Interstitial Channels that Control Band Gaps and Effective Masses in Tetrahedrally Bonded Semiconductors

Yu-ichiro Matsushita and Atsushi Oshiyama

Phys. Rev. Lett. 112, 136403 – Published 2 April 2014

Abstract

We find that electron states at the bottom of the conduction bands of covalent semiconductors are distributed mainly in the interstitial channels and that this floating nature leads to the band-gap variation and the anisotropic effective masses in various polytypes of SiC. We find that the channel length, rather than the hexagonality prevailed in the past, is the decisive factor for the band-gap variation in the polytypes. We also find that the floating nature causes two-dimensional electron and hole systems at the interface of different SiC polytypes and even one-dimensional channels near the inclined SiC surface.

Received 19 November 2013

DOI:https://doi.org/10.1103/PhysRevLett.112.136403

Authors & Affiliations

Yu-ichiro Matsushita

Max-Planck Institute of Microstructure Physics, Weinberg 2, D-06120 Halle, Germany and Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan

Atsushi Oshiyama

Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan

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Vol. 112, Iss. 13 — 4 April 2014

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Images

Figure 1
Energy bands and the contour plot of the Kohn-Sham (KS) orbital of the conduction-band minimum at $M$ in $3 C − SiC$ shown on the ( $0 \bar{1} 1$ ) plane. The orbital is distributed along the [110] channel. The calculation has been done with the hybrid exchange-correlation functional, HSE [3, 4]. White and burgundy balls depict C and Si atoms, respectively. Simple localized-orbital basis sets are incapable of describing the floating nature of the conduction-band states [2, 5].
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Figure 2
Band gaps for 24 representative SiC polytypes calculated in generalized gradient approximation as a function of the hexagonality (left panel) and a function of the channel length (right panel). In each panel, the fitting function (see text) is also shown. The corresponding variance is 355 meV for the left (hexagonality) and 85 meV for the right (channel length).
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Figure 3
Contour plots on the ( $11 \bar{2} 0$ ) plane of the KS orbitals of the conduction band minimum at $M$ for $10 H − SiC$ with the $A B C A B C A B A B$ stacking (left panel) and with the $A B C A C B C A C B$ stacking (right panel). The value for each contour color is relative to the corresponding maximum absolute value. White and burgundy balls depict C and Si atoms, respectively. The broken lines represent the interface where the longest channel in the cubic region is blocked by the hexagonal region.
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Figure 4
(a) Local density of states $D (ϵ, z)$ near the band gap for the $12 H − SiC$ ( $A B C A C A C A C A C B$ ) with its side view of the atomic structure. The shaded region in the side view depicts the cubic stacking region. The lower panel is the extension of $D (ϵ, z)$ near the valence-band top. The ordinate $ϵ$ is the electronic energy, where the Fermi energy is set to be 0, and the abscissa $z$ is the coordinate along the stacking direction. The value of $D (ϵ, z)$ is represented by the color code (yellow: high, black: low). The black region corresponds to the energy gap. (b) Band-gap variation as a function of the thickness of hexagonal-stacking region of SiC polytypes with the thickness of the cubic stacking region fixed with five bilayers.
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Figure 5
Calculated CBM energy at several $k$ points as a function of reduced lattice constant $a / a_{0}$ ( $a_{0}$ : lattice constant without pressure) in $3 C − SiC$ (a) and in GaAs (c). The corresponding pressures (GPa) at $a / a_{0} = 0$ , 0.95, 0.9, 0.85, 0.80, and 0.75 are 0 (0), 31.0 (5.9), 73.7 (12.5), 128.7 (21.3), 202.6 (33.6), and 306.5 (51.5) in SiC (GaAs). (b) The kinetic-energy contribution $ε_{kin} = ⟨ ϕ_{i} | - \nabla^{2} / 2 | ϕ_{i} ⟩$ to the orbital energy of each Kohn-Sham (KS) state at the $M$ point in $3 C − SiC$ . The abscissa represents the $i$ th KS state from the valence-band bottom, and the 25th state is the CBM. (d) Contour plots of the KS orbital of the CBM in GaAs at $a = 0.75 a_{0}$ . The gray and green balls represent Ga and As atoms, respectively.
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Figure 6
One-dimensional electron channel on the inclined SiC surface (xylophone channel). (a) Schematic side view of $5 H − SiC$ near the inclined surface (the inclined angle $θ$ ) on the ( $11 \bar{2} 0$ ) plane. The five arrows represent channels with each number discriminating the channel length. (b) and (d): Contour plots on the ( $11 \bar{2} 0$ ) plane of the CBM KS orbitals near the surface with the inclined angle of $θ = 30.39 °$ and $θ = 16.32 °$ . (c) Top view of the CBM KS orbitals as an isovalue surface at its value of 20% of the maximum value. Blue and green isovalue surfaces represent the positive and negative signs of the KS orbital. White and burgundy balls depict C and Si atoms, respectively.
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