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Thermophysical properties of Pr1−xCaxCoO3
Thermochimica Acta 550 (2012) 53–58
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Thermophysical properties of Pr1−x Cax CoO3 Rasna Thakur ∗ , Rajesh K. Thakur, N.K. Gaur Department of Physics, Barkatullah University, Bhopal 462026, India
a r t i c l e
i n f o
Article history: Received 9 February 2012 Received in revised form 10 September 2012 Accepted 12 September 2012 Available online 3 October 2012 Keywords: Specific heat Thermal expansion Cobaltate Cohesive energy
a b s t r a c t In the present paper, the thermophysical properties of PrCoO3 cobaltates is mapped here with reference to the changed local environment at A-site due to calcium doping for the first time using Rigid Ion Model (RIM). The specific heat and there by thermal expansion for temperature (1 K ≤ T ≤ 1000 K) of Pr1−x Cax CoO3 (x = 0.0–0.5) are presented. The trends of variation of our computed results on specific heat with temperature are in more or less similar with corresponding experimental data for almost all the compositions (x) of Pr1−x Cax CoO3 . Strong electron–phonon interactions are present in these compounds which causes the variation of the lattice specific heat (Cv(lattice) ) with cation doping of varying size and valence. In addition, we have computed the thermal expansion (˛), bulk modulus (B), cohesive energy (), molecular force constant (f ), and Restsrahalen frequency () whose results are discussed in detail. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Recently, physicists are interested in the research of cobaltate, due to a couple of unique properties; namely, the large magnetoresistance (MR) [1], enormous Hall effect [2], the existence of the spin-state transitions [3], and the unusual magnetic ground states of doped cobaltate [4] as well as metal–insulator transitions. The perovskite cobaltates ACoO3 was discovered in 1950s. But they still attracted interest due to a key aspect of cobalt oxides that distinguishes them clearly from the transition metal oxides, i.e. the spin state degree of freedom of the Co3+/III and CoIV ions: it can be low, intermediate and high spin state (S = 0, 1, 2 for Co3+/III and S = 1/2, 3/2, 5/2 for CoIV ). Perovskite cobaltite ACoO3 (A: rare earth element) with a 3D network of corner-sharing CoO6 octahedra often under6 e0 , S = 0) ground goes a spin state change from low-spin state (LS, t2g g 5 e1 , S = 1) or high-spin state state to intermediate-spin state (IS, t2g g 4 e2 , S = 2) with increasing temperature [5]. A spin state tran(HS, t2g g sition was also proposed for PrCoO3 , for which the − T ( denotes magnetic susceptibility; and T denotes temperature) curve exhibits a broad minimum at around 200 K [6]. The parent PrCoO3 shows paramagnetic behavior down to 5 K [7]. Pandey el al. [8] studied the electronic states of PrCoO3 using X-ray photoemission spectroscopy and LDA + U density of states calculation and found that PrCoO3 was a charge transfer
∗ Corresponding author at: Superconductivity Research Lab, Department of Physics, Barkatullah University, Bhopal 462026, India. Tel.: +91 755 2491821; fax: +91 755 2491823. E-mail address: [email protected] (R. Thakur). 0040-6031/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.tca.2012.09.025
insulator. Yoshii and Nakamur [9] studied magnetic behavior and showed that PrCoO3 exhibited no magnetic ordering down to 4.5 K. The properties of these materials are expected to be strongly dependent on the average radius of the A-site cation. Substitution of cations with different ionic radii, at the A-site, distorts the structure, introduces disorder, and enhances antiferromagnetic (AFM) superexchange interactions and change the spin state of Co ions. The compound Pr0.7 Ca0.3 CoO3 has a semiconductor-like and a magnetic cluster-glass behavior [10] below 70 K. The physical properties of these compounds will depend on the average size of
yi ri2 − < rA >2 (where
the A-site (Pr3+ and Ca2+ ) cations, 2 =
i
yi is the fractional occupancy of A-site ions, and ri is the corresponding ionic radius). In the studies of ‘half-doped’ perovskite cobaltates, Pr0.5 Ca0.5 CoO3 behaves anomalously. Pr0.5 Ca0.5 CoO3 is the first compound in cobalt oxides which describes both the cooperative metal–insulator (M–T) and spin state transition in perovskite cobaltates and is considered a strongly correlated spincrossover system [3,11]. At ambient temperature, they appear in 5 ∗ phase as expected but on cooling they undergo a the metallic t2g sharp metal–insulator (M–I) transition at TM–I = 90 K, documented for the first time by Tsubouchi et al. [3,12]. The same transition was observed also on the less doped samples Pr1−x Cax CoO3 (x = 0.3) under high pressures or upon a partial substitution of praseodymium by smaller rare earth cations or yttrium [13]. It has been seen that the formal cobalt valency in Pr0.5 Ca0.5 CoO3 is changed at TM–I from the mixed-valence Co3.5+ toward pure Co3+ with strong preference for LS state, and the praseodymium valence is simultaneously increased from Pr3+ toward Pr4+ . Pr0.5 Ca0.5 CoO3 shows a temperature-induced paramagnetic–paramagnetic
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R. Thakur et al. / Thermochimica Acta 550 (2012) 53–58
spin-state transition accompanied by a simultaneous MIT (TM–I = 70 K). Recent experiments have proved the possibility of generating metallic domains in the insulating low temperature phase of Pr0.5 Ca0.5 CoO3 by ultrafast photoexcitation, making this material of interest in the area of ultrafast optical switching devices [14]. In this compound, a spin state transition from IS Co3+ to diamagnetic LS Co3+ and a charge-ordering of Co3+ (S = 0) and Co4+ sites was proposed as the origin of the MIT transition [3]. The magnetic measurements indicate that the Ca-doped samples (x = 0, 0.3, 0.5) have at low temperatures, similar properties to the frustrated magnetic materials. A semiconducting type behavior and high negative magnetoresistance was found for the Ca-doped compounds [15]. The crystal structure of given cobaltates with space group Pbnm have an orthorhombic structure and four formula units per unit cell [15]. However, the studies regarding the theoretical understanding of these lattice effects on the thermophysical properties have been initiated by very few researchers whereas it is important to know the effects of doping a cation with different charge and size compared to the host cation on heat conduction of these materials. Taking into consideration these results, the Pr1−x Cax CoO3 have been investigated to study their thermophysical properties, temperature dependence of the lattice contribution to the specific heat at constant volume (Cv(lattice) ) and thermal expansion by varying x in the range 0.0 < x < 0.5. To the best of our knowledge no systematic experimental or theoretical study of thermal properties of these calcium-doped cobaltates was undertaken in the past whereas it is well established that strong electron–phonon coupling is present in these compounds and this electron–phonon interaction is one of the most relevant contributions in determining the conduction mechanism in these cobaltates. In order to do a systematic analysis of how the specific heat, thermal expansion, cohesive and thermal properties can be tuned by doping, the series of cobaltites Pr1−x Cax CoO3 (0.0 ≤ x ≤ 0.5) is studied as a function of temperature (0 ≤ T ≤ 1000 K) using RIM. The essentials of RIM formalism and the results obtained from its application are presented in subsequent sections. 2. RIM computations The potential describing the formalism of RIM is expressed as [18–24]: R (r) = − kk
i
+
ni 2
bi [ˇkk exp{(2rk − rkk )/i } + ˇk k exp{(2rk − rk k )/i }] (1)
Here, first term is attractive long-range (LR) coulomb interactions energy and the second term is overlap repulsive energy represented by the Hafemeister–Flygare-type (HF) interaction extended up to the second neighbour. Here, rkk’ represents separation between the nearest neighbours while rkk and rk’k’ appearing in the next terms are the second neighbour separation. rk (rk’ ) is the ionic radii of k (k ) ion. n (n ) is the number of nearest (next nearest neighbour) ions. In ABO3 (such as PrCoO3 ) perovskite structure, k represents cation (A, B) and k denotes the (O1 , O2 ) type of ions. bi and i are the hardness and range parameters for the ith cation–anion pair (i = 1, 2) respectively and (i kk is the Pauling coefficient [25] given by ˇi
kk
=1+
Z Z k k Nk
+
Nk
d
(r) dr
r=r0
=0
(3)
and the bulk modulus: B=
1 9Kr0
d2 (r) dr 2
(4) r=r0
where K is the crystal-structure-dependent constant and r0 is the equilibrium nearest neighbor distance. The cohesive energy for Pr1−x Cax CoO3 (0.0 ≤ x ≤ 0.5) is calculated using Eq. (1) and other thermal properties, the molecular force constant (f), Restsrahalen frequency (), and Debye temperature ( D ) are computed using the expression given in our previous papers [18–24]. The expression for calculating lattice specific heat is D
CV (lattice) = 9R
T 3 T D
ex x 4 (ex
0
− 1)2
dx
(5)
In the above equation R is the universal gas constant and D is the Debye temperature. CP is formulated by taking into account the important role of the ˛2 BT VT term, which links the measured specific heat with the measured physical properties of solids at high temperature. CP = CV + ˛2 BT VT
(6)
where T is the temperature, BT is the isothermal bulk modulus, V is the unit cell volume and ˛ is the volume thermal expansion calculated by ˛=
CV BT V
(7)
Here BT , V, CV are the isothermal bulk modulus, unit formula volume and specific heat at Constant volume respectively and is the Gruneisen parameter. The results thus obtained are presented and discussed below. 3. Results and discussion
e2 −1 Zk Zk rkk ni bi ˇkk exp{(rk + rk − rkk )/i } + 2 kk
Zk (Zk ) and Nk (Nk ) are the valence and the number of electrons in the outermost orbit of k (k ) ion respectively. The model parameters, hardness (b) and range () parameters are determined from the equilibrium condition:
(2)
We have evaluated the model parameters (b, ) from Eqs. (3) and (4) by taking the values of the input data like unit cell parameters (a, b, c) and other interionic distances directly from Refs. [3,15–17]. The values of computed model parameters (b1 , 1 ) and (b2 , 2 ) corresponding to the ionic bonds Co O and Pr/Ca O as a function of temperature are listed in Table 1. The values of b and show decreasing trend with increasing Ca doping. This decreasing trend of hardness parameter (b) with Ca doping indicates the decrease in the strength of the crystal with higher levels of doping. The unit cell parameter (a, b, c) taken from the reported data [3,15–17], are used to calculate the simple perovskie lattice cell parameter (r) in the space group Pbnm for the present compounds [15–17]. The values of ionic radii for Pr3+ , Ca2+ , Co3+ , Co4+ and O2− are taken from ref. [26] and the data on atomic compressibility are taken from ref. [27]. In the ideal cubic perovskite structure, RE3+ occupies the centre of dodecahedron of oxygen and has an ideal unity value of tolerance factor. The Co O Co angle reaches the value 180◦ which facilitates the exchange interactions and electron hopping. But due to orthorhombic deformation of the lattice, the coordination number of Pr reduces to 9, as three oxygen atoms remain essentially non-bonded due to large Pr3+ O distance. So we considered the ionic radii of A-site cations of coordination number 9 [26] in the
R. Thakur et al. / Thermochimica Acta 550 (2012) 53–58
55
Table 1 Values of critical radius, tolerance factor, model parameters of orthorhombic Pr1−x Cax CoO3 (0 ≤ x ≤ 0.5) cobaltate. Doping concentration (x)
rCR (Å)
Tolerance factor (t)
b1 × 10−19 (J)
b2 × 10−19 (J)
1 (Å)
2 (Å)
0.00 0.1 0.2 0.3 0.5
1.026 1.023 1.021 1.017 0.995
0.9377 0.9385 0.9393 0.9400 0.9416
1.47 1.45 1.44 1.43 1.36
1.81 1.80 1.79 1.77 1.66
0.198 0.195 0.193 0.192 0.190
0.349 0.348 0.344 0.343 0.342
present investigation. It is useful to recall the Goldschmidt tolerance factor which is a fundamental parameter determining the kind of structural distortion in perovskites with general formula ABO3 . √ It is defined as t (t = (rA + rO )/ 2 (rB + rO )), where rA , rB , and rO are the radii of the A-site cation, the B-site cation, and the oxygen ion, respectively. The condition that t lies between 0.77 and 1 for stable perovskite structure [28] is satisfied in these compounds (Table 1). Goldschmidt’s tolerance factor (t) has been calculated with the aid of Shannon’s [26] set of ionic radii for series of perovskites with concentration x. t gradually decreases as the cation radius decreases down the series, which also suggests a increase of global distortions. Depending on the composition of the perovskite a critical radius rcr can be calculated which describes the maximum size of the mobile ion to pass through. The critical radius can be calculated by using the following Eq.
√ ao 34 ao − 2rB + rB2 − rA2 rcr = (8) √ 2(rA − rB ) + 2ao where rA and rB are the radius of the A ion and B ion, respectively, and a0 corresponds to the pseudo cubic lattice parameter (V1/3 ). The condition that critical radius does not exceed 1.05 A˚ [29] for typical perovskite materials satisfied in our compounds. 3.1. Bulk modulus Bulk modulus of the compounds decides the equilibrium condition of the lattice, but the experimental values of bulk modulus are not available in literature for the studied compounds. So we thought of making a study of determination of bulk modulus of PrCoO3 and diluted compounds systematically on the basis of formulations of Atoms in Molecules theory (AIM) [30] which emphasize the partitioning of static thermophysical properties in condensed systems into atomic or group distributions. Hence, inverse of the bulk modulus is the simple weighted average of the atomic compressibilities [30]: =
i
fi i and
1 1 = fi B Bi
following relation for the bulk modulus of the distorted perovskite cobaltate: BT =
KS B0 m cos ω exp( JT )C
(10)
where KS is the spin order-dependent constant of proportionality, and its value is less than 1 for the ferromagnetic state and more than 1 for paramagnetic state, B0 is the bulk modulus for undistorted structure calculated on the basis of AIM theory, m is the size mismatch, c is the charge mismatch, JT is JT distortion of CoO6 octahedra and cos ω is the effect of buckling of the Co O Co angle. The values of bulk modulus of the distorted structure are presented in Table 2 as B which takes effects of all the distortions. 3.2. Cohesive properties Using values of bulk modulus, we have computed the cohesive energy () for Pr1−x Cax CoO3 at composition 0.0 ≤ x ≤ 0.5 using Eq. (1) and listed them in Table 2. It is seen from Table 2 that the magnitude of the cohesive energy increases systematically from () = −153.5 eV for x = 0.0 (i.e. PrCoO3 ) to −160.9 eV for x = 0.5. The cohesive energy is, expected to follow the same trend of variation as that of the bulk modulus, and is related to the overall atomic binding properties of a material. In conformity with this, we notice from Table 2 that our results on the cohesive energy () have followed more or less similar trend of variation as that exhibited by the bulk modulus, which represents the resistance to volume change and is related to the overall atomic binding properties of a material. Nevertheless, they are stable compounds as the cohesive energy is negative within the studied range of composition (x). Further, to test the validity of our model, we calculated the cohesive energy of these compounds using the generalized Kapustinskii Equation [31] which uses the ionic strength of the crystal. 1 nk zk2 2 T
I=−
(11)
k
(9)
i
where fi = Vi /V where fi is the fractional volume occupancy due to quantum subsystem i in a unit formula, B is the bulk modulus of the compound and k is its compressibility. We have computed the bulk modulus using Eq. (9) on the basis of AIM theory and in this analysis the octahedral are considered to be undistorted and perfect. When we consider the Ca doping at the A-site in PrCoO3 compound then the local compressibility of impurity with respect to the host ions are governed by the size difference and tuned by the formal charge mismatch between host and guest cations of A-site. We considered the effect of charge, size mismatch along with the octahedral distortions due to Jahn–Teller effects on the bulk modulus of the compounds. These factors will determine the change in the unit cell volume which in turn will change the global bulk modulus of the compound. The formal expression for calculating the cation size and charge mismatch of the distorted perovskite cobaltate is given in our earlier paper [21]. It is now appropriate to propose the
where T is the number of type of ions in the formula unit each of number nk and charge zk . In our calculation the value of ionic strength for PrCoO3 is found to be 16 and this value decreases slightly with Ca doping. According to generalized Kapustinskii equation [31] the lattice energies of crystal with multipole ions is given as: U/kJ mol−1 = −
1213.9 r
1−
r
nk zk 2
(12)
where r = weighted mean cation–anion radius sum (using Goldschmidt radii) and is taken as the average of our model parameters 1 and 2 . Estimated values of lattice energy by Kapustinskii equation for pure and calcium doped PrCoO3 are reported in Table 2 and seems to be in very close agreement with our calculated values which confirms the validity of RIM. The experimental values of the cohesive energy for Pr1−x Cax CoO3 (0.0 ≤ x ≤ 0.5) compounds are not available, but our calculated values are comparable to the reported value of Madelung energy of the valence skip compound BaBiO3 , which is −165 eV [32].
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R. Thakur et al. / Thermochimica Acta 550 (2012) 53–58
Table 2 Values of bulk modulus, cohesive and thermal properties of orthorhombic Pr1−x Cax CoO3 (0.0 ≤ x ≤ 0.5) cobaltate. Doping concentration (x)
B (Gpa)
(eV) (RIM)
(eV) (Kapustinskii equation)
f (N/m)
(THz )
D (K)
0.00 0.1 0.2 0.3 0.5
224.8 230.1 235.3 238.0 248.9
−153.5 −154.2 −155.6 −157.0 −160.9
−153.5 −153.5 −154.8 −155.9 −157.8
43.14 44.11 45.08 45.62 46.95
10.5 10.7 10.9 11.3 12.1
504.43 512.56 527.83 542.18 579.44 630a
Ref. [16].
160
3.4. Specific heat and thermal expansion The specific heat in the normal state of the material is usually approximated by the contributions of the lattice and electronic specific heat. In our model, we have considered only the lattice contributions. We have computed the variation of the lattice contributions to the specific heat of Pr1−x Cax CoO3 (x = 0, 0.1, 0.2, 0.3, and 0.5) compounds in the wide temperature range 1 K ≤ T ≤ 1000 K. The temperature variations of specific heat of these compounds for various compositions are depicted in the main panel of Figs. 1–5. At low temperatures (T < D /50), a meticulous attention was given to the investigation of the assumption that these cobaltate perovskites are Debye-like solids and specific heat obeys T3 law. At intermediate temperature, the temperature dependence of specific heat is governed by the vibrations of atoms. Now, the systematic investigation of the specific heat at 1 K ≤ T ≤ 1000 K of Pr1−x Cax CoO3 (x = 0, 0.1, 0.2, 0.3, and 0.5) are in satisfactorily agreement with the measured data of Tsubouchi et al. [12]. The specific heats in these materials varies almost linearly between 0 and 200 K and later on vary along curvature from 200 to 550 K and finally become almost constant beyond 550 K. The remarkable deviations appearing at temperatures (10–300 K) are due to the anharmonic effects, whose contributions are very well demonstrated by Jindal et al. [33] for other materials. However, at higher temperatures, the anharmonic effects on specific heat is suppressed and it is very close to the
Exp.
PrCoO3
Cal. 120
20
80
40
10
0
0
0
200
0
200
400
400 600 T (K)
T (K)
600
800
800
1000
1000
Fig. 1. Computed specific heat and thermal expansion (inset) of perovskite PrCoO3 as a function of temperature. Present results are shown by line with solid circle (−•−) and experimental values [12] are shown by line with open circle (− −).
Dulong-Petit limit CV = 15 NkB ∼ 125 J K−1 mol−1 in ABO3 perovskites, which is common to all solids at high temperature. It is noticed from Fig. 5 that the computed specific heat curve for Pr0.5 Ca0.5 CoO3 is in good agreement with the experimental data [12] and the trends of variations exhibited by the experimental and theoretical results are almost similar except that an anomaly seen to tail shown by measured data at T = 90 K. This observed tail owes its origin to the occurrence of the simultaneous metal–insulator and paramagnetic–paramagnetic spin-state transition of the firstorder type at 90 K, indicating a cooperative nature. However, this peak could not be reproduced from our model potential calculation due to the exclusion of the magnetic contribution. An anomaly at x = 0.5 around TC (90 K) is observed in experimental curve as shown in the Fig. 5, but on the other hand at x = 0.2 and 0.3 no clear anomaly 160
Exp.
Pr 0.9 Ca 0.1 CoO 3
Cal.
Specific heat (J/mole K)
We have also computed the molecular force constant (f), Reststrahlen frequency (), and Debye temperature ( D ) for pure Ca doped PrCoO3 compounds, and results are reported in Table 2. In the Debye approach we consider the frequency of vibration of positive ion lattice with respect to negative ion lattice obtained from this model is reported here as the Reststrahlen frequency () for Pr1−x Cax CoO3 (x = 0.0–0.5). It can be seen from Table 2 that the values of Restrahalen frequency are increasing with Ca concentration (x), which shows that the bond is becoming stronger with Ca doping. This feature is similar to the increasing trends exhibited by the stability (−) of the present materials. Since the Restrahalen frequency is directly proportional to the molecular force constant (f) therefore both of them vary with the temperature accordingly for these compounds. The values of Debye temperatures ( D ) for Pr1−x Cax CoO3 (0.0 ≤ x ≤ 0.5) have been calculated using these restrahalen frequencies and listed in Table 2. It is noticed that D for PrCoO3 increases with the doping of Ca concentration (x), indicating the higher phonon frequencies of the given system. Present investigation reemphasise that Debye temperature is dependent on the A-site cation radius and it increases with increasing calcium doping. The present results could not be compared due to lack of experimental data on them. At present our results are only of academic interest but will certainly work as a guide to the experimental workers in future.
Specific heat (J/mole K)
3.3. Thermal properties
(10-6 K-1)
a
120
80
40
0
0
200
400
600
800
1000
T (K) Fig. 2. Computed specific heat and thermal expansion (inset) of perovskite Pr0.9 Ca0.1 CoO3 as a function of temperature. Present results are shown by line with solid circle (−•−) and experimental values [12] are shown by line with open circle (− −).
R. Thakur et al. / Thermochimica Acta 550 (2012) 53–58
Specific heat (J/mole K)
160
Exp.
Pr 0.8 Ca
Cal.
0.2
CoO 3
120
80
0
0
200
400
600
800
1000
T (K) Fig. 3. Computed specific heat and thermal expansion (inset) of perovskite Pr0.8 Ca0.2 CoO3 as a function of temperature. Present results are shown by line with solid circle (−•−) and experimental values [12] are shown by line with open circle (− −).
160
Exp.
Pr
Specific heat (J/mole K)
Cal.
0.7
Ca
0.3
CoO 3
(13)
where the first term is the lattice contribution which has been calculated by us using Eq. (7), the second term is the magnetic contribution and the third term is the ˛I–M refers to the anomalous expansion associated with the insulator–metal (I–M) transition. An inspection of Figs. 1–5 (inset) shows that the thermal expansion coefficient ˛ increases with T3 at low temperatures and gradually approaches a linear increase at high temperatures, and then the increasing trend becomes gentler. It is shown that the thermal expansion coefficient ˛ converges to a constant value at high temperatures. The lattice thermal expansion decreases, respectively, as the calcium concentration increases from 0 to 50%. We could not compare our results on thermal expansion, due to non-availability of the experimental data.
120
4. Conclusions 80
40
0
0
200
400
600
800
1000
T (K) Fig. 4. Computed specific heat and thermal expansion (inset) of perovskite Pr0.7 Ca0.3 CoO3 as a function of temperature. Present results are shown by line with solid circle (−•−) and experimental values [12] are shown by line with open circle (− −).
are found (Figs. 3 and 4) because of magnetically inhomogeneous behavior. The change of Co O distance by the substitution of Ca at Pr site increases Debye temperature ( D ) and hence there is a consistent decrease in the specific heat. Hence, the concentration (x) dependence of Debye temperature in Pr1−x Cax CoO3 (0.0 ≤ x ≤ 0.5) suggests that the increased Ca doping drives the
Exp.
160
Specific heat (J/mole K)
system effectively close to the strong electron–phonon coupling regime. We have computed the variation of the lattice contribution to the thermal expansion for temperature range (1 K ≤ T ≤ 1000 K) for Pr1−x Cax CoO3 (0.0 ≤ x ≤ 0.5). The values of the thermal expansion at different temperatures are displayed in the inset of Figs. 1–5. The total contribution to thermal expansion is ˛(T ) = ˛latt + ˛mag + ˛I-M
40
57
Pr 0.5Ca
Cal.
0.5
CoO 3
120 80 40 0
0
200
400
600
800
1000
T (K) Fig. 5. Computed specific heat and thermal expansion (inset) of perovskite Pr0.5 Ca0.5 CoO3 as a function of temperature. Present results are shown by line with solid circle (−•−) and experimental values [12] are shown by line with open circle (− −).
On the basis of overall discussion, it may be concluded that the present RIM is successful and represents a proper theoretical approach to provide the understanding of the thermal properties of Ca doped PrCoO3 cobaltate family of perovskites. Present results on specific heat could be further improved by incorporating the effects of the interactions due to van der Walls, magnetic and anharmonic contributions. Our results are probably the first reports of thermal expansion at these temperatures and compositions. Thus a satisfactory prediction of the Pr1−x Cax CoO3 (0.0 ≤ x ≤ 0.5) attained by us is remarkable in view of the inherent simplicity of RIM and its less parametric nature. Acknowledgements Financial support by Department of Science and Technology (DST) and University Grant Commission (UGC) New Delhi are gratefully acknowledged. References [1] G. Briceno, H.Y. Chang, X.D. Sun, P.G. Schultz, X.D. Xiang, A class of cobalt oxides magnetoresistance materials discovered with combinational synthesis, Science 270 (1995) 273. [2] A.V. Samoilov, G. Beach, C.C. Fu, N.C. Yeh, R.P. Vasquez, Magnetic percolation and giant spontaneous Hall effect in La1−x Cax CoO3 (0.2 ≤ x ≤ 0.5), Phys. Rev. B57 (1998) R14032. [3] S. Tsubouchi, T. Kyomen, M. Itoh, P. Ganguly, M. Oguni, Y. Shimojo, Y. Morii, Y. Yoshi, Simultaneous metal–insulator and spin state transitions in Pr0.5 Ca0. 5CoO3 , Phys. Rev. B66 (2002) 052418. [4] M.J.R. Hoch, P.L. Kuhns, W.G. Moulton, A.P. Reyes, J. Lu, J. Wu, C. Leighton, Evolution of the ferromagnetic and nonferromagnetic phases with temperature in phase-separated La1−x Srx CoO3 by high-field 139 La NMR, Phys. Rev. B70 (2004) 174443. [5] Y. Kobayashi, T. Nakajima, K. Asai, Magnetic and transport properties in La1−x Prx CoO3 single crystals, J. Magn. Magn. Mater. 272 (2004) 83. [6] J.Q. Yan, J.S. Zhou, J.B. Goodenough, Bond-length fluctuations and the spin-state transition in LCoO3 (L = LaPr, and Nd), Phys. Rev. B69 (2004) 134409. [7] H.W. Brinks, H. Fjellvag, A. Kjekshus, B.C. Hauback, Structure magnetism of Pr1−x Srx CoO3−␦ , J. Solid State Chem. 147 (1999) 464. [8] S.K. Pandey, A. Kumar, S.M. Choudhari, A.V. Pimpale, Electronic states of PrCoO3 : x-ray photoemission spectroscopy and LDA + U density of states studies, J. Phys.: Condens. Matter 18 (2006) 1313–1323. [9] K. Yoshii, A. Nakamur, Magnetic properties of Pr1-xSrxCoO3 , Physica B 281 (2000) 514–515. [10] I.G. Deac, A. Vladescu, I. Balasz, A. Tunyagi, R. Tetean, Electrical and magnetic behavior of transition metal oxides Ln0.7 A0.3 TMO3 Ln = La, Pr, A = Ca, Sr and TM = Mn, Co, Int. J. Mod. Phys. B 24 (2010) 762.
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[11] K. Knizek, J. Hejtmanek, P. Novak, Z. Jirak, Charge transfer, valence, and the metal–insulator transition in Pr0.5 Ca0.5 CoO3 , Phys. Rev. B 81 (2010) 155113. [12] S. Tsubouchi, T. Kyomen, M. Itoh, M. Oguni, Electric, magnetic, and calorimetric properties and phase diagram of Pr1−x Cax CoO3 (0 ≤ x ≤ 0.55), Phys. Rev. B 69 (2004) 144406. [13] T. Naito, H. Sasaki, H. Fujishiro, Simultaneous metal–insulator spin-state transition in (Pr1−y REy )1−x Cax CoO3 (RE = Nd, Sm, Gd, and Y), J. Phys. Soc. Jpn. 79 (2010) 034710. [14] Y. Okimoto, X. Peng, M. Tamura, T. Morita, K. Onda, T. Ishikawa, S. Koshihara, N. Todoroki, T. Kyomen, M. Itoh, Ultrasonic propagation of a metallic domain in Pr0.5 Ca0.5 CoO3 undergoing a photoinduced insulator–metal transitions, Phys. Rev. Lett. 103 (2009) 027402. [15] I.G. Deac, R. Tetean, I. Balasz, E. Burzo, Low-temperature magnetic ordering in the perovskites Pr1-xAxCoO3 (A = Ca, Sr), J. Magn. Magn. Mater. 322 (2010) 1185–1188. [16] K. Knizek, J. Hejtmanek, Z. Jirak, P. Tomes, P. Henry, G. Andre, Neutron diffraction heat capacity studies of PrCoO3 and NdCoO3 , Phys. Rev. B 79 (2009) 134103. [17] B. Rivas-Murias, M. Sanchez-Andujar, J. Rivas, A. Fondado, J. Mira, M.A. SenarisRodriguez, Structure, microstructure and magneto-transport properties of Pr1-xBxCoO3 -␦ (B2+ = Ba2+ , Ca2+ ) perovskite materials, Nato Sci. Ser. 61 (2002) 577–584. [18] R. Choithrani, M.N. Rao, N.K. Gaur, S.L. Chaplot, R.K. Singh, Lattice dynamics of manganites RMnO3 (R = Sm, Eu or Gd): instabilities and coexistence of orthorhombic and hexagonal phases, New J. Phys. 11 (2009) 073041. [19] R. Thakur, A. Srivastava, R.K. Thakur, N.K. Gaur, Specific heat of rare earth cobaltates RCoO3 (R = La, Pr and Nd), J. Alloys Compd. 516 (2012) 58–64. [20] N.K. Gaur, R. Choithrani, A. Srivastava, Elastic moduli and thermal properties of La0.83 Sr0.17 MnO3 , Solid State Commun. 145 (2008) 308–311. [21] A. Srivastava, N.K. Gaur, The role of lattice distortions in determining the thermal properties of electron doped CaMnO3 , J. Phys.: Condens. Matter 21 (2009) 096001.
[22] E.G. Rini, M.N. Rao, S.L. Chaplot, N.K. Gaur, R.K. Singh, Phonon dynamics of lanthanum manganite LaMnO3 using an interatomic shell model potential, Phys. Rev. B 75 (2007) 214301–214307. [23] N. Kaur, R. Mohan, N.K. Gaur, R.K. Singh, Influence of La doping on elastic and thermodynamic properties of SrMoO3 , J. Alloys Compd. 509 (2011) 6077–6082. [24] R. Choithrani, M.N. Rao, S.L. Chaplot, N.K. Gaur, R.K. Singh, Structural and phonon dynamical properties of perovskite manganites: (Tb, Dy, Ho) MnO3 , J. Magn. Magn. Mater. 323 (2011) 1627. [25] L. Pauling, Nature of the chemical bond, Z. Kristallogr. 67 (1928) 377. [26] R.D. Shannon, Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides, Acta Crystallogr. A 32 (1976) 751–767. [27] C. Kittel, Introduction to Solid State Physics, 5th edn, Wiley, New York, 2004, www.webelements.com [28] O. Muller, R. Roy, The Major Ternary Structural Families, Springer Heidelberg, 1974. [29] M. Mogensen, D. Lybye, N. Bonanos, P.V. Hendriksen, F.W. Poulsen, Factors controlling the oxide ion conductivity of fluorite and perovskite structured oxides, Solid State Ionics 174 (2004) 279–286. [30] A.M. Pendas, A. Costales, M.A. Blanco, J.M. Recio, V. Luana, Local compressibilities in crystals, Phys. Rev. B 62 (2000) 13970–13978. [31] L. Glasser, Lattice energies of crystals with multiple ions: a generalized Kapustinskii equation, Inorg. Chem. 34 (1995) 4935. [32] I. Hase, T. Yanagisawa, Madelung energy of the valence-skipping compound BaBiO3 , Phys. Rev. B 76 (2007) 174103–174106. [33] V.K. Jindal, R. Righini, S. Califano, Fourth-order relaxation processes in crystalline CO2 , Phys. Rev. B 38 (1988) 4259.
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Thermochimica Acta journal homepage: www.elsevier.com/locate/tca
Thermophysical properties of Pr1−x Cax CoO3 Rasna Thakur ∗ , Rajesh K. Thakur, N.K. Gaur Department of Physics, Barkatullah University, Bhopal 462026, India
a r t i c l e
i n f o
Article history: Received 9 February 2012 Received in revised form 10 September 2012 Accepted 12 September 2012 Available online 3 October 2012 Keywords: Specific heat Thermal expansion Cobaltate Cohesive energy
a b s t r a c t In the present paper, the thermophysical properties of PrCoO3 cobaltates is mapped here with reference to the changed local environment at A-site due to calcium doping for the first time using Rigid Ion Model (RIM). The specific heat and there by thermal expansion for temperature (1 K ≤ T ≤ 1000 K) of Pr1−x Cax CoO3 (x = 0.0–0.5) are presented. The trends of variation of our computed results on specific heat with temperature are in more or less similar with corresponding experimental data for almost all the compositions (x) of Pr1−x Cax CoO3 . Strong electron–phonon interactions are present in these compounds which causes the variation of the lattice specific heat (Cv(lattice) ) with cation doping of varying size and valence. In addition, we have computed the thermal expansion (˛), bulk modulus (B), cohesive energy (), molecular force constant (f ), and Restsrahalen frequency () whose results are discussed in detail. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Recently, physicists are interested in the research of cobaltate, due to a couple of unique properties; namely, the large magnetoresistance (MR) [1], enormous Hall effect [2], the existence of the spin-state transitions [3], and the unusual magnetic ground states of doped cobaltate [4] as well as metal–insulator transitions. The perovskite cobaltates ACoO3 was discovered in 1950s. But they still attracted interest due to a key aspect of cobalt oxides that distinguishes them clearly from the transition metal oxides, i.e. the spin state degree of freedom of the Co3+/III and CoIV ions: it can be low, intermediate and high spin state (S = 0, 1, 2 for Co3+/III and S = 1/2, 3/2, 5/2 for CoIV ). Perovskite cobaltite ACoO3 (A: rare earth element) with a 3D network of corner-sharing CoO6 octahedra often under6 e0 , S = 0) ground goes a spin state change from low-spin state (LS, t2g g 5 e1 , S = 1) or high-spin state state to intermediate-spin state (IS, t2g g 4 e2 , S = 2) with increasing temperature [5]. A spin state tran(HS, t2g g sition was also proposed for PrCoO3 , for which the − T ( denotes magnetic susceptibility; and T denotes temperature) curve exhibits a broad minimum at around 200 K [6]. The parent PrCoO3 shows paramagnetic behavior down to 5 K [7]. Pandey el al. [8] studied the electronic states of PrCoO3 using X-ray photoemission spectroscopy and LDA + U density of states calculation and found that PrCoO3 was a charge transfer
∗ Corresponding author at: Superconductivity Research Lab, Department of Physics, Barkatullah University, Bhopal 462026, India. Tel.: +91 755 2491821; fax: +91 755 2491823. E-mail address: [email protected] (R. Thakur). 0040-6031/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.tca.2012.09.025
insulator. Yoshii and Nakamur [9] studied magnetic behavior and showed that PrCoO3 exhibited no magnetic ordering down to 4.5 K. The properties of these materials are expected to be strongly dependent on the average radius of the A-site cation. Substitution of cations with different ionic radii, at the A-site, distorts the structure, introduces disorder, and enhances antiferromagnetic (AFM) superexchange interactions and change the spin state of Co ions. The compound Pr0.7 Ca0.3 CoO3 has a semiconductor-like and a magnetic cluster-glass behavior [10] below 70 K. The physical properties of these compounds will depend on the average size of
yi ri2 − < rA >2 (where
the A-site (Pr3+ and Ca2+ ) cations, 2 =
i
yi is the fractional occupancy of A-site ions, and ri is the corresponding ionic radius). In the studies of ‘half-doped’ perovskite cobaltates, Pr0.5 Ca0.5 CoO3 behaves anomalously. Pr0.5 Ca0.5 CoO3 is the first compound in cobalt oxides which describes both the cooperative metal–insulator (M–T) and spin state transition in perovskite cobaltates and is considered a strongly correlated spincrossover system [3,11]. At ambient temperature, they appear in 5 ∗ phase as expected but on cooling they undergo a the metallic t2g sharp metal–insulator (M–I) transition at TM–I = 90 K, documented for the first time by Tsubouchi et al. [3,12]. The same transition was observed also on the less doped samples Pr1−x Cax CoO3 (x = 0.3) under high pressures or upon a partial substitution of praseodymium by smaller rare earth cations or yttrium [13]. It has been seen that the formal cobalt valency in Pr0.5 Ca0.5 CoO3 is changed at TM–I from the mixed-valence Co3.5+ toward pure Co3+ with strong preference for LS state, and the praseodymium valence is simultaneously increased from Pr3+ toward Pr4+ . Pr0.5 Ca0.5 CoO3 shows a temperature-induced paramagnetic–paramagnetic
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R. Thakur et al. / Thermochimica Acta 550 (2012) 53–58
spin-state transition accompanied by a simultaneous MIT (TM–I = 70 K). Recent experiments have proved the possibility of generating metallic domains in the insulating low temperature phase of Pr0.5 Ca0.5 CoO3 by ultrafast photoexcitation, making this material of interest in the area of ultrafast optical switching devices [14]. In this compound, a spin state transition from IS Co3+ to diamagnetic LS Co3+ and a charge-ordering of Co3+ (S = 0) and Co4+ sites was proposed as the origin of the MIT transition [3]. The magnetic measurements indicate that the Ca-doped samples (x = 0, 0.3, 0.5) have at low temperatures, similar properties to the frustrated magnetic materials. A semiconducting type behavior and high negative magnetoresistance was found for the Ca-doped compounds [15]. The crystal structure of given cobaltates with space group Pbnm have an orthorhombic structure and four formula units per unit cell [15]. However, the studies regarding the theoretical understanding of these lattice effects on the thermophysical properties have been initiated by very few researchers whereas it is important to know the effects of doping a cation with different charge and size compared to the host cation on heat conduction of these materials. Taking into consideration these results, the Pr1−x Cax CoO3 have been investigated to study their thermophysical properties, temperature dependence of the lattice contribution to the specific heat at constant volume (Cv(lattice) ) and thermal expansion by varying x in the range 0.0 < x < 0.5. To the best of our knowledge no systematic experimental or theoretical study of thermal properties of these calcium-doped cobaltates was undertaken in the past whereas it is well established that strong electron–phonon coupling is present in these compounds and this electron–phonon interaction is one of the most relevant contributions in determining the conduction mechanism in these cobaltates. In order to do a systematic analysis of how the specific heat, thermal expansion, cohesive and thermal properties can be tuned by doping, the series of cobaltites Pr1−x Cax CoO3 (0.0 ≤ x ≤ 0.5) is studied as a function of temperature (0 ≤ T ≤ 1000 K) using RIM. The essentials of RIM formalism and the results obtained from its application are presented in subsequent sections. 2. RIM computations The potential describing the formalism of RIM is expressed as [18–24]: R (r) = − kk
i
+
ni 2
bi [ˇkk exp{(2rk − rkk )/i } + ˇk k exp{(2rk − rk k )/i }] (1)
Here, first term is attractive long-range (LR) coulomb interactions energy and the second term is overlap repulsive energy represented by the Hafemeister–Flygare-type (HF) interaction extended up to the second neighbour. Here, rkk’ represents separation between the nearest neighbours while rkk and rk’k’ appearing in the next terms are the second neighbour separation. rk (rk’ ) is the ionic radii of k (k ) ion. n (n ) is the number of nearest (next nearest neighbour) ions. In ABO3 (such as PrCoO3 ) perovskite structure, k represents cation (A, B) and k denotes the (O1 , O2 ) type of ions. bi and i are the hardness and range parameters for the ith cation–anion pair (i = 1, 2) respectively and (i kk is the Pauling coefficient [25] given by ˇi
kk
=1+
Z Z k k Nk
+
Nk
d
(r) dr
r=r0
=0
(3)
and the bulk modulus: B=
1 9Kr0
d2 (r) dr 2
(4) r=r0
where K is the crystal-structure-dependent constant and r0 is the equilibrium nearest neighbor distance. The cohesive energy for Pr1−x Cax CoO3 (0.0 ≤ x ≤ 0.5) is calculated using Eq. (1) and other thermal properties, the molecular force constant (f), Restsrahalen frequency (), and Debye temperature ( D ) are computed using the expression given in our previous papers [18–24]. The expression for calculating lattice specific heat is D
CV (lattice) = 9R
T 3 T D
ex x 4 (ex
0
− 1)2
dx
(5)
In the above equation R is the universal gas constant and D is the Debye temperature. CP is formulated by taking into account the important role of the ˛2 BT VT term, which links the measured specific heat with the measured physical properties of solids at high temperature. CP = CV + ˛2 BT VT
(6)
where T is the temperature, BT is the isothermal bulk modulus, V is the unit cell volume and ˛ is the volume thermal expansion calculated by ˛=
CV BT V
(7)
Here BT , V, CV are the isothermal bulk modulus, unit formula volume and specific heat at Constant volume respectively and is the Gruneisen parameter. The results thus obtained are presented and discussed below. 3. Results and discussion
e2 −1 Zk Zk rkk ni bi ˇkk exp{(rk + rk − rkk )/i } + 2 kk
Zk (Zk ) and Nk (Nk ) are the valence and the number of electrons in the outermost orbit of k (k ) ion respectively. The model parameters, hardness (b) and range () parameters are determined from the equilibrium condition:
(2)
We have evaluated the model parameters (b, ) from Eqs. (3) and (4) by taking the values of the input data like unit cell parameters (a, b, c) and other interionic distances directly from Refs. [3,15–17]. The values of computed model parameters (b1 , 1 ) and (b2 , 2 ) corresponding to the ionic bonds Co O and Pr/Ca O as a function of temperature are listed in Table 1. The values of b and show decreasing trend with increasing Ca doping. This decreasing trend of hardness parameter (b) with Ca doping indicates the decrease in the strength of the crystal with higher levels of doping. The unit cell parameter (a, b, c) taken from the reported data [3,15–17], are used to calculate the simple perovskie lattice cell parameter (r) in the space group Pbnm for the present compounds [15–17]. The values of ionic radii for Pr3+ , Ca2+ , Co3+ , Co4+ and O2− are taken from ref. [26] and the data on atomic compressibility are taken from ref. [27]. In the ideal cubic perovskite structure, RE3+ occupies the centre of dodecahedron of oxygen and has an ideal unity value of tolerance factor. The Co O Co angle reaches the value 180◦ which facilitates the exchange interactions and electron hopping. But due to orthorhombic deformation of the lattice, the coordination number of Pr reduces to 9, as three oxygen atoms remain essentially non-bonded due to large Pr3+ O distance. So we considered the ionic radii of A-site cations of coordination number 9 [26] in the
R. Thakur et al. / Thermochimica Acta 550 (2012) 53–58
55
Table 1 Values of critical radius, tolerance factor, model parameters of orthorhombic Pr1−x Cax CoO3 (0 ≤ x ≤ 0.5) cobaltate. Doping concentration (x)
rCR (Å)
Tolerance factor (t)
b1 × 10−19 (J)
b2 × 10−19 (J)
1 (Å)
2 (Å)
0.00 0.1 0.2 0.3 0.5
1.026 1.023 1.021 1.017 0.995
0.9377 0.9385 0.9393 0.9400 0.9416
1.47 1.45 1.44 1.43 1.36
1.81 1.80 1.79 1.77 1.66
0.198 0.195 0.193 0.192 0.190
0.349 0.348 0.344 0.343 0.342
present investigation. It is useful to recall the Goldschmidt tolerance factor which is a fundamental parameter determining the kind of structural distortion in perovskites with general formula ABO3 . √ It is defined as t (t = (rA + rO )/ 2 (rB + rO )), where rA , rB , and rO are the radii of the A-site cation, the B-site cation, and the oxygen ion, respectively. The condition that t lies between 0.77 and 1 for stable perovskite structure [28] is satisfied in these compounds (Table 1). Goldschmidt’s tolerance factor (t) has been calculated with the aid of Shannon’s [26] set of ionic radii for series of perovskites with concentration x. t gradually decreases as the cation radius decreases down the series, which also suggests a increase of global distortions. Depending on the composition of the perovskite a critical radius rcr can be calculated which describes the maximum size of the mobile ion to pass through. The critical radius can be calculated by using the following Eq.
√ ao 34 ao − 2rB + rB2 − rA2 rcr = (8) √ 2(rA − rB ) + 2ao where rA and rB are the radius of the A ion and B ion, respectively, and a0 corresponds to the pseudo cubic lattice parameter (V1/3 ). The condition that critical radius does not exceed 1.05 A˚ [29] for typical perovskite materials satisfied in our compounds. 3.1. Bulk modulus Bulk modulus of the compounds decides the equilibrium condition of the lattice, but the experimental values of bulk modulus are not available in literature for the studied compounds. So we thought of making a study of determination of bulk modulus of PrCoO3 and diluted compounds systematically on the basis of formulations of Atoms in Molecules theory (AIM) [30] which emphasize the partitioning of static thermophysical properties in condensed systems into atomic or group distributions. Hence, inverse of the bulk modulus is the simple weighted average of the atomic compressibilities [30]: =
i
fi i and
1 1 = fi B Bi
following relation for the bulk modulus of the distorted perovskite cobaltate: BT =
KS B0 m cos ω exp( JT )C
(10)
where KS is the spin order-dependent constant of proportionality, and its value is less than 1 for the ferromagnetic state and more than 1 for paramagnetic state, B0 is the bulk modulus for undistorted structure calculated on the basis of AIM theory, m is the size mismatch, c is the charge mismatch, JT is JT distortion of CoO6 octahedra and cos ω is the effect of buckling of the Co O Co angle. The values of bulk modulus of the distorted structure are presented in Table 2 as B which takes effects of all the distortions. 3.2. Cohesive properties Using values of bulk modulus, we have computed the cohesive energy () for Pr1−x Cax CoO3 at composition 0.0 ≤ x ≤ 0.5 using Eq. (1) and listed them in Table 2. It is seen from Table 2 that the magnitude of the cohesive energy increases systematically from () = −153.5 eV for x = 0.0 (i.e. PrCoO3 ) to −160.9 eV for x = 0.5. The cohesive energy is, expected to follow the same trend of variation as that of the bulk modulus, and is related to the overall atomic binding properties of a material. In conformity with this, we notice from Table 2 that our results on the cohesive energy () have followed more or less similar trend of variation as that exhibited by the bulk modulus, which represents the resistance to volume change and is related to the overall atomic binding properties of a material. Nevertheless, they are stable compounds as the cohesive energy is negative within the studied range of composition (x). Further, to test the validity of our model, we calculated the cohesive energy of these compounds using the generalized Kapustinskii Equation [31] which uses the ionic strength of the crystal. 1 nk zk2 2 T
I=−
(11)
k
(9)
i
where fi = Vi /V where fi is the fractional volume occupancy due to quantum subsystem i in a unit formula, B is the bulk modulus of the compound and k is its compressibility. We have computed the bulk modulus using Eq. (9) on the basis of AIM theory and in this analysis the octahedral are considered to be undistorted and perfect. When we consider the Ca doping at the A-site in PrCoO3 compound then the local compressibility of impurity with respect to the host ions are governed by the size difference and tuned by the formal charge mismatch between host and guest cations of A-site. We considered the effect of charge, size mismatch along with the octahedral distortions due to Jahn–Teller effects on the bulk modulus of the compounds. These factors will determine the change in the unit cell volume which in turn will change the global bulk modulus of the compound. The formal expression for calculating the cation size and charge mismatch of the distorted perovskite cobaltate is given in our earlier paper [21]. It is now appropriate to propose the
where T is the number of type of ions in the formula unit each of number nk and charge zk . In our calculation the value of ionic strength for PrCoO3 is found to be 16 and this value decreases slightly with Ca doping. According to generalized Kapustinskii equation [31] the lattice energies of crystal with multipole ions is given as: U/kJ mol−1 = −
1213.9 r
1−
r
nk zk 2
(12)
where r = weighted mean cation–anion radius sum (using Goldschmidt radii) and is taken as the average of our model parameters 1 and 2 . Estimated values of lattice energy by Kapustinskii equation for pure and calcium doped PrCoO3 are reported in Table 2 and seems to be in very close agreement with our calculated values which confirms the validity of RIM. The experimental values of the cohesive energy for Pr1−x Cax CoO3 (0.0 ≤ x ≤ 0.5) compounds are not available, but our calculated values are comparable to the reported value of Madelung energy of the valence skip compound BaBiO3 , which is −165 eV [32].
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R. Thakur et al. / Thermochimica Acta 550 (2012) 53–58
Table 2 Values of bulk modulus, cohesive and thermal properties of orthorhombic Pr1−x Cax CoO3 (0.0 ≤ x ≤ 0.5) cobaltate. Doping concentration (x)
B (Gpa)
(eV) (RIM)
(eV) (Kapustinskii equation)
f (N/m)
(THz )
D (K)
0.00 0.1 0.2 0.3 0.5
224.8 230.1 235.3 238.0 248.9
−153.5 −154.2 −155.6 −157.0 −160.9
−153.5 −153.5 −154.8 −155.9 −157.8
43.14 44.11 45.08 45.62 46.95
10.5 10.7 10.9 11.3 12.1
504.43 512.56 527.83 542.18 579.44 630a
Ref. [16].
160
3.4. Specific heat and thermal expansion The specific heat in the normal state of the material is usually approximated by the contributions of the lattice and electronic specific heat. In our model, we have considered only the lattice contributions. We have computed the variation of the lattice contributions to the specific heat of Pr1−x Cax CoO3 (x = 0, 0.1, 0.2, 0.3, and 0.5) compounds in the wide temperature range 1 K ≤ T ≤ 1000 K. The temperature variations of specific heat of these compounds for various compositions are depicted in the main panel of Figs. 1–5. At low temperatures (T < D /50), a meticulous attention was given to the investigation of the assumption that these cobaltate perovskites are Debye-like solids and specific heat obeys T3 law. At intermediate temperature, the temperature dependence of specific heat is governed by the vibrations of atoms. Now, the systematic investigation of the specific heat at 1 K ≤ T ≤ 1000 K of Pr1−x Cax CoO3 (x = 0, 0.1, 0.2, 0.3, and 0.5) are in satisfactorily agreement with the measured data of Tsubouchi et al. [12]. The specific heats in these materials varies almost linearly between 0 and 200 K and later on vary along curvature from 200 to 550 K and finally become almost constant beyond 550 K. The remarkable deviations appearing at temperatures (10–300 K) are due to the anharmonic effects, whose contributions are very well demonstrated by Jindal et al. [33] for other materials. However, at higher temperatures, the anharmonic effects on specific heat is suppressed and it is very close to the
Exp.
PrCoO3
Cal. 120
20
80
40
10
0
0
0
200
0
200
400
400 600 T (K)
T (K)
600
800
800
1000
1000
Fig. 1. Computed specific heat and thermal expansion (inset) of perovskite PrCoO3 as a function of temperature. Present results are shown by line with solid circle (−•−) and experimental values [12] are shown by line with open circle (− −).
Dulong-Petit limit CV = 15 NkB ∼ 125 J K−1 mol−1 in ABO3 perovskites, which is common to all solids at high temperature. It is noticed from Fig. 5 that the computed specific heat curve for Pr0.5 Ca0.5 CoO3 is in good agreement with the experimental data [12] and the trends of variations exhibited by the experimental and theoretical results are almost similar except that an anomaly seen to tail shown by measured data at T = 90 K. This observed tail owes its origin to the occurrence of the simultaneous metal–insulator and paramagnetic–paramagnetic spin-state transition of the firstorder type at 90 K, indicating a cooperative nature. However, this peak could not be reproduced from our model potential calculation due to the exclusion of the magnetic contribution. An anomaly at x = 0.5 around TC (90 K) is observed in experimental curve as shown in the Fig. 5, but on the other hand at x = 0.2 and 0.3 no clear anomaly 160
Exp.
Pr 0.9 Ca 0.1 CoO 3
Cal.
Specific heat (J/mole K)
We have also computed the molecular force constant (f), Reststrahlen frequency (), and Debye temperature ( D ) for pure Ca doped PrCoO3 compounds, and results are reported in Table 2. In the Debye approach we consider the frequency of vibration of positive ion lattice with respect to negative ion lattice obtained from this model is reported here as the Reststrahlen frequency () for Pr1−x Cax CoO3 (x = 0.0–0.5). It can be seen from Table 2 that the values of Restrahalen frequency are increasing with Ca concentration (x), which shows that the bond is becoming stronger with Ca doping. This feature is similar to the increasing trends exhibited by the stability (−) of the present materials. Since the Restrahalen frequency is directly proportional to the molecular force constant (f) therefore both of them vary with the temperature accordingly for these compounds. The values of Debye temperatures ( D ) for Pr1−x Cax CoO3 (0.0 ≤ x ≤ 0.5) have been calculated using these restrahalen frequencies and listed in Table 2. It is noticed that D for PrCoO3 increases with the doping of Ca concentration (x), indicating the higher phonon frequencies of the given system. Present investigation reemphasise that Debye temperature is dependent on the A-site cation radius and it increases with increasing calcium doping. The present results could not be compared due to lack of experimental data on them. At present our results are only of academic interest but will certainly work as a guide to the experimental workers in future.
Specific heat (J/mole K)
3.3. Thermal properties
(10-6 K-1)
a
120
80
40
0
0
200
400
600
800
1000
T (K) Fig. 2. Computed specific heat and thermal expansion (inset) of perovskite Pr0.9 Ca0.1 CoO3 as a function of temperature. Present results are shown by line with solid circle (−•−) and experimental values [12] are shown by line with open circle (− −).
R. Thakur et al. / Thermochimica Acta 550 (2012) 53–58
Specific heat (J/mole K)
160
Exp.
Pr 0.8 Ca
Cal.
0.2
CoO 3
120
80
0
0
200
400
600
800
1000
T (K) Fig. 3. Computed specific heat and thermal expansion (inset) of perovskite Pr0.8 Ca0.2 CoO3 as a function of temperature. Present results are shown by line with solid circle (−•−) and experimental values [12] are shown by line with open circle (− −).
160
Exp.
Pr
Specific heat (J/mole K)
Cal.
0.7
Ca
0.3
CoO 3
(13)
where the first term is the lattice contribution which has been calculated by us using Eq. (7), the second term is the magnetic contribution and the third term is the ˛I–M refers to the anomalous expansion associated with the insulator–metal (I–M) transition. An inspection of Figs. 1–5 (inset) shows that the thermal expansion coefficient ˛ increases with T3 at low temperatures and gradually approaches a linear increase at high temperatures, and then the increasing trend becomes gentler. It is shown that the thermal expansion coefficient ˛ converges to a constant value at high temperatures. The lattice thermal expansion decreases, respectively, as the calcium concentration increases from 0 to 50%. We could not compare our results on thermal expansion, due to non-availability of the experimental data.
120
4. Conclusions 80
40
0
0
200
400
600
800
1000
T (K) Fig. 4. Computed specific heat and thermal expansion (inset) of perovskite Pr0.7 Ca0.3 CoO3 as a function of temperature. Present results are shown by line with solid circle (−•−) and experimental values [12] are shown by line with open circle (− −).
are found (Figs. 3 and 4) because of magnetically inhomogeneous behavior. The change of Co O distance by the substitution of Ca at Pr site increases Debye temperature ( D ) and hence there is a consistent decrease in the specific heat. Hence, the concentration (x) dependence of Debye temperature in Pr1−x Cax CoO3 (0.0 ≤ x ≤ 0.5) suggests that the increased Ca doping drives the
Exp.
160
Specific heat (J/mole K)
system effectively close to the strong electron–phonon coupling regime. We have computed the variation of the lattice contribution to the thermal expansion for temperature range (1 K ≤ T ≤ 1000 K) for Pr1−x Cax CoO3 (0.0 ≤ x ≤ 0.5). The values of the thermal expansion at different temperatures are displayed in the inset of Figs. 1–5. The total contribution to thermal expansion is ˛(T ) = ˛latt + ˛mag + ˛I-M
40
57
Pr 0.5Ca
Cal.
0.5
CoO 3
120 80 40 0
0
200
400
600
800
1000
T (K) Fig. 5. Computed specific heat and thermal expansion (inset) of perovskite Pr0.5 Ca0.5 CoO3 as a function of temperature. Present results are shown by line with solid circle (−•−) and experimental values [12] are shown by line with open circle (− −).
On the basis of overall discussion, it may be concluded that the present RIM is successful and represents a proper theoretical approach to provide the understanding of the thermal properties of Ca doped PrCoO3 cobaltate family of perovskites. Present results on specific heat could be further improved by incorporating the effects of the interactions due to van der Walls, magnetic and anharmonic contributions. Our results are probably the first reports of thermal expansion at these temperatures and compositions. Thus a satisfactory prediction of the Pr1−x Cax CoO3 (0.0 ≤ x ≤ 0.5) attained by us is remarkable in view of the inherent simplicity of RIM and its less parametric nature. Acknowledgements Financial support by Department of Science and Technology (DST) and University Grant Commission (UGC) New Delhi are gratefully acknowledged. References [1] G. Briceno, H.Y. Chang, X.D. Sun, P.G. Schultz, X.D. Xiang, A class of cobalt oxides magnetoresistance materials discovered with combinational synthesis, Science 270 (1995) 273. [2] A.V. Samoilov, G. Beach, C.C. Fu, N.C. Yeh, R.P. Vasquez, Magnetic percolation and giant spontaneous Hall effect in La1−x Cax CoO3 (0.2 ≤ x ≤ 0.5), Phys. Rev. B57 (1998) R14032. [3] S. Tsubouchi, T. Kyomen, M. Itoh, P. Ganguly, M. Oguni, Y. Shimojo, Y. Morii, Y. Yoshi, Simultaneous metal–insulator and spin state transitions in Pr0.5 Ca0. 5CoO3 , Phys. Rev. B66 (2002) 052418. [4] M.J.R. Hoch, P.L. Kuhns, W.G. Moulton, A.P. Reyes, J. Lu, J. Wu, C. Leighton, Evolution of the ferromagnetic and nonferromagnetic phases with temperature in phase-separated La1−x Srx CoO3 by high-field 139 La NMR, Phys. Rev. B70 (2004) 174443. [5] Y. Kobayashi, T. Nakajima, K. Asai, Magnetic and transport properties in La1−x Prx CoO3 single crystals, J. Magn. Magn. Mater. 272 (2004) 83. [6] J.Q. Yan, J.S. Zhou, J.B. Goodenough, Bond-length fluctuations and the spin-state transition in LCoO3 (L = LaPr, and Nd), Phys. Rev. B69 (2004) 134409. [7] H.W. Brinks, H. Fjellvag, A. Kjekshus, B.C. Hauback, Structure magnetism of Pr1−x Srx CoO3−␦ , J. Solid State Chem. 147 (1999) 464. [8] S.K. Pandey, A. Kumar, S.M. Choudhari, A.V. Pimpale, Electronic states of PrCoO3 : x-ray photoemission spectroscopy and LDA + U density of states studies, J. Phys.: Condens. Matter 18 (2006) 1313–1323. [9] K. Yoshii, A. Nakamur, Magnetic properties of Pr1-xSrxCoO3 , Physica B 281 (2000) 514–515. [10] I.G. Deac, A. Vladescu, I. Balasz, A. Tunyagi, R. Tetean, Electrical and magnetic behavior of transition metal oxides Ln0.7 A0.3 TMO3 Ln = La, Pr, A = Ca, Sr and TM = Mn, Co, Int. J. Mod. Phys. B 24 (2010) 762.
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