Summary lecture II. Graphene exhibits a remarkable linear and gapless band structure

Transcription

1 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

2 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

3 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) Bae et al. Nature Nano 5, 574 (2010)

4 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

5 Chapter III III. Electron-electron interaction 1. Coulomb interaction 3. Jellium & Hubbard models 4. Hartree-Fock approximation 5. Screening 6. Plasmons 7. Excitons

6 Chapter III III. Electron-electron interaction 1. Coulomb interaction 3. Jellium & Hubbard models 4. Hartree-Fock approximation 5. Screening 6. Plasmons 7. Excitons

7 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

8 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

9 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

10 Coulomb matrix element 1. Coulomb interaction Inserting momentum conservation, Coulomb interaction reads in second quantization Coulomb-induced scattering can take place intraband or interband

11 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

12 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)

13 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

14 (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

15 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)

16 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

17 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

18 Vacuum state Application of a creation operator with Every state can be constructed from the vacuum state by applying the creation operator

19 Fundamental commutation relations Fundamental commutation relations for bosons (commutator [ ] - ) and fermions ([ ] + ) A detailed derivation of these relations will be done on board

20 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)

21 Important operators Occupation number operator reveals the number of particles occupying the one-particle state Particle number operator reveals the total number of particles

22 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

23 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