Introduction: The Hubbard model (HM) describes particles confined in a lattice (1), see Fig. 1.a. When the particle hoping t dominates, a superfluid phase which exhibits a spatial phase coherence emerges. On the contrary, when the on site interaction U dominates, a Mott insulator (MI) which is an incompressible state with the same integer number of particle per site is formed, see Fig. 1.b.
Nowadays, there is a variety of different HM quantum simulators like cold atomic vapours or arrays of Josephson junctions (2). We have proposed recently a new candidate in the solid state: semiconductor dipolar excitons confined in artificial electrostatic lattices.
Figure 1: a Semiconductor dipolar excitons (purple balls) confined in an artificial electrostatic lattice. They experience hoping (t), on site (U) and nearest neighbouring (V) interactions. b Phase diagram of the Bose HM with insulating phases in blue. Mott Insulator (MI) and checkerboard (CB) arrangements are represented on the right. Superfluid phase (SF) is drawn in grey and the supersolid (SS) expected phase in orange. c GaAs/AlAs heterostructure embedding a double quantum well (DQW), with surface electrodes negatively biased. Dipolar excitons (purple balls) are optically created by a resonant broad laser spot. d SEM image of a square 250 nm period electrode pattern.
Dipolar excitons: Our device is based on a GaAs/AlAs high quality heterostructure grown with MBE by our collaborators of Princeton university. The structure that embeds two GaAs quantum wells (DQW) separated by a thin AlAs barrier is polarized by surface electrodes while the sample back surface is grounded, see Fig. 1.c. A laser resonant with the DQW absorption injects electrons and holes that minimize their potential energy by tunneling to a different quantum well and form spatially indirect excitons (3).
These indirect excitons (X) are composite bosons with strong repulsive dipolar interaction that can be studied at thermodynamic equilibrium at 330 mK. Their physical properties are obtained by studying photons emitted when electrons and holes recombine. The specificity of our photoluminescence (PL) experiments is that the laser excitation is 100ns long, repeated at around 1MHz so that the PL is studied dynamically after the laser extinction (5).
Using a circular electrode, to trap the X, the quasi-condensation of X has been explored in our group (6,7). Using electronic lithography process, we have more recently built 2D square lattice geometries with 3 um (8,9), 800 nm and 400 nm (10) down to 250 nm (11,12) periods.
Bose Hubbard model: To identify insulating phases like MI and CB, we have setup an optical measurement of the compressibility of our exciton fluid, by statistically analysing PL spectra measured in the same experimental conditions (10). The variance of the peak intensity is linked through the fluctuation dissipation theorem to the excitons compressibility. Incompressible phases are signalled by reduced PL fluctuations, see coloured area in Fig. 2.a compared to classical phases that exhibit poissonian fluctuations (grey area in Fig. 2,a).
Taking advantage of the strong repulsive interaction between dipolar excitons, we recently reported excitonic Mott insulating states with 1 (Fig. 2.c) boson per site. As the dipolar interactions are quasi-long ranged, we also observed a checkerboard (CB) phase (Fig. 2.b) which builds up at half filling of the lattice sites (11, Fig. 1.b). It is the signature of the Bose HM extended by nearest neighboring (NN) interactions (V in Fig. 1.a).
Figure 2: a Compressibility of excitons confined in a 250 nm period square lattice as a function of the laser excitation power that controls the excitons density. The Compressibility shows two minima signalling MI and CB phases at unitary and half filling of the lattice sites. The X spectra in MI and CB phases are shown in c and b respectively. K is the compressibility normalized by the poissoninan compressibility. The spectra in e and d are the corresponding MI and CB for charged excitons confined in the 250 nm lattice. a, b ,c are adapted from Ref. (11). d and e are adapted from Ref. (12).
Fermi Hubbard model: At this stage, it is important to note that one challenge when working with X is the presence of excess free charges in the sample. In our heterostructure, we measured, an upper limit of excess charges of 4x107cm-2 in the neutral regime (11) in agreement with transport measurement (13). If now we change by a couple of ‰ the gate voltage as in Fig. 3.b, we observe the emergence of a second PL peak (red) around 1 meV below the excitonic one (blue). We have attributed this peak to the formation of charged excitons (CX) in the lattice, i.e. composite fermions made of one electron and two holes. By tuning the laser excitation power that controls the excitons density and the voltage that controls the holes density, we manage to observe MI (Fig. 2.e) and CB (Fig. 2.d) phases of CX. We have then simulated the Fermi HM extended by NN interaction (12).
Bose Fermi mixture: Finally we went one step further to study mixtures of X and CX with a total density of one particle per site, by varying the CX density from 0 to 1. Both species are subject to intra and inter species repulsive NN interactions, see Fig. 3.a. We recovered MI phases for the majority specie when the minority specie density is below 15%. Otherwise the two species are in normal phases, except for 3 specific CX filling, normally (1/3,1/2,2/3), for which we observed incompressible phases of charged and neutral excitons, see Fig. 3.c (12).
To determine whether these incompressible phases are two separated MI of each species or a more organized two-species insulator, we study the energy spacing between the X and CX PL lines (black points in Fig. 3.d). The energies of the two PL peaks are increased by the mean particle interaction with its NN, which only depends on the specie and on the spatial organization. Having measured all the NN interaction terms, we calculated the expected energy spacing for various phases. Comparing these with our experiments, we concluded that we do not have two separated MI phases but for the three CX fractional fillings, dual alternating stripe patterns, see Extended Data Fig. 4 in Ref. (12).
Figure 3: a Neutral (X, blue balls) and charges (CX, red balls) excitons confined in the 2D square lattice experiencing intra and inter species NN interactions. b Spectrum showing two PL peaks separated by around 1 meV attributed to CX (red) and X (blue). c Compressibility of X (blue points) and CX (red points) for a total unitary site filling as a function of the CX filling. d Energy difference between the X and the CX PL peaks as a function of the CX filling. The expected phases are drawn close to there corresponding energy differences. The shaded area mark the presence of insulating phases in c and d. This figure is adapted from Ref. (12).
Conclusion: In Ref. (12) we have implemented the Bose-Fermi Hubbard model extended by long range interactions. So far, this model has not yet been studied theoretically and our studies evidence that dual density waves are energetically preferred, unlike for mixtures bound to short-range interactions (14-15). Then, we note that our experimental bath temperature (330 mK) is currently too high to expect bosonic superfluid phases to emerge. Nevertheless, it can be reduced by over one order of magnitude (to around 10 mK) where extended spatial coherence possibly arises. In this regime, neutral and charged excitons mixtures could provide a platform to explore the low doping regime of fermionic Mott insulating phases relevant to high temperature superconductivity. In that respect, our ability to design desired lattice geometries, e.g. with a controlled number of lattice confined states, provides a clear asset. Along the same direction, one could even envision to engineer supersolid phases of neutral excitons, stabilised by a robust checkerboard solid of charged excitons.
References:
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