Basics of Optoelectronic pn-Junctions

112
OPTOELECTRONICS Prof. Wei-I Lee 1 Basics of Optoelectronic pn-Junctions

Transcript of Basics of Optoelectronic pn-Junctions

Basics of Optoelectronic pn-Junctions
OPTOELECTRONICS Prof. Wei-I Lee 2
de Broglie Hypothesis (1923) : The motion of a particle is governed by the wave propagation properties of a “pilot” wave called matter wave
λ = h / p λ
( matter wave ) Ψ( r,t ) ,

The probability that the particle will be found in volume dV :
P( r,t ) dV = Ψ2 dV , P( r,t ) = Ψ2 : probability density
Fundamental Principles of Quantum Mechanics
Semiconductor Bonding and Band Structure
Prince Louis-Victor de Broglie
1-dimensional time-dependent Schrodinger equ.
1-dimensional time-independent Schrodinger equ.
Ψ( x,t ) = χ


2m 1
1-dimensional time-independent Schrodinger equ.
χ( x ) dχ
/ dx 1. 2. 3.
E , χ
E : eigenvalues ( of particle total energy )
energy eigenvalues χ( x ) : eigenfunction, eigenstate
quantization of E and other physical quantities
Schrodinger Equation for Atom


1-dimensional time-independent Schrodinger equ.
3-dim. time-independent Schrodinger equ.
Ep ≈
Schrodinger Equation for Atom
Ψ( x,t ) = χ


⇒ –h2/2m •( ∂2χ/∂x2 + ∂2χ/∂y2 + ∂2χ/∂z2 ) + Ep( x,y,z ) χ
= E χ
- 1 Ze2
in spherical coordinate
) = R( r ) Θ( θ
)
must be well-behaved functions 3 quantum numbers must be well-behaved functions 3 quantum numbers
3-Dimensional Schrodinger Equation
Ep( x,y,z ) Ep( r )
use separation of variables : χ( r,θ,φ
) = R( r ) Θ( θ
Φ, Θ, R must be well-behaved functions 3 quantum numbers
n, l, ml decide χ χn l ml to further include electron spin effect
4 quantum #’s to specify a state ( or a wave-function ) of an electron : n : 1, 2, 3, ….. , l : 0, 1, 2, ..., n-1
ml : 0, ±1, ±2, …, ±l , ms : ±
½
n : principle quantum number ( ) ( decide En of H-atom )
l : orbital quantum number ( ) 0, 1, 2, ….., n-1
ml : magnetic quantum number ( ) 0, ±1, ±2, ±3, ….., ±l
spin up spin down
OPTOELECTRONICS Prof. Wei-I Lee 8
n, l, ml decide χ χn l ml each state can accommodate 2 electrons ( ms = ±
½ )
Quantum States of H-Atom
0 1 2 3 l s p d f
n
4
3
2
1 E
( ml =0 ) ( ml = -1 , 0 , 1 ) ( ml = -2 , -1 , 0 , 1 , 2 )
( for H atom )
∑ χ∗n,l,ml χn,l,ml has spherical symmetry
Ex : all 6 of the p state e- wave functions add up
spherical symmetry
all 10 of the d state e- wave functions add up
spherical symmetry
ml
Formation of Periodic Table
( valence shell : the outmost shell with electrons )
OPTOELECTRONICS Prof. Wei-I Lee 11
elements in a group have similar chemical and optical properties ( because they have similar valence e- configurations )
( outermost e- ⇒ valence electron , )
Characters of Elements in Periodic Table
Semiconductor Bonding and Band Structure
OPTOELECTRONICS Prof. Wei-I Lee 12
)
order disorder
OPTOELECTRONICS Prof. Wei-I Lee 13
Ex. molecule hydrogen, H2 e- spend a lot of time between 2 protons
protons attracted toward center and each other covalent bond
Covalent Bond
~ 0.7 Å
covalent bonds in solids : diamond ( C ), Si, Ge C : 1s22s22p2
Si : 1s22s22p63s23p2 need 4 e- to have a full p subshell Ge : 1s22s22p63s23p63d104s24p2
4 covalent bonds to each atom ( tetrahedral structure )
covalent bonds is very strong e.g. 7.4 eV for diamond , 12.3 eV for SiC ( carborandum ) covalent bonding materials are usually insulators or semiconductors
Covalent Bonds in Solids
Crystals With Different Bondings
Course Project A power point presentation about optoelectronic devices Possible topics include, but not limited to,


LED LD Solar Cell Photo Detector

The presentation should include at least 15 slides or pages The presentation file should be self-explanatory Don’t copy from a single source. Be creative Should list all references, including books, papers, websites, and etc. Submit the selected topics for permission at 4/17 (Thur.) Project deadline : 5/21 (Wed.) Mid-term Exam : in the week of 4/21-4/25
OPTOELECTRONICS Prof. Wei-I Lee 17
crystal structure : a 3-dimensional periodic array of atoms or molecules space lattice : regular periodic arrangement of points in space basis : atom or group of atoms ( lattice ) + ( basis ) crystal structure
Crystal Structure
atom A
atom B
OPTOELECTRONICS Prof. Wei-I Lee 18
only 7 categories and 14 possible lattice arrangements : 14 Bravais lattices primitive unit cell : minimum repeated unit
Barvais Lattices
Semiconductor Bonding and Band Structure
OPTOELECTRONICS Prof. Wei-I Lee 19
diamond structure : fcc , basis consists of 2 identical atoms (C, Si, Ge, Sn)
Diamond Structure
Semiconductor Bonding and Band Structure
OPTOELECTRONICS Prof. Wei-I Lee 20
zinc blende structure : fcc , basis consists of 2 unidentical atoms (GaAs, InP)
Zinc Blende Structure
Semiconductor Bonding and Band Structure
OPTOELECTRONICS Prof. Wei-I Lee 21
wurtzite structure : hexagonal, basis consists of 2 different kinds of atoms (GaN)
Wurtzite Structure
Semiconductor Bonding and Band Structure
OPTOELECTRONICS Prof. Wei-I Lee 22
Ex. Plane intersects: a axis at r=2 , b axis at s=4/3 , c axis at t=1/2 How do we symbolically designate this plane in a lattice?
1. Take the reciprocals of r, s, and t. 2. Find the least common multiple that
converts all the reciprocals to integers. 3. Enclose the triple in parentheses.
Miller index notation for this plane is (2 3 8)
Miller Index Notation for Crystal Planes
Semiconductor Bonding and Band Structure
OPTOELECTRONICS Prof. Wei-I Lee 23
unit cell : volume enclosed by vector a1, a2, and c vector addition gives a3 = -(a1 + a2 ). The 4-index notation [m n p q] for the direction labeled D in the figure can be found as follows: 1. Projecting the vector onto the basal plane,
it lies between a1 and a2 ( vector B). 2. The vector B is equal to the vector sum (a1 + a2 ),
so m = 1, n = 1 (a1 & a 2 are independent vectors) 3. a 3 is the dependent vector , and can be found
from a 3 = -(a 1 + a 2 ) p = -2 4. q = 1
The four-index notation for D is [1 1 2 1]
Four-Index Notation for Hexagonal Crystal
Semiconductor Bonding and Band Structure
OPTOELECTRONICS Prof. Wei-I Lee 24
Lattice Planes in Four-Index Notation ( h k i l )
Semiconductor Bonding and Band Structure
procedure for finding Miller indices of a plane in a hexagonal crystal : 1. Find the intersections, r and s, of the plane with any two of the basal
plane axes. 2. Find the intersection t of the plane with the c axis. 3. Evaluate the reciprocals 1/r, 1/s, and 1/t. 4. Find the least common multiple to converts all the reciprocals to
integers. 5. Use the relation i = -(h+ k), where h is
associated with a1 , k is associated with a2 , and i is associated with a3 .
6. Enclose 4 indices in parentheses : (h k i l) Ex. What is the designation of the lattice plane
shaded pink in the figure? designation of this plane is (1 2 1 0)
OPTOELECTRONICS Prof. Wei-I Lee 25
Bloch’s Theorem
Semiconductor Bonding and Band Structure
Bloch’s Theorem describes a general property of the wavefunctions in a periodic potential potential distribution in a crystal can be described as a periodic function with the period of d ( e.g. spacing of ions or atoms = d ) : Ep( x ) = Ep( x+d ) Bloch’s Theorem : for a particle moving in a periodic potential with the period d χ( x ) = uk( x ) • e±ikx , uk( x ) = uk( x+d )
χ*( x ) χ( x ) = uk *( x ) e-ikx uk( x ) e+ikx = uk *( x ) uk( x )
χ*( x+d ) χ( x+d ) = uk*( x+d ) uk( x+d )
= uk *( x ) uk( x ) = χ*( x ) χ( x )
the probabilities of finding the particle at the position of ( x ) and
at the position of ( x+d ) are the same
OPTOELECTRONICS Prof. Wei-I Lee 26
Periodic Potential in a Crystal
Semiconductor Bonding and Band Structure
potential distribution around an atom and a molecule
potential distribution in a one-dimensional crystalline structure
Ep( x ) = – ——— —— 1 qe
4πε0 x
Kronig-Penny Model
Kronig-Penny Model of Ep in a one-dimensional crystalline structure
in region I : Ep = 0 , χI : eigenfunction in region I – —— —— = E χI h2
2m d2χI
uI( x ) = Aei( γ-k )x + Be-i( γ+k )x
—— + 2ik —— + ( γ2-k2 ) uI = 0 d2uI
dx2
duI
dx
Kronig-Penny Model
in region II : Ep = Ep0 χII : eigenfunction in region II
similarly, substitute χII = uII( x ) eikx uII = Ce( ε-ik ) x + De-( ε+ik )x
– —— ——— + EpoχII = E χII h2 d2χII 2m dx2
——— + ε2χII = 0 , ε
Kronig-Penny Model
χ
and dχ/dx must be continuous across a boundary
χI( c/2 ) = χII( c/2 ) , dχI( c/2 ) / dx = dχII( c/2 ) / dx periodicity requirement on u( x ) : uI( -c/2 ) = uII( b+ c/2 ) , duI( -c/2 ) / dx = duII( b+ c/2 ) / dx
4 linear algebraic equations for A, B, C, D acceptable solutions of A, B, C, D, and χ exist only if
P( sinγd / γd ) + cos γd = cos kd
P = ( mEpobd ) / h2 , γ
k is related to the electron’s momentum ( k = 2π/λ
and from de Broglie hypothesis : λ
= h/p p = hk )
This equ. relates the e-’s energy to its wave vector k (and momentum)
OPTOELECTRONICS Prof. Wei-I Lee 30
Dispersion Relation
Semiconductor Bonding and Band Structure
dispersion relation : the relation between a particle’s energy( E ) and its wave vector( k ) Ex. for a free particle E = p2 / 2m p = h / λ
( de Broglie relation ) , λ
E = h2k2 / 2m , E ∝ k2
for an e- moving in a 1-dim array of potential wells
the dispersion relation : P( sinγd / γd ) + cos γd = cos kd γ
= 2mE / h2
P( sinγd / γd ) + cos γd = cos kd
γ
= 2mE / h2
this equation can be solved numerically : ( pick a value of E
obtain a corresponding k )
some values of E imaginary k physically unacceptable these E’s are forbidden allowed and forbidden energy bands created
band discontinuities occur at k = ±

/ d
P( sinγd / γd ) + cos γd = cos kd
γ
= 2mE / h2
this equation can be solved numerically : ( pick a value of E
obtain a corresponding k )
some values of E imaginary k physically unacceptable these E’s are forbidden allowed and forbidden energy bands created
band discontinuities occur at k = ±

/ d
OPTOELECTRONICS Prof. Wei-I Lee 33
Band Structure in 3-Dimensional k-Space
Semiconductor Bonding and Band Structure
Brillouin zone in k-space for diamond and zincblende lattices crystal momentum depends on orientation
complicated band structure ( E vs. k ) energy band structure of zincblende crystal, e.g. GaAs
OPTOELECTRONICS Prof. Wei-I Lee 34
Problems of Kronig-Penney Model
Semiconductor Bonding and Band Structure
problems of Kronig-Penney model : (1) not much physical insight (2) does not give the # of energy states in a band
another approach : tight binding approximation
for a infinite square potential well ( 1-dim. ) if χ1 and χ2 are solutions of Schrodinger equ.
( - χ1 ) and ( - χ2 ) are also solutions of Schrodinger equ.
( - χ1 ) and χ1 same E1, and ( - χ1 )2 = χ1 2
( - χ2 ) and χ2 same E2, and ( - χ2 )2 = χ2 2
-χ1
-χ2
Symmetric and Antisymmetric Combination
two finite potential wells : 2 kinds of combinations :
(1) χS = a ( χB + χC ) ( symmetric ) ( a : introduced for normalization )
(2) χA = a ( χB - χC ) ( antisymmetric )
for two wells far apart χS and χA are degenerate states with the same χ2 and energy
the same
the same
Closely Spaced Potential Wells
when two wells get close enough :
χS is like the ground state for a well of width 2a
χA is like the 1st excited state for a well of width 2a
when 2 wells get close enough degenerate states begin to break up into nondegenerate states
degeneracy of χS and χA begins to disappear E of χS < E of χA , but physically why ?
OPTOELECTRONICS Prof. Wei-I Lee 37
Tight Binding Approximation of Two H-Atoms
Semiconductor Bonding and Band Structure
Ex. consider the 1s state of 2 H-atoms : 2 kinds of combinations : ( 1 ) χS = a( χB + χC )
( 2 ) χA = a( χB – χC )
an e- in χS state spend more time in between the 2 protons stronger negative binding energy an e- in χS state has a lower energy than in χA state
OPTOELECTRONICS Prof. Wei-I Lee 38
Tight Binding Approximation of Multi-Atoms
Semiconductor Bonding and Band Structure
2 atoms brought together two separate energy levels formed from each level of the isolated atom
What if six atoms are brought together ? Start with the six individual 1s states ……
χfirst level = χ1 +χ2 +χ3 +χ4 +χ5 +χ6
χsecond level = (χ1 +χ2 +χ3 ) - (χ4 +χ5 +χ6 )
• •
Formation of Bands
Semiconductor Bonding and Band Structure
N atoms brought together each level N discrete, closely spaced energy levels a quasicontinuous band of energy levels
Ex. width of a band ~ a few eV if N = 1023
separation between adjacent levels ~ 10-23 eV
Ex. Na : 1s2 2s2 2p6 3s1
OPTOELECTRONICS Prof. Wei-I Lee 40
Characteristics of Energy Bands
widths of the bands should not depend appreciably on N
widths of the bands depend mainly on distance between adjacent atoms
atoms closer to each other greater bandwidth
bandwidth of the low-lying levels < bandwidth of the higher energy
levels
Number of Allowed Electrons in Bands
Semiconductor Bonding and Band Structure
2N
6N
2N
2N

# of e- states in the band = 2 • ( 2l + 1 ) • N l : orbital quantum no. N : # of atoms
OPTOELECTRONICS Prof. Wei-I Lee 42
Sodium Crystal
Ex. Na crystal : 1s2 2s2 2p6 3s1
11 electrons/atom, N atoms in the solid total of 11N e-
when ε
applied e- gain energy move into empty, slightly higher energy states current conduction
valence band : the highest band containing e-
conduction band : the band e- in which can conduct net current ( in this case both are the 3s band )
( * in actual case, for Na, 3s band and 3p band overlap )
OPTOELECTRONICS Prof. Wei-I Lee 43
Magnesium Crystal
Ex. Mg crystal : 1s2 2s2 2p6 3s2
12 electrons/atom, N atoms in the solid total of 12N e-
for Mg, 3p band and 3s band overlap
when ε
applied e- gain energy move into empty, slightly higher energy states current conduction Mg crystal is a conductor
OPTOELECTRONICS Prof. Wei-I Lee 44
Carbon Crystal
Ex. C crystal : 1s2 2s2 2p2
6 electrons/atom, N atoms in the solid total of 6N e-
when ε
applied ( @ T = 0 K or low T ) e- has no higher energy level available no electron conduction
conduction band is separated from valence band
band gap ( Eg ) : energy gap between conduction band and valence band
OPTOELECTRONICS Prof. Wei-I Lee 45
Column IV Crystal
Ex. C ( diamond ), Si, and Ge have similar band structures
Eg C (diamond) ~ 6 eV Si ~ 1.1 eV Ge ~ 0.7 eV
6N
2N
4N
4N C : 2s 2p Si : 3s 3p Ge : 4s 4p Sn : 5s 5p Pb : 6s 6p
interatomic distance ,
Semiconducting Si and Ge
Semiconductor Bonding and Band Structure
at high temperatures ⇒ some e- excited into conduction band free e-
⇒ create “holes” in the valence band effective free “+” charge
the probability of e- transition across the bang gap is very sensitive to the magnitude of Eg Eg determines whether a solid is an insulator or a semiconductor T ↑ free e- and holes ↑ conductivity ↑
Ge
Si
C ~ 6 eV insulator @ 300 K (diamond) Si ~ 1.1 eV Ge ~ 0.7 eV semiconductor
OPTOELECTRONICS Prof. Wei-I Lee 47
Holes in Semiconductors
Semiconductor Bonding and Band Structure
an empty state in the valence band hole conduction by e- in the valence band = conduction by positive charge of positive effective mass ( i.e. hole ) the number of holes = the # of empty states in the valence band for perfectly pure semiconductor (intrinsic semiconductor) :
# of free e- = # of holes
no = po = ni (ni :intrinsic carrier concentration)
OPTOELECTRONICS Prof. Wei-I Lee 48
Direct and Indirect Bandgap
Semiconductor Bonding and Band Structure
direct bandgap : crystal momentum at the min. of CB coincide with crystal momentum at the max. of VB indirect bandgap : crystal momentum at the min. of CB never coincide with crystal momentum at the max. of VB
OPTOELECTRONICS Prof. Wei-I Lee 49
Elementary and Binary Compound Semiconductor
Semiconductor Bonding and Band Structure
a general trend : lattice const. ↓
Eg ↑
Vegard’s Law
mix two compound semiconductors ternary or quaternary compound semiconductors physical parameters of this new compound semiconductor vary linearly in proportion to the alloy composition (mole fraction) Vegard’s law
Ex. GaAs + AlAs AlxGa1-xAs
B an
dg ap
(e V)
Density of States
Semiconductor Bonding and Band Structure
ρ(E)dE : number of energy states between E and E+dE ρ(E) : density of states
Reference book for the remaining
material in this chapter :
Extrinsic Semiconductors
Electrical Properties
Ex. group-III or group-V impurity atoms sit on Si lattice sites substitutionally
OPTOELECTRONICS Prof. Wei-I Lee 55
n-Type Semiconductor and Donor Energy Level
Electrical Properties
when Si is doped with donor impurities, such as As or P
the extra e- not tightly bound to its parent nucleus ( ~ 0.01 eV ) can be easily ionized extra free e-, with no corresponding creation of holes more free e- than h+
n- type semiconductor
the extra e- occupies an energy level ED that lies ~ 0.01 eV below conduction band
a donor level @ T = 0 K donor level full, conduction band empty
as T↑ e- from donor level can jump into the empty conduction band and become free e-
OPTOELECTRONICS Prof. Wei-I Lee 56
p-Type Semiconductor and Acceptor Energy Level
Electrical Properties
when Si is doped with acceptor impurities, such as B or Al
extra holes, with no corresponding
creation of free e-
p- type semiconductor
an energy level EA that lies ~ 0.01 eV above the valence band is created
an acceptor level
@ T = 0 K acceptor level empty, valence band full
as T↑ e- from valence band can jump into the empty acceptor level create extra holes in valence band
OPTOELECTRONICS Prof. Wei-I Lee 57
Impurity Ionization Energies in GaAs
Electrical Properties
OPTOELECTRONICS Prof. Wei-I Lee 58
Effective Mass
Electrical Properties
when electric field ε
acts on a free e- a = eε / m , m : mass of e-
what if the e- is in a crystal under the influence of the lattice ion potentials
a = eε / m* , m* : effective mass of e- , m* = ?
vparticle = dE/ dp = d( —— ) / d p = mvparticle / m = vparticle
in quantum mechanics, E is often expressed in terms of wave vector k change dE/dp dE/dk
p = h/λ
, λ
= 2π/k p = h k dp = h dk ⇒ vparticle = ( 1/h ) dE/dk dE = dW ( work done on the particle ) = eε
dx = eε
( dx/dt ) dt
a = dvparticle /dt = — — —— = — — (——)
p2
2m
1 d2E h2 dk2
Electrical Properties
in semi-classical view : when an electric field is applied
( e- with m* > 0 ) and ( e- with m* < 0 ) drift in opposite directions
in semiconductors, electrons and holes have different effective masses
OPTOELECTRONICS Prof. Wei-I Lee 60
Fermi-Dirac Distribution
Electrical Properties
the probability that different energy states being occupied by electrons Fermi-Dirac distribution
if (E – EF) > > kBT
kBT ~ 0.026 eV at 300 K
~kBT
Carrier Concentration
Electrical Properties
Effective Density of States
mass action law :
ni : intrinsic carrier concentration
m0 : electron mass in vacuum mde : electron effective mass mdh : hole effective mass
OPTOELECTRONICS Prof. Wei-I Lee 63
Typical Characteristics of Semiconductors
Electrical Properties
NA - = NA f( EA ) , NA : acceptor conc.
from charge neutrality requirement EF can be determined at a certain T
ex. n-type
Carrier Concentration vs. Temperature
n = ND + = ND
Band Tailing in Heavily Doped Semiconductors
Electrical Properties
conc. of doped impurities ↑↑ EF approached Ec or Ev (Ec – EF) >> 3kBT or (EF – Ev) >> 3kBT not valid degenerate semiconductors
band tailing bandgap narrowing EF enters conduction band or valence band , impurity band overlap with conduction or valence band
all donors and acceptors ionized carrier conc. temp indep. , mass action law not valid
many optoelectronic semiconducting devices have heavily doped regions
OPTOELECTRONICS Prof. Wei-I Lee 67
Electrical Properties
Electrical Properties
Electrical Properties
carrier movement or transport by (1) drift : induced by electric field , or (2) diffusion : induced by non-uniform
carrier concentration drift velocity : vd = μ Ffield , μ
: carrier mobility
scattering ( lattice scattering )
OPTOELECTRONICS Prof. Wei-I Lee 70
Drift Current and Conductivity
ex. lightly doped Si
Carrier Diffusion and Total Current Density
Electrical Properties
Nonradiative Recombination of Carriers
Recombination via deep levels
Light Emission in Direct and Indirect Band-Gaps
Optical Properties
combinations of free e- in C.B with holes in V.B photon emissions radiative recombination
radiative recombination in direct bandgap materials (GaAs, GaN, InP) : momentum conservation can be maintained with only electrons and holes
radiative recombination in indirect bandgap materials (Si, Ge, GaP) : require involvement of phonons to maintain momentum conservation
probability of radiative recombination : in indirect bandgap << in direct bandgap
OPTOELECTRONICS Prof. Wei-I Lee 74
Band-to-Band Spontaneous Emission
two kinds of radiative emissions : spontaneous emission and stimulated emission
band-to-band spontaneous emission wavelength : in direct bandgap : ( Eg : eV )
in indirect bandgap :
band-to-band transition in direct bandgap materials usually have spectral widths from 10 ~ 100 meV, due to distribution of e- and h+ in each band
OPTOELECTRONICS Prof. Wei-I Lee 75
Bandgap and Spontaneous Emission Color
Optical Properties
Spontaneous Emissions
Optical Properties
4 major categories of spontaneous emissions in semiconductors direct bandgap semiconductor : band to band transition more important indirect bandgap semiconductor : radiative transition related to impurity levels more important spontaneous emissions are incoherent radiations spontaneous band to band emission rate in unit volume at excited state : Rsp = Bsp•n•p Bsp : radiative recombination coef. Bsp :10-9~ 10-11 cm3s-1 in direct bandgap semiconductors
10-13~10-15 cm3s-1 in indirect bandgap semiconductors
OPTOELECTRONICS Prof. Wei-I Lee 77
Quasi-Fermi Level at Nonequilibrium
Optical Properties
light emitting devices usually operated at excited states, with large # of e-
and holes nonequilibrium at excited states, n and p still nearly obey Fermi-Dirac distributions by using 2 quasi-Fermi levels in each band electron occupation probability in C.B :
electron occupation probability in V.B :
for nondegenerate semiconductors :
EFc-EFv : a measure of deviation from T.E. ( at T.E. : EFc= EFv=EF )
OPTOELECTRONICS Prof. Wei-I Lee 78
Spontaneous Emission at Nonequilibrium
spontaneous emission rate in unit volume :Rsp = Bsp•n•p
( assuming Δn = Δp )
: radiative recombination rate for excess carriers
under high excitation condition Δn >> n0, p0
( n≈Δn , p≈Δp, Δn=Δp for charge neutrality )
under high excitation condition n, p ↑↑ band-filling effect emission peak energy ↑ and emission wavelength becomes shorter
OPTOELECTRONICS Prof. Wei-I Lee 79
Optical Absorption
Optical Properties
Lambert’s law : (dI / dx) x (1/I ) = -αab ( or dI / dx = -αabI )
[ I : light intensity , αab : absorption coef. (cm-1) ]
αab a function of λ, 1/ αab : absorption length or penetration depth at x = 1/ αab ( absorption length, penetration depth ) I = (1/e) I0
categories of absorptions : 1. fundamental (band-band)
absorption 2. absorption via energy
levels in bandgap 3. intraband absorption 4. free carrier absorption 5. exciton absorption
OPTOELECTRONICS Prof. Wei-I Lee 80
at energies near absorption edge, fundamental absorption coef. : : for direct transition
: for indirect transition ( αcvin < αcv )
heavy doping can cause 2 different effects : (1) impurity band, band tailing
( absorption edge lower energy and longer λ
)
)
Absorption Coefficient
Optical Properties
OPTOELECTRONICS Prof. Wei-I Lee 82
Stimulated Emission
Optical Properties
3 features of stimulated emission : (1) 1 photon in, 2 photons out
light amplification (2) emitted photon in the same
direction as the incoming photon (3) emitted photon in phase (coherent) with the incoming photon
for a useful laser medium : efficiency of stimulated emission > efficiencies of spontaneous emission
and absorption
Optical Properties
(2) nph follows Planck black body radiation distribution law
( Einstein relations ) ( A21 , B12 , B21 : Einstein coef. )
nph(hυ)
Optical Properties
high densities of e- in C.B and holes in V.B. (population inversion)
rstim (E21) > rsp (E21)
large photon concentration nph(E21)
Optical Properties
define net rate of absorption :
Rabs = αcv • (photon flux or # of incident photons per unit time) photon flux = nph / ( c/nr )
when stimulated emission rate is high absorption coef. < 0 , which indicates light amplification
OPTOELECTRONICS Prof. Wei-I Lee 86
Abrupt Homojunctions in Thermal Equilibrium
pn-Junctions and Heterostructures
EF remains constant in both n- and p-regions band structures far from junction remain as in isolated n- and p-type semiconductors built-in potential :
– qVD = Ecn – Ecp = Evn – Evp
= exp ( -qVD /kB T )
pn-Junctions and Heterostructures
Junction Capacitance
positive charge in n-region per unit area of the pn-junction :
differential capacitance per unit area :
if NA >> ND
Forward Bias
pn-Junctions and Heterostructures
“+” bias p-side and “-” bias n-side forward bias large current flow from p-side to n-side
@ T.E.
Forward Bias
pn-Junctions and Heterostructures
“+” bias p-side and “-” bias n-side forward bias large current flow from p-side to n-side
OPTOELECTRONICS Prof. Wei-I Lee 91
Forward Bias Diffusion Current
Forward Bias Current-Voltage Characteristics
( caused mainly by deep levels in the
depletion region )
Reverse Bias
OPTOELECTRONICS Prof. Wei-I Lee 94
Reverse Voltage Breakdown
pn-Junctions and Heterostructures
Heterojunction
heterojunction : junction formed by connecting two different materials 2 major kinds of heterojunctions : semiconductor / metal
semiconductor /semiconductor semiconductor/semiconductor heterojunctions (1) pn-junciton and (2) isojunction ( n-n or p-p junction )
pn-heterojunction :
χ
pn – Heterojunction
: electron affinity
: work function
built-in potential in n-side : qVDn in p-side : qVDp total built-in potential :
χnχp
pn – Heterojunction Under Bias
Vn : bias drop on n-side Vp : bias drop on p-side
applied voltage mainly drop on the less doped side (similar to homojunction)
OPTOELECTRONICS Prof. Wei-I Lee 98
pn – Heterojunction Currents Under Forward Bias
pn-Junctions and Heterostructures
( Egn – Egp ) > 0 e- diffusion from
large Egn side to small Egp side dominant
barrier on Ec < barrier on Ev
OPTOELECTRONICS Prof. Wei-I Lee 99
pn – Heterojunction Currents Under Forward Bias
pn-Junctions and Heterostructures
in general, majority carrier in the wide band-gap semiconductor injected into the narrow band-gap semiconductor dominates the current
qV > Egp Jdiff
pn-Junctions and Heterostructures
1. epitaxial growth of compound semiconductor layers on the substrates
2. process of the epitaxially layered wafers, i.e. epi-wafers, into chips or dies
3. packaging of chips or dies
Step. 1 Epitaxy Step. 2 Process Step. 3 Package ,

Interfacial States at Heterojunction
interfacial states originate primarily from crystal defects caused by lattice mismatch
interfacial states generation/
Isotype Heterojunction
pn-Junctions and Heterostructures
electrons or holes diffuse from the wider band-gap semiconductor to the narrower band-gap semiconductor depletion layer in wider band-gap side narrower band-gap side forms 2-dimensional carrier gas
band offsets :
built-in potential :
Double Heterostructures
double heterostructure most DH in optoelectronic devices : wide Eg/narrow Eg/wide Eg
( >500nm /100~1500nm/ >500nm )
under forward bias V :
V = Vpn + Vpp ≈
Vpn electron diffused into and confined in the narrow band-gap p-region
OPTOELECTRONICS Prof. Wei-I Lee 104
Current Voltage Characteristics of Double Heterostructures
pn-Junctions and Heterostructures
under forward bias V : high conc. e- confined in narrow Eg p-region holes injected from wide Eg p-region for charge neutrality high n and p high radiative recombination rate
OPTOELECTRONICS Prof. Wei-I Lee 105
Quantum Wells
characteristics of many optoelectronic pn-junction devices gradually deteriorate with decreasing thickness of the sandwiched narrow Eg layer
narrow Eg layer thickness de Broglie wavelength of e-/hole ( <10nm ) quantized (quantum) effects occur quantum well (QW) structure
single QW
multiple QW
OPTOELECTRONICS Prof. Wei-I Lee 106
Density of States in Quantum Wells
Low Dimensional Structures
Low Dimensional Structures
Strained Quantum Wells
Low Dimensional Structures
if lattice-mismatched well 1. mismatch less than a few % 2. < critical layer thickness
QW without misfit dislocation strained QW
deformation of lattice in the well variation of band-gap energy emission wavelength and absorption edge can be adjusted by controlling the strain
carrier effective mass reduced in strained QW decrease in density of states
OPTOELECTRONICS Prof. Wei-I Lee 109
Quantum Wire and Quantum Box
Low Dimensional Structures
density of states : quantum box < quantum wire < quantum well
OPTOELECTRONICS Prof. Wei-I Lee 110
Quantum Wire and Quantum Box
Low Dimensional Structures
Low Dimensional Structures
Density of states in structures of various dimensionality, calculated on the assumption that the carrier energy E in the band is a quadratic function of wave vector k.
OPTOELECTRONICS Prof. Wei-I Lee 112
Quantum Wire and Quantum Box
Low Dimensional Structures
Schrodinger Equation
From Atoms to Crystals
Four-Index Notation for Hexagonal Crystal
Lattice Planes in Four-Index Notation ( h k i l )
Bloch’s Theorem
Kronig-Penny Model
Kronig-Penny Model
Kronig-Penny Model
Dispersion Relation
Problems of Kronig-Penney Model
Symmetric and Antisymmetric Combination
Closely Spaced Potential Wells
Tight Binding Approximation of Multi-Atoms
Formation of Bands
Sodium Crystal
Magnesium Crystal
Carbon Crystal
Vegard’s Law
Density of States
Impurity Ionization Energies in GaAs
Effective Mass
Fermi-Dirac Distribution
Carrier Concentration
Carrier Concentration vs. Temperature
QUIZ # 4
Carrier Drift Velocity and Mobility
Drift Current and Conductivity
Nonradiative Recombination of Carriers
Band-to-Band Spontaneous Emission
Spontaneous Emissions
Junction Capacitance
Forward Bias
Forward Bias
Manufacturing of Semiconductor Optoelectronic Devices
Interfacial States at Heterojunction
Quantum Wells
Optical Properties of Quantum Wells
Strained Quantum Wells