Transcript Feature Selection/Extraction for Classification Problems

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,
Princeton University Press (1995)
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Lecture 1 : Overview on Semiconductors and PhCs
2009. 03.
Hanjo Lim
School of Electrical & Computer Engineering
[email protected]
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Overview

Review on the similarity of SCs and PhCs
 Semiconductors: Solid with periodic atomic positions
Photonic Crystals: Structure with periodic dielectric constants

(1,2 )
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
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Review on semiconductors
Solid materials: amorphous(glass) materials, polycrystals,
(single) crystals
- Structural dependence : existence or nonexistence of
translational vector R , 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

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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 : PbS, PbTe, PbSe/ Bi2Te3 , Sb2Te3
- band gap and covalency & ionicity
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 Crystal structure of Si, GaAs and NaCl
- covalent bonding : no preferential bonding direction
- Td symmetry : Si, SiO2
- the so-called 8N rule : 1s 2s 2 p 3s 3 p 3d 4s 4 p 4d 4 f    
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- ionic bond: preferencial bonding direction (NaCl)
 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
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* quantum well, quantum dot, quantum wire
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Crystal Structure and Reciprocal
Latiice
Crystal = (Bravais) lattice + basis


R

N
a
,

- lattice = a geometric array of points,

with integer numbers Ni , a i ; 3 primitive vectors
- Basis = an atom (molecule) identical in composition and arrangement
* lattice points : have a well-defined symmetry
* position of lattice point vs basis ; arbitrary

- primitive unit cell : volume defined by 3 a i 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)



* a1  axˆ, a2  ayˆ , a3  azˆ , a =lattice constant
Report : Obtain the primitive vectors for the bcc and fcc.

3
i 1
i
i
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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
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Diamond structure : Basics of group IV, III-V, II-VI Semiconductors
- C : 2s 2 2 p2  sp3hybridization: diamond, sp 2hybridiztion: graphite
- Diamond : with tetrahedral symmetry, two overlapped fcc structures
a
with tow carbon atoms at points 0, and 4 ( xˆ  yˆ  zˆ )
 Zincblende (sphalerite) structure
- Two overlapped fcc structures with different atoms at 0
a
and 4 ( xˆ  yˆ  zˆ )
- 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 2sp 2 bonding in the plane
weak van der Waals bondding to the vertical direction
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* Graphite : layered structure with hexagonal ring plane
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Symmetry operations in a crystal lattice
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


- Translational symmetry operation R  n1a1  n2a2  n3a3 with integer ni
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 m (mirror)
- Inersion symmetry i (or 1)

def) space group : structure classified by R and point operations
- Difference btw the symm. of diamond (Oh ) and that of GaAs (Td )
* Difference between cubic and hexagonal zincblende
ex) CdS bulk or nanocrystals, Egc  Egh , TiO2 (rutile, anatase)
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Electron motions in a solid
- Nearly free electrons : weak interactions (elastic scattering)
between sea of free e and lattice of the ions (e )
* elastic scattering btw e and e: momentum conservation, why?
- lattice : a perfectly regular array of identical objects


ikz
- free e : represented by plane waves, e , exp(i k r )

- interaction btw e and lattice ↔ optical (x-) ray and grid
* Bragg law (condition) : when 2d sinθ =  with integer  
constructive
interference
 


d

k
k

a2

a1
(2D rectangular lattice)



k  2 / , p  h/ , let k  (2 / )uˆ, k   (2 / )u
 
 
 
2
then p  k , and
2d sin  k[u d  (u ) d ]  2


  


  
d (k k )  2, let K  k k   k , then K  2 / d
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
d
: position vector defining a plane made of lattice sites.




k  k   K  reflection plane, K  2 / d ; inversely proportional to d



With general R  n1a1  n2a2 (positions of real lattice points),
 
 
K  R  2 or exp [iK  R]1 should be satisfied in general.


A set of points R in real space ⟹ a unique set of points with K

K: defined in k -space. → Reciprocal lattice vector,
 3D Crystal with a1  a2  a3 ,      (triclinic)



  
 
 
With R  n1a1 n2a2 n3a3 , a1 K  2h1, a2 K  2h2 , a3 K  2h3 (1)
should be satisfied simultaneously for the integral values of h1,h2 ,h3.




  

Let K  k  h1b1  h2b2  h3b3 (2) and b1, b2 , b3 to be determined.
Then eq. (2) will be solution of eq. (1) if eq. (3) holds
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 
b1 a1  2
 
b1 a2  0
 
b1 a3  0
Note that
  
b1 (a2 ,a3 )
 
b2 a1  0
 
b2 a2  2
 
b2 a3  0
 
b3 a1  0
 
b3 a2  0
 
b3 a3  2
(3)
  
 
 
plane and b2 (a1,a3 ) plane, etc. (a2 a3 )  (a2 ,a3 ) plane
 
 
 

a2 a3 
a3 a1 
a1 a2
Thus b1  2    , b2  2    , b3  2    should be
a1 a2 a3
a1 a2 a3
a1 a2 a3
the fundamental (primitive) vectors of the reciprocal lattice.
    
Note 1) p  k , K  k k  ;scattering vector, crystal momentum, Fourier
  
  
transformed space of R , called as reciprocal lattice. K  k k  or K  k k
Note 2) X-ray diffraction, band structure, lattice vibration, etc.
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Note 3) Reciprocal lattice of a Bravais lattice is also a Bravais lattice.


Report : Prove that K forms a Fourier-transformed space of R.
 Brillouin zone : a Wigner-Seitz cell in the reciprocal lattice.
Elastic scattering of an EM wave by a lattice ; w  w, k   k
  

Scattering condition for diffraction; k   k  K with RLV K
  2 2  
k   (k  K )  k  2k  K  K 2
 
 2k  K  K 2  0 : Bragg law.
 


2
K : RLV   K : a vector in the reciprocal lattice k ( K / 2)  ( K / 2)
2


K and K 

K
( 2)

k2

k1

K
(1)
a given reciprocal lattice
Take
so that they terminate at one
of the RL points, and take (1), (2) planes


so that they bisect normally K and K  ,


respectively. Then any vector k1 or k2 that
terminates at the plane (1) or (2) will
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satisfy the diffraction condition.


The plane thus formed (at K / 2, K  / 2, ...) is a part of BZ boundary.
 
Note 4) An RLV has a definite length and orientation relative to a1 , a2 ,

a3 . Any wave ( x ray, electron) incident to the crystal will be
diffracted if its wavevector has the magnitude and direction
resulting to BZ
and the diffractedwave
 boundary,
 
 will have the
wave vector k   k  K with corresponding K , K  , etc.
  
If K , K  , K are primitive RLVs ⟹ 1st Brillouin zone.
Report : Calculate the RLVs to sc, bcc, and fcc lattices.
st BZ
 Miller indices and high symmetry points in the 1
- (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 (?).
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Basic Concepts of
photonic(electromagnetic) crystals
Electronic crystals (conductor, insulator)
ex) one-dimensional electronics crystals => periodic atomic arrangement

 2 d 2
V  E
Schroedinger equation : 
2
2m dx
If V  Vc  0   0eikx, k  (2mE )1/ 2 /  => plane wave
If V Vc is not a constant,   uk ( x)eikx ; Bloch function
uk (x) ; modulation, eikx ; propagation with k  2 / 
2
  uk ( x)* uk ( x) ,  Total wave  eikx  e  ikx
If k  / a with the lattice constant a
eikx  e  ikx  cos ka 1
eikx  e ikx  sin ka  0
a
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Ek
a

3
2

a
a


a
0

a
2
a
3
a
k
Note) Bragg law of X-ray diffraction
If 2asin  n , 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 a is considered,
(  90, k  2 /  )  2a  n(2 / k )  strong reflection at k  n / a
∴ Strong reflection of electron wave at k  n / a (BZ boundary)
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Optical control
- 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 ni (refr. index n   ).
- photonic frequency bandgap at the BZ boundaries.
ex) DBR (distributed Bragg reflector): 1D photonic crystal
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Strong reflection around 2a  n (k  n / a), a: period.
R
1
- Exist. of complete PBG in 3D PhCs :
theoretically predicted in 1987.
k  / a
 ,
“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.
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