Scaling behaviour of magnetic entropy change in bilayered manganites by two-variable polynomials fitting to magnetization
SScaling behaviour of magnetic entropychange in bilayered manganites bytwo-variable polynomials fitting tomagnetization ∗ Bao Xu a) † a) Key Laboratory of Magnetism and Magnetic Materials at Universities of Inner MongoliaAutonomous Region and Department of Physics Science and Technology, Baotou Teachers’College, Baotou, 014030
Based on the two-variable polynomial model of magnetization, magnetic entropy change of bilayered manganiteswith 327-structure and its scaling behaviour with respect to applied magnetic fields are investigated. It’s found thatthe Curie temperature, which is defined as the point at which the partial derivative of magnetization with respectto temperature reaches its maximum, is different from the temperature of peak magnetic entropy change. Thus amean-field model can not apply to this kind of manganites. In contrast to what has been found in manganites withthe 113-structure, the scaling behaviour at the Curie temperature in manganites with 327-structure is much differentfrom that at the temperature of peak magnetic entropy. It’s also found that the temperature dependence of the scalingexponent under weak fields is distinct from that under strong fields. This difference is attributed to an crossover fromone-step transition under weak fields to two-step transition under strong fields.
Keywords: scaling exponent, magnetic entropy, bilayered manganite
PACS:
For magnetic materials, both physical properties coupled with the magneticdegree of freedom and associated microscopic coupling mechanism can be re-vealed from magnetization data. Following are some examples: First of all, themagnetic transition points, and the type of effective exchange couplings undermolecular field approximation (ferromagnetic, antiferromagnetic, or ferrimag-netic type), can be determined from the temperature dependence of inversesusceptibility. The second example is to identify the order of magnetic transi-tions according to the Banerjee criterion, subsequently qualitatively assess thestrength of spin-lattice couplings, and eventually provide valuable informationfor the construction and test of different microscopic models. The third onementioned here is to estimate the grain distribution of polycrystalline, espe-cially nanocrystalline, samples through investigating the ratio of spontaneousmagnetization at finite temperature with respect to that at zero point. Bycomparing the resultant distribution with that obtained from X-ray diffraction(XRD), one can give clues to optimizing the chemical doping and also controllingthe preparation process. The fourth one is for first-order transition materials. ∗ Project supported by National Natural Science Foundation of China (Grant No. 61565013)and The Scientific Research Project of the Inner Mongolia Autonomous Region Colleges andUniversities (Grant No. NJZZ199). † Corresponding author. E-mail: [email protected] a r X i v : . [ c ond - m a t . s t r- e l ] D ec y measuring its first-order reversal curve (FORC), the contribution to magne-tization from irreversible rotations can be quantitatively analysed, and thereforeinformation about magnetocrystal anisotropy and associated lattice symmetry isprovided. The fifth but not the last one is to calculate magnetic entropy changefrom magnetization data and assess its potential value as a new magnetocaloricmaterial.In spite so much information is revealed by magnetization data, the anal-ysis results quite depend on the particular model and approximation that areemployed. This paper aims at studying the the scaling properties of magneticentropy change ∆ S H based on series approximation of magnetization. Themagnetic entropy change at temperature T and maximum applied field H isestimated as ∆ S H = (cid:90) H (cid:18) ∂M∂T (cid:19) H (cid:48) dH (cid:48) . (1)Its scaling exponent with respect to applied magnetic field n is defined as n = ∂ ln | ∆ S H | ∂ ln H (2)= H ∆ S H (cid:18) ∂M∂T (cid:19) H . (3)In deriving expressions (2) and (3), the magnetic entropy at H = 0 with fixed T is assumed to be a constant number. and the Maxwell relation (cid:0) ∂S H ∂H (cid:1) T = (cid:0) ∂M∂T (cid:1) H is used.To calculate exponent n , the model most often used is the approximatedversion of molecular field theory at M much smaller than its saturated value.Under this approximation, the equation of state can be expressed as [1 , HM = a ( T − T C ) + bM (4)and exponent n can be easily obtained as [3] n ( T ) = 2 M M − M s θ ( T ≤ T C ) + 2 H/MH/M + 2 bM θ ( T > T C ) (5)where, spontaneous magnetization M s equals [ a ( T C − T ) /b ] / at temperatureslower than T C ; θ ( A ) = 1 , A being true andfalse, respectively. Note that n = 1 , , T (cid:28) T C ( M approximatelyequal to M s ), T → T C ( M s approaching zero) and T (cid:29) T C ( M being a smallquantity), respectively [4] . The above approximation applies to materials withweak short-range correlations.When short-range correlations becomes important, the molecular field model(4) fails to give correct results in the vicinity of critical point T C . In this caseone can employ the semi-empirical Arrott-Noakes equation of state [5] (cid:18) HM (cid:19) /γ = a ( T − T C ) + bM /β , (6)with a , b , T C , β and γ are parameters to be determined. Near to the criticalpoint, β and γ correspond to critical exponents defined as M s ∝ ( T C − T ) β and2 ∂M∂H (cid:1) H → ∝ ( T − T C ) γ , respectively. It’s considered that the Arrott-Noakesequation is particularly suitable to deal with magnetization data in the vicinityof the critical point. From equation (6), it’s easy to obtain following relationsat T C (cid:18) ∂M∂T (cid:19) H = − ab βγβ + γ M − β ∝ H β − β + γ , (7)and ∆ S H = − ab − γ βγ β + γ − M γ − β ∝ H β + γ − β + γ . (8)Substituting (8) into (2), exponent n can be obtained as [6 , n ( T C ) = 1 + β − β + γ . (9)The amazing thing is that, by substituting (7), (8) into (3), exponent in (9) isagain obtained. By assigning γ = 1 and β = 1 /
2, equation (9) reduces to themolecular field result 2 /
3. It’s also noted that the scaling exponents of ∆ S H with respect to magnetic fields equals that of H (cid:0) ∂M∂T (cid:1) H .To determine n at T C , one can perform fitting to the field dependence ofpeak magnetic entropy ∆ S pk H ( H ). This method does not strictly distinguish T C from the temperature of peak magnetic entropy, T pk . For estimating exponent n within the whole measuring temperatures, it’s usually to utilize the definitionin (3). This paper not only uses above two methods but also apply (7) toestimating n from magnetization data.The paper is organized as follows: Section 2 provides the model and formulaused in this work; In Section 3 the numerical results and associated discussionscan be found. Conclusions are put into Section 4. Before doing the series estimation, make size transformations as follow: Ap-plied magnetic field H , temperature T and magnetization M are given as H = f H ( x ) = H min + x ( H max − H min ) ,T = f T ( y ) = T min + y ( T max − T min ) ,M = f M ( z ) = M min + z ( M max − M min ) , where, dimensionless variants x, y, z ∈ [0 ,
1] represent reduced magnetic field,temperature and magnetization. For brevity, the µ before H is omitted andthe applied fields are measured in Tesla.Giving up the symmetry restriction connecting magnetization and appliedfields, the reduced magnetization z can be represented as a series in x and yz = S (cid:88) t =0 c t h t ( x, y ) (10)3here h t ( x, y ) = 1 , x, y, x , xy, y , x , · · · and S denotes the maximum indexin above series expression. To determine coefficients c t , a straight method istwo-variable orthogonal polynomial fitting to experimental data. c t Expression (10) can be estimated by rewriting it in normalized orthogonalpolynomials P s ( x, y ) as ˆ z ( x, y ) = S (cid:88) t =0 b t P t ( x, y ) , (11)where, P t ( x, y ) is the t -th orthogonal polynomial and b t the corresponding co-efficient with subscripts t ≥ h t ( x, y ), the orthogonal polynomial P s ( x, y ) can be recursivelygenerated as P s ( x, y ) = a ss h s ( x, y ) + s − (cid:88) t =0 a st P t ( x, y ) . Coefficients a ss and a st are given by summing over N experimentally recordedvalues, ( x i , y i , z i ), as a ss = − (cid:34) N (cid:88) i =1 P ( x i , y i ) h s ( x i , y i ) (cid:35) − , (12) a st = − a ss N (cid:88) i =1 P t ( x i , y i ) h s ( x i , y i ) . (13)By minimizing the fitting error with a regularization term, characterized byparameter λ , as σ = 1 N N (cid:88) i =1 [ˆ z ( x i , y i ) − z i ] + λ (cid:2) ∇ ˆ z ( x i , y i ) (cid:3) , (14)coefficient b t is determined as b t = − λR t Q t + (cid:80) Ni =1 z i P t ( x i , y i )1 + λ ( Q t ) , (15)where, R t = t − (cid:88) r =0 b r Q r ,Q t = N (cid:88) i =1 ∇ P t ( x i , y i ) , with t ≥
0. After a st and b s being determined, the values of coefficient c t canbe readily computed from them. 4 .3 Spontaneous magnetization, partial derivative of mag-netization with respect to temperature, and magneticentropy change By expressing subscript t as t ( m, j ) = 12 m ( m + 1) + j (16)with m ≥ ≤ j ≤ m , reduced magnetization z in (10) can be expressedas z = p S (cid:88) p =0 z p x p (17)where z p = j p (cid:88) j =0 c t ( j + p,j ) y j , (18)and p S and j p are maximum indices of p and j , respectively. Spontaneous magnetization at reduced temperature y is expresses as M = f M ( z ) (19)with z = j (cid:88) j =0 c t ( j,j ) y j . (20)Other coefficients before the powers like x j with 1 ≤ j ≤ j p can be similarlyobtain. At fixed magnetic field H , the partial derivative of M with respect to T canbe estimated as ∂M∂T = A ∂z ( x, y ) ∂y = A S (cid:88) t =0 c t ∂h t ( x, y ) ∂y , (21)with A = ( M max − M min ) / ( T max − T min ) . (22)Representing t as in (16), the summation over t in (21) can be rewritten as S (cid:88) t =0 c t ( m,j ) ∂h t ( m,j ) ( x, y ) ∂y = 1 x k S (cid:88) k =1 (cid:15) k ( y ) x k (23)where, (cid:15) k ( y ) = j k (cid:88) j =1 c t ( k + j − ,j ) · j · y j − , (24)5 k = (cid:36)(cid:114) S + 2 k + 14 − k − (cid:37) , (25)and k S = (cid:36)(cid:114) S − − (cid:37) , (26)with (cid:98)·(cid:99) denoting the rounding down operation. In above derivations, indexreplacement m = k + j − The magnetic entropy change is calculated according to expression (1)∆ S H = A (cid:90) x ∂z ( x (cid:48) , y ) ∂y d x (cid:48) . (27)In above equation, the integral can be computed as (cid:90) x ∂z ( x (cid:48) , y ) ∂y d x (cid:48) = S (cid:88) t =0 c t (cid:90) x ∂h t ( x (cid:48) , y ) ∂y d x (cid:48) . Thus the magnetic entropy change can be estimated as∆ S M = A S (cid:88) t =0 c t (cid:90) x ∂h t ( x (cid:48) , y ) ∂y d x (cid:48) , (28)where the leading factor A is expressed as A = ( M max − M min )( H max − H min ) / ( T max − T min ) . At fixed reduced temperature y , equations (28) can be rewritten as S (cid:88) t =0 c t ( m,j ) (cid:90) x ∂h t ( m,j ) ( x (cid:48) , y ) ∂y d x (cid:48) = k S (cid:88) k =1 k (cid:15) k ( y ) x k (29)By comparing equation (23) and (29), it’s easy to see that T C determinedfrom the maximum magnitude of partial derivative (cid:12)(cid:12) (cid:0) ∂M∂T (cid:1) H (cid:12)(cid:12) max , generally dif-fers from T pk that is identified from the peak value of magnetic entropy change | ∆ S pk H | ; and the difference increases with the applied field. Actually, inspectionsof the recursive relations (36) and (37) give the following relation (cid:90) x ∂h t ( m,j ) ( x (cid:48) , y ) ∂y d x (cid:48) = xm − j + 1 ∂h t ( m,j ) ( x, y ) ∂y . (30) We next calculate exponent n . Substituting (21) and (28) into (3), exponent n can be expressed as n = A A f H ( x ) x (cid:80) St =0 c t x ∂h t ( x,y ) ∂y (cid:80) St =0 c t (cid:82) x ∂h t ( x (cid:48) ,y ) ∂y d x (cid:48) (31)6y fixing H min = 0, the leading factor in (31) equal to 1, we now reach theexpression of exponent n as n = (cid:15) ( y ) + (cid:15) ( y ) x + (cid:15) ( y ) x + · · · + (cid:15) k S ( y ) x k S − (cid:15) ( y ) + (cid:15) ( y ) x + (cid:15) ( y ) x + · · · + k S (cid:15) k S ( y ) x k S − (32)Note that n depends on the reduced temperature at fixed reduced field.For the exponent n in (5), it’s obvious that at weak magnetic fields, i.e., muchsmaller than saturation field and x being small quantity, the exponent n ≈
1. At T much higher than T C , the k = 1 term should be abandoned for the vanishingspontaneous magnetization, and exponent n approaches 2 at weak magneticfields. It’s noted that in the range of T (cid:28) T C or T (cid:29) T C , exponent n is notdependent on y , i.e., reduced temperature. When the applied field increases upto the saturation field, i.e., x →
1, exponent n is expressed as n = (cid:15) ( y ) + (cid:15) ( y ) + (cid:15) ( y ) + · · · + (cid:15) k S ( y ) (cid:15) ( y ) + (cid:15) ( y ) + (cid:15) ( y ) + · · · + k S (cid:15) k S ( y ) . (33) n at T C and T pk When exponent n at the transition point is considered, the situation be-comes complicated as T C generally differers from T pk . T C is determined in thiswork as the temperature where the maximum value of the partial derivative ofmagnetization with respect to temperature, (cid:12)(cid:12) (cid:0) ∂M∂T (cid:1) H (cid:12)(cid:12) max , occurs. Exponent n at T C can be determined as (cid:12)(cid:12)(cid:12) (cid:18) ∂M∂T (cid:19) H (cid:12)(cid:12)(cid:12) max ∝ H n − . (34) T pk is the temperature at which the peak value of magnetic entropy change,∆ S pk H , appears. Exponent n as T pk is determined as (cid:12)(cid:12) ∆ S pk H (cid:12)(cid:12) ∝ H n . (35)It needs stressing that both (34) and (35) are derived from the Arrott-Noakesequation (6), with assumption that parameters a , b , T C , β and γ do not dependon temperatures or applied fields. Another point needs noticing is that exponent n at T C and T pk are usually considered to the same. h ( i ) ← ∂h i ( x,y ) ∂y used to determine partial derivative of magneti-zation with respect to temperature h (0) = 0; h (1) = 0; h (2) = 1; s = 1;For m ≥ { s = s + m ; h ( s + j ) = , j = 0; x · h ( s + j − m ) , ≤ j ≤ m − mm − · y · h ( s − , j = m ; m = m + 1 . } (36)7 .6.2 h ( i ) ← (cid:82) x ∂h i ( x (cid:48) ,y ) ∂y d x (cid:48) used to compute magnetic entropy change h (0) = 0; h (1) = 0; h (2) = x ; s = 1;For m ≥ { s = s + m ; h ( s + j ) = , j = 0; m − jm − j +1 · x · h ( s + j − m ) , ≤ j ≤ m − mm − · y · h ( s − , j = m ; m = m + 1 . } (37) Here, we apply above method to deal with the magnetization data of poly-crystalline samples La . Sr . Mn O obtained with Physical Property Measure-ment System (PPMS) of Quantum Design Company. More details can be foundin reference [8] . Following calculations use the whole data (3789) as the traininggroup. The regularization parameter λ = e − ≈ . × − , is selected outby comprehensive considerations about the overfitting degree γ (with samplingfactor equal to 3) and magnetic entropy change | ∆ S H | . With 96 orthogonalpolynomials ( S = 95), the fitting error reached is 0 . × − . For compari-son, the fitting error with S = 0 is 0 . × − .Fig. 1 summarizes the temperature dependence of magnetic entropy changeunder applied magnetic fields 0 . ≤ H ≤ . λ , we havealso created magnetization data on the uniform mesh from the fitted expression(17); and compared the value of | ∆ S H | , estimated according to equation (28),with that calculated according to finite difference∆ S H (cid:18) H j max , T i + T i +1 (cid:19) = − j max − (cid:88) j =1 M ( H j +1 , T i +1 ) − M ( H j +1 , T i ) T i +1 − T i ( H j +1 − H j ) . (38)At the value of λ given above, no obvious fluctuations appear in the vicinity of T pk , which manifests that overfitting is not significant.Fig. 2 displays the temperature dependence of spontaneous magnetization.Note that in contrast to the 113-structure, spontaneous magnetization in bi-layered manganites does not approaches zero at the Curie temperature T atvanishing magnetic fields, which is considered to be the temperature at whichspontaneous magnetization reaches the minimum. The fact that the minimumis not equal to zero at T , might be attributed to the finite magnetization in theMn-O two-layers above T . Therefore, a two-step magnetization process takesplace in the bilayered manganites. Since experimental recorded data are fewerin this temperature range, the fitting error at T ≥ T C )with that of the temperature of peak magnetic entropy change ( T pk ). It’s noted8hat T C is bigger than T pk in bilayered manganites. This result is in contractto that found in reference, [7] the latter announces that T pk is equal to T C whileusing the mean field model, and larger than T C with the Heisenberg model. It’salso noted that with increasing field intensity, the difference T C − T pk increasesup to the maximum and then lowers down to negative value at about µ H = 5T. The strange field-dependence of T C might suggest spin dimerization occurswithin the Mn-O two-layers, as similar field-dependences are usually found inspin-dimerized insulators. However, it needs to stress that the electronic itin-erating properties and magnetic frustrations in the bilayered manganite mightmask the step-by-step magnetization that is found in other spin-dimerized two-layered compounds. Hence, the perfect step structure of magnetization can notbe observed in two-layered manganite.Shown in Fig. 4 are the magnetic field dependences of magnetic entropychange ∆ S H , the partial derivatives of magnetization with respect to tempera-ture (cid:12)(cid:12)(cid:12) (cid:0) ∂M∂T (cid:1) H (cid:12)(cid:12)(cid:12) , and | H (cid:0) ∂M∂T (cid:1) H | , at T C and T pk . The magnetic entropy ∆ S H ismeasured in J · Kg − · K − and magnetization M in A · m · kg − . We try to deter-mine the scaling exponents n at T C and T pk according to H | (cid:0) ∂M∂T (cid:1) H | max ∝ H n and | ∆ S pk H | ∝ H n . It’s noted that two distinct dependences of | H (cid:0) ∂M∂T (cid:1) H | on the applied field at H ≤ .
75T and H ≥ . n ( T C ) = 1with H ≤ .
75T and n ( T C ) ≈ H ≥ .
75T can explain the obtained re-sults. Hence, the crossover between two different scaling laws happens at bout H = 1 . n ( T pk ) to unify the estimated results in the whole range ofmeasurement. In contrast, a nonlinear dependence is found likeln | ∆ S pk H | = 0 .
19 + 0 .
99 ln H − .
16 ln H. (39)It needs emphasizing that under weak fields, exponent n equal to 1 is a di-rect consequence of the non-vanishing spontaneous magnetization at T C or T pk ,which is the typical characteristic of bilayered compounds. Comparing panel(a) with (c), it is noted that the value of exponent n estimated according to (3)is unreasonably large at T C , and that T pk seems more reliable.Shown in Fig. 5 is the temperature dependence of the scaling exponent ofmagnetic entropy change with respect to applied magnetic field. It is noticedthat two different transitions seem present under strong magnetic fields. It maybe explained by one transition in the vicinity of the transition point occurs underweak magnetic fields, and a magnetic structure rearrangement and thereforetwo-step process happens under strong fields. In conclusion, we apply an two-variable polynomial fit to the magnetizationof bilayered manganites, in order to investigate its magnetic entropy changeand the associated scaling behaviour with respect to applied magnetic fields.It’s found that the Curie temperature is different from the temperature of peakmagnetic entropy change. The difference between these two temperatures aredependent on applied magnetic fields. Therefore a mean-field theory does notapply to bilayered manganites. The field dependence of the Curie temperaturemight imply weak dimerization occurs in the bilayered manganite. In contrast to9 | D SH| (J Kg-1 K-1)
T ( K )
Figure 1: The temperature dependence of magnetic entropy change underapplied magnetic fields 0 . ≤ H ≤ . M0 (A m2 Kg-1)
T ( K )
Figure 2: The temperature dependence of spontaneous magnetization.10 T p k T C Tpk & TC (K)
H ( T )
Figure 3: Comparison between the magnetic-field dependence of Curie tem-perature ( T C ) and that of the peak temperature of magnetic entropy change( T pk ). T C is determined as the temperature at which (cid:12)(cid:12)(cid:12) (cid:0) ∂M∂T (cid:1) H (cid:12)(cid:12)(cid:12) reaches itsmaximum. The strange field-dependence of T C might suggest spin dimerizationoccurs within the Mn-O two-layers.what has been found in manganites with the 113 structure, the scaling behaviourat the Curie temperature in bilayered manganites is much different from thatat the peak temperature. Hence it requires distinguishing the actual transitionpoint from the peak temperature while discussing the scaling law of magneticentropy change. It’s also found that the temperature dependence of the scalingexponent at peak temperatures under weak fields is distinct from that understrong fields. This difference is attributed to an crossover from one-step tran-sition under weak fields to two-step transition under strong fields. To furthertest the validity of the method provided in this paper, other kinds of magneticmaterials will be analysed in the future. References [1] Arrott A 1957
Phys. Rev. Phys. Rev.
A1626.[3] Osterreicher H and Parker F T 1984
J. App. Phys. Acta Phys. Pol. A
Phys. Rev. Lett. J. Phys.:Condens. Matter J. Appl.Phys. a)(c) (b)(d)
Figure 4: The magnetic field dependence of magnetic entropy change ∆ S H , thepartial derivatives of magnetization with respect to temperature (cid:12)(cid:12)(cid:12) (cid:0) ∂M∂T (cid:1) H (cid:12)(cid:12)(cid:12) , and | H (cid:0) ∂M∂T (cid:1) H | , at T C and T pk . T C is defined as the temperature at which (cid:12)(cid:12)(cid:12) (cid:0) ∂M∂T (cid:1) H (cid:12)(cid:12)(cid:12) reaches its maximum, and T pk the temperature at which | ∆ S H | reaches its peakvalue. The magnetic field is measured in Tesla, ∆ S H in J · Kg − · K − andmagnetization M in A · m · kg − . n T ( K )0 . 2 5 T