Numerically "exact" charge transport dynamics in a dissipative electron-phonon model rationalizing the success of the transient localization scenario

TL;DR

The paper tackles the question of how low-frequency optical conductivity and TL behavior arise in molecular semiconductors when carrier–phonon coupling occurs with a realistic, dissipative phonon spectrum. Using numerically exact DEOM in momentum space with a Brownian-oscillator spectral density, the authors show that increasing damping broadens the phonon spectrum and drives the transport dynamics toward TL-like descriptions, with TL emerging already in the underdamped regime for rubrene-like parameters. The results reconcile TL with long-time quantum dynamics, showing that artifacts seen in delta-like (undamped) phonon models are mitigated by finite damping, and that the dc mobility remains within experimental bounds while the low-frequencyUpturns disappear at realistic damping. Additionally, the work highlights the limitations of the Drude–Lorentz bath and demonstrates how Ishizaki’s closure schemes enable stable, accurate transport calculations for dissipative, nonlocal carrier–phonon interactions. Overall, the study provides strong evidence that TLS remains a reliable phenomenology under physically realistic dissipation and offers a quantitative, exact framework for assessing transport in organic semiconductors and related materials.

Abstract

Optical conductivity in molecular semiconductors is suppressed in the terahertz region, featuring the displaced Drude peak that reflects carriers' transient localization (TL) by slow intermolecular vibrations. Meanwhile, recent computations in minimal models evidence optical-conductivity enhancements below the characteristic vibrational frequency, which cannot be captured by the TL phenomenology. These models assume that the carrier's hopping amplitude is modulated by a single undamped vibration. The modulation is, however, by many low-frequency modes, whose net effect can be approximated using a few effective damped oscillators. Here, we employ the dissipaton equations of motion (DEOM) method to compute the finite-temperature real-time current autocorrelation function in a one-dimensional model with Brownian-oscillator spectral density of nonlocal carrier-phonon interaction. We exploit the dissipaton algebra to handle the phonon-assisted current, reduce the method's computational requirements by working in momentum space, and confirm that numerically stable transport dynamics are virtually independent of a specific DEOM closing scheme. With increasing damping, we find that DEOM optical-conductivity profiles become increasingly qualitatively similar to TL predictions. For parameters representative of room-temperature hole transport in single-crystal rubrene, we conclude that the TL phenomenology is established already in the underdamped-oscillator regime. Reasonable variations in the damping constant weakly affect the carrier mobility, which remains within experimental bounds. Overall, our results strongly suggest that optical-conductivity enhancements at very low frequencies are artifacts of the assumed delta-like phonon spectrum and rationalize the success of the TL phenomenology in describing experimental data.

Figures (7)

• Figure 1: Frequency profile of (a) the BO SD [Eq. \ref{['Eq:J_BO_w']}] and (b) the BO frictional spectrum $\mathcal{J}_\mathrm{BO}(\omega)/\omega$ for different values of the damping $\gamma_0$. The curves for $\gamma_0/\omega_0=0.05$ are scaled down by a factor of 4 for visual clarity. The inset in (a) compares the BO SD for $\gamma_0/\omega_0=4/\sqrt{17}$ to the DL SD [Eq. \ref{['Eq:J_DL_omega']}] with $\gamma_\mathrm{DL}=\gamma_0/2$.
• Figure 2: (a) Real (black) and imaginary (red) parts of the bath correlation function $\mathcal{C}(t)$ computed using $K=2$ (symbols) and $K=15$ (lines) terms in the exponential decomposition in Eq. \ref{['Eq:def_mathcalC_t']}. The time-dependent diffusion constant (b) and the dynamical-mobility profile (c) computed using DEOM with $\Gamma(k,\mathbf{n})=0$ in Eq. \ref{['Eq:general_closing']} (label "TNL," thin lines) and with the Markovian-adiabatic [Eqs. \ref{['Eq:MA_closing']} and \ref{['Eq:def_gamma_k']}, label "MA," thick solid line] and derivative-resum [Eq. \ref{['Eq:DR_closing']}, label "DR," dashed line] closing terms with respect to $D$. The inset of panel (b) compares the diffusion-exponent dynamics computed using the same closing schemes. The model parameters are $J=1,\omega_0=0.044,\lambda=0.336,T=0.175,$ and $\gamma_0=1.7\omega_0$, while the maximum DEOM depth is $D=4$.
• Figure 3: (a) Magnitude of the relative deviation of the real part of the carrier's kinetic energy [Eq. \ref{['Eq:E_kin']}, up-triangles connected by a dashed line] and the carrier--bath interaction energy [Eq. \ref{['Eq:E_int']}, down-triangles connected by a dashed line] from the corresponding values at zero damping (denoted by $\langle O\rangle_0$) as a function of the damping parameter. (b) The magnitude of the ratio of the imaginary to the real part of the quantities considered in panel (a) as a function of the damping parameter. In both panels, the quantities on vertical axes are given in units of $10^{-5}$, while solid lines show the appropriate power-law scalings. The propagation of the i-DEOM [Eq. \ref{['Eq:im-time-deom']}] over the interval $[0,\beta]$ used 1000 imaginary-time steps.
• Figure 4: Dynamics of (a) the diffusion constant $\mathcal{D}$ and (b) the diffusion exponent $\alpha$ for different values of the chain length $N$ and maximum hierarchy depth $D$. (c) The dynamical-mobility profile $\mathrm{Re}\:\mu(\omega)$ for different values of $N$ and $D$. The insets in (a) and (b) zoom in the dynamics of $\mathcal{D}$ and $\alpha$, respectively, on intermediate-to-long timescales. The model parameters are $J=1,\omega_0=0.044,\lambda=0.336,T=0.175$, and $\gamma_0/\omega_0=0.05$. Note the logarithmic scale on the horizontal axes.
• Figure 5: Dynamics of (a) the diffusion constant $\mathcal{D}$ and (b) the diffusion exponent $\alpha$ for different values of the friction coefficient $\gamma_0$. (c) The dynamical-mobility profile $\mathrm{Re}\:\mu(\omega)$ for different values of $\gamma_0$. The TLS predictions employ $\alpha_d=2.2$. The insets in (a) and (b) display the dynamics of $\mathcal{D}$ and $\alpha$, respectively, on intermediate-to-long timescales. The model parameters are $J=1,\omega_0=0.044,\lambda=0.336,$ and $T=0.175$. DEOM computations are performed for $N=31,D=4$. Note the logarithmic scale on the horizontal axes.