Umklapp-driven first-order transition into a charge density wave phase

Using phenomenological Landau free energy, we systematically analyze the influence of so-called umklapp contributions on the phase diagram of a solid hosting a commensurate charge density wave phase. The umklapp terms may change the transition into the ordered state from continuous (second order) to discontinuous (first order). Additional elements, such as critical and tricritical points, emerge on the phase diagram under suitable conditions. The proposed mechanism is generalized for the case of a nearly commensurate charge density wave, as well as spin-density wave orders. Several charge-density-wave-hosting alloys are used as experimentally available examples illustrating the formalism.