Two-dimensional Wigner lattice in a magnetic field and in the presence of a random array of pinning centers

We consider a two-dimensional Wigner lattice in the presence of a static external magnetic field perpendicular to the lattice and a random array of pinning centers. The phonons of the electron lattice are treated in the harmonic approximation, and the random distribution of pinning centers is dealt with in the single-site coherent-potential approximation. We obtain the average phonon Green's function and from it the spectral density for the phonons of the electron lattice. The presence of the random array of pinning centers gives rise to a low-frequency gap in the phonon spectral density. Thus the (two-dimensional) mean-square displacement of an electron about its equilibrium lattice site is finite at finite temperatures. The magnitude of the low-frequency gap decreases as the strength of the magnetic field is increased. For sufficiently high fields the gap disappears, but the phonon spectral density vanishes sufficiently fast as $\omega \to 0$ that the mean-square displacement remains finite at finite temperatures. We present detailed numerical results for the phonon spectral density and electron mean-square displacement, both at $T=0\mathrm{}$ K and at finite temperatures. Our results are analyzed from the point of view of the stability of the lattice to thermal fluctuations. We find that at low temperatures ($T\sim 1$ K) both the pinning centers and the magnetic field considerably enhance the localization of the electrons about their lattice sites, thus enhancing the stability of the lattice. At higher temperatures, the electron mean-square displacement depends very weakly on the strength of the magnetic field, and its value is then determined by the concentration and strength of the pinning centers.