Theory of band warping and its effects on thermoelectronic transport properties

Nicholas A. Mecholsky, Lorenzo Resca, Ian L. Pegg, and Marco Fornari

Phys. Rev. B 89, 155131 – Published 24 April 2014

Abstract

Optical and transport properties of materials depend heavily upon features of electronic band structures in proximity of energy extrema in the Brillouin zone (BZ). Such features are generally described in terms of multidimensional quadratic expansions and corresponding definitions of effective masses. Multidimensional quadratic expansions, however, are permissible only under strict conditions that are typically violated when energy bands become degenerate at extrema in the BZ. Even for energy bands that are nondegenerate at critical points in the BZ there are instances in which multidimensional quadratic expansions cannot be correctly performed. Suggestive terms such as “band warping,” “fluted energy surfaces,” or “corrugated energy surfaces” have been used to refer to such situations and ad hoc methods have been developed to treat them. While numerical calculations may reflect such features, a complete theory of band warping has not hitherto been developed. We define band warping as referring to band structures that do not admit second-order differentiability at critical points in $k$ space and we develop a generally applicable theory, based on radial expansions, and a corresponding definition of angular effective mass. Our theory also accounts for effects of band nonparabolicity and anisotropy, which hitherto have not been precisely distinguished from, if not utterly confused with, band warping. Based on our theory, we develop precise procedures to evaluate band warping quantitatively. As a benchmark demonstration, we analyze the warping features of valence bands in silicon using first-principles calculations and we compare those with previous semiempirical models. As an application of major significance to thermoelectricity, we use our theory and angular effective masses to generalize derivations of tensorial transport coefficients for cases of either single or multiple electronic bands, with either quadratically expansible or warped energy surfaces. From that theory we discover the formal existence at critical points of transport-equivalent ellipsoidal bands that yield identical results from the standpoint of any transport property. Additionally, we illustrate with some basic multiband models the drastic effects that band warping and anisotropy can induce on thermoelectric properties such as electronic conductivity and thermopower tensors.

Received 20 August 2013
Revised 24 February 2014

DOI:https://doi.org/10.1103/PhysRevB.89.155131

Authors & Affiliations

Nicholas A. Mecholsky^*, Lorenzo Resca^†, and Ian L. Pegg^‡

Department of Physics and Vitreous State Laboratory, Catholic University of America, Washington, DC 20064, USA

Marco Fornari^§

Department of Physics, Central Michigan University, Mount Pleasant, Michigan 48858, USA

^*Corresponding author: nmech@vsl.cua.edu
^†resca@cua.edu; http://physics.cua.edu/people/faculty/resca.cfm
^‡ianp@vsl.cua.edu
^§marco.fornari@cmich.edu; http://www.phy.cmich.edu/people/fornari

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Vol. 89, Iss. 15 — 15 April 2014

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Figure 1
Selected band plots for the top three valence bands of Si near the $Γ$ point disregarding spin-orbit interactions. Red points represent the energy dispersion calculated according to first principles (QE) along the $Λ$ line starting at $0.05 \frac{2 π}{\sqrt{3} a_{0}} (1, 1, 1)$ and moving toward $Γ$ . Green points move from $Γ$ out to $0.05 \frac{2 π}{a_{0}} (1, 0, 0)$ along the $Δ$ line. Blue points are plotted from $0.05 \frac{2 π}{\sqrt{2} a_{0}} (1, 1, 0)$ along the $Σ$ line toward $Γ$ . The inset to the bottom right shows the three paths in $k$ space selected for the left panels. The solid curves in the left panels represent the radial parabolic approximations to the energy dispersions for each band. Every direction requires a different parabolic fit for each band, as a result of its warping. The curvatures of all the radial parabolas then generate the corresponding nonspherical angular effective mass surfaces shown in the right panel. The top two surfaces correspond to the heavy-hole and light-hole bands. Without spin-orbit coupling, those two bands are degenerate with a third valence band at $Γ$ . As a result, all three effective mass surfaces are warped and nonspherical, particularly for the heavy-hole band.
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Figure 2
Selected band plots for the top three valence bands of Si near the $Γ$ point including spin-orbit interactions. Curve and surface descriptions and definitions are the same for all panels as those given in the captions of Fig. 1. The top two angular effective mass surfaces shown in the right panel correspond to the heavy-hole and light-hole bands, which remain degenerate at $Γ$ . As a result, the effective mass surfaces for the top two bands remain warped and nonspherical, particularly for the heavy-hole band. However, the split-off band becomes nondegenerate on account of spin-orbit interactions. It thus appears to lose its warping, as shown by its spherical effective mass surface in the third subgraph down in the right panel.
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Figure 3
Angular effective mass surface $f (θ, ϕ)$ for the highest or heavy-hole valence band of Si at $Γ$ , including spin-orbit interactions, as calculated from first principles using the $q u a n t u m e s p r e s s o$ (QE) package [48]. (a) 2653 angular data points calculated using a quadratic fit with respect to $k_{r}$ of the energy dispersion for each angular direction. (b) Spherical harmonics fit up to $l = 20$ to the QE data in plot (a). (c) Azimuthal $ϕ = 0$ cross section of the effective mass surface. The black-dashed curve shows an interpolation from the data, the red-dashed curve represents a $l = 20$ spherical harmonics fit, and the black-dotted curve shows a least-squares fit to the “Kittel form,” Eq. (16), with optimal parameters $A = - 4.204 49$ , $B = 0.378 191$ , $C = 5.309$ . (d) Azimuthal $ϕ = π / 4$ cross section of the effective mass surface, with curves drawn as in (c).
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Figure 4
Conductivity and thermopower tensor eigenvalues for a model with two highly anisotropic ellipsoidal bands. The two ellipsoidal bands have masses $m_{v, x} = 0.1, m_{v, y} = 3, m_{v, z} = 5$ , and $m_{c, x^{'}} = 0.2, m_{c, y^{'}} = 7, m_{c, z^{'}} = 11$ , respectively. The principal axes of the conduction band are rotated with respect to those of the valence band through Euler angles $Φ = π / 7$ , $Θ = π / 6$ , and $Ψ = 0$ . Two constant-energy surfaces for the two bands are shown in panel (a). The valence-band maximum is set at $- 0.5$ eV and the conduction-band minimum is set at +0.5 eV. Panels (b) and (c) show three curves each, representing the three eigenvalues of the conductivity tensor, normalized by the relaxation time, $τ$ , and of the thermopower tensor, as functions of the chemical potential, $μ$ . The absolute temperature is set at $T = 500$ K. One-band asymptotic expansions, as derived in Appendix pp4, are represented by black dashed curves.
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Figure 5
Isotropic conductivity and Seebeck coefficient for two models, comprising either one [panels (a) and (b)] or two [panels (c) and (d)] warped valence bands of the “Kittel form,” at three different levels of warping. The maximum of all valence bands is set at $- 0.5$ eV. Both models further comprise a single spherical conduction band with effective mass $m = m_{e}$ and its minimum set at +0.5 eV. Each panel shows three curves, representing a negligible (black), medium (blue), and high (red) level of warping for the valence bands. In panel (a), the insets pointing at the valence-band curves at their three different levels of warping depict their corresponding angular effective mass surfaces. The isotropic conductivity, normalized by the same relaxation time for all bands, $τ$ , and the Seebeck coefficient are shown as functions of the chemical potential, $μ$ . The absolute temperature is set at $T = 500$ K in all cases.
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