_Applied_  
_sciences_  
_Review_  
**Dynamics of Carrier Transport in Nanoscale**  
**Materials: Origin of Non-Drude Behavior in**  
**the Terahertz Frequency Range**  
**[PERSON] 1,*,+ and [PERSON] 2,+**  
Footnote †: journal: Applied Physics Letters B  
Footnote 10:  
nanomaterials. We, therefore, discuss why do free carriers behave as a Lorentz-like resonance, since free carriers should follow the Drude law in usual understanding. A few models have been proposed for the origin of the Lorentz-type resonance behavior [1]. We know that a quantum mechanical electronic (optical) transition between states is well described by the Lorentz resonance. In the following, we state briefly the models so far proposed:  
(1) When a motion of free carriers involves a restoring force, _i.e._, the restoring force is added in the Drude equation, the equation of motion is described by a damped harmonic oscillator, which should leads to the Lorentz oscillator [3]. A surface depletion or accumulation field can be a source of the restoring force. This is called the plasmon model and was applied to semiconductor nanoparticles [1,20]. (2) By assuming only one backscattering of free carriers, complex conductivity shows the Lorentz-type behavior, which is called the generalized Drude model or the Drude-Smith (DS) model [21]. As the DS model has been most widely used to explain the non-Drude behavior [1], we summarize here the DS approach. A complex conductivity given by [PERSON] [21] is given as:  
\[\sigma*(\omega)=\frac{\sigma(0)}{1-i\omega\tau}\left[1+\sum_{n=1}^{\infty}\frac {c_{n}}{\left(1-i\omega\tau\right)^{n}}\right] \tag{1}\]  
where \(\sigma(0)\) is the Boltzmann dc conductivity given by \(\mathrm{e}^{2}n_{\mathrm{f}}\tau/m^{*}\), \(n_{\mathrm{f}}\) is the density of free carrier, \(m^{*}\) the effective mass, \(\tau\) the scattering time, and \(\omega\) the angular frequency of external excitation, assumed to be \(\exp(-i\omega t)\). The coefficient \(c_{n}\) represents the fraction of the carrier's original velocity that is retained after the \(n\)th scattering. Note that \(n=0\) produces just the Drude law. When we take only one scattering (\(n=1\)) and \(-1.0\leq c_{1}<0\) (backscattering), free carrier behavior dramatically changes from the Drude relaxation to the Lorentz-type resonance (see Figure 1). A large issue in the DS model is that there is no proper physical basis, while good fitting to the experimental results is obtained. Note also that the DS model should basically be applied to homogeneous media. (3) When we discuss the electronic transport in inhomogeneous media, such as nanoparticles with grain boundaries, charge transfer can occur from one constituent to another. Thus, the role of interfaces (grain boundaries) may be important. A schematic view for such carrier transport is shown in Figure 2.  
Figure 1: Optical conductivity in the Drude relaxation ((**a**) and (**b**)) and in the Lorentz resonance ((**c**) and (**d**)). We call this a series sequence of the free (intragrain) and tunneling (intergrain; grain boundary) carrier (SSFTC) transport model [2; 22]. Let us review the SSFTC formalism. As stated already, the frequency dependence of complex conductivity for free carrier is given by the Drude law (\(n=0\) in Equation (1)) as:  
\[\sigma_{f}*(\omega)=\frac{\sigma(0)}{1-i\omega\tau} \tag{2}\]  
Note that \(\tau\) in this equation should be viewed as an effective relaxation time that accounts for various scattering processes within the grain, so-called intragrain scattering processes. For the tunneling transport, the so-called Dyre expression, based on an effective medium approach, which should be equivalent to a parallel capacitor-resistor random connection, is adopted [23]. The tunneling conductivity is given as:  
\[\sigma_{t}*(\omega)=\sigma_{t}(0)\frac{-i\omega\tau_{t}}{\ln(1-i\omega\tau_{t })} \tag{3}\]  
where \(\sigma_{t}(0)\) is the dc tunneling conductivity and is given by \(n_{t}(er_{t})^{2}/2 kT\tau_{t}\).