1. Atilla Ozgur Cakmak, PhD
Nanophotonics
Atilla Ozgur Cakmak, PhD

2. Unit 2
Lecture 17: Quantum Confinement Effects in Solids and Quantum Devices- Part1

3. Outline Quantum Confinement in Solids Quantum Wells
Quantum Wires/Nanorods

4. A couple of words…
Now, since we have just covered the fundamentals of semiconductor physics and also covered quantum effects/potentials in the previous lectures, it is the high time we start looking into quantum confinement devices that are used for photodetection or photoemission. This lecture will initiate that analysis.
Suggested reading: Sergey V. Gaponenko, “Introduction to Nanophotonics”, 4th Chapter.

5. Quantum Confinement in Solids
Let us remember de Broglie wavelength from the previous lectures:
(The book uses m0 for me in vaccum)
A thin film, hence would confine the electron motion in 1D => Quantum Wells
If we create a rod, wire, we can confine the electron motion in 2D => Quantum Wires
Crystalline nanoparticles (nanocrystals) can offer confinement in 3D => Quantum dots
Quantum particle should be confined in a device smaller than its de Broglie wavelength.

6. Quantum Confinement in Solids
Remember that we had derived the 3D D(E) as an assignment for particles:

7. Quantum Wells D(E) gets quantized for confined structures:
Quantum wells can be fabricated by means of epitaxial growth of a multilayer semiconductor structure. A narrow band gap material is buried into a wider band gap material => Heterostructure
This is feasible only when the crystal lattices match (ie lattice symmetry, chemical compability)

8. Quantum Wells
We need perfect lattice match at the heterostructure interface. Some examples are:
GaAs-GaAlxAs1-x , ZnS-ZnSxSe1-x, Ge-GexSi1-x, CdSe-CdSexTe1-x. Quantum well devices can be cascaded and will result in quantum well superlattices similar to the case we investigated in quantum mechanics review.
An example with GaAs-GaAlxAs1-x Quantum Wells:
GaAs is sandwiched between two GaAlxAs1-x layers. Band gap of GaAs is 1.42 eV. GaAlxAs1-x has empirically x for x < There is also a relative offset in the conduction band minimum between the two materials: 0.836x. The electron effective mass is given to be m*= x. We can construct quantum wells, superlattices, by changing the composition of the alloy. Likewise, we can create the quantum well using the valance band offsets. Valance band offset is x. The effective mass is m*=-( x).
Let us look at the wavefunctions in these possible structures.

9. Quantum Wells
Conduction band quantum well
Valance band quantum well

10. Quantum Wells
An amazing property is their strong sensitivity of the absorption spectrum to an external electric field. Si-Ge offers an easy and cheap integration with CMOS electronic circuitry. Electro-absorptive multiple quantum wells can be tuned to the 1.55um of the optical networks. Their electro-absorptive properties yield electro-refractive effects (changes in the refractive index).

11. Quantum Wells
A step discontinuity in the conduction and valance band offsets create the key capability for the invention of the room temperature semiconductor lasers. The conduction band offset blocks the flow of the electrons, whereas the valance band offset blocks the flow of holes. Electrons and holes get trapped in the active region.
We can adjust the quantum well width to play with the emission wavelength + play with the stoichiometry .

12. Quantum Wells
Quantum well widths are very small compared to the optical communication wavelengths. Most practical applications stack quantum wells to overlap the size of the cascaded quantum wells with the optical mode.
Band diagram and wavefunctions in cascaded quantum wells

13. Quantum Wells
They can also be used as photodetectors due to the absorption capability. In quantum well lasers, these absorption processes happen based on interband transitions (from valance confined states to the conduction confined states). There can be some intersubband transitions as well. Whereas for the photodetectors, they have to be intersubband (from one confined state to the other in the conduction band).

14. Quantum Wells
We have to conduct current. Therefore, transition to the conduction band confined state is not good enough. There are several ways of doing this: a) c)
Send the electron to the next level of conduction band confined state, which can tunnel through the barrier. Large dark current (unwanted signal) - > Disadvantage
There is only one confined state in the conduction band and electron jumps up to the conduction band continuum. Lowered dark current.
Coupled quantum wells produce a miniband that coincides with the upper confined state in the conduction band. The electron will be conducted just like as in the case of b). b)

15. Quantum Wells
Light polarization (as we will see what it means in the upcoming lectures) will be crucial in the absorption mechanism. Only red colored wave (shown below, not the wavelength!) will be absorbed. There is a dipole moment defined between the confined states of the quantum well (perpendicular to the well) and the direction of this dipole moment has to match the moment of the incoming light.

16. Quantum Wells (problem)
Find the emission wavelength of the InP|In0.53Ga0.4As|InP laser. The bandgap of InP is 1.35eV. The bandgap of In0.53Ga0.4As is 0.75eV. The conduction band and valance band offsets are 0.25eV and 0.35eV, respectively. The InP and In0.53Ga0.4As effective electron masses in the conduction band are 0.08 and 0.041, respectively. Likewise, it is and for the valance band for InP and In0.53Ga0.4As, respectively. The graph of the configuration is shown below. Drawn out of scale.
0.25eV
InP
1.35eV
0.7eV
InP
0.35eV
In0.53Ga0.4As

17. Quantum Wells (solution)
Remember the bound states in TMM solutions. Focus on each well, separately, in conduction and valance band sections. The only difference are the effective mass changes. Define two different electron masses and use the same formulas as before. For the conduction band:

18. Quantum Wells (solution)
Here V0 is 0.25eV and m1, m2 are the effective masses in InP and In0.53Ga0.4As, respectively, multiplied with the electron mass.

19. Quantum Wells (solution)
Here V0 is 0.25eV and m1, m2 are the effective masses in InP and In0.53Ga0.4As, respectively, multiplied with the electron mass.
We are 54.4meV away from the conduction band

20. Quantum Wells (solution)
Here V0 is 0.35eV and m1, m2 are the effective masses in InP and In0.53Ga0.4As, respectively, multiplied with the electron mass for the valance band part. Since the electron is much heavier, we have more states available.

21. Quantum Wells (solution)
If we look at the options. Laser can emit (band gap=0.75eV):
54.04meV+0.75eV+7.04meV=> Convert to nm wavelength => 1529 nm, this is between the first confined states. OR
54.04meV+0.75eV+28.7meV=>1488.8nm
All the other options will result in wavelengths that are outside of the telecommunication wavelength regime (1,55um). Hence, 1527nm seems to be the viable option.

22. Quantum Wires/Nanorods
A one dimensional conductor: Quantum wire exhibits the conductivity named as conductivity quantum.
An electron with a high speed will have a lower density of states (D(E)). The speed of the electron determines the conductivity but speedxD(E) will be constant in this case revealing a constant conductivity.
CdSe nanowires have been synthesized with size dependent optical properties.

23. Quantum Wires/Nanorods