Download Debye temperature

Subgroup meeting 2010.12.07
introduction of thermal transport
members: 王虹之.盧孟珮
introduction of
thermal transport
Phonon effect
Electron effect
Lattice vibration
Debye model of
lattice vibration
Bose-Einstein
model
Dulong-petit
model
Heat capacity
The contribution
of electron of
heat capacity
Debye
temperature
phonon
K space,
Reciprocal lattice
Brillouin zone
Boundary
scattering
Scattering
mechanism
Dispersion
relation
Phonon-phonon
scattering
Normal Process
Umklapp process
Optical and
Acoustic phonons
Outline
•
•
•
•
•
Dispersion relation (review)
Sepcific heat (review)
Different temperature ranges
Low temperature range
Debye temperature
Disparsion relation
ω axis
Optical branch
• At lower energy (lower k)
• Ka << π (near zero)
• The dispersion relation is
regard as linear
Acoustic
branch
0
Electromagnetic wave
• ω/k=c
• c:light speed
• c is constant  linear
dispersion relation
п/a K axis
Disparsion relation
ω axis
Optical branch
Acoustic
branch
0
п/a K axis
We get linear part of dispersion relation.
Disparsion relation
ω axis
Optical branch
Standing wave
k= π/a 2a=λ
(k is fixed, ω is independent from k)
Group velocity
Acoustic
branch
vg = 0
0
п/a K axis
at k=π/a
Energy does not propagate in
the medium (standing wave)
Disparsion relation
• Classical mechanism :
Harmonic oscillator
(conservative force and wave)
• We get dipersion relation by only
classical assumption
Specific heat
Low temperature
Intermediate
High temperature
There are three segments of the experiment
curve of specific heat (Cv).
Specific heat
General form of specific heat
Quatum theory of Harmonic solid
• Bose-Einstein distribution function
• Stationary states of vibration
s: types of phonons
k: wave factor
Frequencies are functions
of wave factor
Specific heat
Low temperature
Intermediate
High temperature
Different approximationa of temperature ranges:
• Low temperature: Debye model
• Intermediate temperature: Debye & Einstein
• High temperature: Dulong-Petit
Specific heat
at Low temperature
D(ω)
ωD
Debye model approximation
ω
Low temperature
Debye model
•The angular frequencies are
diverse values.
•The maximum allowed value is
cut-off frequency ωD
Debye model
Specific heat
Density of states
Energy per phonon
Bose-Einstein
distribution
Cut-off frequency
Debye model
Density of states D(ω)
D(ω)
ωD
The spacing between constant
frequency surface depends on
(group velocity) dω/dk
ω
dω/dk1 > dω/dk2
For simple case that vp=vg
ω/k=constant=sound velocity
ds (ω)
Δk1
dk
ω+dω
constant surface
Δk2
K space
ω surface
The difference between
constant ω surface is Δω
We can get approximation D(ω) of
Debye model (low temperature)
Specific heat at Low temperature
ω axis
Replacing by approximation
Optical
branch
Acoustic
branch
- п/a
0
п/a K axis
Dispersion relation
•Ignored optical branches
•Replace acoustic branch with liner branches
Specific heat at Low temperature
ω axis
When T is small
Optical
branch
High frequencies result in
large
Acoustic
branch
- п/a
0
п/a K axis
The contributions of high
frequencies for LOW
temperature are negligible.
Specific heat at Low temperature
Specific heat at Low temperature
At very low temperature
Specific heat
Which is Debye
law
Condition assumption:
• Only Acoustic modes are thermally excited
• For actual crystals Temperature may necessary
to be below T=θ/50
θ :Debye temperature
D(ω)
Debye temperature
ωD
ω
Determine a cut-off frequency of specimen
E=kB θ
KD
KT
E=kBT
Define Debye temperature θ in terms of cut-off frequency
Debye temperature
L: length of specimen
N: number of ions in
specimen (total
vibration modes)
Compare boundary conditions:
Brillion Zone (atom spacing)
Cut off frequency
(atom packing)
For ideal simple cubic specimen
KD are different with different kinds of lattice packing
n/ L
Debye temperature
What are the factors that determine Debye
temperature?
Theoretical :
v(ρ,K,G): sound velocity in the specimen
N/V: atom concentration (packing density)
Experimental:
The Debye temperature were determined by fitting
the observed specific heats Cv formula at the point of
½ Dulong-Petit value (3NkB/2)
Debye temperature
Element
Crystal structure
Concentration 10^22
/cm^3
Sound speed m/s
Debye temperature
Li
Bcc
4.7
Na
Bcc
2.652
K
Bcc
1.402
Rb
Bcc
1.148
Cs
Bcc
0.905
6000
344
334
3200
158
151.6
2000
91
76.6
1300
56
46.6
~
38
Element
Crystal structure
Concentration 10^22
/cm^3
Sound speed m/s
Debye temperature
Be
Hcp
12.1
Mg
Hcp
4.3
Ca
Fcc
2.3
Sr
Fcc
1.78
Ba
Bcc
1.6
12870
1440
1011.3
4640
400
258.3
3810
230
172.1
~
147
1620
110
64.9
Future work
• Intermediate region
• High temperature region
• Electron Effect
Thanks for your attention