electronic and band structure slater blochl

7/29/2019 Electronic and Band Structure Slater Blochl

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Chapter 7: The Electronic Band Structure of Solids

Bloch & Slater

April 2, 2001

Contents

1 Symmetry of (r) 3

2 The nearly free Electron Approximation. 6

2.1 The Origin of Band Gaps . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Tight Binding Approximation 15

4 Photo-Emission Spectroscopy 24

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Free electrons -FLT Band Structure

E

Ef

D(E)

V(r)

Ef

Ef

metal

"heavy" metal

insulator

E

D(E)

V(r) = V0

Ef

Figure 1: The additional effects of the lattice potential can have a profound effect on

the electronic density of states (RIGHT) compared to the free-electron result (LEFT).

In the last chapter, we ignored the lattice potential and con-

sidered the effects of a small electronic potential U. In this

chapter we will set U = 0, and consider the effects of the ion po-

tential V(r). As shown in Fig. 1, additional effects of the lattice

potential can have a profound effect on the electronic density of

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states compared to the free-electron result, and depending on

the location of the Fermi energy, the resulting system can be

a metal, semimetal, an insulator, or a metal with an enhanced

electronic mass.

1 Symmetry of (r)

From the symmetry of the electronic potential V(r) one may

infer some of the properties of the electronic wave functions

(r).

Due to the translational symmetry of the lattice V(r) is pe-

riodic

V(r) = V(r + rn), rn = n1a1 + n2a2 + n3a3 (1)

and may then be expanded in a Fourier expansion

V(r) =G

VGeiGr, G = hg1 + kg2 + lg3 , (2)

which, since G rn = 2m (m Z) guarantees V(r) =

V(r + rn). Given this, and letting (r) =

k Ckeikr the

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Schroedinger equation becomes

H(r) = h

2

2m2 + V(r)

= E (3)

k

h2k2

2mCke

ikr+

kG

CkVGei(k+G)r = E

k

Ckeikr, k kG

(4)

ork

eikrh

2

k2

2m E

Ck + G

VGCkG = 0r (5)

Since this is true for any r, it must be thath

2k2

2m E

Ck +

G

VGCkG = 0, k (6)

Thus the potential acts to couple each Ck only with its recip-

rocal space translations Ck+G and the problem decouples in to

N independent problems for each k in the first BZ. Ie., each of

the N problems has a solution which is a sum over plane waves

whos wave vectors differ only by Gs. Thus the eigenvalues

may be indexed by k.

Ek = E(k), I.e. k is still a good q.n.! (7)

We may now sum over G to get k with the eigenvector sum

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X

X

X

X

X

X

First B.Z.

Figure 2: The potential acts to couple each Ck with its reciprocal space translations

Ck+G (i.e. x x, , and) and the problem decouples into N indepen-dent problems for each k in the first BZ.

restricted to reciprocal lattice sites k, k + G, . . .

k(r) = G

Ck

Gei(kG)r =

G

Ck

GeiGr eikr (8)

k(r) = Uk(r)eikr, where Uk(r) = Uk(r + rn) (9)

Note that ifV(r) = 0, U(r) = 1V

. This result is called Blochs

Theorem; ie., that may be resolved into a plane wave and a

periodic function. Its consequences as follows:

k+G(r) =G

Ck+GGei(GkG)r =

GCkGeiG

r eikr

= k(r), where G G G (10)

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Ie., k+G(r) = k(r) and as a result

Hk = E(k)k Hk+G = E(k + G)k+G (11)

= Hk = E(k + G)k+G (12)

Thus E(k + G) = E(k) : E(k) is periodic then since both

k(r) and E(k) are periodic in reciprocal space, one only needs

knowledge of them in the first BZ to know them everywhere.

2 The nearly free Electron Approximation.

If the potential is weak, VG 0 G, then we may solve the

VG = 0 problem, subject to our constraints of periodicity, and

treat VG as a perturbation.

When VG = 0, then

E(k) =h2k2

2mfree electron (13)

However, we must also have that (if VG = 0)

E(k) = E(k + G) h

2

2m|k + G|2 (14)

Ie., the possible electron states are not restricted to a single

parabola, but can be found equally well on paraboli shifted by

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First BZ 2/a

E

Figure 3: For small VG, we may approximate the band structure as composed of N

parabolic bands. Of course, it is sufficient to consider this in the first Brillouin zone,

where the parabola centered at finite G cross at high energies. To understand the

effects of the perturbation VG consider this special k at the edge of the BZ. where the

paraboli cross.

any G vector. In 1-d Since E(k) = E(k + G), it is sufficient

to represent this in the first zone only. For example in a 3-D

cubic lattice the energy band structure along kx(ky = kz = 0)

is already rather complicated within the first zone. (See Fig.4.)

The effect ofVG can now be discussed. Lets return to the 1-d

problem and consider the edges of the zone where the [paraboli

intersect. (See Fig. 3.) An electron state with k = a will

involve at least the two G values G = 0, 2a . Of course, the

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a a k

x

First B.Z.

a-a

Figure 4: The situation becomes more complicated in three dimensions since there are

many more bands and so they can cross the first zone at lower energies. For example

in a 3-D cubic lattice the energy band structure along kx(ky = kz = 0) is already

rather complicated within the first zone.

exact solution must involve all G since

h2k22m

Ek

Ck + G

VGCkG = 0 (15)

We can generally take V0 = 0 since this just sets a zero for the

potential. Then, those G for which Ek = EkG h2k2

2mare

going to give the largest contribution since

Ck =G

VG CkGh2k22m

EkG(16)

Ck VG1CkG1

h2k2

2m EkG1

(17)

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CkG1 = G

VGCkG1G

h2k2

2m EkGG1(18)

CkG1 VG1Ck

h2k2

2m Ek

(19)

Thus to a first approximation, we may neglect the other CkG,

and since VG = VG (so that V(r) is real) |Ck| |CkG1|

other GkG

k(r) =G

CkGei(kG)r

(eiGx/2 + eiGx/2) cos xa

(eiGx/2 eiGx/2) sin xa

(20)

The corresponding electron densities are sketched in Fig. 5.

Clearly + has higher density near the ionic cores, and will

be more tightly bound, thus E+ < E. Thus a gap opens in

Ek near k =G2 .

2.1 The Origin of Band Gaps

Now lets reexamine this gap at k = G1/2 in a quantitative

manner. Start with the eigen value equation shifted by G.

CkG

Ek h

2

2m|k G|2

=

GVGCkGG =

G

VGGCkG

(21)

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V(x)

(x)

(x)+

E

k

Gap!

E

D(E)

Figure 5: + cos2(x/a) has higher density near the ionic cores, and will be moretightly bound, thus E+ < E. Thus a gap opens in Ek near k =

G2

.

CkG =

G VGGCkGEk

h2

2m|k G|2 (22)

To a first approximation (VG 0) lets set E =h2k2

2m(a free-

electron energy) and ignore all but the largest CkG; ie., those

for which the denominator vanishes.

k2 = |k G|2 , (23)

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-/a /a0 k

2 VG

Ek

k

e

Figure 7:

And the band structure looks something like Fig. 7. Within this

approximation, the gap, or forbidden regions in which there are

no electronic states arise when the Bragg condition (kf k0 =G) is satisfied.

| k| |k + G| (33)

The interpretation is clear: the high degree of back scattering

for these k-values destroys the electronic states.

Thus, by treating the lattice potential as a perturbation to

the free electron problem, we see that gaps arise due to enhanced

electron-lattice back scattering for k near the zone edge. How-

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Ai

BiB

i

Bi

Bi

Figure 11: A simple cubic tight binding lattice composed of s-orbitals, with overlap

integral Bi.

Bi =

i (r rm)v(r rn)i(r rn)d3r (46)

Bi describes the hybridization of adjacent orbitals.

Ai; Bi > 0, since v(r rn) < 0 (47)

Thus

E(k) Ei Ai Bim

eik(rnrm) sum over m n.n. to n

(48)Now, if we have a cubic lattice, then

(rn rm) = (a, 0, 0)(0, a, 0)(0, 0, a) (49)

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so

E(k) = Ei Ai 2Bi{cos kxa + cos kya + cos kza} (50)

Thus a band centered about Ei Ai of width 12Bi is formed.

Near the band center, for k-vectors near the center of the zone

we can expand the cosines cos ka 1 12 (ka)2 + and let

k2

= k2

x + k2

y + k2

z, so that

E(k) Ei Ai + Bia2k2 (51)

The electrons near the zone center act as if they were free with

a renormalized mass.

h2k2

2m = Bia2

k2

, i.e.1

m curvature of band (52)

For this reason, the hybridization term Bi is often associated

with kinetic energy. This makes sense, from its origins of wave

function overlap and thus electronic transfer.

The width of the band, 12 Bi, will increase as the electronic

overlap increases and the interatomic orbitals (core orbitals or

valance f and d orbitals) will tend to form narrow bands with

high effective masses (small Bi).

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First B.Z.

Fermi surface

Figure 12: Electronic states for a cubic lattice near the center of the B.Z. act like free

electrons with a renormalized mass. Hence, if the band is partially filled, the Fermi

surface will be spherical.

The bands are filled then by placing two electrons in each

band state ( with spins up and down). A metal then forms

when the valence band is partially full. I.e., for Na with a1s22s22p63s1 atomic configuration the 1s, 2s and 2p orbitals

evolve into (narrow) filled bands, but the 3s1 band will only

be half full, and thus it evolves into a metal. Mg 1s22s22p63s2

also metal since the p and s band overlaps the unfilled d-band.

There are exceptions to this rule. Consider C with atomic con-figuration of 1s22s22p2. Its valance s and p states form a strong

sp3 hybrid band which is further split into a bonding and anti-

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a

E

E1

E1

E2

2E

1A

2A

212 B

112 B

k = 0

- /a /a

k

k

k

k

111

111

111

Atomic Potential Tight Binding Bands

Figure 13: In the tight-binding approximation, band form from overlapping orbitals

states (states of the atomic potential). The bandwidth is proportional to the hy-

bridization B (12B for a SC lattice). More localized, compact, atomic states tend to

form narrower bands.

bonding band. (See Fig.14). Here, the gap is not tied to the

periodicity of the lattice, and so an amorphous material of C

may also display a gap.

The tight-binding picture can also explain the variety of fea-

tures seen in the DOS of real materials. For example, in Cu

(Ar)3d

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4s the d-orbitals are rather small whereas the valences-orbitals have a large extent . As a result the s-s hybridization

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a

P

sp antibonding3

sp bonding3

S

a

E

r

Figure 14: C (diamond) with atomic configuration of 1s22s22p2. Its valance s and p

states form a strong sp3 hybrid band which is split into a bonding and anti-bonding

band.

Bssi : is strong and the Bddi is weak.

Bddi Bssi (53)

In addition the s-d hybridization is inhibited by the opposing

symmetry of the s-d orbitals.

Bsdi =

si (r r1)v(r r2)di (r r2)d

3r Bssi (54)

where si is essentially even and di is essentially odd. So B

sdi

Bssi . Thus, to a first approximation the s-orbitals will form a

very wide band of mostly s-character and the d-orbitals will

form a very narrow band of mostly d-character. Since both the

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+ +

+1 2

+ +

+ +

d

s

D(E)

Figure 15: Schematic DOS of Cu 3d104s1. The narrow d-band feature is split due to

crystal fields.

s and d bands are valance, they will overlap leading to a DOS

with both d and s features superimposed.

4 Photo-Emission Spectroscopy

The electronic density of electronic states (especially for oc-

cupied states), and to a less extent band structure, are very

important for illuminating the interesting physics of materials.

As we saw in Chap. 6, an enhanced DOS at the Fermi sur-

face indicates an enhanced electronic mass, and if D(EF) = 0,

we have an insulator (semiconductor). The effective electronic

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hd

detector

synchrotron

r

materialsample

V

e

Figure 16: XPS Experiment: By varying the voltage one may select the kinetic

energy of the electrons reaching the counting detector.

mass also varies inversely with the curvature of the bands. The

density of states away from the Fermi surface can allow us to

predict the properties of the material upon doping, or it can

yield information about core-level states. Thus it is important

to be able to measure D(E). This may be done by x-ray pho-

toemission (XPS), UPS or PS in general. The band dispersion

E(k) may also be measured using angle-resolved photoemission

(ARPES) where angle between the incident radiation and the

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detector is also measured.

The basic idea is that a photon (usually an x-ray) is used

to knock an electron out of the system (See figure 17.) Of

Eh - E

b

h - E -b

D(E)

Intensity(E)

Kinetic Energy of electrons h -

X-ray

Eb

Figure 17: Let the binding energy be defined so that Eb > 0, = work function, then

the detected electron intensity I(Ekin h ) D(Eb)f(Eb)

course, in order for an electron at an energy of Eb below the

Fermi surface to escape the material, the incident photon must

have an energy which exceeds Eb and the work function ofthe material. If h > , then the emitted electrons will have

a distribution of kinetic energies Ekin, extending from zero to

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h . From Fermis golden rule, we know that the probability

per unit time of an electron being ejected is proportional to

the density of occupied electronic states times the probability

(Fermi function) that the electronic state is occupied

I(Ekin) =1

(Ekin) D(Eb)f(Eb)

D(Ekin + h)f(Ekin + h) (55)

Thus if we measure the energy and number of ejected particles,

then we know D(Eb).

phonon Coulombicinteration

Secondary electrons

e

e

Figure 18: Left: Origin of the background in I(Ekin. Right: Electrons excited deep

within the bulk scatter so often that they rarely escape. Thus, most of the signal I

originates at the surface, which must be clean and representative of the bulk.

There are several problems with this procedure. First some

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of the photon excited particles will scatter off phonons and elec-

tronic excitations within the material. Since these processes can

occur over a very wide range of energies, they will produce a

broad featureless background in N(Ek). Second, due to these

0 1 2 3 4

Ekin

0

2

4

6

I(E

kin

)

background subtracted

background

raw data

h-

Figure 19: In Photoemission, we measure the rate of ejected electrons as a function

of their kinetic energy. The raw data contains a background. Once this is subtracted

off, the subtracted data is proportional to the electronic density of states convolved

with a Fermi function I(Ekin) D(Ekin + h)f(Ekin + h).

secondary scattering processes, it is very unlikely that an elec-

tron which is excited deep within the bulk, will ever escape from

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measure

h

Ekin

e

Figure 20: BIS Ekin = h Eb , Eb = h Ekin

the material. Thus, we only learn about D(E) near the surface

of the material. Therefore it is important for this surface to be

clean so that it is representative of the bulk. For this reasonthese experiments are often carried out in ultra-high vacuum

conditions.

We can also learn about the electronic states D(E) above the

Fermi surface, E > FF, using Inverse Photoemmision. Here,

an electron beam is focussed on the surface and the outgoingflus of photons are measured.

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electronic and band structure slater blochl

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