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 sur- face 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 ener- getically favour special surface reconstruc- tions

3.4. Structure of Interfaces General Principle: µ 1 = µ 2 in thermodynamic equilibrium 12 For electrons this means, there should be a common Fermi level !

Metal-Metal Interfaces Adjustment of Fermi levels – Contact potential ΔV 12 = Φ 2 – Φ 1

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. E c1 E c2 E v1 E v2 E F1 E F2 E F

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 z E EcEc EvEv Ideal crystalline interfaces – Epitaxy GaAs/Al x Ga 1-x As

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 - V 0 0 E -a 0 a [ - ħ 2 /2m d 2 /dx 2 + V(x) ] φ(x) = E φ(x) solving by ansatz method A + cos (kx)| x | < a φ + (x) = A + cos (ka) e κ (a - x) x > a A + cos (ka) e κ (a + x) x < - a, A - sin (kx)| x | < a φ - (x) = A - sin (ka) e κ (a - x) x > a - A - sin (ka) e κ (a + x) x < - a κ = √ - 2m E / ħ 2,k = √ 2m {E – (- V 0 )} / ħ 2.

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 C 2 = 2mV 0 / ħ 2 a 2. 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 C 2 = 2mV 0 / ħ 2 a 2

Quantum Dots – Superatoms (spherical symmetry) Can be prepared e.g. by self-organized island growth

E V(x) V0V0 s 4.2. Tunneling Structures Tunneling through a potential well

Tunneling probability Wave function within the wall (classically „forbidden“) φ in wall ~ exp (- κ s); κ = √2m(V 0 -E)/ħ 2 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 double- barrier T goes to 1 !!! Interference effect similar to Fabry-Perot interferometer

I-V characteristics shows negative differential resistance I U NDR