N. Peter Armitage The Johns Hopkins University Riccardo Tediosi, E. - - PowerPoint PPT Presentation

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)

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 1017 e-/cm3

• self-compensated n=p
• Low effective masses: ~ 10-2 me
• EF ~ 23-27 meV
• λF ~ 40 nm --> Quantum confinement
• Interaction energy: e2n1/3/ε = 3 meV

Does something very interesting with modest pressures!

Motivation

Hole Band Electron Band

Pressure!

MITs are of much long term interest What is nature of this one? Fermi surface topological transition --> Lifshitz transition? But is it?

Motivation

• Balla and Brandt JETP (1965)

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 580

• vs. 600 and d/d’=0.9).

Deviation from higher symmetry gives small band

• verlap

Highly anisotropic compressibility in the c and a directions --> pressure pushes it back towards higher symmetry and reduces band overlap

Bismuth Introduction

Liu et al., PRB. 52, 1556 (1995) – Golin 1968

13 26.7 13.7

Band Structure

Band Structure

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). (Lifshitz 1960)

Nature of the Transition?

But what really happens? How does transition proceed? Enhanced interaction effects at low density? Ep ~ 1/rs ~ n1/3 Ek ~ kF

2 ~ 1/rs 2 ~ n2/3

Ep / Ek ~ 1/n1/3 strongly correlated Wigner liquid --> Wigner crystal

Nature of the Transition?

Relative interaction strength diverges @ low density Wigner xtal @ rs~80

Can disorder dominate at low density? Anderson localization at band edges?

Nature of the Transition?

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 Pc~ 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 Pc --> “Wigner pair liquid”

• Significant deviations from a pure Lifshitz transition behavior

SAMPLE GOLD

• )

( 4 ) ( ) ( 1 ) , ( 1 ) , ( ) , (

1 1 2

i T T T R + = +

• =

s p p s

Reflectivity (3.7 meV  0.75 eV) Ellipsometry (0.75 eV  3.7 eV)

R(ω)IR ε1(ω),ε2(ω)VIS

Kramers-Kronig Consistent Drude-Lorentz Fit Optical conductivity, dielectric function, etc.

ρ(T) DC

Optical Spectroscopy

100 300 500 700 50 150 250 Temperature (K) wavenumber (cm-1) Reflectivity

Reflectivity Spectra

Variational Fitting Drude-Lorentz model

Conductivity Spectra

MIR - Features

) ( 4 ) (

2 1

• =

Conductivity Spectra

EDI (67.1 meV) L T 13.7 meV Electron Band Hole Band Energy σ1(ω ) EDI

T = 0 T ≠ 0

Expected decrease of SW upon cooling in the energy range of the gap (also Pauli blocking) Intraband I n t e r b a n d

Band Structure and Optical Conductivity

Experimental Results

Onset of interband expected @ 67.1 meV

• Anomalous mid infrared absorption
• Onset lower than band expectation

and wrong temperature dependence

• Clear “pre-peak” structure

T = 2 K T = 210 K

Extended Drude Analysis

• =
• )

( 1 Im ) ( 1

2 p

• ε∞

has been considered temperature DEPENDENT;

• ωp 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

210 K 150 K 100 K 70 K 20 K

Electron-Plasmon Interaction

We interpret the mid-infrared absorption as a strong-coupling boson `shake-off’, as seen for phonons or magnons. A Holstein side band for plasmons

E

2kF

Electron-Hole Continuum

wp

2

) ( cq q

P

p

+ =

• qc

Can be described using same Holstein Hamiltonian as used for polarons Plasma oscillation + Electron System = “Plasmaron” [1,2] [1] B. Lundqvist, Phys. Kondens. Materie 6, 193 (1967). [2] B. Lundqvist, Phys. Stat. Sol. 32, 273 (1969).

• 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

Hole Band Electron Band

Pressure!

Hydraulic Press Piston Clamping Nut Internal Piston Body Optical Access

Optical Pressure Cell

• Piston-Cylinder Cell;
• Kerosene Filled;
• Pressure up to 2.5 GPa (25 kbar)

Thanks to Richard Gaal (EPFL) Indium Foil 2° Wedged Diamond Window Sample Epoxy Glue

Pressure Effects

• Bismuth is anisotropically

compressible

• Pressure pushes crystal

structure back towards the lattice structure of higher symmetry.

• Transition of the Lifshitz

variety(?) at Pc = 25 kbar

Reflectivity under pressure

Pressure effects on plasma frequency

MIT

SMSC

With absolute reflectivity KK analysis is now possible!

Optical constants

Optical constants

Enhanced interactions on approach to Lifshitz transition

Interaction features become enhanced at unique value

• f 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?

2.2 2.0 1.8 1.6 1.4

m*/m

25 20 15 10 5

Pressure (kbar)

Mass renormalizations

Mass renormalizations from partial sum rule of loss function

(Kraak, Herrmann, and Haupt 1982)

Resistivity Coefficient of T2 Coefficient of T2 term increases

• n approach to MIT (Kraak,

Herrmann, and Haupt 1982)

• Wigner crystallization

(small mass, large εinf)

• 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

Explanations

• -> ambient P