arXiv:1904.11862v2 [cond-mat.mtrl-sci] 19 Nov 2019 t.As,fatl r bevdi efasmldpoly- assembled self in 15], observed [14, are [16–19]. mers fractals transport Also, quantum via in explained etc. levels [13], energy quantum dimensions of fractional splitting fractal exhibits in also [12], effect and magnetic Hall [11] perpendicular [10], under [9], electrons fields butterfly of Hofstadter Apart behaviour as a observed etc. - are [8], fractals imaging these, medical from [5–7], transi as antennas [3], impor- [4], such storage tors hydrogen applications as [2], found emerged electronics have stretchable also which have constructs “Fractals”. tant fractals as these named However, patterns mathematical repeating others. based on or features identify just probability to on or needed be information correlation can of or basis This the counting plot [1]. box log dimension via as log taken implemented the is descriptors where two Hausdorff, Dimension” of by “Hausdorff provided is as a definition is called dimension on alternate co- its point An unit so a coordinates, three. a while three requires For one, require would is line cube dimension a it. its on on hence residing point ordinate, point a a specifying example, for required coordinates nfatldmnin 2] o efrigti work, this performing For triangle [20]. SierpiÅĎski Smith dimensions as (CO) fractal patterned molecules in surface monoxide Cu(111) carbon on of wavefunction the inlter DT a eapidfrdsgigand designing for geometry. fractal applied in be electrons of can characterization using (DFT) that theory demonstrate tional we the Here, of initio understanding required. applica- detailed is new a same developing this, facilitate of To dimen- possibilities tions. non-integer up in opens wavefunctions sions demonstration of This confinement Cu(111). of on CO of manipulation for eety Smith Recently, self explaining for used are dimensions Fractional of number the by defined is object an of Dimension acltoswti h rmwr fdniyfunc- density of framework the within calculations tal. et ofieeto aefnto nFatlgoer,adetecti a geometry, Fractal in wave-function of Confinement s cnigtneln irsoe(STM) microscope tunnelling scanning a use u11 ufc.Ti pn ptepsiiiyo sn high using crystal known of currently possibility from the fracta lattices up Vicsek such discovering opens and whi and This hexaflake lattices the Lieb surface. construct and Cu(111) We graphene molecular fractals. fir the electronthese the study characterize that to and demonstrate method design we investigate, study, to this ha employed In lattice c artificial on lattice. creating work by artificial recent dimensions The fractal pro in properties. its tion interesting defines unambiguously exhibit system can a tem of dimension the While rcasaesl-eetn atrswihhv dimension have which patterns self-repeating are Fractals tal. et eotdta hyhv confined have they that reported oamdGhadiyali Mohammed eateto hsc,Uiest fMumbai, of University Physics, of Department aiaCmu,Snarz(E), Santacruz Campus, Kalina ubi-4008 India. 098, 400 - Mumbai ab s- ∗ n aevChacko Sajeev and iiigahxgnlflk nsvnsget hc may which segments seven by in constructed flake fractal hexagonal iterative a an dividing is hexaflake A study. the images. remove to periodic z-axis the the Å with along 0.15 interactions added of was 10Å mesh of finer vacuum a states, of density a tadsac f02 Å 0.20 of distance a at was etdi E h iei nryctffwssta 50Ry at set was Monkhorst-Pack cutoff imple- a energy with as kinetic The methodology QE. in DFT-D2 mented taken via also im- were For consideration correction 26]. into Waals [25, der library van were SSSP accuracy prove the correlation from available exchange as used (PBEsol) Ernzerhof Perdew- of form Burke- the augmented in [24] project pseudopotentials (PAW) The wave package. based pseudopotential ESPRESSO lo oee,tees n ffiinyo F r excep- are DFT of methodology efficiency other strong. and tionally using ease performed the however, be applica also, can their also This but fractals tions. for on schemes based different investigate lattices to artificial it allow of only competence not and DFT inher- comprehensiveness which the Further, systems geom- etry. fractal of having structures detection crystal topology. demonstrate for their ently on allow based could crystals This of to number techniques throughput large high screen with combined be can it as yDT nohrrcn okb Fabian by work recent other achieved In be DFT. by can spectroscopy performing while al. ieo h ra snal eo[2.Sc ytm a be can systems Such [22]. zero extensively nearly studied is array create the to of side used be can [21]. lattice magnet Halbach-like fractal smallest with the combined create This to used was Holmium of eslc eaae 7 n iskfatl 2]frthis for [27] fractals Vicsek and [7] hexaflakes select We h okpeetdi efre via performed is presented work The sa xml,teapiainpitdotb Smith by out pointed application the example, an As methodology suited better is theory functional Density o sn rca atcsa otoldenvironment controlled a as lattices fractal using for ie iet e osblte fdesigning of possibilities new to rise given s nHudr ieso.W pl the apply We dimension. Hausdorff in s rdvlpn hmaltogether. them developing or s ie yfatosrte hnintegers. than rather fractions by given s tpicpemtoscnb effectively be can methods principle st nnmn fteeetoi wavefunc- electronic the of onfinement sb dopino Omlclson molecules CO of adsorption by ls hogpttcnqe o screening for techniques throughput ete,afatoa iesoa sys- dimensional fractional a perties, hla ofatlltie ae on based lattices fractal to lead ch ra nwihtemgei edo one on field magnetic the which in array ( QE akg 2] ti ln wave plane a is It [23]. package ) nsilico in † k ms,each -mesh, yDFT. by − nuigDFT using on 1 hl,frcalculating for While, . k pito h mesh the of -point − 1 tal. et a sd A used. was , Quantum natom an , nsilico in et - 2 be described as a defective molecular graphene [28]. On However, the density distribution between them is not the other hand, Vicsek fractal is constructed by decompo- very well resolved as they follow the same trend of in- sition of squares in a 3 by 3 grid. This system is described creasing in density as the function of energy. This is as defective Lieb lattice [29]. As CO molecules adsorbed also demonstrated from the STM maps taken at two dif- on Cu(111) surface have been used to construct both ferent biases of -0.3 eV and 0.5 eV, where no change is molecular graphene [28] as well as Lieb lattice [29], it observed. This indicates that the hexaflake so formed is was the obvious choice for this study (see supplementary quite homogeneous. On the other hand, a similar infer- information). Other possible constructs are Fe adatom ence cannot be drawn for the Vicsek fractal. From its on Cu(111) [30], Cl vacancy on chlorinated Cu(100) [31] LDOS, it may be clearly observed that the CO molecules etc [32–34]. For constructing the systems both Hexaflake are indifferent from each other, when being grouped to- and Vicsek fractal, CO molecules are placed on three gether. However, except for two molecules, the density atom thick Cu(111) surface. This surface is cleaved from distribution for most of them is not well resolved. This bulk using ATK VNL Builder [35] package. The bottom can be observed in the STM maps given in figure 1(e) two atomic layers were fixed and the system was allowed and 1(f) (marked with blue and red circle). The maps to relax along the direction of vacuum. This approach are computed for the bias of -0.3 eV and 0.45 eV respec- of creating system is a direct derivative of the approach tively. These same colors are also used to represent the used by Paavilainen et al. [34], here a Kagome lattice of LDOS plots in figures 1(a) and 1(d). CO molecule was created on Cu(111) surface (it is further The dimensional analysis of both the systems was per- explained in supplementary information). formed using box counting method. The Hausdorff di- In this work, we study the second generation of hex- mensions determined by this method are also known as aflake and Vicsek fractals. First generation of hexaflake Minkowski âĂŞ Bouligand dimension. This method is is a single hexagon, while the second generation is collec- well suited as each pixel in the STM maps can be taken as tion of seven, i.e. a graphene ring with a central atom a box of length lbox, and the number of boxes as N(lbox). as illustrated in figure 1(a). Similarly, the second gener- Hence, the Hausdorff dimension would be [1]: ation of Vicsek fractal is illustrated in figure 1(d). The theoretical Hausdorff dimensions of these fractals are 1.77 log(N(lbox)) and 1.46 respectively. The local density of states accom- D = lim lbox→0 log(lbox) panying the STM maps are also given in figure 1. The STM maps are computed using the method proposed by However, as the STM maps are not flat and have gra- Tersoff and Hamann [36] as implemented in QE package. dients, it is difficult to calculate the Hausdorff dimension at the boundaries. To solve this problem we remove the a b c ring gradients by setting binary values of the STM maps: for 0.3 center values above a certain threshold the STM map was set 0.2 to 1, and 0 elsewhere. In figure 2(a-d), the STM maps

ldos (abs) 0.1 for hexaflake are given for threshold percentages of 20%, 50%, 70% and 90% respectively. -0.4 -0.2 0 0.2 0.4 0.6 energy (eV) The STM maps were calculated for the bias from -0.5 d e f to 0.5 eV, the range in which the wavefunction is generally 0.6 well-confined for CO on Cu(111). In figure 2(e) and 2(f),

0.4 the plot of Hausdorff dimension (D) v/s bias is given. The red horizontal line represents the Hausdorff dimen-

ldos (abs) 0.2 sions for hexaflake and Vicsek fractal at values 1.77 and -0.4 -0.2 0 0.2 0.4 0.6 1.46, respectively. The error bars in figure 2 represent energy (eV) the upper and lower bounds of the calculated Hausdorff dimension at different threshold percentages. For hex- FIG. 1: (colour online) Figures (a) and (d) are the local den- sity of states of systems hexaflake and Vicsek fractal respec- aflake, the threshold percentage is 20% for upper bound, tively. The STM maps of hexaflake are given in (b) and (c). 70% for lower bound and 30% for the central range. For Their biases were kept at -0.3 eV and 0.5 eV respectively. Vicsek fractal, the threshold percentage for the central Similarly, (e) and (f) represent STM maps of Vicsek fractal region is 80% while the upper and lower bounds are given at bias of -0.3 eV and 0.45 eV respectively. The colour in both by 65% and 90% respectively. the images represents the unique sites in both the fractals. Although the Hausdorff dimensions calculated differ from the know values of the structure, it can be observed There are the six CO molecules forming the outer ring that they are still non-integer values. This difference is of hexaflake and central CO molecule. From the LDOS partially due to the approximation initially taken while of hexaflake (figure 1(a)), it can be observed that this computing the wavefunctions using DFT. The other rea- structure is a composed of two types of CO molecules. son is the approximate construction of fractal geometry 3

Name of Lowdin Charges tal lattices and characterizing them using techniques such Percentage as DFT. Complete Isolated Change Additionally, the behaviour of fractional wavefunction the System System System (%) of these lattices can be studied under the influence of Benzene on Cu 29.25 29.54 0.991 an electrical field. However, for the case of hexaflake no change in the fractal dimension is observed, see supple- Hexaflake 69.05 69.15 0.144 mentary information for further details. Such approach Vicsek fractal 246.19 245.95 0.0975 could open up several possibility of being utilized in quan- TABLE I: In the table presented above the Lowdin charges tum information sciences, where a system needs to be of all three systems are given. A change is observed in the robust enough to be in entanglement while be probed. Lowdin charges for complete and isolated system, this change Further, changes in wavefunction can be studied due to due to both approximation in DFT and sharing of charges. application of pressure on these systems and could be The percentage difference gives a much clear picture, as the used for screening molecules like porphyrin on metal, as ≈ change for Benzene on Cu 10times higher than other two sensors. Changes in the Hausdorff dimension due to ap- systems. plication of stress-strain on lattices can also be studied.

a b c d via CO on Cu(111) - calculations when performed on higher orders of the fractals may improve the results. Nevertheless, it can be observed that both of the artifi- cial lattices are in Hausdorff dimensions. However, as this method of counting dimension is e 1.6 f based just on the image processing techniques, it can not alone be used with certainty to determine that the 1.5 D electron wavefunction is indeed in Hausdorff dimensions. 1.4 To demonstrated this point we adsorb benzene molecule on Cu(111) surface. Upon the calculation of fractal di- 1.3 -0.4 -0.2 0 0.2 0.4 mension, according to the methodology describe above, Energy (eV) it is found to be in between 1.85 to 1.76 (see supplemen- tary information). Hence, this type of analysis is not FIG. 2: (colour online) In the figure above (a), (b), (c) and (d) sufficient. are the STM maps of hexaflake taken at threshold percentage of 20%, 50%, 70% and 90% respectively, while the plots (e) To determine if there the electron wavefunctions are in and (f) show the dimensional analysis done by box counting fractal dimensions, Lowdin charge analysis is performed. method for hexaflake and Vicsek fractals respectively. The The procedure for doing so is, first Lowdin charges of error bars represent the upper and lower bound of the fractal the complete system is calculated, then a second Lowdin dimension. The red line represents actual fractal dimension charge calculation is performed with the substrate re- for both systems at 1.77 and 1.46 for hexaflake and Vicsek moved. If no change is observed within the accuracy of fractals respectively. DFT, then it can inferred that electron wavefunctions are in Hausdorff dimensions. From the data presented in the table I, a change is observed for the Benzene on ACKNOWLEDGMENT Cu(111), this is an indication of charge transfer between them, while for both the fractals this kind of change is The computational work described here is performed not observed. at the High Performance Computational Facility at In conclusion, we have created artificial lattices based IUAC, New Delhi, India. We would like to express our on fractal geometry through adsorption of CO molecules gratitude to them. Also, we would like to thank the Uni- on Cu(111) surface and studied via DFT. Then LDOS versity Grant Commission of India for providing partial and STM maps were computed for both artificial lattices. funding for the research work through the UGC-BSR Re- Further, Hausdorff dimensions using box Minkowski- search Startup Grant (Ref. No.F.30-309/2016(BSR)). Bouligand method were calculated and observed to be fractions for both systems. We also demonstrated that only Minkowski-Bouligand method is not sufficient for calculation of Hausdorff dimensions with Benzene on Cu ∗ Electronic address: [email protected] and propose that Lowdin charge analysis should also be † Electronic address: [email protected];[email protected] used. Thus, demonstrating that fractals other than Sier- [1] J. Keesling Duvall, P. and A. Vince. The hausdorff di- piński triangle can be used to create artificial lattices. mension of the boundary of a self-similar tile. Journal of We also demonstrated that one can study artificial frac- the London Mathematical Society, 61(3):748–760, 2000. 4

[2] Jonathan A Fan, Woon-Hong Yeo, Yewang Su, Yoshi- [17] Charles N Moorefield, Anthony Schultz, and George R aki Hattori, Woosik Lee, Sung-Young Jung, Yihui Zhang, Newkome. From dendrimers to fractal polymers and Zhuangjian Liu, Huanyu Cheng, Leo Falgout, et al. Frac- beyond. Brazilian Journal of Pharmaceutical Sciences, tal design concepts for stretchable electronics. Nature 49(SPE):67–84, 2013. communications, 5:3266, 2014. [18] Qiang Zhao, Jinwen Qian, Zhangliang Gui, Quanfu An, [3] Deepak P Dubal, Omar Ayyad, Vanesa Ruiz, and Pedro and Meihua Zhu. Interfacial self-assembly of cellulose- Gomez-Romero. Hybrid energy storage: the merging of based polyelectrolyte complexes: pattern formation of battery and supercapacitor chemistries. Chemical Society fractal ?trees? Soft Matter, 6(6):1129–1137, 2010. Reviews, 44(7):1777–1790, 2015. [19] Rajarshi Sarkar, Kai Guo, Charles N Moorefield, [4] Richard Reiner, Patrick Waltereit, Fouad Benkhelifa, Mary Jane Saunders, Chrys Wesdemiotis, and George R S Muller, S Müller, H Walcher, S Wagner, R Quay, Newkome. One-step multicomponent self-assembly of a M Schlechtweg, and O Ambacher. Fractal structures for first-generation sierpiński triangle: From fractal design low-resistance large area algan/gan power transistors. In to chemical reality. Angewandte Chemie International 2012 24th International Symposium on Power Semicon- Edition, 53(45):12182–12185, 2014. ductor Devices and ICs, pages 341–344. IEEE, 2012. [20] SN Kempkes, MR Slot, SE Freeney, SJM Zevenhuizen, [5] Robert G Hohlfeld and Nathan Cohen. Self-similarity D Vanmaekelbergh, I Swart, and C Morais Smith. Design and the geometric requirements for frequency indepen- and characterization of electrons in a fractal geometry. dence in antennae. Fractals, 7(01):79–84, 1999. Nature physics, 15(2):127, 2019. [6] Francesco De Nicola, Nikhil Santh Puthiya Purayil, Da- [21] Fabian D Natterer, Kai Yang, William Paul, Philip vide Spirito, Mario Miscuglio, Francesco Tantussi, An- Willke, Taeyoung Choi, Thomas Greber, Andreas J Hein- drea Tomadin, Francesco De Angelis, Marco Polini, Ro- rich, and Christopher P Lutz. Reading and writing single- man Krahne, and Vittorio Pellegrini. Multiband plas- atom magnets. Nature, 543(7644):226, 2017. monic sierpinski carpet fractal antennas. ACS Photonics, [22] Klaus Halbach. Design of permanent multipole magnets 5(6):2418–2425, 2018. with oriented rare earth cobalt material. Nuclear instru- [7] Effect of FSS ground plane on second iteration of hex- ments and methods, 169(1):1–10, 1980. aflake fractal patch antenna. IEEE, 2012. [23] Paolo Giannozzi, Stefano Baroni, Nicola Bonini, Mat- [8] Audrey Karperien, Herbert F Jelinek, Jorge JG Lean- teo Calandra, Roberto Car, Carlo Cavazzoni, Davide dro, João VB Soares, Roberto M Cesar Jr, and Alan Ceresoli, Guido L Chiarotti, Matteo Cococcioni, Ismaila Luckie. Automated detection of proliferative retinopa- Dabo, et al. Quantum espresso: a modular and open- thy in clinical practice. Clinical ophthalmology (Auck- source software project for quantum simulations of ma- land, NZ), 2(1):109, 2008. terials. J. of Phys. Cond. Matt., 21(39):395502, 2009. [9] Douglas R Hofstadter. Energy levels and wave func- [24] P. E. Blöchl. Projector augmented-wave method. Phys. tions of bloch electrons in rational and irrational mag- Rev. B, 50:17953–17979, Dec 1994. netic fields. Physical review B, 14(6):2239, 1976. [25] Gianluca Prandini, Antimo Marrazzo, Ivano E Castelli, [10] M Morgenstern, J Klijn, Chr Meyer, and R Wiesendan- Nicolas Mounet, and Nicola Marzari. Precision and ef- ger. Real-space observation of drift states in a two- ficiency in solid-state pseudopotential calculations. npj dimensional electron system at high magnetic fields. Computational Materials, 4(1):72, 2018. Physical review letters, 90(5):056804, 2003. [26] Kurt Lejaeghere, Gustav Bihlmayer, Torbjörn Björkman, [11] MO Goerbig, P Lederer, and C Morais Smith. On the Peter Blaha, Stefan Blügel, Volker Blum, Damien Cal- self-similarity in quantum hall systems. EPL (Euro- iste, Ivano E Castelli, Stewart J Clark, Andrea Dal Corso, physics Letters), 68(1):72, 2004. et al. Reproducibility in density functional theory calcu- [12] W Pan, HL Stormer, DC Tsui, LN Pfeiffer, KW Baldwin, lations of solids. Science, 351(6280):aad3000, 2016. and K Wr West. Fractional quantum hall effect of com- [27] Eric W Weisstein. Box fractal. posite fermions. Physical review letters, 90(1):016801, http://mathworld.wolfram.com/BoxFractal.html. 2003. [28] Warren Mar Wonhee Ko Francisco Guinea Gomes, Ken- [13] A. Mishchenko X. Chen S. Pezzini G. H. Auton L. A. jiro K. and Hari C. Manoharan. Designer dirac fermions Ponomarenko U. Zeitler Laurence Eaves V. I. Fal2019ko˘ and topological phases in molecular graphene. Nature, Kumar, R. Krishna and A. K. Geim. High-order frac- 483(7389):306, 2012. tal states in graphene superlattices. Proceedings of the [29] Thomas S. Gardenier Peter H. Jacobse Guido CP van National Academy of Sciences, 115(20):5135–5139, 2018. Miert Sander N. Kempkes Stephan JM Zevenhuizen Cris- [14] Arunava Chakrabarti and Bibhas Bhattacharyya. Sier- tiane Morais Smith Daniel Vanmaekelbergh Slot, Mar- pinski gasket in a magnetic field: electron states lou R. and Ingmar Swart. Experimental realization and and transmission characteristics. Physical Review B, characterization of an electronic lieb lattice. Nature 56(21):13768, 1997. Physics, 12(7):672, 2017. [15] Edo van Veen, Shengjun Yuan, Mikhail I Katsnelson, [30] Christopher P. Lutz Crommie, Michael F. and Donald M. Marco Polini, and Andrea Tomadin. Quantum transport Eigler. Confinement of electrons to quantum corrals on in sierpinski carpets. Physical Review B, 93(11):115428, a metal surface. Science, 262(5131):218–220, 1993. 2016. [31] FE Kalff, Marnix P Rebergen, E Fahrenfort, Jan [16] George R Newkome, Pingshan Wang, Charles N Moore- Girovsky, Ranko Toskovic, Jose L Lado, Joaquín field, Tae Joon Cho, Prabhu P Mohapatra, Sinan Li, Fernández-Rossier, and Alexander F Otte. A kilo- Seok-Ho Hwang, Olena Lukoyanova, Luis Echegoyen, Ju- byte rewritable atomic memory. Nature nanotechnology, dith A Palagallo, et al. Nanoassembly of a fractal poly- 11(11):926, 2016. mer: a molecular" sierpinski hexagonal gasket". Science, [32] C. P. Lutz Manoharan, H. C. and D. M. Eigler. Quan- 312(5781):1782–1785, 2006. tum mirages formed by coherent projection of electronic 5

structure. Nature, 403(6769):512, 2000. of molecular graphene. Nano letters, 16(6):3519–3523, [33] Christopher P. Lutz Hirjibehedin, Cyrus F. and An- 2016. dreas J. Heinrich. Spin coupling in engineered atomic [35] Virtual nanolab version 2017.1, quantumwise a/s. structures. Science, 312(5776):1021–1024, 2006. www.quantumwise.com. [34] Sami Paavilainen, Matti Ropo, Jouko Nieminen, Jaakko [36] Jerry Tersoff and D. R. Hamann. Theory of the scan- Akola, and Esa Rasanen. Coexisting honeycomb and ning tunneling microscope. Physical Review B, 31(2):805, kagome characteristics in the electronic band structure 1985.