We develop a symmetry-based theory of spin-orbit-valley coupling in silicon that reveals an intrinsic source of Dresselhaus spin-orbit coupling independent of interfaces or external electric fields. Treating the valley degree of freedom as a symmetry-carrying quantum degree of freedom, we show that the Dresselhaus interaction is necessarily valley off-diagonal and that a bulk contribution proportional to the valley Pauli matrix $τ_1$ is symmetry allowed. Tight-binding calculations yield a bulk coupling more than an order of magnitude larger than typical interface-induced spin-orbit coupling; achieving the same energy scale through the interface-induced mechanism would require electric fields roughly 50 times larger than typical fields. We further derive the symmetry-allowed spin-valley couplings generated by magnetic-field gradients and show how they account for the valley-dependent Zeeman splitting observed in micromagnet experiments. A slight tilt of the background magnetic field out of the plane produces an additional isotropic contribution linear in $B_z$, providing an experimentally accessible signature of the corresponding coupling constant. Finally, we predict a spin-independent micromagnet-induced valley splitting in the $τ_3$ channel, which is distinct from the $τ_{1,2}$ channels generated by alloy disorder and therefore remains robust against disorder-induced cancellation. These results establish valley symmetry as a fundamental ingredient in the spin-orbit physics of silicon and provide new mechanisms for controlling and probing spin and valley degrees of freedom in silicon quantum devices.
AI Summary
The paper asks whether silicon’s valley symmetry permits spin–orbit coupling (SOC) without interfaces or external electric fields, and how magnetic-field gradients couple spin and valley states. It constructs the double space group at the $X$ point and uses symmetry invariants and time reversal to classify couplings in the combined spin–valley space.
The resulting intrinsic bulk term is $\beta_1(k_x\sigma_x-k_y\sigma_y)\tau_1$, which is off-diagonal in the valley basis. The corresponding interface-induced Dresselhaus term involves $E_z\tau_2$, while Rashba SOC involves $E_z\tau_0$. An $sp^3d^5s^*$ tight-binding model with onsite SOC of approximately $18.5$ meV gives $\beta_1\approx0.87$ meV nm at $X$ and $0.67$ meV nm near the conduction-band minima. Matching this coefficient through interface-induced SOC is estimated to require approximately $0.5$ V/nm, around 50 times typical fields.
The symmetry analysis also identifies three valley-off-diagonal spin couplings to magnetic-field gradients. Applying these perturbatively to published micromagnet data gives an estimated coupling $\lambda_1\approx13\,\mu$eV nm/T. A slight out-of-plane field tilt is predicted to add an in-plane-angle-independent contribution linear in $B_z$ to the valley-dependent Zeeman splitting. A spin-independent term proportional to $(\partial_xB_y-\partial_yB_x)\tau_3$ is also predicted; its valley-space direction prevents cancellation by alloy-disorder contributions in the $\tau_{1,2}$ channels.
The classification is restricted to the specified low-order momentum and field terms. The micromagnet treatment assumes gradient corrections small compared with the homogeneous Zeeman coupling and neglects vertical derivatives for the cited experiment. Reduced device symmetry can permit additional terms.
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Central claim may not hold: The paper does not establish that its common-momentum X-point interband invariant acts as an intervalley coupling between the two distinct low-energy conduction-band minima.
The displayed bulk Hamiltonian retains doubly degenerate eigenvalues at every momentum and therefore does not demonstrate the claimed lifting of spin degeneracy.
Away from X, the displayed bulk term has no first-order intrabranch matrix element and produces a leading spin-independent quadratic correction, invalidating the direct interpretation of its coefficient as a larger equivalent low-energy Dresselhaus interaction.
The quoted coefficients imply a matching interface field of approximately 0.335 V/nm, or 33.5 times the stated typical field, rather than 0.5 V/nm or 50 times.
The predicted spin-independent micromagnet valley-splitting term vanishes in a static, current-free, nonmagnetic quantum-dot region because it is proportional to the curl of the magnetic field.
Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.
We develop a symmetry-based theory of spin-orbit-valley coupling in silicon that reveals an intrinsic source of Dresselhaus spin-orbit coupling independent of interfaces or external electric fields. Treating the valley degree of freedom as a symmetry-carrying quantum degree of freedom, we show that the Dresselhaus interaction is necessarily valley off-diagonal and that a bulk contribution proportional to the valley Pauli matrix τ1 is symmetry allowed. Tight-binding calculations yield a bulk coupling more than an order of magnitude larger than typical interface-induced spin-orbit coupling; achieving the same energy scale through the interface-induced mechanism would require electric fields roughly 50 times larger than typical fields. We further derive the symmetry-allowed spin-valley couplings generated by magnetic-field gradients and show how they account for the valley-dependent Zeeman splitting observed in micromagnet experiments. A slight tilt of the background magnetic field out of the plane produces an additional isotropic contribution linear in Bz, providing an experimentally accessible signature of the corresponding coupling constant. Finally, we predict a spin-independent micromagnet-induced valley splitting in the τ3 channel, which is distinct from the τ1,2 channels generated by alloy disorder and therefore remains robust against disorder-induced cancellation. These results establish valley symmetry as a fundamental ingredient in the spin-orbit physics of silicon and provide new mechanisms for controlling and probing spin and valley degrees of freedom in silicon quantum devices.
The paper asks whether silicon’s valley symmetry permits spin–orbit coupling (SOC) without interfaces or external electric fields, and how magnetic-field gradients couple spin and valley states. It constructs the double space group at the X point and uses symmetry invariants and time reversal to classify couplings in the combined spin–valley space.
The resulting intrinsic bulk term is β1(kxσx−kyσy)τ1, which is off-diagonal in the valley basis. The corresponding interface-induced Dresselhaus term involves Ezτ2, while Rashba SOC involves Ezτ0. An sp3d5s∗ tight-binding model with onsite SOC of approximately 18.5 meV gives β1≈0.87 meV nm at X and 0.67 meV nm near the conduction-band minima. Matching this coefficient through interface-induced SOC is estimated to require approximately 0.5 V/nm, around 50 times typical fields.
The symmetry analysis also identifies three valley-off-diagonal spin couplings to magnetic-field gradients. Applying these perturbatively to published micromagnet data gives an estimated coupling λ1≈13μeV nm/T. A slight out-of-plane field tilt is predicted to add an in-plane-angle-independent contribution linear in Bz to the valley-dependent Zeeman splitting. A spin-independent term proportional to (∂xBy−∂yBx)τ3 is also predicted; its valley-space direction prevents cancellation by alloy-disorder contributions in the τ1,2 channels.
The classification is restricted to the specified low-order momentum and field terms. The micromagnet treatment assumes gradient corrections small compared with the homogeneous Zeeman coupling and neglects vertical derivatives for the cited experiment. Reduced device symmetry can permit additional terms.
Increasing the valley splitting in Si-based heterostructures is critical for improving the performance of semiconductor qubits. This paper demonstrates that the two low-energy conduction band valleys are not independent parabolic bands. Instead, they originate from the X-point of the Brillouin zone, where they are interconnected by a degeneracy protected by the non-symmorphic symmetry of the diamond lattice. This semi-Dirac-node degeneracy gives rise to the Δ1 and Δ2′ bands, which constitute the valley degrees of freedom. By explicitly computing the two-component Bloch functions X1±, using the wave vector group at the X-point, we determine the transformation properties of the object (X1+,X1−). We demonstrate that these properties are fundamentally different from those of a spinor. Consequently, we introduce the term "valleyor" to emphasize this fundamental distinction. The transformation properties of valleyors induce corresponding transformations of the Pauli matrices τ1,τ2 and τ3 in the valley space. Determining these transformations allows us to classify possible external perturbations that couple to each valley Pauli matrix, thereby identifying candidates for valley-magnetic fields, B. These fields are defined by a Zeeman-like coupling B⋅τ to the valley degree of freedom. In this way, we identify scenarios where an applied magnetic field B can leverage other background fields, such as strain, to generate a valley-magnetic field B. This analysis suggests that beyond the well-known mechanism of potential scattering from Ge impurities, there exist additional channels (mediated by combinations of magnetic and strain-induced vector potentials) to control the valley degree of freedom
While bulk silicon has long been understood to exhibit relatively weak spin-orbit coupling (SOC), confinement of electrons to quantum dots (QDs) at a silicon heterointerface results in significantly larger SOC. This is a concern for electron spin qubit performance, as intravalley and intervalley SOC can significantly perturb the operation of electron spin qubits. While these interactions can be harnessed to drive coherent rotations in a singlet-triplet qubit, coupling to low-lying excited valley states can lead to undesirable spin relaxation when valley splitting is on resonance with the Zeeman energy. In this work, we measure the angular dependence of the interfacial spin-orbit interaction as a function of the direction and magnitude of an applied external magnetic field in SiMOS and Si/SiGe heterostructures, two common material platforms for silicon spin qubits. We construct a physical model that accurately infers intra- and inter-valley SOC physics from fits to the data, allowing for a direct comparison between these two material systems. For the devices measured we find that, while the g-factor differences are comparable, the SiMOS QDs exhibit an order of magnitude larger spin-valley coupling than for Si/SiGe. Moreover, we find that the angular dependence of the spin-valley coupling is similar for both devices, with similar magnetic field orientations minimizing the spin-valley coupling. Our work points towards operational schemes for optimizing spin-valley coupling to avoid or exploit this mechanism for qubit operation.
Valley splitting sets the energy scale at which spin and valley degrees of freedom hybridize in silicon quantum-well qubits, but its sensitivity to interface structure makes it difficult to predict. Using the valleyor basis, we formulate a two-band effective-mass model for intervalley coupling in a finite quantum well and develop an analytical framework for treating non-perturbative effects of shear strain and wiggle-well potentials. For a z-independent coupling Vsτ2, corresponding to uniform shear strain, we construct a variational state from the exact plane-wave valleyors and a hard-wall envelope. This yields closed-form expressions for the two lowest states and their splitting, including the characteristic oscillatory ∣sin(kminL)∣ dependence. The variational result agrees with exact diagonalization to within ∼1% for realistic couplings, and provides a simple prescription for tuning the shear strain away from the nodes to enhance the valley splitting. We further show that finite barrier heights primarily renormalize the effective width of the quantum well while preserving the oscillatory dependence. For a wiggle-well coupling Vwcos(kwz)τ1, the valleyor representation reveals a crossover near the kw=2k1 resonance from wiggle-dominated splitting to orbital quantization. Remarkably, at high wiggle-well amplitudes, the empty-box energy scale predominantly dictates the near-resonant valley splitting, while the wiggle-well potential enters only as a sub-leading correction. This behavior can be viewed as asymptotic freedom for spin qubits. Away from resonance, the splitting exhibits a broadly peaked resonance with diffraction-like side-lobes as a function of detuning.
Si/SiGe quantum wells that incorporate Ge concentration oscillations, known as long-period Wiggle Wells, have been shown to enhance the Dresselhaus spin-orbit coupling of conduction-band electrons. Such intrinsic spin-orbit coupling is desirable when performing spin-qubit gate operations based on electric dipole spin resonance (EDSR) because it eliminates the need for external micromagnets. However, random-alloy disorder plays a key role in this materials system by spatially randomizing the valley splitting and the valley phase φs,s, and it has not been fully accounted for in recent EDSR analyses. Here, we show that alloy disorder affects EDSR in two main ways. First, the Rabi frequency Ω acquires a dependence on the valley phase, given by cosφs,s, which causes spatial randomization of Ω. Despite this variability, we show that fast EDSR can be achieved at most locations across a given sample. Second, a new Rabi driving mechanism emerges, enabled by valley dipoles and the hybridization of ground and excited valley states, which arise from alloy disorder and EDSR driving, respectively. This mechanism is dominant in regions of low valley splitting. Alloy disorder can therefore strengthen EDSR, but it can also cause gradients in Ω that lead to dephasing in the rotating frame. We explore this problem by first locating "sweet spots," where EDSR is relatively insensitive to electric-field fluctuations. We then show that high-fidelity Rabi oscillations can be achieved in the presence of realistic charge noise. These results suggest that Wiggle Wells are a promising platform for high-quality, micromagnet-free gate operations.
Controlling the energy scales of a quantum system is essential for defining robust qubits. In silicon spin qubits, the nearly degenerate conduction-band valleys create a leakage channel from the single-spin computational basis, posing a challenge to scaling and to shuttling-based architectures. Here, we measure the relevant energy scales of single-electron spin qubits in buried silicon quantum wells co-designed for low disorder and high valley splitting. Across a linear array of four quantum dots with an average orbital energy of 2.4(2) meV, we report an average single-electron valley splitting of 0.40(6) meV and an average two-electron singlet-triplet splitting of 0.24(7) meV. In three dots, we observe a strong correlation between valley splitting and orbital energy, with an average linear coefficient of ≈0.22 (meV/meV), demonstrating that electrostatic confinement can increase the valley splitting by several hundred microelectronvolts. In contrast, the remaining dot exhibits the highest valley splitting of 0.76(2) meV and low correlation, suggesting excellent characteristics for spin-qubit operation. Our findings demonstrate that strong confinement can be exploited in buried quantum wells to effectively enhance the valley splitting, thereby establishing a viable path toward the realization of shuttling and sparse-occupation-based architectures in low-disorder heterostructures.
This dissertation contains a collection of topics on the spin-orbit coupling of cold atoms and semiconductors. We first consider the effects of an optically induced spin-orbit interaction on a system of Bosonic atoms. The spin-orbit term couples the emergent pseudo-spin- 1/2 degree of freedom to momentum. The single particle energy spectrum has a low energy band with two minima at non-zero momentum. At low temperatures, a many-body system will condense into these minima. In the presence of vanishing interactions, the ground state wavefunction is found to be N00N state of "left" and "right"-moving state. An experimentally observable signature of the condensate is predicted that can be observed using time-of-flight imaging. We then predict a bulk manifestation of the spin-Hall effect in an inhomoge- neous spin-orbit-coupled system. The phenomena is predicted in the framework of the spin diffusion equations generalized to include arbitrary Rashba and linear and cubic Dresselhaus terms. This framework shows that a bulk spin-density wave with a wavevector oriented perpendicular to an applied electric field will induce a charge- density wave characterized by a π/2-phase shift and a non-monotonic time-varying amplitude. The optimal values of spin-orbit coupling for observation of the effect are determined. We propose a scheme for using atom interferometry to measure weak time- dependent accelerations. This proposal uses an ensemble of dilute trapped bosons with two pseudo-spin states coupled to a synthetic magnetic field, but with opposite effective charges. The synthetic field acts to couple spin to momentum continuously, which continuously imparts the acceleration on the phase of the internal states. We use time reversal pulses to reduce noise. The sensitivity of such a system is estimated to be S ∼ 10−7 m/s^2/√Hz Finally, we consider the semiclassical dynamics of a trapped spin-orbit coupled system. We find non-linear dynamics parameterized by the value of the initial displacement of the trap, and the anisotropy of the spin-orbit energy spectrum. We show that the dynamics can give a Berry's phase, and propose an experiment to measure this phase. We then propose a generalization of the 4-level scheme that allows for spin-orbit coupling described by a vector potential that is proportional to the angular momentum operator.