The Nuts and Bolts of First-Principles Simulation

Durham, 6th-13th December 2001

12: Band Structure Calculations

CASTEP Developers’ Group

with support from the ESF ψk Network

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Outline

r Introduction to band structures

r Calculating band structures using Castep

r Examples results

r Some applications

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Band Structuresr Bloch’s theorem introduces a wavevector k.

r It can always be confined to the 1st BZ (any k outside the 1st BZ can be mapped back into it).

r The band index appears in Bloch’s theorem because for each kthere are many solutions.

knknkn EH ,,. Ψ=Ψ

This leads to a description of the energy levels of electrons in a

periodic potential in terms of a family of continuous functions En,k.

This is the band structure of the solid.

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Simple Methods

Free electrons give

parabolic bands

Using a basic basis set for the

wavefunctions improves results slightly

In Castep we use and accurate plane wave basis set.

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Calculating Band Structuresr After calculating the electronic structure of a material we have the

charge density n(r) , the potential V(r) and hence the complete Hamiltonian:

)]([)]([)]([2

1 2 rnVrnVrnVHxceiee +++∇−= −−

n(r) V(r)⇒

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Algorithmr Calculate self-consistent density and potential.

r For a chosen k-point in the band structure calculate it’s eigenvalues and eigenvectors (wavefunctions) as follows:

r Start from random wavefunction.

r For the first band, calculate the search direction using self-consistent potential.

r Assuming parabolic energy curve, step wavefunction coefficients to bottom in given direction.

r Repeat previous two steps (using orthogonal search direction) until no change occurs.

r For subsequent bands at this k-point, perform similar process, ensuring the bands remain orthogonal to all lower bands.

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Silicon Band Structure

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Conventional Cell?r Although the full set of levels can be described with restricted k,

it is often useful to allow k to range over more of k-space.

r The set of wavefunctions and energy levels for 2 values of kdiffering by a reciprocal lattice vector K are identical.

knKkn

knKkn

EE

rr

,,

,, )()(

=

Ψ=Ψ

+

+

By not using a primitive cell, the band structure contains redundant

information.

For each of the k=k+K we can map them back into the 1st Brillouin zone of the primitive cell.

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Non-primitive Cell

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Density of Statesr Many electronic properties depend on the electronic structure

throughout the whole Brillouin zone.

r A band structure usually shows the electronic states along lines of high symmetry.

r Instead, we need to sample the whole Brillouin zone in a method similar to the SCF.

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Example of DOS Calculation

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Magnetic Materials

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Applications – Band Offsetsr Heterojunctions are formed when two different types of

semiconductor are joined together.

r Heterostructures are used extensively in the electronic (e.g. transistors) and optoelectronic (e.g. LED’s) industry.

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Calculating Band Offsets

r Knowledge of the band structures of the two individual materialsis not sufficient to determine the band offset.

r Band energies are determined with respect to the average potential in the solid.

r It is also necessary to perform a supercell calculation to determine how the potentials are lines up with respect to each other.

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The Average Potentialr The potential can be averaged across the x-y planes

(junction normal to z).

r This is the microscopic average potential.

r The macroscopic average potential is calculated over the period over the microscopic potential.

r The band structures are measured relative to the macroscopic average.

∫ ∫+

−

+

−

=2/

2/

2/''

2/''

''')''('

1)(

λ

λ

λ

λλλ

z

z

z

z

tottot dzdzzVzV

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Band Line-up Using Potential

( ) ( )vbmRtotRvbmLtotLv EVEVE +−+=∆

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An Example: FeSi2 on Si

Offset is 0.4eV – useful for LED’s and near-IR detectors

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Applications - Surfacesr We can also use the supercell method to calculate the band

structures of surfaces.

r The wavefunctions at surfaces decay exponentially into the vacuum region.

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Details of GaAs Surface

Surface supercell

There is reconstruction of the surface –

this is calculated first.

A band structure of the supercell is then used for the surface band structure.

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The GaAs Band Structurer K-points are chosen using the

surface Brillouin zone.

r Many regions have a continuous energy spectrum, while gaps still exist.

r The details of the surface band structure depend on the details of the surface reconstruction.

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Applications - Orbitals

r In addition to the energy eigenvalues for a given material, a band structure calculation also gives the eigenvectors (wavefunctions) for any point in the Brillouin zone.

r It should be noted that no proof exists which confirms that the DFT single particle wavefunctions generated here correspond to the many particle wavefunction.

r However, the orbitals generated can lead to useful physics and give further insight in the the nature of the bonding in materials.

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Example of DFT Orbitals

r Instead of summing up the squares of all the wavefunctions from

each electron, we can look at each one individually.

r Each one will be the charge density for a Kohn-Sham orbital.

r This gives a chemistry point of view!

r We can examine the electronic structure electron by electron.

r Note: Kohn-Sham orbitals/one electron orbitals: meaning is not necessarily well defined!

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Example of Orbitals

Example shown is for LaMnO3 – a magnetic material.