sanjay govindjee structural engineering, mechanics, and...
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Phonon dispersion computationsSanjay Govindjee
Structural Engineering, Mechanics, and Materials
Department of Civil Engineering
University of California, Berkeley
Phonon dispersion computations – p. 1/20
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Equations of Motion
mkq̈α(A
k)= −
∑
β,B,n
Kα(A
k)β(B
n)qβ(B
n)
To solve we make a plane-wave plus a periodic (Born –v. Karmen) assumption:
qα(A
k)(t) = q̂αk(ω,f)ei(xA·f−ωt)
qα(N+1
k ) = qα(1
k)
Assuming N lattice cells in each coordinate direction.xA = FXA, if deformation.
Phonon dispersion computations – p. 2/20
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Implications of Periodicity
Consider a fixed lattice direction xA = Aa1. Theperiodic BC implies
ei(N+1)a1·f = eia1·f
Define the (dual) reciprocal vectors bj via bj · ai = δij;i.e. bi = (aj × ak)/[a1,a2,a3], where i, j, k are a cyclicpermutation of {1, 2, 3}. If one expands f asf = f1b1 + f2b2 + f3b3, then
f1 =2πn1
Nn1 ∈ J
In general f ∈(
2πn1
N , 2πn2
N , 2πn3
N
)where n1, n2, n3 ∈ J.
Phonon dispersion computations – p. 3/20
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Restricted Wave Vector Range
The unique (physically) distinct solutions only occur for
ni ∈ {0, 1, · · · , N − 1}
Or any shifted set of integers of “length” N .
This follows since adding N to a wave vector index, say2πn/N → 2π(n + N)/N , yields
ei2π(n+N)/N = ei2πnei2π = ei2πn
Phonon dispersion computations – p. 4/20
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Eigen Equations
Using our expansion in the equations of motion
−mkω2(f)q̂αk(ω,f)eixA·f = −
∑
β,B,n
Kα(A
k)β(B
n)q̂βneixB ·f
The sum over B ranges over all cells in the crystalbut effectively is restricted to just those that interactwith the atom at
(Ak
).
Without loss of generality, let us focus on the cell A = 1and assume xA = 0. Define the dynamical matrix
Dαkβn =∑
B
Kα(1
k)β(B
n)eixB ·f
Phonon dispersion computations – p. 5/20
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Phonon Spectra Equation
We are now left with the eigenproblem∣∣D(αk)(βn)(f) − mkω
2(f)I(αk)(βn)
∣∣ = 0
This is a 3s× 3s eigenvalue problem that needs to besolved for each acceptable f of which there areN3 = Nc.By symmetry considerations one can greatly reducethe number of needed computations.All the binding energy of the crystal is hidden insidethe dynamical matrix which contains the crystalstiffness.The result is valid for full finite deformation states aslong as the stiffness is computed with respect to thedeformed (Cauchy-Born) lattice positions (with basisrelaxation).
Phonon dispersion computations – p. 6/20
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1D example with 2 atom basis
a
M_1 m_2
2 atom basis s = 2
1 dof per atom
N lattice cells
Phonon dispersion computations – p. 7/20
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Equations in 1D
In 1D we can drop the α, β subscripts to give:
mkq̈(A
k)= −
∑
B,n
K(A
k)(B
n)q(B
n)
The corresponding Ansatz is:
q(A
k)(t) = q̂k(ω, f)ei(xAf−ωt)
where xA = (A − 1)a and by the Born – v. Karmenconditions f = 2πn/aN .
Phonon dispersion computations – p. 8/20
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Dynamical Matrix
The dynamical matrix in this setting will be
Dkn(f) =∑
B
eixBfK(1
k)(B
n)
To compute let us assume that φ(lb) = 12C(lb − lo)
2 andthat interactions only occur with respect to neighboringatoms (nearest neighbor approximation).
Phonon dispersion computations – p. 9/20
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Stiffness and Dynamical Matrix
Potential Energy (skip self interaction terms)
V =1
2
∑
A,B,k,n
φ(|r(A
k)− r(B
n)|)
Phonon dispersion computations – p. 10/20
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Stiffness and Dynamical Matrix
Forces
∂V
∂r(C
m)=
1
2
∑
A,B,k,n
φ′(|r(A
k)− r(B
n)|)
r(A
k)− r(B
n)
|r(A
k)− r(B
n)|
[δACδkm − δBCδmn]
=∑
B,n
φ′(|r(C
m) − r(B
n)|)
r(C
m) − r(B
n)
|r(C
m) − r(B
n)|
Phonon dispersion computations – p. 10/20
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Stiffness and Dynamical Matrix
The stiffness
∂2V
∂r(C
m)∂r(D
p)=
∑
B,n
φ′′(|r(C
m) − r(B
n)|)
r(C
m) − r(B
n)
|r(C
m) − r(B
n)|
r(C
m) − r(B
n)
|r(C
m) − r(B
n)|[δCDδmp − δBDδnp]
+ φ′(|r(C
m) − r(B
n)|)
∂
∂r(D
p)
r(C
m) − r(B
n)
|r(C
m) − r(B
n)|
=∑
B,n
φ′′(|r(C
m) − r(B
n)|)[δCDδmp − δBDδnp]
Phonon dispersion computations – p. 10/20
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Stiffness and Dynamical Matrix
The Dynamical matrix
Dmp =∑
D
eixDf
∑
B,n
φ′′(|x( 1
m) − x(B
n)|)[δ1Dδmp − δBDδnp]
=∑
B,n
φ′′(|x( 1
m) − x(B
n)|)δmpe
i·0
−∑
D
eixDfφ′′(|x( 1
m) − x(D
p)|)
Phonon dispersion computations – p. 10/20
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Stiffness and Dynamical Matrix
The Dynamical matrix
Dmp =
[
2C 0
0 2C
]
−
[
0 C
0 0
]
e−iaf
−
[
0 C
C 0
]
ei·0
−
[
0 0
C 0
]
eiaf
Phonon dispersion computations – p. 10/20
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Stiffness and Dynamical Matrix
The Dynamical matrix
Dmp = C
[
2 −(1 + e−iaf )
−(1 + eiaf ) 2
]
Phonon dispersion computations – p. 10/20
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Final eigencompuation
Eigenvalue problem∣∣∣∣∣C
[
2 −(1 + e−iaf )
−(1 + eiaf ) 2
]
− ω2(f)
[
m1 0
0 m2
]∣∣∣∣∣= 0
Solutions
ω2(f) =C(m1 + m2)
m1m2
[
1 ±
√
1 −2m1m2(1 − cos(fa))
(m1 + m2)2
]
Phonon dispersion computations – p. 11/20
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Dispersion Curves
a = 1.2, m1 = 50, m2 = 30, C = 1000
0 0.5 1 1.5 2 2.5 30
2
4
6
8
10
12
f
ω
sqrt(2C/m1)
sqrt(2C/m2)
sqrt(2C(m1+m
2)/m
1m
2)
Optical BranchAcoustic Branch
Phonon dispersion computations – p. 12/20
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Branches
Acoustic branch eigenvector f = π2a → λ = 2π
f = 4a
Optical branch eigenvector f = π2a → λ = 2π
f = 4a
Phonon dispersion computations – p. 13/20
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Remarks
Per space dimension, d, one has 1 acoustic branch([P,SV,SH] in 3D).
(s − 1) · d optical branches.
s · d branches in total
Continuous curves are shown but they are reallydiscrete.
Slopes correspond to the group velocities dω/df whichgovern the velocity of energy transport.
Phonon dispersion computations – p. 14/20
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Multi-D Case
The potential depends on V (FXA + bk + q(A
k)) where
the bk need to be found by minimizing the crystalenergy for the given deformation state.
Assuming such, as well as pair potentials, and noting
r(A
k)=
xA︷ ︸︸ ︷
FXA +bk︸ ︷︷ ︸
x(A
k)
+q(A
k), gives
V =1
2
∑
A,B,k,n
φ(‖r(A
k)− r(B
n)‖)
Phonon dispersion computations – p. 15/20
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Forces
∂V
∂r(C
m)=
1
2
∑
A,B,k,n
φ′(‖r(A
k)− r(B
n)‖)
r(A
k)− r(B
n)
‖r(A
k)− r(B
n)‖
[δACδkm − δBCδmn]
=∑
B,n
φ′(‖r(C
m) − r(B
n)‖)
r(C
m) − r(B
n)
‖r(C
m) − r(B
n)‖
Phonon dispersion computations – p. 16/20
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Stiffness
∂2V
∂r(C
m)∂r(D
p)=
∑
B,n
{
φ′′(‖r(C
m) − r(B
n)‖)
r(C
m) − r(B
n)
‖r(C
m) − r(B
n)‖
⊗r(C
m) − r(B
n)
‖r(C
m) − r(B
n)‖[δCDδmp − δBDδnp]
+ φ′(‖r(C
m) − r(B
n)‖)
[
1δDCδmp − 1δDBδpn
‖r(C
m) − r(B
n)‖
−(r(C
m) − r(B
n)) ⊗ (r(C
m) − r(B
n))
‖r(C
m) − r(B
n)‖3
[δDCδmp − δBDδpn]
]}
Phonon dispersion computations – p. 17/20
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Dynamical Matrix (3 × 3 block)
Dmp(f) =∑
D
eixD·fK( 1
m)(D
p)
= δmp
∑
B,n
{
φ′′(‖β‖)β ⊗ β
‖β‖2+ φ′(β)
[1
‖β‖−
β ⊗ β
‖β‖3
]}
+∑
D
{
eixD·f[
φ′(‖v‖)
(v ⊗ v
‖v‖3−
1
‖v‖
)
− φ′′(‖v‖)v ⊗ v
‖v‖2
]}
β = x( 1
m) − x(B
n)and v = x( 1
m) − x(D
p)
Phonon dispersion computations – p. 18/20
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Final Matrices
Dynamical Matrix
D =
D11 D12 · · · D1s
D21. . . ...
... . . . ...Ds1 · · · · · · Dss
Mass Matrix
I =
m11
m21
. . .
ms1
Phonon dispersion computations – p. 19/20
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Remarks
Self energy terms are to be skipped, i.e. when v = 0
and β = 0.
Reasonable units for such computations are Åand eV.This implies force = eV/Å≈ 1.6 × 10−9 N.
If mass is given in a.m.u. (grams per mole) thencomputed frequencies are in units of roughly 0.49 THz.
Phonon dispersion computations – p. 20/20