Carbon-related materials, such as carbon nanotubes (CNTs), graphenes, and organic semiconductors, have attracted much attention as new functional materials used in electronic devices. For example, the carbon nanotubes (CNTs) and graphenes are expected as high-mobility materials in comparison with silicon semiconductors. On the other hand, the mobility of organic semiconductors is much lower, but organic semiconductors have advantages as low-cost, light-weight, and printable materials. Recently, to improve the low mobility, synthesis of new organic molecules has been actively carried out. Prediction of the charge transport properties of various new materials is very important from a viewpoint of materials development. Furthermore, the systematic understanding of various transport phenomena is a challenging issue in a field of basic science.

Therefore, we have developed a new numerical simulation methodology called the time-dependent wave-packet-diffusion (TD-WPD) method [1], in which we evaluate the mobility from the wavepacket dynamics using Kubo’s formula. To reduce the computational cost, we apply Chebyshev polynomial expansion to the time-evolution operator. As the result, we perform the order-N calculations, which enable to treat up to a hundred million atoms. This indicates that we can directly compute the transport property from an atomic point of view and compare the simulation results with experimental observations of samples which have micron-order length. The transport property around room temperature is dominated by the phonons (the molecular vibrations). The effects of phonons on electronic states are included as the time-dependent electron transfer integrals, since the transfer integrals are dynamically modulated by phonons. We obtain the motion of atoms using the classical molecular dynamics, then evaluate the transfer integrals using the density functional theory (DFT). As the result, we can investigate the transport properties of micron-scale material at room temperature from an atomistic viewpoint within an accuracy of DFT.

As a demonstration, we apply the TD-WPD method to the metallic (5,5)-CNTs and investigate how the transport property at room temperature changes according to the CNT length. Figure 1(a) shows the calculated length dependence of the resistance. We can confirm the crossover from the ballistic transport region to the diffusive band transport region. When the CNT length becomes small to zero, the resistance becomes independent of the length and converges to the inverse of quantized conductance. This is a typical behavior of the quantum ballistic transport limit. On the other hand, the resistance monotonically increases with the length when the CNT becomes longer than 1 micron. This means that the classical diffusive transport is realized in the long CNTs. From our simulation, the mean free path is estimated as 515 nm. We can specify the quasi-ballistic region for the region where the resistance deviates from the diffusive and ballistic transport limits, which is 300 -- 1000 nm for the (5,5)-CNTs. The calculated results quantitatively agree with experimental data.

Then, we investigate the temperature dependence of mobility for representative single-crystal organic semiconductors, pentacene and C8-BTBT. It is well known that the high-quality single-crystal pentacene has the temperature-independent mobility of 1 cm²/Vs around room temperature while the single-crystal C8-BTBT exhibits the diffusive band-like transport with power-law temperature dependence. The mobility of C8-BTBT is about 10 times larger than that of pentacene at room temperature. In Fig. 1(b), circles and squares represent the calculated mobilities as a function of temperature for single-crystal pentacene and C8-BTBT, respectively. The calculated temperature dependences are quantitatively in good agreement with the experimental values for the single crystal devices. The simulations give the theoretically predicted upper limit of intrinsic mobility. Even if the same organic semiconductors are used for devices, the measured mobilities have distributions to some extent depending on the employed device architecture and the crystal quality. Low-quality devices show the low mobility with thermally activated behavior. To take the difference of device quality into account, we introduce the carrier-trap potentials in the simulation. The calculated mobility for pentacene are plotted by triangles. We can confirm the magnitude of mobility dramatically decreases and the temperature dependence changes to thermally activated behavior characterized by the hopping transport nature.

In our presentation, we will present our TD-WPD methodology and discussions about the above-mentioned results for CNTs and organic semiconductors in detail. We also show you how our simulation can provide a systematic understanding for different transport mechanisms, such as the ballistic transport, diffusive-band transport, and hopping transport.

[1] H. Ishii, et al., Phys. Rev. B 82, 085435 (2010); Phys. Rev. B 85, 245206 (2012); Phys. Rev. B 90, 155458 (2014); Phys. Rev. B 95, 035433 (2017).