Simulated coherent electron shuttling in silicon quantum dots

Shuttling of single electrons in gate-defined silicon quantum dots is numerically simulated. A minimal gate geometry without explicit tunnel barrier gates is introduced, and used to define a chain of accumulation mode quantum dots, each controlled by a single gate voltage. One-dimensional potentials are derived from a three-dimensional electrostatic model, and used to construct an effective Hamiltonian for efficient simulation. Control pulse sequences are designed by maintaining a fixed adiabaticity, so that different shuttling conditions can be systematically compared. We first use these tools to optimize the device geometry for maximum transport velocity, considering only orbital states and neglecting valley and spin degrees of freedom. Taking realistic geometrical constraints into account, charge shuttling speeds up to $\sim 300$ m/s preserve adiabaticity. Coherent spin transport is simulated by including spin-orbit and valley terms in an effective Hamiltonian, shuttling one member of a singlet pair and tracking the entanglement fidelity. With realistic device and material parameters, shuttle speeds in the range $10–100$ m/s with high spin entanglement fidelities are obtained when the tunneling energy exceeds the Zeeman energy. High fidelity also requires the interdot valley phase difference to be below a threshold determined by the ratio of tunneling and Zeeman energies, so that spin-valley-orbit mixing is weak. In this regime, we find that the primary source of infidelity is a coherent spin rotation that is correctable, in principle. The results pertain to proposals for large-scale spin qubit processors in isotopically purified silicon that rely on coherent shuttling of spins to rapidly distribute quantum information between computational nodes.

Figure 1

Schematic of a triple linear quantum dot chain using a vertical electrode geometry with no explicit tunnel barrier gates. (a) 3D render of the gate geometry with a plot of a simulated electrostatic potential obtained with a self-consistent Poisson calculation. (b) A 2D top view of the potential with plunger gates outlined. The potential is a 2D slice taken 1-nm below the $\mathrm{Si}/{\mathrm{SiO}}_2$ interface. Darker color indicates a more attractive potential for electrons. The white horizontal dashed line indicates the 1D potential slice used in the shuttling simulations. (c) Side-view of a plunger gate taken along the black line in (b) showing the vertical plunger gate design. The yellow oval indicates electron accumulation in a quantum dot.

Figure 2

A constant-adiabaticity shuttling pulse calculated for the linear triple dot system, with $\xi =0.02$. The electron is initially localized in dot 1 and then shuttled through dots 2 and 3 by sweeping the three plunger gates. When $V_1\approx V_2$ or $V_2\approx V_3$, the detuning between neighboring dots is ${\epsilon }_{i,j}={\epsilon }_i-{\epsilon }_j\approx 0$. The right inset figures show the pulse shapes over the full voltage range, while the main (left) panel is an enlarged view showing the smooth nature of the pulse shape near the upper corners.

Figure 3

Relationship between the adiabatic parameter $\xi$, final orbital state infidelity $1-F\left(T\right)$, and pulse length $T$. Pulses with arbitrary fidelity can be found by reducing $\xi$ at the cost of increased pulse length.

Figure 4

Optimizing the gate electrode geometry for fast shuttling. (a) Shuttling velocity $(G+D)/T$ determined by finding a constant-adiabaticity pulse of duration $T$ for given values of $D$ and $t_c$. The ten curves corresponds to values of $t_c$ ranging from 10 to $100\mu \mathrm{eV}$ in steps of $10\mu \mathrm{eV}$. The shaded (green) region corresponds to the range of ($D$, $t_c$) values achievable in the geometry of (b) with $G\ge 10$ nm. Smaller gap values are considered impractical for realistic device fabrication. (b) Top-down schematic view of the four-electrode model used to calculate $t_c$ as a function of geometrical parameters $G$ and $D$. The two central gates form the double QD used to model shuttling, while the outer gates are set to fixed potentials to make the double dot potential realistic. The dot length $D$ and interelectrode gap $G$ are varied uniformly for all four gates. (c) Dependence of $t_c$ on $D$ and $G$ resulting from a 3D Schrödinger-Poisson calculation of the four-electrode model, with a single electron occupying the central double dot.

Figure 5

Valley-orbit energy spectrum for a double quantum dot with single electron occupation. The Hamiltonian parameters are $|{\mathrm{\Delta }}_L|=200\mu \mathrm{eV}$, $|{\mathrm{\Delta }}_R|=150\mu \mathrm{eV}$, and $\delta \varphi =\pi /3$. The four eigenstates when $\epsilon \ll 0$ are labeled on the left side. The intravalley and intervalley tunnel couplings are labeled $t_{c,+}$ and $t_{c,-}$, respectively.

Figure 6

Shuttling one member of a singlet pair, for $t_c>E_z$. For all panels, the fixed parameters are $\xi =0.005$, $t_c=75\mu \mathrm{eV}$, $E_z=40\mu \mathrm{eV}$, $|{\mathrm{\Delta }}_R|=150\mu \mathrm{eV}$, ${\eta }_1={\eta }_2=2\mu \mathrm{eV}$. (a) Variation of shuttle speed (color scale) with the left QD valley splitting $|{\mathrm{\Delta }}_L|$ and the interdot valley phase difference $\delta \varphi$. These speeds are based on finding constant adiabaticity pulses with $\xi =0.005$. (b) The fidelity of maintaining the spin singlet state versus $|{\mathrm{\Delta }}_L|$ and $\delta \varphi$. Infidelity is plotted in color scale, defined as $1-|\mathrm{Tr}\left[\rho \left(T\right)(I_4\otimes |S\rangle \langle S\left|\right)\right]{|}^2$, where $\rho \left(T\right)$ is the density matrix of the postshuttle state. High (low) fidelity is indicated by dark blue (yellow). (c), (d) Energy spectra of the Hamiltonian in Eq. (3) versus detuning ${\epsilon }_L-{\epsilon }_R$. $|{\mathrm{\Delta }}_L|=200\mu \mathrm{eV}$ in both panels, $\delta \varphi =1$ rad and $\delta \varphi =2$ rad for (c) and (d), respectively. Color indicates the spin state, with red (blue) corresponding to spin down (up). Energy levels are labeled by the corresponding eigenstates on the left when the detuning $\ll 0$ and on the right when the detuning $\gg 0$. Enlarged views near the $t_{c,+}$ (dashed square) and $t_{c,-}$ (dotted square) anticrossings illustrate how SVO mixing varies with $\delta \varphi$.

Figure 7

Error due to single-spin rotation during shuttling. For both panels, $\xi =0.005$, $t_c=75\mu \mathrm{eV}>E_z=40\mu \mathrm{eV}$, $|{\mathrm{\Delta }}_L|=200\mu \mathrm{eV}$, $|{\mathrm{\Delta }}_R|=150\mu \mathrm{eV}$, and ${\eta }_1={\eta }_2=2\mu \mathrm{eV}$. (a) Phase (${\sigma }_z$) rotation of the shuttled spin in the regime $\delta \varphi \in [0,1.67]$ rad. (b) Effect of corrective rotations on the infidelity of the postshuttle state with respect to the singlet, as a function of $\delta \varphi$. (Purple) no corrective rotations are applied; (blue) $R_z\left(\theta \right)$ correction applied; (yellow) $R_z\left(\theta \right)$ and $R_y\left({\theta }^{\prime }\right)$ corrections applied; (green) $R_z\left(\theta \right)$, $R_y\left({\theta }^{\prime }\right)$, and $R_x\left({\theta }^{″}\right)$ corrections applied.