Emission and Absorption Rates in Semiconductors

Emission and Absorption Rates in Semiconductors: • The band structure of semiconductors must be considered to

determine these rates. • With semiconductors transitions occur between states within the

conduction and valence bands. • For a transition to occur:

i. States must exist ii. States must be occupied/unoccupied

• Spontaneous emission in a semiconductor can occur only if the energy

state E2 is occupied by an electron and E1 is unoccupied or occupied by a hole.

• Occupation probabilities for electrons within the valence and

conduction bands are described by Fermi-Dirac distribution functions

1

2 2( ) 1 exp ( ) /c ff E E E k Tc B

− = + −

1

1 1( ) 1 exp ( ) /v ff E E E k Tv B

− = + −

• Efc and Efv are the Fermi energy levels. Expressions give the

probabilities that the states in the conduction bands above or below the Fermi levels will be occupied with electrons.

• At 0o K the probability that an electron occupies an energy level

below the Fermi level is 1 and above it 0. i.e ( ) 1 0of E at T= = K • At E = Ef the probability is taken as 0.5.

• As the temperature increases above absolute zero the probability of occupying energy levels above and below the Fermi energy becomes more gradual as shown in the figure.

Conduction Band

ValenceBand

E FC

E FV

Eg

E

khν

Filled States

VacantStates

holes

(electrons)

F(E)

1.0

0.5

0.0EEF

Fermi-Dirac Probability Distribution

T = 0

T > 0

Density of States Z(E- E )

E

ph

ph- Eg

E )ph- Eg gZ(

• The energy of a photon emitted for a particular transition is

νhEEE ph =−= 12 • There are many transitions involved in the spontaneous emission from

semiconductors. • The density of states Z(E) represents the available unoccupied energy

levels that are jointly vacant in the valence and conduction bands (per unit volume).

( ) 2/13

2/3)(28)( gphr

ph EEhmEZ −=

where mr is the reduced mass

vc

vcr mm

mmm

+=

mc and mv are the effective masses of the electrons and holes in their respective bands. • Notice that the photon energy (Eph) must be greater than Eg • At higher energies there are more states available, however the

probability of being occupied is lower. Total Spontaneous Emission Rate – determined by summing over all allowed transitions between the conduction and valence bands.

[ ] 2211212 ),()(1)(),()( dEEEZEfEfEEAR vE cspc

−= ∫∞

ν

Total Absorption Rate –

[ ] 2212112 )(),()(1)(),()( dEEEZEfEfEEBR phcE vabsc

νρν −= ∫∞

Total Stimulated Emission Rate -

[ ] 2211212 )(),()(1)(),()( dEEEZEfEfEEBR phvE cstimc

νρν −= ∫∞

• Note that the stimulate emission and absorption rates also depend on

the photon density at the transition frequency ν. Population Inversion Condition - Rstim > Rabs

This result is obtained when

( ) ( )12 EfEf vc >

gfvfc EEEEE >−>− 12 • This implies that the separation of Fermi levels must exceed the

bandgap for a population inversion to occur. • At thermal equilibrium in a p-n junction (normal non-excited state)

Efc = Efv (i.e. the Fermi levels coincide)

∴ pumping energy from an external supply is necessary.

Semiconductor Material Properties: • Electron-hole pairs can recombine either radiatively to form photons or

nonradiatively. • Nonradiative recombination mechanisms include recombination at traps,

surface effects, and Auger recombination (kinetic energy imparted to an electron).

• Internal Quantum Efficiency is a measure of the radiative to nonradiative

recombination rates.

nrrr

rr

tot

rr

RRR

RR

+==intη

Rrr is the radiative recombination rate and Rnr is the nonradiative recombination rate. • These rates can be expressed in terms of recombination times τnr and τrr with

Rrr = N/τrr and Rnr = N/τnr

where N is the carrier density.

• τrr is usually quite fast however τnr can be 10-5 times as fast. This results in a

very large Rnr and low ηint. • Materials of this type are poor candidates optical sources. Si and Ge have ηint ~

10-5. Note these are indirect bandgap materials. • The radiative recombination rate consists of:

Rrr = Rspon + Rstim

• In light emitting diodes LEDs spontaneous emission dominates. • A measure of carrier lifetime in the absence of stimulated emission is often a

useful quantity for evaluating materials and is defined as

τc =N/(Rspon + Rnr) .

Semiconductor Compatibility Issues

• Semiconductor sources typically consist of compounds of different

semiconductors.

• In order to have a semiconductor sources that can operate at room temperature

for long periods of time it is necessary to have materials with lattice constants

(atomic spacings) that are matched to less than 0.1%.

• Junctions of this type can be formed using artificially made materials.

Compounds of Al1-xGaxAs and In1-xGaxAsyP1-y are frequently used. The x- and

y- represent fractional concentrations of materials.

LatticeConstant(Angstroms)

Bandgap wavelength (um)

Bandgap energy (eV)1.0 2.0

6.0

5.4

6.4 1.02.0 0.6InSb

GaSb AlSb

AlAs

GaP

GaAs

InP

InAs

AlP

2.50.5

5.6

In the above figure the line connecting GaAs and AlAs show the allowed range

for mixtures of the ternary (3) compound Al1-xGaxAs and the hatched area the

regions of direct bandgap semiconductor mixtures of the quarternary (4)

compound In1-xGaxAsyP1-y.

• There is a limited range of wavelengths that can be synthesized from these

compounds.

• For Al1-xGaxAs the bandgap depends on the fraction x- . In the range (0 < x <

0.45) the change in BG is nearly linear and can be expressed as:

xxEg 247.1424.1)( +=

Eg is in electron volts (eV).

• Similarly for In1-xGaxAsyP1-y quaternary compounds the ratio of fractions must

be chosen so that

x/y = 0.45

• In this case the BG can be expressed in terms of y only such that

212.072.035.1)( yyyEg +−=

with 0 ≤ y ≤ 1.

• The smallest BG for the corresponding ternary compound In0.55Ga0.45As emits

light near 1.65 µm.

• In1-xGaxAsyP1-y sources can be selected so that the emission wavelengths range

from 1.0 – 1.65 µm.

• An additional benefit of these compositions is that the refractive index can also

be modified.

• For Al1-xGaxAs the refractive index changes as

xnn AsAlGaGaAs xx62.0

1=−

−

• This allows the refractive index to change from ~3.1 to 3.6 and provides a mechanism for forming a waveguide and confining the optical field.

pn junctions : • Intrinsic semiconductors have a certain level of free electrons. • These levels can be changed to a certain degree by heating and by

illumination with an optical source with a frequency corresponding to the bandgap of the semiconductor.

• The semiconductors can also be doped with impurities to change the

concentration of electrons and holes (the absence of electrons at an atom site).

• Dopants contributing to an increase in electrons are donors Nd and to an

increase in holes Na. • Doping with Nd → n-type and Na → p-type semiconductors. • Combining n- and p-type semiconductors can be used to form a

x

x

CarrierDensity(logM)

ImpurityConcentration(logM)

p-type n-type

junction

pp

n p

nn

pn

Energy Levels Across a p-n Junction: • The Fermi level of a p-doped semiconductor is close to the valence band

edge and Ef for the n-type semiconductor is near the conduction band edge.

• When the diode is not biased the Fermi levels on both sides are

continuous. • As a result the conduction and valence bands bend forming an energy

barrier of magnitude qVo.

p-type

E

DepletionRegion, W

Vo

Potential

Bulkp-type

Bulkn-type

+ + ++ + ++ + +

- - -- - -- - -

qVoelectronenergy

EF

E FnEFp

E c

Ev

Ecn

Evn

Ecp

Evp

• Barrier results from carrier diffusion across the junction that exposes

fixed charge. Fixed charge forms a Coulomb potential Vo . • Vo prevents further carrier diffusion • A space charge region results that is relatively free of mobile charge

carriers. • The width of the space charge region and the height of the potential

barrier can be changed by applying a bias potential to the junction. • This forms a condition for a population inversion.

p-type n-type

VA+ -

Forward Bias Condition

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

Ec

Ev

Efv Efc

Ec

Ev+ + + + + + + + + +

- - - - - - - - - - - - - - - - -

Unbiased Junction

Forward Biased Junction

I

• An injection current is produced in the forward biased condition is given

by

( )[ ]1/exp −= TkqVII Bs . • A junction with the same semiconductor material used on both sides

represents a homojunction type diode. • A problem with this type of junction is that e-h recombination occurs

over a wide region determined by the diffusion length of the carriers. • In addition the optical field that is generated is not confined. • The result of these factors is that high injection currents are required to

pump the diode to threshold and lasing. • Usually this type of laser cannot be operated in continuous mode at room

temperature.

p n

FieldMode

GainRegion

Loss Region

• This situation can be improved by going to heterojunction and double heterojunction diode configurations.

• The figure below shows a double heterojunction diode p-doped region of

narrow bandgap material surrounded by larger bandgap materials.

p+ GaAs

p - Ga Al As

p -GaAs

n - Ga Al As

n+ - GaAs

GaAs - substrate

10^19

10^18

10^18

10^18

2.5-4x10^18

(0.1 um)

(0.7-1.2 um)

(0.1-0.2 um)

(0.7-1.2 um)

(0.2 um)

1-x x

1-x x

• An added benefit of going to heterojunction diode structures is that the refractive index of the active region is higher than the surrounding region.

• This forms an effective waveguide that guides the optical field and

improves the mode confinement factor. • As a result very low threshold currents are possible with this type of

diode. •

n-type Active p-type

Bandgap

n1 - n2Index

Energy

OpticalIntensity

Conduction Band

Valence Band+ + + + +

- - - - - - -

Mode Profile

OPTICAL GAIN:

• The addition of feedback through an optical resonator provides a mechanism

for reaching the threshold conditions for the semiconductor medium.

• Consider a simple experiment with an optical gain material with two levels N2

and N1

Light at frequency ν is incident on the gain medium of length ∆z.

The gain medium can be considered essentially a two-level system with a band

• g through the gain medium the irradiance is changed by an amount

The light then passes through an optical filter of spectral bandwidth ∆ν

I

FilterGain Medium

Eg

Polarizer Detector

z

N

N1

(ν) I (ν) I (ν)+ ∆

∆

E2

•

•

gap of Eg.

After passin

∆I.

•

followed by a polarizer and a detector.

The optical irradiance is related to the photon density (ρν) by •

( )I chνν ρ ν= .

The change in optical irradiance is given by •

)()( νν hdtdNI =∆ .

• In terms of the two-level atomic system the change in irradiance can be written

as

21 2 21 1 21 21( ) ( ) ( ) ( )2 4

I II B g N zh B g N zh A g N zc c

δν ν ν ν ν νπΩ

= ∆ − ∆ + ∆∆ .

The first term represents stimulated emission, the second absorption, and the

In the last term the factor ½ is due to the polarizer and the fact that

he fraction

•

third spontaneous emission.

•

spontaneous emitted light is randomly polarized. δΩ/4π represents t

of the total emitted light that is collected by the system. Spontaneously emitted

light goes into 4π steradians.

• Reducing ∆ν and ∆Ω makes the contribution from spontaneous emission

relatively small compared to the stimulated emission optical power.

• Therefore the change in irradiance can be written as

zIgNNBc

hI ∆−=∆ )(( )1212 νν

( ) IIgNNBc

hzI

o )()(1212 νγνν=−=

∆∆

,

with g(ν) the lineshape function of the gain medium.

• Integrating this expression results in

[ ]zIzI o )(exp)0()( νγ=

γo(ν) is referred to as the gain coefficient for the optical medium and can be

expressed as

))(()( 12 NNo −= νσνγ .

• When N2 > N1 the material acts as a gain medium and when N2 < N1 it

absorbs light.

For semiconductors the gain is in the form of bands rather than discrete energy

levels.

Ec

Ev

∆

∆

Eb

Ea

h ν

Z(E)

Eb

Ea

Stimulated Emission

( ) ( ) ( )( )21 1rb a c b v a

nR B I Z h f E f Ecν ν→

= −

Stimulated Absorption

( ) ( ) ( )( )12 1ra b v a c b

nR B I Z h f E f Ecν ν→

= −

( ) ( ) ( )ra b c b v a

nR BI Z E f E f Ecν↔ = −

The rate of change of I with z

( ) ( ) ( )rc b v a

ratedI h dzvol

nh BI Z E f E f Ec

Idz

ν

ν

ν

γ

=

= −

=

since ( ) ( )1/ 2

gZ E h Eν∝ −

[ ])()()( 2/1avbcg EfEfEhC −−= νγ .

• The coefficient C must be determined from external absorption measurements.

• fc(Eb) and fv(Ea) represent the probabilities of finding the states near energy

bands Eb and Ea occupied by an electron or hole as shown in the figure.

• Associated with these probability functions are the quasi-Fermi levels.

• They are used to represent the state of occupancy of the different bands.

• They will be different for electrons and holes due to their different masses.

• The occupancy will also depend on the temperature of the semiconductor.

• The density of electrons (n) and holes (p) are determined from

( )( )[ ] dE

kTFEEEm

ncE

n

ce ∫∞

+−−

=

1/exp2

21 2/12/3

2

*

2π

( )( )[ ] dE

kTEFEEmp

cEp

vh ∫∞

+−−

=

1/exp2

21 2/12/3

2

*

2π

• The transitions in semiconductors recombine at a rate that depends on their

concentrations

Rpndtdn

+⋅⋅−= β ,

where β is the recombination rate and R is the pumping rate.

For GaAs β ~ 2 X 10-10 cm3/sec.

• By setting dn/dt = 0 can find the pump power necessary to maintain steady

state.

Example to maintain a carrier density of 2 X 1018 for holes and electrons requires

( ) 2621810 108102102 ×=×⋅×= −R e-h pairs/(cm3-sec)

∴ Large Injection currents are required!

However the volume of the active medium is small:

Volume: 1 µm × 10 µm × 100 µm = 10-9 cm3

Total e-h pairs/sec = 8 × 1017 ~ 125 mA

Current Density to establish specific gain with carrier density N/cm3

qdRJ =

d is the depth of the active medium.

Homojunction laser with d = 1 µm; required current density 1018 carriers/cm3

Requires a generation rate ~ 4 × 1026 carriers/cm3/sec and

J = 6.4 kA/cm2