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)
1
Lecture 1. Overview on Semiconductors and PhCs
2009. 09.
Hanjo Lim
School of Electrical & Computer Engineering
hanjolim@ajou.ac.kr
2
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
3
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

4
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 / Bi 2Te 3 , Sb 2Te 3
Band gap versus covalency & ionicity
- Bond and Bands (Tight binding theory, Feynman model)

5
Schroedinger eq. in time-dependent form
 (  2 m )   V    i     t , or    i     t (partial diff. eq.)
Let    ( t ) ( r ), or generally     ( t ) ( r ) , for a system.
Then Schroedinger eq. becomes      i    d  dt . (why?)
Multiply both sides by  k and then integrate. Then,
2
2
j
j
j
j
j
j
j
j
j
  j   k   j d   i  d  j dt   j  k d 
j
j
Let  ,  ; orthogonal fts, i .e    d   c ( if k  j ) or 0 ( if k  j )
If normalized (orthonomal) fts, c  1, Let H    H  d 
Then i  d  dt   H  (physical meaning?)
If two particles (components or states) are interacting, it should
be i  d  dt  H   H  , i  d  dt  H   H 
j
k
k
j
kj
kj
k
kj
kj
k
j
j
j
1
11
1
12
2
2
21
1
22
2
6
If H  H  0 (physical meaning?) i  d  dt  H  ,
  K exp(  i H t  ) and  ( t )  K ; constant probability
If interaction (or coupling) exists between two identical states
Let H  H  E , H  H   A Then,
12
21
1
2
1
1
11
11
22
0
1
12
i  d  1 dt  E 0  1  A  2 ,
11
1
2
1
21
i  d  2 dt   A  1  E 0  2
Assuming the trial solution in the form of
 1  K 1 exp(  i Et  ),  2  K 2 exp(  i Et  ), couped eqs. become
 K 1 E  E 0 K 1  A K 2
E0  E  A
0
nontrivial
solutions
if
and
only
if

 K 2 E   A K 1  E 0 K 2
 A E0  E
2
2
That is ( E 0  E )  A , E  E 0  A
Conclusion : Whenever there is a coupling, the energy splits.
Bonding for E 0  A and antibonding for E 0  A
7
What happens when n atoms are brought close together to
couple?
Expectation
; There will(b)Interacting
be an n-fold
splitting in energy.
(a)Isolated atoms
atoms
n
:
n 1
n
E2
:
 n
2
Allowed
1
0
:
Forbidden
n
2
E1
Allowed
Consider an 1D arraye of atoms
j  2 ; finite transition (tunneling) probability
j  1happens to the
What
to
atoms and
atoms.
i  d  dt   H 
From
coupled modes
i  d  the
dt rate
E  eq.
 Aof
 Feynman’s
 A
counting only nearest interactions.8

j
kj
j
j
1
j
j 1
j 1
j
Assuming the solution in the form of   K exp(  i Et  ),
EK  E K  A ( K  K ), or EK ( x )  E K ( x )  AK ( x  a )  K ( x  a ) 
Assuming the trial solution as K ( x )  f ( x ) exp( iik x ) [ Bloch wave ft .]
j
j
1
j
j 1
j 1
j
1
j
j
j
j
j
j
E exp( i kx j)  E 1exp( i kx j )  A{exp[ ik ( x j  a )]  exp[ ik ( x j  a )]}
The range of splitted energy from E and E 2
1
E  E 1  2 A cos ka , E  E 2  2 B cos ka
Band widths are 4A and 4B, respectively.
 a
 a
What about a 3D rectangular lattice? (homeworks)
- Formation of energy bands in an ionic crystal ( ex : NaCl )
For the tight-binding model to work well, E  E  4A or 4B .
2
1
9
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 ( ex : Na Cl
e in Na 3s state Cl 3p state in NaCl .
The Na  Cl pair can lower its total energy
compared to the sum of independent free
atom energies by ~7eV.  NaCl  Na Cl
The energy gained in the jump is a measure
of the ‘ionicity’ of the band  E  ~ 7 eV
The broadening of atomic Na 3s (conduction
band) or Cl 3p (VB) state into a band to be small.






g
10

).
 Band

Tight Binding Model; Works well for insulators
Model

 Free Electron Model; Works well for semiconductors
- Tight Binding Model (by time independent Schroedinger eq.)
Consider an array of N atoms seporated by distance a .
Let the electronic wave ft. of this 1D solid in the form of
 ( x )  N  c  ( x  x ), with the condition   ( x ) ( x ) dx  1,
where j labels atoms in the lattice and  ( x  x ) a free-atom wave
ft. at the jth site, i.e.  satisfies the Schroedinger eq. for a free
atom with the electron energy E . If the atoms were entirely
independent, c  1 . But for 1D solid  ( x  R )   ( x ) with R  na
for any integer n . Thus  ( x ) should satisfy
1 / 2
N
j 1

j
j
*

j
at
j
 ( x  R )  exp( ikR ) ( x )
[ Bloch
Theorem ]
11
ex)1 If  is a linear combination of plane waves, i.e. 
k
 A exp( ikx )
 ( x  R )  A exp[ ik ( x  R )]  exp( ikR )[ A exp( ikx )]  exp( ikR ) x ( x)
ex)2 If   f ( x ) e with periodic ft.
ikx
k
fk ( x)
k
[ Bloch
ft .],
 k ( x  R )  f k ( x  R ) [exp( ikR )]  exp( ikR ) exp( ikx ) f k ( x  R )[  f k ( x )]
 exp( ikR ) k ( x)
ex)3 If c  exp( ikx ) in the above wave ft. of liner molecule,
j
j
 k ( x  R )  exp( ikx j ) ( x  R  x j)   exp( ikR ) exp[ ik ( x j  R )]  ( x  ( x j R ))
j
j
 exp( ikR )  exp( ikx l ) ( x  x l )  exp( ikR ) k( x)
l
Actually, it can be proved that any solution of the Schroedinger eq.
with a periodic potential must obey the eq.  ( x  R )  exp( ikR ) ( x ).
Then the Schroedinger eq. for the 1D solid is, with the crystal
potential V   V ( x  x ), which is the sum of the atomic potentials,
k
k
N
c
j 1
at .
j
12
 (
2
2 m ) d dx  Vc  E k  1 N  exp( ikx j) ( x  xj)  0
2
2
1 2
N
j 1
Multiply  ( x ) and integrate over x for the full space. Then,
*
k
N
N

1 / N     [exp(  ikx n ) ( x  x n )[ (   / 2 m ) d /dx  Vc  E k ] exp ( ikx j ) ( x  xj) dx  0
*
2
2
2
n 1 j 1
Assume that only states from adjacent atoms are interacting. Then;
  ( x  xn )  (  2 m ) d dx  Vc  ( x  x j) dx

*

2
2
2
  A,

  E at ,

 0,
for n  j  1
for n  j
otherwise
and   ( x  x )  ( x  x ) dx 1 for n  l , and 0 for n  l .
Then E  E  A[exp(  ika )  exp( ika )] or E  E  2 Acos( ka )
*

n
at
l
k
at
The bandgap E g of a semiconductor is the result of the difference
of two energy levels between the outmost valence electrons and the
broadening. Variation of E g 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.
13
 Crystal structure of Si, GaAs and NaCl
- covalent bonding : no preferential bonding direction
- T d symmetry : Si , SiO
- the so-called 8N rule : 1s 2 s 2 p 3 s 3 p 3 d 4 s 4 p 4 d 4 f    
8
8
8
- ionic bond: preferencial bonding direction ( ex ; 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
14
* quantum well, quantum dot, quantum wire
2
2
2
6
2
6
10
2
6
10
Crystal Structure and Reciprocal
Latiice
Crystal = (Bravais) lattice + basis


R

N
a
,

- lattice = a geometric array of points,

with integer numbers N i , 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)



* a 1  a xˆ , a 2  a yˆ , a 3  a zˆ , a =lattice constant
Report : Obtain the primitive vectors for the bcc and fcc.

3
i 1
i
i
15
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

16
Diamond structure : Basics of group IV, III-V, II-VI Semiconductors
- C : 2 s 2 p  sp hybridizat ion : diamond , sp hybridizti on : 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 2 sp 2 bonding in the plane
weak van der Waals bondding to the vertical direction
17
* Graphite : layered structure with hexagonal ring plane

2
2
3
2
Symmetry operations in a crystal lattice




- Translational symmetry operation R  n1 a1  n 2 a 2  n 3 a 3 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 m ( mirror )
- Inersion symmetry i (or 1 )

def) space group : structure classified by R and point operations
- Difference btw the symm. of diamond ( O ) and that of GaAs (T )
* Difference between cubic and hexagonal zincblende
ex) CdS bulk or nanocrystals, E  E , TiO2 (rutile, anatase)

h
gc
d
gh
18
ni
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




k

d
k


a2

a1
(2D rectangular lattice)




k  2 /  , p  h /  , let k  ( 2  /  ) uˆ , k   ( 2  /  ) u 


 
 
2
then p   k , and
2 d sin   k [ u  d  (  u )  d ]  2  


  


  
d  ( k  k )  2   , let K  k  k    k , then K  2   / d
19

d
: position vector defining a plane made of lattice sites.




k  k   K  reflection plane, K  2   / d ; inversely proportional to d



R
With general  n a  n a (positions of real lattice points),
 
 
K  R  2  or exp [ i K  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 a  a  a ,      (triclinic)



  
 
 
With R  n a  n a  n a , a  K  2 h , a  K  2 h , a  K  2 h (1)
should be satisfied simultaneously for the integral values of h , h , h .




  

Let K   k  h b  h b  h b ( 2 ) and b , b , b to be determined.
Then eq. (2) will be solution of eq. (1) if eq. (3) holds
1
1
2
1
1
1
2
2
3
3
2
1
2
3
1
2
2
3
3
1
1 1
2
2
3 3
1
2
2
3
3
20
 
b1  a 1  2 
 
b1  a 2  0
 
b1  a 3  0
Note that
Thus

 
b1  ( a 2 , a 3 )
 
b 2  a1  0
 
b 2  a 2  2
 
b2  a 3  0
plane and
 
b3  a1  0
 
b3  a 2  0
 
b3  a 3  2

 
b 2  ( a1 , a 3 )
(3)

plane, etc. ( a
 
 
 



a2  a3
a 3  a1
a1  a 2
b1  2     , b 2  2     , b3  2   
a1  a 2  a 3
a1  a 2  a 3
a1  a 2  a 3

 

a
)

(
a
,a )
2
3
2 3
plane
should be
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.
21
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


K : RLV   K :

K
(2)

k2

k1
a given reciprocal
 
2
 2k K  K  0
: Bragg law .
 
lattice  k  ( K / 2 )  ( K / 2 ) 2
a vector in the reciprocal


Take K  and K  so that they terminate at one
of the RL points, and take (1), (2) planes



so that they bisect normally K  and K  ,
K


respectively. Then any vector k 1 or k 2 that
terminates at the plane (1) or (2) will
lattice
22
satisfy the diffraction condition.

(1)
2


( at K  / 2 , K  / 2 , ...)
The plane thus formed
is a part of BZ boundary.
 
Note 4) An RLV has a definite length and orientation relative to a1 , a 2 ,

a 3 . Any wave ( x  ray , electron ) 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 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 (?).
23
Basic Concepts of
photonic(electromagnetic) crystals
Electronic crystals (conductor, insulator)
ex) one-dimensional electronics crystals => periodic atomic arrangement

Schroedinger equation : 

2
d 
2
2 m dx
2
V  E
If V  V c  0    0 e  ikx , k  ( 2 mE )1 / 2 /  => plane wave
If V  V c is not a constant,   u k ( x ) e ikx ; Bloch function
u k ( x ) ; modulation, e ikx ; propagation with k  2  / 

2
 uk ( x) uk ( x) ,
*

Total
If k   / a with the lattice constant a
e
ikx
e
 ikx
 cos ka  1
e
ikx
e
 ikx
 sin ka  0
wave
e
ikx
e
 ikx
a
24
Ek
a

3
a

2

a

0
a

2
3
a
a
a
k
Note) Bragg law of X-ray diffraction
If 2 a sin   n  , constructive reflection of the incident wave (total
reflection)
∴ A wave satisfying this Bragg condition can not propagate through the
structure of the solids.
a
If one-dimensional
(  90  , k  2  /  ) material
 2 a  n (with
2  / k an
) atomic
strong spacing
reflection at isk considered,
  n / a
k   n / a
∴ Strong reflection of electron wave at
(BZ boundary)
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Strong reflection around 2 a  n  ( k   n  / a ), a : period .
R
1
k  /a
- Exist. of complete PBG in 3D
PhCs :
 , theoretically predicted in 1987.
“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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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 n i ( refr . index n   ).
- photonic frequency bandgap at the BZ boundaries.
ex) DBR (distributed Bragg reflector): 1D photonic crystal

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