Real-Time Out-of-Equilibrium Quantum Dynamics in Disordered Materials

TL;DR

The work introduces a real-time, linear-scaling method to simulate nonequilibrium electron dynamics in large, disordered materials using a Chebyshev expansion of spectral functions and time evolution. It evolves two vectors, one for the instantaneous state and one carrying the initial distribution, with a relaxation term toward an instantaneous equilibrium $\hat{N}_{\mathrm{eq}}(t)$ on a timescale $\tau$, linking to a Liouville-like equation for the density matrix. The equilibrium parameters $T(t)$ and $\mu(t)$ are determined at each step to conserve energy and carrier density, with $\hat{N}_{\mathrm{eq}}(t) = \hat{F}(\hat{H}(t),T(t),\mu(t))$. Applying the framework to graphene and nanoporous graphene, the authors reproduce universal and saturable optical absorption, reveal disorder-enhanced absorption and anisotropy, and demonstrate scalability to millions of atoms, enabling exploration of far-from-equilibrium dynamics in realistic disordered systems via time-dependent tight-binding descriptions.

Abstract

We report a linear-scaling numerical method for exploring nonequilibrium electron dynamics in systems of arbitrary complexity. Based on the Chebyshev expansion of the time evolution of the single-particle density matrix, the method gives access to nonperturbative excitation and relaxation phenomena in models of disordered materials with sizes on the experimental scale. After validating the method by applying it to saturable optical absorption in clean graphene, we uncover that disorder can enhance absorption in graphene and that the interplay between light, anisotropy, and disorder in nanoporous graphene might be appealing for sensing applications. Beyond the optical properties of graphene-like materials, the method can be applied to a wide range of large-area materials and systems with arbitrary descriptions of defects and disorder.

Figures (4)

• Figure 1: Time evolution of the graphene carrier occupation for photon energies (a) $\hbar\omega_p = 0.8$ eV and (b) $0.6$ eV, with $\mu_0 = 0$. The dashed lines indicate the expected absorption peak at $\hbar\omega_p/2$. (c) Evolution of $n$- and $p$-doped distributions with $\mu_0 = \pm 0.6$ eV (solid and dashed lines, respectively), indicating Pauli blocking. (d) Absorption efficiency vs. photon energy for different optical intensities, with the universal optical absorption of monolayer graphene $\eta\approx2.3\%$ highlighted by the yellow dashed line. In all cases, $T_0 = 0$ K with a pulse envelope $P(t)=\mathop{\mathrm{sech}}\nolimits\left[ (t-2T_p) / \gamma T_p \right]$, where $T_p = 2\pi / \omega_p$ and $\gamma \approx 0.5673$.
• Figure 2: (a) Energy absorption efficiency vs. light intensity for clean graphene with $\mu_0=-5$ meV and $T_0 = 300$ K, under a 63-fs optical pulse with $\hbar\omega_p = 1.55$ eV. Black symbols are experimental data baek2012efficient, and colored lines are simulations with varying $\tau_\mathrm{ee}$. The solid line and the shaded area for the MD data correspond, respectively, to the mean optical absorption and its standard deviation over $10$ graphene samples. (b) Time-dependent carrier distribution at high intensity ($I_0 = 3.02\times10^4$ MW/cm$^2$) for $\tau_\mathrm{ee} = 25$ fs. (c) Energy absorption efficiency vs. light intensity in the presence of Anderson disorder, for the same conditions as in panel (a), with $\tau_\mathrm{ee} = 25$ fs. (d) Time-dependent carrier distribution at high intensity ($I_0 = 3.02\times10^4$ MW/cm$^2$) for Anderson disorder strength $W=3$ eV.
• Figure 3: (a) Anisotropic energy absorption of NPG for $\mu_0 = -0.1$ eV, $T_0 = 300$ K and $\tau_\mathrm{ee} = 25$ fs under a $97.65$-fs pulse with $\hbar\omega_p = 1$ eV at various optical intensities. Inset: Schematic of NPG used in our simulations. The red arrows are the ribbon axes and the blue arrow is the direction of light polarization. (b) Angle-dependent energy absorption for disordered NPG under the same conditions, for $I_0 = 3.142 \times 10^3$ MW/cm$^2$ and various Anderson disorder strengths $W$.
• Figure S4: (a) Density of states of graphene with $512\times512\times4$ carbon atoms for various Anderson disorder strengths $W$, computed with $M=1024$ Chebyshev polynomials. (b) Density of states of a nanoporous graphene system composed of $64\times128\times80$, using $M=1536$ Chebyshev polynomials.