Magnetization-step studies of antiferromagnetic clusters and single ions: Exchange, anisotropy, and statistics

A magnetic cluster is a group of magnetic ions (“spins”) that interact with each other but which, to a good approximation, do not interact with other magnetic ions. Such clusters are responsible for many of the interesting and useful properties of a large number of molecular crystals, and of dilute magnetic materials below the percolation concentration. In a molecular crystal the magnetic clusters are usually all of one type. In a dilute magnetic material, on the other hand, many cluster types are present. The magnetization-step (MST) method is a relatively new form of spectroscopy for measuring intracluster magnetic interactions, mainly exchange constants and anisotropy parameters. In dilute magnetic materials this method also yields the relative populations of different cluster types. This review focuses on the principles and applications of the MST method to relatively small clusters, no more than a dozen spins or so. It covers only MSTs from spin clusters in which the dominant exchange interaction is antiferromagnetic (AF), and MSTs from isolated magnetic ions. Such MSTs are the result of changes of the magnetic ground state, caused by energy-level crossings in a magnetic field H. At a sufficiently low temperature, each change of the ground state leads to a MST. Magnetic clusters may be classified by size. The smallest is a “single,” consisting of one isolated magnetic ion. Next are “pairs” (dimers), followed by “triplets” (trimers), “quartets” (tetramers), etc. Although the classification by size is useful, clusters of the same size may have different intracluster interactions, and also different geometrical shapes. More detailed classifications of magnetic clusters are therefore also needed. A cluster “type” specifies both the size of the cluster and the set of all intracluster magnetic interactions which are nonzero. Different geometries of clusters of the same type correspond to different “configurations.” MSTs from isolated spins (singles) are discussed first. When subjected to certain types of single-ion anisotropy, e.g., uniaxial hard-axis anisotropy, singles give rise to MSTs. Examples of anisotropy parameters which were determined from such MSTs are presented. An interesting application of MSTs from singles is the determination of the populations of Jahn–Teller distortions which are energetically equivalent at H=0 but are inequivalent at finite H. For clusters larger than singles, the strongest intracluster interaction is usually the isotropic exchange. Using a model with one isotropic exchange constant J, predictions for MSTs from pairs, open and closed triplets, and the six possible types of quartets, are presented. Observations of some of these MSTs, and the exchange constants derived from them, are discussed. Recent studies of MSTs from AF rings in molecular crystals are summarized. The remainder of the review is devoted to a detailed discussion of MSTs in dilute magnetic materials, exemplified by the dilute magnetic semiconductors (DMSs). The theory for MSTs in these materials is based on various cluster models (each specifying the exchange constants that are included), and on the assumption of a random distribution of the magnetic ions. The latter assumption is needed for calculations of the populations of various cluster types. The simplest cluster model includes only the largest isotropic exchange constant, usually J1 between nearest neighbors (NNs). This J1 model accounted for much of the early MST data in Mn-based II–VI DMSs. These early data yielded values of J1, showed that the distribution of the Mn ions was random, and explained the difference between the apparent and true saturation values of the magnetization. Following these early successes the “pure” J1 model was improved in several ways: (1) Some effects of the weaker exchange interactions with distant neighbors (DNs) were treated approximately. (2) Weak anisotropies, and the Dzyaloshinski–Moriya interaction, were added to the model. (3) A spread in the values of J1, due to alloy disorder and/or a lower crystal symmetry, was included. (4) The possibility a nonrandom magnetic-ion distribution was considered, and methods of observing nonrandomness experimentally, and quantifying the degree of nonrandomness, were devised. (5) Cluster probabilities in molecular beam epitaxy (MBE)- grown quantum structures, particularly near interfaces, were considered. Experimental data relating to each of these improvements of the J1 model are presented. Very recent works focused on a direct determination, using MSTs, of the relatively small DN exchange constants. Most of these experiments on DNs required a magnetometer operating in a dilution refrigerator, near 20 mK. The data interpretations were based on cluster models with up to five exchange constants. These models involve hundreds of cluster types, even when clusters with more than four spins are excluded. Clusters with more than four spins were treated approximately. Elaborate computer programs for computing all cluster probabilities and energy levels were required. The results for the DN exchange constants Ji in Mn-based II–VI DMSs disagree with all previous theoretical predictions. Specifically, the next-nearest-neighbor exchange constant J2 is not the second-largest exchange constant. The distance dependence of the Ji is material dependent, unlike the universal behavior predicted by all theories which considered this issue. The experimental results are partially explained by the Yu–Lee and Wei–Zunger theories, which include the directional dependence of the exchange interaction in addition to the distance dependence. The directional dependence leads to a reduction of J2. Electronically accessible tables for cluster types and their probabilities are included as EPAPS. These tables are for all clusters with up to four spins, in both the fcc cation lattice and in the (ideal) hcp cation structure. For fcc the tables include 16 different cluster models with exchange interactions up to the fifth neighbor. For hcp, 64 cluster models with up to eight exchange constants (corresponding to interactions up to the fourth neighbor in fcc) are included. Tables for quintets in the special case of the NN model in fcc and hcp are also included.