Bilayer artificial spin ice: magnetic force switching and basic thermodynamics
Fabio S. Nascimento, Afranio R. Pereira, Winder A. Moura-Melo
MMagnetic van der Waals-like framework in bilayer artificial spin ice
Fabio S. Nascimento, a) Afranio R. Pereira, b) and Winder A. Moura-Melo c)1) Centro de Forma¸c˜ao de Professores, Universidade Federal do Recˆoncavo da Bahia,45300-000 - Amargosa - Bahia - Brazil. Departamento de F´ısica, Universidade Federal de Vi¸cosa,36570-900 - Vi¸cosa - Minas Gerais - Brazil.
We study an artificial spin ice system consisting by two identical layers separatedby a height offset h . For short separation, the layers are shown to attract eachother, provided the whole system is in the ground state. Such an attraction comesabout by means of a power-law force compared to van der Waals forces. Whenevermagnetic monopoles show up in one (or both) layers, the scenario becomes evenmore interesting and these layers may also repel each other. By tuning parameterslike h and monopole distance, switching between attraction and repulsion may beaccomplished in a feasible way. Regarding its thermodynamics, the specific heatpeak shifts to lower temperature as h increases.Keywords: artificial spin ice, geometrical frustration, magnetic monopole, van derWaals forces a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p . INTRODUCTION AND MOTIVATION Geometrical frustration is an interesting phenomenon which has received a lot of attentionrecently . In magnetism, it arises whenever interaction between magnetic degrees of freedomis incongruous regarding the lattice underlying the crystal geometry. Frustration emergesin appealing natural materials , and it can also be created by design . Indeed, artificialsystems have been built in diverse configurations which allow us to control frustration byexperimentally tuning suitable parameters. An important class of such designed systems isprovided by artificial spin ice (ASI) arrangements , which essentially consists of a planar-type regular array of nanosized elongated ferromagnetic rods where geometrical frustrationtakes place at the vertices. By virtue of strong shape anisotropy along the major axis, everynanoisland effectively behaves as an Ising-type dipole. Now, the collective interaction amongall these dipoles yields surprising emergent phenomena, such as fractionalization. Actually,above the ground state the most elementary excitations show up as magnetic monopoles,coupled in pairs by energetic strings , which are flux-carrying magnetized chains. In words,the original degrees of freedom, the usual magnetic dipoles, have been fractionalized intoisolated monopoles emerging at ASI vertices. Although they had been originally named Dirac monopoles and strings , it is more suitable to speak about Nambu monopoles andstrings as claimed in Refs. , after Nambu picture adaptation of Dirac description to aLondon-type framework . Such a magnetic scenery has been observed to occur in distinctASI lattice geometries including square, rectangular, triangular, and kagome arrangements.The interest in their physical properties lies in the fact that such systems are promis-ing candidates for new technologies based upon the control of magnetic charges and theircurrents, something termed as magnetricity and magnetronic. Actually, magnetic chargeflow was firstly realized in Dy T i O compound, an example of natural three-dimensionalspin ice crystal, but at very low temperature ∼ − . At room tem-perature, an ordered magnetic current has been observed in an unidirectional arrangementof patterned nanoislands , where no geometrical frustration takes place at all. In turn,even though theoretical studies regarding three-dimensional (3 D ) ASI appeared more thana decade ago , their experimental realization took place only very recently , whichhas been achieved due to novel advances in patterning 3 D magnetic nanostructures . These2ystems consist of only one ASI built by offsetting one of the sublattices by some height h such that the energy of interaction between all nearest neighbours becomes equivalent, al-lowing this arrangement to undergo a transition to a magnetic Coulomb phase . Inorder to shed further light onto such a scenery, we investigate a rather similar system whoselayers contain nanoislands arranged like a square lattice. Then, we study a bilayer artificialspin ice (BASI, for short), where interactions take place among all the islands of both layers(see Fig. 1). Since each island behaves like an Ising-type dipole, we can envision a frame-work where van der Waals-like magnetic forces will show up as the collective interaction ofthese dipoles.Indeed, van der Waals (vdW) forces arise from the mutual coupling among electric dipolescomposing a system. Although two ideal dipoles interact like r − -potential, whenever effectslike orientation, induction and dispersion are taken into account the net interaction goes like V vdW ( r ) ∼ r − , see Ref. . Such forces are keystones to understand how atoms and moleculescombine to form gaseous, liquid and solid substances . More recently, vdW interactionhas become increasingly important for a better understanding of layered compounds, in-cluding graphene and other 2 D -materials . Indeed, in such materials the ions are held bystrong covalent bonds whereas the layers experience weak out-of-plane vdW forces, makingease their exfoliation. Our main findings give further support to such an envisioning, sincethe layers composing the BASI interact through a potential, V ( h ) ∼ h − . , with power-lawapproximate to vdW potential, V vdW ( r ) ∼ r − . The deviation may be explained by geomet-rical frustration taking place in each layer, which forbids the Ising-type dipoles of achievinga unique ground-state. We also study the interaction between magnetic monopoles showingup as excitations in the Coulomb phase. In Section II we present our model and methods,while results and discussion is left to Section III. We close our paper by presenting ourconclusions and prospects for forthcoming research. II. MODEL AND METHODS
Every square ASI is described by a dipolar Hamiltonian like follows H D = Da (cid:88) i>j (cid:20) ˆ e i · ˆ e j r ij − e i · (cid:126)r ij )(ˆ e j · (cid:126)r ij ) r ij (cid:21) S i S j , (1)3 V ~ h -6.4
GS1GS1 GS2GS1 aa h t1 t2 t3 t4 (a)(b)(c) (d)(e) upperlower t1 - lower t1 - upper FIG. 1. (a) : the four possible classes of vertices in a single monolayer square ASI vertex: t t t t
4. Each class comprises different vertex types that share the same energy. The first twoclasses obey the ice-like rule: two spins point in while the other two point out of the vertex center,in short , while the other topologies violate it. (b) : sketch of BASI system showing thesquare arrangement of each layer with lattice spacing a and separated by height offset h . ( c ): eachlayer displays ground state composed only by t GS d ): now, bottomlayer presents GS
2. ( e ) potential between the layers is plotted as a function of h . It is noteworthythat GS − GS h (cid:46) a ). Indeed, in this case,the layers experience a mutual attraction, while for GS − GS V ∼ h − . , and practicallyfalls off as h (cid:38) . a . where D = µ π µ a is the dipole-dipole coupling constant, ˆ e i is the local Ising axes of thelattice, r ij is the distance between S i and S j , and S i = ± µ ∼ − µ B ( µ B
4s the Bohr magneton) and they are separated by lattice spacing a ∼ nm, so that D ∼ − − − J]. In words, if we have a single ASI layer, then such vectors run overdirections x and y , confined on the layer plane. Whenever a second layer is taken intothe game the mutual interaction between the layers must also be accounted. This is ac-complished in a simple way just allowing a third component for r ij vectors, so that it mayalso compute the interaction between pairs of nanoislands belonging to distinct layers aswell (clearly, vectors ˆ e i S i are kept on layers planes, say, only with x and y components).If one intends to bring a third layer, one simply permits r ij to run over its vertices, and so on.Our simulation is carried out by considering two ASI layers parallel each other and sep-arated by a height offset, h . Each layer comprises 29 ×
29 = 841 vertices comprising a totalof N = 3480 magnetic moments disposed in a square lattice. Our first task is to determinethe combined ground state of the coupled layers as function of h . This is done by startingoff from a disordered state at a very high temperature; later, the system is driven to veryslow dynamics by cooling it to very low temperature, ∼ . D/k B . III. RESULTS AND DISCUSSION
Since t t GS GS
2) and populated only by t t t t t GS
1, then, the same vertex would be t GS
2. The first excited state demands the appearance of t t t ±
1, blue and red spots)joined by an energetic string which is a segment of t t S GS GS
1, theother must be GS
2. In this case, the layers experience a considerable attraction, whosepotential goes like V ( h ) = −
78 ( h/a ) − . ( V is measured in units of the dipolar constant, D ). On the other, if the layers were put side by side exhibiting the same individual groundstate, say, GS GS V ( h ) = +78 ( h/a ) − . (see Fig.1). Other combined ground states are possible, providedthat every vertex in one layer alternate as upper/lower types regarding its nearest neighborlying on the other ASI layer. This is clearly accomplished by the combined GS − GS t t t t t t h ; here, we have determined it for small h . Indeed, as h becomes larger, h (cid:38) . a , BASIis practically decoupled and one has two non-interacting ASI systems. [Indeed, its basicthermodynamics puts an even more stringent value, indicating that for h > a one effectivelyhas two decoupled layers, as discussed later].As a whole, once the original degrees of freedom are dipoles interacting via model Hamil-tonian (1), the above findings suggest that we are faced with a magnetic van der Waals-likeframework. In general both layers are expected to experience the potential V ∼ h − . . Thedeviation from vdW power-law potential, V vdW ∼ (cid:96) − may be attributed to strong Isinganisotropy that just allows each dipole to flip, along with the geometric frustration, comingfrom the rigid lattice geometry arrangement. Besides this numerical deviation in the po-tential, in a BASI not only attraction takes place: layers also repel each other dependingon their dipole configurations. Additionally, the attraction between ASI layers at groundstate (vacuum sate) may be faced as a classical magnetic analog of the famous Casimireffect which describes the attraction of two perfectly conducting neutral parallel platesdue to vacuum fluctuations. At zero temperature, Casimir potential also obeys a power6aw behavior, V C = − ( π (cid:126) c/ A(cid:96) − ∼ − A(cid:96) − ( A is the area of one of the plateswhich are separated by (cid:96) ). Indeed, Casimir pressure is so tiny, F C /A ∼ − (cid:96) − (N / m ),which jeopardized its experimental demonstration for around half-century (for a review, seeRef. ). In a BASI, the force between the layers goes like F ≈ h/a ) − . (in units of D/a ∼ − − − N). In addition, each ASI vertex in typical arrangements comprisesan area ∼ − m (since the major axis of a nanoisland goes around a few a ), then eachASI layer has a total area of A ∼ − m (each layer comprises 29 ×
29 = 841 vertices,as aforementioned), yielding a magnetic pressure F/ A ∼ − ( h/a ) − . (N / m ) whichis generally much higher than its Casimir counterpart discussed above. Actually, if BASIlayers were considered as neutral plates separated (cid:96) = a ∼ nm, then they would attracteach other with a Casimir force F C ∼ − N, while BASI layers at h = a interact withmagnetic force F ( h = a ) ∼ − − − N.For practical purpose, the repulsion between BASI layers can be exploited as a kind ofmagnetic levitation or magnetic damping system at nanoscale. Switching between magneticattraction/repulsion can be achieved whenever one may drive one of the layers from GS GS
2. Additionally, in the realm of a bilayer system, for instance configurations GS GS BASI molecule ’,where the attractive and repulsive forces balance out. For instance, one may conceive aninteresting situation where one layer is fixed while another rotates so that attractive andrepulsive forces alternate yielding oscillation to this molecule . These and other proposalsare appealing nowadays since actual devices are rapidly shrinking to nanometer scale givingthe possibility of tuning the magnitude of BASI magnetic force on demand.Now, we depart to investigate the appearance of excitations above the ground state andhow they change the previous results. At each ASI layer these excitations emerge as mag-netic monopole-antimonopole pairs connected by energetic strings. We then start off byconsidering BASI ground state given by GS − GS
2, say, GS GS GS s = a , with s being the7 a) � E ~ h -2.2 s = 1as = 2a s = 1as = 2a s h (b) � E ~ h -5.8
FIG. 2. ( a ) A monopole pair lying on layer 2 (upper layer is kept at GS
1) reinforce repulsionbetween the layers, which becomes strengthen as the string tension/size is enlarged. In addition,note that the energy difference between this configuration and GS − GS
2, ∆ E , gets higher as thelayers become closer. ( b ) On the other hand, having one pair by layer with opposite poles closer,layers attract each other, as realized before with GS − GS size of the string. Eventually, successive flips of neighbor dipoles move one of the monopolesaway so that the pair is now separated by a larger (higher energetic) string, say s = 4 a , asdepicted in Fig.2(a). The appearance of these excitations imputes in a repulsion betweenthe layers: indeed, even a single monopole pair (along with the smaller string, s = a ) isenough to overcome the attraction so that repulsive regime dominates. Actually, wheneverthe string is enlarged and/or more monopoles take place, the layers repel each other withfurther strength. This is the fact if monopoles and strings appear in one of the layers whileanother is kept in its ground state. How about we consider a monopole pair in each layer?As we shall see in what follows, depending on the configuration of the monopoles, the at-tractive regime can be restored. Let us begin by the simplest case with a monopole pairand a string of size s = a , say, in layer 1 and a similar configuration in layer 2. In addition,let the poles of layer 2 be inverted with respect to those of layer 1, as depicted in Fig.2(b).In this case, the attraction between opposite and closest monopoles (separated by d = h ;north/blue and south/red poles) overcome the whole repulsion brought about by the strings8 a) (b) � E ~ h -4.9 � E ~ h -2.9
FIG. 3. ( a ) Isolated south poles (red spots) in each layer. As expected, their interaction strengthenrepulsion between layers. ( b ) However, if opposite poles are in order, then its mutual attractionovercomes yielding attraction between the layers. and like poles interaction, so that the layers experience an attractive regime once again.If like poles are moved away their repulsion decreases considerably and attraction betweenlayers is strengthen. On the other hand, if the configuration were like poles closer each other,repulsion would be huge.Now, we would like to study the case two isolated monopoles, one placed in each layer and,whether and how the surrounding medium affects their interaction. To isolate a monopolein a layer, we should move its partner far away (what has the cost of enlarging the stringuntil the edge of the layer, effectively expelling the moving pole outside the system). Thisshed further light onto former results and discussion. Indeed, if we consider two isolatedlike poles, one lying at a fixed position of each layer, as shown in Fig. 3(a), then the layersexperience a strong repulsion and tend to keep far apart. On the other hand, if one hasopposite poles, their interactions overcome and the layers experience attraction once again.For the sake of completeness, now the layers are kept at definite and fixed h values andwe intend to study the energetic of the system as one of the monopoles is displaced along.For concreteness, let the monopole at bottom layer fixed at ( x , y , z = 0), while the otheris initially at ( x = x , y = y , z = z + h = h ), but it may be displaced along xy plane.As expected, also in this framework like poles repel whereas opposite monopoles attract9 a) (x ,y , h )(x ,y , h )(x ,y ,0) d (b) h = 0.50 h = 0.60 h = 0.75 h = 1.50 C C
FIG. 4. ( a ) Two like monopoles were initially separated by a vertical distance d = h . Then,the pole in the upper layer is displaced along xy plane, as indicated. The effective potential, V eff = q m d − + κ s + c , against d for a number of height offsets. For short monopole separation,Coulomb term dominates, while as d increases the string size s tends to increase even faster andthe linear potential dominates. For both cases, the bottom graphics display the Coulomb potentialbetween two magnetic monopoles, V C = q m d − , which is clearly repulsive for like charges andattractive for opposite ones. themselves following a Coulomb potential, d − ( d is the spatial distance between the poles,see Fig. 4). The energetic resembles that obtained for a single ASI layer, like below: V eff ( d, s ) = q m d + κ s + E c . (2)where, q m is the charge of an isolated magnetic pole (which may be positive or negative), κ is the string tension of size s which accounts for the energy cost of moving the other two10onopoles far away from the remaining ones; E c is the energy cost to create these excita-tions, monopole pair and string as well. Table I presents how such parameters vary with h :namely, for h = 0 . a one gets q m ≈ . Da (along with κ ≈ D/a and E c ≈ D ). Thismonopole charge is comparable to that for a single square lattice ASI , ∼ . Da , butstring tension and creation energy are much higher, evidencing the strong coupling betweenlayers at this height offset. As a whole, our findings clearly show that field lines producedby magnetic monopoles lying in a layer spread radially throughout the 3 D space followinga Coulomb-like law. TABLE I. Height offset dependence of parameters q m , κ and E c .Equal poles Opposite polesh(a) q m κ E c q m κ E c Finally, we deal with the basic thermodynamics of the BASI system. The specificheat, c = (cid:104) E (cid:105)−(cid:104) E (cid:105) Nk B T ( k B is the Boltzmann constant), has been obtained by standard MonteCarlo technique along with Metropolis algorithm implemented using Boltzmann distribu-tion, ∼ e − ∆ E/k B T , for our original array consisting by 3480 dipoles per layer. We have alsoimplemented 10 Monte Carlo steps to reach a steady state and up to 10 Monte Carlo stepsto obtain the averages of thermodynamic variables, each Monte Carlo step corresponding to3480 single-spin flips. [In order to save time computation, we adopt a cutoff radius r c = 6 a whenever dealing with the dipolar energy. Such a cutoff yields deviations ≤ .
1% in thetotal energy of the system]. Fig. 5 depicts specific heat as a function of temperature fordistinct h values. First, note that the specific heat peak is shifted to lower temperature as h increases. Indeed, as the layers become decoupled the peak temperature is T p ≈ D/k B .This value is slightly smaller than T = T c = 7 . D/k B reported in the works of Refs. for a square ASI at the thermodynamics limit, whereas here we are taking into account the11nite size of BASI layers. (b) decoupled BASI T p ( D / k B ) h(a) (a) hhhh FIG. 5. Simple BASI thermodynamics: ( a )Specific heat, c , as a function of temperature, T .Increasing h the peak is shifted to lower temperature. ( b ) The peak temperature, T p againstheight offset, showing that BASI effectively behaves as two decoupled layers for h > a . IV. CONCLUSION AND PROSPECTS
Whenever in the ground state, the layers composing our BASI system attract each otherwith a force which resembles van der Waals power-law forces. When excitations emerge inthe system, layers still attract if single opposite charges are disposed in each layer (Fig. 3b),or a monopole pair in each layer but with opposite charges close each other. Other situationsfavor repulsion, which by itself could be thought as a kind of magnetic levitation or magneticdamping for nanoscaled systems. The switching between attraction and repulsion may beuseful to design stable
BASI-type molecule by balancing these forces. The dependence ofspecific heat peak with h may be useful to determine optimal heights offset favoring stability.As prospects, we intend to investigate how BASI behave under translation and rotationof one layer (while the other is kept fixed). As fundamental symmetries they are expectedto yield novel effects and to shed further light into system physical properties.12 CKNOWLEDGEMENTS
The authors thank CAPES (Finance Code 001), CNPq and FAPEMIG for partial finan-cial support.
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