Magnetic-field dependence of valley splitting for Si quantum wells grown   on tilted SiGe substrates
  The valley splitting of the first few Landau levels is calculated as a function of the magnetic field for electrons confined in a strained silicon quantum well grown on a tilted SiGe substrate, using a parameterized tight-binding method. For a zero substrate tilt angle, the valley splitting slightly decreases with increasing magnetic field. In contrast, the valley splitting for a finite substrate tilt angle exhibits a strong and non-monotonous dependence on the magnetic field strength. The valley splitting of the first Landau level shows an exponential increase followed by a slow saturation as the magnetic field strength increases. The valley splitting of the second and third Landau levels shows an oscillatory behavior. The non-monotonous dependence is explained by the phase variation of the Landau level wave function along the washboard-like interface between the tilted quantum well and the buffer material. The phase variation is the direct consequence of the misorientation between the crystal axis and the confinement direction of the quantum well. This result suggests that the magnitude of the valley splitting can be tuned by controlling the Landau-level filling factor through the magnetic field and the doping concentration. 
Seungwon Lee and Paul von Allmen