Large Thermospin Effects in Carbon Nanotubes with Vacancy Defects

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C: Energy Conversion and Storage; Energy and Charge Transport

Large Thermo-Spin Effects in Carbon Nanotubes with Vacancy Defects Majid Shirdel-Havar, and Rouhollah Farghadan J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b04369 • Publication Date (Web): 05 Aug 2019 Downloaded from pubs.acs.org on August 5, 2019

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The Journal of Physical Chemistry

Large Thermo-Spin Eects in Carbon Nanotubes with Vacancy Defects Majid Shirdel-Havar and Rouhollah Farghadan

∗

Department of Physics, University of Kashan, Kashan, 87317-53153, Iran E-mail: [email protected]

Abstract We systematically investigated the possibility of improving thermo-spin eects in single-walled carbon nanotubes (SWCNTs) by vacancy engineering. The proposed method possesses several advantages: (I) both the armchair and zigzag SWCNTs with vacancy defects exhibit a spin-semiconducting phase with a localized band structure near the Fermi level, (II) applying a thermal gradient can generate spin-up and spindown currents with the same magnitudes but opposite directions, demonstrating the existence of a pure spin current without any applied bias, (III) a giant spin Seebeck coecient SS , comparable with those of other carbon-based structures, can be achieved, (IV) the phononic thermal conductance in defective SWCNTs is reduced by about 50%, thus strongly enhancing the thermo-spin conversion eciency ZS T , and (V) varying the nanotube diameter can tune the spin gap, such that the spin gap increases as the nanotube becomes narrower, indicating the possible application of SWCNTs in thermospintronic.

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Introduction For the past several decades, there has been signicant interest in exploring a strategy for recycling waste energy. Today, a major portion of the waste energy is dissipated in the form of heat. 1 The thermoelectric (TE) phenomenon, which is dened as the conversion of a temperature gradient into an electrical voltage, can be a useful strategy for recovering some portion of the waste heat and converting it into an electrical power. 2,3 Therefore, exploring materials that have a high TE energy conversion eciency is critical for the recovery of waste heat. The TE eciency of a material is characterized by the gure of merit (FOM),

ZT = S 2 GT /K, where S , G , T , and K are the Seebeck coecient, the electrical conductivity, the temperature, and the thermal conductance, respectively. It has been conrmed both experimentally and theoretically that the TE eciency is signicantly increased by reducing the system dimensions, so that ZT reaches the maximum value in one-dimensional systems mainly due to the change in the density of electronic states. 4,5 Due to its excellent electronic properties, particularly a very high electrical conductivity resulting from its fantastic charge carrier mobility, 6,7 single-walled carbon nanotube (SWCNT), as a one-dimensional material, is a promising candidate for TE applications. 1,819 Moreover, due to its advantages such as its low-cost fabrication, exibility, lightness, nontoxicity, and chemical stability, SWCNT is superior to other materials. It has been shown that semiconducting SWCNTs (s-SWCNTs) have a better eciency than metallic SWCNTs (m-SWCNTs) in converting the thermal energy into an electrical power. 14,17,20 Two recent studies have demonstrated that the band gap widening, due to the nanotube diameter decreasing, increases the Seebeck coecient as well as the TE eciency. 14,17 Despite the extensive studies in this area and the large spin correlation length in carbon nanotube (CNTs), which is about 1 µm, 2123 spin-dependent TE properties of CNTs have been studied less. In order to investigate thermo-spin properties, the existence of a net magnetic moment in the device is essential to create the spin polarization between majority and minority 2


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states. In this regard, few researchers have investigated the possibility of a thermally-induced spin current using partially hydrogenated SWCNTs 24 and a hybrid structure of SWCNT and graphene. 25 Interestingly, it has been shown in the literature that the adsorption of atoms on the nanotube wall and the presence of vacancies in CNTs can induce magnetic properties. 2632 Moreover, the presence of vacancies decreases the thermal conductance, due to the suppression of phononic modes, and leads to an improved TE eciency. 33 On the other hand, experimental studies have conrmed the existence of defects such as vacancies in graphene sheets and CNTs. 34,35 In this study, we systematically investigated the thermo-spin properties of both the armchair and zigzag SWCNTs with vacancy defects. Our results indicated the vacancy-induced magnetism and the creation of a spin semiconducting behavior with localized spin-up and spin-down subbands around the Fermi level separated with a spin gap. These behaviors generate a spin-polarized current when applying a temperature gradient between the left and right electrodes. Furthermore, vacancy defects in SWCNT strongly enhance the thermo-spin conversion eciency due to the suppression of phonons. We also investigated the thermospin properties of SWCNT when varying the nanotube diameter. The results indicated the strong dependence of the spin gap on the nanotube diameter, and thus the spin gap can be tuned by varying the diameter. Furthermore, the maximum spin gap is about 0.28 eV and the corresponding maximum values of the spin Seebeck coecient (SS ) and the thermo-spin eciency (ZS T ) are about 1.6 mV/K and 5, respectively, which originate from the large spin gap and the localized band structure. Interestingly, our results based on SWCNTs are comparable to those based on zigzag-edged graphene nanoribbons with the periodic edge reconstruction. 36 Finally, the large thermo-spin eects can pave the way toward improvements in the design and performance of thermo-spintronic devices based on SWCNTs.

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model and theoretical method Atomic conguration The structure of a particular SWCNT is denoted by two integers (n, m), through which the chiral and translational vectors are dened as Ch = na1 + ma2 and T = [(2m + n)/dR ]a1 −

[(2n + m)/dR ]a2 , respectively, as shown in Figure 1(a1), where a1 and a2 are the lattice vectors of the graphene sheet, as the mother structure, and dR is the greatest common divisor of (2m + n) and (2n + m). 37 Ch and T are perpendicular to each other and dene the circumferential direction and the axis of the nanotube, respectively. Achiral SWCNTs are classied in two groups of (n, 0) and (n, n), representing the zigzag and the armchair SWCNTs, respectively. In this study, we considered (n, n) and (n, 0) SWCNTs with a periodic array of vacancies located along the nanotube axis, as shown in Figures 1(b1) and 1(b2), which represent (6,6) and (8,0) nanotubes with two vacancies in each unit cell; we name them (6,6)-2V and (8,0)2V, respectively. Here, each unit cell consists of ve armchair carbon rings with the length of 12.3 ˚ A for (n, n) nanotube and six zigzag carbon rings with the length of 12.78 ˚ A for (n, 0) nanotube. Two vacancies are created by removing two atoms facing each other from the sublattice of A in each unit cell, as illustrated in Figures 1(c1) and 1(c2) for (6,6) and (8,0) nanotubes, which leads to a nonzero dierence between the number of atoms in the sites of

A and B , i.e. between NA and NB . Now, by considering the electron-electron interactions in the form of a Hubbard repulsion, the total spin, S = |NA − NB |/2, is 1 per unit cell in the ground state, in accordance with the Lieb's theorem. 38 Moreover, Lieb's theorem relies on the Hubbard model and can be used for curvature SWCNTs similar to zigzag-edge graphene nanoribbons. 39 It is worth mentioning that a vacancy in SWCNT may be metastable because there are three dangling bonds (DBs) around it that can be reconstructed through the formation of chemical bond between two of them and the formation of a pentagon ring with an extra DB, 4


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Figure 1: (a1) The conguration of carbon atoms in a graphene sheet, which forms SWCNT when it is rolled into a cylinder. Ch and T are the chiral and translational vectors, and a1 and a2 are the lattice vectors. (a2) The rotations required to move an atom from the graphene sheet to the nanotube surface. Furthermore, (b1) and (b2) are (8,0) and (6,6) SWCNTs with two vacancies facing each other, where (c1) and (c2) illustrate their corresponding unit cells, respectively.

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which is called the 5-1DB defect. 40,41 Depending on the orientation of the new C-C bond with respect to the tube axis, there are two kinds of 5-1DB defects: 5-1DB-P for the perpendicular orientation and 5-1DB-T for the tilt orientation. 41 Note that theoretical studies have shown that the magnetism can still be preserved in SWCNTs with the formation of a 5-1DB defect, especially in 5-1DB-P types, despite it falls in some congurations. 26,27 However, according to our results for SWCNTs with a single vacancy show that even by reducing the magnetization in SWCNTs, the calculated thermo-spin coecients are signicant. Moreover, the three DBs can be saturated by hydrogen passivation that leads to the vacancy stability. 26 Interestingly, the vacancy-induced magnetism is also preserved in this conguration. 26,27,42 Theoretical studies have shown that vacancy defects with zigzag edges primarily tend to have a ferromagnetic phase than other magnetic phases, and they can preserve the magnetic ordering, even above room temperature. 43,44 In detail, the three DBs around that vacancy are terminated by the formation of strong C-H σ -bonds that lie far from the Fermi level and thus ignoring the hydrogen saturation on the vacancy edge does not sensitively change the electronic transport calculations. 45,46 Furthermore, the hydrogen saturation reduces the phononic thermal conductance 47,48 and thus considering the hydrogen saturation could enhance the thermo-spin eciency. To investigate thermo-spin properties, we rst obtained electronic and phononic transmissions by applying the non-equilibrium Green's function (NEGF) method and the LandauerBütticker transport formalism. Then, we used the electronic and phononic transmission functions to describe the thermo-spin coecients.

Electronic formalism To obtain the electronic structure, we have to solve the self-consistent problem for an innite system with the translational symmetry. We divided the system into three regions: the left electrode, the right electrode, and the central region, as depicted in Figure 1. The total

6


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Hamiltonian is composed of some sub-matrices, as follows: (1)

H = HL + HR + HC + VCL + VCR ,

where HL , HR , and HC are the Hamiltonian matrices of the semi-innite left electrode, the semi-innite right electrode, and the central region, respectively. Furthermore, VCL and VCR dene the hopping between the central region and the left and right electrodes, respectively, which is obtained using the tight-binding (TB) method for π -orbitals with the rst-nearest-neighbor hopping of t = 2.7 eV. 49 To calculate the mentioned Hamiltonian matrices, regarding to the size of SWCNTs, 50,51 we applied the single band TB method and the mean-eld Hubbard model with the on-site Coulomb repulsion of U 52 (U = 3 eV), which is given as follows: 53,54

Hσ = −t

X i,j,σ

dˆ†iσ dˆjσ + U

X i,σ

1 hˆ ni,−σ i n ˆ i,σ − hˆ ni,σ i , 2

(2)

where, dˆ†i,σ and dˆj,σ creates and annihilates an electron in the lattice sites i and j with the spin σ (σ =↑, ↓), respectively. Furthermore, hˆ ni,σ i is the expectation value of the occupation operator of electrons on the site i with the spin σ , which can be solved as a self-consistent problem using the numerical iteration method 55,56 and provides the net magnetic moment

mi = µB [hni,↑ i − hni,↓ i] on the site i. Then, the electronic Green's function of the system is determined as

h X GC,σ = (ε + i0+ )I − HC,σ − here,

P

L(R),σ (ε)

L,σ

−

i−1

X R,σ

,

(3)

† = VCL(CR) gL(R),σ (ε)VCL(CR) is the left (right) self-energy, where gL(R),σ is

the surface Green's function of the left (right) electrode, which is calculated by an iterative method. 57 Due to the small strength of the electron-electron interaction (about 10−3 eV2 ) in the momentum conserving regime, 58 we ignored the self-energy of the electron-electron interaction in our transport calculations. Moreover, since the spin diusion length in CNTs

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is of the order of µm, 2123 which is much larger than the length of the considered system, we neglected the spin-ip scattering in this study. Therefore, the spin-polarized transmission function is determined as 57

h i Tσ (ε) = Tr ΓL,σ (ε)GC,σ (ε)ΓR,σ (ε)G†C,σ (ε) , where ΓL(R),σ (ε) = −2Im

(4)

hP

i (ε) is the broadening function, which describes the L(R),σ

coupling between the central region and the left (right) electrode.

Phononic formalism and thermal conductance In this section, we briey describe the method employed in this work to calculate the phononic thermal conductance. Since the large phonon mean free path in CNTs, which is of the order of µm at room temperature 11,59 and reaches 1.5 µm at low temperatures, 60 is very larger than the transport length of the considered system, we considered the phonon transport in a ballistic manner. To calculate the phononic transmission, we dene an equation similar to Eq. (1), where only the Hamiltonian is replaced by a dynamical matrix D. The dynamical matrix consists of small 3 × 3 matrices as {D(i,j) (k)}, i, j = (1, 2, ..., N ), which are dened by 37

! D(ij) (k) =

X

K (ij

00 )

−Mi ω 2 (k)I δij

j 00

−

X

0

K (ij ) eik.∆Rij0 ,

(5)

j0

where k , ω and Mi are the wave vector, the phonon frequency and the atomic mass of the

ith atom, respectively, and ∆Rij = Ri − Rj is the relative coordinate of the ith and jth atoms. Furthermore, K (ij) exhibits the 3 × 3 force-constant tensor (FCT) between the ith and jth carbon atoms. When the direction from a given atom to its nth-nearest neighbor

8


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coincides with the x-axis, the FCT has the diagonal form 37





(n) φr

0

  K=  0  0 (n)

(n)

(n)

φti 0

0   0    (n) φto

(6)

(n)

where φr , φti , and φto represent the force-constant parameters in the radial, in-plane, and out-of plane tangential directions of the nth-nearest neighbors. In this paper, the FCT was constructed in the fourth-nearest-neighbor force-constant (4NNFC) method 37 using the force-constants reported in Ref. 61 Since the atoms in the nanotube are on the cylinder wall, the FCT is determined by applying the unitary rotation matrix on Eq. (6) around the coordinate axes. Here, by applying three rotations, each atom in the graphene sheet can be moved to the nanotube surface. For this purpose, we chose a coordinate system where the

x and z -axes are parallel and perpendicular to the chiral vector, respectively, as shown in Figure 1(a1). We rst have to rotate the Bj atom into a virtual one placed along the x-axis with respect to the Ai atom. This is done by rotating around the y -axis with the angle

α between Ch and Rij (see Figure 1(a1)). In addition, as presented in Figure 1(a2), we have to rotate the tensor around the z -axis with an angle Ψ = (Ri .Ch /|Ch |)2π/|Ch |, which describes the position of the Ai atom with respect to the start of Ch . Furthermore, because of the rotation of Rij when the tube is formed, ϕ/2 = (Rij .Ch /|Ch |)π/|Ch |. For each of the

N atoms in the unit cell, we identify the 18 neighbors within the fourth-nearest-neighbor distance and rotate the tensor of Eq. (6) according to the arguments mentioned above 62

K Ai Bj = Uy−1 (α)Uz−1 (Ψ + ϕ/2)KUz (Ψ + ϕ/2)Uy (α),

(7)

where Uy and Uz are the unitary rotation matrices around the y and z -axes, respectively. Now, by substituting the FCT in Eq. 5, the dynamical matrix is obtained and then, we can

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calculate the phononic Green's function of the central region as

h X GC,ph = (M ω 2 + i0+ )I − DC,ph − where DC,ph and

P

L(R),ph

L,ph

−

i−1

X R,ph

,

(8)

are the dynamical matrix of the central region and the phononic

self-energy of the left (right) electrode. Now, one can dene a phononic transmission function

Tph (ω) similar to Eq. (4) and then calculate the phononic thermal conductance according to the following equation 36

Kph

~2 = 2πkB T 2

Z 0

∞

ω 2 exp(~ω/kB T ) h i2 Tph (ω)dω, exp(~ω/kB T ) − 1

(9)

Thermo-spin formalism Applying a temperature gradient ∆T = TL −TR across the left and right electrodes leads to a dierence in the Fermi-Dirac distribution of the two electrodes, and consequently an electrical current ows from the left electrode (at the higher temperature) to the right electrode (at the lower temperature). In the Landauer-Bütticker formalism, the thermally-induced spinpolarized current through the system is determined as 63

e Iσ = h

Z

+∞

h i Tσ (ε) fL (ε, TL ) − fR (ε, TR ) dε,

(10)

−∞

where fL(R) is the Fermi-Dirac distribution of the left (right) electrode, and e and h are the electron charge and the Plank constant, respectively. The pure spin and charge currents are dened by IS = I↑ − I↓ and IC = I↑ + I↓ , respectively. Assuming ∆T = TL − TR , the spin Seebeck coecient is dened as the spin voltage dierence ∆VS required to neutralize thermally-induced spin and charge currents, is given as below in the linear response regime 64

∆VS SS = − = (S↑ − S↓ ), ∆T IC =0,IS =0

10


(11)

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where, Sσ(↑,↓) = −(L1σ /L0σ )/(|e|T ) is the spin-resolved Seebeck. Furthermore, the spinresolved electrical conductivity and the electronic thermal conductance are obtained according to Gσ = e2 L0σ and Ke,σ = (L2σ − L21σ /L0σ ) /T , respectively, where Lnσ (µ, T ) is dened as 63

1 Lnσ (µ, T ) = − h

Z

(ε − µ)n

∂f (ε, µ, T ) Tσ (ε)dε. ∂ε

(12)

Finally, the thermo-spin conversion eciency is characterized based on the spin FOM as 36

ZS T =

SS2 GS T , Ke + Kph

(13)

where GS = |G↑ − G↓ | and Ke = Ke,↑ + Ke,↓ are the spin-dependent electrical conductivity and electron contribution to the thermal conductance, respectively.

results and discussion Theoretical calculations have indicated that the electronic structure of the (n, n) nanotube has a metallic nature, while the (n, 0) nanotube has a semiconducting nature for n = 3m − 1 and n = 3m−2, with an almost equal gap, and a metallic nature for n = 3m. 65,66 On the other hand, experimental results have demonstrated the existence of an energy gap in the (3m, 0) nanotube that decreases as the nanotube diameter increases. 67 This gap is much smaller than the gap in (3m − 1, 0) and (3m − 2, 0) nanotubes. 51,67 However, the (3m, 0) nanotube has properties dierent from those of (3m − 1, 0) and (3m − 2, 0) nanotubes. 51 Therefore, achiral SWCNTs can be classied into three families: (n, n), (3m − 1, 0), (3m − 2, 0), and (3m, 0) nanotubes. In this study, to investigate thermo-spin properties of SWCNTs, we rst selected

(8, 0), (9, 0), and (6, 6) nanotubes, with diameters of 6.26, 7.05, and 8.14 ˚ A, respectively, in the presence of a vacancy. Then, we studied the eect of changing the nanotube diameter on the thermo-spin properties.

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F

1

1

(9,0)-2V

0

-1

-1

0

1

2

K

3

(6,6)-2V

0

-1

-2

-2

(eV)

(8,0)-2V

2

F

Spin-down

0

2

(eV)

Spin-up

E-E

F

(eV)

1

(c2)

(c1)

(b2)

(b1)

(a2)

E-E

(a1)

2

E-E

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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-2 0

4

1

2

3

K

4

0

1

2

3

4

K

Figure 2: The panels (a1), (b1), and (c1) are the spin-polarized band structures of (8,0)-2V, (9,0)-2V, and (6,6)-2V nanotubes, respectively, and the panels (a2), (b2), and (c2) are the corresponding spin-dependent transmission coecients.

Electronic structure The spin-polarized electronic band structures of the three systems considered in this study are presented in Figures 2(a1)-2(c1) corresponding to (8,0)-2V, (9,0)-2V, and (6,6)-2V nanotubes, respectively. The vacancy-induced magnetism with the spin-semiconducting phase for both armchair and zigzag SWCNTs in the presence of the e − e interaction is quite visible, which leads to the creation of two localized spin-up energy bands below the Fermi level (the red solid line) and two localized spin-down energy bands above the Fermi level (the blue dashed line), separated by the spin gap. The spin-semiconducting behavior and the localized bands in SWCNTs, which have a signicant role in improving the thermo-spin conversion efciency, have been observed in sawtooth-edge and armchair-edge graphene nanoribbons. 36,68 Interestingly, the dierent electronic behaviors of (8, 0), (9, 0), and (6, 6) nanotubes lead to a spin-semiconducting behavior with various spin gaps in the presence of a vacancy, which can be due to the zigzag-edges created in the vacancy and the presence of the e − e interaction. 69 The spin gap values for (8,0)-2V, (9,0)-2V, and (6,6)-2V nanotubes are 0.255, 0.153, and 0.142 eV, respectively, indicating that the decrease in the spin gap is proportional to the increase in the nanotube diameter. Furthermore, the spin-polarized transmission function was calculated using Eq. 4 and the results are presented in Figures 2(a2)-2(c2). The sharp 12


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peaks in the transmission spectrum around the Fermi level are due to the existence of the localized energy bands, which signicantly contribute to the large Seebeck coecient. 36

Phononic structure and thermal conductance To investigate the eect of the vacancy on the phononic structure, the phononic transmission functions Tph and density of states of the considered systems are presented in Figures 3(a)3(f). The results related to the perfect SWCNTs are also shown as red solid lines for comparison. As can be seen, in the presence of the vacancy, the phononic transmission is strongly suppressed over most of frequencies except very low ones. Phonon bands split for SWCNTs with periodic vacancies than for the perfect one, and the degree of degeneracy between phonon modes is largely reduced. 47 Moreover, phonon bands become at or less dispersive generally (not shown here). is largely reduced [46]. Furthermore, phonon bands become at or less disperse generally (not shown here). The tendency towards atness in phonon band structures reects also on the phonon density of states of (8,0)-2V, (9,0)-2V, and (6,6)-2V SWCNTs, in which more peaks appear, as illustrated with blue solid lines in Figures 3(d)3(f). Furthermore, due to the periodic-vacancy-induced redistribution of phonon structures in the Brillouin zones and the reduction of three-phonon modes per vacancy (two in-plane modes and one out-of-plane mode), there are fewer phonon modes for SWCNTs than for perfect SWCNTs, which means that Tph (perfect) > Tph (with vacancy) as shown in Figures 3(a)-3(c). In the proposed spin-caloritronic devices the phonon transport is ballistic and the periodic vacancies redistribute only the phonon branches and reduce the phonon modes rather than the modes in the perfect one, as evidenced by the integer phonon transmission function, shown in Figure 3. Furthermore, other scattering mechanisms, such as the electron-phonon scattering and the phonon-phonon scattering, which are important only when the transport length is comparable with, or larger than, the phonon mean free path, 70,71 are not relevant to the present discussion and were neglected in our calculations. 13



Phonon DOS

(8,0) SWCNT

(8,0)-2V SWCNT

Phonon DOS

(9,0) SWCNT

(9,0)-2V SWCNT

(6,6) SWCNT

Phonon DOS

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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(6,6)-2V

(6,6)-2V SWCNT

0

Frequency (cm

-1

Frequency (cm

)

-1

)

Figure 3: (a)-(c) The phononic transmission of (8,0), (9,0) and (6,6) nanotubes with (the blue-lled area) and without (the red solid lines) vacancy as a function of the phonon frequency, and (d)-(f) corresponding phonon density of states.

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The suppressed phononic transmission leads to a decrease in the thermal conductance, as shown in Figure 4. At the temperature T = 300 K, the thermal conductance of (8,0)-2V, (9,0)-2V, and (6,6)-2V nanotubes are about 1.28, 1.42, and 1.89 nW/K, respectively, much lower than those of the perfect SWCNTs, which are about 2.74, 3.1, and 3.65 nW/K, respectively. Generally the thermal conductance decreases about 50% in the presence of the vacancy. Furthermore, at low temperatures (T < 25 K), the dierences in Kph between the SWCNTs with or without the vacancy are nearly zero, regardless of the nanotube diameter. Therefore, at low temperatures, the major contribution in Kph is exclusively related to phonons in the low-frequency regions, where all of the SWCNTs have approximately the same Tph (see Figure 3). As the temperature increases, the transmission phonons in the highfrequency region contribute to the weight function of Eq. 9 (α2 eα /(eα − 1)2 , α = ~ω/kB T ), which varies with the nanotube diameter and thus leads to a dierent Kph , so that it gradually increases as the nanotube diameter increases both with and without the vacancy, which is quite clear in Figure 4. 5 (nW/K)

4

ph

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(8,0)

(8,0)-2V

(9,0)

(9,0)-2V

(6,6)

(6,6)-2V

3 2 1 0 0

100

200

T

300

400

(K)

Figure 4: Phononic thermal conductivities of (8,0), (9,0), and (6,6) nanotubes with (the dashed lines) and without (the solid lines) vacancy as a function of temperature.

Thermo-spin properties In this section, we present the thermo-spin properties of (8,0)-2V, (9,0)-2V, and (6,6)-2V nanotubes. For this purpose, we rst investigate the thermally-induced spin current and then the spin-dependent Seebeck coecient and FOM. 15



Applying a temperature gradient ∆T across the left and right electrodes leads to a nonzero dierence in their Fermi-Dirac distribution and thus a thermally-induced spin current is generated according to Eq. 10. Figures 5(a)-5(c) shows the thermally-induced spin-up and spin-down currents (I↑ and I↓ ) as a function of TL for dierent ∆T at µ = 0 for (8,0)-2V, (9,0)-2V, and (6,6)-2V nanotubes. The opposite signs of I↑ and I↓ indicate that the spin current in the spin-up and spin-down channels ow in opposite directions and demonstrate spin Seebeck eect (SSE) in SWCNTs in the presence of the vacancy. Furthermore, the threshold temperatures Tth for the generation of the spin-polarized current for (8,0)-2V, (9,0)-2V, and (6,6)-2V nanotubes are about 150, 110, and 100 K, respectively, indicating that

Tth decreases as the nanotube diameter increases. The dierence in Tth can be attributed to the dierence in the bandgap of the individual spin channels for dierent nanotubes (see Figure 2), so that increasing the bandgap increases Tth . As can be seen, there is a high degree of symmetry between I↑ and I↓ due to the existence of symmetry between spin-up and spin-down transmission, which causes that the pure spin current IS and nearly zero charge current IC . Figures 5(e)-5(f) shows IS and IC for (8,0)-2V, (9,0)-2V, and (6,6)-2V nanotubes as a function of TL at µ = 0 for dierent ∆T . The spin current increases as the

∆T or TL increases, while the charge current changes slightly in both cases. Now, we present the spin Seebeck coecients of the SWCNTs with a vacancy, which are illustrated in Figure 6. The spin-resolved Seebeck coecients (S↑ and S↓ ) and the spin Seebeck coecient (SS ) as a function of the chemical potential at T = 300 K for (8,0)-2V, (9,0)-2V, and (6,6)-2V nanotubes are depicted in Figures 6(a)-6(c). In the given energy range, both S↑ and S↓ have the maximum absolute values of about 1.07, 0.6, and 0.7 mV/K for (8,0)-2V, (9,0)-2V, and (6,6)-2V nanotubes, respectively. These values are comparable with those previously reported for armchair and zigzag-edge graphene nanoribbons with values about 0.5-1.4 mV/K 56,68,72 and also for sawtooth-edge graphene nanoribbons with values about 2 mV/K. 36 Consider Figure 6(a) at µ ' 0.008 eV, S↑ = −S↓ , which indicates the existence of a spin voltage and a zero charge voltage and also the perfect symmetry of the 16


Page 16 of 32

0 T=5 K T=10 K T=15 K T=20 K

-30

60

T=5 K T=10 K T=15 K T=20 K

5

T

Spin-Up

th

-24

45

(e)

(6,6)-2V

-12

(9,0)-2V Spin Current

(8,0)-2V

(nA)

0

-15

(d)

10

th

(f) (6,6)-2V

Spin Current

-8

T

12

(c)

Spin-Down

Spin-Up

15

th

-4

15

(9,0)-2V

Spin-Down

T

0

,

(nA)

4

(8,0)-2V

24

(b)

45

30

Spin Current

30

(a)

Spin-Up

8

S,C

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Spin-Down

Page 17 of 32

30

15

15 Charge Current

Charge Current

0

0 100

200

300

T

L

(K)

400

500

100

200

300

T

L

400

(K)

500

Charge Current

0 100

200

300

T

L

400

500

(K)

Figure 5: (a)-(c) Thermally-induced spin-polarized currents of (8,0), (9,0) and (6,6) nanotubes in the presence of vacancy as a function of TL at µ = 0 for dierent ∆T , and (d)-(f) the corresponding pure spin and charge currents. thermally-induced spin-up and spin-down currents. Therefore, a net spin current, without any charge current, is generate at this chemical potential for (8,0)-2V nanotube. This process is also seen for (9,0)-2V and (6,6)-2V nanotubes at µ = 0.004 and −0.001 eV, respectively. For completely symmetric transmission functions, this process can occur at µ = 0 eV. 64 Furthermore, there are chemical potentials where one of the spin-resolved Seebeck is zero and the other is non-zero, indicating the induction of the spin current in only one of the spin channels by the thermal gradient at these points. For example, in Figure 6(a), S↑ at

µ = −0.141 eV and S↓ at µ = 0.161 eV are zero. Furthermore, the spin Seebeck coecients SS are presented in Figure 6 as a wide colored line for the three considered systems. As can be seen, SS in the spin gap region is rather at due to the linear variation of S↑ and S↓ in this region for all classes of SWCNTs with vacancy defects. The maximum value of SS inside the spin gap for (8,0)-2V, (9,0)-2V, and (6,6)-2V nanotubes are 0.98, 0.67, and 0.62 mV/K, respectively, indicating the reduction of SS by increasing the nanotube diameter. However, the maximum absolute value of SS lies outside of the spin gap, with values of 1.5, 0.83, and 1.1 mV/K for (8,0)-2V, (9,0)-2V,

17



(mV/K)

1.0

(a) (8,0)-2V

0.5 0.0

1

-0.5

PS

-1.0 -1.5

S

-1

(b)

(mV/K)

0.5

(9,0)-2V

0.0

-0.5

(c)

0.5

(mV/K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 32

(6,6)-2V

0.0

-0.5

-1.0 -0.5

0.0

0.5

(eV)

Figure 6: (a)-(c) Spin Seebeck coecients of (8,0), (9,0), and (6,6) nanotubes in the presence of vacancy as a function of the chemical potential at T = 300 K. Open and solid symbol curves illustrate the spin-up and spin-down Seebeck (S↑ and S↓ ), respectively, and the wide colored curves illustrate the spin Seebeck (SS ), where the color of SS changes according to the Seebeck polarization.

18


Page 19 of 32

and (6,6)-2V nanotubes, respectively. These values are even larger than those of the hybrid structure of SWCNT and a zigzag-edge graphene nanoribbon, where the maximum value of

SS is about 1 mV/K at room temperature. 25 The color of SS changes according to the Seebeck polarization PS = (|S↑ | − |S↓ |)/(|S↑ | + |S↓ |). The color of SS in the close vicinity of the Fermi level demonstrates the equal contribution of spin-up holes and spin-down electrons in the thermo-spin properties, as seen in the spin-polarized currents in Figure 5. On the other hand, the color of SS on the left and right sides of the Fermi level demonstrates the dominant contribution of spin-down electrons and spin-up holes in the thermo-spin properties, respectively. Finally, we presented the thermo-spin eciencies of the three considered SWCNTs with vacancy defects in Figure 7(a) as a function of the chemical potential at T = 300 K. The maximum values of ZS T for (8,0)-2V, (9,0)-2V, and (6,6)-2V nanotubes are 4.07, 3.47 and 1.68, respectively. 4

S

T

3

2

(a) Periodic Vacancy (two vacancies)

(b) Periodic Vacancy (single vacancy)

(8,0)-2V

(8,0)-1V

(9,0)-2V

(9,0)-1V (6,6)-1V

(6,6)-2V

1

2

T

1 0

0

3

0.6

2

(c) Non-periodic Structures

(d) Perfect Electrodes

(8,0)-2V

(8,0)-2V

(9,0)-2V

(9,0)-2V (6,6)-2V

(6,6)-2V

0.3

S

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1

0

0.0 -0.5

0.0

0.5

-0.5

0.0

0.5

(eV)

(eV)

Figure 7: The spin FOM as a function of the chemical potential at T = 300 K for (8,0), (9,0), and (6,6) nanotubes (a) with two vacancies and (b) with a single vacancy in the unit cell, (c) non-periodic structures (defect separation in the central region is dierent from that of the electrodes), and (d) vacancy defects only in the central region. It is worth mentioning that the results presented in this paper can be reproduced even with a single vacancy defect in the unit cell. In the presence of a single vacancy, the spin19



semiconducting behavior is preserved but the value of the spin gap and the transmission spectra are varied. Furthermore, four localized energy bands around the Fermi level (see Figure 2) in the presence of two vacancies are reduced to two energy bands. Generally, also in the presence of a single vacancy, SWCNTs can generate spin-polarized currents when applying a temperature gradient, indicating the existence of the pure spin current and SSE. However, the values of SSE and the pure spin current are reduced as compare to the case with two vacancies. For more clarication, we calculated the spin FOM, and the results are presented in Figure 7(b). Comparing Figure 7(a) and 7(b) indicates that ZS T for nanotubes with a single vacancy are lower, by about 50%, than that for nanotubes with two vacancies. This is mainly attributed to the more suppression of phonon modes and thus the more reduction of the thermal conductance in nanotubes with two vacancies, than nanotubes with a single vacancy, in each unit cell. To elucidate the importance of periodic defects, we increased the distance between defects in the central region by 33% for (8,0)-2V and (9,0)-2V SWCNTs and 40% for (6,6)-2V SWCNT, without changing the distance in the electrodes, leading to non-periodic structures. Then, we calculated the thermo-spin properties of the structures, and their spin FOMs are plotted in Figure 7(c). The results indicate reductions of ∼ 40% and ∼ 50% of the conversion eciency in (8,0)-2V and (6,6)-2V SWCNTs, respectively, than in the periodic vacancy one. While in (9,0)-2V SWCNT, a decrease of ∼ 14% was observed; however, their values are still remarkable. However, these results demonstrate that the decrease or the increase of the distance between defects as well as the periodic and non-periodic structure can be of signicant importance for producing a large thermo-spin eects. Furthermore, thermo-spin eects in the presence of vacancy defects in only the central region with two perfect semi-innite electrodes were also investigated. In this design, a ferromagnetic phase is induced in the central region, while electrodes have a nonmagnetic phase. Interestingly, the spin-dependent transport is still preserved and leads to a notable thermo-spin eect. The spin FOMs of these congurations are shown in Figure 7(d). Despite 20


Page 20 of 32

Page 21 of 32

the decrease in ZS T values as compared to the previous ones, it is still considerable. Spin Gap (eV)

0.3

(a) (n,0)-2V (n,n)-2V (14,14)

(4,4)

0.2

0.1 (6,0) (18,0)

(12,0)

(24,0)

Maximum

S

(mV/K)

0.0 1.2

(b)

0.8

0.4

0.0

(c)

S

T

5

Maximum

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


4

3

2

1

0 4

6

8

10

12

14

16

A

Nanotube Diameter (

18

)

Figure 8: The diameter dependence of the spin gap and the thermo-spin properties of SWCNTs in the presence of vacancy. (a)-(c) shows the variations of the spin gap and the maximum values of SS and ZS T versus the diameter for (n, 0)-2V (the solid circles) and (n, n)-2V (the open circles) nanotubes, respectively.

Diameter dependence of thermo-spin properties To investigate the thermo-spin properties of SWCNTs based on the changes in the nanotube diameter, we present the diameter dependence of the spin gap, the maximum value of SS in the spin gap region, and the maximum value of ZS T for (n, 0)-2V and (n, n)-2V nanotubes in Figure 8. The smallest and the largest nanotubes are (6, 0) and (14,14) with diameters of about 4.7 and 19 ˚ A, respectively. In Figure 8, the red solid circles are the results obtained for (n, 0)-2V nanotubes that are distinguished from the blue open circles for the (n, n)-2V nanotubes, and each circle represents a nanotube. As seen in Figure 8(a), by increasing the nanotube diameter, the spin gap decreases for both (n, 0)-2V and (n, n)-2V nanotubes, which leads to the decrease of the maximum values of SS and ZS T , as shown in Figures 8(b) and 8(c), so that the same behavior is seen for all the three panels in Figure 8, in particular 21



for the panels (a) and (b). The similar oscillating behavior in the panels (a) and (b) for

(n, 0)-2V nanotubes demonstrates the strong dependence of the maximum value of SS on the spin gap, so that the smallest variation in the spin gap due to the change in the nanotube diameter aects the maximum of SS . The similarity is less in the panel (c) than in the other two panels, which is due to the dependence of ZS T on the parameters other than the spin gap. Furthermore, a sharp decrease in the spin gap and the maximum values of SS and ZS T for the diameters corresponding to the (6n, 0)-2V nanotubes are observed, as specied in the panel (a). To clarify this behavior, the spin-polarized band structure and the transmission spectrum are shown in Figure 9 for (12,0)-2V and (18,0)-2V nanotubes, as an example. As it was seen, despite the spin-semiconducting behavior as well as the existence of four localized energy bands, there are four non-localized (broad) energy bands: two bands below and two bands above the Fermi level that overlap with the localized energy bands near the Fermi level. The non-localized energy bands create a wide transmission coecient in (6n, 0)-2V nanotubes, unlike other nanotubes that only have localized bands (without overlapping with other bands) and a narrow transmission peak around the Fermi level (compare Figure 2 and Figure 9). Furthermore, the spin gaps of the (6n, 0)-2V nanotubes are strongly decreased as compared to those of the other nanotubes, so that the values for (12,0)-2V and (18,0)-2V nanotubes are 0.05 and 0.034 eV, respectively. Therefore, the strong deterioration of the thermo-spin properties of (6n, 0)-2V nanotubes indicates the importance of the localized energy bands and the spin gap for designing and fabricating spin-caloritronic devices. It is important to point out that it is dicult to fabricate experimentally an ensemble of individual SWCNTs with a specic (n, m) structure. In experiments, SWCNTs' samples are formed as a bundle, consisting of a mixture of s-SWCNTs and m-SWCNTs with dierent diameters and chiralities, and each of the parameters is able to modify the electronic structure. 37 Therefore, the thermo-spin eects can be aected in realistic devices based on SWCNTs, leading to a decrease in the conversion eciency because of interactions between dierent nanotubes. 12 These decreases are mainly due to low Seebeck coecient and high 22


Page 22 of 32

Page 23 of 32

(a2)

(a1)

(b1)

(b2)

Spin-Up

(eV)

1

Spin-Down

Spin-Up

(12,0)-2V

F

E-E

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Spin-Down

(18,0)-2V

0

-1

0

2

4

k

0

2

4

k

Figure 9: (a1) and (b1) shows the spin-polarized band structure, and (a2) and (b2) shows the spin-polarized transmission functions for (12,0)-2V and (18,0)-2V nanotubes. thermal conductance, which might result from the presence of dierent SWCNTs as well as impurities with low concentration in samples. 14

conclusions In summary, we theoretically studied the thermo-spin eects for both armchair and zigzag SWCNTs in the presence of vacancy defects by the NEGF method and the LandauerBütticker transport formalism. We indicated that the vacancy-induced magnetism could create a spin-semiconducting behavior with localized energy bands around the Fermi level and a modiable spin gap for dierent diameters of SWCNTs. Furthermore, applying a temperature gradient can generate highly symmetric spin-up and spin-down currents with opposite directions, which demonstrates the existence of the pure thermally-induced spin current without any charge current and SSE in SWCNTs with vacancies. The phononic thermal conductance in the presence of a vacancy was reduced by about 50%, which has a signicant eect on the enhancement of the thermo-spin conversion eciency. We also investigated the diameter dependence of the thermo-spin eects, and the numerical results showed that an increase in the diameter tends to decrease the spin gap, which itself tends to decrease SS and ZS T . Furthermore, the maximum value of SS and ZS T are about 1.6

mV/K and 5, respectively. The results indicate the possible application of SWCNTs in 23



Page 24 of 32

thermo-spintronic.

Acknowledgments This work nancially supported by Iran National Support Foundation: INSF.

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Large Thermospin Effects in Carbon Nanotubes with Vacancy Defects

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