Mapping g-factors and complex intervalley coupling in Si/SiGe by conveyor-mode shuttling

Abstract

As silicon spin qubit chips are increasing in qubit number and area, methods for the screening of qubit related material parameters become vital. Here we demonstrate the two-dimensional mapping of small variations of the electron g-factor of quantum dots formed in planar Si/SiGe quantum wells with precision better than $10^{-3}$ and with nanometer lateral resolution. We scan the electron g-factor across a 40 nm $\times$ 400 nm area and observe two g-factors per QD site which obey a striking symmetry and bimodal distribution across the area. These two g-factors relate to valley states of the electron in the quantum dot in agreement with a recent theoretical model. Using conveyor-belt shuttling of entangled electron spin pairs, complementary to the mapping of the local valley-splitting, we map the g-factor. We compare g-factor and valley splitting maps measured on the same device, and extract the complex intervalley coupling parameter along the shuttle trajectories applying a theoretical model of g-factor dependence on intervalley coupling. These maps will allow unprecedented insights into the spin-valley dynamics during qubit manipulation, readout and shuttling and serve as a benchmark for the engineering of Si/SiGe heterostructures for large-scale quantum chips.

Figures (5)

• Figure 1: Sample and experimental method. (a) Scanning electron micrograph of the device used in this experiment. The scale bar corresponds to 200n m. (b) Schematic illustrating the experiments execution. (c) $\tau_\text{W}$-sweep: Raw data of a measurement where we shuttle the electron to position d and vary the wait time $\tau_\text{W}$ while keeping the magnetic field constant ($B = 1.7T$). (d) Extracted $g$-factor variation by Fourier transformation of (c).(e) Version of (d) marked with found peaks and their average.
• Figure 2: Extraction of the $\Delta g$ values. (a-f) Fourier transformation of the raw data from $y=18n m$ to $y=-12n m$ to extract the local $\Delta g$. (g-l) Fourier transformations with one to two marked peaks (white/grey crosses; white cross represents the dominant component) and their respective averages (white circles; only if two peaks are identified).
• Figure 3: Statistical analysis of the $\Delta g$ measurement. (a) Self correlation of the data points with two $\Delta g$-factor peaks. The circles are shaded by the relative strength of the secondary peak. The Pearson $r$ value is displayed. (b) Histogram of the extracted $\Delta g$ values. (c) Boxplots for the $\left< \overline{\Delta g}(d,y)\right>_d$ averaged over all $d$ for each scanline. The box spans from the first quartile to third quartile, the median is marked by a vertical line and the averages are marked by black points. The whiskers indicate the 5% and 95% mark with outliers indicated by circles. (d) Schematic of the origin of the two components $\Delta g_{1(2)}$ and $\Delta g_l$ and their symmetry. In the upper scenario only the shuttled electron is in a mixed valley state, and in the lower scenario only the inert (static) electrons are in a mixed valley state. The respective valley-dependent $g$-factors differ by $2 \delta g_s$ and $2 \delta g_i$, respectively.
• Figure 4: Comparison of a g-factor map to its corresponding valley splitting map. (a) Map of the absolute value of $|\delta g_s|$ as a function of $d$ and $y$. (b) Map of the dominant component $\Delta g_1$ as a function of $d$ and $y$. (c) Map of the valley splitting as a function of $d$ and $y$. (d-i) Direct comparison of valley splitting scanlines against the extracted $\Delta g$-factors. (j) Simulation of a valley splitting and a $\Delta g$ trace.
• Figure 5: Reconstruction of the intervalley coupling $\Delta$. (a-e) Extracted intervalley coupling from the $g$-factor and $E_\text{VS}$ measurements from $y=12n m$ to $y=-12n m$. (f-g) Corresponding reconstructions of the full trace of the intervalley coupling. (i-j) Histogramm of the real and imaginary part of the all the intervalley couplings $\Delta$, which we calculated together with a least-square fit Gaussian function (black solid line). The mean $\mu$ and the variance $\sigma$ of the fits are given.