Bandstructures and Density of States - TCM Group

1 RecapThe BrillouinzoneBandstructureDOSP hononsBandstructures and Density of HasnipDFT Spectroscopy Workshop 2009 RecapThe BrillouinzoneBandstructureDOSP hononsRecap of Bloch s TheoremBloch s theorem: in a periodic potential, the Density has thesame periodicity. The possible wavefunctions are all quasi-periodic : k(r)= (r).We writeuk(r)in a plane-wave basis as:uk(r)= ,whereGare thereciprocal lattice vectors, defined so BrillouinzoneBandstructureDOSP hononsFirst Brillouin ZoneAdding or subtracting a reciprocal lattice vectorGfromkleaves the wavefunction unchanged in other words oursystem is periodic in reciprocal-space only need to study the behaviour in the reciprocal-spaceunit cell, to know how it behaves everywhere. It isconventional to consider the unit cell surrounding thesmallest vector,G=0 and this is called the first BrillouinzoneBandstructureDOSP hononsFirst Brillouin Zone (2D)The region of reciprocal space nearer to the origin than anyother allowed wavevector is called the1st Brillouin BrillouinzoneBandstructureDOSP hononsFirst Brillouin Zone (2D)The region of reciprocal space nearer to the origin than anyother allowed wavevector is called the1st Brillouin BrillouinzoneBandstructureDOSP hononsE versuskHow does the energy of States vary across the Brillouinzone?

2 Let s consider one particular wavefunction: (r)= (r)We ll look at two different limits electrons with highpotential energy, and electrons with high kinetic BrillouinzoneBandstructureDOSP hononsVery localised electronsIf an electron is trapped in a very strong potential, then wecan neglect the kinetic energy and write: H= VThe energy of our wavefunction is thenE(k)= ?(r)V(r) (r)d3r= V(r)| (r)|2d3r= V(r)|u(r)|2d3rIt doesn t depend onkat all! We may as well do allcalculations atk= BrillouinzoneBandstructureDOSP hononsFree ElectronsFor an electron moving freely in space there is no potential,so the Hamiltonian is just the kinetic energy operator: H= ~22m 2 The eigenstates of the Hamiltonian are just plane-waves except for one wavefunction is now (r)=cGei(k+G).r 2 (r)= (k+G)2 (r)RecapThe BrillouinzoneBandstructureDOSP hononsFree ElectronsE(k)= ~22m ?

3 (r) 2 (r)d3r=~22m(k+G)2 ?(r) (r)d3r=~22m(k+G)2 SoE(k)is quadratic ink, with the lowest energy stateG= BrillouinzoneBandstructureDOSP hononsFree ElectronsRecapThe BrillouinzoneBandstructureDOSP hononsFree ElectronsEach state has an energy that changes withk they formenergybandsin reciprocal that the energies are periodic in reciprocal-space there are parabolae centred on each of the reciprocal BrillouinzoneBandstructureDOSP hononsFree ElectronsRecapThe BrillouinzoneBandstructureDOSP hononsFree ElectronsAll of the information we need is actually in the first Brillouinzone, so it is conventional to concentrate on BrillouinzoneBandstructureDOSP hononsFree ElectronsRecapThe BrillouinzoneBandstructureDOSP honons3 DIn 3D things get complicated. In general the reciprocallattice vectors do not form a simple cubic lattice, and theBrillouin zone can have all kinds of BrillouinzoneBandstructureDOSP hononsBand structureThe way the energies of all of the States changes withkiscalled theband a 3D vector, it is common just to plot theenergies along special high-symmetry directions.

4 Theenergies along these lines represent either maximum orminimum energies for the bands across the whole , in real materials electrons are neither completelylocalised nor completely free, but you can still see thosecharacteristics in genuine band BrillouinzoneBandstructureDOSP hononsBand structureRecapThe BrillouinzoneBandstructureDOSP hononsTransitionsBecause the lowestNestates are occupied by electrons, at0K there is an energy below which all States are occupied,and above which all States are empty; this is theFermienergy. Many band-structures are shifted so that the Fermienergy is at zero, but if not the Fermi energy will usually bemarked semi-conductors and insulators there is a region ofenergy just above the Fermi energy which has no bands in it this is called theband BrillouinzoneBandstructureDOSP hononsBand structureRecapThe BrillouinzoneBandstructureDOSP hononsDensities of StatesThe band structure is a good way to visualise thewavevector-dependence of the energy States , the band-gap,and the possible electronic actual transition probability depends on how manystates are available in both the initial and final energies.

5 Theband structure is not a reliable guide here, since it only tellsyou about the bands along high symmetry BrillouinzoneBandstructureDOSP hononsDensities of StatesWhat we need is the fulldensity of statesacross the wholeBrillouin zone, not just the special directions. We have tosample the Brillouin zone evenly, just as we do for thecalculation of the ground BrillouinzoneBandstructureDOSP hononsDensities of StatesRecapThe BrillouinzoneBandstructureDOSP hononsDensities of StatesRecapThe BrillouinzoneBandstructureDOSP hononsDensities of StatesRecapThe BrillouinzoneBandstructureDOSP hononsDensities of StatesOften the crystal will have extra symmetries which reducethe number ofk-point we have to sample we ve applied all of the relevant symmetries to reducethek-points required, we are left with theirreducible BrillouinzoneBandstructureDOSP hononsDensities of StatesRecapThe BrillouinzoneBandstructureDOSP hononsComputing band structures and DOSC omputing a band structure or a DOS is straightforward.

6 Compute the ground state Density with a goodk-pointsamplingFix the Density , and find the States at the bandstructure/DOSk-pointsBecause the Density is fixed for the band structure/DOScalculation itself, it can be quite a lot quicker than the groundstate calculation even though it may have BrillouinzoneBandstructureDOSP hononsPhononsWhen a sound wave travels through a crystal, it creates aperiodic distortion to the BrillouinzoneBandstructureDOSP hononsPhononsWhen a sound wave travels through a crystal, it creates aperiodic distortion to the BrillouinzoneBandstructureDOSP hononsPhononsThe periodic distortion also has an associated wavevector,which we usually callq. This distortion is of the atomicpositions so is real, rather than complex, and we can write itas:dq(r)=aqcos( )We can plot aphononband structure, though we usuallyplot the frequency againstqrather thanE.

7 This showsthe frequency of different lattice vibrations, from thelong-wavelength acoustic modes to the shorter optical BrillouinzoneBandstructureDOSP hononsPhononsWhen a sound wave travels through a crystal, it creates aperiodic distortion to the atoms.