CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS 2015-6-9Fundamentals of Photonics 1 Chapter 5 Photons in...

CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS

23/4/18Fundamentals of Photonics 1

Chapter 5Photons in Semiconductors



Semiconductors

• A semiconductor is a solid material that has electrical conductivity in between a conductor and an insulator.

• Semiconductors can be used as optical detectors, sources (light-emitting diodes and lasers), amplifiers, waveguides, modulators, sensors, and nonlinear optical elements.



A. Energy bands and A. Energy bands and charge carrierscharge carriers



Energy bands in semiconductors

• The solution of the Schrödinger equation for the electron energy in the periodic potential created by the atoms in a crystal lattice, results in a splitting of the atomic energy levels and the formation of energy bands.

ATOMSOLID

E2’

E1’

Conduction band

Valence band

E21




• Each band contains a large number of finely separated discrete energy levels that can be approximated as a continuum.

• The valence and conduction bands are separated by a “forbidden” energy gap of width Eg bandgap energy

Conduction band

Valence band

BandgapE

Unfilled bands

Filled bands




Materials with a large energy gap (>3eV) are insulators, those for which the gap is small or nonexistent are conductors, semiconductors have gaps roughly in the range 0.1 to 3 eV



ev

1.1ev

Si

Conduction band

Valence band

En

ergy

5

-10

0

-15

-5

ev

1.42ev

GaAs

Conduction band

Valence band

En

ergy

5

-10

0

-15

-5

EgEg

(a) (b)

Figure 15.1-1 Energy bands: (a) in Si, and (b) in GaAs



Electrons and holes

• In the absence of thermal excitations, the valence band is completely filled and the conduction band is completely empty. Thus, the material cannot conduct electricity.

• As the temperature increases, some electrons will be thermally exited into the empty conduction band, result in the creation of a free electron in the conduction band and a free hole in the valence band.



Bandgap energy Eg

Conduction band

Valence band

Electron Hole

Ele

ctro

n e

ner

gy

E

Figure 15.1-2 Electrons in the conduction band and holes in the valence band at T>0.

Electrons and holes



Energy-momentum relations

2 2 2

0 02 2

p kE

m m

pk

p is the magnitude of the momentum

k is the magnitude of the wavevector associated with the electron’s wavefunction

m0 is the electron mass



K

K

[100][111]

[100][111]

E

E

Si

GaAs

Figure 15.1-3 Cross section of the E-K function for Si and GaAs along the crystal directions [111] and [100].

Eg=1.1ev

Eg=1.42ev

The energy of an electron in the conduction band depends not only on the magnitude of its momentum, but also on the direction in which it is traveling in the crystal



Effective mass

Near the bottom of the conduction band, the E-k relation may be approximated by the parabola

2 2

2cc

kE E

m

2 2

2vv

kE E

m

Ec,Ev: the energy at the bottom of the conduction band and at the top of the valence band

mc,mv: effective mass of the electron in the conduction band and the hole in the valence band

The effective mass depends on the crystal orientation and the particular band under consideration



Typical ratios of the average effective masses to the mass of the free electron mass

mc/m0 mv/m0

Si 0.33 0.5

GaAs 0.07 0.5

Effective mass



Direct- and indirect-gap semiconductors

K

K

[100][111]

[100][111]

E

E

Si

GaAs

Figure 15.1-4 Approximating the E-K diagram at the bottom of the conduction band and at the top of the valence band of Si and GaAs by parabols.

Eg=1.1ev

Eg=1.42ev

The direct-gap semiconductors such as GaAs are efficient photon emitters, while the indirect-gap counterparts are not.



B. Semiconducting B. Semiconducting materialsmaterials



Semiconducting materials

• Si: widely used for making photon detectors but not useful for fabricating photon emitters due to its indirect bandgap.

• GaAs, InP GaN etc.: used for making photon detectors and sources.

• Ternary and quaternary semiconductors: AlxGa1-xAs, InxGa1-xAsyP1-y etc.

tunable bandgap energy with variation of x and y.



Semiconducting materials

AlxGa1-xAs is lattice matched to GaAs, means it can be grown on the GaAs without introducing strains.

Solid and dashed curves represent direct-gap and indirect-gap compositions respectively.

We can see that a material may have direct bandgap for one mixing ratio x and an indirect bandgap for a different x.



Doped semiconductors

• Dopants: alter the concentration of mobile charge carriers by many orders of magnitude.

• n-type: predominance of mobile electrons

• p-type: predominance of holes

n p

p n



C. Electron and hole C. Electron and hole concentrationsconcentrations



Density of states

The density of states describes the number of states at each energy level that are available to be occupied.

3/ 21/ 2

2 3

(2 )( ) ( ) ,

2c

c c c

mE E E E E

3/ 21/ 2

2 3

(2 )( ) ( ) ,

2v

v v c

mE E E E E

Density of states near band edges



K

Ec

Ef

Eg

E

Ec

Ef Ef

Ec

E

Density of states

c(E)

v(E)

(a) (b) (c)

Figure 15.1-7 (a) Cross section of the E-K diagram (e.g., in the direction of the K1 component with K2 and K3 fixed). (b) Allowed energy levels (at all K). (c) Density of states near the edges of the conduction and valence bands. Pc(E)dE is the number of quantum states of energy between E and E+dE, per unit volume, in the conduction band. P(E) has an analogous interpretation for the valence band.

E



Probability of occupancy

Under the condition of thermal equilibrium, the probability that a given state of energy E is occupied by an electron is determined by the Fermi function.

1( )

exp[( ) / ] 1f B

f EE E k T

Ef: Fermi level, the energy level for which the probability of occupancy is 1/2.

f(E) is not itself a probability distribution, and it does not integrate to unity; rather it is a sequence of occupation probabilities of successive energy levels.



Figure 15.1-8 The Fermi function f(E) is the probability that an energy level E is filled with an electron; 1-f(E) is the probability that it is empty. In the valence band, 1-f(E) is the probability that energy level E is occupied by a hole. At T=0K, f(E)=1 for E<Ef, and f(E)=0 for E>Ef, i.e., there are no elctrons in the conduction band and no holes in the valence band.

Ec

Ef

Ev

Eg

T>0K T=0K

Ec

Ef

Ev

Ec

Ef

Ev

f(E)

f(E)

0 1 10 0.50.5

f(E)

1-f(E)

EE



Figure 15.1-10 Energy-band diagram, Fermi function f(E), and concentrations of mobile electrons and holes n(E) and p(E) in an n-type semiconductor.

E ED

Ec

Ev

f(E)

Carrier concentration

0 1

EE

Donor level Ef

n(E)

p(E)

The Fermi level is above the middle of the bandgap



Figure 15.1-11 Energy-band diagram, Fermi function f(E), and concentrations of mobile electrons and holes n(E) and p(E) in an p-type semiconductor

E

EA

Ec

Ev

f(E)

Carrier concentration

0 1

EE

Acceptor levelEf

n(E)

p(E)

The Fermi level is below the middle of the bandgap



Exponential approximation of the Fermi function

( ) ( ) ( )cn E E f E ( ) ( )[1 ( )]vp E E f E

( )cE

n n E dE

When the Fermi level lies within the bandgap, but away from its edges by an energy of at least several times , the equations above gives:

exp( )c fc

B

E En N

k T

exp( )f v

vB

E Ep N

k T

exp( )g

c vB

Enp N N

k T

2 3/ 22(2 / )c c BN m k T h2 3/ 22(2 / )v v BN m k T h

As Nc and Nv are

( )vE

p p E dE



Law of mass action

3 3/ 22

2exp( ) 4( ) ( ) exp( )g gB

c v c vB B

E Ek Tnp N N m m

k T h k T

is independent of the location of the Fermi level

Thus in p n for intrinsic semiconductor

We have 1/ 2( ) exp( )gi c v

B

En N N

k T

The law of mass action is useful for determining the concentrations of electrons and holes in doped semiconductors.

Therefore the law of mass action can be written as:

2inp n



D. Generation, D. Generation, recombination, and injectionrecombination, and injection



Generation and recombination in thermal equilibrium

Thermal equilibrium requires that the process of generation of electron-hole pairs must be accompanied by a simultaneous reverse process of deexcitation,.

The electron-hole recombination, ooocurs when an electron decays from the conduction band to fill a hole in the valence band.

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Radiative recombination: the energy released take the form of an emitted photon.Nonradiative recombination: transfer of the energy to lattice vibrations or to another free electron.

Recombination may also occur indirectly via traps or defect centers, they can act as a recombination center if it is capable of trapping both the electron and the hole, thereby increasing their probability of recombining.

rate of combination= npε(cm3/s) a parameter that depends on the characteristics of the material

Generation and recombination in thermal equilibrium




Electron-hole injection

Define G0 as the rate of recombination at a given temperature:

0 0 0=G n p

Now let additional electron-hole pairs be generated at a steady rate R

A new steady state: 0 0,n n n p p p

We get: R=n

With 0 0

1

[( ) ]n p n

τ: the electron-hole recombination lifetime

0 +R=G np As



Electron-hole injection

For an injection rate such that 0 0n n p

0 0

1

( )n p

In a n-type material, where n0>>p0,the recombination lifetime τ≈1/εno is inversely proportional to the electron concentration.

Similarly, for p-type material, τ≈1/εp0.

This simple formulation is not applicable when traps play an important role in the process.



Internal quantum efficiency

The internal quantum efficiency ηi is defined as the ratio of the radiative electron-hole recombination rate to the total recombination.

It determines the efficiency of light generation in a semiconductor material.

nrr ri

r nr r r nr

For low to moderate injection rates, 0 0

1

( )rr n p



Internal quantum efficiency

ε r

(cm3/s)τr τnr τ ηi

Si 10-15 10ms 100ns 100ns 10-5

GaAs 10-30 100ns 100ns 50ns 0.5

The radiative lifetime for Si is orders of magnitude larger than its overall lifetime, mainly due to its indirect bandgap, result in a small internal quantum efficiency

On the other hand, direct bandgap material as GaAs, the decay is largely via radiative transitions, consequently larger internal quantum efficiency, useful for fabricating light-emitting devices.



E. JunctionsE. Junctions



• Homojunctions: junctions between differently doped reguions of a semiconductor material

• Heterojunctions: junctions between different semiconductor materials

The p-n junctionThe p-n junction



Figure 15.1-15 Energy levels and carrier concentrations of a p-type and n-type semiconductor before contact.

Ef

Ef

Ele

ctro

n

ener

gy

Car

rier

co

nce

ntr

atio

n

P

P

n

n

P-type n-type

position



Figure 15.1-16 A p-n junction in thermal equilibrium at T>0K. The depletion-layer, energy-band diagram, and concentrations (on a logarithmic scale) of mobile electrons n(x) and holes p(x) are shown as functions of position x. The built-in potrntial difference V0 corresponds to an energy eV0, where e is the magnitude of the electron charge.




1. The depletion layer contains only the fixed charges, the thickness of the depletion layer in each region is inversely proportional to the concentration of dopants in the region.

2. The fixed charges created a built-in field obstructs the diffusion of further mobile carriers.

3. A net built-in potential difference V0 is established.

4. In thermal equilibrium there is only a single Fermi function for the entire structure so that the Fermi levels in the p- and n- regions must align.

5. No net current flows across the junction.



Energy-band and carrier concentrations in a forward-biased p-n junction.




The biased junction

Forward biased

A misalignment of the Fermi levels in the p- and n-regions

Net current i=isexp(eV/kBT)-is

Reverse biased

Net current ≈-is as V is negative in exp(eV/kBT) and |V|>>kBT/e

[exp( ) 1]sB

eVi i

k T

Acts as a diode with a current-voltage characteristic



The p-i-n junction diode

• Made by inserting a layer of intrinsic (or lightly doped) semiconductor material between the p- and n-type region.

• The depletion layer penetrates deeply into the i-region.

• Large depletion layer: small junction capacitance thus fast response

• Favored over p-n diodes as photodiodes with better photodetection efficiency



The p-i-n junction diode

A p-i-n diode has the depletion layer penetrates deeply into the i-region



Heterojunctions

Junctions between different semiconductor materials are called heterojunctions.

P nP

Eg1 Eg2 Eg3

Electro

n en

ergy

They can provide substantial improvement in the performance of electronic and optoelectronic devices


23/4/18Fundamentals of Photonics 45E

lectron

en

ergy

Ef

Eg1 Eg2 Eg3

Electro

n

energ

y

A potential energy discontinuity provides a barrier that can be useful in preventing selected charge carriers from entering regions where they are undesired.

This property used in a p-n junction can reduce the proportion of current carried by minority carriers, and thus to increase injection efficiency

Junctions between materials of different bandgap create localized jumps in the energy-band diagram



• Discontinuities in the energy-band diagram created by two heterojunctions can be useful for confining charge carriers to a desired region of space

• Heterojunctions are useful for creating energy-band discontinuities that accelerate carriers at specific locations

• Semiconductors of different bandgap type can be used in the same device to select regions of the structure where light is emitted and where light is absorbed

• Heterojunctions of materials with different refractive indices create optical waveguides that confine and direct photons.



Quantum Wells and Super-lattices

• When the Heterostructures layer thickness is comparable to, or smaller than, the de Broglie wavelength of thermalized electrons (≈50 nm in GaAs), the energy-momentum relation for a bulk semiconductor material no longer applies.

• A quantum well is a double heterojunction structure consisting of an ultrathin (≤50 nm) layer of semiconductor material whose bandgap is smaller than that of the surrounding material.

• The sandwich forms conduction- and valence band rectangular potential wells within which electrons in the conduction-band well and holes in the valence-band well are confined.



(a) Geometry of the quantum-well structure. (b) Energy-level diagram for electrons and holes in a quantum well. (c) Cross section of the E-k relation in the direction of k2 or k3. The energy subbands are labeled by their quantum number q1 = 1,2,... The E-k relation for bulk semiconductor is indicated by the dashed curves.



The energy levels Eq in one-dimensional infinite rectangular well are determined by Schrodinger equation:

2 2( / ), 1, 2,....

2q

q dE q

m

The energy-momentum relation:

2 2

1 1, 1, 2,3,...,2c q

c

kE E E q

m

where k is the magnitude of a two-dimensional k = (k2, k3) vector in the y-z plane

For the quantum well, k1 takes on well-separated discrete values. As a result, the density of states associated with a quantum-well structure differs from that associated with bulk material.



In a quantum-well structure the density of states is obtained from the magnitude of the two-dimensional wavevector (k2, k3)

121

1

,( )

0

............

................., ..

cc q

c

c q

mE E E

dE

E E E

1 1,2,...,q



quantum-well structure exhibits a substantial density of quantum-well structure exhibits a substantial density of states at its lowest allowed conduction-band energy level states at its lowest allowed conduction-band energy level and at its highest allowed valence-band energy leveland at its highest allowed valence-band energy level



Multiquantum Wells and Superlattices

Multiple-layered structures of different semiconductor materials that alternate with each other are called multiquantum-well (MQW) structures

A multiquantum-well structure fabricated from alternating layers of AlGaAs and GaAs



Quantum Wires and Quantum Dots

The density of states in different confinement configurations: (a) bulk; (b) quantum well; (c) quantum wire; (d) quantum dot. The conduction and valence bands split into overlapping subbands that become successively narrower as the electron motion is restricted in more dimensions.



A semiconductor material that takes the form of a thin wire of rectangular cross section, surrounded by a material of wider

bandgap, is called a quantum-wire structure2 2

1 2 2c q qc

kE E E E

m

2 2 2 21 1 2 2

1 2 1 2

( / ) ( / ), , , 1, 2,...,

2 2q qc c

q d q dE E q q

m m

1/ 21 2

1 21/ 21 2

1 2

.....

..........................

(1/ )

........

( / 2 ),

( ) ( )

0,

, 1,2,...,

.....

.

cc q q

c c q q

d d mE E E E

E E E E E

otherwise

q q

Density of states



In a quantum-dot structure, the electrons are narrowly confined in all three directions within a box of volume d1d2d3.

1 2 3

2 21 1

1

2 22 2

2

2 23 3

3

1 2 3

( / ),

2

( / ),

2

( / )

2

, , 1, 2,...,

c q q q

qc

qc

qc

E E E E E

q dE

m

q dE

m

q dE

m

q q q

Quantum dots are often called artificial atoms.



INTERACTIONS OF PHOTONS WITH

ELECTRONS AND HOLES

Chapter 15.2



Mechanisms leading to absorption and emission of photons in a semiconductor:

• Band-to-Band (Inter-band) Transitions.• Impurity-to-Band Transitions.• Free-Carrier (Intraband) Transitions• Phonon Transitions• Excitonic Transitions.



Examples of absorption and emission of photons in a semiconductor. (a) Band-to-band transitions in GaAs can result in the absorption or emission of photons of wavelength . (b) The absorption of a photon of wavelength results in a valence-band to acceptor-level transition in Hg-doped Ge (Ge:Hg). (c) A free-carrier transition within the conduction band.

0 / 0.87g ghc E m

0 / 14A Ahc E m



Absorption coefficient for some semiconductor materials



• For photon energies greater than the bandgap energy Eg, the absorption is dominated by band-to-band transitions

• Absorption edge:

The spectral region where the material changes from being relatively transparent to strongly absorbing

• Direct-gap semiconductors have a more abrupt absorption edge than indirect-gap materials



Band-to-Band Absorption and Emission

Bandgap Wavelength:

1.24( )

( )gg

mE eV

The quantity g is also called the cutoff wavelength.



Absorption and Emission

(a) The absorption of a photon results in the generation of an electron-hole pair. This process is used in the photodetection of light. (b) The recombination of an electron-hole pair results in the spontaneous emission of a photon. Light-emitting diodes (LEDs) operate on this basis. (c) Electron-hole recombination can be stimulated by a photon. The result is the induced emission of an identical photon. This is the underlying process responsible for the operation of semiconductor injection lasers.



Conditions for Absorption and Emission

• Conservation of Energy

• Conservation of Momentum

2 1E E h

2 1

2 1

....

....

/ /

/. 2

p p hv c h or

k k



Conditions for Absorption and Emission

• Energies and Momenta of the Electron and Hole with Which a Photon Interacts

2 ( )rc g

c

mE E h E

m

1 2( )rv g

v

mE E h E E h

m



Optical Joint Density of States

3/ 21/ 2

2

(2 )( ) ( ) ,r

g g

mh E h E

The density of states with which a photon of energy interacts under conditions of energy and momentum conservation in a direct-gap semiconductor is determined by:

( ) hv

The density of states with which a photon of energy interacts increases with in accordance with a square-root law

hvghv E

And illustrated as follow:



Photon Emission Is Unlikely in an Indirect-Gap Semiconductor ！

Photon emission in an indirect-gap semiconductor

The recombination of an electron near the bottom of the conduction band with a hole near the top of the valence band requires the exchange of energy and momentum. The energy may be carried off by a photon, but one or more phonons are required to conserve momentum. This type of multiparticle interaction is unlikely.




Photon Absorption is Not Unlikely in an Indirect-Gap Semiconductor！

Photon absorption in an indirect-gap semiconductor

The photon generates an excited electron and a hole by a vertical transition; the carriers then undergo fast transitions to the bottom of the conduction band and top of the valence band, respectively, releasing their energy in the form of phonons. Since the process is sequential it is not unlikely.




Rates of Absorption and Emission

• Occupancy probabilitiesOccupancy probabilities

• Transition probabilitiesTransition probabilities

• Density of statesDensity of states

the probability densities of a photon of energy h being emitted or absorbed by a semiconductor material in a direct band-to-band transition are mainly determined by three factors:



Occupancy Probabilities

• Emission condition: A conduction-band state of energy E2 is filled (with an electron) and a valence-band state of energy E1 is empty (i.e., filled with a hole)

• Absorption condition: A conduction-band state of energy E2 is empty and a valence-band state of energy E1 is filled.

The probabilities are determined from the appropriate Fermi functions and associated with the conduction and valence bands of a semiconductor in quasi-equilibrium

( )cf E ( )vf E



2 1( ) ( )[1 ( )]e e vf f E f E

The probability that the emission condition is satisfied for a photon of energy is:

( )ef hv

The probability that the absorption condition is satisfied is :

( )af

2 1( ) [1 ( )] ( )a c vf f E f E



Transition Probabilities

• Satisfying the emission/absorption occupancy condition does not assure that the emission/absorption actually takes place.

• These processes are governed by the probabilistic laws of interaction between photons and atomic systems

Transition Cross Section:

2

( ) ( )8 r

g



Probability density for the spontaneous emission:

1( ) ( )sp

r

P d g d

Probability density for the stimulated emission:

2

( ) ( ) ( )8i v v

r

W d d g d

If the occupancy condition for absorption is satisfied, the probability density for the absorption is also fit this formula



Overall Emission and Absorption Transition

Rates of Spontaneous Emission 、 Stimulated Emission and Absorption：

1( ) ( ) ( )sp e

r

r f

2

( ) ( ) ( )8st v e

r

r f

2

( ) ( ) ( )8ab v a

r

r f

the occupancy probabilities and is determined by the quasi-Fermi levels Efc and Efv

( )ef v ( )af v



Spontaneous Emission Spectral Density in Thermal Equilibrium

A semiconductor in thermal equilibrium has only a single Fermi function so

2 1( ) ( )[1 ( )] exp( )eB

hf f E f E

k T

then

1/ 20( ) ( ) exp( ),g

sp g gB

h Er D h E h E

k T

where3/ 2

0 2

(2 )exp( )gr

r B

EmD

k T



Spectral density of the direct band-to-band spontaneous emission rate (photons per second per hertz per cm3) from a semiconductor in thermal equilibrium as a function of hv. The spectrum has a low-frequency cutoff at

and extends over a width of approximately .

( )spr v

/gv E h 2 /Bk T h



Gain Coefficient in Quasi-Equilibrium

The net gain coefficient is2

0 ( ) ( ) ( )8 g

r

f

Where the Fermi inversion factor is given by

2 1( ) ( ) ( ) ( ) ( )g e a c vf f f f E f E

The gain coefficient may be cast in the form：1/ 2

0 1( ) ( ) ( ),g g gD h E f h E

with3/ 2 2

1 2

2 r

r

mD

h

The sign and spectral form of the Fermi inversion factor are governed by the quasi-Fermi levels Efc and Efv, which, in turn, depend on the state of excitation of the carriers in the semiconductor.

( )gf v



Absorption Coefficient in Thermal Equilibrium

A semiconductor in thermal equilibrium has only a single Fermi level, so:

1( ) ( ) ( )

exp[( ) / ] 1c vf B

f E f E f EE E k T

Therefore the absorption coefficient:

1/ 21 1 2( ) ( ) [ ( ) ( )]gD h E f E f E

If Ef lies within the band gap but away from the band edges

2 3/ 21/ 2

2

2 1( ) ( )

( )r

gr

c mh E

h



The absorption coefficient resulting from direct band-to-band transitions as a function of the photon energy (eV) and wavelength for GaAs

1( )( )v cm

hv 0 ( )m



Refractive Index

• The ability to control the refractive index of a semi-conductor is important in the design of many photonic devices

• Semiconductor materials are dispersive the refractive index is dependent on the wavelength

• The refractive Index is related to the absorption coe-fficient inasmuch as the real and imaginary parts of the susceptibility must satisfy the Kramers-Kronig relations

• The refractive index also depends on temperature and on doping level

( )v



Refractive index for high-purity, p-type, and n-type GaAs at 300 K, as a function of photon energy (wavelength).



Refractive Indices of Selected Semiconductor Materials at T = 300 K for Photon Energies Near the Bandgap Energy of the Material ghv E

CHAPTER 5-1---PHOTONS IN SEMICONDUCTORS 2015-6-9Fundamentals of Photonics 1 Chapter 5 Photons in...

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