Thermal conductivity of some low-dimensional magnetic systems

Thermal conductivity of some low-dimensional magnetic

systems

Citation for published version (APA):

Buijs, J. A. H. M. (1983). Thermal conductivity of some low-dimensional magnetic systems. Technische

Hogeschool Eindhoven. https://doi.org/10.6100/IR154977

DOI:

10.6100/IR154977

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Published: 01/01/1983

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THERMAL CONDUCTIVITY OF SOME

LOW-DIMENSIONAL MAGNETIC SYSTEMS

\ '

THERMAL CONDUCTIVITY OF SOME·

LOW-DIMENSIONAL MAGNETIC SYSTEMS

PROEFSCHRIFT

ter verkrijging van de graad van doctor in de technische

wetenschappen aan de Technische Hogeschool Eindhoven, op

gezag van de rector magnificus, prof. dr. S.T.M. Ackermans,

voor een commissie aangewezen door het college van dekanen

in het openbaar te verdedigen op dinsdag 14 juni 1983

te

16.00 uur

door

Johannes Antonius Henricus Maria Buijs

geboren te Gilze-Rijen

1983

Offsetdrukkerij Kanters B.V.,

Alblasserdam .

Dit proefschrift is goedgekeurd door de promotoren

TABLE OF CONTENTS

I INTRODUC'l'ION

II REVIEW OF TBERMAL CONDUCTIVITY IN MAGNETIC INSULATORS

2. 1. Int:roduction

2. 2.

Elementax>y rmgnetia exai tátions

.2. 2. 1.

Magnons

2. 2. 2.

Soliton8

2.3.

Phonon saattering p:roaesses

2.3.1.

Non-magnetia saattering

2.3.2.

Re8onant magnon-phonon interaation

2.;;.;;.

T!Jo magnon - one phonon saattenng

/

2.3.4.

Other magnetia saattenng machanisms

2.4.

EarUer e:JJPerimentat !ûOrk

III EXPERIMENTAL METHODS

3.1.

Int-Poduction

3.2. E:JJPe'Pimentai apparatus

3. 3.

T!Jo-step measunng method

IV TBE . .RMAL CONDUC'l'IVITY OF CoBr 2

.6s

2

o

4 .1.

Int:roduation

4.2.

E:JJPerimentai resuits on CoB-P

₂.8H 2

o

1 3 6 13 18 19 24 29 31 39 40 46 5Çl

4. 2.1.

Temperature dependenae of the the'l'mat aonàua-

55

tivity

4.2.2. Magnetia field ,dependenoe of the the:l'mal aon-

56

duativity

4.3.

Interpretation of the e:JJPe'Pimentai result8

4.3.1.

Qualitative interpretation

58

4.

3.

2. Calautations based on the r-esonant magnon:-

62

phonon ,interaation

4.3.3.

Cafoutation8 based on the 2 rmgnon-

65

1 phonon saatte'Ping

.,·

\

4.4. Discuzsion

V THERMAL CONOOCTIVITY OF TMMC AND DMMC 5. 1. I ntrodwtion

67

. 5. 1. 1. Statia properties of TMMC and DMMC 71

5. l.. 2. E:reitatiorw in TMMC and DMMC 74

5.2. E:r:perimentai resuZts on TMMC and DMMC

5.2.1. Terrrperature dependenae of the themrr:t.Z aondw- 79

tivity

5. 2. 2. Magnetia fieZd dependenae of the the1'maZ 81

a<Yniluctivity

5. 3. Inte:rpretation of the e:r:per:imentai resuZts 5.3.1. Introdwtion

5.3.2. Interpretation based on magnorw 5.3.3. Inte:rpretation based on soZitorw 5.4. ConaZusiorw and remarks

APPENDIX A \ APPENDIX B REFERENCES SAMENVATTING 83 84 87 95 97 100 104 109

CHAPTER I INTRODUCTION

The el'ementary excitations in magnetic systems have alreaäy been

studied fora large'number of ye~rs. With respect to the subjects of

investigations one can distinguish between the statie and dynamic properties of magnetic systems.

The first category includes measurements of thermodynamic properties like specific heat, susceptibility, magnetisation and allied quantities. Common resonance techniques like NMR and ESR can aiso be employed. for the study of statie behavior. '!hese resonance techniques, however, can also yield information about the dynamic properties, for instance when relaxation effects are considered. Neutron scattering .and thermal conductivity measurements can likewise serve for the study of dynamic behavior. It is the last mentioned technique that will be of particular

interest in the i~vestigations presented in this thesis.

In thermal conductivity measurements the magnetic system cannot be studied separately. There bas to be a certain amount of magneto-elastic coupling in order to probe the influence of magnetic

excita-tions. An external magnetic field is a powerful tool in examining the

influence of these magnetic excitations on the heat transport. The crucial point in the interpretation of field and temperature dependent thermal conductivity data remains the knowledge of the nature of the

magneto-elastic interactions, which is reflected by the pos~ible

scattering mechanisms of elementary magnetic and latticé excitations. The interest in the dynamic behavior of magnetic excitations bas

increased enormously in the past few years. The discovery of the

occurence of solitons in low-dimensional magnetic systems may be considered as an important factor in this respect. These non-linear basic exci tations can exist together wi th the' well known linear excitations (spinwaves) and manifest themselves primarily in the dynamics of the magnetic system. The solitons are believed to occur only in the paramagneti,c phase of linear chain systems. Therefore we might expect to observe characteristics of both the linear and the non-linear excitations, depending on whether we are measuring in-or outside the three-dimensionally in-ordered phase.

1 1

i

The organisation of this thesis will be as follows. In cllapter II

we will present some introductoeyiconcepts. The basic excitations in

magnetic cfystalline insulators wil,l be outlined and a the.oretical

introduction to some magneto-elastic scattering processes will be

presented. This chapter closes with a brief survey of what bas been reported on the topic of heat transport in magnetic materials so far.

The experimental apparatus will be presented in chapter III, together with the measuring method applied.

In chapter IV we present results of thermal conductivity measure-ments in the compound CoBr

2

.6a

2

o.

In this magnetically quasi

two-dimensional system the spinwaves (magnons) are the relevant magnetic i

excitations throughout the investigated field and temperature range

and consequently this system may serve

at

an indication for magnon

influences on the thermal conductivity. A qualitative interpretation based on spinwave behavior completes this chapter.

The final chapter V will deal with the field and temperature

dependance of the heat transport

ln

two quasi one-dimensional

substances ( (CH

3l 4NMnC13 and (CH3)!2NH2Mncl3) in which soliton influences may be expected. Measurements are performed as well in

a~ outside the ordered phase, which allows a discussion of the

CHAPTER II REVIEW OF THE.RMAL CONDUCTIVITY IN MAGNETIC INSULATORS

2.1.

Int~oduation

In condertsed matter heat can be transported by the various

excita-tions which may appear, Since we will confine ourselves to magnetic insulators, contributions from conduction electrons will not be considered. The remaining excitations can be distinguished to occur either in the lattice or in the magnetic system. The lattice excitations are most commonly described by pláne harmonie waves. The

quantum-mechanical excitiation of a lattice vibrational normal mode is called

a phonon. An introduction to the description of collective lattice

vibrations in terms of phonons can be found in any textbook ön solid state physics. For a review of the description of the process of heat transport by phonons we refer to Klemens (1951), Carruthers (1961), Parrott and Stuckes (1975) and references therein.

The idea that also magnetic excitations (more specifically the magnetic counterpart of the phonon, the magnon} could have a contri-bution to the thermal conductivity was proposed by Sato in 1955 and was further investigated in many articles since then. In all investi-gations the existence of interactions between lattice and magnetic excitations is crucial. These interactions arise from magneto-elastic effects which are ,inevitable in any real solid. Without this

magneto-elastic interaction heat cannot be transferred from the lattice

vibrational system to the magnetic system and vice versa. A consequence of this interaction is, however, that magnetic excitations can also

have a negative influence on the phonon heat transport by the induced scattering. In genera! the strength of the mutual interaction

determines.whether magnetic excitations will have a net positive

or nega~ive contribution to the thermal conductivity.

Thè thermal conductivity is formally defined by the equation:

- À·'i/T, (2 .1}

~ 2 +

tempe-... . 1

rature gradient (K/m) and À deno~es the thermal conductivd.ty tensor

(W/mK). We will confine ourselves to cases where the heat flux is

parallel to .the temperature gradient. This leads to a simplification of equatlon 2. 1 :

(2. 2)

where a indicates a certain spatial direction. The macroscopie thermal conductivd.ty coefficient Àa can be related to thé microscopie parameters of the system within the relaxation time approximation (see for instance Carruthers 1961). Ina non-magnetic insulator, where the excitations

can be regarded as a gas of phonons, this approximation leads to the

expression:

+

+ 2 + +

Àa(T)

=

E

c

(q,T) vp,a (q)Tp(q,T)

<i,p p . (2. 3)

The wavenumber q of the phonons runs over the entire first Brillouin

zone, while the suffix p denotes .the various acoustic and optical

branches with dispersion relation w (q). The specific heat of a certain

p

mode is 9iven by:

+

C {q,T)

p

The phonon veföcity is defined by:

+ + % +

v (q)

=

v..w; {q)

p qp

- 1}2

The remaining and most complicated quantity in equation 2.3 is the

+

(2 .4)

(2.5)

phonon relaxation time T (q,T), which expresses the characteristical _{p .}

time the phonon system needs to recover equilibrium after a disturbance.

The relaxation time can be built up from several independent

contribu-tions by adding the associated relaxation rates:

-1

T

As examples of specïfic scattering processes we mention the crystal boundary scattering, scattering from iatt:ice imperfections and all

kinds of phonon-phonon scattering processes. In the description of

these latter processes the ~il:hharmonic interaction t~rms have to be ·

retained in the hamiltonian under consideration.

In maqnetic insulators we have to include the maqnetic excitations in the model describing the thermal conductivity. In a maqnetic system with wave-like excitations we can also introduce a dispersion

+

re7ation wm(k}. The polarisation suffix m again indicates the possible branches, but is in this case restricted to one for a ferromaqnet and

+

two for a (2-sublattice) antiferromaqnet. The maqnetic field H now

enters the expression for Àa as an external parameter, leading to the extended version of eq. 2.3:

+ 2 + + + C (q,T)v (q)T (q,T,H) P p,a P + + 2 ++ + + + +~ C (k,T,H)v (k,B)< (k,T,B) k,m m m,a . m (2. 7) +

The quantities C and v in this case are formally identical to C

+ m m p

and v , provided the maqnetic excitations have the same boson

,P

description as the phonons. The maqnetic relaxation tiroo 'm as well 'as 'p becoroo fielÇl dependent due to the influence of various

maqneto-elastic scattering processes. Depending on the strenqth of the

spin-lattice interaction one can distinguish several cases (Walton 1972).

For a small but non zero interaction the maqnetic effects on T can

p

be negli;lcted and the two parts of eq. 2. 7 can be oQserved separately. A positive contribution to Àa of the maqnetic excitations may be

. expected in this case (Sanders and Walton 1977a). For large interaction strengths any maqnetic contribution to Àa will be masked by the

enhanced maqnetic scattering of the phonons. The second part of eq.

2. 7 can be omitted in this case and maqnetic scattering will cause

a severe reduction of À compared to the case of the non-maqnetic

system. In all intermediate cases both terms in eq. 2.7 have to be taken into account.

In a real system several methods can be employed to investigate the influence of the magnetic system pn À. One can comparé thermal conducti-vity data.on a magnetic compound with these on a non-magnetic isomorphous system. Assuming the respective lattice systems to behave identical, the difference in magnitude of the observed conductivities is

a

streng indi-cation for thé effect of magnetic excitations. In some cases, however, i t is impossible .to grow a non-(or dia-)magnetic isomorphous crystal and besidés, since À is a delicate function of crystal imperfections the two lattice systems do not have to behave similarly a priori. Therefore frequently insight in the role of spin-lattice scattering processes has been obtained solely from the temperature dependance of À. Much more conclusive evidence can be optained, however, by measuring the magnetic field dependance of À at a certain fixed temperature. In sufficiently high magnetic fields the activation epergy of the itiagnetic excitations will be raised so that they can b.e neglected at low

temperatures and only phonon conductivity and scattering remains. By comparing the high field thermal conductivity with the zero field data one can conclude whether the magnetic excitations have a positive contribution to the heat transport or merely act as an extra scattering mechanism for the phonons~ In the present investigation we will explore this method. '

In the next section we will pay attention to the magnetic excitations of interest. Apart from linear·excitations (magnons), .we shal.l also consider non-linear excitations ("solitons") since i t has been. realized recently that these can play a dominant rele in the dynamics of certain magnetic systems. In section 2. 3 we shall describe the various phonon scattering processes, with the emphasis on magnon-phonon scattering. In section 2.4 we present a survey of earlier experimental werk, indicating what features may be expected in magnon bearing systems.

2, 2. Elementa:x>y magnetic e:ccitations

2.2.1. Magnons

with a predominating Heisenberg or XY-like exchange interaçtion can be descr±bed by linear spinwave theory (see for instctlice Keffer 1966). The temperature range in which this model is applicable depends, among others, on the spatial dimensionality of the magnetic system. For quasi 1-dimensional systems the validity range of the spinwave

'description extends even above the 3d oraering temperature, due to the well developed correlations in the chain. In 2d and 3d systems the

spin~ave concept is only valid well inside the ordered phase.

The compounds of present interest are considered to be good

physical realizations of the 2d XY model (CoBr

2.6H2

o,

chapter IV) and

the ld Heisenberg model (TMMC and DMMC, chapter V) respectively. As -+

·an example, the deri vation of the spinwave dispersion relations

w

(k)

via the Holstein-Primakoff (HP) formalism (1940) for the 2d XY model

will be presented in the remaining part of this section. A small

orthorombic anisotropy will be included by means of anisotropic exchange -+

interac~ions. Results for W(k) in case of the 1d Heisenberg model

including single-ion anisotropy terms will be given in Appendix A. The

spinwave dispersion relations, together with the Base-Einstein dis-tribution function, can provide us with all thermodynamic quantities of the. magnetic system.

We start with the following hamiltonian for a 2-sublattice anti-ferromagnetic system:

(2.8)

m

The exchange constants Ja(a = x,y,z) are all negative and

!Jzl > !Jyl > !Jxl. This condition defines the three principal, magnetic

axes easy (z) , intermediate (y) and hard (x} • The indices il and

m run over "up" and "down" lattice sites respectively. The notation

<!lm> indicates that all nearest neighbour exchange couplings have to

be considered. With respect to the actual situation i~ CoBr

2.6a2

o,

we will restrict ourselves to the appropriate model in this case, i.e.

Fig.2.1.

Schematio representation

of the 2d antiferromagnetic

struoture.

a 2d arrangement of spins on two penetrating sublattices (fig. 2.1).

The magnitude of the effective spin is S =

t.

We will first consider

the case where the magnetic field is directed along the easy axis. At

low temperatures three s~ccessive phases will be realized when the

field is raised, the antiferromagnetic phase, the spin-flop phase and

the paramagnetic phase. The derivation of wtk) in this case is

analogous to the more general treatment by Iwashita and Uryû 0975a). The basic step in the HP formalism is the introduction of rising

+ - +

-and lowering operators a.e,, a!l -and, bm' bm via:

s+ -

₁₂₅

_f.e,a~ s+ _{12S'b+ f} !l - _m m m

sï

=

55'.

a~f.e,

s

-

_m =

v'2s'

f b _mm

-

(2. 9) z + - _sz

₌

₊ b+b-st

=

s - at at -s m mm

with f.e, =

Á -

(a!ai/2S)

1

and fm

=

/t -

(b:b~/2S)

1

•

In linear spinwave

,theory one uses the approximation f_N,m 0 = 1. Inserting 2.9 (f_k,m 0 = 1) in

eq.

2.8 leads to a hamiltonian in terms of spin-deviation operators.

The quadratic part of this hamiltonian is:

'(2. 10)

+ +

Next the spinwave creation and an~ihilation operators

aj;

and b; are

irttroduced by means of the Fourier transformations:

++ ...

a+ =I~

'*-

+ -ik.ri b+ =I~ ~ b; e-ik.rm

9.. N k .~ e m N _{k .} (2 .11) ... + b- =I~ ... + - 2 _~ - ik.r9,,

!1.

b~

eik.rm at =IN ~e _{m N} " k k + ...

1_{In these transformations r9.. and rm denote the respective positions of}

the lattice sites and N is the total number of spins. For convenience

/ ...

the vector signs of k in the indices of the magnon operators are omitted. The lattice vectors comprise a complete set:

+ + ...

E e1(k-k'l.r9,,=

!

Ó(k-k')

JI, 2 (2. 12)

Inserting 2. 11 in to hamiltonian 2. 10 and applying the above completeness , .

relation leads to:

(2. 13)

with

...

·ït"

+ ... +

d Y(k) '>' ei P, where 11 i hb

an

t

P =rJI, - rm runs over a . nearest ne g ours

of one lattice site.In the configuration of fig. 2.1

y(k)

can be written as:

+ .

y(k)

=

4 cos(ka a/2)cos(~ b/2) (2.14)

Hamiltonian H~ {2.13) can be transformed toa diagonal represéntation

by means of a so-called Bogoliubov transformation. 'I'his transformation

is extensively studied by Colpa· (1978) from a mathematica! point of

can be found in Appendix B. The result of the diagonalization is:

(2. 15}

The spinwave spectrum consists of two branches

w

₁ (k) and

w

₂

Ck)

for

which

th~ dispersion relations read as:

(2.16)

q =

Ï<E~

+

E;}

+

E~

- E;

22 21 2 2 l:i

r = [(El +E₂) E

3 + (El -E2} f4(El +E2) - E4})

In zero external magnetic field the spinwave spectrum exhibits two

+

e~rgy gaps at k

=

O, which are given by:

(2. 17)

tiro2 (0)

Át the ZQne-boundary (y (kmax) = 0) the two branches coincide .• In fig.

2. 2 an e~ample of a zero field magnon dispersion is presented.

The antiferromagnetic phase remains stable for fields up to the spinflop critical field HSF given by:

(2.18)

In the spinflop phase we have to describe the system on another coordinate system. In fig. 2. 3 the new set of coordinates is denoted

by a tilde. The canting angle

a,

for which the linear part of the spinwave hamiltonian vanishes, fellows from

z z y z

sin

e

=

g llsH

/-as

(J + J l (2. 19)

. The dispersion curves are now gi ver by (see appendix B) ~

0 kmax

Fig. 2. 2.

Zero fieid magnon dispersion

reiations for a 2-subiattiae

antiferromagnet.

z,~

Fig.2.3.

Loaation of the quantization

a,xes in the spin-fiop phase.

hw

₁,₂

ckl

= /

(B1

:!:

B )2 ₂ - B2 ₃ (2. 20)

with _B1 -8S (J y cos 0 - J sin 2 z . 2 0) + g z ~H z sin e

(-~

- JYs1n2

e

+ Jzcos 2 +

B2 = 0)sy(k)

~ y 2 z 2 +

B3

c-

+ J sin

e-

J cos eisy(kl 1

Th.e paramagnetic phase (complete alignment of spins) is re;;i.ched for 0

=

îf/2. The torque balance equation 2, 19 provides US the value

of the paramagnetic transition field H,,E:

(2. 21)

In the paramagnetic phase the derivation of magnon dispersion is analogous to the AF-phase and leads to:

-hwl , 2

Ck

l

= /

(C + c l 2 - c2

with

In fact the two dispersion curves gi ven by eq. 2. 22 can be mapped into a single curve extending over the double first Brillouin zone; The dispersion is consequently identical to that of a ferromagnet. The complete field dependence of the magnon dispersion curves is mainly

....

reflected by the behavior of the. two k

=

0 modes. In fig. 2. 4 a representative behavior of these energy-gaps is sketched, with the magnetic field along the easy direction;

Expressions foriiw

1,2Ckl with the field along the intermediate or hard direction can be obtained fr0m the equations 2. 19 - 2. 22 by the following interchange of parameters:

Fig.2.4.

Fieid dependence of the

energy-gaps ûlith the fietd

in the easy direction.

Fig.2.5.

Fieid dependence of the

energy-gapá ûlith the fieid

along the intemediate (full

curve) or hard direction

(braken curve).

1

I ,

+ H// inteniiediate axis + . Hl/ "hard axis

The appropriate g-values

(gt'

and. gx respecti vely) should be used. The

field dependance of the energy gaps in these two "perpendicular" cases is schematically given in fig.· 2.5 (gz >

gY

> gx). rn this figure HPI and HPH denote the respective paramagnetic transition fields.

The main features of the dispersion relations as derived in this section have been verified by experiment (Keffer 1966). Especially the behavior of the energy gaps, whi.<rh can be observed wi th anti ferro- · magnetic resonance techniques seems to be established rather well (see for instance Date and Nagata 1963 and Anders. et al. 1978). Therefore · we may conclude that linear spinwave theory offers a satisfactory description, at least in first approximation, of the magnetic exci tations of systems wi th nearly isotropic interactions' near the groundstate •

2.2.2. Solitons

Since a few years it is realized that in one-dimensional systems also another class of excitations may appear under certain circum-stances. In contrast to spinwaves these excitations are soluti9ns of a non-Unea:P equation of motion of the magnetic system. 'l'hey are denoted by the name "solitary waves" owing to their limited spatial

extent. A certain class of non-linear equations, among which the

sine-Gordon (s-G) equation, possesses solitary wave solutions which preserve identity after colliding. This special kind of solitary waves is called soliton. In 1978 Mikeska showed that the equation of motion describing a linear chain Heisenberg system with a small XY anisotropy

and a magnetic field in the XY plane could be transformed into the

s-G equation. Kjems and Steiner (1978) claimed the first experimental observation of solitons in the ferromagnetic chainlike compound CsNiF

solitons in magnetic systems, which is still g0ing on.

One-dimensional Heisenberg systems with XY symmetry can be described by the following hamiltonian

(2. 23) The single-ion anisotropy term (A > 0) fa.vours alignment of the spins in the "easy" XY plane. The spin-system of quasi one-dimensional (anti) ferromagnets in the para.magnetic phase is very adequately described by this m::>del hamiltonian. It has been demonstrated that in the limit of classical spins, the equation of motion associated to this hamiltonian can be transformed, within a continuum approximation, to a second order differential equation:

(2.24)

The transfo:rmation of 2.23 into the so-called sine-Gordon equation 2.24 is given in detail by Riseborough et al. (1981) for ferromagnets and by Mikeska (1980), Leung et al. (1980) and Maki (1980) for anti-ferromagnets. The angle q:l is defined in fig. 2.6. The "magnon veloc;:ity" c and the "mass parameter" m can be expressed in the parameters of hamiltonian 2.23. The results are comprised in table 2.1. The excita-tion spectrum of a sine-Gordon chain consists of three kinds of solutions, magnoris, kinks and breathers (Currie et al. 1980). The 'latter two are solitons.

c

m b.AF

Fig.2.6. Definition of the angte

lP

for

the F (a)

and

AF

(b)

ahain.

(<P

in

xy-ptane)

F AF 2aS

r1ïJ'

fi 4

t-

8

IJl

~g

glla~

a 2JS

4alJls

Table 2.1<' Parameter8 of

the sine-Gordon equation.

The

magnons

represent small amplitude oscillations {sin (j) Fl:l q>) of the spin system as a whole. The plane wave solutions (q> ,..,, exp.[i (WT ;... kzl] l

are characterized by the dispersion relation:

(2. 25)

This expression, which also gives the associated energy hw(k), is a sim-plified case of the.· magnon dispersion that was more generally introduced in the preceding section. Because of the continuum approximation in the derivation of 2.24, relation 2.25 will only be appropriate for small wavenumbers.

The

kirik soliton

solution of eq. 2.24 is given by

q>(z,t)

=

4 arctan exp

±

ym(z - vt- z

0) (2.26)

with

y

The kink velooity v can range from zero to the maximum velocity c. In fig. 2. 7. tP is plotted as a function of ym(z -vt - z l. The anti-kink

0 .

solution (minus sign in eq. 2.~6) is also indicated in this figure. With the definition of the angle q> as shown in fig. 2.6 we conclude that a kink soliton excitation in a ferromagnetic chain corresponds to a full 211' rotation of the spins within a limited distance in the chain direction. In the antiferromagnetic chain, however, the spins rotate over an angle 1l'. This distinction is related to the degeneracy of the antiferromagnetic groundstate. lj'ig. 2.8 shows an illustration of the 1l' and 211'-kink soli ton. For slowly movinq kinks the soli ton lenqth is approximated by 6/m. The energy associated with a kink

lt -4 0

'

. ' 'anti-kink ....

_

--2 ym (z-vt -iol 4 Fig.2.7.

Graphic repreeentation .of the

kink

and

anti-kink eolution

of the eine-Gordon equation.

Fig.2. 8.

2:i-soliton

A kink soliton in

the AF and F ahain.

soliton is given by (see for instance Mikeska 1978, 1980): F. (v) = ~ (0)

K K (2.27)

The kink rest energy EK(O) can be specified by:

F E.K(O)

8(2g~HXJS

3

)

l:i (2. 28)

AF EK (0) ~HXS (2. 29)

The second type of non-linear solutions of the s-G equation 2. 24 is the

breather:

2 , {tan 6 sin [wB y (t - vz/c - t₀} )} 1.p (z, tl = 4 arctan c0sh [my sin e (z - vt - z ) ] 0 (2. 30)

· Thè form of a breather is that of a translating evelope (velocity v < c)

'

with an internal oscillation with frequency

w

8 < w0• The breather energy can b~ written as:

(2. 31)

OWing to this particular energy a breather i·s frequently considered as a pulsating kink-antikink bound pair ..

In a classical treatment of the excitations of a system described by the. s-G equation, · the magnons, kinks and breathers together exhaust the excitation spectrum {Currie et al. 1980). Inclusion of quantum effects, however, yields a different view of the excitation spectrum of s-G chains. Takayama and Maki (1980) showed that in fact kinks and

breathers are the elementary (quantum) excitations and magnons corres-pond to the lowest breather modes. Another effect of the incorporation of quantum corrections is the renormalisation of the kink rest energy

(?Jla.Jl;i 1981). Experimentally a reduction of EK(O) with, about 20% is found for as well the s 1 ferromagnet (Kjems and Steiner 1978) as the S = 5/2 antiferromagnet (Boucher et al. 1981).

In the derivation of the s-G equation 2.24 for the angle ~ the spins were confined to the XY plane. Consequently only one of the two

anti-ferromagnetic spinwave branches is obtained. Mikeska (1980) considered the more genera! case which includes out of plane spin motions. Only in the limiting case of zero applied magnetic field the out of plane spin motions could be described by a s-G equation. The energy of the associated "hard" kink solitons is characterized by:

(2. 32)

which is in fact the corresponding "hard" antiferromagnetic magnon energy multiplied by s. In finite magnetic fields the out of plane motions cannot be described by s-G equations. The excitations under these circumstancés will retain some soliton-like features and appropriate theoretical models are being developed.

the elementary excitations of the sine-Gordon chain, the question remains to what extent each individual excitation contributes to the 'thermodynamical propérties of the system. This problem is, however, ·not

completely solved. The magnon contribution to thermodynamic quantities can be detehnined straightforward. Because of the correspondence ·of

the magnons to the lowest energetic breather modes, further breather

contributions to the thermodynamics are left out of consideration. The total density of kinks (plus anti-kinks) is calculated to be (Currie et al. 1980, Leung et al. 1980) :

n _K

=

(2. 33)

Trullinger and Bishop (1981) have discussed the validity range of this expression for the kink density, ::r'hey conclude that for temperatures

kBT ~ 0,2 EK(O) equation 2.33 adequately describes the kink density.

At higher temperatures kink-kink interactions have to be taken into account. On the low-temperature side the onset of three-dimensional ordering restricts the existence of solitons in real magnetic chain systems. Inside the ordered phase magnons predominantly determine the thermodynamica! behavior. The energy associated with out of plane spin motions (compare eq. 2.32) is generally much larger than the in plane kink energy, so that "hard" solitons play a minor role in the low temperature properties of the magnetic system.

2.3. Phonon saattering pioaesses

2.3.1. Non-magnetia saattering

Pho~on scattering processes in non-magnetic crystals have been

studied extensively ,since they determine the thermal conductivity of

crystalline solids. We refer to the review articles by Klemens (1951)

and carruthers (1961) for a general introduction. Here we shall only give a very brief summary on the most important scattering processes and their influence on the thermal conductivity in dielectric materials.

In general the phonon scattering processes can be divided in two categories. The first category contains phonon scattering by crystal-boundaries and -imperfections with a temperature independent relaxation rate. The second category contains all kinds of phonon-phonon scattering processes which are essentially temperature dependent. The competition between these two categories of scattering processes causes the well known temperatµre dependence of the thermal conductivity of solids with a maximum in the cryogenic region. The limiting low temperature behavior is governed by the boundary scattering with relaxation time:

+ - +

T

8(q)

=

d/v(q) (2.34)

In this expression

d

denotes the average crystal dimension. In real

crystals, however, in ternal grain boundaries may cause a mean free

path which is a few orders of magnitude smaller than

d.

The resulting

limiting low temperture conductivity will be proportional to the

specific heat (see eq. 2.3) of the lattice. (À,..., Tn in the n-dimensional

isotropic lattice). Scattering from lattice imperfections {impurities, isotopes, vacancies, dislocations) may result in a slightly different

low temperature behavior of À·. With increasing temperature

phonon-phonon scattering processes gradually become more important, leading to an ultimate decrease of the thermal conductivity. Our attention

will be focussed on the low temperatures (liquid 4He region) since

the most interesting magnetic effects.on À may be expected in that

region.

2.3.2.

Resonant magnon-phonon interaotion.

The total hamiltonian for a combined spin-lattice system can be written as:

(2. 35)

where the spin hamiltonian B₅ includes all magneto-èlastic interactions.

as:

L

hw

(q) ( c + c - +

.!.. )

+ p q,p q,p 2

q,p

(2.36)

Here c+ and c ~re the phonon creation and annihilation operators

q,p q,p

respectively. The general spin hamiltonian can be written as:

(2.37)

The succes si ve terms denote the gene ral exchange interaction, the single-ion anisotropy and the Zeeiran interaction. Dipole interactions can be included in the first term. Since the exchange and single-ion anisotropy parameters depend on the relative positions of the various ions, it is obvious that the associated energy is modulated by the

+

lattice vibrations. After defining the lattice displacement ufl by

+ +

r + u.₀ fl,O x,

(2. 38)

we can expand the spinhamiltonian in the displacemènts as fellows:

(2.39)

The "pure" magnetic hamiltonian

H~

contains the equilibrium parameters

'"*

-+ + *-+

J ₀_(r0 -r ) and D(r0 ) and can be treated according to the spinwave

Mlt N,O m,o N,O

formalism described in section 2.2.1. The interaction hamiltonian

H'

contains first order terms in the,displacements and the appropriate

"*

"*

gradients of JQJll and Dfl. At low temperatures higher-order terms in

the expansion 2.39 can be omitted. The first-order interaction

hamiltonian H' describes one-phon9n processes, as will be apparent

+ e+ ~

lw

(q} p (2.40L

(see for instance Akhiezer et al. 1961). In this expressi~n

;;+

is the

q,p

unit vector in the direction of the displacement and p and

v

d.enote

the crystal density and volume respectively. Introduction of the spin

d.eviation operators (eq. 2.9} in the interaction .hamiltonian

H'

will

lead to terms of first-order in the phonon operators and of arbitrary order iri the magnon operators. These terms correspond to:the distinct processes one magnon - one phonon, 2 magnon - 1 phonon etc. The one

1 '

magnon~one phonon process, also called resonant interaction, manifests itself essentially different than the higher-order processes. The resonant interaction changes the equilibrium properties of i;he basic excitations, while the higher-order processes reflect dissipative scattering mechanisms. Therefore the resonant interaction requires a distinct approach.

The effect of the resonant interaction on the equilibrium properties of a coupled spin-lattice system is visualized as a hybridization of the 'respective magnon and phonon energies. The resulting cµspersion curves are schematically given in fig. 2.9 for the antiferromagnetic system. The process can be described by the approximated interaction

Fig.2.9.

Hyb'l'idization of magnon

(m₁,m₂J and

phonon (p)

energies as a result of

\

the resonant magnon-phonon

interaation.

! .

hamiltonian (see for instance Boiteux et al. 1972, Laurence and Petitgrand 1973 and White et al. 1965):

+ +

(2.41)

The operators

ak

and

$k

are associated with the pure magnetic system

(eq. 2.15). The interaction parameter Dk can be specified as Dk D

Ik

where D denotes a typical strength: of the repulsion of the branches.

The magneto-elastic modes, as sketched in fig. 2.9, are now obtained

by diagonali~ing the total hamiltonian, which can be written as:

HT =

<{~~)

M

[~]

(2.42)

with M=

[

h~

(k)

li~

(k)

~

l

Die

o

hwm2

(kl

The thermal conductivity of the coupled system can now be obtained by

performing the swmnation 2. 7 over the magneto-elastic modes

w

1 (k),

w

2 (k) and w3 (k), which correspond :to the eigenvalues of M. The main

problem in this procedure is the calculation of the relaxation rates

of. the magneto-elastic modes. An approximation of these relaxation

rates can be obtained by the method of Kittel (Laurence and Petitgrand

1973, Kittel 1958). This method is based on the presupposition that the relaxation rates of the unperturbed magnon and phonon modes

-1 -1 -1 . .

(Tml' Tm

2' Tp ) are known (see section 2.3.1 for phonon scattering

processes and Dyson (1956) for magnon scattering processes). For the unperturbed modes complex energies are introduced by :

liW

"'"hw

+ iÎi T -1 p p p

tw

_ml 11wm1 + i

h

Tml -1 (2.43}

tW

=

hw

m2 .m2

h

-1 + i Tm2

Replacement of the real energies in the total hamiltonian 2.42 by the associated complex ones and following the same diagonalization procedure will lead to the complex magneto-elastic mode energies:

-1

The parameters Tj (j = 1,2,3) provide approximations of the ultimate

relaxation rates.

The method of Kittel, as described above, can be used to calculate all possible influences (positive as well as negative) on the thermal conductivity which may arise from the resonant interaction. In real systems, however, magnons frequently have no positive contribution,to the heat transport. In such èases the calculations of the thermal conductivity (eq. 2.7} can be restricted to the phonon modes while the

resonant interaction is included by the phenomenological exp~ession

for the relaxation rate (Walton et al. 1973, Tiwari and Ram 1980):

-1 T2W2 p

(2.45)

with

wc

the intersection frequency of the unperturbed magnon and phonon

modes. In real systems, wit.h phonon velocities much larger than the maghon velocity, the intersection will occur at very small wave-numbers

.and consequently

wc

.can be approximated by wm(O). In an

antiferro-magnetwtth two magnon branches, this leads to the approximate relaxation rate due to the resonant interaction:

-1 T (x) res 2 x (2.46)

with x =

Îiw

/k

8T, x1

=

k

1 (0)/k8T and x2

=

fiw

2CO)/k8T. The field

. \ p

dependance of this process follows from the field dependent behavior of the energy gaps (figs. 2.4 and 2.5).

2. 3. 3. Two magnon - one phonon saattering

The process of two magnon - one phonon scattering is described by

the terms of the interaction hamiltonian

H'

(2.39) which are quadratic

in the spin-deviation operators. Expressed on the basis of the non

perturbed magnetic system, the second orde~ interaction hamiltonian

will take the following general form:

(2 .47)

The parameters R. . and

c. .

have to be calculated and will be a function

+ + + l.J l.J •

of q, k and k'. The hamiltonian H

2 can be divided in two parts

describing radiation (R) processes (magnon + phonon & magnon) and

' conversion (C) processes (magnon + magnon <1+ phonon) respecti vely" These

processes will contribute to the relaxation rates of as well magnon as phonon modes. Again we shall restrict ourselves to cases in which magnon contributions to the thermal conductivity can be neglected. The two magnon - one phonon scattering in this case provides the simplest dissipa:tive magnon-phonon scattering mechanism which does not effect

equilibrium properties. Hamiltoni~n 2.47 refers to the unperturbed

magnon and phonon modes which implies that the resonant interaction is left out of consider:ation in the present description of the two magnon -one phonon process.

The requirement of conservation of energy and wavevector in two magnon - one phonon scattering processes restricts the contribution of

/

the various terms of hamiltonian 2.47 to the phonon relaxation rate.

Time dependent perturbation theory leads to the foHowing expressions

for the phonon relaxation rate for the two distinct classes of two magnon - one phonon scattering (see for instance Akhiezer et al. 1961, Keffer 1966): -1 + T R . . (q) l.J -1 + T C .. (q) l.J 2'11' ++++1 2 1 + + + I - h 1 R .. (q,k,k+q) n. (k) - n. (k+q)

h

2

_t

l.J l. J 1 + + + 1 +

*

O( W. (k} - W. (k+q) - W (q)) l. J p 27T 1 + + + + 12 + + +

'ti2

i

cij (q,-k,k+q) (ni (-kl

+

nj (k+ql + '1)

*Ö

cw. c-k>

+

w.

ck~>

l. J + - w (q)) p (2.48) (2.49)

These relaxation rates express the net result of a certain proc:ess, i.e. the difference between the phonon creation and the reverse annihilation process. In real physical systems with phonon velocities much larger than magnon velocities, not all processes can conserve energy and momentum. For instance in a 2-sublattice antiferromagnet only four

processes (and reverse ones) remain. These processes (~

₁

, ~

₁₁

, c

22 and c

21l are depicted in fig. 2.10.

Fig.

2.10.

Allowed 2 magnon-1 phonon

proaesses in a typiaal

In actual qalculations of the relaxation rates 2.48 and 2.49 the

magnon spectra 1ll

1{k) and

ro

2

Ck)

are assumed to be isotropic in n-₊

dimensional space. Under this assumption the summation over k can be

+

replaced by an integration over the n-dimensional k-space, leading to

the modified z:elax~tion rates:

(2.50) -1 + I + + + + · 1 2 + + + I ...., + + 1 -1 n-' T C (er) = F Ci. (q,-k ,k +q) {ni {-k ) +n. {k +q) +1) v

1{-k )+vj (k +q) k .

ij ~ n J c c c J c c c c

(2.51)

+

The pre-factor Fn contains all appropriate constants and kc denotes the, wavevector,for which energy is conserved. The only remaining problem

is the càlculation of the matrix ~lements Rij en cij. In the following

we will present an example of this calculation based on the spinwave

treatment of section 2.2.1. In fact this is a generalisation of the

procedure of Ono (1975) and Dixon (1980), who described the uniaxia,l

case.

Adopting the collllllOn simplification to ignore the non-diagon~l

elements of the general spin hamiltonian 2. 37, we start from the

·anisotropic exchange hamiltonian 2.8. The expansion of the exchange

parameters is in first order described by: (see eq, 2.38)

The interaction hamiltonian H1 _{{eq. 2.39) now becomes:}

H' = -2 E

<lm>

(2.52)

+

+

Inserting eq. 2.40 for uR, and um and defining:

(2. 54) leads to: + H' = +E H'(q,p) (2.55) q,p with + H' (q,p) (2.56)

The abbre.viation h.c. denotes the hermittian conjugated term. For the AF-phase we now introduce the spin-deviation operators (eq. 2.9) and retain terms displaying them in second order:

+ + +.+

z + - + - } -~ + - iq. rR,

iq.

rm

-2A _q,p (a0a0 +bb) sw (q)c (e· -e )+h.c.

N N m m p q,p (2.57)

The next step is the transformation to magnon operators via the Fourier

transforms 2.11. The result is:

+ { - - + +++ +- + + +

H 2

1_(q,p)

=

L

_A [~bk _{U(k+q,k) +} ~bk _U(-k+q,-kl]

with A _q -Ax + Ay q,p q,p A+ -Ax - Ay q q,p q,p Az _{-2 Az} q q,p and (compare 2. 13)

From now on i t will be convenient to use a matrix representatiOn of the interaction hamiltonian 1

....

*

H2(q,p)

....

~ _{~ k+ql\~} c + h.c. (2.59) k>O q,p with _~=

*

(~,bk, + + a_k,b-k) (2.60) and 0 0 0 A U(-k,-k-q} - -+ -+-+ q 1 0 (2.61)

Let Ek be the matrix that diagonalizes

H~,

i.e. (Appendix B)

(2.62)

with

'

....

Subjecting the interaction hamiltonian H

transformation (~.6) Will now provide US the desired matrix elements Rij ànd.Cij: (2.63) with + + + + + + + +

..:

+ ...

...

+ + ... R 11 (q,k,k+q) ~12 (q,k,k+q) c11 (q,-k,k+ql c12 (q,-k,k+q) + + + + + + + + + -+ + _,. -+ + + +

<!\!,/

~1\1=

Rz 1 (q,k,k+q) R 22 (q,k,k+q) ·. c21 (q,--k,k+q) c22 (q,-k,k+q)

*

+ + + +

_*

+ ... ++

...

...

..,...,.

....

... + ... c 11 (q,k,-k-ql c12 (q,k,-k-ql R11 (q,-k,--k-q) R12 (q,-k,-k-q}

*

+ + + .... c 21 lq,k,-k-ql

*

+ + + + c 22 (q,k,-k-ql + + + + R21 (q,-k,--k-q) Rz2 (q, -k, -k-q) + + + .... (2.64)

This ultimate form of the interaction hamiltonian is equivalent to the

general hamiltonian 2.47. A numerical calculation of the matrix-elements

of 2.64, together with the approximated expressions 2.50 and 2.51, wi.11

now enable us to determine the 2 magnon - 1 phonon relaxation rates. In

the SF and P phase, as well as the phases with perpendicular field

direction, a straightforward substitution of

J'"

by Aa as described

q,p

abqve, will provide. us the matrix elements 2.64 of the var~ous processes.

2.3.4. OtheP magnetic ecattePing mecha:nismB

After the more elaborate description of two specific magnon-phonon scattering processes, we will now present some brief information on possible other magnetic scattering mechanisms in the compounds of interest. As already pointed out in the beginning of section 2.3.2, all kinds of multi magnon - multi phonon scattering processes may contribute to the phonon relaxation rate. The influence of these higher-order processes will of course strongly depend on temperature. Generally the first-order approximation in the phonon operators will hold for. temperatures small compared to the Debije temperature of

the crystal. This implies that one-phonon processes will dominate

in the liquid helium range. For the magnon operators the situation

is different. It is a well known fact (Keffer 1966) that spinwave~

theory up to second order in the magnon operators only gives satisfactory results in the zero temperature limit. In the inter-pretation of temperature-dependent magnetic properties the inclusion

~f higher-order terms in the magnetic hamiltonian,is essential.

Th,erefore we may expect an increasing contribution of higher-order

scattering processes (3 magnon - 1 phonon etc.) to the phonon relaxation

at'.higher temperatures. In the liquid helium range t.~e influence of

higher-order processes may already be considerable. In thermal

conductivity measurements, no direct evidence for·the influence of higher-order processes has been published thus fä.r. This may, however,

be related to the extremely difficult calculation of the associated

relaxation rates. In ether dynamic properties the rele of multimagnon processes has been established rather well (see, for instance, Nishihara et al. 1975).

Another type of magnetic scattering is the so-called critical scattering which can occur in the immediate vicinity of the ordering tenperature. In this process phonons interact with fluctuations in the magnetic 'system, which are typical for the three- dimensional orde ring phenomenon. The critical scattering is described by Kawaski (1963) and Stern (1965), while also the paper of Dixon (1973) may be consulted.

The critical scattering will manifest itself as a dip in the thermal

conductivity around the ordering temperature. Well inside the ordered phase this mechanism may be ruled out as a noticable scattering source for the phonons.

Finally some remarks will be made op the scattering of phonons by

magnetic solitons •. Since magnons and solitons are elementary excitations of a magnetic system with predomi)nating exchange interactions, which

strongly depend on position, the 1

existence of some kind of soliton-phonon scattering will be evident. One may visualize a soliton-soliton-phonon

scattering by ~oting that a soli ton in the magnetic system is

acconpanied by a local lattice distortion, if a magneto-elastic coupling is present. This lattice distortion acts as a scattering

source for .the phonons. Schöbinger and Jelitto {1981) developed a

simple model for the soliton-phonon interaction in a one-dimensio~al

system. They arrive at the conclusion that a renormalisation of phonon dispersionc arises, in addi tion to all kinds of two phonon -one soliton scattering of the Raman type. These latter processes imply an energy exchange between soliton and phonon modes with the restric-tion that only the kinetic part of the soliton energy is involved. This restriction refiects an essential difference with magnon-phonon scattering •. This distinction is strongly related to the different nature of magnons and (kink-)solitons, as was already pointed out in section

2.2.2. However, it remains to be investigated to what extent the

· conclusion of Schöbinger and Jelitto will be modified by the inclusion of lattice discreteness and quantum effects. We will return to this topic in chapter V •

. · It should be stressed once more that the magnetic scattering processes as described in the preceding sections refer to magnetic systems displaying collective magnetic excitations (Heisenberg and XY-like systems). The investigation of magnetic scattering phenomena in alternative systems (paramagnets, Ising-like systems) is beyond the scope of this thesis.

2.4. Ea:t>lier e:x:perimental

~ork

In this section we will give a review of developments in the field of 1;:hermal conductivity measurements on (anti-) ferromagnetic compounds. Our attention will mainly be airected to those systems exhibiting predominant exchange interactions i.e. systems in which the magnetic excitations in principle may have a positive contribution to the heat transport process. We do not intend to be complete but merely want to show what in genera! can be expected from experiments and what at-tempts have been made to explain the observed behavior. Extensi ve reviews on the same topic have been'given by Sanders and Walton (1977a) .and Dixon (1980). These authors also present rather complete lists

of references on the subject.

1.

with collective magnetic excitations roughly can be divided in two

categories: compounds wi th a posi ti ve magnon contributi.on to À and

'

those in which the scattering of the phonons by magnons predominates.

Oficourse.in any real system both effects will be present, but depending on the magnon-phonon scattering rate the net effect can be

pqsitive or negative. Since Hisano Sato (1955) su~gested that magnons

prob,ably could contribute to the thermal transport, several investiga-tors started to look for experimental evidence. The first results on

MnO (Slack and Newman 1958), on some ferrites (Douthett and Friedberg

1961) and on MnF

2 and CoF2 (Slack 1961) indeed indicated magnetic

effects on À, but positive contributions could not be reported. The

general tendency in these compounds is enhanced scattering of phonons

around the ordering temperature. In 1962 however, Friedberg and Harris

were the first to report rather strong evidence for a substantial lhagnon conductivity in Yttrium Iron Garnet (YIG). Comparable

measure-ments on the same compound appeared shortly afterwards by Lüthi ( 1962)

and Douglas (1963), and in 1973 again by Walton et al. The field and

temperature dependent data of figures 2.11 and 2.12 are taken from

;;:

5

~

.< s.o~--~,----~--~.,,...,.,-_, 2.0-1.0,... 0.5- 0.2-0.1 YIG •

•

Q • 0 Q • 0 • •o

•

e 0 0 • 0 • 0 x x • 0 0 x • 0 0 Xx X x x x • zero field tA

r

> x 40k0e tApH l o Àf- ÀPH 1 1 1 0.2

o.s

1.0 TlK>

-Fig. 2.11.

TheY'mal aond:uativity of

YIG as a function of

temperatUX'e.(Walton et ai.

1973)

0.8 0.6 § ~ ~ 0.4 1 1 \ \ ~ \ \ \

·'

\ \ e'\._

•',

.

;--... YIG • exp. data

- - theory without interii<:tio - - - theory including "

l•O.nK

Fig.2.12.

• .-- .... _ ..:a:::'"::::l...--..---,,r

02

•

· Fieül

dependenae of

the thermal

aonductivity of

YIG at 0.77K.(Walton et

al.1973)

0 2 8 10

this last article.

YIG is a three-dimensional ferromagnet. In ferromagnetic compounds application of a magnetic field results in a uniform shift of the magnon branch to hiqher energies. Ina field of 40 kOe*the lowest

magnon mode in YIG is at about 6 K. Therefore the results of À at

40 kOe in Fig. 2.11 may be identified with the phonon thermal conductivity <>pH> and \ , - \ s corresponds to the positive magnon contribution. As can be seen in fig. 2.12 this magnon contribution amounts to about 70%. The field dependent data can reasonably be explained by a theory comprising phonon as well as magnon contributions

3 2

{proportional to T and T respectively) and an additional resonant magnon-phonon interaction {dashed curve in fig. 2.12).

Further evidence for magnon heat transport has been observed in a series of two-dimensional Heisenberg ferromagnets (CnH

2n+lNH3l2CuC14 (CnèuCl). Reports on thermal conductivity in these systems have been published by Gorter et al. (1969), coenen et al. (1976 and 1977) and

*Throughout this thesis the cgs-unit Oersted (Oe) will be used for the magnetic field strength. In vacuum a magnetic field of 1 Oe

-4

àe

Lang et al. (1977). Figures 2.13 and 2.14 are reproquced from the

latter authors. In Fig. 2.13 it can be seen that in

c

₁cucl the magnons

carry more than 90% of the heat at certain temperatures. This, is the

largest contribution ever reported. The dip in i,.(T) at Te (H

=

0 kOe) in

Fig. 2.14 emphasizes the fact that magnon-phonon scattering can not be

neglected. rurthermore we will mention the magnon corttribution

in

EuS ·(Me Collum et al. 1964) ëµid EuO (Martin and Dixon 1972), two

three-dimensional ferromagnets. There is only one .antiferromagnetic system in which magnon contribution is anticipated: Cu{NH

3J4

so

4

.a

2

o

by Miedema et al. {1964). The evidence, however, is not decisive.

Examples of magnetic systems in which the magnon-phonon scattering dominates, are more numerous. In general, three specific scattering

1.0

•

•

c

1

cuct

•

•

T•3.46K

ê

•

::::

_0.5

_•

s:

';;:

•

•

• •

•

•

• • • •

•

0 5 10 15 H/TCkOe/Kl

Fig, 2.13.

Field dependenae of the

the:r>rnal aond:uativit;y of

c1euct.The solid line

Pepresents the sum of

aalaulated magnon and

phonon aontributions.

(de

Lang

et al.1977)

20 ;;:: E v ~ -< O.l. OJ 0.2 0

Fig.2.14.

Temperature

d.epende~ae

of the theY'fflal aonduativity

of C1CuCl at H=O

and

60k0e.

0.2 0 5 HKl 10 20 Fi,g.2.15.

ThePmal aonduativity

data of FeCl

₂

(open

airales).The

dra:i.un

aurve is the result of

a resonant interaation

model.(Laurenae and

Petitgrand 1973)

mechanisms have been applied succesfully, i.e. the critical scattering the resonant interaction and the 2 magnon -1 phonon scattering. Since the second and third process will have special attention in the present

investigation we will first emphasize on these.

The most succesful demonstration of the influence of a resonant magnon-phonon interaction on the thermal conductivity has been reported by Laurence and Petitgrand (1973) for the antiferromagnetic compound

FeC1₂• The measured temperature dependance and a fit using a

theoreti-cal model for resonant scattering are displayed in Fig. 2.15. The

fact that the anomalous minimum in À (T) at about 1 7 K is reproduced

very well, is rather convincing evidence for the existence of the resonant process. Later reports on the temper at ure dependence in applied magnetic fields (Petitgrand and Laurence 1975) seem to corroborate this interpretation.

Other interpretations based on the resonant process were made on GdCl

3 (Rives et al. 1969) and CoC12.Ga2o (Rives and Bhatia 1974).

These" interpretations, however, are rather questionable as we shall see later on.

The first attempts to incorporate the 2 magnon ,... 1 phonon processes in

the interpretation of thermal conductivity measurements originate

from Nettleton (1964). He tried to explain À(T) in MnF

the theory developed by Upadhyahya and Sinha (1963) for 2 magnon -1 phonon scattering but the agreement was very poor. In -1975 Ono was able to give a satisfactory qualitative explanation of results on GdV0

4 obtained by Metcalfe and Rosenberg (1972) with the aid of a morè detailed theoretical description of the process. The most outstanding reports on this kind of scattering we re gi ven by Dixon and co-workers on the compound MnC1

2,4H2

o.

Measurements of À and a q\.talitative explanation had already been published several times

{Ri.ves and Walton 1968, Ri.ves et. al. 1%9, Ri.ves et al. 1970, Ri.ves f971). Only recently Dixon et al, (1980} succeeded in giving a

remarkably good qualitative and quantitative explanation from as well the temperature

as

the field dependence. In the preceding article .the theoretical background was presented in complete detail (Dixón 1980)

In the figures 2.16 and 2.17 we reproduce some of their results. Since

MnC1₂.4H₂o is a three-dimensional antiferromagnetic compound the spinwaves disappear above TN, resulting in a negligible field effect. At zero field the magnons cause a reduction of the phonon thermal conductivity of at most 30% while .application of a field results in a much larger scattering at the spinflop transition (Fig. 2.17). The

• 0 k()e

6 40 k()e

Q ; Q S Q L - - - ' - - - ' - - - ' 2 - - - ' J

T <Kl

Fig.2.16. Temperatui>e dependenae 1of the thermal aonduativity

of MnCZ2.4H2

o

divided by T3.Curve A is a fit to the

40 kOe data.Curve B ie a model aalaulation inaluiling

1