Introduction to density functional perturbation theory for lattice dynamics

Transcription

1 Introduction to density functional perturbation theory for lattice dynamics SISSA and DEMOCRITOS Trieste (Italy)

2 Outline 1 Lattice dynamic of a solid: phonons Description of a solid Equations of motion The phonon solution 2 ph.x 3 q2r.x Discrete Fourier transform in 1D matdyn.x

3 Description of a solid Equations of motion The phonon solution Let s consider a periodic solid. We indicate with R I = R µ + d s (1) the equilibrium positions of the atoms. R µ indicate the Bravais lattice vectors and d s the positions of the atoms in one unit cell (s = 1,..., N at ). We take N unit cells with Born-von Karman periodic boundary conditions. Ω is the volume of one cell and V = NΩ the volume of the solid. At time t, each atom is displaced from its equilibrium position. u I (t) is the displacement of the atom I.

4 Description of a solid Equations of motion The phonon solution Within the Born-Oppenheimer adiabatic approximation the nuclei move in a potential energy given by the total energy of the electron system calculated (for instance within DFT) at fixed nuclei. We call E tot (R I + u I ) (2) this energy. The electrons are assumed to be in the ground state for each nuclear configuration. If u I is small, we can expand E tot in a Taylor series with respect to u I. Within the harmonic approximation: E tot (R I +u I ) = E tot (R I )+ Iα E tot u Iα u Iα Iα,Jβ 2 E tot u Iα u Jβ u Iα u Jβ +... where the derivatives are calculated at u I = 0 and α and β indicate the three cartesian coordinates. (3)

5 Description of a solid Equations of motion The phonon solution Equations of motion At equilibrium E tot u Iα = 0, so the Hamiltonian of the ions becomes: H = Iα P 2 Iα 2M I Iα,Jβ 2 E tot u Iα u Jβ u Iα u Jβ (4) where P I are the momenta of the nuclei and M I their masses. The classical motion of the nuclei is given by the N 3 N at functions u Iα (t). These functions are the solutions of the Hamilton equations: u Iα = H P Iα Ṗ Iα = H u Iα (5)

6 Description of a solid Equations of motion The phonon solution Equations of motion-ii With our Hamiltonian: u Iα = P Iα M I or: Ṗ Iα = Jβ M I ü Iα = Jβ 2 E tot u Iα u Jβ u Jβ (6) 2 E tot u Iα u Jβ u Jβ (7)

7 Description of a solid Equations of motion The phonon solution The phonon solution We can search the solution in the form of a phonon. Let s introduce a vector q in the first Brillouin zone. For each q we can write: u µsα (t) = 1 Ms u sα (q)e i(qrµ ωqt) (8) where the time dependence is given by a simple exponential e iωqt and the displacement of the atoms in each cell identified by the Bravais lattice R µ can be obtained from the displacements of the atoms in one unit cell, for instance the one 1 that corresponds to R µ = 0: Ms u sα (q).

8 Description of a solid Equations of motion The phonon solution The phonon solution-ii Inserting this solution in the equations of motion and writing I = (µ, s), J = (ν, s ) we obtain an eigenvalue problem for the 3 N at variables u sα (q): ω 2 qu sα (q) = s β D sαs β(q)u s β(q) (9) where: D sαs β(q) = 1 Ms M s ν is the dynamical matrix of the solid. 2 E tot e iq(rν Rµ) (10) u µsα u νs β

9 ph.x Within DFT the ground state total energy of the solid, calculated at fixed nuclei, is: E tot = ψ i ψ i + V loc (r)ρ(r)d 3 r +E H [ρ]+e xc [ρ]+u II i (11) where ρ(r) is the density of the electron gas: ρ(r) = ψ i (r) 2 (12) i and ψ i are the solution of the Kohn and Sham equations. E H is the Hartree energy, E xc is the exchange and correlation energy and U II is the ion-ion interaction. According to the Hellmann-Feynman theorem, the first order derivative of the ground state energy with respect to an external parameter is: E tot λ = Vloc (r) λ ρ(r)d 3 r + U II λ (13)

10 ph.x Deriving with respect to a second parameter µ: 2 E tot λ = + 2 V loc (r) λ ρ(r)d 3 r + 2 U II λ Vloc (r) ρ(r) λ d 3 r (14) So the new quantity that we need to calculate is the charge density induced, at first order, by the perturbation: ρ(r) = i ψi (r) ψ i(r) + ψi (r) ψ i(r) (15) To fix the ideas we can think that λ = u µsα and µ = u νs β

11 ph.x The wavefunctions obey the following equation: [ 1 ] V KS (r) ψ i (r) = ε i ψ i (r) (16) where V KS = V loc (r) + V H (r) + V xc (r). V KS (r, µ) depends on µ so that also ψ i (r, µ), and ε i (µ) depend on µ. We can expand these quantities in a Taylor series: V KS (r, µ) = V KS (r, µ = 0) + µ V KS(r) +... ψ i (r, µ) = ψ i (r, µ = 0) + µ ψ i(r) +... ε i (µ) = ε i (µ = 0) + µ ε i +... (17)

12 ph.x Inserting these equations and keeping only the first order in µ we obtain: [ 1 ] ψi (r) V KS (r) ε i = V KS ψ i(r) + ε i ψ i(r) (18) where: V KS = V loc + V H + Vxc and V H V xc 1 ρ(r ) = r r d 3 r = dv xc ρ(r) dρ (19) depend self-consistently on the charge density induced by the perturbation.

13 ph.x The induced charge density depends only on P c ψ i where P c = 1 P v is the projector on the conduction bands and P v = i ψ i ψ i is the projector on the valence bands. In fact: ρ(r) = i ψi P (r) c ψ i(r) + ψi (r)p ψ i (r) c + i ψi P (r) v ψ i(r) + ψi (r)p ψ i (r) v (20) ρ(r) = i + ij ψi P (r) c ψ i(r) + ψi (r)p c ψ j (r)ψ i(r) ψ i (r) ( ψ i ψ j + ψ i ψ j ) (21)

14 ph.x DFPT Therefore we can solve the self-consistent linear system: [ 12 ] ψ i (r) 2 + V KS (r) ε i P c = P V KS c ψ i(r) (22) where and ρ(r) = i V KS = V loc + V H + V xc ψi P (r) c ψ i(r) + ψi (r)p ψ i (r) c (23) (24)

15 ph.x The program ph.x solves this self-consistent linear system for 3 N at perturbations at a fixed vector q. With ρ(r) for all the perturbations it calculates the dynamical matrix D sαs β(q) (25) at the given q. Diagonalizing this matrix we obtain 3 N at frequencies ω q. By repeating this procedure for several q we could plot ω q as a function of q and display the phonon dispersions. However, it is more convenient to adopt a different approach that requires the calculation of the dynamical matrix in a small set of points q.

16 q2r.x Discrete Fourier transform in 1D matdyn.x The dynamical matrix of the solid: D sαs β(q) = 1 Ms M s ν 2 E tot e iq(rν Rµ) (26) u µsα u νs β is a periodic function of q with D sαs β(q + G) = D sαs β(q) for any reciprocal lattice vector G. Moreover, since in a solid all Bravais lattice points are equivalent, it does not depend on µ. Eq.26 is a Fourier expansion of a three dimensional periodic function. We have Fourier components only at the discrete values R ν of the Bravais lattice and we can write: 1 Ms M s 2 E tot u µsα u νs β = Ω (2π) 3 d 3 qd sαs β(q)e iq(rν Rµ) (27)

17 q2r.x Discrete Fourier transform in 1D matdyn.x We can use the properties of the discrete Fourier transform and sample the integral in a uniform mesh of points q. This will give the interatomic force constants only for a set of R ν neighbors of R µ. The code q2r.x reads a set of dynamical matrices calculated in a uniform mesh of q points and calculates, using Eq. 27, the interatomic force constants for a few neighbors of the point R µ = 0.

18 q2r.x Discrete Fourier transform in 1D matdyn.x Let us consider a one dimensional periodic function f (x + a) = f (x) with period a. This function can be expanded in a Fourier series and will have a discrete set of Fourier components at the point k n = 2π a n, where n is an integer (positive, negative or zero). f (x) = n c n e iknx (28) where the coefficients of the expansion are: c n = 1 a a 0 f (x)e iknx dx (29) In general, if f (x) is a sufficiently smooth function, c n 0 at large n. Now suppose that we know f (x) only in a uniform set of N points x j = j x where x = a/n and j = 0,..., N 1,

19 q2r.x Discrete Fourier transform in 1D matdyn.x then we can calculate: c n = 1 N N 1 j=0 f (x j )e i 2π N nj (30) c n is a periodic function of n and c n+n = c n. So, if N is sufficiently large that c n = 0 when n N/2 then c n is a good approximation of c n for n < N/2 and the function f (x) = n=n/2 n= N/2 c n e iknx (31) is a good approximation of the function f (x) also on the points x different from x j.

20 q2r.x Discrete Fourier transform in 1D matdyn.x Therefore, if the dynamical matrix is a sufficiently smooth function of q and the interatomic force constants decay sufficiently rapidly in real space, we can use Eq. 26 to calculate the dynamical matrix at arbitrary q, limiting the sum to a few R ν neighbors of R µ = 0. The program matdyn.x reads the interatomic force constants calculated by q2r.x and calculates the dynamical matrices at an arbitrary q using Eq. 26.

21 q2r.x Discrete Fourier transform in 1D matdyn.x This procedure fails in two cases: In metals when there are Kohn anomalies. In this case D sαs β(q) is not a smooth function of q and the interatomic force constants are long range. In polar insulators where the atomic displacements generate long range electrostatic interactions and the dynamical matrix is non analytic for q 0. This case, however, can be dealt with by calculating the Born effective charges and the dielectric constant of the material.