Position-dependent Fermi velocity

Question

In models of strained graphene one seems to obtain a position-dependent Fermi velocity. By this I mean that if the original Dirac operator is

$$ H = v_0(\sigma_1 p_1 + \sigma_2 p_2),$$

with constant Fermi velocity $v_0$, then after the straining procedure one obtains, see this reference (20) for instance

$$ H = \sigma_i v_{ij}(r) p_j + \text{ potential terms}$$

I am curious if there is any model in physics that produces a position-dependent Fermi velocity that is the same for both $p_1,p_2$, i.e. I want

$$ H = v(r)(\sigma_1 p_1 + \sigma_2 p_2) + \text{ potential terms}.$$

My understanding is that although the latter is a special case of the strain in graphene, one cannot design any non-trivial examples of strain fields to get an operator of this type.

Answer 1

Strictly, band structure calculations are done for an infinite crystal (graphene sheet, carbon nanotube, etc.), in which case the energy states are labeled by band numbers and psedo-momenta: $\epsilon_n(\mathbf{k})$. In the effective mass approximation these are then expanded in powers of quasi-momenta, producing quadratic dispersion relations near the band minima/maxima or conical dispersion, if the bands touch each other - like in graphene.

If there is a potential or strain superimposed on the periodic lattice, then the above calculation technically breaks down, and we cannot speak of the bands. However one often assumes slowly varying (adiabatic) potential, where the local band structure is supposed to depend parametriclaly on the on the slowly varying potential. One sometimes speaks of envelope approximation, where the wave functions is factorized into of a fast part (the Bloch part) and a slowly varying (in space) coefficients: $$ \psi(\mathbf{x})=\sum_{n,\mathbf{k}}c_{n,\mathbf{k}}(\mathbf{x})e^{i\mathbf{k}\mathbf{x}}u_{n,\mathbf{k}}(\mathbf{x}) $$ The procedure is rarely spelled in practice - one simply takes the dispersion relation and converts it into Hamiltonian by adding the position-dependent potential and often treating quasi-momentum as if it were a real momentum: $$ \epsilon_n(\mathbf{k})\longrightarrow \epsilon_n(-i\nabla) + U(\mathbf{x}) $$ Some examples where it is used are:

• describing pn-junctions
• Zener breakdownn
• Peierls substitution
• Weyl-like Hamiltonian for graphene and carbon nanotubes

Getting to the point of the question formulated in the OP: we could start with a tight-binding approximation for graphene (i.e., hopping hamiltonian on a hexagonal lattice) and assume that the hopping elements vary in space $t_{ij}(\mathbf{x})$, which would readily produce the desired type of the Hamiltonian. The case with strain is actually more complex, since the hopping elements vary differently, depending on their orientation in respect to the strain direction. Physically however, straining is the easiest way of affecting the hopping matrix elements.

Related:
Is there a general math term for the idea behind the WKB and similar methods that assume slowly varying sources?
Peierls substitution vs minimal coupling