Density-dependent two-dimensional optimal mobility in ultra-high-quality semiconductor quantum wells

We calculate using the Boltzmann transport theory the density-dependent mobility of two-dimensional (2D) electrons in GaAs, SiGe, and AlAs quantum wells as well as of 2D holes in GaAs quantum wells. The goal is to precisely understand the recently reported breakthrough in achieving a record 2D mobility for electrons confined in a GaAs quantum well. Comparing our theory with the experimentally reported electron mobility in GaAs quantum wells, we conclude that the mobility is limited by unintentional background random charged impurities at an unprecedented low concentration of $\sim {10}^{13}{\mathrm{cm}}^{-3}$. We find that this same low level of background disorder should lead to 2D GaAs hole and 2D AlAs electron mobilities of $\sim {10}^7$ and $\sim 4\times {10}^7{\mathrm{cm}}^2/\text{V}\text{s}$, respectively, which are much higher theoretical limits than the currently achieved experimental values in these systems. We therefore conclude that the current GaAs hole and AlAs electron systems are much dirtier than the state-of-the-art 2D GaAs electron systems. We present theoretical results for 2D mobility as a function of density, effective mass, quantum-well width, and valley degeneracy, comparing with experimental data.

Figure 1

(a) Experimental mobility (O-markers) for each sample at a different carrier density given in Fig. 2(a) of Ref. [2] along with the theoretically calculated mobilities (X-markers) that best fit each sample mobility data obtained using the Boltzmann transport theory [Eq. (2)] with the impurity density $n_{\mathrm{i}}$ being the tuning parameter. The straight lines represent the power-law relation $\mu \sim n^{0.7}$. The dependence of $\mu$ on $n$ depicted here (and in Ref. [2]) is not a functional dependence since $\mu$ depends on two independent parameters ($n$ and $n_{\mathrm{i}}$.) (b) Plot of the background impurity densities extracted in (a) as a function of the carrier density $n$. Here red and black indicate with and without doping wells, respectively [2].

Figure 2

Calculated mobility as a function of the carrier density for four different materials using (a) the fixed impurity density corresponding to the lowest mobility in Fig. 1 and a somewhat higher impurity density of (b) $n_{\mathrm{i}}=2.13\times {10}^{13}{\mathrm{cm}}^{-3}$. The fitted red straight lines in (a) indicate the power-law exponent $p$ (i.e., $\mu \sim n^p$) at low and high densities. For the calculations, we use a fixed value of the quantum-well width $a=30\mathrm{nm}$. Note that the varying exponent $p$ happens to be $\sim 0.7$ for $n$-GaAs at $n\sim {10}^{11}{\mathrm{cm}}^{-2}$, in rough agreement with Fig. 1.

Figure 3

(a) Calculated mobilities as a function of the inverse of the effective mass at different carrier densities $n$ with the valley degeneracy of $g_v=1$ and 2, showing the mobility dependence on both valley degeneracy and effective mass. Here $m$ is in units of electron mass. (b) Calculated mobilities as a function of the quantum-well width $a$ for four different materials at a fixed carrier density of $n={10}^{11}{\mathrm{cm}}^{-2}$, showing the mobility dependence on the well width. For both results in (a) and (b), we use the lowest best-fit impurity density of the $n$-GaAs sample with a doping-well ($n_{\mathrm{i}}=1.33\times {10}^{13}{\mathrm{cm}}^{-3}$), which we obtain in Fig. 1.

Figure 4

Phase diagram showing the regime where the single subband approximation is valid. The approximation is valid when the carrier density (at a given well width $a$) is below the the plotted curves, which represent the onset density where the second subband occupation occurs. The X-markers indicate the carrier densities (and the corresponding quantum-well widths) of the high-mobility $n$-GaAs samples in Ref. [2], which are used in our calculations of Fig. 1.