Effect of Dirac nodal line on valley splitting of Si quantum wells

• Applicant: Professor Seyed Akbar Jafari, Ph.D.
• Subject Area: Theoretical Condensed Matter Physics
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 567117814

The band structure of bulk Si is characterized by 6 conduction band minima known as Si valleys. It splits into 4 higher energy and 2-fold degenerate low-lying valleys in quantum wells. The two-fold degeneracy limits the scaling of Si-based spin qubits. For this reason, one needs to understand the interplay between external influences and the valley energy levels. This proposal builds on a recently discovered topological aspect of bulk Si according to which a gift-box wrapping pattern of Dirac nodal lines surrounds the cube containing the Brillouin zone of the Si lattice. This aspect can be incorporated into the band structure (i.e. kinetic energy) as a new term via a new coupling constant κ that has the dimension of the inverse effective mass tensor, yet is 15 times larger than the inverse of the longitudinal mass. Our preliminary study shows that the nodal line coupling κ, in an in-plane magnetic field gives rise to a generalization of the celebrated Jaynes-Cummings (JC) model. Here the bosonic states are Landau-levels of electrons and the two-level aspect of the JC model is simply the two-valleys. This encourages us to formulate a complete proposal to study the effect of such nodal line coupling κ on valley-splitting physics and investigate its implications for Si spin qubits. With the above motivation, we aim to use a generalization of effective mass by incorporating the nodal line coupling κ. The objectives of the proposal are: 1. Validation of our nodal-line model and extraction of the numerical values of model parameters such as effective mass and κ for relevant compositions (i.e. various concentrations of Ge) from the tight-binding method and determination of the limits of validity of our model. 2. Understanding the effect of deterministic valley splitting at both continuum and lattice levels for the nodal-line model: The candidate fields to break symmetry and induce valley splitting are an in-plane magnetic field B combined with (i) in-plane E field, (ii) out-of-plane E field. 3. Effects of position-dependent model parameters: (i) the velocity v_0(x) and (ii) Effective mass m_l(x) on deterministic valley splitting for both zero nodal line coupling κ=0 and κ≠0 . 4. Investigation of the effect of random Dirac term and the random longitudinal mass in producing random valley splitting for the nodal-line model with κ≠0: Role of transverse dimensions. 5. Combining the above insights to draw Implications for hetero-structure design and valley-splitting statistics in realistic devices.

• DFG Programme: Research Grants
• Co-Investigator: Professor Dr. Hendrik Bluhm