Spin Lifetime and Charge Noise in Hot Silicon Quantum Dot Qubits

We investigate the magnetic field and temperature dependence of the single-electron spin lifetime in silicon quantum dots and find a lifetime of 2.8 ms at a temperature of 1.1 K. We develop a model based on spin-valley mixing and find that Johnson noise and two-phonon processes limit relaxation at low and high temperature, respectively. We also investigate the effect of temperature on charge noise and find a linear dependence up to 4 K. These results contribute to the understanding of relaxation in silicon quantum dots and are promising for qubit operation at elevated temperatures.

Figure 1

(a) Scanning electron microscope image of a device identical to the one measured. $R$ is the reservoir gate, $P1$, $P2$, $B1$, and $B2$ are the plunger gates, and $C$ confines the electrons in the dots. LB and RB are the left and right barrier of the quantum dot used for sensing, and ST is used both as top gate and reservoir. The ESR line can be used for spin manipulation. (b) Charge stability diagram of the device measured via a double lock-in technique [14] (see Supplemental Material [15]). The transition lines, due to the different slope, can be attributed to three coupled quantum dots. The red arrow shows the ($0\to 1$) charge transition relevant for the experiment. (c) Tunneling rate between the dot and the reservoir as a function of ${\mathrm{V}}_{P2}$. $\mathrm{\Delta }{\mathrm{V}}_{P2}=0$ corresponds to the value set during the experiment. The red line is an exponential fit. (d) Pulsing sequence used to perform single-shot readout of the electron spin [20] in the case $E_z<E_{\text{vs}}$. Above the valley splitting there is also an intermediate level between the ground and excited spin state, corresponding to the spin-down state of the excited valley.

Figure 2

(a) Energy levels in a silicon quantum dot, showing both valley and spin degrees of freedom. As an example, the transition ${\mathrm{\Gamma }}_{\overline{2}\overline{1}}$ is sketched in first order and in second order via virtual and resonant transitions. (b) Relaxation rate as a function of magnetic field. The fittings include contributions from Johnson and phonon mediated relaxation obtained through the model explained in the main text. From the fittings of the magnetic field and temperature dependence we extract $E_{\text{vs}}=275\mu \mathrm{eV}$, ${\mathrm{\Gamma }}_0^J(E_{\text{vs}}/\hslash )=2\times {10}^{-12}\mathrm{s}$, ${\mathrm{\Gamma }}_0^{\mathrm{ph}}(E_{\text{vs}}/\hslash )=6\times {10}^{-12}\mathrm{s}$, and $\mathrm{\Delta }=0.4\mathrm{neV}$.

Figure 3

(a)–(c) Temperature dependence of the relaxation rate at $B_0=1.5\mathrm{T}$ (a), 2 T (b), and 3 T (c). The red line is a fit taking into account Johnson and phonon noise in first and second order. The red dashed line includes possible contributions coming from the coupling with the excited orbital states. First-order processes are shown in the dashed blue line. (d),(e) Relaxation rate as a function of magnetic field and valley splitting for $T=10\mathrm{mK}$ (d) and for $T=1\mathrm{K}$ (e) as extracted from the model discussed in the main text.

Figure 4

(a) Charge noise spectra obtained for three different temperatures. At higher frequencies the $1/\mathrm{f}$ signal is masked by white noise. (b) Charge noise at a frequency of 1 Hz as a function of temperature fitted with a linear function.