Temperature-dependence of the combined effective mass of holes in silicon

Abstract

The complex structure of the silicon valence band leads to great difficulties in studying the transport properties of holes in this material. On the basis of experimental and theoretical data we have calculated in this paper a low-field combined conduct ivi ty effective mass and combined density-of-states effective mass of holes which are temperature-dependent. The model we propose is valid for Ohmic problems in the temperature range from 40 °K to 400 °K. Theoretical calculations (1) of the hole energy band in silicon by the k . p method have established the following features. The valence b~ald can be obtained by overlapping individual atomic p-functions and exibits the max imum at k = 0. In the absence of spin-orbit splitt ing this maximum would be sixfold degenerate corresponding to the product of three p-functions with two spin-up and spin-down functions. Away from k : 0 there are three separate bands (E 1, E 2 and Ea) ~hich are doubly degenerate due to spin. I t is customary to label the bands by their spectroscopic character t)~ and by the magnetic quantum number ± M~ (degenerate in absence of magnetic field). Accordingly the first band E 1 is characterized by P~ and 3I~1⁄2, the second E 2 by P1⁄2 and M~_], the third E a by P1⁄2 and M~1⁄2. Spin-orbit splitting part ial ly removes the above degeneracy at k : 0 by lowering the two (E3) P1⁄2 bands with respect to the four P t bands (E 1 and E~). The coupling of the first two bands (E 1 and E2) leads to equienergetie surfaces of warped structure. For k ~ 0, E 1 is proport ional to k ~, keeping the same warped character as at k _~ 0. /E 2 does not mantain a parabolic character away from k ~ 0, its curvature changing with k 2, and tends to assume the same curvature of E 1. The curvature of E a changes with k 2 in the opposite way. From the above features the following conclusions can be summarized in the effectivemass approximation. The first band E 1 is characterized by holes having an energy* which is directiomdependent. The second band E~ is independent effective mass m 1 characterized by holes with effective mass m* which is increasing with energy and is direction-dependent. Calculations shows that at /¢ ~ 0 m * ~ m ~ , and that at kv~0 m~* approximates m~* for energies above about 0.02 eV from the minimum. The split-off