electron-phonon coupling: a tutorial - magnetismmagnetism.eu/esm/2011/slides/huebner-slides1.pdf ·...
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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 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) The harmonic oscillatorquantization of the oscillator in real space
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
1) The harmonic oscillatorquantization of the oscillator in real space
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
1) The harmonic oscillatorquantization of the oscillator in real space
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
1) The harmonic oscillatorquantization of the oscillator in real space
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
1) The harmonic oscillatorquantization of the oscillator in real space
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
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 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
1) The harmonic oscillatorquantization of the oscillator in energy basis
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
1) The harmonic oscillatorquantization 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 1a C
But eigenvalues non‐negative requirement
no further lowering allowed0 0a
0 0a a 012
1) The harmonic oscillatorquantization of the oscillator in energy basis
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) The harmonic oscillatorquantization of the oscillator in energy basis
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
1) The harmonic oscillatorquantization of the oscillator in energy basis
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) The harmonic oscillatorquantization of the oscillator in energy basis
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
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
( )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
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
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
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 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
3) Electron-phonon interaction: Hamiltonian
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
3) Electron-phonon interaction: Hamiltonian
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
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
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
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
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
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
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
3) Electron-phonon interaction: phonons in metals
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
3) Electron-phonon interaction: phonons in metals
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
3) Electron-phonon interaction: phonons in metals
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
3) Electron-phonon interaction: phonons in metals
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
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) 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
4) SuperconductivityBCS (Bardeen, Cooper, and Schrieffer) theory
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
4) SuperconductivityBCS (Bardeen, Cooper, and Schrieffer) theory
, ' '', , ', ' ,( ) ( ) ( ') ( ) ,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
4) SuperconductivityBCS (Bardeen, Cooper, and Schrieffer) theory
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
4) SuperconductivityBCS (Bardeen, Cooper, and Schrieffer) theory
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
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
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)