spin time-relaxation within strongly coupled paramagnetic systems exhibiting...
TRANSCRIPT
*Corresponding author. Tel.: #212-2-704672; fax: #212-2-704675.
E-mail address: [email protected] (M. Be-nhamou).
Journal of Magnetism and Magnetic Materials 218 (2000) 287}293
Spin time-relaxation within strongly coupled paramagneticsystems exhibiting paramagnetic}ferrimagnetic transitions
M. Chahid, M. Benhamou*
Laboratoire de Physique des Polyme% res et Phe&nome%nes Critiques, B.P. 7955, Faculte& des Sciences Ben M+sik, Casablanca, Morocco
Received 18 January 2000; received in revised form 12 April 2000
Abstract
The purpose of the present work is a quantitative study of the spin time relaxation within superweak ferrimagneticmaterials exhibiting a paramagnetic}ferrimagnetic transition, when the temperature is changed from an initial value¹
*to a "nal one ¹
&very close to the critical temperature ¹
#. From a magnetic point of view, the material under
investigation is considered to be made of two strongly coupled paramagnetic sublattices of respective moments u and t.Calculations are made within a Landau mean-"eld theory, whose free energy involves, in addition to quadratic andquartic terms in both moments u and t, a lowest-order coupling } Cut, where C(0 stands for the coupling constantmeasuring the interaction between the two sublattices. We "rst determine the time dependence of the shifts of the orderparameters du and dt from the equilibrium state. We "nd that this time dependence is completely controlled by twokinds of relaxation times q
1and q
2. The former is a long time and the second a short one, and they are associated,
respectively, with long and local wavelength #uctuations. We "nd that, only the "rst relaxation time is relevant forphysics, since it drives the system to undergo a phase transition. Spatial #uctuations are also taken into account. In thiscase, we "nd an explicit expression of the relaxation times, which are functions of temperature¹, coupling constant C andwave vector q. We "nd that the critical mode is that given by the zero scattering-angle limit, i.e. q"0. Finally, weemphasize that the appearance of these two relaxation times is in good agreement with results reported in recentexperimental work dealt with the Curie}Weiss paramagnet compound Li
xNi
2~xO
2, where the composition x is very
close to 1. ( 2000 Elsevier Science B.V. All rights reserved.
PACS: 05.70.Fh; 75.50.Gg; 05.20.Dd
Keywords: Sublattices; Coupling; Para-ferrimagnetic transition; Relaxation time
1. Introduction
Super-weak ferrimagnetic systems constitutea special class of magnetic materials, which are of
great interest from a theoretical and experimentalpoint of view. They may exhibit a para-ferrimag-netic transition at a critical temperature ¹
#greater
than room temperature. Their common feature isthat they present a small magnetization at lowtemperatures, in contrary to the usual ferrimagneticmaterials. As examples of super-weak ferrimagneticsystems, we can quote (1) certain members of Heu-sler Pauli-paramagnetic alloys [1] based on the
0304-8853/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 3 9 6 - 6
composition X2YZ, with X"Pd,Cu; Y"Ti, V;
Z"Al, In, Sn, and (2) very recent lamellar Curie}Weiss paramagnetic compounds [2], likeAM
xM@
1~xO
2(0)x(1) with A"Li, Na,K;
M,M@"N,Co,2. The lithium}nickel oxides arepromising candidates for electrode materials inlithium batteries [3}5] and electrochromic displays[6].
Recently, in order to describe the super-weakferrimagnetism arising from such materials,Neumann and co-workers [7] have proposeda continuous model based on the Landau theory[8}10]. Such a model assumes that the materialconsists of a lattice made up of two coupled Paulior Curie}Weiss paramagnets sublattices [11,12],with respective local magnetizations u and t.Above the critical temperature ¹
#, these magneti-
zations vanish and the system is a paramagnet.Below this temperature, an anti-parallel con"gura-tion of the magnetizations is favored, but with non-vanishing overall magnetization. One can say thatthe material exhibits a ferrimagnetic state. The ap-pearance of such an order is intimately related tothe existence of a strong coupling between the twosublattices. Quantitatively, this coupling manifestsitself through the introduction of an extra coupling!Cut in the free energy (see below). Negativevalues of the coupling constant C favor the anti-parallel alignment of the local moments u and t,and a ferrimagnetic order appears.
From a static point of view, the theory has beendeveloped, "rst, through a numerical simulation[7], and second, by an exact analytic analysis[13,14], within a mean-"eld context. Afterwards,and in order to take into account the strong #uctu-ations of the local moments near the critical point,we have reconsidered the problem, in a recentpaper [15], using the renormalization}grouptechniques [10,16}18] applied to the "eld modeldescribing such a transition.
The purpose of this work is to investigate theproblem from a kinetics point of view, within theframework of a mean-"eld theory. More precisely,we are interested in time relaxation of the magneti-zation when the temperature is changed from aninitial value ¹
*to a "nal one ¹
&assumed to be very
close to the critical temperature ¹#. In the temper-
ature range ¹*(¹(¹
&, both magnetic moments
u and t relax in time from initial values to "nalones. The natural question to ask is about thequantitative nature of such a relaxation. In fact,close to the critical point, the system exhibits im-portant modi"cations in the dynamic quantities,like transport coe$cients, relaxation time andtime-correlation functions. The existence of conser-vation laws (energy, magnetic moments, etc.) indu-ces e!ective correlations at long distances, and thenleads to a slow evolution towards the "nal equilib-rium state. The study and the classi"cation of suchabnormal behaviors are complex but richer than inthe static case.
The kinetics is especially concerned with thevariation with time (at long scale) of the orderparameters or other quantities varying slowly nearthe critical point. This is the case, in general, forconserved quantities. One can have an intuitiveidea about the meaning of the critical slow-down ofthe order parameters. Indeed, in our case, we havean anti-ferromagnetic interaction between twonearest-neighbor spins (one belongs to the "rstsublattice and the other to the second). Then, in thecritical region, the existence of large domains ofanti-parallel spins is then favored. Each spin havingto be inverted is strongly constrained by the e!ec-tive "eld generated by its neighbors, and a longtime is necessary to reverse these domains by ther-mal agitation. The term &critical slow-down' tra-duces the fact that the long-wavelength #uctuationsof the order parameters have a long relaxationtime.
Our "ndings are the following. Within the frame-work of the mean-"eld theory described below, we"rst determine the time dependence of the shifts ofthe order parameters from the equilibrium state.We "nd that these are governed by two relaxationtimes q
1and q
2. The former is very long, because of
the long wavelength #uctuations of the order para-meters, while the second is a short time and asso-ciated with local #uctuations. We note that theappearance of these two time-scales is in goodagreement with some results reported in a recentexperimental work by Reimers and co-workers[19] on the Curie}Weiss paramagnetic compoundLi
xNi
2~xO
2, where the composition x is very close
to 1. Such a compound has been the subject of someexperiments [20,21], and it was found that it
288 M. Chahid, M. Benhamou / Journal of Magnetism and Magnetic Materials 218 (2000) 287}293
exhibits ferrimagnetic behavior, at low temper-atures. The authors of Ref. [19] have shown thatthe time relaxation of the magnetization when themagnetic "eld is lowered from 0.05 to 0.001T keep-ing the temperature "xed, occurs on more than onetime scale. They proposed an empirical relationshipof the magnetization with two relaxation times.This relationship is similar to that discussed below.Also, we plot some curves showing the dependenceof q
1upon both temperature ¹ and coupling con-
stant C. Spatial #uctuations are also taken intoaccount. In this case, we "nd that the relaxationtimes are not only functions of ¹ and C but alsoa function of the wave vector q. Finally, we "nd thatthe critical mode is that given by the zero scatter-ing-angle limit, i.e. q"0.
This paper is organized as follows. In Section 2,we give a succinct description of the usedmean-"eld theory. Section 3 is devoted to the deter-mination of the time-dependence of the orderparameters and the associated relaxation times.Investigation of the spatial #uctuation e!ects is theaim of Section 4. We draw some concluding re-marks in Section 5.
2. The model
The physical system we consider consists of twosublattices of respective magnetic moments or or-der parameters u and t. The free energy allowingto investigate the para-ferrimagnetic transitionwithin this system reads [7,13,14]
F [u,t]"F0#
a
2u2#
A
2t2!C u t
#
u
4u4#
v
4t4. (1)
In this expression, the coupling constants u andv are taken to be positive, to ensure the stability ofthe free energy. The coe$cients a and A are temper-ature dependent, whose temperature dependence is
a(¹)"a1#b
1¹2, A(¹)"a
2#b
2¹2 (2)
for a Pauli paramagnet [11,12], or
a(¹)"a6 (¹!h1), A(¹)"AM (¹!h
2) (3)
for a Curie}Weiss paramagnet [12,22]. Coe$cientsaiand b
iappearing in Eq. (2) have a simple depend-
ence on both free electron density and Fermi energyrelative to the two sublattices [23]. In relation (3),h1
and h2
stand for the Curie}Weiss temperatures;they are proportional to exchange integrals J
1'0
and J2'0 inside the sublattices [15]. In the free
energy expression (1), the coupling term Cut rep-resents the lowest-order interaction between thetwo sublattices. In fact, this coupling plays the roleof an internal magnetic "eld. Negative values of thecoupling C(C(0) favor the antiparallel con"gura-tions of moments u and t, while positive values(C'0) favor their parallel alignment. In the case ofCurie}Weiss materials, like lamellar compounds[2,24], the coupling C was found to be propor-tional to the exchange integral J
12(0 between the
two sublattices [15]. In this work, we are concernedonly with negative values of C, for which ferri-magnetic order appears at low temperatures. Thiscoupling is assumed to be constant in the wholetemperature range around the critical point.
The equilibrium order parameters u6 and tM canbe obtained by minimizing the above free energy,i.e.
dF
du6"0,
dF
dtM"0 (4)
to obtain the coupled equations
au6 #uu6 3!CtM "0,
AtM #vtM 3!Cu6 "0. (5)
We note that these equations yield the variation ofmoments u6 and tM versus temperature (¹(¹
#) for
"xed coupling constant C. These equations havebeen solved in Ref. [13]. In the disordered phase(¹*¹
#), of course, these moments vanish, i.e.
u6 "0 and tM "0.We recall that the critical temperature ¹
#at
which the system undergoes a phase transition isgiven by the equality [13]
a(¹#)A(¹
#)"C2, (6)
where a and A are temperature-dependent coe$-cients de"ned through relations (2) or (3). Theabove equality implies that a critical temperature isassociated with each coupling constant C.
M. Chahid, M. Benhamou / Journal of Magnetism and Magnetic Materials 218 (2000) 287}293 289
The above considerations are concerned with thestatic study of the paramagnetic}ferrimagnetictransition. The aim of the following sections isprecisely a quantitative study of the kinetics of thepara-ferrimagnetic transition.
3. The relaxation times
Let us assume that, in the beginning, the materialis in an equilibrium state at an initial temperature¹
*di!erent from ¹
#. Suppose that this temperature
is changed to a "nal value ¹&
very close to thecritical point ¹
#. In the interval ¹
*)¹)¹
&, both
order parameters u(t) and t(t) vary with timet from their initial values u6
*and tM
*to their "nal
ones u6&and tM
&. We are interested in how the order
parameters relax from their initial values to their"nal ones.
It is natural to assume that the time-dependentorder parameters satisfy the Langevin equations(without noise) [25,26]
dudt
du,
dtdt
dt, (7)
where p is some positive constant and F the freeenergy, Eq. (1). In fact, quantities of our interest arerather the shifts of these order parameters fromtheir mean values u6 and tM , i.e.
u"u6 #du, t"tM #dt. (8)
Within the framework of the linearized theory,these shifts of the order parameters satisfy the linearcoupled equations
Addu
dtddtdt B"!pA
L2FLu2
L2F
LuLtL2F
LtLuL2FLt2 Br6 ,tM A
du
dtB, (9)
where the second derivative matrix of the free en-ergy is taken at the mean values u6 and tM . We notethat the above equations are obtained from Eqs. (7)expanding to "rst order the free energy F aroundthe static order parameters u6 and tM . Explicitly, thesecond derivative matrix of the free energy, denoted
by ¸, reads
¸"Aa#3uu6 2 !C
!C A#3vtM 2B. (10)
The above linear di!erential system can be sol-ved using the standard method based on the diag-onalization of the matrix ¸. We give simply theresult
du"E1e~t@q1#E
2e~t@q2 , (11a)
dt"F1e~t@q1#F
2e~t@q2 , (11b)
where E*and F
*are known amplitudes. Relation-
ships (11a) and (11b) imply that the #uctuation ofthe overall magnetization, denoted by dM, is givenby
dM"du#dt"He~t@q1#Ge~t@q2 , (11c)
where G and H are known amplitudes. In relations(11), q
1and q
2are the relaxation times we "nd to be
given by
q~11
"pj1"
p2[a6 #AM !J(a6 !AM )2#4C2], (12)
q~12
"pj2"
p2[a6 #AM #J(a6 !AM )2#4C2], (13)
where j*stand for the two eigenvalues of matrix ¸.
We have used the short notations
a6 "a#3uu6 2, AM "A#3vtM 2. (14)
Now, let us comment on these results. We "rst notethat we have two kinds of relaxation times. This isnatural, since the transition is governed by twoorder parameters. A quantitative comparisonshows that the relaxation time q
1is greater than q
2,
whatever the value of the relevant parameters of theproblem, which are the temperature ¹ and thecoupling C. The small time q
2corresponds to local
or short-wavelength #uctuations, and thus, it playsno signi"cant role in the kinetics of the phasetransition. As a matter of fact, when the temper-ature is lowered keeping "xed the coupling con-stant C, spins have at "rst a tendency to formmicrodomains of antiparallel con"gurations. Thus,the relaxation time q
2can be regarded as the neces-
sary time to form such microdomains. We can saythat, over time q
2, the system is controlled by local
290 M. Chahid, M. Benhamou / Journal of Magnetism and Magnetic Materials 218 (2000) 287}293
Fig. 1. The relaxation time pq1
versus temperature at "xedcoupling (C"!4;¹
#+16.4K). Here, a(¹) and A(¹) are chosen
to be of the form a(¹)"2.33#0.02796¹2; A(¹)"0.51#0.00408¹2; u"3; v"1.
dynamics. The relaxation time q1
corresponds tothe necessary time to form macroscopic domains ofantiparallel spins, i.e. a ferrimagnetic order. Indeed,this time increases as the critical point ¹
#is reach-
ed. Thus, the time q1
governs long-scale dynamics.We note that the appearance of these two relax-
ation times is in good agreement with some resultsreported in a recent experiment [19] on theCurie}Weiss paramagnet compound Li
xNi
2~xO
2,
where the composition x is close to 1. Such a com-pound is used as electrode material in secondarylithium batteries [3}5]. The aim of this experimentwas the measurement of the relaxation of the mag-netization with time, after the magnetic "eld waschanged from 0.05 to 0.001T (magnetic "eld cool-ing at ¹"5 K). The main result is that, the intro-duction of one relaxation time is not su$cient formodelling experimental data. To have a good "t toexperiment, Reimers and co-workers proposed anempirical relationship characterized by two relax-ation times [19]. This relationship is analogous torelation (11c) of the present work. We emphasizethat this relation is controlled by temperature andthat it is physically equivalent to that of the experi-ment. Indeed, increasing temperature at vanishingmagnetic "eld has the same e!ect as a decrease inmagnetic "eld keeping the temperature constant.
Combining now expressions of the static orderparameters pointed out in Ref. [13] and relation(12), to "nd, around the critical point, the scalinglaw for the relaxation time q
1
q1&D¹!¹
#D~y, (y"1). (15)
This relaxation time naturally diverges at the criti-cal temperature ¹
#. In relation (15), we have ignor-
ed some known amplitude. In several cases, it ispreferable to express the relaxation time in terms ofthe correlation length m&D¹!¹
#D~l. The Landau
theory allows to exhibit, in a simple way, the uni-versality of kinetics, and to characterize the latterby a new exponent z"l/y. Then, we have
q1&mz, (z"2). (16)
The variation of the relaxation time q1
versus tem-perature around the critical point, at "xed couplingC, is depicted in Fig. 1. Of course, q
1diverges at ¹
#,
but with a di!erent amplitude depending onwhether one is above or below ¹
#.
Now, assume that the temperature is "xed tosome value ¹
0. In this case, the relaxation time
depends only on the coupling C between the twosublattices. Let C
0be some critical value of C re-
lated to ¹0
by relation (6). This means that if C0
isthe true coupling, then, ¹
0will be the critical tem-
perature. The relaxation time q1
increases withincreasing C below C
0. This tendency is inverted
when C becomes greater than C0, and the relax-
ation time decreases. The variation of q1
versus thecoupling C is depicted in Fig. 2.
4. Spatial 6uctuations e4ects
We note that the above considerations ignore thespatial #uctuations of the order parameters. Thepurpose, now, is to show how these can a!ectthe relaxation time. To this end, we "rst replace theLandau free energy of Eq. (1) by the Ornstein}Zer-nike one, de"ned as usual by [9,10]
F[u,t]
kB¹
"GPdr12(+u)2#1
2(+t)2#
a
2u2
#
A
2t2!Cut#
u
4u4#
v
4t4H. (17)
The gradient terms on the right-hand side of rela-tion (17) traduce the spatial variations of the order
M. Chahid, M. Benhamou / Journal of Magnetism and Magnetic Materials 218 (2000) 287}293 291
Fig. 2. The relaxation time pq1
versus coupling !C at a "xedtemperature ¹"25 K. For !C"!C
0+7.78, one is at
¹"¹#"25 K. When !C'!C
0, one is below ¹
#, and for
!C(!C0, one is above ¹
#. Here, a(¹), A(¹), u and v are
those of Fig. 1.
Fig. 3. The characteristic frequency 1/pq1q
upon the wave vectorq, for "xed temperature and coupling (¹"6K(¹
#+25.4K,
C"!8). Here, a(¹), A(¹), u and v are those given in Fig. 1.
parameters u(r) and t(r). There, r is the d-dimen-sional position vector. The classical con"gurationis obtained by minimizing the free energy. Theresult reads as
(!D3#a)u#uu3!Ct"0,
(!D3#A)t#vt3!Cu"0, (18)
where D3is the Laplacian operator. The shifts of the
order parameters from their mean values are de-"ned as above, and they are solutions of the follow-ing linear di!erential system:
Adu
dtdtdt B"pA
!D3#a6 !C
!C !D3#AM BA
du
dtB. (19)
These equations can be easily solved in reciprocalspace. Without details, we give the expressions ofthe Fourier transform of the order parametersshifts
du8 (q, t)"E1q
e~t@q1q#E2q
e~t@q2q , (20a)
dtI (q, t)"F1q
e~t@q1q#F2q
e~t@q2q . (20b)
In these relations, q is the wave vector. As in thehomogeneous case described above, we have two
kinds of relaxation times. We show that the variousmodes are given by
q~11q
"q~11
#pq2, (21a)
q~12q
"q~12
#pq2, (21b)
where q1
and q2
stand for those relaxation timesde"ned above, Eqs. (12) and (13). As above, thelonger relaxation time is q
1q. The critical mode
corresponds to the value of q1q
at q"0. The in-creasing of q~1
1qor characteristic frequency against
the modulus q"DqD of the wave vector q, for "xed¹ and C, is depicted in Fig. 3.
5. Concluding remarks
In this work, we have investigated the spin timerelaxation within superweak ferrimagnetic mat-erials undergoing a para-ferrimagnetic transition,when the temperature is changed from an initialvalue ¹
*towards a "nal one ¹
&very close to the
critical temperature ¹#. We have "rst derived the
time dependence of the shifts of the order para-meters du and dt from the equilibrium state. Wefound that these functions decrease exponentiallywith time t, but with two relaxation times q
1and q
2.
These time scales are physically associated withlong and local-wavelength #uctuations, respective-ly. We have shown that, only the former is relevant
292 M. Chahid, M. Benhamou / Journal of Magnetism and Magnetic Materials 218 (2000) 287}293
for physics, since it drives the material to undergoa phase transition. Indeed, this long-scale timeis found to be divergent as the critical pointis reached. The appearance of these two relaxationtimes is in good agreement with that result reportedin a recent experiment on the lamellar Curie}Weiss compound Li
xNi
2~xO
2, where the composi-
tion x is close to 1. Spatial variations are alsotaken into account, and we gave the explicit expres-sions of the two relaxation times as functions oftemperature ¹, coupling constant C and wave vec-tor q. We have shown that the critical mode corre-sponds to the zero scattering-angle limit, that isqP0.
This paper must be regarded as a natural exten-sion of some previous works [13,14] concernedwith the static aspect of the problem.
Finally, we note that to take into account thestrong #uctuations of the order parameters near thecritical point and in order to get correct kinetics, itwould be of interest to re"ne the present work byusing the renormalization}group techniques. Sucha work is in progress [27].
Acknowledgements
We are much indebted to our referee for hisuseful suggestions and remarks and for carefulreading of the manuscript.
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