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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
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