1

Electrons in weak period potentials II

• Periodic potential means single-electron eigenstates are Bloch states.

• Periodic boundary conditions set the allowed values of k.

• For a given k, there are multiple discrete values of E allowed.

• States with k near a reciprocal lattice vector (having periodicity that’s a harmonic of the lattice) have energies strongly affected by lattice, even when lattice potential is weak.

• Result: energy gaps open up at particular values of k: not all energies are allowed anymore.

What we saw last time looking at the single-particle problem:

Crystal structure I

• In bulk, many solids are crystalline.

• Have discrete translational and rotational symmetries.

• Real-space structure is periodic - repetitions of a single unit cell.

• Smallest unit cell that gives full structure: primitive unit cell

• Can describe structure by a lattice and a basis.

lattice basis

a1

a2 r31

2

Crystal structure II

• Wigner-Seitz primitive cell: all points closer to a single lattice point than any other.

• Type of stacking depends on energetics of bonding.

• Surfaces have different energies per atom than bulk, so nanoscale crystals (high surface to volume ratio) can have different structure than bulk materials!

Crystal structure III - Miller Indices

Crystallographer’s way of labeling planes of atoms:

• Determine the intercepts of the plane along the crystallographic axes, in terms of unit cell dimensions.

• Take reciprocals.

• Write as integers rather than fractions.

• Negatives are written using overlines: (00-1) = (001)

• Triplets: (hkl); quadruplets: (hjkl)

a

a

a

a

a

a

3

Common crystal structures

Simple cubic Face-centered cubic

Body-centered cubic Hexagonal close-packed

a(1,0,0)a(0,1,0)a(0,0,1)

a/2(-1,1,1)a/2(1,-1,1)a/2(1,1,-1)

a/2(0,1,1)a/2(1,0,1)a/2(1,1,0)

Al, Cu, Ni, Sr, Rh, Pd, Ag,Ce, Tb, Ir, Pt, Au, Pb, Th

W, Li, Na, K, V, Cr, Fe, Rb,Nb, Mo, Cs, Ba, Eu, Ta

Mg, Be, Sc, Te, Co, Zn, Y,Zr, Tc, Ru, Gd, Tb, Py, Ho,Er, Tm, Lu, Hf, Re, Os, Tl

a/2(1,-31/2,0)a/2(1, 31/2,0)c(0,0,1)

http://cst-www.nrl.navy.mil/lattice/http://www.chem.ox.ac.uk/icl/heyes/structure_of_solids/Strucsol.htmlhttp://home3.netcarrier.com/~chan/SOLIDSTATE/CRYSTAL/

Common crystal structures II - semiconductors

Diamond

a/2(0,1,1) a/2(1,0,1) a/2(1,1,0)

C, Si, Ge, Sn

Two interpenetrating fcc lattices displaced by 1/4 a.

Result of all sp3 covalent bonds.

Zinc blende

a/2(0,1,1) a/2(1,0,1) a/2(1,1,0)

ZnS, AgI,AlAs, AlP, AlSb, BAs, BN, BP, BeS, BeSe, BeTe,CdS,CuBr, CuCl, CuF, CuI,GaAs, GaP, GaSb, HgS, HgSe, HgTe, InAs,InP, MnS, MnSe,SiC, ZnSe, ZnTe

Two interpenetrating fcc lattices displaced by 1/4 a, each lattice a different species.

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Reciprocal basis vectors

We saw last time that there are special vectors in k-space (also called reciprocal space) that behave like:

ijji πδ2=⋅ab

The bi define a lattice in reciprocal space just as the ai do in real space. In 3d,

)(2

)(2

)(2

213

213

132

132

321

321

aaaaa

b

aaaaa

b

aaaaa

b

×⋅×=

×⋅×=

×⋅×=

π

π

π

Reciprocal lattice vectors

Using the b’s, we can build up a lattice in reciprocal (k) space.

The reciprocal lattice is the set of points in reciprocal space given by integer linear combinations of the reciprocal lattice vectors:

332211 bbbG ccc ++=

where c1, c2, c3 are integers.

b1

b2What’s special about the G’s?

Any function in real space with the periodicity of the (real space) lattice can be written exactly as a sum like:

rG

GGr ⋅∑= ieρρ )(

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Reciprocal lattices

Real space Reciprocal space

a

a

a

SC

FCC

BCC

2π/a

4π/a

4π/a

Brillouin zones I

• Each point in the reciprocal lattice is a reciprocal lattice vector.

• Remember: when k is close to a such a vector, the electronic states are strongly affected by the lattice potential (gaps!).

• All unique k values compatible with b.c. may be written within the first BZ - that is, within the first Wigner-Seitz unit cell of reciprocal space.

What do these Brillouin zones look like?

6

Brillouin zone - FCC

)4

11

4

1(

)04

3

4

3(

)102

1(

)2

1

2

1

2

1(

)010(

)000(

=

=

=

=

=Χ=Γ

U

K

W

L

Image from Marder.

Brillouin zone - BCC

)2

1

2

1

2

1(

)02

1

2

1(

)010(

)000(

=

=

==Γ

P

N

H

Image from Marder.

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Why should we care about Brillouinzones and reciprocal space? Reason #1.

Recall our free-electron gas procedure:

• Find allowed single particle states, labeled by k.

• Using E(k), figure out the energy levels of those states.

• For noninteracting electrons, find many particle ground state by filling those levels from the bottom up, two electrons per single-particle state (spin).

Filling of k states in reciprocal space determines electronic properties of bulk solids.

Can do same thing here, but E(k) no longer simple!

Band diagrams

Images from Blakemore.

Free particle Weak periodic potential

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Band diagrams

Start filling single-particle states from the bottom.

Where do we end up?

EF in middle of band: metal

EF such that integer number of bands exactly full: band insulator

Special case: Eg is small = intrinsic semiconductor.

Eg

Complication:Images from Blakemore.

Lattice spacing depends on direction. Result: bands can overlap in energy.

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Real band structuresImages from Harrison

Germanium Diamond BZ

More about this on Monday.

Why should we care about Brillouin zones and reciprocal space? Reason # 2.

Planes in reciprocal space labeled by Miller indices make diffraction experiments possible!

Bragg planes: the set of all equispaced parallel planes containing all the sites in a lattice.

G1

G2

||

2

1Gπ

||

2

2Gπ

The spacing between (hkl) planes is given by where Ghkl = hb1+kb2+lb3

||

2

hklGπ

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Diffraction

Basic idea:

Constructive interference from periodic planes leads to peaks in diffracted intensity along directions dependent on λ of incoming wave.

d

θ θθ

θ

Total extra distance traveled by bottom ray = 2d sin θ.

Constructive interference requires

λθ nd =sin2Bragg condition

Diffraction and antennae

In many respects, diffraction problems for small numbers of scatterers are very similar to problems about antenna arrays.

Basic methodology:

• Define coordinates nicely.x

yr

a

r’ = r - a

• Assume each scatterer is a source of spherical waves of wavenumberk. Find amplitude at position of interest.

)'exp()exp( 00

21

ikrAikrA

AAAtot

+=+=

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Diffraction and antennae

• Make legitimate approximations.

x

yr

a

r’ = r - a

• Find expression for scattered intensity, proportional to |amplitude|2 :

θcos' arrar −≈→>>θ

22

0

22

0

22

0

2

)cos(exp(1

))cos(exp(1)(exp(

))cos(exp()exp(

~

θ

θ

θ

aikA

aikikrA

arikikrA

AI tot

−+=

−+=

−+=

Diffraction techniques

Scattering of some wave from a sample tells us about the structure of the sample.

x-ray diffraction: electronic density distribution

neutron diffraction: mass density distribution, magnetic ordering

θ θ

Send in k, get out k’ , with |k|=|k’ |.

Bragg condition ends up being ∆k = Ghkl.

Intensity ~ G Fourier component of lattice potential.

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Derivation of Bragg condition

Extra phase for lower pair of rays =

( )'coscos θθ ddk +

θ θ’

d = closest spacingk

k’Assume |k| = |k’|

)'( kkd −⋅=

Since d is an example of a real space lattice vector R, and this phase must be an integer multiple of 2π to get constructive interference, we find from definition of b’s that GkkRkk =−→=⋅− )'(2)'( jπ

The difference in incident and outgoing k must be a reciprocal lattice vector to constructive interference (a diffraction maximum).

Diffracted intensity

Incoming plane waves

Each scattering site = source of outgoing spherical waves

Scattered amplitude ~ rr rk de i∫ ⋅∆− )()(ρIntegral over sample volume

For periodic lattice,

rG

GGr ⋅∑= ieρρ )(

Intensity ~ |Amp|2 ~2)( || rrkG

GG de i∫∑ ⋅∆−−ρ

For G = ∆k, integrand ~ 1; I ~ V2

Otherwise, I ~ 0.

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Types of diffraction θ θ

Single-crystal diffraction:

• For fixed λ, knowing sample orientation, diffraction peaks only at certain specific angles, when Bragg condition is satisfied -- Laue spots. Can deduce structure from spot positions.

• Spacing of spots is inversely prop. to lattice spacings.

Powder diffraction:

• Sample is randomly oriented grains.

• For any θ, some of the grains are going to have an hklmeeting the Bragg condition.

• Each grain produces spots at particular θ,φ, so that adding the spot patterns incoherently produces a set of peaks at particular values of θ.

Powder diffraction

http://www.uni-wuerzburg.de/mineralogie/crystal/teaching/teaching.html

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Diffraction tidbits

• Finite T cuts intensity, but does not affect width of peaks (!)

• Finite size does affect peak widths - possible trouble for nanopowders.

• Amorphous materials / liquids show 2 or 3 very broad rings, indicative of very short-range order (bond lengths / interparticle spacings).

To summarize:

• Define crystal by lattice + basis in real space, unit cell.

• Real space lattice can be used to define reciprocal space lattice.

• Reciprocal space lattice unit cell = Brillouin zone

• Real space planes + reciprocal lattice vectors labeled by Miller indices.

• Because of periodic lattice potential, gaps open up in free electron energy for k near edge of Brillouin zone.

• Details of Brillouin zone & filling determine electronic state of material (more on this next time).

• Lattice vectors in reciprocal space determine locations of diffraction peaks.

• Diffraction is powerful method for structure determination.

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Next time:

• More electronic properties + bands

• “Doping” of semiconductors

• Boundaries & surface states

• Impurities