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Temperature-dependent electronic structure of a quasi-two-dimensional conductor η-Mo4O11

Abstract

The origin of the temperature-dependent physical properties of η-Mo4O11 has been a topic discussed for a couple of decades. By using a photoelectron momentum microscope, we measured the Fermi surfaces and band structures of this material at 150 K, 70 K, and 20 K, i.e., above, between, and below the temperatures where two charge density wave (CDW) transitions were reported to occur in the literature. Three metallic Fermi surfaces with 1D character along the b* and b* ± c* axes were clearly observed at these three temperatures with a negligible difference, suggesting that η-Mo4O11 does not show CDW transition in the measured temperature range. No trace of CDW transition was observed in the spectra of the three metallic bands near the Fermi level as well. Based on these results and theoretical supports, we discuss the origin of the two conflicting results on the temperature-dependent physical property of η-Mo4O11.

Introduction

Owing to the loss of atomic bonding in one- or two-dimensional directions, low-dimensional materials have the potential to show a variety of interesting physical phenomena that do not appear in the bulk phase. The η-Mo4O11, known as a Magnéli phase compound1,2, is one of such materials. The monoclinic η-Mo4O11 belongs to the P21/a space group, and is formed by Mo6O22 layers that are separated by weakly bonded MoO4 tetrahedral layers along the a axis (Fig. 1a). (Mo6O22 layers consist of distorted MoO6 octahedral slabs parallel to the bc plane.) The weak connection to MoO4 tetrahedral layers leads conduction electrons to be confined within the Mo6O22 layers, and thus η-Mo4O11 to have a quasi-two-dimensional (quasi-2D) electronic structure in the bc plane. This quasi-2D electronic structure is reported to possess hidden 1D metallic character along the b and b ± c axes3,4,5,6,7,8,9,10,11,12 (Fig. 1b), which might destabilize the Fermi surface (FS)13 with nesting vectors of q1 = 0.23b* 14,15 and q2 = (0.552a*, 0.47b*, 0.30c*)15, and would consequently cause two charge density wave (CDW) transitions at Tc1 ~ 105 K and Tc2 ~ 30 K14,15,16,17.

Fig. 1

figure 1

The atomic structure of η-Mo4O11. (a) The unit cell and (b) the bc plane consists of the four inner MoO6 slabs. The arrows in (b) indicate the 1D channels of conduction electrons along the b and b ± c directions.

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The change in electronic structure by CDW transitions and the presence of very small closed FSs below Tc1 and Tc2 were predicted from magnetoresistance and Hall effect measurements5,17,18,19,20,21,22. So far, the physical properties of η-Mo4O11 were mainly discussed based on the results obtained by transport and/or diffraction measurements, and there are only few reports obtained by angle-resolved photoelectron spectroscopy (ARPES) measurements7,8,9,10,11,12, though ARPES provides direct information on the band structure and FS and is thus a valuable tool to discuss the CDW transition. These former ARPES studies show the FSs at 150 K7, 125 K10, and 50 K9 only, i.e., at temperatures higher than Tc1 and between Tc1 and Tc2, and discuss the transition based on the temperature-dependent change in the spectra near the Fermi level (EF) from 160 to 20 K10. Furthermore, the FS observed at 50 K9 closely resembles those at 150 K7 and 125 K10, and thus the predicted small FSs, which are supposed to result from the CDW transitions, are not confirmed yet.

In this paper, we report the FSs and band structures of η-Mo4O11 measured at 150 K, 70 K, and 20 K, i.e., at temperatures above Tc1, between Tc1 and Tc2, and below Tc2 by ARPES. The FSs and most of the band structures shown in this paper are obtained by a photoelectron momentum microscope (PMM)23,24,25. The use of PMM allows to obtain the FS in a wide momentum region at once without rotating the sample, and without serious radiation damage that occurs in most molybdenum oxides26. Three metallic FSs with 1D character, which show good agreement with those reported at 50 K in Ref.9, were observed at the measured three temperatures. No gap opening was observed in the temperature-dependent ARPES spectra near EF, and the spectral shape of the three metallic bands at EF can be fitted using Fermi–Dirac functions convoluted with Gaussian functions at the three temperatures. Taking these experimental results and the theoretical supports into account, we conclude that the so-called “three metallic bands” of η-Mo4O11 are not in the Peierls instability scheme at 20 K and above, because they are formed by bunches of more than one metallic band and not all of them can satisfy the nesting vectors simultaneously.

Results

Dimensionality of band dispersion

Normal emission photoelectron spectra taken at 10 K with photon energies (\(h\nu\)) from 60 to 72 eV are shown in Fig. 2a. Each spectrum is obtained by integrating the photoemission intensity in a k// range of 0 ± 0.025 Å−1. The two peaks labeled A and B, whose binding energies (EBs) are approximately 0.47 eV and 0.69 eV, correspond to the bottom of the two bands shown in Fig. 2b. The negligible photon energy dependent dispersion of these two peaks indicates the electronic structure of η-Mo4O11 to have no kz dependence, and thus to have a quasi-2D property. This result agrees well with the electronic structure expected from the quasi-2D layered structure of this material and is consistent with the result obtained using \(h\nu =8\)–20 eV in a former study9.

Fig. 2

figure 2

(a) Normal emission photoelectron spectra, obtained by integrating the photoemission intensity in a k// range of 0 ± 0.025 Å−1 at 10 K, using \(h\nu =60\)–72 eV. (b) Band dispersion obtained around the normal emission at \(h\nu =66\) eV. (c)–(e) Fermi level intensity maps obtained at 150 K, 70 K, and 20 K using \(h\nu =66\) eV. (f) is the intensity map obtained by superimposing the intensity of the Fermi level inside the first Brillouin zone (− 0.561 Å−1 \(\le\) kb \(\le\) 0.561 Å−1, − 0.456 Å−1 \(\le\) kc \(\le\) 0.456 Å−1), inside the second one (− 1.683 Å−1 \(\le\) kb \(\le\) − 0.561 Å−1, − 0.456 Å−1 \(\le\) kc \(\le\) 0.456 Å−1), and inside the fourth one (− 1.683 Å−1 \(\le\) kb \(\le\) − 0.561 Å−1, − 2.278 Å−1 \(\le\) kc \(\le\) − 1.367 Å−1).

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Temperature-dependent Fermi surface

Figure 2c–e show the Fermi level intensity map of η-Mo4O11 obtained at 150 K, 70 K, and 20 K with a photon energy of \(h\nu =\)66 eV. Three pairs of straight FSs, which extend to the c* and b* ± c* directions are clearly observed in these three figures (b* is the direction parallel to kb and c* is parallel to kc). Here, we labeled the three FSs as b, b + c, and b-c, i.e., the same labels as those used in Ref.9. The Fermi level intensity maps within the first Brillouin zone (− 0.561 Å−1 \(\le\) kb \(\le\) 0.561 Å−1, − 0.456 Å−1 \(\le\) kc \(\le\) 0.456 Å−1) of Fig. 2c–e look different from that reported in Ref.9. That is, the b ± c FSs extended from the second Brillouin zone are clearly observed in Ref.9 but hardly observable in Fig. 2. This difference would be due to a final states effect in the photoemission process, which comes from either the use of different photon energy or the use of different light-polarization. In the present case, we conclude the negligible intensity of the FSs from the second Brillouin zone to result by the final state effect that comes by using photons of \(h\nu =66\) eV, because the Fermi level intensity map was not affected by the polarization of the light.

The Fermi level intensity map reported in Ref.9 can be reproduced by superimposing the intensity maps of several Brillouin zones. As shown in Fig. 2f, the k separation between the two b + c bands, which is indicated by an arrow, agrees well with the nesting vectors expected both experimentally14 and theoretically3. However, none of the expected gaps in FSs3,4,9,10,11 is observed in the present study. (Gaps around the crossing point of the b + c and b-c bands between Γ and Z, around the crossing points of the b band and b ± c bands, and gaps of the b ± c bands around the Y point were expected in the former studies3,4,9,10,11.)

Dispersion of the three metallic bands

To obtain more detailed information about the three metallic bands, we show in Fig. 3a–c the band dispersion along the Γ-Y, Γ-Z, and Γ-M directions, i.e., the three directions indicated by dotted lines in Fig. 2e. As shown in Fig. 3a–c, all the three metallic bands have relatively parabolic dispersions around the symmetry points Γ and M, and the presence of at least two parabolic bands at the Γ point is clear from Fig. 3a,c,d. (The temperature-dependent dispersions of the b and b–c bands obtained with a hemispherical electron energy analyzer is shown in Fig. S1 of the supplementary information.) The parabolic dispersions of the three metallic bands were also predicted theoretically by tight binding calculation,3,4 but their behaviors are different from those observed in the present study. According to these former theoretical calculations, the band with lower EB at Γ is predicted to disperse upward along both the Γ-Y and Γ-Z directions, and the band with higher EB at Γ to disperse upward along the Γ-Y direction only and to show a negligible dispersion along the Γ-Z direction. However, these former results are inconsistent with the present ones, where the band with lower EB at Γ (the band labeled b) shows a large upward dispersion along the Γ-Y direction and a small downward dispersion along Γ-Z as shown in Fig. 3b,d, and the bands with higher EB at Γ (the band labeled b ± c) disperse upward along both Γ-Y and Γ-Z as shown in Fig. 3a–d.

Fig. 3

figure 3

Band dispersions along the (a) Γ-Y, (b) Γ-Z, and (c) Γ-M directions at 20 K. (d) and (e) show the band dispersions and MDCs at the cuts A–Aʹ and B–Bʹ of Fig. 2e, respectively. The open circles in (e) are the experimentally obtained intensity at the Fermi level, and the solid line overlapping the open circles are the fitting curves obtained by the Lorentzian functions shown below each spectrum. The bands in (a)–(d) are obtained by integrating the data obtained in a k// region of ± 0.05 Å−1. (f) is a schematic illustration of the dispersion of the three metallic bands of η-Mo4O11 within the first Brillouin zone obtained from the present study. The shaded quadrilateral corresponds to the Fermi level.

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Regarding the FS separation (Δk// at EF), none of the three bands show obvious k// dependence in Figs. 2c–e and 3a–d. Since a small 2D modulation of the b band, and thus a modulation in the FS separation of this band, has been reported to be a crucial factor for the CDW transition at Tc29, we have measured the momentum distribution curves (MDCs) at the Fermi level along the cuts A–Aʹ and B–Bʹ indicated in Fig. 2e. The open circles in Fig. 3e are the experimentally obtained photoelectron intensity at the Fermi level of Fig. 3d, and the solid line overlapping the open circles are the fitting results obtained using the Lorentzian functions shown at the bottom of each spectrum. The peak positions and full width at half maxima (FWHM) of the Lorentzian functions for the two peaks labeled b are 0.845 ± 0.01 Å−1 and 0.062 ± 0.001 Å−1 for the peak with smaller kb and 1.358 ± 0.01 Å−1 and 0.0564 ± 0.001 Å−1 for that with larger kb in the two MDCs. This indicates that the FS separation of the b band hardly changes and thus the modulation of the b band is negligible at least within the k// resolution of the experimental setup used in the present study. By considering the relatively 1D band structures and FSs of all the three metallic bands, we conclude the electron pockets of these bands to disperse as the illustration shown in Fig. 3f.

Discussion

CDW transitions have been proposed to be the origin of the temperature-dependent change in resistivity of η-Mo4O11, and their nesting vectors were predicted based on the theoretically obtained FSs and diffraction measurement. However, the shape of FSs hardly changed from 150 to 20 K, i.e., when changing the temperature from above TC1 to below TC2, and no gap opening was observed in the band even at 20 K. The present results indicate that the three 1D metallic bands of η-Mo4O11 do not contribute to the transition observed in resistivity measurement, and therefore that these bands are robust against the Peierls instability even if it exists. In other words, the electron–phonon coupling is not strong enough to induce a CDW transition in η-Mo4O11. Tomonaga-Luttinger liquid27,28,29 would be another candidate to explain the transition in resistivity. To examine this possibility, we discuss the temperature-dependent ARPES spectra of the three metallic bands near the EF. Figure 4a–c are the spectra of the b, b + c, and b− c bands, respectively, obtained by integrating the photoelectron intensity over each area shown in the inset of Fig. 4c. (The temperature-dependent spectra of the b and b− c bands obtained with a hemispherical electron energy analyzer is shown in Fig. S2 of the supplementary information.) The intensity of each band at EF, which is exactly the half of the peak of the convoluted Fermi–Dirac function, do not show any temperature dependence, and do not follow the Tomonaga-Luttinger type power law either.

Fig. 4

figure 4

Temperature-dependent ARPES spectra of the (a) b band, (b) b + c band, and (c) b− c band near the EF. Open circles are the data obtained by integrating the photoelectron intensity over each area shown in the inset of (c). Solid curves represent fitting results with Fermi–Dirac functions convoluted with 55 meV FWHM Gaussian functions.

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The results shown in Fig. 4 do not match the temperature-dependent ARPES spectra reported in Ref.10, which suggests a gap of up to 40 meV in the b ± c bands and up to 10 meV in the b band by measuring the leading edge shift. (The leading edge shift was absent with an error of ± 5 meV in the present study.) Two reasons can be considered as the origin of this discrepancy. First, considering that the temperature-dependent ARPES spectra have been measured at a certain k// point in Ref.10 while they are obtained by integrating the photoelectron intensity over a finite k// area in the present study, the gap opens at the specific k// point measured in Ref.10 only because the Fermi surfaces are not perfectly straight. Second, both the b ± c bands and b band are suggested to show gap opening at below Tc1 in Ref.10, but this does not fit the presence of two transitions at two different temperatures. Since the surface of a cleaved η-Mo4O11 crystal is usually rugged, and because changing the sample temperature has possibility to change the measuring sample position, the gap opening observed in Ref.10 has possibility to be a result obtained at different k// points, i.e., the measured k// points at low temperature were shifted from the k// of the Fermi surface.

Since defects, such as oxygen vacancy, which can be created at surfaces by photo-irradiation, changes the charge states of the material and thus its Fermi surface, a possible origin of the conflicting results regarding the CDW transition might be the fact that ARPES is a surface sensitive measurement. However, the agreement between the FS separation of two b + c bands observed in the present study (Fig. 2f) and the nesting vectors expected in the literatures, 0.23b* Å−1, denies this possibility. To further discuss the origin of the conflicting results, we show the band dispersion obtained by DFT calculations in Fig. 5 (filled circles connected by thin lines are the theoretical results and the thick colored lines show the band dispersion expected from the present experimental study). Although the theoretical band dispersion in Fig. 5 shows some similarities with the band dispersion predicted in the former theoretical studies3,4, there is one major difference. That is, the number of bands in Fig. 5 is much more than that reported in the former studies. The origin of this discrepancy would be the use of different atomic structures in the calculations. In the former study, Mo6O22 layers made by ideal MoO6 octahedra were used to carry out the tight-binding calculation, while slightly distorted octahedra obtained by optimizing the atomic structure is used in the present study. The distorted octahedra lift the band-degeneracy and split the “b” band into two bands and the “b ± c” bands into four bands. The theoretically obtained larger energy split of the “b ± c” bands compared to the “b” bands around the Γ point explains well the broader “b ± c” band observed experimentally in Figs. 2b and 3a–d. We therefore conclude that the “b and b ± c bands" of η-Mo4O11 with parallel FSs and parabolic dispersions are not within a simple Peierls instability scheme, because they are formed by multiple metallic bands with slightly different dispersions, and the simple nesting vectors picture is no longer effective since even if a nesting vector exists all the metallic bands cannot satisfy the nesting vector simultaneously.

Fig. 5

figure 5

Theoretically band dispersion of the structure-optimized η-Mo4O11 obtained by DFT calculations. The bands obtained in the experiment are superimposed by colored thick lines.

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Conclusion

In conclusion, we have investigated the FSs and band structures of η-Mo4O11 at temperatures above, between and below the two predicted CDW transition temperatures by ARPES. The Fermi level intensity maps show the presence of three pairs of straight FSs not only at 150 K but also at 70 K and 20 K, i.e., at temperatures lower than those predicted for CDW transitions. These metallic bands show parabolic dispersions, and their FS separations hardly show any k// dependence. Furthermore, the intensity of each band at EF does not show any temperature dependence and does not follow the Tomonaga-Luttinger type power law. In our DFT calculations, we found that the slightly distorted MoO6 octahedra structure lifts the band degeneracy and makes the number of metallic bands larger than that reported in former studies. Based on these, we conclude the “three metallic bands” of η-Mo4O11 to be not in either a simple Peierls instability scheme or a Tomonaga-Luttinger scheme, but instead multiple bands with slightly different dispersions form the “1D metallic bands” of η-Mo4O11 and not all of these bands can satisfy the nesting vectors simultaneously even if it exists.

Methods

ARPES measurements

ARPES measurements were performed mainly using a single hemispherical deflection analyzer PMM (SPECS, KREIOS 150 MM)25 at the beamline BL6U of the UVSOR synchrotron radiation facility, Japan, together with a hemispherical electron energy analyzer (MB Scientific, A-1) at the same beamline in UVSOR30 and at the PGM station of the beamline 13 of SAGA Light source (SAGA-LS), Japan31. The energy and momentum resolutions of the PMM analyzer are ΔE ~ 23 meV and Δk// ~ 0.012 Å−1, respectively, in case of using \(h\nu =21.2\) eV, and those of the A-1 are ΔE ~ 10–60 meV and Δk// ~ below 1% of the Brillouin zone. Single η-Mo4O11 crystals, grown by chemical vapor transport method18,32, were cleaved inside ultra-high vacuum chambers under a base pressure of < 3 × 10−8 Pa to obtain a clean surface before all ARPES measurements. The incident angle of the p-polarized light was 68° from the surface normal direction in PMM measurements.

Theoretical calculation

Density functional theory (DFT) calculations were performed using the Quantum-ESPRESSO package33 based on the plane-wave pseudo-potential method34,35. The projected-augmented wave (PAW) was used to describe all electron properties, and the revised Perdew, Burke, and Ernzerhof (PBE) for solids (PBEsol) functional36 was used to treat the electronic structures.

The use of PBEsol functional is reported to show good agreement with the experimental results in Mo–O systems37,38. To include the large Coulomb repulsion between localized d electrons, the DFT with on-site Hubbard U correction (DFT + U) method39 with a U value of 5 eV was applied. We used the experimentally obtained lattice parameters40, the unit cell shown in Fig. 1a, and energy cutoffs of 50 Ry and 500 Ry for wave functions and charge densities. A Monkhorst-Pack41 k-point grid of 6 × 6 × 2, in which the Γ point was centered, was used for Brillouin zone integration. All atomic positions were fully relaxed to an assumed criterion of atomic forces (less than 1 × 10−4 Ry/a.u.).

Data availability

All data needed to evaluate the conclusion of this paper are presented in the paper. Additional data are available from the corresponding author upon reasonable request.

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Acknowledgements

This research is supported by the JSPS Grant-in-Aid for Scientific Research (B) JP22H01957, JP20H02707, and JP19H02592, and the JSPS Grant-in-Aid for Specially Promoted Research JP20H05621, and the Spintronics Research Network of Japan. The experiments at UVSOR were performed under the approval of the Program Advisory Committee (Proposal No. 22IMS1218 and 23IMS1216), and those at SAGA-LS were performed under the approval of the Program Advisory Committee (Proposal No. R2-302P).

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Authors and Affiliations

Department of Material and Life Science, Osaka University, Osaka, 565-0871, Japan

Takahiro Kobayashi

UVSOR Synchrotron Facility, Institute for Molecular Science, Okazaki, 444-8585, Japan

Fumihiko Matsui

Department of Applied Physics, Osaka University, Osaka, 565-0871, Japan

Emi Iwamoto & Kazuyuki Sakamoto

Graduate School of Engineering, Osaka University, Osaka, 565-0871, Japan

Hidetoshi Kizaki

Spintronics Research Network Division, Institute for Open and Transdisciplinary Research Initiatives, Osaka University, Osaka, 565-0871, Japan

Hidetoshi Kizaki & Kazuyuki Sakamoto

Center for Spintronics Research Network, Graduate School of Engineering Science, Osaka University, Osaka, 560-8531, Japan

Hidetoshi Kizaki & Kazuyuki Sakamoto

School of Materials Science, Japan Advanced Institute of Science and Technology, Ishikawa, 923-1292, Japan

Masanobu Miyata & Mikio Koyano

Synchrotron Light Application Center, Saga University, Saga, 840-8502, Japan

Isamu Yamamoto

SANKEN, Osaka University, Osaka, 567-0047, Japan

Shigemasa Suga

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Takahiro Kobayashi

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