Transcript Lecture Energy Bands

Electrons in Solids
Simplest Model: Free Electron Gas
Quantum Numbers E,k
Fermi “Surfaces”
Beyond Free Electrons:
Bloch’s Wave Function
E(k) Band Dispersion
Angle-Resolved Photoemission
Free Electron Gas
Quantum numbers of electrons in a solid: E, kx ,ky , (kz)
Two cuts:
Fermi “surface”: I(kx, ky)
Band dispersion: I(E, kx)
E(k) = ħ2 k2/2m = Paraboloid
E
Fermi circle
Band
dispersion
kx
ky
Fabricating a Two-Dimensional Electron Gas
Lattice planes
V(z)
Inversion Layer
(z)
Bulk Hamiltonian +
boundary conditions
V(z)
Doped Surface State
Surface Hamiltonian
(z)
Measuring E, kx,ky in a Two-Dimensional Electron Gas
Fermi Surface I(ky,kx)
0.012
0.015
0.022
ky
e-/atom: 0.0015
kx
Band Dispersion I(E,kx)
0.086
Superlattices of Metals on Si(111)
1 monolayer Ag is
semiconducting:
3x3
Surface doping:
2  1014 cm-2
Equivalent bulk doping: 3  1021 cm-3
Add extra Ag, Au
as dopants:
21x21
Fermi Surface of a Superlattice
Fermi circles are diffracted by the superlattice.
Corresponds to momentum transfer from the lattice.
ky
Angle-resolved
photoemission
data
ky
kx
Model using
Diffraction
kx
Fermi “surfaces” of two- and one-dimensional electrons
ky
2D
2D +
superlattice
1D
kx
One-Dimensional Electrons at
Semiconductor Surfaces
Beyond the Free Electron Gas
E(k) Band Dispersion
Band Dispersion in a Semiconductor
E
(eV)
Band gap
Empty
lattice
bands
[111]
Density of states
[100]
[110]
Wave vector
Two-dimensional bands of graphene

E [eV]
Empty
EFermi
Occupied
Empty *
K
M
Occupied 
K
=0
M
kx,y
K
“Dirac cones”
in graphene
A special feature of the
graphene -bands is their
linear E(k) dispersion near
the six corners K of the
Brillouin zone (instead of
the parabolic relation for
free electron bands).
In a plot of E versus kx,ky
one obtains cone-shaped
energy bands at the Fermi
level.
Topological Insulators
A spin-polarized version of a “Dirac cone”
occurs in “topological insulators”. These
are insulators in the bulk and metals at
the surface, because two surface bands
bridge the bulk band gap. It is impossible
topologically to remove the surface bands
from the gap, because they are tied to
the valence band on one side and to the
conduction band on the other.
The metallic surface state bands
have been measured by angle- and
spin-resolved photoemission (left).
Photoemission
(PES, UPS, ARPES)

EFermi
• Measures an “occupied state” by creating a hole
• Determines the complete set of quantum numbers
• Probes several atomic layers (surface + bulk)
Measuring the quantum numbers E,k of electrons in a solid
The quantum numbers E and k can be measured by angle-resolved
photoemission. This is an elaborate use of the photoelectric effect,
which was explained as quantum phenomenon by Einstein in 1905.
Photon in
Electron outside
(final state)
Electron inside
(initial state)
Energy and momentum of the emitted photoelectron are measured.
Energy conservation:
Momentum conservation:
Efinal = Einitial + h
k||final = k||initial + G||
h = Ephoton
kphoton  0
Only k|| is conserved (surface!)
Photoemission setup:
D(E)
Einitial
Efinal
h
Photoemission process:
Efinal
Photoemission spectrum:
e counts
EF+h
Core
Valence
Secondary
electrons
W = width
Efinal
Spectrometer with E,kx - multidetection
50 x 50 = 2500 Spectra in One Scan
E,k Multidetection: Energy Bands on a TV Screen
E
Calculated E(k)
4
Ni
(eV)
Measured E(k)
EF=
2
EF= 0
-2
0.7
0.9
1.1
(Å1)
-4
Electrons within ±2kBT of the Fermi level EF
-6
are not locked in by the Pauli principle. This is
the width of the Fermi-Dirac cutoff at EF.
-8
-10

K
Xk
Spin-split Bands
in a Ferromagnet
These electrons determine magnetism, superconductivity, specific heat in metals, …
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