Six-electron semiconductor double quantum dot qubits

We consider a double quantum dot (DQD) qubit which contains six electrons instead of the usual one or two. In this spin qubit, quantum information is encoded in a low-lying singlet-triplet space much as in the case of a two-electron DQD qubit. We find that initialization, manipulation, and readout can be performed similarly to the two-electron case, and that energy gaps remain large enough that these operations can be performed robustly. We consider DQD potentials with parameters chosen to be representative of current experimental capabilities. Results are obtained using two complementary full configuration interaction methods.

Figure 1

DQD potential along $x$ axis, $V(x,0)$, given by Eq. (1) with $L=30\text{nm}$ and $E_0=3.0\text{meV}$. $V$ is the minimum of parabolas centered at $x=\pm L$ with curvature proportional to $E_0^2$.

Figure 2

Convergence of the lowest two $S_z=0$ levels as a function of the cutoff energy using CI-1 and CI-2. CI-2 uses a basis of $n_G=26$ Gaussian functions, arranged on each dot in a 3$\times$3 array with one additional element along each compass direction.

Figure 3

Valence tunneling as a function of half the interdot separation $L$, for $E_0=24.26$, $7.28$, and $1.82\text{meV}$ [shown in (a), (b), and (c), respectively]. A vertical magnetic field $B$ is set such that $\hslash {\omega }_c=2E_0/\sqrt{99}$ (${\omega }_c=eB/m$). Horizontal lines indicate the constant value of the valence tunneling chosen to define the $L$ parameters in each case. Calculations were performed using CI-2 with $n_G=18$ and $E_c=55\text{meV}$.

Figure 4

Lowest 20 six-electron energies vs detuning using CI-1 for small-dot parameters ($E_0=24.26\text{meV}$, $L=20\text{nm}$, $B=2.8\text{T}$) and energy cutoff $E_c=30\hslash {\omega }_c/2=73.15\text{meV}$. The lowest singlet (S0) and lowest triplet (T0) states are labeled. Higher excited states are displayed using solid lines.

Figure 5

Six-electron energies vs detuning using CI-1 for small-dot parameters and energy cutoff $E_c=30\hslash {\omega }_c/2=73.15\text{meV}$, zoomed to each of the three anticrossings (3,3)-(2,4), (2,4)-(1,5), and (1,5)-(0,6) of the lowest singlet state. Only the lowest two singlet states, S0 and S1, and the lowest $S_z=0$ triplet, T0, are shown. The charge sector of the singlet states on either side of the anticrossings is labeled by S($N_1$,$N_2$), where $N_i$ is the number of electrons in dot $i$.

Figure 6

Two-electron energies vs detuning using CI-1 for small-dot parameters ($E_0=24.26\text{meV}$, $L=16.5\text{nm}$, $B=2.8\text{T}$) and energy cutoff $E_c=30\hslash {\omega }_c/2=73.15\text{meV}$. The top panel shows the lowest 20 two-electron energies over a range of detuning; the bottom panel shows a zoomed view of the lowest singlet anticrossing. S0 and S1 label the two lowest singlet state energies, and T0 labels the lowest triplet state energy. The charge sector of the singlet states on either side of the anticrossing is labeled by S($N_1$,$N_2$), where $N_i$ is the number of electrons in dot $i$.

Figure 7

Six-electron energies vs detuning using CI-2 ($n_G=18$, $E_c=55\text{meV}$) for medium-dot parameters ($E_0=7.28\text{meV}$, $L=50\text{nm}$, $B=847\text{mT}$). $S_z=0$ for all states, and labels denote spin (S = singlet, T = triplet) and energy-ordering index.

Figure 8

Six-electron energies vs detuning using CI-2 ($n_G=18$, $E_c=55\text{meV}$) for medium-dot parameters, zoomed to each of the three anticrossings (3,3)-(2,4), (2,4)-(1,5), and (1,5)-(0,6) of the lowest singlet state. $S_z=0$ for all states, and labels denote spin (S = singlet, T = triplet) and energy-ordering index.

Figure 9

Six-electron energies vs detuning using CI-2 ($n_G=10$, no cutoff) for large-dot parameters ($E_0=1.82\text{meV}$, $L=50\text{nm}$, $B=212\text{mT}$). $S_z=0$ for all states, and labels denote spin (S = singlet, T = triplet) and energy-ordering index.

Figure 10

Six-electron energies vs detuning using CI-2 ($n_G=10$, no cutoff) for large-dot parameters, zoomed to each of the three anticrossings (3,3)-(2,4), (2,4)-(1,5), and (1,5)-(0,6) of the lowest singlet state. $S_z=0$ for all states, and labels denote spin (S = singlet, T = triplet) and energy-ordering index.

Figure 11

Energy spectrum of a single dot with six electrons. Dot confinement energy $E_0=1.82\text{meV}$. At a vertical magnetic field $B=212\text{mT}$, the spectrum approximates the (0,6) regime of a DQD with our large-dot parameters. Computed using CI-2 with $n_G=13$ and no energy cutoff. $S_z=0$ for all states, and labels denote spin (S = singlet, T = triplet, Q = quintuplet) and energy-ordering index.

Figure 12

Comparison of the exchange energy vs detuning in 6- and 2-electron DQDs using small-dot parameters (given in Table ). Note that the dot-separation parameter $L$ is different for the two cases in order to equate the valence tunneling of the 6- and 2-electron DQDs as described in Sec. 3 A. We find comparable orders of magnitude, indicating that $z$ rotations of this 6-electron DQD qubit can be controlled in a similar manner to the rotation in the corresponding 2-electron DQD qubit. Calculations were performed with CI-1 using $E_c=30\hslash {\omega }_c/2=73.15\text{meV}$.

Figure 13

Exchange energy of a single dot with confinement energy $E_0$ in magnetic field $B=847\text{mT}$ (the same value used in our set of medium-dot DQD parameters; see Table ) for 2, 4, and 6 electrons. Computed using CI-2 with $n_G=13$ and no energy cutoff.

Figure 14

Comparison of the singlet-triplet energy coupling $h$ in 6-electron (top pane) and 2-electron (bottom pane) DQDs using small-dot parameters (given in Table ). We find comparable order of magnitudes, indicating that $x$ rotations of this 6-electron DQD qubit can be controlled in a similar manner to the rotation in the corresponding 2-electron DQD qubit. Calculations were performed with CI-1 using $E_c=30\hslash {\omega }_c/2=73.15\text{meV}$.

Figure 15

Comparison of the singlet-triplet energy coupling $h$ and exchange energy $J$, vs detuning in the (3,3) region of a 6-electron DQD (top panel) and the (1,1) region of the corresponding 2-electron DQD (bottom panel) using small-dot parameters (given in Table ). The similarity indicates that this 6-electron DQD could be controlled as a qubit in the same way as the corresponding 2-electron DQD.

Figure 16

Comparison of the difference in Coulomb energy [defined as $\int e^2{\left|\Psi \left(r\right)\right|}^2/(r-r_0)dr$] between the singlet and triplet states $50\text{nm}$ away from the DQD for the 6-electron (top) and 2-electron (bottom). Small-dot parameters are used as given in Table and computation was done with CI-1 using $E_c=30\hslash {\omega }_c/2=73.15\text{meV}$.

Figure 17

Exchange energy vs detuning using CI-2 ($n_G=10$) for medium-dot parameters $E_0=7.28\text{meV}$ and $L=50\text{nm}$, and magnetic fields $B=500$, 847, and $1000\text{mT}$. The ability to tune the two exchange “plateaus” in the (2,4) and (0,6) regions with magnetic field is one possible advantage of a 6-electron DQD qubit over its 2-electron counterpart.

Figure 18

Convergence of the lowest singlet energy (S0) of the CI-2 method for a 6-electron DQD in the (3,3) regime (upper panel) and (0,6) regime (lower panel). Small-dot parameters $E_0=24.26\text{meV}$, $L=20\text{nm}$, and $B=2.8\text{T}$ are used and the tilt $\epsilon$ is indicated above each plot.

Figure 19

Lowest singlet level vs detuning of a 6-electron DQD using CI-2 with different numbers of Gaussian basis elements $n_G$. Small-dot parameters are used: $E_0=24.26\text{meV}$, $L=20\text{nm}$, $B=2.8\text{T}$, and for each value of $n_G$ the energy is converged with respect to the cutoff parameter $E_c$. In the 10-Gaussian case, the Gaussian's centers are arranged 5 per dot in a “plus sign” pattern, with one Gaussian in the center of the dot and one along each of the $+/-$ $x$ and $+/-$ $y$ axis directions. In the 18-Gaussian case, the centers are arranged in 3$\times$3 grids on each dot, and in the 26-Gaussian case in addition to a 3$\times$3 grid there is a Gaussian even farther out along each of the $+/-$ $x$ and $+/-$ $y$ directions.

Figure 20

Convergence of the lowest two energies of the CI-1 method for a 6-electron DQD using the “medium” (upper panel) and “large” (lower panel) dot parameters of Table with zero tilt ($\epsilon =0$).

Figure 21

Convergence of the lowest singlet energy (S0) of the CI-2 method for a 6-electron DQD in the (3,3) regime (upper panel) and (0,6) regime (lower panel). Medium-dot parameters $E_0=7.28\text{meV}$, $L=35\text{nm}$, and $B=847\text{mT}$ are used and the tilt $\epsilon$ is indicated above each plot.

Figure 22

Convergence of the lowest singlet energy (S0) of the CI-2 method for a 6-electron DQD in the (3,3) regime (upper panel) and (0,6) regime (lower panel). Large-dot parameters $E_0=1.82\text{meV}$, $L=60\text{nm}$, and $B=212\text{mT}$ are used and the tilt $\epsilon$ is indicated above each plot.

Figure 23

Comparison of six-electron energies produced by the two different CI implementations CI-1 and CI-2 when CI-2 uses $n_G=10$ $s$-type Gaussians in the upper plot and $n_G=26$ in the lower plot. Small-dot parameters ($E_0=24.26\text{meV}$, $L=20\text{nm}$, $B=2.8\text{T}$) are used.

Figure 24

Polysilicon gate pattern of the device used as a representative example throughout this appendix. This image was obtained by tracing an SEM image of an actual device, and so shapes are indicative of the actual gate shapes in the device after nearly all processing has been completed.

Figure 25

Gate voltages used to obtain a few-electron double quantum dot using the example device (refer to the gate labels of Fig. 24). Dot shape and occupation are solved for using a semiclassical Thomas-Fermi solver, QCAD.

Figure 26

A cut along the double quantum dot axis showing the QCAD-generated potential along with the double parabolic DQD potential that results from the fit parameters $E_0=3.5\text{meV}$ and $L=55\text{nm}$.

Figure 27

The $S_z=0$ energy spectrum of a 6-electron DQD with $E_0=3.5\text{meV}$ and $L=55\text{nm}$. Lines are labeled by a letter S, T, or D indicating whether the state is a $S=0$ singlet, a $S=1$ triplet, or a $S=2$ quintuplet, respectively, followed by an index indicating the energy ordering. Insets (a), (b), (c), and (d) show magnified views in the middle of each charge sector (3,3), (2,4), (1,5), and (0,6), respectively. The ground state is a singlet in (2,4) and (0,6) charge sectors, and the energy splittings are so small in the (3,3) and (1,5) sectors that the ground state spin cannot be determined.