A Unification of the Holstein Polaron and Dynamic Disorder Pictures of Charge Transport in Organic Crystals

We present a unified and nonperturbative method for calculating spectral and transport properties of Hamiltonians with simultaneous Holstein (diagonal) and Peierls (off-diagonal) electron-phonon coupling. Our approach is motivated by the separation of energy scales in organic molecular crystals, in which electrons couple to high-frequency intramolecular Holstein modes and to low-frequency intermolecular Peierls modes. We treat Peierls modes as quasiclassical dynamic disorder, while Holstein modes are included with a Lang-Firsov polaron transformation and no narrow-band approximation. Our method reduces to the popular polaron picture due to Holstein coupling and the dynamic disorder picture due to Peierls coupling. We derive an expression for efficient numerical evaluation of the frequency-resolved optical conductivity based on the Kubo formula and obtain the dc mobility from its zero-frequency component. We also use our method to calculate the electron-addition Green’s function corresponding to the inverse photoemission spectrum. For realistic parameters, temperature-dependent dc mobility is largely determined by the Peierls-induced dynamic disorder with minor quantitative corrections due to polaronic band narrowing, and an activated regime is not observed at relevant temperatures. In contrast, for frequency-resolved observables, a quantum-mechanical treatment of the Holstein coupling is qualitatively important for capturing the phonon replica satellite structure.

Popular Summary

Solids made from organic molecules are a promising class of semiconductors, however, their performance in electronic devices is typically inferior to that of inorganic semiconductors. This behavior is largely due to the strong interactions between electrons and nuclei, i.e., electron-phonon coupling, which has historically been modeled using a variety of theoretical approaches. Here, we provide a unification of the two most prominent theories, ultimately providing a new theoretical approach and arguing that only the unified theory is justified when modeling most organic semiconductors.

Our theory is developed for the simulation of electronic spectra and transport behavior at finite temperatures. We carefully combine a number of approximation techniques from many-body theory, including similarity transformations, finite-temperature mean-field theory, and quasiclassical quantum dynamics. Our theory predicts a temperature dependence of the electron mobility that is in good agreement with experimental results and highlights which physical effects are most responsible for the observed behaviors.

Compared to numerically exact techniques, ours is computationally efficient and can thus be used for realistic interactions or combined with techniques for strongly correlated electrons. This approach would also be well suited to address unconventional superconductivity in organic solids.

Figure 1

Momentum-resolved spectral function $A(k,\omega )$ at $T=300\mathrm{K}$ with no dynamic disorder (top row) and strong dynamic disorder ($g_P=2$, bottom row) at different values of Holstein coupling $g_H$. Calculated with electronic transfer integral $\tau =100\mathrm{meV}$, phonon frequencies ${\omega }_P=6\mathrm{meV}$ and ${\omega }_H=150\mathrm{meV}$, and a Lorentzian broadening with $\eta =2.5\mathrm{meV}$.

Figure 2

The per-carrier optical conductivity of the Holstein-Peierls model at different temperatures $T=100$, 300, and 1000 K. Results are shown for the Holstein-only case with $g_P=0$ [(a), (c), and (e)] and strong dynamic disorder similar to that of high-mobility OMCs $g_P=2$ [(b), (d), and (f)]. All other parameters are the same as in Fig. 1. Note that in panels (a), (b), (c), and (e), the zero-frequency conductivity is too large to show on the same scale as the other spectral features. See Fig. 3 for details of the zero-frequency conductivity.

Figure 3

dc (zero-frequency) mobility with zero [$g_P=0$, (a)] Peierls coupling, moderate Peierls coupling [$g_P=1$, (b)], and strong Peierls coupling [$g_P=2$, (c)] and various Holstein coupling. The top axis marks the major energy scales of the model, i.e., Peierls frequency (${\omega }_P$), Holstein frequency (${\omega }_H$), and the bare nearest-neighbor hopping $\tau$. The dashed lines show the coherent component of the mobility, where the Holstein coupling is evident only through the renormalized electronic energy $\tilde{\tau }$. Panel (c) compares to the rubrene $b$-axis Hall effect hole mobility measurements from Ref. [7]. A Gaussian broadening of 0.25 meV is used in all calculations to converge the conductivity at zero frequency; all other parameters are the same as in Fig. 1. Below the key on the top right, we plot idealized power laws $T^{-\gamma }$ for visual reference.

Figure 4

(a) Band-narrowed nearest-neighbor hopping $\tilde{\tau }$ (solid lines) for which the bare value is $\tau =100\mathrm{meV}$. Shaded regions represent 1 standard deviation in the distribution of values that $\tilde{\tau }$ can take under Peierls disorder with moderate coupling ($g_P=1$, darker shading) and strong coupling ($g_P=2$, lighter shading). (b) Energy-dependent localization length (solid lines) for zero Holstein coupling ($g_H=0$, red) and moderate Holstein coupling ($g_H=1$, blue) and thermally occupied density of states (shaded regions). Results are shown for $T=300\mathrm{K}$ and Peierls coupling $g_P=2$. (c) Temperature-dependent average localization length with Peierls coupling $g_P=2$. All other parameters are the same as in Fig. 1.

Figure 5

Momentum-resolved spectral function $A(k,\omega )$ at zero temperature obtained by numerically exact EDVHS results (top row) and approximate Lang-Firsov treatment (bottom row) at two different values of the Holstein electron-phonon coupling constant $g_H$. Results are calculated with the electronic transfer integral $\tau =100\mathrm{meV}$, Holstein phonon frequency ${\omega }_H=150\mathrm{meV}$, and a Lorentzian broadening with $\eta =10\mathrm{meV}$.

Figure 6

(a) Ground-state energy $E_0$ and (b) effective mass enhancement $m^{*}/m_0$ of the Holstein polaron at zero temperature calculated via approximate Lang-Firsov and exact EDHVS. Calculated for the weakly antiadiabatic case where $\tau =100\mathrm{meV}$ and ${\omega }_H=150\mathrm{meV}$.

Figure 7

The per-particle optical conductivity $\sigma (\omega )/n{e}_0$ of the Holstein model at zero temperature obtained by numerically exact EDVHS and approximate Lang-Firsov treatment. Insets show the non-Drude regular conductivity ${\sigma }_{\mathrm{reg}}(\omega )=\sigma (\omega )-D\delta (\omega )$. Parameters are the same as in Fig. 5 except the broadening is $\eta =15\mathrm{meV}$. Note that the vertical scales of (a) and (b) differ by a factor of 10.