Detection of geometric phases in superconducting nanocircuits

by Giuseppe Falci, Rosario Fazio, G. Massimo Palma, Jens Siewert and Vlatko Vedral

Journal club talk by Nakyeon Hwang
Summary by Patrick Morales



This week's talk, given by Nakyeon, was on a proposed experimental setup to detect geometrical phases in a superconducting device. The proposed device is an asymmetric SQUID where the thickness of the tunneling barrier is different in the two arms of the SQUID. The quantum interference due to the geometrical phases could be detected by measuring the charge state of the asymmetric SQUID as the Hamiltonian is adiabatically evolved by varying the offset charge and the flux through the SQUID in a cyclical fashion. Quantum interferometery based on these geometric phases can be used to develop a new design of gates for quantum computation using charge qubits.

When a quantum mechanical system, is evolved adiabatically such that the phase and the amplitude of the wave function describing the system is varied in a cyclical manner, the resulting wave function may differ from the original wave function by a phase factor. This geometrical phase, or Berry phase, results from the geometrical properties of the parameter space of the Hamiltonian. One example of this is the Aharonov-Bohm phase picked up from a charged particle encircling a magnetic field. The Aharonov-Bohm effect results in a periodic modulation of the critical current of a conventional SQUID as a function of magnetic field and provides evidence of macroscopic phase coherence of the superconducting condensate.

The Hamiltonian of an asymmetric SQUID operating in the charging regime, where the temperature is lower than the Josephson coupling energies of the junctions which in turn is much smaller than the charging energy of the SQUID contains two terms: one related to the charge of the SQUID and the other to the flux through it. The Hamiltonian can be swept through its phase space by applying a voltage across the SQUID, varying the charge of the SQUID, and by applying a perpendicular magnetic field to vary the flux through the SQUID. A non-trivial loop in phase space produces a Berry phase due to the asymmetry of the SQUID. The dynamical component of the phase can be subtracted by inverting the state of the SQUID and retracing the same path in phase space in the reverse direction. The resulting phase difference is due only to the geometrical component of the phase.

Ultimately, the geometric phase in an asymmetric SQUID could be used in the design of gates for quantum computation. Two capacitively coupled asymmetric SQUIDS could be used to create a universal two qubit gate. The gate voltages and the magnetic fluxes of each SQUID could be set independently. The effective charging of the target qubit would depend on the state of the control qubit, resulting in a controllable phase shift of the target qubit.

Quantum critical behaviour in the superfluid density of strongly underdoped ultrathin copper oxide films

by Iulian Hetel, Thomas R. Lemberger & Mohit Randeria

Journal club talk by Ganesh Ramachandran
Summary by Brandon Ramakko



This week's talk by Ganesh focused on experimental results for thin films of underdoped YBCO. To begin the discussion, he briefly explained the temperature-doping phase diagram for this HTSC. The transition in the overdoped regime of the superconducting dome can be explained by a mean-field transition but it is not clear what causes the transtion in the underdoped regime. This paper aims to explain the transition in this regime using their experimental results.

The speaker then discussed a plot of superfluid density versus temperature for Helium films. The superfluid density is constant and drops discontinuously to zero at Tc. The size of this drop increases linearly. This is characteristic of a 2D Kosterlitz-Thouless-Berezinski transition. So if the plot of superfluid density for YBCO has the same bahaviour, you could conclude that it is indeed a 2D Kosterlitz-Thouless-Berezinski transition.

The speaker went on to explain the two-coil mutual-inductance method used by the authors. The sample is placed between two coils and an AC current running through the first coil produces a flux which induces an EMF in the sample which drives a current. The flux due to the current in the sample goes through the second coil and by measuring the voltage in the second coil you can calculate the conductivity. Using London's equations you can get the superfluid density.

The speaker then discusses the experiental plot of superfluid density versus temperature. There is no discontinuous drop as expected in a Kosterlitz-Thouless-Berezinski transition. The drop is probably smoothed out due to complexities such as the measurement being taken at a frequency of 50 kHz, inhomogeneities, vortex pinning and possible new physics. A line was added that seems to go through each curve where the density starts to decrease quickly. This seemed to be consistent over a range of doping values. The Tc values were determined from this fit.

It was asked how thick the layers used were and the speaker replied that the thinnest sample was 2 layers thick. It was also asked why the behaviour of the density for T less than Tc was quadratic when it should be linear. This difference in behaviour might be due to disorder or the frequency of 50 kHz (intead of 0) being used.

The speaker then discussed a plot of Tc versus superfluid density(T=0). The measurements show a linear relationship consistent with a 2D Kosterlitz-Thouless-Berezinski transition for small doping or low temperature. It was asked at what temperature it stops behaving linearly. The speaker aswered that at 10K, Tc behaves like the square root of the density. Josephson scaling near a quantum critical point (QCP) implies that the critical point in the underdoped regime is a 2D QCP. Since the transition is mediated by vortex-antivortex pairs, if these could be suppressed you could have a room temperature superconductor.

Patrick Morales commented on the difficulty of the experiment. Having attemped similar experiments he said that the split coil experiment on a thin film is difficult because you have weak signals with large noise. You need a large homogeneous sample. He mentioned it is extremely difficult to get a nice uniform thin sample because YBCO does not grow well in the underdoped regime and it is very hard to remove oxygen in a homogeneous manner.

Dynamics of a Quantum Phase Transition in a Ferromagnetic Bose-Einstein Condensate

by Bogdan Damski and Wojciech H. Zurek

Journal club talk by Edward Taylor
Summary by Ganesh Ramachandran



Ed chose an interesting paper studying dynamics through a quantum phase transition in a spinor condensate. The discussion brought out some general ideas/features of the dynamics of quantum phase transitions.

Ed gave a lightning introduction to spinor condensates and wrote down the energy functional given in the paper. To put it in context, he described previous experimental study of the ferromagnetic-polar transition by Sadler et al.The phase diagram is known to have ferromagnetic, polar and normal(non-condensed) phases. Experiments have probed the dynamics as well as the detailed domain structure formed. The phase diagram is known to have ferromagnetic, polar and normal(non-condensed) phases.

The Hamiltonian possesses three Bogoliubov modes. In the polar phase, one is gapless, corresponding to the broken U(1) symmetry of the BEC. In the ferromagnet, broken U(1) and rotational symmetries give 2 gapless modes. The only energy scale then, is $\Delta$, the excitation gap of the third mode. As we move away from the critical point where even this mode is gapless(?), $\Delta$ rises from zero and eventually saturates. This gives us two time scales which should determine the dynamics - the relaxation time $1/\Delta$ and the transition time taken until saturation.

Tuning the rate of increase of magnetic field, we can explore an impulse regime and an adiabatic regime depending on which time scale dominates. The crossover between regimes occurs where the time scales are equal, which should scale as the one-third power of the 'quench time'. The numerical calculations in the paper do give this precise scaling.

Ed drew a typical plot of the z-magnetization as a function of changing magnetic field or time. The plot showed that the order parameter moved away from zero toward the expected equilibrium value, only after a delay. This was identified as the crossover between impulse and adiabatic regimes. The delay thus read off, showed the expected 1/3 scaling except for values close to zero.

Ed wound up with a neat quick summary, only to make way for a brisk discussion. Igor's question prompted a discussion on domain formation. Drawing a parallel to the early universe where the size of structures was limited by the speed of light, Ed brought out that the size of domains was given by the speed of sound. With two soundlike modes, for some reason, it is the slower mode velocity that plays a role. Kibble-Zurek theory predicts a 1/3 scaling for domain size, which has been observed in this paper.

There was a question from Michael which brought out that the above considerations only hold for intermediate time scales, where the transition is non-adiabatic, but slow enough. Michael also pointed out that it would be interesting to explore the normal region just above the critical point.

A Spin Triplet supercurrent though the half-metallic ferromagnet CrO2

by R. S. Keizer, S. T. B. Goennenwein, T. M. Klapwijk, G. Miao, G. Xiao and A. Gupta

Journal club talk by Patrick Morales
Summary by Fazel Fallah Tafti



Superconductivity and ferromagnetism are competing ordered states
of matter but it appears that they can coexist. In this article we
learned about another example of SC in a FM material which exists in
the half-metallic material CrO2.

The experiment is realized by growing a single crystal thin film of
CrO2 on TiO2 substrate (epitaxial growth) and patterning two
superconducting islands made of NbTiN on top it using organic resist
mask with conventional electron beam lithography.

NbTiN is a conventional S-wave BCS superconductor and CrO2 is a
spin polarized half metallic ferromagnet i.e. it has a finite DOS at
the Fermi level for the up spin polarization but it shows a gap of ~
2eV in the down spin polarization.

The supercurrent tunnels through the Josephson junction
NTN-CrO2-NTN which is shown in IV characteristic curve. Fraunhofer
pattern is observed for Ic as a function of induced magnetic field
which is regarded as another evidence of supercurrent tunneling
through the FM material.

The distance between the two NTN leads is about 300nm which is much
higher than the coherence length expected for a FM metal. This is even
higher than the coherence length in most normal metals (~100 nm) which
is surprising.

Such persistence of a supercurrent through a FM half-metal can be
justified by the idea of spin triplet cooper pairs. In this picture we
can still have the main features of the BCS theory but the pairing
mechanism in unconventional such that instead of singlet spin pairs we
have a pair made of two electrons with parallel spins.

The question to be answered is that what are is the mechanism
behind singlet-triplet conversion at the junction between the normal
s-wave SC and CrO2. In fact there are a number of proposed mechanisms
for such phenomenon but they all agree on the fact that such process
has to be caused by an interplay between the exchange field of the FM
at the FM-SC junction and a non-homogeneous magnetic field across the
junction. In order to make a correct theory to describe such process
one has to include various important factors such as the effect of
domain walls at the interface, spin-orbit interactions in the SC,
Double exchange interaction in CrO2 and perhaps the effect of local
magnetic impurities.

Local Tunneling Spectroscopy across a Metamagnetic Critical Point in the Bi-layer Ruthenate Sr3Ru2O7

by K. Iwaya, S. Satow, T. Hanaguri, N. Shannon, Y. Yoshida, S. I. Ikeda, J. P. He, Y. Kaneko, Y. Tokura, T. Yamada and H. Takagi

Journal Club talk by Hyeonjin Doh (see also his notes !)
Summary below by Igor Fridman



This week's talk by Hyeonjin focused on experimental results of tunneling experiments on the compound Sr3Ru2O7. This compound has garnered recent interest due to possible presence quantum criticalities. The paper discussed presents the first available spectroscopic measurements on this compound using Scanning Tunneling Spectroscopy (STS).

To begin the discussion, Hyeonjin illustrated the physical properties of Sr3Ru2O7. The crystaline structure the compound is described by the A(n+1)B(n)O(3n+1) group, where n is the number of layers. The compound under discussion, with n=2 layers, is an intermediate between Sr2RuO3 (n=1) and SrRuO3 (n=inf). The related compounds are known in their ground states to be a spin triplet superconductor and an itinerant ferromagnet, respectively.

The ground state of Sr3Ru2O7 is a paramagnetic Fermi liquid, with most of the electronic properties coming from the Ru4+ ions. Unlike the n=inf compound, this bi-layer compound shows no evidence of an FM transition. There is, however, a meta-magnetic transition at 5.5T, applied parralel to the ab plane, between states of low and high magnetization. This transition was observed previously in measurements of susceptibility, and also seen in the tunneling data. Another transition occurs when ia field between 7 and 8T is applied parallel to the c axis. In the present tunneling experiment, field was applied parallel to the c axis, and no transitions were observed at 7T and 8T. However, Hyeonjin pointed out that this could be due to the fact that crystals used in the previous study were much cleaner, having a residual resisitivity of Rho_0 ~0.4uOhm-cm, as opposed to the crystals used for this study with Rho_0 ~7uOhm-cm.

The spectroscopy measurements revealed two peaks in the DOS close to the Fermi level, at ~ +/- 7 meV, as measured in zero-field at T = 560 mK. Hyeonjin suggested that this is consistent with the Stoner picture. Indeed, the authors calculate that spectroscopic evidence of Stoner-type metamagnetism should show up on an energy scale of ~1 meV at these temperatures. The authors also collected data over a range of fields from 0 to 11T, and found a change in amplitudes of the DOS peaks with increasing field.

The authors analyze their data by looking at the amplitudes of various spectroscipic peaks as a function of magnetic field. The first effect is a rapid increase of the DOS at the Fermi level above the critical field. This confirms the c-axis metamagnetic transition. The second effect is a shift of the spectral weight from the lower energy peaks at 2 and -1 meV to the slightly higher energies of 4 and -3 meV above the critical field. Thus, there is a change in the field dependency of the DOS at the transition, which is evidence that the transition is a quantum critical point.

Putting the magnetic properties aside, Hyeonjin also discussed an apparent DOS modulation seen in the STM images of the cleaved compound. By changing the Fermi level between 7 and 100 meV, the authors were able to image two different geometries on the surface. At low voltage bias, the authors show an apparent asymmetry in the DOS between neighboring Ru atoms. This is unexpected, since electrons should not see any difference between neighboring sites according to the present model. This modulation gave the authors a reason to believe that the orbital degrees of freedom might have an important role in this cleaved sample, but they are not sure this kind of modulation would survive in bulke. Thus far, this effect is not fully understood.

Dimer-Quadrupolar Quantum Phase Transition in the Quasi-1D Heisenberg model with Biquadratic Interaction

by Kenji Harada, Naoki Kawashima and Matthias Troyer

Journal club talk by Christoph Puetter
Summary below by Thomas Grzesiak



Christoph Puetter introduced the paper "Dimer-Quadrupolar Quantum Phase Transition in the Quasi-1D Heisenberg model with Biquadratic Interaction" by Harada, Kawashima, and Troyer. He explained that the concept of a deconfined quantum critical point (DQCP) is only a few years old, and people are still looking for models which exhibit this kind of phase transition. This paper investigates 2 critical points in the phase diagram of the (anisotropic) bilinear biquadratic Heisenberg model with Jl the bilinear coupling strength and Jq for the biquadratic term. For |Jl|>> |Jq|, we get the familiar ferromagnetic
or Neel state depending on the sign of Jl. When the couplings are comparable in strength, the situation becomes more complex.

To illustrate this point, Christoph began by considering 2 isolated sites and showed that a singlet is preferred for Jl < Jq < 0. He explained that once more sites are included the singlet condition can no longer be satisfied on bonds sharing the same site. The system therefore develops new phases such as the dimer phase (for &lambda << 1) and the spin nematic. (lambda is the anisotropy parameter in the 2d Heisenberg model considered here).

The authors of the paper performed MC simulations for various lattice sizes L, and measured the Quadrupole-Quadrupole (for nematic) and Dimer-Dimer correlation functions, Gq and Gd. They examined the ratio G(L/2)/G(L/4). Tuning lambda they found that these ratios (Rq and Rd) become independent of L at lambda_critical, signalling the phase transition.

It was asked why the above ratio was chosen. Daniel explained this was for the correlation length, to get a constant alpha independent of L (a power law). Christoph then focused on a special point on the nematic-dimer phase boundary (Y in the paper). He explained that with spin-rotational symmetry preserved and translational symmetry broken in the dimer phase, and vice versa in the nematic, a first order transition is expected. It is also expected that Rq and Rd should be 1 (why?), and that there is a cusp in the energy at the transition. However, the fact that this was not observed could be due to a DQCP, although other possibilities cannot be ruled out.

It was asked why DQCP doesn't appear from the Neel side (point X in the paper)? If there are 2 ordered states with no relationship between the orderings, there should be a first order transition, according to Landau, yet this paper found a second order transition. The talk ended with another question, as to what a DQCP actually is (in a nutshell). Michael gave an example: in a magnetic phase has spin waves that are spin 1 excitations in a spin 1/2 system. They can split into 2 spin 1/2 excitations at a DQCP, ie they become deconfined. For the dimer phase, spins are bound into singlet states. At a DQCP, they would become deconfined, breaking the singlet.

Metallic Spin-Liquid Behavior of the Geometrically Frustrated Kondo Lattice Pr2Ir2O7

by S. Nakatsuji, Y. Machida, Y. Maeno, T. Tayama, T. Sakakibara, J. van Duijn, L. Balicas, J. N. Millican, R. T. Macaluso, and Julia Y. Chan

Journal club talk by Daniel Podolsky
Summary below by Jean-Sebastien Bernier



For the first CMP journal club of the fall semester, our discussion leader was Daniel Podolsky who presented a paper by Nakatsuji and coworkers on the possible observation of metallic spin liquid behavior in the geometrically frustrated Kondo lattice Pr$_2$Ir$_2$O$_7$.

Daniel began by presenting the material under study. He pointed out that the conduction electrons come from Iridium (Ir$^{4+}$) and that the local moments (Pr$^{3+}$) occupy the sites of a pyrochlore lattice (network of corner sharing tetrahedra). He noted that due to crystal field splitting the spins of the local moments are effectively Ising-like and point along an axis passing through the center of each tetrahedron. Daniel also explained that the Ising nature of the local moments makes this system unfrustrated if the RKKY interaction is antiferromagnetic. An AF RKKY interaction seems to be indicated by the large temperature susceptibility data. However, if the system is unfrustrated, it is difficult to understand why no long-range-order develops at the Curie-Weiss temperature.

Someone then asked how can we be sure that there is RKKY interaction in this system and not only superexchange. Daniel replied that the paper points out that $T^* = -20K$ is much higher than what should be expected from superexchange.

Daniel continued by presenting susceptibility data. He highlighted the unusual $\chi ~ \ln T$ behavior at low temperature and that no anomalies due to a magnetic transition were detected in the susceptibility except for freezing at $T_f = 120 mK$.

He then presented the resistivity data that shows the usual Kondo minimum and pointed out that $T_{Kondo} = 25 K$ is very large for such a poor metal.

Someone then asked if this material was also a poor metal at high temperatures. Daniel replied that he did not know.

Moving on, he presented that for $T_f = 120 mK < T < T_{CW} = 1.7 K$, the entropy and specific heat vary as $\sqrt{T}$ which leads to a very large entropy.

Someone then asked if $S(0)$ is known. Daniel answered that only $\Delta S$ was presented in this paper.

Finally, Daniel presented the global phase diagram put forward by the authors:

$T_f < T < T_{CW}$: metallic spin-liquid with $S \sim C \sim \sqrt{T}$
$T_{CW} < T < |T^*|$: Kondo screening, but incomplete since susceptibility diverges
$T > |T^*|$: decoupled Ir and Pr

... and he noted that the presence of a metallic spin-liquid did not sound too convincing to him.

Someone asked if any instability of the conduction electrons were observed. Daniel answered that there is no evidence in the data for such a phase transition.