Tuning the Fermi velocity in Dirac materials with an electric field

Dirac materials are characterized by energy-momentum relations that resemble those of relativistic massless particles. Commonly denominated Dirac cones, these dispersion relations are considered to be their essential feature. These materials comprise quite diverse examples, such as graphene and topological insulators. Band-engineering techniques should aim to a full control of the parameter that characterizes the Dirac cones: the Fermi velocity. We propose a general mechanism that enables the fine-tuning of the Fermi velocity in Dirac materials in a readily accessible way for experiments. By embedding the sample in a uniform electric field, the Fermi velocity is substantially modified. We first prove this result analytically, for the surface states of a topological insulator/semiconductor interface, and postulate its universality in other Dirac materials. Then we check its correctness in carbon-based Dirac materials, namely graphene nanoribbons and nanotubes, thus showing the validity of our hypothesis in different Dirac systems by means of continuum, tight-binding and ab-initio calculations.

Here we solve the problem without making use of Feynman-Gell-Mann ansatz and for arbitrary values of the electric field. We begin by considering the envelope-function for a symmetric-gap junction in a uniform electric field along the z-direction, i.e., the growth direction. It satisfies the following Dirac-like equation with the same notation as in the main text. Using the reduced variables introduced therein, We solve the problem for Φ u and then Φ d can be obtained from where κ = |κ| and θ κ = tan −1 (k y /k x ).
The problem can now be solved by considering continuity at z = 0 and applying boundary conditions at ±∞. Although the former is easily applied, the latter are not so straightforward. In order to account for those conditions, we consider our system placed within a very large box in the z-direction, such that the outward component of the current density can be set to zero. Considering a box of size 2L and L d, the condition for a vanishing current where we define x ± = ( ± f d/L)/ √ f . By considering this prescription and continuity at z = 0 we obtain the desired implicit relationship between and κ for any value of the The matrices N ± are given by where with F ± = F (x ± ) and G ± = G(x ± ).
Equation ( at L d allows us to verify that the in-plane dispersion is still a Dirac cone that becomes wider upon increasing the electric field. Additionally, we can numerically confirm that the results given by this full approach match the low-field limit presented in the main text, as

RESONANCES AT LOW ELECTRIC FIELD
Approximate solutions to Eq. (7) of the main text can be obtained in closed form in the low-electric-field limit. The argument of the Airy functions is large if F F C and we make

ELECTRIC FIELD
Graphene allows for a low-energy description in terms of a massless Dirac equation around the high-symmetry points of the Brillouin zone, K and T , where A and B denote the two sublattices. In this basis, the massless Dirac-like Hamiltonian reads where τ z = ±1 acts on the valley degree of freedom K/K . Let us consider a symmetric armchair nanoribbon of width W along the x direction. Then, a uniform electric field across the ribbon, F = Fx can be modelled by adding an electrostatic potential of the form, whereW = W + a, being a = √ 3a CC with a CC = 1.42Å the carbon-carbon distance.
Due to the translational invariance in the y direction we can ask for solutions of the form Φ K (r) = exp(ik y y)Φ(x). This problem can be exactly solved using the following boundary with Γ = exp(−2πi4W /3a). It is not difficult to show that metalic nanoribbons take place whenever Γ = 1, that is, when 4W /3a = n with n ∈ N.
In order to solve the problem, it is more convenient to turn to non-dimensional variables. Let where F W = v F /eW 2 . Then Dirac equation is written as where ∂ ξ ≡ ∂/∂ξ.