1. 1. Modulators using Optical transitions in quantum wells
UCONN ECE (F. Jain) L4
1. Modulators using Optical transitions in quantum wells
2.Photon absorption or emission involves electronic transitions
(Absorption is dependent on probability of transition and density of states; density of states depend on the physical structure of the absorbing layer if it is thick (>10-15nm), or thin).
In terms of thin, we need to know if it comprise of quantum wells, multiple quantum wells, quantum wires, and quantum dots.
Absorbing or emitting layer is made of direct gap or indirect gap semiconductors.
Upward transitions involve photon absorption. These transitions are with and without phonons. In indirect semiconductors, phonon participation is essential to conserve momentum.
Downward transitions result in photon emission.
Transitions involve valence band-to-conduction band, band to impurity band or levels, donor-to-acceptor, and intra-band.
Band-to-band transitions could involve free carriers or excitonic transitions.

2. Optical Modulators using quantum confined Stark Effect
We cover basics of exciton formation in quantum wells and related changes in the absorption coefficient and index of refraction. This phenomena is called confined Stark effect.
Photon energy needed to create an exciton is given by
  h = Eg (Ee Ehh1) - Eex
(Bulk Band gap) (Zero field energy of (Binding energy Electron and holes) of exciton)
If the photon energy is greater than above equation, free electron and hole pair is created. Eg
Ehh1
Fig. 1(a) E = 0
Ee1

3. Application of perpendicular Electric Field
When a perpendicular electric field is applied, the potential well tilts. Its slope is related to the electric field.
Electron wave function
Hole wave function
Fig. 1(b) Quantum well in the presence of E
As E increases Ee & Eh decreases.
As a result photon energy at which absorption peak occurs shifts to lower values (Red shift) .
The interaction between electron and hole wavefunctions (and thus, the value of the optical matrix elements and absorption coefficeint) also is reduced as magnitude E is increased. This is due to the fact that electron, hole wave functions are displaced with respect to each other. Therefore, the magnitude of absorption coefficient decreases with increasing electronic filed E.

4. MQW Modulator based on change in absorption due to quantum confined Stark effect
Fig.3. Responsivity of a MQW diode
(acting as a photodetector or optical modulator).

5. Change in index of refraction

6. Phase Modulator: …………………(2)
A light beam signal (pulse) undergoes a phase change as it transfers an electro-optic medium of length ‘L’
(1)
In linear electro-optic medium
…………………(2)
E= Electric field of the RF driver; E=V/d, V=voltage and d thickness of the layer.
r= linear electro-optic coefficient
n= index

7. MQWs have quadratic electro-refractive effect.
Figure compares linear and quadratic variation Dn as a funciton of field.
Figure shows a Fabry-Perot Cavity which comprises of MQWs whose index can be tuned (Dn) as a funciton of field.

8. Fabry-Perot Modulator (pp. 178-179)

9. Fabry-Perot Modulator (pp. 180-181,ECE 5212

10. Fabry-Perot Modulator (pp. 180-181)

11. Asymmetric InGaAs-GaAs Modulator

12. Mach-Zehnder Modulator
Here an optical beam is split into two using a Y-junction or a 3dB coupler. The two equal beams having ½ Iin optical power. If one of the beam undergoes a phase change , and subsequently recombine. The output Io is related as
Mach-Zehnder Modulator comprises of two waveguides which are fed by one common source at the input (left side). When the phase shift is 180, the out put is zero. Hence, the applied RF voltage across the waveguide modulates the input light.

13. Band to band transitions in nanostructures
Band to band free carrier transitions.
Band to band transitions involving exciton formation.
Light emission and absorption.

14. Energy bands
Fig. 8 and 9. p.119

15. Energy bands Fig. 13(b) and Fig. 13c. p.123-124 V(R) V0 2b<a
-b a a+b
(Ref. F. Wang, “Introduction to Solid State Electronics”, Elsevier, North-Holland)
Fig. 13(b) and Fig. 13c. p

16. Absorption and emission of photons1)
(Transition Probability)
here, (h) is the joint density of states in a volume V (the number of energy levels separated by an energy
Fig. 2 p.138 2)
Pmo probability that a transition has occurred from an initial state ‘0’ to a final state ‘m’ after a radiation of intensity I(h) is ON for the duration t.
The joint density of states is expressed as Eq. 2B) o

17. Time dependent Schrodinger Equation p.68-69

19. Infinite Potential Well energy Levels
Solve Schrodinger Equation
In a region where V(x) = 0, using boundary condition that (x=0)=0, (x=L)=0. Plot the (x) in the 0-L region.

20. In the region x = 0 to x = L, V(x) = 0
Let
(2)
The solution depends on boundary conditions.
Equation 2 can be written as D coskxx + C sinkxx.
At x=0,
(3)
, so D= 0
nx = 1,2,3… (4)
Using the boundary condition (x=L) = 0, we get from Eq. 3 sin kxL = 0, L = Lx

21. Page 52-53: Using periodic boundary condition (x+L) = (x),
we get a different solution:
We need to have kx=
---(6)
The solution depends on boundary conditions. It satisfies boundary condition when Eq.6 is satisfied.
If we write a three-dimensional potential well, the problem is not that much different
ky =
n y= 1,2,3,…
kZ =
, n z = 1,2,3,….
…(7)
The allowed kx, ky, kz values form a grid. The cell size for each allowed state in k-space is
…(8)

22. Density of States in 3D semiconductor film (pages 53, 54)k
k+dk
Density of states between k and k+dk including spin
N(k)dk =
…(10)
…11
Density of states between E and E+dE
N(E)dE=2 VB E
N(E)
E=Ec +
(12)
…(13)
N(E)dE =
…14
N(E)dE =
…15

23. Energy Levels in a Finite Potential Well
AlxGa1-xAs
GaAs
∆EV
∆EC
V(z) z
-EG
∆Eg
-EG+∆EV
∆Ec = 0.6∆Eg
∆Ev = 0.4∆Eg

24. In the barrier region

25. Wavefunction matching at well-barrier boundary

26. Wavefunction matching at well-barrier boundary

28. Electrons
EC1 = 72.5 meV
Heavy Holes
EHH1 = 22.3 meV
Light Holes
ELH1 = 52.9 meV

29. The effective width, Leff can be found by:
kL/2
Electrons
EC1 = 72.5 meV
Heavy Holes
EHH1 = 22.3 meV
Light Holes
ELH1 = 52.9 meV
Where, radius is
Output: Equation (15) and (12), give k, α’ (or α). Once k is known, using Equation (4) we get:
The effective width, Leff can be found by:
E1 is the first energy level.
(4)

30. Summary of Schrodinger Equation in finite quantum well region.
(1)
In the well region where V(x) = 0, and in the barrier region V(x) = Vo = ∆Ec in conduction band, and ∆Ev in the valence band.
The boundary conditions are:
(5) Continuity of the wavefunction
(6) Continuity of the slope
AlxGa1-xAs
GaAs
∆EV
∆EC
V(z) z
-EG
∆Eg
-EG+∆EV
∆Ec = 0.6∆Eg
∆Ev = 0.4∆Eg
Eigenvalue equation:

31. C. 2D density of state (quantum wells) page 84
Carrier density number of states
(1)
Without f(E) we get density of states
Quantization due to carrier confinement along the z-axis.

32. Density of states summary
2D Well
1D Wire
0-D Box
3-D (bulk)
No confinement
2-D (Quantum well)
1-D of confinement
1-D (Quantum wire)
2-D of confinement
0-D (Quantum dot)
3-D of confinement
Density of States N(E)
Plots
N(E)
Energy level

33. Photon Absorption
Absorption coefficient a(hn) depends on type of transitions involving phonons (indirect) or not involving phonons (direct). Generally, absorption starts when photon energy is about the band gap. It increases as hv increases above the band gap Eg. It also depends on effective masses and density of states.(p.140, eqs 1-2)
The absorption coefficient in quantum wires is higher than in wells. It also starts at higher energy than band gap in bulk materials.
Absorption coefficient is related to rate of emission.
Rate of emission in quantum wells is higher than in bulk layers.
Rate of emission in quantum wires is higher than in quantum wells.
Rate of emission in quantum dots is higher than in quantum wires.
Rate of emission has two components:
Spontaneous rate of emission
Stimulated rate of emission.

34. · Direct and Indirect Energy Gap Semiconductors
Semiconductors are direct energy gap or indirect gap. Metals do have not energy gaps. Insulators have above 4.0eV energy gap.
Fig. 10b. Energy-wavevector (E-k) diagrams for indirect and direct semiconductors. Here, wavevector k represents momentum of the particle (electron in the conduction band and holes in the valence band). Actually momentum is = (h/2p)k = k

35. Effect of strain on band gap
Ref: W. Huang, 1995 UConn doctoral thesis with F. Jain
Under the tensile strain, the light hole band is lifted above the heavy hole, resulting in a smaller band gap.
Under a compressive strain the light hole is pushed away from heavy. As a result the effective band gap as well as light and heavy hole m asses are a function of lattice strain. Generally, the strain is +/ %. "+" for tensile and "-" for compressive.
Strain does not change the nature of the band gap. That is, direct band gap materials remain direct gap and the indirect gap remain indirect.

36. Electrons & Holes
Photons
Phonons
Statistics
F-D & M-B
Bose-Einstein
Velocity
vth ,vn
1/2 mvth2 =3/2 kT
Light c or v = c/nr
nr= index of refraction
Sound
vs = 2,865 meters/s in GaAs
Effective Mass
mn , mp
(material dependent)
No mass
Energy
E-k diagram
Eelec=25meV to 1.5eV
ω-k diagram (E=hω)
ω~1015 /s at E~1eV
Ephotons = 1-3eV
ω-k diagram (E= ω)
ω~5x1013/s at E~30meV
Ephonons = meV
Momentum
P= k
k=2π/λ
λ=2πvelec/ω
momentum: 1000 times
smaller than phonons
and electrons
P= k
λ=2πvs/ω

37. Rate of emission in quantum wells is higher than in bulk layers.
Direct and Indirect p.135 .
(phonon absorption) + (phonon emission)
p A=
Absorption coefficient a(hn) depends on type of transitions involving phonons (indirect) or not involving phonons (direct). Generally, absorption starts when photon energy is about the band gap. It increases as hn increases above the band gap Eg. It also depends on effective masses and density of states. (p.140, Eqs 1-2)
The absorption coefficient in quantum wires is higher than in wells. It also starts at higher energy than band gap in bulk materials.
Downward transitions result in photon emission. This is called radiative transition. When there is no photon emission it is called non-radiative transition.
Rate of emission in quantum wells is higher than in bulk layers.
Rate of emission in quantum wires is higher than in quantum wells.
Rate of emission in quantum dots is higher than in quantum wires.
Rate of emission has two components:
Spontaneous rate of emission
Stimulated rate of emission.

38. Quantum efficiency
Absorption coefficient is related to rate of emission via Van Roosbroeck- Shockley. This enables obtaining an expression for radiative transition lifetime tr.
There are ways to compute non-radiative lifetime tnr. The internal quantum efficiency is expressed as follows: Pr is the probability of a radiative transition.
Two types of downward transitions.
1. Radiative transitions.
2. Non-radiative transition:

39. Absorption coefficient is related to rate of emission.
Eq. 7 –page 214
is the rate of emission of photons at  within an interval
Van Roosbroeck - Shockley Relation under equilibrium
Eq. 8 –page 214
Absorption coefficient is related to imaginary component of the complex index of refraction.
Index of refraction nc is complex when there are losses.
nc= nr – i k where extinction coefficient k = α/4pn, here alpha is the absorption coefficient.
Also, nc is related to the dielectric constant. nc = =
In non-equilibrium, we have excess electron-hole pairs. Their recombination gives emission of photons. The non-equilibrium rate of recombination Rc is
In equilibrium,

40. Radiative life time in intrinsic and p-doped semiconductors (p.216)
Nonradiative recombination: Auger Effect and other mechanisms (217)
Nonradiative transitions are processes in which there are no photons emitted. Thus, there are several models by which energy is dissipated.
Experimental observations are in terms of (1) emission efficiency, (2) carrier lifetime (coupled with emission kinetics), (3) behavior (or recombination mechanisms response) to temperature and carrier concentration variations.
Recombination processes which are not associated with an emission of a photon are as follows:
Auger effect (carrier-carrier interaction)
Surface recombination, recombination through defects (Shah-Noyce-Shockley)
Multi phonon emission
and others which may fit with the above proposed criterion.
Since Auger processes involve carrier-carrier interactions, it is typical to assume that probability of occurrence of such a process should increase with the carrier concentrations.

41. Auger Transitions (p217-219)
(n type semiconductor)
(p type semiconductor) e h
First term:
AnP e h
Second Term:
Bp2

42. Defect states caused nonradiative transitions
Recombination via lattice defects or inclusions:
Defects (dislocations, grain boundaries) and inclusions produce a continuum of energy states. These can trap electrons and holes which may recombine via continuum of states. Such a transition would be non-radiative.
The energy states are distributed as:
Nonradiative recombination:
A microscopic defect or inclusion could induce a deformation of the band structure.
e.g.
Density of states
States in the Gap
Density of states due to defects and inclusions E

43. The absorption coefficient relations are in Chapter 3
Transitions involve:
Valence band-to-conduction band
Direct band-to-band
Indirect band-to-band
Band to band involving excitons (hn < Eg)
Band to impurity band or levels,
Donor level-to-acceptor level, and
Intra-band or free carrier absorption.
Similar transitions in downward direction result in emission of photons.

44. Spectral width of emitted radiation
Density of states in bulk is N(E)dE =
The electron concentration ‘n’ in the entire conduction band is given by
(EC is the band edge)
This equation assumes that the bottom of the conduction band is =0.
Electron and hole concentration as a function of energy

45. Graphical method to find carrier concentration in bulk or thick film (Chapter 2 ECE 4211)

46. Graphical method to find carrier concentration in quantum well

47. Spectral width of emitted radiation
Spectral width in quantum well active layer is smaller than bulk thin film active layer

48. DEVICES based on optical transitions Emission: LEDs. Lasers
DEVICES based on optical transitions Emission: LEDs Lasers Photodetectors Absorption: Solar cells Optical modulators and switches Optical logic

49. Transitions in Quantum Wires:
The probability of transition Pmo from an energy state "0" to an energy state in the conduction band "m" is given by an expression (derived in ECE 5212). This is related to the absorption coefficient a. It depends on the nature of the transition.
The gain coefficient g depends on the absorption coefficient a in the following way:
g = -a(1- fe - fh ).
The gain coefficient g can be expressed in terms of absorption coefficient a, and Fermi-Dirac distribution functions fe and fh for electrons and holes, respectively. Here, fe is the probability of finding an electron at the upper level and fh is the probability of finding a hole at the lower level.
Free carrier transitions: A typical expression for g in semiconducting quantum wires, involving free electrons and free holes, is given by:

50. Excitonic Transitions in Quantum Wires
Excitonic Transitions: This gets modified when the exciton binding energy in a system is rather large as compared to phonon energies (~kT). In the case of excitonic transitions, the gain coefficient is:

51. Laser Modulators
The gain coefficient depends on the current density as well as losses in the cavity or distributed feedback structure.
Threshold current density is obtained once we substitute g from the above equation.
For modulators, we need to know:
Absorption and its dependence on electric field (for electroabsorptive modulators)
Index of refraction nr and its variation Dn as a function of electric field (electrorefractive modulators)
Change of the direction of polarization, in the case of birefringent modulators.
In the presence of absorption, the dielectric constant e (e1-je2) and index of refraction n (nr –j k) are complex.
However, their real and imaginary parts are related via Kramer Kronig's relation.
See more in the write-up for Stark Effect Modulators.
Lasers and Modulators:
In the case of lasers, we need to find the gain coefficient and its relationship with fe and fh, which in turn are dependent on the current density (injection laser) or excitation level of the optical pump (optically pumped lasers).

52. 4. Quasi Fermi Levels Efn, Efp, and Dz
p114 notes
4. Quasi Fermi Levels Efn, Efp, and Dz
Quantum Wells
Quantum Wires
Quantum Dots

53. Laser operating wavelength (p 114)
5. Operating Wavelength l:
The operating wavelength of the laser is determined by resonance condition
L=ml/2nr
Since many modes generally satisfy this condition, the wavelength for the dominant mode is obtained by determining which gives the maximum value of the value of the gain. In addition, the index of refraction, nr, of the active layer is dependent on the carrier concentration, and knowing its dependence on the current density or gain is important. For this we need to write the continuity equation.
6. Continuity equation and dependence of index of refraction on injected carrier concentration
The rate of increase in the carrier concentration in the active layer due to forward current density J can be expressed as:

54. Examples of Quantum dot lasers

55. Why quantum well, wire and dot lasers, modulators and solar cells?
Quantum Dot Lasers:
Low threshold current density and improved modulation rate.
Temperature insensitive threshold current density in quantum dot lasers.
Quantum Dot Modulators:
High field dependent Absorption coefficient (α ~160,000 cm-1) : Ultra-compact intensity modulator
Large electric field-dependent index of refraction change (Δn/n~ ): Phase or Mach-Zhender Modulators
Radiative lifetime τr ~ 14.5 fs (a significant reduction from fs).
Quantum Dot Solar Cells: High absorption coefficent enables very thin films as absorbers.
Excitonic effects require use of pseudomorphic cladded nanocrystals (quantum dots ZnCdSe-ZnMgSSe, InGaN-AlGaN)
Table I Computed threshold current density (Jth) as a function of dot size in for InGaN/AlGaN Quantum Dot Lasers (p.11)
(Ref. F. Jan and W. Huang, J. Appl. Phys. 85, pp , March 1999).

56. Quantum Confined Stark Effect (QCSE) in Conventional Field Effect Structure (FES)
Under electric fields, large shifts in optical absorption and change in refractive index are observed.
Result of decreases wave function interaction. (electron states to lower energies, holes to higher energies)
Fig 17. Resulting absorption shift via QCSE.[6]
Fig 18. (a) refractive index increase (higher absorption) in MQW. (b) Refractive index shift increases.

57. QD Modulator Modeling Modeling of II-VI (ZnSe-CdSe) QDs
Electro-Absorption
Greater binding energies = more stable excitons (smaller QD advantageous)
Electro refraction
Index changes occur singular for large QDs
Previously modeled index changes up to .021
Fig 18. Absorption change of 30Å QDs under varying field.[13]
Modeling Hierarchy
Calculation of elastic Strain in the superlattice (mismatch) and calculate new band gap and band offset.
Create Hamiltonian to solve Schrodinger's equation for wavefunction.
Calculate imaginary part of dielectric constant (exciton transitions included)
Calculate real part of the dielectric constant using Kramer-Kronig relations
Calculate index of refraction and absorption coefficients. [13]
Fig 19. Refractive index change of 30Å QDs for varying bias Voltages. [13]

58. Mathematical Representation of QCSE
Under Electric Field, Excitonic binding energies decrease
Fig 20. Refractive index of 5nm QD with 50kV/cm and 100kV/cm E field.