Simulation of Si:P spin-based quantum computer architecture

We present a systematic and realistic simulation for single and double phosphorous donors in a silicon-based quantum computer design. A two-valley equation is developed to describe the ground state of phosphorous donors in a strained silicon quantum well (QW), with the central cell effect treated by a model impurity potential. The valley splitting of the donor ground state as a function of QW width or donor position is calculated and a comparison with the valley splitting of the lowest QW states is presented. Oscillation of the valley splitting is observed as the QW width or donor position is varied at atomic scale. We find that the increase of quantum well confinement leads to shrinking charge distribution in all three dimensions. Using an unrestricted Hartree-Fock method with generalized valence bond (GVB) single-particle wave functions, we are able to solve the two-electron Schrödinger equation with quantum well confinement and realistic gate potentials. The lowest singlet and triplet energies and their charge distributions for a neighboring donor pair in the quantum computer (QC) architecture are obtained at different gate voltages. The effects of QW width, gate voltages, donor separation, and donor position shift are calculated and analyzed. The gate tunability and gate fidelity are defined and evaluated, for a typical QC design. Estimates are obtained for the duration of spin-half-swap gate operation and the required accuracy in voltage control. A strong exchange oscillation is observed as both donors are shifted along [001] axis but with their separation unchanged. Applying a gate potential tends to suppress the oscillation. The exchange oscillation as a function of donor separation along [100] axis is found to be completely suppressed as the donor separation is decreased. The simulation presented in this paper is of importance to the practical design of an exchange-based silicon quantum computer.

Figure 1

(Color online) The composite three-spin “universal exchange” qubit of Si:P donors in a Kane-type architecture with the integrated SET readout. The heterolayers are grown along $z\Vert \left[001\right]$ crystal axis. From bottom to top, the heterostructure cross section consists of a thick, $n$-doped ground layer, a $28\mathrm{nm}$ undoped ${\mathrm{Si}}_{0.7}{\mathrm{Ge}}_{0.3}$ tunnel barrier, a $10\text{−}\mathrm{nm}$ Si quantum well (with phosphorous donor array embedded in the middle), a $7\mathrm{nm}$ undoped ${\mathrm{Si}}_{0.7}{\mathrm{Ge}}_{0.3}$ tunnel barrier, a $5\mathrm{nm}$ ${\mathrm{Si}}_3{\mathrm{N}}_4$ layer and lithographically patterned metallic top gates. The P donors are lined along the $\mathrm{x}\Vert \left[100\right]$ axis of silicon crystal. Accordingly, the $y$ axis is parallel to the [010] crystal axis. The logic qubit is represented by the two-dimensional subspace $\mid \mathbf{S},{\mathbf{S}}_{\mathbf{z}}⟩=\mid \frac{1}{2},\frac{1}{2}⟩$ of the three neighboring donor spins, with logic zero $\mid 0_L⟩=\mid S⟩\mid \uparrow ⟩$ and logic one $\mid 1_L⟩=\sqrt{2∕3}\mid T_{+}⟩\mid \downarrow ⟩-\sqrt{1∕3}\mid T_0⟩\mid \uparrow ⟩$. In this paper, we only activate the gate electrodes labeled by $V_{01}$, $V_{12}$, and $V_{23}$, to control the exchange coupling between donors 1 and 2. Furthermore, we set for the central gate $V_{12}\equiv V_c$ and the side gates $V_{01}=V_{23}\equiv V_g$.

Figure 2

(Color online) The energies of two lowest states of a single Si:P donor and their splitting as a function of quantum well width. For convenience, the valley splitting is plotted in negative values.

Figure 3

(Color online) The comparison of the averaged 1D charge distribution of Si:P ground state in a quantum well with width of $6\mathrm{nm}$ and $10\mathrm{nm}$. (a) and (b) show the charge distribution along the $z$ axis and the $x$ axis, respectively. The inset of (b) shows the charge distribution along the $y$ axis.

Figure 4

(Color online) The valley splitting of (a) Si:P donor ground state and (b) lowest quantum well state, as a function of quantum well width. Lines connecting calculating points are used as guides to the eyes. The dotted lines connect points with a spacing of $a_0∕2$. The inset of (a) shows the valley splitting of Si:P ground state as a function of donor position shift from the quantum well central plane for a QW width fixed at $9\mathrm{nm}$.

Figure 5

(Color online) The gate potential (measured in units of ${\mathrm{Ry}}^{*}$) profile at different gate voltages: (a) $V_c=-2.1\mathrm{V}$, (b) $V_c=-0.8\mathrm{V}$, (c) $V_c=-0.6\mathrm{V}$, and (d) $V_c=-3.2\mathrm{V}$. $V_g=-1.8$ for all these four cases. In each figure, the potential shape is plotted along the $x$ axis, with three different distances (15, 17, and $19\mathrm{nm}$) from the top gates. The doping plane in our QC device is $17\mathrm{nm}$ below the top gates.

Figure 6

(Color online) The exchange coupling of a donor pair as a function of quantum well width. The donor separation is fixed at $10a_B^{*}$. In the main body of this figure the plotted quantum well width does not have its boundaries exactly at lattice sites, as the quantum well width often fluctuates due to complicate interface conditions. The inset shows the change of exchange coupling as the QW width is varied at atomic scale, where the calculating points are spaced by $a_0∕4\simeq 1.36\mathrm{\AA }$.

Figure 7

(Color online) (a) The exchange coupling of a donor pair separated by $10a_B^{*}$ as a function of the central gate voltage. The three curves correspond to three different values of side gate voltages $V_g=-1.4$, $-1.8$, and $-2.0\mathrm{V}$, respectively. The isolated point (filled square) on the $V_c=0$ axis corresponds to the exchange coupling without any gate potential $(V_c=V_g=0)$. (b) The dependence of exchange coupling on central gate voltages with $V_g$ fixed at $-1.8\mathrm{V}$. The three curves correspond to three different donor separations $R=8$, 10, and $12a_B^{*}$, respectively. Both (a) and (b) are plotted in logarithmic scale for the exchange coupling. The quantum well width is fixed at $10\mathrm{nm}$.

Figure 8

(Color online) The averaged 1D two-electron charge distribution of singlet (solid) and triplet (dotted) states along the interdonor axis for different gate voltages. The side gate voltage is $V_g=-1.8\mathrm{V}$ for (b), (c), and (d). The charge distributions for singlet and triplet almost coincide for weakly coupled donor pairs as shown in (a) and (d). The gate-off exchange coupling is very close to the gate-on case with $V_c=-2.1\mathrm{V}$ and $V_g=-1.8\mathrm{V}$, while their charge distributions are apparently different as shown in (a). It is clearly demonstrated that the gate voltages can tune the overlap between neighboring donor electron wave functions and, consequently, the exchange coupling.

Figure 9

(Color online) The exchange coupling as a function of donor position shift away from the quantum well central plane: (a) gate off, (b) gate on ($V_c=-1.6\mathrm{V}$ and $V_g=-1.8\mathrm{V}$). Lines connecting calculating points are used as guides to the eyes. The donor pair remains on the [100] axis, with their separation fixed at $10a_B^{*}$. The inset of (a) plots the donor electron wave function overlap (for the singlet state) as a function of donor position shift, which shows the same pattern as the exchange coupling. The dotted line in (b) is an exponential fitting curve, illustrating the significant effect of gate potential on the exchange coupling for donors at different distances from top gates.

Figure 10

(Color online) The exchange coupling (plotted in logarithmic scale) as a function of donor separation. The donor pair remains on the [100] axis, with their separation varied from 70 to 90, in units of half lattice constant, $a_0∕2$. The solid line connecting calculating points are used as guides to the eyes. The dotted line is an exponential fitting curve, appearing as a straight line in the logarithmic-scale plotting.