T T4!cesn3T42 K
3. Influence of the magnetic structure on the thermal conductivity of lanthanide compounds
120 1.A. SMIRNOV and V.S. OSKOTSKI
3. Influence of the magnetic structure on the thermal conductivity of lanthanide
THERMAL CONDUCTIVITY OF RECs 121 (Barbara et al. 1977). Other compounds are ferromagnetics with T c in the range 6
168 K. The R atoms (excluding Ce, Eu and Yb) have valency + 3. RA12 are metals.
The measured tc is equal to
~tot = ~cL + 1%. (18)
At temperatures T ~ O thermal phonons in metals are mainly scattered by conduc- tion electrons. According to a theory of Klemens (1969), the predominant phonon electron scattering is given by
1 % = b T 2. (19)
At these temperatures ~c¢ is determined by conduction electron scattering on static defects (Klemens 1969) and
~Ce = aT. (20)
So, at T ~ O
~Cto t = a T + b T 2 o r Iqot/r= a + bT. (21)
The coefficients a and b can be evaluated from the dependence/qot/T = f ( T ) . Figure 8 shows such a dependence for YA12 (Bauer et al. 1986). The resulting b shows that the contribution of ~c L to fqot is about 5% at 20 K. F o r higher temperatures a numerical calculation has been done using approximate theoretical equations (Klemens 1969). It turns out that, at 300 K, ~c L contributes only about 2% to ~qot.
So, practically the experimental value of/~tot for RA12 is equal to ~c e.
The influence of the electron-spin disorder scattering on i% in the paramagnetic region (i.e. the magnetic contribution to the thermal conductivity) is estimated by comparing ~qot -~ tre of the magnetic and nonmagnetic compounds of the RA12 family.
YA12 is chosen as a nonmagnetic reference material. In the paramagnetic region (Dekker 1965) p(T) can be represented as
, o ( g ) =- jo o -}- P p h Jr- Dmagn, (22)
where the terms are the residual, the lattice and the magnon electrical resistivity, respectively.
Pmag,1 = Pspd OC const.(g -- 1 ) 2 j ( j + 1) = const. Aa, (23) where Pspa is the electrical resistivity due to spin-disorder scattering (Dekker 1965), g the Land~ factor, J the total angular momentum, and A a the de Gennes factor.
a~
o
O 5 10 15 20 25
~£ T,K
Fig. 8 ~:,oUTas a function of Tin YAl~ (Bauer et al.
D86).
122 I.A. SMIRNOV and V.S. OSKOTSKI
The electronic thermal resistivity We = 1/tee in the paramagnetic region for RAI2 can be written as:
(a) for nonmagnetic compounds (NM) (YAI2, LuAI2)
We = W~,d + W¢,ph, NM NM (24)
(b) for magnetic compounds (M) (GdA12, etc.)
= + wo,ph + Wo,m.+n. M (25)
The indices (e,d), (e,ph), (e,magn) indicate free-carrier scattering by defects, phonons and magnons, respectively.
The difference (W M - W~ M) corresponds to the contribution to the electronic thermal resistivity from electron scattering by the disordered spins
A W ~-- We---We = ( W ~ , d - M NM U We,d ) + (We,ph -- NM M We,ph ) -I- NM W~ m . g n . (26) Let us suppose that We,ph and Wc,ph are equal in all related RA12 compounds, then M NM
AW = (w+Ma - - We,d ) + We,magn. NM (27)
We,a in magnetic and nonmagnetic compounds is proportional to T - a [see eq. (20)]:
W~Md = BM/T, WeN, M = B•M/T, where B = 1/a [see eq. (20)]. According to Bauer et al.
(1986)
We,mag n = Wsp d ~ const.(g -- l)2J(J + 1)/T= const. Aa/T. (28) So, finally we have
A W = (B M - -
BNM)/T+
c o n s t .Aa/T.
(29)Experimentally AW is determined as the difference 1/to M - 1/~c TM, where tc NM is the thermal conductivity of YA12 (see, e.g., fig. 9). Figures 10 and 11 show the dependence of P+pd, eq. (23), and AW'T, eq. (30), on the de Gennes factor
A W ' T = A W T - (B M - B NM) = Wsp d r. (30)
As one can see in these figures, the experimental dependences Pspa(Aa) and W+pa(Aa)
W 103, cm K / r n W
I l ~ ~ t i I I -I I I
4 0 ~ GdAI2 3O
YAI 2
0 I I l
0 1 O0
t
z I I ~ I t I I I
200 T, K
Fig. 9. VV~ = l#c mM and We = 1/K M as functions of T, respectively, for YA12 and GdA12 (Bauer et al.
1986).
T H E R M A L C O N D U C T I V I T Y O F R E C s 123
80 E 60
& u
& 40 20
(g-1)2 .] ( 3 + 1 )
0 5 10 15
Er HO Dy Tb Gd
Fig. 10. The dependence of P~p,1 on tile de Gennes l~.ctor for compounds RA12 (Bauer et al. 1986).
( 9 - ' i ) 2 " J ( J + 1 )
0 5 10 15
8
0 I . , ' ~ " I I I
'~ I I I I [ I
Tm Er Ho Dy Tb Gd
Fig. 11. The dependence of AW'eT on the de Gennes factor for compounds RAI 2 (Bauer et al. 1986).
u
&
14
10
I I I I I I 1 I I I I I I
2 I I I t I I I I I I I I I
LQ Pr Pm Eu Tb HO Tm Lu
Ce Nd Sm Gd Dy Er Yb
Fig. 12. The dependence of W of compounds RA12 on the atomic number of the lanthanides at T = 2 7 0 K (Bauer et al. 1985a, b, 1986).
agree well with the theory for RA12 for the heavy lanthanide elements [cf. eq. (23) (Dekker 1965) and eq. (28) (Bauer et al. 1986)].
Experimental data for W, as shown in fig. 12, show a large influence of electron spin disorder scattering on t~, of RA12. The dotted line is considered as a reference, corresponding to W of a hypothetical nonmagnetic material. Additional scattering of electrons by disordered spins causes an increase in W compared to the reference value. GaAI2 has the largest W. W decreases from GdA12 in both directions to nonmagnetic LaA12 and LuA12. "A jump" of W is observed for CeA12. CeA12, unlike other RAlz, is a K o n d o material with peculiar physical properties. The experimental data shown in fig. 13 confirm the theoretical dependence, eq. (29).
Table 2 shows data for ,Osp d (obtained from electrical conductivity data) (Gratz and
124 I.A. SMIRNOV and V.S. OSKOTSKI
1 0
E u eo
250 150 T,K 100
I .I I _
G d A I /
/
SmAI 2I// 7
t I i I I i I I I I I I I d_.
5 10
103/1,
K -1 Fig. 13. The dependence AW(I/T) for compounds RA12 (Bauer et al. 1985a, b, 1986).TABLE 2
Electrical resistivity due to spin-disorder scattering of electrons.
RA1 z GdAl/ TbA12 DyA12 HoA12 ErA12 TmA12
P*pd (gf~ cm) 43.7 32.8 16.6 13.2 5.5 1.7
Pspa (I "t~ cm) 59.0 35.0 23.5 13.5 6.5 1.5
p*pa: calculated from the Wiedemann Franz law.
Pspd: experimental results.
Z u c k e r m a n 1982a,b, G r a t z 1982, G r a t z and N o v o t n y 1983, G r a t z et al. 1985, Bauer et al. 1985a,b, 1986) and P*pa (calculated using the W i e d e m a n n - F r a n z law from experimental AW')./)spa -~ Lo A W ' T (Bauer et al. 1986). As one can see in table 2, the agreement between Pspa and P*pa is good.
3.2. Electron scattering by spin disorder in the pararnagnetic region in Kondo systems (electrical conductivity, thermal conductivity)
The theoretical dependence p(T) has been obtained by Coornut and Coqblin (1972) for c o m p o u n d s with concentrated K o n d o lattices and carrier scattering by disordered spins (Pspd), taking into account the crystal field effect. According to this calculation the temperature dependence pspa(ln T) could be divided into two regions separated by a m a x i m u m at a temperature approximately equal to the value of the splitting by the crystal field.
CeAI2 is a p r o t o t y p e of materials with concentrated K o n d o lattices. F o r TN > 3.8 K (the paramagnetic region) strong scattering of carriers by disordered spins is observed
THERMAL CONDUCTIVITY OF RECs 125 (Bauer et al. 1986). Figure 14 shows the dependence p~pd=f(ln T) [p~pd = p(CeA12) - p(YA12)], which agrees well with the conclusions of Coornut and Coqblin (1972). It turns out that the dependence AW T = f ( l n T) lAW = W(CeA12)-W(YA12)] for CeA12 has the same shape as Psp,~ (Bauer et al. 1986) (fig. 14). Similar dependences AW(T) have also been obtained in other systems with concentrated K o n d o lattices: CeCu2 (Gratz et al. 1985) and CeCu 6 (Bauer et al.
1987). All this allows us to conclude that the observed behaviour AW T = f ( l n T) is typical for compounds with K o n d o lattices and electron scattering by disordered spins,
So, the influence of electron scattering by spin disorder on tee of RECs has been considered in sections 3.1 and 3.2. We have failed to find data on the influence in the temperature region T > TN, Tc of p h o n o n scattering by spin disorder on teL.
3.3. Magnetic phase transitions (temperature region 2." T ~ T N, T c, T~)
Magnetic phase transitions (MPTs) can be divided in two groups (Belov et al.
1976, 1979, Belov 1972):
group 1: M P T s of the order disorder type. These transitions are connected with the destruction of ferro- and antiferromagnetic ordering (observed transitions:
ferromagnetic-paramagnetic and antiferromagnetic-paramagnetic at temper=
atures Tc and TN, respectively).
group 2: M P T s of the order order type. These transitions are connected with a magnetic structure type change (e.g., ferromagnetic-antiferromagnetic, collinear antiferromagnetic-noncollinear antiferromagnetic and so on). Such transitions resemble structural transitions in crystals and sometimes they are called magneto- structural phase transitions.
The spin-orientation transitions can be distinguished by their unique features from the large family of magneto-structural phase transitions. Unfortunately, the thermal conductivity of RECs has not been investigated for all groups of magnetic phase transitions. Reliable data are available for the M P T of the order disorder type and for the spin-orientation phase transition (SOPT).
E 6 6 u ::L
58
| ~ 2 i i - l - - x J i i - -
7i -, \ \ - -
50~
-- I 1 I l" ~ I I 1 I5 10 2 0 5 0 100 2 0 0 3 0 0
T, K
12 "7 o E
8 % ®
%
xa Fig. 14. Dependence of P~pa (1) and AW~T (2) on In T for CeAI z (Bauer et al. 1986). Pspa- p(Ce AI2) - p(YAla). AWe = We(CeAI2)
- W2 (YAI2).
I 2 6 I.A. S M I R N O V a n d V.S. O S K O T S K I
3.3.1. Thermal conductivity of magnetic lanthanide materials in the region T,,~ TN, Tc in this region there are no specific effects in the thermal conductivity of lanthanide magnetic materials. Their ~c behaviours are analogous to the K of other magnetic materials. Very often the phase transition from the paramagnetic to the ferromagnetic or antiferromagnetic state does not manifest itself in the temperature dependence of tc L and Ke. However, there are cases where at T N and Tc appreciable variations are observed in the temperature dependence and the value of KL and K c. There are qualitative considerations (which, however, are not always confirmed experimentally), why a variation of IcE and ~c e is observed at TN and Tc. It is important to know which scattering and transfer mechanisms prevail near TN and Tc and which variations of these mechanisms are connected with the reconstruction of a magnetic system. F o r example, if for T > T N (Tc) there is a very strong electron scattering by disordered spins in a material with/£tot ' ~ /£e, then an increase of the value/(tot is possible at the transition temperature TN (Tc).
Let us consider some examples of 1% and ~cc behaviour of RECs at Tc and TN:
(1) E u O (insulator, 1(.to t = I(.L)" No variation of ~c L is observed at T c (fig. 15) (Salamon et al. 1974, Martin and Dixon 1972). It is interesting to note that in EuO a strong decrease of the thermal diffusivity, D, at Tc is observed (Salamon et al. 1974). This decrease is fully compensated for in 1(.c by an increase of the heat capacity, C, in this temperature range 0eL = DCd, d is the density of the material). For T ~< T c in E u O there is no additional heat transfer by magnons (because of strong m a g n o n - m a g n o n interaction), p h o n o n - m a g n o n scattering is low and KL is determined mainly by p h o n o n - p h o n o n scattering.
(2) PrA12, SmA12 [metals, ~Cto t = ~c~ (Bauer et al. 1987, (Mfiller et al. 1983)] (see fig. 16). ~Cto t changes at Tc in PrA12 and does not change in SmA12. Possibly, this is connected with a different influence of electron scattering by the spin system on the thermal conductivity (for T > Tc electrons are scattered by disordered spins, for T < Tc by the disordered spin system). This situation has been discussed above.
(3) DyS [ p o o r metal, /(:tot = /£L Jr- lee (Novikov et al. 1975)] (see fig. 17). Anomalies in KL(T) and Kc(T) are observed at T = TN. The behaviour of ~co(T) of DyS near TN is analogous to that in PrA12 (cf. figs. 16 and 17) and could be explained by electron scattering by the spin subsystem. It is more complicated, however, to explain the behaviour of tCL(T ) of DyS near T N.
%
u 6
0.3
0.2
- I E ~ I "{ I I I I
6 0 7 0 8 0
T,K
0.50
0 . 4 5 •
0 . 4 0
0.35
Fig. 15. D e p e n d e n c e Ktot(T)=J%(T ) a n d D(T) for a m o n o c r y s t a l of E u O . T c = 69.33 K ( S a l a m o n et al.
1974).
T H E R M A L C O N D U C T I V I T Y O F R E C s 127
150 E u
~
" 100 E2. o 50
0 "
0
I I I I I ----
I --L I I I
50 100 t50 200 250 300
T,K
Fig. 16. Dependence K,ot(T)~-I%(T) for PrAI 2 (I) and
SmA12 (2). T c is the Curie temperature (Bauer et al. 1986).
0 gO 100 T, K
I I l l l l l 1 I I r t l l ~
O.1 -- "~
0.05 ,/
v / , , , / . / ~-
/ / / TN
/
o . o l ~1 ~ , , , 1 I ~ i I , , ~ , 1 ,
5 10 20 50 t 0 0 T,K
Fig. 17. Temperature dependence of the thermal conductivity of DyS. Curve 1: ~ct,,t, 2: Kc, 3: iq.
(Novikov et al. 1975).
3.3.2. Behaviour of the thermal conductivity of lanthanide materials at a SOPT
At a S O P T the orientation of the magnetic m o m e n t s relative to the crystallographic axes changes under influence of external parameters (temperature, magnetic field, pressure) (Belov et al. 1976, 1979, Belov 1972). If the S O P T occurs at a variation of temperature or magnetic field, it is called, correspondingly, spontaneous or induced transition. Transitions with spin reorientations are characterised by the ordering p a r a m e t e r 00, the angle of magnetic m o m e n t rotation relative to the crystal axes. A S O P T could be of the first (0 changes abruptly) or of the second kind (0 changes gradually). A great n u m b e r of S O P T s are observed in RECs (orthoferrites, ferrites garnets, intermetallics) (Belov et al. 1976, 1979). The nature and characteristic features of the S O P T s have been investigated most completely in the orthoferrites RFeO3.
The crystal symmetry of orthoferrites is described by the o r t h o r h o m b i c space group - Pbnm. The orthoferrites have a distorted perovskite structure. 17
The most frequent reorientational transitions in the lanthanide orthoferrites are transitions with spin and magnetic m o m e n t u m reorientations in the (ac) crystal plane.
At the temperatures Tls and T2~, which correspond to the beginning and to the end of the reorientation process, one observes phase transitions of second kind. The values of TI~ and Tzs in R F e O 3 vary over a wide range depending on the atomic n u m b e r of the lanthanide (Belov et al. 1976). At TI~ and T2~ there are anomalies
128 I,A. S M I R N O V and V.S. OSKOTSKI
of the elastic modulus, the heat capacity, the magnetic susceptibility, the frequencies of soft spin modes become zero, and so on (Belov et al. 1976, 1979). The character of the transitions changes significantly in (ac)-reorientation in an external magnetic field H. When applying H along the a- and c-axes of a crystal the temperature of the first phase transition is shifted and the other disappears (Belov et al. 1976). To our regret, we could not find data on measurements of lc in a magnetic field for this phase transition type.
The influence of a S O P T on ~c has been investigated in single crystals of SmFcO3, Smo.6Gdo.4FeO 3 and Smo.6sEro.15Tbo.2FeO 3 (Barilo et al. 1984). Figure 18 shows data on ~c(r),
Oo(T)
andAv/vt
= f ( T ) (the relative variation of the sound velocity) in Smo.6Gdo.~FeO3. Analogous results are obtained for two above-mentioned composi- tions as well. F o r SmFeO3 and Smo.65Ero.l~Tbo.2FeO3 T~s = 454 and 303 K and T2s = 487 and 358 K, respectively (Barilo et al. 1984).RFeO3 are insulators and the experimentally measured ~cto t = ~CL. As one can see in fig. 18a, at the temperatures T~s and T2~ ~c has minima. At Tt~ spins are oriented along the a-axis, at T2s along the c-axis (fig. 18b). Anomalous behaviour of to(T) of Smo.6Gdo.4FeO 3 in the S O P T region is connected with unusual behaviour of the sound velocity (fig. 18c). Both anomalies are due to a strong spin p h o n o n interaction.
As an example of the S O P T influence on ~cL we have listed the most impressive experimental data. However, for other types of SOPTs t¢e could behave differently at T = ~: from sharp changes of the ~c L value to the complete absence of an effect.
Besides data on the influence of the S O P T on tc L of RFeO3 there is information
I ~ F - - ' I I I - - q
I 'a.
. f i 10 J
~' 6 I i I [ li
I
9 0 . . . . [ - - - -
o~ 60
3 0
0 I I ~ I
L
I I Q- Oxis
-2 x i s
- 4 1 - I I ~ I I
200 3 0 0 4 0 0
T , K
Fig. 18. Properties of Smo.6Gdo.4FeO 3 in the SOPT region as functions of T (Barilo et al.
1984). (a) ~q. (b) 0o, the angle between the c- axis and F, the light magnetisation vector in the spin-reorientation regions. (c) A v / v t. The dashed lines indicate TI~ and T2~.
THERMAL CONDUCTIVITY OF RECs 129 in the literature on the behaviour in the S O P T region in RCu2 (R = Sm, Ho, Dy, Er, Tm) (Gratz et al. 1990, G r a t z and N o v o t n y 1985). Theoretical calculations of KL near the S O P T have been carried out by Buchelnikov et al. (1987).
3.4. Phonon magnon scattering, heat transfer by magnons in lanthanide compounds ( T < TN, Tc)
Magnetic materials insulators and semiconductors - at 7 ' < T c (TN) can display two effects:
(1) A decrease of 1eL as a result of p h o n o n scattering by magnons.
(2) An additional thermal conductivity due to m a g n o n heat transfer (tOm).
Since in metals bc L ~ ~cc, it is difficult to analyse the influence of p h o n o n m a g n o n scattering on ice. In practice, one can consider only the contribution tq,. Electron m a g n o n scattering has to decrease the value tce in these metals. In magnetic lanthanide insulators a p h o n o n - m a g n o n scattering effect or an additional heat transfer by m a g n o n s could prevail. It is possible that b o t h effects are comparable. As a rule, K m contributes at lower temperatures while p h o n o n m a g n o n scattering contributes at higher temperatures (near Tc (TN)).
Let us first consider data on the thermal conductivity of insulating magnetic lanthanide materials. Strong p h o n o n m a g n o n scattering occurs when m a g n o n and p h o n o n dispersion branches intersect. They perturb one another, branches split, and an effective gap A appears between them (fig. 19) (Rives et al. 1969, Sheard 1976, Kittel 1958). As a result, the contribution to the heat transfer of phonons with the energy of the intersection decreases. In an external magnetic field the m a g n o n dispersion branch shifts to higher frequencies. The distribution of phonons carrying the heat has a m a x i m u m near energy 4k0 T. If the intersection of the p h o n o n and m a g n o n branches at H = 0 occurs at an energy less than 4ko T, then the thermal conductivity first decreases and then increases with increasing H (curve l of fig. 20).
At different mutual dispositions of the dispersion curves variant 2 of fig. 20 is possible.
In the works of Rives et al. (1969), Sheard (1976), Dixon (1976, 1981), Dixon et al.
(1974) and Dixon and L a n d a u (1976) some versions of p h o n o n - m a g n o n scattering are considered: one p h o n o n - o n e magnon, two m a g n o n s - o n e phonon.
(0 0
z3 t
, / <
q ( K )
Fig. 19. Schematic shape of magnon (1) and phonon (2) dispersion curves for a hypothetic fcrromagnet (Rives et al. 1969, Sheard 1976).
(3) The region of strong phonon-magnon interaction.
130 I.A. S M I R N O V and V.S. OSKOTSKI x ( H ) / x (o)
Hs
l
- - 2 ( ( H s ) = %L t
/ (magnon-- p h o n o n ... inter'action = O )
H
X ( b t s ) = X L I ( 3 ~ m = O )
Fig. 20. Scheme of the dependence of ~c(H)/~:(O) on magnetic field. H~ is the field in which ~c(H)/K(O)
saturates (gflHs>> koT). Curves 1 and 2: strong p h o n o n - m a g n o n interaction, 3: magnon heat transport.
In a high magnetic field (H > H~) the ratio
~c(H)/~c(O)
saturates (fig. 20), and the influence of p h o n o n - m a g n o n scattering on ~ci~ completely disappears, and ~c(Hs) becomes equal to ~cL. As a rule, ~cL(Hs) is larger than ~cL(H = 0), since at H = 0 p h o n o n - m a g n o n scattering decreases ~i,. A high magnetic field (H > H~) could suppress lc m as well (fig. 20, curve 3) (Rives et al. 1969, Martin and Dixon 1972, McCollum et al. 1964). Then lc(H~)= ~c L. When analysing experimental data on the thermal conductivity of lanthanide magnetic materials in a magnetic field one has to keep in mind that:(l) If heat transfer is to occur by magnons or p h o n o n - m a g n o n scattering it is necessary to reach magnetic fields H >~ H~. At weak magnetic fields one could erroneously take p h o n o n magnon scattering for ~Cn~ (fig. 20, curves 1, 2 at H < H~).
(2) Zeeman splitting of paramagnetic levels occurs in magnetic fields. At low temperatures a decrease of ~c L due to resonance p h o n o n scattering by these levels is possible.
The contribution of /£m to /£ and the influence of p h o n o n - m a g n o n scattering on
~:L have been studied for a great number of lanthanide magnetics. References on pioneer studies are found in Dixon and Landau (1976), Charap (1964), Slack and Oliver (1971), Martin and Dixon (1972). In this review we will consider data on ~c of the lanthanide insulating magnetic materials GdC13 (Tc = 2.2 K), EuO (Tc ~ 69 K), H o P O ~ (TN ~ 1.39 K) and the garnets DyA1G (TN = 2.5 K) and YIG ( 7 c ~ 545- 560 K), and for these materials we will analyse the influence of ~m and p h o n o n - magnon scattering on their thermal conductivity.
In the ferromagnet GdCI 3 and the antiferromagnet H o P O 4 heat transfer by magnons is insignificant. The dependence
~c(H)/~c(O)
(figs. 21a, b) is analogous to curves 1 and 2 in fig. 20.By contrast, in EuO the effect of p h o n o n magnon scattering is small and the heat at low temperatures is transferred mainly be magnons. At T = 0.93 K the contribution of ic m to ~c in EuO is about 75%, fig. 22. This value is a record for the investigated magnetics at the present time. However, simple dependences like those shown in figs.
21 and 22 (Dixon and Landau 1976, Walton et al. 1973, Metcalfe and Rosenberg
T H E R M A L C O N D U C T I V I T Y O F ' R E C s 131
L}'8 U I 1 I 1- I 1
0.4 . . . . . .
.~ 0 a.
0.4
0 I0 20 30 40
H, kOe
F - - T - - T - - T ~ I ~ T - ' T ' ~ 0.5
1.0 . . .
1.5
2.0
0 5 10 15
H, kOe
Fig. 21. (a) Dependence A~c/~(H) for GdC13 (Rives et al. 1969). Curve 1:1.3 K, 2:0.46 K, Tc= 2.2 K, AIc = I~(H) - t~(0). (b) Dependence tc(H)/t¢(0) for HoPO~ (Parsons 1976). T = 0.54 K.
3::
4
1.0 0.8 0.6 0.4 0.2 0
0 2 0 4 0 6 0
H , k O e
Fig. 22. Dependence K(H)/K(O) for EuO (Martin and Dixon 1974). T = 0.93 K.
1.0-- i I I
0.8 \ \ -~° 0.6 .* \ \ B
~
0.4 \ ~]0,2
0 I I I
0 2 4 6
AH
(3.
I
8 10
i i i i
b.
'),
• k B" "-Z"
2
I I I I
2 4 6 8 10
AH
Fig. 23. Dependence K(H)/K(O) for YIG (Walton et al. 1973) at temperatures 0.458 K (a) and 0.273 K (b).
A = #B/l% T. Curve 1: according to a theory without taking into account phonon-magnon scattering and magnon heat transport. Curve 2: according to a theory which takes into account phonon magnon scattering and magnon heat transport. B is the contribution from phonon-magnon scattering.
1979) are n o t always observed in experiments. F o r example, in garnets, parallel with the p r e v a i l i n g h e a t transfer b y m a g n o n s , t h e effect o f m a g n o n - p h o n o n s c a t t e r i n g p l a y s a s i g n i f i c a n t role a n d its c o n t r i b u t i o n increases with t e m p e r a t u r e (figs. 23a, b) (Dixon a n d L a n d a u 1976, W a l t o n et al. 1973).
132 I.A. SMIRNOV and V.S. OSKOTSKI
Peculiarities in the p h o n o n and m a g n o n dispersion curves can have influence o n the considered effects (Slack and Oliver 1971). Figures 24a, b shows dependences
~c(H, T) for GdCI3 (Dixon 1976, 1981) a n d D y A I G (Dixon and L a n d a u 1976), respec- tively, a n d helps to illustrate m o r e clearly o u r considerations on the c o n t r i b u t i o n of 1~ m a n d the role of p h o n o n m a g n o n scattering in t¢ of l a n t h a n i d e magnetic materials.
At low temperatures in a magnetic field H = H~ one observes in GdC13 an increase (+AK) and in D y A 1 G a decrease (-A~c) of the thermal conductivity. In GdC13 p h o n o n m a g n o n scattering is excluded as scattering m e c h a n i s m a n d in D y A 1 G Gn is suppressed.
Theoretical calculations have been carried out by Callaway (1959) of the two parts of ~L(T) with T < T o TN a n d T > To TN in a high magnetic field. It turns o u t that b o t h parts can be described by using c o m m o n c o n s t a n t s in the theoretical formulae for p h o n o n scattering by crystal boundaries, point defects a n d p h o n o n s (with taking into a c c o u n t the U a n d N processes). This points out that tc(H~, T) obtained at low temperatures in both materials is equal to the real ~c L.
Let us consider experimental d a t a on the thermal conductivity of the m a g n e t i c l a n t h a n i d e metals. As it has been noted at the beginning of this section, it is rather complicated to separate Gn from the experimentally measured/£tot. I n these metals
/('tot ~- K:L -[- K e -t- 1C m .
(31)
T h e m a i n difficulties arise in the separation of teL. At very low temperatures (where /c m is m o s t manifested) p h o n o n s are scattered, mainly, by crystal b o u n d a r i e s and i% decreases rapidly with decreasing t e m p e r a t u r e (KLoc T3). Therefore, at these
1
"~ 0.1
O.011- 0.2
I I I I I I I I I-
;o
/ 6 ~ i ,~ x • 3 13.
°1 I 1 I I I 1 1 I
1 10 100 200
T,K
500 200
E 50 u
E 2O
10 5 2 1 O.1
I I I 1 I
2.5K
H=O z ~
--'-,'/I
- - 2
b,
I I I I i
0.5 1.0 5.0 T,K
Fig. 24. (a) The temperature dependence of the thermal conductivity for GdC13 (Dixon 1976, 1981) at H = 0 (1) and H = 3.5 kOe (3). The solid line (2) represents a calculation according to Callaway (1959) taking into account phonon scattering by crystal boundaries, point defects and phonons (U and N processes). The heat flow and magnetic field are directed ahmg the c-axis of the crystal. (b) The temperature dependence of the thermal conductivity for DyA1G (Dixon and Landau 1976) at H - 0 (dotted line) and H = 35 kOe (points 1). H II [111], heat flow II [1001. The solid line (2) represents a calculation taking into account phonon scattering by crystal boundaries, point defects and phonons (U and N processes). The curve for H = 0 is given for averaged experimental values.