PHYSICAL REVIE%' B VOLUME 43, NUMBER 10 1 APRIL 1991 Static scaling in a short-range Ising spin glass K. Gunnarsson, P. Svedlindh, P. Nordblad, and L. Lundgren Uppsala University, Department of Technology, Box 534, S-751 21 Uppsala, Sweden H. Aruga and A. Ito Ochanomizu University, Department of Physics, Faculty of Science, Bunkyo ku, -Tokyo 112, Japan t, 'Received 22 May 1990; revised manuscript received 29 November 1990) The nonlinear ac susceptibility y'„I of the short-range Ising spin glass Feo &Mno, TiO, has been measured using a superconducting-quantum-interference-device magnetometer. The spin-glass tem- perature, Tg, and the critical exponent y were estimated from the temperature dependence of the quadratic field term of y'„&, yielding Tg =20.70 and y=4. 0. Static-scaling analyses, using different scaling equations, gave similar results. Using y and results from previous dynamic-scaling analyses, a number of critical exponents have been obtained through different scaling relations, e.g. , 6=8. 4 and v=1. 7. The results support the existence of a finite-temperature phase transition in a three- dimensional Ising spin glass. M +nl +0 The response to an applied time varying magnetic field, H =H0+h singlet, where co is the angular frequency, can be calculated using Eq. (1). For h/Ho «1, the ampli- tude of the ac component of the magnetization M„can be written: ~~ =+0k + 3+2H0h + 5+4H0h + 7y6H0h + (3) DifFerentiating with respect to the time varying field gives y'(co) =8M /t)h =y +3y H +5y H +7y H + (4) The question whether the spin-glass transition is a true phase transition or a gradual freezing of the magnetic moments has been discussed during two decades of research. To describe critical phenomena in magnetic systems, the Ising model has played a major part. Con- cerning Ising spin-glass systems, no analytical solution to the problem exists. However, utilizing Ising models, both Monte Carlo (MC) simulations' and high-temperature series expansions give strong indications of a phase tran- sition at a finite temperature. Therefore, comparative studies on spin-glass materials that closely image a 3D Is- ing model system are of utmost importance to shed realis- tic light on this problem. The magnetization in a spin-glass system may be ex- pressed in odd powers of the magnetizing field, H: M =yoH+y2H +y4H +y6H + If a phase transition occurs at a finite temperature, T, the linear susceptibility term y0 is nondivergent, whereas the cubic term g2 and higher-order terms diverge in the critical region. ' Thus, to investigate a possible critical behavior in a spin glass, the adequate quantity to measure is the nonlinear susceptibility, g„&, defined as follows: In accordance with Eq. (2), we define the nonlinear ac susceptibility y„'& as aM„ +nl +0 In the low-field limit (HO~0), yz always dominates y'„t. With increasing field the inAuence of higher-order terms becomes significant, and a deviation from an H0 depen- dence will be observed. As the temperature approaches T, this deviation begins at continuously lower fields. In this paper we present measurements of the nonlinear ac susceptibility of the short-range Ising spin-glass Fe0 5Mn0 5TiO3. , Static- and dynamic-scaling analyses in the vicinity of T yields good scaling behaviors with con- sistent values of the critical exponents and support the existence of a phase transition at a finite temperature. Comparisons are made with results from MC simulations on a 3D short-range Ising spin-glass system. ' The magnetic structure of Fe0 5Mn0 ~Ti03 is most con- veniently described by a hexagonal unit cell, with the spins aligned along the c axis. The spin-glass behavior is due to a random mixture of ferro and antiferromagnetic interactions within the hexagonal layers, causing bond disorder. The compound is regarded as a good model system for a 3D Ising spin glass. ' The sample used in this study was a single crystal in the shape of a rectangu- lar parallelepiped, 2X2X5 mm, with its long axis paral- lel to the c axis. The ac susceptibility measurements were made, using a superconducting-quantum-interference-device (SQUID) magnetometer, with a small ac field superimposed on a static field and both fields applied along the c axis of the crystal. The frequency of the ac field, co/2~, was either 1. 7 Hz or 0. 01 Hz. The ac magnetizing coil was wound directly onto the sample, which was glued on a sapphire rod and placed into a third-order gradiometer connected to the signal coil of the SQUID. In order to improve the resolution of the measurement, a second coil was wound 43 8199 1991 The American Physical Society