# Metal-Insulator Transition Temperatures and Excitonic Phases in Vanadium Oxides

Authors: A. L. Pergament
Journal: ISRN Condensed Matter Physics
 (2011)

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## Abstract

The possibility of applying the excitonic insulator model to the description of metal-insulator transitions in vanadium oxide Magneli phases is investigated. Based on the Animalu transition metal model potential, the equation for the constant of Coulomb interaction in the theory of excitonic insulator is modified. It is shown that this theory allows the transition temperatures of all the oxides to be calculated. The conformity of the theory with the experimental data concerning the effective mass values for electrons in vanadium oxides is discussed.


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## Body

## 1. Introduction

Studies of metal-insulator transitions (MITs) challenged the well-established division of materials into metals and dielectrics according to the type of electron spectrum and occupation of a band by collective electrons [1]. At present, one can find quite a few examples of materials “violating” the abovementioned division. Among those are compounds exhibiting MITs and revealing properties of either a metal in certain external conditions (temperature, pressure) or an insulator in other conditions. The transition between these two states is generally followed by a drastic change of electrical conduction (by a few orders of magnitude) as well as of other physical characteristics.A lot of transition- and rare-earth-metal compounds undergo insulator-metal transitions at a particular temperature,Tt. Of vanadium oxides within the narrow composition range, from V2O3 to V6O13 (the “Magneli phases” VnO2n−1, n=2–9), at least eight show such transitions at different temperatures [1]. Vanadium dioxide could be formally included in this pattern with n=∞, whereas V6O13 belongs to another homology series (the so-called “Wadsley phases” V2nO5n−2), the limiting term of which with n=∞ is the insulator V2O5.The surprising thing is that the MIT temperatures in vanadium oxides, as well as in other oxides of transition metals, are not calculated in the context of current theories, unlike superconductivity, where the BCS formula predicts the value ofTc of many superconducting materials with sufficient accuracy, at least, for elementary metals and some compounds, except for the high-Tc superconductors.In the present work I will try to give some particular formulae allowing the MIT temperature to be calculated for transition metal compounds, vanadium oxides, in the first place. As a matter of fact, there is no end of “true stories” in the world, in no way remarkable among them. I would not have bothered the reader with mine, had it not been for papers [1–4] which played a special part in all of this. My intention here is to provide some general formulae for calculating Tt of vanadium oxides on the basis of the excitonic insulator model [2–8], using the model pseudopotential concept [8].


## 2. General Theory of an Excitonic Insulator

To gain an insight into a possible electron spectrum, use could be made of the scheme offered in [1] that enabled one to calculate with a high degree of accuracy the Neel temperature, TN, for all vanadium oxides of the Magneli series [1, 2]. Each vanadium ion can give for a chemical bond 5 electrons, of which two supplement the p-shell of each of the (2n-1) oxygen ions, and, therefore, the number of band electrons per vanadium atom, ρ, is easy to obtain as(1)ρ=5n-2(2n-1)n=n+2n.
It is relation (1) that we will apply to calculate the free electron density (in a metal phase) for each of the compounds in question.The concept is based on the excitonic insulator model often used to describe MITs in various systems, including, for instance, doped chalcogenides of Sm [9], potassium molybdenum bronzes [10], and a number of other markedly different systems [11–14], including vanadium oxides [2–4, 8, 15].In the early studies devoted to the excitonic insulator theory (see, e.g., [5–7] and references therein), it was shown that the energy gap width of an excitonic insulator is given by a simple relation that is actually quite similar to the BCS formula:(2)Δ=ℏω0exp⁡⁡(-1μ),
where the parameter ω0 is proportional to the plasma frequency of free carriers, and μ is the constant of Coulomb attraction. In the weak-interaction approximation in an ideal model, the latter can be written as [5, 7](3)μ=αln⁡⁡α-2,
where α is a nondimensional variable proportional to ρ-1/3; in fact, α=(πaHkF)-1, with aH being the effective Bohr radius, and kF—the Fermi wave vector. Knowing the specific gravity for each oxide, as well as the value of ρ, it is possible to determine the charge carrier density from (1), and kF is just proportional to the latter to the power 1/3.The precise equation for the constant of Coulomb interaction is [16](4)μ=α∫02kFdkkε(k)=αΩ4π∫02kFU(k)kdk.Hereε is the dielectric function of free carriers, and Ω is a normalizing volume. If we take the potential of Coulomb interaction between two-point charges as U(k)=(4π/Ω)/(k2+λ2), then it is straightforward to obtain an equation for μ(α) from expression (4). The result is as follows:(5)μ=αln⁡⁡(1α+1)2.Equation (5) virtually agrees with (3) in the limit of α→0, but at high values of α (corresponding to low electron densities), the gap widths obtained from (2) differ significantly in the cases of μ calculated using (3) and (5), see [8] and curves 1 and 2 in Figure 1.Figure 1
Δ as a function of α. 1: with μ from (3); 2: (5); 3: (7) with R=1 a.u. Curve 3 tends to zero at αc=0.3-0.4 which corresponds to the Mott criterion (8).Now we consider a system in which the normal ground state of each atom is a spherically symmetric one-electrons-state. The exact potential of such an atom is just the empty-core pseudopotential:(6)UEC(k)=4πk2ε(k)Ωcos⁡⁡(kR).
Here R is the radius of the empty core (for ordinary electron-hole interaction, it is equal to the Bohr radius aH if ε and m*/me are equal to unity). The dielectric response ε(k) characterizes the shielding effect in the Hartree-Fock approximation with the Lindhard factor g(k)=1-k2/(2k2+kF2+2kF/π). Replacing the potential in (4) by the form taken from (6), we obtain the final form of the constant μ and, accordingly, the value of the exciton insulator energy gap width from (2). We thus arrive at the following form of the constant of Coulomb interaction (introducing another nondimensional variable x=k/kF):(7)μ=αR∫02xcos⁡(Rx/πα)x2+4(α/R)g(x)dx.The results of numerical calculations ofΔ, with μ taken from (7), are presented in [8] and shown to comply in this case with the Mott criterion [14]:(8)aH(ρΩ)1/3≈0.25,
where aH is the effective Bohr radius, and Ω is the volume per one vanadium atom, that is, (ρ/Ω) is in fact the electron density. This is where the main difference lies between the result obtained here and the ones presented in [5, 7] (see Figure 1 and discussion in [8]).


## 3. Vanadium Oxides (Magneli Phases)

Next, it has been suggested in [8] that for rather complicated systems, such as vanadium oxides, it is necessary to use in (4) a more realistic model pseudopotential (MPP), UMPP(k), as compared to a simple UEC(k). A model potential of a transition metal introduced by Animalu [17], for instance, could act as such a potential, but the effective mass is not determined in this case and should be considered as a parameter. Note that m* appears in expressions for α, ω0, and U(k) in (2) and (4).As was briefly discussed in [8], calculations of the interaction constant for vanadium dioxide using (4), with U(k) in the form of the Animalu MPP for V [17] (see the appendix), yield the values of Δ in the range of 0.1 to ~1 eV (depending on the choice of m* in the range of 1 to 10me).This appears to be quite an acceptable agreement if the correlation energy in VO2 is considered to be ~0.1 eV [15]. It can be shown that in this case the temperature of the MIT for VO2 is Tt=Δ/3.53kB=340 K (which is equal to the experimental value of Tt [1, 14]) for m*~1.3me. This estimation of m* appears to be quite reasonable (the experimentally obtained values for vanadium dioxide vary from 0.5 to 3me [1–4, 18, 19]), and the approach proposed may therefore prove to be an acceptable model for describing a MIT in vanadium and other transition-metal oxides. The same is true for lower oxides VnO2n−1 (except for V7O13): m*~(1–5)me, see Figure 2. The experimental values of Tt for vanadium oxides are also presented in Table 1, along with the calculated values of m*, for the sake of convenience.Table 1
Parameters of vanadium oxides.
OxideTt, Kαm*/meVOMetal——V2O31500.4621.14V3O54500.5050.97V4O72400.5261.19V5O91300.5411.41V6O111700.5511.35V7O13Metal0.5605 (forTt=0.1 K)V8O15700.5651.67VO23400.6161.25V6O131500.7141.67Figure 2
1: experimental transition temperatures for vanadium oxides. 2: calculatedTt’s using the Animalu potential with m*=me, but  with varied valence z (see the appendix). 3: effective masses (in the units of the mass of a free electron) calculated from the experimental values of  Tt’s; see also Table 1.  4: parameter α  as a function of the oxygen stoichiometric index, calculated using (1) and specific gravities of all the oxides.Thus, after knowing the effective mass, one can determine the correlation gap and hence the transition temperature asTt=Δ/3.53kB, using (4) and BCS-like (2). It should be however noted that the effective masses are not necessarily known. Even for the much-studied vanadium dioxide, the experimental values obtained by various authors differ nearly by an order of magnitude, as seen above. At the same time the calculated value of Δ depends rather strongly on m* (see Figure 3). The difficulties and vagueness while determining an effective mass are well known; these are due to both the quality of a sample and the measuring technique. Optical methods (from measurements of the plasma frequency) and transport oscilloscopic measurements yield as a rule two different magnitudes, which is no wonder, since the values of m* of DOS and band m* should not necessarily coincide [19]. Furthermore, in anisotropic substances this physical quantity depends on direction.Figure 3
Transition temperatures calculated with the Animalu MPP as a function of the oxygen content in vanadium oxides withm*=1 (1), 1.5 (2), and 2 of me (3).Figure 4
Transition temperature as a function ofα: 1—for z=5, 2—for z=1; m*=me.


## 4. Conclusion

It has been shown in the present work that using the Animalu’s TMMP (transition metal model pseudopotential) forV, in the context of the excitonic insulator model, it is possible to calculate correlation energies (from BCS-like formula (2)) and hence the MIT temperature, Tt=Δ/3.53kB for vanadium oxides. Fair agreement between the theory and experiment is observed when choosing the values of the carriers’ effective mass in the range from 1 to 5 free electron masses, which does not contradict the well-known experimental data. The sharp increase of m* for V7O13 may be due to the effective mass divergence close to the MIT (which renders the transition unobservable, drastically decreasing its temperature nearly to absolute zero).It is more difficult to apply the technique described to multicomponent compounds of transition metals, for example, to CMR manganites or thiospinel Cu(Ir1-xCrx)2S4 [20], where a few kinds of transition metal atoms are available. In general, it is safe to say that the problem discussed is in a certain sense more complicated than the similar problem of Cooper pairing in the theory of superconductivity, where only two electrons and a virtual phonon are involved. In the case of an excitonic phase transition for a d-metal compound, we have to deal with an electron and not with a point-like positively charged hole (or an empty core potential), but we should work with a complex TMMP.Finally, it should be noted that the problem of the Mott MIT has recently acquired particular significance (or “particular popularity”), mainly because of the intensive studies in the field of MITs in 2D electron gases [21] that are potentially of great importance for the development of nanoelectronics. A purely scientific interest should not be neglected at that. It is important, for instance, to take into account the results obtained here when studying the effect of electric field on the Mott MIT in vanadium oxides [22]. As for the transition of a metal to the excitonic insulator state, it fits well into the framework of a general paradigm for the Mott MITs.



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*Source: 605913-2011-11-17.xml*