Phonon
Mitsuaki Kawamura
September 14, 2022

1

Single harmonic Oscillator

The hamiltonian operator isĤ =C û2p̂2+,2M2(1)where û and p̂ are the position and the momentum operator. These operators satisfies the following
commutation relation:[û, p̂] ≡ ûp̂ − p̂û = i(2)Then we introduce the creation- and anihiration- operator (b̂† and b̂) as follows
1(b̂ + b̂† ),(4M C)1/41/4
MC(−ib̂ + ib̂† ).
p̂ ≡4û ≡(3)
(4)

The commutation relation of b̂† and b̂[b̂, b̂] = 0, [b̂, b̂† ] = 1(5)lead to the orignal commutation relation of û and p̂ as[û, p̂] =i(−[b̂, b̂] + [b̂, b̂† ] − [b̂† , b̂] + [b̂† , b̂† ]) = i,2

and the hamiltonian becomes
1 (M C)1/2
C†
†
Ĥ =
(−ib̂ + ib̂† )(−ib̂ + ib̂† ) +(b̂+b̂)(b̂+b̂)
4
M(M C)1/2
 1/2
1 C1,=(b̂b̂† + b̂† b̂) = ω b̂† b̂ +2 M2(6)(7)

where ω ≡ (C/M )1/22

Coupled Harmonic Oscillator

The hamiltonian operator isĤ =1 X p̂2s1X+ûs Css0 ûs0 ,2 s Ms
2 0ss1(8)where ûs and p̂s are the position and the momentum operator. These operators satisfies the following
commutation relation:[ûs , p̂s0 ] ≡ iδss0 .

(9)Then, we introduce the creation- and anihiration- operator (b̂†ν and b̂ν ) as followsûs ≡vsν(b̂ν + b̂†ν ),(Ms ων )1/2(10)X(Ms ων )1/2 vsν (−ib̂ν + ib̂†ν ),(11)Xνp̂s ≡

ν

where vsν and ων2 are eigenvectors and eigenvalues of rescaled force constant as
Css0vs0 ν = ων2 vsν(Ms Ms0 )1/2(12)[b̂ν , b̂ν 0 ] = 0, [b̂ν , b̂†ν 0 ] = δνν 0(13)Xs0

The commutation relation of b̂†ν and b̂νlead to the orignal commutation relation of ûs and p̂s as[ûs , p̂s0 ] =Xi
vsν vs0 ν = iδss0 ,vsν vs0 ν 0 (−[b̂ν , b̂ν 0 ] + [b̂ν , b̂†ν 0 ] − [b̂†ν , b̂ν 0 ] + [b̂†ν , b̂†ν 0 ]) = i
20ν

X
νν

(14)

and the hamiltonian becomes
X Ms1X(ων ων 0 )1/2 (−ib̂ν + ib̂†ν )(−ib̂ν 0 + ib̂†ν 0 )vsν vsν 02 0Mssνν
X1
Css0
1X
(b̂ν + b̂†ν )(b̂ν 0 + b̂†ν 0 )vsν
vs0 ν 0+1/2
0
2 0(ων ων )(Ms Ms0 )1/2
νν
ss0X
1X
=(ων ων 0 )1/2 (−b̂ν b̂ν 0 + b̂ν b̂†ν 0 + b̂†ν b̂ν 0 − b̂†ν b̂†ν 0 )vsν vsν 02 0sĤ =ννXων20vsν vsν 0
1/22 0(ων ων 0 )sνν
X 11X,ων b̂†ν b̂ν +=ων (b̂ν b̂†ν + b̂†ν b̂ν ) =
2 ν2
ν+31X

(b̂ν b̂ν 0 + b̂ν b̂†ν 0 + b̂†ν b̂ν 0 + b̂†ν b̂†ν 0 )(15)

Periodic Coupled Harmonic Oscillator

The hamiltonian operator isĤ =1 X p̂2Tsα1+2Ms
2TsαX

ûTsα CTsαT0 s0 α0 ûT0 s0 α0 ,(16)

TsαT0 s0 α0

where ûTsα and p̂Tsα are the position and the momentum operator. These operators satisfies thefollowing commutation relation:[ûTsα , p̂T0 s0 α0 ] ≡ iδTT0 δss0 δαα0 .

2(17)The Fourier-transformed operators are defined as follows:
1 X
1 X
Ûqsα ≡ 1/2ûTsα eiq·T , P̂qsα ≡ 1/2p̂Tsα eiq·T
NC T
NC T
1 X
1 X
ûTsα = 1/2Ûqsα e−iq·T , p̂Tsα = 1/2P̂qsα e−iq·T ,NCNqqC

(18)
(19)

where NC is the number of cells within the Born—von Karman boundary condition. They also satisfy
the commutation relation.
1 X i(q−q0 )·T
1 X iq·T −iq0 ·T0ee[ûTsα , p̂T0 s0 α0 ] = i[Ûqsα , P̂q†0 s0 α0 ] =eδss0 δαα0(20)
NCNC0TTT= iδqq0 δss0 δαα0 .

(21)The hamiltonian becomesĤ ==12NCX P̂qsα P̂q0 sα ei(q+q0 )·T
1+Ms
2NC0qq Tsα†P̂qsα
1
1 X P̂qsα+
2 0
Ms2NC
qq Tsα=XqX

X

Ûqsα Ûq0 s0 α0

qq0 sαs0 α0Ûqsα Ûq0 s0 α0

qq0 sαs0 α0X

X0CTsαT0 s0 α0 eiq·T eiq ·T

0TT0000C0sα(T0 −T)s0 α0 eiq ·(T −T) ei(q+q )·TTT0†P̂qsα1 X P̂qsα1 X †Ûqsα C̃qsαs0 α0 Ûqs0 α0+2 sα
Ms20 0!
,(22)

C0sαTs0 α0 eiq·T

(23)

sαs α

whereC̃qsαs0 α0 ≡XTWith the same discussion in the previous section, we obtain the following results:
Xvsαqν
Ûqsα ≡(b̂ + b̂†qν ),1/2 qν(Mω)
s qννXP̂qsα ≡(Ms ωqν )1/2 vsαqν (−ib̂qν + ib̂†qν ),(24)(25)νC̃qsαs0 α0
2vs0 α0 qν = ωqνvsαqν(26)0 )1/2(MMsss0 α0X1(27)Ĥ =ωqν b̂†qν b̂qν +2qνXi
[Ûqsα , P̂q†0 s0 α0 ] =vsαqν vs∗0 α0 q0 ν 0 (−[b̂qν , b̂q0 ν 0 ] + [b̂qν , b̂†q0 ν 0 ] − [b̂†qν , b̂q0 ν 0 ] + [b̂†qν , b̂†q0 ν 0 ])2νν 0X
= iδqq0vsαqν vs∗0 α0 q0 ν = iδqq0 δss0 δαα0 ,(28)Xν4Electron-phonon vertex

Electron-nuclear Hamiltonian in 2nd quantization representation

Ĥen =XZd3 rψ̂σ (r)† ψ̂σ (r)V (r; {R̂Tsα }),σ3(29)where R̂Tsα is the position operator of nuclear as0R̂Tsα = RTsα+ ûTsα ,(30)0RTsα
is equiribrium position.
0We expand V (r; {R̂Tsα }) arround RTsαand obtainZX ∂V (r; {R0 })X
XZ3†
0d3 rψ̂σ (r)† ψ̂σ (r)Ĥen =d rψ̂σ (r) ψ̂σ (r)V (r; {RTsα }) +ûTsα + O(u2 ).
0∂RTsασσTsα(31)The first term is the interaction between electron and fixed nuclear, and the second term is theelectron-phonon interaction Ĥep . By expanding ψ̂σ with Bloch orbitals ϕnkX
ψ̂σ (r) =ϕnk (r)ĉnkσ ,(32)nkwe obtainĤep =ĉ†nkσ ĉn0 k0 σ ûTsαXZd3 rϕ∗nk (r)ϕn0 k0 (r)σnn0 kk0 Tsα∂V (r; {R0 }).
0∂RTsα(33)−1/2

By using Eqs (19, 24) and ϕnk (r) ≡ NC eik·r χnk (r), we obtain
XqνĤep =ĉ†nkσ ĉn0 k0 σ (b̂qν + b̂†qν )gnkn
0 k0 ,(34)σnn0 kk0 qνqν
where gnkn
0 k0 is the electron-phonon vertex asZX0∂V (r; {R0 })
vsαqν
qν
−iq·T
d3 rei(k −k)·r χ∗nk (r)χn0 k0 (r)gnkn≡e00k031/2∂RTsα(NC Ms ωqν )TsαZX∂V (r + T0 ; {R0 })vsαqν
3 i(k0 −k)·(r+T0 ) ∗00dreχ(r)χ
(r)=e−iq·T 3nknk0∂RTsα(NC Ms ωqν )1/2 cell
TT0 sαZXX000∂V (r; {R0 })vsαqν
d3 rei(k −k)·(r+T −T) χ∗nk (r)χn0 k0 (r)=ei(k −k−q)·T
031/2∂R(T−T(NC Ms ωqν )0 )sαcellTT0 sαZXXvsαqν
∂V (r; {R0 })= δk0 ,k+qd3 rχ∗nk (r)χn0 k+q (r)eiq·(r−T)
.
(35)
01/2
∂RTsα
(NC Ms ωqν )cellsαTIf we use VKS alternative
to V , we obtain the screened electron-phonon vertex. The screened deforP0mation potential T eiq·(r−T) ∂VKS (r; {R0 })/∂RTsα
is obtained as a biproduct of DFPT calculation.
This deformation potential has lattice periodicity asXT

50eiq·(r+T −T)∂V (r + T0 ; {R0 }) X iq·(r−T) ∂V (r; {R0 })
=e00
∂RTsα
∂RTsα(36)

TDensity functional perturbation theory for latticeForce constant

Z
∂FTsα
∂
∂V (r; {R0 })∂2EC
3=−=−F+drρ(r)Tsα00000∂RTsα∂RT∂RT∂RT∂RTsα0 s0 α00 s0 α00 s0 α0ZZ
∂ 2 EC∂ 2 V (r; {R0 })∂ρ(r) ∂V (r; {R0 })33=+drρ(r)+dr(37)000000
∂RTsα
∂RT
∂RTsα
∂RT∂RT∂RTsα0 s0 α00 s0 α00 s0 α0

CTsαT0 s0 α0 ≡4Dynamical materix
XC̃qsαs0 α0 ≡C0sαTs0 α0 eiq·T
T=X

eiq·TT∂ 2 EC+0 ∂R0∂RsαT0 s0 α0Z

d3 rρ(r)∂ 2 V (r; {R0 })
0 ∂R0
∂R0sα
Ts0 α0

Z+d3 r∂V (r; {R0 }) X iq·T ∂ρ(r)e00∂R0sα∂RTs0 α0T(38)
Monochromatic perturbation∂ρ(r)eiq·T=2eiq·T0∂RTsα
TTnkXX∂ϕ∗nk (r)∂ϕnk (r)ϕnk (r) + ϕ∗nk (r)
00
∂RTsα
∂RTsα



∂χnk (r)∂χ∗nk (r)∗=2eχnk (r) + χnk (r)
00
∂RTsα
∂RTsα
Tnk

X∂χ∗nk (r)∂χnk (r)iq·riq·(T−r)∗eχnk (r) + χnk (r)= 2e00∂RTsα∂RTsα
Xiq·T

(39)

Tnk



∇2+ VKS (r) − εnk−2∂εnk∂VKS (r)−ϕnk (r)00∂RTsα∂RTsα
(ik + ∇)2∂χnk (r)∂εnk∂VKS (r)+ VKS (r) − εnk−=−χnk (r)
0002
∂RTsα
∂R0sα
∂RTsαX∂VKS (r)∂χnk (r) X iq·(T−r) ∂εnk
(ik + ∇)2+ VKS (r) − εnk=e−χnk (r)eiq·(T−r) −
0002
∂RTsα
∂R0sα
∂RTsα
TT!
XX
(ik + iq + ∇)2∂εnkiq·(T−r) ∂χnk (r)iq·(T−r) ∂VKS (r)−+ VKS (r) − εnke= NC δq0−eχnk (r).
0002
∂RTsα
∂R0sα∂RTsα∂ϕnk (r)=0∂RTsαTT(40)Each component has lattice periodicity
X0eiq·(T−r−T )TXT∂χnk (r + T0 ) X iq·(T−r) ∂χnk (r)
=e00
∂RTsα
∂RTsα(41)∂VKS (r + T0 ) X iq·(T−r) ∂VKS (r)
=e00
∂RTsα
∂RTsα(42)T0eiq·(T−r−T )TDeformation potential is computed as follows:ZXδVH (r) δVXC (r) ∂ρ(r0 )∂VKS (r) X iq·(T−r) ∂V (r)3 0=e+dr+eiq·(T−r)
000
∂RTsα
∂RTsα
δρ(r0 )
δρ(r0 )
∂RTsαTTZX∂V (r)
1∂ρ(r0 )
3 00=eiq·(T−r)+dr+f(r,r)
XC00∂RTsα|r − r0 |∂RTsαT
X
Z0X
1iq·(T−r) ∂V (r)3 0 iq·(r0 −r)
0iq·(T−r0 ) ∂ρ(r )e+dre+f(r,r)
e=XC00∂RTsα|r − r0 |∂RTsαTT(43)

5