Line Edge Roughness, part 1

Transcription

1 Tutor56c.doc: Versio 11/8/06 Lie Edge Roughess, part 1 T h e L i t h o g r a p h E x p e r t (Februar 007) While resolutio is commol discussed relative to optical limits, ad sometimes eve resist cotrast limitatios, it is quite possible that the ultimate limit of resolutio will come from lie edge roughess. Whe variatios i the width of a resist feature occur quickl over the legth of the feature, this variatio is called liewidth roughess (see Figure 1). Whe examiig these variatios alog just oe edge it is called lie edge roughess (LER). LER becomes importat for feature sizes o the order of 100 m or less, ad ca become the most sigificat source of liewidth cotrol problems for features below 50 m. LER is caused b a umber of statisticall fluctuatig effects at these small dimesios such as shot oise (photo flux variatios), statistical distributios of chemical species i the resist (such as photoacid geerators), the radom walk ature of acid diffusio durig chemical amplificatio, ad the o-zero size of resist polmers beig dissolved durig developmet. It is importat to ote that most theoretical descriptios of lithograph make a extremel fudametal (ad mostl ustated) assumptio about the phsical world: the so-called cotiuum approximatio. Eve though light eerg is quatized ito photos ad chemical cocetratios are quatized ito spatiall distributed molecules, the descriptios of aerial images, latet images, ad resist developmet rates usuall igore the discrete ature of these fudametal uits ad use istead cotiuous mathematical fuctios. A cotiuum mathematical model predictig the shape of the acid latet image after exposure is fudametall icapable of predictig how statistical variatios i photo ad molecule umbers might cotribute to LER. Thus, whe describig lithographic behavior at the aometer level, a alterate approach, ad i a ver real sese a more fudametal approach, is to build the quatizatio of light as photos ad matter as atoms ad molecules directl ito the models used. Such a approach is called stochastic modelig, ad ivolves the use of radom variables ad probabilit desit fuctios to describe the situatio. Such a probabilistic descriptio, however, caot state what will happe with certait, but ol give probabilities that deped o circumstaces. I the first part of this series o LER, I ll tackle exposure ad derive the statistics of the resultig acid cocetratio. Photo Shot Noise To begi, cosider a light source that radoml emits photos at a average rate of L photos per uit time ito some area A. Assume further that each emissio evet is idepedet. Over some small time iterval dt (smaller tha 1/L ad small eough so that it is essetiall impossible for two photos to be emitted durig that iterval), either a photo is emitted or it is ot (a biar propositio). The probabilit that a photo will be emitted durig this iterval will be Ldt. Cosider ow some log time T = Ndt (» dt). What ca we expect for the umber of photos emitted durig the period T? This basic problem is called a Beroulli trial ad the resultig probabilit distributio is the well-kow biomial distributio. If NLdt = TL remais

2 fiite as N goes to ifiit, the biomial distributio coverges to a more maageable equatio for the probabilit of fidig () photos called the Poisso distributio: ( TL) TL P( ) = e (1)! The Poisso distributio ca be used to derive the statistical properties of photo emissio. The expectatio value of (that is, the mea umber of photos that will be emitted i a time iterval T) is TL (a ver reasoable result sice L was defied as the average rate of photo emissio). The variace (the stadard deviatio squared) is also TL. Note that the Poisso distributio differs from the familiar ormal (or Gaussia) probabilit distributio; there is ol oe free parameter the average, TL. To use these statistical properties, we must covert from umber of photos to a more useful measure, itesit. If photos cross a area A over a time iterval T, the itesit of light will be hc I =. () TA λ The stadard deviatio of the itesit ca the be computed from the properties of the Poisso distributio. σ I I σ = = 1 = I 1/ λ TA hc (3) As this equatio shows, the fractioal ucertait of the itesit grows as the mea umber of photos ( I TA ) is reduced, a pheomeo kow as shot oise. The shot oise (the relative ucertait i the actual itesit that the resist will see) icreases with decreasig itesit, exposure time, ad area of cocer. As a example, cosider a 193 m exposure of a resist with a dose-to-clear of 10 mj/cm. At the resist edge, the mea exposure eerg ( = I T ) will be o the order of the dose-to-clear. At this wavelegth, the eerg of oe photo, hc/λ, is about 1.03 X J. For a area of 1 m X 1 m, the mea umber of photos durig the exposure, from equatio (), is about 97. The stadard deviatio is about 10, or about 10% of the average. For a area of 10 m square, the umber of photos icreases b a factor of 100, ad the relative stadard deviatio decreases b a factor of 10, to about 1%. Sice these are tpical values for a 193 m lithograph process, we ca see that shot oise cotributes a oticeable amout of ucertait as to the actual dose see b the photoresist whe lookig at legth scales less tha about 10 m. For Extreme Ultraviolet (EUV) lithograph, the situatio will be cosiderabl worse. At a wavelegth of 13 m, the eerg of oe photo will be 1.53 X J, more tha fiftee times greater tha at 193 m. Also, the goal for resist sesitivit will be to have EUV resists that are 4 times more sesitive tha 193 m resists (though it is uclear whether this goal will be

3 achieved). Thus, the umber of photos will be times less for EUV tha 193 m lithograph. A 1 m X 1 m area will see ol 3 photos, ad a 10 m square area will see o the order of 00 photos, with a stadard deviatio of 7%. Chemical Cocetratio Iterestigl, chemical cocetratio exhibits a ucertait ot ulike the icidece of photos. As metioed above, there reall is o such thig as cocetratio at a poit i space sice the chemical species is formed b discrete molecules, ot a cotiuous medium. Cocetratio, the average umber of molecules per uit volume, exhibits coutig statistics idetical to photo emissio. Let C be the average umber of molecules per uit volume, ad dv a volume small eough so that at most oe molecule ma be foud i it. The probabilit of fidig a molecule i that volume is just CdV. For some larger volume V, the probabilit of fidig exactl molecules i that volume will be give b a biomial distributio exactl equivalet to that for photo coutig. Ad, for a reasoabl large volume (CV > 1), this biomial distributio will also be well approximated b a Poisso distributio. The average umber of molecules i the volume will be CV, ad the variace will also be CV. The relative ucertait i the umber of molecules i a certai volume will be, like for our photo statistics, equal to oe over the square root of the umber of molecules i that volume of iterest. As a example, cosider a 193m resist that has a iitial PAG cocetratio of 3% b weight, or a cocetratio of about 0.07 mole/liter (correspodig to a desit of 1. g/ml ad a PAG molecular weight of 500 g/mole). Covertig from moles to molecules with Avogadro s umber, this correspods to 0.04 molecules of PAG per cubic aometer. I a volume of 10 m cubed, the mea umber of PAG molecules will be 4. The stadard deviatio will be 6.5 molecules, or about 15%. For 48 m resists, the PAG loadig is tpicall 3 times higher or more, so that closer to 150 PAG molecules might be foud i a 10 m cubed volume, for a stadard deviatio of 8%. Photo Absorptio ad Exposure of Oe PAG Molecule What is the probabilit that a photo will be absorbed b a molecule of light sesitive material i the resist? Further, what is the probabilit that a molecule of sesitizer will react to form a acid? As discussed above, there will be a statistical ucertait i the umber of photos i a give regio of resist, a statistical ucertait i the umber of PAG molecules, ad additioall a ew statistical ucertait i the absorptio ad exposure evet itself. Cosider a sigle molecule of PAG. First order kietics of exposure ca be used to derive the cocetratio of PAG remaiig after exposure (ad, as well, the cocetratio of acid geerated) i the cotiuum approximatio. From a stochastic modelig perspective, this kietic result represets a probabilit desit fuctio for reactio. Let be a radom variable that represets whether a give sigle PAG molecule was coverted to acid or remais uexposed b the ed of the exposure process (this is a biar propositio either the PAG molecule reacts or it does t). Thus = 1 meas a acid has bee geerated (PAG has reacted), ad = 0 meas the PAG has ot geerated acid (either it was ot bee exposed or reacted differetl). A kietic aalsis of exposure gives us the probabilit for each of these states, give a certai itesit i the resist I:

4 CIt P( = 0 I) = e, CIt P( = 1 I) = 1 e (4) where C is the exposure rate costat ad t is the exposure time. For a give itesit, the mea value ad variace of ca be calculated usig the defiitio of a discrete probabilit expectatio value. However, we kow from our discussio of photo coutig statistics that I is a probabilistic fuctio. Thus, the mea ad variace of must take ito accout this probabilistic ature. Lettig be the umber of photos exposig a give area A over a exposure time t, it will be useful to defie a ew costat i terms of photo umber : CIt = ψ which leads to hc C Φσ ψ = = M abs (5) λ A A The term ψ is the exposure shot oise coefficiet, ad is equal to the acid geeratio quatum efficiec (Φ) multiplied b the ratio of the PAG absorptio cross-sectio (σ M-abs ) to the area of statistical iterest. Sice the quatum efficiec is tpicall i the rage ad the PAG absorptio cross-sectio is o the order of 1 Å for 193 m resists, for most areas of iterest this exposure shot oise coefficiet will be much less tha 1. For EUV resists, the PAG absorptio cross-sectio is expected to be a bit larger, o the order of 30 Å. Usig this exposure shot oise coefficiet to covert itesit to umber of photos ad the emploig the properties of the Poisso distributio, the average probabilit of geeratig acid becomes: ψ ( 1 e ) = 1 e (6) As ofte happes whe takig statistical distributios ito accout, the mea value of the output of the fuctio is ot equal to the fuctio evaluated at the mea value of the iput. The mea value of is alwas less tha the value of the fuctio evaluated at the mea value of the itesit, though the differece becomes small for ψ «1. The impact of photo shot oise o the variace of the acid geeratio probabilit () ca be calculated, givig ( ) σ = = 1 (7) (This result is alwas true for a biar variable, beig a fudametal result for a biomial distributio). PAG Cocetratio After Exposure The mea value ad ucertait of the state of oe acid molecule after exposure ca ow be traslated ito a mea ad ucertait of the overall acid cocetratio after exposure. Cosider a volume V that iitiall cotais some umber 0 of PAG molecules. After exposure, the umber of photogeerated acid molecules Y will be

5 0 i i= 1 Y = (8) where i is the discrete radom variable represetig the exposure state of the i th molecule foud i this volume (all of which are assumed to be idepedet). For a give 0, the mea ad variace of Y ca be readil computed. But 0 itself has a Poisso statistical distributio, as discussed above. The mea value of Y icludig the statistical variatio of 0, is Likewise, the variace of Y ca be computed to give [] Y Y = 0 (9) 0 0 σ = σ + σ (10) Usig equatio (7) ad relatig the umber of acid molecules per uit volume Y to the cocetratio of acid H, σ Y H 1 1 = σ = = Y H (11) Y The fial result, which accouts for photo fluctuatios, ucertait i the iitial cocetratio of photoacid geerator, ad the probabilistic variatios i the exposure reactio itself, is reasoabl ituitive. The relative ucertait i the resultig acid cocetratio after exposure is equal to oe over the square root of the mea umber of acid molecules geerated withi the volume of iterest. For large volumes ad reasoabl large exposure doses, the umber of acid molecules geerated is large ad the statistical ucertait i the acid cocetratio becomes small. For small volumes or low doses, a small umber of photogeerated acid molecules results i a large ucertait i the actual umber withi that volume. For the case of the 10 m cube of 193 m resist above, the 1 σ ucertait i iitial acid cocetratio ear the resist edge will be >0%! I the ext editio of this colum, we ll look at how this ucertait i acid cocetratio propagates to a ucertait i the cocetratio of blocked polmer after the reactio-diffusio of the post-exposure bake. The additio of developmet will complete the stor, allowig us to predict the statistics of LER. 0 Refereces 1. Chris A. Mack, Field Guide to Optical Lithograph, SPIE Field Guide Series Vol. FG06, (Belligham, WA: 006).. The derivatio of this result is quite mess, so I leave the mathematics to m forthcomig lithograph textbook.

6 Figure 1 SEM pictures of photoresist features exhibitig lie edge roughess (from Ref. 1).