Wigner localization in a graphene quantum dot with a mass gap
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Wigner Localization in a Graphene Quantum Dot with a Mass Gap
K. A. Guerrero-Becerra
CNR-NANO Research Center S3, Via Campi 213/a, 41125 Modena, Italy andDipartimento di Scienze Fisiche, Informatiche e Matematiche,Universit`a degli Studi di Modena and Reggio Emilia, Italy
Massimo Rontani
CNR-NANO Research Center S3, Via Campi 213/a, 41125 Modena, Italy
In spite of unscreened Coulomb interactions close to charge neutrality, relativistic massless elec-trons in graphene allegedly behave as noninteracting particles. A clue to this paradox is that bothinteraction and kinetic energies scale with particle density in the same way. In contrast, in a dilutegas of nonrelativistic electrons the different scaling drives the transition to Wigner crystal. Herewe show that Dirac electrons in a graphene quantum dot with a mass gap localize `a la Wignerfor realistic values of device parameters. Our theoretical evidence relies on many-body observablesobtained through the exact diagonalization of the interacting Hamiltonian, which allows us to takeall electron correlations into account. We predict that the experimental signatures of Wigner lo-calization are the suppression of the fourfold periodicity of the filling sequence and the quenchingof excitation energies, which may be both accessed through Coulomb blockade spectroscopy. Ourfindings are relevant to other carbon-based nanostructures exhibiting a mass gap.
PACS numbers: 73.22.Pr, 73.21.La, 31.15.ac, 73.20.Qt
I. INTRODUCTION
The role of electron-electron interactions in graphene isa fundamental and yet open issue that impacts on theoperation of quantum dots (QDs) and other graphene-based nanodevices.
Since the density of states van-ishes at the charge neutrality point, making Coulombinteraction unscreened, one might expect strongly corre-lated behavior at low energies. Indeed, the fine structureconstant α = e / ( ǫ ~ v F )—which is the ratio of Coulombto Fermi energy—is of order unity, much larger than thevalue α = 1 /
137 of quantum electrodynamics, thereforethe many-body problem may not be treated with pertur-bative methods (here ǫ is the background dielectric con-stant and v F the Fermi velocity). As a matter of fact, thepredicted ratio of viscosity to entropy per electron is char-acteristic of an extremely interacting quantum fluid. However, electrons in bulk graphene allegedly behaveas noninteracting particles, except for subtle effects dueto velocity renormalization, coupling with phonons/ plasmons, and a hypothetical excitonic gap. Thekey to this paradox is that the density parameter r s ,which quantifies the impact of electron correlations, does not depend on the electron density n but coincideswith α . In contrast, r s ∼ n − / of the conventional two-dimensional electron gas increases as n decreases dueto the massive dispersion of electrons. An electron solid(Wigner crystal) is even predicted in the dilute limit, as the long-range order induced by Coulomb interactionlocalizes electrons in space. Therefore, a way to disclosethe many-body physics of graphene is to make electronsmassive, invalidating the above scaling argument. Thisoccurs e.g. in the fractional quantum Hall effect and in bilayer graphene, which might be an excitonicinsulator. In this paper we explore theoretically the few-bodyphysics of a graphene QD with a mass gap. Our mo-tivation is twofold: On one side, electrons in semicon-ductor QDs may form Wigner molecules, i.e., finite-size precursors of the Wigner crystal, including carbon-based nanostructures—nanotubes—for which the effectis dramatic. On the other side, a current trend ingraphene QDs is to minimize the roles of disorder andedge states, which are extrinsic sources of localization.These next-generation devices include atomically pre-cise nanoribbons and bilayer QDs—possibly definedthrough gates.
Here we consider a clean, circular QD with a massgap induced by the breaking of sublattice symmetry.This could be realized through the interaction betweengraphene and substrate, such as BN and SiC (but the evidence for these materials is debated ).The presence of the gap allows to electrostatically definethe QD as well as to perform Coulomb blockade spec-troscopy, as sketched in Figs. 1(a)-(b).Other authors already suggested that electrons ingraphene QDs may crystallize. However, some of theseanalyses were limited to degenerate edge states thatare sensitive to interactions as well as to all kindsof perturbation, whereas other theories treatedCoulomb interaction at the mean field level, whichmay artificially enhance localization, or considered onlyvalley-polarized electrons, which artfully breaks time-reversal symmetry. Here we exactly diagonalize the fullinteracting Hamiltonian taking into account correlationsat all orders and the presence of inequivalent K (isospin τ = 1) and K ′ ( τ = −
1) Dirac cones. Through the anal-ysis of the energy spectrum, charge density and pair cor-relation functions we show that electrons form Wignermolecules in realistic devices, exhibiting signatures ofTypeset by REVTEX (a)(c) (d)(b)A B × × T ) defines the dotwhile source (V S ), drain (V D ), and back (V G ) gates allow forCoulomb blockade spectroscopy. (b) Radial QD confinementpotential. The interaction between graphene and substrateopens a mass gap 2∆ in the QD energy spectrum. (c) Low-est noninteracting QD energy levels in the conduction band.Black (red [gray]) lines label states in K (K ′ ) valley. (d) Realpart of sublattice-resolved envelopes whose energies are la-beled by the square and triangle symbols in panel c. crystallization in Coulomb blockade spectra.The structure of this paper is the following: After il-lustrating the low-energy effective-mass Hamiltonian aswell as the exact-diagonalization method we use to solvethe few-body problem (Sec. II), we report our predic-tions for the QD addition energy (Sec. III) and one-body charge density (Sec. IV). These data, together withspin-resolved charge densities (Sec. V), show evidence ofWigner localization in a broad range of device param-eters. By breaking the QD circular symmetry throughangular pair correlation functions (Sec. VI) we are ableto image the formation of Wigner molecules in space.We predict as an experimental signature of the Wignermolecule the quenching of its highly degenerate excita-tion energies (Sec. VII). II. THEORETICAL MODEL
The envelope-function QD Hamiltonian for noninter-acting electrons in the valley τ (Ref. 39) isˆ H τ = − i ~ v F (ˆ τ ∂/∂x + ˆ τ ∂/∂y ) + τ ∆ˆ τ + U ( ρ )ˆ τ . (1)Here v F ≈ m/s is the Fermi velocity, the 2 × τ , ˆ τ , ˆ τ , and the unit matrix ˆ τ act on pseu-dospinors whose components are the A/B sublattice en-velopes, U ( ρ ) = U Θ( ρ − R ) is the circular hard-wall con- finement potential of height U plotted in Fig. 1(b), with R being the QD radius and ρ = ( x + y ) / . The po-tential U , modulated by the top gate shown in Fig. 1(a),confines the electrons in the QD since the Zeeman-liketerm ∆ˆ τ breaks sublattice inversion symmetry, henceinducing a gap 2∆ into the energy spectrum [Fig. 1(b)].In the following we take ∆ = U = 0 .
26 eV.We find numerically the eigenvalues of ˆ H τ followingthe method of Ref. 39. The QD bound states Φ( r ) arepseudospinors of the formΦ( r ) = e i ( j − / ϕ R A ( ρ ) e iϕ R B ( ρ ) , (2)where ϕ is the azimuthal angle, j = ± / ± / . . . is the half-integer quantum number eigenvalue of thetotal angular momentum ˆ z = − i ~ ∂/∂ϕ + ~ ˆ τ /
2, and R A ( ρ ) [ R B ( ρ )] is the radial envelope on sublattice A[B] (Ref. 57). As illustrated in Fig. 1(c) for the low-est conduction-band states, QD orbitals whose quan-tum numbers differ solely in the sign of τ (black or red[gray] lines) have different energies since inversion sym-metry is broken, whereas time reversal simmetry protects ε ( τ, j ) = ε ( − τ, − j ). Overall, including the spin degreeof freedom σ = ↑ , ↓ , QD levels are four-fold degenerate.Both radial profiles and integrated weights of envelopes R ( ρ ) are generically different on the two sublattices, asshown in the example of Fig. 1(d).We consider a few excess interacting charge carrierspopulating the QD conduction band. The presence of thegap 2∆ allows us to ignore the pathologies that plaguethe many-body problem of Dirac electrons due to the un-boundedness of the energy spectrum. The interactingHamiltonian isˆ H = X aτσ ε aτ ˆ c † aτσ ˆ c aτσ + 12 X abcd X ττ ′ X σσ ′ × h aτ, bτ ′ | v ( r − r ′ ) | cτ ′ , dτ i ˆ c † aτσ ˆ c † bτ ′ σ ′ ˆ c cτ ′ σ ′ ˆ c dτσ , (3)where ˆ c † aτσ creates an electron of spin σ in the orbital | aτ i labeled by quantum numbers τ and a ≡ ( j a , n a )whose energy is ε aτ ( n a is the number of radial nodes).Two-body interaction takes the Ohno form v ( r − r ′ ) = v (cid:2) v ǫ/e ) | r − r ′ | (cid:3) − / , where ǫ is the backgroundrelative dielectric constant. Since realistic values of ǫ fallin a wide range between ǫ = 1 . ǫ = 44, depending onthe substrate as well as on nearby gates, here we treat ǫ as a free parameter. At large distances v approachesthe Coulomb potential, whereas its contact limit is theHubbard-like intra-atomic interaction v = 15 eV for the2 p z orbital. Matrix elements h aτ, bτ ′ | v | cτ ′ , dτ i are ob-tained from tight-binding states neglecting interatomicorbital overlaps as well as small intervalley exchangeterms. The many-body states are superpositions of the Slaterdeterminants obtained by filling the lowest 68 spin-valley-orbitals with N electrons in all possible ways (aka full Figure 2. Coulomb blockade linear spectroscopy. Chemicalpotential µ ( N ) vs electron number N for different backgrounddielectric constants ǫ , with radius R = 250 ˚A. Inset: Chargingenergy ∆ µ ( N ) vs N . Lines are guides to the eye. ∆ µ may bemeasured as electrons are added to the quantum dot one byone tuning the backgate shown in Fig. 1(a). configuration interaction ). This size of the truncatedsingle-particle basis set was chosen after checking that thecomputed many-body ground-state energy is well con-verged. In the Fock basis of Slater determinants ˆ H is asparse matrix, with blocks labeled by the total angularmomentum and (iso)spin. The maximum linear size ofthe matrix is 2,187,712, which we diagonalize with thehome-built parallel code DONRODRIGO. Thisprovides highly accurate energies and wave functions ofboth ground and excited states, in contrast to other high-level methods, such as quantum Monte Carlo, addressingground state properties only.
III. COULOMB BLOCKADE SPECTROSCOPY
A key quantity we obtain from the computed groundstate energies E ( N ) is the chemical potential µ ( N ) = E ( N ) − E ( N − N th electron injected into the QD contain-ing N − In Fig. 2 we artificially modulate the back-ground screening ǫ to highlight the effect of Coulomb in-teraction on the filling sequence (here R = 250 ˚A). Inthe absence of interactions ( ǫ = 100, dotted line), µ ( N )is constant except for a step when adding the fifth elec-tron, which corresponds to a peak in the charging energy∆ µ ( N ) = µ ( N + 1) − µ ( N ) (see inset). This finite value∆ µ ( N = 4) ≈
10 meV is the orbital energy cost requiredto add an electron to the second shell after the first onehas been filled with four electrons. This fourfold period- 〈 n 〉 ( un i t s o f R - ) ρ ( units of R ) ε = 2 ε = 5 ε = 25 ε = 100 Figure 3. Emergence of radial correlations in the wave func-tion. One-body density h n ( r ) i vs radial coordinate ρ for dif-ferent values of dielectric constant ǫ and electron number N ,with radius R = 1250 ˚A. Realistically screened mutual inter-actions push electrons against the QD potential wall. icity is generic for all fillings, as clear from Fig. 1(c).As the interaction strength is turned on, the shellstructure of µ ( N ) is progressively washed out. In con-trast with circular QDs in ordinary semiconductors, the charging energy ∆ µ shown in the inset of Fig. 2 nei-ther exhibits half-shell peaks linked to Hund’s rule nordecreases with N . The former feature, shared by carbon-nanotube QDs, is due to the spin-valley multicom-ponent nature of the wave function. In fact, at the non-interacting level the four-fold degenerate spin-valley pro-jections are linked to a single orbital state, hence thereis no Hund’s rule, which is associated with the partialfilling of a degenerate manifold of separate orbital states.The latter feature is peculiar to the hard-wall confine-ment potential, as in the case of ordinary semiconductorsthe potential is soft so the dot size L increases with N whereas the charging energy ∆ µ = e /C decreases with N ( C ∼ L is the QD capacitance).For realistic values of ǫ the Coulomb energy over-whelms the kinetic energy, making µ increase almost lin-early with N (dashed and solid lines in Fig. 2 for ǫ = 5and 2, respectively). IV. EMERGENCE OF RADIALCORRELATIONS
To clarify how interactions affect the wave function wecompute the—circularly symmetric—one-body density h n ( r ) i = 1 N N X i =1 h δ ( r − r i ) i , (4)where h . . . i is the quantum statical average for vanish-ing temperature. In practice, we average h n ( r ) i over the 〈 n σ 〉 ( un i t s o f R - ) ε = 2 ε = 5 ε = 25 R = 250 Å R = 500 Å R = 1250 Å ρ ( units of R ) Figure 4. (color online). Suppression of exchange interac-tions. Spin-resolved density h n σ ( r ) i vs radial coordinate ρ fordifferent values of dielectric constant ǫ (left panel, R = 500˚A) and radius R (right panel, ǫ = 5), with N = 5 and spinprojection S z = 1 /
2. Solid and dashed lines point to h n ↑ i and h n ↓ i , respectively. Wigner localization depletes the probabil-ity weight in the regions halfway an electron and its neighborsand hence suppresses exchange interactions, inducing largespin degeneracies. ground-state multiplet, whose large angular-momentumdegeneracy is protected by symmetry against the effectof interaction. After the averaging h n ( r ) i is the same onboth sublattices, unspecified in the following.Figure 3 shows the evolution of the radial profile of h n ( r ) i with the interaction strength. Whereas for largescreening (dotted lines) the probability weight is spreadall over the QD, as ǫ is decreased the central region isdepleted with its weight being moved towards the dotwall. For realistic screening (dashed and solid lines) h n i is a ring with electrons pushed against the potential wallby Coulomb repulsion, which hints to the formation ofa Wigner molecule. This trend is generic for differentelectron numbers and dot radii, the larger R the higher ǫ at which the ring structure sets in (data not shown). V. SUPPRESSION OF EXCHANGEINTERACTIONS
A fingerprint of Wigner localization is provided by thespin-resolved one-body density h n σ ( r ) i , h n σ ( r ) i = 1 N σ N X i =1 h δ σσ i δ ( r − r i ) i S z . (5)Here N σ is the number of electrons with spin σ so h n σ ( r ) i is normalized to one, and h . . . i S z is the average takenover the manifold of states with fixed total spin projec-tion S z = ( N ↑ − N ↓ ) /
2. For odd electron numbers h n ↑ ( r ) i and h n ↓ ( r ) i generically differ, as illustrated in Fig. 4 forfive electrons and S z = 1 /
2. However, as the interactionstrength is increased by either suppressing screening (leftpanel) or increasing the dot size (right panel), the radialprofiles of h n ↑ i (solid lines) and h n ↓ i (dashed lines) tendto overlap and form the same probability density ring. The rationale is that Coulomb forces localize electronsin space, depleting the probability weight in the regions halfway an electron and its neighbors. Therefore, ex-change interactions between pairs of electrons are sup-pressed, making spin degrees of freedom redundant.
VI. EMERGENCE OF ANGULARCORRELATIONS
To detect whether angular correlations are enforced byinteractions we break the circular symmetry of the one-body density introducing the pair correlation function P ( r , r ), i.e., the conditional probability of finding anelectron at r provided another electron is located at thefixed position r displaced from the origin, P ( r , r ) = 1 N ( N − X σ ,σ ,...,σ N Z d r d r . . . d r N | ψ ( r , σ ; r , σ ; r , σ ; . . . ; r N , σ N ) | . (6)For the sake of simplicity, here we take the quantum av-erage over a selected pure quantum state ψ belonging tothe ground-state multiplet and show the sublattice com-ponent with the largest weight.The insets of Fig. 5(b) show how the contour plotsof P ( r , r ) for four electrons evolve in the xy plane asscreening is suppressed. The black dots highlight the po-sitions r of the fixed electron, located at the maximum ofof the one-body density with arbitrary angle. As the in-teraction strength increases [panels from left ( ǫ = 100) toright ( ǫ = 2)], we see—beyond the onset of the correlationhole around the fixed particle—a strong rearrangementof the probability weight: a non-trivial structure emergesmade of three peaks located at the vertices of a squarewhose last vertex is placed at r . Overall, the three peaksplus the fixed electron realize a square Wigner molecule,which rotates together with r .Cutting the contour plots of P ( r , r ) along a ring ofradius | r | allows us to appreciate the role of interactionsin driving spatial order and localization, as we show inFig. 5(b). For weak correlations (dotted line) P vs ϕ isfeatureless, exhibiting a minor depression close to ϕ = 0,2 π , which realizes the exchange hole around the fixedelectron position. Increasing the interaction (up to ǫ = 2,solid line) the three peaks of the square Wigner moleculeemerge together with a deep correlation hole around r ,the peak-to-valley ratio increasing with decreasing ǫ .Figure 5(a) shows the generic behavior of N electronsin the strongly correlated limit, here enforced with ǫ = 2and R = 2250 ˚A. The electrons realize Wigner moleculeswhose forms are regular polygons with N vertices, asillustrated by the three-dimensional plots of P ( r , r ) forthe dimer ( N = 2), the triangle ( N = 3), the square( N = 4), and the pentagon ( N = 5). VII. EXCITATION SPECTRUM
The excitation spectrum of a Wigner molecule maybe measured by either non-linear Coulomb blockade (a)(b)Figure 5. (color online). Polygonal Wigner molecules. (a)Three-dimensional contour plots of pair correlation functions P ( r , r ) for ε = 2 and R = 2250 ˚A. Black dots point to thelocations r of fixed electrons. (b) Pair correlation function P ( r , r ) vs angle ϕ with | r | = | r | for four electrons and dif-ferent values of dielectric constant ǫ , with R = 500 ˚A. Inset:corresponding contour plots of P ( r , r ) in the xy plane. In-creasing the interaction strength leads to the formation of thecorrelation hole as well as the development of angular corre-lations, which enforce a square Wigner molecule. spectroscopy —opening the source-drain bias window inthe setup of Fig. 1(a)—or inelastic light scattering. Figure 6 shows the dependence of low-lying excitationenergies on the interaction strength for four electrons.For weak interactions ( ǫ = 100), the spectrum remindsus of the single-particle ladder of levels of Fig. 1(c), asto excite the ground state one moves an electron fromthe lowest completely filled shell to higher orbital states.Whereas in this specific case the ground state is nondegenerate, the excited multiplets exhibit large degen-eracies (labeled by numbers) linked to different (iso)spinorientations. For stronger interactions, the lowest excita-tion energies are strongly quenched as the system turnsinto a square Wigner molecule. Comparing the one-bodydensity n ( r ) of the ground-state (black curves in the in-sets) with n ( r ) averaged over the lowest excited multiplet(red [gray] curves), we see that the two curves overlap for strong interactions (left inset, ǫ = 2). In fact, in the limitof perfect localization the Wigner-molecule ground state ________________ _________________ ___________ _____________________ __ _____ ε E xc i t a t i on ene r g y ( m e V ) ε = 2 ε = 100 Figure 6. (color online) Excitation spectrum of a Wignermolecule. Low-lying excitation energies vs dielectric constant ǫ for N = 4 and R = 500 ˚A. Numbers label degeneracies ofselected multiplets. Insets: density n ( r ) vs radial coordinate ρ averaged over the ground state (black curve) and the firstexcited multiplet (red [gray] curve). The Wigner-moleculeground state is highly degenerate as localized electrons mayindependently flip their (iso)spins. exhibits a huge degeneracy since localized electrons mayindependently flip their (iso)spins, as exchange interac-tions are completely suppressed. Therefore, the energyspectrum of the Wigner molecule is a ladder of highly-degeratate rotovibrational quanta. VIII. CONCLUSIONS
In conclusion, electrons in a disorder-free graphenequantum dot with a mass gap form Wigner moleculesfor a broad range of device parameters. The signaturesof Wigner localization may be traced in Coulomb block-ade and other electron spectroscopies. We expect ourfindings to be generic to clean carbon-based nanostruc-tures exhibiting a mass gap, including atomically preciseribbons and bilayer-graphene quantum dots.
ACKNOWLEDGMENTS
We thank Andrea Secchi, Elisa Molinari, DeborahPrezzi, Marco Polini, Andrea Candini, Andrea Ferretti,Vittorio Pellegrini, Stefano Corni, Stefan Heun, andPino D’Amico for stimulating discussions. This work issupported by MIUR-PRIN2012 MEMO, EU-FP7 MarieCurie initial training network INDEX, MIUR ABNAN-OTECH, CINECA-ISCRA grants IscrC TUN1DFEW,IscrC TRAP-DIP, and IscrC PAIR-1D. A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S.Novoselov, and A. K. Geim, Rev. Mod. Phys. , 109(2009) D. S. L. Abergel, V. Apalkov, J. Berashevich, K. Ziegler,and T. Chakraborty, Adv. Phys. , 261 (2010) S. Das Sarma, S. Adam, E. H. Hwang, and E. Rossi, Rev.Mod. Phys. , 407 (2011) V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, andA. H. Castro Neto, Rev. Mod. Phys. , 1067 (2012) P. Recher and B. Trauzettel, Nanotechnology , 302001(2010) A. V. Rozhkov, G. Giavaras, Y. P. Bliokh, V. Freilikher,and F. Nori, Phys. Rep. , 77 (2011) J. G¨uttinger, F. Molitor, C. Stampfer, S. Schnez, A. Ja-cobsen, S. Dr¨oscher, T. Ihn, and K. Ensslin, Rep. Prog.Phys. , 126502 (2012) D. Prezzi, D. Varsano, A. Ruini, A. Marini, and E. Moli-nari, Phys. Rev. B , 041404(R) (2008) O. V. Yazyev, Rep. Prog. Phys. , 056501 (2010) X. Wang, Y. Ouyang, L. Jiao, H. Wang, L. Xie, and J. Wu,Nature Nanotech. , 563 (2011) D.-K. Ki and A. F. Morpurgo, Phys. Rev. Lett. ,266601 (2012) M. M¨uller, J. Schmalian, and L. Fritz, Phys. Rev. Lett. , 025301 (2009) D. C. Elias, R. V. Gorbachev, A. S. Mayorov, S. V. Mo-rozov, A. A. Zhukov, P. Blake, L. A. Ponomarenko, I. V.Grigorieva, K. S. Novoselov, F. Guinea, and A. K. Geim,Nature Phys. , 701 (2011) J. Chae, S. Jung, A. F. Young, C. R. Dean, L. Wang,Y. Gao, K. Watanabe, T. Taniguchi, J. Hone, K. L. Shep-ard, P. Kim, N. B. Zhitenev, and J. A. Stroscio, Phys. Rev.Lett. , 116802 (2012) D. A. Siegel, W. Regan, A. V. Fedorov, A. Zettl, andA. Lanzara, Phys. Rev. Lett. , 146802 (2013) A. Bostwick, F. Speck, T. Seyller, K. Horn, M. Polini,R. Asgari, A. H. MacDonald, and E. Rotenberg, Science , 999 (2010) D. V. Khveshchenko, Phys. Rev. Lett. , 246802 (2001) J. E. Drut and T. A. L¨ahde, Phys. Rev. Lett. , 026802(2009) M. Rontani and L. J. Sham, “Novel superfluids volume 2,”(Oxford University Press, Oxford, UK, 2014) Chap. 19,preprint at arXiv:1301.1726 N. W. Ashcroft and N. D. Mermin,
Solid State Physics (Holt-Saunders Intnl, New York, NY, 1976) H. P. Dahal, Y. N. Joglekar, K. S. Bedell, and A. V. Bal-atsky, Phys. Rev. B , 233405 (2006) T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. ,437 (1982) E. Wigner, Phys. Rev. , 1002 (1934) X. Du, I. Skachko, F. Duerr, A. Luican, and E. Y. Andrei,Nature (London) , 192 (2009) K. I. Bolotin, F. Ghahari, M. D. Shulman, H. L. Stormer,and P. Kim, Nature (London) , 196 (2009) E. McCann and M. Koshino, Rep. Prog. Phys. , 056503(2013) H. Min, R. Bistritzer, J. Su, and A. H. MacDonald, Phys.Rev. B , 121401(R) (2008) S. M. Reimann and M. Manninen, Rev. Mod. Phys. ,1283 (2002) C. Ellenberger, T. Ihn, C. Yannouleas, U. Landman,K. Ensslin, D. Driscoll, and A. C. Gossard, Phys. Rev.Lett. , 126806 (2006) S. Kalliakos, M. Rontani, V. Pellegrini, C. P. Garcia,A. Pinczuk, G. Goldoni, E. Molinari, L. N. Pfeiffer, andK. W. West, Nature Phys. , 467 (2008) A. Singha, V. Pellegrini, A. Pinczuk, L. N. Pfeiffer, K. W.West, and M. Rontani, Phys. Rev. Lett. , 246802(2010) S. Pecker, F. Kuemmeth, A. Secchi, M. Rontani, D. C.Ralph, P. L. McEuen, and S. Ilani, Nature Phys. , 576(2013) P. Ruffieux, J. Cai, N. C. Plumb, L. Patthey, D. Prezzi,A. Ferretti, E. Molinari, X. Feng, K. M¨ullen, C. A.Pignedoli, and R. Fasel, ACS Nano , 6930 (2012) J. Milton Pereira Jr., P. Vasilopoulos, and F. M. Peeters,Nano Letters , 946 (2007) M. T. Allen, J. Martin, and A. Yacoby, Nature Commun. , 934 (2012), doi:10.1038/ncomms1945 A. M. Goossens, S. C. M. Driessen, T. A. Baart, K. Watan-abe, T. Taniguchi, and L. M. K. Vandersypen, Nano Lett. , 4656 (2012) A. M¨uller, B. Kaestner, F. Hohls, T. Weimann, K. Pierz,and H. W. Schumacher, J. App. Phys. , 233710 (2014) M. Zarenia, B. Partoens, T. Chakraborty, and F. M.Peeters, Phys. Rev. B , 245432 (2013) P. Recher, J. Nilsson, G. Burkard, and B. Trauzettel, Phys.Rev. B , 085407 (2009) G. Giovannetti, P. A. Khomyakov, G. Brocks, P. J. Kelly,and J. van den Brink, Phys. Rev. B , 073103 (2007) F. Amet, J. R. Williams, K. Watanabe, T. Taniguchi,and D. Goldhaber-Gordon, Phys. Rev. Lett. , 216601(2013) B. Hunt, J. D. Sanchez-Yamagishi, A. F. Young,M. Yankowitz, B. J. LeRoy, K. Watanabe, T. Taniguchi,P. Moon, M. Koshino, P. Jarillo-Herrero, and R. C.Ashoori, Science , 1427 (2013) C. R. Woods, L. Britnell, A. Eckmann, R. S. Ma, J. C. Lu,H. M. Guo, X. Lin, G. L. Yu, Y. Cao, R. V. Gorbachev,A. V. Kretinin, J. Park, L. A. Ponomarenko, M. I. Kat-snelson, Y. N. Gornostyrev, K. Watanabe, T. Taniguchi,C. Casiraghi, H. Gao, A. K. Geim, and K. S. Novoselov,Nature Phys. , 451 (2014) S. Y. Zhou, G. H. Gweon, A. V. Fedorov, P. N. First,W. A. de Heer, D. Lee, F. Guinea, A. H. Castro Neto, andA. Lanzara, Nature Mater. , 770 (2007) C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sor-genfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shep-ard, and J. Hone, Nature Nanotech. , 722 (2010) J. Xue, J. Sanchez-Yamagishi, D. Bulmash, P. Jacquod,A. Deshpande, K. Watanabe, T. Taniguchi, P. Jarillo-Herrero, and B. J. LeRoy, Nature Materials , 282 (2011) R. Decker, Y. Wang, V. W. Brar, W. Regan, H. Tsai,Q. Wu, W. Gannett, A. Zettl, and M. F. Crommie, NanoLetters , 2291 (2011) L. Vitali, C. Riedl, R. Ohmann, I. Brihuega, U. Starke,and K. Kern, Surface Science , L127 (2008) E. Rotenberg, A. Bostwick, T. Ohta, J. L. McChesney,T. Seyller, and K. Horn, Nature Mater. , 258 (2008) S. Y. Zhou, D. A. Siegel, A. V. Fedorov, F. El Gabaly,A. K. Schmid, A. H. Castro Neto, D. Lee, and A. Lanzara,
Nature Mater. , 259 (2008) B. Wunsch, T. Stauber, and F. Guinea, Phys. Rev. B ,035316 (2008) I. Romanovsky, C. Yannouleas, and U. Landman, Phys.Rev. B , 075311 (2009) A. D. G¨u¸cl¨u, P. Potasz, O. Voznyy, M. Korkusinski, andP. Hawrylak, Phys. Rev. Lett. , 246805 (2009) P. Potasz, A. D. G¨u¸cl¨u, A. W´ojs, and P. Hawrylak, Phys.Rev. B , 075431 (2012) T. Paananen, R. Egger, and H. Siedentop, Phys. Rev. B , 085409 (2011) N. Yang and J.-L. Zhu, J. Phys.: Condens. Matter ,215303 (2012) D. P. DiVincenzo and E. J. Mele, Phys. Rev. B , 1685(1984) W. Greiner, B. M¨uller, and J. Rafelski,
Quantum Electro-dynamics of Strong Fields (Springer Verlag, Berlin, 1985) W. H¨ausler and R. Egger, Phys. Rev. B , 161402(R)(2009) A. L. Walter, A. Bostwick, K.-J. Jeon, F. Speck, M. Ostler,T. Seyller, L. Moreschini, Y.-J. Chang, M. Polini, R. As-gari, A. H. MacDonald, K. Horn, and E. Rotenberg, Phys.Rev. B , 085410 (2011) C. Hwang, D. A. Siegel, S.-K. Mo, W. Regan, A. Ismach,Y. Zhang, A. Zettl, and A. Lanzara, Sci. Rep. , 590 (2012) K. Ohno, Theor. Chim. Acta , 219 (1964) A. Secchi and M. Rontani, Phys. Rev. B , 035417 (2010) A. Secchi and M. Rontani, Phys. Rev. B , 125403 (2013) M. Rontani, C. Cavazzoni, D. Bellucci, and G. Goldoni, J.Chem. Phys. , 124102 (2006) L. P. Kouwenhoven, C. M. Marcus, P. L. McEuen,S. Tarucha, R. M. Westervelt, and N. S. Wingreen, “Elec-tron transport in quantum dots,” (Kluwer, 1997) p. 105 S. Tarucha, D. G. Austing, T. Honda, R. J. van der Hage,and L. P. Kouwenhoven, Phys. Rev. Lett. , 3613 (1996) A. Secchi and M. Rontani, Phys. Rev. B80