Quantum dot spin qubits in silicon: Multivalley physics

Research on Si quantum dot spin qubits is motivated by the long spin coherence times measured in Si, yet the orbital spectrum of Si dots is increased as a result of the valley degree of freedom. The valley degeneracy may be lifted by the interface potential, which gives rise to a valley-orbit coupling but the latter depends on the detailed structure of the interface and is generally unknown a priori. These facts motivate us to provide an extensive study of spin qubits in Si double quantum dots, accounting fully for the valley degree of freedom and assuming no prior knowledge of the valley-orbit coupling. For single-spin qubits, we analyze the spectrum of a multivalley double quantum dot, discuss the initialization of one qubit, identify the conditions for the lowest energy two-electron states to be a singlet and a triplet well separated from other states, and determine analytical expressions for the exchange splitting. For singlet-triplet qubits, we analyze the single-dot spectrum and initialization process, the double-dot spectrum, singlet-triplet mixing in an inhomogeneous magnetic field, and the peculiarities of spin blockade in multivalley qubits. We review briefly the hyperfine interaction in Si and discuss its role in spin blockade in natural Si, including intravalley and intervalley effects. We study the evolution of the double-dot spectrum as a function of magnetic field. We address briefly the situation in which the valley-orbit coupling is different in each dot due to interface roughness. We propose a new experiment for measuring the valley splitting in a single quantum dot. We discuss the possibility of devising other types of qubits in Si QDs, and the size of the intervaley coupling due to the Coulomb interaction.

Figure 1

(Color online) Conduction-band valleys in silicon in the bulk and in 2D. (a) In bulk Si there are six equivalent conduction-band minima positioned along the Cartesian axes. (b) Uniaxial strain at the interface causes the valleys parallel to the interface (i.e., in the $xy$ plane here) to be higher by several tens of millielectron volt than the valleys perpendicular to the interface. The interface potential further splits the lower-energy valleys by an amount $\Delta$, which present experimental work is striving to determine unambiguously. Based on measurements to date we expect $\Delta <1\text{meV}$.

Figure 2

(Color online) Two-electron energy levels on a single dot at finite $\Delta$ and magnetic field such that $2\left|\Delta \right|>E_Z$ as introduced in Ref. 83. In this case the lowest-energy state is the singlet $S_{--}$, followed by the triplet $T_{+-}^{\uparrow \uparrow }$, the degenerate singlet $S_{+-}$/triplet $T_{+-}^{\uparrow \downarrow }$ and triplet $T_{+-}^{\downarrow \downarrow }$, and finally the singlet $S_{++}$. Electron-spin orientations in $T$ are indicated by arrows.

Figure 3

Energy-level spectrum for a $\text{Si}/{\text{SiO}}_2$ DQD with $a=8.2\text{nm}$, $b=3\text{nm}$, $d=2.45$ and $\left|\tilde{\Delta }\right|=0.1\text{meV}$, $E_Z=0.05\text{meV}$, $\tilde{t}=0.02\text{meV}$. The solid lines represent singlet and unpolarized triplet levels $\tilde{S}$ and ${\tilde{T}}^0$ while the dashed lines represent polarized triplet levels ${\tilde{T}}^{+}$ and ${\tilde{T}}^{-}$. The top and bottom anticrossings each consist of two singlets and one triplet In the middle anticrossing each of the three dashed lines represents a degenerate singlet/triplet level.

Figure 4

Energy-level spectrum for a $\text{Si}/{\text{SiO}}_2$ DQD with $a=8.2\text{nm}$, $b=3\text{nm}$, $d=2.45$ and $\left|\tilde{\Delta }\right|=0.01\text{meV}$, $E_Z=0.1\text{meV}$, $\tilde{t}=0.02\text{meV}$. Here the valley splitting $\left|\tilde{\Delta }\right|$ has been set as the smallest energy scale, the Zeeman energy as the largest scale, and the tunnel coupling in between. This figure illustrates the opposite scenario to Fig. 3, showing the qualitatively different structure of the energy spectrum when $\left|\tilde{\Delta }\right|<\tilde{t}$.

Figure 5

Energy-level spectrum for a $\text{Si}/{\text{SiO}}_2$ DQD with $a=8.2\text{nm}$, $b=3\text{nm}$, $d=2.45$ and $\left|\tilde{\Delta }\right|=0.05\text{meV}$, $E_Z=0.05\text{meV}$, $\tilde{t}=0.1\text{meV}$. In this figure the tunnel coupling has been set as the largest energy scale. Although the magnitude of $\tilde{t}$ in this graph has been exaggerated for clarity and exceeds what one expects to measure experimentally, this figure illustrates the complications inherent in experiments seeking to distinguish parameters of comparable magnitude.

Figure 6

Energy level spectrum for a $\text{Si}/{\text{SiO}}_2$ DQD with $a=8.2\text{nm}$, $b=3\text{nm}$, $d=2.45$ and $\left|\tilde{\Delta }\right|=0.01\text{meV}$, $E_Z=0.2\text{meV}$, $\tilde{t}=0.02\text{meV}$. In this figure the Zeeman energy has been set as the largest energy scale and the valley splitting as the lowest. Notice the qualitative similarity of this figure to Fig. 3. As the magnetic field increases, in effect the Zeeman field and the valley-orbit coupling trade places.

Figure 7

Energy-level spectrum for a $\text{Si}/{\text{SiO}}_2$ DQD with $a=8.2\text{nm}$, $b=3\text{nm}$, $d=2.45$ and $\left|\tilde{\Delta }\right|=0.01\text{meV}$, $E_Z=0.05\text{meV}$, $\tilde{t}=0.2\text{meV}$. This figure illustrates the case when the tunnel coupling is the largest energy scale in the problem.