fys3410 - vår 2015 (kondenserte fasers fysikk) · 23/2 free electron gas (feg) versus free...
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FYS3410 - Vår 2015 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/FYS3410/v15/index.html
Pensum: Introduction to Solid State Physics
by Charles Kittel (Chapters 1-9 and 17, 18, 20)
Andrej Kuznetsov
delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO
Tel: +47-22857762,
e-post: [email protected]
visiting address: MiNaLab, Gaustadaleen 23a
FYS3410 Lectures (based on C.Kittel’s Introduction to SSP, chapters 1-9, 17,18,20)
Module I – Periodic Structures and Defects (Chapters 1-3, 20) 26/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 2h
27/1 Laue condition, Ewald construction, interpretation of a diffraction experiment
Bragg planes and Brillouin zones 4h
28/1 Elastic strain and structural defects in crystals 2h
30/1 Atomic diffusion and summary of Module I 2h
Module II – Phonons (Chapters 4 and 5) 09/2 Vibrations, phonons, density of states, and Planck distribution 2h
10/2 Lattice heat capacity: Dulong-Petit, Einstien and Debye models
Comparison of different models 4h
11/2 Thermal conductivity 2h
13/2 Thermal expansion and summary of Module II 2h
Module III – Electrons (Chapters 6, 7 and 18) 23/2 Free electron gas (FEG) versus Free electron Fermi gas (FEFG) 2h
24/2 Effect of temperature – Fermi- Dirac distribution
FEFG in 2D and 1D, and DOS in nanostructures 4h
25/2 Origin of the band gap and nearly free electron model 2h
27/2 Number of orbitals in a band and general form of the electronic states 2h
Module IV – Semiconductors and Metals (Chapters 8, 9, and 17) 09/3 Energy bands; metals versus isolators 2h
10/3 Semiconductors: effective mass method, intrinsic and extrinsic carrier
generation 4h
12/3 Carrier statistics 2h
13/3 p-n junctions and optoelectronic devices 2h
Lecture 6: Vibrations and phonons
• Examples of phonon-assisted processes
• Infinite 1D lattice with one or two atoms in the basis;
• Examples of dispersion relations in 3D;
• Finite chain of atoms, Born – von Karman boundary conditions;
• Phonon density of states in 1-D;
• Collective crystal vibrations – phonons;
Lecture 10: Vibrations and phonons
• Examples of phonon-assisted processes
• Infinite 1D lattice with one or two atoms in the basis;
• Examples of dispersion relations in 3D;
• Finite chain of atoms, Born – von Karman boundary conditions;
• Phonon density of states in 1-D;
• Collective crystal vibrations – phonons;
Ghkl
k′
-k
k
Diffraction
k K k G
Photoluminescence
CB
VB
ED
EA hn
hn
hn
EXCITATION •Photo generation •Electrical injection
Eg
Photons
Photoluminescence
Lecture 10: Vibrations and phonons
• Examples of phonon-assisted processes
• Infinite 1D lattice with one or two atoms in the basis;
• Examples of dispersion relations in 3D;
• Finite chain of atoms, Born – von Karman boundary conditions;
• Phonon density of states in 1-D;
• Collective crystal vibrations – phonons;
longitudinal wave transverse wave
Vibrations of crystals with monatomic basis
Vibrations of crystals with monatomic basis
a
Spring constant, g Mass, m
xn xn+1xn-1
Equilibrium Position
Deformed Position
us: displacement of the sth atom from its equilibrium position
us-1 us us+1
M
1 1s s s s sF C u u C u u Force on sth plane =
Equation of motion: 2
1 122s
s s s
d uM C u u u
dt
i t
s su t u e → 2
1 1 2s s s sM u C u u u
0
iK as
su u e → 2 2i K a i K aM C e e 2 21 cos
CKa
M
Dispersion relation
2 24 1sin
2
CKa
M
4 1sin
2
CKa
M
(only neighboring planes interact )
Vibrations of crystals with monatomic basis
G
dv
d K
vG = 0 at zone boundaries
2 1cos
2
CaKa
M
g K
v
1-D:
Group velocity:
4 1sin
2
CKa
M
Vibrations of crystals with monatomic basis
2
1 12
2
2 12
2
2
ss s s
ss s s
d uM C v v u
dt
d vM C u u v
dt
i sK a i t
su ue
i sK a i t
sv ve →
2
1
2
2
1 2
1 2
i K a
i K a
M u Cv e Cu
M v Cu e Cv
2
1
2
2
2 10
1 2
i K a
i K a
C M C e
C e C M
Vibrations of crystals with two atoms per basis
Ka → π:
(M1 >M2 )
22
1
2 /
2 /
C M optical
acousticalC M
4 2 2
1 2 1 22 2 1 cos 0M M C M M C Ka
=
=
1 22
2 2
1 2
1 12
2
C opticalM M
CK a acoustical
M M
Ka → 0:
Vibrations of crystals with two atoms per basis
Lecture 10: Vibrations and phonons
• Examples of phonon-assisted processes
• Infinite 1D lattice with one or two atoms in the basis;
• Examples of dispersion relations in 3D;
• Finite chain of atoms, Born – von Karman boundary conditions;
• Phonon density of states in 1-D
• Collective crystal vibrations – phonons;
p atoms in primitive cell → d p branches of dispersion.
d = 3 → 3 acoustical : 1 LA + 2 TA
(3p –3) optical: (p–1) LO + 2(p–1) TO
E.g., Ge or KBr: p = 2 → 1 LA + 2 TA + 1 LO + 2 TO branches
Ge KBr
Number of allowed K in 1st BZ = N
Vibrations of crystals with two atoms per basis
Phonon dispersion in real crystals: aluminium FCC lattice with 1
atom in the basis
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.
Phonon dispersion in real crystals: FCC lattice with 1 (Al) and 2
(Diamond) atoms in the basis
Characteristic points of the reciprocal space – Γ, X, K, and L points are
introduced at the center and bounduries of the first Brillouin zone
Lecture 10: Vibrations and phonons
• Examples of phonon-assisted processes
• Infinite 1D lattice with one or two atoms in the basis;
• Examples of dispersion relations in 3D;
• Finite chain of atoms, Born – von Karman boundary conditions;
• Phonon density of states in 1-D;
• Collective crystal vibrations – phonons;
Calculating phonon density of states – DOS – in 1-D
A vibrational mode is a vibration of a given wave vector (and thus ),
frequency , and energy . How many modes are found in the
interval between and ?
E
k
),,( kE
),,( kdkdEEd
# modes kdkNdEENdNdN
3)()()(
We will first find N(k) by examining allowed values of k. Then we will be
able to calculate N() and evaluate CV in the Debye model.
First step: simplify problem by using periodic boundary conditions for the
linear chain of atoms:
x = sa x = (s+N)a
L = Na
s
s+N-1
s+1
s+2
We assume atoms s
and s+N have the
same displacement—
the lattice has periodic
behavior, where N is
very large.
Lecture 10: Vibrations and phonons
• Examples of phonon-assisted processes
• Infinite 1D lattice with one or two atoms in the basis;
• Examples of dispersion relations in 3D;
• Finite chain of atoms, Born – von Karman boundary conditions;
• Phonon density of states in 1-D;
• Collective crystal vibrations – phonons;
• Thermal equilibrium occupancy of phonons – Planck distribution.
Calculating phonon density of states – DOS – in 1-D
This sets a condition on
allowed k values: ...,3,2,12
2 nNa
nknkNa
So the separation between
allowed solutions (k values) is:
independent of k, so
the density of modes
in k-space is uniform
Since atoms s and s+N have the same displacement, we can write:
Nss uu ))(()( taNskitksai ueue ikNae1
Nan
Nak
22
Thus, in 1-D: 22
1 LNa
kspacekofinterval
modesof#
Lecture 10: Vibrations and phonons
• Examples of phonon-assisted processes
• Infinite 1D lattice with one or two atoms in the basis;
• Examples of dispersion relations in 3D;
• Finite chain of atoms, Born – von Karman boundary conditions;
• Phonon density of states in 1-, 2-, and 3-D;
• Collective crystal vibrations – phonons;
• Thermal equilibrium occupancy of phonons – Planck distribution.
Energy level diagram for a chain of
atoms with one atom per unit cell and a
lengt of N unit cells
Energy level
diagram for one
harmonic oscillator
FYS3410 - Vår 2015 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/FYS3410/v15/index.html
Pensum: Introduction to Solid State Physics
by Charles Kittel (Chapters 1-9 and 17, 18, 20)
Andrej Kuznetsov
delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO
Tel: +47-22857762,
e-post: [email protected]
visiting address: MiNaLab, Gaustadaleen 23a
FYS3410 Lectures (based on C.Kittel’s Introduction to SSP, chapters 1-9, 17,18,20)
Module I – Periodic Structures and Defects (Chapters 1-3, 20) 26/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 2h
27/1 Laue condition, Ewald construction, interpretation of a diffraction experiment
Bragg planes and Brillouin zones 4h
28/1 Elastic strain and structural defects in crystals 2h
30/1 Atomic diffusion and summary of Module I 2h
Module II – Phonons (Chapters 4, 5 and 18) 09/2 Vibrations, phonons, density of states, and Planck distribution 2h
10/2 Lattice heat capacity: Dulong-Petit, Einstien and Debye models
Comparison of different models 4h
11/2 Thermal conductivity 2h
13/2 Thermal expansion and summary of Module II 2h
Module III – Electrons (Chapters 6 and 7) 23/2 Free electron gas (FEG) versus Free electron Fermi gas (FEFG) 2h
24/2 Effect of temperature – Fermi- Dirac distribution
FEFG in 2D and 1D, and DOS in nanostructures 4h
25/2 Origin of the band gap and nearly free electron model 2h
27/2 Number of orbitals in a band and general form of the electronic states 2h
Module IV – Semiconductors and Metals (Chapters 8, 9, and 17) 09/3 Energy bands; metals versus isolators 2h
10/3 Semiconductors: effective mass method, intrinsic and extrinsic carrier
generation 4h
12/3 Carrier statistics 2h
13/3 p-n junctions and optoelectronic devices 2h
Lecture 7: Lattice heat capacity: Dulong-Petit, Einstien and Debye models
• Repetition of phonon DOS
• Classical theory for heat capacity of solids treating atoms as classical harmonic
oscillators - Dulong-Petit model – success and problems
• Einstein model for heat capacity considering quantum properties of oscillators
constituting a solid – success and problems
• Debye model
• Comparison of different models
Lecture 11: Lattice heat capacity: Dulong-Petit, Einstien and Debye models
• Repetition of phonon DOS
• Classical theory for heat capacity of solids treating atoms as classical harmonic
oscillators - Dulong-Petit model – success and problems
• Einstein model for heat capacity considering quantum properties of oscillators
constituting a solid – success and problems
• Debye model
• Comparison of different models
Calculating phonon density of states – DOS – in 1-D
This sets a condition on
allowed k values: ...,3,2,12
2 nNa
nknkNa
So the separation between
allowed solutions (k values) is:
independent of k, so
the density of modes
in k-space is uniform
Since atoms s and s+N have the same displacement, we can write:
Nss uu ))(()( taNskitksai ueue ikNae1
Nan
Nak
22
Thus, in 1-D: 22
1 LNa
kspacekofinterval
modesof#
Energy level diagram for a chain of
atoms with one atom per unit cell and a
lengt of N unit cells
Energy level
diagram for one
harmonic oscillator
Lecture 7: Lattice heat capacity: Dulong-Petit, Einstien and Debye models
• Repetition of phonon DOS
• Classical theory for heat capacity of solids treating atoms as classical harmonic
oscillators - Dulong-Petit model – success and problems
• Einstein model for heat capacity considering quantum properties of oscillators
constituting a solid – success and problems
• Debye model
• Comparison of different models
Classical (Dulong-Petit) theory for heat capacity
For a solid composed of N such atomic oscillators:
Giving a total energy per mole of sample:
TNkENE B31
RTTkNn
TNk
n
EBA
B 333
So the heat capacity at
constant volume per mole is: KmolJ
V
V Rn
E
dT
dC 253
This law of Dulong and Petit (1819) is approximately obeyed by most
solids at high T ( > 300 K).
Calculating phonon density of states – DOS – in 3-D
Now for a 3-D lattice we can apply periodic boundary
conditions to a sample of N1 x N2 x N3 atoms:
N1a N2b
N3c
)(8222 3
321 kNVcNbNaN
spacekofvolume
modesof#
Now we know from before
that we can write the
differential # of modes as:
kdkNdNdN
3)()( kdV
3
38
We carry out the integration
in k-space by using a
“volume” element made up
of a constant surface with
thickness dk:
dkdSdkareasurfacekd )(3
Calculating phonon density of states – DOS – in 3-D
A very similar result holds for N(E) using constant energy surfaces for the
density of electron states in a periodic lattice!
dkdSV
dNdN
38
)(Rewriting the differential
number of modes in an interval:
We get the result: k
dSV
d
dkdS
VN
1
88)(
33
Temperature dependence of experimentally measured heat capacity
Lecture 7: Lattice heat capacity: Dulong-Petit, Einstien and Debye models
• Repetition of phonon DOS
• Classical theory for heat capacity of solids treating atoms as classical harmonic
oscillators - Dulong-Petit model – success and problems
• Einstein model for heat capacity considering quantum properties of oscillators
constituting a solid – success and problems
• Debye model
• Comparison of different models
Einstein model for heat capacity accounting for quantum
properties of oscillators constituting a solid
Planck (1900): vibrating oscillators (atoms) in a solid have quantized
energies ...,2,1,0 nnEn
[later showed is actually correct] 21 nEn
...,2,1,0 nnEn
Einstein (1907): model a solid as a collection of 3N independent 1-D
oscillators, all with constant , and use Planck’s equation for energy levels
occupation of energy level n:
(probability of oscillator
being in level n)
0
/
/
)(
n
kTE
kTE
n
n
n
e
eEf classical physics
(Boltzmann factor)
Average total
energy of solid:
0
/
0
/
0
3)(3
n
kTE
n
kTE
n
n
nn
n
n
e
eE
NEEfNUE
Boltzmann factor is a weighting factor that determines the relative
probability of a state i in a multi-state system in thermodynamic
equilibrium at tempetarure T.
Where kB is Boltzmann’s constant and Ei is the energy of state i.
The ratio of the probabilities of two states is given by the ratio of
their Boltzmann factors.
kTEie/
Boltzmann factor determines Planck distribution
Einstein model for heat capacity accounting for quantum
properties of oscillators constituting a solid
0
/
0
/
3
n
kTn
n
kTn
e
en
NU
Using Planck’s equation: Now let
kTx
0
03
n
nx
n
nx
e
en
NU
0
0
0
0 33
n
nx
n
nx
n
nx
n
nx
e
edx
d
N
e
edx
d
NU Which can
be rewritten:
Now we can use
the infinite sum: 1
1
1
0
xforx
xn
n
1
3
1
3
1
13
/
kTx
x
x
x
x
e
N
e
N
e
e
e
e
dx
d
NU
To give: 11
1
0
x
x
xn
nx
e
e
ee
So we obtain:
Einstein model for heat capacity accounting for quantum
properties of oscillators constituting solids
Differentiating:
Now it is traditional to define
an “Einstein temperature”:
Using our previous definition:
So we obtain the prediction:
1
3/ kT
A
V
Ve
N
dT
d
n
U
dT
dC
2/
/2
2/
/
1
3
1
3 2
kT
kT
kT
kT
kT
kT
A
V
e
eR
e
eNC
kE
2/
/2
1
3)(
T
T
TV
E
EE
e
eRTC
Einstein model for heat capacity accounting for quantum
properties of oscillators constituting solids
Low T limit:
These predictions are
qualitatively correct: CV 3R
for large T and CV 0 as T 0:
High T limit: 1T
E
RR
TC
T
TTV
E
EE
311
13)(
2
2
1T
E
T
TT
T
TV
EE
E
EE
eRe
eRTC
/2
2/
/2
33
)(
3R
CV
T/E
Energy level diagram for a chain of
atoms with one atom per unit cell and a
lengt of N unit cells
Energy level
diagram for one
harmonic oscillator
High T limit: 1T
E
Low T limit: 1T
E
Correlation with energy level diagram for a harmonic oscillator
Problem of Einstein model to reproduce the rate of heat capacity
decrease at low temperatures High T behavior:
Reasonable
agreement with
experiment
Low T behavior:
CV 0 too quickly
as T 0 !
Lecture 7: Lattice heat capacity: Dulong-Petit, Einstien and Debye models
• Repetition of phonon DOS
• Classical theory for heat capacity of solids treating atoms as classical harmonic
oscillators - Dulong-Petit model – success and problems
• Einstein model for heat capacity considering quantum properties of oscillators
constituting a solid – success and problems
• Debye model
• Comparison of different models
More careful consideration of phonon occupancy modes
as a way to improve the agreement with experiment
Debye’s model of a solid:
• 3N normal modes (patterns) of oscillations
• Spectrum of frequencies from = 0 to max
• Treat solid as continuous elastic medium (ignore details of atomic structure)
This changes the expression for CV
because each mode of oscillation
contributes a frequency-dependent
heat capacity and we now have to
integrate over all :
dTCDTC EV ),()()(max
0
# of oscillators per
unit , i.e. DOS
Distribution function
Debye model
3
3
3
4
2k
LNk
3
126
V
N
k
v
k
B
BD
Density of states of acoustic phonos for 1 polarization
Debye temperature θ
32
3
6 v
VN D
N: number of unit cell
Nk: Allowed number of k
points in a sphere with a
radius k
/vk
32
3
3
33
63
4
2)(
v
V
v
LN
32
2
2
)()(
v
V
d
dND
Thermal energy U and lattice heat capacity CV : Debye model
D
D
D
x
x
x
BV
B
B
BV
V
B
e
exdx
TNkC
Tk
Tkd
Tkv
V
T
UC
Tkv
VdnDdU
0
2
43
0
2
4
232
2
0
32
2
)1(9
]1)/[exp(
)/exp(
2
3
1)/exp(23)()(3
3 polarizations for acoustic modes
Debye model
Universal behavior
for all solids!
Debye temperature
is related to
“stiffness” of solid,
as expected
Better agreement
than Einstein
model at low T
Debye model
Quite impressive
agreement with
predicted CV T3
dependence for Ar!
(noble gas solid)
Energy level diagram for a chain of
atoms with one atom per unit cell and a
lengt of N unit cells
More careful consideration of phonon occupancy modes
as a way to improve the agreement with experiment
Lecture 7: Lattice heat capacity: Dulong-Petit, Einstien and Debye models
• Repetition of phonon DOS
• Classical theory for heat capacity of solids treating atoms as classical harmonic
oscillators - Dulong-Petit model – success and problems
• Einstein model for heat capacity considering quantum properties of oscillators
constituting a solid – success and problems
• Debye model
• Comparison of different models
Energy level diagram for a chain of
atoms with one atom per unit cell and a
lengt of N unit cells
Ensemble of 3N independent harmonic oscillators modeling vibrations in a solid
Quantum
oscillators
Classical
oscillators En
erg
y
TNkENE B31
Any energy state is accessible for any
oscillator in form of kBT, i.e. no
distribution function is applied and
the total energy is
Any energy state is accessible for
any oscillator in form of kBT, i.e.
no distribution function is
necessary, so that
Energy level diagram for a chain of
atoms with one atom per unit cell and a
lengt of N unit cells
Ensemble of 3N independent harmonic oscillators modeling vibrations in a solid
Quantum
oscillators
Classical
oscillators En
erg
y
TNkENE B31
Any energy state is accessible for any
oscillator in form of kBT, i.e. no
distribution function is applied and
the total energy is
Not all energies are accessible, but only those
in quants of ħωn, and Planck distribution is
employed to calculate the occupancy at
temperature T, so that nNE 3
1
133)(3
/
0
/
0
/
0Tk
n
TkE
n
TkE
n
n
nnB
Bn
Bn
eN
e
eE
NEEfNE
Ensemble of 3N independent harmonic oscillators modeling vibrations in a solid
Dulong-Petit model is valid
only at high temperatures
Einstein model is in a good
agreement with the experiment,
except for that at low temperatures
Energy level diagram for a chain of
atoms with one atom per unit cell and a
lengt of N unit cells
Energy level
diagram for one
harmonic oscillator
nNE 3
max
min
)(3 nDdE
FYS3410 - Vår 2015 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/FYS3410/v15/index.html
Pensum: Introduction to Solid State Physics
by Charles Kittel (Chapters 1-9 and 17, 18, 20)
Andrej Kuznetsov
delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO
Tel: +47-22857762,
e-post: [email protected]
visiting address: MiNaLab, Gaustadaleen 23a
FYS3410 Lectures (based on C.Kittel’s Introduction to SSP, chapters 1-9, 17,18,20)
Module I – Periodic Structures and Defects (Chapters 1-3, 20) 26/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 2h
27/1 Laue condition, Ewald construction, interpretation of a diffraction experiment
Bragg planes and Brillouin zones 4h
28/1 Elastic strain and structural defects in crystals 2h
30/1 Atomic diffusion and summary of Module I 2h
Module II – Phonons (Chapters 4 and 5) 09/2 Vibrations, phonons, density of states, and Planck distribution 2h
10/2 Lattice heat capacity: Dulong-Petit, Einstien and Debye models
Comparison of different models 4h
11/2 Thermal conductivity 2h
13/2 Thermal expansion and summary of Module II 2h
Module III – Electrons (Chapters 6, 7 and 18) 23/2 Free electron gas (FEG) versus Free electron Fermi gas (FEFG) 2h
24/2 Effect of temperature – Fermi- Dirac distribution
FEFG in 2D and 1D, and DOS in nanostructures 4h
25/2 Origin of the band gap and nearly free electron model 2h
27/2 Number of orbitals in a band and general form of the electronic states 2h
Module IV – Semiconductors and Metals (Chapters 8, 9, and 17) 09/3 Energy bands; metals versus isolators 2h
10/3 Semiconductors: effective mass method, intrinsic and extrinsic carrier
generation 4h
12/3 Carrier statistics 2h
13/3 p-n junctions and optoelectronic devices 2h
Lecture 8: Thermal conductivity
• We understood phonon DOS and occupancy as a function of temperature, but what
about transport properties?
• Phenomenological description of thermal conductivity
• Temperature dependence of thermal conductivity in terms of phonon properties
• Phonon collisions: N and U processes
• Comparison of temperature dependence of κ in crystalline and amorphous solids
Lecture 8: Thermal conductivity
• We understood phonon DOS and occupancy as a function of temperature, but what
about transport properties?
• Phenomenological description of thermal conductivity
• Temperature dependence of thermal conductivity in terms of phonon properties
• Phonon collisions: N and U processes
• Comparison of temperature dependence of κ in crystalline and amorphous solids
Understanding phonons as «harmonic waves» can not explain thermal
restance since harmonic wafes perfectly move one through another
Lecture 8: Thermal conductivity
• We understood phonon DOS and occupancy as a function of temperature, but what
about transport properties?
• Phenomenological description of thermal conductivity
• Temperature dependence of thermal conductivity in terms of phonon properties
• Phonon collisions: N and U processes
• Comparison of temperature dependence of κ in crystalline and amorphous solids
When thermal energy propagates through a solid, it is carried by lattice waves
or phonons. If the atomic potential energy function is harmonic, lattice waves
obey the superposition principle; that is, they can pass through each other
without affecting each other. In such a case, propagating lattice waves would
never decay, and thermal energy would be carried with no resistance (infinite
conductivity!). So…thermal resistance has its origins in an anharmonic
potential energy.
Classical definition of
thermal conductivity vCV
3
1
VC
wave velocity
heat capacity per unit volume
mean free path of scattering
(would be if no anharmonicity)
v
high T low T
dx
dTJ
Thermal
energy flux
(J/m2s)
Phenomenological description of thermal conductivity
Lecture 8: Thermal conductivity
• We understood phonon DOS and occupancy as a function of temperature, but what
about transport properties?
• Phenomenological description of thermal conductivity
• Temperature dependence of thermal conductivity in terms of phonon properties
• Phonon collisions: N and U processes
• Comparison of temperature dependence of κ in crystalline and amorphous solids
Temperature dependence of thermal conductivity in terms of
phonon prperties
Mechanisms to affect the mean free pass (Λ) of phonons in periodic crystals:
2. Collision with sample boundaries (surfaces)
3. Collision with other phonons deviation from
harmonic behavior
1. Interaction with impurities, defects, and/or isotopes
VC 11 / kT
ph
en
ThighR
TlowT
3
3
ThighkT
Tlow
VC v
To understand the experimental dependence , consider limiting values
of and (since does not vary much with T).
)(T
deviation from
translation
symmetry
1) Please note, that the temperature dependence of T-1 for Λ at the high temperature limit results
from considering nph , which is the total phonon occupancy, from 0 to ωD. However, already
intuitively, we may anticipate that low energy phonons, i.e. those with low k-numbers in the vicinity
of the center of the 1st BZ may have quite different appearence conparing with those having bigger
k-naumbers close to the edges of the 1st BZ.
1)
Thus, considering defect free, isotopically clean sample having limited size D
CV
low T
T3
nph 0, so
, but then
D (size)
T3
high T
3R
1/T
1/T
How well does this match experimental results?
Temperature dependence of thermal conductivity in terms of
phonon prperties
T3
However, T-1 estimation for κ in
the high temperature limit has a
problem. Indeed, κ drops much
faster – see the data – and the
origin of this disagreement is
because – when estimating Λ –
we accounted for all excited
phonons, while a more correct
approximation would be to
consider “high” energetic
phonons only. But what is “high”
in this context?
Experimental (T)
T-1 ?
T3 estimation
for κ the low
temperature
limit is fine!
Temperature dependence of thermal conductivity in terms of
phonon prperties
Better estimation for Λ in high temperature limit
NaNak
2121
aNa
NkN
22
NaNak
4222
𝝎𝟐 𝝎𝑫 𝝎𝟏
1/2
«significant» modes «insignificant» modes
The fact that «low energetic phonons» having k-values << 𝝅/𝒂 do not
participate in the energy transfer, can be understood by considering so called N-
and U-phonon collisions readily visualized in the reciprocal space. Anyhow, we
account for modes having energy E1/2 = (1/2)ħωD or higher. Using the definition
of θD = ħωD/kB, E1/2 can be rewritten as kBθD/2. Ignoring more complex
statistics, but using Boltzman factor only, the propability of E1/2 would of the
order of exp(- kB θD/2 kB T) or exp(-θD/2T), resulting in Λ exp(θD/2T).
estimate in terms of affecting Λ!
CV
low T
T3
nph 0, so
, but then
D (size)
T3
high T
3R
exp(θD/2T)
exp(θD/2T)
Temperature dependence of thermal conductivity in terms of
phonon prperties
Thus, considering defect free, isotopically clean sample having limited size D
Lecture 8: Thermal conductivity
• We understood phonon DOS and occupancy as a function of temperature, but what
about transport properties?
• Phenomenological description of thermal conductivity
• Temperature dependence of thermal conductivity in terms of phonon properties
• Phonon collisions: N and U processes
• Comparison of temperature dependence of κ in crystalline and amorphous solids
Phonon collisions: N and U processes
How exactly do phonon collisions limit the flow of heat?
2-D lattice 1st BZ in k-space:
1q
2q
3q
a2
a2
321 qqq
No resistance to heat flow
(N process; phonon momentum conserved)
Predominates at low T << D since
and q will be small
What if the phonon wavevectors are a bit larger?
2-D lattice 1st BZ in k-space:
1q
2qa
2
a2
Gqqq
321
Two phonons combine to give a net phonon
with an opposite momentum! This causes
resistance to heat flow.
(U process; phonon momentum “lost” in
units of ħG.)
More likely at high T >> D since and
q will be larger
21 qq
G
3q
Umklapp = “flipping over” of
wavevector!
Phonon collisions: N and U processes
Explanation for κ exp(θD/2T) at high temperature limit
11 / kT
ph
en
ThighT
Tlow
1
The temperature dependence of T-1 for
Λ results from considering the total
phonon occupancy, from 0 to ωD.
However, interactions of low energy
phonons, i.e. those with low k-
numbers in the vicinity of the center
the 1st BZ, are not changing energy.
These are so called N-processes having
little impact on Λ.
vCV3
1
1q
2q
3q
a2
a2
1q
2qa
2
a2
21 qq
G
3q
U-process , i.e. to turn over the
wavevector by G, from the German
word umklappen.
A more correct approximation for Λ (in high temperature limit) would be to
consider “high” energetic phonons only, i.e those participating in U- processes.
Explanation for κ exp(θD/2T) at high temperature limit
NaNak
2121
aNa
NkN
22
NaNak
4222
𝝎𝟐 𝝎𝑫 𝝎𝟏
1/2
«significant» modes «insignificant» modes
The fact that «low energetic phonons» having k-values << 𝝅/𝒂 do not
participate in the energy transfer, can be understood by considering so called N-
and U-phonon collisions readily visualized in the reciprocal space. Anyhow, we
account for modes having energy E1/2 = (1/2)ħωD or higher. Using the definition
of θD = ħωD/kB, E1/2 can be rewritten as kBθD/2. Ignoring more complex
statistics, but using Boltzman factor only, the propability of E1/2 would of the
order of exp(- kB θD/2 kB T) or exp(-θD/2T), resulting in Λ exp(θD/2T).
estimate in terms of affecting Λ!
Explanation for κ exp(θD/2T) at high temperature limit
Temperature dependence of thermal conductivity in terms of
phonon prperties
CV
low T
T3
nph 0, so
, but then
D (size)
T3
high T
3R
exp(θD/2T)
exp(θD/2T)
Thus, considering defect free, isotopically clean sample having limited size D
Lecture 8: Thermal conductivity
• We understood phonon DOS and occupancy as a function of temperature, but what
about transport properties?
• Phenomenological description of thermal conductivity
• Temperature dependence of thermal conductivity in terms of phonon properties
• Phonon collisions: N and U processes
• Comparison of temperature dependence of κ in crystalline and amorphous solids
Comparison of temperature dependence of κ in crystalline and amorphous solids
FYS3410 - Vår 2015 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/FYS3410/v15/index.html
Pensum: Introduction to Solid State Physics
by Charles Kittel (Chapters 1-9 and 17, 18, 20)
Andrej Kuznetsov
delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO
Tel: +47-22857762,
e-post: [email protected]
visiting address: MiNaLab, Gaustadaleen 23a
FYS3410 Lectures (based on C.Kittel’s Introduction to SSP, chapters 1-9, 17,18,20)
Module I – Periodic Structures and Defects (Chapters 1-3, 20) 26/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 2h
27/1 Laue condition, Ewald construction, interpretation of a diffraction experiment
Bragg planes and Brillouin zones 4h
28/1 Elastic strain and structural defects in crystals 2h
30/1 Atomic diffusion and summary of Module I 2h
Module II – Phonons (Chapters 4 and 5) 09/2 Vibrations, phonons, density of states, and Planck distribution 2h
10/2 Lattice heat capacity: Dulong-Petit, Einstien and Debye models
Comparison of different models 4h
11/2 Thermal conductivity 2h
13/2 Thermal expansion and summary of Module II 2h
Module III – Electrons (Chapters 6, 7 and 18) 23/2 Free electron gas (FEG) versus Free electron Fermi gas (FEFG) 2h
24/2 Effect of temperature – Fermi- Dirac distribution
FEFG in 2D and 1D, and DOS in nanostructures 4h
25/2 Origin of the band gap and nearly free electron model 2h
27/2 Number of orbitals in a band and general form of the electronic states 2h
Module IV – Semiconductors and Metals (Chapters 8, 9, and 17) 09/3 Energy bands; metals versus isolators 2h
10/3 Semiconductors: effective mass method, intrinsic and extrinsic carrier
generation 4h
12/3 Carrier statistics 2h
13/3 p-n junctions and optoelectronic devices 2h
Lecture 9: Thermal expansion and repetition of Module II
• Thermal expansion
• Repetition
dxe
dxxe
xkTxU
kTxU
/)(
/)(
In a 1-D lattice where each atom experiences the same potential energy
function U(x), we can calculate the average displacement of an atom from its
equilibrium position:
Thermal expansion
I
Thermal Expansion in 1-D
Evaluating this for the harmonic potential energy function U(x) = cx2 gives:
dxe
dxxe
xkTcx
kTcx
/
/
2
2
Thus any nonzero <x> must come from terms in U(x) that go beyond x2. For
HW you will evaluate the approximate value of <x> for the model function
The numerator is zero!
!0x independent of T !
),0,,()( 43432 kTfxgxandfgcfxgxcxxU
Why this form? On the next slide you can see that this function is a reasonable
model for the kind of U(r) we have discussed for molecules and solids.
Potential Energy of Anharmonic Oscillator
(c = 1 g = c/10 f = c/100)
0
2
4
6
8
10
12
14
16
-5 -3 -1 1 3 5
Displacement x (arbitrary units)
Po
ten
tia
l E
ner
gy
U (
arb
. u
nit
s)
U = cx2 - gx3 - fx4 U = cx2
Lattice Constant of Ar Crystal vs. Temperature
Above about 40 K, we see: TxaTa )0()(
Usually we write: 00 1 TTLL = thermal expansion coefficient