Electron-phonon coupling: a tutorial W. Hbner, C. D. Dong, and G. - PowerPoint PPT Presentation

Electron-phonon coupling: a tutorial W. Hübner, C. D. Dong, and G. Lefkidis University of Kaiserslautern and Research Center OPTIMAS, Box 3049, 67653 Kaiserslautern, Germany Targoviste, 29 August 2011

Outline 1. The harmonic oscillator  real space  energy basis 2. 1D lattice vibrations  one atom per primitive cell  two atoms per primitive cells 3. Electron ‐ phonon interactions  localized electrons  small ‐ polaron theory  phonons in metals 4. Superconductivity 5. A numerical example: CO 6. Literature

Outline 1. The harmonic oscillator  real space  energy basis 2. 1D lattice vibrations  one atom per primitive cell  two atoms per primitive cells 3. Electron ‐ phonon interactions  localized electrons  small ‐ polaron theory  phonons in metals 4. Superconductivity 5. A numerical example: CO 6. Literature

1) The harmonic oscillator quantization of the oscillator in real space Eigenvalues of Projection of eigenvalue equation to X basis (Substitution by differential operators) leads to 2 2  d 1       2 2 ( m x ) E 2 2 m dx 2

1) The harmonic oscillator quantization of the oscillator in real space 1) Dimensionless variables   2 2 2 2 4 2 d mEb m b y       2 x by 0 leads to 2  2  2 dy 1 2   2  mEb E      and  b m     2   10 5       '' 2 (2 y ) 0 - 10 - 5 5 10 - 5 2 2        '' 2 m y 0 with solution y Ay e since     2 m 1 m m ( 1)     2       2   '' m 2 y 2 m 2 y 2 2 Ay e  1  Ay e y  2 4 y   y y

1) The harmonic oscillator quantization of the oscillator in real space          '' with solution A cos[ 2 y ] B sin[ 2 y ] 2 0 consistency requires 1.0      2 A cy O y ( )  y 0 0.5 thus (3) - 10 - 5 5 10 - 0.5 2 2    y ansatz: ( ) y u y e ( ) - 1.0      '' ' leads to u 2 yu (2 1) 0

1) The harmonic oscillator quantization of the oscillator in real space    4) Power ‐ series expansion: n u y ( ) C y n  n 0          n 2 n n C n n [ ( 1) y 2 ny (2 1) y ] 0 inserted into differential equation n  n 0     n 2 C n n ( 1) y n  n 2   m n 2 with index shift          m n C ( m 2)( m 1) y C ( n 2)( n 1) y   m 2 n 2   m 0 n 0          n y C [ ( n 2)( n 1) C (2 1 2 )] n 0 we get  n 2 n  n 0    (2 1 2 ) n  feeding back in the original leads to recursion: C C    n 2 n ( n 2)( n 1)

1) The harmonic oscillator quantization of the oscillator in real space so we have                     2 4 3 5 (1 2 ) y (1 2 ) (4 1 2 ) y (2 1 2 ) y (2 1 2 ) (6 1 2 ) y             u y ( ) C 1 C y             0 1     (0 2)(0 1) (0 2)(0 1) (2 2)(2 1) (1 2)(1 1) (1 2)(1 1) (3 2)(5 1) Problems => way out: termination of series required       2 4 n C C y C y ... C y       2 2 y 2 y 2  0 2 4 n  ( ) y u y e ( ) e     3 5 n   C C y C y ... C y 1 3 5 n

1) The harmonic oscillator quantization of the oscillator in real space consequence energy quantization of the harmonic oscillator by backwards substitution 1     E ( n ) n 2 Examples: y  H 0 ( ) 1  H 1 ( ) y 2 y      2 H 2 ( ) y 2 1 2 y   2    3   H ( ) y 12 y y 3   3   4    2 4 ( ) 12 1 4   H y y y 4   3

Outline 1. The harmonic oscillator  real space  energy basis 2. 1D lattice vibrations  one atom per primitive cell  two atoms per primitive cells 3. Electron ‐ phonon interactions  localized electrons  small ‐ polaron theory  phonons in metals 4. Superconductivity 5. A numerical example: CO 6. Literature

1) The harmonic oscillator quantization of the oscillator in energy basis Oscillator in energy basis   2 P 1    2 2   m X E E E   2 m 2 Direct way: Fourier transform from real to momentum space No savings compared to direct solution of Schrödinger equation in real space

1) The harmonic oscillator quantization of the oscillator in energy basis commutator       X P , i I i definition and adjoint  1 2 1 2  1 2 1 2         m 1 m 1              a X i P a X i P               2 2 m 2 2 m further    a a  , 1   New operator (dimensionless) H    ˆ H ( a a 1 2)  

1) The harmonic oscillator quantization of the oscillator in energy basis Commutator of creation and annhiliation operators with Hamiltonian             ˆ a H , a a a , 1 2 a a a , a             ˆ a , H a   Raising and lowering properties                ˆ ˆ ˆ ˆ Ha aH [ , a H ] aH a ( 1) a      a C  1  1 But eigenvalues non ‐ negative requirement a   0 0 1    a a   no further lowering allowed 0 0 0 2

1) The harmonic oscillator quantization of the oscillator in energy basis 1            ( n 1 2), n 0,1,2,... E ( n 1 2) , n 0,1,2,... 0 n 0 2 A possible second family must have the same ground state, thus it is not allowed      * a n C n 1 and adjoint equation 1 n a n C n n form scalar product of both equation     * n a a n n 1 n 1 C C n n   ˆ * n H 1 2 n C C n n  2 2  n e   1 2 i n n n C C n C n n n

1) The harmonic oscillator quantization of the oscillator in energy basis   1 2 a n n n 1     1 2 a n ( n 1) n 1       1 2 1 2 1 2 a a n a n n 1 n n n n n   with number operator N a a   ˆ H N 1 2 further    n  1 2 1 2 n a n ' n n n ' 1  n n ', 1        1 2 1 2 n a ' n ( n 1) n n ' 1 ( n 1)  n n ', 1

1) The harmonic oscillator quantization of the oscillator in energy basis position and momentum operators 1 2 1 2                 X ( a a ) P ( a a ) m  m      2 2    n 0 n 1 n 2 ...   1 2 0 1 0 0 ...   0 0 0 ...    0   n 1 2 0 0 2 0   1 2 1 0 0        n 1  1 2 a a   0 0 0 3 1 2 0 2 0      n 2   . 1 2   0 0 3   .   . .

1) The harmonic oscillator quantization of the oscillator in energy basis What do we learn?      1 2 1 2 0 1 0 0 ... 0 1 0 0 ...      1 2 1 2 1 2 1 2 1 0 2 0 1 0 2 0          1 2  1 2   1 2 1 2   1 2 1 2  0 2 0 3  0 2 0 3 m           X P i m       1 2   1 2  2 2 0 0 3 0 0 0 3 0     . .             . . Analogously for derived operators   1 2 0 0 0 ...   0 3 2 0 0        H 0 0 5 2   .       .

Outline 1. The harmonic oscillator  real space  energy basis 2. 1D lattice vibrations  one atom per primitive cell  two atoms per primitive cells 3. Electron ‐ phonon interactions  localized electrons  small ‐ polaron theory  phonons in metals 4. Superconductivity 5. A numerical example: CO 6. Literature

2) 1D lattice vibrations (phonons) 1 atom per primitive cell    force on one atom F C ( u u )  s p s p s p 2 d u    s M C ( u u ) equation of motion of atom  p s p s 2 dt p solution in the form of traveling wave      i s ( p Ka ) i t u ue e s p EOM reduces to                2 ipKa 2 isKa i t i s ( p Ka ) isKa i t ( 1) M C e Mue e C ( e e ) ue p p p p translational symmetry 2      2       C (1 cos pKa ) 2 ipKa ipKa finally leads to M C ( e e 2) p M p  p 0  p 0