Disentangling orbital and confinement contributions to $g$-factor in Ge/SiGe hole quantum dots

TL;DR

The study discriminates pure Zeeman and orbital contributions to the hole $g$-factor in Ge/SiGe quantum dots by comparing addition spectroscopy (CBAS) and excited-state spectroscopy (PESS) on two devices. The authors show that orbital effects can modify the measured $g$-factor by up to about $10\%-20\%$ across occupancies and orbital states, helping explain method- and state-dependent discrepancies. They demonstrate gate-tunability of the $g$-factor by up to $\sim$15% through confinement control and wavefunction relocation, highlighting a pathway for all-electric qubit manipulation. The work also emphasizes the limitations of simple Fock–Darwin models in hole dots and underscores the role of confinement, screening, exchange, and HH–LH mixing (via the Luttinger–Kohn framework) in shaping $g$-factor behavior, with implications for consistent cross-study comparisons and quantum-dot qubit design.

Abstract

Spin qubits are typically operated in the lowest orbital of a quantum dot to minimize interference from nearby states. In valence-band hole systems, strong spin-orbit coupling links spin and orbital degrees of freedom, strongly influencing the hole $g$-factor, a key parameter for qubit control. We investigate the out-of-plane $g$-factor in Ge quantum dots using excitation (single-particle) and addition (many-body) spectra. Excitation spectra allow us to distinguish the pure Zeeman $g$-factor from orbital contributions to the magnetic field splitting of states despite the strong spin-orbit coupling. This distinction clarifies discrepancies between $g$-factors extracted with the two methods, for different orbital states and different hole numbers. Furthermore, we find gate-tunability of $g$-factors at the level of 15%, highlighting its relevance for all-electric qubit manipulation.

Figures (19)

• Figure 1: Operation of the double quantum dot (DQD) system and Coulomb blockade addition spectroscopy.(a) Schematic of the device. The DQD is defined beneath plunger gates P$_1$ and P$_2$ and is monitored via a proximal charge sensor. Interdot coupling is tuned via the central barrier gate B$_{12}$, while coupling to the reservoirs is controlled by gates RB$_1$ and RB$_2$. The charge sensor signal is extracted from the differential current between source and drain, $I_{\mathrm{SD}}$, as indicated by the black arrow. (b) Charge stability diagram of the DQD as a function of virtual plunger gates $V_{\overline{\mathrm{P{1}}}}$ and $V_{\overline{\mathrm{P{2}}}}$. Charge configurations are labeled as $(N,M)$, denoting the hole occupation of each dot. (c) Magnetospectroscopy of Coulomb blockade addition energies, revealing spin and orbital structure. Charge states are again labeled as $(N,M)$. (d) Extracted addition energies (black), charging energies (gray), and single-particle energies (orange) as a function of total dot occupation. Error bars are smaller than the marker size. (e) Single-particle energy spectrum obtained by subtracting the charging energy $E_C$ from the addition energies in panel (d). The dashed box highlights the region analyzed in Fig. \ref{['fig:fig2']}b. (f) Linear fit to the spin-split energy levels used to extract the CBAS $g$-factor, based on the slope prior to the first level crossing.
• Figure 2: Comparison of CBAS analysis to PESS.(a) Energy spectrum obtained via CBAS. Ground-states of d from magnetospectroscopy are analyzed analogously. The yellow line marks the discontinuity at the intersection of $\epsilon_2$ and $\epsilon_3$. (b) Linear fits to the spin-split energy levels used to extract the CBAS and PESS $g$-factor. All data are shown in absolute values. For clarity, an offset is added to all traces except those corresponding to pure spin transitions ($\uparrow_\text{o1}-\downarrow_\text{o1}$ and $T_0-T_{-}$). (c) CBAS and PESS $g$-factor for both devices with $3\sigma$-error bars. (d) Derivative of the charge sensor current, $\mathrm{d} I_\textnormal{det}/\mathrm{d} V_{\overline{\textnormal{P1}}}$, plotted as a function of pulse amplitude $\delta V_{\mathrm{P1}}$ and DC gate voltage $V_{\overline{\mathrm{P{1}}}}$ at $B_\perp = 0$. Colored regions indicate where specific hole numbers can be dynamically loaded in response to the applied pulses. (e)$\mathrm{d} I_\textnormal{det}/\mathrm{d} V_{\overline{\textnormal{P1}}}$ as a function of $V_{\overline{\mathrm{P{1}}}}$ and $B_\perp$ at fixed $\delta V_{\overline{\mathrm{P{1}}}}$. The hole number $N$ is labeled, and regions sensitive to excited-state spectra are highlighted. Extracted energy levels are overlaid as colored lines.
• Figure 3: Voltage-tunable $g$-factor.(a) Single-particle energies at $B_\perp=0T$ as a function of quantum dot occupation for different values of $V_{\overline{B{12}}}$. (b) CBAS $g$-factor extracted via CBAS as a function of the interdot barrier voltage $V_{\overline{B{12}}}$. (c) PESS $g$-factor as a function of the plunger gate voltage applied to the neighboring quantum dot. The lower plot shows the corresponding charge stability diagram, with arrows indicating the direction of plunger gate tuning. The charge configuration is labeled as $(N,M)$.
• Figure S1: Extraction of the lever arm from temperature-broadened transitions.(a) Example dataset showing eight charge transitions measured via the sensor current $I_{\mathrm{SD}}$ for device 1 at $V_{\overline{\mathrm{B{12}}}}$ is -0.23V. (b) Zoom-in of the first transition, measured on device 2. The sensor current is normalized, and the transition is fit using a Fermi–Dirac distribution at various temperatures, illustrating the thermal broadening.
• Figure S2: Extraction of the lever arm from temperature-broadened transitions for device 1.(a, c, e) Width of the Fermi–Dirac distribution extracted from the data in Fig. \ref{['fig:lever_arm_Tsweep']}, plotted as a function of fridge temperature for the $N$th transition, and fitted for different values of $V_{\overline{\mathrm{B{12}}}}$. (b, d, f) Resulting lever arms plotted as a function of hole number $N$ for the corresponding $V_{\overline{\mathrm{B{12}}}}$ values.