Short-length carbon nanotubes as building blocks for high dielectric constant materials in the terahertz range

Due to the high polarizability of finite-length carbon nanotubes (CNTs) in the quasi-static regime, they can be considered as building blocks for the fabrication of high dielectric constant material. Our theoretical estimations, based on an effective medium approach and solutions of a boundary value problem for individual CNT, predict that composite materials comprising short-length CNTs can have very high dielectric constants (up to 300) and low dielectric loss tangents (below 0.03) in the terahertz range. In order to prove this, 500–1000 nm thick films comprising single- and multi-walled CNTs of both long (0.5–2 μm) and short (0.1–0.4 μm) lengths have been fabricated. The analysis, based on the time-domain terahertz spectroscopy in the range 0.2–1.0 THz, demonstrated a decrease in the dielectric loss tangents of the CNT-based materials with a reduction in CNT length. In the terahertz range, the films comprising short-length CNTs had a relative effective permittivity with a large real part (25–136) and dielectric loss tangent (0.35–0.60).

As discussed in [7], there are three regimes in which CNT interact with electromagnetic fields: quasi-static (at frequency range f f s , f s being the frequency of the first antenna resonance), intermediate (or resonant, ≈ f f s ), and dynamic ( f > f s ). In the dynamic regime, the finite-length effect is negligibly small, the CNT absorption cross-section is proportional to the tube conductivity, and the imaginary part of the tube polarizability exceeds its real part. In the quasi-static regime, the real part of the carbon nanotube polarizability is practically the same as that at zero frequency, and energy dissipation is strongly suppressed by the screening effect. The latter is caused by a depolarizing field that almost compensates the axial component of the incident field [7].
For short-length ( ≈ L 250 nm) and long-length ( ≈ L 1 μm) SWCNTs, the terahertz resonances are expected at ≈ f 12 s THz and ≈ f 3 s THz, respectively [10]. This means that the quasistatic regime for short length tubes includes the terahertz range, f < 2 THz.
In the present study we report for the first time an observation of the quasi-static regime in the terahertz range for both short-length single-and multi-walled CNTs. Our investigation includes fabrication, measurement and analysis of the terahertz effective permittivity of thin films comprising both long ( ⩾ µ L 0.5 m) and short ( ⩽ µ L 0.3 m) CNTs. Previously, the alternating-current conductivity and the effective permittivity of CNT films in the terahertz range have mainly only been studied experimentally [12][13][14][15][16][17]. The real part of the relative permittivity ( ) ε Re eff and the dielectric loss Im Re eff e ff of SWCNT films have been evaluated to be in the ranges 10-10 4 and 1-5, respectively [12,13,18]. A high value of δ tan can be explained by two factors: (i) the use of long-length tubes (>0.5 μm) for film prep aration; (ii) electrical contact between tubes resulting in the formation of a conductive network and partial suppression of the finite-length effect.
The fabrication of high constant dielectric terahertz materials is a real and important task. They concentrate the energy of terahertz frequency light into a small volume, thus enabling a reduction in the size of passive elements for terahertz optics. Moreover, these materials may be useful in the production of high-contrast periodic materials (or metamaterials) in the terahertz range. For these purposes, media with Re( ) ε eff maximized and δ tan minimized are needed. Our investigation aims at demonstrating the potential for application of shortlength CNTs as building blocks for high dielectric constant terahertz materials.

Theoretical consideration
In this section, we will theoretically consider a CNT-based composite material, where a high density of nanotubes are dispersed homogeneously throughout the material, and where each tube is electrically isolated from others. In this case, the electromagnetic parameters of the composite material will be primarily determined by the properties of the individual tubes.
Following the work of Slepyan [8], we used the adopted Waterman-Truell formula to estimate the relative effective permittivity of composite material containing CNT inclusions: where ε h is the relative permittivity of the host material; ε = × − 8.85 10 0 12 F m −1 ; the function N j (L) describes the number density of the inclusions of type j with radius R j and length L; the factor 1/3 in (1) is due to the random orientations of the CNTs; and α j is the axial polarizability of inclusion, calculated using the integral equation technique described in [6], [19] and [7] for SWCNT, SWCNT bundle and MWCNT, respectively. Equation (1) ignores the electromagnetic interactions between inclusions in the composites, i.e. it is valid when all the inclusions can be considered to be electrically isolated from each other.
In our calculations we used ε = 1 h and the volume fraction occupied together by the inclusions was F = 20%. For simplicity, each composite material considered contained identical inclusions. To calculate the axial conductivity of the metallic SWCNTs, the doped semiconducting SWCNTs, and the metallic shells of the MWCNTs, we used the Drude formula [1] with an electron relaxation time of 40 fs for the SWCNTs [8] and 10 fs for the MWCNTs [20]. Figure 1 shows the calculated frequency dependence of the real part of the relative effective permittivity ( ) ε Re eff and the dielectric loss tangent δ tan of composites with different, but consistent within each sample, nanotube lengths. Let us note that figure 1 shows the relative effective permittivity in the frequency range below the LPR of the SWCNT, and the lines are discontinuous at the LPR. Due to the finite length effect [6], an individual, isolated SWCNT has a polarizability with a large real part ( ) α Re and a small imaginary part ( ) α Im in the quasi-static regime. This regime occurs up to 1 THz for L = 400 nm and 10 THz for L = 100 nm.
As one can see in figure 1, in the terahertz range, SWCNTbased composite materials behave like a high-dielectric constant material. In particular, for composites containing CNTs with a tube length of L = 400 nm, ( Figure 2 shows the frequency dependence of ( ) ε Re eff and δ tan for composites comprising identical metallic SWCNTs, undoped and doped bundles of SWCNTs, and individual MWCNTs at the same length and volume fraction of the inclusions. After comparing the parameters of composites containing different CNT inclusions, we can conclude that the individual SWCNTs are the most appropriate for producing high dielectric contrast materials in the terahertz range. All the considered CNT inclusions have approximately the same polarizability; however, the number density of SWCNTs is much higher than that of other types of CNTs. This explains one advantage SWCNTs have over other types of tubes.
The parameters of the composite comprising MWCNTs of different diameters are described theoretically in [7]. Let us note that since the number density of MWCNTs in the film is higher for thinner tubes, and MWCNT polarizability slightly depends on the number of walls in the quasi-static regime [7], a higher terahertz permittivity is expected for the films comprising thinner MWCNTs.
As shown in figure 2(b), for composites comprising doped and undoped bundles of SWCNTs, doping leads to an increase of ( ) ε Re eff and to a decrease of δ tan . The LPR frequency is higher for doped bundles [3] resulting in the observed lower energy dissipation in doped bundles than in undoped ones. As the electron relaxation time is shorter for MWCNTs than for bundle of SWCNTs, the value of δ tan is lower for SWCNT bundle than for MWCNT-based composite (see figure 2(b)).
Let us note that the fabrication of composites with nonpercolating tubes with high tube volume fraction is a nontrivial task, as SWCNTs tend to aggregate even at F = 1% due to electrostatic interaction. Meanwhile, the tendency to the high ( ) ε Re eff and low δ tan predicted in figures 1 and 2 occurs even in the network of short-length CNTs. In section 3 we will experimentally show that SWCNT-based films comprising short length CNTs (with L < 300 nm) behave like high dielectric constant materials in the terahertz range, i.e. they have ( )⩾ ε Re 100 eff and ⩽ δ tan 0.5.

Experimental results and discussion
In our experiment we used different CNT materials: (i) purified SWCNTs (Nanointegris Technology Inc., Batch PO568) produced by gas-phase catalysis (HiPCO process) with an average diameter of 1 nm, and a purity exceeding 95%; and (ii) MWCNTs produced by CVD (Bayer Baytubes) with a length of <1 μm, an inner diameter of ∼4 nm and an outer diameter of ∼13 nm. We also did experiments with double-walled CNTs; all the data concerning this material are presented in supplementary data (stacks.iop.org/JPhysD/50/08LT01/ mmedia) 8 . Short-length CNTs (50-300 nm) were prepared using a soft cutting approach developed in [21]. CNT films were fabricated via the vacuum filtration technique [22]. The film thicknesses d, measured with the profilometer Veeco Dektak 6 M, were in the range of 300-1000 nm. In order to  . Consequently, they possess two consecutive semiconducting shells, which are followed by a metallic shell. Each inclusion has length of 250 nm. dope SWCNT thin films, droplets of 1 M HNO 3 were placed on samples and allowed to dry for 12 h. This process leads to p-type doping of SWCNTs [23]. Details of the sample preparation and characterization are presented in supplementary data 9 . The length and diameter of SWCNT bundles and individual MWCNTs, obtained using atomic force microscopy (AFM, Solver P47 PRO, NT-MDT, Inc.), are collected in table 1. One can see from table 1 that the applied cutting strategy effectively reduced the CNT lengths. Cutting also led to partial debundling of the SWCNTs: i.e. the maximal bundle diameter was reduced after cutting.   Raman spectroscopy was used to estimate the crystalline quality of the CNTs. Measurements were performed using the Raman spectrometer combined with the confocal microscope Nanofinder High End (Tokyo Instruments) with 532 nm wavelength laser excitation. A ratio of the integrated area of the D-band (1300-1350 cm −1 ) to that of the G-band (1520-1600 cm −1 ) / I I D G was taken as a measure of the CNT crystalline quality [21]: the lower the value of / I I D G , the lower the concentration of sidewall defects in the CNTs. The value / I I D G is presented in table 1 for short-and long-length CNTs. Small changes in the ratio / I I D G indicate that cutting slightly disturbs the crystalline structure of the CNTs.
The measurements of the complex transmission were conducted under normal incidence in the frequency range ( ) ∈ f 0.2, 1 THz by means of time-domain terahertz spectrometry (EKSPLA, Vilnius Lithunia) with a femtosecond laser (wavelength 1 μm, with a pulse duration of less than 150 fs) and a GaBiAs photoconductive switch. Typical terahertz pulses with and without the samples, as well as the transmission spectra are shown in supplementary data 10 . Spectra of the effective permittivity of the CNT films were calculated using the Fourier transform of the measured time-domain signal. Figures 3 and 4 show the frequency dependencies of ( ) ε Re eff , ( ) ε Im eff , δ tan , and the real part of the effective conductivity, ( ) ( ) σ ε ω ε = Re Im eff 0 eff , of the samples comprising short and long SWCNTs and MWCNTs. As shown in figures 3 and 4, all the values of ( ) ε Re eff , ( ) ε Im eff , δ tan , and ( ) σ Re eff decrease drastically as the average length of the tubes decreases by about three times (from 0.5-1 μm to 100-300 nm).
In order to quantitatively compare the parameters of the different types of CNT films, the values ( ) ε Re eff , δ tan , and ( ) σ Re eff for different samples at 0.75 THz are given in table 2. One can see from table 2 that, for SWCNT and MWCNT samples respectively, cutting leads to a reduction in the value of ( ) σ Re eff by about 17 and 6 times and in the value ( ) ε Re eff by about 5 and 1.9 times. As a result, the loss tangent decreases by 4 and 2.8 times after cutting of SWCNTs and MWCNTs, respectively.
Simultaneously with quantitative variations of the terahertz parameters we observed their qualitative changes. Resonance and dynamic regime of long-length CNT interaction with the terahertz field [7] leads to high energy dissipation in the CNTs. Long-length CNT samples could be associated with metallic films for which the value of ( ) σ Re eff is high and δ tan 1. Quasi-static regime of short-length CNT interaction with terahertz radiation causes small energy dissipation preserving high value of the real part of the CNT polarizability. The parameters of short-length CNT samples are close to that of high dielectric constant terahertz materials with high ( ) ε Re eff and δ tan 1. As follows from table 2, out of all CNT types, the SWCNTs are the most perspective nanoparticles for fabrication of high dielectric constant material for the terahertz range. Thin film comprising short SWCNTs demonstrates ( ) ε = Re 73 eff and δ = tan 0.4 at 0.75 THz. We believe that loss tangent could be reduced if the SWCNTs were electrically isolated from each other in the film. Figure 5 shows the frequency dependence of ( ) ε Re eff , δ tan , and ( ) σ Re eff for SWCNT films before and after the doping by nitric acid. Acid complexes attached to the tube walls cause p-type doping effects [23] which leads to the 'metallization' of the semiconducting SWCNTs and a blue shift of the LPR in SWCNT bundles [19]. As shown in figures 5(a) and (c), doping leads to an increase of both Re(ε eff ) and ( ) σ Re eff . Since doping reduces intertube contact conductance, the depolarizing field diminishes and terahertz energy dissipation in SWCNTs increases. This explains the increase in the loss tangent δ tan at lower frequencies (<0.5 THz) after doping (see figure 5(b)). Comparison of the terahertz parameters given in table 2 for SWCNT films before and after doping demonstrates a doubling of the value of ( ) ε Re eff and a slight reduction of δ tan at 0.75 THz due to the doping process. So, for doped SWCNT film, we obtained ( ) ε = Re 136 eff and δ = tan 0.35 at 0.75 THz.