Quantum gates with donors in germanium

Recent work has shown that electron spins in germanium (Ge) nanoscale transistors can be electrically tuned and have encouraging coherence times. Based on a complete and validated theory of Ge-donor electron states, we propose that Ge spin qubits could have significant advantages over silicon (Si) in the implementation of a donor-based quantum processor architecture. Our work shows that the intrinsic features of the Ge band structure allow for a speedup of selective (local) one-qubit gates of up to two orders of magnitude as compared to Si. Further, we find that fast, robust two-qubit gates in Ge pose less stringent fabrication constraints than in Si devices: Ge donors spaced three times farther apart than in Si show comparable exchange couplings, allowing more space for readout and control gates. In addition, for realistic position uncertainty in donor placement, Ge:P spin couplings have a $33\%$ chance of being within an order of magnitude of the largest coupling, compared with only $10\%$ in Si:P. It is therefore possible that a Ge-based platform would enable fast, parallel, and robust architectures for quantum computing.

Figure 1

Density plot of the Stark-detuned Zeeman coupling $[g\left(E\right)-g_0]{\mu }_{\mathrm{B}}{B}_0$ induced by a nonzero electric field E, aligned with the $\langle 111\rangle$ crystallographic direction, with an angle $\varphi$ between $\mathbf{E}$ and the magnetic field ${\mathbf{B}}_0$. For realistic applied voltages, the detuning can be changed over the range $\{-3$ GHz, 3 $\text{GHz}\}$, allowing for nanosecond selective manipulation of locally detuned spins. $B_0=0.4$ T corresponds to a donor spin coherence $T_2\approx 1$ ms [11].

Figure 2

Distribution of $J$ couplings (on a log scale) of a statistical ensemble of Si:P (top) and Ge:P (bottom) donor pairs, with pair separations ${\mathbf{d}}_0+{\mathbf{d}}_r$, where the nominal separations are ${\mathbf{d}}_0^{\text{Si}}=[20\text{nm},0,0], {\mathbf{d}}_0^{\text{Ge}}=[50/\sqrt{2}\text{nm},50/\sqrt{2}\text{nm},0]$ for Si and Ge, respectively. These combine with a random separation ${\mathbf{d}}_r$ that is uniformly distributed over cubes with edge length 5 nm. These cubes are illustrated in the insets and show all the possible positions of donor partners that could couple to a fixed donor at the origin of the ${\mathbf{d}}_0$ vector. The Ge:P distribution spreads across more orders of magnitude overall, but it is more skewed towards larger couplings.

Figure 3

Comparison of the cumulative distributions of the ensembles of Ge:P and Si:P couplings (on a log scale) across the top four orders of magnitude of $J$. As explained in the text, the larger skew of the Ge:P distribution in Fig. 2 implies higher probabilities for Ge:P couplings to lie within the top two orders of magnitude. Larger couplings are desirable for faster two-qubit gates, and current schemes for uniform control of such gates in a cluster require that $J$ values vary at most within two orders of magnitude in the cluster. In this sense, Ge:P donors provide a significant advantage over Si:P.

Figure 4

Density plots of exchange couplings $J$ between neighboring donor spins as a function of misplacements ${\mathbf{d}}_{r1},{\mathbf{d}}_{r2}$ in the plane orthogonal to the nominal donor separation ${\mathbf{d}}_0$, i.e., ${\mathbf{d}}_0=[20\text{nm},0,0]$ for Si:P (left) and ${\mathbf{d}}_0=[50/\sqrt{2}\text{nm},50/\sqrt{2}\text{nm},0]$ for Ge:P (right). $\{{\mathbf{d}}_{r1},{\mathbf{d}}_{r2}\}$ are sampled with steps of 0.2 nm along each of the two directions; these calculated values are then interpolated for the purpose of illustration. Top panels refer to a situation of no parallel misplacement along ${\mathbf{d}}_0$, bottom panels have a parallel misplacement of 5 nm. It is clear that the two-dimensional oscillatory pattern is preserved intact between bottom and top pictures, as explained in the text. The period of the oscillations is slightly larger in Si:P, as a result of the different conduction band structure. Remarkably, higher $J$ values survive for longer misplacement vectors in the plane for Ge:P than for Si:P.