Critical Behavior of La0.75Sr0.25Mno3

PHYSICAL REVIEW B, VOLUME 65, 214424

Critical behavior of La0.75Sr0.25MnO3

D. Kim, B. L. Zink, and F. Hellman Department of Physics, University of California, San Diego, La Jolla, California 92093

J. M. D. Coey Physics Department, Trinity College, Dublin 2, Ireland ͑Received 17 November 2000; published 7 June 2002͒ Magnetization, susceptibility, and speciﬁc heat measurements were made on a single crystal of

La0.75Sr0.25MnO3. The ferromagnetic-to-paramagnetic phase transition was found at 346 K and four critical exponents were measured as: ␣ϭ0.05Ϯ0.07, ␤ϭ0.40Ϯ0.02, ␥ϭ1.27Ϯ0.06, and ␦ϭ4.12Ϯ0.33. The values of critical exponents are all between mean-ﬁeld values and three-dimensional-͑3D͒-Ising-model values. The scaling behavior is well obeyed for all measurements, and the associated exponent relations are well satisﬁed, validating the critical analysis. Although the cubic crystal structure of this material makes the 3D Heisenberg the expected model, uniaxial magnetic anisotropy arising from the shape of the sample causes the 3D Ising model to be important within the experimental temperature range.

DOI: 10.1103/PhysRevB.65.214424 PACS number͑s͒: 75.40.Cx, 75.30.Vn, 64.60.Fr

Ϫ␣ I. INTRODUCTION Cϩ͑T͒ϳA͉t͉ /␣, tϾ0,

͑ Ϫ␣Ј Interest in the perovskite manganites A1ϪxBxMnO3 A, CϪ͑T͒ϳAЈ͉t͉ /␣Ј, tϽ0, trivalent rare earth; B, divalent metal͒ has revived since the 1 ͑ ͒ ͒ϳ ͉ ͉␤ Ͻ discovery of colossal magnetoresistance CMR near the M S͑T B t , t 0, Curie temperature T for x near 0.3. Some manganites are C Ϫ␥ also believed to possess strongly spin-split bands, leading to ␹͑T͒ϳC͉t͉ , tϾ0, what is called half-metallicity ͑complete spin polarization of 1/␦ the charge carriers at the Fermi level͒. The magnetotransport M͑H͒ϳDH , tϭ0. behavior and the possibility of applications utilizing CMR or Here A, AЈ, B, C, and D are constants. Scaling requires that half-metallicity motivated intensive studies on the underly- ␣ϭ␣Ј. In addition, there are two exponent relations that 2,3 ing physics. Above TC , CMR perovskite manganites are limit the number of independent variables to two, paramagnetic insulators ͑insulating in the sense that the temperature dependence of the resistivity d␳/dTϽ0͒.As ␣ϩ2␤ϩ␥ϭ2, the temperature is decreased below TC , they become ferromagnetic metals. Among the perovskite manganites, ␥ϭ␤͑␦Ϫ1͒. ͑ ͒ ϳ La1ϪxSrxMnO3 LSMO has the highest TC 370 K. Re- Separate studies of magnetization and specific heat critical placement of La by Pr, Nd, and Sm or of Sr by Ca, Ba, and phenomena have been reported, both on single crystals and 3,8–10 Pb results in distortions of MnO6 octahedra in the pseudocu- on polycrystalline samples. These have indicated incon- bic structure ͑hence increased electron-phonon coupling͒, sistent results, with magnetization critical exponents varying lower TC , and higher resistivity by several orders of fairly widely. For characterization of critical phenomena, 4 magnitude. The lower-TC materials are reported to exhibit high-quality samples with a large grain size and preferably a phase separation of metallic and insulating regions below single crystal are needed to avoid smearing of TC . For epi- 5 6 TC , leading to a possible first-order phase transition at TC . taxial thin-film samples, the small signal makes it hard to get La1ϪxSrxMnO3 however is the most metallic in the mangan- data of the quality needed for critical analysis, given the ite family and therefore has the most itinerant electrons. signal-to-noise ratio. Also as T gets near TC , the correlation There is a report of phase separation on the surface of LSMO length can be of the order of the thickness of the sample, thus by scanning-tunnel-microscope experiment, but other bulk switching from three-dimensional ͑3D͒ to 2D behavior. A 7 experiments indicate no phase separation. single crystal of La0.7Sr0.3MnO3 was measured by Ghosh For a second-order phase transition near the critical tem- et al., who found ␤ϭ0.37Ϯ0.04, ␥ϭ1.22Ϯ0.03, and ␦ perature, where a correlation length can be defined, the spe- ϭ4.25Ϯ0.2, fit with a reduced temperature t range of 0.002 ϵ ϭ 3 cific heat, spontaneous magnetization ͓M S M(H 0)͔, and to 0.03. Mohan et al. measured a polycrystalline sample of ␤ ͉ ץ ץ␹ϵ initial magnetic susceptibility ( M/ H Hϭ0) show La0.8Sr0.2MnO3 and found significantly different values: power-law dependence on the reduced temperature, tϵ(T ϭ0.50Ϯ0.02, ␥ϭ1.08Ϯ0.03, and ␦ϭ3.13Ϯ0.2 ͑all much Ϫ ͒ ͉ ͉Ͻ 9 TC)/TC . Also at TC , M has a power-law dependence closer to mean-field values with t 0.008. It is not clear if on H, the differences reflect the polycrystalline nature, the different

composition, or the different t range. Neutron-scattering studies on a single crystal of La0.7Sr0.3MnO3 yielded a value of ␤ϭ0.295 but the t range used was somewhat far from the transition.10 Lin et al. measured magnetization exponents of ␤ϭ ␦ϭ La0.7Sr0.3MnO3 and found 0.31 and 5.1, fit in a non- specified t range, and a specific heat critical exponent ␣, which is anomalously large ͑ϩ0.16͒ and inconsistent with their magnetization data.8 For these reasons, we report here a study of the specific heat, magnetic susceptibility, and spontaneous magnetization on a single sample of single crystal La0.75Sr0.25MnO3 very near TC .

II. EXPERIMENT A. Speciﬁc-heat measurement

Single crystal La0.75Sr0.25MnO3 was made by a floating- zone process. Viret et al. previously made small-angle, po- larized neutron-scattering measurements11 on a sample from the same batch. The sample used in the specific heat measurement is a thin flake of approximate dimensions 0.2 ϫ0.5ϫ0.5 mm3, weighing 212 ␮g. To make this measurement, we used a microcalorimeter originally developed by FIG. 1. Side view of microcalorimetry device and sample show- ͑ ͒ our group to measure the specific heat of thin films of thick- ing steps used to measure C P(T). a Measure C P1; includes Au ness 200–400 nm weighing 2–20 ␮g.12 For this experiment, conduction layer, Pt heater and thermometer, and SiN membrane. ͑ ͒ ␮ Ϫ b Measure C P2 after attaching 72 gofIn.C P2 C P1 we modified the design of the microcalorimeter to allow us ϭ ␮ ͑ ͒ heat capacity of 72- g In. c Measure C P3 after attaching an- to measure a bulk sample attached by indium. The modified ␮ ␮ Ϫ ␮ other 96 g of In and 212 gofLa0.75Sr0.25MnO3. C P3 C P2 calorimetry device consists of a 1.5- m-thick amorphous ϭ ␮ ϩ ␮ ϫ 2 heat capacity of 96 gofIn 212 gofLa0.75Sr0.25MnO3. Sub- silicon-nitride membrane of 5 5mm lateral size. The cen- ͓ϭ * Ϫ ͔ ϫ 2 ␮ traction of In contribution 96/72 (C P2 C P 1) gives C P of tral 2.5 2.5-mm sample area is covered by a 1.5- m-thick La Sr MnO evaporated gold film, which has high thermal conductivity. 0.75 0.25 3 Indium was used to thermally and physically attach the ␮ ␮ sample to the gold layer. Due to the high thermal conductiv- capacity of 96 g of In and 212 gofLa0.75Sr0.25MnO3. ity of the gold layer and the poor thermal conductivity of the Since we have already measured the specific heat of In in ␮ amorphous SiN membrane, the sample area is linked to the C P2, the heat capacity of 96 g of In can be subtracted, Si frame only by a very weak thermal link and is effectively yielding the heat capacity of La0.75Sr0.25MnO3. isothermal on the experimental time scale. On the front side of the membrane, platinum thermometers and heaters were B. Magnetization measurements evaporated and patterned by photolithography. The relax- ation method was used to measure the heat capacity. Refer- Another thin flake ͑0.2ϫ0.7ϫ0.8 mm3; 750 ␮g͒ of single ence 12 describes the experiment in more detail. crystal La0.75Sr0.25MnO3, grown and cut from the same batch Figure 1 shows a schematic of the series of experiments as the sample used in the specific heat experiment, was used performed to measure the specific heat C P of the single crys- for magnetization measurements. Room-temperature torque ͑ ͒ ͑ ͒ tal La0.75Sr0.25MnO3. Figures 1 a –1 c are side views of the magnetometry was first used to characterize the magnetic micro-calorimetry device in three different stages of the ex- anisotropy. Shape anisotropy was found to dominate the in- ͑ ͒ periment. Fig. 1 a shows the C P1 measurement; this in- trinsic cubic anisotropy. The sample has an easy plane of ͑ ϭ cludes the heat capacity of the Au layer, the Pt heater and magnetization perpendicular hard axis with KH 5.1 thermometer, and the silicon-nitride membrane. Seventy-two ϫ105 dyne/cm2͒, and within the plane, the easy axis is along micrograms of In is then bonded to the gold layer, using a the longest body diagonal of the slightly rhombic shape ϭ ϫ 4 2 hot plate to melt In, after which C P2 was measured as shown (KE 8.1 10 dyne/cm ). in Fig. 1͑b͒. The heat capacity of the 72 ␮g of In was deter- A Quantum Design dc superconducting quantum interfer- mined by subtracting C P1 from C P2. The single-crystal ence device magnetometer was used to measure the magne- manganite sample did not stick well to the In ͑even above the tization. All measurements were made with the magnetic In melting temperature͒ without applying pressure, which field applied along the easy axis. Figure 2 shows the magne- has a danger of breaking the SiN membrane. Instead we tization versus applied field ͓M(Ha)͔ curves at different pressed the sample into an additional 96 ␮g of In. This temperatures. At very low temperatures, the saturated mag- sample/In piece can then be attached easily to the existing In netization reached the expected spin-only magnetization of ͑ ͒ ␮ on the device, once again using a hot plate. Figure 1 c 3.75 B /Mn atom. The reduction of magnetization due to shows the device with additional In and bulk sample, mea- spin canting reported elsewhere for a polycrystalline 13 ͑ sured as C P3. Subtracting C P2 from C P3 yields the heat sample was not observed. The coercivity is very low less

214424-2 CRITICAL BEHAVIOR OF La0.75Sr0.25MnO3 PHYSICAL REVIEW B 65 214424

FIG. 2. M vs Ha hysteresis curves along the easy axis at differ- FIG. 3. Specific heat of La0.75Sr0.25MnO3 obtained following ent temperatures. Low-field susceptibility gives demagnetization procedure outlined in Fig. 1. Sample mass was 212 ␮g. 1 mole factor D, due to small shape anisotropy of slab-shaped sample. weighs 229 g. Heat capacity of the sample was of the order of 50% of the total measured heat capacity C P3. than 10 Oe͒ and no hysteresis was observed at fields Ͼ100 Oe, most likely due to the sample quality and single-crystal neity of this single-crystal sample. After an initial specific ϭ nature. The steep rise in M near Ha 0 is due to domain-wall heat measurement to locate the peak temperature, we varied movement. M(Ha) is linear until M is very near the sponta- the temperature near the peak by small amounts to get many data points near the critical temperature to do scaling analy- neous magnetization M S for each temperature. From the sis. C (T) near T is composed of a smooth background slope of the M(Ha) curve near zero field, the demagnetiza- P C ϭ plus a singular part, C , due to the magnetic phase transition factor D was calculated, D Ha /M. The demagnetiza- S ϭ tion. The smooth background comes primarily from phonons tion field HD DM is small compared to the high applied fields ͑less than 4% at 10 000 Oe͒, and becomes more sig- and free electrons, but can also include a non-singular con- nificant at low applied fields reaching 50% around 400 Oe. tribution to the magnetic specific heat. As is commonly done, In order to minimize the error arising from the determination we assumed that this background in the small region near TC between 321 and 354 K is linear in temperature. We used the of D, only data with HD less than 50% of Ha were used in least-squares method to fit C (T) data. Following the the critical analysis below. Throughout this article, Ha stands P ϭ Ϫ ϭ Ϫ method of Kornblit and Ahlers,18 the functions we used to fit for the applied field and H Ha HD Ha DM is the cor- rected magnetic field after subtraction of the demagnetization C P(T) in this region are field. ϩ͑ ͒ϭ͑ ␣͉͒ ͉Ϫ␣ϩ ϩ Ͼ C P t A/ t B Ct for t 0,

Ϫ͑ ͒ϭ͑ Ј ␣͉͒ ͉Ϫ␣ϩ ϩ Ͻ III. RESULTS C P t A / t B Ct for t 0. A. Speciﬁc heat Here C represents the slope of the underlying physical

Figure 3 shows the specific heat of La0.75Sr0.25MnO3 be- smooth background. C has to be the same above and below tween 80 and 430 K. Above the transition temperature, the TC because the smooth background should not have a dis- absolute value of the specific heat is just above the Dulong- continuous derivative at TC . B consists of the specific heat Petit limit, 125 J/mol K 1 mole of La0.75Sr0.25MnO3 weighs of the underlying smooth background at TC (B1) plus a con- 229 g and has one acoustic mode and four optical modes per stant B2 . B2 is used to fit the critical region in conjunction 18 ϭ ϩ formula unit. Since the phonon contribution is dominant at with a simple power-law term. Therefore, B B1 B2 . ϩ this high temperature, almost all phonon modes must be ex- B1 Ct is the physical smooth background containing the cited to account for this large value of the specific heat, i.e., noncritical contributions of phonons, electrons, and magnetic Debye and Einstein temperatures less than 400 K. In addition excitations. The alternative approach, which is often used, is to the phonon contribution, there will be an electronic spe- to fit the physical smooth background through TC using the 1 Ϫ1 cific heat ␥Tϳ3JK mol and a positive dilation term, measured C P(T) far from TC or using high fields to suppress Ϫ ϳ Ϫ1 Ϫ1 ϩ Ϫ (C P CV) 0.4JK mol , due to the expansion of the the fluctuations. In this case, a different form for C P and C P sample. These two contributions are less than 3% of the total is used. The merits of these two approaches have been dis- ␥ 18 C P , based on low-temperature determinations of cussed by Kornblit and Ahlers. It is certain that the same Ϫ2 Ϫ1 ϳ3–6mJK mol ͑Refs. 4 and 14͒ and estimates of value of B1 should be used for both above and below TC , Ϫ (C P CV) from thermal-expansion measurements of LSMO because the smooth background has to be continuous at TC . ͑ ͒ 16,17 Ref. 15 and bulk modulus of La1.8Sr0.2CuO4. If we assume the continuity of total specific heat C P(T)at ␭ ␣Ͻ ͑ ͒ The most obvious feature in C P(T) is the -shaped peak the critical temperature when 0 cusp , then the same ͑ ͒ located near 347 K associated with the phase transition. The value of B2 hence the same value of B above and below TC sharpness of the peak indicates the uniformity and homoge- is necessary. Note that in the case of ␣Ͻ0, B is larger than

214424-3 D. KIM, B. L. ZINK, F. HELLMAN, AND J. M. D. COEY PHYSICAL REVIEW B 65 214424

Ϫ ϳϪ B1 125. The value of C is large compared to the expected background slope of ϳ50. Forcing the background to fit the apparent slope of 50 changes ␣ to 0.06 with a slightly worse fit; this value of ␣ is within the range we find below for ␣, hence the difference is not significant. Defining the singular part of the specific heat, ͒ϭ ͒Ϫ ϩ ͒ϭ ␣͉͒ ͉Ϫ␣ CS͑t C P͑t ͑B Ct ͑A/ t , ͉ ͉ Ͻ log10 CS vs log10 t will yield two straight lines for t 0 and tϾ0 with the slope of Ϫ␣. Figure 4 shows the log-log plot with the parameters mentioned above. The critical exponent ␣ was 0.05. ␹2 contour plots were used to determine the range of ␣ within the two standard-deviation contour for 95% confidence levels. From this, the range of ␣ was determined as ͑Ϫ0.022, 0.11͒. The reduced temperature range for ͉ ͉ Ͻ Ͼ Ϫ Ϫ FIG. 4. log10(CS)vslog10( t ) fit for both t 0 and t 0. this determination is Ϫ7.3ϫ10 2ϽtϽϪ3.6ϫ10 4. ␣ Ͼ ϭ␣Ј Ͻ ϭ Ϫ␣ (t 0) (t 0) 0.05. The slope is . The amplitude ratio A/AЈ is also a universal quantity with ϩ large variation between different universality classes. The C P(T), and B Ct lies above the specific heat curve al- value 0.81 in Table I is the result of the best fit described ϩ though of course B1 Ctlies below. Whether B2 is continu- above. The range of A/AЈ is obtained using the same two ␣Ͼ ous for 0 is not as obvious. Kornblit and Alhers dis- standard-deviation contour for ␹2. The values for A/AЈ ͑and cussed this issue in their work and chose to constrain it to be ͒ ␣ 18 the error bars are anticorrelated to the values of , e.g., for continuous. We have followed this method and also have ␣ϭϪ0.022, A/AЈϭ1.1 and for ␣ϭϩ0.12, A/AЈϭ0.63. constrained it to be continuous. We assumed ␣(tϾ0) ϭ␣Ј(tϾ0), adopting scaling theory. B and C were then var- B. Magnetization ied over a wide range and we adjusted TC for each set of parameters ͑B, C͒ to satisfy ␣ϭ␣Ј. We defined ␹2 as We used a modified Arrot plot scheme to determine the critical exponents ␤ and ␦, as reviewed by Kaul.20 The scal- ͓C Ϫ͕͑A/␣͉͒t ͉Ϫ␣ϩBϩCt ͖͔2 ing equation of state ␹2ϭ P i i ͚ 2 Ͼ ͑0.005C ͒ 1/␥ 1/␤ ti 0 P ͑H/M ͒ ϭatϩbM Ϫ␣Ј ͓C Ϫ͕͑AЈ/␣Ј͉͒t ͉ ϩBϩCt ͖͔2 causes isothermal curves of M(H) data to fall into a set of ϩ P i i ͚ 2 , 1/␤ 1/␥ ϭ ͑ ͒ parallel straight lines in a plot of M vs (H/M) if the ti 0 0.005C P correct values of ␤ and ␥ are chosen. For the mean-field ␹2 ␤ϭ ␥ϭ where ti denotes each experimental reduced temperature. values of 0.5 and 1, this is the well-known Arrot- is a measure of the quality of the fit. In calculating ␹2, each Kouvel plot, but departures from mean-field values necessi- 2 data point had a weight of 1/(0.005C P) , where we esti- tate the use of this modified Arrot plot. Note that the inter- ␹ 1/␥ mated 0.5% error in C P(T) data. Absolute accuracy in C P is cepts of the isotherms on the x and y axes are (1/ ) for Ϯ Ͼ 1/␤ Ͼ estimated to be 2%, dominated by uncertainty in the ad- t 0 and M S for t 0, respectively. The isothermal line that ϭ denda subtraction and in the absolute temperature scale of passes through the origin is the critical isotherm at T TC . the commercially calibrated thermometers. Relative preci- To find the correct values of ␤ and ␥, an initial choice of ␤ sion is estimated at 0.5% and is dominated by noise. We used and ␥ is made, yielding quasi-straight lines in the modified the grid-search method19 to get the best fit with the lowest Arrot plot. From these initial values of ␤ and ␥, linear fits to ␹2 ϭ . The parameters with the best fit were B 5.26, C the isotherms are made to get the intercepts giving M S(T) ϭ ϭ ␹ 147.55, and TC 347.22 K. Assuming the smooth physi- and (T), and an initial value of TC was determined from ϳ ϭ cally relevant background B1 130 from Fig. 3, B2 B the isotherm that passes through the origin. From these

TABLE I. Comparison of measured critical exponents with different theoretical models.

␣␤␥␦ A/AЈ

This work 0.05Ϯ0.07 0.40Ϯ0.02 1.27Ϯ0.06 4.12Ϯ0.33 0.81ϯ0.18/0.29a t range ͑this work͒͑0.000 36, 0.073͒͑0.000 006, 0.0004͒͑0.0006, 0.006͒ N/A MF theory 0 ͑discontinuity͒ 0.5 1.0 3.0 3D Ising model 0.11 0.325 1.24 4.82 0.524 3D XY model Ϫ0.007 0.333 1.34 1.03 3D Heisenberg model Ϫ0.115 0.365 1.336 4.80 1.521

aValues for ␣ and A/AЈ are correlated. A/AЈϭ0.63 when ␣ϭ0.12; A/AЈϭ1.1 when ␣ϭϪ0.022.

214424-4 CRITICAL BEHAVIOR OF La0.75Sr0.25MnO3 PHYSICAL REVIEW B 65 214424

FIG. 5. Modiﬁed Arrot plot after 30 iterations for the conver- gence of ␤ and ␥. Isothermal curves fall into a set of parallel ␹ straight lines. Intercepts give M S(T) and 1/ (T).

␤ M S(t) data, a new value of is obtained from a ln͓MS(t)͔ vs ln(͉t͉) plot by fitting the data to a straight line, the slope of ͑ ͒ which is 1/␤. Similarly a new value of ␥ is obtained from the FIG. 6. a M S vs T, obtained from X intercepts of Fig. 5. Inset shows ln(M )vsln(͉t͉); slope gives ␤ϭ0.40Ϯ0.02. ͑b͒ 1/␹ vs T, ln(1/␹) vs ln(t) plot. These new values of ␤ and ␥ are then S obtained from Y intercepts of Fig. 5. Inset shows ln(1/␹) vs ln(͉t͉); used to make a new modified Arrot plot. From this new slope gives ␥ϭ1.27Ϯ0.06. modified Arrot plot, improved values of TC , M S(T), and ␹ ␤ ␥ (T) are obtained, yielding better values of and .By ␤ ͑␤ϩ␥͒ ␤ ␥ M͑t,H͒/t ϭ f Ϯ͓H/t ͔, iterating this method, the values of TC , , and converged to stable values. Figure 5 is the final result of these iterations Ͻ Ͻ ␤ for 400 Oe Ha 10 000 Oe. As previously discussed, fields where f ϩ is for tϾ0, and f Ϫ is for tϽ0. By plotting M/t vs below 400 Oe were not included because the uncertainty in (␤ϩ␥) H/t , all data points below TC are expected to fall on the demagnetization correction becomes large. The isotherms f Ϫ , whereas the data points above TC will be on f ϩ . Figure in Fig. 5 are quite parallel with slopes within 1% of each 8 shows such a scaling plot both on a linear scale, which other. TC was determined from Fig. 5 to be 345.6 K by emphasizes the high-field data and on a log scale, which finding the isotherm passing through the origin. This value of emphasizes the low-field data. Indeed all data points over the ϳ ͑Ͻ ͒ TC differs by 1.5 K 0.5% from the value obtained from entire range of the variables fall on two branches of curves specific-heat measurement, due most likely to slightly differ- depending on the sign of t. ent thermometer calibrations for each experimental setup Finally, the two predicted exponent relations were since the absolute value of commercial thermometers is not checked. ␣ϩ2␤ϩ␥ϭ2 ͑theory͒, 2.12Ϯ0.13 ͑experiment͒ is calibrated better than 1%. ͑The relative values of temperature, required for the critical analysis, are however far better ͒ than 1%. From the x and y intercepts of Fig. 5, M S(T) and 1/␹(T) were obtained as shown in Figs. 6͑a͒ and 6͑b͒. The insets show the ln-ln plots used to get the final values of ␤ and ␥. Error bars come from the deviation of the least- squares fit analysis. The value of ␤ϭ0.40Ϯ0.02 is near the 3D Heisenberg ferromagnet value of 0.37. The value of ␥ ϭ1.27Ϯ0.06 is near but smaller than the 3D Heisenberg value 1.39. The reduced-temperature range used for these fits is 6ϫ10Ϫ4 –6ϫ10Ϫ3(tϾ0) and 6ϫ10Ϫ6 –4ϫ10Ϫ3(tϽ0). The M(H) curve at the critical temperature, TC ϭ345.6 K, is plotted in Fig. 7. From the ln-ln plot shown in the inset, the critical exponent ␦ϭ4.12, significantly smaller than the 3D Heisenberg value of 4.8 but larger than the mean-field value of 3.0. ␦ determined from neighboring temperatures, 345.4 and 345.8 K, were 4.09 and 4.45, respectively. From this the error of ␦ was set as ␦ϭ4.12Ϯ0.33. FIG. 7. Critical isothermal curve of M(H), taken from Fig. 5 As a further test, the static-scaling hypothesis predicts that zero intercept, which gives TC . Inset shows ln(M)vsln(H) plot at ␦ϭ Ϯ M(t,H) is a universal function of t and H, TC ; slope gives 4.12 0.2.

214424-5 D. KIM, B. L. ZINK, F. HELLMAN, AND J. M. D. COEY PHYSICAL REVIEW B 65 214424

XY. Likewise, the crossover from 3D XY to 3D Ising was ϳ ϭ ϫ 4 2 estimated as tx2 0.004 by using KE 4.1 10 dyne/cm instead of KH . tx1 is within the experimental temperature ␣ range for critical exponent . tx2 is within the experimental temperature range for critical exponents ␣ and ␥ ͑see Table I͒. Therefore the anisotropy plays a big role in the critical analysis and we do not expect to see pure 3D Heisenberg behavior within our experimental temperature range. Fur- thermore, the amplitude ratio A/AЈ in Table I is closer to the 3D Ising ͑or 3D XY͒ model than the 3D Heisenberg model. All our results point towards the 3D Ising model rather than the 3D Heisenberg model. There are many similarities between manganites and high-TC superconductors. The short correlation length is an example. In the high-TC superconductors, there was an

␤ ␤ϩ␥ analysis of crossover effects on the heat capacity of FIG. 8. Scaling plot of M/t vs H/t( ) with ␤, ␥, ␦, and T 21 C YBa2Cu3O7Ϫx from the mean-field to the 3D XY model. values from Figs. 5–7. Different symbols represent different tem- They estimated the temperature at which the fluctuation con- peratures. Linear plot emphasizes the difference in high H, whereas tribution to C P becomes as significant as the mean field. This log plot emphasizes the difference in low H. ϭ ␲2 ⌬ ␰3 2 Ginzburg criterion, tG (1/32 )(kB / C 0) was estimated Ϫ3 ␰ ϭ to be of the order of 10 , using 0 10 Å. If this is also the satisfied within experimental error. The second relation ␥ case with LSMO, then we can expect the crossover from the Ϫ␤ ␦Ϫ ϭ ͑ ͒ Ϯ ͑ ͒ ( 1) 0 theory , 0.02 0.07 experiment is very mean-field to 3D Ising model since tG is of the same order as well satisfied. tx above. Our results are consistent with this analysis. In other words, the temperature region suitably describable by Ͻ Ͻ 3D Heisenberg model can be estimated as tG t tx . Be- IV. DISCUSSION cause tG is of the same order as tx due to the short correlation length and large anisotropy, essentially we see a cross- The critical exponents for this experiment and theoretical over from the mean-field model directly to the 3D Ising values for mean field, 3D Heisenberg, 3D XY, 3D Ising model, skipping 3D Heisenberg. In this view, the mean-field model are listed in Table I. The critical exponents ␣, ␤, ␥, behavior reported by Mohan et al. can also be understood. If and ␦ obtained in the present experiment are all between we assume that the average crystallite size of their polycrys- 3D-Ising-model and mean-field values. The values we find talline sample is 500 Å, then as the temperature gets closer to are completely consistent with those reported by Ghosh et al. ϳ ␰ϭ␰ ͉ ͉Ϫv TC ,att 0.001 the correlation length, 0 t , would on a single-crystal sample, and significantly different than exceed the grain size, therefore the fluctuation effects will be those found by Mohan et al. The mean-field-like values replaced by mean-field behavior again. Assuming the domain found by Mohan et al. suggested that long-range coupling size of our single-crystal sample is 5 ␮m as observed by could be important in the manganites; the values found here SEM,11 we have to get to tϳ10Ϫ6 to observe similar effects. and by Ghosh et al. suggest that shorter-range coupling is a Since our reduced-temperature range is larger than this, fluc- better model. The deviations from 3D Heisenberg could be tuation effects are still important for the single-crystal due to uniaxial anisotropy causing a crossover effect to the sample. 3D Ising model with the final critical behavior visible only at still lower reduced temperatures. We estimated the temperature range at which crossover occurs from the 3D Heisenberg V. CONCLUSIONS model to the 3D XY model. The criterion for this crossover We have made magnetization and specific-heat measure- ␰3ϳ temperature was KH kBTC , i.e., the hard-axis anisotropy ments on a small ͑sub-mg͒ single crystal of energy within the correlated volume is on the order of the La0.75Sr0.25MnO3. From the specific heat above 400 K, all ␰ϭ␰ ͉ ͉Ϫv thermal energy. 0 t , where the correlation-length ex- phonon modes appear to be excited, implying Debye and ponent vϭ(2Ϫ␣)/dϭ0.65 is predicted by an exponent re- Einstein temperatures of less than 400 K. The phase transi- lation with the experimental value for ␣ϭ0.05. The shape- tion is very sharp, suggesting the homogeneity of the sample ϭ anisotropy energy at reduced temperature t, KH(t) KH with no phase separation. Good homogeneity was possible ͑ ͒ 2 2͑ ͒ 300 K M S(t)/M S 300 K was calculated using the sponta- partly due to the small size of the sample we looked at. The neous magnetization from the experiment. We assumed the critical exponent ␣ was found to be ϩ0.05Ϯ0.07. At low ␰ ϳ zero-temperature correlation length 0 5 Å, from the neu- temperatures, the saturation magnetization reached the spin- 11 ␰ ␮ tron scattering data of the same sample. The error in 0 only value of 3.75 B , indicating no canting of the spins. ϭ5ϳ10 Å is not as important as the error in v. We did not Critical exponents derived from modified Arrot plots ␤ use their value of vϭ0.4 because this value of v yields ␣ ϭ0.40Ϯ0.02, ␥ϭ1.27Ϯ0.06, and ␦ϭ4.12Ϯ0.2 are consis- ϭ0.8, too large compared to our experiment. As a result, we tent with the equalities expected from the scaling hypothesis, ϳ get tx1 0.02 for the crossover from 3D Heisenberg to 3D also supporting a second-order phase transition in this mate-

214424-6 CRITICAL BEHAVIOR OF La0.75Sr0.25MnO3 PHYSICAL REVIEW B 65 214424 rial. All critical exponent values are between the mean-field ACKNOWLEDGMENTS and 3D Ising model. The 3D Ising model is more relevant than 3D Heisenberg due to anisotropy. Fluctuations are im- We would like to thank M. Salamon and R. Fisch for their portant when crystallite and magnetic domain sizes are large. insights and discussions on the critical-exponent theory, M. Further studies of the crossover analysis will be useful to Viret and E. N. Abarra for their help with the experiments, confirm this view. Although no simple model explanation is and the NSF for support. We thank Alex Revcolevschii for possible, the exponent relations are well satisfied. kindly providing the crystals used in the experiment.

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