Clamped and relaxed ion elastic constants

This example demonstrates relaxation and property evaluation for cubic structures of silicon. In particular, it illustrates how one can compute both the ‘clamped ion’ elastic constant \(c_{44}^0\) and the ‘relaxed ion’ elastic constant \(c_{44}\).

Location

examples/elastic_constants_Si

Input files

  • main.xml: main input file

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    <?xml version="1.0" encoding="iso-8859-1"?>
    <job>
    	<name>Relaxation of structures and elastic constants for silicon</name>
    
    	<verbosity>medium</verbosity>
    
    	<atom-types>
    	  <species>Si</species>
    	</atom-types>
    
    	<potentials>
    	  <tersoff id="Si potential" species-a="*" species-b="*">
    	    <param-file>SiC.tersoff</param-file>
    	  </tersoff>
    	</potentials>
    	
    	<structures>
    
    	  <fcc-lattice id="A1 (fcc)">
    	    <atom-type>Si</atom-type>
    	    <lattice-parameter>3.6</lattice-parameter>
    	    <properties>
    	      <atomic-energy/>
    	      <lattice-parameter/>
    	      <bulk-modulus/>
    	    </properties>
    	  </fcc-lattice>
    
    	  <bcc-lattice id="A2 (bcc)">
    	    <atom-type>Si</atom-type>
    	    <lattice-parameter>3.0</lattice-parameter>
    	    <properties>
    	      <atomic-energy/>
    	      <lattice-parameter/>
    	      <bulk-modulus/>
    	    </properties>
    	  </bcc-lattice>
    
    	  <hcp-lattice id="A3 (hcp)">
    	    <atom-type>Si</atom-type>
    	    <lattice-parameter>2.8</lattice-parameter>
    	    <ca-ratio>1.6</ca-ratio>
    	    <properties>
    	      <atomic-energy/>
    	      <lattice-parameter/>
    	      <bulk-modulus/>
    	    </properties>
    	  </hcp-lattice>
    
    	  <diamond-lattice id="A4 (diamond), relaxed ion elastic constants">
    	    <atom-type>Si</atom-type>
    	    <lattice-parameter>5.</lattice-parameter>
    	    <relax-dof>
    	      <atom-coordinates/>
    	    </relax-dof>
    	    <properties>
    	      <atomic-energy/>
    	      <lattice-parameter/>
    	      <bulk-modulus/>
    	      <C11/>
    	      <C12/>
    	      <C44/>
    	    </properties>
    	  </diamond-lattice>
    
    	  <diamond-lattice id="A4 (diamond), clamped ion elastic constants">
    	    <atom-type>Si</atom-type>
    	    <lattice-parameter>5.</lattice-parameter>
    	    <properties>
    	      <atomic-energy/>
    	      <lattice-parameter/>
    	      <bulk-modulus/>
    	      <C11/>
    	      <C12/>
    	      <C44/>
    	    </properties>
    	  </diamond-lattice>
    
    	  <B4-lattice id="B4 (wurtzite), relaxed ion elastic constants">
    	    <atom-type-A>Si</atom-type-A>
    	    <atom-type-B>Si</atom-type-B>
    	    <lattice-parameter>3.</lattice-parameter>
    	    <ca-ratio>1.6</ca-ratio>
    	    <u-parameter>0.375</u-parameter>
    	    <relax-dof>
    	      <atom-coordinates/>
    	    </relax-dof>
    	    <properties>
    	      <atomic-energy/>
    	      <lattice-parameter/>
    	      <ca-ratio/>
    	      <u-parameter/>
    	      <bulk-modulus/>
    	      <C11/>
    	      <C12/>
    	      <C13/>
    	      <C33/>
    	      <C44/>
    	      <C66/>
    	    </properties>
    	  </B4-lattice>
    
    	  <B4-lattice id="B4 (wurtzite), clamped ion elastic constants">
    	    <atom-type-A>Si</atom-type-A>
    	    <atom-type-B>Si</atom-type-B>
    	    <lattice-parameter>3.</lattice-parameter>
    	    <ca-ratio>1.6</ca-ratio>
    	    <u-parameter>0.375</u-parameter>
    	    <properties>
    	      <atomic-energy/>
    	      <lattice-parameter/>
    	      <ca-ratio/>
    	      <u-parameter/>
    	      <bulk-modulus/>
    	      <C11/>
    	      <C12/>
    	      <C13/>
    	      <C33/>
    	      <C44/>
    	      <C66/>
    	    </properties>
    	  </B4-lattice>
    
    	</structures>
    </job>
    
  • SiC.tersoff: initial parameter set in Lammps/Tersoff format

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    # Si and C mixture, parameterized for Tersoff potential
    # this file is from ulomek(at)imm.rwth-aachen.de
    # values are from Erhart/Albe - Phys Rev B, 71, 035211 (2005)
    
    # m=n=beta=1
    # R,D,gamma,c,d ->same
    # lambda1=beta*sqrt(2*S)
    # lambda2=beta*sqrt(2/S)
    # lambda3=2*mu=0
    # costheta0=-h
    # A=D0/(S-1)*exp(lambda1*r0)
    # B=S*D0/(S-1)*exp(lambda2*r0)
    
    # Tersoff parameters for various elements and mixtures
    # multiple entries can be added to this file, LAMMPS reads the ones it needs
    # these entries are in LAMMPS "metal" units:
    #   A,B = eV; lambda1,lambda2,lambda3 = 1/Angstroms; R,D = Angstroms
    #   other quantities are unitless
    
    # format of a single entry (one or more lines):
    #   element 1, element 2, element 3,
    #   m, gamma, lambda3, c, d, costheta0, n,
    #   beta, lambda2, B, R, D, lambda1, A
    
    C   C   C    1 0.11233  0 181.91  6.28433 -0.5556 1 1 1.93090093 175.426651 2    0.15 4.18426232 2019.8449
    Si  Si  Si   1 0.114354 0 2.00494 0.81472 -0.259  1 1 1.53810493 219.521624 2.82 0.14 2.83318929 2145.7128
    Si  Si  C    1 0.011877 0 273987  180.314 -0.68   0 0 0          0          2.4  0.2  0          0
    Si  C   C    1 0.011877 0 273987  180.314 -0.68   1 1 1.76807421 225.189481 2.4  0.2  3.26563307 1779.36144
    C   Si  Si   1 0.011877 0 273987  180.314 -0.68   1 1 1.76807421 225.189481 2.4  0.2  3.26563307 1779.36144
    C   Si  C    1 0.11233  0 181.910 6.28433 -0.5556 0 0 0          0          2    0.15 0          0
    C   C   Si   1 0.011877 0 273987  180.314 -0.68   0 0 0          0          2.4  0.2  0          0
    Si  C   Si   1 0.114354 0 2.00494 0.81472 -0.259  0 0 0          0          2.82 0.14 0          0
    

Output

  • The final properties (as well as parameters) are written to standard output.

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    This is program version 0.1.6
    Reading job file main.xml
    -------------------------------------------------------
    Parsing input file(s)
    -------------------------------------------------------
    -------------------------------------------------------
    Computing structure properties
    Structure 'A1 (fcc)':
       total-energy: -3.30192 eV
       atomic-energy: -3.30192 eV/atom
       total-volume: 11.664 A^3
       atomic-volume: 11.664 A^3/atom
       bulk-modulus: 510.506 GPa
       lattice-parameter: 3.6 A [0:]
    Structure 'A2 (bcc)':
       total-energy: -4.10857 eV
       atomic-energy: -4.10857 eV/atom
       total-volume: 13.5 A^3
       atomic-volume: 13.5 A^3/atom
       bulk-modulus: 193.516 GPa
       lattice-parameter: 3 A [0:]
    Structure 'A3 (hcp)':
       total-energy: -7.36226 eV
       atomic-energy: -3.68113 eV/atom
       total-volume: 30.4176 A^3
       atomic-volume: 15.2088 A^3/atom
       bulk-modulus: 1818.47 GPa
       lattice-parameter: 2.8 A
    Structure 'A4 (diamond), relaxed ion elastic constants':
       total-energy: -8.3398 eV
       atomic-energy: -4.1699 eV/atom
       total-volume: 31.25 A^3
       atomic-volume: 15.625 A^3/atom
       bulk-modulus: 256.382 GPa
       C11: 320.231 GPa
       C12: 224.457 GPa
       C44: 53.6712 GPa
       lattice-parameter: 5 A [0:]
    Structure 'A4 (diamond), clamped ion elastic constants':
       total-energy: -8.3398 eV
       atomic-energy: -4.1699 eV/atom
       total-volume: 31.25 A^3
       atomic-volume: 15.625 A^3/atom
       bulk-modulus: 256.382 GPa
       C11: 320.231 GPa
       C12: 224.457 GPa
       C44: 210.497 GPa
       lattice-parameter: 5 A [0:]
    Structure 'B4 (wurtzite), relaxed ion elastic constants':
       total-energy: 6.62742 eV
       atomic-energy: 1.65686 eV/atom
       total-volume: 37.4123 A^3
       atomic-volume: 9.35307 A^3/atom
       bulk-modulus: 14074 GPa
       C11: 3505.34 GPa
       C12: 3893.32 GPa
       C13: 8849.11 GPa
       C33: 29374.3 GPa
       C44: 6987.52 GPa
       C66: -196.228 GPa
       lattice-parameter: 3 A [0:]
       ca-ratio: 1.6
       u-parameter: 0.375
    Structure 'B4 (wurtzite), clamped ion elastic constants':
       total-energy: 6.6917 eV
       atomic-energy: 1.67292 eV/atom
       total-volume: 37.4123 A^3
       atomic-volume: 9.35307 A^3/atom
       bulk-modulus: 14277.4 GPa
       C11: 5315.51 GPa
       C12: 2670.65 GPa
       C13: 8286.95 GPa
       C33: 31388.1 GPa
       C44: 8498.48 GPa
       C66: 1828.9 GPa
       lattice-parameter: 3 A [0:]
       ca-ratio: 1.6
       u-parameter: 0.375
    -------------------------------------------------------
    -------------------------------------------------------