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FYS3400 - Vår 2020 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/FYS3410/v17/index.html
Pensum: Introduction to Solid State Physics
by Charles Kittel (Chapters 1-9 and 17 - 20)
Andrej Kuznetsov
delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO
Tel: +47-22857762,
e-post: andrej.kuznetsov@fys.uio.no
visiting address: MiNaLab, Gaustadaleen 23a
2020 FYS3400 Lecture Plan (based on C.Kittel’s Introduction to SSP, Chapters 1-9, 17-20 + guest lectutes)
Module I – Periodity and Disorder (Chapters 1-3, 19, 20) calender week
To 16/1 12-13 Introduction.
On 22/1 10-12 Crystal bonding. Periodicity and lattices. Lattice planes and Miller indices. Reciprocal space. 4
To 23/1 12-13 Bragg diffraction and Laue condition
On 29/1 10-12 Ewald construction, interpretation of a diffraction experiment, Bragg planes and Brillouin zones 5
To 30/1 12-13 Surfaces and interfaces. Disorder. Defects crystals. Equilibrium concentration of vacancies
On 5/2 10-12 Mechanical properties of solids. Diffusion phenomena in solids; Summary of Module I 6
Module II – Phonons (Chapters 4, 5, and 18 pp.557-561)
To 6/2 12-13 Vibrations in monoatomic and diatomic chains of atoms; examples of dispersion relations in 3D
On 12/2 10-12 Periodic boundary conditions (Born – von Karman); phonons and its density of states (DOS) 7
To 13/2 12-13 Effect of temperature - Planck distribution;
On 19/2 10-12 Lattice heat capacity: Dulong-Petit, Einstein, and Debye models 8
To 20/2 12-13 Comparison of different lattice heat capacity models
On 26/2 10-12 Thermal conductivity and thermal expansion 9
To 27/2 12-13 Summary of Module II
Module III – Electrons (Chapters 6, 7, 11 - pp 315-317, 18 - pp.528-530, 19, and Appendix D)
On 4/3 10-12 Free electron gas (FEG) versus free electron Fermi gas (FEFG); DOS of FEFG in 3D 10
To 5/3 12-13 Effect of temperature – Fermi-Dirac distribution; Heat capacity of FEFG in 3D
On 11/3 10-12 DOS of FEFG in 2D - quantum wells, DOS in 1D – quantum wires, and in 0D – quantum dots 11
To 12/3 12-13 Transport properties of electrons; Summary of module III.
teaching free week 12
Module IV – Disordered systems (guest lecture slides)
On 25/3 10-12 Thermal properties of glasses: Model of two level systems (Joakim Bergli) 13
To 26/3 12-13 Experiments in porous media (Gaute Linga)
On 1/4 10-12 Electron transport in disordered solids: wave localization and hopping (Joakim Bergli) 14
To 2/4 12-13 Theory of porous media (Gaute Linga)
Easter 15
On 15/4 10-12 Advanced theory of disordered systems (??) 16
To 16/4 12-13 Advanced theory of disordered systems (??)
Module V – Semiconductors (Chapters 8, 9 pp 223-231, and 17, 19)
On 22/4 10-12 Origin of the band gap; Nearly free electron model; Kronig-Penney model 1
To 23/4 12-13 Effective mass method for calculating localized energy levels for defects in crystals
On 29/4 10-12 Intrinsic and extrinsic electrons and holes in semiconductors 18
To 30/4 12-13 Carrier statistics in semiconductors
On 6/5 10-12 p-n junctions 19
To 07/5 12-13 Optical properties of semiconductors (Inhwan Lee)
On 13/5 10-12 Advanced optoelectronic devices (Inhwan Lee) 20
Summary and repetition
To 14/5 12-13 Repetition - course in a nutshell
Exam: oral examination
May 28th – 29th
Condensed Matter Physics
Solid State Physics of Crystals
Properties of Waves in Periodic Lattices
Electron waves in lattices
Free electrons
Electron DOS
Fermi-Dirac distribution
Elastic waves in lattices
Vibrations
Phonon DOS
Planck distribution
Elecronic properties: Electron concentration and transport,
contribution to the heat capacity
Thermal properties: heat capacity and conductance,
thermal expansion
Advanced theory and novel materials properties
Condensed Matter Physics
Solid State Physics of Crystals
Properties of Waves in Periodic Lattices
Electron waves in lattices
Free electrons
Electron DOS
Fermi-Dirac distribution
Elastic waves in lattices
Vibrations
Phonon DOS
Planck distribution
Elecronic properties: Electron concentration and transport,
contribution to the heat capacity
Thermal properties: heat capacity and conductance,
thermal expansion
Advanced theory and novel materials properties
Disordered
systems
Disordered
systems
Disordered
systems
Disordered
systems
Lectures 5-6: 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;
Lectures 5-6: 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;
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
Lectures 5-6: 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;
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
Lectures 5-6: 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;
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
Lectures 5-6: 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;
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 5-6: 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;
• 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 5-6: 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-, 2-, and 3-D;
• Collective crystal vibrations – phonons;
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
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