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

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

I

U

NDR