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The classic model used to demonstrate a quantum well is to confine particles, which were initially free to move in three dimensions, to two dimensions, by forcing them to occupy a planar region. The effects of [[quantum confinement]] take place when the quantum well thickness becomes comparable to the [[de Broglie wavelength]] of the carriers (generally [[electron]]s and [[electron hole|holes]]), leading to energy levels called "energy subbands", i.e., the carriers can only have discrete energy values.
The concept of quantum well was proposed in 1963 independently by [[Herbert Kroemer]] and by [[Zhores Alferov]] and R.F. Kazarinov.<ref name="kroemer">{{cite journal | last=Kroemer | first=H. | title=A proposed class of hetero-junction injection lasers | journal=Proceedings of the IEEE | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=51 | issue=12 | year=1963 | issn=0018-9219 | doi=10.1109/proc.1963.2706 | pages=1782–1783}}</ref><ref name="alferov">Zh. I. Alferov and R.F. Kazarinov, Authors Certificate 28448 (U.S.S.R) 1963.</ref>
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== Description and overview ==
One of the simplest quantum well systems can be constructed by inserting a thin layer of one type of semiconductor material between two layers of another with a different band-gap. Consider, as an example, two layers of [[Aluminium gallium arsenide|AlGaAs]] with a large bandgap surrounding a thin layer of [[Gallium arsenide|GaAs]] with a smaller band-gap. Let’s assume that the change in material occurs along the ''z''-direction and therefore the potential well is along the ''z''-direction (no confinement in the ''x–y'' plane.). Since the bandgap of the contained material is lower than the surrounding AlGaAs, a quantum well (Potential well) is created in the GaAs region. This change in band energy across the structure can be seen as the change in the potential that a carrier would feel, therefore low energy carriers can be trapped in these wells.<ref name=":1" /> [[File:MCM QW1 GaAs.png|thumb|The band structure diagram in a quantum well of GaAs in between AlGaAs. An electron in the conduction band or a hole in the valence band can be confined in the potential well created in the structure. The available states in the wells are sketched in the figure. These are "particle-in-a-box-like" states.]]
Within the quantum well, there are discrete [[Stationary state|energy eigenstates]] that carriers can have. For example, an electron in the [[Valence and conduction bands|conduction band]] can have lower energy within the well than it could have in the AlGaAs region of this structure. Consequently, an electron in the conduction band with low energy can be trapped within the quantum well. Similarly, holes in the valence band can also be trapped in the top of potential wells created in the valence band. The states that confined carriers can be in are [[Particle in a box|particle-in-a-box]]-like states.<ref name=":0" />
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=== Infinite well model ===
The simplest model of a quantum well system is the infinite well model. The walls/barriers of the potential well are assumed to be infinite in this model.
Consider an infinite quantum well oriented in the ''z''-direction, such that carriers in the well are confined in the ''z''-direction but free to move in the ''x–y'' plane. we choose the quantum well to run from <math>z = 0</math> to <math>z = d</math>. We assume that carriers experience no potential within the well and that the potential in the barrier region is infinitely high.
The [[Schrödinger
:<math>-\frac{\hbar^2}{2m{_w}^*}\frac{\partial^2\psi(z)}{\partial z^2} = E\psi(z)</math>
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=== Superlattices ===
[[File:GaAs-AlAs SL.JPG|thumb|A heterostructure made of AlAs and GaAs arranged in a superlattice configuration.
A superlattice is a periodic heterostructure made of alternating materials with different band-gaps. The thickness of these periodic layers is generally of the order of a few nanometers. The band structure that results from such a configuration is a periodic series of quantum wells. It is important that these barriers are thin enough such that carriers can tunnel through the barrier regions of the multiple wells.<ref name=":3">Odoh, E. O., & Njapba, A. S. (2015). A review of semiconductor quantum well devices. ''Adv. Phys. Theor. Appl'', ''46'', 26-32.</ref> A defining property of superlattices is that the barriers between wells are thin enough for adjacent wells to couple. Periodic structures made of repeated quantum wells that have barriers that are too thick for adjacent wave functions to couple, are called multiple quantum well (MQW) structures.<ref name=":0" />
Since carriers can tunnel through the barrier regions between the wells, the wave functions of neighboring wells couple together through the thin barrier, therefore, the electronic states in superlattices form delocalized minibands.<ref name=":0" /> Solutions for the allowed energy states in superlattices is similar to that for finite quantum wells with a change in the boundary conditions that arise due to the periodicity of the structures. Since the potential is periodic, the system can be mathematically described in a similar way to a one-dimensional crystal lattice.
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=== Solar cells ===
Quantum wells have been proposed to increase the efficiency of [[solar cell]]s. The theoretical maximum efficiency of traditional single-junction cells is about 34%, due in large part to their inability to capture many different wavelengths of light. [[Multi-junction solar cell]]s, which consist of multiple p-n junctions of different bandgaps connected in series, increase the theoretical efficiency by broadening the range of absorbed wavelengths, but their complexity and manufacturing cost limit their use to niche applications. On the other hand, cells consisting of a p-i-n junction in which the intrinsic region contains one or more quantum wells, lead to an increased photocurrent over dark current, resulting in a net efficiency increase over conventional p-n cells.<ref>{{cite journal|title=Quantum well solar cells | first1 = K. | last1 = Barnham | first2 = A. | last2 = Zachariou | year = 1997 |journal=Applied Surface Science |volume=113-114 |pages=722–733 |doi=10.1016/S0169-4332(96)00876-8|bibcode=1997ApSS..113..722B}}</ref> Photons of energy within the well depth are absorbed in the wells and generate electron-hole pairs. In room temperature conditions, these photo-generated carriers have sufficient thermal energy to escape the well faster than the [[Carrier generation and recombination|recombination rate]].<ref>{{cite journal|title=Modeling of multiple-quantum-well solar cells including capture, escape, and recombination of photoexcited carriers in quantum wells | first1 = S. M. | last1 = Ramey | first2 = R. | last2 = Khoie | year = 2003 |journal=IEEE Transactions on Electron Devices |volume=50 |issue=5 |pages=1179–1188 |doi=10.1109/TED.2003.813475 |bibcode=2003ITED...50.1179R }}</ref> Elaborate multi-junction quantum well solar cells can be fabricated using layer-by-layer deposition techniques such as molecular beam epitaxy or chemical vapor deposition. It has also been shown that metal or dielectric
==== Single-junction solar cells ====
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:<math>E_{G,\text{Eff}} = E_{G}^\text{well,relaxed} + \Delta E_G^\text{Strain} + \Delta E_G^\text{QSE} + \Delta E_G^\text{QCSE}</math>
The strain of the material causes two effects to the bandgap energy. First is the change in relative energy of the conduction and valence band. This energy change is affected by the strain, <math>\epsilon</math>, elastic stiffness coefficients, <math>C_{11}</math> and <math>C_{12}</math>, and hydrostatic deformation potential, <math>a</math>.<ref name=":4" /><ref>{{Cite journal|last1=Asai|first1=Hiromitsu|last2=Oe|first2=Kunishige|date=1983|title=Energy
:<math>\Delta E=-2a\left(\frac{C_{11}-C_{12}}{C_{11}}\right)\epsilon</math>
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In the 1.1-1.3 eV range, Sayed et al.<ref name=":4" /> compares the [[Quantum efficiency|external quantum efficiency]] (EQE) of a metamorphic InGaAs bulk subcell on Ge substrates by Spectrolab<ref>King, R., Law, D., Fetzer, C., Sherif, R., Edmondson, K., Kurtz, S., ... & Karam, N. H. (2005, June). Pathways to 40%-efficient concentrator photovoltaics. In ''Proc. 20th European Photovoltaic Solar Energy Conference'' (pp. 10-11).</ref> to a 100-period In<sub>0.30</sub>Ga<sub>0.70</sub>As(3.5 nm)/GaAs(2.7 nm)/ GaAs<sub>0.60</sub>P<sub>0.40</sub>(3.0 nm) QWSC by Fuji et al.<ref>{{cite journal|doi=10.1002/pip.2454|title=100-period, 1.23-eV bandgap InGaAs/GaAsP quantum wells for high-efficiency GaAs solar cells: Toward current-matched Ge-based tandem cells|journal=Progress in Photovoltaics: Research and Applications|volume=22|issue=7|pages=784–795|year=2014|last1=Fujii|first1=Hiromasa|last2=Toprasertpong|first2=Kasidit|last3=Wang|first3=Yunpeng|last4=Watanabe|first4=Kentaroh|last5=Sugiyama|first5=Masakazu|last6=Nakano|first6=Yoshiaki|s2cid=97467649 }}</ref> The bulk material shows higher EQE values than those of QWs in the 880-900 nm region, whereas the QWs have higher EQE values in the 400-600 nm range.<ref name=":4" /> This result provides some evidence that there is a struggle of extending the QWs' absorption thresholds to longer wavelengths due to strain balance and carrier transport issues. However, the bulk material has more deformations leading to low minority carrier lifetimes.<ref name=":4" />
In the 1.6-1.8 eV range, the lattice-matched AlGaAs by Heckelman et al.<ref>{{cite journal|doi=10.1109/jphotov.2014.2367869|title=Investigations on Al<sub>x</sub>Ga<sub>1-x</sub>As Solar Cells Grown by MOVPE|journal=IEEE Journal of Photovoltaics|volume=5|issue=1|pages=446–453|year=2015|last1=Heckelmann|first1=Stefan|last2=Lackner|first2=David|last3=Karcher|first3=Christian|last4=Dimroth|first4=Frank|last5=Bett|first5=Andreas W.|s2cid=41026351|url=https://publica.fraunhofer.de/handle/publica/239680 }}</ref> and InGaAsP by Jain et al.<ref>{{cite journal|doi=10.1109/jphotov.2017.2655035|title=Enhanced Current Collection in 1.7 eV GaInAsP Solar Cells Grown on GaAs by Metalorganic Vapor Phase Epitaxy|journal=IEEE Journal of Photovoltaics|volume=7|issue=3|pages=927–933|year=2017|last1=Jain|first1=Nikhil|last2=Geisz|first2=John F.|last3=France|first3=Ryan M.|last4=Norman|first4=Andrew G.|last5=Steiner|first5=Myles A.|osti=1360894|s2cid=20841656|doi-access=free}}</ref> are compared by Sayed<ref name=":4" /> with the lattice-matched InGaAsP/InGaP QW structure by Sayed et al.<ref>{{cite journal|doi=10.1063/1.4993888|title=100-period InGaAsP/InGaP superlattice solar cell with sub-bandgap quantum efficiency approaching 80%|journal=Applied Physics Letters|volume=111|issue=8|pages=082107|year=2017|last1=Sayed|first1=Islam E. H.|last2=Jain|first2=Nikhil|last3=Steiner|first3=Myles A.|last4=Geisz|first4=John F.|last5=Bedair|first5=S. M.|bibcode=2017ApPhL.111h2107S|osti=1393377}}</ref> Like the 1.1-1.3eV range, the EQE of the bulk material is higher in the longer wavelength region of the spectrum, but QWs are advantageous in the sense that they absorb a broader region in the spectrum. Furthermore, they can be grown in lower temperatures preventing thermal degradation.<ref name=":4" />
The application of quantum wells in many devices is a viable solution to increasing the [[Efficient energy use|energy efficiency]] of such devices. With lasers, the improvement has already lead to significant results like the LED. With QWSCs harvesting energy from the sun become a more potent method of cultivating energy by being able to absorb more of the sun's radiation and by being able to capture such energy from the charge carriers more efficiently. A viable option such as QWSCs provides the public with an opportunity to move away from greenhouse gas inducing methods to a greener alternative, solar energy.
==See also==
*[[Modulating retro-reflector]]▼
*[[Particle in a box]]▼
*[[Quantum wire]], carriers confined in two dimensions.▼
*[[Quantum dot]], carriers confined in all three dimensions.
*[[Quantum well laser]]
▲*[[Quantum wire]], carriers confined in two dimensions.
▲*[[Modulating retro-reflector]]
▲*[[Particle in a box]]
*[[Finite potential well]]
==References==
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{{Quantum mechanics topics}}
{{Authority control}}
{{DEFAULTSORT:Quantum Well}}
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