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Summary lecture 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

1. Consider kinetic energy of ions perturbatively (large mass)

with

2. Schrödinger equation for electron motion in static ion potential

3. Develop the wavefunction of the total system as linear combinationof 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

where

Born-Oppenheimer approximation

II. Electronic properties of solids

1. Bloch theorem

2. Electronic band structure

3. Density of states

Chapter II

Learning Outcomes Chapter II

Explain the Bloch theorem and its derivation

Recognize the concept of electronic band structure in effective mass and tight-binding approximation

Describe the remarkable band structure of graphene

Calculate the density of states of low-dimensional nanomaterials

1. Bloch theorem

Focus on non-interacting electrons in a rigid ion lattice with a strictly periodic arrangement (ideal crystal)

Goal: Solution of the eigenvalue problem to the Hamilton operator

with as zeroth term of Taylor expansion of describing electron motion in a static potential of ions

• For non-interacting particles, one-particle Schrödinger equation sufficient (sum of eigenvalue, product of eigenfunctions for many-particle systems)

Non-interacting electrons

1. Bloch theorem

The potential is translational invariant with respect to lattice vectors

According to Noether's theorem, space translational symmetry is equivalent to the momentum conservation law

Translation operator commutes with the Hamilton operator

H and TR have the same eigenfunctions

Normalization of the wave function requires

Eigenvalue of TR corresponds to a phase factor where k isthe wave vector and element of the reciprocal lattice

Translational symmetry

problem set 1

problem set 1

1. Bloch theorem

Direct spatial lattice is spanned by basis lattice vectors ai

Unit cell is the smallest cell that can be periodically expanded spanning the entire crystal

In the case of graphene, the direct lattice is hexagonal and the unit cell consists of 2 atoms (A and B atom)

Direct spatial lattice

1. Bloch theorem

Due to periodicity

size of the 1. BZ

To each direct lattice a reciprocal lattice can be ascribed with reciprocal lattice vectors ki that are orthogonal to ai

The unit cell of the reciprocal lattice is called first Brillouine zone (BZ)

Reciprocal lattice

1. Bloch theorem

H and TR have the same eigenfunctionswith

Eigenfunctions are not periodic and can differ through the phase factor from one unit cell to another

Ansatz for wave function Bloch function

with the periodic Bloch factor

Bloch theorem: Eigenfunctions 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

Bloch theorem

1. Bloch theorem

Bloch function with periodic Bloch factor

Bloch functions are orthonormal

Schrödinger equation for Bloch factors

Since is periodic with respect to lattice translations, solutions are restricted to one unit cell (boundary problem): for every k, there are discrete eigenvalues and eigenfuctions with the band index λ

Schrödinger equation for Bloch factors

problem set 1

2. Electronic band structure

Free electrons are characterized by and

Eigenenergies

are parabolic in k, where curvature is given my the inverse electron mass m

Eigenfunctions correspond to plane waves

Energy of free electrons

2. Electronic band structure

Consider the impact of the periodic lattice potential perturbatively through harmonic approximation of the band structure at the minimum

with the inverse effective mass

determining the band curvature and reflecting the impact of the lattice

Effective mass approximation

valence band

conduction band

Tight-binding (TB) approximation is based on the assumption that electrons are tightly bound to their nuclei

Start from isolated atoms, their wave functions overlap and lead to chemical bonds forming the solid, when the atoms get close enough

Due to the appearing interactions, electronic energies broaden and build continuous bands

Tight-binding aproach

2. Electronic band structure

(a) levels in isolated atoms (b) band structure in solids

The electronic band structure of graphene iscalculated with TB wave functions

with 2pz-orbital functions takenfrom hydrogen atom with an effective atomic number

• TB wave functions are based on superposition of wave functions for isolated atoms located at each atomic site

• Solve the eigenvalue problem

Band structure of graphene

2. Electronic band structure

Multiply with and separately and integrate over r leads to a set of coupled equations

that can be solved by evaluating the secular equation

with and

2. Electronic band structure

Band structure of graphene

Exploit the equivalence of the A and B atoms withand assume the nearest-neighbour approximation with

2. Electronic band structure

Band structure of graphene

Band structure of graphene

2. Electronic band structure

Electronic band structure of graphene reads

with σc = -1 and σv = +1

problem set 1

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 (semi-metal)

with the Fermi velocity υF

2. Electronic band structure

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

problem set 1

Summary Chapter II

Bloch theorem: eigenfunctions 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

Electronic band structure is material-specific and illustrates all possible electronic states. It can be calculated in and effective mass or tight-binding approximation

Graphene exhibits a remarkable linear and gapless band structure opening up novel relaxation channels for non-equilibrium electrons

Density of states reveals the number of states in a certain energy interval and strongly depends on material dimensionality

Learning Outcomes Chapter II

Explain the Bloch theorem and its derivation

Recognize the concept of electronic band structure in effective mass and tight-binding approximation

Describe the remarkable band structure of graphene

Calculate the density of states of low-dimensional nanomaterials