Nanophysics III

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Nanophysics III

• Michael Hietschold
• Solid Surfaces Analysis Group
• Electron Microscopy Laboratory
• Institute of Physics

Portland State University, May 2005
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Remark 1 concerning the nature of the
one-dimensional states   ?(r) f(z)
exp i(kxx kyy) Eµ Ei h2/2m (kx2
ky2)
EIII
E
EII
EI
The 1d levels correspond to the bottoms of
(sub-)bands
k-
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Resonance tunneling structure 50 periods
of 72 Å n-GaAs quantum well 39 Å undoped
Al0.33Ga0.67As barrier 18 Å undoped GaAs quantum
well, 154 Å undoped Al0.33Ga0.67As
barrier Barriers asymmetrically designed to
have the same transmittivity for a forward bias
IR illuminating causes a photo- current
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3rd Lecture 4.3. Superlattices5.
2-Dimensional Electron Gas6. Quantum
Interference, Molecular Devices, and
Self-Assembling7. Outlook
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4.3.Superlattices
1970 Esaki Tsu Periodic nm-scale
superperiodicity superimposed on the atomic-scale
lattice periodicity Kronig-Penney model !
contravariant covariant
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  A1 e ? x B1 e ? x
( - l a x lt 0 ) f(x)
A2 e ik x B2 e ik x
( 0 x lt a ) V(x l) V(x) ? f(x
l) ? f(x) ? 1, i.e. ? e
iKl e iK l A1 e ? (x-l) B1 e
-? (x-l) ( a x lt l ) f(x)

e iK l A2 e ik (x-l) B2 e ik
(x-l) ( l x lt l a ) Continuous
1st derivative cos Kl cos ka cosh ?b
(?2 k2) / (2?k) sin ka sinh ?b  
cos Kl lt 1 ? energy bands -
subbandstructure inside the ordinary
bands
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Sub-band structure engineering
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http//techtransfer.nrl.navy.mil/exhibits/pdfs/Inf
o20Sheet20pdfs/Nanotechnology/STM_Superlattice_2
.pd
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http//www.pa.uky.edu/straley/poster/Xu-DeLong200
4.pdf
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5. 2-Dimensional Electron Gas
Free-Electron Gases noninteracting free
particles   ?k,s ?(sz) fk(r)   fk(r)
1/vV exp ikr, k (kx,ky,kz) 3-dimensional  
1/vA exp ikr, k (kx,ky) 2-dimensional  
Single-particle energies   e(k) e0 h2k2/2m
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• Single-particle density
•  
• n N/V
•  
• N/A

• Wigner-Seitz radius rs
•  
• V N 4p/3 rs3
•  
• rs 3v3/4pn
•  
• A N p rs2
• rs 1/vpn

  Density of states in k-space   Z(k) 2
V/(2p)3   Z(k) 2 A/(2p)2
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Energy density of states D(e) dN/de   D(e)
Z(k)/V ?3k(e)/de   1/4p3
4pk-2dk-/de   1/p2 2m/h2 (e e0)
v2m/h2 1/2v(e e0)   1/2p2
(2m/h2)3/2 v(e e0)   D(e) Z(k)/A
?2k(e)/de 1/2p2 2pk-dk-/de
1/p v2m/h2 v(e e0) v2m/h2 1/2v(e e0)
2m/ph2 ?(e e0),
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D(e)   1/2p2 (2m/h2)3/2 v(e e0) 3d D(e)
2m/ph2 ?(e e0) 2d
D(e)
e
eF
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Dimensionality and density of states
http//www.mtmi.vu.lt/pfk/funkc_dariniai/nanostruc
tures/superlattice.htm
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The single-particle states are filling the
k-space according to Paulis-principle till to a
maximum value of kF and a maximum energy eF
(Fermi energy).EF E0 h2kF2/2m
 Occupied Fermi sphereFermi circle
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2 DEG in accumulation layers
Advantage carriers from heavily doped areas
collected in weakly doped regions ? increase of
mobility
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Electron gases in a magnetic field
Classical Hall effect
Ey RH B jx, Cross section area A b d (d
depth of channel) UH / b RH B Ix / b d  
I B / e ns(Ugs) RH 1 / Ugs
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Quantum-Hall Effect
Measurement in the gate channel of a FET
structure at low temperature and strong magnetic
field
  h / 2e2   h / 3e2   h /
4e2
RH
Ugs
Quantization of Hall resistance in fractions
of h/e2 25.812,8 O with extreme accuracy
K.v.Klitzing et al. 1981
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Electron gases in a magnetic field
3d Fermi sphere splits off into concentrical
Landau cylinders
Instead of a homogeneous occupation in k-space
there are only electron states on the cylinder
surfaces
2d Fermi circle splits of into concentrical
Landau halos
If there is such a change in the magnetic field
that the number of cylinders/halos inside the
occupied region changes than there is a major
redistribu- tion of all the electron states !!!
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Density of states in the magnetic field
D(E)
H 0
E
En E0 h ?c (n 1/2) h ?L sz
Number of states per circle ? e B /
h In 2d  RH UH / I B / e ns B /
e i ? h / e2 i plateaus correspond to
maximal filling
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2DEG and measurement of QHE
FET structure
2 DEG Landau halos DOS Energies
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Fractional QHE
Still lower T and still higher fields (Störmer,
Tsui, Gossard 1982)
Plateaus corresponding to i 1/3 2/3
1/5 2/5 3/5 4/5 1/7 2/7 3/7 ...
1/9 ... 1/11 ... 1/13 ...
1/15 ... Electrons form a new state of matter
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6. Quantum Interference, Molecular Devices, and
Self-Assembling
Aharonov-Bohm Effect Interferometer for
electron wave functions in magnetic
field external magnetic field
interpenetrating a loop phase shift
between left- and right- hand going electron
wave ? dr A(r) - ? dr A(r) ? dS
?/?r x A(r) ? dS B(r) F (1)
(2) Constructive interference for F
2 p n h / e n F0
B
S
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Experimental periodic fluctuations of
conductance on nanostructured Au loops which
depend on cross sectional area
G
H
? Logical circuits switched by magnetic fields
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Molecular devices
Single molecules (coronene) rotating in a
self-assembled molecular mesh (interconnected
trimesic acid on a graphite surface) Molecular
wires molecular diodes ... Help from Life
Sciences assembling reproduction repair ...
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7. Outlook
Nanoscience has a big future (very complex tasks
can be solved with very little amount of
material and energy) Nanoscience is complex and
interdisciplinary (physics, chemistry,
engineering, life sciences, ...) Nanoscience
needs revolutionary ideas and enthuisasm Nanoscie
nce should attract the best young people
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References F.Capasso Physics of Quantum
Electron Devices, Springer 1990 H.Lüth Surfaces
and Interfaces of Solids, Springer
1993 Y.Murayama Mesoscopic Systems, Wiley VCH
2001 H.-J.Butt, K.Graf, M.Kappl Physics and
Chemistry of Interfaces, Wiley VCH 2003 New
books of Bushan