraphenesheets
comb lattice Brillouin-zone. The energy bands near e.g. depend[15, 16] on the dimensionless fine-structure cou-
the K point are described at low energies by the spin- pling constant α = e2/((cid:15)(cid:126)v) (which is defined as the
ee
independent massless Dirac Hamiltonian Hˆ introduced ratio between the Coulomb energy scale e2k /(cid:15) and the
D F
3
kinetic energy scale (cid:126)vk ) and on density via the ultra- the RPA[23, 24, 25] result for the plasmon dispersion
F
violet cut-off Λ = k /k . The fine-structure constant at long wavelengths. The RPA Drude weight D =
max F
α can be tuned experimentally by changing the dielec- 4ε σ where,restoringPlanck’sconstantforamoment,
ee F uni
tricenvironmentsurroundingthegrapheneflake[17, 18]. σ = e2/(4(cid:126)) is the so-called universal[26, 27, 28, 29]
uni
Theultravioletcut-offΛvariesfrom∼10foraveryhigh- frequency-independent interband conductivity of a neu-
densitygraphenesystemwithn∼1014cm−2to∼100for tral graphene sheet. We show that both ω (q →0) and
pl
a density n∼1012 cm−2 just large enough to screen out D are substantially altered by electron-electron interac-
unintended[19] inhomogeneities. From now on Planck’s tions.
constanthdividedby2πwillbesetequaltounity,(cid:126)=1.
The collective (plasmon) modes of the system can be
found by solving the following equation[6], PSEUDOSPIN RESPONSE AND DIRAC-MODEL
PLASMONS
1−V χ (q,ω)=0 (1)
q(cid:101)ρρ
whereχ (q,ω)istheso-calledproper[20]density-density By using the equation of motion for the density
(cid:101)ρρ
response function. In the q →0 limit of interest here we operators[6] which appear in the density-density re-
can neglect the distinction between the proper and the sponse function, χρρ(q,ω) can be reexpressed in terms
full causal response function χ (q,ω). We show below of the longitudinal current-current response function.
ρρ
that When this procedure is applied to a Galilean-invariant
system with mass m it leads immediately to the well-
v2q2
lim lim(cid:60)e χ (q,ω)=A (2) known result lim lim(cid:60)e χ (q,ω)=nq2/(mω2). In the
ω→0q→0 ρρ ω2 ω→0q→0 ρρ
case of graphene, however, the current operator (defined
where A is a density-dependent constant which has the
asthederivativeoftheHamiltonianwithrespecttok)is
value A = gε /(4πv2) for α → 0. Here g = g g = 4
0 F ee s v directlyproportionaltothepseudospinoperator[30]and
accounts for spin and valley degeneracy and ε =vk is
F F we obtain instead
theFermienergy. NotetheorderoflimitsinEq.(2)and
below: the limit ω →0 is always taken in the dynamical vq v2q2
χ (q,ω)= (cid:104)[σˆx,ρˆ ](cid:105)+ χ (q,ω) , (7)
sense, i.e. vq (cid:28) ω (cid:28) 2εF. Using Eq. (2) in Eq. (1) and ρρ ω2 q −q ω2 σxσx
solving for ω we find that, to leading order in q,
whereσˆx isthecomponentofthepseudospinfluctuation
2πe2v2A q
ω2(q →0)= q . (3) operator along the direction of q, which we assume to
pl (cid:15) be the xˆ direction, and χ (q,ω) is the longitudinal
σxσx
Inthesamelimittheimaginary-partofthelow-frequency pseudospin-pseudospin response function. The latter de-
conductivity σ(ω)=ie2ωχ (ω)/q2 has the form scribes the response of σˆx to a pseudomagnetic field B
ρρ q q
which enters the Hamiltonian with a term of the form
(cid:61)m σ(ω)→ e2v2A . (4) σˆ−xqBq (notice that this has the opposite sign compared
ω to the usual Zeeman coupling).
ItthenfollowsfromastandardKramers-Kr¨oniganalysis Because of the presence of the infinite sea of nega-
that the real-part of the conductivity has a δ-function tive energy states Eq. (7) must be handled with great
peakatω =0: (cid:60)eσ(ω)=Dδ(ω)wheretheDrudeweight care[31]. In the noninteracting case we have
D =πe2v2A . (5) (cid:104)[σˆx,ρˆ ](cid:105)= (cid:88) [cos(ϕ )n(0) −cos(ϕ )n(0) ] ,
q −q k k,− k−q k−q,−
In the presence of disorder the δ-function peak is broad- |k|<ΛkF
ened into a Drude peak, but the Drude weight is pre- (8)
served. The Drude weight D defines an effective f-sum where ϕk is the angle between k and the xˆ axis and
rule (cf. Ref.[21]) in the dynamical regime vq (cid:28) ω (cid:28) n(0) = 1 is the occupation of the lower band. For
k,−
2ε [22]. q (cid:28) Λk this can be rewritten as the sum of cos(ϕ )
F F k
WethusseefromEqs.(3)and(5)thatthequantityA over the region comprised between the circles |k|<Λk
F
completelycontrolstheplasmondispersionatlongwave- and |k − q| < Λk : only states deep in the negative
F
lengthsandtheDrudeweight. Inthefollowingsectionwe energy Dirac sea contribute. When interactions are in-
firstrelateAtothelongitudinalpseudospinsusceptibility cludedn(0) isreplacedbyexactoccupationnumbersand
k,−
and then carry out a self-consistent microscopic calcula- additional terms associated with pseudospin-orientation
tionofAwhichdemonstratesthatitsvalueissuppressed fluctuationsappear. Inthenextsectionwepresentami-
by electron-electron interactions. When this renormal- croscopic time-dependent Hartree-Fock theory of A that
ization is neglected A→A0 and is valid to first order in αee. Since this theory neglects
gα q ground-state pseudospin and occupation-number fluctu-
ω2(q →0)=ε2 ee , (6)
pl F 2 kF ations, which are of second order in αee, the anomalous
4
commutator can be consistently evaluated in the nonin- transformation[33]. Atfinitefrequency, however, andno
teracting electron ground state and we find that[21] matter how small the frequency, the states are unable to
repopulate(k isaconstantofthemotioninthepresence
lim vq (cid:104)[σˆx,ρˆ ](cid:105)= v2q2gεmax . (9) of the perturbing field) and pseudospin reorientation is
q→0 ω2 q −q ω2 4πv2 the only effect that is left. This leads to a finite regu-
larized pseudospin response in the zero-frequency limit.
As we explain below, gε /(4πv2) is the negative of
max And this is clearly the limit of interest here[34].
thepseudospinsusceptibilityofanoninteractingundoped
The HF theory is usually described as an approxi-
graphene sheet, i.e.
mate factorization of the two-body interaction Hamilto-
gε nian into a product of simpler one-body terms. In the
max =− lim lim(cid:60)e χ(0u) (q,ω) , (10)
4πv2 ω→0q→0 σxσx presentcase,thetotalHFHamiltoniancanbewrittenin
the following physically transparent form:
whereχ(0u) (q,ω)isthepseudospin-pseudospinresponse
functionσoxfσxthenoninteractingundopedsystem. Combin- Hˆ = (cid:88) ψˆ† [δ B (k)+σ ·B(k)]ψˆ , (12)
HF k,α αβ 0 αβ k,β
ing the two terms on the r.h.s. of Eq. (7) we arrive at k,α,β
the following expression for A:
where the HF fields B (k) and B(k) are defined by
0
(cid:104) (cid:105)
A≡ lim lim(cid:60)e χ (q,ω)−χ(0u) (q,ω) . (11)
ω→0q→0 σxσx σxσx (cid:90) d2k(cid:48)
B (k)=− V f (k(cid:48)) , (13)
0 (2π)2 k−k(cid:48) +
ThusAhasaveryclearphysicalmeaning: itisthepseu-
dospin susceptibility of the interacting system regular-
and
ized by subtracting the pseudospin susceptibility of the
reference noninteracting undoped system. (cid:90) d2k(cid:48)
B(k)=B xˆ+vk− V f (k(cid:48))nˆ(k(cid:48)),(14)
On quite general grounds it is possible to express the ext (2π)2 k−k(cid:48) −
fully interacting value of A in terms of a small set of di-
mensionless parameters by adapting to doped graphene with f (k) = (n(0) ± n(0) )/2, where the n(0) are
sheetstheoriginalmacroscopicphenomenologicaltheory ± k,+ k,− k,λ
noninteracting band occupation factors, and nˆ(k) =
of Landau[6], which is usually applied to normal Fermi
B(k)/|B(k)| is the unit vector in the direction of B(k).
liquids. In what follows, however, we will present a mi-
Wehavetakenthelimitq →0inEq.(14)byconsidering
croscopic theory of A, which we believe to be accurate
a spatially homogeneous external pseudomagnetic field
at weak coupling and which enables us to draw quanti-
applied along the xˆ direction. According to the previ-
tative conclusions on the impact of electron-electron in-
ous discussion the occupation factors in k-space are not
teractions on the plasmon dispersion and Drude weight
affected by the perturbation. The second term in this
of doped graphene sheets. equationisthebandpseudomagneticfield(seeHˆ ), and
D
thelasttermistheexchangefield. TheHamiltonianHˆ
HF
has two bands with energies ε(±)(k) = B (k)±|B(k)|.
SELF-CONSISTENT HARTREE-FOCK HF 0
MEAN-FIELD THEORY OF THE PSEUDOSPIN In time-dependent HF theory electrons respond to the
SUSCEPTIBILITY external field and to the induced change in the exchange
field.
Wenowproceedtoaquantitativemicroscopiccalcula- In the absence of the external field nˆ(k) = kˆ, so that
tion of the pseudospin susceptibility A which goes be- there is no total pseudospin polarization, and |B(k)| →
yond the RPA. Specifically, we will take into account |Beq(k)| depends only on k = |k| [16]. Our aim here
exactly the self-consistent exchange field which accom- is to calculate the additional exchange field that arises
panies pseudospin polarization. To this end we set up from the polarization of the pseudospin when Bext (cid:54)=
the time-dependent Hartree-Fock (HF) theory of the re- 0, since this determines the exchange correction to the
sponse of the system to a uniform pseudomagnetic field pseudospin susceptibility.
B =B xˆ oriented along the xˆ direction. For q = 0 the response is due to vertical interband
ext ext
It is important to realize that the zero-frequency limit transitionsatwavevectorswith|k|>k ;transitionswith
F
oftheuniformpseudospinsusceptibilityissingularinthe |k| < k are Pauli blocked. Because only the transverse
F
following sense. When B is truly time-independent component δB (∝ zˆ×kˆ) of the pseudospin field con-
ext T
then its effect is simply to shift the occupied states in tributes to these matrix elements, it is evident that only
k-space while reorienting the pseudospins. Changes due δB is relevant to the calculation of the susceptibility.
T
to k-state repopulation and pseudospin reorientation at We find that
a given k cancel each other as required by gauge invari-
ance, since a constant B can be eliminated by a gauge δB (k)=−[δB (k)sin(ϕ )] zˆ×kˆ , (15)
ext T T,1 k
5
FIG. 2: The exchange contribution to the equilib- FIG. 3: The uniform pseudospin susceptibility of a
riumpseudomagneticfieldandtheinducedtransverse doped graphene sheet. The data labeled by filled sym-
pseudospin field. The data shown in this figure (as well as bols refer to the Hartree-Fock value of the ratio A/A , as
0
thosereportedinFig.3)havebeenobtainedusingaThomas- calculated from Eq. (20), as a function of electron density n
FermiscreenedpotentialV =2πe2/[(cid:15)(gα k +q)]. Thedata (in units of 1012 cm−2) for various values of graphene’s fine-
q ee F
labeledbythebluesolidlinerefertothequantityΣ¯ (x)plot- structureconstantα . Thedashedhorizontallinerepresents
Λ ee
ted as a function of x = k/k (for wavevectors k up to the the prediction of the RPA[23, 24, 25], for which A/A = 1
F 0
ultravioletcut-offk )forα =0.2andn∼2×1013 cm−2 for every value of n and α .
max ee ee
(Λ = 20). The data labeled by the red solid line refer the
solutionu (x)=δB /B ofEq.(16)plottedasafunction
Λ T,1 ext
of x = k/k for the same physical parameters. The dashed After straightforward manipulations we arrive at the
F
verticallineindicatesthepointk=kF. NotethatuΛislarger following HF expression for the ratio between A and its
thanunity,thusgivinganinducedtransversepseudospinfield noninteracting value A in terms of Σ¯ (x) and u :
0 Λ Λ
δB that is larger than the bare external field B .
T,1 ext
A (cid:90) Λ (cid:20) u (x) (cid:21)
= 1− dx Λ −1 . (20)
A 1+Σ¯ (x)/x
where δB (k)/B ≡u solves the integral equation: 0 1 Λ
T,1 ext Λ
AplotoftheratioA/A asafunctionofelectrondensity
(cid:90) Λ 0
u (x) = 1+ dx(cid:48)K(x,x(cid:48)) u (x(cid:48)) (16) forvariousvaluesofαeehasbeenreportedinFig.3. From
Λ Λ
1 this plot we clearly see that A/A0 is substantially lower
than unity. According to Eqs. (3) and (5) this implies a
with
strongreductionoftheplasmonfrequencyandtheDrude
1 V¯ (x,x(cid:48))+V¯ (x,x(cid:48)) weight.
K(x,x(cid:48))= α 0 2 . (17)
4 ee 1+Σ¯ (x(cid:48))/x(cid:48) In order to explain this result we show in Fig. 4 the
Λ
response of the pseudospins to a pseudomagnetic field
Here we have introduced the dimensionless Coulomb in the +xˆ direction. Notice that the pseudospins in the
pseudopotentials lowerbandaretiltedawayfromthepseudomagneticfield,
whilethoseintheupperbandaretiltedtowardsthepseu-
(cid:15)k (cid:90) 2π dθ
V¯ (k,k(cid:48))= F e−imθ V | , (18) domagneticfield. Becausetherearemanymoreparticles
m 2πe2 0 2π q q=|k−k(cid:48)| in the lower band than in the upper band we see that
the total pseudospin response is negative. It is only after
θ being the angle between k and k(cid:48), and all wavevectors
subtractingthenoninteractingundopedresponsethatwe
have been scaled with k . The quantity
F obtainthepositivequantityA. Itshouldbeevidentfrom
1 (cid:90) Λ thisdescriptionthatanymany-bodyeffectthatenhances
Σ¯Λ(x)= 2αee dx(cid:48)x(cid:48)V¯1(x,x(cid:48)) (19) the total pseudospin response will reduce the value of
1 A, while any many-body effect that suppresses the total
is the exchange contribution to the equilibrium pseudo- pseudospin response will increase the value of A.
magnetic field Beq(k). Illustrative numerical results for InthepresenttheoryinteractionsaffectthevalueofA
Σ¯ andu [thelatterasobtainedfromtheself-consistent in two competing ways. (i) The exchange field enhances
Λ Λ
solution of Eq. (16)] for α =0.2 and Λ=20 are shown δB relative to B (u = δB /B > 1), thus en-
ee T,1 ext Λ T,1 ext
in Fig. 2. hancing the total pseudospin response. From what we
6
will be observable in experiments of the kind discussed
in Refs.[26, 27, 28, 29].
Finally let us comment on the broader implications
of our results. Effects similar to those described in this
article are also expected in graphene bilayers and other
few-layersystems. ThelackofGalileaninvarianceshould
also affect the cyclotron resonance frequency and oscil-
lator strength when the 2D sheet of graphene is placed
in a perpendicular magnetic field. Undoubtedly much
interesting physics, potentially useful for applications in
optoelectronics, has still to be learned from the study of
graphene and other non-Galilean invariant systems.
M.P. acknowledges partial financial support from the
CNR-INFM“SeedProjects”andwishestothankAndrea
Tomadinformanyinvaluableconversationsrelatedtothe
numerical solution of Eq. (16). Work in Austin was sup-
ported by the Welch Foundation and by NSF Grant No.
0606489. G.V. acknowledges support from NSF Grant
No. 0705460. The authors thank Rosario Fazio and Vit-
torio Pellegrini for a critical reading of the manuscript
and the FSB for helpful support.
∗ Electronic address: [email protected]
FIG.4: Response of the pseudospins to a pseudomag-
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netic field. Panel4a): Pseudospinsinthehigh-energyband
Phys. Rev. 33, 195 (1929).
tilt towards the pseudomagnetic field B applied along the
ext [2] Pines, D. & Bohm, D. A Collective Description of Elec-
+xˆ direction. Panel4b): Pseudospinsinthelow-energyband
tronInteractions: II.CollectivevsIndividualParticleAs-
tilt away from the pseudomagnetic field. The reason for this
pects of the Interactions. Phys. Rev. 85, 338 (1952).
unusual response is easy to understand. Pseudospins in the
[3] Pines,D.&Nozi´eres,P.TheTheoryofQuantumLiquids
lower band are in their ground state: because of the anoma-
(W.A. Benjamin, Inc., New York, 1966).
lous sign of the pseudospin-pseudomagnetic coupling men-
[4] Ebbesen, T.W., Genet, C. & Bozhevolnyi, S.I. Surface-
tioned in the main body of this article, they are anti-aligned
plasmon circuitry. Phys. Today 61(5), 44 (2008).
withtheintrinsicDiracbandpseudomagneticfield. Whenan
[5] Maier,S.A.Plasmonics–FundamentalsandApplications
additional external pseudomagnetic field is applied they will
(Springer, New York, 2007).
simplytiltawayfromittominimizetheenergy. Theoccupied
[6] Giuliani, G.F. & Vignale, G. Quantum Theory of the
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