Plasmons and Coulomb drag in Dirac/Schroedinger hybrid electron systems

Alessandro Principi,1 Matteo Carrega,1 Reza Asgari,2, ∗ Vittorio Pellegrini,1, † and Marco Polini1, ‡

1NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56126 Pisa, Italy2School of Physics, Institute for Research in Fundamental Sciences (IPM), Tehran 19395-5531, Iran

(Dated: October 31, 2018)

We show that the plasmon spectrum of an ordinary two-dimensional electron gas (2DEG) hostedin a GaAs heterostructure is significantly modified when a graphene sheet is placed on the surface ofthe semiconductor in close proximity to the 2DEG. Long-range Coulomb interactions between mas-sive electrons and massless Dirac fermions lead to a new set of optical and acoustic intra-subbandplasmons. Here we compute the dispersion of these coupled modes within the Random Phase Ap-proximation, providing analytical expressions in the long-wavelength limit that shed light on theirdependence on the Dirac velocity and Dirac-fermion density. We also evaluate the resistivity ina Coulomb-drag transport setup. These Dirac/Schroedinger hybrid electron systems are experi-mentally feasible and open new research opportunities for fundamental studies of electron-electroninteraction effects in two spatial dimensions.

PACS numbers: 71.45.-d,71.45.Gm,72.80.Vp,71.10.-w

I. INTRODUCTION

Plasmons are ubiquitous high-frequency collective den-sity oscillations of an electron liquid, which occur in met-als and semiconductors1. Their importance across differ-ent fields of basic and applied physics is by now wellestablished. They play, for example, a key role in plas-monics2 and in the photodetection of far-infrared radia-tion based on field-effect transistors3,4.

Plasmons in doped graphene sheets (hereafter dubbed“Dirac plasmons”) have been intensively investigatedby electron energy-loss spectroscopy (EELS)5. EELS,however, does not have the sufficient energy and wave-number resolution to address e.g. the impact of bro-ken Galilean invariance on Dirac plasmons6. Moreover,its application is limited to the collective modes in thecharge channel with no information on the spin degreesof freedom, since the method probes only the “loss”function, i.e. the imaginary part of the inverse di-electric function ε(q, ω). Angle-resolved photoemissionspectroscopy7 and scanning tunneling spectroscopy8 alsogive precious information on Dirac plasmons, albeit ina slightly-indirect way. Finally, Dirac plasmons havebeen probed by engineering directly their coupling to far-infrared light in a number of intriguing ways9. On thecontrary, the application of inelastic light (Raman) scat-tering to probe plasmons in graphene, has been, so far,limited to the investigation of magneto-plasmons in highmagnetic fields10.

The situation is very different for the case of plas-mons in 2DEGs realized, for example, in GaAs/AlGaAsmodulation-doped semiconductor heterostructures. In-deed, these excitations have been successfully studied,both in zero magnetic field and in the quantum Hallregime, by resonant inelastic light scattering11,12. Thismethod has a much better energy resolution than EELS(typically below 0.1 meV) and can be straightforwardlygeneralized to probe spin excitations. The cross sectionfor scattering of photons by the electronic subsystem is,

FIG. 1: (Color online) Schematics of the double-layer mass-less Dirac/Schroedinger hybrid electron system studied inthis work. A graphene sheet is deposited on the surface ofa semiconductor, underneath which a GaAs quantum well(at a distance d from the surface) hosts a high-mobility two-dimensional electron gas (2DEG). Carriers in the two layersare induced by conventional gating techniques (the + signsbelow the quantum well indicate the doping layer). In thepresence of Coulomb coupling between the two subsystems,optical and acoustic hybrid collective modes emerge, whichcan be probed by resonant inelastic light scattering.

however, tiny. An ingenious resonance condition can beused in a light scattering experiment to enhance this crosssection dramatically12. The frequency of the incoming orscattered photon can indeed be tuned close to the semi-conductor band gap (e.g. ≈ 1.5 eV for GaAs at lowtemperature) making the energy denominators appear-ing in second-order perturbation theory13 small and thusyielding a much higher sensitivity to collective electronicmodes.

Graphene14 is a gapless material though, and thuspresents immediately an obstacle for the application ofresonant inelastic light scattering. The van Hove singu-larity at the M point of graphene’s Brillouin zone canbe used to achieve resonance15, although the band gapat the M point is in the ultraviolet region of the electro-magnetic spectrum (≈ 4 eV) where tunable lasers having

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In this Article we propose a hybrid double-layer systemcomposed by a doped graphene sheet that is Coulombcoupled to an ordinary 2DEG in a GaAs/AlGaAs semi-conductor heterostructure, as schematically illustratedin Fig. 1. We demonstrate that precious informationon Dirac plasmons is encoded in the dispersion of thenew collective in-phase and out-of-phase plasmon exci-tations of this massless Dirac/Schroedinger hybrid sys-tem (MDSHS). These collective modes can in principlebe probed via inelastic light scattering by exploiting, asexplained above, the resonance condition offered by theGaAs band gap. In passing, we mention that a simi-lar resonant enhancement mechanism has been recentlyobserved in the case of the Raman phonon lines in epi-taxial graphene on copper substrates with incident pho-ton energies overlapping the copper substrate photolumi-nescence16. We also study transport in a MDSHS in aCoulomb-drag setup17, demonstrating that, in the low-temperature Fermi-liquid regime and at strong coupling,the drag resistivity displays an intriguing dependence oncarrier densities and inter-layer separation, which is dif-ferent from that known17 for coupled 2DEGs in semicon-ductor double quantum wells.

Our manuscript is organized as follows. In Sect. II weintroduce the model Hamiltonian and some basic defi-nitions. In Sect. III we present analytical and numericalresults, obtained by means of the Random Phase Approx-imation, for the collective plasmons modes of MDSHSs.Our main analytical results on the Coulomb drag resis-tivity are summarized and discussed in Sect. IV. Finally,in Sect. V we present a short summary of our main resultsaccompanied by a discussion of the potential of MDSHSs.

II. MODEL HAMILTONIAN AND IMPORTANTDEFINITIONS

The system depicted in Fig. 1 can be modeled by thefollowing Hamiltonian:

H = ~vD∑k,α,β

ψ†k,α(σαβ · k)ψk,β +1

2S

∑q 6=0

Vtt(q)ρ(t)q ρ

(t)−q

+∑k

~2k2

2mbφ†kφk +

1

2S

∑q 6=0

Vbb(q)ρ(b)q ρ(b)−q

+1

2S

∑q

Vtb(q)(ρ(t)q ρ(b)−q + ρ(b)q ρ

(t)−q) . (1)

The terms in the first and second lines of the previousequation describe two 2D electron systems with masslessDirac and parabolic Schroedinger bands, respectively.The last term describes Coulomb interactions betweenthe two subsystems.

In the first line of Eq. (1), vD is the Dirac velocity,σ = (σx, σy) is a 2D vector of Pauli matrices, S is the

sample area, and ψ†k,α (ψk,α) creates (destroys) an elec-tron with momentum k and sublattice-pseudospin indexα = A,B in graphene. The Dirac density operator is de-

fined by ρ(t)q =

∑k,α ψ

†k−q,αψk,α. In the second line, mb

is the bare band mass of the 2DEG (mb ≈ 0.067 me in

GaAs, me being the electron mass in vacuum), φ†k (φk)creates (destroys) an electron with momentum k in the

2DEG, and ρ(b)q =

∑k φ†k−qφk is the usual 2DEG density

operator.In writing Eq. (1) we have intentionally hidden “flavor”

indices labeling spin and valley degrees-of-freedom in thegraphene part of the Hamiltonian and spin degrees-of-freedom in the 2DEG part of the Hamiltonian. Theseflavors will just appear as degeneracy factors in the lin-ear response functions χt(q, ω) and χb(q, ω), as we willsee below. We have also completely neglected inter-layertunneling, which is strongly suppressed by the AlGaAsbarrier between graphene and the quantum well.

The bare intra- and inter-layer Coulomb interactions,Vij(q) (with i, j ∈ {t,b}), are influenced by the lay-ered dielectric environment, modeled by two dielectricconstants - ε1 and ε2 in Fig. 1. A simple electro-static calculation18,19 implies that the Coulomb inter-action in the graphene sheet (top layer) is given byVtt(q) = 4πe2/[q(ε1 + ε2)], The Coulomb interaction inthe 2DEG (bottom layer) is instead

Vbb(q) =4πe2

qD(q)[(ε2 + ε1)eqd + (ε2 − ε1)e−qd] , (2)

where D(q) = 2ε2(ε1 + ε2)eqd. Finally, the inter-layerinteraction is given by

Vtb(q) = Vbt(q) =8πe2

qD(q)ε2 . (3)

Note that in the “uniform” ε1 = ε2 ≡ ε limit we recoverthe familiar expressions Vtt(q) = Vbb(q)→ 2πe2/(εq) andVtb(q) = Vbt(q) = Vtt(q) exp(−qd).

For the following analysis we introduce the electrondensity in the top (bottom) layer nt (nb) and the corre-

sponding Fermi wave numbers kF,i =√

4πni/Ni, whereNi is a degeneracy factor (Nt = 4 in the graphenelayer and Nb = 2 in the 2DEG layer). We also in-troduce the Fermi energies εF,t = ~vDkF,t and εF,b =~2k2F,b/(2mb) in the top and bottom layers, respec-

tively, αee = e2/(~vD) ≈ 2.2, and rs = (πnba2B)−1/2,

aB = ~2/(mbe2) being the Bohr radius calculated with

the semiconductor band mass mb (note the absence ofany dielectric constant in the definition of aB).

III. COLLECTIVE OPTICAL AND ACOUSTICPLASMON MODES

The collective modes of the system described by theHamiltonian (1) can be determined1 by locating the poles

3

of the linear-response function χ(q, ω). Within the Ran-dom Phase Approximation (RPA) these functions satisfythe following matrix equation1:

χ−1(q, ω) = χ−10 (q, ω)− V (q) , (4)

where χ0(q, ω) is a 2×2 diagonal matrix whose elementsχt(q, ω) and χb(q, ω) are the well-known non-interacting(Lindhard) response functions of each layer at arbitrarydoping ni. The mathematical and physical propertiesof χt(q, ω) are discussed in Ref. 20, while expressionsfor <e [χb(q, ω)] and =m [χb(q, ω)] can be found inRef. 1. The off-diagonal (diagonal) elements of the ma-trix V = {Vij}i,j=t,b represent inter-layer (intra-layer)Coulomb interactions.

A straightforward inversion of Eq. (4) yields the fol-lowing condition for collective modes:

ε(q, ω) = [1− Vtt(q)χt(q, ω)][1− Vbb(q)χb(q, ω)]

− V 2tb(q)χt(q, ω)χb(q, ω) = 0 . (5)

Zeroes of ε(q, ω) occur above the intra-band particle-holecontinuum where χi is real, positive, and a decreasingfunction of frequency. Eq. (5) admits two solutions21,a higher frequency solution at ωop(q → 0) ∝ √q whichcorresponds to in-phase oscillations of the densities inthe two layers (optical plasmon), and a lower frequencysolution at ωac(q → 0) ∝ q which corresponds to out-of-phase oscillations (acoustic plasmon).

A summary of our main results for ωop(q), obtainedfrom the numerical solution of Eq. (5), is reported inFigs. 2-3. In these figures we plot the quantity ∆(q) =~ωop(q)−~ωpl(q), which physically represents the energyof the MDSHS optical plasmon measured from that ofthe plasmon of an isolated 2DEG, ~ωpl(q). We clearlysee that ∆(q) is positive and ≈ 10 meV at q ∼ 105 cm−1.

To leading order in q in the long-wavelength q → 0limit, the frequency of the MDSHS optical plasmon modecan be found analytically. The result is

ω2op(q → 0) =

(Nte

2vDkF,t2~ε

+2πnbe

2

mbε

)q , (6)

where we have introduced ε ≡ (ε1 + ε2)/2. We havechecked that Eq. (6) is in perfect agreement with thenumerical results displayed in Figs. 2-3. The first termon the r.h.s. of Eq. (6) can be easily recognized20 to bethe square of the RPA plasmon frequency of the elec-tron gas in an isolated graphene sheet separating twomedia with dielectric constants ε1 and ε2. The sec-ond term is the well-known RPA plasmon frequency ofa 2D parabolic-band electron gas1. A measurement ofωop(q) in a MDSHS at sufficiently small q [more pre-cisely, for q � min(kF,t, kF,b)] thus allows to access di-rectly the Dirac velocity vD and the Dirac-fermion den-sity nt ∝ k2F,t.

In Fig. 4 we show illustrative results for the acous-tic plasmon ωac(q) as obtained from the numerical so-lution of Eq. (5) for different values of the 2DEG car-rier densities nb. The dashed line in this figure repre-sents the upper bound ω = vDq of the Dirac-fermion

0 1 2 3 4 5 6

q [105 cm−1 ]

0

10

20

30

40

50

60

70

∆(q

)[m

eV]

nb =2×1011 cm−2

nb =5×1011 cm−2

nb =1×1012 cm−2

nb =2×1012 cm−2

FIG. 2: (Color online) Optical plasmon dispersion in a mass-less Dirac/Schroedinger hybrid electron system. In this fig-ure we plot the quantity ∆(q) = ~ωop(q) − ~ωpl(q) (in meV)as a function of the wave number q (in units of 105 cm−1).Different line styles refer to different values of the 2DEGcarrier density nb, ranging from nb = 1 × 1011 cm−2 tonb = 1 × 1012 cm−2. The Dirac-fermion density is fixed atthe value nt = 1012 cm−2, while the inter-layer distance isd = 30 nm. With reference to Fig. 1, the dielectric constantshave been fixed to ε1 = 1 and ε2 = 13. Note that ∆(q) ispositive, implying a blue shift of the optical plasmon mode ofthe hybrid system with respect to the usual plasmon mode ofan isolated 2DEG.

intra-band electron-hole continuum20. Acoustic plasmondispersions lying above the dashed line are not damped(within RPA) since they cannot decay by exciting intra-band electron-hole pairs in the graphene sheet. When thedispersion hits the dashed line, Landau damping occursand the acoustic plasmon pole acquires a finite lifetime.Note that when the 2DEG density is low (empty circlesin Fig. 4) the acoustic-plasmon dispersion stays out theDirac-fermion intra-band particle-hole continuum for avery small range of wave numbers q. It will be thus dif-ficult to observe acoustic plasmons in MDSHSs in whichthe 2DEG carrier density is low.

The acoustic-plasmon group velocity cs =limq→0 ωac(q)/q can be calculated analytically fol-lowing the procedure explained in Ref. 22. Aftersome tedious but straightforward algebra we find thefollowing equation for x = cs/vD, the ratio betweenthe acoustic-plasmon group velocity cs and the Diracvelocity vD:

2NtαeeξΓf(x)g(x)− ε2Γg(x)√x2 − 1

−2ε2Ntαeef(x)√x2Nbr2s − 4α2

ee = 0 , (7)

where ξ = dkF,t, Γ = 2Nb/(kF,taB), f(x) = x−√x2 − 1,

and g(x) = x√Nbrs −

√x2Nbr2s − 4α2

ee. Note that thesolution of Eq. (7) depends only on ε2, while, as we have

4

0.0 0.5 1.0 1.5 2.0 2.5

q [105 cm−1 ]

0

5

10

15

20

25

30

35

40

45

∆(q

)[m

eV]

nt =1×1011 cm−2

nt =2×1011 cm−2

nt =5×1011 cm−2

nt =1×1012 cm−2

FIG. 3: (Color online) Same as in Fig. 2 but for differentvalues of the Dirac-fermion density nt and for a fixed valueof the 2DEG carrier density nb = 2× 1011 cm−2. Notice thenon-monotonic behavior of ∆(q): for each q, ∆(q) reaches itsminimum when the Dirac-fermion density matches the 2DEGdensity (nt = nb).

seen above, ωop(q → 0) depends on the average ε.An undamped acoustic plasmon emerges for x > xcrit,

where the threshold xcrit can be directly found fromEq. (7). When the Dirac velocity vD is larger [smaller]than the Fermi velocity vF,b = ~kF,b/mb in the 2DEG,xcrit = 1 [xcrit = vF,b/vD = 2αee/(rs

√Nb)]. The ex-

istence of a solution of Eq. (7) thus depends on threeparameters, namely d, nt, and nb (when the dielectricconstants ε1 and ε2 are fixed). For example, given ntand nb, the acoustic plasmon emerges out of the contin-uum for d > d(crit) with

d(crit) =

ε2√Nbr2s − 4α2

ee

ΓkF,tg(1), if vD > vF,b

ε2vF,b√

4α2ee −Nbr2s

4Ntα2eevDkF,tf(vF,b/vD)

, otherwise

.

(8)We remind the reader that the functions f(x) and g(x)have been defined right after Eq. (7). Similarly, given ntand d, the acoustic plasmon emerges out of the contin-

uum for n > n(crit)b , where

n(crit)b =

ε2Nb(ε2 + 2ξΓ)

4π2a2Bα2ee(ε2 + ξΓ)2

, if vD > vF,b

Nb(ε2 + 2ξNtαee)2

4πε2a2Bα2ee(ε2 + 4ξNtαee)

, otherwise

.

(9)Note that the critical values of d and nb do not dependon nt when vD > vF,b. They indeed depend only on theproducts ΓkF,t and ξΓ, which are independent of kF,t.

The theoretical prediction based on the solution of

0.0 0.5 1.0 1.5 2.0

q [105 cm−1 ]

0

2

4

6

8

10

12

14

ωac(q)

[meV

]

nb =5×1011 cm−2

nb =1×1012 cm−2

0.0 0.1 0.2 0.3 0.40

1

2

3

FIG. 4: (Color online) The acoustic plasmon mode of a mass-less Dirac/Schroedinger hybrid system. The acoustic plasmondispersion ~ωac(q) (in meV) is plotted as a function of thewave number q (in units of 105 cm−1). Empty symbols labeldata for two different values of the 2DEG carrier density nb.In this figure we have fixed nt = 1×1011 cm−2 and d = 60 nm.The other parameters are identical to those used in Figs. 2-3.In the inset we illustrate the comparison between numericalresults for the acoustic plasmon dispersion (empty symbols)and the analytical result ωac(q → 0) = csq (solid lines) withthe velocity cs obtained from the solution of Eq. (7). Thedashed line marks the upper boundary of the Dirac fermionintra-band single particle continuum.

Eq. (7) is compared with the numerical results in theinset to Fig. 4.

IV. COULOMB DRAG RESISTIVITY

In a drag experiment17 a constant current is imposedon one layer (the “active” or “drive” layer). If no currentis allowed to flow in the other one (the “passive” layer),an electric field develops, whose associated force cancelsthe frictional drag force exerted by the electrons in theactive layer on the electrons in the passive one. The dragresistivity ρD, is defined as the ratio of the induced volt-age in the passive layer to the applied current in the drivelayer and reads ρD = −σD/(σtσb − σ2

D) ' −σD/(σtσb)in terms of the intra-layer conductivities σt (σb) of thetop (bottom) layer and the drag conductivity σD. Noticethat in writing the last equality of the previous equationwe have assumed σD � σt, σb.

Within the Kubo formalism the drag conductivity atthe second order in the screened inter-layer interactionUtb(q, ω) reads23 (~ = 1)

σD =βe2

8π

∫d2q

(2π)2

∫ ∞0

dω|Utb(q, ω)|2

sinh2(βω/2)Γt(q, ω)Γb(q, ω) ,

(10)

5

where β = (kBT )−1 and Γi(q, ω) is the so-called “non-linear susceptibility” of the i-th layer. The dynamically-screened inter-layer interaction that appears in Eq. (10)is Utb(q, ω) = Vtb(q)/ε(q, ω), where ε(q, ω) is the RPAdielectric function defined in Eq. (5).

Here we focus only on the low-temperature limit, i.e.kBT � min(εF,t, εF,b). In this regime we can do thefollowing approximations24: (i) use the low-temperatureexpressions for the intra-layer conductivities σi, (ii) sub-stitute Utb(q, ω) in Eq. (10) with the statically-screenedinter-layer interaction Utb(q, ω = 0), and (iii) expand thenon-linear susceptibilities Γi(q, ω → 0) up to the lowestorder in ω.

Following Ref. 24, we find the following results for thetop layer (graphene sheet):

limT→0

σt =e2

4πNtτt(kF,t)εF,t (11)

and

limT→0

Γt(q, ω → 0) = −Ntωτt(kF,t)

2πvDΘ(2kF,t − q)

×

(1− q2

4k2F,t

)1/2

cos(ϕq) , (12)

and the following results for the bottom layer (2DEG):

limT→0

σb =e2

2πNbτb(kF,b)εF,b =

nbe2τb(kF,b)

mb(13)

and

limT→0

Γb(q, ω → 0) = −Nbωτb(kF,b)

4πvF,bΘ(2kF,b − q)

×

(1− q2

4k2F,b

)−1/2cos(ϕq) ,(14)

where vF,b is the 2DEG Fermi velocity. Note the sin-gularity at q = 2kF,b in the non-linear susceptibility inEq. (14). On the contrary, Γt(q, ω → 0) vanishes atq = 2kF,t, as a consequence of the absence of backscat-tering for massless Dirac fermions14.

Using Eqs. (11)-(14) we find that the drag resistivityin the low-temperature limit is given by

limT→0

ρD = − 1

24e21

vDvF,b

(kBT )2

εF,tεF,b

∫ qmax

0

q dq|Utb(q, 0)|2

× F(

q

2kF,t,

q

2kF,b

), (15)

where qmax = min(2kF,t, 2kF,b) and F(x, y) =√(1− x2)/(1− y2).It is convenient at this stage to introduce a dimension-

less expansion parameter η ≡ d√kF,tkF,b. The MDSHS

is weakly (strongly) coupled if η � 1 (η � 1). Similar

reasoning to the one employed in Ref. 24 yields the fol-lowing result for ρD in the weak-coupling limit (restoringPlanck’s constant for clarity):

limη→∞

limT→0

ρD = − h

e2πζ(3)ε22

16α2eeN

2t N

2b

(kBT )2

~2v2Dk3F,tk3F,bd4

∝ − h

e2T 2

n3/2t n

3/2b d4

. (16)

We emphasize that the functional dependence of ρD onthe carrier densities ni and on the inter-layer distance din this limit is the same as in the case of two 2DEGs25

and two graphene sheets24,26.In the strong-coupling limit, we find

limη→0

limT→0

ρD = − h

e2π

3α2ee

vDvF,b

(kBT )2

εF,tεF,b

∫ xmax

0

dx x

×F

(√kF,bkF,t

x

2,

√kF,tkF,b

x

2

)[2εx+ 2(qTF,t + qTF,b)/

√kF,tkF,b

]2 ,

(17)

where xmax = min(2√kF,t/kF,b, 2

√kF,b/kF,t) and we

have introduced the Thomas-Fermi screening wave num-

bers qTF,t = NtαeekF,t and qTF,b = N3/2b rskF,b/2.

The quadrature on the r.h.s. of Eq. (17) can be carriedout analytically in the special case kF,t = kF,b. Due tothe different degeneracies in two layers, this conditionimplies a density imbalance between the two layers: nb =Nbnt/Nt ≡ n. In this case, Eq. (17) yields

limη→0

limT→0

ρD∣∣kF,t=kF,b

= − h

e2π

12

vDvF,b

(kBT )2

εF,bεF,t

α2ee

ε2

×

{ln

[1 +

2ε

Ntαee +N3/2b rs/2

]

− 2ε

2ε+Ntαee +N3/2b rs/2

}.

(18)

Eqs. (15)-(18) are the most important results of this Sec-tion27. As in the case of drag between two graphenesheets24, Eq. (18) does not depend on the inter-layer dis-tance d. In the limit n→ 0 it yields a dependence of ρDon carrier density of the form ρD ∝ n−1, analogously towhat found by Carrega et al.24 for two graphene sheetswith equal density [note that in the limit n→ 0 one hasto find the asymptotic behavior of the expression in curlybrackets in Eq. (18) for rs →∞]. Finally, we emphasizethat Eq. (18) does not contain the well-known17 ln(T )factor which appears in the drag resistivity at strong cou-pling in the case of two 2DEGs in semiconductor doublequantum wells. This factor, which stems from the con-tribution to drag from momenta q of the order of 2kF, iscompletely suppressed in the present case by the absenceof backscattering for massless Dirac fermions14 [compareEq. (12) with Eq. (14)].

6

V. SUMMARY AND DISCUSSION

In this work we have proposed that masslessDirac/Schroedinger hybrid double layers be used to probeDirac plasmons in a graphene sheet by employing res-onant inelastic light scattering12. A natural resonantcondition in this system is offered by the band gapof the semiconductor (e.g. GaAs) hosting an ordinaryparabolic-band 2D electron gas, thus bypassing the non-trivial issue of the absence of a gap in an isolatedgraphene sheet. We have demonstrated that informa-tion on Dirac plasmons can be extracted in a rather di-rect manner from the measurement of the optical plas-mon mode of the hybrid double layer. The latter sup-ports also a soft mode which disperses linearly as afunction of wave number and whose observation requiresa sufficiently-high electron density in the semiconduc-tor quantum well. Finally, we have calculated the low-temperature Coulomb-drag resistivity in the Boltzmann-transport limit, showing that in the strong-coupling limitit displays a dependence on carrier densities that is notshared by conventional all-semiconductor double quan-tum wells17.

The system depicted in Fig. 1 is amenable to exper-imental investigations and paves the way for the studyof many other intriguing phenomena. We expect, for ex-ample, that massless Dirac/Schroedinger hybrid doublelayers will display interesting correlated states (and as-

sociated transport anomalies) induced by inter-layer in-teractions deep in the quantum Hall regime. Anothervery appealing subject of investigation could be “hybridexciton condensates”. Exciton condensates are elusivemany-particle systems in which electron-hole pairs con-dense below a certain critical temperature in a coher-ent superfluid28. Double-layer structures are extremelyuseful to spatially separate electrons and holes in twodifferent layers, thereby suppressing electron-hole recom-bination29. Realizing all-semiconductor-based electron-hole double layers is a rather difficult task30. On thecontrary, holes can be trivially induced in the graphenesheet depicted in Fig. 1 by gating techniques. MasslessDirac/Schroedinger hybrid double layers thus offer un-precedented opportunities to realize, probe, and manip-ulate novel electron-hole quantum liquids and, hopefully,hybrid exciton condensates at sufficiently low tempera-tures.

Acknowledgments

Work in Pisa was supported by MIUR through theprogram “FIRB - Futuro in Ricerca 2010”, Grant no.RBFR10M5BT (“Plasmons and terahertz devices ingraphene”). We thank Allan MacDonald, Leonid Lev-itov, Vincenzo Piazza, Aron Pinczuk, and Giovanni Vig-nale for useful discussions.

∗ Electronic address: asgari@ipm.ir† Electronic address: vp@sns.it‡ Electronic address: m.polini@sns.it1 G. F. Giuliani and G. Vignale, Quantum Theory of theElectron Liquid (Cambridge University Press, Cambridge,2005).

2 S. A. Maier, Plasmonics – Fundamentals and Applications(Springer, New York, 2007); M. I. Stockman, Phys. Today64(2), 39 (2011).

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19 A form factor related to the finite width of the quantumwell where the 2DEG is hosted has been neglected for thesake of simplicity and has the effect of weakening Vbb(q).

Its introduction does not pose any conceptual difficultyand does not change our results qualitatively.

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