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

Convenional materials graphene

valence band

conduction band

Graphene has a linear and gapless electronic band structure around Dirac points (K, K’ points) in the Brillouine zone

with the Fermi velocity υF

2. Electronic band structure

Excellent conductor of current and heat (ballistic transport)

Very strong, however also light and flexible at the same time (sp2 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)

Properties of graphene

www.extremetech.com Bae et al. Nature Nano 5, 574 (2010) www.free-stock-illustration.com

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

Density of states

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

Chapter III

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

Chapter III

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

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 interaction

1. Coulomb interaction

Coulomb matrix element for 3D materials reads (problem set 2)

Coulomb matrix element for 2D materials reads

Coulomb matrix element

Momentum conservation only spin-conserving processes

Feynman diagram for electron-electron interaction

1 4

2 3

q

k, s k+q, s

k’-q, s’k’, s’

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

Coulomb matrix element

2. 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)

Second quantization

2. Second quantization

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

Indistinguishable particles

2. Second quantization

Wave function of a system of identical particles has to be symmetric orantisymmetric 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:

withwith p as number of transpositions building the permutation operator

(Anti-)symmetric many-particle states

2. Second quantization

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)

Fermions and bosons

2. Second quantization

The normalization factor C_ for fermions and C+ for bosons reads

with ni as the occupation number of the identical one-particle state αi

(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

Occupation number basis

2. Second quantization

Introduction of creation and annihilation operators directly changing the occupation number of states

Bosons

Fermions

with Ni as number of pair-wise exchanges to move the created/ annihilated particle to the right place in the N-particle state

Creation and annihilation operators

2. Second quantization

Application of a creation operator

with

Every state can be constructed from the vacuum state by applying the creation operator

Vacuum state

2. Second quantization

Fundamental commutation relations for bosons (commutator […]-) and fermions ([…]+)

A detailed derivation of these relations will be done on board

Fundamental commutation relations

2. Second quantization

In most cases the observables can be expressed as a sum of one- and two-particle operators

General one-particle operators read in second-quantization

General two-particle operators read in second quantization (problem set 2)

Observables in second quantization

2. Second quantization

Occupation number operator

reveals the number of particles occupying the one-particle state

Particle number operator

reveals the total number of particles

Important operators

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