Course homepage - Göteborgs universitet
Transcript of Course homepage - Göteborgs universitet
Course homepage
The course homepage is online http://physics.gu.se/~tfkhj/CMP2016.html
There you can find powerpoint parts of the lectures, problem sets, office hours, email addresses, information on exams, literature, etc.
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