SYNTHETIC HELICAL LIQUID IN A QUANTUM WIRE George I. Japaridze
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Transcript of SYNTHETIC HELICAL LIQUID IN A QUANTUM WIRE George I. Japaridze
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SYNTHETIC HELICAL LIQUID IN A QUANTUM WIREGeorge I. Japaridze
Ilia state University and Andronikashvili Institute of
Physics
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What is a helical liquid (HL)?
Unconventional State of matter with spin-momentum locking
S
p
S
p
SYNTHETIC HELICAL LIQUID IN A QUANTUM WIRE
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What is a helical liquid (HL)?
Unconventional State of matter with spin-momentum locking
SYNTHETIC HELICAL LIQUID IN A QUANTUM WIRE
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Helical modes: where do they occur?
topological insulators(edges):
quasi- ‐1d Super Condsuperconductors:
Semiconducingnanowires: -magnetic
field
carbon nanotubes:
Pankratov, Pakhomov & Volkov, SSC 61, 93 (1987) Hasan and Kane, RMP 82, 3045 (2010)
Pott er and Lee, PRL 105, 227003 (2010)
Streda and Seba, PRL 90, 256601 (2003)
Braunecker, Japaridze, Klinovaja, DL, PRB 82, 045127 (2010)
Klinovaja, Schmidt, Braunecker, DL, PRL 106 156809 (2011), and PRB 84, 085452 (2011)
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Insulators with Strong Spin-OrbitInteraction:Spin-Hall Effect: Edge states
L. Fu, C. L. Kane and E. J. Mele PRL 98, 106803 (2007).
J. E. Moore and L. Balents, PRB 75, 121306(R) (2007).
S.-C. Zhang, Nature Physics 1, 6 (2008).
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Exotic quantum states Majorana Fermions in 1d
Alicea, PRB 81, 125318 (2010) Oreg, Refael, and von Oppen, PRL 105, 177002 (2010) Lutchyn, Sau, and Das Sarma, PRL 105, 077001 (2010) Kitaev, Phys.- Usp. 44, 313 (2001)‐ Alicea, Oreg, Refael, v.Oppen, Fisher, Nature Phys.7,412 (2011) Gangadharaiah, Braunecker, Simon, DL, PRL 107, 036801 (2011)
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Semiconducting Nanowires
Various materials:ZnO, InAs, InP, GaAs, AlAs, Ge, Si, SiGe, GaN, GaP, CdS,
…
Operate both in theconduction band (CB) and valence band (VB)
Charge: similarSpin: very different
ELECTRONS HOLESParticularly characteristic
for semiconductors
Holes turn out to be advantageous in many aspects!
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Ge/Si Core/Shell Nanowires
Xiang et al., Nature (2006); Hu et al., Nat. Nano (2007); Hu et al., preprint (2011)
Nanowiregrown along [110]
Large Ge/Si valence band offset of ~ 0.5 eV, narrow interfaces→ replace with hard wall at core radius Rc ≡ R
Lauhon et al., Nature (2002), Lu et al., PNAS USA(2005)
Ex
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v
Lorenz transformation
v=0H
•c
E
E
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• The 1D lattice Hamiltonian:
H l a t = − t,
(c†
n,σn,σcn+1,σ +h.c.)
+αR ,
[(c† n
n,↑cn+1,↓ − c† cn+1,↑) +h.c.]n,↓
+, f
Uρn,
n
↑ n,↓ρ +V ρ ρn n+1 + J S · S ...n n+1
V. Gritsev, G. Japaridze, M. Pletyukhov, and D. Baeriswyl , Phys. Rev. Lett. 94, 137207 (2005).
S. Gangadharaiah, J . Sun, and O. A. Starykh, Phys. Rev. B 78, 054436 (2008).
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Part I: Magnetic field induced quasi-helical liquid state
in a disordered 1D electron system
with strong spin-orbit interaction
Anders Ström, Bernd Braunecker and G.J. PRB 87, 075151 (2013).
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04.pptx
Helical mode: spin coupled to momentum
Rashba spin-orbit field along z-axis
HR px z
e.g. InAs nanowire λSO ~ 100 nm, Fasth et al., PRL 98, 266801 (2007)
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Helical Hole States
Rc = 5.0 nmRs = 6.5 nm No RSOI!
Kloeffel, Trif, DL, arXiv:1107.4870
E-fieldalong x:
Ex = 6 V/µm strongRashba SOI (~ 1-10 meV)
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Helical Hole States
Rc = 5.0 nmRs = 6.5 nm
Bx opens a gap 0.8 T:~
0.25 meV0.3 T: ~ 0.10 meV
E-field along x: Ex = 6 V/µm strong Rashba SOI (~ 1-10 meV)
Kloeffel, Trif, DL, arXiv:1107.4870
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2kF2kF
Effect of Interaction : strongrenormalization of Zeeman gap!
see also, Stoudenmire, Alicea, Starykh, Fisher, PRB 84, 014503 (2011)
Braunecker, Japaridze, Klinovaja, Loss, PRB 82, 045127 (2010)
2kF
local spin basis transformation: spin-dependent shift in k-space
Helical States from Rotating Zeeman Field
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Band structuresSpins look anti-
parallell for opposite velocities.
Helical?
Corrupted by the magnetic field HS!
Let’s calculate the spin overlaps!
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Band structuresSpin Overlaps
Limits:
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Spin Overlaps
Band structures
Detail for aa overlap. 0.3 meV (ca 1 T) yields overlap squared of about 0.01.
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Overlaps:
Limits:
Disorder: Anderson Localization in 1D !!
(absent in the Helical Liquid State!!)
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DisorderRG
Scaling equations!
For full Luttinger liquid with spin: For spinless or helical Luttinger liquid:
Above gap
Below gap
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DisorderLocalisation length
Can have loc. from ab disorder, if gap is smaller than ca 6ehjhjhjee5 eV, almost 1 K.
Below the gap, we only need to consider disordered aa backscattering.
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01.pptx
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Part II: SYNTHETIC HELICAL LIQUID IN A QUANTUM WIRE
Mariana Malard, Henrik Johannesson and GJ
PRB 89, 201403(R) (2014).
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)(k
k
+_
F
+_
0qkF 0qkF 0qkF 0qkF
)(2 0qkQ F
insulating phase
branch 2
Band picture
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)(k
k
+_
)(2 0qkQ F
F
+_
0qkF 0qkF 0qkF 0qkF
branch 2
conducting helical
electron liquid S
p
S
p
Band picture
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HL in a quantum wire using electrical fields only and standard nanoscale semiconductor technology!
quantum wire
Semiconductor
heterostructure
top nanogat
es
Our proposal
modulation of the spin-orbit
Rashba interaction+ spin-orbit
Dresselhaus + e-e interactions
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The MODEL
hopping and chemical potential
uniform Rashba and Dresselhaus
interactions
+
modulated Rashba
interaction
+
e-e interacti
on
+
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Effective theory
sine-Gordon potential
relevant for
K < ½
helical Dirac hamiltonian
branch-mixing
potential
gap
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For future success in building of German-
Georgian Science Bridge
Thank you for your attention!