Cross-hatch strain effects on SiGe quantum dots for qubit variability estimation

TL;DR

This work addresses qubit variability in Si/SiGe quantum dots arising from cross-hatch strain and interface disorder in virtual substrates. It employs a multi-technique workflow—Raman strain mapping, AFM surface imaging, and cross-sectional HAADF-STEM—across 25 CVD wafers, paired with a strain-driven surface-diffusion model and valley-splitting ensemble calculations to quantify how strain inhomogeneity translates into roughening and valley-splitting variability. The findings show that cross-hatch strain fluctuations and pregrowth anneal-driven roughening primarily set the disorder landscape, with interface steps modestly reducing valley splitting and thicker buffers plus lower processing temperatures mitigating roughening. These insights provide quantitative benchmarks for qubit-yield projections and practical fabrication guidance, including strategies to decouple qubits from strain through thicker buffers and lower-temperature growth. The work thus connects microscopic strain and interfacial disorder to macroscopic qubit energy scales, informing scalable SiGe quantum-dot qubit architectures.

Abstract

SiGe heterostructures integrated with Si via virtual substrate (VS) growth are promising hosts for spin qubits. While VS growth targets plastic relaxation, residual cross-hatch strain inhomogeneity propagates into heterostructure overgrowth. To quantify strain inhomogeneity's influence on interface structure and qubit properties, we measure strained-silicon (s-Si)/Si$_{0.7}$Ge$_{0.3}$ heterostructures on 25 wafers processed via standard commercial chemical vapor deposition. Spatially-aligned images of strain (Raman microscopy) and interface structure (atomic force microscopy and cross-sectional scanning transmission electron microscopy) reveal strain-roughness interplay. A strain-driven surface diffusion model predicts the roughness and its temperature dependence. Measured strains suggest spurious double-dot qubit detunings of 0.1 meV over 100 nm distances may result. Modeling shows that interface roughness (atomic steps), when convolved with alloy disorder, only modestly reduces valley splitting (70$\pm$13 vs. 77$\pm$14 $μ$eV on average). Our findings point to thicker VS buffer layers beneath heterostructures and lower-temperature growth (T $\le$ 700 $^{\circ}$C) to limit roughening.

Figures (6)

• Figure 1: The growth experiment process showing virtual substrate (VS) preparation and heterostructure growth.a A schematic cross-section showing the VS and heterostructure layers. b The process flow for the VS with AFM images tracking surface evolution: (wafer #1) Si(100) substrate, (wafer #3) relaxed Si$_{0.7}$Ge$_{0.3}$ on graded growth, (wafer #5) surface post chemical mechanical polishing. c The heterostructure growth with AFM images: (wafer #7) epitaxy-ready Si$_{0.7}$Ge$_{0.3}$ relaxed buffer (post annealing), (wafer #13) 225 nm SiGe regrowth, (wafer #14) 6.1 nm Si QW, and (wafer #15) 3 nm s-Si cap. Imaging of the relaxed Si$_{0.7}$Ge$_{0.3}$ heterostructure layer was not undertaken; given the thin s-Si cap overlayer, it is assumed to be nominally similar. Heterostructure deposition, T = 600 ° C. Epitaxy-ready surfaces were prepared by dilute HF and H$_2$ anneal at 900 ° C for 2 min prior to heterostructure growth. AFM images show 20 $\mu$m $\times$ 20 $\mu$m area. RMS roughness values were obtained from each processed wafer and from different areas, rather than from a single identical location. Raman microscopy imaging over a 20 $\mu$m $\times$ 15 $\mu$m area, shown above the schematic drawings, reveals the underlying strain inhomogeneity in the layered structures.
• Figure 2: Cross-correlating heterostructure surface and interface cross-hatch roughness with bulk elastic inhomogeneity using multimodal coincident-site measurements from two sites, 1 (left column) and 2 (right column).a Heterostructure surface topography images (AFM) measured along with, b, Raman microscopy images, and the corresponding strain, $\epsilon_{||}$. Heavy black scale bars between images are 20 µm and apply to both AFM and Raman data. The colored lines on the AFM and Raman images indicate the locations of cross-sectional lamellae for interface structure analysis (HAADF-STEM). c Plots showing spatially-aligned interface structure, and Raman (strain) measurements at the locations indicated by colored lines in the AFM and Raman images. d Interface nanoscale structure measured using cross-sectional HAADF-STEM imaging. In both cases, the lateral scale bars are 1 µm, and the vertical scale bars are 10 nm. Inset panels e show the atomic-scale interface structure and image intensity line trace across the interface. The 6.0 nm vertical scale bar applies to the image and vertical line trace. f Misfit dislocation bunches, the likely source of the cross-hatch strain fluctuation seen in the Raman data, are distributed randomly throughout the VS SiGe graded layer as shown by coincident-site bright-field TEM images. Horizontal and vertical scale bars are 3 µm for images from both sites.
• Figure 3: AFM-measured surface profiles with simulated counterparts for two selected sites. Comparison between the simulated profile and the AFM experiments in the first selected region a and in the second selected region b. Comparison between the statistical descriptors of the surface profile, i.e. surface roughness $h_{\text{rms}}$ and correlation length $L_{\text{corr}}$ for the first profile c and the second profile d. The simulation time selected for the best fit with the surface profiles reported in panels a and b is highlighted in red in both plots.
• Figure 4: Strain and conduction band variation inferred from Raman data.a In-plane biaxial strain $\epsilon_{\parallel}(x,y)$ over the measured sample area, b Conduction band shift $\Delta E_{\mathrm{CB}}(x,y)$ based on deformation theory, c One-dimensional cut through the middle of the measurement region ($y_{0}$=17.5 $\mu$m) of the relative detuning bias shift of the conduction band between points in-plane separated by $\Delta x = 100$ nm along the $x$-axis, $\Delta E_{\mathrm{CB}}(x+\Delta x/2, y_{0}) - \Delta E_{\mathrm{CB}}(x-\Delta x/2, y_{0})$. This quantifies the inter-quantum dot bias that may be expected due to strain inhomogeneity assuming typical inter-dot separations Burkard2021.
• Figure 5: Valley splitting distributions resulting from interface disordera STM image of interface steps on the top of the Si layer. Boxed region used for the calculation in b with dashed green circle of diameter of 65.4 nm, corresponding to the $4\sigma$ width of the ground state of an electron in a harmonic confinement with 1.5 meV spacing. b Map of the valley splitting as a function of the center of a QD for a realization of alloy disorder using the interface steps from a. Note that we consider a conformal step configuration for which both lower and upper interfaces have steps at the same in-plane coordinates. c Similar map to b, but with a flat interface between the Si and SiGe layers. d Histogram of the valley splittings from b and c. Calculated valley splitting distribution dependence on e electric field strength and f conduction band offset. Inset: Sampling of Ge atomic locations around $y=0$ at the Si/SiGe interface for the system with (top) and without (bottom) interface steps. The conduction band offset is applied to the Si layers to match the extremes of the expected shift from measured strains, and an electric field of 0.5 mV/nm is assumed for all three instances. Solid curves are fits to Rice distributions, with the associated parameters in the legend. Histograms of e and f are calculated from 500 alloy realizations, and all valley splitting calculations use a 6.11 nm well thickness, 3.05 nm of the top/bottom SiGe layers included in the computational domain, and assuming a 1.5 meV harmonic in-plane confinement.