Anomalous satellite inductive peaks in alternating current response of defective carbon nanotubes

Cu interconnects have achieved great success in traditional large-scale integrations (LSIs). With the continued downscaling, however, electromigration in Cu interconnects becomes crucial. In addition, the electron scattering from grain boundaries and surfaces also increases with the downscaling, which in turn increases the resistivity. Therefore, an alternative metal to the Cu is required for next-generation electronic devices.

Metallic carbon nanotubes (CNTs) are promising for the above purpose because they have a large allowable current density (>10⁹ A/cm²).¹ Moreover, because of their high carrier mobility (>10⁵ cm²/(Vs)),² they have also attracted much attention as components in ultrahigh-frequency electronic devices.

Towards the application of CNT-based ultrahigh-frequency electronic devices, it is important to understand the AC transport properties as well as the DC transport ones. So far, the AC response of pristine metallic single-walled carbon nanotubes (SWNTs) has theoretically been investigated.^3,4 It was shown that with increasing the frequency, the conductance oscillates and the susceptance repeats transitions between an inductive and a capacitive responses. The sub-THz AC response, which is important for next-generation electronic devices, is inductive, and this inductive response becomes stronger with the CNT length. This can be understood from the increase of the kinetic inductance, which can be ascribed to the inertia of electron,⁵ with the CNT length. The effect of contact between a metallic electrode and the pristine metallic SWNT on the AC response was also analyzed,^6–8 and the results indicate that the AC response properties depend on the contact-coupling strength and correlate with the DC response: as the DC conductance decreases, the low-frequency admittance changes from the inductive response to the capacitive one.^7,8 The correlation between low-frequency admittance and DC conductance is well-known in other systems.^9–12 

Very recently, based on nonequilibrium Green's function (NEGF) method and tight-binding approximation, we examined the influence of a single atomic vacancy on the AC response and clarified that electron scattering by the vacancy state around the Fermi level induces capacitive response^13,14 in contrast to pristine metallic CNTs which behave inductively.

The vacancy state is composed of π-orbitals left around a single atomic vacancy^15–18 as can be seen in Fig. 1(a). According to more accurate electronic structure calculations based on the density functional theory (DFT), it is known that another scattering state, composed of σ-orbitals left after removing a carbon atom (Fig. 1(b)), is also induced near the Fermi level,¹⁸ which is called dangling-bond state. The dangling-bond state involves the decrease of DC conductance similar to the vacancy state. Regarding the AC response, a natural guess would be that the capacitive response appears at the dangling-bond state similarly to the vacancy state. However, this is not obvious because the AC transport properties give richer information on materials than the DC transport ones. In fact, we showed that the AC response depends on the position of a defect, while the DC response does not.^13,14

In this paper, we examine the AC response of metallic SWNTs with a single defect from first principles with the combination of the DFT and the NEGF method. As typical examples, we take the single atomic vacancy, 5-6 defect and Stone-Wales defect. We clearly see an unexpected feature of the AC response, the appearance of satellite inductive peaks, which apparently seems to violate the correlation between the DC conductance and the low-frequency susceptance shown by many theoretical studies. On the basis of a simple resonant scattering model, we show that the sharpness of features of the DC conductance spectra, which is directly related to the dwell time of electrons, is the key to understand the above behavior of nanoscale AC responses.

Our simulation method is based on the DFT combined with the NEGF method (DFT + NEGF method).^19–21 The DFT + NEGF method enables us to analyze AC transport properties of a two-terminal system at an atomic level from first principles. In this method, the two-terminal system is separated into three parts: a scattering region and left/right lead regions. In this study, all of the scattering region and the left/right lead regions are modeled by (10,10) metallic SWNTs. A single defect is introduced in the scattering region.

The admittance matrix is written as

where G and B are the conductance and susceptance, respectively. The susceptance B determines the sign of a phase difference θ_I−V between a current and a bias voltage. We focus our attention on the low-frequency admittance as previous studies^4,7–14

where G_DC is the DC conductance and $ℏ$ and ω are the Dirac's constant and the frequency of AC signals, respectively, and the subscripts indicate indices of the leads. Here, E is called an emittance,^9,10 which corresponds to the phase difference similar to the susceptance. Because a power factor $η=cos θI−V$⁠, which determines the efficiency of power transmission, is related to the phase difference $θI−V=tan−1(Eℏω/GDC)$⁠, the understanding and controlling the emittance are crucial in a nanoscale circuit.

We now consider the case where AC bias voltages $VL/RAC(ω)=±V cos ωt$ are applied in the left (L) and the right (R) lead regions, respectively. Here, V is an amplitude of the AC bias voltage. In this work, a linear AC response to V is considered. In the NEGF formalism, the DC conductance is written within the wide-band-limit approximation²² as

where $Gr/a(εF)$ are retarded and advanced Green's functions in the scattering region²³ and $Γα=−2ImΣα$ is a level-broadening function, where Σ_α is energy-independent self energy due to contact between the scattering region and α lead, and ε_F is the Fermi level of the system. The emittance is expressed as¹⁴ 

where

In our DFT + NEGF code, we use an efficient first-principles calculation package SIESTA²⁴ to construct Kohn-Sham Hamiltonian of each region with a single zeta basis set, Troullier-Martins norm-conserving pseudo-potential,²⁵ and Ceperley-Alder exchange-correlation potential.²⁶ In addition, we use an iterative scheme to calculate the self energy for the contact between the scattering region and left/right lead regions²⁷ and the Green's function in the scattering region,²⁸ and an efficient integration method based on a continued fraction representation of the Fermi-Dirac distribution function to calculate the electron density of the scattering region.²⁹ 

In this study, we focus on the case where the single defect is at the center of the scattering region.³⁰ The length of the scattering region is 4.674 nm, which corresponds to 19 unit cells. It is noted that the length is not important for our findings, as discussed in Ref. 14. In our simulation, a 1 × 1 × 1 k point is taken in the first Brillouin zone, where a direction of electronic transport is set as the z-axis. Here, we do not need multiple k-point along the z-axis since there is no periodicity along the transport direction. In the following discussions, the influence of magnetic inductance is not considered since it is much smaller than that of the kinetic inductance for CNTs.⁵ 

First, we simulate the AC transport properties of (10,10) metallic SWNT with a single atomic vacancy seen in Fig. 1. Figure 1(c) shows the Fermi-level dependence of the DC conductance of (10,10) metallic SWNT with a single atomic vacancy, where the structure optimization is not performed. Although it is known that the atomic vacancy shown in Fig. 1 is not the most stable,^31–34 this defect is useful for clarifying the effect of defect on physical properties.^13–18 We can see two conductance dips: a broad dip around ε_F = −0.25 eV and a sharp dip around ε_F = 0.05 eV.^18,35,36 These dips are attributed to the electron scattering by the vacancy state (Fig. 1(a)) and that by the dangling-bond state (Fig. 1(b)), respectively. The latter cannot be obtained from the simple nearest-neighbor π-orbital tight-binding calculation.^13–18 The asymmetric feature of conductance dip at the dangling-bond state is due to the Fano resonance.³⁷ 

Next, let us examine the emittance behavior (Fig. 1(d)). First, we can see the capacitive response around the vacancy state (ε_F = −0.25 eV). This has been discussed in our previous papers.^13,14 The capacitive response also appears at the dangling-bond state (ε_F = 0.05 eV). These capacitive responses are attributed to the electron scattering.^13,14 The most notable feature seen in Fig. 1(d) is the appearance of anomalous satellite inductive peaks adjacent to the capacitive dip at the dangling-bond state. The inductive peaks are puzzling on the following two points. First, the peaks are visible only around the dip due to the dangling-bond state. Second, the peaks appear to violate the correlation between the DC conductance and the emittance shown by many theoretical studies.^7–14 

As have been mentioned, the atomic vacancy shown in Fig. 1 is not stable. When it is introduced, structural reconstruction occurs, so that no dangling bonds of σ-orbitals remain. The most stable defect structure obtained by the structure optimization is shown in Fig. 2. This defect is composed of two pentagons and two hexagons: hereafter it is called 5-6 defect.^31,32 Figure 2(c) shows the Fermi-level dependence of the DC conductance. It has two dips at ε_F = −0.7 eV and 0.6 eV due to the electron scattering by the defect states shown in Figs. 2(a) and 2(b), respectively. The scattering states at ε_F = −0.7 eV and 0.6 eV resemble the dangling-bond state and the vacancy state seen in Fig. 1, respectively. Actually, the state at ε_F = −0.7 eV is more localized than that at ε_F = 0.6 eV as can be understood from the conductance spectra in Fig. 2(c). Corresponding to the electron scattering, the capacitive responses are seen at these levels (Fig. 2(d)). We also observe the satellite inductive peaks adjacent to the capacitive dips at ε_F = −0.7 eV, although these peaks are not remarkable in contrast to the case of single atomic vacancy seen in Fig. 1(d).

We also analyze the effect of the Stone-Wales (SW) defect^38,39 on the AC response behavior. The SW defect is pentagon-heptagon pair defect that is created by a 90° rotation of two carbon atoms with respect to the midpoint of the bond as seen in Fig. 3. Figure 3 shows the AC transport properties of (10,10) metallic SWNT with the SW defect. Two conductance dips at ε_F = −0.7 eV and 0.55 eV are seen in Fig. 3(c) due to the electron scattering by the defect states shown in Figs. 3(a) and 3(b), respectively.^3,39 From Fig. 3(c), it is understood that the scattering state at ε_F = −0.7 eV is a little bit localized compared to that at ε_F = 0.55 eV. Indeed, we see the state like a dangling-bond one in Fig. 3(a). The capacitive responses also appear at these Fermi levels, which are induced by the electron scattering. Here, it should be noted that there are no remarkable satellite inductive peaks differently from previous two cases (see Figs. 1 and 2).

We have seen the capacitive peaks and the satellite inductive peaks adjacent to some of them in the above three simulations. In our previous studies,^13,14 we clarified that the capacitive peaks are induced by the electron scattering. Now, let us investigate the origin of the satellite inductive peaks. The fact that the satellite inductive peaks appear regardless of defect structures (see Figs. 1 and 2) suggests that this behavior can be understood in a unified manner. Thus, we introduce a simple resonant scattering model shown in the inset of Fig. 4(b). This model is composed of a one-dimensional chain (denoted by red circles) and an adatom (denoted by a purple circle), where the transfer integral between nearest-neighbor sites in the chain is −t and that between the adatom and its neighboring site is $−t′$⁠, and the on-site energy is set to be zero for all the atoms. Note that this model was adopted to examine the effect of a single defect on the DC conductance of graphene nanoribbon.⁴⁰ As can be seen in the following, we can elucidate the effect of sharpness of conductance dips on the emittance using this model.

Figure 4(a) shows the energy dependence of the DC conductance of the system. The DC conductance of this model exhibits a dip around ε = 0 eV. It is also seen that the conductance dip becomes sharper with decreasing $t′$⁠.⁴¹ This corresponds to the fact that once electron moves to the adatom site, it is difficult for electron to come back to the chain for small $t′$⁠, which leads to a long dwell time of electron at the adatom site. This trapping causes the conductance decrease.

Figure 4(b) shows the emittance behavior of this system. First, we see that regardless of the value of $t′$⁠, the capacitive response appears around ε = 0 eV. More importantly, the satellite inductive peaks are induced adjacent to the capacitive dip for any value of $t′$ and these peaks are enhanced with decreasing the value of $t′$⁠. In other words, the satellite peaks are enhanced with the sharpness of capacitive dip, which also relates the sharpness of the DC conductance dip.

This emittance behavior is quite similar to those seen in Figs. 1–3. Around the broad scattering states such as those at ε_F = −0.25 eV in Fig. 1, ε_F = 0.6 eV in Fig. 2, and ε_F = −0.7 eV and 0.55 eV in Fig. 3, no satellite inductive peaks are visible. On the other hand, adjacent to the localized scattering states seen at ε_F = 0.05 eV in Fig. 1 and ε_F = −0.7 eV in Fig. 2, the satellite inductive peaks can be clearly observed. In fact, the estimated half widths at half maximum of the DC conductance dips are 0.12 eV at ε_F = −0.25 eV and 0.006 eV at ε_F = 0.05 eV in Fig. 1, 0.0085 eV at ε_F = −0.7 eV and 0.045 eV at ε_F = 0.6 eV in Fig. 2, and 0.015 eV at ε_F = −0.7 eV and 0.02 eV at ε_F = 0.55 eV in Fig. 3. This confirms the similarity between the emittance behaviors seen in Figs. 1–3 and that of the resonant scattering model in Fig. 4. The above analysis strongly suggests that the apparently different emittance behaviors are not essential, and can be understood in a unified manner. That is, this difference originates just from the difference in the degree of sharpness of conductance dips.

Since the capacitive response caused by electron scattering has already been elucidated in our previous studies,^13,14 the remaining feature of the emittance behavior that has to be clarify is the satellite inductive response. This can be understood as follows. In the resonant scattering model, an electron is trapped in the adatom site for a while. This causes the increase of the dwell time in the scattering region, which involves the increase of the kinetic inductance. Therefore, we can expect that whenever an electron is trapped at the defect site an inductive response is involved, and thus competition between the inductive and capacitive responses should be considered.

According to conventional transport theory,⁴² the dwell time of a system with defects, τ, is expressed in terms of the dwell time of the corresponding defect-free system, τ_pure, as

where D and D_pure are density of states of the systems with and without defects, respectively. Therefore, the dwell time is proportional to the ratio D/D_pure. The behavior of D/D_pure in the resonant scattering model is shown in Fig. 4(c) together with the emittance behavior in the case of $t′=0.1 eV$⁠. D/D_pure and thus the dwell time become large around ε = 0 eV because of the electron trapping as expected. Moreover, we can see that the behaviors of the dwell time and the emittance are similar except in the immediate vicinity of ε = 0 eV. This means that the dwell time, i.e., the kinetic inductance, also has strong influence on the emittance behavior in addition to the electron scattering related to the correlation between the DC conductance and the emittance.

We have also examined the AC transport properties of metallic SWNT with two scattering centers.⁴³ In this case, resonant tunneling due to the interference between incident and reflected wave functions appears instead of the resonant scattering discussed above. Although detailed analysis is in progress, the results in this case can also be understood from the degree of sharpness of features of conductance spectra, that is, the dwell time, and the competition between the capacitive and inductive responses.

We have investigated the AC response of metallic SWNT with a single defect, such as the atomic vacancy, the 5-6 defect, and the SW defect, from first principles. We found the satellite inductive peaks adjacent only to the sharp capacitive dip in spite of the fact that no satellite features are seen in the DC conductance, which apparently seems to violate the correlation between the DC conductance and the emittance. From the analysis based on the simple resonant scattering model, we revealed that this is a consequence of the competition between the capacitive and inductive responses both involved with the introduction of a defect plus correlation of the sharpness between the DC conductance dip and the satellite inductive peaks. From this point of view, we can also explain the apparently different AC response behaviors around the broad scattering states and the localized ones. Judging from the simple systems and model used in the present study, we can expect that our findings are not specific to the limited systems but general in various nanostructures. This study stresses that the degree of sharpness of features of conductance spectra, which corresponds to the dwell time, is an important key for understanding the AC transport properties at nanoscale as well as the correlation between the DC conductance and the emittance.

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.