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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
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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
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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?
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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.