Tunable High-Frequency Acoustoelectric Current Oscillations in Fluorine-Doped Single-Walled Carbon Nanotubes

,


Introduction
Terahertz (THz) radiation indeed has a myriad of applications in various felds, specifcally science and industry.Te THz region of the electromagnetic (EM) spectrum, located between the microwave and infrared frequencies with wavelength 1mm to about 100μm (i.e.frequency range of 300 GHz to 3 THz), ofers unique opportunities for studying molecular resonances and interactions [1].In other to create a functional coherent THz radiation source or amplifer at ambient temperature, researchers have explored diferent approaches.Employing superlattices (SLs) and the nonlinear interaction of high-frequency electric feld with miniband carriers in dc-biased SLs is one intriguing technique.Tis idea expands on past research by Ktitorov et al. [1] and Esaki-Tsu [2].In this system, miniband carrier Bloch oscillations can potentially lead to the amplifcation of THz radiation [3].Bloch oscillations, which refer to the periodic motion of carriers in the crystalline lattice being infuenced by a static electric feld, are a fundamental quantum phenomenon.By carefully designing the SL structure and applying static electric feld, it is possible to exploit this efect and generate THz radiation [1,2].
Buttiker et al. demonstrated the phenomenon of negative diferential velocity (NDV) caused by Bragg scattering in a semiconductor [4].Tey demonstrated this efect over decades ago, showing that NDV can lead to the propagation of electric feld domains in a semiconductor and the generation of self-sustained current oscillations without the need for an external resonator [4][5][6][7].Te phenomenon of self-sustaining current oscillations in an SL has been experimentally observed by Le Person et al. [8][9][10].Tey utilised picosecond light pulses to generate and observe photocurrent oscillations in an SL, which decayed over time.Furthermore, reports in [11] indicate the observance of selfsustaining current oscillations up to a frequency of 20 MHz in a doped GaAs/AlAs SL.Tese oscillations were achieved under voltage control at a temperature of 5 K, specifcally in the region of sequential tunneling where Bragg-refected carriers are unlikely to be detected.
However, despite achieving this milestone, there are still some challenges associated with realising these THz amplifers.One signifcant obstacle is the development of highfeld electric domains inside the SL [9,[12][13][14][15][16][17][18][19][20][21][22][23][24][25].Tese electric domains are necessary for achieving NDC, which is crucial for the amplifcation process [9,12].Overcoming these challenges and realising practical, ambient temperature THz amplifers are an active area of research.Advances in semiconductor technology, device fabrication techniques, and understanding of quantum phenomena are contributing to the progress in this feld.Successful development of compact and efcient THz radiation sources and amplifers can have signifcant implications for a wide range of applications, including communications, spectroscopy, imaging, and sensing.
A single-walled carbon nanotube (SWCNT), a novel material with an exceptional carrier conductivity and inherent nonlinearity due to their one-dimensional structure, makes SWCNT a potential for RF (radio frequency) electronic application [26][27][28].SWCNT-based RF transistors have shown impressive transport characteristics and density of states, leading to advancements in the feld over the last decade.SWCNT-FETs (SWCNT-feld-efect transistors) have achieved signifcant milestones in terms of their intrinsic maximum oscillation frequency (f T,int ) and maximum frequency (f max,int ).For instance, certain SWCNT-FETs have demonstrated f T,int values as high as 153 GHz, while other SWCNT-FETs have attained f max,int values as high as 70 GHz.Tese accomplishments have cleared the path for the creation of analog radio systems and high-speed circuits based on SWCNT-FETs [29][30][31][32][33].In addition to this, SWCNT flm-based radio-frequency transistors (SWCNTFBRFTs) have been reported to exhibit maximal hypersound oscillation frequencies exceeding 100 GHz [29][30][31][32][33].
As to why we chose fuorine-doped SWCNTs to elements like Cl, Br, and I was due to FSWCNT's semiconducting properties.Taborowska et al. conducted investigation into the doping of SWCNTs with halogenated solvents (i.e., dichloromethane, chloroform, and bromoform) [34].Tey found out that using halogenated solvents as the medium signifcantly enhances the carrier conductivity at room temperature.For SWCNT, flms made in an acetone/toluene mixture had a carrier conductivity of 853 ± 62 Scm −1 [35], but when dichloromethane and chloroform were used, the carrier conductivity increased to 1652 ± 186 Scm −1 and 1966 ± 425 Scm −1 , respectively.Te Br-SWCNT specimen created in bromoform shows an even more noticeable rise.Te Br-SWCNT's carrier conductivity increases were quadrupled in comparison to the material made utilising nonhalogen solvents, reaching 3819 ± 241 Scm −1 .Te measured values were quite high despite the mild chemical nature of these molecules [36].
Te carrier conductivity, which reduces as temperature rises, indicated that the SWCNT networks were basically metallic in nature [36].Due to this, the Seebeck coefcient values of variously produced SWCNT flms were adequate but not outstanding.Te doped-SWCNT room temperature Seebeck coefcient of 46 ± 3μV/K matched the performance of other undoped SWCNT-based thermogenerators [37].In contrast, the Seebeck coefcients for the ensembles made in dichloromethane (SWCNT/DCM), chloroform (SWCNT/ CF), and bromoform (SWCNT/BF) were 26 ± 2, 22 ± 3, and 19 ± 1μV/K, respectively.Terefore, the observed drop in Seebeck coefcients strongly suggests that the material was doped when processed using these halogenated aromatic solvents on a regular basis.Doping changed the material from nondegenerate to degenerate character, which had a signifcant impact on the carrier conductivity but a detrimental impact on the thermoelectric power [38].Doping also increased the carrier concentration and added more state to the already existing band structure.Te increase in carrier conductivities brought on by doping was responsible for the decline in Seebeck coefcients.
Although changing chirality can also yield a semiconducting tube as doping, doping with fuorine is a quick, easier, fast, and more economical way to enhance the carrier density and also modify the carrier band structure from metallic to semiconducting p-type FSWCNT.Such an internal chirality is inherent and included during the band structure derivation of carbon nanotubes [2,3].Te onedimensional semiconducting FSWCNT benefts from the ADC efect because a "high-frequency-induced phasedependent dc current by Bloch oscillator nonohmicity" using an external feld has been observed by Seeger et al. in a 3-dimensional superlattice (SL) [39], Seidu et al. in a quasione dimensional SWCNT [40], and Mensah et al. in a nonparabolic semiconductor [41].Moreover, due to the high nonparabolicity and carrier density of FSWCNT, the dc generated was much higher than in the other materials.
In this study, we focus on calculating the high-frequency ADC of FSWCNTs in the presence of acoustic phonons.Te characteristics under investigation, such as the ADC and carrier momenta, are signifcantly infuenced by various factors, including the geometric chiral angle (GCA, θ h ), temperature (T), and real overlapping integrals for jumps along the tubular axis (Δ z ) and base helix (Δ s ).FSWCNTs are being explored as potential candidates for THz applications, and their ADC generation becomes particularly interesting when these parameters undergo variation.However, it is worth noting that prior to this study, no research had been conducted on the high-frequency ADC oscillations of FSWCNTs to the best of our knowledge.Terefore, the primary goal of this study was to investigate and analyse the high-frequency ADC oscillations in FSWCNTs.Understanding the acoustoelectric response of FSWCNTs can contribute to the development of novel technologies by utilising these materials in the THz frequency range.

Theory
To solve the problem in the semiclassical regime, the following conditions were utilised: (i) Δ s,z ≫ τ −1 (ℏ � 1): Tis condition implies that the energy gap between the spin-up and spin-down states, denoted by Δ s,z , is larger than the inverse of the scattering time τ.It suggests that scattering processes between the spin states are negligible compared to the energy diference.(ii) ω ≫ 1/τ: Tis condition states that the frequency of the electric feld ω should be much larger than the inverse scattering time 1/τ, indicating that the system responds quickly to the external feld.(iii) ω q ≪ ϑ(p): Tis condition assumes that the characteristic phonon frequency ω q is much smaller than the energy (ϑ(p)) of the carriers.In other words, the energy carried by the carriers dominates over the energy associated with phonons.(iv) ω ≫ Δ s,z : Te frequency of the external electric feld should be much larger than the energy gap between the spin states.(vii) Te phonons, which represent the lattice vibrations, are in a state of thermal equilibrium.Tis implies the distribution of phonons follows a thermal distribution corresponding to the temperature of the FSWCNT.(viii) For a frequency range of 100 GHz to 3 THz and a nanotube of 100 nm at a high temperature > rbin 50K (or 5 meV), low-temperature quantum efects such as Coulomb blockade become irrelevant: Under this condition, the frequency range and size of the system, along with the temperature, are such that quantum efects like Coulomb blockade can be ignored.In other words, the classical approximation is valid under these conditions.(ix) Wave phenomena such as refection and tunneling are negligible in the hypersound regime (ql ≫ 1 and ωτ ≫ 1).In this regime, high-frequency acoustic phonons can be treated as particles with energy (and momentum), allowing for a semiclassical treatment.Te absence of refections is explained by approximating the slowly varying potential as a large number of small potentials.Te small refections at each interface interfere destructively, resulting in no net refection.Te carriers (phonons in this case) can then be described semiclassically, following Newton's laws.(x) If the energy gained by the carrier from the external feld is much smaller than the overlapping integral (the energy scale associated with the slowly varying potential, Δ s,z ) along the characteristic length (d s,z ) of the system, and the scattering rate (]) is small compared to the energy picked up by the carrier from the electric feld, then the carrier will oscillate within the frst miniband with Bloch frequency (ω B � eEd s,z /ℏ).Te carrier's energy and group velocity become periodic functions of time.In this scenario, the semiclassical approximation (i.e.Δ s,z ≫ eEd s,z ) is satisfed, allowing for the carrier to be treated semiclassically using classical equations of motion.(xi) Under the semiclassical condition, the carrier wave packet, representing the carrier, is treated as a particle.Te uncertainty in the electron's momentum is assumed to be minimal, making the carrier's energy sharply defned.In addition, the uncertainty in the carrier's position is considered to be minimal compared to the spatial variations of the applied and built-in potentials.Te motion of the center of the wave packet is described by the equation ℏk/ dt � −∇ϑ � F, which resembles the classical relation between force and momentum.(xii) Te relaxation time (τ) is the characteristic time it takes for a system to return to its equilibrium state after being perturbed.In doped SWCNTs, the relaxation time is much smaller than that of undoped SWCNTs.Tis suggests that doped SWCNTs exhibit faster relaxation dynamics.Te wavelength of the wave is denoted as � 2π/q (l � 10 −6 cm), which is much less than that of the carrier free path length (λ � 10 −6 cm).In this context, the condition ql ≫ 1 is satisfed which implies that the wave's characteristic length scale is much smaller than the average distance a carrier can travel without scattering.
We consider a single longitudinal acoustic wave that travels along a uniform FSWCNT tube that has electrical insulation at both ends.In order to guarantee that there is no wave refection at the termination, the travelling acoustic wave is created by driving one end of the tube with a vibrator or interdigital transducer (IDT 1) and matching the other end to an appropriate acoustic impedance (IDT 2).Because the tube's ends are electrically isolated, the acoustic wave would pull carriers to one end, leaving the other end lacking in carriers.Te conventional electric current produced by the resulting electric feld along the tube precisely cancels out the current related to the acoustoelectric efect.Terefore, by measuring the electric potential diference between the tube's tyro ends, the acoustoelectric efect may be quantifed.Surprisingly, this process is similar to how temperature diferences cause voltage to be generated in an open circuit during the thermoelectric efect.One way to think of the net fow of travelling acoustic waves over a temperature gradient is as a net fow of phonons.
Te acoustoelectric current density is determined as follows [42][43][44][45]: where Φ, Λ, ρ, v s , and ω q are the acoustic phonon fux density, deformation potential constant, FSWCNT's density, velocity of sound, and frequency of the acoustic waves, respectively.As a result, we describe the FSWCNT's energy dispersion relation as in [46][47][48][49][50]: where and b s,z is the c − c bond length along the base helix and tubular directions, respectively.Te Δ s and Δ z (supposed for clarity to be real) are the overlap integrals for transitions along the spiral (base helix) and between the coils (tubular).Te quantities (Δ s and Δ z ) are phenomenological adjustable parameters to be determined for a real SWCNT by frst-principle numerical calculations; their estimates are given in [51].
We adhere to the proposition of Romanov and Kibis' phenomenological spiral (helicoidal) model [51][52][53].Let us consider a system of atoms arranged periodically with b s,z intervals along the spiral line, which twines at an angle θ h round a circular cylinder of radius R ≫ b s,z and contains N ≫ 1 atoms per coil.For the sake of simplicity, we assume the nearest atoms of neighbouring coils to be situated (b s,z � 2πR tan θ h ) just at the same element of the cylinder (i.e., 2πR/b s,z cos θ h integer and equal to N). Te true honeycomb crystalline structure of graphene is ignored in favour of treating FSWCNTs as a periodic chain of carbon atoms strung together on a helix.By taking into account a periodic structure of coaxial atomic rings, with the same distance between atoms in each ring as between two nearby rings, the spiral model simplifes the analysis.Tis approach allows for analytical calculations and provides insights into the properties of doped nanotubes like FSWCNTs and BC2N.It is noteworthy that the spiral model is specifcally applicable to doped nanotubes, such as FSWCNTs and BC2N, rather than pure carbon nanotubes [53][54][55].
FSWCNT exhibits a two-scale periodicity as a result of its chiral geometry: one scale is caused by the helical pitch and the other by the interatomic lengths along the base helix.Te carrier momenta p s and p z are along the base helix and tubular axis, respectively.b s is the distance between the site n and n + 1 along the base helix, and b z is the distance between the site n and n + N along the tubular axis.Due to the transverse quantisation of carrier motion not being taken into account before, both p s and p z change arbitrarily within the frst Brillouin zone [55].Here, we emphasise that the overall equality between N jumps along the base helix and one hop along the tubular axis is not guaranteed.Te carrier velocities at the ending sites are diferent for the two leaps despite the fact that both jumps begin at the same site (l) with the same initial velocity and conclude at the same site (N + l).Te carrier velocity has a circumferential component for travel along the base helix but not for motion along the tubular axis.Tis demonstrates that an FSWCNT is totally a quantum structure and that the classical minimal action principle cannot be used since quantum theory permits multiple ways for atoms to connect together.By doing this, we ignore the interference between the axial and helical paths that connect two atoms and consider the Z− and S− components of momentum to be independent of one another.Actually, this indicates that transverse motion quantisation is not something we take into account [53][54][55].
Te carrier miniband velocity along the S and Z coordinates was calculated using (2) as v � zϑ(p)/zp [46,47,50]; Te Boltzmann transport equation for carriers interacting with an acoustic wave of frequency (ω q ) and wavenumber (q) in the presence of a high-frequency electric feld was quoted as Te preceding approach is semiclassical, wherein the carriers are treated as classical particles, except for their adherence to quantum principles (Fermi-Dirac).Te initial requirement for this semiclassical approach to be applicable is that the wavelength of phonons needs to be signifcantly greater than the de Broglie wavelength of the particles, i.e., λ ph ≫ λ el .If this scenario holds true, the movement of the carrier can be approached in a classical manner, except for their statistics.By doing this, it becomes possible to treat the ions as a spread-out continuous positive background, rather than needing to consider the lattice's discrete nature.Tis preceding condition is also crucial if we plan to take into account the efects of lattice deformation through a deformation potential.
Te equation describing the motion of the carriers in FSWCNT when subjected to a high-frequency external electric feld, along with the given initial conditions t ′ � t and p ′ � p, is obtained as which has a solution given as One issue that needs to be addressed prior to obtaining a solution for the BTE is how to manage the collision term located on the RHS of (4).Te standard approach involves employing the relaxation time ansatz.In this context, the assumption is that when all external forces are deactivated, the distribution function will gradually return to a suitable equilibrium distribution F o (p), within a time interval τ.Tis ansatz has been examined and found to be quite efective when the primary scattering processes are elastic, as is the case with impurity scattering, and when τ is independent of the electric and magnetic felds.Tis ansatz has extensive application in the computation of the absorption coefcient for acoustic waves interacting with carriers, yielding favourable outcomes when compared to experimental data.Tus, employing the relaxation time ansatz, (4) becomes Te distribution function to which the carriers relax in the presence of the acoustic wave F S (p) is not necessarily the same as the equilibrium distribution function in the absence of the wave F o (p).If the dominant scattering mechanism for the carriers is from impurities, as is usually the case at low temperatures, or from acoustic phonons, as is the case at high temperatures, then the carriers relax to an equilibrium distribution in the rest frame of the moving lattice.
Te BTE is solved by using Chambers' approach.In this method, a carrier contributes to the distribution function F(r, p, t) only if it is at a point (r, p) in phase space at time t.Te number of carriers scattered in a time dt ′ and at a point (r ′ , p ′ ) on a trajectory that goes through this point in phase space is F S (r ′ , p ′ , t ′ )dt ′ .To obtain the probability that the carrier will reach this point in phase space, we multiply the number scattered onto a trajectory going through this point by the probability that the carrier will not scatter again and integrates over all time up to t.By ignoring the spatial distribution, the nonequilibrium distribution function is obtained as where F S (p, t ′ ) is the distribution in the presence of acoustic phonons and electric feld.Performing the transformation p ⟶ p − p ′ , the solution to the BTE, is found by assuming τ to be constant as When the acoustic phonons and electric feld are switched of, the Fermi-Dirac function reduces Boltzmann's function for a nondegenerate carrier gas where the Fermi level is several times below the energy of the band edge ϑ c (i.e., T ≪ ϑ c ) as follows: where T � kT is the temperature in energy unit, k is Boltzmann's constant, and E o , E 1 , and 9 represent the constant dc feld, ac feld, and carriers' electrochemical potential, respectively.A † is the normalisation constant determined using the normalisation condition to be [46,47,50]:

Journal of Nanotechnology
where n o is the carrier concentration and I n (x) is a modifed Bessel function of order n.Substituting ( 9)-( 11) into (1) yields Within the frst Brillouin zone, we use the following transformation to change the summation over p into an integral over p as and the acoustoelectric current density takes the form and the carrier current densities along the base helix (S) and tubular (Z) directions are obtained as follows [46,47,50]: 6 Journal of Nanotechnology and Te carrier momenta for the base helix and tubular directions in the frst and second quadrants of the frst Brillouin zone in the presence of the acoustic phonons are determined as in [52]: Equations ( 17) and ( 18) are substituted into (15) and ( 16), and conventional integrals are used to obtain the acoustoelectric carrier current densities along the base helix (S) and tubular (Z): where θ is the Heaviside step function.For T ≫ Δ s and T ≫ ω q , Journal of Nanotechnology and where Equations ( 20) and ( 21) further reduce to Making use of the identity, cos 2 (x) � 1/2(1 + cos(2x)) yields Solving explicitly gives where Te development of the chiral current, a distinctive feature distinguishing FSWCNTs from zigzag and armchair SWCNTs, is a key characteristic [53,54].An axial electric feld compels the current to follow a helical path.We use the labels Z and S to denote the axial and azimuthal components of the surface current density, respectively.Miyamoto et al. conducted a frst-principle numerical simulation of the chiral current, while elsewhere, researchers employed phenomenological modeling [53][54][55][56].Both approaches reached the same conclusion: chiral conductivity is relatively low compared to axial conductivity.To maintain generality, we decompose the high-frequency acoustoelectric current density into axial and circumferential components as follows: → q cos θ h , respectively.Te expression for the axial carrier thermal current density is given as and where z ′ � 2z � 2eE 1 d s /ω and Ω o � Ω � eE o d s,z .
(i) For z ′ ≪ 1 and z ≪ 1, then J ± ≈ (z/2) 2 and J o ≈ 1 − z 2 /2; thus, where Journal of Nanotechnology when Ωτ ≪ 1, i.e., in a linear approx.to E 0 Furthermore, when Ωτ ≫ 1, We can set only k � 0: Te high-frequency acoustoelectric current density relies on the amplitude of the ac electric feld in an oscillatory fashion.In the limit z ′ ≫ 1 and z ≫ 1, Tis is similar to that of acoustoelectric current in a quantised electric feld.Tis is not accidental because in a quantised electric feld, carriers undergo Stark oscillations, while in our case, an ac feld undergoes harmonic oscillations [57].

Results and Discussion
Te acoustic phonons to be examined in this investigation have a wavelength denoted as λ � 2π/q, which is shorter than the mean-free path (l) of FSWCNT carriers within the hypersound region where ql ≫ 1.In this context, the acoustic wave is treated as the packet of coherent phonons, essentially monochromatic phonons, characterised by a δfunction distribution given as Note that k → represents the electron wavevector, ℏ denotes the reduced Planck's constant, Φ → signifes the sound fux density, and ω q and v s correspond to the frequency and group velocity of sound waves characterised by the wavevector q → .Equations ( 27) and ( 28) present the general equations describing the acoustoelectric current density in FSWCNT with a primary focus directed toward the axial component outlined in (27).A numerical analysis of ( 27) was conducted using the following set of parameters: ω q � 10 11 s −1 , v s � 2.5 × 10 3 m/s, Φ � 10 5 Wb/m 2 , q � 10 6 cm −1 , λ � 10 −5 cm −1 , and l � 10 −4 cm.Te acoustic current generated in both cases is observed to be strongly dependent on the acoustic wavenumber (q), frequency (ω q ), temperature (T), and external feld (E).A transparency window is observed when ω q ≫ 2Δ s sin(qd s /2) and ω q ≫ 2Δ s sin(qd z /2), which is a consequence of the conservation laws of energy and momentum.It follows then that only carriers with momenta ℏq/2 interact with the acoustic phonons.If the phonon fux passing through the FSWCNT has a frequency which is extremely high, there will be no absorption of phonons and the acoustoelectric current density (J ae zz /J ae o ) will be zero.
Figure 1 shows the normalised acoustoelectric direct current density (J ae zz /J ae o ) dependency on the dimensionless electric feld (Ωτ) in the region of ωτ ≪ 1 and z < 1. Te current density is observed to be complex and highly anisotropic.
To understand how the various parameters afects J ae zz /J ae o dependency on Ωτ, we look at the physics of the carrier behaviour exhibited in Figure 1.When E o is negative, the carriers are trapped by the acoustic waves, dragged, and move in the direction of the acoustic waves, while the acoustic waves interact strongly with the holes to generate the hole current.Tis interaction results in the generation of a current carried by either electrons or holes, depending on the polarity of the electric feld.Conversely, when the external electric feld E o is positive, the opposite behaviour occurs.Te carriers (either electrons or holes) are trapped and dragged by the acoustic waves in the direction of the waves.Tis time, the acoustic waves interact strongly with the opposite type of carrier (electrons or holes) compared to the previous case, which results in the generation of a current carried by the opposite type of carrier.Te basic idea is undoubtedly the proof that for a sufciently large dc force amplitude, the diferential dc conductivity becomes positive over a range of negative dc bias values where a range of positive dc bias conductivity is negative.Terefore, there is a good likelihood that domain suppression and acoustic Bloch gain will be applied in purely dynamic dc stabilisation.
It is inferred from this behaviour that the dynamic complex current density is initially positive and becomes more positive with increasing dc electric feld (Ωτ), until it reaches resonance maximum enhancement at a frequency (i.e., Bloch frequency) somewhat below the critical dc electric feld (Ωτ), which turns negative and becomes more negative, until it again reaches negative resonant minimum enhancement and turns positive again.Tis observed behaviour is attributed to Bragg refection of the carriers at the band edges due to the Umklapp process the carriers are subjected to and results in Bloch oscillations of the carriers.
In order to suppress the electric feld domains cause by the onset of NDV (and NDC), the negative minimum resonant enhancement should make it possible to suppress any domain instabilities induced by any NDV (and NDC) at low frequencies with positive conductance that is sufciently high to make the acoustoelectric dc conductivity positive, without destruction of NDV (and NDC) just below the critical dc feld E c o .Space charge instabilities can also be suppressed if the ac part of the drive feld is kept so strong (ωτ ≫ 1) that the overall feld dips, during each cycle, to very low values at which the static velocity-feld-characteristic has a steep positive slope.Under steady-state operation at such negative dc felds, the FSWCNT will be an "ordinary" conductor with positive conductivity, and any space charges will decay when dc is negative rather than build up.Given a suitable combination of the dc and ac operating conditions, domains are unable to build up much.Similar dynamic stabilisation can also be achieved when the high-frequency component of the drive feld builds up to values that are no longer small compared to the dc component.

Journal of Nanotechnology
We however observe from Figure 1(a) that when Δ z is varied and Δ s is fxed (Δ s � 0.25eV), the acoustoelectric current increases in both directions than in Figure 1(b) when Δ s is varied and Δ z is fxed (Δ s � 0.25eV) [59-61].Tis again is attributed to scattering of carriers which is low along Δ s (see Figure 1(a)) than Δ z (see Figure 1(b)).Te FSWCNT's current-voltage characteristic (see Figure 1) is essentially antisymmetric.Te current is proportionate to the voltage in both bias directions until it reaches a resonance maximum (or minimum).A further increase in voltage results in a drop in ADC, which indicates the onset of NDV (and NDC).NDV (and NDC) begins at a critical voltage U c � 1.0 V/ − 1.07 V (see Figures 1(a) and 1(b)) where E o � U/L.ADC displays both gradual and sudden variations above U, indicating a space charge instability and a nonstatic electric feld.Te current reduces signifcantly in the NDV (and NDC) ranges, as expected for a homogenous electric feld distribution [1].We suggest that this can also be due to the suppression of space charge instability by the ac component of the external feld.We however observe THz emissions from the FSWCNT throughout the NDV (and NDC) regions.
We display in Figure 2 the normalised acoustoelectric current density (J ae zz /J ae o ) dependency on the dimensionless electric feld (Ωτ) in the region of strong ac feld (i.e., ωτ ≫ 1) but not strong enough to quantise the FSWCNT band structure.In the presence ωτ ≫ 1 and z > 1, the peak values of J ae zz /J ae o begin to fall (see Figure 2(a)) but oscillate harmonically with more enhancements as the dc electric feld (Ωτ) becomes more positive when Δ z is varied and Δ s is fxed (Δ s � 0.25eV).However, when Δ s is varied and Δ z is fxed (Δ s � 0.25eV), J ae zz /J ae o dependency on Ωτ increases but with similar behaviour as observed in Figure 2(b).Te ac feld in this case acts as a modulator and modulates the dc feld by updating the momenta and kinetic energies of carriers that has less energy to interact with the acoustic phonons.To enhance J ae zz /J ae o , the majority of carriers with requisite momenta and energies interacted strongly with the acoustic phonons and performed intraminiband transition and generated a large intraminiband current (see Figures 2(a) and 2(b)).ADC shows both smooth and abrupt changes giving evidence for a strong space charge instability and a nonstatic electric feld in the nondegenerate FSWCNT (i.e., ωτ ≫ 1).
Herein, we examine in Figure 3 the normalised acoustoelectric current density (J ae zz /J ae o ) dependency on the dimensionless electric feld (Ωτ) when z is varied and ωτ is fxed.In Figures 3(a) and 3(b), we display J ae zz /J ae o dependency on Ωτ when ωτ � 0 (see Figure 3(a)) and ωτ � 0.5 (see Figure 3(b)).Tere is minimal or no change observed in J ae zz /J ae o when ωτ � 0 and z is varied between z � 0 and z � 0.5.However, J ae zz /J ae o starts to decrease drastically from z � 1.5 − 3.0 (see Figure 3(a)).Similar behaviour is observed in Figure 3(b), but the change observed when z is varied for fxed ωτ � 0.5 is a bit clearer in Figure 3(b) than in 3(a).Furthermore, when z ≫ 0, ωτ � 1, and ωτ � 1.5 for Figures 3(c) and 3(d), respectively, we observe that J ae zz /J ae o dependency on Ωτ begins to fall gradually but oscillates much more strongly harmonically in Figure 3(b) than observed in 3(a).
Figure 4 displays the normalised acoustoelectric current density (J ae zz /J ae o ) dependency on the dimensionless electric feld (Ωτ) for varied values of n o in diferent dynamic regimes.Te J ae zz /J ae o dependency on Ωτ increases gradually when the impurity concentration n o increases in the region of Δ s � Δ z � 0.25eV, ωτ � 0.5, and z � 0.5.Tis high J ae zz /J ae o observed is due to the fact that as the level of doping grows, the FSWCNT starts fipping from nondegenerate to degenerate character (see Figure 4(a)).Tis is consistent with experimental data, which demonstrates that a rise in fuorine atom concentration leads to an increase in n o , which causes the Fermi level to begin moving into the conduction band  and change the FSWCNT from nondegenerate to degenerate state.By introducing the additional carrier band structure and nonlinearly modifying the carbon π− bonds around the Fermi level, this leads to chemical activation of the passive SWCNT surface and produces a band structure with a width of two periods.Due to fuorine's extreme electronegative nature and the π− electrons attached to it, the SWCNT's walls weakens, which reduces the interaction of free charge carriers with the felds.Te density of free carriers reduces as a result of the bonding charge change.In other words, addition of an impurity band close to the Fermi level alters the chemical potential and reduces the band gap from nondegenerate to degenerate character.Tus, there are more carriers that interact strongly with the acoustic waves to generate a high intraminiband carrier current.As ωτ � 1 and z ≫ 1, the amplitude of the ac feld starts to modulate the net current density and the carrier-phonon interaction starts to fall gradually and so as the current density (see Figure 4(b)).Moreover, when ωτ � 1 and z ≫ 1, the peak of J ae zz /J ae o begins to fall, and there is an observance of a strong harmonic oscillations in this regime (see Figure 4(c)).Furthermore, in the region when ωτ � 2 and z ≫ 2, J ae zz /J ae o continues to decrease further with strong harmonic oscillations observed (see Figure 4(d)).
We determined that the self-current oscillation is caused by the emergence of space charge instabilities.For a specifc bias voltage in the NDV (and NDC) region, the space charge and electric feld distribution in the FSWCNT is no longer uniform along the FSWCNT axis.It is predicted that there will be a space charge instability if the FSWCNT's carrier concentration (n o ) and length (L) product is sufciently high, i.e., nL ≥ 7ϵϵ o E c /e, where ϵ is the dielectric constant, ϵ o is the permittivity of free space or the electric feld constant, and E c � ℏ/ed s,z τ is the critical/threshold feld at which

14
Journal of Nanotechnology NDV (and NDC) emerges.We fnd that the observance of the strong space charge instability is indicative of the fact that n o L exceeds the threshold by an order of magnitude, and thus, it is observance.However, in our FSWCNT, the NDV (and NDC) is due to Bloch oscillation miniband carriers [23].At electric felds larger than E c , the carriers sufer Bragg refection at the miniband boundary, which lead to an electric feld induced localisation of carriers and consequently to a decrease of carrier velocity with increasing feld [2].At the critical feld E c , the average mean free path of a carrier along the FSWCNT axis is given by the length x � Δ s,z /E c of a trajectory, which corresponds to the path a carrier traverses when it streams from a state of minimum to a state of maximum miniband energy.At typical hypersound fux, the value of J o at room temperature for a transistor-based single-walled carbon nanotube can be as high as 0.4mA/μm for V gs of −2.5 to −0.5V [50].Moreover, this value can be high for −0.5V to 5V.

Conclusion
Te semiclassical carrier dynamics was used to examine the axial acoustoelectric direct current (ADC) density of a nondegenerate FSWCNT under the infuence of a highfrequency feld in the hypersound regime.We demonstrated theoretically that the generation of THz radiation solely due to the dynamics of space charge instabilities (i.e., without resonant system) occurred due to Bragg refection of Bloch oscillating carriers in the FSWCNT's miniband.Te FSWCNT parameters n o , Δ s ,and Δ z , as well as E o and E 1 , were found to have a signifcant impact on the axial ADC (J ae zz /J ae o ).Varying (Δ s , Δ z , E 1 , and n o could be used to generate and tune the FSWCNT to a high carrier drift velocity and a high fundamental frequency of the ADC selfoscillations at 300K.Interestingly, the main fnding of the study was undoubtedly the proof that for a sufciently large dc force amplitude (E o ), ADC becomes positive over a range of negative bias (-E o ) values, and for a range of positive bias (+E o ), the ADC was negative.Tus, the FSWCNT can therefore function as an active device up to very high frequencies, possibly up to frequencies that correspond to the submillimeter wavelength range, and it is likely that domain suppression and acoustic Bloch gain might be obtained in purely dynamic ADC stabilisation.
represents the nonequilibrium carrier distribution function, v(p) represents the carrier miniband velocity, p represents the carrier quasi-momentum, and τ represents the carrier relaxation time.Te LHS of (4) results from a change in the distribution function due to time dependence, spatial gradients, and external forces, whereas the RHS arises from a change in the distribution due to the collision processes.