Towards the Lifshitz transition in elemental bismuth: Light electrons gone heavy at the metal-insulator transition? * N. Peter Armitage The Johns Hopkins University Riccardo Tediosi, E. Giannini, and D. van der Marel University of Geneva L. Forro and R. Gaal Ecole Polytechnique Federale de Lausanne * R. Tediosi et al. PRL (2007) R. Tediosi et al. to be sub. (2008)
Motivation Correlation effects in semimetals are a hot topic... graphene, fractional excitations in 3D Bismuth, plasmarons etc. Bismuth should be described by Boltzmann/Drude transport, but in practice very different energy scales and parameters than typical give strong deviations. •Semi-metallic •Low carrier concentration: ~ n = 3 x 10 17 e - /cm 3 •self-compensated n=p •Low effective masses: ~ 10 -2 m e •E F ~ 23-27 meV • λ F ~ 40 nm --> Quantum confinement •Interaction energy: e 2 n 1/3 / ε = 3 meV Does something very interesting with modest pressures!
Motivation Electron Band Hole Band Pressure! MITs are of much long term interest What is nature of this one? Fermi surface topological transition --> • Balla and Brandt JETP (1965) Lifshitz transition? But is it?
Bismuth Introduction Crystal is rhombohedral A7 symmetry (space group R3m) with two atoms per unit cell. Can be viewed as a simple cubic with slight distortion along diagonal (out of vertical angle 58 0 vs. 60 0 and d/d ’ =0.9). Deviation from higher symmetry gives small band overlap Highly anisotropic compressibility in the c and a directions --> pressure pushes it back towards higher symmetry and reduces band overlap
Band Structure 13 13.7 26.7 Liu et al. , PRB. 52 , 1556 (1995) – Golin 1968
Band Structure
Nature of the Transition? (Lifshitz 1960) Rigid bands --> Lifshitz? --> Fermi surface topological transition 2nd derivatives of electronic thermodynamic potentials have non-diverging square-root singularities and 3rd derivatives have diverging inverse-square-root singularities; “Transitions of the 2.5 order” based on Ehrenfest terminology (Lifshitz JETP 1960).
Nature of the Transition? But what really happens? How does transition proceed? Enhanced interaction effects at low density? E p ~ 1/r s ~ n 1/3 E k ~ k F 2 ~ 1/r s 2 ~ n 2/3 E p / E k ~ 1/n 1/3 Relative interaction strength diverges @ low density Wigner xtal @ r s ~80 strongly correlated Wigner liquid --> Wigner crystal
Nature of the Transition? Can disorder dominate at low density? Anderson localization at band edges? Conduction Valence Localized states
“In this paper we show ….” • Evidence for a strongly coupled electron-plasmon feature --> a plasmaron • Consistency with a novel Metal-Insulator transition near P c ~ 25 kbar • A vanishing charge density with increasing pressure • An enhanced mass over a broad region of finite pressure • Large correlation effects that indicate a novel strongly correlated metallic state on the approach to P c --> “Wigner pair liquid” • Significant deviations from a pure Lifshitz transition behavior
Optical Spectroscopy 2 ( , T ) 1 � � � R ( , T ) � = ( , T ) 1 � � + 4 ( ) �� � GOLD 1 ( ) ( ) i � � = � � + 1 � SAMPLE Reflectivity (3.7 meV 0.75 eV) ε 1 ( ω ), ε 2 ( ω ) VIS R( ω ) IR ρ (T) DC p p Kramers-Kronig Consistent Drude-Lorentz Fit s s Optical conductivity, Ellipsometry (0.75 eV 3.7 eV ) dielectric function, etc.
Reflectivity Spectra Reflectivity 50 Temperature (K) 150 250 100 300 500 700 wavenumber (cm -1 )
Conductivity Spectra Variational Fitting Drude-Lorentz model
Conductivity Spectra � ( ) ( ) MIR - Features � � = � � 1 2 4 �
Band Structure and Optical Conductivity L σ 1 ( ω ) T Intraband I n t e E DI (67.1 meV) r b a n Hole d Band 13.7 meV T ≠ 0 T = 0 Energy Electron E DI Band Expected decrease of SW upon cooling in the energy range of the gap (also Pauli blocking)
Experimental Results K 0 2 Onset of interband expected = T @ 67.1 meV T = 210 K • Anomalous mid infrared absorption • Onset lower than band expectation and wrong temperature dependence • Clear “pre-peak” structure
Extended Drude Analysis 2 1 1 � � � p Im � � = � � � ( ) ( ) � � � � � � � � � � • ε ∞ has been considered temperature DEPENDENT ; • ω p extracted from the Drude- Lorentz model • The increase of the scattering rate coincides with the position of prepeak ’ s onset !
Evidence in the EEL Function
Temperature Dynamics 100 K 210 K 70 K 150 K 20 K
Electron-Plasmon Interaction We interpret the mid-infrared absorption as a 0 2 ( q ) cq strong-coupling boson `shake-off ’ , as seen � = � + p for phonons or magnons. P E A Holstein side band for plasmons Electron-Hole Continuum Can be described using same Holstein Hamiltonian as used for polarons w p Plasma oscillation + Elect ron System = “ Plasmaron” [1,2] [1] B. Lundqvist, Phys. Kondens. Materie 6, 193 (1967). q c [2] B. Lundqvist, Phys. Stat. Sol. 32, 273 (1969). 2k F • Effect that always exists in metals, but usually almost irrelevant! • In semi-metals, the vastly different energy scales mean that this effect can dominante! Enters into low energy physics.
Pressure Effects Electron Band Hole Band Pressure!
Optical Pressure Cell Hydraulic Press Piston Thanks to Richard Gaal (EPFL) • Piston-Cylinder Cell; • Kerosene Filled; Clamping Nut • Pressure up to 2.5 GPa (25 kbar) 2° Wedged Sample Internal Piston Diamond Window Epoxy Glue Indium Foil Body Optical Access
Pressure Effects •Bismuth is anisotropically compressible •Pressure pushes crystal structure back towards the lattice structure of higher symmetry. •Transition of the Lifshitz variety(?) at P c = 25 kbar
Reflectivity under pressure
Pressure effects on plasma frequency SMSC MIT
Optical constants With absolute reflectivity KK analysis is now possible!
Optical constants
Enhanced interactions on approach to Lifshitz transition
Interaction features become enhanced at unique value of N/m.
Signatures of strong interactions T ~ 15K el-plasmon feature in loss function enhanced at high pressures
Unique anti-crossing behavior of plasmaron and plasmon peak in loss function?
Mass renormalizations 2.2 Mass renormalizations from partial sum rule of 2.0 loss function m*/m 1.8 1.6 1.4 0 5 10 15 20 25 Pressure (kbar)
Coefficient of T 2 term increases on approach to MIT (Kraak, Herrmann, and Haupt 1982) Resistivity Coefficient of T 2 (Kraak, Herrmann, and Haupt 1982)
Explanations --> ambient P • Wigner crystallization (small mass, large ε inf ) • Excitonic insulator (suppressed for unequal masses) • CDW (Overhauser) (Poor nesting efficiency in 3D) • e(h)-plasmon induced Wigner liquid Electron-hole interaction is very effective in semimetal systems ’ Wigner pair liquid ’ state, which significantly enhances character of metal insulator transition Future: More transport under pressure!, Thermodynamics; compressibility!, Charge scattering at finite q. Optical measurements closer to Pc
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