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A room temperature organometallic ferromagnet of approximate composition V(TCNE)x·y solvent (where TCNE - 1 - stands for tetracyanoethylene - a well known organic electron acceptor; x » 2 and y depends on the type of the solvent)
supplied by about one half of the solvent molecule CH2Cl2 per formula unit had been synthesized yet in the beginning of the 1990's Manriquez as an amorphous moisture sensitive precipitate. Its most remarkable property was the nonvanishing spontaneous magnetization persistent almost up to the decomposition temperature of ca. 350 K which allowed to estimate the critical temperature of the ferromagnetic transition (the Curie temperature) to be of ca. 400 K, i.e., higher than the decomposition temperature itself. The following years witnessed many analogs of the above compound both in terms of extending variety of involved organic acceptors (Ref. [] - tetracyanopyrazine - 2; Ref. [] - 7,7,8,8-tetracyano-p-quinodimethane - 3; Ref. [] - tetracyanobenzene - 4 ) and in terms of the metals (Ref. Pokhodnya - iron), although none of them manifested as fascinating magnetic properties as the first V(TCNE)2 compound (the Curie temperatures ranged from 44 K for Ni and Co compounds through 107 K for the Mn to 121 K for the Fe compound - all with TCNE). Varying the solvent also affects the Curie temperature, e.g., replacement of CH2Cl2 presented in the original compound Ref. [] by tetrahydrofuran (THF) reduces the Curie temperature of V(TCNE)x·y solvent to 210 K and by MeCN to ca. 100 K.
Generally one has to say that not only the Curie temperature, but also other properties of the compounds of the considered class are sensitive to the details of the preparation procedure. For example, the V-TCNE compound is known in two forms - the original of Ref. [] coming from the reaction of V(C6H6)2 with TCNE and another obtained from vanadium hexacarbonyl V(CO)6. The latter exhibits a magnetization that is almost twice as strong at the zero temperature compared to the original one.
For more than one decade the amorphousity of the compound of interest did not allow to make any definite conclusions concerning its structure so that the statement made in Ref. [] 'almost nothing is known about the actual structure of the V-TCNE ferromagnet' remained true during these years. Nevertheless, the reasonable limitations upon the tentative structure could be formulated yet in Ref. Tchougreeff-Hoffmann. These conditions are as follows. The acceptable structure has to (i) correspond to the observed composition V(TCNE)2, (ii) form a three-dimensional network of V atoms and TCNE molecules without pronounced anisotropy to ensure the proliferation of the magnetic order through the sample, and (iii) be loose enough to be able to accommodate solvent molecules. These simple ideas allowed the authors of Tchougreeff-Hoffmann to propose a structure compatible with the magnetic data available at that time. It is presented in Fig. , where one can see rather spacious channels capable to accommodate solvent molecules. The V atoms experiences 8-fold coordination by the N atoms; the V-N distance being ca. 2.06 Å .
In the frame of that model it was possible to discuss the tentative electronic structure of the new material with the purpose to understand the most intriguing - the magnetic properties. It was assumed that two unpaired electrons occupy the LUMO b3g( p*) -orbitals of the two TCNE units forming respective anion radicals TCNE[(-)\dot] and three more of them reside in the d-shell of the vanadium (II) cations acquiring there a high-spin configuration with the total spin 3/2. In contrast with the description of the V(TCNE)2 compound as a "ferromagnet" the effective magnetic interaction between the local spins 1/2, 1/2, and 3/2 has an äntiferromagnetic" sign i.e. the spins 3/2 located on vanadium ions tend to be oriented in the opposite direction to that of the spins 1/2 located on the TCNE[(-)\dot] units, so that the spontaneous magnetic momentum corresponds to overall value of one unpaired electron per formula unit (or per vanadium atom):
Picture Not Created.
This view was supported by the electronic structure model proposed in Tchougreeff-Hoffmann which is characterized by complete filling of
involved bands by electrons of only one spin projection extensively used by
other authors []. The model []
represents an unrestricted Hartree-Fock (UHF) band model spanned by three d-orbitals of vanadium ions and two acceptor orbitals per unit cell of the
hypothetical structure in Fig. 1. The calculations with use of this
model produce five very flat bands
(in fact two of the three predominantly d- and one of the two
predominantly acceptor bands have zero dispersion.
These bands are strongly split into subbands
corresponding to different projections of spins of electrons occupying these
subbands which yields the model density of states (DoS) shown in Fig. 2.
The fact that the bands obtained in the calculations are extremely narrow and the subbands corresponding to different projections of electron spins are completely filled indicates that the local description may be more adequate. It reduces to the Heisenberg Hamiltonian describing the interactions of the above local spins 1/2 and 3/2. The interactions are of the äntiferromagnetic" sign so that the interacting spins are aligned in the opposite directions.
The main objection against the model Ref. [] was the eight-fold coordination of the vanadium atom by nitrogen atoms coming from eight respective TCNE[(-)\dot] units, being in contrast with six-coordination which one would usually expect. The latter viewpoint found experimental support [] from the EXAFS and XANES experiments which have shown the average coordination number of vanadium to be 6.04±0.25 and have strongly emphasized almost octahedral environment of the latter although have not been able to completely resolve the structure issues. The critical breakthrough came with the recent EXAFS work [] where the authors were able to establish the structure of the Fe2+ analog (presented in Fig. 3)
of the V(TCNE)2 compound and revealed some of its remarkable features. It had been shown that the dimer form of the TCNE[(-)\dot] radical-anion: [ TCNE] 22- = C4(CN)82- plays an important role in shaping the loose three-dimensional structure satisfying the general conditions of Ref. [] and that of the octahedral coordination of the metal ion.
Because of the existence of the dimer form of the TCNE[(-)\dot] radical-anion with doubled charge, we note that the synthesis of a chemically even simpler class of magnetic materials containing bridging N atoms has recently been accomplished. The so called 3d carbodiimides incorporate divalent magnetic transition metal (e.g. Mn2+, Fe2+, Co2+, Ni2+) that are connected to each other by the NCN2- molecular anions, the basic form of carbodiimides (or cyanamide) molecule, forming a three-dimensional structure. In all cases the transition metal cations experience the octahedral coordination whereas the NCN2- units may be also octahedrally coordinated (following the [NaCl] type such as in MnNCN compound []) or coordinated according to the motif of a trigonal prism (quasi [NiAs] or delafossite type as in CoNCN/NiNCN []). Antiferromagnetic coupling between the magnetic centers is characteristic for this novel class of compounds which may be looked upon as nitrogen-containing analogs of the 3d-metal oxides.
We see that the chemical composition of the örganic" part of the organic ferromagnet resembles that of the well-studied carbodiimides. Next, since the ionic radii of V2+ and Fe2+ almost coincide (respectively 0.79 Å and 0.78 Å for the high-spin form of Fe2+) [] and also due to the fact that the solid solutions of the composition FexV1-x(TCNE)2 in a wide range of x form readily it seems to be possible to use the structure [] (Fig. 3) as a starting point for further consideration.
In the present work we continued our efforts directed to concert elucidation
of the spatial and electronic structure of the room temperature
organometallic ferro(ferri)magnet V(TCNE)2. Close analogy with
transition metal carbodiimides allows the use of the solid-state
electronic-structure package VASP []. We performed calculations on
several structures which could be considered as relevant to the problem. The
calculation had been performed with use of the LSDA+U functional where the
U term had been added for the d-shells of vanadium ions and p-shells
of carbon and nitrogen atoms to ensure the convergence of calculations to a
spin-polarized ground state. The LSDA+U functional had been taken in the
form:
| (1) |
| (2) |
With these values the following numerical experiments have been undertaken. First, we performed the energy optimization for the original structure model of Ref. [] (Fig. 1). This model belongs to the P4/mmm space group (No. 123). Taking somewhat arbitrary parameters of the unit cell a = b = 6.413 \mathringA, c = 8.736 \mathringA (V = 359 \mathringA3) corresponding to dVN = 1.987 Å results in the number of unpaired electrons of 1.011 per unit cell (formula unit) which fairly corresponds to the picture when three unpaired electrons in the vanadium d-shell are partially compensated by two unpaired electrons residing each in two TCNE radical anions. The corresponding density of states are shown in Fig. 4 (left). The magnetic moments are predominantly concentrated in the vanadium d-shells (1.025). The rest comes from the organogenic atoms. However, the distribution of magnetization coming from the VASP modeling does not allow to single out any local momenta in the organic part. This can be possibly interpreted as a trend towards pairing the electrons in the TCNE radical anions on one hand and to pairing of electrons in the d-shells of vanadiums, which can be expected at that small unit cell volume.
When the lattice parameters are optimized they relax towards the values a = b = 8.557 \mathringA, c = 7.214 \mathringA (V = 528 \mathringA3); dVN = 2.111 Å , which is fairly close to the values of Ref. [] a = b = 8.54 \mathringA, c = 7.20 \mathringA being in its turn in agreement with the experimental density []. At the optimized geometry the magnetization amounts 2.456 per formula unit. The magnetic moments are predominantly concentrated in the vanadium d-shells (1.959) The rest comes from the p-orbitals of the organogenic atoms. However, in this case either the distribution of magnetization coming from the VASP modeling does not allow to single out any local momenta in the organic part. The density of states for the relaxed structure of Ref. Tchougreeff-Hoffmann is represented in Fig. 4 (right). It manifests some spin polarization of the bands in the range -1¸-5 eV below the Fermi level. Meanwhile the uppermost filled bands in the range of 1 eV right below the Fermi level do not manifest almost any spin polarization. In analogy with the compressed structure one can expect that the trend towards pairing the electrons in the TCNE radical anions persists over the geometry relaxation, but on the other hand the tendency to pairing of electrons in the d-shells of vanadiums significantly reduces. These results indicate that the hypothetical structure of Ref. Tchougreeff-Hoffmann does not get an immediate support in the numerical experiment. Thus the situation requires more thorough investigation.
For this end we notice that setting the central C=C bonds of the TCNE units orthogonally to each other in the same plane in the öriginal" structure was (see Fig. 1) not the unique possibility. Alternatively one might consider a similar structure differing from that on Fig. 1 by rotating one of the TCNE units entering the unit cell presented on Fig. 1 by 90° in its plane. The result of such a rotation is presented in Fig. 5. In this structure the central C=C bonds of the TCNE units are as previously orthogonal, but now they lie in orthogonal planes as well so that the axes of these bonds do not intersect rather cross each other. For the reason which will be clear later we call this structure the "principal" structure.
The principal structure with initial lattice parameters a = b = c = 7.14 Å (V = 364 Å 3) yields the number of unpaired electrons to be 0.426 per formula unit. It can be characterized as a "poor metal". The density of states in two spin channels is given in Fig. 6 (left).
One can see some (expectedly weak) spin polarization of the upper filled bands as well as a noticeable density of states at the Fermi level in both spin channels (in fact the Fermi level is close to the maxima of DOS of the corresponding bands). When the principal structure optimizes with the above parameters of the VASP calculation it relaxes to a = 8.181, b = 9.219, c = 7.484 Å (V = 564 Å 3) with two nonequivalent dVN = 2.18,2.509 Å distances and the number of unpaired electrons 2.519 per unit cell (formula unit). The volume of the unit cell fairly corresponds to the observed density of the material []. Its electronic structure considerably differs from that of the "compressed" principal structure described above. It still can be characterized as that of a "poor metal". The density of states in two spin channels is given in Fig. 6 (right). In variance with that of Fig. 6 (left) the spin polarization is well developed below the Fermi level, where one can see a completely occupied subband with no obvious occupied counterpart of the opposite spin projection. It must be identified (on the basis of the DOS projection analysis) with the subband strongly hybridizing the p-shell of TCNE units and the d-shells of vanadium (see below). Two shoulders above and below the main maximum apparently have filled counterparts of the opposite spin projection visible as well pronounced maxima of DOS. The highest completely filled band is strongly spin polarized as well: one of its subbands filled by electrons with spin projection coinciding with those in the presumed strongly hybridized p-d-band is seen as an isolated peak right below the Fermi level. The counterpart of the former is the upper of two maxima slightly below it. This strongly spin-polarized band can be thought to be spanned by the acceptor states of one of the two TCNE molecules. This is manifested by the considerable asymmetry between them, which develops throughout the optimization procedure. As it has been mentioned in the optimized structure one can observe two V-N separations. The TCNE[(-)\dot] radical anion which is most strongly coordinated to vanadium (one with the shorter V-N interatomic separation) incidentally acquires a strong deformation of its own geometry: ethylenic C=C bond becomes 1.761 Å long, the single CC bond does not change too much (1.543 Å ) as is the triple C º N bond (1.209 Å ). That strong elongation C=C bond may be a prerequisite for the spin polarization of the band stemming from it. At the same time the most weakly coordinated TCNE radical anion largely maintains its original structure: the ethylenic C=C bond is only 1.570 Å long, the triple C º N bond is 1.205 Å long, and the main deformation is concentrated on the single CC bond which becomes 1.765 Å long. Turning back to the band structure of the relaxed principal structure of V(TCNE)2 we mention only that there is a significant density of states at the Fermi level in both spin channels coming from the bands spanned by the TCNE anion radicals. This density at the Fermi level may serve as a prerequisite for consequent instabilities leading to more stable structures (see below).
The Fermi level in the relaxed "principal" structure lays close to the local DOS maximum. Thus it can be assumed that such a band might be sensitive to various symmetry breaking perturbations of the electronic and/or crystal structure. Indeed, authors of Ref. [] suggest that such a band as being half filled might undergo an antiferromagnetic metal-insulator transition. It is not completely true. The transitions accompanied by doubling of unit cell parameters in crystals with half-filled bands are not mandatory of whatever magnetic nature. These may be also structure transitions. What type of transition actually takes place is determined by the relative magnitude of the energy gain acquired throughout the transition of each respective type.
Bearing this in mind we notice that the principal structure can be relatively easily put in the connection with the experimental one. Indeed, as one can see the quadrupled unit cell 2a,2b,c of the principal structure as presented on Fig. 5 transforms to that of Fig. 3 if one allows four TCNE units extended in the b direction and weakly coordinating V atoms to rotate pairwisely towards each other so that a C-C bond between respective ethylenic carbon atoms is formed yielding the [TCNE]22-=C4(CN)82- dimers. Intermediate structures along this hypothetical reaction path are presented in Fig. 7.
For these structures we performed the VASP calculations of the respective electronic structures and energies. The calculations have been performed in two variants: first the principal structure with the equalized lattice parameters have been quadrupled and connected with the experimental structure by a straight line in the configuration space. These form the first sequence of studied structures. The second sequence is obtained by the same procedure but it starts at the quadruped relaxed principal structure and ends in the experimental structure as well.
The results are presented in Table 2. As one can see after overcoming a relatively small energy barrier the quadrupled principal structure goes down in energy along the "reaction path" to some intermediate shallow minimum which corresponds to structures in the middle row of Fig. 7. Next another barrier occurs after which the structure arrives to the experimental one of the iron compound (Fig. 3). The possibility that it can be considered as a prototype structure for the entire M(TCNE)2 family will be studied elsewhere.
The density of states for the spin-up and spin-down electrons for the last structure is presented in Fig. 9 in such a way that the spin asymmetry of the DOS is better seen.
In picture Fig. 9 one can clearly see weakly spin polarized doubly occupied bands below -5 eV responsible for general bonding in the material. On the other hand, above -5 eV one can see several spin-polarized bands. Right above and below the Fermi level shown by the horizontal dashed line two symmetrical subbands appear one of which is completely filled whereas another is empty. The latter can be attributed to the (sub)bands spanned by LUMOs of the TCNE units. Below that, between two noticeably polarized bands one more subband of approximately triple intensity as compared to the above TCNE LUMO subband can be seen which is also completely filled but by electrons with the spin projection opposite to those filling the above TCNE LUMOs spanned subband. It can be attributed to the d-bands of vanadium ions. The magnetization in the last structure (that of the relaxed Fe(TCNE)2) acquires the value of 8.016 unpaired electrons per unit cell which fairly corresponds to the picture of four vanadium atoms bearing the momenta of 3/2 pointing in one direction which are partially compensated by four 1/2 momenta located in four TCNE units and pointing in the opposite direction. On the stoichiometric grounds one can expect four more 1/2 momenta, which are, however, cancelled due to formation of two diamagnetic [TCNE]22-=C4(CN)82- groups in each unit cell. Incidentally the Fe(TCNE)2 structure substituted by V ions optimizes (as previously - first the unit cell parameters, then the positions of atoms in the relaxed unit cell) to a very close structure with the parameters a = 14.373, b = 17.472, c = 7.282, Å b = 90°, space group C2/m (group No 12). No remarkable difference between the local structures of the iron (experimental) and vanadium (hypothetical) compounds has been found: dFeN = 2.16 (axial), 2.19, 2.183 Å ;dVN = 2.16 (axial), 2.184, 2.19 Å .
One can see on the pictures of the density of states Fig. 8 corresponding to the subsequent structures along the "reaction path" how the maxima of the spin DOS and other features at the Fermi level of the initial (quadrupled relaxed principal one) develop along the "reaction coordinate" going from it to the experimental one. The final electronic structure as compared to that initial one can be characterized as not one antiferromagnetically ordered, rather as a result of a structural transition driven by the C-C bond formation between TCNE[(-)\dot] radical-anions extended in the b direction. From the point of view of the band theory it is nothing but a Peierls-like rather than a Mott transition.
Magnetic properties of the intermediate structures are also of interest. The magnetization values corresponding to the structures depicted in Fig. 7 are given in Table 2. It is of interest to compare the magnetizations obtained along the first sequence of structures (starting at the quadrupled compressed principal structure) and those obtained for the second sequence (starting at the quadrupled relaxed principal structure). As one could expect from the results of the calculation on the principal structure its relaxation results in a much stronger spin polarization. However, already small deformation towards the ëxperimental" structure along the corresponding "reaction path" (second sequence) switches the system to a state characterized by the values of magnetization close to those found along the first sequence of structures. It is remarkable that in the area of the intermediate shallow minimum (three structures in the middle row of Fig. 7) the magnetization values are close to those characteristic for the material obtained from V(C6H6)2. At the end of both transition sequences (both referring to the ëxperimental" structure) the magnetization reaches the value of 8.016. That is something one could expect if a cell, containing one localized momentum 3/2 antiferromagnetically interacting with two localized momenta of 1/2, is quadrupled and then two of the four pairs of localized momenta 1/2 form a nonmagnetic state.
Namely the sign of the effective spin-spin interactions explains the reason why the total magnetic momentum per unit cell increases while going from the principal structure to the experimental one despite the simultaneous decrease of the number of magnetic centers in a unit cell.
The distribution of spin polarization in the direct space is also of importance. It is found that for all structures depicted in Figs. 7, 8 the magnetic moments residing on vanadium ions are almost constant and range from 2.615 for the first structure to 2.582 for the last one in the second sequence. The observed (in our numerical experiment) significant variation of magnetic moment along the "reaction path" must be almost entirely attributed to variation of the magnetic moments residing in the örganic" part of the organometallic magnet. This is precisely what one should expect within the general picture including formation of [TCNE]22- dimers.
The results of numerical modeling obtained in the previous Section call for some qualitative discussion. It can be given in terms of an effective model Hamiltonian representing the electronic structure of the V(TCNE)2 magnet (in its ëxperimental" structure), similar to that proposed yet in Ref. []. The general methodology of constructing such Hamiltonians is based on a heuristic search of the most important one-electron states contributing to the energy bands in the vicinity of the Fermi level. The physical reason for the search for such a construct is rather obvious: namely the electron transitions between the bands in the close vicinity of the Fermi level are the low-energy excitations of the solid responsible for its observable properties controlled by its response to the low-energy perturbations. From certain point of view one may think that heavy numerical modeling tools (like VASP) merely provide parameters for such effective states and for the Hamiltonians describing their interactions. The original model Hamiltonian for the V(TCNE)2 magnet proposed in Ref. [] used such a restricted set of one-electronic states as described in the Introduction: namely the d-states of vanadium ions and the b3g( p*) LUMOs of the TCNE (singly occupied in the radical anion). The electronic structure calculations by VASP for the original structure fairly manifest the characteristic features of the required effective model (the p- and d-states). The effective Hamiltonian for the "principal" model constructed on the same set of one-electron states differs from the öriginal" model by phase factors (see below). The VASP calculations of the electronic structure of the "principal" arrangement of the building blocks of the organometallic ferromagnet also manifests the characteristic one-electron states required for constructing the effective model.
On the other hand the ëxperimental" structure differs quite strongly from the "principal" structure. The main difference is the formation of diamagnetic C4(CN)82- units which leads to effective isolation of the V(TCNE) sheets. Thus of the primary importance must be the Hamiltonian for the separate ruffled V-TCNE plane. One can employ for constructing the Hamiltonian for this plane the unit cell of the principal model dropping from it the TCNE unit extended to the b direction (and finally engaged in formation of the [TCNE]22-= C4(CN)82- dimers) and extending the rest in the a and c directions (neglecting the ruffling). In each such a sheet each metal ion is surrounded (coordinated) by four TNCE units which are in their turn coordinated to (surrounded by) four metal atoms.
On the vanadium sites it suffices to consider only the d-shells of the metal ions. The overlap of the d-shell with the s-orbitals of the TCNE's (including those implied in the model) ensures the standard two-over-three splitting of the d-shell characteristic for the octahedral environment. In case of vanadium it means that the three unpaired electrons in the d-shell occupy respectively three orbitals in the t2g-manifold. The dxy-, dxz-, and dyz-orbitals can be characterized by the normal to the plane in which each of the orbitals lays - z, h, and x - and subsequently used in the notation.
The model Hamiltonian for the V(TCNE)2 magnet, formulating the above
ideas, has the general form:
| (3) |
| |||||||||||||||||||||||||||||||||
The operator Hd(r) describes electrons in the d-shell of the
vanadium ion in the r-th unit cell:
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
The spin operators and spin-operator product terms are
defined by the well-known relations:
| |||||||||||||||||||||||||||||||||||||||
Operator Hda(r) describes electron hopping interaction between
the d-states of vanadium ions and the acceptor states. The dxy-state
represented by the zrs+ (zrs)
operators being of the (approximate) s-symmetry with respect to the ac plane (ruffled V-TCNE plane) has no overlap with the LUMO's of TCNE's
which are of the p-symmetry with respect to the same plane. Two others (dxz- and dyz-states represented respectively by the hrs+ (hrs) and xrs+ (xrs) operators) overlap with the LUMOs of two neighbor TCNE
units each. The phase relations between the orbitals involved in the model
lead to the following distribution of signs at the one-electron hopping
integrals between the orbitals. This produces the contribution to the
Hamiltonian of the form:
| (6) |
The sum of the contributions to the effective Hamiltonian in fact form that for isolated V-TCNE layers. Meanwhile, one should assume that certain indirect delocalization of the d-states in the b-direction is possible through the mediation of the [TCNE]22- units. On the symmetry grounds one may expect the following additional term in the Hamiltonian coupling different V-TCNE layers:
| (7) |
The suggested Hamiltonian defined by eqs. (3) - (7) suffices for analysis of
the basic results of the above numerical experiments performed with use of
the VASP package []. The latter implements in a way the general
self consistent field (SCF) approximation to the ëxact" ground state of the
Hamiltonian for the crystal. The SCF approximation reduces to factorizing
the terms of electron-electron interaction in the Hamiltonian according to
the rule:
| (8) |
In order to find the ground state for the crystal we apply the spin unrestricted Hartree-Fock (UHF) version of the SCF approximation, which produces the dispersion laws for the electron bands of the crystal and other quantities relevant for the solution.
The parameter
| (9) |
| ||||||||||||||||||||||||
The above (nonvanishing) effective bandwidths must be compared with the
on-site electron-electron interaction parameters Uaa and Udd which
play the rôles of the Stoner factors for the respective bands. Since
|
Now let us address the interaction between the completely spin-polarized (ferromagnetically ordered) bands spanned respectively by the TCNE LUMOs and the vanadium d-shells. As in the model [] the resonance (one-electron hopping) between the acceptor states and the d-states dominates. These interactions are described by the Hda(r) operators. Due to the structure of the spin-polarized SCF theory the corresponding hopping parameter tda appears in the result only through the corresponding effective band widths. Correspondingly the relative orientation of the electronic spins in the occupied d- and acceptor subbands is determined by the relative position of the acceptor subband corresponding to the spin projection t = ±s on the energy scale, provided the d-subbands corresponding to the spin projection s are occupied. This is determined by the requirement of minimizing the energy per formula unit also given in the Appendix A which yields that namely the acceptor spanned subband with t = -s is lower in energy and thus is populated.
The DOS coming from the model band Hamiltonian is depicted in Fig. 10. It fairly reproduces the most important features of numerical (VASP) DOS as shown in Fig. 9.
Finally we can address the relation between the Hamiltonian proposed in the present paper and that of Ref. []. One can notice that the key difference between the two is the presence of one more acceptor band in the model of Ref. [] coming from the second acceptor unit in the minimal unit cell required by stereochemistry. The following events can be described as follows. First of all we notice that despite the fact that the unit cell of the experimental structure is quadrupled as compared to that of the principal one it is enough to consider the unit cell doubled in the a-direction: 2a,b,c. Then according to the band folding scheme (see []) the number of bands doubles. The electron count is such that the acceptor bands which are almost flat (small dispersion) are half filled. This is a prerequisite for these bands to be sensitive to whatever perturbation mixing the states with the same value of k relative to the doubled unit cell. As one can check introducing a modulation of the one-electron hopping integrals between the acceptor orbitals localized on the TCNE units extended in the b-direction (in the principal structure they are assumed to be zero) can serve as a required perturbation. It splits the folded acceptor band into two bands such that one of them sinks among other occupied bands. In plain words this can be interpreted as pairing of electrons located on the TCNE[(-)\dot] units which upon pairing - forming the C-C bonds cannot be used to compensate the momenta located on the V ions.
The spin polarization of the solution is assured by the fact that if the a-subbands of three predominantly d-bands are completely filled the
b-subband of the predominantly acceptor bands is lower in energy and
for that reason is populated first. The total contribution of electron
hopping between the acceptor orbitals and the d-orbitals is obtained by
the integration of the contribution
| (11) |
In the present paper we performed numerical studies of the structural models of V-TCNE room temperature organometallic "ferromagnet" and analyzed the result in terms of the effective Hamiltonian for a selection of interacting atomic/molecular states and related band picture. It turns out that the results of numerical study can be fairly interpreted in terms of the proposed model.
It makes thus sense to apply the proposed models to analysis of a wider collection of experimental data available on this fascinating objects. First of all we notice that the spin polarization per unit cell (number of electrons with spin up minus that with spin down) which can be related with observed magnetization per formula of the compound at hand. We see that the calculation performed for the experimental structure 2 (spin polarization ca. 8 spins-1/2 per unit cell corresponding to two net unpaired electrons per V atom) is in a fair agreement with the magnetization measured in the V-TCNE compound prepared from V(CO)6. On the other hand the original material prepared from V(C6H6)2 manifests a weaker saturation magnetization, namely corresponding to ca. one unpaired electron per V atom. This allows to think about certain differences in the structures of two materials, described below, as dependent on the way of their fabrication. Nevertheless both experimentally observed values (ca. 10·103 emu·Oe·mol-1 and 6·103 emu·Oe·mol-1, respectively) both deviate from the theoretical values of 11.2 and 5.6 giving the magnetization produced by the integer number of net spin-polarized electrons in an assumption of the Landé being equal to 2. Remarkable is that in a more magnetized material the experimental magnetization seems to be smaller than the theoretical value whereas in a less magnetized material it is larger than the theoretical one.
Turning to analysis of the obtained band structure in the vicinity of the Fermi level we notice that the main contribution to relative shifts of the subbands corresponding to different spin-projections comes from the on-site electron-electron repulsion integrals Udd and Uaa. That means that the empty d-subbands are shifted upwards by Udd as compared to the filled ones, whereas the empty acceptor subband is shifted upwards by Uaa relative to the occupied one. The pair of the acceptor subbands is well seen in the DOS picture Fig. 9 which allows us to estimate the latter value to be 0.9 eV. This is in fair agreement with the estimate of Ref. [] based on magnetoresitivity data. One has to notice only that the formula used in Ref. [] to estimate the concentration of carriers is corrected to take into account the fact that the holes in the occupied subband arising when the electrons are excited to the conduction subband serve as carriers as well. This results in the factor of 1/2 (see Ref. []) relating the one-electron gap Uaa with the thermal activation energy entering the exponential function for the carriers' concentration.
When trying to extend their model [] to analysis of the V-TCNE compounds these authors argue that the vanadium compound must have somewhat different structure since the saturation magnetization in it is lower and approximately corresponds to two spins 1/2 compensating (interacting antiferromagnetically with) one spin 3/2 per formula unit. From this observation the authors of [] conclude that the interlayer interactions must be mediated by m4-TCNE radical-anions, as it has been suggested yet in []. From the point of view of the spin-Hamiltonian model Ref. [] the existence of the effective spins SV = 3/2 on the vanadium sites is assured by the strong intrashell interaction Jdd having order of 103¸104 K. In this respect our results agree with the suggestions of Ref. Her. On the other hand our numerical experiment shows that for the relaxed experimental structure the calculated magnetization fairly corresponds to experimental value obtained on V-TCNE material derived from V(CO)6. Incidentally, the magnetization values obtained numerically at intermediate structures on the "reaction path" depicted in Fig. 7 allows us to assume that some similar structures may present in the V-TCNE material derived from V(C6H6)2.
In the present paper we performed extensive numerical studies on thinkable structures of room-temperature organometallic magnet V(TCNE)2 and obtained corresponding densities of one-electron states, magnetizations and other characteristics of the considered structures. Model band Hamiltonian is developed for analysis and interpretation of numerical results and experimental data. Careful analysis of magnetic data in terms of the models is performed. A remarkable correspondence between experimental (structural and magnetic) data on V(TCNE)x·y solvent and numerical model is already observed: magnetization corresponding to two unpaired electrons per formula unit in remarkable agreement with experiment on V-TCNE material derived from V(CO)6 is obtained numerically for V(TCNE)2 taken in the relaxed experimental Fe(TCNE)2 structure; magnetization approximately corresponding to one unpaired electrons per formula unit in fair agreement with experiment on V-TCNE material derived from V(C6H6)2 is obtained numerically for intermediate structures between that of Fe(TCNE)2 and the structure proposed in Ref. [].
This work is performed with the partial support of the RFBR grants Nos 05-07-90067, 07-03-01128 extended to ALT. The generous support of the visit and stay of ALT at RWTH - Aachen University by DFG through the grant No DR342/18-1 is gratefully acknowledged. ALT is thankful to Drs B. Eck, J. von Appen, and A. Houben and to Mr. M. Gilleß en for their help in mastering the solid state electronic structure software at the IAC of the RWTH. The authors are thankful to Prof. J.S. Miller for his valuable comments on reported experimental data.
Applying the spin-unrestricted Hartree-Fock approximation to the model
Hamiltonian eq. (3) leads to the Fockian composed of two
individual components for the spatial states occupied by electrons of each
spin projection. The Fockian for a given spin-projection depends on the
population of the states with the opposite spin projection. In the
solid-state (translationally invariant) context these Fockian blocks
significantly simplify by going to the Fourier transforms (the Bloch sums)
of the Fermi operators aks and gks, g = z,h,x:
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First three rows of each 4×4 matrix block correspond to the Bloch
sums of the atomic z-, h-, and x-states, respectively, and
the fourth row corresponds to the Bloch sum of the acceptor states. The
Fockian dependence on k involves the form-factors:
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| (19) |
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Keeping in mind the symmetry-conditioned restrictions on the form of the averages imposed above, we obtain, after some algebra, the electronic energy of the crystal per unit cell in the UHF approximation:
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| (20) |