Linear Chain Magnetism Richard L. Carlin University of Illinois, Chicago, IL 60680 The study of magnetic systems that display large amounts of order in but one or two dimensions has been one of the most active areas recently in solid state physics and chemistry. This article provides a hrief introduction to the subject, which is also called lower dimensional magnetism. We limit the discussion, of both theory and experimental results, to the coordination compounds of the iron series ions and to a lattice dimensionality of one. The physicist calls these materials insulators. A system with unpaired electron spins and in which the wins. . . or maenetic moments. have a random orientation with respect to one another, and are therefore uncorrelated, is called varamaenetic. When such a svstem has its maximum magneiicdiso;der, it also has its ma&num possibleentropy. This situation gives rise to the Curie law, whirh states that the magnetic susceptibility of a paramagnetic system varies inversely with absolute temperature but is independent of the applied field ( I ) . The susceptibility is derived from the force an applied field exerts on a sample and is a measure of the deeree bv whirh the measurine field tends to alien the spin s&m k i t h it. The energy f o f a spin in an external field H is E = (1/2)gp~H,where p~ is the Bohr magneton, a fundamental constant that equals lelhl4rmc. The quantity g is characteristicof eachsystem; it is 2.0023 for a free electron and is often close to 2 for many metal ions. When there is crystal fieldanisotropy, thevalue ofgmay vary withorientation, being gl when H is parallel t o the principal symmetry axis of the crystal and g, when the two are perpendicular. That the susceptibility of a sample can be anisotropic is the major reason why susceptibility measurements are best made on oriented sinele crvstal samoles. The magnetic moments on the m&l ions in a sample can interact with each other in several wavs. one of which is magnetic dipolar. This phenomenon vaAe$ with the magnitude of the moments and on the distance separatine them and is generally important only at temperatures of l%r 2 K and below. A more chemically interesting kind of interaction is called superexrhange, in which the electron spins on one metal ion interact with the spins on other metal ions via a minute transfer of spin density by means of the intervening ligands. Any magnetic interactions between the spins on the metal ions at most temperatures are small compared to thermal energies. As the temperature of a given system is decreased sufficiently, the interaction energy eventually becomessignifirant compared to the thermal energy, klrT. The situation is much like that of an ideal gas condensing to a liquid as the temperature is lowered. The interactions lead eventually to a cooperative phase transition, indicative of long-range magnetic order, at some temperature that is rharacteristic for each material. The entropy of the spin system also decreases as i t becomes ordered: The critkal temperature for long-range order is labeled T,, and substances which order in the liquid 4Hetemperature region are particularly amenable for further studies. The ordering most commonly found is antiferromagnetic, which means that the spins on adjoining sites tend to align oppositely and tend to cancel out each others' moments. Most real systems are three-dimensional (hereafter, 3-D), in that the magnetic interactions extend more or leas equally
in all three soatial directions. These are, of course. the commonest situtkions and are those found most oftenin nature (2). In addition to the maenetic suscevtibilitv, the meauurement of the specific heat provides an excellent way to probe magnetic phase transitions. That is hecause, in addition to the well-known Debye lattice specific heat, there is a contribution from the ordering of the unpaired electron spins as well. For exarnole. the soecific heat of H 3-D svstem. dotted as a function of tempeiature, usually exhibits a );-ihaped anomalv a t T,. The lattice soecific heat usuallv decreases dependence a t temperatures below 20 K and bewith a comes neelieible below 1K. This result makes the maenetic contribution a larger fraction of the whole, making iteasier to determine i t accurately, which in turn is one of the reasons that magnetism is so often studied a t low temperatures. The calculation of the physical properties of a 3-D maanetic system is exceedingiy difficult; however, there are in fact no exact solutions available for the thermodynamic properties (say, the specific heat as a function of temperature) of the 3-D magnetic phase transition. We know a lot about the nature of these systems, nevertheless, from numerical (computer) approximations and from the calibration of theorv. bv. exveriment. I t is easier to carrv out statistical mechanical calculations on problems in two dimensions or, better vet, one dimension. These are of course svnthetic systems, b u t t h e remarkable thing is that there a;e many physical realizations of these artificial models that indeed come close to representing this lower dimensional hehavior (I). The concept of short-range order arises in discussing the specific heat of a magnetic system, particularly the magnetic contribution to the specific heat which occurs above T,. More specifically, by short-range or lower dimensional order we mean that the magnetic interactions are taking place in less than three spatial dimensions. This gives rise to the accumulation of entrowv -.bv- a mametic svstem as it is cooled but remains ahove the long-range ordering temperature, T,. Though this occurs to a certain deeree with all mametic systems, the physical meaning impl6d here by the concept of lower dimensional marnetism is that maenetic ions are assumed to interact onlywith their n e a ~ e s t h e i ~ h b o in rs a particular spatial sense. The term "lower dimensional" is thereby restricted to magnetic interactions in one and two lattice dimensions. In that regard, i t is interesting to note that the discussion is therebv restricted to the oaramaenetic region, ahove T,. Let us stless the fact that'the ma2netic hehavior follows directlv from the structure of the various compounds. The discu&on centers on phenomena which are field-indeoendent: field-deoendent ohenomena are reviewed in detail elsewhere ( I ) . -
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One-Dlmenslonal or Linear Chaln Systems There are two separate and independent dimensionalities with which we are dealing here. The first, and more familiar, is the lattice or spatial dimensionality, being one, two, or three. A well-known example of a lower dimensional (but nonmagnetic) material is 2-D graphite. The less familiar dimensionalitv. that of soin. arises from the interaction of the spins ofthkieveral interacting ions. This may be defined by the expansion of what is called the exchange term in the Volume 68
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spin Hamiltonian (energy term) for a magnetic ion, which is usually written as:
H = - 2 a S i . S,
(1)
in which S; and S j are the spin operators of the interacting ions i and i, and J is called the exchanee constant. This measures the strength of the magnetic interaction. For our Durposes here, the descri~tionof magnetic linear chains, the Auimation over lattice sites i and I is restricted to adjacent sites in a line. The dot product may he expanded as Several approximations lead to the common models that are in use. Thus, when a equals 1 and B = 0, only the z component of spin is used, and that is called the Ising model; the soin dimensionalitv is one. When a is zero. the term in z is dropped and we are left with the terms in and y and thus the so-called XY Hamiltonian, of spin dimensionality two. When a = B = 1, the three terms enter equally, that is, the spin dimensionality is three. This is an isotropic model that is called the Heisenberg model. (There is no contradiction in having, for e x a m ~ l ea, substance which has a three-dimensionaiiattice that is nevertheless, say, an XY (two-dimensional) spin system.) Equation 2 can also be written more explicitly in the following way,
r
I t can be shown (3) that there is a relationship between theg values and theexchange constants; for a s p i n s = 112 system, the relationship is J&, = [g,,/g,]2. Thus the g-value anisotropy is a guide t o the exchange anisotropy. We can also equate g, with gll, and g,, with gl, when the crystal and molecular lattices are congruent. There are verv eood theories available that describe the thermodynamicp;operties of 1-D magnetic systems, a t least for S = 112: what mav he more surorisine is that there are extensive experimental data as well of metal ions linked into uniform chains. Some examples will be d~acussedbelow. The next point of interest, discovered long ago by Ising, is that an infinitelv lone 1-D svstem (the idealized lsine linear chain. with no interchain interactions) undergoes long-range mag: netic order only a t the temperature of absolute zero. This turns out to be true in general, no matter what degree of magnetic anisotropy is present. In reality, of course, longrange interchain interactions become more important as the temperature is lowered, and all known 1-D systems ultimately interact and undergo long-range order.
Figure 1. The -MCIz- chain ol CoC12.2py. The N atoms are from the pyridine rings. pound CoClzdpy, where py is pyridine, offers a good examole of an I s i n e s ~ i nS = 1/2 linear chain (1.2).The structure bf this subst&& is sketched in Figure 1, wheke i t will he seen to consist of a chain of cobalt atoms hrideed hv oairs of chlorine atoms. In trans position are found k e twb &ridine molecules, forming a distorted octahedral configuration. This basic structure is quite common and occupies a large part of the exposition of 1-D systems. The chloride ligands provide an efficient superexchange path, which means that they provide the medium by which the electron spins on one metal atom interact with those of the neiehbors. --.-~ - - ~ The more polarizable the ligand hetween two metal ions, generally, the stroneer the maenetic interactions between those metal ions. " (The nature of the intervening ligand is more important than the resultine metal-metal distance.) The nvridine molecules, on the other hand, act as insulation'and tend to isolate one chain from another. This is one of the maior lessons confirmed by the work on low-dimensional systeks, that the magnetic phenomena follow from the srructure of the materials. ~
~
~
~
-
/sing Systems
The Ising model is a very anisotropic one that assumes that the soins on a metal ion are s~ontaneouslveither uo or down w i t i respect to the principalaxis; that is,ihe electronic s ~ i n are s assumed t o have only z com~onents.That means that one requires t h a t g >>gl.The specific heat andsusceptihilities of an Ising S = 112 chain have been calculated exactly in zero field (the absence of an applied magnetic field.) The molar specific heat is found to he where R is the molar gas constant and JJke is again the exchange constant, in temperature units. t he curve is featureless, with a broad maximum. Equation 3 is an even function of the exchange constant J,,and thus the measured specific heat c of an Ising system does not allow one to distineuish ferromaenetic (J.> 0). . . with the snins . aliened paralk, from antiferromagndtic (J,< 01behavior. Cobaltrll) is the most anisotropic iron-series ion, whether in octahedral or tetrahedral coordination. It thus provides the best examples ot'the king model. For example, the com-
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Figure 2. Magnetic specific heat of CoCI..Zpy. The curve is the theoretical prediction for ths S = 1/2lsing chain calculated with J4ks = 9.5 K. Repro. dwed with permission from de Jongh. L. J.; Miedema. A. R. Adv. Phys. 1974, 23,1.
Figure 3. Entropy vs. (reduced) temperature curves for two cobalt salts. The large enhancement of the 1-0 character gained by substituting the pyridine molecules for the water molecules can be clearly seen from the large reduction in the amount of entropy galned below T,. Reproduced with permission from de Jongh, L. J.: Miedema, A. R. Adv. Phys. 1974, 23,1. Figure 4. Structure of CsCoCI,~2H20.Only one set of hydrogen atoms and hydrogen bonds is shown.
Susceptibility measurements indicate that the compound undergoes short-range (that is, intrachain) ferromagnetic interaction. The specific heat of the material is illustrated in Figure 2. The long-range (3-D) ordering, indicated by the sharp peak a t T, = 3.15 K, complicates the situation but the breadth of the full curve suggests that a large amount of (Isine) short-ranee order is nresent. . Now, the magnetlc entropy of the system at temperature Tisobtained bv intematine L d In Tfrom T = 0 to T, where c is the empirical magneticspecific heat contribution. The one-dimensional effect is illustrated nicely upon comparing in Figure 3, on a reduced temperature scale, the magnetic ~ entropies obtained from the specific heats of C o C l z . 2 ~and COCI.. 2 ~ ~As0with . all ordering phenomena, the e i r i i p y of R In (2.5 r 1) must heacquired as the thespin system ~ S =M system is cooled, and lower dimensional systems acquire much of this entropy ahove T,.Octahedral cohalt(I1) at low temperatures is an effective spin-1R iun, because of crystal field effects ( I ) . The hydrate has a structure similar to that of the nvridine adduct. but with water molecules in d a c e of the pGidine molecules. Replacement of water by &dine would be exnected to enhance the mametic 1-D character, for the larger pyridine molecules should cause an increased interchain separation. Anv hvdrogen bonding interactions between chains that provide additional sGperexchange oaths should also be diminished. Not only does T,drop from i7.2 K for the aquo complex to 3.15 K for the pyridine compound, but i t will be seen that 60% of the total entropy (R in 2) is obtained by the hydrate below T,, while only 15% of the entropy is acquired below T, for CoClz.2py. (The reader should also consider the system from another point of view: what fraction of the entropy is acquired ahove T,?) The exchange interaction within the chains is essentially the same in the two compounds (because both chains consist of the same r e ~ e a t i n eCoCb unit and thereby have the same superexchange path), and so these results must be ascribed pyrito a more ideal 1-D (that is. less 3-D) character in the . . dine adduct. The exchange constant is 9.5 K. This entropy argument is an important one. An ideal Ising chain with no interactions between chains, which would order only a t T, = OK, would necessarily acquire all its entropy of ordering above T,. This phenomenon is what we mean by short-range order. Another cobalt compound that exhibits a high degree of short range order is CsCoC134H20 (4), whose molecular structure is illustrated in Figure 4. Chains of chloride-
-
bridged cobalt atoms run along parallel to the a axis; the octahedra are cis-diaquotetrachlorocobalt(II)moieties. The cobalt systems described are excellent examples of the Ising model. Another is provided by [(CHhNH]CoC13. 2H70, which also contains the same chain of cobalt atoms bridged by two chlorides, with two water molecules in the trans nositions. The chains are infinitelv long: another chloride ion in the lattice bridges the chains a t tbe water molecules by hydrogen bonding (5). This contributes a small amount oftwo-dimensionai character to the system, while the trimethylammonium ions isolate the planes from one another. XY Systems
There are as yet only a few examples known of systems of anv lattice dimensionalitv which follow this model. Aeain. - . coGalt(11) provides the oniy experimental examples. Consider~h~eanisotropy in theg values, withg- >>g in thiscase, is the prerequisite for theapplicability of the XYmodel. l h e g values are defined once again with respect to the principal crystal field symmetry axis. Single crystals of CsnCoCld provide the most thoroughly studied example of the XY linear chain, S = 112 model (6). The system consists of discrete [CoCLI2- tetrahedra, yet the interactions along a CoCl.. .C1-Co superexchange path in one lattice direction dominate the magnetic interactions. The substance is described by the exchange constant of J , , l k ~ = -1.35 K. HelsenbergSystems
Manganese(I1) is the most isotropic iron series ion; its g values are usually isotropic and equal to 2. I t therefore provides the most ideal examples of the Heisenberg magnetic model system. The best example of an antiferromagnetic Heisenberg chain compound is [N(CH3)4]MnC13,familiarly called TMMC, part of the structure of which is illustrated in Figure 5. The crystal consists of chains of S = 512 manganese atoms bridged by three chloride ions; the closest distance between manganese ions in different chains is 0.915 um. The intrachain exchange constant IJllke is about 6.7 K, which is large enough to cause a broad peak in the magnetic heat capacity a t high temperatures. The tetramethylammonium ions serve to isolate the chains very well, so that the trmsition to long range order, characterized by T,, is below 1K. The important calculations for the antiferromagnetic HeiVolume
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Figure 6. The Bonner-Fisher ousceptlbllity f a an amifemmagnetic spin-112 linear chain, ploned using reduced unlts. h negative signs are intraduced
because J < 0.
F w r e 5. Sketch of the linear cha ns found in TMMC. [N(CH,l,]MnCI,. The octahedral environment aboffl me manganese atom 0s sllghtly dislwted, mrresponding to a lengthening along the chain.
senberg model are those of Bonner and Fisher (7). Since the model is an isotropic one, the susceptibility is calculated to be isotropic. A broad maximum in the susceptibility is predicted, as illustrated in Figure 6, where the (reduced) susceptibility is plotted against the (reduced) temperature. (This curve may he appreciated more easily by comparison with the Curie law for a paramagnet, for which x = C/T; thus, such a plot of x vs. T increases with decreasing temperature.) The theory has been applied to many sets of exnerimental data (1.2). b n the other hand,'the specific heat for the ferromagnetic Heisenhere chain is calculated to be rather flat and differs substantiahy from that for the antiferromagnetic chain. The lone example to date of data illustrating this situation is provided by measurements on [(CH3)3NH]CuClr2H20 (8). The structure of this svstem is similar t o that of the cobalt substance of this stoicbiometry described above, but the copper system is distorted. Copper always exhibits a smallgvd"e an-isotropy, yet it still furnishes many good examples of the Heisenberg model. The comnound CsMnCL-2H10 -~~~ " - is another eood examnle of the ~ e i s e n b e model r~ (9) and exhibits a stricture ohskrved freouentlv. That is. it has a u-chloro (sinele chloride) bridee between 2 s octahedra, as illustrated for h e cobalt analogue in Fieure 4. Suwe~tibilitvand s~ecificheat data have heen analyzed in termsof an exchange constant of -3.00 K. Two substances which are isostructural with CsMnClr2H20 are a-RbMnClr2Hz0 and CsMnBrr2H20. These salts hehave similarly to the Cs/C1 salt as expected, with exchange constants of -3 K and -2.6 K, respectively. ~~
Concluslons Linear chain magnet8 have been well characterized, and many experimental examples exist. The structural chemis-
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try of the systemdetermines, by andlarge, the lattice dimensionality, while the single-ion properties of the metal ion determine the character of the collective magnetic properties. This is an area of research in which ex~erimentalinorganic chemists have worked profitably, hand in hand, with theoreticalphysicists. Few simplifiedmodels, nomatter how outlandishthey might appear a t first, have not yielded to the synthetic chemist. Current work in the field centers on attempts to synthesize ferromagnetic chains, as well as other chains in which the exchange alternates with position down the chain. This would happen if, for some reason, the internuclear separation alternates. Mixed-metal ("bimetallic") systems, which should exhibit ferrimagnetism, are currently of especial interest. The latter phenomenon occurs when the metal atoms are linked antiferromagnetically but the spins are of unequal magnitude. The system then retains apermanent moment even in the ordered state. An e x a m ~ l ewould be provided by a substance in which, say, a manganese atom (S = 5/2) alternated remlarlv with a comer ion (S = 1/2) in the chain. Acknowledgment The author's research has been supported bv a succession of grants from the Solid State Chem&try ~ r o & a m Division , of Materials Research, National Science Foundation, most recently by DMR-8815798. 1. Carlin. R.L. Magnetochrmisfry: Springer: Berlin, 1986. 2. de Jongh, L. J.; Miedems,A. R.Ad". Phys. 1374.23.1. 3. Carlin, R.L.;de J0ngh.L. J. ChemRmr. 1986,86,659. 4. Herweijer, A.; de Jongr,W. J. M.: Botterman, A. C.: Bongaarts, A. I.M.;Cowen, J. A. Phys. Reu. 1972,85.4618. 5. Losee.D.B.;McEleemey, J.N.;Shankle,G.E.:Carlin,R.L.:Cresswell,P. J.;Robinaon, W. T. Phya. Re". 1973.88,2185. 6. Algra. H.A.;deJongo,L.J.;Blate,H.W. J.:Huiaksmp.W.J.:Csriin.R. L.Physica 1976,
