Stability of Tip in Adhesion Process on Atomic Force Microscopy Studied by Coupling Computational Model

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1 Appl. Sc. Converg. Technol. 26(1): 6-10 (2017) Research Paper Stablty of Tp n Adheson Process on Atomc Force Mcroscopy Studed by Couplng Computatonal Model Yasuhro Senda a *, Janne Blomqvst b, and Rsto M. emnen b a Department of Appled Scence, Yamaguch Unversty, Yamaguch, , Japan b COMP Centre of Excellence, Department of Appled Physcs, Aalto Unversty, P.O. Box 11100, Aalto, Fnland Receved ovember 14, 2016; revsed ovember 30, 2016; accepted December 2, 2016 Abstract We nvestgated the stablty of onc confguratons of the tp of the cantlever n non-contact AFM.; For ths, we used a computatonal model that couples the onc moton of the MgO surface and the oscllatng cantlever. The moton of ons was connected to the oscllatng cantlever usng a couplng method that had been recently developed. The adhesve process on the onc MgO surface leads to energy dsspaton of the cantlever. It s shown that lmted types of onc confguratons of the tp are stable durng the adhesve process. Based on the present computatonal model, we dscuss the adhesve mechansm leadng to energy dsspaton. Keywords: Atomc force mcroscopy, Molecular dynamcs I. Introducton Atomc force mcroscopy (AFM) has shown promse as a tool for observng atomc-scale mages of varous materals [1]. In the non-contact AFM, an atomc force between the tp of the cantlever and the surface gves the varaton n the frequency of the oscllaton of the cantlever. The observed frequency offers an atomcresoluton mage of the surface. In ths experment, the ampltude of the cantlever s oscllaton was reduced as the tp approached the surface. The dsspaton of the oscllaton s mechancal energy also provdes an atomc mage [2-9]. The mechansm that causes ths dsspaton has been nvestgated by many theoretcal and computatonal studes [10-25]; nevertheless, t s stll under dscusson. Although a number of mechansms have been proposed for ths dsspaton, the adhesve hysteress mechansm s consdered to be the most acceptable. As the cantlever tp approaches and moves away from the surface, the adhesve behavor of the atoms of the tp and the surface arses, whch provdes the hysteress n force actng on cantlever; ths non-conservatve force results n the energy dsspaton of the oscllaton. Ths adhesve mechansm s based on the exstence of stable atomc structures of the tp, and the structure of the tp and the surface remans ntact even after adhesve atomc contact; ths s because they reversbly reconstruct durng the adheson process. Ths adhesve mechansm has been extensvely nvestgated *Correspondng author E-mal: senda@yamaguch-u.ac.jp through theoretcal and computatonal means [10-17], and these studes have calculated the atomc nteracton between the tp and the surface through many dfferent atomc models. One such model uses the molecular dynamcs (MD) method, and the equaton of moton for the cantlever s solved wth the nteracton force that was obtaned from the MD calculaton, n whch the motons of the atoms and the cantlever are ndependently calculated. We consder that the couplng of the atomc motons and the cantlever s oscllaton s necessary to trace the adhesve behavor of the atoms and understand the mechansm of the energy dsspaton. The oscllaton of the cantlever consderably effects on the atomc-scale reacton and adhesve moton of the atoms. In our prevous studes [23-25], we proposed a computatonal AFM model that couples the atomc moton between the tp and the surface wth the oscllaton of the cantlever. We constructed such a couplng model accordng to the MD-contnuum couplng method that we had been developed [26-28]. In the present work, ths computatonal model was appled to the AFM of the surface of an MgO onc crystal, whch has been nvestgated n the computatonal studes [13-15]. We am to confrm the adhesve mechansm of the energy dsspaton, as well as the stablty of the onc confguraton of the tp, usng ths couplng computatonal model. In the followng, we explan the computatonal model that couples the moton of the MgO ons and the cantlever of the AFM. We then show the adhesve process of the ons on the surface and the energy dsspaton of the cantlever that s assocated wth the adhesve process. The stablty of the onc confguraton of the tp n the adhesve process s 6

2 Stablty of Tp n Adheson Process on Atomc Force Mcroscopy Studed by Couplng Computatonal Model 7 nvestgated, and the mechansm of the energy dsspaton s subsequently dscussed. II. Method The computatonal AFM model s composed of the atomc system and a sprng. The atomc system descrbes the atomc force between the tp of the cantlever and the surface. The oscllaton of the sprng represents that of the cantlever n AFM. In the followng, we explan the atomc system, and subsequently the couplng system that combnes ths atomc system wth the sprng. 1. Atomc system The tp ons faced the surface of the MgO crystal, as shown n Fg. 1(a). The nter-atomc potentals between Mg-Mg, O-O and Mg-O ons were set as used n the reference [29], respectvely, n whch each potental conssts of parts of Coulombc, dsperson, and the repulsve nteractons between the ons. The potental energy of the entre atomc system was assumed to be the sum of the above parwse atomc potental. The perodc boundary condton (PBC) was appled along the x and y drectons, whle t was not appled along the z-drecton. At ths pont, the system was solated along z drecton. We then ntroduced frozen atoms and the Langevn atom nto the atomc system. The frozen atoms dd not move durng the MD calculaton and remaned at the ntal Fugre 1. (a) Atomc system under the solated condton n z drecton. The upper atoms corresponds to the tp atoms, and bottom atoms are the surface. Whte and black balls represent normal MD atoms and frozen atoms, respectvely. Gray balls ndcate Langevn atoms. (b) Atomc system under the perodc boundary condton n the z drecton usng the unt cells of the orgnal unt cell (sold lne) and the coped unt cell, so-called mage cell (broken lne). The gray atoms are not llustrated here. (c) A couplng system. The sprng s connected to the unt cell along the z drecton. (d) The unt cell becomes longer as the sprng shrnks, n whch the tp atoms n the orgnal cell s far from the surface n the mage cell. ote that the frozen atoms are always fxed at ther ntal postons. postons. They were postoned at the boundary regon n the atomc system as llustrated n Fg. 1(a). The moton of the Langevn atom, meanwhle, obeys the Langevn equaton, n whch the atoms are affected by an nteratomc force, a frctonal force, and random force components. The other atoms are referred to as normal MD atoms, and they are only affected by the nter-atomc force mentoned above. The Langevn atoms were located between the frozen atoms and the normal MD atoms as shown by the gray spheres n Fg. 1(a). Due to the exstence of the Langevn atoms, the temperature of the atomc system was kept constant durng the MD smulaton. The phonons n the fnte atomc system were reflected at the boundary of the frozen atoms; ths reflecton was absorbed at the layers of the Langevn atoms, n whch the frctonal force and the random force components absorbed the reflecton of the phonons. Thus far, the atomc system has been assumed to be under the solaton condton along the z drecton. At ths pont, we appled the PBC n the z drecton, as shown n Fg. 1(b). The atomc system s stll solated: the unt cell was long enough n the z drecton such that the atomc system n the orgnal cell s separated from that n the neghborng mage cell. As llustrated n Fg. 1(b), the tp atoms n the orgnal cell get close to the atoms of the surface n the neghborng mage cell, whch gves the attractve nteracton between the tp and the surface through the PBC. 2. Couplng system A sprng s attached to the atomc system as shown n Fg. 1(c). Although the atomc system was stll under the PBC, we assumed that the sprng was mechancally connected to the unt cell of the atomc system, n whch the dsplacement of the sprng have the same varatons n magntude of the z-length of the unt cell. If the sprng shrnks, the unt cell becomes longer n the z drecton by a dstance that equaled the dsplacement of the sprng (Fg. 1(d)), n whch the frozen atoms were fxed at ther ntal postons even when the z-length of the unt cell was vared. In ths case (Fg.1(d)), the dstance between the tp atoms n the orgnal cell and the surface atoms n the neghborng mage cell ncreases, and the attractve nteracton between the tp and the surface decreased. When the sprng oscllated, the z-length of the unt cell oscllated, and the separaton between the tp and the surface n the unt cell also oscllated. The sprng representng a realstc cantlever n the AFM was, n general, of a macroscopc scale. In the present couplng model, we could connect the sprng to the whole atomc system, n whch the sprng s connected to the unt cell, nstead to each atom n the unt cell. We can defne the Lagrange functon for the above couplng system. The Lagrange functon of the atomc system, L atom, s defned as, Appl. Sc. Converg. Technol. Vol. 26, o. 1 January 2017

3 8 Yasuhro Senda, Janne Blomqvst, and Rsto M emnen L atom [{ x, x, y, y, z', z' }] = V( { x, y, lz' }) m( 2 x + y + l 2 z' 2 ) Here, l s the z-length of the unt cell of the atomc system wth atoms of mass m. In ths functon, x and y are the x and y coordnaton of atom, respectvely, and z s the scaled z coordnaton, such that z s z =lz. The frst term on the r.h.s. s the knetc energy of the atomc system, and the velocty, z n t s defned as z = lz' ; the second term V({x, y, lz }), meanwhle, s the potental energy whch s the sum of the nteratomc potental n the atomc system. At ths pont, we ntroduced the unt cell s new degree of freedom, l, nto the atomc system. The magntude of the varaton n l corresponds to that of the dsplacement of the sprng, therefore the degree of freedom of l was equvalent to those of the sprng. We added the elastc and knetc energes of the sprng to the above L atom. and defned a new Lagrange functon for the couplng model, L coupl. : L coupl. x x y z' z' [{,,,, }, ll, ] = --m( 2 x + y + l 2 z' 2 ) 1 V( { x, y, lz' }) + --W k ( l l 0 ) 2 W s the nertal mass of the l, equvalent to the sprng s mass, whch corresponds to the cantlever s mass n the experment. In ths couplng model, we assume that cantlever s mass W does not nclude the atomc mass of the tp. The poston l 0 s the equlbrum poston of the sprng wth sprng constant k. The equatons of moton for the atoms and the sprng are derved from the functon L coupl. : V mx = , x (2) my = V, y (3) mz V = mz, z - l (4) 2 V mz z Wl z = kl ( l 0 ) (5) l The equaton (4) s obtaned from the Euler-Lagrange equaton for z, and the second dervatve of z =lz wth respect to tme. The moton of the normal MD atoms obey the equaton (2)-(4). The frctonal and random forces also acts on the Langevn atom n addton. The equaton of moton of the sprng s equaton (5), where the frst term on the r.h.s. s the sprng force, and the second and thrd terms, 2 V m, on the r.h.s. correspond to the nternal force z z l z wthn the atomc system. The nternal force s appled to the sprng, and t can be treated as the nteracton force between the tp and the surface. These equatons are ntegrated by a standard numercal method and we obtan the trajectores for the atoms and the sprng. The unts of energy, length, and mass n the calculaton are ev, Å, and the mass of an Mg on (m Mg ), respectvely. The tme unt s τ = A m Mg ev = s. The tme step n the MD calculaton s set to 0.02 τ. We set the sprng constant k=0.041 evå 2, and ts frequency ω= (2Π/τ). The temperature of atomc sytem s 300 K. We set three types of atomc confguraton of the tp n the atomc system. For type 1, one layer that conssts of fve O ons and four Mg ons was located on the surface of the MgO crystal, and one Mg on was mounted on ths layer at ts center. For type 2, one Mg on and another O on were mounted on the surface of the MgO crystal, and the ons were separated from one another. For type 3, one layer that conssts of eght O ons and eght Mg ons was located at the surface of the MgO crystal, and a par of Mg and O ons was mounted onto ths layer. For all of these cases, these onc structures were optmzed before the numercal ntegraton of the equatons of moton was performed, and the optmzed onc structures were used as an ntal onc confguraton of the tp. The tp ons n all cases also faced the flat MgO(001) surface,n whch the O on of the flat surface was postoned so as to be just below the Mg on of the tp apex. III. RESULTS 1. Adheson process and the energy dsspaton of the cantlever For type 1, durng one oscllaton cycle of the sprng, the tp atoms approached the surface atoms and retract from the surface, as llustrated n Fg. 2. When the tp approached the surface at the sprng dsplacement Δl= 3.5 Å and tme T=200 τ, the Mg on at the tp apex was separated from the surface O on. As the tp got closer to the surface, the Mg on attracted the O on, and an onc bond formed between them at T=310 τ. When the tp retracted from the surface, the Mg on was stll bonded to the O on at Δl=3.5 Å and T=420 τ, and the Mg on was separated from the O on when the dstance ncreased. Ths ndcates that there was an adhesve behavor n the bondng between the Mg on of the tp and the O on of the surface. Ths adhesve behavor leads to the hysteress property n the nteracton force between the tp and the surface. The onc confguraton of the tp at Δl= 3.5 Å durng the approach of the sprng dffered from that at the same Δl durng retracton. The dfferent atomc confguratons gave a dfferent magntude of the nteracton between the tp and the surface, and the nteracton force vared as the tp approached and retracted from the surface. Fg. 3 shows such a hysteress, n whch the nteracton s calculated as explaned n secton 2-2. The force dd not depend on the dsplacement of the

4 Stablty of Tp n Adheson Process on Atomc Force Mcroscopy Studed by Couplng Computatonal Model 9 Fgure 2. Tme (T) convoluton of the onc confguraton of the tp and the surface for type 1 of the tp. The Mg and O ons are represented by the black and whte spheres, respectvely. The dsplacements of the sprng, Δl, at tme T are also ndcated. Fgure 3. Interacton force as a functon of the sprng dsplacement, Δl. Fgure 5. The onc confguraton of the tp and the surface both before and after the formaton of the Mg-O onc bond for (a) type 2 and (b) type was defned as E = 1 2W + 1 2k ( l l0 ). The ampltude of the oscllaton was reduced and the energy E was also reduced by about ΔE ~ 0.02 ev per cycle of the oscllaton. Fgure 4. (a) Dsplacement of the sprng and (b) ts energy as a functon of tme. sprng. Such a non-conservatve force resulted n the dsspaton of the sprng's oscllaton. Fgure 4 shows the oscllaton of the sprng, where the energy of the sprng 2. Stablty of the atomc arrangements of the tp and the surface For type 1, the ntal onc confguraton of the tp and the surface remaned after the tp had approached and retracted from the surface as llustrated n Fg. 2. Durng the tp approach and retracton, the Mg on of the tp formed an onc bond wth the O on and separated from the O on, respectvely. The ntal onc confguraton of the tp and the surface dd not change, not even after the formaton and breakng of the Mg-O onc bond; n fact, t stll perssted, even after many cycles of the sprng oscllaton. A stable reconstructon of the onc confguraton s therefore shown for type 1. For types 2 and 3, however, once the ons of the tp formed the onc bond wth the ons on the surface, the onc confguraton of the Appl. Sc. Converg. Technol. Vol. 26, o. 1 January 2017

5 10 Yasuhro Senda, Janne Blomqvst, and Rsto M emnen tp and the surface never returned to ther ntal states. For type 2, as the sprng retracted from the surface, the Mg on of the tp pulled the O on up from the surface. As the dstance between the tp and the surface ncreased, the tp pulled up strngs of ons from the surface, as llustrated n Fg. 5 (the O on of the tp apex postoned far from the Mg on also pulled the on up from the surface and formed the strngs). For type 3, the par of the Mg and O ons at the tp apex pulled the ons up from the surface and formed the stngs of ons, as had been the case for type 2. For both these cases, the onc structures of the tp and the surface changed rreversbly after an ntal cycle of the oscllaton of the sprng. These rreversble changes were n contrast to the stable reconstructon of the tp that had been observed for type 1. IV. Dscusson As the tp approached the surface, the ons at the tp apex formed onc bonds wth the ons on the surface for all types of the tp. As the sprng retracted from the surface, the ntal onc confguraton of the tp remaned ntact only for type 1, whle t changed rreversbly for types 2 and 3. The stablty of the tp strongly depended on ts onc confguraton of the tps. In addton to the present three cases, we nvestgated the stabltes of other types of onc confguratons of the tp, for example, the tp of two Mg ons and two O ons at the tp apex. For these other types, the onc confguratons also changed rreversbly, as had been the case for types 2 and 3. Based on the present calculaton, t seems that the adhesve behavor of atoms results n an rreversble change n the tp and surface for most confguratons of the tp. Atomc adheson has been thought of as a promsng mechansm for energy dsspaton n non-contact AFM, as mentoned n the ntroducton secton. Ths adhesve mechansm s based on the exstence of a tp that would have a stable atomc confguraton even after adheson between the atoms of the tp and the surface. In the present calculaton, such a stable tp occurs only for type 1, n whch the energy dsspaton came from the adhesve behavor of the ons between the tp and the surface. Ths stablty, however, was not observed for most types of onc confguratons of the tp. There are many types of onc confguratons at the tp apex of a realstc tp n an AFM experment, and, based on our results, t would seem that there are only a lmted number of types of onc confguratons that would be stable durng the adheson process. V. Conclusons A computatonal model for an AFM experment that couples the atomc moton wth the oscllaton of the AFM s cantlever was appled to the AFM of an onc MgO surface. It was shown that the adhesve behavor of the MgO ons resulted n an energy dsspaton durng oscllaton of the cantlever. 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