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:

\[ZT=\frac{G S^{2}T}{k_{e}k_{ph}},\]

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.

../../_images/agnr-supercell.svg

Fig. 6.2.1 The 8AGNR supercell \(1\times 1\times 5\) reconstruction.

The calculation steps are:

  1. Build:

    • 8AGNR unit cell;

    • Expand the primitive 8-acGNR unit cell.

  2. Self-consistent calculation;

  3. Band structure calculation;

  4. Hessian matrix calculation;

  5. Phonon band and density of states calculation;

  6. 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:

  1. 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.

  2. 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.

Tabs, example output

Fig. 6.2.2 Graphical user interface of Device Studio to build and expand the unit cell.

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:

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:

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:

  1. 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.

  2. 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:

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.

  1. In the Device Studio navigate to the menu bar, click on Simulator ➟ NanoDCAL ➟ Analysis to set physical quantity

    • Navigate 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

Tabs, example output

Fig. 6.2.3 Electronic band structure of 8AGNR (a) without -H and (b) with -H.

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.

  1. 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 ➟ Analysis to pop up the interface shown in Fig. 6.2.4.

  1. 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.

  2. Set the values according to the requirements of the atomic structure and click Generate files.

../../_images/hessian-settings.svg

Fig. 6.2.4 The Hessian matrix module simulator.

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:

  1. 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.

  2. 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.

  3. 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.

../../_images/phBS-8AGNR.svg

Fig. 6.2.5 Phonon band structure of 8AGNR supercell of \(1\times 1\times 5\) reconstruction.

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.

../../_images/phdos-8agnr.svg

Fig. 6.2.6 Phonon density of states of 8AGNR supercell of \(1\times 1\times 5\) reconstruction.

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:

  1. 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.

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

  • In SystemObectFiles ➟ primitive cell click on the bottom to navegate and select the NanodcalObject.mat files generated during the scf and Hessian matrix calculation for 8AGNR.

  • Go to the k-point sampling set n1=1, n2=1, and n3=40.

  • In the q-point sampling set n1=1, n2=1, and n3=40, and click Generate files.

../../_images/zt.svg

Fig. 6.2.7 The ZT module simulator.

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].

../../_images/zt-mu.svg

Fig. 6.2.8 The \(ZT\) as a function of chemical potential for different values os temperature.

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.

../../_images/G-S-mu.svg

Fig. 6.2.9 The Conductance (a), and Seebeck coefficient (b) as a function of chemical potential for different values os temperature.

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).

../../_images/kp-ke.svg

Fig. 6.2.10 The \(k_{e}\) (a), and \(k_{ph}\) (b) as a function of chemical potential for different values os temperature.

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 \(N=3p+2\).

[CDH]

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

[YZH]

  1. 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.

[CJK]

  1. 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.