SYNTHETIC HELICAL LIQUID IN A QUANTUM WIREGeorge I. Japaridze

Ilia state University and Andronikashvili Institute of

Physics

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

What is a helical liquid (HL)?

Unconventional State of matter with spin-momentum locking

SYNTHETIC HELICAL LIQUID IN A QUANTUM WIRE

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)

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).

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)

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!

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

v

Lorenz transformation

v=0H

•c

E

E

• 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).

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).

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)

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)

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

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

Band structuresSpins look anti-

parallell for opposite velocities.

Helical?

Corrupted by the magnetic field HS!

Let’s calculate the spin overlaps!

Band structuresSpin Overlaps

Limits:

Spin Overlaps

Band structures

Detail for aa overlap. 0.3 meV (ca 1 T) yields overlap squared of about 0.01.

Overlaps:

Limits:

Disorder: Anderson Localization in 1D !!

(absent in the Helical Liquid State!!)

DisorderRG

Scaling equations!

For full Luttinger liquid with spin: For spinless or helical Luttinger liquid:

Above gap

Below gap

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.

01.pptx

Part II: SYNTHETIC HELICAL LIQUID IN A QUANTUM WIRE

Mariana Malard, Henrik Johannesson and GJ

PRB 89, 201403(R)  (2014).

)(k

k

+_

F

+_

0qkF 0qkF 0qkF 0qkF

)(2 0qkQ F

insulating phase

branch 2

Band picture

)(k

k

+_

)(2 0qkQ F

F

+_

0qkF 0qkF 0qkF 0qkF

branch 2

conducting helical

electron liquid S

p

S

p

Band picture

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

The MODEL

hopping and chemical potential

uniform Rashba and Dresselhaus

interactions

+

modulated Rashba

interaction

+

e-e interacti

on

+

Effective theory

sine-Gordon potential

relevant for

K < ½

helical Dirac hamiltonian

branch-mixing

potential

gap

For future success in building of German-

Georgian Science Bridge

Thank you for your attention!