Drift-insensitive distributed calibration of probe microscope scanner in nanometer range: Virtual mode

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1 Dif-insensiie disibued alibaion of pobe miosope sanne in nanomee ange: Viual mode Rosisla V. Lapshin 1, 2 1 Solid Nanoehnolog Laboao, Insiue of Phsial Poblems, Zelenogad, Mosow, , Russian Fedeaion 2 Mosow Insiue of Eleoni Tehnolog, Zelenogad, Mosow, , Russian Fedeaion lapshin@gmail.om A mehod of disibued alibaion of a pobe miosope sanne is suggesed whih main idea onsiss in a seah fo a ne of loal alibaion oeffiiens (LCCs) in he poess of auomai measuemen of a sandad sufae, wheeb eah poin of he moemen spae of he sanne an be haaeized b a unique se of sale faos. Feaue-oiened sanning (FOS) mehodolog is used as a basis fo implemenaion of he disibued alibaion pemiing o elude in siu he negaie influene of hemal dif, eep and hseesis on he obained esuls. Possessing he alibaion daabase enables oeing in one poedue all he spaial disoions aused b nonlineai, nonohogonali and spuious ossalk ouplings of he miosope sanne piezomanipulaos. To poide high peision of spaial measuemens in nanomee ange, he alibaion is aied ou using naual sandads onsans of sal laie. One of he useful modes of he deeloped alibaion mehod is a iual mode. In he iual mode, insead of measuemen of a eal sufae of he sandad, he alibaion pogam makes a sufae image measuemen of he sandad, whih was obained ealie using onenional ase sanning. The appliaion of he iual mode pemis simulaion of he alibaion poess and deail analsis of ase disoions ouing in boh onenional and oune sufae sanning. Moeoe, he mode allows o esimae he hemal dif and he eep eloiies aing while sufae sanning. PACS: Cz, Ef, Lh, Ps, Dz, , uf, d Kewods: STM, AFM, SPM, sanne, alibaion, dif, eep, nonlineai, nonohogonali, ossalk oupling, gaphie, HOPG, feaue, eogniion, feaue-oiened sanning, FOS, oune-sanning, oune-sanned images, CSI, nanomeolog, sufae haaeizaion, nanoehnolog 1. Inoduion B using seeal ehniques embedded ino he feaue-oiened sanning (FOS) mehodolog [1, 2], a new disibued appoah o alibaion of he pobe miosope sanne is suggesed [3, 4]. The essene of he deeloped appoah is ha insead of haaeizing he whole moemen spae of a pobe miosope sanne b hee alibaion oeffiien K, K, K z [5], eah poin (,, z) of his spae is haaeized b is own unique iple of loal alibaion oeffiiens (LCCs) K (,, z), K (,, z), K z (,, z) [4]. As a esul, i is possible o oe all spaial disoions aused b nonlineai, nonohogonali and spuious ossalk ouplings of he miosope sanne X, Y, Z piezomanipulaos. In he eal mode [6], appliaion of he FOS appoah [1, 2] and he mehods of ounesanning [7] pemis eliminaing in siu he negaie influene of hemal dif, eep, and hseesis on he disibued alibaion esuls. A efeene sufae used fo alibaion should onsis of elemens, alled heeinafe feaues, suh ha he disanes beween hem o hei sizes ae known wih a high peision. The oeed oodinae of a poin on he disoed image of an unknown sufae is obained b summing up he LCCs elaed o he poins of he moemen

2 Dif-insensiie disibued alibaion of pobe miosope sanne ajeo of he sanne [4]. Viual disibued alibaion is suh a alibaion ha he phsial sufae of a sandad is subsiued wih a opogaph image unde oeion. Of ouse, he hemal dif and eep of he miosope an no possibl hae been eluded duing suh so of alibaion, whih is efleed in he obained LCC and obliqui angle disibuions. Afe he iual alibaion daabase (CDB) of he image unde oeion has been eaed, he nonlinea oeion of he image is aied ou. I does no mae fo his mode whih ea faos hae disoed he san unde oeion, e. g., hemal dif, eep, o sai piezosanne nonlineaiies aing ogehe o sepaael. The iual mode is inended fo simulaing he poess of alibaion and alidaing he analial soluions found in Ref. 4. The iual mode of disibued alibaion allows hoough analsis of pobe miosope sanne opeaion. In paiula, he mode pemis deeminaion of he alues and he haae of ase disoions fo boh egula and oune pes of sanning. Moeoe, he iual mode an be used o esimae he hemal dif and eep eloiies, fo moié deeion, and fo auomai haaeizaion of sal sufaes. 2. Measuemen ondiions The aomi opogaph of he basal plane (0001) of highl oiened poli gaphie (HOPG) monosal was used as a sandad sufae. The measuemens wee aied ou a he sanning pobe miosope (SPM) Sole P4 (NT-MDT Co., Russia) b mehod of sanning unneling miosop (STM) in he ai a oom empeaue. In ode o minimize hemal defomaion of he sample, a gaphie sal of small dimensions mm was used. Thee adjaen abon aoms (o inesies) foming an equilaeal iangle ABC (see Fig. 3(a) in Ref. 4) wee seleed as a loal alibaion suue (LCS). Aoding o he neuon diffaion mehod, he HOPG laie onsan a (i. e., side lengh of he ABC iangle) makes 2.464±0.002 Å [8]. As he ip, a mehaniall u 0.3 mm NiC wie was used. To poe he miosope agains seismi osillaions, a passie ibaion isolaion ssem was emploed. Moeoe, he miosope was housed unde a hemoinsulaion hood, whih also seed as an absobe of eenal aousi disubanes. The pial noise leel of he unneling uen in he ouse of he measuemens made abou 20 pa (peak-o-peak). Duing he ase sanning, he pobe moemen eloi a he eae sweep was se he same as a he fowad ae. Immediael befoe he ase sanning begins, sanne aining was aied ou [7]. The sanne aining is a epeaed moemen along he fis line, whih allows o deease eep a he beginning of he san [9]. While aining, he aual sanning eloi was also deemined. Some of he alues in he seions pesened below ae inenionall gien wih a edundan numbe of signifian digis. Tha will pemi o ompae hem wih he simila alues obained unde diffeen measuemen ondiions o in diffeen measuemen modes. 3. Almos linea ase disoions 3.1. Analsis, oeion, ompaison of eos of die and oune images STM-sans of HOPG sufae disoed b dif ae shown in Fig. 1. A egula (die) sufae san and a san oune o i [7] ae gien in Fig. 1(a) and Fig. 1(b), espeiel. Viual FOS [1] of he pesened images allows deemining a mean aomi laie spaing as 2.734±0.25 Å and 2.199±0.20 Å, whene i is eas o esimae a elaie measuemen eo as 11.0% and 10.8%, espeiel. Mean spaings deemined b inesies made 2.731±0.25 Å and 2.197±0.20 Å, elaie measuemen eo is 10.8% and 10.9%, espeiel. 2

3 R. V. Lapshin (a) Fig. 1. Dif-disoed sans of aomi sufae of poli gaphie (a) die image, (b) oune image. Measuemen mode: STM, onsan-heigh, U un =85 mv, I un =750 pa. Numbe of poins in he ase: Sanning sep size: =0.306 Å, =0.307 Å. Numbe of aeagings a he ase poin is 15. Sanning eloi = =223.1 Å/s (is deemined while aining). CSI sanning ime T CSI =6 min. Mean onsan of aomi laie equals o (a) 2.7 Å, (b) 2.2 Å whih oesponds o he elaie measuemen eo of (a) 11%, (b) 11%. In Fig. 2, shown ae: paiioning of he gaphie san b an inege-alued ne haing squae ells; pobe aellings duing he iual alibaion (ompae wih he ajeo of he eal mode in Ref. 6); and LCS posiions fo whih LCCs and loal obliqui angles wee deemined. The images in Fig. 2 oespond o he ase when abon aoms ae used as feaues; using inesies would gie a simila esul. In ode o deease he influene of edge effes [9, 10], feaues loaed along he image edges wee eluded fom onsideaion b seing oesponding aea of disibued alibaion. Apeues of poins and segmens of poins wee used while alibaing b he die image; and while alibaing b he oune one apeues wee sanned on he die image and 1849 apeues on he oune one. All LCSs found in he apeue wee used duing he alibaion. The alibaion pogam eealed no defeie LCSs. The numbe of LCSs found, he mean alues of laeal LCCs and he obliqui angle of he iual CDB ae gien in Table 1. I is seen well fom he able pesened ha alibaion b LCSs onsised of abon aoms and alibaion b LCSs onsised of inesies esul in e lose alues. Theefoe, a leas wih he aua of enh of a peen, he inesies ma be used as feaues fo sanne alibaion along wih he abon aoms and he CDBs obained b boh pes of feaues ma be ombined ino a single CDB. I should be noed ha despie of a noieable diffeene in mean oeffiiens < K > of he die and he oune sans fom he iniial lumped alibaion oeffiien =0.307 Å [7], we an speak abou appoimael equal deiaion of hese oeffiiens fom oeffiien, i. e., he elaion /< K > fo he die san is appoimael equal o he elaion < K >/ fo he oune san. (b) Seond ode egession sufaes K, K, α buil b he iual CDB of he die image ae shown in Figs. 3(a)-() (CDB obained b LCSs onsising of abon aoms and CDB obained b LCSs onsising of ine- 3

4 Dif-insensiie disibued alibaion of pobe miosope sanne (a) Fig. 2. LCS posiions (designaed as ) fo whih LCCs and loal obliqui angles wee obained duing he iual disibued alibaion b (a) die, (b) oune image. Tajeo of pobe aeling is imaged wih a solid line. Iniial ne (a) 36 36, (b) nodes, sep (a) 6, (b) 5 posiions. Ne nodes ae designaed wih a + smbol. Cabon aoms ae used as feaues. Numbe of poessed apeues (a) 1296, (b) Numbe of found LCSs (a) 1503, (b) The inse shows a zoomed aea loaed in he lowe lef one. sies ae inegaed ino a single CDB). As he sans unde analsis hae small sizes, suh disoions as spuious ouplings beween manipulaos and sai piezosanne nonlineai [6] would be insignifian on he egession sufaes. Indeed, egession sufae K (see Fig. 3(a)) is paallel o ais indiaing ha he sanne moemen along he slow san dieion does no impa he sanne moemen along he fas san dieion (LCCs hanging equall in all ase lines). Regession sufae K ae K (see Fig. 3(b)) is paallel o ais indiaing ha he sanne moemen along he fas san dieion does no impa he sanne moemen along he slow san dieion (LCCs K ae hanging equall in all ase olumns). Almos hoizonal posiion of egession sufae α, whih is paallel o ais and slighl iled o ais (see Fig. 3()), eeals a weak dependene of he obliqui angle on he sanne moemens in he laeal plane. The simila siuaion is obseed fo he oune image (b) Table 1. Mean alues of laeal LCCs and obliqui angles of iual CDBs. The fis alue is a esul of alibaion b abon aoms, he seond one b inesies of sal laie. HOPG san Numbe of LCSs in CDB < K > (Å) < K > (Å) <α> (deg) Die, Fig. 1(a) 1503, , , , Coune, Fig. 1(b) 2274, , , , 7.02 Die, Fig. 6(a) 1233, , , , Coune, Fig. 6(b) 2248, , , ,

5 R. V. Lapshin (a) (b) () (d) (e) (f) Fig. 3. Regession sufaes of 2nd ode dawn hough (a), (d) LCCs (see Figs. 3(d)-(f)). Thus, he eeps eied in he X segmens and Y segmens of he piezoube sanne hae paiall no ouplings wih eah ohe, a leas in he moemen sales being onsideed. The esimae of oo-mean-squae deiaions of LCCs and he obliqui angle shown in Fig. 3 is aied ou b fomulae (14) gien in Ref. 4 and he obained alues ae pesened in Table 2. In paiula, i is lea fom he able ha, egadless of whehe i is a die san o oune, he dispesion of LCCs of LCCs K is less han he dispesion K [11]. Suh aio of dispesions an be easil eplained an hange in dif eloi duing he ase sanning has a geae impa in he slow san dieion han in he fas one [11, 12]. Alhough, ompaing he mean alues of LCCs K, (b), (e) LCCs K, (), (f) loal obliqui angles α of he die (uppe ow) and he oune (lowe ow) images. Join CDB is used whih inludes LCSs of aoms and LCSs of inesies. Veial sales of he oesponding figues in uppe and lowe ows ae he same. K of he die and he oune images (see Table 1) indiaes Table 2. Sandad deiaions of LCCs and obliqui angles of he iual CDBs. The fis alue oesponds o he egession sufae of he 1s ode, he seond 2nd ode. HOPG san σ K (Å) σ K (Å) σ α (deg) Die, Fig. 1(a) , , , 3.18 Coune, Fig. 1(b) , , , 2.46 Die, Fig. 6(a) , , , 3.12 Coune, Fig. 6(b) , , ,

6 Dif-insensiie disibued alibaion of pobe miosope sanne Table 3. Esimae of nonlinea disoions of he oune-sanned images epessed in ems of he maimal deiaion of a egession sufae fom he hoizonal plane. The fis alue oesponds o he 1s ode egession sufae, he seond alue 2nd ode egession sufae. The esimae peenage is gien in paenheses. HOPG san ma K (Å) ma (Å) ma (deg) K α Die, Fig. 1(a) (100%), (100%) (100%), (100%) Coune, Fig. 1(b) (93%), (93%) (65%), (87%) Die, Fig. 6(a) (100%), (100%) (100%), (100%) Coune, Fig. 6(b) (87%), (82%) (80%), (83%) 2.16 (100%), 2.42 (100%) 0.23 (11%), 0.93 (38%) (100%), (100%) 2.36 (9%), 2.85 (9%) ha he eep equall disos hese images in ase line, ompaing he K disibuions in Fig. 3(a) and Fig. 3(d) (see also Table 2) indiaes ha he aiane of hese oeffiiens in he oune image is noabl less. This fa, in is un, means ha LCS in he oune image and, heefoe, he image as a whole ae less subjeed o disubing faos, mainl, suh as eep. As o he aiane of K disibuion, onaiwise, i is somewha lage in he oune image han in he die one. Mos likel his is due o he unequal epesenaion of he feaues in hose images along he slow san dieion. The fa is ha he image of aom/inesie in he oune san has fewe poins along he slow san dieion (see Figs. 1, 2 and 2nd olumn of Table 1). Nonohogonaliies of he sanne shown in Table 1 ae mainl onneed wih he dif-aused ase disoion (shif of lines elaie o eah ohe). The aual nonohogonali of he sanne is ens of imes less (α=0.4 [7], see also Ref. 6). Thus, he alue of obliqui angle an disinl poin ou he pesene of a dif. The able shows ha he absolue alue of mean obliqui angle of he oune image is less han he mean obliqui angle of he die image. This means ha he oune image is less disoed. Moeoe, like wih he disibuion of LCCs K, he aiane of he disibuion of he obliqui angles in he oune image is also less (p. Fig. 3() wih Fig. 3(f), see Table 2). The sufae K enes ino ansfomaions (5) gien in Ref. 4 lineal, heefoe, a sligh hange of his sufae would lead o a sligh nonlinea onibuion o he ase disoion along. The sufae K also enes ino ansfomaions (5) lineal bu i has a moe disin hange, heefoe, is influene along and dieions is nonlinea o a geae een. The obseed il of he egession sufaes is linked wih eep (see Se. 3.2) [7]. The sufae α enes ino ansfomaions (5) nonlineal. Neeheless, onsideing ha α is paiall onsan all oe he image field and is quie small (sin( α ) α, os( α ) 1), is onibuion o he nonlinea disoion along and is insignifian. Visuall, he egession sufaes of he oune image (see Figs. 3(d)-(f)) look like he sufaes of he die 6

7 R. V. Lapshin (a) Fig. 4. Coeed images of gaphie sufae: (a) die, (b) oune. Mean onsan of aomi laie a boh images is equal o he nominal alue Å. Numbe of image poins: (a) , (b) Size of an image poin = = Å. (b) image bu he ae iled o he opposie side. In ode o quanif disinions beween die and oune images, pa of whih has been disussed aboe (see Tables 1, 2), he maimum diffeenes (see Table 3) wee alulaed fo egession sufaes shown in Fig. 3 b fomulae (13) gien in Ref. 4. Aoding o he able daa, he oune san is disoed less han he die one (he esimae fo disoion of he die san is aken fo 100%). The oeed oune-sanned images (CSIs) of gaphie sufae ae shown in Fig. 4. Coeion of he die image is aied ou b fomulae (8) gien in Ref. 4 wih he use of he 1s ode egession sufaes As egads he 2nd ode of he egession sufaes in Fig. 3, i was inoled mainl jus o poin ou he fa ha hese sufaes ae lose o planes. Coeion of he oune image is aied ou b fomulae (8), whee he 2nd [ ] ode egession sufaes (11) gien in Ref. 4 ae used insead of K (, ) sin α ( ), K ( ) os ( ) ode of sufae, K, K, [ ], α., α ; he K is equal o 1. Wih onl he menioned pes and powes of he egession polnomials, he nonlinea oeion poides he leas esidual eos. The esimae of he esidual eos is aied ou b a mean alue of he laie onsan, whih is deemined duing iual FOS [1] of he oeed image (see Table 4). I should be noed ha o obain suh small deiaions of he mean laie onsan fom is nominal alue, i is neessa o use ahe lage samples (see olumn 2 onaining he numbe of inoled aoms), i. e., he sans should be of size no less han he ones shown in Fig. 1. Sine he oune image onains appoimael 1.5 imes as man feaues as he die one, in ode fo he esidual eos of hese images o be ompaed oel, he mean laie onsan of he oune image should be esimaed wih using he same numbe of abon aoms as in he die image. Howee, onsideing ha he abon aoms ae epesened in he oune image wih less numbe of poins and heefoe hei posiions ae deemined less auael, all aailable aoms in he oune image wee used o esimae he mean laie onsan. Beside he equali of he mean laie onsan a in he die and oune images o he nominal alue, ohe eidenes of he alidi of he implemened oeion ae (see Fig. 4): he same oienaion of he sallogaphi dieions on he die and he oune images and he absene of uaue of hese dieions. When neessa, he oeed CSIs in Fig. 4 an be supeposed ino a single image b using a oinidene poin [7]. While supe- 7

8 Dif-insensiie disibued alibaion of pobe miosope sanne Table 4. Mean alues of laie onsan in he oeed CSIs of pogaphie (in paenheses, he elaie oeion eos ae gien). Mean alues of LCCs (nomalized o he laeal size of poin of he oeed image) and obliqui angles of CDBs obained duing he iual alibaion b he oeed CSIs. HOPG san Numbe of C aoms Mean laie onsan (Å) Numbe of LCSs in CDB -1 < K > -1 < K > <α> (deg) Die, Fig. 4(a) Coune, Fig. 4(b) Die, Fig ±0.12 (0.002%) 1454, ±0.09 (0.001%) 2255, ±0.07 (0.001%) 1007, Coune ±0.11 (0.052%) 2330, posed, he opogaph is aeaged wihin he oelap egion of hese images esuling in addiional noise leel eduion in he obained image. The alidi of he suggesed nonlinea oeion was addiionall onfimed b pefoming he iual disibued alibaions b he oeed images of Fig. 4 (self-onsisen es). The eidenes of he alidi (see Table 4) ae a nea-uni elaion of he mean alue of laeal LCC o he laeal size = = Å of a poin of he oeed image, a nea-zeo mean alue of he obliqui angle, and he egession sufaes appoahing hoizonal planes [4] Deeminaion of hemal dif and eep eloiies B subaing fom he, sizes of he oeed image (see Fig. 4) he, sizes of he iniial san (see Fig. 1), espeiel, and diiding i b he known sanning ime of he image T=T CSI /2, he, omponens of mean eloi of oal dif (hemal dif + eep), ae o be found (see Table 5). Numeiall, he obained dif eloiies oespond o he dif eloiies obseed duing he eal alibaion afe sanne alming down in he uen ne node (waiing sae + a sequene of aahmens, see Ref. 6). Sine he obained dif eloiies hae he same dieion egadless of whehe he san is die o oune, he hemal omponen makes he main onibuion o he dif [7]. Also, he obained dif eloiies appoimael oespond o he dif eloiies pesened ealie in Ref. 7. Sine doubling he san sizes in Fig. 1 elaie o he san sizes in Figue 3 of Ref. 7 did no hange he dif eloi, he sanne epeiening heaing due o inenal losses duing sanning [13] is no he main soue of hemal dif in he gien miosope while sanning wih aomi esoluion. Suppose ha he hemal dif eloi is no hanging while sanning [1, 7]. The fa (see Fig. 3(b), Fig. 5(b) and equaions (8) in Ref. 4) ha he alues belonging o he egession sufae K of he die image ae less han he speified lumped oeffiien =0.307 Å [14] means ha duing he sanning he aual miosope sep while moing fom line o line was less han he speified one. Sep deease ous when hemal dif is dieed along he moemen. Sep deease leads o image sehing along (p. Fig. 1(a) wih Fig. 4(a)) [7]. The alues 8

9 R. V. Lapshin Table 5. Mean eloiies of oal dif, hemal dif, and eep. The alues in paenheses ae obained b summing he -omponens of eloiies of hemal dif and eep. Veloiies ae gien elaie o he oodinae ssem of he die image. HOPG san Mean oal dif eloi (Å/s) Mean hemal dif eloi (Å/s) Mean eep eloi (Å/s) Die, Fig. 1(a) Coune, Fig. 1(b) Die, Fig. 6(a) Coune, Fig. 6(b) (0.081) (0.114) (0.142) (0.137) belonging o he egession sufae K of he oune image (see Fig. 3(e), Fig. 5(b)), on he ona, eeed he speified oeffiien. In his ase, he aual miosope sep is geae han he speified one, hemal dif opposie o he moemen, and he image is undegoing shinkage along (p. Fig. 1(b) wih Fig. 4(b)) [7]. In Fig. 5, 1s ode egession ues ae used sine, as i was deemined ealie, he 1s ode is quie suffiien fo he obseed disoions o be peisel desibed. [ ] The eo aused b hemal dif and eep along he slow san dieion (see Fig. 5(b)) an be found as a dif- feene beween he known oeffiien and he egession sufae pofile K (, ) os ( 0 ), is 0 α. Assuming ha eep is finished b he end of he die san, he mean hemal dif of he die san an be alulaed as follows [ ] { K (, ) os α ( 0, )} 0 ma ma ma =. (1) T The numeao of he fomula is an aea of he hahed eangle in he figue. Aodingl, he mean eep eloi in dieion an be deemined b = ma = 0 whee T( ) ( 2 ) [ ] ( ) [ ( )] { K ( 0, ) os α ( 0, ) K 0, os α 0, } ma T ma ( 0, ) ma, = + ma + is he ime sine he beginning of he san ill he ime he pobe has eahed he ase poin wih oodinaes, ;, ae he moemen eloiies in he ase line and fom line o line of he ase, espeiel (as a ule, = =). In ase of he 1s ode egession sufaes, he fomula (2) an be simplified o [ ] ( ) [ ( )] { K (, ) os α ( 0, ) K 0,0 os α 0,0 } 0 ma ma ma =. (3) 2T The numeao of he fomula is a doubled aea of he hahed iangle in Fig. 5(b). The fomulae fo alulaion of, (2) 9

10 Dif-insensiie disibued alibaion of pobe miosope sanne (a) Fig. 5. Regession ues of he 1s ode showing he hange in miosope sep while moing along he ase (a) fis line, (b) fis olumn. Suh ues allow eaing infomaion abou mean eloiies of hemal dif and eep. he mean hemal dif and he mean eep of he oune san ae buil he same wa; he found alues ae gien in Table 5. Cheking b fomulae = +, = + shows ha he obained alues (pesened in paenheses in he able) ae lose o he oal dif deemined aboe. Appaenl, he small diffeenes ae linked o he eep whih does no finished b he end of he die (oune) san (he siuaion is ondiionall shown in Fig. 5(b) wih doed lines) as well as wih he iual alibaion whih omied peipheal aeas of he san (o aoid influene of edged disoions). Mean hemal dif along dieion is alulaed aoding o he fomula = ma = 0 K [ ] ( 0, ) sin α ( 0, ) T The obained alues (see Table 5) ae e lose o he alues of he oal dif. This implies ha he mean eep along is e small (he ae eep is ompensaed b eae eep). Mean eep in he fas san dieion is deemined as follows = 0 [ K (,0) ] T (,0) whee =0.306 Å is a lumped alibaion oeffiien along [14]. The eep alues alulaed using fomula (4) sine alulaions using fomula (6) lead o a lage eo. ma The fa of ossing beween he egession line = ma (b),. (4) (5) (6) pesened in Table 5 ae K and he hoizonal line (poin A) means ha he diso- 10

11 R. V. Lapshin (a) Fig. 6. Sans of aomi sufae of poli gaphie disoed b song dif (a) die image, (b) oune image. Well noieable disoions in he beginning of he die san wee inenionall pooked pio o he sanning b an abup laeal sanne offse ino he iniial ase poin. The eep eied b he offse had sopped b he momen he oune san began. Measuemen mode: STM, onsan-heigh, U un =85 mv, I un =750 pa. Numbe of poins in he ase: Sanning sep size: =0.306 Å, =0.307 Å. Numbe of aeagings a he ase poin is 15. Sanning eloi = =223.1 Å/s. CSI sanning ime T CSI =6 min. Mean laie onsan equals o (a) 2.9 Å, (b) 2.2 Å whih oesponds o he elaie measuemen eo of (a) 19%, (b) 13%. The maimal alue of he laie onsan a he beginning of he die san is almos 2.5 imes geae han he nominal alue. ions aused b eep in dieion in his plae of he san ae ompleel absen. One should pa aenion o he signifian disoninui of he egession sufaes a he end of he die line/san and in he beginning of he oune line/san ha poins ou a lage dnami eo ha oued a hese ase poins [10, 15]. 4. Subsaniall nonlinea ase disoions 4.1. Analsis, oeion, ompaison of eos of die and oune images A die san of HOPG sufae subsaniall disoed b dif is shown in Fig. 6(a). The song disoion was inenionall pooked pio o he sanning b an abup laeal sanne offse o he iniial ase poin 0, 0 [9]. The disoion in he beginning of he san is so lage ha he laie onsan along one of he lose-paked dieions is neal 2.5 imes geae han he nominal alue. The oune san is gien in Fig. 6(b). Aoding o he figue, he song eep aused b he offse has paiall eminaed b he momen he oune-sanning began. As i was alead menioned aboe, he oune san is geneall less disoed han he die one (see Table 3) sine is eep is being paiall ompensaed b he opposiel oiened eep of he die san. Thus, in some ases, despie of wie inease in sanning ime, he oune-sanning is o be pefomed insead of onenional sanning een if no following CSI oeion [7] is planned. Sine all oeffiiens < K > in Table 1 ae appoimael equal and lose o he lumped oeffiien =0.306 Å in spie of diffeen degee of he image disoions, he ase line iself is disoed jus insignifianl duing he ase sanning. The fa is ha he moemen along a line is ompaaiel fas, heefoe he slow dif has no ime o diso i noabl; and so he eep indued in he line a fowad ae is ompensaed b he eep indued a eae. The bulk of he dif-indued disoions along dieion is elaed o small elaie shifs of (b) 11

12 Dif-insensiie disibued alibaion of pobe miosope sanne Fig. 7. LCS posiions fo whih duing he iual disibued alibaion LCCs and loal obliqui angles wee obained. The iniial ne nodes, sep 6 posiions. Cabon aoms ae used as feaues. Numbe of poessed apeues Numbe of found LCSs lines being aumulaed, whih is aouned fo b he obliqui angle α. I is ineesing o noe ha he oeffiien < K > fo suh heail eep-disoed die image as he one shown in Fig. 6(a) has a e small disinion fom he oeffiiens < K > of ohe muh less disoed images. Fom he aboe follows he known ule of alibaion of an ohogonal sanne b a 1D diffaion gaing. Fis, b fiing gaing lines aoss he fas sanning dieion, he gaing pofile should be measued b whih he alibaion oeffiien K should be deemined. Then, afe oaing he gaing b 90 degees and swihing he fas sanning dieion fom o, he gaing pofile should be measued again followed b deemining he alibaion oeffiien K. In Fig. 7, LCS posiions ae shown fo whih LCCs and loal obliqui angles wee obained duing he iual alibaion b he die image of Fig. 6(a). Despie of e song disoions in he beginning of he die san, he algoihm of disibued alibaion eognized he gaphie uni ells wihou eos and oel deemined LCCs wihin his pa of he san. In Fig. 8, 2nd ode egession sufaes buil wih he use of he join CDB of he die san ae shown. In aodane wih he esuls obained befoe, a song uaue of he egession sufae K migh hae been epeed. Howee, auall he uaue of his sufae uned ou o be een noabl less han he one of he sufae (see Fig. 3(b)) obained fo he die san of Fig. 1, whee he pial alues of disoions of ase sanning ae obseed (o simplif he ompaison, he sales in Figs. 8(a), (b) ae se he same as in oesponding Figs. 3(a), (b)). If he egession sufaes K, K ae weakl ued and he obseed disoion degee of he die san in (a) (b) () Fig. 8. Regession sufaes of 2nd ode dawn hough (a) LCCs he die image. 12 K, (b) LCCs K, () loal obliqui angles α of

13 R. V. Lapshin Fig. 6 is high, hen he uaue of he egession sufae α is epeed o be song. Indeed, in Fig. 8() we obsee suh a uaue (see also Table 3). Moeoe, in ompaison wih he sufae in Fig. 3(), his uaue is no onl signifianl lage b absolue alue bu also has an opposie sign. Thus, in he onsideed disoion i is he obliqui angle ha undegoes he main hange, whih is efleed in he image as a song nonlinea shif of he ase lines elaiel o eah ohe. The oeed die san is shown in Fig. 9. The oeion of he die san is aied ou b fomulae (12) gien in Ref. 4 b using egession sufaes of he 1s ode fo Fig. 9. Coeed die image of gaphie sufae. Mean onsan of aomi laie is equal o he nominal alue Å. Numbe of image poins Size of an image poin = = Å. K and he 2nd ode fo K, he oune san is aied ou b fomulae (12) b using egession sufaes of he 1s ode fo K. Coeion of K and he 4h ode fo K, K. Conol of he oeion esuls b mean alue of he laie onsan was ondued b means of he iual FOS and i poed ha in an pa of he die image eah abon aom has si neighboing aoms foming a egula heagon wih a side equaled o he laie onsan se fo alibaion (see Table 4). A somewha highe eo of laie onsan deeminaion in he oune image ould be eplained b he fa ha he feaues wee epesened in he oune image b a smalle numbe of poins han in Fig. 1(b) beause of a songe squeezing of he image along. The alidi of he pefomed nonlinea oeion is onfimed b iual disibued alibaions b he oeed images (see Table 4). Map of LCS posiions, egession sufaes, and a oeed opogaph of he oune san ae no gien beause he hae insignifian disinions fom he images obained ealie fo he oune san in Fig. 1. Mean eloiies of difs and eeps duing he sanning ae gien in Table Disussion The iual mode an be applied fo oeion of nonlinea disoions of sufae opogaph when he sufae image onains elemens quie eenl disibued oe he san aea and wih a pioi known sizes o disanes beween hem [5, 16]. Fo eample, b using he sufae of a sandad as a subsae, one ma deposi deahed objes unde inesigaion on his sufae, a ou egula sanning, pefom iual disibued alibaion b he known obseable elemens of he sandad, and hen oe he obained image aoding o he aquied CDB. Moeoe, b pefoming a egula san of a sandad sufae unde he same ondiions and wih he same paamees as a paiula unknown sufae, i beomes possible o oe he unknown sufae. The oeion is aied ou b he CDB aquied duing iual disibued alibaion b he obained image of he sandad. Aknowledgmens This wok was suppoed b he Russian Foundaion fo Basi Reseah (poje ) and b he 13

14 Dif-insensiie disibued alibaion of pobe miosope sanne Minis of Eduaion and Siene of Russian Fedeaion (onas , ). I hank O. E. Lapin and Asso. Pof. S. Y. Vasilie fo hei iial eading of he manusip; D. A. L. Gudko, Pof. E. A. Ilihe, Asso. Pof. E. A. Feiso, and lae Pof. E. A. Poloask fo hei suppo and simulaion. Refeenes [1] R.V. Lapshin, Feaue-oiened sanning mehodolog fo pobe miosop and nanoehnolog, Nanoehnolog 15 (2004) (aailable a [2] R.V. Lapshin, Feaue-oiened sanning pobe miosop, in: H.S. Nalwa (Ed.), Enlopedia of Nanosiene and Nanoehnolog, Ameian Sienifi Publishes, ol. 14, 2011, pp (aailable a [3] R.V. Lapshin, Mehod of auomai disibued alibaion of pobe miosope sanne, Russian Paen , 2005 (aailable a R.V. Lapshin, Auomai disibued alibaion of pobe miosope sanne, J. Suf. Ines., no. 11 (2006) (in Russian, aailable a R.V. Lapshin, Disibued alibaion of pobe miosope sanne in nanomee ange, Po. XVII Russian Smposium on Sanning Eleon Miosop and Analial Mehods of Inesigaion of Solids, Chenogoloka, Russian Fedeaion, 2011, p. 94 (in Russian, aailable a [4] R.V. Lapshin, Dif-insensiie disibued alibaion of pobe miosope sanne in nanomee ange: Appoah desipion, axi: [ond-ma.ml-si], [5] R.V. Lapshin, Auomai laeal alibaion of unneling miosope sannes, Re. Si. Insum. 69 (1998) (aailable a [6] R.V. Lapshin, Dif-insensiie disibued alibaion of pobe miosope sanne in nanomee ange: Real mode, 2015, pepaed fo submission. [7] R.V. Lapshin, Auomai dif eliminaion in pobe miosope images based on ehniques of oune-sanning and opogaph feaue eogniion, Meas. Si. Tehnol. 18 (2007) (aailable a [8] P. Tuano, R. Chen, Suue of gaphie b neuon diffaion, Naue 258 (1975) [9] E.P. Soll, Resoaion of STM images disoed b ime-dependen piezo die afeeffes, Ulamiosop (1992) [10] M.J. Ros, L. Cama, P. Shakel, E. an Tol, G.B.E.M. an Velzen-Williams, C.F. Oegauw, H. e Hos, H. Dekke, B. Okhuijsen, M. Senen, A. Vijfigshild, P. Han, A.J. Kaan, K. Shoos, R. Shumm, W. an Loo, T.H. Oosekamp, J.W.M. Fenken, Sanning pobe miosopes go ideo ae and beond, Re. Si. Insum. 76 (2005) [11] J. Ganaes, L. Nielsen, K. Dishel, J.F. Jøgensen, J.B. Rasmussen, P.E. Lindelof, C.B. Søensen, Twodimensional nanomee-sale alibaion based on one-dimensional gaings, Appl. Phs. A 66 (1998) S831-S835. [12] V.Y. Yuo, A.N. Klimo, Sanning unneling miosope alibaion and eonsuion of eal image: dif and slope eliminaion, Re. Si. Insum. 65 (1994) R. Saub, D. Alliaa, C. Niolini, Dif eliminaion in he alibaion of sanning pobe miosopes, Re. Si. Insum. 66 (1995)

15 R. V. Lapshin [13] R.V. Lapshin, Analial model fo he appoimaion of hseesis loop and is appliaion o he sanning unneling miosope, Re. Si. Insum. 66 (1995) (aailable a [14] he oeffiien was obained duing alibaion b he image whih disoions indued b hemal dif and eep had been eliminaed, see Ref. 7. [15] L. Xu, X. Tian, X. Li, G. Shang, J. Yao, Geomei disoion oeion fo sinusoidall sanned images, Meas. Si. Tehnol. 22 (2011) [16] J. F. Jøgensen, L. L. Madsen, J. Ganaes, K. Caneio, K. Shaumbug, Calibaion, dif eliminaion, and moleula suue analsis, J. Va. Si. Tehnol. B 12 (1994) D. Alliaa, C. Ceoni, C. Niolini, A simple mehod fo pepaing alibaion sandads fo he hee woking aes of sanning pobe miosope piezo sannes, Re. Si. Insum. 67 (1996)