3.3.5 Effective Density of States

The effective density of states (DOS) in the conduction and the valence bands are expressed by the following theoretical expressions [86]:

$$N_{C} = 2\cdot M_C\cdot{\left(\frac{2\cdot \pi\cdot m_n \cdot k_B\cdot T_{{\mathrm{L}}}}{h^{2}}\right)}^{3/2}$$ (3.91)

$$N_{V} = 2\cdot{\left(\frac{2\cdot \pi\cdot m_p \cdot k_B\cdot T_{{\mathrm{L}}}}{h^{2}}\right)}^{3/2}$$ (3.92)

$M_C$ represents the number of equivalent energy minima in the conduction band.

Table 3.19: Parameter values for energy minima in the DOS model

Material	$M_C$	Material	$M_C$
Si	6	InAs	1
Ge	4	InP	1
GaAs	1	GaP	3
AlAs	3

For $\mathrm{Si}$ an alternative model based on data after Green [120] is implemented, which is based on a second order polynomial fit.

$$N_{C,V} = N_{0,\nu} + N_{1,\nu}\cdot \left(\frac{T_{\mathrm{L}}}{\mathrm{30... ...}}\right)+ N_{2,\nu}\cdot \left(\frac{T_{\mathrm{L}}}{\mathrm{300 K}}\right)^2$$ (3.93)

Table 3.20: Parameter values for modeling the effective carrier masses

Material	$N_{0,n}$[cm$^{-3}$]	$N_{1,n}$[cm$^{-3}$]	$N_{2,n}$[cm$^{-3}$]	$N_{0,p}$[cm$^{-3}$]	$N_{1,p}$[cm$^{-3}$]	$N_{2,p}$[cm$^{-3}$]
Si	-0.14e19	1.56e19	1.44e19	-0.17e19	0.93e19	2.34e19

In the model for alloy materials effective carrier masses of the constituents are used in the expressions (3.91) and (3.92).

In the case of a transition between a direct and an indirect bandgap in III-V ternary compounds the valley degeneracy factor $M_C$ is modeled by an expression equivalent to the one proposed in [157].

$$M_{C} = M_{C}^\mathrm {d}\cdot\exp\left(\frac{E_{\mathrm{g}}... ...E_{\mathrm{g}}-E_{\mathrm{g,X}}}{\mathrm{k_B}\cdot T_L}\right)$$ (3.94)

The superscripts $\mathrm {d}$ and $\mathrm {i}$ denote direct and indirect, respectively.

In the case of SiGe the splitting of the valley degeneracy due to strain is modeled accordingly as in [158].

$$M_{C} = M_{C}^\mathrm {Si}\cdot\exp\left(-{\frac{\delta E_{C... ...rac{\delta E_{C}}{\mathrm{k_B}\cdot T_L}\cdot x}\right)\right)$$ (3.95)

Here, $\delta E_{C}$ denotes the energy difference between the valleys shifted down and up in energy, respectively. It is set equal to 0.6 eV as given in [158].