Enrico Rossi: Neutron resonance in electron-doped cuprates

In his talk, Enrico Rossi discussed the work with J-P Ismer,
Ilya Eremin and Dirk Morr on the neutron resonance in electron-doped cuprates. His main idea is that the resonance is a spin exciton, shifted to a higher frequency by a finite fermionic damping rate. Enrico started his presentation with a brief review of the excitonic scenario for the resonance. He then argued that, in distinction to hole-doped cuprates, where the resonance is well below 2 \Delta_[max}, the resonance in electron-doped PrCeCuO and NdCeCuO is observed at 11 meV, which might be larger than 2\Delta (the measured gap
maximum is less than 5 meV). as determined by ARPES experiments Enrico presented RPA-type calculations of the resonance, which include a finite broadening of the fermionic linewidth. He argued that due to a finite broadening, the resonance shifts to a higher frequency, which may exceed 2\Ddelta_{max}. Enrico then argues that in the presence of a magnetic field, the resonance is split into three distinct peaks. Due to the smaller magnitude of the gap, and a resonance frequency which is much smaller in the electron-doped cuprates than in the hole-doped ones, the experimental resolution in INS experiments is sufficiently good to resolve a splitting of the resonance in field of about 8 T, a splitting which is of the order of 1 meV.

Finally, Enrico argued that in those electon-doped cuprates, in which superconductivity exists with antiferromagnetism and T_c>T_N, the resonance shifts down to lower frequencies as T_N is approached, and reaches zero frequency at T_N.

The discussion after the talk focused on the intensity and the linewidth of the peak.
Enrico was asked whether a large width of the peak may prevent the development of three sub-peaks in a field. Enrico replied that even if a large quasi-particle damping prevents the resolution of the three peaks, the resonance will become highly asymmetric.

Maxim Vavilov: Quantum Disorder in Andreev Billiards

Maxim Vavilov discussed the effects of quantum disorder in Andreev
Billiards. These billiards consist of a small grain of normal state
material that is brought into contact with a superconducting
reservoir. These systems are realized, for example, by connecting a
quantum dot to a superconducting leads

Maxim first discussed the various energy scales that are relevant
for this problem. The largest energy scale is set by the (isotropic)
superconducting gap, Delta_sc, which implies perfect Andreev
reflection at the interface between the normal and superconducting
systems. The next smaller energy scale is set by the Thouless energy
E_T=hbar/tau_f where tau_f=L/v_F is the flight time of the
electrons, and L is the size of the normal state grain. Another
energy scale is set by E_g=hbar/tau_d where tau_d=tau_f*L/b is the
dwell time of the electrons, and b is the length of the interface
between the normal and superconducting systems. The last energy
scale is set by the mean level spacing, delta_I, of the normal state
system. The relative order of energy scales for the system that
Maxim studied is given by

Delta_sc >> E_T >> E_g >> delta_I

The objective of Maxim's work was to study the properties of the
electrons in the normal state grain, which are reflected in the
averaged density of states (DOS). Of particular interest is the
question of whether Andreev scattering off the interface leads to a
suppression of the normal state DOS at low energies. Maxim then
proceeded to outline a calculation using Random matrix theory (RMT) (see "Induced superconductivity distinguishes chaotic from integrable billiards", J. A. Melsen, P. W. Brouwer, K. M. Frahm, C. W. J. Beenakker Europhys. Lett. 35 (1996) 7) and a Gaussian Orthogonal Ensemble, which can be exactly solved in the
limit hbar/(tau_f * delta_I) -> 00. In this case, the DOS opens up a
hard gap at low energies up to an energy scale set by E_g, and
increases as DOS ~ sqrt(w - E_g) for energies w>E_g. At this point
Daniel Sheehy asked whether this result is achieved by averaging
over ensembles. Maxim answered that in the case he considered,
averaging over ensembles is equal to averaging over many energy
levels. Hence the RMT result should be valid for the average DOS of a
single normal grain.

Maxim then proceeded to outline a different calculation based on the
Eilenberger equations developed with Anatoly Larkin ("Quantum Disorder and Quantum Chaos in Andreev Billiards", M.G. Vavilov, A.I. Larkin, Phys. Rev. B 67, 115335 (2003)). This approach corresponds to the semiclassical approximation only if impurity scattering is not taken
into account. Without disorder, this approach yields an averaged DOS
in the normal grain that is suppressed at low energies (below E_g),
but does not show a hard gap, in contrast to the results of the
random matrix theory. Finally, Maxim considered the effects of
disorder, as realized by a distribution of short range impurities.
In the limit of strong disorder, when the scattering time is
comparable with the dwell time, the Eilenberger approach recovers
the RMT result, and a hard gap opens in the DOS up to a frequency of
E_g. However, even in the case of weak disorder, a gap opens in the
DOS.

Daniel Sheehy asked whether the Andreev reflection at the interface
is perfect. Maxim answered that this is the case as long as the
superconducting gap is the largest energy scale in the problem, and
in particular, as long as Delta_sc >> E_T.

Andrey Chubukov asked whether this averaged DOS can be measured
experimentally. Maxim pointed out that in general, it can be
measured by studying quantum dots connected to superconducting
leads. However, the main experimental problem seems to be the
interface between the superconducting and normal state materials.
Finally, Maxim remarked that while his theory was developed for
two-dimensional grains, the effect might be more easily observable
in three dimensional systems.

A. Auerbach: "Quantum Tunneling of vortices in underdoped cuprates: theory and experiment"

Assa Auerbach (Technion, Israel) told us about a new experiment performed by the group of G. Koren also at the Technion (cond-mat/0707.284) where variable range hopping (VRH) of vortices was observed in a special YBCO film. The film was 1m long and 14 um wide wire arranged in the form of a meander. This allowed the experimentalist to perform magneto-resistance (MR) measurements at low currents
(1uA) and low fields (up to 6T) where most of the contribution to the MR comes from single vortex tunneling.
The major finding in the experiment is a VRH type temperature dependence of the MR at low T, namely,

1) MR~exp[-(T_0/T)^1/3] .

Assa argued that VRH is not expected in a conventional BCS type superconductor, with large coherence length.
However, for underdoped cuprate superconductors, where the ratio of carrier density n_s to pinning site density n_pin can be low, Auerbach, Arovas Ghosh (PRB 74 2006) have predicted Eq. 1 based on a interacting boson model, and have calculated that

T0~(n_s/n_pin)^2(1/n_lay)*dV,

where n_lay is the layer density and dV is the fluctuations of the pinning potential. Reasonable values of the parameters agree with the measured T_0.

In the context of tunneling vortices, Assa also described recent studies of vortices in a model of half filled, hard core, lattice bosons (Lindner et. el. cond-mat/0701571).
He reported numerical estimates which found the vortex mass to be quite low (~3 times the boson mass), and hence the critical melting density of the vortex lattice was estimated to be of order 10^{-3} vortices per lattice site. This implies that quantum vortex liquid phases could be achieved by relatively weak rotations (in an optical lattice) or magnetic field (in e.g. cuprates), much lower than Hc_2.

Victor Galitski: Mesoscopic disorder fluctuations in a d-wave superconductor

Thursday, Aug 23th

Victor Galitski started our Patio Discussion by returning to the recent STM experiments by Ali Yazdani showing an inhomogeneous spatial gap distribution above the superconducting transition temperature Tc in the cuprates. Taking these as motivation for his today's presentation, he first pointed out that the important features seen by Ali Yazdani are that the experimental gapmaps are static and reproducible when varying temperature. In particular this means no phase separation takes place.

Victor went on by stressing the break-down of Anderson's theorem in d-wave superconductors in the presence of disorder potentials, leading to a dependence of Tc on the disorder. As the density of impurities is random, there are fluctuations in real space. These are according to Victor associated with a local Tc larger than the Tc for a corresponding homogeneous state. A picture of paddles of superconductivity within a normal background emerges, where each of the paddles have their private Tc. Victor now continued with an overview over what is known from s-wave superconductors, in which case Tc does not depend on disorder in leading order in accordance to Anderson's theorem. In this case fluctuations are not important.
Victor proceeded by reminding us that in s-wave superconductors with magnetic impurities there is an Abrikosov-Gorkov formula
ln (Tc0/Tc) = \Psi(1/2 + \Gamma/[2\pi Tc]) - \Psi(1/2)
that determines the actual Tc in terms of the critical temperature for a system without disorder, Tc0. The crucial parameter in this formula is the pair breaking parameter Gamma. In a magnetic field and in the diffusive limit it is proportional to D*H, where D is the diffusion constant and H the magnetic field. This leads to the well known Hc2(T) curve. Victor draw our attention to the fact that for s-wave superconductors this theoretical curve is smooth at low temperatures, whereas experimentally often an upturn of the Hc2-curve is observed. A possible explanation would then be that Tc depends on disorder via the diffusion constant D, and thus Hc2(0)~n_imp. Dan Sheehy asked the question what happens for n_imp=0, and Victor stressed that he restricts his discussions to the dirty limit, so that Tc0 \tau <<>

Next Victor draw a picture of superconducting islands connected by the Josephson effect and mentioned the works about Josephson networks by Spivak/Zhou PRL '95 and by Larkin/Galitski PRL 2002. At this point a specific model in terms of a Ginzburg-Landau action followed, in which spatial fluctuations of the order parameter where taken into account.

Victor mentioned in passing that in cuprates in principle Tc0 depends on doping, such that Tc is determined by an interplay between the intrinsic x-dependence of Tc0 and the induced one by the spatial disorder. This leads to a superconducting dome resembling very roughly that of the cuprates.

The spatial randomness of the gaps introduces via the eigenvalue equation

(1/v) \int C(r,r') \Delta(r') = (Tc/Tc0) \Delta(r)

also a random Tc. The random operator C(r,r') is the Cooperon. The statistics of C(r,r') can be expressed diagrammatically, and leads to a distribution of Tc's as function of coherence length, mean free path and (Tc-Tc0)/Tc0. Victor finished his talk with developing a picture of underdoped cuprates in terms of superconducting islands separated by normal regions, however with a fluctuation gap. This also implies a reduced local density of states in the normal regions.

In the discussion part, Phil Anderson commented that all this does not seem to be related to high-Tc cuprates, but to d-wave BCS superconductors. The nature of the phase transition in cuprates is that of an x-y model, where Tc~\rho_s, not ~\Delta. Thus, fluctutating gaps are not related to fluctuation Tc's. Victor basically agreed and mentioned that he studied a BCS model, not an x-y model. Andrei Chubukov commented that Tc in the calculations should be related to the pseudogap temperature T*.

Dirk Morr asked how the distribution of local Tc's is related to the global Tc. Victor answered that the distribution of Tc's is related to disorder, but that there were no direct relation to a global Tc. Claudio Castellani commented at this point that he thinks Tc as a local quantity is only a technical parameter of the BCS model, any real Tc has to be global. Victor disagreed in the sense that if the puddles are in size larger that a coherence volume, it makes sense to talk about a local Tc for each puddle.

Yong-Baek Kim: "Heisenberg Antiferromagnet on the Hyper-Kagome Lattice: Application to Na4Ir3O8"

Yong-Baek Kim discussed very interesting recent experiments on a new three dimensional antiferromagnetic compound [Okamoto et al., "Spin liquid state in S=1/2 hyper-kagome antiferromagnet Na4Ir3O8"] and recent theoretical work by his group directed at understanding the magnetic behavior of the material [Hopkinson, Isakov, Kee, and Kim, "Classical antiferromagnet on a hyperkagome lattice," PRL 99, 037201 (2007) and "Topological spin liquid on the hyper-kagome lattice Na4Ir3O8"].

Stoichiometry shows that the Ir ions are in a 4+ valence state and the five 5d electrons form a spin-1/2 state in the t2g level. The 3D lattice is like the better known pyrochlore lattice except for the fact that the Ir ions occupy only 3 of the 4 sites of each tetrahedron; the (spinless) Na ions occupy the 4th site. The Ir ions on the decimated pyrochlore lattice form a network of corner-sharing triangles that has been dubbed a "hyperkagome" lattice. As the unit cell contains 12 spins, the material doesn't have an odd number of spin-1/2's per unit cell (in contrast to several 2D candidate spin liquids that may support gapless spin excitations). Nevertheless, magnetic susceptibility measurements find a large Curie-Weiss temperature of -650K and no sign of ordering down to 2K. Due to the large nuclear charge of Iridium (Z=77) there could be a sizable spin-orbit interaction, but as the lattice has inversion symmetry, Dzyaloshinsky-Moriya interactions are apparently forbidden.

Kim's group first investigated the behavior of a classical Heisenberg antiferromagnet on the hyperkagome lattice. Assuming nearest-neighbor exchange, the Hamiltonian can be rewritten as a sum over all the triangles of:


(J/2) * (S_triangle)^2


where S_triangle is the sum of the three spins on each triangle. At zero temperature the ground state is specified simply by setting S_triangle = 0 on each triangle; as there are many choices of the spins that satisfy the constraint, there is a macroscopic degeneracy. Here Andrey Chubukov asked if the constraint could be implemented independently on each triangle, and Yong-Baek clarified that the triangles are not independent, but nevertheless the ground state does have macroscopic degeneracy.

The question then arises as to whether or not there is an order--by-disorder transition induced by classical thermal fluctuations at non-zero temperatures; Hopkinson et al. addressed this by classical Monte Carlo calculations and found a transition from a "cooperative paramagnet" at high temperatures to a spin-nematic phase at low temperatures (below 0.3 to 1.5 K for an exchange constant of about 300K). The rather unusual spin-nematic order parameter is given in terms of cross-products of pairs of spin operators on the triangles. Correlations are found numerically to become long-ranged in the ordered phase, and the change in the entropy is consistent with the formation of spin-nematic order.

Analysis of the corresponding quantum Heisenberg antiferromagnet by Read and Sachdev's bosonic Sp(N) method finds two other phases: An ordered co-planar state and a Z2 spin liquid with massive deconfined spinon excitations. Thus quantum fluctuations have a markedly different effect than classical thermal fluctuations. I noted that spin-nematic order would be difficult to find within the Sp(N) method, as the order parameter doesn't generalize to Sp(N) in a natural way.

John Mydosh suggested experimental investigation by neutron scattering would be interesting as it could detect a spin-ordered phase, and Yong-Baek agreed but pointed out that the common isotope of Iridium strongly absorbs neutrons. Amit Keran suggested that NMR measurements on the sodium atoms would be the next logical step, especially as lower temperatures can be reached.

Phil Anderson: Nernst Effect in the Cuprates

Phil Anderson presented an experimental/theoretical talk entitled "Theory of the Nernst Effect in the Cuprates: Is not Black Hole Physics". Here Phil gave an overview of the anomalous behavior of the Nernst effect above Tc in the so-called vortex liquid state. First and foremost was his phase diagram (temperature versus hole doping) which contained an additional phase line, concave in form, above the usual superconducting Tc - dome. This new phase represents the vortex liquid where vortices form due to the charge pairing and a diamagnetic response is found. The pseudogap appears above these domes with its monotonic decreasing structure from the antiferromagnetic region to an intersection with the supoerconducting dome.In a superconductor the Nernst effect tracks the vortex motion due to a temperature gradient and from the second Josephson equation a transverse voltage develops as a function of the perpendicular applied magnetic field, i.e., a phase slip voltage. In a type 2 superconductor a large Nernst signal results below Tc up Bc2. Now for the generic high Tc superconductors the Nernst signal remains far above Tc and according to the physical model it is proportional to the vortex velocity. Phil's theory enables one to relate the Nernst and Ettingshauser coefficients to the order parameter of the vortex fluid phase, i.e., the energy gap of preformed pairs which now appears as a distribution of gap sizes. Since phase coherence is broken the material in not in a conventional superconducting state.Phil showed that the distribution of energy gaps in the vortex liquid phase is related to the Nernst coefficient minus the field derivative of the Nernst signal. Thus one can now determine the distribution of gap sizes and the probability distribution of order parameters. Note that the pseudogap, as usually determined from NMR, ARPES, optical conductivity, etc., is distinct from the vortex liquid phase. And for certain materials there seems to be no correlation between the pseudogap temperature and the onset of the Nernst signal. One needs further experimental studies to map out the the vortex liquid phase boundary and to fully establish its properties in a variety of high Tc materials.

Amit Keren: Magnetic "Isotope Effect" in Cuprates

Amit Keren (Technion, Israel) told us about an accumulation of 10 years of
research by his group, of a family of YBCO-like high Tc cuprates called
CLBLCO, = (Ca_x La_{1-x})(Ba_{1.75-x}La_{0.25+x})Cu_3O_y.

While most systematic studies
of cuprates involve changing just one doping parameter, such as the oxygen
concentration y, CLBLCO presents a unique opportunity to continuously vary TWO
parameters, x (family index) and y (oxygen concentration), without
significantly disturbing the structure or varying the disorder in the CuO_2
planes. In fact, the primary effect of changing x on the CuO plane is to
slightly vary the copper-oxygen buckling angle which is known to change the
magnetic superexchange constant.

Amit showed his group's mu-SR data for the superconducting transition
temperature Tc(x,y), the spin freezing temperature Tg(x,y) at intermediate
doping, and Neel temperature T_N(x,y) at low doping. Moreover, the 2D AFM
exchange J(x) was extracted from the Neel temperature (by fitting the
T-dependent staggered magnetization to estimate the interlayer exchange).

At first, the data seems scattered on the (T,y) phase diagram. Amit chose to
collapse the data by rescaling all transition temperatures by T_c^{max}(x), and
also rescaling the y axis by an "effective doping" Delta p = K(x) (y-y_max),
which collapsed all the Tc(Delta p) "domes" onto one universal curve.
Collapsing the Tc domes is hardly surprising. However the same axes rescaling
completely collapses the -magnetic- freezing transitions onto one curve as
well!

The conclusion is that T_c^{max}(x) \propto J(x).

Apparently, the data collapse indicates that a single energy scale determines
both antiferromagnetic and SC ordering temperatures!

This blogger feels that this finding, although simple, is far from obvious. It
puts a serious constraint on theoretical mechanisms of cuprate
superconductivity: One would naively expect more than just J, to determine T_c (say e.g. some
additional kinetic or interaction energy might be important). These scales have
no apparent reason to stay proportional to each other as the two material parameters x, and y,
are independently varied.


Amit also showed uniform susceptibility data, which was used to define
the "pseudogap temperature" T*(x,y). While this energy scale did not precisely
collapse by rescaling the axes, it seemed to follow for some reason the 3D Neel
temperature T_N, which depends on magnetic interactions both in and out of the CuO planes.

Frank Marsiglio "Issues Concerning the Optical Sum Rule Anomaly below Tc in the Cuprates"

TOC
1. Conventional Theory
2. Experiment
3. Phenomenological Explanation
4. Issues

Conventional Theory

Kubo sum rule
Integral of the real part of conductivity over frequency is
constant -- temperature independent.


Single band sum rule (theoretical construction)
Integral of the real part of conductivity coming from a single band
over frequency (denoted by W(T)) measures the average second
derivative of energy over momentum.

In conventional cases it is proportional to the minus average energy.


In Fermi Gas as temperature increases the distribution function smears and
particles get transferred to higher energy, so W(T) goes down.

If one now decreases the temperature the Superconducting transition occurs,
the distribution function gets smeared, kinetic energy increases and W(T)
goes down.

So W(T) has a maximum at T_c.


Experiment

Experiment shows the decrease W(T) as one decreases temperature through T_c
in overdoped materials, but in optimally doped and underdoped materials
it goes up.

It means that in optimally and underdoped materials the kinetic energy
decreases in superconducting state. That gives us
"kinetic energy driven superconductors".


Phenomenological Explanation

Norman & Pepin (2002) showed that interactions decrease W(T).
Microwave experiments show that there is a collapse of the scattering
rate (scattering rate is due to interactions) below T_c

Taking together those two statements mean that above T_c W(T) is suppressed
by the interactions while below T_c interactions are suppressed and W(T)
goes over to the one of the noniteracting case -- increases.


Issues

There is no issue of the low energy cutoff as Kuzmenko et al explained that
although they cannot measure conductivity at low frequencies accurately
enough, they can measure the contribution to the sum rule.


The main issue is the upper cutoff.
In order to measure the sum rule for the single band one has to introduce
an upper frequency cutoff which is below the frequency of the interband transitions.

Imagine that we have a simple Drude behaviour of the conductivity.
The Drude peak sharpens up as one lowers temperature. If one then checks the
sum rule up to some upper cutoff in frequency one finds that it is more
weight below this frequency. So although the total weight is conserved
the total weight below a frequency cutoff is temperature dependent.

So the normal state ~T^2 behavior can be explained by a mundane upper cutoff
effect. We are currently investigating whether the anomalous rise of W(T)
below T_c can be attributed to a mundane cutoff effect as well.



Questions

Assa: What should the high frequency cutoff be in order to recover full temperature independent sum rule?

F.M. Large, depends what "full" means.


Chubukov: comment, the increase or decrease of W(T) due to cutoff
depends on the valueof \Delta\tau


Pepin: Has anyone investigated the influence of the van Hove singularity
on W(T)?

F.M. Theoretically, last year in a PRB paper we showed that the change below
T_c can be anomalous, using just a BCS approach. As far as I know no one
has measured this same quantity in High T_c samples that are doped beyond the van Hove singularity.

Konstantin Efetov: Transport in Graphene

Konstantin Efetov started by drawing parallels between high-T_cs (which
dominated earlier discussion) and graphene. He emphasized that both are 2D
systems which have been lauded as materials of the future for energy and
nanoelecronics respectively. Konstantin reminded the audience that grapehene
has 2D honeycomb crystal lattice, and therefore contains two sublattices. The
Brillouin Zone has two valleys with linear, Dirac-like, energy dispersion, so
that the effective hamiltonian for pure graphene is a 4x4 block-diagonal
matrix. Gating the substrate with graphene film on top allows changing the
filling fraction easily.

Konstantin then described the effects of impurity scattering on transport in a
system with such an energy spectrum following his recent work with Igor
Aleiner [PRL 97, 236801 (2006)]. The work was motivated by experimental claims
of delocalized low energy states and universal metallic resistivity in
graphene (which, it seems, is no longer universal), and theoretical analyses
of weak localization corrections.

Efetov and Aleiner considered general purely potential impurity scattering
(spin-orbit interaction is graphene is weak), which replaces zeros in the
block-diagonal Hamiltonian with finite values. Symmetries of the problem
(time-reversal, translation, etc.) dictate that there are 5 independent
parameters that characterize disorder. Konstantin pointed out that the
self-consistent Born approximation does not work for Dirac spectrum as there
are many logarithmic corrections that need to be resummed using the RG
methods. The main conclusion is that all 5 impurity constants grow under RG flow.

The work considers a finite filling fraction and proceeds by looking at the
free energy functional using the
supersymmetry approach. The main conclusions are that, if one neglects the
scattering between bands, the system maps onto a symplectic ensemble,
resulting in antilocalization: increase in conductivity upon lowering the
temperature. However, upon lowering the temperature, intervalley scattering
becomes important, and one finds an orthogonal ensemble for which all states
are known to be localized.

The prediction is for a non-monotonous behavior of the conductivity with
temperature. As T is decreased, first the conductivity is reduced in accord
with the log corrections. At lower T antilocalization kicks in and the
conductivity increases when the temperature is lowered. At yet lower T, the
intervalley scattering takes over, and the conductivity drops to zero as the
states become localized. Konstantin concluded by saying that there is no
chance for minimal metallic conductivity due to generic disorder.

Assa Auerbach asked whether this behavior had been found numerically since
this is a non-interacting theory that lends itself easily to modeling.
Konstantin replied that exploring the phase space of 5 parameters is hard, and
that the localization length is expected to be large.

Claudio Castellani and Andrey Chubukov both asked what happens when graphene
is tuned very close to the Dirac point, i.e. filling fraction is small. The
system is right away in the strong disorder limit, but Konstantin believes the
conductivity still goes to zero at T=0 since the states are almost localized
already.

John Mydosh asked what are the potential impurities, and apparently these are
mostly charged impurities on the substrate.

Victor Galitskii asked what determines the crossover scale between
antilocalization and localization. There is no unique answer, this is related
to how different components of the impurity scattering vary with T.

Catherine Pepin asked whether interaction correction have been considered and
what they do. There seems to be some work done on that, but lunch truck was
about to arrive, and we stopped.