Dispersive readout of valley splittings in cavity-coupled silicon quantum dots

The band structure of bulk silicon has a sixfold valley degeneracy. Strain in the Si/SiGe quantum well system partially lifts the valley degeneracy, but the materials factors that set the splitting of the two lowest lying valleys are still under intense investigation. Using cavity input-output theory, we propose a method for accurately determining the valley splitting in Si/SiGe double quantum dots embedded in a superconducting microwave resonator. We show that low lying valley states in the double quantum dot energy level spectrum lead to readily observable features in the cavity transmission. These features generate a “fingerprint” of the microscopic energy level structure of a semiconductor double quantum dot, providing useful information on valley splittings and intervalley coupling rates.

Figure 1

(a) A cavity-coupled DQD is probed using a small input field $a_{\mathrm{in}}$ with frequency ${\omega }_R$. Charge dynamics in the DQD result in changes in the transmitted field $a_{\mathrm{out}}$. The DQD is coupled to source (S) and drain (D) electrodes. (b) DQD energy levels plotted as a function of energy level detuning $\epsilon$. In general, the left dot valley splitting $E_L$ is different than the right dot valley splitting $E_R$. For this plot, $E_L=76\mu \mathrm{eV},E_R=58\mu \mathrm{eV},t=25\mu \mathrm{eV}$, and $t^{\prime }=13\mu \mathrm{eV}$.

Figure 2

(a) Microwave transmission coefficient ${\left|A\right|}^2$ as a function of the DQD detuning $\epsilon$ and temperature $T$. The vertical red dotted lines indicate the anticrossings in the DQD spectrum (cf. Fig. 1). The horizontal yellow lines indicate two cuts at temperatures $T=1\mathrm{K}$ and $T=250\mathrm{mK}$ for which the transmission coefficient is plotted separately in (b) and (c). The tunneling matrix elements between the QDs are assumed to be $t=25\mu \mathrm{eV}$, while those between opposite valleys are $t^{\prime }=13\mu \mathrm{eV}$. The microwave resonator frequency is $f_0={\omega }_0/2\pi =7.8\mathrm{GHz}=32\mu \mathrm{eV}$ and the probe field is on resonance with the cavity ${\omega }_R={\omega }_0$. The valley splittings are chosen as $E_L=76\mu \mathrm{eV}$, and $E_R=58\mu \mathrm{eV}$.

Figure 3

(a) Cavity transmission as a function of the interdot bias $\epsilon$ and the right dot valley splitting $E_R$, with $E_L=76\mu \mathrm{eV},f_0={\omega }_0/2\pi =7.8\mathrm{GHz}=32\mu \mathrm{eV},{\omega }_R={\omega }_0,t=25\mu \mathrm{eV},t^{\prime }=13\mu \mathrm{eV}$, and $T=250\mathrm{mK}$. The horizontal yellow lines indicate two cuts at valley splittings: (b) $E_R=100\mu \mathrm{eV}$ and (c) $E_R=10\mu \mathrm{eV}$. The vertical red dotted lines indicate the anticrossings in the DQD spectrum (cf. Fig. 1) at $\epsilon =-E_L$ and $\epsilon =E_R$; the other avoided crossings are not shown due to their vicinity to the $|L\rangle -|R\rangle$ avoided crossing at $\epsilon =0$.

Figure 4

(a) Cavity transmission as a function of $\epsilon$ and the intervalley matrix element $t^{\prime }$. The horizontal yellow lines indicate two cuts at intervalley tunnel couplings: (b) $t^{\prime }=12\mu \mathrm{eV}$ and (c) $t^{\prime }=4\mu \mathrm{eV}$. Here we use a fixed $t=25\mu \mathrm{eV},E_L=E_R=60\mu \mathrm{eV}$, and $f_0={\omega }_0/2\pi =7.8\mathrm{GHz}=32\mu \mathrm{eV},{\omega }_R={\omega }_0,T=250\mathrm{mK}$. The vertical red dotted lines indicate the anticrossings in the DQD spectrum (cf. Fig. 1).

Figure 5

Cavity transmission as a function of $\epsilon$ and the intervalley matrix element $t^{\prime }$ for $t\sim 0$. Here we use $E_L=E_R=60\mu \mathrm{eV}$, and $f_0={\omega }_0/2\pi =7.8\mathrm{GHz}=32\mu \mathrm{eV},{\omega }_R={\omega }_0,T=250\mathrm{mK}$. The yellow horizontal line indicates $t^{\prime }=f_0/2$ where the double lines merge into single lines. The vertical red dotted lines indicate the anticrossings in the DQD spectrum (cf. Fig. 1).