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Low-temperature heat transport in α-HgI2 single crystals
0022~3@7182/030311Jl77$03.0010 Pergamon Press Ltd.
1. Phys Chn Solids Vol. 43. No. 3. pp. 311-317. 1982 Printed in Great Britain.
LOW-TEMPERATURE HEAT TRANSPORT IN a-HgIz SINGLE CRYSTALS A. M. DEG~ER and M. LOCATELLI Servicedes Basses Temperatures, Laboratoire de Cryophysique, Centre d’Etudes Nucleaires, 85 X-38041 Grenoble Cedex, France
I. F. NICOLAU Laboratoire d’Electronique et de Technologie de I’Informatique, Laboratoire de Cristallogedse et Recherche sur les Materiaux, Centre d’Etudes Nucleaires, 85 X-38041Grenoble Cedex, France (Received 20 May 1981;accepted in revised form 20 August 1981) Abstract-The thermal conductivity of several samples from cr-HgIzcrystals grown by two different methods has been measured from 50 mK to 200K. The thermal conductivity is found to be intrinsic but aniSotropic above 15K: it is smaller along c-axis than along n-axis, the anisotropy ratio being about 5 between 15 and 200K. Below I5 K, the thermal conductivity is sample dependent and the calculated Casimir limit is not reached at the lowest temperatures. The results have been interpreted considering phonon scattering by structural defects. A simple quantitative analysis of the curves suggests that phonons are scattered mainly by large clusters of interstitial defects due to the lack of stoichiometry of the crystals; the typical dimensions of these clusters are not smaller than 1 pm perpendicular to c-axis and 0.3 gm along c-axis. The presence of plane defects is also detected. Point defect scattering is relatively small and exolained bv residual metallic impurities and carbon at interstitial sites. The
intrinsic anisotropyis briefl; discussed.
’
1. INTRODUCTION
o-Hg12 is a wide band-gap semiconductor, which was used as a photoconductor[l]. Nowadays it is a potential nuclear detector at room temperaturel21, so that there is a need to obtain stoichiometric single crystals of high purity and perfection. Therefore a great deal of work has been done during the last years to grow such crystals using different methods ([3] and Ref. therein) and to characterize these crystals. It is well known that low temperature thermal conductivity is a useful tool to study many defects in insulating or semi-conducting crystals [4,5]. We have used this technique in the present work as, to our knowledge, no measurements of the thermal properties of a-Hg12 below room temperature have been made so far. Lattice dynamics studies have been performed by Raman experiments ([6] and Ref. therein), infrared reflectivity[‘l] and inelastic neutron scattering[8]. Therefore there is a fundamental interest to study heat transport in this layered crystal for which the phonon dispersion curves are already known[8] as well as the elastic constants[9]. 2. FXPERtMJPrrAL Several crystals have been studied and the geometrical characteristics of the used parallelepipedic samples are given in Table 1. The crystals labelled LETI-13, -14 and -16 have been grown from solution at 25-3X using the dimethylsulfoxide as a complexing agent[lO]. EGG S3-7 and EGG S7-24 crystals have been grown from the vapor phase at 12o”C[ll] using the temperature oscillation method. Two samples have been cut from crystal LETI16, one parallel to the a-axis (LETI-16) and the other parallel to the c-axis. This second sample LETI-16-(a) was broken after the first run and a part of it, labelled 311
LETI-16-(b) was remeasured. All the other samples were cut as LETI- so that most of the results concern heat flow parallel to a-axis. The measured crystals proved to be nonstoichiometric. The deviation from stoichiometry of the LET&crystals was studied by measurements of lattice constants and density at room temperature and also transition temperature to the yellow form (-131“C)[12]. In the case of the EGG-crystals it was qualitatively appreciated knowing the deviation of stoichiometry of the mother phase and the growth conditions. Therefore, the deviation from stoichiometry of the crystals may be ranged in the following row: (Hg-rich crystals) LETI- > LETI- > stoichiometric composition < S7-24 < LETI- < S3-7 (I-rich crystals). The total impurity content is within 100 -500ppm and crystal dependent, the carbon being the main characteristic impurity[ 131. The thermal conductivity measurements have been carried out by the constant heat flow method, and a dilution fridge was used to extend the measurements below 1 K in some cases. In view of the brittleness of the crystals, a special mounting technique has been used [ 141. The uncertainty on the absolute value of the thermal conductivity is about 7%. 3. RESULTS AND QUALITATIVE
DISCUSSION
The results of the thermal conductivity (K) as a function of temperature are given in Fig. 1 for all samples except EGG-S7-24. The values for this sample are omitted in view of clarity as they are very close to those of
312
A. M. DE
GOER
et a/.
Table I. Characteristics of the samples 1)
Sample
1
KtW.cm.'K'I
(cm)
0.33
0.27
2.0
0.71
0.30
3.1
1.05
LETl-14
0.37
0.39
3.5
1.54
EGG-S3-7
0.30
0.46
1.65
1.40
EGG-57-24
0.22
0.282
2.1
0.99
LETI-16-(a)
0.38
0.35
2.6
1.48
LET]-&(b)
0.29
0.39
1.0
1.17
A
.“._“..-...-
10-2
10-2
10'
(K'T')‘IIC. (W cm-'K-')
LETI-
t
1066
L (cm)
LETI-
..:
1tF
12
(cm)
-....,,,
'...T(K) -nTrj/[r+-F-
,m
lo"
10-l
1
1
10
lo2
T(K)
Fig. 1. Thermal conductivity of a-HgIz samples as a function of temperature: LETI- 0; LETI- X; LETI-13, 0; EGG-S3-7, A; LETI-16-(a), A; LETI-16-(b), A. Solid line is calculated for sample LETI- supposed to be perfect (see text).
specimen LETI-16. The main features of these results are: (i) the anisotropy of the thermal conductivity in the whole temperature range; (ii) the sample dependence of K below 10 K, despite the fact that the dimensions of the different specimens are comparable (see Table 1); (iii) the departure from the expected T’ dependence at very low temperatures (except for sample LETI-16-(b)) which is illustrated in the K/T’ plots (insert of Fig. 1). The anisotropy in the intrinsic region (T > 15 K) is to be related to the structure of H& and will be compared later to that observed in other layered crystals (see Section 5).
1.22
The sample dependence of the thermal conductivity below 10 K indicates that phonons are scattered by structural defects and impurities. In a perfect infinite crystal of diameter 2R, the phonon scattering at the boundaries leads to a limiting value of K/RT3, which is constant at very low temperatures and can be calculated from the elastic constants[lS]. We have used the values of the elastic constants measured by ultrasonic methods [9] to calculate this factor respectively for crystals parallel to c-axis and parallel to u-axis: the anisotropy ratio is found to be only 1.25. The values of K/T’ for all samples are given in Table 1. These theoretical values are all about 1 W/cm K4 but, even for the crystal LETIhaving the highest conductivity, the experimental value of K/ T3 at 100 mK is still smaller by a factor larger than 5. This fact and the shape of the K/T’ curves, which suggests the possibility of a resonant phonon scattering at very low temperatures for the samples LETIand LETI-13, give some evidence of the presence of large clusters of defects[5]. The maximum interaction would occur when the wavelength of the most important phonons is of the order of the diameter d of the cluster (supposed to be spherical), that is A = hu/4 kT = d. As the maximum effect, which should be .observed as a minimum of the K/T3 vs T plot towards low temperatures, is not reached at the lowest temperature = 5 x 10e2 K, the diameter d must be of the order of 0.3 pm or larger (using u = 1.2 lo5 cm/s, the mean value of the sound velocity calculated from elastic constants). Phonon scattering by other structural defects (dislocations, stacking faults, subgrain boundaries, glide bands, etc.. .) and by point defects can also play a role and it is necessary to achieve a quantitative analysis of the K(T) curves to determine their relative importance. 4. QUANTITATIVE
ANALYSIS
The thermal conductivity is calculated in a simplified isotropic model, using the Debye approximation of the specific heat, which leads to the well known integral[4].
Low-temperatureheat transport in a-HgI*singlecrystals
313
(the symbols having their usual meaning). Such a model completely neglects the elastic anisotropy but it is known from lattice dynamics studies that this anisotropy is relatively small in a-H& compared to other layered crystals [g, 161. The inverse relaxation time T-‘(x, T) includes all resistive processes and we have neglected the correction due to the special role of normal three phonon processes[l’l] as HgI, is far from isotopically pure. The total inverse relaxation time is written as:
Fig. 1. It is clear that additional phonon scattering must be included to fit the experimental results. Therefore we have adjusted for each sample the point defect scattering coefficient A, and also the frequency independent term r;’ to take into account the phonon scattering by large clusters. The interesting parameters are A-A, and r;’ = ( ri’).xl,- (ri’Llc. Satisfactory fits have been obtained for samples LETI-14, LETI- and EGG-S3-7, as illustrated in Figs. 2 and 3. The corresponding parameters are given in Table 3 and we discuss them in turn.
r-‘(o, 7’)= rs’ + Au4 + T;; + 7;’
4.1.1 Independent frequency scattering. As can be seen in Figs. 2 and 3, the calculated conductivities are too high at very low temperatures. This may be due to the fact that we have used a frequency independent limit of the exact cross-section for the scattering by clusters; this geometrical limit 7;’ is valid only at high frequencies and is given by:
(1)
including the classical terms: boundary scattering (Q,-‘), point defect scattering (Au4), phonon-phonon interactions (TVXP-‘); the additional term TV-’corresponds to phonon scattering by structural defects. In principle it is possible to calculate the first two terms. r;’ is obtained from the theoretical NT3 limit already discussed, and the values for the different samples are given in Table 3; the coefficient A due to isotope scattering is given by [4]: A = VJ4av3 x fi[AM-JM]’
rE--1= N,v?rd=/4
where N, is the number of clusters per cm3[5]. These contributions T,-’ are given in Table 3 and we have
(2)
where V, is the molecular volume and fi the fraction of isotope i which introduces a difference of mass AMi of the molecular cell of mass M The appropriate parameters are given in Table 2 and A0 is the contribution of the isotopes. V, has been calculated from the value of the density p corresponding to the stoichiometric composition [ 121.The Debye temperature B is calculated from the mean sound velocity, supposing that the molecule HgI, is the vibrating unit.t
K(W.cm’.K-1) t l_
10-1_
4.1 Thermal conductivity parallel to a-axis In a first step, the intrinsic curve above 15 K obtained for all samples parallel to a-axis has been fitted. It was found that the best form of the phonon-phonon term was: up:,, = B.o*T exp (-0loT)
(4)
x)-2_
lo-J_ ._
(3) lv_
with B. = 2.0 lo-” cgs and a = 5.5 The theoretical curve calculated for the sample LETI14 supposed to be perfect (7;’ = 0, A = A,) is drawn in tThe corresponding atomic Debye temperature should be 105K, in good agreement with the value of !99K obtained recently from specific heat measurements[llI].
10-5 , I II*Illll 10-2 W’
/ ‘IllIll 1
I 1 II1IIIl la
~1llu!.L lo* T(K)
Fig. 2. Comparisonbetween calculatedcurves (solid lines) and experimental results for the samples LETI-13,0; EGGS-7, A.
Table 2. Parameters used to calculate the thermal conductivity integral and the isotope point defect scattering coefficient A0
A. M. DE GOER
314
Fig. 3. Comparisonbetween calculated curves (solid lines) and experimentalresults for the samplesLETI-14, 0;LETI-16-(a),A; and LETI-16-(b),A.
calculated the corresponding values of N,, supposing the diameter d = 0.3 pm (see Section 3). N, is of the order of few 10“‘/cm3 so that the total cluster volume per cm3 is less than 10m3, which seems reasonable. It is worth noticing that the value of N, is larger for the vaporphase sample EGG-S3-7; this result still holds if we introduce an additional scattering due to isolated dislocations: r;’ = Go (see Table 3). The fit such obtained is of comparable quality to the first one with G = 0. 4.1.2 Point defect scattering. The coefficients A needed to fit the experimental results are given in Table 3. They are of the same order of magnitude (some 1O-43s’) for all samples and larger than the value A,, calculated for isotope scattering. The additional contributions A-A, (which are also given in Table 3) must
et al.
be due to chemical impurities and (or) to interstitials which are present in view of the imperfect stoichiometry of the crystals[lO]. Another possible source of Rayleigh scattering would be small precipitates, in the limit of very low frequencies (A % d)[5]. The contribution of about 1OOppmatomic of metallic impurities supposed to be substituted for Hg atoms (AM- 160) is calculated to be A,,, -7 x lo-“?, considering the mass defect alone, eqn (2). Now an important impurity is carbon and its state in HgIz crystals is unknown. Isolated carbon atoms can replace Hg atoms or occupy interstitial sites of the Hg-sublattice. Carbon could also be present as hydrocarbon groups such as CH,, C2H, etc.. , and CH, could substitute for iodine atoms. A typical concentration of carbon is 1OOppmby weight, and would lead to A, = 2 x lo-@ s’, 1.4 x lo-” s3 and 1.2x 10m4* s3 respectively in the three cases just considered. It is clear that carbon in place of Hg or CH, in place of I gives too large values of A,. Interstitial carbon atoms give smaller A, values so that the total contribution of chemical impurities A,,, t A, is about 8.4 x lo-” s3 and compares favourably with the experimental values of A-A,,. The fact that carbon atoms occupy preferentially insterstitial sites is probably related to the structure, as it is known that graphite readily intercalates between its basal planes bromine, iodine monochloride and ferric chloride [ 191. Now the stoichiometry of the crystals measured in the present work is known[l2] (see Section 2). From this study, it was suggested that the defects due to the lack of stoichiometry were Hg interstitials and iodine molecules giving I; ions, in Hg-rich and I-rich crystals respectively. It is possible to calculate the concentrations of these defects in the crystals which are close to stoichiometry, such as the I-rich crystal LETI-13. In this case, the concentration of 1, ions is given by eqn (5) of [12]:
C,? = 0.6 [Aplp,,,+ A V/ VJ. Using the experimental values Ap/pm = 1.9 x 10m4and A V/ V,,, = 1.5 X 1O-3 (Table 1 of [12]), we obtain Crs = 1.O10-j. corresponding to a number of I, interstitials per
Table 3. Parametersused to fit the thermalconductivitycurves Sample
TB-:xp,w)
LETI-
1.10’
LETI-
1.9
LETI-& LETI-16-(b)
TB-;,,,!‘-‘I
rc-l(s-‘)
2.2
105
7.8
2.8
10’
1.10’
2.3
1.5 10’
2.9
106
10’
NC (cm-‘)
A(s’l
A-A,
(s-‘1
G’
n*
9 2 109
1.4
lo-*]
6.10-”
0
-
1.6 10’
1.9
2.7
IO“’
1.910-*‘
3
-
101
9.7
1.1 10”
5.1
lo-“
4.3
105
1.47
1.7
1.7 lo-”
10’ 10’
10’0
10”
IO-*’
9.10-“’
5.8
lS_”
2
2.3
10‘”
*
I *rd-’= GO”(n = 1 or 2).
Low-temperatureheat transport in a-H& single crystals cm3 n = 2.5 x lo”?; the related contribution to point defect scattering is A, = 5.2 x 10F4*s3, which is about 30 times larger than the observed A-A0 value. This result clearly shows that the 1; defects are not isolated and that some clustering must occur, even in the best crystals. 4.1.3 Other scattering mechanisms. The possible contribution of scattering by dislocations has been tested by including a term r;’ = GW in eqn (1) (see Section 4.1.1). It was possible to achieve a fit only for the sample EGG-S3-7. The order of magnitude of the corresponding dislocation density N,+can be estimated from the relation [20]: G = 6 lo-* Ndb2yZ. With the values y = 2 of the Griineisen constant and b = 6” A of the Burgers vector, the experimental G value leads to N,, ~2.7 X 10p/cmZ,which appears too large to be physically meaningful. As the corresponding fit is not better than with G = 0, it is clear that the thermal conductivity is nearly insensitive to the presence of dislocations, the reason being that it is mainly determined by the other scattering mechanisms previously described. 4.2 Thermal conductivity along c-axis Above 15 K the conductivity along c-axis is about 5 times smaller than the conductivity perpendicular to c-axis: the anisotropy ratio is nearly constant, as illustrated in Fig. 4, so that we have used the same expression of the phonon-phonon scattering term as before, eqn (3) to analyze the curves. It was necessary to change the values of both B, and (2 and the best fits were achieved with B, = 1.3 x lo-l6 cgs and (I = 4.7. In contrast with the case of crystals perpendicular to c-axis, it was not possible to obtain a good description of the curves in the whole temperature range by adjusting only the frequency independent term ri’ and the point defect coefficient A. Additional scattering was needed and satisfactory fits have been achieved only with TV-’= GO*, which corresponds to phonon scattering by plane defects such as stacking faults[20]. The parameters used are given in Table 3 and the fits are illustrated in Fig. 3. We recall that the sample LETI-16-(b) was a part of the crystal LETI-16-(a) after it was broken; it appears that the crystal LETI-16-(b) contains more structural defects than LETI16-(a) and especially a larger number of plane defects (see Table 3). The density Ns of stacking faults can be estimated from the value of G, using Klemens’ expression[20] G = 0.7 NSa2y*lv. With y=2 and a =;/V,=4.9~ Ns = 100 to 4OO/cm.
lo-*cm,
we obtain
5. DISCUSSION The previous analysis of the experimental results appears to be consistent and leads to the following destCri is defined as the ratio n/N, N being the total number of lattice sites[l21.
315
cription of the detected defects in the cy-Hg12crystals investigated. It was shown in Section 4.1.2 that the point-defect scattering terms A-A, could be entirely explained by the chemical impurities present in the crystal (including interstitial carbon atoms). Therefore the defects due to the lack of stoichiometry do not scatter phonons as isolated point defects nor as small aggregates, and we suggest that the large defects considered in Section 4.1.1 are in fact clusters rich in Hg interstitials or 1; ions. It can be seen from Table 3 that the independent frequency term r;’ is larger by a factor from 2 to 10 for samples parallel to c-axis compared to these for samples perpendicular to c-axis. As this term is proportional to N,d*, eqn (4), it follows that the clusters look somewhat anisotropic (the phonons “see” preferentially the typical dimension in the plane perpendicular to the heat flow direction). The experimental results of the sample LETI16 perpendicular to c-axis have not been quantitatively analyzed as they were only slightly below those of the sample LETIand the measurements have not been extended below 1.3 K (see Fig. 1). From the thermal conductivity value at 1.5 K, we estimate 7,’ = 9 x 10’ s-’ for the sample LETI-16, so that the ratio of the cluster typical dimensions perpendicular and parallel to c-axis is about VlO. This means that, if the dimension along c-axis is 0.3 pm or larger as previously supposed (Section 4.1.1), the dimension perpendicular to c-axis would be about 1 pm or larger, and the cluster density NC= 1.1 x 10’“/cm3. Within these assumptions, the total cluster volume per cm3 is calculated to be =3 x 10e3.If there are two interstitial defects per unit cell within the cluster volume, this total perturbed volume would correspond to 2.5 x 10” defects/cm3. The exact agreement with the number of interstitials estimated from the lack of stoichiometry in the sample LETI(see Section 4.1.2) is evidently fortituous: it is not the same crystal and moreover very crude assumptions have been made to obtain these estimates. Nevertheless we think that the orders of magnitude are correct and that the presence of large clusters is effectively related to the non-stoichiometry of the crystals. The anisotropy of these clusters is not surprising as it is known that point defects condense preferentially in the closest packed crystallographic plane, namely the (001) basal plane for a-Hg12. These large defects could be the same as those observed by scanning electron microscopy operated in cathodoluminescent mode and by chemical etching[21]. Indeed clusters of various shapes having a mean diameter running from 1 to lO@m have been observed using these techniques. The other structural defects detected in the present work are plane like defects (see Section 4.2) perpendicular to c-axis, which could be glide bands as observed by X-ray topography[22]. The last point to be discussed shortly is the anisotropy of the thermal conductivity above 15 K. The ratio KJK,, is plotted in Fig. 4 and it appears that this ratio is independent of temperature in a large range, including the Debye temperature. Larger but temperature dependent anisotropy ratios have been measured for hcp He’
316
A. M. DE GOER
et al.
t K,‘K//
Ga Se__________
__._._ _____._._ _-^__ le
L
1 ..-I 1
Se
1 I LlU-Y_ 10
100
Fig. 4. Anisotropy ratio K,/K,f of the thermal conductivity perpendicular to c-axis compared to that parallel to c-axis as a function of temperature: 0 cr-HgI, (present work; solid line is not calculated); GaSe [26];Se and Te [17](the ratio in these cases are &/1<, as the chains are along c-axis).
(about 20 at 1 K[23]) and graphite below room temperature (>300 at 300 K[24]). Temperature independent
anisotropy has been observed in pyrolytic graphite above room temperature1251 and in the cases of layered GaSe[26] and of the chain-structured trigonal Se and Te[27], as illustrated in Fig. 4. A theoretical approach given by Simons[28] leads to a temperature independent anisotropy, in the limit of low temperatures only, which depends on the elastic anisotropy and also of the anisotropy of the Brillouin zone. Both factors are larger for GaSe compared to Hg12,in qualitative agreement with the experimental results. The only other theoretical work on the anisotropy in the Umklapp region is that by Benin[29], which is restricted to the case of hcp He4. As noted in a recent review by Slack[30], much more work is needed in this area. Finally we note that Lanyi et a/.(31) have measured the thermal conductivity of Hgi, above room temperature and found a value of about 4 10m3W/cm K, nearly temperature independent between 320 and 390 K, but which is dependent of the application of an electric polarisation. The agreement with our results for the sample LETIparallel to c-axis is not bad and we suggest that an electronic contribution to the thermal conductivity becomes noticeable near room temperature.
6. CONCLUSION
The qualiative and quantitative analyses of the thermal
conductivity results lead to the following conclusions. Point defect scattering is relatively weak and is explained by the presence of residual chemical impurities and principally carbon in interstitial sites. Large anisotropic clusters of defects strongly scatter phonons in the low temperature range: their dimensions are not less than 0.3 pm along c-axis and I pm perpendicular to c-axis. There is also evidence of the presence of plane defects from the experimental results of the thermal conductivity parallel to c-axis. The results are in qualitative agreement with those obtained from other techniques[21] and [22]. The large
defects are attributed to a condensation of the interstitials Hg or 1; ions due to the lack of stoichiometry of the crystals. The nearly temperature independent anisotropy of the thermal conductivity in the intrinsic range (above 15 K) could give informations on the phonon-phonon interactions in such layered crystals if a theoretical analysis becomes available.
Acknowledgements-We acknowledge the E.G.G. Co. for providing some of the crystals used in this work. We are very grateful to Prof. L. J. Challis (University of Nottingham) for useful comments on the manuscript.
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7. Biellmann B. and Prevot B., Infrared Physics 24.99 (1980). 8. Prevot B., Schwab C. and Dorner B.. Phvs. _ 8at. Solidi (b) 88, 327 (1978).
9. Haussiihl S. and Sholz H., Kristall und Technik, lO(1I), 1175 (1975). 10. Nicolau I. F. and Joly J. P., 1. Cryslal Growth 48.61 (1980). 11. Schieber M., Schnepple W. F. and Van den Berg L., _f. Crystal Growth 33, 125(1976). 12 Nicolau 1. F. and Rolland G., Mat. Res. Bull. 16,759 (1981). 13. Scharager C.. Tadjine A., Toulemonde M., Gros I. J. and Siffert P., Proc. of Nuclear Physics, 7th Divisional Conf., Nuclear Physics Methods in Materials Research, 26/U9/1980. Darmstadt. West Germany (1980).
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14. Locatelli M., Note ‘Technique &A-CENG-SBT-178-78: Mesure de la Conductivite thermique aux basses tempCratures sul le systtme KCI-Pb dans difftrents ttats thermiques (1978). 15. Doulat J., Locatelli M. and Rivallin J., Phonon Scattering in So/ids (editedby L. J. Challis et al). p. 383. Plenum Press, New York (1976). 16. Nakashima S., Daimon M. and Mitsvishi A., I. Phys. Chem. Solids 40,39 (1979). 17. Callaway J., Phys. Rev. 119, 1046(1959).
Low temperature heat transport in a-HgI, single crystals 18. Khattak G. D., Akbarzadeh H. and Keesom P. H., Phys. Rev. 23,291l (1981). 19. Heerschap M., Delavignette P. and Amelinckx S., Carbon 1, 235 (1964). 20. Klemens P. G., Solid State Physics (Edited by F. Seitz and D. Turnbull) Vol. 7, p. 1. Academic Press, New York (1958). 21. Nicolau 1. F. and Dupuy M., in preparation. 22. Nicolau I. F. and Rolland G.. in orenaration. 23. Hogan E. hf., Guyer R. A. and Fairbank H. A., Phys. Rev. 185,356 (1%9). 24. de Combarieu A., 1. Phys. 28,951 (1%7).
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25. Taylor R., Phil. Msg., 13, 157(1966). 26. Guseinov G. D. and Rasulov A. I., Phys. Stat. Solidi 18,911 (1966). 27. Adams A. R., Baumann F. and Stucke J., Phys. Stat. Solidi 23, K 99 (1%7). 28. Simons S., Proc. Phys. Sot. 76,458 (1960). 29. Benin D., Phys. Rev. 7,334 (1973). 30. Slack G. A.. Solid State Physics (Edited by H. Ehrenreich et al.), Vol. 34, p. 1. Academic Pres;, New York (1979). 31. Lanyi S., Dikant I. and Ruzicka M., Acta Phys. Slou. 28,210 (1978).
1. Phys Chn Solids Vol. 43. No. 3. pp. 311-317. 1982 Printed in Great Britain.
LOW-TEMPERATURE HEAT TRANSPORT IN a-HgIz SINGLE CRYSTALS A. M. DEG~ER and M. LOCATELLI Servicedes Basses Temperatures, Laboratoire de Cryophysique, Centre d’Etudes Nucleaires, 85 X-38041 Grenoble Cedex, France
I. F. NICOLAU Laboratoire d’Electronique et de Technologie de I’Informatique, Laboratoire de Cristallogedse et Recherche sur les Materiaux, Centre d’Etudes Nucleaires, 85 X-38041Grenoble Cedex, France (Received 20 May 1981;accepted in revised form 20 August 1981) Abstract-The thermal conductivity of several samples from cr-HgIzcrystals grown by two different methods has been measured from 50 mK to 200K. The thermal conductivity is found to be intrinsic but aniSotropic above 15K: it is smaller along c-axis than along n-axis, the anisotropy ratio being about 5 between 15 and 200K. Below I5 K, the thermal conductivity is sample dependent and the calculated Casimir limit is not reached at the lowest temperatures. The results have been interpreted considering phonon scattering by structural defects. A simple quantitative analysis of the curves suggests that phonons are scattered mainly by large clusters of interstitial defects due to the lack of stoichiometry of the crystals; the typical dimensions of these clusters are not smaller than 1 pm perpendicular to c-axis and 0.3 gm along c-axis. The presence of plane defects is also detected. Point defect scattering is relatively small and exolained bv residual metallic impurities and carbon at interstitial sites. The
intrinsic anisotropyis briefl; discussed.
’
1. INTRODUCTION
o-Hg12 is a wide band-gap semiconductor, which was used as a photoconductor[l]. Nowadays it is a potential nuclear detector at room temperaturel21, so that there is a need to obtain stoichiometric single crystals of high purity and perfection. Therefore a great deal of work has been done during the last years to grow such crystals using different methods ([3] and Ref. therein) and to characterize these crystals. It is well known that low temperature thermal conductivity is a useful tool to study many defects in insulating or semi-conducting crystals [4,5]. We have used this technique in the present work as, to our knowledge, no measurements of the thermal properties of a-Hg12 below room temperature have been made so far. Lattice dynamics studies have been performed by Raman experiments ([6] and Ref. therein), infrared reflectivity[‘l] and inelastic neutron scattering[8]. Therefore there is a fundamental interest to study heat transport in this layered crystal for which the phonon dispersion curves are already known[8] as well as the elastic constants[9]. 2. FXPERtMJPrrAL Several crystals have been studied and the geometrical characteristics of the used parallelepipedic samples are given in Table 1. The crystals labelled LETI-13, -14 and -16 have been grown from solution at 25-3X using the dimethylsulfoxide as a complexing agent[lO]. EGG S3-7 and EGG S7-24 crystals have been grown from the vapor phase at 12o”C[ll] using the temperature oscillation method. Two samples have been cut from crystal LETI16, one parallel to the a-axis (LETI-16) and the other parallel to the c-axis. This second sample LETI-16-(a) was broken after the first run and a part of it, labelled 311
LETI-16-(b) was remeasured. All the other samples were cut as LETI- so that most of the results concern heat flow parallel to a-axis. The measured crystals proved to be nonstoichiometric. The deviation from stoichiometry of the LET&crystals was studied by measurements of lattice constants and density at room temperature and also transition temperature to the yellow form (-131“C)[12]. In the case of the EGG-crystals it was qualitatively appreciated knowing the deviation of stoichiometry of the mother phase and the growth conditions. Therefore, the deviation from stoichiometry of the crystals may be ranged in the following row: (Hg-rich crystals) LETI- > LETI- > stoichiometric composition < S7-24 < LETI- < S3-7 (I-rich crystals). The total impurity content is within 100 -500ppm and crystal dependent, the carbon being the main characteristic impurity[ 131. The thermal conductivity measurements have been carried out by the constant heat flow method, and a dilution fridge was used to extend the measurements below 1 K in some cases. In view of the brittleness of the crystals, a special mounting technique has been used [ 141. The uncertainty on the absolute value of the thermal conductivity is about 7%. 3. RESULTS AND QUALITATIVE
DISCUSSION
The results of the thermal conductivity (K) as a function of temperature are given in Fig. 1 for all samples except EGG-S7-24. The values for this sample are omitted in view of clarity as they are very close to those of
312
A. M. DE
GOER
et a/.
Table I. Characteristics of the samples 1)
Sample
1
KtW.cm.'K'I
(cm)
0.33
0.27
2.0
0.71
0.30
3.1
1.05
LETl-14
0.37
0.39
3.5
1.54
EGG-S3-7
0.30
0.46
1.65
1.40
EGG-57-24
0.22
0.282
2.1
0.99
LETI-16-(a)
0.38
0.35
2.6
1.48
LET]-&(b)
0.29
0.39
1.0
1.17
A
.“._“..-...-
10-2
10-2
10'
(K'T')‘IIC. (W cm-'K-')
LETI-
t
1066
L (cm)
LETI-
..:
1tF
12
(cm)
-....,,,
'...T(K) -nTrj/[r+-F-
,m
lo"
10-l
1
1
10
lo2
T(K)
Fig. 1. Thermal conductivity of a-HgIz samples as a function of temperature: LETI- 0; LETI- X; LETI-13, 0; EGG-S3-7, A; LETI-16-(a), A; LETI-16-(b), A. Solid line is calculated for sample LETI- supposed to be perfect (see text).
specimen LETI-16. The main features of these results are: (i) the anisotropy of the thermal conductivity in the whole temperature range; (ii) the sample dependence of K below 10 K, despite the fact that the dimensions of the different specimens are comparable (see Table 1); (iii) the departure from the expected T’ dependence at very low temperatures (except for sample LETI-16-(b)) which is illustrated in the K/T’ plots (insert of Fig. 1). The anisotropy in the intrinsic region (T > 15 K) is to be related to the structure of H& and will be compared later to that observed in other layered crystals (see Section 5).
1.22
The sample dependence of the thermal conductivity below 10 K indicates that phonons are scattered by structural defects and impurities. In a perfect infinite crystal of diameter 2R, the phonon scattering at the boundaries leads to a limiting value of K/RT3, which is constant at very low temperatures and can be calculated from the elastic constants[lS]. We have used the values of the elastic constants measured by ultrasonic methods [9] to calculate this factor respectively for crystals parallel to c-axis and parallel to u-axis: the anisotropy ratio is found to be only 1.25. The values of K/T’ for all samples are given in Table 1. These theoretical values are all about 1 W/cm K4 but, even for the crystal LETIhaving the highest conductivity, the experimental value of K/ T3 at 100 mK is still smaller by a factor larger than 5. This fact and the shape of the K/T’ curves, which suggests the possibility of a resonant phonon scattering at very low temperatures for the samples LETIand LETI-13, give some evidence of the presence of large clusters of defects[5]. The maximum interaction would occur when the wavelength of the most important phonons is of the order of the diameter d of the cluster (supposed to be spherical), that is A = hu/4 kT = d. As the maximum effect, which should be .observed as a minimum of the K/T3 vs T plot towards low temperatures, is not reached at the lowest temperature = 5 x 10e2 K, the diameter d must be of the order of 0.3 pm or larger (using u = 1.2 lo5 cm/s, the mean value of the sound velocity calculated from elastic constants). Phonon scattering by other structural defects (dislocations, stacking faults, subgrain boundaries, glide bands, etc.. .) and by point defects can also play a role and it is necessary to achieve a quantitative analysis of the K(T) curves to determine their relative importance. 4. QUANTITATIVE
ANALYSIS
The thermal conductivity is calculated in a simplified isotropic model, using the Debye approximation of the specific heat, which leads to the well known integral[4].
Low-temperatureheat transport in a-HgI*singlecrystals
313
(the symbols having their usual meaning). Such a model completely neglects the elastic anisotropy but it is known from lattice dynamics studies that this anisotropy is relatively small in a-H& compared to other layered crystals [g, 161. The inverse relaxation time T-‘(x, T) includes all resistive processes and we have neglected the correction due to the special role of normal three phonon processes[l’l] as HgI, is far from isotopically pure. The total inverse relaxation time is written as:
Fig. 1. It is clear that additional phonon scattering must be included to fit the experimental results. Therefore we have adjusted for each sample the point defect scattering coefficient A, and also the frequency independent term r;’ to take into account the phonon scattering by large clusters. The interesting parameters are A-A, and r;’ = ( ri’).xl,- (ri’Llc. Satisfactory fits have been obtained for samples LETI-14, LETI- and EGG-S3-7, as illustrated in Figs. 2 and 3. The corresponding parameters are given in Table 3 and we discuss them in turn.
r-‘(o, 7’)= rs’ + Au4 + T;; + 7;’
4.1.1 Independent frequency scattering. As can be seen in Figs. 2 and 3, the calculated conductivities are too high at very low temperatures. This may be due to the fact that we have used a frequency independent limit of the exact cross-section for the scattering by clusters; this geometrical limit 7;’ is valid only at high frequencies and is given by:
(1)
including the classical terms: boundary scattering (Q,-‘), point defect scattering (Au4), phonon-phonon interactions (TVXP-‘); the additional term TV-’corresponds to phonon scattering by structural defects. In principle it is possible to calculate the first two terms. r;’ is obtained from the theoretical NT3 limit already discussed, and the values for the different samples are given in Table 3; the coefficient A due to isotope scattering is given by [4]: A = VJ4av3 x fi[AM-JM]’
rE--1= N,v?rd=/4
where N, is the number of clusters per cm3[5]. These contributions T,-’ are given in Table 3 and we have
(2)
where V, is the molecular volume and fi the fraction of isotope i which introduces a difference of mass AMi of the molecular cell of mass M The appropriate parameters are given in Table 2 and A0 is the contribution of the isotopes. V, has been calculated from the value of the density p corresponding to the stoichiometric composition [ 121.The Debye temperature B is calculated from the mean sound velocity, supposing that the molecule HgI, is the vibrating unit.t
K(W.cm’.K-1) t l_
10-1_
4.1 Thermal conductivity parallel to a-axis In a first step, the intrinsic curve above 15 K obtained for all samples parallel to a-axis has been fitted. It was found that the best form of the phonon-phonon term was: up:,, = B.o*T exp (-0loT)
(4)
x)-2_
lo-J_ ._
(3) lv_
with B. = 2.0 lo-” cgs and a = 5.5 The theoretical curve calculated for the sample LETI14 supposed to be perfect (7;’ = 0, A = A,) is drawn in tThe corresponding atomic Debye temperature should be 105K, in good agreement with the value of !99K obtained recently from specific heat measurements[llI].
10-5 , I II*Illll 10-2 W’
/ ‘IllIll 1
I 1 II1IIIl la
~1llu!.L lo* T(K)
Fig. 2. Comparisonbetween calculatedcurves (solid lines) and experimental results for the samples LETI-13,0; EGGS-7, A.
Table 2. Parameters used to calculate the thermal conductivity integral and the isotope point defect scattering coefficient A0
A. M. DE GOER
314
Fig. 3. Comparisonbetween calculated curves (solid lines) and experimentalresults for the samplesLETI-14, 0;LETI-16-(a),A; and LETI-16-(b),A.
calculated the corresponding values of N,, supposing the diameter d = 0.3 pm (see Section 3). N, is of the order of few 10“‘/cm3 so that the total cluster volume per cm3 is less than 10m3, which seems reasonable. It is worth noticing that the value of N, is larger for the vaporphase sample EGG-S3-7; this result still holds if we introduce an additional scattering due to isolated dislocations: r;’ = Go (see Table 3). The fit such obtained is of comparable quality to the first one with G = 0. 4.1.2 Point defect scattering. The coefficients A needed to fit the experimental results are given in Table 3. They are of the same order of magnitude (some 1O-43s’) for all samples and larger than the value A,, calculated for isotope scattering. The additional contributions A-A, (which are also given in Table 3) must
et al.
be due to chemical impurities and (or) to interstitials which are present in view of the imperfect stoichiometry of the crystals[lO]. Another possible source of Rayleigh scattering would be small precipitates, in the limit of very low frequencies (A % d)[5]. The contribution of about 1OOppmatomic of metallic impurities supposed to be substituted for Hg atoms (AM- 160) is calculated to be A,,, -7 x lo-“?, considering the mass defect alone, eqn (2). Now an important impurity is carbon and its state in HgIz crystals is unknown. Isolated carbon atoms can replace Hg atoms or occupy interstitial sites of the Hg-sublattice. Carbon could also be present as hydrocarbon groups such as CH,, C2H, etc.. , and CH, could substitute for iodine atoms. A typical concentration of carbon is 1OOppmby weight, and would lead to A, = 2 x lo-@ s’, 1.4 x lo-” s3 and 1.2x 10m4* s3 respectively in the three cases just considered. It is clear that carbon in place of Hg or CH, in place of I gives too large values of A,. Interstitial carbon atoms give smaller A, values so that the total contribution of chemical impurities A,,, t A, is about 8.4 x lo-” s3 and compares favourably with the experimental values of A-A,,. The fact that carbon atoms occupy preferentially insterstitial sites is probably related to the structure, as it is known that graphite readily intercalates between its basal planes bromine, iodine monochloride and ferric chloride [ 191. Now the stoichiometry of the crystals measured in the present work is known[l2] (see Section 2). From this study, it was suggested that the defects due to the lack of stoichiometry were Hg interstitials and iodine molecules giving I; ions, in Hg-rich and I-rich crystals respectively. It is possible to calculate the concentrations of these defects in the crystals which are close to stoichiometry, such as the I-rich crystal LETI-13. In this case, the concentration of 1, ions is given by eqn (5) of [12]:
C,? = 0.6 [Aplp,,,+ A V/ VJ. Using the experimental values Ap/pm = 1.9 x 10m4and A V/ V,,, = 1.5 X 1O-3 (Table 1 of [12]), we obtain Crs = 1.O10-j. corresponding to a number of I, interstitials per
Table 3. Parametersused to fit the thermalconductivitycurves Sample
TB-:xp,w)
LETI-
1.10’
LETI-
1.9
LETI-& LETI-16-(b)
TB-;,,,!‘-‘I
rc-l(s-‘)
2.2
105
7.8
2.8
10’
1.10’
2.3
1.5 10’
2.9
106
10’
NC (cm-‘)
A(s’l
A-A,
(s-‘1
G’
n*
9 2 109
1.4
lo-*]
6.10-”
0
-
1.6 10’
1.9
2.7
IO“’
1.910-*‘
3
-
101
9.7
1.1 10”
5.1
lo-“
4.3
105
1.47
1.7
1.7 lo-”
10’ 10’
10’0
10”
IO-*’
9.10-“’
5.8
lS_”
2
2.3
10‘”
*
I *rd-’= GO”(n = 1 or 2).
Low-temperatureheat transport in a-H& single crystals cm3 n = 2.5 x lo”?; the related contribution to point defect scattering is A, = 5.2 x 10F4*s3, which is about 30 times larger than the observed A-A0 value. This result clearly shows that the 1; defects are not isolated and that some clustering must occur, even in the best crystals. 4.1.3 Other scattering mechanisms. The possible contribution of scattering by dislocations has been tested by including a term r;’ = GW in eqn (1) (see Section 4.1.1). It was possible to achieve a fit only for the sample EGG-S3-7. The order of magnitude of the corresponding dislocation density N,+can be estimated from the relation [20]: G = 6 lo-* Ndb2yZ. With the values y = 2 of the Griineisen constant and b = 6” A of the Burgers vector, the experimental G value leads to N,, ~2.7 X 10p/cmZ,which appears too large to be physically meaningful. As the corresponding fit is not better than with G = 0, it is clear that the thermal conductivity is nearly insensitive to the presence of dislocations, the reason being that it is mainly determined by the other scattering mechanisms previously described. 4.2 Thermal conductivity along c-axis Above 15 K the conductivity along c-axis is about 5 times smaller than the conductivity perpendicular to c-axis: the anisotropy ratio is nearly constant, as illustrated in Fig. 4, so that we have used the same expression of the phonon-phonon scattering term as before, eqn (3) to analyze the curves. It was necessary to change the values of both B, and (2 and the best fits were achieved with B, = 1.3 x lo-l6 cgs and (I = 4.7. In contrast with the case of crystals perpendicular to c-axis, it was not possible to obtain a good description of the curves in the whole temperature range by adjusting only the frequency independent term ri’ and the point defect coefficient A. Additional scattering was needed and satisfactory fits have been achieved only with TV-’= GO*, which corresponds to phonon scattering by plane defects such as stacking faults[20]. The parameters used are given in Table 3 and the fits are illustrated in Fig. 3. We recall that the sample LETI-16-(b) was a part of the crystal LETI-16-(a) after it was broken; it appears that the crystal LETI-16-(b) contains more structural defects than LETI16-(a) and especially a larger number of plane defects (see Table 3). The density Ns of stacking faults can be estimated from the value of G, using Klemens’ expression[20] G = 0.7 NSa2y*lv. With y=2 and a =;/V,=4.9~ Ns = 100 to 4OO/cm.
lo-*cm,
we obtain
5. DISCUSSION The previous analysis of the experimental results appears to be consistent and leads to the following destCri is defined as the ratio n/N, N being the total number of lattice sites[l21.
315
cription of the detected defects in the cy-Hg12crystals investigated. It was shown in Section 4.1.2 that the point-defect scattering terms A-A, could be entirely explained by the chemical impurities present in the crystal (including interstitial carbon atoms). Therefore the defects due to the lack of stoichiometry do not scatter phonons as isolated point defects nor as small aggregates, and we suggest that the large defects considered in Section 4.1.1 are in fact clusters rich in Hg interstitials or 1; ions. It can be seen from Table 3 that the independent frequency term r;’ is larger by a factor from 2 to 10 for samples parallel to c-axis compared to these for samples perpendicular to c-axis. As this term is proportional to N,d*, eqn (4), it follows that the clusters look somewhat anisotropic (the phonons “see” preferentially the typical dimension in the plane perpendicular to the heat flow direction). The experimental results of the sample LETI16 perpendicular to c-axis have not been quantitatively analyzed as they were only slightly below those of the sample LETIand the measurements have not been extended below 1.3 K (see Fig. 1). From the thermal conductivity value at 1.5 K, we estimate 7,’ = 9 x 10’ s-’ for the sample LETI-16, so that the ratio of the cluster typical dimensions perpendicular and parallel to c-axis is about VlO. This means that, if the dimension along c-axis is 0.3 pm or larger as previously supposed (Section 4.1.1), the dimension perpendicular to c-axis would be about 1 pm or larger, and the cluster density NC= 1.1 x 10’“/cm3. Within these assumptions, the total cluster volume per cm3 is calculated to be =3 x 10e3.If there are two interstitial defects per unit cell within the cluster volume, this total perturbed volume would correspond to 2.5 x 10” defects/cm3. The exact agreement with the number of interstitials estimated from the lack of stoichiometry in the sample LETI(see Section 4.1.2) is evidently fortituous: it is not the same crystal and moreover very crude assumptions have been made to obtain these estimates. Nevertheless we think that the orders of magnitude are correct and that the presence of large clusters is effectively related to the non-stoichiometry of the crystals. The anisotropy of these clusters is not surprising as it is known that point defects condense preferentially in the closest packed crystallographic plane, namely the (001) basal plane for a-Hg12. These large defects could be the same as those observed by scanning electron microscopy operated in cathodoluminescent mode and by chemical etching[21]. Indeed clusters of various shapes having a mean diameter running from 1 to lO@m have been observed using these techniques. The other structural defects detected in the present work are plane like defects (see Section 4.2) perpendicular to c-axis, which could be glide bands as observed by X-ray topography[22]. The last point to be discussed shortly is the anisotropy of the thermal conductivity above 15 K. The ratio KJK,, is plotted in Fig. 4 and it appears that this ratio is independent of temperature in a large range, including the Debye temperature. Larger but temperature dependent anisotropy ratios have been measured for hcp He’
316
A. M. DE GOER
et al.
t K,‘K//
Ga Se__________
__._._ _____._._ _-^__ le
L
1 ..-I 1
Se
1 I LlU-Y_ 10
100
Fig. 4. Anisotropy ratio K,/K,f of the thermal conductivity perpendicular to c-axis compared to that parallel to c-axis as a function of temperature: 0 cr-HgI, (present work; solid line is not calculated); GaSe [26];Se and Te [17](the ratio in these cases are &/1<, as the chains are along c-axis).
(about 20 at 1 K[23]) and graphite below room temperature (>300 at 300 K[24]). Temperature independent
anisotropy has been observed in pyrolytic graphite above room temperature1251 and in the cases of layered GaSe[26] and of the chain-structured trigonal Se and Te[27], as illustrated in Fig. 4. A theoretical approach given by Simons[28] leads to a temperature independent anisotropy, in the limit of low temperatures only, which depends on the elastic anisotropy and also of the anisotropy of the Brillouin zone. Both factors are larger for GaSe compared to Hg12,in qualitative agreement with the experimental results. The only other theoretical work on the anisotropy in the Umklapp region is that by Benin[29], which is restricted to the case of hcp He4. As noted in a recent review by Slack[30], much more work is needed in this area. Finally we note that Lanyi et a/.(31) have measured the thermal conductivity of Hgi, above room temperature and found a value of about 4 10m3W/cm K, nearly temperature independent between 320 and 390 K, but which is dependent of the application of an electric polarisation. The agreement with our results for the sample LETIparallel to c-axis is not bad and we suggest that an electronic contribution to the thermal conductivity becomes noticeable near room temperature.
6. CONCLUSION
The qualiative and quantitative analyses of the thermal
conductivity results lead to the following conclusions. Point defect scattering is relatively weak and is explained by the presence of residual chemical impurities and principally carbon in interstitial sites. Large anisotropic clusters of defects strongly scatter phonons in the low temperature range: their dimensions are not less than 0.3 pm along c-axis and I pm perpendicular to c-axis. There is also evidence of the presence of plane defects from the experimental results of the thermal conductivity parallel to c-axis. The results are in qualitative agreement with those obtained from other techniques[21] and [22]. The large
defects are attributed to a condensation of the interstitials Hg or 1; ions due to the lack of stoichiometry of the crystals. The nearly temperature independent anisotropy of the thermal conductivity in the intrinsic range (above 15 K) could give informations on the phonon-phonon interactions in such layered crystals if a theoretical analysis becomes available.
Acknowledgements-We acknowledge the E.G.G. Co. for providing some of the crystals used in this work. We are very grateful to Prof. L. J. Challis (University of Nottingham) for useful comments on the manuscript.
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