K K’

Berry phase

p

- Berry phase

-p Graphene electronic spectrum

1

Gilles Montambaux Laboratoire de Physique des Solides, Orsay CNRS, Université Paris-Sud, France

Manipulation of Dirac cones in artificial graphenes

Quy-Nhon avenues

J.-N. Fuchs, M. Goerbig, F. Piéchon P. Dietl, P. Delplace, R. De Gail (PhDs) Lih-King Lim (post-doc)

p -p

« Life and death of Dirac points »

Manipulation of Dirac cones in artificial graphenes

« Artificial » graphenes

p -p 0

Outline

Motion and merging of Dirac points Modified graphene as a toy model A universal Hamiltonian, spectrum at the merging Physical realizations : 1) Microwaves in a honeycomb lattice of dielectric discs Mortessagne’s group, Nice (2012) 2) Graphene-like lattice of cold atoms in an optical lattice Esslinger’s group, ETH (2012) Landau-Zener tunneling as a probe of Dirac points Conclusion and other artificial graphenes

p -p

Graphene

*

0 ( )

( ) 0

f kH

f k

1 2( ) 1ik a ik a

f k t e e

-

( ) ( )k f k

1a2

a

K K’

A B

K K’

Graphene

1a2

a

K K’

*

0 ( )

( ) 0

f kH

f k

1 2( ) 1ik a ik a

f k t e e

-

( ) ( )k f k

A B

DOS

Tight-binding problem on honeycomb lattice

1 2'ik a ik a

te eE tt

Y. Hasegawa et al., 2006

't

tt

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

't t

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

' 1.5t t

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3

' 2t t

1 2

2

3K a K a

p -

0

't t ' 1.5t t

' 2t t ' 2.3t t

Motion and merging of Dirac points

2

2

4 42 arctan 1

3 'D

tq

t -

3'

2xc t

2 23 ' / 4yc t t -

* 2

3m

t0 for ' 2yc t t

“hybrid” “semi-Dirac”

Hybrid 2D electron gas : a new dispersion relation

xqyq

4

2 2

24

y

x

qc q

m

2/3

1/ 2n eB

P. Dietl, F. Piéchon, G.M., PRL 100, 236405 (2008)

G. M., F. Piéchon, J.N. Fuchs, M.O. Goerbig, Eur. Phys. J. B 72, 509 (2009),

Phys. Rev. B 80, 153412 (2009)

Schrödinger Dirac Semi-Dirac

' 2t t

't t 1.5't t 2't t

Berry phase

p

-p

p

-p

1 | . .

2k k k k k

C

B

C

i u u dk dk p

k

.

( )

11

2

ik r

i ke

eu

k

Topological transition

Two component wavefunction

0

General description of the motion of Dirac points (with time reversal symmetry)

When changes,

*

0 ( )

( ) 0

f kH

f k

-

.

,

( ) mnik R

mn

m n

f k t e-

move

Where is the merging point?

D D -0

2

GD

mnt D D-and

4 possible positions in space k

(0,0) (0,1) (1,0) (1,1)

X Y M

0

2

( )2

y

x

qf D q icq

m - Expansion near 0D

2

2

02

( )

02

y

x

y

x

qicq

mH q

qicq

m

-

,

( 1) mn

mnmn

m n

t Rcx

-

12

*,

( 11

) mn

mn mn

m n

tm

R

-

,

* ( 1) mn

mn

m n

t

- *

*

2

2

02

( )

02

y

x

y

x

qicq

mH q

qicq

m

-

At the merging transition :

Near the transition :

*( ' 2 )t t -

* 0 * 0

yq

This Hamiltonian describes the topological transition, the coupling between valleys

and the merging of the Dirac points

The parameter drives the topological transition

* 0 * 0

*

« universal Hamiltonian » *

*

2

2

02

( )

02

y

x

y

x

qicq

mH q

qicq

m

-

*

*2yq m - p

-p 0

* 0

B B2/3B

By varying band parameters, it is possible to manipulate the Dirac points. They can move in k-space and they can even merge. The merging transition is a topological transition: 2 Dirac points evolve into a single hybrid « semi-Dirac » point and eventually a gap opens and the Fermi surface disappears. Universal description of motion and merging of Dirac points.

First summary: Manipulation of Dirac points and merging

Honeycomb

Brick wall

G. M., F. Piéchon, J.N. Fuchs, M.O. Goerbig,

Phys. Rev. B 80, 153412 (2009)

it

Physical realizations of the merging transition

* Strained graphene * Microwaves * Ultracold atoms in optical lattices

' 2t t

strain ~ 23%

merging is unreachable

Pereira, Castro Neto, Peres, PRB 2009 See also Goerbig, Fuchs, Piéchon, G.M., PRB 2008

Referee A The paper is well written and accessible to a broad audience. The results might be applicable to optical lattice systems. I would recommend publishing the paper in PRL as it is. Referee B It is a nice simple toy model... The authors propose that merging of Dirac points might be possible with cold atoms in optical lattices. I think that it is a very long shot, given that ... the systems are yet to be realized experimentally. I do not recommend publication of this paper in PRL. Referee C While the physical system is certainly interesting, its relevance to current experiments is rather tenuous…

Merging of Dirac points in a 2D crystal G. M., F. Piéchon, J.N. Fuchs, M.O. Goerbig, Phys. Rev. B 80, 153412 (2009)

Topological transition of Dirac points in a microwave experiment M. Bellec et al. (2012) Creating moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice, L.Tarruell et al. Nature (2012)

Graphene physics with microwaves !

Honeycomb lattice of dielectric resonators Evanescent propagation between the dots -> Tight-binding description Measure of the reflexion coefficient LDOS

High flexibility 1) Look for the merging transition 2) Probe the edge states

M. Bellec, U. Kuhl, F. Mortessagne (NICE)

~ 250 « atoms »

(2nd and 3rd nearest neighbor couplings not negligeable)

2 3/ 0.091 / 0.071t t t t

~ 50cm

The merging transition seen in the microwave experiment

. 3' 2 3critt t t -

Uniaxial strain increase t’

The merging transition seen in the microwave experiment

Armchair edges

't t

' 1.8 t t

' 3.5 t t

. 3' 2 3critt t t -

Gapped phase + new edge states

The merging transition seen in the microwave experiment

Armchair edges

' 3.5 t t

t’/t n

exp.

th.

't t

' 1.8 t t

. 3' 2 3critt t t -

« armchair » have no edge states. But by under strain, new edge states are predicted along certain armchair edges.

Probing the edge states…

Increasing uniaxial strain

P. Delplace, G.M.

Atoms are trapped in an optical lattice potential and form an artifical crystal

Nature 483, 302 (2012)

Honeycomb Brick wall

'i i

E te te t 1 2k.a k.a ( ) ( )'x y x yi k k a i k k a

E te tte -

Honeycomb Brick wall

't t

t’=2 t’=1.414 t=1

t’=2 t’=1.414 t=1

Bloch oscillations

Nature 483, 302 (2012)

dkF

dt

l l 40K

l l

How to manipulate and merge Dirac points ?

Anisotropy of the optical potential

How to detect and localize Dirac points ?

Bloch oscillations + Landau-Zener Tunneling

Measurement of the proportion of atoms in the upper band

2

4

gE

c F

ZP e

p-

Landau-Zener transition

1 ZP-

F tk

gE

c

ETH experiment

Measured transfered fraction of atoms: directions of motion

Single Dirac cone Double Dirac cone

Merging line

gapped phase merging gapless phase

Lih King Lim, Jean-Noel Fuchs, G. M., PRL 108, 175303 (2012)

Bloch-Zener oscillations across a merging transition of Dirac points

Explain the experimental data using Universal Hamiltonian

1) Ab-initio band structure from optical lattice potential: VX, VXb, VY (laser intensities). 2) Tight-binding model on an anisotropic square lattice: t,t',t'' (hopping amplitudes) 3) Universal hamiltonian describing the merging transition: Δ*, cx, m*

1 ) Relate the parameters of the optical lattice to the parameters of the Universal Hamiltonian

Explain the experimental data using Universal Hamiltonian

Anisotropic square lattice t-t'-t'': Dirac points, merging, gapped phase,

''t

't

Mapping tight-binding model on universal hamiltonian

xc

*

2 (1 )t Z ZP P P -

Single Zener tunneling

Double Zener tunneling

ZP

xq

xq

yq

yq

2 ) Compute the inter-band tunneling probability within the Universal Hamiltoninan

Single Dirac cone: single atom tunneling Transfer probability as a function of qy and Δ*

*

*2yq m -

E

xq

yq

xq

yq

22

**( )2

y

x

q

m

c Fx

ZP ep

-

Single Dirac cone: Fermi sea tunneling Transfer probability as a function of qy and Δ*

22

**( )2

y

x

q

m

c Fx

ZP ep

-

Transfer probability for a cloud of size kFy

*

*2yq m -

Maximum slightly inside the Dirac phase

E

xq

yq

xq

yq

yq

Transfer probability for a cloud of size kFy

Single Dirac cone: Fermi sea tunneling

Maximum slightly inside the Dirac phase

* 0

* 0

Double Dirac cone: Fermi sea tunneling

E

xq

yq

xq

yq

xq

Non-monotonous function of PZ. Maximum for Pz=1/2

Double Dirac cone: Fermi sea tunneling

* 0 * 0

Gapped phase Dirac

phase

0y

ZP

1y

ZP

1/ 2y

ZPxc

*

xq

Non-monotonous function of PZ. Maximum for Pz=1/2

Single Dirac cone Double Dirac cone

Experiment

Theory

Summary

Conclusions and perspectives Universal description of motion and merging of Dirac points in 2D crystals

(-p,p) merging : hybrid semi-Dirac spectrum Cold atoms : Landau-Zener probe of the Dirac points Interference effects Condensed matter : New thermodynamic and transport properties Interaction effects : from Dirac to Schrödinger

-p p 0

p p 2p

Other universality class : (p,p) merging