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

2. 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. Introduction and key concepts

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

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 ~5 eV so barrier can be made very thin without electrical leakage and particle-hole recombination, D eh << a B *.

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

7. Disadvantages of 2MLG Fermi energy E F = ( ℏ v F √  / g v ) √n for carrier density n Average interaction = e 2 /  r 0 = (e 2 √  /  ) √n. (hBN dielectric  =4) Ratio r s = /E F that measures importance of interactions.  r s = e 2 /(  ℏ v F ) < 1 small and fixed ! Conclusion: For monolayer graphene, r s is a small constant, r s <1, so the two monolayers graphene 2MLG system is always weakly interacting

8. Indeed, recent Coulomb drag experiments (Manchester) show no evidence of e-h superfluidity in two monolayer graphene with very thin barriers D eh ~ 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 graphene sheets (2BLG) separated by a dielectric barrier. Biases on the top and bottom metal gates independently control the electron and hole densities. A.Perali, D. Neilson, and A. R. Hamilton, Phys. Rev. Lett. 110, 146803 (2013). Australian Provisional Patent No. 2012904903 High temperature superfluidity system Electrons Holes hBN insulating barrier Metal gate Electric contact

11. m* ~ 0.04 m e Over a wide range of n, each symmetrically biased graphene bilayer is a zero gap semiconductor with parabolic dispersion he With this quadratic energy dispersion, Fermi Energy E F in each bilayer sheet now depends linearly on the density n. This makes r s = (e 2 g v m*/ ℏ 2 √  ) (1/√n) dependent on density. r s can be made as large as r s = 9 The strongly interacting region in bilayer graphene can be reached simply by reducing the density n. EFEF

12.  Restrict barrier widths much larger than bilayer sheet thicknesses, D eh >> D h, D eh >> D e.  Then system is well approximated by single layer of electrons with parabolic energy bands  k e ± = ± ℏ 2 k 2 / 2m e * - μ e μ e  k e ± = ± ℏ 2 k 2 / 2m e * - μ e with chemical potential μ e coupled to a single layer of holes with parabolic bands  k h ± = ± ℏ 2 k 2 / 2m h * - μ h μ h  k h ± = ± ℏ 2 k 2 / 2m h * - μ h with chemical potential μ h

13. BCS + RPA approach: gap and Tc

14. We calculate the static screened V eh k-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 Δ in the excitation spectrum at 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. = (Curves are restricted to densities n > n min = 1 x 10 11 cm -2 so that the Fermi energy lies in the quadratic part of the energy band. At n= 5 x 10 11 cm -2, the Fermi temperature is T F = 175 K.) D eh (D eh ) Results for the superfluid gap at T=0 without screening

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

18. Determination of the Superfluid transition temperature In 2D, T c is given by the Kosterlitz-Thouless temperature T KT =  ρ s (T KT )/2 above which a proliferation of vortices and antivortices occurs.  s (T KT ) 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). rsrs Condensate fraction T=0 condensate fraction of the superfluid state as a function of r s (density) and of interlayer distance (d) Onset of superfludity Bi-Excitons See also: De Paolo et al., Phys. Rev. Lett. 88, 206401 (2002).

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

22. Comparison between DQMC and different screening approximations: onset of 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 graphene can 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 low densities, because of bosonic pairs, screening becomes completely ineffective.

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 graphene can 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, key concepts and possible applications. Advantages of graphene: double monolayer graphene (2MLG) and double bilayer graphene (2BLG). High-Tc superfluidity of electron-hole pairs in 2BLG: mean field + RPA calculations for the superfluid gap. Selfconsistent suppression of Coulomb screening. Electron-hole bilayers: 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. De Paolo, Rapisarda, Senatore, Excitonic Condensation in a Symmetric Electron-Hole Bilayer, Phys. Rev. Lett. 88, 206401 (2002). 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. Quantum Monte Carlo simulations: «old» results

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

31. Quantitative comparison between DQMC and different screening approximations: onset of the condensate fraction NS SSC US BCS+RPA (SSC), Unscreened (US), Normal State Screening (NS)

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

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 n min = 1 x10 11 cm -2 Tc ≥ 4 K at n min = 1 x10 11 cm -2

34.  The large control over T c 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-T c 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 T c 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 T c 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. By using bilayer graphene over a wide range of densities, 1 x 10 11 < n< 4 x10 12 cm -2, symmetrically biased graphene bilayers behave as a zero gap semiconductor with a parabolic dispersion at the Fermi level given by, Four valley-degenerate bands

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 MgB 2 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. μμ  k e- and  k h-  For μ > 0 and for superfluid energy gaps Δ < μ, we can neglect the negative branches of the electron and hole bands,  k e- and  k h-. h h e

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. h e

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, D h ~ lattice spacing. Lower graphene bilayer is an analogous electron-electron bilayer of stacked closely coupled electron layers h e

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