Melting of electronic and excitonic crystals in 2D semiconductor moir\'e patterns: a perspective from the Lindemann criterion

Using the Lindemann criterion, we analyzed the quantum and thermal melting of electronic/excitonic crystals recently discovered in two-dimensional (2D) semiconductor moir\'e patterns. We show that the finite 2D screening of the atomically thin material can suppress (enhance) the inter-site Coulomb (dipolar) interaction strength, thus inhibits (facilitates) the formation of the electronic (excitonic) crystal. Meanwhile, a strong enough moir\'e confinement is found to be essential for realizing the crystal phase with a wavelength near 10 nm or shorter. From the calculated Lindemann ratio which quantifies the fluctuation of the site displacement, we estimate that the crystal will melt into a liquid above a critical temperature ranging from several tens Kelvin to above 100 K (depending on the system parameters).