Linear vs. non-linear magnetic and charge relaxation in itinerant ferromagnets: magnetoresistive manganites

3 June 2002

Physics Letters A 298 (2002) 199–206 www.elsevier.com/locate/pla

Linear vs. non-linear magnetic and charge relaxation in itinerant ferromagnets: magnetoresistive manganites A. Solontsov a,b,c,∗ , C. Lacroix c a State Center for Condensed Matter Physics, Rogova str. 5, 123060 Moscow, Russia b A.A. Bochvar Institute for Inorganic Materials, 123060 Moscow, Russia c Laboratoire Louis Néel, CNRS, BP166, 38042 Grenoble cedex 9, France

Received 3 April 2002; accepted 11 April 2002 Communicated by V.M. Agranovich

Abstract An interplay between possible mechanisms of magnetic and charge relaxation in itinerant electron ferromagnets is investigated. It is shown that non-linear ones due to emission (absorption) of spin or charge fluctuations by magnons may dominate giving rise to non-linear quasielastic spin and charge fluctuations observed in the colossal magnetoresistive manganites.  2002 Elsevier Science B.V. All rights reserved. PACS: 71.27.+a; 75.30.Vn; 75.40.Gb; 76.20.+w Keywords: Magnetic and charge relaxation; Magnons; Spin fluctuations; Magnetoresistance

After the rediscovery in 1994 of colossal magnetoresistive (CMR) effects in manganites [1] La1 − x Bx MnO3 (B = Sr, Ca, Ba), these oxides are in the focus of the research activity in itinerant electron magnetism. Besides possible applications, these materials possess unusual physics. In this Letter we discuss the central quasielastic peak which is observed in the spin fluctuations (SF) spectrum in the ferromagnetic doping range, at temperatures T above T ∼ 0.6TC . This peak increases with temperature and dominates over the conventional magnon peaks on approaching the Curie temperature Tc [2]. Similar quasielastic SF with * Corresponding author.

E-mail addresses: [email protected], [email protected] (A. Solontsov).

a longitudinal polarization were discovered in ferromagnetic Pd–Fe alloy [3] and amorphous Invar Fe–B [4] and non-Invar Fe–Ni–P–B [5] systems, and in antiferromagnets UN [6] and RuMnF3 [7]. Surprisingly, quasielastic SF were not found in most of itinerant magnets aside the critical region though they must exist at least due to linear magnetic relaxation caused by the Landau damping or spin diffusion [8,9]. This behaviour of quasielastic SF strongly suggests that their dynamics cannot be described within a simple linear approximation used in the existing SF theory [8–11] and must account for non-linear effects of SF coupling or spin anharmonicity. Recently within a phenomenological mode–mode coupling approach [12,13], microscopic Heisenberg and t–J models [14] for isotropic ferromagnets it was shown that three-

0375-9601/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 4 7 6 - 0

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mode couplings with emission (absorption) of a SF by a magnon may give rise to strongly temperature and magnetic field dependent quasielastic longitudinal SF. This may explain the observed features of the quasielastic SF peak [3–6], especially in manganites [2], where it can be directly related to the CMR phenomenon [13]. Besides the central SF peak the Ca-doped manganites possess a charge density wave (CDW) developing at a finite wavevector Q0 = 0 [15] with a direction changing with doping [16]. CDW is usually related to some charge ordering but it has also been attributed to a polaron formation or charge stripes [15]. There is an empirical correlation between temperature dependencies of intensities of the quasielastic SF, CDW and electrical resistivity [15], suggesting that the mechanism of CMR may be due to SF or dynamic charge fluctuations (CF) developing near Q0 , which probably were not observed due to an insufficient experimental resolution. In the present Letter we analyze the interplay between different mechanisms of spin and charge relaxation in itinerant ferromagnets, in the CMR manganites in particular. Following the previously developed phenomenological approach [12,13] we show that non-linear mechanisms due to emission (or absorption) of SF or CF by a magnon may dominate, giving rise to non-linear quasielastic spin and charge fluctuations, which represent a novel class of collective excitations described by a set of appropriate normal variables and may fairly good explain the observed properties of manganites. Our model is based on the analysis of non-linear dynamical equations for the normal variables: amplitudes of spin, mν (k), and charge, n(k), fluctuations. Here ν = t, l denotes transverse and longitudinal spin polarizations, k = (ω, k), ω and k are the frequency and wavevector of fluctuations. In order to get dynamical equations of this type one should use certain microscopic models accounting for electron–electron and electron–lattice interactions in different classes of magnetic materials and integrate out the variables related to individual electrons, spin and charge of different electron orbitals and to the crystal lattice (cf. [17]). However, this procedure is rather complicated and up to now was realized only for a single electron band model of isotropic itinerant ferromagnets [18,19], which cannot be applied to the CMR man-

ganites, usually described in terms of the “doubleexchange” model for almost localized t2g and fully polarized itinerant eg electrons of Mn atoms [20,21]. Fortunately, the general form of the dynamical equations for spin and charge fluctuations is not dependent on a particular microscopic model and is defined only by symmetry considerations which for isotropic ferromagnets yields, χt−1 (k)mt (k)
 Wlt t (k, k1 , k2 )ml (k1 ) =− k1 +k2 =k

 + Wet t (k, k1 , k2 )n(k1 ) mt (k2 ) + · · · , (1)

χl−1 (k)ml (k) =−




Wt t l (k, k1 , k2 )mt (k1 )m∗t (−k2 ) + · · · ,

k1 +k2 =k

χe−1 (k)n(k) = −




(2) Wt t e (k, k1 , k2 )

k1 +k2 =k

× mt (k1 )m∗t (−k2 ) + · · · , (3) where χν (k) are dynamical magnetic transverse (ν = t) and longitudinal (ν = l) susceptibilities, χe (k) is the dynamical charge susceptibility, the kernels Wˆ describe tree-mode couplings oftransverse with a  SF +∞ longitudinal one or a CF, and k = k −∞ ( dω 2π ). It should be noted that Eqs. (1)–(3) are expansions in terms of powers of fluctuation amplitudes mν (k) and n(k). Below we shall neglect all other terms in the r.h.s. of (1)–(3), assuming that they renormalize the static susceptibilities χν (ω = 0, k) = χν (k) (where ν = t, l, e) and kernels Wˆ so, that they can be treated as measurable quantities [13]. Being interested in the quasielastic spin and charge fluctuations developing near the wavevectors k = 0 and Q0 = 0, respectively, we neglect the ω and k dependencies of the kernels. As we shall see later this will allow us to find their values basing on the phenomenological arguments, without using any microscopic models. Spectrum and relaxation of SF and CF in isotropic itinerant ferromagnets are characterized by dynamical magnetic, χt,l (k), and charge, χe (k), susceptibilities [8]. In the non-relaxational spin-wave regime the transverse susceptibility χt (k) has poles at the magnon

A. Solontsov, C. Lacroix / Physics Letters A 298 (2002) 199–206

frequencies ω = ωm (k, T ) ≈ D(T )k2 : 2µB M 1 , (4) h¯ ωm (k, T ) − ω + iτ −1 (k, T ) where D(T ) is the temperature dependent magnon stiffness, τ −1 (k, T ) = ωR(k, T ) is the inverse magnon lifetime, R(k, T ) depends on the scattering mechanism [13], and M is the magnetization. In the relaxational regime for the longitudinal susceptibility χl (k) in the low frequency long wavelength limit, ω, k → 0, we assume the conventional form [8,9]: ω , χl−1 (k) = χl−1 (k) − i (5) Γl0 (k)

χt (k, T ) =

with χl−1 (k) = χl−1 + cl k2 , where χl is the static temperature dependent longitudinal susceptibility, cl accounts for the spatial dispersion, and Γl0 (k, T ) is the linear relaxation rate for longitudinal SF, not accounting for the non-linear effects of mode–mode coupling. We can write an analogous expression for the charge susceptibility χe (k, T ). In view of the observed CDW at finite k = Q0 = 0, it is reasonable to assume that χe (k, T ) has a maximum near this wavevector, and in the low frequency limit is given by ω , (6) χe−1 (Q0 + k, ω, T ) = χe−1 (Q0 + k) − i Γe0 (k) where χe−1 (Q0 + k) = χe−1 (Q0 ) + ce k2 , χe (Q0 ) is the static charge susceptibility at Q0 , ce accounts for the spatial dispersion, and Γe0 (k) for the relaxation rate of CF in the linear approximation. In the present approach we focus on the spin and charge excitations described by a set of normal variables, amplitudes of SF and CF, all other are considered to be integrated out. This would result in additional ω and k dependencies of the susceptibilities χν (k, T ). However here we focus on the quasielastic fluctuations with frequencies less than those of phonons, optical magnons, excitons, etc. This means that the lattice distortions, motions of spin and charge of different electron orbits will adiabatically follow spin and charge fluctuations, which must be viewed as coupled spin–lattice and electron density–lattice excitations rather than the spin density and charge fluctuations of a pure electron Fermi liquid, as was shown in Refs. [18,19]. At the same time the static susceptibilities χν and parameters cν and Γν0 (k), are assumed to incorporate effects of the crystal lattice, localized spins in manganites, etc. in a phenomenological way.

201

Recently we have reviewed the main mechanisms of relaxation of transverse SF, resulting in a finite lifetime of magnons due to their scattering by electrons and SF [13]. Here we concentrate on the relaxation of longitudinal SF and CF, taking into account both linear relaxation processes and non-linear ones due to mode–mode couplings of fluctuations. The latter should renormalize the inverse relaxation rates −1 (k) (ν = l, e) in Eqs. (5) and (6), which in the Γν0 Born approximation should be replaced by a sum of −1 a linear relaxation term Γν0 (k) and an additive contribution arising from different mode–mode scattering processes [12], 1 1 1 1 → = + + ···. Γν0 (k) Γν (k, T ) Γν0 (k) Γνn (k, T ) (7) Here Γν0(k) defining linear relaxation without taking into account non-linear effects of mode–mode couplings describes two relaxational regimes: the collisionless (ballistic) or Landau damping regime and the diffusion one. For long wavelength SF Γl0 (k) = Γz |k|z−2 (where z is the dynamical exponent). The collisionless regime, for which z = 3, takes place when [8] |k|l  4 (where l is the electron mean free path). In the random phase approximation (RPA) for an isotropic Fermi surface we may estimate: Γ3 ≈ (2/π)χp vF , where χp = µ2B νF is the Pauli susceptibility, νF is the density of states at the Fermi level, and vF is the Fermi velocity. For the spin diffusion regime, when [4] |k|l  4 one has z = 4, and in the RPA Γ4 ≈ (1/6)χp lvF . It should be noted that according to the theory of Nagaev [22] the mean free path l has a strong temperature dependence due to the temperature dependent Coulomb screening of impurity potentials and exhibits a rather sharp minimum near the Curie temperature Tc . For CF near k = Q0 Γe0 (k) is nearly independent on k, and the RPA gives Γe0 ≈ (2/π)νF vF Q0 and Γe0 ≈ (1/6)νF lvF Q20 for the collisionless and diffusion regimes, respectively. For the non-linear relaxation rate Γνn (k, T ) due to anharmonic couplings, as in the critical dynamics [23], we shall take into account the lowest order threemode scattering processes, among which we consider the main processes in the ferromagnetic phase, i.e., emission or absorbsion of a longitudinal SF or a CF

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by a magnon. Thus we obtain [12]     2h¯
−1 Γνn (k, T ) = |Wνt t |2 Im χt k  Im χt k + k  ω  k

× (Nω − Nω +ω ),

(8)

where ν = l, e. Here Wνt t are the matrix elements which according to Eqs. (1)–(3) describe three-mode processes of emission (absorbtion) of a longitudinal SF or CF. In our phenomenological approach to get Wνt t we use the effective Ginzburg–Landau Hamiltonian: 2 1
−1  Heff = χν0 (k) mν (k, t) 2 k 2 1
−1  + χe0 (k) n(k, t) 2 k
+ γ0 M0 ml (k, t)mt (k1 , t) k+k1 +k2 =0

× m∗t (−k2 , t) +

1 2


k+k1 +k2 =0

∂χt−1 0 (k) ∂n

n(k, t)mt (k1 , t)

× m∗t (−k2 , t),

(9)

where χν0 (k) (ν = l, e) and χt 0 (k) are static susceptibilities, γ0 is the SF coupling constant, M0 is the magnetization without SF effects, n is the density of itinerant electrons, and m(k, t) and n(k, t) are the Fourier transforms of the SF and CF amplitudes. According to the neutron scattering data in the Cadoped manganites [15,16] we assume that the coupling of CF with magnons takes place near k = Q0 . With account of the effects of renormalization due to spin anharmonicity [10] from (9) we have: Wlt t = 2Wt t l = 2γ M, Wet t = 2Wt t e =

∂χt−1 (Q0 ) , ∂n

(10)

where γ , M and χt−1 (Q0 ) account for SF and CF effects and are measurable quantities, which we treat as phenomenological parameters. It should be mentioned that at the Curie temperature Tc the matrix element Wlt t in (10) vanishes. However, it must be finite as it follows, e.g., from the critical dynamics [21]. This means that near Tc next terms in ω and k besides the k = ω = 0 value should be taken into account which

needs to use particular microscopic models, e.g., the “double-exchange” one for manganites [20,21]. Besides that the Ginzburg–Landau formalism used to derive Eq. (10) break down in the critical region near Tc . However, this is out of the scope of the present approach, and here we assume that T is not very close to Tc , so that we can use Eq. (10). With the use of (4) and (10) the integration in (8) over ω and k can be easily performed provided that magnons are weakly damped, and we get (cf. [12,13]) −1 Γνn (k, T ) =

1 kB T L(k, T ) Γνn |k| hω ¯

(ν = l, e),

(11)

where Γνn = 4π h¯

D µB MWνt t

2 ,

L(k, T )    (k,T ))2 +h¯ τ −2 (k,T )  − 1  exp − h¯ (ω+ωm4k B T ωm (k) ,  = ln   (k,T ))2 +h¯ τ −2 (k,T )  − 1 exp − h¯ (ω−ωm4k B T ωm (k)

(12)

(13)

and Wνt t (ν = l, e) are given by (10). Now we discuss the possible mechanisms of magnetic relaxation and the SF spectrum. According to Nagaev [22], we can assume that the mean free path of electrons l has a minimum near Tc and estimate it to be about 3 lattice constants a [20]. For the neutron scattering experiments in La0.67 Ca0.33 MnO3 with k = 0.07 [2] we get |k|l/4 ≈ 0.2, showing that |k|l  4 and the spin diffusion mechanism dominates over the collisionless one in the long wavelength limit. To compare the spin diffusion and non-linear mechanisms of magnetic relaxation we consider the ratio (k) 2 ξl = ΓΓlnl0(k,T ) , where, Γl0 (k) = Γ4 k is defined by spin diffusion, and Γln (k, T ) is given by (11). In the limit ξl  1 non-linear effects are negligible, and the magnetic relaxation is assumed to be due to the spin diffusion. In the opposite limit ξl  1, non-linear mechanism of magnetic relaxation will dominate. Below we estimate ξl for itinerant ferromagnets and apply then this estimation to manganites. In the low frequency limit, when h¯ ω  h¯ ωm (k, T )  kB T , we use the first term in the expansion of L(k, T ) in (13), L(k, T ) = 4



2 1 ω ω +··· , 1+ ωm (k, T ) 3 ωm (k, T ) (14)

A. Solontsov, C. Lacroix / Physics Letters A 298 (2002) 199–206

to obtain ξl in the following form ξl = αl (k, T )

h¯ ωSF (k) , 4kB T

(15)

where ωSF (k) = Γl0 (k)χl−1 (k) is the characteristic SF frequency in the linear theory [4–7], αl (k, T ) = 4

kB T , h¯ ωln (k)

(16)

and ωln (k) = Γln (k)χl−1 (k) characterize non-linear SF. As we shall see below, αl (k, T ) is the key parameter defining the SF spectrum in the presence of non-linear effects. We shall estimate the parameters αl and ξl for strong itinerant ferromagnets near Tc assuming that the longitudinal susceptibility χl near Tc has the Landau form, χl−1 = 2γ M 2 , and the zero-temperature susceptibility of the fully polarized electrons is of the order of the Pauli susceptibility χp . In our estimates we assume for simplicity that the Fermi surface is isotropic and use the following relations between the Fermi momentum pF , electron density of states at the Fermi level νF , and the density of electrons n: pF /h¯ ∝ π/a, νF ∝ pF2 /2π 2 h¯ 3 vF and n ∝ pF3 /6π 2 h¯ 3 , which qualitatively hold not only for magnets with isotropic Fermi surfaces but also for more complicated materials including manganites [21]. Finally, we get  εF kB T D0 2 αl (k, T ) = 4π h¯ ωmax h¯ ωmax D(T )

M 2 pF χl (k) × , (17) M0 h¯ |k| χl and ξl (k, T ) =

4π 3



εF h¯ ωmax

2 

D0 D(T )

2

M M0

4 l|k|, (18)

where D0 = D(T = 0), ωmax = D0 (pF ¯ and εF is the Fermi energy. From the last formula we see that non-linear magnetic relaxation mechanism in itinerant ferromagnets is favoured for long wavelength SF and high temperatures. Let us estimate (18) for La0.67 Ca0.33 MnO3 under the conditions of the neutron scattering experiment [24]. Taking into account that in this compound itinerant electrons at T = 0 are fully polarized, magnon /h)2

203

stiffness D(T ) is not vanishing near Tc , D0 /D(T ) ≈ 2, and using estimates εF ≈ 0.5 eV, h¯ ωmax ≈ 50 meV, S ≈ 1 (see [20,24]), for |k| = 0.07 Å−1 , we find ξl ≈ 3 ×102 (M/M0 )4 . Using the measured temperature dependence of M(T ) we see that ξl in the La-manganites is quite large practically up to Tc ; thus non-linear magnetic relaxation caused by emission (absorption) of longitudinal SF by a magnon is dominating. Now we discuss the shape of the longitudinal SF spectrum provided that the main relaxation mechanism is due to non-linear SF couplings, and that ξl  1. The main features of the spectrum were already discussed in our previous paper [13], so here we focus on some important details, which were out of the scope of the previous work. It is convenient to describe the spectrum by Il (k) = ω1 Im χl (k) which in the low frequency limit kB T  h¯ ω defines the cross-section of inelastic neutron scattering (see, e.g., [25]). From Eq. (5) taking into account only non-linear relaxation given by (11)–(13) we have Il (k) = χl (k)

ωln (k)ωT (k, T ) , 2 (k) + ω2 (k, T )]ω [ωln T

(19)

where ωT (k, T ) = (kB T /h¯ )L(k, T ). In the low temperature limit the intensity of longitudinal SF is exponentially small, Il (k) ∼ ωT (k, T ) → 0. At higher temperatures Il (k) exhibits two broad maxima near the magnon frequencies ±ωm (k, T ) with a quasielastic dip between them. Using the expansion (14) one finds that as the temperature is raising, a dip transfers into a √ quasielastic peak provided the condition αl (k, T ) > 3 is satisfied, where the parameter αl is given by (17). It should be noted that when the height of the quasielastic scattering is comparable with the height of the resonances at ±ωm (k), the latter transfers into antiresonances with narrow dips at ±ωm (k). At high temperatures kB T  hω ¯ m (k) the spectrum of longitudinal SF is dominated by the central quasielastic peak. The height of the peak and its width are given by 

Il (k)

 max

= χl (k)

αl (k, T ) , ωm (k, T )

hω ¯ m (k, T )ωln (k) √ 8 kB T √ = 2 ωm (k, T )/αl (k, T ),

(20)

Λl (k) =

(21)

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It is also worth mentioning a comparison of the height of the central quasielastic peak [Il (k)]max with the conventional magnon one [It (k)]max = χt (k) × R(k, T )/ωm (k) following from (4). Their ratio is given by [Il (k)]max αl (k, T ) , = cl k2 χl (k) [It (k)]max R(k, T )

Fig. 1. Characteristic temperature dependence of the non-linear spin (ν = l) and charge (ν = e) fluctuations: solid curve at the √ temperature T far below T0 defined by αν (k, T ) = 3; dashed curve at T close to T0 ; dotted curve at temperatures T0  T  Tc , and dashed dotted curve at temperatures near Tc .

which in the long wavelength limit behave as [Il (k)]max ∼ 1/|k|3 and Λ(k) ∼ |k|3 . Here we emphasize that for the non-linear magnetic relaxation mechanism which holds when ξl  1, the central quasielastic peak is absent at low temperatures and √ develops only at high temperatures when αl > 3. If we assume that for most of the ferromagnetic metals ξl  1, this would probably explain why the central quasielastic peak was not observed up to now by inelastic neutron scattering in itinerant magnets far below Tc . The quasielastic spectrum of SF is schematically illustrated by Fig. 1. On the contrary, the conventional theory, based on the linear relaxation mechanism, for ξl  1 gives a weakly temperature dependent central peak in the longitudinal SF spectrum in the whole temperature range of the ferromagnetic phase; this peak has a 2 + Lorentzian shape [8–11] Il (k) = χl (k)ωSF (k)/[ωSF ω2 ], with the height and width given by [Il (k)]max = χl (k)/ωSF (k) and Λ(k) = 2ωSF (k). In the long wavelength limit they are proportional to |k|2−z and |k|z−2 , respectively. The collisionless (z = 3), spin diffusion (z = 4) and non-linear relaxation mechanisms result in different wavevector dependencies of the width of the central peak, Λ(k) ∼ |k|, |k|2 and |k|3 , respectively, which opens a possibility to distinguish experimentally between these three mechanisms, e.g., using inelastic neutron scattering.

which near Tc varies as αl (k, T ) provided ck2 χl (k) ≈ 1 and R(k, T ) ∼ 1. The parameter αl (k, T ) also defines the quantum and classical regimes of SF in thermodynamics. According to (14) and (19) estimating the characteristic frequency of quasielastic SF as ωf l = h¯ ωm (kC T ) × ωln (kC )/4kB T , where kC is a cut-off wavevector (for which we use kC ≈ pF /h¯ ≈ π/a), we can measure the hω 1 ¯ max quantum effects by the ratio r = ¯kB fTl = hω kB T αl (k,T ) [10], which is large in the quantum and small in the classical regimes. From the definition (17) of the parameter αl (k, T ) it follows that at high temperatures r is usually much larger than unity, even very close to Tc . Here, using the parameters for La0.67 Ca0.33 MnO3 that we have estimated above, we get αl ≈ 0.5 × 104 (M/M0 )2 χl (k)/χl . Assuming χl (k) ∼ χl and using the measured temperature dependence of M(T ) [24] we can conclude that in this manganite system the parameter αl remains much larger than unity practically up to Tc . Using the estimates of ξl and αl for La0.67 Ca0.33 MnO3 , we can conclude, that (i) at high temperatures a central quasielastic peak developing in the spectrum of longitudinal SF has a non-linear nature and is absent at low temperatures; (ii) in the long wavelength limit the width of the peak (21) must vary as |k|3 ; (iii) at high temperatures the quasielastic longitudinal peak in the SF spectrum dominates over the conventional transverse magnon peaks; (iv) near Tc where h¯ ωmax ∼ kB Tc quasielastic longitudinal SF can be treated in thermodynamics as classical. Using the last conclusion and the SF theory [10] one can easily estimate Tc in La0.67 Ca0.33 MnO3 using the equation 3δm2l (Tc ) = M 2 (T = 0), where effects of transB T kc verse SF are neglected, δm2l (T ) ≈ k2π 2 c is the squared l averaged amplitude of longitudinal SF in the classical limit [8,9] and kc is the cut-off wavevector. Estimating for La1−x Cax MnO3 (x = 1/3) the magnetization and number of itinerant electrons per Mn atom as

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[20] µB (4 − x) and 1 − x, for kc = pF /h¯ = π/a one gets Tc ≈ 177 K, which is close to the observed value [15] Tc ≈ 257 K. Using a smaller cut-off wavevector kc ≈ 0.67pF /h¯ we shall get the observed value of Tc . These simple estimates strongly suggest that quasielastic longitudinal SF observed in a variety of La manganites may be a driving force of the ferromagnetic transition as was anticipated previously by Lynn et al. [24]. The analysis of the charge relaxation and the spectrum of CF near the wavevector k = Q0 = 0 is very similar to the one presented above for SF though some estimates will be different due to the differences in wavevectors and matrix elements (10). First, we estimate the parameter Q0 l/4, allowing to differentiate between the collisionless and diffusion mechanisms of charge relaxation. For La0.67 Ca0.33 MnO3 taking l ∼ 3a and Q0 = ( 14 , 14 , 0) we get Q0 l/4 ≈ 5, satisfying Q0 l/4  1 (collisionless charge relaxation regime). To compare the latter with the non-linear charge ree0 (Q0 ) . laxation, we must estimate the ratio ξe = ΓenΓ(k=Q 0 ,T ) Analogously to ξl we get



2 M0 2 εF ξe = 0.3 × 10 h¯ ωmax µB n 3

2   M χt (Q0 ) D0 . × D(T ) M0 χl (T = 0)

2

(22)

Under the same assumptions as above, for La0.67 Ca0.33 MnO3 near Tc Eq. (22) yields ξe ≈ 0.2 × 104 (M/M0 )2 , provided χt−1 (Q0 )χl (T = 0) ∼ 1, which shows that ξe  1 almost up to Tc ; thus the non-linear mechanism of charge relaxation is dominating in this system. As follows from (22) the same situation holds for other metals if the temperature is not too low. The spectrum of CF near k = Q0 has the same form as the SF spectrum provided the main mechanisms are similar, and is given by the same formulae, where the spin susceptibilities χl (k), χl (k) and the SF frequency ωln (k) should be replaced by the charge susceptibilities χe (k), χe (k) and the CF frequency ωen (k) = Γen |k|χe−1 (k). Similar to the SF spectrum the behavior of the CF spectrum is governed by BT the parameter αe (k, T ) = 4 h¯ ωken (k) . Similar estimates give: αe (k, T ) = π

kB T pF χe (Q0 ) ξe . εF h¯ |Q0 | χe (T = 0)

205

√ If αe < 3, the CF spectrum consists of two peaks near the magnon frequencies ω = ±ωm (k) with a quasielastic dip between them. When the temperature is high enough, a dip transfers into a quasielastic peak centered at ω = 0 and k = Q0 . Like for SF, with the raise of temperature resonances at ω = ±ωm (k) transfer into antiresonances. Finally, at high temperatures the quasielastic peak is dominating. The estimation of αe for La0.67 Ca0.33 MnO3 near Tc yields αe ≈ 0.5 × 104 (M/M0 )2 (χe (Q0 )/χe (T = 0)) which shows that near TC this system must possess a well defined quasielastic peak around the wavevector k = Q0 . The height and width of this quasielastic peak are given by formulae (20) and (21), where χl (k), ωln (k) and αl (k, T ) should be replaced by χe (k), ωen (k) and αe (k, T ). It should be noted that the width of the quasielastic CF peak may be smaller than that of SF. This may explain that quasielastic CF near k = Q0 in the La0.67 Ca0.33 MnO3 system are observed by neutron scattering as static fluctuations [15] probably due to an insufficient energy resolution. It is quite obvious that even if there are static CDW due to a possible maximum of χe (k) near k = Q0 , inevitably there exist charge relaxation, which we have shown to be mainly of non-linear origin, giving rise to dynamic CF near k = Q0 . The quasielastic spectrum of CF is similar to the SF one (Fig. 1). The results presented above clearly shows that in a wide temperature range below Tc the non-linear spin and charge relaxation mechanisms are dominating in manganites and probably in other itinerant magnets giving rise to the observed by neutron scattering quasielastic spin and charge fluctuations [2–6,15,24] which must have a non-linear nature. They also shed some light on a new mechanism of electron scattering by quasielastic fluctuations which could give rise to the CMR phenomena in manganites.

Acknowledgements This work was partially supported by Université Joseph Fourier, Grenoble. A.S. also acknowledges the support of Minatom of Russia and the Program of support of leading scientific schools of Russia, and would like to thank J. Lynn for sending the results prior to publication, N. Bernhoeft, A. Ivanov and A. Vedyaev

206

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for stimulating discussions, and A. Vasil’ev for computer assistance.

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