Nanophysics II
Michael Hietschold
Solid Surfaces Analysis Group &
Electron Microscopy Laboratory
Institute of Physics
Portland State University, May 2005
2nd Lecture
3b. Surfaces and Interfaces – Electronic
Structure
3.3. Electronic Structure of Surfaces
3.4. Structure of Interfaces
4. Semiconductor Heterostructures
4.1. Quantum Wells
4.2. 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 surface normal
k┴ not any longer a good
quantum number
- Projected bulk bands
- Surface state bands
Surface States
Two types of
electronic states:
- Truncated bulk states
- Surface states
Surface states splitting from semiconductor bulk
bands may act as additional donor or acceptor states
Interplay with Surface Reconstruction
The appearance and
occupation of surface
state bands may energetically favour special
surface reconstructions
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
Ec2
Ec1
EF1
EF2
Ev1
EF
Ev2
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.
4. Semiconductor Heterostructures
4.1. Quantum Wells
4.2. Tunnelling Structures
4.3. Superlattices
4.1. Quantum Wells
Effective potential structures consisting of well defined
semiconductor-semiconductor interfaces
E
Ideal crystalline interfaces –
Epitaxy
Ec
Ev
z
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
[ - ħ2/2m d2/dx2 + V(x) ] φ(x) = E φ(x)
0
E
- V0
solving by ansatz method
-a
0
a
φ+(x) =
A+ cos (kx)
A+ cos (ka) eκ (a - x)
A+ cos (ka) eκ (a + x)
|x| < a
x > a
x < - a,
φ-(x) =
A- sin (kx)
A- sin (ka) eκ (a - x)
- A- sin (ka) eκ (a + x)
|x| < a
x > a
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 represenation

discrete 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-organized
island growth
4.2. Tunneling Structures
Tunneling through a potential well
V(x)
V0
E
s
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 eV
there 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 doublebarrier
T goes to 1 !!!
Interference effect
similar to Fabry-Perot
interferometer
I-V characteristics shows negative differential resistance
I
NDR
U