Observation of Inter-valley Gap Anomaly in Two Dimensional Electrons in Si

We report a systematic study of the energy gaps at the odd-integer quantum Hall states $\nu$=3 and 5 under tilted magnetic (B) fields in the two-dimensional electron system (2DES) in Si/Si$_{1-x}$Ge$_x$ heterostructures [1]. Consistent with previous studies, we find that out of the coincidence region, the valley splitting is independent of the in-plane B-field. However, the $\nu$=3 valley gap appears to be highly asymmetric and differs significantly on different sides of the coincidence. Similar behaviors were observed in both high (20m$^2$/Vs) and low (6m$^2$/Vs) mobility samples. More surprisingly, instead of reducing to zero at coincidence, as expected in the independent-electron model, the inter-valley gaps at $\nu$=3 and 5 rise rapidly towards the coincidence angles. We will discuss our results in the framework of two known models, level coupling with random-matrix elements and quantum Hall ferromagnetism, and show that the anomaly is related to the strong couplings of the Landau levels close in energy in the coincidence region. [1] K. Lai $et$ $al$., cond-mat/0510599.