Theory of Phonon Shakeup Effects on Photoluminescence from the Wigner Crystal in a Strong Magnetic Field

We develop a method to compute shakeup effects on photoluminescence from a strong magnetic field induced two-dimensional Wigner crystal. Only localized holes are considered. Our method treats the lattice electrons and the tunneling electron on an equal footing, and uses a quantum-mechanical calculation of the collective modes that does not depend in any way on a harmonic approximation. We find that shakeup produces a series of sidebands that may be identified with maxima in the collective mode density of states, and definitively distinguishes the crystal state from a liquid state in the absence of electron-hole interaction. In the presence of electron-hole interaction, sidebands also appear in the liquid state coming from short-range density fluctuations around the hole. However, the sidebands in the liquid state and the crystal state have different qualitative behaviors. We also find a shift in the main luminescence peak, that is associated with lattice relaxation in the vicinity of a vacancy. The relationship of the shakeup spectrum with previous mean-field calculations is discussed.