Ch3 Semiconductor Science and Light Emitting...

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Chapter 3 Semiconductor Science and

Light Emitting Diodes

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Outline3.1 Semiconductor Concepts and Energy Bands3.2 Direct and Indirect Bandgap Semiconductor3.3 pn Junction Principles3.4 The pn Junction Band Diagram3.5 Light Emitting Diodes3.6 LED Materials3.7 Heterojunction High Intensity LEDs3.8 LED Characteristics3.9 LEDs for Optical Fiber Communications

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3-1 Semiconductor Concepts and Energy Bands

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2 s Band

Overlapping energybands

Electrons2 s2 p

3 s3 p

1 s 1sSOLIDATOM

E = 0

Free electronElectron Energy, E

2 p Band

3s BandVacuum

level

In a metal the various energy bands overlap to give a single bandof energies that is only partially full of electrons. There are stateswith energies up to the vacuum level where the electron is free.?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

A. Energy Band Diagrams

Li : 3 e- (1023 Li atoms)

Metal: partially filled band

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Electron energy, E

Conduction Band (CB)Empty of electrons at 0 K.

Valence Band (VB)Full of electrons at 0 K.

Ec

Ev

0

Ec+χ

(b)

Band gap = Eg

(a)

Covalent bond Si ion core (+4e)

(a) A simplified two dimensional view of a region of the Si crystalshowing covalent bonds. (b) The energy band diagram of electrons in theSi crystal at absolute zero of temperature.

SemiconductorValence band (VB)Conduction band (CB)Electron affinity χBandgap (Eg)

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e–hole

CB

VB

Ec

Ev

0

Ec+χ

EgFree e–hυ > Eg

Hole h+

Electron energy, E

(a) A photon with an energy greater than Eg can excite an electron from the VB to the CB.(b) Each line between Si-Si atoms is a valence electron in a bond. When a photon breaks aSi-Si bond, a free electron and a hole in the Si-Si bond is created.

hυ

(a) (b)

Effective mass:Electron in CB: me*Hole in VB: mh*

(take into account the e- in the CB interacts with a “periodic potential” as it movers through the crystal)

viewed as free carriers with an effective mass

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B. Semiconductor Statistics

( )21

=FEf However, there may be no state.

1( )1 exp F

B

f EE E

k T

=⎛ ⎞−

+ ⎜ ⎟⎝ ⎠

(1)Probability of finding e- with energy E(Fermi-Dirac function)

Density of States (DOS) ( ) 21)( cCB EEEg −∝

Probability of finding h+ with energy E )(1 Ef−

eVEF =Δ (2)Electric work input or output per e-

In equilibrium, no applied field, under dark environment:

FE( : uniform)0=Δ FE

Under applied potential V:

(# of e- states per unit energy per unit volume)

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E

g(E) fE)

EF

nE(E) or pE(E)

E E

Forelectrons

For holes

[1–f(E)]

nE(E)

pE(E)

Area = p

Area = � nE(E)dE = n

Ec

EvEv

Ec

0

Ec+χ

EF

VB

CB

(a) (b) (c) (d)g(E) ∝ (E–Ec)

1/2

(a) Energy band diagram. (b) Density of states (number of states per unit energy perunit volume). (c) Fermi-Dirac probability function (probability of occupancy of astate). (d) The product of g(E) and f(E) is the energy density of electrons in the CB(number of electrons per unit energy per unit volume). The area under nE(E) vs. E isthe electron concentration.

∫ == ndEEnArea E )(

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( ) ( ) ( ) StatisticsBoltzmann exp ⎟⎠⎞

⎜⎝⎛ −−≈→>>− Tk

EEEfTkEEB

fBfc

g ( ) ( )cc

E

CBEn E f E dE

χ+= ∫ (3)

Electron concentration in CB

exp c FcB

E En Nk T

⎡ ⎤−= −⎢ ⎥

⎣ ⎦(4)

[ ] 2/32* /22 hTkmN Bec π=Effective DOS at the CB edge

Non-degenerate Semiconductor(# of e-h+ from Ed (B- site) into VB

at room temperature

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e hen epσ μ μ= +2i

d e h d ed

neN e eNN

σ μ μ μ⎛ ⎞

= + ≈⎜ ⎟⎝ ⎠

(8)

(9)

C. Extrinic Semiconductorsn-type semiconductor: add pentavalent impurity.Donor impurity: e.g. As in Si, --> As+ +e-Nd>>ni

2

, idnn N pn

≈ =

Semiconductor conductivityn-type conductivity

P-type semiconductor: add trivalent impurityAcceptor impurity: e.g. B in Si, --> B- +h+Na>>ni

pnnNp ia

2

, ==p-type conductivity haeN μσ ≈

Electrons: majority carrier; Holes: minority carrier

nono ppnn == ,

,po pon n p p= =

(in equilibrium)

(in equilibrium)

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Ec

Ev

EFi

CB

EFp

EFnEc

Ev

Ec

EvVB

(a) (c)(b)

Energy band diagrams for (a) intrinsic (b) n-type and (c) p-typesemiconductors. In all cases, np = ni2. Note that donor and acceptorenergy levels are not shown.

?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

The level of EF determines the electron and hole concentrations.

⎥⎦

⎤⎢⎣

⎡ −−=

TkEENn

B

FnCC exp ⎥

⎦

⎤⎢⎣

⎡ −−=

TkEE

NpB

VFpV exp

1 1 ln2 2

cFi v g

v

NE E E kTN

⎛ ⎞= + − ⎜ ⎟

⎝ ⎠

(quasi-Fermi level: EFn & EFp)

exp Fn FiiB

E En nk T

⎡ ⎤−= ⎢ ⎥

⎣ ⎦exp Fi Fpi

B

E Ep p

k T−⎡ ⎤

= ⎢ ⎥⎣ ⎦

ni

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D. Compensation Doping

Doping of semiconductor with both donors and acceptors to control the properties.

Electrons (from donors) recombine with holes (from acceptors)

nnp

nNNn

i

iad2

=

>>−=

pnn

nNNp

i

ida2

=

>>−=

p-type n-type n-type p-type

n p => recombination

Finally, 2inp n=

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CB

g(E)

E

Impuritiesforming a band

(a) (b)

EFp

Ev

EcEFn

Ev

Ec

CB

VB

(a) Degenerate n-type semiconductor. Large number of donors form aband that overlaps the CB. (b) Degenerate p-type semiconductor.

E. Degenerate and Non-degenerate SemiconductorsNon-degenerate semiconductor:

The electron statistics ~the Boltzmann statistics (4)Pauli exclusion principle can be neglected.

vc NpNn

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In degenerate semiconductor:n ≠ Ndp ≠ Nanp ≠ ni2

When dopant concentration is large

dopants interact with each other

not all dopants can become ionized

Saturation of carrier concentration ~ 1020/cm3

For Si:

(heavy doping)

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F. Energy BandDiagrams in an Applied Field

V

n-Type Semiconductor

Ec

EF − eV

A

B

V(x), PE (x)

x

PE (x) = � eV

Energy band diagram of an n-type semiconductor connected to avoltage supply of V volts. The whole energy diagram tilts becausethe electron now has an electrostatic potential energy as well

EElectron Energy

Ec − eV

Ev− eV

V(x)

EF

Ev

-eV(potential energy of e-)

EF uniform ΔEF=0

Applied field ΔEFe- concentration uniform⇒EC-EV constant⇒CB, VB, and EF bend

by the same amount

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Example 3.1.1 Fermi levels in semiconductors(1) n-type doping of Si wafer: 1016/cm3 antimony (Sb) atoms

Find: Fermi energy w. r. t. Fermi energy EFi in the intrinsic Si

(2) Compensated doping of Si wafer: 1016/cm3 antimony (Sb) atoms (n-type) and 2×1017/cm3 boron (B) atoms (p-type)

Find: (a) Fermi energy w. r. t. Fermi energy EFi in the intrinsic Si at room temperature (300K)

(b) Fermi energy w. r. t. Fermi energy in the n-type case in (1)

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(1) n-type doping of Si wafer: 1016/cm3 antimony (Sb) atomsFind: Fermi energy w. r. t. Fermi energy EFi in the intrinsic Si

( )

( )[ ]( )[ ]

( )[ ]( ) ( ) ( ) eVeVnNTkEE

TkEEnNNTkEENn

TkEENncmNn

cmnNcmN

idBFiFn

BFiFnid

dBFncc

BFicci

d

id

d

348.01045.1/10ln0259.0/ln

/exp/

/exp/exp

10

1045.1

,10

1016

316

310

316

=×==−

−=

=−−=−−=

==

×=>>

=

−

−

−

exp Fn FiiB

E En nk T

⎡ ⎤−= ⎢ ⎥

⎣ ⎦

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(2) Compensated doping of Si wafer: 1016/cm3 antimony (Sb) atoms (n-type) and 2×1017/cm3 boron (B) atoms (p-type)

Find: (a) Fermi energy w. r. t. Fermi energy EFi in the intrinsic Si at room temp. (b) Fermi energy w. r. t. Fermi energy in the n-type case in (1)

( )( )[ ]

( )[ ]( )[ ]

( ) ( ) ( ) eVeVnpTkEETkEEnp

NNTkEENpTkEENnp

cmNNpcmNcmN

iBFiFp

BFiFpi

daBvFpv

BvFivi

da

da

424.01045.1/109.1ln0259.0/ln

/exp/

/exp

/exp109.110102

10102

1017

3171617

316317

−=××−=−=−

−−=

−=−−=

−−==×=−×=−=

=>×=−

−− (P-type)

exp Fi FpiB

E Ep n

k T−⎡ ⎤

= ⎢ ⎥⎣ ⎦

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Example 3.1.2 ConductivityGiven: (a) N-type Si crystal with doping 1016/cm3 phosphorus (P)

(b) Drift mobility of electrons is ~ 1350 cm2V-1s-1

Find: conductivity of the n-type Si

• Since

• The electron concentration

• We can neglect the hole concentration

• Thus

( )

( )( )( ) 1111231619

2

310316

16.21350100.1106.1

,/

1045.110

−−−−−−

−−

Ω=××==

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r

PE(r)PE of the electron around anisolated atom

When N atoms are arranged to formthe crystal then there is an overlapof individual electron PE functions.

x

V(x)

x = Lx = 0 a 2a 3a

0aa

Surface SurfaceCrystal

PE of the electron, V(x), insidethe crystal is periodic with aperiod a.

The electron potential energy (PE), V(x), inside the crystal is periodic with the sameperiodicity as that of the crystal, a. Far away outside the crystal, by choice, V = 0 (theelectron is free and PE = 0).

?1999 S O Kasap Optoelectronics (Prentice Hall)

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0)]([2 222

=−+ ψxVEmdxψd e

( ) ( ) ; 1 , 2 , 3 ,V x V x ma m= + =

(1)

( ) ( ) exp( )k kx U x jkxψ =

(2)(3)

Schrodinger Equation

Periodic Potential

Bloch Wavefunctions

The electron in a crystal

( ) wavetraveling)(

)(on dependfunction periodic

→⋅

→− Etjjkx

k

ee

xVxU

, ( , ) exp( / ) ( )nn k n kx t jE t xψ ψ= −Overall electron wavefunction

kn: a state in Ek

nkCrystal momentum:( ) / extd k dt F=

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Ek

kš /a–š /a

Ec

Ev

ConductionBand (CB)

Ec

Ev

CB

The E-k Diagram The Energy BandDiagram

Empty ψk

Occupied ψkh+

e-

Eg

e-

h+

hυ

VB

hυ

ValenceBand (VB)

The E-k diagram of a direct bandgap semiconductor such as GaAs. The E-kcurve consists of many discrete points with each point corresponding to apossible state, wavefunction ψk(x), that is allowed to exist in the crystal.The points are so close that we normally draw the E-k relationship as acontinuous curve. In the energy range Ev to Ec there are no points (ψk(x)solutions).

E-k Diagram = 0K, all fill the lower states> 0K, from VB to CB (thermal)

-π/a π/a

kCrystal momentumof a electron:

( ) / extd k dt F=

many discrete points=> continuous curve

Bandgap: no solution

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E

CB

k–k

Direct Bandgap

(a) GaAs

E

CB

VB

Indirect Bandgap, Eg

k–k

kcb

(b) Si

E

k–k

Phonon

(c) Si with a recombination center

EgEc

EvEc

Evkvb VB

CB

ErEc

Ev

Photon

VB

(a) In GaAs the minimum of the CB is directly above the maximum of the VB. GaAs istherefore a direct bandgap semiconductor. (b) In Si, the minimum of the CB is displaced fromthe maximum of the VB and Si is an indirect bandgap semiconductor. (c) Recombination ofan electron and a hole in Si involves a recombination center .

?1999 S O K O l (P i H ll)

Direct bandgap semiconductor: e.g. GaAs

Indirect bandgap semiconductor: e.g. Sirecombination via a recombination center: defect or impurityphonons (lattice vibrations, quantized, waves)

photon momentum: negligible as compared with electron momentum

kcrystal momentum:

( ) / extd k dt F=

Conservation: energy + momentum

(defects or impurities)

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3-3 pn Junction Principles

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nno

xx = 0

pno

ppo

npo

log(n), log(p)

-eNa

eNd

M

x

E (x)

B-

h+

p n

M

As+

e–

Wp Wn

Neutral n-regionNeutral p-region

Space charge region Vo

V(x)

x

PE(x)

Electron PE(x)

Metallurgical Junction

(a)

(b)

(c)

(e)

(f)

x

–WpWn

(d)

0

eVo

x (g)

–eVo

Hole PE(x)

–Eo

Eo

M

ρnet

M

Wn–Wp

ni

A. Open circuit

(abrupt discontinuity)

NdNa

NdNa

2inp n=

no applied fieldor photoexcitation

equilibrium

concentration gradient=> diffusion I

electrical field=> drift I

Equilibrium: diffusion I = drift I

ndpa WNWN =charge neutrality

(Depletion region)(layer)

(SCL)

Banddiagram

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ndpa WNWN = (1)

(2)21

2 2 ( )a d o

o o oa d

eN N WV E WN Nε

= − =+ (3)

Depletion widths

Built-in field

Built-in potential

2lna dB

oi

N Nk TVe n

⎛ ⎞= ⎜ ⎟

⎝ ⎠

(4)

(5)

(6)

⎟⎟⎠

⎞⎜⎜⎝

⎛=

po

noBo n

nlneTkV

⎟⎟⎠

⎞⎜⎜⎝

⎛=

no

poBo p

peTkV ln

Build-in potential

( )ε

ρ xdx

dE net=

dxdVE −=

( )TkeVpp Bopono −= exp

( )TkeVnn Bonopo −= expMinority/majority

( ) energy potential is where,exp 121

2 ETk

EEnn

B⎥⎦

⎤⎢⎣

⎡ −−=

By Boltzmann statistics

εεpand WeNWeNE −=−=0

2exp( )gc v iB

Enp N N n

k T= − =

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Neutral n-regionNeutral p-regionEo – E

Log (carrier concentration)

Holediffusion

Electrondiffusion

np(0)

Minute increase

pn(0)

pnonpo

pponno

V

Excess holes

Excess electrons

x′x

(a)

W

e(Vo–V)eVo M

x

Wo

Hole PE(x)

(b)

SCL

Forward biased pn junction and the injection of minority carriers (a) Carrierconcentration profiles across the device under forward bias. (b). The holepotential energy with and without an applied bias. W is the width of the SCLwith forward bias

B. Forward BiasVoltage drop across SCL or depletion region W

Reduce the built-in potential against diffusion Result in the injection of excess minority carrier and small increase in the majority carrier

(charge neutrality)

Banddiagram

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(0 ) ex pn n oB

eVp pk T

⎡ ⎤= ⎢ ⎥

⎣ ⎦ (8)Law of the junction(just outside Wn)

( )(0 ) ex p on p oB

e V Vp pk T

⎡ ⎤− −= ⎢ ⎥

⎣ ⎦(7)Excess holes at x’=0

(0 ) ex pp p oB

eVn nk T

⎛ ⎞= ⎜ ⎟

⎝ ⎠ (9)Law of the junction(just outside Wp)

( ') (0 ) ex p ( '/ )n n hp x p x LΔ = Δ − (10)Excess minority carrier concentration, : diffusion coefficient

: mean hole recomb. lifetimeh h h h

h

L D Dττ

=Hole diffusion length

(due to reduction in built-in potential barrier)

2 12 1 exp

B

E En nk T

⎛ ⎞−= −⎜ ⎟

⎝ ⎠

V=0 =>

e-

e-h+ e

-

(p and n-regions are longer than diffusion length of minority carriers)

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Jelec

x

n-region

J = Jelec + Jhole

SCL

Minority carrier diffusioncurrent

Majority carrier diffusionand drift current

Total current

Jhole

Wn–Wp

p-region

J

The total currentanywhere in the device isconstant. Just outside thedepletion region it is dueto the diffusion ofminority carriers.


e-

e-h+ e

-

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,( ') ( ')

' 'n n

D hole h hdp x d p xJ eD eD

dx dxΔ

= − = −

,'(0) exphD hole n

h h

eD xJ pL L

⎛ ⎞ ⎛ ⎞= ⋅ Δ −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠(11)

2, , exp 1h eD hole D elec i

h d e a B

eD eD eVJ J J nL N L N k T

⎡ ⎤⎛ ⎞ ⎛ ⎞= + = + −⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠ ⎣ ⎦

exp 1soB

eVJ Jk T

⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦(12)or

Hole diffusion current density

Assume electron and hole currents don’t change across the SCL.

Shockley diode equation

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛= 1

2

TkeVexp

NLneDJ

Bdh

ihhole,DAt x’=0

At x’=0

Jso: reverse saturation current densitysoBr JJmVeTkV −=≈>− ),25/( :bias reverse

(0 ) ex pn n oB

eVp pk T

⎡ ⎤= ⎢ ⎥

⎣ ⎦

The total current density

(narrow junction and neglect e-h recombination)

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C

WnWp

Log (carrier concentration)

np(0)pnonpo

ppo nno

V

x

p-sideSCL

pn(0)

pM

M

nM

n-side

B

HolesElectrons

A D

Forward biased pn junction and the injection ofcarriers and their recombination in the SCL.

Some of the minority carriers will recombine in the depletion region.

a symmetrical pn junction, at the junction C MM np =

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exp2M i B

eVp nk T

⎛ ⎞= ⎜ ⎟

⎝ ⎠

exp2 2

pi nrecom

e h B

Wen W eVJk Tτ τ

⎡ ⎤ ⎛ ⎞= + ⎜ ⎟⎢ ⎥

⎝ ⎠⎣ ⎦

exp 1oB

eVI Ik Tη

⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

(14)

(15)

1 12 2p M n M

recome h

e W n e W pJ

τ τ≈ + (13)

Diode Equation

Recombination current

Consider a symmetrical pn junction, at the junction C,Electrons recombine in Wp and holes recombine in Wn.

MM np =

n, pWin ion timerecombinat (electron) holemean :e h,τdt

dQJ =

η=1, diffusion controlledη=2, SCL recombination controlled

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛= 1

2exp

TkeVJJ

Brorecom

Diode ideality factor

(Vo-V)/2 at M

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nA

I

Shockley equation

Space charge layergeneration.

V

mAReverse I-V characteristics of apn junction (the positive andnegative current axes havedifferent scales)

I = Io[exp(eV/ηkBT) − 1]

exp 1oB

eVI Ik Tη

⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦Diode Equation η=1, diffusion controlled

η=2, SCL recombination controlled

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛= 1

2exp

TkeVJJ

Brorecomexp 1diffusion so

B

eVJ Jk T

⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

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WoWo

Neutral n-regionNeutral p-region

x

W

HolesElectrons

DiffusionDrift

x

(a)(b)

ThermallygeneratedEHP

pnonpo

Vr

Eo+E

Minority CarrierConcentration

e(Vo+Vr)eVo

W(V = –Vr)

MHole PE(x)

Reverse biased pn junction. (a) Minority carrier profiles and the origin of thereverse current. (b) Hole PE across the junction under reverse bias


C. Reversed Bias

Banddiagram

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g

ii

ae

e

dh

h eWnnNL

eDNL

eDJτ

+⎟⎟⎠

⎞⎜⎜⎝

⎛+= 2rev (17)

Total Reverse Current

g

ieWnJτ

=gen (16)EHP thermal generation in SCL

:gτ mean thermal generation time

Shockley diode equation

direction reverse:""

2

−

⎟⎟⎠

⎞⎜⎜⎝

⎛+−=−=

in

NLeD

NLeDJJ

ae

e

dh

hso

exp 1soB

eVJ Jk T

⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦soBr JJmVeTkV −=≈>>− ),25/( :bias reverse

(indep. of Vr)reverse saturation current density

(cf. photodetectors and solar cells: EHP generation also by photons )

(J

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nA

I

Shockley equation

Space charge layergeneration.

V

mAReverse I-V characteristics of apn junction (the positive andnegative current axes havedifferent scales)

I = Io[exp(eV/ηkBT) − 1]

g

ii

ae

e

dh

h eWnnNL

eDNL

eDJτ

+⎟⎟⎠

⎞⎜⎜⎝

⎛+= 2rev

exp 1oB

eVI Ik Tη

⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦Diode Equation η=1, diffusion controlled

η=2, SCL recombination controlled

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛= 1

2exp

TkeVJJ

Brorecomexp 1diffusion so

B

eVJ Jk T

⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦

42

901 37500 光電導論

0.002 0.004 0.006 0.0081/Temperature (1/K)

10 -16

10 -14

10 -12

10 -10

10 -8

10 -6

10 -4

Reverse diode current (A) at V = −5 V

Ge Photodiode323 K

238 K0.33 eV

0.63 eV

Reverse diode current in a Ge pnjunction as a function of temperature ina ln( Irev ) vs. 1/ T plot. Above 238 K, Irevis controlled by ni2 and below 238 K itis controlled by ni. The vertical axis isa logarithmic scale with actual currentvalues. (From D. Scansen and S.O.Kasap, Cnd. J. Physics. 70 , 1070-1075,1992.)


( )

( )

2

238 K1ln v.s. 0.63 eV~ 0.66 eV ,

238 K11ln v.s. 0.33 eV~2

rev g rev i

rev g rev i

T

I E I nTT

I E I nT

>

= = → ∝

<

= → ∝

exp( )2

gi c v

B

En N N

k T= −

⎥⎥⎦

⎤

⎢⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

g

ii

ae

e

dh

h eWnnNL

eDNL

eDAIτ

2rev

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901 37500 光電導論

1/ 2

1/ 2

( )( ) 2( )

a ddep

o a d

e N NA ACW V V N N

εε ⎡ ⎤= = ⎢ ⎥− +⎣ ⎦

(20)

D. Depletion Layer Capacitance

d e pd QCd V

= (18)Depletion Layer Capacitance

1/ 22 ( )( )a d o

a d

N N V VWeN N

ε⎡ ⎤+ −= ⎢ ⎥

⎣ ⎦(19)SCL width

and voltage

212 2 ( )

a d oo o o

a d

eN N WV E WN Nε

= − =+ (3)

pF

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/p p p thermaln t Bn p G∂Δ ∂ = − +

( )p p p po pon

B n p n pt

∂Δ= − −

∂p p

e

n nt τ

∂Δ Δ= −

∂

/p a pn t BN n∂Δ ∂ = − Δ

(21)

(22)

(23)

(24)

E. Recombination Lifetime

1/e aBNτ =

2/ ( )p p p pn t B p n B n∂Δ ∂ = Δ Δ = Δ

(25)

(26)

B: direct recombination capture coefficient

Rate of change due to recombination

Excess minority carrier recombination time (lifetime)

In weak injection, apoppppop Nppnnpn ≈≈Δ≈→Δ

majority ) (

minority

ppppop

ppop

npnppnnn

Δ=ΔΔ+=

Δ+=Forward bias

steady state

Banddiagram

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901 37500 光電導論

Example 3.3.1: A direct bandgap pn junctionSymmetric GaAs pn junction with a cross section A = 1 mm2, given

(1) Na (p-side doping) = Nd (n-side doping) = 1023 m-3

(2) B = 7.21 x 10-16 m3s-1

(3) ni = 1.8 x 1012 m-3

(4) εr = 13.2(5) μh (in the n-side) = 250 cm2V-1s-1

(6) μe (in the p-side) = 5000 cm2V-1s-1

(7) Diffusion coefficients (Einstein relation)Dh = μhkBT/e & De = μekBT/e

(8) Forward voltage = 1 V

Find: (a) diode current due to the minority carrier diffusion at 300K

(assume direct recombination)(b) Estimate the recombination component of the current

(if the mean minority carrier recombination time in the depletion region is ~ 10ns) 46

901 37500 光電導論

• Assuming weak injection, we can readily calculate the recombination times τe and τh for electrons and holes recombining in the neutral p and n-regions.

( )( )( )( )( )( )

[ ] ( )( )[ ][ ] ( )( )[ ]

( ) ( )( ) ( )( ) ( )( )

( ) AV

VATk

eVII

A

msm

msm

enNL

DNL

DAI

mssmDL

mssmDL

smeTkDsmeTkD

smsmBN

VeTk

Bsodiff

iae

e

dh

hso

eee

hhh

Bee

Bhh

aeh

B

421

21

21219235

122

236

1246

2

55.081225.0

65.081245.0

1224

1244

83231316

109.302585.0

0.1exp1013.6exp

1013.6

108.1106.1101034.1

1029.1101000.3

1046.610

1034.11039.11029.1

1000.31039.11046.6

1029.11050002585.0/

1046.6102502585.0/

1039.1100.11021.7

1102585.0/

−−

−

−−

−−

−

−−−

−−−−

−−−−

−−−

−−−

−−−−

×=⎟⎠⎞

⎜⎝⎛×=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

×=

××⎥⎦

⎤⎢⎣

⎡××

+××

=

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

×=××==

×=××==

×=×==

×=×==

×=××

===

=

τ

τ

μ

μ

ττ

47

901 37500 光電導論

( ) ( )( )( )

( )( )( )( )( )( )

( )( )( )

( ) ( ) AVVA

TkeVII

ACWAenWWAenI

nsWWW

mmmmC

VmFm

NeNVVNNW

VVnNN

eTkV

Brorecom

r

i

h

n

e

piro

rhe

np

da

oda

i

daBo

412

129

812196

85.0

32332319

32323112

5.0

212

2323

2

103.302585.02

0.1exp103.12

exp

103.11010109

2108.1106.110

22

10

5.0

09.0100.91010106.1

128.110101085.82.132

2

28.1108.11010ln02585.0ln

−−

−−

−−−

−−−−

−−−

×=⎟⎟⎠

⎞⎜⎜⎝

⎛×=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

×=⎟⎟⎠

⎞⎜⎜⎝

⎛××××

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

≈==

==

=×=⎥⎦

⎤⎢⎣

⎡×

−+××=

⎥⎦

⎤⎢⎣

⎡ −+=

=⎟⎟⎠

⎞⎜⎜⎝

⎛

×=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

τττ

τττ

μ

ε

In this example, the diffusion and recombination components are about the same order of magnitude.

48

901 37500 光電導論

3-4 The pn Junction Band Diagram

49

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EcEv

Ec

EFp

M

EFn

eVo

p nEo

Evnp

(a)

VI

np

Eo–E

e(Vo–V)

eV

EcEFn

Ev

Ev

Ec

EFp

(b)

(c)

Vr

np

e(Vo+Vr)

EcEFn

Ev

Ev

Ec

EFp

Eo+E (d)

I = Very SmallVr

np

Thermalgeneration Ec

EFnEv

Ec

EFp

Ev

e(Vo+Vr)

Eo+E

Energy band diagrams for a pn junction under (a) open circuit, (b) forwardbias and (c) reverse bias conditions. (d) Thermal generation of electron holepairs in the depletion region results in a small reverse current.

SCLdiffusion efor

eV barrier potentialpotentialin -built variationh and e

:bended is Banduniform is

-0

-

→

+

FE( )

TkeV

Be

VVe

∝

→

−

current forwardionconcentratcarrier minority

barrier potential 0

( )

current reversesmallEHP of generation thermalthin carrier wiminority small

barrier potential

or

0

→+

+

he

r

LVVe

50

901 37500 光電導論 Introduction of optoelectronics 50

Forward bias

hhhhhhh hh h

Apply positive bias at lefthand side

hhhhhhh hh h

hhhhhhh h

h hhhhhh hhhh

+V

V0

⎥⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎥

⎦

⎤⎢⎣

⎡−⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+ 1exp1exp~ 2

TkeVJ

TkeVn

NLeD

NLeDJ

Bso

Bi

ae

e

dh

h

If electron and hole do not recombine (Si), the current can be expressed as :

lifetimeion recombinat : t coefficiendiffusion :

ττ DDL =

51


hhhhhhh hh h h hhhhhh hhhh

+V

1 12 2p M n M

recome h

e W n e W pJ

τ τ≈ +

Recombination Current

Consider a symmetrical pn junction, at the junction C,Electrons recombine in Wp and holes recombine in Wn.

MM np =

dtdQJ =

n, pWin ion timerecombinat (electron) holemean :e h,τ

V0-V

52


Reverse bias

hhhhhhh hh hhhhhh h hhhhhhhh hh hhhhh hhh-VR

h

g

ieWnJτ

=gen

EHP thermal generation in SCL

g

ii

ae

e

dh

h eWnnNL

eDNL

eDJτ

+⎟⎟⎠

⎞⎜⎜⎝

⎛+= 2rev

Total Reverse Current

h

1exp , and 0when

~1exp~ 22

53

901 37500 光電導論

53

3-5 Light Emitting Diodes

54

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Introduction of optoelectronics 54

An actual configuration of a LED

55

901 37500 光電導論

LED = 半導體?什麼是半導體?導電性介於導體與非導體之間

半導體的組成

週期表上第四族，三五或二六族化合物週期性排列的結構

依據參入不同特定之雜質又分為p型與 n型半導體

(a) (b) (c)

帶電的載子n型 – electronp型 – hole

55 56901 37500 光電導論

半導體如何發光?

• Absorption

• Electron transition

• Light emission

56

57

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57

hυ - Eg

Eg (b)

V

(a)

p n+

EgeVo

EF

p n+

Electron in CBHole in VB

Ec

Ev

Ec

Ev

EF

eVo

Electron energy

Distance into device

(a) The energy band diagram of a p-n+ (heavily n-type doped) junction without any bias.Built-in potential Vo prevents electrons from diffusing from n+ to p side. (b) The appliedbias reduces Vo and thereby allows electrons to diffuse, be injected, into the p-side.Recombination around the junction and within the diffusion length of the electrons in thep-side leads to photon emission.?1999 S.O. Kasap, Optoelectronics (Prentice Hall)

LED: pn junction diodedirect band gap semiconductor, e.g. GaAsEHP recombination photon emission (injection electroluminescence)

spontaneous emission in active region (SCL & within Le in p region)

A. Principles

(Random directions)

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58

Light output

Insulator (oxide)p

n+Epitaxial layer

A schematic illustration of typical planar surface emitting LED devices. (a) p-layergrown epitaxially on an n+ substrate. (b) First n+ is epitaxially grown and then p regionis formed by dopant diffusion into the epitaxial layer.

Light output

pEpitaxial layers

(a) (b )

n+

Substrate Substrate

n+

n+

Metal electrode

B. Device Structuresn+ region: heavily doped, electron injection

photon absorbed or reflectionp region: light emitting, narrow (a few microns) to avoid photon absorption

Epitaxial (a) or diffused (b) junction planar LED Substrate-n+ region: Lattice matching

Lattice mismatch radiationless EHP recombination

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59

Light output

p

Electrodes

LightPlastic dome

Electrodes

Domedsemiconductor

pn Junction

(a) (b) (c)

n+n+

(a) Some light suffers total internal reflection and cannot escape. (b) Internal reflectionscan be reduced and hence more light can be collected by shaping the semiconductor into adome so that the angles of incidence at the semiconductor-air surface are smaller than thecritical angle. (b) An economic method of allowing more light to escape from the LED isto encapsulate it in a transparent plastic dome.

Substrate


TIR low e.g. GaAs-air, θc~16o

Domed semiconductorDifficult process in fabrication

Plastic Domeinexpensive and widely used

Cf. overcome TIR by roughness and photonic crystals

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60

3-6 LED Materials

61

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什麼材料 - 發什麼光1λ

2λ

‧藍綠光: 氮化銦鎵

‧紅黃光: 磷化鋁銦鎵 (四元)

61

InGaN

AlInGaP

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The Material Properties – Zinc Blend structure

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Wurtzite Nitrides

Wu et. Al. Solid State Comm., 127:411-414, 2003.

The bandgap of nitrides cover the whole visible light spectrum.

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64

Ec

Ev

EN

(b) N doped GaP

Eg

(a) GaAs1-yPyy < 0.45

(a) Photon emission in a direct bandgap semiconductor. (b). GaP is anindirect bandgap semiconductor. When doped with nitrogen there is anelectron trap at EN. Direct recombination between a trapped electron at ENand a hole emits a photon. (c) In Al doped SiC, EHP recombination isthrough an acceptor level like Ea.

Ec

EvEa

(c) Al doped SiC

?1999 S O Kasap Optoelectronics (Prentice Hall)

Direct bandgap:

Indirect Bandgap:

1

1

0.49 0.51

1 1

0.45, 630)

0.43, 640)0.17)

( 3.4 )

(0 870 nm

(0 870 nm (0.058

y y

x x

x x

x x y y

g

GaAs P yAl Ga As xIn Al Ga P xIn Ga As PGaN E eV

λ

λ−

−

−

− −

≤ < > >

≤ < > >

≤ <

=

ternary (3 elements)

AlSNyPGaAs yy

with doped , iC with doped ),45.0( 1 >−

(isoelectronic impurities,nucleus of N is less shielded)

acceptor type localized energy level

quarternary (4 elements)(High intensity)

(Optical communication)

(EHP recomb. thrurecomb. centers and Involve lattice vibrations)

GaAs1-yPyN doped indirect Egwidely used in inexpensive green, yellow, and orangeLEDs

Blue LED: InGaN (Eg = 2.7 eV)

(GaN, Eg = 3.4 eV)Al-SiCZnSe

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65

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6λ

1.7Infrared

GaAs1-yPy

InP

In1-xGaxAs1-yPyAlxGa1-xAs

x = 0.43

Indirectbandgap

Free space wavelength coverage by different LED materials from the visible spectrum to theinfrared including wavelengths used in optical communications. Hatched region and dashedlines are indirect Eg materials.

In0.49AlxGa0.51-xP


= internal efficiency of radiative recombination process and photon extraction

(Generally, ext. efficiency < 1% for indirect Eg materials)

( ) 100%outexternalP optical

IVη = ×External efficiency

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66

, 100%out opticalexternal internal extractionP

IVη η η= ⋅ = ⋅

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67

3-7 Heterojunction High Intensity LEDs and Quantum

Well LEDs

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68

2 eV

2 eVeVo

Holes in VB

Electrons in CB1.4 eV

No bias

Withforwardbias

Ec

EvEc

Ev

EFEF

(a)

(b)

(c)

(d)

pn+ p

ΔEc

GaAs AlGaAsAlGaAs

ppn+

~ 0.2 μm

AlGaAsAlGaAs

(a) A doubleheterostructure diode hastwo junctions which arebetween two differentbandgap semiconductors(GaAs and AlGaAs)

(b) A simplified energyband diagram withexaggerated features. EFmust be uniform.

(c) Forward biasedsimplified energy banddiagram.

(d) Forward biased LED.Schematic illustration ofphotons escapingreabsorption in theAlGaAs layer and beingemitted from the device.


GaAs

Homojunction: narrow p region EHP recombine through surface defectslong p region re-absorption of photons increases

Heterojunction: a junction between two diff. Eg semiconductors

Double heterojunction device:Thin p-GaAs layer between p and nAlGaAs layers: a confining layer

Photon will not be absorbed with the wide bandgapmaterial AlGaAs.

∆Ec: potential energy barrier against electron in CB (p-GaAs)to CB of p-AlGaSa

Additional advantage of AlGaAs/GaAs hetero-junction:small lattice mismatch between two crystal materials

negligible strain induced defects(Homo-junction: defects at the surface )

large nsmall n small n

69


Double Heterojunction LEDsHomojunction: narrow p region EHP recombine through surface defects

long p region re-absorption of photons increasesHeterojunction: a junction between two diff. Eg semiconductors

Eg1 Eg2

Heterojunction

Type I semiconductor

Eg1Eg2

Type II

Eg1

Eg2

Type IIIvaccum

sχ

70


Alloy -> Using two materials with different composition to change the bandgap.

GaAsInAs

InxGa1-xAs

( ) ( )( ) ( )xInGaExGaAsEAsGaInE ggxxg +−− 1~1

( ) ( ) ( )InAsmx

GaAsmx

AsGaInm xx**

1*

1~1 +−

−

71


Double Heterojunction LEDs

Eg1 Eg2 Eg1

~0.1-0.2um

Eg1

Eg1

n

Eg2hhhhhhh hh hhhhh hhhhhhh hhhhhhhh

p

Eg2 Eg1

p

Eg1n

+v

hhhhhh h

hhh

hhh

hhh

h

hhh

hh

h

hh

72


Band diagrams of different designs of Band diagrams of different designs of heterojunctionsheterojunctions

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901 37500 光電導論

Carrier loss in double Carrier loss in double heterostructuresheterostructures

Leakage increases exponentially with temperatureRadiative recombination decreases with the increase of temperature.

74


Eg2 Eg1

p

Eg1n

+v

hhhhhh h

hhh

hhh

hhh

h

hhh

hh

h

hh

1. Due to the potential barrier, carrier is much harder to overflow.2. Weaker carrier reabsorption. The material on both with larger bandgap will not re-absorb the emitted light.

1. The active layer is still to thick that the electron and hole are still free to move. electron might concentrate at left side while hole might accumulate at right hand side

75


Quantum Well LED A electron or a hole is a wave, and the typical wave length of electron or hole wave in the semiconductor is around 1nm~10nm

3nm~20nm

e

h

1. Electron and hole are confined in the quantum well and cannot move freely. Electron and hole are staying in the quantum well and enhance radiative recombination chance.

2. Since the quantum well is very thin, it cannot hold to many electrons and holes, not good for high current cases.

76


The confined electric wave will have corresponding eigenEnergy level, where the effective band gap will change.

Quantum Well LED

W

Effective bandgap

2*

222

2*

222

22~ effective

Wmn

WmnEE

hegg

ππ++

By changing the quantum well width, we can fine tune the effective bandgap.

77


Multiple Quantum Well LED e

h

1. Ef1. n-type

1. i-type

1. p-type

1. Blocking layer for electron

Opt

ical

pow

ercurrent

1 well

3wells2wells

+ ++ ++-- - -- - - --

78


Multiple Quantum Well LED e

h

eee eee

hhh h h h

1. Using a multiple quantum well, the total carrier staying in the active layer will enhance.

2. More quantum well, the total internal resistant from the barrier between two quantum wells will increase, so the electron may not be able to the last quantum well. So the numbers of quantum well is limited.

79


Optical emission with different number of quantum wellsOptical emission with different number of quantum wells

80


Electron blocking layerElectron blocking layer

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81

3-8 LED Characteristics

5/07/2009

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82

E

Ec

Ev

Carrier concentrationper unit energy

Electrons in CB

Holes in VBhυ

1

0

Eg

hυ1

hυ2

hυ3

CB

VBλ

Relative intensity

1

0λ

1λ

2λ

3

ΔλΔhυ

Relative intensity

(a) (b) (c) (d)

Eg + kBT

(2.5-3)kBT

1/2kBT

Eg1 2 3

2kBT

(a) Energy band diagram with possible recombination paths. (b) Energy distribution ofelectrons in the CB and holes in the VB. The highest electron concentration is (1/2)kBT aboveEc . (c) The relative light intensity as a function of photon energy based on (b). (d) Relativeintensity as a function of wavelength in the output spectrum based on (b) and (c).

Wavelengths of photon emission depends on the e- and h+ distribution.The min. photon energy = EgThe max. emission at photon energy ~ Eg+kBT= Eg+0.5kBT+0.5kBTThe linewidth ~ 2.5-3 kBT

For heavily doped semiconductor narrow impurity band overlaps CBthe minimum photon energy < Eg

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83

V

2

1

(c)

0 20 40I (mA)0

(a)

600 650 7000

0.5

1.0

λ

Relativeintensity

24 nm

Δλ

655nm

(b)

0 20 40I (mA)0

Relative light intensity

(a) A typical output spectrum (relative intensity vs wavelength) from a red GaAsP LED.(b) Typical output light power vs. forward current. (c) Typical I-V characteristics of ared LED. The turn-on voltage is around 1.5V.

Example: GaAsP LED (center at 655 nm)- Spectrum: less asymmetric than ideal spectrum- Linewidth ~24 nm (2.7kBT)- The output light power is not linear with LED current as in (b)- The turn-on or the cut-in voltage ~ 1.5V (depends on material, and increases as Eg increases e.g. blue 3.5~4.5V, yellow ~2V)

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901 37500 光電導論

84

Example 3.8.1: LED output spectrumGiven: the width of the relative light intensity vs. photon energy spectrum of an

LED is ~ 3kBTFind: the linewidth Δλ1/2 in the output spectrum in terms of wavelength?

( )

nmnmnmnm

nmnmhc

TkTkhvE

EEhc

dEd

EEhc

dEd

Ehcvc

ph

phph

phphphph

ph

149,1550105,1300

47,870

3,3

,

//

2

2

2

=Δ==Δ=

=Δ=

≈Δ≈Δ=Δ

Δ≈Δ

≈ΔΔ

−=

==

λλλλ

λλ

λλ

λ

λλλ

λ

BB

We note that the emitted wavelength λ is related to the photon energy Eph by

The linewidths are typical values and the exact values depend on the LED structure.

85

901 37500 光電導論

85

Example 3.8.2: GaAs LED output wavelength variationsGiven: GaAs Eg (300K) = 1.42 eV

dEg / dT = -4.5 x 10-4 eV K-1 (decreases with temp.)Find: the change in the emitting wavelength if the temperature change is 10°C

( )( )( ) ( )

( )( ) nmKmnKTdTd

nmKormKdTd

dTdE

Ehc

dTd

Ehctaking

TkNeglecting

g

g

g

B

87.210277.0

.277.0,1077.2

106.1105.4106.142.1

10310626.6

/

1

1110

194219

834

2

≈=Δ=Δ

×=

×××−××

××−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

=

−

−−−

−−

−

−

λλ

λ

λ

λ

The wavelength increases with temperature. This calculated change is within 10% of typical values for GaAs LEDs quoted in data books.

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901 37500 光電導論

86

Example 3.8.3: InGaAsP on InP substrateBackground: ternary alloy In1-xGaxAsyP1-y on InP substrate

for infrared wavelength LED and laser diodeRequirement: InGaAsP is lattice matched to In P

(avoid crystal defects in InGaAsP)y ≈ 2.2 x

Given: Eg of the ternary alloy (empirical relationship)Eg ≈ 1.35 – 0.72y + 0.12 y2 0 ≤ x ≤ 0.47

Find: the compositions of InGaAsP ternary alloys for peak emission at a wavelength of 1.3 μm

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• We first note that we need the required bandgap Eg at the wavelength of interest. The photon energy at peak emission is hc/λ=Eg+kBT. Then in electron volts,

( )( )( )( )

34.066.03.07.0

2

619

348

6

3.02.2/66.066.0

12.072.035.1928.0

928.00259.0103.1106.110626.6103

300,103.1

/

PAsGaInxy

yy

eVeVE

KTm

eTkehcE

g

Bg

⇒==

=+−=

=−××××

=

=×=

−=

−−

−

−λ

λ

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3-9 LED for Optical Fiber Communications

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(a) Surface emitting LED (b) Edge emitting LED

Doubleheterostructure

Light

Light


Short haul communication: use LEDsimpler to drive, more economic, longer lifetime, wider output spectrum

Long haul and wide bandwidth communication: use LDnarrow linewidth, high output power, higher signal BW capacity

Surface emitting LED (SLED)Edge emitting LED (ELED)

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ElectrodeSiO2 (insulator)

Electrode

Fiber (multimode)

Epoxy resin

Etched wellDouble heterostructure

Light is coupled from a surface emitting LEDinto a multimode fiber using an index matchingepoxy. The fiber is bonded to the LEDstructure.

(a)

Fiber

A microlens focuses diverging light from a surfaceemitting LED into a multimode optical fiber.

Microlens (Ti2O3:SiO2 glass)

(b)


Burrus type device

Coupling of SLED to a fiber

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Schematic illustration of the the structure of a double heterojunction stripecontact edge emitting LED

InsulationStripe electrode

SubstrateElectrode

Active region (emission region)

p+-InP (Eg = 1.35 eV, Cladding layer)

n+-InP (Eg = 1.35 eV, Cladding/Substrate)

n-InGaAs (Eg = 0.83 eV, Active layer)

Currentpaths

L

60-70 μm

Light beam

p+-InGaAsP (Eg = 1 eV, Confining layer)

n+-InGaAsP (Eg = 1 eV, Confining layer) 12 3200-300 μm

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

InGaAs: active regionInGaAsP: confining layerInP: cladding layer

Edge emitting LED: greater light intensity, more collimated beam

Op. @ 1.5μm

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Multimode fiberLens

(a)

ELED

Active layer

Light from an edge emitting LED is coupled into a fiber typically by using a lens or aGRIN rod lens.

GRIN-rod lens

(b)

Single mode fiberELED


Coupling of ELED to a fiber

Output spectra from SLED and ELED are not necessarily the samebecause (a) different doping levels,

(b) self-absorption guided along active layer in ELED

Typical linewidth of ELED < linewidth of SLEDe.g. InGaAsP operated at 1300 nm

linewidth (ELED) = 75 nm linewidth (SLED) = 125 nm

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Ch3 Semiconductor Science and Light Emitting...

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