Transition of Iron Ions from High-Spin to Low-Spin State and Pressure-Induced Insulator Metal Transition in Hematite Fe 2 O 3

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1 ISSN , Journal of Experimental and Theoretical Physics, 7, Vol. 5, No. 5, pp. 35. Pleiades Publishing, Inc., 7. Original Russian Text A.V. Kozhevnikov, A.V. Lukoyanov, V.I. Anisimov, M.A. Korotin, 7, published in Zhurnal Éksperimental noœ i Teoreticheskoœ Fiziki, 7, Vol. 3, No. 5, pp ELECTRONIC PROPERTIES OF SOLIDS Transition of Iron Ions from High-Spin to Low-Spin State and Pressure-Induced Insulator Metal Transition in Hematite Fe O 3 A. V. Kozhevnikov a, A. V. Lukoyanov b, V. I. Anisimov a, and M. A. Korotin a a Institute of Metal Physics, Ural Division, Russian Academy of Sciences, ul. S. Kovalevskoi 8, Yekaterinburg, 6 Russia b Ural State Technical University, Yekaterinburg, 6 Russia mkorotin@optics.imp.uran.ru, michael.a.korotin@gmail.com Received April 6, 7 Abstract For the analysis of the electron structure of hematite under pressure, methods of the generalized transition state and local electron density approximation combined with the dynamical mean-field theory have been used. The transition of iron ions from the high-spin to low-spin state and the insulator metal transition observed in Fe O 3 at high pressures have been considered. It is shown that in the low-symmetry crystal structure of Fe O 3 experimentally revealed at high pressures a low-spin metallic state is also preferable. The theoretical results obtained agree well with experimental data. PACS numbers: 7.7.+a, 7.3.+h, 9.6.Pn DOI:.3/S INTRODUCTION Under the standard conditions, the hematite α-fe O 3, which is a component of the terrestrial mantle, crystallizes in the rhombohedral structure of the corundum type (space group R3 c) and exhibits properties of an antiferromagnetic insulator []. Under pressure, first a phase transition into an orthorhombic structure occurs at GPa [. At a pressure of approximately 5 GPa (which corresponds to the unit-cell volume that constitutes 8% of the equilibrium volume), there occurs a first-order phase transition in which the volume of the unit cell decreases by almost % and the crystal-lattice symmetry is reduced (a structure of the Rh O 3 -II type is formed). The transition is accompanied by the appearance of metallic conductivity, the stabilization of the low-spin state of the Fe 3+ ions, and the disappearance of the long-range magnetic order [3]. The nature of this transition has not yet been determined. The experimental data [3, ] obtained using Mössbauer spectroscopy, X-ray diffraction, and measurements of electrical resistance depending on pressure and temperature (R(P, T)) were interpreted as follows. The reason for first-order transition in Fe O 3 at a pressure on the order of 5 GPa is a considerable breakdown of d d correlations at Fe sites and the appearance of a metallic nonmagnetic state. This Mott transition is accompanied by a sharp decrease in volume. The structural transition in this case is not determinative, since in both crystal modifications (both at normal and high pressures) the Fe 3+ ion has an octahedral coordination. The scenario proposed in [3, ] was checked in [5] on the basis of an analysis of data obtained by X-ray emission spectroscopy and X-ray diffraction. This analysis indicates the opposite scenario; i.e., first, there occurs a structural transition with a decrease in volume and a change in the lattice symmetry, and only afterwards this transition induces a change in the electronic properties of Fe O 3. The theoretical investigations of the electronic structure of hematite were performed within various theoretical approaches and gave different (frequently controversial) results. Thus, calculations in the local spin density (LSDA) and generalized gradient correction (GGA) approximations qualitatively incorrectly reproduce the characteristics of even the ground state under the standard conditions, giving underestimated values of the local magnetic moment at the iron atom (.5µ B lower than the experimental values) and of the energy gap (six times less than the experimental data). Within the framework of the GGA, the structural transition together with the transition into the metallic ferromagnetic state with Fe ions in the low-spin state is predicted at a pressure of GPa [6], which is approximately four times less than the experimental value. Since narrow 3d bands of iron are present in the electronic structure of hematite, it is necessary to use numerical methods, which include the explicit allowance for correlation effects. The results of studies within the framework of the LSDA with explicit allowance for correlation effects (LSDA + U) or other similar approximations are presented in [6 9]. According to [7], the LSDA + U poten- 35

2 36 KOZHEVNIKOV et al. (a) e g a g e g tial describes many basic electronic properties of hematite better than the LSDA potential; at the value U = ev of the parameter describing the on-site Coulomb interaction of d electrons of Fe, a better agreement is reached with all experimental data; the orbital angular momentum of the d shell of iron is small (. µ B ). At the same time, the authors of [7] restricted themselves to the study of the electronic structure of Fe O 3 with the Fe 3+ ion only in the high-spin state and only under the standard conditions. The results of calculations in the GGA + U approximation are in best agreement with the experimental photoemission and inverse photoemission spectra of hematite at U = ev [6]. In [6], the transition of the iron ion from the high-spin to low-spin state under pressure was reproduced, and the O p Fe 3d regime of the formation of a semiconducting gap was obtained (chargetransfer gap). In spite of a detailed analysis of the structural and magnetic changes occurring with increasing pressure, the authors left opened the problem of the change in the orbital composition of the valence band. Moreover, at U > ev, the transitions are not reproduced at all. In [8, 9], a value U = 5 ev was used. Note also that the Coulomb parameter U determined from the fitting of theoretical curves to experimental photoemission spectra is equal to 8 ev according to [] or to 7 ev according to []. Thus, the theoretical studies of the phase transition in Fe O 3, in our opinion, is still far from the completion. The unresolved problems are as follows: how strong are the electronic correlations after structural transition at high pressures; what is the magnitude of the on-site Coulomb interaction U in the d shell of Fe and how it depends on pressure; whether it is possible to describe the low-spin state of Fe ions in the paramagnetic phase. In the corundum-type structure, the Fe 3+ ions are located in the centers of distorted octahedra formed by oxygen ions. The resulting symmetry group C 3i (point subgroup of the space group R3c of the corundum-type structure) causes a splitting of the fivefold degenerate d level into two doubly degenerate levels, which are transformed according to the and representa- e g Fig.. A schematic of (a) low-spin and high-spin states of the d shell of iron ions in Fe O 3. e g tions, and one nondegenerate level, which is transformed according to the a g representation. The schematic arrangement of d levels is displayed in Fig.. Since the magnitude of the trigonal distortion of octahedra is low, the designations of the cubic representations t g (= a g + e g ) and e g (= e g ) are frequently used. The electron configuration d 5 of iron ions can be realized in a low-spin or a high-spin state shown schematically in Fig.. Which of the two spin states is realized in iron ions depends on the relationship between the magnitudes of the parameters of splitting of the sublevels t g and e g by the octahedral crystalline field oct and on the exchange Hund interaction J. Neglecting the trigonal splitting trig (this quantity is approximately. ev, whereas oct = ev), it can be found from the ionic model that upon the transition from a high-spin to a low-spin state the energy of the electronic system changes by a value 6J oct caused by the transition of two electrons from the e g onto e g sublevels (see Fig. ). Thus, for the realization of the low-spin state it is required that a condition oct > 3J be fulfilled. The exchange Hund parameter J is a characteristic of an ion and remains constant even under pressure, whereas the splitting by the crystalline field can substantially change because of a decrease in the interatomic distance and, as consequence, because of the strengthening of the hybridization of O p Fe 3d orbitals under pressure. The spin state of iron ions in Fe O 3 can be different, depending on the applied pressure. Under the standard conditions, the exchange interaction is superior to the crystalline field (6J > oct ) and a high-spin state is realized, which is confirmed by the high values of the magnetic moments of the iron ions M =.6.8 µ B []. At high pressures, the exchange interaction is inferior to the crystalline field (6J < oct ) and a lowspin state is realized in iron ions, which is also predicted by numerous experimental data [3]. In this work, the above-described scenario of the spin transition from a high-spin into a low-spin state of iron ions in Fe O 3 and the insulator metal transition are investigated within the framework of two mutually complementing theoretical methods of calculation of the electronic structure. The calculations performed made it possible to determine the magnitude of the onsite Coulomb interaction in Fe O 3, magnetic moments, the total-energy curves, and spectral functions in the corundum-type structure at various applied pressures, and also the spectral functions of the real structure of the Rh O 3 -II type.. THEORY The method of generalized transition state (GTS) used for calculating the electronic structure and the total energy of hematite is presented in detail in [3]. In this method, the one-electron energies, which are the Lagrange multipliers in the minimization problem for JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 5 7

3 TRANSITION OF IRON IONS FROM HIGH-SPIN TO LOW-SPIN STATE 37 the electron-density functional [], are modified in such a way that their differences would correspond to the observed excitation energies of the system calculated with the use of the Slater approach [5]. Slater has demonstrated that the ionization energy of the system i (the energy of the removal of an electron to infinity) can be calculated through the one-electron energy of an orbital in the so-called half-filled transition state: i = Eq [ i = ] Eq [ i = ] ε i ( q i =.5). () Here, E[q i ] is the total energy of the system with an occupancy q i of the orbital i, and ε i is the one-electron energy of the corresponding orbital. The excitation energy of the system ij (the energy of electron transition from the orbital i onto the orbital j) can be calculated through the difference in the total energies of the system in the excited and ground states ij = Eq [ i =, q j = ] Eq [ i =, q j = ]. () Adding and subtracting the total energy of the system without an electron E[q i =, q j = ] and using equality (), the excitation energy of the system can be written through the difference in the one-electron energies in the transition state ij = ε j (.5) ε i (.5). (3) To calculate the energy of the orbital in the transition state, we leave two first members in the expansion of the one-electron energy in a Taylor series in the vicinity of the point q =.5 ε i (.5) = ε i ( q i ) + (.5 q i ) ε i q i q i =.5 () The transition-state correction (.5 q) ε/ q to the one-electron energies is negative for the occupied states and is positive for the empty states. Thus, the transitionstate correction increases the splitting between the empty and occupied states. The derivative of the oneelectron energy of the orbital with respect to the appropriate occupancy is calculated using an auxiliary calculation with the condition of a change in the occupancy of a selected orbital by.5 and a constant occupancy of the remaining orbitals. In the method of the generalized transition state [3], the Slater approach is extended to the case of infinite periodic systems. In this case, it is assumed that the excitations in the crystal occur on localized states which are selected in the form of Wannier functions W n [6] close to maximally localized ones [7, 8]. The method of constructing Wannier functions, their energies and occupancies used in this work is presented in detail in [3, 9]. For the correction of one-electron energies of Wannier functions, a functional of the generalized transition state is introduced in the following form:. E GTS [ ρ, q i,, q N ] = E LDA [ ρ] N -- (.5 q n ) ε n q n n = q n =.5 (5) Here, ε n and q n are the energy and the occupancy of the corresponding Wannier functions, respectively; N is the number of Wannier functions for which a correction for the transition state is introduced; and E LDA [ρ] is the functional of electron density in the local electron density approximation []. The one-electron energies of the Wannier functions, which are the derivatives of the total energy of the system with respect to the appropriate occupancy, are equal to ε n E GTS = = ε q n n LDA + δv n, (6) where δv n is the difference in the energies of the Wannier functions in the transition and ground states, δv n = (.5 q n ) ε n q n The variation of the functional of the generalized transition state leads to the following Hamiltonian: (7) The solution to the problem with this Hamiltonian is found iteratively until the fully self-consistent state is reached with respect to the total charge density ρ, occupancies q n of the Wannier functions, and derivatives of the energies of the Wannier functions with respect to the appropriate occupancies ε n / q n. For the investigation of the paramagnetic metallic state at high pressures, the LDA + DMFT method was used [], which combines the local-density approximation (LDA) with the dynamical mean-field theory (DMFT) []. The scheme of the LDA + DMFT method employed in this work was described in detail in [9]. From the results of self-consistent LDA calculations, the energies of the Wannier functions are calculated and a Hamiltonian is constructed for the problem which is further solved by the DMFT method. For the solution of the Anderson impurity problem which appears in the DMFT method, the quantum Monte Carlo (QMC) method is used. The computational difficulties of this method are mainly determined by the temperature value in question and by the number of orbitals. The total number of orbitals in the case in question can be substantially reduced by using a Wannier-function basis [3, 9].. q n =.5 Ĥ GTS = Ĥ LDA + W n δv n W n. N n =. JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 5 7

4 38 KOZHEVNIKOV et al. E, Ry % M, µ B.98 (a) V, %. Fig.. Dependence of the total energy E on the unit-cell volume for Fe O 3 with a corundum-type structure calculated by the GTS method for () high-spin and () low-spin configurations of iron ions Fe 3+. The volume at which there occurs a transition of the iron ions from the high-spin to the low-spin state is marked by the vertical dashed line. The % point on the abscissa axis corresponds to the experimental value of the unit-cell volume of Fe O 3 under standard conditions. The zero point on the ordinate axis is selected arbitrarily. 3. RESULTS AND DISCUSSION As was mentioned above, for calculating the oneelectron spectrum and the total energy of Fe O 3 with high-spin and low-spin configurations of the Fe 3+ ions for various volumes of the unit cell (which corresponds to the application of an external pressure in the model of uniform compression) we used the GTS method. Under the standard conditions, Fe O 3 has a rhombohedral structure of the corundum type with two formula units in the unit cell. This cell is sufficiently large for the calculations by the GTS method. Under the term sufficiently large, we mean here that the Wannier functions do not overlap upon translations into adjacent cells and, thus, the effect of the self-action of the Wannier functions in the calculation of the derivative of the one-electron energy with respect to the corresponding occupancy is excluded. The rhombohedral cell was used in all the calculations with all different volumes. The GTS method is realized within the framework of the method of linearized muffin-tin orbitals (LMTO) in the atomic-sphere approximation (ASA) []. The following parameters of the LMTO method were used for all the calculations. The radii of the iron atoms were taken to be equal to.85 au and remained constant for all values of the volume. As the basis functions, s, p, and 3d orbitals of iron were taken. The radii of the oxygen atoms and of the empty spheres were scaled equally in order to satisfy the condition of the equality of the sum of the volumes of atomic spheres to the volume of the unit cell. As the basis functions, s and p orbitals of oxygen were selected. For the calculation of Fe O 3, the following set of Wannier functions was determined: five functions with the symmetry of d orbitals from each of the four atoms of iron and three functions with the symmetry of p orbitals from the six atoms of oxy V, % Fig. 3. Dependence of the magnetic moment M of the iron ion Fe 3+ on the unit-cell volume for (a) low-spin and high-spin states of configurations in the corundum-type structure. gen, which give 38 Wannier functions in total. The selected set of functions describes the band structure of Fe O 3 in a wide energy range near the Fermi level from to +6 ev. For the calculations by the GTS method, Wannier functions [3, 9] of the 3d states of iron with the symmetry e g, e g, and a g were constructed. The high-spin and low-spin states were specified using the starting values for the additional potential δv (7) for the Wannier functions. In the case of the high-spin state, the potential for five functions of one spin was assigned negative, and for five functions of another spin, positive. The low-spin state was specified as follows: for three functions with the symmetry e g and a g for one spin and for two functions with the symmetry e g for the second spin, the starting potential δv was selected negative; for the remaining five functions, the value of δv was assigned positive. The use of a self-consistent procedure results in two stable solutions for the low-spin and high-spin states. Figure displays the results of GTS calculations of the total energy E for Fe O 3 in the corundum-type rhombohedral structure as functions of the unit-cell volume V for the high-spin and low-spin configurations. The high-spin state has a minimum energy in the vicinity of the point %, which indicates the correct description of the total energy within the framework of the GTS method. The value of the forbidden JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 5 7

5 TRANSITION OF IRON IONS FROM HIGH-SPIN TO LOW-SPIN STATE 39 DOS, state/ev f.u. 8 (a) DOS, state/ev f.u. 8 (a) DOS, state/ev spin atom (d) DOS, state/ev spin atom 8 6 (d) Fig.. Densities of states (DOS) of Fe O 3 in the corundumtype structure for the high-spin configuration of iron ions calculated by the GTS method: (a) total DOS of iron ions; (b, c) partial densities of states of t g and e g states of iron ions; and (d) the p state of the oxygen ion. The dark portions of the graphs refer the % unit-cell volume; lines show the results corresponding to 8% unit-cell volume. The curves for different volumes have been superimposed with the Fermi level taken as the zero point in the abscissa axis. energy gap at normal pressure (at the unit-cell volume of %) is equal to 3. ev for the high-spin configuration and.8 ev for the low-spin one. The experimental value of the band gap is equal to.88 ev [3] for the indirect transition and.75 ev [3] or..7 ev [] for the direct gap. When the unit-cell volume composes 78% of the volume V at the normal pressure, the intersection of the curves of the total energy for the highspin and low-spin configurations can be interpreted as a phase transition. A shortcoming of the GTS calculation is the prediction of a dielectric state of the low-spin configuration of Fe O 3 below the transition point (for the unit cell volume equal to 75% of V, the value of the energy gap is equal to. ev in the case of the lowspin configuration). The reason for this can be the ferromagnetic long-range magnetic order taken in the calculation, while according to the experimental data, Fe O 3 is a paramagnetic material below the transition point. As has already been discussed in the Introduction, the criterion for the realization of the low-spin state of iron ions that follows from the simplest ionic model is the fulfillment of a condition oct > 3J. The exchange parameter J in this work was determined from the calculations with fixed occupancies and was found to be ev; therefore, the critical value for the splitting by the Fig. 5. Same as in Fig. for the low-spin configuration of iron ions. octahedral crystalline field oct is a value of 3 ev. This quantity was determined when calculating the energies of the Wannier functions as the difference in the energies of the functions with the t g and e g symmetries and was found to depend on the unit-cell volume and the hematite-type crystal structure of Fe O 3. For the corundum-type phase at normal pressure (V = %), the value oct is equal to approximately.5 ev (in this case, oct + trig =.6 ev); at V = 8%, oct =.55 ev; and at V = 75%, oct =.8 ev, which assumes that it is the high-spin state of the iron ion that is more advantageous. However, for the phase with the corundum-type structure with a volume of the unit cell equal to 7% of the volume at the normal pressure, the value of oct exceeds the critical value of 3. ev and is approximately 3. ev. Thus, the ionic model qualitatively well agrees with the results on the total energy obtained by the GTS method (see Fig. ). A small quantitative deviation (less than 8%) of the critical volume at which spin transition occurs can be explained by essential simplifications assumed in the model, in particular, by the neglect of the trigonal splitting trig. Figure 3 displays the dependence of the magnetic moment of the iron ion calculated by the GTS method on the unit-cell volume. For the high and low-spin configurations, the value of the magnetic moment M monotonically decreases with decreasing volume, which is explained by increasing p d hybridization as a result of the contraction of the nearest oxygen environment of iron ions. For the equilibrium volume of the phase with the corundum-type structure, the calculated value of the magnetic moment (.5 µ B ) agrees well with the experimental values (.6.8) µ B []. JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 5 7

6 KOZHEVNIKOV et al. DOS, state/ev atom (a) Spectral function, ev. (a) Fig. 6. Partial densities of electronic 3d states (DOS) of iron ions for Fe O 3 with a structure of the corundum type derived from LDA calculations for the a g (solid line), e g (dashed line), and e g (dark regions) subbands. The unitcell volume is (a) 7% (the width W of the subband t g = a g + e g is.57 ev), 75% (W =.8 ev), and 8% (W =.5 ev). The Fermi level is shown by the vertical dashed line. The graphs of the densities of states for the highstate and low-spin configurations for the unit-cell volumes which compose and 8% of V presented in Figs. and 5 indicate that a decrease in the unit-cell volume leads to a significant broadening of the energy bands, which can be one of the reasons for phase transition. It is evident from the above data that Fe O 3 in the high-spin configuration is a charge-transfer insulator, while in the low-spin configuration, it is a d d transition that corresponds to the minimum excitation. When investigating the magnetic and spin states of Fe O 3 at a high pressure, the application of the GTS and LSDA + U methods does not give reliable results, since it requires that a long-range magnetic order (ferromagnetic or antiferromagnetic) be specified and, in view of this limitation, cannot yield a paramagnetic metallic state. The experimental data [3] testify to the low-spin state of iron ions at high pressures. From a microscopic viewpoint, this indicates the filling of the t g subband of the iron ion by five d electrons (see Fig. ). At the same time, the e g subband proves to be unoccupied, and the energy corresponding to the center of gravity of the subband is located considerably higher than the Fermi level (by ev according to LDA calculations, see Figs. 6 and 8 below). Therefore, to describe the lowspin state in the DMFT method it suffices to use only Fig. 7. Spectral functions obtained from the LDA + DMFT results for Fe O 3 in the phase with a corundum-type structure for three unit-cell volumes ((a) 7; 75; and 8%) for the e g (dashed lines) and a g (continuous lines) subbands. As the method of solving the impurity problem, the quantum Monte Carlo method was used. The calculations were performed for the temperature of 6 K. The position of the Fermi level is shown by the vertical dashed line. orbitals of t g symmetry. For the selected set of orbitals and configurations of shielding electron shells, it is necessary to determine the parameters of the direct (U) and exchange (J) interactions which are required in the DMFT method. This can be made on the basis of LDA calculations with fixed occupancies of the orbitals [5]. The calculations carried out for Fe O 3 show that for the partially filled t g subband the value of the parameter of direct Coulomb interaction U is.3 ev for all the volumes used for the corundum-type phase; the exchange parameter J is equal to ev. For the investigation of the spin transition, we examined three values of the unit-cell volumes of Fe O 3 in the corundum-type structure close to the experimentally determined critical volume (which composes 8% of the equilibrium volume V ), in the model of uniform compression. Figure 6 displays the partial densities of the a g and e g states and also the contributions of e g states (dark fields) as determined from LDA calculations. In Fig. 6, the quantity W denotes the width of the correlated t g subband. It turns out that a change in the unit-cell volume by % leads to a broadening of the t g subband by more than %. For the metal insulator transition, in a number of cases one of the main signs of the regime of metal, insulator, or a boundary state (with the formation of a quasi-particle peak at the Fermi level) is the relationship between the parameters U and W for the correlated band []. An increase in the width W of the band leads to the appearance of a metallic state with a greater spectral density at the Fermi level, which, apparently, is observed in the corundum structure of Fe O 3 under pressure. Figure 7 shows the results of cal- JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 5 7

7 TRANSITION OF IRON IONS FROM HIGH-SPIN TO LOW-SPIN STATE DOS, state/ev spin atom DOS, state/ev f.u. 8 (a) 8 6 (d) Fig. 8. (a) Total and (b, c) partial densities of states (DOS) of t g and e g states of iron ions and (d) of the p state of oxygen ion for the experimentally observed structure of Fe O 3 of the Rh O 3 -II type obtained in an LDA calculation. The position of the Fermi level corresponds to the zero point of the abscissa axis. culations by the LDA + DMFT method. It was found that in given volumes there indeed exists a preferable low-spin state with an almost completely filled e g bands, partially filled a g subband, and the metallic nature of the spectral functions at the Fermi level. Note that as the unit-cell volume decreases from 8 to 7% of the equilibrium volume, there is observed an increase in the metallic nature of the obtained paramagnetic (low-spin) solution due to an increase in the spectral density of the a g orbital at the Fermi level. However, the experiments show that, the compound undergoes, besides a spin transition, a structural transition from the phase with a corundum-type structure into the phase with a structure of the Rh O 3 -II type [3. Additional calculations were carried out to check the possible influence of this crystal structure on the lowspin state. Below, we present results obtained by the LDA + DMFT for Fe O 3 with the structure of Rh O 3 -II type. Figure 8 shows the total and partial densities of states derived from LDA calculations. On the whole, the t g and e g subbands for these structures are similar. However, in the Rh O 3 -II phase, the width of the t g subband is.7 ev, which exceeds the appropriate value for the corundum-type structure and is caused by its lower symmetry. The spectral functions obtained by the LDA + DMFT method (Fig. 9) show that the spectral density at the Fermi level increases in comparison with the case of the corundum-type structure (cf. Fig. 7); in this case, two e g orbitals are almost completely occupied, as in the case of the corundumtype phase, and the a g orbital is occupied only partially and gives a large metallic contribution at the Fermi Spectral function, ev Fig. 9. Spectral functions obtained from the LDA + DMFT results for the Rh O 3 phase of the Fe O 3 compound. As the method of solving the impurity problem, the quantum Monte Carlo method was used. The calculations were performed for the temperature of 6 K. The continuous and dashed lines correspond to the e g and e g subbands, respectively; the dark region, to the a g state. The position of the Fermi level is shown by the vertical solid line. level. In the spectral density of this orbital, both the quasi-particle peak at the Fermi level and the lower and upper Hubbard bands can clearly be identified. The results obtained confirm the presence of a low-spin state in Fe O 3 in the structure of the Rh O 3 -II type, whose electron spectrum has a high density at the Fermi level characteristic of metals.. CONCLUSIONS The transition from the high-spin to low-spin state and the insulator metal transition in hematite Fe O 3 under pressure were investigated by the method of generalized transition state (GTS) and also within the framework of the local electron density approximation (LDA) in combination with the dynamical mean-field theory (LDA + DMFT). As a result of calculations of the corundum-type phase at different pressures by the GTS method in the Wannier-function basis, it has been shown that as the unit-cell volume of hematite contracts to 78% of the equilibrium volume characteristic of Fe O 3, it is the low-spin state that becomes more advantageous in terms of the total energy, whereas at lower pressures it is the high-spin state that is realized in the compound. The values of the magnetic moment and of the forbidden energy gap obtained for the phase with the corundum structure within the framework of the GTS method agree well with the experimental data. The value of the Coulomb parameter U for these volumes remains virtually constant (equal to.3 ev) for the t g shell of the iron ion, which confirms the presence of strong electron correlations in the high-pressure phases of Fe O 3. For three different unit-cell volumes of Fe O 3 in the corundum-type structure at pressures lower than that corresponding to the transition into the metallic state, the LDA + DMFT calculations give a low-spin metallic paramagnetic solution, which also agrees with experiment. With a further decrease in volume, the metallic nature of the state is strengthened due JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 5 7

8 KOZHEVNIKOV et al. to the broadening of the correlated t g bands. In the calculations for the phase with the corundum-type structure, the metallic paramagnetic state is retained up to a volume equal to 8% of the unit-cell volume at normal pressure. It follows from the calculations of the electron structure that the structural transition of Fe O 3 into a phase with the structure of the Rh O 3 -II type is not the basic factor responsible for the insulator metal and low-spin high-spin state transitions. However, the results of calculations by the LDA + DMFT method show that, due to the larger width of the correlated t g bands, this structure additionally stabilizes the paramagnetic state. ACKNOWLEDGMENTS This work was supported in part by the Russian Foundation for Basic Research (project nos , 7--, NWO 7.6.5) within the framework of Project 8, Subprogram no. 3, Program no. 9 of the Presidium of the Russian Academy of Sciences. One of us (A.V.L.) is grateful to the Dynasty Foundation and the International Center of Basic Physics (Moscow); M.A.K. is grateful to the Russian Science Support Foundation. REFERENCES. C. G. Shull, W. A. Strauser, and E. O. Wollan, Phys. Rev. 83, 333 (95).. H. Liu, W. A. Caldwell, L. R. Benedetti, et al., Phys. Chem. Miner. 3, 58 (3). 3. M. P. Pasternak, G. Kh. Rozenberg, G. Yu. Machavariani, et al., Phys. Rev. Lett. 8, 663 (999).. G. Kh. Rozenberg, L. S. Dubrovinsky, M. P. Pasternak, et al., Phys. Rev. B 65, 6 (). 5. J. Badro, G. Fiquet, V. V. Struzhkin, et al., Phys. Rev. Lett. 89, 55 (). 6. G. Rollmann, A. Rohrbach, P. Entel, and J. Hafner, Phys. Rev. B 69, 657 (). 7. M. P. J. Punkkinen, K. Kokko, W. Hergert, and J. J. Väyrynen, J. Phys.: Condens. Matter, 3 (999). 8. A. Bandyopadhyay, J. Velev, W. H. Butler, et al., Phys. Rev. B 69, 79 (). 9. V. V. Mazurenko and V. I. Anisimov, Phys. Rev. B 7, 83 (5).. A. Fujimori, M. Saeki, N. Kimizuka, et al., Phys. Rev. B 3, 738 (986).. A. E. Bocquet, T. Mizokawa, T. Saitoh, et al., Phys. Rev. B 6, 377 (99).. J. M. D. Coey and G. A. Sawatzky, J. Phys. C, 386 (97). 3. V. I. Anisimov and A. V. Kozhevnikov, Phys. Rev. B 7, 755 (5).. W. Kohn and L. J. Sham, Phys. Rev. [Sect. A], A33 (965). 5. J. C. Slater, Quantum Theory of Molecules and Solids (McGraw-Hill, New York, 97), Vol.. 6. G. H. Wannier, Phys. Rev. 5, 9 (937). 7. N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 87 (997). 8. K. Wei, H. Rosner, W. E. Pickett, and R. T. Scalettar, Phys. Rev. Lett. 89, 67 (). 9. V. I. Anisimov, D. E. Kondakov, A. V. Kozhevnikov, et al., Phys. Rev. B 7, 59 (5).. K. Held, I. A. Nekrasov, G. Keller, et al., Phys. Status Solidi B 3, 599 (6).. A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 3 (996).. O. K. Andersen, Phys. Rev. B, 36 (975). 3. F. P. Koffyberg, K. Dwight, and A. Wold, Solid State Commun. 3, 33 (979).. Keu Hong Kim, Sung Han Lee, and Jae Shi Choi, J. Phys. Chem. Solids 6, 33 (985). 5. W. E. Pickett, S. C. Erwin, and E. C. Ethridge, Phys. Rev. B 58, (998). Translated by S. Gorin JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 5 No. 5 7