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Specific heat studies on Ru substituted FeSi Kondo Insulator
Solid State Communications 146 (2008) 391–394
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Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Specific heat studies on Ru substituted FeSi Kondo Insulator Awadhesh Mani ∗ , J. Janaki, Soubhadra Sen, A. Bharathi, Y. Hariharan Materials Science Division, Indira Gandhi Centre for Atomic Research, Kalpakkam - 603 102, Tamil Nadu, India
article
info
Article history: Received 5 January 2008 Received in revised form 4 March 2008 Accepted 25 March 2008 by E.V. Sampathkumaran Available online 29 March 2008 PACS: 71.28.+d 71.23.An 72.15.Qm
a b s t r a c t The temperature dependent specific heat studies of a Ru substituted Fe1−x Rux Si Kondo Insulating system has been carried out in the 77 to 300 K range. The specific heat has been analyzed based on contributions from an electronic part and a lattice part. For the electronic part a Gaussian density of states model, which incorporates the effect of correlation, band structure and disorder in its defining parameters is used. The lattice part of specific heat is described in the Debye model. Parameters that define the electronic density of states and the Debye temperature are extracted from fits of the temperature dependent specific heat data and are compared with earlier resistivity measurements that employed a similar analysis. © 2008 Elsevier Ltd. All rights reserved.
Keywords: A. FeSi D. Heat capacity D. Kondo insulator
1. Introduction Binary compound FeSi crystallizing in a B20 type structure is a narrow gap semiconductor, which belongs to a class of strongly correlated system termed as Kondo Insulator (KI) on account of its unusual physical properties [1–6]. In the band semiconductors or band insulators an energy gap emerges in the density of states (DOS) spectrum due to the usual filling of bands complying with the one electron band picture, where electron correlations has little role to play. On the other hand, in KI systems the energy gap arises due to Kondo interaction where electron correlations modify the Bloch states resulting in a quasi-particle DOS spectrum. The net effects of the correlations are to strongly renormalize the DOS, which causes a reduction in the energy gap and a narrowing of the band width [7]. Therefore, in general, the energy gap and the band width of correlated insulators are smaller than that of band semiconductors/insulators [1]. The evidence for the presence of a strong electron correlation in the FeSi system, which distinguish it from the band semiconductors, has been indicated by several experiments such as optical conductivity [8], photoemission [9] and tunneling spectroscopy [10]. For instance, optical studies found the disappearance of an optical gap at a temperature about ∼3.5 times smaller than the actual gap
∗ Corresponding author. Tel.: +91 44 27480081; fax: +91 44 27480081. E-mail address: [email protected] (A. Mani). 0038-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2008.03.031
value, which is not expected based on simple thermal activation applicable to a band semiconductor, but it is indicative of collective behavior of a correlated system [8]. Photoemission and tunneling spectroscopy had revealed an unusual temperature variation of the gap and sharp features of DOS near the valence-band edge with a narrow band of width ∼25 meV. A bandwidth of such a small magnitude cannot be accounted for by the band structure calculations. However, narrow band width can be realized in the presence of strong correlation. The transport properties of FeSi also exhibit remarkable temperature dependence. For example, the electrical conductivity, σ(T ), shows metallic [4,5] conduction above room temperature, while an activated behavior akin to a narrow gap semiconductor in about 100–200 K range [10–12]. In the low temperature regime of 5–40 K, the transport is governed by a variable range hopping (VRH) mechanism indicating the presence of localized states inside the gap [10–12]. In order to account for these features, several models such as interplay of electron correlation and hybridization within the two-band Hubbard model [7], the periodic Anderson model [13] and a narrow band - small energy gap type density of states model [3–6] have been used. While the high temperature (T > 100 K) behavior of FeSi could be understood fairly well within the purview of these models, the low temperature behavior (T < 40 K) which is influenced by the disorder could not be rationalized. In an attempt to understand the evolution of the ground state and Kondo insulating (KI) gap in FeSi, we had carried out series of investigations encompassing the isoelectronic substitutions of Ge at a Si site in FeSi1−x Gex and Ru at a Fe site in Fe1−x Rux Si systems
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followed by high-pressure studies on pristine and substituted samples via transport measurements [14–17]. The validity of the aforementioned existing models had also been examined in our studies. The high pressure-low temperature electrical conductivity studies on FeSi1−x Gex [14] had brought about a complicated pressure and composition dependence of the KI gap, which could not be explained solely within the framework of the one-electron band picture or a hybridization gap model. For example, expansion of the lattice by Ge-substitution resulted in decreasing gap, consistent with the band structure prediction and a hybridization gap model. However, compression of the lattice under high pressure in FeSi was also found to decrease the gap, contrary to what was expected from a band structure or hybridization gap model. Further, a non-monotonic variation of gap was seen for a 5% Ge substituted sample, while an increase in the gap was observed in a 20% Ge substituted sample with increasing pressure. In the low temperature regime (5–40 K), a pressure induced insulator to metal transition (MIT) arising due to delocalization of the localized gap states was observed in these systems. Such a complex pressure dependent behavior could not be understood without invoking effects of disorder. Our Fe1−x Rux Si system had provided an ideal avenue for testing a correlation cum hybridization model [15]. The substitution of Ru at the Fe site affects the correlation and hybridization simultaneously by reducing the on-site Coulomb repulsion and by expanding the lattice respectively. Another interesting feature of the Fe1−x Rux Si system was that its end compounds, namely, FeSi and RuSi belong to two different classes of materials—FeSi is a KI with a modest value of gap ∼700 K, while RuSi is a band semiconductor with a larger gap of ∼3600 K [18]. Therefore, under Ru substitution a transformation from KI to band semiconductor was expected, which indeed could be observed near x ∼ 0.06. This transition could be realized based on the observed non-monotonic evolution of energy gap (also an anomaly in other parameters see Ref. [15]) as a function of x with a minimum at x ∼ 0.06. Within the KI regime (x ≤ 0.06), the decrease of the gap with increasing x could be attributed to the combined effect of correlation and hybridization. On the other hand, in the semiconductor regime (x > 0.06), an increase in gap with x complies with the predictions of band structure calculations [15]. It was brought about from the above studies that the physical properties of the FeSi system were primarily dictated by the combined effects of band structure, electron correlation and disorder. Based on these studies, a new density of states model, namely, the Gaussian density of states (GDOS) model [15,16] which incorporates these effects in its defining parameters, had been proposed and used for a quantitative understanding of the electrical conductivity behavior of this system. This model had consistently explained the transport behavior of FeSi system as a function of temperature, pressure and composition. The GDOS model is an improvisation over the narrow band–small energy gap DOS model of Jaccarino et al. [3– 6]. In the later model, a DOS consisting of two rectangular bands with extremely narrow width separated by a small energy gap was proposed. This feature of DOS was believed to arise due to the renormalization of the corresponding non-interacting bands in the presence of correlation. Incorporation of the disorder effects in this model had resulted in our GDOS model. Further details of the GDOS model are presented in the analysis section of this paper. In this paper, we report the low temperature specific heat studies on Fe1−x Rux Si with x = 0.0 to 0.2. The motivation for carrying out the present study is based on the following reasons. As mentioned in the previous paragraph, Fe1−x Rux Si exhibit transformation from a KI to band semiconductor around x ∼ 0.06. Therefore, it is of interest to investigate whether similar behavior can be observed via specific heat studies in this system. In addition, it is also aimed to extend the GDOS model for the electronic part of the specific heat for a quantitative analysis of specific heat data and find its applicability in understanding the thermal properties of the FeSi system.
2. Experimental details Fe1−x Rux Si (0 ≤ x ≤ 0.3) and CoSi samples were prepared by arc melting and were characterized by X-ray diffraction (XRD) followed by the Rietveld refinements to obtain the lattice constant and the internal atomic position parameters for the elemental constituents of the samples [15,17]. Among these, samples with x = 0.0, 0.03, 0.06 and 0.2 have been used for specific heat studies. Selection of these compositions is based on the fact that samples below x = 0.06 are in the KI regime, and above are in the band semiconductor regime, while x = 0.06 is at the verge of transformation [15]. The specific heat measurements in the 77–300 K temperature range were carried out in a home made cryostat, using a quasi-adiabatic heat pulse calorimeter. A small amount of sample (∼100–200 mg) is mounted on a sample holder assembly comprising of thin sapphire disc covered with Ni–Cr thin film as a heating element and a platinum resistance thermometer for temperature measurements. A heat pulse for a short known duration is applied to the sample assembly at a fixed temperature and the rise in temperature is deduced from analysis of the time dependent temperature drift curve [19]. The heat capacity of sample holder assembly is measured before mounting the sample and that is subtracted from the total heat capacity in order to extract the specific heat of the sample under study. 3. Analysis of specific heat data In general the specific heat of a system is given by [20] C = Cel + Cph + Cmag + Csch
where Cel is the electronic part, Cph the lattice part, Cmag the magnetic part while Csch the Schottky part of specific heat that arises from the two level systems. In a previous analysis of specific heat in FeSi, the Cph was substituted by CCoSi , where CCoSi is the measured specific heat in the non-magnetic isostructural compound CoSi [3,4,21]. The excess specific heat thus obtained by subtracting CCoSi was thought to be Cmag + Cel while Csch was taken to be absent. Further Cel was either taken to be of the form γ T [3] or included in term Cmag [21]. In the study carried out by Mandrus et al. [4], Cph is taken to be CCoSi and for the electronic part they used narrow band–small energy gap type density of state function. In the present study, we use the general form of the Debye model to describe the lattice contribution and for the electronic part we employed a Gaussian density of states (GDOS) previously used to fit our resistivity [15,16]. A schematic diagram of the GDOS proposed for electronic structure of the FeSi system [15,16] is reproduced in Fig. 1. In this model, the sharp features of the DOS arise due to the electron correlation, while the effect of the band structure is captured by Eg , the energy separation between valence band edge (VBE) and conduction band edge (CBE). The effect of correlation/disorder appears in the form of the variation in width W of both bands having Gaussian distribution. Disorder shifts the position of the mobility edge (ME) residing above and below the Fermi energy EF by introducing localized states inside the gap. The ME separates localized states from the extended states (see Fig. 1). In this model, the temperature dependent specific heat C of the FeSi system is obtained as follows: ∂U + Cph (1) C=
∂T
ele
where first term on right hand side takes into account the electronic specific heat and the second term represents the lattice specific heat. The internal electronic energy U is given as: Z ∞ U= ED(E)f (E)dE. (2) Eµ
A. Mani et al. / Solid State Communications 146 (2008) 391–394
Fig. 2. Variation of specific heat of FeSi and CoSi as a function of temperature. Inset shows variation of the anomalous electronic contribution to specific heat of FeSi as described in the text along with fit to GDOS model.
Fig. 1. A schematic representation of GDOS model.
Here D(E) is a Gaussian DOS which is expressed as " 2 ! Ng E − Eg /2 D(E) = √ exp −2 W 2π(W /2) 2 !# E + Eg /2 . + exp −2 W
393
(3)
For the lattice specific heat Cph , we use following general Debye expression which is valid at all temperatures T ; Cph = 9Nk(T /θ)3
Z θ/T 0
x4 ex
( − 1)2 ex
dx.
(4)
Here Eµ is the energy separation between EF and ME, f (E) is the Fermi distribution function, N is the density of unit cells containing a four FeSi formula unit, g is the total number of states per unit cell, k is Boltzmann’s constant, θ is the Debye temperature and x = hν/kT [h—Planck’s constant, ν—phonon frequency]. Experimental data has been analyzed using the above expressions and Eg , W , Eµ and θ are used as the fit parameters. It should be noted that, as such, there is no sharply defined energy gap in the GDOS model. An effective energy gap, ∆eff , has been deduced by computing the number density of carriers n at several temperatures above 100 K using the parameters Eg , Eµ and W extracted from the fits to the specific heat data using the following formula Z ∞ n= D(E)f (E)dE. (5)
Fig. 3. Variation of specific heat as a function of temperature along with fits for (a) FeSi, (b) Fe0.97 Ru0.03 Si, (c) Fe0.94 Ru0.06 Si and (d) Fe0.8 Ru0.2 Si.
Eµ
The set of n(T ) thus obtained is then fitted to a thermal activated formula of type n ∝ exp(−∆eff /2kB T ) to extract ∆eff . It should be mentioned that experimentally we measure specific heat at constant pressure, CP (T ), however Eq. (1) is defined for specific heat at constant volume, CV (T ). Since the difference between values of CP (T ) and CV (T ) is very small for solids [19], therefore we use CP (T ) data as such for analysis. 4. Results & discussion The variation of CP (T ) of pristine FeSi and CoSi in the temperature range of 77 to 300 K is shown in Fig. 2. CoSi is an iso-structural diamagnetic counterpart of FeSi, therefore the lattice specific heat of the former is expected to be similar to that of the latter as suggested by Jaccarino et al. [3]. Adapting the procedure described in Ref. [3], we extract the anomalous magnetic
contribution of the specific heat of FeSi by taking the difference [CP (FeSi) − γFeSi T ] − [CP (CoSi) − γCoSi T ] with the Sommerfeld −1 coefficients γFeSi = 1.5 × 10−4 Cal mol K−2 and γCoSi = −1 −4 −2 2.4 × 10 Cal mol K for FeSi and CoSi respectively. The resulting excess specific heat δCexcess of FeSi is shown in the inset of Fig. 2. We fit this data using the GDOS model excluding the phonon part of Eq. (1). The fit is shown in the inset of Fig. 2. The parameters Eg , W and Eµ deduced from this fit are used as initial guess parameters for fitting the experimental CP (T ) data of Fe1−x Rux Si using Eq. (1) by taking into account the total specific heat comprising of both electronic and lattice contributions. The results of these analyses are shown in Fig. 3(a)–(d) for x = 0.0, 0.03, 0.06 and 0.2 respectively. The χ2 of the fits are found to be in the range of the order of 10−3 –10−4 . This indicates that data fits very well to our model. Parameters Eg , W , Eµ and θ have been extracted from these fits for all the samples. For pristine FeSi, the
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values of these parameters are found to be Eg = 828 K, W = 500 K, Eµ = 516 K and θ = 605 K. The value of θ found from our analysis is in very good agreement with the reported value of 600 K [22]. Using GDOS fit parameters, the carrier concentration n at various temperatures T ≥ 100 K have been calculated to deduce the effective energy gap ∆eff . For pristine FeSi, ∆eff is found to be ∼630 K, which is larger than the value of ∼560 K obtained from σ(T ) [15]. This discrepancy is understandable because in transport, electrons travel along the path of least resistance which gives rise to a smaller effective gap. On the other hand, a thermodynamic measurement averages over the entire sample leading to larger effective gap [22]. The value of carrier concentration n300 K ∼ 1.26 × 1028 m−3 deduced for FeSi is in good agreement with the value of 2 × 1028 m−3 obtained from the Hall measurements [5]. The variation of Eg , W , Eµ , θ and ∆eff are shown in Fig. 4(a)–(e) respectively as a function of Ru concentration x. The parameter Eg exhibits a monotonic increase with increasing x. This is consistent with band structure calculations which predict an increase in band gap under Ru substitution in FeSi [15]. W shows a marginal increase up to x ≤ 0.06, while a drastic increase for x = 0.2. This sudden increase in broadening of bandwidth W should indicate reduction in the correlation effect. Parameters Eµ and ∆eff show non-monotonic variation i.e., a decrease up to x = 0.06 followed by an increase. The variation of Eµ is consistent with similar trend seen in the hopping parameter in σ(T ) studies [15], which indicates delocalization/localization of the gap states. It is known that RuSi is a band semiconductor which has a larger energy gap than that of FeSi [18]. Therefore, if the nature of FeSi and RuSi had been similar, then under Ru substitution ∆eff should have shown a monotonic increase with increasing x. The fact that ∆eff exhibits a non-monotonic variation (see Fig. 4(e)), must indicate a change in nature of the samples beyond a Ru concentration of x ≥ 0.06. The observed monotonic decrease in θ complies with the fact that lattice parameter increases with Ru substitution [15] which leads to phonon softening with increasing x. It is worth noting that the parameters Eµ and ∆eff deduced from σ(T ) analyses exhibit similar variation with x [15]. Therefore, intuitively, the present result should be consistent with the inference drawn from our previous σ(T ) studies. 5. Summary and conclusions The specific heat of Fe1−x Rux Si with x = 0.0–0.2 have been measured using quasi adiabatic heat pulse calorimetry in the 77–300 K temperature range. The GDOS model in conjunction with the general form of the Debye expression have been used to analyze the experimental CP (T ) data. The parameters extracted from these analyses clearly indicate the change in nature of the sample beyond x > 0.06. It should be mentioned that band structure calculations predict a monotonic increase of gap with increasing Ru concentration in Fe1−x Rux Si [15]. In contrast, ∆eff is seen to decrease in the regime of x ≤ 0.06 implying that the one electron band picture is not applicable in this regime. However, the observed increase in ∆eff beyond x > 0.06 complies with the prediction of the band structure calculations. The sudden increase of band width W for x > 0.06 is an indication of the diminishing of the correlation effect. Therefore, the non-monotonic variation of ∆eff with x and the sudden increase of band width W observed beyond x ∼ 0.06 give an indirect indication of a transformation from a KI to a band semiconductor. However, a more direct method to verify this transition would be photoemission or tunneling spectroscopy in which an actual feature of DOS and broadening of band may be seen in the semiconducting regime in contrast to the KI regime. It should be mentioned that in our previous studies [15] such anomalies were found in all experimentally deduced parameters such as energy gap, hopping parameter, low
Fig. 4. Variation of fit parameters (a) Eg , (b) W, (c) Eµ , (d) θ, (e) ∆eff as a function of Ru concentration, x, in Fe1−x Rux Si. Solid lines connecting the data points are guide to the eye.
temperature conductivity, lattice parameters etc. near x ∼ 0.06, the detailed analysis of which had indicated a transition from the KI to the Band insulator regime. Therefore, the present result turns out to be complementary to our previous study. References [1] G. Aeppli, Z. Fisk, Comments Condens. Matter Phys. 16 (1992) 155. [2] Z. Fisk, J.L. Sarrao, S.L. Cooper, P. Nyhus, G.S. Boebinger, A.S. Passner, P.C. Canfiled, Physica B 223–224 (1996) 409. [3] V. Jaccarino, G.K. Wertheim, J.H. Wernick, L.R. Walker, S. Arajs, Phys. Rev. 160 (1967) 476. [4] D. Mandrus, J.L. Sarrao, A. Migliori, J.D. Thompson, Z. Fisk, Phys. Rev. B 51 (1995) 4763. [5] B.C. Sales, E.C. Jones, B.C. Chakoumakos, J.A. Fernandez-Baca, H.E. Harmon, J.W. Sharp, E.H. Volckmann, Phys. Rev. B 50 (1994) 8207. [6] M.B. Hunt, M.A. Chernikov, E. Felder, H.R. Ott, Z. Fisk, P.C. Canfield, Phys. Rev. B 50 (1995) 14933. [7] C. Fu, S. Doniach, Phys. Rev. B 51 (1995) 17439. [8] Z. Schlesinger, Z. Fisk, H.T. Zhang, M.B. Maplem, J.D. Di-Tusa, G. Aeppli, Phys. Rev. Lett. 71 (1993) 1748. [9] C.-H. Park, Z.-X. Shen, A.G. Loeser, D.S. Dessau, D.G. Mandrus, A. Migliori, J. Sarrao, Z. Fisk, Phys. Rev. B 52 (1995) R16981. [10] M. Fath, J. Aarts, A.A. Minovsky, G.J. Nieuwenhuys, J.A. Mydosh, Phys. Rev. B 58 (1998) 15483. [11] A. Bharathi, Awadhesh Mani, G.V. Narasimha Rao, C.S. Sundar, Y. Hariharan, Physica B 240 (1997) 1. [12] S. Takagi, H. Yasouka, S. Ogawa, J.H. Wernick, J. Phys. Soc. Japan 50 (1981) 2539. [13] M.J. Rozenberg, G. Kotliar, H. Kajueter, Phys. Rev. B 54 (1996) 8452. [14] Awadhesh Mani, A. Bharathi, Y. Hariharan, Phys. Rev. B 63 (2001) 115103. [15] Awadhesh Mani, A. Bharathi, S. Mathi Jaya, G.L.N. Reddy, C.S. Sundar, Y. Hariharan, Phys. Rev. B 65 (2002) 245206. [16] Awadhesh Mani, Solid State Commun. 132 (2004) 551. [17] Awadhesh Mani, Ph.D. Thesis: Isoelectronic substitution and high pressure studies on FeSi Kondo insulator, submitted to the University of Madras, India, 2003. [18] B. Buschinger, W. Guth, M. Weiden, C. Geibel, F. Steglich, V. Vescoli, L. Degeorgi, C. Wassilew-Reul, J. Alloys Compounds 262–263 (1997) 238. [19] K.V. Devadhasan, et al., Solid State Commun. 92 (1994) 715. [20] E.S.R. Gopal, Specific Heats at Low Temperatures, Plenum Press, New York, 1966. [21] Y. Takahashi, T. Kanomata, R. Note, T. Nakagawa, J. Phys. Soc. Japan 69 (2000) 4018. [22] S. Yeo, S. Nakatsuji, A.D. Bianchi, P. Schlottmann, Z. Fisk, L. Balicas, P.A. Stampe, R.J. Kennedy, Phys. Rev. Lett. 91 (2003) 046401.
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Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Specific heat studies on Ru substituted FeSi Kondo Insulator Awadhesh Mani ∗ , J. Janaki, Soubhadra Sen, A. Bharathi, Y. Hariharan Materials Science Division, Indira Gandhi Centre for Atomic Research, Kalpakkam - 603 102, Tamil Nadu, India
article
info
Article history: Received 5 January 2008 Received in revised form 4 March 2008 Accepted 25 March 2008 by E.V. Sampathkumaran Available online 29 March 2008 PACS: 71.28.+d 71.23.An 72.15.Qm
a b s t r a c t The temperature dependent specific heat studies of a Ru substituted Fe1−x Rux Si Kondo Insulating system has been carried out in the 77 to 300 K range. The specific heat has been analyzed based on contributions from an electronic part and a lattice part. For the electronic part a Gaussian density of states model, which incorporates the effect of correlation, band structure and disorder in its defining parameters is used. The lattice part of specific heat is described in the Debye model. Parameters that define the electronic density of states and the Debye temperature are extracted from fits of the temperature dependent specific heat data and are compared with earlier resistivity measurements that employed a similar analysis. © 2008 Elsevier Ltd. All rights reserved.
Keywords: A. FeSi D. Heat capacity D. Kondo insulator
1. Introduction Binary compound FeSi crystallizing in a B20 type structure is a narrow gap semiconductor, which belongs to a class of strongly correlated system termed as Kondo Insulator (KI) on account of its unusual physical properties [1–6]. In the band semiconductors or band insulators an energy gap emerges in the density of states (DOS) spectrum due to the usual filling of bands complying with the one electron band picture, where electron correlations has little role to play. On the other hand, in KI systems the energy gap arises due to Kondo interaction where electron correlations modify the Bloch states resulting in a quasi-particle DOS spectrum. The net effects of the correlations are to strongly renormalize the DOS, which causes a reduction in the energy gap and a narrowing of the band width [7]. Therefore, in general, the energy gap and the band width of correlated insulators are smaller than that of band semiconductors/insulators [1]. The evidence for the presence of a strong electron correlation in the FeSi system, which distinguish it from the band semiconductors, has been indicated by several experiments such as optical conductivity [8], photoemission [9] and tunneling spectroscopy [10]. For instance, optical studies found the disappearance of an optical gap at a temperature about ∼3.5 times smaller than the actual gap
∗ Corresponding author. Tel.: +91 44 27480081; fax: +91 44 27480081. E-mail address: [email protected] (A. Mani). 0038-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssc.2008.03.031
value, which is not expected based on simple thermal activation applicable to a band semiconductor, but it is indicative of collective behavior of a correlated system [8]. Photoemission and tunneling spectroscopy had revealed an unusual temperature variation of the gap and sharp features of DOS near the valence-band edge with a narrow band of width ∼25 meV. A bandwidth of such a small magnitude cannot be accounted for by the band structure calculations. However, narrow band width can be realized in the presence of strong correlation. The transport properties of FeSi also exhibit remarkable temperature dependence. For example, the electrical conductivity, σ(T ), shows metallic [4,5] conduction above room temperature, while an activated behavior akin to a narrow gap semiconductor in about 100–200 K range [10–12]. In the low temperature regime of 5–40 K, the transport is governed by a variable range hopping (VRH) mechanism indicating the presence of localized states inside the gap [10–12]. In order to account for these features, several models such as interplay of electron correlation and hybridization within the two-band Hubbard model [7], the periodic Anderson model [13] and a narrow band - small energy gap type density of states model [3–6] have been used. While the high temperature (T > 100 K) behavior of FeSi could be understood fairly well within the purview of these models, the low temperature behavior (T < 40 K) which is influenced by the disorder could not be rationalized. In an attempt to understand the evolution of the ground state and Kondo insulating (KI) gap in FeSi, we had carried out series of investigations encompassing the isoelectronic substitutions of Ge at a Si site in FeSi1−x Gex and Ru at a Fe site in Fe1−x Rux Si systems
392
A. Mani et al. / Solid State Communications 146 (2008) 391–394
followed by high-pressure studies on pristine and substituted samples via transport measurements [14–17]. The validity of the aforementioned existing models had also been examined in our studies. The high pressure-low temperature electrical conductivity studies on FeSi1−x Gex [14] had brought about a complicated pressure and composition dependence of the KI gap, which could not be explained solely within the framework of the one-electron band picture or a hybridization gap model. For example, expansion of the lattice by Ge-substitution resulted in decreasing gap, consistent with the band structure prediction and a hybridization gap model. However, compression of the lattice under high pressure in FeSi was also found to decrease the gap, contrary to what was expected from a band structure or hybridization gap model. Further, a non-monotonic variation of gap was seen for a 5% Ge substituted sample, while an increase in the gap was observed in a 20% Ge substituted sample with increasing pressure. In the low temperature regime (5–40 K), a pressure induced insulator to metal transition (MIT) arising due to delocalization of the localized gap states was observed in these systems. Such a complex pressure dependent behavior could not be understood without invoking effects of disorder. Our Fe1−x Rux Si system had provided an ideal avenue for testing a correlation cum hybridization model [15]. The substitution of Ru at the Fe site affects the correlation and hybridization simultaneously by reducing the on-site Coulomb repulsion and by expanding the lattice respectively. Another interesting feature of the Fe1−x Rux Si system was that its end compounds, namely, FeSi and RuSi belong to two different classes of materials—FeSi is a KI with a modest value of gap ∼700 K, while RuSi is a band semiconductor with a larger gap of ∼3600 K [18]. Therefore, under Ru substitution a transformation from KI to band semiconductor was expected, which indeed could be observed near x ∼ 0.06. This transition could be realized based on the observed non-monotonic evolution of energy gap (also an anomaly in other parameters see Ref. [15]) as a function of x with a minimum at x ∼ 0.06. Within the KI regime (x ≤ 0.06), the decrease of the gap with increasing x could be attributed to the combined effect of correlation and hybridization. On the other hand, in the semiconductor regime (x > 0.06), an increase in gap with x complies with the predictions of band structure calculations [15]. It was brought about from the above studies that the physical properties of the FeSi system were primarily dictated by the combined effects of band structure, electron correlation and disorder. Based on these studies, a new density of states model, namely, the Gaussian density of states (GDOS) model [15,16] which incorporates these effects in its defining parameters, had been proposed and used for a quantitative understanding of the electrical conductivity behavior of this system. This model had consistently explained the transport behavior of FeSi system as a function of temperature, pressure and composition. The GDOS model is an improvisation over the narrow band–small energy gap DOS model of Jaccarino et al. [3– 6]. In the later model, a DOS consisting of two rectangular bands with extremely narrow width separated by a small energy gap was proposed. This feature of DOS was believed to arise due to the renormalization of the corresponding non-interacting bands in the presence of correlation. Incorporation of the disorder effects in this model had resulted in our GDOS model. Further details of the GDOS model are presented in the analysis section of this paper. In this paper, we report the low temperature specific heat studies on Fe1−x Rux Si with x = 0.0 to 0.2. The motivation for carrying out the present study is based on the following reasons. As mentioned in the previous paragraph, Fe1−x Rux Si exhibit transformation from a KI to band semiconductor around x ∼ 0.06. Therefore, it is of interest to investigate whether similar behavior can be observed via specific heat studies in this system. In addition, it is also aimed to extend the GDOS model for the electronic part of the specific heat for a quantitative analysis of specific heat data and find its applicability in understanding the thermal properties of the FeSi system.
2. Experimental details Fe1−x Rux Si (0 ≤ x ≤ 0.3) and CoSi samples were prepared by arc melting and were characterized by X-ray diffraction (XRD) followed by the Rietveld refinements to obtain the lattice constant and the internal atomic position parameters for the elemental constituents of the samples [15,17]. Among these, samples with x = 0.0, 0.03, 0.06 and 0.2 have been used for specific heat studies. Selection of these compositions is based on the fact that samples below x = 0.06 are in the KI regime, and above are in the band semiconductor regime, while x = 0.06 is at the verge of transformation [15]. The specific heat measurements in the 77–300 K temperature range were carried out in a home made cryostat, using a quasi-adiabatic heat pulse calorimeter. A small amount of sample (∼100–200 mg) is mounted on a sample holder assembly comprising of thin sapphire disc covered with Ni–Cr thin film as a heating element and a platinum resistance thermometer for temperature measurements. A heat pulse for a short known duration is applied to the sample assembly at a fixed temperature and the rise in temperature is deduced from analysis of the time dependent temperature drift curve [19]. The heat capacity of sample holder assembly is measured before mounting the sample and that is subtracted from the total heat capacity in order to extract the specific heat of the sample under study. 3. Analysis of specific heat data In general the specific heat of a system is given by [20] C = Cel + Cph + Cmag + Csch
where Cel is the electronic part, Cph the lattice part, Cmag the magnetic part while Csch the Schottky part of specific heat that arises from the two level systems. In a previous analysis of specific heat in FeSi, the Cph was substituted by CCoSi , where CCoSi is the measured specific heat in the non-magnetic isostructural compound CoSi [3,4,21]. The excess specific heat thus obtained by subtracting CCoSi was thought to be Cmag + Cel while Csch was taken to be absent. Further Cel was either taken to be of the form γ T [3] or included in term Cmag [21]. In the study carried out by Mandrus et al. [4], Cph is taken to be CCoSi and for the electronic part they used narrow band–small energy gap type density of state function. In the present study, we use the general form of the Debye model to describe the lattice contribution and for the electronic part we employed a Gaussian density of states (GDOS) previously used to fit our resistivity [15,16]. A schematic diagram of the GDOS proposed for electronic structure of the FeSi system [15,16] is reproduced in Fig. 1. In this model, the sharp features of the DOS arise due to the electron correlation, while the effect of the band structure is captured by Eg , the energy separation between valence band edge (VBE) and conduction band edge (CBE). The effect of correlation/disorder appears in the form of the variation in width W of both bands having Gaussian distribution. Disorder shifts the position of the mobility edge (ME) residing above and below the Fermi energy EF by introducing localized states inside the gap. The ME separates localized states from the extended states (see Fig. 1). In this model, the temperature dependent specific heat C of the FeSi system is obtained as follows: ∂U + Cph (1) C=
∂T
ele
where first term on right hand side takes into account the electronic specific heat and the second term represents the lattice specific heat. The internal electronic energy U is given as: Z ∞ U= ED(E)f (E)dE. (2) Eµ
A. Mani et al. / Solid State Communications 146 (2008) 391–394
Fig. 2. Variation of specific heat of FeSi and CoSi as a function of temperature. Inset shows variation of the anomalous electronic contribution to specific heat of FeSi as described in the text along with fit to GDOS model.
Fig. 1. A schematic representation of GDOS model.
Here D(E) is a Gaussian DOS which is expressed as " 2 ! Ng E − Eg /2 D(E) = √ exp −2 W 2π(W /2) 2 !# E + Eg /2 . + exp −2 W
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(3)
For the lattice specific heat Cph , we use following general Debye expression which is valid at all temperatures T ; Cph = 9Nk(T /θ)3
Z θ/T 0
x4 ex
( − 1)2 ex
dx.
(4)
Here Eµ is the energy separation between EF and ME, f (E) is the Fermi distribution function, N is the density of unit cells containing a four FeSi formula unit, g is the total number of states per unit cell, k is Boltzmann’s constant, θ is the Debye temperature and x = hν/kT [h—Planck’s constant, ν—phonon frequency]. Experimental data has been analyzed using the above expressions and Eg , W , Eµ and θ are used as the fit parameters. It should be noted that, as such, there is no sharply defined energy gap in the GDOS model. An effective energy gap, ∆eff , has been deduced by computing the number density of carriers n at several temperatures above 100 K using the parameters Eg , Eµ and W extracted from the fits to the specific heat data using the following formula Z ∞ n= D(E)f (E)dE. (5)
Fig. 3. Variation of specific heat as a function of temperature along with fits for (a) FeSi, (b) Fe0.97 Ru0.03 Si, (c) Fe0.94 Ru0.06 Si and (d) Fe0.8 Ru0.2 Si.
Eµ
The set of n(T ) thus obtained is then fitted to a thermal activated formula of type n ∝ exp(−∆eff /2kB T ) to extract ∆eff . It should be mentioned that experimentally we measure specific heat at constant pressure, CP (T ), however Eq. (1) is defined for specific heat at constant volume, CV (T ). Since the difference between values of CP (T ) and CV (T ) is very small for solids [19], therefore we use CP (T ) data as such for analysis. 4. Results & discussion The variation of CP (T ) of pristine FeSi and CoSi in the temperature range of 77 to 300 K is shown in Fig. 2. CoSi is an iso-structural diamagnetic counterpart of FeSi, therefore the lattice specific heat of the former is expected to be similar to that of the latter as suggested by Jaccarino et al. [3]. Adapting the procedure described in Ref. [3], we extract the anomalous magnetic
contribution of the specific heat of FeSi by taking the difference [CP (FeSi) − γFeSi T ] − [CP (CoSi) − γCoSi T ] with the Sommerfeld −1 coefficients γFeSi = 1.5 × 10−4 Cal mol K−2 and γCoSi = −1 −4 −2 2.4 × 10 Cal mol K for FeSi and CoSi respectively. The resulting excess specific heat δCexcess of FeSi is shown in the inset of Fig. 2. We fit this data using the GDOS model excluding the phonon part of Eq. (1). The fit is shown in the inset of Fig. 2. The parameters Eg , W and Eµ deduced from this fit are used as initial guess parameters for fitting the experimental CP (T ) data of Fe1−x Rux Si using Eq. (1) by taking into account the total specific heat comprising of both electronic and lattice contributions. The results of these analyses are shown in Fig. 3(a)–(d) for x = 0.0, 0.03, 0.06 and 0.2 respectively. The χ2 of the fits are found to be in the range of the order of 10−3 –10−4 . This indicates that data fits very well to our model. Parameters Eg , W , Eµ and θ have been extracted from these fits for all the samples. For pristine FeSi, the
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values of these parameters are found to be Eg = 828 K, W = 500 K, Eµ = 516 K and θ = 605 K. The value of θ found from our analysis is in very good agreement with the reported value of 600 K [22]. Using GDOS fit parameters, the carrier concentration n at various temperatures T ≥ 100 K have been calculated to deduce the effective energy gap ∆eff . For pristine FeSi, ∆eff is found to be ∼630 K, which is larger than the value of ∼560 K obtained from σ(T ) [15]. This discrepancy is understandable because in transport, electrons travel along the path of least resistance which gives rise to a smaller effective gap. On the other hand, a thermodynamic measurement averages over the entire sample leading to larger effective gap [22]. The value of carrier concentration n300 K ∼ 1.26 × 1028 m−3 deduced for FeSi is in good agreement with the value of 2 × 1028 m−3 obtained from the Hall measurements [5]. The variation of Eg , W , Eµ , θ and ∆eff are shown in Fig. 4(a)–(e) respectively as a function of Ru concentration x. The parameter Eg exhibits a monotonic increase with increasing x. This is consistent with band structure calculations which predict an increase in band gap under Ru substitution in FeSi [15]. W shows a marginal increase up to x ≤ 0.06, while a drastic increase for x = 0.2. This sudden increase in broadening of bandwidth W should indicate reduction in the correlation effect. Parameters Eµ and ∆eff show non-monotonic variation i.e., a decrease up to x = 0.06 followed by an increase. The variation of Eµ is consistent with similar trend seen in the hopping parameter in σ(T ) studies [15], which indicates delocalization/localization of the gap states. It is known that RuSi is a band semiconductor which has a larger energy gap than that of FeSi [18]. Therefore, if the nature of FeSi and RuSi had been similar, then under Ru substitution ∆eff should have shown a monotonic increase with increasing x. The fact that ∆eff exhibits a non-monotonic variation (see Fig. 4(e)), must indicate a change in nature of the samples beyond a Ru concentration of x ≥ 0.06. The observed monotonic decrease in θ complies with the fact that lattice parameter increases with Ru substitution [15] which leads to phonon softening with increasing x. It is worth noting that the parameters Eµ and ∆eff deduced from σ(T ) analyses exhibit similar variation with x [15]. Therefore, intuitively, the present result should be consistent with the inference drawn from our previous σ(T ) studies. 5. Summary and conclusions The specific heat of Fe1−x Rux Si with x = 0.0–0.2 have been measured using quasi adiabatic heat pulse calorimetry in the 77–300 K temperature range. The GDOS model in conjunction with the general form of the Debye expression have been used to analyze the experimental CP (T ) data. The parameters extracted from these analyses clearly indicate the change in nature of the sample beyond x > 0.06. It should be mentioned that band structure calculations predict a monotonic increase of gap with increasing Ru concentration in Fe1−x Rux Si [15]. In contrast, ∆eff is seen to decrease in the regime of x ≤ 0.06 implying that the one electron band picture is not applicable in this regime. However, the observed increase in ∆eff beyond x > 0.06 complies with the prediction of the band structure calculations. The sudden increase of band width W for x > 0.06 is an indication of the diminishing of the correlation effect. Therefore, the non-monotonic variation of ∆eff with x and the sudden increase of band width W observed beyond x ∼ 0.06 give an indirect indication of a transformation from a KI to a band semiconductor. However, a more direct method to verify this transition would be photoemission or tunneling spectroscopy in which an actual feature of DOS and broadening of band may be seen in the semiconducting regime in contrast to the KI regime. It should be mentioned that in our previous studies [15] such anomalies were found in all experimentally deduced parameters such as energy gap, hopping parameter, low
Fig. 4. Variation of fit parameters (a) Eg , (b) W, (c) Eµ , (d) θ, (e) ∆eff as a function of Ru concentration, x, in Fe1−x Rux Si. Solid lines connecting the data points are guide to the eye.
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