15. Phonons
Transcript of 15. Phonons
Institute of Solid State PhysicsTechnische Universität Graz
15. Phonons
May 15, 2018
Normal modes
The motion of the atoms can be described in terms of normal modes.
In a normal mode, all atoms oscillate at the same frequency .
The energy in a normal mode is quantized,
n is the number of phonons in that normal mode.
12( ).E n
Linear Chain
2
1 1 1 12 ( ) ( ) ( 2 )ss s s s s s s
d um C u u C u u C u u udt
21
22
23
24
25
26
2 0 0 02 0 0 0
0 2 0 0=0
0 0 2 00 0 0 2
0 0 0 2
AC m C CAC C m CAC C m CAC C m CAC C m CAC C C m
a
s = 0 s = 1 s = 2 s = 3 s = 4s = -1s = -2s = -3
Assume every atom oscillates with the same frequencyi t
s su A e
2 12 I T T 0.C m C A
Normal modes are eigen functions of T
solutions are eigenfunctions of the translation operator
a
( 1 ) ( )i k s a t ika i ksa t ikas k k sTu A e e A e e u
s = 0 s = 1 s = 2 s = 3 s = 4s = -1s = -2s = -3
N atoms, N normal modes, N eigenvectors of the translation operator, N allowed values of k in the first Brillouin zone.
( )iksa i t i ksa ts k ku A e e A e
solutions:
4 sin2
C kam
2
1 12 ( 2 )ss s s
d um C u u udt
( )i ksa ts ku A e
1 12
2
2
( 2 )( 2 )
2 (1 cos( ))
i k s a t i k s a ti ksa t i ksa t
ika ika
me C e e em C e em C ka
2 1sin 1 cos2 2ka ka
a
Linear Chains = 0 s = 1 s = 2 s = 3 s = 4s = -1s = -2s = -3
Linear Chain - dispersion relation
4 sin2
C kam
2
1 12 ( 2 )ss s s
d um C u u udt
( )i ksa ts ku A e
C am
speed of sound =
Max. freq.
Linear Chain - density of states
Determine the density of states numerically 4 sin2
C kam
Linear Chain - density of states
for every k calculate the frequency
4 sin2
C kam
4 sin2
C kam
1( )D k
cos2
C kad a dkm
( ) ( ) dkD D kd
2
1
14
DC mam C
This case is an exception where the density of states can be determined analytically.
density of states
D()
van Hove singularity
2
1
14
DC mam C
4 sin2
C kam
1( )D k
cos2
C kad a dkm
( ) ( )D k dk D d
maximum frequency
Linear chain M1 and M2
2
1 12 ( 2 )ss s s
d uM C v u vdt
( )i ksa ts ku u e
Newton's law:
2N modes
assume harmonic solutions
2
2 12 ( 2 )ss s s
d vM C u v udt
( )i ksa ts kv v e
21
22
(1 exp( )) 2
(1 exp( )) 2k k k
k k k
M u Cv ika Cu
M v Cu ika Cv
a
Linear chain M1 and M2
21
22
(1 exp( )) 2
(1 exp( )) 2k k k
k k k
M u Cv ika Cu
M v Cu ika Cv
a
21
22
2 (1 exp( ))0
(1 exp( )) 2k
k
uM C C ikavC ika M C
4 2 21 2 1 22 2 1 cos( ) 0M M C M M C ka
dispersion relation 2
22
1 2 1 2 1 2
4sin1 1 1 1 2
ka
C CM M M M M M
Optical phonon branch
Acoustic phonon branch
http://lampx.tugraz.at/~hadley/ss1/phonons/1d/1d2m.php
normal modes
density of states
2 22
1 2 1 2 1 2
1 1 1 1 4sin kaC CM M M M M M
Linear chain M1 and M2
2a
The branches of the dispersion curves can be translated by a reciprocal lattice vector G.
fcc
2
1 1 1 12
1 1 1 1 1 1 1 1
1 1 1 1 1 1
2
xx x x x x x x xlmnl mn lmn l mn lmn lm n lmn lm n lmn
x x x x x x x xl mn lmn l mn lmn lm n lmn lm n lmn
y y y y y y y yl mn lmn l mn lmn lm n lmn lm n lmn
lm
d u Cm u u u u u u u udt
u u u u u u u u
u u u u u u u u
u
1 1 1 1 1 1z z z z z z z z
n lmn lm n lmn l mn lmn l mn lmnu u u u u u u
and similar expressions for the y and z motion
1 2 3expx xlmn ku u i lk a mk a nk a t
These are eigenfunctions of T.
Normal modes are eigenfunctions of T
1 2 3expy ylmn ku u i lk a mk a nk a t
1 2 3expz zlmn ku u i lk a mk a nk a t
1 1 2 2 3 3
1 2 3 1 2 3
1 2 3
exp
exp exp
exp
x xpqr lmn k
xk
xlmn
T u u i lk a pa mk a qa nk a ra t
i lpk a qmk a rnk a u i lk a mk a nk a t
i lpk a qmk a rnk a u
fcc
1 2 3exp exp .
2 2 2yx zx x x
lmn k k
l n k al m k a m n k au u i lk a mk a nk a u i
Substitute the eigenfunctions of T into Newton's laws.
http://lamp.tu-graz.ac.at/~hadley/ss1/phonons/fcc/fcc.html
For every k there are 3 solutions for .
Phonon dispersion AuFr
om S
prin
ger M
ater
ials
: Lan
dhol
t Boe
rnst
ein
Dat
abas
e
Materials with the same crystal structure will have similar phonon dispersion relations
Cu
Au
Phonon DOS fcc
D()
From
Spr
inge
r Mat
eria
ls: L
andh
olt B
oern
stei
n D
atab
ase
Phonon dispersion Fe
From Springer Materials: Landholt Boernstein Database
H P N
Phonon DOS Fe
D()
From Springer Materials: Landholt Boernstein Database
Next nearest neighbors (bcc)
D() C2/C1 = 0 C2/C1 = 1/6 C2/C1 = 1/3
C2/C1 = 1/2
D()C2/C1 = 1
The normal modes remain the same (the translational symmetry is the same).
Phonon DOS Fe
D()
From Springer Materials: Landholt Boernstein Database
C2/C1 = 0
C2/C1 = 1/6
D()