Magnetic excitations 327 Table 1Ordering temperatures TC, spin wave stiffness D, zone-boundary magnon energy EZB and criteria for itinerancy D/kTC and EZB/kTC of the infinite layer and b
Trang 1Magnetic excitations 327 Table 1
Ordering temperatures TC, spin wave stiffness D, zone-boundary magnon energy EZB and criteria for itinerancy
D/(kTC) and EZB/(kTC) of the infinite layer and bilayer manganites along with those of other known itinerant
and localized magnetic materials [52,87]
tems like EuO and EuS Table 1 gives the ordering temperature TC, spin wave stiffness D, zone-boundary magnon energy EZBand criteria for itinerancy D/(kTC) and EZB/(kTC) of
the infinite-layer and bilayer manganites along with those of other known itinerant and calized magnetic materials Comparing the ratios kT D
lo-C and EZB
kTC of different materials fromTable 1 we see that the doped ferromagnetic manganites, which are close to the compo-sition at which CMR effects are maximum, are of rather itinerant character Especiallythe bilayer manganite is more itinerant that the infinite-layer manganites In fact, the ratio
D/kTCfor La1.2Sr1.8Mn2O7is as high as 15.3 compared to 10.1 of Ni The ratio EZB/kTCfor La1.2Sr1.8Mn2O7is 3.35, which is lower than that for Ni (6.43), is higher than that ofPd2MnSn (1.83) and Pt3Mn (2.05)
6 Concluding remarks
We have covered only a part of the huge field of magnetic excitations investigated byinelastic neutron scattering during the past half a century The research on magnetic ex-
Trang 2citations is still very much alive and is contributing enormously to our understanding ofthe properties of magnetic materials in general and the recently discovered high temper-ature superconducting and colossal magnetoresistive materials in particular Because ofspace limitations we have considered only some typical examples of magnetic excitationsstudied by inelastic neutron scattering We have considered as introductory examples theinsulating magnetic systems that can be understood by a localized Heisenberg model, viz.the so-called Heisenberg ferro-, antiferro- and ferrimagnets Then we considered more dif-ficult itinerant metallic systems such as transition metallic elements Fe and Ni A completeunderstanding of these itinerant ferromagnets is still to be achieved Finally we consid-ered ferromagnetic strongly correlated transition metal oxide CMR manganites which arealso metallic and itinerant However to understand these materials one has to consideralso the orbital and lattice degrees of freedom interacting with the spin system The the-ory of CMR manganites has progressed enormously but still is far from being completelysatisfactory We have left out the interesting topic of low dimensional quantum spin sys-tems Also left out is the topic of strongly correlated high temperature and heavy fermionsuperconductors in which magnetic fluctuations are considered to be crucial for the su-perconducting Cooper-pair formation We have also completely left out the huge field ofcrystal-field excitations in rare earth and actinide magnetic systems However there existsome good review articles covering these topics and the interested readers should consultthose For magnetic excitations in low-dimensional magnetic systems the excellent bookedited by De Jong [93] is recommended The magnetic excitations in cuprate supercon-ducting materials the book edited by Furrer [94] can be consulted The book by Jensenand Mackintosh [95] discusses the magnetic excitations in rare earth metals in great depth.The article by Stirling and McEwen [96] is a good introduction to magnetic excitations ingeneral and also for magnetic excitations in rare earth and actinide compounds not treated
in the present chapter
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Trang 6Paramagnetic and Critical Scattering
Tapan Chatterji
Institut Laue–Langevin, B.P 156X, 38042 Grenoble cedex, France
E-mail: chatt@ill.fr
Contents
1 Introduction 335
2 Universal critical phenomenon and static critical exponents 335
3 Magnetic critical scattering 337
4 Magnetic correlations above Tc 340
4.1 Magnetic correlations in localized systems 341
4.2 Diffuse magnetic scattering from metallic magnetic system 348
4.3 Paramagnetic excitations in itinerant electron systems 350
4.4 Diffuse magnetic scattering from quasi-2D ferromagnetic La1.2Sr1.8Mn2O7 355
5 Concluding remarks 359
Acknowledgments 359
References 360
NEUTRON SCATTERING FROM MAGNETIC MATERIALS
Edited by Tapan Chatterji
© 2006 Elsevier B.V All rights reserved
333
Trang 81 Introduction
We have considered spin waves in magnetic materials at low temperatures in some details
in Chapter 6 We also considered the temperature dependence of the spin wave excitationsbelow the ordering temperature Here we shall consider magnetic excitations above theordering temperature in the paramagnetic state Also we will discuss the critical magneticscattering close to the transition temperature The magnetic excitations above the order-ing temperature have not been investigated as intensively as those at low temperatures Thetheory of magnetic excitations above the ordering temperature is also much less developed.Most of the experiments are focused to study the universal critical behavior for which thetheory is well developed The energy integrated diffuse scattering of several magnetic sys-tems has been measured and has been used to evaluate the exchange interactions by usingthe expression of wave vector dependent susceptibility calculated using a mean field ap-proximation Also the validity of quasistatic approximation has been assumed but seldomchecked properly A complete investigation including the energy analysis of the magneticexcitations above the ordering temperature has been done only in a few magnetic systems
2 Universal critical phenomenon and static critical exponents
One of the most important discoveries in condensed matter physics is that of universalbehavior of a system close to the second-order or continuous phase transition [1,2] At
a second-order, continuous or critical phase transition all systems show the following threeimportant properties:
1 There is a symmetry that is broken at the critical temperature Tc The degree of broken symmetry is represented by a parameter η, which is called the order parameter The order parameter η is a continuous function of temperature, is nonzero below Tcand is
zero above Tc For a ferromagnetic phase transition, for example, the order parameter
is the magnetization The order parameter may be a scalar, a vector or a tensor with
D components.
2 Fluctuating microregions with correlation length ξ exist in both phases close to Tc
The correlation length tends to become infinite as one approaches Tceither from thehigher or from the lower temperature side
3 The response time of the system tends to become infinite as Tc is approached fromeither side This is known as “critical slowing” down
It has been observed for continuous magnetic phase transitions that the susceptibility,the specific heat, the correlation length and the magnetization obey a simple power law
If we define a reduced temperature t = (T − Tc)/Tc, then close to the critical point the
isothermal susceptibility χT, for example, in small field and just above Tcobey the powerlaw
where a and γ are constant and γ is called the critical exponent of susceptibility for
T > Tc In Table 1 we give the definitions of some magnetic critical exponents and their
Trang 9Specific heat, cH α ( −t) −α T < Tc, H= 0 Discontinuous −0.3–0.3
Table 2
Approximate values of critical exponents for various models (after Collins [1])
to the following hypothesis of the universality of the continuous phase transition
The critical exponents in a continuous phase transition depend on the following threeproperties and on nothing else:
(1) the dimensionality of the system d,
(2) the dimensionality of the order parameter D,
(3) whether the interactions are of short or long range
There exist inequality relationships between the critical exponents defined in Table 1
We give these inequalities below without proof:
−α+ γ 2(β − 1),
(2)
α + 2β + γ 2.
These are known as the Rushbrooke inequalities For the Ginzburg–Landau model,
α= 0, β = 1/2 and γ= 1, therefore the inequality becomes an equality
Trang 103 Magnetic critical scattering
In analogy with the phenomenon of critical opalescence at the liquid–gas critical points,one expects strong scattering of neutrons from the magnetic fluctuations near the criticaltemperature of a continuous magnetic phase transition Indeed such scattering is experi-
mentally observed near Tc in many magnetic phase transitions Van Hove [3] developedthe theory of scattering from magnetic fluctuations close to the phase transition by us-ing the Ornstein–Zernike [4] theory to describe fluctuations For a ferromagnetic Bravaislattice and Heisenberg Hamiltonian the generalized susceptibility can be written as
The first term under the square bracket is the Bragg scattering proportional toSα2which
is small close to Tc The second term is related to the magnetization fluctuation which
becomes anomalously large as T → Tc Under the static approximation t→ 0 and for
Q→ 0 this term (cr) gives
where M is the magnetization fluctuation and χs is the static susceptibility For T → Tc,
M and therefore χsbecomes very large and diverges at T = Tc Consequently the neutron
Trang 11338 T Chatterji
scattering cross-section also becomes very large for Q= 0 in the case of a ferromagnet
For an antiferromagnet we can show similarly that the neutron scattering cross-section
becomes anomalously large at Q= τ One can express χ
αα,cr in terms of an isothermal
staggered susceptibility χ α (Q) and a spectral weight function F x (Q, E) having unit area
when integrated over E
χ
Assuming instantaneous spin correlations decay exponentially with distance (in analogy
with the Ornstein–Zernike theory [4] of critical opalescence) one can express χ α (Q) as
in which Λ is the diffusion constant One expects Lorentzian Q dependence of diffuse
scattering in the critical region of temperature
Neutron scattering has played a major role in testing the theories of critical phenomena.Many simple magnetic – rather than structural – systems can be treated as ideal systems fortesting theoretical models This is because magnetic systems often have relatively simpleexchange interactions between the near neighbors only and very simple crystal structures.Moreover the magnetic properties are often only weakly coupled with the lattice and there-fore strains and crystallographic defects do not play important roles Many ideal magneticsystems have been investigated to test the predictions of the theories of critical phenom-ena Here we give an example of a two-dimensional Ising system for which there exists
an exact solution first derived by Onsager [5] The materials K2CoF4 and Rb2CoF4 are
good examples of two-dimensional Ising system with S = 1/2 Figure 1 shows the
tem-perature variation of the sublattice magnetization M(T ) of K2CoF4 obtained from thesquare root of the intensities of the elastic antiferromagnetic reflections [6] The solid lineshows the prediction of the exact Onsager solution The agreement is excellent Figure 2
shows the Q-dependent critical scattering observed [7] in K2CoF4 at two temperatures
above Tc= 107.72 K The solid lines are fits to Lorentzian form convoluted with the
in-strumental resolution function Table 3 gives the measured values of critical exponents oftwo-dimensional Ising systems, which have been compared with the theoretical values
Trang 12Fig 1 Temperature variation of the sublattice magnetization M(T ) of K2 CoF 4 obtained from the square root
of the intensities of the elastic antiferromagnetic reflections [6] The solid line shows the prediction of the exact
solution due to Onsager The agreement is excellent (after Ikeda and Hirakawa [6]).
Fig 2 Q-dependence critical scattering observed [7] in K2CoF4 at two temperatures above Tc= 107.72 K.
The solid lines are fits to Lorentzian form convoluted with the instrumental resolution function (after Cowley
et al [7]).
Trang 134 Magnetic correlations above Tc
Magnetic correlations have been investigated by neutron scattering mainly at temperaturesclose to the phase transition, i.e., in the critical region At higher temperatures the magneticcorrelations are no longer given by the theory of Ornstein and Zernike [4] If we assumethe validity of the quasistatic approximation, i.e., the energy width of the magnetization
fluctuations is much smaller than the energy equivalent kBT of the corresponding
temper-ature of the sample and also much smaller than the energy of the incident neutrons, thenthe energy integrated neutron scattering cross-section can be written in the form
dσ
dΩ =(γ r0)2kBT
Nmg2µ2 B
d,d
Jd,dSd· Sd, (13)
where Jd,d is the exchange interaction between a spin Sd in the unit cell with level d
and another spin Sd The wave vector dependent susceptibility χ αβ dd(Q) of (12) can be
calculated for the model Hamiltonian (13) in the framework of molecular field theory The
molecular field theory is exact in the limit T → 0, but it is expected to give sufficiently
good results as long as short range order can be neglected The mean-field expression forthe wave vector dependent susceptibility [8] in the paramagnetic state is given by
Trang 14whereJ dd(Q) is the Fourier transform of the exchange integrals.
4.1 Magnetic correlations in localized systems
4.1.1 Magnetic excitations of EuO and EuS above TC. We have discussed the magneticexcitations in EuO and EuS in Chapter 6 in some detail EuO and EuS are considered to
be very good realizations of the three-dimensional isotropic Heisenberg model So muchexperimental and theoretical investigations have been undertaken on the magnetic excita-
tions of these ideal Heisenberg ferromagnets above the Curie temperature TCand also theircritical behavior Mook [9] measured the shape of magnetic excitations in EuO near and
above TC Figure 3 shows measurements at the[111] zone boundary The solid lines are
fits to data in which a Gaussian line shape was used below TC, and above TCa dampedharmonic oscillator form given by
F α (Q, ω)= ¯h2γ2ω20
was used Here γ is the width and ω0is the position This form for the line shape in ω fits the data better than the Gaussian or Lorentzian line shapes The width γ and the posi- tion ω0 are denoted in Figure 3 as width W and position P One notes that the scat-
tering does not peak at ¯hω = 0 This means that at the zone boundary some type of
propagating excitation is present However, the energy width is too large to suggest it
to be spin-wave like However, one has to distinguish this type of excitations from theparamagnetic excitations which peak at ¯hω = 0 The dotted lines represent calculations
by Hubbard [10] Better theoretical calculations of the line shape F α (Q, ω) above TCarenow available for Heisenberg-type systems like EuO and were done by Young and Shas-try [11], Lindgard [12] and Takahashi [13] The agreement of these theoretical calculationswith the experiment is satisfactory Bohn et al [14] have measured magnetic excitations
in EuS at temperatures exceeding TC They observed distinct peaks at nonzero energy
for momentum transfers near the zone boundary at T = 1.08TC The excitations can bethought to be as broad spin waves For smaller momentum transfers and higher temper-atures the peaks become less pronounced and the scattering becomes more intense near
¯hω = 0 One may conclude that some kind of broad “spin waves” can be observed even in
localized Heisenberg ferromagnets These conclusions have been supported by the MonteCarlo molecular dynamical calculations [15,16] on EuO and EuS The main point whichemerges of these Monte Carlo molecular dynamical calculations is that the structure func-
tion S(q, ω) has interesting and nontrivial structure in the paramagnetic phase and departs greatly from the Lorentzian or semi-Lorentzian (spin diffusion) shape forced at small q by the global spin conservation laws There exist clear shoulders at finite values of ω for large enough q and may be interpreted as (damped) propagating modes These are considered
to arise from the nonlinearity of the equations of spin dynamics rather than from cant equilibrium (static) correlations The frequencies of the propagating modes obtained
signifi-in these Mone Carlo molecular dynamical calculations are quite similar to those obtasignifi-ined
in the approximate analytical calculations of Young and Shastry [11]
Trang 15342 T Chatterji
Fig 3 Measurements of magnetic excitations of EuO at temperatures near and above TC(Mook [20]).
4.1.2 Diffuse magnetic scattering from MnF 2 We have discussed the magnetic structureand spin dynamics of MnF2in previous chapters Here we will consider spin correlations
close to TN= 67.3 K We have shown that the wave vector dependent susceptibility can
be directly measured in the static approximation Schulhof et al [17] verified the validity
of the static approximation in MnF2in the temperature range 0.04 K T − TN 8 K and
for incident neutron energies 56 meV E 134 meV They have measured
quasielas-tic magnequasielas-tic scattering from MnF2in the critical region T > TN Because of the uniaxialanisotropy in MnF2it is necessary to assign to it two Néel temperatures, one for the longitu-dinal and the other for the transverse properties The true Néel temperature is that at which
the susceptibility first diverges as T decreases from the high temperature side For MnF2 the longitudinal transition determines the true Néel temperature TN= 67.3 K because it
is the longitudinal component which diverges whereas the transverse component does not.The longitudinal and the transverse components of susceptibility can be separated by mea-suring the differential neutron scattering cross-sections around the reciprocal lattice points
(001) and (100) We now turn to the dynamical effects near TNmeasured on MnF2also
by Schulhof et al [18] They confirm the effect of slowing down of the spin fluctuationsbut the experimental data do not agree quantitatively with the results of mean-field calcu-
lations Also there are two important features: (a) for T < TNbut close to the critical pointthere exists a central peak in the longitudinal susceptibility and (b) the spin wave peak per-
sists above TNfor measurements at (001) and finally merges with the central peak as the
temperature is further increased Figures 4 and 5 give the measured neutron cross-sections
around (100) and (001), respectively.