Increasing valley splitting in Si/SiGe by practically achievable heterostructure profiles

Abstract

Silicon spin qubits are marred by the valley degeneracy of the conduction band. In a nanodevice, the degeneracy is lifted by interfaces and alloy disorder, but the arising valley splitting is small, of order 100 $μ$eV in Si/SiGe quantum wells. Substantial efforts were invested both in theory and experiments to overcome the valley issue. Unfortunately, the existing recipes either rely on atomistic details of the interface that are beyond experimental control, or demand heterostructure profiles beyond current state-of-the-art heterostructure epitaxy. We revisit the valley splitting induced by non-trivial Ge profiles and advocate a novel view of the intervalley coupling as a backscattering on point-like impurities realized by crystal planes containing Ge atoms. This perspective reveals that enhancing the backscattering amplitude, which sets the valley splitting, requires constructive interference of multiple scatterers. % We arrive at a remarkable prediction, that the Ge content along the heterostructure growth direction does not have to have any specific periodicity, including the practically unreachable $2π/(2k_0)$ period, to significantly increase the valley splitting. This statement is corroborated with numerical evidence from tight-binding simulations and intuitive physical interpretations. We devise profiles that seem within the capabilities of current MBE growth techniques and boost the valley splitting beyond the 1\,meV scale.

Figures (18)

• Figure 1: Ge-doped monolayers (colored circles) with distance 12 (violet), 7(red) and 5(green) are approximately commensurate with the valley interference pattern $|\psi_+(z) + \psi_-(z)|^2 \propto 1 + \cos(2k_0z +\phi)$.
• Figure 2: Valley splitting as obtained from tight binding for delta-like Ge spikes (single Ge-doped monolayers) with a Ge concentration of $10\,\%$ separated by $p-1$ pure Si layers (except for the value in the blue diamond, which alternates 5 and 7 monolayer separations). For each $p$, we evaluate 100 realizations of the heterostructure with nominally identical doping profiles and randomly distributed Ge atoms. The statistical ensemble of the resulting valley splitting is depicted by its median value (orange lines) and its maximum and minimum (error bars; mostly too small to be clearly visible with the figure resolution). The electric field is $E=5\,\mathrm{mV/nm}$. The inset shows the histogram of values for $p=5$.
• Figure 3: The valley splitting as a function of $S$ for various types of profiles. In all cases, the profile contains $M$ spikes, $M$ ranges from 1 to 14, and we sample 20 structures (except for for yellow with 49 structures; and for black with 1 structure). In the selected monolayers, the Ge doping is $\rho_m=10\%$. The colors mean the following. Yellow: the spikes are distributed completely randomly (but not on top of each other). Black: after fixing $M$, we obtain the profile by maximizing $S$ numerically (see App. \ref{['app:optimization']} for the optimization algorithm). The remaining structures are constrained such that Ge spikes can only fall onto every 5-th Si monolayer ($p=5$ in Fig. \ref{['fig:commensurability']}). Green: random on the $p=5$ lattice. Red: periodic (with period $q$) on the $p=5$ lattice. The period $q$ is taken as the largest integer multiple of $p$ such that $M$ spikes fit into the quantum well. There remains one degree of freedom, a shift of the whole arrangement, which is chosen at random. Purple: periodic (with the period $q=p$) on the $p=5$ lattice. Again, the arrangement is randomly shifted around in the well.
• Figure 4: DFT results for a profile with Ge-doping spikes placed periodically at every $p$-th monolayer. The green curve shows the quantum well potential $U_\mathrm{qw}$ obtained by convoluting the Hartree potential with a Gaussian filter with a $0.25$ nm width. The unfiltered potential is shown in blue in the right upper inset. The black solid (dashed) curve shows the electron density of the lowest (second lowest) valley state, $|\Psi_{1,2}|^2$. In the main panel, $p=5$ and the positions of three Ge monolayers are denoted by vertical black lines. In the upper left inset, analogous quantities are plotted for $p=6$. Due to the limited size of the supercell (18 atoms per plane), the smallest nonzero Ge content of a monolayer is 5.5%, which has been used for the doped monolayers inside the quantum well. The barrier monolayers contain 5 Ge atoms (27.8%).
• Figure 5: Tight-binding results for realistic heterostructures. We analyze four instances of a quantum well built by repeating one of the blocks plotted on the left over the total width of 72 monolayers (about 10 nm) sandwiched by Si$_{0.3}$Ge$_{0.7}$ barriers. The four instances include three periodic spikes (5, 7, and 12 monolayer steps apart; the lower three panels on the left) and one more complex pattern, built from a 7-5-5-7 block (the top panel on the left). The electric field is $E=5\,\mathrm{mV/nm}$. The resulting valley splitting $E_\mathrm{VS}$ is shown in the right panel in corresponding colors. The IQR (inter quartile range) is shown by a box, the extreme outliers by the bars, and the median is in black. Compared to the values for ideal single-layer dopings, given in Fig. \ref{['fig:ideal']}, the valley splittings are suppressed by a factor 0.24, 0.29, 0.2, and 0.15, respectively, for the four profile choices plotted. These numbers can be explained by Eq. \ref{['eq:suppression']} which, using, $k_0 \approx 0.83$, gives $\rho_{3,2}=0.26$, $\rho_{3,3}=0.17$, $\rho_{4,3}=0.24$, $\rho_{4,4}=0.21$, and $\rho_{5,4}=0.14$, for a few profiles similar to the chosen ones.