Dynamical conductivity of lithium-intercalated hexagonal boron nitride films: A memory function approach

This work is a study of dynamical conductivity of a quasi-two-dimensional heterostructure $\mathrm{Li}{\left(\mathrm{BN}\right)}_8$. The conducting electrons have a free-electron-like parabolic dispersion and are assumed to scatter only on acoustic phonons. The approach used to derive the generalized Drude relation with frequency and temperature dependent relaxation consists of defining the induced current density as a function of the electron-hole operator. This element is proportional to the perturbating electric field and is decomposed into the zeroth and second order of the electron-phonon interaction. The electron-hole operator is then evaluated using the Heisenberg equation. The connection between the electron-hole operator and the memory function is shown. The analytical properties of the imaginary and real part of the frequency and temperature dependent memory function, that describe electron scattering on acoustic and optical phonons, are examined in detail. Finally, the dynamical conductivity of $\mathrm{Li}{\left(\mathrm{BN}\right)}_8$ is calculated, with the explanation of all relevant features originating from the memory function.

Figure 1

(a) Top view and (b) side view of the three layered structure $\mathrm{Li}{\left(\mathrm{BN}\right)}_8$. The boron, nitrogen, and lithium atoms are represented by the colors red, yellow, and blue, respectively. The distance between the Li atoms is $a=5$ Å, while the thickness of the structure is $6.4\AA$. The two BN layers are shifted with respect to each other, so that the boron atom in the bottom BN layer lies below a nitrogen atom in the top BN layer and vice versa.

Figure 2

Diagrammatic representation of an expansion of the fully dressed irreducible electron-hole operator. Diagram $A$ is the zero order and diagrams $B_1$ to $C_2$ are the second order of the electron-phonon interaction denoted by a red wavy line. $C_1$ and $C_2$ describe the electron and hole single-particle self-energies. $B_1$ and $B_2$ describe the current vertex renormalization or phonon mediated electron-hole scattering.

Figure 3

(a) Imaginary and (b) real part of the memory function describing the scattering of an electron on an acoustic phonon. The real and imaginary parts of the memory function are calculated at various temperatures. The parameters used in the calculation are ${\varepsilon }_F=0.96$ eV, sound speed $c=3500$ m/s, $D_A=0.5$ eV, and Debye energy $\hslash {\omega }_D=20$ meV, with equivalent Debye temperature $T_D=230$ K. The imaginary part is calculated directly from Eq. (4.12), while the real part is obtained from Eq. (4.12) using the Kramers-Kronig relations.

Figure 4

Real (dashed line) and the imaginary part (solid line) of the memory function describing the scattering of an electron on an optical phonon, calculated for various temperatures. The parameters used in the calculation are ${\varepsilon }_F=0.96$ eV, $\hslash {\omega }_O=30$ meV, and $D_O=0.5$ eV.

Figure 5

(a) Real part of the dynamical conductivity Eq. (3.27) in units of ${\sigma }_0=e^2/\left(4\hslash \right)$ calculated at various temperatures with the residual scattering relaxation constant $\hslash \gamma =0.1$ meV. The conductivity contains the memory function for acoustic phonon Eq. (4.12) shown in Fig. 3. At $T=0$, a shark-fin shape dominates the intraband channel. The inset shows the temperature dependence of the dc resistivity. The characteristic temperature is $T_M=60$ K. (b) The $1/\tau \left(\omega \right)$ at $T=0$ (solid black line) and the $M_2(\omega ,0)$ (dashed line) calculated from Eqs. (4.20) and (4.21). The inset shows the dynamical mass for two temperatures.