ab initio Lattice Vibrations: Calculating the Thermal Expansion Coeffcient Felix Hanke & Martin Fuchs June 30, 2009 This afternoon s plan


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1 ab initio Lattice Vibrations: Calculating the Thermal Expansion Coeffcient Felix Hanke & Martin Fuchs June 3, 29 This afternoon s plan introductory talk Phonons: harmonic vibrations for solids Phonons: how Thermodynamics with phonons practical exercise Obtain & assess phonons for Si Calculate & understand thermal expansion
2 Recap: Molecular vibrations (nonperiodic) Newton s equations for small displacements of atom!, coordinate i about PES minimum: m α ẍ α,i = F α,i ({x β,j }) m α ẍ α,i + 2 E (δx α,i ) (δx β,j ) δx β,j β,j ω 2 [ m α δx α,i ]= β,j In symmetric matrix form 1 mα m β 2 E (δx α,i ) (δx β,j ) [ m j δx β,j ] The harmonic lattice Translational symmetry (PBC s) leads to kdependence One eigenvalue problem for each k, giving 3Natoms modes ω 2 n(k)ξ n (k) =D(k)ξ n (k) D is the dynamic matrix, contains information from all supercells R, all pairs of atoms! and ", and all pairs of coordinates i and j: D αβ,ij = R e ik R mi m j 2 E r (R=),αi r (R),βj
3 Phonon dispersion relation Diamond fcc conventional cell 2 atoms per primitive cell = 6 phonon branches a = 3.5Å Recall diatomic molecules: 2 atoms = 6 degrees of freedom! =! =! =!vib 3 translations 2 rotations 1 vibration How about solids? 1 5 Frequency " (cm 1 )15 Phonon dispersion relation Diamond fcc conventional cell 2 atoms per primitive cell = 6 phonon branches optical branches! X W K! L kvector acoustic branches a = 3.5Å DOS
4 Phonon density of states DOS = number of phonon modes per unit frequency per unit cell g(ω) = δ(ω ω n (k)) exp [ (ω ω n(k)) 2 ] 2σ 2 n,k n,k smearing # makes it feasible to Converge smearing vs plot (same as electronic DOS!) kpoint sampling! numerical computation: frequency interval [$+%$/2,$%$/2] with discrete kpoint sampling g(ω) = w ω+ ω/2 k dω exp [ (ω ω n(k)) 2 ] 2π 2σ 2 n,k ω ω/2 The Direct Method approximation: finite interaction distance, use finite supercells D αβ,ij = R e ik R mi m j 2 E r (R=),αi r (R),βj
5 Harmonic approximation: Thermodynamics Approximate thermal effects with independent harmonic oscillators: one for each mode F = k,n = k,n ln Z(k,N) [ ωk,n 2 ] + k B T ln[1 exp( ω k,n /k B T )] Practically, one tends to use the density of states: = dω g(ω) [ ] ω 2 + k BT ln[1 exp( ω/k B T )] Quasiarmonic approximation: Changing lattice constants Alternative to explicit treatment of each phonon at all a s Anharmonic properties implicitly through dependence on lattice constant Treat many lattice constants harmonically & minimize free energy over all a s
6 Lattice Constants: Zero Point Vibrations how does the DOS change as fct of lattice constant? does this have any effect on the lattice? Density of states a = 3.3 Å a = 3.5 Å a = 3.7 Å diamond a Phonon frequency! (cm 1 ) Energy (ev) 1.5 Lattice Constants: Zero Point Vibrations Decreasing phonon free energy at larger a: Lattice constant including ZPE is larger than T= result ZPE = Å Å dω g(ω) ω 2 diamond Lattice constant (Å) no ZPE ZPE only sum
7 Lattice constants: Temperature effects Murnaghan fits of Ftot(a,T) at different temperatures lattice constant & Bulk modulus a (Å) diamond quasiharmonic no phonons B (MBar) quasiharmonic no phonons Temperature (K) " (K 1 ) Thermal expansion coefficient need to minimize F(a,T) at a given temperature to find lattice constant F (a, T )=E el (a)+f ph (a, T ) Differentiate to find α(t )= 1 a Watch the scale!!! 4!16 3!16 2!16 1!16 da dt diamond Temperature (K)
8 Heat capacity: cv Computed from free energy c v (T ) = T ds dt = T 2 F V T 2 V = dω g(ω) ( ω)2 exp( ω/k B T ) k B T 2 (exp( ω/k B T ) 1) 2 Grüneisen parameter: coupling to! α = γc v 3B c v (k B /unit cell) diamond Temperature (K) Breakdown of the harmonic approximation High temperatures: anharmonic excitations soft modes  dynamically stabilized structures V ( ξ) imaginary frequency Quantum effects, eg in hydrogen bonded crystals ξ
9 Beyond the quasiharmonic approximation MD to describe complete PES beyond quasiharmonic approximation F = E el + F qh + F anh + F vac thermodynamic integration: slowly switching on anh terms from qh solution Determine from selfconsistently optimizing vacancy volume Grabowski, Ismer, Hickel & Neugebauer, Phys. Rev. B (29) Today s tutorial: Thermal properties of Silicon this talk afternoon exercise
10 Schedule for this tutorial (A) simple phonon dispersion relation (B) converging the supercell 45 min 45 min (C) zero point vibrations 8 min (D) thermal expansion 4 min Exercise A: calculations of phonons with FHIaims use Si (diamond structure) lattice constant = 5.44 Å, light species defaults Supercell size 2x2x2 Compute: phonon dispersion relation for the directions "XWK"L density of states, specific heat cv converge DOS see also: FHIaims manual, section 4.5
11 Exercise A: calculations of phonons with FHIaims Phonon calculation works similar to vibrations: run aims.phonons.workshop.mpi.pl in the directory containing your control.in and geometry.in files. Output phonon_band_structure.dat  same format as ebands phonon_dos.dat  frequency, density of states phonon_free_energy.dat  T, F(T), U(T), cv(t), Svib(T) output stream  status reports phonon_workdir/  working files & restart info Attention Please specify a SINGLE PRIMITIVE CELL ONLY! The script does all the necessary copying & displacing. Exercise A: calculations of phonons with FHIaims Phonon dispersion calculation completely driven by phonon keyword in control.in (see FHIaims manual, section 3.5) supercell size  phonon supercell (for now) DOS specification  phonon dos Free energy  phonon free_energy kgrid: hand set to match supercell size, i.e. k_grid Band structure: phonon band <start> <end> <Npoints> <sname> <ename> working directory: /usr/local/aimsfiles/tutorial6/exercise_a
12 Exercise A: Solutions Frequency " (cm 1 ) ! X W K! L c v (k B /unit cell) kvector Temperature (K) DOS 4 2 Exercise B: Supercell convergence Compute phonon dispersion and DOS for the supercell sizes 4x4x4, 6x6x6 Use the converged DOS settings from the last exercise. All necessary files are in directory /usr/local/aimsfiles/tutorial6/exercise_b NOTE: We have provided partial control.in and geometry.in files as well as ALL of the required DFT output for the remainder of this tutorial
13 Exercise B: solutions Phonon frequency! (cm 1 ) x2x2 4x4x4 6x6x6! X W K! L kvector Exercise B: solutions DOS (arb units) 2x2x2 4x4x4 6x6x6 2 4 Phonon frequency (cm 1 )
14 Exercise C: zero point vibrations Using data for the lattice constants provided in the directory /usr/local/aimsfiles/tutorial6/exercise_c+d For each lattice constant provided, calculate the total energy of a single unit cell using a 12x12x12 kpoint grid. Store the result in the format 4x4x4_a*.***/output.single_point_energy Again, for each lattice constant, calculate the phonon free energy & specific heat for a temperature range from K to 8K in 81 steps (***) Extract the ZPE from the T= free energies from file phonon_free_energy.dat in each directory produce a Murnaghan fit with and without ZPE and plot the two fits & the resulting lattice constants (***) These exact settings are important for exercise D Exercise C: solutions Energy (mev) ZPE E(a), no ZPE E(a), incl ZPE Lattice constant a (Å)
15 Exercise D: compute a(t), cv &! Again, using the data provided directory /usr/local/aimsfiles/tutorial6/exercise_c+d use the script eval_alpha.sh to calculate the lattice constant at different temperatures & the thermal expansion coefficient eval_alpha.sh has to be started in the directory exercise_c+d and requires the EXACT specifications of phonon free_energy from the last exercise. find cv(t) at the optimal lattice constant a=5.44å Compare with previously calculated values Exercise D: solutions c v (k B /unit cell) Specific heat at constant volume Temperature (K)
16 Exercise D: solutions lattice constant a(t) & thermal expansion coefficient! " (K 1 ) a (Å) ! Temperature (K) Density of states Density oof states (arb units) a = 5.4 Å a = 5.45 Å a = 5.5 Å frequency reordering 25 5 Phonon frequency! (cm 1 ) a = 5.4 Å a = 5.45 Å a = 5.5 Å
17 Phonon frequency " (cm 1 ) Phonon band structure! X W K! L kvector DOS Negative thermal expansion coefficient: Why optical phonon bands become softer acoustic bands become harder in some part of the BZ
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