Supersolidity in electron-hole bilayers with a large density - PowerPoint PPT Presentation

Supersolidity in electron-hole bilayers with a large density imbalance Francesca Maria Marchetti ICSCE-5, 8 February 2011

In collaboration with Meera Parish & Peter Littlewood Cavendish Laboratory, University of Cambridge, UK [1] arxiv > cond.mat > 1009.1420 Previous related work on imbalanced Fermi gases: [2] Nature Physics 3, 124 (2007) [3] Phys. Rev. Lett. 98 98, 160402 (2007) Supersolidity in imbalanced electron-hole bilayers

Background: Two component Fermi gases ✧ Tunable interactions ✧ BEC-BCS crossover BEC BCS Supersolidity in imbalanced electron-hole bilayers

Background: Two component Fermi gases ✧ Tunable interactions ✧ BEC-BCS crossover [Zwierlein et al. Nature (2005)] BEC BCS Supersolidity in imbalanced electron-hole bilayers

Background: Imbalanced Fermi gases ✧ Tunable interactions ✧ BEC-BCS crossover ✧ Density imbalance frustrates pairing Can superfluidity persist in presence of a population imbalance? BEC BCS Supersolidity in imbalanced electron-hole bilayers

Background: T=0 phase diagram Polarised SF or 1FS breached pairing BCS BEC (space homogeneous) Bose-Fermi mixture Macroscpic spatial phase separation between SF and N (inhomogeneous) N SF N [Shin et al . PRL (2006)] Fully paired SF phase [Parish, Marchetti et al. Nature Physics (2007)] [Sheehy et al. PRL (2006)] Supersolidity in imbalanced electron-hole bilayers

Background: T=0 phase diagram FFLO phase SUPERSOLID PHASE Polarised SF or 1FS breached pairing BCS BEC (space homogeneous) Bose-Fermi mixture Macroscpic spatial phase separation between SF and N (inhomogeneous) N SF N FFLO [Shin et al . PRL (2006)] Fully paired SF phase [Parish, Marchetti et al. Nature Physics (2007)] [Sheehy et al. PRL (2006)] Supersolidity in imbalanced electron-hole bilayers

Background: FFLO unlikely in 3D Fermi gases FFLO phase  Displacement of Fermi surface: …allows the system to polarise …allows Fermi surface nesting  However …it costs kinetic energy …nesting is partial 
  Phase separation dominates over FFLO 
 FFLO ✧ Experiments (Rice & MIT): FFLO elusive [Liao et al. Nature (2010)] Supersolidity in imbalanced electron-hole bilayers

Motivation: Relevance to other systems  Magnetised superconductors  QCD and neutron stars Supersolidity in imbalanced electron-hole bilayers

Motivation: Relevance to other systems e -  Magnetised superconductors  QCD and neutron stars  Electron-hole systems h + (N.B. here the density is varied) BEC of bound excitons exciton insulator e - e - h + e - h + h + h + e - BCS BEC e - h + e - h + e - h + h + e - Supersolidity in imbalanced electron-hole bilayers

Motivation: Relevance to other systems e -  Magnetised superconductors  QCD and neutron stars  Electron-hole systems h + (N.B. here the density is varied) BEC of bound excitons exciton insulator e - e - h + e - e - h + e - h + h + e - BCS BEC e - h + e - e - h + e - h + h + e - Supersolidity in imbalanced electron-hole bilayers

Electron-hole bilayers: FFLO more ‘likely’ h + electron-hole bilayers: h + h + better route to achieve FFLO! e - e - e - h + 1 h + e - 1. Enhanced Fermi surface nesting e - e - 3D 2D 1D QW 1 QW 2 2. No phase separation on macro-scales conduction band valence band position Supersolidity in imbalanced electron-hole bilayers

Electron-hole bilayers: FFLO more ‘likely’ h + electron-hole bilayers: h + h + better route to achieve FFLO! e - e - e - h + 1 h + e - 1. Enhanced Fermi surface nesting e - e - 3D 2D 1D QW 1 QW 2 2. No phase separation on macro-scales conduction band valence band Novel ‘bosonic’ limit of FFLO position Supersolidity in imbalanced electron-hole bilayers

Electron-hole bilayers: Experimental realisations h + h + h +  Optically pumped coupled quantum wells e - e - e - h + 1 h + e - e - e - [Butov et al. Nature (2002)] [Snoke et al. Nature (2002)] [ … and many more … ]  Individually contacted doped layers QW 1 QW 2 conduction band [Croxall et al. PRL (2008)] valence [Seamons et al. PRL (2009)] band ⇒ Equilibrium phase diagram position Supersolidity in imbalanced electron-hole bilayers

Electron-hole bilayers: Hamiltonian 2 2 2 1 1 2 1 1 2 1 1 1 bare intra-layer Coulomb interaction bare inter-layer ⇒ No spin (spin polarised) Supersolidity in imbalanced electron-hole bilayers

Limit of large imbalance: Variational ground state 2  single particle in the 2 nd layer + 1 1 Fermi see in the 1 st layer 1 1 1 relative k CoM Q 1 1 (SF) (FFLO)  Good description of both low and high density limit Supersolidity in imbalanced electron-hole bilayers

Limit of large imbalance: Variational ground state 2  single particle in the 2 nd layer + 1 1 Fermi see in the 1 st layer 1 1 1 relative k CoM Q 1 1 Interpolate by screening the interactions (within RPA) = dress with density fluctuations, including particle-hole excitations (N) (SF)  Good description of both low and high density limit Supersolidity in imbalanced electron-hole bilayers

Phase diagram 1. Minimise 2. Eigenvalue equation 3. Unbinding transition  Read also as mean-field (linearised) gap equation ⇒ Current theories [Yamashita et al. J Phys Soc Japan (2010)] ⇒ … some neglect screening [Pieri et al. PRB (2007)] ⇒ … some neglect finite Q FFLO (next talk) [Subasi et al. PRB (2010)]  SF & FFLO  Parameters: ⇒ mass ratio ⇒ interlayer distance exciton Bohr radius 
 ⇒ dimensionless density Supersolidity in imbalanced electron-hole bilayers

Phase diagram 1. Minimise 2. Eigenvalue equation 3. Unbinding transition  Read also as mean-field (linearised) gap equation ⇒ Current theories [Yamashita et al. J Phys Soc Japan (2010)] ⇒ … some neglect screening [Pieri et al. PRB (2007)] ⇒ … some neglect finite Q FFLO (next talk) [Subasi et al. PRB (2010)]  SF & FFLO  Parameters: ⇒ mass ratio ⇒ interlayer distance exciton Bohr radius 
 ⇒ dimensionless density Supersolidity in imbalanced electron-hole bilayers

Phase diagram of fully imbalanced EH bilayer high density low density  GaAs: e - h + h + e - FFLO region enhanced if minority particle lighter ⇒ mass ratio ⇒ interlayer distance ⇒ dimensionless density [arxiv/cond.mat/1009.1420] Supersolidity in imbalanced electron-hole bilayers

Effect of screening on the LOFF phase  Unscreened case ⇒ Numerically ⇒ Analytically Supersolidity in imbalanced electron-hole bilayers

Effect of screening on the LOFF phase  Unscreened case ⇒ Numerically ⇒ Analytically Supersolidity in imbalanced electron-hole bilayers

Which kind of FFLO phase? Supersolidity in imbalanced electron-hole bilayers

Dilute gas of minority particles: Interactions  Normal phase: 2
 effective interaction between two unbound minority particles (RPA) 1
 ⇒ repulsive and dipolar : the dilute gas is a Fermi liquid  Excitonic phase: 2
 1
 well separated excitons (dipoles) ⇒ also repulsive and dipolar : no phase separation (biexciton formation suppressed for single spin species & ) Supersolidity in imbalanced electron-hole bilayers

Phenomenology of ‘bosonic’ FFLO 2
  Landau theory for = exciton density 1
 chemical potential minimum energy at Supersolidity in imbalanced electron-hole bilayers

Phenomenology of ‘bosonic’ FFLO 2
  Landau theory for = exciton density (weak crystalisation theory) 1
 chemical potential minimum energy at  Complex order parameter  Minimal energy solution ⇒ supersolid: exciton condesate with 2D spatial modulation (diagonal and off-diagonal order) Supersolidity in imbalanced electron-hole bilayers

Observing FFLO 2
 1. Light scattering off the spatial modulations 1
 2. Photon angular emission (electron hole recombining): finite momentum pairing ( ) Exciton binding energy in GaAs = upper bound for the FFLO critical temperature GaAs Supersolidity in imbalanced electron-hole bilayers

Conclusions  Electron-hole bilayers: promising for observing exotic pairing phenomena  Evidence of FFLO phase at large imbalance: finite-Q exciton in presence of a fermi sea  Dilute gas of finite-Q excitons: condensation with 2D spatial modulation (a supersolid) ⇒ bosonic limit of FFLO  Prospects for experimental observation Supersolidity in imbalanced electron-hole bilayers

Additional slides

Motivation: Relevance to other systems  Magnetised superconductors Zeeman term SC Supersolidity in imbalanced electron-hole bilayers

Include p-h excitation (in within RPA)  Interpolate between high and low 2 density via RPA 1 1 1 1  Full imbalance limit: 1 polarisation operator 1 1 (Linhard function) 1
 = + + … = 1
 1
 + 1
  Screening between 12 through 11 1
 = + + … = 1
 1
 + 1
 Supersolidity in imbalanced electron-hole bilayers

Phase diagram 1. Minimise 2. Eigenvalue equation 3. Unbinding transition: compare with  Read also as mean-field (linearised) gap equation (BEC-BCS crossover) [Yamashita et al. J Phys Soc Japan (2010)] [Pieri et al. PRB(2007)] [Subasi et al. PRB (2010)] ⇒ minority particle chemical potential  SF & FFLO Supersolidity in imbalanced electron-hole bilayers