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STUDY OF CLUSTERS IN Pb DOPED KCl SINGLE CRYSTALS BY THERMAL CONDUCTIVITY MEASUREMENTS AT LOW AND VERY LOW
TEMPERATURES
M. Locatelli
To cite this version:
M. Locatelli. STUDY OF CLUSTERS IN Pb DOPED KCl SINGLE CRYSTALS BY THERMAL CONDUCTIVITY MEASUREMENTS AT LOW AND VERY LOW TEMPERATURES. Journal de Physique Colloques, 1976, 37 (C7), pp.C7-322-C7-326. �10.1051/jphyscol:1976775�. �jpa-00216935�
JOURNAL DE PHYSIQUE CoZZoque C7, suppltment au no 12, Tome 37, Dtcembre 1976
STUDY OF CLUSTERS IN Pb DOPED KC1 SINGLE CRYSTALS BY THERMAL CONDUCTIVITY MEASUREMENTS
AT LOW AND VERY LOW TEMPERATURES
M. LOCATELLI
Service des Basses Temptratures, CEN Grenoble, 85 X, 38041 Grenoble Cedex, France
RBsumB. - Nous avons mesure la conductivitk thermique, entre 60 mK et 50 K, de monocris- taux de KC1 contenant du Pb pour ktudier la precipitation du plomb. Nous observons les cine tiques de trois defauts en fonction des traitements thermiques. A partir d'une analyse numerique de nos rksultats nous pouvons caracteriser ces defauts :
- modes quasilocalis6s dus aux complexes PbZ+ lacune,
- prkcipitks de 20 A de diametre,
gros prkcipitks de phases de Suzuki, avec la plus grande dimension de 200 a 500 8.
Ces rksultats confirment et complktent ceux obtenus par la mesure de Courants Therrno-Ioniques par d'autres groupes.
Abstract. -We have measured the thermal conductivity, between 60 mK and 50 K, of Pb doped KC1 single crystals to study the precipitation of lead. We observe the kinetics of three types of defects versus the thermal treatments. From a numerical analysis of our results we can cha- racterize these defects :
- quasilocalized modes due to Pbz+ vacancy dipoles, - clusters of 20 diameter,
- big clusters of Suzuki phases with the largest dimension 200 A to 500 A.
These results confirm and complete those obtained by Ionic Thermal Currents (I. T. C.) measu- rements by other groups.
1. Introduction. - Using the results of S. W. Schwartz on C. T. Walker [I], K. Guckelsberger and K. Neumaier [2,3,4] have shown the usefulness of thermal conductivity measurements in the detection and characterization of clusters embedded in dielectric single crystals, with diameter in the range from 10
A
to 1 000
A.
W e have used this method in order to complete the results obtained by Professor R. Capelletti et al. [5] on the precipitation of lead in Pb doped KC1 samples ; these authors use ionic thermal currents measurements (I. T. C.) so that they cannot determine the size of the clusters.2. Short review of thermal conductivity. - As in the kinetic gaz theory the thermal conductivity of a phonon gaz in a dielectric can be expressed by the
following relation :
K = 113 Cph x v x 1 (1) where Cph is the specific heat of phonons, v the velocity of sound and I the mean free path of phonons. When several phonon scattering processes are taking place one supposes that :
C,, and I depend on phonon frequency. Therefore the right hand side in equation (1) becomes an integral over the entire range of phonon frequencies :
in a pure dielectric crystal one observes three regions in the thermal conductivity curve in function of tempe- rature, see figure 1.
- At very low temperatures I is determined only by the crystal dimensions and the specific heat varies with the temperature as T ~ , therefore K is proportional to T 3 (Boundary scattering [6]).
- A t high temperatures I is determined by phonon- phonon interaction processes, thus K decreases with the temperature [7, 81.
- In the intermediate region a maximum is observed which depends on the concentration of points defects (isotopes, residual impurities) and dislocations [9] in the crystal ; this is illustrated by the curve for a pure KC1 sample (Fig. 1). In this case the reciprocal of the total mean free path is written :
A W ~ Gw 1 l/l,o,,, = 111
+
-+
-+
---V lphonon phonon (4)
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1976775
CLUSTERS IN Pb DOPED KC1 SINGLE CRYSTALS C7-323 two apparatuses were used ; one apparatus monitored by a minicomputer [J. Doulat and M. Locatelli] for T > 1.5 K and a He3-He4 dilution refrigerator [lo] for T < 1.5 K.
Samples were thermally treated in our laboratory.
The main experimental difficulty was the brittleness of samples, which increases with cluster size.
4. Experimental results and qualitative discussion. -
A pure sample and two samples doped with lead (about 1017 at/cm3) have been measured. The latter two have been subjected to two different thermal treatments.
They were :
FIG. 1. - Thermal conductivity versus the temperature : -
theoretical pure sample ; (yy) - - - pure sample experimental points and calculated curves.
111 corresponds to boundary scattering [6], where 1 is the sample size (diameter or mean width) ; Am4 is due to point defects and G o to dislocations [9].
One can see that the region at very low temperature is very well defined by crystal dimensions. Any supple- mentary phonon scattering should by easily detectable and it introduces an additional term 1; in the expres- sion of the inverse mean face path :
1 - 1 - 1-1 + l - 1 total - pure 2 '
In the case of phonon scattering by clusters, two limiting cases characterized by the phonon wavelength 1 and the cluster diameter d can be easily described theoretically. When 1 9 d,
12'
K 04 x d6 (Rayleigh scattering) and when 1 4 d, 12 cc o0 d 2 (geometrical scattering). A transition arises for 1 E d, and a dip is observed on the thermal conductivity curve.3. Experimental. - Our samples have been kindly provided by Professor R. Capelletti. They have been sandblasted to eliminate specular reflexion at very low temperatures 161.
In order to perform measurements of thermal conductivity we employed the standard double heat flux method. Because of the temperature range studied,
- quenched after annealing in argon at 500 0C during 1 hour,
- annealed in argon at 250 OC during 24 hours, and then slowly cooled (10 OC/hour).
Results on the pure sample are given in figure 1, and on the two doped samples are shown in figure 2.
The two doped samples display the same behaviour versus the thermal treatments, that is : the thermal
FIG. 2. -Thermal conductivity versus the temperature for doped sample 1 and 2 experimental points and calculated curves. Sample 1 : ( 0 )
-.-.
as received, (A) - quench- ed from 500 OC, (y) - -- annealed for 24 hours at 250 OC.Sample 2 : -. .-.
.
pure calculated curve, ( 0 )-.-.
as received, ( f ) - quenched from 500 OC, (A) -- - anneal-ed for 24 hours at 250 OC.
C7-324 M. LOCATELLI conductivity K increases at the maximum after quench-
ing, indicating that some defects have disappeared, and after annealing K decreases in the low temperature range. The results at low temperature on sample 2 are more clearly shown on figure 3 where we have plotted KIT3 in function of temperature : we observe a reso- nance due to large clusters, especially after annealing.
FIG. 3. - KIT3 versus the temperature for sample 2, experi- mental points and calculated curves :- pure calculated,
(e)
as received, (+) quenched from 500 OC, (A) annealed for 24 hours at 250 OC.5. Quantitative analysis. - The results are analyzed with the expression (3) of the thermal conductivity, using Deby's model [ll]. The expression (4) of the inverse mean free path gives a good fit for the pure sample (Fig. I), with the parameters given in table I (we use the phonon phonon term from Schwartz and Walker [12]). For the doped samples, three additional terms due to the presence of lead are introduced in the inverse mean free path. The first one corresponds to the resonant scattering by impurity-vacancy dipole, and is given by Pohl [13] :
where D is proportional to the defect concentration and o0 is the resonant frequency related to the mass difference between the defect and the matrix [12].
The other two terms correspond to phonon scattering on spherical clusters, in a form simplified from that proposed by S . W. Schwartz and C. T. Walker [I] :
(
(1+
R exp(- 113)) x (ro/v)' for - rw<
12,
r o for- > 1
v
where N is the number of clusters per cm3, r is the cluster radius, o is the phonon frequency, v is the velocity of sound ; R is a parameter which depends on the density of both the clusters and the matrix. Its magnitude determines whether the transition between the two regions where ru/v ,< 1 and ru/v > 1 is reso- nant or not. In our case we have a resonant transition and R = 2.1.
The values of the parameters are given in table I.
The calculated curves are plotted on figures 2 and 3, and a good agreement with experimental points is observed.
The experimental boundary scattering terms are in good agreement with the calculated curves.
The dislocation term G and point defects term A obtained from the fits are significant only for the pure sample ; however the values of A are always, larger than the contribution of natural isotopes 1121.
Depending on the thermal history of the doped samples we obtain the following results :
1) For samples as received :
- quasilocalised mode with To = 30 K due to Pb vacancy dipoles,
- clusters of 20
A
diameter,- clusters of 200
A
diameter.2) For samples annealed in argon at 500 O C and rapidly cooled :
- quasilocalised mode with To = 30 K, - more clusters of 200
A,
- the 20
A
clusters have disappeared.3) For samples annealed for 24 hours in argon at 250 OC and slowly cooled :
- quasilocalised mode with To = 30 K,
- more clusters of 200
A
and clusters of 500A.
The experimental values of the different parameters given in table I are more significant for sample 2 measured in a larger temperature range (see Fig. 2 and 3).
In figure 4 we give an example of the numerical analysis in the case of sample 2 in the as received state.
CLUSTERS IN Pb DOPED KC1 SINGLE CRYSTALS
Experimental values Sample
-
Thermal treatment Boundary scattering
s-l
x 10-4Dislocations G x lo7 Pt defects A x S3 N, (@ : 20 A)/cm3 x 10-
C2 at/cm3 x 10-l7
N, (@ : 200 A)/cm3 x 10- l o
C3 at/cm3 x 10-17
N, (@ : 500 A)/cm3 x lo-'' C4 at[cm3 x 10-l7
C, at/cm3 x 10-l7
C,
+
C2+
C,+
C4 atlcm3x 10-l7
Thermal treatment : 1) as received,
2) quenched from 500 OC,
Pure
-
1 61
8.4 6.9 0 0 0 0 0 0 0 0
Sample 1
- -
1 2
3) annealed at 250 OC for 24 hours.
FIG. 4. - K calculatedlK experimental versus the temperature : a) pure, b) a
+
clusters 200 A, c) b+
quasilocalised modes,d) c
+
clusters 20 A.The ratio K calculated/K experimental is plotted in function of the temperature at the different steps of the analysis and the contributions of the three types of defects are illustrated.
6. Discussion. - From the comparison of our results with those obtained by Professor R. Capelletti et al. [5] on samples of the same origine, we can deduce the following facts :
- the dipole Pb2+ vacancy, responsable for IV peak in ITC measurements, induces a quasi localised mode which gives rise to resonant scattering (To = 30 K) detected by the thermal conductivity measurements, - in the as received samples the ITC peak observed below the I. V. peak is due to the small clusters of 20
A
diameter detected by the thermal conductivity measure-
Sample 2 -
2 68 2 0 0 40
0.27 0 0 4.3 4.6
ments. These clusters disappear after quenching as it is shown by the two types of measurements,
- the B Band seen by ITC in samples annealed at 200 OC, corresponds to the large clusters of Susuki phase [14] of the order of 200
A
to 500A
observed by thermal conductivity measurements.A small number of these clusters is also present, in our doped samples, in the as received and quenched states ; this is probably due to the fact that the anneal- ing time at 500 OC was too short or the cooling rate was not rapid enough. We note that the thermal conducti- vity is very sensitive to this type of defects.
Because of the brittleness of the samples containing large clusters, especially those annealed for long periods, we could'nt perform a sufficient number of experiments on the same sample, to study the evolution of the large clusters with anneal time.
Nevertheless, from this first series of experiments, we can deduce the order of magnitude of the lead concen- tration in the various defects (see Table I) : C, corres- ponds to the dipoles, C , to the clusters of 20 A, C3 and C4 to the clusters of 200
A
and 500A.
C,, C3, C4 can be immediatly calculated from the corresponding numbers of clusters N,, N3, N4 in expression (7), as they were supposed to be of spherical shape.
C, can only be estimated from the value of C2 and D in expression (6) for sample I , with the hypothesis that only dipoles are present in the quenched state.
We see that the order of magnitude of the concen- tration is about some 1017 at/cm3, as expected from the doping level ; C, does not change very much with the thermal treatment, so that only a small part of lead must have precipitated. Nevertheless in the annealed
22
state the values of C , and C, corresponding to the big - not spherical big clusters of Suzuki phases, with clusters seem to increase, and begin to be large. We the largest dimension 200
A
to 500A.
have overestimated these values because the clusters are
These results are in good agreement and complete supposed to be spherical in our analysis ; in fact
those of ITC measurements by R. Capelletti et al.
R. Capelletti et al. have shown, that these clusters
We conclude that it is possible to study the divalent correspond to Suzuki phases in extended cylinders. In
cation impurity precipitation in alkali halides from the thermal conductivity experiments we only see the
first steps and continuously along the whoIe process by projection of these phases, that is the largest dimension.
measuring the thermal conductivity in a large tempera- ture range especially at very low temperature.
Conclusion. - By thermal conductivity measure-
The author wishes to thank Professor R. Capelletti, ments we have detected in KCIPb2+ system :
Dr. A. M. de Goer, Dr. K. Guckelsberger and - resonant scattering due to dipole Pb2+ vacancy, Dr. K. Neumaier for their interest in this work and - small clusters 20
A
diameter, many helpful discussions.References
[I] SCHWARTZ, J. W. and WALKER, C. T., Phys. Rev. 155 (1967) 969.
[2] NEUMAIER, K., J. LOW Temp. Phys. 1-2 (1969) 27.
131 GUCKELSBERGER, K., thesis (1973) Grenoble University.
[4] GUCKELSBERGER, K. and NEUMAIER, K., J. Phys. & Chem.
Solids 36 (1975) 1353.
[53 CAPELLETTI, R., GAINoTTI, A. and PARETI, L., J. Physique Colloq. 37 (1976) this issue.
[6] CASIMIR, H. B. C., Physica 5 (1938) 485.
171 WALKER, C. T. and POHL, R. O., Phys. Rev. 131 (1963) 1433.
[8] WALKER, C. T., Phys. Rev. 132 (1963) 1963.
[9] KLEMENS, P. C., Proc. Phys. Soc. (London) A 68 (1955) 133.
1101 LOCATELLI, M., Internal Note SBT 233172 (1972).
[ I l l DEBYE, see KITTEL, C., Introduction ri la Physique de I'itat solide (Dunod, Paris).
[12] SCHWARTZ, 5. W. and WALKER, C. T., Phys. Rev. 155 (1967) 959.
[13] BAUMANN, F. C. and POHL, R. O., Report f 671, 1967, Cornell University.
[I41 SUZUKI, R., J. Phys. SOC. Japan 10 (1955) 794.
