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
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