Toward Hole-Spin Qubits in $\mathrm{Si}$ $p$-MOSFETs within a Planar CMOS Foundry Technology

Hole spins in semiconductor quantum dots represent a viable route for the implementation of electrically controlled qubits. In particular, the qubit implementation based on $\mathrm{Si}$ $p$-MOSFETs offers great potentialities in terms of integration with the control electronics and long-term scalability. Moreover, the future down scaling of these devices will possibly improve the performance of both the classical (control) and quantum components of such monolithically integrated circuits. Here, we use a multiscale approach to simulate a hole-spin qubit in a down-scaled $\mathrm{Si}$-channel $p$-MOSFET, the structure of which is based on a commercial 22-nm fully depleted silicon-on-insulator device. Our calculations show the formation of well-defined hole quantum dots within the $\mathrm{Si}$ channel and the possibility of a general electrical control, with Rabi frequencies of the order of $100\mathrm{MHz}$ for realistic field values. A crucial role of the channel aspect ratio is also demonstrated, as well as the presence of a favorable parameter range for the qubit manipulation.

Figure 1

(a) The schematics of the simulated $p$-MOSFET FDSOI transistor. (b) The composition and geometry of the high-$\kappa$ metal-gate stack. (c)–(e) Top (c), side (d), and front (e) views of the device; in the latter, we highlight the nonuniform Gaussian-doping profile in the source and drain.

Figure 2

The confining potential $U(\mathbf{r})$ induced by the top and back gates along the symmetry axes of the silicon channel: (a) $y=z=0\mathrm{nm}$, (b) $x=z=0\mathrm{nm}$, and (c) $x=y=0\mathrm{nm}$. In (a), the shaded areas ($|x|\ge 4.98\mathrm{nm}$) correspond to the source and drain regions, while the dotted lines delimit the region below the nitride spacers ($7.96\mathrm{nm}\ge |x|\ge 4.98\mathrm{nm}$). In (b) and (c), the shaded areas identify the $\mathrm{Si}\mathrm{O}$${}_2$ regions ($|y|\ge 5\mathrm{nm}$ and $|z|\ge 2.5\mathrm{nm}$, respectively). The device temperature in the simulation is $T=2\mathrm{K}$.

Figure 3

A map of the confining potential $U(\mathbf{r})$ at $y=0\mathrm{nm}$, for different values of the gate potential ($V_G=-0.8\mathrm{V}$ and $V_G=-0.6\mathrm{V}$ in the top and bottom panels, respectively). The black contour lines show the charge density ${\rho }_{1,\xi }(\mathbf{r})=|⟨\mathbf{r}|1,\xi ⟩{|}^2$, corresponding to the hole ground states at zero magnetic field, from a value of 0.1 (outer line) to 0.95 (inner line), with an incremental step of 0.05 (all in atomic units).

Figure 4

The Rabi frequencies (a) $f_R^X$ and (b) $f_R^Z$ as a function of the magnetic field intensity $B$ ($\theta ={45}^{\circ }$ and $\varphi =0^{\circ }$). The Rabi frequencies (c) $f_R^X$ and (d) $f_R^Z$ as a function of ${\mathrm{\Delta }}_{\mathrm{SO}}$, ranging from $0.044\mathrm{eV}$ (its actual value in $\mathrm{Si}$, blue squares) to $1\mathrm{eV}$ ($B=1\mathrm{T}$, $V_G=-0.8\mathrm{V}$). The limit ${\mathrm{\Delta }}_{\mathrm{SO}}=\mathrm{\infty }$ (dotted lines) corresponds to calculations performed without including the split-off band.

Figure 5

The dependence of the Rabi frequencies $f_R^X$ (upper panel) and $f_R^X$ (middle), and of $f_R^{XZ}$ (lower panel) on the orientation of the static magnetic field. The plotted quantities are computed by means of the $g$-matrix formalism ($V_G=-0.8\mathrm{V}$, $\delta V_G=10\mathrm{mV}$, and $B=1\mathrm{T}$).

Figure 6

(a) The energy splitting $\mathrm{\Delta }E$ between the logical state $|1,\Uparrow ⟩$ and the lowest nonlogical state $|2,\Downarrow ⟩$ as a function of the magnetic field intensity $B$, for different values of the channel width $W$ (the field is oriented along the $z$ direction). (b) The dependence of $\mathrm{\Delta }E$ on the channel width $W$ at zero magnetic field and $V_G=-0.6\mathrm{V}$. (c) The occupation probabilities of the $lh$, $hh$, and $so$ bands in the ground state in the weak-field limit ($B={10}^{-2}\mathrm{T}$, with $\theta =0^{\circ }$).

Figure 7

The dependence on the channel width $W$ of (a) the $g$ factors and (b) their derivatives with respect to $V_G$, for $V_G=-0.6\mathrm{V}$. The dotted line in (a) gives the values of $g_z$ obtained by neglecting the paramagnetic and diamagnetic contributions in the Hamiltonian ($H_B=H_{B,Z}$).

Figure 8

The dependence on the magnetic field orientation of the Rabi frequencies (a) $f_R^X$, (b) $f_R^Z$, and (c) $f_R^{XZ}$ for different values of the channel width $W$, for $V_G=-0.6\mathrm{V}$, $\delta V_G=10\mathrm{mV}$, $B=1\mathrm{T}$, and $\varphi ={90}^{\circ }$. The plots include both the results derived from the $g$-matrix approach (solid lines) and the numerical results obtained directly from the $\mathbf{k}\cdot \mathbf{p}$ calculations (symbols).