Remaining Issues
•
•
•
•
•
Electron wave picture has fixed some thermal/electrical properties and
electron velocity issues
Still can not explain:
–Hall coefficients
–Colors of metals
–Insulators, Semiconductors
Can not ignore the ions (i.e. everything else but the valence electrons
that we have been dealing with so far) any longer!
Whatever we modify, can not change the electron wave picture that is
now working well for many materials properties
HOW DO THE VALENCE ELECTRON WAVES INTERACT
WITH THE IONS AND THEIR POTENTIALS?
1
©1999 E.A. Fitzgerald
Improvements? What are ion cores doing...
• Scattering idea seems to work
• any effect of crystal (periodic) lattice?
• Diffraction
– proves periodicity of lattice
– proves electrons are waves
– proves strong interaction between crystal and electrons (leads to band structures=semiconductors and insulators)
– useful characterization technique
• Course: bias toward crystalline materials: many applications: materials
related to either end of spectrum (atomic/molecular or crystalline)
extended
localized
Point defects,
atoms, molecules
Polymers,
α Si
Bands; properties of many
solids with or without extended defects
Diffraction is a useful characterization in all these materials
©1999 E.A. Fitzgerald
2
Electrons in a Periodic Potential
• Rigorous path: HΨ=EΨ
• We already know effect: DeBroglie and electron
diffraction
• Unit cells in crystal lattice are 10-8 cm in size
• Electron waves in solid are λ=h/p~10-8 cm in size
• Certain wavelengths of valence electrons will diffract!
3
©1999 E.A. Fitzgerald
Diffraction Picture of the Origin of Band Gaps
• Start with 1-D crystal again
λ~a
1-D
a
nλ = 2d sin θ
d=a,
sinθ=1
Take lowest order, n=1, and
consider an incident valence
electron moving to the right
π
i x
π
ki = ;ψ i = e a
a
Reflected wave to left: ko = −
nλ = 2a
k=
k=
a
;ψ o = e
π
−i x
a
2π
a
Total wave for electrons with diffracted wavelengths:
ψ = ψ i ±ψ o
ψ s = ψ i + ψ o = 2 cos
π
ψ a = ψ i − ψ o = i2sin
©1999 E.A. Fitzgerald
a
Δk = ki − k o =
2π
λ
πn
π
a
π
a
x
x
4
Diffraction Picture of the Origin of Band Gaps
Probability Density=probability/volume of finding electron=|ψ|2
ψ a = 4sin 2
2
ψ s = 4 cos 2
2
π
a
π
a
x
x
a
a
•Only two solutions for a diffracted wave
•Electron density on atoms
•Electron density off atoms
•No other solutions possible at this wavelength: no free traveling wave
5
©1999 E.A. Fitzgerald
Nearly-Free Electron Model
•
•
•
Assume electrons with wave vectors (k’s) far from diffraction
condition are still free and look like traveling waves and see ion
potential, U, as a weak background potential
Electrons near diffraction condition have only two possible solutions
–electron densities between ions, E=Efree-U
–electron densities on ions, E= Efree+U
Exact solution using HΨ=EΨ shows that E near diffraction conditions
is also parabolic in k, E~k2
6
©1999 E.A. Fitzgerald
Nearly-Free Electron Model (still 1-D crystal)
states
h 2k 2 p 2
=
E=
2m 2m
E
Away from k=nπ/a,
free electron curve
Near k=nπ/a,
band gaps form, strong
interaction of e- with
U on ions
Eg=2U
dE h 2 k
=
dk
m
h 2k
ΔE =
Δk
m
Diffraction,
k=nπ/a
Quasi-continuous
Δk=2π/L
-π/a
0
Δk=2π/a=G=reciprocal lattice vector
©1999 E.A. Fitzgerald
π/a
k
7
Consequences of Diffraction on E vs. k curves
•
•
•
At k=π/a, there must be also a k=-π/a wave, since there is absolute
diffraction at this k
Standing wave at
True for every k=nπ/a
diffraction
condition
Creates a parabola at every nG
E
k=-π/a in
reference
parabola
−2π/a
-π/a
G
©1999 E.A. Fitzgerald
0
−2π/a
k=π/a in
reference
parabola
π/a
-π/a
2π/a k
08
Extended-Zone Scheme
•
•
Bands form, separated by band gaps
Note redundancy: no need for defining k outside +-π/a region
E
Band 2
Band gap Eg
Band 1
−2π/a
-π/a
0
π/a
2π/a
k
9
©1999 E.A. Fitzgerald
Reduced-Zone Scheme
• Only show k=+-π/a since all solutions represented there
−π/a
©1999 E.A. Fitzgerald
π/a
10
Real Band Structures
•
•
GaAs: Very close to what we have derived in the nearly free electron model
Conduction band minimum at k=0: Direct Band Gap
E, eV
2.0
1.0
[111]
Conduction
band
(0,0,0)
-1.0
-2.0
-3.0
[100]
k
Valence
band
-4.0
Figure by MIT OpenCourseWare.
11
©1999 E.A. Fitzgerald
6
Real Band Structures
Ge
5
4
3
0.36eV
5.4eV
1
0
1.1eV
Ek 0.9eV
2.0eV
0.29eV
-1
Energy in eV
• Ge: Very close to GaAs,
except conduction band
minimum is in <111>
direction, not at k=0
• Indirect Band Gap
3.5eV
2
1.01eV
3.8eV
-2
-3
-4
-5
-6
-7
-8
-9
-10
-11
-12
-13
2π 1 1 1
k= a (2 2 2 )
[111]
k=0
[100]
2π
k = a (100)
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©1999 E.A. Fitzgerald
Figure by MIT OpenCourseWare.
Trends in III-V and II-VI Compounds
Energy Gap and Lattice Constants
2.6
AlP
0.517
GaP
2.2
0.563
AlAs
0.620
2.0
Energy gap (eV)
1.8
0.689
AlSb
1.6
0.775
GaAs (D)
1.4
1.2
0.885
InP(D)
1.033
Si
1.0
1.240
0.8
2.067
Indirect band gap
0.4
1.550
GaSb (D)
Ge
0.6
3.100
InAs (D)
Direct band gap (D)
0.2
0
0.54
Wavelength (microns)
2.4
0.477
6.200
InSb (D)
0.55
0.56
0.57
0.58
0.59
0.60
0.61
0.62
0.63
0.64
0.65
Lattice constant (nm)
Figure by MIT OpenCourseWare.
©1999 E.A. Fitzgerald
Lattice Constant (A)
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