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Local structure of the thermoelectric material Mg2Si using XRD
Journal of Alloys and Compounds 479 (2009) 26–31
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom
Local structure of the thermoelectric material Mg2 Si using XRD R. Saravanan a,∗ , M. Charles Robert b a b
Department of Physics, The Madura College, Madurai 625 011, Tamil Nadu, India Department of Physics, H.K.R.H. College, Uthamapalayam - 625 533, Tamil Nadu, India
a r t i c l e
i n f o
Article history: Received 2 September 2008 Received in revised form 22 December 2008 Accepted 24 December 2008 Available online 4 January 2009
a b s t r a c t The electron density distribution and local structure of the high temperature thermoelectric material Mg2 Si have been studied and analyzed. Powder X-ray data of Mg2 Si is analyzed in terms of local structure, thermal vibration parameters, 1D, 2D and 3 dimensional electron density distributions. The bonding between the atoms has been studied using maximum entropy method (MEM) and the bond-length distribution using pair distribution function (PDF). © 2008 Elsevier B.V. All rights reserved.
PACS: 61.05.C− 61.05.cp 71.20.−b 36.40.Cg 85.80.Fi Keywords: Mg2Si Thermoelectric MEM PDF X-ray Powder
1. Introduction The research interests on thermoelectric materials and devices based on them are growing enormously as they can be the potential alternate power sources in the near future. Also, thermoelectric devices are highly reliable, silent, light-weighted etc. Numerous efforts have been made to find and prepare materials with high Seebeck coefficient and hence high figure of merit given by ZT = S2 T/, where is the electrical conductivity, is the thermal conductivity, S is the Seebeck coefficient and T is the temperature and ZT represents the thermoelectric figure of merit. Materials with high Seebeck coefficient, high electrical conductivity and low lattice thermal conductivity are the candidates for producing high thermoelectric power. Materials with loosely bound atoms and high thermal parameters can scatter phonons much effectively than electrons.
∗ Corresponding author. Tel.: +91 452 2673354x211. E-mail address: [email protected] (R. Saravanan). URL: http://www.saraxraygroup.net (R. Saravanan). 0925-8388/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2008.12.117
It was proved that the Mg2 Si is an intermediate thermoelectric power generator material with maximum operating temperature of around 600 K and hence it plays an important role, in high temperature applications. The preparation and characterization of Mg2 Si has been reported in numerous research articles. For example, Lee et al. [1] has reported the preparation of Mg2 Si by mechanical alloying methods; Zang et al. [2] and Tani and Kido [3] have studied the thermoelectric properties of Sb doped Mg2 Si; Song et al. [4] reported the synthesis and thermoelectric properties of Mg2 Si1−x Gex (x = 0, 0.2, 0.4, 0.6, 0.8, 1); Bose et al. [5] have studied the thermoelectric properties of Mg2 Si; Kamilov et al. [6] report the thin film preparation and characterization of Mg2 Si; Bashenov et al. [7] report the calculation of density of states in Mg2 Si; and Akasaka et al. [8] report the growth of Mg2 Si using vertical Bridgman method and its characterization. Though there is much experimental work on the growth and physical characterization of thermoelectric materials, only limited information about the local structure, electron density distribution and bonding is available, particularly for Mg2 Si. Hence, the present work is aimed at the structural analysis in terms of the local and average structural properties, the electron density distribution between atoms and hence the bonding of the thermoelectric materials Mg2 Si.
R. Saravanan, M.C. Robert / Journal of Alloys and Compounds 479 (2009) 26–31
27
Fig. 1. Rietveld refined X-ray powder profile of Mg2 Si. Crosses represent the observed points and the continuous line represents the calculated powder profile. The small vertical lines represent the calculated Bragg positions.
The raw X-ray powder intensity data of the crystal Mg2 Si has been refined in this study using the Rietveld refinements [9] and also studied for the thermal vibration parameters. The refined parameters in the maximum entropy method (MEM) studies have also been reported. The qualitative and quantitative electron distribution of the atoms in crystalline materials was studied using maximum entropy formalism, because it provides less biased information on the electron densities of the crystals compared to conventional Fourier synthesis [10–15]. Fourier synthesis of electron densities can be of useful in picturizing the bonding between two atoms. But, it suffers from the major disadvantage of series termination error and negative electron densities which prevent the clear understanding of the bonding between atoms, the factor which has been intended to be analyzed. The MEM is a method to derive the most probable electron density map given a set of experimental data. In crystallography, the MEM is used to determine the electron density in the unit cell that provides the best fit to the scattering data. It is a model-free approach in contrast to structure refinements in which the positions of spherical atoms are determined. Collins [16] formalism is based on the entropy expression S, given by S=−
(r) ln
(r) (r)
(1)
(r) is the electron density at a certain pixel, r and (r), the prior electron density for (r). We introduce the soft constraint as, 1 C= N
2
Fcal (K) − Fobs (K) 2 (Fobs (K))
(2)
Where N is the number of observed reflections, (r) is the standard deviation for Fobs (K), V is the volume of the unit cell and Fcal (K) = V
(r) exp(2i k.r)dv
(3)
We use the Lagrange’s method of undetermined multiplier to constrain C to be unity while we maximize the entropy. Then we have Q () = −
(r) ln
(r) C (r) 2
(4)
By setting dQ()/d(r) = 0 and solving for (r)using approximation ln(X) = X−1,
(r) = exp
ln (r) + Fcal (0) N
(Fobs (K) − Fcal (K)) exp(−i2ik.r)
2 (K) (5)
This is refined until C = 1, the final result is (r) = MEM (r)
(6)
The electron density can provide information about the nature of charges in any system, particularly in thermoelectric materials. Whether the charges are localized or not can be understood through the electron densities and the nature of bonding, which is a highly essential characteristic to be analyzed as far as the thermoelectric materials are concerned. The local structural analysis has been carried out using pair distribution function [17–20]. Pair distribution function (PDF) is the method to analyze the powder diffraction data in the real space. It gives very useful information in doped systems about the bondlength distribution and hence the lattice constants, since one can analyze the differences in the structural properties due to doping. Many studies are available reporting the implementation of this fruitful method [21–28]. The PDF reflects the short range ordering in a material. This approach has been widely used for studying the local structure of many materials. More recently, it has been applied to disordered crystalline and partially crystallized materials. Quantitative structural information on nanometer length scales can be obtained by fitting a model directly to the PDF. The PDF method shows that when there are no short range deviations from the average structure, the PDF agrees well with the inter-atomic distances computed from a crystallographic model. This method is also known to take into account both the Bragg as well as diffuse scattering which is due to short range effects come from the localized additional phases. We have carried out an analysis using the above two methods, i.e., MEM and PDF to understand the electronic and local structure of the thermoelectric material Mg2 Si to explore the nature of distribution of the electrons.
28
R. Saravanan, M.C. Robert / Journal of Alloys and Compounds 479 (2009) 26–31
Table 1 The observed and calculated structure factors obtained from the powder data refinement of Mg2 Si refined using JANA 2006. hkl
Fobs
Fcal
(Fobs )
111 002 202 113 222 004 313 204 224 115 333 404 315 424 006 206 335 226 444 515 117
44.38 31.57 97.94 31.78 23.21 80.27 27.07 17.60 67.78 23.99 23.99 58.57 19.98 9.08 9.08 41.15 15.55 7.35 41.07 14.16 14.16
44.42 31.21 98.63 32.57 24.12 78.79 27.16 18.09 64.82 23.20 23.26 53.94 20.07 9.95 9.95 45.25 17.38 7.41 38.23 15.11 15.11
0.521 0.621 0.552 0.408 0.760 1.094 0.593 0.616 0.645 0.613 0.613 1.199 0.536 0.862 0.862 0.727 0.543 0.965 1.479 0.621 0.621
Fobs = observed structure factor; Fcal = calculated structure factor; (Fobs ) = standard error in the observation.
2. X-ray characterization of electron density In order to analyze the bonding and structural behavior of Mg2 Si, powder X-ray intensity data was collected using Cu-K␣ radiation from high quality Mg2 Si sample, which was a commercial sample from Alfa–Aesar with purity 99.99%. The powder X-ray data set was collected in the 2 range from 10◦ to 120◦ with step size 0.05◦ at Regional Research Laboratory (RRL), CSIR, Trivandrum, India, using X’-PERT PRO (Philips, Netherlands) X-ray diffractometer with a monochromatic incident beam of wavelength 1.54056 Å, offering pure Cu-K␣1 radiation. A standard software package [29] was used to refine the cell parameters of Mg2 Si using the observed 2 values. The refined cell parameter is found to be 6.3899 (0.0001) Å whereas the reported [30] value is 6.390 Å.
Table 3 The parameters used and obtained in MEM electron density analysis. Parameter
Value
Number of cycles, N Lagrange parameter () No. of electrons/unit cell (F000 ) RMEM (%) wRMEM (%)
39 0.0266 152 1.79 1.59
RMEM = reliability index from MEM refinement. wRMEM = weighted reliability index from MEM refinement. = The undetermined multiplier in the MEM methodology.
2.2. Electron density using MEM For the numerical MEM computations, the software package PRIMA [32] was used. For the 2D and 3D representation of the electron densities, the program VESTA [33] package was used. The MEM refinements were carried out by dividing the unit cell into 128 × 128 × 128 pixels. The initial electron density at each pixel is fixed uniformly as F000 /a0 3 = 0.583 e/Å3 . Where F000 is the total number of electrons in the unit cell and a0 is the cell parameter. The Lagrange parameter () is suitably chosen so that the convergence criterion C = 1 is reached after minimum number of iterations. The MEM parameters have been given in Table 3. The 3D electron density distribution in the form of iso-surface in the unit cell has been represented in Fig. 2. The 2D electron density distribution on the (1 0 0) and (1 1 0) planes has been given in Figs. 3 and 4 respectively. The one dimensional electron density profiles along [1 0 0], [1 1 0] and [1 1 1] directions are represented in Fig. 5a and b. The numerical values of the electron densities along different directions are given in Table 4. 2.3. Local structure In order to understand the local structure of Mg2 Si using the real space analysis of the X-ray powder data, the observed pair distribution function was obtained using the software package PDFGETX [34]. The observed PDF was fitted with the calculated one using the software PDFGUI [35]. The fitted PDFs are given in Fig. 6. The
2.1. Rietveld refinement The raw intensity data of Mg2 Si was refined using Rietveld [9] refinement. It is a well-known powder profile fitting method for the structural refinement. The cell parameters and other structural parameters were refined using this method, by JANA2006 [31]. The fitted profile and the positions of Bragg peaks for Mg2 Si have been shown in Fig. 1. The refined structure factors with (Fobs ) values are represented in Table 1. In this table, Fobs represents the observed structure factor, Fcal the calculated structure factor and (Fobs ) the standard error in the measurements. The refined structural parameters are given in Table 2.
Table 2 The Rietveld refined structural parameters of Mg2 Si using JANA 2006. Parameter
Value
Cell constant (Å) BMg (Å2 ) BSi (Å2 ) Robs (%) RP (%)
6.3706 (0.0131) 1.802 (0.121) 1.439 (0.123) 3.79 8.27
BMg = Debye–Waller factor of Mg atom. BSi = Debye–Waller factor of Si atom. Robs = reliability index from Rietveld refinement. RP = weighted reliability index from Rietveld refinement.
Fig. 2. 3D iso-surface of the electron density of Mg2 Si in the unit cell superimposed on the structure of Mg2 Si. (Si atom is at the origin). The eight Mg atoms are seen inside the cubic unit cell at ±(1/4 1/4 1/4).
R. Saravanan, M.C. Robert / Journal of Alloys and Compounds 479 (2009) 26–31
29
Table 4 MEM Bond electron densities of Mg2Si obtained from one dimensional electron density profiles. Direction
Position (Å) (distance from origin)
Electron density (e/Å3 )
Comment
[1 0 0] [1 0 0] [1 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1]
2.288 3.184 4.079 1.618 2.251 2.880 0.000 1.465 2.757 4.136 4.566 5.514 6.462 6.893 8.271
0.102 0.003 0.102 0.143 0.190 0.143 98.299 0.116 59.655 0.108 0.164 0.003 0.164 0.108 59.655
Hump Mid-bond (Si–Si) Hump Dip Hump Dip Peak Si Mid-bond (Si–Mg) Peak Mg (1) Dip Hump Dip Hump Dip Peak Mg (2)
3. Results and discussion Mg2 Si has a fluorite structure with space group Fm3m. The Si atoms are at the origin and Mg atoms at ±(1/4 1/4 1/4) and equivalent positions. There are four Mg2 Si molecules in the unit cell. This cubic structure has a cell constant of 6.39 Å [30]. The cell difference between reported [30] and that from PDF method is 0.033 Å. Since, PDF analysis relies on the local structure, the local undulations in the bond charges and the interactions
Fig. 3. MEM Electron density map of Mg2 Si on (1 0 0) plane. Contour range is from 0.01 to 0.8 e/Å3. The step size is 0.01 e/Å3 . Si atom is at the origin.
refined cell constant from PDF analysis is 6.3568 (0.0001) Å which is slightly different from that reported, 6.39 Å [30]. Some neighbor atom distances obtained from the PDF analysis have been given in Table 5.
Fig. 4. MEM electron density map of Mg2 Si on (1 1 0) plane. Contour range is from 0.04 to 0.8 e/Å3. The step size is 0.01 e/Å3 . Si atom is at the origin.
Fig. 5. (a) One dimensional peak MEM electron density profiles of Si and Mg atoms in Mg2 Si along [1 0 0, [1 1 0] and [1 1 1]. The Si atom is at the origin. (b) One dimensional low MEM electron density profiles of Si and Mg atoms showing the bonding electron densities in Mg2 Si along [1 0 0, [1 1 0] and [1 1 1]. The Si atom is at the origin.
30
R. Saravanan, M.C. Robert / Journal of Alloys and Compounds 479 (2009) 26–31
Table 5 The inter atomic distances of Mg2 Si obtained from PDF analysis. Atom pair
Si–Mg
Mg–Mg; Si–Si Mg–Mg; Si–Si
Inter-atomic distances from PDF analysis (Å) Observed
Calculated
2.94 5.36 6.90 8.06 9.38 10.26 11.98 4.36 13.38
2.88 5.32 7.00 8.10 9.36 10.26 11.98 4.40 13.40
between atoms lead to deviations in the cell parameter from that obtained only from the average Bragg positions. Since the deviation in the cell constant in PDF analysis is very small compared to that reported, it is obvious that the there are not much local distortions in the structure of the pure Mg2 Si, except those resulting from bonding interactions between atoms. The Rietveld refinement [9] gives the thermal vibration parameters of the individual atoms which are large for both the atoms Mg and Si (Table 2). This also signifies the use of Mg2 Si as a thermoelectric material. The 3D MEM electron density distribution shows (Fig. 2) that the electronic charges in the unit cell are symmetric for both atoms. This figure has been constructed superimposing the structural parameters and the radii obtained from the present study and the MEM electron densities. No apparent difference in the size of the core regions is visible due to the comparatively same number of core electrons of Si and Mg. The 2D electron density map on (1 0 0) plane (Fig. 3) shows the electronic charge distribution of silicon atom. Elongations in the valence regions show attractive character of silicon with the neighbors. Small charge islands are visible at (1/4 1/4 1/4) and equivalents positions due to the inner Mg atoms. The (1 1 0) map (Fig. 4) clearly shows attractive character between Si and Mg atoms (slightly loose bound outer contour lines in Si and Mg) and covalent character between Mg atoms (sharing of charge clouds). The 1D profiles in Fig. 5a show enhanced peak heights for Si atom compared to Mg atom. Fig. 5b shows the low density 1D profiles. The bond strength between Si–Si bond amounts to 0.087 e/Å3 at a distance of 2.252 Å. The mid-bond electron density between Si and Mg atoms along [1 1 1] direction is 0.132 e/Å3 at 1.465 Å. The mid-bond electron density along [1 0 0] direction is found to be 0.033 e/Å3 at a distance of 3.185 Å from the origin. The profile electron density value along [1 1 1] direction is not too high or not too
Calculated inter-atomic distances (Å)
2.75 5.27 6.92 8.25 9.39 10.41 11.88 4.49 13.47
low. Hence, the bonding between Si and Mg atoms might be having a mixed character (ionic and covalent). Though the 1D electron density values are not too different from each other, one can make a comparison of values to elucidate type of bonding in conjunction with the 2D maps in addition. Fig. 6 shows PDF peaks representing the interaction between Si–Si, Mg–Mg and Si–Mg atomic pairs. PDF peaks up to a distance of 13 Å are suitably indexed as shown in Fig. 6. The PDF peak at 2.94 Å corresponds to the Si–Mg bond length along [1 1 1] direction. The mid-bond position turns out to be (2.94/2) 1.47 Å. This position is almost the same as the mid-bond position along [1 1 1] direction (1.465 Å), obtained from the MEM analysis. Similarly, from the MEM analysis along [1 1 0] direction, a small hump is seen at 2.251 Å. From the PDF analysis, the Si–Si bond length turns out to be 4.36 Å. Thus the Si–Si mid-bond position is 2.18 Å, which compares with the hump seen in MEM analysis at 2.252 Å. Thus all the PDF peaks can be suitably indexed. Hence, the MEM technique combined with PDF analysis can give much more information about the electron density and local structure of the system under consideration.
4. Conclusion The MEM and PDF analyses have been applied to the high temperature thermoelectric material Mg2 Si to extract the maximum possible information X-ray powder data. In the present study, the mid-bond electron density is very low along the bonding direction [1 1 1], indicating that the bonding in Mg2 Si is not sufficiently strong to prevent the movement of the electronic charges. This is an essential criterion for a thermoelectric material for charge movement. Moreover, the thermal vibration parameter of Mg atom, BMg is larger than that of Si atom. Hence, the thermoelectric properties in Mg2 Si depend more on the Mg atom than Si atom in Mg2 Si. These methodologies are highly useful when systems like Mg2 Si are doped with other metal atoms to increase its figure of merit and efficiency.
References
Fig. 6. The fitted observed and calculated pair distribution functions (PDF) of Mg2 Si.
[1] C.H. Lee, S.H. Lee, S.Y. Chun, S.J. Lee, J. Nanosci. Nanotechnol. 6 (2006) 3429. [2] L.M. Zang, C.B. Wang, H.Y. Jiang, Q. Shan, in: 22nd International Conf. on ICT, Thermoelectrics (2003) P146. [3] J.I. Tani, H. Kido, Intermetallics 15 (2007) 1202. [4] R. Song, L. Yazheng, A. Tatsuhiko, J. Mat. Sci. Technol. 21 (2005) 618. [5] S. Bose, H.N. Acharya, H.D. Banerjee, J. Mat. Sci. 28 (1993) 5461. [6] T.S. Kamilov, D.K. Kabilov, R.Kh. Kamilova, M.E. Azimov, V.V. Kleechkovskaya, A.S. Orekhov, E.I. Suvorova, in: 25nd International Conf. on ICT, Thermoelectrics (2006) P468. [7] V.K. Bashenov, A.M. Mutal, V.V. Timofeenko, Phy. Stat. Solidi (b) 87 (1978) K77. [8] M. Akasaka, T. Iida, T. Nemoto, J. Soga, J. Kato, K. Makino, M. Fukano, Y. Takanashi, J. Cryst. Growth 304 (2007) 196. [9] H.M. Rietveld, J. Appl. Cryst. 2 (1969) 65. [10] S. Israel, R. Saravanan, N. Srinivasan, R.K. Rajaram, J. Phys. Chem. Solids 64 (2003) 43. [11] S. Israel, R. Saravanan, R.K. Rajaram, Physica B349 (2004) 390. [12] M. Sakata, M. Sato, Acta Cryst. A42 (1990) 263.
R. Saravanan, M.C. Robert / Journal of Alloys and Compounds 479 (2009) 26–31 [13] K. Yamamoto, Y. Takahashi, K. Ohshima, F.B. Okamura, K. Yukino, Acta Crystallogr. A52 (1996) 606. [14] M. Sakata, T. Uno, M. Takata, J. Appl. Crystallogr. 26 (1993) 159. [15] T. Kajitani, R. Saravanan, Y. Ono, K. Ohno, M. Isshiki, J. Crystal Growth 229 (2001) 130. [16] D.M. Collins, Nature 298 (1982) 49. [17] P. Debay, H. Menki, Phys. Z 31 (1930) 797. [18] B.E. Warren, X-ray Diffraction, Addison-Wesley, New York, 1969. [19] D.T. Bowran, J.L. Finney, J. Chem. Phys. 118 (2003) 8357. [20] T. Egami, S.J.L. Billinge, Underneath the Bragg peaks: Structural Analysis of Complex Material, Oxford University Press, London, 2003. [21] W. Dmowski, B.H. Toby, T. Egami, M.A. Subramanian, J. Gopalakrishan, A.W. Sleight, Phys. Rev. Lett. 61 (1988) 2608. [22] T. Proffen, S.J.L. Billinge, J. Appl. Crystallogr. 32 (1999) 572. [23] T. Proffen, R. Neder, J. Appl. Crystallogr. 30 (1997) 171. [24] T. Egami, Mater. Trans. Jpn. Inst. Met. 31 (3) (1990) 163. [25] V. Petkov, S.J.L. Billinge, P. Larsan, S.D. Mahanthi, T. Vogt, K.K. Rangen, M.G. Kanatzidis, Phys. Rev. B65 (2002) 092105.
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B.H. Toby, T. Egami, Acta Crystallogr. A48 (1992) 336. L.M. Peng, G. Ren, S.L. Dudarev, M. Whelan, Acta Cryst. A52 (1996) 456. P.F. Peterson, E.S. Bozin, T. Proffen, S.J.L. Billinge, J. Appl. Cryst. 36 (2003) 53. T.J.B. Holland, S.A.T. Redfern, Mineral. Mag. 61 (1997) 65. R.W.G. Wyckoff, Crystal Structures, vol. 1, Inter-Science publishers, London, 1963. JANA, in: V. Petˇríˇcek, M. Duˇsek, L. Palatinus (Eds.), The Crystallographic Computing System, Institute of Physics, Academy of sciences of the Czech Republic, Praha, 2000. F. Izumi, R.A. Dilanian, Recent Research Developments in Physics, vol.3 Part II, Transworld Research Network, Trivandrum, 2002, pp. 699–726. K. Momma and F. Izumi. Commission on Crystallogr. Comput., IUCr Newslett., No. 7:106, 2006. I. K. Jeong, J. Thompson, TH. Proffen, A.Perez, S.J.L Billinge, “PDFGetX, A Program for Obtaining the Atomic Pair Distribution Function from X-ray Powder Diffraction Data”, 2001. C.L. Farrow, P. Juhás, J.W. Liu, D. Bryndin, E.S. Bozin, J. Bloch, Th. Proffen, S.J.L. Billinge, J. Phys.: Condens Matter 19 (2007) 335219.
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jallcom
Local structure of the thermoelectric material Mg2 Si using XRD R. Saravanan a,∗ , M. Charles Robert b a b
Department of Physics, The Madura College, Madurai 625 011, Tamil Nadu, India Department of Physics, H.K.R.H. College, Uthamapalayam - 625 533, Tamil Nadu, India
a r t i c l e
i n f o
Article history: Received 2 September 2008 Received in revised form 22 December 2008 Accepted 24 December 2008 Available online 4 January 2009
a b s t r a c t The electron density distribution and local structure of the high temperature thermoelectric material Mg2 Si have been studied and analyzed. Powder X-ray data of Mg2 Si is analyzed in terms of local structure, thermal vibration parameters, 1D, 2D and 3 dimensional electron density distributions. The bonding between the atoms has been studied using maximum entropy method (MEM) and the bond-length distribution using pair distribution function (PDF). © 2008 Elsevier B.V. All rights reserved.
PACS: 61.05.C− 61.05.cp 71.20.−b 36.40.Cg 85.80.Fi Keywords: Mg2Si Thermoelectric MEM PDF X-ray Powder
1. Introduction The research interests on thermoelectric materials and devices based on them are growing enormously as they can be the potential alternate power sources in the near future. Also, thermoelectric devices are highly reliable, silent, light-weighted etc. Numerous efforts have been made to find and prepare materials with high Seebeck coefficient and hence high figure of merit given by ZT = S2 T/, where is the electrical conductivity, is the thermal conductivity, S is the Seebeck coefficient and T is the temperature and ZT represents the thermoelectric figure of merit. Materials with high Seebeck coefficient, high electrical conductivity and low lattice thermal conductivity are the candidates for producing high thermoelectric power. Materials with loosely bound atoms and high thermal parameters can scatter phonons much effectively than electrons.
∗ Corresponding author. Tel.: +91 452 2673354x211. E-mail address: [email protected] (R. Saravanan). URL: http://www.saraxraygroup.net (R. Saravanan). 0925-8388/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2008.12.117
It was proved that the Mg2 Si is an intermediate thermoelectric power generator material with maximum operating temperature of around 600 K and hence it plays an important role, in high temperature applications. The preparation and characterization of Mg2 Si has been reported in numerous research articles. For example, Lee et al. [1] has reported the preparation of Mg2 Si by mechanical alloying methods; Zang et al. [2] and Tani and Kido [3] have studied the thermoelectric properties of Sb doped Mg2 Si; Song et al. [4] reported the synthesis and thermoelectric properties of Mg2 Si1−x Gex (x = 0, 0.2, 0.4, 0.6, 0.8, 1); Bose et al. [5] have studied the thermoelectric properties of Mg2 Si; Kamilov et al. [6] report the thin film preparation and characterization of Mg2 Si; Bashenov et al. [7] report the calculation of density of states in Mg2 Si; and Akasaka et al. [8] report the growth of Mg2 Si using vertical Bridgman method and its characterization. Though there is much experimental work on the growth and physical characterization of thermoelectric materials, only limited information about the local structure, electron density distribution and bonding is available, particularly for Mg2 Si. Hence, the present work is aimed at the structural analysis in terms of the local and average structural properties, the electron density distribution between atoms and hence the bonding of the thermoelectric materials Mg2 Si.
R. Saravanan, M.C. Robert / Journal of Alloys and Compounds 479 (2009) 26–31
27
Fig. 1. Rietveld refined X-ray powder profile of Mg2 Si. Crosses represent the observed points and the continuous line represents the calculated powder profile. The small vertical lines represent the calculated Bragg positions.
The raw X-ray powder intensity data of the crystal Mg2 Si has been refined in this study using the Rietveld refinements [9] and also studied for the thermal vibration parameters. The refined parameters in the maximum entropy method (MEM) studies have also been reported. The qualitative and quantitative electron distribution of the atoms in crystalline materials was studied using maximum entropy formalism, because it provides less biased information on the electron densities of the crystals compared to conventional Fourier synthesis [10–15]. Fourier synthesis of electron densities can be of useful in picturizing the bonding between two atoms. But, it suffers from the major disadvantage of series termination error and negative electron densities which prevent the clear understanding of the bonding between atoms, the factor which has been intended to be analyzed. The MEM is a method to derive the most probable electron density map given a set of experimental data. In crystallography, the MEM is used to determine the electron density in the unit cell that provides the best fit to the scattering data. It is a model-free approach in contrast to structure refinements in which the positions of spherical atoms are determined. Collins [16] formalism is based on the entropy expression S, given by S=−
(r) ln
(r) (r)
(1)
(r) is the electron density at a certain pixel, r and (r), the prior electron density for (r). We introduce the soft constraint as, 1 C= N
2
Fcal (K) − Fobs (K) 2 (Fobs (K))
(2)
Where N is the number of observed reflections, (r) is the standard deviation for Fobs (K), V is the volume of the unit cell and Fcal (K) = V
(r) exp(2i k.r)dv
(3)
We use the Lagrange’s method of undetermined multiplier to constrain C to be unity while we maximize the entropy. Then we have Q () = −
(r) ln
(r) C (r) 2
(4)
By setting dQ()/d(r) = 0 and solving for (r)using approximation ln(X) = X−1,
(r) = exp
ln (r) + Fcal (0) N
(Fobs (K) − Fcal (K)) exp(−i2ik.r)
2 (K) (5)
This is refined until C = 1, the final result is (r) = MEM (r)
(6)
The electron density can provide information about the nature of charges in any system, particularly in thermoelectric materials. Whether the charges are localized or not can be understood through the electron densities and the nature of bonding, which is a highly essential characteristic to be analyzed as far as the thermoelectric materials are concerned. The local structural analysis has been carried out using pair distribution function [17–20]. Pair distribution function (PDF) is the method to analyze the powder diffraction data in the real space. It gives very useful information in doped systems about the bondlength distribution and hence the lattice constants, since one can analyze the differences in the structural properties due to doping. Many studies are available reporting the implementation of this fruitful method [21–28]. The PDF reflects the short range ordering in a material. This approach has been widely used for studying the local structure of many materials. More recently, it has been applied to disordered crystalline and partially crystallized materials. Quantitative structural information on nanometer length scales can be obtained by fitting a model directly to the PDF. The PDF method shows that when there are no short range deviations from the average structure, the PDF agrees well with the inter-atomic distances computed from a crystallographic model. This method is also known to take into account both the Bragg as well as diffuse scattering which is due to short range effects come from the localized additional phases. We have carried out an analysis using the above two methods, i.e., MEM and PDF to understand the electronic and local structure of the thermoelectric material Mg2 Si to explore the nature of distribution of the electrons.
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R. Saravanan, M.C. Robert / Journal of Alloys and Compounds 479 (2009) 26–31
Table 1 The observed and calculated structure factors obtained from the powder data refinement of Mg2 Si refined using JANA 2006. hkl
Fobs
Fcal
(Fobs )
111 002 202 113 222 004 313 204 224 115 333 404 315 424 006 206 335 226 444 515 117
44.38 31.57 97.94 31.78 23.21 80.27 27.07 17.60 67.78 23.99 23.99 58.57 19.98 9.08 9.08 41.15 15.55 7.35 41.07 14.16 14.16
44.42 31.21 98.63 32.57 24.12 78.79 27.16 18.09 64.82 23.20 23.26 53.94 20.07 9.95 9.95 45.25 17.38 7.41 38.23 15.11 15.11
0.521 0.621 0.552 0.408 0.760 1.094 0.593 0.616 0.645 0.613 0.613 1.199 0.536 0.862 0.862 0.727 0.543 0.965 1.479 0.621 0.621
Fobs = observed structure factor; Fcal = calculated structure factor; (Fobs ) = standard error in the observation.
2. X-ray characterization of electron density In order to analyze the bonding and structural behavior of Mg2 Si, powder X-ray intensity data was collected using Cu-K␣ radiation from high quality Mg2 Si sample, which was a commercial sample from Alfa–Aesar with purity 99.99%. The powder X-ray data set was collected in the 2 range from 10◦ to 120◦ with step size 0.05◦ at Regional Research Laboratory (RRL), CSIR, Trivandrum, India, using X’-PERT PRO (Philips, Netherlands) X-ray diffractometer with a monochromatic incident beam of wavelength 1.54056 Å, offering pure Cu-K␣1 radiation. A standard software package [29] was used to refine the cell parameters of Mg2 Si using the observed 2 values. The refined cell parameter is found to be 6.3899 (0.0001) Å whereas the reported [30] value is 6.390 Å.
Table 3 The parameters used and obtained in MEM electron density analysis. Parameter
Value
Number of cycles, N Lagrange parameter () No. of electrons/unit cell (F000 ) RMEM (%) wRMEM (%)
39 0.0266 152 1.79 1.59
RMEM = reliability index from MEM refinement. wRMEM = weighted reliability index from MEM refinement. = The undetermined multiplier in the MEM methodology.
2.2. Electron density using MEM For the numerical MEM computations, the software package PRIMA [32] was used. For the 2D and 3D representation of the electron densities, the program VESTA [33] package was used. The MEM refinements were carried out by dividing the unit cell into 128 × 128 × 128 pixels. The initial electron density at each pixel is fixed uniformly as F000 /a0 3 = 0.583 e/Å3 . Where F000 is the total number of electrons in the unit cell and a0 is the cell parameter. The Lagrange parameter () is suitably chosen so that the convergence criterion C = 1 is reached after minimum number of iterations. The MEM parameters have been given in Table 3. The 3D electron density distribution in the form of iso-surface in the unit cell has been represented in Fig. 2. The 2D electron density distribution on the (1 0 0) and (1 1 0) planes has been given in Figs. 3 and 4 respectively. The one dimensional electron density profiles along [1 0 0], [1 1 0] and [1 1 1] directions are represented in Fig. 5a and b. The numerical values of the electron densities along different directions are given in Table 4. 2.3. Local structure In order to understand the local structure of Mg2 Si using the real space analysis of the X-ray powder data, the observed pair distribution function was obtained using the software package PDFGETX [34]. The observed PDF was fitted with the calculated one using the software PDFGUI [35]. The fitted PDFs are given in Fig. 6. The
2.1. Rietveld refinement The raw intensity data of Mg2 Si was refined using Rietveld [9] refinement. It is a well-known powder profile fitting method for the structural refinement. The cell parameters and other structural parameters were refined using this method, by JANA2006 [31]. The fitted profile and the positions of Bragg peaks for Mg2 Si have been shown in Fig. 1. The refined structure factors with (Fobs ) values are represented in Table 1. In this table, Fobs represents the observed structure factor, Fcal the calculated structure factor and (Fobs ) the standard error in the measurements. The refined structural parameters are given in Table 2.
Table 2 The Rietveld refined structural parameters of Mg2 Si using JANA 2006. Parameter
Value
Cell constant (Å) BMg (Å2 ) BSi (Å2 ) Robs (%) RP (%)
6.3706 (0.0131) 1.802 (0.121) 1.439 (0.123) 3.79 8.27
BMg = Debye–Waller factor of Mg atom. BSi = Debye–Waller factor of Si atom. Robs = reliability index from Rietveld refinement. RP = weighted reliability index from Rietveld refinement.
Fig. 2. 3D iso-surface of the electron density of Mg2 Si in the unit cell superimposed on the structure of Mg2 Si. (Si atom is at the origin). The eight Mg atoms are seen inside the cubic unit cell at ±(1/4 1/4 1/4).
R. Saravanan, M.C. Robert / Journal of Alloys and Compounds 479 (2009) 26–31
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Table 4 MEM Bond electron densities of Mg2Si obtained from one dimensional electron density profiles. Direction
Position (Å) (distance from origin)
Electron density (e/Å3 )
Comment
[1 0 0] [1 0 0] [1 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1] [1 1 1]
2.288 3.184 4.079 1.618 2.251 2.880 0.000 1.465 2.757 4.136 4.566 5.514 6.462 6.893 8.271
0.102 0.003 0.102 0.143 0.190 0.143 98.299 0.116 59.655 0.108 0.164 0.003 0.164 0.108 59.655
Hump Mid-bond (Si–Si) Hump Dip Hump Dip Peak Si Mid-bond (Si–Mg) Peak Mg (1) Dip Hump Dip Hump Dip Peak Mg (2)
3. Results and discussion Mg2 Si has a fluorite structure with space group Fm3m. The Si atoms are at the origin and Mg atoms at ±(1/4 1/4 1/4) and equivalent positions. There are four Mg2 Si molecules in the unit cell. This cubic structure has a cell constant of 6.39 Å [30]. The cell difference between reported [30] and that from PDF method is 0.033 Å. Since, PDF analysis relies on the local structure, the local undulations in the bond charges and the interactions
Fig. 3. MEM Electron density map of Mg2 Si on (1 0 0) plane. Contour range is from 0.01 to 0.8 e/Å3. The step size is 0.01 e/Å3 . Si atom is at the origin.
refined cell constant from PDF analysis is 6.3568 (0.0001) Å which is slightly different from that reported, 6.39 Å [30]. Some neighbor atom distances obtained from the PDF analysis have been given in Table 5.
Fig. 4. MEM electron density map of Mg2 Si on (1 1 0) plane. Contour range is from 0.04 to 0.8 e/Å3. The step size is 0.01 e/Å3 . Si atom is at the origin.
Fig. 5. (a) One dimensional peak MEM electron density profiles of Si and Mg atoms in Mg2 Si along [1 0 0, [1 1 0] and [1 1 1]. The Si atom is at the origin. (b) One dimensional low MEM electron density profiles of Si and Mg atoms showing the bonding electron densities in Mg2 Si along [1 0 0, [1 1 0] and [1 1 1]. The Si atom is at the origin.
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Table 5 The inter atomic distances of Mg2 Si obtained from PDF analysis. Atom pair
Si–Mg
Mg–Mg; Si–Si Mg–Mg; Si–Si
Inter-atomic distances from PDF analysis (Å) Observed
Calculated
2.94 5.36 6.90 8.06 9.38 10.26 11.98 4.36 13.38
2.88 5.32 7.00 8.10 9.36 10.26 11.98 4.40 13.40
between atoms lead to deviations in the cell parameter from that obtained only from the average Bragg positions. Since the deviation in the cell constant in PDF analysis is very small compared to that reported, it is obvious that the there are not much local distortions in the structure of the pure Mg2 Si, except those resulting from bonding interactions between atoms. The Rietveld refinement [9] gives the thermal vibration parameters of the individual atoms which are large for both the atoms Mg and Si (Table 2). This also signifies the use of Mg2 Si as a thermoelectric material. The 3D MEM electron density distribution shows (Fig. 2) that the electronic charges in the unit cell are symmetric for both atoms. This figure has been constructed superimposing the structural parameters and the radii obtained from the present study and the MEM electron densities. No apparent difference in the size of the core regions is visible due to the comparatively same number of core electrons of Si and Mg. The 2D electron density map on (1 0 0) plane (Fig. 3) shows the electronic charge distribution of silicon atom. Elongations in the valence regions show attractive character of silicon with the neighbors. Small charge islands are visible at (1/4 1/4 1/4) and equivalents positions due to the inner Mg atoms. The (1 1 0) map (Fig. 4) clearly shows attractive character between Si and Mg atoms (slightly loose bound outer contour lines in Si and Mg) and covalent character between Mg atoms (sharing of charge clouds). The 1D profiles in Fig. 5a show enhanced peak heights for Si atom compared to Mg atom. Fig. 5b shows the low density 1D profiles. The bond strength between Si–Si bond amounts to 0.087 e/Å3 at a distance of 2.252 Å. The mid-bond electron density between Si and Mg atoms along [1 1 1] direction is 0.132 e/Å3 at 1.465 Å. The mid-bond electron density along [1 0 0] direction is found to be 0.033 e/Å3 at a distance of 3.185 Å from the origin. The profile electron density value along [1 1 1] direction is not too high or not too
Calculated inter-atomic distances (Å)
2.75 5.27 6.92 8.25 9.39 10.41 11.88 4.49 13.47
low. Hence, the bonding between Si and Mg atoms might be having a mixed character (ionic and covalent). Though the 1D electron density values are not too different from each other, one can make a comparison of values to elucidate type of bonding in conjunction with the 2D maps in addition. Fig. 6 shows PDF peaks representing the interaction between Si–Si, Mg–Mg and Si–Mg atomic pairs. PDF peaks up to a distance of 13 Å are suitably indexed as shown in Fig. 6. The PDF peak at 2.94 Å corresponds to the Si–Mg bond length along [1 1 1] direction. The mid-bond position turns out to be (2.94/2) 1.47 Å. This position is almost the same as the mid-bond position along [1 1 1] direction (1.465 Å), obtained from the MEM analysis. Similarly, from the MEM analysis along [1 1 0] direction, a small hump is seen at 2.251 Å. From the PDF analysis, the Si–Si bond length turns out to be 4.36 Å. Thus the Si–Si mid-bond position is 2.18 Å, which compares with the hump seen in MEM analysis at 2.252 Å. Thus all the PDF peaks can be suitably indexed. Hence, the MEM technique combined with PDF analysis can give much more information about the electron density and local structure of the system under consideration.
4. Conclusion The MEM and PDF analyses have been applied to the high temperature thermoelectric material Mg2 Si to extract the maximum possible information X-ray powder data. In the present study, the mid-bond electron density is very low along the bonding direction [1 1 1], indicating that the bonding in Mg2 Si is not sufficiently strong to prevent the movement of the electronic charges. This is an essential criterion for a thermoelectric material for charge movement. Moreover, the thermal vibration parameter of Mg atom, BMg is larger than that of Si atom. Hence, the thermoelectric properties in Mg2 Si depend more on the Mg atom than Si atom in Mg2 Si. These methodologies are highly useful when systems like Mg2 Si are doped with other metal atoms to increase its figure of merit and efficiency.
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Fig. 6. The fitted observed and calculated pair distribution functions (PDF) of Mg2 Si.
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