Fundamental solution of the stationary Dirac equation with a linear potential
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FUNDAMENTAL SOLUTION OF THE STATIONARY DIRAC EQUATION WITH A LINEAR POTENTIAL I. A. Bogaevsky∗
We explicitly express the fundamental solution of the stationary two-dimensional massless Dirac equation with a constant electric field in terms of Fourier transforms of parabolic cylinder functions. This solution describes the flux of quasiparticles in graphene emitted by a pointlike source of electrons that are partially converted into holes (antiparticles). Using our explicit formula, we calculate its semiclassical asymptotic behavior in the hole region.
Keywords: massless Dirac equation, semiclassical asymptotic behavior, fundamental solution, Green’s matrix, parabolic cylinder function, graphene, quasiparticle DOI: 10.1134/S0040577920120016
1. Introduction We study the fundamental solution of the stationary two-dimensional massless Dirac equation Hψ(x, y) = h3/2 δ(x + a, y)w,
= xσ0 + σx pˆx + σy pˆy , H
where (x, y) ∈ R2 is a point on the plane, ψ : R2 → C2 spinor, 1 0 0 , σx = σ0 = 0 1 1
a > 0,
(1)
is an unknown spinor field, w ∈ C2 is a known 1 0
,
σy =
0
−i
i
0
is are Pauli matrices, δ is the Dirac delta function, the coefficient x of the matrix σ0 in the Hamiltonian H the potential energy of a particle in a constant electric field, pˆx = −ih ∂x and pˆy = −ih ∂y are momentum operators, and h > 0 is a small real parameter. If the limit-absorption principle is satisfied, then the solution of Eq. (1) is unique and describes the stationary flux of quasiparticles in graphene emitted with zero energy from a source located at the point x = −a < 0, y = 0. Graphene has a constant electric field parallel to the x axis, and the potential energy of a quasiparticle in this field is U (x, y) = x. Quasiparticles in graphene (electrons for U < 0 and holes for U > 0) are fermions with a zero effective mass and the Fermi velocity vF ≈ 106 mps (effective speed of light in graphene), but we assume that vF = 1 in the used system of units. The source described by Eq. (1) emits ∗
Lomonosov Moscow State University, Moscow, Russia; Scientific Research Institute for System Analysis of the Russian Academy of Sciences, Moscow, Russia, e-mail: [email protected]. This research was supported in part by the Russian Foundation for Basic Research and the Japanese Society for the Advancement of Science in the framework of scientific project No. 19-51-50005. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 205, No. 3, pp. 349–367, November, 2020. Received July 15, 2020. Revised July 15, 2020. Accepted August 6, 2020. c 2020 Pleiades Publishing, Ltd. 0040-5779/20/2053-1547
1547
Fig. 1.
Propagation of rays from a pointlike source.
a|w|2 /2 electrons per unit time, where |w|2 is the squared Hermitian spinor in the right-hand side of the equation, as shown in the appendix. The limit-absorption principle physically means that quasiparticles do not come from infinity, and its mathematical formulation is given in Sec. 3.3. The graphene theory is outlined in [1]. In Theorem 1, we propose an explicit formula for solu
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