Abstract
All known superfluid and superconducting states of condensed matter are enabled by composite bosons (atoms, molecules and Cooper pairs) made of an even number of fermions. Temperatures where such macroscopic quantum phenomena occur are limited by the lesser of the binding energy and the degeneracy temperature of the bosons. Highcritical temperature cuprate superconductors set the present record of ~100 K. Here we propose a design for artificially structured materials to rival this record. The main elements of the structure are two monolayers of a transition metal dichalcogenide separated by an atomically thin spacer. Electrons and holes generated in the system would accumulate in the opposite monolayers and form bosonic bound states—the indirect excitons. The resultant degenerate Bose gas of indirect excitons would exhibit macroscopic occupation of a quantum state and vanishing viscosity at high temperatures.
Introduction
Coherent states of excitons have been a subject of intense theoretical studies^{1,2,3}. A general framework for creation and manipulation of degenerate gases of indirect excitons has been established in prior experimental studies of GaAsbased coupled quantum wells (CQWs) where electrons and holes are confined in GaAs quantum wells separated by a thin AlGaAs barrier^{4,5}. Here we apply similar principles to the design of a CQW from atomically thin materials stacked on top of each other. Research on such van der Waals heterostructures is gaining momentum in the last few years^{6}, and their quality and availability is steadily improving.
In the proposed device (Fig. 1a), an indirect exciton is composed from an electron and a hole located in two different MoS_{2} layers separated by a hexagonal boron nitride (hBN) insulating barrier and surrounded by hBN cladding layers. The zdirection electric field is controlled by voltage applied to external electrodes. The applied field modifies the band structure in a way that it becomes advantageous for optically excited electrons and holes to reside in the opposite MoS_{2} monolayers and form indirect excitons (Fig. 1b).
The design of the MoS_{2}/hBN structure is similar to that of GaAs/AlGaAs CQW we studied previously^{4,5}, except GaAs is replaced by MoS_{2} and AlGaAs by hBN. Here we predict the phase diagram of such a device using the results of numerical calculations and scaling arguments. Our most intriguing finding is that in the proposed MoS_{2}/hBN structures degenerate Bose gas of indirect excitons can be realized at recordhigh temperatures.
Results
Degeneracy temperature
The characteristic temperature T_{d} at which excitons become degenerate is determined by their density n_{x} per flavour (spin and valley), and effective mass m_{x}=m_{e}+m_{h}:
Here m_{e} and m_{h} are the electron and hole effective masses. For n_{x}~10^{10} cm^{−2}, and m_{e}=0.07, m_{h}=0.15, m_{x}=0.22 representative of GaAs CQW, we find T_{d}~3 K. (Here and below all the masses are in units of the bare electron mass.) At such temperatures longrange spontaneous coherence of indirect excitons is observed^{5}.
To explain why MoS_{2}based CQW would possess much higher T_{d} than GaAsbased ones, we consider the characteristic length and energy scales in the problem, which are the exciton Bohr radius a_{x}=ħ^{2}ε_{1}/μe^{2} and the Rydberg energy defined in terms of reduced mass μ=m_{e}m_{h}/m_{x}. For estimates we use the effective static dielectric constant ε_{1}=4.9 of hBN. We also use the calculated value μ=0.25 (refs 7, 8) for MoS_{2}. On the basis of the measurements^{9} done on a related compound MoSe_{2}, this value should be accurate to within 20%. We find a_{x}≈1.0 nm, much shorter than 15 nm in GaAs CQW. In turn, Ry_{x}≈140 meV in the proposed device, about forty times larger than in GaAs.
Since it enters the denominator of equation (1), one may think that having larger m_{x} is unfavourable for attaining higher T_{d}. In fact, when n_{x} can be controlled, the opposite is true. Indeed, equation (1) can be rewritten as
The upper limit on is imposed by quantum dissociation of excitons that occurs when the ratio of the exciton size (the inplane gyration radius) r_{x} and the mean interexciton distance reaches the critical value^{10,11,12} of about 0.3. In the case of our primary interest where interlayer centretocentre distance c≈a_{x} and m_{e}=m_{h}=2μ, we estimate r_{x}=2.4a_{x} (Fig. 2), so that the corresponding Mott critical density n_{M} is set by the condition
Substituting this into equation (2), we obtain
Hence, the key to high T_{d} is the enhanced value of Ry_{x}. The exciton binding energy E_{ind}≈0.6 Ry_{x} (Fig. 2) does not pose further fundamental limitations. Actually, the conclusion that the theoretical maximum of T_{d} is proportional to Ry_{x} follows from dimensional analysis. If c~a_{x}, m_{e}~m_{h} and excitons are treated as an equilibrium Bose gas, then Ry_{x} is the only relevant energy scale in the problem.
In reality, only quasiequilibrium state is possible because of exciton recombination, and so limitations posed by the finite exciton lifetime τ must be discussed. Radiative recombination of indirect excitons requires interlayer tunnelling. The rate of this process can be made exponentially small by adjusting the barrier width. In GaAs CQW, τ can be routinely made in the range of microseconds and can be controllably varied over several orders of magnitude. The large τ enables creation of indirect exciton gases of highdensity n_{x}=Pτ with low optical excitation power P that does not cause overheating of either the lattice or the excitons. This is why for the task of achieving cold dense exciton gases, indirect excitons are superior to conventional bulk excitons or twodimensional (2D) direct excitons (bound states of electrons and holes in the same layer).
In the proposed device, the interlayer tunnelling rate would decay exponentially with the number N of layers in the hBN spacer. For N=2, the tunnelling rate is comparable to that in the GaAs CQW^{4,5} (see Methods) indicating that longlife indirect excitons can be realized in MoS_{2}/hBN structures. The tunnellinglimited lifetime can be enhanced by possible rotational misalignment of the MoS_{2} layers, which makes the excitons indirect not only in real but also in momentum space, similar to excitons in MoS_{2} multilayers^{9,13}. Hence, sufficiently long τ may perhaps be achieved with a monolayer hBN spacer. Working with very small N, one has to worry about possibility of dielectric breakdown of the hBN spacer. To make the indirect exciton more energetically favourable than the direct one, a voltage equal or larger than (E_{dir}−E_{ind})/e must be applied between the MoS_{2} layers. Assuming the direct exciton binding energy E_{dir} of approximately^{8,14} 0.5 eV, for N=2 hBN spacer, the required voltage is about 0.4 V, which is safely below the breakdown limit^{15}.
A schematic phase diagram of a neutral electron–hole system in the proposed heterostructure is shown in Fig. 3. The solid line represents the Mott transition. It emanates from the T=0 critical point computed according to equation (3). At the transition there is a discontinuous jump in the degree of exciton ionization^{16}. Above the Mott critical temperature T_{M} the transition changes to a smooth crossover. Such a crossover has been studied experimentally in photoexcited singlewell GaAs and InGaAs structures^{17,18}. We assume that T_{M} should be lower than and use in Fig. 3 for illustration. The quantum degeneracy line (equation 1) shown by the dasheddotted line in Fig. 3 demarcates a crossover from classical to degenerate Bose gas of excitons. At n=n_{M} this line extrapolates to the temperature (equation 4).
Whereas the Mott transition represents quantum dissociation of excitons, the thermal dissociation occurs above the Saha temperature , which is shown by the dashed line near the T axis in Fig. 3. This line marks a crossover from the exciton phase to a classical plasma. Since their binding energy is very large, indirect excitons in MoS_{2}/hBN structure remain stable against the thermal dissociation well above the room temperature in a broad range of electron densities.
Formation of quantum degenerate Bose gas of longlifetime repulsively interacting indirect excitons leads to local exciton superfluidity below T_{d}. The superfluidity spreads over a macroscopic area at the Berezinskii–Kosterlitz–Thouless transition temperature^{19}
Superfluid transport of neutral indirect excitons produces dissipationless charge currents in the opposite directions in the two layers^{2}. To observe and utilize this effect, one can, for example, make separate contacts to each layer and form a closed circuit for the hole layer. The electric current in the electron layer will then be dissipationless. This is referred to as the counterflow superconductivity^{20}.
The most intriguing conclusion we draw from Fig. 3 is that in the proposed MoS_{2}/hBN structures degenerate Bose gas of indirect excitons can be realized at recordhigh temperatures.
Discussion
In the remainder of this paper, we overview phenomena analogous to a coherent state of indirect excitons in other 2D systems. We choose not to survey systems where electrons and holes reside in the same layer, such as quantum wells in microcavities^{21}. They are interesting in their own right but counterflow superconductivity therein is impossible.
First, evidence for brokensymmetry phases conceptually similar to condensates of indirect excitons have been reported in GaAs CQW^{22}. These phases arise in the presence of a quantizing magnetic field at subKelvin temperatures. Next, a number of brokensymmetry states, some of which are similar to exciton condensates, have been predicted to form in bilayer graphene^{23}. The doublelayer graphene (DLG) systems must be specially mentioned because a hightemperature coherent state in DLG was theoretically discussed recently. The fundamental obstacle to exciton condensation in DLG is that monolayer graphene is a zerogap semimetal with a linear quasiparticle dispersion. Although in such a system phases of weakly bound excitons may exist, the corresponding critical temperature is extremely sensitive to the effective strength of the electron–hole attraction. Calculation of the latter requires accounting for screening of the longrange Coulomb interaction as well as shortrange correlation effects, both of which are challenging problems. Theoretical estimates of the critical temperature in DLG range from hundreds of Kelvin^{24,25,26} to a few milliKelvin^{27}. We wish to stress that there is no room for such an enormous uncertainty in our proposal based on MoS_{2}, a semiconductor with a significant bandgap, modest dielectic constant and exceptionally stable exciton state. The dependence of the characteristic temperatures T_{d} and T_{BKT} on electron density is linear rather than exponential. All pertinent numerical factors are constrained by numerous prior studies of similar semiconducting systems. Therefore, the error in the estimated temperatures should be small for the system considered in our paper.
The scaling law of equation (4) elucidates a general principle for realizing hightemperature coherent states of excitons. The proposed design is an initial blueprint and is amenable to further optimization. For example, one^{28} or both of MoS_{2} layers can be substituted by a different transition metal dichalcogenide (TMD), such as WS_{2} or WSe_{2}. The two MoS_{2} monolayers may be replaced by a single severalmonolayerthick TMD encapsulated by hBN in which indirect excitons would be composed of electrons and holes confined at the opposite sides of the TMD layer. The outlined principle for realizing hightemperature superfluidity can also be extended to layered materials other than TMD and hBN.
Besides providing a new platform for exploring fundamental quantum phenomena, indirect excitons in van der Waals structures can be also utilized for the development of optoelectronic circuits^{4}. In such devices, inplane potential landscapes for excitons are created and controlled by external electric fields that couple to the permanent dipole moment p=ec of the indirect excitons. The operation temperature for excitonic circuits in the van der Waals structures is expected to exceed by an order of magnitude the ~100 K record^{29} set by GaAsbased CQWs.
Finally, we note that Fig. 3 was constructed following the example of the MonteCarlo simulations^{10,11,12} in which only exciton phases with two possible spin flavours were considered. In TMDs, spin and orbital degeneracies of indirect excitons may have a more intriguing structure. Such degeneracies can be controlled by strong spin–orbit coupling^{30}, polarization of the excitation beam^{7,31} or manybody interactions^{32}.
Experimental realization of superfluidity and counterflow superconductivity as well as excitonic circuits and spintronic/valleytronic devices in atomically thin heterostructures may have farreaching implications for science and technology.
Methods
Interlayer tunnelling
The action S for tunnelling of quasiparticles across the hBN spacer can be estimated from the usual formula for the rectangular potential barrier of height U_{b}:
Here c_{b}=Nc_{1} is the tunnelling length, N is the number of hBN layers, c_{1}=0.333 nm is the thickness of one hBN layer, a_{B}=0.0529, nm is the hydrogen Bohr radius and m_{b} is the effective carrier mass inside the barrier. If the chemical potential is in the middle of the hBN energy gap, we expect U_{b}≈3 eV. Assuming also m_{b}≈0.5 in units of the bare electron mass, we obtain S≈5N, which is in agreement with S=4.6N deduced from the tunnelling conductance measured in graphene/hBN/graphene structures^{15}. In comparison, in GaAs/AlGaAs CQW structures with m_{b}≈0.35, U_{b}≈0.15 eV and d_{b}=4.0 nm we get S=9.4. Hence, for N=2 the tunnelling rate is comparable to that in the GaAs CQW^{4,5}. More detailed estimates would have to include the trapezoidal shape of the barrier (Fig. 1b) and precise nature of the band alignment in the TMD/hBN structures. Nevertheless, it is clear that by a minor adjustment of N in the range from, say, 2 to 4, sufficiently long interlayer tunnelling lifetimes can be achieved for indirect excitons.
Binding energy of indirect excitons
We model both the electron and the hole that compose an indirect exciton as 2D quantum particles confined in the midplanes of two separate MoS_{2} layers. The Schrödinger equation for the relative motion reads
where μ is the reduced mass. We model the potential U(r) of electron–hole Coulomb interaction using the continuummedium electrostatics^{8}. This simplified approach neglects frequency dependence of the dielectric functions of the materials involved. We approximate each of MoS_{2} layers as a uniaxial dielectric slab of thickness c_{2}=0.312 nm with principal dielectric tensor components^{8} and in the directions perpendicular and parallel to the z axis, respectively. In turn, the hBN spacer is modelled as a slab of thickness Nc_{1}, c_{1}=0.333 nm, with the dielectric constants^{33} and . We define the anisotropy parameter κ_{j} and the effective dielectric constant ε_{j} of each material by
so that κ_{2}=1.44, ε_{2}=9.91 (MoS_{2}) and κ_{1}=1.37, ε_{1}=4.89 (hBN). The interaction potential can be found by solving the Poisson equation for this layered system, which is done by reducing it to a set of linear equations for the Fourier component and using the inverse Fourier transform:
By symmetry, depends only on the magnitude q=q of vector q not on its direction. After simple algebra, we arrive at the result , where functions C(q) and D(q) are given by
Here we introduced the shorthand notations c=Nc_{1}+c_{2} (the centretocentre distance) and k_{j}=κ_{j}q (the zdirection wavenumber of the evanescent Fourier harmonics in medium j). From numerical calculations using equations (9)–(11), , we found that potential U(r) is accurately approximated by the following analytical expression:
With a suitable choice of coefficients 0<A<1 and B>0 this form produces asymptotically exact results for U(r) at both small and large r.
To solve equation (7), we discretized it on a realspace 2D grid (typically, 75 × 75). The resultant linear eigenvalue problem was diagonalized by standard numerical methods yielding the binding energy E_{ind}(N) and the gyration radius r_{x}(N) of indirect excitons. The latter is defined in terms of a normalized groundstate wavefunction φ(r) by means of the integral
The results are shown in Fig. 2. For N=2 we find r_{x}=2.50 nm=2.43a_{x} and E_{ind}=87 meV≈0.6 Ry_{x}. This binding energy is an order of magnitude larger than E_{ind}=4–10 meV typical for excitons in GaAs/AlGaAs CQW structures^{29,34}.
Zerotemperature phases
It is instructive to complement the above discussion of the finiteT phase diagram (Fig. 3) with commenting on the T=0 phases. Such phases include electron–hole Fermi gas, exciton Bose gas and exciton solid. The approximate phase boundaries based on available MonteCarlo calculations^{10,11,12,35,36} are shown in Fig. 4. Although those simulations were done for electron–hole bilayers in vacuum, an approximate correspondence with our dielectric environment can be achieved if we neglect the correction term in the parenthesis in equation (12). The mapping is then obtained by setting the distance between the layers to d=κ_{1}c and including the effective dielectric function ε_{1} in the definition of the electron Bohr radius:
On the horizontal axis in Fig. 4 we plot the dimensionless intralayer distance parameter , where n_{e} is the total electron density. Solid phases form at large r_{s} and d/a_{e}. The exciton dipole solid crosses over to the interlocked Wigner crystals of electrons and holes as the interlayer distance becomes larger than the characteristic intralayer one, d≳r_{s}a_{e}.
The considered MoS_{2}/hBN heterostructure with N=2 corresponds to d/a_{e}≈2.8 for which the ground state is never a solid phase. Instead, at large r_{s}, that is, at low electron density, the ground state of the system is an exciton gas. As r_{s} decreases, the Mott transition to electron–hole Fermi gas occurs. At this transition the excitons dissociate owing to screening and phase space filling^{1}. The latest estimates^{12} of the T=0 Mott transition at d=2.8a_{e} give r_{s}≈6, which corresponds to equation (3). Interestingly, a Bardeen–Cooper–Schriefferlike excitonic state of a dense electron–hole gas predicted by the earlier theory^{37} was not found in the cited MonteCarlo calculations^{10,12}.
Unless the effective interlayer separation d is small, the interaction between indirect excitons is dominated by the classical dipole repulsion term^{38,39}. However, as shown previously^{40,41,42}, at d<0.87a_{e} quantum exchange–correlation effects cause the change of repulsion to attraction leading to appearance of biexcitons phases^{12,39}. Narrow ranges of modulated phases (stripes, bubbles or supersolids) may exist near any of the firstorder phase transition lines^{43}.
Additional information
How to cite this article: Fogler, M. M. et al. Hightemperature superfluidity with indirect excitons in van der Waals heterostructures. Nat. Commun. 5:4555 doi: 10.1038/ncomms5555 (2014).
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Acknowledgements
This work was supported by the US ONR & UCOP (M.M.F.), NSF (L.V.B.) and also by the European Research Council and ECFET European Graphene Flagship (K.S.N.). We are grateful to André K. Geim for comments.
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Fogler, M., Butov, L. & Novoselov, K. Hightemperature superfluidity with indirect excitons in van der Waals heterostructures. Nat Commun 5, 4555 (2014). https://doi.org/10.1038/ncomms5555
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DOI: https://doi.org/10.1038/ncomms5555
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