Electricity and Magnetism,Volume 1: Third Edition,Electricity and Magnetism,Volume 2: Third Edition,by B.I. Bleaney and B. Bleaney

During the last decade the term ‘spin glass’ has become prominent in the literature on magnetism. It refers to magnetic alloys where the spins on the impurities become locked or frozen into random orientations below a characteristic temperature T ₀. In this article the properties of spin glasses are described with particular reference to the two archetypal examples AuFe and CuMn. Interest in spin glasses was mainly stimulated by some a.c. susceptibility measurements which showed sharp, cusp-like peaks, accurately defining T ₀ and suggesting that some type of phase transition was occurring. The Mossbauer effect and the anomalous Hall effect also showed clear features at T ₀ supporting this viewpoint. But measurements of the electrical resistivity and ‘specific heat’, here usually meaning the molar heat capacity, also the remanence, magnetic hysteresis and time-dependent effects observed in spin glasses were difficult to reconcile with a phase transition approach. This article discusses the results obtained from the very wide variety of experimental techniques which have been used to investigate spin glasses, and also deals with some of the important theoretical concepts which have arisen out of these studies. Then follows a short account of the many systems which have been found to exhibit spin glass behaviour and which suggest that it is a widespread magnetic state of matter. Lastly, an example is given which shows that some of the ideas of spin glasses are applicable to problems outside the sphere of magnetic alloys.     

The spin dynamics of the cluster‐glass Fe0.25Mg0.75TiO3 probed by the muon spin relaxation method

A cluster‐glass (CG) Fe_0.25Mg_0.75TiO₃ is investigated by muon spin relaxation measurements. It is found that the local magnetic fields at the muon stopping site are established at temperatures five times as high as the freezing temperature T_f. It is also found that the electron spins fluctuate even below T_f. The critical region around T_f in Fe_0.25Mg_0.75TiO₃ is extremely wide in contrast to that of ordinary spin‐glass (SG) systems. It is clearly demonstrated that the microscopic and dynamical properties of CG are remarkably different from those of SG. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical mean field studies of the site disordered spin glassCuMn

Numerical mean field studies ofCuMn alloys have been performed for different concentration and external magnetic field strengths in a wide temperature range using a cube with 12*12*12 lattice points and periodic boundary conditions. The calculations include only RKKY interaction based on hyperfine field measurements by Slichter and coworkers. They result in realistic values for the spin glass transition temperturesT _f . The temperature was varied from the paramagnetic regime down to the deeply frozen state far belowT _f . The exchange and hyperfine field distributions have been calculated. The copper NMR spectra of the main line and the satellites are shown. Moreover, the temperature behavior of the order parameter, the probe magnetization, the magnetic heat capacity and the NMR line width are given.     

Magnetoresistance of the semiconducting spinelide Cu0.625Ga0.375Cr2Se4 having a low-temperature long-range magnetic order-spin glass transition

A study is reported of the dependence of magnetoresistance Δρ/ρ on the square of magnetization σ ² of the semiconducting spinelide Cu_0.625Ga_0.375Cr₂Se₄, which exhibits a low-temperature transition from long-range magnetic order (LRMO) to the spin glass (SG) state in strong magnetic fields. It is shown that at the freezing temperature T _f the Δρ/ρ(σ ²) relations change their slope, and that below T _f this slope is about one half that for T>T f. This finding, together with the earlier observation that the freezing temperature does not depend on the frequency of the ac magnetic field in which it was measured, suggests that the spin-glass phase consists of spins of individual Cr³⁺ ions, and that the SG-LRMO crossover is a phase transition. Fiz. Tverd. Tela (St. Petersburg) 40, 315–317 (February 1998)     

Model glasses coupled to two different heat baths

In a p-spin interaction spherical spin-glass model both the spins and the couplings are allowed to change with time. The spins are coupled to a heat bath with temperature T, while the coupling constants are coupled to a bath having temperature T_J. In an adiabatic limit (where relaxation time of the couplings is much larger that of the spins) we construct a generalized two-temperature thermodynamics. It involves entropies of the spins and the coupling constants. The application for spin-glass systems leads to a standard replica theory with a non-vanishing number of replicas, n=T/T _J . For p>2 there occur at low temperatures two different glassy phases, depending on the value of n. The obtained first-order transitions have positive latent heat, and positive discontinuity of the total entropy. This is an essentially non-equilibrium effect. The dynamical phase transition exists only for n<1. For p=2 correlation of the disorder (leading to a non-zero n) removes the known marginal stability of the spin glass phase. If the observation time is very large there occurs no finite-temperature spin glass phase. In this case there are analogies with the non-equilibrium (aging) dynamics. A generalized fluctuation-dissipation relation is derived. Received 12 July 1999 and Received in final form 8 December 1999     

Theoretical study of the magnetism in the incommensurate phase of TiOCl

Going beyond a recently proposed microscopic model [D. Mastrogiuseppe, A. Dobry, arXiv:0810.3018v1] for the incommensurate transition in the spin-Peierls TiOX (X=Cl, Br) compounds, in the present work we start by studying the thermodynamics of the model with XY spins and adiabatic phonons. We find that the system enters an incommensurate phase by a first order transition at a low temperature T_c1. At a higher temperature T_c2 a continuous transition to a uniform phase is found. Furthermore, we study the magnetism in the incommensurate phase by density matrix renormalization group (DMRG) calculations on a one-dimensional Heisenberg model where the exchange is modulated by the incommensurate atomic position pattern. When the wave vector q of the modulation is near π, we find local magnetized zones (LMZ) in which spins abandon their singlets as a result of the domain walls induced by the modulated distortion. When q moves far away enough from π, the LMZ disappear and the system develops incommensurate magnetic correlations induced by the structure. We discuss the relevance of this result regarding previous and future experiments in TiOCl.     

Interplays ofμSR,susceptibility and neutron studies on dilute-alloy spin glasses

Static spin polarization of Fe moments in a spin glassAuFe, determined by zero-fieldSR, is compared to an ac-susceptibility measurement below the cusp temperatureT _g, and a rather uniform amplitude of the static polarization is pointed out for frozen spins. Completely random orientation of frozen spins is revealed by neutron scattering in a dilute 1%CuMn, and an importance of comparing results of these different methods is demonstrated.     

Effects of Fe doping on ac susceptibility of Pr0.75Na0.25MnO3

Ac susceptibility at low temperatures of Pr_0.75Na_0.25Mn_1-xFe_xO₃ (0 ≤ x ≤ 0.30) is investigated. The peak value of the real component of ac susceptibility χ' at the freezing temperature T_f is suppressed with the increasing frequency. The peak value of χ' shows a linear relation between T_f and the logarithm of the frequency ω. The normalized slope P = ΔT_f/T_fΔlgω, which is much lower than canonical insulating spin glass systems in which 0.06 ≤ P ≤ 0.08. The peak value of the imaginary component of the ac susceptibility χ' at T_f for the x = 0, 0.02, 0.30 samples increases with increasing frequency, suggesting a cluster glass ground state with a coexistence of charge-ordered phase and correlated ferromagnetic clusters in spin glass matrix. The peak value of χ' at T_f for the x = 0.10 sample decreases with increasing frequency, suggesting a phase separation ground state. The peak value of χ' at T_f for the x = 0.05 sample decreases with increasing frequency for ω ≤ 52 Hz and increases subsequently till 701 Hz, and then decreases with further increasing frequency for ω ≥ 1501 Hz. This complex behaviour is ascribed to the competition between the effects of large and little ferromagnetic clusters in the sample. The ground state of x = 0.05 sample is a transition state from cluster glass to phase separation.     

The influence of fixed spins on the thermodynamic behavior of an Ising ferromagnet

We study the thermodynamic behavior of a ferromagnetic Ising system on a Bethe lattice in the presence of given boundary conditions. More specifically, we study the interface of the system when the spins on half of the surface are fixed opposite to the spins on the other half. We find an interface width that remains finite in the whole range (0,T _c ), a feature due to the special topology of the Bethe lattice. We also study the case where the spin on a certain lattice site belonging to a domain is fixed in a direction opposite to the domain magnetization at all temperaturesT _c . We obtain the influence of that spin on the local magnetization, and we find that the fixed spin nucleates a local domain that extends over a distance of only a few lattice sites from it at all temperaturesT _c .     

Phonon-assisted spin diffusion in solids

The spin flip-flop transition rate is calculated for the case of spectral spin diffusion within a system of dipolarly coupled spins in a solid where the lattice vibrations are present. Long-wavelength acoustic phonons time-modulate the interspin distance r_ij and enhance the transition rate via the change of the 1/r³_ij term in the coupling dipolar Hamiltonian. The phonon-assisted spin diffusion rate is calculated by the golden rule in the Debye approximation of the phonon density of states. The coupling of the spins to the phonons introduces temperature dependence into the transition rate, in contrast to the spin diffusion in a rigid lattice, where the rate is temperature-independent. The direct (one-phonon absorption or emission) processes introduce a linear temperature dependence into the rate at temperatures not too close to T = 0. Two-phonon processes introduce a more complicated temperature dependence that again becomes simple analytical for temperatures higher than the Debye temperature, where the rate is proportional to T², and in the limit T → 0, where the rate varies as T⁷. Raman processes (one-phonon absorption and another phonon emission) dominate by far the phonon-assisted spin flip-flop transitions.     

Phase transition from spin glass to long-range magnetic order in semiconductor spinels CuCr1.5+0.5x Sb0.5−0.5x S4 (x=0.34 and 0.4)

Reentrant behavior to a spin glass state is discovered in the solid solution system CuCr_{1.5+0.5x Sb x0.5−0.5 x S 4} with x=0.34 and 0.4. The spin-glass transition temperature _{T f} determined from the kink in the temperature dependence of the initial susceptibility in an alternating magnetic field depends on the measurement frequency ω. It is shown that the frequency dependence _{T f} (ω) is a power-law function 1/ω=(1/ω₀)[T _f /( _{T f} − _T *)]^zv with _zv =7.7 for both compositions. For the composition with _x =0.34 a maximum is observed near _T * in the temperature dependence of the resistivity. These facts indicate that the transition from the spin glass to long-range magnetic order is a phase transition. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 4, 265–269 (25 August 1996)     

Spin-glass-like ordering in the spinel ZnFe2O4 ferrite

In the present study, spin-glass-like ordering has been observed in the spinel ZnFe₂O₄ ferrite. Field cooled (FC) and zero-field cooled (ZFC) DC magnetizations display divergence at low temperature, which indicates a frozen state with the freezing temperature of T_f=21 K. Frequency dependence of AC susceptibility measurement was performed on the sample. It shows a peak at around T_f, with the peak position shifting as a function of driving frequency, indicating a spin-glass-like transition of the sample. The sample shows a typical spin-glass behavior with a manifestation of non-equilibrium dynamics of the spin glass, such as aging, rejuvenation and memory effects. These experimental findings indicate that Zn-ferrite exhibits a spin-glass-like phase at low temperature and it is not canted antiferromagnetic.     

Concentration dependence of the transition temperature in spin glasses

The transition temperature T_f of randomly distributed spins interacting via the RKKY interaction (spin glasses) has been calculated numerically within the spherical approximation. The dependence of T_f on the concentration c of magnetic impurities and on the damping of the interaction due to a finite mean free path of the conduction electrons is investigated for 10^?5 < c < 10^?1.     

Nuclear-spin-lattice relaxation in a bilayer quantum Hall system

We examined the electron spin degree of freedom around the total Landau-level filling factor ν=1 in a bilayer system via nuclear spins. In a balanced bilayer system, nuclear-spin-lattice relaxation rate 1/T₁, which probes low-energy electron spin fluctuations, increases gradually as the system is driven from the quantum Hall (QH) state through a phase transition to the compressible state. This result demonstrates that the electron spin degree of freedom is not frozen either in the QH or compressible states. Furthermore, as the density difference between the two layers is increased from balanced bilayer to monolayer configurations, 1/T₁ around ν=1 shows a rapid yet smooth increase. This suggests that pseudospin textures around the bilayer ν=1 system evolves continuously into the spin texture for the monolayer system.     

Mössbauer study on the magnetic field-induced insulator-to-metal transition in perovskite Eu0.6Sr0.4MnO3

We have investigated the spin dynamics of a distorted perovskite Eu_0.6Sr_0.4MnO₃ by means of Mössbauer spectroscopy. Below 70 K the exchange interaction grows gradually, and below 42 K the spins turn into a cluster glass state. The magnetic field-induced insulator-to-metal (IM) transition at low temperature is a transition from cluster glass to ferromagnet. The induced metallic phase seems to be still in non-uniform electronic state. On the other hand, at 80 K, just above T _c of the induced ferromagnet, a metamagnetic transition was observed.     

Pore size distribution in porous glass: fractal dimension obtained by calorimetry

By differential Scanning Calorimetry (DSC), at low heating rate and using a technique of fractionation, we have measured the equilibrium DSC signal (heat flow) J _q ⁰ of two families of porous glass saturated with water. The shape of the DSC peak obtained by these techniques is dependent on the sizes distribution of the pores. For porous glass with large pore size distribution, obtained by sol-gel technology, we show that in the domain of ice melting, the heat flow J_q is related to the melting temperature depression of the solvent, ΔT _m , by the scaling law: J _q ⁰∼ΔT _m ^{- (1 + D)}. We suggest that the exponent D is of the order of the fractal dimension of the backbone of the pore network and we discuss the influence of the variation of the melting enthalpy with the temperature on the value of this exponent. Similar D values were obtained from small angle neutron scattering and electronic energy transfer measurements on similar porous glass. The proposed scaling law is explained if one assumes that the pore size distribution is self similar. In porous glass obtained from mesomorphic copolymers, the pore size distribution is very sharp and therefore this law is not observed. One concludes that DSC, at low heating rate ( q? 2^°C/min) is the most rapid and less expensive method for determining the pore distribution and the fractal exponent of a porous material. Received 23 July 1999 and Received in final form 16 February 2001     

Quantum fluctuations in a scale-free network-connected Ising system

We study the effect of quantum fluctuations in an Ising spin system on a scale-free network of degree exponent γ>5 using a quantum Monte Carlo simulation technique. In our model, one can adjust the magnitude of the magnetic field perpendicular to the Ising spin direction and can therefore control the strength of quantum fluctuations for each spin. Our numerical analysis shows that quantum fluctuations reduce the transition temperature T_c of the ferromagnetic-paramagnetic phase transition. However, the phase transition belongs to the same mean-field type universality class both with and without the quantum fluctuations. We also study the role of hubs by turning on the quantum fluctuations exclusively at the nodes with the most links. When only a small number of hub spins fluctuate quantum mechanically, T_c decreases with increasing magnetic field until it saturates at high fields. This effect becomes stronger as the number of hub spins increases. In contrast, quantum fluctuations at the same number of “non-hub” spins do not affect T_c. This implies that the hubs play an important role in maintaining order in the whole network.     

Specific heat of the spin-Peierls compound CuGeO3

The specific heat of the quasi-onedimensional magnetic compound CuGeO₃ shows a sharp anomaly at the spin-Peierls transition temperatureT _sp . The experimental decrease of the magnetic specific heat belowT _sp indicates the presence of a spin gap as observed previously with inelastic neutron scattering. A magnetic field of 6T suppressesT _sp only slightly but reduces the spin gap by a factor of two.