Michael Fuhrer Director, FLEET Monash University 24th Canberra International Physics Summer School – Topological Matter Lecture 1

Aims of this lecture series Introduce electronic transport measurements What do we measure? How do we interpret the measurements? What are (some of) the signatures of topology in electronic transport? 2D topological insulators Quantum Hall effect and the “edge state picture” Quantum spin Hall effect Interpreting conductance within Landauer-Buttiker formalism and edge state picture 2D Dirac surface state of 3D TIs Graphene as an example of a 2D Dirac system Conductivity of a disordered 2D Dirac system Non-linear screening and electron-hole “puddling” Weak localization and weak anti-localization Landau levels of the 2D Dirac system

Longitudinal and Hall resistivity Vx Vy L Ix W Ix 2D Longitudinal resistivity: Note: In 2D, [resistivity] = Ω 2D Hall resistivity:

Relaxation-time approximation kx ky δk Fermi surface Relaxation-time approximation: Steady state:

Note: v is average velocity i.e. “drift velocity” Cyclotron frequency

Resistivity Tensor Resistivity in 2D does not depend on B “Hall effect” – depends only on B and n, can be used to measure n

What do we measure at higher magnetic field? Strong deviations from semi-classical expectation Shubnikov-de Haas oscillations, and eventually quantum Hall effect: Resistivity: Conductivity: ν is integer, quantized to approx 1 in 109 no materials parameters appear in conductivity, only fundamental constants

2D electrons in a strong magnetic field Choose gauge: “Landau gauge” Take U = 0 harmonic oscillator centered on

Landau levels, U = 0 +kx states Real space: -kx states E Energy gaps: momentum space E Energy gaps: System with filled LL is insulator Group velocity is zero k real space y +kx states Real space: x States of greater kx are located at larger +y -kx states

Add an electric field harmonic oscillator centered on: Eigenenergies:

Landau levels, constant E-field (U = -eEyy) kx All electrons have same velocity, in –v X B direction Same as classical result for electron in crossed E, B fields: v E = Eyy B = Bzz degeneracy of Landau level Topological insulator: carries current perpendicular to E Quantised Hall effect!

Harmonic confining potential Again, complete the square to recast as harmonic oscillator: States centred on:

Landau levels, parabolic U kx y +kx states have vx > 0 x -kx states have vx < 0

Thought experiment: hard-walled confining potential kx EF U E kx EF y y x

the “edge state picture” of quantum Hall effect y E Confining potential pushes up Landau levels at edges of sample Fermi energy crosses Landau levels Edges are metallic – charge can move in/out 1D edge states carry current in E x B direction B E kx EF E kx EF

What is the conductance of the 1D edge state? kx EF

1D band (spin non-degenerate) EF True for any E(k)! k

Conductance Quantization in 1D Apply voltage V, Calculate current I: lead 1D Channel E k Left-moving states Right-moving ΔE=eV lead 1D Channel E k Left-moving states Right-moving Including spin degeneracy: quantum of conductance: G0 = 2e2/h = (12.9 kΩ)-1

Conductance Quantization B.J. van Wees et al., (1988)

Conductance and Hall Effect in Quantum Hall regime ν = # of edge modes = # of full Landau levels V V potential = V I I ν = 2 potential = 0 Current = ? ν modes carry (e2/h)V each E k Left-moving states Right-moving ΔE=eV

A multi-terminal device potential is same for incoming/outgoing modes Potential same here and here Voltage probe has I = 0 V ΔVx=0 Vy Ix ΔVy=V

Is the edge-state picture “real”? Scanning capacitance microscopy of AlGaAs/GaAs quantum well in quantum Hall regime New J. Phys. 14, 083015 (2012) Current does flow along 1D modes highly confined to sample edges in QHE and QSHE Scanning SQUID image of current in an HgTe quantum well in the quantum spin Hall effect regime Nature Materials 12, 787 (2013)

“Bulk-edge correspondence” Gapless states must exist at interface between different topological phases Quantised Hall conductivity is both bulk and edge property! e.g. light shining on bulk of sample shows a quantised Faraday rotation – induced current at angle to polarisation e.g. transport current is carried along edges of sample

Landau levels, constant E-field (U = -eEyy) kx Same as classical result for electron in crossed E, B fields: degeneracy of Landau level Quantised Hall effect!

Quantum spin Hall effect

The trio of quantum Hall effects Science 340, 153 (2013) Quantum Hall and quantum anomalous Hall effects: edge modes propagate in a single direction, so no possibility of scattering if bulk is sufficiently wide and insulating Quantum spin Hall effect: edge modes propagate in both directions (with opposite spin). Scattering is possible with flip of spin.

1D “Contact Resistance” Drops in average electrochemical potential occur at contacts Dissipation (relaxation) occurs in contacts S. Datta, 1995

Landauer Formula 1-T 1 T E E DE=eV k k (Ti = transmission probability of ith mode) 1-T 1 T E E DE=eV k k

Resistance of a Barrier Inevitable quantized “contact” resistance Resistance of the barrier

Two Barriers - Incoherent Addition “Incoherent” = ignore interference terms (classical particles) For incoherent transmission Rbarrier adds like Ohm’s law

Mean free path Define a length le of 1D conductor such that T = ½ (and R = ½): A conductor of length L has (L/le such sections): and 1D resistivity: Note: le called “mean free path”, “elastic length”, “backscattering length”, etc.

Mean free path in QSHE: experiments HgTe quantum well in the QSHE regime: le ~ 7 μm Nature Materials 12, 787 (2013) QSHE in WTe2: le ~ 150 nm Nature Physics 13, 677–682 (2017)

Questions?