nanophysics ii michael hietschold solid surfaces analysis group & electron microscopy laboratory...
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3b. Surfaces and Interfaces – Electronic Structure 3.3. Electronic Structure of Surfaces 3.4. Structure of InterfacesTRANSCRIPT
Nanophysics IINanophysics IIMichael Hietschold
Solid Surfaces Analysis Group &Electron Microscopy Laboratory
Institute of Physics
Portland State University, May 2005
2nd Lecture
3b. Surfaces and Interfaces – Electronic Structure3.3. Electronic Structure of Surfaces3.4. Structure of Interfaces4. Semiconductor Heterostructures4.1. Quantum Wells4.2. Tunnelling Structures
3b. Surfaces and Interfaces – Electronic Structure
3.3. Electronic Structure of Surfaces
3.4. Structure of Interfaces
3.3. Electronic Structure of Surfaces
Projected Energy Band Structure:
Lattice not any longer periodic along the sur-face normal
k┴ not any longer a goodquantum number
- Projected bulk bands
- Surface state bands
Surface States
Two types of electronic states:
- Truncated bulk states
- Surface states
Surface states splitting from semiconductor bulkbands may act as additional donor or acceptor states
Interplay with Surface Reconstruction
The appearance and occupation of surfacestate bands may ener-getically favour specialsurface reconstruc-tions
3.4. Structure of Interfaces
General Principle:µ1 = µ2 in thermodynamic equilibrium
1 2
For electrons this means, there should be a common Fermi level !
Metal-Metal Interfaces
Adjustment of Fermi levels –
Contact potential
ΔV12 = Φ2 – Φ1
Metal – Semiconductor Interfaces
Small density of free electrons in the semiconductor –
Considerable screening length (Debye length) –
Band bending
Schottky barrier at the interface
Semiconductor-Semiconductor Interfaces
Within small distances from the interface (and at low doping levels)
- band bending may be neglected
- rigid band edges; effective square-well potentials for the electrons and holes.
Ec1 Ec2
Ev1
Ev2
EF1 EF2 EF
4. Semiconductor Heterostructures
4.1. Quantum Wells
4.2. Tunnelling Structures
4.3. Superlattices
4.1. Quantum Wells
Effective potential structures consisting of well definedsemiconductor-semiconductor interfaces
z
E
Ec
Ev
Ideal crystalline interfaces –Epitaxy
GaAs/AlxGa1-xAs
Preparation by Molecular Beam Epitaxy (MBE)
Allows controlled deposition of atomic monolayers and complex structures consisting of them
- UHV- slow deposition (close to equilibrium)- dedicated in-situ analysis
One-dimensional quantum well – from a stupid exercise inquantum mechanics (calculating the stationary bound states)for a fictituous system to real samples and device structures
- V0
0
E
-a 0 a
[ - ħ2/2m d2/dx2 + V(x) ] φ(x) = E φ(x)
solving by ansatz method
A+ cos (kx) | x | < aφ+(x) = A+ cos (ka) eκ (a - x) x > a
A+ cos (ka) eκ (a + x) x < - a,
A- sin (kx) | x | < aφ-(x) = A- sin (ka) eκ (a - x) x > a - A- sin (ka) eκ (a + x) x < - a
κ = √ - 2m E / ħ2, k = √ 2m {E – (- V0)} / ħ2 .
From stationary Schroedinger`s equation (smoothly matching the ansatz wave functions as well as their 1st derivatives):
| cos (ka) / ( ka ) | = 1 / C tan (ka) > 0
| sin (ka) / (ka) | = 1 / C tan (ka) < 0
C2 = 2mV0 / ħ2 a2 .
Graphical represenationdiscrete stationary solutions
1 / C
Finite number of stationary bound states
Eigenfunctions and energy level spectrum
Dependence of the energy spectrum on the parameter
C2 = 2mV0 / ħ2 a2
Quantum Dots – Superatoms (spherical symmetry)
Can be prepared e.g.by self-organizedisland growth
E
V(x)V0
s
4.2. Tunneling Structures
Tunneling through a potential well
Tunneling probability
Wave function within the wall (classically „forbidden“)
φin wall ~ exp (- κ s); κ = √2m(V0-E)/ħ2
Transmission probability
T ~ |φ(s)|2 ~ exp (- 2 κ s)
For solid state physics barrier heights of a few eVthere is measurable tunneling for s of a few nm only.
Resonance tunneling
double-barrier structure
If E corresponds to the energy of a (quasistationary)state within the double-barrierT goes to 1 !!!
Interference effectsimilar to Fabry-Perotinterferometer
I-V characteristics shows negative differential resistance
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