Theory of the effective Hamiltonian for degenerate bands in an electric field

Abstract

Abstract.Recent experiments have generated a renewed interest in the properties of the degenerate valence states in semiconductors under the influence of a uniform external electric field. In response, a number of authors have proposed that the standard Luttinger-Kohn effectivemass Hamiltonian should be modified to include the energy of interaction between the electric field and the dipole matrix elements of the relevant zone-centre Bloch functions. This article examines these proposals by comparing the proposed dipole interaction with rigorous derivations of the fielddependent Hamiltonian in the Bloch and Luttinger-Kohn representations. It is shown that the dipole matrix element of a unit cell is not a suitable foundation for a Hamiltonian because it depends on the choice of unit cell, which is arbitrary. Moreover, the correct Luttinger-Kohn Hamiltonian has no wave-vector-independent matrix elements that are linear in the applied field E, except for small terms of relativistic origin. Therefore, the proposed modifications to the Luttinger-Kohn theory are not valid. The correct form of the Luttinger-Kohn Hamiltonian is derived here through terms of order k 3 , kE, and E 2 , along with the momentum matrix to first order in k and E. In addition, the recent discovery of field-induced mixing at the Brillouin zone boundary in the Luttinger-Kohn representation is studied in detail.

First-principles study of Dirac and Dirac-like cones in phononic and photonic crystals

“…Therefore, it is clear that the coupling of the degenerate Bloch states leads to the frequency repulsion effect, which in turn gives rise to linear dispersions of a Dirac/Dirac-like cone. It should be pointed out that the above perturbative approach is exact to the first order in k. 23 Thus, the existence of linear dispersions requires both the degeneracy of Bloch states at some high-symmetry point k 0 and the corresponding p lj = 0, independent of whether the degeneracy is accidental or due to lattice symmetry. The matrix elements of p lj and q lj can be easily evaluated by performing numerical integration of Eqs.…”

“…II, our theory is exact to the first order in k (Ref. 23) so that it can correctly predict the linear slopes of the Dirac cone. By solving the secular equation, we find…”

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“…The sum runs over all conduction and valence-band states |n s except n = n. Here,P = 1 h ∂ kĤ (k) is the eight-band momentum operator obtained by the Helman-Feynman theorem 28 and the quantization axis z is taken to be 001 , the direction of the applied static magnetic field. While we evaluated Eq.…”

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“…This expression can only be applied as long as ∆E i j (k) in the denominator is not equal to zero. The case of degenerate bands presents additional mathematical challenges [27,35,36]; in particular, the transition matrix elements ξ ξ ξ i j (k) are singular at degeneracies [36]. For reference, we also give the relation between ξ ξ ξ i j (k) and the matrix elements of the position operator between the Bloch functions [15,37,38]:…”