Spin thermoelectric properties of the Al$_{12}$N$_{12}$ molecule

Spin thermoelectric properties of the Al

N

molecule

N DEHGHAN, M YAGHOBI ^∗and M R NIAZIAN Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran

∗Corresponding author. E-mail: mojtabayaghobii@gmail.com

MS received 26 March 2020; revised 13 October 2021; accepted 29 October 2021

Abstract. In this paper, the effect of the direction of a magnetic field has been studied in leads (electrodes with limited width) on thermoelectric properties of Al12N12cage with and without considering inelastic electron–phonon interactions. Results showed that the biggest (smallest) value of total conductivity is observed in the case of the maximum value of elastic conductivity (G^M_σ,σ) with respect to the direction of the magnetic field for antiparallel (↑↓ or ↓↑), parallel configurations (↓↓ or ↑↑)for the left (σ =↑↓)and the right (σ =↑↓)lead spins, i.e., G_↓,↑^M (G^M_↑,↑). The maximum number of conduction heights was seen forσ =↓andσ =↑. With respect to the direction of magnetic field in the leads, the maximum value of electronic thermal conductance(K_e^max_,σσ)can be arranged in the following order:K_e^max_,↓↑>K_e^max_,↓↓>K_e^max_,↑↓>K_e^max_,↑↑. The direction of the magnetic field in the leads affects the numbers, height and width peaks and valleys of the first derivative of thermopower (TP). We observe the highest (deepest) peak (valley) whenσ =↑andσ=↓(σ =↑andσ =↑). Also, the effect of temperature on current and tunnel magnetoresistance (TMR) was considered in this paper.

Keywords. Quantum dot; spin transport; magnetic–semiconductor structures.

PACS Nos 65.20.+w; 65.40.-b; 65.80.+n; 65.80.Ck; 66.70.+f; 72.15.Eb

1. Introduction

In recent years, molecular systems have attracted much attention of experimental and theoretical researchers and important and novel properties were seen in these sys- tems. Molecular electronics is the study of the molecular structure for the construction of electronic components.

Due to the reduction in size in electronics, molec- ular electronics have received much attention from researchers [1–9]. The nature and electronic structure of the device and the contacts and the strength and type of the coupling electrode device have a significant impact on the transport properties of molecular devices.

Spintronics or spin electronics is a field of electron- ics with respect to the magnetic moment which depend on the intrinsic spin of the electron. Due to the rela- tive orientation of magnetisations in the ferromagnetic electrodes, the transport characteristics of molecules are spin-dependent when the molecule is sandwiched between two ferromagnetic electrodes [10–14].

In addition to the difference in the chemical potential of leads, a current between two leads is due to the differ- ence in temperature of the two leads. Also, the studies indicated that the thermoelectric effect of materials and

nanostructures is an important factor in molecular elec- tronics [15–18]. Kochet alfound that the thermopower contains information on the electronic and phononic excitations of the molecule. A study showed that at high temperatures, the thermoelectric potential affects the position of the Fermi energy relative to the molecu- lar states [19]. Using Green’s function theory within the framework of polaron transformation (GFT-PT), Wal- czak showed that strong electron–phonon coupling has an impact on the thermoelectric characteristics [20].

The thermopower and thermal conductance of several molecular junctions, atomic-sized metallic contacts, 1D wires and double quantum well were studied using experimental and theoretical methods [21–29].

After the discovery of carbon fullerenes and nanotubes, many investigations of the electron transport and thermoelectric properties of fullerene and fullerene- like cages have been reported [30]. The novel physical and chemical properties of aluminum nitride (AlN) nanostructures are essential in electronic and opto- electronic applications. For example, nitride nanocages have low electron affinity and large band gap com- pared to other nanocages [31]. Lei et al suggested a growth mechanism of AlN nanostructures with urchin

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shapes [32]. Using DFT/B 3LYP/LANL2DZp method, Wu et al investigated the structure and stability of (AlN)n clusters (n = 2–41) [33]. The theoretical cal- culations indicated that the Th symmetrical Al12N12

cluster is the most stable cage among different types of (AlN)n clusters. Applying density functional the- ory (DFT), the adsorption and decomposition of formic acid (HCOOH) on the surface of Al₁₂N₁₂fullerene-like nanocage was studied [34]. Farrokhpouret alindicated that the Al12N12 nanocage can be used as the adsor- bent for HCN and ClCN gases [35]. Baeiet alreported that the Al12N12 nanocage can be used as a sensor for phosgene molecules [36]. Magnetic resistance is a phe- nomenon in which the application of a magnetic field changes material resistance. The effects of magnetic resistance were observed in carbon nanotubes at low temperatures [37]. Currently, if two ferromagnets are segregated by a few layers of atomic insulation, and the distance between the two layers is very low (nm), the electron is transferred from one layer to another layer, which is called the tunnelling magnetoresistance (TMR). Therefore, tunnelling is a complete quantum effect, which is prohibited in classical physics [38,39].

This research is based on the formulation of Green’s function theory within the framework of polaron trans- formation. The results clearly show that the effects of the difference in temperature and magnetic field of two lead molecules Al12N12 are important factors in spin thermoelectric properties.

2. Theoretical method

In this section, we consider that Al12N12 molecule is sandwiched between two ferromagnetic electrodes with finite cross-sections (see figure1). The goal of this work is to calculate the thermoelectric properties of Al₁₂N₁₂ molecule.

The interaction between the electrodes is ignored (H_{L R} = H_{R L} = 0) since the central region is big enough. Then, Hamiltonian (Hˆ) of a molecular junc- tion that is sandwiched between two FM electrodes can be shown as

Hˆ = ˆHL+ ˆHR+ ˆHC + ˆHT, (1) where HˆL(HˆR), HˆC and HˆT are the Hamiltonians of the left (right) electrodes, the channel and the molecule- FM electrodes interaction, respectively. The electrodes Hamiltonian with non-interacting electrons are written as

Hˆ_α^σ = (ⁱα,jα)^,σ

(εαδi_α,j_α −ti_α,j_α)c⁺_iασcj_ασ(α=L,R).

(2)

Here,ε_αis the on-site energy of the electrodes which is equal to the gate voltage.

The molecule-FM electrode interaction is given as follows:

Hˆ_T =

α={L,R}

(i_α,jC),σ

(−t_α)(c⁺_i

ασd_jC +h.c.). (3) The Hamiltonian of the conductor can be described as [20,39]

HˆC =

i

εid_i⁺di+ ˆHSSH (4) where

HSSH=

i,j

−t0+

k

λ^k_i,j(a⁺_k +ak)

×(d_i⁺di +h.c.)+

k

hωka_k⁺ak. (5) Here the nearest-neighbour hopping integrals are indi- cated ast₀ andt,d_i⁺ orc⁺_i,σ(d_i orc_i,σ)is the creation (annihilation) operator of theπelectron at the siteiwith spinσ, whileak anda_k⁺are phonon creation and anni- hilation. The sum is given over the nearest-neighbour junctioni, j.λ^k_i,j andωk are the junction strength in electron–phonon interaction and phonon frequency in phononic energy.

The electron states of the electrodes and central region are expanded onto the direct product states composed of single-electron states andm-phonon Fock states, respec- tively as [20]

|i,m =c⁺_i,σ(a⁺)^m

m! |0, |i,m =d_i⁺(a⁺)^m

m! |0. (6) In the new representation, HˆC and HˆT can be re- structured in the following forms [28]:

HC =

i,m

(εi +m )|i,mi,m|

+

i,j

−t0|i,m j,m| +

i,m

λ√ m+1

×(|i,m+1j,m| + |i,mj,m+1|)

. (7) and

Hˆ_T =

α={L,R}

(i_α,jc),m

−t_α |i_α,mj_c,m|

. (8)

The energy-dependent electrical conductance at zero temperature is given by the Landauer–Biittiker formula:

G^δ_m,n^,δ(E)= dI(E)

dV(E) =G₀L₀, (9)

whereG0 =2e²/his the quantum conductance.

The coefficientsLn(n =0,1, . . .) are defined as L^σ_n^,σ = 2

h

∞

−∞

m,n

−εn

−∂f

∂ε

×P_mT_m^σ_,n^,σ(ε)dε. (10)

Here Pm =

1−exp(−βRhω)

exp(−βRmhω) is the Boltzmann distribution function, f_α^m(ε) = exp[βα(ε+mhω−μα)] +1

(α = L,R) are the Fermi–Dirac distribution functions,μL,R =εF are the electrochemical potentials of the right and left elec- trodes,σ =↑↓ andσ =↑↓are the left and right lead spins, e is the electron charge and h is the Planck’s constant. When the incoming channel m with outgo- ing channel n is connected, transmission probability (Tm^σ,n^,σ(ε))is written as [29,39]:

T_m^σ_,n^,σ(ε)=tr[ ˆ^σ_LGˆ^σ_m+1,n+1^,σ,R (ε)ˆ^σ_RGˆ^σ_m+1,n+1^,σ,A (ε)]

with

^σ_α=L,R =2πt_α²ρ_α^σ, (11)

where _α=^σ _L,R are the linewidth functions, t_α is the nearest-neighbour hopping integrals between the elec- trodes and the central part of the system andρ_α^σ is the density of states in theα-electrode in the Fermi energy.

G_C^R,A are the retarded and advanced Green’s functions and are estimated as [20,38]

G^σ,σ,R/A(ε)

=

⎡

⎣(ε±iη)Iˆ− ˆHC −

σ

L

(ε)− σ

R

(ε)

⎤

⎦

−1

. (12) Hereη=0⁺is a positive infinitesimal constant. Using the wide band approximation, the self-energy matrices are given as [21,39]

σ α

=2i_α^σ = − ˆτ_αC⁺ gˆ_α^στˆ_αC(α=L,R), (13) wheregˆ_α^σ =iπρ_α^σ is the surface Green’s function of the uncoupledα-electrode.

The thermopowerSand thermal conductance of elec- trons at zero electron current can be calculated as follows [20,40]:

S^σ,σ = kBβRL^σ₁^,σ

eL^σ₀^,σ (14)

and κ^σ,σ =e

−S^σ,σL^σ₁^,σ +kB

e βRL^σ₂^,σ

. (15)

3. Results and discussion

First, we optimised the structure of Al12N12 molecule using B3LYP/6-31G*(d) density functional level of the- ory in Gaussian 98 [41]. For different nanostructures, the B3LYP/6-31G* approximation was indicated as a com- mon and reliable level of theory [42–44]. With respect to the gap energy and bond lengths of the Al₁₂N₁₂ molecule by using the B3LYP/6-31G*(d), we select the following parameters [20]:

εF =0, ε0 =0, ρ_L^↑=ρ^↑_R =0.734, ρ^↓_L =ρ^↓_R =0.174, t_L =t_R =0.24, t0 =2.5, β_L⁻¹=kBθL =0.015, β⁻_R¹ =k_BθR =0.025

and

λ^ki,j =λ=0.2.

Here kB is the Boltzmann constant and θR/L is the limit temperature of the right and left electrodes. All computations are done in a weak coupling regime (tL/R t0) by applying wide-band (WB) approxima- tion. Here, interactions between charge carriers are ignored, and our concentrate is due to the effect of the inelastic scattering process precisely. Thus, this work has been used in the tunnelling method based on Green’s function theory within the framework of polaron trans- formation (GFT-PT) [20,45]. Currents are expressed with antiparallel and parallel spins and spins are con- sidered conserved during tunnelling.

Figures 4a and 4b show the energy levels of the Al12N12molecule using the B3LYP/6-31G*(d) and the Su–Schrieffer–Heeger (SSH) approximations, respec- tively. Comparison of figures4b and4a shows that the energy levels and difference between HOMO (highest occupied molecule orbital) and LOMO (lowest unoccu- pied molecular orbital) of the Al₁₂N₁₂ molecule, using the B3LYP/6-31G*(d) and SSH approximations, are in agreement with each other.

The elastic and total electrical conductivity against energy of Al12N12molecule is depicted in figures3a–3d for antiparallel and parallel configurations. Figure3dis- plays the maximum value of elastic conductivity (G_σ,σ^M ) with respect to the direction of the magnetic field in left (σ =↑↓)and right(σ =↑↓)leads. It is clear that the maximum value (number peaks) of elastic conductivity is smaller (lesser) than that of total conductivity. The biggest value of total conductivity belongs to the case ofG^M_↓,↑. The least value of total conductivity is observed inG_↑,↑^M . In the case ofσ =↓andσ =↑, the maximum number of total conductivity heights is greater.

Figure 1. Geometric configuration of FM/Al12N12/FM junction when (a)σ = ↑andσ = ↑, (b)σ = ↑andσ = ↓, (c) σ = ↓andσ= ↓and (d)σ = ↓andσ= ↑

To understand the thermoelectric properties of Al12N12molecule better, electronic thermal conductiv- ities (K_e) as a function of electron energy is presented in figures 4a–4d. The large oscillations of parabolic behaviour are seen for electronic thermal conductance for Al₁₂N₁₂ molecule. Based on figure 4, the num- bers and heights of peaks of thermal conductivities increase in the case of inelastic interactions. Also, the comparison of thermal conductance of molecule shows that the number of major peaks change with respect to

the direction of the magnetic field in the leads. With respect to the direction of the magnetic field in the leads, the maximum value of electronic thermal conduc- tance (K_e,σσ^max)can be arranged in the following order:

K_e^max_,↓↑ > K_e^max_,↓↓ > K_e^max_,↑↓ > K_e^max_,↑↑. For example, the maximum value of electronic thermal conductance for the case ofσ = ↓andσ = ↑is about three times larger than that for the case ofσ = ↑ and σ = ↑. Results indicate that electronic thermal conductance reaches a minimum value of Fermi energy. The behaviour via

Figure 2. Energy level structures of the Al12N12molecule for (a) DFT/B3LYP/6-31G* and (b) SSH methods

energy and values of thermal conductance for Al₁₂N₁₂ molecule are similar to the same system [28,29,46–49].

With respect to the direction of the magnetic field in the leads, thermopower (TP) against energy is calculated in this work. To better understand the effect of the direc- tion of the magnetic field in the leads on the staircase structure of TP, the first derivatives of TPs with respect to energy are presented in figure5. The first derivative of TP and any small changes in the slope of the linear trend are seen as an abrupt jump and hump. Therefore, the numbers of abrupt humps are equal to the number of stairs.

Results indicated that a linear graph via energy with staircase structure is seen for TP for elastic and inelastic interactions with respect to the direction of the mag- netic field in the leads. The effect of the direction of the magnetic field in the leads is small on TP values. For negative energies, sign of TP is positive while for pos- itive energies, the sign is negative. These results are in accordance with similar systems [28,29]. The direction of the magnetic field in the leads affects the staircase structure of TP.

Comparisons of figures 2a–2d with figures 5a–5d, respectively, indicate that positions of stairs against energy are proportional to that of conductance peaks.

Therefore, the staircase structure of thermopower reveals that conductance increases. Results indicated that inelas- tic interactions affect the number and width of stairs of the TP graph. The number of stairs in the case of elastic interaction is fewer than those of inelastic interactions.

According to figure 5, numbers, height, width of the peaks and the valleys of the first derivative of TP depend on the direction of the magnetic field in the leads. We

observe the highest (deepest) peak (valley) in the case of σ =↑andσ =↓(σ =↑andσ=↑). These effects also are in accordance with similar systems given in [50,51].

Finally, the effect of temperature on current and TMR (tunnel magnetoresistance) is studied in this paper. We fixed the temperature of the right lead as 180 K and changed the temperature of the left lead from 190 to 350 K. Figure6indicates parallel current(IP = I_↓↓+ I_↑↑)and antiparallel current(Ia = I_↑↓+I_↓↑)against temperature difference by considering e–p interactions and zero voltage bias. Behaviours of both parallel and antiparallel currents are Ohmic. Therefore, the current increases with temperature.

The effect of temperature on TMR of Al12N12

molecule is shown in figure7after considering e–p inter- actions and zero voltage bias. Gate and bias voltages can affect the TMR values. For zero gate and bias voltages, the maximum value of TMR is seen when the temper- ature difference is 50 K. After a temperature difference of 50 K, the ratio of TMR decreases.

4. Conclusions

The effect of the direction of the magnetic field in the leads on thermoelectric properties of FM–Al12N12–FM junctions was investigated with and without considering inelastic electron–phonon interactions by applying WB approximation and GFT-PT. Our results indicate that the effect of the direction of the magnetic field in the leads is an important factor in the thermoelectric properties of Al12N12molecule. The values and behaviour of electri- cal conductance, thermal conductance and thermopower

Figure 3. Inelastic (straight line) and inelastic (dash line) conductance values as a function of electron energy when (a) σ = ↑andσ = ↑, (b)σ = ↑andσ = ↓, (c)σ = ↓and σ= ↓and (d)σ = ↓andσ= ↑

Figure 4. Inelastic (straight line) and inelastic (dash line) electronic thermal conductivity values as a function of elec- tron energy when (a)σ = ↑ andσ = ↑, (b)σ = ↑ and σ= ↓, (c)σ = ↓andσ= ↓and (d)σ = ↓andσ= ↑

Figure 5. Inelastic (straight line) and inelastic (dash line) first derivative of thermopower values as a function of electron energy when (a)σ = ↑andσ= ↑, (b)σ = ↑andσ= ↓, (c)σ = ↓andσ= ↓and (d)σ = ↓andσ= ↑

Figure 6. The parallel current (straight line) and antiparallel current (dash line) against temperature difference between the left and right leads

Figure 7. The TMR ratio against temperature difference between the left and right leads

of Al12N12 molecule can change with respect to the direction of the magnetic field in the leads. Results also indicated that inelastic interactions affect the numbers, the width of peaks of electrical conductivity, electronic thermal conductance and TP graphs.

References

[1] Y Wada,Pure Appl. Chem.71, 2055 (1999)

[2] C Joachim, J K Gimzewski and A Aviram,Nature408, 541 (2000)

[3] C Dekker and M Ratner,Phys. World14, 29 (2001) [4] Y Xue, S Datta and M A Ratner,Chem. Phys.281, 151

(2002)

[5] A W Ghosh and S Datta,J. Comp. Electr.1, 515 (2002) [6] A Nitzan and M A Ratner,Science300, 1384 (2003)

[7] J R Heath and M A Ratner,Phys. Today56, 43 (2003) [8] A H Flood, J F Stoddart, D W Steuerman and J R Heath,

Science306, 2055 (2004)

[9] C R Kagan and M A Ratner,MRS Bull.29, 376 (2004) [10] E G Emberly and G Kirczenow,Chem. Phys.281, 311

(2002)

[11] M Zwolak and M Di Ventra,Appl. Phys. Lett.81, 925 (2002)

[12] W I Babiaczyk and B R Bułka,Phys. Status Solidi A 196, 169 (2003)

[13] W I Babiaczyk and B R Bułka,J. Phys.: Condens. Matter 16, 4001 (2004)

[14] K Walczak,Physica B365, 193 (2005)

[15] M Paulsson and S Datta,Phys. Rev. B67, 241403 (2003) [16] J Koch, F von Oppen, Y Oreg and E Sela,Phys. Rev. B

70, 195107 (2004)

[17] X Zheng, W Zheng, Y Wei, Z Zeng and J Wang,J. Chem.

Phys.121, 8537 (2004)

[18] D Segal,Phys. Rev. B72, 165426 (2005)

[19] Y Dubi and M Di Ventra,Phys. Rev. B79, 119902 (2009) [20] Kamil Walczak,Physica B392, 173 (2007)

[21] P Reddy, S Y Jang, R A Segalman and A Majumdar, Science315, 1568 (2007)

[22] S H Ke, W Yang, S Curtarolo and H U Baranger,Nano Lett.10, 3257 (2008)

[23] K A Matveev and A V Andreev,Phys. Rev. B66, 045301 (2002)

[24] T Smith, M Tsaousidou, R Fletcher, P T Coleridge, Z R Wasilewski and Y Feng, Phys. Rev. B67, 155328 (2003)

[25] E N Bogachek, A G Scherbakov and U Landman,Phys.

Rev. B60, 11678 (1999)

[26] B Ludolph and J M van Ruitenbeek, Phys. Rev. B59, 12290 (1999)

[27] N J Appleyard, J T Nicholls, M Pepper, W R Tribe, M Y Simmons and D A Ritchie,Phys. Rev. B62, R16275 (2000)

[28] M Yaghobi and F A Larijani, Indian J. Phys.89, 257 (2015)

[29] M Yaghobi and F A Larijani,Pramana — J. Phys.84, 155 (2015)

[30] F Zhang, Q Wu, X Wang, N Liu, J Yang, Y Hu, L Yu, X Wang, Z Hu and J Zhu,J. Phys. Chem. C113, 4053 (2009)

[31] Q Wu, F Zhang, X Wang, C Liu, Z Hu and Y Lu,J. Phys.

Chem. C111, 12639 (2007)

[32] W Lei, D Liu, J Zhang, P Zhu, Q Cui and G Zou,Cryst.

Growth Des.9, 1489 (2009)

[33] H-S Wu, F-Q Zhang, X-H Xu, C-J Zhang and H Jiao,J.

Phys. Chem. A107, 204 (2003) [34] M D Esrafili,R. Nurazar.75, 17 (2014)

[35] H Farrokhpour, H Jouypazadeh and S V Sohroforouzani, Mol. Phys.118, 1626506 (2020)

[36] M T Baei, A Soltani, S Hashemian and H Mohamma- dian,Can. J. Chem.92, 605 (2014)

[37] G D Guttman, E Ben-Jacob and D J Bergman, Phys.

Rev. B51, 17758 (1995)

[38] K Walczak,Physica B 392, 173 (2007)

[39] M R Niazian, L F Matin, M Yaghobi and A A Masoudi, Indian J Phys.95, 1131 (2021)

[40] M R Niazian and M Yaghobi,IJPAP54, 123 (2016) [41] M J Frisch, G W Trucks, H B Schlegel, G E Scuseria,

M A Robb and J A Pople,Gaussian 98(Gaussian, Inc., Pittsburgh, PA, 1998)

[42] J Jia, H Wang, X Pei and H Wu,Appl. Surf. Sci.253, 4485 (2007)

[43] H Wu, X Xu, F Zhang and H Jiao, J. Phys. Chem. A107, 6609 (2003)

[44] H Wang, J Jia, X Pei and H Wu,Chem. Phys. Lett.118, 423 (2006)

[45] J Dong, J Jiang, Z Wang and D Xing,Phys. Rev. B51, 1977 (1995)

[46] Y E Putri, C Wan, Y Wang, W Norimatsu, M Kusunoki and K Koumoto,Scr. Mater.66, 895 (2012)

[47] C Chang, W Han and A Zettl,J. Vac. Sci. Technol. B 23, 1883 (2005)

[48] Y Park,Synth. Met.45, 173 (1991)

[49] J Vavro, M Llaguno, B Satishkumar, D Luzzi and J Fis- cher,Appl. Phys. Lett.80, 1450 (2002)

[50] K Baheti, J Malen, P Doak, P Reddy, S Jang, T Tilley, A Majumdar and R Segalman,Nano Lett.8, 715 (2008) [51] A Shakouri,Ann. Rev.41, 399 (2011)