Quantum transport of 2d

topological insulator edge states

Sven Essert, Viktor Krückl,

and Klaus Richter

Introduction

• 2d topological insulators can be

realized in HgTe/CdTe quantum wells

• Edge transport should be protected

from elastic backscattering for TR-

invariant Hamiltonians

dissipationless transport

König et al., Science 308, 776 (2007)

Introduction

• 2d topological insulators can be

realized in HgTe/CdTe quantum wells

• Edge transport should be protected

from elastic backscattering for TR-

invariant Hamiltonians

dissipationless transport

König et al., Science 308, 776 (2007)

• BUT:

Experiments on longer samples observe backscattering

Backscattering not temperature dependent

Grabecki et al., PRB 88, 165309 (2012), Nowack et al., Nat. Mater. 12, 787 (2013),

Gusev et al., PRB 89, 125305 (2014)

Introduction

• 2d topological insulators can be

realized in HgTe/CdTe quantum wells

• Edge transport should be protected

from elastic backscattering for TR-

invariant Hamiltonians

dissipationless transport

König et al., Science 308, 776 (2007)

• How can we understand the temperature-independent

backscattering?

• BUT:

Experiments on longer samples observe backscattering

Backscattering not temperature dependent

Grabecki et al., PRB 88, 165309 (2012), Nowack et al., Nat. Mater. 12, 787 (2013),

Gusev et al., PRB 89, 125305 (2014)

On-edge backscattering in 2d TIs

• Possible mechanisms:

• Magnetic impurities

• 2-particle backscattering

• Phonons

• Dephasing and subsequent elastic backscattering

Maciejko et al., PRL 102, 256803 (2009), ….

Wu et al., PRL 96, 106401 (2006),

Xu et al., PRB 73, 045322 (2006),

Väyrynen et al., PRL 110, 216402 (2013),

Geissler et al., PRB 89, 235136 (2014), …

Budich et al., PRL 108, 086602 (2012), ….

Jiang et al., PRL 103, 036803 (2009),

Roth et al., Science 325, 294 (2009), ….

On-edge backscattering in 2d TIs

• Possible mechanisms:

• Magnetic impurities

• 2-particle backscattering

• Phonons

• Dephasing and subsequent elastic backscattering this talk

Maciejko et al., PRL 102, 256803 (2009), ….

Wu et al., PRL 96, 106401 (2006),

Xu et al., PRB 73, 045322 (2006),

Väyrynen et al., PRL 110, 216402 (2013),

Geissler et al., PRB 89, 235136 (2014), …

Budich et al., PRL 108, 086602 (2012), ….

Jiang et al., PRL 103, 036803 (2009),

Roth et al., Science 325, 294 (2009), ….

• Event that breaks coherent wave function evolution

(interactions with other quantum systems)

• Sources: • other carriers

• trapped charges

• phonons

Dephasing

• Event that breaks coherent wave function evolution

(interactions with other quantum systems)

• Sources: • other carriers

• trapped charges

• phonons

• Spin-conserving dephasing

Dephasing

No backscattering on edge Backscattering in puddle

Backscattering due to interplay of lifetime in the puddle and dephasing time

Lifetimes in puddles

Time evolution of density in a puddle:

Lifetimes in puddles

Time evolution of density in a puddle:

electrons leave puddle in random direction after the event

1. Heuristic model treatment of

dephasing:

• independent events with time

constant

• spin-conserving

• event lead to full dephasing

Model treatment

electrons leave puddle in random direction after the event

Simple expression in case of linear density decay with decay time :

1. Heuristic model treatment of

dephasing:

• independent events with time

constant

• spin-conserving

• event lead to full dephasing

Model treatment

Results of model treatment

Hier Fig. 3

Saturation

Results of model treatment

Hier Fig. 3

Saturation

Saturation may explain temperature-independent transmission

Einselection

• Include dephasing in dynamics inspired by “environment-

induced superselection” (einselection)

• What is einselection?

Zurek, RMP 75, 715 (2003)

Einselection

• Include dephasing in dynamics inspired by “environment-

induced superselection” (einselection)

• What is einselection?

• Related to the measurement problem:

Zurek, RMP 75, 715 (2003)

Einselection

• Include dephasing in dynamics inspired by “environment-

induced superselection” (einselection)

• What is einselection?

• Related to the measurement problem:

Zurek, RMP 75, 715 (2003)

measure

Einselection

• What about a system in an environment?

• Einselection: measurement due to environment for

is called pointer basis

interference between pointer

states is suppressed

Einselection

What is a “typical” pointer basis?

• For very weak coupling to the environment:

eigenstates of the system Hamiltonian

• For very strong coupling:

eigenstates of the coupling Hamiltonian

• For intermediate coupling and local interaction:

states which are localized in phase space

(both in position and energy)

• Ideal implementation (wishful thinking):

propagate full density matrix

Dynamical calculations with dephasing

• Ideal implementation (wishful thinking):

propagate full density matrix too costly

• Instead:

coherent propagation of ,

dephasing at discrete (Poisson process with )

Dynamical calculations with dephasing

• Ideal implementation (wishful thinking):

propagate full density matrix too costly

• Instead:

coherent propagation of ,

dephasing at discrete (Poisson process with )

• At the dephasing event ( ):

Suppresses interference between pointer states

Decompose in

pointer basis

Randomize

coefficients

Recompose to

yield .

Dynamical calculations with dephasing

Dynamical calculations with dephasing

What to choose as pointer basis?

• We choose “local energy eigenstates”:

for a set of eigenenergies in the gap

• Basis both local in energy and in space (finite propagation)

• In the limit , we recover the weak coupling limit

• Conserves average spin and density

• Disadvantage: auxilliary propagation required

Dynamical calculations with dephasing

• Limit of weaker dephasing accessible

• Allows study of modified carrier dynamics:

Dynamical calculations with dephasing

• Limit of weaker dephasing accessible

• Allows study of modified carrier dynamics:

Results for 400 nm puddles:

Dynamical calculations with dephasing

Results for 400 nm puddles:

Dynamical calculations with dephasing

Kozlov et al., JETP Lett. 96, 730 (2013)

Conclusions

• Dephasing in puddles explains temperature-independent resistance if

mainly large (>500 nm) badly coupled puddles

• Good control experiment: Artificial small puddles which should show

temperature dependence

extract dephasing time with our predictions

• Magnetoconductance also hints to the existence of charge puddles

• We developed a new scheme to include dephasing in time evolution

algorithms.

• Transport framework: TQT by V. Krückl (www.krueckl.de/en/tqt.php)

Essert et al., 2D Mater. 2, 024005 (2015)

Conclusions

• Dephasing in puddles explains temperature-independent resistance if

mainly large (>500 nm) badly coupled puddles

• Good control experiment: Artificial small puddles which should show

temperature dependence

extract dephasing time with our predictions

• Magnetoconductance also hints to the existence of charge puddles

• We developed a new scheme to include dephasing in time evolution

algorithms.

• Transport framework: TQT by V. Krückl (www.krueckl.de/en/tqt.php)

Thank you for your attention!

Essert et al., 2D Mater. 2, 024005 (2015)

Magnetoconductance of 2d-TI edges

König et al., Science 308, 776 (2007) Essert et al., 2D Mater. 2, 024005 (2015)

Dynamics with dephasing

• Dephasing opens extra decay channel

decreased lifetime

Dephasing algorithm