Marder M.P. Condensed Matter Physics

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Tightly Bound Electrons

223

The Wannier functions form an orthonormal set, as may be seen by computing

J drw

(R,

r)w*

(R',

r) = J dr £ £ ^^^^M^^) (8-44)

k V

kk'

= δ—

,δ„

The sum is normalized because as shown in (8 46)

R,R

"."*· Section 7.2.4, the number of tin the first Bril-

louin zone equals

TV.

If one should happen to know the Wannier functions, the Bloch functions can

be recovered from them by computing

-j= Σ

»(R, r)e

= φ

ηϊ

(7). (8.47)

Ambiguity in Definition of

Wannier

Functions.

At first it seems that the Wannier

functions are completely determined by the prescription given in Eq. (8.43). How-

ever, there is a subtle ambiguity. Each Bloch function is determined only within an

overall phase factor, so the Wannier functions can be rewritten

w„(R, r) = -j= Ç

e-^+^Hjf),

(8.48)

where

</>(&)

is a completely arbitrary real function. Addressing this ambiguity re-

quires an investigation of the phases of wave functions.

8.4.3 Geometric Phases

Geometrical phases have been discovered and rediscovered many times. The ear-

liest paper is probably by Pancharatnam (1956). In the special case of electro-

magnetic vector potentials, they appeared in Aharonov and Böhm (1959), and then

with much greater generality in a paper of Berry ( 1984) after whom they are widely

known as Berry phases.

These phases appear very generally in the study of quantum mechanical prob-

lems.

Suppose one has a Hamiltonian Ή^ that depends upon some collection of

parameters λ = (λι (t), Xiit) . . . λ„(ί). These parameters change in time, but they

change very slowly, meaning one can treat λ as small. These slow changes are

also called

adiabatic.

What are some specific examples? An experimentalist might

very slowly lower the temperature of a sample, in which case X = T. An electron

within a solid might move through electric and magnetic fields that vary weakly in

space, in which case the electron's wave vector k = λ can be viewed as as slowly

changing.

To see how geometric phases arise, let |Φ^) be an eigenstate of the Hamiltonian

ίΚ^ with eigenvalue £^. According to the

adiabatic

theorem — conventional proof

224

Chapter 8. Nearly Free and Tightly Bound Electrons

in Landau and Lifshitz (1977) and later refinements discussed by Zhao (2008) —

changes slowly enough, a particle once placed in state |Φ

) will remain in

state |Φ

). Saying that a particle is in an eigenstate, however, says nothing about

the phase. Indeed, the collection of wave functions

(Γ|Φ

) will in general have

arbitrary phases, just like the Wannier functions of Eq. (8.48).

One feature of this arbitrary phase is wise to demand at all times; require that

the phase be continuous and differentiable. This requirement is not quite as trivial

as it sounds, because it extends to cases where the parameter space for

is periodic.

For example, if the parameter is the wave vector k, one should require Φ^ to be

continuous and differentiable as a function of k for k lying in the first Brillouin

zone,

as in Figure 8.3.

Now examine what the time-dependent Schrödinger equation says about phases:

In the adiabatic limit where

changes slowly, guess a solution of the form

|φ(

))

(ί

/β)

/ο

Λ(,0^(0|

φχ(ο

)

;

(8.50)

the wave function equals Φ

up to a phase factor, and the phase factor has two

pieces. The first is what one would first guess based upon the time dependence of

eigenfunctions, and the second φ is a correction.

Inserting Eq. (8.49) into Eq. (8.50) gives

/λ·(Φ

|-^|Φ

) (8.52)

dt ~"

x χ

λΙ

3Α

r\(t) The line integral is performed over the circuit

=^φ{ΐ)

— φ{ϋ)= I dX-Ckx λ follows from time 0 to/; there is no depen- (8.53)

J'XfQ)

dence on the speed of

traversal

so long as it is

slow

enough.

where α

= /(φ-|ίφ

) (8.54)

ολ

The quantity 3ί

is called the Berry connection because it connects wave functions

for slightly differing values of the parameter A. In many applications |Φ

) is a

periodic function of

Then in Eq. (8.53), one can choose a circuit such that A

starts and ends at the same point; the result is in general nonzero, and known as the

geometrical phase or Berry phase

Τ=φάλ·ΰΙ^. (8.55)

Tightly Bound Electrons 225

Application to

Wannier

Functions.

For Wannier functions to serve their intended purpose as localized basis func-

tions,

they should have as small a spatial spread as possible. This condition can be

applied equally well to any of

the

functions indexed by R, and so will be applied to

the one centered at the origin R = 0. Dropping for the moment reference to R = 0

to avoid cluttering the notation, the center of the Wannier function is given by

dr W*{?) r W*(r) Suppressing notation R = 0. (8.56)

A measure of the spread of the Wannier function is

—=-

f _. _\ i-, -aio +/->\ ~ϊ ι=*ι2 The definition of

should be

δΐ~= / dr W^(f) \r — r\ W

(r) = r„ — \r

\ . clear. Obtain the last identity by (8.57)

J multiplying out the square.

las

multiplying out the square.

As shown in Problem 9,

r„ = \ ^ !R - The sum is over the N wave (8.58)

Z—/

nk vectors in a single Brillouin zone.

where in this case the Berry connection involves the periodic Bloch functions of

Eq. (7.45)

The integral is over one unit cell as in

\l = i J

u^^iu^f)

>·

(8.59)

Also,

^=-El

<k(

(8-60)

Take advantage now of the freedom expressed in Eq. (8.47) to modify

^ by

an arbitrary real function φ

(k)

of k, periodic over the Brillouin zone. Problem

9 shows that by manipulating φ one can minimize the spread 6r^ of the Wannier

function through the condition

{k)

= V

. (8.61)

In general, this is as far as one can go without proceeding to numerical solutions of

the Poisson equation, of which examples can be found in Marzari and Vanderbilt

(1997).

However, in one dimension, one can write immediately

φ(1ή-φ(0)= / dk' R

,. (8.62)

Problem 10 shows that for a one-dimensional lattice of spacing a, Wannier func-

tions defined with this choice of phase are eigenfunctions of the position operator,

provided that the positions R are quantized according to

Γα

2π

R = la-\ ; Γ = / dk

I* an integer and Γ is defined also in Eq. (8.55). (8.63)

2π Jo

226 Chapter 8. Nearly

Free

and

Tightly

Bound Electrons

8.4.4 Tight Binding Model

Thus,

by manipulating the phase φ, one can optimize the Wannier functions and

make them drop off as fast as possible when r moves away from R. Kohn (1959)

first proved that Wannier functions in one dimension decay exponentially away

from their centers, and Nenciu (1993) generalized the proof to three dimensions.

The precise rate of decay is not easy to determine without detailed computation,

but it is related to the size of the energy gap separating different bands of Bloch

wave functions from each other. The larger the energy gap separating a band from

all other bands, the more completely localized its Wannier functions can be.

Looking forward to the band structures in Section 10.4, this observation im-

plies that electrons in insulators can safely be regarded as sitting in localized Or-

bitals around specific atoms. However, while Wannier functions can be defined for

metals, they are not guaranteed to be localized at atomic sites.

If the Wannier function centered at R does decay exponentially once it leaves

site R, then it is very useful to write Schrödinger's equation in terms of a Wannier

function basis. This calculation is much more concise than the one with atomic

orbitals. Denote the state vector corresponding to w

(R, r) by \R), and write

Ìi = ^2\R'){R

\'k\R){R\. (8.64)

RR'

The Hamiltonian in (8.64) is not the full Hamiltonian, but has been restricted to the

nth band. It is not difficult to add an extra index and consider many bands, but that

is not needed for the following discussion, so the index n will be dropped.

Expression (8.64) is useful if the Wannier function centered at R becomes small

enough once it is further away from R than the nearest neighbors of

In this case,

one can neglect the matrix element

•K

, = {R'\K\R} = I dfwl (/?', r) [-^- + u(f)]w„{R, r) (8.65)

unless R and R' are nearest neighbors. Problem 6 shows that

MRR'

= Σ

~^nl

eil{

)

is the energy ot a Bloch state of wave

(

)

7 '

number k in band n.

Therefore,

depends only upon the difference between R and K. Furthermore,

when R and κ are nearest neighbors, symmetry often dictates that "K^, equal a

single constant t, while when R = R', one can denote Ή^ by a constant U. In this

case the Hamiltonian (8.64) becomes

_, _, _ ^ _, _, δ is again a set of vectors pointing from R

τΒ

= 22 \R) t (R + S\ +22

)

(

\·

t0 its nearest nei

hbors

This

notation helps (8.67)

suggest the idea of hopping from site R + δ to

Rö R H

The Hamiltonian appearing in Eq. (8.67) is the tight-binding Hamiltonian. The

first term on the right hand side of (8.67) is the hopping term that allows electrons

Problems

227

to move from one site to another. The second term is an on-site term that describes

the energy of placing an electron at a lattice site.

The tight-binding Hamiltonian has a simple exact solution. Dehne

for k in the first Brillouin zone, so that one has the inversion formula

\R) = —j= y e \k), Sum only over k in the first Brillouin zone. (8.69)

Inserting Eq. (8.69) into Eq. (8.67) gives

ÄTB

= Σ - Σ

\k)ie-

iU+ik

iii+

^(k'\

Y^ - Σ

\k)Ue-

iM+il

(k'\ (8.70)

RS kk' R kk'

= ΣΦ(ί\

(8.71)

with

-k

u+t

(

)

This expression is identical to Eq. (8.39) if one identifies U with £

+ It.

Although Eqs. (8.72) and (8.39) are identical, they represent a different phi-

losophy. In the first case, there is a series of approximations, and eventually an

approximate result is derived in a particular instance. In the second case, gener-

ality is maintained whenever possible, and there is only a single approximation,

which consists in neglecting interactions except between nearest neighbors. It is

easy to pose new questions by modifying the tight-binding Hamiltonian. For ex-

ample, what happens if the on-site energies U in Eq. (8.67) vary from site to site?

This question leads to the theory of localization (Section 18.5). Or what happens if

one moves beyond the single-electron approximation, and modifies Eq. (8.72) so

that the Hamiltonian has an additional repulsive energy when more than one elec-

tron sits on a given site /?? This question leads to studies of electron correlation

and magnetism in Chapter 26.

Problems

Nearly free electrons:

(a) Consider a two-dimensional square lattice of lattice constant a and take

t/(r) =-4C/

cos — cos—. (8.73)

a a

Find the Fourier transform

from Eq. (7.26).

228

Chapter 8. Nearly Free and

Tightly

Bound Electrons

(b) For ^ι = (π/α, π/α), ip(k\) will couple strongly to three other components

of φ, ipfa)

■ ■ ■

I/JCU).

What are fo · · · &4? Identify the values of K that one

must include when doing perturbation theory to find φ(ϋ\) . . . ip(kn) to first

order in

UQ.

only for one value of K. Therefore perturbation theory can be reduced to the

subspace involving only ψ(ϋ\) and, say, ^(fo)·

(d) Write down Schrödinger's equation in the subspace involving only ψ(ΐί\)

and

?/>(&2)·

(e) Solve the resulting 2x2 system of equations and find the two allowed ener-

gies at Bloch index k\.

(f) Sketch £^ for the lowest two bands along the line T-T, and indicate the size

of the energy gap.

Smoothness of Fermi surface:

(a) Show that the unit normal to the Fermi surface must be continuous in the

reduced zone scheme, if the energy £ ^ is continuous and differentiable.

(b) Show that in the nearly free electron approximation, the Fermi surface is

perpendicular to the Brillouin zone boundary.

Reciprocal lattice:

(a) Consider a two-dimensional lattice with primitive vectors

fl(l,0),

α(^,Ι).

(8.74)

Find primitive vectors for the reciprocal lattice, and draw pictures of the first

and second Brillouin zones.

(b) Find the areas of the first and second Brillouin zones.

Two-dimensional Fermi surface: Find the Fermi surface for a triangular

lattice in two dimensions, with six noninteracting spin 1/2 electrons per site.

(a) Calculate the size of the Fermi sphere.

(b) Draw the reciprocal lattice of

the

triangular lattice on a piece of tracing paper,

and indicate the first three Brillouin zones.

Problems 229

the Brillouin zones, and by moving the tracing paper over the Fermi sphere,

find the Fermi surface.

Tight-Binding Band Structure

Graphene: Graphene

is a

honeycomb

lattice (Eq. (1.5)) with lattice spacing

a =

2.46 Â (Table 2.1). The electronic

band structure

graphene

can be

computed

reasonable approximation

from

s and p

orbitals. The bands break into two groups. There

is a

first

set of bands that come from the interactions of the

orbital and the

and

orbital, assuming graphene sits in the

x-y

plane. These are the

bands, which

require solution of a 6

6 matrix problem, described by Saito et al. (1998).

second set of bands comes from the

orbitals, called the

bands. These can

computed completely independently from the

bands because all integrals of

the form

dra^

(r)af(r)

or J

drdfiffla^r) vanish by symmetry. Thus

for

the

7Γ

bands there is only one orbital

to consider, cylindrically symmetrical

in the

—

y plane, and with symmetry like

z in

the

direction. Because the

honeycomb lattice is built from two basis vectors, the matrix

Eq. (8.32)

2x2.

(a) Refer to the two Bravais lattices corresponding

the two basis vectors

the honeycomb lattice as the A and

sublattices. Show that the matrices

and S have the form

χ=

„r s= :r r

75)

Use Eq. (8.42).

(b) Show more specifically

ignoring overlap integrals between

any

atoms

farther apart than nearest neighbors that

where

t/(Z)\ (1 af(k)\

trCk)

)

Ur(*)

i )

(8J6)

f(k)

Ê>-'V/V3

<V/(2v

cos

/2).

(8.77)

Find the definitions

and

£p±tw(k)

= -z- y-

(8.78)

1 ±aw(k)

where

w(k)

^\f(k)\

(8.79)

(d) Setting

= 0

because

is just

constant, and taking

t =

-3.033 eV,

a =

0.129, plot Eq. (8.78) between Γ and

(Figures 7.10 and 7.11). Show that in

the vicinity of

Ä^

the two bands are degenerate and have the form

±\k

—

ÌCK\

230

Chapter 8. Nearly

Free

and Tightly Bound Electrons

6. Wannier functions:

(a) Show that matrix elements of the Hamiltonian between Wannier function

coming from different bands must vanish, and that it is therefore legitimate to

write down Eq. (8.64).

(b) Show that the matrix element (R\K\R') depends only upon R

—

R', where (R\

is the Wannier function defined in Eq. (8.43).

Tight-binding model: Consider a tight-binding Hamiltonian that acts upon a

single band of localized states in one dimension,

Ä = 2t^

{Ì(|/)(/ + 1| + |/)(/-1|)+COS[27T/T

]|/)(/|} (8.80)

with

= f. (8.81)

The integer / should be thought of as indexing sites along a chain of atoms;

the state |/) locates an electron on atom /.

(a) What is the periodicity of the Hamiltonian?

(b) Use Bloch's theorem to reduce the eigenvalue problem associated with

Eq. (8.80) to the solution of a small finite matrix equation.

will have to be carried out numerically.

8. Berry phases: The definition of

the

Berry connection of

Eq.

(8.54) appears to

require knowledge of the wave function Φ^ for adjacent values of λ in order

to compute the derivative. However, the expression can be recast to involve

wave functions at only one point in λ space.

(a) Let m and n index separate eigenfunctions and eigenvalues of the Hamilto-

nian in Eq. (8.49). Show that

<*«xlV

l*»x> = ^Λ^-Α^

(b) Specialize to the case where λ has three components. Return to the definition

the

Berry phase Eq. (8.55). Let

be any surface whose boundary is the line

around which one integrates λ. Using Stoke's Theorem, show that

Γ = - [

dh-V

(8.83)

where « is a unit normal integrated over surface A and

» =

L·

(

ε ,_

,)2

84)

References

231

9. Localization

Wannier functions:

(a) Verify Eqs. (8.58)

and

(8.60).

(b) Replace u

-(r) with u

^(r)e'^

\ where

<j>(k)

periodic over

the

Brillouin

zone. Show that

unchanged when

the

phase φ(1ί) changes,

and

find

how

r\ changes

as a

functional

φ{Κ).

Sr%

with respect

to φ(κ) and

show that

the

condition

φ(ί) is

Eq. (8.61)

10.

Wannier functions

one dimension:

The

phases

Wannier functions

can

be determined

general

the prescription that the spread

the functions

minimized. For one-dimensional systems, there

another way

arrive

at the

same conclusion, proposed

Kivelson (1982).

In one

dimension, Wannier

functions with phases given

by Eq.

(8.62)

are

eigenfunctions

the position

operator.

Dropping the band index

n and

specializing

to a

one-dimensional crystal with

N unit cells

length

a, let w(R, r) be

Wannier functions, satisfying

w(R, Ό

= 4^ Σ e-

M+im

Mr),

(8.85)

yJy

where φ(Κ) satisfies Eq. (8.62),

and

Eq. (8.59) becomes

= Ì

drul{r)^u

{r).

(8.86)

one

slight departure from other calculations, assume that

the

Bloch func-

tions îpk{r) rather than w&(r) have their phases arranged

so as to be

periodic

functions

of k

over the Brillouin zone [0,

27r/a].

Making use

Eq. (8.85)

and

■0/t(r)

exp[ikr]uk(r) show that

(a) Show that

rw{R,

r)=Rw(R,

(■

-(2πί·/α)Λ+ί0(2π/α)

Λφ{0)

Mr)

V^V (8-87)

(b) Multiply

the

last term

Eq. (8.87)

w*(R',

r) and

integrate over

Show

that the result vanishes

for

all

R' and

that therefore this last term

zero.

the

condition

for the

second term

Eq. (8.87)

vanish

is the

quantization condition

on R

Eq. (8.63).

232 Chapter 8. Nearly

Free

and Tightly Bound Electrons

References

Y. Aharonov and D. Böhm (1959), Significance of electromagnetic potentials in quantum theory,

Physical Review, 115,

485-491.

M. V. Berry (1984), Quantal phase factors accompanying adiabatic changes, Proceedings of the

Royal Society of London, A392, 45-57.

Bloch (1928), The quantum mechanics of electrons in crystal lattices, Zeitschrift für

Physik,

52,

555-600. In German.

A. V. Gold (1958), An experimental determination of the Fermi surface in lead, Philosophical

Trans-

actions of the Royal Society of London, A251, 85-112.

R. Hartree (1928), The wave mechanics of an atom with a non-coulomb central field, Proceedings

of the Cambridge Philosophical Society, 24, 89-132.

S. Kivelson (1982), Wannier functions in one-dimensional disordered systems: Application to frac-

tionally charged solitons, Physical Review B, 26, 4269^277.

W. Kohn (1959), Analytic properties of Bloch waves and Wannier functions, Physical Review, 115,

809-821.

L. D. Landau and E. M. Lifshitz (1977), Quantum Mechanics (Non-relativistic Theory), 3rd ed.,

Pergamon Press, Oxford.

Marzari and D. Vanderbilt (1997), Maximally localized generalized Wannier functions for com-

posite energy bands, Physical Review B, 56, 12 847-12 865.

G. Nenciu (1993), Existence of the exponentially localised Wannier functions, Communications in

Mathematical Physics, 91, 81-85.

S. Pancharatnam (1956), Generalized theory of interference, and its applications, part i. coherent

pencils, Proceedings of the Indian Academy of Sciences, A44, 247-262.

R. Peierls (1930), The theory of

the

electrical and thermal conductivity of

metals,

Annalen der

Physik

(Leipzig),

121.

In German.

R. Saito, G. Dresselhaus, and M. S. Dresselhaus (1998), Physical Properties of

Carbon

Nanotubes,

Imperial College Press, Singapore.

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Review, 52, 191-197.

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