Loading in 2 Seconds...
Loading in 2 Seconds...
Lectures on the Basic Physics of Semiconductors and Photonic Crystals. References 1. Introduction to Semiconductor Physics, Holger T. Grahn, World Scientific (2001) 2. Photonic Crystals, John D. Joannopoulos et al, 2nd Ed. Princeton University Press (2008) . 2009. 09. Hanjo Lim
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
1. Introduction to Semiconductor Physics, Holger T. Grahn, World Scientific (2001)
2. Photonic Crystals, John D. Joannopoulos et al, 2nd Ed. Princeton University Press (2008)
School of Electrical & Computer Engineering
Lecture 1. Overview on Semiconductors and PhCs
- Structural dependence : existence or nonexistence of translational vector , depends on how to make solids
- main difference between liquid and solid; atomic motion
* liquid crystals (nematic, smetic, cholestoric)
- (superconductors), conductors(metals), (semimetals), semiconductors, insulators
- Difference of material properties depending on the structure
* metals, semiconductors, insulators : different behaviors
- metals, semiconductors, insulators
* temperature dependence of electrical conductivity,
conductivity dependence on doping
- Wide bandgap SC, Narrow bandgap SC,
- Elemental semiconductors : group IV in periodic table
- Compound semiconductor : III-V, II-VI, SiGe, etc
* binary, ternary, quaternary : related to 8N rule(?)
* IV-VI/V-VI semiconductors :
Multiply both sides by and then integrate. Then,
Let orthogonal fts, or 0
Then (physical meaning?)
If two particles (components or states) are interacting, it should
and constant probability
If interaction (or coupling) exists between two identical states
Conclusion : Whenever there is a coupling, the energy splits.
Bonding for and antibonding for
Expectation ; There will be an n-fold splitting in energy.
Consider an 1D array of atoms
What happens to the ; finite transition (tunneling) probability to atoms and atoms.
From the rate eq. of Feynman’s coupled modes
counting only nearest interactions.
Assuming the trial solution as
What about a 3D rectangular lattice? (homeworks)
- Formation of energy bands in an ionic crystal
For the tight-binding model to work well, 4 or 4 .
The range of splitted energy from and
Band widths are 4 and 4 , respectively.
For the valence electron in a solid to remain tightly bound to the atom (atomic-like), it must either accomplish a complete jump to a nearest-neighbor atomic state or stay in its free-atom state
in 3s state 3p state in
The pair can lower its total energy
compared to the sum of independent free
atom energies by ~7eV.
The energy gained in the jump is a measure
of the ‘ionicity’ of the band
The broadening of atomic 3s (conduction
band) or 3p (VB) state into a band to be small.
Tight Binding Model; Works well for insulators
- Tight Binding Model (by time independent Schroedinger eq.)
Consider an array of N atoms seporated by distance
with the condition
where j labels atoms in the lattice and a free-atom wave
ft. at the jth site, i.e. satisfies the Schroedinger eq. for a free atom with the electron energy If the atoms were entirely independent, But for 1D solid with
for any integer Thus should satisfy
ex)2 If with periodic ft.
ex)3 If in the above wave ft. of liner molecule,
Actually, it can be proved that any solution of the Schroedinger eq.
with a periodic potential must obey the eq.
Then the Schroedinger eq. for the 1D solid is, with the crystal potential which is the sum of the atomic potentials,
Multiply and integrate over for the full space. Then,
Assume that only states from adjacent atoms are interacting. Then;
and for and for
The bandgap of a semiconductor is the result of the difference
of two energy levels between the outmost valence electrons and the broadening. Variation of in different semiconductors made by
atoms at the same row and the same period of the periodic table.
Formation of semimetals for compounds with heavy ions.
- covalent bonding : no preferential bonding direction
- symmetry :
- the so-called 8N rule :
- ionic bond: preferencial bonding direction
- electronic era or IT era : opened from Ge transitor
* Ge transistor, Si DRAMs, LEDs and LDs
- merits of Si on Ge
- where quantum effects dominate
* quantum well, quantum dot, quantum wire
* lattice points : have a well-defined symmetry
* position of lattice point basis ; arbitrary
- primitive unit cell : volume defined by 3 vectors, arbitrary
- Wignez-Seitz cell : shows the full symmetry of the Bravais lattice
- simple cubic(sc), body-centered cubic(bcc), face-centered (fcc)
* =lattice constant
Report : Obtain the primitive vectors for the bcc and fcc.
- fcc : a rhombic dodecahedron, * Confer Fig. 2.2
- Packing density of close-packed cubics
- hexagonal lattice = two dimensional (2D) triangular lattice + c axis
- Wignez-Seitz cell of hcp : a hexagonal column (prism)
Diamond structure : Basics of group IV, III-V, II-VI Semiconductors
- Two overlapped fcc structures with different atoms at 0
- Most III-V (parts of II-VI) Semiconductors : Cubic III-V, II-VI
- Concept of sublattices : group III sub-lattice, group V sub-lattice
- Rotational symmetry n, defined by 2π/n (n=1~6 not 5)
- Reflection symmetry
- Inersion symmetry
* elastic scattering btw :momentum conservation. (why?)
- lattice : a perfectly regular array of identical objects
- free : represented by plane waves,
- interaction btw and lattice ↔ optical (x-) ray and grid
* Bragg law (condition) : when 2d sinθ = with integer constructive interference
(2D rectangular lattice)
Note that plane and plane, etc. plane
Note 3) Reciprocal lattice of a Bravais lattice is also a Bravais lattice.
The plane thus formed isa part of BZ boundary.
; modulation, ; propagation with
- concept of PhCs: based on electromagnetism & solid-state physics
- solid-state phys.; quantum mechanics
Hamiltonian eq. in periodic potential.
- photonic crystals; EM waves (from Maxwell eq.) in periodic dielectric materials single Hamiltonian eq.
- multiple reflection (scattering) of electrons near the BZ boundaries.
- electronic energy bandgap at the BZ boundaries.