Download 3. Born-Oppenheimer approximation

Condensed matter physics – FKA091
• Ermin Malic
• Department of Physics
• Chalmers University of Technology
• Henrik Johannesson
• Department of Physics
• University of Gothenburg
• Teaching assistants: Roland Jago & Oleksandr Balabanov
Organization of the course
The course is divided in two parts
Electronic and optical properties of solids (1 -22 November)
Ermin Malic ([email protected])
office hours: Tuesday 4-5 pm, Origo 7115
Roland Jago as teaching assistant ([email protected])
office hours: Mondays and Wednesdays 3.30 - 4.30 pm
Soliden 3022
Lattice properties of solids (24 November – 15 December)
Henrik Johannesson (henrik. [email protected])
Oleksandr Balabanov ([email protected])
Requirements to pass the course
The course will be accompanied by 4 problem sets that should be solved in
groups of 3-4 students
1. Problem set: 3 November, deadline 11 November
2. Problem set: 11 November, deadline 22 November
3. Problem set: 24 November, deadline 6 December
4. Problem set: 6 December, deadline 15 December
• The course will end with oral exams. All students with more than 50% of
possible points in problem sets can take part.
Individual 30 min oral exams take place in the week of
9 -13 January 2017
Contents
I. Introduction
1. Main concepts
2. Theoretical approaches
3. Born-Oppenheimer approximation
II. Electronic properties of solids
1. Bloch theorem
2. Electronic band structure
3. Density of states
III. Electron-electron interaction
1. Coulomb interaction
2. Second quantization
3. Jellium & Hubbard models
4. Hartree-Fock approximation
5. Screening
6. Plasmons
7. Excitons
Contents
IV. Density matrix theory
1. Statistic operator
2. Bloch equations
3. Boltzmann equation
V. Density functional theory (guest lecture by Paul Erhart)
VI. Optical properties of solids
1. Electron-light interaction
2. Absorption spectra
3. Differential transmission spectra
4. Statistics of light
Learning Outcomes
Recognize the main concepts of condensed matter physics including
introduction of quasi-particles (such as excitons, plasmons)
Realize the importance of Born-Oppenheimer, Hartree-Fock, and
Markov approximations
Explain the Bloch theorem and calculate the electronic band structure
Define Hamilton operator in the formalism of second quantization
Realize the potential of density matrix and density functional theory
Explain the semiconductor Bloch and Boltzmann equations
Recognize the optical finger print of nanomaterials
Chapter I
I. Introduction
1. Main concepts
2. Theoretical approaches
3. Born-Oppenheimer approximation
Learning Outcomes Chapter I
Recognize the main concepts of condensed matter physics including
introduction of quasi-particles (such as excitons, plasmons)
Realize the importance of the Born-Oppenheimer approximation
Solid state physics
1. Main concepts
Condensed matter physics:
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Focus on solid state physics describing electronic, optical, and
thermal properties of solids
Solid as accumulation of many atomic-scale systems that are
chemically bound and localized around equilibrium positions
Crystalline solids (periodically arranged, translational symmetry) as
thermodynamically most stable state of matter (lowest entropy)
Reveal elementary microscopic atomic-scale processes behind
macroscopic large-scale phenomena
Comprises methods of quantum mechanics (electrons, ions),
electrodynamics (fields), and statistical physics (many-particle)
Quasi particles
1. Main concepts
Central concepts
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Interacting many-particle systems challenging to model (only
hydrogen problem exactly solvable)
Introduction of the concept of quasi-particles or collective excitations
(original particle + parts of its environment / interaction)
non-interacting quasi-particles (easy to model!)
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Examples: excitons (Coulomb-bound electron-hole pairs)
phonons (collective lattice vibrations)
plasmons (collective plasma oscillations)
polarons (electrons moving in lattice)
polaritons (electron interacting with photons,
also exciton- or phonon-polaritons)
Effective Hamilton operator
1. Main concepts
Central concepts
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Description of specific phenomena, focus on the relevant part of a
general problem
Many-particle Hamilton operator
Effective Hamilton operator for the relevant problem
Equations of motion for relevant observables
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Understanding of elementary many-particle processes, such as
electron-light, electron-electron, electron-lattice interaction
technological application of nanomaterials
Density matrix theory
2. Theoretical approaches
Density matrix formalism (Chapter IV)
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Introduction of a statistic operator (density matrix)
with
with the probability
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to find the system in the state
Diagonal elements of the density matrix
correspond to the carrier occupation probability
while non-diagonal elements describe the
transition probability between two states
second quantization
Chapter III
Density matrix theory
2. Theoretical approaches
Density matrix formalism (Chapter IV)
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Derive equations of motion for diagonal and
off-diagonal elements of the density matrix
Hamilton operator H including all matrix
elements describing interactions of electrons
(electronic wavefunctions as input needed!)
Chapter II
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Semiconductor Bloch equations
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Bridge to macroscopic quantities, such as optical absorption α(ω)
Chapter VI
Density functional theory
2. Theoretical approaches
Density functional theory (Chapter V, guest lecture)
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Calculation of the quantum mechanic ground state
of a many-particle system
Ground state can be unambiguously determined from the electron
density
(Hohenberg-Kohn Theorem)
Full Schrödinger equation with N3 degrees of freedom does not
need to be solved!
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Electron density is solved through Kohn-Sham equations assuming
an effective one-particle Hamilton operator
Advantage: First-principle calculation of the ground state,
Disadvantage: Limited to systems of less than 100 atoms
Decoupling of electron and lattice dynamics
3. Born-Oppenheimer approximation
Many-particle systems in a solid are not exactly solvable
Solids can be divided in subsystems of lattice ions and electrons
Electron and lattice dynamics can be decoupled due to the much larger
mass of lattice ions (104 times larger)
electronic system is much faster and can almost instantaneously
adapt to the new ion positions (adiabatic approximation)
Procedure: 1) describe electron motion in static ion lattice
2) describe ion motion in a homogenous electron sea
3) perturbative description of electron-ion interaction
Born-Oppenheimer approximation
3. Born-Oppenheimer approximation
Goal: Separation of electron and lattice dynamics
Atoms
Atom nuclei
core electrons
filled shells
Lattice ions
Electrons
valence electrons
outer shell, chemical bonds
Electrons
Hamilton operator
3. Born-Oppenheimer approximation
Hamilton operator describes the total energy of the solid and can now be
separated in an electronic and ionic part plus their interaction
He describes electrons moving in the potential of lattice ions
Bloch electrons (quasi electrons with effective mass)
Chapter II
Hi describes localized ions oscillating around their ground state positions
Phonons (collective oscillations for strong ion-ion interaction)
He-i describes electron-ion interaction
Part II of the course
Polarons as new quasi particles (electron plus polarisation cloud)
Electronic part of the Hamiltonian
3. Born-Oppenheimer approximation
Electronic part of the Hamilton operator can be separated in kinetic
energy and Coulomb-induced electron-electron interaction
Ne : number of valence electrons
me : electronic mass
ε0 : vacuum permittivity (electric constant)
momentum of electron i determined by its mass and velocity vi
Lattice part of the Hamiltonian
3. Born-Oppenheimer approximation
Lattice part of the Hamilton operator can be separated in kinetic energy of
ions and ion-ion interaction
Nα : number of lattice ions, mα : ion mass
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Considering only atom nuclei, ion-ion interaction can be expressed as
with atomic numbers Zα , Zβ
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Free Coulomb potential is not a very good approximation, since core
electrons give rise to an effective screened potential
Lattice part of the Hamiltonian
3. Born-Oppenheimer approximation
Ion-ion potential needs to have a minimum as a function of distance
between ions to have a stable solid (eg Lennard Jones potential)
repulsive part
attractive part
equilibrium
distance
atomsinmotion.com
Electron-ion interaction
3. Born-Oppenheimer approximation
Electron-ion interaction
0
0. term of Taylor expansion
electron motion in static
potential of ions
1. term of Taylor expansion
interaction of electrons with timedependent potential of ions
lattice distortions
electron-phonon interaction
Mathematical justification
3. Born-Oppenheimer approximation
Detailed derivation performed on board; here brief sketch:
1.
Consider kinetic energy of ions perturbatively (large mass)
2.
Schrödinger equation for electron motion in static ion potential
3.
Develop the wavefunction of the total system as linear combination
of eigenfunction of the electron motion
4.
Schrödinger equation for ion motion in an effective potential
determined by electronic energies
5.
Estimation of the validity of the Born-Oppenheimer approximation
Summary Chapter I
Since interacting many-particle systems are challenging to model,
introduction of non-interacting quasi-particles (excitons, phonons) is
an important concept of condensed matter physics
Main theoretical approaches include density matrix (Bloch functions)
and density functional theory (Hohenberg-Kohn theorem)
In Born-Oppenheimer approximation, electron and ion dynamics is
separated based on the much larger mass and slower motion of ions
Learning Outcomes Chapter I
Recognize the main concepts of condensed matter physics including
introduction of quasi-particles (such as excitons, plasmons)
Realize the importance of the Born-Oppenheimer approximation