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	<updated>2026-09-21T10:04:33Z</updated>
	<subtitle>User contributions</subtitle>
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	<entry>
		<id>https://en.formulasearchengine.com/w/index.php?title=Trommel_screen&amp;diff=292404</id>
		<title>Trommel screen</title>
		<link rel="alternate" type="text/html" href="https://en.formulasearchengine.com/w/index.php?title=Trommel_screen&amp;diff=292404"/>
		<updated>2014-12-19T02:16:17Z</updated>

		<summary type="html">&lt;p&gt;67.166.91.49: /* Grizzly screen */&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;Before you determine whether stainless-steel cookware is price buying, lets first discuss what stainless steel cookware is. Stainless steel is made from an alloy, or a combination of metals. Mostly, primary iron with chromium, nickel or some other minor metals. The chromium provides rust protection and provides your cookware sturdiness. The nickel provides rust protection as well, and adds a sophisticated look. Most effectively made chrome steel cookware has copper or aluminum added to the underside of the pan or pot. That is completed to will increase the power of the pot or pan to conduct warmth.&amp;lt;br&amp;gt;The very best stainless steel cookware is the main class, but still it&#039;s divided into several subcategories based mostly on the quality and the price range. It can be complicated to choose the very best stainless steel cookware out of the classes that will meet your necessities. This is where we took a step ahead to clarify you all the data that will likely be useful for you to understand how to choose the very best stainless-steel cookware. One of the best stainless steel cookware set is manufactured from low-cost to costly and quality constructed pots and pans. &amp;lt;br&amp;gt;You will see that magnetic stainless-steel in the layer on the skin of some high quality items of stainless steel. This is to make it suitable with induction stovetops, which contain the usage of a rapidly charging electromagnetic field to warmth cookware. High-quality stainless steel, like All-Clad , makes use of three layers of steel—the austenite layer of metal on the inside, ferrite metal on the surface, and a layer of aluminum sandwiched between the 2 for optimum warmth conductivity (metal alone does not conduct warmth evenly). Lesser-high quality stainless steel is normally just one layer of austenitic stainless steel.&amp;lt;br&amp;gt;Aesthetically talking, stainless steel is a clever alternative for those who desire to display or hold pots or pans. The clear, crisp look of all stainless-steel kitchenware can transform a mishmash of cookware into a complicated décor assertion. Stainless-steel kettles, such because the Cuisinart Tea Kettle will mix individual kitchenware right into a cohesive and nice entity. Contemplate purchasing stainless steel utensils as properly. Already acquired a gorgeous chrome steel cookware collection? The Cuisinart Chef’s Assortment stainless pot rack is perhaps the of entirety for a kitchen, freeing up house and making these pots and pans readily accessible. Get the chrome steel cookware of your culinary goals at Macy’s!&amp;lt;br&amp;gt;Onerous-anodized aluminum cookware is likely one of the most popular forms of material, even though many people do not fairly understand the development. Arduous-anodized aluminum is obvious aluminum that has been processed in a sequence of chemical baths charged with an electrical current. The result&#039;s a cloth that has the identical superior warmth conductivity as aluminum but is non-reactive with acidic meals, reminiscent of tomatoes, and twice as hard as chrome steel. Two drawbacks to arduous-anodized cookware are that it&#039;s not dishwasher-protected and, as a result of it isn&#039;t magnetic, it will not work with induction range tops.&amp;lt;br&amp;gt;The enamel over metal approach creates a bit that has the warmth distribution of carbon metal and a non-reactive, low-stick floor.  When you have any concerns relating to where by and tips on how to employ [http://cookwarehq.drupalgardens.com/best-stainless-steel-cookware-top-reviews-2014 stainless steel cookware reviews], it is possible to contact us from the web site. Such pots are much lighter than most other pots of similar measurement, are cheaper to make than stainless steel pots, and don&#039;t have the rust and reactivity problems with forged iron or carbon metal.  quotation wanted  Enamel over metal is good for large stockpots and for different giant pans used largely for water-based cooking. Because of its mild weight and simple cleanup, enamel over metal is also in style for cookware used while camping. Clad aluminium or copper  edit&amp;lt;br&amp;gt;Unique specialty cookware items served a la carte to go with any cookware set are constructed of a sturdy Stainless Steel with a brushed exterior end. Designed with an impact bonded, aluminum disk encapsulated base which distributes warmth shortly and evenly to permit exact temperature control. Handles are riveted for durability and performance. The New Specialty Cookware is appropriate for all vary sorts together with induction. Along with the multi use perform, one other distinctive characteristic is bottom to high interior volume markings in both quarts and metric measurement; and each piece comes with a tempered glass lid, oven protected to 350°F.&amp;lt;br&amp;gt;Whether you&#039;re a cooking fans, a professional chef or just cooking for your family you already know the importance of getting a completely stocked kitchen. Not only do you need the best elements, however you additionally want the proper instruments to get the job performed. In any type of basic cooking coaching lesson, you&#039;ll be taught that chrome steel is your new finest good friend with regards to kitchen cookware. What you will also study is that quality cooking tools doesn&#039;t normally come at a discounted price. For that reason, it is very important take excellent care of your cookware! Listed here are some basics for stainless-steel care. &amp;lt;br&amp;gt;To fight the uneven heating problem, most chrome steel pans are laminations of aluminum or copper on the underside to unfold the warmth round, and stainless-steel contained in the pan to provide a cooking surface that is impervious to whatever you would possibly put inside. In my experience, this stainless-steel surface is still too sticky to fry on, and when you ever burn it you get a everlasting hassle spot. However, typically a stainless steel cooking floor is useful when you may&#039;t use aluminum (see under) so I hold some around. Select something with a reasonably thick aluminum layer on the underside.&amp;lt;br&amp;gt;Well, until you’re a metals professional and go examine the manufacturing facility the place the metal is made to see whether or not their manufacturing process creates a pure austenite without corrosive supplies shaped, you’re not going to know for positive whether or not the craftsmanship of your stainless is of the best quality. I believe your finest wager is to simply buy high-high quality stainless-steel from the start, from a brand with a reputation for good quality. But, I think I have found out a technique that you can determine if the stainless cookware you already have is potentially reactive.&lt;/div&gt;</summary>
		<author><name>67.166.91.49</name></author>
	</entry>
	<entry>
		<id>https://en.formulasearchengine.com/w/index.php?title=User:Daablced/sandbox&amp;diff=310907</id>
		<title>User:Daablced/sandbox</title>
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		<updated>2013-12-08T22:49:08Z</updated>

		<summary type="html">&lt;p&gt;67.166.147.139: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;Singapore has increased a tax on international property consumers as part of new non permanent measures to cool its residential housing market which has seen continued robust demand despite earlier efforts to curb costs.&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;c)(i) SC who already own # one residential property must pay ABSD of 7% on the purchase or acquisition of the second residential property. Asian buyers seemed to be on the forefront, with the biggest chunk comprising investors from Singapore, whose demand has overshot that of consumers from Hong Kong. This increase has been documented by Jones Lang LaSalle, a leading consultant in the actual property sector. The housing loan market in Singapore has evidently been rosy, these previous few years. A mixture of recent ambiance and an expensive aura of nature, Haus @ Serangoon Gardens is the first in Singapore to obtain the Green Mark Platinum award. The Santorini @ Tampiness Condominium by MCC Land 29 March The Glades @ Tanah Merah open for sale Bedok Reservoir Condominium @ Bedok&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;Rates of interest will stay low though a rise is expected when it is certain that the American economy is on the highway to recovery. As it&#039;s, some banks have upped their housing loan charges by 0.1 per cent currently. Regardless of the rise, the general charges are nonetheless low. This will remain so long as the rise in rates of interest is gradual over the following one to 2 years, bearing any unexpected circumstances. In view of the expected rise in interest rates, debtors may want to lock within the decrease rate now, for instance, within the type of a hard and fast charge for the subsequent three years or on the present rates pegged to the Singapore Interbank Supplied Rate (SIBOR).&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;We assist shoppers in figuring out websites, locations and suggestion of development combine and sizes. We also assist clients establish particular market niches for the mission and careful evaluation of market conditions; recommend the most effective solution for his or her property needs. Most importantly, we specialised in promoting many successful Huttons Initiatives in Singapore. I&#039;m very impressed by the speed and effectivity of Lille within the sale of my unit at Forest Hills Apartment. I known as her on Friday and she or he introduced a buyer to view my unit on Saturday. City Development and Ho Bee Land&#039;s pursuit of opportunities overseas is way from being remoted. In fact, the trend&#039;s clear – builders in Singapore are more and more wanting toward foreign shores. Safety of property rights.&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;In its assertion, the Singapore government claimed that most citizens shopping for their first residence is not going to be harm by the brand new measures. Some concessions will also be prolonged to chose teams of buyers, such as married couples with at the least one Singaporean spouse who are buying their second property as long as they intend to sell their first residential property. Lower the LTV restrict on housing loans granted by monetary institutions regulated by MAS from 70% to 60% for [http://www.sonomasoroptimist.org/?q=node/349425 Property Agent In Singapore] purchasers who&#039;re individuals with a number of outstanding housing loans at the time of the new housing purchase. Singapore Property Measures - 30 August 2010 The most well-liked seek for the variety of bedrooms in Singapore is 4, adopted by 2 and three. Lush Acres EC @ Sengkang&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;Even if Congress cannot extend the mortgage aid exemption this 12 months, many notice that a deal in 2013 would nonetheless be doable and could apply retroactively. However the heightened uncertainty would deter some underwater householders from promoting their properties, and their unfavorable fairness would proceed to pull on the housing market, Goold warned. Rents have been freely negotiated for the reason that passage of Management of Rent (Abolition) Act 2001. Subsidized rents are provided to poor Singaporeans by the Housing and Growth Board (HDB). Deposits Changi / Pasir Ris (D17-18) Greenview Crescent- Model New Semi Indifferent House @6.5mil neg Freehold cluster landed nestled within exclusive Seletar Hills! Oxley Tower is a 32-storey industrial growth located in Robinson Road&amp;lt;br&amp;gt;&amp;lt;br&amp;gt;BuyRentSingaporeProperty.com is dedicated in providing the latest residential launches, Developer&#039;s gross sales, new launches, resale, PRIME initiatives and rental of properties in Singapore. Then there are new launch property condos in the mid-tier range. These are typically within the neighborhood of the town fringe, in areas equivalent to Newton and Novena Tanjong Rhu. These new launch initiatives are well-liked because of their proximity to the city center, the place the business and leisure centers and all the facilities they offer, but they&#039;re normally inexpensive than the City core area projects. Why submit cheques to register for brand new property launches in Singapore? Riversails Rental @ Upper Serangoon LakeVille @ Jurong Lakeside Apartment by MCL Ferra Rental @ Leonie Hill&lt;/div&gt;</summary>
		<author><name>67.166.147.139</name></author>
	</entry>
	<entry>
		<id>https://en.formulasearchengine.com/w/index.php?title=Thiele_modulus&amp;diff=27831</id>
		<title>Thiele modulus</title>
		<link rel="alternate" type="text/html" href="https://en.formulasearchengine.com/w/index.php?title=Thiele_modulus&amp;diff=27831"/>
		<updated>2013-05-26T02:49:32Z</updated>

		<summary type="html">&lt;p&gt;67.166.145.23: &lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;{{Orphan|date=October 2012}}&lt;br /&gt;
&lt;br /&gt;
The &#039;&#039;&#039;molecular-scale temperature&#039;&#039;&#039; is the defining property of the [[U.S. Standard Atmosphere#1962 version|U.S. Standard Atmosphere, 1962]]. It is  defined by the relationship:&lt;br /&gt;
&lt;br /&gt;
: &amp;lt;math&amp;gt;T_m(z)=\frac{M_0}{M(z)}\cdot{T(z)}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
: &#039;&#039;T&amp;lt;sub&amp;gt;m&amp;lt;/sub&amp;gt;(z)&#039;&#039; is molecular-scale temperature at [[altitude]] &#039;&#039;z&#039;&#039;;&lt;br /&gt;
: &#039;&#039;M&amp;lt;sub&amp;gt;0&amp;lt;/sub&amp;gt;&#039;&#039; is [[molecular mass|molecular weight]] of air at [[sea level]];&lt;br /&gt;
: &#039;&#039;M(z)&#039;&#039; is molecular weight of air at altitude &#039;&#039;z&#039;&#039;;&lt;br /&gt;
: &#039;&#039;T(z)&#039;&#039; is [[Thermodynamic temperature|absolute temperature]] at altitude &#039;&#039;z&#039;&#039;.&lt;br /&gt;
&lt;br /&gt;
This is citation of the Technical Report of [[United States Air Force|USAF]] from 1967.&lt;br /&gt;
&amp;lt;ref&amp;gt;{{cite book&lt;br /&gt;
| last          = Brackbill&lt;br /&gt;
| first         = Jeremiah U.&lt;br /&gt;
| first2        = Edmund A.&lt;br /&gt;
| last2         = Nawrocki&lt;br /&gt;
| first3        = William A.&lt;br /&gt;
| last3         = Whitaker&lt;br /&gt;
| title         = Atmospheric models for use in hydrodynamic computer codes&lt;br /&gt;
| url           = http://www.dtic.mil/cgi-bin/GetTRDoc?AD=AD0818407&lt;br /&gt;
| format        = pdf&lt;br /&gt;
| accessdate    = 2012-05-30&lt;br /&gt;
| series        = AFWL-TR-67-75&lt;br /&gt;
|date=August 1967&lt;br /&gt;
| publisher     = Air Force Weapons Laboratory. Research And Technology Division. Air Force Systems Command&lt;br /&gt;
| location      = [[Kirtland Air Force Base]], [[New Mexico]] (USA)&lt;br /&gt;
}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
==References==&lt;br /&gt;
{{reflist}}&lt;br /&gt;
&lt;br /&gt;
[[Category:Atmosphere| ]]&lt;br /&gt;
[[Category:Temperature]]&lt;br /&gt;
&lt;br /&gt;
{{climate-stub}}&lt;/div&gt;</summary>
		<author><name>67.166.145.23</name></author>
	</entry>
	<entry>
		<id>https://en.formulasearchengine.com/w/index.php?title=Heaviside_cover-up_method&amp;diff=21895</id>
		<title>Heaviside cover-up method</title>
		<link rel="alternate" type="text/html" href="https://en.formulasearchengine.com/w/index.php?title=Heaviside_cover-up_method&amp;diff=21895"/>
		<updated>2013-01-02T19:47:56Z</updated>

		<summary type="html">&lt;p&gt;67.166.28.236: /* Method */&lt;/p&gt;
&lt;hr /&gt;
&lt;div&gt;&#039;&#039;&#039;Multi-junction solar cells&#039;&#039;&#039; are [[solar cell]]s with multiple p-n junctions made of different semiconductor materials.  Each material&#039;s p-n junction will produce electric current in response to a different [[Electromagnetic spectrum|wavelength of light]].   A multi-junction cell solar cell will produce electric current at multiple wavelengths of light, increasing the conversion efficiency of the no-cost solar light power to usable electric power.&lt;br /&gt;
&lt;br /&gt;
Traditional single-junction cells have a maximum theoretical efficiency of 34%.  A theoretical infinite number of junctions, multi-junction cell efficiency would be 87% under highly concentrated sunlight.&lt;br /&gt;
&lt;br /&gt;
Currently, the best lab examples of traditional silicon solar cells have efficiencies around 25%,&amp;lt;ref&amp;gt;{{cite web|url=http://www.nsinnovations.com.au/news/solar_cell_record.html |title=New South Innovations News - UNSW breaks solar cell record |publisher=NewSouth Innovations |date=2008-11-18 |accessdate=2012-06-23}}&amp;lt;/ref&amp;gt; while lab examples of multi-junction cells have demonstrated performance over 43%.&amp;lt;ref&amp;gt;[http://news.cnet.com/8301-11128_3-20053851-54.html &amp;quot;Solar Junction Breaks Concentrated Solar World Record with 43.5% Efficiency&amp;quot;]. Cnet.com.&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;[http://cleantechnica.com/2012/05/31/sharp-hits-concentrator-solar-cell-efficiency-record-43-5/ &amp;quot;Sharp Hits Concentrator Solar Cell Efficiency Record, 43.5%&amp;quot;]&amp;lt;/ref&amp;gt; Commercial examples of tandem, two layer, cells are widely available at 30% under one-sun illumination,&amp;lt;ref&amp;gt;[http://www.emcore.com/assets/photovoltaics/ZTJ_datasheet.pdf &amp;quot;ZTJ Space Solar Cell&amp;quot;], emcore&amp;lt;/ref&amp;gt; and improve to around 40% under concentrated sunlight. However, this efficiency is gained at the cost of increased complexity and manufacturing price. To date, their higher price and higher [[price-to-performance ratio]] have limited their use to special roles, notably in [[aerospace]] where their high [[power-to-weight ratio]] is desirable. In terrestrial applications these solar cells have been suggested for use in [[concentrated photovoltaics]] (CPV),&amp;lt;ref&amp;gt;[http://www.nrel.gov/csp/concentrating_pv.html &amp;quot;Concentrating Photovoltaic Technology&amp;quot;], NREL&amp;lt;/ref&amp;gt; with numerous small test sites around the world.&amp;lt;ref&amp;gt;[http://qualenergia.it/sites/default/files/articolo-doc/Mappa%20fotovoltaico%20a%20concentrazione%20nel%20mondo.pdf CPV World Map 2011]&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Tandem fabrication techniques are used to improve the performance of existing many designs. In particular, the technique can be applied to lower cost [[thin-film solar cell]]s using [[amorphous silicon]], as opposed to mono-crystalline silicon, to produce a cell with about 10% efficiency that is lightweight and flexible. This approach has been used by several commercial vendors,&amp;lt;ref&amp;gt;[http://www.uni-solar.com/wp-content/uploads/pdf/Energy_Production_EN.pdf &amp;quot;Uni-Solar Energy Production&amp;quot;], Uni-Solar&amp;lt;/ref&amp;gt; but these products are currently limited to certain niche roles, like roofing materials.&lt;br /&gt;
&lt;br /&gt;
==Description==&lt;br /&gt;
&lt;br /&gt;
===Basics of solar cells===&lt;br /&gt;
[[File:Effetphotovoltaic.jpg|right|thumb|325px|Figure A. [[Band diagram]] illustration of the [[photovoltaic effect]]. [[Photon]]s give their energy to electrons in the depletion or quasi-neutral regions. These move from [[conduction band]] to [[valence band]]. Depending on the location, [[electron]]s and [[Electron hole|holes]] are accelerated by &#039;&#039;E&amp;lt;sub&amp;gt;drift&amp;lt;/sub&amp;gt;&#039;&#039;, which gives generation [[photocurrent]], or by E&amp;lt;sub&amp;gt;scatt&amp;lt;/sub&amp;gt;, which gives scattering photocurrent.&amp;lt;ref name=a7&amp;gt;R.Delamare, O.Bulteel, D.Flandre, Conversion lumière/électricité: notions fondamentales et exemples de recherche&amp;lt;/ref&amp;gt;]]&lt;br /&gt;
&lt;br /&gt;
Traditional photovoltaic cells are commonly composed by [[doping (semiconductor)|doped]] [[silicon]] and then depositing metallic contacts on the top and bottom. The doping is normally applied to a thin layer on the top of the cell, producing a [[pn-junction]] with a particular [[bandgap]] energy, E&amp;lt;sub&amp;gt;g&amp;lt;/sub&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
[[Photon]]s that hit the top of the solar cell are either reflected or transmitted into the cell. Transmitted photons have the potential to give their energy &#039;&#039;hν&#039;&#039; to an [[electron]] if &#039;&#039;hν&#039;&#039; ≥ E&amp;lt;sub&amp;gt;g&amp;lt;/sub&amp;gt;, generating an electron-[[electron hole|hole]] pair.&amp;lt;ref&amp;gt;[http://books.google.com/books?id=_Xs6QgAACAAJ &amp;quot;Basic Photovoltaic Principles and Methods&amp;quot;], Technical Information Office, Solar Energy Research Institute (1982)&amp;lt;/ref&amp;gt; In the depletion region, the drift electric field &#039;&#039;E&amp;lt;sub&amp;gt;drift&amp;lt;/sub&amp;gt;&#039;&#039; accelerates both electrons and holes towards their respective n-doped and p-doped regions (up and down, respectively). The resulting [[electric current|current]] &#039;&#039;I&amp;lt;sub&amp;gt;g&amp;lt;/sub&amp;gt;&#039;&#039; is called the generation [[photocurrent]]. In the quasi-neutral region, the scattering electric field E&amp;lt;sub&amp;gt;scatt&amp;lt;/sub&amp;gt; accelerates holes (electrons) towards the p-doped (n-doped) region, which gives a scattering photocurrent &#039;&#039;I&amp;lt;sub&amp;gt;pscatt&amp;lt;/sub&amp;gt;&#039;&#039; (&#039;&#039;I&amp;lt;sub&amp;gt;nscatt&amp;lt;/sub&amp;gt;&#039;&#039;). Consequently, due to the accumulation of [[electric charge|charge]]s, a potential &#039;&#039;V&#039;&#039; and a photocurrent &#039;&#039;I&amp;lt;sub&amp;gt;ph&amp;lt;/sub&amp;gt;&#039;&#039; appear. The expression for this photocurrent is obtained by adding generation and scattering photocurrents: &#039;&#039;I&amp;lt;sub&amp;gt;ph&amp;lt;/sub&amp;gt; = I&amp;lt;sub&amp;gt;g&amp;lt;/sub&amp;gt; + I&amp;lt;sub&amp;gt;nscatt&amp;lt;/sub&amp;gt; + I&amp;lt;sub&amp;gt;pscatt&amp;lt;/sub&amp;gt;&#039;&#039;.&lt;br /&gt;
&lt;br /&gt;
The &#039;&#039;J-V&#039;&#039; characteristics (J is current density, i.e. current per unit area) of a solar cell under illumination are obtained by shifting the &#039;&#039;J-V&#039;&#039; characteristics of a [[diode]] in the dark downward by &#039;&#039;I&amp;lt;sub&amp;gt;ph&amp;lt;/sub&amp;gt;&#039;&#039;. Since solar cells are designed to supply power and not absorb it, the power &#039;&#039;P = V·I&amp;lt;sub&amp;gt;ph&amp;lt;/sub&amp;gt;&#039;&#039; must be negative. Hence, the operating point &#039;&#039;(V&amp;lt;sub&amp;gt;m&amp;lt;/sub&amp;gt;, J&amp;lt;sub&amp;gt;m&amp;lt;/sub&amp;gt;)&#039;&#039; is located in the region where &#039;&#039;V&#039;&#039;&amp;gt;0 and &#039;&#039;I&amp;lt;sub&amp;gt;ph&amp;lt;/sub&amp;gt;&#039;&#039;&amp;lt;0, and chosen to maximize the [[absolute value]] of the power |&#039;&#039;P&#039;&#039;|.&amp;lt;ref name=a9&amp;gt;{{cite book|url=http://sunlab.site.uottawa.ca/pdf/whitepapers/HiEfficMjSc-CurrStatus&amp;amp;FuturePotential.pdf|author=N.V.Yastrebova|title=High-efficiency multi-junction solar cells: current status and future potential|year=2007}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
===Loss mechanisms===&lt;br /&gt;
[[File:ShockleyQueisserFullCurve.svg|thumb|The [[Shockley-Queisser limit]] for the efficiency of a single-junction solar cell. It is essentially impossible for a single-junction solar cell, under unconcentrated sunlight, to have more than ~34% efficiency. A multijunction cell, however, can exceed that limit.]]&lt;br /&gt;
&lt;br /&gt;
The theoretical performance of a solar cell was first studied in depth in the 1960s, and is today known as the [[Shockley–Queisser limit]]. The limit describes several loss mechanisms that are inherent to any solar cell design.&lt;br /&gt;
&lt;br /&gt;
The first are the losses due to [[blackbody radiation]], a loss mechanism that affects any material object above [[absolute zero]]. In the case of solar cells at [[standard temperature and pressure]], this loss accounts for about 7% of the power. The second is an effect known as &amp;quot;recombination&amp;quot;, where the [[electron]]s created by the [[photoelectric effect]] meet the [[electron hole]]s left behind by previous excitations. In silicon, this accounts for another 10% of the power.&lt;br /&gt;
&lt;br /&gt;
However, the dominant loss mechanism is the inability for a solar cell to extract all of the power in the [[photon]], and the associated problem that it cannot extract any power at all from certain photons. This is due to the fact that the electrons must have enough energy to overcome the bandgap of the material.&lt;br /&gt;
&lt;br /&gt;
If the photon has less energy than the bandgap, it is not collected at all. This is a major consideration for conventional solar cells, which are not sensitive to most of the [[infrared]] spectrum, although that represents almost half of the power coming from the sun. Conversely, photons with more energy than the bandgap, say blue light, initially eject an electron to a state high above the bandgap, but this extra energy is lost through collisions in a process known as &amp;quot;relaxation&amp;quot;. This lost energy turns into heat in the cell, which has the side-effect of further increasing blackbody losses.&amp;lt;ref&amp;gt;{{cite book |title=Third Generation Photovoltaics|last=Green|first=M.A.|year=2003|publisher=Springer-Verlag|isbn=3-540-26562-7}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Combining all of these factors, the maximum efficiency for a single-bandgap material, like conventional silicon cells, is about 34%. That is, 66% of the energy in the sunlight hitting the cell will be lost. Practical concerns further reduce this, notably reflection off the front surface or the metal terminals, with modern high-quality cells at about 22%.&lt;br /&gt;
&lt;br /&gt;
Lower, also called narrower, bandgap materials will convert longer wavelength, lower energy photons.  Higher, or wider bandgap materials will convert shorter wavelength, higher energy light   An analysis of the [[AM1.5]] spectrum, shows the best balance is reached at about 1.1&amp;amp;nbsp;eV, in the near infrared, which happens to be very close to the natural bandgap in silicon and a number of other useful semiconductors.&lt;br /&gt;
&lt;br /&gt;
===Multi-junction cells===&lt;br /&gt;
&lt;br /&gt;
Cells made from multiple materials have multiple bandgaps. So, it will respond to multiple light wavelengths and some of the energy that would otherwise be lost to relaxation as described above, can be captured and converted.&lt;br /&gt;
&lt;br /&gt;
For instance, if one had a cell with two bandgaps in it, one tuned to red light and the other to green, then the extra energy in green, cyan and blue light would be lost only to the bandgap of the green-sensitive material, while the energy of the red, yellow and orange would be lost only to the bandgap of the red-sensitive material. Following analysis similar to those performed for single-bandgap devices, it can be demonstrated that the perfect bandgaps for a two-gap device are at 1.1 eV and 1.8 eV.&amp;lt;ref name=super&amp;gt;[http://www.superstrate.net/pv/limit/tandem.html &amp;quot;Tandem solar cells&amp;quot;]&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Conveniently, light of a particular wavelength does not interact strongly with materials that are not a multiple of that wavelength. This means that you can make a multijunction cell by layering the different materials on top of each other, shortest wavelengths on the &amp;quot;top&amp;quot; and increasing through the body of the cell. As the photons have to pass through the cell to reach the proper layer to be absorbed, [[Transparent conducting film|transparent conductors]] need to be used to collect the electrons being generated at each layer.&lt;br /&gt;
&lt;br /&gt;
[[File:StructureMJetspectre.png|thumb|550px|Figure C. (a) The structure of an MJ solar cell. There are six important types of layers: pn junctions, back surface field (BSF) layers, window layers, tunnel junctions, [[anti-reflective coating]] and metallic contacts. (b) Graph of spectral irradiance E vs. wavelength λ over the [[Air mass coefficient|AM]]1.5 solar spectrum, together with the maximum electricity conversion efficiency for every junction as a function of the wavelength.&amp;lt;ref name=a9 /&amp;gt;]]&lt;br /&gt;
&lt;br /&gt;
Producing a tandem cell is not an easy task, largely due to the thinness of the materials and the difficulties extracting the current between the layers. The easy solution is to use two mechanically separate [[thin film solar cell]]s and then wire them together separately outside the cell. This technique is widely used by [[amorphous silicon]] solar cells, [[Uni-Solar]]&#039;s products use three such layers to reach efficiencies around 9%. Lab examples using more exotic thin-film materials have demonstrated efficiencies over 30%.&amp;lt;ref name=super/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The more difficult solution is the &amp;quot;monolithically integrated&amp;quot; cell, where the cell consists of a number of layers that are mechanically and electrically connected. These cells are much more difficult to produce because the electrical characteristics of each layer has to be carefully matched. In particular, the photocurrent generated in each layer needs to be matched, otherwise electrons will be absorbed between layers. This limits their construction to certain materials, best met by the III-V semiconductors.&amp;lt;ref name=super/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
====Material Choice====&lt;br /&gt;
&lt;br /&gt;
The choice of materials for each sub-cell is determined by the requirements for lattice-matching, current-matching, and high performance opto-electronic properties.&lt;br /&gt;
&lt;br /&gt;
For optimal growth and resulting crystal quality, the crystal lattice constant &#039;&#039;a&#039;&#039; of each material must be closely matched, resulting in lattice-matched devices. This constraint has been relaxed somewhat in recently developed [[Solar cell research#Metamorphic multijunction solar cell|metamorphic solar cells]] which contain a small degree of lattice mismatch. However, a greater degree of mismatch or other growth imperfections can lead to crystal defects causing a degradation in electronic properties.&lt;br /&gt;
&lt;br /&gt;
Since each sub-cell is connected electrically in series, the same current flows through each junction. The materials are ordered with decreasing [[bandgap]]s, E&amp;lt;sub&amp;gt;g&amp;lt;/sub&amp;gt;, allowing sub-bandgap light (&#039;&#039;hc/λ &amp;lt; e·E&amp;lt;sub&amp;gt;g&amp;lt;/sub&amp;gt;&#039;&#039;) to transmit to the lower sub-cells. Therefore, suitable bandgaps must be chosen such that the design spectrum will balance the current generation in each of the sub-cells, achieving current matching. Figure C(b) plots [[spectral irradiance]] &#039;&#039;E&#039;&#039;(λ), which is the source power density at a given [[wavelength]] λ. It is plotted together with the maximum conversion efficiency for every junction as a function of the wavelength, which is directly related to the number of photons available for conversion into photocurrent.&lt;br /&gt;
&lt;br /&gt;
Finally, the layers must be electrically optimal for high performance. This necessitates usage of materials with strong absorption coefficients α(λ), high minority carrier lifetimes τ&amp;lt;sub&amp;gt;minority&amp;lt;/sub&amp;gt;, and high mobilities µ.&amp;lt;ref&amp;gt;{{cite journal|last1=Miles|first1=R|title=Photovoltaic solar cells: Choice of materials and production methods|journal=Vacuum|volume=80|issue=10|page=1090|year=2006|doi=10.1016/j.vacuum.2006.01.006 }}&amp;lt;/ref&amp;gt; &lt;br /&gt;
&lt;br /&gt;
The favorable values in the table below justify the choice of materials typically used for multi-junction solar cells: [[InGaP]] for the top sub-cell (E&amp;lt;sub&amp;gt;g&amp;lt;/sub&amp;gt; = 1.8 - 1.9 eV), [[InGaAs]] for the middle sub-cell (E&amp;lt;sub&amp;gt;g&amp;lt;/sub&amp;gt; = 1.4 eV), and [[Germanium]] for the bottom sub-cell (E&amp;lt;sub&amp;gt;g&amp;lt;/sub&amp;gt; = 0.67 eV). The use of Ge is mainly due to its lattice constant, robustness, low cost, abundance, and ease of production. &lt;br /&gt;
&lt;br /&gt;
Because the different layers are closely lattice-matched, the fabrication of the device typically employs [[metalorganic vapour phase epitaxy|metal-organic chemical vapor deposition]] (MOCVD). This technique is preferable to the [[molecular beam epitaxy]] (MBE) because it ensures high [[crystal]] quality and large scale production.&amp;lt;ref name=a9 /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
{| class=&amp;quot;wikitable&amp;quot;&lt;br /&gt;
|-&lt;br /&gt;
! Material&lt;br /&gt;
! E&amp;lt;sub&amp;gt;g&amp;lt;/sub&amp;gt;, eV&lt;br /&gt;
! &#039;&#039;a&#039;&#039;, nm&lt;br /&gt;
! absorption &amp;lt;br/&amp;gt;(λ = 0.8 μm), 1/µm&lt;br /&gt;
! µ&amp;lt;sub&amp;gt;n&amp;lt;/sub&amp;gt;, cm²/(V·s)&lt;br /&gt;
! τ&amp;lt;sub&amp;gt;p&amp;lt;/sub&amp;gt;, µs&lt;br /&gt;
! Hardness &amp;lt;br/&amp;gt;(Mohs)&lt;br /&gt;
! α, µm/K&lt;br /&gt;
! &#039;&#039;S&#039;&#039;, m/s&lt;br /&gt;
|-&lt;br /&gt;
| c-Si&lt;br /&gt;
| 1.12&lt;br /&gt;
| 0.5431&lt;br /&gt;
| 0.102&lt;br /&gt;
| 1400&lt;br /&gt;
| 1&lt;br /&gt;
| 7&lt;br /&gt;
| 2.6&lt;br /&gt;
| 0.1–60&lt;br /&gt;
|-&lt;br /&gt;
| InGaP&lt;br /&gt;
| 1.86&lt;br /&gt;
| 0.5451&lt;br /&gt;
| 2&lt;br /&gt;
| 500&lt;br /&gt;
| –&lt;br /&gt;
| 5&lt;br /&gt;
| 5.3&lt;br /&gt;
| 50&lt;br /&gt;
|-&lt;br /&gt;
| GaAs&lt;br /&gt;
| 1.4&lt;br /&gt;
| 0.5653&lt;br /&gt;
| 0.9&lt;br /&gt;
| 8500&lt;br /&gt;
| 3&lt;br /&gt;
| 4–5&lt;br /&gt;
| 6&lt;br /&gt;
| 50&lt;br /&gt;
|-&lt;br /&gt;
| Ge&lt;br /&gt;
| 0.65&lt;br /&gt;
| 0.5657&lt;br /&gt;
| 3&lt;br /&gt;
| 3900&lt;br /&gt;
| 1000&lt;br /&gt;
| 6&lt;br /&gt;
| 7&lt;br /&gt;
| 1000&lt;br /&gt;
|-&lt;br /&gt;
| InGaAs&lt;br /&gt;
| 1.2&lt;br /&gt;
| 0.5868&lt;br /&gt;
| 30&lt;br /&gt;
| 1200&lt;br /&gt;
| –&lt;br /&gt;
| –&lt;br /&gt;
| 5.66&lt;br /&gt;
| 100–1000&lt;br /&gt;
|}&lt;br /&gt;
&lt;br /&gt;
===Structural elements===&lt;br /&gt;
&lt;br /&gt;
====Metallic contacts====&lt;br /&gt;
The metallic contacts are low-resistivity [[electrodes]] that make contact with the semiconductor layers.  They are often [[aluminum]].  This provides an electrical connection to a load or other parts of a solar cell array.  They are usually on two sides of the cell. And are can be important on the back face of so that shadowing on the lighting surface is reduced.&lt;br /&gt;
&lt;br /&gt;
====Anti-reflective coating====&lt;br /&gt;
&lt;br /&gt;
[[Anti-reflective coating|Anti-reflective]] (AR) coating is generally composed of several layers in the case of MJ solar cells. The top AR layer has usually a [[NaOH]] surface texturation with several [[pyramid]]s in order to increase the transmission coefficient &#039;&#039;T&#039;&#039;, the trapping of the light in the material (because photons cannot easily get out the MJ structure due to pyramids) and therefore, the path length of photons in the material.&amp;lt;ref name=a7 /&amp;gt; On the one hand, the thickness of each AR layer is chosen to get destructive interferences. Therefore, the reflection coefficient &#039;&#039;R&#039;&#039; decreases to 1%. In the case of two AR layers &#039;&#039;L&amp;lt;sub&amp;gt;1&amp;lt;/sub&amp;gt;&#039;&#039; (the top layer, usually {{chem|SiO|2}}) and &#039;&#039;L&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt;&#039;&#039; (usually {{chem|TiO|2}}), there must be &amp;lt;math&amp;gt;n_{L2} = n_{AlInP}^{1/2} \cdot n_{L1}&amp;lt;/math&amp;gt; to have the same amplitudes for reflected fields and &#039;&#039;n&amp;lt;sub&amp;gt;L1&amp;lt;/sub&amp;gt;d&amp;lt;sub&amp;gt;L1&amp;lt;/sub&amp;gt;&#039;&#039; = 4λ&amp;lt;sub&amp;gt;min&amp;lt;/sub&amp;gt;,&#039;&#039;n&amp;lt;sub&amp;gt;L2&amp;lt;/sub&amp;gt;d&amp;lt;sub&amp;gt;L2&amp;lt;/sub&amp;gt;&#039;&#039; = λ&amp;lt;sub&amp;gt;min&amp;lt;/sub&amp;gt;/4 to have opposite phase for reflected fields.&amp;lt;ref&amp;gt;{{cite journal|last1=Strehlke|first1=S|last2=Bastide|first2=S|last3=Guillet|first3=J|last4=Levyclement|first4=C|title=Design of porous silicon antireflection coatings for silicon solar cells|journal=Materials Science and Engineering B|volume=69-70|page=81|year=2000|doi=10.1016/S0921-5107(99)00272-X }}&amp;lt;/ref&amp;gt; On the other hand, the thickness of each AR layer is also chosen to minimize the reflectance at wavelengths for which the photocurrent is the lowest. Consequently, this maximizes &#039;&#039;J&amp;lt;sub&amp;gt;SC&amp;lt;/sub&amp;gt;&#039;&#039; by matching currents of the three subcells.&amp;lt;ref&amp;gt;{{cite journal|doi=10.1002/1099-159X(200011/12)8:6&amp;lt;563::AID-PIP327&amp;gt;3.0.CO;2-8|author=Daniel J.Aiken|journal=Progress in Photovoltaics: Research and Applications|volume=8|title=Antireflection coating design for multi-junction, series interconnected solar cells|year=2000|issue=6|pages=563–570|url=http://photovoltaics.sandia.gov/docs/PDF/aikencell.pdf}}&amp;lt;/ref&amp;gt; As example, because the current generated by the bottom cell is greater than the currents generated by the other cells, the thickness of AR layers is adjusted so that the infrared (IR) transmission (which corresponds to the bottom cell) is degraded while the [[ultraviolet]] transmission (which corresponds to the top cell) is upgraded. Particularly, an AR coating is very important at low wavelengths because, without it, &#039;&#039;T&#039;&#039; would be strongly reduced to 70%.&lt;br /&gt;
&lt;br /&gt;
====Tunnel junctions====&lt;br /&gt;
[[File:Tunneljunction.jpg|thumb|300px|Figure D: Layers and [[band diagram]] of the tunnel junction. Because the length of the depletion region is narrow and the band gap is high, electrons can tunnel.]]&lt;br /&gt;
&lt;br /&gt;
The main goal of [[tunnel junction]]s is to provide a low [[electrical resistance]] and optically low-loss connection between two subcells.&amp;lt;ref name=a1&amp;gt;{{cite journal|last1=Yamaguchi|first1=M|last2=Takamoto|first2=T|last3=Araki|first3=K|title=Super high-efficiency multi-junction and concentrator solar cells|journal=Solar Energy Materials and Solar Cells|volume=90|issue=18–19|page=3068|year=2006|doi=10.1016/j.solmat.2006.06.028}}&amp;lt;/ref&amp;gt; Without it, the p-doped region of the top cell would be directly connected with the n-doped region of the middle cell. Hence, a pn junction with opposite direction to the others would appear between the top cell and the middle cell. Consequently, the [[surface photovoltage|photovoltage]] would be lower than if there would be no parasitic [[diode]]. In order to decrease this effect, a tunnel junction is used.&amp;lt;ref name=a2&amp;gt;J.F.Klem, S.Park, J.C.Zolper, Semiconductor tunnel junction with enhancement layer, {{US Patent|5679963}}(1997)&amp;lt;/ref&amp;gt; It is simply a wide band gap, highly doped diode. The high doping reduces the length of the depletion region because&lt;br /&gt;
:&amp;lt;math&amp;gt;l_{depl} = \sqrt{\frac{2 \epsilon (\phi_0 - V)}{q} \frac{N_A+N_D}{N_A N_D}}&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Hence, electrons can easily tunnel through the depletion region. The J-V characteristic of the tunnel junction is very important because it explains why tunnel junctions can be used to have a low electrical resistance connection between two pn junctions. Figure D shows three different regions: the tunneling region, the negative differential resistance region and the thermal diffusion region. The region where electrons can tunnel through the barrier is called the tunneling region. There, the voltage must be low enough so that energy of some electrons who are tunneling is equal to energy states available on the other side of the barrier. Consequently, current density through the tunnel junction is high (with maximum value of &amp;lt;math&amp;gt;J_P&amp;lt;/math&amp;gt;, the peak current density) and the slope near the origin is therefore steep. Then, the resistance is extremely low and consequently, the [[voltage]] too.&amp;lt;ref&amp;gt;{{cite journal|author=J.F.Wheeldon&#039;&#039;et al.&#039;&#039;|title=AlGaAs Tunnel Junction for high efficiency multi-junction solar cells: simulation and measurement of temperature-dependent operation|url=http://sunlab.site.uottawa.ca/pdf/conferences/PVSC2009_JWheeldon_AlGaAsTjHiEfficMjSc-Sim&amp;amp;MeasTDep.pdf|year=2009}}&amp;lt;/ref&amp;gt; This is why tunnel junctions are ideal for connecting two pn junctions without having a voltage drop. When voltage is higher, electrons cannot cross the barrier because energy states are no longer available for electrons. Therefore, the current density decreases and the differential resistance is negative. The last region, called thermal diffusion region, corresponds to the J-V characteristic of the usual diode:&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;J = J_S \left(\exp\left(\frac{qV}{kT}\right) - 1\right)&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
In order to avoid the reduction of the MJ solar cell performances, tunnel junctions must be transparent to wavelengths absorbed by the next photovoltaic cell, the middle cell, i.e. E&amp;lt;sub&amp;gt;gTunnel&amp;lt;/sub&amp;gt; &amp;gt; E&amp;lt;sub&amp;gt;gMiddleCell&amp;lt;/sub&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
====Window layer and back-surface field====&lt;br /&gt;
[[File:Bsfwindow.jpg|left|thumb|500px|Figure E: (a) Layers and [[band diagram]] of a window layer. The surface recombination is reduced. (b) Layers and band diagram of a BSF layer. The scattering of carriers is reduced.]]&lt;br /&gt;
&lt;br /&gt;
A window layer is used in order to reduce the surface recombination velocity &#039;&#039;S&#039;&#039;. Similarly, a back-surface field (BSF) layer reduces the scattering of carriers towards the tunnel junction. The structure of these two layers is the same: it is a [[heterojunction]] which catches electrons (holes). Indeed, despite the [[electric field]] &#039;&#039;E&amp;lt;sub&amp;gt;d&amp;lt;/sub&amp;gt;&#039;&#039;, these cannot jump above the barrier formed by the heterojunction because they don&#039;t have enough energy, as illustrated in figure E. Hence, electrons (holes) cannot recombine with holes (electrons) and cannot diffuse through the barrier. By the way, window and BSF layers must be transparent to wavelengths absorbed by the next pn junction i.e. E&amp;lt;sub&amp;gt;gWindow&amp;lt;/sub&amp;gt; &amp;gt; E&amp;lt;sub&amp;gt;gEmitter&amp;lt;/sub&amp;gt; and E&amp;lt;sub&amp;gt;gBSF&amp;lt;/sub&amp;gt; &amp;gt; E&amp;lt;sub&amp;gt;gEmitter&amp;lt;/sub&amp;gt;. Furthermore, the lattice constant must be close to the one of InGaP and the layer must be highly doped (&#039;&#039;n&#039;&#039; ≥ 10&amp;lt;sup&amp;gt;18&amp;lt;/sup&amp;gt; cm&amp;lt;sup&amp;gt;−3&amp;lt;/sup&amp;gt;).&amp;lt;ref name=a3&amp;gt;{{harvnb|Luque|Hegedus|2003|p=390 ff}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
===J-V characteristic===&lt;br /&gt;
&lt;br /&gt;
For maximum efficiency, each subcell should be operated at its optimal J-V parameters, which are not necessarily equal for each subcell. If they are different, the total current through the solar cell is the lowest of the three. By approximation,&amp;lt;ref&amp;gt;{{cite journal|last1=Peharz|first1=G.|last2=Siefer|first2=G.|last3=Bett|first3=A.W.|title=A simple method for quantifying spectral impacts on multi-junction solar cells|journal=Solar Energy|volume=83|issue=9|page=1588|year=2009|doi=10.1016/j.solener.2009.05.009 }}&amp;lt;/ref&amp;gt; it results in the same relationship for the short-circuit current of the MJ solar cell: &#039;&#039;J&amp;lt;sub&amp;gt;SC&amp;lt;/sub&amp;gt; = min (J&amp;lt;sub&amp;gt;SC1&amp;lt;/sub&amp;gt;, J&amp;lt;sub&amp;gt;SC2&amp;lt;/sub&amp;gt;, J&amp;lt;sub&amp;gt;SC3&amp;lt;/sub&amp;gt;)&#039;&#039; where &#039;&#039;J&amp;lt;sub&amp;gt;SCi&amp;lt;/sub&amp;gt;&#039;&#039;(λ) is the short-circuit current density at a given wavelength λ for the subcell &#039;&#039;i&#039;&#039;.&lt;br /&gt;
&lt;br /&gt;
Because of the impossibility to obtain &#039;&#039;J&amp;lt;sub&amp;gt;SC1&amp;lt;/sub&amp;gt;, J&amp;lt;sub&amp;gt;SC2&amp;lt;/sub&amp;gt;, J&amp;lt;sub&amp;gt;SC3&amp;lt;/sub&amp;gt;&#039;&#039; directly from the total J-V characteristic, the quantum efficiency &#039;&#039;QE&#039;&#039;(λ) is utilized. It measures the ratio between the amount of electron-hole pairs created and the incident photons at a given wavelength λ. Let φ&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;(λ) be the photon flux of corresponding incident light in subcell &#039;&#039;i&#039;&#039;and&#039;&#039;QE&#039;&#039;&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;(λ) be the quantum efficiency of the subcell &#039;&#039;i&#039;&#039;. By definition, this equates to:&amp;lt;ref&amp;gt;{{cite journal|last1=Liu|first1=Lei|last2=Chen|first2=Nuofu|last3=Bai|first3=Yiming|last4=Cui|first4=Ming|last5=Zhang|first5=Han|last6=Gao|first6=Fubao|last7=Yin|first7=Zhigang|last8=Zhang|first8=Xingwang|title=Quantum efficiency and temperature coefficients of GaInP/GaAs dual-junction solar cell|journal=Science China Technological Sciences|volume=52|issue=5|page=1176|year=2008|doi=10.1007/s11431-008-0203-9}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
:&amp;lt;math&amp;gt;QE_i(\lambda) = \frac{J_{SCi}(\lambda)}{q \phi_i(\lambda)} \Rightarrow J_{SCi} = \int_{0}^{\lambda2} q \phi_i(\lambda) QE_i(\lambda) \, d \lambda&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The value of &amp;lt;math&amp;gt;QE_i(\lambda)&amp;lt;/math&amp;gt; is obtained by linking it with the absorption coefficient &amp;lt;math&amp;gt;\alpha(\lambda)&amp;lt;/math&amp;gt;, i.e. the number of photons absorbed per unit of length by a material. If it is assumed that each photon absorbed by a subcell creates an electron/hole pair (which is a good approximation), this leads to:&amp;lt;ref name=a3 /&amp;gt; &lt;br /&gt;
:&amp;lt;math&amp;gt;QE_i(\lambda) = 1 - e^{-\alpha(\lambda) d_i}&amp;lt;/math&amp;gt; where &#039;&#039;d&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;&#039;&#039; is the thickness of the subcell &#039;&#039;i&#039;&#039; and &amp;lt;math&amp;gt;e^{-\alpha(\lambda) d_i}&amp;lt;/math&amp;gt; is the percentage of incident light which is not absorbed by the subcell &#039;&#039;i&#039;&#039;.&lt;br /&gt;
&lt;br /&gt;
Similarly, because&lt;br /&gt;
:&amp;lt;math&amp;gt;V = \sum_{i=1}^3 V_i&amp;lt;/math&amp;gt;, the following approximation can be used: &amp;lt;math&amp;gt;V_{OC} = \sum_{i=1}^3 V_{OCi}&amp;lt;/math&amp;gt;.&lt;br /&gt;
&lt;br /&gt;
The values of &amp;lt;math&amp;gt;V_{OCi}&amp;lt;/math&amp;gt; are then given by the J-V diode equation:&lt;br /&gt;
&lt;br /&gt;
:&amp;lt;math&amp;gt;J_i = J_{0i} \left(e^{\frac{qV_i}{kT}}-1\right)-J_{SCi} \Rightarrow V_{OCi} \approx \frac{kT}{q} \ln(\frac{J_{SCi}}{J_{0i}})&amp;lt;/math&amp;gt;&lt;br /&gt;
&lt;br /&gt;
== Materials ==&lt;br /&gt;
&lt;br /&gt;
The majority of multi-junction cells that have been produced to date use three layers. These cells require the use of semiconductors that can be tuned to specific frequencies, which has led to most of them being made of [[gallium arsenide]] (GaAs) compounds, often germanium for the bottom-, GaAs for the middle-, and GaInP&amp;lt;sub&amp;gt;2&amp;lt;/sub&amp;gt; for the top-cell.&lt;br /&gt;
&lt;br /&gt;
=== Gallium arsenide substrate ===&lt;br /&gt;
&lt;br /&gt;
Dual junction cells can be made on Gallium arsenide wafers.  Alloys of [[Indium gallium phosphide]] in the range In&amp;lt;sub&amp;gt;.5&amp;lt;/sub&amp;gt;Ga&amp;lt;sub&amp;gt;.5&amp;lt;/sub&amp;gt;P through In&amp;lt;sub&amp;gt;.53&amp;lt;/sub&amp;gt;Ga&amp;lt;sub&amp;gt;.47&amp;lt;/sub&amp;gt;P serve as the high band gap alloy.  This alloy range provides for the ability to have band gaps in the range of 1.92eV to 1.87eV.  The lower [[GaAs]] junction has a band gap of 1.42eV.{{Citation needed|date=March 2011}}&lt;br /&gt;
&lt;br /&gt;
=== Germanium substrate ===&lt;br /&gt;
&lt;br /&gt;
Triple junction cells consisting of [[Indium gallium phosphide]], [[Gallium arsenide]] or [[Indium gallium arsenide]] and [[Germanium]] can be fabricated on germanium wafers.  Early cells used straight gallium arsenide in the middle junction.  Later cells have utilized In&amp;lt;sub&amp;gt;0.015&amp;lt;/sub&amp;gt;Ga&amp;lt;sub&amp;gt;0.985&amp;lt;/sub&amp;gt;As, due to the better lattice match to Ge, resulting in a lower defect density.{{Citation needed|date=March 2011}}&lt;br /&gt;
&lt;br /&gt;
Due to the huge band gap difference between GaAs (1.42eV), and Ge (0.66eV), the current match is very poor, with the Ge junction operated significantly current limited.{{Citation needed|date=March 2011}}&lt;br /&gt;
&lt;br /&gt;
Current efficiencies for commercial InGaP/GaAs/Ge cells approach 40% under concentrated sunlight &amp;lt;ref&amp;gt;http://www.azurspace.com/index.php/en/products/products-cpv/cpv-solar-cells&amp;lt;/ref&amp;gt;&amp;lt;sup&amp;gt;,&amp;lt;/sup&amp;gt;.&amp;lt;ref&amp;gt;http://www.spectrolab.com/dataSheets.htm&amp;lt;/ref&amp;gt; Lab cells (partly using additional junctions between the GaAs and Ge junction) have demonstrated efficiencies above 40%.&amp;lt;ref&amp;gt;M.A. Green, K. Emery, Y. Hishikawa, W. Warta, E.D. Dunlop, Solar cell efficiency tables (version 40). Progress in Photovoltaics: Research and Applications, 2012. 20(5): pp. 606-14. DOI: [http://dx.doi.org/10.1002/pip.2267 10.1002/pip.2267]&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
=== Indium phosphide substrate ===&lt;br /&gt;
[[Indium phosphide]] may be used as a substrate to fabricate cells with band gaps between 1.35eV and 0.74eV.  Indium Phosphide has a band gap of 1.35eV.  [[Indium gallium arsenide]](In&amp;lt;sub&amp;gt;0.53&amp;lt;/sub&amp;gt;Ga&amp;lt;sub&amp;gt;0.47&amp;lt;/sub&amp;gt;As) is lattice matched to Indium Phosphide with a band gap of 0.74eV.  A quaternary alloy of Indium gallium arsenide phosphide can be lattice matched for any band gap in between the two.{{Citation needed|date=March 2011}}&lt;br /&gt;
&lt;br /&gt;
Indium phosphide-based cells have the potential to work in tandem with gallium arsenide cells. The two cells can be optically connected in series (with the InP cell below the GaAs cell), or in parallel through the use of spectra splitting using a [[Dichroic filter]].{{Citation needed|date=March 2011}}&lt;br /&gt;
&lt;br /&gt;
==Performance improvements==&lt;br /&gt;
&lt;br /&gt;
===Structure===&lt;br /&gt;
&lt;br /&gt;
All MJ photovoltaic cells use [[III-V semiconductor]] materials. GaAsSb-based heterojunction tunnel diodes, instead of conventional InGaP highly doped tunnel diodes described above, have a lower tunneling distance. Indeed, in the heterostructure formed by GaAsSb and InGaAs, the valence band of GaAsSb is higher than the valence band of the adjoining p-doped layer.&amp;lt;ref name=a2 /&amp;gt; Consequently, the tunneling distance &#039;&#039;d&amp;lt;sub&amp;gt;tunnel&amp;lt;/sub&amp;gt;&#039;&#039; is reduced and so the tunneling current, which exponentially depends of &#039;&#039;d&amp;lt;sub&amp;gt;tunnel&amp;lt;/sub&amp;gt;&#039;&#039;, is increased. Hence, the voltage is lower than that of the InGaP tunnel junction.&lt;br /&gt;
GaAsSb heterojunction tunnel diodes offer other advantages. The same current can be achieved by using a lower doping.&amp;lt;ref&amp;gt;{{cite journal|author=J.C. Zolper &#039;&#039;et al.&#039;&#039;|title=GaAsSb-based heterojunction tunnel diodes for tandem solar cell interconnects|doi=10.1109/WCPEC.1994.520724|year=1994|page=1843|volume=2|url=http://www.osti.gov/bridge/servlets/purl/10107906-3FIrKs/webviewable/}}&amp;lt;/ref&amp;gt; Secondly, because the lattice constant is larger for GaAsSb than Ge, one can use a wider range of materials for the bottom cell because more materials are lattice-matched to GaAsSb than to Ge.&amp;lt;ref name=a2/&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Chemical components can be added to some layers. Adding about one percent of [[Indium]] in each layer better matches lattice constants of the different layers.&amp;lt;ref name=a6&amp;gt;{{cite journal|last1=Yamaguchi|first1=M|last2=Takamoto|first2=T|last3=Araki|first3=K|last4=Ekinsdaukes|first4=N|title=Multi-junction III–V solar cells: current status and future potential|journal=Solar Energy|volume=79|issue=1|page=78|year=2005|doi=10.1016/j.solener.2004.09.018 }}&amp;lt;/ref&amp;gt; Without it, there is about 0.08 percent of mismatching between layers, which inhibits performance. Adding aluminium to the top cell increases its band gap to 1.96 eV,&amp;lt;ref name=a6 /&amp;gt; covering a larger part of the solar spectrum and obtain a higher open-circuit voltage &#039;&#039;V&amp;lt;sub&amp;gt;OC&amp;lt;/sub&amp;gt;&#039;&#039;.&lt;br /&gt;
&lt;br /&gt;
The theoretical efficiency of MJ solar cells is 86.8% for an infinite number of pn junctions,&amp;lt;ref name=a9/&amp;gt; implying that more junctions increase efficiency. The maximum theoretical efficiency is 37, 50, 56, 72% for 1, 2, 3, 36 pn junctions, respectively, with the number of junctions increasing exponentially to achieve equal effiency increments.&amp;lt;ref name=a3 /&amp;gt; The exponential relationship implies that as the cell approaches the limit of efficiency, the increase cost and complexity grow rapidly. Decreasing the thickness of the top cell increases the transmission coefficient &#039;&#039;T&#039;&#039;.&amp;lt;ref name=a3 /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Finally, an InGaP hetero-layer between the p-Ge layer and the InGaAs layer can be added in order to create automatically the n-Ge layer by scattering during MOCVD growth and increase significantly the quantum efficiency &#039;&#039;QE&#039;&#039;(λ) of the bottom cell.&amp;lt;ref name=a6 /&amp;gt; InGaP is advantageous because of its high scattering coefficient and low solubility in Ge.&lt;br /&gt;
&lt;br /&gt;
===Spectral variations===&lt;br /&gt;
&lt;br /&gt;
Solar spectrum at the Earth surface changes constantly depending on the weather and sun position. This results in the variation of φ(λ), &#039;&#039;QE&#039;&#039;(λ), α(λ) and thus the short-circuit currents &#039;&#039;J&amp;lt;sub&amp;gt;SCi&amp;lt;/sub&amp;gt;&#039;&#039;. As a result, the current densities &#039;&#039;J&amp;lt;sub&amp;gt;i&amp;lt;/sub&amp;gt;&#039;&#039; are not necessarily matched and the total current becomes lower. These variations can be quantified using the average photon energy (APE) which is the ratio between the spectral irradiance G(λ) (the power density of the light source in a specific wavelength λ) and the total photon flux density. It can be shown that a high (low) value for APE means low (high) wavelengths spectral conditions and higher (lower) efficiencies.&amp;lt;ref name=a4&amp;gt;{{cite journal|doi=10.1109/WCPEC.2003.1306273|title=Impact of spectral effects on the electrical parameters of multijunction amorphous silicon cells|url=http://hdl.handle.net/2134/8216}}&amp;lt;/ref&amp;gt; Thus APE is a good indicator for quantifying the effects of the solar spectrum variations on performances and has the added advantage of being independent of the device structure and the absorption profile of the device.&amp;lt;ref name=a4 /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
===Use of light concentrators===&lt;br /&gt;
&lt;br /&gt;
Light concentrators increase efficiencies and reduce the cost/efficiency ratio. The three types of light concentrators in use are refractive lenses like [[Fresnel lens]]es, reflective dishes (parabolic or cassegraine), and [[light guide optics]]. Thanks to these devices, light arriving on a large surface can be concentrated on a smaller cell. The intensity concentration ratio (or “suns”) is the average intensity of the focused light divided by 0.1 W/cm². If its value is &#039;&#039;X&#039;&#039; then the MJ current becomes &#039;&#039;X&#039;&#039; higher under concentrated illumination.&amp;lt;ref&amp;gt;{{harvnb|Luque|Hegedus|2003|pp= 61 ff}}&amp;lt;/ref&amp;gt;&amp;lt;ref name=a5&amp;gt;{{harvnb|Luque|Hegedus|2003|pp=449 ff}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Using concentrations on the order of 500 to 1000, meaning that a 1&amp;amp;nbsp;cm² cell can use the light collected from 0.1 m² (as 1 m² equal 10000 cm²), produces the highest efficiencies seen to date. Three-layer cells are fundamentally limited to 63%, but existing commercial prototypes have already demonstrated over 40%.&amp;lt;ref name=cnet&amp;gt;Michael Kanellos, [http://news.cnet.com/Solar-cell-breaks-efficiency-record/2100-11395_3-6141527.html &amp;quot;Solar cell breaks efficiency record&amp;quot;], &#039;&#039;CNET News&#039;&#039;, 6 December 2006&amp;lt;/ref&amp;gt;&amp;lt;ref&amp;gt;[http://www.nrel.gov/news/press/2008/625.html &amp;quot;NREL Solar Cell Sets World Efficiency Record at 40.8 Percent&amp;quot;], National Renewable Energy Laboratory, 13 August 2008&amp;lt;/ref&amp;gt; These cells capture about 2/3 of their theoretical maximum performance, so assuming the same is true for a non-concentrated version of the same design, one might expect a three-layer cell of 30% efficiency. This is not enough of an advantage over traditional silicon designs to make up for their extra production costs. For this reason, almost all multi-junction cell research for terrestrial use is dedicated to concentrator systems, normally using mirrors or fresnel lenses.&lt;br /&gt;
&lt;br /&gt;
Using a concentrator also has the added benefit that the number of cells needed to cover a given amount of ground area is greatly reduced. A conventional system covering 1 m² would require 625 16&amp;amp;nbsp;cm² cells, but for a concentrator system only a single cell is needed, along with a concentrator. The argument for concentrated Multi-junction cells has been that the high cost of the cells themselves would be more than offset by the reduction in total number of cells. However, the downside of the concentrator approach is that efficiency drops off very quickly under lower lighting conditions. In order to maximize its advantage over traditional cells and thus be cost competitive, the concentrator system has to track the sun as it moves to keep the light focused on the cell and maintain maximum efficiency as long as possible. This requires a [[solar tracker]] system, which increases yield, but also cost.&lt;br /&gt;
&lt;br /&gt;
==Fabrication==&lt;br /&gt;
&lt;br /&gt;
Multi-junction cells are expensive to produce, using techniques similar to [[semiconduction|semiconductor device fabrication]], usually [[metalorganic vapour phase epitaxy]] but on &amp;quot;chip&amp;quot; sizes on the order of centimeters. In cases where outright performance is the only consideration, these cells have become common, they are widely used in [[satellite]] applications for instance, where the [[power-to-weight ratio]] overwhelms practically every other cost.&lt;br /&gt;
&lt;br /&gt;
==Comparison with other technologies==&lt;br /&gt;
&lt;br /&gt;
There are four main categories of photovoltaic cells: c-Si solar cells, [[thin film solar cell]]s, MJ solar cells and new technologies (including organic solar cells).&lt;br /&gt;
&lt;br /&gt;
{| class=&amp;quot;wikitable&amp;quot;&lt;br /&gt;
|-&lt;br /&gt;
! Technology&lt;br /&gt;
! η (%)&lt;br /&gt;
! &#039;&#039;V&amp;lt;sub&amp;gt;OC&amp;lt;/sub&amp;gt;&#039;&#039; (V)&lt;br /&gt;
! &#039;&#039;I&amp;lt;sub&amp;gt;SC&amp;lt;/sub&amp;gt;&#039;&#039; (A)&lt;br /&gt;
! W/m²&lt;br /&gt;
! t (µm)&lt;br /&gt;
|-&lt;br /&gt;
| u c-Si&lt;br /&gt;
| 24.7&lt;br /&gt;
| 0.5&lt;br /&gt;
| 0.8&lt;br /&gt;
| 63&lt;br /&gt;
| 100&lt;br /&gt;
|-&lt;br /&gt;
| p c-Si&lt;br /&gt;
| 20.3&lt;br /&gt;
| 0.615&lt;br /&gt;
| 8.35&lt;br /&gt;
| 211&lt;br /&gt;
| 200&lt;br /&gt;
|-&lt;br /&gt;
| a-Si&lt;br /&gt;
| 11.1&lt;br /&gt;
| 6.3&lt;br /&gt;
| 0.0089&lt;br /&gt;
| 33&lt;br /&gt;
| 1&lt;br /&gt;
|-&lt;br /&gt;
| CdTe&lt;br /&gt;
| 16.5&lt;br /&gt;
| 0.86&lt;br /&gt;
| 0.029&lt;br /&gt;
| –&lt;br /&gt;
| 5&lt;br /&gt;
|-&lt;br /&gt;
| CIGS&lt;br /&gt;
| 19.5&lt;br /&gt;
| –&lt;br /&gt;
| –&lt;br /&gt;
| –&lt;br /&gt;
| 1&lt;br /&gt;
|-&lt;br /&gt;
| MJ&lt;br /&gt;
| 40.7&lt;br /&gt;
| 2.6&lt;br /&gt;
| 1.81&lt;br /&gt;
| 476&lt;br /&gt;
| 140&lt;br /&gt;
|}&lt;br /&gt;
&lt;br /&gt;
MJ solar cells and other photovoltaic devices have significant differences (see the table above). Physically, the main property of a MJ solar cell is having more than one pn junction in order to catch a larger photon energy spectrum while the main property of the [[thin film solar cell]] is to use thin films instead of thick layers in order to decrease the cost efficiency ratio. As of 2010, MJ solar panels are more expensive than others. These differences imply different applications: MJ solar cells are preferred in space and c-Si solar cells for terrestrian applications.&lt;br /&gt;
&lt;br /&gt;
[[File:Best Research-Cell Efficiencies.png|thumb|500px]]&lt;br /&gt;
The efficiencies of solar cells and Si solar technology are relatively stable, while the efficiency of solar modules and multi-junction technology are progressing.&lt;br /&gt;
&lt;br /&gt;
Measurements on MJ solar cells are usually made in laboratory, using light concentrators (this is often not the case for the other cells) and under standard test conditions (STCs). STCs prescribe, for terrestrial applications, the AM1.5 spectrum as the reference. This air mass (AM) corresponds to a fixed position of the sun in the sky of 48° and a fixed power of 833 W/m². Therefore, spectral variations of incident light and environmental parameters are not taken into account under STC.&amp;lt;ref&amp;gt;{{cite journal|last1=Albuflasa|first1=H|last2=Gottschalg|first2=R|last3=Betts|first3=T|title=Modeling the effect of varying spectra on multi junction A-SI solar cells|journal=Desalination|volume=209|issue=1–3|page=78|year=2007|doi=10.1016/j.desal.2007.04.012}}&amp;lt;/ref&amp;gt;&lt;br /&gt;
&lt;br /&gt;
Consequently, performance of MJ solar cells in terrestrial environment is inferior to that achieved in laboratory. Moreover, MJ solar cells are designed such that currents are matched under STC, but not necessarily under field conditions. One can use &#039;&#039;QE&#039;&#039;(λ) to compare performances of different technologies, but &#039;&#039;QE&#039;&#039;(λ) contains no information on the matching of currents of subcells. An important comparison point is rather the output power per unit area generated with the same incident light.&lt;br /&gt;
&lt;br /&gt;
==Applications==&lt;br /&gt;
&lt;br /&gt;
As of 2010, the cost of MJ solar cells was too high to allow use outside of specialized applications. The high cost is mainly due to the complex structure and the high price of materials. Nevertheless, with light concentrators under illumination of at least 400 suns, [[Concentrated photovoltaics|MJ solar panels]] become practical.&amp;lt;ref name=a3 /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
MJ cells are currently being utilized in the Mars rover missions.&amp;lt;ref&amp;gt;{{cite journal | doi = 10.1016/S0094-5765(02)00287-4| title = The performance of gallium arsenide/germanium solar cells at the Martian surface  | year = 2004 | author = D. Crisp, , a, A. Pathareb and R. C. Ewell | journal = Progress in Photovoltaics: Research and Applications | volume = 54 | issue = 2 | pages = 83–101 }}&amp;lt;/ref&amp;gt;  &amp;lt;!-- &amp;quot;The rovers have outlived their predicted life spans and functioned since for over two years.&amp;quot; Editors note: this is going to be out of date continuously. Please provide power-up date and allow users to calculate. --&amp;gt;&lt;br /&gt;
&lt;br /&gt;
The environment in space is quite different. Because there is no atmosphere, the solar spectrum is different (AM0). The cells have a poor current match due to a greater photon flux of photons above 1.87eV vs. those between 1.87eV and 1.42eV.  This results in too little current in the GaAs junction, and hampers the overall efficiency since the InGaP junction operates below MPP current and the GaAs junction operates above MPP current.  To improve current match, the InGaP layer is intentionally thinned to allow additional photons to penetrate to the lower GaAs layer.{{Citation needed|date=March 2011}}&lt;br /&gt;
&lt;br /&gt;
In terrestrial concentrating applications, the scatter of blue light by the atmosphere reduces the photon flux above 1.87eV, better balancing the junction currents.Radiation particles that are no longer filtered can cause damage the cell. There are two kinds of damage: [[ionisation]] and atomic displacement.&amp;lt;ref&amp;gt;{{harvnb|Luque|Hegedus|2003|pp=414 ff}}&amp;lt;/ref&amp;gt; Still, MJ cells offer higher radiation resistance, higher efficiency and a lower temperature coefficient.&amp;lt;ref name=a3 /&amp;gt;&lt;br /&gt;
&lt;br /&gt;
==References==&lt;br /&gt;
&lt;br /&gt;
{{reflist|2}}&lt;br /&gt;
&lt;br /&gt;
==Further reading==&lt;br /&gt;
&lt;br /&gt;
* {{cite book |editor1-last=Luque |editor1-first=Antonio |editor2-last=Hegedus |editor3-first=Steven&lt;br /&gt;
 |url=http://books.google.com/books?id=u-bCMhl_JjQC&amp;amp;pg=PT390 |title=Handbook of Photovoltaic Science and Engineering&lt;br /&gt;
 |publisher=[[John Wiley &amp;amp; Sons]] |year=2003 |isbn=0-471-49196-9 |ref=harv }}&lt;br /&gt;
* {{cite book |last=Yarris |first=Lynn |coauthors= |editor= |publication-date=7 Nov 2011&lt;br /&gt;
 |title=Berkeley Lab Research Sparks Record-Breaking Solar Cell Performance |work=News Center&lt;br /&gt;
 |publisher=[[Lawrence Berkeley National Laboratory]] |accessdate=10 Dec 2011&lt;br /&gt;
 |url=http://newscenter.lbl.gov/feature-stories/2011/11/07/record-breaking-solar-cell-performances/&lt;br /&gt;
 |quote=Theoretical research by scientists with the U.S. Department of Energy (DOE)’s Lawrence Berkeley National Laboratory (Berkeley Lab) has led to record-breaking sunlight-to-electricity conversion efficiencies in solar cells. }} ([http://www.rdmag.com/News/2011/11/Energy-Solar-Energy-Berkeley-Lab-Research-Sparks-Record-Breaking-Solar-Cell-Performances/ reprinted in &#039;&#039;R&amp;amp;D Magazine&#039;&#039;])&lt;br /&gt;
&lt;br /&gt;
==See also==&lt;br /&gt;
{{Portal|Renewable energy|Energy}}&lt;br /&gt;
&lt;br /&gt;
*[[List of semiconductor materials]]&lt;br /&gt;
*[[Organic photovoltaic cell]]&lt;br /&gt;
*[[PIN diode]]&lt;br /&gt;
&lt;br /&gt;
{{Photovoltaics}}&lt;br /&gt;
&lt;br /&gt;
{{DEFAULTSORT:Multijunction Solar Cell}}&lt;br /&gt;
[[Category:Solar cells]]&lt;br /&gt;
[[Category:Energy conversion]]&lt;br /&gt;
[[Category:Semiconductor devices]]&lt;br /&gt;
[[Category:Electronics]]&lt;/div&gt;</summary>
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		<title>Schrödinger equation</title>
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