T T4!cesn3T42 K
6. Influence of paramagnetic lanthanide ions on the thermal conductivity of ordered and disordered systems
As it has been noted in the Introduction, a principal special feature of ions with partly filled inner shells (d- and f-elements are related here) is the existence of nonzero spin (S), orbital (L) and total (J) m o m e n t a (see table 4). A crystal electric field removes the degeneration of the orbital momentum orientations and instead of one energy level in a crystal there is a system of levels. The magnitude of the splitting is determined by the value of the electric field gradient, the electron shell configuration and its location inside the ion. In a magnetic field, H, additional splitting of the levels (the so-called Zeeman splitting) is possible. The energy difference between the split levels equals AE = g#uH, where g is the Land6 factor and #3 the Bohr magneton.
Lattice vibrations can transfer an ion from one level to another with the absorption of a phonon. Thus paramagnetic rare earth ions with unfilled inner shells are defects which reduce the thermal conductivity of a crystal lattice. There are two approaches for an evaluation of the variation of the thermal conductivity of compounds with
TABLE 4
E l e c t r o n c o n f i g u r a t i o n s , s p i n (S), o r b i t a l (L) a n d t o t a l ( J ) m o m e n t a o f l a n t h a n i d e i o n s R 3 ~, R 4 + a n d R 2~ .
I o n 4f n S L J G r o u n d t e r m
L a 3 . , C e 4+ 4f ° 0 0 0 a So
C e 3+ 4f 1 ½ 3 ~ 2F5/2
P r 3 + 4f 2 i 5 4 3 H 4
N d 3 + 4f3 3 6 9 4j9/2
p m 3 + 4f * 2 6 4 5 J4
S m 3 + 4f5 5 5 5 6H5/2
E u 3+, S m z+ 4f 6 3 3 0 7 F o
G d 3+, E u 2+ 4f 7 -~ 0 7 8Sv/2
T b 3 + 4f ~ 3 3 6 v F6
D y 3+ 4f 9 25- 5 1@ 6H15/2
H o 3+ 4f t° 2 6 8 5J 8
E r 3 + 4 f l l 3 6 wls ~J15/2
T m 3+ 4f 12 1 5 6 3 H 6
Y b 3+ 4f a3 ½ 3 ~ ~F7/2
L u 3 ~ , Y b 2 + 4 f 1~ 0 0 0 1 So
T H E R M A L C O N D U C T I V I T Y O F RECs 153
paramagnetic ions: a resonance approach (Orbach 1960, 1962, McClintok ct al. 1967, McClintok and Rosenberg 1968, Oskotski and Smirnov 1971, 1972, Oskotski et al.
1972, 1982, Luguev et al. 1975b, Vasil'ev et al. 1984b, Smirnov et al. 1985, 1989) and a coherent one taking into account the formation of spin p h o n o n excitations in the crystal spectrum (Elliot and Parkinson 1967, Iolin 1970, Kokshenev 1985). Both theoretical approaches give roughly the same result - essentially a decrease of the thermal conductivity of crystals containing paramagnetic ions. The resonance is more obvious and we will use it in the following.
We will also consider separately the influence of internal crystal field and external magnetic field on tc of compounds with lanthanide ions.
6.1. Phonon scattering by paramagnetic levels split by a lattice crystal field
Let us consider the two-level scheme (fig. 60) (Smirnov et al. 1989) and a phonon energy distribution function which is a product of the Planck function and the p h o n o n density function (phonon spectrum) (fig. 61). A p h o n o n with a resonance energy he) = A can be absorbed in an intermediate process (fig. 60). Because of that a narrow band of phonons (shaded in fig. 61) is in practice no longer involved in the heat transport process, decreasing ~c by (-AGe~) (fig. 62). To absorb a phonon it is necessary to have A less than OD (i.e. the energy A lies inside the p h o n o n spectrum).
The f-shells are found deeply in atoms, screened by outer shells and their splitting by a crystal field is small (~100 K), which is just inside the p h o n o n spectra.
Phonons can be scattered as well by the levels of the d-shell split by the crystal field. But interactions of the crystal field with the d- and f-shells are significantly different. The d-shells are external and the crystal field acts on them stronger than the spin orbit interaction. The d-levels in crystals are split according to the following scheme. T h e orbital eigenstates formed by a crystal field from states with different
- -Ii _
Fig. 60. The two-level system. A is the splitting by the lattice crystal field, ~ is the spread of the levels with temperature, ~ oc T - L/2 (Oskotski et al. 1972, Luguev et al. 1975b).(~)
/
+--~
T
L ~ ~
koT Fig. 61. The phonon energy distribution function.
154 I.A. S M I R N O V a n d V.S. O S K O T S K I (-~ xres)
TmQx
/ i
'7' !%',""
/ i \
/ i \"
T Fig. 62. Theoretical temperature dependence of -A~cre ~ for the two-level model.projections of the orbital momentum are distant from one another by ~1000 K.
These levels are degenerate according to the orientations of the total spin of the d-shell. The spin-orbit interaction removes this degeneration and splits the levels by
~100 K (this is inside the p h o n o n spectrum). For this to occur an eigenstate has to have a nonzero average orbital momentum. So, the paramagnetic levels influence tc L only in special cases. It must as well be noted that crystals with d-elements are often ferromagnetics with high Tc and the crystal field does not split the levels.
In f-elements the spin-orbit interaction is essentially stronger than the l a t t i c e orbit one. This leads to the splitting of states with a total momentum Y = L + S and to a more simple level structure than in the case of the d-elements. Only ions with L # 0 and Y # 0 can interact with phonons. This condition is not satisfied for the ions La 3+, Ce 4+, Lu 3+, Yb 2+ (L = J = 0), EU 3+, Sm 2+ ( J = 0), G d 3+ and Eu 2+ ( L = 0 ) (see table 4). First studies of the influence of paramagnetic lanthanide ions (PLnIs) on ~c L were performed at low temperatures on holmium and cerium ethylsulphates with hexagonal symmetry (McClintock et al. 1967, McClintock and Rosenberg 1968, M o r t o n and Rosenberg 1962). In this case the paramagnetic levels are split in the lattice crystal field by some degrees and could be split further by an external magnetic field (for details see section 6.1.6.6). The typical splitting in cubic crystals is about 100 K, so the effect on ~ci~ is essential at medium and high temperatures, in this section we are mainly considering the experimental data concerning the influence of PLnIs on ~L in crystals of cubic symmetry.
6.1.1. T e m p e r a t u r e dependence o f - A~cr~ ~
The temperature dependence of - AGes - the reduction of the thermal conductivity due to resonance p h o n o n scattering due to PLnIs has a resonance form (fig. 62) in the two-level model (fig. 60) (Oskotski and Smirnov 1971, Oskotski et al. 1972, Luguev et al. 1975b, Smirnov et al. 1989, Smirnov 1972, Golubkov et al. 1973, Vasil'ev et al. 1978). The temperature region T > Tin,x, T > O is the most informative (Luguev et al. 1975b, Smirnov et al. 1989). The temperature dependence and absolute value of -A~cr~ ~ depends on:
(1) The lanthanide ion concentration.
T H E R M A L C O N D U C T I V I T Y O F RECs 155 TABLE 5
The temperature dependence of - AKro ~ in the high-temperature region on the concentrations and arrange- ments of the PLnIs in the lattice.
Small concentration High concentration of PLnIs
of PLnIs
Ordered arrangement of PLnIs in the lattice
Disordered arrangement of PLnIs in the lattice P h o n o n - p h o n o n
scattering is greater than p h o n o n - i m p u r i t y one
P h o n o n - p h o n o n scattering is less than phonon-impurity one
I II III IV
T-2 T-O.5 T-1 T-O.5
(2) The arrangement of lanthanide ions in the crystal lattice.
(3) The relative contribution of the different mechanisms of p h o n o n scattering (table 5).
6.1.2. M e t h o d s f o r separating - A G e ~
There are two methods for separating - A G e ~ : a theoretical one and an experimen- tal one. Theoretically - A G , ~ is determined in the Callaway model as the difference of ~CL(1 ) and ~Ce(2), where XL(1) is calculated taking into account p h o n o n scattering by sample boundaries, defects and phonons (N and U processes), and tce(2) is calculated taking into account the same processes plus resonance scattering of phonons by the split paramagnetic levels of the lanthanide ions (Oskotski et al. 1982, Vasil'ev et al. 1978, 1984b, Neelmani and Verma 1972, Arutyunyan et al. 1986, 1987):
- AG~ s = teL(2 ) -- 1eL(1 ). (43)
Experimentally -- AGes is determined as differences between ~c L of crystals containing PLnIs and that of crystals with the same concentration of lanthanide ions with zero L or J. F o r example:
- - A K r ~ ~ = K L ( P r S ) - - x L ( L a S ) ,
- A G ~ = ~cL(TbS) - tcL(GdS), - Axre~ = xe(ErS) - XL(LuS).
All lanthanide elements can be grouped around three reference elements: around La - - Ce, Pr, N d (group I); around G d - - Sm, Eu (group II) and Tb, Dy, Ho (group III); and around Lu - - Er, Tm, Yb (group IV). The errors in the evaluation of --AlCre s are small, because the properties of the reference elements and the corre- sponding compounds do not differ too much, and the differences in their masses and ionic radii are small. The maximum differences in the atomic mass and ionic radii are:
156 1.A. SMIRNOV and V.S. OSKOTSKI G r o u p l
G r o u p II G r o u p I I I G r o u p IV
A M = M(Nd) - M(La) = 5.34 g, Ar = r(Nd) - r(La) = -- 0.066 ,~.
A M = M(Sm) - M(Gd) = - 6 . 8 5 g, Ar = r(Sm) - r(Gd) = 0.026 A.
A M = M(Ho) -- M ( G d ) = 7.68 g, Ar = r(Ho) - r(Gd) = - 0.044 A.
A M = M(Er) -- M(Lu) = - 7.74 g, Ar = r(Er) - r(Lu) = 0.033 A.
F o r example, for the substitution of Pr by Sc in PrS the values of A M a n d Ar are m u c h higher: A M = 95.951 g, Ar = 0.183 ~ .
6.1.3. Arrangement of PLnls in a crystal lattice
P L n l s can be a r r a n g e d in a crystal lattice in different ways (fig. 63). in all cases the a p p e a r a n c e of -Alcros can be expected. Crystals where P L n l s are the m a i n ions, and n o t defects, are of special interest. Here, in an ideal lattice w i t h o u t impurities an additional thermal resistance W2 appears due to p h o n o n scattering by the para- magnetic levels of the PLnIs.
6.1.4. Choice of materials for investigation
Let us analyse all cases of table 5. F o r this p u r p o s e experimental d a t a on ~(T) are presented for the different g r o u p s of materials:
- Small c o n c e n t r a t i o n s of P L n I s (these ions are impurities): Y~ ~LnxA15012 (Ln = Gd, Tb, Dy, Er, Tm, Lu) (Oskotski et al. 1972, Vasil'ev et al. 1984a, b, S m i r n o v et al. 1985, A r u t y u n y a n et al. 1987, D z h a b b a r o v et al. 1978, P a r f e n ' e v a et al. 1979, Smirinov et al. 1988). These c o m p o u n d s are insulators 0qot = 1eL), and they have a cubic lattice with the space g r o u p O~°-Ia3d.
Large c o n c e n t r a t i o n of P L n I s - they f o r m the matrix of a c o m p o u n d with ordered a r r a n g e m e n t of P L n I s in a lattice:
(a) PrS (LaS is the reference material) (Vasil'ev et al. 1978, Oskotski et al. 1982, S m i r n o v et al. 1988, 1989). This is a metal ('~tot = 1£L +
1£e)
with a cubic lattice of NaCl-type.(b) PrTe~.33 (LaTe1.33 is the reference material) (Luguev et al. 1975b, S m i r n o v et al. 1985, 1988, 1989, Oskotski et al. 1982, Vasil'ev et al. 1976). This is a degenerated s e m i c o n d u c t o r (tqo t = tc L + tcc) with a cubic lattice of Th 3 P~-type.
Large c o n c e n t r a t i o n of P L n I s disordered a r r a n g e m e n t of P L n I s in a lattice:
(a) PrTe~.5 (LaTel.s is the reference material) a defect lattice, where the p h o n o n p h o n o n scattering is stronger than the p h o n o n - d e f e c t one (Luguev et al. 1975b, 1978, Oskotski et al. 1982, S m i r n o v et al. 1985, 1988, 1989, Vasil'ev et al. 1976). These are insulators (/¢tot ~-KL) with cubic structure of Th3P4-type.
(b) Glasses a solid where there is complete disorder. T h e p h o n o n p h o n o n scattering is less t h a n the p h o n o n - i m p u r i t y one. P r 2 S 3 ( G a 2 0 3 ) 2 [ L a a S 3 ( G a 2 0 3 ) 2 as the reference m a t e r i a l ] (Smirnov et al. 1985, 1988, 1989, 1990, Parfen'eva et al. 1990a); P r P s O 1 4 ( L a P s O 1 4 as the reference material) (Parfen'eva et al. 1990b).
THERMAL CONDUCTIVITY OF RECs 157
l Impurities A+B+R
- - F -
W :I/xL=WI+W 2
PLn ][ I
I Regutar tot tice