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 22d 1 () mx 22  E 22mdx2 1) The harmonic oscillator quantization of the oscillator in real space

1) Dimensionless variables 22224 dmEbmb2 2 xb y leads to 22  2y 0 dy  

12 2  mEb E b   and  2 m  10

5  '' (2y 2 ) 0

-10 -5 5 10

-5 2  '' y 2  0 with solution   Aymy e 2

since

''22my2 21mmm (1) my  2222  Aye1 24y Aye y yy 1) The harmonic oscillator quantization of the oscillator in real space

 ''  20  with solution  Acos[ 2yB ] sin[ 2 y ] consistency requires 1.0  A cy  O() y2 y0 0.5 thus (3) -10 -5 5 10

-0.5 2 ansatz:  ()yuye () y 2 -1.0 leads to uyu''2(21)0 '  1) The harmonic oscillator quantization of the oscillator in real space

 4) Power‐series expansion: n uy()  Cyn n0

 nn2 n inserted into differential equation Cnnn [( 1) y 2 ny (2 1)] y 0 n 0

 n 2 Cnnn (1) y n 2

with index shift mn  2

 mn Cmmn22( 2)( my 1) Cn (  2)( ny 1) mn00

 n we get  yC[(2)(1)(212)]0nn2 n n C  n  n0

(2n  1 2 ) feeding back in the original leads to recursion: CC nn2 (2)(1)nn  1) The harmonic oscillator quantization of the oscillator in real space

so we have

 (12)yyyy2435 (12)  (41  2)  (21  2) (21  2)(61  2)  uy() C0  1     C1 y      (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

24 n yy2 22CCyCy02 4 ... Cyn 2 ()yuye () 35 n  e CCyCy13 5 ... Cyn 1) The harmonic oscillator quantization of the oscillator in real space

consequence energy quantization of the harmonic oscillator by backwards substitution 1 En() n 2 

Examples:

Hy0 () 1

Hy1() 2 y

2 Hy2 () 212  y

2 3 Hy3 () 12 y  y 3

244 H4 ()yyy 121 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 22 mX E EE 22m

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 , PiIi  definition and adjoint 12 12 12 12 m 1   m 1  aXiP   aXiP   22 m  22 m 

further  

 aa,1 

New operator (dimensionless)

H Haaˆ (12)  1) The harmonic oscillator quantization of the oscillator in energy basis

Commutator of creation and annhiliation operators with Hamiltonian

ˆ  aH,,12, aaa aaa a  ˆ  aH,  a

Raising and lowering properties

Haˆˆˆˆ  aH[, a H ]  aH a  ( 1) a

 aC 1 1

But eigenvalues non‐negative requirement

a 0  0  1 no further lowering allowed aa  0  0  0 2 1) The harmonic oscillator quantization of the oscillator in energy basis

1  0  0  (12),0,1,2,...nn Enn  (12), n 0,1,2,... 2 A possible second family must have the same ground state, thus it is not allowed

an C n 1  * n and adjoint equation na n1 Cn form scalar product of both equation

 * naan n11 n  CCnn

ˆ * nH12 n CCnn

2 2 12 i nnn Cn Cnn  Cnen  1) The harmonic oscillator quantization of the oscillator in energy basis

an n12 n1

an  (1)1 n12 n

aanann121 nnnnn 12 12 with number operator Naa  HNˆ 12

further

12 12 nan''1 n nn nnn', 1

 12 12 na'(1)'1(1) n n nn  n nn', 1 1) The harmonic oscillator quantization of the oscillator in energy basis position and momentum operators

12 12     X ()aa P ()aa 2m 2m

nnn012... 12 000... 01 0 0... n  0  12  10012 002 0    12 n 1 12   a  02 0 a  00 03 n  2   12 . 003   .    .  . 1) The harmonic oscillator quantization of the oscillator in energy basis What do we learn?

0112 0 0...  01 12 0 0... 12 12  12 12  1020 1020  12 12 12 12  12 12   02 03 m 02 0 3 X  Pi   2m 00312 0 2  00312 0  .  .    .  . 

Analogously for derived operators 12 0 0 0 ... 03200   H   0052    .  .  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 FCuus  psps() p 2 dus equation of motion of atom M 2 Cupsps() u dt p solution in the form of traveling wave is() pKa i t uueesp  EOM reduces to

2()isKa i t i s p Ka isKa i t 2MCe (1)ipKa  Mue e Cp() e  e ue  p p p

translational symmetry

2 2 ipKa ipKa 2 finally leads to  MCee (2) CpKap (1 cos )  p M  p 0 p0 2) 1D lattice vibrations (phonons) 1 atom per primitive cell

2 w d 2 1.0 since paCp sin pKa 0  0.8 dK M p  0 0.6 0.4 nearest‐neighbor interaction only 0.2 221 K   (4CM )sin ( Ka ) 0.5 1.0 1.5 2.0 2.5 3.0 1 2 2 (2CM )(1 cos Ka ) 1 1  (4CM )12 sin( Ka ) dispersion relation  1 2

w K, dw dK 0.4 0.2  K 1 2 3 4 5 6 -0.2 -0.4 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) 2 atoms per primitive cell

du2 dv2 2 EOMs M s Cv(2) v u and M s Cu(2) u v 11dt 2 s ss 21dt 2 s ss

isKa i t isKa i t ansatz uuees  and vvees 

2 iKa 2 iKa substituting  M1uCve(1 ) 2 Cu and  M 2vCue(1)2  Cv

11  2 2C 2 iKa  2(1)CM C e MM12 leads to 1  0 Ce(1iKa ) 2 CM  2 C 2 2 222 Ka MM  12 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 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 3) Electron-phonon interaction: Hamiltonian

The basic interaction Hamiltonian is HH peei H H

2  pi 112    Haapq  12 HVrVrR() ( )  qq Hee  ei  i ei i j q   iij22mrij iij

Taylor series expansion for the displacements

the electron‐phonon interaction reads

and the Fourier transform of the potential

1 iq r 1 iq r Vrei()  Vqe ei () Vrei() i qVqe ei () N q N q 3) Electron-phonon interaction: Hamiltonian

we need to calculate

  (0) i iq r  iq Rj Vr ()  qVqeei ()  Qe j N qj by using

 ii iq R(0)  j  12  Qe Q  () a a jqG12 qG   qG qG   NNjGG2 MNqG

and

MN  

we can write the Hamiltonian in the form

 ir() q G    12   Vr () e VqGqGei ()()()    q  a  a  2 qq qG,  q 3) Electron-phonon interaction: Hamiltonian

by integrating the potential over the charge density of the solid

3      12  H drrVr()() ( qGVqGqG ) (  )( ) ( ) a a ep  ei q  qq  qG, 2 q or in an abbreviated form

1    H   ()qGM  a  a ep 12  qG  qq  qG,   with

   12 M   VqGqG()()()     qG ei q 2 q 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 3) Electron-phonon interaction: localized electrons

If the electrons are localized the Hamiltonian becomes

iq r  e i     iG ri HH H  aa 12  a  a ( qGMe  )   pepq qq12  q q 0 qG  qiG  here the electron density operator is the Fourier transform the localized charge density

  2  ()qG   dre3()ir q G () rr dre 3() ir  q G ()  qG  00 0 i     2 ()qG  dre3()ir q G () r 00  i rearranging terms

1  iq ri  HaaaaeFr12 ( ) qqiqq12  q  q  qi  with the periodic function      iG r F ()rqGMe ( )  q  0 qG G 3) Electron-phonon interaction: localized electrons we now transform the creation and annhiliation operators  *  Fr ()   Fr ()   1 q iq ri 1 q  iq ri Aae Aa e qq12  and qq12  q i q i and rewrite the Hamiltonian

2 2  Fq  1 iq ri HAAe 12 q qq  qqi q which has the eigenstates and eigenvalues

n  q 2 2 A  Fq q 1 iq ri 0 and En12 e 12 qq qqi q nq ! 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   HD()() 12  qqaa  ep   qq  q 2 q 3) Electron-phonon interaction: piezoelectric interaction

The electric field is proportional the the stress

Ek MS ijk ij ij Stress is the symmetric derivative of the displacement field

11Q Q   i j   12  iq i r Sqqaaeij() i j  j i    qq 222xxjiq  q The field is longitudinal and can hence be written as the gradient of a potential

  1 iq r Er()  iqe kkq12   xk q  This potential is proportional to the displacement

     12  iq r ()rQr () ()ri ( ) Mqea () a    qq   q 2 q leading to       12    Hi( ) Mqqaa ()() ep    qq  q 2 q 3) Electron-phonon interaction: polar coupling

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

  iq r DqEPe 0(4)  qq   q induced field   EPqq 4  The polarization is proportional to the displacement

     12  and EUeQUeiqaa44() ˆ  PUeQqq   qq  qq  2 LO

 

  iq r E  iiqe q q

 4Ue ()re iq r ( )12 aa   qq   q q 2 LO

  3) Electron-phonon interaction: polar coupling

The interaction of two fixed electrons is 3 iq r 2 2 dq e Vr()  4 Ue  R  32  LO2(2) LO q

Fourier transforming

22 e2 4U 2 ee1 Vr() with and   R 2 rr r LO 0 

2 2  11 U LO  4  0 and the interaction Hamiltonian

M  11 H ()qa a 22 ep 12    with Me2  q qq  LO  q   0 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 3) Electron-phonon interaction: Fröhlich Hamiltonian describes the interaction between a single electron in a solid and LO phonons

2 p M 0 1  Hccaaccaa   pp0 qq12 pqp  q  q  pqq 2mq where 

32 2 4 0 M0  12 2m and 12 em2 11   2 00

For a single electron it can be rewritten as

  pe2 M iq r H  aa 0 a  a 0qq12 q  q  2mqqq

 3) Electron-phonon interaction: small polaron theory – large polarons

Transform to collective coordinates

 1 ikRj CCe  k 12  j N j

  HzJCC  aa CCMaa     kk k0 qq kqk q  q  q   kkqq  optical absorption

  1 ik    e k  z  the polaron self‐energy in first‐order Rayleigh‐Schrödinger perturbation theory becomes

 (1) 2 NnqF1()kk Nn qF  () ()kM  RS q      q kkqqq kkq 3) Electron-phonon interaction: small polaron theory – small polarons

Canonical transformation

  M H  eHeSS with SneaaiqRj q    j qq  jq q leads to

H  JCCXX aa  n  jjjj0 qq j jqj

2 M     M  with polaron self‐energy  q and Xeaaexp iqRj q  j  qq    q q  q q  finally we write

H  HV0  with Haan  and VJCCXX  00qq j  jjj j qj j 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 3) Electron-phonon interaction: phonons in metals

The Hamiltonian is  2 2 p Pj 11   HeVrRVRRi 2(0)     ei i  ii    ii222mMj ijirrij  first we neglect phonons

p2  11e2  HVrRVRRi  (0)  (0)  (0) 0eeiiii   ii222m ijrrij within the harmonic approximation for the phonons

 (0)  HH  H H RRQ  00epep     (0) HQVrRep ei i      RVRii    j with the bare‐phonon Hamiltonian 2 P 1   (0) (0) HQQQQRR0   p 24M    

 3) Electron-phonon interaction: phonons in metals

expand displacements and conjugate momenta in a set of normal modes

     1  ik R (0)  1  ik R (0)     QQe  12 PPe  12  k  k Ni k Ni k thus

11   HPPQQk0 p     kk kk k 22m

where

   11(0)ik R(0) (0) ik  R (0)       keeeeik R ik R  RR(0) (0)  GkG       2 0 G

 and

  3  iq R  qdRRe      

 3) Electron-phonon interaction: phonons in metals if the ions were point charges we would have   22 22 4 Z 22eqq  Z e Ze    3RR   VR   q ii   35 q2 i R i RR i find the normal modes of the bare‐phonon system through

2  detMk  0 k     and use the frequencies and eigenstates to define a set of creation and annihilation operators

12      Qaa kk   2   kk k

 1  Haa0 p     k kk k 2 3) Electron-phonon interaction: phonons in metals the same set can be used as a basis for the electron‐ion interaction

  1   iG q r  HMGqeaaep    12  qq  qG , where

12  M Gqqei  GqVGq   2     k

in second quantization   

1 Mq    HCCCCCCaa      CCaa   qpqp   q12      kk k kq''   k qq  kqk q q kqkp2 ' q nqk 

Note: the phonon‐states basis is unrealistic and serves only as starting point for a Green´s function calculation



 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 v Mq Vqi,, q  Dqi  eff   2 i qi, qi,  where v  qi,1q P qi , i  and the phonon Green´s function

(0)  D Dqi,    1,, M 2(0)DPqi  qi 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 4) Superconductivity BCS (Bardeen, Cooper, and Schrieffer) theory

Screened interaction of two scattering electrons

vM2 (2 ) Vqw, qqq s  qw, 2 22 q  q   V0 qD Vqws ,   0   qD

The Hamiltonian takes the form

1 HCCVqCCCC pp p() pq,','',', pq p p pqpp2 v ''

 4) Superconductivity BCS (Bardeen, Cooper, and Schrieffer) theory consider the EOMs

 1    () , ()  CHCCpppppqpp,,,  VqCCC ','',',  v qp'' first derivative of the equation for the Green´s function 

   Gp,'  [  ' C C '  '  C '  C ]    pp,,  p , p ,

      ',CC  T C  ()' C  pp,,    p , p ,

  Gp,'  ' T Cpp,, ()'  C  

leads after some math to    1    ppqppqpGp ,     '  Vq ()  TC',' () C  ',' ()  C , () C , (')   '  v qp'' 4) Superconductivity BCS (Bardeen, Cooper, and Schrieffer) theory

   TCppqp',() C , () TC ',  (') pqC '  , () ,'ppq '  F p q ,0 F p , ' and

   TCpq,',',',() pC  () TC p q  () pC  (') ,'ppq '  F p q ,0 F p , '  thus we get

11  () ()  ()  ()  (') () , '  ,0 , ' Vq TCpq',' C  p ','  C pq , C p ,  Vq G  p n pq  F p  q F p vvqp '' q 1 1  defining pVqFpq()  , 0 Fp,0    with  22  v q ip pnpE  gives the EOM for the Green´s function 4) Superconductivity BCS (Bardeen, Cooper, and Schrieffer) theory we sum over frequencies by the contour integral

and get    E Fp ,0 tanh p  22Ep  which, in turn, gives the equation for the gap function   1 pq  E   pVq  () tanhpq   vEq 22pq 

2  12 D tanh E  2 22  pq and E  where  NVF 0 d   2  D E 4) Superconductivity BCS (Bardeen, Cooper, and Schrieffer) theory

 12 D 22 2D 1lnNVln  NVF 0  F 0  0 which, solved, produces the energy gap

1 NVF 0 EegD24

The energy gap decreases as the temperature increases. The critical temperature is

1 NVF 0 kTcD1.14 e

BCS predicts

E 4.0 g 3.52 kTc 1.14 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 A numerical example: CO

Static nonrelativistic Hamiltonian

SOC, external magnetic field, and eletron‐phonon coupling involved 5) A numerical example: CO The Hellmann-Feynman theorem

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 calculating the electron-phonon coupling

Wavefunctions of the harmonic oscillator 5) A numerical example: CO calculating the electron-phonon coupling

When

With coupling

Diagonalizing

Result in 5) A numericalExample: example: CO CO calculating the electron-phonon coupling

(GAUSSIAN 03) 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)