ch3 lattice waves - snu open courseware
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Chapter 3. Lattice Waves
Lattice wavesโข Lattice waves
: Vibrational motion of the atoms in a crystalline solid in terms of a wave passing through the atoms of the crystal as they are displaced by their thermal energy from their rest positions.
โข The thermal properties of solids are strongly related to the lattice waves
โข The movement of electrons (mobility) are hindered due to scattering by lattice waves.
โข Lattice waves have their particle-like counterpart, called phonons: quanta of energy ฤงฯn, ฯn: normal vibrational modes
โข Energy exchanging interactions with lattice waves occur in integral multiple of ฤงฯn.
Lattice wavesโข Two examples:
1. Vibrations associated with a one-dimensional crystal in which all the atoms have the same mass and the same atomic spacing.
โAcoustic modesโ : long wavelength longitudinal vibration corresponds to the sound wave.
2. Vibrations with two or more different kinds of atoms in a one-dimensional crystals.โ Two different masses with a common atomic spacing
โก Two different atomic spacings for atoms with the same mass
๐
๐
๐๐-1 ๐๐ ๐๐+1
๐-1 ๐ ๐๐
displacement
position๐+1 ๐
โOptical modesโ : long wavelength transverse vibrations characterized by neighboring atoms being displaced in opposite directions. The long wavelength vibration can be excited by interaction with light if the material is at least partially ionic.
โข Transverse waves in a one dimensional infinite string
๐: displacement away from the x-axis
Transverse waves in a 1-D infinite lattice
-11up
+11downward
Force at
: ~
: ~
r rrr
r rrr
x ra
F F Fa
F F Fa
๐
๐
๐๐-1 ๐๐ ๐๐+1
๐-1 ๐ ๐๐
displacement
position๐+1 ๐
Assumption 1. Restrict the forces between nearest neighbor atoms 2. The force is an attractive force ๐น 3. ๐น is constant and in the direction of the nearest neighbor atoms
(Assumption: ๐ โช ๐
Transverse waves in a 1-D infinite lattice
โข The harmonic solution
: mathematical wave passing through the displaced atoms.
Such a wave has physical reality only at the locations of atoms, i.e., only at x=ra. Then,
๐ ๐ฅ, ๐ก ๐ด exp ๐ ๐๐ฅ ๐๐ก
๐ ๐๐, ๐ก ๐ด exp ๐ ๐๐๐ ๐๐ก
O
โด The net upward force on the atom at ๐ฅ ๐๐
๐น ๐น ๐๐ ๐๐๐ก โ
๐ ๐๐๐ก ๐๐ 2๐๐ ๐๐ where ๐ ๐น/๐๐
๐ ๐ด๐ ๐๐ ๐ด๐ ๐
-11 ~ r rrrF F
a
+11 ~ r rrrF F
a
โข Dispersion relationship
Lattice wave has a dispersive system: The velocity varies with frequency and wave length Reducible to 0 โค k โค ฯ/a (The first Brillouin zone)
๐ 4๐ sin ๐ ๐/2 โ ๐ 2๐ / sin ๐ ๐/2
Transverse waves in a 1-D infinite lattice
๐ 2๐ ๐ ๐ ๐
2๐ 1 cos ๐ ๐
The shortest wavelength
2, (1 cos 2 ) 2sincf
/F ma
Dispersion relation between ๐ and ๐
Transverse waves in a 1-D infinite lattice
๐โฒ ๐ด๐
๐ด๐ ๐ด๐ / โ ๐ ๐๐ โ / โ ๐ exp ๐๐2๐๐ ๐
๐ 2๐ / sin ๐ ๐/2
sin ๐ ๐/2 โ ๐๐/2
๐ฃ~
๐ฃ~
๐๐ ๐ / ๐ ๐น๐/๐ /
small ๐ region velocity becomes constant
cf) Displacement is identical for any ๐ and ๐ ๐
For small ๐ (long wavelength)
Transverse waves in a 1-D infinite latticeโข Note:
โ ๐ space is the reciprocal latticeโก
๐max 2 ๐น/๐๐Debye frequency
โ 2 a
Infinite wavelength
All atoms displaced by the same amount in the same direction.
โข Neighboring atoms are displaced by the same distance in opposite directions.
โข The shortest wavelength; โข The dashed wave (n>2) has shorter wavelength. However it does not
give any new information on the position of atoms.โข Equivalent to the Bragg reflection condition
โข Cannot propagate: group velocity at k=/a is equivalent to zero
Acoustical Branch
๐ 2๐
๐๐ 2๐ sin ๐ ๐ 2๐, ๐ 1, ๐ ๐
(โต ๐ 2๐ / sin ๐ ๐/2 )
Transverse waves in a 1-D finite latticeโข General solution for transverse waves
1, 2, , ( 1)mk m nL
๐ ๐ sin๐๐๐
2๐ฟ
๐ sin๐๐๐
2 ๐ 1 ๐ ๐: # of moving atoms
โด ๐ ๐ sin๐๐
2 ๐ 1 : normal modes
sin( / ( 1))The general solution for the atom at is
i trm m
r m rmm
A m r n ex ra
A
Finite set of discrete (,k) values
๐ ๐ฅ, ๐ก ๐ด exp ๐ ๐๐ฅ ๐๐ก ๐ต exp ๐ ๐๐ฅ ๐๐ก
๐ 2 : total number of atoms
(boundary conditions)
(allowed frequencies)
(displacement at ๐ฅ ๐๐ and for ๐ )
๐ 0 ๐ ๐ฟ 0
Longitudinal waves in a 1-D infinite lattice
โข The restoring force for longitudinal displacement depends on the spatial variation of the force (F) between atoms
โข Let F(a) represent the force between atoms when separated by a normal lattice spacing (a), then the net force on the rth
atom is
๐-1 ๐ ๐๐ ๐+1 ๐
๐๐-1 ๐๐ ๐๐+1
๐น ๐น ๐ ๐ ๐ ๐น ๐ ๐ ๐
๐น ๐ ๐ ๐ ๐น ๐ ๐ ๐๐๐น๐๐ โฏ
๐น ๐ ๐ ๐ ๐น ๐ ๐ ๐๐๐น๐๐ โฏ
For very small displacement
โด ๐น ๐ 2๐ ๐ 1๐๐น๐๐ ๐
๐ ๐๐๐ก
Longitudinal waves in a 1-D infinite lattice
๐น ๐ 2๐ ๐๐๐น๐๐ ๐
๐ ๐๐๐ก
Let ๐โฒ1๐
๐๐น๐๐
Then ๐ ๐๐๐ก ๐โฒ๐ 2๐โฒ๐ ๐โฒ๐
L: longitudinal acoustic waveT1, T2: transverse acoustic wavesT1=T2 for isotropic crystal structure
long ฮป longitudinal wave โก sound waves
3D
Crystallographic direction
Freq
uenc
y (T
Hz)
Q: Why are the frequencies for L greater than T?
๐ ๐๐๐ก ๐๐ 2๐๐ ๐๐
simiar to transverse waves except for ๐ โ ๐โฒ
Longitudinal waves in a 1-D infinite lattice
โข Long wavelength longitudinal wave โก sound wavesโข Velocity is given by the slope at k=0
๐ฃ๐๐ ~
๐ / ๐
๐ฃ๐๐ ~
๐โฒ / ๐
๐น โ ๐
๐ฃ ๐๐ ๐ ๐ ๐ฃ
๐1๐
๐๐น๐๐ ,
The longitudinal waves are ๐ times faster than transverse waves.
๐โฒ ๐๐๐๐น
๐๐ ,
Density of states for lattice wavesโข Density of states: # of allowed vibrational modes, N(), per
unit frequency interval, d.
max
max
1/22
max
max
sin2( +1)
cos2( +1) 2( +1)
1 : frequency spacing2( +1)
mn
d mdm n n
n
1/22
max max
2( +1)( ) 1nN d d
In a frequency interval d, there are d/ states. Therefore,
N(v) starts with a value of [2(n+1)/vmax] at v=0 and then increases with increasing v to a large value as v approaches vmax.
cf) cos ๐ = 1 sin ๐
between allowed modes
From ๐ 2๐ / sin
Lattice waves for two kinds of atoms
โขโ Lattice parameter is the same (a)โก Masses are different (m and M)
ex) compound
โขโ Lattice parameters are different (a and b)โก Mass is the same (m)
ex) more atoms in a unit cell
m M
a b
The first case (different masses & same spacing)
โข Follow the same procedure of transverse wave except that near atoms are different kinds.
m M
Zr-1 Zr Zr+1๐๐-1 ๐๐ ๐๐+1
(2r-1)a 2ra (2r+1)a r=0, 1, 2, โฏ
โ . ๐ ๐๐๐ก ๐ ๐ 2๐ ๐ ๐ ๐ where ๐
๐น๐๐
โ ก. ๐ ๐๐๐ก ๐ ๐ 2๐ ๐ ๐ ๐ where ๐
๐น๐๐
๐ ๐ด๐ for ๐ฅ 2๐ 1 ๐๐ ๐ต๐ for ๐ฅ 2๐๐
โข Assume that harmonic waves with the same values of ๐ and ๐ for both types of atoms
M M M
M M M
2 2
2
2 02 0
ika ika ika
ika ika
e A e A B e BA A e B e B
๐ ๐ด๐ ๐ ๐ด๐
M M
m m
2
2
( 2 ) 2 cos 02 cos ( 2 ) 0
A B kaA ka B
M m M m M m
M m M m
4 2 2
4 2 2
(2 2 ) 4 4 cos 02( ) 4 sin 0
kaka
๐ ๐ ๐ ๐ ๐ 4๐ ๐ sin ๐ ๐ (dispersion relations)
M M
m m
2
2
2 2 cos0
2 cos 2ka
ka
: two separate branches in the vibration spectrum
๐ด๐ต
2๐ cos ๐ ๐๐ 2๐
for ๐ฅ 2๐ 1 ๐
for ๐ฅ 2๐๐๐ ๐ต๐
๐ ๐ด๐ต ๐ ๐
๐ ๐๐๐ก ๐ ๐ 2๐ ๐ ๐ ๐
๐ ๐๐๐ก ๐ ๐ 2๐ ๐ ๐ ๐
The first case (different masses & same spacing)
1) For k = 0
< optical mode >
< acoustic mode >
ฯ+
ฯ-
The first case (different masses & same spacing)
๐ด๐ต
2๐ cos ๐ ๐๐ 2๐
2๐2๐ 1
โต 1 ๐ฅ / 1 ๐ฅ/2 for ๐ฅ<<1)
<<1
From previous solution,
๐ ๐ ๐ ๐ ๐ 14๐ ๐ ๐ ๐
๐ ๐
/
๐ ๐ ๐ ๐ ๐ 12๐ ๐ ๐ ๐
๐ ๐
๐ ๐ ๐ ๐
2๐ ๐ ๐ ๐
๐ ๐
โต cos ๐๐ 1 and ๐ 0 at k = 0
:Equal displacement of neighboring atoms
๐ด๐ต
๐๐
๐ 2 ๐ ๐
a)
b)
:Opposite displacement: In the long wavelength mode(k=0), neighboring atoms are displaced in opposite directions.
2) Near k = ฯ / 2a, sin2ka โ 1
๐ ๐ ๐ ๐ 4๐ ๐๐ ๐ ๐ ๐
โ ๐ 2๐
๐ด๐ต 0
โก ๐ 2๐
๐ด๐ต โ
gap opens for mโ M
๐ด๐ต
2๐ cos ๐ ๐๐ 2๐
The first case (different masses & same spacing)
Large mass
Small mass
AB
AB
๐ ๐ ๐ ๐ ๐ 4๐ ๐ sin ๐ ๐
๐2๐ ๐ ๐ ๐
๐ ๐
๐ 2 ๐ ๐ ๐ 2๐
๐ 2๐
k = ฯ / 2ak = 0
Vibration spectrum pf CdTe
LO: Longitudinal optical vibrationTO: Transverse optical vibration
LA: Longitudinal acoustic vibrationTA: Transverse acoustic vibration
An EM wave that propagates the lattice displaces the oppositely charged ions in opposite directions and forces them to vibrate at the frequency of the wave. Most of the energy is then absorbed from the EM wave and converted to lattice vibrational energy (heat).
Reststrahlen absorption: (German: residual rays) โข long wavelength transverse modes in partially ionic crystals could be
directly excited by light of a suitable.โข Strong interaction between a light wave and a lattice wave under the
unusual conditions for resonance.extinction coefficient K versus wavelength