Quantum disordered state of magnetic charges in nanoengineered honeycomb lattice
George Yumnam, Yiyao Chen, Jiasen Guo, Jong K. Keum, Valeria Lauter, Deepak K. Singh
QQuantum disordered state of magnetic charges in nanoengineered honeycomb lattice
G. Yumnam , Y. Chen , J. Guo , J. Keum , V. Lauter , ∗ , and D. K. Singh , ∗ Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211 Oak Ridge National Laboratory, Oak Ridge, TN 37831 and ∗ email: [email protected], [email protected] A quantum magnetic state due to magnetic charges is never observed, even though they are treatedas quantum mechanical variable in theoretical calculations. Here, we demonstrate the occurrenceof a novel quantum disordered state of magnetic charges in nanoengineered magnetic honeycomblattice of ultra-small connecting elements. The experimental research, performed using spin resolvedneutron scattering, reveals a massively degenerate ground state, comprised of low integer and ener-getically forbidden high integer magnetic charges, that manifests cooperative paramagnetism at lowtemperature. The system tends to preserve the degenerate configuration even under large magneticfield application. It exemplifies the robustness of disordered correlation of magnetic charges in 2Dhoneycomb lattice. The realization of quantum disordered ground state elucidates the dominance ofexchange energy, which is enabled due to the nanoscopic magnetic element size in nanoengineeredhoneycomb. Consequently, an archetypal platform is envisaged to study quantum mechanical phe-nomena due to emergent magnetic charges.
The transformation of magnetic moment to mag-netic charges[1] has spurred the exploration of emergentphenomena, such as avalanche of magnetic monopoles,spin solid order of vortex loops and Wigner crystal-lization of magnetic charges, in two-dimensional arti-ficial magnetic lattice.[2–6] Among the many variantsof two-dimensional magnetic lattices,[7–10] a honey-comb structure is of special interest due to its nat-ural manifestation of strong geometric frustration be-tween magnetic moments, replicating the quasi-spin iceconfiguration.[11–13] The governing Hamiltonian in amagnetic honeycomb lattice consists of three competingenergy terms:[11, 14] the magnetic dipolar interaction,nearest and next nearest neighbor exchange interactions( J and J , respectively). The dipolar magnetic interac-tion is comparable to the nearest neighbor exchange en-ergy in a large element size, typically of sub-micrometerlength (fabricated using electron-beam lithography tech-nique), honeycomb.[11] Reducing the element size ofmagnetic honeycomb reduces the dipolar energy signifi-cantly. Thus, a honeycomb made of nanoscopic elementsfosters a new energetic regime where exchange interac-tions, J - J terms, dictate the underlying mechanism be-hind magnetic ground state. This scenario renders anexciting research venue to explore the occurrence of the-oretically formulated, but never observed, quantum dis-ordered magnetic ground state with massive degeneracy.The novel state was originally predicted to arise in two-dimensional bulk material with exchange interaction cou-pled atomic Ising moments on a honeycomb motif.[15–18]The one-to-one correspondence between magnetic mo-ment and magnetic charge makes the exploration viablein magnetically frustrated artificial system. The mag-netic charges, in essence, are quantum mechanical enti-ties, represented by Pauli matrices.[14, 19] Yet, they areobserved as classical variables in local and macroscopicprobes.[2, 12] In this communication, we report the existence of aquantum disordered state of magnetic charges in nano-engineered honeycomb lattice of connecting permalloy(Ni . Fe . ) elements, with typical dimension of (cid:39) × ×
10 nm (thickness). We haveemployed a hierarchical large throughput nanoengineer-ing technique (described in the Experimental Section)to create an artificial honeycomb lattice of nanoscopicelements. The ultra-small size of nanoscopic element,smaller than the size of a typical permalloy domain ∼ a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b +Q-Q +3Q -3Q+Q-Q-3Q +3Q-3Q+3Q +Q -Q +3Q -3Q+Q-Q μ m Height J = -0.5 - J / J Layer Thickness (nm)
L = 10 nmL = 12 nmL = 14 nm n m a) b) c) d) e) FIG. 1: Degenerate arrangements of magnetic charges and variation of exchange constants as functions of geometrical parametersin a honeycomb lattice. (a-c) Schematic description of degenerate configuration of magnetic charges on honeycomb vertices: ±
3Q charges in (a), ± Q charges in (b) and ± Q and ±
3Q charges in (c). In a large ensemble of vertices, the degeneracy willtransform into macroscopic characteristic. Magnetic charge pattern across the lattice transforms by releasing or absorbing a netcharge defect of magnitude ±
2Q between neighboring vertices. (d) Atomic force micrograph (AFM) of a typical nanoengineeredhoneycomb lattice, created by the new top down fabrication scheme (see Experimental Section for detail). (e) Plots of J / J asa function of thickness, calculated using Monte-Carlo simulations, for honeycomb element of varying length. J and J beingthe nearest neighbor and the next nearest neighbor exchange interactions. and one moment is pointing away from it; also termedas the quasi-ice rule.[13]The individual moment (treatedas Ising spin) can be considered as a pair of ’+’ and’-’ magnetic charges of magnitude Q (directly relatedto the net moment M under the dumbbell representa-tion by Q = M / l where l being the length of the con-necting element) that interact via magnetic Coulomb’sinteraction.[1, 3, 20] Consequently, a ’two-in & one-out’(or vice-versa) moment arrangement imparts the netcharge of +Q (-Q) to a given vertex, whereas the ’all-inor all-out’ configurations give rise to more energetic ± ±
2Q charge, under external tuning param-eters of applied magnetic field or temperature variationin a thermally tunable system.[2, 3, 22, 23] Additionally,the charges can arrange themselves in multiple differentconfigurations across the honeycomb vertices without af-fecting the overall energy of the system, schematicallydescribed in Fig. 1a-c.[19, 20] For a macroscopic ensem-ble of the charge hosting vertices, it causes very largedegeneracy in the system.The exchange interaction terms become dominant asthe size of the constituting elements of a magnetic hon-eycomb lattice reduces to the nanoscopic level. Unlikethe very large magnetic dipolar interaction energy, (cid:39) K, between micrometer size elements,[11] the strengthof dipolar interaction is significantly smaller, (cid:39)
10 K,in our nanoengineered honeycomb lattice. The reducedinter-elemental dipolar energy increases the occurrenceprobability of ±
3Q charges at low temperature; typi-cally forbidden under the dipolar interaction model. In the case of artificial square lattice of permalloy elements,Monte-Carlo (MC) simulations have shown that J canbe comparable to J in moderately thick sample.[20, 24]Therefore, the next nearest neighbor exchange interac-tion can be the second competing term in the Hamilto-nian (besides the nearest neighbor exchange J ), describ-ing the phase transition process in a nanoengineered mag-netic lattice. Accordingly, the governing Hamiltonian ofthe system can be written as:[14] H = − J (cid:88) nn σ i σ j − J (cid:88) nnn σ i σ j (1)where J < J > σ i and σ j are quantum Pauli matrices ofIsing variable ( S i = σ i e i and e i . e j = 1/2) on neighbor-ing vertices, related to the net magnetic charges by Q α = ± (cid:80) σ i where σ i = ± J (see Experimental Section for detail). In Fig.1e, we plot J / J as a function of thickness for differentelement size of the honeycomb lattice. For the initializa-tion value of J between 0.13 - 0.6 (previously used fornumerical simulation to determine the validity of J - J model in geometrically frustrated lattice),[16] compara-ble strengths of J and J are inferred to arise in the mod-erately thick, (cid:39)
10 nm, honeycomb lattice (also see Fig.S1-S2 in Supporting Information). Thus, our nanoscopichoneycomb provides an ideal setup to realize the J - J model, which can lead to a macroscopically degenerateground state of magnetic charges with disordered param-agnetic configuration.[15–17] In the following paragraphs,we discuss experimental results to this effect.We have performed detailed neutron scattering mea-surements on (cid:39) H = 20 Oe to maintain thepolarization of incident and scattered neutron (see Ex-perimental Section for detail). Schematic of the mea-surement procedure is shown in Fig. 2a. Experimen-tal data were collected at multiple temperatures between T = 300 K and T = 5 K in zero and saturating mag-netic field of H = 0.5 T, applied in-plane to the sample.Both spin polarized specular reflectivity and off-speculartwo dimensional (2D) scattering patterns were obtainedat each temperature and field. We show the plots ofspecular reflectivities R + and R − (’+’ and ’-’ correspondto neutron with spin parallel and anti-parallel to guidemagnetic field) at a characteristic temperature of T =5 K in Fig. 2b (see Fig. S5 for other temperatures).Two features are clearly observed in the plots: (a) thereflectivities R + and R − are separated from each other,indicating the presence of net magnetic moment in thesystem, and (b) the splitting between R + and R − grad-ually increases as temperature reduces to T = 5 K (seeFig. S5). The quantitative determination of the changein the overall magnetization at different temperatures isobtained by analyzing the experimental results using ageneric LICORNE-PY program, which generates the re-flectivity pattern for a given set of physical parameters ofthe system e.g. layer thickness, density, interface rough-ness and magnetic moment of magnetic layer. The fittingwas performed simultaneously to the data sets measuredat different temperatures and fields, including the X-rayreflectivity (XRR, Fig. S4). This procedure ensured thehigh accuracy of obtained parameters. Nuclear and mag-netic scattering length densities (NSLD and MSLD), cor-responding to the depth profiles of chemical structureand in-plane magnetization vector distributions, respec-tively, are depicted in Fig. 2c. The magnetization ofpermalloy honeycomb lattice is manifested by the spinasymmetry SA, given by (R + - R − )/( R + + R − ). Ex-perimental plot of spin asymmetry SA as a function ofwave vector transfer Q at T = 5 K is shown in Fig. 2d(see Fig. S5 for other temperatures). Estimated mag-netization at different temperatures is shown in Fig. 2e.Expectedly, the net magnetization increases when tem-perature is reduced. But there is no divergence at lowtemperature, which suggests the absence of long range or-der in the system. Additionally, the magnetization doesnot change much between an intermediate temperature of T = 30 K and T = 5 K.The net magnetization nearlydoubles in magnitude between T = 200 K and T = 5K. Overall, such a trend is indicative of the cooperativeparamagnetic behavior.[27] The paramagnetic behavioris independently verified using bulk magnetization mea-surement as a function of temperature at low field, seeFig. S3. It is arising due to an increase in the popu-lation density of ± Q charges at lower temperature. Athigh temperature, the honeycomb vertices are occupiedby both ± Q and ±
3Q charges. As temperature reduces,the population of energetic ±
3Q charges decreases. Wenote that ± Q charges give rise to the finite magnetiza-tion due to the short-range correlation of ’2-in & 1-out’(or vice-versa) moment arrangement. On the other hand,vertices with ±
3Q charges have zero net magnetization.Hence, there are two possibilities behind the paramag-netic behavior: a, the energetic high integer charges arestill present at low temperature with significant popu-lation density i.e. not all vertices contribute to the netmagnetization as if it is a fragmented state or b, the hon-eycomb vertices are occupied by only +Q and -Q chargesat low temperature and the system has a preference forone of the polarities. We have employed the analysisof off-specular PNR data to understand the underlyingmechanism.Off-specular PNR results at few characteristic temper-atures are shown in Fig. 3a-c. Here, y-axis represents theout-of-plane scattering vector ( Q z = πλ (sin α i + sin α f )),while the difference between the z-components of theincident and the outgoing wave vectors ( p i − p f = πλ (sin α i − sin α f )) is drawn along the x-axis (as shownschematically in Fig. 2a). Thus, vertical and horizontaldirections correspond to the out-of-plane and in-planecorrelations, respectively. [22, 28]. The specular reflec-tivity lies along the x = 0 line. At T = 300 K, verysmall off-specular scattering is detected, which could bearising due to the ferromagnetic nature of permalloy filmand the structure of honeycomb lattice. Most of the scat-tering is confined to the specular line. As temperatureis reduced, magnetic diffuse scattering starts developingalong the horizontal direction. The broad diffuse scatter-ing in the off-specular data at intermediate temperatureof T = 30 K manifests significant in-plane correlation ofmagnetic moments along honeycomb elements. At fur-ther reduced temperature of T = 5 K, the diffuse scat-tering becomes more prominent. We also observe a pe-culiar inverted cone-shape in the off-specular PNR dataat intermediate temperature between Q z = 0.09 and 0.05˚ A − , which becomes stronger as temperature decreases.Numerical modeling of experimental data is performedusing the Distorted Wave Born Approximation (DWBA),as utilized in the BornAgain [29]platform (see Experi-mental Section for detail). Using DWBA simulations,we generate reflectometry patterns of both specular andoff-specular reflectivity. A thin layer of air condensationon the surface of the sample, typical in low tempera- p i α i α f k i → Q z + - H Q x Q → k f → p f Specular O ff - S pe c u l a r a) b)c) d) e) -5 -4 -3 -2 -1
0 0.03 0.06 0.09 . R e f l e c t i v i t y ( a . u . ) Q (Å - ) R + R - SA ( a . u . ) Q (Å -1 ) S L D ( - Å - ) Depth (Å)
NuclearMagnetic x
5 30 90 200 M ( A / m ) T (K)
FIG. 2: Spin resolved neutron scattering measurements of nanoengineered honeycomb. (a) Schematic design of grazing incidencepolarized neutron scattering (PNR) experiment on an artificial magnetic honeycomb lattice. A small guide field of H = 20Oe is applied to keep neutron polarized. The schematic description depicts the scattering profiles of neutrons, leading to a2D pattern of specular and off-specular intensities. (b) Measured and fitted (solid curves) reflectivity curves for neutron withspin-up (R + ) and spin-down (R − ) polarizations as a function of wave vector transfer at a characteristic temperature of T = 5K (see Fig. S5 for other temperatures). (c) Nuclear and magnetic scattering length density profiles, obtained from the fit toexperimental data, as a function of depth of honeycomb element. (d) Plot of spin asymmetry SA at T = 5 K, obtained fromthe experimental and fitted reflectivity in fig. c. (e) Estimated net magnetization from PNR measurements as a function oftemperature, revealing weak paramagnetic-type correlation in permalloy honeycomb. The lack of divergence in magnetizationdata indicates the absence of long range order in the system. Error bar in experimental data, in Figs b, d and e, representsone standard deviation. PNR measurements were repeated on two different samples, fabricated under identical conditions.Experimental results were found to be similar. ture PNR measurement, is accounted for in the numericalcalculation.[30] As we can see in Fig.3d, the experimen-tal data at T = 300 K is well-described by the scatter-ing from the honeycomb structure. When measurementtemperature is reduced to T = 30 K, the numericallysimulated reflectometry pattern, Fig. 3e, for the mag-netic charge configuration, comprised of both ± Q and ±
3Q charges as shown in Fig 3h, is found to be in goodagreement with experimental result. Numerical simula-tions suggest that a significant number of honeycomb ver-tices are occupied by the ±
3Q charges. The short-rangemagnetic moment correlation, associated to ± Q charges,contribute to the overall magnetization, as shown in Fig.2e. Most interestingly, the magnetic charge configura-tion manifests massive degeneracy. Both ± Q and ± T = 5 K, the simulated pat-tern due to ± Q and ±
3Q charges reproduces the shapeof off-specular scattering data. Besides the shape andintensity of off-specular scattering profile, the reflectivitybelow Yoneda lines (at low Q) is reproduced reasonablywell in the simulated plot. The horizontal width of simu-lated profile at T = 5 K appears slightly larger than theexperimental data, as the actual sample is much larger insize than the simulated honeycomb lattice. The popula-tion density of ± Q charges is enhanced at the expense of ±
3Q charges, yet occur in large numbers. Consequently,the overall magnetization increases by (cid:39)
10% between T = 30 K to T = 5 K. For completeness, we have alsoperformed numerical modeling of the theoretically pre-dicted pure states, such as spin ice and long range spinsolid states (see Fig. S7). Compared to the pure states,mixed phases of ± Q and ±
3Q charges manifest distinctreflectometry profiles that are in congruence with exper-imental data. Q z ( Å - ) -5 -4 -3 -2 -1 a) T = 300 K Q z ( Å - ) b) T = 30 K -3 -2 -1 0 1 2 3 p i - p f (10 -2 Å -1 ) Q z ( Å - ) c) T = 5 K d) e) -3 -2 -1 0 1 2 3 p i - p f (10 -2 Å -1 ) f) g)h)i) FIG. 3: Evolution of quantum disordered state in nanoengineered permalloy honeycomb lattice. (a-c) Experimental two-dimensional (2D) maps of neutron intensity as a function of ( k i − k f ) at characteristic temperatures. Here k i and k f are theperpendicular components of incoming and outgoing neutron, respectively (see schematic in Fig. 2a. The 2D maps are obtainedby summing both (+ -) and (- +) scattering components. With decreasing temperature, diffuse scattering in the off-specularchannel becomes stronger. (d-f) Numerically simulated 2D maps for the corresponding charge configurations shown in Fig.(g-i). Numerical calculation of magnetic structure factor is performed using the DWBA formalism (see text). The highlydegenerate disordered charge correlation, involving both ± Q and ± T = 30 K) and low temperature (also see Fig. S6). Blue and red balls indicate the presence of +3Q and -3Q charges, respectively. There are two important findings that need to be high-lighted here: (a) ±
3Q charges persist to the lowest tem-perature in the lattice. It is a major departure from theprevailing understanding that the high integer chargescannot occur at low temperature in artificially frustratedlattice.[11] This becomes possible due to the nanoscopicelement size in nanoengineered honeycomb, which re-duces the inter-elemental energy to (cid:39)
10 K. (b) Numer-ical modeling does not indicate any large change in themagnetic charge configuration pattern at low tempera-ture, compared to the intermediate temperature, in thehoneycomb lattice. We argue that the massive degener- acy in magnetic charge configuration forbids the devel-opment of a stable ground state. The presence of ± -3 -2 -1 0 1 2 3 p i - p f (10 -2 Å -1 ) Q z ( Å - ) -5 -4 -3 -2 -1 a) -3 -2 -1 0 1 2 3 p i - p f (10 -2 Å -1 ) b)c) -5 -4 -3 -2 -1
0 0.03 0.06 0.09 0.12 R e f l e c t i v i t y ( a . u . ) Q (Å -1 ) R + R - d) FIG. 4: Magnetic field effect on degenerate charge configura-tion. (a) In applied magnetic field of H = 0.5 T, larger thanthe coercivity of the permalloy honeycomb lattice ( (cid:39) T = 5 K, as manifested by the spec-ular reflection (Fig.d), is nominally stronger than zero field(magnetic field is applied inplane to the sample). The diffusescattering in the off-specular data remains mostly unaffectedto field application. It is also confirmed by numerically sim-ulated pattern, Fig. b, for the degenerate charge correlationshown in Fig. c. Field application does not lift the degener-acy. (See Fig. S8 for spin asymmetry plot). Error bar in Fig.d represents one standard deviation. short-range correlation between magnetic moments asso-ciated to ± Q charges on the non-correlated vertices. Thisobservation is in good agreement with the theoretical pre-diction of a disordered state in J - J Ising model on thehoneycomb lattice.[15] Here, it is worth mentioning theeffect of disorder on magnetic properties in artificial spinice. In µ m element size honeycomb lattice, magnetic do-mains are often pinned by inhomogeneity or disorder inthe lattice.[13] In some cases, disorder is used as a tun-ing agent to explore novel properties. For instance, arecent research work by Saccone et al. has shown thatthe artificially induced disorder of Gaussian form in lat-tice construction can be used to tune magnetic proper-ties between ferromagnetic and spin glass type behaviorin arrays of Ising nanomagnets.[31] A modest thicknessfluctuation, to the tune of 1-2 nm (see Fig. S9), is ob-served across the plaquette in our honeycomb sample.Yet, the overall thickness is smaller than the size of atypical permalloy domain, ∼
18 nm. Hence, the effect ofdisorder due to modest thickness variation in experimen-tal observation can be ruled out in this case.A quantum disordered magnetic state would reject ex-ternal influences, such as magnetic field application, thatcan reduce the degrees of freedom. We test the mag- netic field effect on the prevailing disordered state offragmented ± Q and ±
3Q charges at low temperature.Additional measurements were performed in an inplaneapplied magnetic field of H = 0.5 T, which is significantlylarger than the coercivity ( H c (cid:39) + and R − becomes nominallystronger, as shown in Fig. 4d, compared to H = 0 T.However, the broad feature in diffuse scattering and itsinverted cone shape remain intact. Numerical modelingof experimental PNR pattern takes into account the fieldeffect on Ising moments. The best fit to experimentaldata is obtained for the degenerate charge configurationwith higher density of ± Q charges (see Fig. 4b-c). Itexplains the enhanced splitting between R + and R − inspecular reflectivity pattern in Fig. 4d (see Fig. S8 inSupporting Information for spin asymmetry and MSLDplots). The numerical simulation for the aforementionedmagnetic charge configuration, Fig. 4c, reproduces theshape and the intensity of the diffuse scattering. Mag-netic field application does not lift the degeneracy, albeitnominally reduces it. The system defies external influ-ences and still maintains a very high level of degeneracy.This suggests that the disordered state of mixed chargesis indeed robust.Two-dimensional nanoengineered magnetic systemwith geometrically frustrated motif not only provides analternative platform to study the novel and emergentmagnetic properties of bulk materials, but also allowsus to explore new phenomena in the reduced degrees offreedom. In this communication, we have presented ex-perimental evidences to the occurrence of quantum dis-ordered state of magnetic charges in artificial magnetichoneycomb lattice of nanoscopic element, smaller thanthe size of a permalloy domain.[33] The disordered state,comprised of both ± Q or ±
3Q charges, is characterizedby macroscopic degeneracy, which persists to low tem-perature. Typically, the high multiplicity charges, ± ±
2Q charge defects between vertices. Yet, alarge number of vertices are still occupied by ±
3Q chargesat low temperature, even in applied magnetic field. Inprinciple, magnetic field application to the tune of H = 0.5 T is capable to align moments to field direction.Consequently, the lattice will be free of high multiplic-ity charges. This does not seem to be the case here. Itfurther suggests that the observed disorder, due to degen-erate distribution of ± Q and ±
3Q charges, is intrinsic innature. The persistent disorderness to the lowest mea-surement temperature points to the quantum mechanicalnature of magnetic charge quasi-particles. After all, mag-netic charges are related to the quantum Pauli matricesvia the charge sum rule under the dumbbell model.[1, 14]Our observation is also consistent with a previous theo-retical study, which predicts the occurrence of a quantumdisordered state in two-dimensional Ising system.[15, 18]According to the theoretical researches, an Ising Kagomesystem with antiferromagnetic nearest neighbor and fer-romagnetic next nearest neighbor interactions is expectedto depict three key properties: a disordered ground statewith massive degeneracy where strong fluctuation nullifythe development of any long range order, the persistenceof disordered state from moderate to low temperatureand magnetic field independence of the ground state de-generacy. All three premises of the theoretical predictionare fulfilled in two-dimensional nanoengineered honey-comb lattice of connected permalloy elements. The pres-ence of disordered ground state in an artificial honeycomblattice elucidates the quantum mechanical properties ofmagnetic charges. Future experimental research workson the investigation of dynamic properties of magneticcharges using microwave excitation or optical pumpingmethod are highly desirable to further understand the en-ergetic profile of the quantum state of magnetic chargesin permalloy honeycomb lattice.The new honeycomb lattice design has spurred the ex-ploration of emergent magnetic phenomena that are atthe forefront of research. Besides the observation of quan-tum disordered state in moderately thick lattice, previousresearch works on thinner analogue of artificial honey-comb have demonstrated the development of spin solidtype loop state at low temperature.[22, 32] Unlike the col-lective paramagnetic behavior in magnetization data atlow temperature in the disordered state, loop state tendsto achieve a near zero magnetization as T → EXPERIMENTAL SECTION
Sample Fabrication and Characterizations : Fabrica-tion of artificial honeycomb lattice involves the synthe-sis of porous hexagonal diblock template on top of asilicon substrate, calibrated reactive ion etching (RIE)using CF gas to transfer the hexagonal pattern to theunderlying silicon substrate and the deposition of mag-netic material (permalloy) on top of the uniformly ro-tating substrate in near-parallel configuration ( (cid:39) o )to achieve the two-dimensional character of the system.The sample fabrication process utilized diblock copoly-mer polystyrene(PS)-b-poly-4-vinyl pyridine (P4VP) ofmolecular weight 29K Dalton with the volume frac-tion of 70% PS and 30% P4VP. The diblock copoly-mer tends to self-assemble, under right condition, in ahexagonal cylindrical structure of P4VP in the matrix of polystyrene (PS).[34] A 0.5% PS-b-P4VP copolymersolution in toluene was spin casted onto cleaned siliconwafers at 2500 rpm for 30 s and placed in vacuum for 12hours to dry. The samples were solvent annealed at 25 o C for 12 hours in a mixture of THF/toluene (80:20 v/v)environment. The process results in the self-assembly ofP4VP cylinders in a hexagonal pattern within a PS ma-trix. The average diameter of a P4VP cylinder is (cid:39)
15 nmand the center-to-center distance between two cylindersis (cid:39)
30 nm, also consistent with that reported by Park etal.[34]. Submerging the samples in ethanol for 20 minutesreleases the P4VP cylinders yielding a porous hexagonaltemplate. The diblock template is used as a mask totransfer the topographical pattern to the underlying sil-icon substrate. The top surface of the reactively etchedsilicon substrate resembles a honeycomb lattice pattern.This property is exploited to create metallic honeycomblattice by depositing permalloy, Ni . Fe . , in near par-allel configuration in an electron-beam evaporation. Forthis purpose, a new sample holder was designed and setupinside the e-beam chamber. The substrate was rotateduniformly about its axis during the deposition to createuniformity. This allowed evaporated permalloy to coatthe top surface of the honeycomb only, producing the de-sired magnetic honeycomb lattice with a typical elementsize of 12 nm (length) × ×
10 nm (thickness).Atomic force micrograph of a typical honeycomb latticeis shown in Fig. 1a. The center-to-center spacing be-tween neighboring honeycombs is (cid:39)
30 nm. Thus, eachhoneycomb is about 30 nm wide.
Neutron Scattering Measurements : Neutron scatteringmeasurements were performed on a 20 ×
20 mm surfacearea sample at the Magnetism Reflectometer, beam lineBL-4A of the Spallation Neutron Source (SNS), at OakRidge National Laboratory. Neutron measurements werealso reproduced on another permalloy honeycomb sam-ple with similar geometrical dimensions, nanofabricatedunder the identical conditions. The instrument utilizesthe time of flight technique in a horizontal scattering ge-ometry with a bandwidth of 5.6 ˚ A (wavelength varyingbetween 2.6 - 8.2 ˚ A ). The beam was collimated usinga set of slits before the sample and measured with a 2Dposition sensitive He detector with 1.5 mm resolution at2.5 m from the sample. The sample was mounted on thecopper cold finger of a close cycle refrigerator with a basetemperature of T = 5 K. Beam polarization and polariza-tion analysis was performed using reflective super-mirrordevices, achieving better than 98% polarization efficiencyover the full wavelength band. For reflectivity and off-specular scattering the full vertical divergence was usedfor maximum intensity and a 5% ∆ θ / θ (cid:39) ∆ q z / q z relativeresolution in horizontal direction. Monte-Carlo simulation : We have performed Monte-Carlo simulation based on the loop algorithms [35], as im-plemented in ALPS [36]. We have employed the Heisen-berg exchange Hamiltonian: H = − J (cid:80) (cid:104) i,j (cid:105) σ i σ j − J (cid:80) (cid:104) i (cid:48) ,j (cid:48) (cid:105) σ i σ j , where σ i is the Pauli spin-1/2 operatorat the lattice vertex sites i (see the main text), J , J arenearest and next-nearest neighbor exchange interactionparameters. For numerical simulations, we have used akagome lattice as the host sites for σ , which arises dueto the moment aligned along the length of honeycombelement. The near-zero temperature properties of thespin-system was calculated at sufficiently low tempera-ture, or large β = 1 /K B T such that we can consider oursystem as a converged ground-state with 5 × sweepsper thermalization step. The ground state energy E calculation is repeated for different values of J , J . Modeling of off-Specular scattering using DistortedWave Born Approximation formalism : The simulatedOff-Specular reflectivity profiles were generated by usingDistorted-Wave Born-Approximation (DWBA) as imple-mented in the BornAgain [29] software.[22]. The basis forthe model used in our simulation are based on the specu-lar neutron reflectivity data at 300 K, 30 K, and 5 K (asshown in Fig. 2), fitting using Licorne-Py [37] software.In our model, we defined our sample as a multilayer con-sisting of the substrate with 3 layers ( l ) on top, for whichthe scattering matrix elements can be expressed by: (cid:104) ψ i | δv | ψ f (cid:105) = (cid:88) l (cid:88) ± i (cid:88) ± f (cid:104) ψ ± il | δv | ψ ± fl (cid:105) where, δv is the first order perturbation expansion termof scattering length density ( v ( r )), and ψ i , ψ f denotesthe incident, and final wavefunctions, respectively.The forward or backward traveling wavefunction inreal-space is given by ψ + and ψ − , respectively. Thebottom-layer is composed of nanostructured silicon withhoneycomb patterns, and the next layer, permalloy-layer, comprises of nanostructured permalloy. Thetop-layer is formed of a thin layer of partially oxidizedpermalloy. An ambient air layer was placed on topof the permalloy-layer. We have introduced a layoutof honeycomb patterns of permalloy-hexagons withcylinders cut-out from the center, by using a hexagonallattice with a = 31 nm within the permalloy-layer.The form factor for the cylindrical cut-out is definedas: F = 2 πR H sinc (cid:18) q z H (cid:19) exp (cid:18) iq z H (cid:19) J ( q || R ) q || R ,where, q || ≡ (cid:113) q x + q y , J is a Bessel function ofthe first-kind, and radius, R = 11.2 nm, height, H= 13 nm. The magnetic phases were constructedby using rectangular elements with fixed magneti-zation directed along its length. The form factorof these rectangular elements is defined as: F = LW H sinc (cid:18) q x L (cid:19) sinc (cid:18) q y W (cid:19) sinc (cid:18) q z H (cid:19) exp (cid:18) iq z H (cid:19) ,where L , W , and H are 12.5 nm, 5 nm, and 13 nm,respectively. These magnetized-elements are placed asa part of the hexagonal lattice to incorporate long- range correlations with inter-cluster interference. Thescattering matrix elements can be written as: (cid:104) ψ i | δv | ψ f (cid:105) = (cid:88) j exp (cid:0) i q || R j || (cid:1)(cid:90) d r || exp (cid:0) i q || r || (cid:1) (cid:90) dzφ ∗ i ( z ) F ( r − R j || ; T j ) φ f ( z )where, F ( r − R ij ; T j ) is the form-factor for j th particle,such that v p ( r ) = (cid:80) j F ( r − R j || ; T j ). The elastic scat-tering cross-section is given by: dσd Ω = | (cid:104) ψ i | δv | ψ f (cid:105) | . Toaccount for the finite-size effect, we have used a 2D latticeinterference function with a large isotropic 2D-Cauchydecay function with lateral structural correlation lengthof λ x,y = 1000 nm. The position-correlation is givenas: ρ S G ( r ) = (cid:80) m,n δ ( r − m a − n b ) − δ ( r ), with latticebasis ( a, b ) and also introduced the effects of natural-disorder of the system by applying a small Debye-Wallerfactor corresponding to a position-variance of (cid:104) x (cid:105) = 0 . . The interference function can be written as: S ( q ) = ρ S (cid:88) q i ∈ Λ ∗ πλ x λ y (cid:0) q x λ x + q y λ y (cid:1) / For the simulation with just the structure, the magne-tized elements were not introduced, and the sample con-tained only the coherent nuclear components. For thesimulation with an external magnetic field, we allowedan external field commensurate with 0.5T and allowedhigher magnetization in the magnetized-elements corre-sponding to the magnetic scattering length density ob-tained from specular reflectivity fitting.
Statistical Analysis : 1. Pre-processing of data: Exper-imental results are evaluated for statistical accuracy. Forthis purpose, neutron scattering measurements were per-formed for sufficiently longer amount of time to obtainstatistically significant results. Experimental data is pre-sented as it is, without any processing. 2. Data presen-tation: An error bar of one standard deviation is addedin the data wherever applicable. It is mentioned in thefigure caption of relevant figures. 3. Sample size: Exper-imental results were reproduced on two samples, fabri-cated under identical conditions. 4. Statistical methods:Neutron scattering method is a statistical probe. Mea-surements in different experimental conditions, e.g. dif-ferent temperatures and fields, is used to reveal intrinsicground state magnetic properties with sufficient detailsin magnetic honeycomb lattice. 5. Software used for sta-tistical analysis: We have used LICORNE-PY softwareto analyze neutron reflectivity data.
SUPPORTING INFORMATION
Supporting Information is available from the WileyOnline Library or from the author.
ACKNOWLEDGEMENTS
We thank Artur Glavic, S. K. Kim and Giovanni Vig-nale for helpful discussion. The research at MU is sup-ported by the U.S. Department of Energy, Office of Ba-sic Energy Sciences under Grant No. DE-SC0014461. Aportion of this research used resources at the SpallationNeutron Source, a DOE Office of Science User Facilityoperated by the Oak Ridge National Laboratory.
CONFLICT OF INTEREST
The authors declare no conflict of interest.
KEYWORDS
Artificial magnetic honeycomb lattices, Geometricfrustration, Magnetic charges, Degenerate states, Neu-tron reflectometry measurements.
REFERENCES [1] C. Castelnovo, R. Moessner, S. Sondhi,
Nature , , 42.[2] E. Mengotti, L. Heyderman, A. Rodriguez, F. Nolting,R. Hugli, H. Braun, Nature Phys. , , 68.[3] W. Branford, S. Ladak, D. Read, K. Zeissler, L. Cohen, Science , , 1597.[4] O. Sendetskyi, et al. , Phys. Rev. B , , 224413.[5] N. Rougemaille, et al. , Phys. Rev. Lett. , ,057209.[6] Y. Chen, B. Summers, A. Dahal, V. Lauter, G. Vignale,D. K. Singh, Adv. Mat. , , 224413.[7] Y. Perrin, B. Canals, N. Rougemaille, Nat. Phys. , , 410.[8] C. Reichhardt, A. Libal, C. Reichhardt, New J. Phys. , et al. , Nat. Phys. , , 162.[10] C. Nisoli, V. Kapaklis, P. Schiffer, Nat. Phys. , ,200.[11] C. Nisoli, R. Moessner, P. Schiffer, Rev. Mod. Phys. , , 1473.[12] M. Tanaka, E. Saitoh, H. Miyajima, T. Yamaoka, Y. Iye, Phys. Rev. B , , 052411.[13] Y. Qi, T. Brintlinger, J. Cumings, Phys. Rev. B , , 094418[14] G. Chern, O. Tchernyshyov, Phil. Trans. Royal Soc. A , , 5718.[15] A. Wills, R. Ballou, C. Lacroix, Phys. Rev. B , ,144407.[16] Y. Iqbal, W. Hu, R. Thomale, D. Poilblanc, F. Becca, Phys. Rev. B , , 144411. [17] R. Siddharthan, B. Shastry, A. Ramirez, A. Hayashi, R.Cava, S. Rosenkranz, Phys. Rev. Lett. , , 1854.[18] T. Takagi, M. Mekata, J. Phys. Soc. Jpn. , , 3943.[19] G. Chern, P. Mellado, O. Tchernyshyov, Phys. Rev. Lett. , , 207202.[20] G. Moller, R. Moessner, Phys. Rev. B , ,140409(R).[21] Y. Perrin, B. Canals and N. Rougemaille, Phys. Rev. B , , 224434.[22] A. Glavic, B. Summers, A. Dahal, J. Kline, W. VanHerck, A. Sukhov, A. Ernst, D. K. Singh, Adv. Sci. , , 1700859.[23] O. Sendetskyi, L. Anghinolfi, V. Scagnoli, G. Moller, N.Leo, A. Alberca, J. Kohlbrecher, J. Luning, U. Staub andL. Heyderman, Phys. Rev. B , , 224413.[24] G. Moller, R. Moessner, Phys. Rev. Lett. , ,237202.[25] M. Kashem, et al. , Macromolecules , , 6202.[26] V. Lauter-Pasyuk, J. Phys.
IV France , 1, 221-240;DOI: https://doi.org/10.1051/sfn:2007024[27] J. Gardner, M. Gingras, J. Greedan,
Rev. Mod. Phys. , , 53.[28] V. Lauter, H. Lauter, A. Glavic, B. Toperverg, ReferenceModule in Materials Science and Materials Engineering .[29]
BornAgain - Software for simulating and fitting X-rayand neutron small-angle scattering at grazing incidence
Results in Physics , , 263.[31] M. Saccone, A. Scholl, S. Velten, S. Dhuey, K. Hofhuis, C.Wuth, Y. Huang, Z. Chen, R. Chopdekar, Alan Farhan, Phys. Rev. B , , 224403.[32] B. Summers, L. Debeer-Schmitt, A. Dahal, A. Glavic, P.Kampschroeder, J. Gunasekera, D. Singh, Phys. Rev. B , , 014401.[33] Spin Electronics, Edited by Michael Zeiss and MartinThornton, Springer-Verlag Berlin Heidelberg .[34] S. Park, B. Kim, O. Yavuzcetin, M. Tuominen, T. Rus-sell, ACS Nano , , 1363.[35] S. Todo, K. Kato, Phys. Rev. Lett. , , 047203.[36] B. Bauer, et al. , , J. Stat. Mech.
P05001.[37] V. Lauter, A. Savici, S. Hahn,
Licorne-Py: A polar-ized neutron specular reflectivity fitting software. (2019)(2019)