15. phonons
TRANSCRIPT
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()