Interacting electrons in graphene nanoribbons in the lowest Landau level
We study the effect of electron-electron interaction and spin on electronic and transport properties of gated graphene nanoribbons (GNRs) in a perpendicular magnetic field in the regime of the lowest Landau level (LL). The electron-electron interaction is taken into account using the Hartree and Hubbard approximations, and the conductance of GNRs is calculated on the basis of the recursive Greens function technique within the Landauer formalism. We demonstrate that, in comparison to the one-electron picture, electron-electron interaction leads to the drastic changes in the dispersion relation and structure of propagating states in the regime of the lowest LL showing a formation of the compressible strip and opening of additional conductive channels in the middle of the ribbon. We show that the latter are very sensitive to disorder and get scattered even if the concentration of disorder is moderate. In contrast, the edge states transport is very robust and can not be suppressed even in the presence of a strong spin-flipping.
pacs:72.80.Vp, 73.22.Pr, 73.43.Cd, 73.43.Qt
Due to Dirac spectrum graphene possesses a number of unique electronic and transport propertiesCastro_Neto_review (). In a magnetic field perpendicular to a graphene layer the spectrum is modified into a series of Landau levels (LL) with the energies , where the cyclotron frequency , cm/s is the Fermi velocity and Gusynin () One of interesting features of this spectrum is a presence of 0’th LL with , which is equally shared between electrons and holes. In contrast to conventional two-dimensional electron gas systems the position of this level does not depend on a value of the magnetic field. An experimental manifestation of these unusual series is the anomalous quantum Hall effect with the Hall conductivity given by , where the factor comes from the spin and valley degeneracyNovoselov (); Kim (). In a strong magnetic field the four-fold degeneracy of the lowest (i.e. 0’th) LL is lifted leading to the insulating behavior characterized by the presence of additional plateaux in the Hall conductivity accompanied by the increase of the longitudinal resistivity .Checkelsky (); Zhang (); Zaliznyak () The large energy gaps identified experimentally suggests that electro-electron interactions along with the Zeeman spin-splitting play an important role in understanding of this phenomena.
Most of the experiments reported so far are performed in the four-terminal geometry which allows a direct measurement of the components of the resistivity and conductivity tensorsCheckelsky (); Zhang (); Zaliznyak (); Giesbers (). However in some cases the only possible experimental setup is the two-terminal measurements, even though the results obtained in this geometry are not as straightforward to interpret as the four-terminal onesWilliams (). A difference between the longitudinal and the Hall resistances is not clearly defined in such the measurements. The unexpected consequence of analysis of the two-terminal geometry is that a measured two-terminal resistance between the right and left contacts, corresponds, in the four-terminal setup, to the Hall resistance rather than to the longitudinal one . This stems from the fact that the voltage drop between the sample edges, , equals to the difference of the chemical potentials in the contacts .Kramer (); Goerbig_Lecture ()
An explanation of the quantum Hall effect is often based on the picture of edge states. In a strong enough magnetic field the right and left propagating states are localized at different edges of the system due to the Lorentz force. It leads to exponentially small overlap between counterpropagating states which, in turn, greatly suppresses a possibility of backscattering and is eventually manifested by developing pronounced plateaux in the conductance. In graphene another transport regime can be realizedAbanin (). Coupling of spin to magnetic field due to the Zeeman effect leads to a splitting of the lowest LLs into two sublevels. For the neutral graphene the chemical potential coincides with the zero energy and lies in the center of the spin-gap, see Fig. 2 (d) below. At a given Fermi energy there are two forward propagating states of opposite spins located near different edges. The most prominent feature of this transport regime is that forward and backward propagating states of the different spins are not spatially separated. If a scattering event leads to a spin flipping, the overlap between the counterpropagating states of different spins might be sufficient to cause the backscattering. This backscattering of the spin-polarized states is used for an explanation of the non-zero behavior of near the Dirac pointAbanin_Dissipative ().
It should be stressed that the structure of edge states derived within a one-electron approximation and shown on Fig.(2) (d) corresponds to the case of neutral graphene when the electron density is zero. It is often used for understanding and explanation of a variety of experiments on magnetotransportOng (). The conditions for existence of the edge states in GNRs was discussed in details in Ref.[Gusynin_edge ()]. Also, self-consistent Hartee-Fock calculations were performed for GNR in magnetic fieldJung (). It was shown than the favored ground states in the neutral graphene are charge- or spin-density waves. Neither of these orderings support the edge states. However, in experiments one usually needs to tune a charge concentration by means of the gate voltage. It was shown before that electron-electron interaction between induced charges can dramatically change the energy dispersion and the structure of propagating statesSilvestrov (); Shylau_Cap (); Shylau_Magnet (). The magnetosubband structure and the character of propagating states in the lowest LL for the case of nonzero induced charge density remain largely unexplored and represent the main focus of the present study. We investigate the effect of the electron-electron interaction, spin and global gate electrostatics on magnetotransport properties of GNR in the vicinity of the charge neutrality point. Based on the calculated self-consistent magnetosubband structure we compute and analyze the two-terminal conductance of realistic GNRs focussing on the effect of disorder, surface warping and spin flipping on the robustness of the edge state transmission.
We consider a system consisting of a GNR of the width nm and the length nm, located on an insulating substrate and connected to ideal graphene leads of the same width, see Fig. 1. A metallic back gate is used to tune the Fermi energy and the charge concentration in the GNR. The whole system is subjected to a uniform magnetic field, T, perpendicular to the graphene plane. Such the high value of the magnetic field allows us to use relatively narrow GNR (in order to reduce computational time) and choose the ratio in an accordance with typical experiments (with being the magnetic length). In order to model magnetotrasport in a GNR we use the tight-binding Hubbard Hamiltonian in the mean-field approximation which is shown to describe carbon electron systems in good agreement with first-principles calculationsPalacios (),
where , describe two opposite spin states , ; the summation runs over all sites of the graphene lattice, includes the nearest neighbors only, with eV, being the magnetic flux quantum which is calculated using the Pierel’s substitution
with being the vector potential. (In our calculations we use the Landau gauge, ). The first term in the Hamiltonian (1) describes hopping of electrons through the graphene lattice. The second term describes the spin-splitting in the magnetic field due to the Zeeman effect. Both these terms correspond to the non-interacting part of the Hamiltonian. The interaction between electrons of the opposite spins is described by the Hubbard term with Yazyev (); Jung (), and is a Hartree term describing the long-range Coulomb interaction between induced charges in the GNRShylau_Cap (); FR (),
where is the distance between the GNR and the gate, and the second term describes a contribution from the mirror charges. In order to calculate the electron number at site , Eq.(1) is solved using the recursive Green’s function techniqueXu ()
where is the Green function in the real space representation of an electron of the spin residing on the site and is the Fermi energy which position is adjusted by the gate voltage. Equations (1)-(4) are solved self-consistently in order to compute the band structure, the charge density and the potential distribution. For a given potential distribution we compute the conductance using the Landauer formula
where is a total transmission coefficient, is the conductance quantum, and is the Fermi-Dirac distribution.
Iii Results and discussion
Figure 2 shows the charge density distribution (a)-(c) and the magnetosubband structure (d)-(f) of a GNR for several gate voltages, , 0.1, and 0.5 respectively in the regime of the lowest LL, ( being the filling factor and is the average charge concentration). An application of the gate voltage induces extra charges in the system, Fig. 2 (b),(c), which leads to an asymmetry between the electron and hole branches in the energy spectrum, see Fig. 2 (e),(f). For small gate voltages, see Fig. 2 (e), the lowest unoccupied spin-up branch deforms and gets pinned to the Fermi energy. As a result, a region with compressible densities is formed in the center of the ribbon. We define the compressible strip as a region where the dispersion lies within the energy window .Shylau_Magnet () The compressible strips form because in the above energy window the states are partially filled, i.e. and hence the system has a metallic character. The formation of compressible strips in graphene has been recently discussed in Silvestrov (); Shylau_Magnet (). Further increase of the gate voltage leads to increase of the total electron density, which, in turn, causes even stronger pinning of the lowest LL to the Fermi energy such that the compressible strip gets extended almost over the entire ribbon, see Fig. 2 (f). Figures 2 (a)-(c) show a density distribution for spin-up and spin-down electrons, and the total density distribution, Due to the strong Zeeman splitting the electron branch is predominantly spin-up polarized, whereas the hole branch is spin-down polarized. The total charge distribution shows a density enhancement towards ribbon’s edges. This feature is related to the effect of the electrostatic Coulomb repulsion in a structure with a hard-wall confinement and was discussed (for the spinless case) in Silvestrov (); Shylau_Cap ().
Compressible strips in GNRs are characterized by the existence of counterpropagating states at the same edgeShylau_Magnet (). This can play an important role in understanding of the magnetotransport of interacting electrons in the vicinity of the Dirac point. In one-electron picture the number of propagating states in the region always equals to two with electrons of different spins propagating near different boundaries, see Fig. 2 (d). (Note that a number of propagating states at a given energy is given by a number of intersections of the line with the dispersion bands and the direction of the propagation is determined by a sign of .) The backscattering is probable only in events accompanied by a spin-flipping since the counterpropagating states with the same spin are located at the different edges. However in graphene a spin-relaxation length is rather large and can reach several micrometersTombros (), therefore backscattering is expected to be rather unimportant. However, the situation is different when electron-electron interaction is taken into account. In this case there are a number of additional transport channels at the Fermi energy where electrons of the same spin propagate in the opposite directions at the same boundary, see Fig. 2 (f). In this case no spin-flipping is apparently needed for the backscattering and the latter can be caused by e.g. impurity scattering of electrons of the same spin. To summarize, our self-consistent calculations demonstrate that, in comparison to the one-electron picture, electron-electron interaction leads to the drastic changes in the dispersion relation and structure of propagating states in the regime of the lowest LL such as a formation of the compressible strip and opening of additional conductive channels in the middle of the ribbon.
We proceed to the discussion of the conductance of GNRs in the regime of the lowest LL taking into account the electron interaction and spin effects. We will study the effect of different types of disorder on GNR’s conductance, focussing on the robustness of respectively edge and bulk state transmission. Let us start with the case of an ideal GNR (i.e. without any disorder) whose conductance is shown in Fig. 3 as a red (solid) curve. In the range there are only two states in the energy interval and the transport is determined by the edge states only, see Fig. 2(e) (note that in this range of the states in the compressible regions practically do not contribute to the conductance). As a result, the conductance of the ideal ribbon equals to . Further increase of the filling factor leads to the enhancement of the conductance due to the additional states forming the compressible strip, see Fig. 2(f). The conductance of the ideal ribbon is therefore not quantized and develops a bump-like structure with the increase of the filling factor above . The suppression of conductance quantization in an ideal GNR in a perpendicular magnetic field for was discussed in details in Shylau_Magnet ().
Let us now study the effect of impurities on the conductance, see Fig. 3 (a). Recent two-terminal measurements show that the quantized conductance plateaux are not observed even in high magnetic fieldRibeiro (); Oostinga (); Poumirol (). In particular, this behavior is attributed to the effect of impurities which can crosslink the chiral currents flowing at opposite edges.Ribeiro () We utilize the model of short-range impurities which are uniformly distributed over the GNR. They are modeled by adding to the self-consistent potential a term which is randomly chosen in the energy interval , where is an impurity strength. The concentration of impurities is chosen in the range nm. As expected, the states forming the compressible strip are the most sensitive to disorder. They propagate in the center of the ribbon and have a significant spatial extension in the transverse direction. When the concentration of impurities is moderate (green dotted and blue solid lines), the conductance decreases but remains larger than since there are at least two edge states which transmit the current. The edge states are very robust and can be suppressed only if the impurity concentration greatly exceeds typical experimental values (green dashed line).
Another source of disorder can arise from a surface warping which is an inherent property of grapheneMeyer (). Even though the magnetic field is uniform, a normal to the surface component of the magnetic field fluctuates due to the corrugated geometry of graphene. It results in a spatial-dependent hopping integrals. It has been shown that these spatial correlations can affect the quantum Hall (QH) transition at leading to anomalously abrupt behaviourKawarabayashi () or result in a field-driven topological transition from the QH-metal state to the QH-insulator state in the vicinity of .Zhu () In order to take this effect into account we first generate a corrugated surface (see inset in Fig. 3 (b)) using a method described in Refs. Fasolino (); Klos (). Then graphene lattice is projected on the surface and a hopping integrals between two adjacent carbon atoms are calculated by integration of the vector potential along a line linking these atoms according to Eq. (2). We find that the conductance of the warped GNR is practically the same as that one of the ideal ribbon, see Fig. 3 (b). We therefore conclude that for realistic nanoribbons the spatial modulation of the magnetic field due to the warping has a negligible effect on the conductance.
Finally we investigate the effect of spin-flipping on the magnetotransport of GNRs. It is modeled by introducing in the Hamiltonian an additional term , where describes the local spin-flip rate. This term will in generally admix the counter-propagating states of different spins leading to backscattering. In our calculations the value of is varied in the range Qiao (). The conductance of GNRs in the presence of spin-flipping is shown in Fig. 3(c). When (dotted line) the states constituting the compressible strips get scattered and the total conductance is only due to the edge states. As increased up to (dashed line) the conductance decreases below the value of due to stronger scattering but still remains mostly dominated by the edge states. Backscattering of spin-polarized states due to spin-flipping was suggested as the way to manipulate spin currentsAbanin (). However our results show that even in the case of strong admixture of counterpropagating states, the edge state transmission remains practically unaffected.
The magnetotransport properties of GNRs in the regime of the lowest LL was studied taking into account electron interaction, gate electrostatic and spin effects. When the gate voltage is applied extra charges are induced in the ribbon leading to an asymmetry between the electron and hole branches in the energy spectrum. The lowest LL gets pinned to the Fermi energy and a compressible strip is formed in the middle of the GNR. There are two types of current carrying states in the lowest LL, the conventional edge states and the bulk states in the middle of the ribbon constituting the compressible strip. The later are very sensitive to disorder and get scattered even if the concentration of disorder is moderate. In contrast, the edge states are very robust and can not be suppressed even in the presence of a strong spin-flipping.
Acknowledgements.We acknowledge a support of the Swedish Research Council (VR).
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