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Governing ultrafast the conductivity of correlated materials

Final Report Summary - GO FAST (Governing ultrafast the conductivity of correlated materials)

Executive Summary:
The GO FAST project, launched the 1st April 2012, officially ended the 31st March 2015. The results were presented in Brussels the 10-11 March 2015, at the EC premises in occasion of the Cluster Review of three femtodynamics projects GO FAST, CRONOS and FEMTOSPIN, funded under the NMP.2011.2.1-2 topic Modelling of ultrafast dynamics in materials.
The aim of this research project was to develop novel schemes to study electronic, optical and structural properties of correlated materials driven out of equilibrium, in view of achieving an ultrafast optical control of their electronic properties. In particular, the most advanced theoretical techniques for correlated systems have been extended, i.e. Dynamical Mean Field Theory (DMFT) and the Gutzwiller variational approach, to model the temporal evolution after high-energy excitations are impulsively photoinjected by ultrafast laser pulses.
Realistic modelling was achieved through validation against the outcomes of different ad-hoc time-resolved experimental techniques. The possibility to optically switch on and off the metallic phase in a model Mott insulator (vanadium sesquioxide) and the superconducting phase in model high-temperature superconductors (BISCCO) have been investigated and tested.
The mutual and effective collaboration between the theoretical and experimental groups ensured the skill and the expertise needed to develop and validate realistic models of the ultrafast dynamics in complex materials, where the electronic, structural and magnetic degrees of freedom are strongly intertwined. The important achievements of the project are testified by the number of publications in high impact factor journals: 44 articles of which 4 published in Nature journals, 4 in Physical Review Letters, 2 in Scientific Reports, 15 in Physical Review B, and an invited review in Advanced in Physics, to appear in 2015.
GO FAST was funded with EUR 1,673,200.00 (one million six hundred and seventy three thousand two hundred EURO) granted by the European Commission in the 7th Framework Programme, coordinated by Prof. Michele Fabrizio of the Scuola Internazionale di Studi Superiori Avanzati – SISSA (Italy), and carried out by a consortium of 7 leading research institutions of four different Countries, each of them with specific roles and different levels of involvement. The project was carried out under supervision of the EU Project Officer (PO) Anne De Baas and of the Project Technical Advisor (PTA) Richard Ball. For more information on the project, please go to www.gofastproject.eu.

Project Context and Objectives:
Over the last decade, impressive experimental progresses made it possible to access the early temporal (femtoseconds to picoseconds) evolution of materials driven out of equilibrium by ultra-short laser pulses. Such technology was believed to have two major potentials: it could represent a tool to investigate dynamics directly in the time domain and much beyond the linear response regime, thus providing relevant physical information otherwise inaccessible; it might allow changing material properties much faster than usual and studying their properties along the way.
Among the materials these novel experimental tools were most frequently applied to, correlated transition metal compounds had a prominent role. These materials display incredibly rich phase diagrams, where even a tiny change of pressure, temperature, chemical composition or other external perturbations can drive a transition between completely different phases. Such noteworthy property when faced with conventional semiconductors makes correlated materials promising candidates for “phase-change” ultrafast devices. In addition, one may even foreshadow that these compounds, just because of their wealth of phases, could be driven by a sudden perturbation into transient metastable phases not present in the equilibrium phase diagram, a potentially exciting scenario. Time-resolved femtosecond spectroscopy was just the right tool to study correlated materials along the way of an electric-field-driven or optically driven phase-change, and assess all their potentials.

The GO FAST project started when this new field was just to explode, with leading experimental and theoretical groups localized mostly in Europe. The purpose was to assemble together a strong team with complementary expertise to understand the physics of ultrafast photo-excited correlated materials better than what each member could individually achieve. In other words, GO FAST aimed at steering theoretical and experimental efforts from different places all over Europe towards a single and shared target, thus increasing the probability of successful achievements.

In particular, the project focused on two representative compounds: vanadium sesquioxide (V2O3), likely the first Mott insulator ever discovered back in 1960, and copper-oxide superconductors in the BISCCO family, the materials that 30 years ago piqued the interest of the scientific community in strong correlations. V2O3 was chosen as representative of Mott insulator transition metal oxides with a d-d charge gap, while cuprates as representatives of charge-transfer Mott insulators. The goal was to explore experimentally and model theoretically their non-equilibrium phase diagrams to ascertain whether intense and ultra short laser pulses indeed allow driving phase-changes faster than by usual means, e.g. rising temperature or pressure, and studying their non-equilibrium dynamics. Particular emphasis was placed to photoinduced transitions of technological interest, specifically between phases with completely different conducting properties, Mott insulating versus metallic/superconducting, with the objective of identifying non-equilibrium pathways between them, i.e. the external parameters that can be tailored to drive more efficiently those transitions (laser frequency, polarization and incidence angle, or sample thickness).

To achieve these goals, a multidisciplinary network was organized comprising the condensed-matter theory group at SISSA (Trieste), with expertise in strongly correlated systems, and well established European experimental groups in the field of ultrafast spectroscopies, with expertise in time-resolved optical and photoemission spectroscopies, time resolved X-ray and electron diffraction. The theoretical task undertaken by SISSA was to develop and refine tools for studying models of correlated materials in out-of-equilibrium conditions, and apply those methods to simulate realistic situations. Inputs and feedbacks from experiments performed by the other partners were mandatory to validate modeling and results, and to better orient the theoretical activity. Such a symbiotic effort by theoretical and experimental groups has been the fingerprint and key of success of the project.

Project Results:
Major Scientific Results (MSA)
Among the many fundamental scientific results obtained within the GOFAST consortium, we summarize here the most relevant innovative aspects, which are grouped into three Major Scientific Achievements (MSA):
MSA1 Out-of-equilibrium Gutzwiller method
The development of an efficient and flexible scheme to simulate the non-equilibrium dynamics of lattice models for correlated electron systems based on the Gutzwiller variational wavefunction and approximation. This method is exceptionally fast and efficient while providing results in accordance with much more expensive rigorous techniques.

MSA2 First-order insulator-to-metal phase transition
In the course of the project, it has become clear to us and to the whole scientific community that the gap between the lower and the upper Hubbard bands in Mott insulators is as robust as the gap in conventional band insulators. Therefore, despite early expectations, the gap between Hubbard sidebands cannot be collapsed down; hence the “metallic” carrier density of Mott insulators cannot be made available by electric or light-driven breakdown. On the contrary, a feature that really differentiates Mott from band insulators is that a Mott transition is generically first order so that, within the coexistence region on the insulating side of that transition there exists a metastable metal phase besides the stable insulating one. One can thus imagine driving the insulator into the metastable metal phase. Thanks to the knowledge developed within the GO-FAST project, we have been among the first to propose that this remarkable physics requires internal degrees of freedom not present in the simple single band Hubbard model, but common to most known narrow-gap Mott insulators. The main issue in this context is to identify the extensive variable M that controls the first order transition (the equivalent of the volume in the liquid-vapour phase transition) and the optical excitation channel, if any, that can change M from its equilibrium value. We have accomplished this program in a model mimicking V2O3, and shown that such a non-equilibrium trapping into a metastable metal phase can be realized. We also performed a pump-probe time-resolved photoemission experiment on V2O3, and found that indeed the gap collapses down and recovers back on time scales of few picoseconds, much longer than typical electronic relaxation times but not unrealistic for the time a nucleus of a wrong phase would take to dissolve within the coexistence region of a first order phase transition. We have indirect support that the mechanism underneath the observed gap-collapse is the predicted one by the evidence of photo-induced phonon hardening in V2O3 and its critical dependence upon laser polarization and incidence geometry.

MSA3 Non-thermal ultrafast effects in superconducting copper oxides
Copper-oxide high-Tc superconductors are one of the most interesting class of materials. The superconducting temperature exceeding the liquid nitrogen opens important routes towards the all-optical control of the superconducting-to-normal phase transition. In particular, the possibility of optically switching on and off a superconducting phase would have a dramatic impact in the field of ultrafast electronics and THz devices. The major complexity in modelling the ultrafast radiation-matter interaction is the strong k-space anisotropy of the physical properties of copper oxides, which is inherently related to the strong short-range coulomb repulsion between the charges in the CuO2 plane. Within the GOFAST project, we were able to develop the basic model to capture the intrinsic dichotomy of the excitations photo-injected with wavevectors parallel to or at 45° with respect to the Cu-O bonds. We demonstrated that the non-equilibrium population that is created in the first femtoseconds has a strongly non-thermal character both in the k- and energy-distribution. These findings pave the way to optical-control the state of superconducting cuprates in a novel fashion, which goes beyond the simple tuning of the temperature of the system.
We summarize here two major outcomes of the joint theoretical-experimental efforts carried out during the project. First, by combining modelling, femto-ARPES and time-resolved spectroscopy, we have demonstrated that the optical excitations of doped cuprates induces a non-thermal overpopulation of anti-nodal states, characterized by a lifetime longer than that in equilibrium conditions. In simple terms, at specific doping concentrations and low temperature, the photo-excitation creates a transient conducting state that cannot be achieved in equilibrium conditions.
Secondly, cuprates are inherently particle-hole asymmetric. For example, in the simplest case of the undoped compounds that are typical charge-transfer insulators, the photo-doped holes go to the oxygen atoms while the electrons to copper atoms. In such a situation one can envisage that the optical excitation may appreciably alter the chemical potential and effectively dope the material. We have indeed discovered such a remarkable effect by time-resolved femto-ARPES on Bi2Sr2CaCu2O8+δ. The above experimental observation could represent a real breakthrough since it opens the possibility of transiently photodoping the system and entering a superconducting phase in a non-thermal fashion.

Key Exploitable Results
During the project, the consortium had the opportunity to cooperate with a consultant belonging to the Exploitation Strategy and Innovation Consultants (ESIC) Agency. The Exploitation Strategy Seminar (ESS), help in SISSA, was very helpful to identify, within the Major Scientific Achievements, the Key Exploitable Results (KERs), listed in the Table below, which comprise: (1) practical side-products (specifically a new software and the design of a 3D cryostat managing tool); and (2) theoretical results in terms of methods and theoretical concepts.

No. WP MSA TYPE COMMENT RESPONSIBLE PARTNER
1 4 3 method METHOD of photodoping and its application to transistor-like devices Uwe/ Ping Zhou UDE
2 1 1 method METHOD to study out-of-equilibrium models Michele Fabrizio SISSA
3 1 3 code CODE to implement the time dependent dynamical mean theory t-DMFT Michele Fabrizio SISSA
4 3 2 theory THEORY on the reasons why the Mott gap collapses for several ps after an ultra short laser pulse Michele Fabrizio SISSA
5 1 3 code CODE for the analysis of pump-probe THz spectroscopy experiments Claudio Giannetti UNICATT
6 1 3 code CODE for the dynamics of the order parameter in a symmetry-broken phase Claudio Giannetti UNICATT
7 1 3 tool DESIGN of 3D cryostat managing system – by PRODUCT Claudio Giannetti UNICATT

Interacting Quantum Systems (TRIQS), and Wien2k (Wien2dynamics). Except Wien2k all of the above codes are licensed under GPL. During the project new modules for the multi-physics software COMSOL have been developed. The GO FAST project has also paved the way for future code developments such as a time-dependent dynamical mean-field theory (t-DMFT) code that could be joined with existing programs such as Quantum Espresso.
The GO FAST project had a significant experimental component and interesting exploitable side-products have emerged from this activity, including the modelling of THz pump-probe experiments (Menlo Systems, a German spin-off from the Max-Planck-Institute for Quantum Optics) and the development of a low-vibration mechanical handling system for a closed-cycle cryostat (5Pascal, an Italian vacuum and cryogenic systems company; ARSCryo, a US company for cryogenic systems). Menlo Systems is looking out for new laser materials, but does not envisage to adopt the modelling developed. It is thus concluded that GO FAST was set-up and remained a rather academic project with exploitation potentials only in the far future.
The reached TRL is judged to be 2.


Description of the MSA and related KERs

MSA1 Out-of-equilibrium Gutzwiller method
The MSA1 relates to a method that has been developed by GO FAST to approximate the out-of-equilibrium time-evolution of lattice models for correlated electrons. This Major Scientific Achievement is also reputed one of the Key Exploitable Results, indeed KER2 in the Table above. We present the method in detail in what follows.

KER2: METHOD to study out-of-equilibrium models
Correlated materials are difficult to model by all methods based on independent particle schemes, like DFT in local or semi-local approximations or Hartree-Fock. The reason is that the Mott’s localization characterizing these systems is a collective phenomenon, which evidently escapes any representation in terms of independent particles. Over the years, several ad-hoc techniques have been developed to deal with strong correlations. The most rigorous are also the most numerically demanding, like Variational or Quantum Monte Carlo, exact diagonalization, Density Matrix Renormalization Group, or Dynamical Mean Field Theory (DMFT). In the late 1960 Martin Gutzwiller proposed a very simple variational wavefunction for strongly correlated systems and an approximation to calculate average values on that wavefunction, which were named Gutzwiller wavefunction and approximation after him. The Gutzwiller wavefunctions consists of a variational Slater determinant that is modified by the action of variational linear operators, one at each site, whose role is to reduce the weights of local configurations that cost too much repulsive energy. This class of wavefunctions is broader than that of the Hartree-Fock mean-field theory, which includes only Slater determinants, hence it is variationally more accurate.
The Gutzwiller approximation, which later was found to become exact in the limit of infinite coordination lattices, allowed to uncover over the years many basic and popular concepts in the physics of strongly correlated systems, like the Brinkman-Rice description of the Mott transition, or the RVB scenario for High-Tc superconductivity. In reality, one may regard the results of the Gutzwiller approximation as the limiting expression of the results obtained in finite-coordination lattices by Jastrow variational wavefunctions when the lattice coordination tends to infinity. Since Jastrow wavefunctions are known to describe very accurately correlated systems, the Gutzwiller approximation is believed to provide sensible results also when used in finite-coordination lattices; the Mott’s physics is local hence there should not be sensible differences between e.g. a cubic lattice and a hypercubic one in infinite dimensions.
It is well known that the variational principle can be generalized also in the time domain, where, instead of minimizing the total energy, one has to extremize an action defined by the integral over time of the average value of the Schrödinger equation, the so-called Dirac-Frenkel variational principle. We have shown that a generalized Gutzwiller approximation allows working out analytically the time-dependent variational principle for Gutzwiller wavefunctions in infinite coordination lattices. This result is far from being trivial. Indeed, the Gutzwiller approximation relies on a self-consistency condition that, in the temporal evolution, should be enforced at any instant of time, a constrained time-evolution extremely hard to implement. We found that a proper extension of the class of Gutzwiller wavefunctions still manageable analytically leads to the remarkable result that, if the self-consistency condition is satisfied at the initial time, it will remain fulfilled at any subsequent time without requiring any constraint on the dynamics. This makes our time-dependent Gutzwiller approximation method exceptionally fast and efficient while providing results in accordance with much more expensive rigorous techniques, like time-dependent DMFT. We have described the method in a monograph appeared in the book “New Materials for Thermoelectric Applications: Theory and Experiments”, NATO Science for Peace and Security Series B: Physics and Biophysics 2013, pp. 247-273, published by Springer, see http://link.springer.com/chapter/10.1007/978-94-007-4984-9_16 and downloaded 748 times so far. We were also invited to present the method at the APS 2015 March Meeting held in San Antonio, one of the most prestigious condensed-matter conferences worldwide.
The method is relatively easy to implement; one just needs some clever routine to integrate first-order non-linear differential equations, plenty of which are freely available in open source numerical libraries. Therefore we believed not worthy exploiting the method by a software tool. The important result here was the method itself, rather than its implementation in a code.
In fact, the method has the nice feature that it can describe on equal footing both the dynamics of quasiparticles, i.e. the variational parameters of the Slater determinant, as well as the dynamics of the Hubbard sidebands, represented by the variational parameters of the local linear operators. These two sets of degrees of freedom are coupled to each other in a mean-field fashion, i.e. each evolves in a time-dependent potential generated by the other. Such limitation implies that the method cannot describe a full relaxation to a steady state. However the values of time-integrated observables compare well with those obtained by more expensive rigorous tools, like the time-dependent DMFT.
Within the GO FAST project we have applied the t-GA to a variety of relevant case studies. For instance we have found that the surface dead layer with poor metallic behaviour that appears at the surface of a correlated metal can be driven dynamically in a Mott insulator, the dynamical counterpart of a surface Mott transition, by the analogous of a photo-excitation of the surface layers.
More recently we have studied in detail by t-GA the dynamical properties of a correlated slab in different non-equilibrium conditions. For instance we have investigated the dielectric breakdown of a Mott insulating slab in the presence of an applied electric field and found that it occurs exactly as in conventional semiconductors by Landau-Zener tunnelling across lower and upper Hubbard bands, see the figure above. This also implies that the threshold field Eth ≈ EGAP/ξ, where EGAP is the Mott gap and ξ the insulating correlation length.
Another methodology achievement obtained within GO FAST has been the extension at finite temperature of the Gutzwiller approximation. The variational principle at finite temperature states that the actual free-energy is the absolute minimum of the functional F(ρ) = Tr(ρ H) – T S(ρ), where ρ is a density matrix probability distribution, H the Hamiltonian, and S(ρ) = - Tr(ρ lnρ) the Von Neumann entropy of the distribution. Evidently, if one limits the search within a subset of density matrices ρ, one obtains an upper estimate of the actual free energy. Exploiting mathematical trace inequalities, we have been able to find a simple analytic expression that provides an upper estimate of the free energy functional for Gutzwiller-type density matrices. This allows explicitly calculating a variational free energy within the Gutzwiller approximation, hence phase diagrams at finite temperature. The comparison with more rigorous DMFT is quite satisfactory.
In the future it will be important to extend the time-dependent Gutzwiller approximation at finite temperature, in order to obtain more realistic results. In addition, in view of recent successful attempts to combine DFT-LDA+U with the Gutzwiller approximation, it would be of interest to extend such tool at finite temperature, which could allow describing by first principles Mott insulators that are not accompanied by any symmetry breaking, like in V2O3 above the Nèel temperature. In addition, it might be worth exploring the possibility to join time-dependent DFT (TDDFT) with t-GA, which could describe accurately the early time evolution of a correlated material in view of the next generation of attosecond time-resolved spectroscopies.

MSA2 First-order Mott insulator-to-metal phase transition
This theoretical result was actually achieved in the last year of GO FAST and stimulated by very recent exciting experiments and by the awareness that theoretical modelling was missing some crucial fact and thus could not explain the experimental data. We hereafter discuss thoroughly the theory that we developed and that we consider a key exploitable result, KER4 in the list.

KER4: THEORY on the reasons why the Mott gap collapses for several ps after an ultra-short laser pulse.
Mott insulators are “unsuccessful metals” where electron motion is impeded by strong Coulomb repulsion. Their use in microelectronics started to be seriously considered in the 1990s, when first reports of field-effect switches appeared. These attempts were motivated by the expectation that the dielectric breakdown in Mott insulators, which have metallic-like electron density, could all of a sudden release all formerly localized carriers, a significant potential for nanometer scaling. This was the general belief when GO FAST started. However, experiments on hard gap Mott insulators, like Ni or Cu oxides, as well as theoretical results have partly failed to meet such expectation. For instance we, as well as other groups worldwide, have shown that the dielectric breakdown of a simple Mott insulator described by the single-band Hubbard model occur via Landau-Zener tunneling across lower and upper Hubbard bands, as if the latter were as rigid as valence and conduction bands in semiconductors. We have also shown that excitations across the preformed Mott-gap drive correlated metals towards a Mott insulating phase, rather then making them more metallic. In addition, a photo-doped single-band Mott insulator, with holes in the lower Hubbard band and electrons in the upper one, appears qualitatively close to a regular photo-doped semiconductor: the gap is essentially unaffected and the mobile carriers are just the poorly coherent photo-injected holes and electrons.
However, noteworthy experimental discoveries on narrow-gap Mott insulators, which intervened over the very last years, have changed this state of affairs and suggested that the conventional portrait of a Mott transition was overlooking some crucial elements. These experiments have actually materialized the original expectation showing that narrow-gap Mott insulators can be driven out-of-equilibrium in metal phases that last anomalously long time, and, furthermore, that such phase change does not require strong electric fields or intense laser pulses, i.e. comparable with the Mott gap.
We have paid lot of efforts during the last year of GO FAST to understand what theoretical modeling was missing, and we believe we finally got an answer that actually opens very interesting conceptual and practical perspectives. We started from observing that all known Mott transitions are strongly first order, which entails an extended metal-insulator coexistence. Traditionally, the first order nature has been considered as accidental and secondary at best in most present discussions of equilibrium and especially non-equilibrium metal-insulator transitions. Instead we now believe that this is just the key to rationalize the experiments: it is not the Mott insulator that is continuously turned into a metal by the external perturbation, but it is a totally different metal phase that emerges, which was only metastable at equilibrium. The other observation that guided our analysis was that the gap had to be different from that between lower and upper Hubbard bands. In fact, in the latter case the behavior should be similar to a single-band Hubbard model, which we mentioned behaves just like a normal semiconductor. Remarkably, in all known Mott insulators the gap is never the one between lower and upper Hubbard bands. In fact it is either a charge-transfer gap or a gap between occupied and unoccupied d-orbitals.
In conclusion, our working hypotheses were: (1) a strong first-order Mott transition; (2) a gap different from that between lower and upper Hubbard bands. We argued that both (1) and (2) are actually manifestation of the same physics. While the main drive to the metal-insulator transition is electron repulsion, on the way to Mott’s localization other mechanisms come into play and contaminate the otherwise ideal Mott transition. Magnetism is the best-known example, but not the only one. For instance, the coupling to the lattice, which controls the crystal field and the degree of bonds covalency, and the Coulomb exchange splitting, responsible of Hund’s rules, are other major actors. It is not difficult to realize that these new actors support and boost the first order character of the Mott transition and as well they open a gap that is not the one, already preformed and much bigger, between lower and upper Hubbard bands. We disclosed this scenario in a toy-model for V2O3, which consists of two orbitals split by a crystal field with a density of one electron per site. In this model the Mott insulator describes an empty higher orbital and a half-filled lower one, which is Mott localized and antiferromagnetically ordered.
We indeed observed this intriguing metallization in pump-probe femto-ARPES on V2O3, as shown in the figure. We believe that the physical phenomenon that we discovered has a more general validity than the simple model where we disclosed it, and raises a lot of interesting conceptual questions with practical consequences worth being addressed in the future. First of all, one should re-examine known narrow-gap Mott insulators to uncover how such physics can be realized, i.e. to find the observable that plays the role of m above and that can be varied experimentally. This information is crucial to evaluate the potentials of each material for applications. In addition one has to understand how typical phenomena that appears at first order transitions, like nucleation or wetting at interfaces, manifest themselves in such a novel context of metal-insulator transitions.
MSA3 Non-thermal ultrafast effects in superconducting copper oxides

All the time-resolved experiments carried out on copper oxides evidenced a dramatic difference between the simply heating of the system and the excitation through an ultrashort light pulse. While at the thermal equilibrium the largest number of excitations should be in the nodal region of the Brillouin zone (where the superconducting gap is zero), time-resolved techniques evidenced a very effective increase of the ANTINODAL excitations. This result suggests the possibility of using this non-thermal distribution as an additional knob to control the electronic properties of the material and, eventually, to define the most promising strategies to optically switch the system from the normal to the superconducting phase and viceversa.These findings triggered an intense activity within the WP4 of the GO FAST project, to measure and model the effects of this non-thermal photo-excited distribution on the macroscopic electronic properties of different families of prototypical copper oxides. Among the different results, we briefly discuss here, as an example, the physical picture emerging from the time-resolved study [Cilento2014] of the pseudogap phase of copper oxides, which provides a very clear example of the novel conducting properties that emerge in non-equilibrium conditions.

The pseudogap state of copper oxides is defined by a T*(p) line in the doping-temperature (p-T) phase diagram that progressively vanish as the doping increases. The pseudogap is characterized by a Fermi surface that is gapped at the antinodes, even though the superconductivity is not observed. As far as the ultrafast dynamics is concerned, it is mandatory to understand the role of the antinodal (pseudo)-gap in the relaxation dynamics of the photoexcited population. To address this issue, we combined broadband pump-probe optical spectroscopy to time-resolved photoemission spectroscopy. We performed the experiments on Bi2212 and and HgBa2CuO4+δ (Hg1201) single crystals at different hole concentrations. Time-resolved photoemission unveiled the creation of a non-thermal electron distribution characterized by an excess of antinodal excitations. On the other hand, broadband pump-probe spectroscopy evidenced an anomalous decrease in the scattering rate of the charge carriers in a pseudogap-like region of the phase diagram. In other terms, when excited the system becomes more conducting than in the equilibrium condition. These findings suggest the following scenario:
• In the pseudogap region, delimited by a well-defined T*neq(p) line, the non-thermal photoexcitation process triggers the evolution of antinodal excitations from gapped (localized) to delocalized quasiparticles characterized by a longer lifetime. As a consequence, a decrease of the scattering rate is observed as long as the non-thermal antinodal population is maintained.

The transient increase of the conductivity observed in the pseudogap state of copper oxides suggests that the T*(p) line delimits a region in which the antinodal states evolve into more metallic ones upon photoexcitation with the pump pulses. The generality of the results obtained called for a general model that accounted for the phase diagram unveiled by the non-equilibrium optical spectroscopy. Considering that the measured transient decrease in the carrier scattering rate was faster than the complete heating of the lattice, we focused on the 2D Hubbard Hamiltonian, i.e. the minimal model that neglects electron-phonon coupling and retains the genuine physics of correlations. To compute the temperature-dependent electronic self-energy in different positions of the Brillouin zone, we used the dynamical cluster approximation, a cluster extension of Dynamical Mean Field Theory (DMFT) that captures the k-space differentiation of the electronic properties between different k-space regions [Cilento2014]. In the first stage, the increase in energy related to the pump excitation was mimicked by selectively increasing the effective temperature of the nodal and antinodal self-energies. The results obtained through the modeling, suggest a strong dichotomy between the scattering rate of nodal and antinodal excitations. In contrast to nodal QPs, whose scattering rate increases with temperature, the scattering rate of antinodal excitations exhibits a completely different evolution, decreasing as the effective temperature rises. This striking dichotomy of the nature of the elementary excitations in the k-space is the consequence of a momentum-space selective opening of a correlation-driven gap, which eventually evolves into the full Mott gap at p=0. When the hole doping is increased, the k-space differentiation of the nodal-antinodal fundamental excitations is washed out and a more conventional metallic behaviour is recovered. These calculations confirm an intrinsic U-driven momentum-space differentiation of the electronic properties of cuprates at finite hole concentrations and temperatures: the nature of the antinodal states is similar to that of a Mott insulator in the sense that the scattering rate of AN states decreases when the internal energy of the system is increased. These results explain the experimentally observe pseudogap-related photoinduced increase of conductivity when a non-thermal antinodal population is created [Cilento2014].

These results perfectly exemplify how to exploit the non-thermal distribution created by an ultrashort light pulse to manipulate the conductivity of copper oxides and transiently create artificial electronic properties that do not exist at equilibrium. These observations constitute the background from which various KERs (KER1,3,5,6,7) were developed. Besides the KERS, we also summarize the other most relevant scientific results that are strictly related to the study of the non-equilibrium physics of copper oxides:

• During WP4, we performed time-resolved photoemission spectroscopy with Vacuum UltraViolet (VUV) pulses on copper oxides. This experiment, performed at the beamline ARTEMIS at the Rutherford Laboratories (UK) allowed us to track, for the first time, the photoexcitation process in copper oxides within the entire Brillouin zone. The results evidenced a giant and unexpected increase of electrons at the antinodes and a long-lived modification of the oxygen bands at 1.5 eV binding energy.
• We performed high-temporal resolution (<10 fs) pump-probe spectroscopy on different families of copper oxides to directly measure, for the first time, the ultrafast coupling of the photo-excited charge carriers to magnetic bosons. [DalConte2015]
• We performed pump-probe optical spectroscopy on La2CuO4 and Bi2201 that evidenced the transient collapse of the charge-transfer gap in the underdoped region of the cuprate phase diagram. [Novelli2014]

KER1: METHOD of photodoping and its application to transistor-like devices

The possibility of photo-injecting a non-thermal distribution of excitations open interesting routes toward the optical control of the electronic phase in superconducting copper oxides. The possibility of using ultrashort light pulses to photodope the material has been recently demonstrated by time-resolved ARPES experiments.
The dynamics of the transient occupation of the electronic bands in copper oxides has been measured by time-resolved UV photoemission [Rameau2014]. The fourth harmonics (6 eV) of the output of an amplified Ti:sapphire oscillator has been employed to photo-emit electrons. A 40 fs pulse at 1.5 eV photon energy is used to excite the system. By changing the orientation of the sample and the delay t between the pump and probe pulses, the dynamics of the electronic bands in the k-space, i.e. E(k,t) is reconstructed. This technique has been applied to optimally doped Bi2Sr2CaCu2O8+δ (Bi2212) single crystals with Tc=96 K. Figure 1 (taken from Ref. [Rameau2014]) displays the relative variation of the photoemitted spectra at different delays. The Fermi energy is indicated by a dashed line. The data demonstrate a transient depletion of the states at EEF. From these data we can draw two main conclusions:
• The effective mass of the charge carriers (holes) transiently decreases suggesting that the optical excitation directly modifies the electronic interactions that are responsible of the kinks in the bare band. This dynamics has the same timescale of the experimental temporal resolution (100 fs), thus pointing to a possible role of the magnetic excitations (antiferromagnetic fluctuations) that are expected to interact with the charge carriers on a very fast timescale (<20 fs).
• After the photoexcitation, the fermi energy progressively moves to smaller values of the momentum k. This behaviour is analogous to the effect of an increase of the chemical doping. These results suggest that, within 500 fs, the photoexcitation process can be considered as a transient photodoping that drives the physical properties of the system along a horizontal line in the phase diagram.
While the possibility of completely quenching the superconducting phase through an ultrashort light pulse is a well established experimental fact, the opposite process, i.e. the creation of a transient superconducting state is still a subject of intensive research. This would unlock the gate to the design of solid state switches (or transistor-like devices) that could change their conducting properties on the sub-picosecond timescale (i.e. 1-10 THz frequencies). The huge conductivity contrast between the normal state and the superconducting state, along with the relatively high critical current Jc of the superconducting state would make it possible a very high instantaneous current flow. A promising strategy to pursue this objective, emerges from the femto-ARPES data. Furthermore, the electron-hole asymmetry in the cuprates also plays a fundamental role during the recovery dynamics. While the excitation process create electron-hole pairs, the following dynamics can be very different for the electrons and the holes. This gives rise to a shift of the chemical potential, as observed by time-resolved photoemission, that can simulate a transient increase of doping that is recovered within <1ps. These observations open very intriguing scenarios in which the hole doping p could be transiently increased in order to enter the superconducting dome and trigger a transient superconducting state. This seems to constitute a promising scheme for a realistic ultrafast (1 THz bandwidth) normal-to-superconducting switch that would operate at low temperature. The optical excitation could be provided through a fiber tip in order to miniaturize the device size down to approximately 1 μm. A thickness of 100 nm of the superconducting film should guarantee a homogeneous excitation profile that should limit possible problems related to the transversal diffusion of the excitation.
As far as the chemical composition of the copper oxides is concerned, Bi2Sr2CaCu2O8+δ seems the most simple system to start with, even though YBCO could have some advantages since it is already more used at the industrial level and YBCO thin films are commercially available. The doping and the operating temperature of the system should be tuned considering the shape of the superconducting dome, that is reproduced by the phenomenological formula: Tc/Tc,max=1–82.6(p−0.16)2 where Tc,max=96 K for Bi2212. For example, considering that the contrast of the switch is expected to be larger in the underdoped region and at low temperature, we can fix Tc=4 K and we obtain an equilibrium hole doping of about p=0.05. On the other hand, it could result useful to operate at higher temperatures, even though the superconducting dome becomes progressively parallel to the p axis when approaching optimal doping. Considering the liquid nitrogen temperature, Tc=77 K, we obtain p=0.11.
[Rameau2014] J. D. Rameau et al. Photoinduced changes in the cuprate electronic structure revealed by femtosecond time- and angle-resolved photoemission. Phys. Rev. B 89, 115115 (2014)

KER3: CODE to implement the time dependent dynamical mean theory t-DMFT

One of the backbones for the theoretical investigation of non-equilibrium correlated materials has been the development of a code implementing the time-dependent dynamical mean-field theory (t-DMFT) both for simple models (Hubbard model) and for realistic descriptions of correlated materials.
From a formal point of view, t-DMFT does not require conceptual extensions with respect to equilibrium, and it amounts to solve iteratively an impurity model subject to a self-consistency condition. However, both aspects become much less obvious out of equilibrium. Our development of a t-DMFT code has worked in both directions using a modular approach.

We have developed a completely general Fortran driver which solves the self-consistency equations (essentially the Baym-Kadanoff-Keldysh equations with a local self-energy) in an effective and optimized way. This part of the code has been designed in order to be completely general, in order to study normal metallic phases, superconductivity, magnetic ordering, it is interfaced with the popular software Wannier90, which implies that it can be used with most Density-Functional Theory packages and codes, and it introduces non standard features, unavailable in other implementations of t-DMFT, like inhomogeneous states (by means of real-space DMFT algorithms). The modular and flexible structure of the code allows to include other features, like electron-phonon interaction including a full phonon dynamics, multiorbital and cluster structures.

This part of the code works like a black box which can be interfaced with different “impurity solvers”, which solve the impurity model out-of-equilibrium. About this part, it has become clear that none of the available solvers can be considered more effective than others irrespective of the physical problem, regime of temperature and other model-specific aspects. It is therefore crucial to implement different solvers to be used in different regimes (and to be benchmarked one against the other in the regimes where they simultaneously work).

We have successfully implemented the simple, yet approximate, Iterated Perturbation Theory IPT, where particular effort has been devoted to the superconducting state, which has not been included in previous implementations. The full IPT code is completely operational and it is currently applied to a range of problems. The code is close to a level where it can be distributed and provide a simple solution to most strongly correlated models out of equilibrium.

The Continuous-Time Quantum Monte Carlo has been implemented in the hybridization expansion. The code is operational and “numerically exact”, but it can not be used for times longer than 100 fs because of numerical noise. We are currently working on minimizing the influence of such noise on the actual observables. Finally, we are developing a new kind of solver based on exact diagonalization, but this tool is still in the development stage.

KER5: CODE for the analysis of pump-probe THz spectroscopy experiments

As a fall-out of the main scientific tasks of the GO-FAST project, the Exploitation Board (EB) individuated some modelling tools that could be of relevance for technology-oriented applications. As far as the THz business of MenloSystems is concerned, the THz line (optical antennas and detectors and time-domain THz kit) is supported by the TeraLyzer software that allows extracting the optical constants from time-domain THz measurements. In this perspective, some of the GO-FAST partners tackled the problem of reconstructing the change of the THz optical conductivity, after the excitation with an optical ultrashort light pulse. In this configuration, called optical pump-THz probe, the large mismatch between the penetration depths of the optical and THz pulses makes the dynamics of the pump-probe experiment intrinsically inhomogeneous. As shown in the figure, the THz electric field reconstructed from the electro-optical sampling strongly depends on the profile of the excitation pulse. Therefore, the deconvolution between the measured signal and the spatial profile of the pump pulse is mandatory to extract the dynamics of the optical constants which can be successively used for extracting the physical informations about the process under study. Considering that the optical pump-THz probe configuration is raising increasing commercial interest, the EB suggested the possibility of developing a GO-FAST tool for the analysis of pump probe THz experiments, which could be eventually considered as tool of the TeraLyzer software.

KER6: CODE for the dynamics of the order parameter in a symmetry-broken phase

In order to describe the spatial propagation of the impulsive perturbation of the superconducting order parameter, we used the Ginzburg-Landau functionals. The motion equation that is obtained is a wave equation with a damping term. This partial differential equation can be solved through the finite-elements approach, taking into account the realistic geometry of the system. In the limit of large damping, that corresponds to the thermodynamic limit, the order parameter relaxes back to equilibrium without oscillating and the dynamics reduces to a diffusion process. As a first step we modelled the dynamics of the order parameter in a thin film of finite thickness and we studied the role of the laser penetration depth in destroying the superconducting phase. In the figure we report the average value of the order parameter probed by the optical pulse, as a function of the intensity (arb. units) and penetration depth of the pump pulse. In the homogeneous limit (i.e. the pump penetration length is significantly larger than the thickness), the average value of the order parameter is linear with the pump fluence until the threshold corresponding to the complete destruction of the superconducting phase is reached. In the opposite case (i.e. the pump penetration length is significantly smaller than the thickness) we observe a non linear regime, in which the phase transition has already happened in a small superficial volume of the system.
This model has been used to analyze pump-probe data on a bulk high-temperature superconductor (Y-Bi2212) and precisely estimate the critical fluence necessary to optically quench the superconducting phase in the first layer of the system. The tool developed during this activity is very versatile and can be extended to any kind of symmetry-broken phase that is locally and impulsively perturbed.
The EB suggested the possibility of developing a GO-FAST tool, based on commercially-available finite-elements software, that allow the solution of GL equations in symmetry-broken phases in a simple and versatile fashion. This software could be further extended to study supelattices constituted by alternate layers of materials with different order parameters, which constitute a promising route towards the manipulation of the physical properties of solid-state systems.


KER7: DESIGN of 3D cryostat managing system

During the project we tackled the problem of developing a 3-degrees-of-freedom-mechanical handling system (manipulator) to control and move the closed-cycle cryostat employed for the low-temperature optical measurements. Since the cryostat (ARScryo) is constituted by two mechanically-decoupled parts that should be moved in a synchronized way, we developed a 6-motors synchronized manipulator that avoids transmitting the vibrations of the compressor to the sample, while leaving the freedom of moving the samples. The manipulator has been designed and assembled in the labs of the UNICATT partner. The manipulator is controlled via a software written in the Labview (National Instruments) code. The design of this manipulator raised the interest of Italian distributor of ARSCryo, that is available to jointly develop a commercial prototype that could be offered to the customers interested in this solution. In view of the possible commercialization, the consortium is studying the most suitable form of design protection.

Potential Impact:
Socio-economic impact

The aforementioned results, which have been obtained during the three years of the project, point towards several research directions worth to be further pursued in the future. In particular, the non-thermal photo-induced behaviour observed in vanadium- and copper-oxides superconductors foreshadows quite remarkable phenomena; a great challenge that we do intend to take up. In addition, the preliminary and so far feeble evidence that one can optically nucleate droplets of a metastable metal phase within a Mott insulator discloses interesting perspectives to exploit the first order nature of known Mott transitions, which are hitherto unexplored.
The new concepts and modelling tools introduced by the GO FAST project and related to the non-equilibrium properties of correlated materials, can impact on the next-generation of solid-state devices, such as ultrafast memories and switches and THz devices, that are emerging as fundamental tools in the field of diagnostics and imaging. Even though the state-of-the-art technology is still based on “conventional” semiconductor devices, a huge effort is currently ongoing to investigate the capabilities of novel materials that could replace current technology. In this perspective, the knowledge developed by the GO FAST project in the field of non-equilibrium correlated materials is strategic in view of establishing funding concepts for the potentially immense market of next-generation ultrafast solid-state devices. The possibility of modelling the ultrafast properties of correlated oxides, demonstrated by GO FAST on the prototypical vanadates and cuprates, reinforces the leading role of the European science in the worldwide race towards the development of novel materials whose functionalities can be artificially manipulated with unprecedented speed.

Dissemination

Several actions have been undertaken during the whole course of the project to promote the project’s results both within the consortium, and in the European industrial and universal scientific communities, in order to increase public awareness of the project. The project could rely on a website, which has been constantly updated during the whole course of the project lifetime, and will be kept alive for other 5 years and will continue updating the external audience on news related to the project’s partners and on their research works.
In order to spread information on the project, a periodical newsletter have been sent out to promote the results achieved so far. External users could subscribe to the newsletter service through the form available in the homepage of the project website. Finally, the Project Office acted as interface with the external users through the email info@gofastproject.eu.
Moreover, the project has been widely disseminated mostly during internationals events such as conferences, workshops, and through a significant number of publications and articles. The scientific results achieved within the project have been make known to the scientific community through articles that have been published in the most prestigious international journals. Moreover, in occasion of the project meetings or when articles were published in particularly prestigious journals, dissemination initiatives towards the non-scientific community have been undertaken.
The success of the project is demonstrated by the important number of publications in high impact factor journals, such as: 49 publications of which 4 in Nature journals, 4 in Physical Review Letters, 2 in Scientific Reports, 15 in Physical Review B, and an invited review in Advanced in Physics, to appear in 2015. The full list of publications is reported in the Annex II to this report. The complete list of articles is included in the Annex I to this document.
In addition, all members of the consortium have presented their main achievements within GO FAST in important international conferences. During the whole course of the project, the partners made more than 70 dissemination activities, of which most are oral presentations to a scientific event, followed by poster presentations.
On October 2013 SISSA organized an international conference held at the International Centre for Theoretical Physics in Trieste that was very successful, with more than 100 participants. All talks of the conference were recorded and made available on-line. The conferences program is available in the II periodic report of the GO FAST project.
Then, in collaboration with the FEMTOSPIN project, a joint summer school was organized at Nijmegen, in the Netherlands. The Summer School Multiscale Dynamics in Condensed Matter - Non-equilibrium spectroscopy of correlated materials and superconductors, was organized by RU in and took place in August 2014. And five more workshops and a seminar have been organized during the whole course of the project lifetime, as shown in the table hereunder.
Finally, the Editor of Advanced in Physics recently contacted us to write a review paper, which is going to appear this year with the title “Ultrafast optical spectroscopy of strongly-correlated materials: a non-equilibrium approach”.
Finally, an important part of the dissemination undertaken by the GO FAST consortium, was aimed at interfacing with other running research projects at EU and national level to achieve effective interactions with other current project at the EU level. The projects potentially interested in GO FAST were already identified during the proposal preparation. In particular, a strong interaction with the projects CRONOS and FEMTOSPIN of the 7th Framework Programme have been established. The first activity planned was the organization of the Cluster Workshop “Theory meets Industry”, which was held the 27th-28th November 2014, in Dublin, and was hosted by the coordinator of the CRONOS project, the Trinity College of Dublin. Then, as a result of a deep activity done by the three consortia to identify the key exploitable results among the big mole of knowledge generated by the projects themselves, the Cluster Exploitation Workshop of three femtodynamics projects was held on 10th and 11th of March 2015 in Brussels, with the aim of reviewing these projects in view of the industrial exploitation of the newly developed modelling and simulation capabilities.
The review was done against the FP7 contract issued by a programme called “Industrial Technologies". However, as pointed out by Anne de Baas in her introductory remarks, as F7 is transitioning to H2020 there is an increasing emphasis by the European Commission on the impact of modelling software and know-how on the global competitiveness of the European industry. This is in accordance with the Roadmap for Materials Modelling as formulated by the European Materials Modelling Council (EMMC). The reviews were thus requested to focus on industrial exploitation potential of the three projects, that demonstrated a high level of scientific achievements and significant progress in the development of new software, thus reinforcing the scientific leadership of European research groups in the field of computational materials science. While the industrial exploitation of these new software capabilities has started, the full potential of these innovative software achievements remains to be developed.

Exploitation

In view of the possible exploitation of some of GO FAST outcomes, we report the list of the KERS but now discussing details of the possible IPR protection and the form that the exploitation of these results can take (direct industrial use; patenting, technology transfer, license agreement, publication, standards). The KERs, which have been described in detail earlier, have been organized in:
GROUP 1: CODE related issues (KER 3,5,6)
GROUP 2: THEORY/METHODS related issues (KER 1,2,4)
GROUP 3: Product model related issues (KER 7)


GROUP 1: CODES

KER3: CODE to implement the time dependent dynamical mean theory t-DMFT
This KER concerns the implementation of time-dependent DMFT codes flexible enough to be implemented in popular platforms. This action is still underway, although several important progresses have been achieved and applied to model experimental data that have been included in relevant publications. The detail of the KER is reported in the attachement to this report.

KER5: CODE for the analysis of pump-probe THz spectroscopy experiments
During the project we developed a new tool for the analysis of pump-probe THz spectroscopy experiments. The tool is based on the transfer matrix and allows the user to extract the variation of the optical conductivity, dielectric function and refraction index of the sample investigated, accounting for the penetration depth of the optical pump pulse. The detail of the KER is reported in the attachement to this report.

KER6: CODE for the dynamics of order parameters in symmetry-broken phases
During the project we developed a tool for reproducing the spatial and temporal dynamics of the order parameter in a broken-symmetry phase driven out-of-equilibrium. The tool is based on the numerical solution obtained by the finite-element method of the equations of motion within a time-dependent Ginzburg-Landau (GL) functional. The tool developed can model the propagation of a local perturbation within a nanostructured system, and compute the propagation of a local perturbation. However, this tool can be easily extended to any system characterized by a broken-symmetry phase, such as charge-density wave systems and metal-to-insulator transitions. These systems are relevant for technological applications since they can be potentially employed for ultrafast switching applications. The detail of the KER is reported in the attachement to this report.

GROUP 2: THEORY and MODELS

KER 1: METHOD of photodoping and its application to transistor-like devices
As aforementioned, this KER concerns the possibility of exploiting the non-thermal self-doping observed in photo-excited cuprates for transistor-like devices. The detail of the KER is reported in the attachement to this report.
KER2: METHOD to study out-of-equilibrium models
This KER concerns a new variational method developed by GO FAST that allows to simulate the time-evolution of lattice models for correlated electron systems. The method is extremely simple and flexible as well as easy to implement. The detail of the KER is reported in the attachement to this report.
KER4: THEORY on the reasons why the Mott gap collapses for several ps after an ultra short laser pulse
This is a KER that consists of a theoretical concept developed by the Consortium that we believe will have important practical consequences.
The detail of the KER is reported in the attachement to this report.


GROUP 3: PRODUCTS

KER7: DESIGN of 3D cryostat managing system
During the project we tackled the problem of developing a 3-degrees-of-freedom-mechanical handling system (manipulator) to control and move the closed-cycle cryostat employed for the low-temperature optical measurements. Since two mechanically decoupled parts that should be moved in a synchronized way constitute the cryostat, we developed a 6-motors synchronized manipulator that avoids transmitting the vibrations of the compressor to the sample, while leaving the freedom of moving the samples. The manipulator has been designed and assembled in our labs. The manipulator is controlled via homemade software written in the Labview (National Instruments) code.
The detail of the KER is reported in the attachement to this report.

List of Websites:

www.gofastproject.eu
Project e-mail: info@gofastproject.eu


Coordinator's contact:
Michele Fabrizio
International School for Advanced Studies SISSA
Via Bonomea 265, I-34136, Trieste, Italy
phone: +39 0403787457
fax: +39 0403787528
E-mail: fabrizio@sissa.it


Project Office's contact
Laura Martinelli

IN SRL
Via A. Manzini, 21
33100 Udine (UD) - Italy
E-mail: l.martinelli@insrl.eu
phone: +39 04321573359
fax: +39 04321481732