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Electron-beam lithography
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==Electron energy deposition in matter== [[File:Electron Beam scattering.svg|right|thumb|200px|'''Electron trajectories in resist:''' An incident electron (red) produces secondary electrons (blue). Sometimes, the incident electron may itself be backscattered as shown here and leave the surface of the resist (amber).]] The primary electrons in the incident beam lose energy upon entering a material through [[inelastic scattering]] or collisions with other electrons. In such a collision the momentum transfer from the incident electron to an atomic electron can be expressed as<ref name="feldman">{{cite book|author1=L. Feldman|author2=J. Mayer|title=Fundamentals of Surface and Thin Film Analysis|volume=54|year=1986|pages=130โ133|publisher=North-Holland |isbn=978-0-444-00989-0}}</ref> <math>dp=2e^2/bv</math>, where ''b'' is the distance of closest approach between the electrons, and ''v'' is the incident electron velocity. The energy transferred by the collision is given by <math>T = (dp)^2/2m = e^4/Eb^2</math>, where ''m'' is the electron mass and ''E'' is the incident electron energy, given by <math>E=(1/2) mv^2</math>. By integrating over all values of ''T'' between the lowest binding energy, ''E<sub>0</sub>'' and the incident energy, one obtains the result that the total [[cross section (physics)|cross section]] for collision is inversely proportional to the incident energy <math>E</math>, and proportional to ''1/E<sub>0</sub> โ 1/E''. Generally, ''E >> E<sub>0</sub>'', so the result is essentially inversely proportional to the binding energy. By using the same integration approach, but over the range ''2E<sub>0</sub>'' to ''E'', one obtains by comparing cross-sections that half of the inelastic collisions of the incident electrons produce electrons with kinetic energy greater than ''E<sub>0</sub>''. These [[secondary electrons]] are capable of breaking bonds (with binding energy ''E<sub>0</sub>'') at some distance away from the original collision. Additionally, they can generate additional, lower energy electrons, resulting in an [[electron avalanche|electron cascade]]. Hence, it is important to recognize the significant contribution of secondary electrons to the spread of the energy deposition. In general, for a molecule AB:<ref>{{cite web|last1=Mason|first1=Nigel J|last2=Dujardin|first2=G|last3=Gerber|first3=G|last4=Gianturco|first4=F|last5=Maerk|first5=T.D.|title=EURONanochem โ Chemical Control at the Nanoscale|url=http://www.arrs.gov.si/sl/medn/esf/gradivo/dokum/INC/EuroNANOCHEM/Gaberscek_januar08.ppt|website=Slovenian Research Agency|publisher=European Space Foundation|archiveurl=https://web.archive.org/web/20110720012703/http://www.arrs.gov.si/sl/medn/esf/gradivo/dokum/INC/EuroNANOCHEM/Gaberscek_januar08.ppt|archivedate=2011-07-20|url-status=live|language=English|date=January 2008}}</ref> :e<sup>โ</sup> + AB โ AB<sup>โ</sup> โ A + B<sup>โ</sup> This reaction, also known as "electron attachment" or "dissociative electron attachment" is most likely to occur after the electron has essentially slowed to a halt, since it is easiest to capture at that point. The cross-section for electron attachment is inversely proportional to electron energy at high energies, but approaches a maximum limiting value at zero energy.<ref>{{cite journal|doi=10.1088/0963-0252/10/2/321|year=2001|last1=Stoffels|first1=E|last2=Stoffels|first2=W W|last3=Kroesen|first3=G M W|journal=Plasma Sources Science and Technology|volume=10|page=311|bibcode = 2001PSST...10..311S|issue=2|title=Plasma chemistry and surface processes of negative ions |citeseerx=10.1.1.195.9811|s2cid=250916447 }}</ref> On the other hand, it is already known that the mean free path at the lowest energies (few to several eV or less, where dissociative attachment is significant) is well over 10 nm,<ref>{{cite journal|doi=10.1002/sia.740010103|title=Quantitative electron spectroscopy of surfaces: A standard data base for electron inelastic mean free paths in solids|year=1979|last1=Seah|first1=M. P.|last2=Dench|first2=W. A.|journal=Surface and Interface Analysis|volume=1|page=2}}</ref><ref name=tanuma>{{cite journal|doi=10.1002/sia.740210302|title=Calculations of electron inelastic mean free paths. V. Data for 14 organic compounds over the 50โ2000 eV range|year=1994|last1=Tanuma|first1=S.|last2=Powell|first2=C. J.|last3=Penn|first3=D. R.|journal=Surface and Interface Analysis|volume=21|page=165|issue=3|url=https://mdr.nims.go.jp/pid/c28159ad-54ac-42de-9bc0-6e62fef074c8 }}</ref> thus limiting the ability to consistently achieve resolution at this scale. ===Resolution capability=== [[File:Electron_travel_(Monte_Carlo).png|thumb|300px|left|'''Low energy electron migration.''' The distance (r) traveled by a low energy electron affects the resolution and can be at least several nanometers.]] With today's electron optics, electron beam widths can routinely go down to a few nanometers. This is limited mainly by [[Optical aberration|aberration]]s and [[space charge]]. However, the feature resolution limit is determined not by the beam size but by forward scattering (or effective beam broadening) in the [[resist]], while the pitch resolution limit is determined by [[secondary electrons|secondary electron]] travel in the [[resist]].<ref name="broers">{{cite journal|title=Electron beam lithographyโResolution limits|author=Broers, A. N.|journal=Microelectronic Engineering|volume=32|issue=1โ4|year=1996|pages=131โ142|doi=10.1016/0167-9317(95)00368-1|display-authors=etal}}</ref><ref>{{cite journal|bibcode=2009JKPS...55.1720L |title=Secondary electron generation in electron-beam-irradiated solids:resolution limits to nanolithography |url=http://www.kps.or.kr/home/kor/journal/library/downloadPdf.asp?articleuid=%7BDBBE4A45-2FC0-4127-BF2D-3081FAE2CE37%7D |author=K. W. Lee |journal= J. Korean Phys. Soc. |volume=55 |page=1720 |year=2009 |doi=10.3938/jkps.55.1720 |issue=4 |url-status=dead |archiveurl=https://web.archive.org/web/20110722141849/http://www.kps.or.kr/home/kor/journal/library/downloadPdf.asp?articleuid=%7BDBBE4A45-2FC0-4127-BF2D-3081FAE2CE37%7D |archivedate=2011-07-22 |url-access=subscription }}</ref> This point was driven home by a 2007 demonstration of double patterning using electron beam lithography in the fabrication of 15 nm half-pitch zone plates.<ref>[http://spie.org/x8367.xml SPIE Newsroom: Double exposure makes dense high-resolution diffractive optics]. Spie.org (2009-11-03). Retrieved on 2011-08-27.</ref> Although a 15 nm feature was resolved, a 30 nm pitch was still difficult to do due to secondary electrons scattering from the adjacent feature. The use of double patterning allowed the spacing between features to be wide enough for the secondary electron scattering to be significantly reduced. The forward scattering can be decreased by using higher energy electrons or thinner resist, but the generation of [[secondary electrons]] is inevitable. It is now recognized that for insulating materials like [[Poly(methyl methacrylate)|PMMA]], low energy electrons can travel quite a far distance (several nm is possible). This is due to the fact that below the [[ionization potential]] the only energy loss mechanism is mainly through [[phonon]]s and [[polaron]]s. Although the latter is basically an ionic lattice effect,<ref>{{cite journal|author=Dapor, M.|journal= J. Micro/Nanolith. MEMS MOEMS|volume=9|issue= 2|page= 023001|year=2010|doi=10.1117/1.3373517|title=Monte Carlo modeling in the low-energy domain of the secondary electron emission of polymethylmethacrylate for critical-dimension scanning electron microscopy |display-authors=etal}}</ref> polaron hopping can extend as far as 20 nm.<ref>{{cite journal|author=P. T. Henderson|journal= Proc. Natl. Acad. Sci. U.S.A. |volume=96|pages= 8353โ8358|doi=10.1073/pnas.96.15.8353 |year=1999|display-authors=1|title=Long-distance charge transport in duplex DNA: The phonon-assisted polaron-like hopping mechanism|issue=15|pmid=10411879|last2=Jones|first2=D|last3=Hampikian|first3=G|last4=Kan|first4=Y|last5=Schuster|first5=GB|pmc=17521|bibcode = 1999PNAS...96.8353H |doi-access= free }}</ref> The travel distance of [[secondary electrons]] is not a fundamentally derived physical value, but a statistical parameter often determined from many experiments or [[Monte Carlo simulations]] down to < 1 eV. This is necessary since the energy distribution of secondary electrons peaks well below 10 eV.<ref name="seiler">{{cite journal|title=Secondary electron emission in the scanning electron microscope|author=H. Seiler|journal=J. Appl. Phys.|volume=54|year=1983|pages=R1โR18|doi=10.1063/1.332840|bibcode = 1983JAP....54R...1S|issue=11 }}</ref> Hence, the resolution limit is not usually cited as a well-fixed number as with an optical diffraction-limited system.<ref name="broers" /> Repeatability and control at the practical resolution limit often require considerations not related to image formation, e.g., resist development and intermolecular forces. A study by the College of Nanoscale Science and Engineering (CNSE) presented at the 2013 EUVL Workshop indicated that, as a measure of electron blur, 50โ100 eV electrons easily penetrated beyond 10 nm of resist thickness in PMMA or a commercial resist. Furthermore dielectric breakdown discharge is possible.<ref>{{cite web |first1=G. |last1=Denbeaux |first2=J. |last2=Torok |first3=R. |last3=Del Re |first4=H. |last4=Herbol |first5=S. |last5=Das |first6=I. |last6=Bocharova |first7=A. |last7=Paolucci |first8=L.E. |last8=Ocola |first9=C. |last9=Ventrice Jr. |first10=E. |last10=Lifshin |first11=R.L. |last11=Brainard |title=Measurement of the role of secondary electrons in EUV resist exposures |date=2013 |work=International Workshop on EUV Lithography |url=https://www.euvlitho.com/2013/P29.PDF}}</ref> More recent studies have indicated that 20 nm resist thickness could be penetrated by low energy electrons (of sufficient dose) and sub-20 nm half-pitch electron-beam lithography already required double patterning.<ref>[https://semiwiki.com/lithography/294565-the-complexities-of-the-resolution-limits-of-advanced-lithography/ Complexities of the Resolution Limits of Advanced Lithography]</ref><ref>[https://www.linkedin.com/pulse/complexities-resolution-limits-advanced-lithography-frederick-chen Resolution Limits]</ref> As of 2022, a state-of-the-art electron multi-beam writer achieves about a 20 nm resolution.<ref>{{cite video |url=https://www.youtube.com/watch?v=WWF3VdI2E7A |title=Electron Blur Impact on Electron Beam and EUV Lithography |first=Frederick |last=Chen |date=2023}}</ref><ref>{{cite conference |first1=M. |last1=Chandramouli |first2=B. |last2=Liu |first3=Z. |last3=Alberti |first4=F. |last4=Abboud |first5=G. |last5=Hochleitner |first6=W. |last6=Wroczewski |first7=S. |last7=Kuhn |first8=C. |last8=Klein |first9=E. |last9=Platzgummer |title=Multibeam mask requirements for advanced EUV patterning |book-title=Photomask Technology |volume=12293 |series=SPIE Proceedings |date=2022 |doi=10.1117/12.2645895 |pages=122930O }}</ref> ===Scattering=== In addition to producing secondary electrons, primary electrons from the incident beam with sufficient energy to penetrate the resist can be multiply scattered over large distances from underlying films and/or the substrate. This leads to exposure of areas at a significant distance from the desired exposure location. For thicker resists, as the primary electrons move forward, they have an increasing opportunity to scatter laterally from the beam-defined location. This scattering is called '''forward scattering'''. Sometimes the primary electrons are scattered at angles exceeding 90 degrees, i.e., they no longer advance further into the resist. These electrons are called '''[[backscattering|backscattered electrons]]''' and have the same effect as long-range [[Lens flare|flare]] in optical projection systems. A large enough dose of backscattered electrons can lead to complete exposure of resist over an area much larger than defined by the beam spot. ===Proximity effect=== The smallest features produced by electron-beam lithography have generally been isolated features, as nested features exacerbate the [[Proximity effect (electron beam lithography)|proximity effect]], whereby electrons from exposure of an adjacent region spill over into the exposure of the currently written feature, effectively enlarging its image, and reducing its contrast, i.e., difference between maximum and minimum intensity. Hence, nested feature resolution is harder to control. For most resists, it is difficult to go below 25 nm lines and spaces, and a limit of 20 nm lines and spaces has been found.<ref name="liddle">{{cite journal|title=Resist Requirements and Limitations for Nanoscale Electron-Beam Patterning|author=J. A. Liddle|journal=Mater. Res. Soc. Symp. Proc.|volume=739|issue=19|year=2003|pages=19โ30|url=http://www.mrs.org/s_mrs/sec_subscribe.asp?CID=2557&DID=118066&action=detail|display-authors=1|author2=<Please add first missing authors to populate metadata.>}}{{Dead link|date=March 2024 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> In actuality, though, the range of secondary electron scattering is quite far, sometimes exceeding 100 nm,<ref>{{cite journal|doi=10.1016/S0167-9317(02)00531-2|title=The inclusion of secondary electrons and Bremsstrahlung X-rays in an electron beam resist model|year=2002|last1=Ivin|first1=V|journal=Microelectronic Engineering|volume=61โ62|page=343 }}</ref> but becoming very significant below 30 nm.<ref>{{cite journal|doi=10.1143/JJAP.36.7552|title=Novel Proximity Effect Including Pattern-Dependent Resist Development in Electron Beam Nanolithography|year=1997|last1=Yamazaki|first1=Kenji|last2=Kurihara|first2=Kenji|last3=Yamaguchi|first3=Toru|last4=Namatsu|first4=Hideo|last5=Nagase|first5=Masao|journal=Japanese Journal of Applied Physics|volume=36|issue=12B|page=7552 |bibcode = 1997JaJAP..36.7552Y |s2cid=250783039 }}</ref> The proximity effect is also manifest by secondary electrons leaving the top surface of the resist and then returning some tens of nanometers distance away.<ref>{{cite journal|doi=10.1088/0953-8984/10/26/010|year=1998|last1=Renoud|first1=R|last2=Attard|first2=C|last3=Ganachaud|first3=J-P|last4=Bartholome|first4=S|last5=Dubus|first5=A|journal=Journal of Physics: Condensed Matter|volume=10|page=5821|bibcode = 1998JPCM...10.5821R|issue=26|title=Influence on the secondary electron yield of the space charge induced in an insulating target by an electron beam |s2cid=250739239 }}</ref> Proximity effects (due to electron scattering) can be addressed by solving the [[inverse problem]] and calculating the exposure function ''E(x,y)'' that leads to a dose distribution as close as possible to the desired dose ''D(x,y)'' when [[convolution|convolved]] by the scattering distribution [[point spread function]] ''PSF(x,y)''. However, it must be remembered that an error in the applied dose (e.g., from shot noise) would cause the proximity effect correction to fail.
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