An Introduction to Disordered Elastic Systems. T. Giamarchi

Transcription

1 An Introduction to Disordered Elastic Systems T. Giamarchi

2 Many Physical Systems Interfaces Classical Crystals Quantum crystals

3 Interfaces Magnetic domain walls Ferroelectrics Contact line in wetting Epitaxial growth

4 Magnetic domain wall S. Lemerle et al. PRL (98)

5 Ferroelectrics T. Tybell et al. PRL (02); P. Paruch et al., Annal. Physik (2004) Pb Pb 500 nm v / A Ec=E ¹

6 Classical Crystals Vortex Lattice Magnetic bubbles Charges spheres Charge density waves

7 Magneto-optics NbSe 2 Vortex lattice in type II superconductors Y. Baselevitch T. Johansen Oslo Bitter decoration NbSe 2 Scanning SQUID Nb C. Veauvy, D. Mailly, & K Hasselbach CRTBT Grenoble M. Marchevsky, J. Aarts, P.H. Kes (Kamerlingh Onnes Laboratorium, Leiden University)

8 Classical crystals Charged spheres: M. Saint Jean, GPS (Jussieu), 2000 Magnetic Bubbles: R. Seshadri et al.

9 Quantum Crystals Spin density waves Luttinger liquids (1D interacting electrons) Two dimensional Wigner crystal

10 Quantum systems V I Strong repulsion : Wigner crystal Quantum fluctuations instead (in addition to) thermal fluctuations

11 Wigner Crystal E.Y. Andrei, et al PRL (1988) R.L. Willett, et al. PRB 38 R7881 (1989)

12 C.-C. Li, et al. PRB (2000)

13 Basic Features : (Thermal, quantum) fluctuations `Elasticity Disorder

14 New type of physics Very controlled (e.g. magnetic field) Can pull on on the system Plunged in an external disorder

15 Questions Competition ``Order / ``Disorder Melting Glassyphases Statics Dynamics

16 Statics T. Klein et al. Nature 413, 404 (2001) P. Kim PRB 60 R12589 (99)

17 Dynamics Competition between disorder and elasticity: glassy properties Dynamics? v T 0 Large v: Nature of moving phase? Creep: v=???? Fc T=0 F Depinning: v ( ) β F - F c

18 Elastic description How to model

19 Elastic description of crystals 1 2 R 0 i : crystal u i : displacements n=2 d=3 vortices Elastic hamiltonian H = c ( q) u ( q) u ( q) dq αβ αβ α β

20 Limitations Interfaces (overhangs, bubbles) Periodic dislocations, etc. J. P Jamet, V. Repain M. Marchevsky, J. Aarts, P.H. Kes

21 What to measure (statics) r u B( r) = [ u( r) u(0)] 2 Positional order

22 S( q) = ρ( q) ρ( q) Structure Factor Fourier transform of: C( x) = e ik 0 Decorations u( r) ik0u(0) e K0 Neutrons q

23 Should we care about disorder?

24 S. Lemerle et al. PRL (98) ς u L

25 Loss of translational order (Larkin) u( R ) a a H el = c u r 2 d d ( ( )) 2 r H dis = V ( r) ρ( r) d d r cr d a 2 a 2 d VR a / 2 ρ 0 R a c a V 2 2 a ρ d 2 0 1/(4 d ) No crystal below four spatial dimensions

26 Very difficult stat-mech problem Optimization : many solutions Glass Ε

27 Disorder (point like defects) ) ( ) ( ) ( ) ( 0 i i R i u x x dx x x V H = = δ ρ ρ + = K x u x ik e x u x )) ( ( ) ( ) ( ρ ρ ρ ρ

28 Larkin Model H el = c u r 2 d d ( ( )) 2 r H dis = f ( r) u( r) d d r Exactly solvable B( r) = B th + c 2 r 4 d Exponential loss of translational order C( r) e r 4 d Not valid at large distance

29 ρ ik( x ( )) 0 e u x V( x) f ( x) u( x) K Not valid when : K MAX u 1 u( R c ) ξ New length Rc Larkin model has no metastable states and pinning Rc is related to pinning c R 2 c F c ξ

30 General Model + = K x u x ik d c x V e x d u H ) ( ) ( )) ( ( ρ Classical systems x d x u x u K x d u c H d b a K b a a d a =, 0 2 ))) ( ) ( ( cos( ) ( ρ Quantum problem (disorder is time independent) ' '))), ( ), ( ( cos( ) (, τ τ τ τ ρ d xd d x u x u K x d u c S d b a K b a a d a = +

31 How to solve? Two main methods : Variational approach Renormalization (functional RG)

32 Interfaces: only one length Larkin length u( R ) c ξ R < R c ; u(r) = R (4-d)/2 R > R c ; u(r) =?????

33 Interfaces H el = c u r 2 d d ( ( )) 2 r H dis = V ( r, u( r)) d V ( z, x) V ( z', x') = Dδ ( x x') δ ( z z') d r u u 2 d 2 cu L 1/ 2 d / 2 m / 2 RB RF L L 4 d 4+ m 4 d 4 m D L u Flory argument (mean field)

34 u RB ς L ζ : roughness exponent d = 1 ; ζ = 2/3 (random bond)

35 Crystals: Two crucial lengthscales Positional order Larkin length u( R ) a a u( R ) c ξ C(r) S(q) d R a Ra r 1/R a 1/R a q

36 Crystals Identical to interfaces? u» L ζ Above R c C r / e L ³ Exponential loss of positional order??

37 Naive vision of a D.E. crystal Loss of translational order beyond Ra (Wrong) argument: disorder induces dislocations at Ra Ra Crystal broken in crystallites of size Ra

38 Periodic systems: new universality class u ~ ς L u ~ Log( L 1/ ) 2

39 B(r) Larkin r d Random manifold r 2ν 4 Asymptotic A d log(r) Rc Ra r S(q) K0 q Bragg glass T.G. + P. Le Doussal Phys. Rev. B (1995)

40 T. Klein et al. Nature (2001)

41 Dynamics

42 TAFF vs Creep v T 0 Fc T=0 F x TAFF : typical barrier Linear response v e β F

43 Creep Glassy system Slow dynamics determined by statics qty H el = c u r 2 d d ( ( )) 2 r = d H el Fu( r) d r cr d 2+2ς FR d + ζ L opt F ς 2 v e βu ( L opt ) U ( L opt ) F d + 2ς 2 ς 2 (Ioffe + Vinokur; Nattermann)

44 D.T. Fuchs et al. PRL (98) S. Lemerle et al. PRL (98)

45 FRG: other scales than naive version of creep P. Chauve + T.G. + P. Le Doussal Phys. Rev. B (2000). Molecular dynamics: (A. Kolton, A. Rosso, TG, cond-mat/ )

46 Wigner crystal (2DEG) R. Chitra, TG, P. Le Doussal PRL (1998); PRB (2001)

47 Questions

48 Ferroelectics P. Paruch et al., Annal. Physik (2004) v/ A Ec=E ¹

49 Roughness of the irradiated layer : (V. Repain et al. (Orsay)) 210 µm Pt/Co(0,5 nm)/pt/sio 2

50 References on DES Classical systems : G. Blatter et al. Rev. Mod. Phys (1994). T.G. + P. Le Doussal, In ``Spin Glasses and Random Fields'', ed. A.P. Young, World Scientific 1998, cond-mat/ T. Nattermann and S. Scheidl Adv. Phys (2000) TG + S. Bhattacharya, In ``High Magnetic Fields, ed. C. Berthier et al., Springer 2002, cond-mat/ Quantum systems : T.G. + E. Orignac In ``Theoretical Methods for Strongly Correlated Electrons", D. Senechal et al. ed, Springer (2004), cond-mat/ T.G. In ``Quantum phenomena in mesoscopic systems" (Varenna school CLI) IOS Press (2003), cond-mat/ And references therein..