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Exchange narrowing of the phonon contribution to the electron spin resonance line width in exchange-coupled magnetic insulators

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 056002 (http://iopscience.iop.org/0953-8984/26/5/056002) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 192.236.36.29 This content was downloaded on 14/06/2017 at 00:56 Please note that terms and conditions apply.

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 056002 (3pp)

doi:10.1088/0953-8984/26/5/056002

Exchange narrowing of the phonon contribution to the electron spin resonance line width in exchange-coupled magnetic insulators D L Huber Department of Physics, University of Wisconsin-Madison, Madison, WI 53706, USA E-mail: [email protected] Received 10 November 2013, revised 17 December 2013 Accepted for publication 23 December 2013 Published 17 January 2014

Abstract

In this paper we extend earlier calculations of the phonon contribution to the electron spin resonance line width at high temperatures in exchange-coupled magnetic insulators. We show that the one-phonon contribution is exchange-narrowed, similar to the static anisotropy contribution. The effect of the exchange narrowing is to limit contributing phonons to those modes whose energies are less than a cutoff, γ max , that is proportional to the exchange interaction. Linear-T behavior in the line width occurs when kB T is greater than γ max . Keywords: electron spin resonance, exchange coupled insulators, phonon contribution to linewidth, exchange narrowing (Some figures may appear in colour only in the online journal)

1. Introduction

much less than K T . In this note we will show that the cutoff in the phonon contribution to the line width is a direct consequence of the exchange interaction and represents a different example of exchange narrowing. The focus of this paper on the linear behavior as opposed to the magnitude of the temperature-dependent term which is dependent on the details of the spin-phonon coupling and the phonon spectrum.

In many instances, the electron spin resonance (ESR) line width in exchange-coupled paramagnets is dominated by anisotropy effects, approaching a constant value at temperatures T  Tc where Tc denotes the critical temperature associated with a transition to a magnetically ordered phase. The limiting value of the line width, instead of being on the 2 i1/2 is significantly order of the rms anisotropy field, hHanis 2 i/hH i where hH i denotes smaller, i.e. on the order hHanis ex ex the exchange field. The reduction in the line width by the 2 i1/2 /hH i is referred to as exchange narrowing. factor hHanis ex However, in studies of CrBr3 [1], it was found that the line width did not approach a constant at high temperatures (T  Tc = 32.5 K) but above 100 K, varied as constant plus a term linear in the temperature. The origin of the linear temperature was attributed to the interaction between the magnetic ions and the lattice vibrations through a process involving the creation or annihilation of a single phonon. The linear temperature dependence is indicative of the fact that the phonons contributing to the line width had energies 0953-8984/14/056002+03$33.00

2. Analysis

In the appendix to [1], there is a model calculation of the one-phonon contribution to the zero-field linewidth. Equation (A5) in [1] is a formal expression for the line width involving a sum over phonon modes of the time integral of the phonon factor {(Nq + 1) exp[iωq t] + N q , exp[−iωq t]}, multiplied by the spin–spin correlation function denoted by hR(−q, 0)R(q, t)i. Here q denotes the phonon wave vector and polarization effects are neglected in the interest of 1

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Printed in the UK

J. Phys.: Condens. Matter 26 (2014) 056002

D L Huber

simplicity. The symbol Nq denotes the Bose population factor 1/[exp(ωq /kB T ) − 1]. In the model calculation, the spin function R(q, t) is expressed as   X y y R(q, t) = A e−iq·r j S xj (t)S j (t) + S j (t)S xj (t) , (1) j

where j labels the lattice sites. The important point is that R(q, t), unlike the total spin, 6 j S j , does not commute with the isotropic exchange interaction 6 j>i Ji j Si ·S j . As a consequence the correlation function hR(−q, 0)R(q, t)i decays rapidly in a time set by the reciprocal of the nearestneighbor exchange integral. To see the consequences of the rapid decay, we make a Gaussian approximation for the correlation function, i.e.

Figure 1. The function f (kB T /γ max ), defined in equation (8), versus kB T /γ max .

hR(−q, 0)R(q, t)i ≈ hR(−q, 0)R(q, 0)i exp[−(γq t) /2], (2) where γq is the decay rate for hR(−q, 0)R(q, t)i and the brackets denote a thermal average. In the Gaussian approximation, γq is expressed as 2

γq =



h(dR(−q, 0)/dt)(dR(q, 0)/dt)i hR(−q, 0)R(q, 0)i

1/2

.

Making use of equation (5), γ max can be written as γ

(3)

−∞ Q.1

(4)

The critical factor in equation (4) is the exponential cutoff. Phonon modes whose frequencies are much greater than the decay rate of the spin correlation function do not contribute significantly to the line width. As a consequence, the resonance line is ‘exchange narrowed’. At high temperatures, γq and hR(−q, 0)R(q, 0)i approach constant values, weakly dependent on q, whereas the phonon population term varies linearly with temperature and inversely with frequency. In this limit, the phonon contribution to the line width involves an integral over a band of modes with the upper cutoff γ max where γ max can be approximated by the high temperature limit of the long wavelength spin correlation decay rate γ0 . When kB T > γ max , the contributions from the modes with frequencies below the cutoff are proportional to T. An estimate of the decay rate can be obtained from the long wavelength limit of the spectral moments of the correlation function of the z component of the total spin in the presence of the static distortion A6 j (S xj )2 : P

j

dt

S zj

 = −i

S zj , A

j

=A

f (kB T /γ

X

X y

(6)

(7)

)=

∞

Z

dx x 3 coth(xγ max /2K T ) (8)

and is displayed in figure 1. Note that f (kB T /γ max ) is close to a linear function of the temperature for kB T /γ max > 1. Although the result for γ max given in (7) is obtained from a particularly simple model of the spin-phonon coupling (strain-dependent crystal field anisotropy) which is appropriate only for S > 1/2, the qualitative features are also present in strain-dependent anisotropic spin–spin interactions with arbitrary spin as, for example, the Dzialoshinsky–Moriya coupling. In place of equation (1) one has a general expression

j y

max

× exp(−x 2 /2)

(S xj )2 

[S xj S j + S j S xj ] = A R(0, 0).

,

0

 X

1/2

where n 0 is the number of nearest neighbors and Jnn is the nearest-neighbor exchange integral in the Heisenberg coupling J nn Si ·S j . The isotropic exchange interaction between nearest neighbors is assumed to be the dominant term in the spin Hamiltonian. Equation (7) shows that the cutoff is proportional to the exchange integral thus justifying the use of the term ‘exchange narrowing’. Since only phonon modes with frequencies below the cutoff contribute significantly to the line width, linear-T behavior in the phonon contribution will be present when kB T > γ max . Treating the phonons in the Debye approximation, the line width at high temperatures can be expressed as A(χ0 (T )/χ (T )) f (kB T /γ max ), where A is a temperature-independent constant, χ (T ) is the susceptibility of the resonating ions and χ0 is the corresponding Curie susceptibility. The function f (kB T /γ max ) takes the form

+ (Nq + 1) exp[iωq t]) = (2π/γq2 )1/2 hR(−q, 0)R(q, 0)i

d

=

M4 M2

2 1/2 γ max = (2S(S + 1)n 0 Jnn ) .

dthR(−q,0) R(q, 0)i exp[−(γq t)2 /2](Nq exp[−iωq t]

×(2Nq + 1) exp[−(ωq /γq )2 /2].



where M2 and M4 are the spectral moments of the correlation function of the z component of the total spin. Using the results in [2] for the moments, we find

With the Gaussian approximation, the time integral of the product of the correlation factor and the phonon factor is immediately evaluated with the result Z ∞

max

(5)

j 2

J. Phys.: Condens. Matter 26 (2014) 056002

D L Huber

of the form " R(q, t) =

X j

e−iq·r j

Korringa mechanism). Recently, linear-T behavior has been observed at high temperatures in the manganites YBaMn2 O6 (T > 520 K) [5] and La0.7 Ca0.3 MnO3 (T > 300 K) [6]. Since the conductivity of manganites often arises from correlated hopping rather than conventional band transport, it is not yet established whether the linear-T behavior of the line width in these materials comes from phonon interactions or interactions with mobile carriers [7]. In YBaMn2 O6 the most plausible interpretation of the resonance is that it is associated with the array of Mn4+ ions that are weakly coupled due to nearly offsetting super exchange and double exchange interactions [7] in which case the condition kB T > γ max is readily satisfied (γ max /kB ≤ 300 K). In La0.7 Ca0.3 MnO3 the resonance is also associated with the Mn4+ ions, but there is a strong super exchange interaction between them. One can use the paramagnetic Curie temperature of the antiferromagnet CaMnO3 to obtain an upper limit for the Mn4+ –Mn4+ exchange interaction in the doped material [8]. With this approximation, one finds γ max /kB < 260 K, a result also compatible with the phonon mechanism. Earlier line width measurements by Lofland et al [9] in La[x]MnO3 , x = Sr, Ca and Bi, also showed linear behavior over a wide range of temperature for both conducting and insulating samples.

# X

g jk (S j (t), Sk (t)) ,

(9)

k

where g jk (S j , Sk ) is a function of spins S j and Sk that does not commute with the isotropic exchange interaction. Assuming the isotropic exchange interaction is dominant, we can neglect anisotropic terms in the Hamiltonian in evaluating the time evolution of hR(−q, 0)R(q, t)i. In the long wavelength-infinite temperature limit, the correlation function decays at a rate on the order of [S(S + 1)6 j Ji2j ]1/2 . It should be noted that the analysis outlined here was anticipated in earlier studies of the linear temperature dependence of the ESR line width of the spin-1/2 system Cu(HCOO)2 · 4H2 O by Seehra and Castner [3] who attributed the linear term to the strain dependence of the Dzialoshinsky–Moriya interaction. They obtained an estimate for the line width based on the analysis of an exchange-coupled pair that neglects correlations between different pairs. Their analysis differs from the present approach where correlation effects involving the Mn4+ array are included in the Gaussian approximation. 3. Discussion

References

We use the theory outlined above to analyze the line width data for CrBr3 reported in [1]. A plot of the product T χ (T )1H (T ) showed linear behavior over the range 125 K ≤ T ≤ 500 K. In this material, the dominant exchange interaction is the in-plane coupling with the three nearest neighbors. In [4], the result Jnn = 1.45 meV was obtained from measurements of spin wave dispersion curves. Evaluating equation (7) with S = 3/2 and n 0 = 3, we obtain γ max /kB = 80 K which is well below the temperature marking the onset of linear-T behavior. In conducting materials, the ESR line width associated with the magnetic ions can also have a linear-T term coming from exchange interactions with the mobile carriers (the

[1] [2] [3] [4] [5] [6] [7] [8] [9]

3

Huber D L and Seehra M S 1975 J. Phys. Chem. Solids 36 723 Huber D L et al 1999 Phys. Rev. B 60 12155 Seehra M S and Castner T G 1968 Phys. Kondens. Mater. 7 185 Samuelsen E J, Silberglitt R, Shirane G and Remeika J P 1971 Phys. Rev. B 3 157 Schaile S et al 2012 Phys. Rev. B 85 205121 Auslender M, Rozenberg E, Shames A I and Mukovskii Y A 2013 J. Appl. Phys. 113 17D705 Huber D L 2013 arXiv:1309.6353 Winkler E L, Tovar M and Causa M T 2013 J. Phys.: Condens. Matter 25 296003 Lofland S E et al 1997 Phys. Lett. A 333 476