Manipulation of Dirac cones in artificial graphenes
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1 Manipulation of Dirac cones in artificial graphenes Gilles Montambaux Laboratoire de Physique des Solides, Orsay CNRS, Université Paris-Sud, France - Berry phase Berry phase K K -p Graphene electronic spectrum p 1
2 Manipulation of Dirac cones in artificial graphenes «Life and death of Dirac points» p -p p -p 0 «Artificial» graphenes J.-N. Fuchs, M. Goerbig, F. Piéchon P. Dietl, P. Delplace, R. De Gail (PhDs) Lih-King Lim (post-doc) Quy-Nhon avenues
3 Outline Motion and merging of Dirac points Modified graphene as a toy model A universal Hamiltonian, spectrum at the merging Physical realizations : p -p 1) Microwaves in a honeycomb lattice of dielectric discs Mortessagne s group, Nice (01) ) Graphene-like lattice of cold atoms in an optical lattice Esslinger s group, ETH (01) Landau-Zener tunneling as a probe of Dirac points Conclusion and other artificial graphenes
4 Graphene H f A * B 0 f( k) ( k) 0 a a 1 ik a f ( k ) -t 1 e e ik a 1 ( k ) f ( k ) K K K K
5 Graphene H f A * 0 f( k) ( k) 0 ik a B f ( k ) -t 1 e e ik a 1 a a 1 ( k ) f ( k ) DOS K K
6 Tight-binding problem on honeycomb lattice Y. Hasegawa et al., 006 t t E t ' t ' te ika te ika t ' t 3 t ' 1.5t K a -K a 1 p t' t
7 Motion and merging of Dirac points t' t t ' 1.5t hybrid semi-dirac t ' t t'.3t 4 4t qd arctan -1 3 t ' y c t -t 3 ' / 4 cx 3 t ' c 0 for t ' t y m 3t *
8 Hybrid D electron gas : a new dispersion relation t ' t Schrödinger Semi-Dirac Dirac q y q x q 4 y 4m cq x n 1/ eb P. Dietl, F. Piéchon, G.M., PRL 100, (008) G. M., F. Piéchon, J.N. Fuchs, M.O. Goerbig, Eur. Phys. J. B 7, 509 (009), Phys. Rev. B 80, (009) /3
9 Berry phase Two component wavefunction u k 1 1 e i ( k) e ik. r B 1 i u u. dk. dk p k k k k k C C k t ' t t ' 1.5t t ' t p -p p -p 0 Topological transition
10 General description of the motion of Dirac points (with time reversal symmetry) H 0 f( k) ( k) 0 - * f When tmn changes, D and -D f ( k ) tmne - move mn, ik. Rmn Where is the merging point? D-D D0 G 4 possible positions in k space (0,0) (0,1) (1,0) (1,1) X Y M Expansion near D0 f ( D q) -icq 0 x q y m
11 At the merging transition : Hq ( ) icq x 0 q m y -icq x 0 q y m cx 1 * m tmn Rmn - mn, t R (-1 mn, mn mn ( 1) mn 1mn ) Near the transition : q y Hq ( ) * 0 icq x q y m * -icq x 0 q y m * tmn( -1) mn * mn, ( t' - t) * 0 * 0
12 «universal Hamiltonian» Hq ( ) * 0 icq x q y m * -icq x 0 q y m The parameter * drives the topological transition * 0 * 0 * 0 p -p qy * -m * 0 B /3 B This Hamiltonian describes the topological transition, the coupling between valleys and the merging of the Dirac points B
13 First summary: Manipulation of Dirac points and merging 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: 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. Honeycomb Brick wall t i G. M., F. Piéchon, J.N. Fuchs, M.O. Goerbig, Phys. Rev. B 80, (009)
14 Physical realizations of the merging transition * Strained graphene t' t strain ~ 3% merging is unreachable Pereira, Castro Neto, Peres, PRB 009 See also Goerbig, Fuchs, Piéchon, G.M., PRB 008 * Microwaves * Ultracold atoms in optical lattices
15 Merging of Dirac points in a D crystal G. M., F. Piéchon, J.N. Fuchs, M.O. Goerbig, Phys. Rev. B 80, (009) 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 Topological transition of Dirac points in a microwave experiment M. Bellec et al. (01) Creating moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice, L.Tarruell et al. Nature (01)
16 Graphene physics with microwaves! M. Bellec, U. Kuhl, F. Mortessagne (NICE) Honeycomb lattice of dielectric resonators Evanescent propagation between the dots -> Tight-binding description Measure of the reflexion coefficient LDOS ~ 50cm ~ 50 «atoms» High flexibility 1) Look for the merging transition ) Probe the edge states ( nd and 3 rd nearest neighbor couplings not negligeable) t / t t / t
17 The merging transition seen in the microwave experiment t ' t -3t crit. 3 Uniaxial strain increase t
18 The merging transition seen in the microwave experiment t' t Armchair edges t' 1.8 t t ' t -3t crit. 3 t' 3.5 t
19 The merging transition seen in the microwave experiment t' t Armchair edges n t /t exp. t' 1.8 t t ' t -3t crit. 3 th. t' 3.5 t Gapped phase + new edge states
20 Probing the edge states «armchair» have no edge states. But by under strain, new edge states are predicted along certain armchair edges. P. Delplace, G.M. Increasing uniaxial strain
21 Nature 483, 30 (01) Atoms are trapped in an optical lattice potential and form an artifical crystal
22 Honeycomb Brick wall
23 Honeycomb Brick wall i i E te 1 te t ' k.a k.a i( k k ) a i( k -k ) a x y x y E te te t ' t ' t
24 t=1 t =1.414 t = t=1 t =1.414 t =
25 Nature 483, 30 (01) 40 K Bloch oscillations dk dt F
26 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
27 Landau-Zener transition c PZ e E g p - 4 cf E g 1- PZ k Ft
28 Measured transfered fraction of atoms: directions of motion ETH experiment Single Dirac cone Double Dirac cone Merging line gapped phase merging gapless phase
29 Explain the experimental data using Universal Hamiltonian Bloch-Zener oscillations across a merging transition of Dirac points Lih King Lim, Jean-Noel Fuchs, G. M., PRL 108, (01)
30 Explain the experimental data using Universal Hamiltonian 1 ) Relate the parameters of the optical lattice to the parameters of the Universal Hamiltonian 1) Ab-initio band structure from optical lattice potential: V X, V Xb, V Y (laser intensities). ) Tight-binding model on an anisotropic square lattice: t,t',t'' (hopping amplitudes) 3) Universal hamiltonian describing the merging transition: Δ *, c x, m*
31 Anisotropic square lattice t-t'-t'': Dirac points, merging, gapped phase, t ' t ''
32 Mapping tight-binding model on universal hamiltonian * c x
33 ) Compute the inter-band tunneling probability within the Universal Hamiltoninan Single Zener tunneling P Z q y q x Double Zener tunneling P P (1 -P ) t Z Z q y q x
34 Single Dirac cone: single atom tunneling Transfer probability as a function of q y and Δ* qy * -m * E q y q x q y x PZ y * q ( * ) - m cf x e p q x
35 Single Dirac cone: Fermi sea tunneling Transfer probability as a function of q y and Δ* qy * -m * E q y q x Transfer probability for a cloud of size k Fy q y x PZ y * q ( * ) - m cf x e p q y q x Maximum slightly inside the Dirac phase
36 Single Dirac cone: Fermi sea tunneling Transfer probability for a cloud of size k Fy * 0 * 0 Maximum slightly inside the Dirac phase
37 Double Dirac cone: Fermi sea tunneling q x E q y Non-monotonous function of P Z. Maximum for P z =1/ q x q y q x
38 Double Dirac cone: Fermi sea tunneling q x Non-monotonous function of P Z. Maximum for P z =1/ * * 0 Gapped phase y P Z 0 * 0 Dirac phase c x y P Z y P Z 1/ 1
39 Summary Single Dirac cone Double Dirac cone Experiment Theory
40 Conclusions and perspectives Universal description of motion and merging of Dirac points in D crystals -p p 0 (-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 Other universality class : (p,p) merging p p p
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