Condensed matter physics –...

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

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])

Organization of 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

Requirements to pass the course

I. Introduction1. Main concepts2. Theoretical approaches3. Born-Oppenheimer approximation

II. Electronic properties of solids1. Bloch theorem2. Electronic band structure 3. Density of states

III. Electron-electron interaction1. Coulomb interaction2. Second quantization3. Jellium & Hubbard models4. Hartree-Fock approximation5. Screening6. Plasmons7. Excitons

Contents

IV. Density matrix theory1. Statistic operator2. Bloch equations3. Boltzmann equation

V. Density functional theory (guest lecture by Paul Erhart)

VI. Optical properties of solids1. Electron-light interaction2. Absorption spectra 3. Differential transmission spectra4. Statistics of light

Contents

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, andMarkov 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

I. Introduction

1. Main concepts

2. Theoretical approaches

3. Born-Oppenheimer approximation

Chapter I

Learning Outcomes Chapter I


Realize the importance of the Born-Oppenheimer approximation

1. Main concepts

Condensed matter physics:

• 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)

Solid state physics

1. Main concepts

Central concepts

• 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!)

• 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)

Quasi particles

1. Main concepts

Central concepts

• 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

• Understanding of elementary many-particle processes, such as electron-light, electron-electron, electron-lattice interaction

technological application of nanomaterials

Effective Hamilton operator


Density matrix formalism (Chapter IV)

• Introduction of a statistic operator (density matrix)

with

with the probability 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

Density matrix theory

second quantization Chapter III


Density matrix formalism (Chapter IV)

• Derive equations of motion for diagonal andoff-diagonal elements of the density matrix

• Hamilton operator H including all matrix elements describing interactions of electrons (electronic wavefunctions as input needed!)

• Semiconductor Bloch equations

• Bridge to macroscopic quantities, such as optical absorption α(ω)

Density matrix theory

Chapter II

Chapter VI


Density functional theory (Chapter V, guest lecture)

• 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!

• 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

Density functional theory


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

Decoupling of electron and lattice dynamics


Goal: Separation of electron and lattice dynamics

Born-Oppenheimer approximation

Electrons

valence electrons

Atom nuclei

Atoms

core electrons

filled shells outer shell, chemical bonds

ElectronsLattice ions


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)

Hi describes localized ions oscillating around their ground state positions

Phonons (collective oscillations for strong ion-ion interaction)

He-i describes electron-ion interaction

Polarons as new quasi particles (electron plus polarisation cloud)

Hamilton operator

Chapter II

Part II of the course


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

Electronic part of the Hamiltonian


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

• Considering only atom nuclei, ion-ion interaction can be expressed as

with atomic numbers Zα , Zβ

• Free Coulomb potential is not a very good approximation, since core electrons give rise to an effective screened potential

Lattice part of the Hamiltonian


Ion-ion potential needs to have a minimum as a function of distance between ions to have a stable solid (eg Lennard Jones potential)

Lattice part of the Hamiltonian

atomsinmotion.com

attractive part

repulsive part

equilibrium distance


Electron-ion interaction

0. term of Taylor expansion 1. term of Taylor expansionelectron motion in static interaction of electrons with time-potential of ions dependent potential of ions

lattice distortions

electron-phonon interaction

Electron-ion interaction

0


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 combinationof 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

Mathematical justification

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


Realize the importance of the Born-Oppenheimer approximation

Condensed matter physics –...

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