Spin-Orbit Coupling, Quantum Dots, and Qubits in Monolayer Transition Metal Dichalcogenides

We derive an effective Hamiltonian that describes the dynamics of electrons in the conduction band of monolayer transition metal dichalcogenides (TMDC) in the presence of perpendicular electric and magnetic fields. We discuss in detail both the intrinsic and the Bychkov-Rashba spin-orbit coupling induced by an external electric field. We point out interesting differences in the spin-split conduction band between different TMDC compounds. An important consequence of the strong intrinsic spin-orbit coupling is an effective out-of-plane $g$ factor for the electrons that differs from the free-electron $g$ factor $g\simeq 2$. We identify a new term in the Hamiltonian of the Bychkov-Rashba spin-orbit coupling that does not exist in III-V semiconductors. Using first-principles calculations, we give estimates of the various parameters appearing in the theory. Finally, we consider quantum dots formed in TMDC materials and derive an effective Hamiltonian that allows us to calculate the magnetic field dependence of the bound states in the quantum dots. We find that all states are both valley and spin split, which suggests that these quantum dots could be used as valley-spin filters. We explore the possibility of using spin and valley states in TMDCs as quantum bits, and conclude that, due to the relatively strong intrinsic spin-orbit splitting in the conduction band, the most realistic option appears to be a combined spin-valley (Kramers) qubit at low magnetic fields.

Popular Summary

Monolayers of transition metal dichalcogenides (TMDC), such as ${\mathrm{MoS}}_2$, have recently emerged as a new class of materials, both of fundamental scientific interest and as potential platforms for nanoelectronic, spintronic, and optoelectronic devices. Unlike graphene, the so-far best-known atomically thin material, these monolayers have in their electronic structures an energy gap (the so-called “band gap”) of a size comparable to the photonic energies of the visible light. This optically relevant band gap creates an extremely interesting fundamental possibility: a new platform of quantum dots in truly two-dimensional materials that can be created by use of external electrical gates. The low dimensionality of such quantum dots should offer new opportunities for miniaturization. Moreover, these dots can be used for applications in both electronics and optics—a dual functionality that has been missing in the existing quantum-dot platforms.

In this theoretical paper, we take the first few fundamental steps in this new direction, including a necessary theoretical framework for studying quantum dots in monolayer TMDCs.

Confinement of charge carriers (for example, electrons) within a quantum dot leads to a spectrum of localized electronic states of discrete energies, which can then be exploited for nanoelectronics, optoelectronics, and also as qubits in quantum computing. The physics of a quantum dot in a monolayer TMDC has a new twist: Electrons in the material show significant coupling between their spins and their spatial motion (i.e., spin-orbit coupling). In the presence of a magnetic field, the interplay between this coupling and the field allows the spectrum of the discrete electronic states to be modified magnetically. We have provided a concrete theoretical model that enables the determination of the spectrum in a material-dependent manner. As an application of this framework, we have identified, in the spectrum of a generic monolayer TMDC, a number of two-level systems that can be manipulated with small magnetic fields and thus serve as qubits.

The proposed new quantum-dot platform exhibits rich physics, and its fabrication should be within the current experimental reach. We expect our work to motivate further studies along this direction, into topics such as possible mechanisms for qubit control, relaxation processes, and the role of electron-electron interaction.

Figure 1

Schematics of a QD defined with the help of four top gates in a monolayer TMDC. S and D denote the source and the drain, respectively.

Figure 2

Spin-resolved band structure of ${\mathrm{MoS}}_2$ from DFT calculations. The qualitative features of the band structure are the same for all TMDCs. An enlargement of the region in the black frame is shown in the upper panel of Fig. 3.

Figure 3

Upper panel: Spin-split DFT CB of ${\mathrm{MoS}}_2$ in the vicinity of the $K$ point, which is indicated by a vertical dashed line. Lower panel: The same for ${\mathrm{WS}}_2$. A band crossing, which can be seen in the case of ${\mathrm{MoS}}_2$, is absent for ${\mathrm{WS}}_2$. The small asymmetry in the figures with respect to the $K$ point, especially in the case of the band-crossing points in the upper panel, is due to the fact that the calculations were performed along the $\mathrm{\Gamma }KM$ line.

Figure 4

(a) Spectrum of a ${\mathrm{MoS}}_2$ QD of radius $R_d=40\mathrm{nm}$ as a function of the perpendicular magnetic field $B_z>0$. Black (purple) lines: spin $\downarrow$ ($\uparrow$) in the $K$ valley; red (blue) lines: spin $\uparrow$ ($\downarrow$) in the $K^{\prime }$ valley. States up to $|l|=2$ and $n=2$ are shown. (b) Part of the spectrum shown in (a) for small magnetic fields and low energies. Labels show the valley, orbital quantum number $l$, and spin state for each level. The values of $m_{\text{eff}}^{\tau ,s}$, $g_{\mathrm{vl}}$, and $g_{\mathrm{sp}}^{\perp }$ used in the calculations can be found in Tables 1 and 2.

Figure 5

Spectrum of a 40 nm ${\mathrm{WS}}_2$ QD as a function of the perpendicular magnetic field $B_z>0$. Black (red) lines show the spin $\uparrow$ ($\downarrow$) states from valley $K$ ($K^{\prime }$). The values of $m_{\text{eff}}^{\tau ,s}$ can be found in Table 1, whereas $g_{\mathrm{vl}}=1.6$ and $g_{\mathrm{sp}}^{\perp }=1.99$ (see Table 2).

Figure 6

Spectrum of a 40 nm ${\mathrm{WS}}_2$ QD as a function of the perpendicular magnetic field $B_z>0$. The values of $m_{\text{eff}}^{\tau ,s}$ can be found in Table 1 and we used $g_{\mathrm{vl}}=2.31$ and $g_{\mathrm{sp}}^{\perp }=1.84$ (cf. Fig. 5). Black (red) lines show spin $\uparrow$ ($\downarrow$) states from the $K$ ($K^{\prime }$) valley.