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S. Biermann, A.I. Poteryaev, A. Georges, A.I. Lichtenstein, Phys. Rev. Lett. 94 026404 (2005)
Vanadium dioxide is a prominent
example for a material
exhibiting a metal-insulator transition as a function of temperature:
it undergoes a first-order transition from a high-temperature metallic
phase to a low-temperature insulating phase at almost room-temperature
(T=340 K). The resistivity jumps by several orders of magnitude through
this transition, and the crystal structure changes from rutile
(R-phase, see Fig. 1) at high-temperature to monoclinic (so-called
M1-phase, see Fig. 2) at low-temperature. The latter is characterized
by a dimerization of the vanadium atoms into pairs, as well as a
tilting of these pairs with respect to the c-axis.
Fig. 1: Rutile structure (picture taken from V. Eyert)
Whether these structural changes are solely responsible for the insulating nature of the low-T phase, or whether correlation effects also play a role, has been a subject of much debate. The strong dimerization, as well as the fact that this phase is non-magnetic suggests that VO2 might be a typical case of a Peierls insulator. However, pioneering experimental work by Pouget et al. showed that minute amounts of Cr-substitutions, as well as, remarkably, uniaxial stress applied to pure VO2 lead to a new phase ("M2"), in which only half of the V-atoms dimerize, while the other half form chains of equally-spaced atoms behaving as spin-1/2 Heisenberg chains. That this phase is also insulating strongly suggests that the physics of VO2 is very close to that of a Mott-Hubbard insulator. Zylbersztejn and Mott, Sommers and Doniach and Rice et al. suggested that Coulomb repulsion indeed plays a major role in opening the gap.
Electronic structure calculations using density functional theory within the local density approximation (LDA) fail indeed in opening a band gap: the top of the bonding band is found to overlap slightly with the bottom of the pi-band (only for a hypothetical structure with larger dimerizations would the band gap open). The discrepancies between the LDA density of states and experimental spectra are, however, not limited to the low-temperature M1 phase. Photoemission spectra of the rutile phase not only show a pronounced peak at the Fermi level but also exhibit significant spectral weight at higher binding energies, in a so-called lower Hubbard band.
Fig. 2 Spectral function for the R-phase as calculated within DMFT with U = 4 eV, J = 0.68 eV (solid lines) in comparison to the LDA density of states (dashed lines). The cyan (blue) lines show the partial contributions of the a1g (eg^{pi}) bands.
In our LDA+DMFT calculations we reproduce this transfer of spectral weight, and the resulting spectral function is in good agreement with the photoemission experiments, see Fig.3. More importantly, we also propose a theory for the (somewhat more difficult) M1 phase: In this phase, an accurate description of short-range spatial correlations is crucial, while local on-site interactions still play an important role. For this reason, a single-site implementation of LDA+DMFT fails in describing this phase: Fig.4 (solid lines) shows the results of single-site LDA+DMFT calculations which exhibit a pronounced peak at the Fermi level. However, a cluster extension of dynamical mean field theory combined with DFT-LDA is able to handle this problem. In this approach, a cluster of sites (a dimer in the case of VO2) instead of a single site is taken to be the key unit which is then embedded in a self-consistent bath. Non-local intra-dimer fluctuations are then correctly taken into account. Fig.5 shows our results: the gap opens and its magnitude is in agreement with the experimental value.
Fig. 3 Spectral function for the M1-phase as calculated within cluster-DMFT with U = 4 eV, J = 0.68 eV (solid lines) in comparison to the LDA density of states (dashed lines). The cyan (blue) lines show the partial contributions of the a1g (eg^{pi}) bands.
Mott or Peierls? The mere fact that introducing strong electronic Coulomb correlations into the theory cures the problem that LDA encounters in the description of the insulating phase could make you think that the M1 phase is indeed a Mott insulator. However, a careful look at Fig.4 which represents the self-energy of the monoclinic phase shows you that the local component does not diverge at low frequencies as it should in a conventional Mott insulator! Indeed, the gap opens up due to a tricky renormalisation and shifting effect, but the sharp peak just below should not be interpreted as a lower Hubbard band -- it's just the quasiparticles which have been gapped out. Indeed, we have observed a charge transfer into the a1g orbital (the one directed into the dimerization direction) which carries nearly one electron and behaves in a similar way as in a "VO2-molecule" where electrons pair up in singlets. Of course, in the solid the different molecules are coupled and electrons can hop between them: the singlets are becoming "dynamical". Cluster-DMFT embeds a VO2 molecule into a self-consistent bath, which describes precisely these hopping processes between different singlets in this "molecular solid". To our knowledge it is the only method that is able to describe this kind of physics from first principles!
Fig. 4 Self-energies of the M1-phase within cluster-DMFT with U = 4 eV, J = 0.68 eV at low (imaginary) frequency. Circles [triangles]: imaginary part of the
on-site diagonal a1g [eg^pi] component. Diamonds [stars]: real [imaginary] part of the inter-site a1g component.
If you want to read more about this, please check out our paper about "Dynamical Singlets and Correlation-assisted Peierls Transition in VO2".