Calculation of band structure using group theory

Sudeep Kumar Ghosh, Department of Physics, Indian Institute of Science, Bangalore.

Plan of the talk

Brief overview of the representation theory and the

character system of a group.

Brief overview of the Tight Binding Model and the Nearly

Free Electron Model of band structure calculation.

Discussion of the tight binding model and the formation

of the secular determinant.

Application of the Group Theory in simplifying the

secular determinant for Beta Brass and explicit calculation.

Calculation of the band structure of Tellurium using the

Nearly Free Electron Approximation using the symmetry

arguments from Group Theory.

Concluding remarks.

Brief overview of the representation theory and the character system of a group

Representation

Reducible Irreducible

Characters

Uniquely describes Within a representation

a representation uniquely describes an

Equivalence Class

Useful theorems of Group Theory 1) Relation b/w dimension of irreps and order of a group :

2) Relation b/w functions associated with irreps and atomic orbitals:

Where,

is the function associated the ith irrep,

are the atomic orbitals,

is the character associated with the operator R in the ith irrep.

Basic idea of the Tight Binding model(TBM) and the Nearly Free Electron model(NFE)

TBM:

• Atoms in lattice are far apart electrons in an atom are only affected by nearest or next nearest neighbors.

• Most useful in describing energy bands of transition metals(having partially filled d-shells) and electronic structure of insulators.

NFE:

• Electrons are nearly free and perturbed only by a weak periodic potential of the ion-cores.

• Applicable to metals in groups of the periodic table(i.e. having atomic structure consisting of ‘s’ and ‘p’ electrons outside of a closed shell noble gas configuration.)

Generic way of TBM calculation • Aim of energy band structure calculation determine

E(k).One needs to solve the one electron Schroedinger eqn:

where, is the periodic crystalline potential, are the Bloch wave functions with energy E & are the primitive translation vectors.

• Bloch Sum: Considering a simple crystal with one atom per unit cell the Bloch Sum of wave vector k is:

where,

is the AO of quantum no. ‘i’ in the reference unit cell; is the same orbital of the atom in the unit cell and

N is the no. of unit cells in the crystal.

Continued… • LCAO: Try the solution of the Schroedinger equation to be:

where, are the coefficients of expansion.

• Secular Determinant:

where,

Now considering V(r)= + we can write:

where, the AO is Eigen function of the H with energy .

Application of Group Theory to simplify the secular determinant for Beta Brass

• Unit cell has one Zinc atom and one Copper atom and SC lattice.

Table:-1

Continued…

Table:-2

• Ref:E.Wigner,Phys.Rev.,vol.50,pp.58-67,July 1,1936

Motivation to the Character Table • Matrix representing ‘E’ has the form and hence its

order will equal the dimensionality of that representation. • The cubic system has 10 classes and 48 operations; hence

using Theorem-2 the characters for ‘E’ are

This is the only combination of 10 integers whose squares add up to 48. • Interesting feature of character tables involving the inversion

operation is that if the character table is divided into 4 blocks then 3 are identical and the 4th one is the negative of the others.

This leads to considerable simplification in the computation of the characters.

Explanation to the associated atomic orbitals in some of these representation • The atomic orbitals can be expressed as:

• Conveniently expressed as:

• The five d functions can also be expressed as l.c of (x,y,z).

• Cubic symmetry operations

on a point with coordinates

(x,y,z) will lead to the table

as shown.

Table:-3

Continued… • Rerep. of E and must have chars 3 & -1 respectively. Since we can

write for the matrix have the following form:

• Similarly, the full set of red. chars is: 3,-1,1,-1,0,-3,1,-1,1,0 which is identical to irrep.

• Since the orbitals are prop. to x,y,z respec. we say they belong to and are triply degenerate.

• Using the same logic we can see s,( ) and ( ) belong to

and representations respectively.

• N.B: Due to crystal field splitting the 5 degenerate d-states have been split into a 2-fold and a 3-fold state.

Symmetry in the B.Z • Along the axis( ) ;there only 4 symmetry operations in the

group and the associated coord. transformations are:

Table:-4

• The character table is determined

in the same way as before to be:

Table:-5

The symmetry argument in B.Z for Beta-Brass(B.B) • TBM calculation for B.B bases on the use of d functions since Cu

and Zn both have ten 3d valence electrons. Neglecting Zn-Zn n.n int. ,since the dist. b/w these 2 atoms in the alloy is much larger than that b/w 2 Cu atoms or b/w Cu-Zn neighbours,then the only nonvanishing field integrals are:

• Ref: J.C.Slater,Phys. Rev.,vol.94,pp.1498-1524,June 15,1954.

The 1010 Secular Determinant

Rearranging Rows and Columns

Further simplification by group theory • Table:-4 Table:-6

• From Table:-6 applying

Theorem-2 for the rep.

of Table:-5 we have:

Table:-6

So, xy and belong to .Further, functions belonging to diff. irrep do not mix which is seen from the following vanishing integral:

Reduced Secular Determinant

Determination of degeneracies of the energy bands calculated by NFE for Te • Cubic crystalline field splits the 5 fold d levels into a 3 fold and a 2

fold level.

• Expect removal of some degeneracies when a crystal field applied to the energies calculated using FE.

• Can be understood using group theory.

For example, consider Tellurium(Te):

Consideration of the symmetries of the Te B.Z

• At the top of the B.Z the point A(0,0,π/c) the full symmetry group is:

1) E , the Identity;

2) , a rotation about the z-axis , followed by a translation c/3 along the axis.

3) A

4) B ,followed by a translation c/3.

5) ,followed by no translation.

6) ,followed by a translation c/3.

7) T , a displacement of c along the z-axis.

• Note:

Cayley table and character table

Table:-7

Table:-8

Symmetry along the Δ-axis of Te

• Both rotational and translational symmetries are present.

• For the rotational part the symmetry operations are , and

rotations. They form an abelian group and the characters associated with them are the cube roots of unity, which are where

. .

• The translational characters have the form .

• Hence for the direct product representation the character table is:

Table:-9

Concept of compatibility • Basic Idea: The symmetry of the wave functions at ‘Γ’ and at

some adjacent point on the Δ-axis must bear some relation to one another, since it is necessary that the functions be continuous in the B.Z.

• The symmetry along the axis should be similar to, but of lower order than that at the center.

• The compatibility relations are somewhat complicated to obtain but we can verify for example at ‘Γ’ the quantity in table-9 is 1 and then using it is seen that:

The energy bands of Te • Note:-

1) The crystal field removes the degeneracy at the intersection of the and (200) curve for the point A. The higher energy curves for NFE are sketched based on the curves for FE and the use of the compatibility relations.

2) In this case, since and ,so that the curves joining a and can only belong to .

Concluding Remarks

We saw group theory does not give quantitative answers but it merely assisted in the band structure calculation in the following 2 ways:

1) To simplify the evaluation of secular determinants,

2) To determine the degeneracies of energy bands.

However, the group theoretical framework can be extended(by taking into account dynamical symmetries) to give quantitative answers to dispersion relations, transfer matrices and energy bands.

Ref: D.Kusnezov, arXiv:cond-mat/9907202v1

References

oAllen Nussbaum, Solid State Physics,Vol.18,1966,p.165-272; Science Direct.

oGroup Theory in Physics,Vol.1 by J.F.Cornwell.

Thanks to A. Sinha for useful discussions.