Valley-orbit Coupling Research Articles

Searching for Unconventional Superfluid in Excitons of Monolayer Semiconductors.

It is well known that two-dimensional (2D) bosons in homogeneous space cannot undergo real Bose-Einstein condensation, and the superfluid to normal phase transition is Berezinskii-Kosterlitz-Thouless (BKT) type, associated with vortex-antivortex pair unbinding. Here we point out a 2D bosonic system whose low energy physics goes beyond conventional paradigm of 2D homogeneous bosons, i.e., intralayer excitons in monolayer transition metal dichalcogenides. With intrinsic valley-orbit coupling and valley Zeeman energy, exciton dispersion becomes linear at small momentum, giving rise to a series of novel features. The critical temperature of Bose-Einstein condensation of these excitons is nonzero, suggesting true long-range order in 2D homogeneous system. The dispersion of Goldstone mode at long wavelength has the form ϵ(q)∼sqrt[q], in contrast to conventional linear phonon spectrum. The vortex energy deviates from the usual logarithmic form with respect to system size, but instead has an additional linear term. Superfluid to normal phase transition is no longer BKT type for system size beyond a characteristic scale, without discontinuous jump in superfluid density. With the recent experimental progress on exciton fluid at thermal equilibrium in monolayer semiconductors, our work points out an experimentally accessible system to search for unconventional 2D superfluids beyond BKT paradigm.

Single Photon Emitters with Polarization and Orbital Angular Momentum Locking in Monolayer Semiconductors.

Excitons in monolayer transition metal dichalcogenide are endowed with intrinsic valley-orbit coupling between their center-of-mass motion and valley pseudospin. When trapped in a confinement potential, e.g., generated by strain field, we find that intralayer excitons are valley and orbital angular momentum (OAM) entangled. By tuning the trap profile and external magnetic field, one can engineer the exciton states at the ground state and realize a series of valley-OAM entangled states. We further show that the OAM of excitons can be transferred to emitted photons, and these novel exciton states can naturally serve as polarization-OAM locked single photon emitters, which under certain circumstance become polarization-OAM entangled, highly tunable by strain trap and magnetic field. Our proposal demonstrates a novel scheme to generate polarization-OAM locked/entangled photons at the nanoscale with a high degree of integrability and tunability, pointing to exciting opportunities for quantum information applications.

Valleytronic full configuration-interaction approach: Application to the excitation spectra of Si double-dot qubits

The influence of strong electron-electron interactions and Wigner-molecule (WM) formation on the spectra of $2e$ singlet-triplet double-dot Si qubits is presented based on a full configuration interaction (FCI) approach that incorporates the valley degree of freedom (VDOF) in the context of the continuous (effective mass) description of semiconductor materials. Our FCI treats the VDOF as an isospin in addition to the regular spin. Our treatment is able to assign to each energy curve in the qubit's spectrum a complete set of good quantum numbers for both the spin and the valley isospin. This reveals an underlying SU(4) $\supset$ SU(2) $\times$ SU(2) group-chain organization in the Si double-dot spectra. With parameters in the range of actual experimental situations, we demonstrate in a double-dot qubit that, in the (2,0) charge configuration and compared to the expected large, and dot-size determined, single-particle (orbital) energy gap, the strong $e-e$ interactions drastically quench the spin-singlet$-$spin-triplet energy gap, $E_{\rm ST}$, within the same valley, making it competitive to the small energy gap, $E_V$, between the two valleys. We present results for both the $E_{\rm ST} < E_V$ and $E_{\rm ST} > E_V$ cases. We investigate the spectra as a function of detuning and demonstrate the strengthening of the avoided crossings due to a lowering of the interdot barrier and/or the influence of valley-orbit coupling. We further demonstrate, as a function of an applied magnetic field, the emergence of avoided crossings in the (1,1) charge configuration due to the spin-valley coupling. The valleytronic FCI formulated here, and implementeded for two electrons confined in a tunable double quantum dot, offers also a most effective tool for analyzing the spectra of Si qubits with more than two wells and/or more than two electrons.

Measurement of Tunnel Coupling in a Si Double Quantum dot Based on Charge Sensing

Tunnel coupling is a key parameter in coupled semiconductor quantum dots, and is a crucial ingredient in various device applications, such as exchange gates and spin shuttling in quantum information processing. A widely used charge-sensing technique to extract the tunnel coupling of a double quantum dot accounts for only the ground orbital state in each dot, but the authors show that in a Si double dot, valley-orbit coupling must be included in the analysis. With their more complete model, one can not only extract the $i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}v\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}l\phantom{\rule{0}{0ex}}l\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}y$ (ground-state) tunnel coupling more accurately, but also obtain information on $i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}v\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}l\phantom{\rule{0}{0ex}}l\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}y$ (ground-to-excited-state) tunnel coupling.

Impact of the valley orbit coupling on exchange gate for spin qubits in silicon

The mixing of conduction band valleys plays a critical role in determining electronic spectrum and dynamics in a silicon nanostructure. Here, we investigate theoretically how valley–orbit coupling affects the exchange interaction in a silicon double quantum dot. We find that exchange splitting can be strongly suppressed at finite valley phase differences between the dots because of the valley-phase-dependent dressing of the ground states and Coulomb exchange integrals, and a small valley splitting can render the exchange Hamiltonian incomplete in describing low-energy dynamics due to nearby excited valley states. The higher orbital states are also vital in calculating the exchange splitting, which is crucial for applications such as exchange gates for spin qubits.

Quantitative Analysis of Valley–Orbit Coupling in Germanium Doped with Group-V Donors

Quantitative Analysis of Valley–Orbit Coupling in Germanium Doped with Group-V Donors

Долинно-орбитальное взаимодействие в германии, легированном донорами V группы: количественный анализ

In the framework of the envelope function approximation, the wave functions of electrons localized at shallow donors P, As, Sb in Ge are calculated taking into account the valley-orbit coupling caused by the donor short-range potential. It is proposed an approach that makes it possible to include inter-valley mixing in the equation for a multi-component envelope function. The calculation of the effects of the valley-orbit interaction was carried out according to the perturbation theory, while the "bare" single-valley functions were found using the Ritz method. The parameters of the short-range part of the potential and the coefficient of inter-valley mixing were found individually for each donor, making it possible to obtain the best agreement with the results of experimental measurements of the energies of the singlet and triplet states. The envelope functions of the 1s(A1) and 1s(T2) states are calculated. The parameters of the valley-orbit interaction are found for each donor. It is also shown how the functions of the excited 2s, 2p0, 2p±, 3p0 states should be modified in order to remain orthogonal to the singlet and triplet functions within the framework of a more rigorous multivalley model.

Direct Measurement of Electron Intervalley Relaxation in a Si/Si - Ge Quantum Dot

The presence of non-degenerate valley states in silicon can drastically affect electron dynamics in silicon-based heterostructures, leading to electron spin relaxation and spin-valley coupling. In the context of solid-state spin qubits, it is important to understand the interplay between spin and valley degrees of freedom to avoid or alleviate these decoherence mechanisms. Here we report the observation of relaxation from the excited valley state to the ground state in a Si/SiGe quantum dot, at zero magnetic field. Valley state read-out is aided by a valley-dependent tunneling effect, which we attribute to valley-orbit coupling. We find a long intervalley relaxation time of 12.0 $\pm$ 0.3 ms, a value that is unmodified when a magnetic field is applied. Furthermore, we compare our findings with the spin relaxation time and find that the spin-valley "hot spot" relaxation is roughly four times slower than intervalley relaxation, consistent with established theoretical predictions. The precision of this technique, adapted from electron spin read-out via energy-dependent tunneling, is an improvement over indirect valley relaxation measurements and could be a useful probe of valley physics in spin and valley qubit implementations.

Two-axis quantum control of a fast valley qubit in silicon

Quantum dots in silicon are a promising architecture for semiconductor quantum computing due to a high degree of electric control and compatibility with existing silicon fabrication processes. Although electron charge and spin are prominent methods for encoding the qubit state, valley states in silicon can also store quantum information via valley-orbit coupling with protection against charge noise. By observing coherent oscillations between valley states in a Si/SiGe double quantum dot device tuned to the two-electron charge configuration, we measure the valley energy splitting in both quantum dots individually. We further demonstrate two-axis quantum control of the valley qubit using gated pulse sequences with X and Z rotations occurring within a fast operation time of 300 ps. This control is used to completely map out the surface of the Bloch sphere in a single phase-space plot that is subsequently used for state and process tomography.

Effects of interface steps on the valley-orbit coupling in a Si/SiGe quantum dot

Valley-orbit coupling is a key parameter for a silicon quantum dot in determining its suitability for applications in quantum information processing. In this paper we study the effect of interface steps on the magnitude and phase of valley-orbit coupling for an electron in a silicon quantum dot. Within the effective mass approximation, we find that the location of a step on the interface is important in determining both the magnitude and the phase of the valley-orbit coupling in a Si/SiGe quantum dot. Specifically, our numerical results show that the magnitude of valley orbit coupling can be suppressed up to 75\% by a step of one atomic monolayer, and its phase can change by almost $\pi$. When two steps are present, the minimum value of the valley-orbit coupling can even approach zero. We also clarify the effects of an applied external magnetic field and the higher orbital states on the valley-orbit coupling. Overall, our results illustrate how interface roughness affect the valley-orbit coupling in silicon, and how spin qubits in silicon may be affected.

Valley Phase and Voltage Control of Coherent Manipulation in Si Quantum Dots

With any roughness at the interface of an indirect-bandgap semiconducting dot, the phase of the valley-orbit coupling can take on a random value. This random value, in double quantum dots, causes a large change in the exchange splitting. We demonstrate a simple analytical method to calculate the phase, and thus the exchange splitting and singlet-triplet qubit frequency, for an arbitrary interface. We then show that, with lateral control of the position of a quantum dot using a gate voltage, the valley-orbit phase can be controlled over a wide range, so that variations in the exchange splitting can be controlled for individual devices. Finally, we suggest experiments to measure the valley phase and the concomitant gate voltage control.

Electronic states and valley-orbit coupling in linear and planar molecules formed by coupled P donors in silicon

Using the effective mass theory and the multi-valley envelope function representation, we have developed a theoretical framework for computing the single-electron electronic structure of several phosphorus donors interacting in an arbitrary geometrical configuration in silicon taking into account the valley-orbit coupling. The methodology is applied to three coupled phosphorus donors, arranged in a linear chain and in a triangle, and to six donors arranged in a regular hexagon. The results of the simulations evidence that the valley composition of the single-electron states strongly depends on the geometry of the dopant molecule and its orientation relative to the crystallographic axes of silicon. The electron binding energy of the triatomic linear molecules is larger than that of the diatomic molecule oriented along the same crystallographic axis, but the energy gap between the ground state and the first excited state is not significantly different for internuclear distances from 1.5 to 6.6 nm. Three donor atoms arranged in a triangle geometry have larger binding energies than a triatomic linear chain of dopants with the same internuclear distances. The planar donor molecules are characterized by a strong polarization in favor of the valleys oriented perpendicular to the plane of the molecule. The polarization increases with number of atoms forming the planar molecule.

Valleytronics and phase transition in silicene

Magnetic and transport properties of silicene in the presence of perpendicular electromagnetic fields and a ferromagnetic material are studied. It is shown that for small exchange field, the magnetic moment associated with each valley is opposite for the other and it gives a shift in band energy, by a Zeeman-like coupling term. Thus opening a new horizon for valley–orbit coupling. Magnetic proximity effect is seen to adjust the spintronics of each valley. Valley polarization is calculated using the semi classical formulation of electron dynamics. It can be modified and measured due to its contribution in Hall conductivity. Quantum phase transitions are observed in silicene, providing a tool to control the topological state experimentally. The strong dependence of the physical properties on valley degree of freedom is an important step towards valleytronics.

Multivalley envelope function equations and effective potentials for phosphorus impurity in silicon

We propose a system of real-space envelope function equations without fitting parameters for modeling the electronic spectrum and wave functions of a phosphorus donor atom embedded in silicon. The approach relies on the Burt-Foreman envelope function representation and leads to coupled effective-mass Schroedinger equations containing smooth effective potentials. These potentials result from the spatial filtering imposed on the exact potential energy matrix elements in the envelope function representation. The corresponding filter function is determined from the definition of the envelope function. The resulting effective potentials and the system of envelope functions jointly reproduce the valley-orbit coupling effect in the doped silicon. Including the valley-orbit coupling not only of the 1s, but also for 2s atomic orbitals, as well as static dielectric screening is found crucial to accurately reproduce experimental data. The measured binding energies are recovered with a maximum relative error of 1.53 %. The computed wave functions are in a good agreement with experimental measurements of the electron density provided by scanning tunneling microscopy.

Single-valley engineering in graphene superlattices

The two inequivalent valleys in graphene preclude the protection against inter-valley scattering offered by an odd-number of Dirac cones characteristic of Z2 topological insulator phases. Here we propose a way to engineer a chiral single-valley metallic phase with quadratic crossover in a honeycomb lattice through tailored \sqrt{3}N *\sqrt{3}N or 3N *3N superlattices. The possibility of tuning valley-polarization via pseudo-Zeeman field and the emergence of Dresselhaus-type valley-orbit coupling are proposed in adatom decorated graphene superlattices. Such valley manipulation mechanisms and metallic phase can also find applications in honeycomb photonic crystals.

Exchange coupling between silicon donors: The crucial role of the central cell and mass anisotropy

Donors in silicon are now demonstrated as one of the leading candidates for implementing qubits and quantum information processing. Single qubit operations, measurements and long coherence times are firmly established, but progress on controlling two qubit interactions has been slower. One reason for this is that the inter donor exchange coupling has been predicted to oscillate with separation, making it hard to estimate in device designs. We present a multivalley effective mass theory of a donor pair in silicon, including both a central cell potential and the effective mass anisotropy intrinsic in the Si conduction band. We are able to accurately describe the single donor properties of valley-orbit coupling and the spatial extent of donor wave functions, highlighting the importance of fitting measured values of hyperfine coupling and the orbital energy of the $1s$ levels. Ours is a simple framework that can be applied flexibly to a range of experimental scenarios, but it is nonetheless able to provide fast and reliable predictions. We use it to estimate the exchange coupling between two donor electrons and we find a smoothing of its expected oscillations, and predict a monotonic dependence on separation if two donors are spaced precisely along the [100] direction.

Dirac cones and Dirac saddle points of bright excitons in monolayer transition metal dichalcogenides.

In monolayer transition metal dichalcogenides, tightly bound excitons have been discovered with a valley pseudospin optically addressable through polarization selection rules. Here, we show that this valley pseudospin is strongly coupled to the exciton centre-of-mass motion through electron-hole exchange. This coupling realizes a massless Dirac cone with chirality index I = 2 for excitons inside the light cone, that is, bright excitons. Under moderate strain, the I = 2 Dirac cone splits into two degenerate I = 1 Dirac cones, and saddle points with a linear Dirac spectrum emerge. After binding an extra electron, the charged exciton becomes a massive Dirac particle associated with a large valley Hall effect protected from intervalley scattering. Our results point to unique opportunities to study Dirac physics, with exciton's optical addressability at specifiable momentum, energy and pseudospin. The strain-tunable valley-orbit coupling also implies new structures of exciton condensates, new functionalities of excitonic circuits and mechanical control of valley pseudospin.

Electronic states and wavefunctions of diatomic donor molecular ions in silicon: multi-valley envelope function theory

Using the Burt–Foreman envelope function theory and effective mass approximation, we develop a theoretical model for an arbitrary number of interacting donor atoms embedded in silicon which reproduces the electronic energy spectrum with high computational efficiency, taking into account the effective mass anisotropy and the valley–orbit coupling. We show that the variation of the relative magnitudes of the electronic coupling between the donor atoms with respect to the valley–orbit coupling as a function of the internuclear distance leads to different kinds of spatial interference patterns of the wavefunction. We also report on the impact of the orientation of the diatomic phosphorus donor molecular ion in the crystal lattice on the ionization energy and on the energy separation between the ground state and the lowest excited state.

Magnetic control of the valley degree of freedom of massive Dirac fermions with application to transition metal dichalcogenides

We study the valley-dependent magnetic and transport properties of massive Dirac fermions in multivalley systems such as transition metal dichalcogenides. The asymmetry of the zeroth Landau level between valleys and the enhanced magnetic susceptibility can be attributed to the different orbital magnetic moment tied with each valley. This allows the valley polarization to be controlled by tuning the external magnetic field and the doping level. As a result of this magnetic-field-induced valley polarization, there exists an extra contribution to the ordinary Hall effect. All these effects can be captured by a low-energy effective theory with a valley-orbit coupling term.

Coulomb interaction and valley-orbit coupling in Si quantum dots

The valley-orbit coupling in a few-electron Si quantum dot is expected to be a function of its occupation number $N$. We study the spectrum of multivalley Si quantum dots for $2\ensuremath{\le}N\ensuremath{\le}4$, showing that, counterintuitively, electron-electron interaction effects on the valley-orbit coupling are negligible. For $N=2$ they are suppressed by valley interference, for $N=3$ they vanish due to spinor overlaps, and for $N=4$ they cancel between different pairs of electrons. To corroborate our theoretical findings, we examine the experimental energy spectrum of a few-electron metal-oxide-semiconductor quantum dot. The measured spin-valley state filling sequence in a magnetic field reveals that the valley-orbit coupling is definitively unaffected by the occupation number.