Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 7-Day Trial for You or Your Team.

Learn More →

High-temperature superfluidity with indirect excitons in van der Waals heterostructures

High-temperature superfluidity with indirect excitons in van der Waals heterostructures ARTICLE Received 5 May 2014 | Accepted 27 Jun 2014 | Published 28 Jul 2014 DOI: 10.1038/ncomms5555 High-temperature superfluidity with indirect excitons in van der Waals heterostructures 1 1 2 M.M. Fogler , L.V. Butov & K.S. Novoselov 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. High-critical temperature cuprate superconductors set the present record of B100 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. 1 2 Department of Physics, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA. School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK. Correspondence and requests for materials should be addressed to M.M.F. (email: [email protected]). NATURE COMMUNICATIONS | 5:4555 | DOI: 10.1038/ncomms5555 | www.nature.com/naturecommunications 1 & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5555 oherent states of excitons have been a subject of intense and energy scales in the problem, which are the exciton Bohr 1–3 2 2 2 2 theoretical studies . A general framework for creation radius a ¼ : E /me and the Rydberg energy Ry ¼ ‘ 2ma x 1 Cand manipulation of degenerate gases of indirect excitons defined in terms of reduced mass m¼ m m /m . For estimates we e h x has been established in prior experimental studies of GaAs-based use the effective static dielectric constant E ¼ 4.9 of hBN. We also coupled quantum wells (CQWs) where electrons and holes are use the calculated value m¼ 0.25 (refs 7,8) for MoS . On the basis confined in GaAs quantum wells separated by a thin AlGaAs of the measurements done on a related compound MoSe , this 4,5 barrier . Here we apply similar principles to the design of a value should be accurate to within 20%. We find a E1.0 nm, CQW from atomically thin materials stacked on top of each much shorter than 15 nm in GaAs CQW. In turn, Ry E140 meV other. Research on such van der Waals heterostructures is gaining in the proposed device, about forty times larger than in GaAs. momentum in the last few years , and their quality and Since it enters the denominator of equation (1), one may think availability is steadily improving. that having larger m is unfavourable for attaining higher T .In x d In the proposed device (Fig. 1a), an indirect exciton is fact, when n can be controlled, the opposite is true. Indeed, composed from an electron and a hole located in two different equation (1) can be rewritten as MoS layers separated by a hexagonal boron nitride (hBN) 2 2 kT ¼ 4p m m m n a Ry : ð2Þ d e h x x x x insulating barrier and surrounded by hBN cladding layers. The z-direction electric field is controlled by voltage applied to The upper limit on n a is imposed by quantum dissociation of external electrodes. The applied field modifies the band structure excitons that occurs when the ratio of the exciton size (the in- in a way that it becomes advantageous for optically excited plane gyration radius) r and the mean inter-exciton distance pffiffiffiffiffi 10–12 electrons and holes to reside in the opposite MoS monolayers 1 n reaches the critical value of about 0.3. In the case of and form indirect excitons (Fig. 1b). our primary interest where interlayer centre-to-centre distance The design of the MoS /hBN structure is similar to that of cEa and m ¼ m ¼ 2m, we estimate r ¼ 2.4a (Fig. 2), so that x e h x x 4,5 GaAs/AlGaAs CQW we studied previously , except GaAs is the corresponding Mott critical density n is set by the condition replaced by MoS and AlGaAs by hBN. Here we predict the phase n a  0:02: ð3Þ diagram of such a device using the results of numerical x calculations and scaling arguments. Our most intriguing finding Substituting this into equation (2), we obtain is that in the proposed MoS /hBN structures degenerate Bose gas max kT  0:06Ry : ð4Þ of indirect excitons can be realized at record-high temperatures. d x Hence, the key to high T is the enhanced value of Ry . The d x Results exciton binding energy E E0.6 Ry (Fig. 2) does not pose ind x Degeneracy temperature. The characteristic temperature T at further fundamental limitations. Actually, the conclusion that the which excitons become degenerate is determined by their density theoretical maximum of T is proportional to Ry follows from d x n per flavour (spin and valley), and effective mass m ¼ m þ m : x x e h dimensional analysis. If cBa , m Bm and excitons are treated x e h as an equilibrium Bose gas, then Ry is the only relevant energy 2p‘ kT ¼ n : ð1Þ d x scale in the problem. In reality, only quasi-equilibrium state is possible because of Here m and m are the electron and hole effective masses. e h exciton recombination, and so limitations posed by the finite 10  2 For n B10 cm , and m ¼ 0.07, m ¼ 0.15, m ¼ 0.22 repre- x e h x exciton lifetime t must be discussed. Radiative recombination of sentative of GaAs CQW, we find T B3 K. (Here and below all the d indirect excitons requires interlayer tunnelling. The rate of this masses are in units of the bare electron mass.) At such process can be made exponentially small by adjusting the barrier temperatures long-range spontaneous coherence of indirect excitons is observed . Centre−to−centre distance (nm) To explain why MoS -based CQW would possess much higher 0.511.5 2 T than GaAs-based ones, we consider the characteristic length 4.0 0.15 3.5 z Energy 0.1 x 3.0 hBN MoS 2.5 0.05 hBN 2.0 hBN 0 1.5 MoS 2 02 4 6 # hBN layers hBN Figure 2 | Estimated exciton parameters. E (left axis) is the binding ind energy and r (right axis, equation (13)) is the in-plane gyration radius Figure 1 | Schematics of the proposed device. (a) Geometry and of indirect excitons. The bottom axis is the number of the hBN spacer (b) band structure. The ellipse indicates an indirect exciton composed layers and the top axis is the corresponding centre-to-centre distance of an electron ( ) and a hole (þ ). between the MoS layers in the heterostructure. 2 NATURE COMMUNICATIONS | 5:4555 | DOI: 10.1038/ncomms5555 | www.nature.com/naturecommunications & 2014 Macmillan Publishers Limited. All rights reserved. E (eV) ind r (nm) x Mott NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5555 ARTICLE width. In GaAs CQW, t can be routinely made in the range of degenerate Bose gas of excitons. At n¼ n this line extrapolates max microseconds and can be controllably varied over several orders to the temperature kT (equation 4). of magnitude. The large t enables creation of indirect exciton Whereas the Mott transition represents quantum dissociation gases of high-density n ¼ Pt with low optical excitation power of excitons, the thermal dissociation occurs above the Saha E =kT ind S P that does not cause overheating of either the lattice or the temperature kT  p‘ n m e , which is shown by the S x x excitons. This is why for the task of achieving cold dense exciton dashed line near the T axis in Fig. 3. This line marks a crossover gases, indirect excitons are superior to conventional bulk excitons from the exciton phase to a classical plasma. Since their binding or two-dimensional (2D) direct excitons (bound states of energy is very large, indirect excitons in MoS /hBN structure electrons and holes in the same layer). remain stable against the thermal dissociation well above the In the proposed device, the interlayer tunnelling rate would room temperature in a broad range of electron densities. decay exponentially with the number N of layers in the hBN Formation of quantum degenerate Bose gas of long-lifetime spacer. For N¼ 2, the tunnelling rate is comparable to that in the repulsively interacting indirect excitons leads to local exciton 4,5 GaAs CQW (see Methods) indicating that long-life indirect superfluidity below T . The superfluidity spreads over a excitons can be realized in MoS /hBN structures. The tunnelling- macroscopic area at the Berezinskii–Kosterlitz–Thouless transi- limited lifetime can be enhanced by possible rotational tion temperature misalignment of the MoS layers, which makes the excitons indirect not only in real but also in momentum space, similar to 2 ‘ n 9,13 excitons in MoS multilayers . Hence, sufficiently long t may kT  1:3 : ð5Þ 2 BKT perhaps be achieved with a monolayer hBN spacer. Working with very small N, one has to worry about possibility of dielectric Superfluid transport of neutral indirect excitons produces breakdown of the hBN spacer. To make the indirect exciton more dissipationless charge currents in the opposite directions in the energetically favourable than the direct one, a voltage equal or two layers . To observe and utilize this effect, one can, for larger than (E  E )/e must be applied between the MoS dir ind 2 example, make separate contacts to each layer and form a closed layers. Assuming the direct exciton binding energy E of dir 8,14 circuit for the hole layer. The electric current in the electron layer approximately 0.5 eV, for N¼ 2 hBN spacer, the required will then be dissipationless. This is referred to as the counterflow voltage is about 0.4 V, which is safely below the breakdown superconductivity . limit . The most intriguing conclusion we draw from Fig. 3 is that in A schematic phase diagram of a neutral electron–hole system the proposed MoS /hBN structures degenerate Bose gas of in the proposed heterostructure is shown in Fig. 3. The solid line 2 indirect excitons can be realized at record-high temperatures. represents the Mott transition. It emanates from the T¼ 0 critical point computed according to equation (3). At the transition there is a discontinuous jump in the degree of exciton ionization . Discussion Above the Mott critical temperature T the transition changes to In the remainder of this paper, we overview phenomena a smooth crossover. Such a crossover has been studied analogous to a coherent state of indirect excitons in other 2D experimentally in photo-excited single-well GaAs and InGaAs systems. We choose not to survey systems where electrons and 17,18 max structures . We assume that T should be lower than T holes reside in the same layer, such as quantum wells in max 21 and use kT ¼ 0:5kT  0:03Ry in Fig. 3 for illustration. The microcavities . They are interesting in their own right but d x quantum degeneracy line (equation 1) shown by the dashed- counterflow superconductivity therein is impossible. dotted line in Fig. 3 demarcates a crossover from classical to First, evidence for broken-symmetry phases conceptually similar to condensates of indirect excitons have been reported in GaAs CQW . These phases arise in the presence of a Classical electron-hole gas quantizing magnetic field at sub-Kelvin temperatures. Next, a number of broken-symmetry states, some of which are similar to exciton condensates, have been predicted to form in bilayer graphene . The double-layer graphene (DLG) systems must be specially mentioned because a high-temperature coherent state in DLG was theoretically discussed recently. The fundamental obstacle to exciton condensation in DLG is that monolayer Degenerate 80 Classical graphene is a zero-gap semimetal with a linear quasiparticle electron– exciton dispersion. Although in such a system phases of weakly bound hole Fermi gas excitons may exist, the corresponding critical temperature is gas extremely sensitive to the effective strength of the electron–hole Degenerate attraction. Calculation of the latter requires accounting for exciton 40 screening of the long-range Coulomb interaction as well as Bose gas short-range correlation effects, both of which are challenging problems. Theoretical estimates of the critical temperature in 24–26 27 DLG range from hundreds of Kelvin to a few milli-Kelvin . We wish to stress that there is no room for such an enormous uncertainty in our proposal based on MoS , a semiconductor with a significant bandgap, modest dielectic constant and exceptionally 0 1 2 3 stable exciton state. The dependence of the characteristic 12 –2 n (10 cm ) temperatures T and T on electron density is linear rather d BKT Figure 3 | Schematic phase diagram of the system studied. T is the than exponential. All pertinent numerical factors are constrained temperature and n is the electron density per flavour. The dashed lines by numerous prior studies of similar semiconducting systems. indicate crossovers and the solid lines mark the phase transitions. The dot Therefore, the error in the estimated temperatures should be is the Mott critical point. small for the system considered in our paper. NATURE COMMUNICATIONS | 5:4555 | DOI: 10.1038/ncomms5555 | www.nature.com/naturecommunications 3 & 2014 Macmillan Publishers Limited. All rights reserved. BKT T(K) ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5555 Here c ¼ Nc is the tunnelling length, N is the number of hBN layers, b 1 c ¼ 0.333 nm is the thickness of one hBN layer, a ¼ 0.0529 nm is the hydrogen 1 B Solid Bohr radius and m is the effective carrier mass inside the barrier. If the chemical potential is in the middle of the hBN energy gap, we expect U E3 eV. Assuming also m E0.5 in units of the bare electron mass, we obtain SE5N, which is in agreement with S¼ 4.6N deduced from the tunnelling conductance measured in graphene/hBN/graphene structures . In comparison, in GaAs/AlGaAs CQW structures with m E0.35, U E0.15 eV and d ¼ 4.0 nm we get S¼ 9.4. Hence, for b b b 4,5 N¼ 2 the tunnelling rate is comparable to that in the GaAs CQW . More detailed Electron-hole estimates would have to include the trapezoidal shape of the barrier (Fig. 1b) and Fermi gas 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. Exciton Bose gas 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 mid- planes of two separate MoS layers. The Schro¨dinger equation for the relative motion reads Bi-exciton r fðrÞþ UðrÞfðrÞ¼  E fðrÞ; ð7Þ ind phases 2m where m is the reduced mass. We model the potential U(r) of electron–hole Coulomb interaction using the continuum-medium electrostatics . This simplified 0.5 10 200 approach neglects frequency dependence of the dielectric functions of the materials involved. We approximate each of MoS layers as a uniaxial dielectric slab of 8 ? thickness c ¼ 0.312 nm with principal dielectric tensor components E ¼ 14:29 Figure 4 | Schematic T¼ 0 phase diagram. The system is an and E ¼ 6:87 in the directions perpendicular and parallel to the z axis, electron–hole bilayer in which the particles have spin 1/2 but no valley respectively. In turn, the hBN spacer is modelled as a slab of thickness Nc , 33 ? c ¼ 0.333 nm, with the dielectric constants E ¼ 6:71 and E ¼ 3:56. We define degeneracy. The MoS /hBN device with the N¼ 2 layer hBN spacer 1 1 the anisotropy parameter k and the effective dielectric constant E of each j j corresponds to d/a E2.8. At such d/a the ground state of the system e e material by at r \6 is the superfluid Bose gas of indirect excitons. qffiffiffiffi qffiffiffiffi qffiffiffiffiffi qffiffiffiffiffi k k ? ? k ¼ E E ; E ¼ E E ; j ¼ 1; 2; ð8Þ j j j j j j The scaling law of equation (4) elucidates a general principle so that k ¼ 1.44, E ¼ 9.91 (MoS ) and k ¼ 1.37, E ¼ 4.89 (hBN). The interaction 2 2 2 1 1 for realizing high-temperature coherent states of excitons. The potential can be found by solving the Poisson equation for this layered system, proposed design is an initial blueprint and is amenable to further which is done by reducing it to a set of linear equations for the Fourier component optimization. For example, one or both of MoS layers can be UðqÞ and using the inverse Fourier transform: substituted by a different transition metal dichalcogenide (TMD), Z d q iqr such as WS or WSe . The two MoS monolayers may be replaced UðÞ r ¼ e UðÞ q : ð9Þ 2 2 2 2 ðÞ 2p by a single several-monolayer-thick TMD encapsulated by hBN By symmetry, UðÞ q depends only on the magnitude q¼ |q| of vector q not on in which indirect excitons would be composed of electrons and its direction. After simple algebra, we arrive at the result UqðÞ¼ CqðÞ=DqðÞ, where holes confined at the opposite sides of the TMD layer. The functions C(q) and D(q) are given by outlined principle for realizing high-temperature superfluidity 8pe 2 c k =2 c k =2  ck 2 2 2 2 1 CqðÞ¼ EðÞ E  E e ðÞ E  E e e ; ð10Þ can also be extended to layered materials other than TMD 1 1 2 1 2 and hBN. 2 2 Besides providing a new platform for exploring fundamental  c k c k 2 2 2 2 DqðÞ¼ðÞ E  E e ðÞ E þ E e 1 2 1 2 ð11Þ quantum phenomena, indirect excitons in van der Waals 2 2  2ðÞ c c k 2 2 1 4 E  E e sinh c k : 2 2 1 2 structures can be also utilized for the development of opto- Here we introduced the short-hand notations c¼ Nc þ c (the centre-to-centre 4 1 2 electronic circuits . In such devices, in-plane potential landscapes distance) and k ¼ k q (the z-direction wavenumber of the evanescent Fourier j j iqr  k z for excitons are created and controlled by external electric fields j harmonics e in medium j). From numerical calculations using equations (9)–(11) we found that potential U(r) is accurately approximated by the following that couple to the permanent dipole moment p¼ ec of the analytical expression: indirect excitons. The operation temperature for excitonic circuits in the van der Waals structures is expected to exceed by an order e 1 A pffiffiffiffiffiffiffiffiffiffiffiffiffiffi UðrÞ¼ 1 ; d ¼ k c: ð12Þ 2 2 2 of magnitude the B100 K record set by GaAs-based CQWs. E 1þ Br 1 r þ d Finally, we note that Fig. 3 was constructed following the With a suitable choice of coefficients 0oAo1 and B40 this form produces 10–12 asymptotically exact results for U(r) at both small and large r. example of the Monte-Carlo simulations in which only To solve equation (7), we discretized it on a real-space 2D grid (typically, exciton phases with two possible spin flavours were considered. 75 75). The resultant linear eigenvalue problem was diagonalized by standard In TMDs, spin and orbital degeneracies of indirect excitons may numerical methods yielding the binding energy E (N) and the gyration radius ind have a more intriguing structure. Such degeneracies can be r (N) of indirect excitons. The latter is defined in terms of a normalized ground- state wavefunction f(r) by means of the integral controlled by strong spin–orbit coupling , polarization of the 7,31 32 excitation beam or many-body interactions . 2 2 2 2 r ¼ r f ðÞ r d r: ð13Þ Experimental realization of superfluidity and counterflow The results are shown in Fig. 2. For N¼ 2 we find r ¼ 2.50 nm¼ 2.43a and superconductivity as well as excitonic circuits and spintronic/ x x E ¼ 87 meVE0.6 Ry . This binding energy is an order of magnitude larger than ind x valleytronic devices in atomically thin heterostructures may have 29,34 E ¼ 4–10 meV typical for excitons in GaAs/AlGaAs CQW structures . ind far-reaching implications for science and technology. Zero-temperature phases. It is instructive to complement the above discussion Methods of the finite-T phase diagram (Fig. 3) with commenting on the T¼ 0 phases. Interlayer tunnelling. The action S for tunnelling of quasiparticles across the hBN Such phases include electron–hole Fermi gas, exciton Bose gas and exciton spacer can be estimated from the usual formula for the rectangular potential barrier solid. The approximate phase boundaries based on available Monte-Carlo of height U : 10–12,35,36 calculations are shown in Fig. 4. Although those simulations were done for electron–hole bilayers in vacuum, an approximate correspondence with our 1=2 S ¼ 2ðÞ c =aðÞ m =m U ðÞ Ry : ð6Þ b B b 0 b dielectric environment can be achieved if we neglect the correction term in the 4 NATURE COMMUNICATIONS | 5:4555 | DOI: 10.1038/ncomms5555 | www.nature.com/naturecommunications & 2014 Macmillan Publishers Limited. All rights reserved. d / a e NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5555 ARTICLE parenthesis in equation (12). The mapping is then obtained by setting the distance 20. Su, J.-J. & MacDonald, A. H. How to make a bilayer exciton condensate flow. between the layers to d¼ k c and including the effective dielectric function E in the 1 1 Nat. Phys. 4, 799–802 (2008). definition of the electron Bohr radius: 21. Littlewood, P. B. et al. Models of coherent exciton condensation. J. Phys.: 2 2 Condens. Matter 16, S3597–S3620 (2004). ‘ E 1 ‘ E a 1 1 x a ¼ ¼ ¼ : ð14Þ e 22. Eisenstein, J. P. & MacDonald, A. H. Bose-Einstein condensation of excitons in 2 2 m e 2 me 2 bilayer electron systems. Nature 432, 691–694 (2004). On the horizontal axis in Fig. 4 we plot the dimensionless intralayer distance pffiffiffiffiffiffiffiffiffiffiffiffi 23. McCann, E. & Koshino, M. The electronic properties of bilayer graphene. Rep. parameter r ¼ 1 pn a , where n is the total electron density. Solid phases form s e e Prog. Phys. 76, 056503 (2013). at large r and d/a . The exciton dipole solid crosses over to the interlocked Wigner s e 24. Min, H., Bistritzer, R., Su, J.-J. & MacDonald, A. H. Room-temperature crystals of electrons and holes as the interlayer distance becomes larger than the superfluidity in graphene bilayers. Phys. Rev. B 78, 121401 (2008). characteristic intralayer one, d\r a . s e 25. Perali, A., Neilson, D. & Hamilton, A. R. High-temperature superfluidity in The considered MoS /hBN heterostructure with N¼ 2 corresponds to d/a E2.8 2 e double-bilayer graphene. Phys. Rev. Lett. 110, 146803 (2013). for which the ground state is never a solid phase. Instead, at large r , that is, at low 26. Neilson, D., Perali, A. & Hamilton, A. R. Excitonic superfluidity and screening electron density, the ground state of the system is an exciton gas. As r decreases, in electron-hole bilayer systems. Phys. Rev. B 89, 060502(R) (2014). the Mott transition to electron–hole Fermi gas occurs. At this transition the 1 27. Kharitonov, M. Y. & Efetov, K. B. Electron screening and excitonic excitons dissociate owing to screening and phase space filling . The latest condensation in double-layer graphene systems. Phys. Rev. B 78, 241401 (2008). estimates of the T¼ 0 Mott transition at d¼ 2.8a give r E6, which corresponds e s 28. Komsa, H.-P. & Krasheninnikov, A. V. Electronic structures and optical to equation (3). Interestingly, a Bardeen–Cooper–Schrieffer-like excitonic state of a properties of realistic transition metal dichalcogenide heterostructures from dense electron–hole gas predicted by the earlier theory was not found in the cited 10,12 first principles. Phys. Rev. B 88, 085318 (2013). Monte-Carlo calculations . 29. Grosso, G. et al. Excitonic switches operating at around 100 K. Nat. Photonics 3, Unless the effective interlayer separation d is small, the interaction between 38,39 577–580 (2009). indirect excitons is dominated by the classical dipole repulsion term . However, 40–42 30. Korma´nyos, A., Zo´lyomi, V., Drummond, N. D. & Burkard, G. Spin-orbit as shown previously ,at do0.87a quantum exchange–correlation effects cause coupling, quantum dots, and qubits in monolayer transition metal the change of repulsion to attraction leading to appearance of bi-excitons 12,39 dichalcogenides. Phys. Rev. X 4, 011034 (2014). phases . Narrow ranges of modulated phases (stripes, bubbles or supersolids) may exist near any of the first-order phase transition lines . 31. Ross, J. S. et al. Electrical control of neutral and charged excitons in a monolayer semiconductor. Nat. Commun. 4, 1474 (2013). 32. Ben-Tabou de Leon, S. & Laikhtman, B. Mott transition, biexciton crossover, References and spin ordering in the exciton gas in quantum wells. Phys. Rev. B 67, 235315 1. Keldysh, L. V. & Kozlov, A. N. Collective properties of excitons in (2003). semiconductors. J. Exp. Theor. Phys. 27, 521 (1968). 33. Cai, Y., Zhang, L., Zeng, Q., Cheng, L. & Xu, Y. Infrared reflectance spectrum 2. Lozovik, Y. E. & Yudson, V. I. A new mechanism for superconductivity: pairing of BN calculated from first principles. Solid State Commun. 141, 262–266 between spatially separated electrons and holes. J. Exp. Theor. Phys. 44, 389 (2007). (1976). 34. Szymanska, M. H. & Littlewood, P. B. Excitonic binding in coupled quantum 3. Fukuzawa, T., Kano, S., Gustafson, T. & Ogawa, T. Possibility of coherent light wells. Phys. Rev. B 67, 193305 (2003). emission from Bose condensed states of SEHPs. Surf. Sci. 228, 482–485 (1990). 35. Astrakharchik, G. E., Boronat, J., Kurbakov, I. L. & Lozovik, Y. E. Quantum 4. High, A. A., Novitskaya, E. E., Butov, L. V., Hanson, M. & Gossard, A. C. phase transition in a two-dimensional system of dipoles. Phys. Rev. Lett. 98, Control of exciton fluxes in an excitonic integrated circuit. Science 321, 060405 (2007). 229–231 (2008). 36. Bu¨chler, H. P. et al. Strongly correlated 2D quantum phases with cold polar 5. High, A. A. et al. Spontaneous coherence in a cold exciton gas. Nature 483, molecules: controlling the shape of the interaction potential. Phys. Rev. Lett. 98, 584–588 (2012). 060404 (2007). 6. Geim, A. K. & Grigorieva, I. V. Van der Waals heterostructures. Nature 499, 37. Keldysh, L. V. & Kopaev, Y. V. Possible instability of the semimetallic state 419–425 (2013). toward Coulomb interaction. Sov. Phys. Solid State 6, 2219 (1965). 7. Xiao, D., Liu, G.-B., Feng, W., Xu, X. & Yao, W. Coupled spin and valley 38. Zhu, X., Littlewood, P. B., Hybertsen, M. S. & Rice, T. M. Exciton condensate in physics in monolayers of MoS and other group-VI dichalcogenides. Phys. Rev. semiconductor quantum well structures. Phys. Rev. Lett. 74, 1633–1636 (1995). Lett. 108, 196802 (2012). 39. Lozovik, Y. E. & Berman, O. L. Phase transitions in a system of two coupled 8. Berkelbach, T. C., Hybertsen, M. S. & Reichman, D. R. Theory of neutral and quantum wells. JETP Lett. 64, 573–579 (1996). charged excitons in monolayer transition metal dichalcogenides. Phys. Rev. B 40. Meyertholen, A. D. & Fogler, M. M. Biexcitons in two-dimensional systems 88, 045318 (2013). with spatially separated electrons and holes. Phys. Rev. B 78, 235307 (2008). 9. Zhang, Y. et al. Direct observation of the transition from indirect to direct 41. Tan, M. Y. J., Drummond, N. D. & Needs, R. J. Exciton and biexciton energies bandgap in atomically thin epitaxial MoSe . Nat. Nanotechnol. 9, 111–115 in bilayer systems. Phys. Rev. B 71, 033303 (2005). (2013). 42. Schindler, C. & Zimmermann, R. Analysis of the exciton-exciton interaction in 10. De Palo, S., Rapisarda, F. & Senatore, G. Excitonic condensation in a symmetric semiconductor quantum wells. Phys. Rev. B 78, 045313 (2008). electron-hole bilayer. Phys. Rev. Lett. 88, 206401 (2002). 43. Spivak, B. & Kivelson, S. A. Phases intermediate between a two-dimensional 11. Schleede, J., Filinov, A., Bonitz, M. & Fehske, H. Phase diagram of bilayer electron liquid and Wigner crystal. Phys. Rev. B 70, 155114 (2004). electron-hole plasmas. Contrib. Plasma Phys. 52, 819–826 (2012). 12. Maezono, R., Lo´pez Ros, P., Ogawa, T. & Needs, R. J. Excitons and biexcitons in symmetric electron-hole bilayers. Phys. Rev. Lett. 110, 216407 (2013). Acknowledgements 13. Mak, K. F., Lee, C., Hone, J., Shan, J. & Heinz, T. F. Atomically thin MoS : 2 This work was supported by the US ONR & UCOP (M.M.F.), NSF (L.V.B.) and also by a new direct-gap semiconductor. Phys. Rev. Lett. 105, 136805 (2010). the European Research Council and EC-FET European Graphene Flagship (K.S.N.). We 14. Chernikov, A. et al. Non-hydrogenic exciton Rydberg series in monolayer WS . 2 are grateful to Andre´ K. Geim for comments. Preprint at http://arxiv.org/abs/1403.4270 (2014). 15. Britnell, L. et al. Field-Effect tunneling transistor based on vertical graphene Author contributions heterostructures. Science 335, 947–950 (2012). All the authors contributed to the ideas, discussion, and writing of the paper. M.M.F. 16. Nikolaev, V. V. & Portnoi, M. E. Mott transition of spatially indirect excitons. carried out numerical simulations. Proc. SPIE 5509, 187–193 (2004). 17. Butov, L. V., Kulakovskii, V. D., Lach, E., Forchel, A. & Gru¨tzmacher, D. Magnetoluminescence study of many-body effects in homogeneous quasi-two- Additional information dimensional electron-hole plasma in undoped In Ga As/InP single quantum x 1–x Competing financial interests: The authors declare no competing financial interests. wells. Phys. Rev. B 44, 10680–10688 (1991). 18. Kappei, L., Szczytko, J., Morier-Genoud, F. & Deveaud, B. Direct observation of Reprints and permission information is available online at http://npg.nature.com/ the Mott transition in an optically excited semiconductor quantum well. Phys. reprintsandpermissions/ Rev. Lett. 94, 147403 (2005). 19. Filinov, A., Prokof’ev, N. V. & Bonitz, M. Berezinskii-Kosterlitz-Thouless How to cite this article: Fogler, M. M. et al. High-temperature superfluidity with transition in two-dimensional dipole systems. Phys. Rev. Lett. 105, 070401 indirect excitons in van der Waals heterostructures. Nat. Commun. 5:4555 (2010). doi: 10.1038/ncomms5555 (2014). NATURE COMMUNICATIONS | 5:4555 | DOI: 10.1038/ncomms5555 | www.nature.com/naturecommunications 5 & 2014 Macmillan Publishers Limited. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Nature Communications Springer Journals

High-temperature superfluidity with indirect excitons in van der Waals heterostructures

Download PDF
5 pages

Loading next page...
 
/lp/springer-journals/high-temperature-superfluidity-with-indirect-excitons-in-van-der-waals-vhvblq0LAR

References (51)

Publisher
Springer Journals
Copyright
Copyright © 2014 by Nature Publishing Group, a division of Macmillan Publishers Limited. All Rights Reserved.
Subject
Science, Humanities and Social Sciences, multidisciplinary; Science, Humanities and Social Sciences, multidisciplinary; Science, multidisciplinary
eISSN
2041-1723
DOI
10.1038/ncomms5555
Publisher site
See Article on Publisher Site

Abstract

ARTICLE Received 5 May 2014 | Accepted 27 Jun 2014 | Published 28 Jul 2014 DOI: 10.1038/ncomms5555 High-temperature superfluidity with indirect excitons in van der Waals heterostructures 1 1 2 M.M. Fogler , L.V. Butov & K.S. Novoselov 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. High-critical temperature cuprate superconductors set the present record of B100 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. 1 2 Department of Physics, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093, USA. School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK. Correspondence and requests for materials should be addressed to M.M.F. (email: [email protected]). NATURE COMMUNICATIONS | 5:4555 | DOI: 10.1038/ncomms5555 | www.nature.com/naturecommunications 1 & 2014 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5555 oherent states of excitons have been a subject of intense and energy scales in the problem, which are the exciton Bohr 1–3 2 2 2 2 theoretical studies . A general framework for creation radius a ¼ : E /me and the Rydberg energy Ry ¼ ‘ 2ma x 1 Cand manipulation of degenerate gases of indirect excitons defined in terms of reduced mass m¼ m m /m . For estimates we e h x has been established in prior experimental studies of GaAs-based use the effective static dielectric constant E ¼ 4.9 of hBN. We also coupled quantum wells (CQWs) where electrons and holes are use the calculated value m¼ 0.25 (refs 7,8) for MoS . On the basis confined in GaAs quantum wells separated by a thin AlGaAs of the measurements done on a related compound MoSe , this 4,5 barrier . Here we apply similar principles to the design of a value should be accurate to within 20%. We find a E1.0 nm, CQW from atomically thin materials stacked on top of each much shorter than 15 nm in GaAs CQW. In turn, Ry E140 meV other. Research on such van der Waals heterostructures is gaining in the proposed device, about forty times larger than in GaAs. momentum in the last few years , and their quality and Since it enters the denominator of equation (1), one may think availability is steadily improving. that having larger m is unfavourable for attaining higher T .In x d In the proposed device (Fig. 1a), an indirect exciton is fact, when n can be controlled, the opposite is true. Indeed, composed from an electron and a hole located in two different equation (1) can be rewritten as MoS layers separated by a hexagonal boron nitride (hBN) 2 2 kT ¼ 4p m m m n a Ry : ð2Þ d e h x x x x insulating barrier and surrounded by hBN cladding layers. The z-direction electric field is controlled by voltage applied to The upper limit on n a is imposed by quantum dissociation of external electrodes. The applied field modifies the band structure excitons that occurs when the ratio of the exciton size (the in- in a way that it becomes advantageous for optically excited plane gyration radius) r and the mean inter-exciton distance pffiffiffiffiffi 10–12 electrons and holes to reside in the opposite MoS monolayers 1 n reaches the critical value of about 0.3. In the case of and form indirect excitons (Fig. 1b). our primary interest where interlayer centre-to-centre distance The design of the MoS /hBN structure is similar to that of cEa and m ¼ m ¼ 2m, we estimate r ¼ 2.4a (Fig. 2), so that x e h x x 4,5 GaAs/AlGaAs CQW we studied previously , except GaAs is the corresponding Mott critical density n is set by the condition replaced by MoS and AlGaAs by hBN. Here we predict the phase n a  0:02: ð3Þ diagram of such a device using the results of numerical x calculations and scaling arguments. Our most intriguing finding Substituting this into equation (2), we obtain is that in the proposed MoS /hBN structures degenerate Bose gas max kT  0:06Ry : ð4Þ of indirect excitons can be realized at record-high temperatures. d x Hence, the key to high T is the enhanced value of Ry . The d x Results exciton binding energy E E0.6 Ry (Fig. 2) does not pose ind x Degeneracy temperature. The characteristic temperature T at further fundamental limitations. Actually, the conclusion that the which excitons become degenerate is determined by their density theoretical maximum of T is proportional to Ry follows from d x n per flavour (spin and valley), and effective mass m ¼ m þ m : x x e h dimensional analysis. If cBa , m Bm and excitons are treated x e h as an equilibrium Bose gas, then Ry is the only relevant energy 2p‘ kT ¼ n : ð1Þ d x scale in the problem. In reality, only quasi-equilibrium state is possible because of Here m and m are the electron and hole effective masses. e h exciton recombination, and so limitations posed by the finite 10  2 For n B10 cm , and m ¼ 0.07, m ¼ 0.15, m ¼ 0.22 repre- x e h x exciton lifetime t must be discussed. Radiative recombination of sentative of GaAs CQW, we find T B3 K. (Here and below all the d indirect excitons requires interlayer tunnelling. The rate of this masses are in units of the bare electron mass.) At such process can be made exponentially small by adjusting the barrier temperatures long-range spontaneous coherence of indirect excitons is observed . Centre−to−centre distance (nm) To explain why MoS -based CQW would possess much higher 0.511.5 2 T than GaAs-based ones, we consider the characteristic length 4.0 0.15 3.5 z Energy 0.1 x 3.0 hBN MoS 2.5 0.05 hBN 2.0 hBN 0 1.5 MoS 2 02 4 6 # hBN layers hBN Figure 2 | Estimated exciton parameters. E (left axis) is the binding ind energy and r (right axis, equation (13)) is the in-plane gyration radius Figure 1 | Schematics of the proposed device. (a) Geometry and of indirect excitons. The bottom axis is the number of the hBN spacer (b) band structure. The ellipse indicates an indirect exciton composed layers and the top axis is the corresponding centre-to-centre distance of an electron ( ) and a hole (þ ). between the MoS layers in the heterostructure. 2 NATURE COMMUNICATIONS | 5:4555 | DOI: 10.1038/ncomms5555 | www.nature.com/naturecommunications & 2014 Macmillan Publishers Limited. All rights reserved. E (eV) ind r (nm) x Mott NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5555 ARTICLE width. In GaAs CQW, t can be routinely made in the range of degenerate Bose gas of excitons. At n¼ n this line extrapolates max microseconds and can be controllably varied over several orders to the temperature kT (equation 4). of magnitude. The large t enables creation of indirect exciton Whereas the Mott transition represents quantum dissociation gases of high-density n ¼ Pt with low optical excitation power of excitons, the thermal dissociation occurs above the Saha E =kT ind S P that does not cause overheating of either the lattice or the temperature kT  p‘ n m e , which is shown by the S x x excitons. This is why for the task of achieving cold dense exciton dashed line near the T axis in Fig. 3. This line marks a crossover gases, indirect excitons are superior to conventional bulk excitons from the exciton phase to a classical plasma. Since their binding or two-dimensional (2D) direct excitons (bound states of energy is very large, indirect excitons in MoS /hBN structure electrons and holes in the same layer). remain stable against the thermal dissociation well above the In the proposed device, the interlayer tunnelling rate would room temperature in a broad range of electron densities. decay exponentially with the number N of layers in the hBN Formation of quantum degenerate Bose gas of long-lifetime spacer. For N¼ 2, the tunnelling rate is comparable to that in the repulsively interacting indirect excitons leads to local exciton 4,5 GaAs CQW (see Methods) indicating that long-life indirect superfluidity below T . The superfluidity spreads over a excitons can be realized in MoS /hBN structures. The tunnelling- macroscopic area at the Berezinskii–Kosterlitz–Thouless transi- limited lifetime can be enhanced by possible rotational tion temperature misalignment of the MoS layers, which makes the excitons indirect not only in real but also in momentum space, similar to 2 ‘ n 9,13 excitons in MoS multilayers . Hence, sufficiently long t may kT  1:3 : ð5Þ 2 BKT perhaps be achieved with a monolayer hBN spacer. Working with very small N, one has to worry about possibility of dielectric Superfluid transport of neutral indirect excitons produces breakdown of the hBN spacer. To make the indirect exciton more dissipationless charge currents in the opposite directions in the energetically favourable than the direct one, a voltage equal or two layers . To observe and utilize this effect, one can, for larger than (E  E )/e must be applied between the MoS dir ind 2 example, make separate contacts to each layer and form a closed layers. Assuming the direct exciton binding energy E of dir 8,14 circuit for the hole layer. The electric current in the electron layer approximately 0.5 eV, for N¼ 2 hBN spacer, the required will then be dissipationless. This is referred to as the counterflow voltage is about 0.4 V, which is safely below the breakdown superconductivity . limit . The most intriguing conclusion we draw from Fig. 3 is that in A schematic phase diagram of a neutral electron–hole system the proposed MoS /hBN structures degenerate Bose gas of in the proposed heterostructure is shown in Fig. 3. The solid line 2 indirect excitons can be realized at record-high temperatures. represents the Mott transition. It emanates from the T¼ 0 critical point computed according to equation (3). At the transition there is a discontinuous jump in the degree of exciton ionization . Discussion Above the Mott critical temperature T the transition changes to In the remainder of this paper, we overview phenomena a smooth crossover. Such a crossover has been studied analogous to a coherent state of indirect excitons in other 2D experimentally in photo-excited single-well GaAs and InGaAs systems. We choose not to survey systems where electrons and 17,18 max structures . We assume that T should be lower than T holes reside in the same layer, such as quantum wells in max 21 and use kT ¼ 0:5kT  0:03Ry in Fig. 3 for illustration. The microcavities . They are interesting in their own right but d x quantum degeneracy line (equation 1) shown by the dashed- counterflow superconductivity therein is impossible. dotted line in Fig. 3 demarcates a crossover from classical to First, evidence for broken-symmetry phases conceptually similar to condensates of indirect excitons have been reported in GaAs CQW . These phases arise in the presence of a Classical electron-hole gas quantizing magnetic field at sub-Kelvin temperatures. Next, a number of broken-symmetry states, some of which are similar to exciton condensates, have been predicted to form in bilayer graphene . The double-layer graphene (DLG) systems must be specially mentioned because a high-temperature coherent state in DLG was theoretically discussed recently. The fundamental obstacle to exciton condensation in DLG is that monolayer Degenerate 80 Classical graphene is a zero-gap semimetal with a linear quasiparticle electron– exciton dispersion. Although in such a system phases of weakly bound hole Fermi gas excitons may exist, the corresponding critical temperature is gas extremely sensitive to the effective strength of the electron–hole Degenerate attraction. Calculation of the latter requires accounting for exciton 40 screening of the long-range Coulomb interaction as well as Bose gas short-range correlation effects, both of which are challenging problems. Theoretical estimates of the critical temperature in 24–26 27 DLG range from hundreds of Kelvin to a few milli-Kelvin . We wish to stress that there is no room for such an enormous uncertainty in our proposal based on MoS , a semiconductor with a significant bandgap, modest dielectic constant and exceptionally 0 1 2 3 stable exciton state. The dependence of the characteristic 12 –2 n (10 cm ) temperatures T and T on electron density is linear rather d BKT Figure 3 | Schematic phase diagram of the system studied. T is the than exponential. All pertinent numerical factors are constrained temperature and n is the electron density per flavour. The dashed lines by numerous prior studies of similar semiconducting systems. indicate crossovers and the solid lines mark the phase transitions. The dot Therefore, the error in the estimated temperatures should be is the Mott critical point. small for the system considered in our paper. NATURE COMMUNICATIONS | 5:4555 | DOI: 10.1038/ncomms5555 | www.nature.com/naturecommunications 3 & 2014 Macmillan Publishers Limited. All rights reserved. BKT T(K) ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5555 Here c ¼ Nc is the tunnelling length, N is the number of hBN layers, b 1 c ¼ 0.333 nm is the thickness of one hBN layer, a ¼ 0.0529 nm is the hydrogen 1 B Solid Bohr radius and m is the effective carrier mass inside the barrier. If the chemical potential is in the middle of the hBN energy gap, we expect U E3 eV. Assuming also m E0.5 in units of the bare electron mass, we obtain SE5N, which is in agreement with S¼ 4.6N deduced from the tunnelling conductance measured in graphene/hBN/graphene structures . In comparison, in GaAs/AlGaAs CQW structures with m E0.35, U E0.15 eV and d ¼ 4.0 nm we get S¼ 9.4. Hence, for b b b 4,5 N¼ 2 the tunnelling rate is comparable to that in the GaAs CQW . More detailed Electron-hole estimates would have to include the trapezoidal shape of the barrier (Fig. 1b) and Fermi gas 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. Exciton Bose gas 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 mid- planes of two separate MoS layers. The Schro¨dinger equation for the relative motion reads Bi-exciton r fðrÞþ UðrÞfðrÞ¼  E fðrÞ; ð7Þ ind phases 2m where m is the reduced mass. We model the potential U(r) of electron–hole Coulomb interaction using the continuum-medium electrostatics . This simplified 0.5 10 200 approach neglects frequency dependence of the dielectric functions of the materials involved. We approximate each of MoS layers as a uniaxial dielectric slab of 8 ? thickness c ¼ 0.312 nm with principal dielectric tensor components E ¼ 14:29 Figure 4 | Schematic T¼ 0 phase diagram. The system is an and E ¼ 6:87 in the directions perpendicular and parallel to the z axis, electron–hole bilayer in which the particles have spin 1/2 but no valley respectively. In turn, the hBN spacer is modelled as a slab of thickness Nc , 33 ? c ¼ 0.333 nm, with the dielectric constants E ¼ 6:71 and E ¼ 3:56. We define degeneracy. The MoS /hBN device with the N¼ 2 layer hBN spacer 1 1 the anisotropy parameter k and the effective dielectric constant E of each j j corresponds to d/a E2.8. At such d/a the ground state of the system e e material by at r \6 is the superfluid Bose gas of indirect excitons. qffiffiffiffi qffiffiffiffi qffiffiffiffiffi qffiffiffiffiffi k k ? ? k ¼ E E ; E ¼ E E ; j ¼ 1; 2; ð8Þ j j j j j j The scaling law of equation (4) elucidates a general principle so that k ¼ 1.44, E ¼ 9.91 (MoS ) and k ¼ 1.37, E ¼ 4.89 (hBN). The interaction 2 2 2 1 1 for realizing high-temperature coherent states of excitons. The potential can be found by solving the Poisson equation for this layered system, proposed design is an initial blueprint and is amenable to further which is done by reducing it to a set of linear equations for the Fourier component optimization. For example, one or both of MoS layers can be UðqÞ and using the inverse Fourier transform: substituted by a different transition metal dichalcogenide (TMD), Z d q iqr such as WS or WSe . The two MoS monolayers may be replaced UðÞ r ¼ e UðÞ q : ð9Þ 2 2 2 2 ðÞ 2p by a single several-monolayer-thick TMD encapsulated by hBN By symmetry, UðÞ q depends only on the magnitude q¼ |q| of vector q not on in which indirect excitons would be composed of electrons and its direction. After simple algebra, we arrive at the result UqðÞ¼ CqðÞ=DqðÞ, where holes confined at the opposite sides of the TMD layer. The functions C(q) and D(q) are given by outlined principle for realizing high-temperature superfluidity 8pe 2 c k =2 c k =2  ck 2 2 2 2 1 CqðÞ¼ EðÞ E  E e ðÞ E  E e e ; ð10Þ can also be extended to layered materials other than TMD 1 1 2 1 2 and hBN. 2 2 Besides providing a new platform for exploring fundamental  c k c k 2 2 2 2 DqðÞ¼ðÞ E  E e ðÞ E þ E e 1 2 1 2 ð11Þ quantum phenomena, indirect excitons in van der Waals 2 2  2ðÞ c c k 2 2 1 4 E  E e sinh c k : 2 2 1 2 structures can be also utilized for the development of opto- Here we introduced the short-hand notations c¼ Nc þ c (the centre-to-centre 4 1 2 electronic circuits . In such devices, in-plane potential landscapes distance) and k ¼ k q (the z-direction wavenumber of the evanescent Fourier j j iqr  k z for excitons are created and controlled by external electric fields j harmonics e in medium j). From numerical calculations using equations (9)–(11) we found that potential U(r) is accurately approximated by the following that couple to the permanent dipole moment p¼ ec of the analytical expression: indirect excitons. The operation temperature for excitonic circuits in the van der Waals structures is expected to exceed by an order e 1 A pffiffiffiffiffiffiffiffiffiffiffiffiffiffi UðrÞ¼ 1 ; d ¼ k c: ð12Þ 2 2 2 of magnitude the B100 K record set by GaAs-based CQWs. E 1þ Br 1 r þ d Finally, we note that Fig. 3 was constructed following the With a suitable choice of coefficients 0oAo1 and B40 this form produces 10–12 asymptotically exact results for U(r) at both small and large r. example of the Monte-Carlo simulations in which only To solve equation (7), we discretized it on a real-space 2D grid (typically, exciton phases with two possible spin flavours were considered. 75 75). The resultant linear eigenvalue problem was diagonalized by standard In TMDs, spin and orbital degeneracies of indirect excitons may numerical methods yielding the binding energy E (N) and the gyration radius ind have a more intriguing structure. Such degeneracies can be r (N) of indirect excitons. The latter is defined in terms of a normalized ground- state wavefunction f(r) by means of the integral controlled by strong spin–orbit coupling , polarization of the 7,31 32 excitation beam or many-body interactions . 2 2 2 2 r ¼ r f ðÞ r d r: ð13Þ Experimental realization of superfluidity and counterflow The results are shown in Fig. 2. For N¼ 2 we find r ¼ 2.50 nm¼ 2.43a and superconductivity as well as excitonic circuits and spintronic/ x x E ¼ 87 meVE0.6 Ry . This binding energy is an order of magnitude larger than ind x valleytronic devices in atomically thin heterostructures may have 29,34 E ¼ 4–10 meV typical for excitons in GaAs/AlGaAs CQW structures . ind far-reaching implications for science and technology. Zero-temperature phases. It is instructive to complement the above discussion Methods of the finite-T phase diagram (Fig. 3) with commenting on the T¼ 0 phases. Interlayer tunnelling. The action S for tunnelling of quasiparticles across the hBN Such phases include electron–hole Fermi gas, exciton Bose gas and exciton spacer can be estimated from the usual formula for the rectangular potential barrier solid. The approximate phase boundaries based on available Monte-Carlo of height U : 10–12,35,36 calculations are shown in Fig. 4. Although those simulations were done for electron–hole bilayers in vacuum, an approximate correspondence with our 1=2 S ¼ 2ðÞ c =aðÞ m =m U ðÞ Ry : ð6Þ b B b 0 b dielectric environment can be achieved if we neglect the correction term in the 4 NATURE COMMUNICATIONS | 5:4555 | DOI: 10.1038/ncomms5555 | www.nature.com/naturecommunications & 2014 Macmillan Publishers Limited. All rights reserved. d / a e NATURE COMMUNICATIONS | DOI: 10.1038/ncomms5555 ARTICLE parenthesis in equation (12). The mapping is then obtained by setting the distance 20. Su, J.-J. & MacDonald, A. H. How to make a bilayer exciton condensate flow. between the layers to d¼ k c and including the effective dielectric function E in the 1 1 Nat. Phys. 4, 799–802 (2008). definition of the electron Bohr radius: 21. Littlewood, P. B. et al. Models of coherent exciton condensation. J. Phys.: 2 2 Condens. Matter 16, S3597–S3620 (2004). ‘ E 1 ‘ E a 1 1 x a ¼ ¼ ¼ : ð14Þ e 22. Eisenstein, J. P. & MacDonald, A. H. Bose-Einstein condensation of excitons in 2 2 m e 2 me 2 bilayer electron systems. Nature 432, 691–694 (2004). On the horizontal axis in Fig. 4 we plot the dimensionless intralayer distance pffiffiffiffiffiffiffiffiffiffiffiffi 23. McCann, E. & Koshino, M. The electronic properties of bilayer graphene. Rep. parameter r ¼ 1 pn a , where n is the total electron density. Solid phases form s e e Prog. Phys. 76, 056503 (2013). at large r and d/a . The exciton dipole solid crosses over to the interlocked Wigner s e 24. Min, H., Bistritzer, R., Su, J.-J. & MacDonald, A. H. Room-temperature crystals of electrons and holes as the interlayer distance becomes larger than the superfluidity in graphene bilayers. Phys. Rev. B 78, 121401 (2008). characteristic intralayer one, d\r a . s e 25. Perali, A., Neilson, D. & Hamilton, A. R. High-temperature superfluidity in The considered MoS /hBN heterostructure with N¼ 2 corresponds to d/a E2.8 2 e double-bilayer graphene. Phys. Rev. Lett. 110, 146803 (2013). for which the ground state is never a solid phase. Instead, at large r , that is, at low 26. Neilson, D., Perali, A. & Hamilton, A. R. Excitonic superfluidity and screening electron density, the ground state of the system is an exciton gas. As r decreases, in electron-hole bilayer systems. Phys. Rev. B 89, 060502(R) (2014). the Mott transition to electron–hole Fermi gas occurs. At this transition the 1 27. Kharitonov, M. Y. & Efetov, K. B. Electron screening and excitonic excitons dissociate owing to screening and phase space filling . The latest condensation in double-layer graphene systems. Phys. Rev. B 78, 241401 (2008). estimates of the T¼ 0 Mott transition at d¼ 2.8a give r E6, which corresponds e s 28. Komsa, H.-P. & Krasheninnikov, A. V. Electronic structures and optical to equation (3). Interestingly, a Bardeen–Cooper–Schrieffer-like excitonic state of a properties of realistic transition metal dichalcogenide heterostructures from dense electron–hole gas predicted by the earlier theory was not found in the cited 10,12 first principles. Phys. Rev. B 88, 085318 (2013). Monte-Carlo calculations . 29. Grosso, G. et al. Excitonic switches operating at around 100 K. Nat. Photonics 3, Unless the effective interlayer separation d is small, the interaction between 38,39 577–580 (2009). indirect excitons is dominated by the classical dipole repulsion term . However, 40–42 30. Korma´nyos, A., Zo´lyomi, V., Drummond, N. D. & Burkard, G. Spin-orbit as shown previously ,at do0.87a quantum exchange–correlation effects cause coupling, quantum dots, and qubits in monolayer transition metal the change of repulsion to attraction leading to appearance of bi-excitons 12,39 dichalcogenides. Phys. Rev. X 4, 011034 (2014). phases . Narrow ranges of modulated phases (stripes, bubbles or supersolids) may exist near any of the first-order phase transition lines . 31. Ross, J. S. et al. Electrical control of neutral and charged excitons in a monolayer semiconductor. Nat. Commun. 4, 1474 (2013). 32. Ben-Tabou de Leon, S. & Laikhtman, B. Mott transition, biexciton crossover, References and spin ordering in the exciton gas in quantum wells. Phys. Rev. B 67, 235315 1. Keldysh, L. V. & Kozlov, A. N. Collective properties of excitons in (2003). semiconductors. J. Exp. Theor. Phys. 27, 521 (1968). 33. Cai, Y., Zhang, L., Zeng, Q., Cheng, L. & Xu, Y. Infrared reflectance spectrum 2. Lozovik, Y. E. & Yudson, V. I. A new mechanism for superconductivity: pairing of BN calculated from first principles. Solid State Commun. 141, 262–266 between spatially separated electrons and holes. J. Exp. Theor. Phys. 44, 389 (2007). (1976). 34. Szymanska, M. H. & Littlewood, P. B. Excitonic binding in coupled quantum 3. Fukuzawa, T., Kano, S., Gustafson, T. & Ogawa, T. Possibility of coherent light wells. Phys. Rev. B 67, 193305 (2003). emission from Bose condensed states of SEHPs. Surf. Sci. 228, 482–485 (1990). 35. Astrakharchik, G. E., Boronat, J., Kurbakov, I. L. & Lozovik, Y. E. Quantum 4. High, A. A., Novitskaya, E. E., Butov, L. V., Hanson, M. & Gossard, A. C. phase transition in a two-dimensional system of dipoles. Phys. Rev. Lett. 98, Control of exciton fluxes in an excitonic integrated circuit. Science 321, 060405 (2007). 229–231 (2008). 36. Bu¨chler, H. P. et al. Strongly correlated 2D quantum phases with cold polar 5. High, A. A. et al. Spontaneous coherence in a cold exciton gas. Nature 483, molecules: controlling the shape of the interaction potential. Phys. Rev. Lett. 98, 584–588 (2012). 060404 (2007). 6. Geim, A. K. & Grigorieva, I. V. Van der Waals heterostructures. Nature 499, 37. Keldysh, L. V. & Kopaev, Y. V. Possible instability of the semimetallic state 419–425 (2013). toward Coulomb interaction. Sov. Phys. Solid State 6, 2219 (1965). 7. Xiao, D., Liu, G.-B., Feng, W., Xu, X. & Yao, W. Coupled spin and valley 38. Zhu, X., Littlewood, P. B., Hybertsen, M. S. & Rice, T. M. Exciton condensate in physics in monolayers of MoS and other group-VI dichalcogenides. Phys. Rev. semiconductor quantum well structures. Phys. Rev. Lett. 74, 1633–1636 (1995). Lett. 108, 196802 (2012). 39. Lozovik, Y. E. & Berman, O. L. Phase transitions in a system of two coupled 8. Berkelbach, T. C., Hybertsen, M. S. & Reichman, D. R. Theory of neutral and quantum wells. JETP Lett. 64, 573–579 (1996). charged excitons in monolayer transition metal dichalcogenides. Phys. Rev. B 40. Meyertholen, A. D. & Fogler, M. M. Biexcitons in two-dimensional systems 88, 045318 (2013). with spatially separated electrons and holes. Phys. Rev. B 78, 235307 (2008). 9. Zhang, Y. et al. Direct observation of the transition from indirect to direct 41. Tan, M. Y. J., Drummond, N. D. & Needs, R. J. Exciton and biexciton energies bandgap in atomically thin epitaxial MoSe . Nat. Nanotechnol. 9, 111–115 in bilayer systems. Phys. Rev. B 71, 033303 (2005). (2013). 42. Schindler, C. & Zimmermann, R. Analysis of the exciton-exciton interaction in 10. De Palo, S., Rapisarda, F. & Senatore, G. Excitonic condensation in a symmetric semiconductor quantum wells. Phys. Rev. B 78, 045313 (2008). electron-hole bilayer. Phys. Rev. Lett. 88, 206401 (2002). 43. Spivak, B. & Kivelson, S. A. Phases intermediate between a two-dimensional 11. Schleede, J., Filinov, A., Bonitz, M. & Fehske, H. Phase diagram of bilayer electron liquid and Wigner crystal. Phys. Rev. B 70, 155114 (2004). electron-hole plasmas. Contrib. Plasma Phys. 52, 819–826 (2012). 12. Maezono, R., Lo´pez Ros, P., Ogawa, T. & Needs, R. J. Excitons and biexcitons in symmetric electron-hole bilayers. Phys. Rev. Lett. 110, 216407 (2013). Acknowledgements 13. Mak, K. F., Lee, C., Hone, J., Shan, J. & Heinz, T. F. Atomically thin MoS : 2 This work was supported by the US ONR & UCOP (M.M.F.), NSF (L.V.B.) and also by a new direct-gap semiconductor. Phys. Rev. Lett. 105, 136805 (2010). the European Research Council and EC-FET European Graphene Flagship (K.S.N.). We 14. Chernikov, A. et al. Non-hydrogenic exciton Rydberg series in monolayer WS . 2 are grateful to Andre´ K. Geim for comments. Preprint at http://arxiv.org/abs/1403.4270 (2014). 15. Britnell, L. et al. Field-Effect tunneling transistor based on vertical graphene Author contributions heterostructures. Science 335, 947–950 (2012). All the authors contributed to the ideas, discussion, and writing of the paper. M.M.F. 16. Nikolaev, V. V. & Portnoi, M. E. Mott transition of spatially indirect excitons. carried out numerical simulations. Proc. SPIE 5509, 187–193 (2004). 17. Butov, L. V., Kulakovskii, V. D., Lach, E., Forchel, A. & Gru¨tzmacher, D. Magnetoluminescence study of many-body effects in homogeneous quasi-two- Additional information dimensional electron-hole plasma in undoped In Ga As/InP single quantum x 1–x Competing financial interests: The authors declare no competing financial interests. wells. Phys. Rev. B 44, 10680–10688 (1991). 18. Kappei, L., Szczytko, J., Morier-Genoud, F. & Deveaud, B. Direct observation of Reprints and permission information is available online at http://npg.nature.com/ the Mott transition in an optically excited semiconductor quantum well. Phys. reprintsandpermissions/ Rev. Lett. 94, 147403 (2005). 19. Filinov, A., Prokof’ev, N. V. & Bonitz, M. Berezinskii-Kosterlitz-Thouless How to cite this article: Fogler, M. M. et al. High-temperature superfluidity with transition in two-dimensional dipole systems. Phys. Rev. Lett. 105, 070401 indirect excitons in van der Waals heterostructures. Nat. Commun. 5:4555 (2010). doi: 10.1038/ncomms5555 (2014). NATURE COMMUNICATIONS | 5:4555 | DOI: 10.1038/ncomms5555 | www.nature.com/naturecommunications 5 & 2014 Macmillan Publishers Limited. All rights reserved.

Journal

Nature Communications – Springer Journals

Published: Jul 28, 2014

There are no references for this article.