A Package for calculating elastic tensors of cubic Phases by using second-order derivative with WIEN2k Package

Transcription

1 IR E L A S T + W I E N 2k A Package for calculating elastic tensors of cubic Phases by using second-order derivative with WIEN2k Package User s guide, Cubic-elastic_13.2 (Release ) Morteza Jamal Ghods City-Tehran-Iran

2 MANDATORY CONDITIONS: In any publication in the scientific literature please reference the program as follows: M. Jamal, Cubic-elastic, (2012). ACKNOWLEDGMENT I gratefully appreciate B.Z. Yanchitsky for fruitful discussions, P. Blaha and S. Jalali Asadabadi for suggestions, and Carol Phillips for editing. For suggestions or bug reports please contact the author by m_jamal57@yahoo.com

3 1- Introduction Cubic-elastic is a Package for finding elastic constants of cubic symmetries with WIEN2k. This Package calculates elastic constants by second-order derivative ( E ( δ) ) of Polynomial fit ( E=E(δ) ) of Energy vs. strains ( δ) at zero strain (δ =0). This called energy approach [1]. 2- Cubic-elastic and Thomas Charpin Packages I think there will be a question for users of this Package. What is difference between Cubic-elastic and Thomas Charpin [2] Package? Since for cubic symmetry, the bulk modulus given by: So in Thomas Charpin Package, it calculates (C 11 +2C 12 ) by using Equation Of State (EOS) and as a result bulk modulus. Also it calculates (C 44 ) by a rhombohedral strain which results: While Cubic-elastic calculates (C 11 +2C 12 ) and (C 44 ) without any detours. For more information see background theory section. So, a new source of error may come from bulk modulus calculation in Thomas Charpin Package. 3- Background theory (energy approach) Elastic constants are defined by means of a Taylor expansion of the total energy for the system, with respect to a small strain ( ) of the lattice. If we consider the bravais lattice vectors of a cubic crystal structure as a matrix form the distortion of the lattice ( ) is expressed by multiplying with a symmetric ( ) distortion matrix i.e. ( ), which is written as,

4 And in Voigt notation ( It is often convenient to change to the Voigt notation to reduce the number of indices. The Voigt notation replaces 1, 2, 3, (and ) 4, (and ) 5, (and ) 6 ) We express the energy of the strained system by means of a Taylor expansion in the distortion parameters, The linear terms vanish if the strain causes no changes in the volume of the crystal. Otherwise, are related to the strain on the crystal and are elastic constants and is the volume of unstrained cubic system and we use it to evaluate the elastic constants. To obtain elastic constants of cubic structure I have used the strain method (energy approach). In this method elastic constants were calculated by applying small strains to the unstrained lattice. There are three independent elastic constants for a cubic symmetry, called C 11, C 12, and C 44. Since we have three independent elastic constants, we need three different strains to determine these elastic constants. The three distortions used

5 in the Cubic-elastic Package are described below. These three distortions are written as [3-5] and D 1 is volume-conserving orthorhombic distortion and D 2 changes the lattice parameter in the a-lattice direction. The symmetry of the strained lattice is therefore still cubic, however the volume of the distortion lattice changes by using D 2. D 3 is volume-conserving monoclinic distortion and the energy for these distortions can be obtained as And Respectively,

6 4- File structure and program flow The following table describes input and output files for each program of the Cubic-elastic Package. Program needs generates C_set_elast_lapw case.struct init.struct runcommand1 runcommand2 pwdname command_init_lapw C_command_run_lapw C_setup11m12 C_setupc1112 C_setupc44 getcalljobc makestructc init.struct pwdname runcommand1 runcommand2 init.struct.styp CUBIC.job auto_init_lapw runcommand1 runcommand2.vper.styp CUBIC.job number.strain StypX_Y.struct vol.optimize C_modifyjob_lapw C_calljob_lapw CUBIC.job VstVene C_fitdivELC number.strain ELCorder.fit vol.optimize ELC.output.styp ELC.fit VstVene C_ana_elastc_lapw VstVene case.outputeos ELC.fit vol.optimize ELC.ps C_ana_elast_lapw ELC.output ELC-matrix case.output_elastic C_InverseELC ELC-matrix INVELC-matrix MassRho init.struct.rho vol.optimize C_ana_elastorder_lapw ELCorder.fit output-order sgroupcheck_lapw StypX_Y.struct case.struct command_intso_lapw.infso command_initu_lapw.infldau auto_initso_lapw.infso case.inso auto_initu_lapw.infldau case.inorb case.indm/c Bold font is OPTIONAL Italic bold font means it is the user s choice

7 4-2- Short description for input and output files case.struct init.struct pwdname runcommand1/2 Is a WIEN2k standard struct file. Is a copy of the case.struct file. Contains the name of present work directory. Contains the run commands for running. It looks similar to: run_lapw ec p in1new 2 auto_init_lapw A C-shell program which automatically runs the initialization. It looks similar to:.vper.styp CUBIC.job #!/bin/csh -f set RM = not if ( $RM == 'not' ) then init_lapw -vxc 13 -ecut -6 -mix 0.2 -numk b else init_lapw -red 0 -vxc 13 -ecut -6 -mix 0.2 -numk b endif Defines the percent of changes for different strains. Defines the type of strain. A C-shell program which calculates the energy for each strain by using the WIEN2k Package. It looks similar to: #!/bin/csh -f #STRAIN TYPE IS 1 #Modify this script according to your needs unalias rm set co = 1 set name set bj set file = `pwd` set file = $file:t if (-e VstVene ) then set i=`/bin/ls VstVene* wc ` echo " saving pervious VstVene to VstVene_$i[1]" cp VstVene VstVene_$i[1] rm VstVene endif # # to reuse previous scf runs (without a new scf run) set answscf=y # and use the same "savename". # When you make modifications (RKmax, k-mesh, XC-potentials) choose # answscf=no and a new savename (eg. "_pbe_rk8_1000k"). set answscf=y

8 set savename= if (-e cscl.clmsum && \! -z cscl.clmsum) then x dstart -super endif if (-e cscl.clmup && \! -z cscl.clmup ) then x dstart -super -up x dstart -super -dn endif foreach i ( \ ) echo "*******************************" echo $i set name=`echo "$name $i"` echo "*******************************" Styp1_-3.0 \ Styp1_-2.0 \ Styp1_-1.0 \ Styp1 0.0 \ Styp1 1.0 \ Styp1 2.0 \ Styp1 3.0 \ StypX_Y.struct VstVene number.strain case.outputeos A WIEN2k struct file for each value of changes and for each strain type where X and Y denote type of strain and value of changes, respectively. The main information file, contains values of changes (strains) and energies for each type of strain, for the calculation of the elastic constant. Contains the number of strains. A WIEN2k output of equation of states(eos). For finding the best values of elastic constants, find EOS and then copy the case.outputeos in the "case" directory Within the c11+2c12, c11-c12 and c44 directories. Otherwise, it sets the optimized volume from the original struct file i.e. case.struct vol.optimize Contains the optimize volume. ELCorder.fit Contains the elastic constants for different values of order of fit. ELC.output Contains the elastic constants for order of fit =2 ELC.fit Contains the data to plot curve of energy vs value of changes (strains) for each strain type. ELC-matrix Defines the elastic constant matrix for each symmetry. case.output_elastic Contains the final elastic constant values. INVELC-matrix Defines the inverse of elastic constant matrix.

9 .rho.infso.infldau Contains density of mass and atomic volume. Contains information for making case.inso file for running spin-orbit coupling. Contains information for making case.inorb and case.indm/c files for LDA+U calculations Flow and short description for programs The Cubic-elastic Package consists of several FORTRAN and SHELL SCRIPTS which are described below. A flowchart of the program is shown in the following diagram. C_set_elast_lapw : Makes an elast-constant directory in present work directory ( PWD ) and c11+2c12, c11-c12 and c44 directories in the elast-constant directory. The C_set_elast_lapw program also copies information of "PWD" into the c11+2c12, c11-c12 and c44 directories and calls "command_init_lapw", C_command_run_lapw, C_setupc1112, C_setupc11m12 and C_setupc44 programs. command_init_lapw : Gets informations for making "auto_init_lapw". C_command_run_lapw : Gets the run commands for making CUBIC.job. C_setupcX (X=1112, 11m12, 44) : Gets the type of strain and calls the getcalljobc program. getcalljobc : Calls makestructc program and makes the CUBIC.job file. makestructc : Makes StypX_Y.struct files where X and Y stand for type of strain and value of changes, respectively and the vol.optimize file. C_modifyjob_lapw : Edits the job files according to the user s needs. C_calljob_lapw : Calls the CUBIC.job files for running. C_ana_elast_lapw : Calls the C_ana_elastc_lapw program for calculation elastic constants then calculates

10 the Voigt, Reuss, and Hill bulk, shear, and Young modulus as well as the Poisson ratio. After that it calls the C_InverseELC and MassRho programs and calculates sound velocity and Debye temperature then makes two output files in the elastconstant directory with the name case.output_elastic and the INVELC-matrix which is elastic compliance constants generated by inverting the elastic constant matrix. At the end it calls C_ana_elastorder_lapw program. C_ana_elastc_lapw : Calls the C_fitdivELC program with appropriate libraries for calculating C11-C12, C11+2C12, and C44. C_InverseELC : Makes the Elastic compliance constants generated by inverting the elastic constant matrix. MassRho : Finds density of mass and atomic volume. C_ana_elastorder_lapw : Checks the sensitivity of the elastic constants to the order of fit. sgroupcheck_lapw : Finds the best value of tol in sgroup [6] program and copies case.struct_sgroup as case.struct.

11 C_set_elast_lapw command_init_lapw generates auto_init_lapw C_command_run_lapw generates commandrun1/2 C_setupcX X=1112, 11m12, 44 getcalljobc generates CUBIC.job makestructc generates StypX_Y.struct sgroupcheck_lapw CUBIC.job generates VstVene auto_init_lapw C_ana_elastorder_lapw generates ELCorder.fit C_ana_elast_lapw C_ana_elastc_lapw C_InverseELC MassRho ELASTIC CONSTANTS IS READY generates case.output_elastic and INVELC-matrix C_fitdivELC calls Libraries generates C 11 -C 12, C 11 +2C 12, C 44 Program flow in Cubic-elastic Dash arrow means user must run

12 5 Elastic constants calculation 1. Create a struct file and validate it by running "sgroupcheck_lapw". 2. If Spin-Orbit calculations are required run "command_initso_lapw. 3. If LDA+U calculations are required run "command_initu_lapw" and then "auto_initu_lapw". 4. Run "C_set_elast_lapw". 5. Now you must adapt the job files according to your needs (you can run "C_modifyjob_lapw" in Terminal ). It is not necessary to do step 5 if you define the COMMAND RUN commands in step Now you must run the job files ( you can run "C_calljob_lapw" in Terminal ). It will take time. 7. Run "C_ana_elast_lapw". This package calculates elastic constants by second-order derivative ( E (δ) ) of Polynomial fit ( E=E(δ) ) of Energy vs. strains (є) at zero strain ( δ =0) so, you must use values of strain around zero and from the viewpoint of fit convergence, we usually expect to see a minimum when we plot Energy vs. strain ( this Package plots it ). It is recommended that the sensitivity of the results is checked to the order of fit. This program shows them. 5-1 Notes about elastic constants calculation After using distortions for calculation C11-C12 and C44 the symmetry of the cubic compound changes and usually the number of atoms change. So when you run "command_initso_lapw" or "command_initu_lapw", in the section name of an atom, type "all <name of atom>" ( for example: all Mn). With this command, you use SO or LDA+U calculations for example for all Mn atoms. When you want to rerun job files with modifications in (RKmax, k-mesh, XC-potentials ) call command_init_lapw and after that choose "answscf=no" in the CUBIC.job files and a new "savename" (eg. "_use_pbe_rk8"). Optionally you can specify more cases by rerunning C_setupcX (X=1112, 11m12, 44 see section 5-3 ). Specify also your old cases. The old results will then be taken

13 automatically into account without recalculation ( unless you modify job files i.e: set answscf=no ). For the calculation of the best values of elastic constants, please find EOS and then copy case.outputeos in "case" directory within c11+2c12, c11-c12 and c44 directories. Otherwise, it sets the optimized volume from the original struct file i.e. case.struct. 5-2 One calculation To calculate C 11 +2C 12 or C 44 the following steps should be performed for example for calculation of C 44 : 1. Make a directory for example c Make a "case" directory in c44 directory. 3. Make a "case.struct" file in "case" directory and name it "init.struct". Creat "pwdname" file and write in it "case." and save it. 4. Run the command_init_lapw 5. chmod +x auto_init_lapw 6. For SO calculations, run the command_initso_lapw. 7. For LDA+U calculations, run the command_initu_lapw and auto_initu_lapw To avoid step 10, you can run C_command_run_lapw for setting the COMMAND RUN commands for making CUBIC.job. 8. Run the C_setupc44 program. 9. chmod +x CUBIC.job file. 10. Modify the CUBIC.job file. 11. Call CUBIC.job 12. Call C_ana_elastc_lapw 5-3 Run with more data points Optionally you can specify more data points, for calculation of the elastic constants, by rerunning C_setupcX (X=1112, 11m12, 44 ). Specify also your old data points. The old results will then be taken automatically into account without recalculation ( unless you modify job files i.e: set answscf=no ). Please do the following steps for this goal for example for c cd to the elast-constant directory. 2. cd to the c44 directory. 3. cd to the case directory. 3-1) To avoid step 6, you can run C_command_run_lapw for setting the COMMAND RUN commands for making the CUBIC.job. 4. Run C_setupc44 program. 5. If you want to rerun job files with modifications in (RKmax, k-mesh, XC-

14 potentials ) call command_init_lapw and then choose "answscf=no" in CUBIC.job files and a new "savename" (eg. "_use_pbe_rk8"). 6. Modify the CUBIC.job file. 7. Call CUBIC.job 8. Call C_ana_elastc_lapw 5-4 Elastic constants calculation for CsCl CsCl compound is a test case for elastic constants calculation. The CsCl structure is described in detail in the following: CsCl P LATTICE,NONEQUIV.ATOMS: 2 MODE OF CALC=RELA unit=bohr ATOM 1: X= Y= Z= MULT= 1 ISPLIT= 0 Cs NPT= 781 R0= RMT= Z: 55.0 LOCAL ROT MATRIX: ATOM 2: X= Y= Z= MULT= 1 ISPLIT= 0 Cl NPT= 781 R0= RMT= Z: 17.0 I selected the following parameters for calculation: Xc = PBE-sol, R_Kmax = 9, L_max = 10 and nkpoint = 4000 After calculation, we have : ================================================================================ Atom name = Cs Atomic Mass from Periodic table = (gr/mol) Atomic Mass from Periodic table = *10^(-23) (gr) Atom name = Cl Atomic Mass from Periodic table = (gr/mol) Atomic Mass from Periodic table = *10^(-23) (gr) Volume in unit of cm^3 = *10^(-24) (cm^3) Mass of Compound : *10^(-23) (gr) Density of Compound : (gr/cm^3) ================================================================================ C11 = GPa C12 = GPa C44 = GPa ================================================================================

15 Prediction VOIGT Bulk modulus by using elastic constant values Prediction REUSS Bulk modulus by using elastic constant values Prediction HILL Bulk modulus by using elastic constant values = (GPa) = (GPa) = (GPa) Prediction VOIGT Shear modulus by using elastic constant values = (GPa) Prediction REUSS Shear modulus by using elastic constant values = (GPa) Prediction HILL Shear modulus by using elastic constant values = (GPa) Prediction VOIGT Young modulus by using elastic constant values = (GPa) Prediction REUSS Young modulus by using elastic constant values = (GPa) Prediction HILL Young modulus by using elastic constant values = (GPa) Prediction VOIGT Poisson's coefficient by using elastic constant values =.270 Prediction REUSS Poisson's coefficient by using elastic constant values =.284 Prediction HILL Poisson's coefficient by using elastic constant values =.277 ================================================================================ By using HILL data Transverse elastic wave velocity = (m/s) Longitudinal elastic wave velocity = (m/s) The average wave velocity = (m/s) Debye Temperature = (K) ================================================================================ Melting Temperature = K or K ================================================================================ CsCl (our Cal) (Exp) 7 Method FPLAPW - Exchange PBE-sol - a (bohr) C C C

16 6 Installation of the Cubic-elastic package The Cubic-elastic package comes as a compress and tar file namely cubic2-elasticderiative-method-2012-p.tar.gz or cubic2-elastic.tar.gz. To install the package firstly copy the file to a directory of your choice. Now, uncompress and expand it as: tar zxvf cubic2-elastic-deriative-method-2012-p.tar.gz cd cubic2-elastic

17 Run buildc2irelast_lapw This program helps you to create the "Makefile" and then compile cubic2-elastic. By default, the Makefile expects the lapack_lapw and blas_lapw to be in the location../src_lib. This should be changed to the correct location by modifying the FOPT parameter as shown below. This Program helps you to define Fortran compiler, Fortran options, and Library options if you have installed WIEN2k. As you can see here this program defines Fortran compiler, Fortran options, and Library options as automatically. Otherwise you can define compiler and linker options as well as the path of mkl library depending on the your selected system. /home/mylib/mkl/lib/em64t is the path of my mkl library. To make Makefile by the lapack_lapw and blas_lapw libraries in the location../src_lib and gfortran use the following options: Fortran compiler: gfortran Fortran options: -ffree-form Library options (Lapack and BLAS): $(FOPT) L/home/physicsprogram/SRC_lib lpthread static llapack_lapw lblas_lapw the location../src_lib should be changed to the correct location by modifying the FOPT parameter as shown above. PS: To install with -ffree-form, you should compile the lapack_lapw and blas_lapw libraries with -ffree-form options. Otherwise it might was caused error.

18 If you view the OPTIONS file of the WIEN2k package you can use the FOPT, LDFLAGS, and R_LIBS of it for compiling. After defining the Fortran compiler, Fortran options, and Library options press Enter key.

19 The Environment Variable ELASTC2_PATH is then defined and added to the end of the.bashrc file. Thus you will be able to call cubic2-elastic s programs for any location. If you edit.bashrc file. You can see :

20 If you view the.bashrc file you can see Now logout from your Linux system and then login.

21 7 References [1] R. Stadler, W. Wolf, R. Podloucky, G. Kresse, J. Furthmller, J. Hafner, Phys. Rev. B 54 (1996) [2] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, WIEN2k, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties (Karlheinz Schwarz, Techn. Universität Wien, Austria), ISBN , [3] Michael J. Mehl, Phys. Rev. B 47 (1993) [4] D. Connétable, O. Thomas, Phys. Rev. B 79 (2009) [5] O. Beckstein, J. E. Klepeis, G. L. W. Hart, and O. Pankratov, Phys. Rev. B 64 (2001) [6] B. Z. Yanchitsky, A. N. Timoshevskii, Determination of the space group and unit cell for a periodic solid, Comp. Phys. Comm. 139 (2001) [7] D.D. Slagle, H. A. McKinstry, J. Appl. Phys. 38 (1967) 446. [8] V. Ganesan, K. Girirajan, Pramāna-J. Phys. 27 (1986) 472.