Quantum-statistical theory for laser-tuned transport and optical conductivities of dressed electrons in $\alpha -{\mathcal{T}}_3$ materials

In the presence of external off-resonance and circularly polarized irradiation, we have derived a many-body formalism and performed a detailed numerical analysis for both the conduction and the optical currents in $\alpha -{\mathcal{T}}_3$ lattices. The calculated complex many-body dielectric function as well as conductivities of displacement and transport currents display strong dependence on the lattice-structure parameter $\alpha$, especially approaching the graphene limit with $\alpha \to 0$. Unique features in dispersion and damping of plasmon modes are observed with different $\alpha$ values, which are further accompanied by a reduced transport conductivity under irradiation. The discovery in this paper can be used for designing novel multifunctional nanoelectronic and nanoplasmonic devices.

Figure 1

Frequency $\left(\omega \right)$ and wave-vector- $(q$-) dependent polarization function ${\mathrm{\Pi }}_0(q,\omega |\varphi )$ (in units of $k_F^2/E_F)$ for the nonirradiated $\alpha -{\mathcal{T}}_3$ model $\left({\lambda }_0=0\right)$ as a function of $\hslash \omega$ with various phase values $\varphi$. Here, each panel corresponds to a chosen wave-vector $q=q_i$, and each curve is for a specific phase $\varphi$ as labeled. Two upper panels [(a) and (b)] present the imaginary part of ${\mathrm{\Pi }}_0(q,\omega |\varphi )$, whereas the two lower ones [(c) and (d)] are its real part. The inset in plot (b) illustrates all possible single-particle transitions (red double-arrow curves) contributing to ${\mathrm{\Pi }}_0(q,\omega |\varphi )$ at $T=0\mathrm{K}$ with a horizontal red line for the Fermi energy $E_F$.

Figure 2

Plasmon damping regions [(a) $\varphi =\pi /6$ and (b) $\varphi =\pi /4]$ and plasmon branches [(c) and (d)] for various types of $\alpha -{\mathcal{T}}_3$ lattices in the absence of external irradiation $\left({\lambda }_0=0\right)$. Two upper panels display $\mathrm{Im}\left[{\mathrm{\Pi }}^{\left(0\right)}(q,\omega |\varphi )\right]$, which indicate regions for single-particle excitations. Here, two partially damping regions above the $\hslash \omega =E_F$ line are highlighted by red circles. Panel (c) presents plasmon dispersions for fixed $\varphi =\pi /6$ but different ${\alpha }_0$ values, whereas panel (d) exhibits plasmon modes for fixed ${\alpha }_0=3.0$ and several phase values of $\varphi$. Plots (e) and (f) demonstrate the damping rates $\gamma \left[{\omega }_p\left(q\right)\right]$ of the plasmons shown in panels (c) and (d), correspondingly.

Figure 3

${\mathrm{\Pi }}_0(q,\omega |{\lambda }_0)$ in units of $k_F^2/E_F$ [(a)–(c)] and plasmon dispersions (d) for a dice lattice $\left(\varphi =\pi /4\right)$ under a circularly polarized laser field with various ${\lambda }_0$ values. Panel (a) shows the $q$ dependence of $\mathrm{Re}\left[{\mathrm{\Pi }}_0(q,\omega =0|{\lambda }_0)\right]$ in the static limit $\omega =0$. Plot (b) presents the $\hslash \omega$ dependence of $\mathrm{Im}\left[{\mathrm{\Pi }}_0(q,\omega =0|{\lambda }_0)\right]$ at $q/k_F=0.7$, whereas its inset displays the $\hslash \omega$ dependence for $\mathrm{Re}\left[{\mathrm{\Pi }}_0(q,\omega =0|{\lambda }_0)\right]$. Plot (c) demonstrates the particle-hole modes $\mathrm{Im}\left[{\mathrm{\Pi }}_0(q,\omega |\varphi ,{\lambda }_0)\right]\ne 0$ at ${\lambda }_0=0.5$.

Figure 4

(a) Real and (b) imaginary parts of optical-current conductivity ${\sigma }_O\left(\omega \right|{\lambda }_0)$ in units of $e^2/\hslash$ for an irradiated dice lattice $\left(\varphi =\pi /4\right)$ as a function of $\hslash \omega$. Here, each curve corresponds to a specific ${\lambda }_0$ value for fixed $\varphi$ and $T$.

Figure 5

Transport conductivity ${\sigma }_T\left({\lambda }_0\right)$ for an irradiated dice lattice. Panel (a) displays the relaxation time $\tau (k,{\lambda }_0)$ in units of ${\tau }_0=2\hslash E_F^{\left(0\right)}/\left(\pi n_i{\alpha }_0^2\right)$ as a function of wave-vector $k$ for various electron-light coupling constants ${\lambda }_0$ as labeled, where $E_F^{\left(0\right)}$ represents the Fermi energy for the case with ${\mathcal{E}}_0=0$. The dashed curves show the corresponding results without taking into account the laser-induced modification to the static dielectric function. Plot (b) presents the ratio ${\sigma }_T\left({\lambda }_0\right)/{\sigma }_T({\lambda }_0=0)$ as a function of ${\lambda }_0$ for different Fermi energies.