Spin-state smectics in spin crossover materials
SSpin-state smectics in spin crossover materials
J. Cruddas, G. Ruzzi, and B. J. Powell a) School of Mathematics and Physics, The University of Queensland, Brisbane, Queensland 4071,Australia (Dated: 3 February 2021)
We show that a simple two dimensional model of spin crossover materials gives rise to spin-state smectic phases wherethe pattern of high-spin (HS) and low-spin (LS) metal centers spontaneously breaks rotational symmetry and trans-lational symmetry in one direction only. The spin-state smectics are distinct thermodynamic phases and give rise toplateaus in the fraction of HS metal centers. Smectic order leads to lines of Bragg peaks in the x-ray and neutron scat-tering structure factors. We identify two smectic phases and show that both are ordered in one direction, but disorderedin the other, and hence that their residual entropy scales with the linear dimension of the system. This is intermediateto spin-state ices (examples of ‘spin-state liquids’) where the residual entropy scales with the system volume, and anti-ferroeleastic ordered phases (examples of ‘spin-state crystals’) where the residual entropy is independent of the size ofthe system.
I. INTRODUCTION
Spontaneous symmetry breaking: the emergence, at low-temperatures, of long-range order from a high-temperaturedisordered state is one of the foundations of condensed mat-ter physics.
For example in crystals translational and rota-tional symmetries, that are present in liquids and gases, arespontaneously broken. This has profound consequences: forexample it leads to the rigidity of crystals and predicts the ex-istance of massless (gapless, linearly dispersing) low-energyexcitations (acoustic phonons).However, it has become increasingly clear that when dis-ordered states survive to low temperatures in strongly inter-acting systems it is often a sign that something extremely in-teresting is happening. Key examples include the Tomonaga-Luttinger liquid in one-dimension and quantum spin liquids intwo and three dimensions.
Here quantum fluctuations aresufficiently strong to entirely suppress long-range order. Butclassical physics can also lead to low-temperature disorderedstates. In classical systems, entropy rather than quantum fluc-tuations, is responsible for the stabilization of a disorderedstate. For example, in ice-states there are a macroscopicallylarge number of microstates with the same energy. Thus, thereis a large residual (zero temperature) entropy and the systemremains disordered a low temperatures. The two best knownexamples are proton disorder in I h and I c water ice and themagnetic disorder in spin ices, but recently ice phases havebeen discovered or predicted in many other systems, includ-ing spin crossover (SCO) materials. Disorder does not imply that the system is completely ran-dom. In gas-like states randomness emerges from the weak-ness of the interactions between the constituents. But, inliquid-like states the low-energy physics is dictated by thestrong correlations between the constituents. The classicexample is the Bernal-Fowler ice rules in water ice, whichdictate that locally every oxygen forms covalent bonds withexactly two protons. There are an extremely large num-ber of microstates consistent with the ice rules meaning that a) Electronic mail: [email protected] the instantaneous configurations look random on large length-scales. Nevertheless the behavior of each proton is stonglycorrelated with those of other protons nearby. This, and sim-ilar ice rules in other systems, result in distinctive signaturesin diffraction experiments.
Liquid crystals host phases of matter intermediate betweencrystalline and liquid phases – rotational order is sponta-neously broken in both nematic and smectic phases, and trans-lational symmetry is also spontaneously broken in one di-rection (but not in the perpendicular directions) of smecticphases. Thus crystals are more ordered than smectics, smec-tics are more ordered than nematics, and nematics are moreordered than liquids. This endows liquid crystals with prop-erties that have proved incredibly technologically useful. Liquid-crystal-like phases – where rotational symmetry isbroken, but translational symmetry is (partially) preserved –have been identified in other systems, including electronicnematics, electronic smectics, spin nematics, andspin smectics. Here we demonstrate that smectic phases occur in a two-dimensional model of SCO materials. We show that the in-terplay between competing crystalline orders and the entropyof mixing can lead to phases, in which rotational symmetry isbroken and translational symmetry is broken in one direction,but preserved in the perpendicular direction. Thus, spin-statesmectics are intermediate between spin-state liquids (of whichspin-state ice is the only concrete example proposed so far)and spin-state crystals (where the pattern of spin-states spon-taneously breaks translations symmetry in all directions, seeFig. 1 for some examples). We show that this hierarchy canbe quantified by the scaling behavior of the residual entropywith the system size.We show that spin-state smectic phases lead to plateaus inthe fraction of high-spin states, n HS (which is experimentallymeasurable via χ T , where χ is the magnetic susceptibilityand T is the temperature). There are several materials whereplateaus in χ T have been observed, but no long-range orderin the spin-states has been resolved. Smectic phases offera possible explanation for these experiments. We discuss thesignatures of smectic phases in x-ray and neutron scatteringexperiments, which would allow for a definitive identificationof a smectic phase. a r X i v : . [ c ond - m a t . s t r- e l ] F e b FIG. 1. The selected spin-state crystalline orderings, with long-range order in both directions, observed in our Monte Carlo simulations. TheLS (HS) sites are indicated by blue (yellow) circles. For clarity, we have retained the labels used in our previous work . Where n HS (cid:54) = / II. MODEL AND METHODS
We consider a square lattice of SCO molecules coupled bysprings (Fig. 2) and described by the Hamiltonian H = ∑ i ( ∆ H − T ∆ S )+ ∑ n = k n ∑ (cid:104) i , j (cid:105) n (cid:8) r i , j − η n (cid:2) R + δ ( σ i + σ j ) (cid:3)(cid:9) , (1)where ∆ H = H HS − H LS is the enthalpy difference between theHS and LS states of an individual molecule, ∆ S = S HS − S LS is the entropy distance between the HS and LS states of an in-dividual molecule, k n is the spring constant between n th near-est neighbors, r i , j is the instantaneous difference between the i th and j th molecules, η n = , √ , , √ , √ , . . . is the ra-tio of distances between the n th and 1st nearest-neighbor dis-tance on the undistorted square lattice, R = ( R HS + R LS ) / δ = ( R HS − R LS ) / R HS ( R LS ) is the average distance be-tween the centers of nearest neighbor molecules in the HS(LS) phases, and the pseudospin degrees of freedom are σ i = −
1) if the i th molecule is HS (LS). This, and closely relatedmodels, have been widely studied previously and providedmany insights into the physics of SCO materials. We make the symmetric breathing mode approximation(SBMA), that is, we assume that for all nearest neighbors,
FIG. 2. Examples of Hofmann frameworks topologically equivalentto the square lattice. The metal centres are marked by yellow circlesand in-plane ligands are marked by black lines. The elastic interac-tions between the metal centres, k n , are marked for the n th nearestneighbours, where n ∈ { , ,..., } . The spring constants ( k n ) areexpected to be typically positive for through-bond interactions (bluelines and circles) and typically negative for through-space interac-tions (grey lines and circles) . r i , j = x , and that the topology of the lattice is not altered by thechanges in the spin-states. In this approximation the Hamilto-nian (1) becomes an Ising-Husimi-Temperley model in a lon-gitudinal field: H ≈ ∑ n = J n ∑ (cid:104) i , j (cid:105) n σ i σ j − J ∞ N ∑ i , j σ i σ j + ∑ i ∆ G i σ i , (2)where, J n = k n η n δ is the pseudospin-pseudospin interactionbetween the n th nearest-neighbors, J ∞ = δ ∑ mn = ( k n z n η n ) isthe long-range strain, ∆ G = ∆ G HS − ∆ G LS = ∆ H − T ∆ S is thefree energy difference between the HS and LS states of the i th molecule, z n is the coordination number for n th nearestneighbors, and N is the number of sites.The spring constants should be typically positive forthrough-bond interactions, but will often be negative forthrough space interactions. This has profound consequencesfor the long-range order observed in different materials. How-ever, the possible range of spring constants is constrained bythe fact that the lattice described by Hamiltonian (1) must bestable, i.e., we must have ∂ H / ∂ x ∝ J ∞ > ∑ n k n z n η n > T =
0. That is, we consider all possiblestates with a unit cell no larger than 4 × T = T > N = ×
60 lattice with periodic boundary conditions.We perform three types of Monte Carlo simulations: heat-ing, cooling and parallel tempering. For cooling (heating)runs we initialize the calculation in a random configuration(the T = The loopalgorithm works by creating a Monte Carlo update that mapsbetween two states with the same local correlations. For thesmectic phases in this paper, the loop update is always a line offlipped spins. The worm algorithm works by creating a MonteCarlo update that removes two defects (violations of the localcorrelations) of the opposite polarization by annihilating themwith one another.The spin flip, loop, and worm algorithms are integrated intoa single update. We choose a single spin flip update with prob-ability 1 − / N and a combined loop/worm update with prob-ability 1 / N . The latter picks a chain of neighboring sites andcalculates the enthalpy change if the spin-states of all of thesesites are changed, starting with a pair of sites and expandingfrom the end of the chain. This process terminates when ei-ther an energetically favorable update is generated or if theBoltzmann probability for the move is less than a randomlyselected number, or if the chain terminates on itself creatinga chain and a loop. The Boltzmann probability of the loopupdate occurring is then checked.The spin-state structure factor is given by S ( q ) = N ∑ i , j (cid:104) σ i σ j (cid:105) e − i q · ( r i − r j ) , (3) FIG. 3. An example of the calculated fraction of high spins, n HS , ina four-step transition. Results are for ∆ S = ∆ H = . k δ , k > k = k / k = − . k , k =
0, and k = . k . Thered and blue lines indicate heating and cooling respectively. where r i is the position of the i th site. The average was eval-uated over 100 configurations, each separated by N MonteCarlo steps, during a parallel tempering calculation.
III. RESULTSA. Disordered phases due to competing orders
We have previously shown that a wide range of long-rangeantiferroelastic ordered phases can occur in an Ising-Husimi-Temperley model of SCO materials, Eq. (2). These phasesshow long-range order in both directions, which we willhenceforth refer to as spin-state crystals. In this paper wewill demonstrate that smectic phases, which are ordered inone spatial direction but disordered in the perpendicular di-rection, are also natural and that, at non-zero temperatures –where all experiments are carried out, achieving these phasesdoes not require fine tuning.An example of a four-step SCO transition with intermediateplateaus at n HS = .
75, 0 . .
25 is shown in Fig. 3. Wehave previously reported several examples of similar transi-tions in this model, in those cases the plateaus are associatedwith spin-state crystals. However, when we plot snapshots,Fig. 4, of the Monte Carlo calculations reported in Fig. 3 wefind long-range order in only one direction in the intermediateplateaus (labeled W H , W L , and X). This is highly reminiscentof smectic phases in liquid crystals.To better understand the origin of these smectic phases weshow a slice of the T = ; those discussed in this paper are shown in Fig. 1.Note that for the parameters studied in Fig. 5 the Néel andstripe phases are degenerate (have the same free energy). Thisis an accidental degeneracy due to the parameters chosen.This fine tuning is not necessary for the physics described FIG. 4. Truncated snapshots of the spin-state smectic W L and Xphases from our Monte Carlo simulations. LS (HS) sites are indi-cated by blue (yellow) circles. below, rather we have chosen to plot this slice of the (eight-dimensional) phase diagram as it simplifies the discussion be-low. In the regions marked “Néel, stripe” in Fig. 5 we find thateither the Néel or the stripe phase is the lowest energy state;since the domain walls between the Néel and stripe phasescarry a non-zero energy cost coexistence does not occur at T =
0. However, in Monte Carlo calculations at finite temper-atures, on cooling into these regions of the phase diagram, wesometimes find domains of both phases.At T = T = FIG. 5. A slice of the T = n HS . The solid black lines are first ordertransitions. Here k > k = k / k =
0, and k = . k ; at the points marked X (on the boundary between the Néel,stripe and J phases) or W (on the boundaries between the Cand I phases) in Fig. 5. By definition, at the phase boundarythe energies of the two competing phases are equal. There-fore, the energetic cost of forming a mixture is set only by theenergy required to form domain walls. It has been demon-strated previously that, in the Ising model at T =
0, if one finetunes to certain points along the phase boundary, then one canfind a disordered state with a residual entropy. Experimen-tally such fine tuning would be difficult to achieve in SCOmaterials. Nevertheless, we will see below that at non-zerotemperatures the entropy of mixing becomes important andthe single point is expanded to a phase spanning a broad re-gion in parameter space.We plot a finite temperature slice of the phase diagram inFig. 6. All of the spin-state crystal phases that are shown inthe T = ∆ S , which, following Wajnflasz and Pick, we have absorbedinto the Hamiltonian [Eq. (2)], which means that sweeping T varies ∆ G . But, there is another contribution to the entropy– the configurational entropy associated with the pattern ofIsing pseduospins ( { σ i } ). This has a dramatic effect on thephase diagram, driving the emergence of semectic W L , W H ,and X phases, which are not found at zero temperature exceptprecisely at first order lines. In these phases we see preciselythe same smectic order that we found in the W L , W H , and Xplateaus respectively (Figs. 3 and 4).A simple demonstration that the configurational entropyplays a key role in stabilizing spin-state smectics can be givenby setting ∆ S =
0. This is no longer a realistic model ofan SCO material, but nevertheless provides insight into thephysics at play in them. In this limit varying the temperaturedoes not change the free energy difference between HS andLS molecules ( ∆ G = ∆ H ) and the configurational entropy isthe only game in town. Given a set of elastic interactions one FIG. 6. Finite temperature phase diagram for ∆ S = k > k = k / k =
0, and k = . k . Shading represents the frac-tion of high spins, n HS ∼ χ T , where χ is the magnetic susceptibility,calculated via parallel tempering. The solid black line represents thefirst order transitions, dashed black lines mark second order transi-tions, and the dotted line appears to be a crossover although could bea second order transition.FIG. 7. Finite temperature phase diagram for ∆ S = ∆ H = k > k = k / k = − . k , k =
0, and k = . k . Shadingrepresents the calculated heat capacity, c V , which has a maximum atthe second order transition between the spin-state crystal phases andthe spin-state smectic. The solid black lines are first order transi-tions and dashed black lines (overlaying the blue shaded regions) aresecond order transitions. can, by an appropriate choice of ∆ H , ensure that, for example,only phases with, say, n HS = / × × × × . It can be seen from the snapshotthat this rule results in a state that is ordered in one direction(along the x -axis in Fig. 4) with alternating HS-LS-HS-LS-. . . . At first sight this is surprising as there is no long-rangeorder in the one-dimensional Ising model for T > How-ever, our findings are not inconsistent with this as we are deal-ing with a two dimensional system that is partially disordered.Thus, not only a 3 × × L column,where L is the linear dimension of the system. This state nec-essarily breaks rotational symmetry as only one direction canbe long-ranged ordered. Thus, the competition between threecrystalline spin-state orders naturally gives rise to long-rangesmectic-X order.The W phases have an important difference from the Xphase – they do not occur near tricritical points. Instead theyoccur near the ordinary critical points between the I and Cphases. The I and C phases are both characterized by contain-ing minority spin-states that are not nearest or next nearestneighbors – neighboring minority spin-states are third nearestneighbors in the C phase and third and forth nearest neighborsin the I phase. Thus, domain walls between C phases dis-placed by one lattice constant from one another induce localconfigurations with I order and vice versa . Therefore, the Wphase only needs two phases to be enthalpically competitivein order for the configurational entropy to stabilize a smecticphase.The W phases are specified by two rules: (a) no minorityspin-state may have other minority spin-states as first or sec-ond nearest neighbors and (b) n HS = .
25 (0.75) for W L (W H ).As with the X phase this leads to long-range order in one di-rection (the y -direction in Fig. 4), it also leads to partial orderin the perpendicular direction, with an alternating pattern ofcolumns with equal numbers of both spin-states (ordered in FIG. 8. Pseudo-spin structure factors, S σσ for the W L and X disor-dered antiferroelastic spin-states observed in our Monte Carlo simu-lations. Labels have the same meaning as 4. The structure factor forW L (X) shows distinct lines of Bragg peaks at q y ± π ( q x = ± π ) indi-cating the existence of long-range order in one direction and disorderin the other. The direction which is ordered is not related to whichphases occurs, but is a spontaneous breaking of rotational symme-try. The calculations were done at a) ∆ H = k δ and b) ∆ H = ∆ S = k B ( ) , k B T = . k > k = k / k = − . k , k = k = . k . S σσ .Depending on how rotational symmetry is broken Bragg peaksoccur along either ( q x , q y ) = ( ± π , q ) or ( q x , q y ) = ( q , ± π ) , forarbitary q , providing direct evidence for smectic order. Thefeaturelessness of the structure factor in one direction indi-cates that there is little or no correlation between the antifer-roelastically ordered columns of spin-states. The pseudo-spinstructure factors can be directly mapped onto the spin scatter-ing structure factor, and is closely related to the positional structure factor and is hence directly measurable by neutronor x-ray scattering experiments.The rules for the W and X phases allow us to make aPauling-like estimate of the residual entropy in these smecticphases. In the X phase there is long-range order in one di-rection, but in the perpendicular direction each may either bealigned or misaligned with its nearest neighbor (say the col-umn to its left in Fig. 4). Thus, neglecting boundary effects,on an N = L × L lattice there are 2 L possible microstates, and S residual = L k B ln 2. Similarly, in the W phases there is long-range order in one direction, but in the perpendicular directioneach column containing both spin states may either be alignedor misaligned with its second nearest neighbor. Thus, neglect-ing boundary effects, there are 2 L / possible microstates, and S residual = L k B ln 2. Thus, in the thermodynamic limit the spe-cific entropy ( s = S / N ) vanishes. This means that the W and Xphases are less disordered than spin-state ices (which are con-crete examples of spin-state liquid phases) where s is a con-stant in the thermodynamic limit [ s Pauling = ( / ) k B ln 2 on thekagome lattice and k B ln ( / ) on the pyrocholre lattice], but more disordered than spin-state crystals where s =
0. Thisquantifies the degree to which these spin-state smectic phasesare intermediate between spin-state liquid and spin-state crys-tals.
IV. CONCLUSIONS
In summary, we have shown that the competition betweenspin-state crystalline orders can give rise to partially disor-dered spin-state smectic phases in SCO materials. We haveidentified two spin-state smectics in a well-known model ofSCO materials. Spin-state smectics display long-range spin-state order in one direction, but the spin-states are disorderedin the perpendicular direction. Thus, the spin-state smecticsare more ordered than spin-state liquids (such as spin-stateice), but less ordered than spin-state crystals (where there islong-range order in both directions, e.g., the phases shownin Fig. 1). These different degrees of disorder are quantifiedby the residual entropies of the phases: the residual entropyof a spin-state crystal is independent of the system size; theresidual entropy of a spin-state smectic scales with the lineardimension of the system; and the residual entropy of a spin-state ice scales with the total system size.We showed that smectic phases give rise to plateaus in thefraction of HS molecules, and hence χ T , similar to the in-termediate plateaus that are caused by spin-state crystals. Wehave shown that spin-state smectics cause lines of Bragg peaksin the pseudo-spin structure factor, which should provide adefinitive test for their existence. ACKNOWLEDGMENTS
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