TUTORIAL 8: PHONONS, LATTICE EXPANSION, AND BAND-GAP RENORMALIZATION

TUTORIAL 8: PHONONS, LATTICE EXPANSION, AND BAND-GAP RENORMALIZATION 1

INVESTIGATED SYSTEM: Silicon, diamond structure Electronic and 0K properties see W. Huhn, Tutorial 2, Wednesday August 2 2

THE HARMONIC APPROXIMATION E ({R 0 + R}) E ({R 0 })+ X i @E @R i R 0 R i + 1 2 X i,j @ 2 E @R i @R j R0 R i R j 3

THE HARMONIC APPROXIMATION E ({R 0 + R}) E ({R 0 })+ X i @E @R i R 0 R i + 1 2 X i,j @ 2 E @R i @R j R0 R i R j Static Equilibrium Energy from DFT Hessian Φij 4

THE HARMONIC APPROXIMATION E ({R 0 + R}) E ({R 0 })+ X i @E @R i R 0 R i + 1 2 X i,j @ 2 E @R i @R 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). 5

THE HARMONIC APPROXIMATION E ({R 0 + R}) E ({R 0 })+ X i @E @R i R 0 R i + 1 2 X i,j @ 2 E @R i @R 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) & S. A. Baroni, Togo, F. et Oba, al., Rev. and Mod. I. Tanaka, Phys. Phys. 73, 515 Rev. (2001). B 78, 134106 (2008). 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). 6

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, 134106 (2008). Finite differences using normalized displacements d: ij = @2 E @R i @R j R 0 = @ F j F j(r 0 i + " d i) @R i R 0 " Example: Diamond Si (2 atoms in the basis): 0 B @ xx yx zx xx yx zx xy yy zy xy yy zy xz yz zz xz yz zz xx yx zx xx yx zx xy yy zy xy yy zy xz yz zz xz yz zz 1 C A Hessian has 36 entries: 6 displacements d required 8

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, 134106 (2008). Finite differences using normalized displacements d: ij = @2 E @R i @R j R 0 = @ F j F j(r 0 i + " d i) @R i R 0 " 0 B @ xx yx zx xx yx zx xy yy zy xy yy zy xz yz zz xz yz zz Example: Diamond Si (2 atoms in the basis): xx yx zx xx yx zx xy yy zy xy yy zy Space Group Analysis xz yz zz xz yz zz 1 C A 0 B @ 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 9

VIBRATIONS IN A CRYSTAL 101 e.g. N. W Ashcroft and N. D. Mermin, Solid State Physics (1976) also see Oliver Hofmann, Tuesday August 1 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!

(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

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 Rjj 0 = Rj 0 - Rj 0 phonon displacement 0.01 phonon supercell 1 1 1 k_grid 4 4 4 phonon displacement 0.01 phonon supercell 2 2 2 k_grid 2 2 2 To achieve convergence, it is essential to have a consistent description of the electronic structure for all supercell sizes: #atoms #k-points constant 13

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 Rjj 0 = Rj 0 - Rj 0 Oblique Cell: Not all Cartesian directions are treated consistently! Cubic ( spherical ) Cell: Consistent assessment of all cartesian directions! 14

VIBRATIONAL BAND STRUCTURE # control.in : Plot vibrational band structure phonon band 0 0 0 0.00 0.25 0.25 100 Gamma Delta phonon band 0.00 0.25 0.25 0 0.5 0.5 100 Delta X phonon band 0 0.5 0.5 0.25 0.50 0.75 100 X W phonon band 0.25 0.50 0.75 0.375 0.375 0.75 100 W K phonon band 0.375 0.375 0.75 0 0 0 100 K Gamma phonon band 0 0 0 0.25 0.25 0.25 100 Gamma Lambda phonon band 0.25 0.25 0.25 0.5 0.5 0.5 100 Lambda L 600 500 400 ω (cm -1 ) 300 200 100 0 Γ X W K Γ Λ L

VIBRATIONAL DENSITY OF STATES g(!) = X s Z dq (2 ) 3 (!!(q)) = X s Z!(q)=! ds 1 (2 ) 3 r!(q) # control.in : Plot vibrational density of states phonon dos 0 800 800 3 45 0.06 g(ω) (a.u.) 0.04 0.02 0 0 200 400 600 ω (cm -1 )

VIBRATIONAL DENSITY OF STATES g(!) = X s Z dq (2 ) 3 (!!(q)) = X s 600 Z!(q)=! ds 1 (2 ) 3 r!(q) 500 400 0.06 ω (cm -1 ) 300 200 100 g(ω) (a.u.) 0.04 0.02 0 Γ X W K Γ Λ L 0 0 200 400 600 ω (cm -1 )

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 18

FREE ENERGY AND HEAT CAPACITY F ha (ev) 0.2 0-0.2-0.4 c V (k B ) 6 5 4 3 @S @ 2 F (T ) C V = T = T 2 @T @T V 2 V 1 0 0 200 400 600 800 1000 T (K) 19

THE QUASI-HARMONIC APPROXIMATION 20

THE HARMONIC APPROXIMATION H = X i T i + 1 2 X i,j ij R i R j ) @H @V =0 Lattice expansion vanishes in the harmonic approximation. THE QUASI-HARMONIC APPROACH H = X i T i + 1 2 X i,j ij(v ) R i R j ) @H @V 6=0 Assess lattice expansion by explicitly accounting for the volume dependence of the Hessian.

THE QUASI-HARMONIC APPROACH (free) energy Lattice constant a0 can be determined from Birch-Murnaghan fit of E(a0) cf. William Huhn, Practical Session 2 EDFT lattice constant a0

THE QUASI-HARMONIC APPROACH (free) energy +Fha(0K) Lattice constant a0 can be determined from Birch-Murnaghan fit of E(a0) cf. William Huhn, Practical Session 2 Add vibrational free energy for each individual value of a0 EDFT lattice constant a0 23

THE QUASI-HARMONIC APPROACH (free) energy a0(t) EDFT lattice constant a0 +Fha(T2) +Fha(T1) +Fha(0K) Lattice constant a0 can be determined from Birch-Murnaghan fit of E(a0) cf. William Huhn, Practical Session 2 Add vibrational free energy for each individual value of a0 Repeat for each temperature 0K < T1 < T2 Birch-Murnaghan fits for each individual temperature allow to determine temperature dependence of lattice constant a0(t). 24

EXERCISE 3 LATTICE EXPANSION @a 5e-06 (T )= 1 a @T p 4e-06 3e-06 α (1/K) 2e-06 1e-06 0-1e-06 0 200 400 600 800 1000 Temperature (K) 25

EXERCISE 3 LATTICE EXPANSION @a 5e-06 (T )= 1 a @T p 4e-06 3e-06 α (1/K) 2e-06 1e-06 < 0? 0-1e-06 0 200 400 600 800 1000 Temperature (K) 26

ELECTRON-PHONON COUPLING 27

BAND GAP RENORMALIZATION Electronic band gaps often exhibit a distinct temperature dependence Linear extrapolation yields the bare gap at 0K, i.e., the gap for immobile nuclei (classical limit) Linear Extrapolation Actual band gap at 0K differs from the bare gap: Band gap renormalization 28 M. Cardona, Solid State Comm. 133, 3 (2005).

ELECTRON-PHONON COUPLING conduction band "(k) bare band gap valence band 29

ELECTRON-PHONON COUPLING conduction band conduction band "(k) bare band gap valence band "(k) band gap valence band 30

BAND GAP TEMPERATURE DEPENDENCE What is the physical mechanism? Exercise 4: Lattice Expansion? Exercise 5: Atomic Motion? Use results from exercise 3 Molecular Dynamics 31

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 300.000 MD_time_step 0.001 MD_schedule MD_segment 5.0 NVT_parrinello 300.000 0.050 harmonic_potential_only fc_constants.dat MD_segment 20.0 NVT_parrinello 300.000 0.050 harmonic_potential_only fc_constants.dat # Equilibration # Sample phase space 32

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! Christian Carbogno Mohsen Yarmohammadi 33 Maja Lenz Florian Knoop

LATTICE EXPANSION S. Biernacki and M. Scheffler, Phys. Rev. Lett. 63, 290 (1989). Z Free energy definition: F ha (T! 0) / d! g(!) ~! 2 600 500 equilibrium V Frequencies lowered at frequenncy (cm -1 ) 400 300 200 expanded V larger volumes! 100 0 34