Simulated non-contact atomic force microscopy for GaAs surfaces based on real-space pseudopotentials

Applied Surface Science 303 (2014) 163–167

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Simulated non-contact atomic force microscopy for GaAs surfaces based on real-space pseudopotentials Minjung Kim a , James R. Chelikowsky a,b,c,∗ a b c

Department of Chemical Engineering, The University of Texas at Austin, Austin, TX 78712, USA Departments of Physics, The University of Texas at Austin, Austin, TX 78712, USA Center for Computational Materials, Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712, USA

a r t i c l e

i n f o

Article history: Received 13 November 2013 Received in revised form 19 February 2014 Accepted 21 February 2014 Available online 3 March 2014 Keywords: Noncontact atomic force microscopy Density functional theory GaAs(1 1 0) surface

a b s t r a c t We simulate non-contact atomic force microscopy (AFM) with a GaAs(1 1 0) surface using a real-space ab initio pseudopotential method. While most ab initio simulations include an explicit model for the AFM tip, our method does not introduce the tip modeling step. This approach results in a considerable reduction of computational work, and also provides complete AFM images, which can be directly compared to experiment. By analyzing tip-surface interaction forces in both our results and previous ab initio simulations, we find that our method provides very similar force profile to the pure Si tip results. We conclude that our method works well for systems in which the tip is not chemically active. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Scanning probe microscopy (SPM) is a powerful technique to explore surface structures and nano-scaled materials [1]. During the last few decades, it has broadened our understanding of not only surface physics and chemistry, but also structure of molecules. The history of SPM began with the invention of the scanning tunneling microscope (STM), which measures quantum tunneling current between a probe and a sample [2]. As the tunneling current is highly sensitive to the tip-sample distance, STM is able to create very high resolution images. While STM has been successful in many applications including conducting and semiconducting materials, it does not always provide direct structural information of the sample as it probes the electronic states close to the fermi level and not the atomic structure. Also, it requires conducting materials for the sample to create tunneling current, which becomes a barrier to the use of STM in insulators. Atomic force microscopy (AFM) is one mode of SPM that has a few significant advantages over STM. The most important aspect of AFM is its applicability to a wide range of materials as it measures forces acting on a probe from the sample and not a tunneling current. AFM is capable of atomic resolution imaging, which has

∗ Corresponding author at: Center for Computational Materials, Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712, USA. Tel.: +1 512 232 9083; fax: +1 512 471 8694. E-mail addresses: [email protected], [email protected] (J.R. Chelikowsky). http://dx.doi.org/10.1016/j.apsusc.2014.02.127 0169-4332/© 2014 Elsevier B.V. All rights reserved.

enabled researchers to reveal many complicated surface structures such as the Si-(1 1 1) 7 × 7 reconstructed surface [3,4]. Moreover, recent AFM studies have been used to investigate complex nanostructures [5–7]. As a result, it has become the most widely used technique in studying insulating surfaces, nano-structured materials, and biomaterials [8–10]. Although AFM has been applied successfully in studying many surface structures, there can be uncertain and ambiguous results owing to various experimental conditions. As such, the interpretation of experimental AFM images should be examined carefully by coupling with theoretical simulations. However, first-principles based AFM simulations are usually arduous since the force scanning process involves a three-dimensional motion of the tip, i.e., a raster scan and a vertical vibration. The interaction energy has to be calculated for each position with a small spacing in the path of the tip, and accurate energies require a large number of calculations even for a small system [11]. Another difficulty in theoretical simulations may stem from modeling the tip [12,13] since the exact morphology of the tip is unknown. Recently, an efficient first-principles simulation method for non-contact AFM was proposed by Chan et al. [14]. This method can be used to calculate the tip-surface force without explicit modeling the tip. With the formalism, the force and frequency shift are directly calculated from only the electronic structure of the sample. This not only eliminates the arbitrary model for the tip, but also dramatically reduces the total number of calculations since the three-dimensional motion of the tip is not required. This method has been successfully applied to large surface structures such

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where Fts , f0 , k0 , A, and d are the tip-surface force, resonance frequency, spring constant, oscillating amplitude, and distance from the surface to the tip turning point, respectively. Most nc-AFM simulations compute the tip-surface force by calculating the gradient of the tip-surface total energy with respect to the tip-surface distance. If the electronic influence of the surface on the tip is treated as a small perturbation, the tip-surface interaction energy can be written as



Ets (r) = Fig. 1. A simple diagram of the tip motion. q = 0 indicates the equilibrium tip-surface position, d is the tip turning point, and A is the oscillation amplitude of the tip.

√ √ as Si(1 1 1)-7 × 7 reconstructed surface, Ag/Si(1 1 1)-( 3 × 3)R30◦ surface, and Ge/Si(1 0 5)–(1 × 2) surface, and demonstrated qualitatively comparable AFM images. In spite of the success, some systems may require a careful investigation of the tip. Implicit in the theory of Chan et al. is the assumption of a “chemically inert” tip. In some cases, this assumption may not be valid. For example, AFM measurement for the TiO2 (1 1 0) surface exhibits three qualitatively different contrast modes, and it is not rare to observe the contrast change even within the same AFM experiment [15]. This implies the AFM tip can be contaminated by surface atoms or impurities [12]. In such cases, our method can be used to assess the role of the tip by providing images that are tip independent. Here we examine non-contact AFM imaging by following a similar approach to Chan et al. [14] based on a real-space implementation of pseudopotentials constructed using density functional theory [16]. Our real-space method is especially useful for partially periodic system such as molecules, wires, surfaces, and interfaces because no artificial periodicity in the non-periodic direction is required [17]. We choose to study the GaAs(1 1 0) surface because it holds a simple structure with no surface reconstruction that has been extensively studied.

2. AFM theory True atomic resolution in AFM can be obtained by the frequency modulation operating mode in an ultra-high vacuum environment [4,18]. With these conditions, the morphology of the atomic structure is captured by a small frequency change of the oscillating tip. The theory of motion for the tip has been reported in many previous studies [19–24]. A common approach is to assume the system is a one-dimensional oscillator with an additional force that depends on the tip-surface distance. Fig. 1 shows the schematic of the tip motion. The tip oscillates with the resonance frequency f0 , which is a material-dependent parameter. In most experiments, the amplitude of the motion of the tip is larger than a typical tip-surface interaction distance [25]. In this case, the frequency change can be considered to be a small perturbation. By following the HamiltonJacobi formalism, a general expression for the frequency shift can be written [19,26]:

f = −

=

f2 f0 < Fts z >= − 0 2 k0 A 2k0 A

f0 k0 A2

f0 = k0 A2





A

q

dq

A2 − q2

−A A

− −A

Fts (d + A − A cos(2f0 t)) cos(2f0 t)dt

0

Fts (d + A − q) 



1 f0

∂Fts (d + A − q) ∂q



A2 − q2 dq

(1)

|(r − r)|2 Vts (r )dr ,

(2)

where r, Vts , and  are the position of the tip, the potential on the tip generated by surface atoms, and the electronic state of the tip, respectively [14]. We assume the tip does not affect surface structure as expected in the limit of a small perturbation. Fts can be described by expanding the potential to the first-order around the tip position r:



Fts = −∇ Ets (r)  −∇ Vts (r) − ∇ [∇ Vts (r)

|(r − r)|2 (r − r)dr ]

= −∇ Vts (r) − ∇ [∇ Vts (r) · p] = −∇ Vts (r) − ˛∇ (|∇ Vts (r)|2 ),

(3)

where p is the polarization of the tip, assumed to have a linear relation to ∇ Vts by ˛, the polarizability of the tip material. The first term is a monopole caused by the electrons of the tip, which is canceled by the ionic potential of the tip if the tip is neutral. In this setting, the force acting on the tip is now proportional to the dipole interaction, Fts (r) ≈ −˛∇ (|∇ Vts (r)|2 ).

(4)

The tip-surface potential is obtained by the hartree and the ionic potential from DFT calculations: Vts = Vhart + Vion . The exchangecorrelation potential is not included in Vts because the tip is treated as a classical object. The calculated tip-surface force from Eq. (4) can be inserted to Eq. (1) to compute the frequency shift. Since Vts decays very fast in vacuum, the most relevant integration region is only a couple of angstroms near the tip-surface turning point (d in Fig. 1). Therefore, the integration region in Eq. (1) can be reduced from (−A, A) to (−A, −A+2), with  chosen to be small compared to A. 3. Computational details All calculations were performed with real-space pseudopotentials with the local density approximation for the exchange-correlation functional from Ceperley and Alder [27]. The Kohn-Sham equation is solved on a grid using a high-order finite difference method [28,29]. The convergence of the total energy is controlled by the grid spacing, which was taken to be 0.35 a.u. We applied the grid spacing of 0.25 a.u. for composing AFM images in order to increase the resolution. We used Troullier-Martins pseudopotentials [30] with a partial core-correction for the Ga atom. The parameters used to generate the pseudopotentials were from previous work by Kim and Chelikowsky for vacancies in the GaAs surface [31]. With these pseudopotentials, we obtained a lattice ˚ which agrees within 1% of the experimental parameter of 5.59 A, value [32]. Our simulation cell for the GaAs(1 1 0) surface contains a 1 × 2 surface unit with five atomic layers that corresponds to the slab ˚ and 21 A˚ of a vacuum was inserted. In order to thickness of 17.79 A, passivate the dangling bonds on the opposite side of the surface, we generated hydrogen-like pseudopotentials with a 1s0.75 and 1s1.25 ionic configuration for Ga and As atom, respectively [33]. The kpoints were generated by a Monkhorst-Pack scheme [34] with a 4 × 3 mesh. Geometry optimization was performed until the force

M. Kim, J.R. Chelikowsky / Applied Surface Science 303 (2014) 163–167

Fig. 2. A side view (a) and a top view (b) of the relaxed GaAs(1 1 0) surface. Black (magenta) and gray (yellow) indicate Ga and As, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

acting on each atom is less than 0.004 Ry/a.u. After structural relaxation, the Ga atom moved inward and the As atom moved outward with 0.69 A˚ of buckling as illustrated in Fig. 2(a). This value agrees very well with the observed value of 0.69 A˚ in the experiment [32]. To create the simulated AFM image, we set three parameters in Eq. (1): A, , and d. We tested the sensitivity of these parameters to the AFM images by varing the values for A and  from 5 nm ˚ respectively, but have not found to 50 nm and from 0.5 A˚ to 2.5 A, significant changes in image contrast. A is chosen to be similar to the typical oscillation amplitude (10 nm) used in experiment [25] and  is selected to be 1 A˚ for all calculations. 4. Results and discussion The parameter that has the most significant effect on the AFM image is the tip turning point d. Fig. 3 shows our simulation images ˚ 4 A, ˚ and 5 A˚ for (a), (b), and (c), with three different values for d: 3 A,

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respectively. In this figure, we observe the opposite trend of how the Ga and As atoms respond. As the tip turning point increases, the bright spots which correspond to the Ga atom disappear while the As atom becomes bigger and brighter. Typically, most AFM experiments on III–V(1 1 0) semiconducting surfaces have shown only the anion image which is similar to Fig. 3(c). This seems natural because the III–V(1 1 0) surface has a surface buckling, i.e., the anion is displaced to the vacuum. However, some recent experimental studies have reported simultaneous observation of both the anion and cation sublattices. In the case of the GaAs(1 1 0) surface, Uehara and coworkers studied image contrast by using several frequency shift values and Si tips [35]. Among the several AFM images they obtained, Ga sublattice was appeared in the large frequency shift AFM images, which is shown in Fig. 3(d) and (e). The larger frequency shift corresponds to the smaller tip-surface distance and vice versa. Our simulation results are consistent with this experiment. Also, the contrast of our simulated images is very similar to that of the experimental images. Previously, an ab initio AFM study on the GaAs(1 1 0) surface with several Si-cluster tips was conducted by Ke and coworkers [36,37]. As the tip can be contaminated by the sample material, which is Ga and As in this case, they calculated tip-surface energy and force with Si, Ga, and As apexes. To make a comparison between our method and their results, we evaluated −∇ (∇ Vts )2 from Eq. (4), which is directly related to the tip-surface force. We do not expect to calculate absolute values for the force as the polarizability of the tip, ˛, is unknown. It is unnecessary to calculate the exact force to obtain AFM images as we only need to know the relative difference of the frequency shift at each point. Fig. 4 presents the calculated forces from our simulation (top panels) and Si-cluster tip simulations (bottom panels). We chose the tip-surface distance to be 3.41 A˚ for line A and 4.21 A˚ for line B, and these values are calculated from the surface As plane. The corresponding tip-surface distances of the reference data are 3.38 A˚ and 4.15 A˚ for line A and B, respectively. In this way, we can maintain a similar distance between the tip apex and the surface As or Ga atom for each line, and can compare the closest set of the force curve from the previous

˚ (b) 4 A, ˚ and (c) 5 A. ˚ The images are overlaid Fig. 3. Simulated AFM images with respect to the tip turning point (d).  is set to be 1 A˚ for (a)–(c), and the values for d are: (a) 3 A, with the surface Ga (magenta) and As (yellow) atom. Black and white indicate low and high frequency shift values, respectively, and the gray scale is adjusted independently. (d)–(f) Non-contact AFM images of GaAs(1 1 0) from experiment [35]. The frequency shift is −137 Hz, −188 Hz, −218 Hz for (d)–(f). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Fig. 4. Comparison of tip-surface forces. (a) A top view of the GaAs(1 1 0) surface and the black color indicates top layer atoms. Dashed-line A and B correspond to graph (b) and (c). Top panels in graph (b) and (c) show our results calculated from Eq. (4). Other three panels are previous ab initio results simulated by Si-cluster tip with Si, Ga, and As apexes (Refs. [36,37]). The tip-surface distances are 3.41 A˚ and 4.21 A˚ for (b) and (c), respectively.

simulations. In both graphs, our results are in good agreement with the Si apex simulation results. Our simulation and the Si apex simulation show the strong force when the tip is close to the surface atoms. On the other hand, Ga and As apexes detect only one surface atom: Ga apex responds to the surface As atom while As apex responds to the Ga surface atom. Only the pure silicon tip images both Ga and As sublattices at relatively small tip-sample distances while the larger tip-sample distance images the As sublattice. This is the same observation that we made in Fig. 3. This result implies that our method should be suitable for simulating AFM images obtained with a Si tip. Our method is based on a simple approximation: the tip does not affect the electronic structure of the sample as expected in the limit of a weak interaction between the tip and the surface. Our simulation may not be applicable in a certain cases where this limit is not satisfied, e.g., when the tip and the surface atom form a chemical bond. Since our simulations match well with previous calculations employing a Si apex tip model [36], we believe the Si tip does not saturate a dangling bond of either Ga or As atom. The pure Si tip does not exhibit a strong relaxation effect with the surface Ga or As atoms if the tip-sample distance is larger than 3 A˚ [36]. The

calculations found for shorter distances the electronic structure of the surface is changed by the tip. In Fig. 4, we note that the position of the force peak is slightly shifted from the actual surface atom positions. In line A, the peak is moved toward [0 0 1] direction from Ga atom while the As atom ¯ direction in line B. These shifts are related signal moves to [0 0 1] to the dangling bond of the unperturbed surface atom as presented in Fig. 5. We plot the electron density of the surface that is close to the fermi level. Fig. 5(a) shows the fully occupied dangling bond of the As atom and 5(b) represents the empty dangling bond of the Ga atom. These position shifts were also observed in experiment. Fig. 5(c) shows measured atomic structure from the AFM experiment along the line X–X and Y–Y that correspond to the line B and line A in Fig. 4, respectively. In this figure, ˇ is expected to be 1.13 A˚ in the ideal situation. However, the AFM measurement determined the value between 2.4 and 2.6 A˚ which is more than twice larger. In ˚ our simulation, the measured value is about 2.7 A. Our method will provide a comparable AFM image as is obtained by a silicon tip in the region where the tip does not undergo considerable changes in their electronic structure. Our approach will be

Fig. 5. (a) The dangling bonds of the surface As atom. The electron density within 1 eV energy window below the Fermi level is visualized. Black and light gray represent Ga and As, respectively. (b) The empty dangling bonds of the surface Ga atom (1 eV energy window above the Fermi level). (c) Ga and As signals from AFM experiments. (Adapted from Ref. [35].) Dashed and solid lines indicate X–X and Y–Y , respectively.

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especially useful not only because most commercial AFM tips are made of Si, but also the Si tip can serve as a reference for interpreting AFM images since it supports the similar contrast mechanism for many surfaces without regard to physical and electronic structures [38]. 5. Conclusions We performed non-contact AFM simulations based on realspace electronic structure calculations with ab initio pseudopotentials. By implementing a new simulation method, which is much simpler than previously performed AFM simulations, we studied image contrast in the GaAs(1 1 0) surface. Our simulated AFM images are in a good agreement with experiments, and calculated tip-surface forces are similar to previous ab initio simulations with a Si-cluster tip. Our AFM simulation tool will provide qualitatively comparable AFM images without intensive computation. Acknowledgments We would like to acknowledge partial support from the Department of Energy for work on nanostructures from grant DEFG02-06ER46286. We also wish to acknowledge support provided by the Scientific Discovery through Advanced Computing (SciDAC) program funded by U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research and Basic Energy Sciences under award number DESC0008877 on algorithms. References [1] L. Gross, Recent advances in submolecular resolution with scanning probe microscopy, Nat. Chem. 3 (2011) 273. [2] G. Binnig, H. Rohrer, C. Gerber, E. Weibel, Surface studies by scanning tunneling microscopy, Phys. Rev. Lett. 49 (1982) 57–61. [3] F.J. Giessibl, C.F. Quate, Exploring the nanoworld with atomic force microscopy, Phys. Today 59 (2006) 44. [4] F.J. Giessibl, Atomic resolution of the silicon (1 1 1)–(7 × 7) surface by atomic force microscopy, Science 267 (1995) 68. [5] C.H. Lui, L. Liu, K.F. Mak, G.W. Flynn, T.F. Heinz, Ultraflat graphene, Nature 462 (2009) 339. [6] R. Pawlak, S. Kawai, S. Fremy, T. Glatzel, E. Meyer, Atomic-scale mechanical properties of orientated C60 molecules revealed by noncontact atomic force microscopy, ACS Nano 5 (2011) 6349. [7] D.G. de Oteyza, P. Gorman, Y.-C. Chen, S. Wickenburg, A. Riss, D.J. Mowbray, G. Etkin, Z. Pedramrazi, H.-Z. Tsai, A. Rubio, M.F. Crommie, F.R. Fischer, Direct imaging of covalent bond structure in single-molecule chemical reactions, Science 340 (2013) 1434. [8] F.J. Giessibl, Advances in atomic force microscopy, Rev. Mod. Phys. 75 (2003) 949. [9] C. Barth, A.S. Foster, C.R. Henry, A.L. Shluger, Recent trends in surface characterization and chemistry with high-resolution scanning force methods, Adv. Mater. 23 (2011) 477. [10] D.J. Müller, J. Helenius, D. Alsteens, Y.F. Dufrêne, Force probing surfaces of living cells to molecular resolution, Nat. Chem. Biol. 5 (2009) 383. [11] L. Gross, F. Mohn, N. Moll, B. Schuler, A. Criado, E. Guitian, D. Pena, A. Gourdon, G. Meyer, Bond-order discrimination by atomic force microscopy, Science 337 (2012) 1326. [12] A. Yurtsever, D. Fernández-Torre, C. González, P. Jelínek, P. Pou, Y. Sugimoto, M. Abe, R. Pérez, S. Morita, Understanding image contrast formation in TiO2 with force spectroscopy, Phys. Rev. B 85 (2012) 125416.

167

[13] V. Caciuc, H. Hölscher, S. Blügel, H. Fuchs, First-principles study of the atomicscale structure of clean silicon tips in dynamic force microscopy, Phys. Rev. B 74 (2006) 165318. [14] T.-L. Chan, C.Z. Wang, K.M. Ho, J.R. Chelikowsky, Efficient first-principles simulation of noncontact atomic force microscopy for structural analysis, Phys. Rev. Lett. 102 (2009) 176101. [15] R. Bechstein, C. Gonzàlez, J. Schütte, P. Jelínek, R. Pérez, A. Kühnle, ‘All-inclusive’ imaging of the rutile TiO2 (1 1 0) surface using nc-afm, Nanotechnology 20 (2009) 505703. [16] L. Kronik, A. Makmal, M.L. Tiago, M.M.G. Alemany, M. Jain, X. Huang, Y. Saad, J.R. Chelikowsky, Parsec—the pseudopotential algorithm for real space electronic structure calculations: recent advances and novel applications to nano-structures, Phys. Status Solidi 243 (2006) 1063. [17] A. Natan, A. Benjamini, D. Naveh, L. Kronik, M.L. Tiago, S.P. Beckman, J.R. Chelikowsky, Real-space pseudopotential method for first principles calculations of general periodic and partially periodic systems, Phys. Rev. B 78 (2008) 075109. [18] K. Fukui, H. Onishi, Y. Iwasawa, Atom-resolved image of the TiO2 (1 1 0) surface by noncontact atomic force microscopy, Phys. Rev. Lett. 79 (1997) 4202–4205. [19] F.J. Giessibl, Forces and frequency shifts in atomic-resolution dynamic-force microscopy, Phys. Rev. B 56 (1997) 16010. [20] H. Hölscher, U.D. Schwarz, R. Weisendanger, Calculation of the frequency shift in dynamic force microscopy, Appl. Surf. Sci. 140 (1999) 344. [21] U. Dürig, Relations between interaction force and frequency shift in largeamplitude dynamic force microscopy, Appl. Phys. Lett. 75 (1999) 433. [22] B. Gotsmann, C. Seidel, B. Anczykowski, H. Fuchs, Conservative and dissipative tip-sample interaction forces probed with dynamic AFM, Phys. Rev. B 60 (1999) 11051–11061. [23] A.I. Livshits, A.L. Shluger, A.L. Rohl, A.S. Foster, Model of noncontact scanning force microscopy on ionic surfaces, Phys. Rev. B 59 (1999) 2436–2448. [24] N. Sasaki, M. Tsukada, Theory for the effect of the tip-surface interaction potential on atomic resolution in forced vibration system of noncontact AFM, Appl. Surf. Sci. 140 (1999) 339. [25] A. Schwarz, W. Allers, U.D. Schwarz, R. Wiesendanger, Dynamic-mode scanning force microscopy study of n-InAs(1 1 0) at low temperatures, Phys. Rev. B 61 (2000) 2837–2845. [26] F.J. Giessibl, A direct method to calculate tip-sample forces from frequency shifts in frequency-modulation atomic force microscopy, Appl. Phys. Lett. 78 (2001) 123. [27] D.M. Ceperley, B.J. Alder, Ground state of the electron gas by a stochastic method, Phys. Rev. Lett. 45 (1980) 566–569. [28] J.R. Chelikowsky, N. Troullier, Y. Saad, Finite-difference-pseudopotential method: electronic structure calculations without a basis, Phys. Rev. Lett. 72 (1994) 1240–1243. [29] J.R. Chelikowsky, The pseudopotential-density functional method applied to nanostructures, J. Phys. D: Appl. Phys. 33 (2000) R33–R50. [30] N. Troullier, J.L. Martins, Efficient pseudopotentials for plane-wave calculations, Phys. Rev. B 43 (1991) 1993. [31] H. Kim, J.R. Chelikowsky, Electronic and structural properties of the As vacancy on the (1 1 0) surface of GaAs, Surf. Sci. 409 (1998) 435. [32] J.L.A. Alves, J. Hebenstreit, M. Scheffler, Calculated atomic structures and electronic properties of GaP, InP, GaAs, and InAs (1 1 0) surfaces, Phys. Rev. B 44 (1991) 6188. [33] T. Ohno, K. Shiraishi, First-principles study of sulfur passivation of GaAs(0 0 1) surfaces, Phys. Rev. B 42 (1990) 11194–11197. [34] H.J. Monkhorst, J.D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13 (1976) 5188–5192. [35] N. Uehara, H. Hosoi, K. Sueoka, K. Mukasa, Tip-sample distance dependency of non-contact atomic force microscopy images on a GaAs(1 1 0) surface, Nanotechnology 15 (2) (2004) S97. ˇ [36] S.H. Ke, T. Uda, R. Pérez, I. Stich, K. Terakura, First-principles investigation of tip-surface interaction on a GaAs(1 1 0) surface: implications for atomic force and scanning tunneling microscopies, Phys. Rev. B 60 (1999) 11631–11638. ˇ [37] S.H. Ke, T. Uda, I. Stich, K. Terakura, First-principles simulation of atomic force microscopy image formation on a GaAs(1 1 0) surface: effect of tip morphology, Phys. Rev. B 63 (2001) 245323. [38] A.S. Foster, A.Y. Gal, J.M. Airaksinen, O.H. Pakarinen, Y.J. Lee, J.D. Gale, A.L. Shluger, R.M. Nieminen, Towards chemical identification in atomic-resolution noncontact AFM imaging with silicon tips, Phys. Rev. B 68 (2003) 195420.