Methods for Calculating Band Structure

SELF-CONSISTENT POTENTIAL Solution of the single-particle Schrödinger equation in the independent electron approximation can lead

to a more realistic set of single-particle levels if a clever choice of potential is made, taking into

consideration not only the periodic ion core contributions, but also the periodic effects due to

interactions with all other electrons. The latter, however, requires solution of the single-particle

Schrödinger equation in order to determine the electronic density and potential. In turn, the

Schrödinger equation can only be solved if the potential is known. This calls for the search for a self-

consistent potential that reproduces itself.

In order to find a self-consistent potential, we start with a trial potential, U0, that we believe it

represents the interactions. We use this potential to solve the single-particle Schrödinger equation for

the occupied electronic levels, and use these solutions to recompute the potential U1. If U1 is sufficiently

close to U0 (to within an acceptable deviation), then self-consistency is achieved and the actual potential

is taken to be U = U1. If this is not the case, we repeat the above procedure with U1, and determine U2,

and so on, until finally we hopefully arrive at a self-consistent potential. The accuracy of the band

structure we finally obtain is limited not only by the accuracy of the solutions to the Schrödinger

equation, but more so by the accuracy of finding the correct potential that describes the electron

interactions. Fig. 1 shows the band structures of vanadium for two different potentials: assuming

3d34s2 configuration, and assuming 3d44s1 configuration.

Γ X

ℰ

Fig. 1: band structure of vanadium

V (3d34s2)

Γ X

V (3d44s1)

ℰ

VALENCE-BAND WAVE FUNCTIONS The low-lying core levels are determined by the tight binding method. The higher-lying valence bands

are derived numerically using various techniques. The nearly-free electron model cannot be applied to

these bands in real solids. This conclusion is understood on the following basis:

1. The potential, especially within the core, is not small.

( )

( )

To evaluate UK we introduce the factor e-αr to the potential, and in the final result we set α = 0.

(

)(

)

( ) (

) ( )

This indicates that the potential is several electron volts, comparable to the kinetic energy of the

free electrons, for a very large number of reciprocal lattice vectors. The assumption that the

potential is small compared to the kinetic energy on which the nearly free electron model is based is

not valid.

2. The core levels are appreciable in the core region, where they have the characteristic

oscillations of atomic levels. The valence wave functions with higher energies are even more

oscillatory than core wave functions. Thus the expansion of the wave functions involves a large

number of components with high wave vectors (large number of K). The nearly free-electron

model assuming a small number of components is thus not feasible.

THE CELLULAR METHOD The Wigner-Seitz cellular method is based on solution of the Schrödinger equation within a single

primitive cell, and then using Bloch relation to determine the wave function in any other cell.

( ) ( ) ( )

The wave function and its derivative must be continuous as the cell boundary is crossed. Thus from Fig.

2 we obtain:

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

Fig. 2: normals are opposite

��(𝒓)

��(𝒓 𝑹)

C0

These boundary conditions introduce the wave vector k into the cellular solution, and yield a discrete

set of energies as a result of confinement to the primitive cell. These are the band energies.

The problem then reduces to solving the Schrödinger equation within the cell C0 subject to the above

boundary conditions. The primitive cell is a Wigner-Seitz cell centered on the lattice point R = 0.

The first approximation of the method is the replacement of the periodic crystal potential by a

spherically symmetric potential V(r) within the cell (taken as that of a single ion at the origin). The

solution of the Schrödinger equation is then of the form:

( ) ( ) ( ) ( )

Where the radial solution is obtained from the radial equation:

(ℰ ( )

( )

) ( )

Consequently, the general solution to Schrödinger equation is:

( ) ∑ ( ) ( )

( )

These solutions must satisfy the boundary conditions. The second approximation is made at this point.

We take in the above expansion (8) as many terms as computationally feasible. The finiteness of the

number of coefficients makes it sufficient to fit the boundary conditions at a number of points on the

surface, leading to a finite number of boundary conditions (chosen to be as many as the number of

unknown coefficients). These lead to a set of simultaneous k-dependent equations for the coefficients

Alm. The energy values for which the determinant of these equations vanishes then determines the

energy bands. The constant energy surfaces can also be mapped by fixing the energy and searching for

the wave vectors for which the determinant vanishes.

Lowest energy level in the valence band of metallic sodium Wigner and Seitz applied the cellular method for this simple problem with k = 0. They assumed spherical

primitive cell of radius r0 (= 0.492 a) having the same volume of the W-S cell. Thus they could demand

that the solution is spherically symmetric (l = 0 = m). Under these conditions we have:

( ) (

)

( )

The solution is shown in Fig. (3), together with the atomic wave function (dashed line).

The obtained wave function is very similar to the atomic one in the core region, but appreciably larger in

the interstitial regions.

Difficulties 1. Numerical solutions satisfying boundary conditions over a fairly complex polyhedral surface.

2. The approximation of the potential within the cell by an ionic potential. This has a discontinuous

derivative at the boundaries, whereas the actual potential is flat in such regions.

To overcome these difficulties, the muffin-tin potential is used, which represents the potential of an

isolated atom within a sphere of radius r0, and is zero elsewhere (Fig. 4 (a)).

( ) ∑ (| |)

| | ( )

Fig. 4 (b) shows the cellular potential for comparison.

Two methods are adopted for band calculation in a muffin-tin potential: the augmented plane wave

(APW) method, and the Korringa-Kohn-Rostoker (KKR) method.

4 2

r/a0

ψ

Fig. 3: ground state of Na

THE APW METHOD The Slater approach (1937) represents the crystal wave function as a superposition of APW φk,ε (plane

waves in the interstitial region, with a rapid oscillatory behavior in the core region). The APWs have the

following characteristics:

1. In the interstitial region

ℰ ( )

Fig. 4 (a): muffin-tin Fig. 4 (b): cellular

There is no constraint between the energy and wave number (Fig. 5). Thus, a given APW does not

satisfy Schrödinger equation for a given energy in the interstitial region (where (

) ).

2. The APW is continuous across the boundary.

3. In the core region, the APW satisfies the atomic Schrödinger equation.

In general, the APW will have a discontinuous derivative at the boundary, leading to delta function

singularities. And since the APW satisfies Bloch condition, the crystal wave function can be

expanded in terms of APWs with wave vectors differing by reciprocal lattice vectors.

∑ ( )

( )

The energy of the APW is taken to be that of the Bloch level, and thus ψk satisfies the crystal

Schrödinger equation. We work with the variational technique (since the wave function is no twice

integrable), which leads to a set of simultaneous equations in the coefficients upon applying the

equations:

ℰ( ) ∫(

| | | | )

∫| | ( )

ℰ

( )

Fig. 5: APW’s

For exact calculation, an infinite number of terms in the expansion (12) is needed. However, for

computational convenience, a finite number of terms is considered, and thus truncation in K is

required, and the wave function is approximated in the interstitial region. Fig. 6 shows the band

structure for cupper and iron calculated by APW method along specified symmetry directions.

THE KKR METHOD: GREEN’S FUNCTION METHOD Based on the work of Korringa (1947) and Kohn and Rostoker (1954) an alternative solution to the

muffin-tin problem is devised. Schrödinger equation reads:

( ) ( ) [

ℰ] ( ) ( )

Defining Green’s function which satisfies:

[

ℰ] ℰ( ) ( ) ( )

We obtain an integral form of the Schrödinger equation:

( ) ∫ ℰ( )( ) ( ) ( ) ( )

Γ X

ℰ

ℰ𝐹

Fig. 6: APW calculations

fcc copper

Γ H

bcc iron ℰ

ℰ𝐹

The functional form of Green’s function is found from (15) (see problem 17.3 A & M):

ℰ( ) | |

| | ( )

√ ℰ

ℰ

√ ( ℰ)

ℰ ( )

Inserting the muffin-tin potential (10) in (17) we obtain:

( ) ∑∫ ℰ( )( ) ( ) ( )

( )

With the use of Bloch condition, this last equation can be rewritten as:

( ) ∫ ℰ( )( ) ( ) ( ) ( )

with:

ℰ( )( ) ∑ ℰ( )

( )

Equation (21) is the integral equation for a Bloch level in a periodic muffin-tin potential, where the

integral is over the core region.

All the dependence on wave vector and crystal structure is contained in the function given by (22).

This can be calculated for a given structure, leading to the dispersion relation defining the band

structure of the solid. Since the wave function is continuous at the boundary surface of the muffin-

tin potential, the wave function must have the atomic form (equations (6) to (8)), and tus the wave

function must satisfy the integral equation:

∫ [ ℰ( )( ) ( )

|

( )

ℰ( )( )

|

] ( )

Up to this point, the treatment is exact, and requires an infinite number of terms in the expansion of the

wave function. The first approximation is to truncate the expansion by keeping a finite number (N) of

terms. We then substitute this truncated expansion in (23), multiply by a spherical harmonic at a time,

and integrate. This gives a set of N simultaneous equations in the coefficients Alm. The dispersion

relation is then obtained by setting the appropriate NXN determinant equal to zero.

Notice that, in contrast to the APW method, this approach approximates the atomic wave function in

the core region. When the same muffin-tin potential is adopted, the two methods give results in

excellent agreement. However, the KKR method requires less terms than the APW method, which saves

some computer time deemed necessary for lengthy calculations.

The KKR method applied to aluminum (three electrons outside a closed-shell neon core) gives a result in

substantial agreement with the free- electron model (dotted lines, Fig. 7). The Fermi energy is about

11.4 eV, which is also in excellent agreement with the free-electron Fermi energy of 11.6 eV. The band

gap is about 1.2 eV. Thus, metals with few s and p electrons outside a closed shell noble metal

configuration have almost a free electron behavior (Na, Mg, Al, K, Ca are good examples).

ORTHOGONALIZED PLANE-WAVE METHOD The approximation of a valence wave function by a few plane waves (as in the nearly free electron

model) fails to reproduce the rapid oscillations in the core region. Herring (1940) recognized that the

problem can be taken care of by working with plane waves orthogonalized to the core levels. The OPW

is thus defined as:

( ) ∑ ( )

( )

⟨ | ⟩ ( )

X k

ℰ

Fig. 7: Al Bands (KKR)

ℰ𝐹

Γ

Thus, the OPW has the following characteristics.

1. It has the required oscillatory behavior in the core region.

2. Due to core-level localization, the second term in (24) is rather small in the interstitial region,

leaving the wave function there very nearly a single plane wave eik.r.

3. It satisfies the Bloch condition, since the plane wave and the core levels do.

The full solution to the crystal Schrödinger equation is a linear combination of OPW’s all with the same

energy, as in the APW method.

( ) ∑ ( )

( )

The dispersion relation and band structure is then obtained by substituting this expression in the

variational principle (13) and using condition (14) on the derivatives of the calculated energy with

respect to all coefficients.

Although the plane-wave matrix elements of U are large, the OPW matrix elements are small. This leads

to much faster convergence than when expanding in terms of plane waves as in the nearly free electron

model. The OPW approach can be used by carrying out a first principle OPW calculation, starting with an

atomic potential, calculating its OPW matrix elements, and then working with large enough

determinants to insure convergence. On the other hand, one can use nearly free-electron band

calculation in which the Fourier components are used as adjustable parameters to fit empirical data or

bands calculated by another method. For example, the KKR band structure of aluminum can be

reproduced by using four plane waves (one for each level) and two adjustable parameters: U200 and U111

(see problem 9.3). In this case, the Fourier components are OPW rather than plane-wave matrix

elements. The bands along ΓW can be shown to have the general form in Fig. 8. I have also calculated

the free electron bands for comparison.

W

ℰ

Fig. 8: Al Bands (OPW)

Γ W

ℰ

Γ

OPW Free

THE PSEUDOPOTENTIAL This method (Antoncik, 1959) refines the OPW, and provides a partial explanation for the success of the

nearly free electron model. Notice that due to periodicity of the Bloch wave function in the reciprocal

space we have from (24):

( ) ∑

( )

( ( ) ) ( )

The exact valence wave function (26) can be written in terms of the pseudo wave function defined by:

∑

( )

( )

The exact valence wave function is thus:

∑(

)

( )

The Schrödinger equation then reads:

ℰ

ℰ

( )

Thus we obtain:

∑(

)

ℰ

( ∑(

)

) ( )

This last equation can be rewritten as:

( ) ℰ

( )

The operator V R is defined by:

∑(ℰ ℰ

)(

)

( )

Since the pseudo wave function can be approximated in terms of linear combination of a few plane

waves, the nearly free electron model (Ch. 8 and 9) is expected to be applicable for the evaluation of

the eigenstates of (33), treating the pseudopotential (35) as a perturbation.

( )

This last statement, however, needs justification. Since the matrix elements of the periodic crystal

potential U are negative due to attractive core potential, and those of V R are positive due to the

fact that the valence levels are above the core levels, one might hope that partial cancellation leaves

the matrix elements of the pseudopotential small enough to be treated as perturbation. Then the

central equation of Ch. 9 is applied, namely:

(ℰ ℰ

) ∑

( )

Difficulties 1. The pseudopotential is nonlocal

2. The pseudopotential depends on the valence energy of interest.