Alternating current response of carbon nanotubes with randomly distributed impurities

In light of the recent advancements in electronic nanodevices, it is important to obtain a better understanding of the dynamical response of nanomaterials. Our present understanding of non-steady-state charge transport is lacking in comparison to our understanding of steady-state transport in nanomaterials. The AC response behaviors of nanomaterials have been examined using simple transport models,^1,2 and these findings provide a good basis for understanding coherent AC transport; however, in-depth theoretical studies on the AC response of nanomaterials have only recently attracted attention.

Carbon nanotubes (CNTs) are promising candidates as components of next-generation ultrahigh-frequency electronic devices owing to their high carrier mobility.³ Previous studies on the AC transport properties of pristine metallic CNTs have shown that the magnitude of the imaginary component (the susceptance) of the admittance becomes comparable to that of the real component when the AC frequency is in the sub-terahertz range.^4,5 In addition, it has been shown that the sub-terahertz frequency susceptance behaves inductively and its inductive response becomes prominent with increase in CNT length. This is attributed to the fact that the kinetic inductance increases with the dwell time of electrons, which is due to the inertia of the electrons.⁶ Investigations of the effect of contact between a pristine metallic CNT and metallic electrodes on the AC transport properties^7–9 have shown that the sub-terahertz frequency susceptance changes from an inductive response to a capacitive response with decreasing DC conductance.^8,9 This indicates that there is a correlation between the DC conductance and the sub-terahertz frequency susceptance. This correlation has also been reported for resonant double-barrier structures,¹ quantum point contacts,² and bridged structures.^10,11 In addition, we have also examined the influence of a single defect on the AC response, which strongly decreases the DC conductance,^12–15 and we have clarified that electron scattering by the defect state induces a capacitive response.^16–18 

The effects of the kinetic inductance and electron backscattering on the AC response have been studied and understood for the above simple cases. However, it is unclear whether the findings obtained from such simple systems are sufficient to understand real systems, which are more complicated. Real systems often have randomly distributed impurities. Regarding the DC transport properties, a number of studies have been conducted for systems with randomly distributed impurities, and it has been clarified that due to the peculiar structures of CNTs, the DC transport properties of metallic CNTs are sensitive to the range of the impurity potential, d. In the case wherein the potential range is larger than the lattice constant, $a = 2.46 Å$⁠, electron backscattering is suppressed; electron backscattering occurs in the case of a short-range potential (d < a), which leads to decrease in the DC conductance.^19–24 However, with regard to the AC transport properties, electron backscattering behaviors in systems with randomly distributed impurities have not thus far been elucidated.

In this study, we theoretically examine the AC transport properties of metallic CNTs with randomly distributed impurities, focusing on the dependence of the AC transport on the impurity potential's range and amplitude and the impurity concentration by utilizing the nonequilibrium Green's function (NEGF) method. We show that multiple electron scattering by long-range impurity potentials affects only the emittance behavior and increases it, which is surprising since electron scattering has generally been considered to decrease emittance.

We express the Hamiltonian of CNTs with randomly distributed impurities as

Here, $H CNT = t ∑ ⟨ i , j ⟩ | i ⟩ ⟨ j |$ denotes the nearest-neighbor tight-binding Hamiltonian of a pristine CNT, where $t = − 2.75 eV$ represents the hopping integral between the nearest-neighbor carbon atoms and $⟨ i , j ⟩$ indicates the sum over the pairs of the nearest-neighbor carbon atoms. The impurity potential $U imp$ is described as

where $u imp , n imp , r i$⁠, and d denote the amplitude of the impurity potential, the number of impurities in the CNT, the impurity-substituted sites, and the range of the impurity potential, respectively. The impurity-substituted sites $r i$ are randomly chosen. In real systems, the distribution of impurities is different from system to system. Therefore, in our simulation, we examine the averages and standard deviations of the AC transport behaviors for over 10 000 different impurity configurations.

We focus our attention on a two-terminal system composed of a scattering region connected with left (L) and right (R) leads. The low-frequency admittance of the system is expressed as

where $G DC$ and E denote the DC conductance and the emittance,¹ respectively, and $ℏ$ and ω denote Dirac's constant and the frequency of AC signals, respectively, and the subscripts indicate the indices of leads. We note that the emittance corresponds to the phase difference between the current and bias voltage, $θ I − V$⁠. When the sign of emittance is positive (negative), the low-frequency response is inductive (capacitive). An understanding of the emittance behavior is crucial in a nanoscale device in the sense that it is related to the power factor $η = cos θ I − V$⁠, which determines the efficiency of power transmission, through the phase difference $θ I − V = tan − 1 ( E ℏ ω / G 0 )$⁠. Here, we consider that AC bias voltages $V L / R AC ( ω ) = ± V cos ω t$ are applied at the left- and the right-lead regions, respectively. Here, V denotes the amplitude of the AC bias voltage.

In the NEGF formalism, the CNT is divided into three regions, the scattering (central) region, and the semi-infinite left- and right-lead regions. In this work, the impurity potential $U imp$ is introduced in the scattering region, and we focus our attention on the linear AC response to V. Our calculation of the admittance is based on our previous investigations.^17,18

Figure 1 shows the tube-length dependence of the DC conductance and emittance of a (10,10) metallic CNT with impurity potentials of (a) $d = 0.1 a , n c = 1 %$ and (b) $d = 1.5 a , n c = 1 %$ for several potential strengths $u imp$⁠. Here, the concentration of impurities in the CNT is represented by $n c$⁠. Potentials, whose ranges are $d = 0.1 a$ and $d = 1.5 a$⁠, are referred to as short-range and long-range impurity potentials, respectively, following Refs. 19–22. We discuss the AC response properties by focusing on the potential ranges. It is noted that the mean-free paths, $L MFP$⁠, estimated by fitting our simulation results with the expression, $G DC / G 0 = 2 / ( 1 + L / L MFP )$⁠, where $G 0 = e 2 / h$ represents the conductance quantum (where h denotes Planck's constant), are longer than the CNT lengths, L, considered here, which implies that we are focusing on the ballistic transport regime in the study.

We first examine the case of (a) $d = 0.1 a , n c = 1 %$⁠, for which the corresponding plots are shown in the left panel of Fig. 1. The DC conductance decreases with the tube length and potential strength. This length dependence can be understood from the fact that with increasing tube length, the number of scatterings also increases. It is also reasonable that the DC conductance decreases with the potential amplitude.

Next, we examine the emittance behaviors. For tube lengths less than L = 200 nm, the emittance increases nearly in proportion to the tube length and hardly depends on the potential strength. However, for the tube lengths greater than L = 200 nm, the potential-amplitude dependence of the emittance gradually appears with increasing tube length, and the emittance decreases with the potential amplitude. This suggests that there is a competition between two dominant factors that determine the emittance behaviors: the kinetic inductance, which contributes to the inductive response^4,6,8,9,18 and is proportional to the transport time of electrons, and the electron scattering, which leads to the capacitive response.^16–18 Based on this fact, we can understand the emittance responses as follows. When the CNT length is small (<200 nm), the effect of electron scattering is weak as can be observed in the DC conductance behaviors. Thus, the emittance increases with the length, which is attributed to an increase in the kinetic inductance with the tube length. On the other hand, when the tube length is large, the effect of electron scattering becomes large, which leads to larger deviation from the linear behavior and smaller values of the emittance with an increase in potential strength. From these results, we can conclude that the AC transport behaviors for short-range impurity scattering can be understood in a manner similar to the cases of simpler systems examined in our previous investigations.^4,16–18

We next examine the effect of long-range impurity scattering on the AC response. Figure 1(b) shows the AC response of (10,10) metallic CNTs with randomly distributed long-range impurity potentials. The DC conductance hardly decreases with the tube length and potential amplitude. This can be understood from the fact that electron backscattering by long-range impurity potentials is suppressed.^19–22 However, we note that the DC conductance slightly decreases when $u imp = 0.75 eV$⁠. This is because the energy dispersion becomes nonlinear at energies away from the Dirac point by such a large amount (∼0.75 eV), which behavior is called the trigonal warping effect.²⁵ The linear dispersion is key to the suppression of electron backscattering.²² 

The emittance behavior hardly depends on the potential amplitude when compared with those in metallic CNTs with short-range impurity potentials. Owing to the suppression of electron backscattering, we observe an increase in the emittance proportional to the CNT length, which originates from the increase in the kinetic inductance with the length.

We next closely examine the emittance behaviors of metallic CNTs with impurity potentials. Figure 2 shows the potential-amplitude dependence of the ratio, $( E imp − E pristine ) / E pristine$⁠, of the emittance of a (10,10) metallic CNT with long-range impurity potentials, $E imp$⁠, to that of pristine (10,10) CNT, $E pristine$⁠, for several CNT lengths, where $n c = 1 %$⁠. We observe that the emittance increases with the potential amplitude. This is surprising and unexpected in the following ways. As discussed above, there are two factors that affect the emittance behavior, i.e., the kinetic inductance and electron backscattering. Since the almost length-independent DC conductance (upper panel of Fig. 1(b)) suggests the suppression of electron backscattering, the only remaining factor that affects the emittance behaviors is the kinetic inductance, which is proportional to the length of the CNT. However, the change in emittance with respect to the potential amplitude does not appear to be related to the kinetic inductance. Therefore, there must be another factor that affects the emittance. Moreover, it is interesting that these emittance values are larger than those of the pristine metallic CNT, as can be seen in Fig. 2.

Next, we attempt to explain the above phenomenon. In our previous studies,^16–18 we have clarified that electron scattering induces a capacitive response. Here, it should be noted that the electron scattering in these cases involves a decrease in the DC conductance, which is in sharp contrast with the present case of long-range impurity scattering. However, we posit that electron scattering actually occurs although this does not affect the DC transport properties. Therefore, because of electron scattering (multiple scattering in particular) the transport time of electrons from the left lead to the right lead increases. As a result, the kinetic inductance increases in proportion to the transport time of electrons,⁶ which in turn leads to an increase in the emittance, as seen in Fig. 2.

In order to confirm the above hypothesis, we estimated the dwell time of electrons using an equivalent circuit derived in a previous study.⁴ According to the study, the emittance of long pristine metallic CNTs can be expressed as follows:

where $τ circuit$ denotes the dwell time of electrons. The dwell time in pristine metallic CNTs with lengths of 1 μm obtained using Eq. (4) is 1.123 ps, which is exactly equal to the value obtained from the relation $τ = L / v F$⁠, where the Fermi velocity $v F = 8.9 × 10 5 m / s$⁠. We used Eq. (4) for estimating the dwell time of electrons in metallic CNTs with long-range impurity potentials since the DC conductance hardly changes due to the suppression of electron backscattering, as can be seen in Fig. 1, in which case it is similar to that of pristine CNTs. Subsequently, we obtained the dwell time as $τ circuit = 1.126 , 1.134$⁠, and 1.147 ps for $u imp = 0.25 , 0.5$⁠, and 0.75 eV, respectively. As expected, we note that the dwell time increases with the amplitude of the impurity potential.

Based on the conventional transport theory,²⁶ we arrived at the following expression for the dwell time:

where D and $D pure$ denote the density of states in a system with and without scattering, respectively. From this expression, we obtained $τ trans = 1.126 , 1.134$⁠, and 1.151 ps for $u imp = 0.25 , 0.5$⁠, and 0.75 eV, respectively. These values are in agreement with those obtained using Eq. (4). From the above analyses, we can conclude that multiple scattering by the impurities increases the dwell time of electrons, which in turn leads to an increase in the kinetic inductance. As a result, the emittance increases with increasing potential amplitude, as can be seen in Fig. 2.

Here, we plot the impurity-concentration dependence of the emittance of a (10,10) metallic CNT of the length of $1 μ m$ with impurity-potential ranges of (a) $d = 0.1 a$ and (b) $d = 1.5 a$ for several potential amplitudes in Fig. 3. In the case of (a) $d = 0.1 a$⁠, the emittance decreases with the impurity concentration. This can be explained in terms of the electron scattering as discussed above. On the other hand, in the case of (b) $d = 1.5 a$⁠, the emittance increases with the impurity concentration, which trend is opposite to case (a). This is attributed to an increase in the dwell time of electrons due to their multiple scattering, which in turn increases the kinetic inductance as mentioned above.

Finally, we discuss the AC response fluctuation, which is defined by the standard deviation $Δ f = ⟨ f 2 ⟩ − ⟨ f ⟩ 2$⁠, where f denotes $G DC$ or E and $⟨ f ⟩$ indicates the average over all the samples we used. The following discussion is focused on metallic CNTs with a length of 1000 nm with an impurity concentration of $n c = 1 %$⁠. We obtained $Δ G / G 0 = 0.026$⁠, 0.089, and 0.164 for short-range impurity potentials with potential amplitudes of $u imp = 0.25$⁠, 0.5, and 0.75 eV, respectively, and $Δ G / G 0 = 8.0 × 10 − 5 , 8.42 × 10 − 4$⁠, and $1.03 × 10 − 2$ for long-range impurity potentials with potential amplitudes of $u imp = 0.25$⁠, 0.5, and 0.75 eV, respectively. The fluctuations in the case of the long-range impurity potential are negligibly small, which is consistent with the suppression of electron backscattering. Here, it is to be noted that the universal conductance fluctuation is not observed because the transport regime considered in this study is not the weak localization regime, but the ballistic transport regime. On the other hand, with regard to the emittance fluctuation, we obtained $Δ E / G 0 = 51.53$⁠, 170.20, and 298.19 for the short-range impurity potentials with the potential amplitudes of $u imp = 0.25$⁠, 0.5, and 0.75 eV, respectively, and $Δ E / G 0 = 0.187$⁠, 1.73, and 18.32 for the long-range impurity potentials with the potential amplitudes $u imp = 0.25$⁠, 0.5, and 0.75 eV, respectively. In contrast to the DC conductance fluctuation, the emittance fluctuation is not suppressed to a large degree even in the case of the long-range impurity potential. This is attributed to the fact that the transport time is different from system to system, which leads to the appearance of emittance fluctuation.

In summary, we have investigated the effect of impurity scattering on the AC transport properties of metallic CNTs. We showed that the short-range impurity scattering causes a decrease in the emittance as well as the DC conductance. This is attributed to electron scattering, as clarified in our previous studies. On the other hand, we found that the long-range impurity scattering causes the emittance to increase with respect to the potential amplitude and impurity concentration. This is interesting in the sense that the suppression of this scattering leaves the DC conductance unchanged, while it changes the emittance behavior. We showed that the above behaviors can be understood from the change in the dwell time of electrons and their multiple scattering. That is, the multiple scattering increases the dwell time, which in turn increases the kinetic inductance and thus also the emittance. In addition, we discussed the fact that the emittance fluctuation in the case of the long-range impurity potentials is related to the transport-time difference from system to system, which leads to kinetic inductance difference from system to system. Our study clarified that the electron scattering is important to understand the inductive response as well as the capacitive response of metallic CNTs. We believe that our findings can significantly contribute to the development of CNT-based nanodevices.

We acknowledge partial financial support from Grants-in-Aid for Scientific Research on Innovative Areas “Materials Design through Computics: Complex Correlation and Non-Equilibrium Dynamics” and for JSPS Fellows, and the Global COE Program “Global Center of Excellence for Mechanical Systems Innovation” from the Ministry of Education, Culture, Sports, Science and Technology of Japan.