Disorder-induced valley-orbit hybrid states in Si quantum dots

Quantum dots in silicon are promising candidates for the implementation of solid-state quantum information processing. It is important to understand the effects of the multiple conduction band valleys of silicon on the properties of these devices. Here we present a systematic effective mass theory of valley-orbit coupling in disordered silicon systems. This theory reveals valley-orbit hybridization effects that are detrimental for storing quantum information in the valley degree of freedom, including nonvanishing dipole matrix elements between valley states and altered intervalley tunneling.

Figure 1

Calculation of low-lying energy eigenstates and electric dipole moments for a three-dimensional (3D) quantum dot in a quantum well with interface disorder, demonstrating that valley-orbit mixing induces a substantial dipole moment. The simulation geometry (inset) has a quantum well thickness of 10 nm and a barrier height of 150 meV. The parabolic quantum dot is circular in the $x-y$ plane, with a diameter of $L=28.3$ nm. Disorder is introduced as a rectangularly shaped bump (black region) in the quantum well barrier, with an $x$ width of $W_x=2L$, a $y$ width of $W_y=4L$, a height of a single atom, and a center position at $(x_0,y_0)=(-0.7L,-0.7L)$. The top two curves show the components of the dipole moment $\mathbf{p}$, defined in Eq. (5), along the $\widehat{x}$ (dashed) and $\widehat{y}$ (dash-dotted) directions, as a function of the electric field applied along $\widehat{z}$. The lower set of solid curves shows the lowest six energy levels in the quantum dot, measured relative to the ground state. At low fields, these form two manifolds: a lower, $S$-like doublet and an upper $P$-like quadruplet. Disorder introduces two distinct effects: valley mixing (VM) between states in the same valley doublet, and valley-orbit hybridization (VOH) between states in different valley doublets. The dipole moment is typically substantial, but it is suppressed near the VM-induced anticrossing indicated by the vertical dashed line. A second anticrossing occurs at a higher field, where the first excited state changes from valley-like to orbital-like, and is accompanied by a large change in the dipole moment.

Figure 2

The intervalley tunnel rate ${\tau }_{-+}=\langle L_{-}\left|H\right|R_{+}\rangle$ between two sides of a double quantum dot in the presence of a bump at the quantum well interface, as a function of the bump position $x_0$. Here, the height of the bump is one atom, $L_{-}$ refers to the lowest left-localized state, and $R_{+}$ refers to the first excited right-localized state. A schematic of the calculation geometry is shown in the inset. In the absence of disorder, the $\pm$ indices refer to unperturbed valley states. Results are shown for bumps with widths $W_x=L/2$, $L$, and $2L$ in the $x$ direction, and infinite widths in the $y$ direction. The solid curves are computed using the disorder-expansion framework described in the main text, with a basis of size $50\left(x\right)\times 1\left(y\right)\times 10\left(z\right)$, while the dashed line is an alternative result for $W=2L$, with a basis of size $1\left(x\right)\times 1\left(y\right)\times 2\left(z\right)$, which does not admit VOH effects. For the calculations shown here, the dimensions of the individual dots are the same as in Fig. 1, and we assume an electric field of $2\times {10}^5$ V/m applied perpendicularly to the quantum well. The intervalley tunneling rate is substantial over a wide range of bump positions and widths.

Figure 3

(a) Comparison of disorder-expansion calculations for a dipole moment with 2D tight-binding calculations. A bump one atom high in $\widehat{z}$ with a width in $\widehat{x}$ of 300 atoms is centered at the lateral position $x_0$. As in the main text, an electric field is applied along the $\widehat{z}$ direction with strength $2\times {10}^5$ V/m, the dot has width $L=28.3$ nm, and the quantum well is 10 nm thick with barrier height 150 meV. The solid curve is the tight-binding result, while the dashed lines correspond to different numbers of $z$-basis functions: the most inaccurate curve is 2, second most is 10, third most is 20, and and most accurate is 40. In all cases, five $x$-basis functions were used. Assuming the tight binding results reflect exact solutions, the inset shows the percent error in the left peak of the main plot as a function of $m_z$, the number of $z$-basis functions used. The curve fit for large $m_z$ indicates that the percent error scales like $m_z^{-1.005\pm 0.007}$ for large $m_z$. The cross ($\times$) indicates the percent error obtained using eight $z$ functions from an augmented basis set described in Appendix . (b) Comparison of disorder-expansion calculations for intervalley tunneling with 2D tight-binding calculations. The disorder and system parameters are identical to panel (a), except that two dots are used, and are separated by a distance $d=150.3$ nm. As in panel (a), the solid line indicates tight-binding results, while the dashed lines correspond to successively more $z$-basis functions. Even though tunneling is very sensitive to small tails of the wave functions along $\widehat{x}$, the 35 $x$-basis functions used here are sufficient to ensure stability such that the number of $z$-basis functions used limits the accuracy.