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THE THERMAL CONDUCTIVITY OF
PARAMAGNETIC CRYSTALS AT LOW
To cite this version:
J O U R N A L D E P H Y S I Q U E Colloque C 1, Supplément au no 2, Tome 28, Février 1967, page C 1-135
THE THERMAL CONDUCTIVITY OF PARAMAGNETIC CRYSTALS
AT LOW TEMPERATURES
The Clarendon Laboratory, Oxford
Résumé. - Bref coup d'œil sur les conductivités thermiques des cristaux paramagnétiques aux basses températures. On discute, en fonction des recherches récentes, les effets d'un champ magné- tique et de sa rotation par rapport à l'échantillon.
Abstract. - This is a short review of the thermal conductivities of paramagnetic crystals at low temperatures. The effects of applying a magnetic field, and also of rotating it relative to the specimen are discussed with reference to current research work.
In this review we discuss the manner in which spin- phonon interactions can be investigated from the interpretation of thermal conductivity measurements a t low temperatures. There are several types of inves- tigation, and the ones on which we shall concentrate are those in which experiments have been made on paramagnetic crystals.
Apart from the experiments made by Bijl a t Leyden many years ago, the present work on the heat conduc- tivity of paramagnetic crystals started with the measu- rements on crystals of zinc sulphate which had small admixtures (5 atomic percent) of Fe2* or Mg2+ ions (Rosenberg and Sujak, [Il (Fig. 1). It was found that the effect of 5
%Fe2+ decreased the thermal conduc- tivity very much more than did 5
%Mg2+. However, if the decrease had been merely an impurity effect then, since this is due to the square of the mass difference, the crystal containing the Mg2+ should have had the lower conductivity. Specific heat measurements sug- gested that the Fe2* had an excited level separated from the ground state level by about 4.6 OK and it seemed likely that the phonons were being scattered by causing transitions between these levels. Since the effect was observed in the liquid hydrogen region it was unlikely that a process of direct scattering of phonons of energy corresponding to 4 . 6 OK was involved. It
seemed more probable that a Raman process was occurring in which a phonon of energy hy, was absor- bed and another of energy hy, was emitted, such that
-hy, = AE, where AE is the energy between the two levels. This process would be much more effective in scattering a larger part of the phonon spectrum, and Orbach  showed that this mechanism could account for the experimental results.
1 , l , l * , i l 1 .
2 5 !O 15 20
FIG. 1. - The thermal conductivity of crystals of pure zinc sulphate and also of mixed crystals containing 5.77 atomic percent MgS04 and up to 5 . 6 atomic percent FeS04. Note that the decrease in conductivity is much greater for the crystals containing Fe than for that containing Mg (Rosenberg and Sujak [Il).
-136 H. M. ROSENBERG
Another possible scattering mechanism has been suggested by Seiden  which he calls coherent magne- tic scattering. In this a phonon of energy hy is absorbed and another phonon, of the same energy hy is emitted, but the emitted phonon has a different wave vector from the original phonon. This will give rise to a scat- tering process which does not actually affect the relaxa- tion of the paramagnetic ions. Papoular and Seiden  show that this process could account for a conside- rable amount of the extra thermal resistivity observed with the mixed Fe -!- ZnSO, crystals.
Experiments on paramagnetic crystals are, however, far more convincing if the levels can be altered by an applied magnetic field, and the more recent investi- gations have been on materials which have a ground state doublet or triplet whose splitting can be changed by a field. When this is done it is possible to observe corresponding changes in the thermal conductivity. It is clear that these levels must have a reasonable breadth, otherwise only an infinitely narrow band of phonons would be scattered, and the effect would be negligible. In the simplest possible case, if we have a specimen with a ground state doublet, then assuming a direct process interaction, there will be a minimum in the thermal conductivity at a temperature T, when the levels are split by an energy of the order of kT (actually it is usually between 3 and 4 kT). Unfortu- nately paramagnetic crystals with an isolated ground state doublet are nearly always Kramers' salts (with an odd number of electrons) and phonon induced tran- sitions between such Kramers' levels can only occur if these levels contain admixtures of other states such as can be obtained by applying a magnetic field. This means that with Kramers' ions the change in thermal conductivity on applying a magnetic field is usually small. Also the transition probability for interaction with the phonons is field dependent, because the higher the field the greater will be the admixture of other states.
Nevertheless quite large effects have been observed in some Kramers' salts-notably in the work of Ber- man, Brock and Huntley  on La,Co,(NO,),,,
24 H 2 0 . Brock  has been able to account quite reasonably for the observed field dependence by assu- ming a field dependent transition probability (propor- tional to H3) and a linewidth for the energy levels which was obtained from electron spin resonance experiments. However, some other Kramers' salts have produced no observable effect.
With non-Krarners' ions the energy level structure is nearly always more complicated. The simplest type of behaviour is obtained with Fe2+ or Ni2+ in a cubic
environment. This has a ground state triplet which is split by a magnetic field into a singlet ( M j = 0) and a doublet ( M j = f 1) (Fig. 2). Transitions can occur bet-
FIG. 2. - The energy level scheme typical of the ground state
of Niz+ or Fez+ in a cubic crystal.
ween al1 three levels, but since the singlet tends to be broadened by the crystalline field more than the doublet, the AM =
+1 transitions are usually more effective in scattering phonons. Two main types of cubic crys- ta1 have been used as a host in these experiments : M g 0 (Challis and Williams ,  ; McClintock, Morton and Rosenberg ) and KMgF, (McClintock and Rosenberg, ). An example of 4 % Ni2' in KMgF, is shown in figure 3. The general features of a mini-
,H K G
FIG. 3. - The relative change in thermal conductivity as a
function of magnetic field of a KMgF3
+4 % Niz+ crystal (Mc Clintock and Rosenberg [IO]).
THE THERMAL CONDUCTIVITY OF PARAMAGNETIC CRYSTALS C 1
than it was in zero field. This is due to the fact that, because there is some zero field splitting, some phonons will be scattered even before the field is applied. Thus the true « spin free » thermal conductivity will only be achieved in very high fields where the energy split- ting of the levels is so large that no phonons of that energy are available. By assuming reasonable values for the linewidths and lineshapes and by taking account of the change in population factors of the levels as the field is applied, Orbach (unpublished) has been able to demonstrate that the type of result shown in figure 3 can be accounted for quite well.
A more complicated situation arises when the triplet shown in figure 2 is split into a ground state doublet and an excited singlet, even in zero magnetic field (each with their own characteristic broadening). This is the case in some rare earth ethylsulphates. The most detailed investigations have been made on holmium ethylsulphate (Mc Clintock, Morton, Orbach and Rosenberg, [l l]), in which the excited singlet is 8 . 6 OK
above the doublet (Fig. 4). The general form of the
FIG. 4. -The energy level scheme for holmium ethylsulphate.
rence in the population of the various levels as the field is changed and also of the relative transition probabilities between the various levels. The other scattering mechanisms (e. g. boundary and impurity scattering) must also be included. By using reaso- nable values for these quantities and for the line- widths of the doublet and the singlet (with Gaussian shape) it has been possible to fit the shape of the experimental curves quite well [Il]. An example is shown in the upper part of figure 5.
variation of resistance with field is shown in the lower curve of figure 5 in which it can be seen that two mini- ma are observed in the thermal resistance - a very pronounced one at about 5 . 5 kG and a shallow one at about 17 kG. These may be understood by assuming that at any arbitrary field two bands of phonons will be removed, due to transitions between the doublet levels themselves, and also due to levels between a doublet and the excited singlet level (marked A and B respectively on figure 4). The first minimum occurs when A = B. In this region only one band of phonons will be scattered. The second minimum occurs where the upper level of the doublet intersects the singlet level at C . Here again, bands of phonons overlap
and the effect on the thermal resistivity will be smal- ler. To calculate the form of the thermal resistivity it is of course necessary to take account of the diffe-
FIG. 5. - Lower figure : typical experimental results for hol- miumethylsulphate of thevariation in thermal resistivity at 4.3 OK as a function of magnetic field. Upper figure : a theoretical attempt to fit the lower curve (McClintock, Morton, Orbach, and Rosenbarg [Il]).
-138 H. M. ROSENBERG g, = 4.4), but for the Y sites g is very anisotropic,
going from g,, = 7.3 to g, = 2.4.
Thus for any arbitrary orientation of the field, two bands of phonons will be scattered since the doublets of the X and Y sites will be split by different amounts. At 590, however, the g values of both types of site are equal and hence when the field is at this angle only one band of phonons will be removed. Figure 6 shows a set
- 1 3
ORIENTATION - MAGNE1 SCALE
FIG. 6. - The variation of thermal resistance as a function of magnetic field orientation of lanthanum cobalt nitrate (Brock
of results in which this effect is observed. The two sharp minima in the thermal resistivity were found to occur when the field was at 560 to the trigonal axis, which is in quite good agreement with the theoretical value of 590.
The effect of the orientation of the applied field has also been investigated by Challis and Williams (to be published) on M g 0 doped with 0 . 3
%Cr. They found a very strong anisotropy which has a cubic symmetry but since their investigations are not yet completed it would be inappropriate to discuss them further in this review.
Challis and Williams ,  have also obtained some very interesting results on the effect of a magnetic field on the thermal resistivity of nominally pure MgO.
They find that a small peak in the resistivity at 15 kG is enhanced after the crystal has been X-irradiated for three hours and then heated at 150 OC until thermolu- minescence has been completed (Fig. 7). They ascribe
-The thermal magneto-resistance ratio of nominally pure M g 0 for
(a) untreated specimen, T = 1 .34 OK,
(b) X-irradiated and heat treated specimen, T = 1.33 OK (Challis and Williams ).
this feature to a phonon scattering between levels associated with the Cr2+ ion, although Cr2' has not been observed in a conventional paramagnetic reso- nance experiment. Analysis of the specimen shows 20 p.p.m. of iron and less than 0.05 p.p.m. Cr3+. It is known that X-irradiation can produce the reaction Cr3+
+Fe2+ -t Cr2+
+Fe3+ and they suggest that
the enhancement of the 15 kG peak after X-irradia- tion is due to more Cr2+ being present. If this is cor- rect then the interaction of phonons with the Cr2* ion must be extremely strong since their concentration is so small. A similar (although larger) peak did indeed occur at 15 kG for M g 0 which was actually doped with 0 . 3
THE THERMAL CONDUCTIVITY OF PARAMAGNETIC CRYSTALS C 1
doped with various transition group ions (de Goer, Doulat, Dreyfus and Zadworny, unpublished).
I t is clear that the measurement of the thermal conductivity of paramagnetic crystals, particularly as a function of applied magnetic field is now an esta- blished method of investigating spin-phonon interac- tions. Detailed interpretation of experiments however is only just beginning and it seems certain that this is a field in which there is still a considerable amount of research to be done.
1 should like to thank several of the authors who
have been quoted, for informing me of their experi- mental results prior to publication.
This work was partly supported by the United States Army, European Research Office.
[l] ROSENBERG (H. M.) and SUJAK (B.), Phil. Mag., 1960,
 ORBACH (R.), Phil. Mag., 1960, 5, 1303.
 SEIDEN (J.), C. R. Acad. Sc., 1963, 256, 393.
 PAPOULAR (M.) and SEIDEN (J.), C. R. Acad. Sc., 1963,
 BERMAN (R.), BROCK (J. C. F.) and HUNTLEY (D. J.),
Physics Letters, 1963, 3, 310.
 BROCK (J. C. F.), 1965, D. Phil. Thesis, Oxford Uni- versity.
 CHALLIS (L. J.) and WILLIAMS (D. J.), Proc. 9th Intern. Conf. Low Temp. Physics, Columbus, Ohio, 1965, Part B., p. 1135 ; Plenum Press.
[SI CHALLIS (L. J.) and WILLIAMS (D. J.), Proc. Phys. Soc.,
1966, 88, 131.
 Mc CLINTOCK (P. V. E.), MORTON (1. P.) and ROSEN-
BERG (H. M.), Proc. Intern. Conf. Magnetism,
Nottingham, 1965, p. 455 ; Institute of Physics and the Physical Society.
[IO] Mc CLINTOCK (P. V. E.) and ROSENBERG (H. M.), Supplément au Bulletin de l'Institut du Froid, annexe 1965-2, 107.
[Il] Mc CLINTOCK (P. V. E.), MORTON (1. P.), ORBACH (R.)