Low-frequency spin qubit detuning noise in highly purified ${}^{28}$Si/SiGe

The manipulation fidelity of a single electron qubit gate-confined in a ${}^{28}$Si/SiGe quantum dot has recently been drastically improved by nuclear isotope purification. Here, we identify the dominant source for low-frequency qubit detuning noise in a device with an embedded nanomagnet, a remaining ${}^{29}$Si concentration of only 60$$ppm in the strained ${}^{28}$Si quantum well layer and a spin echo decay time $T_2^{\text{echo}}=128\mu$s. The power spectral density (PSD) of the charge noise explains both the observed transition of a $1\mathrm{/}{f}^2$- to a $1\mathrm{/}f$-dependence of the detuning noise PSD as well as the observation of a decreasing time-ensemble spin dephasing time from $T_2^{\ast }\approx 20\mu$s with increasing measurement time over several hours. Despite their strong hyperfine contact interaction, the few ${}^{73}$Ge nuclei overlapping with the quantum dot in the barrier do not limit $T_2^{\ast }$, as their dynamics is frozen on a few hours measurement scale. We conclude that charge noise and the design of the gradient magnetic field is the key to further improve the qubit fidelity.