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Lectures on the Basic Physics of Semiconductors and Photonic Crystals

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## Lectures on the Basic Physics of Semiconductors and Photonic Crystals

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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

School of Electrical & Computer Engineering

hanjolim@ajou.ac.kr

Lecture 1. Overview on Semiconductors and PhCs

Overview

- Review on the similarity of SCs and PhCs
- Semiconductors: Solid with periodic atomic positions
- Photonic Crystals: Structure with periodic dielectric constants
- Semiconductor: Electron characteristics governed by the atomic potential. Described by the quantum mechanics (with wave nature).
- Photonic Crystals: Electomagnetic(EM) wave propagation governed by dielectrics. EM wave, Photons: wave nature
- Similar Physics. ex) Energy band ↔ Photonic band

Review on semiconductors

- Solid materials: amorphous(glass) materials, polycrystals, (single) crystals

- 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)

- Classification of solid materials according to the electrical conductivity

- (superconductors), conductors(metals), (semimetals), semiconductors, insulators

- Difference of material properties depending on the structure

* metals, semiconductors, insulators : different behaviors

So-called “band structure” of materials

- metals, semiconductors, insulators

* temperature dependence of electrical conductivity,

conductivity dependence on doping

- Classification of Semiconductors

- 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 :

- Band gap versus covalency & ionicity
- - Bond and Bands (Tight binding theory, Feynman model)

Schroedinger eq. in time-dependent form

- or (partial diff. eq.)
- Let or generally for a system.
- Then Schroedinger eq. becomes (why?)

Multiply both sides by and then integrate. Then,

Let orthogonal fts, or 0

- If normalized (orthonomal) fts, Let

Then (physical meaning?)

If two particles (components or states) are interacting, it should

be

and constant probability

If interaction (or coupling) exists between two identical states

Let Then,

- Assuming the trial solution in the form of
- couped eqs. become
- nontrivial solutions if and only if

That is

Conclusion : Whenever there is a coupling, the energy splits.

Bonding for and antibonding for

What happens when n atoms are brought close together to couple?

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.

(a)Isolated atoms

(b)Interacting atoms

Allowed

Forbidden

Allowed

Assuming the solution in the form of

or

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

- Free Electron Model; Works well for semiconductors

- Tight Binding Model (by time independent Schroedinger eq.)

Consider an array of N atoms seporated by distance

- Let the electronic wave ft. of this 1D solid in the form of

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

- Band Model

ex)1 If is a linear combination of plane waves, i.e.

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

Then or

for

for

otherwise

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.

Crystal structure of Si, GaAs and NaCl

- covalent bonding : no preferential bonding direction

- symmetry :

- the so-called 8N rule :

- ionic bond: preferencial bonding direction

- Importance of semiconductors in modern technology (electrical industry)

- electronic era or IT era : opened from Ge transitor

* Ge transistor, Si DRAMs, LEDs and LDs

- merits of Si on Ge

- IT era: based on micro-or nano-electronic devices

- where quantum effects dominate

* quantum well, quantum dot, quantum wire

CrystalStructure and Reciprocal Latiice

- Crystal = (Bravais) lattice + basis
- - lattice = a geometric array of points,
- with integer numbers 3 primitive vectors
- - Basis = an atom (molecule) identical in composition and arrangement

* 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

- Cubic lattices

- simple cubic(sc), body-centered cubic(bcc), face-centered (fcc)

* =lattice constant

Report : Obtain the primitive vectors for the bcc and fcc.

Wignez-Seitz cells of cubic lattices (sc, bcc, fcc)

- - sc : a cube - bcc : a truncated octahedron

- fcc : a rhombic dodecahedron, * Confer Fig. 2.2

- Packing density of close-packed cubics

- Hexagonal lattice

- hexagonal lattice = two dimensional (2D) triangular lattice + c axis

- Wignez-Seitz cell of hcp : a hexagonal column (prism)

- Note that semiconductors do not have sc, bcc, fcc or hcp structures.
- - SCs : Diamond, Zinc-blende, Wurtzite structures
- - Most metals : bcc or fcc structures

Diamond structure : Basics of group IV, III-V, II-VI Semiconductors

- - C :
- - Diamond : with tetrahedral symmetry, two overlapped fcc structures with tow carbon atoms at points 0, and
- Zincblende (sphalerite) structure

- Two overlapped fcc structures with different atoms at 0

and

- Most III-V (parts of II-VI) Semiconductors : Cubic III-V, II-VI

- Concept of sublattices : group III sub-lattice, group V sub-lattice

- Graphite and hcp structures
- - Graphite : Strong bonding in the plane
- weak van der Waals bondding to the vertical direction
- * Graphite : layered structure with hexagonal ring plane

Symmetry operations in a crystal lattice

- - Translational symmetry operation with integer
- def) point group : collection of symmetry operations applied at a point which leave the lattice invariant ⟹around a given point

- Rotational symmetry n, defined by 2π/n (n=1~6 not 5)

- Reflection symmetry

- Inersion symmetry

- def) space group : structure classified by and point operations
- - Difference btw the symm. of diamond and that of GaAs
- * Difference between cubic and hexagonal zincblende
- ex) CdS bulk or nanocrystals, TiO2 (rutile, anatase)

- - Nearly free electrons : weak interactions (elastic scattering)
- between sea of free and lattice of the ions

* 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)

: position vector defining a plane made of lattice sites.

- reflection plane, ; inversely proportional to
- With general (positions of real lattice points),
- should be satisfied in general.
- A set of points in real space ⟹ a unique set of points with
- : defined in -space. → Reciprocal lattice vector,
- 3D Crystal with (triclinic)
- With
- should be satisfied simultaneously for the integral values of
- Let to be determined.
- Then eq. (2) will be solution of eq. (1) if eq. (3) holds

Note that plane and plane, etc. plane

- Thus should be
- the fundamental (primitive) vectors of the reciprocal lattice.
- Note 1) ;scattering vector, crystal momentum, Fourier-
- transformed space of , called as reciprocal lattice.
- Note 2) X-ray diffraction, band structure, lattice vibration, etc.

Note 3) Reciprocal lattice of a Bravais lattice is also a Bravais lattice.

- Report : Prove that forms a Fourier-transformed space of
- Brillouin zone : a Wigner-Seitz cell in the reciprocal lattice.
- Elastic scattering of an EM wave by a lattice ;
- Scattering condition for diffraction;
- : a vector in the reciprocal lattice
- Take so that they terminate at one
- of the RL points, and take (1), (2) planes
- so that they bisect normally
- respectively. Then any vector that
- terminates at the plane (1) or (2) will
- satisfy the diffraction condition.

The plane thus formed isa part of BZ boundary.

- Note 4) An RLV has a definite length and orientation relative to
- Any wave incident to the crystal will be diffracted if its wavevector has the magnitude and direction resulting to BZ boundary, and the diffracted wave will have the wave vector with corresponding
- If are primitive RLVs ⟹ 1stBrillouin zone.
- Report : Calculate the RLVs to sc, bcc, and fcc lattices.
- Miller indices and high symmetry points in the 1st BZ
- - (hkl) and {hkl} plane, [hkl] and <hkl> direction
- - see Table 2.4 and Fig. 2.7 for the 1st BZ and high symm. points.
- - Cleavage planes of Si (111), GaAs (110) and GaN (?).

Basic Concepts of photonic(electromagnetic) crystals

- Electronic crystals (conductor, insulator)
- ex) one-dimensional electronics crystals => periodic atomic arrangement
- Schroedinger equation :
- If => plane wave
- If is not a constant, ; Bloch function

; modulation, ; propagation with

- If with the lattice constant

Note) Bragg law of X-ray diffraction

- If constructive reflection of the incident wave (total reflection)
- ∴ A wave satisfying this Bragg condition can not propagate through the structure of the solids.
- If one-dimensional material with an atomic spacing is considered,
- ∴ Strong reflection of electron wave at (BZ boundary)

- “Photonic (Electromagnetic) crystals”

- 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.

- - Exist. of complete PBG in 3D PhCs :
- theoretically predicted in 1987.

- - wave guiding (reflector, internal reflection)
- - light generation (LED, LD)
- - modulation (modulator), add/drop filters
- PhCs comprehend all these functions => Photonic integrated ckt.
- Electronic crystals: periodic atomic arrangement.

- multiple reflection (scattering) of electrons near the BZ boundaries.

- electronic energy bandgap at the BZ boundaries.

- Photonic (electromagnetic) crystals: periodic dielectric arrangement.
- - multiple reflection of photons by the periodic
- - photonic frequency bandgap at the BZ boundaries.
- ex) DBR (distributed Bragg reflector): 1D photonic crystal

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