J. Phys. Chem. 1996, 100, 5863-5867
5863
Band Structure Calculations of Ferromagnetic Chromium Tellurides CrSiTe3 and CrGeTe3 B. Siberchicot* CEA, Centre d’Etudes de Limeil Valenton, 94195 VilleneuVe-St-Georges Cedex, France
S. Jobic, V. Carteaux, P. Gressier, and G. Ouvrard Institut des Mate´ riaux de Nantes, UMR 110 CNRS-UniVersite´ de Nantes, Laboratoire de Chimie des Solides, 2 rue de la Houssinie` re, 44072 Nantes Cedex 03, France ReceiVed: August 1, 1995; In Final Form: January 10, 1996X
Band structure calculations using the spin-polarized ASW method were performed on the title compounds. Nonmetallic behavior is attributed to spin-polarization-induced opening of a gap at the Fermi level. Curie temperatures were calculated and found to be in reasonable agreement with the real evolution of silicon and germanium derivatives.
1. Introduction For the quest of new compounds with interesting magnetic properties, we have embarked in the synthesis of chromium ternary chalcogenides. Our choice of chromium as a transition element comes from its d3 high-spin (HS) configuration state when surrounded by S, Se, or Te anions. Hence, we can expect strong cooperative magnetic behavior at low temperatures, inducing ferromagnetism, antiferromagnetism, or helimagnetism that originates from Cr-Cr interactions.1 In addition, from our knowledge of the lamellar transition element chalcogenophosphates MII2(P2)VIIIX6 (X ) S, Se),2 we first tried to obtain similar chromium derivatives corresponding to the CrIII cation in a tellurium environment with a marked bidimensional character of the magnetic interactions. We ended at the synthesis of CrSiTe3 and CrGeTe3 semiconductor compounds,3,4 which showed a ferromagnetic ordering below 32.9(5) and 61(1) K, respectively.3-5 In this paper we correlate the physical properties of these new ternary chromium chalcogenides with electronic band structure calculations, taking into account the exchange energy. We used the self-consistent ASW (augmented spherical waves) method6 in the framework of the local approximation of spin-density functional theory. 2. Structural Description and Physical Properties CrSiTe3 and CrGeTe3 are isostructural. Their lamellar character is evidenced in Figure 1. The structure is properly described in the centrosymmetric space group R3h with the hexagonal cell parameters a ) 6.7578(6) Å and c ) 20.665(3) Å for CrSiTe3 (V ) 817.3(2) Å3) and a ) 6.8275(4) Å and c ) 20.5619(9) Å for CrGeTe3 (V ) 830.1(1) Å3). The atomic positions are given in Table 1. Tellurium atoms build up a hexagonal close packing with an AB sequence in which chromium atoms and main group element pairs fill up the octahedral sites in every other cationic sheet along the c-axis perpendicular to the slabs. Two successive CrATe3 (A ) Si, Ge) sandwiches are then separated by a van der Waals gap with Te-Te interslab distances that range from 4.055(2) to 4.195(2) Å and from 3.99(1) to 4.21(1) Å for CrSiTe37 and CrGeTe3,4 respectively. Such anion-anion distances are in good agreement with the sum of the van der Waals radii (2 × 2.10 Å)8 and affirm the bidimensional character of the materials. * To whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, March 15, 1996.
0022-3654/96/20100-5863$12.00/0
Figure 1. Schematic perspective view of CrATe3 compounds. Black and white octahedra emphasize the sulfur environment of Cr atoms and A2 pairs, respectively.
From a discussion on interatomic distances and magnetic measurements,3,4,7 a charge balance (CrIII2)(A2)VITe-II6 can be deduced. Main distances and angles in CrSiTe3 and CrGeTe3 are gathered in Table 2. A schematic CrATe3 (A ) Si, Ge) slab is shown in Figure 2. Such a sandwich can be viewed as a layered CdI2 one in which one-third of the cation sites is occupied by main group element pairs (Si2 and Ge2) and two-thirds by transition metal atoms. Notice that owing to the refinement of the crystal structure in a hexagonal cell, two successive slabs are translated by a (2/3a + 1/3b + 1/3c) vector. Although magnetic ordering is a cooperative phenomenon that develops in the crystal, it is well known that the basis for magnetic interactions comes from the symmetry and the nature of the cation-cation and cation-anion-cation short-range associations.9 For edge-sharing CrX6 octahedra (X)chalcogen) and CrIII cations with a d3HS configuration, the direct t2g-t2g © 1996 American Chemical Society
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TABLE 1: Cell Parameters and Atomic Positions of CrSiTe3 and CrGeTe3 with the Sphere Radii Used in Band Structure Calculations (ES ) Empty Sphere) from References 4 and 7 CrSiTe3
CrGeTe3
space group: R3h a ) 6.7678(6) Å, c ) 20.665(3) Å Cr (6c): 0/0/0.333 94(20) rWS ) 2.329 75 au Si (6c): 0/0/0.054 88(31) rWS ) 2.329 75 au Te (18f): 0.669 02(15)/ -0.025 41(13)/0.249 34(4) rWS ) 3.881 36 au ES (6c): 0/0/0.1667 rWS ) 2.329 75 au ES (3b): 0/0.0.5 rWS ) 2.329 75 au
space group: R3h a ) 6.8275(4) Å, c ) 20.5619(9) Å Cr (6c): 0/0/0.3302(1) rWS ) 2.388 43 au Ge (6c): 0/0.0.0590(6) rWS ) 2.388 43 au Te (18f): 0.663(1)/-0.033(1)/ 0.2482(5) rWS ) 3.881 36 au ES (6c): 0/0/0.1667 rWS ) 2.388 43 au ES (3b): 0/0/0.5 rWS ) 2.388 43 au
TABLE 2: Main Distances in CrSiTe3 and CrGeTe3 at Room Temperature from References 4 and 7 CrSiTe3 dCr-Te
dA-A (Å) dA-Te (Å)
(CrTe6) Octahedra 2.779(3) (×3) 2.775(3) (×3)
CrGeTe3
TABLE 3: Magnetic Data and Conductivity in the Paramagnetic and Ferromagnetic States
2.819(8) (×3) 2.769(8) (×3)
(A2Te6) ethane-like group (A ) Si, Ge) 2.268(9) 2.43(2) 2.505(2) (×3) 2.546(8) (×3)
Te-Te Distances (Å) across the van der Waals gap 4.055(2) (×2) 4.195(2) across the slab 4.237(1) (×2) 3.952(1) intralayer 3.777(1) (×2) 3.734(2) (×2) 4.213(2) (×2)
Figure 2. Schematic projection of a CrATe3 slab along c-axis. Thick and thin circles represent sulfur atoms in a triangular array above and below the (Cr, A2) plane, respectively.
3.99(1) (×2) 4.21(1) 4.29(1) (×2) 3.96(1) 3.737(2) (×2) 3.81(1) (×2) 4.31(1) (×2)
Relevant for Magnetic Interactions dCr-Cr (Å) 3.9017(3) (×3) 3.9440(2) (×3) Cr-Te-Cr (deg) 89.26(8) 89.8(3)
interaction has an antiferromagnetic character while the superexchange interaction via chalcogen atoms is essentially ferromagnetic. We therefore expect antiferromagnetic total interactions when Cr-Cr distances are short, with ferromagnetic interactions being favored by an increase of Cr-Cr distances and the covalent character of the Cr-X bond.10 The choice of tellurium as the anion always leads to ferromagnetic behavior with chromium. The magnetic structures have been determined by neutron powder diffraction4,5 and show that the spins are all aligned along the c-axis (µ(CrIII) ) 2.73(5)µB at T ) 1.2 K and CrSiTe3 and µ(CrIII) ) 2.80(4)µB at T ) 5 K for CrGeTe3). To a first approximation, the difference in the ferromagnetic temperatures of the Si and Ge derivatives can be attributed to a size effect of the main group elements. This is reflected in the interatomic distances (Table 2): the Ge-Ge distances in CrGeTe3 (2.43(2) Å) are much larger than the Si-Si distances in CrSiTe3 (2.268(9) Å), as is the Ge-Te distance (2.546(8) Å) compared with the Si-Te distance (2.505(2) Å). Going from CrSiTe3 to CrGeTe3, the Cr-Cr distance increases (3.9017(3) and 3.9440(2) Å, respectively), while the Cr-Te-Cr angles do not significantly spread out from 90° in both compounds (89.26(8)° and 89.8(3)° in CrSiTe3 and CrGeTe3, respectively). Such a difference in the Cr-Cr distances would reduce the antiferromagnetic interactions in the germanium derivative, as compared with those of the silicon derivative. Hence, the total ferromagnetic behavior will be reinforced. Resistivity measurements have been performed on plateletlike crystals using the Montgomery method or the van der Pauw method from 500 to 10 K for CrGeTe34 and from room
CrSiTe3 (Tc ) 32.9(5) K) from high temperature to Tc
from Tc to low temperature
F ) 30 Ω cm at RT A ) 0.3 eV µCr ) 2.73(5)µB at 1.2K
CrGeTe3 (Tc ) 61(1) K) θF ≈ 60 K µeff ) 3.77µB F ) 0.02 Ω cm at RT A ) 0.2 eV µCr ) 2.92(5)µB at 5 K A ) 7 meV
temperature to 180 K for CrSiTe3.5 At room temperature, CrSiTe3 and CrGeTe3 present, respectively, a resistivity of 30 and 0.02 Ω cm. Above Tc, the thermal evolution of the resistivity can be fitted as F ∝ exp[A/kT], where A ) 0.3 eV for CrSiTe3 and 0.2 eV for CrGeTe3, suggesting a hopping conduction. At the ferromagnetic transition a drastic change in the curve slope is observed for CrGeTe3 (activation energy A falls from 0.2 eV to about 7 meV). This phenomenon can be related to either a change in the electronic conduction mechanism or the modifications of the cell parameters upon magnetic ordering (a ) 6.809(1) Å and c ) 20.444(3) Å at 270 K and a ) 6.8196(7) Å and c ) 20.371(3) Å at 5 K for CrGeTe3).4 Because of the much higher resistivity at low temperature, a similar experiment has not been performed for CrSiTe3. Table 3 sums up the experimental results on CrSiTe3 and CrGeTe3. Extended Hu¨ckel semiempirical calculations were already performed on these compounds4 as well as on similar materials.11 Despite their efficiency in bond characterization, they cannot be used for the study of magnetic properties because of their nonspin-dependent approach. However, the ASW method is well suited for the study of magnetic properties of materials from the point of view of their electronic structures. It is interesting to perform such calculations on well-characterized isostructural compounds in order to shed some light on the relationship among structure, properties, and the evolution of these. 3. Computational Details The ASW method uses the atomic sphere approximation (ASA), i.e., for each atom a sphere radius is chosen such that the sum of the volumes of all overlapping spheres (WignerSeitz spheres) equals the unit cell volume. Within WignerSeitz spheres the potential is spherical. Although the structures are rather compact, empty spheres (ES) were added in the van
CrSiTe3 and CrGeTe3
J. Phys. Chem., Vol. 100, No. 14, 1996 5865
der Waals vacancies in order to represent the 2D character of the crystals (Table 1). The following ASW basis functions were chosen for the different atomic species:
Cr: 4s, 4p, 3d Si: 3s, 3p, 3d Ge: 4s, 4p, 4d Te: 5s, 5p, 5d ES: 1s, 2p For each atom, (lmax + 1) functions were included in the internal summation of the three-center contributions to the matrix elements. The charges associated with (lmax + 1) correspond to the residues from all higher l states nonexplicitly accounted for. This is due to the fact that the ASW basis functions set is limited.6 A spin-polarized calculation allows the simultaneous determination of two band structures, spin up (v) and spin down (V), with a possible charge transfer between the two spin systems. The potentials seen by majority and minority spins are different. As a matter of fact, exchange and correlation energies lower the majority spins’ energy and increase the minority spins’. The two calculations are coupled by their Fermi levels, which must be the same for the two systems. Spin polarization is taken into account in the Schro¨dinger equation in the following way:
r )0 {-∇2 + Vs - ci,s}ψi,s(b)
Figure 3. Site-decomposed densities of states of CrGeTe3.
(1)
Fv(b′) r + FV(b′) r 3 2Z Vs(b) r )r + µxc (Fs(b)) r (2) +2∫ d b′ r |b r - b| r
aligned with the c-axis of the structure. Scalar relativistic effects were included,12 and spin-orbit effects were neglected in our calculations.
where the charge density for spins s (v or V) is
4. Results of Calculations and Discussion
Fs(b) r )
r i,s (b) r ∑ ψ*i,s (b)ψ e i
(3)
F
and µxc is the exchange-correlation potential. For each atom, starting with a spin-independent potential calculated for Z electrons, a band structure is obtained. The spin-polarized calculation is initiated by allowing a different charge population for each spin direction. Energy bands are filled with Zv and ZV electrons (Zv * ZV and Zv + ZV ) Z) up to two different Fermi levels Fv and FV. Equation 3 now generates two charge densities Fv and FV and then two potentials Vv and VV (eq 2). The Schro¨dinger equation (eq 1) is then solved for these potentials, and two different band structures are obtained. Now, both systems are filled up to a unique Fermi energy as Zv + ZV ) Z. Two new spin-polarized charge densities are computed, and the process is going on to self-consistency. The magnetization is then given by
r - FV (b))d r 3 (b) r M ) ∫ (Fv (b)
(4)
The two compounds belong to the R3h space group, and calculations were performed with hexagonal axes. Atomic coordinates and Wigner-Seitz radii are given in Table 1. Note that used atomic positions and cell parameters were determined at room temperature. Electronic band structure calculations were performed for a colinear ferromagnetic arrangement of the Cr magnetic moments
The site-decomposed densities of states are presented for the two compounds in Figures 3 and 4. The valence density of states inside the -14 to -6 eV range is dominated by the main group element contribution and the Te 5s, 5p orbitals. However, in the vicinity of the Fermi level (between -6 and -2 eV) the contributions of the Cr d band and the Te 5p levels are predominant. Below -6 eV, the Te and A (Si, Ge) atomic contributions are superimposed. Such an observation indicates the strong covalency of the main group element tellurium bonds. By analogy to the transition metal phosphorus trichalcogenides MPX3 (M ) first row transition element; X ) S, Se), the CrATe3 series can be considered as salts of Cr3+ and [A2Te6]6ions. Hence, in a first and oversimplified view, the electronic structure of CrSiTe3 and CrGeTe3 can be approximated by 3d, 4s, and 4p metals bands superimposed on the s, p block bands of the [A2Te6]6- anions. In view of the high covalency of the A-A and A-Te bonds, the ethane-like entity [A2Te6]6- gives rise to sharp and localized peaks, which can be assimilated into molecular-like states as in an [A2Te6] cluster. Such features were already observed in CdPS313 and antiferromagnetic MPS3 (M ) Mn, Fe, Ni)14 containing P2 pairs. In this kind of crystal many of the low-energy crystalline states can be considered as molecular states slightly broadened by k-dispersion. Since chromium atoms are octahedrally coordinated, a splitting of the metal 3d block is expected with three bands deriving from the t2g block lying below two bands deriving from the eg levels. Such a phenomenon is observed in Figures 3 (top) and 4 (top) where the t2g and eg blocks are separated by a gap
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Figure 4. Site-decomposed densities of states of CrSiTe3.
TABLE 4: Magnetic Results of Band Structure Calculations
CrSiTe3 CrGeTe3
Cr (µB)
A (µB)
Te (µB)
ES (µB)
Mtot (µB)
Tc (K)
gap (eV)
3.086 3.041
0.010 0.023
-0.029 -0.019
-0.005 -0.005
3.000 3.000
25.2 130.7
0.04 0.06
for both derivatives. In addition to the effect of the ligand field, the magnetic exchange splits the Cr 3d states, which sit mainly below EF for the spin up direction and above EF for spin down. Exchange splittings are, respectively, 2.91 and 2.83 eV for CrSiTe3 and CrGeTe3. These values are very close to those of other tellurides CrTe, Cr2Te3, and Cr3Te4.15 Cr 3d states are strongly mixed with Te 5p states, leading to a broadening of chromium bands and a slight polarization of the Te moment, which is antiparallel to the Cr (Table 4). Similar results are observed in Cr2Te3 and in CdCr2S4.16 The main, striking feature induced by spin-polarization is the opening of a gap at the Fermi level. When nonspin-polarized calculations are performed (see the previously performed extended-Hu¨ckel calculations4), high density of states at Fermi level are observed. In a Stoner approach17 such results would lead to a product IN(EF) . 1 (where I is the Stoner integral) for chromium atoms, and then an intraband polarization of chromium would be expected. This is the case here, where chromium is in a III+ oxidation state so that the Fermi level is found between the t2g and eg bands, which are well separated by an octahedral crystal field. Insulating behavior is then observed for CrSiTe3 and CrGeTe3 with low gaps of, respectively, 0.04 and 0.06 eV (compared with the 0.3 and 0.2 eV measured activation energies above Tc and 7 meV for CrGeTe3 below Tc). Nevertheless, it may be pointed out that the gap values are not very meaningful because it is well-known that LDA underestimates gaps (see, for example, ref 18). The
mechanism responsible for the change in activation energy still remains to be understood. Calculated magnetic moments are presented in Table 4. The total magnetization is an integer (3.0µB per formula unit) and is mainly due to Cr, which carries a moment of 3.09µB for CrSiTe3 and 3.04µB for CrGeTe3. These values corroborate the III+ valence of chromium and agree well with experimental values (2.73(5)µB for CrSiTe3 and 2.92(5)µB for CrGeTe3). An indication of the ionicity of Cr-Te bonds could be given by the difference of magnetizations between Cr and Te. We obtain, from Table 4, ∆µ ) 3.115µB for CrSiTe3 and 3.060µB for CrGeTe3. The higher value of the former would point out the higher ionicity of the Cr-Te bond, if we admit that it reflects a larger separation between Cr-type bands and Te-type bands. The higher covalent character of the Cr-Te bonds in the Ge derivative leads to a reinforcement of the ferromagnetic interaction by super exchange in CrGeTe3, consistent with the higher Curie temperature observed in the tellurogermanate of chromium. Our ASW package19 allows us to simulate noncolinear arrangements of spins. In the framework of the Heisenberg model it is possible to determine exchange integrals by the calculation of energy differences between various magnetic structures and then by an ordering temperature. Although an accurate analysis of magnetic exchange requires calculations of incommensurate spin-spiral magnetic structures in order to obtain all exchange integrals,20 a rough Curie temperature can be estimated in simple structures. Rather reasonable estimates of Curie temperatures were thus obtained in metals and alloys.21,22 The elementary cell contains three cationic planes, each one bearing two chromium atoms. Energy differences have been calculated for magnetic configurations where chromium spins deviate by an angle θ from the c-axis. The magnetic configuration was chosen in order to maintain all the symmetries of the space group. Considering only the first nearest-neighbors (NN) of a spin this energy difference can be written as
∆E ) zNJS2 (1- cos θ) where z is the number of neighbors, J is the exchange integral, N is the number of magnetic atoms per unit cell, and S ) 3/2 for CrIII. For small values of θ, ∆E is a linear function of (1 - cos θ) and a value of J can be calculated. Since the anisotropy is supposedly high, we could use a 2D-Ising description of the magnetism. In the formalism of Diep,23 we obtain (from Diep’s eq 7)
kBTC )
JS2 0.6585
For CrSiTe3, a value J ) 6.72 K is calculated. This result is close to the experimental nearest-neighbours exchange integral J1 ) 9.2 K.24 We obtain Curie temperatures of 23 and 119 K for CrSiTe3 and CrGeTe3, to be compared with experimental values of 32.9(5) and 61(1) K,4,5 respectively. A greater number of k points of the first Brillouin zone would have made these results more accurate. This was not possible because of CPU time requirements. Nevertheless, the increase of Tc from silicate to germanate, although overestimated, is verified. Since the electronic structures are very close to each other, the explanation for different ordering temperatures lies in size and covalency effects as explained previously. Such an argument could be extended to the chalcogenide CrSnTe3, which has not been synthesized yet. Since the size
CrSiTe3 and CrGeTe3 of Sn2 pairs is greater than those of Si2 and Ge2, the Curie temperature should increase. Although this material is yet hypothetical, a similar theoretical calculation could be done assuming the same crystallographic structure. Acknowledgment. We acknowledge the participation of Dr. V. Zhukov at the beginning of this work and thank G. Zerah and H. T. Diep for fruitful discussions and for providing ref 23. References and Notes (1) Bertaut, E. F.; Roult, G.; Aleonard, R.; Pauthenet, R.; Chevreton, M.; Jansen, R. J. Phys. (Paris) 1964, 25, 582. (2) Brec, R. Solid State Ionics 1986, 22, 3. (3) Ouvrard, G.; Sandre´, E.; Brec, R. J. Solid State Chem. 1988, 73, 27. (4) Carteaux, V. Thesis, University of Nantes, Nantes, France, 1992. Carteaux, V.; Brunet, D.; Ovrard, G.; Andre´, G. J. Phys.: Condens. Matter 1995, 7, 69. (5) Carteaux, V.; Ouvrard, G.; Grenier, J. C.; Laligant, Y. J. Magn. Magn. Mater. 1991, 94, 127. (6) Williams, A. R.; Ku¨bler, J.; Gelatt, C. D., Jr. Phys. ReV. B. 1979, 19, 6094. (7) Marsh, R. E. J. Solid State Chem. 1988, 77, 190. (8) Shannon, R. D. Acta Crystallogr. 1976, A32, 751.
J. Phys. Chem., Vol. 100, No. 14, 1996 5867 (9) Goodenough, J. B. Magnetism and the Chemical Bond; J. Wiley and Sons, Intersciences: New York, 1963. (10) Colombet, P.; Danot, M. Solid State Commun. 1983, 45, 311. (11) Whangbo, M.-H.; Brec, R.; Ouvrard, G.; Rouxel, J. Inorg. Chem. 1985, 24, 2459. (12) Methfessel, M.; Ku¨bler, J. J. Phys. C: Solid State Phys. 1982, 12, 141. (13) Zhukov, V.; Boucher, F.; Alemany, P.; Evain, M.; Alvarez, S. Inorg. Chem. 1995, 34, 1159. (14) Zhukov, V.; Gressier, P.; Ouvrard, G. In preparation. (15) Dijkstra, J.; Weitering, H. H.; van Bruggen, C. F.; Haas, C.; de Groot, R. A. J. Phys.: Condens. Matter 1989, 1, 9141. (16) Siberchicot, B. IEEE Trans. Magn. 1993, 29, 3249. (17) Stoner, E. C. Proc. R. Soc. London 1938, 165, 372. Stoner, E. C. Proc. Roy. Soc. London 1939, 169, 3339. (18) Godby, R. W.; Schlu¨ter, M.; Sham, L. J. Phys. ReV. Lett. 1986, 56, 2415. (19) Ku¨bler, J.; Ho¨ck, M.; Sticht, J.; Williams, A. R. J. Appl. Phys. 1988, 63, 3482. (20) Luchini, M. U.; Heine, V. J. Phys.: Condens. Matter 1989, 1, 8961. (21) Uhl, M. Diploma thesis, TH Darmstadt, 1991, unpublished. (22) Uhl, M.; Siberchicot, B. J. Phys.: Condens. Matter 1995, 7, 4227. (23) Diep, H. T.; Debauche, M.; Giacomini, H. Phys. ReV. B 1991, 43, 8759. (24) Carteaux, V.; Moussa, F.; Spiesser, M. Europhys. Lett. 1995, 29, 257.
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