Valley Splitting Correlations Across a Silicon Quantum Well Containing Germanium

TL;DR

This work addresses the variability of valley splitting $E_V$ in a Ge-containing Si quantum well, a key factor limiting spin qubit fidelity in Si/SiGe devices. It combines device-scale modeling with detuning axis pulsed spectroscopy (DAPS) to map $E_V$ across a 1.3 µm, 21-dot channel and to characterize both nanometer-scale fluctuations and device-scale correlations. The results show that fluctuations are consistent with alloy disorder-dominated (ADD) theory, with a nanometer-scale correlation length $\,\ell_C\approx 19$ nm and meaningful device-scale correlations across the channel, and they provide a framework to predict $E_V$ at unmeasured sites using correlations. These findings inform disorder-aware design and modeling for scalable Si/SiGe qubit devices, highlighting the need for uniform valley splitting to enable reliable qubit operation during shuttling and Pauli blockade.

Abstract

Quantum dots in SiGe/Si/SiGe heterostructures host coherent electron spin qubits, which are promising for future quantum computers. The silicon quantum well hosts near-degenerate electron valley states, creating a low-lying excited state that is known to reduce spin qubit readout and control fidelity. The valley energy splitting is dominated by the microscopic disorder in the SiGe alloy and at the Si/SiGe interfaces, and while Si devices are compatible with large-scale semiconductor manufacturing, achieving a uniformly large valley splitting energy across a many-qubit device spanning mesoscopic distances is an outstanding challenge. In this work we study valley splitting variations in a 1D quantum dot array, formed in a Si$_{0.972}$Ge$_{0.028}$ quantum well, manufactured by Intel. We observe correlations in valley splitting, at both sub-100nm (single gate) and >1$μ$m (device) lengthscales, that are consistent with alloy disorder-dominated theory and simulation. Our results develop the mesoscopic understanding of Si/SiGe heterostructures necessary for scalable device design.

Figures (6)

• Figure 1: Ge-containing quantum well devices and modeling. (a) Top-down SEM of quantum dot and single electron transistor (SET), plunger (P), barrier (B), accumulation (Acc.), and screening (SG) gates in an Intel Tunnel Falls device nominally identical to the device studied in this paper. Screening gates buried underneath the qubit and SET finger gates are outlined in white. The symmetry between plunger and barrier gates allows formation of 21 separate quantum dots (purple circles) in the top channel. (b) Side-on view of a SiGe heterostructure with a 4.6nm-thick Si quantum well and finger gates. The well in this work is Si_0.972Ge_0.028 (the image is of a pure Si well). (c) Electron density in a tuned up double dot demonstrating dot elongation along the 1D channel ($x$). (d) Distribution of simulated valley splitting energies $E_V$ for the device heterostructure according to the alloy disorder-dominated theory in Ref. losert_practical_2023. Overlaid fit is to a Rayleigh distribution. (e) Simulated cross capacitance of an electron under a plunger gate to the outer barrier gate. (f) Electron center position $x$ versus simulated cross capacitance along the linecut in (e), used to calibrated dot position $x_{dot}$ in Fig. \ref{['fig:fig3']}.
• Figure 2: Detuning axis pulsed spectroscopy (DAPS) valley splitting measurement. (a) Single-electron energy level diagram for the example P5-P6 double dot, with anti-crossings arising between ground and excited valley states. (b) When the electron (purple circle) is far-detuned from an anti-crossing it sits in the ground valley of the DAPS source (Src.) dot. At an anti-crossing the valley states hybridize and the electron can tunnel into DAPS target (Tar.) dot. (c) Double dot charge stability diagram focused on the single electron polarization line. Teal arrow shows the direction of the voltage pulse in (d) and (e). (d) DAPS measurement diagram with a variable-time ($t_{hold}$), variable-energy voltage pulse followed by a 30µs SET current measurement and a 300µs delay to allow the electron to relax back to the source dot. (e) DAPS spectrum of the P5 dot at an electron temperature of $T_e=150mK$, revealing valley-orbital states. Valley splitting $E_V$ is extracted from fitting the two lowest peaks in line cuts at one $t_{hold}$ value, and orbital energy $E_O$ is extracted from fitting the first excited orbital (top).
• Figure 3: Continuous electron valley probe. (a-c) Charge stability diagrams (left) of B2-B3 double dot single electron polarization line and corresponding DAPS measurements (right) as the B3 dot's outer barrier (P4) is depleted. DAPS measurements are fit with double-Lorentzians to extract valley splitting energy ($E_V$). Teal arrows show the direction of the DAPS voltage pulse. Triangle, square, and diamond markers in the DAPS plot correspond to the three points in (d). (d) $E_V$ versus center position of the electron $x_{dot}$ accumulated under P3, B3, and P4 dots, relative to the center of gate B3, where error is calculated from the DAPS peak linewidth chen_detuning_2021. An interpolated curve with 2nm gaussian smoothing is overlaid. Inset is the autocorrelation $C_{E_V}$ (solid black) calculated from the interpolated curve and the fit (dashed blue) from theory. The scale bar is the correlation length from the inset figure. On top is a calculation of the thermal population $\rho_e$ at 150mK in the excited valley state for each measured $E_V$. Electron radii are calculated from the orbital energy extracted from DAPS.
• Figure 4: Expected $E_V$ variations. The valley splitting difference $\Delta E_V$ (purple filled in circles) is plotted alongside the 5%--95% range $R(\Delta E_V)$ of valley splitting at some distance $\Delta x$ (teal lines). The mean expected range for each distance $\overline{R(\Delta E_V)}$ is plotted atop the data (open teal circles), fit to an exponential decay (gray line).
• Figure 5: Valley splitting in full array. (a) Valley splitting $E_V$ measured in 21 separate dots in the Tunnel Falls device, with positions extracted from the capacitance model in Fig. \ref{['fig:fig1']}(e). (b) $E_V$ histogram revealing deviations from theoretical Rayleigh distribution, consistent with finite random sampling. (c) $E_V$ versus electron radius along the channel axis, showing no overall dependence. Error bars in (a) and (c) are calculated from the DAPS peak linewidth chen_detuning_2021. (d) Fourier transform of $E_V$ along the array, with no components consistent with long-range correlations. (e) Autocorrelation $C_{VV,n}$ versus distance $n$ with large negative $C_{VV}$ at $n=6$ and moderate positive $C_{VV}$ at $n=12$. (f) Statistical significance (p-value) of observed $C_{VV,n}$ values calculated with a permutation test. Null distributions for permutation tests on $n=6,12$ are shown in (g,h), respectively, with shaded area showing the calculated p-values from (f).