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Phonons in solids
https://wiki.fysik.dtu.dk/ase/ase/phonons.html
www.csun.edu/~rdconner/227/slides/Thermal%20Properties.pp
When wave propagates in the solid, there are one longitudinal and two transverse polarizations .
s-3 s-2 s-1 s s+1 s+2
a
s-1 s s+1 s+2 s+3 s+4
a
us+2 us-2 us-1 us us+1
us-1 us+1 us+2 us+3 us+4
k
k
https://wiki.fysik.dtu.dk/ase/ase/phonons.html
m
ky kx
kz
2-D
3-D
https://wiki.fysik.dtu.dk/ase/ase/phonons.html
in solid-state physics, a quantized particle-like unit of vibrational energy arising from the oscillations of the atoms within a crystal. Any solid crystal, such as ordinary table salt (sodium chloride), consists of atoms bound into a specific repeating three-dimensional spatial pattern called a lattice. Because the atoms have thermal energy, the lattice vibrates in response to applied forces and generates mechanical waves that carry heat and sound through the crystal. In quantum mechanics a packet of these waves constitutes a phonon, which travels within the crystal with particle-like properties.
A phonon is a quantum of vibrational mechanical energy, just as a photon is a quantum of light energy.
Enciclopedia Britannica
The number of normal vibration modes for discrete systems
See Kittel for this
, ,
k
sound velocity
Aschroft
2
http://www.wikiwand.com/en/Phonon
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.0
0.2
0.4
0.6
0.8
1.0
ω [
(4C/
M)1/2
sec-1
]
k [(2π /a) m-1]
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.00.0
0.2
0.4
0.6
0.8
1.0
ω [(4
C/M
1/2 s
ec-1 )]
k [(2π /a) m-1]
The displacement can always be described by a wavevector within the first BZ.
k1 k3 k4 Reduced zone scheme
k2
λ=4a
λ=4a/3 λ=4a/7
2aπ
4a2πk ,a4λ 11 ===
2aπ
a2π
2a3πk ,
34aλ 22 −===
2aπ
a2π
2a5πk ,
54aλ 33 +===
2aπ
a2π2
2a7πk ,
74aλ 44 −
===
Freq
uenc
y, w
Wave vector, K 0 π/a
Longitudinal Acoustic (LA) Mode
Transverse Acoustic (TA) Mode
from Ibach and Luth Solid-State Physics An Introduction to Principles of Materials Science
-acoustic branch (k, ω) -optical branch (k, ω)
why optical branch?
http://www.chemistry.uoguelph.ca/educmat/chm729/phonons/optmovie.htm
For a 3 D crystal there are 2 additional transverse waves
q
ω
π/a
Brillouin zone boundary
acoustic
optic
Phonon dispersion
Acoustic phonon at q=0: Rigid translation of the crystal: ω=0!
3s phonon-branches
s atoms per primitive cell 3s vibration branches Acoustic (3) : LA (longitudinal) TA1 (transverse) TA2 (transverse) Optic (3s-3) : LO (longitudinal) TO (transverse)
NaCl – two atoms per primitive cell
6 branches:
1 LA
1 LO
2 TA
2 TO
For each mode in a given propagation direction, the dispersion relation yields acoustic and optical branches:
• Acoustic • Longitudinal (LA) • Transverse (TA)
• Optical • Longitudinal (LO) • Transverse (TO)
https://www2.warwick.ac.uk/fac/sci/physics/current/postgraduate/regs/mpags/ex5/phonons/
Dispersion Relations: Theory vs. Experiment In a 3-D atomic lattice we expect to observe 3 different branches of the dispersion relation, since there are two mutually perpendicular transverse wave patterns in addition to the longitudinal pattern we have considered.
Along different directions in the reciprocal lattice the shape of the dispersion relation is different. But note the resemblance to the simple 1-D result we found.
Aschroft
http://www.phonon.fc.pl/index.php
Dispersion in Si
http://www.phonon.fc.pl/index.php
0 0.2 0.4 0.6 0.8 1.00.20.40
2
4
6
8
(111) Direction (100) DirectionΓ XL Ka/π
LA
TATA
LA
LO
TO
LO
TO
Freq
uenc
y (1
0 H
z)12
Dispersion in GaAs (3D)
http://physics.stackexchange.com/questions/81097/what-determines-phonon-phonon-collisions
Review
Fundamental concepts needed to understand the vibratory motion of atoms:
Normal mode of vibration : all atoms oscillate with the same frequency.
Only atomic vibrations with certain frequencies, determined by interatomic forces, occur in any given solid. --- periodic arrangement of atoms
Normal mode displacements for these materials have an especially simple form and are relatively easy to discuss.
If the displacements of atoms from their equilibrium sites are small, the forces they exert on each other are proportional to their displacements, as if the atoms were connected by ideal spring.
Atomic motions are simple harmonic.
substitute expressions for the forces into Newton’s 2nd law
generate a set of differential equations, one for each atom
seek solutions for which all atomic displacements have the same frequency
The energy of an elastic mode of angular frequency ω is It is quantized, in the form of phonons, similar to the quantization of light, as both are derived from a discrete harmonic oscillator model. Elastic waves in crystals are made up of phonons. Thermal vibrations are thermally excited phonons.
En = n + 12 ( ) ω
Kittel
http://phy.ntnu.edu.tw/~changmc/Teach/SS/SS_note/
Kittel from Kittel
DDBk ωθ ⋅=⋅
http://phy.ntnu.edu.tw/~changmc/Teach/SS/SS_note/
http://phy.ntnu.edu.tw/~changmc/Teach/SS/SS_note/
ωωωω= ∫ω
=ω
dTnDTU ),()()(max
0
http://phy.ntnu.edu.tw/~changmc/Teach/SS/SS_note/
http://phy.ntnu.edu.tw/~changmc/Teach/SS/SS_note/
htt // h t d t / h /T h/SS/SS t /
Debye temperature ?
• above θD no suplimentary vibration modes are excited.
•Below θD the vibration modes start to freeze.
Debye Model: Theory vs. Expt.
Universal behavior for all solids!
Debye temperature is related to “stiffness” of solid, as expected
Better agreement than Einstein model at low T
from Kittel
Debye Model at low T: Theory
vs. Expt.
Quite impressive agreement with predicted CV ∝ T3 dependence for Ar! (noble gas solid)
Solid argon
Phonons and thermal properties of solids
https://courses.physics.illinois.edu/phys460/fa2006/handouts/460-lect11.pdf
https://courses.physics.illinois.edu/phys460/fa2006/handouts/460-lect11.pdf
https://courses.physics.illinois.edu/phys460/fa2006/handouts/460-lect11.pdf
A thermal current only arise in a temperature
gradient, and the thermal current density is
proportional with the gradient TAQq ∇κ−== /
..
The most important interaction process between phonons is the three-phonon process in which two phonons merge into a single phonon, or a single phonon decays into two phonons. Conservation of energy and quasimomentum requires:
Depending on the signs, these equations reflect the creation or annihilation of a phonon in the collision process. A characteristic feature of quasimomentum conservation is the occurrence of a reciprocal lattice vector G in this equation. Processes that do not involve a reciprocal lattice vector are called normal processes, whereas those that do, are called umklapp processes.
see Enns
see Enns
see Kittel
1st BZ in k-space
Phonon-phonon scattering phonon displaces atom which changes the force constant C (anharmonic terms) scatter other phonons
Normal processes : all ks are in BZ
1k
2k3k
321 kkk =+
crystal momentum is conserved
Umklapp processes : k3 is outside BZ
1st BZ in k-space
1k
2k*3
k3k
G *
21 3kkk =+ outside BZ
*3 3
kGk =+
Gkkk 321 +=+crystal momentum is not conserved
three phonon process
such processes can reverse the direction of heat transfer
Mean free path Σ⋅
=Λn
1n- the density of scattering centers
Σ –scattering cross section
As already mentioned, the sum of the quasimomenta of the colliding phonons is conserved in N-processes, and consequently the total quasimomentum P of all phonons is also conserved. nq represents the number density of the phonons with the wave vector q.
•normal collisions do not diminish the transport of momentum •change the frequency of the colliding phonons and thus contribute to the establishment of the local thermal equilibrium.
3T∝Σ
2Tn ∝
5−∝Λ T
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
< n(
ω, T
) >
x-1 = kBT / ω
At very low temperatures: collisions with defects
Λ = constant i ~ T3
Λ⋅⋅= vc31
At high temperatures, the overwhelming number of excited phonons are phonons with a frequency close to the Debye frequency ωD and a wave vector comparable with that of the zone boundary. As a consequence, virtually every collision leads to a final state outside the Brillouin zone and is therefore an umklapp process. At T > Θ the number of thermally excited phonons and hence the density of scattering centers n rises proportional to T. Since the frequency of the dominant phonons is ωD that does not change with temperature, the cross section Σ for the phonon–phonon collisions is constant.
1−∝Λ T Λ⋅⋅= vC31
1−∝ T
C ~ const.
Enns
Intermediate temperature
Te 2/Θ∝Λ∝
Enns
Phonon mean free path Λ ( ∝ τ )
T (K)
Log-log plot
Exponential
Slope -1
High T , Λ=vgτ ∝ T-1
∝ number of phonons No distinction between N and U processes
Intermediate T , Λ=vgτ ∝ (1/T)exp(1/T)
dominated by U process
Very low T , Λ=vgτ =constant
Below 5K, enriched Ge74 shows T3 dependence of κ
due to boundary scattering
At low temperatures, Λ → L (sample’s size) Phonon propagation ~ ballistic κ =(1/3)vgΛCV ~ vgLCV κ ∝ CV ∝ T3 Debye
Log-log plot of κ(T)
At intermediate temperatures, κ=(1/3)vgΛCV =(1/3)vg
2τCV
/TΘ
B
BV
DeTk
1~τ
3NkconstantC ==
/TΘ
B
DeTk
1~κ
U-processes
κ (W
att m
-1K-
1 )
T(K)
Thermal conductivity of LiF crystal bar
Different cross sectional area
(a)1.33mm × 0.91mm
(b)7.55mm × 6.97mm
Data show
1. Below 10K, κ ∝ T3
2. As temperature increases, κ increases and reaches a maximum around 18K.
3. Above 18K, κ decreases w/. increasing temperature and follows that exp(1/T).
4. Cross sectional area influences κ below 20K. Bigger area crystal has, larger κ it has.
see Enss
Phonon Thermal Conductivity
lslsll vCvCk τ2
31
31
=Λ=Kinetic Theory
Λ
Temperature, T/θD
Boundary Phonon Scattering Defect
Decreasing Boundary Separation
Increasing Defect Concentration
Phonon Scattering Mechanisms • Boundary Scattering • Defect & Dislocation Scattering • Phonon-Phonon Scattering
0.01 0.1 1.0
Temperature, T/θD
0.01 0.1 1.00.01 0.1 1.0
kl
dl Tk ∝
BoundaryPhononScatteringDefect
Increasing DefectConcentration
DIELECTRICS
Impurity scatterings
Defect scatterings
Log-log plot
κ (Watt/m/K)
T (K)
Slope -1
Exponential
Slope 3
break periodicity Other effects
Electronic specific heat
Conduction electrons in a metal ≡ free electron gas
• classical statistical physics (Drude-Lorentz model) : Cel=constant = 3/2 R
• quantum physics : e- = fermions ⇒ Pauli exclusion principle ⇒ Fermi-Dirac statistics
Cel = γ T with γ=2/3 π² kB² n(EF)
Metal : Ctotal= Cphonons+ Célectrons= β T³ + γ T
C / T = β T² + γ
nickel
γ ≈ 9 mJ K-2mole-1
β → θDebye
T² (K²)
C/T
(J K
-2 m
ol-
1
Palladium
Kittel
Hot Th
Cold Tc
L
Q (heat flow)
dxdTkA
LTTkAQ ch =
−=
Thermal conductivity Thermal Properties of Materials, Li Shi, Texas Materials Institute
The University of Texas at Austin
dxdTkq el=
For metals:
( )dEEDdTdfE
dTdC e
e ∫∞
=∈
=0
BF
Be nk
ETkC
=
2
2π
eFeFeel vCvCk τ2
31
31
=Λ=
Specific Heat
Thermal Conductivity
Mean free time: τe = Λ / vF
in 3D
Λ=Λ= FF
Bvel v
mvTnkvCk 2
22
31
31 π
TTekk Bel L=
=
22
3π
σ
τσm
ne2
=
Enns
Kittel
electrical Conductivity
At high temperatures, assumptions of the Wiedemann–Franz law are fulfilled.
At moderately low temperatures, processes with small momentum changes
dominate.
The Wiedemann–Franz law would fail when the temperature is lowered below 273 K,
since the inelastic scattering effect becomes substantial. This leads to a greater
degradation of the thermal current than the electrical current.
at very low temperatures impurity scattering dominates both electrical and thermal transport, and the Wiedemann–Franz law generally recovers.
Tkel
σdiffers from the Lorenz number
from Enss
from Enss
Electron Scattering Mechanisms • Defect Scattering • Phonon Scattering • Boundary Scattering (Film Thickness, Grain Boundary)
Grain Grain Boundary
e
Temperature, T
Defect Scattering
Phonon Scattering
Increasing Defect Concentration
Bulk Solids
Thermal Properties of Materials, Li Shi, Texas Materials Institute The University of Texas at Austin
10 310 210 110 010 0
10 1
10 2
10 3
Temperature, T [K}
Ther
mal
Con
duct
ivity
, k [
W/c
m-K
]
Copper
Aluminum
Defect Scattering Phonon Scattering
11
eFeeFee vCvCk τ2
31
31
=Λ=Matthiessen Rule:
Thermal Conductivity of Cu and Al
phononboundarydefecte
phononboundarydefecte
Λ+
Λ+
Λ=
Λ
++=
1111
1111ττττ
Electrons dominate thermal conductivity in metals
Thermal Properties of Materials, Li Shi, Texas Materials Institute The University of Texas at Austin
