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

SLIDE 1

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

SLIDE 2

Outline

• 1. The harmonic oscillator

 real space  energy basis

• 2. 1D lattice vibrations



• ne 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

SLIDE 3

Outline

• 1. The harmonic oscillator

 real space  energy basis

• 2. 1D lattice vibrations



• ne 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

SLIDE 4

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 2 2 2

1 ( ) 2 2 d m x E m dx       

SLIDE 5

1) The harmonic oscillator

quantization of the oscillator in real space

1) Dimensionless variables

x by 

1 2

b m        

2 2

mEb E      

'' 2

(2 ) y      

'' 2

y    

2 2 2 2 4 2 2 2 2

2 d mEb m b y dy          leads to and

2 2

m y

Ay e 





with solution since

2 2

'' 2 2 2 2 2 2 4

2 1 ( 1) 1

m y m y y

m m m Ay e Ay e y y y  

    

             

• 10
• 5

5 10

• 5

5 10

SLIDE 6

1) The harmonic oscillator

quantization of the oscillator in real space

with solution

''

2     cos[ 2 ] sin[ 2 ] A y B y     

consistency requires

2

( )

y

A cy O y 



    thus (3)

2 2

( ) ( )

y

y u y e 





ansatz: leads to

'' '

2 (2 1) u yu     

• 10
• 5

5 10

• 1.0
• 0.5

0.5 1.0

SLIDE 7

1) The harmonic oscillator

quantization of the oscillator in real space

4) Power‐series expansion: ( )

n n n

u y C y

 



2

[ ( 1) 2 (2 1) ]

n n n n n

C n n y ny y 

  

    



2 2

( 1)

n n n

C n n y

  





2 m n  

2 2

( 2)( 1) ( 2)( 1)

m n m n m n

C m m y C n n y

     

    

 

inserted into differential equation with index shift we get

2

(2 1 2 ) ( 2)( 1)

n n

n C C n n 



     feeding back in the original leads to recursion:

2

[ ( 2)( 1) (2 1 2 )]

n n n n

y C n n C n 

  

     



SLIDE 8

1) The harmonic oscillator

quantization of the oscillator in real space

so we have Problems => way out: termination of series required

2 2

2 4 2 2 2 4 3 5 1 3 5

... ( ) ( ) ...

n y y n n n

C C y C y C y y u y e e C C y C y C y 

 

                

2 4 3 5 1

(1 2 ) (1 2 ) (4 1 2 ) (2 1 2 ) (2 1 2 ) (6 1 2 ) ( ) 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) y y y y u y C C y                                                

SLIDE 9

1) The harmonic oscillator

quantization of the oscillator in real space

consequence energy quantization of the harmonic oscillator by backwards substitution 1 ( ) 2

n

E n     Examples:

0( )

1 H y 

1( )

2 H y y 

 

2 2( )

2 1 2 H y y   

3 3

2 ( ) 12 3 H y y y         

2 4 4

4 ( ) 12 1 4 3 H y y y         

SLIDE 10

Outline

• 1. The harmonic oscillator

 real space  energy basis

• 2. 1D lattice vibrations



• ne 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

SLIDE 11

1) The harmonic oscillator

quantization of the oscillator in energy basis

Oscillator in energy basis

2 2 2

1 2 2 P m X E E E m          Direct way: Fourier transform from real to momentum space No savings compared to direct solution of Schrödinger equation in real space

SLIDE 12

1) The harmonic oscillator

quantization of the oscillator in energy basis

commutator

 

, X P i I i    

1 2 1 2

1 2 2 m a X i P m                  

1 2 1 2

1 2 2 m a X i P m  



                , 1 a a      definition and adjoint further New operator (dimensionless) ˆ ( 1 2) H H a a 



  

SLIDE 13

1) The harmonic oscillator

quantization of the oscillator in energy basis

Commutator of creation and annhiliation operators with Hamiltonian Raising and lowering properties ˆ , , 1 2 , a H a a a a a a a

 

                ˆ , a H a

 

     

   

ˆ ˆ ˆ ˆ [ , ] ( 1) Ha aH a H aH a a           

1

1 a C  

 

  But eigenvalues non‐negative requirement no further lowering allowed a   a a 



 1 2  

SLIDE 14

1) The harmonic oscillator

quantization of the oscillator in energy basis

1 2   ( 1 2), 0,1,2,... n n     ( 1 2) , 0,1,2,...

n

E n n     

A possible second family must have the same ground state, thus it is not allowed

1

n

a n C n  

*

1

n

n a n C

 



*

1 1

n n

n a a n n n C C



  

*

ˆ 1 2

n n

n H n C C  

2 n

n n n C 

2 n

C n 

1 2 i n

C n e  

and adjoint equation form scalar product of both equation

SLIDE 15

1) The harmonic oscillator

quantization of the oscillator in energy basis

1 2

1 a n n n  

1 2

( 1) 1 a n n n



  

1 2 1 2 1 2

1 a a n a n n n n n n n

 

    N a a



 ˆ 1 2 H N  

with number operator further

1 2 1 2 ', 1

' ' 1

n n

n a n n n n n 



  

1 2 1 2 ', 1

' ( 1) ' 1 ( 1)

n n

n a n n n n n 

 

    

SLIDE 16

1) The harmonic oscillator

quantization of the oscillator in energy basis

position and momentum operators

1 2

( ) 2 X a a m



        

1 2

( ) 2 P a a m



        

1 2 1 2 1 2

1 2 ... ... 1 1 2 2 3 . . n n n n n a n



                  

1 2 1 2 1 2

1 ... 2 3 . . a                 

SLIDE 17

What do we learn? Analogously for derived operators

1) The harmonic oscillator

quantization of the oscillator in energy basis

1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 ... 1 2 2 3 2 3 . . X m                            

1 2 1 2 1 2 1 2 1 2 1 2 1 2

1 ... 1 2 2 3 2 3 . . m P i                                 1 2 ... 3 2 5 2 . . H                   

SLIDE 18

Outline

• 1. The harmonic oscillator

 real space  energy basis

• 2. 1D lattice vibrations



• ne 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

SLIDE 19

2) 1D lattice vibrations (phonons)

1 atom per primitive cell

( )

s p s p s p

F C u u



 



2 2

( )

s p s p s p

d u M C u u dt



 



( ) i s p Ka i t s p

u ue e

    

force on one atom equation of motion of atom solution in the form of traveling wave EOM reduces to

2 ( )

( )

isKa i t i s p Ka isKa i t p p

Mue e C e e ue

 



  

  



2

( 1)

ipKa p p

M C e    



translational symmetry

2

( 2)

ipKa ipKa p p

M C e e 

 

   



2

2 (1 cos )

p p

C pKa M 



 



finally leads to

SLIDE 20

2) 1D lattice vibrations (phonons)

1 atom per primitive cell

since

2

2 sin

p p

d paC pKa dK M 



 



2 1

(2 )(1 cos ) C M Ka   

2 2 1 1 2 1

1 (4 )sin ( ) 2 1 (4 ) sin( ) 2 C M Ka C M Ka     nearest‐neighbor interaction only dispersion relation

0.5 1.0 1.5 2.0 2.5 3.0 K 0.2 0.4 0.6 0.8 1.0 w

1 2 3 4 5 6 K

• 0.4
• 0.2

0.2 0.4 wK, dwdK

SLIDE 21

Outline

• 1. The harmonic oscillator

 real space  energy basis

• 2. 1D lattice vibrations



• ne 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

SLIDE 22

2) 1D lattice vibrations (phonons)

2 atoms per primitive cell

2 1 1 2

( 2 )

s s s s

d u M C v v u dt



  

2 2 1 2

( 2 )

s s s s

d v M C u u v dt



  

isKa i t s

u ue e

 



isKa i t s

v ve e

 



2 EOMs ansatz and and substituting

2 1

(1 ) 2

iKa

M u Cv e Cu 



   

2 2

( 1) 2

iKa

M v Cu e Cv      and leads to

2 1 2 2

2 (1 ) (1 ) 2

iKa iKa

C M C e C e C M  



      

2 1 2

1 1 2C M M         

2 2 2 1 2

2 C K a M M   

SLIDE 23

2) 1D lattice vibrations (phonons)

2 atoms per primitive cell

Lattice with 1 atom per primitive cell gives only 1 acoustic branch Lattice with 2 atom per primitive cell gives 1 acoustic and 1 optical branch http://dept.kent.edu/projects/ksuviz/leeviz/phonon/phonon.html

SLIDE 24

Outline

• 1. The harmonic oscillator

 real space  energy basis

• 2. 1D lattice vibrations



• ne 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

SLIDE 25

3) Electron-phonon interaction: Hamiltonian

The basic interaction Hamiltonian is

p e ei

H H H H   

 

1 2

q q

p q q

H a a

 

 





 



2 2

1 1 2 2

i e i ij ij

p H e m r  

 

( ) ( )

ei i ei i j i ij

H V r V r R   

 

    Taylor series expansion for the displacements the electron‐phonon interaction reads and the Fourier transform of the potential 1 ( ) ( ) iq r

ei ei q

V r V q e N







 

  1 ( ) ( ) iq r

ei ei q

V r i qV q e N



 



 

  

SLIDE 26

3) Electron-phonon interaction: Hamiltonian

we need to calculate by using

(0)

( ) ( )

j

iq R iq r ei j q j

i V r qV q e Q e N

  

     

 

   

    

 

(0)

1 2 1 2

( ) 2

j q G q G

iq R j q G q G j G G q G

i i Q e Q a a N N MN  

  

    

   

  

   

     

   and we can write the Hamiltonian in the form

 

( ) 1 2 ,

( ) ( )( ) ( ) 2

q q

ir q G ei q q G q

V r e V q G q G a a   



  

     



 

   

       MN  

SLIDE 27

3) Electron-phonon interaction: Hamiltonian

by integrating the potential over the charge density of the solid

 

3 1 2 ,

( ) ( ) ( ) ( )( ) ( ) 2

q q

ep ei q q G q

H d r r V r q G V q G q G a a     





       

 

 



         

• r in an abbreviated form

1 2

( )( ) ( ) 2

ei q q G q

M V q G q G  

  

  

  

    

 

1 2 ,

1 ( )

q q

ep q G q G

H q G M a a  



 

  



 

 

  with

SLIDE 28

Outline

• 1. The harmonic oscillator

 real space  energy basis

• 2. 1D lattice vibrations



• ne 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

SLIDE 29

3) Electron-phonon interaction:

localized electrons

If the electrons are localized the Hamiltonian becomes

   

1 2

1 2 ( )

i i q q q q

iq r iG r p ep q q G q i G

e H H H a a a a q G M e   



    

            

  

 

      

  here the electron density operator is the Fourier transform the localized charge density

2 3 ( ) 3 ( )

( ) ( ) ( )

ir q G ir q G i

q G d re r r d re q G   

   

    

  

     

     

2 3 ( )

( ) ( )

ir q G i

q G d re r  

 

 

 

  

  rearranging terms

   

1 2

1 1 2 ( )

i q q q q

iq r q q i q i

H a a a a e F r  



  

         

 

   

    

 with the periodic function ( ) ( )

iG r q q G G

F r q G M e 

 

 



    

  

SLIDE 30

3) Electron-phonon interaction:

localized electrons

we now transform the creation and annhiliation operators and rewrite the Hamiltonian

 

2 2

1 1 2

i q q

q iq r q q q i q

F H A A e   

 

      

  

 

      

which has the eigenstates and eigenvalues

 

 

1 2 0

!

q q

n q

A n



 



 

2 2

1 1 2

i

q iq r q q q q i q

F E n e   



      

  

       

and and

1 2

( ) 1

i

q iq r q q i q

F r A a e  



 



     



* 1 2

( ) 1

i

q iq r q q i q

F r A a e  

   

 



     



SLIDE 31

3) Electron-phonon interaction:

deformation potential

Traditionally in semiconductors one parametrizes electron‐phonon interactions (long wavelengths)

• deformation‐potential coupling to acoustic phonons
• piezoelectric coupling to acoustic phonons
• polar coupling to optical phonons

the deformation‐potential coupling takes the form

 

1 2

( ) ( ) 2

q q

ep q q

H D q q a a   





 



 



  

SLIDE 32

3) Electron-phonon interaction:

piezoelectric interaction

The electric field is proportional the the stress

k ijk ij ij

E M S   Stress is the symmetric derivative of the displacement field

 



1 2

1 1 ( ) 2 2 2

i q q

j iq r i ij i j j i q j i q

Q Q S q q a a e x x    



 

                



 

   

 ( ) ( ) r Q r     The field is longitudinal and can hence be written as the gradient of a potential This potential is proportional to the displacement

 

1 2

( ) ( ) ( ) 2

q q

iq r q q

r i M q e a a

 

  

  



 

 



 

   

  

 

1 2

( ) ( ) ( ) 2

q q

ep q q

H i M q q a a

 

  

  





 



 

 

   leading to

1 2

1 ( )

iq r k k q q k

E r iq e x   



     



   



SLIDE 33

3) Electron-phonon interaction:

polar coupling

The coupling is only to LO (TO do not set up strong electric fields) ( 4 ) iq r

q q q

D q E P e 



   



    

    4

q q

E P   

 

 

q q

P UeQ 

 

 

 

1 2 ˆ

4 4 ( ) 2

q q

q q LO

E UeQ Ue iq a a    





    

 

 

  

 

1 2

4 ( ) ( ) 2

q q

iq r q LO

Ue r e a a q    



 

 



 

  

  induced field The polarization is proportional to the displacement and

iq r q q

E i iq e  



    

   

 

SLIDE 34

3) Electron-phonon interaction:

polar coupling

 

3 2 3 2

2 ( ) 4 2 (2 )

iq r R LO LO

d q e V r Ue q    



       

 

 

2

( )

R

e V r r  

2 2

4

LO

U    

2 2

1 e e r r            The interaction of two fixed electrons is Fourier transforming with and

2 2

1 1 4

LO

U    



       

 

1 2

( )

q q

ep q

M H q a a q  





 



 



2 2

1 1 2

LO

M e    



         and the interaction Hamiltonian with

SLIDE 35

Outline

• 1. The harmonic oscillator

 real space  energy basis

• 2. 1D lattice vibrations



• ne 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

SLIDE 36

3) Electron-phonon interaction:

Fröhlich Hamiltonian  

2 1 2

1 2

q q q q

p p p q p p q q

M p H c c a a c c a a m q



 



    

   

  

   

     

   

3 2 2 1 2

4 2 M m     

1 2 2

1 1 2 e m    



               

 

2 1 2

2

q q q q

iq r q q

M p e H a a a a m q



 



  

   

 

   

 

describes the interaction between a single electron in a solid and LO phonons where and For a single electron it can be rewritten as

SLIDE 37

3) Electron-phonon interaction:

small polaron theory – large polarons

1 2

1

j

ikR j k j

C C e N 



 

 

q q q q

q k k k k q k q k kq

H zJ C C a a C C M a a  



    

   

  

   

         

1

ik k

e z

 





 

    (1) 2

1 ( ) ( ) ( )

q F q F k k q RS q q q k k q k k q

N n N n k M        

 

                

 

             

Transform to collective coordinates the polaron self‐energy in first‐order Rayleigh‐Schrödinger perturbation theory becomes

• ptical absorption

SLIDE 38

3) Electron-phonon interaction:

small polaron theory – small polarons

S S

H e He 

 

j q q

iqR q j jq q

M S n e a a 





  



 

  

q q

j j j j j j q j

H J C C X X a a n

  



    

   

  

 

2 q q q

M   

 

 

exp

j q q

iqR q j q q

M X e a a 





         



 

  

H H V  

q q

j q j

H a a n 



  

 

 

j j j j j

V J C C X X

      

  Canonical transformation with leads to with polaron self‐energy and finally we write with and

SLIDE 39

Outline

• 1. The harmonic oscillator

 real space  energy basis

• 2. 1D lattice vibrations



• ne 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

SLIDE 40

3) Electron-phonon interaction:

phonons in metals

The Hamiltonian is

   

2 2 2 (0)

1 1 2 2 2

j i ei i ii i i ij i j i j

P p H e V r R V R R m M r r

    

       

    

     

   

2 2 (0) (0) (0)

1 1 2 2 2

i e ei i ii i i ij i j

p e H V r R V R R m r r

    

      

   

      first we neglect phonons

(0)

R R Q

  

    

e p ep

H H H H   

     

2 (0) (0)

1 2 4

p

P H Q Q Q Q R R M

            

     

 

     

 

(0) ep ei i j

H Q V r R

  

  



  

   

ii

R V R

  

      within the harmonic approximation for the phonons with the bare‐phonon Hamiltonian

SLIDE 41

3) Electron-phonon interaction:

phonons in metals

expand displacements and conjugate momenta in a set of normal modes

 

(0)

1 2

1

ik R k k i

Q Q e N



 





   

 

 

(0)

1 2

1

ik R k k i

P P e N



  





   

 

 

1 1 2 2

p k k k k k

H P P Q Q k m

   

        



    

    

 

  

     

(0) (0) (0) (0)

(0) (0)

1 1 2

ik R ik R ik R ik R G

k e e e e R R G k G

   

      

 

     

              

 

       

     

 

 

3 iq R

q d R R e

  

  



 

  thus where and

SLIDE 42

3) Electron-phonon interaction:

phonons in metals

 

2 2 ii i

Z e V R R    

2 2 3 5

3

i

R R Z e R R

   

           

 

2 2 2

4

i

Z e q q q q

  

     if the ions were point charges we would have find the normal modes of the bare‐phonon system through and use the frequencies and eigenstates to define a set of creation and annihilation operators

 

2

det

k

M k

  

       





 

1 2

2

k k

k k k

Q a a

 

  

 





          



 

  

  1 2

k k

p k k

H a a

 

  

        



 

 

SLIDE 43

3) Electron-phonon interaction:

phonons in metals  

  



1 2 ,

1

q q

i G q r ep q G

H M G q e a a

 

 





  

  



 

    

 

     

1 2

2

q ei k

M G q G q V G q

  

              

 

      

 

 

' '

1 2 '

1 2

q q q q

q p q p q k k k k q k k q k q k qkp nq k

M q H C C C C C C a a C C a a

         

     

  



        

     

   

   

                   

the same set can be used as a basis for the electron‐ion interaction where in second quantization Note: the phonon‐states basis is unrealistic and serves only as starting point for a Green´s function calculation

SLIDE 44

3) Electron-phonon interaction:

phonons in metals

If the electron‐plasma frequency is much larger than the phonon frequency we write the interaction between to electrons as a screened Coulomb interaction and screened phonon interaction

         

2 2

, , , ,

q eff i

v M q V q i D q i q i q i

  

            

   

, 1 ,

q i

v q i P q i        

     

(0) 2 (0)

, 1 , , D D q i M D P q i q i

   

         where and the phonon Green´s function

SLIDE 45

Outline

• 1. The harmonic oscillator

 real space  energy basis

• 2. 1D lattice vibrations



• ne 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

SLIDE 46

4) Superconductivity

BCS (Bardeen, Cooper, and Schrieffer) theory

      



2 2 2 2

(2 ) , ,

q q q s q

v M V q w q w q

 

       

 

,

q D s q D

V V q w             Screened interaction of two scattering electrons The Hamiltonian takes the form

, ' , ' ', ' , ' '

1 ( ) 2

p p p p q p q p p p qp p

H C C V q C C C C v

      



    

 

 

           

SLIDE 47

4) Superconductivity

BCS (Bardeen, Cooper, and Schrieffer) theory

consider the EOMs

     

, ,

, ' ' ( ) '

p p

G p T C C

  

        



              

 



, , , ' , ' ', ' , ' '

1 ( ) , ( )

p p p p p q p p qp

C H C C V q C C C v

      

  

 

         



        

             

, , , ,

, ' [ ' ' ' ' ]

p p p p

G p C C C C

   

             

 

          

   

 first derivative of the equation for the Green´s function

  



 

, , , ,

' , ( ) '

p p p p

C C T C C

    

     

 

           

   

leads after some math to

   

' , ' ', ' , , ' '

1 , ' ( ) ( ) ( ) ( ) ( ') '

p p q p p q p qp

G p V q T C C C C v

     

          

   

              



       



SLIDE 48

4) Superconductivity

BCS (Bardeen, Cooper, and Schrieffer) theory

   

, ' ' ', , ', ' ,

( ) ( ) ( ') ( ) ,0 , '

p p q p p q p p q

T C C T C C F p q F p

   

       

           

    

        

  

   

, ' ' , ', ' , ',

( ) ( ) ( ) ( ') ,0 , '

p p q p q p p q p

T C C T C C F p q F p

   

       

           

      

        

        

' , ' ', ' , , ' '

1 1 ( ) ( ) ( ) ( ) ( ') ( ) , ' ,0 , '

p q p p q p p q qp q

V q T C C C C V q G p n F p q F p v v

     

       

     

         

 

         

   

and thus we get defining

   

1 ( ) ,

q

p V q F p q v      





   gives the EOM for the Green´s function with

 

2 2

1 ,

ip n p

F p p E      





SLIDE 49

4) Superconductivity

BCS (Bardeen, Cooper, and Schrieffer) theory

we sum over frequencies by the contour integral and get

 

, tanh 2 2

p p

E F p E               which, in turn, gives the equation for the gap function

   

1 ( ) tanh 2 2

p q q p q

E p q p V q v E 

 

            



    

  

 

1 2 2 2

E    

 

2

tanh 2 2

D D

p q F

E N V d E

 

 

 

  



 

where and

SLIDE 50

4) Superconductivity

BCS (Bardeen, Cooper, and Schrieffer) theory

 

1 2 2 2 2 1 ln 0 ln

D

D F

N V F

N V





 

        

       

1

2 4

F

N V g D

E e 



  

1

1.14

F

N V c D

kT e 



 4.0 3.52 1.14

g c

E kT   which, solved, produces the energy gap The energy gap decreases as the temperature increases. The critical temperature is BCS predicts

SLIDE 51

Outline

• 1. The harmonic oscillator

 real space  energy basis

• 2. 1D lattice vibrations



• ne 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

SLIDE 52

Static nonrelativistic Hamiltonian SOC, external magnetic field, and eletron‐phonon coupling involved

A numerical example: CO

SLIDE 53

In quantum mechanics, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter, to the expectation value of the derivative of the Hamiltonian with respect to that same parameter.

5) A numerical example: CO

The Hellmann-Feynman theorem

SLIDE 54

Wavefunctions of the harmonic oscillator

5) A numerical example: CO

calculating the electron-phonon coupling

SLIDE 55

With coupling When Diagonalizing Result in

5) A numerical example: CO

calculating the electron-phonon coupling

SLIDE 56

Example: CO

5) A numerical example: CO

calculating the electron-phonon coupling

(GAUSSIAN 03)

SLIDE 57

Literature

• 1. N. W. Ashcroft and D. N. Mermin,

Solid state physics, Holt, Rinehart and Winston (1976)

• 2. G. D. Mahan,

Many particle physics, Springer (2000)

• 3. R. Shankar,

Principles of quantum mechanics, Kluwer academic, Plenum publishers (1994) 4.

• C. Kittel,

Introduction to solid state physics, John Wiley & Sons, inc. (2005)