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 Camerino
Alex R. Hamilton UNSW, Sydney, Australia
E-mail: andrea.perali@unicam.it
Web: http://www.supercondmat.org/perali
MBT17 – Rostock - September 2013
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.
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).
Electron-hole two monolayers graphene (2MLG)
has been proposed to observe this superfluid.
h
Deh
e
Room-temperature superfluidity in graphene bilayers,
Min, Bistritzer, Su, MacDonald,
Phys. Rev. B 78, 121401(R) (2008)
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, Deh << aB*.
Disadvantages of 2MLG graphene
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.
Lozovik et al., Phys. Rev. B 86, 045429 (2012).
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/( ℏ vF) < 1
small and fixed !
Conclusion:
For monolayer graphene, rs is a small constant, rs <1, so
the two monolayers graphene 2MLG system
is always weakly interacting
Indeed, recent Coulomb drag experiments (Manchester) show
no evidence of e-h superfluidity in two monolayer graphene with very thin
barriers Deh ~ 1 nm.
R.V. Gorbachev et al., Nature Physics 8, 896 (2012).
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?
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.
Metal gate
Electrons
hBN insulating barrier
Holes
Metal gate
A. Perali, D. Neilson , and A. R. Hamilton,
Phys. Rev. Lett. 110, 146803 (2013).
Australian Provisional Patent
No. 2012904903
High temperature superfluidity system
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 EF in each bilayer sheet
now depends linearly on the density n.
EF
This makes rs = (e2gv m*/ℏ2√ ) (1/√n)
dependent on density.
rs can be made as large as rs = 9
The strongly interacting region in
bilayer graphene can be reached
simply by reducing the density n.
h
e
 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
BCS + RPA approach: gap and Tc
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.
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
Results for the superfluid gap at T=0
Deh
(Deh)
without
screening
(Curves are restricted to densities n > nmin = 1 x 1011 cm-2 so that the Fermi
energy lies in the quadratic part of the energy band. At n= 5 x 1011 cm-2, the
Fermi temperature is TF = 175 K.)
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.
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
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.
Diffusion Quantum Monte Carlo simulations:
2013 results
Condensate fraction
T=0 condensate fraction
of the superfluid state
as a function of rs (density)
and of interlayer
distance (d)
Ryo Maezono et al.
Phys. Rev. Lett. 110,
216407 (2013).
Onset of superfludity
Bi-Excitons
rs
See also: De Paolo et al.,
Phys. Rev. Lett. 88, 206401 (2002).
Comparison between DQMC and different screening
approximations for 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
Comparison between DQMC and different screening
approximations:
onset of the condensate fraction
BCS+RPA (SSC), Unscreened (US), Normal State Screening (NS)
US
SSC
NS
Superfluid excitation Gap (T=0)
Different approximations give gaps which are different
by order of magnitudes.
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.
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
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
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.
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.
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).
Quantitative comparison between DQMC and different
screening approximations for 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.
Quantitative comparison between DQMC and different
screening approximations:
onset of the condensate fraction
BCS+RPA (SSC), Unscreened (US), Normal State Screening (NS)
US
SSC
NS
Quantitative comparison between D-QMC and
BCS+RPA for the condensate fraction
BCS+RPA
DQMC
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 K at nmin= 1 x1011 cm-2
 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.
 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.
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,
Four valley-degenerate bands
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)
Each bilayer consists of two strongly coupled lattices
including inequivalent sites A; B and à ; B̃
in layers arranged in (A-B̃ ) stacking.
h
e
Strongly coupled and
hybridised layers so
each bilayer forms a
single conducting unit
h
h
e
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- .
e
h
BCS vs BCS+RPA vs D-QMC
Very satisfactory agreement between BCS+RPA and DQMC
for the onset of superfluidity
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
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.