Solid state physics is the study of how atoms arrange themselves into solids and what properties these solids have. Calculate the macroscopic properties from the microscopic structure. Institute of Solid State Physics Advanced Solid State Physics Technische Universität Graz Peter Hadley

Quantization Review: Photons (noninteracting bosons), photonic crystals Review: Free electrons (noninteracting fermions), electrons in crystals Electrons in a magnetic field Fermi surfaces Quantum Hall effect Linear response theory Dielectric function / optical properties Transport properties Quasiparticles (phonons, magnons, plasmons, exitons, polaritons) Mott transition, Fermi Liquid Theory Ferroelectricity, pyroelectricity, piezoelectricity Landau theory of phase transitions Superconductivity Institute of Solid State Physics Advanced Solid State Physics Technische Universität Graz

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Institute of Solid State Physics Student projects Technische Universität Graz Something that will help other students pass this course 2VO + 1UE Improve the script (1 student needed to compile) Solutions to exam questions Example calculations (phonon dispersion relation for GaAs) Javascript calculations Lecture videos You can decide not to put your project on the webpage.

Institute of Solid State Physics Examination Technische Universität Graz 1 hour written exam half of the questions will be from the website Oral exam Student project Mistakes on written exam General questions about the course

Quantization Start with the classical equations of motion Find the normal modes Construct the Lagrangian From the Lagrangian determine the conjugate variables Perform a Legendre transformation to the Hamiltonian Quantize the Hamiltonian Institute of Solid State Physics Technische Universität Graz

Harmonic oscillator Newton's law: Lagrangian (constructed by inspection) Conjugate variable: Euler - Lagrange equations: Legendre transformation: Quantize:

L L L LC circuit Classical equations Lagrangian (constructed by inspection) Conjugate variable: LL Euler - Lagrange equation: Legendre transformation:

L LC circuit Conjugate variable: Hamiltonian: Quantize:

Transmission line L inductance/m C capacitance/m normal mode solution: Each normal mode moves independently from the other normal modes

Transmission line An infinite transmission line is resistive, typically ~ 50 . Substituting the normal mode solution into the wave equation yields the dispersion relation

Transmission line Wave equation Not clear what mass we should use in the Schrödinger equation c is the speed of waves

L L L The Schrödinger equation is for amateurs Classical equations of motion (Newton's law) Lagrangian (constructed by inspection) Conjugate variable: Euler - Lagrange equations: Legendre transformation: Quantize: LL

LL Lagrangian Construct the Lagrangian 'by inspection'. The Euler-Lagrange equation and the classical equation of motion for a normal mode are, The Lagrangian is, L classical equation for the mode k

L Hamiltonian The conjugate variable to V is, The Hamiltonian is constructed by performing a Legendre transformation, To quantize we replace the conjugate variable by L L

Quantum solutions This equation is mathematically equivalent to the harmonic oscillator. j is the number of photons. mass - spring wave mode spring constant

Dissipation in Quantum mechanics An infinite transmission line is resistive Transmission line =

Quantum coherence is maintained until the decoherence time. This depends on the strength of the coupling of the quantum system to other degrees of freedom. Dissipation in quantum mechanics Decay is the decoherence time.

Dissipation in solids At zero electric field, the electron eigen states are Bloch states. Each Bloch state has a k vector. The average value of k = 0 (no current). At finite electric field, the Bloch states are no longer eigen states but we can calculate the transitions between Bloch states using Fermi's golden rule. The final state may include an electron state plus a phonon. The average value of k is not zero (finite current). The phonons carry the energy away like a transmission line.

Institute of Solid State Physics Technische Universität Graz The quantization of the electromagnetic field Wave nature and the particle nature of light Unification of the laws for electricity and magnetism (described by Maxwell's equations) and light Quantization of fields Planck's radiation law Serves as a template for the quantization of noninteracting bosons: phonons, magnons, plasmons, electrons, spinons, holons and other quantum particles that inhabit solids.

Maxwell's equations

The vector potential Maxwell's equations in terms of A Coulomb gauge

The wave equation normal mode solutions have the form: Using the identity Substituting the normal mode solution in the wave equation results in the dispersion relation wave equation

EM waves propagating in the x direction The electric and magnetic fields are

Lagrangian To quantize the wave equation we first construct the Lagrangian 'by inspection'. The Euler-Lagrange equation and the classical equation of motion are, The Lagrangian is, classical equation for the normal mode k

Hamiltonian The conjugate variable to A s is, The Hamiltonian is constructed by performing a Legendre transformation, To quantize we replace the conjugate variable by

Quantum solutions This equation is mathematically equivalent to the harmonic oscillator. j s is the number of photons in mode s.

Partition function At constant temperature the relevant partition function is the canonical partition function The sum over all microstates is a sum over each mode s (polarization and k) containing j s photons.: The sum in the exponent can be converted to a product e a+b+c+... = e a e b e c...