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2015-01-11T16:18:27Z

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2014-01-29T21:43:04Z

50.167.224.81:

{{Thermodynamics|cTopic=[[Thermodynamic equations|Equations]]}}
{{For|electromagnetic equations|Maxwell's equations}}

[[file:Thermodynamic map.svg|400px|right|thumb|Flow chart showing the paths between the Maxwell relations. ''P'' = [[pressure]], ''T'' = [[temperature]], ''V'' = [[volume]], ''S'' = [[entropy]], ''α'' = [[coefficient of thermal expansion]], ''κ'' = thermal [[compressibility]], ''CV'' = [[heat capacity]] at constant volume, ''CP'' = [[heat capacity]] at constant pressure.
</div>
]]

'''Maxwell's relations''' are a set of equations in [[thermodynamics]] which are derivable from the definitions of the [[thermodynamic potentials]]. These relations are named for the nineteenth-century physicist [[James Clerk Maxwell]].

==Equation==

The Maxwell relations are statements of equality among the second derivatives of the thermodynamic potentials. They follow directly from the fact that the order of differentiation of an [[analytic function]] of two variables is irrelevant. If Φ is a thermodynamic potential and ''xi'' and ''xj'' are two different [[Thermodynamic potential#Natural variables|natural variables]] for that potential, then the Maxwell relation for that potential and those variables is:

{{Equation box 1
|title = '''Maxwell relations''' ''(general)''
|indent =:
|equation = <math>\frac{\partial }{\partial x_j}\left(\frac{\partial \Phi}{\partial x_i}\right)=
\frac{\partial }{\partial x_i}\left(\frac{\partial \Phi}{\partial x_j}\right)
</math>
|border colour = #50C878
|background colour = #ECFCF4}}

where the [[partial derivatives]] are taken with all other natural variables held constant. It is seen that for every thermodynamic potential there are ''n''(''n'' − 1)/2 possible Maxwell relations where ''n'' is the number of natural variables for that potential.

== The four most common Maxwell relations ==

The four most common Maxwell relations are the equalities of the second derivatives of each of the four thermodynamic potentials, with respect to their thermal natural variable ([[temperature]] ''T''; or [[entropy]] ''S'') and their ''mechanical'' natural variable ([[pressure]] ''P''; or [[volume]] ''V''):

{{Equation box 1
|title = '''Maxwell's relations''' ''(common)''
|indent =:
|equation =

<math> \begin{align}
+\left(\frac{\partial T}{\partial V}\right)_S &=& -\left(\frac{\partial P}{\partial S}\right)_V &=& \frac{\partial^2 U }{\partial S \partial V}\\

+\left(\frac{\partial T}{\partial P}\right)_S &=& +\left(\frac{\partial V}{\partial S}\right)_P &=& \frac{\partial^2 H }{\partial S \partial P}\\
+\left(\frac{\partial S}{\partial V}\right)_T &=& +\left(\frac{\partial P}{\partial T}\right)_V &=& -\frac{\partial^2 A }{\partial T \partial V}\\

-\left(\frac{\partial S}{\partial P}\right)_T &=& +\left(\frac{\partial V}{\partial T}\right)_P &=& \frac{\partial^2 G }{\partial T \partial P}
\end{align}\,\!</math>

|border colour = #0073CF
|background colour=#F5FFFA}}

where the potentials as functions of their natural thermal and mechanical variables are the [[internal energy]] ''U''(''S, V''), [[Enthalpy]] ''H''(''S, P''), [[Helmholtz free energy]] ''A''(''T, V'') and [[Gibbs free energy]] ''G''(''T, P''). The [[thermodynamic square]] can be used as a [[mnemonic]] to recall and derive these relations.

=== Derivation ===
Maxwell relations are based on simple partial differentiation rules, in particular the [[Total derivative|total]] [[differential of a function]] and the symmetry of evaluating second order partial derivatives.
:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Derivation
|-
|Derivation of the Maxwell relations can be deduced from the differential forms of the [[thermodynamic potentials]]:
:<math>\begin{align}
dU &=& TdS-PdV \\
dH &=& TdS+VdP \\
dA &=& -SdT-PdV \\
dG &=& -SdT+VdP \\
\end{align}\,\!</math>
These equations resemble [[Total derivative|total differentials]] of the form
:<math>dz = \left(\frac{\partial z}{\partial x}\right)_y\!dx +
\left(\frac{\partial z}{\partial y}\right)_x\!dy</math>
And indeed, it can be shown for any equation of the form
:<math>dz = Mdx + Ndy \,</math>
that
:<math>M = \left(\frac{\partial z}{\partial x}\right)_y, \quad
N = \left(\frac{\partial z}{\partial y}\right)_x</math>
Consider, as an example, the equation <math>dH=TdS+VdP\,</math>. We can now immediately see that
:<math>T = \left(\frac{\partial H}{\partial S}\right)_P, \quad
V = \left(\frac{\partial H}{\partial P}\right)_S</math>
Since we also know that for functions with continuous second derivatives, the mixed partial derivatives are identical ([[Symmetry of second derivatives]]), that is, that
:<math>\frac{\partial}{\partial y}\left(\frac{\partial z}{\partial x}\right)_y =
\frac{\partial}{\partial x}\left(\frac{\partial z}{\partial y}\right)_x =
\frac{\partial^2 z}{\partial y \partial x} = \frac{\partial^2 z}{\partial x \partial y}</math>
we therefore can see that
:<math> \frac{\partial}{\partial P}\left(\frac{\partial H}{\partial S}\right)_P =
\frac{\partial}{\partial S}\left(\frac{\partial H}{\partial P}\right)_S </math>
and therefore that
:<math>\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P</math>
Each of the four Maxwell relationships given above follows similarly from one of the [[Gibbs equations]].
|}
:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Extended derivation
|-
|Combined form first and second law of thermodynamics,
:<math>TdS = dU+PdV</math> (Eq.1)
U, S, and V are state functions.
Let,
:<math>U = U(x,y)</math>
:<math>S = S(x,y)</math>
:<math>V = V(x,y)</math>
:<math>dU = \left(\frac{\partial U}{\partial x}\right)_y\!dx +
\left(\frac{\partial U}{\partial y}\right)_x\!dy</math>
:<math>dS = \left(\frac{\partial S}{\partial x}\right)_y\!dx +
\left(\frac{\partial S}{\partial y}\right)_x\!dy</math>
:<math>dV = \left(\frac{\partial V}{\partial x}\right)_y\!dx +
\left(\frac{\partial V}{\partial y}\right)_x\!dy</math>
Substitute them in Eq.1 and one gets,
:<math>T\left(\frac{\partial S}{\partial x}\right)_y\!dx +
T\left(\frac{\partial S}{\partial y}\right)_x\!dy = \left(\frac{\partial U}{\partial x}\right)_y\!dx +
\left(\frac{\partial U}{\partial y}\right)_x\!dy + P\left(\frac{\partial V}{\partial x}\right)_y\!dx +
P\left(\frac{\partial V}{\partial y}\right)_x\!dy</math>
And also written as,
:<math>\left(\frac{\partial U}{\partial x}\right)_y\!dx +
\left(\frac{\partial U}{\partial y}\right)_x\!dy = T\left(\frac{\partial S}{\partial x}\right)_y\!dx +
T\left(\frac{\partial S}{\partial y}\right)_x\!dy - P\left(\frac{\partial V}{\partial x}\right)_y\!dx -
P\left(\frac{\partial V}{\partial y}\right)_x\!dy</math>
comparing the coefficient of dx and dy, one gets
:<math>\left(\frac{\partial U}{\partial x}\right)_y = T\left(\frac{\partial S}{\partial x}\right)_y - P\left(\frac{\partial V}{\partial x}\right)_y</math>
:<math>\left(\frac{\partial U}{\partial y}\right)_x = T\left(\frac{\partial S}{\partial y}\right)_x - P\left(\frac{\partial V}{\partial y}\right)_x</math>
Differentiating above equations by y, x respectively 
:<math>\left(\frac{\partial^2U}{\partial y\partial x}\right) = \left(\frac{\partial T}{\partial y}\right)_x \left(\frac{\partial S}{\partial x}\right)_y + T\left(\frac{\partial^2 S}{\partial y\partial x}\right) - \left(\frac{\partial P}{\partial y}\right)_x \left(\frac{\partial V}{\partial x}\right)_y - P\left(\frac{\partial^2 V}{\partial y\partial x}\right)</math> (Eq.2)
:and
:<math>\left(\frac{\partial^2U}{\partial x\partial y}\right) = \left(\frac{\partial T}{\partial x}\right)_y \left(\frac{\partial S}{\partial y}\right)_x + T\left(\frac{\partial^2 S}{\partial x\partial y}\right) - \left(\frac{\partial P}{\partial x}\right)_y \left(\frac{\partial V}{\partial y}\right)_x - P\left(\frac{\partial^2 V}{\partial x\partial y}\right)</math> (Eq.3)
U, S, and V are exact differentials, therefore,
:<math>\left(\frac{\partial^2U}{\partial y\partial x}\right) = \left(\frac{\partial^2U}{\partial x\partial y}\right)</math>
:<math>\left(\frac{\partial^2S}{\partial y\partial x}\right) = \left(\frac{\partial^2S}{\partial x\partial y}\right)
:\left(\frac{\partial^2V}{\partial y\partial x}\right) = \left(\frac{\partial^2V}{\partial x\partial y}\right)</math>
Subtract eqn(2) and (3) and one gets 
:<math>\left(\frac{\partial T}{\partial y}\right)_x \left(\frac{\partial S}{\partial x}\right)_y - \left(\frac{\partial P}{\partial y}\right)_x \left(\frac{\partial V}{\partial x}\right)_y = \left(\frac{\partial T}{\partial x}\right)_y \left(\frac{\partial S}{\partial y}\right)_x - \left(\frac{\partial P}{\partial x}\right)_y \left(\frac{\partial V}{\partial y}\right)_x</math>
:''Note: The above is called the general expression for Maxwell's thermodynamical relation.''
;Maxwell's first relation
:Allow x = S and y = V and one gets
:<math>\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V</math>
;Maxwell's second relation
:Allow x = T and y = V and one gets
:<math>\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V</math>
;Maxwell's third relation
:Allow x = S and y = P and one gets
:<math>\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P</math>
;Maxwell's fourth relation
:Allow x = T and y = P and one gets
:<math>\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P</math>
;Maxwell's fifth relation
:Allow x = P and y = V
:<math>\left(\frac{\partial T}{\partial P}\right)_V \left(\frac{\partial S}{\partial V}\right)_P</math><math>-\left(\frac{\partial T}{\partial V}\right)_P \left(\frac{\partial S}{\partial P}\right)_V</math> = 1
;Maxwell's sixth relation
:Allow x = T and y = S and one gets
:<math>\left(\frac{\partial P}{\partial T}\right)_S \left(\frac{\partial V}{\partial S}\right)_T -\left(\frac{\partial P}{\partial S}\right)_T \left(\frac{\partial V}{\partial T}\right)_S</math> = 1
|}

== General Maxwell relationships ==

The above are not the only Maxwell relationships. When other work terms involving other natural variables besides the volume work are considered or when the [[Particle number|number of particles]] is included as a natural variable, other Maxwell relations become apparent. For example, if we have a single-component gas, then the number of particles ''N''  is also a natural variable of the above four thermodynamic potentials. The Maxwell relationship for the enthalpy with respect to pressure and particle number would then be:

:<math>
\left(\frac{\partial \mu}{\partial P}\right)_{S, N} =
\left(\frac{\partial V}{\partial N}\right)_{S, P}\qquad=
\frac{\partial^2 H }{\partial P \partial N}
</math>

where μ is the [[chemical potential]]. In addition, there are other thermodynamic potentials besides the four that are commonly used, and each of these potentials will yield a set of Maxwell relations.

Each equation can be re-expressed using the relationship

:<math>\left(\frac{\partial y}{\partial x}\right)_z
=
1\left/\left(\frac{\partial x}{\partial y}\right)_z\right.</math>

which are sometimes also known as Maxwell relations.

== See also ==
* [[Table of thermodynamic equations]]
* [[Thermodynamic equations]]

==External links==
* [http://theory.ph.man.ac.uk/~judith/stat_therm/node48.html a partial derivation of Maxwell's relations]

[[Category:Thermodynamics]]
[[Category:Concepts in physics]]
[[Category:James Clerk Maxwell]]
[[Category:Thermodynamic equations]]

Propeller

2014-01-29T21:08:26Z

50.167.224.81: /* Screw propellers */

{{About||earlier uses of photolithography in printing|Lithography|the same process applied to metal|Photochemical machining}}

'''Photolithography''', also termed '''optical lithography''' or '''UV lithography''', is a process used in [[microfabrication]] to pattern parts of a thin film or the bulk of a [[Substrate (electronics)|substrate]]. It uses light to transfer a [[geometry|geometric pattern]] from a [[photomask]] to a [[photosensitive|light-sensitive]] chemical "[[photoresist]]", or simply "resist," on the substrate. A series of [[Chemical engineering|chemical treatments]] then either engraves the exposure pattern into, or enables deposition of a new material in the desired pattern upon, the material underneath the photo resist. For example, in complex [[integrated circuit]]s, a modern [[CMOS]] [[Wafer (electronics)|wafer]] will go through the photolithographic cycle up to 50 times.

''Photolithography'' shares some fundamental principles with [[photography]] in that the pattern in the [[resist|etching resist]] is created by exposing it to [[light]], either directly (without using a mask) or with a projected image using an [[optical mask]]. This procedure is comparable to a high precision version of the method used to make [[printed circuit board]]s. Subsequent stages in the process have more in common with [[etching]] than with [[lithographic printing]]. It is used because it can create extremely small patterns (down to a few tens of nanometers in size), it affords exact control over the shape and size of the objects it creates, and because it can create patterns over an entire surface cost-effectively. Its main disadvantages are that it requires a flat substrate to start with, it is not very effective at creating shapes that are not flat, and it can require extremely clean operating conditions.

== Basic procedure ==
[[Image:Photolithography etching process.svg|thumb|140px|Simplified illustration of dry etching using positive photoresist during a photolithography process in semiconductor microfabrication (not to scale).]]
A single iteration of photolithography combines several steps in sequence. Modern cleanrooms use automated, [[industrial robot|robot]]ic wafer track systems to coordinate the process. The procedure described here omits some advanced treatments, such as thinning agents or edge-bead removal.<ref>{{cite book |last=Jaeger |first=Richard C. |title=Introduction to Microelectronic Fabrication |edition=2nd |year=2002 |publisher=Prentice Hall |location=Upper Saddle River |isbn=0-201-44494-1 |chapter=Lithography}}</ref>

=== Cleaning ===
If organic or inorganic contaminations are present on the wafer surface, they are usually removed by wet chemical treatment, e.g. the [[RCA clean]] procedure based on solutions containing [[hydrogen peroxide]].

=== Preparation ===
The wafer is initially heated to a temperature sufficient to drive off any moisture that may be present on the wafer surface. Wafers that have been in storage must be chemically cleaned to remove [[contamination]]. A [[liquid]] or [[gas]]eous "adhesion promoter", such as [[Bis(trimethylsilyl)amine|Bis(trimethylsilyl)amine ("hexamethyldisilazane", HMDS)]], is applied to promote adhesion of the photoresist to the wafer. The surface layer of silicon dioxide on the wafer reacts with HMDS to form tri-methylated silicon-dioxide, a highly water repellent layer not unlike the layer of wax on a car's paint. This water repellent layer prevents the aqueous developer from penetrating between the photoresist layer and the wafer's surface, thus preventing so-called lifting of small photoresist structures in the (developing) pattern.,

=== Photoresist application ===
The wafer is covered with [[photoresist]] by [[spin coating]]. A viscous, liquid solution of photoresist is dispensed onto the wafer, and the wafer is spun rapidly to produce a uniformly thick layer. The spin coating typically runs at 1200 to 4800 rpm for 30 to 60 seconds, and produces a layer between 0.5 and 2.5 micrometres thick. The spin coating process results in a uniform thin layer, usually with uniformity of within 5 to 10 nanometres. This uniformity can be explained by detailed fluid-mechanical modelling, which shows that the resist moves much faster at the top of the layer than at the bottom, where viscous forces bind the resist to the wafer surface. Thus, the top layer of resist is quickly ejected from the wafer's edge while the bottom layer still creeps slowly radially along the wafer. In this way, any 'bump' or 'ridge' of resist is removed, leaving a very flat layer. Final thickness is also determined by the evaporation of liquid solvents from the resist. For very small, dense features (< 125 or so nm), lower resist thicknesses (< 0.5 micrometres) are needed to overcome collapse effects at high aspect ratios; typical aspect ratios are < 4:1.

The photo resist-coated wafer is then prebaked to drive off excess photoresist solvent, typically at 90 to 100 °C for 30 to 60 seconds on a hotplate.

=== Exposure and developing ===
After prebaking, the photoresist is exposed to a pattern of intense light. The exposure to light causes a chemical change that allows some of the photoresist to be removed by a special solution, called "developer" by analogy with [[photographic developer]]. Positive photoresist, the most common type, becomes soluble in the developer when exposed; with negative photoresist, unexposed regions are soluble in the developer.

A post-exposure bake (PEB) is performed before developing, typically to help reduce [[standing wave]] phenomena caused by the destructive and constructive [[Interference (wave propagation)|interference]] patterns of the incident light. In deep ultraviolet lithography, chemically amplified resist (CAR) chemistry is used. This process is much more sensitive to PEB time, temperature, and delay, as most of the "exposure" reaction (creating acid, making the polymer soluble in the basic developer) actually occurs in the PEB.<ref>{{cite web | first=Omkaram, et al. | last=Nalamasu | title=An Overview of Resist Processing for DUV Photolithography | url=http://www.journalarchive.jst.go.jp/jnlpdf.php?cdjournal=photopolymer1988&cdvol=4&noissue=3&startpage=299&lang=en&from=jnlabstract
}}</ref>

The develop chemistry is delivered on a spinner, much like photoresist. Developers originally often contained [[sodium hydroxide]] (NaOH). However, [[sodium]] is considered an extremely undesirable contaminant in [[MOSFET]] fabrication because it degrades the [[Electrical insulation|insulating]] properties of gate oxides (specifically, sodium ions can migrate in and out of the gate, changing the threshold voltage of the transistor and making it harder or easier to turn the transistor on over time). Metal-ion-free developers such as [[tetramethylammonium hydroxide]] (TMAH) are now used.

The resulting wafer is then "hard-baked" if a non-chemically amplified resist was used, typically at 120 to 180 °C{{Citation needed|date=May 2007}} for 20 to 30 minutes. The hard bake solidifies the remaining photoresist, to make a more durable protecting layer in future [[ion implantation]], [[Chemical milling|wet chemical etching]], or [[plasma etching]].

=== Etching ===
:{{Main|Etching (microfabrication)}}

In etching, a liquid ("wet") or [[plasma (physics)|plasma]] ("dry") chemical agent removes the uppermost layer of the substrate in the areas that are not protected by photoresist. In [[semiconductor fabrication]], dry etching techniques are generally used, as they can be made [[anisotropic]], in order to avoid significant undercutting of the photoresist pattern. This is essential when the width of the features to be defined is similar to or less than the thickness of the material being etched (i.e. when the aspect ratio approaches unity). Wet etch processes are generally isotropic in nature, which is often indispensable for [[microelectromechanical systems]], where suspended structures must be "released" from the underlying layer.

The development of low-defectivity anisotropic dry-etch process has enabled the ever-smaller features defined photolithographically in the resist to be transferred to the substrate material.

=== Photoresist removal ===
After a photoresist is no longer needed, it must be removed from the substrate. This usually requires a liquid "resist stripper", which chemically alters the resist so that it no longer adheres to the substrate. Alternatively, photoresist may be removed by a plasma containing [[oxygen]], which oxidizes it. This process is called [[plasma ashing|ashing]], and resembles dry etching.

== Exposure ("printing") systems ==
[[Image:Wafertraksystem.jpg|thumb|200px|The wafer track portion of an aligner that uses 365 nm ultraviolet light]]
Exposure systems typically produce an image on the wafer using a [[photomask]]. The light shines through the photomask, which blocks it in some areas and lets it pass in others. ([[Maskless lithography]] projects a precise beam directly onto the wafer without using a mask, but it is not widely used in commercial processes.) Exposure systems may be classified by the optics that transfer the image from the mask to the wafer.

=== Contact and proximity ===
{{Main|Contact lithography}}

A contact printer, the simplest exposure system, puts a photomask in direct contact with the wafer and exposes it to a uniform light. A proximity printer puts a small gap between the photomask and wafer. In both cases, the mask covers the entire wafer, and simultaneously patterns every die.

Contact printing is liable to damage both the mask and the wafer, and this was the primary reason it was abandoned for high volume production. Both contact and proximity lithography require the light intensity to be uniform across an entire wafer, and the mask to align precisely to features already on the wafer. As modern processes use increasingly large wafers, these conditions become increasingly difficult.

Research and prototyping processes often use contact or proximity lithography, because it uses inexpensive hardware and can achieve high optical resolution. The resolution in proximity lithography is approximately the square root of the product of the wavelength and the gap distance. Hence, except for projection lithography (see below), contact printing offers the best resolution, because its gap distance is approximately zero (neglecting the thickness of the photoresist itself). In addition, nanoimprint lithography may revive interest in this familiar technique, especially since the cost of ownership is expected to be low; however, the shortcomings of contact printing discussed above remain as challenges.

=== Projection ===
{{See also|Stepper}}

[[Very-large-scale integration]] (VLSI) lithography uses projection systems. Unlike contact or proximity masks, which cover an entire wafer, projection masks (known as "reticles") show only one die or an array of dice (known as a "field"). Projection exposure systems (steppers or scanners) project the mask onto the wafer many times to create the complete pattern.

== Photomasks ==
{{Main|Photomask}}

The image for the mask originates from a computerized data file. This data file is converted to a series of polygons and written onto a square [[fused quartz]] substrate covered with a layer of [[chromium]] using a photolithographic process. A laser beam (laser writer) or a beam of electrons (e-beam writer) is used to expose the pattern defined by the data file and travels over the surface of the substrate in either a vector or raster scan manner. Where the photoresist on the mask is exposed, the chrome can be etched away, leaving a clear path for the illumination light in the stepper/scanner system to travel through.

== Resolution in projection systems ==
{{Main|Optics#Diffraction and optical resolution|Diffraction|Optical resolution}}
[[Image:Yellow fluorescent light spectrum.png|thumb|right|250px|The filtered [[fluorescent lighting]] in photolithography cleanrooms contains no ultraviolet or blue light in order to avoid exposing photoresists. The spectrum of light emitted by such fixtures gives virtually all such spaces a bright yellow color.]]
The ability to project a clear image of a small feature onto the wafer is [[diffraction limit|limited]] by the [[wavelength]] of the light that is used, and the ability of the reduction lens system to capture enough diffraction orders from the illuminated mask. Current state-of-the-art photolithography tools use [[deep ultraviolet]] (DUV) light from excimer lasers with wavelengths of 248 and 193 [[Nanometre|nm]] (the dominant lithography technology today is thus also called "excimer laser lithography"), which allow minimum feature sizes down to 50 nm. Excimer laser lithography has thus played a critical role in the continued advance of the so-called Moore’s Law for the last 20 years (see below<ref name=LaFontaine>La Fontaine, B., “Lasers and Moore’s Law”, SPIE Professional, Oct. 2010, p. 20; http://spie.org/x42152.xml</ref>).

The minimum feature size that a projection system can print is given approximately by:

:<math>CD = k_1 \cdot\frac{\lambda}{NA}</math>
where

<math>\,CD</math> is the '''minimum feature size''' (also called the '''critical dimension''', ''target design rule''). It is also common to write 2 ''times'' the ''half-pitch''.

<math>\,k_1</math> (commonly called ''k1 factor'') is a coefficient that encapsulates process-related factors, and typically equals 0.4 for production. The minimum feature size can be reduced by decreasing this coefficient through [[Computational lithography]].

<math>\,\lambda</math> is the wavelength of light used

<math>\,NA</math> is the [[numerical aperture]] of the lens as seen from the wafer

According to this equation, minimum feature sizes can be decreased by decreasing the wavelength, and increasing the numerical aperture (to achieve a tighter focused beam and a smaller spot size). However, this design method runs into a competing constraint. In modern systems, the [[depth of focus]] is also a concern:

:<math>D_F = k_2 \cdot\frac{\lambda}{{NA}^2}</math>

Here, <math>\,k_2</math> is another process-related coefficient. The depth of focus restricts the thickness of the photoresist and the depth of the topography on the wafer. [[Chemical mechanical polishing]] is often used to flatten topography before high-resolution lithographic steps.

==Light sources==
[[Image:Lithography Wavelength vs Resolution.PNG|thumb|right|300px|One of the evolutionary paths of lithography has been the use of shorter wavelengths. It is worth noting that the same light source may be used for several technology generations.]]
Historically, photolithography has used ultraviolet light from [[gas-discharge lamp]]s using [[mercury (element)|mercury]], sometimes in combination with [[noble gas]]es such as [[xenon]]. These lamps produce light across a broad spectrum with several strong peaks in the ultraviolet range. This spectrum is filtered to select a single [[spectral line]]. From the early 1960s through the mid-1980s, Hg lamps had been used in lithography for their spectral lines at 436 nm ("g-line"), 405 nm ("h-line") and 365 nm ("i-line"). However, with the semiconductor industry’s need for both higher resolution (to produce denser and faster chips) and higher throughput (for lower costs), the lamp-based lithography tools were no longer able to meet the industry’s requirements.

This challenge was overcome when in a pioneering development in 1982, excimer laser lithography was proposed and demonstrated at I.B.M. by Kanti Jain,<ref>Jain, K. ''“Excimer Laser Lithography”'', SPIE Press, Bellingham, WA, 1990.</ref><ref>Jain, K. et al., “Ultrafast deep-UV lithography with excimer lasers”, IEEE Electron Device Lett., Vol. EDL-3, 53 (1982): http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1482581</ref><ref>Lin, B. J., ''"Optical Lithography"'', SPIE Press, Bellingham, WA, 2009, p. 136.</ref><ref>Basting, D., et al., “Historical Review of Excimer Laser Development,” in ''"Excimer Laser Technology"'', D. Basting and G. Marowsky, Eds., Springer, 2005.</ref> and now excimer laser lithography machines (steppers and scanners) are the primary tools used worldwide in microelectronics production. With phenomenal advances made in tool technology in the last two decades, it is the semiconductor industry view<ref name=LaFontaine /> that excimer laser lithography has been a crucial factor in the continued advance of Moore’s Law, enabling minimum features sizes in chip manufacturing to shrink from 0.5 micrometer in 1990 to 45 nanometers and below in 2010. This trend is expected to continue into this decade for even denser chips, with minimum features approaching 10 nanometers. From an even broader scientific and technological perspective, in the 50-year history of the laser since its first demonstration in 1960, the invention and development of excimer laser lithography has been highlighted as one of the major milestones.<ref>American Physical Society / Lasers / History / Timeline; http://www.laserfest.org/lasers/history/timeline.cfm</ref><ref>SPIE / Advancing the Laser / 50 Years and into the Future; http://spie.org/Documents/AboutSPIE/SPIE%20Laser%20Luminaries.pdf</ref><ref>U.K. Engineering & Physical Sciences Research Council / Lasers in Our Lives / 50 Years of Impact; http://www.stfc.ac.uk/Resources/PDF/Lasers50_final1.pdf</ref>

The commonly used deep ultraviolet [[excimer laser]]s in lithography systems are the [[krypton fluoride]] laser at 248 nm wavelength and the [[argon fluoride laser]] at 193 nm wavelength. The primary manufacturers of excimer laser light sources in the 1980s were Lambda Physik (now part of Coherent, Inc.) and Lumonics. Since the mid-1990s [[Cymer Inc.]] has become the dominant supplier of excimer laser sources to the lithography equipment manufacturers, with [[Gigaphoton Inc.]] as their closest rival. Generally, an excimer laser is designed to operate with a specific gas mixture; therefore, changing wavelength is not a trivial matter, as the method of generating the new wavelength is completely different, and the absorption characteristics of materials change. For example, air begins to absorb significantly around the 193 nm wavelength; moving to sub-193 nm wavelengths would require installing vacuum pump and purge equipment on the lithography tools (a significant challenge). Furthermore, insulating materials such as [[silicon dioxide]], when exposed to photons with energy greater than the band gap, release free electrons and holes which subsequently cause adverse charging.

Optical lithography has been extended to feature sizes below 50 nm using the 193 nm ArF excimer laser and liquid immersion techniques. Also termed [[immersion lithography]], this enables the use of optics with numerical apertures exceeding 1.0. The liquid used is typically ultra-pure, deionised water, which provides for a [[refractive index]] above that of the usual air gap between the lens and the wafer surface. The water is continually circulated to eliminate thermally-induced distortions. Water will only allow ''NA'''s of up to ~1.4, but materials with higher [[refractive indices]] will allow the effective ''NA'' to be increased further.

[[Image:Photon Energy vs Resolution.PNG|thumb|left|300px|Changing the lithography wavelength is significantly limited by absorption. Air absorbs below ~ 185 nm.]]

Experimental tools using the 157 nm wavelength from the F2 excimer laser in a manner similar to current exposure systems have been built. These were once targeted to succeed 193 nm lithography at the 65 nm feature size node but have now all but been eliminated by the introduction of immersion lithography. This was due to persistent technical problems with the 157 nm technology and economic considerations that provided strong incentives for the continued use of 193 nm excimer laser lithography technology. High-index immersion lithography is the newest extension of 193 nm lithography to be considered. In 2006, features less than 30 nm were demonstrated by IBM using this technique.<ref>{{cite web | first=Aaron | last=Hand | title=High-Index Lenses Push Immersion Beyond 32 nm | url=http://www.reed-electronics.com/semiconductor/article/CA6319061}}</ref>

An option, especially if and when wavelengths continue to decrease to extreme UV or X-ray, is the [[free-electron laser]] (or one might say xaser for an X-ray device). These can produce high quality beams at arbitrary wavelengths.

== Experimental methods ==
{{See also|Nanolithography}}
Photolithography has been defeating predictions of its demise for many years. For instance, by the early 1980s, many in the semiconductor industry had come to believe that features smaller than 1 micrometer could not be printed optically. Modern techniques using excimer laser lithography already print features with dimensions a fraction of the wavelength of light used – an amazing optical feat. New tricks such as [[immersion lithography]], dual-tone resist and [[multiple patterning]] continue to improve the resolution of 193 nm lithography. Meanwhile, current research is exploring alternatives to conventional UV, such as [[electron beam lithography]], [[X-ray lithography]], [[extreme ultraviolet lithography]] and [[ion projection lithography]].

== See also ==
* Alternative types of lithography
** [[Extreme ultraviolet lithography]]
** [[Nanoimprint lithography]]
** [[Dip-pen nanolithography]]
** [[Chemical lithography]]
** [[Soft lithography]]
** [[Magnetolithography]]
** [[Nanochannel glass materials]]
* [[Computational lithography]]
* [[Excimer laser]]
* [[Moore's law]]
* [[Photography]] "first permanent photoetching" (1822CE).
* [[Stereolithography]], a macroscale process used to produce three-dimensional shapes
* [[Wafer foundry]]

== References ==
{{Reflist}}

== External links ==
* [http://www.ee.byu.edu/cleanroom/lithography.phtml BYU Photolithography Resources]
* [http://www.lithoguru.com/scientist/lithobasics.html Semiconductor Lithography] — Overview of lithography
* [http://www.research.ibm.com/journal/rd/411/chiu.html Optical Lithography Introduction] — IBM site with lithography-related articles
* [http://sst.pennnet.com/Articles/Article_Display.cfm?Section=ARCHI&Subsection=Display&ARTICLE_ID=205024&p=28 Immersion Lithography Article] — Shows how depth-of-focus is increased with immersion lithography
*[http://www.imtecacculine.com Photolithography Equipment] – UV Light Shields & Information
*[http://microphoto.net Photo Etching Experts]

[[Category:Lithography (microfabrication)]]
[[Category:Microtechnology]]