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 (ermin.malic@chalmers.se) office hours: Tuesday 4-5 pm, Origo 7115 Roland Jago as teaching assistant (roland.jago@chalmers.se) office hours: Mondays and Wednesdays 3.30-4.30 pm Soliden 3022 Lattice properties of solids (24 November 15 December) Henrik Johannesson (henrik. johannesson@physics.gu.se) Oleksandr Balabanov (oleksandr.balabanov@physics.gu.se)
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: 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 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)
Effective Hamilton operator 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
Density matrix theory 2. Theoretical approaches 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 second quantization Chapter III
Density matrix theory 2. Theoretical approaches Density matrix formalism (Chapter IV) 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!) Semiconductor Bloch equations Chapter II 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) 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 N 3 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
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 (10 4 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 Electrons core electrons filled shells valence electrons outer shell, chemical bonds Lattice ions 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 H e describes electrons moving in the potential of lattice ions Bloch electrons (quasi electrons with effective mass) Chapter II H i describes localized ions oscillating around their ground state positions Phonons (collective oscillations for strong ion-ion interaction) H e-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 N e : number of valence electrons m e : electronic mass ε 0 : vacuum permittivity (electric constant) momentum of electron i determined by its mass and velocity v i
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 N α : number of lattice ions, m α : ion mass Considering only atom nuclei, ion-ion interaction can be expressed as with atomic numbers Z α, Z β 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 1. term of Taylor expansion electron motion in static interaction of electrons with timepotential of ions dependent 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