First-Principles Atomic Force Microscopy Image Simulations with

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First-principles atomic force microscopy image simulations with density embedding theory Yuki Sakai, Alex J. Lee, and James R. Chelikowsky Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.6b00741 • Publication Date (Web): 06 Apr 2016 Downloaded from http://pubs.acs.org on April 10, 2016

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First-principles atomic force microscopy image simulations with density embedding theory Yuki Sakai,⇤,† Alex J. Lee,‡ and James R. Chelikowsky†,‡,¶ Center for Computational Materials, Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas 78712, USA, Department of Chemical Engineering, The University of Texas at Austin, Austin, Texas 78712, USA, and Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA E-mail: [email protected] Phone: 512-232-1856. Fax: 512-471-8694

Cu2N!

Pentacene!

Table of Contents Graphic Abstract We present an efficient first-principles method for simulating non-contact atomic force microscopy (nc-AFM) images using a “frozen density” embedding theory. Frozen density embedding theory enables one to efficiently compute the tip-sample interaction ⇤

To whom correspondence should be addressed Center for Computational Materials, Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas 78712, USA ‡ Department of Chemical Engineering, The University of Texas at Austin, Austin, Texas 78712, USA ¶ Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA †

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by considering a sample as a frozen external field. This method reduces the extensive computational load of first-principles AFM simulations by avoiding consideration of the entire tip/sample system and focusing on the tip alone. We demonstrate that our simulation with frozen density embedding theory accurately reproduces full density functional theory simulations of freestanding hydrocarbon molecules while the computational time is significantly reduced. Our method also captures the electronic e↵ect of a Cu(111) substrate on the AFM image of pentacene and reproduces the experimental AFM image of Cu2 N on a Cu(100) surface. Our simulation method is applicable for theoretical imaging applications on large molecules, two-dimensional materials, and materials surfaces.

Keywords atomic force microscopy, density functional theory, frozen density embedding theory, pentacene, Cu2 N, CO tip Characterization of surfaces is important for investigating the physical properties of materials. Recent progress in the field of two-dimensional (2D) materials such as graphene extends the applicable areas of characterization methods from surfaces to materials themselves. Non-contact atomic force microscopy (nc-AFM), an imaging technique that measures the frequency shift of an oscillating cantilever tip, is one of the most widely used surface characterization methods. 1 A major benefit of AFM compared to other surface characterization methods such as scanning tunneling microscopy (STM) is its flexibility and ease of application: AFM can be used to image virtually any flat solid surface without sample preparation whereas in STM a conducting sample is required. 1 Recently, carbon monoxide (CO) functionalized tips have been successfully used to obtain high resolution of chemical bonds in organic molecules 2–11 and 2D materials such as graphene 12,13 and an insulating Cu2 N monolayer on Cu(100). 14

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The interpretation of microscopy images is important for surface characterization. Theoretical simulations play an important role in relating an acquired image to its actual atomic structure. For example, AFM simulations have been used to distinguish bond order in fullerene. 3 STM simulations based on density functional theory (DFT) can often be constructed from a single total energy calculation through post-processing. In contrast, nc-AFM simulations within the framework of DFT 15,16 can require thousands of calculations to obtain an image and therefore take up a huge computational cost. 17,18 The frequency shift measured in nc-AFM ( f ) is related to the second derivative of the interaction energy between the tip and sample (Ets ) with respect to the tip-sample distance q: 19,20 f0 f (z) = ⇡k0 A2

Z

A A

@ 2 Ets (z @q 2

q) p 2 A

q 2 dq.

(1)

Here f0 , A, k0 , and z are the resonant frequency, oscillation amplitude, spring constant, and operating height of the AFM tip respectively. Equation 1 can be simplified by assuming the amplitude is small, i.e., by assuming @ 2 Ets /@z 2 is constant within the oscillation:

f (z) =

f0 @ 2 Ets (z) . 2k0 @z 2

(2)

To construct a simulated nc-AFM image, we create a 2D raster grid of the tip over the sample and compute Ets at several tip heights to numerically obtain a

f map. Creating

a fine enough raster grid to obtain high resolution often entails thousands of total energy calculations for a single AFM image. Chan et al. proposed a simple method to estimate the frequency shift map by approximating the tip-sample interaction using the electrostatic potential of the sample without explicitly modeling the tip. 21 This virtual-tip method has been successfully applied to semiconductor surfaces with or without an overlayer. 21,22 In these simulations, the chemistry of the tip is assumed to play an unimportant role. In contrast, explicit tip-sample interactions have been found to be important for simulations of nc-AFM using a CO-functionalized tip, 18 and where model potentials have been adopted 23,24 3


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for nc-AFM simulations. In such cases, a computationally efficient first-principles nc-AFM simulation is desirable. The intense computational load of first-principles nc-AFM simulations can be dramatically reduced if we approximate the sample as an external field. This leaves only the tip to be explicitly modeled and drastically shrinks the size of the computational system. For this purpose, we adopt a “frozen density” embedding theory (FDET). 25,26 FDET divides a target system into two (or in principle, multiple) subsystems and the electronic charge density of a target subsystem is self-consistently determined by embedding the frozen charge density and potential of the other subsystem as an input. In this description, we assume the tip does not significantly a↵ect the structural and electronic properties of the sample. In other words, here we do not consider situations with strong tip-sample interactions such as atomic manipulation with an AFM tip. 27 It is important to note that the descriptions of the embedded charge density and potential, as well as the interaction between the two subsystems are based on DFT. FDET-based AFM simulations are expected to provide a quantum mechanical and first-principles description with less computational cost than full DFT calculations. Nevertheless, the applicability of FDET to such AFM simulations is not trivial, which we address in this letter. A similar formalism has been applied to CO adsorption to an MgO surface and a zeolite. 28,29 Here, we demonstrate a computationally efficient method for simulating nc-AFM images with FDET. We first test our method on a system of benzene and CO as sample and tip respectively. We find that FDET qualitatively reproduces the intermolecular interaction energies as a function of tip-sample distance, although we find a consistent underestimation of the intermolecular distances and an overestimation of interaction energies. Despite these di↵erences, we can acquire simulated nc-AFM images similar to those obtained with full DFT provided that the operating tip height is o↵set for the underestimation of intermolecular distance. We also demonstrate that this FDET-based method can be applied to pentacene on Cu(100) and Cu2 N monolayer on Cu(111). The FDET-based method is an efficient

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and accurate method for first-principles nc-AFM simulation of both organic and inorganic systems with large numbers of atoms. In our FDET calculations, we divide the total charge density [ntot (r)] of a system into two subsystems of the tip and sample: ntot (r) = nt (r) + ns (r). The total energy functional (Etot [nt , ns ]) in atomic units is given by

Etot = +

Z Z

Z

{nt (r) + ns (r)} {nt (r0 ) + ns (r0 )} drdr0 0 2|r r |

t s Vnuc (r) + Vnuc (r)

nt (r) + ns (r) dr

+ Ts [nt ] + Ts [ns ] + Tsnadd [nt , ns ] + Exc [nt + ns ] + Enuc ,

(3)

where Vnuc , Ts , Exc , and Enuc represent the nuclear potential, kinetic energy functional, exchange-correlation energy functional, and nuclear-nuclear interaction energy respectively. s Here Vnuc (r) and ns (r) are fixed during FDET calculations. The non-additive kinetic energy

functional (Tsnadd ), defined by Tsnadd [nt , ns ] = Ts [nt , ns ]

Ts [nt ]

Ts [ns ],

(4)

is the key term in FDET. We approximate this term by an analytic form by adopting the analytic form of the kinetic energy functional proposed by Tran and Weslowszki (PBE-TW), 30 which has a similar analytic form to the exchange functional of Perdew, Burke, and Ernzerhof (PBE). 31 In our tests the PBE-TW functional gives the most reasonable results compared to the Thomas-Fermi and Lembarki-Chermette kinetic energy functionals. 32 Consequently, the Schrödinger-like equation in FDET for a set of Kohn-Sham eigenvalues and wave functions ({✏ti ,

t i (r)})

is



r2 + Veft f (r) + Vemb (r) 2

t i (r)

= ✏ti

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t i (r),

(5)

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where Veft f (r)

=

Z

nt (r0 ) 0 Exc [n] t dr + Vnuc (r) + 0 |r r | n

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(6) n=nt

is an ordinary Kohn-Sham potential of the tip and Z

ns (r0 ) 0 Exc [n] s dr + Vnuc (r) + 0 |r r | n n=ntot Exc [n] Ts [n] Ts [n] + n n=nt n n=ntot n n=nt

Vemb (r) =

(7)

is an embedded potential. Here, the functional derivatives are approximated by analytic forms of the exchange-correlation and non-additive kinetic potentials. We implemented FDET into a real-space pseudopotential DFT code PARSEC 33–36 (see Supporting Information for computational details). FDET calculations of the tip were performed following a full DFT run of a sample system with local density approximation (LDA) exchange-correlation functional, by using the Hartree and nuclear potentials and the charge density of the sample system as an input. In general, the total energy functional should be optimized with respect to the charge density of every subsystem. 37 Nevertheless, here we optimize the charge density of the tip alone and this is found to be sufficient for our purpose. As first test, we compute the interaction energy (Ets ) between a CO tip and benzene as a function of the oxygen-benzene distance at three representative lateral tip positions: hollow (h), carbon (C ), and hydrogen (H ) sites, to see whether FDET is able to distinguish between these sites (Figure 1). The interaction energy is defined by Ets (z) = E tot (z) E t

E s , where

E tot , E t , and E s are the total energies of the tip-sample system, isolated tip and isolated sample respectively. The oxygen atom of CO is directed toward the molecule, and tip atom coordinates are not relaxed in all calculations because the tip relaxation should mainly sharpens the contrast as demonstrated in Ref. 17. We model only the CO part of the tip without a tip apex because we confirmed that the addition of Cu dimer to CO does not qualitatively a↵ect the distance-energy curves discussed below (see Figure S1 in Supporting Information). 6


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The general tendency of FDET is the overestimation of intermolecular interaction energies and the underestimation of distances in the benzene/CO case1 . Compared with DFT, FDET-optimized intermolecular distances are underestimated by approximately 0.3-0.4 ˚ A (Table 1). Notably, FDET consistently underestimates the distance by 0.3 ˚ A when the lateral tip position is sufficiently far from the carbon atoms whereas the underestimation is larger by 0.1 ˚ A when the tip is close to the carbons (C and CC sites). The overestimation of the interaction energies is 13-22 meV, and the relative ordering of the interaction energies at di↵erent sites is almost identical in both DFT and FDET calculations. Only the order of C and CC is exchanged, where the energy di↵erence is only 1 meV. FDET does not give a complete rigid shift of full DFT calculations, but the maximum errors in relative distances and relative interaction energies among di↵erent lateral positions are about 0.1 ˚ A and 10 meV respectively. The FDET qualitatively reproduced the features of full DFT energy curves (Figure 1a) such as the crossing points between di↵erent curves, as can be seen in Figure 1b. The LDA exchange-correlation functional used here does not include long-range van der Waals interactions which can be remedied by using van der Waals corrections or non-local functionals. 38 However, such long-range interactions should not strongly depend on atomic sites. We expect that van der Waals corrections will not a↵ect the qualitative features obtained here. In such weakly-interacting systems, the reproducibility of the interaction energy curve with FDET can be improved by using empirical repulsive corrections. 39 In Figure 2, we compare AFM images of benzene acquired using three di↵erent methods: full DFT, FDET, and the virtual-tip method. We use a smaller tip height for the FDET simulation (3.07 ˚ A) compared with full DFT (3.39 ˚ A) as per our previous discussion about the underestimation of intermolecular tip distances. The AFM image simulated with FDET (Figure 2b) well reproduces that of full DFT (Figure 2b). Positions closer to atoms and bonds have higher frequency shifts (represented by brightness) compared to the hollow region of benzene (represented by a dark spot). By contrast, the simple virtual-tip method gives a 1 The tendency should be functional dependent in a similar fashion to exchange-correlation functionals. In fact, Thomas-Fermi kinetic energy functional yields an opposite tendency.

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Figure 1: Intermolecular interaction energies between CO and benzene as a function of CO tip height for three di↵erent lateral positions (h, C, and H ). (a) Blue and (b) red lines indicate full DFT and FDET calculations, respectively. The distance region plotted in (b) is shifted by 0.3 ˚ A for a comparison purpose. The horizontal dashed line is the sum of the total energies of freestanding CO and benzene molecules. The three lateral positions are illustrated in Figure 2d.

Table 1: DFT- and FDET-equilibrium intermolecular distances (d in ˚ A) and interaction energies (E in meV) at six di↵erent sites obtained by fitting to an equation of state. The o↵sets of the equilibrium distances ( d) are also listed. The CC and CH sites are at the centers of the C-C and C-H bonds respectively. The CCh site is at the midpoint between the center of the C-C bond and h. Site h C H CC CH CCh

DFT d E 2.96 -52 3.28 -33 2.96 -28 3.18 -32 3.17 -25 3.07 -44

FDET d E 2.65 -74 2.86 -47 2.65 -41 2.86 -48 2.75 -40 2.75 -64

d 0.31 0.42 0.31 0.32 0.42 0.32

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circular bright region rather than the hexagonal shape observed in the DFT and FDET cases as shown in Figure 2c. Furthermore, the contrast between the dark hollow region and the bright bonding regions is reduced. This example demonstrates that FDET better reproduces the full DFT AFM image compared to the virtual-tip method.

Figure 2: Simulated AFM images of benzene using the (a) full DFT, (b) FDET, and (c) virtual-tip method, 21 where the tip heights are set to 3.39, 3.07, and 3.39 ˚ A, respectively. 2 White and black colors indicate high and low frequency shifts (i.e. @ Ets /@z 2 ), respectively. The size of the simulated region is 5.4 ˚ A⇥4.8 ˚ A. (d) Ball-and-stick model of a benzene molecule. Gray and white spheres represent carbon and hydrogen atoms, respectively. The lateral tip positions discussed (h, C, H, CC, and CH) are marked in the figure. We used XCrySDen to visualize the structures. 40 Next we demonstrate a more impressive example of FDET-based AFM image simulation using a larger molecule, pentacene. The simulated nc-AFM images of a freestanding pentacene molecule with di↵erent methods are shown in Figure 3. The image using FDET (Figure 3b) again reproduces the qualitative features of the image with full DFT (Figure 3a) as can be seen from the rescaled line profiles shown in Figure 3f. The FDET image captures the enhanced brightness at the left and right ends of the molecule, where non-parallel C-H bonds exist as shown in Figure 3g. The inner C-C bonds appear less bright than at the ends but are clearly resolved as in the full DFT image. These key features agree with those observed in the experimental nc-AFM image shown in Figure 3e. 2 A notable di↵erence is the thickness of the bond, but this should be sharpened by considering the tilting of the tip (e.g. using the method proposed by Guo et al.). 17 On the other hand, the inner C-C bonds are not resolved in the virtual-tip case (Figure 3c) although the brightness at the left and right ends is observed. This result indicates the importance of the explicit treatment of the tip-sample interaction along with the usefulness of FDET-based AFM simulations.

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Important targets of FDET-based AFM simulations include materials surfaces, molecules on substrates, and 2D materials because such systems tend to possess enormous numbers of atoms. We implement FDET to 2D periodic boundary conditions following the recent study 41 and apply the present method to pentacene on a Cu(111) substrate as in the experiment. 2 We model pentacene on Cu(111) by placing the center of pentacene on a hollow site of a bilayer of Cu(111). 42 Here we optimize only the molecule-substrate distance to see the electronic e↵ect of the substrate. The simulated AFM image (Figure 3d) shows that carbon atoms at the vertical C-H bonds are slightly brighter than those in Figure 3b (remarkable at the carbon atoms at the center). The di↵erence charge density shows that those carbon atoms gain electrons (Figure S3 in Supporting Information). The brighter carbon atoms can be ascribed to this electron gain because the increase of electron density should induce more repulsive interaction with the CO tip and result in a higher frequency shift and enhanced brightness. In contrast, the line profile (green line in Figure 3f) is scarcely a↵ected by the substrate, corresponding to the little di↵erence charge density at the inner C-C bonds. Therefore, our method can capture the the substrate e↵ect on the simulated image. We further demonstrate that FDET-based AFM simulations can be applied to an inorganic system. We adopt a Cu2 N monolayer on a Cu(100) surface 14 as an example, where one half of the hollow sites of Cu(100) is occupied with nitrogen as shown in Figure 4c. We take a square 2⇥2 supercell for the AFM (force map) simulation. Figure 4a shows the simulated @Ets /@z map of the Cu2 N monolayer with a CO tip using the FDET-based method. The nitrogen and hollow sites are brightest and darkest, respectively, while the brightness of the copper site is in between them. Our simulation here reproduces the experimental force map shown in Figure 4b. The FDET-based method can be applied to 2D inorganic systems, with including the substrate. We finally comment on the reduction in computational time the FDET method a↵ords. The computational savings in FDET come from the reduction of the number of valence electronic states to be computed. For example, the number of the valence states to be

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Figure 3: Simulated AFM images of pentacene using the (a) full DFT, (b) FDET, and (c) virtual-tip method, where tip heights are set to 3.39, 3.07, and 3.39 ˚ A respectively. The size ˚ ˚ of the simulated region is 14.3 A⇥5.4 A. (d) Simulated AFM image of pentacene on Cu(111) using FDET with the tip height of 3.07 ˚ A for consistency. (e) Experimental AFM image of pentacene on Cu(111), taken from Ref. 2. (f) Line profile along the direction indicated by the arrow in Figure 3g. The zero of the x is the center of the molecule. The line profiles are rescaled for qualitative comparison purpose. (g) Ball-and-stick model of pentacene.

Figure 4: (a) Simulated force map of Cu2 N monolayer using FDET where the tip height is 2.75 ˚ A. (b) Experimental force map of Cu2 N monolayer with CO tip corresponding to Figure 4a, cut from Ref. 14 for comparison. (c) Ball-and-stick model of the top view of Cu2 N monolayer on Cu(100). Orange and yellow spheres represent copper atoms at and below the surface, respectively, whereas green spheres represent nitrogen atoms. The black square region shows the simulated region in Figure 4a.

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computed in the case of pentacene on Cu(111) is 403 for full DFT sample calculations while that in the CO tip with FDET is only 5. This shrinks the size of the Hamiltonian matrix, and the computational time for the diagonalization of the Hamiltonian matrix is significantly reduced. In addition, we do not use k-point sampling in FDET total energy calculations since the embedded charge density and potentials are k independent. In the full DFT calculation of Cu2 N we included 20 k points of which there are 328 valence states for each k point. The practical reduction of the number of states to be computed goes from 6560 to 5 s.

2

Therefore, our method provides more substantial time savings for AFM simulations of large systems. In summary, we have demonstrated an efficient method for nc-AFM simulations using frozen density embedding theory (FDET). FDET is found to reproduce the qualitative features of intermolecular distances and interaction energies of a full DFT calculation. The nc-AFM image simulated with FDET resolves the inner C-C bonds of pentacene, which cannot be observed in the image with the simple virtual-tip method. We have found a substantial reduction in computational time for the simulations of 2D periodic systems. The FDET-based method presented here is a potentially powerful tool suitable for first-principles nc-AFM simulations of large molecules, molecules on substrates, surfaces, and 2D materials, as well as investigations of tip e↵ects on nc-AFM images.

Acknowledgement We acknowledge support from the U.S. Department of Energy (DoE) for work on nanostructures from grant DE-FG02-06ER46286, support from the Welch Foundation under Grant F-1837, and support provided by the Scientific Discovery through Advanced Computing (SciDAC) program funded by U.S. DoE, Office of Science, Advanced Scientific Computing Research and Basic Energy Sciences under Award No. DE-SC0008877 on algorithms. Computational resources are provided in part by the National Energy Research Scientific Com2 A similar kind of speed up is also possible even when we optimize the total energy functional with respect to all subsystems. 43

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puting Center (NERSC) and the Texas Advanced Computing Center (TACC). We thank Dr. Minjung Kim for helpful discussions. The authors declare no competing financial interests.

Supporting Information Available Details of the computational method, comparison of the Cu2 CO and CO tips, the structural details of pentacene on Cu(111) and Cu2 N on Cu(100), and the di↵erence charge density of a flat pentacene on Cu(111) (PDF). This material is available free of charge via the Internet at http://pubs.acs.org/.

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