Effective mass in semiconductors

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 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.

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 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

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:

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

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.

Hernandez Caballero Indiana M. CI: 15.242.745

Asignatura: EES

Fuente: http://ecee.colorado.edu/~bart/book/effmass.htm

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