Disorder Effects on Helical Edge Transport in Graphene under a Strong, Tilted Magnetic Field
In a recent experiment, Young et al. [Nature 505, 528 (2014)] observed a metal to insulator transition as well as transport through helical edge states in monolayer graphene under a strong, tilted magnetic field. Under such conditions, the bulk is a magnetic insulator which can exhibit metallic conduction through helical edges. It was found that the two-terminal conductance of the helical channels deviates from the expected quantized value ( per edge, at zero temperature). Motivated by this observation, we study the effect of disorder on the conduction through the edge channels. We show that, unlike for helical edges of topological insulators in semiconducting quantum wells, a disorder Rashba spin-orbit coupling does not lead to backscattering, at least to leading order. Instead, we find that the lack of perfect anti-alignment of the electron spins in the helical channels to be the most likely cause for backscattering arising from scalar (i.e. spin-independent) impurities. The intrinsic spin-orbit coupling and other time-reversal symmetry breaking and/or sublattice-parity breaking potentials also lead to (sub-leading) corrections to the channel conductance.
The quantum spin hall effect (QSHE) Kane and Mele (2005a); Maciejko et al. (2011); Kane and Mele (2005b); Bernevig et al. (2006); Qi and Zhang (2011); König et al. (2007); Hasan and Kane (2010) is an interesting phenomenon related to the emergence of an insulating electronic state in the bulk of a two-dimensional material whose one-dimensional edge remains metallic and exhibits quantized conductance at zero temperature. As first discussed by Kane and Mele for graphene,Kane and Mele (2005a) in the QSHE state, the helical edges can transport a spin current without dissipation as long as time-reversal symmetry is not broken. Unfortunately, the spin-orbit coupling is very small in graphene Huertas-Hernando et al. (2006); Yao et al. (2007); Min et al. (2006); Gmitra et al. (2009) ( eV) and therefore the effect is not experimentally accessible in this material.
Nevertheless, in a recent experiment, Young et al. (2014) transport through helical edges has been observed in monolayer graphene when the latter is submitted to a strong, tilted magnetic field . Under such conditions, the bulk becomes a magnetic insulator whilst the edge can remain metallic. By increasing the strength of the component of parallel to the graphene plane (), the magnetic state can be tuned from anti-ferromagnet (AF), to canted anti-ferromagnet (CAF), and finally to ferromagnet (FM). Kharitonov (2012a, b); Young et al. (2014) As shown by Kharitonov, Kharitonov (2012a, b) (see also Ref. Murthy et al., 2014) the nature of the magnetic insulating is determined by the competition of the Zeeman and interaction energies. At the edge of the magnetic insulator in the CAF and FM states, electrons propagate in opposite directions with (roughly) opposite spins, that is, a helical edge is formed. Fertig and Brey (2006); Abanin et al. (2006); Kharitonov (2012a); Murthy et al. (2014) Although time-reversal symmetry is broken explicitly by the external magnetic field, the two-terminal conductance of the the helical edge approaches the quantized conductance at the largest magnetic field: The conductance per edge is , where and at the lowest accessible temperatures in the FM state. Young et al. (2014)
The physical origin of deviation is not entirely clear and it is the main purpose of this work to investigate how disorder can contribute to it. To this end, we assume that a potential scatterer is located near an otherwise perfect armchair graphene edge (see Fig. 1). Our model is not intended to be a faithful description of the experimental situation found in Ref. Young et al., 2014, but we believe that it serves well to the purpose of investigating the effects of disorder-induced dissipation at the helical edges. However, if necessary, the model can be easily generalized to extended disorder potentials, a task that we shall not pursue here.
Nevertheless, the situation envisaged here can exist in an actual experimental device, where the scatterer may be a contact Young et al. (2014) or a metallic cluster located near the edge of the device, for instance. If the cluster/contact contains heavy atoms (e.g. gold Young et al. (2014)), the latter can locally induce by proximity a spin-orbit coupling (SOC) as well as providing a source for potential scattering. Weeks et al. (2011); Ferreira et al. (2014); Balakrishnan et al. (2014); Calleja et al. (2015) Therefore, one of the important goals of this work is to investigate the role of the SOC-induced dissipation at the edge. Indeed, in the case of semiconducting quantum wells, it is known that Rashba-like disorder SOC has been identified as one of the possible causes for backscattering, Strom et al. (2010). In addition, it was suggested by the authors of Ref. Young et al., 2014 that Rashba-like disorder could be an explanation for the absence of perfect quantization of the edge conductance.
In this article, we derive the low-energy effective description of the above model accounting for electron-electron interactions in the helical edge channels, which are crucial in determining the temperature dependence of . The effective low-energy model turns out to be a version of the Kane-Fisher model for an impurity in a Tomonaga-Luttinger liquid. Kane and Fisher (1992) Our analysis shows that the leading contributions to are potential scattering and, to a lesser extent, the so-called intrinsic (or Kane-Mele) type SOC. However, we find that, to leading order, the Rashba SOC does not lead to backscattering. In addition, our analysis indicates that other sources of backscattering sources are potentials that explicitly violate the sublattice inter-change (parity) symmetry and/or time-reversal symmetry.
The rest of this article is organized as follows. In Sec. II, we review Kharitonov’s results Kharitonov (2012a) for the the energy dispersion of the armchair edge. Hence, we derive the effective low energy model for the edge states, which turns out to be, accounting for electron-electron interaction, a Tomonaga-Luttinger liquid with two counter-propagating edge modes of different Fermi velocity. In Sec. III, we discuss the properties of a general scatterer, which may correspond to a large metal cluster located near the edge of the sample. In Sec. IV, we obtain the finite temperature conductance corrections within linear response theory. A summary and a brief discussion of the possible extensions to this work are provided in Sec. V. The most technical details of the calculation have been relegated to the Appendices. In what follows, we work in units where .
Ii Low-energy description of the edge
In order to make the article more self-contained, we will review the main results of Kharitonov’s calculation for a semi-infinite graphene layer submitted to a strong, tilted magnetic field. Within the continuum description, the low-energy effective Hamiltonian of graphene under a tilted magnetic field can be written as a sum of three terms, , where
is the kinetic energy, the Coulomb interaction, and the Zeeman term; the couplings , and are the Landé g-factor, the Bohr magneton and the magnitude of the applied (total) magnetic field, respectively. The Fermi velocity of electrons in graphene is ( being the speed of light), and is the kinetic momentum ( is the electron charge). The matrices () act upon the sublattice (valley) pseudo-spin and act upon the spin degree of freedom. Note that the -axis is perpendicular to the graphene plane and the -axis will be taken along the armchair edge). The Zeeman energy is proportional to the total magnetic field , where () is the magnetic field along the perpendicular (parallel) direction to graphene plane.
Since the largest energy scale of quantum hall system is set by the cyclotron energy, where is the magnetic length, it is useful to neglect the Zeeman and interaction terms in the Hamiltonian and diagonalize the kinetic term first. Furthermore, in the study of edge effects, two kinds of edge terminations are commonly used, namely the armchair and zig-zag edge. Castro Neto et al. (2009); Katsnelson (2012); Brey and Fertig (2006) In this work, for the sake of simplicity, we use armchair edge, which we assume to be located at . The energy spectrum of the kinetic energy part of the Hamiltonian for a semi-infinite armchair edge is shown in Fig. 2. Note that, within this approximation, all the energy levels spin-degenerate and therefore the ground state is non-magnetic.
For what follows, it is useful to express the fermionic field operators and in the basis of Landau level orbitals that diagonalize the kinetic energy :
where is the momentum along the edge (i.e. -axis) and is the Landau level quantum number. In the Landau gauge where ,
for . The states are the solutions of the Landau level problem of the Schrödinger equation in the Landau gauge. In particular,
where is the magnetic length and the length of the (armchair) edge. Finally, the spinors:
Note that in the Landau level (0LL), the sublattice and valley (pseudo-)spins are aligned. This feature will probe extremely important in the discussion that follows. Accounting for the spin degeneracy, each single-particle state in the 0LL is four-fold degenerate corresponding to and .
ii.1 Zeeman and interaction energy effects
In order to fully determine the ground state, we next take into account the Coulomb interaction and the Zeeman term. In neutral (i.e. undoped) graphene, the Landau level of zero energy (0LL) is half filled and therefore all Landau levels below (above) 0LL are completely filled (empty). At temperatures far below the cyclotron energy (i.e. ) and close to half-filling, the low energy dynamics is determined by the electrons in the 0LL. As a result, we project the Coulomb interaction (Eq. 2) onto the subspace of states belonging to the 0LL. Thus, in what follows, we shall suppress the Landau level index as it is implicitly understood that . In addition, we will suppress the sublattice index because in the 0LL the pseudo-spin and valley spin are aligned. The Coulomb interaction in the 0LL subspace is therefore formally described by Kharitonov (2012a)
where stands for normal ordering. The Coulomb amplitudes have been discussed in Refs. Goerbig, 2011; Kharitonov, 2012a and we will not dwell on them here. Nevertheless, it is worth mentioning that the Coulomb amplitudes that do not conserve valley quantum number are exponentially suppressed by an exponential factor where is the distance between the carbon atoms in graphene. Thus, the intra-valley amplitude, , is dominant one. Goerbig (2011) In fact, if only is retained in Eq. (11), the Hamiltonian and the ground state would exhibit SU symmetry (see Ref.Goerbig, 2011 and references therein).
However, when the sub-leading amplitudes that violate the SU symmetry are also included, together with the Zeeman energy, a quantum hall magnet can be stabilized as the ground state. Ezawa and Hasebe (2002); Nomura and MacDonald (2006); Kharitonov (2012a) Its magnetic order can be uncovered by relying upon mean-field theory for which the order parameter is . Kharitonov (2012a, b) This order parameter can be written as follows:
where and are the trial spin-valley spinors. For the semi-infinite system, we shall neglect any deviation of the order parameter from its bulk value near the edge. Fertig and Brey (2006); Gusynin et al. (2009); Kharitonov (2012a) We shall also neglect the dynamics of the order parameter, which can lead to additional dissipation mechanisms in clean systems (via coupling with the bulk Goldstone modes), different from those discussed here. Murthy et al. (2014) In spin space, the spinor accepts the following parametrization:
where is the angle between the spin polarization and the total magnetic field ; is the azimuthal angle around the total magnetic field. The positive sign applies to and the negative sign to . Kharitonov (2012b, a) The above parametrization assumes that the spin-quantization axis points along the direction of the total magnetic field . Note that this choice is different from the one assumed for the operator , which measures the spin-projection perpendicular to the plane of graphene and , which measure the in-plane spin projection. Both sets of spin operators and () are related by a rotation:
By minimizing the kinetic energy and Zeeman term together with the Coulomb interaction within the mean-field approach, the single-particle spectrum shown in Fig. 3 is obtained. Kharitonov (2012a) The Zeeman term in the Hamiltonian lifts the degeneracy between the spin up and spin down bands. However, due to the exchange part of the Coulomb interaction, the 0LL bands near the edge anti-cross (for ) and develop an energy gap. The magnitude of the gap is given by, Kharitonov (2012a)
where is angle subtended by with the -axis perpendicular to the graphene plane, is the Zeeman energy, and ; is the critical angle for which a quantum phase transition from the canted anti-ferromagnet (CAF) and ferromagnet (FM) takes place when the Zeeman energy takes over the exchange energy.
In the CAF state, when the chemical potential lies above (below) the gap in the upper (lower) band, two metallic channels appear at the edge (corresponding to the two Fermi points in Fig. 3 where the energy band intersects the chemical potential). The corresponding states are not eigenstates of and are not mutually orthogonal. This has a significant effect on the suppression of conductance from its perfect quantized value , as we will see in Sec. III.
As the angle is tuned across its critical value, , and the system becomes a FM, the two conducting channels along the edge still exists. Unlike in the CAF, they are eigenstates of (cf. Eq. 14) and therefore mutually orthogonal. Nevertheless, we note that this is the result of the approximation made in Ref. Kharitonov, 2012a, which neglects the variation of the order parameter near the edge. The more sophisticated treatment provided in Ref. Murthy et al., 2014 indicates that spin orientation of the edge states remains slight canted in the FM, only becoming perfectly anti-aligned at large values of . However, to the extent they can be assumed to be perfectly spin anti-aligned, the FM edge channels can transport spin and charge current without dissipation if there were no impurities. However, different from the QSHE in semiconducting quantum wells, the QSHE in the FM quantum hall insulator is not protected by time-reversal symmetry but the U symmetry generated by the total .
In both the CAF and FM states, at , the low-energy physics is controlled by the fermionic degrees of freedom near the Fermi energy . Upon linearlizing the band around the Fermi points defined by the intersects ( and ) of the bands with (see Fig. 3), an effective low-energy Hamiltonian for the edge modes can be obtained:
where and for and and
The creation and annihilation operator obey the usual fermion anti-commutation relation where . Note that, unlike the QSHE in semiconductor quantum wells for which the states at the two channels are a Kramer’s pair, Strom et al. (2010) in the present case the right and left movers can have different Fermi velocities (i.e. ).
In addition, since the degrees of freedom near the edge are gapless, we need to take into account the effect of electron-electron interactions. In experimental conditions, due to the presence of a nearby metallic gate used to tune the Fermi energy , the Coulomb interaction between electrons is screened to yield a short-range interaction:
Note that is not necessarily repulsive as the Coulomb interaction may be over screened at low energies and become an effective attractive interaction. As a consequence, below we have considered both signs for . Thus, the complete low-energy Hamiltonian for a clean armchair edge reads:
Iii Scattering Potentials
iii.1 General form of a localized scatterer potential
We next analyse how the presence of a scatterer affects conduction along the edge. To this end, it is instructive to consider the most general form of the potential created by a generic scatterer located near the edge of the sample. Generally, the latter can be written as follows:
where is a potential matrix acting on both sublattice, valley, and spin degrees of freedom. Since a tilted magnetic field break most symmetries of the graphene Hamiltonian, the most general form of is a linear combination of the matrices , where ( referring to the identity matrix):
where the relative strength of each potential depends on the microscopic details of the scatterer, the strength of the external magnetic field and , as well as other perturbations (boundary conditions at the edge). Therefore, will contain terms that break the symmetries of the zero-field Hamiltonian. Those terms are generated by the projection onto the low-energy subspace containing the zero energy Landau levels. In what follows, we shall discuss which terms are expected to yield a dominant contribution to . Rather than trying to be exhaustive, our choice will be physically motivated. However, we shall first take a detour to discuss some general properties of the 0LL projection.
iii.2 Projection of sublattice-valley Operators onto the 0LL
In order to obtain a low-energy effective description of the edge channels in the presence of the scatterer, its potential must be projected onto the 0LL. To leading order, the scatterer potential in the 0LL is given by
where is the projection operator in the 0LL. The terms contained in the ellipsis represent corrections arising from virtual transitions to Landau levels with . Since the scatterer potential introduced above acts upon the sub-lattice, valley and spin degrees of freedom, it is worth dwelling a bit on the explicit form of the 0LL projection operator and how it affects the Pauli matrices .
The operator can can be written as follows:
The first term in the right hand-side of Eq. (25) () acts on the sub-lattice and valley pseudo-spin degrees of freedom, whereas the second term () acts on the other orbital degrees of freedom. Since sub-lattice spin and valley spin are aligned in 0LL, when projecting the Pauli matrices using , we obtain the following:
where we have introduced the operators:
Thus, behave effectively as Pauli matrices in the subspace of 0LL states.
iii.3 Time-Reversal Invariant Scattering Potentials
Although the most general impurity potential should not respect time-reversal symmetry as the latter is broken by the external magnetic field, it is worth beginning our analysis by considering various time-reversal symmetric (TRS) potentials. The reason is that TRS potentials are present in the microscopic Hamiltonian of graphene with impurities at zero field. However, non-TRS potentials need to be generated by means of virtual transitions to higher energy states in the presence of the time-reversal symmetry breaking perturbations like the magnetic field. Therefore, according to the discussion in previous section, TRS potentials are expected to have larger strength relative to the non-TRS ones. In addition, the discussion that follows is also motivated by the suggestion made in Ref. Young et al., 2014 attributing the absence of conductance quantization to the existence of scatterers proximity-induced Rashba spin-orbit coupling (SOC). Weeks et al. (2011); Ferreira et al. (2014)
TRS potentials can be divided into two classes depending on how they act on the spin degree of freedom: Those proportional to , which are termed ‘scalar’ potentials, and those proportional to , which correspond to SOC. These two classes can be further subdivided into those inducing only intra-valley scattering and those inducing intervalley scattering depending on whether they commute with or not. Note that, typically, the strength of the scalar potentials is much larger than the strength of SOC potentials Weeks et al. (2011); Ferreira et al. (2014); Balakrishnan et al. (2014).
An example for a TRS scalar potential is:
where the second and third terms and others similar to them not explicitly written describe intervalley scattering. Strong inter-valley scattering requires that a rapid spatial variation of and on the scale of the inter-carbon distance in graphene. In other words, it requires atomic scale disorder (i.e. point defects, edge roughness, etcetera). It is possible to estimate, within the Born approximation, the reflection coefficient at the edge from such small-size scatterers, Guinea et al. (2010) which is . However, for the experimentally accessible values of , nm nm, implying that the intervalley scattering terms in the above equation can be neglected. For the same reason, we shall also neglect the inter-valley SOC scattering terms below.
Next, we consider the (intra-valley) SOC scattering potentials, which can take the following general form:
where the first term on the right hand-side corresponds to the intrinsic (or Kane-Mele) type of SOC, and the third and fourth terms correspond to the Rashba-like SOC. Note that we have considered a generalized anisotropic Rashba, for which . Such SOC potentials can be induced when the hexagonal point-group symmetry of graphene is broken down to a smaller symmetry group by proximity to substrates and clusters of heavy-metals. Calleja et al. (2015); Ochoa et al. (2012) The second term is absent in the microscopic Hamiltonian because it is forbidden by the parity symmetry that exchanges of the A and B sub-lattices. However, it can be generated by the application of parity-breaking fields like a tilted magnetic field or by the edge potential. Since this term is not present at the zero-field (for which parity symmetry is not broken), in a first approximation, we shall neglect it compared to the scalar and standard SOC terms. However, in the following section we shall see that this term can yield backscattering in the FM state.
|Scattering Operator||0LL Projection|
where () is the annihilation (creation) field operator in the 0LL, respectively. Note that, in the above expression, measures the projection of the electron spin along the axis perpendicular to the graphene layer and must be expressed in terms of and (cf. Eq. (14)).
It is noteworthy that Rashba SOC of the form given in Eq. (37) gives no contribution to the 0LL projected (cf. table 1). Therefore, it alone cannot suppress the conductance from its perfect quantized value, unlike the case of the QSHE in two-dimensional TRS topological insulators. (Strom et al., 2010; Ilan et al., 2012) This is a direct consequence of the structure of 0LL orbitals in graphene, for which the sub-lattice and valley pseudo-spins are locked. Landau level mixing arising from interactions, etc, can modify this conclusion slightly, but the corrections are expected to be small and suppressed at the high magnetic fields. On the other hand, the scalar and intrinsic SOC potential are not affected by the 0LL projection and therefore can lead to backscattering.
To make further progress towards obtaining the effective Hamiltonian of a TRS scatterer at the edge, let us assume the size of the scatterer (e.g. a metal cluster of radius ) to be comparable to the magnetic length, . Thus, we can approximate and by Dirac-delta functions, which, upon further projection on the low-energy modes near the edge, leads to
where has energy units. When combined with Eq. (21), the Hamiltonian, , is a version of the model studied by Kane and Fisher Kane and Fisher (1992) for an impurity in a Tomonaga-Luttinger liquid. The only difference is that, in our model, right and left movers have different Fermi velocities. Note that, after projecting on the edge, a term of the form has been dropped since it does not contribute to the channel resistance. Kane and Fisher (1992); Giamarchi (2003) The scattering potential coupling contains contributions from both scalar and the intrinsic SOC potentials, i.e. , where
is the single particle energy, which is displayed in Fig. 2 and is the normalization of the single particle wave function; is the polar angle of the applied magnetic field, and are the combination of the mean-field parameters that favour ferromagnetic and anti-ferromagnetic order, respectively. The relative strength of these potentials and has been plotted in Fig. 4. Note that . In addition, typically, Ferreira et al. (2014); Balakrishnan et al. (2014) , which means that the main contribution stems from the scalar part of the scatterer potential.
iii.4 Backscattering in the Ferromagnet
As shown in Fig. 4, the effective potential strength arising from the scalar potential and intrinsic SOC vanishes in the FM state. However, to understand the vanishing of better, let us first recall the valley and spin structure of the single-particle orbitals in the FM state:
Note that the in the FM the orbitals are eigenvectors of and therefore orthogonal. This means that no scalar (intra- or inter-valley) is effective in causing backscattering. Thus, only spin-dependent potentials can cause backscattering. However, the orbital part of the intrinsic SOC (), cannot couple two edge states even if the spin operator does (recall that they are eigenstates of , not ).
However, we need to stress that the above conclusions rely on the orbital structure of the single-particle orbitals in the FM described by Eqs. (44) and (43). The structure is the result of the assumption (made in Ref. Kharitonov, 2012a and which we follow here) that the magnitude of the order parameter does not deviate from its bulk value. This approximation is expected to be accurate Murthy et al. (2014) for large values of . However, a more sophisticated mean-field treatment like the one provided in Ref. Murthy et al., 2014, which allows for the spatial variation of the order parameter, would not automatically imply a vanishing impurity (back)scattering in the FM. The reason is that, near the edge, the order parameter can deviate from its bulk value and, as a consequence, the spins near the edge of the FM can be slightly canted. Murthy et al. (2014) This implies that the backscattering due to the scalar and intrinsic SOC potentials is suppressed relative to its values in the CAF but not they are completely eliminated. On the other hand, a Rashba SOC potential yields no backscattering to leading order. The latter result is independent of detailed form of the single-particle orbitals as it is entirely a consequence of the 0LL projection.
Nevertheless, it is also interesting to consider additional scattering potentials that have been neglected so far, which can also lead to backscattering. If we restrict ourselves to TRS potentials, as we did in the previous section, one possible candidate, which by no means exhausts all the possibilities, is the following parity-breaking intrinsic SOC:
Other possible terms (included in the ellipsis) are proportional to . Upon projection on the 0LL, Eq. (46) yields:
Note that the second term on the right generates backscattering in the FM whereas the first one, which commutes with , does not. In the CAF, both terms lead to backscattering because the spins are canted. However, as pointed out above, this term is not a standard type of SOC as it requires sublattice exchange (parity) symmetry to be broken. This may be caused by the substrate (BN), the tilted magnetic field, or the boundary conditions at the edge. However, estimating the strength of this term is beyond the scope of this work as this will require a detailed microscopic model of the system.
iii.5 Non-time reversal symmetric scatterers
Other possible potentials causing backscattering can break TRS. In particular, a potential of the form:
which describes the exchange interaction with a locally magnetized scatterer (like a magnetized edge Lado and Fernández-Rossier (2014)), gives a non-zero contribution to the backscattering potential the spins of the edge electrons are canted and therefore do not need to be completely flipped in order to backscatter.
However, at large values of , for which canting is negligible, will not produce backscattering in the FM state. The reason is that backscattering requires the potential to act both on the sublattice-valley and spin degree of freedom. In other words, the potential must flip between (approximate) eigenstates of and , meaning that the orbital part must be proportional and . From the results in subsection III.2, we see that arises from potentials proportional to or . Note that is odd under TRS, and since flipping the spin requires the potential be proportional to , a term of the form is TRS. The latter results in the parity-symmetry breaking SOC discussed above. On the other hand, is TR even, and including the spin-part, leads to non-TRS potential of the form:
Since , this potential does not produce intervalley scattering. In addition, it also breaks sublattice exchange parity symmetry like from Eq. (46).
Finally, potentials containing can arise from terms of the from with , which describe inter-valley scattering and are therefore are suppressed by a factor as discussed above.
Iv Corrections to the Edge Conductance
In this section, we calculate the linear conductance of the edge channel in the presence of the scatterer. We focus on the doped CAF, for which the leading sources of backscattering have been identified in subsection III.3 and their strength can be estimated from their values in the zero-field Hamiltonian. The strength of the terms induced by TRS- and/or sublattice parity symmetry-breaking terms is much more difficult to estimate because they are generated upon integrating high-energy states and will also depend on the microscopic details of the scatterer. In what follows, we shall use perturbation theory to lowest order in the scattering potential strength, , assuming that the latter is weak, i.e. .
As explained in Appendix B the correction to the channel conductance within linear response theory is given by:
where is the backscattered current,
being the total number of right moving electrons. Within linear response theory, the (steady-state) backscattered current is Mahan (2000)
where the retarded correlation function is and .
We first consider the non-interacting case for which . The previous expression can be evaluated to yield:
The non-interacting lesser and greater Green’s functions are and , where is the Fermi-Dirac distribution and . Hence,
For , we recover the linear conductance of non-interacting 1D channel with an impurity. Giamarchi (2003)
However, note that the conductance of the non-interacting edge is independent of the temperature, in contradiction with the the experimental observations. Young et al. (2014) This problem can be solved by accounting for interaction effects (i.e. ), which yield a temperature dependence of , in qualitative agreement with the experimental observation. Young et al. (2014) Accounting for a non zero is also necessary because interactions in 1D have a dramatic effect on the stability of fermionic quasi-particles . (Giamarchi, 2003; Cazalilla et al., 2011; Gogolin et al., 2004) Thus, we calculate the correction to the conductance using the bosonization method, which allows us to treat interactions non-perturbatively (see Appendices). The result of this calculation reads:
where is the Luttinger parameter, which characterizes the strength of the interactions: for and () for repulsive (attractive) interactions. The parameters are the velocity of the eigenmodes (see Appendix A for the definitions of and ); is the beta function, is the digamma function. Note that the temperature dependence is a power-law, with an interaction dependent exponent, Kane and Fisher (1992) which vanishes in the non interacting case (i.e. for ), in agreement with Eq. (54).
In Fig. 5, we have plotted the two-terminal conductance per edge against the temperature (in cyclotron frequency units) for different values of Luttinger parameter . For (i.e. attractive interactions), the edge conductance approaches the quantized value as . On the other hand, for (repulsive interactions), the deviation from perfect quantization increases with decreasing . For linear response theory breaks down and we must rely on other methods, but this problem will not be analysed here.
In Fig. 6, the two-terminal conductance per edge is plotted against the deviation of the polar angle of the applied magnetic from the critical polar angle at constant temperature, for different Luttinger parameter .
Finally, let us briefly discuss the angular and temperature dependence of some of the parity-breaking SOC term discused in Sec. III.4. This term is expected to yield a sizeable contribution at large values of in the FM state, when the canting Murthy et al. (2014) of the edge channel spins is small. Upon projection onto the 0LL, Eq. (46) yields a term , where is the magnetic field polar angle. Hence, we obtain the following contribution of the conductance:
where is the absolute temperature and the Luttinger parameter. However, since this term is not present in the zero-field Hamiltonian, we cannot estimate its strength, although we expected it to be small.
V Summary and Outlook
Using a simple single-scatterer model, we have discussed the effects of potential and spin-orbit disorder on conduction through helical edges recently observed in monolayer graphene under a strong, tilted magnetic field. Young et al. (2014) We have shown that the deviation from perfect quantization in the canted anti-ferromagnetic and ferromagnetic states can be understood in terms of potential (i.e. scalar) disorder and to a lesser extent spin-orbit disorder of the intrinsic type. Other types of disorder, such like sub-lattice exchange (parity) breaking spin-orbit coupling and time-reversal symmetry-breaking spin-orbit coupling terms are expected to yield smaller contributions. Interestingly, unlike the situation encountered in semiconducting quantum wells, Strom et al. (2010) we find that Rashba spin-orbit coupling does not lead to backscattering, at least to leading order.
We have also investigated the temperature and magnetic field tilt angle dependence of a single-edge conductance assuming that the main scattering sources are the scalar potential and the (intrinsic) spin-orbit coupling induced by the scatterer. In addition, the other scattering sources discussed in Sec. III may be generated via virtual transitions to different Landau levels, which makes estimating their magnitude rather difficult. However, we expect them to be sub-dominant at large magnetic fields and therefore we have neglected them in a first approximation.
We hope our work will shed light on the origin of the channel dissipation observed in the experiment. In particular, we have found that disorder-induced backscattering in combination with electron-electron interactions leads to a power-law temperature dependence of the conductance which may be observed experimentally. In this regard, we note that, even in the ferromagnet, for which the backscattering effects are milder, the presence of impurities, combined with repulsive interactions can lead to an insulating-like behaviour of the edge conductance.
The model studied here can be fairly easily generalized to disorder potentials with more complicated spatial dependence. Such a study will not be pursued here. However, we can anticipate Giamarchi (2003) it will yield to a different exponent in the temperature dependence of the conductance.A thorough understanding of the origin of dissipation in the helical edges of graphene under a strong, magnetic field will require a more detailed microscopic model of the edge of the device. Such model should account of the existence of multiple channels Gusynin et al. (2008) and also the coupling with the gapless bulk (Goldstone) modes. Murthy et al. (2014)
Acknowledgements.C.H thanks V.M. Pereira and M. Milletari for discussions and the Graphene Research Center of the National University Singapore for support during the early stages of this work. We also gratefully acknowledge discussions with P. Jarillo-Herrero. C.H’s work was supported in part by the Singapore National Research Foundation grant No. NRFF2012-02. MAC’s work is supported by the Ministry of Science and Technology (MoST) of Taiwan.
Appendix A Bosonization
In this and the following Appendices, we collect some of the most important results of bosonization technique Giamarchi (2003) applied to a one-dimensional (1D) channel with two different carrier velocities, which is relevant to the situation studied in the main text.
The Hamiltonian of the 1D edge can be expressed in terms of the Hamiltonian density, , as , and can be fully expressed in terms of the fermion chiral density operators,
where and stands for normal order. The chiral densities obey the Kac-Moody algebra: Giamarchi (2003)
where . Thus, the Hamiltonian density can be written as: Giamarchi (2003)
where () is the Fermi velocity of the right (left) channel and is an effective short-range electron-electron interaction. It is also useful to introduce the bosonic fields , which are defined by
The Hamiltonian density can be brought to diagonal form:
by means of the following linear transformation:
Note that the transformation in Eq. (62) has the following properties:
The new chiral fields obey a modified Kac-Moody algebra:
where and . Using these commutation relations, the chiral densities are found to obey the following equations of motion:
In the stationary state (i.e. for ), when coupled to two different (chiral) chemical potentials, the chiral densities must minimize , which is possible (upon completing the square) if we choose:
Hence the stationary current is given by . Note that the chemical potentials, coupling to the densities are physically different from the chemical potentials, , coupling to the electrons. The latter couple to and are equal to the chemical potentials of the source and drain lead reservoirs, , prespectively. However, it is possible to relate both sets of chemical potentials linearly, Pham et al. (2003) so that the voltage bias . Hence, the quantized conductance for a clean 1D edge channel can be recovered: Maslov and Stone (1995); Kawabata (2007); Pham et al. (2003)
which is the value of the quantized conductance for a single channel in units. The quantized conductance of a perfect non-chiral Tomonaga-Luttinger liquid is the result of the finite contact resistance.
Appendix B Accounting for the Contact Resistance
In this Appendix, we shall follow the approach of Pham and coworkers Pham et al. (2003) to account for the contact resistence of the 1D interacting edge channel in the presence of a scatterer located at . As it has been discussed in the main text, the Hamiltonian for such system is , where the Hamiltonian density is given by:
We are interested in computing the two terminal conductance, which is defined from the current passing through the device, , as follows:
where is the voltage bias between the source and drain leads. The current is given by the following equations (essentially Ohm’s law): Pham et al. (2003)