Ab initio phonon calculations in mixed systems

Transcription

1 Ab initio phonon calculations in mixed systems Andrei Postnikov Outline: Experiment vs. ab initio theory Ways of theory: linear response and frozen phonon approaches Applications: Be x Zn 1 x Se (cooperation with O. Pagès et al.) LiNbO 3 with defects (coooperation with P. Bourson et al., Uni Metz-Supélec) Molecular dynamics; application to TiC clusters (work at Uni Duisburg with P. Entel) Osnabrück, November 22, 2004

2 Experimental study of vibrations Raman effect: inelastic scattering of photons Infrafed spectroscopy (in molecules; Γ modes in crystals) Excited states Stokes anti Stokes I IR ( ) 2 µ q 0 µ: dipole moment, q: normal coordinate Electronic ground state Vibrational levels infrared I Raman ( ) 2 α q 0 α: polarizability, µ ε Electron Energy Loss Spectroscopy (EELS), for surface vibrations Inelastic neutron scattering probes ω(q) in crystals directly the motion of nuclei, not mediated by electrons

3 Lattice dynamics on the Born-Oppenheimer surface classical movement of atoms in the electrostatic force field from core charges and relaxed electron density (as an adiabatic process, different from the Car-Parrinello approach!) Treat the crystal as a system of coupled oscillators, α β mass m α u α u β force constants F ij αβ = 2 E u i α u j β H = α m α 2 Ways to get force constants: 3 ( u i α) i=1 αβ 3 i,j F ij αβ ui αu j β 1) frozen phonon schemes, 2) response theories 3

4 Lattice dynamics on the Born-Oppenheimer surface A general case (cluster, molecule, aperiodic crystal) yields 3N (N: number of atoms in system) coupled equations: m α ü i α = N β 3 j F ij αβ uj β In case of translational invariancy, Ansatz u α u q e i(q r α ωt) and Fourier-transformation of force constants decouple the equation in q, m s ü i sq = yielding 3n (n: number of atoms per unit cell) coupled equations:. F ij ss (q) ms m. ω2 δ s ss δ ij u j s q ms = 0.. n s 3 j F ij ss (q)u j s q det(...)=0 frequencies ω 2 ; eigenvector/ m displacement pattern 4

5 Linear response: The Kohn-Sham equation [ ] h2 2m 2 + V SCF (r) V SCF (r) = e 2 α ρ(r) = ϕ i (r) 2 i (occupied) ϕ i (r) = ε i ϕ i (r) ; Z α r R α + e2 is linearized, introducing small parameter λ: ρ(r ) r r dr + δe XC δρ(r) ; ϕ i (r) = ϕ (0) i (r) + λ ϕ (1) i (r) ; V SCF (r) = V (0) SCF (r) + λ V (1) SCF (r). 5

6 [ Linear response h2 2m 2 + V (0) SCF (r) ε i ] ϕ (1) i (r) = V (1) SCF (r) ϕ (0) i (r) ; (Sternheimer equation) V (1) SCF (r) = e 2 α Z α w α (r R α ) ρ r R α 3 + e 2 (1) (r ) r r dr + ρ (1) (r) ρ (1) (r) = 2 Re ϕ (0) (1) i (r) ϕ i (r). i (occupied) Perturbation λw α : e.g., a phonon q with polarization A, w α = Ae iqr α + A e iqr α. Recent review: Baroni et al., Rev. Mod. Phys. 73, 515 (2001). [ ] dvxc (r) dρ ρ (0) ; 6

7 Linear response S. Savrasov, PRB 54, (1996) Recent calculation by J. Kortus: B-doped diamond, virtual crystal E(meV) 80 Black circles: calculation; triangles: experiment Review by Baroni et al., Rev. Mod. Phys. 73, 515 (2001) 40 undoped 1 % doping 5 % doping 10 % doping 0 L Γ X W

8 Non-linear response generalization of the Sternheimer equation. In the density functional theory (DFT), the 2n+1 theorem holds: the knowledge of (electron) wave functions perturbed to the n th order suffices to calculate (2n + 1) s derivatives of the total energy: X. Gonze and J.-P. Vigneron, Density-functional approach to nonlinear-response coefficients of solids, Phys. Rev. B 39, (1989). Systematic expansion towards higher orders: X. Gonze, Adiabatic density-functional perturbation theory, Phys. Rev. A 52, 1096 (1995). For example Raman intensities: I e i A e s 2 e hω kt ( ω ekt hω 1 ) e i, e s polarisation of initial and scattered waves, A lm = kγ 3 E tot E l E m u kγ ω kγ Mγ E electric field, u kγ displacement of the atom γ in the direction k. Recent calculation: M. Lazzeri and F. Mauri, First-principles calculation of vibrational Raman spectra (...) in crystalline SiO 2, Phys. Rev. Lett. 90, (2003).

9 Zn 1 x Be x Se Experiment: backscattering Raman spectra, nm Ar + excitation, of 1 µm thick Zn 1 x Be x Se layers, deposited on (001) GaAs by MBE. Excitation-detection along [110]. Observed for 0.2 < x < 0.8: two-mode behaviour, typical for random-element isodisplacement, but also an additional mode between 400 and 600 cm 1, that is not due to structural disorder (ZnSe BeSe segregation) the intensities of nominal and anomalous modes change with x and are coupled. Tentative explanation: Contrast in elastic properties related to presence or absence of percolation. Appl. Phys. Lett. 77, 519 (2000), Phys. Rev. B 65, (2001), Phys. stat. sol. (b) 229, 25 (2002). Computational modelling: First-principle supercell calculations of lattice dynamics, using DFT (Siesta) and frozen phonon approach.

10 Zn 1 x Be x Se

11 Zn 1 x Be x Se

12 Explanation of the anomalous mode 1. ZnSe and BeSe exhibit large contrast in mechanical properties, therefore isodisplacement picture does not work in the mixed crystal. 2. Crucial for the formation of anomalous mode is the onset of... -Be-Se-Be-... chains (percolation). 3. Be-Se bonds undergo larger tensile strain in such chains than around isolated Be impurities in ZnSe weaker coupling softening of TO mode. 4. With x, the distribution of Be between chains and isolated regions varies the intensities of normal and anomalous modes are coupled. This is consistent with experimental observations. Can this be confirmed microscopically? Be Zn Se relaxed supercell, containing infinite -Be-Se-Be- chains and isolated Be atoms 12

13 E tot - E min (mev) E tot - E min (mev) ZnSe BeSe The total energy vs. lattice constant curves (left) and the energy profiles corresponding to the Γ-TO phonon (right) as calculated for ZnSe and BeSe. x indicates the magnitude of the cation (or anion) displacement along [100] from the equilibrium. Open squares: WIEN2k (LDA) results; connected black dots: Siesta (LDA) results lattice parameter (Å) x (Å)

14 Elastic properties of ZnSe and BeSe from expt and first-principles calcs. ZnSe BeSe Method a B ω TO a B ω TO (Å) (Kbar) (cm 1 ) (Å) (Kbar) (cm 1 ) WIEN2k (LDA) WIEN2k (GGA) Siesta (LDA) exp other calc

15 Vibrational density of states in small supercells Be Zn Se BeZn 3 Se 4 "impurity mode" I ℵ (ω) = α ℵ A α i (ω) 2 i Zn Be Se "impurity mode" Be 3 ZnSe 4 Γ-TO acoustical zone-boundary Frequency (cm -1 ) 15 TO LO zone-boundary

16 Frozen phonon in a supercell displace this one How to obtain real-space force constants F ij αβ : or, Fouriertransformed ones F ij ss (q): measure the force measure the force here or here Γ phonon in a supercell scans different q values: reciprocal lattice cell single cell Q 1 Q 2 Γ supercell x 2 Q 4 Q 3 Q 3 Q 4 x 3 Q 1 Brillouin zone Q 2

17 Equilibrium lattice constant and anion-cation bond lengths 5.6 chain Be - chain Se chain Be - non-chain Se isolated Be - Se Zn - chain Se Zn - non-chain Se 5.5 lattice constant (Å) Be 6 Zn 26 Se Be 6 Zn 26 Se BeSe Be 0.75 Zn 0.25 Se Be 0.25 Zn 0.75 Se ZnSe 2.5 Be 4 Zn 28 Se 32 Be 4 Zn 28 Se 32 bond lengths (Å) Zn-Se Be-Se Be 3 Zn 29 Be 2 Zn 30 Se 32 Be 3 Zn 29 Se 32 Be 2 Zn 30 Se 32 Be 6 Zn 26 Se 32 BeSe Be 0.75 Zn 0.25 Se Be 0.25 Zn 0.75 Se ZnSe BeZn 31 Se Be-Se bonds (Å) BeZn 31 Se Zn-Se bonds (Å)

18 3 Elements of dynamical matrix (ev/å 2 ) Be-SeZn 3 isolated Be -Be-SeZn 2 -Be- chains Be-Se bonds Zn-Se bonds Bond length (Å) Diagonal elements of force constant matrix F ij αβ between nearest neighbors

19 anion-anion 2 Elements of ddynamical matrix (ev/å ) cation-cation Distance between next nearest neighbors (Å) ij Diagonal elements of force constant matrix Fαβ between next-nearest neighbors

20 chain Be Zn isolated Be Vibrational density of states for the Be 6 Zn 26 Se 32 supercell, resolved over different groups of atoms, calculated for q=0 of the supercell and broadened by 10 cm 1. The vertical scaling for different groups is arbitrary. I ℵ (ω) = α ℵ A α i (ω) 2. i Se with Zn neighbors only Se in the Be chains Se with 1 Be and 3 Zn neighbors A α i (ω): eigenvectors, ℵ: selected group of atoms Frequency (cm -1 ) For N atoms in the supercell, a discrete spectrum of (3N 3) lines (acoustical modes removed), broadened to 10 cm 1.

21 441 cm cm cm cm 1 Vibration patterns of four selected modes with substantial contribution of the chain Be atoms.

22 Phonon spectral function in the Be 6 Zn 26 Se 32 supercell, for three q values of the zincblende lattice. Be in the chains Zn q=( ) q=( 0 0 1/2) q=( ) isolated Be Se I ℵ (ω, q) = α ℵ A α i (ω) exp(qr α ) 2 A α i (ω): eigenvectors, ℵ: selected group of atoms i q=( ) q=( 0 0 1/2) q=( ) Frequency (cm -1 )

23 LiNbO 3 perfect lattice and ilmenite-type defect LNbO 3, Nb<->Li substitution: phonon DOS Shadowed: phonon DOS in perfect crystal Li in nominal sites Nb site Nb in nominal sites Li site Exp. Raman spectra (Bourson) O average Frequency (cm -1 )

24 Molecular dynamics Verlet algorithm: r(t + δt) = 2r(t) r(t δt) + (δt) 2F(t) M Velocity autocorrelation function: C v (τ) = 1 N N i=1 1 t max t max t 0 =1 [v i (t 0 ) v i (t 0 + τ)] Vibrational of states: density I(ω) = G(ω) 2, G(ω) = dτ C v (τ)e iωτ 24

25 MD simulation for clusters Velocity autocorrelation function Velocity autocorrelation function Ti 4 C Time (fs) Ti 14 C 13 Vibrational density of states (arb. units) TiC bulk (8-at. scell) Ti 4 C 4 cluster Ti 14 C 13 cluster Time (fs) Vibrational DOS extracted from MD simulations for bulk TiC (8-atom supercell, thick line in the Frequency (THz) top panel), Ti 4 C 4 cluster (thin line in the top panel) and the Ti 14 C 13 cluster (bottom panel). Phase Transitions 77, 149 (2004).

26 Molecular dynamics vs. frozen phonons (+) anharmonic effects automatically included (+) straightforward treatment of temperature effects (e.g., Nosé thermostat) ( ) total simulation time is limited from below by frequency resolution, t MD run 1/( ν) ; simulation time step is limited from above by the highest characteristic frequency, t 1/ν max., ( ) many simulation steps needed. at low temperatures mostly harmonic behaviour, poor ergodicity. 26