Model dependence of AFM simulations in non-contact mode

Transcription

1 Surface Science 457 (2) Model dependence of AFM simulations in non-contact mode I.Yu. Sokolov a,b,*, G.S. Henderson a, F.J. Wicks a,c a Department of Geology, University of Toronto, Toronto, Ontario, Canada M5S 31 b Department of Physics, University of Toronto, Toronto, Ontario, Canada M5S 1A7 c Department of Mineralogy, Royal Ontario Museum, Toronto, Ontario, Canada M5S 2C6 Received 26 October 1998; accepted for publication 1 December 1999 Abstract Pairwise summation of a Lennard Jones type potential is, because of its simplicity, an attractive method for theoretical simulation of scan images in atomic force microscopy (AFM). However, a serious problem for such simulations is the uncertainty in the AFM tip sample interaction. While the interatomic repulsion is more or less universal, the attractive component of the interaction can include van der Waals, ionic, dipole, chemical, electrical and other kinds of forces. In the present paper, the pairwise summation method for the simulation of AFM scans is analyzed against the uncertainties in the attractive component of the interaction potential. It is shown that the simulation method produces, qualitatively, similar images of an atomic surface for a broad range of possible contributions to the attractive force interaction. This indicates that the numerical method, while relatively insensitive to the explicit attractive force contributing to the tip sample interaction, is suitable for theoretical simulations of atomic resolution in the AFM. 2 Elsevier Science.V. All rights reserved. Keywords: Atomic force microscopy; Atom solid interactions; Computer simulations; Surface defects 1. Introduction for the theoretical simulation of AFM scans in non-contact mode, while scanning with constant Using an atomic force microscope (AFM) it is force gradient. Qualitatively, the results of the possible to obtain atomic resolution images of the paper remain the same for contact mode while surfaces of metals, dielectrics and semiconductors scanning with constant force. However, attaining [1,2]. However, achieving such high resolution is true atomic resolution in contact mode is much a non-trivial problem due to noise, averaging more problematic and needs a specific alteration effects, and the need for a clean surface. of the attractive part of the AFM tip sample Furthermore, once high-resolution images have interactions (see e.g. Ref. [2]). The numerical been achieved, image interpretation remains probis based on pairwise additive summations of a method has been described previously [3 6 ] and lematic and is not simple, particularly since the AFM image is a 2D surface projection of the bulk Lennard Jones type potential, acting between sample structure. the atoms of the AFM tip and sample. While the In the present paper we analyze a simple method method is suitable for numerical simulation of the tip sample force interactions, the model results could be susceptible to the uncertainties inherent * Corresponding author. Fax: in estimations of the attractive force contribution address: sokolov@physics.utoronto.ca (I.Yu. Sokolov) to the overall tip sample interaction. These //$ - see front matter 2 Elsevier Science.V. All rights reserved. PII: S ( ) 384-8

2 268 I.Yu. Sokolov et al. / Surface Science 457 (2) uncertainties arise because the attractive part of the tip sample force interaction may include van der Waals, ionic, dipole, chemical, electrical and other kinds of forces, which are hard to control and simulate. In the present paper we explore the suitability of our method for such simulations in light of the uncertainties noted above for the attractive compo- nent. We show that the method produces, qualitatively, the same image of a surface at atomic resolution for a broad range of parameters. This implies that the numerical method is relatively robust and suitable for theoretical simulations of atomic resolution AFM images. 2. The numerical method Atomic force microscopy scan simulations are performed with the tip surface geometry shown in Fig. 1. The tip is considered to be a paraboloid of rotation with a sharp apex made up of a silicon- type (cubic face-centered) lattice. The apex of the tip is considered to be the {111} corner of the cubic cell, and the orientation of the tip during scans is such that the cell edge is oriented in the direction of the rows of atoms. The radius of the tip curvature (of the paraboloid) is 1 nm, which corresponds to the best commercial tips and the lattice constant of the apex is.54 nm, which is the cell dimension for silicon. The apex itself is considered to consist of four atomic layers (approximately 5 atoms), while the paraboloid is assumed to be a continuous medium. The sample is considered to be a plane surface with a cubic lattice. The surface is considered to be four atomic layers thick, while the rest of the sample is treated as a continuous medium. Such a configuration is convenient for numerical calculation and the error of using a continuous medium instead of discrete atomic structure is less than 4%, depending on the tip sample distance. A range of lattice constants from.3 to.6 nm is considered for the surface. The lower limit of.3 nm is chosen because it is theoretically impossible to do nondestructive contact mode AFM scans of materials containing atomic defects and with lattice constants below this value [7]. The interatomic interaction between the tip and sample is described by a Lennard Jones type potential n(r)=e CAr r N A r r N (1) D where e is the binding energy between atoms, r approximates the equilibrium distance between bound atoms, r is the interatomic distance, and N is an integer between 5 and 11, which describes the power laws for the different types of attractive contributions to the overall attractive part of the interaction. N cannot be less than 5, because the effective range of attractive force interaction is too large and the AFM tip sample potential would then include the macroscopic size of the tip and sample, resulting in an enormously large force of attraction. If N is greater than 11 it would overcompensate for the repulsive force, resulting in an unstable material that could easily collapse. To find the force gradient of interaction between Fig. 1. The tip surface geometry used in the numerical an AFM tip and a sample, one needs to simulations. integrate/sum the potential ( 1) over the

3 I.Yu. Sokolov et al. / Surface Science 457 (2) volume/atoms of the sample and tip. A simple van der Waals interaction. It is given by (see e.g. additive summation for all the atoms of the tip Refs. [8 1]) and sample is a good approximation for repulsive force [the first term of Eq. ( 1)]. However, the e= H 1 H 2 (2) attractive force (the second term) is not an additive 2p2n n r3 r3 one. For example, the van der Waals attraction 1 2 (N=6) is not a simple sum of the pairwise inter- where H, H are the Hamaker constants, n, n actions, and if treated as such is usually greater are atomic densities, r, r are the equilibrium than the actual force between the macrobodies of distances between surface atoms, and between tip interest. Nevertheless, as was shown in Ref. [7], atoms, respectively. We set r and r to be equal the effect of non-additivity results in a change of to the lattice dimensions of the surface material and the interaction constant by a factor of not more the tip material, respectively. This is a good approxi- than ~2%. For the purposes of the present study, mation for the simple cubic lattice of the surface this error is negligible, because we shall consider material, but is not very precise for pure silicon. the change in the interatomic interaction constant This means that our assumption of r =.54 nm, 2 (e) over three orders of magnitude. is suitable only if we are considering a silicon-type To start with, we are considering the minimum material rather than silicon. To get a silicon tip, possible energy (e) to be the energy associated with one should set r #.3 nm, which corresponds to 2 Fig. 2. Plots for the absolute contrast as functions of r, for different a and N: (a) the contrast as a function of r, for different N, a=1; (b) the contrast as a function of N, for different a, N =6; (c) the contrast as a function of both N and a.

4 27 I.Yu. Sokolov et al. / Surface Science 457 (2) the covalent bond distance between two silicon where z is the difference between the z-position of atoms. We discuss this case at the end of this paper. a tip atom and the z-position of the continuous The energy of strong interatomic interactions part of the sample. The three-dimensional integral such as ionic, metal interactions can exceed the in the third term can easily be reduced to a onevan der Waals force by a factor of 13. To take dimensional term and treated numerically therethis into account, we introduce a constant a that after. We now substitute the parameters for our multiplies the energy (2). One then arrives at the tip sample configuration noted above, following formula for the interaction potential r =.54 nm and H = J [11]. The n between the atoms of the tip and sample parameter is set at m 3 and is calculated H H as the ratio of mass density and mass of one silicon U(d )= a 1 2 r 1 r 2p2n n atom. For the sample, we vary r 1 from.3 to.6 nm, and since we consider a cubic structure for the sample, n 2 =(r 1 ) 3. The Hamaker constant A r3 1 r3 2 (3) of the sample is set to 1 19 J. Scans are simulated as surfaces of the same where r refer to the vector positions of the force gradient, i.e. for the common case of nonsample and tip atoms, respectively. contact mode of AFM scanning. 1,2 This is the interaction potential that is used for the additive summation hereafter. The total interaction between the tip and sample, taking into account the discrete (atomic) and continuous parts of the tip and sample, is now given by U(d )= a H 1 H 2 r 1 (atomic part) r 2 (atomic part) 2p2n n 1 2 A r3 1 r3 2 r r N P dr a H 1 H 2 1 2p2n r 2 (atomic part) continuous part 2 A r3 1 r3 2 r r N P dr a H 1 H 2 2 r 1 (atomic part) 2p2n continuous part 1 A r3 1 r3 2 r r N 1 2 rn/2 3 rn/2 3 r r N 1 2 rn/2 3 rn/2 3 r r N 1 2 rn/2 3 rn/2 3 r r N P dr dr a 1 P H 1 H 2 2 2p2 continuous part continuous part A r3 1 r3 2. (4) ecause of the short-range character of the repulsive force, repulsive contributions in the second and fourth terms of Eq. (4) are negligible. We can then treat these two terms as integrals of attractive forces only. Taking Eq. (4), and using the results of Refs. [4 6 ], one gets U(d )= a H 1 H 2 r 1 (atomic part) r 2 (atomic part) 2p2n n 1 2 A r3 1 r3 2 a 12rN/2 3 rn/2 3 H H r 2 (tip atomic part) (N 2)(N 3) pn zn 3 N 2 + P dr a H 1 H 2 2 r 1 (surface atomic part) 2p2n continuous part 1 A r3 1 r3 2 a H 1 H 2 R 24 rn/2 3 rn/2 3 (5) N dn 5 (N 2)(N 3)(N 4)(N 5)

5 I.Yu. Sokolov et al. / Surface Science 457 (2) Results and discussion the interaction strength, i.e. parameter a=1 1, the sample lattice constant r =.3.6 nm and the First, we consider a defect-free cubic surface attraction power law N=5 11. Fig. 2 shows the consisting of uniform atoms with r from.3 to 1 results of our calculations for the height contrast.6 nm and a scan direction parallel to the rows as functions of r, for different a and N. One can of atoms. Height contrast is defined as the change see from Fig. 2c that the contrast changes only of height between when the AFM tip scans over within ~.9 nm for small r (=.3 nm in our an atom and when the tip is exactly between two case), and within ~.5 for large r (=.6 nm). It neighbor atoms: is interesting to note that almost all the change in contrast=height over an atom contrast comes from variation of the energy height between two atoms. ( Fig. 2b), whereas all changes in contrast because of N are within 1%. It should be noted that there We study this contrast change as a function of is some universal behavior, for all values of Fig. 3. Simulated images of a cubic lattice surface with one vacancy and one atom of different sort: (a) a=1, N=5, r =.3 nm; (b) a=1, N=11, r =.3 nm; (c) a=1, N=5, r =.6 nm; (d) a=1, N=11, r =.6 nm. In all cases the scan force gradient is set to 1 4 N/m.

6 272 I.Yu. Sokolov et al. / Surface Science 457 (2) is intuitively clear. It must increase because the tip has a more compact atomic arrangement and can penetrate deeper into the structure in the region between two atoms. The force gradient, and conse- quently height above an atom, is primarily determined by the atom at the tip apex. The arrangement of the rest of the atoms of the tip only influences the interaction force weakly, and consequently the contrast increases. As to the contrast change, our calculations show that it remains about the same. Therefore, we can conclude that for a silicon tip, the simulated image is qualitatively the same for all parameters a and N considered here. r the contrast increases for lower N (Fig. 2a), and decreases with increasing a (Fig. 2b). Furthermore, the contrast change can further decrease. Any change of a can be compensated to some extent by a change in the scan force gradient. The AFM scan is a set of the tip positions, which correspond to the same force gradient. To find these positions one needs to solve the following equation F (a, N, x, y, z)=f scanning. ecause the factor a is multiplicative, one can extract it from the force, and arrive at the following equation F (a=1, N, x, y, z)=f /a. scanning Consequently, an increase of a by 1 is equiva- 4. Conclusions lent to a decrease of the scan force gradient by 1 and, furthermore, because the scan force is Our suggested method for AFM simulation of essentially a variable parameter during the scan, a atomic resolution in non-contact mode is tested with a variety of molecular forces in addition to the universal repulsive part of the scan force. It is shown that the uncertainty in the simulations is small for the purpose of qualitative analysis of the surfaces, and even these differences can be elimi- nated by an increase in the scan force gradient. change in a can be compensated by the appropriate choice of scan force. Let us now address the question of how these extreme cases of contrast change, shown in Fig. 2, look like for a complex structure that consists of different types of atoms. We simulate different atoms by assigning them different a parameters. We add to our cubic lattice a vacancy (a=) and one atom of different type, with a five times larger than a for the other atoms, which corresponds to References adifferent chemical bond. According to Fig. 2, the [1] G. innig, C.F. Quate, Ch. Gerber, Atomic force micromaximum contrast change is between a=1, N=5 scope, Phys. Rev. Lett. 56 (1986) 93. and a=1, N=11. We simulate these conditions [2] I.Yu. Sokolov, G.S. Henderson, F.J. Wicks, G. Ozin, Appl. in Fig. 3. Phys. Lett. 7 (1997) 844. [3] V. Koutsos, E. Mania, G. ten rinke, G. Hadzioannou, Fig. 3a shows the case of a=1, N=11, Europhys. Lett. 26 (1994) 13. r =.3 nm, Fig. 3b the case of a=1, N=5, [4] I.Yu. Sokolov, G.S. Henderson, F.J. Wicks, Surf. Sci. 381 r =.3 nm, Fig. 3c the case of a=1, N=11, (1997) L558. r =.6 nm, and Fig. 3d the case of a=1, N=5, [5] Yu.N. Moiseev, V.M. Mostepanenko, V.I. Panov, I.Yu. r =.6 nm. In all cases the scan force gradient is Sokolov, Phys. Lett. A 132 (1988) 354. set to 1 4 N/m, about the value used in experi- [6] I.Yu. Sokolov, Surf. Sci. 311 (1994) 287. [7] I.Yu. Sokolov, G.S. Henderson, F.J. Wicks, Scanning ments on atomic resolution with non-contact (2) in press. AFM. It is clear that the images are the same, [8].V. Deryaguin, N.V. Churaev, V.M. Muller, Surface qualitatively, for all cases. The vacancy is visible Forces, Nauka, Moscow, under all conditions and the unique atom is almost [9] H. Krupp, W. Schnabel, G. Walfer, The Lifshitz van der indistinguishable except for the case in Fig. 2a. Waals constant, J. Colloid Interface Sci. 39 (1972) 421. [1] F. Varnier, G. Desrousseaux, A. Carlan, Appl. Surf. Sci. As noted previously, in order to calculate simu- 5 (198) 338. lated scans with a silicon tip, one needs to set [11] V.M. Mostepanenko, I.Yu. Sokolov, Sov.-Phys. Dokl. 33 r #.3 nm. What should happen to the contrast 2 (1988) 14.