CASTEP tutorials > Predicting the thermodynamic properties of germanium

Predicting the thermodynamic properties of germanium

Purpose: Introduces the use of CASTEP for calculating linear response and thermodynamic properties.

Modules: Materials Visualizer, CASTEP

Time:  

Prerequisites: Predicting the lattice parameters of AlAs from first principles

Background

Linear response, or density functional perturbation theory (DFPT), is one of the most popular methods of ab initio calculation of lattice dynamics. However, potential applications of the method extend beyond the study of vibrational properties. Linear response provides an analytical way of computing the second derivative of the total energy with respect to a given perturbation. Depending on the nature of this perturbation, a number of properties can be calculated. A perturbation in ionic positions gives the dynamical matrix and phonons; in magnetic field - NMR response; in unit cell vectors - elastic constants; in an electric field - dielectric response, etc. The basic theory of phonons, or lattice vibrations, in crystals is well understood and has been described in detail in several textbooks. The importance of the phonon interpretation of lattice dynamics is illustrated by the large number of physical properties that can be understood in terms of phonons: infrared, Raman, and neutron scattering spectra; specific heat, thermal expansion, and heat conduction; electron-phonon interaction and thus resistivity and superconductivity, etc. Density Functional Theory (DFT) methods can be used to predict such properties and CASTEP provides this functionality.

DFPT phonon calculations using ultrasoft pseudopotentials are not yet supported and nor are DFPT calculations for spin-polarized systems. Nevertheless phonon spectra and related properties can be calculated with those settings in the framework of the finite difference technique.

Introduction

In this tutorial, you will learn how to use CASTEP to perform a linear response calculation in order to calculate phonon dispersion and density of states as well as predict thermodynamic properties such as enthalpy and free energy.

This tutorial covers:

In order to ensure that you can follow this tutorial exactly as intended, you should use the Settings Organizer dialog to ensure that all your project settings are set to their Accelrys default values. See the Creating a project tutorial for instructions on how to restore default project settings.

1. Getting started

Begin by starting Materials Studio and creating a new project.

Open the New Project dialog and enter Ge_phonon as the project name, click the OK button.

The new project is created with Ge_phonon listed in the Project Explorer.

Begin by importing the Ge structure, which is included in the structure library provided with Materials Studio.

Select File | Import... from the menu bar to open the Import Document dialog. Navigate to Structures/metals/pure-metals and select Ge.msi.

2. To optimize the structure of the germanium cell

It is often possible to get significant speedup by converting the structure to a primitive cell.

Select Build | Symmetry | Primitive Cell from the menu bar.

The primitive germanium cell is displayed.

primitive germanium cell

Now optimize the geometry of the Ge structure using CASTEP.

Click the CASTEP button CASTEP on the Modules toolbar and select Calculation or choose Modules | CASTEP | Calculation from the menu bar.

This opens the CASTEP Calculation dialog.

CASTEP Calculation setup dialog
CASTEP Calculation dialog, Setup tab

The default values for geometry optimization do not include optimization of the cell.

On the Setup tab change the Task from Energy to Geometry Optimization and the Functional to LDA. Uncheck the Metal checkbox.

Click the More... button to open the CASTEP Geometry Optimization dialog, check the Optimize cell checkbox and close the dialog.

Select the Electronic tab of the CASTEP Calculation dialog and set the Energy cutoff to Ultra-fine, the SCF tolerance to Ultra-fine, the k-point set to Fine and Pseudopotentials to Norm conserving.

On the Job Control tab select the Gateway location on which you want to run the job and set the Runtime optimization to Speed.

Click the Run button to start the job.

The job is submitted and starts to run. It should take about 2 minutes, depending on the speed of your computer. The results are placed in a new folder called Ge CASTEP GeomOpt.

3. To calculate phonon dispersion and phonon density of states (DOS)

DFPT gives an opportunity to accurately calculate phonon frequencies at any given point in reciprocal space. However, a calculation for each q-point can be expensive. An alternative approach may be used for calculations that require phonon frequencies for large numbers of q-points, for example, for phonon DOS and thermodynamic properties. This alternative scheme takes advantage of the relatively short range of effective ion-ion interactions in crystals. Interpolation can be used to reduce computation time without loss of accuracy. The accurate DFPT calculations are performed only at a small number of q-vectors, and then a cheap interpolation procedure is used to obtain frequencies at other q-points of interest. One advantage of using an interpolation scheme instead of the exact calculation is that thermodynamic properties at low temperatures depend strongly on the number of points in the phonon DOS grid. Using the interpolation approach, this number can be increased at no computational cost.

In order to calculate the phonon dispersion and phonon density of states you must perform a single point energy calculation, after selecting the appropriate properties from the Properties tab on the CASTEP Calculation dialog.

Ensure that Ge.xsd in the Ge CASTEP GeomOpt folder is the active document.

On the Setup tab of the CASTEP Calculation dialog set the Task to Energy.

On the Properties tab choose Phonons and request Density of states and Dispersion by selecting the Both option.

Click the More... button, to display the CASTEP Phonon Properties Setup dialog. Ensure the Method is Linear response and the Use interpolation checkbox is checked. Ensure that q-vector grid spacing for interpolation is 0.05 1/Å. Set the Quality both for Dispersion and Density of states to Fine and close the dialog.

Click the Run button and close the CASTEP Calculation dialog.

The job is submitted and starts to run. This is a more time-consuming job which could take couple of hours, depending on the speed of your computer. A new folder, called Ge CASTEP Energy, is created in the Ge CASTEP GeomOpt folder. When the energy calculation is finished two new results files are placed in this folder, Ge_PhonDisp.castep and Ge_PhonDOS.castep.

4. To display phonon dispersion and density of states

Phonon dispersion curves show how phonon energy depends on the q-vector, along high symmetry directions in the Brillouin zone. This information can be obtained experimentally from neutron scattering experiments on single crystals. Such experimental data are available for only a small number of materials, so theoretical dispersion curves are useful for establishing the validity of a modeling approach to demonstrate the predictive power of ab initio calculations. In certain circumstances it is possible to measure the density of states (DOS) rather than the phonon dispersion. Furthermore, the electron-phonon interaction function, which is directly related to the phonon DOS can be measured directly in the tunneling experiments. It is therefore important to be able to calculate phonon DOS from first principles. Materials Studio can produce phonon dispersion and DOS charts from any .phonon CASTEP output file. These files are hidden in the Project Explorer but a .phonon file is generated with every .castep file that has a PhonDisp or PhonDOS suffix.

When evaluating phonon DOS, use only the results of phonon calculations on the Monkhorst-Pack grid.

Now use the results of the previous calculation to create a phonon dispersion chart.

Select Modules | CASTEP | Analysis from the menu bar to open the CASTEP Analysis dialog. Choose Phonon dispersion from the list of properties. Ensure that the Results file selector displays Ge_PhonDisp.castep.

Select cm-1 from the Units dropdown list and Line from the Graph style dropdown list.

Click the View button.

A new chart document, Ge Phonon Dispersion.xcd, is created in the results folder. It should look something like the chart shown below:

Ge Phonon Dispersion

The experimental Phonon Dispersion is shown below:

Ge Phonon dispersion

The predicted frequencies are available in the Ge_PhonDisp.castep file.

Double-click on Ge_PhonDisp.castep in the Project Explorer. Press the CTRL + F keys and search for Vibrational Frequencies.

The following portion of the results file is displayed:

 ==============================================================================
 +                           Vibrational Frequencies                          +
 +                           -----------------------                          +
 +                                                                            +
 + Performing frequency calculation at  40 wavevectors (q-pts)                +
 + ========================================================================== +
 +                                                                            +
 + -------------------------------------------------------------------------- +
 +  q-pt=    1 (  0.500000  0.250000  0.750000)     0.048780                  +
 + -------------------------------------------------------------------------- +
 +  Acoustic sum rule correction <   0.069997 cm-1 applied                    +
 +     N      Frequency irrep.                                                +
 +                (cm-1)                                                      +
 +                                                                            +
 +     1     114.896630   a                                                   +
 +     2     114.896630   a                                                   +
 +     3     204.785753   b                                                   +
 +     4     204.785753   b                                                   +
 +     5     274.975444   a                                                   +
 +     6     274.975444   a                                                   +
 + .......................................................................... +
 +        Character table from group theory analysis of eigenvectors          +
 +                           Point Group =  32, Oh                            +
 + .......................................................................... +
 +  Rep  Mul |    E   2   2   m  -4                                           +
 +           | --------------------                                           +
 + a       2 |    2   0   0   0  -1                                           +
 + b       1 |    2   0   0   0   1                                           +
 + -------------------------------------------------------------------------- +
....

The results you obtain may vary slightly from those shown because of minor differences in the structure of the starting model.

The frequencies for every q-point and for every branch (Longitudinal Optical or Acoustical (LO/LA), Transverse Optical or Acoustical (TO/TA)) are given in cm-1, as well as the positions of the q-points, in the reciprocal space. The high symmetry points Γ, L and X are at reciprocal space positions (0 0 0), (0.5 0.5 0.5) and (0.5 0 0.5) respectively.

The predicted and experimental frequencies in cm-1 are:

  Predicted Experimental
ΓTO 302 304
ΓLO 302 304
ΓTA 0 0
ΓLA 0 0
LTA 62 63
LLA 223 222
LTO 243 245
LLO 288 290
XTA 80 80
XLA 241 241
XTO 272 276

Overall, the accuracy of the calculation is acceptable. Better agreement with the experimental results may be obtained by running the calculation with a better SCF k-point grid.

Now create a phonon DOS chart.

On the CASTEP Analysis dialog select Phonon density of states from the list of properties. Make Ge.xsd the active document and ensure that the Results file selector displays Ge_PhonDOS.castep.

Set Display DOS to Full. Click the More... button to open the CASTEP Phonon DOS Analysis Options dialog. Select Interpolation from the Integration method dropdown list and set the Accuracy level to Fine. Click the OK button and on the CASTEP Analysis dialog click the View button to create the DOS chart.

Interpolation scheme was chosen to get the best representation of DOS; an alternative setting, smearing, produces DOS with too few fine details.

A new chart document, Ge Phonon DOS.xcd, is created. It should look something like the chart shown below:

Ge_Phonon_DOS

5. To display thermodynamic properties

Phonon calculations in CASTEP can be used to evaluate the temperature dependence of the enthalpy, entropy, free energy, and lattice heat capacity of a crystal in a quasi-harmonic approximation. These results can be compared with experimental data (for example heat capacity measurements) or used to predict phase stability of different structural modifications or phase transitions.

All energy-related properties are plotted on one graph, and the calculated value of the zero-point energy is included. The heat capacity is plotted separately on the right.

The entropy is present as a TS product to allow comparison with the enthalpy.

Now use the results of the phonon calculation to create a thermodynamic properties chart.

On the CASTEP Analysis dialog choose Thermodynamic properties from the list of properties. Make Ge.xsd the active document and ensure that the Results file selector displays Ge_PhonDOS.castep.

Check the Plot Debye temperature checkbox and click the View button.

Two new chart documents, Ge Thermodynamic Properties.xcd and Ge Debye Temperature.xcd, are created in the results folder:

Ge thermo properties
Ge Debye Temp

Experimental results without anharmonicity (Flubacher et al., 1959) show that the Debye temperature at the high temperature limit is 395(3) K. The simulated Debye temperature is 392 K, in excellent agreement with the experimental value.

Overall, the experimental plot is qualitatively very similar to the one generated by CASTEP. There is a dip at about 25 K, with the lowest value of Debye temperature of the order of 255 K, exactly as predicted by CASTEP results. The exact shape of the curve at very low temperatures is not accurate with the calculation settings used in this tutorial; a better sampling of low-frequency acoustic modes is required and it can be achieved by using fined Monkhorst-Pack grid in the phonon density of states calculation.

This is the end of the tutorial.

References

Flubacher, P.; Leadbetter, A. J.; Morrison, J. A. "The heat capacity of pure silicon and germanium and properties of their vibrational frequency spectra", Phil. Mag., 4, 273-294 (1959).

Accelrys Materials Studio 8.0 Help: Wednesday, December 17, 2014
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