Phonons: Bandstructure, thermal transport, thermo-electrics Tutorial Version 2014.0
Phonons: Bandstructure, thermal transport, thermo-electrics: Tutorial Version 2014.0 Copyright 2008 2014 QuantumWise A/S Atomistix ToolKit Copyright Notice All rights reserved. This publication may be freely redistributed in its full, unmodified form. No part of this publication may be incorporated or used in other publications without prior written permission from the publisher.
TABLE OF CONTENTS 1. Introduction... 1 2. Phonon bandstructure of a graphene nanoribbon... 2 Setting up the geometry... 2 Defining the phonon calculation... 3 Analyzing the results... 4 Algorithmic details of the phonon calculator... 8 3. Calculating electrical and heat transport for a graphene nanoribbon... 9 Setting up the graphene nanoribbon device... 9 Defining the transmission calculations... 9 Analyzing the results... 10 Bibliography... 13 iii
CHAPTER 1. INTRODUCTION The purpose of this tutorial is to illustrate the use of the ATK phonon module. The tutorial is applied to small toy problems such that the calculations are very fast. In the first chapter you will calculate the phonon bandstructure and density of states, and learn about how these modules work. In the second chapter you will study a device system and calculate the electronic and phonon thermal transport coefficients, as well as the electrical transport coefficients. From those coefficients you obtain the thermoelectric figure of merit ZT. Note You will primarily use the graphical user interface Virtual NanoLab (VNL) for setting up and analyzing the results. To familiarize yourself with VNL, it is recommended to go through the VNL Tutorial. The underlying calculation engines for this tutorial is ATK-Classical and ATK-SE. A complete description of all the parameters, and in many cases a longer discussion about their physical relevance, can be found in the ATK Reference Manual. In order to run this tutorial, you must have a license for ATK-SE. If you do not have one, you may obtain a time-limited demo license by contacting QuantumWise via our website. You are now ready to begin the tutorial. 1
CHAPTER 2. PHONON BANDSTRUCTURE OF A GRAPHENE NANORIBBON SETTING UP THE GEOMETRY Start VNL and create a new project and give it a name, then select it and click Open. Launch the Builder by clicking the Builder icon on the toolbar. In the builder, click Add From Plugin Nanoribbon. Remove the check box from Passivate dangling bonds with hydrogen. Select Armchair edge. Select the structure 8 atoms wide. Press the Build button to build the structure. In order to make sure that the phonon module can detect that the structure is one dimensional, you need to have ~ 7 Å vacuum on both sides of the structure. To this end open Bulk Tools Lattice Parameters... Increase A-x to 20 Å. Increase B-y to 30 Å. and close down the widget. Next Open Coordinate Tools Center, and press Apply, to center the nanoribbon in the unit cell. 2
Next send the structure to the Script Generator, by using the "Send To" icon right-hand corner of the Builder window. in the lower DEFINING THE PHONON CALCULATION In the following you will optimize the geometry and calculate the Phonon bandstructure and DOS of the nanoribbon using the Tersoff potential. In the Script Generator Add a "New Calculator" Add an "Optimization>OptimizeGeometry" Add an "Analysis>PhononBandstructure" Add an "Analysis>PhononDensityOfStates" Change the output file to armchair.nc ADJUSTING SCRIPT COMPONENTS To set up the calculation parameters, make the following changes: Double-click the New Calculator and select the ATK-Classical calculator and the Tersoff_C_2010 potential. Note that depending on the elements present in your configuration you will be able to choose among several potentials. A literature reference is also shown, giving you the possibility to check exactly how the potentials have been generated. It is very important that the potential you will use has been generated for a similar system or for similar conditions as your system. 3
In this case you can immediately see that the Tersoff_C_2010 potential is well suited for your phonon analysis: "Optimized Tersoff and Brenner empirical potential parameters for lattice dynamics and phonon thermal transport in carbon nanotubes and graphene". Double-click the OptimizeGeometry and set maximum force to 0.01 ev/å. Double-click the PhononBandstructure and set Points pr. segment to 100. Double-click the PhononDensityOfStates, uncheck sync, and set nc=51. Transfer the script to the Job Manager (again using the "Send To" icon) and run the calculation. ANALYZING THE RESULTS The generated output file armchair.nc, visible under Project Files in the main VNL window, now contains a PhononBandstructure object. Select the PhononBandstructure object on the LabFloor and press Show 2D Plot... in the plugin panel. Zoom into the low frequency area to get the figure below. 4
Figure 2.1: Phonon bandstructure of the armchair nanoribbon. Note the negative bands arising from stress in the nanoribbon which makes it unstable perpendicular to the plane. ORIGIN OF NEGATIVE PHONON BANDS The negative bands seen in the phonon spectrum show that the structure is at a saddle point and can gain energy by further relaxing. The problem is that you only performed a force optimization and not a stress optimization, thus there is a large stress in the structure and it would like to release the stress by relaxing perpendicular to the graphene plane. RELAXING THE STRESS IN THE CONFIGURATION In order to relax the stress in the configuration, reopen the Script Generator. Tip All open tools are available from the Windows menu at the top of each VNL tool. Next open the OptimizeGeometry block. Set Maximum stress to 0.0001 ev/å. Remove check from z under constrain cell. 5
Press OK to close the widget, transfer the script to the Job Manager and run the calculation. When you open the phonon bandstructure you will see that the the negative bands are now gone. Figure 2.2: Phonon bandstructure of the armchair nanoribbon after stress relaxation. Note that there are no longer negative bands. 6
THE DENSITY OF STATES The armchair.nc also contains the phonon density of state (DOS) of the armchair nanoribbon with and without stress relaxation. Below is shown the phonon DOS of the stress relaxed system. Figure 2.3: Phonon density of states of the armchair nanoribbon after stress relaxation. From the Phonon DOS it is possible to calculate the entropy of the system. In ATK this is implemented in the utility function entropy() of the PhononDensityOfStates. The script below evaluates the entropy and uses that to get the free energy of the system. # Read the last configuration in the output file configuration = nlread('armchair.nc', BulkConfiguration)[-1] # Read the last phonon DOS in the output file phonon_dos = nlread('armchair.nc', PhononDensityOfStates)[-1] # Calculate the total energy total_energy = TotalEnergy(configuration) e = total_energy.evaluate() # Define the temperature t = 300*Kelvin # Calculate the entropy s = phonon_dos.entropy(t) # Print results print 'Energy = ', e print 'Entropy = ', s print 'Free energy at 300K = ', e - t*s Execute the script by dropping it on the Job Manager, and get the output Energy = -124.297968287 ev Entropy = 3.33141143492 mev/k Free energy at 300K = -125.297391718 ev 7
ALGORITHMIC DETAILS OF THE PHONON CALCULATOR The phonons are calculated from the dynamical matrix of the system. The dynamical matrix is the second derivative of the system, corresponding to the first derivative of the forces. The first derivative of the forces are calculated using a finite difference scheme, where the system is displaced along each degree of freedom in the system, also called frozen phonon calculations. Before the calculation of a phonon quantity, it is checked if the configuration contains a dynamical matrix. If this is not the case the dynamical matrix will be calculated and stored on the object. When the configuration is subsequently stored, the dynamical matrix will be stored together with the configuration. The details of the calculation of the dynamical matrix is controlled by DynamicalMatrixParameters object, which can be given as a parameter of the calculator. dynamical_matrix_parameters = DynamicalMatrixParameters(repeats = (1,1,5), atomic_displacement = 0.01*Ang) potentialset = Tersoff_C_2010() calculator = TremoloXCalculator(parameters=potentialSet, dynamical_matrix_parameters=dynamical_matrix_parameters) In order to take into account the dynamical matrix entries with neighbour cells the system is repeated, controlled by the repeat parameter. If the value is automatic, the system will try to determine the relevant repetition and this is reported in the output file. For the current system it is reported Phonon: Automatically detected repetitions = [1 1 5] The atomic_displacement parameter controls the atomic displacements in the frozen phonon calculation. K-POINTS USED FOR ATK-DFT AND ATK-SE It is also possible to use the phonon module with ATK-DFT and ATK-SE. In this case be aware that the k-point sampling of the calculator will be used for the repeated cell that is used for calculating the dynamical matrix. I.e. in the above example let us say a k-point sampling of (1, 1, 21) is used for optimization of the geometry. For the dynamical matrix calculation you will use a geometry that is repeated 5 times in the z direction. Thus, to obtain the same accuracy for the dynamical matrix calculation, use fewer k-points in the z-direction and thereby save time. The following script shows how to change the k-point sampling of the calculator. # Change k-point sampling of a calculator calculator = bulk_configuration.calculator() numerical_accuracy_parameters = calculator.numericalaccuracyparameters() new_numerical_accuracy_parameters = numerical_accuracy_parameters(k_point_sampling = (1, 1, 5)) new_calculator = calculator(numerical_accuracy_parameters=new_numerical_accuracy_parameters) bulk_configuration.setcalculator(new_calculator) 8
CHAPTER 3. CALCULATING ELECTRICAL AND HEAT TRANSPORT FOR A GRAPHENE NANORIBBON In this section we will calculate the phonon transmission coefficient for the armchair nanoribbon from which the phonon thermal conductance can be obtained. This will be combined with a calculation of the electron transmission coefficient from which the conductance, Peltier coefficient and electron thermal conductance can be obtained. The different parameters are combined for calculating the thermoelectric figure of merit, ZT. The methodology for the calculations are described in Ref. [1]. SETTING UP THE GRAPHENE NANORIBBON DEVICE The first step is to build a device geometry from the relaxed armchair nanoribbon. In the VNL main window select the last configuration in the armchair.nc file and drag and drop the configuration onto the builder. In the builder perform the following operations Bulk Tools Repeat: repeat 5 times in C direction. Device Tools Device From Bulk...: Default values are fine, press OK Next send the structure to the Script Generator. DEFINING THE TRANSMISSION CALCULATIONS In the following you will setup the phonon and electron transmission calculations. For the phonon calculation you will use the Tersoff potential while using a carbon tight-binding model for the electron transmission calculation. Add a "New Calculator" Add a "Analysis>PhononTransmissionSpectrum" Add a "New Calculator" Add a "Analysis>TransmissionSpectrum" Change the output file to armchair_device.nc 9
ADJUSTING SCRIPT COMPONENTS To set up calculation parameters, make the following changes: Double-click the New Calculator and select the ATK-Classical calculator and the Tersoff potential. Double-click the PhononTransmissionSpectrum and set the energy range to 0 ev to 0.5 ev Double-click the New Calculator and select the ATK-SE: Slater-Koster calculator and choose the Hancock.C pppi basis set. No changes need to be made to the TransmissionSpectrum Transfer the script to the Job Manager and run the calculation. ANALYZING THE RESULTS Now inspect the result file armchair_device.nc in the VNL main window. You may select the Phonon and the electron transmission spectrum and plot them using the Transmission Analyzer... or Show 2D plot analysis plugins. Figure 3.1: Phonon transmission spectrum of the armchair nanoribbon. 10
Figure 3.2: Electron transmission spectrum of the armchair nanoribbon zoomed into the energy range -1 to 1 ev. THE THERMO-ELECTRIC FIGURE OF MERIT In this section you will calculate the linear response transport coefficients of an applied voltage difference ( ) or temperature difference ( ) between the two electrodes. The coefficients are The conductance, The Peltier coefficient, The Seebeck coefficient, The heat transport coefficient of electrons,, and phonons, where is the heat current. From these coefficients you can obtain the thermoelectric figure of merit which quantifies how efficient a temperature difference can be converted into a voltage difference in a thermoelectric material. These linear response coefficients are calculated by the Thermoelectric Coefficients plugin. To perform the calculation, select the PhononTransmissionSpectrum and the ElectronTransmissionSpectrum simultanously and press the Calculate button of the plugin. 11
For an undoped nanoribbon, the Fermi level is in the middle of the band gap. To get a significant Peltier coefficient the Fermi level must be at a band edge, corresponding to a doped nanoribbon. The plugin allows for shifting the Fermi level, and Figure 3.3 shows the result for ev, corresponding to an n-doped nanoribbon. Figure 3.3: The linear response electron and phonon transport coefficients for a Fermi level shift of 0.08 ev, corresponding to a Fermi level at the conduction band edge. The Thermoelectric Coefficients plugin allows you to plot the different thermoelectric coefficients as a function of the energy. For example, the figure below reports the figure of merit ZT at 300K as a function of the energy. The peaks corresponding to the band edges are clearly visible. Figure 3.4: ZT at 300 K for the armchair nanoribbon. 12
BIBLIOGRAPHY [1] T. Markussen, A.-P. Jauho, M. Brandbyge, Phys. Rev. Lett., 103, 055502, 2009. 13