Effective mass in semiconductors

1. Introduction
2. Energy-wavenumber (E-k) diagram of silicon
3. Detailed parameters for Ge, Si and GaAs
4. Density of states mass
5. Conductivity mass
6. Short list of parameters for Ge, Si and GaAs

Introduction

Most semiconductors can be described as having one band minimum at k = 0 as well as several equivalent anisotropic band minima at k ¹ 0. In addition there are three band maxima of interest which are close to the valence band edge.

Band structure of silicon

As an example we consider the band structure of silicon as shown in the figure below:

Shown is the E-k diagram within the first brillouin zone and along the (100) direction. The energy is chosen to be to zero at the edge of the valence band. The lowest band minimum at k = 0 and still above the valence band edge occurs at E_c,direct = 3.2 eV. This is not the lowest minimum above the valence band edge since there are also 6 equivalent minima at k = (x,0,0), (-x,0,0), (0,x,0), (0,-x,0), (0,0,x), and (0,0,-x) with x = 5 nm^-1. The minimum energy of all these minima equals 1.12 eV = E_c,indirect. The effective mass of these anisotropic minima is characterized by a longitudinal mass along the corresponding equivalent (100) direction and two transverse masses in the plane perpendicular to the longitudinal direction. In silicon the longitudinal electron mass is m_e,l^* = 0.98 m₀ and the transverse electron masses are m_e,t^* = 0.19 m₀, where m₀ = 9.11 x 10^-31 kg is the free electron rest mass.

Two of the three band maxima occur at 0 eV. These bands are refered to as the light and heavy hole bands with a light hole mass of m_lh^* = 0.16 m₀ and a heavy hole mass of m_hh^* = 0.46 m₀. In addition there is a split-off hole band with its maximum at E_v,so = -0.044 eV and a split-off hole mass of m_v,so^* = 0.29 m₀.

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Effective mass and energy band minima and maxima of Ge, Si and GaAs

m₀ = 9.11 x 10^-31 kg is the free electron rest mass.

Effective mass for density of states calculations

(f24a)

(24b)

for instance for a single band minimum described by a longitudinal mass and two transverse masses the effective mass for density of states calculations is the geometric mean of the three masses. Including the fact that there are several equivalent minima at the same energy one obtains the effective mass for density of states calculations from:

(f65)

where M_c is the number of equivalent band minima. For silicon one obtains:

m_e,dos^* = (m_l m_t m_t)^1/3 = (6)^2/3 (0.89 x 0.19 x 0.19)^1/3 m₀ = 1.08 m₀.

Effective mass for conductivity calculations

As the conductivity of a material is inversionally proportional to the effective masses, one finds that the conductivity due to multiple band maxima or minima is proportional to the sum of the inverse of the individual masses, multiplied by the density of carriers in each band, as each maximum or minimum adds to the overall conductivity. For anisotropic minima containing one longitudinal and two transverse effective masses one has to sum over the effective masses in the different minima along the equivalent directions. The resulting effective mass for bands which have ellipsoidal constant energy surfaces is given by:

(f66)

provided the material has an isotropic conductivity as is the case for cubic materials. For instance electrons in the X minima of silicon have an effective conductivity mass given by:

m_e,cond^* = 3 x (1/m_l + 1/m_t + 1/m_t)^-1 = 3 x (1/0.89 + 1/0.19 +1/0.19)^-1 m₀ = 0.26 m₀.

Effective mass and energy bandgap of Ge, Si and GaAs

m₀ = 9.11 x 10^-31 kg is the free electron rest mass.

¹ Due to the fact that the heavy hole band does not have a spherical symmetry there is a discrepancy between the actual effective mass for density of states and conductivity calculations (number on the right) and the calculated value (number on the left) which is based on spherical constant-energy surfaces. The actual constant-energy surfaces in the heavy hole band are "warped", resembling a cube with rounded corners and dented-in faces.