Derivation of effective-mass expressions for electrons and holes in the anisotropic multiband semimetals Ar, Sb, and Bi

Original publisher: Adelphi, MD : Army Research Laboratory, [2000] OCLC Number: (OCoLC)707344936 Subject: Anisotropy. Excerpt: ... 3. Calculations of Sums Over k-Space Minima for High-Symmetry Band Structures 3.1 Simple Cubic Band Structure A simple cubic band structure is appropriate for elemental silicon ( see fig. 1 ). In order to calculate the effective masses for this method, we assume that the densities in the carrier pockets are the same, and that the principal-axis projections are xˆxˆ, yˆyˆ, and zˆzˆ. Since there are six pockets, each holds 1 / 6 of the electrons. Let us write the effective masses as follows:����− 1 − 1 1 1 ( 1 ) ( − 1 ) m = m = xˆxˆ + ( yˆyˆ + zˆzˆ ), m m t l����− 1 − 1 1 1 ( 2 ) ( − 2 ) m = m = yˆyˆ + ( xˆxˆ + zˆzˆ ), and m m t l����− 1 − 1 1 1 ( 3 ) ( − 3 ) m = m = zˆzˆ + ( yˆyˆ + xˆxˆ ). ( 55 ) m m t l Then the sum is simple:�������↔ 1 1 2 1 1 2 ( j ) − 1 n m = + 2 [ xˆxˆ + yˆyˆ + zˆzˆ ] = + n n, ( 56 ) 1 0 0 0 j 6 m m 3 m m t t l l j ↔ where is the identity matrix. This shows that the mass required is just the 1 usual isotropic optical mass. Figure 1. Conduction band minima ( pockets ) of silicon in k-space. 10