6.1.3 Subband Structure

Figure 6.11 shows the three lowest energy levels, $E_n^v({\ensuremath{\mathitbf{k}}} = 0)$, and the populations of the subband ladders of Si inversion layers for substrate orientation (001), (110), and (111) as a function of the inversion layer concentration at a substrate doping concentration of 1e16 cm$^{-3}$. The energy levels are plotted relative to the lowest subband energy level. Two subband ladders are formed in (001)- and (110)-oriented substrate, whereas in (111)-oriented substrate the six $\Delta$-valleys form only one subband ladder. It can be seen that the subband splitting increases with the inversion layer concentration. Furthermore, the Fermi level crosses the lowest subband level. This happens at an inversion layer concentration of 5e12 cm$^{-2}$ for (001) oriented substrate and above 1e13 cm$^{-2}$ for (110) and (111)-oriented substrate. Therefore, the 2DEG is degenerate and the Pauli exclusion principle has to be included in simulations.

Figure: Three lowest subband energies and subband ladder populations for substrate orientation (001), (110), and (111). $N_\mathrm{sub}=1e16$cm$^{-3}$ and $T=300$K.

[(001)] [(001)]

[(110)] [(110)]

[(111)] [(111)]

The subband populations for the three substrate orientations are compared in Figure 6.11b. The population of the primed ladder is the largest for (001)-oriented substrate. Keeping in mind that for this substrate orientation the transport masses of the primed ladder are larger than for the unprimed ladder, one can anticipate the large potential of strain to reduce the primed ladder population.

In Figure 6.12 the three lowest subband levels and subband populations for a substrate with (001)-orientation are given at three stress configurations. Figure 6.12a and 6.12b show that a tensile stress along [100] causes the primed ladder, being fourfold degenerate in the unstrained case, to split into two ladders with twofold degeneracy. A stress of 1 GPa is able to shift up in energy one of these ladders (the unprimed one) such that the population of this ladder is below 2% at all inversion layer concentrations.

Stress along [110] does not change the degeneracy of the subband ladders. Figure 6.12d shows that the population of the primed ladder decreases below 20% for this type of stress. Additionally, the effective mass change induced by stress along the [110]-direction does not significantly change the subband levels and subband populations at 1 GPa stress. The small effect of stress on the unprimed ladder is not surprising since the subband energies and populations are determined by the quantization mass $\smash{\ensuremath{\mathrm{m}}_{[001]}}$ and the density of states mass $\smash{\sqrt{m_{[110]}m_{[1\bar{1}0]}}}$, respectively, which are not largely affected by the shear component ${\ensuremath{\varepsilon_{xy}}}=0.314\%$ resulting from 1 GPa stress.

Biaxial tensile stress in the (001)-plane can be seen to be the most efficient configuration to reduce the population of the primed subband ladder below 5% at 1 GPa (Figure 6.12f).

Figure: Three lowest subband energies and subband populations for (001)-oriented substrate at 1 GPa tensile stress and three stress directions. $N_\mathrm{sub}=1e16$cm$^{-3}$ and $T=300$K.

[] [Stress along [100]]

[] [Stress along [110]]

[] [Biaxial stress]

In Figure 6.13 the lowest subband levels and subband populations for a substrate with (110)-orientation and 1 GPa tensile stress along [001] are shown. The stress moves the primed subband ladder up in energy, thereby reducing its occupation below 2%.

Figure: Three lowest subband energies and subband populations for (110)-oriented substrate at 1 GPa tensile stress along [001]. $N_\mathrm{sub}=1e16$cm$^{-3}$ and $T=300$K.

E. Ungersboeck: Advanced Modelling Aspects of Modern Strained CMOS Technology