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Quantum electron liquid and its possible phase transition


Purely quantum electron systems exhibit intriguing correlated electronic phases by virtue of quantum fluctuations in addition to electron–electron interactions. To realize such quantum electron systems, a key ingredient is dense electrons decoupled from other degrees of freedom. Here, we report the discovery of a pure quantum electron liquid that spreads up to ~3 Å in a vacuum on the surface of an electride crystal. Its extremely high electron density and weak hybridization with buried atomic orbitals show the quantum and pure nature of the electrons, which exhibit a polarized liquid phase, as demonstrated by our spin-dependent measurement. Furthermore, upon enhancing the electron correlation strength, the dynamics of the quantum electrons change to that of a non-Fermi liquid along with an anomalous band deformation, suggestive of a transition to a hexatic liquid crystal phase. Our findings develop the frontier of quantum electron systems and serve as a platform for exploring correlated electronic phases in a pure fashion.

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Fig. 1: Evolution of pure 2D quantum electrons.
Fig. 2: Floating quantum electrons on the 2D electride.
Fig. 3: Spin-polarized nature of quantum electron liquid.
Fig. 4: Phase transition of quantum electrons.

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All data supporting the findings of this work are included in the main text, extended data and Supplementary Information. These are available from the corresponding authors upon reasonable request.


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This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (Ministry of Science and ICT; grant nos 2021R1A6A1A03039696, 2022M3H4A1A01010832, 2020R1A4A2002828, 2020K1A3A7A09080366, 2021R1A2C1013119 and 2019R1A6A3A01091336), Samsung Science and Technology Foundation under project number SSTF-BA2101-04 and Institute for Basic Science (grant no. IBS-R011-D1). The Advanced Light Source is supported by the Office of Basic Energy Sciences of the US Department of Energy under contract no. DE-AC02-05CH11231. S.H. and C.K. acknowledge support from the Institute for Basic Science in Korea (grant no. IBS-R009-G2). Part of this study has been performed by using facilities at the Institute for Basic Science Center for Correlated Electron Systems, Seoul National University, Korea.

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Authors and Affiliations



S.W.K. and Y.K. conceived the project. S.Y.L., J.B. and S.W.K. synthesized and characterized the single crystals. S.K., C.-y.L., J.H., G.L., Y.L., J.D.D. and Y.K. performed the ARPES measurements. S.K., J.B., S.H., C.K. and Y.K. carried out the spin-resolved ARPES experiments. S.Y.S. and J.S. conducted the STM/S measurements. S.K., J.B., C.-y.L, S.-G.K., Y.K. and S.W.K. analysed the ARPES data. J.B., D.T., S.-G.K., Y.H.L., Y.K. and S.W.K. delivered the DFT calculations. All authors discussed the results. S.K., J.B., Y.K. and S.W.K. prepared the manuscript with contributions from all authors.

Corresponding authors

Correspondence to Yeongkwan Kim or Sung Wng Kim.

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Nature Materials thanks Stephen Dale, Atsushi Fujimori and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Crystal structure of the 2D [Gd2C]2+·2e electride.

a, Schematic illustration of the 2D quantum electron liquid on the surface of [Gd2C]2+·2e electride. It has anti-CdCl2-type layered structure with a space group of R\(\bar 3\)m. The cationic slab [Gd2C]2+ is composed of an edge-sharing octahedra structure and is separated by the interlayer space. IAEs (brown arrow) are confined between the positively charged [Gd2C]2+ layers. The 2D electrons distinct from IAEs are floated on the topmost [Gd2C]2+ layer. The purple lines indicate the unit cell. b,c, Photographs of single crystal [Gd2C]2+·2e rod with 50 mm length grown by floating zone melting method (b) and as-cleaved crystal by 3 M Scotch tape (c). d, Lattice parameters obtained by the XRD measurements19 and the DFT calculations.

Extended Data Fig. 2 The detailed electronic structure of [Gd2C]2+·2e obtained by DFT calculations.

a, Relaxed crystal structure of the nine-slab model for the slab calculations. a × b × 3c supercell with a vacuum of 20 Å along the c-axis was used. For the relaxation of the nine-slab model, the optimised bulk structure was used for the central three-layer, while the three layers at both ends of the slab were relaxed. b,c, Calculated band structure and the Fermi surface of [Gd2C]2+·2e using the nine-slab model, respectively. Blue and red colours indicate spin up and spin down components, respectively. Band thickness represents the contribution of the surface electrons. The thicker line reflects a contribution mainly from the surface character. d,e, Band structure and the Fermi surface (top view along kz direction) of [Gd2C]2+·2e obtained by bulk calculation, respectively. In contrast to the result from the nine-slab model, the bulk calculation does not reproduce the parabolic band dispersion with cylindrical 2D Fermi surface observed by ARPES (Fig. 2a and Extended Data Fig. 3a–c).

Extended Data Fig. 3 Fermi surface of surface electrons on [Gd2C]2+·2e and photon energy dependence.

a, In-plane (kxky) Fermi surface of [Gd2C]2+·2e. The black dashed line indicates the BZ. Circular Fermi surface near BZ centre corresponds to the parabolic band dispersion exhibited in Fig. 2a. b, Fermi surface in the kxkz plane. The inner straight surface, overlaid by dashed white lines, is from the parabolic band shown in Fig. 2a and from the circular Fermi surface in a. c, Band dispersion observed with various photon energies from 80 to 108 eV with a 4 eV step to obtain kxkz plane Fermi surface in b. d, Photon energy-dependent ARPES intensity map extracted from the Fermi level taken with photon energies from 143 to 149 eV, which includes both on- and off-resonant conditions of the Gd 4d core level. The dashed grey lines are overlaid on the surface 2D electron band as a guide for the eye. The intensity of IAE band at 148 eV greatly increased, a clear resonance behaviour while changing in the intensity of the surface 2D electron band is far weak through the whole photon energies. e, The detailed band dispersions at each photon energy in d are exhibited.

Extended Data Fig. 4 Determination of K coverage.

a, Core-level spectra of K 3p with different K coverage on the cleaved [Gd2C]2+·2e surface. The K 3p core-level peak starts to grow with K deposition near the binding energy of 19 eV and eventually saturates, where we estimate the coverage as 1 ML (thick green curve). Above 1 ML, chemically shifted additional peaks emerge at lower binding energy close to 18 eV. b, A 3D representation of the band evolution with increasing K coverage. c, Extracted band dispersion at several different K coverages indicated by red arrows in b. Surface 2D electron band (1) evolves via K deposition (2) and disappears (3). The complete disappearance of the electron band implies that the surface electron density is actually reduced by K deposition. In the rightmost panel (4), corresponding to K 1 ML, which we estimate with core-level spectra in a, K band appears near Fermi level, which well agrees with the core level estimation.

Extended Data Fig. 5 Analysis for the hexagonal topology.

a, Zoomed-in band dispersion of 0.25 ML K coverage, corresponding to deposition step no. 6 in Fig. 4a. b, Schematic band structure of the ARPES result in a. Blue solid lines schematically show constant energy contours (CECs) at different energies. c, CECs obtained at binding energies of 0, 20, 60, and 80 meV, indicated with the dashed red lines (1–4) in a. d, Symmetrised CECs with 3-fold rotation after removing the intensity from the outer band in c, which reveals the shape of energy contours from the quantum electrons more clearly. Overlaid dashed black lines in c and d are guides for the topology of each CEC. The guidelines evidence that the original shape of contours is not deformed by three-fold symmetrisation. e, Autocorrelation analyses performed with CECs of corresponding energies in d, which clearly visualise the angular dependence of the band along with the in-plane momentum.

Extended Data Fig. 6 Scenarios of W-shape band deformation.

ad, Fermi surface with 0.25 ML K coverage. BZs of several possible surface superstructures with sub-ML coverage of K, \(\sqrt 3 \times \sqrt 3 R\,30^ \circ\) (a), 2 × 2 (b), \(\sqrt 7 \times \sqrt 7 R19.1^ \circ\) (c), and 3 × 3 (d), respectively, are overlaid with a different colour. The outermost hexagon with solid black line represents the BZ of 1 × 1 unit cell of [Gd2C]2+·2e. eh, Observed band dispersion along the Γ–Ma,b,c,d direction for new BZs of assumed superstructures at 0.25 ML K coverage. The new zone boundaries by assumed K orderings in ad are overlaid (dashed lines with corresponding colours). Absence of W-shape band at higher momentum above the assumed new zone boundaries, which can be induced by band folding, evidences that W-shaped band deformation is not due to the ordering of deposited K. i,j, Schematic drawings of conceivable scenarios for band deformation (hybridisation) by K deposition with assumed zone folded band (i) and upward convex band (j), respectively. Solid and dashed bands show the original and assumed bands, respectively. Both scenarios may exhibit band deformation; however, they should form the fragment of the original electron band at higher binding energy near BZ centre, which is absent in ARPES results (Fig. 4a).

Extended Data Fig. 7 Band evolution induced by underlying lattice potential.

a, Crystal structure of the nine-slab model for the slab calculations. The vacuum layer with the thickness x along the c-axis is introduced in addition to the original interlayer space (d). be, Calculated band structures depending on the vacuum thickness x of 20.0, 3.5, 1.0 and 0.0 Å, where the result of 0.0 Å is obtained by bulk calculation. Black arrows indicate the gradual shift of the surface electron band (blue) downward to the higher binding energy with a decrease in the vacuum thickness x. This exhibits that the surface electron state follows V-shaped IAE band when it is strongly affected by underlying lattice.

Extended Data Fig. 8 Resonant ARPES measurement after K deposition.

a,b, Band dispersions taken at off- (144 eV, a) and on-resonant (148 eV, b) conditions of the Gd 4d core level after K deposition (deposition step no. 6 in Fig. 4a). Lower panels are magnified images of the enclosed area by black dashed box in the upper panels. c, Normalised intensity (right) of the surface 2D electrons (ISurface, blue) and trapped IAEs (IIAE, brown), and their ratio (left, gray) as a function of photon energy. ISurface and IIAE are obtained by integrating the intensity of the shaded red area in the lower panels and by integrating the area except for ISurface in the upper panels of a and b, respectively.

Extended Data Fig. 9 Phase transition via decreasing temperature.

a, Core-level spectra of K 3p with different K coverage at 40 K. b, Surface electron band dispersion with respect to K deposition. Corresponding deposition steps to the core-level spectra in a are indicated in the panel (#0–#2). Despite the broadening of the spectra and the slight reduction of the band minimum energy after the K deposition, the entire band dispersion still preserves nearly parabolic close to the pristine case. c, Enlarged phase diagram of pure 2D electron phase taken from Fig. 1a. ARPES measurements were performed by following the process marked in the phase diagram by black and blue arrows. d, Core-level spectra of K 3p observed at different temperatures. The preserved intensity of the K 3p core level indicates the absence of K desorption during the cooling process. e, Temperature-dependent surface electron band dispersion observed after K deposition (deposition step #2). Temperatures were set to 40, 25, and 10 K. f, Peak positions (markers) obtained from e by fitting EDCs. Overlaid solid lines are guides to the eyes for the band dispersion. This exhibits the band deformation from parabola (red, 40 K) to W-shape (blue, 10 K) with decreasing the temperature.

Extended Data Fig. 10 Real space imaging of pure 2D electrons.

a, Topograph of the cleaved [Gd2C]2+·2e electride. The sample bias voltage (VB) was set to VB = –25 mV. b, Differential conductance (dI/dV) map taken at same area with a. The periodic pattern shows standing wave of surface 2D electrons due to quasiparticle interference. ce, dI/dV maps observed at several different VB of (c) –25, (d) –100, and (e) –200 mV. The observed periodicity of standing wave increases with decrease of VB. fh, Corresponding Fourier transformed (FT) images of ce. The scattering wavevector q becomes smaller with decrease of VB. i, Surface electron band observed by ARPES. Superimposed red markers are the half of the scattering wavevector (q/2) obtained from FT images at several different VB of –25, –50, –100, and –150 mV. j, dI/dV spectrum of cleaved [Gd2C]2+·2e electride. Dashed line shows that the energy of surface band minimum in i and a peak appeared in j match well. These results further confirm the pure 2D electrons formed on the top of [Gd2C]2+·2e electride.

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Supplementary Figs. 1–4 and Table 1.

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Kim, S., Bang, J., Lim, Cy. et al. Quantum electron liquid and its possible phase transition. Nat. Mater. 21, 1269–1274 (2022).

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