10p PhD Course 18 Lectures Nov-Dec 2011 and Jan – Feb 2012 Literature Semiconductor Physics – K. Seeger The Physics of Semiconductors – Grundmann Basic Semiconductors Physics - Hamaguchi Electronic and Optoelectronic Properties of Semiconductors - Singh Quantum Well Wires and Dots – Hartmann Wave Mechanics Applied to Semiconductor Heterostructures - Bastard Fundamentals of Semiconductor Physics and Devices – Enderlein & Horing Examination Homework Problems (6p) Written Exam (4p) Additionally Your own research area. Background courses (Solid State Physics, SC Physics, Sc Devices)

Semiconductor Physics

1. Introduction 2. Crystal and Energy Band structure 3. Semiconductor Statistics 4. Defects and Impurities 5. Optical Properties I : Absorption and Reflection 6. Optical Properties II : Recombinations 7. Carrier Diffusion 8. Scattering Processes 9. Charge Transport 10. Surface Properties 11. Low Dimensional Structures 12. Heterostructures 13. Quantum Wells/Dots 14. Organic Semiconductors 15. Graphene 16. Reserve and Summary

Course Layout

Based on : Semiconductor Physics, K. Seeger, Chapter 11. Semiconductor Optics, C.F. Klingshirn, Ch. 9-14 Fundamentals of Semiconductors, Yu, Cardona, Chapter 6. The Physics of Semiconductors, Grundmann, Chapter 9.

Lecture Layout Basic Theory Light Interaction with Materia Dielectricity function Refraction and Absorption Lattice Vibrations Excitons Absorbtion Fundamental Bandedge Excitonic Defect Related Free carrier Non-linear Effects

Electromagnetic Theory Maxwells equation

E = Electric Field ; D = Electric Displacement; ρ = Charge density H = Magnetic Field ; B = Magnetic Flux density ; j = electrical current density

Electromagnetic Theory Phase and group velocities

Electromagnetic wave in matter

P = Polarisation Density, M = Magnetisation Density

Dielectric function

Electromagnetic Theory

Reflection and Absorption

The dielectric function determines with the Fresnels Equation the conditions for refraction and diffracted waves during different conditions. Distinction or absorbtion of the intensity in a wave Is given by

α(ω) = 4π/k I(ω,t) = Io exp(- α(ω) x)

Optical Properties of Semiconductors

The optical properties of matter is determined by coupling of various oscillators to the electromagnetic field. In semiconuctors these oscillators are phonons Excitons Polaritons The amplitude of these oscillations is depend on the

angular frequency ω of the incident field, the coupling strength between the field and the oscillator f the eigenfreuency of the oscillator ω0 and its damping γ

Optical Properties of Semiconductors

Lattice Vibrations Phonons

Adiabatic Approximation : Electronic and Lattice movementys is de-coupled. The movement of an atom around its equilibrium position is approximated with a harmonic oscillator.

Lattice Vibrations Phonons Dispersion relation for a diatomic linear chain. The lower branch is called acoustic branch. The upper branch is called optical branch. In the acoustic branch the two atoms oscillate in the same direction. For the optical branch the two atoms oscillate in opposite directions. If the atom is at least partly ionic the atoms carry charge and connected to an electric dipole which can couple to an electromagnetic light fileld.

Lattice Vibrations : Phonons

In three dimensions: Similar to the linear chain. 3 acoustic branches, LA, TA1 and TA2 3s-3 optical branches S is number of atoms per unit cell. Phonons are bosons anf follows Bose-Einstein statistics

Localized Phonons

Around crystal defects, Point defects Complex defects Structural defects

Local crystal vibration modes can occur. These involved the nearest atoms and are localized in the brillouin zone. They have often higher energies than lattice phonons and are important in order to identify local defect structure and symmetri

Excitons Exciton is a coupled electron-hole pair. Energy is reduced compared to free particles du to the coulomb effect. Hydrogen-like models. Exciton Bohr Radius

Excitons Wannier excitons : Exciton Bohr radies larger than size of the unit cell. Frenkel Excitons: When electron-hole pair wavefunctions localized within

one unit cell, the effective mass approach is not possible.

Biexcitons: Two electron-hole pairs. Comparable to Hydrogenic

molecules. Trions : Charge excitions. For example, Two electrons and one

hole.

Polaritons

The coupling between an exciton and an electromagnetic field is called a Polariton. There is an anti-crossing between the dispersion curve of the photon and the exciton. UPB upper polariton branch LPB lower polariton branch Polariton Bottleneck

Plasmons A semiconductor with a high amount of carriers (1017 – 1019 cm-3) A displacement of the electron gas causes a surface charge density ρs = ne dx An electric field is formed towards the remaining negative part. Which is reflected back to the electrons. Leading to an harmonic oscillation of plasma oscillations, called plasmons. Plasmons and phonons interact via their electric field. Causing non-crossing between the transverse and longitudinell optical phono branch.

Bandgap Absorbtion

Absorbtion : Direct Bandgap

Bandgap Absorbtion : Direct Bandgap

Rapid increase of absorption at the bandgap. The bandedge is easily extracted from the absorption curve. The increase of absorption follows the increase in DOS in the bands.

Values of absorption in the order of 104 cm-1

Urbach Tail

Deviations from the (E-EG)1/2

With an exponential tail is called the Urbach Tail

E0 is the Urbach parameter which increases with temperature. The Urbach tail is attributed to band tails below the bandedges. These originates from disorder of the perfect crystal, from defects and doping, or from due to change of electronic energy bands to to lattice vibrations.

Exciton Absorption

Theoretical absorption spectra of GaAs excluding and including the excitonic effect.

Exciton Absorption

At increasing temperature the excitonic influence decreases due to the small EX binding energy, and is wept out by phonons. For GaAs EX = 3.4 meV.

Absorption: Bound Excitons

High Purity GaAs: Absorbtion due to Free excitons, and their excited states, as well as Bound excitons are separated at low temperature.

Absorption: Excitons

Absorption: Bound Excitons Excitons can also be localized at impurities, forming Bound Excitons. One example is GaP:N where an exciton is bound at the isolectronic N at a P-site, the A-exciton. At higher doping Nitrogen also forms complexes with two or more Nitrogen atoms.

Absorbtion : Indirect Bandgaps

Absorption in indirect bandgap

semiconductors less efficient, in the order of 10 cm-1

Absorption close to linear with energy due to the absence of pure photon absoption. More difficult to extract actual bandgap.

Absorbtion : Indirect Bandgaps

Summary