Spin-valley qubits in gated quantum dots in a single layer of transition metal dichalcogenides

There is currently interest in developing quantum circuits based on electron spin qubits Brum and Hawrylak (1997); Loss and DiVincenzo (1998); Tarucha et al. (1996); Ciorga et al. (2000); Meier et al. (2003); Elzerman et al. (2004); Hanson et al. (2007); Kyriakidis et al. (2002); Klinovaja and Loss (2013) in gated quantum dots in gallium arsenide and silicon Hsieh et al. (2012); Nowack et al. (2011); Botzem et al. (2018); West et al. (2019); Ito et al. (2018). In these structures, electrons are localised in a volume containing millions of atoms, hence the nuclear spins and atomic vibrations contribute to the decoherence of electron spins. Recent realization of semiconductor layers with atomic thickness Castro Neto et al. (2009); Kadantsev and Hawrylak (2012); Geim and Grigorieva (2013); Ihn et al. (2010); Mak et al. (2010); Manzeli et al. (2017); Güçlü et al. (2014); Scrace et al. (2015); Tu et al. (2019); Peng et al. (2019); Klein et al. (2019); Jadczak et al. (2019) opens the possibility of confining single electrons to a few-atom thick layers, potentially significantly increasing the operating temperature and the coherence of electron spin qubits.

Recently, quantum dots (QDs) in transition metal dichalcogenides (TMDCs), graphene, and bilayer graphene have been realized Güttinger et al. (2010); Güçlü et al. (2014); McGuire (2016); Wang et al. (2017, 2018); Pisoni et al. (2018) by creating electrostatic confinement with lateral metal electrodes Volk et al. (2011); Allen et al. (2012); Eich et al. (2018); Pisoni et al. (2018); Wang et al. (2018); Kurzmann et al. (2019). Several groups reported the creation of finite-size electron droplets using metallic gates Wang et al. (2018); Pisoni et al. (2018); Brotons-Gisbert et al. (2019). Gated quantum dots combined with large trion binding energies allowed for electrical probing of excitons in TMDC QDs Lu et al. (2019); Pisoni et al. (2018); Wang et al. (2018); Brotons-Gisbert et al. (2019); Chakraborty et al. (2018). For example, a local tunable confinement potential has been realized by Kim and co-workersWang et al. (2018), and gate tuning of QD molecules have been shown by Guo and co-workers Zhang et al. (2017). There has also been significant progress in theoretical understanding of TMDC QDs. Stability and electronic properties of small QDs with various composition, orientation, and edge type have been studied within DFT (Density Functional Theory) Lauritsen et al. (2003); Pei et al. (2015); Javaid et al. (2017); Lauritsen et al. (2007); McBride and Head (2009); Li and Galli (2007). In particular, Galli and co-workers Li and Galli (2007) studied the electronic properties of triangular MoS${}_{\textrm{2}}$ quantum dots as a function of the number of layers and predicted a transition to a direct gap semiconductor in a single layer.

Nevertheless, ab-initio approaches are limited to small structures, and to describe quantum dots with lateral sizes up to tens of nanometers, one can make use of hybrid DFT based tight-binding modelsLiu et al. (2013); Rostami et al. (2013); Cappelluti et al. (2013); Zahid et al. (2013); Fang et al. (2015); Ho et al. (2014); Liu et al. (2015); Ridolfi et al. (2015); Shanavas and Satpathy (2015); Silva-Guillen et al. (2016); Pearce et al. (2016); Dias et al. (2018). Using a 3-band tight-binding model limited to metal orbitals, Peeters, and co-workers analyzed the effect of quantum dot shape and external magnetic field on the single-particle energy spectrum Pavlović and Peeters (2015); Chen et al. (2018). Using an atomistic tight-binding approach, spin-valley qubits have been described in small quantum dots by Bednarek and co-workers Pawłowski et al. (2018); Pawłowski (2019), Szafran and co-workers Zebrowski et al. (2013); Szafran et al. (2018); Szafran and Zebrowski (2018) and Guinea and co-workers Chirolli et al. (2019). Using such an approach, two valley-qubit operations have also been recently proposed by some of us Pawłowski et al. (2021). In order to understand the size dependence of the electronic states in quantum dots for realistic sizes involving millions of atoms, $k\cdot p$ and effective massive Dirac fermion models were also applied Kormányos et al. (2014); Liu et al. (2014); Brooks and Burkard (2017); Széchenyi et al. (2018); Qu et al. (2017); Dias et al. (2016).

In our previous work, an ab-initio based tight-binding model combining metal and chalcogen orbitals, applicable to multi-million atom quantum dots in TMDCs, has been developed Bieniek et al. (2018). We note that in a tight binding model the correct level degeneracies occur, but their direct identification with valleys is difficult. By working in reciprocal space, the valleys were explicitly taken into account. The effect of valley, spin, and band nesting on the electronic properties of gated quantum dots in a single layer of transition metal dichalcogenides was described Bieniek et al. (2020), along with valley- and spin-polarized broken-symmetry many-body states discussed in Ref. Szulakowska et al., 2020. It was shown that the lowest electronic state confined in a quantum dot is a doublet of spin and valley locked states. Hence, such a doublet could serve as a qubit. In order to realize a spin-valley qubit, a way to control spin and valley properties of electrons in these QDs is needed. Several means of manipulating the valley index in quantum dots have been already studied: strain Chirolli et al. (2019), magnetic field Kormányos et al. (2014); Brooks and Burkard (2017); Széchenyi et al. (2018), and coupling to impurity Széchenyi et al. (2018). Valley mixing by the confining potential has also been analyzed by Yao and co-workers Liu et al. (2014) and the magnetic control of the spin-valley coupled states in TMDC QDs has been shown by Qu and co-workers Qu et al. (2017); Dias et al. (2016).

In this paper, building on our previous work, we expand our microscopic theory of electron spin-valley qubits and provide a prescription of how to manipulate the two qubit states. The microscopic tight-binding model developed here is suitable for accurate description of multi-million atom nanostructures compatible with existing experiments. The two degenerate qubit states, belonging to the two non-equivalent valleys, each with the opposite-spin, are built out of conduction band states of even parity with respect to the metal plane. The rotation of the qubit, the logical $\sigma_{x}$ operation, requires simultaneous transition between opposite spin states in each valley and between the two nonequivalent valleys. The understanding of the orbital composition of conduction band states as a linear combination of even parity metal orbitals and even parity sulfur dimer orbitals allows us to show that the qubit rotation is accomplished by applying both a time dependent vertical electric field and a time dependent highly localized lateral potential. The electric field couples primarily to the two sulfur layers, and activates odd conduction bands, which enables in turn spin flips on metal atoms due to the spin-orbit interaction. The admixture of an opposite-spin orbital and application of a lateral local potential enables transition to the opposite valley and spin qubit state. This process is illustrated in Figure 1. Figure 1 shows a cross section of a schematic device consisting of a single TMDC layer, with metallic gates (shown in yellow ) producing a lateral potential confining a single electron to a quantum dot in a single TMDC layer , illustrated with a thick arrow below. In addition, a metallic vertical gate, implemented here with two graphene layers, generates an on demand vertical electric field. The local gate, implemented here with an STM (Scanning Tunneling Microscope) tip, generates an on demand valley mixing potential. The suggested set up shown in Figure 1 is compatible with experimental designs and implementation of gated quantum dot in a single layer of WSe2 (Boddison-Chouinard et al., 2021). We will show that turning these two gates on for a finite time rotates spin valley qubit from logical qubit 0 to logical qubit 1.

The paper is organized as follows. In section 2, we describe logical quantum bits encoded in two lowest degenerate states of an electron confined in a lateral gated quantum dot in TMDC. In section 3, we describe the effect of two external gates allowing for flipping of the spin and flipping of the valley, necessary for logical qubit quantum operations. In section 4, we summarise our results.

Here we identify and analyse the logical quantum bits encoded in quantum states of an electron in a gated quantum dot in a single layer of TMDC as shown in Figure 1. This is necessary, since valleys and the spin-orbit coupling prevent us from identifying qubits with electron spin states only.

Following Ref. Bieniek et al. (2020), the Hamiltonian of an electron in a single layer of TMDC is a sum of the bulk Hamiltonian $H_{b}$ and quantum-dot confinement potential $V_{QD}$ Bieniek et al. (2018, 2020). The potential $V_{QD}(\vec{r})$ is approximated here by a Gaussian potential $V_{QD}(\vec{r})=-V_{0}\exp\left(-r^{2}/R_{QD}^{2}\right)$, where $V_{0}$ is the potential depth and $R_{QD}$ is the quantum dot radius. The electron quantum dot wavefunction $|\Phi^{s}\rangle$ for the electron state $s$ satisfies the Schrödinger equation Bieniek et al. (2018, 2020):

$$(H_{b}+V_{QD}(\vec{r}))\left|\Phi^{s}\right\rangle=E^{s}\left|\Phi^{s}\right\rangle.$$

(1)

As explained in Ref. Bieniek et al., 2020, we define a large computational rhombus consisting of millions of metal atoms (sublattice A), and two layers of upper and lower chalcogen atoms (sublattice B). We retain only even metal orbitals and form an even combination of the upper and lower chalcogen $p$-orbital. We wrap the computational rhombus on a torus, apply the periodic boundary conditions and obtain a set of allowed k-vectors over which we diagonalize the bulk Hamiltonian $H_{b}$. The sublattice A wavefunctions are expressed as a linear combination of even metal $d$-orbitals, with angular momentum two and $m_{d}=0,\pm 2$ and even combination of two, top and bottom, sulfur dimer $p$-orbitals with angular momentum one and $m_{p}=0,\pm 1$. The conduction band (CB) even wavefunction at each wavevector is a linear combination of simple even Bloch functions on the metal and sulfur sublattices $l$ ($l=1,..,6$)

$$\left|\phi^{\textrm{CB,ev}}_{k\sigma}\right\rangle=\sum_{l=1}^{6}A^{\textrm{CB,ev}}_{k\sigma,l}\left|\phi^{\textrm{ev}}_{k,l}\right\rangle\otimes\left|\chi_{\sigma}\right\rangle,$$

(2)

where $\left|\chi_{\sigma}\right\rangle$ represents the spinor part of wavefunction and

$$\left|\phi^{\textrm{ev}}_{k,l}\right\rangle=\frac{1}{\sqrt{N_{\textrm{UC}}}}\sum_{\vec{R_{l}}=1}^{N_{\textrm{UC}}}e^{i\vec{k}\vec{R_{l}}}\varphi^{ev}_{l}\left(\vec{r}-\vec{R_{l}}\right)$$

(3)

are simple Bloch functions built with even orbitals $\varphi^{ev}_{l}$ . $N_{UC}$ is the number of unit cells and $R_{l}$ define the position of even orbitals in the computational box. By diagonalising the 6 by 6 bulk Hamiltonian we obtain the bulk even energy bands $E_{k\sigma}^{\textrm{CB,ev}}$ and wavefunctions $A^{\textrm{CB,ev}}_{k\sigma,l}$.

Figure 2 shows the energy $E_{k\sigma}^{\textrm{CB}}$ of the lowest even conduction band (CB) as well as the map of the conduction band energies on the rhombus of $k$-space over which computations are carried out, including the $+K$ and $-K$ valley minima. The figure also contains the even conduction bands at a higher energy, to be discussed shortly.

In next step we expand the quantum dot wavefunction $\left|\Phi^{s}\right\rangle$ in terms of even lowest energy conduction band states given by Eq. 2

$$\left|\Phi^{s}\right\rangle=\sum_{\vec{k}}\sum_{\sigma}B^{\textrm{s,CB,ev}}_{\vec{k}\sigma}\left|\phi^{\textrm{CB,ev}}_{\vec{k}\sigma}\right\rangle.$$

(4)

The electron Schrödinger equation now converts to an integral equation for coefficients $B^{\textrm{s,CB,ev}}_{\vec{k}\sigma}$

$$E_{q\sigma}^{\textrm{CB,ev}}B_{q\sigma}^{\textrm{s,CB,ev}}+\sum_{\vec{k}\sigma^{\prime}}V_{q,k}A_{q\sigma,k\sigma^{\prime}}B_{k\sigma^{\prime}}^{\textrm{s,CB,ev}}=E^{s}B_{q\sigma}^{\textrm{s,CB,ev}}.$$

(5)

We see that the quantum dot confining potential in wavevector space turns out to be a product of the lateral confinement $V_{q,k}$ and band contribution $A_{q\sigma,k\sigma^{\prime}}$, with

$\displaystyle V_{q,k}=-V_{0}\frac{S}{4\pi}R_{QD}^{2}\exp\left(-\frac{(k-q)^{2}}{4}R_{QD}^{2}\right)$

(6)

being the Fourier transform of the confining potential, with $R_{QD}$ being the radius of the quantum dot, $V_{0}$ $-$the depth of the gate potential, and $S$ $-$the reciprocal lattice unit-cell area. The band structure contribution to the scattering potential $A_{q\sigma,k\sigma^{\prime}}$ is given by

$$A_{q\sigma,k\sigma^{\prime}}=\sum_{l}\left(A^{\textrm{CB,ev}}_{q\sigma,l}\right)^{\dagger}\left(A^{\textrm{CB,ev}}_{k\sigma^{\prime},l}\right).$$

(7)

Solving the integral equation, Eq. 5, we obtain the quantum dot energy levels and wavefunctions.

Figure 3 shows the energy levels of an electron confined in our quantum dot. We see that the levels are grouped into shells. The lowest energy shell consists of 4 low-energy states, related to 2 spin, up and down, states and two valleys, $+K$ and $-K$. The 4 states are split into pairs of levels by the spin-orbit interaction. The splitting, fully characterised in Ref. Bieniek et al. (2020), is limited by the bulk value. From ab-initio calculations, the splitting is $~{}3$ meV for MoS₂ but splitting is greater than room temperature, $30$ meV, for WSe₂Kadantsev and Hawrylak (2012); Bieniek et al. (2020). We hence identify the two logical qubits, $\left|0\right\rangle$ and $\left|1\right\rangle$, with the two lowest energy levels, $\left|0\right\rangle=\left|+K,\sigma=\downarrow\right\rangle$ and $\left|1\right\rangle=\left|-K,\sigma=\uparrow\right\rangle$, shown in Figure 3. Additionally, the energy of the s states depends on the depth of the confining potential of the order of 300 meV used here. This guarantees stability of the qubit. The dependence of the energy separation of the qubit states from excited states on potential depth and radius were already addressed by our previous workBieniek et al. (2020). We want to note that, in our unpublished work we also studied the effect of impurity on the qubit states in the valence band . We found that the impurity shifted the energy of qubit states but preserved the valley-spin locking and degeneracy.

In this section we discuss the necessary steps for single logical qubit manipulation. Before rotating qubit states, the degeneracy of the valley-spin locked system should be lifted so that one can use the doublet state as a qubit, as can be seen from Figure 3. The degeneracy of spin-valley locked system can be lifted by applying a magnetic field ( for $\sigma_{z}$ operation), which is known as valley-Zeeman splitting Wu et al. (2018); Zajac et al. (2018). This procedure will prepare and initialize the qubit states for $\sigma_{x}$ rotation. Hence, in order to rotate the qubit we need to be able to turn on both the $\sigma_{z}$ and $\sigma_{x}$ operations in the space of the logical qubit. To rotate the levels we need to be able to turn on the $\sigma_{x}$ operation. This operation needs to flip the logical qubit, i.e., induce a transition changing the spin and changing the valley. We will discuss this operation as composed of two steps, spin flipping and valley flipping.

To summarise, we developed here a theory of valley-spin based qubits in gated quantum dots in a single layer of transition metal dichalcogenide. The qubits were identified with the two degenerate locked spin and valley states in a gated quantum dot. The two qubit states were accurately described using a multi-million atom tight-binding model solved in k-space. The spin-valley locking and strong spin orbit coupling results in two degenerate states, one of the states of the qubit being spin-down located at the +K valley, and the other state located at the -K valley with spin up. We describe the gates necessary to rotate the spin-valley qubit as a combination of the applied vertical electric field enabling the spin orbit coupling in a single valley combined with a lateral strongly localised valley mixing gate. We note that suggested set up shown in Figure 1 can be readily implemented for one qubit operation. On the other hand, to be able to study manipulation of two or more qubit realizations, one can introduce impurity centers to mimic the role of the STM set up proposed in Figure 1. In addition, the aim of present work is to show how one can manipulate a single spin-valley qubit. The universal quantum computation requires also a two-qubit gate. Hence, our future work will focus on a microscopic description of a two-qubit gate operations where we will build on this and previous works David et al. (2018).

The authors thank L. Szulakowska, M. Cygorek, Y. Saleem, J. Manalo , J. B. Chouinard, A. Bogan, A. Luican-Mayer and L. Gaudreau for valuable discussions. This research was supported by the NSERC Strategic grant QC2DM, NSERC Discovery grant and University of Ottawa Research Chair in Quantum Theory of Quantum Materials, Nanostructures and Devices. M.B acknowledges financial support from Polish National Agency for Academic Exchange (NAWA), Poland, grant PPI/APM/2019/1/00085/U/00001. Computing resources from Compute Canada are gratefully acknowledged.