Electron-phonon coupling电子声子耦合

Outline
1D case:
Etot (x)E0 E1x E2x 2 .....
total energy
Harmonic approximation
M
2
2Etot x
2
2 E2
Calculate energies
displacement x
Fit expansion coefficients
A1g
Ba Cu(2) O(2)-O(3) O(2)+O(3) O(4)
B2g
Ba Cu(2) O(4) O(3) O(2)
B3g
Ba Cu(2) O(4) O(2) O(3)
Raman Active Phonons
Theory
Spectral density [10 sr ]
100 80 60 40 20 0 0 100 200 300 400
total classical Coulomb force acting on the nucleus a stemming from all other charges of the system = electrostatic force stemming from all other nuclei + electrostatic force stemming from the electronic charges
Phonons & electron-phonon coupling
Claudia Ambrosch-Draxl Department fü r Materialphysik, Montanunversitä t Leoben, Austria Institut fü r Physik, Universitä t Graz, Austria
3N degrees of freedom
Set of 3N coupled equations
The Harmonic Approximation
N atoms per unit cell
# displacements Total energy Forces
N=2 YBa2Cu3O7: N=13 5 Ag modes
10 703 21
3 19 4
Harmonic case only! Interpolation only – no fit!
Computational Effort
Many particle Schrödinger equation
electronic coordinates
ionic coordinates
Forces in YBa2Cu3O7
Ag modes
Mode Ba Cu(2) O(2)-O(3) O(2)+O(3) O(4) LDA
103 130 327 387 452
GGA
115 144 328 405 452
Exp. [4]
118 145 335 440 500
rel. dev. LDA
F(x) E1 2E2 x .....
The Frozen-Phonon Approach
General case:
Force constant:
change of the force acting on atom a in unit cell n in direction i, when displacing atom b in unit cell m in direction j.
position [ a b c] Y Cu(1) O(1) Ba Cu(2) O(2) O(3) O(4) (½ (0 (0 (½ (0 (½ (0 (0 ½ 0 ½ ½ 0 0 ½ 0 ½ ) 0) 0) z) z) z) z) z) point symmetry mmm mmm mmm mm mm mm mm mm
-13% -10% -2% -12% -9%
rel. dev. GGA
-3% -1% -2% -8% -9%
YBa2Cu3O7: Phonon Frequencies
Ag modes
Mode Ba Cu(2) O(2)-O(3) O(2)+O(3) O(4) Ba
0.85 0.53 0.00 0.02 0.03
O(1) Cu(1)
Ba O(4)
Cu(2)
Y
O(2)
O(3)
Example: YBa2Cu3O7
Factor group analysis: q=0
5 Ag + 8 B1u + 5 B2g + 8 B2u + 5 B3g + 8 B3u
Dynamical matrix
Ag
B1u B2g
B2u
Infrared-active
Raman Scattering Intensities
O(4) mode
10 8 6 4 2 0 -2 1.0 1.5
Im ezz
60 40 20 0 -20 -40 -60 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 Energy [eV]
d(Reezz)/dq
d(Imezz)/dq
IBS force: incomplete basis set correction due to the use of a finite number of position-dependent basis functions
Core correction: contribution due to the fact that for core electrons only the spherical part of the potential is taken into account
groundstate wavefunction with respect to fixed ions
The Hellmann-Feynman Theorem
component of the electric field caused by the nuclear charge
Hellmann-Feynman force:
oxygen modes
YBa2Cu3O7: Lattice Vibrations
Phonon frequencies [cm-1]
Mode Theory
Ref. [1-2] optimized
123 147 338 422 487 65 142 222 389 593 79 141 293 372 546 105 / 103 127 / 130 312 / 327 361 / 387 513 / 452 57 133 185 365 568 72 133 257 335 524
Re ezz
2.0 2.5 3.0 3.5 4.0
Energy [eV]
Experiment
Ref. [3-7]
116-119 145-150 335-336 435-440 493-500 70 142 210 370 579 83 140 303 526
[1] R. E. Cohen et al., Phys. Rev. Lett. 64, 2575 (1990). [2] R. Kouba et al., Phys. Rev. B 56, 14766 (1997). [3] T. Strach et al., Phys. Rev. B 51, 16460 (1995). [4] G. Burns et al., Solid State Commun. 66, 217 (1988). [5] K. F. McCarty et al., Phys. Rev. B 41, 8792 (1990). [6] V. G. Hadjiev et al., Physica C 166, 1107 (1990). [7] B. Friedl et al., Solid State Commun. 76, 217 (1990). [8] K. Syassen et al., Physica C 153-155, 264 (1988).
Some important phenomena
Charge transport
Heat transport Thermal expansion Electron (hole) lifetimes Superconductivity
effective electron-electron interaction
Forces in the LAPW Basis
Interstitial: planewave basis Atomic spheres: atomic-like basis functions site-dependent!
The LAPW Method
Orthorhombic cell: Pmmm
Displacement wave:
The Harmonic Approximation
Equation of motion:
Vibrational frequ百度文库ncies :
by diagonalization of the dynamical matrix D N atoms per unit cell
The Hellmann-Feynman Force
Total energy:
Atomic force:
Pulay corrections
Forces in DFT
Hellmann-Feynman force: classical electrostatic force excerted on the nucleus by the other nuclei and the electronic charge distribution
k q k' k+q
?
k'-q
Aspects of e-ph Coupling
Basics
The frozen phonon approach Lattice dynamics Atomic forces
Phonons and electron-phonon coupling
Symmetry Vibrational frequencies Normal vectors Raman scattering Linear-response theory Comparison with experiment LAPW / WIEN2k specific aspects and examples
-1
Experiment
(zz)
100 80 60 40 20 0
500 600
-1
= 35K Ba,Cu = 18K
A1g
-7
0
100
200
300
400
-1
500
600
Raman shift [cm ]
Raman shift [cm ]
CAD, H. Auer, R. Kouba, E. Ya. Sherman, P. Knoll, M. Mayer, Phys. Rev. B 65, 064501 (2002).
Cu(2)
0.52 -0.84 0.02 0.09 -0.11
O(2)
0.05 -0.08 -0.81 -0.51 0.28
O(3)
0.05 -0.07 0.59 -0.74 0.32
O(4)
0.00 0.06 -0.05 -0.43 -0.90
YBa2Cu3O7: Normal Vectors
Ba / Cu modes
Cu(2)
0.3530 123.82 -35.22 -88.30 0.30
O(2)
0.3740 0.76 6.50 6.45 13.71
O(3)
0.3787 -8.77 7.91 0.75 -0.09
O(4)
0.1540 290.52 -75.75 -188.87 25.90
position [c] FHF [mRy / a.u.] FIBS [mRy / a.u.] Fcore[mRy / a.u.] F [mRy / a.u.]
Raman-active
B3g B3u
Bilbao Crystallographic Server: http://www.cryst.ehu.es/
Symmetry
Force contributions for a mixed distortion:
-O(2), -O(4)
Ba
0.1815 46.66 -13.03 -36.08 -2.45