High-Resolution Valley Spectroscopy of Si Quantum Dots

We study an accumulation mode $\mathrm{Si}/\mathrm{SiGe}$ double quantum dot (DQD) containing a single electron that is dipole coupled to microwave photons in a superconducting cavity. Measurements of the cavity transmission reveal dispersive features due to the DQD valley states in Si. The occupation of the valley states can be increased by raising the temperature or applying a finite source-drain bias across the DQD, resulting in an increased signal. Using the cavity input-output theory and a four-level model of the DQD, it is possible to efficiently extract valley splittings and the inter- and intravalley tunnel couplings.

Figure 1

(a) Optical image of the device. (b) Schematic representation of the experiment and the DQD energy levels. $E_L$ and $E_R$ denote the valley splittings in the left and right dots, respectively, and $\epsilon$ is the interdot level detuning. (c) Tilted angle false-colored scanning electron microscope image of the DQD [red outline in (a)]. (d) DQD charge stability diagram acquired by measuring the cavity transmission amplitude $A/A_0$ as a function of the plunger gate voltages $V_{\mathrm{LP}}$ and $V_{\mathrm{RP}}$, with fixed interdot barrier gate voltage $V_{\mathrm{MB}}=150\mathrm{mV}$.

Figure 2

(a) Cavity transmission amplitude $A/A_0$ plotted as a function of $V_{\mathrm{LP}}$ and $V_{\mathrm{RP}}$ near the $(1,0)\leftrightarrow (0,1)$ interdot charge transition with $T_{\mathrm{lat}}=10\mathrm{mK}$. The black arrow denotes the detuning parameter $\epsilon$. (b) DQD energy level diagram with $E_L=E_R=51\mu \mathrm{eV}$, $t=20.9\mu \mathrm{eV}$, and $t^{\prime }=15.0\mu \mathrm{eV}$. The gray dashed lines show the uncoupled energy levels ($t=t^{\prime }=0$). (c) $A/A_0$ measured as a function of $\epsilon$ for three values of $V_{\mathrm{MB}}$ and fits to the theory (black lines).

Figure 3

(a) $A/A_0$ plotted as a function of $V_{\mathrm{LP}}$ and $V_{\mathrm{RP}}$ with $V_{\mathrm{MB}}=323\mathrm{mV}$ and $T_{\mathrm{lat}}=250\mathrm{mK}$. (b) $A/A_0$ as a function of $\epsilon$ for three values of $T_{\mathrm{lat}}$, with $V_{\mathrm{MB}}=323\mathrm{mV}$. Black solid lines are theory predictions. The data and fit for $T_{\mathrm{lat}}=10\mathrm{mK}$ [Fig. 2] are reproduced here for a direct comparison.

Figure 4

(a) $A/A_0$ measured with $V_{\mathrm{MB}}=323\mathrm{mV}$ and $V_{\mathrm{SD}}=0.3\mathrm{mV}$. Upper inset: Theory. Lower inset: Measured DQD current $I$. (b),(c) DQD energy level diagrams at $V_{\mathrm{RP}}=622\mathrm{mV}$ for (b) $\epsilon \approx -50\mu \mathrm{eV}$ and (c) $\epsilon \approx 50\mu \mathrm{eV}$. (d),(e) $A/A_0$ measured at (d) $V_{\mathrm{RP}}=624\mathrm{mV}$ and (e) $V_{\mathrm{RP}}=622\mathrm{mV}$. Black solid lines are fits to the theory.