Optoelectronics Devices & Circuits discusses key optical processes in semiconductors including electron-hole pair formation and recombination, absorption, and conductance processes. It describes radiative and non-radiative recombination mechanisms and how the electron-hole pair recombination rate is calculated for low-level and high-level injection cases. It also discusses absorption in semiconductors including indirect intrinsic transitions, exciton absorption, and donor-acceptor and impurity-band absorption.

1. Optoelectronics Devices & Circuits
(MEC-166)
UNIT-II
By
Dr. POOJA LOHIA
Department of Electronics & Communication
Madan Mohan Malaviya University of Technology, Gorakhpur

2. SYLLABUS

3. Key Points
Optical Processes In Semiconductors
Electron-Hole Pair formation and recombination
Radiative and Nonradiative Recombination
Band to band Recombination
Absorption in semiconductors
Effect of electric field on absorption: Franz-Keldysh and stark Effects
Deep level transition
Auger Recombination
Conductance process in semiconductor
Bulk and surface recombination phenomena

4. Electron-Hole Pair Recombination Rate
 All the processes occur at the same time in a material but at different rates.
 If we have device with low field and non-degenerate semiconductor then the
recombination-generation mechanism at room temperature that dominates is-
1. Band-to- Band Transition
2. R-G Center Transition
 For example,
• In a direct band gap semiconductor, band-to-band transitions dominate.
• In indirect band gap semiconductors, R-G center transitions dominate.

5. Electron-Hole Pair Recombination Rate
 In general, the total recombination rate is given by the sum of recombination
rate due to all the processes-
𝑹 = 𝑩𝟏𝒏𝒑 + 𝑩𝟐𝒏𝒑 + 𝑩𝟑𝒏𝒑+. . … … . .
 Not all of them will dominate so one can simplify the expression by looking
at only the dominating recombination mechanism.
 There are two special cases in device physics-
1. Low Level Injection
2. High Level Injection or Excitation

6. Electron-Hole Pair Recombination Rate
Case I- Low Level Injection :
Let in equilibrium we have the equilibrium concentration of electrons and holes
is 𝒏𝟎 & 𝒑𝟎 and 𝒏𝒊 be the intrinsic carrier concentration then-
𝒏𝟎𝒑𝟎 = 𝒏𝒊
𝟐
Now, we create excess electrons and holes such that new carrier concentration
be-
𝒏 = 𝒏𝟎 + ∆𝒏 ; 𝒑 = 𝒑𝟎 + ∆𝒑
Where ∆𝒏 𝒂𝒏𝒅 ∆𝒑 are the excess number of electrons and holes generated
respectively.
(1)
(2)

7. Electron-Hole Pair Recombination Rate
 At t = 0, in optical and thermal generation,
∆𝒏𝟎 = ∆𝒑𝟎 ; ∆𝒏 = ∆𝒑
 Low level injection means the excess carrier generated at any time must be
much smaller than the majority carrier concentration of semiconductor.
∆𝒑, ∆𝒏 ≪ 𝒏𝟎 for n-type
∆𝒑, ∆𝒏 ≪ 𝒑𝟎 for p-type
(3)
(4)
(5)

8. Electron-Hole Pair Recombination Rate
 Now we want to see how the carrier concentration is changing with time at any
time ‘t’-
𝒅𝒏
𝒅𝒕
=
𝒅𝒑
𝒅𝒕
= 𝑩𝒏𝒑 − 𝑮𝒕𝒉
the carrier concentration at any time ‘t’ is-
𝒅 𝒏𝟎 + ∆𝒏
𝒅𝒕
= 𝑩 𝒏𝟎 + ∆𝒏 𝒑𝟎 + ∆𝒑 − 𝑩𝒏𝟎𝒑𝟎
Recombination
Rate
Thermal Generation
Rate (B 𝒏𝟎𝒑𝟎)
(6)
(7)

9. Electron-Hole Pair Recombination Rate
𝒅 𝒏𝟎 + ∆𝒏
𝒅𝒕
= 𝑩 𝒏𝟎 + ∆𝒏 𝒑𝟎 + ∆𝒑 − 𝑩𝒏𝟎𝒑𝟎
𝒅 ∆𝒏
𝒅𝒕
= 𝑩 𝒏𝟎𝒑𝟎 + 𝒑𝟎∆𝒏 + 𝒏𝟎∆𝒑 + ∆𝒏∆𝒑 − 𝑩𝒏𝟎𝒑𝟎
∆𝑛∆𝑝 is negligible at low level injection
𝒅 ∆𝒏
𝒅𝒕
= 𝑩 𝒑𝟎∆𝒏 + 𝒏𝟎∆𝒑
Since ∆𝑛 = ∆𝑝
(8)
(9)

10. Electron-Hole Pair Recombination Rate
𝒅 ∆𝒏
𝒅𝒕
= 𝑩∆𝒏 𝒑𝟎 + 𝒏𝟎
At t=0 , ∆𝒏 = ∆𝒏𝟎 , therefore after solving above equation we get the equation
for change in excess carrier concentration at any time t-
∆𝒏 = ∆𝒏𝟎𝒆−
𝒕
𝝉
Where ′𝝉′ is the ‘carrier life time’ which is defined as
𝝉 =
𝟏
𝑩 𝒏𝟎 + 𝒑𝟎
(10)
(11)
(12)

11. Electron-Hole Pair Recombination Rate
 From equation 10 and 12 we get, the recombination rate R is given by-
𝑹 =
𝒅𝒏
𝒅𝒕
=
𝒅 ∆𝒏
𝒅𝒕
= 𝑩∆𝒏 𝒑𝟎 + 𝒏𝟎 =
∆𝒏
𝝉
Since 𝝉 =
𝟏
𝑩 𝒏𝟎+𝒑𝟎
 In a n-type device majority carrier concentration is 𝒏𝟎 so, 𝝉 =
𝟏
𝑩 𝒏𝟎
 In a p-type device majority carrier concentration is 𝒑𝟎 so, 𝝉 =
𝟏
𝑩 𝒑𝟎
(13)
(15)
(14)

12. Electron-Hole Pair Recombination Rate
 For a single recombination process
𝑹 =
∆𝒏
𝝉
 If more processes of recombination are involved then the recombination rate is
given by-
𝑹 =
∆𝒏
𝝉𝟏
+
∆𝒏
𝝉𝟐
+
∆𝒏
𝝉𝟑
+. . . . . . . . . . . .
• This expression of recombination is for low level injecction
(16)

13. Electron-Hole Pair Recombination Rate
Case I- High Level Injection :
The excess carrier generated in this case is very much larger than the total
equilibrium concentration of electrons an holes-
∆𝒏 ≫ 𝒏𝟎 + 𝒑𝟎
Simillar to low level injection eq (8) is given by-
𝒅 ∆𝒏
𝒅𝒕
= 𝑩 𝒏𝟎𝒑𝟎 + 𝒑𝟎∆𝒏 + 𝒏𝟎∆𝒑 + ∆𝒏∆𝒑 − 𝑩𝒏𝟎𝒑𝟎
Here, in this case 𝑛0𝑝0 is negligible as compared to other and ∆𝑛 = ∆𝑝 .
(17)
(18)

14. Electron-Hole Pair Recombination Rate
𝒅 ∆𝒏
𝒅𝒕
= 𝑩 ∆𝒏 𝒑𝟎 + 𝒏𝟎 + ∆𝒏𝟐
After solving this gives change in carrier concentration,
∆𝒏 𝒕 =
𝟏
𝑩𝒕 + ∆𝒏−𝟏
Thus, the rate of recombination at high level injection is given by-
𝑹 = −
𝒅𝒏
𝒅𝒕
= −
𝑩
𝑩𝒕 + ∆𝒏𝟎
−𝟏 𝟐
(21)
(20)
(19)

15. Summary Recombination Rate
Case I- Low Level Injection Case II- High Level Injection
∆𝒏(𝒕) = ∆𝒏𝟎𝒆−
𝒕
𝝉 ∆𝒏 𝒕 =
𝟏
𝑩𝒕 + ∆𝒏−𝟏
𝝉 =
𝟏
𝑩 𝒏𝟎 + 𝒑𝟎
𝑹 = −
𝒅𝒏
𝒅𝒕
= −
𝑩
𝑩𝒕 + ∆𝒏𝟎
−𝟏 𝟐
𝑹 =
𝒅𝒏
𝒅𝒕
== 𝑩∆𝒏 𝒑𝟎 + 𝒏𝟎 = ∆𝒏
𝝉

16. Absorption in Semiconductors
• The measurement of absorption and emission spectra in semiconductors
constitutes an important aspect of materials characterization.
• They not only provide information on the bandgap, but the measurements also
provide information on direct and indirect transitions, the distribution of
states, defects and impurities.
• The absorption spectrum spans a wide energy (or wavelength) range,
extending from the near bandgap energies to the low energy transitions
involving free carriers and lattice vibrations (near bandgap transitions).

17. Absorption in Semiconductors
Indirect intrinsic transitions
Exciton absorption
Donor Acceptor and impurity-band absorption
Low energy(long Wavelength) absorption

18. Indirect Intrinsic transition
• The momentum or wavevector change required in an indirect transition may
be provided by single or multiple phonons, although the probability of the
latter to occur is very small.
• There are optical and acoustic phonons. Each of these has transverse and
longitudinal modes of vibrations, with characteristic energy and momentum.
• In indirect transition process conservation of momentum requires:
k" ± kp = k' + kph (3.51)
Where k" and k’ are the electron wavevectors for the final and initial states, kp is
the wavevector of the phonon, and kph is the wavevector of the absorbed photon.

19. • Since the latter is small, the conservation of momentum for an indirect transition can be
expressed as
k" - k' = ± kp (3.52)
• Similarly, the conservation of energy for the two cases of phonon emission and absorption
can be expressed as
ħωe = εC - εV + εp (3.53)
ħωa = εC - εV – εp (3.54)
Where the left- hand side represents the energy of the photon absorbed.
• From this energy state the electron finally reaches the indirect valley by phonon scattering.
• The intermediate energy state of the electron is termed a virtual state, in which the carrier
resides until a phonon of the right energy and momentum is available for the scattering
process.

20. • Indirect transition probabilities involving virtual state can be calculated using a second-order time-
dependent perturbation theory.
• The total probability is obtained by a summation over these energy states, as long as each particular
transition conserves energy between initial and final states.
• For a transition with phonon absorption,
αa (ħω) ∞
( ħ𝜔− 𝜀𝑔+ 𝜀ₚ )2
𝑒
𝜀ₚ
𝐾𝐵𝑇−1
(3.55)
• For a photon energy ħω > ( εg – εp ). Similarly, for a transition with phonon emission the absorption
coefficient is given by
αe (ħω) ∞
( ħ𝜔− 𝜀𝑔− 𝜀ₚ )2
1− 𝑒
−𝜀ₚ
𝐾𝐵𝑇
(3.56)

21. Figure 3.5: Energy dependent absorption coefficient due to phonon emission and absorption as a function
of temperature.
• The temperature dependence of the absorption coefficient is illustrated in fig 3.5. At very
low temperature the density of phonons available for absorption become small and therefore
αa is small with increase of temperature, αa increases.
• The shift of curves to lower energies with increase to temperature reflects the temperature
dependence of Eg. This is a convenient technique to experimentally determine the bandgap.

22. Exciton Absorption
• In a very pure semiconductors, where the screening effect of free carriers is almost absent.
• Electrons and holes produces by the absorption of a photon of near- bandgap energy pair to
form an exciton. This is the free exciton.
• The binding energy of the exciton, εex , is calculated by drawing analogy with the Bohr atom
for an impurity center, and is quantized. It is therefore expressed as
ɛ𝑒𝑥
𝑖 =
−𝑚𝑟
∗ 𝑞4
2 (4𝜋𝜖𝑟𝜖𝑜ħ)2 .
1
𝑙2 , l = 1,2,3……..
=
− 13.6
𝑙2
𝑚𝑟
∗
𝑚𝑜
1
𝜖𝑟
2
𝑒𝑉 (3.58)
Here 𝑚𝑟
∗ is the reduced effective mass of the exciton given by Eq.3.36 and l is the
integer.
• Such excitons are also known as effective mass or Wannier excitons.

23. • The optical excitation and formation of excitons usually manifest themselves
as a series of sharp resonances (peaks) at the low energy side of the band edge
in the absorption spectra of direct bandgap semiconductors.
• The total energy of the exciton is given by:
εex =
ħ2𝑘𝑒𝑥
2
2 (𝑚𝑒
∗ + 𝑚ℎ
∗
)
− 𝜀𝑒𝑥
𝑙
(3.59)
Where the first term on the right is the kinetic energy of the exciton.
• The kinetic energy contributes to a slight broadening of the exciton levels. For
a direct transition conservation of momentum requires that 𝑘𝑒𝑥
2 ≅ 0.

24. Donor-accepter and Impurity-band absorption
• Intentionally or unintentionally, both the donor and accepter levels are simultaneously present in
the semiconductor.
• Depending upon the temperature and the state of occupancy of the impurity levels, it is possible
to raise the electron from the accepter to the donor level by absorption of photon.
• The energy of the photon is given by:
ℏ𝝎 = 𝜺𝒈-𝜺𝑫-𝜺𝑨 +
𝒒𝟐
𝜺𝟎𝜺𝒓𝒓
(1)
Where the last term on right hand side accounts for coulomb interaction between the donor and
acceptor atoms in substitutional sites, which result in a lowering of their binding energies.
• Another absorption transition can occur between ionized impurity level and opposite band edge
called impurity-band transition.

25. • This can be understood as follow. Assume that at very low temperatures the donor and acceptor atoms
are neutral.
• If they are brought closer together the additional orbiting electron of the donor becomes “shared” by
the acceptor as in a covalent bond become more ionize, resulting in a lowering of their binding energy.
• Since the donor and acceptor are located at discrete substitutional sites in the lattice, r varies in finite
increments, being the smallest for nearest neighbors.
Therefore for ground state of impurities, the energies 𝜀𝐷 and 𝜀𝐴 corresponds to the most distant pairs and
ℏ ≌ 𝜀𝑔-𝜀𝐷-𝜀𝐴
Fig: Illustration of photon absorption due to donor-acceptor transition. The separation between the
impurity centers, r, is shown in (b)

26. • The absorption spectrum is largely altered if the doping level is increased and gradually taken to the point of
degeneracy.
• For example, in a degenerately doped n-type semiconductor is direct the fermi level 𝜀𝑓𝑛, is above the conduction
bandedge.
• If the semiconductor is direct bandgap then for the conservation of momentum, the transition resulting from the
absorption of a photon will involve states in the conduction band that are at higher than 𝜀𝑔+𝜀𝑓𝑛.
• This shift of the absorption to higher energies due to doping-induced band filling is called the Burstein-Moss
shift.
• An indirect band semiconductor will be similarly affected, except that phonons need not be involved in the
transition. Momentum is conserved by impurity scattering.
Fig: Illustration of (a) donor band and (b) acceptor-band absorption transition.

27. Long-Energy(Long-wavelength) Absorption
• Several types of transitions involving shallow impurity, bandedges, split bands, and free
carriers give rise to resonances at very small energies in the absorption spectra
• These are observed as steps or peaks in the long wavelength region of absorption spectra.
• The different processes are briefly described below:
 Impurity-Band Transition-
• Impurity transition that have energies close to the bandgap.
• These higher-energy impurity-band transition usually require that the impurity levels are ionized(or
empty).
• At low temperature, when these shallow impurity levels are usually filled with their respective carrier,
those carrier can be excited to the respective bandedge by photon as shown in figure.
• For this absorption process the energy of the photon must be at least equal to the ionization energy of
impurity.
• This energy usually correspond to the far infrared region of the optical spectrum.

28.  Intraband -Transition-
• At the zone center the valence band structure of most semiconductors consists of the light-hole(LH),
the heavy-hole(HH) bnads, and the split-off(SO) band.
• The three subbands are separated by spin-orbit interaction. In a p-type semiconductor the valence band
is filled with the hole and the occupancy of the different band depend on the degree of doping and the
position of the fermi level.
• Absorption of photons with the right energy can result in transition from LH to HH, SO to HH, and
SO to LH bands, depending on the doping and temperature of the sample. These transition have been
observed experimentally. They are normally not observed in n-type semiconductors.
Figure: Low Energy (a) donor-band and (b) acceptor-band absorption transition.

29. Free carrier Absorption
• This mechanism involves the absorption of photon by the interaction of a free carrier within a
band, which is consequently raised to a higher energy.
• The transition of the carrier to higher energy within the same valley must conserve momentum.
This momentum change is provided by optical or acoustic phonons or by impurity scattering.
• Free-carrier absorption usually manifest in the long-wavelength region of the spectrum as a
monotonic increase in absorption with a wavelength dependence of the from 𝜆𝑝
, where p range
from 1.5 to 3.5.
• The value of p depends on the nature of the momentum-conserving scattering( i.e. the
involvement of acoustic phonons, optical phonons or ionized impurities).
• The absorption coefficient due to free-carrier absorption can be expressed as.
𝛼 =
𝑁𝑞2𝜆2
4𝜋2𝑚𝑛𝑟𝑐3𝜖0
1
Τ
• Where N is the free-carrier concentration, 𝑛𝑟 is the refractive index of the semiconductor, and
1
Τ
is the average value of the inverse of the relaxation time of scattering process.

30. Effect of Electric Field on Absorption: FRANZ-KELDYSH
AND STARK EFFECTS
 The change in absorption in a semiconductor in the
presence of strong electric field is the Franz-Keldysh
effect, which results in the absorption of photons with
energy less than the band gap of the semiconductor.
 The energy bands of semiconductor in the presence of
electric field E and with an incident photon of energy
ℏ𝜔 < 𝜀𝑔 are shown in figure (a) and (b).
 In figure (a) shows the bending of bands due to
applied electric field and (b) shows the absorption of
photon with ℏ𝝎 < 𝜺𝒈 due to carrier tunneling
(Franz-Keldysh effect).

31. Franz-Keldysh Effect
 The classical turning points are marked as A and B, the electron wave
functions change from oscillatory to decaying behaviour.
 Thus electron in the energy gap is described by an exponentially decaying
function 𝒖𝒌𝒆𝒋𝒌𝒙
, where k is imaginary.
 With increase of electric field, the distance AB decreases and the overlap
of the wave functions within the gap increases.
 In the absence of a photon, the valance electron has to tunnel through a
triangular barrier of height 𝜀𝑔 and thickness d, given by
𝒅 =
𝜺𝒈
𝒒𝑬

32. Franz-Keldysh Effect
 With the assistance of an absorbed photon of energy ℏ𝝎 < 𝜺𝒈, it is evident
that the tunnelling barrier thickness is reduced to
𝒅′ =
(𝜺𝒈−ℏ𝝎)
𝒒𝑬
And the overlap of the wave function increases further and the valance electron
can easily tunnel to the conduction band.
 The net result is that a photon of energy ℏ𝝎 < 𝜺𝒈 is absorbed.
 In this case, the transverse component of the momentum is conserved.
 The Franz-Keldysh effect is therefore, in essence, photon assisted
tunnelling.

33. Franz-Keldysh Effect
 The electric field dependent absorption coefficient is given by-
𝜶 = 𝑲(𝑬′)
𝟏
𝟐 𝟖𝜷 −𝟏𝒆𝒙𝒑 −
𝟒
𝟑
𝜷
𝟑
𝟐
Here, 𝑬′ =
𝒒𝟐𝑬𝟐ℏ𝟐
𝟐𝒎𝒓
∗
𝟏 𝟑
, 𝜷 =
𝜺𝒈−ℏ𝝎
𝑬′ and K is a material dependent parameter
and has value of 𝟓 × 𝟏𝟎𝟒
𝒄𝒎−𝟏
𝒆𝑽
−𝟏
𝟐 in GaAs.
 The exponential term is the tunnelling probability of an electron through a
triangular barrier of height (𝜺𝒈−ℏ𝝎) and can be obtained from the well
known Wentzel-Kramers-Brillouin (WKB) approximation.

34. Franz-Keldysh Effect
 The other factors are related to the upward transition of an electron due to
photon absorption.
 Substituting appropriate values for the different parameters, it is seen that in
GaAs 𝜶 = 𝟒 𝒄𝒎−𝟏
at a photon energy of 𝜺𝒈 − 𝟐𝟎 𝒎𝒆𝑽 with electric field
E ~ 𝟏𝟎𝟒 V/cm.
 This value of absorption coefficient is much smaller than the value of 𝛼 at
the band edge at zero field.
 Therefore, Franz-Keldysh effect will be small unless 𝑬 ≥ 𝟏𝟎𝟓 𝑽/𝒄𝒎.

35. Stark Effect
 The Stark effect refers to the change in atomic energy upon the application
of an electric field.
 The electric field affects the higher order, or outer, orbits of the precessing
electrons so that the center of gravity of the elliptical orbit and the focus are
displaced to each other and linearly aligned in the direction of the electric field.
 As a result, there is splitting of the energy of the outer 2s or 2p states, and the
energy shift is simply given by ∆𝜺 = 𝒒𝒅𝑬, where d is the eccentricity of the
orbit. This is linear Stark effect.
 The effect of electric field on ground state orbits also leads to an energy shift
of the state, and that is the quadratic or second-order Stark effect.