Interwell relaxation times in $p\text{−}\mathrm{Si}∕\mathrm{Si}\mathrm{Ge}$ asymmetric quantum well structures: Role of interface roughness

We report the direct determination of nonradiative lifetimes in $\mathrm{Si}∕\mathrm{Si}\mathrm{Ge}$ asymmetric quantum well structures designed to access spatially indirect (diagonal) interwell transitions between heavy-hole ground states, at photon energies below the optical phonon energy. We show both experimentally and theoretically, using a six-band $\mathbf{k}\bullet \mathbf{p}$ model and a time-domain rate equation scheme, that, for the interface quality currently achievable experimentally (with an average step height $⩾1\mathrm{\AA }$), interface roughness will dominate all other scattering processes up to about $200\mathrm{K}$. By comparing our results obtained for two different structures we deduce that in this regime both barrier and well widths play an important role in the determination of the carrier lifetime. Comparison with recently published experimental and theoretical data obtained for mid-infrared $\mathrm{Ga}\mathrm{As}∕{\mathrm{Al}}_x{\mathrm{Ga}}_{1-x}\mathrm{As}$ multiple quantum well systems leads us to the conclusion that the dominant role of interface roughness scattering at low temperature is a general feature of a wide range of semiconductor heterostructures not limited to IV-IV materials.

Figure 1

Annular dark field scanning transmission electron micrograph showing the first eight sets of pairs of QWs and the barriers for the first structure (BF1499).

Figure 2

(Color online) Calculated subband structure of BF1499 showing the energy levels and the square of the wave functions of the first eight hole states (we use an electronlike picture where the energy scale is reversed: the ground state is the lowermost). The slightly different depths of the two wells shown reproduce the slight difference in Ge content between them found in our HAADF measurements in all periods imaged. The interwell nonradiative transition between level 2 (red) and level 1 (blue) is indicated by a red arrow.

Figure 3

(Color online) Power (a) and temperature (b) dependence of the interwell transition lifetimes, measured at $T=4\mathrm{K}$ for BF1499. In (a) the power decreases with attenuation from $0\text{to}30\mathrm{dB}$.

Figure 4

(Color online) Comparison of experimentally measured (symbols) and theoretically predicted (lines) interwell transition lifetimes as a function of lattice temperature $T_{\mathit{latt}}$ for our two samples BF1499 (red) and BF1500 (blue). The red symbols are extracted from single exponential fits to the data in Fig. 3b. The dashed line represents the calculated lifetime for a hypothetical structure with the wells of BF1499 and the barrier of BF1500. Inset: lifetimes calculated for BF1500 and BF1499 in the case of perfect interfaces, i.e., in the absence of IFR scattering (the experimental data are the same as in the main frame). All curves were obtained using $\Delta =1.2\mathrm{\AA }$, $\Lambda =60\mathrm{\AA }$, and $T_h=T_{\mathit{latt}}+20\mathrm{K}$.

Figure 5

(Color online) Absolute (a) and relative (b) IFR scattering rates as a function of the IFR parameters $\Delta$ and $\Lambda$ calculated for BF1499 at $T_{\mathit{latt}}=20\mathrm{K}$, assuming a carrier temperature of $40\mathrm{K}$.

Figure 6

Total scattering rate calculated for BF1499 as a function of the lattice temperature $T_{\mathit{latt}}$ for $\Delta =1.2\mathrm{\AA }$, $\Lambda =60\mathrm{\AA }$, and $\delta T=20\mathrm{K}$, decomposed into absolute (a) and percentile (b) contributions from the different scattering mechanisms: acoustic (ac) and optical ($\mathrm{op}1=\mathrm{Ge}\text{−}\mathrm{Ge}$, $\mathrm{op}2=\mathrm{Si}\text{−}\mathrm{Si}$, and $\mathrm{op}3=\mathrm{Si}\text{−}\mathrm{Ge}$ branches) phonons, alloy disorder, IFR, and hole-hole interaction.

Figure 7

(Color online) Absolute (a) and relative (b) IFR scattering rates as a function of the IFR parameters $\Delta$ and $\Lambda$ calculated for BF1499 at $T_{\mathit{latt}}=300\mathrm{K}$, assuming a carrier temperature of $320\mathrm{K}$.