PHYSICAL REVIEW B 87, 195106 (2013)

Correlation-assisted phonon softening and the orbital-selective Peierls transition in VO2

Sooran Kim, Kyoo Kim, Chang-Jong Kang, and B. I. MinDepartment of Physics, PCTP, Pohang University of Science and Technology, Pohang, 790-784, Korea

(Received 5 July 2012; published 6 May 2013)

To explore the driving mechanisms of the metal-insulator transition (MIT) and the structural transition inVO2, we have investigated phonon dispersions of rutile VO2 (R-VO2) in the density functional theory (DFT) andthe DFT+U (U : Coulomb correlation) band calculations. We have found that the phonon softening instabilitiesoccur in both cases, but the softened phonon mode only in the DFT+U describes properly both the MIT andthe structural transition from R-VO2 to monoclinic VO2 (M1-VO2). The present ab initio phonon dispersioncalculations clearly demonstrate that the Coulomb correlation effect plays an essential role of assisting thePeierls transition in R-VO2 and producing the spin-Peierls ground state in M1-VO2.

DOI: 10.1103/PhysRevB.87.195106 PACS number(s): 63.20.D−, 71.10.Fd, 71.30.+h

I. INTRODUCTION

Vanadium dioxide (VO2) is one of the most exploredtransition metal oxides due to its intriguing metal-insulatortransition (MIT) and the concomitant structural transitionupon cooling. At ambient pressure and high temperature, VO2

has a tetragonal rutile-type structure (R-VO2) with metallicnature. Upon cooling below 340 K, R-VO2 undergoes thestructural transition to a monoclinic structure (M1-VO2) withinsulating nature.1,2 The mechanism of MIT in VO2 has been alongstanding subject of controversy. In the structural transitionfrom R-VO2 to M1-VO2, V ions construct the dimerizationand the zigzag distortion. In R-VO2, V ions are centered atthe distorted O6 octahedra, which are edge-shared along the c

axis. Due to the crystal field, V 3d states are split into a1g (d‖),eπg (π∗), and eσ

g states in order of energy, and so one electronof V4+ ion occupies the lowest a1g state. While the zigzagdistortion increases the energy of eπ

g bands, the dimerizationof V-V causes the splitting of a1g bands to open the gap atthe Fermi level (EF ).3 This kind of structural distortion isexplained by a typical Peierls transition.

However, the density functional theory (DFT) band ap-proach fails to describe the insulating nature of M1-VO2

properly.3–5 The energy gap at EF can be obtained only whenthe extra Coulomb correlation U effect of V 3d electrons isconsidered, which indicates that M1-VO2 is a Mott-Hubbardtype insulator. Hence the Mott-Hubbard transition was pro-posed as the mechanism of MIT in VO2.6–8 Beyond the DFTband approach, the GW 9–11 or the hybrid functional bandmethod12,13 were employed to describe the insulating natureof M1-VO2 properly. Also the dynamical mean-field theory(DMFT) approach attempted to explain the insulating nature ofM1-VO2 by considering the Mott-Hubbard U explicitly.5,14–18

Recently, the ground states and the insulating gap of VO2

were explained by the modified Becke-Johnson potential.19

Therefore the consensus at the moment is that some amountof Coulomb correlation is necessary to describe the insulat-ing nature of M1-VO2.20,21 However, whether R-VO2 is astrongly correlated system or whether the Coulomb correlationeffect is necessary for the MIT has not been clarified yet,despite several existing studies using the DFT+U 22,23 and theDMFT.5,16,17,24

As described above, there have been extensive electronicstructure studies on VO2. By contrast, there have been only

several phonon studies on VO2.25–29,32 Especially, there hasbeen neither experimental nor ab initio theoretical reporton the phonon dispersion curve for VO2 yet. Since thePeierls transition is closely related to the phonon softeninginstability, the study of phonon dispersions of VO2 is expectedto give a clue to understanding the mechanism of MIT inVO2.25,29–33

In this paper, we have revisited the MIT and the structuraltransition of VO2 by investigating the phonon dispersionsof R-VO2 and M1-VO2. We have examined the U effecton the phonon properties within the DFT+U . In fact, theDFT+U provides the proper nonlocal static screening,15

which is essential to produce the dimerized state in M1-VO2

and thereby captures properly the ground-state physics ofphonon properties in VO2. We have found that R-VO2 is astrongly correlated system with U � 4.0 eV, and the Coulombcorrelation effect plays an essential role in the MIT and thestructural transition from R-VO2 to M1-VO2.

II. COMPUTATIONAL DETAILS

For the electronic structure and phonon dispersion calcu-lations, the pseudopotential band method and the supercellapproach that are implemented in VASP34 and PHONOPY35 areused. The full-potential band method implemented in WIEN2K

is also employed to double-check electronic structures andenergetics.36 In the supercell approach for the phonon calcu-lation, the force constants are calculated for a supercell withsmall lattice displacements, by using the Hellmann-Feynmantheorem.37 The utilized exchange-correlation functional isthe generalized gradient approximation (GGA). The adoptedvalues of U and J in the DFT+U are 4.2 and 0.8 eV,respectively.5 All the phonon calculations were carried outafter the full relaxation of the volume and atomic positions.38

The initial structural parameters of R-VO2 and M1-VO2 beforethe relaxation are taken from experiments by McWhan et al.30

and Longo et al.,39 respectively. After the full-relaxation,the c/a and the volume of R-VO2 become smaller thanthose before the relaxation, and similarly, the volume ofM1-VO2 decreases by relaxation.40 As shown below, the fullrelaxation is essential to get the insulating state of M1-VO2 inthe DFT.

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SOORAN KIM, KYOO KIM, CHANG-JONG KANG, AND B. I. MIN PHYSICAL REVIEW B 87, 195106 (2013)

FIG. 1. (Color online) (a) The phonon dispersion curves of R-VO2

in the DFT. The inset shows the normal mode of softened phononat �. (b) The phonon dispersion curves of R-VO2 in the DFT+U .Both figures show the negative phonon frequencies (imaginaryfrequencies) reflecting the phonon softening instability.

III. RESULTS AND DISCUSSIONS

We have first obtained the electronic structures of R-VO2

in the DFT and the DFT+U . In the DFT, stable metallicstates are obtained both in the nonmagnetic and the magneticband structure calculations, in agreement with the experiment.However, in the DFT+U , a more stable insulating state isobtained in the magnetic band structure calculation, which isseemingly in disagreement with the experiment. In fact, theinsulating state of R-VO2 in the DFT+U had been reportedbefore.22,23 This discrepancy between the calculation and theexperiment can be resolved by considering the competitionbetween the magnetic instability and the structural instabilityin R-VO2. Of course, in nature, the structural instability winsover the magnetic instability, and so, upon cooling, R-VO2

undergoes the structural transition to M1-VO2 in advance ofthe magnetic transition within R-VO2.

Figure 1 shows the phonon dispersion curves of R-VO2

in the DFT and DFT+U . Four Raman modes (B1g , Eg ,A1g , and B2g) are obtained in both cases, in agreementwith experiments.26–28 Interestingly, the phonon softeninginstabilities are obtained both in the DFT [see Fig. 1(a)] andin the DFT+U [see Fig. 1(b)], which imply that R-VO2 is nota stable structure. These results are expected because R-VO2

is stable only at high temperature. The most noteworthy is themarked difference in the phonon dispersion curves betweenthe DFT and the DFT+U . In the DFT, the phonon softeningsoccur at q = �, M , and X [see Fig. 1(a)], while, in the DFT+U ,they occur at q = R, A, and Z [see Fig. 1(b)]. The phononsoftening at q = R was once suggested to be responsible forthe transformation from R-VO2 to M1-VO2.25,29–33

We have examined the normal modes of the softenedphonons. The normal mode at q = � in the DFT is plottedin the inset of Fig. 1(a), and those at q = R in the DFT+U areplotted in Figs. 2(a) and 2(b). Recall that the main structural

FIG. 2. (Color online) The normal modes of the softened phononsof R-VO2 in the DFT+U at q = R. The blue and red balls representV and O, respectively. There are two degenerate modes, as shownin (a) and (b). (c) A linearly superposed mode at q = R using twodegenerate modes of (a) and (b). (d) Schematic picture of V-V pairingand distortion in M1-VO2. Red, blue, and black dots denote V ions inR-VO2, M1-VO2, and M2-VO2 (Cr-doped VO2 phase), respectively.

changes from R-VO2 to M1-VO2 are the dimerization andzigzag distortion of V ions, as shown in Fig. 2(d). It is evidentthat those lattice distortions cannot be described by the normalmode at � in the DFT, which corresponds to just collineardisplacements of ions along the c direction. The softenedmodes at M and X in the DFT do not describe the structuraldistortions either.40 In contrast, the normal modes at R in theDFT+U are seen to be consistent with the lattice distortionsof VO2. As shown in Figs. 2(a) and 2(b), there are twodegenerate softened phonon modes at R. The first mode inFig. 2(a) represents the dimerizations of half of V ions andthe orthogonal displacements of the other half. The secondmode in Fig. 2(b) is just the reverse of the first one. Notethat the mode predicted by Gervais et al.32 is close to the firstmode. Indeed, a linearly superposed mode using these twonormal modes in Fig. 2(c) reveals simultaneous dimerizationsand zigzag distortions of V ions, in perfect agreement withthe observed lattice distortions in M1-VO2 [see Fig. 2(d)].The other independent superposed mode also reveals similardisplacements of V ions. This agreement indicates that thesoftened mode at R in the DFT+U describes the structuraltransition of VO2 properly. The softened modes at A and Z inthe DFT+U are also related to the dimerizations of V ions,but not directly to the zigzag distortions of V ions.40

The distinct phonon properties between the DFT and theDFT+U clearly demonstrates that the Coulomb correlationin R-VO2 facilitates the Peierls-type structural transition.Namely, in R-VO2, the Coulomb correlation effect and thePeierls distortion are mutually cooperating in driving the MITand the structural transition. To examine the U effect inmore detail, we have checked the phonon dispersion curveswith varying U . With increasing U , we have found thatthe softenings that occur in the DFT at q = �, M , and X

disappear, whereas the softening at q = A, R, and Z emergefor U � 4.0 eV (the larger U , the more negative of the phonon

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FIG. 3. (Color online) (a) and (b) The Fermi surface of R-VO2 inthe DFT and in the DFT+U , respectively. (c) and (d) χ (q) of R-VO2

in the DFT and in the DFT+U . (e) and (f) DFT and DFT+U localcharge densities near EF around the V ion of R-VO2 in the ac plane(integrated over ∼−1.2 eV to EF ) in e/A3 (x and z labels are inangstroms).

frequencies at q = A, R, and Z). This feature indicates that largeenough U is required to produce the right Peierls distortionsin VO2. Below we analyze in more detail why the DFT andthe DFT+U yield distinct phonon properties.

Figure 3 shows the Fermi surface, the electric susceptibilityχ (q), and the local charge density of R-VO2 in the DFT andthe DFT+U calculations. By comparing the Fermi surfacesin Figs. 3(a) and 3(b), one can notice that the Fermi surfacebecomes flatter in the DFT+U , suggesting a possible nestingeffect along the c axis.41 This change is more apparent in χ (q).As compared to χ (q) in the DFT, χ (q) in the DFT+U exhibitshigh values at q = (x,0,0.5) (0 � x � 0.5), which explainsthe origin of phonon softenings at q = (x,0,0.5), that is, at Z

and R in Fig. 1(b). Local charge density plots in Figs. 3(e)and 3(f) also support the proper description of the DFT+U forthe structural transition of R-VO2. As compared to the case ofthe DFT, the charge density in the DFT+U is seen to be moreanisotropic, reflecting that more charges are accumulated inthe bonding region along the c-axis. This phenomenon arisesfrom the correlation-induced orbital redistribution.

The partial density of states (PDOS) shown in Figs. 4(a) and4(b) provides more clear evidence for the correlation-inducedorbital redistribution. The orbital occupancies of a1g(dx2−y2 ),eπg (dyz), and eπ

g (dxz) in the DFT are in the ratio of 0.46, 0.32,and 0.22, whereas those in the DFT+U become 0.73, 0.14,and 0.13. Thus, due to U effect in R-VO2, the occupancy ofa1g state is dramatically enhanced with respect to other eπ

g

states, whereby the system becomes more one dimensional-

FIG. 4. (Color online) (a) The PDOS of the nonmagnetic (NM)state of R-VO2 in the DFT. (b) The PDOS of the NM state of R-VO2 in the DFT+U . (c) The PDOS of the NM state of M1-VO2 inthe DFT+U . (d) The PDOS of the dimerized-singlet (DS) state ofM1-VO2 in the DFT+U .

like along the c axis, and so more susceptible to the Peierlstransition. Therefore, by means of the correlation-inducedorbital redistribution, the orbital-selective Peierls transitionoccurs in the a1g orbital. This feature is again consistent withthe phonon softening in the DFT+U manifested at A, Z, andR, which are located in the z direction from �.

Now let us discuss the electronic structure of M1-VO2.In the nonmagnetic DFT calculation for M1-VO2, we haveobtained the insulating state, even though the energy gapis very small (∼0.03 eV).40 This result is different fromexisting nonmagnetic DFT results in which metallic stateswere obtained for M1-VO2.3–5 This difference is thought tocome from the full relaxation of volume and ionic positions inour band calculations, which was not taken into account in theprevious calculations. Without the relaxation, we also get themetallic state for M1-VO2 in the nonmagnetic DFT calculation.On the other hand, in the nonmagnetic DFT+U calculationfor M1-VO2, we have obtained the insulating state with largeenergy gap of ∼0.67 eV between a1g and eπ

g states, as shownin Fig. 4(c), which appears to agree with the experimentalgap of 0.6–0.7 eV.4,42 It is clearly seen that the a1g (dx2−y2 )state is separated into bonding and antibonding states. We havecalculated the phonon dispersions of M1-VO2 both in the DFTand the DFT+U (not shown here). In both cases, there are nophonon softening instabilities. This indicates that M1-VO2 isa stable ground-state structure, which is consistent with theexperiment.

Note, however, the ground state of M1-VO2 in the DFT+U

is not the nonmagnetic state but the dimerized-singlet (DS)state, which has an antiferromagnetic spin configuration withineach V-V dimer. The energy gap size of the DS state isobtained to be ∼1.33 eV, and the a1g antibonding state isshifted further above, as shown in Fig. 4(d). The stableDS state is reminiscent of the spin-Peierls state, which hasbeen suggested to be a ground state of M1-VO2.14,16,18,43–46

M1-VO2 has been considered to be a nonmagnet, because ofthe absence of magnetic response and magnetic order below

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the MIT.47 The absence of magnetic response in M1-VO2 canbe understood by the spin-gap feature of the spin-Peierls state,as in other typical spin-Peierls systems, such as CuGeO3

48 andNaV2O5.49 In the DS state, the estimated spin-gaps amount tobe ∼0.32 and ∼0.38 eV at the experimental and the theoreticalequilibrium volumes, respectively.50 Hence the spin-gap sizedepends slightly on the volume of the DS state, but it is muchsmaller than the charge-gap of 1.33 eV in Fig. 4(d). Theobtained DS state with the different spin and the charge gapssupports that the ground state of M1-VO2 is the spin-Peierlsstate.

IV. CONCLUSION

Based on the phonon dispersion studies in the DFT and theDFT+U , we have demonstrated that the driving mechanism

of the MIT and the structural transition in VO2 is the orbital-selective Peierls transition, subsequent to the correlation-induced orbital redistribution. The Coulomb correlation effectplays an essential role of assisting the Peierls transition inR-VO2 and producing the spin-Peierls state in M1-VO2. Thephonon softening instability manifested at q = R in theDFT+U provides direct evidence of Peierls mechanism ofthe structural transition from R-VO2 to M1-VO2.

ACKNOWLEDGMENTS

This work was supported by the NRF (Nos. 2009-0079947and 2011-0025237), POSTECH Physics BK21 fund, and theKISTI supercomputing center (No. KSC-2012-C2-18). S.K.acknowledges the support from the NRF project of GlobalPh.D. Fellowship (No. 2011-0002351).

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