Tailoring the nature and strength of electron phonon interactions in the SrTiO 3 (001) 2D electron liquid

Transcription

1 DOI: 1.138/NMAT4623 Tailoring the nature and strength of electron phonon interactions in the SrTiO 3 (1) 2D electron liquid Z. Wang, 1, 2 S. McKeown Walker, 2 A. Tamai, 2 Y. Wang, 3, 4 Z. Ristic, 5 F.Y. Bruno, 2 A. de la Torre, 2 S. Riccò, 2 N.C. Plumb, 1 M. Shi, 1 P. Hlawenka, 6 J. Sánchez-Barriga, 6 A. Varykhalov, 6 T.K. Kim, 7 M. Hoesch, 7 P.D.C. King, 8 W. Meevasana, 9 U. Diebold, 1 J. Mesot, 1, 5, 11 B. Moritz, 3 T.P. Devereaux, 3, 12 M. Radovic, 1, 13 2, 1, 8 and F. Baumberger 1 Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland 2 Department of Quantum Matter Physics, University of Geneva, 24 Quai Ernest-Ansermet, 1211 Geneva 4, Switzerland 3 Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, Menlo Park, California 9425, USA 4 Department of Applied Physics, Stanford University, Stanford, CA 9435, USA 5 Institute of Condensed Matter Physics, École Polytechnique Fédérale de Lausanne (EPFL), CH-115 Lausanne, Switzerland 6 Helmholtz-Zentrum Berlin für Materialien und Energie GmbH 7 Diamond Light Source, Harwell Campus, Didcot, United Kingdom 8 SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews, Fife KY16 9SS, United Kingdom 9 School of Physics, Suranaree University of Technology, Nakhon Ratchasima, 3, Thailand 1 Institute of Applied Physics, Vienna University of Technology, Wiedner Hauptstrasse 8-1/134, A-14 Vienna, Austria 11 Laboratory for Solid State Physics, ETH Zürich, CH-893 Zrich, Switzerland 12 Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 9435, USA 13 SwissFEL, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland NATURE MATERIALS 1

2 DOI: 1.138/NMAT Sample characterization. We prepared the Nb-doped (.5 %) and La-doped (.75 wt%) SrTiO 3 (1) substrates by Ar + sputtering (6 ev, 2 µa, 5min) followed by annealing in mbar oxygen for.5 h at different temperatures varying from 7 C to 1 C. This results in a long-range ordered (2 1) surface reconstruction with two equivalent domains, as can be inferred from the low-energy electron diffraction (LEED) pattern shown in Fig. S1 a [1]. The absence of an in-gap state on the as-prepared surface indicates a low defect density. Following exposure of the surface to synchrotron UV light, metallic 2DEL states appear concomitant with an in-gap state that can be attributed to localized excess electrons due to the creation of oxygen vacancies as reported by us and other groups [2 5]. The influence of the (2 1) surface reconstruction on the 2DEL electronic structure is negligible, since the occupied states of the 2DEL have small wave vectors and do not reach the vicinity of the superlattice Brillouin zone boundary. a b x5 c In-gap x1 Metallic (1,) (1/2,) Intensity Intensity (,1/2) (1,1) (,1) -8-4 E-E F (ev) E-E F (ev) FIG. S1. a, LEED pattern of a SrTiO 3 (1) surface prepared in situ by sputtering and annealing at 7 C. The surface exhibits the typical two-domain (2 1) reconstruction [1]. b-c, Angle-integrated valence band spectra measured at the as-prepared SrTiO 3 (1) surface and after exposure to synchrotron UV light for 1 minutes. The inset shows the absence of in-gap and metallic states in the band gap region on the as-prepared surface (b), whereas pronounced in-gap and metallic states appears in (c). 2. Carrier density. Systematic photon energy dependent measurements as shown in Fig. S1 f of the main text and Fig. S2 a-c confirm the absence of band dispersion along the surface normal (k z ) in the xy band for samples with different carrier densities and support the interpretation of our data as a quantum confined 2DEL. 2 NATURE MATERIALS

3 DOI: 1.138/NMAT4623 SUPPLEMENTARY INFORMATION a b c 5. k z (Å -1 ) k z (Å -1 ) k z (Å -1 ) E-E F (ev) -.2 E-E F (ev) -.2 E-E F (ev) k x (Å -1 ) k x (Å -1 ) k x (Å -1 ) FIG. S2. Photon energy dependent measurements for the 2DELs at SrTiO 3 surfaces with varying carrier density of cm 2 (a), cm 2 (b) and cm 2 (c), respectively. No dispersion is observed along the surface normal k z, confirming the two-dimensional nature of the electronic states. All data were measured in the second Brillouin zone. The carrier densities n 2D given in the main manuscript correspond to the Luttinger volume of the first light subband (L1) with circular Fermi surface and the two symmetryequivalent heavy subbands (H1) with elliptical Fermi surfaces with their long axes extending along k x and k y, respectively. We neglect contributions from the higher quantum well states with smaller Fermi surfaces. As shown in Ref. [4], n 2D defined in this way represents approximately 7% of the total Luttinger volume of all occupied subbands. Since we do not have direct measurements of the Fermi surface of the heavy xz/yz bands.6 k F,H1 (Å -1 ) k F,L1 (Å -1 ) FIG. S3. Experimentally determined Fermi wave vectors of the first light and the first heavy subband for various carrier densities (black dots). The black line shows a linear fit with k F,H1 =1.7k F,L1. 3 NATURE MATERIALS 3

4 DOI: 1.138/NMAT4623 for all carrier densities, we use the empirical relation k F,H1 =1.7k F,L1 between the Fermi wave vectors of light and heavy bands derived from the data shown in Fig. S3. Neglecting the small hybridization between these bands, the carrier density can then be written as n 2D =(kf,l1 F,H1 2 +2k2 m L /m H )/2π, with a ratio m H /m L = 12.8 of the effective masses. 3. Replica bands and Franck-Condon fits. In Fig. S4 a-f, we reproduce the density dependent dispersion plots shown in the main text and supplement the raw data with curvature plots (see Ref. [7] for details). For densities up to cm 2, the latter show a clearly defined dispersive replica band due to small q coupling to the LO 4 phonon mode with a b c d e.2 2.9x x x x x1 14 f 1.9x g h i j k l m n o p q r Intensity (arb.units) FIG. S4. a-f, Raw energy-momentum intensity maps of 2DELs with increasing carrier concentration indicated in units of cm 2. The corresponding curvature plots are shown in g-l. m-r, Raw energy distribution curves at the Fermi wave vector indicated by a dashed white line in the corresponding image plots. In m-p we show the results of a Franck-Condon fit as described in the main text and Supplementary Information. In r, we compare a spectral function calculated for the Migdal-Eliashberg self-energy from Ref. [6] with the EDC at k F from the highest density 2DEL. The exponential background in all spectra is indicated by a green line. 4 4 NATURE MATERIALS

5 DOI: 1.138/NMAT4623 SUPPLEMENTARY INFORMATION Ω LO,4 1 mev. At a density of cm 2 a remnant of the replica band remains discernible but starts to merge gradually with the main band. At the highest density of cm 2, we no longer detect traces of a replica band. Instead we find a well defined quasiparticle band with a large band width of 25 mev and a kink in the dispersion at 3 mev from coupling to softer phonon modes. This behavior is characteristic of short range electron phonon coupling with a negligible dependence on momentum transfer q. In order to estimate the strength of electron-phonon coupling in the low-density regime, we estimate the coherent fraction Z from fits to a Franck-Condon model. To this end, we describe the coherent quasiparticle by a Lorentzian and the phonon satellites by Gaussians separated by 1 mev. The peak integrals are restricted to follow the Poisson distribution I n /I QP = a 2n c /n!, where I n and I QP are the integrated intensities for the n-th phonon satellite and the quasiparticle peak and a c is a constant, respectively. The background arising predominantly from the in-gap state at 1.1 ev is approximated by an exponential and fitted simultaneously. The raw data, background (green line) and Franck-Condon fits are shown in Fig. S4. The coupling constant α is then estimated from Z = I QP /(I tot I bkg ) using the numerical results for the 3D Fröhlich model from Ref. [8]. Here, I tot and I bkg are the integrated intensities of the raw spectra and fitted background, respectively. 4. Spectral function calculations. In Fig. S5, we show results from calculations of the single-particle spectral function for various electron concentrations which includes a small q, forward electron-phonon interaction. The 2DEL with electron-phonon coupling can be described by the Hamiltonian H = kσ ε k c kσ c kσ g q (c kσ c k q,σa q + c kσ c k+q,σa q)+ k,q,σ N q Ω q a qa q, (1) in which ε k is the bare band structure; c kσ (c kσ) is the electron creation (annihilation) operator at momentum k of spin σ; a q (a q ) and Ω q are the phonon creation (annihilation) operator and frequency; g q represents the electron-phonon coupling. In the dilute limit of 2DEL system, the band structure can by approximated by parabolic ε k = t k 2 E and the phonon is treated as Einstein mode Ω q Ω. Importantly we consider electrons in STO to be coupled to high energy optical oxygen phonons that couple via electrostatic modulation. To capture this long-wavelength coupling, we assume g q = g exp( q /q ) where q.16å 1 is the interval in momentum space. The value of q sets the overall forwardness of the NATURE MATERIALS 5

6 DOI: 1.138/NMAT4623 A(k,ω ) a b c < s s > =.12 < s s > =.551 < s s > =.575 k=(,) k=(π /5,) k=(2π /5,) k=(3π /5,) ω [ev] ω [ev] ω [ev].1 FIG. S5. Single-particle spectral function A(k,ω) for a 1%, b 2% and c 3% filling calculated by exact diagonalization in a 1 1 cluster. In each figure, the four curves with different colors denote the EDC for various momenta near the band bottom and the dashed line represents the Fermi level. The arrows indicate q = phonon shake-off of the band bottom. interaction whose value is limited by the granularity of the cluster size. A small value of q limits the momentum space of the scattered electron from its initial momentum state, yielding a more faithful dispersion of the replica bands. The spectral functions for various electron densities (1%, 2% and 3%) are shown in Fig. S5 a-c, respectively. In the 1% filled system, the replica bands show up clearly. With the same g q but increased carrier density, the replica/main ratio decreases for increased doping. This tendency is consistent with the experimental results shown in Fig. S2 of the maintext. This can be thought to arise because the Midgal parameter - the ratio of the lattice energy to electron kinetic energy - decreases with increasing electron density, yielding a net smaller effective electron-phonon coupling. In order to connect the electronphonon coupling to superconductivity, we measure the s-wave pairing density s s where s = 1 N k c k c -k. Although the superconducting susceptibility is still increasing in this range of doping, a rapid drop of the slope is observed. In this sense, one expects a saturation and perhaps even a suppression, leading to a superconducting dome, for larger electron fillings, which are not accessible in this calculation. The simulations were performed on a 1 1 cluster assuming uncorrelated electrons. To limit the Hilbert space dimension for the electronic degrees of freedom, we truncated the effective Hilbert space by forbidding scattering outside the region k 1 3q, which was set by estimating the largest k F ( q ) for the carrier densities considered in our calculation. The spectral weight outside k 1 > 3q is weak considering the value of electron-phonon 6 NATURE MATERIALS

7 DOI: 1.138/NMAT4623 SUPPLEMENTARY INFORMATION coupling in the calculation, which further justifies the truncation. A finite maximum phonon occupation P max is required to limit the Hilbert space dimension associated with the bosonic degrees of freedom. Here we set P max = 1 for 1% and 2% filling, while P max = 6 for 3% filling due to the exponential increase of electronic Hilbert space dimension with filling. We primarily are interested in an energy range which includes the first few replicas and have checked that the spectral functions do not change substantively when varying P max from 4 to 6. Therefore, we are confident that the spectral functions have saturated and a further increase of P max would not change them qualitatively. In our calculation, the coupling parameter g = 6 mev, Ω = 1 mev and t = 176 mev. 5. Effective masses. In Fig. S6, we modify Fig. S3 of the main text to show the evolution of the mass enhancement m /m rather than the effective mass m with carrier density. The solid blue symbols show m /m obtained from the experimental quasiparticle FIG. S6. Comparison of the mass enhancement m /m obtained from the QP dispersion (solid blue symbols) and from the relation m /m =1/(1 α/6) for 3D Fröhlich polarons (open blue symbols). The QP residue Z in the polaronic regime is shown in red. Different symbols indicate data taken on substrates annealed at different temperature. The background color encodes the bare band width of the first light subband calculated from the experimentally determined k F shown in the top-axis, assuming a bare mass of m =.6 m e. Dashed lines are guides to the eye. The dome shaped superconducting phase observed at the LaAlO 3 /SrTiO 3 interface is indicated in grey. NATURE MATERIALS 7

8 DOI: 1.138/NMAT4623 dispersion assuming a bare band mass of m =.6 m e. Open blue symbols show m /m = 1/(1 α/6) for the Fröhlich model with weak to intermediate coupling [9]. The latter underestimates the direct measurement of m but reproduces its trend as a function of density. As discussed in the main text, we attribute this behavior to the effect of electronelectron interactions, which are not fully treated in the analysis of Z we use to estimate α. [1] Erdman, N. et al. The structure and chemistry of the TiO 2 -rich surface of SrTiO 3 (1). Nature 419, (22). [2] Meevasana, W. et al. Creation and control of a two-dimensional electron liquid at the bare SrTiO 3 surface. Nat. Mater. 1, (211). [3] Santander-Syro, A. F. et al. Two-dimensional Electron Gas with Universal Subbands at the Surface of SrTiO 3. Nature 469, (211). [4] McKeown Walker, S. et al. Carrier-Density Control of the SrTiO 3 (1) Surface 2D Electron Gas studied by ARPES. Adv. Mater. 27, (215). [5] Wang, Z. et al. Anisotropic Two-dimensional Electron Gas at SrTiO 3 (11). Proc. Natl. Acad. Sci. U.S.A. 111, (214). [6] King, P. D. C. et al. Quasiparticle dynamics and spin-orbital texture of the SrTiO 3 twodimensional electron gas. Nat. Commun. 5, 3414 (214). [7] Zhang, P. et al. A precise method for visualizing dispersive features in image plots. Rev. Sci. Instrum. 82, (211). [8] Mishchenko, A. S., Prokof ev, N. V., Sakamoto, A. & Svistunov, B. V. Diagrammatic quantum Monte Carlo study of the Fröhlich polaron. Phys. Rev. B 62, (2). [9] Devreese, J. T. & Alexandrov, A. S. Fröhlich polaron and bipolaron : recent developments. Rep. Prog. Phys. 72, 6651 (29). 8 NATURE MATERIALS