Giant Thermoelectric Figure of Merit in Fluorine Doped Single Walled Carbon Nanotube

Herein, we report on a giant thermoelectric ﬁgure of merit ( ZT ) of a non-degenerate ﬂuorine-doped single-walled carbon nanotube (FSWCNT) using a tractable analytical approach and the phonon lattice Boltzmann model (LBM). We investigate the inﬂuence of the doping concentration ( n o ), and the overlapping integrals (∆ s and ∆ z ) on the ZT . The ZT and the temperature range of operation can be tuned using the doping (impurity) concentration and the overlapping integrals, respectively. The lattice thermal conductivity obtained using the phonon LBM was calculated to be; κ (cid:96) = 107 . 2 W/mK which yielded a ZT greater than 20. Interestingly, the ZT obtained is higher than what has been reported in superlattices, ( ZT ≈ 1 . 4) and superlattice nanowire, ( ZT ≈ 4) at 300 K , making FSWCNT a potential candidate for thermoelectric applications.


Introduction
Conversion of energy between heat and electricity based on thermoelectric effects is indispensable in the energy mix in mitigating the global energy needs and climate change due to the potential applications in the area of waste heat recovery and refrigeration [1,2]. Practically, the phenomenon is used for electric power generation (Seebeck effect) or for cooling (Peltier effect) without mechanical moving parts [3][4][5]. The performance of a thermoelectric device is usually determined by its dimensionless figure-of-merit (FoM), ZT = α 2 σT /κ, where α, σ, T , κ are the Seebeck coefficient (thermoelectric power), the electrical conductivity, the absolute temperature, and the thermal conductivity (which comprises the electronic (κ e ) and the lattice (κ ) components), respectively [3][4][5][6][7][8][9]. Enhancements in the ZT have been achieved in nanostructures, mainly due to the thermal conductivity reduction by phonon-boundary scattering as the amount of interfaces increases [10].
Theoretical studies and experimental investigations of the thermoelectric properties of low dimensional (1−D and 2−D) systems have been pursued extensively for various materials [8,[11][12][13]. Quantum confinement of electrons are expected to lead to improved ZT value in low-dimensional materials, due to the anticipated enhancement of FoM parameters. 0−D structures hold even greater promise than 1−D and 2−D systems [8]. However, unlike 1−D and 2−D systems, where at least one of the directions is not confined and thus, can provide electrical conduction, 0−D structures such as quantum dots are confined in all directions and may pose difficulties for some applications [14][15][16]. For this reason, several novel structures based on quantum dots that capture the essence of 0−D structures have been proposed and synthesized, such as quantum dot array superlattices [17] and superlattice nanowires [18]. These new structures have shown great promise. Harman et al. have measured an impressive thermoelectric performance with ZT ≈ 2 at 300K for P bT eSe based quantum dot superlattices [17], compared to the highest ZT of ≈ 1 for conventional bulk materials. A possible mechanism for this enhancement is attributed to the mi-niband formation in coupled 3−D quantum dot arrays [19].
1−D nanostructures such as single-walled carbon nanotubes (SWCNTs) has attracted much attention due to their outstanding electrical conductivity. The dominant contribution to the thermoelectric power is purely electronic, though in some materials, such as graphite, an electron-phonon scattering contribution or phonon drag effect also becomes prominent at certain temperature ranges, especially at low temperature. However, even though the thermal properties of nanotubes are dominated by phonons, we expect that the thermoelectric power will be primarily governed by the electronic drift contribution, the electronic band-structure of the nanotubes, and the electron scattering mechanisms [7][8][9]20]. SWCNTs skeletal structure enables the attachment of large number of foreign atoms to the surface without destroying the tubular structure. Doping an armchair-SWCNT with fluorine atoms changes its nature from metallic to semiconducting [8,[21][22][23][24][25].
In this study, we calculate the Peltier coefficient, the zero-current electron density, the thermal conductivity and the FoM of the FSWCNT. We noted that the properties mentioned above strongly depend on the geometric chiral angle (GCA, θ h ), temperature (T ) and real overlapping integrals for jumps along the tubular axis (∆ z ) and the base helix (∆ s ) [7][8][9]. The variation of these parameters can give rise to a giant thermoelectric power, unusual low electron and lattice thermal conductivities and a ZT greater than 6 (and as high as 20), making FSWCNT a potential candidate for thermoelectric applications.
To the best of our knowledge, no study has investigated the FoM in FSWCNTs.
This study aims to investigate the thermoelectric FoM of FSWCNT with a non-parabolic double periodic band.

Carrier Current Densities
We consider the problem in the semi-classical regime using the Boltzmann transport equation (BTE) written as [6][7][8]: Here, f ( r, p, t) is the non-equilibrium carrier distribution function, f o ( p) is the local equilibrium distribution function, v( p) is the carrier miniband velocity, E is the applied electric field, r is the carrier position, p is the quasi-momentum of the carrier, τ is the relaxation time and e is the carrier charge. In the linear approximation, where the second term on the left hand side is considered to be a weak perturbation, the solution to the BTE is expressed as [6][7][8][9]: where µ is the electrochemical potential which ensures carrier conservation. For a p−type FSWCNT with a periodicity of 3b, the energy band is given as [8,[21][22][23][24][25]: where u = √ 3/2 and w = 3 √ 3/2. ε o is the energy of an outer-shell electron in an isolated carbon atom, ∆ s and ∆ z are the real overlapping integral for jumps along the respective coordinates, p s and p z are the carrier momentum along the base helix and the 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. The carrier charge ( j) and thermal ( q) current densities are defined as [6][7][8][9]: The axial component of the carrier charge is obtained as [8]: sin θ h ∇ s T. (5) and the circumferential component is expressed as [8]: The carrier thermal current density is also obtained as: We resolve Eq.(7) along the base helix ( S) and the tubular axis ( Z), by neglecting the interference between the axial and the helical paths connecting a pair of atoms, so that the transverse motion quantizatoin is ignored [7][8][9]. The carrier thermal components yields: and For a non-degenerate electron gas, we use the Boltzmann equilibrium distribution function f o (p) expressed as [8]: where n o is the electron concentration, I n (x) is the modified Bessel function of order n and k B is Boltzmann's constant. The carrier miniband velocity components v s and v z are given in ref. [8], where the integration is carried out respectively. Solving explicitly, we express the thermal current density into axial and circumferential components without a lost of generality as: q z = Z q + S q sin θ h , q s = S q cos θ h , respectively. The axial carrier thermal current density is given as: and circumferential thermal component yields: where E = ∇ z (µ/e − φ). The representation of q in terms of E in Eq.(11) and Eq. (12) is not convinient for comparing theory with experiment; it is therefore worthy to express q in terms of j and ∇T . Combining Eq.(11) and Eq.(12) and solving explicitly yields: and Eq.(13) and Eq. (14) are in the form of the Onsager relation as: where κ is the carrier thermal conductivity when the carrier current density is zero ( j = 0) and Π is the Peltier coefficient (Π = αT ). Comparing Eq.(13) and Eq. (14) to Eq. (15), the axial (Π zz ) and circumferential (Π cz ) components of the Peltier coefficients are respectively, given as: and The expression and evaluation of the axial and circumferential component of the electrical conductivity (σ), and the thermoelectric power (α) of the FSWCNT were reported in ref. [8]. The axial (κ zz ) and the circumferential component (κ cz ) of the carrier thermal conductivities are expressed as: and Lattice Thermal Conductivity (κ ) Following the approach in ref. [26], we establish the phonon lattice Boltzmann model (LBM) which is obtained from the BTE for the kinetic theory of phonons.
The solution to the LBM is conducted using Chapman-Enskog expansion. In the diffusive limit, the macroscopic heat transport equation is obtained and the correlation between the numerical parameters and the bulk material properties are established for 1D, 2D and 3D situations, respectively. In the hydrodynamic regime, q 1, the phonon LBM is obtained in discrete form [26,[27][28][29][30] as: where c i (i = 1, 2.....n) are the discrete velocities. The spatial and time derivatives of i ( x, t) on the left side of Eq.(20) are approximated to a dimensionless first order finite difference by replacing δ x = c i δt, where δ x is the spatial step, δt is the time step and τ = τ R /δt which yields: The discrete equilibrium phonon energy densities are obtained as: with D(ω) as the phonon density of states which incooperates transverse, longitudinal and in-plane branches. Employing Debye's approximation and after much simplication, the equilibrium phonon energy density reduces to: The phonon contribution is obtained by integrating over the phonon density of states with a convolution factor that reflects the energy and occupation of each phonon state. For nonzero temperatures, the convolution factor is 1 at ω = 0, and decreases smoothly to a value ≈ 0.1 athω = k B T /6, so that the specific heat rises with temperature as the phonon states are occupied. This is because D(ω) is a complicated function of ω and thus, the specific heat at moderate temperatures, cannot be calculated analytically. At low temperature (T Θ D ), however, the temperature dependence of the specific heat is in general much simpler. In this regime, the upper bound in Eq.(24) can be taken as infinity, and D(ω) is dominated by acoustic phonon modes, i.e. those with ω → 0 as q → 0. If we consider a single acoustic mode in d dimensions that obeys a dispersion relation ω ∝ q α , then from Eq.(24) it follows that: Thus, the low-temperature specific heat contains information about both the dimensionality of the system and the phonon dispersion [31][32][33][34].
The different types of lattice structures for the phonon LBM simulation are shown in Fig. 1 with their probability distribution functions and their velocities.
The discrete velocities are used to determine appropriate weight factors, w i , through tensor moment expansions [27]. Along with maintaining consistent phonon propagation in each direction, weight factors are needed to preserve rotational invariance, meaning the lattice can be rotated in 90 o increments without changing the results. This is a necessary feature in LBM because the streaming operation is the same in each direction. A rotationally invariant cartesian lattice must satisfy the conservation equations, which are moments about the lattice speed [31][32][33][34].
In order to recover the macroscopic heat transport equation, we apply a Chapman-Enskog expansion to solve the phonon lattice Boltzmann equation in Eq. (21). The central-point component with vanishing lattice velocity is not accounted for in this formalism. This distinction originates from the fact that the conservation laws of particle number and mass in hydrodynamics are no longer ensured in phonon transport [35,36] since phonons can be easily created and destroyed. However, phonons are bosons with zero static mass and so their energy as well as velocity can never reach zero; otherwise they will disappear, which is a basic precept of the classical harmonic oscillator: =hω/2 (n + 1/2).
For a typical D2Q8 lattice, the Chapman-Enskog expansion can be done through similar procedures for other lattice structures. Expanding i ( x + c i δt, t + δt) around i ( x, t) using Taylor series within second order for small lattice time step, we obtain: Substituting Eq.(26) into Eq.(21) yields: Expandng ∂/∂t and introducing two time scales t 1 and t 2 and one spatial scale where ε is a small factor often chosen as the Knudson number, Kn, (ratio of the phonon mean free path, , to the characteristic length of the FSWCNT size L).
The discrete phonon energy densities are asymptotically expanded within first i ( x, t) + ..... Substituting this expression into Eq.(28) yields the subsequent orders of magnitude i.e. ε 0 , ε 1 , ε 2 , respectively as: The heat flux q, is asymptotically expanded within first order as: Summing over i of Eq. (29) and Eq.(31), respectively and combining the two equations in terms of space and time t 1 , t 2 and x 1 yields the energy balance equation which is expressed as: Multiplying both sides of Eq.(30) and Eq.(31) by c i , and summing over i, and combining the two equations gives rise to the heat flux: where d = c v dT and the lattice thermal conductivity is given as: where the lattice speed and the phonon group velocity are related as: c = 2v g /3.

Results and Discussion
We analyzed the ZT of a non-degenerate FSWCNT using the BTE and the phonon LBM. We obtained the carrier thermal conductivity (κ e ), and the Peltier coefficient (Π) based on the analysis of the BTE and simulated the lattice thermal conductivity (κ ) based on the phonon LBM. Additionally, we investigated the influence of the doping concentration (n o ) and the overlapping integral for jumps (∆ s and ∆ z ) on the behavior of the ZT . The ZT was highly nonlinear depending on the doping concentration n o , temperature T , and real overlapping integrals for jumps ∆ s and ∆ z . We evaluated numerically the ZT defined by the relation: where κ(κ + κ e ) is characterized by the carrier thermal conductivity (κ ) and the lattice thermal conductivity (κ e ). The latter plays more important role in the thermal conducativity of FSWCNT. In general, the power factor becomes the main determining parameter in enhancing the ZT when the thermal conductivity does not vary much.  We display in Fig.2 the thermal conductivity (κ zz ) dependence on temperature for different doping concentrations, n o . When the doping concentration is in the range 10 18 cm −3 , the thermal conductivity is dominated by boundary and phonon-phonon scattering in the high temperature region (Fig.2(a)). The scattering of phonons by impurities (doping), indicates a larger reduction in the thermal conductivity for low doping concentrations for materials like FSWCNT with greater mass and radius differences between the foreign (F ) and host atoms (C). This explains the stronger reduction in the thermal conductivity of FSW-CNT as observed in Fig.2(a). The strength of the phonon-impurity scattering rate diminishes at lower temperatures. The thermal conductivities of the FSW-CNT at low doping concentrations is low, indicating that the phonon-boundary scattering and Umklampp processes dominates over the phonon-impurity scat-tering at high temperatures [37][38][39][40][41]. Increasing the doping concentration to the range ≈ 10 19 cm −3 , the thermal conductivity start to show linear behavior in the low temperature regions where scattering is dominated by both phononimperfection and phonon-boundary scattering as shown in Fig.2(b). Phononboundary scattering also contributes significantly to the reduction in the thermal conductivity of the FSWCNT regardless of the level of doping concentration.
The peak values of the conductivities for the FSWCNT is determined by the level of impurities in the FSWCNT which is marginal and shift towards high temperatures as the doping concentration increases. However, in doping levels of 10 19 cm −3 the peak values of the thermal conductivity remains the same. It shows no increase at all as the doping concentration increases, but it is a little bit higher at room temperature for 10 19 cm −3 than 10 18 cm −3 . This slight increase in the former is due to the increase in doping level as the FSWCNT starts switching from non-degenerate to degenerate character ( Fig.2(b)).
In simulating the lattice component of the thermal conductivity (κ ) dependence on temperature, we used the phonon LBM. Both thermally excited transverse and longitudinal phonons that interact through normal (N ), and Umklapp (U ) processes were considered in the simulation. The collision terms were approximated to the relaxation time model with two constants, τ U and τ N representing the time(s) required for U and N −processes to relax the distribution of phonons to their corresponding equilibrium distributions. Treatments of isothermal, isoflux, adiabatic boundary conditions and periodic temperature gradient boundary conditions were considered during the simulation. Figure (Fig.2 shows specifically a linear temperature dependence at low T which is consistent with one-dimensional band-structure of FSWCNT, with linear acoustic bands contributing to the thermal transport at low temperatures and optical subbands at higher temperatures. The peak of the thermal conductivity for the FSWCNT occurs near 100K which separates the low temperature region (where phonon scattering is dominated by imperfections and surfaces), and high temperature regions (where phonon-phonon scattering is dominant). The phonon-phonon (or three-phonon) processes in the FSWCNT are anharmonic effects caused by third-order terms in the lattice potential energy. The scattering is a result of the presence of one phonon that causes a periodic elastic strain which through anharmonic interactions modulates in space and time, the elastic constant of the crystal. A second phonon perceives the modulation of the elastic constant and thereafter is scattered to produce a third phonon. The lattice thermal conductivity obtained for FSWCNT is significantly lower than that of pristine samples (SWCNTs) due to the much stronger reduction of phonon mean free path by boundary scattering. A D1Q2 simulation executed for a 15, 000 time-step calculated the lattice thermal conductivity for the FSWCNT to be 107.2W/mK at 300K for doping level of 10 19 cm −3 . It is observed from Fig.2 that, the electronic contribution is minimal and negligible since most of the heat is carried by the phonons. A ratio of the lattice (phonon) to the electronic contribution evaluated shows that, the lattice contribution is about 5-orders (10 5 ) of magnitude higher than its electronic component.
The dependence of the axial thermal conductivity (κ zz ) on temperature for various values of ∆ z when ∆ s is fixed are presented in Fig.3. It is observed that the thermal conductivity is strongly dominated by the lattice contribution (κ ) since FSWCNT is a non-degenerate semiconductor. In the presence of ∇T , the thermal energy is propagated by means of wave packets consisting of various normal modes. The phonon scattering processes included in the Debye approximation for the phonon LBM are resistive and thus, constitute U −processes. A strong dependence of the axial thermal conductivity on low temperature U −processes (exp(Θ/T )), (where the phonon relaxation time is determined by boundary scattering) and high temperature U −processes (T −1 ), (where phonon-phonon scatterings are dominant in determining the phonon relaxation time) for ∆ z = 0.013, 0.014 and 0.015eV are observed (see Fig.3(a)).
The U −process is observed to be very strong which signifies a strong phononphonon interactions. The total phonon momenta are not conserved during this interaction (phonon-phonon) because the U −processes tend to restore nonequilibrium phonon distribution to equilibrium in the FSWCNT and thus, give rise to thermal resistance [42]. However, the N −process is observed to be non-a constant value (hyperbolic in nature). This behavior has been observed in semiconducting tubes where α zz ≈ T −1 [6][7][8][43][44][45]. The FSWCNT behaved purely as a p−type semiconductor initially but switches to n−type behavior at higher temperatures as the ∆ s increases [8].     The ZT dependence on temperature for various values of ∆ s when ∆ z is fixed is displayed in Fig.4. The ZT rises linearly and saturates as the temperature increases above 100K for ∆ z = 0.015eV when ∆ s is varied. The saturation of the ZT is due to the maximum velocity (saturation velocity) the carriers in FSWCNT attains in the presence of high fields. The FSWCNT carriers move at an average drift velocity proportional to the strength of ∇T with time. The higher the mobility of the FSWCNT carriers, the higher the drift velocity and consequently higher intraminiband current values for the given field strength.
A limit is reached around 100K where further high field values do not allow the FSWCNT carriers to move any faster, having reached its saturation velocity, due to several mechanisms that eventually limit the movement of the carriers in the FSWCNT. As the applied field ∇T increases from this point (T = 100K), the FSWCNT carrier velocity no longer increases because the carriers lose energy through increased levels of interaction with the lattice, by emitting phonons and even photons since the carrier energy is large enough to do so. Thus, at high fields (i.e. high ∇T ) scattering of intraminiband electrons by phonons increases for fixed ∆ z which induces a high-frequency carrier dynamics [8,23] that depend critically on the magnitude of ∇T . The carrier velocity is drifted through the FSWCNT and allowed to perform drifting periodic orbits (in THz frequencies) [8,23] leading to a resonant enhancement of ZT which saturates (i.e. ∇T ≈ 0). In Fig.4(a), we observe the ZT value to be very high i.e ZT > 6. This value of ZT starts saturating gradually as the value of ∆ z increases from 0.015eV to 0.025eV ( Fig.4(a) and Fig.4(b)). In this case, the electron scattering along the axial direction decreases for increasing values of ∆ s when ∆ z is fixed and low. The number of intraminiband electrons scattered by the lattice vibrations are less and so a greater number of the intraminiband electrons contributes to the electrical conductivity (σ) and consequently the power factor (α 2 σ). Thus, α 2 σT κ and so the ZT keeps rising as high as ZT > 6. However, the electron scattering along the axial direction start increasing for increasing values of ∆ s when ∆ z is fixed. This means the number of intraminiband electrons scattered by the lattice vibrations are more and so a lesser number of intraminiband electrons contributes to the electrical conductivity (σ) and consequently the 0.015eV to 0.024eV (Fig.5(a)-Fig.5(f)). Thus, α 2 σT κ and so the ZT keeps rising (i.e. ZT > 20).

Conclusion
The thermoelectric figure of merit (ZT ) of a non-degenerate FSWCNT was investigated using a tractable analytical approach and the phonon LBM. The FSWCNT parameters ∆ s , ∆ z , and n o were found to influence the ZT strongly.
At high fields (high ∇T ), there is an increase in scattering for fixed ∆ z which induces a high-frequency carrier dynamics that depend critically on the magnitude of ∇T . The carrier velocity is drifted through the FSWCNT and allowed to perform drifting periodic orbits (in THz frequencies) leading to a resonant enhancement of ZT which saturates at high temperatures (i.e. ∇T ≈ 0). Optimizing ∆ s , ∆ z , T and n o can yield a ZT greater 20 (ZT > 20) for high ∆ s .
Interestingly, varying the n o , ∆ s and ∆ z can be used to tune the FSWCNT to operate at higher temperatures. However, based on the ZT obtained at weak electric field, we propose FSWCNT as a potential candidate for thermoelectric application. Furthermore, contemporary techniques can be employed to reduce κ .