1. Solid Surfaces Analysis Group & Electron Microscopy Laboratory
Nanophysics III
Michael Hietschold
Solid Surfaces Analysis Group &
Electron Microscopy Laboratory
Institute of Physics
Portland State University, May 2005

2. Remark 1: concerning the nature of the „one-dimensional“ states:
ψ(r) = φ(z) exp i(kxx + kyy)
Eµ = Ei + ħ2/2m (kx2 + ky2)
EIII E
EII EI
The „1d levels“ correspond
to the bottoms of (sub-)bands k┴

3. 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

4. 3rd Lecture 4. 3. Superlattices 5. 2-Dimensional Electron Gas 6
3rd Lecture Superlattices Dimensional Electron Gas 6. Quantum Interference, Molecular Devices, and Self-Assembling 7. Outlook

5. 4.3.Superlattices 1970 Esaki & Tsu: Periodic nm-scale superperiodicity
superimposed on the atomic-scale
lattice periodicity
Kronig-Penney model !
contravariant covariant

6. inside the „ordinary“ bands
A1 e κ x + B1 e – κ x ( - l + a ≤ x < 0 )
φ(x) =
A2 e ik x + B2 e – ik x ( 0 ≤ x < a )
V(x + l) = V(x)  φ(x + l) = λ φ(x) | λ | = 1, i.e. λ = e iKl
e iK l [ A1 e κ (x-l) + B1 e -κ (x-l) ] ( a ≤ x < l )
e iK l [ A2 e ik (x-l) + B2 e –ik (x-l) ] ( l ≤ x < l + a )
Continuous 1st derivative:
cos Kl = cos ka cosh κb + [(κ2 – k2) / (2κk)] sin ka sinh κb
| cos Kl | < 1
 „energy bands“ - subbandstructure
inside the „ordinary“ bands

7. Sub-band structure engineering:

8. http://techtransfer. nrl. navy

10. 5. 2-Dimensional Electron Gas
Free-Electron Gases: noninteracting free particles
ψk,s = χ(sz) φk(r)
φk(r) = 1/√V exp ikr, k = (kx,ky,kz) 3-dimensional
= 1/√A exp ikr, k = (kx,ky) 2-dimensional
Single-particle energies
ε(k) = ε0 + ħ2k2/2m

11. Single-particle density n = N/V = N/A Wigner-Seitz radius rs
Single-particle density
n = N/V
= N/A
Wigner-Seitz radius rs
V = N 4π/3 rs3
rs = 3√3/4πn
A = N π rs2
rs = 1/√πn
Density of states in k-space
Z(k) = 2 V/(2π)3
Z(k) = 2 A/(2π)2

12. D(ε) = Z(k)/A Δ2k(ε)/dε Energy density of states D(ε) = dN/dε
D(ε) = Z(k)/V Δ3k(ε)/dε
= 1/4π3 4πk┴2dk┴/dε
= 1/π2 2m/ħ2 (ε – ε0) √2m/ħ2 1/2√(ε – ε0)
= 1/2π2 (2m/ħ2)3/2 √(ε – ε0)
D(ε) = Z(k)/A Δ2k(ε)/dε
= 1/2π2 2πk┴dk┴/dε
= 1/π √2m/ħ2 √(ε – ε0) √2m/ħ2 1/2√(ε – ε0)
= 2m/πħ2 θ(ε – ε0),

13. D(ε) = 1/2π2 (2m/ħ2)3/2 √(ε – ε0) 3dεF

14. Dimensionality and density of states

15. The single-particle states are filling the k-space according to Pauli’s-principle till to a maximum value of kF and a maximum energy εF (Fermi energy). EF = E0 + ħ2kF2/2m   Occupied Fermi sphere Fermi circle

16. 2 DEG in accumulation layers
Advantage:
carriers from heavily doped
areas collected in weakly
doped regions
 increase of mobility

17. 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

18. 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 = ,8 Ω with extreme accuracy
K.v.Klitzing et al. 1981

19. 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 !!!

20. Density of states in the magnetic field
En = E0 + ħ ωc (n + 1/2) + ħ ωL σz ;
Number of states per circle: η = e B / ħ
In 2d:  RH = | UH | / I = B / e ns = B / e i η = h / e2 i
plateaus correspond to maximal filling

21. 2DEG and measurement of QHE
FET structure
2 DEG
Landau halos
DOS
Energies

22. 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

23. 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’) = Φ
(1) (2)
Constructive interference for Φ = 2 π n ħ / e = n Φ0 B S

24. Experimental: periodic fluctuations of conductance on
nanostructured Au loops which depend
on cross sectional area G H
 Logical circuits switched by magnetic fields

25. 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; ...

26. 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
Nanoscience should attract the best young people

27. 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