TUTORIAL 6: PHONONS, LATTICE EXPANSION, AND BAND-GAP RENORMALIZATION

Transcription

1 Hands-On Tutorial Workshop, July 29 th 2014 TUTORIAL 6: PHONONS, LATTICE EXPANSION, AND BAND-GAP RENORMALIZATION Christian Carbogno & Manuel Schöttler Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin Work Group on Statistical Physics, Institut für Physik, Universität Rostock

2 I. THE HARMONIC CRYSTAL

3 THE HARMONIC APPROXIMATION The total energy E is a 3N-dimensional surface: Total Energy E R i 0 Atomic Coordinate R i E = V (R 1, R 2,, R N ) Approximate by Taylor Expansion around the Static Equilibrium Ri 0 E ({R 0 + R}) E ({R 0 })+ Static Equilibrium Energy

4 THE HARMONIC APPROXIMATION The total energy E is a 3N-dimensional surface: Total Energy E R i 0 Atomic Coordinate R i E = V (R 1, R 2,, R N ) Approximate by Taylor Expansion around the Static Equilibrium Ri 0 E ({R 0 + R}) E ({R 0 })+ i R 0 R i + Forces vanish at R0

5 THE HARMONIC APPROXIMATION Total Energy E 0 R i Harmonic Potential Atomic Coordinate R i The total energy E is a 3N-dimensional surface: E = V (R 1, R 2,, R N ) Approximate by Taylor Expansion around the Static Equilibrium Ri 0 E ({R 0 + R}) E ({R 0 })+ i R 0 R i X 2 j R0 R i R j Hessian Φij

6 THE HARMONIC APPROXIMATION Total Energy E 0 R i Harmonic Potential Atomic Coordinate R i The total energy E is a 3N-dimensional surface: E = V (R 1, R 2,, R N ) WARNING: Harmonic Approximate Approximation by Taylor is only Expansion valid for around small the displacements Static Equilibrium from RRi 0! 0 E ({R 0 + R}) E ({R 0 })+ i R 0 R i X 2 j R0 R i R j

7 THE HARMONIC APPROXIMATION E ({R 0 + R}) E ({R 0 })+ i R 0 R i X 2 j R0 R i R j Static Equilibrium Energy from DFT Hessian Φij Determine Hessian aka the Harmonic Force Constants Φij: from Density-Functional Perturbation Theory S. Baroni, P. Giannozzi, and A. Testa, Phys. Rev. Lett. 58, 1861 (1987) & S. Baroni, et al., Rev. Mod. Phys. 73, 515 (2001). from Finite Differences K. Kunc, and R. M. Martin, Phys. Rev. Lett. 48, 406 (1982) & K. Parlinski, Z. Q. Li, and Y. Kawazoe, Phys. Rev. Lett. 78, 4063 (1997).

8 THE HARMONIC APPROXIMATION E ({R 0 + R}) E ({R 0 })+ i R 0 R i X 2 j R0 R i R j Static Equilibrium Energy from DFT Hessian Φij Determine Hessian aka the Harmonic Force Constants Φij: from Density-Functional Perturbation Theory phonopy-fhi-aims S. Baroni, P. Giannozzi, and A. Testa, Phys. Rev. Lett. 58, 1861 (1987) & A. Togo, F. Oba, and I. Tanaka, Phys. Rev. B 78, (2008). S. Baroni, et al., Rev. Mod. Phys. 73, 515 (2001). from Finite Differences K. Kunc, and R. M. Martin, Phys. Rev. Lett. 48, 406 (1982) & K. Parlinski, Z. Q. Li, and Y. Kawazoe, Phys. Rev. Lett. 78, 4063 (1997).

9 THE FINITE DIFFERENCE APPROACH K. Parlinski, Z. Q. Li, and Y. Kawazoe, Phys. Rev. Lett. 78, 4063 (1997). A. Togo, F. Oba, and I. Tanaka, Phys. Rev. B 78, (2008). Finite differences using normalized displacements d: ij j R 0 F j F j(r 0 i + " d i R 0 " Example: Diamond Si (2 atoms in the basis): 0 xx yx zx xx 21 yx 21 zx 21 xy yy zy xy 21 yy 21 zy 21 xz yz zz xz 21 yz 21 zz 21 xx yx zx xx 22 yx 22 zx 22 xy yy zy xy 22 yy 22 zy 22 xz yz zz xz 22 yz 22 zz 22 1 C A Hessian has 36 entries: 6 displacements d required

10 THE FINITE DIFFERENCE APPROACH K. Parlinski, Z. Q. Li, and Y. Kawazoe, Phys. Rev. Lett. 78, 4063 (1997). A. Togo, F. Oba, and I. Tanaka, Phys. Rev. B 78, (2008). Finite differences using normalized displacements d: ij j R 0 F j F j(r 0 i + " d i R 0 " 0 xx yx zx xx 21 yx 21 zx 21 xy yy zy xy 21 yy 21 zy 21 xz yz zz xz 21 yz 21 zz 21 Example: Diamond Si (2 atoms in the basis): xx yx zx xx 22 yx 22 zx 22 xy yy zy xy 22 yy 22 zy 22 Space Group Analysis xz yz zz xz 22 yz 22 zz 22 1 C A 0 xx xy xx 0 xy 0 xx xy xx 0 xy 0 xz yz xx xz yz xx xx yz xz xx yz xz xy 0 xx 0 xy xx xy 0 xx 0 xy xx 1 C A Hessian has 5 unique, non-zero entries: Only 1 displacement d required

11 (a) Edit your file control.in so that it contains the following lines phonon displacement 0.01 (b) Run phonopy-fhi-aims by typing phonopy-fhi-aims (c) Change into the directory phonopy-fhi-aims-displacement-01 and run FHI-aims: cd phonopy-fhi-aims-displacement-01 mpirun -np 4 aims.x > phonopy-fhi-aims-displacement-01.out (d) Change into parent directory and run phonopy-fhi-aims again cd.. phonopy-fhi-aims

12 THE HARMONIC APPROXIMATION...in Molecules:...in Crystalline Solids: N... Number of atoms Degrees of Freedom: 3N Dimension of Hessian: 9N 2 N... Number of atoms Degrees of Freedom: 3N Dimension of Hessian: 9N 2 Tuesday July 22: Björn Lange, Nuts and Bolts of DFT II O. Hofmann & L. Nemec, Tutorial 1 BUT: N

13 PERIODIC BOUNDARY CONDITIONS cf. Christian Ratsch, Electronic Structure Theory for Periodic Systems: The Concepts, Tuesday July 22 Periodic Images Unit Cell with Np atoms Periodic Images Lattice vector: E 0 Real Space: Hessian Φij with i,j Fourier Transform D i 0 j 0 (q) = X j e i (q (R 0 j R 0 j 0 )) p Mi 0M j 0 i 0 j Reciprocal Space: Dynamical Matrix Di j (q) with i,j Np

14 VIBRATIONS IN A CRYSTAL 101 K. Parlinski, Z. Q. Li, and Y. Kawazoe, Phys. Rev. Lett. 78, 4063 (1997). Real Space: Hessian Φij with i,j Fourier Transform D i 0 j 0 (q) = X j e i (q (R 0 j R 0 j 0 )) p Mi 0M j 0 i 0 j Reciprocal Space: Dynamical Matrix Di j (q) with i,j Np Fourier Transform can be truncated since Φij = 0 for large Rj 0 - Rj 0 Hessian Φij with finite number of non-zero entries Dynamical Matrix Di j (q) known for the whole reciprocal space

15 VIBRATIONS IN A CRYSTAL 101 e.g. N. W Ashcroft and N. D. Mermin, Solid State Physics (1976) also see Björn Lange, Nuts and Bolts of DFT II, Tuesday July 22 Dynamical matrix: D i 0 j 0 (q) = X j e i (q (R 0 j R 0 j 0 )) p Mi 0M j 0 i 0 j Equation of Motion becomes an Eigenvalue Problem: D(q) [ (q)] =! 2 (q) [ (q)] Analytical Solution in Real Space: Superposition of Harmonic Oscillations R j (t) = R 0 j + Re X s A s p Mi e i (q (R 0 j R 0 j 0 )! s (q)t) [ s (q)] j 0!

16 VIBRATIONS IN A CRYSTAL 101 e.g. N. W Ashcroft and N. D. Mermin, Solid State Physics (1976) Björn Lange, Nuts and Bolts of DFT II, Tuesday July 22 Dynamical matrix: D i 0 j 0 (q) = X j e i (q (R j R j 0 )) p Mi 0M j 0 i 0 j! Eigenvalue problem: X q D(q) [ (q)] =! 2 (q) [ (q)]

17 VIBRATIONS IN A CRYSTAL 101 e.g. N. W Ashcroft and N. D. Mermin, Solid State Physics (1976) Björn Lange, Nuts and Bolts of DFT II, Tuesday July 22 Dynamical matrix: " D i 0 j 0 (q) = X j =1 e i (q (R j R j 0 )) p Mi 0M j 0 i 0 j! X q Eigenvalue problem: " " " " D(q) [ (q)] =! 2 (q) [ (q)]

18 VIBRATIONS IN A CRYSTAL 101 e.g. N. W Ashcroft and N. D. Mermin, Solid State Physics (1976) Björn Lange, Nuts and Bolts of DFT II, Tuesday July 22 Dynamical matrix: X D i 0 j 0 (q) = X j e i (q (R j R j 0 )) p Mi 0M j 0 i 0 j! X q Eigenvalue problem: X X X X D(q) [ (q)] =! 2 (q) [ (q)]

19 VIBRATIONS IN A CRYSTAL 101 e.g. N. W Ashcroft and N. D. Mermin, Solid State Physics (1976) Björn Lange, Nuts and Bolts of DFT II, Tuesday July 22 For Np atoms in the unit cell there are: 3 Acoustic modes:! - Atoms in unit cell in-phase - Acoustic modes vanish at " - Strong (typically linear) dispersion close to " X q (3Np - 3) Optical modes: - Atoms in unit cell out-of-phase - ω > 0 at "#(and everywhere else) - Weak dispersion

20 CONVERGING THE SUPERCELL Fourier Transform can be truncated since Φij = 0 D i 0 j 0 (q) = X j e i (q (R 0 j R 0 j 0 )) p Mi 0M j 0 i 0 j for large Rij 0 = Rj 0 - Rj 0 phonon displacement 0.01 phonon supercell k_grid phonon displacement 0.01 phonon supercell k_grid 2 2 2

21 CONVERGING THE SUPERCELL Fourier Transform can be truncated since Φij = 0 D i 0 j 0 (q) = X j e i (q (R 0 j R 0 j 0 )) p Mi 0M j 0 i 0 j for large Rij 0 = Rj 0 - Rj 0 phonon displacement 0.01 phonon supercell k_grid phonon displacement 0.01 phonon supercell k_grid To achieve convergence, it is essential to have a consistent description of the electronic structure for all supercell sizes: #atoms #k-points constant

22 CONVERGING THE SUPERCELL Fourier Transform can be truncated since Φij = 0 D i 0 j 0 (q) = X j e i (q (R 0 j R 0 j 0 )) p Mi 0M j 0 i 0 j for large Rij 0 = Rj 0 - Rj 0 Oblique Cell: Not all Cartesian directions are treated consistently! Cubic ( spherical ) Cell: Consistent assessment of all cartesian directions!

23 VIBRATIONAL BAND STRUCTURE # control.in : Plot vibrational band structure phonon band Gamma Delta phonon band Delta X phonon band X W phonon band W K phonon band K Gamma phonon band Gamma Lambda phonon band Lambda L ω (cm -1 ) Γ X W K Γ Λ L

24 VIBRATIONAL DENSITY OF STATES g(!) = X s Z dq (2 ) 3 (!!(q)) = X s Z!(q)=! ds (2 ) 3 1 r!(q) # control.in : Plot vibrational density of states phonon dos g(ω) (a.u.) ω (cm -1 )

25 THE HARMONIC FREE ENERGY F ha (T ) = E({R 0 }) Z Static Equilibrium Energy + + Z d d g( ) ~ 2 Zero-point vibration g( ) k B T ln 1 e ~ k B T Thermally induced vibrations

26 FREE ENERGY AND HEAT CAPACITY # control.in : Plot harmonic contribution to F(T) phonon free_energy F ha (ev) c V (k B ) C V T (K) V = 2 F (T 2 V

27 THE QUASI-HARMONIC APPROXIMATION

28 THE HARMONIC APPROXIMATION H = X i T i X i,j ij R i R =0 Lattice expansion vanishes in the harmonic approximation. THE QUASI-HARMONIC APPROACH H = X i T i X i,j ij(v ) R i R 6=0 Assess lattice expansion by explicitly accounting for the volume dependence of the Hessian.

29 THE QUASI-HARMONIC APPROACH (free) energy +Fha(0K) Lattice constant a0 can be determined from Birch-Murnaghan fit of E(a0) cf. L. Nemec & B. Bienik, Tutorial 2 Add vibrational free energy for each individual value of a0 EDFT lattice constant a0

30 THE QUASI-HARMONIC APPROACH Lattice constant a0 can be (free) energy a0(t) +Fha(T2) +Fha(T1) +Fha(0K) determined from Birch-Murnaghan fit of E(a0) cf. L. Nemec & B. Bienik, Tutorial 2 Add vibrational free energy for each individual value of a0 EDFT Repeat for each temperature lattice constant a0 0K < T1 < T2 Birch-Murnaghan fits for each individual temperature allow to determine temperature dependence of lattice constant a0(t).

31 ELECTRON-PHONON COUPLING

32 BAND GAP RENORMALIZATION Electronic band gaps often exhibit a distinct temperature dependence Linear Extrapolation Linear extrapolation yields the bare gap at 0K, i.e., the gap for immobile nuclei (classical limit) Actual band gap at 0K differs from the bare gap: Band gap renormalization M. Cardona, Solid State Comm. 133, 3 (2005).

33 BAND GAP TEMPERATURE DEPENDENCE What is the physical mechanism? Exercise 4: Lattice Expansion? Exercise 5: Atomic Motion? Use results from exercise 3 Molecular Dynamics

34 MOLECULAR DYNAMICS Numerical Integration of the equations of motion L. Verlet, Phys. Rev. 159, 98 (1967). M I RI (t) = r Ri E DFT cf. Mariana Rossi & Luca Ghiringhelli, Tutorial 4: Molecular Dynamics

35 HARMONIC MOLECULAR DYNAMICS M I RI (t) = r Ri E DFT M I RI (t) = X j ij R j Harmonic Approximation # Molecular Dynamics MD_MB_init MD_time_step MD_clean_rotations.false. MD_schedule MD_segment 5.0 NVT_parrinello harmonic_potential_only fc_constants.dat MD_segment 20.0 NVT_parrinello harmonic_potential_only fc_constants.dat # Equilibration # Sample phase space

36 WARNING: In the following exercises, the computational settings, in particular the reciprocal space grid (tag k_grid), the basis set, and supercells, have been chosen to allow a rapid computation of the exercises in the limited time and within the CPU resources available during the tutorial session. In a real production calculation, the reciprocal space grid, the basis set, and the supercells would all have to be converged with much more care, but the qualitative trends hold already with the present settings Happy Computing! Manuel Schöttler Uni Rostock Bryan Goldsmith UCSB Daniel Berger TU Munich Björn Lange Duke Patrick Rinke FHI