Nanophysics IIINanophysics III

Michael Hietschold

Solid Surfaces Analysis Group &

Electron Microscopy Laboratory

Institute of Physics

Portland State University, May 2005

Remark 1: concerning the nature of the „one-dimensional“ states:  ψ(r) = φ(z) exp i(kxx + kyy)

Eµ = Ei + ħ2/2m (kx2 + ky

2) E

k┴

EI

EII

EIII

The „1d levels“ correspondto the bottoms of (sub-)bands

Resonance tunneling structure:

50 periods of72 Å n-GaAs quantum well39 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

3rd Lecture

4.3. Superlattices

5. 2-Dimensional Electron Gas

6. Quantum Interference, Molecular Devices, and Self-Assembling

7. Outlook

4.3.Superlattices

1970 Esaki & Tsu:Periodic nm-scale superperiodicitysuperimposed on the atomic-scale lattice periodicity

Kronig-Penney model !

contravariant covariant

  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 )

φ(x) = 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

Sub-band structure engineering:

http://techtransfer.nrl.navy.mil/exhibits/pdfs/Info%20Sheet%20pdfs/Nanotechnology/STM_Superlattice_2.pd

http://www.pa.uky.edu/~straley/poster/Xu-DeLong2004.pdf

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

  Single-particle density n = N/V  = N/A

Wigner-Seitz radius rs

 V = N 4π/3 rs

3

 rs = 3√3/4πn

 A = N π rs

2

rs = 1/√πn

 

Density of states in k-space 

Z(k) = 2 V/(2π)3

 Z(k) = 2 A/(2π)2

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),

D(ε)  = 1/2π2 (2m/ħ2)3/2 √(ε – ε0) 3d

D(ε) = 2m/πħ2 θ(ε – ε0) 2d

D(ε)

ε

εF

http://www.mtmi.vu.lt/pfk/funkc_dariniai/nanostructures/superlattice.htm

Dimensionality and density of states

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

2 DEG in accumulation layers

Advantage:

carriers from heavily doped areas collected in weakly doped regions

increase of mobility

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

Quantum-Hall EffectMeasurement in the gate channel of a FET structure at low temperature and strong magnetic field

K.v.Klitzing et al. 1981

RH

Ugs

 h / 2e2

 

h / 3e2

  h / 4e2

Quantization of Hall resistance in fractions of

h/e2 = 25.812,8 Ω with extreme accuracy

Electron gases in a magnetic field

3d: Fermi sphere splits off into concentrical Landau cylinders

Instead of a homogeneousoccupation 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 thatthe number of cylinders/halos inside the occupied region changes – than there is a major redistribu-tion of all the electron states !!!

Density of states in the magnetic field

H = 0

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

E

D(E)

2DEG and measurement of QHE

FET structure

2 DEG

Landau halos

DOS

Energies

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

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

Experimental: periodic fluctuations of conductance on nanostructured Au loops which depend on cross sectional area

H

G

Logical circuits switched by magnetic fields

Molecular devices:

Single molecules (coronene) rotating in a self-assembled molecular mesh (interconnected trimesic acid on a graphitesurface)

Molecular wires; molecular diodes; ...

Help from Life Sciences: assembling; reproduction; repair; ...

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

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