Intersubband carrier scattering in $n$- and $p\text{−}\mathrm{Si}∕\mathrm{Si}\mathrm{Ge}$ quantum wells with diffuse interfaces

Scattering rate calculations in two-dimensional $\mathrm{Si}∕{\mathrm{Si}}_{1-x}{\mathrm{Ge}}_x$ systems have typically been restricted to rectangular Ge profiles at interfaces between layers. Real interfaces, however, may exhibit diffuse Ge profiles either by design or as a limitation of the growth process. It is shown here that alloy disorder scattering dramatically increases with Ge interdiffusion in (100) and (111) $n$-type quantum wells, but remains almost constant in (100) $p$-type heterostructures. It is also shown that smoothing of the confining potential leads to large changes in subband energies and scattering rates, and a method is presented for calculating growth process tolerances.

Figure 1

${\Delta }_2$ conduction band edge for a $10\mathrm{nm}$ Si QW between two $5\mathrm{nm}$ ${\mathrm{Si}}_{0.5}{\mathrm{Ge}}_{0.5}$ barriers, with varying Ge diffusion length. The ${\Delta }_4$ conduction band edge for an abrupt interface is shown to be $120\mathrm{meV}$ higher in energy.

Figure 2

Energy separation between the lowest pair of subbands in a $\mathrm{Si}∕{\mathrm{Si}}_{0.5}{\mathrm{Ge}}_{0.5}$ QW as a function of well width. Results are shown for $\Delta$ valleys in $n$-type structures (Si wells) in the (100) and (111) orientations and for $p$-type structures (Si barriers) in the (100) orientation. In all cases, the results are shown for interface diffusion lengths of 0, 1, and $2\mathrm{nm}$, with arrows denoting the increasing diffusion length.

Figure 3

Average scattering rates from the second to first subband in a $n$-type (100) $10\mathrm{nm}$ Si QW between two $5\mathrm{nm}$ ${\mathrm{Si}}_{0.5}{\mathrm{Ge}}_{0.5}$ barriers with $T=4\mathrm{K}$, as a function of electron temperature. Rates are shown for electron-electron (E̱E), intravalley acoustic phonon (AC), interface roughness (IFR), ionized impurity (ION), $g$-type phonon emission, and alloy disorder (ADO) processes. ADO is strongly dependent on diffusion length and is shown at $L=0\mathrm{nm}$ (no symbols), $1\mathrm{nm}$ (circles), and $2\mathrm{nm}$ (stars). All other rates are almost independent of interdiffusion and are shown only for $L=0\mathrm{nm}$.

Figure 4

Average scattering rates from second to first subband in a (100) $n$-type Si QW as a function of subband separation at electron temperature of $T_e=24\mathrm{K}$. The ADO rate is shown at diffusion lengths $L=0\mathrm{nm}$ (no symbols), $1\mathrm{nm}$ (circles), and $2\mathrm{nm}$ (stars). All other rates are almost independent of $L$ and are shown only for $L=2\mathrm{nm}$ for simplicity.

Figure 5

Average scattering rates from second to first subband in a (111) $n$-type Si QW as a function of subband separation with $T_e=24\mathrm{K}$. All rates from Fig. 4 are shown (using the same legend) as well as the $f$-phonon rates.

Figure 6

Average scattering rate from second to first HH subband in a $p$-type QW as a function of hole temperature, with subband separation fixed at $10\mathrm{meV}$. The legend is the same as previous figures, with the optical phonon branches as labeled. The alloy disorder scattering rate is shown at diffusion lengths $L=0\mathrm{nm}$ (no symbols), $L=1\mathrm{nm}$ (circles), and $L=2\mathrm{nm}$ (stars). All other rates are shown at $L=0\mathrm{nm}$ only.

Figure 7

Average scattering rates from second to first subband in a (100) $n$-type double QW with Si wells of width 5 and $3\mathrm{nm}$ separated by a $1\mathrm{nm}$ ${\mathrm{Si}}_{0.5}{\mathrm{Ge}}_{0.5}$ barrier as a function of Ge interdiffusion. The legend is the same as that in Fig. 4. The subband separation as a function of interdiffusion is overlaid (crosses).

Figure 8

${\Delta }_2$ conduction band edge in a two-well heterostructure, nominally comprising 50% Ge barriers and Si wells. The left and right well widths are 5 and $3\mathrm{nm}$, respectively, and the separating barrier width is $1\mathrm{nm}$. The potential is shown for diffusion lengths of 0, 0.7, and $1.5\mathrm{nm}$, which coincide with the uncoupled, weakly coupled, and single-well operating regimes, respectively.