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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
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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
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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
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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’
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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
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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
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'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”
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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
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'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
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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
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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
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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
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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
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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
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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)
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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
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The merging transition seen in the microwave experiment
. 3' 2 3critt t t -
Uniaxial strain increase t’
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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 -
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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 -
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« 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.
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Atoms are trapped in an optical lattice potential and form an artifical crystal
Nature 483, 302 (2012)
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Honeycomb Brick wall
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'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
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t’=2 t’=1.414 t=1
t’=2 t’=1.414 t=1
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Bloch oscillations
Nature 483, 302 (2012)
dkF
dt
l l 40K
l l
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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
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2
4
gE
c F
ZP e
p-
Landau-Zener transition
1 ZP-
F tk
gE
c
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ETH experiment
Measured transfered fraction of atoms: directions of motion
Single Dirac cone Double Dirac cone
Merging line
gapped phase merging gapless phase
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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
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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
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Anisotropic square lattice t-t'-t'': Dirac points, merging, gapped phase,
''t
't
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Mapping tight-binding model on universal hamiltonian
xc
*
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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
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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
-
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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
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Transfer probability for a cloud of size kFy
Single Dirac cone: Fermi sea tunneling
Maximum slightly inside the Dirac phase
* 0
* 0
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Double Dirac cone: Fermi sea tunneling
E
xq
yq
xq
yq
xq
Non-monotonous function of PZ. Maximum for Pz=1/2
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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
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Single Dirac cone Double Dirac cone
Experiment
Theory
Summary
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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