Electron-phonon coupling: a tutorial - Magnetismmagnetism.eu/esm/2011/slides/huebner-slides1.pdf · · 2017-06-20Electron-phonon coupling: a tutorial W. Hübner, C. D. Dong, and

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






6. Literature

1) The harmonic oscillatorquantization of the oscillator in real space

Eigenvalues of

Projection of eigenvalue equation to X basis

(Substitution by differential operators)

leads to2 2

2 22

1( )2 2

d m x Em dx


1) Dimensionless variables

x by

1 2

bm

2

2

mEb E

'' 2(2 ) 0y

'' 2 0y

2 2 2 2 42

2 2 2

2 0d mEb m b ydy

leads to

and

2 2m yAy e with solution

since2 2'' 2 2 2 2 2

2 4

2 1 ( 1)1m y m yy

m m mAy e Ay e yy y

-10 -5 5 10

-5

5

10


with solution'' 2 0 cos[ 2 ] sin[ 2 ]A y B y

consistency requires2

0 ( )y A cy O y

thus (3)

2 2( ) ( ) yy u y e ansatz:

leads to '' '2 (2 1) 0u yu

-10 -5 5 10

-1.0

-0.5

0.5

1.0


4) Power‐series expansion: 0

( ) nn

nu y C y

2

0[ ( 1) 2 (2 1) ] 0n n n

nn

C n n y ny y

2

2( 1) n

nn

C n n y

2m n

2 20 0

( 2)( 1) ( 2)( 1)m nm n

m nC m m y C n n y

inserted into differential equation

with index shift

we get

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

nC Cn n

feeding back in the original leads to recursion:

20

[ ( 2)( 1) (2 1 2 )] 0nn n

ny C n n C n


so we have

Problems

=> way out: termination of series required

2 22 4

2 20 2 43 5

1 3 5

...( ) ( )

...

ny yn

nn

C C y C y C yy u y e e

C C y C y C y

2 4 3 5

0 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 yu y C C y


consequence energy quantization of the harmonic oscillator by backwards substitution

1( )2nE n

Examples:

0 ( ) 1H y

1( ) 2H y y

22 ( ) 2 1 2H y y

33

2( ) 123

H y y y

2 44

4( ) 12 1 43

H y y y

Outline






6. Literature

1) The harmonic oscillatorquantization of the oscillator in energy basis

Oscillator in energy basis

22 21

2 2P m X E E Em

Direct way: Fourier transform from real to momentum space

No savings compared to direct solution of Schrödinger equation in real space


commutator

,X P i I i

1 2 1 212 2ma X i P

m

1 2 1 212 2ma X i P

m

, 1a a

definition and adjoint

further

New operator (dimensionless)

ˆ ( 1 2)HH a a


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 1a C

But eigenvalues non‐negative requirement

no further lowering allowed0 0a

0 0a a 012


012

0 ( 1 2), 0,1, 2,...n n ( 1 2) , 0,1,2,...nE n n

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

1na n C n *1 nn a n C

*1 1 n nn a a n n n C C

*ˆ 1 2 n nn H n C C

2nn n n C 2

nC n 1 2 inC n e

and adjoint equation

form scalar product of both equation


1 2 1a n n n

1 2( 1) 1a n n n

1 2 1 2 1 21a a n a n n n n n n n

N a a

ˆ 1 2H N

with number operator

further

1 2 1 2', 1' ' 1 n nn a n n n n n

1 2 1 2', 1' ( 1) ' 1 ( 1) n nn a n n n n n


position and momentum operators

1 2

( )2

X a am

1 2

( )2

P a am

1 2

1 2

1 2

0 1 2 ...0 0 0 ...

01 0 0

10 2 0

20 0 3

.

.

n n n

nnan

1 2

1 2

1 2

0 1 0 0 ...0 0 2 00 0 0 3..

a

What do we learn?

Analogously for derived operators


1 2

1 2 1 2

1 2 1 2 1 2

1 2

0 1 0 0 ...1 0 2 00 2 0 3

2 0 0 3 0..

Xm

1 2

1 2 1 2

1 2 1 2 1 2

1 2

0 1 0 0 ...1 0 2 00 2 0 3

2 0 0 3 0..

mP i

1 2 0 0 0 ...0 3 2 0 00 0 5 2..

H

Outline






6. Literature

2) 1D lattice vibrations (phonons)1 atom per primitive cell

( )s p s p sp

F C u u 2

2 ( )sp s p s

p

d uM C u udt

( )i s p Ka i ts pu 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 tp

pMue e C e e ue 2 ( 1)ipKa

pp

M C e

translational symmetry

2

0( 2)ipKa ipKa

pp

M C e e

2

0

2 (1 cos )pp

C pKaM

finally leads to

2) 1D lattice vibrations (phonons)1 atom per primitive cell

since2

0

2 sin 0pp

d paC pKadK M

21(2 )(1 cos )C M Ka

2 21

1 21

1(4 )sin ( )21(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 3.0K

0.2

0.4

0.6

0.8

1.0w

1 2 3 4 5 6K

-0.4-0.2

0.20.4

wK, dwdK

Outline






6. Literature

2) 1D lattice vibrations (phonons)2 atoms per primitive cell

2

1 12 ( 2 )ss s s

d uM C v v udt

2

2 12 ( 2 )ss s s

d vM C u u vdt

isKa i tsu ue e isKa i t

sv ve e

2 EOMs

ansatz

and

and

substituting 21 (1 ) 2iKaM u Cv e Cu 2

2 ( 1) 2iKaM v Cu e Cv and

leads to 2

12

2

2 (1 )0

(1 ) 2

iKa

iKa

C M C eC e C M

2

1 2

1 12CM M

2 2 2

1 2

2C K aM M

2) 1D lattice vibrations (phonons)2 atoms per primitive cell

Lattice with 1 atom per primitive cell gives only 1 acoustic branchLattice with 2 atom per primitive cell gives 1 acoustic and 1 optical branch

http://dept.kent.edu/projects/ksuviz/leeviz/phonon/phonon.html

Outline






6. Literature

3) Electron-phonon interaction: Hamiltonian

The basic interaction Hamiltonian is p e eiH H H H

1 2q qp q

qH a a

2

21 12 2

ie

i ij ij

pH em r

( ) ( )ei i ei i ji 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 rei ei

qV r V q e

N 1( ) ( ) iq r

ei eiq

V r i qV q eN


we need to calculate

by using

(0)

( ) ( ) jiq Riq rei j

q j

iV r qV q e Q eN

(0) 1 21 2 ( )

2j

q G q G

iq Rj q Gq G

j G G q G

i iQ e Q a aN N MN

and

we can write the Hamiltonian in the form

( ) 1 2

,( ) ( )( ) ( )

2 q q

ir q Gei q

q G q

V r e V q G q G a a

MN


by integrating the potential over the charge density of the solid

3 1 2

,( ) ( ) ( ) ( )( ) ( )

2 q qep ei qq G q

H d r r V r q G V q G q G a a

or in an abbreviated form

1 2( )( ) ( )2ei qq G

q

M V q G q G

1 2,

1 ( )q qep q G

q GH q G M a a

with

Outline






6. Literature

3) Electron-phonon interaction: localized electrons

If the electrons are localized the Hamiltonian becomes

01 21 2 ( )i

i

q q q q

iq riG r

p ep q q Gq i G

eH H H a a a a q G M e

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

23 ( ) 3 ( )0 0 0( ) ( ) ( )ir q G ir q G

i

q G d re r r d re q G

23 ( )0 0( ) ( )ir q G

i

q G d re r

rearranging terms

1 2

11 2 ( )i

q q q q

iq rq q i

q i

H a a a a e F r

with the periodic function

0( ) ( ) iG rq q G

GF r q G M e

3) Electron-phonon interaction: localized electrons

we now transform the creation and annhiliation operators

and rewrite the Hamiltonian

22

11 2 i

q q

qiq rq

q q i q

FH A A e

which has the eigenstates and eigenvalues

1 2 0

!

q

q

n

q

A

n

22

11 2 i qiq rq q

q q i q

FE n e

and

and

1 2

( )1iq iq r

q qiq

F rA a e

*

1 2

( )1iq iq r

q qiq

F rA a e

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 qep

q q

H D q q a a

3) Electron-phonon interaction: piezoelectric interaction

The electric field is proportional the the stress

k ijk ijij

E M SStress is the symmetric derivative of the displacement field

1 21 1 ( )2 2 2

i

q q

j iq riij i j j i

qj i q

QQS q q a a ex 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 qep

q q

H i M q q a a

leading to

1 2

1( ) iq rk k q

qk

E r iq ex

3) Electron-phonon interaction: polar coupling

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

0 ( 4 ) iq rq q

q

D q E P e

4q qE P

q qP UeQ

1 2 ˆ4 4 ( )2 q qq q

LO

E UeQ Ue iq a a

1 24( ) ( )2 q q

iq r

q LO

Uer e a aq

induced field

The polarization is proportional to the displacement

and

iq rq

q

E i iq e

3) Electron-phonon interaction: polar coupling

3

23 2

2( ) 42 (2 )

iq r

RLO LO

d q eV r Ueq

2

( )ReV rr

2

2

4

LO

U

2 2

0

1e er r

The interaction of two fixed electrons is

Fourier transforming

with and

22

0

1 14

LOU

1 2 ( )q qep

q

MH q a aq

2 2

0

1 12 LOM e

and the interaction Hamiltonian

with

Outline






6. Literature

3) Electron-phonon interaction: Fröhlich Hamiltonian

2

00 1 2

12 q q q qp p p q p

p q q

MpH c c a a c c a am q

3 202

0 1 2

42

Mm

1 22

0 0

1 12

e m

2

00 1 22 q q q q

iq r

q q

Mp eH a a a am 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

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

1 2

1 jikRjk

jC C e

N

0 q q q qqk k k k q kqk kq

H zJ C C a a C C M a a

1 ikk e

z

(1) 2 1 ( ) ( )( ) q F q Fk k

qRSq q qk k q k k q

N n N nk M

Transform to collective coordinates

the polaron self‐energy in first‐order Rayleigh‐Schrödinger perturbation theory becomes

optical absorption

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

S SH e He j

q q

iqR qj

jq q

MS n e a a

0 q qj j j j jj q j

H J C C X X a a n

2q

q q

M

exp j

q q

iqR qj

q q

MX e a a

0H H V

0 0 q q jq j

H a a n j j j jj

V J C C X X

Canonical transformation

with

leads to

with polaron self‐energy and

finally we write

with and

Outline






6. Literature

3) Electron-phonon interaction: phonons in metals

The Hamiltonian is

22

2 (0)1 12 2 2

jiei i ii

i i ij ij i j

PpH e V r R V R Rm M r r

2 2

(0) (0) (0)0

1 12 2 2

ie ei i ii

i i ij i j

p eH V r R V R Rm r r

first we neglect phonons

(0)R R Q

0 0e p epH H H H

2

(0) (0)0

12 4pPH Q Q Q Q R RM

(0)ep ei i

jH Q V r R

iiR V R

within the harmonic approximation for the phonons

with the bare‐phonon Hamiltonian


expand displacements and conjugate momenta in a set of normal modes

(0)

1 21 ik R

kki

Q Q eN

(0)

1 21 ik R

kki

P P eN

01 1

2 2p k k k kk

H P P Q Q km

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

0

1 12

ik R ik Rik R ik R

Gk e e e e R R G k G

3 iq Rq d R R e

thus

where

and


2 2

iii

Z eV RR

2 2

3 5

3

i

R RZ eR R

2 2

2

4

i

Z e q qq

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

2det 0kM k

1 2

2 k kk kk

Q a a

012k kp k

k

H a a


1 2,

1q q

i G q rep

q G

H M G q e a a

1 2

2 q eik

M G q G q V G q

' ' 1 2'

12 q q q qq p q p qk k k k q k k q k

qk qkp nq k

M qH 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 fora Green´s function calculation


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

2

2, ,, ,

qeff

i

v M qV q i D q i

q i q i

, 1 ,q

i

vq i P q i

(0)

2 (0),1 , ,

DD q iM D P q i q i

where

and the phonon Green´s function

Outline






6. Literature

4) SuperconductivityBCS (Bardeen, Cooper, and Schrieffer) theory

2

2 2 2

(2 ),

,q q q

sq

v MV q w

q w q

0,

0

q D

s

q D

VV q w

Screened interaction of two scattering electrons

The Hamiltonian takes the form

, ' , ' ', ' ,' '

1 ( )2p p p p q p q p p

p qp pH C C V q C C C C

v


consider the EOMs

, ,, ' ' ( ) 'p pG p T C C

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

1( ) , ( )p p p p p q p pqp

C H C C V q C C Cv

, , , ,, ' [ ' ' ' ' ]p p p pG p C C C C

first derivative of the equation for the Green´s function

, , , ,' , ( ) 'p p p pC C T C C

leads after some math to

' , ' ', ' , ,' '

1, ' ( ) ( ) ( ) ( ) ( ') 'p p q p p q pqp

G p V q T C C C Cv


, ' '', , ', ' ,( ) ( ) ( ') ( ) ,0 , 'p p qp p q p p qT C C T C C F p q F p

, ' ', ', ' , ',( ) ( ) ( ) ( ') ,0 , 'p p qp q p p q pT C C T C C F p q F p

' , ' ', ' , ,' '

1 1( ) ( ) ( ) ( ) ( ') ( ) , ' , 0 , 'p q p p q p p qqp q

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

and

thus we get

defining 1 ( ) , 0q

p V q F p qv

gives the EOM for the Green´s function

with 2 2

1, 0ip n p

F pp E


we sum over frequencies by the contour integral

and get

, 0 tanh2 2

p

p

EF p

E

which, in turn, gives the equation for the gap function

1 ( ) tanh2 2

p q

q p q

Ep qp V q

v E

1 22 2E 2

0

tanh 22

D

D

p qF

EN V d

E

where and


0

0

1 2 22 21 ln0 lnD

DFN VFN V

012 4 FN Vg DE e

011.14 FN Vc DkT e

4.0 3.521.14

g

c

EkT

which, solved, produces the energy gap

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

BCS predicts

Outline






6. Literature

Static nonrelativistic Hamiltonian

SOC, external magnetic field, and eletron‐phonon coupling involved

A numerical example: CO

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: COThe Hellmann-Feynman theorem

Wavefunctions of the harmonic oscillator

5) A numerical example: COcalculating the electron-phonon coupling

With coupling

When

Diagonalizing

Result in

5) A numerical example: COcalculating the electron-phonon coupling

Example: CO5) A numerical example: COcalculating 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)

Documents

Electron-phonon coupling: a tutorial - Magnetismmagnetism.eu/esm/2011/slides/huebner-slides1.pdf · · 2017-06-20Electron-phonon coupling: a tutorial W. Hübner, C. D. Dong, and