p-n Junction and Transport in Semiconductors · Abrupt p-n Junction Space charge region (a)...

CMR

p-n Junction and Transport in Semiconductors

Scope

•Mobility•Current in Semiconductors•p-n Junction•Basic Characteristics of p-n Junction•Manufacturing

Lecture 8, OEN-630

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

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

Band Diagram in Applied Field

Drift Velocity

In electric field appears additional velocity component:

Mobility

EEmev

EEmev

eEtFvm

pp

Cp

nn

Cn

Cnn

μτ

μττ

−=−=

−=−=

−=Δ=

Definition of mobility through the life time:

Mobility and Scattering

Cp

p

Cn

n

me

me

τμ

τμ

≡

≡The mobility through the mean free time (average time between two collisions (C)):

IL

i iCimpClatticeCC

w

μμμ

ττττ

111

1111

,,,

+=

=++== ∑KThe overall probability of collision is a sum of the probabilities of every single collision mechanism:

Mobility

Mobility of electrons and holes in Silicon at room temperature:

Electric Conductivity and Resistance

The electric current which flows in an intrinsic semiconductor consists of both electron and hole currents


The conductivity of a semiconductor can be modeled in terms of the band theory of solids. The band model of a semiconductor suggests that at ordinary temperatures there is a finite possibility that electrons can reach the conduction band and contribute to electrical conduction


( )

dxdE

dxdEF

rVEeF

iC =−=

∇−=−=rrrIn external electric field every

electron experiences a force from the field:

Bottom of the conduction band corresponds to the potential energy

Electric field

Electrostatic potential

( )

( )xeVEdx

xdVE

i −=

−≡

Drift Current

( )

pn

pp

nn

nn

n

ii

nn

JJJ

EepJEenJ

EenenvevAIJ

+=

==

=−=−== ∑=

μμ

μ1

The transport of carriers under the influence of an applied electric field produces a current called the drift current:

Total current is the sum of electron and hole current components


Total current is determined through the geometry of the conductor

Conductivity and Resistivity

The total drift current is the sum of electron and hole current components:

( )( )

( )pn

pn

pnpn

pne

pne

EEpneJJJ

μμσρ

μμσ

σμμ

+=≡

+=

=+=+=

11

Temperature as an Energy

kTvmKE thn 23

21 2 ==

Average kinetic energy of the electrons:

Thermal Velocity

( )

( ) sec4010

2500104~~

30010~3~

,23

21

15

6

152

2

=×

≈

===

==

−

−

msmim

vLt

KTmsmkTvv

kTvmKE thn

Estimated cost-to-cost travel time of electrons

CMR

Direct and Indirect Semiconductors

E-k Diagrams

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).


Potential Energy in Crystals

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

Direct Bandgap Semiconductors

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

Eg

Ec

Ev

Ec

Ev

kvb 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 .


Indirect Bandgap Semiconductors

(a) A bare n-type Si wafer. An oxidized Si wafer by dry or wet oxidation. Application of resist. Resist exposure through the mask.

p-n Junction Fabrication

a. The wafer after the development. The wafer after SiO2removal.The final result after a complete lithography process. A p-n junction is formed in the diffusion or implantation process. The wafer after metalization. A p-n junction after the compete process.

p-n Junction Fabrication

CMR

Thermal Equilibrium ConditionsBand Diagram

(a) Uniformly doped p-type and n-type semiconductors before the junction is formed. (b) The electric field in the depletion region and the energy band diagram of a p-n junction in thermal equilibrium.

CMR

Equilibrium Conditionp-type Semiconductor

( ) ( )

01=−⎟

⎠⎞

⎜⎝⎛=

−=

+=

dxdpkT

dxdE

epe

dxdpeDpe

diffusionJdriftJJ

pi

p

pp

ppp

μμ

μ E

relationEinsteine

kTD

fieldelectricdxdE

e

pp

i

μ=

=1E

There is no electric current at steady-state condition

CMR

Equilibrium Fermi Level

( ) kTFEi

ienp /−= ⎟⎠⎞

⎜⎝⎛ −=

dxdF

dxdE

kTp

dxdp i

constFdxdF

dxdFpJ pp

==

==

,0

0μ

Fermi energy is constant at steady-state condition

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

–Wp

Wn(d)

0

eVo

x (g)

–eVo

Hole PE(x)

–Eo

Eo

M

ρnet

M

Wn–Wp

ni

Properties of the pn junction.


P-n Junction

Abrupt p-n Junction

(a) A p-n junction with abrupt doping changes at the metallurgical junction. Energy band diagram of an abrupt junction at thermal equilibrium.Space charge distribution.Rectangular approximation of the space charge distribution.

CMR

Abrupt p-n JunctionNeutral regions

( )npNNedxdE

dxd

ADss

−+−−=−=−≡Ψ

εερ

2

2

Poisson’s equation

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−=−−≡Ψ −≤

i

Axxip n

Ne

kTFEe p

ln|1p-type neutral region:Assume p>>n, ND=0

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−=−−≡Ψ ≥

i

Dxxin n

Ne

kTFEe n

ln|1n-type neutral regionAssume n>>p, NA=0

⎟⎟⎠

⎞⎜⎜⎝

⎛=Ψ−Ψ= 2ln

i

DApnbi n

NNe

kTV Built-in potential

CMR

Built-in Potentials Depend on Doping Concentration

Built-in potentials on the p-side and n-side of abrupt junctions in Si and GaAs as a function of impurity concentration.

CMR

Abrupt p-n JunctionSpace charge region

Electric field from Poisson’s equation

0,2

2

<≤−=Ψ xxforeN

dxd

ps

A

ε

( ) ( )0, <≤−

+−=

Ψ−= xxfor

xxeNdxdxE p

s

pA

ε

ns

D xxforeNdx

d≤<−=

Ψ 0,2

2

ε

( ) ( )n

s

nAm xxforxxeN

dxdExE ≤<

−=

Ψ+−= 0,

ε

s

pA

s

nDxm

xeNxeNEεε

===0|

Neutrality conditionnDpA xNxN =

WExeNxeNV m

s

nD

s

pAbi 2

122

22

=+=εε

( ) ( ) ⎟⎠⎞

⎜⎝⎛ −−=+−=

WxExWeNxE m

s

B 1ε

DB NN = for p+-n junction

( )( )

⎟⎠⎞

⎜⎝⎛ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=−=Ψ ∫

=ΨWx

WxV

WxxEEdxx bix

m 220

00

2

Abrupt p-n JunctionSpace charge region

(a) One-sided abrupt junction (with NA >> ND) in thermal equilibrium. (b) Space charge distribution. (c) Electric-field distribution. (d) Potential distribution with distance, where Vbi is the built-in potential.

Neutral n-regionNeutral p-regionEo – ELog (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© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Biased p-n Junction

CMR

I-V CharacteristicsBasic assumptions

•Space charge region has abrupt boundaries .•Outside the boundaries the material is neutral.•Carrier densities at the boundaries are related to the potential difference across the junction.•No generation and/or recombination in depletion regions.

Low-injection condition:

00

00

pp

nn

pnnp

<<<<

Ec

Ev

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.

SCL


Total depletion layer thickness

biDA

DAs VNNNN

eW ⎟⎟

⎠

⎞⎜⎜⎝

⎛ +=

ε2

pn xxW +=

( )B

bis

NVV

eW −

=ε2

Biased p-n Junction

Schematic representation of depletion layer width and energy band diagrams of a p-n junction under various biasing conditions. a) Thermal-equilbrium condition. (b) Forward-bias condition. (c) Reverse-bias condition

CMR

Biased p-n JunctionBuilt-in potential

0

02

00

2

lnln

ln

p

n

i

np

i

DAbi

nn

ekT

nnp

ekT

nNN

ekTV

==

=

kTeV

np

kTeV

pn

bi

bi

epp

enn

00

00

=

=

CMR

Biased p-n Junction

At low-injection condition:

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛−=−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=−=

==≈−

1,

1,

000

000

00

kTeV

nnnkTeV

nn

kTeV

pppkTeV

pp

kTeV

pkT

VVe

pn

eppporepp

ennnorenn

enennnbibi

For injected electronsat the boundary x=-xp

For injected holesat the boundary x=xn

CMR

Biased p-n Junction

At the condition of no generation and no electric fields in depletion region:

( )

( )p

p

p

n

Lxx

kTeV

ppp

Lxx

kTeV

nnn

pp

nn

eennn

eeppp

Dpp

dxpd

+

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=−

=−

−

1

1

0

00

00

02

2

τ

Concentration of injected holes

Concentration of injected electrons

CMR

Ideal Diode Equation

Currents at the boundaries:

( )

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−==−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=−=

− 1|

,1|

0

0

kTeV

n

pnx

pnpn

kTeV

p

npx

npnp

eL

neDdx

dneDxJ

eL

peDdxdpeDxJ

p

n

Total current:

( ) ( )

n

pn

p

nps

kTeV

spnnp

LneD

LpeD

J

eJxJxJJ

00

,1

+≡

⎟⎟⎠

⎞⎜⎜⎝

⎛−=−+=

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.


Total Current of p-n Junction

C

WnWp

Log (carrier concentration)

np(0)pnonpo

ppo nno

V

x

p-side

SCL

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

CMR

I-V Characteristics of p-n Junction

Cartesian plot

Semilog plot

CMR

I-V Characteristic of p-n Junction

Current-voltage characteristic of p-n junction

CMR

Current-Voltage Characteristics

Current-voltage characteristics of a typical silicon p-n junction.

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]


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


0.002 0.004 0.006 0.0081/Temperature (1/K)

10 -1 6

10 -1 4

10 -1 2

10 -1 0

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 n i2 and below 238 K itis controlled by n i. The vertical axis isa logarithmic scale with actual currentvalues. (From D. Scansen and S.O.Kasap, Cnd. J. Physics. 70 , 1070-1075,1992.)


b – equal incremental charges dQon both n- and p-sides

c – increase in the electric field dEcaused by the incremental charge dQ

a – increase of p-n junction width due to the incremental bias voltage dV

(a) p-n junction with an arbitrary impurity profile under reverse bias. (b) Change in space charge distribution due to change in applied bias. (c) Corresponding change in electric-field distribution.

Depletion Capacitance

CMR

p-n Junction Capacitance

dVdQC = General definition of capacitance.

WdEdV ⋅= Change in the applied voltage.

ss

dQdxdEεε

ρ==

Electric field in p-n junction is defined by Poisson’s equation.Here ρ is charge density in the depletion layer

( )VVNe

WdQW

dQdVdQC

bi

Bss

s

−====

2εε

ε

Capacitance of abruptp-n junction is dependent on reverse-bias voltage

CMR

Varactor

Doping profile in n-region

m

D xxBN ⎟⎟⎠

⎞⎜⎜⎝

⎛∝

0

Hyper abrupt p-n junction

2−∝VCFrequency of varactor-L circuit

VLC

∝=1ω

Recombination

Recombination

ionrecombinatdirectpnR 00β=

ballanceionrecombinatgenerationpnRG thth −== 00β

( )( )ppnnpnR nn Δ+Δ+== 00ββ

lightwithpppnnn nn ,00 −=Δ=−=Δ

statesteadyRGGRGdt

dpththL

n ,0=−+=−=

Recombination

( ) pnpppnGGRU nnLth Δ≈ΔΔ++==−= ββ 0

p

nnL

pppnGUτ

β 0−=Δ≈= timelife

np ,1

0βτ =

p

nth

n ppURGdt

dpτ

0−−=−=−=

( ) ,exp0 ⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

pLpn

tGptpτ

τ

( ) ,exp0 ⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

pLpn

tGptpτ

τ

Boundary condition:

0,0 =−

= tppGp

nL τ

Lpn Gpp τ+= 0

Generation by Light

Surface Recombination

Real atomic configurations on solid surfaces


Real atomic configurations on solid surfaces with ad-atoms


Surface structure, defects, and oxidation-reduction at the atomic scale can be studied using SPM techniques. (a) An ultrahigh vacuum scanning tunneling microscope (STM) image shows iron atoms at an iron oxide (hematite) surface (1). (b) An STM image of the same type of hematite surface as in part (a) taken in air. Note that the unit cell in parts (a) and (b) are virtually identical. The cells are marked, and the edge length in both views is 0.5 nm. (c) An STM image of a TiO2 surface (2) shows areas of ordered surface structure (box A), steps (box B), and both point (box C), see also inset) and line (box D) defects (the scale bar represents 10 nm). (d) An atomic force microscopy (AFM) image taken in aqueous solution shows a step with kink sites on a calcite (CaCO3) surface (3). (e) An AFM image of gypsum (4) shows one kink site along the step (arrow). (f) In the STM image of protoporphyrin molecules on a graphite surface, the bright spots are Fe atoms complexed within these molecules in a 1:4 mixture of Fe-containing to Fe-free molecules (The scale bar represents 5 nm.)

Continuity Equation

( ) ( ) ( )AdxRGe

AdxxJe

AxJAdxtn

nnnn −+⎥⎦

⎤⎢⎣⎡

−+

−−

=∂∂

( ) ( ) ...+∂∂

+=+ dxxJxJdxxJ n

nn

( ) CEelectronRGxJ

etn

nnn ,1

−+∂∂

=∂∂ ( ) CEholeRG

xJ

etp

ppp ,1

−+∂∂

−=∂∂

Continuity Equation (2)

.dxdneDnEeJ nnn += μ .

dxdpeDpEeJ ppp −= μ

rateionrecombinatnetppRnn

Rp

nnp

n

ppn ,, 00

ττ−

=−

=

,02

2

n

ppn

pn

pnnp

p nnG

xn

Dx

nE

xEn

tn

τμμ

−−+

∂

∂+

∂

∂+

∂∂

=∂

∂

,02

2

p

nnp

np

nppn

n ppGxpD

xpE

xEp

tp

τμμ −

−+∂∂

+∂∂

+∂∂

=∂∂

CMR

1. Problems 3.32. Problem 3.4 a, b

Ref. [1], Chapter 3

Deadline: September 26

Home workOEN-630 L8:

CMR

References

The following references include recommended advanced reading for the subject:

1. S. O. Kasap, Optoelectronics and Photonics, Prentice Hall, 2001, Ch. 3

2. S. O. Kasap, Principles of Electronic Materials and Devices, McGraw Hill, 2006, Chapters 1, 2

3. P. Yu, M. Cardona. Fundamentals of Semiconductors. Springer, 2001

4. C. Kittel, Introduction to Solid State Physics, Benjamin Inc., NY 1986

Website: http://vigyan.nsu.edu/~cmr/vgavrilenko.htm

6. Light emitting devices

6.1 The light emitting diode

6.1.1 Introduction

A light emitting diode consist of a p-n diode which is designed so that radiative

recombination dominates. Homojunction p-n diodes, heterojunction p-i-n diodes where the

intrinsic layer has a smaller bandgap (this structure is also referred to as a double-hetero-

structure) and p-n diodes with a quantum well in the middle are all used for LEDs. We will

only consider the p-n diode with a quantum well because the analytical analysis is more

straight forward and also since this structure is used often in LEDs and even more frequently

in laser diodes.

6.1.2 Rate equations

The LED rate equations are derived from the continuity equations as applied to the p-n

diode:

∂n∂t

= 1q

∂Jn ∂x

- R + G [6.1.1]

where G is the generation rate per unit volume and R is the recombination rate per unit

volume. This equation is now simplified by integrating in the direction perpendicular to the

plane of the junction. We separate the integral in two parts: one for the quantum well, one

for the rest of the structure.

⌡⌠

qw

∂n∂t

+ ⌡⌠

p-n

∂n∂t

= Jq -

JSHR q -

Jbb q -

Jideal q -

Σk

(NkPk - Nik2)Bk - (

NP - Ni12

N + P + 2Ni1)

1τnr

[6.1.2]

where k refers to the quantum number in the well. If we ignore the carriers everywhere

except in the quantum well and assume that only the first quantum level is populated with

electrons/holes and that the density of electrons equals the density of holes, we obtain:

∂N∂t

= Jq - B1N2 -

N2τnr

+ S

τab[6.1.3]

where the last term is added to include reabsorption of photons. The rate equation for the

photon density including loss of photons due to emission (as described with the photon

lifetime τph) and absorption (as described with the photon absorption time τab) equals:

∂S∂t

= B1N2 - S

τph -

Sτab

[6.1.4]

The corresponding voltage across the diode equals:

Principles of Electronic Devices 6.1 © Bart J. Van Zeghbroeck 1996

Va = Egqw

q + Vt ln [(eN/Nc - 1) (eN/Nv* - 1)] [6.1.5]

Where the modified effective hole density of states in the quantum well, Nv*, accounts for

the occupation of multiple hole levels as described in section 4.4.3.d. The optical output

power is given by the number of photons which leave the semiconductor per unit time,

multiplied with the photon energy:

Pout = hν S

τph A (1-R)

Θc2

4 [6.1.6]

where A is the active area of the device, R is the reflectivity at the surface and Θc is the

critical angle for total internal reflection1

R = ( n1 - n2n1 + n2

)2 and Θc = sin-1( n1 n2

) [6.1.7]

The reflectivity and critical angle for a GaAs Air interface are 30 % and 16º respectively.

6.1.3. DC solution to the rate equations

The time independent solution in the absence of reabsorption, as indicated with the subscript

0, is obtained from:

0 = J0 q - B N0

2 - N0 2τnr

[6.1.8]

0 = B N02 -

S0 τph

[6.1.9]

where B is the bimolecular recombination constant. Solving these equations yields:

N0 = 1

4Bτnr [ 1 +

16 τnr2 B J0q -1] [6.1.10]

for small currents this reduces to: (J << q/16τnr2B)

N0 = 2τnr J0

q [6.1.11]

which indicates that SHR recombination dominates, whereas for large currents one finds: (J

>> q/16τnr2B)

1See Appendix A.7 for the derivation of the reflectivity at dielectric interfaces.


N0 = J0

q B[6.1.12]

The DC optical output power is:

P0 = hν B N02 A(1-R)

Θc2

4 [6.1.13]

This expression explains the poor efficiency of an LED. Even if no non-radiative

recombination occurs in the active region of the LED, most photons are confined to the

semiconductor because of the small critical angle. Typically only a few percent of the

photons generated escape the semiconductor. This problem is most severe for planar surface

emitting LEDs. Better efficiencies have been obtained for edge emitting, "superluminescent"

LEDs (where stimulated emission provides a larger fraction of photons which can escape the

semiconductor) and LEDs with curved surfaces.

6.1.4 AC solution to the rate equations

Assume that all variables can be written as a sum of a time independent term and a time

dependent term (note that n(t) is still a density per unit area):

N = N0 + n1(t) J = J0 + j1(t)

S = S0 + s1(t) P = P0 + p1(t)

Va = Va + va(t) [6.1.14]

The rate equations for the time dependent terms the given by:

∂n1∂t

= -B 2N0 n1(t) - B n12(t) -

n1(t)2τ0

+ j1(t)

q [6.1.15]

∂s1∂t

= B 2N0 n1(t) + B n12(t) -

s1τph

[6.1.16]

Assuming the AC current of the form j1 = j10ejωt and ignoring the higher order terms we can

obtain a harmonic solution of the form:

n1 = n10 ejωt s1 = s10 ejωt p1 = p10 ejωt [6.1.17]

yielding:

s10 = 1q

B 2N0 j10 τph τeff (1 + jω τph) (1 + jω τeff)

[6.1.18]

where τeff depends on N0 as:


τeff = 1

2B N0 + 1

2τ0

[6.1.19]

and the AC responsivity is:

p10j10

= hνq (1-R)

πΘc2

4π B 2N0 τeff

(1 + jω τph) (1 + jω τeff)[6.1.20]

at ω = 0 this also yields the differential quantum efficiency (D.Q.E)

D.Q.E. = p10j10

qhν =

(1-R)Θc2B N0 τeff2 =

(1-R)Θc2 B N0 τ0(4 B N0 τ0 + 1)

[6.1.21]

6.1.5 Equivalent circuit of an LED

The equivalent circuit of an LED consist of the p-n diode current source parallel to the diode

capacitance and in series with a linear series resistance R. The capacitance, C, is obtained

from

1C =

dVa dQ =

dVa q dN =

1q Vt

[1

Nc eN/Nc (eN/Nv - 1) +

1Nv

eN/Nv (eN/Nc - 1)]

(eN/Nc - 1) (eN/Nv - 1)

[6.1.22]

or

C = qN0mVt

[6.1.23]

with

m = N0 e

N/Nc

Nc (eN/Nc - 1)

+ N0 e

N/Nv

Nv (eN/Nv - 1)

[6.1.24]

for N0 << Nc and/or Nv m = 2 while for N0 >> Nc and/or Nv m = N0 (Nc+Nv)

NcNv


6.2 The laser diode

6.2.1 Emission, Absorption and modal gain

In order to find the modal gain one first has to calculate the photon absorption spectrum as

well as the spontaneous photon emission spectrum from the quasi-Fermi levels. Energy and

momentum conservation requires that

Eph = En - Ep [6.2.1]

with

En = Ec + E1n + / h2 kn

2

2 mn* [6.2.2]

Ep = Ev - E1p - / h2 kp

2

2 mp* [6.2.3]

and

kn = kp = k [6.2.4]

if we assume that the photon momentum is negligible2. These equations can be reduced to

Eph = Egqw1 + / h2 k2

2 mr* , with

1mr

* = 1

mn* +

1mp

* [6.2.5]

where Egqw1 is the energy between the lowest electron energy in the conduction band and

the lowest hole energy in the valence band. En and Ep then become:

En = Ec + E1n + (Eph - Egqw1) mr

* mn

* [6.2.6]

Ep = Ev - E1p - (Eph - Egqw1) mr

* mp

* [6.2.7]

The emission and absorption spectra (β(Eph) and α(Eph)) are obtained from:

β(Eph) = βmax Fn (En) [1-Fp(Ep)] [6.2.8]

α(Eph) = αmax [1 - Fn(En)] Fp(Ep) [6.2.9]

Stimulated emissions occurs if an incoming photon triggers the emission of another photon.

The spectrum equals that for spontaneous emission, minus that for absorption since both are

competing processes:

g(Eph) = β(Eph) - α(Eph) = gmax [Fn(En) - Fp(Ep)] [6.2.10]

2This assumption causes an error of a few percent.


The normalized gain spectrum is shown in figure 6.1 for different values of the carrier

density. The two staircase curves indicate the maximum possible gain and the maximum

possible absorption in the quantum well.

Photon Energy [eV]

Ise/

Ise,

max

1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9

Fig.6.1. Normalized gain versus photon energy of a 10nm GaAs quantum well for a carrier

density of 1012 (lower curve), 3 x 1012 ,5 x 1012 ,7 x 1012 and 9 x 1012 (upper

curve) cm-2.

The theoretical gain curve of figure 6.1 exhibits a sharp discontinuity at Eph = Egqw1. The

gain can also be expressed as a function of the carrier density when assuming that only one

electron and one hole level are occupied:

g(Eph) = gmax [1 - e-N/Nc

1 + e-N/Nv(exp{Eph - Eqw1

kT mr

* mn

* } - 1)

- e-P/Nv

exp{Eqw1 - Eph

kT mr

* mp

* }(1 - e-P/Nv) + e-P/Nv

] [6.2.12]

The peak value at Eph = Eqw1, assuming quasi-neutrality (N = P) is then:

gpeak = g (Eqw1) = gmax (1 - e-N/Nc - e-N/Nv ) [6.2.13]

The maximum gain can be obtained from the absorption of light in bulk material since the

wavefunction of a free electron in bulk material is the same as the wavefunction in an infinite

stack of infinitely deep quantum wells, provided the barriers are infinitely thin and placed at

the nodes of the bulk wavefunction. This means that for such a set of quantum wells the

absorption would be the same as in bulk provided that the density of states is also the same.


This is the case for Eph = Eqw1 so that the maximum gain per unit length is given by:

gmax = K Eqw1-E1 = K2

h2

2 mr*

1Lx

[6.2.14]

where Lx is the width of the quantum well. This expression shows that the total gain of a

single quantum well due to a single quantized level is independent of the width3. The

corresponding value for GaAs quantum wells is 0.006 or 0.6%.

Experimental gain curves do not show the discontinuity at Eph = Eqw1 due to inter-carrier

scattering which limits the lifetime of carriers in a specific state. The line width of a single set

of electron and hole level widens as a function of the scattering time which disturbs the

phase of the atomic oscillator. Therefore, an approximation to the actual gain curve can be

obtained by convoluting [6.2.10] with a Lorenzian line shape function:

g(Eph) = ⌡⌠ gmax [Fn(En) - Fp(Ep)] ∆ν

2π[(ν - Eph/h)2 + ( ∆ν

2 )2] dν [6.2.15]

with ∆ν = 1π

1τ , where τ is the carrier collision time in the quantum well. The original and

convoluted gain curve are shown in Fig. 6.2.

Photon Energy [eV]

Ise/

Ise,

max

1

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

1.4 1.6 1.8 2 2.2 2.4

Fig.6.2. Original and convoluted gain spectrum of a 10 nm GaAs quantum well with a

3There is a weak dependence of m* on the width of the well.


carrier density of 3 x 1012 cm-2 and a collision time of 0.09 ps.

For lasers with long cavities such as edge emitter lasers, one finds that the longitudinal

modes are closely spaced so that lasing will occur at or close to the peak of the gain

spectrum. It is therefore of interest to find an expression for the peak gain as a function of

the carrier density4. A numeric solution is shown in Fig.6.3 where the peak gain is

normalized to the maximum value of the first quantized energy level. Initially, the gain peak

is linear with carrier concentration but saturates because of the constant density of states,

until the gain peak associated with the second quantized level takes over. Since the peak

gain will be relevant for lasing we will consider it more closely. As a first order

approximation we will set the peak gain g(N) equal to:

g(N) = (N - Ntr) [6.2.16]

where is the differential gain coefficient. This approximation is only valid close to N = Ntr,

and even more so for quantum well lasers as opposed to double-hetero-structure lasers. An

approximate value for the differential gain coefficient of a quantum well can be calculated

from [6.2.13] yielding:

= gmax [ e-Ntr/Nc

Nc +

e-Ntr/Nv Nv

] [6.2.17]

4Experimental values for the gain versus current density can be found in: G. Hunziker, W. Knop and C.

Harder, "Gain Measurements on One, Two and Three Strained GaInP Quantum Well Laser Diodes", IEEE

Trans. Quantum Electr., Vol. 30, p 2235-2238, 1994.


Carrier density [1E12 cm-2]

Pea

k ga

in [c

m-1

]

0

1000

2000

3000

4000

5000

6000

0 2 4 6 8 10 12 14 16 18 20

Fig.6.3. Calculated gain versus carrier density for a 10 nm GaAs quantum well (solid line)

compared to equation [6.2.13]

From Fig. 6.3 one finds that the material becomes "transparent" when the gain equals zero

or:

g(Ntr) = 0 = gmax [Fn(En)(1 - Fp(Ep))-(1 - Fn(En))Fp(Ep)] [6.2.18]

which can be solved yielding:

Eph = En - Ep = Efn - Efp = qVa [6.2.19]

The transparency current density is defined as the minimal current density for which the

material becomes transparent for any photon energy larger than or equal to Egqw1. This

means that the transparency condition is fulfilled for Va = Egqw1

q . The corresponding carrier

density is referred to as Ntr, the transparency carrier density. The transparency carrier

density can be obtained from by setting gmax = 0, yielding

Ntr = - Nc ln(1 - eNtr/Nv) [6.2.20]

This expression can be solved by iteration for Nv > Nc. The solution is shown in Figure 6.4.


Nv/Nc

Ntr

/Nc

0

0.5

1

1.5

2

2.5

0 5 10 15 20

Fig.6.4. Normalized transparency carrier density versus the ratio of the effective density of

states in the valence and conduction band.

To include multiple hole levels one simply replaces Nv by Nv* as described in section

4.4.3.d.

6.2.2 Principle of operation of a laser diode

A laser diode consists of a cavity, defined as the region between two mirrors with reflectivity

R1 and R2, and a gain medium, in our case a quantum well. The optical mode originates in

spontaneous emission which is confined to the cavity by the waveguide. This optical mode is

amplified by the gain medium and partially reflected by the mirrors. The modal gain depends

on the gain of the medium, multiplied with the overlap between the gain medium and the

optical mode which we call the confinement factor, Γ, or:

modal gain = g(N)Γ [6.2.21]This confinement factor will be calculated in section 6.2.5. Lasing occurs when for light

traveling round trip through the cavity the optical gain equals the losses. For a laser with

modal gain g(N)Γ and wave guide loss α this condition implies:

R1R2 e2(g(N)Γ-α)L = 1 [6.2.22]

where L is the length of the cavity. The distributed loss of the mirrors is therefore:

mirror loss = 1L ln

1

R1R2[6.2.23]

6.2.3 Longitudinal modes in the laser cavity.


Longitudinal modes in the laser cavity correspond to standing waves between the mirrors. If

we assume total reflection at the mirrors this wave contains N/2 periods where N is an

integer. For a given wave length λ and a corresponding effective index, neff, this yields:

N = 2 neff L

λ [6.2.24]

Because of dispersion in the waveguide, a second order model should also include the

wavelength dependence of the effective index. Ignoring dispersion we find the difference in

wavelength between two adjacent longitudinal modes from:

N = 2 L neff

λ1[6.2.25]

N + 1 = 2 L neff

λ2[6.2.26]

∆λ = 2 L neff (1N -

1N + 1) ≅

λ12

2 L neff [6.2.27]

Longer cavities therefore have closer spaced longitudinal modes. An edge emitting (long)

cavity with length of 300 µm, neff = 3.3, and λ = 0.8 µm has a wavelength spacing ∆λ of

0.32 nm while a surface emitting (short) cavity of 3 µm has a wavelength spacing of only 32

nm. These wavelength differences can be converted to energy differences using:

∆E = - Eph ∆λλ [6.2.28]

so that 0.32 nm corresponds to -6.2 meV and 32 nm to 620 meV. A typical width of the

optical gain spectrum is 60 meV, so that an edge emitter biased below threshold can easily

contain 10 longitudinal modes, while for a surface emitter the cavity must be carefully

designed so that the longitudinal mode overlaps with the gain spectrum.

A more detailed analysis of a Fabry-Perot etalon is described in A.7.3, providing the

reflectivity, absorption and transmission as a function of photon energy

6.2.4 Waveguide modes5

The optical modes in the waveguide determine the effective index used to calculate the

longitudinal modes as well as the confinement factor which affects the modal gain. Starting

from Maxwell's equations in the absence of sources:

5A detailed description of modes in dielectric waveguides can be found in Marcuse, "dielectric waveguides",

2nd ed.


∇∇×H = ε0 n2(x,y,z) ∂∂t

[6.2.29]

∇∇× = - µ0 ∂H∂t

[6.2.30]

and assuming a propagating wave in the z-direction and no variation in the y-direction weobtain the following one-dimensional reduced wave equation for a time harmonic field, =

x ejωt, of a TM mode:

∂2x

∂x2 + (n2(x) k2 - β2) x = 0 [6.2.31]

with the propagation constant given by β = ωc neff, and k =

ωc, this equation becomes:

d2x

dx2 + (n2(x) - neff2)

ω2

c2 x = 0 [6.2.32]

this equation is very similar to the Schrödinger equation. In fact previous solutions for

quantum wells can be used to solve Maxwell's equation by setting the potential V(x) equal to

-n2(x) and replacing ω2

c2 by 2m*/ h2 . The energy eigenvalues, E, can then be interpreted as

minus the effective indices of the modes: -n2eff, . One particular waveguide of interest is a

slab waveguide consisting of a piece of high refractive index material, n1, with thickness d,

between two infinitely wide cladding layers consisting of lower refractive index material, n2.

From Appendix A.1.3. one finds that only one mode exists for:

V0 = -n22 + n1

2 E10 = c2

ω2 (πd)2 [6.2.33]

or d ≤ cω π

1

n12-n2

2 =

λ2 n1

2 - n22

[6.2.34]

For λ = 0.8 µm, n1 = 3.5 and n2 = 3.3 one finds d ≤ 0.34 µm.

6.2.5 The confinement factor

The confinement factor is defined as the ratio of the modal gain to the gain in the active

medium at the wavelength of interest:

Γ = modal gain

g =

⌡⌠

-∞

∞ g(x) | x|2 dx

⌡⌠

-∞

∞ | x|2 dx

[6.2.35]


for a quantum well with width Lx, the confinement factor reduces to

Γ =

⌡⌠

-Lx/2

Lx/2 | x|2 dx

⌡⌠

-∞

∞ | x|2 dx

≅ 0.02...0.04 for a typical GaAs single quantum well laser [6.2.36]

6.2.6 The rate equations for a laser diode.Rate equations for each longitudinal mode, λ, with photon density Sλ and carrier density Nλwhich couple into this mode are:

∂Nλ∂t

= Jλq - Bλ Nλ2 -

Nλ2τ0

+

Σ k

Nkτkλ

-

Σ k

Nkτλk

- ∂Sλ∂x

∂x∂t

, λ = 1, 2, ..., λmax [6.2.37]

∂Sλ∂t

= βλ BλNλ2 - Sλ

τphλ +

∂Sλ∂x

∂x∂t

, λ = 1, 2, ..., λmax [6.2.38]

Rather than using this set of differential equations for all waveguide modes, we will only

consider one mode with photon density S, whose photon energy is closest to the gain peak.

The intensity of this mode will grow faster than all others and eventually dominate. This

simplification avoids the problem of finding the parameters and coefficients for every single

mode. On the other hand it does not enable to calculate the emission spectrum of the laser

diode. For a single longitudinal mode the rate equations reduce to:

dNdt =

Jq - BN2 -

N2τ0

- vgr Γ (N - Ntr) S [6.2.39]

dSdt = β BN2 -

Sτph

+ vgr Γ (N - Ntr) S [6.2.40]

P1 = vgr S W ln 1

R1[6.2.41]

a) DC solution to the rate equations

The time independent rate equations, ignoring spontaneous emission are:

0 = J0q - B N0

2 - N02τ0

= vgr Γ (N0 - Ntr)S0 [6.2.42]


0 = - S0τph

+ vgr Γ (N0 - Ntr)S0 [6.2.43]

where the photon life time is given by:

1τph

= 1S

∂x∂t

∂S∂x

= vgr (α + 1L ln

1

R1R2) [6.2.44]

from which we can solve the carrier concentration while lasing:

N0 = Ntr + 1

τph vgr Γ [6.2.45]

which is independent of the photon density6. The threshold current density is obtained when

S0 = 0

J0

|| (S0 = 0)

= Jth = q (B N02 +

N02 τ0

) [6.2.46]

The photon density above lasing threshold, and power emitted through mirror R1, are given

by:

S0 = J0 - Jth

q 1

vgr Γ (N0-Ntr)[6.2.47]

and the power emitted through mirror 1 is:

P10 = hν S0 W vgr ln 1

R1[6.2.48]

The differential efficiency of the laser diode is:

D.E. = dP0dI0

= hνq

ln 1

R1

ln1

R1R2 + αL

[6.2.49]

and the quantum efficiency is:

η = qhν

dP0dI0

=

ln 1

R1

ln1

R1R2 + αL

[6.2.50]

Efficient lasers are therefore obtained by reducing the waveguide losses, increasing the

reflectivity of the back mirror, decreasing the reflectivity of the front mirror and decreasing

6a more rigorous analysis including gain saturation reveals that the carrier concentration does increase with

increasing current, even above lasing. However this effect tends to be small in most laser diodes.


the length of the cavity. Decreasing the reflectivity of the mirror also increases the threshold

current and is therefore less desirable. Decreasing the cavity length at first decreases the

threshold current but then rapidly increases the threshold current.

b) AC solution to the rate equations

Assuming a time-harmonic solution and ignoring higher order terms (as we did for the LED)

the rate equations become:

jω n1 = j1q -

n1τeff

- Γ (N0-Ntr) s1 vgr - Γ n1 S0 vgr [6.2.51]

jω s1 = Γ (N0-Ntr) s1 vgr - s1

τph + Γ n1 S0 vgr [6.2.52]

where τeff is the same as for an LED and given by equation [6.1.19]. Using Γ (N0-Ntr) vgr

= 1

τph these equations can be solved yielding:

j1 = jω q n1 + q n1 (1

τeff + Γ S0 vgr) + q n1

Γ S0 vgrjω τph

[6.2.53]

replacing n1 by relating it to the small signal voltage v1

v1 =mVtn1

N0[6.2.54]

The equation for the small signal current i1 can be written as

i1 = (jω C + 1R +

1jω L) v1 [6.2.55]

with C = q N0 AmVt

, and m = N0 e

N/Nc

Nc (eN/Nc - 1) +

N0 eN/Nv

Nv (eN/Nv - 1), where A is the area of the

laser diode.

1R = C (

1τeff

+ Γ S0 vgr) [6.2.56]

and

L = 1C

τphΓ S0 vgr

[6.2.57]

c) Small signal equivalent circuit

Adding parasitic elements and the circuit described by the equation [6.2.48] we obtain the

following equivalent circuit, where LB is a series inductance, primarily due to the bond wire,

Rs is the series resistance in the device and Cp is the parallel capacitance due to the laser

contact and bonding pad.


R

R

C

s

d

LRC

L B

p

Fig.6.5 Small signal equivalent circuit of a laser diode

The resistor, Rd, in series with the inductor, L, is due to gain saturation7 and can be obtained

by adding a gain saturation term to equation [6.2.16]. The optical output power is

proportional to the current through inductor L, i1L, which is given by:

i1L = q A s1

τph = q A s1 vgr (α +

1L ln

1

R1R2) [6.2.58]

and the corresponding power emitted from mirror R1

p1 = s1 hν vgr W ln 1

R1 [6.2.59]

Ignoring the parasitic elements and the gain saturation resistance, Rd, one finds the ac

responsivity p1/i1 as:

p1i1

= hνq

ln 1

R1

(αL + ln 1

R1R2)

1

1 + jω LR + (jω)2LC

[6.2.60]

from which we find the relaxation frequency of the laser:

ω0 = 1

LC =

Γ S0 vgrτph

= Γ P0

τph hν W ln 1

R1

[6.2.61]

or the relaxation frequency is proportional to the square root of the DC output power. The

amplitude at the relaxation frequency relative to that at zero frequency equals:

7for a more detailed equivalent circuit including gain saturation see: Ch. S. Harder et al. High-speed

GaAs/AlGaAs optoelectronic devices for computer applications, IBM J. Res. Develop., Vol 34, No. 4, July

1990, p. 568-584.


p1j1

|| ω = ω0

p1j1

|| ω = 0

= R

L ω0 =

11

ω0 τeff + τph ω0

[6.2.62]

6.2.7 Threshold current of multi-quantum well laser

Comparing threshold currents of laser diodes with identical dimensions and material

parameters but with a different number of quantum wells, m, one finds that the threshold

currents are not simple multiples of that of a single quantum well laser.

Let us assume that the modal gain, g, is linearly proportional to the carrier concentration in

the wells and that the carriers are equally distributed between the m wells. For m quantum

wells the modal gain can be expressed as:

g = m (N - Ntr) = ∆N m [6.2.63]

where is the differential gain coefficient and Ntr is the transparency carrier density. Since

the total modal gain is independent of the number of quantum wells we can express the

carrier density as a function modal gain at lasing8.

N = g m + N0 =

∆Nm [6.2.64]

The radiative recombination current at threshold is then

Jtr = q B1 m (Ntr +∆Nm )2 = q B1 (Ntr2 m + 2Ntr ∆N +

∆N2

m ) [6.2.65]

This means that the threshold current density is a constant plus a component which is

proportional to the number of quantum wells. The last term can be ignored for m>>1 and

∆N<<Ntr.

6.2.8 Large signal switching of a laser diode

Because of the non-linear terms in the rate equations the large signal switching of a laser

diode exhibits some peculiar characteristics. The response to a current step is shown in the

figure below. The carrier density initially increases linearly with time while the photon

density remains very small since stimulated emission only kicks in for N > N0.

8We assume here that we are comparing identical lasers which only differ by the number of quantum wells.


t [ns]

P [m

W] ,

N/N

tr

0

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3

Fig.6.6 Optical power and normalized carrier concentration versus time when applying a

step current at t = 0 from I = 0.95 Ith to I = 1.3 Ith.

Both the carrier density and the photon density oscillate around their final value. The

oscillation peaks are spaced by roughly 2π/ω0, where ω0 is the small signal relaxation

frequency at the final current. The photon and carrier densities are out of phase as carriers

are converted into photons due to stimulated emission, while photons are converted back

into electron-hole pairs due to absorption. High speed operation is obtained by biasing close

to the threshold current and driving the laser well above the threshold. In addition one can

use the non-linear behavior to generate short optical pulses. By applying a current pulse

which is long enough to initiate the first peak in the oscillation, but short enough to avoid

the second peak, one obtains an optical pulse which is significantly shorter that the applied

current pulse. This method is referred to as gain switching or current spiking.


Intro. to Fibre Optic Communications Systems OE & Recvrs - 51

Introduction to photodetectors

• Photodetector: A device which converts opticalpower into an electrical signal

• Desirable properties include:– high responsivity/sensitivity to light– fast response time, i.e. large bandwidth– low noise, i.e. minimally degrades the SNR– insensitive to temperature

• Numerous types of detectors, some of which are:– photomultiplier tubes– pyroelectric detectors

reverse biased diode photodetectors(PIN, Avalanche (APD), MSM)

– phototransistors• The semiconductor photodetectors are commonly

used because of their low cost, high reliability andhigh performance (Note: Fibre optic communicationsystems at 1.3 and 1.55 µm use InGaAs pindevices almost exclusively. For wavelengths < 1µm, Si photodetectors are generally used.)


Photodetectors cont’d:

Basic Concepts: Absorption and photocurrent

Distance into photodetector [m]

Iinc

Inte

nsity

[W/m

2 ]Absorption: Photons with an energy greater than the bandgapenergy of the semiconductor can be absorbed to create an ehp.

( ) exp[ ( ) ]inc sz zα λ= −I I

0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8

Wavelength (µm)

In0.53Ga0.47As

Ge

Si

In0.7Ga0.3As0.64P0.36

InPGaAs

a-Si:H

12345 0.9 0.8 0.7

1×103

1×104

1×105

1×106

1×107

1×108

Photon energy (eV)

Absorption coefficient (α) vs. wavelength (λ) for various semiconductors(Data selectively collected and combined from various sources.)

α (m-1)

1.0


Optical power absorbed by semiconductor of width, w:

0( ) (1 exp[ ( ) ])sP w P wα λ= − −

P0=Iinc*Area


The primary photocurrent, IP, that flows from the photodetector isthe result the absorbed optical power.

0 (1 exp[ ( ) ])(1 )P s feI P w Rh

α λν

= − − −

Takes reflections at surfaceinto account … this term isfrequently ignored.

It is also possible to define the primary photocurrent, IP, as beingdirectly proportional to the incident optical power P0.

0PI P=R

where R is the responsivity in [A/W].

The quantum efficiency, η, is defined as

0 0

/electron generation rate 1 exp[ ( ) ](1 )photon incidence rate /

p abss f

I e P w RP hv P

η α λ= = = = − − −

1.24ehη ηλν

= ≈R

Basic Concepts: Absorption and photocurrent



Basic Concepts: Time response

Fig. 6-10 Schematic representation of a reverse biased pindiode.

Time response of photodetector (photodiode) and its associatedcircuitry depend mainly on 3 things:

1) carrier transit time through depletion region

2) diffusion time of carriers outside generated outside of depletionregion

3) RC time constant of diode and circuitry.

Transit time: fundamental limit to response time of photodiode.

Transit time is the time required for the generated carriers totravel across the depletion/active region (drift).

dd

wtv

=



Basic Concepts: Time responseThe bias field in the depletion region is usually large enough forcarrier to reach their saturated velocities…electron and holesvelocities are typically different. (Example Si: electrons - 8.4 × 106

cm/s, holes 4.4 × 106 cm/s.)

Diffusion: these processes are very slow compared to the drift ofcarriers in the depletion region (~ 1ns to drift 1 µm).

The electric field outside of the depletion region is quite small sothat the generated carriers do not drift. The contribution ofdiffusion to the time response can be minimized by increasing thedepletion region width. Can generally be ignored if the diffusiontime is much larger than the width of the optical pulse.

Capacitance effects: due to parasitic capacitance of the diodepackaging and the capacitance of the diode junction.

Junction capacitance:

Generally, Rload >> Rd, such that the RC time constant associatedwith the circuit is: trc = CjRload.

sj

ACwε

=

IphCj

Rd

Rload



Basic Concepts: Time response

Fig. 6-11 Photodiode response time to an optical input pulseshowing the 10%-90% rise time and fall time. Note that the riseand fall times are not necessarily equal. Usually just specify PDby a risetime, tr.

InputPulse

w >> 1/αSmall Cj

w >> 1/αLarge Cj

w < 1/αSmall Cj

w↑ ⇒ td↑

w↑ ⇒ η↑Tradeoff between speed andefficiency.

w↓ ⇒ td ↓

w↓ ⇒ trc↑

Reduce area of device so that capacitance doesn’tbecome excessive.


Fig. 6-1 Schematic representation of a pin photodiode circuit withan applied reverse bias, and photocurrent Ip. Width of intrinsicregion on order of 3 -5 µm.

PIN photodetector

Fig. 6-2 Simplified energy-banddiagram for a pin photodiode. Ifa photon has an energy > Egthen an electron-hole pair will begenerated.

Vbias

Iph

No Popt

Popt > 0

I-V characteristic for a diode.Dark current exists whenthere is no light. Silicon haslow dark current, Ge high. III-Vs between.



PIN photodetector

Fig. 6-4 Comparison ofthe responsivity andquantum efficiencyas a function ofwavelength for PINdiodes fabricatedfrom differentmaterials.


0PI P=R

0 0

/electron generation rate 1 exp[ ( ) ](1 )photon incidence rate /

p abss f

I e P w RP hv P

η α λ= = = = − − −

1.24ehη ηλν

= ≈R

pin photodiode


Avalanche photodiodes

Fig. 6-5 Reach-throughavalanche photodiodestructure and the electricfields in the depletion andmultiplication regions.Electrons and holes havedifferent ionization rates.Photocurrent has a gainor multiplication factor, M.

Incident photons

Substrate

Fig. 6-7 Typical room temperaturecurrent gains of a Si reach-throughavalanche photodiode.


Carrier multiplication:

M

p

IMI

=

0PI P=R

APDe M Mhην

= =R R

APD

0M APDI P=R



Avalanche photodiodes

Figure 6-6 Carrierionization rates forvarious semiconductormaterials.


Metal-semiconductor-metal (MSM) photodetectors (not inbook)

Typical structure: Side, top and symbol views for an MSMphotodetector. The device is bipolar and is identified by thecharacteristic interdigitated fingers. Simplest MSM requiresfingers, absorption layer and a substrate. Speeds up tp 300 GHzhave been recorded!

MSM Arrays. 25x25 µm MSMs attached to a common electricalbus. The common bus can be used to accept signals fromseveral different optical sources.




Photodetector Noise and Receivers

Notes: Rs << RL → Ignore Rs

Ra >> RL → Ignore Ra

Photodetector Avalanche Gain Load/Bias Resistor Preamplifier

APD Only

Quantum NoiseDark Current Noise

Excess noise due torandom nature ofavalanche process

Thermal Noise Thermal NoiseNoise in Transistors

Noise Mechanisms

Shot Noise - due to the quantum nature of light (photons) andcurrent (electrons/holes) → granular quality to what appears to bea continuous signal.

quantum noise(light)

dark current noise(thermal generation of electrons)

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