Resonant phonon scattering in a coupled spin-phonon system

Solid State Communications,

Vol. 12, PP. 713—7 16, 1973.

Pergamon Press.

Printed in Great Britain

RESONANT PHONON SCATTERING IN A COUPLED SPIN—PHONON SYSTEM F.W. Sheard* CSIRO, National Standards Laboratory, Division ofPhysics, Chippendale, NSW 2008, Australia and G.A. Toombs Department of Physics, University of Nottingham, Nottingham NG7 2RD, England (Received 5 January 1973 In’ R. Loudon)

From a perturbation treatment of phonon scattering by a spin system of arbitrary concentration, we derive the frequency and concentration depen. dence of the phonon relaxation time, which are important for the analysis of thermal conductivity experiments. Comparison is made with theories of coupled spin—phonon modes based on Green-function methods.

As we have pointed out,’2 the results of these calculations are not all in agreement. However, at sufficiently low concentrations the paramagnetic ions must behave independently, so that the damping on the phonon.like parts of the coupled-mode spectrum must reduce to that obtained from the scattering b~ a single ion. Although it is clearly important for thermal conduction this consistency requirement has been neglected in previous treatments of the spin— phonon modes and cannot in any case be applied to those calculations restricted to c = 1. Here we show that a perturbation calculation may be carried out for a spin system of arbitrary concentration and hence we are able to extend the above criterion to include the c = 1 case. For the spinwave-like modes comparison with perturbation theory has already been discussed and is important for the thermodynamic properties of the system.13

THERE have been several theoretical treatments of phonon scattering by paramagnetic ions (spins) in insulating crystals and the effect upon the lattice thermal conductivity. One approach is to calculate the scattering cross-section for a single ion using per. turbation theory and simply multiply by the ionic concentration to obtain the phonon mean free path. However the single-ion scattering has also been viewed in two alternative ways; either as first-order emission and absorption of single phonons, usually referred to as the direct process,”2 or as second-order elastic scattering.~5The other approach takes account of the coherent mixing of the phonon and spin excitations near the crossover point of the unperturbed dispersion curves. Thermal resistance then arises from the finite lifetimes of the coupled spin—phonon modes. The coupled-mode spectrum was first obtained by Jacobsen and Stevens.6 Subsequently the lifetimes were derived by a variety of Green-function methods.7~2Some papers7”1 were restricted to the case of a spin at each lattice site when the fractional concentration c = 1.

We first consider scattering by a single spin, taking as Hamiltonian the spin-½model (Ii = 1) H 14p

*

Visiting Research Scientist. Permanent address: Department of Physics. University of Nottingham. England.

=

I wqpatqpaqp + w

0S,,.~+ H5~. N”2 I (A qpSn+ +A.~’qpS~ _Xaqp +a~qp)exp(i~R~).

where aqp. at,, are operators for the phonon with wave vector q. polarisation index p and frequency 713

714

RESONANT PHONON SCATTERING IN A COUPLED SPIN—PHONON SYSTEM

(i.~qp,S~,+,S,~,S,,~are spin operators for lattice site R~,Aqp is the coupling parameter and N is the number of atoms. In first order the transition rate for one-phonon absorption and emission is zero if w~* w0 (the spin frequency) since energy cannot be conserved. One then commonly includes the broadening of the spin levels by introducing phenomenologically a lineshape function. For lifetime broadening due to spin—lattice relaxation we may use a Lorentzian lineshape. The direct-process phonon relaxation time is 1 then given by 1 (r 8L/~-) (p_p+) 2 Tqp (Wqp wo)2 + FSL direct 2nN~lA~~i (1) —

—

where p, p.. are population factors for the upper and lower spin states. But this procedure is strictly equivalent to summing an infinite series of terms in the perturbation series3’14 since, off resonance, we may expand the lineshape function in powers of r~L/(WQP w 2. 0) Retaining only the first term in the expansion corresponds to second-order perturbation theory, which we now discuss more rigorously. However, if spin—spin interactions dominatethe broadening, employing an empirical lineshape is the only feasible method of analysing thermal conductivity data and has met with some success.2 In second order, the phonon q may be elastically scattered into modes q’ via intermediate states in which the spin has flipped. Similar processes have been discussed by Griffin and Carruthers3 in connection with phonon scattering by impurity levels in semiconductors. Noting that absorption of phonon q may occur before or after emission of q’, the second-order matrix element for scattering by the spin at site R~,when initially in the lower state, is M~(qp,q’p’) = 1/2

flqp (n~’~’ + 1)1/2 exp {i(q where ~ S(qp, qp )

—

q’) R~}N’S(qp, q’p’)

_________

~qp’-~O

We now generalise the calculation to a system of N5 spins for which the fractional concentration c = N5/N In the initial state the spins are distributed randomly over the lattice sites and each may be in the upper or lower states. But we now have the possibility that absorption of q and emission of q’ may be by different spins. However, upon examining the matrix element for such a process we find the contributions from the intermediate states cancel identically. It is this feature which makes possible a straightforward perturbation calculation of phonon scattering in a final many-spin system. Wethe need therefore only consider states for which configuration of an spins is identicalstate to that the initial Since in intermediate onlyinone spingives hasstate. flipped. summing over all intermediate states the matrix element M(qp, q’p’) = In v,, M~(qp,q’p’), where the sum may be extended to all lattice sites b~ defining the variable ~‘+1(occupied site, spin down) =

—l (occupied site, spin up)

(2)

0 (unoccupied site) The transition rate is then W(qp. q’p’) = 2lrnqp(nqp’ + l)N2 S(qp, q’p’)12 tS(wqp X nm I

~n ~m

—

exp {i(q —q’)’(R~ Rm)}

To calculate the phonon relaxation time we take an ensemble average over all possible initial spin states. This may be accomplished by regarding v,, as a random variable whose average value is, from equation (2), = c(.p_— p+). Evaluating the ensemble average of the sum in equation (3) gives

Nc

I

{1

—

exp {i(q—q’)(R~~Rm) = c(.p_—p+)2 } + N2 C2 (p.— p..)2 6q~q’

The latter term is the coherent forward scattering A~qpA_q’p’

— —

Vol. 12, No.7

—

~qp~~o

and nw,, ~ are the phonon occupation numbers in the initial state. We have used the energy conservation condition ~qp = L~g’p’here. If the spin is initially in the upper state the scattering amplitude S is just reversed in sign.

which modifies the phonon energy but does not contribute to the scattering cross-section. For a spin system in thermal equthbrium at temperature T, p_—p.. = tanh(l/2~wo,where ~a= 1/kBT. From the usual rate equation we then obtain the phonon relaxation time r~~:

Vol. 12, No.7 RESONANT PHONON SCATTERING IN A COUPLED SPIN—PHONON SYSTEM c tanh2 2 I3wo)N’ S(qp, q’p’)12 ‘-‘n.’ ~

2irc(l

=

I

••~~

—

—

(4)

715

Elliott Parkinson’s theory does give correctly the energy and shifts and damping of the spin-wave modes.’3 The diagrammatic treatments7’11 restricted to c = also agree with our result.

q’p’

For simplicity the coupled-mode theories have taken an average coupling for all phonon branches.* Since the coupling is proportional to the local lattice strain we set A 2 where ~ is a constant. 4,, = a(wqj,/12)” Then 1 ~ca4o~c~,, 2w~(w~ w~)2~ c tanh2 ~ ,

—

__________________ —

(

—

(5) where we have used the Debye model to evaluate the sum over wave vectors and WD is the Debye frequency. The factor (1 c tanh2 (1 /2)pwo) arises from interference between the scattering amplitudes from different ions. For c = I the phonon damping vanishes at T = 0, since, when the spins are all in the same state, there is only coherent scattering, —

From the coupled-mode dispersion relations given by the Green-function theories we may’ obtain an approximate expression for the damping which is valid for frequencies off-resonance on the phonon-like parts of the excitation spectrum. The perturbation result (5) is then found to agree with that obtained by Toombs and Sheard’2 from a diagrammatic treatment and Stevens and Van Eekelen9 by a decoupling method. Roundy and Mills’°find the same frequency dependence but their temperature dependent factor is (1 tanh2 (1/2)13w The result from Elliott and Parkinson’s0).theory8 has aobtained rather different frequency dependence and only gives approximate agreement as w 4,, expression —p wo. Indoes particular, at low concentrations, their not show the w~ behaviour for w 4,, <
scattering of light. It should be noted however that *

The coupled-mode theories also assume the three phonon branches are degenerate. Polarisation mixing then occurs in second-order giving two uncoupled modes and one damped mode. But in general such mixing only occurs in those symmetry directions for which the transverse phonons are degenerate.

For a single ion the relaxation time is given by equation (5) with c = 1/N, whence the temperature dependent factor becomes unity.’5 Equation (1) shows that provided “iL is neglected in the denomid nator, TQ~~ is also temperature independent since, for ourbymodel, the spin-lattice coth relaxation time is F~L= given r~= (3a2~/2w~) (lI2~woand 1/2 ~TsL. However the Lorentizian approximation does not lead to the correct frequency dependence in the wings of the resonance line. In thermal conduction the role of the spin-phonon interaction is to effectively remove from the heat current a band of phonons for which the scattering rate r~,,due to the spins is much greater than the scattering rate Tj1 due to other processes. Semi-quantitative analyses of experimental data have been made based on this idea.’6 For strong spin-phonon interactior and at low temperatures when Tj1 is small, this band may be sufficiently wide that the phonons effective in the conduction process lie in the wings of the resonance line. The Lorentzian lineshape is then clearly inadequate. Also, for c << 1, our result shows that calculation of the thermal conductivity from the single-ion scattering cross-section is justified althoug~ dispersion in the phonon velocity should properly be included.’2 The coupled-mode analysis of thermal 10 suggests that for conduction by Roundy andisMills strong coupling, dispersion relatively unimportant but no direct comparison with the perturbation 7 treatment was made.’ Finally we remark that the frequency dependence of r4,, given in equation (5) is only strictly applicable to our simple model of the coupling. In discussing the effect of spin-phonon interaction theaEPR 18 alsoon used simplified lineshape but Huber and Van Vleckpolarisation vectors coupling included phonon eq,, by taking a coupling energy 0A 4(e41, Sn). In our notation this corresponds toA~,,= 1/2 (4,, ie~,)A4.By virtue of the closure relation obeyed by the polarisation vectors several terms vanish in evaluating the sum over p’ in equation (4). This leads to a —

relaxation time of the form

716

RESONANT PHONON SCATTERING IN A COUPLED SPIN—PHONON SYSTEM Vol. 12, No.7 I

~

w~) (w~,, w~)2 +

—

(6)

which differs substantially from equation (4), although the low-frequency w~,,behaviour is retained as expected. Of course this form of coupling is still over-simplified. However, the resonant lineshape (6) was also

found by Tucker’9 in a calculation of the phonon Green function (for c = I) using a general form of spin—phonon interaction applicable to a Kramers doublet in a magnetic field with only the restrictions imposed by octahedral symmetry. Acknowledgements We are grateful to Professor M.V. Klein for correspondence on this subject. —

REFERENCES 1. 2.

ORBACH R.,Phys. Rev. Lett. 8, 393 (1962). MCCLINTOCK P.V.E., MORTON I.P., ORBACH R. and ROSENBERG H.M.,Proc. R. Soc. A298, 357 (1967).

3. 4.

GRIFFIN A. and CARRUTHERS P.,Phys. Rev. 131, 1976(1963). KLEIN M.V..Phys. Rev. 186, 839(1969).

5. 6.

CHALLIS L.J.. DE GOER A.M.. GUCKELSBERGER K. and SLACK G.A.. Proc. R. Soc. A330, 29(1972). JACOBSEN E.H. and STEVENS K.W.H.,Phvs. Rev. 129, 2036 (1963).

7.

IOLIN E.M.,.&oc. Phvs. Soc. 85, 759(1965).

8.

ELLIOTT R.J. and PARKINSON J.B.,Proc P/n’s. Soc. 92, 1024(1967).

9. 10.

STEVENS K.W.H. and VAN EEKELEN H.A.M., Proc. P/n’s. Soc 92, 680 (1967). ROUNDYV. and MILLS D.L,Phys. Rev. Bl,3703(l970).

11.

FIDLER F.B. and TUCKER J.W.,Solid State Comniun. 8,2055(1970).

12.

TOOMBS G.A. and SHEARD F.W. [Proc.Jut. Conf on Phonon Scattering in Solids, Paris (1972), (Editor ALBANY H.), p. 247. Saclay] give a diagrammatic treatment valid for arbitrary concentrations. A full account, including discussion of the analysis of thermal conductivity experiments, will be published in I Phys. C (1973). SHEARD F.W. and TOOMBS G.A.,J. Phys~C. 2, 1644(1969) andJ. Phys. C’. 4,313(1971).

13. 14. 15.

18.

KWOK P.C., Phys Rei: 149, 666 (1966). The single-ion perturbation theory of reference 4 is in error in that the scattering amplitudes (rather than probabilities) are added for the spin up and spin down states. which leads to a temperature dependent Tqp. HUBER DL., Phys~Lett. 20, 230(1966).; FOX G.T.. WOLFMEYER M.W., DILLINGER J.R. and HUBER D.L.,Phj’s~Rev. 181, 1308 (1969). DE GOER A.M., C.E.N.G. Report SBT 243/71 (1971) unpublished, has briefly compared some features of the coupled-mode and perturbation treatments of thermal conductivity. HUBERD.L. and VAN VLECKJ.H.,Rev. mod. Phis. 38,187(1966).

19.

TUCKER J.W.,J. Phy~C 5,2064(1972).

16. 17.

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