fys3410 - vår 2014 (kondenserte fasers fysikk) · lecture schedule (based on p.hofmann’s solid...
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FYS3410 - Vår 2014 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/FYS3410/v14/index.html
Pensum: Solid State Physics by Philip Hofmann (Chapters 1-7 and 11)
Andrej Kuznetsov
delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO
Tel: +47-22857762,
e-post: [email protected]
visiting address: MiNaLab, Gaustadaleen 23c
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Lecture schedule (based on P.Hofmann’s Solid State Physics, chapters 1-7 and 11)
Module I – Periodic Structures and Defects 20/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 4h
21/1 Laue condition, Ewald construction, interpretation of a diffraction experiment 2h
22/1 Bragg planes and Brillouin zones (BZ) 2h
23/1 Elastic strain and structural defects 2h
23/1 Atomic diffusion and summary of Module I 2h
Module II - Phonons 03/2 Vibrations, phonons, density of states, and Planck distribution 4h
04/2 Lattice heat capacity: Dulong-Petit, Einstien and Debye models 2h
05/2 Comparison of different models 2h
06/2 Thermal conductivity 2h
07/2 Thermal expansion and summary of Module II 2h
Module III – Free Electron Gas 24/2 Free electron gas (FEG) versus Free electron Fermi gas (FEFG) 4h
25/2 Effect of temperature – Fermi- Dirac distribution 2h
26/2 Heat capacity of FEFG 2h
27/2 Electrical and thermal conductivity in metals 2h
28/2 FEFG in nanostructures 2h
Module IV – Nearly Free Electron Model 10/3 Bragg reflection of electrons at the boundary of BZ, energy bands 4h
11/3 Empty lattice approximation; number of orbitals in a band 2h
12/3 Semiconductors, effective mass method, intrinsic carriers 2h
13/3 Impurity states in semiconductors and carrier statistics 2h
14/3 p-n junctions and optoelectronic devices 2h
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Lecture 14: Free electron gas and free electron Fermi gas
• Free electron gas (FEG) - Drude model
- FEG explaing Ohm’s law
- FEG expaling Hall effect
- FEG explaing Wiedemann-Franz law
- FEG heat capacity
•Free electron Fermi gas (FEFG) – a gas of electrons subject to Pauli principle
• One electron system – wave functions – orbits; FEFG in 1D in ground state
• FEFG in 3D in ground state
• Fermi-Dirac distribution and electron occupancy at T>0
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Lecture 14: Free electron gas and free electron Fermi gas
• Free electron gas (FEG) - Drude model
- FEG explaing Ohm’s law
- FEG expaling Hall effect
- FEG explaing Wiedemann-Franz law
- FEG heat capacity
•Free electron Fermi gas (FEFG) – a gas of electrons subject to Pauli principle
• One electron system – wave functions – orbits; FEFG in 1D in ground state
• FEFG in 3D in ground state
• Fermi-Dirac distribution and electron occupancy at T>0
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There could be different opinions what particular discovery was the main breakthrough for the
acceleration of the condensed matter physics but a very prominent kick-off was by the discovery
of the electron by J.J. Thompson in 1897. Soon afterwards (1900) P. Drude used the new concept
to postulate a theory of electrical conductivity. Drude was considering why resistivity in
different materials ranges from 108 m (Ag) to 1020 m (polystyrene)?
Drude was working prior to the development of quantum mechanics, so he began with a
classical model, specifically:
(i) positive ion cores within an electron gas that follows Maxwell-Boltzmann statistics;
(ii) following the kinetic theory of gases - the electrons ae in form of free electron gas
(FEG), so that individual electrons move in straight lines and make collisions only with
the ion cores; electron-electron interactions are neglected;
(iii) Electrons lose energy gained from the electric field in colisions and the mean free path
was approximately the inter-atomic spacing.
Free electron gas (FEG) - Drude model
Drude (or FEG) model successfully exlpained Ohm amd Wiedemann-Franz laws,
but failed to explain, e.g., electron heat capacity and the magnetic susceptibility of
conduction electrons.
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Free electron gas (FEG) - Drude model
Velocity of electrons in FEG
e.g. at room temperature
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calculate as
#valence
electrons
per atom
density atomic
mass
#atoms
per
volume
Free electron gas (FEG) - Drude model
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Apply an electric field. The equation of motion is
integration gives
remember:
and if is the average time between collisions then the
average drift speed is
for we get
FEG explaining Ohm’s law
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we get
Ohm’s law
number of electrons passing in unit time
and with
current density
current of negatively charged electrons
FEG explaining Ohm’s law
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FEG explaining Ohm’s law
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line
FEG explaining Ohm’s law
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• Accumulation of charge leads to Hall field EH.
• Hall field proportional to current density and B field
is called Hall coefficient
FEG explaining Hall effect
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for the steady state
FEG explaining Hall effect
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FEG explaining Hall effect
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Later it was found by L. Lorenz that this constant is proportional to temperature
constant
FEG explaining Wiedemann-Franz law
Wiedemann and Franz found in 1853 that the ratio of thermal and electrical
conductivity for ALL METALS is constant at a given temperature (for room
temperature and above).
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estimated thermal conductivity
(from a classical ideal gas)
the actual quantum mechanical result is
this is 3, more or less....
FEG explaining Wiedemann-Franz law
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L = 2.45 10-8 Watt Ω K-2
at 273 K
metal 10-8 Watt Ω K-2
Ag 2.31
Au 2.35
Cd 2.42
Cu 2.23
Mo 2.61
Pb 2.47
Pt 2.51
Sn 2.52
W 3.04
Zn 2.31
FEG explaining Wiedemann-Franz law
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• an average thermal energy of an ideal gas particle, e.g. an electron from FEG,
moving in 3D at some temperature T is:
• Then for a total nr of N electrons the total energy is:
• And the electronic heat capacity would then be:
• If FEG approximation is correct this Cel should be added to phonon-related heat
capacitance, however, if we go out and measure, we find the electronic contribution
is only around one percent of this, specifically at high temperature is still Dulong-
Petit value, , that is valid.
FEG heat capacity
TkE B23
TNkU B23
Bel NkdT
dUC
23
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• Why does the Drude model work so relatively well when many of its
assumptions seem so wrong?
• In particular, the electrons don’t seem to be scattered by each other. Why?
• How do the electrons sneak by the atoms of the lattice?
• What are mysterious ”positive” charges realed by Hall effect measuremt in
semiconductors?
• Why do the electrons not seem to contribute to the heat capacity?
Free electron gas (FEG) - Drude model
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Lecture 14: Free electron gas and free electron Fermi gas
• Free electron gas (FEG) - Drude model
- FEG explaing Ohm’s law
- FEG expaling Hall effect
- FEG explaing Wiedemann-Franz law
- FEG heat capacity
•Free electron Fermi gas (FEFG) – a gas of electrons subject to Pauli principle
• One electron system – wave functions – orbits; FEFG in 1D in ground state
• FEFG in 3D in ground state
• Fermi-Dirac distribution and electron occupancy at T>0
![Page 21: FYS3410 - Vår 2014 (Kondenserte fasers fysikk) · Lecture schedule (based on P.Hofmann’s Solid State Physics, chapters 1-7 and 11) Module I – Periodic Structures and Defects](https://reader030.vdocuments.net/reader030/viewer/2022040702/5d66dd7988c99364418b8f9c/html5/thumbnails/21.jpg)
Free electron Fermi gas – a gas of electrons subject to Pauli
principle
• At low temperature, free mean path of a conduction electron in metal can be as long as 1 cm! Why is it not affected by ion cores or other conduction electrons? (30 seconds discussions)
– Motion of electrons in crystal (matter wave) is not affected by periodic structure such as ion cores.
– Electron is scattered infrequently by other conduction electrons due to the Pauli exclusion principle
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Lecture 14: Free electron gas and free electron Fermi gas
• Free electron gas (FEG) - Drude model
- FEG explaing Ohm’s law
- FEG expaling Hall effect
- FEG explaing Wiedemann-Franz law
- FEG heat capacity
•Free electron Fermi gas (FEFG) – a gas of electrons subject to Pauli principle
• One electron system – wave functions – orbits; FEFG in 1D in ground state
• FEFG in 3D in ground state
• Fermi-Dirac distribution and electron occupancy at T>0
![Page 23: FYS3410 - Vår 2014 (Kondenserte fasers fysikk) · Lecture schedule (based on P.Hofmann’s Solid State Physics, chapters 1-7 and 11) Module I – Periodic Structures and Defects](https://reader030.vdocuments.net/reader030/viewer/2022040702/5d66dd7988c99364418b8f9c/html5/thumbnails/23.jpg)
One electron system – wave functions - orbits
• Neglect electron-electron interaction, infinite potential well, simple QM solution
- The Pauli exclusion principle
– n: quantum number
– m(=1/2 and -1/2): magnetic quantum number
- degeneracy: # of orbitals with the same energy
- Fermi energy (EF): energy of the topmost filled
level in the ground state of the N electron system
In this simple system, every quantum state
holds 2 electrons => nF = N/2 Fermi energy:
Standing wave B. C.
n = 1, 2, …
Great, if we know the electron density, we know
the Fermi energy!
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Lecture 14: Free electron gas and free electron Fermi gas
• Free electron gas (FEG) - Drude model
- FEG explaing Ohm’s law
- FEG expaling Hall effect
- FEG explaing Wiedemann-Franz law
- FEG heat capacity
•Free electron Fermi gas (FEFG) – a gas of electrons subject to Pauli principle
• One electron system – wave functions – orbits; FEFG in 1D in ground state
• FEFG in 3D in ground state
• Fermi-Dirac distribution and electron occupancy at T>0
![Page 25: FYS3410 - Vår 2014 (Kondenserte fasers fysikk) · Lecture schedule (based on P.Hofmann’s Solid State Physics, chapters 1-7 and 11) Module I – Periodic Structures and Defects](https://reader030.vdocuments.net/reader030/viewer/2022040702/5d66dd7988c99364418b8f9c/html5/thumbnails/25.jpg)
FEFG in 3D
• Invoking periodic boundary condition instead of the infinite potential wall (standing wave) boundary condition, we get traveling waves as solutions:
Only if we know how much space one k
point occupies (dkxdkydkz= (2π/L)3)
Fermi wave vector
Fermi energy
Fermi velocity
Due to spin
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Lecture 14: Free electron gas and free electron Fermi gas
• Free electron gas (FEG) - Drude model
- FEG explaing Ohm’s law
- FEG expaling Hall effect
- FEG explaing Wiedemann-Franz law
- FEG heat capacity
•Free electron Fermi gas (FEFG) – a gas of electrons subject to Pauli principle
• One electron system – wave functions – orbits; FEFG in 1D in ground state
• FEFG in 3D in ground state
• Fermi-Dirac distribution and electron occupancy at T>0
![Page 28: FYS3410 - Vår 2014 (Kondenserte fasers fysikk) · Lecture schedule (based on P.Hofmann’s Solid State Physics, chapters 1-7 and 11) Module I – Periodic Structures and Defects](https://reader030.vdocuments.net/reader030/viewer/2022040702/5d66dd7988c99364418b8f9c/html5/thumbnails/28.jpg)
Fermi-Dirac distribution and electron occupancy at T>0
• Describes the probability that an orbit at energy E will be occupied in an ideal electron gas under thermal equilibrium
• is chemical potential, f( )=0.5; at 0K, F
In comparison,
High energy tail – approximation
Boltzmann-Maxwell distribution
E EF 3kBT
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• # of states between E and E+dE
Note here:
D(E) has a unit of ev-1.
N N()
Fermi-Dirac distribution and electron occupancy at T>0
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FYS3410 - Vår 2014 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/FYS3410/v14/index.html
Pensum: Solid State Physics by Philip Hofmann (Chapters 1-7 and 11)
Andrej Kuznetsov
delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO
Tel: +47-22857762,
e-post: [email protected]
visiting address: MiNaLab, Gaustadaleen 23c
![Page 31: FYS3410 - Vår 2014 (Kondenserte fasers fysikk) · Lecture schedule (based on P.Hofmann’s Solid State Physics, chapters 1-7 and 11) Module I – Periodic Structures and Defects](https://reader030.vdocuments.net/reader030/viewer/2022040702/5d66dd7988c99364418b8f9c/html5/thumbnails/31.jpg)
Lecture schedule (based on P.Hofmann’s Solid State Physics, chapters 1-7 and 11)
Module I – Periodic Structures and Defects 20/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 4h
21/1 Laue condition, Ewald construction, interpretation of a diffraction experiment 2h
22/1 Bragg planes and Brillouin zones (BZ) 2h
23/1 Elastic strain and structural defects 2h
23/1 Atomic diffusion and summary of Module I 2h
Module II - Phonons 03/2 Vibrations, phonons, density of states, and Planck distribution 4h
04/2 Lattice heat capacity: Dulong-Petit, Einstien and Debye models 2h
05/2 Comparison of different models 2h
06/2 Thermal conductivity 2h
07/2 Thermal expansion and summary of Module II 2h
Module III – Free Electron Gas 24/2 Free electron gas (FEG) versus Free electron Fermi gas (FEFG) 4h
25/2 Effect of temperature – Fermi- Dirac distribution 2h
26/2 Heat capacity of FEFG 2h
27/2 Electrical and thermal conductivity in metals 2h
28/2 FEFG in nanostructures 2h
Module IV – Nearly Free Electron Model 10/3 Bragg reflection of electrons at the boundary of BZ, energy bands 4h
11/3 Empty lattice approximation; number of orbitals in a band 2h
12/3 Semiconductors, effective mass method, intrinsic carriers 2h
13/3 Impurity states in semiconductors and carrier statistics 2h
14/3 p-n junctions and optoelectronic devices 2h
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Lecture 15: Electrons at T>0: DOS + Fermi-Dirac distribution
(i) Repetition of DOS for FEFG
(ii) Fermi-Diract distribution
(iii)Estimate for the FEFG heat capacity
Repetition of DOS for FEFG
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Lecture 15: Electrons at T>0: DOS + Fermi-Dirac distribution
(i) Repetition of DOS for FEFG
(ii) Fermi-Diract distribution
(iii)Estimate for the FEFG heat capacity
Repetition of DOS for FEFG
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Consider electrons
as quantum particles in a box
FEFG model means that
Repetition of DOS for FEFG
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boundary conditions provide restrictions the wavevector k
~
Repetition of DOS for FEFG
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Repetition of DOS for FEFG
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The volume of kx1·ky1·kz1= (2π/L)3
corresponds to only one k-state,
accommodating 2 electrons
While kmax or kF accommodate N/2!
kx1·ky1·kz1= (2π/L)3
Repetition of DOS for FEFG
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For any E
Repetition of DOS for FEFG
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Lecture 15: Electrons at T>0: DOS + Fermi-Dirac distribution
(i) Repetition of DOS for FEFG
(ii) Fermi-Diract distribution
(iii)Estimate for the FEFG heat capacity
Repetition of DOS for FEFG
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At T=0 all the states are filled up to the highest occupied
state. This state is called the Fermi energy EF. It is equal to
the chemical potential μ at T=0.
The Fermi-Dirac distribution
μ
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• Describes the probability that an orbit at energy E will be occupied in an ideal electron gas under thermal equilibrium
• is chemical potential, f( )=0.5; at 0K, F
High energy tail – approximation
Boltzmann-Maxwell distribution
E EF 3kBT
The Fermi-Dirac distribution
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~ kBT
The Fermi-Dirac distribution
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Lecture 15: Electrons at T>0: DOS + Fermi-Dirac distribution
(i) Repetition of DOS for FEFG
(ii) Fermi-Diract distribution
(iii)Estimate for the FEFG heat capacity
Repetition of DOS for FEFG
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kBT
The fraction of electrons that we “transfer” to higher energies ~ kBT/EF, the energy increase for
these electrons ~ kBT. Thus, the increase of the internal energy with temperature is
proportional to n(kBT/EF) (kBT) ~ n (kBT)2 /EF. Note, EF = kBTF
To calculate the heat capacity, we need to know how
the internal energy of the Fermi gas, Et(T), depends on
temperature. By heating a Fermi gas, we populate
some states above the Fermi energy EF and deplete
some states below EF. This modification is significant
within a narrow energy range ~ kBT around EF.
2t B
V
F
dE T k TC N
dT E
The Fermi gas heat capacity is much smaller (by kBT/EF<<1) than that of a classical ideal
gas with the same energy and pressure. The small heat capacity is a direct consequence
of the Pauli principle: most of the electrons cannot change their energy, only a small
fraction kBT/EF or T/TF of the electrons are excited out of the ground state.
3
2V BC nk for an ideal gas
t
V V const
dQ T dE TC
dT dT
2
2
Be B
F
k TC nk
E
One of the greatest successes of the free electron model
and FD statistics is the explanation of the T dependence of
the heat capacity of a metal.
compare
Estimate for the heat capacity of FEFG
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• two contributions: lattice and electrons
• electrons unimportant at high T but dominating and
sufficiently low T
Heat capacity of a metal: lattice + electrons
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FYS3410 - Vår 2014 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/FYS3410/v14/index.html
Pensum: Solid State Physics by Philip Hofmann (Chapters 1-7 and 11)
Andrej Kuznetsov
delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO
Tel: +47-22857762,
e-post: [email protected]
visiting address: MiNaLab, Gaustadaleen 23c
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Lecture schedule (based on P.Hofmann’s Solid State Physics, chapters 1-7 and 11)
Module I – Periodic Structures and Defects 20/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 4h
21/1 Laue condition, Ewald construction, interpretation of a diffraction experiment 2h
22/1 Bragg planes and Brillouin zones (BZ) 2h
23/1 Elastic strain and structural defects 2h
23/1 Atomic diffusion and summary of Module I 2h
Module II - Phonons 03/2 Vibrations, phonons, density of states, and Planck distribution 4h
04/2 Lattice heat capacity: Dulong-Petit, Einstien and Debye models 2h
05/2 Comparison of different models 2h
06/2 Thermal conductivity 2h
07/2 Thermal expansion and summary of Module II 2h
Module III – Electrons 24/2 Free electron gas (FEG) versus Free electron Fermi gas (FEFG) 4h
25/2 Effect of temperature – Fermi- Dirac distribution 2h
26/2 FEFG in 2D and 1D, and DOS in nanostructures 2h
27/2 Origin of the band gap and nearly free electron model 2h
28/2 Number of orbitals in a band and general form of the electronic states 2h
Module IV – Applications 10/3 Energy bands in real solids and transport properties 4h
11/3 Calculation of energy bands 2h
12/3 Semiconductors, effective mass method, intrinsic carriers 2h
13/3 Impurity states in semiconductors and carrier statistics 2h
14/3 p-n junctions and optoelectronic devices 2h
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Lecture 16
FEFG in 2D and 1D, and DOS in nanostructures
(i) Motivation: intriguing carrier recombination in QW resulting in
photons with energies above the band gap;
(ii) DOS for low-dimentional FEFG
(iii) DOS in quantum wells
(iv) DOS in nanowires
(v) DOS in quantum dots
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MQWs
Single quantum well - repetitions of
ZnO/ZnCdO/ZnO periods
Motivation: intriguing carrier recombination in QW resulting in
photons with energies above the band gap
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DOS for low-dimentional FEFG
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DOS in quantum wells
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DOS in quantum wells
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Density of states per unit volume and energy for a 3-D semiconductor (blue
curve), a 10 nm quantum well with infinite barriers (red curve) and a 10 nm
by 10 nm quantum wire with infinite barriers (green curve).
DOS in nanowires and quantum dots
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DOS in nanowires and quantum dots
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FYS3410 - Vår 2014 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/FYS3410/v14/index.html
Pensum: Solid State Physics by Philip Hofmann (Chapters 1-7 and 11)
Andrej Kuznetsov
delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO
Tel: +47-22857762,
e-post: [email protected]
visiting address: MiNaLab, Gaustadaleen 23c
![Page 56: FYS3410 - Vår 2014 (Kondenserte fasers fysikk) · Lecture schedule (based on P.Hofmann’s Solid State Physics, chapters 1-7 and 11) Module I – Periodic Structures and Defects](https://reader030.vdocuments.net/reader030/viewer/2022040702/5d66dd7988c99364418b8f9c/html5/thumbnails/56.jpg)
Lecture schedule (based on P.Hofmann’s Solid State Physics, chapters 1-7 and
11)
Module I – Periodic Structures and Defects 20/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space
4h
21/1 Laue condition, Ewald construction, interpretation of a diffraction experiment
2h
22/1 Bragg planes and Brillouin zones (BZ)
2h
23/1 Elastic strain and structural defects
2h
23/1 Atomic diffusion and summary of Module I
2h
Module II - Phonons 03/2 Vibrations, phonons, density of states, and Planck distribution
4h
04/2 Lattice heat capacity: Dulong-Petit, Einstien and Debye models
2h
05/2 Comparison of different models
2h
06/2 Thermal conductivity
2h
07/2 Thermal expansion and summary of Module II
2h
Module III – Electrons 24/2 Free electron gas (FEG) versus Free electron Fermi gas (FEFG)
4h
25/2 Effect of temperature – Fermi- Dirac distribution
2h
26/2 FEFG in 2D and 1D, and DOS in nanostructures
2h
27/2 Origin of the band gap and nearly free electron model
2h
28/2 Number of orbitals in a band and general form of the electronic states
2h
Module IV – Applications 10/3 Energy bands in real solids and transport properties
4h
11/3 Calculation of energy bands
2h
12/3 Semiconductors, effective mass method, intrinsic carriers
2h
13/3 Impurity states in semiconductors and carrier statistics
2h
14/3 p-n junctions and optoelectronic devices
2h
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Lecture 17
Origin of energy bands and nearly free electron model
• Recap of the one-electron approximation
• Intuitive picture
• Bragg-reflection for nearly free electrons
• Gap opening for nearly free electrons
• Solution of the Schrödinger equation
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58
Recap of one electron approximation Solving the Schrödinger equation for an all-
electron wave function in a rigid lattice is still
hopeless. We assume an effective one electron
potential.
we know for sure that
...but not much more
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59
The idea of energy bands: Na consider one atom of Na: 11 electrons
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60
• Focus only on the valence (outer)
electrons (3s).
• What happens when we move them
together?
consider tow Na atoms / a Na2
molecule: 22 electrons
big
separation
The idea of energy bands: Na
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61
• Levels split up in bonding and anti bonding
molecular orbitals and are occupied
according to the Pauli principle.
• The distance between the atoms must be
such that there is an energy gain.
consider two Na atoms / a Na2
molecule: 2 3s electrons
molecular
energy levels
capacity: 2x2=4 electrons
61
The idea of energy bands: Na
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62
• N levels with very similar energies, like in a
super-giant molecule. We can speak of a
“band” of levels.
• Every band has N levels. We can put 2N
electrons into it (but we have only N electrons
from N Na atoms).
consider many (N) Na atoms (only 3s level)
62
The idea of energy bands: Na
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63
4 electrons per atom
2 atoms per unit cell
sp3: 8 states
4 sp3 per cell
8 states per atom
16 states per unit
cell
The idea of energy bands: Si
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64
free electron model naive band picture
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65
Bragg-reflection for nearly free electrons
electron traveling perpendicular to crystal
planes
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66
free electron wave function
this gives a Bragg condition for
electron waves:
consider only one direction (x)
with a de Broglie wavelength
Bragg condition with
Bragg-reflection for nearly free electrons
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67
Bragg reflection results in standing, not traveling electron waves
Bragg condition for
electron waves:
two possible linear combinations of
Bragg-reflection for nearly free electrons
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68
Gap opening for nearly free electrons
68
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Schrödinger equation for a periodic potential
solutions have the form of Bloch waves
plane wave lattice-periodic function
General form of Bloch theory