Effect of Berry Phase on Nonlinear Response of Two-dimensional Fermions

We develop a theory of nonlinear response to an electric field of two-dimensional (2D) fermions with topologically non-trivial wave functions characterized by the Berry phase $\Phi_n = n \pi, n = 1,2,...$. In particular, we find that owing to suppression of backscattering at odd $n$, Hall field-induced resistance oscillations, which stem from elastic electron transitions between Hall field-tilted Landau levels, are qualitatively distinct from those at even $n$: their amplitude decays with the electric field and their extrema are phase-shifted by a quarter cycle. The theory unifies the cases of graphene ($n = 1$) and graphite bilayer ($n = 2$) with the case of conventional 2D electron gas ($n = 0$) and suggests a new method to probe backscattering in topological 2D systems.

We develop a theory of nonlinear response to an electric field of two-dimensional (2D) fermions with topologically non-trivial wave functions characterized by the Berry phase Φn = nπ, n = 1, 2, ... . In particular, we find that owing to suppression of backscattering at odd n, Hall field-induced resistance oscillations, which stem from elastic electron transitions between Hall field-tilted Landau levels, are qualitatively distinct from those at even n: their amplitude decays with the electric field and their extrema are phase-shifted by a quarter cycle. The theory unifies the cases of graphene (n = 1) and graphite bilayer (n = 2) with the case of conventional 2D electron gas (n = 0) and suggests a new method to probe backscattering in topological 2D systems.
The topological property of two-dimensional (2D) massless Dirac fermions, expressed in terms of the Berry phase Φ 1 = π, is responsible for peculiar Landau quantization manifesting itself in the phase-shifted Shubnikov-de Haas oscillations (SdHO) and unconventional quantum Hall effect [1][2][3][4]. This makes the Dirac fermions in graphene fundamentally distinct from both the conventional 2D electron gas (2DEG) in quantum wells (Φ 0 = 0) and topologically non-trivial fermions in graphite bilayers (Φ 2 = 2π) [5,6]. Another immediate consequence of the topological nature of wave functions of massless Dirac fermions is the absence of elastic backscattering off scalar potentials, which has various manifestations in electronic properties. In particular, it lies at the origin of Klein tunneling [7] and its implications for resistivity of graphene-based n-p-n junctions [8][9][10]. Furthermore, it removes a sharp cusp of static polarizability of degenerate carriers at doubled Fermi wavenumber (characteristic for conventional 2DEG and graphite bilayer) [11], leading to enhanced spatial decay of Friedel oscillations [12].
Another well known effect which crucially depends on backscattering is Hall field-induced resistance oscillations (HIRO) [13][14][15] which emerge in differential resistivity r of a 2DEG subjected to elevated current density j and perpendicular magnetic field B. These oscillations appear due to the property of enhanced phase space for elastic transitions in 2DEG in the vicinity of backscattering. In the presence of classically strong B, the backscattering means a spatial shift of the cyclotron orbit guiding center twice the cyclotron radius R c , so the transition rate increases each instant when 2R c equals integer multiple of spatial separation between Landau levels (LLs) tilted by the electric field E. Therefore, r oscillates with the dimensionless parameter ǫ = 2R c |e|E/ ω c , where E ≃ E H , E H is the Hall electric field, and ω c is the cyclotron frequency. As a result, HIRO are periodic in 1/B with the frequency (ǫ ≡ B 1 /B) where v F is the group velocity at the Fermi level and g is the degree of band degeneracy.
In this paper we demonstrate that the nonlinear response of topologically non-trivial (n ≥ 1) 2D systems crucially depends on the parity of n. For even n, backscattering is not suppressed and the behavior is qualitatively the same as in conventional 2DEGs. However, suppression of backscattering in massless Dirac systems, as well as in any other system with odd n, leads to a decay of the HIRO amplitude (as 1/ǫ) and produces a characteristic quarter-cycle phase shift of the oscillations towards larger ǫ. In particular, for Dirac fermions (n = 1) we obtain [see the definition of τ π for this case after Eq. (32)] which describes HIRO with the frequency given by Eq.
(1) and the maxima near ǫ = m + 1/4. We also find that the HIRO amplitude increases with n for odd n. Our conclusion that the absence of backscattering leads to a phase shift of nonlinear magnetoresistance oscillations can become a basis for a new method to probe backscattering in topological 2D systems. Our theory of nonlinear magnetotransport is developed for the regime of large occupation factors (high LLs), classically strong magnetic fields, and overlapping LLs, which is relevant for observation of HIRO [18]. We consider spin-degenerate 2D systems described by the Hamiltonian where ϕ is the angle of the wave vector k, the energy spectrum ±ε(k) is isotropic, and the winding number n gives the Berry phase Φ n = nπ. The HamiltonianĤ describes fermions in graphene (K valley) at n = 1 [ε(k) = v F k with constant Fermi velocity v F ≃ 10 8 cm/s] and in graphite bilayer at n = 2 [ε(k) = ( k) 2 /2m ⋆ with constant effective mass m ⋆ ≃ 0.037 of free electron mass]. For these particular systems, n can be viewed as the degree of chirality in the carbon sublattice space [6]. ThoughĤ formally produces a two-band spectrum, we consider only intraband excitations and the topologically trivial case (n = 0) can be equally applied to 2DEG in quantum wells.
To derive expressions for the resistivity, we use Eq. (4) and consider elastic scattering of fermions by impurities. Adopting the methods developed for 2DEG with parabolic spectrum [18,19], in particular, using the reference frame moving with the drift velocity [15,19], we can write the steady-state Boltzmann equation for the distribution function f ε,ϕ of a 2DEG placed in perpendicular magnetic field B = (0, 0, B) and inplane electric field E = (E x , E y , 0) in the following form: The right-hand side is the collision integral describing elastic scattering of electrons by impurities within a single valley.
is the density of states per spin and valley, and w kk ′ is the Fourier transform of the correlation function of the impurity potential. It is assumed below that γ ≪ ε so that |k ′ | ≃ |k| ≡ k ε . The cyclotron frequency ω c = |e|B/m ⋆ c is determined by the effective (cyclotron) mass m ⋆ = k ε /v F , which, in general, is energy-dependent (for graphene, k ε = ε/ v F ). The function is the squared overlap integral of columnar eigenstates ofĤ with wave vectors k and k ′ , and θ = ϕ − ϕ ′ is the scattering angle. It is seen directly that backscattering (θ = π) is suppressed for odd n but not for even n.
For overlapping LLs, the oscillatory density of states at ε ≫ ω c follows from the Bohr-Sommerfeld quantization rule corrected by the Berry phase [20]: where ν ε is the density of states at B = 0 (for graphene, ν ε = ε/2π 2 v 2 F ), l B = c/|e|B is the magnetic length, and λ ε is the Dingle factor at energy ε. The quantity k 2 ε l 2 B /2 is the number of magnetic flux quanta inside the cyclotron orbit of electron with energy ε. Equation (7) can also be derived from the self-consistent Born approximation (for graphene, see, e.g., Ref. 21). The density of states has maxima at the Landau quantization energies, 2N (N = 0, 1, 2, ... ) for graphene and ε N = ω c (N − 1/2) for graphite bilayer. In the latter case, ε N approximates the exact spectrum ε N = ω c N (N − 1) [6] at N ≫ 1. The length of q and its angle ϕ q can be expressed as and the work γ can be conveniently rewritten as where χ is the angle of the electric field E. The assumed strong inequality γ ≪ ε always holds at v D ≪ v F . Equation (5) is easily solved in the regime of classically strong magnetic fields (v F τ ≫ k ε l 2 B ), when one can replace the distribution functions under the integral by an isotropic distribution f ε . Substituting such a solution into the expression for the current density, where n s is the carrier density, and taking into account that the impurity potential correlator depends only on the absolute value of q, w kk ′ = w(q), we obtain the dissipative conductivity (j (d) = σ d E): The isotropic part of the distribution function standing in Eq. (11) can be represented as a sum of quasiequilibrium Fermi distribution f where T e is the temperature of carriers and k B is the Boltzmann constant, and a small non-equilibrium contribution δf ε caused by the field-induced redistribution of electrons in the energy domain in the presence of Landau quantization [15,22]. The function δf ε shows rapid oscillations similar to those in ν ε , Eq. (7), and can be found from Eq. (5) averaged over ϕ. To describe relaxation of the isotropic distribution, the inelastic relaxation term −δf ε /τ in (τ in is the inelastic relaxation time), approximating the linearized collision integral for electronelectron scattering, should be added to the right-hand side of Eq. (5). To the first order in λ ε , where k F = 4πn s /g is the Fermi wavenumber. The quantum and the transport scattering rates are given by respectively, with Next, where J α denotes a Bessel function of the first kind, and Assuming degenerate carriers, we have set ε = ε F , k ε = k F in all quantities whose energy dependence is weak.
Substituting f ε = f (0) ε + δf ε into Eq. (11) and calculating the integrals over ε and φ ′ analytically, we represent the result as an expansion in powers of the Dingle factor, is the term describing SdHO, where The influence of the electric field on SdHO is described by as δf ε does not contribute to σ (1) d [23]. SdHO are strongly suppressed by temperature at X T ≫ 1, owing to rapid oscillations of ν ε . In the last term, σ (2) d , we retain only the contributions that survive at X T ≫ 1. We find where the parts and respectively, come from substitution of f (0) ε and δf ε into Eq. (11) and are often referred to as the displacement and the inelastic contributions.
The oscillations of H 1 (ζ) describe HIRO for massless Dirac fermions, Eq. (3), which differ from HIRO for conventional 2DEG, Eq. (2), by a phase shift and by a decay 1/ǫ. These changes are the consequences of modification of the angular dependence of scattering probability and of its considerable reduction in the vicinity of backscattering (θ ≃ π), owing to the factor F (1) θ . A similar phase shift and a stronger decay, for the same reasons, are also present in S 1 (ζ) describing field effect on SdHO. It is unlikely, however, if S 1 (ζ) can be accessed experimentally because heating of electrons by current, leading to a dependence of T e on j, becomes important already at ζ ≪ 1 and the nonlinearity associated with the first term in braces of Eq. (30) prevails [24].
In contrast to HIRO, the weak-field (ζ ≪ 1) nonlinear response of Dirac fermions, dominated (in the absence of SdHO) by G 1 (ζ), is very similar to the one for conventional 2DEG. This response is rather insensitive to smoothness of disorder, but sensitive to the ratio τ /τ in . At ǫ > 1, the function G 1 (ζ) rapidly decreases as ǫ −3 , much faster than ǫ −1 in conventional 2DEG.
It is worth noting that there exists another phenomenon which crucially depends on backscattering, the magnetophonon oscillations of linear resistance due to interaction of 2DEG with acoustic phonons (also known as the phononinduced resistance oscillations, PIRO) [25]. Recent observations of this phenomenon in graphene [26,27] show that PIRO do not shift their phase and behave just like those in conventional 2DEG with parabolic band. This happens because the electron-phonon interaction in graphene is dominated by the gauge-field mechanism for which the interaction potential is not a scalar in the sublattice space and, as a result, backscattering is not suppressed. The microwave-induced resistance oscillations [28], which have not yet been observed in graphene, are not expected to change their phase either, as they are not sensitive to backscattering. Therefore, among the magneto-oscillatory phenomena specific for 2DEG in the regime of high LLs [18] only HIRO is expected to show profound changes in graphene, which makes them a special and promising tool for experimental probing of backscattering.
In summary, we have developed a theory of nonlinear magnetoresistance for 2D fermions with Berry phase Φ n = nπ, based on a model unifying the conventional 2DEG, massless Dirac fermions in graphene, and fermions in graphite bilayer. We have shown that the amplitude and the phase of nonlinear magnetoresistance oscillations of degenerate 2D fermion gas crucially depends on the parity of n. Such a distinction is a consequence of suppression of backscattering off impurity potential in the case of odd n. We believe that our results will simulate nonlinear magnetotransport experiments in graphene and in other topological materials.
We thank I. Dmitriev and M. Khodas for discussions. The work at Minnesota was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award # ER 46640-SC0002567.