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Large structural and magnetic phase stability of VS and VSe
Journal of Alloys and Compounds Z&i (1997) Ihh-176
Using lhc full potential (LAPW) band structure method. a detailed investigation al the structural and magnetic phase stability of VS and VSr has been performed. In full accordance with experiment, the MnP-structure was found to be the equilibrium phaseforVS at low temperature. The changesin the electronicstructurethat accompanythe phase transitionfrom the orthorhombicMnP-structure to the hexagonal NiAs-structure is monitored using the dispersionrelations, the density of states wrve~ ar well as the electronic charge density distribution. FIXVSe in the NiAs-Struciure the magnetic phase diagram as a function of the hexagonal IattiFe parameters a and c has been determined. In agreement with experiment. the antiferromagnciic phase was found to be energeticallythe most favourabie one among the paw-. ferro- and anti-ferromagnetic phases. Kcywonir:
Vanadiumsulphide:Vanadiumselenide; Linear augmented planr wave method; Magnetic phase diagram
1. Introduction Transition-metal-chalcogen
compounds
show an
extremely wide variety in their physical properties depending strongly on the specific stoichiometry. In addition, in many casss structural or magnetic phase transitions occur upon varying the temperature, the or the composition, i.e. preparing non-stoichiometric samples. A very prominent example for this behaviour is the compound V’S,. Together with
pressure
several ckwely related transition-metal-cbalcogen systems,VS, has recently been reinvestigated expcrimentally in great detail by Lewis and Goodenough 111.For the atomic ratio x between 0.85 and 1.06 VS, takes the MnP-structure at room temperature, while for x outside this range. the NiAsstructure is adopted. With increasing temperature the range of x for which the MnP-structure is the stable one gets narrower, and above 903 K the NiAs-structure is adopted even for the stoichiometric compound. A deep understanding of the phase transitions occurring for VS, upon varying the composition or the temperature can be obtained only on the basis of a detailed knowledge of its electronic stmeture. Correspcmding band structure ~&culations have been done in the past by several groups [Z-S]. Owing to the ‘Concupondingauthor. c9xs-83881411$17.~ PII
0
s092s-a388(96)02480-2
broken Bloch symmetry, band structu e calculations for non-stoichiametric compounds are very dit%cult to perform and have been done in a rigorous way only for VS, very recently by using the Korringa-KohnPostoker band structure method in connection with .ix coherent potential approximation alloy theory (KKK-CPA] (91. Apart from this work, aon-stoichiometry was accounted for by means of the so-catled rigid-band model. As this model gives onfy a very crude estimate for the changes in the electronic structure upon changing the composition by shifting the Permi energy according to the average number of electrons, nearly all previous theoretical investigations of the phase stability of VS, were restricted to the stoichiometric case. Based on corresponding band stracture calculations, Liu et al. [5] discussed the Peierls lattice that takes the NiAa-structure. to the MnP-structure in terms of charge density wave instability. A quite general microscopic theory of the structural phase transition of compounds with NiAs-structure has been presented by Katoh and Moth&i [lo] using the concept of the generalized electronic susceptibility x(q,A). This quantity gives the change in the free energy proportional to IQ J* due to the linear electron-lattice interaction with Q,, the normal coordinate of a phonon with wave vector q and mode A. Apart from the electronic dispcrsjon relation E,(k)
1937 Ebevicr Scicncs SA. AU ri@a rCjdmd
the susceptibility x(q,A) is determined by the
electron
the strength of the coupling between two electronic states (xk) and (n’k’) caused by a displaccmcnt of the vth atom in the direction IX. Compared with these approaches, the characteristics of the structural phase transition of VS has been studied here by calculations of the total energy as a function of the lattice distortion using the highly accurate linear augmented plane wave (LAPW) method of band structure calculation in its full potential form. The structural phase transition of VS was explained by Silvestre el al. [4] by a relatively high density of states at the Fermi energy for the NiAs-structure that is lowered as a consequence of a Peierls distortion. Another way to avoid such an energetically unfavourable situation for a solid is to undergo a magnetic phase transition. This seems to be the case for the compound VSe in the NiAS-SrNCtUIe. A da&!ed theoretical investigation of the conditions for such a transition has been performed by Dijkstra et al. [II] for CrSe and other related compounds. In contrast to this. a more extended study is presented here for VSe. By calculating the total energy of the three competing phases with non- or para-, ferro- and anti-ferromagnetic spin ordering for a wide range of Ihe hexagonal lattice parameters 11 and c. the complete magnetic phase diagram of this system has been determined. The paper is organized as follows. The Section 2 Section 3 describe rhe crystal structures studied and the band struUduw method used for the various investigations. In Section 4 the results for the structural phase transition of VS are presented and discussed. Section 5 deals with the magnetic phase stability of VSe in a corresponding way. All results of this investigations are summarized at the end of this contribution. lattice
matrix
element
i,&,.
that
represents
2. SINctural p1operGes VS and VSe crystallize in the NiAs-structure (space group P6,lmmc). This structure, shown in Fig. 1. consists of hexagonal close packed layers occupied alternately by V- or cbalcogen atoms. As can LY seen from Fig. 1. the layers repeated along the c-axis follow the typical hexagonal stacking sequence with V in the octahedral holes [12]. Representative experimental lattice parameters taken from the literature are summarized in Table I. As can be seen. for VS and We rhe c/a-ratio is around 1.76 and 1.63 respectively. Compared with the ideal close pecked hexagonal structure. the lattice is strongly strewbed within the basal planes leading to a V-V distance (3.33 A for VS) within the planes Ihat is about
tb)
10% larger than that for adjacent V-layers (2.94 A for VS). The MnP-structure that is adopted by VS at low temperatures is ckly related to the NiAs&ructure. To demonstrate this, in addition 10 the proper hexagonal unit cell of the NiAs-structure. an alternative orthorhombic unit cell is shown in Fig. 1 containing four instead of only two formula units. For VS, the MnP-sIructLue (space group %M) derives from Lois Table
I
vs lead .vs (W VSe (exp) vse (the)
(A)
e (A)
E
3.332
5.860
1.761
3.152
Lzn
1.974
3.660
5950
L.626
3.918
1528
1.411
cln
The full-potential LAPW is based on the muffin-tin construction that replaces the Wigner-Seitz-spheres by non-overlapping muffin-tin spheres. Within these
spheres of radius rnr,n (a =V, S, or Se) the angular dependence of the electronic potential V,(r) is expanded in spherical harmonics Experiment nlcnrv
5.856 5.868
3.3tJ.5 3.214
T.83” 5.808
0.018 0.018
“.23Y 0.238
In the interstitial region between the spheres the pctcntial is represented by a plane wave expansion orthorhombic unit cell by a pronounced shift of the V-atoms along the orthorhombic c-axis with the shift in opposite directions for adjacent V-layers. In addition, B minor shift along the orthorhombic n-axis occurs. The resulting lattice parameters and atomic positions are given in Table 2. The primary effect of the shifts of the V-atoms wifh respectto the originalNiAs-structureis to destroy the triangular network of these atoms, to form in the basal planes more or less separated zig-zag chains running parallel to one0 another. As a consequence, the V-V distance (2.76A) within these chains is strongly reduced. being nearly 10% smaller than the V-V layer distance (X3 A) that is hardly affected compared wilh the NiAs-slruclure (2.94 A).
(21 with K* the reciprocal lattice vetiors. Owing to the greal flexibility and accuracy of the chosen expansion for the potential and electronic charge density, a very high numerical accuracy is achieved for the LAPW method. In contrast, a relatively high numerical efficiency is provided by the linear approximation for the energy dependence of the electronic wave function. This approximation leads to a generalized algebraic eigenvalue problem to determine for a given wave vector k the corresponding Bloch wave functions +,,&&,) and eigcnenirgies E,, with n the band index. Further technical details on the LAPW-method can be founci in Ref. [21] and the monograph of Singh
P-4.
3. Calculalionof the sleclronk
structure
During the first stage of this work the linear-muflintin-orbital (LMTO) method of band structure calculation [14,15] has been used to investigatethe electronic structure of VS and VSe. In its conventional form the LMTO makes use of the atomic sphere approximation (ASA). This geometric approximation implies as&bing to every atom in the system a region bounded by the so-called Wigner-S&z-sphere in such a way that the sum of the volumes of these spheres is identical to the volume of the unit cell. The eleclmnic potential
As is done nowadays in most electronic structure calculations for solids. the exchange and correlation effects have been treated within the spin density functional theory formalism. This means that the electronic potential V(r) in Eqs. (1;L) is composed of the electrostalic Coulomb or Hartree part and the exchange and correlation part V,,(r). To ensure the most reliable results for the lotal energy of the studied systems, the parametrization of L’_(r) according to the generalized gridien! approximation(CGA) given by Perdew et al. [24 has been used.
and the chargedistributionare taken to be spherically symmetric within these spheres, i.e. these quantities depend only on the distance from the nucleus. The ASA-construction has the drawback that there is no unambiguous way to fix the ratio of the sphere radii of the various atom types (here V and S or Se). Furlhermr)re, the various Wigner-S&x-spheres overlap but leave some uncovered region in between. This seems to he especially problematic for the rather open crystal structures investigated here (see Section 2 and Fig. 1). Although the results obtained using the LMTO method [16-H] turned out to be very reasonable and in line with previous theoretical work [19,2,6,20,4], the full potential version of the LAPW method of band StNctUre calculation 1211has been adopted IO avoid any problems probably caused by the AS.4.
4. Results and discussion lor VS
Based on the experimental lattice parameters given in Tables 1 and 2, the electronic structure of VS in the NiAsslructure as Hell as the MnP-struflure has been calculated in a self-consistent way using the LAPW method. Calculating the dispersion relation E(k) for the NiAs-structurc. shown in Fig. 2(a), the orthorhombic unit cell has been used to allow for a direct comparison with results obtained for the MnP-strocture. To yet the dominating character for the various bands displayed in Fig. 2(a) it is sufficient to look at the corresponding component-resolved density of
(-8
Fig. 1. Diupxdon n-lahon E_(t) Ar (a) VS in the NiAs-structure w&e an orthurhombic unit cell iscr Fig. l-middle). and lb) for VS
states (DOS) curves that are given in Fig. 3(a). From these DOS curves one finds ihat the narrow band complex at amund - 15 eV [not shown in Fig. 2(a) and Fig. 3(a)) is derived from atomic S 3s-states. As Fig. 3(a) shows, the bands from -8 to -2.5 eV are dominated by S p-states. Nevertheless, there is a pronounced mixing or hybridization with V derived states - roughly in the ratio 3~1. The bands in the region from -2.5 to 4eV however, are dominated by V. However, again there is a hybridization with S derived states with a weight of about 6-7~1. These ratios of course primarily reflect the number of available S 3p and V 3d-states. This is indeed confirmed by a furthex analysis of the DOS curves, i.e. a decompositiou according to their angular momentum character. For S one finds that between -8 and -2.5 eV the p-character is by far the most dominating. For the bands between -2.5 and 4 eV there is also some s-character contributing that gets even more important for the bands above 4 eV Analogously, the d-character dmni?ates ? in the case of V. Only for the lcw eoeTgy region
to - 2.5 eV) are there some E and pcontributions thst get stronger and of appreciable weight for the bands above 4 eV. Obviously, ail these findings are completely in line wirh the expectations based on a naive molecular orbital (MO) picture. Starting from atomic S 3s-levels that are much lower in energy than the V 3plevek, one expects MO fevek that have dominating S and V character respectively. with a corresponding separation with respect to theii energy. Owing to the solid state, the various MO-sublevels are spread over a wide range of energy (here from -8 to -2SeVand -2.5 fo 4 eV respectively). The electronic simclure of VS represented by Fig. 2(a) and Fig. 3(a) is in full agreemen tith the corresponding experimental Epectroacopic investigations. For example, rhe XPS measurements of Franzen and Sawatzky [24] probing the occupied part of the DOS curves gave a broad and rather featureless band complex in the range of 9 to 2.5 eV binding energy. A second one. ranging from 2.5 eV binding energy to the emission threshold. i.e. the Fermi energy, was separated by a very pronounced dip in the XPS spectrum. This spectrum could be reprcduced recently in a very satisfying way cm the basis of the corresponding component- and angular-momentum-resolved DOS curves of VS taking properly into account the relevant XPS matrix elements [25]. Furthermore. one should mention that qualitatively, similar results were also found using the less rigorous extended Hiickel methodwithin the tight binding approximation [4]. However, the main feature of the results prexnted in Fig. 2ta) and Fig. 3(a) is in full accordance with our previous work using the LMTOmethod, as well as with that of other workers uing ab i&o band snwcture methods [19,2,6,20]. However. the present resuhs can be rendered more accurate because of the use of a full potential band structure method. while previous investigations of the electronic structure of transhiin metal chakogen compounds used the atomic sphere or the simple muffin-tin approximation. This aspect seems to be especially important for the total energy cakulations presented below. Comparing the dispersion relation E(k) for the MnRstructure (Fig. Z(b)) with that for the NiAs-
structwe (Fig. 2(a) - as mentionedfor cakulatingthe latter, the iarger unit cell ‘of the MnP-shucture has been used to allow for this direct com@sun) a number of pronounced changes can be found. First of all the reduced symmetry of the I&P-structure corn-pared with the NiAs-structure is reflected by lifting of degewracies as well as replacementof band crossin.@ by avoiding of tmods. As expected, ihis is most pronounced for the energy region - 2.5 to 4 eV daminated by V, because only these atoms change their relative posilimss within the unit cell signiitly.
a Pig,. 3. Total and compuncnl-resulvrd DOSn(E)for (a) VS in the NiAsstructure. The full line gives the total DOS, while the dashed liner give the V- and S-contribution nspectivcly. Part (b) shows the results lor VS in the MnP-structure.
Nevertheless, there are also remarkable changes found for the S-dominated energy region (-8 to -2.5 cV). For example, the degeneracies marked with A and B, at the r- end X-points respectively. are lifted, while the band crossings marked with C and D are replaced by band avoidings. As noticed, all the-e changes are dictated by the lowering in symmetry and can be followed in detail by setting up a corresponding Walsh diagram [4]. However, the most important point to note is that the changes arc not restricted to the V-dominated energy region or even to the states in the vicinity of the Fermi energy. From the impact of the lattice distortion that takes the NiAs-structure to the MnP-structure on the dispersion relation E(k), it is obvious that no dramatic changes have to be expected for the corresponding DOS curves. Fig. 3(a) and Fig. 3(b) indeed demonstrate that the overall features of the DOS curve, discussed above for the NiAs-structure are. not altered by the lattice distortion. A further diiussion of the dispersion relations and the DOS curves till be given below in context with the relative stability of the both structures investigated.
4.2.
Hecrmnic
charge
density disfribtiion
The use of the full potential version of the LAPW method g&es a direct and reliable access to the
and dotted
electronic charge density distribution in solids. Corresponding results for the total charge density of VS in the 14’4s-structure : as well as in the MnP-structure are given m Fig. 4 and Fig. 5. where the NiAs-structure is again represented using the orthorhombic unit cell of the MnP-structure with lower symmetry. In Fig. 4 the density distribution in the (OlO)-plane (see Fig. 1) is shown for both structures. This plane is perpendicu!ar to the V-layers and contains two S-atoms per unit cell. Owing to the symmetry, only half of the unit cell is shown in Fig. 4, with V-atoms at the comers of the displayed rectangle and a S-atom in the middle of its left half. As can be seen, both structures are quite open with a rather wide region of low electronic density right to the S-atom. The mirror symmetry present for the NiAs-structure is obviously destroyed by the lattice distortion, which leads to a further decrease of the minimum charge density in the open interstitial region by about 10%. A slight decrease in the charge density can also be seen between the S-
atom and WV-atom that is shifted away (in the lower left comer of Fig. 4(b)). For the approaching V-atom (upper left comer) the interatomic charge density increases. In addition, the anisotropic charge density distribution of the S-atoms is slightly reoriented towards this V-atom due to the atomic shifts. QuaI& tatively. these features can be interpreted in terms of weakened and enhanced, respectively, V-S bonding for the two neighbouring V-atoms of the S-atoms. As mentioned above, the shiis of the V-atoms when
going from the NiAs-structure to the MnP-structure are much more pronounced within the V-piancs than perpendicular to them. While the charge rearrangement within the (010)-planes was found to he relatively small {Fig. 4). the impact of the lattice distortion for the V-planes. i.e. the (LOO)-planes. is accordingly much stronger for that reason. For the NiAs-structure the charge density of course reflects the perfect triangular arrangement of the V-atoms (see Figs. 1 and 5(a)). This highly symmetric charge distribudon is sirongly distorted by the formation of the V zig-zag chains, as can be seen in Fig. 5(b). As for the (010)~plane. the atomic rearrangement reduces the minimum charge density in the enlarged interstitial region (middle of the left half of Fig. 5(b)) by around 10%. However, compared with the (010)plane, its absolute value is still an order of magnitude higher - primarily because of the much higher packing density. As a consequence of the formadon of the V zig-zag chains, the interatomic charge density increases for the approaching V-atoms by about 30% (right half of Fig. 5(b)). Again, this can qualitatively be interpreted as a strengthening of the bonding for the V-atoms within the chains.
With the detailed charge density distribution available. it is ot course tempting 10 determine the corresponding atomic charges and from that the charge transfer. This would allow one lo specify the ionic character of the bonding within the solid as well as to dclerminc - at least in a formal way - ihe valency of both the components. However. owing to the geomclrical representation of charge density. in analogy with Eqs. (1.2).within the LAPW formalism. it is, in the first instance. only possible to calculate the charges in the muffin-tin spheres. These charges can be ascribed approximately to the corresponding atoms occupying that sphere, while the interstitial chars remains unrelated to a specific atom. Table 3 gives these muffin-tin charges of Vand S inVS for the Niisand MnP-structures. First of all, one notes that the muffin-tin charges Q, are nearly identical for both structures, implying that the phase transition gives rise primarily to a charge rearrangemen within the ittterstitiai region. For V and S the, charges Q, are below
the atomic number by 1.90 and 1.43 respectively. These numbers already give some indication for the proper atom-related charges. However. to use the muffin-tin charges Q, as a first estimate for these quantities, the muflin-tin radii r,,, should not be i&d to the same vahte for V and S, as done here. The charge density distribuGon shovm in Fig. 4 suggests that using the position of the chargedensityminimum
paramctcrs arc in rather satisfying agreemenl with the experimental data. One of the reasons for the deviation from experiment - 5% and 6% for LXand c
in between the neighbouring V- and S-atoms would be a much more appropriale way lo fix the muffin-tin radii, As a consequence of this improved choice, the resulting muffin-tin charges would more strongly rcfleet the expected ionic charge distribution among V and S. However, the high interstitial charge Qu makes clear that even this procedure would also lead to only a rather crude estimate for the proper atom-related charges. A more satisfying and unambiguous way to determine atomic charges in a solid from the given electronic charge densiry distribution is supplied by the transfer of the concept of an atom in a molecule of Bader et al. [26] to the solid state. First steps in this direction have bern done recently leading to atomic shapes that are very similar to what one expects from a Wigner-Seitz-construction [27]. Further numerical developments will allow for an unambiguous determination of atomic charges in a solid and, based on thal. a corresponding discussion of the various abovc-mentioned aspects.
4.3. Gromrrry
optimizntion
respectively - is that the experimental data have been determined at room temperature. Furthermore. one has to mention that it is obviously quite hard to produce perfect stoichiometric samples (see also below). To perform corresponding calculations for VS in lhc MnP-structure does not seem to be very reasonable, because in this case one would have to optimize four parameters {see Table 2). For this reason. the lattice distortion that takes the NiAs-structure to the MnPstructure has been used as some kind of reaction coordinate. This means that the total energy has been calculated as a function of a dimensionless parameter I that represents the shift of the V-atoms. Here I = 0 corresponds to no shift at all, i.e. the NiAsatru,ture is adopted, while I = 1 corresponds to a shift according to the experimental data for the MnP-structura.Varying t. the lattice parameters a and c have been varied linearly in parallel; that means, for t = 0 and 1 = 1 the parameters for lhe NiAs-structure and the MnP-structure respectively have been assumed. Fig. 6 shows corresponding results for the lotal energy as a function of the distortion parameter 1. First of all one notes that the MnP-structure is found - in full agreement with experiment - to he the more stable phase. The lattice distortion predicted by the calculation (t = 1.08) is only slightly larger than the experimental one corresponding to I = 1 (see also Table 2). As one can see, the total energy I$) curve is obviously not symmetric around r = 0. This is simply caused by the fact that the lattice parameters a and c have been varied in parallel with t as described above. It is interesting to note that
and phase sfuhility
Calculating the total energy of a solid as a function of its lattice paramelers allows one -. among various physical properties - to determine its theoretical equilibrium lattice parameters. In the past it turned out that it is quite crucial for these types of calculation to apply so-called gradient corrections within the density functional formalism [ZS]. For this reason. a corresponding parametrization for the exchange and correlation energy and potential. respectively, has been used [23]. If. for a given system, there are several lattice parameters and atomic positions within the unit cell to be optimized, the corresponding calculations are quite computer time consuming or even impossible. For VS in the NiAs-structure, only the two lattice parameters (I and E have to be optimized by minimization of the total energy. For this purpose the total energy as a function of II and c is approximated by a paraboloid that has been fitted to nine tabulated data points. As can be seen from Table 1, the theoretical lattice
-Ior@175
1
i
the NiAs-structure,
corresponding lo I = 0. is situated at a local minimum of the E(t)-curve. i.e. it realizes a met&able state. This sheds some doubt on the applicability of the approach of Katoh and Mothuki [lo] mentioned above. Altogether. the energetic situation is obviously very similar to that for the thermal activated phase transition from the hexagonal w-phase of Zr to its high temperarure b.c.c. p-phase [ZY]. In complete analogy with this rather simple case, deeper insight into the characteristics of the NiAs-MnP phase transition could be obtained from an experimenral and theoretical study of the relevantpart of the phonon spectrum. In a previous investigation. using the extended Hiickel method [4], the phase transition of VS has heen explained by the creation of a dip in the DOS curve around the Fermi energy EF for the MnPstructure. Comparing the DOS curves n(E) in Fig. 3. one finds that the opposite happens. i.e. the DOS n(E,) at Ef increases from 0.50 to 0.55 (st.leVspinFU). This finding seems to be in cooflicl with V-Knight shift measurements that revealed a decrease of the shift when ‘going from the N&-structure to the MnP-structure [30]. However, one has to note that the Knight shift is composed of several contributions with only some of them directly related to a part of the I-resolved DOS at E,. In line with the mentioned experiment. the direct Fermi-contact contribution one of the most important ones - can be expected to decrease during the phase transition. This prediction is based on the finding that, in contrast to the total DOS. the relevant V s-DOS decreases by more than 30%. As is obvious from the dispersion relations. and also from the charge density distributions. the phase transition of VS primarily involves the V-dominalrd part of the band structure in the energy range of - 2.5 to 4 eV. A decomposition of the V d-DOS according to the symmetry character shows that its part that is related to the d-orbitals lying within the basal plane are most affected by the lattice distonion. It seems that the shift of the corresponding occupied bands towards higher binding energy due to the stronger overlap of the in-plane V d-orbitals delivers the most substantial contribution to fhe stabilization of the MnP-structure. This could be confbmed by estimating the change in total energy upon the phase transition by using :he so-called force theorem 1311. Such calculations, that are in progress at the moment. could also tell whether the contribution of the S-dominated states between -8 and -2.5 eV is negligible or not.
As one expects,
for VS in the Nis-structure.
replacing the S-atoms by Se-atoms gives rise to a very similar band structure. This can be seen by comparing the DOS curve for VSe in ihe paramagnetic state in Fig. 7(a) with the comqonding
curve for VS in Fig.
3(a). The main difference between both curves is that the chalcogen-dominated s&band complex is 3t higher energies for VSe (-7 to -2eV) and that tbe Vsubband is less spread in energy. This means that the E(k) bands in the vicinity of the Fermi energy E, are rather dispersionless, i.e. flat, giving rise to a very high DOS at EF. This property can be traced back to the lattice expansion of VSe with respect to VS due to the larger chalcogen atoms. This lattice expansion, how-
ever. primarily takes place within the basal plane. increasing the lattice parameter a by about 10% while
c increases only slightly. The relatively high DOS at Er for VSc in the paramagnetic state points out that 11 I system might be in an energetically Iess favourable stale which could be removed hy a structural or magnetic phase transition. Guided by the cxpcrimental properties of VSe. its electronic structure was investigated for the ferro- and anti-ferromagnetic state as well for that reason. The resulting spin-resolved DOS curves for VSe in the ferromagnetic stale are shown in Fig. 7(b). At first sight the DOS curves for the ferromagnetic state seem to be created from those for the paramagnetic one by rigidly shifting the curves for different spin character against one another. F’urther inspection, however. shows that fine details of the DOS curves change as well. [n addition, one notices that the average exchange splitting AE,, of ibe bands is more pronounced for the V-dominated than for the S-dominated band complex. While for the V-region one finds AE,, = 1.2eV the splitting within the S-region is at most 0.35eV. As a consequence of this different exchange splitting. the overlap of both the. band complexes is
slightly increased for the majority spin system, while a band gap opens for the minority spin system. According to the exchange splitting and the DOS curves, the resulting moments for V and Se are also quite different: 1.83 cc, and -0.025 & respectively. The magnitude and sign of the Se-moment inVSe can be traced back to the stronger hybridization for the Se-dominated band complex with majority spin character. compared with that for minority spin character (see Fig. 7(b)). Using a simple MO-scheme, this in turn can he explained by the relative positions of the relevant atomic V 3d- and Se 4p-levels on the energy axis,
taking the exchange splitting into account. Here. one should notethat the calculated moment of SC in VSe is quite different from that found by Dijkstra et al. [Il] for G-Se. These authors find 3.4 h for the Cr-moment and 0.24 k for the Se-moment. The big difference of the latter value compared with that for Se in VSe given
here seems primarily to be of technical origin. While for the present work a full potential method has been used and the moment evaluated within the Se muffintin sphere, the investigation of the system CrSe has been performed using the ASA-approximation with the moment evaiuated within the Wigner-Seitz-sphere of Se. To perform baud structure calculations for an antiferromagnetically ordered state of VSe the magnetic contiguration found by neutron scattering in the case of CrSb (321 has been assumed (the actual experimental spin configuration of VSe in the anti-ferromagnetic state has not yet been investigated). This configuration consists of V-layers in the basal plane with all atoms having the same magnetic moment and oriented along
the same direction. The adjacent V-layers again Scar the same moments: they are. however. oriented in the opposite direction. Owing to this spezihc. highly symmetric spin configuration. there are no induced spin magnetic moments present on the Se-sites. The resulting magnetic unit cell is therefore identical to the crystallographic one. The DOS curves that emerge from that anti-ferromagnetic configuration are shown in Fig. 7(c) for the sublattice wnsisting of the A- and B-layers. Those for the second sublattice, built up by the C- and D-layers, are the same but with spin up and down exchanged. In contrast to the ferromagnetic spin configuration. the anti-ferromagnetic has a very strong influence on the V-dominated part of the DOS CUNB. This can be understood from the fact that for a given spin orienta-
tion the potentials on the neighbouring V-layers differ strongly because of the opposite exchange splitting At?,;,. Taking the difference in position of the main peaks as a measure for A&,, one finds AE,, = 2.4 eV. This value is higher than for the ferromagnetic state, is found to be and accordingly the V-moment [ 1.97 b)
somewhat higher.
For VSe. in the past. very different magnetic proper&s have been found experimentally depending on the preparation and, with this, on the actual composition of the samples (see below). Indeed. within a previous investigation of the electronic structure al VSe using the LMTO-method [Yj it was found that, for a certain range of the c/o-ratio, the generalized Stoner criterion was pointing to a ferromagnetic instability. This means that for the paramagnetic state the condition f&E,:). I >
I
(3)
is fultilled. Here the global exchange
parameter
I is
given by
with n,,(Er) DOS for the exchange or curve in Fig.
the angular momentum, i.e. /-resolved component o (a =V. Se) and /_,,,. the Stoner integrals [33]. From the DOS
7(a) it is obvious that if the condition in Eq. (3) is fulfilled, tXs is only due to the d-electrons of V via nv&). As is obvious from the comparison of VS and VSe, this contribution to the DOS is very sensitive CD the V-V distances. Owing to the very different V-V distance witbin the basal plane (3.7 A) and along the c-axis [XC A), one can conclude that the dispersion of the V-dominated bands is strongest for bands derived from orhitals oriented along the c-axis.
Indeed, a symmetry analysis of the DOS-curve for the paramagnetic state of We (Fig. 7(a)) reveals &at the d+ontribution is spread over the whole V-dominated energy range (-2 to 3 eV). In contrast IO this. the d,,and d,,-derived bands give rise to the pronounced peak in the DOS curve al 1 eV. The d-orbitals oriented within the basal plane, however, are responsible for the sharp peak at the Fermi energy. From this analysis one expects that the product In(&) in Eq. (3) is primarily dependent on the lattice parameter 0. The reason for this is that decreasing or increasing u will respectively increase or decrease the overlap of the in-plane oriented d-orbitals. Thii in turn will decrease or increase the corresponding bandwidth and. as a consequence. will increarrr:or decrease the DOS at E,. The lattice parameter c, however, may be expected lo have a much smaller influence on the magnetic prop erties according to the analysis of the DOS curve hr the paramagnetic state of VSe. In line with this is the previous finding that decrwding the c/a-ratio leads to an increased tendency towards spontaneous magnetism according 10 the Stoner criterion [91. In the form given above (Eqs. (3.4)). this criterion first of ail concerns ferromagnetic ordering and. therefore. does not rule out an anti-ferromagnetic ordering. Which type of ordering is energetically more favourable can only be decided by application of the so-called force theorem 1321 or - in a more reliable way - by performing total energy calculations. This approach has been used here for a detailed imestigation of the interrelation of the magnetic properties of VSe and its lattice parameters by calculating the corresponding phase diagram Cor the para-, ferro- and anti-ferromagnetic states. For this purpose. it turned out in the past to be very important to use gradient
able. One can easily recognize from that figure that the lattice parameter o is much more importan for the magnetic properties of VSe as the parameter c. investigation of the ferromagnetic phase in additiou to the para- and anti-ferromagw ticoaesdoesnot introduce a new global minimum w the energy surface, jusl a Io4 one. Thii means chatthe antiferromagnetic state is left as the energetically most favourabte one. However, this does not nde out the exstence of a more complicated anti-ferromagnetic or non-collinear spin structure that is still lower in energy. In fact the spin structure reported for C&E is quite complicated [38]. As can be seen, the ferrmnagnetic phase tamed out to separate the pan- from the anti-ferromagnetic phase for the investigated RngC of lattice parameters_ As for the para- aad a&fe~netic cases, all phase bormdaries are P~~IIIU& facd by the lateral parameter a. This result s.tro& dev&es from the expectations of Kamimura 1391. based on a survey of fzxperimental data of the magoctie prop6ea of a number of tmn&c4t metal-dmtmgcn compormcir with the Niis-smrture, this author cnacl& that it is primarily the lattiee parameter c that determines the magnetic state and type of ordering. However, ibis conclusionisbynomeansforccdbythcprcsentdata compilation. In addition to tha theoretical phase boun&r& several oxperima~tal data pGts have been addad to Fig. 8. The rhrae points to the rig&, that llurty coincide with each otlwr, represent rrmpkvtht were reported to be -tic. The tw0 dau points to tbe right belong to pumrgvticsampkzLTbewida spread of these experimental data make ckar that
sample imperfections have a lattice
parameters
as well
85 on
strongimpact on the Ihe
magnetic
References
rcatures
of VSe. Nevertheless. they are in full accordance with the central theoretical findings: a lateral lattice expansion takes VSe from the para- tn the anti-ferrmnagnetic state. while an expansion along lhe c-axis has much less influence on the magnetic slate.
4 Summary
The structural phase transition of VS from the low temperature MnP-structure to the high temperature NiAs-structure has been investigated by detailed calculations of the electronic structure for both strttctures. using the futl potential LAPW band structure
method. Calculation of the total energy as a function of the lattice distortion revealed - in full accordance with experiment - the MnP-structure as the stable one. The NiAs-structure, however, was found to be situated in a local energy minimum. The changes in the electronic properties upon the phase transition have been monhorcd in reciprocal space, in the energy domain and in real space by means of the dispersion relations, the DOS curves and the charge distribution. From these results it was concluded tbat the phase stability of the MnP-structure is primarily due to a strengthening of the V-V bonding within the basal plane via their d-orbitals. Further investigations to corroborate this interpretation are in Progress at the momenl. For We. in the NiAs-structure, the magnetic phase diagram has been investigated by performing total energy calculations. Here the anti-ferromagnetic state has been found, in agreement with experiment, to be the mast stable one. However, the other magnetic states were found to provide near-by local minima in energy; this explains why in the past, depending on ;he specific sample preparation. VSe was found experimentally to be para- or anti-ferromagnetic. The calculated phase boundaries that separate the para-. ferro- and anti-ferromagnetic phases from one another were found to be primarily determined by the lateral lattice parameter a. This result is in variance to earlier expectations but can be explained by the fact that the magnetic instability of paramagnetic VSe is due to the high DOS at the Fermi energy produced by the dorbit& oriented within the basal plane.
Financial support provided by the Deutsche Forschungxgemeinschaft within the prograrnme Ungewiibnliche Valenzzustiinde in Festkiirpem for this work is gratefully acknowledged.
III L.H. Lewis (IW) M.
and J-B. Goodenough.
[2] J. Nakaharn. H. Franrcn (1982) 411x(,.
J. Solid Slaw Chum.. 114
and D.K. Mircmw.
J. Chuar. Phys.,
76
IS] K.J. Kematick and C.E. Myers. ln~rg. CIzcm.. 31 (199292)3568. 141 J. Silvestre. W. Tremel and R. Hoffmann. 1. 1.r.wCummun Met.. II6 (19nh) 113. IS] S.H. Liu. W.B. England and H.W. Myron. S&i Starr Cwwwm.. 14 (1974) lW3. 161 K. Motizuki. 49s.
i9j
K. Katah
and A. Yanare,
1. Phjr.
H. Ebert and W. Benssh. in Picparatlon.
1161 W. Bensch 1171 W. Bensch [IS] W. Bensch
C:, IP (1986)
’
‘-
’
and H. Ebert. S&d Srme Conmun.. # (1994) 235. zmd H. Ebert. Solid St& Conzmufz., 92 (1994) 639. et al., J. Phys. C:+ Pp (1995) 33326.
[I91 W.B. Englsnd. (1975) 37a.
S.H. Liu and
H.W. Myron.1. Chem. Php., 60
[ZOO K. Katoh. A. Yanase and K. Molizuki, J. Mogn. Megn. Morrr.. W-S? (19%) 9.59. (211 F! Blaha. I(. S&wan. P. Dutek and R. Augwty”. WIEH~S. Technical University 01 Vienna. 1995. (Improved and “dated Unixversionof the original copyrighted w&ade. P. Bl;ha. K. khwarz. P. Sorantin and S. B. Mckev. Comour. Phvs. Cornnwrl.. 59 (1999) 399.
,.
.
,
1221D. Sing. P/me Waver. Pscudopwenrirrls and the LAPW Meshad, Kluwer Academic. Amsterdam. 1994. [zil J.P. Perdew et al., Phys. Rev. B. 46 (1992) 6671. 1241 H.F. Fnnten and G.A. Sawatzky. 1. Solid Smrc Chum.. 15 (1975) 229. 12.51 H. Ebert, in preparation. [261 RF.W. Bader, T.T.N. Ilang and Y. Tal. Rep. Prq. Phys.. 44 (1981) 893. [271 H. Eberl and A. Vcres. ,,I preparation. IZSIP. Dulek. P. B1aha.V. Sliwko and K. khwarz. fhvs. Rev. 8. 49 (1994) 10170. 14 K.-M. Ho. C. Fu and 8. Harmon, firs. Rev. 8.29 (1984) 1575. I3Ltl H.F. Franzen. D.M. Strwhan and R.G. Barnes. 1. Solid Swc CAtm.. 7 (1973) 374. 1311 A.R. Mackintosh anf O.K. Andsrssn. The slectmnic structure of transition metals. in M. Svrin&rd fed.). El=crronr at fhp Fermi Surface,Cambridge U&&y P&‘Cambridgc, 1960. [32] A.I. Snow, Phys. Rev.. 85 (1952) 34.5. [33l J.P. Janak. Phys. Rev. 6. I6 (1977) 255. I341 D. Singh et al.. PJtys. Rev. B. 44 (1991) 77Ol. 1351 R 8seoo. 0. Jepren and 0. Gunnaraon. Phya. Rev. B. dU (lsa9) 1997. [=] E. Hwhek and W. Klemm. 2. Anorg. Al/g. Chm., 24.7 (1939) 131) :: Elensch. tmpubliahed results. [38] L.hl. Corlis, N. Elliot. J.M. Hastings 1.72 (1961) 14@2.
and R.L. Sass. P#tys. Rev..
Using lhc full potential (LAPW) band structure method. a detailed investigation al the structural and magnetic phase stability of VS and VSr has been performed. In full accordance with experiment, the MnP-structure was found to be the equilibrium phaseforVS at low temperature. The changesin the electronicstructurethat accompanythe phase transitionfrom the orthorhombicMnP-structure to the hexagonal NiAs-structure is monitored using the dispersionrelations, the density of states wrve~ ar well as the electronic charge density distribution. FIXVSe in the NiAs-Struciure the magnetic phase diagram as a function of the hexagonal IattiFe parameters a and c has been determined. In agreement with experiment. the antiferromagnciic phase was found to be energeticallythe most favourabie one among the paw-. ferro- and anti-ferromagnetic phases. Kcywonir:
Vanadiumsulphide:Vanadiumselenide; Linear augmented planr wave method; Magnetic phase diagram
1. Introduction Transition-metal-chalcogen
compounds
show an
extremely wide variety in their physical properties depending strongly on the specific stoichiometry. In addition, in many casss structural or magnetic phase transitions occur upon varying the temperature, the or the composition, i.e. preparing non-stoichiometric samples. A very prominent example for this behaviour is the compound V’S,. Together with
pressure
several ckwely related transition-metal-cbalcogen systems,VS, has recently been reinvestigated expcrimentally in great detail by Lewis and Goodenough 111.For the atomic ratio x between 0.85 and 1.06 VS, takes the MnP-structure at room temperature, while for x outside this range. the NiAsstructure is adopted. With increasing temperature the range of x for which the MnP-structure is the stable one gets narrower, and above 903 K the NiAs-structure is adopted even for the stoichiometric compound. A deep understanding of the phase transitions occurring for VS, upon varying the composition or the temperature can be obtained only on the basis of a detailed knowledge of its electronic stmeture. Correspcmding band structure ~&culations have been done in the past by several groups [Z-S]. Owing to the ‘Concupondingauthor. c9xs-83881411$17.~ PII
0
s092s-a388(96)02480-2
broken Bloch symmetry, band structu e calculations for non-stoichiametric compounds are very dit%cult to perform and have been done in a rigorous way only for VS, very recently by using the Korringa-KohnPostoker band structure method in connection with .ix coherent potential approximation alloy theory (KKK-CPA] (91. Apart from this work, aon-stoichiometry was accounted for by means of the so-catled rigid-band model. As this model gives onfy a very crude estimate for the changes in the electronic structure upon changing the composition by shifting the Permi energy according to the average number of electrons, nearly all previous theoretical investigations of the phase stability of VS, were restricted to the stoichiometric case. Based on corresponding band stracture calculations, Liu et al. [5] discussed the Peierls lattice that takes the NiAa-structure. to the MnP-structure in terms of charge density wave instability. A quite general microscopic theory of the structural phase transition of compounds with NiAs-structure has been presented by Katoh and Moth&i [lo] using the concept of the generalized electronic susceptibility x(q,A). This quantity gives the change in the free energy proportional to IQ J* due to the linear electron-lattice interaction with Q,, the normal coordinate of a phonon with wave vector q and mode A. Apart from the electronic dispcrsjon relation E,(k)
1937 Ebevicr Scicncs SA. AU ri@a rCjdmd
the susceptibility x(q,A) is determined by the
electron
the strength of the coupling between two electronic states (xk) and (n’k’) caused by a displaccmcnt of the vth atom in the direction IX. Compared with these approaches, the characteristics of the structural phase transition of VS has been studied here by calculations of the total energy as a function of the lattice distortion using the highly accurate linear augmented plane wave (LAPW) method of band structure calculation in its full potential form. The structural phase transition of VS was explained by Silvestre el al. [4] by a relatively high density of states at the Fermi energy for the NiAs-structure that is lowered as a consequence of a Peierls distortion. Another way to avoid such an energetically unfavourable situation for a solid is to undergo a magnetic phase transition. This seems to be the case for the compound VSe in the NiAS-SrNCtUIe. A da&!ed theoretical investigation of the conditions for such a transition has been performed by Dijkstra et al. [II] for CrSe and other related compounds. In contrast to this. a more extended study is presented here for VSe. By calculating the total energy of the three competing phases with non- or para-, ferro- and anti-ferromagnetic spin ordering for a wide range of Ihe hexagonal lattice parameters 11 and c. the complete magnetic phase diagram of this system has been determined. The paper is organized as follows. The Section 2 Section 3 describe rhe crystal structures studied and the band struUduw method used for the various investigations. In Section 4 the results for the structural phase transition of VS are presented and discussed. Section 5 deals with the magnetic phase stability of VSe in a corresponding way. All results of this investigations are summarized at the end of this contribution. lattice
matrix
element
i,&,.
that
represents
2. SINctural p1operGes VS and VSe crystallize in the NiAs-structure (space group P6,lmmc). This structure, shown in Fig. 1. consists of hexagonal close packed layers occupied alternately by V- or cbalcogen atoms. As can LY seen from Fig. 1. the layers repeated along the c-axis follow the typical hexagonal stacking sequence with V in the octahedral holes [12]. Representative experimental lattice parameters taken from the literature are summarized in Table I. As can be seen. for VS and We rhe c/a-ratio is around 1.76 and 1.63 respectively. Compared with the ideal close pecked hexagonal structure. the lattice is strongly strewbed within the basal planes leading to a V-V distance (3.33 A for VS) within the planes Ihat is about
tb)
10% larger than that for adjacent V-layers (2.94 A for VS). The MnP-structure that is adopted by VS at low temperatures is ckly related to the NiAs&ructure. To demonstrate this, in addition 10 the proper hexagonal unit cell of the NiAs-structure. an alternative orthorhombic unit cell is shown in Fig. 1 containing four instead of only two formula units. For VS, the MnP-sIructLue (space group %M) derives from Lois Table
I
vs lead .vs (W VSe (exp) vse (the)
(A)
e (A)
E
3.332
5.860
1.761
3.152
Lzn
1.974
3.660
5950
L.626
3.918
1528
1.411
cln
The full-potential LAPW is based on the muffin-tin construction that replaces the Wigner-Seitz-spheres by non-overlapping muffin-tin spheres. Within these
spheres of radius rnr,n (a =V, S, or Se) the angular dependence of the electronic potential V,(r) is expanded in spherical harmonics Experiment nlcnrv
5.856 5.868
3.3tJ.5 3.214
T.83” 5.808
0.018 0.018
“.23Y 0.238
In the interstitial region between the spheres the pctcntial is represented by a plane wave expansion orthorhombic unit cell by a pronounced shift of the V-atoms along the orthorhombic c-axis with the shift in opposite directions for adjacent V-layers. In addition, B minor shift along the orthorhombic n-axis occurs. The resulting lattice parameters and atomic positions are given in Table 2. The primary effect of the shifts of the V-atoms wifh respectto the originalNiAs-structureis to destroy the triangular network of these atoms, to form in the basal planes more or less separated zig-zag chains running parallel to one0 another. As a consequence, the V-V distance (2.76A) within these chains is strongly reduced. being nearly 10% smaller than the V-V layer distance (X3 A) that is hardly affected compared wilh the NiAs-slruclure (2.94 A).
(21 with K* the reciprocal lattice vetiors. Owing to the greal flexibility and accuracy of the chosen expansion for the potential and electronic charge density, a very high numerical accuracy is achieved for the LAPW method. In contrast, a relatively high numerical efficiency is provided by the linear approximation for the energy dependence of the electronic wave function. This approximation leads to a generalized algebraic eigenvalue problem to determine for a given wave vector k the corresponding Bloch wave functions +,,&&,) and eigcnenirgies E,, with n the band index. Further technical details on the LAPW-method can be founci in Ref. [21] and the monograph of Singh
P-4.
3. Calculalionof the sleclronk
structure
During the first stage of this work the linear-muflintin-orbital (LMTO) method of band structure calculation [14,15] has been used to investigatethe electronic structure of VS and VSe. In its conventional form the LMTO makes use of the atomic sphere approximation (ASA). This geometric approximation implies as&bing to every atom in the system a region bounded by the so-called Wigner-S&z-sphere in such a way that the sum of the volumes of these spheres is identical to the volume of the unit cell. The eleclmnic potential
As is done nowadays in most electronic structure calculations for solids. the exchange and correlation effects have been treated within the spin density functional theory formalism. This means that the electronic potential V(r) in Eqs. (1;L) is composed of the electrostalic Coulomb or Hartree part and the exchange and correlation part V,,(r). To ensure the most reliable results for the lotal energy of the studied systems, the parametrization of L’_(r) according to the generalized gridien! approximation(CGA) given by Perdew et al. [24 has been used.
and the chargedistributionare taken to be spherically symmetric within these spheres, i.e. these quantities depend only on the distance from the nucleus. The ASA-construction has the drawback that there is no unambiguous way to fix the ratio of the sphere radii of the various atom types (here V and S or Se). Furlhermr)re, the various Wigner-S&x-spheres overlap but leave some uncovered region in between. This seems to he especially problematic for the rather open crystal structures investigated here (see Section 2 and Fig. 1). Although the results obtained using the LMTO method [16-H] turned out to be very reasonable and in line with previous theoretical work [19,2,6,20,4], the full potential version of the LAPW method of band StNctUre calculation 1211has been adopted IO avoid any problems probably caused by the AS.4.
4. Results and discussion lor VS
Based on the experimental lattice parameters given in Tables 1 and 2, the electronic structure of VS in the NiAsslructure as Hell as the MnP-struflure has been calculated in a self-consistent way using the LAPW method. Calculating the dispersion relation E(k) for the NiAs-structurc. shown in Fig. 2(a), the orthorhombic unit cell has been used to allow for a direct comparison with results obtained for the MnP-strocture. To yet the dominating character for the various bands displayed in Fig. 2(a) it is sufficient to look at the corresponding component-resolved density of
(-8
Fig. 1. Diupxdon n-lahon E_(t) Ar (a) VS in the NiAs-structure w&e an orthurhombic unit cell iscr Fig. l-middle). and lb) for VS
states (DOS) curves that are given in Fig. 3(a). From these DOS curves one finds ihat the narrow band complex at amund - 15 eV [not shown in Fig. 2(a) and Fig. 3(a)) is derived from atomic S 3s-states. As Fig. 3(a) shows, the bands from -8 to -2.5 eV are dominated by S p-states. Nevertheless, there is a pronounced mixing or hybridization with V derived states - roughly in the ratio 3~1. The bands in the region from -2.5 to 4eV however, are dominated by V. However, again there is a hybridization with S derived states with a weight of about 6-7~1. These ratios of course primarily reflect the number of available S 3p and V 3d-states. This is indeed confirmed by a furthex analysis of the DOS curves, i.e. a decompositiou according to their angular momentum character. For S one finds that between -8 and -2.5 eV the p-character is by far the most dominating. For the bands between -2.5 and 4 eV there is also some s-character contributing that gets even more important for the bands above 4 eV Analogously, the d-character dmni?ates ? in the case of V. Only for the lcw eoeTgy region
to - 2.5 eV) are there some E and pcontributions thst get stronger and of appreciable weight for the bands above 4 eV. Obviously, ail these findings are completely in line wirh the expectations based on a naive molecular orbital (MO) picture. Starting from atomic S 3s-levels that are much lower in energy than the V 3plevek, one expects MO fevek that have dominating S and V character respectively. with a corresponding separation with respect to theii energy. Owing to the solid state, the various MO-sublevels are spread over a wide range of energy (here from -8 to -2SeVand -2.5 fo 4 eV respectively). The electronic simclure of VS represented by Fig. 2(a) and Fig. 3(a) is in full agreemen tith the corresponding experimental Epectroacopic investigations. For example, rhe XPS measurements of Franzen and Sawatzky [24] probing the occupied part of the DOS curves gave a broad and rather featureless band complex in the range of 9 to 2.5 eV binding energy. A second one. ranging from 2.5 eV binding energy to the emission threshold. i.e. the Fermi energy, was separated by a very pronounced dip in the XPS spectrum. This spectrum could be reprcduced recently in a very satisfying way cm the basis of the corresponding component- and angular-momentum-resolved DOS curves of VS taking properly into account the relevant XPS matrix elements [25]. Furthermore. one should mention that qualitatively, similar results were also found using the less rigorous extended Hiickel methodwithin the tight binding approximation [4]. However, the main feature of the results prexnted in Fig. 2ta) and Fig. 3(a) is in full accordance with our previous work using the LMTOmethod, as well as with that of other workers uing ab i&o band snwcture methods [19,2,6,20]. However. the present resuhs can be rendered more accurate because of the use of a full potential band structure method. while previous investigations of the electronic structure of transhiin metal chakogen compounds used the atomic sphere or the simple muffin-tin approximation. This aspect seems to be especially important for the total energy cakulations presented below. Comparing the dispersion relation E(k) for the MnRstructure (Fig. Z(b)) with that for the NiAs-
structwe (Fig. 2(a) - as mentionedfor cakulatingthe latter, the iarger unit cell ‘of the MnP-shucture has been used to allow for this direct com@sun) a number of pronounced changes can be found. First of all the reduced symmetry of the I&P-structure corn-pared with the NiAs-structure is reflected by lifting of degewracies as well as replacementof band crossin.@ by avoiding of tmods. As expected, ihis is most pronounced for the energy region - 2.5 to 4 eV daminated by V, because only these atoms change their relative posilimss within the unit cell signiitly.
a Pig,. 3. Total and compuncnl-resulvrd DOSn(E)for (a) VS in the NiAsstructure. The full line gives the total DOS, while the dashed liner give the V- and S-contribution nspectivcly. Part (b) shows the results lor VS in the MnP-structure.
Nevertheless, there are also remarkable changes found for the S-dominated energy region (-8 to -2.5 cV). For example, the degeneracies marked with A and B, at the r- end X-points respectively. are lifted, while the band crossings marked with C and D are replaced by band avoidings. As noticed, all the-e changes are dictated by the lowering in symmetry and can be followed in detail by setting up a corresponding Walsh diagram [4]. However, the most important point to note is that the changes arc not restricted to the V-dominated energy region or even to the states in the vicinity of the Fermi energy. From the impact of the lattice distortion that takes the NiAs-structure to the MnP-structure on the dispersion relation E(k), it is obvious that no dramatic changes have to be expected for the corresponding DOS curves. Fig. 3(a) and Fig. 3(b) indeed demonstrate that the overall features of the DOS curve, discussed above for the NiAs-structure are. not altered by the lattice distortion. A further diiussion of the dispersion relations and the DOS curves till be given below in context with the relative stability of the both structures investigated.
4.2.
Hecrmnic
charge
density disfribtiion
The use of the full potential version of the LAPW method g&es a direct and reliable access to the
and dotted
electronic charge density distribution in solids. Corresponding results for the total charge density of VS in the 14’4s-structure : as well as in the MnP-structure are given m Fig. 4 and Fig. 5. where the NiAs-structure is again represented using the orthorhombic unit cell of the MnP-structure with lower symmetry. In Fig. 4 the density distribution in the (OlO)-plane (see Fig. 1) is shown for both structures. This plane is perpendicu!ar to the V-layers and contains two S-atoms per unit cell. Owing to the symmetry, only half of the unit cell is shown in Fig. 4, with V-atoms at the comers of the displayed rectangle and a S-atom in the middle of its left half. As can be seen, both structures are quite open with a rather wide region of low electronic density right to the S-atom. The mirror symmetry present for the NiAs-structure is obviously destroyed by the lattice distortion, which leads to a further decrease of the minimum charge density in the open interstitial region by about 10%. A slight decrease in the charge density can also be seen between the S-
atom and WV-atom that is shifted away (in the lower left comer of Fig. 4(b)). For the approaching V-atom (upper left comer) the interatomic charge density increases. In addition, the anisotropic charge density distribution of the S-atoms is slightly reoriented towards this V-atom due to the atomic shifts. QuaI& tatively. these features can be interpreted in terms of weakened and enhanced, respectively, V-S bonding for the two neighbouring V-atoms of the S-atoms. As mentioned above, the shiis of the V-atoms when
going from the NiAs-structure to the MnP-structure are much more pronounced within the V-piancs than perpendicular to them. While the charge rearrangement within the (010)-planes was found to he relatively small {Fig. 4). the impact of the lattice distortion for the V-planes. i.e. the (LOO)-planes. is accordingly much stronger for that reason. For the NiAs-structure the charge density of course reflects the perfect triangular arrangement of the V-atoms (see Figs. 1 and 5(a)). This highly symmetric charge distribudon is sirongly distorted by the formation of the V zig-zag chains, as can be seen in Fig. 5(b). As for the (010)~plane. the atomic rearrangement reduces the minimum charge density in the enlarged interstitial region (middle of the left half of Fig. 5(b)) by around 10%. However, compared with the (010)plane, its absolute value is still an order of magnitude higher - primarily because of the much higher packing density. As a consequence of the formadon of the V zig-zag chains, the interatomic charge density increases for the approaching V-atoms by about 30% (right half of Fig. 5(b)). Again, this can qualitatively be interpreted as a strengthening of the bonding for the V-atoms within the chains.
With the detailed charge density distribution available. it is ot course tempting 10 determine the corresponding atomic charges and from that the charge transfer. This would allow one lo specify the ionic character of the bonding within the solid as well as to dclerminc - at least in a formal way - ihe valency of both the components. However. owing to the geomclrical representation of charge density. in analogy with Eqs. (1.2).within the LAPW formalism. it is, in the first instance. only possible to calculate the charges in the muffin-tin spheres. These charges can be ascribed approximately to the corresponding atoms occupying that sphere, while the interstitial chars remains unrelated to a specific atom. Table 3 gives these muffin-tin charges of Vand S inVS for the Niisand MnP-structures. First of all, one notes that the muffin-tin charges Q, are nearly identical for both structures, implying that the phase transition gives rise primarily to a charge rearrangemen within the ittterstitiai region. For V and S the, charges Q, are below
the atomic number by 1.90 and 1.43 respectively. These numbers already give some indication for the proper atom-related charges. However. to use the muffin-tin charges Q, as a first estimate for these quantities, the muflin-tin radii r,,, should not be i&d to the same vahte for V and S, as done here. The charge density distribuGon shovm in Fig. 4 suggests that using the position of the chargedensityminimum
paramctcrs arc in rather satisfying agreemenl with the experimental data. One of the reasons for the deviation from experiment - 5% and 6% for LXand c
in between the neighbouring V- and S-atoms would be a much more appropriale way lo fix the muffin-tin radii, As a consequence of this improved choice, the resulting muffin-tin charges would more strongly rcfleet the expected ionic charge distribution among V and S. However, the high interstitial charge Qu makes clear that even this procedure would also lead to only a rather crude estimate for the proper atom-related charges. A more satisfying and unambiguous way to determine atomic charges in a solid from the given electronic charge densiry distribution is supplied by the transfer of the concept of an atom in a molecule of Bader et al. [26] to the solid state. First steps in this direction have bern done recently leading to atomic shapes that are very similar to what one expects from a Wigner-Seitz-construction [27]. Further numerical developments will allow for an unambiguous determination of atomic charges in a solid and, based on thal. a corresponding discussion of the various abovc-mentioned aspects.
4.3. Gromrrry
optimizntion
respectively - is that the experimental data have been determined at room temperature. Furthermore. one has to mention that it is obviously quite hard to produce perfect stoichiometric samples (see also below). To perform corresponding calculations for VS in lhc MnP-structure does not seem to be very reasonable, because in this case one would have to optimize four parameters {see Table 2). For this reason. the lattice distortion that takes the NiAs-structure to the MnPstructure has been used as some kind of reaction coordinate. This means that the total energy has been calculated as a function of a dimensionless parameter I that represents the shift of the V-atoms. Here I = 0 corresponds to no shift at all, i.e. the NiAsatru,ture is adopted, while I = 1 corresponds to a shift according to the experimental data for the MnP-structura.Varying t. the lattice parameters a and c have been varied linearly in parallel; that means, for t = 0 and 1 = 1 the parameters for lhe NiAs-structure and the MnP-structure respectively have been assumed. Fig. 6 shows corresponding results for the lotal energy as a function of the distortion parameter 1. First of all one notes that the MnP-structure is found - in full agreement with experiment - to he the more stable phase. The lattice distortion predicted by the calculation (t = 1.08) is only slightly larger than the experimental one corresponding to I = 1 (see also Table 2). As one can see, the total energy I$) curve is obviously not symmetric around r = 0. This is simply caused by the fact that the lattice parameters a and c have been varied in parallel with t as described above. It is interesting to note that
and phase sfuhility
Calculating the total energy of a solid as a function of its lattice paramelers allows one -. among various physical properties - to determine its theoretical equilibrium lattice parameters. In the past it turned out that it is quite crucial for these types of calculation to apply so-called gradient corrections within the density functional formalism [ZS]. For this reason. a corresponding parametrization for the exchange and correlation energy and potential. respectively, has been used [23]. If. for a given system, there are several lattice parameters and atomic positions within the unit cell to be optimized, the corresponding calculations are quite computer time consuming or even impossible. For VS in the NiAs-structure, only the two lattice parameters (I and E have to be optimized by minimization of the total energy. For this purpose the total energy as a function of II and c is approximated by a paraboloid that has been fitted to nine tabulated data points. As can be seen from Table 1, the theoretical lattice
-Ior@175
1
i
the NiAs-structure,
corresponding lo I = 0. is situated at a local minimum of the E(t)-curve. i.e. it realizes a met&able state. This sheds some doubt on the applicability of the approach of Katoh and Mothuki [lo] mentioned above. Altogether. the energetic situation is obviously very similar to that for the thermal activated phase transition from the hexagonal w-phase of Zr to its high temperarure b.c.c. p-phase [ZY]. In complete analogy with this rather simple case, deeper insight into the characteristics of the NiAs-MnP phase transition could be obtained from an experimenral and theoretical study of the relevantpart of the phonon spectrum. In a previous investigation. using the extended Hiickel method [4], the phase transition of VS has heen explained by the creation of a dip in the DOS curve around the Fermi energy EF for the MnPstructure. Comparing the DOS curves n(E) in Fig. 3. one finds that the opposite happens. i.e. the DOS n(E,) at Ef increases from 0.50 to 0.55 (st.leVspinFU). This finding seems to be in cooflicl with V-Knight shift measurements that revealed a decrease of the shift when ‘going from the N&-structure to the MnP-structure [30]. However, one has to note that the Knight shift is composed of several contributions with only some of them directly related to a part of the I-resolved DOS at E,. In line with the mentioned experiment. the direct Fermi-contact contribution one of the most important ones - can be expected to decrease during the phase transition. This prediction is based on the finding that, in contrast to the total DOS. the relevant V s-DOS decreases by more than 30%. As is obvious from the dispersion relations. and also from the charge density distributions. the phase transition of VS primarily involves the V-dominalrd part of the band structure in the energy range of - 2.5 to 4 eV. A decomposition of the V d-DOS according to the symmetry character shows that its part that is related to the d-orbitals lying within the basal plane are most affected by the lattice distonion. It seems that the shift of the corresponding occupied bands towards higher binding energy due to the stronger overlap of the in-plane V d-orbitals delivers the most substantial contribution to fhe stabilization of the MnP-structure. This could be confbmed by estimating the change in total energy upon the phase transition by using :he so-called force theorem 1311. Such calculations, that are in progress at the moment. could also tell whether the contribution of the S-dominated states between -8 and -2.5 eV is negligible or not.
As one expects,
for VS in the Nis-structure.
replacing the S-atoms by Se-atoms gives rise to a very similar band structure. This can be seen by comparing the DOS curve for VSe in ihe paramagnetic state in Fig. 7(a) with the comqonding
curve for VS in Fig.
3(a). The main difference between both curves is that the chalcogen-dominated s&band complex is 3t higher energies for VSe (-7 to -2eV) and that tbe Vsubband is less spread in energy. This means that the E(k) bands in the vicinity of the Fermi energy E, are rather dispersionless, i.e. flat, giving rise to a very high DOS at EF. This property can be traced back to the lattice expansion of VSe with respect to VS due to the larger chalcogen atoms. This lattice expansion, how-
ever. primarily takes place within the basal plane. increasing the lattice parameter a by about 10% while
c increases only slightly. The relatively high DOS at Er for VSc in the paramagnetic state points out that 11 I system might be in an energetically Iess favourable stale which could be removed hy a structural or magnetic phase transition. Guided by the cxpcrimental properties of VSe. its electronic structure was investigated for the ferro- and anti-ferromagnetic state as well for that reason. The resulting spin-resolved DOS curves for VSe in the ferromagnetic stale are shown in Fig. 7(b). At first sight the DOS curves for the ferromagnetic state seem to be created from those for the paramagnetic one by rigidly shifting the curves for different spin character against one another. F’urther inspection, however. shows that fine details of the DOS curves change as well. [n addition, one notices that the average exchange splitting AE,, of ibe bands is more pronounced for the V-dominated than for the S-dominated band complex. While for the V-region one finds AE,, = 1.2eV the splitting within the S-region is at most 0.35eV. As a consequence of this different exchange splitting. the overlap of both the. band complexes is
slightly increased for the majority spin system, while a band gap opens for the minority spin system. According to the exchange splitting and the DOS curves, the resulting moments for V and Se are also quite different: 1.83 cc, and -0.025 & respectively. The magnitude and sign of the Se-moment inVSe can be traced back to the stronger hybridization for the Se-dominated band complex with majority spin character. compared with that for minority spin character (see Fig. 7(b)). Using a simple MO-scheme, this in turn can he explained by the relative positions of the relevant atomic V 3d- and Se 4p-levels on the energy axis,
taking the exchange splitting into account. Here. one should notethat the calculated moment of SC in VSe is quite different from that found by Dijkstra et al. [Il] for G-Se. These authors find 3.4 h for the Cr-moment and 0.24 k for the Se-moment. The big difference of the latter value compared with that for Se in VSe given
here seems primarily to be of technical origin. While for the present work a full potential method has been used and the moment evaluated within the Se muffintin sphere, the investigation of the system CrSe has been performed using the ASA-approximation with the moment evaiuated within the Wigner-Seitz-sphere of Se. To perform baud structure calculations for an antiferromagnetically ordered state of VSe the magnetic contiguration found by neutron scattering in the case of CrSb (321 has been assumed (the actual experimental spin configuration of VSe in the anti-ferromagnetic state has not yet been investigated). This configuration consists of V-layers in the basal plane with all atoms having the same magnetic moment and oriented along
the same direction. The adjacent V-layers again Scar the same moments: they are. however. oriented in the opposite direction. Owing to this spezihc. highly symmetric spin configuration. there are no induced spin magnetic moments present on the Se-sites. The resulting magnetic unit cell is therefore identical to the crystallographic one. The DOS curves that emerge from that anti-ferromagnetic configuration are shown in Fig. 7(c) for the sublattice wnsisting of the A- and B-layers. Those for the second sublattice, built up by the C- and D-layers, are the same but with spin up and down exchanged. In contrast to the ferromagnetic spin configuration. the anti-ferromagnetic has a very strong influence on the V-dominated part of the DOS CUNB. This can be understood from the fact that for a given spin orienta-
tion the potentials on the neighbouring V-layers differ strongly because of the opposite exchange splitting At?,;,. Taking the difference in position of the main peaks as a measure for A&,, one finds AE,, = 2.4 eV. This value is higher than for the ferromagnetic state, is found to be and accordingly the V-moment [ 1.97 b)
somewhat higher.
For VSe. in the past. very different magnetic proper&s have been found experimentally depending on the preparation and, with this, on the actual composition of the samples (see below). Indeed. within a previous investigation of the electronic structure al VSe using the LMTO-method [Yj it was found that, for a certain range of the c/o-ratio, the generalized Stoner criterion was pointing to a ferromagnetic instability. This means that for the paramagnetic state the condition f&E,:). I >
I
(3)
is fultilled. Here the global exchange
parameter
I is
given by
with n,,(Er) DOS for the exchange or curve in Fig.
the angular momentum, i.e. /-resolved component o (a =V. Se) and /_,,,. the Stoner integrals [33]. From the DOS
7(a) it is obvious that if the condition in Eq. (3) is fulfilled, tXs is only due to the d-electrons of V via nv&). As is obvious from the comparison of VS and VSe, this contribution to the DOS is very sensitive CD the V-V distances. Owing to the very different V-V distance witbin the basal plane (3.7 A) and along the c-axis [XC A), one can conclude that the dispersion of the V-dominated bands is strongest for bands derived from orhitals oriented along the c-axis.
Indeed, a symmetry analysis of the DOS-curve for the paramagnetic state of We (Fig. 7(a)) reveals &at the d+ontribution is spread over the whole V-dominated energy range (-2 to 3 eV). In contrast IO this. the d,,and d,,-derived bands give rise to the pronounced peak in the DOS curve al 1 eV. The d-orbitals oriented within the basal plane, however, are responsible for the sharp peak at the Fermi energy. From this analysis one expects that the product In(&) in Eq. (3) is primarily dependent on the lattice parameter 0. The reason for this is that decreasing or increasing u will respectively increase or decrease the overlap of the in-plane oriented d-orbitals. Thii in turn will decrease or increase the corresponding bandwidth and. as a consequence. will increarrr:or decrease the DOS at E,. The lattice parameter c, however, may be expected lo have a much smaller influence on the magnetic prop erties according to the analysis of the DOS curve hr the paramagnetic state of VSe. In line with this is the previous finding that decrwding the c/a-ratio leads to an increased tendency towards spontaneous magnetism according 10 the Stoner criterion [91. In the form given above (Eqs. (3.4)). this criterion first of ail concerns ferromagnetic ordering and. therefore. does not rule out an anti-ferromagnetic ordering. Which type of ordering is energetically more favourable can only be decided by application of the so-called force theorem 1321 or - in a more reliable way - by performing total energy calculations. This approach has been used here for a detailed imestigation of the interrelation of the magnetic properties of VSe and its lattice parameters by calculating the corresponding phase diagram Cor the para-, ferro- and anti-ferromagnetic states. For this purpose. it turned out in the past to be very important to use gradient
able. One can easily recognize from that figure that the lattice parameter o is much more importan for the magnetic properties of VSe as the parameter c. investigation of the ferromagnetic phase in additiou to the para- and anti-ferromagw ticoaesdoesnot introduce a new global minimum w the energy surface, jusl a Io4 one. Thii means chatthe antiferromagnetic state is left as the energetically most favourabte one. However, this does not nde out the exstence of a more complicated anti-ferromagnetic or non-collinear spin structure that is still lower in energy. In fact the spin structure reported for C&E is quite complicated [38]. As can be seen, the ferrmnagnetic phase tamed out to separate the pan- from the anti-ferromagnetic phase for the investigated RngC of lattice parameters_ As for the para- aad a&fe~netic cases, all phase bormdaries are P~~IIIU& facd by the lateral parameter a. This result s.tro& dev&es from the expectations of Kamimura 1391. based on a survey of fzxperimental data of the magoctie prop6ea of a number of tmn&c4t metal-dmtmgcn compormcir with the Niis-smrture, this author cnacl& that it is primarily the lattiee parameter c that determines the magnetic state and type of ordering. However, ibis conclusionisbynomeansforccdbythcprcsentdata compilation. In addition to tha theoretical phase boun&r& several oxperima~tal data pGts have been addad to Fig. 8. The rhrae points to the rig&, that llurty coincide with each otlwr, represent rrmpkvtht were reported to be -tic. The tw0 dau points to tbe right belong to pumrgvticsampkzLTbewida spread of these experimental data make ckar that
sample imperfections have a lattice
parameters
as well
85 on
strongimpact on the Ihe
magnetic
References
rcatures
of VSe. Nevertheless. they are in full accordance with the central theoretical findings: a lateral lattice expansion takes VSe from the para- tn the anti-ferrmnagnetic state. while an expansion along lhe c-axis has much less influence on the magnetic slate.
4 Summary
The structural phase transition of VS from the low temperature MnP-structure to the high temperature NiAs-structure has been investigated by detailed calculations of the electronic structure for both strttctures. using the futl potential LAPW band structure
method. Calculation of the total energy as a function of the lattice distortion revealed - in full accordance with experiment - the MnP-structure as the stable one. The NiAs-structure, however, was found to be situated in a local energy minimum. The changes in the electronic properties upon the phase transition have been monhorcd in reciprocal space, in the energy domain and in real space by means of the dispersion relations, the DOS curves and the charge distribution. From these results it was concluded tbat the phase stability of the MnP-structure is primarily due to a strengthening of the V-V bonding within the basal plane via their d-orbitals. Further investigations to corroborate this interpretation are in Progress at the momenl. For We. in the NiAs-structure, the magnetic phase diagram has been investigated by performing total energy calculations. Here the anti-ferromagnetic state has been found, in agreement with experiment, to be the mast stable one. However, the other magnetic states were found to provide near-by local minima in energy; this explains why in the past, depending on ;he specific sample preparation. VSe was found experimentally to be para- or anti-ferromagnetic. The calculated phase boundaries that separate the para-. ferro- and anti-ferromagnetic phases from one another were found to be primarily determined by the lateral lattice parameter a. This result is in variance to earlier expectations but can be explained by the fact that the magnetic instability of paramagnetic VSe is due to the high DOS at the Fermi energy produced by the dorbit& oriented within the basal plane.
Financial support provided by the Deutsche Forschungxgemeinschaft within the prograrnme Ungewiibnliche Valenzzustiinde in Festkiirpem for this work is gratefully acknowledged.
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