quantum Hall effects Other - KITP Online Talks ...

KITP, Dec 2008

Other quantum Hall effects

• Part 1: Quantum Valley Hall

– No interactions

– Valley = “emergent” spin

– “Edge” states

Ivar Martin, Ya. M. Blanter, A. F. Morpurgo, PRL 100, 036804 (2008)

• Part 2: Spontaneous Hall effect

– Interactions – yes (weak coupling)

– Broken Continuous and Ising symmetries

Ivar Martin and C. D. Batista, PRL 101, 156402 (2008)

K K’

KITP, Dec 2008

Single layer graphene

H =

0 vF( p

xip

y) 0 0

vF( p

x+ ip

y) 0 0 0

0 0 0 vF( p

xip

y)

0 0 vF( p

x+ ip

y) 0

= px x z + py y

Valley K

Valley K’

Linear (Dirac) dispersion

K

K’

KITP, Dec 2008

Graphene with “mass” EA – EB=2m 0 (G. Semenoff 1984)

H =

m vF( p

xip

y) 0 0

vF( p

x+ ip

y) m 0 0

0 0 m vF( p

xip

y)

0 0 vF( p

xip

y) m

= px x z + py y +m z g

Valley K

Valley K’

How about quantum Hall?

There is quantum Valley Hall effect! BUT! 1. is suspicious 2. Need sublattice imbalance, EA-EB=2m

g

px py

KITP, Dec 2008

Quantum spin hall effect (Kane + Mele, 2005)

• Another possibility (T-invariant) for the mass term:

Still, dispersion

But instead of valley quantum Hall obtain Quantum Spin Hall

T-inv – spin-orbit

KITP, Dec 2008

Quantum spin Hall effect: (Molenkamp, SC Zhang, et al, 2007)

Quantized conductance!

I: trivial insulator

II, III, IV: quantum spin hall insulators I & II – (20 x 13) um2

III – (1 x 1) um2 & IV (1 x 0.5) um2

M. Konig et al Science 318, 766 (2007)

KITP, Dec 2008

What about valley Hall effect?

Possible to observe in the conventional Bilayer graphene!

1. Can generate mass term 2. Can create a smooth “edge”

KITP, Dec 2008

K

K’

Bilayer graphene

v

F= 3ta / 2

t

Bilayer, one of the valleys (e.g. K):

H =

0 vF( p

xip

y) 0 0

vF( p

x+ ip

y) 0 t 0

0 t 0 vF( p

xip

y)

0 0 vF( p

xip

y) 0

Layer 1

Layer 2

vF

2

t

0 ( px

ipy)2

( px+ ip

y)2 0

| E |< t

E ± p2

Interlayer coupling Quadratic dispersion!

KITP, Dec 2008

Biased graphene bilayers

Field-induced gap in the spectrum McCann ‘06

H =v

F

2

t

( px

ipy)2

( px+ ip

y)2

g( px, p

y)i

BLG

KITP, Dec 2008

Topology of insulating BLG

H =v

F

2

t

( px

ipy)2

( px+ ip

y)2

g( px, p

y)i

g = ( py

2p

x

2 ,2 pxp

y, )

Quantum Valley Hall effect?

xy=

e2

4 hdp

xdp

yg [

px

gp

y

g] =e2

hsgn( ) z

Mass !

Biased bilayer:

Single layer with mass:

g

g

Quantized Valley Conductance, similar to charge and spin Hall effects

px py

px py

p

g

KITP, Dec 2008

How about the quantum Valley Hall effect? Is it observable?

Edge states

QH edge states, Halperin ‘82

OK: any disorder (V, Vso, S)

OK: V and Vso

X: S - magnetic

OK: any, Unless it scatters

X: K K’

.

.

K

K’

QHE (charge) QSHE (spin) QVHE (valley)

..

..

.

.

KITP, Dec 2008

Observation of Valley Hall “Internal” edges

K

K’

..

..

V(x) > 0 V

x

V(x) > 0 V

x

0 0

V(x) < 0

X X X X

Usual edge states are destroyed by disorder. Not the internal ones!

KITP, Dec 2008

Topological confinement

Equations of motion for V(x)

(x)u + (x+ p

y)2

v = u

(x)v + (x

py)2

u = v

Martin, Blanter, Morpurgo ‘07

KITP, Dec 2008

Valleytronics

Proposals to use the valley degree of freedom for classical or quantum manipulaton (intervalley relaxation can be slow)

Use zig-zag ribbon edge state or Aharanov-Bohm (Rycerz, Tworzydlo, Beenakker ’07, Recher et al ’07)

Both require ideal interfaces: Hard with the existing technology

+V

top gate

-V

top gate

K K’

Alternative: use INTERNAL valley-polarized states:

I

K’ K

Valley filter:

+V -V

KITP, Dec 2008

Valley Valve

Direction of propagation of K-valley states

+V -V

-V +V

Iin

K I I

0

-V +V

-V +V

Vtr I

in

K

Iout

= Iin

0 0

KITP, Dec 2008

Open questions

• Interesting phenomena in more complicated structures? Aharonov-Bohm, non-ocal transport,

+ Superconductivity, + Magnetism?

• Interesting physics upon inclusion of interactions? Luttinger liquid?

• Valley Hall vs. Spin Hall: Spin <~> Valley

KITP, Dec 2008

Spontaneous CHARGE quantum Hall effect

S1

S2

S3

Hopping in a texture Hopping in an orbital field

H = t ci† cj

<ij>

JH Sii

ici† ci

ci† itinerant electrons

Si local moments (classical)

S1i[S2 S3]

= / 2

KITP, Dec 2008

Weak-coupling instability

• Chiral spin ordering:

• Ordering with Q1 & Q2 & Q3 fully gaps Fermi surface • for classical spins “tetrahedral” order is the best

At n = 1.5 el/site

KITP, Dec 2008

Answers and Questions

Band structure Observable: Hall conductivity (mean field Hubbard)

Questions: - role of fluctuations? - topological defects (Z2 vortices, domain walls)? - topological doping, superconductivity? - connection to quantum limit (S >>1 S~1)?

L = [ r + 2 (q Q )2 ] | S |2 u | S |4 +v[Sa i(Sb Sc )]

2