Slowing-down processes, energy deposition, sputtering and desorption in ion and electron interactions with solids

Transcription

1 Slowing-down procsss, nrgy dposition, sputtring and dsorption in ion and lctron intractions with solids J. Schou Risø National Laboratory, DK-4000 Roskild, Dnmark Abstract Th slowing-down procsss of ions and lctrons in mattr ar rviwd with spcial mphasis on th stopping cross-sction. Th rlationships btwn lastic (knock-on) sputtring and nuclar stopping as wll as btwn lctronic sputtring and lctronic stopping ar discussd. 1 Introduction Many phnomna which tak plac during ion, lctron, and photon bombardmnt of solids ar closly connctd. Ths phnomna includ nrgy loss of chargd particls in solids, nrgy dposition and jction of scondary particls during bombardmnt by chargd particls. Whil th trm sputtring usually compriss particls jctd as a rsult of momntum transfr to targt particls or lctronic transitions, dsorption usually rfrs to th rmoval of lss than a monolayr by lctronic transitions at th vry surfac. Sputtring as mass rmoval from th cathod during a gas discharg was discovrd around 1850 [1], but th procss was not undrstood, sinc th concpt of chargd particls was not yt stablishd. Th discovry of radioactivity ld to a big stp forward towards wll-dfind ion sourcs []. Ruthrford s famous work on th structur of th atom immdiatly stimulatd work on particl slowing down and a numbr of paprs on nrgy loss basd on classical thory appard during th following yars. Aftr quantum mchanical tratmnts bcam availabl, stimats on light ion and lctron nrgy loss in mattr could also b prformd, but rfinmnts of th thortical framwork ar still continuing [ 3]. Sputtring of nutrals by kv ions was largly xplaind in th yars [4], whras lctronic sputtring (of insulators) is still a partly unrsolvd issu for nonlmntal solids. Both typs of sputtring could b xplaind by a rlationship to th rlvant part of th stopping cross-sction (to b xplaind blow). Scondary lctron mission inducd by ions was discovrd around 1900, but th thortical tratmnts did not appar until [5]. Prcis masurmnts of th dsorption rats wr not possibl until rliabl ultra-high vacuum systms wr availabl. Th lading thoris wr mad around 1960 by Mnzl, Gomr, and Rdhad [6]. Stopping forc (de/dx).1 Ion slowing down Th ky quantity in nrgy loss considrations is th stopping forc de/dx (which in th past was namd stopping powr). This can b considrd as th forc that th mdium xrts on th pntrating particl: de/dx = NS(E), (1) whr N is th numbr dnsity of atoms in th mdium and S(E) th stopping cross-sction, in which th dpndnc on th kintic nrgy E of th primary particl is xplicitly writtn. Th stopping cross-sction is a dnsity-indpndnt quantity xcpt for ultrarlativistic nrgis and has th 169

2 J. SCHOU dimnsion [nrgy ara]. In fact, many tabls do not distinguish btwn stopping forc (powr) with th dimnsion [nrgy/lngth] and stopping cross-sction,.g., Rf. [7]. Th thortical framwork for a gnral tratmnt of nrgy loss to th nucli as wll as to th lctrons, i.., th lctronic systm, was comprhnsivly tratd by Bohr [8] and Lindhard t al., [9] in a numbr of important paprs. Th collisions btwn th primary ions and th atoms in a solid can b dividd into collisions btwn th primary particl and th nucli, and thos btwn th primary and th lctrons. Th first collisions tak plac for small impact paramtrs and lad to a larg-angl scattring procss, whras th lattr ons lad to nrgy loss without any significant dflction of th primary particl. Th stopping cross-sction S(E) can b split up into S(E) = S n (E) + S (E) () () whr S n is th nuclar stopping cross-sction and S th lctronic stopping cross-sction. Th trm nuclar is mislading, sinc th primary ion intracts with th scrnd nucli rathr than th bar nucli. Th two stopping cross-sctions drawn from th tabulations in Rf. [10] ar shown in Fig. 1. Th four rgims covr th low-nrgy (I), intrmdiat (II), high-nrgy (III) and ultrarlativistic rgim (IV) (not shown in Fig. 1). Th nuclar stopping is dominant in rgim (I), but dcrass gradually to lss than a factor of th ordr M p /m ~ 000 than th lctronic stopping (M p is th proton mass) in rgim (II). In rgim (III) th lctronic cross-sction dcrass monotonously. An important scaling paramtr is Lindhard s rducd nrgy ε : 3.53ME ε = ZZ M M Z Z /3 /3 1/ 1 ( 1+ )( 1 + ) (3) whr E is th nrgy in kv and th masss M 1 and M ar in amu. Lindhard and coauthors [11] showd that th nuclar stopping for all targt bam combinations could b xprssd as πalγn NSn( E) = sn( ε ) ( ε / E) (4) whr γ = 4M 1 M /(M 1 + M ) (5) is dtrmind by th maximum nrgy transfr γe from a particl of mass M 1 with an nrgy E to a particl at rst with mass M and by Lindhard s scrning paramtr /3 /3 ( ) 1/ a = a Z + Z. (6) L B 1 Th absolut magnitud is dtrmind partly by th factor (ε/e) in th dnominator, which mans that for havy atoms on a havy targt th factor bcoms small and, in turn, lads to a larg nuclar stopping. Equation (5) also dmonstrats that ε dcrass with incrasing atomic numbr (and mass) of th projctil. Th maximum of th nuclar stopping, which for th classical Thomas Frmi modl occurrd at ε 0.3, is thrfor shiftd to highr nrgis for havy projctils. Th rducd nuclar stopping powr is usually calculatd from a rprsntativ potntial, th so-calld KrC potntial, for which svral rfind approximations xist [1]. Th nuclar stopping shown in Fig. 1 is computd on th basis of th potntial givn in Rf. [1], which also is th undrlying basis for Rf. [10]. 170

3 SLOWING-DOWN PROCESSES, ENERGY DEPOSITION, SPUTTERING AND DESORPTION IN ION AND ELECTRON INTERACTIONS WITH SOLIDS Ar Cu Stopping Cross Sction [ Vcm /10 15 Cu-atoms ] Elctronic Nuclar Rgim I Rgim II Rgim III 0 1E Enrgy [MV] Fig. 1: Elctronic and nuclar stopping cross-sction for Ar ions incidnt on Cu. Data from Rf. [10]. Th lctronic stopping in th low-nrgy and intrmdiat rgims (I and II) can b dtrmind by Lindhard Scharff tratmnt [13]. Th prdictions from th Lindhard Scharff modl ar usually corrct within a factor of two, and can still b considrd as a convnint rfrnc standard. In rducd units th lctronic stopping can b xprssd as S (ε) = k L ε 1/ (7) whr th constant of proportionality k L dpnds on th atomic numbr and th mass of th bam atom and th targt atom [9,13]. Excpt for vry light ions on havy targts, k L Th lctronic stopping cross-sction is thus proportional to th vlocity and can b convrtd to ral units with th sam factor as th rducd nuclar stopping cross-sction in Eq. (4). A numbr of tratmnts with improvd accuracy hav appard [7,10,1,14], but th tratmnt of slow ions is still a difficult task [15]. With incrasing vlocity th projctil loss th lctrons and at high vlocitis is compltly strippd. At intrmdiat vlocitis in rgim (II) th projctil lctrons with vlocitis xcding th projctil vlocity v will stick to th projctil, whil th slowr ons, blonging to th outr shlls, will b strippd. Sigmund [16] has stimatd th broad maximum of th lctronic stopping crosssction to b approximatly at a vlocity of v B Z /3 1, whr v B is th Bohr vlocity (v B m/s). For a proton (Z 1 = 1) this corrsponds to a primary nrgy of 5 kv. At high nrgis (rgim III) th lctronic stopping is dtrmind by Bth s or Bohr s tratmnt []. Th dcisiv point is whthr or not th problm can b tratd by classical thory. This is possibl for larg valus of Bohr s paramtr [8] Z1vB k = (8) ( h/ π ) v whr h is Planck s constant. In th quantum mchanical limit k 1 Bth s formula basd on a firstordr prturbation tratmnt givs th high-nrgy xprssion in th non-rlativistic cas valid for light ions: 171

4 J. SCHOU 4π ZZ mv S ( ) = ln (9) I 4 1 E mv and I Z 10 V is th man ionization potntial. For rlativistic corrctions th radr is rfrrd to Rf. []. On nots that th condition for quantum tratmnt cannot b xpctd to hold for slow projctils or projctils of high atomic numbr. In this cas th logarithm in Eq. (9) is rplacd by Bohr s xprssion: S 4π ZZ Cm v =. (10) 4 Z 3 1 ( E) ln mv ν = 1 Z 1 ων Th major diffrnc is that Z 1 ntrs into th argumnt of th logarithm, and thrfor influncs th position of th stopping maximum. Hr ω ν is a charactristic frquncy of ach lctron lvl of th targt atom, sinc th drivation is prformd for a harmonic oscillator []. A complicating fatur is th charg stat of th incoming ions. According to Bohr s stimat [8], th avrag charg stat of havy ions is Z ν = (11) ν * 1/3 1 Z1 for th vlocity rang whr 1 < Z 1 * < Z 1 /. It mans that th avrag ion charg stat incrass with th ion vlocity. Howvr, rgardlss of th initial ion charg stat, which may b vry far from th quilibrium charg stat, an ion bam will approach charg stat quilibrium aftr having pntratd a fw layrs from th surfac. Thr ar dviations from th lmntal stopping powrs, in particular for slow ions. Th gas solid diffrnc is largst for low-lmnts, but xcds rarly a 10% corrction. For chmical componnts th stopping forc is clos to th wightd sum of th componnts, but also hr only minor dviations may appar []. Th printd tabulations, Rfs. [7] and [1], hav bn rplacd by th popular cod SRIM [10]. SRIM is basd on an ffctiv charg stat which can b adjustd to xprimntal valus but is not supportd by a firmly stablishd thortical basis. Also Paul [14] and Paul and Schinnr [17] hav publishd xtnsiv tabulations basd on xprimntal stopping cross-sctions.. Elctron slowing down In contrast to ion slowing down, lctron slowing down has not undrgon any substantial improvmnt sinc th 1980s xcpt for dvlopmnt of lctron cods for microscopy [18]. A numbr of computations hav bn incorporatd in Fig.. For th simpl cas of a positron with kintic nrgy E th Bth tratmnt givs 4 B ln mv 4π S = Z. (1) mv I For an lctron, whr on cannot distinguish btwn a scattrd lctron and a tru scondary on, th stopping cross-sction bcoms slightly modifid: whr a = in th non-rlativistic limit. S 4 4π amv = Z ln, (13) mv I 17

5 SLOWING-DOWN PROCESSES, ENERGY DEPOSITION, SPUTTERING AND DESORPTION IN ION AND ELECTRON INTERACTIONS WITH SOLIDS Th stopping forc up to 10 kv has bn compild by Schou in Fig. [19]. Th Bth limit is sn clarly at high nrgis at which S ~ Z /E. Howvr, at low nrgy th Bth formula (13) is no longr valid, and th stopping is largly dtrmind by dnsity of fr lctrons in th matrial (for xampl, aluminium has th largst stopping bcaus of th high dnsity of fr (conduction) lctrons). Fig. : Stopping cross-sction for lctrons in slctd lmnts. From J. Schou [0]. (A part of th figur is publishd in Rf. [19].) Rang and stopping forc calculations for lctrons of nrgy highr than 10 kv hav bn prformd by Brgr t al. [1]. For nrgis abov 10 kv, rlativistic ffcts play a major rol and th stopping cross-sction falls off until about 1 MV. Abov 1 MV th radiativ componnt of th stopping forc bcoms dominant, and th total stopping incrass strongly (Fig. 3). Th nrgy loss by collisions is proportional to Z and incrass logarithmically with th nrgy, whras th loss to radiation largly is proportional to Z and incrass practically linarly with th nrgy E []. 100 Stopping cross sction [V cm /10 15 particls] Collisional stopping H O Total stopping H O Collisional stopping Au Total stopping Au Enrgy [MV] Fig. 3: Stopping of MV lctrons in liquid watr and gold on th basis of th data in Rf. [1] 173

6 J. SCHOU.3 Enrgy dposition Th nrgy dpositd by an ion is distributd in lctronic xcitations and kintic nrgy that nds up in atomic motion and damag. Sinc th rcoils may hav a considrabl rang, th nrgy may b transportd far away from th collision point. This mans that th nrgy dposition can b substantially diffrnt from an nrgy-loss distribution. Th thortical work on th spatial distribution F D (E) of nrgy dpositd in atomic collisions was initially prformd by Wintrbon t al. [3], and latr publishd as xtnsiv tabulations [4]. Distributions D (E) for th lctronically dpositd nrgy by ions and thir rcoils wr computd by Schou [5]. Th nrgy dpositd by lctrons,.g., in th kv rang, is usually quit broad with a Gaussian profil, bcaus of th pronouncd scattring of th lctrons. Dposition profils hav bn publishd by Valkalahti t al., [6]. Similar profils for MV lctrons hav bn computd by Andro [7]. 3 Sputtring 3.1 Erosion procsss Matrial jction from a solid can b inducd by a numbr of procsss which may vn oprat simultanously (Fig. 4): a) bam-inducd vaporation, b) collisional (lastic, knock-on) sputtring, c) lctronic sputtring, and d) dsorption of thin layrs. Sputtring is th rosion of matrial by singl-particl impact in contrast to vaporation producd by many particls in bam-inducd hating. Dsorption is strictly spaking th rmoval of parts of a monolayr or full monolayrs dpositd on a diffrnt substrat. It is closly rlatd to sputtring by lctronic transitions, i.., lctronic sputtring. In th following only b) and c) will b tratd. Bam-inducd vaporation is tratd by Schou in Rf. [8] and dsorption in Rf. [6]. Fig. 4: A survy of diffrnt rosion procsss. From Schou [9]. 174

7 SLOWING-DOWN PROCESSES, ENERGY DEPOSITION, SPUTTERING AND DESORPTION IN ION AND ELECTRON INTERACTIONS WITH SOLIDS Rfrnc [1] is still a classical rfrnc for sputtring, whil a mor rcnt collction of paprs on collisional and lctronic sputtring can b found in Rf. [4]. Th nrgy E dpositd by a primary ion can b dividd into th nrgy which is dpositd into lctronic xcitations η(e) and into atomic motion ν(e) according to Rf. [9]. Ths quantitis can b xprssd as th intgral ovr th distributions F D (E,x) and D (E,x): D(, )d (, )d ν( ) η( ). (14) E = F E x x+ D E x x= E + E As mntiond abov, th quation rflcts th fact that th primary particl may gnrat rcoils which undrgo nuclar stopping as wll, such that nrgy is transportd away from th point of collision. It also mans that th surfac valu F D (E,0) << NS n (E) in many cass. In th cas of vry light projctils th probability for backscattring incrass strongly, and in this cas F D (E,0) >> NS n (E). 3. Collisional (lastic) sputtring This typ of sputtring is inducd by th momntum transfr from th primary particl to th targt atoms. Sinc th atoms ar hit dirctly, th trms knock-on and ballistic sputtring hav bn usd. A pictur of th procsss is shown in Fig. 4. Th primary particl initiats a collision cascad via scondary or highr ordr collisions. Sinc th kintic nrgy of th atoms st in motion is much largr than th binding nrgy of th matrial xcpt for th last stag of th cascad, th standard tratmnt for mtals can b largly xtndd to smiconductors and insulators. Th standard thory for sputtring is Sigmund s analytical thory [30] which is basd on Boltzmann transport thory. Th solution is obtaind in th limit for high primary nrgy compard with th instantanous nrgy of th cascad atoms. This corrsponds to isotropic motion of th atoms in th solid. Th tratmnt accounts for low collision dnsitis in which th struck atoms in th cascad ar at rst bfor th collision. 100 Sputtring Yild [atoms/au + ] Au + projctil Gold 0 Silvr Thory Gold Thory Silvr Enrgy [kv] Fig. 5: Sputtring yild Y for Au ions on Au (black squars and lin) and Ag (rd circls and lin). Th solid lins ar th prdictd sputtring yild from Eq. (18). Th dviations from thory occur in th so-calld spik rgion, whr th nuclar stopping is largst, in particular for th cas Au+ ions on Au. Data from Bounau t al. [31]. Th (back)sputtring yild Y from a plan surfac is givn as Y =Λ F D ( E, x) = 0, (15) 175

8 J. SCHOU whr F D (E,0) is th surfac valu of th spatial distribution of nrgy dpositd into atomic motion. Th constant Λ dpnds only on th matrial proprtis, th atomic dnsity N, and th sublimation nrgy U 0 : Λ = 3/(4π NC 0 U 0 ). (16) Th cross-sction C 0 ( m ) originats from a low-nrgy intratomic Born Mayr potntial and is common for all targt matrials in this approximation [1,30]. From dimnsional argumnts on may writ F D (E, 0) = αns n (E), (17) whr α is a function of th mass ratio M /M 1 (and of th angl θ of incidnc which is not mntiond xplicitly hr). Hr α is rlativly insnsitiv for variations in th primary nrgy E, and can b wll approximatd by α 0.17 for most bam atom targt combinations, whr M 1 >> M. This low valu compard with unity also mans that most of th nrgy dpositd by th primary ion is transportd away from th targt surfac by rcoil atoms as discussd in th prvious sction. By combining Eqs. (16) and (17) on arrivs at th wll-known formula for th sputtring yild, in which no adjustabl paramtrs ntr [1,30] Y = ΛαNS n (E). (18) Th linar-cascad thory has convincingly mad it possibl to prdict important faturs of sputtring fairly wll, and no othr thortical tratmnt has rachd a comparabl lvl [1,4,30]. Two bam atom targt atom combinations basd on rcnt data ar shown in Fig. 5. An important xcption from linar cascad thory occurs if th rcoil cascads bcom too dns. This cas, th spik rgim, is charactrizd by a high collision dnsity such that a moving atom in a cascad hits othr atoms which hav alrady bn st in motion. Th spik yild is typically much largr than that from a linar collision cascad [1]. This is also sn in Fig. 5 for Au ions incidnt on th havist targt matrial, gold, from 80 kv up to 500 kv. 3.3 Elctronic sputtring Sputtring by lctronic transitions, i.., lctronic sputtring, taks plac primarily for insulating matrials. A typical cas is a volatil, frozn gas bombardd by light ions or lctrons, but roomtmpratur insulators may also xhibit lctronic sputtring [8,3]. Th atomic motion is gnratd from rpulsiv potntials that aris during lctronic d-xcitation. Th procsss dpnd spcifically on ach matrial and can b xtrmly complicatd for chmical compounds. Th mchanism of lctronic sputtring for th solid rar gass is now wll undrstood, but th procsss hav not bn fully idntifid vn for a comparativly simpl frozn matrial such as watr ic [33]. In analogy to Eq. (16) th yild can b xprssd by Y = Λ(1/)D (E, x = 0)(E s /W), (19) whr D (E, 0) is th surfac valu (x = 0) of th nrgy dpositd into lctronic transitions, E s th nrgy producd by rpulsiv procsss, typically a fw V, pr ion gnratd by th primary ion, and W th nrgy rquird to produc an lctron ion pair (which is usually about 30 V). Sinc th nrgy rlas from ths non-radiativ transitions is compltly isotropic, on arrivs at th factor (1/) in Eq. (19). Th quation is basd on th assumption that th nrgy E s is sufficintly larg to gnrat a low-nrgy cascad. Th factor (E s /W) is usually much lss than unity, sinc a considrabl fraction of th nrgy is lost by luminscnc from radiativ transitions. Formula (19) cannot dscrib sputtring from a solid with mobil xcitations,.g., solid rar gass, but can giv a qualitativ trnd for othr frozn gass [8]. 176

9 SLOWING-DOWN PROCESSES, ENERGY DEPOSITION, SPUTTERING AND DESORPTION IN ION AND ELECTRON INTERACTIONS WITH SOLIDS Rfrncs [1] Sputtring by Particl Bombardmnt, Ed. R. Bhrisch (Springr, Brlin Hidlbrg, 1981). [] P. Sigmund, Particl Pntration and Radiation Effcts (Springr, Brlin Hidlbrg, 006). [3] P. Sigmund, Stopping of Havy Ions (Springr, Brlin Hidlbrg, 004). [4] Fundamntal Procsss in Sputtring of Atoms and Molculs (SPUT 9), Mat. Fys. Mdd. Dan. Vid. Slsk. 43 (1993) 1, Ed. P. Sigmund. [5] W. O. Hofr, in: Fundamntal Elctron and Ion Bam Intractions with Solids for Microscopy, Microanalysis and Microlithography, Eds. J. Schou, P. Kruit and D.E. Nwbury (Scanning Microscopy Intrnational, AMF O Har Chicago, 1990), p. 65. [6] Dsorption Inducd by Elctronic Transitions, DIET1, Eds. N. H. Tolk, M. M. Traum, J. C. Tully and T. E. Mady (Springr, Brlin Hidlbrg, 1983). [7] H. H. Andrsn and J. F. Ziglr, Hydrogn Stopping Powrs and Rangs in All Elmnts (Prgamon, Nw York, 1977). [8] N. Bohr, Mat. Fys. Mdd. Dan. Vid. Slsk. 18, no. 8 (1948). [9] J. Lindhard, V. Nilsn, M. Scharff and P. V. Thomsn, Mat. Fys. Mdd. Dan. Vid. Slsk. 33, no. 10 (1963). [10] J. F. Ziglr, SRIM ( [11] J. Lindhard, V. Nilsn and M. Scharff, Mat. Fys. Mdd. Dan. Vid. Slsk. 36, no. 10 (1968). [1] J. F. Ziglr, Handbook of Stopping Cross Sction for Enrgtic Ions in All Elmnts (Prgamon, Nw York, 1980). [13] J. Lindhard and M. Scharff, Phys. Rv. 14 (1961) 18. [14] H. Paul, Stopping powr graphs ( [15] P. Sigmund, to b publishd. [16] P. Sigmund, in: Radiation Damag Procsss in Matrials, Ed. C. H. S. Dupuy (Nordhoff, Lydn,1975), p. 3. [17] H. Paul and A. Schinnr, Atomic Data and Nuclar Data Tabls, 85 (003) 377. [18] R. Gauvain, CASINO - (mont CArlo SImulation of lctrons in solids) ( [19] J. Schou, Scan. Micr. (1988) 607. [0] J. Schou, Thsis, Risø National Laboratory (1979) (unpublishd). [1] M. J. Brgr, J. S. Coursy, M. A. Zuckr and J. Chang (005), ESTAR, PSTAR, and ASTAR: Computr Programs for Calculating Stopping-Powr and Rang Tabls for Elctrons, Protons, and Hlium Ions (vrsion 1..3). [Onlin] Availabl: [006, Sptmbr 5]. National Institut of Standards and Tchnology, Gaithrsburg, MD. Stopping-powr and rang tabls for lctrons, protons and hlium ions (physics.nist.gov/physrfdata/star/txt/) 006. [] H. A. Bth and J. Ashkin, in: Exprimntal Nuclar Physics, Ed. E. Sgré (Wily, Nw York, 1953), pp [3] K. B. Wintrbon, P. Sigmund and J. B. Sandrs, Mat. Fys. Mdd. Dan. Vid. Slsk. 37, no. 14 (1970). [4] K. B. Wintrbon, Ion Implantation Rang and Enrgy Distributions, Ed. D. K. Bric (Plnum, Nw York, 1975). [5] J. Schou, Phys. Rv. B (1980) 141. [6] S. Valkalahti, J. Schou and R. M. Niminn, J. Appl. Phys. 65 (1989) 58. [7] P. Andro, Nucl. Instrum. Mthods, B 51 (1990) 107. [8] J. Schou, Nucl. Instrum. Mthods, B 7 (1987) 188. [9] J. Schou, in: Stuctur-Proprty Rlationships in Surfac-Modifid Cramics, Eds. C. J. McHargu, R. Kossowsky and W. O. Hofr (Kluwr, Dordrcht, 1989), p

10 J. SCHOU [30] P. Sigmund, Phys. Rv. 184 (1969) 383 and 187 (1969) 768. [31] S. Bounau, A. Brunll, J. Dpauw, D. Jaqut, Y. L. Byc, M. Pautrat, M. Fallavir, J. C. Poizat and H. H. Andrsn, Phys. Rv. B 65 (00) [3] R. E. Johnson and J. Schou, Mat. Fys. Mdd. Dan. Vid. Slsk. 43 (1993) 403. [33] R.A. Baragiola, R. A. Vidal, W. Svndsn, J. Schou, M. Shi, D. A. Bahr and C.L. Attbrrry, Nucl. Instrum. Mthods, B 09 (003)