ROUGH DRAFT authorea.com/91094

# Charge Density Waves, their Transport in Layered Materials, and Potential Applications

In this project, the effects of electron transport due to the Charge Density Wave (CDW) transition will be investigated. Following a review of how CDWs are formed, a review of several publications that discuss possible applications of CDW-based devices, and practical value of CDWs for electo-optical applications will be discussed. Specifically, pristine and Ca-intercalated bilayer graphene will be considered for photodetector and bolometer applications. Using transport formalism developed in class, figures of merit for such devices using CDWs will be discussed quantitatively.

Theory and Identification of CDW

Charge Density Waves occur due to a reconfiguration of both the electronic and lattice structures as a result of the interactions of the electrons with both themselves, and the lattice. Given that a certain amount of coupling between electrons and phonons exists, the phonon dispersion will undergo a strong change at lower temperatures. This causes the phonon mode to form a standing wave, (that is to say, a certain nonzero wavevector of energy zero.) Consequently, the electron density, which is coupled to the phonon density, also undergoes oscillations. The CDW state occurs when such a reconfiguration is energetically preferable.

Even though an exact, overlapping model for CDW occurence from any mechanism, and of any dimensionality is yet elusive, a simple 1D model exists that illustrates the CDW occurence at $$T=0K$$. Following the derivation proposed in the work by Rossnagel [1], which closely follows the mean-field theory approach taken by Gruner [2], an electronic susceptibility can be extracted.
Upon observing the behavior of this susceptibility as $$T->0$$, we can see that it features a singularity. This means that the regular ground state becomes unstable past a certain temperature, and a phase change will occur. It is possible to write down the total amount of energy change and also include the Coulomb and screening interactions that occur between electrons to obtain [1]:
$$\frac{4g^2_q}{\hbar\omega_q}-2U_q+V_q\geq\left(\frac{1}{\chi_0\left(q\right)}\right)$$
Here, $$g_q$$ represents the electron-phonon coupling strength, $$U_q$$ is the Coulomb interaction, and $$V_q$$ is the electron-electron screening potential. By using this criterion, it is possible to extract a well-defined phase-change temperature for when the CDW state emerges. CDW states are usually accompanied by a full or partial gap opening. In the case of the 1D Frohlich Hamiltonian, we can see that the band gap that forms around $$k_F$$ also changes the electron density of states around it: