LanTraP: Tutorial Cases

Transcription

1 LanTraP: Tutorial Cases Kyle Conrad, Jesse Maassen, Mark Lundstrom This document is a tutorial for using the thermoelectric transport calculator tool. The purpose of these examples is to show how to use the tool, compare the results of simple electron/phonon dispersions versus accurate full band descriptions, and to demonstrate the accuracy of the tool with published results. For more information about the input parameters used to run the tool, see the user manual. All of the examples that are covered in this tutorial are available for download at under supporting documents. Table of Contents A. Electron Transport... 2 A.1 3D Parabolic Conduction Band Ballistic Transport... 2 A.2 3D Parabolic Conduction Band Diffusive Transport... 3 A.3 3D Parabolic Valence Band... 4 A.4 3D Ellipsoidal Conduction Band... 5 A.5 2D Parabolic Conduction Band... 6 A.6 1D Parabolic Conduction Band... 7 A.7 Full Dispersion of Si... 8 A.8 Full Dispersion of Bi2Te B. Phonon Transport B.1 Debye Phonon Dispersion B.2 Full Phonon Dispersion of Si References Appendix A: 3D Parabolic Conduction Band - Ballistic Case Appendix C: 3D Parabolic Valence Band Appendix F: 3D Ellipsoidal Conduction Band Appendix D: 2D Parabolic Conduction Band Appendix E: 1D Parabolic Conduction Band Appendix G: Full Dispersion of Si Appendix H: Full Dispersion of Bi2Te

2 A. Electron Transport Download and unzip 'electron_dispersions.zip' from under the supporting documents at for the files used in these examples. A.1 3D Parabolic Conduction Band Ballistic Transport 1. Load Data Slide 1.1. Select 'Upload' from the drop down box and navigate to the 'Ek_3D_conduction.txt' file Review the data in the 'Data File' text box. Be sure that nothing extra has been typed in the box. If there is text, be sure that it is always preceded by a % so that it is treated as a comment and not read by the tool Select 'TE & Modes' from the 'What to calculate' drop down menu Click the 'Modes Options' button in the lower right corner to continue. 2. Modes Options 2.1. The Monkhurst-Pack k-grid?' boolean should be selected as yes From the 'Dimensionality' drop down box select Type '0.5' into Lx, Ly, and Lz. These are the dimensions of the simulation box in nm Select '51' for the number of kx, ky, and kz points Select 1 for the Number of Bands Select 'Z' for the 'Transport Direction'. Note that transport may be in any direction and produce the same results, since this example is an isotropic conduction band Select '2' for the 'Spin Degeneracy' Specify ' -1 ev' for 'Emin', '0.001 ev' for 'de', and '1 ev' for 'Emax'. These three boxes display the units (ev) of the numbers being entered Click on the 'TE Options' button to move to the next slide. 3. TE Options 3.1. Select electron from the 'Particle' drop down menu Set the 'Temperature' to 300K From the 'Transport Type' drop down select 'Ballistic' Specify 'Ef min (ev)' to be '-0.5 ev', 'delta Ef (ev)' to be '0.001 ev', and 'Ef Max (ev)' to be '0.5 ev' Click on the 'Simulate' button in the lower right corner.

3 4. Results 4.1. After clicking the simulate button, the screen will show what the tool is currently calculating. Upon completion the 'Results' slide will appear, with plots of the outputs Distribution of modes the distribution of modes for a parabolic conduction band (page 23 of [1]) is given by: where m* is the distribution of modes effective mass, E C is the minimum energy of the conduction band, A is the cross sectional area of the conductor and H(x) is the Heaviside function. As we can see from Figure 1, the distribution of modes is linear with a slope of. Note that increasing the k-point density will lead to a smoother curve. Figure 1. The distribution of modes for a 3D parabolic conduction band All the thermoelectric properties are shown in Appendix A. 5. Download the data 5.1. Download the 'Modes Table' and 'TE Table' from the tool by selecting the download option, to the right of the 'Results' drop down menu In MATLAB, by using the command load('tetable.txt'), the text file data can be loaded as a matrix for plotting and/or data manipulation. A.2 3D Parabolic Conduction Band Diffusive Transport 1. Follow the 3D Parabolic Conduction Band - Ballistic tutorial to step 4 (TE Options) 2. TE Options 2.1. Select electron from the 'Particle' drop down menu.

4 2.2. Set the 'Temperature' to 300K From the 'Transport Type' drop down select 'Diffusive' Specify 'Ef min' to be '-0.5 ev', 'delta Ef' to be '0.001 ev', and 'Ef Max' to be '0.5 ev' Set 'MFP CB' to '10' (the units are nanometers) Set 'CB scattering parameter' to '0' (case of acoustic phonon scattering) Set 'Conduction Band Minimum' to '0 ev' Set 'Valence Band Maximum' to '-1 ev'. Since there are no electronic states below the conduction band edge (0 ev), the user can set this input option to anything less than 0 ev. In this case, 'MFP VB' and 'VB scattering parameter' can take any value, and will not affect the calculation Click on the 'Simulate' button in the lower right corner. 3. Results 3.1. After clicking the simulate button, the screen will show what the tool is currently calculating. Upon completion the 'Results' slide will appear, with plots of the outputs Distribution of modes The modes will be the same as those for the 3D Parabolic Conduction Band - Ballistic Case All the thermoelectric properties are shown in Appendix B. 4. Easier use 4.1. If the Ballistic results were calculated in the same instance of the tool, rather than going back to the beginning select 'TE Options' and then make the appropriate changes If the distribution of modes for the 3D Conduction band have been calculated before and are saved, then simply upload the modes data file rather than the dispersion relation and select 'TE' from 'What to calculate'. This will take you straight to the 'TE Options Slide' where all options can be set. A.3 3D Parabolic Valence Band 1. Load Data Slide - select 'Ek_3D_valence.txt' as the file to upload. 2. Modes Options 2.1. 'Monkhorst-Pack k-grid?' should be 'yes' Set 'Dimensionality' to '3' Set 'Lx', 'Ly', and 'Lz' to be '0.5 nm' Set the 'Number of k-points' to '51' in all directions Set 'Number of Bands' to '1' Set 'Transport Direction' to 'Z'.

5 2.7. Set 'Spin Degeneracy' to '2' Set 'Emin' to '-1 ev', 'de' to '0.001 ev', and 'Emax' to '1 ev'. 3. TE Options 3.1. Set 'Particle' to 'Electron' Set 'Temperature' to '300K' Select 'Diffusive' for 'Transport type' Set 'Ef min' to '-0.5 ev', 'def' to '0.001 ev', and 'Ef max' to '0.5 ev' Set 'Conduction Band Minimum' to '1 ev'. (By setting this value to 1 ev, the scattering options for the conduction band will not affect the calculations.) 3.6. Set 'MFP VB' to '10' (the units are nanometers) Set 'VB scattering parameter' to '0' (case of acoustic phonon scattering) Set 'Valence Band Maximum' to '0 ev'. 4. Results 4.1. Distribution of modes note how Figure 2 is the same as the modes for the conduction band (Figure 1), but increasing with negative energy All the thermoelectric properties are shown in Appendix C. Figure 2. The distribution of modes for a 3D parabolic valence band. A.4 3D Ellipsoidal Conduction Band It is common for many semiconductors to have ellipsoidal bands instead of spherical bands, e.g. silicon. In this example the E(k) data set corresponds to,, and. Follow the same directions as the 3D Parabolic Conduction Diffusive Transport example, but upload 'Ek_3D_conduction_ellipsoidal.txt'. By running the tool, one will find the

6 distribution of modes presented in Figure 3, and all the thermoelectric characteristics shown in Appendix D. Notice how the distribution of modes has the same shape as the 3D spherical case (Figure 1), but the slope is smaller due to the smaller distribution of modes effective mass ( [1]). Figure 3. The distribution of modes for a 3D ellipsoidal conduction band. A.5 2D Parabolic Conduction Band Follow the same directions for the 3D Parabolic Conduction Band Diffusive Transport example, with the following changes: 1. Load Data Slide- upload 'Ek_2D_conduction.txt'. 2. Modes Options 2.1. Set 'Dimensionality' to '2' Set 'Number of kx points' to '1' (since we only have periodicity along the y-z directions) Set 'Number of ky points' and 'Number of kz points' to '151'. 3. Results 3.1. Distribution of Modes for a 2D parabolic conduction band, the distribution of modes (page 23 of [1]) is given by: where W is the width of the conductor. The square-root dependence on energy is seen in Figure 4. Note to achieve a smoother function, one needs to simply increase the number k- points.

7 Figure 4. The distribution of modes for a 2D parabolic conduction band All the thermoelectric properties are shown in Appendix E. A.6 1D Parabolic Conduction Band Follow the same directions for the 3D Parabolic Conduction Band Diffusive Transport example, with the following changes: 1. Load Data Slide - upload ' Ek_1D_conduction.txt '. 2. Modes Options 2.1. Set 'Dimensionality' to '1' Set 'Number of kx points' to '1' (only periodic along z) Set 'Number of ky points' to '1' (only periodic along z). 3. Results 3.1. Distribution of Modes for a 1D parabolic conduction band, the distribution of modes (page 23 of [1]) is given by the Heaviside function (Figure 5):.

8 Figure 5. The distribution of modes for a 1D parabolic conduction band All the thermoelectric properties are shown in Appendix F A.7 Full Dispersion of Si In this next example, we compute the distribution of modes using the full band dispersion of silicon (Figure 6). Figure 6. The full electronic dispersion of Si. Follow the same directions for the 3D Parabolic Conduction Band Diffusive Transport example, with the following changes:

9 1. Load Data Slide - upload 'Ek_Si.txt'. 2. Modes Options 2.1. Set 'Lx', 'Ly', and 'Lz' to '0.543 nm' Set 'Number of kx points', 'Number of ky points', 'Number of kz points' to '55' Set 'Number of Bands' to '17' Set 'Emin' to '-2 ev', 'de' to '0.001 ev', and 'Emax' to '2 ev'. 3. TE Options 3.1. Set 'Ef min' to '-1.5 ev', 'de' to '0.001 ev', and 'Ef Max' to '1.5 ev' Set 'MFP CB' to '10' (10 nm is a typical mean-free-path for backscattering for Si) Set 'CB scattering parameter' to '0' Set 'Conduction Band Minimum' to '0.55 ev' Set 'MFP VB' to '10' Set 'VB scattering parameter' to '0' Set 'Valence Band Maximum' to '-0.55 ev'. 4. Results 4.1. Distribution of Modes The calculated modes should look like that shown in Fig. 7 (compare to those in [2]). To test the validity of the parabolic approximation, a linear fit to the Si modes is shown in Fig. 7. We can see that the parabolic approximation works relatively well near the band edges for Si. Figure 7. The distribution of modes for a full dispersion of Si, with a parabolic fit for the valence and conduction bands All the thermoelectric properties are shown in Appendix G.

10 A.8 Full Dispersion of Bi2Te3 Next, we compute the distribution of modes of Bi 2 Te 3, one of the best known bulk thermoelectric materials, from its full band dispersion (Figure 8). Figure 8. The full dispersion of Bi 2 Te 3. Follow the same directions for the 3D Parabolic Conduction Band-Diffusive, with the following changes: 1. Load Data - Upload 'Ek_Bi2Te3.txt'. 2. Modes Options 2.1. Set 'Lx' to '3.049 nm', 'Ly' to ' nm', and 'Lz' to '0.438 nm' Set 'Number of kx points' to '19', 'Number of ky points' to '45', and 'Number of kz points' to '75' Set 'Number of Bands' to '49' Set 'Emin' to '-2 ev', 'de' to '0.001 ev', and 'Emax' to '2 ev'. 3. TE Options 3.1. Set 'Ef min' to '-1.5 ev', 'de' to '0.001 ev', and 'Ef Max' to '1.5 ev' Set 'MFP CB' to 21' (21 nm is a typical value for the conduction band of Bi 2 Te 3 ) Set 'CB scattering parameter' to '0' Set 'Conduction Band Minimum' to '0.08 ev' 3.5. Set 'MFP VB' to '15' (15 nm is a typical value for the valence band of Bi 2 Te 3 ).

11 3.6. Set 'VB scattering parameter' to '0' Set 'Valence Band Maximum' to '-0.08 ev'. 4. Results 4.1. Distribution of Modes running the tool you should obtain a plot similar to Fig. 9 (you can compare to those in [3]). Contrary to the case of Si, the complex electronic structure of Bi 2 Te 3 is not so well represented by a simple parabolic approximation, as shown with the linear fit in the vicinity of the band edges. Figure 9. The distribution of modes for a full dispersion of Bi 2 Te 3, with a parabolic fit for the valence and conduction bands All the thermoelectric properties are shown in Appendix H. B. Phonon Transport Download and unzip 'phonon_dispersions.zip' from under the supporting documents at for the files used in these examples. B.1 Debye Phonon Dispersion We next consider phonon transport, where we begin with the case of a simple Debye model (i.e. three linear dispersion phonon bands with maximum energy cutoff of ).

12 1. Load Data Slide- Upload 'Eq_Debye_phonon.txt'. 2. Modes Options 2.1. Set 'Dimensionality' to '3' Set 'Lx', 'Ly', and 'Lz' to '0.5 nm' Set 'Number of k points' to '51' in all directions Set 'Number of Bands' to '3' Set 'Transport Direction' to 'Z' Set 'Spin Degeneracy' to '2' Set 'Emin' to '0 ev', 'de' to '1e-05 ev', and 'Emax' to '0.025 ev'. 3. TE Options 3.1. Select 'Phonon' from the 'Particle' menu Set 'Temperature' to '300K' Select 'Diffusive' from the 'Transport Type' drop down menu Set 'MFP Phonon' to 10 nm Set 'Scattering Parameter' to Results 5.3. Distribution of Modes the distribution of modes for a linear dispersion band (page 183 of [1]) is given by: The parabolic energy dependence of the distribution of modes is shown in Figure 10. Figure 10. The distribution of modes for a Debye phonon dispersion The lattice thermal conductivity of this example is calculated to be.

13 B.2 Full Phonon Dispersion of Si Next we consider a realistic phonon dispersion, that of silicon (Figure 11). Figure 11. The full phonon dispersion of Si. 1. Load Data Slide- Upload 'Eq_Si_phonon.txt'. 2. Modes Options 2.1. Set 'Dimensionality' to '3' Set 'Lx', 'Ly', and 'Lz' to '0.547 nm' Set 'Number of k points' to '55' in all directions Set 'Number of Bands' to '24' Set 'Transport Direction' to 'Z' Set 'Spin Degeneracy' to '2' Set 'Emin' to '0 ev', 'de' to '1e-05 ev', and 'Emax' to '0.07 ev'. 3. TE Options 3.1. Select 'Phonon' from the 'Particle' menu Set 'Temperature' to '300K' Select 'Diffusive' from the 'Transport Type' drop down menu Set 'MFP Phonon' to 141 nm Set 'Scattering Parameter' to Results 4.1. Distribution of Modes The modes computed from an accurate full band description deviates significantly from a simple Debye model which is quadratic in energy (compare to those in [4]), except for the low energy phonons.

14 Figure 12. The distribution of modes for a full phonon dispersion of Si The calculated lattice thermal conductivity is. References [1] M. Lundstrom and C. Jeong, Near-Equilibrium Transport Fundamentals and Applications, World Scientific, [2] J. Maassen, C. Jeong, A. Baraskar, M. Rodwell and M. Lundstrom, "Full band calculations of the lower limit of contact resistivity," Appl. Phys. Lett., vol. 102, no. 11, [3] J. Maassen and M. Lundstrom, "A computational study of thermoelectric performance of ultrathin Bi 2 Te 3 films," Appl. Phys. Lett., vol. 102, no. 9, [4] C. Jeong, S. Datta and M. Lundstrom, "Full dispersion versus Debye model evaluation of lattice thermal conductivity with a Landauer approach," J. Appl. Phys., vol. 109, no. 7, 2011.

15 Appendix A: 3D Parabolic Conduction Band - Ballistic Case Conductance Seebeck Coefficient

16 Electron Thermal Conductance Power Factor Electronic ZT

17 Appendix B: 3D Parabolic Conduction Band - Diffusive Case Conductivity Seebeck Coefficient

18 Electron Thermal Conductivity Power Factor Electronic ZT

19 Appendix C: 3D Parabolic Valence Band Conductivity Seebeck Coefficient

20 Electron Thermal Conductivity Power Factor Electronic ZT

21 Appendix F: 3D Ellipsoidal Conduction Band Conductivity Seebeck Coefficient

22 Electron Thermal Conductivity Power Factor Electronic ZT

23 Appendix D: 2D Parabolic Conduction Band Conductivity Seebeck Coefficient

24 Electron Thermal Conductivity Power Factor Electronic ZT

25 Appendix E: 1D Parabolic Conduction Band Conductivity Seebeck Coefficient

26 Electron Thermal Conductivity Power Factor Electronic ZT

27 Appendix G: Full Dispersion of Si Conductivity Seebeck Coefficient

28 Electron Thermal Conductivity Power Factor Electronic ZT

29 Appendix H: Full Dispersion of Bi2Te3 Conductivity Seebeck Coefficient

30 Electron Thermal Conductivity Power Factor Electronic ZT