Density Collective Modes of Exciton Superfluidity in Bilayer Systems

Abstract

We propose a new way to establish the existence of a superfluid phase in an exciton bilayer system by exploiting the properties of its collective modes. We focus on the density collective modes and treat them within Random Phase Approximation. By comparing results for the normal and superfluid states, we are able to identify unambiguous fingerprints of the exciton superfluid phase. We compare the collective modes of the exciton system and cold atom systems, and we discuss the collective modes of the exciton superfluid order parameter.

1. Introduction

Recent reports of the probable observation of superfluidity in exciton semiconductor bilayer systems in zero magnetic fields are currently attracting a lot of interest [1,2,3,4,5,6,7]. The excitons are in the interlayer: bound electron–hole pairs that form across the spatially separate electron and hole conducting layers. The spatial separation suppresses electron–hole recombination, thus opening the way to stable exciton superfluids in equilibrium [8,9]. The strong binding energy of the excitons leads to predictions of high superfluid transition temperatures.

The long-range pairing interaction makes this system [10] very different from familiar superconductors and superfluids in which the pairing interaction is contact or short-range. Changing the charge-carrier density with top and bottom metal gates tunes the strength of the screening of the electron–hole pairing attraction [11]. This allows a continuous scan of the BEC-BCS crossover from strongly bound pairs forming an exciton Bose–Einstein condensate (BEC) to weakly bound electron–hole pairs in the crossover regime [12,13].

In the last decade, with the discovery of new materials and the rapid advances in experimental techniques, there has been a huge effort to search for and identify the existence of stable exciton normal liquid and superfluid phases in these bilayer systems. Techniques include interlayer tunneling [1], electroluminescence [2], counterflow measurements [3], and Coulomb drag [6]. However, to date, while these works have been able to establish the existence of a strongly correlated excitonic phase, a definitive claim for the existence of a superfluid state is still missing. Observation of dissipationless currents and measurements of phase coherence are particularly challenging because, even though excitons contain charged electrons and holes, they are overall neutral particles.

Here, we establish that collective modes are a powerful tool to unambiguously identify exciton superfluidity in these systems. Collective modes describe the macroscopic response to an external perturbation acting on a many-body system. We propose a complete description of the density collective modes using a superfluid model without any restriction on the localization of electrons and holes. Previously, Ref. [14] has treated these modes within a “quasi-localized approximation” where the electrons and holes are assumed to be highly localized. This is valid only at low densities and large interlayer separations. We present results for the superfluid and normal states, searching for fingerprints of exciton superfluidity.

2. Materials and Methods

The response functions that describe the collective modes of macroscopic observables can be measured by externally perturbing the system at a given wavenumber q and frequency $\omega$ [15]. We explore the response of the system to particle density perturbations using Random Phase Approximation (RPA) in the normal liquid and superfluid exciton phases. RPA has proved to be a quantitatively good approximation to treat density fluctuations in the zero temperature exciton bilayer system [16,17].

Within RPA, the Dyson equation gives the density response function [15,18]:

$${\Pi }^{RPA}(q,\omega )={\displaystyle \frac{{\Pi }_0(q,\omega )}{1-{\Pi }_0(q,\omega )V\left(q\right)}}.$$

(1)

${\Pi }_0(q,\omega )$ is the zero-order polarization function matrix in the superfluid state [19]. $V\left(q\right)$ is the bare Coulomb interaction matrix:

$$\begin{array}{cc}\hfill V\left(q\right)& =\left(\begin{array}{cc}{V}_S\left(q\right)& V_D\left(q\right)\\ V_D\left(q\right)& V_S\left(q\right)\end{array}\right),\hfill \end{array}$$

(2)

where $V_S\left(q\right)=1/4\pi ϵq$ and $V_D\left(q\right)=-e^{-qd}/4\pi ϵq$ are the bare intralayer repulsive and interlayer attractive interactions, respectively. $ϵ$ is the dielectric constant of the insulating barrier between the conducting layers, and d is the interlayer separation.

We consider only equal electron and hole effective masses and densities, making the ${\Pi }^{RPA}(q,\omega )$ symmetric:

$$\begin{array}{cc}\hfill {\Pi }^{RPA}(q,\omega & )=\left(\begin{array}{cc}{\Pi }_S^{RPA}(q,\omega )& {\Pi }_D^{RPA}(q,\omega )\\ & \\ {\Pi }_D^{RPA}(q,\omega )& {\Pi }_S^{RPA}(q,\omega )\end{array}\right).\hfill \end{array}$$

(3)

${\Pi }_S^{RPA}(q,\omega )$ is the density response in a layer from a density perturbation in the same layer, while ${\Pi }_D^{RPA}(q,\omega )$ is the density response in one layer due to a density perturbation in the opposite layer.

The zero-order polarization matrix can be written as

$$\begin{array}{cc}\hfill {\Pi }_0(q,\omega )& =\left(\begin{array}{cc}{\Pi }_0^N(q,\omega )& {\Pi }_0^A(q,\omega )\\ {\Pi }_0^A(q,\omega )& {\Pi }_0^N(q,\omega )\end{array}\right).\hfill \end{array}$$

(4)

The ${\Pi }_0^N(q,\omega )$ and ${\Pi }_0^A(q,\omega )$ are the normal and anomalous zero-order polarizations [20]:

$$\begin{array}{cc}\hfill {\Pi }_0^N(q,\omega )& ={\displaystyle \frac{1}{\mathrm{S}}}\sum _k{\displaystyle \frac{u_k^2{v}_{k^{\prime }}^2}{\omega -E\left(k\right)-E\left(k^{\prime }\right)}}-{\displaystyle \frac{u_{k^{\prime }}^2{v}_k^2}{\omega +E\left(k\right)+E\left(k^{\prime }\right)}},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\Pi }_0^A(q,\omega )& ={\displaystyle \frac{1}{\mathrm{S}}}\sum _k{\displaystyle \frac{u_{k^{\prime }}{v}_{k^{\prime }}{u}_k{v}_k}{\omega -E\left(k\right)-E\left(k^{\prime }\right)}}-{\displaystyle \frac{u_{k^{\prime }}{v}_{k^{\prime }}{u}_k{v}_k}{\omega +E\left(k\right)+E\left(k^{\prime }\right)}},\hfill \end{array}$$

(6)

where $k^{\prime }=\left|\mathbf{k}+\mathbf{q}\right|$, S is the layer surface area, and $u_k$ and $v_k$ are the Bogoliubov amplitudes:

$$\begin{array}{c}\hfill u_k^2={\displaystyle \frac{1}{2}}\left(1+{\displaystyle \frac{\xi \left(k\right)}{E\left(k\right)}}\right),v_k^2={\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{\xi \left(k\right)}{E\left(k\right)}}\right).\end{array}$$

(7)

The superfluid energy dispersion relation is

$$E\left(k\right)=\sqrt{\xi {\left(k\right)}^2+\Delta {\left(k\right)}^2},\xi \left(k\right)=\epsilon \left(k\right)-\mu -\Sigma \left(k\right),\Sigma \left(k\right)={\displaystyle \frac{1}{\mathrm{S}}}\sum _p{V}_S^{sc}(p-k)v_p^2,$$

(8)

with $\epsilon \left(k\right)={\hslash }^2{k}^2/2m^{*}$ the electronic parabolic dispersion relation, and $\Sigma \left(k\right)$ is the intralayer correlation energy term evaluated within the Hartree–Fock approximation [17]. The superfluid gap $\Delta \left(k\right)$ and the single-particle chemical potential $\mu$ are evaluated as a function of the density n by solving the energy gap and number equations:

$$\begin{array}{cc}\hfill & \Delta \left(k\right)=-{\displaystyle \frac{1}{\mathrm{S}}}\sum _q{V}_D^{sc}\left(\right|\mathbf{k}-\mathbf{q}\left|\right){\displaystyle \frac{\Delta \left(q\right)}{2E\left(q\right)}},\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & n={\displaystyle \frac{g_s{g}_v}{\mathrm{S}}}\sum _k{v}_k^2.\hfill \end{array}$$

(10)

$g_s$ and $g_v$ are the spin and valley degeneracies. In Equations (8) and (9), $V_S^{sc}$ is the static screened repulsive intralayer interaction [11,21] and $V_D^{sc}$ the static screened attractive interlayer interaction [19,22]:

$$\begin{array}{cc}\hfill V_D^{sc}\left(q\right)& ={\displaystyle \frac{V_D\left(q\right)+{\Pi }_0^A(q,0)\mathcal{A}\left(q\right)}{1-2\left({\Pi }_0^N(q,0)V_S\left(q\right)+{\Pi }_0^A(q,0)V_D\left(q\right)\right)+\mathcal{A}\left(q\right)\mathcal{B}\left(q\right)}},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill V_S^{sc}\left(q\right)& ={\displaystyle \frac{V_S\left(q\right)-{\Pi }_0^N(q,0)\mathcal{A}\left(q\right)}{1-2\left({\Pi }_0^N(q,0)V_S\left(q\right)+{\Pi }_0^A(q,0)V_D\left(q\right)\right)+\mathcal{A}\left(q\right)\mathcal{B}\left(q\right)}},\hfill \end{array}$$

(12)

where $\mathcal{B}(q,\omega )={\Pi }_0^N{(q,\omega )}^2-{\Pi }_0^A{(q,\omega )}^2$ and $\mathcal{A}\left(q\right)=V_S{\left(q\right)}^2-V_D{\left(q\right)}^2$.

We solve Equations (9) and (10) self-consistently as a function of the density [10,17]. Then, we calculate the density response functions (Equation (3)):

$$\begin{array}{c}\hfill {\Pi }_S^{RPA}(q,\omega )={\displaystyle \frac{{\Pi }_0^N(q,\omega )-V_S\left(q\right)\mathcal{B}(q,\omega )}{1-2\left({\Pi }_0^N(q,\Omega )V_S\left(q\right)+{\Pi }_0^A(q,\omega )V_D\left(q\right)\right)+\mathcal{A}\left(q\right)\mathcal{B}(q,\omega )}},\\ \hfill \end{array}$$

(13)

$$\begin{array}{c}\hfill {\Pi }_D^{RPA}(q,\omega )={\displaystyle \frac{V_D\left(q\right)\mathcal{B}(q,\omega )+{\Pi }_0^A(q,\omega )}{1-2\left({\Pi }_0^N(q,\omega )V_S\left(q\right)+{\Pi }_0^A(q,\omega )V_D\left(q\right)\right)+\mathcal{A}\left(q\right)\mathcal{B}(q,\omega )}}.\end{array}$$

(14)

The energy spectra of the density collective modes correspond to the $\omega \left(q\right)$ which satisfies

$$1-2({\Pi }_0^N(q,\omega )V_S\left(q\right)+{\Pi }_0^A(q,\omega )V_D\left(q\right))+\mathcal{A}\left(q\right)\mathcal{B}(q,\omega )=0.$$

(15)

This equation contains the poles in Equations (13) and (14).

The collective modes are stable only if Equation (15) gives a real solution $\omega \left(q\right)$. When the solution is complex, $\Omega \left(q\right)=\omega \left(q\right)+i\Gamma \left(q\right)$, the collective modes have a lifetime defined by the damping parameter $\Gamma \left(q\right)$. The solution is complex if and only if ${\Pi }_0^N(q,\omega )$ and ${\Pi }_0^A(q,\omega )$ have poles for real $\omega \left(q\right)$ [23]. This occurs in the so-called pair-breaking continuum region. Equation (15) is invariant under the transformation $\Omega$ to $-\Omega$ so we need consider only $Re[\Omega ]=\omega >0$.

Here, we examine the stable collective modes that lie outside the pair-breaking continuum region. The branch cut in which ${\Pi }_0^N(q,\omega )$ and ${\Pi }_0^A(q,\omega )$ diverge in the superfluid phase is given by $\omega \in {\mathit{C}}_q=\left\{E\right(\left|\mathbf{k}+\mathbf{q}\right|)+E\left(k\right),k\in R^2\}$, originating from the denominator $\omega -E\left(k\right)-E\left(\right|\mathbf{k}+\mathbf{q}\left|\right)$ in the zero-order polarization functions. To locate the branch cut and hence the lower boundary $\overline{\omega }$ of the pair-breaking continuum region, we must minimize $\omega -E\left(k\right)-E\left(\right|\mathbf{k}+\mathbf{q}\left|\right)$ as a function of k. The dispersion relation $E\left(k\right)$, and thus the position of the pair-breaking continuum in the ($\omega$-q) space, depends on the density.

3. Results

We consider a system with $g_s{g}_v=2$ and interlayer distance $d=0.2a_B^{*}$, where $a_B^{*}$ is the effective Bohr radius. For the BEC regime, we set the average interparticle distance $r_0=3a_B^{*}$ [17], for the crossover regime $r_0=1.5a_B^{*}$, and for the normal state $r_0=0.5a_B^{*}$. The particle density in each layer is $n=1/\pi r_0^2$.

3.1. Density Collective Modes in the BEC Regime

Figure 1 shows the pair-breaking continuum region for the exciton superfluid state at a low density in the deep BEC and for the normal state. Appendix A presents the calculation and results for the collective modes in the normal state.

Remarkably, in the superfluid phase, Figure 1a, the pair-breaking region with $\omega >\overline{\omega }$ has no upper boundary for the following reason. In the numerators of the superfluid polarization functions, Equations (5) and (6), the Bogoliubov amplitudes smoothly approach zero for large momenta. There is no cutoff to the sum in k, and thus for any q and for $\omega >\overline{\omega }$ there is always a value of k for which the denominators of the polarization functions are zero. Thus, there is no upper bound of the continuum region in the superfluid phase.

In contrast, in the normal liquid state, Figure 1b, the continuum has an upper ${\omega }_{+}$ boundary. This is because the zero temperature Fermi step functions in Equation (A3) impose a cutoff to the sum in k, so for a fixed q and for $\omega >{\omega }_{+}$, as well as for the lower boundary $\omega <{\omega }_{-}$, there exists no k-value for which the denominator of ${\Pi }_0^N(q,\omega )$ vanishes.

In the superfluid state outside the continuum, Equation (15) has stable solutions only when ${\Pi }_0^N(q,\omega )-{\Pi }_0^A(q,\omega )$ is real and positive. Figure 2a shows ${\Pi }_0^N(q,\omega )-{\Pi }_0^A(q,\omega )$ for $q=k_F$. The imaginary part is zero for $\omega <\overline{\omega }$, the boundary of the pair-breaking excitation region, while the real part is negative. At the $\overline{\omega }$ boundary, the imaginary part is a maximum and the real part becomes positive. With increasing $\omega >\overline{\omega }$, the imaginary part remains non-zero. Thus, outside the continuum for $\omega <\overline{\omega }$, Equation (15) has no real solutions (Figure 3), and we conclude here that there are no stable density collective modes in the BEC regime of exciton bilayer superfluidity. In the superfluid BEC regime, the effect of density fluctuations is negligible because the excitons are well spaced (very large $r_0$) so they interact minimally with each other [11]. For this reason, density fluctuations do not induce any stable collective response in the system in the BEC regime.

This is in sharp contrast with the collective mode properties in the normal liquid state for which we recall that above the continuum, $\omega >{\omega }_{+}$, the real part of the polarization function is positive and the imaginary part is zero (Figure 2b). For $\omega >{\omega }_{+}$, Equation (A4) has two real $\omega \left(q\right)$ solutions, the acoustic and optic density collective modes (Figure A2) [16,24].

3.2. Density Collective Modes in the Crossover Regime

As the density is increased, the system passes from the BEC regime into the BCS-BEC crossover regime. In this regime, the interlayer and intralayer interactions become significantly affected by density fluctuations and by screening [17]. We find that, as in the BEC regime, in the crossover regime there are no real solutions for Equation (15) lying outside the continuum since the denominator of the polarization functions is always greater than zero (Figure 4). Figure 4 shows that the intralayer correlations lower the boundary of the continuum, from the dashed blue line down to the solid blue line. This is because the gap energy is significantly suppressed by the correlations [17]. The small peak in $\overline{\omega }$ reflects the maximum in the gap energy $\Delta \left(k\right)$ that appears near $k_F$ in the crossover regime. As in the BEC regime, ${\Pi }_0^N(q,\omega )-{\Pi }_0^A(q,\omega )$ outside the continuum is always negative. This again means Equation (15) has no real solutions.

References [11,25] have shown that very strong screening blocks the exciton superfluid from entering the weakly interacting BCS regime. Reference [11] found that, as a result, the gap energy in this system is always larger than the Fermi energy.

We consider only density fluctuations and we find that the optic and acoustic modes in the normal state disappear at the superfluid transition by tuning the density at temperature zero. An optic out of phase mode in the exciton superfluid state is not possible because of the large binding energy and rigidity of the excitons in the superfluid state [26]. For the acoustic mode, its energy in the normal liquid state is found to lie below the Fermi energy. Consequently, in the superfluid state it would lie within the forbidden region inside the energy gap of the excitation spectrum. The disappearance of the acoustic and optic branches at the normal liquid to superfluid transition is a striking characteristic of the transition.

4. Superfluid Collective Modes in Ultracold Atoms and Exciton Bilayers Compared

We recall features of the collective modes in ultracold atoms. Since experimentally it is not possible to isolate one perturbation channel, perturbations of the three fields (density, amplitude, and phase of the order parameter) are generally simultaneously included [27]. Thus a perturbation of the density will couple with the amplitude and phase response of the order parameter [28,29].

In zero temperature ultracold atoms, there is a stable acoustic Anderson–Bogoliubov (AB) mode lying outside the continuum [30] throughout the entire BCS-BEC crossover. This mode is driven by both density and phase modulations but it is not coupled to amplitude of the order parameter. In neutral systems, the AB mode is gapless [31]. We recall that in 3D charged systems, due to the repulsive Coulomb interactions, the AB mode acquires a gap of the order of the plasmon frequency [32]. In the BCS regime of 3D cold atoms systems, the AB mode has been shown to appear in the density and phase response functions outside the continuum [33]. Moving toward the crossover and BEC regimes, the AB mode disappears from the density response function but remains in the phase response function [32].

In 2D charged systems there is as yet no investigation of the AB mode. The plasmon frequency in 2D goes to zero at $q=0$ (Figure A2), so it is reasonable to expect that the AB mode for a 2D charged system will be gapless too.

In the exciton bilayer system, strong screening prevents the system from entering the BCS regime. The absence of a BCS regime in the exciton superfluid implies that the AB modes are governed solely by phase fluctuations of the order parameter. Here, we investigate modes that couple only to density fluctuations, and we indeed find no stable collective modes outside the continuum in the crossover and BEC regimes of the exciton superfluid system.

By incorporating phase fluctuations, acoustic AB modes are expected to emerge in the exciton superfluid outside the continuum, analogous to those observed in ultracold atoms, where the transition from normal to BCS superfluidity (small energy gap) leads to a smooth behavior of the AB modes [34,35,36]. A non-smooth transition of the AB modes may be expected from the normal liquid to the exciton superfluid phase, reflecting the sudden shift in the collective mode’s nature: density-driven in the normal state and phase-driven in the superfluid state.

Inside the continuum, in the BCS-Crossover regime of ultracold atoms, a pair-breaking mode related to fluctuations of the amplitude of the order parameter has been reported [27,37]. This pair-breaking mode does not couple with density fluctuations. In the exciton bilayer system, we have noted that the long-range interactions have a strong dependence on screening and particle density. The density fluctuations can significantly affect the amplitude of the order parameter in the crossover regime. This should generate additional and relatively long-lived density collective mode branches inside the continuum.

There have been as yet no experimental investigations of the collective modes in exciton bilayer superfluids. Exciton bilayers in Transition Metal Dichalcogenides (TMDs) would be particularly attractive in this regard. The binding energies of excitons in these systems are large [38] and the expected transition temperatures to the superfluid state are high [39]. Reference [40] has investigated phonon propagation in the normal state in a TMD exciton bilayer system using different spectroscopic techniques including Raman. Raman spectroscopy has been successfully used to study collective modes in superconductors [41] and could be adapted for the collective modes in the exciton bilayer superfluid phase.

5. Conclusions

In the liquid normal state of the bilayer system, the response of the system to small external density perturbations is through the low-energy optic and acoustic density collective modes. When the system enters the exciton superfluid phase, the response changes significantly. Due to the strong screening associated with the long-range Coulomb interactions, the exciton superfluid always features a large energy gap $\Delta \left(k\right)$ and this prevents the propagation of stable acoustic and optic density collective modes. Unlike with ultracold atoms, the long-range attractive interaction in exciton bilayers will cause an order parameter amplitude response from density fluctuations. It is likely that this will result in additional long-lived unstable collective modes in the amplitude channel appearing in the continuum at the normal to superfluid transition. At the transition, the disappearance of the optic mode, the probable change in behavior of the acoustic mode outside the continuum, and the likely appearance of amplitude-density modes in the continuum are all experimentally detectable. These should act as signatures of the superfluid phase in the exciton bilayer system.