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Anomalous transport properties in thorium arsenosulphide crystals
PERGAMON
Solid State Communications 122 (2002) 1±6
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Anomalous transport properties in thorium arsenosulphide crystals Z. Henkie*, R. Wawryk Department of Electron Transport, W. Trzebiatowski Institute of Low Temperature and Structure Research, Polish Academy of Sciences, ul. OkoÂlna 2, P.O. Box 1410, 50-950 Wrocøaw, Poland Received 18 February 2002; received in revised form 18 February 2002; accepted 21 February 2002 by A. Zawadowski
Abstract Single crystals of thorium arsenosulphide have been grown and their electron transport properties have been examined to understand better the origin of the Kondo-like behaviour of the transport properties of ferromagnetic UAsSe and diamagnetic ThAsSe. The thorium arsenosulphide is a metallic conductor with the Hall carrier concentration equal to 1.5 £ 10 22e 2/cm 3. Its room temperature resistivity and thermoelectric power equal 32 mV cm and 23.7 mV/K, respectively. There is a Kondo-like temperature dependence of the resistivity and Hall coef®cient observed with the Kondo components scaled with Kondo temperature < 20 K. The Kondo resistivity of thorium arsenosulphide is lower by an order than that in ThAsSe. It is consistent with the hypothesis that this Kondo effect originates from anions' substitutional disorder. q 2002 Elsevier Science Ltd. All rights reserved. PACS: 72.10.Fk; 72.15.Eb; 72.15.Gd; 72.15.Jf Keywords: B. Crystal growth; D. Kondo effects; D. Electronic transport
1. Introduction This work continues the studies of uranium and thorium pnictidochalcogenides motivated by the discovery of a Kondo-like component in the resistivity of ferromagnetic compound UAsSe [1,2]. Coexistence of the ferromagnetism and the Kondo effect may be possible here due to nonmagnetic origin of the latter. Such suggestion is consistent with low temperature magnetoresistivity of UAsSe that is weak and of speci®c anisotropy [3]. Furthermore, similar Kondo-like effect is observed for diamagnetic ThAsSe [4,5]. Therefore, two level system (TLS) Kondo model [6] is considered as the one that might describe properly a mechanism of conduction electron scattering in UAsSe [3,5]. In the simplest TLS model, the defect atom tunnels between two displaced positions in double well potential whose energy levels of the two wells are split by D. In the case of TLS Kondo effect one should think of the
* Corresponding author. Tel.: 148-71343-50-21; fax: 14871344-10-29. E-mail address: [email protected] (Z. Henkie).
atom position in one well or the other as a pseudo spin variable. Electrons may `¯ip' the spin by assisting the tunnelling between the wells. Scattering of electrons by the TLS Kondo centre leads to similar consequences as the scattering by magnetic Kondo impurity [6,7]. A large anisotropic displacement factor (ADF), reaching Ê 2 for As u11 in ThAsSe and decreasing together 0.03 A with the Kondo-like resistivity when x increase in the UAs12xSe11x series, speaks forward of this type of interpretation [4,5]. The ADF represents the average value of displacement of atoms vibrating around lattice positions and equals mean-square displacements along the Cartesian axes. Large ADF is related to a disordered occupation of As and Se sublattices detected in UAsSe either by X-ray or neutrons [8]; about 6% of As positions is occupied by Se and vice versa, in UAsSe crystals with T C 108 K: The X-ray does not detect the anion disorder in UAsS [8] and both the ADF and the Kondo resistivity are by the factor of about 1/3 lower than those in UAsSe [5]. One can think that a bigger difference of the anions' size leads to more ordered structure, decreasing ADF and diminishing the Kondo effect. It will be shown here that it is also true for the thorium arsenosulphide
0038-1098/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0038-109 8(02)00098-4
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Fig. 1. (a) The ab plane resistivity vs. temperature for two thorium arsenosulphide single crystals. (b) Temperature dependence of the r /r 300 K ratio for sample no. 1 (circles), sample no. 2 (triangles) of thorium arsenosulphide crystals. The dashed line 3 schematically represents the r /r 300 K data for ThAsSe taken from Ref. [12]. The solid lines represent Eq. (3)) ®tted to experimental data of resistivity at positive r 0.
whose resistivity r (T ), thermoelectric power S(T ) and the Hall coef®cient RH(T ) have been examined.
2. Experimental details Crystals of the thorium arsenosulphide were grown by the chemical vapour transport method with bromine, in temperatures 1020 8C ! 970 8C and in a silica tube covered inside with a layer of pyrolytic carbon. The X-ray diffraction examination showed that the crystals have a composition given by formula ThAs1.23S0.77 and tetragonal structure of the PbFCl type (P4/nmm space group) with the lattice para They can be meters: a 4:0225 A and c 8:483 A: compared with a(c) parameter that is equal to 4.012 Ê for ThAsS and ThAs2, respec(8.464) and 4.086 (8.575) A tively, synthesised previously in a powder form by reaction of the elements [9]. We estimate that the composition of the crystals corresponds to formula ThAs1.15S0.85 if we assume a linear dependence of the unit cell volume on a composition in hypothetical ThAsS±ThAs2 solid solution. This estimation proves that the crystals contain the excess of As at the
expense of S in comparison to stoichiometric composition. The crystals grew in a plate-like form. Three of them were selected for the examination. Their length along a-axis was between 0.85 and 0.95 mm and their thickness along c-axis was between 0.12 and 0.25 mm. Electrical resistivity in temperatures 2±315 K has been measured by a conventional dc method with electrodes placed along a-axis. The geometrical factor of the samples was determined with the accuracy of about 18%. The thermoelectric power along a-axis has been measured by a modi®ed method of Ref. [10]; a Cu±constantan thermocouple has been replaced by an AuFe±chromel one to extend the temperature range for measurements down to 5.2 K. The sample was placed between two blocks of Ag97Au3 alloy [11]. Electrical and thermal contacts of sample with the blocks were improved by wetting the contacts with a liquid In±Ga alloy. The measurements were taken at the constant difference of the blocks' temperatures equal to 2 K. The Hall coef®cient has been determined for Hall voltage measured along the b-axis at current along a-axis and magnetic ®eld of 1.25 T along c-axis. The possibility of the fast alternation of the ®eld direction produced by electromagnet allowed us an increase of the measurements' accuracy [2]. The magnetic susceptibility has been measured with SQUID magnetometer (Quantum Design MPMS-5) using 11 mg sample composed of several single crystals.
3. Analysis of the results 3.1. Resistivity All the selected crystals of ThAs1.23Se0.77 showed the same type of the r (T ) dependence; the maximum of the r (T ) is observed at sample dependent temperature Tmax followed by a resistivity minimum at about 200 K. The Tmax equals to 124, 134 and 149 K for the three crystals, respectively. The r (T ) data for the ®rst two crystals are presented in Fig. 1a. Their mean value of the resistivity at 300 K equals 32 mV cm and it is considerably lower than the value of about 220 mV cm observed for ThAsSe. [12]. The temperature dependence of the r (T )/r (300) for the two compounds is shown in Fig. 1b. One can expect that conduction electrons in ThAs1.23S0.77, like those in ThAsSe [5], are scattered by static impurities, phonons, and the TLS centres contributing to the resistivity r 0, r ph(T ) and r K(T ) components, respectively. According to Matthiessen's rule the total resistivity is the sum of these components. We write the phonon resistivity r ph(T ) in the form of product CphFB±G(T ) where Cph is a coef®cient dependent on a band structure and an electron±phonon coupling and FB±G(T ) is the well known Bloch±GruÈnaisen (B±G) function, universally dependent on T/Q D ratio. We have taken Q D 175 K [13], determined for ThAsSe. An attempt of description of our data by assuming
Z. Henkie, R. Wawryk / Solid State Communications 122 (2002) 1±6
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Fig. 2. Temperature dependence of the phonon resistivity r ph/r 300 K and the sum of r 0 and Kondo resistivity (r 0 1 r K)/r 300 K determined by ®tting of Eq. (3) to experimental data of resistivity for thorium arsenosulphide sample of nos. 1 and 2. The ®tting was done without constraining of any parameter. Experimental data of resistivity diminished with the ®tted phonon resistivity for samples nos. 1 and 2 are shown by circles and triangles, respectively.
r K(T ),±ln T [14] leads to unphysical data both the r ph(T ) and r K(T ) near the room temperature. The resistivity of the Kondo systems cannot be given in an analytical form for the whole range of temperature. The nonperturbative results of calculation for T @ TK are quite reliable. We choose the results of Arai [15] to make an explicit comparison: rK T rK 0{1 1 p2 =3a T=TK 22 }21 ;
1
where a (T ) is a universal function of T/TK and is given by solution of: a T 2 TL =Tln{1 1 a2 T1=2 T=TL } 2 p ln 2 0;
2
where TL TK =0:624 and Kondo temperature TK is a characteristic temperature scaling for the particular Kondo system. The total resistivity:
r T r0 1 rph T 1 rK T
3
has been ®tted to experimental resistivity data for temperature range T . 1.1Tmax of two ThAs1.23S0.77 crystals. The ®tted parameters are r 0, Cph, r K(0) and TK. For positive but negligible small values of r 0 one gets r ph(300); r K(0); TK equal to 19.3; 68.6 mV cm; 24.8 K and 14.0; 55.5 mV cm; 25.7 K for the crystal 1 and 2, respectively. The above parameters determine the r (T ) dependence shown by the solid lines in Fig. 1b. Eq. (3) has been also used for the approximation of the resistivity data of ThAsSe from Ref. [12] shown schematically by a broken line and the calculated r (T ) by the solid line in Fig. 1b. The latter was obtained for r 0, r ph(300), r K(0) and TK equal to 0, 6.6, 1057 mV cm and 19.4 K, respectively. The analysis presented above shows that anomalous behaviour of the resistivity either ThAs1.23S0.77 or ThAsSe could be under-
stood in terms of the same model. The main difference between these two compounds consists in the amplitude of the Kondo component; the r K(0) for the ThAsSe is an over one order higher than that for the ThAs1.23S0.77 while the other ®tted parameters are comparable. The analysis also shows that the parameters were obtained by the ®tting done in the T/TK range 7±15. For this range, either calculations of Arai or the second order perturbative calculations of Neal and Collins [16] give r K(T )/r K(0) ranging between 0.2 and 0.35. But a relative difference, between the r K(T )/r K(0) of the two theories in the considered range, reaches 30% both in value and a slop of the r K(T )/r K(0) vs. T/TK dependence. We may also expect a similar difference between the experimental data and those theoretically predicted ones, especially that we use results of the theory for the single impurity to interpret the presumably TLS system. Though the theory [6,7] predicts similarity of the both systems behaviour, a high uncertainty in the determination of r 0 is possible. Therefore, we think that the physically correct ®tted parameters are also those giving positive high temperature values of r 0 1 r K(T ). A ®tting done without any constraining of the parameters of Eq. (3) gives TK equal to 20.3 and 22.5 K for crystal 1 and 2, respectively. The calculated r ph(T )/r (300) and the [r 0 1 r K(T )]/r (300) are shown in Fig. 2 by the curves denoted r ph and r K, respectively. Circles and triangles present experimental data of resistivity reduced by the phonon component determined for crystal 1 and 2, respectively. Above Tmax there is a very good agreement between the experimentally determined and the calculated one of the Kondo-like resistivity behaviour. The quantities determined by the last ®t do not differ suf®ciently enough from that
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Fig. 3. Hall coef®cient vs. temperature for the thorium arsenosulphide. Magnetic ®eld equal to 1.25 T was parallel to c-axis of sample no. 1. The inset presents Hall coef®cient as a function of r K/T ratio.
obtained in the previous ®t to question the previous interpretation. The temperature derivative of r ph(T ) and r K(T ) components which is comparable in values but opposite in sign generates the minimum of the resistivity of ThAs1.23S0.77 at about 200 K. Because of the much larger component of the Kondo-like resistivity, the ThAsSe shows the minimum of the r (T ) shifted to about 500 K [12]. 3.2. Hall and thermoelectric power coef®cients Experimental data of the RH(T ) for the no. 1 crystal of
ThAs1.23S0.77 is shown in Fig. 3. The Hall coef®cient is negative and it varies from 21.15 £ 10 23 cm 3/C at 107 K to 26 £ 10 24 cm 3/C at 336 K. These values of the RH(T ) determine the carrier concentration that in one band model is equal to 5.4 £ 10 21 and 10.4 £ 10 21e 2/cm 3, respectively. The well-known semimetal, antimony has conduction electron concentration equal to 3.7 £ 10 19e 2/cm 3 [17] and it is suf®ciently high to, like in normal metals, temperature independent in considered temperature range. On the other hand, the magnetic Kondo systems are known as the ones showing temperature dependent Hall coef®cient despite a high carrier
Fig. 4. Thermoelectric power, measured along a-axis is plotted vs. temperature for the thorium arsenosulphide sample no. 1.
Z. Henkie, R. Wawryk / Solid State Communications 122 (2002) 1±6
concentration [18]: R H T R0 1 gx TrK T
4
where R0 is the ordinary Hall constant, gx (T ) r K(T ) is the skew scattering contribution that at high temperatures decreases linearly with product of reduced magnetic susceptibility x (T ) and the Kondo resistivity. Linear extrapolation of RH vs. xr K dependence from the range where it was determined to xrK 0 determines the R0. The ThAs1.23S0.77 crystals show weak and temperature independent diamagnetism. Their susceptibility measured in ®eld 5 T transverse to ab plane equals 21.2 £ 10 28 emu/mol at temperatures 20±300 K. So we should exclude the presence of magnetic Kondo impurities from our consideration. Following predictions [6,7] of great similarity of the TLS Kondo system behaviour to that of magnetic Kondo one, we implicitly assume that the Eq. (4) type dependence can be used for the interpretation of the RH(T ) for the thorium arsenosulphide. But there is a question about a quantity that should replace the x (T ). However, it was found that the RH(T ) for the arsenosulphide shows linear dependence on r K(T )/T ratio at T . 150 K. This is shown in set of Fig. 3. The linear extrapolation gives R0 24:3 £ 1024 cm3 =C: It corresponds to the one band carrier density 1.45 £ 10 22e 2/ cm 3 and the Hall mobility 13 cm 2/V s. The thermoelectric power for the no. 1 crystal of ThAs1.23S0.77 is negative except for narrow temperature range around < 50 K as Fig. 4 shows. A linear S(T ) dependence is observed above 150 K characterised by the measured value S 316 K 23:9 mV=K and determined by the linear extrapolation value S 0 21:2 mV=K: For the estimation of the conduction band parameters, we use the well-known Mott formula for diffusion thermoelectric power in the form: S T x p2 k2 T=3eEF
5
where k and e are the Boltzmann constant and electron charge, respectively. The parameter x depends on the mechanism of the electron scattering and experiment shows that at high temperatures it assumes values between 2 and 3 [19]. Taking x 3 and S at 316 K we estimate the Fermi energy EF 6:0 eV and effective mass equal to 0.37 free electron mass. Anomalies of S(T ), seen below 150 K in Fig. 4, can hardly be explained at the moment. 4. Discussion and conclusions The crystals of ThAs1.23S0.77 are free of magnetic impurities and, like in metals, have high conduction electron concentration. In spite of this, the crystals show anomalous behaviour of resistivity and Hall coef®cient. The r (T ) above 150 K has a shape of a shallow valley and can be resolved into phonon component and the Kondo one. The latter is consistent with that predicted in the Kondo limit of the Anderson model [15] and determines the Kondo tempera-
5
ture equal to 20.3 and 22.5 K for the crystals 1 and 2, respectively. The temperature dependent Hall coef®cient can be resolved into temperature independent normal Hall coef®cient and the temperature dependent skew scattering-like component characteristic for the magnetic Kondo systems. It is assumed that the Kondo like behaviour of the transport properties of the ThAs1.23S0.77 is due to scattering of the current carriers by the TLS centres created by the anion disorder, as it is in uranium and thorium arsenoselenides [5,8]. The simplest TLS model predicts deviation of the Kondo resistivity from logarithmic behaviour below T < D and saturation at T ! 0 [6]. If this was the reason for the observed discrepancy between the experimentally determined and the calculated Kondo resistivity below Tmax then we should assume D < 40 K: But Cox and Zawadowski [6] suggest that the Kondo state should not be developed for TK , D . The original TLS Kondo model also cannot explain the Kondo temperature as high as TK < 20 K (see Refs. [20,21]). However, as realised recently, electron±hole symmetry breaking present in all realistic band-structure-based density of states can increase TK substantially and is very promising to resolve the long standing problem of the experimentally observed high TK [21,22]. Undoubtedly more studies are needed to explain the origin of the unpredicted behaviour of the r (T ) below 140 K. The normal Hall constant of ThAs1.23S0.77 corresponds to 1e 2/f.u. (f.u.Ðformula unit) and is consistent with the odd number of the valence electrons in the stoichiometric ThAsS. Two ThAsS molecules per one Brillouine zone are in this structure and ThAsS should be a compensated metal, i.e. it should have the equal number of holes and electrons. In such a case the one band model can lead to the correct number of electrons if their mobility is clearly higher than the mobility of the holes. This seems to be the case of the thorium arsenoselenide where neither Hall coef®cient nor the thermoelectric power change sign despite the decrease of the valence electrons in the ThAs1.23S0.77 resulting from replacing part of the S atoms by the As ones. We expect that the total number of holes and electrons be of order 2 per f.u. It can be compared to 2.6 carriers per f.u. derived from room temperature optical conductivity of ThAsSe [23] and free electron mass of the carriers. Schoenes et al. [12] have found Hall coef®cient for ThAsSe decreasing monotonously with increasing temperature. In the one-band model the Hall concentration increases from 0.08e 2/f.u. at 100 K to 0.6e 2/f.u. at room temperature. They suggest the low effective mass of the carrier ( < 0.2m0) explains the discrepancy between the Hall carrier concentration and that derived from optical conductivity. They also suggest that a strong electron±phonon coupling may be responsible for low carrier mobility that equals merely 3.5 cm 2/V s. It seems to be too low for so light carriers. Finally they discuss several propositions of the origin of the varying number of carriers and propose more experiments before the ®nal explanation. We propose another
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explanation of the anomalous behaviour of the transport properties of the thorium arsenoselenides. The thermoelectric power of the ThAs1.23S0.77 crystals is negative, small and weekly varying with temperature above 150 K. It approves expectation that the main charge transport in the thorium arsenosulphide is controlled by a wide electron band. Its estimated parameters such as the carrier concentration n, EF, and the effective mass ratio m p/m0, are equal < 1.5 £ 10 22e 2/cm 3; < 6 eV; and < 0.4, respectively. The scattering of the electrons is characterised by mobility equal < 13 cm 2/V s at room temperature. The above parameters and the ®tted value of rK 0 ù 60 mV cm allows us to estimate the scatterers concentration with formula derived from [24,25]: h i c r K 0nN EF e2 h =2mp 8:75 £ 1019 rK 0=EF m0 =mp n22 2 cm23
6
where h is the Planck constant, N(E) the density of states and n22 is the carrier concentration in 10 22 cm 23 units n n22 1022 cm23 : Putting EF in eV and r K(0) in mV cm we obtain the concentration of scatterers c 5 £ 10 21 cm23 ; i.e. 1/3 per f.u. For such density of the scatterers we might consider another mechanism for an explanation of the observed below 140 K deviation of the measured r (T ) from that predicted by the TLS Kondo mechanism. One of possible thing might be some sort of electron mean free path `saturation' or interaction effects between TLSs, which certainly must be present at such a high concentration. In conclusion, the dominant contributions to the electron transport properties of the ThAs1.23S0.77 crystals originate from wide electron band carriers. The electron transport coef®cients show anomalous temperature behaviour of the type known for the Kondo-impurity system. This behaviour is ascribed to scattering of the electrons not only by phonons and the static impurities but also by dynamic centres, i.e. TLS centres, created by disordered occupation of the anion sublattices positions. This last scattering adds Kondo-like components to the electron transport coef®cients. However, the intensity of this scattering in ThAs1.23S0.77 is lower than that in ThAsSe and UAsSe. It is consistent with our hypothesis that a bigger difference of the anions' size leads to diminishing the Kondo effect, presumably due to favouring the more ordered structure. Such behaviour is common either for uranium or for thorium compounds. Acknowledgements The authors thank Prof. A. Pietraszko for X-ray analysis
and M.Sc. R. Gorzelniak for technical assistance with the magnetic susceptibility measurements. This work was supported by the Polish Committee for Scienti®c Research, Grant KBN-2 P03B 062 18, for years 2000±2001.
References [1] Z. Henkie, R. Fabrowski, A. Wojakowski, Acta Phys. Pol., A 85 (1994) 249. [2] Z. Henkie, R. Fabrowski, A. Wojakowski, J. Alloys Compd 219 (1995) 248. [3] Z. Henkie, T. Cichorek, R. Fabrowski, A. Wojakowski, B.S. Kuzhel, J. Tadla, Physica B 281& 282 (2000) 226. [4] Z. Henkie, A. Pietraszko, A. Wojakowski, L. KepinËski, T.  Cichorek, M. Szczepaniak, Visnik Lviv Univ. Ser. Khim. 39 (2000) 173. [5] Z. Henkie, A. Pietraszko, A. Wojakowski, L. KepinËski, T.  Cichorek, J. Alloys Compd 317&318 (2001) 52. [6] D.L. Cox, A. Zawadowski, Adv. Phys. 47 (1998) 599. [7] G. ZaraÂnd, K. VladaÂr, Phys. Rev. Lett. 76 (1996) 2133. [8] Z. Henkie, T. Cichorek, A. Pietraszko, R. Fabrowski, A. Wojakowski, B.S. Kuzhel, L. KepinËski, L. Krajczyk, A. Guka sov, P. WisÂniewski, J. Phys. Chem. Solids 59 (1998) 385. [9] F. Hulliger, J. Less-Common. Met. 16 (1968) 113. [10] Z. Henkie, P.J. Markowski, A. Wojakowski, Ch. Laurent, J. Phys. E: Sci. Instrum. 20 (1987) 40. [11] R. Wawryk, Z. Henkie, Philos. Mag. B 81 (2001) 223. [12] J. Schoenes, W. Bacsa, F. Hulliger, Solid State Commun. 68 (1988) 287. [13] T. Cichorek, Z. Henkie, P. Gegenvart, M. Lang, A. Wojakowski, M. Dischner, F. Steglich, 30 iemes JourneÂes des Actinides, Dresden, Abstract SO4, May 4±6, 2000. [14] J. Kondo, Prog. Theor. Phys. 32 (1964) 37. [15] T. Arai, J. Appl. Phys. 57 (1985) 3161. [16] H.L. Neal, D.J. Collins, Phys. Rev. B 48 (1993) 4299. [17] J-P. Issi, Aust. J. Phys. 32 (1979) 585. [18] A. Hamzic, A. Fert, J.M. Broto, M. Miljak, E. Bauer, E. Gratz, de Physique (Colloque C8, Suppl. no. 12) 49 (1988) C8±777. [19] N. Cusack, The Electrical and Magnetic Properties of Solids 1958 London, p. 116. [20] I.L. Aleiner, B.L. Altshuler, Y.M. Galperin, T.A. Shutenko, Phys. Rev. Lett. 86 (2001) 2629. [21] A. Zawadowski, G. ZaraÂnd, cond-mat/0009283.  jsaÂghy, G. ZaraÂnd, A. Zawadowski, Solid State Commun. [22] O. U 117 (2001) 167. [23] W. Reim, J. Magn. Magn. Mater. 58 (1986) 1. [24] G. GruÈner, A. Zawadowski, Rep. Prog. Phys. 37 (1974) 1497. [25] Y. Nagaoka, Phys. Rev. 138 (1965) A1112.
Solid State Communications 122 (2002) 1±6
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Anomalous transport properties in thorium arsenosulphide crystals Z. Henkie*, R. Wawryk Department of Electron Transport, W. Trzebiatowski Institute of Low Temperature and Structure Research, Polish Academy of Sciences, ul. OkoÂlna 2, P.O. Box 1410, 50-950 Wrocøaw, Poland Received 18 February 2002; received in revised form 18 February 2002; accepted 21 February 2002 by A. Zawadowski
Abstract Single crystals of thorium arsenosulphide have been grown and their electron transport properties have been examined to understand better the origin of the Kondo-like behaviour of the transport properties of ferromagnetic UAsSe and diamagnetic ThAsSe. The thorium arsenosulphide is a metallic conductor with the Hall carrier concentration equal to 1.5 £ 10 22e 2/cm 3. Its room temperature resistivity and thermoelectric power equal 32 mV cm and 23.7 mV/K, respectively. There is a Kondo-like temperature dependence of the resistivity and Hall coef®cient observed with the Kondo components scaled with Kondo temperature < 20 K. The Kondo resistivity of thorium arsenosulphide is lower by an order than that in ThAsSe. It is consistent with the hypothesis that this Kondo effect originates from anions' substitutional disorder. q 2002 Elsevier Science Ltd. All rights reserved. PACS: 72.10.Fk; 72.15.Eb; 72.15.Gd; 72.15.Jf Keywords: B. Crystal growth; D. Kondo effects; D. Electronic transport
1. Introduction This work continues the studies of uranium and thorium pnictidochalcogenides motivated by the discovery of a Kondo-like component in the resistivity of ferromagnetic compound UAsSe [1,2]. Coexistence of the ferromagnetism and the Kondo effect may be possible here due to nonmagnetic origin of the latter. Such suggestion is consistent with low temperature magnetoresistivity of UAsSe that is weak and of speci®c anisotropy [3]. Furthermore, similar Kondo-like effect is observed for diamagnetic ThAsSe [4,5]. Therefore, two level system (TLS) Kondo model [6] is considered as the one that might describe properly a mechanism of conduction electron scattering in UAsSe [3,5]. In the simplest TLS model, the defect atom tunnels between two displaced positions in double well potential whose energy levels of the two wells are split by D. In the case of TLS Kondo effect one should think of the
* Corresponding author. Tel.: 148-71343-50-21; fax: 14871344-10-29. E-mail address: [email protected] (Z. Henkie).
atom position in one well or the other as a pseudo spin variable. Electrons may `¯ip' the spin by assisting the tunnelling between the wells. Scattering of electrons by the TLS Kondo centre leads to similar consequences as the scattering by magnetic Kondo impurity [6,7]. A large anisotropic displacement factor (ADF), reaching Ê 2 for As u11 in ThAsSe and decreasing together 0.03 A with the Kondo-like resistivity when x increase in the UAs12xSe11x series, speaks forward of this type of interpretation [4,5]. The ADF represents the average value of displacement of atoms vibrating around lattice positions and equals mean-square displacements along the Cartesian axes. Large ADF is related to a disordered occupation of As and Se sublattices detected in UAsSe either by X-ray or neutrons [8]; about 6% of As positions is occupied by Se and vice versa, in UAsSe crystals with T C 108 K: The X-ray does not detect the anion disorder in UAsS [8] and both the ADF and the Kondo resistivity are by the factor of about 1/3 lower than those in UAsSe [5]. One can think that a bigger difference of the anions' size leads to more ordered structure, decreasing ADF and diminishing the Kondo effect. It will be shown here that it is also true for the thorium arsenosulphide
0038-1098/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0038-109 8(02)00098-4
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Z. Henkie, R. Wawryk / Solid State Communications 122 (2002) 1±6
Fig. 1. (a) The ab plane resistivity vs. temperature for two thorium arsenosulphide single crystals. (b) Temperature dependence of the r /r 300 K ratio for sample no. 1 (circles), sample no. 2 (triangles) of thorium arsenosulphide crystals. The dashed line 3 schematically represents the r /r 300 K data for ThAsSe taken from Ref. [12]. The solid lines represent Eq. (3)) ®tted to experimental data of resistivity at positive r 0.
whose resistivity r (T ), thermoelectric power S(T ) and the Hall coef®cient RH(T ) have been examined.
2. Experimental details Crystals of the thorium arsenosulphide were grown by the chemical vapour transport method with bromine, in temperatures 1020 8C ! 970 8C and in a silica tube covered inside with a layer of pyrolytic carbon. The X-ray diffraction examination showed that the crystals have a composition given by formula ThAs1.23S0.77 and tetragonal structure of the PbFCl type (P4/nmm space group) with the lattice para They can be meters: a 4:0225 A and c 8:483 A: compared with a(c) parameter that is equal to 4.012 Ê for ThAsS and ThAs2, respec(8.464) and 4.086 (8.575) A tively, synthesised previously in a powder form by reaction of the elements [9]. We estimate that the composition of the crystals corresponds to formula ThAs1.15S0.85 if we assume a linear dependence of the unit cell volume on a composition in hypothetical ThAsS±ThAs2 solid solution. This estimation proves that the crystals contain the excess of As at the
expense of S in comparison to stoichiometric composition. The crystals grew in a plate-like form. Three of them were selected for the examination. Their length along a-axis was between 0.85 and 0.95 mm and their thickness along c-axis was between 0.12 and 0.25 mm. Electrical resistivity in temperatures 2±315 K has been measured by a conventional dc method with electrodes placed along a-axis. The geometrical factor of the samples was determined with the accuracy of about 18%. The thermoelectric power along a-axis has been measured by a modi®ed method of Ref. [10]; a Cu±constantan thermocouple has been replaced by an AuFe±chromel one to extend the temperature range for measurements down to 5.2 K. The sample was placed between two blocks of Ag97Au3 alloy [11]. Electrical and thermal contacts of sample with the blocks were improved by wetting the contacts with a liquid In±Ga alloy. The measurements were taken at the constant difference of the blocks' temperatures equal to 2 K. The Hall coef®cient has been determined for Hall voltage measured along the b-axis at current along a-axis and magnetic ®eld of 1.25 T along c-axis. The possibility of the fast alternation of the ®eld direction produced by electromagnet allowed us an increase of the measurements' accuracy [2]. The magnetic susceptibility has been measured with SQUID magnetometer (Quantum Design MPMS-5) using 11 mg sample composed of several single crystals.
3. Analysis of the results 3.1. Resistivity All the selected crystals of ThAs1.23Se0.77 showed the same type of the r (T ) dependence; the maximum of the r (T ) is observed at sample dependent temperature Tmax followed by a resistivity minimum at about 200 K. The Tmax equals to 124, 134 and 149 K for the three crystals, respectively. The r (T ) data for the ®rst two crystals are presented in Fig. 1a. Their mean value of the resistivity at 300 K equals 32 mV cm and it is considerably lower than the value of about 220 mV cm observed for ThAsSe. [12]. The temperature dependence of the r (T )/r (300) for the two compounds is shown in Fig. 1b. One can expect that conduction electrons in ThAs1.23S0.77, like those in ThAsSe [5], are scattered by static impurities, phonons, and the TLS centres contributing to the resistivity r 0, r ph(T ) and r K(T ) components, respectively. According to Matthiessen's rule the total resistivity is the sum of these components. We write the phonon resistivity r ph(T ) in the form of product CphFB±G(T ) where Cph is a coef®cient dependent on a band structure and an electron±phonon coupling and FB±G(T ) is the well known Bloch±GruÈnaisen (B±G) function, universally dependent on T/Q D ratio. We have taken Q D 175 K [13], determined for ThAsSe. An attempt of description of our data by assuming
Z. Henkie, R. Wawryk / Solid State Communications 122 (2002) 1±6
3
Fig. 2. Temperature dependence of the phonon resistivity r ph/r 300 K and the sum of r 0 and Kondo resistivity (r 0 1 r K)/r 300 K determined by ®tting of Eq. (3) to experimental data of resistivity for thorium arsenosulphide sample of nos. 1 and 2. The ®tting was done without constraining of any parameter. Experimental data of resistivity diminished with the ®tted phonon resistivity for samples nos. 1 and 2 are shown by circles and triangles, respectively.
r K(T ),±ln T [14] leads to unphysical data both the r ph(T ) and r K(T ) near the room temperature. The resistivity of the Kondo systems cannot be given in an analytical form for the whole range of temperature. The nonperturbative results of calculation for T @ TK are quite reliable. We choose the results of Arai [15] to make an explicit comparison: rK T rK 0{1 1 p2 =3a T=TK 22 }21 ;
1
where a (T ) is a universal function of T/TK and is given by solution of: a T 2 TL =Tln{1 1 a2 T1=2 T=TL } 2 p ln 2 0;
2
where TL TK =0:624 and Kondo temperature TK is a characteristic temperature scaling for the particular Kondo system. The total resistivity:
r T r0 1 rph T 1 rK T
3
has been ®tted to experimental resistivity data for temperature range T . 1.1Tmax of two ThAs1.23S0.77 crystals. The ®tted parameters are r 0, Cph, r K(0) and TK. For positive but negligible small values of r 0 one gets r ph(300); r K(0); TK equal to 19.3; 68.6 mV cm; 24.8 K and 14.0; 55.5 mV cm; 25.7 K for the crystal 1 and 2, respectively. The above parameters determine the r (T ) dependence shown by the solid lines in Fig. 1b. Eq. (3) has been also used for the approximation of the resistivity data of ThAsSe from Ref. [12] shown schematically by a broken line and the calculated r (T ) by the solid line in Fig. 1b. The latter was obtained for r 0, r ph(300), r K(0) and TK equal to 0, 6.6, 1057 mV cm and 19.4 K, respectively. The analysis presented above shows that anomalous behaviour of the resistivity either ThAs1.23S0.77 or ThAsSe could be under-
stood in terms of the same model. The main difference between these two compounds consists in the amplitude of the Kondo component; the r K(0) for the ThAsSe is an over one order higher than that for the ThAs1.23S0.77 while the other ®tted parameters are comparable. The analysis also shows that the parameters were obtained by the ®tting done in the T/TK range 7±15. For this range, either calculations of Arai or the second order perturbative calculations of Neal and Collins [16] give r K(T )/r K(0) ranging between 0.2 and 0.35. But a relative difference, between the r K(T )/r K(0) of the two theories in the considered range, reaches 30% both in value and a slop of the r K(T )/r K(0) vs. T/TK dependence. We may also expect a similar difference between the experimental data and those theoretically predicted ones, especially that we use results of the theory for the single impurity to interpret the presumably TLS system. Though the theory [6,7] predicts similarity of the both systems behaviour, a high uncertainty in the determination of r 0 is possible. Therefore, we think that the physically correct ®tted parameters are also those giving positive high temperature values of r 0 1 r K(T ). A ®tting done without any constraining of the parameters of Eq. (3) gives TK equal to 20.3 and 22.5 K for crystal 1 and 2, respectively. The calculated r ph(T )/r (300) and the [r 0 1 r K(T )]/r (300) are shown in Fig. 2 by the curves denoted r ph and r K, respectively. Circles and triangles present experimental data of resistivity reduced by the phonon component determined for crystal 1 and 2, respectively. Above Tmax there is a very good agreement between the experimentally determined and the calculated one of the Kondo-like resistivity behaviour. The quantities determined by the last ®t do not differ suf®ciently enough from that
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Z. Henkie, R. Wawryk / Solid State Communications 122 (2002) 1±6
Fig. 3. Hall coef®cient vs. temperature for the thorium arsenosulphide. Magnetic ®eld equal to 1.25 T was parallel to c-axis of sample no. 1. The inset presents Hall coef®cient as a function of r K/T ratio.
obtained in the previous ®t to question the previous interpretation. The temperature derivative of r ph(T ) and r K(T ) components which is comparable in values but opposite in sign generates the minimum of the resistivity of ThAs1.23S0.77 at about 200 K. Because of the much larger component of the Kondo-like resistivity, the ThAsSe shows the minimum of the r (T ) shifted to about 500 K [12]. 3.2. Hall and thermoelectric power coef®cients Experimental data of the RH(T ) for the no. 1 crystal of
ThAs1.23S0.77 is shown in Fig. 3. The Hall coef®cient is negative and it varies from 21.15 £ 10 23 cm 3/C at 107 K to 26 £ 10 24 cm 3/C at 336 K. These values of the RH(T ) determine the carrier concentration that in one band model is equal to 5.4 £ 10 21 and 10.4 £ 10 21e 2/cm 3, respectively. The well-known semimetal, antimony has conduction electron concentration equal to 3.7 £ 10 19e 2/cm 3 [17] and it is suf®ciently high to, like in normal metals, temperature independent in considered temperature range. On the other hand, the magnetic Kondo systems are known as the ones showing temperature dependent Hall coef®cient despite a high carrier
Fig. 4. Thermoelectric power, measured along a-axis is plotted vs. temperature for the thorium arsenosulphide sample no. 1.
Z. Henkie, R. Wawryk / Solid State Communications 122 (2002) 1±6
concentration [18]: R H T R0 1 gx TrK T
4
where R0 is the ordinary Hall constant, gx (T ) r K(T ) is the skew scattering contribution that at high temperatures decreases linearly with product of reduced magnetic susceptibility x (T ) and the Kondo resistivity. Linear extrapolation of RH vs. xr K dependence from the range where it was determined to xrK 0 determines the R0. The ThAs1.23S0.77 crystals show weak and temperature independent diamagnetism. Their susceptibility measured in ®eld 5 T transverse to ab plane equals 21.2 £ 10 28 emu/mol at temperatures 20±300 K. So we should exclude the presence of magnetic Kondo impurities from our consideration. Following predictions [6,7] of great similarity of the TLS Kondo system behaviour to that of magnetic Kondo one, we implicitly assume that the Eq. (4) type dependence can be used for the interpretation of the RH(T ) for the thorium arsenosulphide. But there is a question about a quantity that should replace the x (T ). However, it was found that the RH(T ) for the arsenosulphide shows linear dependence on r K(T )/T ratio at T . 150 K. This is shown in set of Fig. 3. The linear extrapolation gives R0 24:3 £ 1024 cm3 =C: It corresponds to the one band carrier density 1.45 £ 10 22e 2/ cm 3 and the Hall mobility 13 cm 2/V s. The thermoelectric power for the no. 1 crystal of ThAs1.23S0.77 is negative except for narrow temperature range around < 50 K as Fig. 4 shows. A linear S(T ) dependence is observed above 150 K characterised by the measured value S 316 K 23:9 mV=K and determined by the linear extrapolation value S 0 21:2 mV=K: For the estimation of the conduction band parameters, we use the well-known Mott formula for diffusion thermoelectric power in the form: S T x p2 k2 T=3eEF
5
where k and e are the Boltzmann constant and electron charge, respectively. The parameter x depends on the mechanism of the electron scattering and experiment shows that at high temperatures it assumes values between 2 and 3 [19]. Taking x 3 and S at 316 K we estimate the Fermi energy EF 6:0 eV and effective mass equal to 0.37 free electron mass. Anomalies of S(T ), seen below 150 K in Fig. 4, can hardly be explained at the moment. 4. Discussion and conclusions The crystals of ThAs1.23S0.77 are free of magnetic impurities and, like in metals, have high conduction electron concentration. In spite of this, the crystals show anomalous behaviour of resistivity and Hall coef®cient. The r (T ) above 150 K has a shape of a shallow valley and can be resolved into phonon component and the Kondo one. The latter is consistent with that predicted in the Kondo limit of the Anderson model [15] and determines the Kondo tempera-
5
ture equal to 20.3 and 22.5 K for the crystals 1 and 2, respectively. The temperature dependent Hall coef®cient can be resolved into temperature independent normal Hall coef®cient and the temperature dependent skew scattering-like component characteristic for the magnetic Kondo systems. It is assumed that the Kondo like behaviour of the transport properties of the ThAs1.23S0.77 is due to scattering of the current carriers by the TLS centres created by the anion disorder, as it is in uranium and thorium arsenoselenides [5,8]. The simplest TLS model predicts deviation of the Kondo resistivity from logarithmic behaviour below T < D and saturation at T ! 0 [6]. If this was the reason for the observed discrepancy between the experimentally determined and the calculated Kondo resistivity below Tmax then we should assume D < 40 K: But Cox and Zawadowski [6] suggest that the Kondo state should not be developed for TK , D . The original TLS Kondo model also cannot explain the Kondo temperature as high as TK < 20 K (see Refs. [20,21]). However, as realised recently, electron±hole symmetry breaking present in all realistic band-structure-based density of states can increase TK substantially and is very promising to resolve the long standing problem of the experimentally observed high TK [21,22]. Undoubtedly more studies are needed to explain the origin of the unpredicted behaviour of the r (T ) below 140 K. The normal Hall constant of ThAs1.23S0.77 corresponds to 1e 2/f.u. (f.u.Ðformula unit) and is consistent with the odd number of the valence electrons in the stoichiometric ThAsS. Two ThAsS molecules per one Brillouine zone are in this structure and ThAsS should be a compensated metal, i.e. it should have the equal number of holes and electrons. In such a case the one band model can lead to the correct number of electrons if their mobility is clearly higher than the mobility of the holes. This seems to be the case of the thorium arsenoselenide where neither Hall coef®cient nor the thermoelectric power change sign despite the decrease of the valence electrons in the ThAs1.23S0.77 resulting from replacing part of the S atoms by the As ones. We expect that the total number of holes and electrons be of order 2 per f.u. It can be compared to 2.6 carriers per f.u. derived from room temperature optical conductivity of ThAsSe [23] and free electron mass of the carriers. Schoenes et al. [12] have found Hall coef®cient for ThAsSe decreasing monotonously with increasing temperature. In the one-band model the Hall concentration increases from 0.08e 2/f.u. at 100 K to 0.6e 2/f.u. at room temperature. They suggest the low effective mass of the carrier ( < 0.2m0) explains the discrepancy between the Hall carrier concentration and that derived from optical conductivity. They also suggest that a strong electron±phonon coupling may be responsible for low carrier mobility that equals merely 3.5 cm 2/V s. It seems to be too low for so light carriers. Finally they discuss several propositions of the origin of the varying number of carriers and propose more experiments before the ®nal explanation. We propose another
6
Z. Henkie, R. Wawryk / Solid State Communications 122 (2002) 1±6
explanation of the anomalous behaviour of the transport properties of the thorium arsenoselenides. The thermoelectric power of the ThAs1.23S0.77 crystals is negative, small and weekly varying with temperature above 150 K. It approves expectation that the main charge transport in the thorium arsenosulphide is controlled by a wide electron band. Its estimated parameters such as the carrier concentration n, EF, and the effective mass ratio m p/m0, are equal < 1.5 £ 10 22e 2/cm 3; < 6 eV; and < 0.4, respectively. The scattering of the electrons is characterised by mobility equal < 13 cm 2/V s at room temperature. The above parameters and the ®tted value of rK 0 ù 60 mV cm allows us to estimate the scatterers concentration with formula derived from [24,25]: h i c r K 0nN EF e2 h =2mp 8:75 £ 1019 rK 0=EF m0 =mp n22 2 cm23
6
where h is the Planck constant, N(E) the density of states and n22 is the carrier concentration in 10 22 cm 23 units n n22 1022 cm23 : Putting EF in eV and r K(0) in mV cm we obtain the concentration of scatterers c 5 £ 10 21 cm23 ; i.e. 1/3 per f.u. For such density of the scatterers we might consider another mechanism for an explanation of the observed below 140 K deviation of the measured r (T ) from that predicted by the TLS Kondo mechanism. One of possible thing might be some sort of electron mean free path `saturation' or interaction effects between TLSs, which certainly must be present at such a high concentration. In conclusion, the dominant contributions to the electron transport properties of the ThAs1.23S0.77 crystals originate from wide electron band carriers. The electron transport coef®cients show anomalous temperature behaviour of the type known for the Kondo-impurity system. This behaviour is ascribed to scattering of the electrons not only by phonons and the static impurities but also by dynamic centres, i.e. TLS centres, created by disordered occupation of the anion sublattices positions. This last scattering adds Kondo-like components to the electron transport coef®cients. However, the intensity of this scattering in ThAs1.23S0.77 is lower than that in ThAsSe and UAsSe. It is consistent with our hypothesis that a bigger difference of the anions' size leads to diminishing the Kondo effect, presumably due to favouring the more ordered structure. Such behaviour is common either for uranium or for thorium compounds. Acknowledgements The authors thank Prof. A. Pietraszko for X-ray analysis
and M.Sc. R. Gorzelniak for technical assistance with the magnetic susceptibility measurements. This work was supported by the Polish Committee for Scienti®c Research, Grant KBN-2 P03B 062 18, for years 2000±2001.
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