Bound states for massive Dirac fermions in graphene in a magnetic step field

We calculate the spectrum of massive Dirac fermions in graphene in the presence of an inhomogeneous magnetic field modeled by a step function. We find an analytical universal relation between the bandwidths and the propagating velocities of the modes at the border of the magnetic region, showing how by tuning the mass term one can control the speed of these traveling edge states.


INTRODUCTION
Graphene is generally described by massless Dirac fermions 1 , nevertheless different techniques have been developed for nanotechnological applications and for exploring non-trivial topological properties, in order to generate a gap at the Dirac points 2-4 so to include a mass term in the Dirac-Weyl Hamiltonian, which describes the low energy physics in graphene 1,[5][6][7] . Mass terms, confining scalar potentials or magnetic fields can spoil the simple linear dispersion of the original massless Dirac fermions. [8][9][10][11][12][13][14][15] In particular, applying inhomogeneous magnetic fields perpendicularly to the graphene sheet one can produce bound states trapped in the vicinity of the discontinuity of the magnetic field and propagating along the magnetic edges. [16][17][18][19] Several bound state spectra have been already obtained and scattering problems solved for massless Dirac fermions in graphene embedded in discontinuous magnetic fields employing boundary conditions. [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] What we are going to present in this work, instead, is the bound state spectrum for massive Dirac fermions in the simplest inhomogeneous magnetic pattern described by a step function. In the massless limit we recover the known results. 16,18 We observe some universal behaviors, in particular we show analytically that, in this magnetic structure, for each band, the bound state threshold does not depend on the mass term even if the energy levels do depend on it. Moreover we show that the maximum of the velocity of the edge modes propagating along the magnetic boundary, as a function of the mass term, seems to be proportional to the bandwidth, therefore the ratio between the bandwidth and the corresponding maximum velocity is mass-independent quantity. In summary, by this analysis, we show how, by tuning the mass term, one can control the propagating speed of the modes located at the edge of a magnetic region, relevant for future tunable graphene-based mesoscopic devices.

MAGNETIC STEP
Let us consider a magnetic field perpendicular to the plane of graphene, z-direction, and with a step profile along one direction in the plane of graphene, B z (x) = B θ(x), where θ(x) is the Heaviside theta function. The potential vector is, therefore, A = (0, A(x), 0), defined by where = h/2π with h the Planck constant, c the speed of light, e the elementary charge, and l B = c/eB the magnetic length. We are supposing that the length scale over which B(x) significantly varies, say λ B , is assumed much larger than the lattice spacing so that, at low energy scales, the two Dirac points in the massless limit are not coupled by the magnetic field and can be treated separately. We assume also that λ B is much smaller than the quasiparticle Fermi wavelength so that we can safely arXiv:2103.14345v1 [cond-mat.mes-hall] 26 Mar 2021 approximate B(x) as a step function. The Dirac-Weyl equation is, then given by υ F (σ x π x + σ y π y ) + ∆σ z ψ(x, y) = εψ(x, y) (2.2) with υ F the Fermi velocity, σ x , σ y , σ z Pauli matrices and π the momentum operator π = −i ∇ + e c A. Because of the translational invariance along the y-direction, the spinor can be written as Let us split the space in two regions, I and II.
Region I. For x > 0, from Eq. (2.2) we can write the following equations for the two components From these equations, putting one into the other, we obtain Using Eq. (2.1), we can write the following equation, for x > 0, Notice that this is an effective one-dimensional Schrödinger equation where for k y < 0 the potential develops a minimum within the magnetic region for which there exist bound state solutions with energy |k y | > (ε 2 − ∆ 2 )/ υ F , as we will verify in what follows. Making the change of variables Eq. (2.7) becomes simply where we defined the quantity, which is generally a real number, The normalizable solution of Eq. (2.9) is where D η (z) is a parabolic cylinder function and c I a constant value. Notice that if η = n is a nonnegative integer number, one can write D n ( √ 2ξ) = 2 −n/2 e −ξ 2 /2 H n (ξ), namely in terms of the Hermite polynomials H n (ξ) = (−1) n e ξ 2 d n dξ n e −ξ 2 . To find the second component of the spinor we can write and using the recursive relation we get the complete spinorial wavefunction (2.14) Region II. For x < 0, we obtain the following equation by placing so that, analogously to what done in the other case, we can write whose solution is can be written as with r a constant value and where we defined The bound state solutions are those with k 2 y > ε 2 − ∆ 2 / 2 υ 2 F so that, for x < 0 the only normalizable contribution in Eq. . (2.20) Imposing the matching condition at x = 0, from Eqs. (2.14) and (2.20) we find where we used Eq. (2.10) and where k x is defined in Eq. (2.19). If we put ∆ = 0 in Eq. (2.21), the matching condition reduces to that of the gapless graphene 16 . In this case in addition to the finite-energy states, solution of the above equation, there is also the zero energy state,˜ 0 = 0, for k y < 0, whose wavefunction is In order to find the finite-energy spectrum defined by Eq. (2.21), in the general case of gapped graphene, it is convenient to introduce the dimensionless parameters whose solutions are quantized,˜ n , with n = 1, 2, 3, .... Notice that Eq. (2.26) is valid for˜ = −∆. Also in this case there is an extra-state with a completely flat band at˜ 0 = −∆ whose wavefunctions is described by Eq. (2.22), localized at the edge, where the discontinuity of the magnetic field is located, but whose band is not dispersive.

RESULTS
The dispersive energy levels are obtained by solving the matching condition Eq. (2.26). We verified that the bound states exists for k y < 0 and for The negative-energy spectrum is specular with, in addition, the flat zeroth energy level˜ 0 = −∆.
located at the threshold,k y = p n ≡ − ˜ 2 n −∆ 2 , solution of the equation therefore p n does not depend on the mass term, as shown in Fig. 2 (first two plots). For instance, numerically, we get p 1 ≈ −1.31325, p 2 ≈ −1.92427, p 3 ≈ −2.38626 and so on. We have then min ˜ n (k y ) =˜ n (p n ) = ∆2 + p while for modes with |k y | > |p n |, |ṽ n (k y )| are smaller, see Fig. 2 (last plot) for n = 1, 2. In particular, for∆ = 0 we have |ṽ n (p n )| = 1, while increasing the mass term the modulus of the velocity decreases. Surprisingly we find that the ratios between the bandwidths and the maximum velocities, although both functions of the mass term, are universal quantities, namely, in their turn, the ratios do not depend on the mass, but are equal to the bandwidths in the massless case, at∆ = 0. Actually we observe numerically, at least for the first three levels reported in Fig. 3, that We checked that the curves in the second and third plots of Fig. 3  with p n solution of Eq. (3.29). We finally notice that all curves representingṽ n (k y ) in log-scale reported in the last plot of Fig. 3 collapse into a single curve after a rescaling,ṽ n (k y )[∆] =ṽ n (k y )[0]|ṽ n (p n )|[∆]. Calling, for each band,ṽ o n (k y ) ≡ṽ n (k y )[∆ = 0] the velocity in the massless case, we have the following simple scaling law for the velocities in the massive ones v n (k y ) =ṽ o n (k y ) √ 2n − |p n | ∆2 + 2n − ∆2 + p 2 n . (3.36)

CONCLUSION
In this paper we derived the bound state spectrum for massive Dirac fermions in graphene subjected to a perpendicular magnetic field with a step function profile. We showed that the energy levels approaches the relativistic Landau levels while the dispersive parts of the bands exhibit some universal behaviors. We find that the mass term modifies the bulk spectrum while reducing the number and the speed of the traveling modes at the border of the magnetic region, however the threshold of each bound states does not depends on the mass term and the ratio between the maximum propagating velocities and the bandwidths is also a mass-independent quantity. In conclusion, we show how, by tuning the mass term, one can control the speed of the edge modes traveling along the boundary of the magnetic region, paving the way for novel tunable graphene-based mesoscopic devices.