Charge and Spin Order in Doped Mott Insulators Many strongly correlated (quantum) systems exhibit spatially inhomogeneous and anisotropic phases. ◮ Stripe and Nematic phases of cuprate superconductors ◮ Stripe phases of manganites and nickelates ◮ Anisotropic transport in 2DEG in large magnetic fields and in Sr 3 Ru 2 O 7 Common underlying physical mechanism : � effective short range attractive forces Competition ⇒ long(er) range repulsive (Coulomb) interactions Frustrated phase separation ⇒ spatial inhomogeneity (Kivelson and Emery (1993), also Di Castro et al)
Charge and Spin Order in Doped Mott Insulators Many strongly correlated (quantum) systems exhibit spatially inhomogeneous and anisotropic phases. ◮ Stripe and Nematic phases of cuprate superconductors ◮ Stripe phases of manganites and nickelates ◮ Anisotropic transport in 2DEG in large magnetic fields and in Sr 3 Ru 2 O 7 Common underlying physical mechanism : � effective short range attractive forces Competition ⇒ long(er) range repulsive (Coulomb) interactions Frustrated phase separation ⇒ spatial inhomogeneity (Kivelson and Emery (1993), also Di Castro et al) ◮ Examples in classical systems: blockcopolymers, ferrofluids, etc.
Charge and Spin Order in Doped Mott Insulators Many strongly correlated (quantum) systems exhibit spatially inhomogeneous and anisotropic phases. ◮ Stripe and Nematic phases of cuprate superconductors ◮ Stripe phases of manganites and nickelates ◮ Anisotropic transport in 2DEG in large magnetic fields and in Sr 3 Ru 2 O 7 Common underlying physical mechanism : � effective short range attractive forces Competition ⇒ long(er) range repulsive (Coulomb) interactions Frustrated phase separation ⇒ spatial inhomogeneity (Kivelson and Emery (1993), also Di Castro et al) ◮ Examples in classical systems: blockcopolymers, ferrofluids, etc. ◮ Astrophysical examples: “Pasta Phases” (meatballs, spaghetti and lasagna!) of neutron stars “lightly doped” with protons (G. Ravenhall et al,1983)
Charge and Spin Order in Doped Mott Insulators Many strongly correlated (quantum) systems exhibit spatially inhomogeneous and anisotropic phases. ◮ Stripe and Nematic phases of cuprate superconductors ◮ Stripe phases of manganites and nickelates ◮ Anisotropic transport in 2DEG in large magnetic fields and in Sr 3 Ru 2 O 7 Common underlying physical mechanism : � effective short range attractive forces Competition ⇒ long(er) range repulsive (Coulomb) interactions Frustrated phase separation ⇒ spatial inhomogeneity (Kivelson and Emery (1993), also Di Castro et al) ◮ Examples in classical systems: blockcopolymers, ferrofluids, etc. ◮ Astrophysical examples: “Pasta Phases” (meatballs, spaghetti and lasagna!) of neutron stars “lightly doped” with protons (G. Ravenhall et al,1983) ◮ Analogues in lipid bilayers intercalated with DNA (Lubensky et al, 2000)
Soft Quantum Matter or Quantum Soft Matter
Electron Liquid Crystal Phases Nematic Crystal Smectic Isotropic
Schematic Phase Diagram of Doped Mott Insulators Isotropic (Disordered) Temperature Nematic Superconducting Crystal Smectic C 3 C C 1 2 h ω � ¯ ω measures transverse zero-point stripe fluctuations of the stripes. Systems with “large” coupling to lattice displacements ( e. g. manganites) are “more classical” than systems with “primarily” electronic correlations (e. g. cuprates); nickelates lie in-between.
Phase Diagram of the High T c Superconductors T antiferromagnet bad metal pseudogap superconductor 1 x 8 Full lines: phase boundaries for the antiferromagnetic and superconducting phases. Broken line: phase boundary for a system with static stripe order and a “1 / 8 anomaly” Dotted line: crossover between the bad metal and pseudogap regimes
Order Parameter for Charge Smectic (Stripe) Ordered States
Order Parameter for Charge Smectic (Stripe) Ordered States ◮ unidirectional charge density wave (CDW)
Order Parameter for Charge Smectic (Stripe) Ordered States ◮ unidirectional charge density wave (CDW) ◮ charge modulation ⇒ charge stripe
Order Parameter for Charge Smectic (Stripe) Ordered States ◮ unidirectional charge density wave (CDW) ◮ charge modulation ⇒ charge stripe ◮ if it coexists with spin order ⇒ spin stripe
Order Parameter for Charge Smectic (Stripe) Ordered States ◮ unidirectional charge density wave (CDW) ◮ charge modulation ⇒ charge stripe ◮ if it coexists with spin order ⇒ spin stripe ◮ stripe state ⇒ new Bragg peaks of the electron density at k = ± Q ch = ± 2 π ˆ e x λ ch
Order Parameter for Charge Smectic (Stripe) Ordered States ◮ unidirectional charge density wave (CDW) ◮ charge modulation ⇒ charge stripe ◮ if it coexists with spin order ⇒ spin stripe ◮ stripe state ⇒ new Bragg peaks of the electron density at k = ± Q ch = ± 2 π ˆ e x λ ch ◮ spin stripe ⇒ magnetic Bragg peaks at k = Q spin = ( π, π ) ± 1 2 Q ch
Order Parameter for Charge Smectic (Stripe) Ordered States ◮ unidirectional charge density wave (CDW) ◮ charge modulation ⇒ charge stripe ◮ if it coexists with spin order ⇒ spin stripe ◮ stripe state ⇒ new Bragg peaks of the electron density at k = ± Q ch = ± 2 π ˆ e x λ ch ◮ spin stripe ⇒ magnetic Bragg peaks at k = Q spin = ( π, π ) ± 1 2 Q ch ◮ Charge Order Parameter: � n Q ch � , Fourier component of the electron density at Q ch .
Order Parameter for Charge Smectic (Stripe) Ordered States ◮ unidirectional charge density wave (CDW) ◮ charge modulation ⇒ charge stripe ◮ if it coexists with spin order ⇒ spin stripe ◮ stripe state ⇒ new Bragg peaks of the electron density at k = ± Q ch = ± 2 π ˆ e x λ ch ◮ spin stripe ⇒ magnetic Bragg peaks at k = Q spin = ( π, π ) ± 1 2 Q ch ◮ Charge Order Parameter: � n Q ch � , Fourier component of the electron density at Q ch . ◮ Spin Order Parameter: � S Q spin � , Fourier component of the electron density at Q spin .
Order Parameter for Charge Smectic (Stripe) Ordered States ◮ unidirectional charge density wave (CDW) ◮ charge modulation ⇒ charge stripe ◮ if it coexists with spin order ⇒ spin stripe ◮ stripe state ⇒ new Bragg peaks of the electron density at k = ± Q ch = ± 2 π ˆ e x λ ch ◮ spin stripe ⇒ magnetic Bragg peaks at k = Q spin = ( π, π ) ± 1 2 Q ch ◮ Charge Order Parameter: � n Q ch � , Fourier component of the electron density at Q ch . ◮ Spin Order Parameter: � S Q spin � , Fourier component of the electron density at Q spin .
Nematic Order
Nematic Order ◮ Translationally invariant state with broken rotational symmetry
Nematic Order ◮ Translationally invariant state with broken rotational symmetry ◮ Order parameter for broken rotational symmetry: any quantity transforming like a traceless symmetric tensor
Nematic Order ◮ Translationally invariant state with broken rotational symmetry ◮ Order parameter for broken rotational symmetry: any quantity transforming like a traceless symmetric tensor ◮ Order parameter: a director, a headless vector
Nematic Order ◮ Translationally invariant state with broken rotational symmetry ◮ Order parameter for broken rotational symmetry: any quantity transforming like a traceless symmetric tensor ◮ Order parameter: a director, a headless vector In D = 2 one can use the static structure factor
Nematic Order ◮ Translationally invariant state with broken rotational symmetry ◮ Order parameter for broken rotational symmetry: any quantity transforming like a traceless symmetric tensor ◮ Order parameter: a director, a headless vector In D = 2 one can use the static structure factor Z ∞ d ω S ( � 2 π S ( � k ) = k , ω ) −∞ to construct
Nematic Order ◮ Translationally invariant state with broken rotational symmetry ◮ Order parameter for broken rotational symmetry: any quantity transforming like a traceless symmetric tensor ◮ Order parameter: a director, a headless vector In D = 2 one can use the static structure factor Z ∞ d ω S ( � 2 π S ( � k ) = k , ω ) −∞ to construct k = S ( � k ) − S ( R � k ) Q � S ( � k ) + S ( R � k ) where S ( k , ω ) is the dynamic structure factor , the dynamic (charge-density) correlation function, and R = rotation by π/ 2.
Nematic Order ◮ Translationally invariant state with broken rotational symmetry ◮ Order parameter for broken rotational symmetry: any quantity transforming like a traceless symmetric tensor ◮ Order parameter: a director, a headless vector In D = 2 one can use the static structure factor Z ∞ d ω S ( � 2 π S ( � k ) = k , ω ) −∞ to construct k = S ( � k ) − S ( R � k ) Q � S ( � k ) + S ( R � k ) where S ( k , ω ) is the dynamic structure factor , the dynamic (charge-density) correlation function, and R = rotation by π/ 2.
Nematic Order and Transport Transport: we can use the resistivity tensor to construct Q
Nematic Order and Transport Transport: we can use the resistivity tensor to construct Q � ρ xx − ρ yy � ρ xy Q ij = ρ xy ρ yy − ρ xx Alternatively, in 2D the nematic order parameter can be written in terms of a director N ,
Nematic Order and Transport Transport: we can use the resistivity tensor to construct Q � ρ xx − ρ yy � ρ xy Q ij = ρ xy ρ yy − ρ xx Alternatively, in 2D the nematic order parameter can be written in terms of a director N , N = Q xx + i Q xy = | N | e i ϕ Under a rotation by a fixed angle θ , N transforms as
Nematic Order and Transport Transport: we can use the resistivity tensor to construct Q � ρ xx − ρ yy � ρ xy Q ij = ρ xy ρ yy − ρ xx Alternatively, in 2D the nematic order parameter can be written in terms of a director N , N = Q xx + i Q xy = | N | e i ϕ Under a rotation by a fixed angle θ , N transforms as N → N e i 2 θ Hence, it changes sign under a rotation by π/ 2 and it is invariant under a rotation by π . On the other hand, it is invariant under uniform translations by R .
Nematic Order and Transport Transport: we can use the resistivity tensor to construct Q � ρ xx − ρ yy � ρ xy Q ij = ρ xy ρ yy − ρ xx Alternatively, in 2D the nematic order parameter can be written in terms of a director N , N = Q xx + i Q xy = | N | e i ϕ Under a rotation by a fixed angle θ , N transforms as N → N e i 2 θ Hence, it changes sign under a rotation by π/ 2 and it is invariant under a rotation by π . On the other hand, it is invariant under uniform translations by R .
Charge Nematic Order in the 2DEG in Magnetic Fields Energy 2 DEG 5/2 h ω c 3/2 h ω c B E F 1/2 h ω c Angular Momentum bulk edge Al As − Ga As heterostructure ◮ Electrons in Landau levels are strongly correlated systems, K.E.=“0”
Charge Nematic Order in the 2DEG in Magnetic Fields Energy 2 DEG 5/2 h ω c 3/2 h ω c B E F 1/2 h ω c Angular Momentum bulk edge Al As − Ga As heterostructure ◮ Electrons in Landau levels are strongly correlated systems, K.E.=“0” ◮ For N = 0 , 1 ⇒ Fractional (and integer) Quantum Hall Effects
Charge Nematic Order in the 2DEG in Magnetic Fields Energy 2 DEG 5/2 h ω c 3/2 h ω c B E F 1/2 h ω c Angular Momentum bulk edge Al As − Ga As heterostructure ◮ Electrons in Landau levels are strongly correlated systems, K.E.=“0” ◮ For N = 0 , 1 ⇒ Fractional (and integer) Quantum Hall Effects ◮ Integer QH states for N ≥ 2
Charge Nematic Order in the 2DEG in Magnetic Fields Energy 2 DEG 5/2 h ω c 3/2 h ω c B E F 1/2 h ω c Angular Momentum bulk edge Al As − Ga As heterostructure ◮ Electrons in Landau levels are strongly correlated systems, K.E.=“0” ◮ For N = 0 , 1 ⇒ Fractional (and integer) Quantum Hall Effects ◮ Integer QH states for N ≥ 2 ◮ Hartree-Fock predicts stripe phases for “large” N (Koulakov et al, Moessner and Chalker (1996))
Transport Anisotropy in the 2DEG M. P. Lilly et al (1999), R. R. Du et al (1999)
Transport Anisotropy in the 2DEG M. P. Lilly et al (1999), R. R. Du et al (1999)
Transport Anisotropy in the 2DEG K. B. Cooper et al (2002)
Transport Anisotropy in the 2DEG This is not the quantum Hall plateau transition!
Transport Anisotropy in the 2DEG This is not the quantum Hall plateau transition! Is this a smectic or a nematic state?
Transport Anisotropy in the 2DEG This is not the quantum Hall plateau transition! Is this a smectic or a nematic state? ◮ The effects gets bigger in cleaner systems ⇒ it is a correlation effect
Transport Anisotropy in the 2DEG This is not the quantum Hall plateau transition! Is this a smectic or a nematic state? ◮ The effects gets bigger in cleaner systems ⇒ it is a correlation effect ◮ The anisotropy is finite as T → 0
Transport Anisotropy in the 2DEG This is not the quantum Hall plateau transition! Is this a smectic or a nematic state? ◮ The effects gets bigger in cleaner systems ⇒ it is a correlation effect ◮ The anisotropy is finite as T → 0 ◮ The anisotropy has a pronounced temperature dependence ⇒ it is an ordering effect
Transport Anisotropy in the 2DEG This is not the quantum Hall plateau transition! Is this a smectic or a nematic state? ◮ The effects gets bigger in cleaner systems ⇒ it is a correlation effect ◮ The anisotropy is finite as T → 0 ◮ The anisotropy has a pronounced temperature dependence ⇒ it is an ordering effect ◮ I − V curves are linear (at low V ) and no threshold electric field ⇒ no pinning
Transport Anisotropy in the 2DEG This is not the quantum Hall plateau transition! Is this a smectic or a nematic state? ◮ The effects gets bigger in cleaner systems ⇒ it is a correlation effect ◮ The anisotropy is finite as T → 0 ◮ The anisotropy has a pronounced temperature dependence ⇒ it is an ordering effect ◮ I − V curves are linear (at low V ) and no threshold electric field ⇒ no pinning ◮ No broad-band noise is observed in the peak region
Transport Anisotropy in the 2DEG This is not the quantum Hall plateau transition! Is this a smectic or a nematic state? ◮ The effects gets bigger in cleaner systems ⇒ it is a correlation effect ◮ The anisotropy is finite as T → 0 ◮ The anisotropy has a pronounced temperature dependence ⇒ it is an ordering effect ◮ I − V curves are linear (at low V ) and no threshold electric field ⇒ no pinning ◮ No broad-band noise is observed in the peak region ◮ The 2DEG behaves as a uniform anisotropic fluid: it is a nematic charged fluid
Transport Anisotropy in the 2DEG This is not the quantum Hall plateau transition! Is this a smectic or a nematic state? ◮ The effects gets bigger in cleaner systems ⇒ it is a correlation effect ◮ The anisotropy is finite as T → 0 ◮ The anisotropy has a pronounced temperature dependence ⇒ it is an ordering effect ◮ I − V curves are linear (at low V ) and no threshold electric field ⇒ no pinning ◮ No broad-band noise is observed in the peak region ◮ The 2DEG behaves as a uniform anisotropic fluid: it is a nematic charged fluid
Transport Anisotropy in the 2DEG The 2DEG behaves like a Nematic fluid! Classical Monte Carlo simulation of a classical 2D XY model for nematic order with coupling J and external field h , on a 100 × 100 lattice Fit of the order parameter to the data of M. Lilly and coworkers, at ν = 9 / 2 (after deconvoluting the effects of the geometry.) Best fit: J = 73 mK and h = 0 . 05 J = 3 . 5 mK and T c = 65 mK . E. Fradkin, S. A. Kivelson, E. Manousakis and K. Nho, Phys. Rev. Lett. 84 , 1982 (2000). K. B. Cooper et al. , Phys. Rev. B 65 , 241313 (2002)
Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields ◮ Sr 3 Ru 2 O 7 is a strongly correlated quasi-2D bilayer oxide
Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields ◮ Sr 3 Ru 2 O 7 is a strongly correlated quasi-2D bilayer oxide ◮ Sr 2 Ru 1 O 4 is a quasi-2D single layer correlated oxide and a low T c superconductor ( p x + ip y ?)
Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields ◮ Sr 3 Ru 2 O 7 is a strongly correlated quasi-2D bilayer oxide ◮ Sr 2 Ru 1 O 4 is a quasi-2D single layer correlated oxide and a low T c superconductor ( p x + ip y ?) Sr 3 Ru 2 O 7 is a paramagnetic metal with metamagnetic behavior at ◮ low fields
Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields ◮ Sr 3 Ru 2 O 7 is a strongly correlated quasi-2D bilayer oxide ◮ Sr 2 Ru 1 O 4 is a quasi-2D single layer correlated oxide and a low T c superconductor ( p x + ip y ?) Sr 3 Ru 2 O 7 is a paramagnetic metal with metamagnetic behavior at ◮ low fields ◮ It is a “bad metal” (linear resistivity over a large temperature range) except at the lowest temperatures
Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields ◮ Sr 3 Ru 2 O 7 is a strongly correlated quasi-2D bilayer oxide ◮ Sr 2 Ru 1 O 4 is a quasi-2D single layer correlated oxide and a low T c superconductor ( p x + ip y ?) Sr 3 Ru 2 O 7 is a paramagnetic metal with metamagnetic behavior at ◮ low fields ◮ It is a “bad metal” (linear resistivity over a large temperature range) except at the lowest temperatures ◮ Clean samples seemed to suggest a field tuned quantum critical end-point
Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields ◮ Sr 3 Ru 2 O 7 is a strongly correlated quasi-2D bilayer oxide ◮ Sr 2 Ru 1 O 4 is a quasi-2D single layer correlated oxide and a low T c superconductor ( p x + ip y ?) Sr 3 Ru 2 O 7 is a paramagnetic metal with metamagnetic behavior at ◮ low fields ◮ It is a “bad metal” (linear resistivity over a large temperature range) except at the lowest temperatures ◮ Clean samples seemed to suggest a field tuned quantum critical end-point ◮ Ultra-clean samples find instead a new phase with spontaneous transport anisotropy for a narrow range of fields
Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields ◮ Sr 3 Ru 2 O 7 is a strongly correlated quasi-2D bilayer oxide ◮ Sr 2 Ru 1 O 4 is a quasi-2D single layer correlated oxide and a low T c superconductor ( p x + ip y ?) Sr 3 Ru 2 O 7 is a paramagnetic metal with metamagnetic behavior at ◮ low fields ◮ It is a “bad metal” (linear resistivity over a large temperature range) except at the lowest temperatures ◮ Clean samples seemed to suggest a field tuned quantum critical end-point ◮ Ultra-clean samples find instead a new phase with spontaneous transport anisotropy for a narrow range of fields
Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields Phase diagram of Sr 3 Ru 2 O 7 in the temperature-magnetic field plane. (from Grigera et al (2004).
Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields R. Borzi et al (2007)
Transport Anisotropy in Sr 3 Ru 2 O 7 in magnetic fields R. Borzi et al (2007)
Charge and Spin Order in the Cuprate Superconductors ◮ Stripe charge order in underdoped high temperature superconductors(La 2 − x Sr x CuO 4 , La 1 . 6 − x Nd 0 . 4 Sr x CuO 4 and YBa 2 Cu 3 O 6 + x ) (Tranquada, Ando, Mook, Keimer) ◮ Coexistence of fluctuating stripe charge order and superconductivity in La 2 − x Sr x CuO 4 and YBa 2 Cu 3 O 6 + x (Mook, Tranquada) and nematic order (Keimer). ◮ Dynamical layer decoupling in stripe ordered La 2 − x Ba x CuO 4 and in La 2 − x Sr x CuO 4 at finite fields (transport, (Tranquada et al (2007)), Josephson resonance (Basov et al (2009))) ◮ Induced charge order in the SC phase in vortex halos in La 2 − x Sr x CuO 4 and underdoped YBa 2 Cu 3 O 6 + x (neutrons: B. Lake, Keimer; STM: Davis) ◮ STM Experiments: short range stripe order (on scales long compared to ξ 0 ), possible broken rotational symmetry (Bi 2 Sr 2 CaCu 2 O 8 + δ ) (Kapitulnik, Davis, Yazdani) ◮ Transport experiments give evidence for charge domain switching in YBa 2 Cu 3 O 6 + x wires (Van Harlingen/Weissmann)
Charge and Spin Order in the Cuprate Superconductors Static spin stripe order in La 2 − x Ba x CuO 4 near x = 1 / 8 in neutron scattering (Fujita et al (2004))
Charge and Spin Order in the Cuprate Superconductors Static charge stripe order in La 2 − x Ba x CuO 4 near x = 1 / 8 in resonant X-ray scattering (Abbamonte et. al.(2005))
Induced stripe order in La 2 − x Sr x CuO 4 by Zn impurities La 1.86 Sr 0.14 Cu 0.988 Zn 0.012 O 4 La 1.85 Sr 0.15 CuO 4 � 300 ∆ E = 1.5 meV ∆ E = 2 meV Intensity (arb. units) (a) (c) I 7K (arb. units) 200 100 100 0 0 0.2 -0.2 0.0 0.2 -0.2 0.0 800 1200 I 7K -I 80K (arb. units) Intensity (arb. units) (b) (d) ∆ E = 0 ∆ E = 0 400 800 T = 1.5 K T = 50 K 400 0 -0.2 0.0 0.2 -0.2 0 0.2 (0.5+h,0.5,0) (0.5+h,0.5,0) Magnetic neutron scattering with and without Zn (Kivelson et al (2003))
Electron Nematic Order in High Temperature Superconductors (i) (j) Temperature-dependent transport anisotropy in underdoped La 2 − x Sr x CuO 4 and YBa 2 Cu 3 O 6 + x ; Ando et al (2002)
Charge Nematic Order in underdoped YBa 2 Cu 3 O 6 + x ( y = 6 . 45) Intensity maps of the spin-excitation spectrum at 3, 7,and 50 meV, respectively. Colormap of the intensity at 3 meV, as it would be observed in a crystal consisting of two perpendicular twin domains with equal population. Scans along a ∗ and b ∗ through Q AF . (from Hinkov et al (2007).
Charge Nematic Order in underdoped YBa 2 Cu 3 O 6 + x ( y = 6 . 45) a) Incommensurability δ (red symbols), half-width-at-half-maximum of the incommensurate peaks along a ∗ ( ξ − 1 a , black symbols) and along b ∗ ( ξ − 1 b , open blue symbols) in reciprocal lattice units. (from Hinkov et al 2007)).
Static stripe order in underdoped YBa 2 Cu 3 O 6 + x at finite fields Hinkov et al (2008)
Charge Order induced inside a SC vortex “halo” Induced charge order in the SC phase in vortex halos: neutrons in La 2 − x Sr x CuO 4 (B. Lake et al , 2002), STM in optimally doped Bi 2 Sr 2 CaCu 2 O 8 + δ (S. Davis et al , 2004)
STM: Short range stripe order in Dy-Bi 2 Sr 2 CaCu 2 O 8 + δ (k) Left (l) Right Left: STM R-maps in Dy-Bi 2 Sr 2 CaCu 2 O 8 + δ at high bias: R ( � r , 150 mV ) = I ( � r , + 150 mV ) / I ( � r , − 150 mV ) Right: Short range nematic order with ≫ ξ 0 ; Kohsaka et al (2007)
Optimal Degree of Inhomogeneity in La 2 − x Ba x CuO 4 ARPES in La 2 − x Ba x CuO 4 : the antinodal (pairing) gap is largest , even though T c is lowest , near x = 1 8 ; T. Valla et al (2006), ZX Shen et al (2008)
Dynamical Layer Decoupling in La 2 − x Ba x CuO 4 near x = 1 / 8 Dynamical layer decoupling in transport (Li et al (2008))
Dynamical Layer Decoupling in La 2 − x Ba x CuO 4 near x = 1 / 8 Kosterlitz-Thouless resistive transition (Li et al (2008)
Dynamical Layer Decoupling in La 2 − x Sr x CuO 4 in a magnetic field Layer decoupling seen in Josephson resonance spectroscopy: c-axis penetration depth λ c vs c-axis conductivity σ 1 c (Basov et al, 2009)
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