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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, Princeton University Press (1995)
2
2009. 03.
Hanjo Lim
School of Electrical & Computer Engineering
Lecture 1 : Overview on Semiconductors and PhCs
Overview
3
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
),( 21
4
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
R
5
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 and covalency & ionicity3232 ,/,, TeSbTeBiPbSePbTePbS
6
Crystal structure of Si, GaAs and NaCl
- covalent bonding : no preferential bonding direction
- symmetry :
- the so-called 8N rule :
- 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
* quantum well, quantum dot, quantum wire
dT
888
4444333221 10621062622 fdpsdpspss2, SiOSi
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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.
,3
1
i
iiaNR
azaayaaxaa ,ˆ,ˆ,ˆ 321
ia
;, ii aN
Crystal Structure and Reciprocal Latiice
vs
8
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 :
- 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
graphiteonhybridiztispdiamondionhybridizatspps :,:22 2322
)ˆˆˆ(4
zyxa
)ˆˆˆ(4
zyxa
22sp
10
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)
332211 anananR
)1(ori
R
,ghgc EE
)( hO )( dT
in
)(mirrorm
11
Electron motions in a solid
- 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
e )( e
e )exp(, rkieikz
ukuklethpk
)/2(,ˆ)/2(,/,/2
2])([sin22
, dudukdandkpthen
,,2)( kkkKletkkd
(2D rectangular lattice)
e
eande
dKthen
/2
k
k
d
2a 1a
12
: 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
dK
/2
d
Kkk
2211 ananR
1][exp2 RKiorRK
R
K
K
,321 aaa
)1(2,2,2, 332211332211 hKahKahKaanananR
321332211 ,,)2( bbbandbhbhbhkK
k
.,, 321 hhh
d
13
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.
0
0
2
31
21
11
ab
ab
ab
),( 321 aab
),( 312 aab
321
213
321
132
321
321 2,2,2
aaa
aab
aaa
aab
aaa
aab
kkKkp
,
R
kkKorkkK
),()( 3232 aaaa
2
)3(0
0
33
23
13
ab
ab
ab
0
2
0
32
22
12
ab
ab
ab
14
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.
kkww ,
KRLVwithKkk
.R
KRLVK
:2)2/()2/( KKk
KandK
, KandK
21 kork
latticereciprocalgivena
K
2222 2)( KKkkKkk
.:02 2 lawBraggKKk
2k 1k
K
K
)2(
)1(
15
The plane thus formed is a 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 ⟹ 1st Brillouin 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 (?).
,, 21 aa
Kkk
...),2/,2/( KKat
.,, etcKK
),( electronrayx.3a
KKK
,,
16
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
EV
dx
d
m
2
22
2
Basic Concepts of photonic(electromagnetic) crystals
/)2(,0 2/10 mEkeVV ikx
c cVV ikx
k exu )()(xuk
ikxe /2kikxikx
kk eewaveTotalxuxu ,)()( *2
aak /
0sin
1cos
kaee
kaeeikxikx
ikxikx
a
17
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)
ankatreflectionstrongknak /)/2(2)/2,90( ank /
a
kE
k
a
3
a
2
a
a
0a
2
a
3
a
,sin2 na
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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
- photonic frequency bandgap at the BZ boundaries.
ex) DBR (distributed Bragg reflector): 1D photonic crystal
)..( nindexrefrni
19
Strong reflection around
“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.
.:),/(2 periodaankna
1
ak /
R
,
- Exist. of complete PBG in 3D PhCs :
theoretically predicted in 1987.
