Bose-Einstein condensation of quasiparticles in graphene
The collective properties of different quasiparticles in various graphene based structures in high magnetic field have been studied. We predict Bose-Einstein condensation (BEC) and superfluidity of 2D spatially indirect magnetoexcitons in two-layer graphene. The superfluid density and the temperature of the Kosterlitz-Thouless phase transition are shown to be increasing functions of the excitonic density but decreasing functions of magnetic field and the interlayer separation. The instability of the ground state of the interacting 2D indirect magnetoexcitons in a slab of superlattice with alternating electron and hole graphene layers (GLs) is established. The stable system of indirect 2D magnetobiexcitons, consisting of pair of indirect excitons with opposite dipole moments, is considered in graphene superlattice. The superfluid density and the temperature of the Kosterlitz-Thouless phase transition for magnetobiexcitons in graphene superlattice are obtained. Besides, the BEC of excitonic polaritons in GL embedded in a semiconductor microcavity in high magnetic field is predicted. While superfluid phase in this magnetoexciton polariton system is absent due to vanishing of magnetoexciton-magnetoexciton interaction in a single layer in the limit of high magnetic field, the critical temperature of BEC formation is calculated. The essential property of magnetoexcitonic systems based on graphene (in contrast, e.g., to a quantum well) is stronger influence of magnetic field and weaker influence of disorder. Observation of the BEC and superfluidity of 2D quasiparticles in graphene in high magnetic field would be interesting confirmation of the phenomena we have described.
pacs:03.75.Hh, 73.20.Mf, 71.36.+c
The production of graphene, a two-dimensional (2D) honeycomb lattice of carbon atoms that form the basic planar structure in graphite, has been achieved recently (1); (2). Electronic properties of graphene, caused by unusual properties of the band structure, became the object of many recent experimental and theoretical studies (1); (2); (3); (4); (5); (6); (7). Graphene is a gapless semiconductor with massless electrons and holes, described as Dirac-fermions (8). The various studies of unique electronic properties of graphene in a magnetic field have been performed recently (9); (10); (11); (12). The energy spectrum and the wavefunctions of magnetoexcitons, or electron-hole pairs in a magnetic field, in graphene have been calculated in interesting works (13); (14).
The 2D electron system was studied in quantum wells (QWs) (15). The systems of spatially-indirect excitons (or pairs of electrons and holes spatially separated in different QWs) in the system of coupled quantum wells (CQWs), with and without a magnetic field were studied in Refs. [(16); (17); (18); (19); (20); (21); (22); (23)]. The experimental and theoretical interest to study these systems is particularly caused by the possibility of the BEC and superfluidity of indirect excitons, which can manifest in the CQWs as persistent electrical currents in each well and also through coherent optical properties and Josephson phenomena (16); (18); (19); (20); (22); (23). The outstanding experimental success was achieved now in this field (24); (25); (26); (27). The electron-hole pair condensation in the graphene-based bilayers have been studied in [(28); (29); (30); (31)].
The collective properties of Bose quasiparticles such as excitons, biexcitons, and polaritons in various graphene-based structures in high magnetic field are very interesting in the relevance to the BEC and superfluidity, since the random field in graphene is weaker than in a QW, particularly, because in a QW the random field is generated due to the fluctuations of the width of a QW. Let us mention that if the interaction of bosons with the random field is stronger, the BEC critical temperature is lower (32). In this paper we study the superfluidity of magnetoexcitons in bilayer graphene, instability of the system of magnetoexcitons in superlattice formed by many GLs, superfluidity of magnetobiexcitons in graphene superlattice, and BEC of polaritons in GL embedded in optical microcavity in a trap. Let us mention that all these systems of quasiparticles are considered in high magnetic field. The BEC of magnetoexcitons in graphene structures can exist at much lower magnetic field than in QWs, because the distance between electron Landau levels in graphene is much higher than in a QW at the same magnetic field, and, therefore, the lower magnetic field is required in graphene than in a QW to neglect the electron transitions between the Landau levels.
Ii Effective Hamiltonian of magnetoexcitons and photons in microcavity in high magnetic field
Recently, Bose coherent effects of 2D exciton polaritons in a quantum well embedded in an optical microcavity have been the subject of theoretical (33). and experimental (34); (35); (36) studies. To obtain polaritons, two mirrors placed opposite each other form a microcavity, and quantum wells are embedded within the cavity at the antinodes of the confined optical mode. The resonant exciton-photon interaction results in the Rabi splitting of the excitation spectrum. Two polariton branches appear in the spectrum due to the resonant exciton-photon coupling. The lower polariton (LP) branch of the spectrum has a minimum at zero momentum. The effective mass of the lower polariton is extremely small, and lies in the range of the free electron mass. These lower polaritons form a 2D weakly interacting Bose gas. The extremely light mass of these bosonic quasiparticles, which corresponds to experimentally achievable excitonic densities, result in a relatively high critical temperature for superfluidity, of or even higher. The reason for such a high critical temperature is that the 2D thermal de Broglie wavelength is inversely proportional to the mass of the quasiparticle.
While at finite temperatures there is no true BEC in any infinite untrapped 2D system, a true 2D BEC can exist in the presence of a confining potential (37); (38). Recently, the polaritons in a harmonic potential trap have been studied experimentally in a GaAs/AlAs quantum well embedded in a GaAs/AlGaAs microcavity (39). In this trap, the exciton energy is shifted using stress. In this system, evidence for the BEC of polaritons in a QW has been observed (40). The theory of the BEC and superfluidity of excitonic polaritons in a QW without magnetic field in a parabolic trap has been developed in Ref. [(41)]. The Bose condensation of polaritons is caused by their bosonic character (40); (41); (42). However, while the exciton polaritons have been studied in a QW, the formation of the polaritons in graphene in high magnetic field have not yet been considered. Moreover, the magnetopolaritons formed as superposition of magnetoexcitons and cavity photons in magnetic field have not yet been studied. We consider a 2D system of polaritons in graphene layers (GLs) embedded in a microcavity from the point of view of the existence of the BEC within it.
Lets us consider the most general case when the superlattice with alternating electronic and hole parallel GLs in the external field is embedded in an optical microcavity in high magnetic field. At the small densities the system of indirect excitons at low temperatures is the two-dimensional weakly nonideal Bose gas with normal to wells dipole moments in the ground state (, is the charge of an electron, is the interlayer separation). In contrast to ordinary excitons, for the low-density spatially indirect magnetoexciton system the main contribution to the energy is originated from the dipole-dipole interactions and of magnetoexcitons with opposite (see Fig. 1) and parallel dipoles, respectively. The potential energy of interaction between two indirect magnetoexcitons with parallel and opposite dipoles is a function of the distance between indirect magnetoexcitons along GLs and is given as
where is the charge of an electron, is the dielectric constant.
The Hamiltonian of magnetoexcitons and photons in the strong magnetic field is given by
where is a magnetoexcitonic Hamiltonian, is a photonic Hamiltonian, and is the Hamiltonian of magnetoexciton-photon interaction.
Let us analyze each term of the Hamiltonian (2). The effective Hamiltonian and the energy dispersion for magnetoexcitons in graphene layers in a high magnetic field in the infinite system was derived in Ref. [(43)]. The effective Hamiltonian of the low-density system of the indirect magnetoexcitons in high magnetic field in the superlattice in the subspace of the lowest Landau level is given by (43) (we neglect the electrons transitions between different Landau levels due to electron-hole Coulomb attraction)
Here is the effective Hamiltonian of the system of noninteracting trapped magnetoexcitons in high magnetic field:
where , , , are creation and annihilation operators of the magnetoexcitons with up and down dipoles. In Eq. (4), is the band gap energy which can depend on the position of the magnetoexciton in space in the presence of the trap, is the binding energy of a magnetoexciton, and , where is the effective magnetic mass of a magnetoexciton. Similarly to the Bose atoms in a trap in the case of a slowly varying external potential (44), we can make the quasiclassical approximation, assuming that the effective magnetoexciton mass does not depend on a characteristic size of the trap and it is a constant within the trap. This quasiclassical approximation is valid if . The harmonic trap is formed by the two-dimensional planar potential in the plane of graphene. The potential trap can be produced in two different ways. One way is when the potential trap can be produced by applying an external inhomogeneous electric field. The spatial dependence of the external field potential is caused by shifting of magnetoexciton energy by applying an external inhomogeneous electric field. The photonic states in the cavity are assumed to be unaffected by this electric field. In this case the band energy is given by ( is the band gap energy, which is the difference between the Landau levels and , is the Fermi velocity of electrons in graphene (45), is the magnetic length). Near the minimum of the magnetoexciton energy, can be approximated by the planar harmonic potential , where is the spring constant, is the distance between the center of mass of the magnetoexciton and the center of the trap. Note that a high magnetic field does not change the trapping potential in the effective Hamiltonian (46). Let us mention that, while the quasiparticles in GLs and QWs in high magnetic field are described by the same effective Hamiltonian with the only difference in the effective magnetic mass of magnetoexciton. This difference is caused by the four-component spinor structure of magnetoexciton wave function in GL, while magnetoexciton in a QW and CQWs is characterized by the one-component scalar wave function.
The Hamiltonian which describes the interaction between magnetoexcitons is
where and are the 2D Fourier images of and , respectively, and is the surface of the system.
The Hamiltonian and the energy spectrum of non-interacting photons in a semiconductor microcavity are given by (47):
where and are photonic creation and annihilation Bose operators. In Eq. (6), is the length of the cavity, is the effective refractive index and is the dielectric constant of the cavity. We assume that the length of the microcavity has the following form:
corresponding to the resonance of the photonic and magnetoexcitonic branches at , i.e. . As it follows from the energy spectra in (4) and (6), and Eqs. (9) and (7), the length of the microcavity, corresponding to a magnetoexciton-photon resonance, decreases with the increment of the magnetic field as . The resonance between magnetoexcitons and cavity photonic modes can be achieved either by controlling the spectrum of magnetoexcitons by changing magnetic field or by choosing the appropriate length of the microcavity .
Alternatively to the case considered above, now the trapping of magnetopolaritons is caused by the inhomogeneous shape of the cavity when the length of the cavity is determined by Eq. (7) with the term added to , where is the distance between the photon and the center of the trap. In case 2, the is the curvature characterizing the shape of the cavity. The Hamiltonian and the energy spectrum of the photons in this case are shown by Eq. (6), and the length of the microcavity is given by Eq. (7). In this case, for the photonic spectrum in the effective mass approximation is given by substituting the slowly changing shape of the length of cavity depending on the term into Eq. (6) representing the spectrum of the cavity photons. This quasiclassical approximation is valid if , where is the size of the magnetoexciton cloud in an ideal magnetoexciton gas and . The Hamiltonian and energy spectrum of magnetoexcitons in this case are given by (4).
The projection of the electron-hole Hamiltonian in magnetic field for the CQWs and GLs onto the lowest Landau level results in the effective Hamiltonian (15) with renormalized mass and where term related to the vector potential is missing. The magnetic field in the effective Hamiltonian (15) enters in the renormalized mass of the magnetoexciton . Therefore, Hamiltonian for the spatially separated electrons and holes in two-layer system for the CQWs and for the bilayer graphene can be reduced in high magnetic field to the effective Hamiltonian (15). Magnetic field is reflected by the effective Hamiltonian (15) only through the effective magnetic mass of a magnetoexciton in the expression for in the first term of . The only difference in the effective Hamiltonian (15) for the CQWs and bilayer graphene realizations of two-layer systems is that for the bilayer graphene is four times less than for the CQWs due to the four-component spinor structure of the wavefunction of the relative motion for the isolated non-interacting electron-hole pair in magnetic field (52).
Transitions between Landau levels due to the Coulomb electron-hole attraction for the large electron-hole separation can be neglected, if the following condition is valid: for the QWs and for the GLs, where is the Fermi velocity of electrons (45), and are the magnetoexcitonic binding energy and the cyclotron frequency, respectively. This corresponds to the high magnetic field , the large interlayer separation and large dielectric constant of the insulator layer between the GLs.
Iii Bose-Einstein condensation of polaritons in graphene in a high magnetic field
In this Section we consider trapped polaritons in a single graphene layer embedded into an optical microcavity in high magnetic field. When an undoped electron system in graphene in a magnetic field without an external electric field is in the ground state, half of the zeroth Landau level is filled with electrons, all Landau levels above the zeroth one are empty, and all levels below the zeroth one are filled with electrons. We suggest using the gate voltage to control the chemical potential in graphene by two ways: to shift it above the zeroth level so that it is between the zeroth and first Landau levels (the first case) or to shift the chemical potential below the zeroth level so that it is between the first negative and zeroth Landau levels (the second case). In both cases, all Landau levels below the chemical potential are completely filled and all Landau levels above the chemical potential are completely empty. Based on the selection rules for optical transitions between the Landau levels in single-layer graphene (53), in the first case, there are allowed transitions between the zeroth and the first Landau levels, while in the second case there are allowed transitions between the first negative and zeroth Landau levels. Correspondingly, we consider magnetoexcitons formed in graphene by the electron on the first Landau level and the hole on the zeroth Landau level (the first case) or the electron on the zeroth Landau level and the hole on the Landau level (the second case). Note that by appropriate gate potential we can also use any other neighboring Landau levels and .
For the relatively high dielectric constant of the microcavity, the magnetoexciton energy in graphene can be calculated by applying perturbation theory with respect to the strength of the Coulomb electron-hole attraction analogously as it was done in [(17)] for 2D quantum wells in a high magnetic field with non-zero electron and hole masses. This approach allows us to obtain the spectrum of an isolated magnetoexciton with the electron on the Landau level and the hole on the Landau level in a single graphene layer, and it will be exactly the same as for the magnetoexciton with the electron on the Landau level and the hole on the Landau level . The characteristic Coulomb electron-hole attraction for the single graphene layer is . The energy difference between the first and zeroth Landau levels in graphene is . For graphene, the perturbative approach with respect to the strength of the Coulomb electron-hole attraction is valid when (17). This condition can be fulfilled at all magnetic fields if the dielectric constant of the surrounding media satisfies the condition . Therefore, we claim that the energy difference between the first and zeroth Landau levels is always greater than the characteristic Coulomb attraction between the electron and the hole in the single graphene layer at any if . Thus, applying perturbation theory with respect to weak Coulomb electron-hole attraction in graphene embedded in the microcavity () is more accurate than for graphene embedded in the microcavity (). However, the magnetoexcitons in graphene exist in high magnetic field. Therefore, we restrict ourselves by consideration of high magnetic fields.
Polaritons are linear superpositions of excitons and photons. In high magnetic fields, when magnetoexcitons may exist, the polaritons become linear superpositions of magnetoexcitons and photons. Let us define the superpositions of magnetoexcitons and photons as magnetopolaritons. It is obvious that magnetopolaritons in graphene are two-dimensional, since graphene is a 2D structure. The Hamiltonian of magnetopolaritons in the strong magnetic field is given by Eq. (2). It can be shown that the interaction between two direct 2D magnetoexcitons in graphene with the electron on the Landau level and the hole on the Landau level can be neglected in a strong magnetic field, in analogy to what is described in Ref. [(17)] for 2D magnetoexcitons in a quantum well. Thus, the Hamiltonian (2) does not include the term corresponding to the interaction between two direct magnetoexcitons in a single graphene layer. So in high magnetic field there is the BEC of the ideal magnetoexcitonic gas in graphene. Therefore, in a single graphene layer is high mahnetic field we assume in Eq. (3).
The binding energy and effective magnetic mass of a magnetoexciton in graphene obtained using the first order perturbation respect to the electron-hole Coulomb attraction similarly to the case of a single quantum well (17) are given by
We obtain the effective Hamiltonian of polaritons by applying the standard procedure (48); (49); (50); (51), when we diagonalize the Hamiltonian (2) by using Bogoliubov transformations. If we measure the energy relative to the lower magnetopolariton energy , we obtain the resulting effective Hamiltonian for trapped magnetopolaritons in graphene in a magnetic field. At small momenta () and weak confinement , this effective Hamiltonian is
and the effective magnetic mass of a magnetopolariton is given by
According to Eqs. (11) and (9), the effective magnetopolariton mass increases with the increment of the magnetic field as . Let us emphasize that the resulting effective Hamiltonian for magnetopolaritons in graphene in a magnetic field for the parabolic trap is given by Eq. (10) for both physical realizations of confinement represented by case 1 and case 2.
Neglecting anharmonic terms for the magnetoexciton-photon coupling, the Rabi splitting constant can be estimated quasiclassically as
where , and are Pauli matrices, is the Hamiltonian of the electron-photon interaction corresponding to the electron in graphene described by Dirac dispersion, is the electric field corresponding to a single cavity photon, is the volume of microcavity, is the photon frequency. The initial electron state corresponds to the completely filled Landau level and completely empty Landau level . The final electron state corresponds to creation of one magnetoexciton with the electron on the Landau level and the hole on the Landau level . The transition dipole moment corresponding to the process of creation of this magnetoexciton is given by . Let us note that in Eq. (12) the energy of photon absorbed at the creation of the magnetoexciton is given by (we assume that ). Substituting the photon energy and the transition dipole moment from into Eq. (12), we obtain the Rabi splitting corresponding to the creation of a magnetoexciton with the electron on the Landau level and the hole on the Landau level in graphene: .
Thus, the Rabi splitting in graphene is related to the creation of the magnetoexciton, which decreases when the magnetic field increases and is proportional to . Therefore, the Rabi splitting in graphene can be controlled by the external magnetic field. It is easy to show that the Rabi splitting related to the creation of the magnetoexciton, the electron on the Landau level and the hole on the Landau level will be exactly the same as for the magnetoexciton with the electron on the Landau level and the hole on the Landau level . Let us mention that dipole optical transitions from the Landau level to the Landau level , as well as from the Landau level to the Landau level , are allowed by the selection rules for optical transitions in single-layer graphene (53).
Although Bose-Einstein condensation cannot take place in a 2D homogeneous ideal gas at non-zero temperature, as discussed in Ref. [(37)], in a harmonic trap the BEC can occur in two dimensions below a critical temperature . In a harmonic trap at a temperature below a critical temperature (), the number of non-interacting magnetopolaritons in the condensate is given in Ref. [(37)]. Applying the condition , and assuming that the magnetopolariton effective mass is given by Eq. (11), we obtain the BEC critical temperature for the ideal gas of magnetopolaritons in a single graphene layer in a magnetic field:
where is the total number of magnetopolaritons, and are the spin and graphene valley degeneracies for an electron and a hole, respectively, is the Boltzmann constant. At temperatures above , the BEC of magnetopolaritons in a single graphene layer does not exist. as a function of magnetic field and spring constant is presented in Fig. 2. In our calculations, we used . According to Eq. (13), the BEC critical temperature decreases with the magnetic field as and increases with the spring constant as . Note that we assume that the quality of the cavity is sufficiently high, so that the time of the relaxation to the Bose condensate quasiequilibrium state is smaller than the life time of the photons in the cavity.
Above we discussed the BEC of the magnetopolaritons in a single graphene layer placed within a strong magnetic field. What would happen in a multilayer graphene system in a high magnetic field? Let us mention that the magnetopolaritons formed by the microcavity photons and the indirect excitons with the spatially separated electrons and holes in different parallel graphene layers embedded in a semiconductor microcavity can exist only at very low temperatures . For the case of the spatially separated electrons and holes, the Rabi splitting is very small in comparison to the case of electrons and holes placed in a single graphene layer. This is because and the matrix element of magnetoexciton generation transition is proportional to the overlapping integral of the electron and hole wavefunctions, which is very small if the electrons and holes are placed in different graphene layers. Therefore, we cannot predict the effect of relatively high BEC critical temperature for the electrons and holes placed in different graphene layers.
Iv Superfluidity of magnetoexcitons in Bilayer Graphene
We consider two parallel graphene layers (GLs) separated by an insulating slab of dielectric (for example, SiO) (52). The spatial separation of electrons and holes in different GLs can be achieved by applying an external electric field. Besides, the spatially separated electrons and holes can be created by varying the chemical potential by applying a bias voltage between two GLs or between two gates located near the corresponding graphene sheets. The equilibrium system of local pairs of spatially separated electrons and holes can be created by varying the chemical potential by using a bias voltage between two GLs or between two gates located near the corresponding graphene sheets (case 1) (for simplicity, we also call these equilibrium local e-h pairs in magnetic field as indirect magnetoexcitons). In the case 1 a magnetoexciton is formed by an electron on the Landau level and hole on the Landau level . Magnetoexcitons with spatially separated electrons and holes can be created also by laser pumping (far infrared in graphene) (case 2) and by applying perpendicular electric field as for CQWs(24); (25); (27). In the case 2 a magnetoexciton is formed by an electron on the Landau level and hole on the Landau level . We assume the system is in quasiequilibrium state. Below we assume the low-density regime for magnetoexcitons, i.e. magnetoexciton radius , where is the 2D magnetoexciton density.
In a strong magnetic field at low densities, ( is the magnetic length, is the electron charge, is the speed of light), indirect magnetoexcitons repel as parallel dipoles, and we have for the pair interaction potential:
where is the interlayer separation, is the dielectric constant of the insulator between two layers, are radius vectors of the center of mass of two magnetoexcitons. Since typically, the value of is , and in this approximation, the effective Hamiltonian in the magnetic momentum representation in the subspace the lowest Landau level has the same form (compare with Ref.[(19)]) as for two-dimensional boson system without a magnetic field, but with the magnetoexciton magnetic mass (which depends on and ; see below) instead of the exciton mass (), magnetic momenta instead of ordinary momenta. We can obtain the effective Hamiltonian for bilayer graphene without a confinement if we keep considering only two graphene layers in the magnetoexciton effective Hamiltonian without a trap (3):
where the matrix element is the Fourier transform of the pair interaction potential and for the lowest Landau level we denote the spectrum of the single exciton . For an isolated magnetoexciton on the lowest Landau level at the small magnetic momenta under consideration, , where is the effective magnetic mass of a magnetoexciton in the lowest Landau level and is a function of the distance between – and – layers and magnetic field (see Ref. [(21)]). In strong magnetic fields at the exciton magnetic mass is for the QWs (21) and for the GLs (52).
We study the magnetoexciton-magnetoexciton scattering applying the theory of weakly-interacting 2D Bose-gas (16); (19). The chemical potential of two-dimensional dipole magnetoexcitons in graphene bilayer system, in the ladder approximation, has the form (compare to Refs. [(16); (19)]):
where is the spin and valley degeneracy factor for a magnetoexciton in graphene bilayer, is the 2D density of magnetoexcitons.
At small momenta the collective spectrum of magnetoexciton system is the sound-like ( is the sound velocity) and satisfied to Landau criterium for superfluidity. The density of the superfluid component for two-dimensional system with the sound spectrum can be estimated as:(32)
where is the Riemann zeta function, and . The second term in Eq.(17) is the temperature dependent normal density taking into account gas of phonons (”bogolons”) with dispersion law , is given by Eq.(16).
In a 2D system, superfluidity of magnetoexcitons appears below the Kosterlitz-Thouless transition temperature (54): . The dependence of on and for the cases 1 and 2 is represented in Fig. 3 (since in the case 1 the binding energy two times higher and the effective magnetic mass is two times smaller than in the case 2 (52), the magnetoexcitons in the case 1 are expected to be twice more stable and in the case 1 is expected to be approximately twice higher than in the case 2 at fixed , and ). The temperature for the onset of superfluidity due to the Kosterlitz-Thouless transition at a fixed magnetoexciton density decreases as a function of magnetic field and interlayer separation . This is due to the increased as a functions of and . The decreases as at or as when .
V Instability of dipole magnetoexcitons and superfluidity of magnetobiexcitons in graphene superlattices
We consider magnetoexcitons in the superlattices with alternating electronic and hole GLs. We suppose that recombination times can be much greater than relaxation times due to small overlapping of spatially separation of electron and hole wave functions in GLs. In this case electrons and holes are characterized by different quasi-equilibrium chemical potentials. Then in the system of indirect excitons in superlattices, as in CQW (16); (19), the quasiequilibrium phases appear. No external field applied to a slab of superlattice is assumed. If ”electron” and ”hole” quantum wells alternate, there are excitons with parallel dipole moments in one pair of wells, but dipole moments of excitons in another neighboring pairs of neighboring wells have opposite direction. This fact leads to essential distinction of properties of system in superlattices from one for coupled quantum wells with spatially separated electrons and holes, where indirect exciton system is stable due to dipole-dipole repelling of all excitons. This difference manifests itself already beginning from three-layer or system. We assume that alternating layers can be formed by independent gating with the corresponding potentials which shift chemical potentials in neighboring layers up and down or by alternating doping (by donors and acceptors, respectively).
Let us show that the low-density system of weakly interacting two-dimensional indirect magnetoexcitons in superlattices is instable, contrary to the two-layer system in the CQW. At the small densities the system of indirect excitons at low temperatures is the two-dimensional weakly nonideal Bose gas with normal to wells dipole moments in the ground state (, is the interlayer separation). In contrast to ordinary excitons, for the low-density spatially indirect magnetoexciton system the main contribution to the energy is originated from the dipole-dipole interactions and of magnetoexcitons with opposite (see Fig. 1) and parallel dipoles, respectively.
The behavior of the potential energies and as the functions of the distance between two excitons is shown in Fig. 4. We suppose that and , where is the mean distance between dipoles normal to the wells. We consider the case, when the number of quantum wells in superlattice is restricted . This is valid for small or for sufficiently low exciton density.
We can obtain the effective Hamiltonian for graphene superlattice without a confinement if we keep considering only the magnetoexciton effective Hamiltonian (3) without a trap. Let us apply the Bogolubov approximation to analyze a stability of the ground state of the weakly nonideal Bose gas of indirect excitons in superlattices. We assume and are the 2D Fourier images of and at , respectively, and is the surface of the system. Let us mention that the appropriate cut-off parameter for this Fourier transform is the classical turning point of the dipole-dipole interaction. Let us mention that the appropriate cut-off parameter for this Fourier transform is the classical turning point of the dipole-dipole interaction. Note that the cut-off parameter for the potential is much greater than for (the cut-off parameters for the both potentials can be represented in Fig. 4 by the points where the curves corresponding to and are crossed by the chemical potential represented by the horizontal straight line placed right above but close to ). Therefore, we claim that , , and .
Applying the unitary Bogoliubov transformations to the magnetoexciton operators , , , and , we diagonalize the Hamiltonian in the Bogoliubov approximation (55). Finally, we obtain
with the spectrum of quasiparticles :
Since and , we have at . Therefore, at low temperatures the quasiparticles only with the spectrum will be excited, since the excitation of these quasiparticles requires less energy than for the quasiparticles with the spectrum . Since , it is easy to see from Eq. (V) that for the small momenta the spectrum of excitations becomes imaginary. Hence, the system of weakly interacting indirect magnetoexcitons in the slab of the superlattice is unstable. It can be seen that the condition of the instability of magnetoexcitons becomes stronger as magnetic field higher, because increases with the increment of the magnetic field, and, therefore, the region of resulting in the imaginary collective spectrum increases as increases.
As the ground state of the system we consider the low-density weakly nonideal gas of two-dimensional indirect magnetobiexcitons, created by indirect magnetoexcitons with opposite dipoles in neighboring pairs of wells (Fig. 1). The mean dipole moment of indirect magnetobiexciton is equal to zero. However, the quadrupole moment is nonzero and equal to (the large axis of the quadrupole is normal to quantum wells/graphene layers). So indirect magnetobiexcitons interact at long distances as parallel quadrupoles: .
We apply the theory of weakly-interacting 2D Bose gas (16) to study the magnetobiexciton-magnetobiexciton repulsion. The chemical potential of two-dimensional biexcitons, repulsed by the quadrupole law, in the ladder approximation, has the form (compare to Refs. [(16); (19)]):
where is the density of magnetobiexcitons in graphene layers and we considered that the magnetic mass of magnetobiexciton is twice of the magnetic mass of magnetoexciton, i.e. .
At small momenta the collective spectrum of magnetobiexciton system is the sound-like ( is the sound velocity) and satisfied to Landau criterion for superfluidity. The density of the superfluid component for two-dimensional system with the sound spectrum is (32)
In a 2D system, superfluidity of magnetobiexcitons appears below the Kosterlitz-Thouless transition temperature . Employing for the superfluid component, we obtain an equation for the Kosterlitz-Thouless transition temperature . The dependence of on the density of magnetoexcitons at different magnetic field for superlattice consisting of quantum wells and graphene layers is represented on Fig. 5. Let us mention, that we have apply the same effective Hamiltonian to describe magnetobiexcitons in high magnetic field in the superlattice of QWs and GLs with the only difference in the effective magnetic mass of magnetoexciton .
According to Fig. 5, the temperature for the onset of superfluidity due to the Kosterlitz-Thouless transition at a fixed magnetoexciton density decreases as a function of magnetic field and interlayer separation . This is due to the increased effective magnetic mass of magnetoexcitons as a functions of and . The decreases as at or as when . According to Fig. 5, the Kosterlitz-Thouless temperature is higher for the superlattice consisting of graphene layers than for the superlattice consisting of the quantum wells, and this difference is as stronger as the magnetic field is smaller.
We have obtained the effective Hamiltonian of the quasiparticles in graphene structures in high magnetic field: indirect magnetoexcitons in graphene bilayer, magnetobiexcitons in graphene superlattices and magnetopolaritons in a graphene layer embedded in optical microcavity. It was shown that the gas of magnetoexcitons in graphene superlattice is instable due to the attraction between magnetoexcitons with parallel dipoles, while the system of magnetobiexcitons in the graphene superlattice is stable. We have shown that, while the quasiparticles in GLs and QWs in high magnetic field are described by the same effective Hamiltonian with the only difference in the effective magnetic mass of magnetoexciton. This difference is caused by the four-component spinor structure of magnetoexciton wave function in GL, while magnetoexciton in a QW and CQWs is characterized by the one-component scalar wave function. Besides, we show that the magnetoexciton system graphene bilayer and magnetobiexciton system in graphene superlattice can be described as a 2D weakly-interacting Bose gas, which is superfluid below the Kosterlitz-Thouless phase transition temperature. We have calculated the density of the superfluid component and the Kosterlitz-Thouless temperature for the systems of magnetoexcitons and magnetobiexcitons as functions of magnetic field , interlayer separation , and magnetoexciton density . In contrast to the magnetoexcitons in graphene bilayer and magnetobiexcitons in graphene superlattice, the magnetopolaritons in GL embedded in an optical cavity in the limit of high magnetic field is ideal Bose gas without interparticle interactions, and, therefore, magnetopolariton gas is not superfluid. However, there is is BEC at the temperatures below critical one in this system in a trap. We have calculated the critical temperature of magnetopolariton BEC in GL embedded in an optical microcavity in a trap as a function of magnetic field and the curvature of the trap . Note that taking into account the virtual transitions of electrons and holes between Landau levels results in weak (at large ) interactions between magnetoexcitons (17). In turn, this leads to the possibility of the superfluidity of the magnetopolariton system.
- K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666 (2004).
- Y. Zhang, J. P. Small, M. E. S. Amori, and P. Kim, Phys. Rev. Lett. 94, 176803 (2005).
- K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, and S. V. Dubonos, Nature (London) 438, 197 (2005).
- Y. Zhang, Y. Tan, H. L. Stormer, and P. Kim, Nature (London) 438, 201 (2005).
- K. Kechedzhi, O. Kashuba, and V. I. Fal’ko, Phys. Rev. B77, 193403 (2008).
- M. I. Katsnelson, Europhys. Lett. 84, 37001 (2008).
- A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
- S. Das Sarma, E. H. Hwang, and W.- K. Tse, Phys. Rev. B75, 121406(R) (2007).
- K. Nomura and A. H. MacDonald, Phys. Rev. Lett. 96, 256602 (2006).
- C. Tőke, P. E. Lammert, V. H. Crespi, and J. K. Jain, Phys. Rev. B74, 235417 (2006).
- V. P. Gusynin and S. G. Sharapov, Phys. Rev. B71, 125124 (2005).
- V. P. Gusynin and S. G. Sharapov, Phys. Rev. Lett. 95, 146801 (2005).
- A. Iyengar, J. Wang, H. A. Fertig, and L. Brey, Phys. Rev. B75, 125430 (2007).
- Z. G. Koinov, Phys. Rev. B79, 073409 (2009).
- T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437 (1982).
- Yu. E. Lozovik and V. I. Yudson, JETP Lett. 22, 26(1975); JETP 44, 389 (1976); Solid State Commun. 18, 628 (1976); 19, 391 (1976); 21 211 (1977); Physica A 93, 493 (1978).
- I. V. Lerner and Yu. E. Lozovik, JETP 51, 588 (1980); JETP 53, 763 (1981); A. B. Dzyubenko and Yu. E. Lozovik, J. Phys. A 24, 415 (1991).
- S. I. Shevchenko, Phys. Rev. Lett. 72, 3242 (1994).
- Yu. E. Lozovik and O. L. Berman, JETP Lett. 64, 573 (1996); JETP 84, 1027 (1997).
- Yu. E. Lozovik and A. V. Poushnov, Phys. Lett. A 228, 399 (1997).
- Yu. E. Lozovik and A. M. Ruvinsky, Phys. Lett. A 227, 271 (1997); JETP 85, 979 (1997).
- Yu. E. Lozovik, O. L. Berman and V. G. Tsvetus, Phys. Rev. B59, 5627 (1999).
- Yu. E. Lozovik and I. V. Ovchinnikov, Phys. Rev. B66, 075124 (2002).
- D. W. Snoke, Science 298, 1368 (2002).
- L. V. Butov, J. Phys.: Condens. Matter 16, R1577 (2004).
- V. B. Timofeev and A. V. Gorbunov, J. Appl. Phys. 101, 081708 (2007).
- J. P. Eisenstein and A. H. MacDonald, Nature 432, 691 (2004).
- Yu. E. Lozovik and A. A. Sokolik, JETP Lett. 87, 55 (2008); Yu. E. Lozovik, S. P. Merkulova, and A. A. Sokolik, Physics-Uspekhi, 51, 727 (2008).
- R. Bistritzer and A. H. MacDonald, Phys. Rev. Lett. 101, 256406 (2008).
- M. Yu. Kharitonov and K. B. Efetov, Phys. Rev. B78, 241401(R) (2008).
- C. -H. Zhang and Y. N. Joglekar, Phys. Rev. B77, 233405 (2008).
- O. L. Berman, Yu. E. Lozovik, D. W. Snoke, and R. D. Coalson, Phys. Rev. B70, 235310 (2004).
- A. Kavokin and G. Malpeuch, Cavity Polaritons, Elsevier, 2003.
- J. Kasprzak et. al., Phys. Rev. Lett. 100, 067402 (2008).
- S. Utsunomiya, L. Tian, G. Roumpos, C. W. Lai, N. Kumada, T. Fujisawa, M. Kuwata-Gonokami, A. Loeffler, S. Hoefling, A. Forchel and Y. Yamamoto, Nature Physics 4, 700 (2008).
- J. J. Baumberg et. al., Phys. Rev. Lett. 101, 136409 (2008).
- V. Bagnato and D. Kleppner, Phys. Rev. A44, 7439 (1991).
- P. Nozières, in Bose-Einstein Condensation, A. Griffin, D. W. Snoke, and S. Stringari, Eds. (Cambridge Univ. Press, Cambridge, 1995), p.p. 15-30.
- R. Balili, D. W. Snoke, L. Pfeiffer, and K. West, Appl. Phys. Lett. 88, 031110 (2006).
- R. Balili, V. Hartwell, D. W. Snoke, L. Pfeiffer and K. West, Science 316, 1007 (2007).
- O. L. Berman, Yu. E. Lozovik, and D. W. Snoke, Phys. Rev. B77, 155317 (2008).
- J. Kasprzak, et. al., Nature 443, 409 (2006).
- O. L. Berman, R. Ya. Kezerashvili, and Yu. E. Lozovik, Phys. Rev. B78, 035135 (2008); Phys. Lett. A 372 6536 (2008).
- L. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Clarendon Press, Oxford (2003).
- V. Lukose, R. Shankar, and G. Baskaran, Phys. Rev. Lett. 98, 116802 (2007).
- O. L. Berman, Yu. E. Lozovik, D. W. Snoke, and R. D. Coalson, Phys. Rev. B73, 235352 (2006).
- S. Pau, G. Björk, J. Jacobson, H. Cao and Y. Yamamoto, Phys. Rev. B51, 14437 (1995).
- J. J. Hopfield, Phys. Rev. 112 1555 (1958).
- V. M. Agranovich, Theory of Excitons, (in Russian) Nauka, Moscow (1968).
- C. Ciuti, P. Schwendimann, and A. Quattropani, Semicond. Sci. Technol. 18, S279 (2003).
- V. M. Agranovich, Excitations in Organic Solids, (Oxford University Press, Oxford, England, 2008).
- O. L. Berman, Yu. E. Lozovik, and G. Gumbs, Phys. Rev. B77, 155433 (2008).
- V. P. Gusynin, S. G. Sharapov, and J. P. Carbotte, J. Phys.: Condens. Matter 19, 026222 (2007).
- J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973); D. R. Nelson and J. M. Kosterlitz, Phys. Rev. Lett. 39, 1201 (1977).
- A. A. Abrikosov, L. P. Gor’kov, and I. E. Dzyaloshinskii, Methods of Quantum Field Theory in Statistical Physics, Prentice-Hall, Englewood Cliffs, N.J. (1963).