# 6.2. Thermoelectric properties of materials

The thermoelectric performance (for either power generation or cooling) depends on the efficiency of the thermoelectric material for transforming heat into electricity. Thermoelectric materials are those capable of converting temperature differences into electrical current (Seebeck effect) or, conversely, generating temperature differences when an electric current is applied (Peltier effect). The thermoelectric figure of merit (FOM), often denoted as \(ZT\), is a metric that combines important properties of a thermoelectric material to determine its efficiency in converting heat to electricity. The general formula for \(ZT\) is:

where:

\(G\) is the electric conductance,

\(S\) is the Seebeck coefficient (which measures a material’s ability to generate an electric voltage in response to a temperature gradient),

\(T\) is the absolute temperature in kelvins,

\(k_{e}\) is the electronic thermal conductivity, and

\(k_{ph}\) is the lattice thermal conductivity (phononic) of the thermoelectric material.

A higher \(ZT\) value indicates a more efficient thermoelectric material. Improving \(ZT\) is crucial for the development of more efficient thermoelectric devices, such as thermoelectric coolers or thermoelectric power generators. Researchers continually seek materials with specific electrical and thermal properties to improve \(ZT\) and enhance the performance of these devices [CDH].

In this tutorial, we’ll show how to calculate the \(ZT\) in an armchair graphene nanoribbon (AGNR) supercell (Fig. 6.2.1) using NanoDCAL.

The calculation steps are:

Build:

8AGNR unit cell;

Expand the primitive 8-acGNR unit cell.

Self-consistent calculation;

Band structure calculation;

Hessian matrix calculation;

Phonon band and density of states calculation;

Thermoelectric properties calculation.

## 6.2.1. Build 8AGNR and Expand the primitive cell

The AGNRs can be divided into three subfamilies of \(N=3p\), \(N=3p+1\) and \(N=3p+2\), where \(p\) is an integer and \(N\) denotes the rows of carbon atoms across the ribbon width. The AGNRs that fall into the \(N=3p+2\) subfamily are expected to possess excellent electrical properties. For practical purposes of this tutorial, we will consider a pristine 8AGNR, periodic along the z direction.

This device can be built using Device Studio modeling as shown in Fig. 6.2.2:

Open Device Studio software, go to the menu bar and click

**Build**➟**Gr_Nanoribbon**;In the

**Gr_Nanoribbon**window, keep uncheck the*Passivate dangling bonds with hydrogen*checkbox option, set**n = 0**and**m = 4**, click**Preview**and**Build**the structure.

In the menu bar click

**Build**➟**Redefine Crystal**;In the cell vector set

**u = (1a, 0b, 0c)**,**v = (0a, 1b, 0c)**, and**w = (0a, 0b, 5c)**, click**Preview**or**Build**to expand the cell**5**times in the c direction, click on**Preview**and**Build**the structure.

Important

In the context of Density Functional Theory (DFT) calculations, passivating atoms with hydrogen (H) is a common practice to simulate realistic conditions for certain systems and ensuring that the system is stable. It’s important to note that the choice of passivation strategy depends on the specific system being studied and the research objectives.

In the proposal of this tutorial, we choose not to passivate dangling bonds with hydrogen atoms to reduce the computational cost. However, this decision can introduce artificial electronic states in electronic energy band diagram.

The atomic coordenates for 8AGNR primitive cell can be download here:

The 8AGNR primitive cell

*0_4Gr_Nanoribbon.hzw*file (`0_4Gr_Nanoribbon.hzw`

).

0_4Gr_Nanoribbon.hzw

```
% The number of probes
0
% Uni-cell vector
32.67804859 0.00000000 0.00000000
0.00000000 20.00000000 0.00000000
0.00000000 0.00000000 4.26000000
%Total number of device_structure
16
%Atom site
C 11.41999999 10 2.83999999
C 11.41999999 10 1.41999999
C 12.64975607 10 0.71000000
C 12.64975607 10 3.55000000
C 13.87951214 10 2.84000000
C 13.87951214 10 1.41999999
C 15.10926822 10 0.71000000
C 15.10926822 10 3.54999999
C 16.33902429 10 2.83999999
C 16.33902429 10 1.42000000
C 17.56878036 10 0.71000000
C 17.56878036 10 3.55000000
C 18.79853644 10 2.84000000
C 18.79853644 10 1.42000000
C 20.02829251 10 0.71000000
C 20.02829251 10 3.55000000
```

Tip

To perform the phonon calculation the atomic coordinates and the cell-vectors of the structure have to be well relaxed, which means that force and stress in the structure have to be extracted from the global minimum energy.

The atomic structure of graphene nanoribbons generated by Device Studio software exhibits optimum lattice vectors and \(C-C\) bond length (1.44 Å).

The generated atomic coordenates file for 8AGNR-supercell could be downloaded here:

The 8AGNR supercell

0_4Gr_Nanoribbon_Rede.hzwfile (`0_4Gr_Nanoribbon_Rede.hzw`

).

0_4Gr_Nanoribbon_Rede.hzw

```
% The number of probes
0
% Uni-cell vector
32.67804859 0.00000000 0.00000000
0.00000000 20.00000000 0.00000000
0.00000000 0.00000000 21.30000000
%Total number of device_structure
80
%Atom site
C 11.41999999 10 2.84000000
C 11.41999999 10 7.09999999
C 11.41999999 10 11.36000000
C 11.41999999 10 15.62000000
C 11.41999999 10 19.87999999
C 11.41999999 10 1.42000000
C 11.41999999 10 5.68000000
C 11.41999999 10 9.93999999
C 11.41999999 10 14.19999999
C 11.41999999 10 18.45999999
C 12.64975607 10 0.71000000
C 12.64975607 10 4.96999999
C 12.64975607 10 9.22999999
C 12.64975607 10 13.48999999
C 12.64975607 10 17.75000000
C 12.64975607 10 3.55000000
C 12.64975607 10 7.81000000
C 12.64975607 10 12.07000000
C 12.64975607 10 16.32999999
C 12.64975607 10 20.58999999
C 13.87951214 10 2.84000000
C 13.87951214 10 7.10000000
C 13.87951214 10 11.36000000
C 13.87951214 10 15.62000000
C 13.87951214 10 19.87999999
C 13.87951214 10 1.42000000
C 13.87951214 10 5.68000000
C 13.87951214 10 9.93999999
C 13.87951214 10 14.19999999
C 13.87951214 10 18.45999999
C 15.10926822 10 0.71000000
C 15.10926822 10 4.96999999
C 15.10926822 10 9.23000000
C 15.10926822 10 13.49000000
C 15.10926822 10 17.75000000
C 15.10926822 10 3.54999999
C 15.10926822 10 7.81000000
C 15.10926822 10 12.07000000
C 15.10926822 10 16.32999999
C 15.10926822 10 20.58999999
C 16.33902429 10 2.84000000
C 16.33902429 10 7.09999999
C 16.33902429 10 11.36000000
C 16.33902429 10 15.62000000
C 16.33902429 10 19.87999999
C 16.33902429 10 1.42000000
C 16.33902429 10 5.68000000
C 16.33902429 10 9.93999999
C 16.33902429 10 14.19999999
C 16.33902429 10 18.46000000
C 17.56878036 10 0.71000000
C 17.56878036 10 4.97000000
C 17.56878036 10 9.23000000
C 17.56878036 10 13.49000000
C 17.56878036 10 17.75000000
C 17.56878036 10 3.55000000
C 17.56878036 10 7.81000000
C 17.56878036 10 12.07000000
C 17.56878036 10 16.33000000
C 17.56878036 10 20.58999999
C 18.79853644 10 2.84000000
C 18.79853644 10 7.10000000
C 18.79853644 10 11.36000000
C 18.79853644 10 15.62000000
C 18.79853644 10 19.87999999
C 18.79853644 10 1.42000000
C 18.79853644 10 5.68000000
C 18.79853644 10 9.93999999
C 18.79853644 10 14.19999999
C 18.79853644 10 18.46000000
C 20.02829251 10 0.71000000
C 20.02829251 10 4.97000000
C 20.02829251 10 9.23000000
C 20.02829251 10 13.49000000
C 20.02829251 10 17.75000000
C 20.02829251 10 3.55000000
C 20.02829251 10 7.81000000
C 20.02829251 10 12.07000000
C 20.02829251 10 16.33000000
C 20.02829251 10 20.58999999
```

## 6.2.2. SCF for 8AGNR

Once the 8AGNR supercell structure is created, we’ll perform the self-consistent calculation. The required input file could be generated using Device Studio with NanoDCAL module simulator, as follows:

In the menu bar of Device Studio, click

**Simulator**➟**NanoDCAL**➟**SCF Calculation**to pop up the interface.Set the parameter k-point Sampling to

**n1 = 1**,**n2 = 1**, and**n3=3**.The Users can choose the calculation parameters related to SCF method, or use the default parameters.

Click

**Generate files**;The Device Studio will create a new folder that contain the

*scf.input*files of**Nanodcal-Crystal**.

The generated input file for 8AGNR supercell could be downloaded here:

8AGNR supercell

scf.inputfile (`8AGNR-supercell.scf.input`

).

Scf input file

```
%%What quantities should be calculated
calculation.name = scf
%Basic setting
calculation.occupationFunction.temperature = 300
calculation.realspacegrids.E_cutoff = 80 Hartree
calculation.xcFunctional.Type = LDA_PZ81
calculation.k_spacegrids.number = [ 1 1 3 ]'
system.centralCellVectors = [[32.678 0 0]' [0 20 0]' [0 0 21.3]']
system.spinType = NoSpin
%Iteration control
calculation.SCF.monitoredVariableName = {'rhoMatrix','hMatrix','totalEnergy','bandEnergy','gridCharge','orbitalCharge'}
calculation.SCF.convergenceCriteria = {1e-04,1e-04,[],[],[],[]}
calculation.SCF.maximumSteps = 200
calculation.SCF.mixMethod = Pulay
calculation.SCF.mixRate = 0.1
calculation.SCF.mixingMode = H
calculation.SCF.startingMode = H
%calculation.SCF.donatorObject = NanodcalObject.mat
%Basic set
system.neutralAtomDataDirectory = '../'
system.atomBlock = 80
AtomType OrbitalType X Y Z
C LDA-DZP 11.42000000 10.00000000 2.84000000
C LDA-DZP 11.42000000 10.00000000 7.10000000
C LDA-DZP 11.42000000 10.00000000 11.36000000
C LDA-DZP 11.42000000 10.00000000 15.62000000
C LDA-DZP 11.42000000 10.00000000 19.88000000
C LDA-DZP 11.42000000 10.00000000 1.42000000
C LDA-DZP 11.42000000 10.00000000 5.68000000
C LDA-DZP 11.42000000 10.00000000 9.94000000
C LDA-DZP 11.42000000 10.00000000 14.20000000
C LDA-DZP 11.42000000 10.00000000 18.46000000
C LDA-DZP 12.64975607 10.00000000 0.71000000
C LDA-DZP 12.64975607 10.00000000 4.97000000
C LDA-DZP 12.64975607 10.00000000 9.23000000
C LDA-DZP 12.64975607 10.00000000 13.49000000
C LDA-DZP 12.64975607 10.00000000 17.75000000
C LDA-DZP 12.64975607 10.00000000 3.55000000
C LDA-DZP 12.64975607 10.00000000 7.81000000
C LDA-DZP 12.64975607 10.00000000 12.07000000
C LDA-DZP 12.64975607 10.00000000 16.33000000
C LDA-DZP 12.64975607 10.00000000 20.59000000
C LDA-DZP 13.87951215 10.00000000 2.84000000
C LDA-DZP 13.87951215 10.00000000 7.10000000
C LDA-DZP 13.87951215 10.00000000 11.36000000
C LDA-DZP 13.87951215 10.00000000 15.62000000
C LDA-DZP 13.87951215 10.00000000 19.88000000
C LDA-DZP 13.87951215 10.00000000 1.42000000
C LDA-DZP 13.87951215 10.00000000 5.68000000
C LDA-DZP 13.87951215 10.00000000 9.94000000
C LDA-DZP 13.87951215 10.00000000 14.20000000
C LDA-DZP 13.87951215 10.00000000 18.46000000
C LDA-DZP 15.10926822 10.00000000 0.71000000
C LDA-DZP 15.10926822 10.00000000 4.97000000
C LDA-DZP 15.10926822 10.00000000 9.23000000
C LDA-DZP 15.10926822 10.00000000 13.49000000
C LDA-DZP 15.10926822 10.00000000 17.75000000
C LDA-DZP 15.10926822 10.00000000 3.55000000
C LDA-DZP 15.10926822 10.00000000 7.81000000
C LDA-DZP 15.10926822 10.00000000 12.07000000
C LDA-DZP 15.10926822 10.00000000 16.33000000
C LDA-DZP 15.10926822 10.00000000 20.59000000
C LDA-DZP 16.33902429 10.00000000 2.84000000
C LDA-DZP 16.33902429 10.00000000 7.10000000
C LDA-DZP 16.33902429 10.00000000 11.36000000
C LDA-DZP 16.33902429 10.00000000 15.62000000
C LDA-DZP 16.33902429 10.00000000 19.88000000
C LDA-DZP 16.33902429 10.00000000 1.42000000
C LDA-DZP 16.33902429 10.00000000 5.68000000
C LDA-DZP 16.33902429 10.00000000 9.94000000
C LDA-DZP 16.33902429 10.00000000 14.20000000
C LDA-DZP 16.33902429 10.00000000 18.46000000
C LDA-DZP 17.56878037 10.00000000 0.71000000
C LDA-DZP 17.56878037 10.00000000 4.97000000
C LDA-DZP 17.56878037 10.00000000 9.23000000
C LDA-DZP 17.56878037 10.00000000 13.49000000
C LDA-DZP 17.56878037 10.00000000 17.75000000
C LDA-DZP 17.56878037 10.00000000 3.55000000
C LDA-DZP 17.56878037 10.00000000 7.81000000
C LDA-DZP 17.56878037 10.00000000 12.07000000
C LDA-DZP 17.56878037 10.00000000 16.33000000
C LDA-DZP 17.56878037 10.00000000 20.59000000
C LDA-DZP 18.79853644 10.00000000 2.84000000
C LDA-DZP 18.79853644 10.00000000 7.10000000
C LDA-DZP 18.79853644 10.00000000 11.36000000
C LDA-DZP 18.79853644 10.00000000 15.62000000
C LDA-DZP 18.79853644 10.00000000 19.88000000
C LDA-DZP 18.79853644 10.00000000 1.42000000
C LDA-DZP 18.79853644 10.00000000 5.68000000
C LDA-DZP 18.79853644 10.00000000 9.94000000
C LDA-DZP 18.79853644 10.00000000 14.20000000
C LDA-DZP 18.79853644 10.00000000 18.46000000
C LDA-DZP 20.02829251 10.00000000 0.71000000
C LDA-DZP 20.02829251 10.00000000 4.97000000
C LDA-DZP 20.02829251 10.00000000 9.23000000
C LDA-DZP 20.02829251 10.00000000 13.49000000
C LDA-DZP 20.02829251 10.00000000 17.75000000
C LDA-DZP 20.02829251 10.00000000 3.55000000
C LDA-DZP 20.02829251 10.00000000 7.81000000
C LDA-DZP 20.02829251 10.00000000 12.07000000
C LDA-DZP 20.02829251 10.00000000 16.33000000
C LDA-DZP 20.02829251 10.00000000 20.59000000
end
```

The keywords were specified in the previous tutorial. It is can be check in keywords-scf.

Run the *scf.input* file from the **Device** (`8AGNR-supercell.scf.input`

).
The *NanodcalObject.mat* files are required for the band structure calculation.

## 6.2.3. The electronic band structure

Now, we can verify the physical properties of 8AGNR by calculating the electronic band structure.

In the Device Studio navigate to the menu bar, click on

**Simulator**➟**NanoDCAL**➟**Analysis**to set physical quantityNavigate to the

**Analysis**panel (left side) ➟ select**BandStructure**➟ click on the*Right arrow*button to transfer**BandStructure**calculator to the**Calculation Selected**panel.The

**Symmetry k-points**and**Number of k-points**are show on the right side, click on**path**to see the k-paths along the high-symmetry point in the Brillouin zone and press**Apply**. Returning to the**Analysis**window, click**Generate files**to create the*BandStructure.input*file.

Band structure input file:

```
system.object = NanodcalObject.mat
calculation.name = bandstructure
calculation.bandStructure.symmetryKPoints = {'G','Z'}
calculation.bandStructure.coordinatesOfTheSymmetryKPoints = [0 0 0;0 0 0.5]'
calculation.bandStructure.numberOfKPoints = 200
%calculation.bandStructure.plot = true
calculation.control.xml = true
```

The user can visit the previous tutorial to run the *BandStructure.input* file (`8AGNR-bands.input`

).
Once the calculation finish the .xml or .mat data files will return to the **Nanodcal-crystal** folder, and could be loaded in Device Studio or MATLAB platform to plot the band structures.

Band structure analyses of 8AGNR supercell

The AGNRs in subfamilies of \(N=3p+2\) (such as \(N = 8, 11, 14, ...\)) present a low bandgap [WSG]. The unsatisfied valence at the edge of the 8AGNR produces states around the Fermi energy owing to unpaired electrons black lines in Fig. 6.2.3 (a).

Note

The 8AGNR passivated with hydrogen has a small bandgap of about 0.52 eV and exhibits a nonlinear dispersion around the \(\Gamma\) point, as shown Fig. 6.2.3 (b).

**Users might redo the calculation passivating the edge of 8AGNR structure with H atoms as an exercise.**

To perform the calculation download the input file `8AGNR-H-supercell.scf.input`

## 6.2.4. Hessian matrix calculation

The Hessian matrix is calculated using a finite difference scheme, where each unique atom of the symmetry is displaced along all cartesian directions. It is a well-known approach called frozen phonons.

FIn the frozen phonon aproach the monochromatic perturbation is frozen with a finite amplitude in the system. The Fourier transform of force constants at the wave vector is calculated from finite differences of forces induced on all the atoms of the supercell by the monochromatic perturbation.

Once the Hamiltonian from the 8AGNR supercell unpassivated is calculated (*NanodcalObject.mat*), we can move to Hessian matrix calculation.

The required input file for this procedure could be generated using Device Studio with NanoDCAL simulator module, as follows:

In the menu bar of Device Studio, click on

Simulator➟NanoDCAL➟Analysisto pop up the interface shown in Fig. 6.2.4.

Navigate to the

**Analysis**panel (left side) ➟ select**Hessian**➟ click on the*Right arrow*button to transfer**Hessian**matrix calculator to the**Calculation Selected**panel.Set the values according to the requirements of the atomic structure and click

**Generate files**.

Attention

In this tutorial, the default values are enough to describe the structure. However, the user should note that more acurate calculations can be peformed by NanoDCAL code. Please, look at the documentation Input Reference

Now, we are ready to start the Hessian matrix calculation. Before doing that, we’ll briefly present the simulation parameters specified in the generated files.

Hessian matrix input file

```
system.object = NanodcalObject.mat
calculation.name = hessian
calculation.hessian.delta = 0.02
calculation.hessian.order = 2
calculation.hessian.primitiveCellVectors = [[ 32.678 0 0]' [0 20 0]' [0 0 12.78]']
calculation.control.xml = true
```

`calculation.hessian.delta`

This input parameter specifies that a small displacement of atomic position will be used to calculate the hessian matrix by numerical derivatives of forces. The default value is 0.03 au;`calculation.hessian.order`

The number of points used to calculate the derivative of force numerically. The possible values are 2, 3, or 5. The default value is 3 au.`calculation.hessian.primitiveCellVectors`

Hessian matrix is normally calculated with a large supercell (the current cell), and this parameter defines a smaller cell. The Hessian matrix being consistent with the smaller cell will be given after the calculation.

*Run* the simulation with `hessian-matrix.input`

file.

After the calculation finish, the following output files are generated: *CalculatedResults.mat*, *log.txt*, *NanodcalObject.mat* and *Hessian.mat*.

The *Hessian.mat* file is used to calculate the phonon frequencies at different points in the Brillouin zone.
After this step we’ll calculate the phonon band structure and phonon density of states, which provides information about how phonon frequencies vary with wave vector, and provides insights into the material’s vibrational properties.

## 6.2.5. Phonon band structure and phonon density of states calculation

For the phonon band structure and phonon density of states calculation the user should repeat the procedure discribed in Hessian matrix calculation section as following:

As shown in Fig. 6.2.4, navigate to the

**Analysis**panel (left side) ➟ select**PhononBandStructure**➟ click on the*Right arrow*button to transfer**PhononBandStructure**calculator to the**Calculation Selected**panel.The

**Symmetry k-points**and**Number of k-points**are show on the right side panel, click on**path**to see/edit the K-paths along the high-symmetry points in the Brillouin zone and press**Apply**.

Returning to the

**Analysis**window, navigate to the**Analysis**panel (left side) ➟ select**PhononDensityofStates**➟ click on the*Right arrow*button to transfer**PhononDensityofStates**calculator to the**Calculation Selected**panel.Set the

**k-points sampling**to**n1 = 1**,**n2 = 1**, and**n3=3**;Set the

**Energy range****0**to**100****meV**;In the

**Projected**keep uncheck.

click

**Generate files**to create the*PhononBandStructure.input*and*PhononDensityofStates.input*files.

we’ll briefly present the simulation parameters specified in the generated files.

Phonon band structure input file:

```
system.object = NanodcalObject.mat
calculation.name = phononBandStructure
calculation.phononBandStructure.symmetryKPoints = {'G','Z'}
calculation.phononBandStructure.coordinatesOfTheSymmetryKPoints = [0 0 0;0 0 0.5]'
calculation.phononBandStructure.numberOfKPoints = 200
calculation.control.xml = true
```

Phonon density of states input file:

```
system.object = NanodcalObject.mat
calculation.name = 'pdos'
calculation.phononDensityOfStates.qSpaceGridNumber = [1 1 5]'
calculation.phononDensityOfStates.energyRange =[0,100] meV
calculation.phononDensityOfStates.numberOfEnergyPoints = 101
calculation.phononDensityOfStates.whatProjected = 'None'
%calculation.phononDensityOfStates.plot = true
calculation.control.xml = true
```

The user should be able to *Run* the phonon band structure calculations by `phBands-8AGNR.input`

and `phDos-8AGNR.input`

files following the procedure previously described at run BS.

After the calculation is finished, the following output files are generated:

For phonon band structure calculation:

*CalculatedResults.mat*,*log.txt*,*PhononBandStructure.mat*,*PhononBandStructure.xml*.For phonon density of states calculation:

*CalculatedResults.mat*,*log.txt*,*PhononDensityOfStates.mat*,*PhononDensityOfStates.xml*.

Those results could be plotted with MATLAB or loaded in Device Studio software:

In menu bar of Device Studio, click on

**Simulator**➟**NanoDCAL**➟**Analysis Plot**➟ import*PhononBandStructure.xml*or*PhononDensityOfStates.xml*file.

Phonon band structure analysis

The phonon spectrum for the 8AGNR supercell is shown in Fig. 6.2.5.

The key features and interpretations of the phonon band structure of the 8AGNR supercell include:

Vibrational modes and frequencies: The phonon spectrum provides information about the frequencies of vibrational modes in the material. This helps in understanding the types of vibrational motions (e.g., stretching, bending) and their associated energies.

Thermodynamic stability: The system has no imaginary frequency at \(\Gamma\)-point (G) meaning that the 8AGNR structure is stable.

Thermal properties: It is crucial for calculating thermal properties such as specific heat capacity, thermal conductivity, and heat capacity at constant volume or pressure. This information is essential for understanding how heat is conducted and stored in a material.

Phonon Dispersion: The horizontal axis of the band structure plot represents the crystal’s reciprocal lattice vectors (typically denoted as k-points), while the vertical axis represents the phonon frequencies. Different branches or curves correspond to different vibrational modes, and their slopes provide information about the speed or group velocity of the phonons.

Thermal expansion: Information about the low-frequency vibrational modes can be used to understand the thermal expansion behavior of the material.

Density of states analysis

The phonon density of states calculations are valuable for understanding the vibrational behavior and thermal properties of materials, providing crucial insights into their physical characteristics and potential applications.

The key features and interpretations of the phonon density of states of the 8AGNR supercell include:

The phonon density of states in conjunction with phonon dispersion relations gives a comprehensive picture of the vibrational properties in different directions of the crystal lattice.

Changes in the phonon density of states, such as the appearance of new peaks or shifts in peak positions, can indicate phase transitions in the material.

Phonon lifetimes: By examining the width of peaks in the phonon density of states, one can estimate the phonon lifetimes. Longer lifetimes are associated with sharper peaks, indicating more stable vibrational modes.

## 6.2.6. Calculation of thermoelectric properties

After completing the hessian-matrix calculations, it is time to compute the thermoelectric properties. The calculation of thermoelectric properties, includes the \(ZT\), the Seebeck coefficient \(S\), the electricconductance \(G\), thermal conductivity \(k_{ph}\) with both lattice and electronic thermal conductance \(k_{e}\) contributions.

The *ZT.input* file can be generated using Device Studio, as we shown in Fig. 6.2.7:

Select the

*8AGNR-supercell.hzw*structure, go the the menu bar, click on**Simulator**➟**NanoDCAL**➟**Analysis**.Navigate to the

**Analysis**panel (left side) ➟ select**ZT**➟ click on the*Right arrow*button to transfer**ZT**calculator to the**Calculation Selected**panel.

The

**ZT**options are shown in the right side of the**Analysis**window. Edit the parameters to fit the studied system:

In

SystemObectFiles➟primitive cellclick on the bottom to navegate and select theNanodcalObject.matfiles generated during the scf and Hessian matrix calculation for 8AGNR.Go to the

k-point samplingsetn1=1,n2=1, andn3=40.In the

q-point samplingsetn1=1,n2=1, andn3=40, and clickGenerate files.

The Device Studio will create new folder that contain the *ZT.input* file.
The generated input file could be downloaded here: (`ZT.input`

).
Before starting the calculation, we’ll briefly present the simulation parameters specified in the *ZT* file.

ZT input file:

```
calculation.name = 'ZT'
calculation.ZT.systemObjectFiles = {'../scf-8AGNR/NanodcalObject.mat','../hessian-matrix/NanodcalObject.mat'}
calculation.ZT.temperature = [50:50:300]
calculation.ZT.chemicalPotentialEnergy =[-1:0.01:1] eV
calculation.ZT.whatDirection = 3
calculation.ZT.kSpaceGridNumber = [1 1 40]'
calculation.ZT.qSpaceGridNumber = [1 1 40]'
%calculation.ZT.plot = true
calculation.control.xml = true
```

The keywords specify the following:

`calculation.ZT.systemObjectFiles`

It is supposed that the calculations for the electronic Hamiltonian (SCF) and the phononic Hamiltonian (Hessian) have already been completed seperatly. This parameter gives in order the file names of the calculated Nanodcal objects for the electronic and the phononic calculations, or the calculated transmissions for the electron and the phonon.

`calculation.ZT.temperature`

The electronic and phononic temperature of the system, in the unit of Kelvin. The value of Boltzmann constant \(k = 8.61733326 \times 10^{-5}\) eV/K, \(k = 27.2113834\) eV/Hartree, \(k = 3.1668151 \times 10^{-5}\) Hartree/Kelvin.

`calculation.ZT.chemicalPotentialEnergy`

The thermoelectric figure of merit \(ZT\) and other related properties will be calculated respectively when the chemical potential energy is set to those input values. Note that the chemical potential energy is measured from the calculated Fermi energy and is in the unit of meV.

`calculation.ZT.whatDirection`

The transmission direction of the system, which need to simulate. It is necessary to choose one and only one of the three central cell vectors as the lead direction (i.e. transmission direction); the transverse plane is spanned by the other two directions.

`calculation.ZT.kSpaceGridNumber`

The number of small \(k-\) space grids in each direction, which are used in the electronic \(k-\) space integration. The default value: the value of calculation.k_spacegrids.number which was used in the electronic Hamiltonian calculation.

`calculation.ZT.qSpaceGridNumber`

The number of small \(q-\) space grids in each direction, which are used in the phononic \(q-\) space integration.

Now we can *Run* the simulation with the \(ZT\) input file.
After the calculation, the user should perform the analyses of the results from the produced ZT dataset (*.xml* and *.mat*) which can be plotted using MATLAB or Device Studio as follows:

In menu bar of Device Studio, click on

**Simulator**➟**NanoDCAL**➟**Analysis Plot**➟ import*ThermoelectricProperties.xml*file.select various thermoelectric properties \(ZT\), \(S\), \(G\), \(k_{ph}\), \(k_{e}\) at different temperatures or different chemical potentials.

Thermoelectric properties as a function of chemical potential

The thermoelectric properties of a material can be influenced by its chemical potential, which is a measure of the energy required to add or remove a particle (e.g., an electron) from the material.

The 8AGNR shows that the thermoelectric figure of merit, \(ZT\) , exhibits a peak at the subband edges that increase with the temperature increase, as shown in Fig. 6.2.8. The \(ZT\) value reached 0.08 at 300K, which is in agreement with the results of [CJK].

On the Device Studio mudule change the **plot type** for conductance \(G\) and Seebeck coefficient \(S\) features.
As we can see in Fig. 6.2.9 (a) and (b) for all the different temperature values the conductance has a two broad region that rise around the Fermi-level.
Such a small change is reflected in the low Seebeck coefficient.

The thermal conductivity \(k_{e}\) and lattice thermal conductance (phononic) \(k_{ph}\) increased with increasing temperature , as we can see in Fig. 6.2.10 (a) and (b).

Tip

As an additional task the users should be able to:

Explore the thermoelectric properties of this 8AGNR as a function of temperature.

Calculate the thermoelectric properties for 8AGNR passivated with hydrogen atom.

Explore thermoelectric properties for different ribbon width \(N=3p\), \(N=3p+1\) and \(N=3p+2\).

Chen, D. Liu, W. Duan, H. Guo. Photon-assisted thermoelectric properties of noncollinear spin valves Phys. Rev. B (2013) p. 085427.

Yao; H. Zhang; F. Kong; A. Hinaut; R. Pawlak; M. Okuno; R. Graf; P. N. Horton; S. J. Coles; E. Meyer, L. Bogani, M. Bonn, H. I. Wang, K. Müllen, A. Narita N=8 Armchair Graphene Nanoribbons: Solution Synthesis and High Charge Carrier Mobility Angew. Chem. Int. 62 (2023) p. 202312610.

Chen, T. Jayasekera, A. Calzolari, K. W. Kim, M. B. Nardelli Thermoelectric properties of graphene nanoribbons, junctions and superlattices J. Phys.: Condens. Matter 22 (2010) p. 372202.