Optimal Thermoelectric Power Factor of Narrow-Gap Semiconducting Carbon Nanotubes with Randomly Substituted Impurities

We have theoretically investigated thermoelectric (TE) effects of narrow-gap single-walled carbon nanotubes (SWCNTs) with randomly substituted nitrogen (N) impurities, i.e., N-substituted (20,0) SWCNTs with a band gap of 0.497 eV. For such a narrow-gap system, the thermal excitation from the valence band to the conduction band contributes to its TE properties even at the room temperature. In this study, the N-impurity bands are treated with both conduction and valence bands taken into account self-consistently. We found the optimal N concentration per unit cell, $c_{\rm opt}$, which gives the maximum power factor ($PF$) for various temperatures, e.g., $PF=$0.30$\rm{W/K^2m}$ with $c_{\rm opt}=3.1\times 10^{-5}$ at 300K. In addition, the electronic thermal conductivity has been estimated, which turn out to be much smaller than the phonon thermal conductivity, leading to the figure of merit as $ZT\sim 0.1$ for N-substituted (20,0) SWCNTs with $c_{\rm opt}=3.1\times 10^{-5}$ at 300K.


Introduction
In 1993, Hicks and Dresselhaus proposed that significant enhancement of the thermoelectric (TE) performance of materials could be realized by employing one dimensional (1D) semiconductors. 1) Single-walled carbon nanotubes (SWC-NTs) are of particular interest as high-performance, flexible and lightweight TE 1D materials. [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] Both n-and p-type semiconducting SWCNTs are required to develop SWCNT-based TE devices. A great deal of effort has been put into the carrier doping of SWCNTs using various chemical [5][6][7][8][9][10][11][12] and field-effect doping methods. [13][14][15] In the case of field-effect doping, the present authors (T.Y and H.F) have theoretically clarified that an SWCNT exhibits the bipolar TE effect (i.e., the sign inversion of Seebeck coefficient from positive (p-type) to negative (n-type) by changing the gate voltage) within the constant-τ approximation and the self-consistent Born approximation. 16) On the other hand, in the case of chemical doping, such as with nitrogen (N) and boron (B) doping, the impurity-doped SWCNTs are regarded as strongly disordered systems of which the TE properties cannot, in principle, be theoretically described by the conventional Boltzmann transport theory (BTT). The present authors (T.Y. and H.F.) have recently succeeded in describing the TE properties of N-substituted SWCNTs using the linear response theory (Kubo-Lüttinger formula 21,22) ) combined with the thermal Green's function technique. 17) In Ref. 17, the authors reported that a decrease in the N concentration of a (10,0) SWCNT increases both the electrical conductivity and the Seebeck coefficient at room temperature (T = 300 K), and eventually the room-T thermoelectric power factor of the SWCNTs increases monotonically as the N concentration decreases down to an extremely low concentration of 10 −5 atoms per unit cell.
In the case of a (10,0) SWCNT with small diameter of d t = 0.78 nm, the influence of thermal excitation from the valence band to the conduction band on the room-T TE effects is negligible because of the large band gap E g = 0.948 eV. On the other hand, when the diameter is larger, the electron-hole excitation probability (∼ e −E g /k B T ) becomes much larger than that for a (10,0) SWCNT. For example, the electron-hole excitation probability at T = 300 K for a (20,0) SWCNT with a diameter of d t = 1.57 nm and a band gap of E g = 0.497 eV, which are a typical diameter and band gap in experiments, 23) is much larger (3.80 × 10 7 times larger) than that for a (10,0) SWCNT. The influence of electron-hole excitation on the TE properties of SWCNTs determines the performance of SWCNT-based TE devices.
To clarify the objective of the present study, we here briefly summarize the above-mentioned two our previous studies. 16,17) In Ref. 16, overall trends of bipolar TE effects have been studied by incorporating the both conduction and valence bands, but the impurity band was not incorporated. In Ref. 17, we focus on the impurity-band effects on TE properties of (10,0) SWCNTs with N concentration from c = 10 −2 to c = 10 −5 . Here, we neglect the presence of valence band because the electron-hole excitation probability is negligible even at a high temperature of 400 K (see Appendix A). In this situation, the thermoelectric power factor increases with decreasing the N impurity concentration (see Fig.8 in Ref. 17). On the other hand, for the (20,0) SWCNT, the contribution of valence band to the TE effects cannot be neglected even at ∼300 K because of the small band gap of E g = 0.497 eV and it is not clarified yet. Thus, in this study, we incorporate both the valence and the conduction bands of N-substituted (20,0) SWCNTs and treat precisely the N-induced impurity band in the band gap using the self-consistent t-matrix approximation. As a result, we found the power factor exhibits the maximum value at a certain concentration of N atoms for a fixed temperature. In addition, we also estimate the temperature dependence of electronic thermal conductivity λ e of N-substituted SWCNTs to be compared to that of phonons λ ph and then estimate the figure of merit, ZT .

Linear Response Theory for Thermoelectric Effects
In the presence of both an electric field E and a temperature gradient of dT/dz along the z-direction in a material (e.g., the tube axis of an SWCNT), the electrical current density J is generally given by within the linear response with respect to E and dT/dz. 24) Here, L 11 and L 12 are the electrical conductivity and the thermoelectrical conductivity, respectively. Using L 11 and L 12 , the Seebeck coefficient S is expressed as (2) and the power factor PF, which is one of the figures of merit for TE materials, is described by The expression of L 11 and L 12 is given by in terms of the spectral conductivity α(E). Here, e is the elementary charge, µ is the chemical potential and f (E − µ) = 1/(exp((E − µ)/k B T ) + 1) is the Fermi-Dirac distribution function. S in Eq. (2) and PF in Eq. (3) can thus be determined from Eqs. (4) and (5) once α(E) is known. To the best of our knowledge, the expression of L 11 and L 12 in Eqs. (4) and (5) was first proposed by Sommerfeld and Bethe in 1933, 25) subsequently by Mott and Jones,26) and then by Wilson. 27) Recently, the authors (T.Y. and H.F.) applied the Sommerfeld-Bethe (SB) relation expressed as Eqs. (4) and (5) to disordered N-substituted SWCNTs using a simple tight-binding model combined with a self-consistent t-matrix approximation. 17) Also, Akai and co-workers adopted the SB relation to treat the disordered metal alloys using the density functional theory combined with a coherent potential approximation (CPA) . 28,29) More recently, Ogata and Fukuyama clarified the range of validity of the SB relation, even for correlated systems including electron-phonon coupling and electron correlations. 30)

Effective-Mass Hamiltonian of SWCNTs
In this subsection, we briefly review the electronic structure of semiconducting SWCNTs with zigzag-type edges (z-SWCNTs). In our previous paper in Ref. 16, we gave a onedimensional Dirac Hamiltonian with an energy dispersion for the effective Hamiltonian of semiconducting (n, 0) SWC-NTs near the conduction (+) and valence (−) band edges, where k is the wavenumber along the tube-axial direction and q specifies the two pairs of lowest-conduction and highest-valence bands: and ∆ q is a half of the band gap (i.e., E g ≡ 2∆) and v q is a group velocity, expressed as and v q = − a z γ 0 cos πq n where γ 0 = 2.7 eV is the hopping integral between nearestneighbor carbon atoms and a z = 0.426 nm is the unit-cell length for an (n, 0) SWCNT. 31,32) The energy origin (E = 0 eV) in Eq. (6) is set at the middle of the band gap, E g . In the small-k region that obeys k 2 (∆/ v) 2 , the energy dispersion in Eq. (6) is reduced to with the effective mass m * q = ∆ q /v 2 q for both conduction and valence bands. Thus, the effective Hamiltonian is also given by with Eq. (11), where c † k and d † k (c k and d k ) are the creation (annihilation) operators for the conduction and valence band electrons, respectively. The spin and orbital degrees of freedom q are omitted from Eq. (12).
At this point, we take account of the random potential term in H 0 in Eq. (12) such that to examine the effects of N-doping on SWCNTs. Here, V 0 is the attractive potential (V 0 < 0) for an N atom in an SWCNT. For example, V 0 = −0.91 eV for (20,0) SWCNTs (see Sec. 2.3 for details). In Eq. (13), c † j (c j ) is the creation (annihilation) operator of an electron at the jth impurity position, and j represents the sum with respect to randomly distributed impurity positions for a fixed average concentration of c = N imp /N unit , where N imp is the total number of impurity positions and N unit is the number of unit cells in a pristine SWCNT with the length L.
We also confirm that the small-k condition of | vk| ∆ is satisfied within the temperature region of 0 < T < 500K discussed in this paper.

Self Energy due to Impurity Potential
The modification of thermoelectric effects by randomly distributed impurities will be studied based on the thermal Green's function formalism through self-energy corrections of the Green's functions. In this study, the influence of random N potential on the conduction-and valence-band electrons is incorporated into the retarded self energy using the self-  Diagram of a self-consistent t-matrix approximation for the retarded self-energy. The crosses, dashed lines and solid double lines with arrows respectively denote the impurity sites, the impurity potential and the one-particle retarded Green's function to be determined self-consistently.
consistent t-matrix approximation as shown in Fig. 1, 17,[33][34][35] which corresponds to the dilute limit of CPA for binary alloys. 28,29,36,37) Within the self-consistent t-matrix approximation, both the self energies Σ c/v (E) for conduction/valence-band electrons in an N-substituted SWCNT are independent of k because of the short-range of the impurity potential in Eq. (13) and are determined by the requirement of self-consistency, as with where ± corresponds to c/v, respectively. The k-summation in Eq. (15) can be analytically performed by substituting Eq. (11) into Eq. (15), and we obtain where Im ±(x − σ c/v (x)) − δ > 0, and equal to the characteristic energy of an SWCNT. From Eqs. (14) and (16), the self-consistent equation for σ c/v (x) is given by Equation (18) can also be rewritten as or as the cubic equation for σ c/v , with a 3 = 1, a 2 = −(x∓δ+2cv 0 ±v 2 0 /4), a 1 = cv 0 {2(x∓δ)+cv 0 }, and a 0 = −(x ∓ δ)(cv 0 ) 2 , where the upper/lower sign is for the conduction/valence-band electrons. Equation (20) indicates that for each energy,  mined using the condition dx/dσ c/v = 0. In addition, we can see in Fig. 2(a) that the impurity band appears just below the conduction band. In the limit of c → 0, the binding energy of bound state can be calculated from a pole of t-matrix T c (E) = V 0 /(1 − X c (E)) for the conduction-band electron as Thus, once E b is given, the attractive potential V 0 can be determined by Eq. (21). For the (20,0) SWCNT, E b is known to be E b = 0.068 eV, 38) and eventually the attractive potential is V 0 = −0.91 eV. In other words, the single impurity level is located at E = 0.18 eV (x = 0.060). Note that, in CPA methods, including the present selfconsistent t-matrix approximation, the spectral conductivity α(E) becomes finite once DOS becomes finite (see Sec. 2.6), since CPA ignores the effects of Anderson localization due to the interference effects of scattered waves, which can lead to finite DOS even in the energy region where the conductivity is zero. It is known that every state is localized in one and two dimensions in the presence of finite scattering. 39) However, once the system size or temperature becomes finite, the effects of Anderson localization are greatly reduced. This situation is assumed in the present study and hence the band edges in the CPA are used to represent the effective mobility edges.

Density of States ρ
Once the self energies Σ c (E) and Σ v (E) are obtained via the above procedure, the density of states (DOS) can be determined as follows. Within the present approximations, the DOS per the unit cell for each spin and each orbital consists of two parts, as where ρ c (E) is the DOS including the contribution from the conduction and impurity bands, and ρ v (E) is the DOS of the valence band, which are given by where Im ± E − Σ c/v (E) − ∆ > 0. The signs + and -correspond to ρ c and ρ v , respectively. Equation (24) indicates that for the region of x with complex solutions of σ c/v (x), the DOS will be finite, i.e., in the shaded region in Figs. 2(a) and 2(b). One caution is that the DOS can be finite even for real   a sharp peak near E = ±∆, which implies that the electrons in the conduction and valence electronic states are not significantly disordered by N impurities. We can see that as c increases, the DOS near E = ±∆ decreases strongly in comparison with that in the conduction and the valence bands.

Chemical Potential µ
At T = 0, the chemical potential (Fermi energy) lies in the impurity band. We now explain how to determine the Tdependence of the chemical potential µ(T ). Once the DOS in Eq. (24) is obtained, the T -dependence of the chemical potential µ(T ) can be determined with respect to the total electron density: where the left and right hand sides indicate the total amount of carriers per unit cell of the system at finite T and zero T , respectively. The factor 4 originates from the spin and orbital degeneracy of z-SWCNTs and E v = −|E v | is the valence-band top, which can be determined by the condition dx/dσ v = 0. Figure 4 presents the T -dependence of µ(T ) for Nsubstituted (20,0) SWCNTs with c = 10 −3 (blue solid curve), 10 −4 (red solid curve) and 10 −5 (black solid curve). The dashed curves are µ(T ) where the valence band is not taken into account in the calculation, which was previously discussed in Ref. 17. We now focus on the case of c = 10 −5 (black solid curve) as an example. The black solid curve shows characteristic changes around T ∼ 80 K and T ∼ 250 K, as indicated by the arrows. In the ionization region of T 80 K (see Appendix B), µ(T ) lies in the impurity band and decreases slowly with an increase in T . As T increases further, the system shows a crossover from the ionization region to the exhaustion region (see Appendix B). In this crossover region of 80 K T 250 K, µ(T ) decreases rapidly. Over T ∼ 250K, the black solid curve deviates upward from the black dashed curve and approaches the center of the band gap (E = 0) in the high-T limit because the valence band electrons begin to be thermally excited from the valence band to the conduction band. This temperature region, T 250K, is the so-called in- E=0.18eV trinsic region (see Appendix B). Similar features are evident in the red and blue solid curves. It should be noted that the two characteristic temperatures indicated by the arrows in Fig. 4 shift toward higher T as c increases.

Spectral Conductivity α(E)
Similar to the expression of the DOS in Eq. (22), the spectral conductivity α(E) can also be divided into two parts, as within the present approximation. Here, α c (E) is the spectral conductivity of conduction and impurity band electrons, and α v (E) is the spectral conductivity of valence band electrons. α c (E) and α v (E) are given by Refs. 16, 17, 40 where the factor 4 comes from the spin degeneracy and the orbital degeneracy of z-SWCNTs, 31,32) v k is the group velocity of an electron with wavenumber k, and V is the volume of the system. Here, G c/v (k, E) is the retarded Green's function Furthermore, within the effective-mass approximation for z-SWCNTs in Eq. (11), the k-summation in Eq. (27) can be performed analytically and α c/v (x) are given by where the signs + and − correspond to α c and α v , respectively, and A is the cross-sectional area of an SWCNT (A ≡ πd t δ w is conventionally used as the effective cross-sectional area of an SWCNT, where d t = 1.57 nm is the diameter of a (20,0) SWCNT and δ w = 0.34 nm is the van der Waals diameter of carbon). Figure 5(a) shows α(E) for N-substituted (20,0) SWCNTs with various concentrations of N impurities (c = 10 −3 (blue solid curve), 10 −4 (red solid curve) and 10 −5 (black solid curve)). α(E) has finite value once DOS is finite. With a decrease in c, α(E) in the energy regions of E ≥ E c and E ≤ E v is proportional to 1/c, as shown in Fig. 5(b). This can be understood by the BTT expression α c/v (E) ∝ τ c/v with the relaxation time τ c/v . Since τ c/v is proportional to 1/c within the t-matrix approximation, we obtain α c/v (E) ∝ 1/c (see Appendix C). In contrast to the conduction/valence-band energy region, α c (E) for the impurity-band energy region, which cannot be described by the BTT, is proportional to c, as shown in Fig. 5(c). This is because that the averaged distance of N impurities becomes short in proportion to c. 11 We now discuss the T -dependence of L 11 for N-substituted (20,0) SWCNTs, which can be calculated by the substitution of Eq. (29) into Eq. (4). Figure 6 shows the T -dependence of L 11 for c = 10 −3 (blue solid curve), 10 −4 (red solid curve) and 10 −5 (black solid curve). The dashed curves are L 11 where the valence band is not taken into account in the calculation. 17) Here, we focus on L 11 for c = 10 −5 (black solid curve) as an example. In Fig. 6, the black solid curve exhibits two rapid increases at T ∼ 40 K (see also the inset of Fig. 6) and T ∼ 250 K. The increase at T ∼ 40 K originates from the change in the transport regime of this system from impurity band con- duction to conduction-band conduction. On the other hand, at T ∼ 250 K where electrons begin to be excited from the valence band to the conduction band, the black solid curve begins to deviate upward from the dashed curve. This is because the valence-band holes contribute to L 11 in addition to the conduction band electrons at T 250 K. In the intermediate temperature region of 170 K T 250 K, which corresponds to the exhaustion region, the conduction band electron density is almost constant with T , as shown in Appendix B, and therefore the T -dependence of L 11 is weak. The T -dependence of L 11 in the exhaustion region is discussed in Appendix D.

Electrical Conductivity L
In the last part of this section, we consider the other solid curves (the red and blue solid curves in Fig. 6) to clarify the c-dependence of L 11 . In the extremely low-T region shown in the insets of Fig. 6, L 11 increases with c, in contrast to the high-T L 11 . This is due to the opposite tendency of the c-dependency of α c (E) for the conduction-band and impurityband energy regions [see Figs. 5(b) and 5(c)].

Thermoelectrical Conductivity L 12
In this section, we discuss the T -dependence of L 12 for Nsubstituted (20,0) SWCNTs, which can be calculated by the substitution of Eq. (29) into Eq. (5). Figure 7 shows the Tdependence of the L 12 value of N-substituted (20,0) SWC-NTs with c=10 −3 (blue solid curve), 10 −4 (red solid curve) and 10 −5 (black solid curve). The dashed curves are L 12 where the valence band is not taken into account in the calculation. Here, we focus on the case of c = 10 −5 (black solid curve) as an example. In Fig. 7, the black solid curve shows a rapid increase at T ∼ 30 K (see also the inset of Fig. 7) and deviates downward from the black dashed curve at T ∼ 250 K. The rapid increase at T ∼ 30 K is due to the contribution to L 12 from the conduction band electrons becoming more dominant than that from the impurity band electrons. It should be noted that the crossover temperature (T ∼ 30 K) of L 12 is lower than that of L 11 (T ∼ 40 K) shown by the black solid curve in the inset of Fig. 6. This difference implies that L 12 is more sensitive to the thermal excitation of carriers than L 11 , and the difference determines the low-T behavior of the Seebeck coefficient, as explained in Sec. 3.3. On the other hand, the deviation of the black solid curve from the dashed curve at T ∼ 250 K is due to cancellation between the contributions from conduction band electrons and valence band holes to L 12 . In addition, we discuss the T -dependence of L 12 in the intermediate T region of 170K T 250 K in Appendix D.
Before closing this section, we consider the c-dependence of L 12 . In the extremely low-T region where the impurityband conduction dominates as seen in the inset of Fig. 7, |L 12 | increases with c, in contrast to the high-T |L 12 |, which originates from the opposite tendency of the c-dependence of α c (E) for the conduction-band and impurity-band energy regions [see Figs. 5(b) and 5(c)].

Seebeck Coefficient S
The Seebeck coefficient S can be calculated using the relation of S = 1 T L 12 L 11 in Eq. (2). Figure 8 shows the T -dependence of the S value of N-substituted (20,0) SWCNTs for c = 10 −3 (blue solid curve), 10 −4 (red solid curve) and 10 −5 (black solid curve). The dashed curves are S where the valence band is not taken into account in the calculation. 17) Here, we explain S with a focus on the case of c = 10 −5 (black solid curve) as an example. In the low-T region, |S | increases sharply near 30 K due to the rapid increase in |L 12 | near 30 K, as shown in the inset of Fig. 7. At extremely low T , much lower than 30 K, |S | is proportional to T in accordance with the Mott formula, 41) despite the impurity band conduction that cannot be described by the BTT. 17) As T increases, |S | deviates rapidly upward from the Mott formula and has a large peak at T ∼50 K and then decreases with further increase in T . The large peak originates from the thermal excitation from the impurity band to the conduction band with a small m * , which is a different mechanism from a large S of 1D semiconductors with pudding-mold-type band, i.e, a large m * . 42,43) The T dependence of |S | can be explained in terms of the T dependence of L 12 /L 11 . As shown in the inset of Fig. 6, L 11 begins to increase sharply at T ∼40 K. The |S | peak at T ∼50 K indicates that L 12 /L 11 is proportional to T , i.e., dS /dT = 0. Beyond T ∼50 K, the T -dependence of L 12 /L 11 becomes weaker than T linear and |S | decreases with T . In the region of 170K T 250 K, which corresponds to the exhaustion region, |S | is insensitive to T because the conduction band electron density is almost constant. Over T ∼250 K entering the intrinsic region, |S | decreases rapidly and approaches zero at the high-T limit where the chemical potential µ is located at the center of the band gap, as represented in Fig. 4. This is because S due to conduction band electrons is perfectly cancelled out by valence-band holes in the limit T → ∞. Similar features are evident in the red and blue solid curves.

Power Factor PF
The power factor (PF) can be calculated using the relation of PF = L 11 S 2 in Eq. (3). Figure 9 shows the T -dependence of the PF for N-substituted (20,0) SWCNTs with c = 10 −3 (blue solid curve), 10 −4 (red solid curve) and 10 −5 (black solid curve). Here, we explain the PF with a focus on the case of c = 10 −5 (black solid curve) as an example. Figure 9 shows that the PF increases rapidly from T ∼30 K, at which S rises sharply, as shown in Fig. 8. In the exhaustion region of 170K T 250 K, the PF shows weak dependence with respect to T because the T -dependency of L 11 and S are weak in this region. When T exceeds approximately 250 K, entering the intrinsic region, the PF drops rapidly and goes to zero due to the sharp decrease in S . Similar T -dependence of the PF can be observed for c = 10 −4 (red solid curve) and 10 −3 (blue solid curve), as represented in Fig. 9. In addition, the characteristic temperatures at which the solid curves deviate from the dashed curves shift toward high T as c increases. Due to this shift, the optimal concentration c opt that gives the maximum PF is dependent on T .
To show c opt at a fixed temperature, we present the cdependence of the PF within 10 −6 ≤ c ≤ 10 −2 at various temperatures in Fig. 10(a). At 200 K, the PF increases monotonically with a decrease in c within the present range of c. This is because the thermal excitation from the valence band to the conduction one is negligible and the monotonic increase of the PF is given by PF ∝ (ln c) 2 (c 1) as discussed for N-substituted (10,0) SWCNTs in our previous report. 17) In contrast, at T =250 K, 300 K, 350 K and 400 K, the PFs exhibit the maximum values at c opt = 4.7 × 10 −6 , 3.1 × 10 −5 , 1.2 × 10 −4 and 3.4 × 10 −4 , respectively (see Table I). Thus, we can see that c opt increases with increasing T . In order to clarify what determines the value of c opt , we show PF as a function of c/n hole in Fig. 10(b). Here, n hole is the number of valence-band holes, which is defined by where E v (< 0) is the valence-band top and the factor 4 comes from the spin degeneracy and the orbital degeneracy. As shown in Fig. 10(b), each PF curve exhibits a peak at c opt /n hole ∼ 20, which means that PF becomes the maximum when the N concentration reaches about 20 times the number of thermally excited holes. Note that this condition (c opt /n hole ∼ 20) is not satisfied for N-substituted (10,0) SWC-NTs with c = 10 −2 ∼ 10 −5 at T ≤ 400 K. As a result, PF does not show the maximum for the condition of c = 10 −2 ∼ 10 −5 and T ≤ 400 K as shown in Appendix A.

Electronic Thermal
Conductivity λ e Thermal conductivity is known to be due to electrons and phonons: the former, electronic thermal conductivity, λ e , is  defined as follows, where L (e) 22 is given by (32) in the present case of N-substituted SWCNTs (See Appendix E). Figure 11 shows the T -dependence of L (e) 22 for N-substituted (20,0) SWCNTs with c = 10 −3 (blue solid curve), 10 −4 (red solid curve) and 10 −5 (black solid curve). As seen in Fig. 11, L (e) 22 increases monotonically with increasing T for all c. At a fixed T , L (e) 22 increases as c decreases except for extremely low T at which the impurity-band conduction is dominant. This is due to the fact that α(E) in the conduction band contributing at finite T because of thermal excitations is proportional to 1/c as shown in Fig. 5 (b). On the other hand, L (e) 22 /c is proportional to T 2 and is independent of c in the limit of low T as seen by the Sommerfeld expansion as with α(E F ) ∝ c at the Fermi energy E F lying in the impurity band (See Fig. 5(c)). Figure 12 displays the T -dependence of λ e for c = 10 −3 (blue solid curve), 10 −4 (red solid curve) and 10 −5 (black solid curve). Similar to L (e) 22 , λ e increases as T increases and as c decreases except for extremely low T , while at an extremely low T , λ e /c is proportional to T and is independent of c as shown in the inset of Fig. 12. This can be understood by the

Sommerfeld expansion as
and α(E F ) ∝ c as shown in Fig. 5(c). As seen from Fig. 9, the contribution of second term L 12 L 21 /(T L 11 ) = PF × T in Eq. (31) is negligible in comparison to the first term L (e) 22 /T except for the exhaustion region. Figure 13 illustrates the low-T behavior of electronic contribution to the Lorenz ratio L e (T ) ≡ λ e (T )/(T L 11 (T )) scaled by the universal Lorenz number L 0 ≡ π 2 k 2 B /(3e 2 ) for c = 10 −3 (blue solid curve), 10 −4 (red solid curve) and 10 −5 (black solid curve). All curves in Fig. 13 approach unity in the low-T limit. This means that the Wiedemann-Franz law holds even for the impurity-band conduction. As T increases, L e (T ) deviates downward from L 0 in proportion to T 2 as From Fig. 12, we note λ e is much smaller than the phonon thermal conductivity, λ ph . It is known that room temperature λ ph of SWCNTs without N impurities is of the order of 1,000 W/Km, [44][45][46][47][48][49][50][51] which is comparable to that of Nsubstituted SWCNT with dilute N concentration. 52,53) Hence in the figure of merit ZT = (PF/λ)T , PF is determined by electrons while λ by phonons, ZT ≈ (PF/λ ph )T , and then the analysis of optimal condition for PF applies also for ZT resulting in ZT ∼ 0.1 for N-substituted (20,0) SWCNTs with c opt = 3.1 × 10 −5 at 300K.

Summary
The thermoelectric effects of N-substituted SWCNTs were investigated using the Kubo-Lüttinger theory combined with the Green's function technique. We have clarified the temperature dependence of the electrical conductivity L 11 and thermoelectrical conductivity L 12 , as well as the Seebeck coefficient S and power factor PF for a wide temperature range from the ionization region to the intrinsic region through the exhaustion region. S and PF decrease rapidly toward zero around a crossover temperature from the exhaustion region  to the intrinsic region, and the crossover temperature shifts toward higher temperature with an increase in the impurity concentration. Due to this doping dependence of the shift of the crossover temperature, the optimal impurity concentration c opt that gives the maximum PF changes depending on temperature. As shown in Table I, we have determined c opt for various temperature for N-substituted (20,0) SWCNTs. In addition, using the Sommerfeld-Bethe expression of L (e) 22 , we elucidate the temperature dependence of λ e ≡ (L (e) 22 − L 12 L 21 /L 11 )/T and show that the Wiedemann-Franz law for λ e /L 11 is valid in the limit of low T even for the impurity-band conduction. The optimal condition for PF applies also for the figure of merit ZT because the electronic thermal conductivity λ e is much smaller than the phonon thermal conductivity λ ph . We estimate ZT ∼ 0.1 for N-substituted (20,0) SWCNTs with c opt = 3.1×10 −5 and λ ph = 1, 000W/Km at room temperature.
Finally, we note that the results obtained in the present study can also be applied to boron-substituted SWCNTs by replacement of the impurity potential from an attractive potential to a repulsive potential.
In this Appendix, we discuss the contribution of thermal excitation from the valence band to the conduction band to the PF of an N-substituted (10,0) SWCNT. Figure A·1 shows the c-dependence of PF of the N-substituted (10,0) SWCNT at T = 200 K, 300 K and 400 K. The solid curves indicate PFs for systems including N-impurity bands with both conduction and valence bands self-consistently as discussed in Sec. 2.3 and the dashed curves are PFs shown in Fig.8(c) of the previous paper 17) where the valence band is not incorporated into electronic states of N-substituted SWCNTs. As seen in Fig. A·1, the solid curves fit the dashed curves c 10 −5 even at T = 400 K. This means that the valence band does not contribute to the PF of N-substituted (10,0) SWCNT. In addition, all the solid curves increase with decreasing c within c ≥ 10 −6 and do not exhibit the maximum, which is different from the N-substituted (20,0) SWCNT discussed in this paper. Figure B·1 shows the T -dependence of the conduction band electron number n c per unit cell for each spin and each orbital of the N-substituted (20,0) SWCNT with c = 10 −5 . Here, n c is determined by

Appendix B: Temperature Dependence of Electron Number in Conduction Band
where E c is the conduction-band bottom, which can be determined by the condition dx/dσ c = 0. The T -dependence of n c has three regions: the ionization region at low T , the exhaustion region at middle T and the intrinsic region at high T . In the ionization region, most N atoms (donors) still capture valence electrons, i.e., they are not thermally ionized, and the T -dependence of n c is given by n c ∼ exp(−E b /k B T ) (see Fig. B·1). In the exhaustion region, most N atoms are thermally ionized, and the valence band electrons are still frozen out. In this case, n c is almost equal to the density of N atoms and is independent of T (see Fig. B·1). In the intrinsic region, the valence band electrons are thermally excited from the valence bands to the conduction bands, and the T -dependence of n c is given by n c ∼ exp(−∆/k B T ) (see Fig. B·1). and with µ = k B T ln( c 2 πt k B T ). We have confirmed that these analytical results are in good agreement with the numerical results for L 11 and L 12 based on the Kubo-Lüttinger theory shown in Figs. 6 and 7.

Appendix E: Electronic Thermal Conductivity
Under the electric field E and the temperature gradient dT/dz, the thermal current density J Q is expressed as within the linear response with E and dT/dz. Here, L 21 is called electrothermal conductivity and is connected to L 12 by Onsager's reciprocal relation L 21 = L 12 . 54 1)). According to Kubo's linear response theory, L (e) 22 can be obtained as where χ 22 (iω λ ) is the J Q -J Q correlation function, expressed as where β ≡ 1/(k B T ) is the inverse temperature, T τ is the imaginary-time-ordering operator, · · · denotes the thermal average in equilibrium, and V is the volume of a system with J Q being the thermal current density. In the present case, where electrons are scattered by elastic impurities, J Q is given by where v (±) k = ± k/m * , u(q) = V 0 j e −iq j /N, Φ k (τ) = e τH c † k e −τH , e τH d † k e −τH (E·7) and Φ k (τ) = e τH c k e −τH e τH d k e −τH .
Substituting Eq. (E·6) into Eq. (E·5) and performing a similar procedure as Ref. 40 by Jonson and Mahan, we can straightforwardly obtain the SB type expression of L (e) 22 in Eq. (32).