Quantum Transport in Disordered Topological Insulators

Vincent Sacksteder IV, Royal Holloway, University of London

Quansheng Wu, ETH Zurich

Liang Du, University of Texas Austin

Tomi Ohtsuki and Koji Kobayashi, Sophia University

Stefan Kettemann, Jacobs University, Bremen

Ivan Shelykh and Kristin Arnardottir, Nanyang Technological University

arXiv:1605.02203 (AB-AAS), Physical Review Applied v. 3 p. 064006 (engineering TI surface conduction). PRB v. 90 p. 2035148 (topological effects via sidewalls). PRB v. 90 p. 045408 (topological protection in 3-D). PRB 88 041307 (2-D TIs with edge disorder). See also my PRB 88 045429 (experiment on magnetoconductance/WAL of 3-D TIs), PRB 85 195140.

Self-Introduction   Topological research threads:

  Nonperturbative numerical study of strongly disordered TIs and Chern insulators; integer quantum Hall effect and analogs.

  Magnetoconductivity (weak antilocalization) in 3-D TIs, using analytical calculations of the Cooperon.

  Other research threads:

  Field theory (path integrals, sigma models, perturbation theory, numerics) of disordered conduction.

  Accurate calculation of spin diffusion equations in realistic materials, starting from realistic k.p and tight-binding Hamiltonians. Both analytical and numerical techniques.

  Scientific computing and algorithms, especially parallel computing.   Implementation of GW/MBPT calculation of electronic structure, with

Keith Refson, CASTEP. Also magnetic susceptibility, GW+DMFT.

Topic 1: AB effect in 3-D TI wires   The first article: Yi Cui’s group in Stanford, 2009.

  Incontrovertible proof of surface transport on the wire surface.

  15+ experiments since then. 6 theory papers.

AB illustration from De Kismalac - Trabajo propio, CC BY-SA 3.0

AB effect in ballistic TI wires

Figures from Dufouler et al, Dresden.

•  Much Lower Temperature •  Many Harmonics Visible, because the dephasing length is much longer. •  Three regimes:

•  Ballistic, if the scattering length l is larger than the wire dimensions. •  Diffusive, if l is shorter than the wire dimensions. •  Localized, if the conductance G is less than 1 G_0.

AB Effect: Our Focus   Single wires. (This is what is measured experimentally, not average behavior.)

  Zero temperature, where quantum interference is strongest.

  Tuning a sample between ballistic, diffusive, and localized regimes.

  Integrated picture of the zoo of periodic conductance features:

  The first harmonic - called “AB oscillations”

  The second harmonic – Altshuler-Aronov-Spivak oscillations.

  Universal conductance fluctuations – sensitive dependence to sample details, Fermi level, etc. “Noise.”

  The Perfectly Conducting Channel - a topologically protected conductance quantum; the conductance plateaus at 1 G_0 and never goes below that.   In TI wires, requires a tuned longitudinal magnetic field.

  3-D analogue of the Integer Quantum Hall Effect.

  The only incontrovertible, unambiguous way of showing topological protection from disorder.

Our TI model   Z2 tight binding on a tight binding

cubic lattice.

  No bulk conduction.

  Uniform cross-section.

  No penetration of surface state into the bulk.

  Perfectly parallel magnetic field.

  Disorder only on the TI surface.

Disorder Averaged Conductance   Left shows the energy levels, right shows the conductance as a function of

Fermi energy and magnetic flux.   Diffusive regime (0.35 < E): vertical stripes = AAS oscillations.   Ballistic regime (0.07 < E < 0.35) : cross-hatched = AB oscillations.   Localized regime (E < 0.07) : localized regime, with PCC.

  Almost all theoretical work on the AB-AAS effect is based on disorder averages; AB taken as sign of ballistic physics, AAS as sign of diffusive physics.

Single-wire conductance very different from averaged conductance

  All parameters the same; the diffusive-ballistic-localized break down is the same.

  AB and AAS oscillations present in all three regimes, with similar weights!

  The only way to distinguish diffusive from ballistic, in a single wire, is by varying E and seeing whether the AB component is periodic (ballistic regime) or random (diffusive.)

  AAS signal is always positive (weak antilocalization.)

  Strong UCFs.

Long wires and the PCC

  Wire length L = 403, localization length = 200.

  A very crisp, spectacular signal.

  Almost noise-free.

  The PCC extends through most of the band gap; no tuning needed.

Strong UCFs in typical magnetoconductance traces

  Thin lines show the disorder average; thick lines show single wires.

  L=21,39 are ballistic, L=126 is diffusive, L=403 is localized.

  Except in long localized wires, UCFs are dominant in single wires.

Topic 2: Controlling Surface State Properties with disorder depth   The idea is to introduce disorder on the

surface and to systematically control the disorder depth d.

  The surface state is always topologically protected.

  If surface disorder is tuned to a resonant value near the topological phase transition, the topological state will occupy the entire disordered region to depth d. Depinning. Soft edges.

  At very very large disorder the surface states move into the clean bulk and their plane-wave character is restored.

  Controlled by a resonance between disorder and the bulk band.

3-D Pinned and Depinned States   (a) is pinned to the

outer surface and is a good plane wave.

  (b) Is depinned and topological. It has a much smaller volume and is so strongly disordered that it should be localized.

  (c) Shows non-topological states in the disordered region, coexisting with a topological state at the inner boundary with the clean bulk.

probability density

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(b) Topological & depinned�

(c) Non-topological�

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(b) Should be a kind of strongly disordered Weyl fermion.

Transport Properties are Strongly Dependent on Disorder Depth

  Depend either on d or on 1/d.

  Can be deduced from energy scaling, because scattering length scale is pinned to the lattice spacing.

  Allows patterning and design of devices on the surface of a TI.

Applications

  Disorder can be patterned on the TI surface using masking and ion implantation.

  When d is increased, the increased DOS will decrease sensitivity to contaminants and push the Fermi level toward the Dirac point. Sensitivity to light and to static and ac voltages will be increased, as will the state’s self-interaction. This may favor strongly interacting topological phases.

  The density of states will focus in regions of increased d. Lines of increased d will guide current along specific channels on the TI surface. Incoming pulses will be redirected and focused (lensed) into these channels.

  External gates at control points will move the Fermi level, cause the state to reroute around the disordered layer, and therefore block the channels at these control points.

  Channeling + gate-controlled flow => topological integrated circuits.

Topic 3: 2-D topological metals on the surface of 3-D TIs.

  A Single Dirac Cone in two dimensions. (1/4 of graphene.)

  In the Symplectic (AII) Universality Class – Spin-orbit interaction, but no magnetic field.

  The topological metal is expected to be robust against any disorder strength, even in infinitely big systems.

  There are only TWO experimental transport signatures of topological protection:   (1) in long samples: a perfectly conducting channel with perfectly quantized

conductance.   (2) in short samples with conductance G>1: an always-increasing conductivity,

which is a universal model-independent function of sample length

  The issues:   Surface states may make excursions into the bulk, which could affect their

conduction.   The bulk hosts bulk states which could assist in tunneling between the TI

surfaces and kill the surface states.

Testing the Topological Metal on a 3-D TI, including the bulk

  In 3-D TIs (as opposed to graphene) topological metals require extensive parallel computing. The following results required extensive use of a large supercomputer in Nanjing.

  Our main results:   We show that the topological metal is robust against bulk effects.

  We show that conduction on the surface does not follow the universal conductivity curve; non-universal conduction. This will change the shape and magnitude of the magnetoconductance.

The Perfectly Conducting Channel in Long Wires

  Verified PCC in very long TI wires.

  Found decay length as a function of wire width W, which scales as W^3. (Previous analytical prediction was W^4.)   This decay is not caused by tunneling through the wire,

which is exponentially small.

  Found effects of penetration into the bulk, sensitivity to magnetic field fine tuning.

Universal Conductivity Curve   The conductivity Sigma = G (L/W), where G is the conductance and L,W are the length, width.

Measured in the diffusive regime with G > 1; i.e. samples not too long.

  The conductivity always increases with L,W, and follows a universal curve:   A consequence of diffusive conduction, with many scatterings.   Model-independent: The first figure was obtained on the continuum, while the second one

was on a lattice. Both verify perturbation theory.   At large sigma, strong theoretical arguments require logarithmic growth: sigma = (1/pi) \ln

L, where L is the sample size.   The coefficient (1/pi) is universal and determines the magnetoconductivity’s magnitude. It is

tied to the 1 / 4 pi in the Hikami Larkin Nagaoka formula.

Note that corrections like 1/ sigma^2 are numerically zero.

  PRL 99 106801, J Condensed Matter 22 273201, PRL 99 14680

The Conductivity in TIs

  We show four disorder strengths and two Fermi levels.

  The decreases at small L are caused by disorder-assisted tunneling between the leads.

  Always-increasing; topologically protected.

  Logarithmic, as seen in another calculations.

  Universality: the pink U=7.2, E=0 curve matches the (1/pi) ln L curve seen in other models.

  Non-universality, maybe not diffusion: at other values of U,E the coefficient is not 1/pi => variable. This implies that the magnetoconductivity magnitude is also non-universal in TIs.

Thank you!   Vincent Sacksteder

  vincent@sacksteder.com