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Quantum Confinement in Nanostructures
Confined in:
1 Direction: Quantum well (thin film)
Two-dimensional electrons
2 Directions: Quantum wire
One-dimensional electrons
3 Directions: Quantum dot
Zero-dimensional electrons
Each confinement direction converts a continuous k in a discrete quantum number n.
kx
nz
ny
ny
nz
nx
kx
ky
nz
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N atomic layers with the spacing a = d/n
N quantized states with kn ≈ n /d ( n = 1,…,N )
Quantization in a Thin Crystal
An energy band with continuous k
is quantized into N discrete points kn
in a thin film with N atomic layers.
n = 2d / n
kn = 2 / n = n
/d
d
E
0 /a/d
EFermi
EVacuum
Photoemission
Inverse Photoemission
Electron Scattering
k= zone
boundary
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N atomic layers with spacing a = d/n :
N quantized states with kn ≈ N /d
Quantization in Thin Graphite FilmsE
0 /a/d
EFermi
EVacuum
Photoemission
Lect. 7b, Slide 11
k
1 layer = graphene
2 layers
3 layers
4 layers
layers = graphite
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Quantum Well States
in Thin Films
discrete for small N
becoming continuous for N
Paggel et al.Science 283, 1709 (1999)
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10
16
16
16
16
16
16
13
14
14
11.5
13
1413
14
h (eV)Ag/Fe(100)
Binding Energy (eV)
012
Photo
emiss
ion In
tensity
(arb.
units)
1
2
3
4
5
6
7
8
9
10
11
13
14
15
12
N
Temperature (K)
100 200 300
Line W
idth (e
V)
0.1
0.2
0.3
(N, n')
(3, 1)(7, 2)(12, 3)(13, 3)
(2, 1)
1
3
2
4
10
16
16
16
16
16
16
13
14
14
11.5
13
1413
14
h (eV)Ag/Fe(100)
Binding Energy (eV)
012
Photo
emiss
ion In
tensity
(arb.
units)
1
2
3
4
5
6
7
8
9
10
11
13
14
15
12
N
Temperature (K)
100 200 300
Line W
idth (e
V)
0.1
0.2
0.3
(N, n')
(3, 1)(7, 2)(12, 3)(13, 3)
(2, 1)
1
3
2
4
Periodic Fermi level crossing of quantum well states with
increasing thickness
Counting Quantum Well States
Number of monolayers N
n
Thickness N (ML)0510152025Wo
rk F
un
ctio
n (
eV)
4.34.4
Bin
din
g E
ner
gy
(eV
) 01212345678(a) Quantum Well States for Ag/Fe(100)
(b)n
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Kawakami et al.Nature 398, 132 (1999)
HimpselScience 283, 1655 (1999)
Quantum Well Oscillations in Electron Interferometers
Fabry-Perot interferometer model: Interfaces act like mirrors for electrons. Since electrons have so short wavelengths, the interfaces need to be atomically precise.
n
12
34
56
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The Important Electrons in a Metal
Energy EFermi
Energy Spread 3.5 kBT
Transport (conductivity, magnetoresistance, screening length, ...)
Width of the Fermi function:
FWHM 3.5 kBT
Phase transitions (superconductivity, magnetism, ...)
Superconducting gap:
Eg 3.5 kBTc (Tc= critical temperature)
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Energy Bands of Ferromagnets
States near the Fermi level cause
the energy splitting between
majority and minority spin bands
in a ferromagnet (red and green).-10
-8
-6
-4
-2
0
2
4
XK
Ni
En
erg
y R
ela
tiv
e t
o E
F [
eV
]
0.7 0.9 1.1
k|| along [011] [Å-1 ]
Calculation Photoemission data
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(Qiu, et al. PR B ‘92)
Quantum Well States and Magnetic Coupling
The magnetic coupling between layers plays a key role in giant magnetoresistance (GMR), the Nobel prize winning technology used for reading heads of hard disks.
This coupling oscillates in sync with the density of states at the Fermi level.
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Minority spins discrete,Majority spins continuous
Magnetic interfaces reflect the two spins differently, causing a spin polarization.
Spin-Polarized Quantum Well States
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Filtering mechanisms
• Interface: Spin-dependent Reflectivity Quantum Well States
• Bulk: Spin-dependent Mean Free Path Magnetic “Doping”
Parallel Spin Filters Resistance Low
Opposing Spin Filters Resistance High
Giant Magnetoresistance and Spin - Dependent Scattering
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Giant Magnetoresistance (GMR): (Metal spacer, here Cu)
Tunnel Magnetoresistance (TMR): (Insulating spacer, MgO)
Magnetoelectronics
Spin currents instead of charge currents
Magnetoresistance = Change of the resistance in a magnetic field