Summary lecture II Bloch theorem: eigen functions of an electron in a perfectly periodic potential have the shape of plane waves modulated with a Bloch factor that possess the periodicity of the potential Eigen energies are summarized in the material-specific electronic band structure that can be calculated in and effective mass or tight-binding approximation Graphene exhibits a remarkable linear and gapless band structure
Band structure of graphene 2. Electronic band structure conduction band valence band Convenional materials graphene Graphene has a linear and gapless electronic band structure around Dirac points (K, K points) in the Brillouine zone with the Fermi velocity υ F
Properties of graphene Excellent conductor of current and heat (ballistic transport) Very strong, however also light and flexible at the same time (sp 2 bonds) Almost transparent (absorbs only 2.3 % of visible light) Extremely sensitive to changes in environment (atomically thin material) Ultrafast carrier dynamics (extraordinary electronic band structure) www.extremetech.com Bae et al. Nature Nano 5, 574 (2010) www.free-stock-illustration.com
Density of states 2. Density of states While band structure provides the complete information about possible electronic states in a solid, often it is sufficient to know the number of states in a certain energy range density of states corresponds to number of states with energy in the interval
Chapter III III. Electron-electron interaction 1. Coulomb interaction 3. Jellium & Hubbard models 4. Hartree-Fock approximation 5. Screening 6. Plasmons 7. Excitons
Chapter III III. Electron-electron interaction 1. Coulomb interaction 3. Jellium & Hubbard models 4. Hartree-Fock approximation 5. Screening 6. Plasmons 7. Excitons
Learning Outcomes Write down the Hamilton operator for Coulomb interaction in second quantization and explain electron-electron scattering Recognize the advantage of the formalism of second quantization Be able to construct a many-particle state by applying creation and annihilation operators Write down the fundamental commutation relations
Coulomb interaction 1. Coulomb interaction Electron-electron interaction is driven by repulsive Coulomb potential between equally charged particles (expressed in first quantization) In a real solid, the Coulomb potential is screened due to the presence of many electrons Section 5 of this chapter It is more convenient to deal with the Coulomb interaction in the formalism of second quantization
Coulomb matrix element 1. Coulomb interaction Coulomb matrix element for 3D materials reads (problem set 2) 1 4 k, s k+q, s Momentum conservation only spin-conserving processes Coulomb matrix element for 2D materials reads k, s q 2 3 k -q, s Feynman diagram for electron-electron interaction
Coulomb matrix element 1. Coulomb interaction Inserting momentum conservation, Coulomb interaction reads in second quantization Coulomb-induced scattering can take place intraband or interband
Coulomb matrix element 1. Coulomb interaction Inserting momentum conservation, Coulomb interaction reads in second quantization Coulomb-induced scattering can take place intraband or interband Auger processes including Auger recombination (AR) and impact ionization (II) bridge the valence and conduction band
Second quantization Second quantization is a formalism to describe many-particle systems Its advantage is that the tedious (anti-)symmetrisation of many-particle wave functions is not needed All physics ins included in fundamental commutation relations between creation and annihilation operators (+ for fermions, - for bosons)
Indistinguishable particles Distinguishable particles can be numbered and the wave function of an N-particle system is just a product of one-particle wave functions Identical particles (e.g. electrons) are indistinguishable and their numbering does not make sense exchange of particles must not change observables
(Anti-)symmetric many-particle states Wave function of a system of identical particles has to be symmetric or antisymmetric with respect to an exchange of a pair of particles with the transposition operator since (Anti-)symmetric wave functions can be constructed from not symmetrized one-particle wave functions: with with p as number of transpositions building the permutation operator
Fermions and bosons Spin-statistics theorem relates the spin of a particle to its statistics: : Hilbert room of symmetric states of N identical particles with integer-spin (bosons), such as photons, phonons, etc. : Hilbert room of antisymmetric states of N identical particles with half-integer-spin (fermions), such as electrons, protons etc. For fermions the antisymmetric wave function can be expressed as Slater determinant If 2 sets of quantum numbers are identical, 2 rows in the determinant are identical and the wave function is 0 (Pauli principle)
Occupation number basis The normalization factor C_ for fermions and C + for bosons reads with n i as the occupation number of the identical one-particle state α i (Anti-)symmetric N-particle states can be unambiguously determined by the occupation number of each single-particle state second quantization: each state is represented by the occupation number basis (Fock state) Fock states are orthonormal and build a complete set of functions
Creation and annihilation operators Introduction of creation and annihilation operators directly changing the occupation number of states Bosons Fermions with N i as number of pair-wise exchanges to move the created/ annihilated particle to the right place in the N-particle state
Vacuum state Application of a creation operator with Every state can be constructed from the vacuum state by applying the creation operator
Fundamental commutation relations Fundamental commutation relations for bosons (commutator [ ] - ) and fermions ([ ] + ) A detailed derivation of these relations will be done on board
Observables in second quantization In most cases the observables can be expressed as a sum of one- and twoparticle operators General one-particle operators read in second-quantization General two-particle operators read in second quantization (problem set 2)
Important operators Occupation number operator reveals the number of particles occupying the one-particle state Particle number operator reveals the total number of particles
Summary lecture II Electron-electron interaction is driven by repulsive Coulomb potential between equally charged particles Identical particles are indistinguishable and their many-particle wave function needs to be (anti-)symmetric with respect to particle exchange (Anti-)symmetric many-particle states can be unambiguously determined by the occupation number of each single-particle state Second quantization avoid (anti-)symmetrisation of many-particle states and mirrors the physics in fundamental commutation relations
Learning Outcomes Write down the Hamilton operator for Coulomb interaction in second quantization and explain electron-electron scattering Recognize the advantage of the formalism of second quantization Be able to construct a many-particle state by applying creation and annihilation operators Write down the fundamental commutation relations