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Graphene bilayer structures with superfluid magnetoexcitons
Nanoscale Research Lettersvolume 7, Article number: 145 (2012)
Abstract
In this article, we study superfluid behavior of a gas of spatially indirect magnetoexcitons with reference to a system of two graphene layers embedded in a multilayer dielectric structure. The system is considered as an alternative of a double quantum well in a GaAs heterostructure. We determine a range of parameters (interlayer distance, dielectric constant, magnetic field, and gate voltage) where magnetoexciton superfluidity can be achieved. Temperature of superfluid transition is computed. A reduction of critical parameters caused by impurities is evaluated and critical impurity concentration is determined.
1 Introduction
Recent progress in creation of heterostructures with two graphene layers separated by a thin dielectrics [1] opens possibilities to use graphene for creation of multiple quantum well structures with separately accessed conducting layers. In [1], SiO_{2} substrate and Al_{2}O_{3} internal dielectric layer were used. Another promising dielectric is hexagonal BN [2]. It has a number of advantages, such as an atomically smooth surface that is free of dangling bonds and charge traps, a lattice constant similar to that of graphite, and a large electronic bandgap.
The attention to graphene heterostructures is caused, in some part, by the idea to use them for a realization of superfluidity of spatially indirect excitons [3–9]. Bound electronhole pairs cannot carry electrical charge, but in bilayers they can provide a flow of oppositely directed electrical currents. Therefore, exciton superfluidity in bilayers should manifest itself as a special kind of superconductivitythe counterflow one, that means infinite conductance under a flow of equal in modulus and oppositely directed currents in the layers.
The idea on counterflow superconductivity with reference to electronhole bilayers was put forward in [10, 11]. The attempts to observe counterflow conductivity directly were done [12–14] for bilayer quantum Hall systems realized in GaAs heterostructures. In the latter systems superconducting behavior might be accounted for magnetoexcitons [15, 16]. The effect is expected for the filling factors of Landau levels ${\nu}_{i}=2\pi {\ell}^{2}{n}_{i}(\ell =\sqrt{\hslash c/eB}$ is magnetic length, n_{ i }is the electron density in the i th layer) satisfying the condition ν_{1} + ν_{2} = 1. The role of holes is played by empty states in zero Landau level. In experiments [12–14], an exponential increase of the counterflow conductivity under lowering of temperature was observed, but zeroresistance state was not achieved. The latter can be explained by the presence of unbound vortices [17–19]. Such vortices may appear due to spatial variation of the electron density caused by disorder.
To demonstrate counterflow superconductivity quantum Hall bilayers should have the parameters that satisfy two additional conditions: $d\lesssim \ell $ and $\ell \lesssim {a}_{B}^{*}$, where d is the interlayer distance, and ${a}_{B}^{*}=\epsilon {\hslash}^{2}/{e}^{2}{m}^{*}$ is the effective Bohr radius (ε is the dielectric constant of the matrix, and m* is the effective electron mass). The first inequality comes from the dynamical stability condition. For balanced bilayers (ν_{1} = ν_{2}) the meanfields theory yields d < 1.175 ℓ. The second inequality is the condition for the Coulomb energy e^{2}/ε ℓ be smaller than the energy distance between Landau levels. In GaAs ${a}_{B}^{*}\approx 10\phantom{\rule{0.3em}{0ex}}\mathsf{\text{nm}}$ and the condition $\ell \lesssim {a}_{B}^{*}$ is fulfilled at rather strong magnetic fields $B\gtrsim 6\phantom{\rule{0.3em}{0ex}}T$ (actually, the experiments [12–14] were done at smaller fields). At $d\lesssim 10\phantom{\rule{2.77695pt}{0ex}}\mathsf{\text{nm}}$ the interlayer tunneling is not negligible small and may result in a locking of the bilayer for the counterflow transport at small input current [20, 21]. At larger input current the system unlocks, but the state becomes nonstationary one [22–24] that is accompanied by a dissipation (the power of losses is proportional to the square of the amplitude of the interlayer tunneling [22, 24]).
The idea to use graphene for the realization of electronhole superfluidity in quantum Hall bilayers [6–9] looks very attractive. The distance between Landau Levels in monolayer graphene is proportional to the inverse magnetic length, magnetic field does not enter into the condition of smallness of the Coulomb energy, and small magnetic fields can be used. Smaller magnetic fields correspond to smaller critical temperature, but, at the same time, they correspond to larger critical d. Use of large d allows to suppress completely negative effects caused by interlayer tunneling.
In this article, we concentrated on three questions. First, we determine, in what range of internal parameters and external fields magnetoexciton superfluidity can be realized. Second, we evaluate critical temperature for pure system. Third, we consider its reduction caused by electronimpurity interaction. Our study extends the results of [8], where a system of two graphene layers embedded into a bulk dielectric matrix was considered. Here we investigate structures with one and two graphene layers situated at the surface.
2 Conditions for the electronhole pairing in zero Landau level
Quantum Hall effect in graphene is characterized by unusual systematics of Landau levels and the additional fourfold degeneracy connected with two valleys and two spin projections [25]. The energies of Landau levels in graphene are ${E}_{\pm N}=\pm \frac{\hslash {v}_{F}}{\ell}\sqrt{2\leftN\right}$, where N = 0, 1, 2, ..., and v_{ F }≈ 10^{6} m/s is the Fermi velocity. In a free standing graphene, the N = 0 Landau level is halffilled. A state with only completely filled Landau levels corresponds to a plateau at the Hall conductivity plot (dependence of σ_{ xy }on electron density). A free standing graphene is just between two plateaus [26]. A given quantum states in zero Landau level is characterized by the guiding center index X and the combination of the spin and valley indexes. Below we call four possible combinations, the components, and numerate them by the index β = 1, 2, 3, 4.
We describe electronhole pairing in zero Landau level in graphene by the wave function that is a generalization of the wave function [15] to the multicomponent case
Here ${c}_{i\beta X}^{+}$ is the electron creation operator (the operator that fills a given state in N = 0 Landau Level), 0〉 is the state with empty zero level, i is the layer index. The u  v coefficients satisfy the condition u_{ β }^{2} + v_{ β }^{2} = 0. The function (1) can be rewritten in the form
where ${h}_{1\beta X}^{+}={c}_{1\beta X}$ is the hole creation operator, and the vacuum state is defined as $vac1\u3009={\displaystyle {\prod}_{\beta}{\displaystyle {\prod}_{X}{c}_{1\beta X}^{+}}}0\u3009$. One can see that the function (2) is an analog of the BCS function in the BardinCooperSchrieffer theory of superconductivity.
The quantity ${\stackrel{\u0303}{\nu}}_{\beta}={\left{u}_{\beta}\right}^{2}{\left{v}_{\beta}\right}^{2}$ gives the filling factor imbalance for the component β. The order parameter of the electronhole pairing reads as ${\Delta}_{\beta}={u}_{\beta}^{*}{v}_{\beta}=\sqrt{1{\stackrel{\u0303}{\nu}}_{\beta}^{2}}{e}^{i\phi}/2$. If a given component is maximally imbalanced $\left({\stackrel{\u0303}{\nu}}_{\beta}=\pm 1\right)$ the order parameter Δ_{ β }is equal to zero.
If a one component bilayer system is balanced, the order parameter for the electronhole pairing is maximum. But if the number of components is even, the balance ${\sum}_{\beta}{\stackrel{\u0303}{\nu}}_{\beta}=0$ can be reached at ${\stackrel{\u0303}{\nu}}_{\beta}=1$ for half of the components and ${\stackrel{\u0303}{\nu}}_{\beta}=1$ for the other half. In the latter case all Δ_{ β }= 0. As is shown below, just such a state corresponds to the energy minimum. In other words, in balanced graphene bilayers electronhole pairing does not occur.
At nonzero imbalance ${\sum}_{\beta}{\stackrel{\u0303}{\nu}}_{\beta}\ne 0,\pm 2,\pm 4$ at least for one component ${\stackrel{\u0303}{\nu}}_{\beta}\ne \pm 1$, and electronhole pairing may occur. Nonzero imbalance can be provided by electrical field directed perpendicular to the layers. Such a field can be created by a voltage difference applied between top and bottom gates (see, Figure 1).
We consider the general structure "dielectric 1graphene 1dielectric 2graphene 2dielectric 3" with three different dielectric constants ε_{1}, ε_{2}, and ε_{3}. Dielectrics 1 and 3 are assumed to be thick (much thicker than the distance between graphene layers d). Solving the standard electrostatic problem we obtain the Fourier components of the Coulomb interaction V_{ ii' }for the electrons located in i and i' graphene layers
For electrons in N = 0 Landau level in graphene the Hamiltonian of Coulomb interaction has the form
where S is the area of the system. The interaction with the gate field is described by the Hamiltonian
where V_{ g }is the interlayer voltage created by the external gate (bare voltage).
Rewriting the wave function (1) in the form
and computing the energy in the state (8) we obtain
where W = e^{2}d/ε_{2}ℓ^{2} is the energy of direct Coulomb interaction. The exchange interaction energies
determine the parameters J_{0} = (J_{11} + J_{22})/2  J_{12} and J_{ z }= J_{11}  J_{22}. The relation between θ_{ β }and ${\stackrel{\u0303}{\nu}}_{\beta}$ is given by equation ${\stackrel{\u0303}{\nu}}_{\beta}=\text{cos}{\theta}_{\beta}$.
Taking into account the inequalities W > J_{0}, and J_{11}, J_{22} > J_{12} (that can be checked directly) we find that at V_{ g }= 0 the minimum of (9) is reached at ${\stackrel{\u0303}{\nu}}_{1}={\stackrel{\u0303}{\nu}}_{2}=1,{\stackrel{\u0303}{\nu}}_{3}={\stackrel{\u0303}{\nu}}_{4}=1$. It indicates the absence of electronhole pairing in balanced systems.
If V_{ g }≠ 0 and belongs to one of the intervals
where n = 4, 2, 0, 2, the energy minimum is reached at ${\stackrel{\u0303}{\nu}}_{{\beta}_{a}}\ne \pm 1$ for one of the components. We will call such a component the active one.
Let us, for instance, consider the interval (10) with n = 0. Then the energy minimum is reached at
The case ${\stackrel{\u0303}{\nu}}_{{\beta}_{a}}=0$ (with maximum order parameter) corresponds to the voltage
Equation (11) determines the relation between magnetic field and the gate voltage V_{ g }. To keep ${\stackrel{\u0303}{\nu}}_{{\beta}_{a}}=0$ the gate voltage should be varied synchronically with B. In particular, at J_{ z }= 0 (ε_{1} = ε_{3}) the quantities V_{ g }and B are linearly related:
where α ≈ 1/137 is the fine structure constant (the relation (12) is given in SI units).
If only the gate voltage or magnetic field is varied, the order parameter (and the critical temperature) changes nonmonotonically reaching the maximum at the point determined by (11).
3 Collective mode spectrum and phase diagram
The components that belong completely to one layer do not take part in the pairing. In what follows we consider the dynamics of only the active component.
We describe the active component by the wave function
(here and below we omit the component index). Equation (13) describes the state with nonzero counterflow currents. To illustrate this statement we neglect for a moment the order parameter fluctuations $\left({\stackrel{\u0303}{\phi}}_{X}=0,{\theta}_{X}={\theta}_{a}\right)$.
The order parameter is determined by the equation
where
is the singleparticle wave function in the coordinate representation, L_{ y }is the width of the system.
Substitution (13) into (14) yields
One can see from equation (15) that Q = (Q_{ x }, Q_{ y }) is the gradient of the phase of the order parameter.
Computing the energy in the state (13) and neglecting the fluctuations we obtain
where
and
Electrical currents can be found from a variation of the energy caused by a variation of the vectorpotential
Here A_{ i }is the inplane component of the vectorpotential in the layer i. To obtain the explicit expression for the variation (19) we replace the phase gradient in (16) with the gaugeinvariant expression $\mathbf{Q}\frac{e}{\hslash c}\left({\mathbf{A}}_{pl,1}{\mathbf{A}}_{pl,2}\right)$, where A_{ pl,i }is the parallel to the graphene layers component of the vector potential in the layer i. Then, using (19) one finds the currents
At small gradients Q ℓ ≪ 1 equation (20) is reduced to
where coefficient of proportionality between the current and the phase gradient
is called the zero temperature superfluid stiffness (the definition is given in the following section). Since we neglect fluctuations, the expression (20) yields the current at T = 0.
Implying the fluctuations of the amplitude and the phase of the order parameter are small one can present the energy as
The quadratic in fluctuations term can be diagonalized:
where
are the Fourier components of the fluctuations.
Equation (24) yields the energy of fluctuations with the wave vector directed along the x axis. The component of the matrix K can be presented in form independent of the choice of the direction of the coordinate axes
where
The quantities K_{ αβ }(q) in (24) are expressed in terms of (26) as ${K}_{\alpha \beta}(q)={K}_{\alpha \beta}(\mathbf{q},\mathbf{Q})\mathbf{q}=q{\mathbf{i}}_{x}.$.
The quantity ħ cos θ_{ X }/2 can be treated as a zcomponent of the pseudospin and it is canonically conjugated with the phase φ_{ X }. The Fourier transformed quantities (25) are defined as canonical variables as well. The equations of motion for the quantities m_{ z }(q) and φ(q) read as
Equation (31) yield the collective mode spectrum $\Omega \left(q,\mathbf{Q}\right)=\sqrt{{K}_{\phi \phi}\left(q\right){K}_{zz}\left(q\right)}+{K}_{z\phi}\left(q\right)$. Rotating the axes one obtains the excitation spectrum at general q
At Q = 0 the spectrum (33) is isotropic. It can be presented in the Bogolyubov form
In equation (34)
is the kinetic energy (ε_{ q }≈ ħ^{2}q^{2}/2M at q ℓ ≪ 1, where M is the magnetoexciton mass, see, for instance [27]), and
has the sense of the excitonexciton interaction energy (that includes the direct and exchange parts).
The condition for the dynamical stability of the state (13) is the real valueness of the excitation spectrum (34). This condition determines the diapason of d/ℓ and ε_{ i }where superfluid magnetoexciton state can be realized. To be more concrete we consider three types of heterostructures. Type A is a graphenedielectricgraphene sandwich with two graphene layers at the surface, Type B is a graphenedielectricgraphenedielectric structure with one such a layer, and Type C is a system of two graphene layers embedded in a dielectric matrix (Figure 2). For simplicity, we imply the same dielectric constants ε for the interfacial layer and the substrate.
The dynamical stability condition is fulfilled at $0<d/\ell <{\stackrel{\u0303}{d}}_{c}\left(\epsilon \right)$, where ${\stackrel{\u0303}{d}}_{c}\left(\epsilon \right)$ depends on the imbalance parameter ${\stackrel{\u0303}{\nu}}_{{\beta}_{a}}\equiv {\u1e7d}_{a}$. The dependence ${\stackrel{\u0303}{d}}_{c}\left(\epsilon \right)$ at ${\stackrel{\u0303}{\nu}}_{a}=0$ is shown in Figure 3.
The requirement for the Coulomb energy be smaller than the distance between Landau levels yields the restriction on ε. Since we study the pairing in N = 0 Landau level we compare the Coulomb energy with the energy distance between N = 0 and N = 1 levels ${\omega}_{c}=\sqrt{2}\hslash {v}_{F}/\ell $.
We have four parameters that characterize the Coulomb energy W, J_{11}, J_{22}, and J_{12}. At $d/\ell <{\stackrel{\u0303}{d}}_{c}$ the largest of them is J_{11} (the intralayer exchange interaction in the graphene layer at the surface). Therefore, it is natural to consider the condition
as the additional restriction on the parameters. Equation (37) can be rewritten as ε > ε_{ c }(d/ℓ). The quantity ε_{ c }can be understood as a critical dielectric constant. The dependence ε_{ c }(d/ℓ) is also shown in Figure 3.
Two conditions $d/\ell <{\stackrel{\u0303}{d}}_{c}\left(\epsilon \right)$ and ε > ε_{ c }(d/ℓ) determine the range of parameters where one can expect a realization of electronhole pairing and magnetoexciton superfluiduty in graphene bilayer systems.
4 Critical temperature
In a bilayer graphene heterostructure with a fixed d the magnetoexciton superfluidity can be realized in a wide range of magnetic field. Variation of B at fixed gate voltage results in a change of imbalance of the active component. Simultaneous tuning of V_{ g }allows to keep zero imbalance ${\stackrel{\u0303}{\nu}}_{a}=0$ and maximum order parameter under variation of B. In this section, we study the dependence of critical temperature on magnetic field implying such a simultaneous tuning.
Superfluid transition temperature is given by the BerezinskiiKostelitzThouless equation [15]
where ρ_{ s }(T) is the superfluid stiffness at finite temperature. The superfluid stiffness is defined as the coefficient in the expansion of the free energy in the phase gradient $F={F}_{0}+\int {d}^{2}r{\rho}_{s}{(\nabla \phi )}^{2}/2$. In a weakly nonideal Bose gas it is equal to ρ_{ s }= ħ^{2}n_{ s }/m, where n_{ s }is the superfluid density. As was shown in previous section, superfluid stiffness determines also the supercurrent.
Taking into account linear excitations we present the free energy F = E_{0}  TS in the following form
Expansion of equation (39) yields the following expression for the superfluid stiffness
It follows from (40) and (33) that ρ_{ s }(T) < ρ_{s 0}(thermal fluctuations reduce the superfluid stiffness).
For the spectrum Ω(q) = E(q) + ħ qv (where v = ħ∇φ/m is the superfluid velocity) (40) yields the wellknown answer for the superfluid density [28]. Equation (40) generalizes the results [28] for the general case.
The dependence of critical temperature on magnetic field at ${\stackrel{\u0303}{\nu}}_{a}=0$ and ε = 4 is shown in Figure 4. One can see that the maximum critical temperature is reached approximately at B ≈ 0.5B_{ d }, where B_{ d }= ϕ/πd^{2} with ϕ = hc/2e, the magnetic flux quantum.
5 Influence of impurities on the critical parameters
In the previous section, we have determined the influence of thermal fluctuations on the superfluid stiffness. In this section, we consider the effect of reduction of the superfluid stiffness caused by the interaction of magnetoexcitons with impurities.
The Hamiltonian of the interaction of the active component with impurities can be presented in the form
where U_{ z }(q) = U_{1}(q)  U_{2}(q), U_{ i }(q) is the Fouriercomponent of the impurity potential in the layer i, and
is the Fourier component of the electron density operator for the active component.
In the state (13), the energy of interaction with the impurities expressed in terms of m_{ z }(q) reads as
where
The interaction (43) induces the fluctuations of the density and the phase of the order parameter.
Their values can be obtained from the EulerLagrange equations
where E is the energy of the system, described by the Hamiltonian H = H_{ C }+ H_{ G }+ H_{imp} in the state (13).
Equations (44) solved in linear in impurity potential approximation yield
Substituting (45), (46) into the expression for the energy one finds the correction to the energy caused by the electronimpurity interaction
In equation (47), the contribution of fluctuations with the wave vectors directed along x is taken into account. Summing the contribution for all wave vectors one obtains
For simplicity, we specify the case where impurities are located in graphene layers. Then the Fouriercomponent of the impurity potential can be presented in the form
where r_{ a }are the impurity coordinates, and u_{ z,i }(q) = u_{1,i}(q)  u_{2,i}(q) with u_{ k,i }(q), the potential in the layer k of a single impurity centered at r = 0 in the layer i.
Averaging over impurities yields
where n_{imp} is the impurity concentration in a layer.
At Q ℓ ≪ 1 the energy (50) can be expanded in series as
where
is the correction of the superfluid stiffness. One can check that the correction Δρ_{ s }is negative. Thus, the interaction with impurities results in decrease of critical parameters.
At ${\stackrel{\u0303}{\nu}}_{a}=0$ equation (52) is reduced to
where ρ_{s 0}(equation (22)) is taken at θ_{ a }= π/2.
The shift of critical temperature is evaluated as ΔT_{ c }/T_{ c }≈ Δρ_{ s }/ρ_{s 0}.^{a} We define the critical impurity concentration ${n}_{\mathsf{\text{imp}}}^{c}$ as a concentration at which Δρ_{ s }/ρ_{s 0}= 1. We consider charged impurities with the potential u_{ z,i }(q) = (1)^{i}(V_{12}(q)  V_{ ii }(q)). The dependence of critical impurity concentration on magnetic field at ε = 4 and ${\stackrel{\u0303}{\nu}}_{a}=0$ is shown in Figure 5. We also evaluated critical concentrations for neutral impurities. These concentrations are much larger, and the influence of neutral impurities can be neglected.
6 Conclusion
In conclusion, we present some estimates. Let us specify the type B structure (the one used in [1]) with d = 20 nm and ε = 4. For this structure the maximum critical temperature T_{ c }≈ 3 K (in pure case) is reached in magnetic field B ≈ 0.8 T. At such B the critical impurity concentration is ${n}_{\mathsf{\text{imp}}}^{c}\approx 2\cdot 1{0}^{9}\mathsf{\text{c}}{\mathsf{\text{m}}}^{\mathsf{\text{2}}}$. The gate voltage determined by equation (11) is V_{ g }≈ 6 mV, that corresponds to electrostatic field E ≈ 3 kVcm^{1}.
Basing on the results of our study we may state the following.

1.
Graphene bilayer structures are perspective objects for the observation of magnetoexciton superfluidity. The advantages are smaller magnetic fields and no restriction from above on physical interlayer distance, that means the possibility to suppress completely interlayer tunneling.

2.
Gate voltage should be created between graphene layers for a realization of magnetoexciton superfluidity.

3.
Certain conditions on dielectric constant and on the ratio between interlayer distance and magnetic length should be satisfied.

4.
Structures with graphene layers situated at the surface have larger critical parameters.

5.
Neutral impurities are not dangerous for the magnetoexciton superfluidity, but the concentration of charged impurities should be controlled.
Endnote
^{a}Since in our approach we assume smallness of Δp_{ s }/p_{s 0}it is just an estimate.
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Acknowledgements
This study was supported by the Ukraine State Program "Nanotechnologies and nanomaterials" Project No. 1.1.5.21.
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AAP carried out the calculation and took part in the manuscript preparation. DVF designed and coordinated of the study and prepare the manuscript. All authors read and approved the final manuscript.
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Keywords
 graphene
 exciton superfluidity
 multilayer heterostructures