Andrea Perali University of Camerino Italy

1. High temperature e-h superfluidity and self-consistent suppression of screening in double bilayer graphene Andrea Perali University of Camerino Italy In collaboration with: David Neilson, University of CamerinoAlex R. Hamilton UNSW, Sydney, Australia E-mail: andrea.perali@unicam.it Web: http://www.supercondmat.org/perali MBT17 – Rostock - September 2013

2. Introduction and key concepts There has been interest for a long time to create a superfluid condensate of electron-hole pairs in semiconductor systems.Superfluidity of separated electrons and holes, Lozovik & Yudson, JETP Lett. 22, 274 (1975). The high-transition temperatures should be larger than those of conventional superconductors because condensation is driven by Coulomb interactions over the full bandwidth, rather than by phonon mediated interactions between electrons in a narrow shell around the Fermi surface.

3. Semiconductor electron-hole bilayers offer the possibility of getting the electrons and holes close enough for them to bind to form exciton bound states while a potential barrier keeps them apart to prevent them from recombining. • Yet in spite of long-standing theoretical predictions (Lozovik 1975) and large experimental efforts (Cavendish (2008) and Sandia (2009)), BCS-type superfluidity in semiconductor electron-hole bilayers has not been observed, even though interesting anomalies and deviations from the normal state behavior have been detected by Coulomb drag experiments with magnetic field (Cavendish). Lozovik& Yudson, Superfluidity of separated electrons and holes JETP Lett.22, 274 (1975).A. Croxall et al., Coulomb drag in e-h Bilayers, Phys. Rev. Lett. 101, 246801 (2008). Seamons et al.,Coulomb drag in Exciton region of eh Bilayers Phys. Rev. Lett. 102, 026804 (2009).

4. h e Electron-hole two monolayers graphene (2MLG) has been proposed to observe this superfluid. Deh Room-temperature superfluidity in graphene bilayers, Min, Bistritzer, Su, MacDonald, Phys. Rev. B 78, 121401(R) (2008)

5. Advantages of graphene • Graphene is atomically flat gapless semiconductor with • near identical conduction and valence bands. • A hexagonal boron nitride (hBN) dielectric barrier in graphene • has a large bandgap~5eV so barrier can be made very thin • without electrical leakage and particle-hole recombination, Deh << aB*.

6. Arises from its linear energy dispersion for electrons or holes E±(k) = ±ℏvFk This makes it extremely difficult to experimentally access the region of strong interactions. Localised excitons do not even form because of the massless carriers. Disadvantages of 2MLG graphene Lozovik et al., Phys. Rev. B 86, 045429 (2012).

7. Disadvantages of 2MLG Fermi energy EF = (ℏvF√ / gv)√n for carrier density n Average interaction <VCoul> = e2/  r0 = (e2√/)√n. (hBN dielectric  =4) Ratio rs= <VCoul> /EF that measures importance of interactions. rs= e2/( ℏvF) < 1 small and fixed ! Conclusion: For monolayer graphene, rsis a small constant, rs <1, so the two monolayers graphene2MLG system is always weakly interacting

8. Indeed, recent Coulomb drag experiments (Manchester) show no evidence of e-h superfluidity in two monolayer graphenewith very thin barriers Deh ~ 1 nm. R.V. Gorbachev et al., Nature Physics 8, 896 (2012).

9. This poses an exciting challenge: Can a new and experimentally practicable Graphene-type structure be designed using atomically thin crystals? A new structure that will carry the system into the superfluid state?

10. We propose a system consisting of a pair of parallel bilayer graphenesheets (2BLG) separated by a dielectric barrier.Biases on the top and bottom metal gates independently control the electron and hole densities. Metal gate Electrons Electric contact hBN insulating barrier Holes Metal gate • Perali, D. Neilson , and A. R. Hamilton, Phys. Rev. Lett. 110, 146803 (2013). Australian Provisional Patent No. 2012904903 High temperature superfluidity system

11. Over a wide range of n, each symmetrically biased graphene bilayer is a zero gap semiconductor with parabolic dispersion m* ~ 0.04 me With this quadratic energy dispersion, Fermi Energy EFin each bilayer sheet now depends linearly on the density n.This makes rs = (e2gv m*/ℏ2√) (1/√n)dependent on density. rscan be made as large as rs = 9 The strongly interacting region in bilayer graphene can be reached simply by reducing the density n. EF h e

12. Restrict barrier widths much larger than bilayer sheet thicknesses, Deh >> Dh, Deh >> De. • Then system is well approximated by single layer of electrons with parabolic energy bands ke± = ±ℏ2k2/2me* - μe with chemical potential μe coupled to a single layer of holes with parabolic bands kh± = ±ℏ2k2/2mh* - μh with chemical potential μh

13. BCS + RPA approach: gap and Tc

14. We calculate the static screened Vehk-k’ (T) within the RPA where is the unscreened Coulomb interaction. is the sum of the normal and anomalous polarisabilities of the graphene bilayer sheet.

15. Screening is suppressed because the energy gap Δin the excitation spectrumat the Fermi surface suppresses particle-hole excitations of energy less than Δ. It is these excitations which screen the long-range Coulomb interaction. When there is a gap Δ, the sum of diagonal and off-diagonal Green function contributions to the total screening bubble exactly vanishes for q = 0. Δ/μ

16. Results for the superfluid gap at T=0 Deh without screening (Deh) (Curves are restricted to densities n > nmin= 1 x 1011cm-2 so that the Fermi energy lies in the quadratic part of the energy band. At n= 5 x 1011cm-2, the Fermi temperature is TF = 175 K.)

17. Wave-vector dependence of the superfluid gap e-h pairs delocalized in wave-vector space and localized in real space • At density nmin , Δk is constant out to k/kF ~ 4, indicating pair radii ~ r0 , the average particle spacing, so the BEC region is close. • As density increases, peak remains centred on k=0. • A peak at kF should eventually form at high density, but before this the screening kills the gap and so the BCS limit is not reached.

18. Determination of the Superfluid transition temperature • In 2D, Tc is given by the Kosterlitz-Thouless temperature • TKT = ρs(TKT)/2 • above which a proliferation of vortices and antivortices occurs. • s(TKT) is superfluid stiffness

19. Density – temperature phase diagram System (A) is for two bilayer sheets embedded in a hBN dielectric. System (B) is for two bilayer sheets separated by a hBN barrier but suspended in air.

20. Diffusion Quantum Monte Carlo simulations: 2013 results Ryo Maezono et al. Phys. Rev. Lett. 110, 216407 (2013). T=0 condensate fraction of the superfluid state as a function of rs (density) and of interlayer distance (d) Bi-Excitons Condensate fraction See also: De Paolo et al., Phys. Rev. Lett. 88,206401 (2002). rs Onset of superfludity

21. ComparisonbetweenDQMC and different screening approximationsfor the T=0 condensate fraction c=N0/(N/2) US SSC NS BCS+RPA (SSC), Unscreened (US), Normal State Screening (NS) D. Neilson, A. Perali, A.R. Hamilton, preprint 2013 arXiv: 1308.0280

22. ComparisonbetweenDQMC and different screening approximations: onsetof the condensate fraction BCS+RPA (SSC), Unscreened (US), Normal State Screening (NS) US SSC NS

23. Superfluid excitation Gap (T=0) Different approximations give gaps which are different by order of magnitudes.

24. Conclusions • Double bilayer graphenecan generate equilibrium electron-hole superfluidity at high temperatures. • The necessary experimental sample parameters have already been attained in similar graphene devices. • The appearance of a superfluid gap strongly suppresses screening. • However, at high densities screening eventually kills superfluidity. • At lowdensities, because of bosonicpairs, screening becomescompletelyineffective.

25. MultiSuper2014 International Conference on Multi-Condensate Superconductiviy and Superfluidity in Solids and Ultracold Gases, Camerino, Italy, 24-27 June 2014. http://www.multisuper.ml1.net

26. Conclusions • Double bilayer graphenecan generate equilibrium electron-hole superfluidity at readily attainable temperatures. The necessary sample parameters have already been attained in existing graphene devices. • The quantitative agreement for different barriers, between our screened mean field and DQMC calculations for the superfluid condensate fraction demonstrates that : (i) the appearance of a superfluid gap strongly suppresses screening, and (ii) our normal plus anomalous polarisation screening terms dominate the effective pairing interaction in the superfluid phase. This indicates a negligible role for additional (vertex) correction terms. • At lower densities, our results for the mean field with screening and without screening converge. The e-h pair radii ~average particle spacing (local e-h pairs and pseudogap precursor effects). This explains the lack of screening at low density. • MultiSuper2014 International Conference on Multi-Condensate Superconductiviy and Superfluidity in Solids and Ultracold Gases,Camerino, Italy, 24-27 June 2014. http://www.multisuper.ml1.net

27. Outline Introduction, keyconcepts and possibleapplications. Advantages of graphene: double monolayergraphene (2MLG) and double bilayergraphene (2BLG). High-Tcsuperfluidity of electron-holepairs in 2BLG: meanfield + RPA calculationsfor the superfluid gap. Selfconsistentsuppression of Coulomb screening. Electron-holebilayers: condensate fraction of the superfluid state and comparison with diffusion Quantum Monte Carlo.

28. Applications ? (High) e-h supercurrents can be carried in a counterflow set up, with opposite electric fields acting on the electron and hole layers (importance of separare electrical contacts). Devices based on the Coulomb drag, which is predicted to show a large enhancement in the superfluid state (for instance, DC transformers). New Josephson junctions (Super-Normal-Super) using neutral superfluids, just increasing the density in a stripe of the graphene layers. Coherent light emission when the coherent electron-hole system is allowed to recombine.

29. Quantum Monte Carlo simulations: «old» results Our conclusions from mean field + RPA screening at T = 0 are consistent with DQMC calculations for the ground state of one electron-hole bilayer with quadratic dispersion in semiconductors. De Paolo, Rapisarda, Senatore, Excitonic Condensation in a Symmetric Electron-Hole Bilayer, Phys. Rev. Lett. 88,206401 (2002).

30. Quantitative comparisonbetweenDQMC and different screening approximationsfor the T=0 condensate fraction c=N0/(N/2) US SSC NS BCS+RPA (SSC), Unscreened (US), Normal State Screening (NS) D. Neilson, A. Perali, A.R. Hamilton, preprint 2013 arXiv: 1308.0280 Submitted to Phys. Rev. Lett.

31. Quantitative comparisonbetweenDQMC and different screening approximations: onsetof the condensate fraction BCS+RPA (SSC), Unscreened (US), Normal State Screening (NS) US SSC NS

32. Quantitative comparison between D-QMC and BCS+RPA for the condensate fraction BCS+RPA DQMC

33. Density – temperature phase diagram System (A) is for two bilayer sheets embedded in a hBN dielectric. System (B) is for two graphene bilayer sheets separated by a hBN barrier but suspended in air. Tc ≥ 4 Kat nmin= 1 x1011cm-2

34. The large control over Tc is highlighted by nearly an order of magnitude density range over which superfluidity can be observed in the single device. • This result is in contrast with high-Tc superconductors, where superconductivity manifests itself only in a narrow 30% band of doping centered at optimal doping.

35. There will be strong superfluid signatures below Tc The ability to make separate electrical contacts to each bilayer permits Coulomb drag & inter-layer tunnelling measurements.These will show enhancements as T drops below Tc Counterflow measurements can directly probe the superflow. • Precursor effects signalled by the pseudogap opening in the compressibility, specific heat and spin susceptibility will remain at least up to room temperature.

36. Four valley-degenerate bands By using bilayer graphene over a wide range of densities, 1 x 1011 < n< 4 x1012 cm-2, symmetrically biased graphene bilayers behave as a zero gap semiconductor with a parabolic dispersion at the Fermi level given by,

37. This system is the first multiband superfluid in the clean limit with just the one condensate. • It is an opposite case to multiband superconductors such as MgB2 where pairing is only within the bands, and multiple coupled condensates appear, with each band having its own gap parameter† • †Komendova’, Chen, Shanenko, Milovsevic’ & Peeters,PRL108, 207002 (2012)

38. Each bilayer consists of two strongly coupled lattices including inequivalent sites A; B and à ; B̃ in layers arranged in (A-B̃) stacking. Strongly coupled and hybridised layers so each bilayer forms a single conducting unit h e

39. h e h • For μ > 0 and for superfluid energy gaps Δ < μ, we can neglect the negative branches of the electron and hole bands, ke-and kh-.

40. In contrast to other multiband system, here pairing is only possible between the different hole and electron bands, leading in to a single condensate and gap. • Our approach applies to other small gap semiconductors that can be made into atomically thin flakes and used with dielectrics such as hBN. e h

42. BCS vs BCS+RPA vs D-QMC Very satisfactory agreement between BCS+RPA and DQMC for the onset of superfluidity

43. One graphene bilayer is a hole-hole bilayer of stacked closely coupled layers. The hole-hole layer separation is fixed, Dh ~ lattice spacing . h e Lower graphene bilayer is an analogous electron-electron bilayer of stacked closely coupled electron layers

44. The two bilayer sheets are separated by a hBN insulating barrier of width Deh < aB* . The barrier prevents tunnelling from one bilayer to the other bilayer. h e The two bilayers have separate electric contacts. Biases on the top and bottom metal gates independently control the electron and hole densities.