Markus Garst (Köln) -- Multiscale quantum criticality: Nematic instability in metals

References for what Markus will talk about:
M. Zacharias, P. Wölfle and M. Garst, PRB 80, 165116 (2009)
M. Garst and A. Chubukov, PRB 81, 235105 (2010)


Markus began by apologizing to the experimentalists for giving a (possibly) technical theoretical talk, but there was no mass exit from the lecture theatre so all should be OK.

Introduction: Quantum Phase Transitions (QPT) and multiple scales

A 2nd order QPT is an instability in the ground state at T=0, as a function of some control parameter we will call r (e.g. magnetic field, pressure, doping, etc...) While this transition happens strictly only at T=0, it has a strong effect on finite-T properties due to an abundance of low-energy fluctuations. So on phase diagram, should have T too - region in phase space where properties controlled by Quantum Critical Point (QCP)

Some technical things:
--Correlation length exponent \nu : \xi \sim |r|^{-\nu} (\xi = correlation length, r=control parameter)
--dynamical exponent z : spectrum of critical fluctuations goes as \omega \sim k^z which gives a vanishing characteristic energy scale \epsilon\sim\xi^-z
--enhanced dimensionality: correlation volume in space: \xi^d and time \xi^z gives you an effective dimensionality d_eff = d + z

This set of exponents will give you a scaling ansatz of the critical free energy
F(r,T) = b^{-(d+z)} F ( r b^{1/\nu}, T b^z)
for some arbitrary scaling variable b.
Such scaling behavior are widely used to interpret a wide range of experiments where you don't know exactly the microscopic details of what happens, but the scaling may still work.

Quantum-to-classical crossover: we have the correlation volume in time \xi^z, but this is limited by the temperature \xi^z < t="0),">0). However, the flow is such that for T>0, a crossover temperature (\xi_T) may be defined where the flow leaves the T=0 path to divert to the classical critical point.
(see e.g. Nelson 1975, Millis, 1993)

As a brief summary: the relation between the thermal length \xi_T and correlation length, \xi gives us a crossover in the phase diagram.

In many systems, there is a coexistence of low-energy fluctuations.
For example, near a magnetic instability, have critical magnetic fluctuations as well as ballistic electrons - so can find sometimes two dynamical exponents z_1 and z_2.

With two dynamical exponents, this raises a lot of points:
--coexistence and interacting fluctuations -> entanglement!
--identification of proper critical degrees of freedom?
--two dynamical exponents-> breakdown of scaling and power laws?
--crossovers in phase diagram
--fluctuation driven first order transitions
Markus tells us that these are the `big questions' in this topic, many of which are unanswered. The remainder of this talk will be about a simple specific case.

Nematic Instability of Fermi liquid

Pomeranchuk instability: instability of FS towards development of a quadropole moment
Order parameter is a tensor object, similar to nematics. The traceless part of the strain tensor gives us the shear modes of the Fermi surface. In d=2, there are two shear modes.

We will look at a simple model Hamiltonian introduced by Oganesyan, Kivelson and Fradkin (2001). They developed an effective bosonic model which acts as the Ginzburg-Landau theory in the usual way by Hubbard-Stratonovich and integrating out fermions. More complicated than regular GL due to tensor order parameter, but otherwise standard. We note that in d=2, no cubic term is allowed, which means a 2nd order transition is possible.

OKF analyzed this model, finding a Pomeranchuk instability, with a criterion basically identical to the Stoner criterion as a function of some control parameter r depending on the density of states and the coupling constant (a control parameter Piers referred to as pretentious, but this story can wait for a rainy day...)

Markus then says the most interesting point about the instability is the dynamics - which he will now discuss.

One can have polarization of the excitations both longitudinal and transverse to the quadrupolar momentum q tensor - but because of different phase spaces for exciting particle hole pairs, one finds
--longitudinal polarization, Landau damping z=3 dynamics
--transverse polarization, ballistic z=2 dynamics.
This is the multi-scale property that was introduced early.

Two energy scales, two modes. Which mode is more important?

Naive answer: longitudinal z=3 mode has larger phase space: \Omega_n \sim q^z, so dominates the specific heat. This led OKF to claim that the z=2 mode plays no role in the critical theory.

However in reality, things are much more interesting. Transverse d=2 mode has smaller effective dimension d+z=4 - so generates logarithmic singularities in loop corrections, and it is the interplay of both modes that determines the critical properties.


We now become a bit more technical and look at the structure of the theory at T=0.
The transverse z=2 fluctuations allows us to write a logarithmic RG flow, giving mass renormalization. This flow is marginally irrelevant, but introduces a logarithmic scale dependence of the correlation length
\xi^{-2}(\epsilon) \sim r / [log(1/\epsilon)]^{4/9} for \epsilon>\xi^{-2} (z=2 energy scale)
This exponent 4/9 differs from Ising and XY universality, and is characteristic for Pomeranchuk.

What about theory at finite temperature?
multiple z -> multiple thermal lengths (\xi_T \sim T^{-1/z} )

For each thermal length, there is a quantum to classical crossover - so can have coexistence of quantum and classical fluctuations because of the multi-scale dynamics. (This is Markus' answer to Piers' question at the beginning).

This overlap regime in fact controls the thermodynamics in a wide range of the phase diagram.

Lets look at this again, in terms of RG: on the RG phase diagram, one mode wants to push the system away from the primary quantum critical point, while the other competes with it pushing in the other direction; and it is this wide range of scales that is important for thermodynamics in a wide range of the phase diagram.

Now look at the temperature boost of the RG flow -> some technical calculation, but the answer ultimately is that there is a universal correlation length at criticality (r=0):
\xi^{-2} = cT
which is universal in that it is generated, but independent of, the bare quartic coupling, u. This feature is entirely due to the competition of these different z=2 and z=3 modes.
In other words: multi-scale dynamics leads to a new kind of universality.

These multiple scales are also present in the phase diagram:
two thermal lengths -> two crossover lines in the phase diagram.
The sensitivity of something on the crossover depends strongly on specific thermodynamic quantity that you are looking at. Means that there is no scaling in terms of a single dynamical exponent; things are just a bit more complicated.

Markus then briefly shows us the results for a few thermodynamic quantities (specific heat, etc..) , which all have `funny' logs in them.


Electron spectral function

One-loop self-energy from z=3 mode (see Oganesyan Kivelson and Fradkin, 2001; Metzner, Rohe and Andergassen 2003, etc...) gives singular correction.
Also look at one-loop self-energy from the transverse z=2 mode; which gives interesting contribution to off-mass shell part, and singular correction to Z.

Sum up more logs, RG, ... find that the combination \Gamma Z = 1 is invariant. Hence the polarizations are unaffected by the electrons, and the critical dynamics are preserved.

Question (Piers): Is this a Ward identity? Answer: Not strictly speaking (but unfortunately I missed why - the slides are getting mode and more technical and difficult to blog...).

Hence the electron propagator has three important parts:
-- non-Fermi liquid frequency dependence at z=3 energy scale
--anomalous dimension at z=1 energy scale (z=1 is electrons)
--interesting correlation length dependence at z=2 energy scale.

This is still no the full story though, as there are further logarithms, including some that appear only in 3rd loop order (see Mross, McGreevy, Liu and Senthil, 2010)

Summary

--Nematic quantum criticality in metals: multiple energy scales
--extended quantum-to -classical crossover where quantum and classical fluctuations coexist and interact, and may lead to new forms of universality.
--In thermodynamics, all energy scales can appear, depending on quantity in question

Questions:

Q) Does this theory obey sum rules?
A) It should, no good reason why it shouldn't although must take into account all correct crossover scales to make them work. Chubukov extended this by commenting that this is a low energy calculation, high energy modes will adjust to make sum rules work

Q (Nevidomsky) In many cases, longitudinal modes don't couple to things, is that also true here?
A) (mostly given by Chubukov) - longitudinal and transverse for this quadropolar order are a different notation to what we are used to - should be careful trying to draw analogies

Q) In which (real) materials might you expect this to occur?
A) Crystal lattice may make big differences in this theory e.g. z=2 mode may become gapped (did I hear that correctly?), so at the moment, this work is without reference to real materials.

Q) (missed)

Q (Schofield) In the Stoner criterion, the Pomeranchuk instability condensed around q=0, rather like a ferro-magnet. If the condensation was about finite q (like SDW), would the structure of the theory change?
A) Yes, dramatically. No z=3 mode, lots of other stuff

Comment (Nersesyan) There are also cases where this multi-scale criticality can arise, even without different z's (e.g. in spin-charge separation in Luttinger-Liquid)
Answer: Absolutely true, although these sorts of single z cases are somewhat simpler, as you know how to rescale momenta, etc... Lots of new stuff when more than one z present.

Satoru NAKATSUJI: Quantum criticality and spin liquid in Kondo Lattices (YbAlB4, Pr2Ir2O7)

Satoru begins by presenting the classic Doniach type phase diagram (of course in a its more "modern" form) showing a quantum critical point between the magnetism and Fermi liquid. The question is whether the magnetic order can be suppressed using geometric frustration for the f-electron system. This could reveal a quantum spin-liquid. Here the case to be presented is Pr2Ir2O7. The alternative is to use the itinerant electrons to weaken the order (perhaps by mediating a feromagnetic interaction) and this is the case in YbAlB4 where a chiral spin-liquid is stabilized.

Time reversal symmetry is fundamental in physics and can be broken say with magnetic dipole order. It can also be broken without dipolar order via a chiral spin state (where there is a net S.S x S around a plaquette) as proposed in the curpates in the distant past. It is difficult to detect this order, but the Hall effect may provide a signal. The anomalous Hall effect (AHE) in a ferromagnet would be an example (with a contribution proportional to the magnetization). However a chiral state would also give a contribution to the Hall effect because the Berry phase accumulates as electrons move in the presence of chirality. Here you get a Hall effect without M or B.

Pr2Ir2O7: (S. Nakatsuji, PRL 96, 2006) has Pr3+ 4f2 localized Ising moments in a pyrochlore lattice of corner sharing tetrahedra. It is highly frustrated with no order down to 0.3K much less than the Heisenberg scale of 2K. The Ir4+ 5d5 provide the conduction electrons. Piers asks about the crystal fields and in response to Satoru's comment, Paul Canfield: asks about the point symmetry and worries about the declared doublet groundstate since this is non Kramers. Satoru parks the question about the evidence for a magnetic doublet. The pyrochlore lattice has the ice rules physics (2 moments in, 2 moments out on each tetrahedra along the local 111; direction of a tetrahedra) with the residual entropy classically. Evidence that this is the case in the Pr2Ir2O7 is the magnetic anisotropy which is consistent and neutrons. There is also a metamagnetic transition only for B fields along 111 which is when the system switches to a 3in 1 out state. S-W Cheong: says why is there no magnetization plateaux then? The numbers seems to match expectations. The claim is that there are ferromagnetic correlations with a Jff of 1.4K as seen as a peak in the specific heat. Chi3 has a steep negative increase and saturates to a large negative value. Andriy: why negative? The explanation comes in the form of the M(H) curve being convex not concave (which seems like a restatement of the facts to me). Normally you would expect spin freezing at around 1.4K but this does not happen.

Below 1.5K there is an enhancement and hysteresis in rho_xy between zero field cold and field cooled. Yet muSR shows no freezing of the moments down to 20mK. Field is along [111] and current along [110]. There is also a remnant Hall effect when B is returned to zero, though no evidence in the magnetization of a net moment. So this points to a spontaneous breaking of time reversal symmetry (TRS) at 1.5K. S-W Cheong as a question and demands to see the next slide(!). He means previous of course to much amusement. Is it really a zero field state on domain related state? Answer: domains are being alligned which need to be trained by the field. Paul asks about a specific heat signature. Answer: there is a peak but no jump.
There is anisotropy in this hysteresis effect: largest with B//[111] and smallest with B//[100]. So the 3-in, 1-out state may be stabalized not only at B>B_c but also may have some overlap with the groundstate.

Zero field quantum criticality in beta YbAlB4

We now turn to the second material which may show a spin-liquid groundstate. In beta YbAlB4 the Yb ions are organized into a honeycomb lattice which is slightly stretched and the B ions form a lattice of 5 and 7 fold coordinated rings. Here the ordered magnetism is suppressed apparently by the enhanced hybridization of the local moments with the conduction electron sea. Hard X-ray XPS reveals that this is ain intermediate valence compound with Yb averaging 2.75 rather than either 3+ or 2+.

Application of pressure enhances the hybrization as seen in the resistivities. The evidence for quantum criticality comes from the magnetic Gruniesen parameter which diverges but with two distinct power-laws. T^{-1.5} for 0.4K < t="0.">300). The critical field is quite anisotropic and seems to be paramagnetically limited. SdH oscillations are seen '(mean free path of order 1 micron) anbd show a 3D Fermi surface, and a mass of 30m. It is a combination of 2D cylinders and 1D sheets. A surprising thing is that X-ray PES shows that Yb3+ and Yb2+ coexists (roughly 2.75) . This mixed valence would normally yield a Pauli paramgnetic suscpetibility because of the screening/itinerancy. However in this material you actually see shows 1/T. Paul says, no it does not look like 1/T. Coleman says: is there one or two regions of Curie-Weiss. Answer: there are two. Resistivity exponent (rho ~ T^2) colour maps suggest a quantum critical point at B=0. Moreover it is a rare example of M diverging as a quantum critical point is approached. With interesting power laws. Rafael Fernandes asks is there something in the specific heat? Answer: yes but wait.... The interesting thing is that there is scaling of the magnetization (dM/dT) at B< f="B^\alpha" b="0". Oh no... it looks like my attempt to put in the scaling form as deleted the rest of my blog....help. Is there anyway of recovering it? Here is my attempt to recall the other later aspects of the blog and the questions.

Yuri Grin and Satoru Nakatsuji discusss alpha and beta YAlB4

There was a nice experimental contrast with the alpha phase of this material which differs by having a chequerboard arrangement of the distortions in the honeycomb lattice. It seems to be a Fermi liquid. So the claim is that this (ie the beta material) system is right on the border (within 1 gauss) of a quantum critical point. This seems an unlikely fine tuning so Satoru offered an argument that this is part of a quantum critical phase where magnetism has become a spin liquid. The mechanism was argued to be ferromagnetic interactions effectively frustrating the system.
Question time (apologies to those I have not remembered)
Amy Briffa and Rafael Fernandes: both asked about the evidence for the first material being chiral as opposed to a ferromagnet. The fact that there is no hysteresis in the magnetization but only in the Hall precludes ferromagnetism. There is no microscopic signature yet of the chirality - just the Hall anomaly.
Juri and Thamizhavel: both asked about the growth conditions of the alpha and beta phases. They can be discriminated by colour and morphology from the flux. However, why such similar materials emerge in pure form from a flux did not seem to be understood.

Val Rosandra

Sunday - a walking trip towards Slovenia with Sam Carr as a trusty local guide! When I have worked out how to do it I will post the walk using google earth/maps. Here it is:

View Larger Map

Aquileia and Grado

Saturday sees some of the conference participants off to Aquileia and Grado. Aquileia was once a Roman city of 100,000 people - though now is a shadow of its former glory at a mere 12000. The old forum is an evocative ruin though the white pavement can still be seen. The ruins have been recycled into later buildings. The city boasts an impressive Basilica whose present building dates from the 11th century but within is a mosaic found below the 11th century floor level which is 4th century.

While our guide explained the history, your faithful blogger made an astonishing observation - the Romans had already discovered the Iron superconductors and have published, in the mosaic, the phase diagram! Two features will be of immediate interest. First there is clear evidence of coexistence of magnetism and superconductivity. Second, there is a mysterious "hidden order" phase which lies on the other side of the magnetism and the authors have labelled with a rooster. Clearly much more still be be discovered in that system.

Then on to the seas side resort of Grado. Here we explored the old fishermen's houses before venturing onto the crowded beaches. A great day out!

Girsh Blumberg (Rutgers): Raman spectroscopy of multiband superconductors with competing order parameters

Girsh started by a remark that his talk will be devoted to the many-body aspects of multiband superconductivity as seen from Raman spectroscopy.

First example given is a Raman study of layered system, NbSe_2, by Sooryakumar & Klein, PRL 45, 660 (1980), the material with coexisting CDW (T_CDW ~ 9K) and (multiband) superconductivity (T_c ~ 2K). The first challenge for Raman spectroscopy was to keep the material below the sc transition temperature as laser heats the sample [remark about the Burch's talk earlier today].
On the theory side the story began with the study by Abrikosov and Falkovskii in 1961.

What is interesting on the side of multiband superconductors? Following Littlewood and Varma in 1982 there are two possible modes in a usual superconductor: amplitude mode (amplitudon) and the phase (Bogolyubov-Anderson ) mode. In a multiband SC there is an additional Legget's mode associated with the phase difference of the sc gaps on different bands.

Next Girsh has reviewed the classical work by Klein and Dierker, PRB 1984. In the normal state
the only electronic excitations Raman spectroscopy is able to see is a particle-hole continuum with a small (almost zero) momentum transfer. In a superconductor due to renormalization of the electron bands the continuum will be gaped up to a twice of the sc gap. Observed by Hackl (at a time graduate student) in V_3Si for the first time. HOWEVER: this is all for non-interacting electrons: in case the residual interaction is involved the peak at 2\Delta could be shifted to the lower energies and whether it happens or not depends on the symmetry of the OP, Raman vertices and so on. Special example is a long-range Coulomb interaction. The excitation driven by light are electron charge densities which should be screened by the long-range Coulomb interaction especially in the fully symmetric A_1g channel.

Then he moved to cuprates, and has shown experimental examples of Raman scattering in the various (B_2g, B_1g, and A_1g) symmetry channels. Due to the fact that Raman vertices are momentum dependent and have maximums and minimums at the various parts of the Fermi surface you realize that B_1g probes mostly quasiparticles around the M point of the BZ, while in B_2g you taking mostly the excitations around the diagonal of the BZ. Naturally if the gap is d-wave the largest effect comes in the B_1g channel [few curved from YBCO]. Then he moved to the story of interaction and reminded about the work by Chubukov and Deveraux on the final-state interaction effects in the B_1g channel (below T_c). Comments by Chubukov: first work is with Girsh. Comment by Nevidomskii yielded the understanding among the audience that what we hear today is only the electronic Raman scattering. [no two magnon, no phonons]. Hirschfied pointed out that in the data the peak is hardly distinguishable from usual 2\Delta peak, Chubukov replied that the story actually is that the peak in Raman is BELOW 2\Delta and the latter is measured by other techniques and this is what effect of the interaction is: shifting the peak to energies below 2\Delta.

Then Girsh turned to the electron-doped cuprates (PCCO) and the story of electron and hole pockets there. Blogger is busy typing but notices that the story itself is quite interesting and for non-expert probably heavy to follow with one slide that Girsh has shown. Anyway if you accept that there are pockets you understand the asymmetry of Raman spectroscopy in electron and hole-doped cuprate superconductors.

Ok now we are back at the Legget's mode as Girsh comes up with the paper by Klein, PRB 2010 where he
analyzes Raman in MgB_2 (by now classical two-band superconductor). By looking in E_2g symmetry we notice two separate 2\Delta structures associated with pair-breaking of the correspodning gaps [temperature dependece, coupling to phonons]. Now if you look at the A_1g you see an extra (unexpected) feature at 9.4 meV which, as Girsh argues, is a collective mode effect. To understand this you look into the theory of a two-band superconductors:
For two order parameters the phases of the order parameters are arbitrary but if there is an interaction you can have in phase locking for an attractive interaction (V_interband>0) and out-of-phase locking of the phases for the repulsive interaction (V_interband<0). Klein pointed out that in MgB_2 the interband interaction is large and positive. The former shifts the collective mode associated with this phase locking to higher energies (into continuum) which yields damping.

Now iron-based supercondctors (from 2 to 5 bands). The idea about collective mode goes back to Chubukov, Eremin, Korshunov, PRB79 (2009) [thanks Girsh!]. They story is that if the symmetry is extended s-wave and these are electron and hole pockets involved then the effect of the final state interaction will be to create a collective mode below 2\Delta. Now the apparent contradiction [after discussion of Chubukov, Hirshfield, Fernandes, and Nevidmoskii, blogger also liked to contribute but had to continue typing] was that the collective modes Girsh has introduced earlier had nothing to do with the collective mode in B_1g in cuprates and in the A_1g channel of iron-based superconductors which he discussed now. These are all density-density excitations (excitons) shifted to lower energies due to interference of the the gap and Raman vertices. So these are neither Leggett's or amplitudon modes.

Finally in the last few minutes Girsh comes back to NbSe_2. He discussed some STM and ARPES data by Hanaguri et al., and Borisenko et al., respectively. What about Raman? Well, it seems that you see 2\Delta features only (sc and CDW). However, it is bit controversial because Girsh suggests that it might be an amplitude mode.

Questions: 1 ) blogger: why one should believe that the intreband intreaction in NbSe_2 is repulsive, Answer: the system is complicated not that much is known about the system.
2) questions raised by Nevidmoskii produced some discussion which has moved to a lunch and basically concerned the symmetry representation of the various channels in the B1g, B2g and A1g symmetry representations for iron-based superconductors. It seems that to correctly interprete the data of Muschel et al you have to work in the unfolded BZ.

Yoshihiro Iwasa (Tohoku U): Electric field induced superconductivity with electric double layer transistors

Outline:

• electric field induced SC
• double layer Transistor
• oxides
• increase of carrier density
• layered ZrNCl
• summary

Intro:

charge accumulation in capacitor lead to MOSFET by replacement of one electrode structure, however, difficult to make in the past

at the same time (Glover 1960) electric field control of SC was investigated (In)

field effect on high-Tc cuprates (YBCO): by applying gate voltage increase of Tc

dream: gate control of localization-SC transition (e.g. a-Bi, Parendo PRL 2005)

best would be SC in an insulator by gating

conventional FET with weak electric field ~1 MV/cm, 10^13 cm^-2 which is too little, therefore attempt to use liquid gate (ionic), this leads to charge accumulation devices like Li battery or electric double layer capacitor

comparison of capacitance: double layer capacitor b/w electrolytic capacitor and battery, in double layer capacitor voltage drop right at the electrode

application of EDLT to organic conductors: sharp raise of conductance --> low operation voltage, but conductance in order of 4 microS therefore switch to inorganics


made of ZnO and patterned, ZnO shows large increase of sheet conductance at about 1.7 V, carrier densities are several 10^13 cm^-2, the capacitance is 7.8 microF/cm^2, e. i. half that as on Au

low T measurements;

crossover from insulator to metallic behavior at 0.7 V
now STO/PEO(KClO4) is being used, current two order of magnitude larger, insulator to metal transition at around a few volts

electric field induced SC was then found at V_g = 3 V, first field induced SC without doping, critical magnetic field about 30 mT, H_c2 is smaller than in the bulk due to interface which is not working as a pinning center

increase of carrier density:

uniqueness of STO:
STO is lowest carrier density SC, also atomically flat surfaces can readily be made

higher carrier density in ionic liquids: polymer electrolyte (solvent + salt) but large solvent molecules are bad, therefore solvent is to be removed --> melting at RT (organic material)

now accumulated carrier density is in range of 10^14 cm^-2, gate voltages of 0.5 V needed at room temperature, gate V goes up with T --> enhanced electric charging at low temperature, that should be advantageous for SC



Device:


ZrNCl:

discovered about ten years ago, layered material, cleaveable and interesting phase diagram

very small specific heat capacity if compared to other SC but relatively high Tc of ~15 K (goes down with Li-doping, pairing interaction increases which is strange, magnetic fluctuation are present but it is not clear where they come from, crossover from isotropic gap to an anisotropic one

try to explain these strange properties by a fluctuation exchange approx. theory, afm fluctuation develops causing d pairing even in doped band insulator







device fabrication:

exfoliating single crystals, electrodes by e-beam lithography, TiAu electrodes, single crystal size some 10 micrometer,

sigma kicks in at around 1,5 V at 220 K, goes up to 1.5 mS, by increasing the gate voltage remarkable decrease of resistivity, at about 4 V SC is found below ~14 K, however it is not clear how much volume of the sample turns SC

doping study: Tc goes down to below 12 K but there seems to be a maximum at ~0.04 as shown by recent low-doping studies

novel materials:

also, KTaO_3 turns SCbelow 0.04 K

• new tunable 2D systems
• challenges: increase Tc, discover new SC
• new states at interface which are inaccessible by conventional chemistry

Ken BURCH (U. Toronto): Tuning Materials with Mechanical Exfoliation

In order to tune properties - how to tune carrier densities without affecting purity, homogeneity etc.

OUTLINE:

• Introduction
  
• Method: exfoliation. A very expensive glove-box is (almost) all one needs.

• Materials: BSCCO, Bi2Se3. Using spectroscopy (Raman) in order to characterize the tickness of the layers.

E.g. in BSCCO. Characteristic features in the Raman spectrum: a low and a higher energy peaks. Changing the doping level enhances the intensity of the higher-energy (3-phonon) peak.

• Future directions

I. Introduction.
Disorder vs. doping.
E.g. STM work in BSCCO-2212 (McElroy, Science 309, 1048 (2005)) shows a lot of disorder on the surface (and presumably the bulk too). One would like to tune the doping without affecting the disorder. Exfoliation technique is promising.

Spectroscopy is a powerful tool to study materials, e.g.:
- ps-gap in the cuprates. Optical studies (Basov and Timusk, RMP 77, 721 (2005)) were one of the first to show the existence of the ps-gap. Also Fermi-arcs were observed in ARPES (Tanaka, Science 314, 1910 (2006)). Raman studies (Blumberg, Science'1997).
- Fe-pnictides - a new exciting direction.

Interfaces and surfaces (epitaxially grown)
- Ohmoto et al, thin films of BaTiO3/ Nature ()
- ferroelectrics

Field-effect transistor.
Consider SiO2 grown on top of Si:P.
There are few examples of combined thin layer materials (e.g. LaAlO3/LaVO3 etc), but material and growth issues are daunting. Lattice mismatch is important and prevents one from growing arbitrary combination of the film/substrate.
The exfoliation technique, on the other hand, is free from this drawback - the same thin film can be exfoliated onto a number of different substrates.

II. A new way: exfoliation technique
Pioneered by Andrew Geim for graphene. A.K.A. "skotch-tape" technique

- graphene
E.g. Tony Heinz's group showed that if grown on MICA, the graphene layer is very flat, unlike if grown on e.g. silica.
(The blogger did not know what MICA was - as far as I understood (please correct in incorrect) it is a substrate typically used in AFM for calibration. The surface of it is covered with Na aroms, leaving an atomically flat surface underneeth it)

- topological insulators
See Peng, Nat. Mater 9, 225 (2009); D. Hsieh, Science (2009)
e.g. Bi2Se3, Bi2Te3
Since MICA is transparent, the speaker used it for exfoliation, and then measured Raman on it.

- NbSe2
Raman A1g mode shifts depending on the thickness of the layers.

- MoS2
A. Splendiani.., Nano Lett. 4, 1271 (2010)

III. Results on BSCCO-2212
Bi2Sr2Ca0.6Dy0.4Cu2O(8+x)
x = 0.3, 0.4, in the ps-gap regime

Depending on the film thickness, the colour of the crystal is changing. Where does it come from? Chemistry is unchanged! Turns out - interference effect by reflection from different layers. Plot so-called contrast depending on the layer thickness - convincing.

Lab Tour: 9 T magnet, ellipsometer, Raman scattering ...

History of Raman: sunlight from a telescope, passed through a polarizer, filters to a single wave-length, and scatters it from the sample. The shift in frequency (Raman shift) is what measured.

Sharp features in Raman - 2-phonon joint density of states.
Broad features: two-magnon excitations. Review: T. Deveraux and R. Hackl RMP 79, 175 (2007)
E(1 magnon) = 2J;
E(2 magnon)=3J - most pronounced signal, since one breaks 6 spin-spin bonds by flipping two adjacent spins, paying energy cost J/2 for each.
Note: Other frequencies are possible (e.g. E(2 magnon) = 4.5 J).

The 2-magnon excitations have definite selection rules. E.g. B1g excitation corresponds to XX or X'X' polarization. The intensity of the Raman signal from thin films (exfoliated direcltly on the SiO2 substrate) greatly enhances compared to the bulk (See data in: Sandilands, PRB (2010)).

NOTE: Raman signal, even from the bulk, always has interference correction from reflection off different layers. (See Y. Wang, APL 92, 043121 (2008) ).

So what is the reason for much enhanced intensity of the B1g 2-magnon peak?
It turns out - this is the effect of the changed doping level. In fact, the bulk data show that reducing concentration of holes (under-doping) hardens the 2-magnon peak (i.e. shifts it to higher frequency). This is precisely what we observe in our exfoliated thin film flakes.

SUMMARY:

• using exfoliation technique allows to create very thin films without worrying about the interaction with the substrate, lattice mismatch etc.
• optical studies are a very powerful method
  
• thinning out the BSCCO films seems to result in increasing under-doping, judging from the 2-magnon Raman peak frequency.

Questions:
Q: P. Hirschfeld. Presumably, the reason for under-doping in exfoliated films is that you lose O atoms from the surface. However, don't you think that only few top layers would be affected?
A: It may well be true. Our data show however that the effective doping level, as measured by Raman 2-magnon peak position, reduces. We are not quite sure at present, due to what microscopic mechanism.

Q: H. Alloul. Are data taken at room temperature
A: Yes. The low-temperature data can be different

Q: A. Chubukov. Were all the measurements done for one doping level?
A: Yes, so far we started from overdoped O-concentrations. We would like in future to do exfoliation starting from optimally doped samples. We have also experimented with different Dy doping (not shown in this talk).

Q: G. Blumberg.
What was the laser power
A: a few miliwatts.
Blumberg's comment: concentrating this much power on a tiny surface area would lead to heating of the sample, and hence oxygen diffusion off the surface.

Q: P. Coleman: what do you know about the dependence of exfoliation on temperature?
A: we heat the substrate before exfoliation. We haven't yet done the exfoliation at low temperatures.
Q: Is the size of the flakes sufficient to attach leads?
A: The flakes are too small. It would be great to be able to make them larger.

Q: A. Schofield. What are the prospects of this exfoliation technique?
A: Exploring the phase diagram of e.g. cuprates, without affecting the purity of the crystals.

Q: Y. Grin. What do you know about the real structure of the spacer? Can you do TEM?
A: TEM proves hard. We do AFM, showing very flat surfaces. We'd like to do other structural probes in the future.

Q: [someone]. Can one see a gradual change in the Raman B1g position from bulk to thin film, by gradually increasing the film thinkness?
A: No, it's very hard to control the film thickness during exfoliation process. Somewhat of a black art.

Q: P. Hirschfeld. How important is lattice mismatch with the substrate? E.g. other studies on epitaxially grown films?
A: We hope that we are not stretching the film, but we cannot really be sure. We tried different substrates to try to answer this question.