SPONTANEOUS QUANTUM HALL EFFECT AND FRACTIONALIZATION IN CHIRAL MAGNETS

Transcription

1 SPONTANEOUS QUANTUM HALL EFFECT AND FRACTIONALIZATION IN CHIRAL MAGNETS Ivar Martin Los Alamos National Laboratory S 2 s S 1 S 3 f Collaborators: C. D. Batista Y. Kato D. Solenov D. Mozyrsky R. Muniz A. Rahmani A. Morpurgo Y. Blanter M. Buttiker J. Li Spontaneous Hall effect

2 QUANTUM COHERENT EFFECTS IN COMPLEX MAGNETS (and exotic metals) Ivar Martin Los Alamos National Laboratory S 2 s S 1 S 3 f Collaborators: C. D. Batista Y. Kato D. Solenov D. Mozyrsky R. Muniz A. Rahmani A. Morpurgo Y. Blanter M. Buttiker J. Li Spontaneous Hall effect

3 ~ 200 BC, China Magnetism

4 Novel applications of ferromagnetism 1980 s 2007 Grunberg Fert Magnetic Tunnel Junctions Parkin Giant Magnetoreistance Racetrack MRAM

5 New frontier: Quantum coherent effects in complex magnets

6 Complexity of order: Chiral Magnetism collinear coplanar non-coplanar (e.g. spiral) (chiral) skyrmion

7 Frustrated magnetism Skyrmions in MnSi (Muhlbauer, Science 2009) Pr 2 Ir 2 O 7 Monopoles in spin ice (Castelnovo, Nature 2008) Strange supercond. in Frustrated magnet (Takada, Nature 2003) Hall effect w/o B or M (Nakatsuji, Nature 2010)

8 Outline of the lectures Part I: Quantum Hall effect Topological reasons Hall effect without Landau levels Applications to topological insulators and graphene Part II: Noncoplanar magnets as a playground for anomalous (quantum) Hall effect Model systems, energetic reasons Promising material realizations Part III: Fractionalization Semiconductor fractional quantum Hall systems Polyacetylene Spontaneous quantum Hall magnets

9 Part I: Quantum Hall effects Topological reasons Hall effect without Landau levels Applications to topological insulators and graphene

10 Hall Effect B z 2d classical Hall effect Magnetic field Integer quantum Hall effect (K. von Klitzing) Hall conductivity is quantized ( s xy = ne 2 / h ) only when an integer number of cyclotron bands (Landau levels) are occupied B Images from

11 Quantum Hall effect and Topology How does a system with fully filled bands conduct? Formally, use the Kubo formula s xy = i å n p j x m p m p j y n p [E n (p) - E m (p)] -{x «y} 2 n-occup m-empty pîbz BZ = ie 2 å px n p py n p -{x «y} n-occup pîbz dp = ie 2 å ò Ñ (2p ) 2 p A n (p), A n = n p Ñ p n p n-occup BZ Properties of vector potential A: M. Kohmoto, 1985 (1) complex function (2) defined up to A A + iñ p c p (3) if A is non-singular in BZ, then xy = 0 (Stokes thm) (4) if A is singular, then dp p An( p) 2 int Requires T-breaking! BZ Thus, in the insulating state ONLY QUANTIZED xy is allowed!

12 An example: 2x2 Simplest example: (2x2) Hamiltonian in 2D: + empty band - occupied band Insulator at half filling if g is non-zero for any p Torus p (BZ) Sphere g ie n n { x y} 2 xy px p p n occup p BZ 2 e dp 垐 [? xdpy p ] x py 4 h g g g BZ y p p g Degree of mapping wrapping number ( p x, p y ) (g x,g y,g z )

13 Single layer graphene K K K Linear (Dirac) dispersion æ ç ç H = ç ç è ç 0 v F (p x - ip y ) 0 0 v F (p x + ip y ) v F (-p x - ip y ) 0 0 v F (-p x + ip y ) 0 = p x s x t z + p y s y : sublattice ( T C) valley ( T i C) : y ö ø Valley K Valley K E v F p

14 Graphene with mass E A E B =2m 0 (G. Semenoff 1984) Valley K Valley K E p m 2 2 How about quantum Hall? 2 2 e e 1 dp dp 垐 [?] sgn( m) x y 4 h g g g h 2 xy x y p p z There is quantum Valley Hall effect! BUT! 1. ½ is strange 2. Need sublattice imbalance, E A -E B =2m p x g p y

15 Quantum spin hall effect (Kane + Mele, 2005) Another possibility (T-invariant) for the mass term: æ s z m v F (p x - ip y ) 0 0 ö ç ç v F (p x + ip y ) -s z m 0 0 H = ç ç 0 0 -s z m v F (p x - ip y ) è ç 0 0 v F (p x + ip y ) s z m ø T-inv spin-orbit = p x s x t z + p y s y + ms z s z t z Still, dispersion E = ± p 2 + m 2 s xy = å t z e 2 h in every valley But instead of valley quantum Hall obtain Quantum Spin Hall 1 2 sgn(m)t z (t z s z ) = e2 h sgn(m)s z

16 Is quantum Valley Hall effect observable? Edge states QH edge states, Halperin 82 QHE (charge) QSHE (spin) QVHE (valley).. K.. OK: any disorder (V, V so, S).. OK: V and V so X: S - magnetic. K OK: any, Unless it scatters X: K K.

17 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) um 2 III (1 x 1) um 2 & IV (1 x 0.5) um 2 M. Konig et al Science 318, 766 (2007)

18 .... Observation of Valley Hall Internal edges K X X X X K V V(x) > 0 V V(x) > 0 V(x) < x Usual edge states are suppressed by disorder. Not the internal ones! x

19 Topological confinement Equations of motion for V(x) -j(x )u + ( x + p y ) 2 v = eu j(x )v + ( x - p y ) 2 u = ev Martin, Blanter, Morpurgo 07 j(x) = -j(-x) Symmetries: for v (x ) = ±u(-x ) Þ ( u(x ),u(-x )),(v (x ),-v (-x )) p - p (u,v) (v,-u), e -e a j 0 a

20 Step-like potential j(x) = j 0 sign x K - valley Solutions: -j(x)u + ( x + p y ) 2 v = eu j(x)v + ( x - p y ) 2 u = ev p y = - ±e + j / 2 0 ±e + j 0 2 Cross zero energy! Same velocity for the same valley (opposite in the other valley) Possibility to filter valleys

21 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 Alternative: use INTERNAL valley-polarized states: Valley filter: +V -V I +V -V K K K K top gate top gate

22 Free spread, SLG Injection from kink, SLG potential mass potential mass

23 Valley Valve V2 V2 V 1 V 1 V tr I in I in 0 -V +V K 0 ½ I -V +V K ½ I -V +V +V -V I out = I in 0 Direction of propagation of K-valley states

24 Valley valve - CLOSED, SLG Valley valve - OPEN, SLG potential mass potential mass

25 Interferometer, SLG Interferometer, BLG potential mass potential mass

26 Summary Quantum Hall effect Topological reasons Hall effect without Landau levels Applications to topological insulators and graphene

27 Part II: Noncoplanar magnetets as a playground for anomalous (quantum) Hall effect Model systems, energetic reasons Promising material realizations

28 Hall Effect B z 2d classical Hall effect Magnetic field Integer quantum Hall effect (K. von Klitzing) Hall conductivity is quantized ( s xy = ne 2 / h ) only when an integer number of cyclotron bands (Landau levels) are occupied B Images from

29 Spontaneous Hall effect Definition: Hall effect in the absence of magnetic field (H = 0) or uniform magnetization (M = 0) B z + C B z c Magnetic field

30 T f ~0.2 K Pr 2 Ir 2 O 7 : Geometrically Frustrated Kondo Lattice 2 K T CW ~20 K T (K) Spin Freezing 4f of Pr Spin Liquid Spin Ice Correlation Finite AHE in the spin liquid state at B = 0 Residual M Residual r xy From: S. Nakatsuji (Nature 2010) 30

31 Mechanisms of Anomalous Hall Effect Mechanisms: spin-orbit + magnetization Geometrical Hall effect Berry s phase in chiral magnet control electron phase control electron motion Naoto Nagaosa, et al, Anomalous Hall Effect, Rev. Mod. Phys. 82, 1539 (2010)

32 The key concept: Controlling electron phase <=> control electron transport W S 1 S 2 S 3 Aharonov-Bohm f = W / 2 Berry Phase Spin texture can induce effective magnetic field of 10 5 Tesla Frustration is the key for generating non-coplanar texture 32

33 Berry phase

34 Free-electron mediated magnetism + + e e (Low electron density) (high electron density) Ferromagnet Anti-Ferromagnet

35 Weak-Coupling (nesting) AF on square & triangular lattice H = -tåc ia c ja - J H ås i ic ia s ab c ib or H = -tå c ia c ja +Uån i n i <ij> i <ij> i Square lattice, n = 1 Q = (p,p) Triangular lattice, n = vectors Q a,b,c k y Q Q a Q b Qc k x IM, C. D. Batista, PRL 101, (2008)

36 Chiral order Quantum Hall Uniform chiral ordering S 2 s S 1 S 3 j j j j j j j j j j j j j j j = W / 2 = p / 2 S i = (S a cosq a r i,s b cosq b r i,s c cosq c r i ) Hofstadter problem => Q. Hall! Ivar Martin and C. D. Batista, PRL 101, (2008)

37 3Q AF state with uniform scalar spin chirality! Uniform spin chirality -> uniform Berry curvatre -> Hall effect IF insulator (filling ½*int) -> QUANTUM Hall effect! n = 2: s xy = 0 n = 1.5: s xy = e2 h n = 1: s xy = 0 n = 0.5: s xy = e2 h No magnetic field, no uniform magnetization cf. Pr 2 Ir 2 O 7 (Nakatsuji) IM and C. D. Batista, PRL 101, (2008)

38 Non-coplanar structures away from ¾ filling (n = 1.5) T = 0 phase diagram H = -tåc ia c ja - J H ås i ic ia s ab c ib <ij> i Very rich variety of phases! Akagi and Motome, 2010

39 P(!) c Finite temperature transition: Monte-Carlo simulations ¼ filling! L=8 L=10 L=12 Spin configuration E- distribution likely 1 st order T/t= T/t= T/t= T/t= a b c d T/t Specific heat and chirality Y. Kato, IM, C.D. Batista, PRL Energy per site

40 Chiral magnetism (J. Villain, J. Phys. 1977) Ising argument: Chirality remains broken up to Finite temperature T ~ J (energy per unit length of chiral domain wall) Topological defects: Extended: domain walls Point-like: vortices Cf. p 1 (SO(3)) = Z 2 SO(3) ~ RP Nematic LQ: OP RP 2

41 Materials with magnetism and itinerant electrons Mott ins band ins Three-Dimensional Spin Structure on a Two-Dimensional Lattice: MnCu(111), Ph. Kurz, G. Bihlmayer, K. Hirai *, and S. Blügel, Phys. Rev. Lett. 86, (2001) Cobaltates -Triangular lattice -Right filling (n = 1.5) -Correct magnetic ordering vector from neutrons -Hall effect anomaly - quantum spins (1/2)

42 Summary & high field phase diagram (Na 0.5 CoO 2 ) CO region (< 52 K) show a phase transition from high to low resistivity at high magnetic fields. Region of lo w ρ xy region is observed in t he CO phase. Resistivity is larger than the h/e 2 value by 2.5 to 20 times depending on the sample preparation (or thickness). Charge order Planned experiments 1) Temperature dependent measurement of cleaved samples: recent exps failed. 2) Fabrication of smaller samples using FIB: NG contact 3) Efforts to detect non-coplanar spin structure with neutron diffraction (SANS?). J. W. Kim, R. D. McDonald, V. Zapf, I. Martin (work in progress)

43 Na x CoO 2 Cleaved samples with patterned contacts From U. of Geneve From P. Moll (ETH, Zurich) Future plans 1) High field exps with patterned samples. 2) c-axis transport measurement with bulk? 3) I-V measurement

44 Anomalous Hall in UCu 5 H KLM t c i c j c j c i J K c i c i S i, i, j i H CS J 1 S i S j J 2 S i S j K S i S j H S i, i, j i, j i, j 2 i B. G. Ueland 1*, C. F. Miclea 1,2, Yasuyuki Kato 1, O. Ayala Valenzuela 1, R. D. McDonald 1, R. Okazaki 3, P. H. Tobash 1, M. A. Torrez 1, F. Ronning 1, R. Movshovich 1, Z. Fisk 1,4, E. D. Bauer 1, Ivar Martin 1, and J. D. Thompson 1 -- Nature Comm (2012)

45 UCu 5 Theory & Expt B. G. Ueland 1*, C. F. Miclea 1,2, Yasuyuki Kato 1, O. Ayala Valenzuela 1, R. D. McDonald 1, R. Okazaki 3, P. H. Tobash 1, M. A. Torrez 1, F. Ronning 1, R. Movshovich 1, Z. Fisk 1,4, E. D. Bauer 1, Ivar Martin 1, and J. D. Thompson 1 -- Nature Comm (2012)

46 Continuum: + e (Low electron density) Or is it??? Ferromagnet

47 Low-density 2DEG + magnetic dopants H = åx ò dr c x (r){-ñ 2 / 2m d xx ' - JS(r) s xx ' }c x ' (r) Can easily integrate out electron to 2 nd order in J (RKKY) F(S) = -J 2 ò dq ( 2p ) 3 c(q)s(q) S(-q) However, c(q) c(q) 3D 2q F standard ferromagnetism q 2D 2q q F large degeneracy! higher orders are crucial!

48 Magnetic state in dilute 2D magnets Sr ( ) (sin Ky cos Qx,sin Kysin Qx,cos Ky) Unconstraint numerical optimization K ~ Q ~ q 1 J / S xs ys KQsin Ky Solenov, Mozyrsky, IM, PRL (2012) Experiment: mono-atomic Fe film on Ir(111, fcc) evidence of incommensurate square magnetic lattice F Roland Wiesendanger, STM group Institute of Applied Physics, University of Hamburg ( 2010)

49 Summary Hall Effect due to Berry phase 2D triangular Kondo lattice: Spontaneous quantum Hall effect system Materials: NaCoO2, UCu5 2D continuum magnetism: skyrmion crystal states Anomalous quantum Hall effect in spin-ice systems (GW Chern, A. Rahmani, I. Martin, CD Batista condmat 2013) Open questions: quantum fluctuations, relation to superconductivity, fractional spontaneous Hall, magnetic and density quasi-crystals..

50 Part III: Fractionalization Semiconductor fractional quantum Hall systems Polyacetylene Spontaneous quantum Hall magnets

51 Fractional Quantum Hall effect s xy = e2 mh B What can be so special about partially occupied lowest Landau Level?? Correlations make it special: incompressible Laughlin state Same filling! Seems floppy compressible!

52 Fractional quantum Hall states and quasiparticles In fractional QHE, insertion of flux quantum brings in a fraction of electron! q = e / m f = 2p

53 Fractional statistics To change statistics just add (attach) flux! Naïve consideration: q = e / m φ φ q, m q, m f = 2p Statistical angle = ½ of Aharonov-Bohm phase Θ = π/m Same result as from Berry phase, Plasma Analogy, Chern-Simons

54 Polyacethylene (Su-Schrieffer-Heeger, PRB 1980)

55 Soliton in polyacethylene (topological defect in 1D) conduction electron per two C atoms. Shift by 1 unit cell transfer of ½ electrons from infinity => Soliton charge ½ e (per spin)

56 Topological defects in 2D Vortices in SO(2) [same as U(1)] XY magnets, superconductors, super-fluids More accurately, can represent point defects in 2D of line defects in 3D codimenstion = 2 (for solitons, codimension = 1)

57 Non-collinear isotropic magnets Vortices in SO(3)

58 Chiral order Quantum Hall Uniform chiral ordering S 2 s S 1 S 3 j j j j j j j j j j j j j j j = W / 2 = p / 2 S i = (S a cosq a r i,s b cosq b r i,s c cosq c r i ) Hofstadter problem => Q. Hall! Ivar Martin and C. D. Batista, PRL 101, (2008)

59 P(!) c Finite temperature transition: Monte-Carlo simulations ¼ filling! L=8 L=10 L=12 Spin configuration E- distribution likely 1 st order T/t= T/t= T/t= T/t= a b c d T/t Specific heat and chirality Y. Kato, IM, C.D. Batista, PRL Energy per site

60 Chiral magnetism (J. Villain, J. Phys. 1977) Ising argument: Chirality remains broken up to Finite temperature T ~ J (energy per unit length of chiral domain wall) Topological defects: Extended: domain walls Point-like: vortices Cf. p 1 (SO(3)) = Z 2 SO(3) ~ RP Nematic LQ: OP RP 2

61 Edge states/vortices Chiral domain wall: codim = 1 Vortex: codim = 2 Chiral edge modes Z2 fractionalization (half-odd int charge) R. Muniz, A. Rahmani, and IM, submitted

62 Effective theory of inhomogeneous state If we integrating out fermions: L= + R. Muniz, A. Rahmani, and IM, submitted

63 Half-vortex ϕ To make transform around vortex single-valued Laughlin argument: half-flux quantum -> q = σ xy /2 R. Muniz, A. Rahmani, and IM, submitted

64 Vortex charge and spin H = -tåc ia c ja - håc ia s abc z ib - J å H S i ic ia s ab c ib <ij> i i Application of the Laughlin s argument (using linear response conductivities) agrees with numerically obtained accumulated spin and charge at the vortex core R. Muniz, A. Rahmani, and IM, submitted

65 Fractional statistics Numerical value of exchange Phase φ φ q, m q, m Theoretical expectation:

66 Quantum Hall in Kondo kagome lattice

67 The end Summary classical fractional Quantum Hall effect in Semiconductors fractional charge and statistics Fractionalization in Polyacethylene Different flavors of topological defects Fractionalization in integer spontaneous quantum Hall systems

68 The End