1 Tuning Phononic and Electronic Contributions of Thermoelectric in defected S-Shape Graphene Nanoribbon s

2025-04-30 3 0 2.63MB 26 页 10玖币

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Tuning Phononic and Electronic Contributions of Thermoelectric

in defected S-Shape Graphene Nanoribbons

M.Amir Bazrafshan, Farhad Khoeini*

Department of Physics, University of Zanjan, P.O. Box 45195-313, Zanjan, Iran

Abstract

Thermoelectrics as a way to use waste heat, is essential in electronic industries, but its low performance at

operational temperatures makes it inappropriate in practical applications. Tailoring graphene can change its

properties. In this work, we are interested in studying the transport properties of S-shape graphene structures

with the single vacancy (SV) and double vacancy (DV) models. The structures are composed of a chiral

part, which is an armchair graphene nanoribbon, and two zigzag graphene ribbons. We investigate the

changes in the figure of merit by means of the Seebeck coefficient, electronic conductance, and electronic

and phononic conductances with the vacancies in different device sizes. The transport properties of the

system are studied by using the non-equilibrium Green’s function method, so that the related Hamiltonians

(dynamical matrices) are obtained from the tight-binding (force constant) model. The maximum figure of

merit (ZT) obtains for the DVs in all lengths. Physical properties of such a system can be tuned by

controlling various parameters such as the location and the type of the defects, and the device size. Our

findings show that lengthening the structure can reduce phononic contribution, and single vacancies than

double vacancies can better distinguish between electronic thermal conductance behavior and electronic

conductance one. Namely, vacancy engineering can significantly increase thermoelectric performance. In

the large devices, the SVs can increase the ZT up to 2.5 times.

Keywords: Phononic conductance, Electronic conductance, Graphene structure, Vacancy defect, Green’s

function, Tight-binding.

1. Introduction

Modern life is entangled with computers, and transistors are the heart of the computer processors. As the

number of transistors in an electronic chip rises, the power density also increases, decreasing the integrated

circuits performance reliability [1]. In electronic devices, maintaining the temperature in an appropriate

range is very important [2]. Lowering their temperature can be achieved by dissipating the heat or

converting it into another form of energy and using it. For heat dissipation, high thermal conductance is

essential, and to convert heat into another form of energy, such as electricity, thermoelectric is one of the

solutions, but it requires low thermal conductance. Researchers are trying to find the best material for

thermoelectrics [3]. They face some challenges. For example, electron scattering in junctions produces heat,

so the material must be a good electronic conductor to produce less heat, and to enhance the thermoelectric

performance, the temperature difference should be maintained [4].

Many materials can be fabricated, thanks to nowadays facilities. But this is not enough if someone wants

to synthesize materials for a specific goal with trial and error. Theoretical approaches can provide an easy

way to discover the underlying factors involved in the properties of materials. Based on theoretical

approaches, researchers are trying to find the best material for this purpose. Theoretically, using an

algorithm, in an edge defected long ZGNR, ZT is reported four in Ref. [5] at room temperature in a 

long ZGNR. Other works also reported a figure of merit of six for narrow AGNRs at room temperature [6],

0.88 for a bent structure with 24 pores [7] at 500 K, up to three in a graphene structure with different

percentage of carbon isotopes doping, and vacancies [8], ZT>2 for a  long ZGNR with extended line

defect [9], and  at 77 K for ZGNR based devices with a length of >1 containing two special

nonperiodic nanopores with different diameters [10]. As mentioned in Ref. [11], for long GNRs, the

electron-phonon interaction may not be neglected since it might be significant in long GNRs, though we

neglect it since our structures are small enough [12]. Besides, the Umklapp scattering is not considered for

the same reason [13].

Graphene is the first 2D successfully synthesized material [14]. It has the highest thermal conductance until

now [15], which makes it the best candidate to conduct heat for dissipation applications [11,16]. However,

as the dimensions decrease, quantum confinement effects become important [17], which can help maximize

ZT [18] by manipulating physical properties. This can be accompanied by phonon scattering due to

nanostructure boundaries [19]. In 2D materials, especially hexagonal structures, the edge geometry of a

ribbon provides a degree of freedom to tailor its physical properties [16,20,21]. Introducing defects, doping,

and applying mechanical strain can also alter the physical properties of graphene [22–25].

Since the shape and geometry of the nanodevices are important in tuning physical properties at the

nanoscale, we are interested in studying the S-Shape graphene structures with three different lengths. S-

Shape structure is a mix-up of zigzag and armchair edge geometries, which can help to tune physical

properties. In an S-Shape graphene nanoribbon (GNR), electronic contributions can be significantly altered

due to the quantum confinement and edge effects [26]. In this work, the temperature is considered 350 K,

close to what is to be controlled in processor units [27,28]. To evaluate the thermoelectric performance, the

figure of merit, a dimensionless parameter, is investigated. The figure of merit can be calculated as

 

 , with  as electronic conductance,  as Seebeck coefficient,  as electronic thermal

conductance,  as phononic thermal conductance, and  as absolute temperature. These parameters are

individually plotted for each of the studied structures.

The two experimentally observed vacancies, single vacancy (SV) and divacancy (DV) [29], are introduced,

and their impact on both electronic and phononic contributions related to thermoelectric performance is

studied.

We have used the non-equilibrium Green’s function (NEGF) method to calculate the interested quantities.

Hamiltonians are obtained from the tight-binding (TB) approach by considering up to third nearest-neighbor

(3NN) interactions. To be more accurate [30], overlap integrals are taken into account. For phononic

thermal conductance, force tensor matrices obtained via the force constant (FC) model by considering up

to 4NN.

Vacancies are introduced and named as indicated in figure 1, e.g., a single vacancy located at the eleventh

atomic position in the armchair direction and the fifteenth one in the zigzag direction is identified by SV-

11-15; the number of atomic positions is also presented for each direction.

The article is arranged as follows; in the next section, we will describe the model, with a brief introduction

on the TB, and FC formulations. Results and discussion are in section three. In the last section, we conclude

our study.

2. Model and Method

In this section, we describe a system consisting of left and right contacts and a central device connected to

them.

To start, a schematic of the structure of system is presented in figure 1. The system is divided into three

parts with black boxes; the device section is an S-Shape graphene structure, the right and left contacts are

two semi-infinite ZGNRs with a width of 12 atoms. The grey dashed boxes show the unit cells in contacts.

Also, the first, second, and third nearest neighbors are displayed with concentric circles in the left contact,

so that magenta dashed circle shows the first nearest-neighbor domain, the cyan dashed circle shows the

2NN domain, and red dashed circle indicates the 3NN domain. Vacancies are identified by their position in

the armchair and the zigzag edge geometries. The numbers on the left and the bottom of the device section

are for easy identification of those vacancies. Three examples of how to identify vacancies are shown.

Vacancies are identified with the general form of VT-m-n-or, in which the VT is the vacancy type, here it

can be SV or DV, m and n are atomic positions in the armchair and zigzag directions, respectively, and the

last part indicates respective orientation to the nanoribbon axis. For single vacancies, it is omitted, but for

divacancies, the relative orientation of the hypothetical line between two removed atoms determines the

last part. If a DV is perpendicular to the ribbon axis (or parallel to the ribbon width), it is indicated with

‘pr’, it is indicated with ‘or’. To be more precise, the cyan box in figure 1 indicates a divacancy DV-7-8-

or, in which its first atom (as numbers, from left to right) is in the seventh atomic position in the armchair

direction, and the eighth atomic position in the zigzag direction, this divacancy is oriented respect to the

ribbon axis which is indicated by “or” in the name of the DV. The direction, in which vacancies move in

the structure is marked by the green arrow. The dashed lines are bonds that are affected throughout the

study. The hatched area shows the zone where the vacancies are introduced.

Figure 1 A schematic of the model implemented in the NEGF method with all Hamiltonians and overlap matrices. Vacancies are

relocated in the hatched area, and four vacancies with their names are shown. The thick purple dashed line indicates the middle of

the device section. Three concentric circles show the first, second, and third nearest-neighbor domains, respectively, by magenta,

cyan, and red colors for a selected atom. Vacancies were considered in different places in the hatched area (the green arrow shows

this direction). Numbers in the device were used for this purpose as described in the text.

To employ the NEGF method for electronic and phononic contributions, matrices that describe electron and

phonon energies and their interaction with nth nearest neighbors, are essential. To form matrices for

electrons (i.e., Hamiltonians) in the tight-binding approach, the unit cell should be defined (as depicted in

figure 1 with dashed gray rectangles). In the non-orthogonal tight-binding approach, the Hamiltonian of the

system, its elements, and the elements of overlap matrix are as [7]:



  







(1a)

(1b)

where  is the on-site energy and , and  are the interatomic and overlap parameters, respectively.

There are several sub and superscripts that LC means the left contact, RC means the right contact, and D is

the device. As indicated in figure 1, 

 is the Hamiltonian of the unit cell 0 in the left contact, and 



is the coupling Hamiltonian between the unit cell number -1 and 0 in the left contact.

Hopping and overlap parameters are presented in table 1 as reported in Ref. [31]. The electronic energy

dispersion for a periodic system, like the left contact, can be obtained by solving the eigenvalue

problem [32]:

,

(2)

where  and  are given by:





 

 ,





 

 ,

(3 a)

(3 b)

in which  and are the wave vector and the lattice constant, respectively. The transmission probability

(for electrons , and for phonons it is shown by ) can be calculated using Green’s function method [33];

details are provided in the supplementary materials.

By having transmission probability, one can calculate the electronic conductance , the Seebeck

coefficient , and the electronic thermal conductance  as [11,34]:

















(4a)

(4 b)

(4 c)

here  is the elementary charge, and  is given by:















 

(5)

which its numerical form is as follows:















(6)

with  as plank constant and  as the Boltzmann constant. This is the discrete form of integral. The

summation is over the whole energy range. By considering  as total steps, the integration element for

numerical integration in the rectangular method is .

The secular equation for phonons, which derives from Newton’s second law, is:



(7)

in which,  is the matrix containing the vibrational amplitude of all atoms,  is the angular frequency, and

 is the dynamical matrix:





























 









(8)

where  is the mass of the ith atom, and  represents  force tensor between the ith and jth atoms:

 



(9)

with  as the angle between the and the atom. The unitary matrix  is defined by the rotation

matrix in a plane as:

  

  

  

(10)

also 

 is given by:



 

 

  

(11)

where are force constant parameters in the radial, in-plane, and out of plain directions of the

 atom, respectively. To be more clear about these matrices, e.g., for the , which represents the

dynamical matrix of the device section, regarding Eq. 7, and to write what each atom feels (or when ),

one must consider all 4 NN effects in the summation, including atoms in the neighboring unit cells, i.e.,

   

. For coupling terms like the  elements, the interaction

between the first atom of the device and the first atom of the right neighbor (i.e., diagonal elements) is

already accounted, so one can safely set this to zero. Force constants [35], and other essential parameters

are presented in Table 1.

Phononic band structure can be obtained by solving the following eigenvalue problem [36]:







(12)

with   as the distance between the ith and jth atoms, and  as the wave vector. To calculate the

phononic density of states or vDOS, with v as vibrational, one can use 



within Green’s function method [37,38], or by using Gaussian smearing of the Dirac Delta [7]:

 







(13)

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1TuningPhononicandElectronicContributionsofThermoelectricindefectedS-ShapeGrapheneNanoribbonsM.AmirBazrafshan,FarhadKhoeini*DepartmentofPhysics,UniversityofZanjan,P.O.Box45195-313,Zanjan,IranAbstractThermoelectricsasawaytousewasteheat,isessentialinelectronicindustries,butitslowperformanceatoperation...

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