Marder M.P. Condensed Matter Physics

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References

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10.

Realistic Calculations in Solids

10.1 Introduction

The attempt to construct approximations accurate enough so that properties of

solids could be deduced from Schrödinger's equation was initiated seriously by

Wigner and Seitz (1933), who computed properties of metallic sodium. They re-

garded the attempt to proceed much further along these lines with a degree of

skepticism, and two decades after their original work they commented that

If one had a great calculating machine, one might apply it to the problem

of solving the Schrödinger equation for each metal and obtain thereby

the interesting physical quantities, such as the cohesive energy, the lat-

tice constant, and similar parameters. It is not clear, however, that a great

deal would be gained by this. Presumably the results would agree with

the experimentally determined quantities and nothing vastly new would

be learned from the calculation. It would be preferable instead to have a

vivid picture of the behavior of the wave functions, a simple description

of the essence of the factors which determine cohesion and an under-

standing of the origins of variation in properties from metal to metal.

—Wigner and Seitz (1955), p. 97

Now that the great calculating machines exist, not all agree that Wigner and

Seitz's advice should be adopted. Slater spoke with slight disdain of those "trying

to get valid results relating to energy bands from simplified models, rather than

through the direct types of calculations which one can make with the methods

now in use" [Slater (1975), p. 191]. However, even if the resolution to proceed

with large calculations is taken, it is not at all easy to obtain numerical results in

agreement with experiment. Many approximations must always be employed, and

decades have been devoted to finding the schemes that are most successful and

effective. This collection of approximations and numerical techniques is referred

to as band structure calculation. It has achieved many successes and has reached

the stage where many properties of

wide variety of solids can be computed almost

automatically, with knowledge of nothing but atomic numbers. The methods can

also experience qualitative failures, which only experience and comparison with

experiment reveal.

265

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

266

Chapter 10. Realistic Calculations in Solids

10.2 Numerical Methods

10.2.1 Pseudopotentials and Orthogonalized Planes Waves (OPW)

Band structure calculations all operate within the single-electron framework, mean-

ing that they assume it possible to study the behavior of solids by choosing a poten-

tial and studying the behavior of a single electron in that potential. Almost always

the potential will be taken to depend self-consistently upon solutions for electron

states in the potential, but in the end one solves a problem with only one electron in

it. The reason is simply a matter of

practicality.

The size of the calculation goes up

as the power of the number of particles that are treated in it simultaneously. On a

lattice of M sites with N particles, the difference between solving the true interact-

ing Schrödinger equation in all its glory and a reduction to one-particle problems is

the difference between working with M

variables as opposed to M x N. The cal-

culations of the previous chapter have shown how to obtain one-particle potentials

incorporating information about interactions of electrons with themselves.

Yet even after reduction to a periodic one-electron problem, the task one faces

in solving Schrödinger's equation is daunting. Take the case of gold. A variety of

experimental probes, such as the low-temperature specific heat in Table 6.2, reveal

that gold greatly resembles a nearly empty box containing one electron per atom.

Yet gold's atomic number is 79. In some fashion the first 78 electrons per atom

simply crowd about the nucleus, screening it from the outside. How then should

they be treated? What potential do they leave behind? The answer is provided by

pseudopotentials, due to Phillips and Kleinman (1959).

In fact, pseudopotentials serve two different purposes simultaneously. On the

one hand they provide crucial conceptual justification for the nearly free electron

model of solids, showing explicitly how the problem of finding wave functions in

the presence of ionic Coulomb potentials can be mapped onto equivalent problems

where the potentials are much weaker. Simultaneously, they provide a calculational

tool that substantially increases the range of problems in solids that can be brought

within computational reach.

The best illustration of the idea is provided by orthogonalized plane waves,

(OPW) although they have not found much practical use. Suppose one has a pe-

riodic solid in which the electrons can safely be divided into two groups, the core

states and the conduction states. The core states are localized around particular

atomic sites. Denote core states by \ψ

) and plane waves (exp[/£·?]) by |it). An

orthogonalized plane wave indexed by k is defined to be

_ _,

_ The subscript "ps" indicates that this is a "pseudo"

|&p

) = \k) — 22 \Ψΰ) (lpc\k), k state. The following derivation works for any ex- (10.1)

c tended states \k).

where the sum is over all occupied core levels

\tp

Look at the effect on these

states of the Coulomb potential for an atom of valence Z,U = Z/R. It is

Û\k

)

ΰ\%)-Σ

ΰ(ψ

\%)\ψ

€

(10.2)

Numerical Methods

267

If \ip

) were a complete set of states, then

Û\k

would vanish. As it is, the hope is

that all matrix elements of Û \k

pii

) will be small. Examine the Schrödinger equation

in a basis of

pii

states. It is

ÇK-£)ft,

> = (—+*/-ε)|*ρ,)

where

V2m

^2m

(—+ΰ-ή\\ϊ)-Σ(^\ΐ)\Α)

|*) +

(tf-£)|*)-£(£

-£)|VO<rf>

-E)\k)

= ^k

-E)\k),

is the eigenvalue of

\ip

Note that if \ip

)

( £

— £ ) 11p

) (tpc I.

were a

complete set of states, t/

would flat-

ten nut tn thp r*r*nct!inf y

ten out to the constant δ.

(10.3)

(10.4)

(10.5)

(10.6)

(10.7)

The summary of Eqs. (10.3) through (10.7) is

In particular, if one finds an eigenvector

) = J^

\k)

(!JÎ — £) jfcps) = ÇJips — £,)\k). causing the right hand side to vanish, then |Φ) = y^Cj:|t

)(10.8)

is an eigenvector for the left hand side.

Thus there is a complete correspondence between the original Schrödinger equa-

tion and a new one with a weak pseudopotential (7

whose solutions may be close

to plane waves, as in Section 8.2. There is some price to be paid for this simplifi-

cation. The pseudopotential Ù

is nonlocal; that is, to compute its action upon a

general state ψ(7), one has to perform integrals. In addition, t/

depends upon the

unknown energy eigenvalue £, and the eigenvalue problem one needs to solve is

more involved than for the usual quantum-mechanical problems.

Empirical and First-Principles Pseudopotentials. Development of the pseudopo-

tential beyond this first suggestive calculation has proceeded along two routes. On

the one hand, it could be argued that the most important lesson to take away is

that the apparent existence of weak potentials is explained in principle, so one may

as well experiment with simple weak potentials, chosen so as to match important

features of experiment. One example is the empty-core potential, due to Ashcroft

(1966),

depicted in Figure 10.1. The three free parameters of this potential—its

Figure 10.1. The Ashcroft

empty core pseudopotential

is zero up to a critical ra-

dius R

, and it equals a

screened Coulomb potential

—UQ

exp[—r/i/]/r thereafter.

Distance r

268 Chapter 10. Realistic Calculations in Solids

0.0

•(A)

2.0

Figure 10.2. Real and pseudo wave functions for the 5s, 5p, and Ad levels of silver. The

5p level is not much occupied in the ground state of silver, but it can be included in the

pseudopotential nevertheless.

magnitude i/o, the cutoff R

, and the exponential decay length d—can be adjusted

to fit measurements taken from optical or magnetic experiments, as shown in Sec-

tion 23.4. Potentials of this sort are good enough to permit a quantitative compar-

ison with experiment for some of the simpler metals, such as the alkali metals or

aluminum. Their use is reviewed by Heine (1970), and the empty core potential

will be used for the cohesion of metals in Section 11.4.

On the other hand, it is possible to construct first-principles pseudopotentials to

encode information about atomic wave functions in a form particularly convenient

for transportation into solids. The name given these pseudopotentials promises too

much, because numerous approximations are needed to produce them, but the basic

idea behind them is worth appreciating. It proceeds in three steps.

Choose an atom, and write down the Kohn-Sham equations for it. Employ

the approximation that the electron density n(r) = Σ

1^/(7)|

be spherically

symmetric about the nucleus, and impose this requirement in replacing n

by its spherical average if necessary. Once the equations Eq. (9.104) have

been forced to be spherically symmetric, all of their solutions are of the form

i(r)Yi

, where Y[

is a spherical harmonic and 3i„/ is a radial wave function.

The equation for the radial functions is

2m rdr

1(1

+ 1)

]Xn!

n{r')

Z δΕ

Χ£

r δη

■•ni

Ä„/(r)=0.

(10.9)

Numerical Methods 269

Solve these equations for all the electrons belonging to the atom. Denote the

energies of these states by £„/.

Consider all the outermost s, p, d, and / states lying in partially filled shells.

These are the states that contribute to bonding between atoms in molecules

and solids, and they will be singled out for special treatment. Drawings of

these radial wave functions appropriate to silver appear in Figure 10.2. To

form pseudo wave functions 3^ deriving from solutions of

Eq.

(10.9), simply

take each radial function, pick a point beyond its rightmost node, and draw a

smooth curve in to the origin. This smooth curve has few constraints to satisfy,

and it is easy to construct. It should be without nodes, except at the origin

where it should vanish as r

. It should join on to the original radial function

with at least two derivatives continuous. And finally, the wave functions built

from these new radial functions should remain correctly normalized. Any

function with these properties produces an acceptable pseudo wave function,

as illustrated in Figure 10.2.

Finally, replace the original Coulomb potential U with a pseudopotential Û

built so that solving the Kohn-Sham equations gives the functions Ä

rather

than

"R.

This task can be accomplished directly from Eq. (10.9). Just write out

the equation for the radial wave functions as

£//»

1 d

r^ /(/+!)

2mrX5 dr

2 J

(Vi . HF.„ 1

(10.10)

yy) ÔE

—

r\ onP

In other words, by brute force, take Uf

(r) to be whatever is necessary to have

the desired

as the solution. Potentials constructed in this way for silver

appear in Figure 10.3. Because there is a different pseudopotential for each

angular momentum state, the way the pseudopotential acts upon an arbitrary

function ψ(?) is through first breaking φ down into its angular momentum

components

φ{7) = Υ^Υ

1πι

{θ,φ)φ

1ιη

{ν)- φι

{ν)=

j'αθάφ

siner^e, φ)ψ{7), (10.11)

and then multiplying ipi

(r) by £//

(r) in forming the Hamiltonian. The need

for these integrals means that the pseudopotential is nonlocal.

Analytical results can also be employed to improve the form of the pseu-

dopotential. As remarked in Section 9.2.2, the Hartree-Fock approximation does

not properly take into account the many-electron phenomenon of screening, with

the consequence of giving incorrect results for electrons near the Fermi surface.

The electrons surrounding each nucleus should screen the Coulomb potential, but

270

Chapter 10. Realistic Calculations in Solids

100

-50

Figure 10.3. Pseudopotentials for the

5s,

5p, and

states of silver.

Hartree-Fock and its descendants have no guarantee of handling this phenomenon

correctly. An unscreened Coulomb potential Ze

jr behaves as AixZe

once

Fourier transformed; the main effect of screening is to eliminate the singularity at

—

0, producing a form such as

f/P

4πΖ<?

is a screening length.

(10.12)

Section 20.4.2 presents a general theory of dielectric functions that can be used to

address this problem, and according to a model calculation presented as Problem

20.7,

an electron cloud will cover each bare nucleus so that

U**(q

= 0) = --E

This constraint applies to the Fourier trans-

form of

the

/ = 0 component of

the

pseudopo-

tential. Ω is the volume of a unit cell.

(10.13)

Relation (10.13) can be used as a constraint on pseudopotentials—for example, to

estimate κ in Eq. (10.12).

For heavy atoms, relativistic effects become important because electrons near

the nuclei move at speeds that are a significant fraction of the speed of light. The

electron wavefunctions near the nuclei must therefore be described by the Dirac

equation. However, the pseudopotential method can also be applied here. To de-

termine the radial wavefunctions, one must generalize the Kohn-Sham equations

(9.104)so that they correspond to the Dirac equation. Suppose this is done and

wave functions and energies have been determined. Then the techniques described

above can be employed again; in fact, one can construct a pseudopotential so that

Numerical Methods 271

a solution of the ordinary Schrödinger equation gives a wave function that resulted

originally from a solution of the Dirac equation. In this way, a code that solves the

Schrödinger equation everywhere can secretly account for relativistic effects.

The pseudopotential procedure is not completely without

risks.

One risk is that

it will not work quite as advertised, and the pseudopotential will have in addition

to its ground state some low-lying states that enter into later calculations, but bear

no resemblance to any state of the original potential. Such states are called ghosts,

and as discussed by Gonze et al. (1991) they can lead to inaccurate computations.

10.2.2 Linear Combination of Atomic Orbitals (LCAO)

Given a collection of atoms arranged in some fashion on a lattice and their asso-

ciated pseudopotentials, the next task is to construct corresponding solutions of

Schrödinger's equation. One must therefore choose a set of basis functions and

devise a procedure that keeps the calculation within reasonable bounds.

One of the first methods employed historically to obtain moderately realistic

solutions of Schrödinger's equation is the Linear Combination of Atomic Orbitals

introduced in Section

8.4.1.

Application of the method begins with construction of the matrices described

by Eqs. (8.32) and (8.42). The minimal number of overlap integrals that needs

to be computed, notation to describe them, and detailed calculations for the most

common lattices were first outlined by Slater and Koster (1956). These integrals,

the Slater-Koster

parameters

can be viewed in two ways. On the one hand they

could be the calculational goal of a numerical effort, which finds the atomic wave

functions and calculates all the integrals. On the other hand, they can be treated as

adjustable parameters that describe the structure of

the

bands, but whose values are

to be determined by any means possible, including a fit to experiment. It was for

this latter purpose that Slater and Koster originally developed the formalism.

While methods of this sort were used extensively in early band structure cal-

culations, they have largely been displaced by more general numerical methods

described in the following sections. However tight-binding calculations can lead

to much more rapid calculations than the more general methods, and therefore may

provide the only way to approach quantum-mechanical problems involving large

numbers of atoms arranged in a non-crystalline manner. An example of such usage

is provided by Bernstein and Hess (2003).

10.2.3 Plane Waves

There is a method for conducting a systematic attack upon Schrödinger's equation

that is conceptually even simpler than summing up atomic orbitals. Write down

Bloch's theorem in the form Eq. (7.33) and solve it directly. For many years,

this strategy was regarded as impractical because computer memories were not

large enough and the full benefits of pseudopotentials had not been realized. The

situation has changed now that computer memories over 100 MB are no longer

exceptional, but it is worth explaining why the problem used to exist.

272

Chapter 10. Realistic Calculations in Solids

How many terms does one need in (7.33) to get reasonable accuracy? Suppose

that one were not employing a pseudopotential, so that the potential U(r) varied

like \/r in each unit cell. Consider the case of silver, whose outermost electron

is conventionally assigned to the 5s state. In the vicinity of the nucleus, such an

electron wave function has four nodes within a distance on the order of

angstrom

(see Figure 10.2), so to represent this function accurately in a Fourier basis would

require Fourier components with wave vectors up to at least 2π x 4/(

1Â)

~ 24 Â

Because silver has a lattice constant of a = 4 Â, its reciprocal lattice vectors start

with a magnitude of around Αττ/α ~ 3 Â~ '. Thus in the sum over reciprocal lattice

vectors

K = 1^+1202 + ^3 (10.14)

one needs to allow /), I2, and

IT,

to vary from around —8 to around 8. This sim-

ple estimate says that one needs around 5000 plane waves to have any hope of

accuracy, and it seems to require trying to diagonalize

5000 x

5000 nonsparse ma-

trices.

This task is by no means impossible, but it is sufficiently time-consuming

that plane waves seem at a disadvantage in useful calculations.

Although this argument has been used to justify the neec " r more sophisticated

basis sets, such as augmented plane waves, it is not entirely correct. There are

two flaws. The first flaw lies in dealing with real atomic wave functions rather

than pseudo wave functions. The pseudo wave functions lack the fine wiggles of

their predecessors, and the number of Fourier components needed to describe them

diminishes accordingly. For example, in silicon allowing l\ . . .

1-$

to vary from

—4

to 4 (leading to 800 plane waves) is enough. The second flaw lies in the hasty

conclusion that there is a need to diagonalize an 800 x 800 matrix. In fact, no one

is at all interested in finding the energy levels of 800 bands. All one actually needs

in order to calculate anything of physical interest is to find the energy levels below,

and within a few electron volts above, the Fermi surface; that is, one needs to find

the lowest-lying five to ten energy bands.

There are methods of varying degrees of sophistication to extract low-lying

eigenvalues and the corresponding eigenvectors from large matrices. One method

involves little more than multiplying the matrix repeatedly into a randomly chosen

initial vector. This method is rather crude, and it does not converge nearly as well

as more sophisticated approaches if

the

only task at hand is diagonalizing a matrix.

However, band structure calculations involve not only diagonalizing a matrix, but

also determining the entries in the matrix self-consistently, either because they de-

pend upon charge density through density functional theory or even because ions

are being allowed to move in a search for the structural ground state. In the context

of such self-consistent problems, Car and Parrinello (1985) found that the crude

method of matrix multiplication holds its own rather well, and because this method

is so simple it will be described below.

Suppose that one has a HermitianN xN matrix M for which the lowest eigen-

value λι is some large negative number and for which the highest eigenvalue λ/ν

is zero. Because the matrix is Hermitian, its eigenvectors ê\ . . . ê^ can be chosen

orthonormal. Pick a random vector a\ with N components, multiply M into a\,