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
Recap: Molecular vibrations (non-periodic) 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 k-dependence 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
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 k-vector acoustic branches a = 3.5Å DOS 15 1 5
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!) k-point sampling! numerical computation: frequency interval [$+%$/2,$-%$/2] with discrete k-point 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
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
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 5 1 15 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 = 3.532 Å 3.519 Å dω g(ω) ω 2 diamond 3.4 3.5 3.6 3.7 Lattice constant (Å) no ZPE ZPE only sum
Lattice constants: Temperature effects Murnaghan fits of Ftot(a,T) at different temperatures lattice constant & Bulk modulus a (Å) 3.54 3.53 3.52 diamond quasiharmonic no phonons B (MBar) 4.7 4.6 quasiharmonic no phonons 4.5 1 2 3 4 5 6 7 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!1-6 3!1-6 2!1-6 1!1-6 da dt diamond 1 2 3 4 5 6 7 Temperature (K)
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) 6 4 2 diamond 1 2 3 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
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 self-consistently optimizing vacancy volume Grabowski, Ismer, Hickel & Neugebauer, Phys. Rev. B 79 13416 (29) Today s tutorial: Thermal properties of Silicon this talk afternoon exercise
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 FHI-aims use Si (diamond structure) lattice constant = 5.44 Å, light species defaults Supercell size 2x2x2 Compute: phonon dispersion relation for the directions "-X-W-K-"-L density of states, specific heat cv converge DOS see also: FHI-aims manual, section 4.5
Exercise A: calculations of phonons with FHI-aims 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 e-bands 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 FHI-aims Phonon dispersion calculation completely driven by phonon keyword in control.in (see FHI-aims manual, section 3.5) supercell size - phonon supercell 2 2 2 (for now) DOS specification - phonon dos 6 6 5 2 Free energy - phonon free_energy 8 81 2 k-grid: hand set to match supercell size, i.e. k_grid 6 6 6 Band structure: phonon band <start> <end> <Npoints> <sname> <ename> working directory: /usr/local/aimsfiles/tutorial6/exercise_a
Exercise A: Solutions Frequency " (cm -1 ) 5 4 3 2 1! X W K! L c v (k B /unit cell) 6 4 2 k-vector 2 4 6 8 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
Exercise B: solutions Phonon frequency! (cm -1 ) 5 25 5 25 5 25 2x2x2 4x4x4 6x6x6! X W K! L k-vector Exercise B: solutions DOS (arb units) 2x2x2 4x4x4 6x6x6 2 4 Phonon frequency (cm -1 )
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 k-point 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) 6 4 2 ZPE E(a), no ZPE E(a), incl ZPE 5.3 5.35 5.4 5.45 5.5 5.55 Lattice constant a (Å)
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) 6 4 2 Specific heat at constant volume 2 4 6 8 Temperature (K)
Exercise D: solutions lattice constant a(t) & thermal expansion coefficient! " (K -1 ) 5.428 a (Å) 5.432 5.424 2!1-6 1 2 3 4 5 6 7 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
Phonon frequency " (cm -1 ) 6 5 4 3 2 1 Phonon band structure! X W K! L k-vector DOS Negative thermal expansion coefficient: Why optical phonon bands become softer acoustic bands become harder in some part of the BZ