Carrier transport calculations of organic semiconductors with static and dynamic disorder

Recently, organic materials have attracted much attention as low-cost semiconducting materials for flexible and printed electronics, because the intrinsic mobility of single-crystal organic semiconductors exceeds that of amorphous silicon. However, technological progress in the field of organic electronics requires a much better fundamental understanding of the nature of their charge transport mechanisms.

Up to the 1950s, organic materials such as carbon compounds were recognized as being insulators. However, graphite consisting of benzene rings has delocalized π electrons and is a good electrical conductor. In 1954, Akamatu and Inokuchi reported that a molecular material consisting of violanthrone molecules with π-electrons exhibits some electrical conductivity.^1,2) Eley and Vartanyan also reported the electrical conductivities of the organic molecular materials phthalocyanine³⁾ and cyanine dye,⁴⁾ These electrical conductivities were quite low at about 10⁻⁸ S m⁻¹, because of low carrier concentrations. Furthermore, a large amount of charge scattering occurred doe to extrinsic static disorder, such as low crystallinity and the existence of impurities.

In the 1970s, Ferraris and Shirakawa reported that charge-transfer complexes⁵⁾ and chemically doped polyacetylene film⁶⁾ showed high conductivities, comparable to that of metals. Then, in 1984, Kudo determined the mobility of organic semiconductors using a field effect transistor (FET) of merocyanine dye and showed that the mobility of the organic semiconductors ranges from 10⁻⁵ to 10⁻⁷ cm² V⁻¹ s⁻¹.⁷⁾ Subsequently, FET mobilities have increased year by year due to improvements in crystallinity from amorphous materials to polycrystals with large domains. In the late 1990s, the mobility of pentacene polycrystals reached 1 cm² V⁻¹ s⁻¹, comparable to that of amorphous silicon.⁸⁾ The maximum FET mobility of rubrene single crystals has reached in excess of 10 cm² V⁻¹ s⁻¹.^9,10) Thermally activated behavior $\exp \left\{-{E}_{a}/({k}_{{\rm{B}}}T)\right\}$ of the mobility is changed to a power-law behavior ${T}^{-n}$ as the mobility increases.⁹⁾ Therefore, it is believed that coherent band-like transport, and not hopping transport, is realized in high-mobility organic semiconductors. In fact, the Hall effect^11,12) and band structure^13–15) of organic semiconductors have been experimentally observed.

Since the effect of extrinsic static disorder can be significantly reduced in single-crystal FET devices, it is believed that we can access the intrinsic charge transport properties, which are dominated by dynamic disorder due to interactions between electrons and molecular vibrations. Molecular vibrations give rise to large dynamic disorder in the intermolecular transfer integrals, whose amplitudes can be theoretically evaluated and can reach several 10s of meV.^16–19) On the other hand, it has been reported that extrinsic static disorder produces a carrier-trap potential in organic semiconductors.^20,21) Even in a crystalline domain of pentacene thin films, there exists a trap potential with a depth of at least ~10 meV.²²⁾ The depth of the trap potential of static disorder is a similar energy to fluctuation width of the transfer integral due to dynamic disorder around room temperature. These results imply that the competition between intrinsic disorder and dynamic disorder is important for understanding the experimentally observed charge transport properties.

In the present paper, we investigate the roles of static and dynamic disorder on the charge transport properties of organic semiconductors, using the Kubo formula based on the quantum mechanical calculations of hole wavepacket dynamics. We show that transport properties such as the temperature dependence of mobility, are determined by the subtle balance between static disorder and dynamic disorder.

The mobility of a carrier with charge q is obtained from the following Einstein relation

$$\begin{eqnarray}&&{\mu }_{x}=\displaystyle \frac{{{qD}}_{x}}{{k}_{{\rm{B}}}T},\end{eqnarray} \tag{ 1 }$$

where T is the temperature. In our approach based on quantum dynamics, called the time-dependent wavepacket-diffusion (TD-WPD) method, we first calculate the diffusion constant D_x for a charge carrier along the x direction by ${D}_{x}\equiv {\mathrm{lim}}_{t\to +\infty }{D}_{x}(t)$ using Kubo's linear response theory.^23,24) D_x(t) is called the time-dependent diffusion coefficient and is defined using the velocity correlation function as

$$\begin{eqnarray}&&{D}_{x}(t)\equiv \displaystyle \frac{1}{\beta }{\int }_{0}^{t}{ds}{\int }_{0}^{\beta }d\lambda \ {Tr}\{\hat{\rho }{\hat{v}}_{x}(-i{\hslash }\lambda ){\hat{v}}_{x}(s)\}.\end{eqnarray} \tag{ 2 }$$

Here, the density operator is defined as $\hat{\rho }\,\equiv \exp (-\beta \hat{H})/{Tr}\{\exp (-\beta \hat{H})\}$ using the Hamiltonian for charge carriers $\hat{H}$ and the inverse temperature $\beta \equiv 1/({k}_{{\rm{B}}}T)$. The velocity operator at time s in the Heisenberg representation is given by ${\hat{v}}_{x}(s)\equiv {\hat{U}}^{\dagger }(s){\hat{v}}_{x}\hat{U}(s)$, where $i{\hslash }{\hat{v}}_{x}\equiv [\hat{H},\hat{x}]$ and $\hat{U}(s)$ is the time-evolution operator from the initial time 0 to s. Equation (2) is computed by our previously reported order-N method.²⁴⁾ D_x(t) has different time-dependent behavior depending on the different charge transport phenomena.

In the case of a diffusive band-transport mechanism, the correlation disappears gradually through scattering, and is given by $\langle {\hat{v}}_{x}(0){\hat{v}}_{x}(s)\rangle ={v}_{{\rm{F}}}^{2}{e}^{-s/\tau }$. Here, v_F and τ represent the group velocity at the Fermi energy and scattering time, respectively. The time-dependent behavior of D(t) is obtained as ${D}_{x}(t)={v}_{{\rm{F}}}^{2}\tau \left(1-{e}^{-t/\tau }\right)$ and is shown by the green curve in Fig. 1. The diffusion constant D_x has a finite value of ${v}_{{\rm{F}}}^{2}\tau$. On the other hand, when the system has a large amount of static disorder, charge carriers are spatially localized in a phenomenon known as Anderson-localization. Using simple algebra, D_x(t) can be rewritten as ${D}_{x}(t)=\langle \{\hat{x}(t)-\hat{x}(0)\}{}^{2}\rangle /(2t)\simeq {L}_{\xi }^{2}/(2t)$, where L_ξ is the time-independent localization length. As shown by the red curve in Fig. 1, the diffusion constant becomes zero.

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The time-dependent Hamiltonian for electrons interacting with dynamic disorder and static disorder is written as

$$\begin{eqnarray}&&\hat{H}(t)=\displaystyle \sum _{n\gt m}{\gamma }_{{nm}}(t)\left({\hat{a}}_{n}^{\dagger }{\hat{a}}_{m}+{\hat{a}}_{m}^{\dagger }{\hat{a}}_{n}\right)+\displaystyle \sum _{n}{V}_{n}{\hat{a}}_{n}^{\dagger }{\hat{a}}_{n},\end{eqnarray} \tag{ 3 }$$

where ${\hat{a}}_{n}$ represents the annihilation operator for an electron in the nth orbital. The dynamic disorder originating from molecular vibrations (phonons) induces the dynamically fluctuating transfer integrals γ_nm(t) because the spatial overlap between neighboring orbitals is modulated as a function of time. The effects of static disorder, such as impurities, can be included in the on-site potential energy V_n. In the TD-WPD method, we carry out charge transport calculations of covalent-bond systems and van der Waals bond systems, including dynamic and static disorder on an equal footing without perturbative treatment.

We first focus on the difference in the magnitude of the dynamic disorder between covalent-bond systems and van der Waals bond systems. Figure 2 show the calculated electronic band structure and phonon-band structure of a (10,0) carbon nanotube (CNT) as an example of a covalent-bond system and those of dinaphtho[2,3-b:2',37-d]thiophene (DNT-V)²⁵⁾ as an example of a high-mobility organic semiconductor. Weak intermolecular van der Waal interactions of organic semiconductors bring narrower electronic- and phononic band widths compared with covalent-bond systems.

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We compare the magnitude of electron–phonon coupling of organic semiconductors to that of covalent-bond systems. The vibration amplitude of carbon atoms in CNTs at 300 K can be estimated as ${\rm{\Delta }}{r}_{c}=\sqrt{{\hslash }{n}_{{ph}}/({m}_{c}\omega )}\,\sim \sqrt{{k}_{{\rm{B}}}T/({m}_{c}{\omega }^{2})}\sim 0.095\,\mathring{\rm A}$, where ${m}_{c}{\omega }^{2}$ corresponds to the elastic constant between atoms, and m_c = 12 u and hω = 1600 cm⁻¹. When the fluctuation amplitude of the transfer integrals Δγ due to phonons is defined by $\gamma ={\gamma }^{0}\pm {\rm{\Delta }}\gamma$, where γ⁰ = 2.5 eV is the transfer integral at the equilibrium position, we obtain Δγ ~ 0.12 eV from Harrison's rule of $\gamma ={\gamma }^{0}/| ({r}^{0}\pm {\rm{\Delta }}{r}_{c})/{r}^{0}{| }^{2}$.²⁶⁾ Here, r⁰ represents the equilibrium distance between neighboring carbon atoms and is set to 1.44 Å. A perturbative treatment of electron–phonon coupling can be valid for covalent-bond systems since the fluctuation amplitude Δγ is much smaller than γ⁰. Therefore, as discussed in our previous studies,^27,28) the charge transport properties of CNTs calculated by the TD-WPD method agrees very well with the results using ideal coherent band-transport theory based on the deformation potential model.²⁹⁾ The calculated D(t) exhibits typical diffusive band-transport behavior, as shown by the green curve in Fig. 1.²⁷⁾

For organic semiconductors, the molecular vibration amplitude is estimated as follows, within a hard sphere approximation for a single molecule. The amplitude at 300 K is estimated as ${\rm{\Delta }}R\sim \sqrt{{k}_{{\rm{B}}}T/(M{\omega }^{2})}\sim 0.15\,\mathring{\rm A}$, where the mass of a single molecule is M = 284 u and hω = 200 cm⁻¹. A change in the intermolecular transfer integral is obtained as Δγ ~ 0.016 eV, which is comparable to γ⁰ = 0.050 eV. The calculated results indicate that electron–phonon coupling in organic semiconductors significantly influences the charge transport and requires a non-perturbative treatment.

To clarify the basic nature of charge transport in organic semiconductors, we employ a simplified one-dimensional model with parameters of typical organic semiconductors.^24,30,31) The calculated time-dependent diffusion coefficient at 300 K is shown by the red solid line in Fig. 3. The obtained mobility using Eq. (1) is 4 cm² V⁻¹ s⁻¹. The time-dependent behavior of D(t) has a negative slope from 20 to 100 fs, which is similar to Anderson-localization behavior, but has a constant finite value in the long-time limit, which resembles diffusive band-transport behavior. Anderson-localization originating from quantum interference between clockwise and counterclockwise scattering paths can be observed when charge carriers are scattered by static disorder, but not dynamic disorder, because phase coherence should be conserved in scattering events. To clarify the role of the dynamic nature of molecular vibration on D_x(t), we calculate D_x(t) in the presence of static disorder. Here, static disorder is produced using a snapshot at t = 0 of the geometry of molecular dynamics. The calculated D_x(t) shown by the red broken line exhibits typical Anderson-localization behavior. The vertical broken line corresponds to an oscillating period of 1/ω₀ = 60 fs of molecular vibrations. A comparison between the two calculated results shows that molecular vibrations localize the charge carriers in a short time regime of 0 to 1/ω₀, before the charge carriers start moving diffusively by destroying the localization state due to the dynamic nature of the molecular vibrations. The quantum correction beyond the Boltzmann transport theory is very important for understanding the intrinsic charge transport mechanism of organic semiconductors. This phenomenon is called transient localization.³⁰⁾

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The huge variety of organic molecules is a feature of organic semiconductors and implies that we can discover new materials with higher mobility or novel functions. The ability to quantitatively predict the charge transport properties of different organic semiconductors based on their crystal structures is a powerful tool for material screening when developing new materials. In Sect. 3, a "toy model" is used, namely a one-dimensional model for electronic states and a hard sphere approximation for molecular vibrations, which is too simple to quantitatively evaluate the intrinsic mobility including the temperature dependence. Recently, we reported the TD-WPD method for two-dimensional organic semiconductors, for numerically evaluating the mobilities of charge coupling with all normal phonon modes using wavepacket dynamics and all-atom normal-mode analysis.²⁴⁾

4.1. Treatment of electron–phonon coupling of organic semiconductors beyond the toy model

Although the hard sphere approximation gives only six phonon modes (translational modes) with frequencies less than 200 cm⁻¹ because there are two molecules in a unit cell of organic semiconductors with a Herringbone structure, as discussed in this paper, organic semiconductors have $(3N-3)$ phonon modes ranging from 0 to 1600 cm⁻¹³²⁾ where N is the number of atoms in a unit cell. These phonon modes are mainly classified into low-frequency intermolecular vibrational modes and high-frequency intramolecular vibrational modes. The electron propagation in organic semiconductors is determined by the interactions with the inter- and intramolecular vibrations.

To properly consider these interactions, we must separate the slow and fast interactions with respect to the characteristic time of electron dynamics.³³⁾ The time for Bloch wave formation has been estimated to be longer than h/γ⁰ 40 fs since the bare transfer integrals γ⁰ of typical organic semiconductors are smaller than 0.1 eV. The intramolecular vibrations strongly coupled with the π electron states of a molecule have frequencies ω^intra between 1000 and 1600 cm⁻¹ (0.12 and 0.20 eV),³⁴⁾ which correspond to intramolecular carbon stretching modes. Since these fast interactions arise prior to the formation of the Bloch wave, they have the effect of dressing the charge with an intramolecular distortion cloud, consequently leading to renormalization of the bare transfer integrals, called the band narrowing effect, by polaron formation.^33,35) On the other hand, intermolecular vibrations such as the translational mode of rigid molecules, have a characteristic time much longer than h/γ⁰ because these frequencies ω^inter range from 0 to 100 cm⁻¹ (0 to 0.012 eV). The slow and large intermolecular vibrations scatter the Bloch waves and induce transient localization, as shown in Fig. 3.³⁰⁾

The transfer integral between the Nth and Mth molecular orbitals of a hole coupled with inter- and intramolecular vibrations is written as

$$\begin{eqnarray}&&{\gamma }_{{NM}}(t)={\alpha }^{{\rm{intra}}}\{{\gamma }_{{NM}}^{0}+{\alpha }_{{NM}}^{{\rm{inter}}}(t)\},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{\alpha }_{{NM}}^{{\rm{inter}}}(t)=\displaystyle \sum _{l,{\bf{q}}}{\rm{\Delta }}{\gamma }_{{NMl}{\bf{q}}}^{{\rm{inter}}}\sin ({\omega }_{l{\bf{q}}}^{{\rm{inter}}}t+{\bf{q}}\cdot {{\bf{R}}}_{\xi }+{\phi }_{l{\bf{q}}}),\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{\alpha }^{{\rm{intra}}}=\exp \left\{-(\lambda /2{\hslash }{\omega }^{{\rm{intra}}})\coth \left(\beta {\hslash }{\omega }^{{\rm{intra}}}/2\right)\right\},\end{eqnarray} \tag{ 6 }$$

where γ⁰ represents the bare transfer integral. The dynamic disorder due to intermolecular vibrations is included in ${\alpha }_{{NM}}^{{\rm{inter}}}(t)$ as a function of the frequency ${\omega }_{l{\bf{q}}}^{{\rm{inter}}}$ of the lth mode with wavenumber q and initial random phase ${\phi }_{l{\bf{q}}}$. Here we employ the all-atom normal-mode analysis beyond the hard sphere approximation, namely, the amplitude of the dynamic disorder is given by ${\rm{\Delta }}{\gamma }_{{NMl}{\bf{q}}}^{{\rm{inter}}}=g({\omega }_{l{\bf{q}}})(\partial {\gamma }_{{NM}}^{0}/\partial {{\bf{u}}}^{l{\bf{q}}}){\rm{\Delta }}{{\bf{r}}}^{l{\bf{q}}}$, where the component of ${{\bf{u}}}^{l{\bf{q}}}$ with respect to the nth atom is defined by ${{\bf{e}}}_{n}^{l{\bf{q}}}/\sqrt{{m}_{n}}$ using the eigenvectors ${{\bf{e}}}^{l{\bf{q}}}$ of the dynamical matrix based on the all-atom normal-mode analysis and the mass m_n of the nth atom. The amplitude of the displacement is given by ${\rm{\Delta }}{{\bf{r}}}^{l{\bf{q}}}=\sqrt{{\hslash }{n}_{l{\bf{q}}}/2{\omega }_{l{\bf{q}}}}{{\bf{u}}}^{l{\bf{q}}}$, where ${n}_{l{\bf{q}}}$ is the phonon number excited at temperature T. Taking into account only molecular vibrations slower than the Bloch wave formation, we introduce a filtering function g(ω).²⁴⁾ The fast intramolecular vibrations renormalize the bare transfer integral by small-polaron formation. The renormalization factor is obtained as ${\alpha }^{{\rm{intra}}}=\exp \left\{-(\lambda /2{\hslash }{\omega }^{{\rm{intra}}})\coth \left(\beta {\hslash }{\omega }^{{\rm{intra}}}\,/2\right)\right\}$, where λ and ω^intra represent the reorganization energy and the frequency of the normal-mode with the most significant contribution, respectively.³⁶⁾

We can compute the bare transfer integrals ${\gamma }^{0}$ from the experimentally observed crystal structure using the Wannier method^37,38) or dimer method^39,40) based on density functional theory. To reduce the computational cost, we employ a dimer method using the PBE functional in this work. We take the highest occupied molecular orbitals (HOMOs) and the second HOMOs (SHOMOs) as the basis set of $\hat{H}$ because we found that the dispersion at the HOMO band top of some materials is affected by the transfer integrals between the HOMO and SHOMO.⁴¹⁾ The normal vibrational modes of the crystal are obtained from the dynamical matrix constructed by the force field MMFF94s using CONFLEX.⁴²⁾ The quantity of $\partial {\gamma }^{0}/\partial {{\bf{u}}}^{l{\bf{q}}}$ for each normal vibrational mode can be computed by numerical differentiation. The λ in α^intra is calculated by the adiabatic potential energy surface method⁴³⁾ using the B3LYP/6-31G(d) level⁴⁴⁾ derived by GAMESS.⁴⁵⁾ We employ 0.15 eV as a typical value of ω^intra in this work.³⁴⁾ The mobility is obtained using Eqs. (1) and (2) within the framework of the TD-WPD method. We employ a monolayer consisting of 200 × 200 unit cells, which corresponds to a single-crystal with an area larger than 1 × 1 μm² containing 1.6 × 10⁵ molecular orbitals. The carrier dynamics is computed up to 2 ps with a time step of 0.5 fs.

4.2. Effects of dynamic disorder induced by molecular vibrations on transport properties

We investigate the intrinsic hole mobilities restricted by dynamic disorder induced by the molecular vibrations. We apply the TD-WPD method to single crystals of pentacene (LT phase)⁴⁶⁾ and [1]benzothieno[3,2-b][1]benzothiophene (C₈-BTBT).⁴⁷⁾ We then investigate the recently reported high-mobility organic semiconductor, decyl-substituted dinaphtho[2,3-d:2',3'-d']benzo[1,2-b:4,5-b']dithiophene (C₁₀-DNBDT stand phase).⁴⁸⁾ This molecule has another crystal phase called the sleep phase. Figures 4(a) and 4(b) show the calculated temperature dependence of the mobility of these materials along the column direction. For comparison, we also show the experimental FET mobilities of single-crystal devices. Our simulation method can reproduce the difference in magnitude of mobility among these materials. Furthermore, the power-law exponents of C₁₀-DNBDT (stand) and C₈-BTBT are −0.88 and −1.44, in very good agreement with the experimental values of −0.85 and −1.1, shown as open symbols, respectively.^49,50) In contrast to the above, single-crystal pentacene is known to exhibit an intermediate character between hopping and band-transport, resulting in temperature-independent mobility.⁵²⁾ The present simulations also successfully reproduce this behavior, as shown in Fig. 4(b) in the temperature range from 200 to 300 K. We note that the mobility of the C₁₀-DNBDT stand phase has been experimentally determined but not that of the sleep phase. Our simulation method can predict the hole mobility at 300 K of the sleep phase using the experimentally obtained single-crystal structure and we find that the sleep phase has a much lower mobility of 1.5 cm² V⁻¹ s⁻¹ with an activation energy of 9.5 meV, than the value of 25.0 cm² V⁻¹ s⁻¹ for the stand phase. The large difference between the two phases is notable. Experimental results for the sleep phase are anticipated.

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To clarify the origin of the different temperature dependences of the mobility for these materials, we investigate the time-dependent behavior of D_x(t). Figure 4(c) shows the calculated results for pentacene, C₈-BTBT, and the stand and sleep phases of C₁₀-DNBDT. The transient localization behavior for C₁₀-DNBDT (sleep) is clearly observed, resulting in thermally activated behavior at low mobility. On the other hand, we do not observe a negative slope of D_x(t) for C₁₀-DNBDT (stand) and C₈-BTBT. The time-dependent behavior of D_x(t) resembles that of ideal band-transport, but we confirm that the HOMO band-edge states are spatially localized by the large dynamic disorder owing to the intermolecular vibrations. Therefore, the mobilities calculated by the TD-WPD method are lower than those obtained from the band-transport theory without quantum corrections.²⁴⁾

4.3. Competition between intrinsic dynamic disorder and extrinsic static disorder

We next introduce the effect of extrinsic static disorder on the charge transport properties, because there will inevitably be impurities and defects even in single-crystal organic semiconductors. The effect has been recognized as the creation of gap states and trap potentials in previous studies.^20–22) It is different from intrinsic dynamic disorder, and it is difficult to quantitatively evaluate the trap potentials because its microscopic origin has not yet been specified and may depend on each device.

Electron spin resonance (ESR) measurements can detect the spatial extent of the charge carriers of organic semiconductors.⁵³⁾ The analysis of the ESR spectra of pentacene thin-film transistors at a low enough temperature of 20 K showed that the major trap levels comprise localized wavefunctions spanning around 1.5 and 5 molecules and a broad feature at 6–20 molecules.⁵⁴⁾

To reproduce the experimentally identified spatial extent of charge carriers, we introduce a trap potential to the on-site potential energy V_n of Eq. (3). The trap potential consists of the spatially uniform disorder potentials V_n^C with a Gaussian distribution having an energy width of 0.010 eV, the delta-function potential V_n^B with a potential depth of 0.090 eV, and the delta-function potential V_n^A with a deeper potential depth of 0.300 eV. Here, the spatial extent of the nth eigenfunction Ψ_n(r) can be evaluated by ${N}_{{\rm{extent}}}\equiv {\left({\sum }_{i}| {C}_{i}^{n}{| }^{4}\right)}^{-1}$, where the eigenfunction is written as a linear combination of molecular orbitals Φ_i(r), namely ${{\rm{\Psi }}}_{n}({\bf{r}})={\sum }_{i}{C}_{i}^{n}{{\rm{\Phi }}}_{i}({\bf{r}})$. The eigenfunctions are computed using the numerical diagonalization of $\hat{H}$ with 60 × 60 unit cells (7200 molecules). The delta-function potentials, V_n^A and V_n^B, create strongly localized states of around 1.5 and 5 molecules in pentacene crystal, respectively. The Gaussian disorder potential V_n^C produces weakly bound states at the edge of the HOMO band, whose spatial extent is larger than 10 molecules.

The black dots of Fig. 5(a) represent the calculated HOMO band structure renormalized by polaron formation. For comparison, the bare-band structure is shown by gray dots. When we set the concentrations of the delta-function potentials V^A and V^B to 0.05% and 0.2%, respectively, the density of states (DoS) and spatial extent of the eigenfunctions as a function of energy are obtained as shown in Figs. 5(b) and 5(c). We can confirm that V_n^A and V_n^B create localized states at E = 0.17 and 0.015 eV, as shown by red arrows. Our calculated energy distribution of the spatial extent is consistent with previous theoretical work using a square lattice model.⁵⁵⁾ The statistical distribution of the spatial extent of the wavefunctions of pentacene crystal with trap potential V is shown in Fig. 5(d). The result is averaged over 80 trap potentials V with different geometries. The obtained spatial extent of Fig. 5(d) quantitatively agrees very well with the experimentally identified extent (Fig. 4 of Ref. 54).

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Finally, using the obtained trap potentials, we investigate the competition between intrinsic dynamic disorder and extrinsic static disorder on the charge transport properties around room temperature. Figure 5(e) represents the calculated mobility as a function of temperature for the pentacene crystal with static and dynamic disorder. The introduction of static disorder, namely the trap potential V, significantly changes the temperature dependence of the mobility, from a temperature-independent behavior to thermally activated behavior. Furthermore the magnitude of mobility at low temperature is considerably reduced, for example, to 10⁻³ cm² V⁻¹ s⁻¹ at 200 K. Even if the same organic semiconductors are employed to fabricate FETs, a significantly different temperature dependence of mobility for each FET is often observed in experiments.⁸⁾ Our result implies each device has different static disorder.

To understand the origin of the change in temperature dependence of the mobility, we investigate the DoS for the case of coexistence of static and dynamic disorder. In Fig. 5(b), we discussed the DoS dominated by trap potentials at low temperature. When the temperature increases to 300 K, the DoS is changed as shown in Fig. 5(f). The peak due to carriers trapped at V^B is buried in the tail of the HOMO band induced by dynamic disorder on the transfer integrals, while the peak for V^A remains above the HOMO band. The carrier concentration is calculated by $n=\int {dE}(-{df}/{dE})\nu (E,T)$, where f and ν(E, T) represent the Fermi-distribution function for holes and the DoS. When the carrier concentration is fixed to 10¹² cm⁻², energy distributions of charge carriers (holes) at 200 and 300 K are obtained as shown in Figs. 5(g) and (h). Here, the energy distribution of charge carriers is expressed by $(-{df}/{dE})\nu (E,T)$. The results for the energy distribution of the charge carriers show that most of the charge carriers are located in localized states induced by the trap potential V^A at 200 K. As the temperature increases, some of the charge carriers can be thermally excited to mobile states consisting of HOMO-band states, resulting in thermally activated behavior of the mobility.

To summarize, we review the effects of static and dynamic disorder on the charge transport properties of organic semiconductors using our TD-WPD method based on quantum dynamics.

The molecular vibrations give rise to a large dynamic disorder in the intermolecular transfer integrals, whose amplitude reaches several 10 meV, which is comparable to the magnitude of the transfer integrals. The large dynamic disorder induces transient localization of the charge carriers, which determines the intrinsic transport properties. This result implies that the quantitative evaluation of intrinsic mobility requires a calculation method that includes quantum corrections beyond conventional Boltzmann transport theory. We showed that the TD-WPD method based on quantum theory quantitatively reproduces the experimentally observed temperature dependence of the mobility of representative organic semiconductors such as pentacene, C₈-BTBT, and C₁₀-DNBDT, systematically.

Furthermore, we investigated the effects of static disorder on the transport properties, since impurity and defects inevitably exist in organic semiconductors. In fact, several experimental studies imply that the depth of the trap potential induced by static disorder has a similar energy value as dynamic disorder. We produced a realistic trap potential from the experimental ESR data for a pentacene crystal. We showed that the introduction of static disorder significantly changes the magnitude of the mobility and the temperature dependence from those for the intrinsic mobility. To understand the transport properties of organic devices, it is important to evaluate the transport properties using quantum theory and consider the competition between static and dynamic disorder.

Since the TD-WPD method enables quantitative predictions of the transport properties of various soft materials using first-principles calculations, we believe that it will become a powerful tool for developing new materials.

We wish to thank Prof. S. Yanagisawa, Prof. I. Hamada, Dr. S. Obata, Prof. H. Goto, Prof. T. Okamoto, and Prof. J. Takeya for their valuable contributions of data, comments, and suggestions. We acknowledge JSPS KAKENHI Grants No. JP18H01856, No. JP17H02780, No. JP17H06123, and No. JP26105011. The numerical calculations were performed at the Center for Computational Sciences, University of Tsukuba and the Supercomputer Center, the Institute for Solid State Physics, the University of Tokyo.