Magnetic field dependence of valley splitting in realistic Si∕SiGe quantum wells

Abstract

The authors investigate the magnetic field dependence of the energy splitting between low-lying valley states for electrons in a Si/SiGe quantum well tilted with respect to the crystallographic axis. The presence of atomic steps at the quantum well interface may explain the unexpected, strong suppression of the valley splitting observed in recent experiments. The authors find that the suppression is caused by an interference effect associated with multiple steps, and that the magnetic field dependence arises from the lateral confinement of the electronic wave function. Using numerical simulations, the authors clarify the role of step disorder, obtaining quantitative agreement with the experiments.Qubits in silicon are leading candidates for scalable quantum computing, owing to their favorable and well studied materials properties.1,2 Indeed, because of its prominence in the electronics industry, silicon may be the best understood semiconducting material. However, as devices continue to shrink in size, approaching the quantum regime, important questions arise. Unlike direct gap semiconductors, the conduction band structure in silicon possesses six symmetric minima or "valleys" that are not at the Brillouin zone center. Consequently, the minima are degenerate, and must be described by a valley index which competes with the spin index as a relevant quantum number in the qubit Hilbert space.3 Therefore, to construct spin qubits in silicon, it is necessary to lift all valley degeneracy.A silicon quantum well grown on the [001] surface of strain-relaxed silicon-germanium is under tensile strain, causing the four lateral valleys to rise significantly in energy.4 At low temperatures, only the two low-lying valleys are populated. The remaining two-fold degeneracy can be removed by the sharp confinement potential of the quantum well interface.5 Theoretical estimates suggest that the resulting valley splitting can be of the order of 1 meV ≈ 12 K, 6 which is sufficiently large for quantum computing. However, recent experiments in SiGe 7,8,9,10 measure a valley splitting much smaller than the theoretical prediction. There is currently no explanation for this discrepency.11 Indeed, a prevalent theory 12 predicts an enhancement of the valley splitting in a magnetic field that is different from the experimental observations.In this letter, we describe a single-electron valley splitting theory for silicon quantum wells grown on a vicinal substrate, building upon an initial suggestion by Ando. 13Such miscuts are often incorporated into Si/SiGe heterostructures to ensure uniform growth surfaces and to avoid step bunching. The resulting quantum well, obtained by conformal epitaxial deposition, is misaligned with respect to the crystallographic z axis, as shown in Fig. 1. We now describe the effective mass theory and explain how the presence of interfacial atomic steps suppresses valley splitting.For silicon strained in the [001] direction, the effective mass wavefunction 14 can be written as a sum of contributions from the two z va...

Fractional quantum Hall effect of two-dimensional electrons in high-mobility Si/SiGe field-effect transistors

“…The ν = 4/5 state may be viewed as the particle-hole conjugate of the state at ν = 1/5 in the first level; the ν = 6/5 state is the FQH state at 1/5 filling of the second level. This observation implies that the two-fold valley degeneracy of the 2D electrons is lifted in the high magnetic fields at which the two states are observed, consistent with the well known fact that the valley-splitting gap in (100) Si 2D electrons is dependent on the host device structure [28] and on the external magnetic field [29,30]. These two states can be seen as the IQH states of CFs at ν = 1/4, formed by attaching four magnetic flux quanta to one electron.…”

Signatures of the valley Kondo effect in Si/SiGe quantum dots

“…The appearance and persistence of the center peak with B > 0 is a key prediction of the theory of the valley Kondo effect and arises physically from an interference of valley conserving and valley non-conserving processes. This suggests that valley index is not always conserved during tunneling, a subject of some debate, 12,13,15,[22][23][24][25][26][27] and in accord with recent effective mass calculations of the effects of atomic step disorder at the quantum well interface. 17 Secondly, given a non-zero ∆, the QD electrons must occupy an excited state rather than the ground state viewed in terms of single-particle levels.…”

Electric dipole spin resonance in systems with a valley-dependentgfactor

“…The discrepancy between our theory and the experiment may be due the fact that the product of valley coupling strength and the square of the wavefunction at the position of the Si/SiGe interface v v ξ 2 (z 0 ), is a free parameter. v v depends on the abundance of Ge x in the Si/Si x Ge 1−x quantum well and can be estimated from tight binding theories [29]. On the other hand, ξ 2 (z 0 ) depends on the thickness of the Si layer and the exact type of the confinement in the Si/SiGe quantum well.…”