Effective mass in semiconductors

Contents - Glossary - Study Aids - 1 2 3 4 5 6 7 8 9
In this section:

Introduction

The effective mass of a semiconductor is obtained by fitting the actual E-k diagram around the conduction band minimum or the valence band maximum by a parabola. While this concept is simple enough the issue turns out to be substancially more complex due to the multitude and the occasional anisotropy of the minima and maxima. In this section we first describe the different relevant band minima and maxima, present the numeric values for germanium, silicon and gallium arsenide and introduce the effective mass for density of states calculations and the effective mass for conductivity calculations.

Most semiconductors can be described as having one band minimum at k = 0 as well as several equivalent anisotropic band minima at k ą 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 Ec,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 = Ec,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 me,l* = 0.98 m0 and the transverse electron masses are me,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 mlh* = 0.16 m0 and a heavy hole mass of mhh* = 0.46 m0. In addition there is a split-off hole band with its maximum at Ev,so = -0.044 eV and a split-off hole mass of mv,so* = 0.29 m0.

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

The values of the energy band minima and maxima as well as the effective masses for germanium, silicon and gallium arsenide are listed in the table below:

Name Symbol Germanium Silicon Gallium Arsenide
Band minimum at k = 0
Minimum energy Eg,direct [eV] 0.8 3.2 1.424
Effective mass me*/m0 0.041 ?0.2? 0.067
Band minimum not at k = 0
Minimum energy Eg,indirect [eV] 0.66 1.12 1.734
Longitudinal effective mass me,l*/m0 1.64 0.98 1.98
Transverse effective mass me,t*/m0 0.082 0.19 0.37
Wavenumber at minimum k [1/nm] xxx xxx xxx
Longitudinal direction (111) (100) (111)
Heavy hole valence band maximum at E = k = 0
Effective mass mhh*/m0 0.28 0.49 0.45
Light hole valence band maximum at k = 0
Effective mass mlh*/m0 0.044 0.16 0.082
Split-off hole valence band maximum at k = 0
Split-off band valence band energy Ev,so [eV] -0.028 -0.044 -0.34
Effective mass mh,so*/m0 0.084 0.29 0.154

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

Effective mass for density of states calculations

The effective mass for density of states calculations equals the mass which provides the density of states using the expression for one isotropic maximum or minimum or: (f24a)
for the density of states in the conduction band and: (24b)
for the density of states in the valence band.

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

me,dos* = (ml mt mt)1/3 = (6)2/3 (0.89 x 0.19 x 0.19)1/3 m0 = 1.08 m0.

Effective mass for conductivity calculations

The effective mass for conductivity calculation is the mass which is used in conduction related problems accounting for the detailed structure of the semiconductor. These calculations include mobility and diffusion constants calculations. Another example is the calculation of the shallow impurity levels using a hydrogen-like model.

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:

me,cond* = 3 x (1/ml + 1/mt + 1/mt)-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

Name Symbol Germanium Silicon Gallium Arsenide
Smallest energy bandgap at 300 K Eg (eV) 0.66 1.12 1.424
Effective mass for density of states calculations
Electrons me*,dos/m0 0.56 1.08 0.067
Holes mh*,dos/m0 0.29 0.57/0.811 0.47
Effective mass for conductivity calculations
Electrons me*,cond/m0 0.12 0.26 0.067
Holes mh*,cond/m0 0.21 0.36/0.3861 0.34

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.

© Bart J. Van Zeghbroeck, 1997