Metal-Insulator Transition Temperatures and Excitonic Phases in Vanadium Oxides

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.

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, 𝑇_𝑡. Of vanadium oxides within the narrow composition range, from V₂O₃ to V₆O₁₃ (the “Magneli phases” V_nO_2n−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” V_2nO_5n−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 of 𝑇_𝑐 of many superconducting materials with sufficient accuracy, at least, for elementary metals and some compounds, except for the high-𝑇_𝑐 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 𝑇_𝑡 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, 𝑇_𝑁, 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 as𝜌=5𝑛−2(2𝑛−1)𝑛=𝑛+𝑛2.(1) 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:Δ=ℏ𝜔₀exp−1𝜇,(2) 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]𝜇=𝛼ln𝛼⁻²,(3) where 𝛼 is a nondimensional variable proportional to 𝜌^−1/3; in fact, 𝛼=(𝜋𝑎_H𝑘_F)⁻¹, with 𝑎_H being the effective Bohr radius, and 𝑘_F—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 𝑘_F is just proportional to the latter to the power 1/3.

The precise equation for the constant of Coulomb interaction is [16]𝜇=𝛼^2𝑘F₀𝑑𝑘𝑘𝜀(𝑘)=𝛼4Ω𝜋^2𝑘F₀𝑈(𝑘)𝑘𝑑𝑘.(4)

Here 𝜀 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:𝜇=𝛼ln1𝛼+1².(5)

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-electron s-state. The exact potential of such an atom is just the empty-core pseudopotential:𝑈_EC(𝑘)=4𝜋𝑘²𝜀(𝑘)Ωcos(𝑘𝑅).(6) Here 𝑅 is the radius of the empty core (for ordinary electron-hole interaction, it is equal to the Bohr radius 𝑎_H if 𝜀 and 𝑚^∗/𝑚_𝑒 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+𝑘²_F+2𝑘_F/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/𝑘_F):𝜇=𝛼𝑅²⁰𝑥cos(𝑅𝑥/𝜋𝛼)𝑥²+4(𝛼/𝑅)𝑔(𝑥)𝑑𝑥.(7)

The results of numerical calculations of Δ, with 𝜇 taken from (7), are presented in [8] and shown to comply in this case with the Mott criterion [14]:𝑎_H𝜌Ω^1/3≈0.25,(8) where 𝑎_H 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), 𝑈_MPP(𝑘), as compared to a simple 𝑈_EC(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 10𝑚_𝑒).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 𝑇_𝑡=Δ/3.53𝑘_B=340 K (which is equal to the experimental value of 𝑇_𝑡 [1, 14]) for 𝑚^∗∼1.3𝑚_𝑒. This estimation of 𝑚^∗ appears to be quite reasonable (the experimentally obtained values for vanadium dioxide vary from 0.5 to 3𝑚_𝑒 [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 V_nO_2n−1 (except for V₇O₁₃): 𝑚^∗∼(1–5)𝑚_𝑒, see Figure 2. The experimental values of 𝑇_𝑡 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.

Figure 2 

1: experimental transition temperatures for vanadium oxides. 2: calculated 𝑇𝑡’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 as 𝑇_𝑡=Δ/3.53𝑘_B, 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 with 𝑚∗=1 (1), 1.5 (2), and 2 of 𝑚𝑒 (3).

Figure 4 

Transition temperature as a function of 𝛼: 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) for 𝑉, 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, 𝑇_𝑡=Δ/3.53𝑘_B 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(Ir_1-xCr_x)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.

Appendix

For construction of TMMP, Animalu [17] used the atomic potential in a modified form of the well-known Heine-Abarenkov MPP [23]:𝑉_MPP(𝑟)=⎧⎪⎨⎪⎩−_𝑘−𝑧A_𝑘P_𝑘,for𝑟≤𝑅_𝑚,𝑟,for𝑟>𝑅^𝑚,(A.1) where 𝑃_𝑘 is a component of the angular momentum projection operator, and 𝐴_𝑘 is a parameter representing the depth of a potential well. These parameters, as well as 𝑅_𝑚 and some other constants, were either found ab initio or fitted from the experimental phonon spectra of transition metals [17].

The Fourier-image of this potential, 𝑈_MPP(𝑘) represents a rather long expression containing the parameters 𝐴_𝑘(𝑘=1,2,3), 𝑅_𝑚, and a number of other constants which we will not write here completely. All of these parameters for 𝑉 were used in the present work, except for 𝑚^∗, that is taken in [17] to be equal to unity for all the metals. Also, the valence 𝑧 (which is obviously taken to be equal to 5 for vanadium metal in [17]) is varied in the present work in compliance with the real value of 𝑧 of a vanadium ion in a particular V_nO_2n−1 compound (see Figure 2, curve 2, and Figure 4).

The calculating procedure consisted in computing Δ and then 𝑇_𝑡 for each oxide, using formulae (2) and (4) with 𝑈(𝑘) being the Animalu TMMP, varying 𝑚^∗ so that the corresponding adjustment yielded finally the real (experimental) value of 𝑇_𝑡. Note that, in principle, one could get some unrealistic magnitudes of an effective mass, for example, 0.001 or 100 of 𝑚_𝑒. Nevertheless, this procedure yielded, as was said, 𝑚^∗∼1𝑚_𝑒.

Acknowledgments

This work was supported in part by the Ministry of Education and Science of Russian Federation through the “Scientific and Educational Community of Innovation Russia (2009–2013)” Program, contracts no. 16.740.11.0562, 14.740.11.0895, 02.740.11.5179, 14.740.11.0137, P1156, P1220, and 02.740.11.0395, and the “Development of Scientific Potential of High School” Program, Project no. 12871. The author also is grateful to V. P. Novikova and D. V. Artyukhin for the help with paper preparation.