Abstract
Electron–hole pairing can occur in a dilute semimetal, transforming the system into an excitonic insulator state in which a gap spontaneously appears at the Fermi surface, analogous to a Bardeen–Cooper–Schrieffer (BCS) superconductor. Here, we report optical spectroscopic and electronic transport evidence for the formation of an excitonic insulator gap in an inverted InAs/GaSb quantumwell system at low temperatures and low electron–hole densities. Terahertz transmission spectra exhibit two absorption lines that are quantitatively consistent with predictions from the pairbreaking excitation dispersion calculated based on the BCS gap equation. Lowtemperature electronic transport measurements reveal a gap of ~2 meV (or ~25 K) with a critical temperature of ~10 K in the bulk, together with quantized edge conductance, suggesting the occurrence of a topological excitonic insulator phase.
Introduction
It was predicted several decades ago^{1,2} that Coulomb interactions in an electron–hole (e–h) coexisting system can make the normal semimetallic state unstable against the spontaneous appearance of excitons, or bound e–h pairs, inducing a phase transition into an insulator, called the excitonic insulator (EI) or Bardeen–Cooper–Schrieffer (BCS)like excitonic condensation. The EI phase emerges below a densitydependent critical temperature (Fig. 1a), where a gap opens at the energy of the original Fermi surface of the semimetal, in a manner analogous to the BCS gap in a superconductor^{3,4,5,6,7,8,9}. In the density regime where the EI phase is expected to occur from semimetal (Fig. 1b), the spatial extent of the exciton wavefunction is larger than the average interexciton distance, i.e., electrons and holes are only weakly bound (Fig. 1c), similar to Cooper pairs. Conceptually, this density regime is distinct from the dilute limit where a quantumdegenerate gas of tightly bound e–h pairs (Fig. 1d) is transformed into a Bose–Einstein condensate (BEC) with macroscopic coherence (Fig. 1e). The possibility of excitonic ground state formation is enhanced in twodimensional (2D) systems due to reduced screening. There have been extensive experiments on the possible appearance of nonequilibrium BEC in photoexcited 2D e–h systems probed by photoluminescence^{10,11}, and equilibrium BEC state in quantum Hall electronelectron doublelayers probed by counterflow measurements^{12} and light scattering^{13}. Note that these two types of BEC states have distinctions, besides the way to prepare BEC state, the former state is a short lived metastable superfluid supporting interference and vortices, whereas the latter is a ground state of the system. However, the EI side of the phase diagram has not been experimentally explored in any 2D system.
The originally proposed EI was based on a low density, equilibrium e–h gas that exists in certain semimetals, and the possibility of the EI phase was systematically studied in the vicinity of a pressuretuned semimetalsemiconductor transition^{5}. Highly controllable 2D semiconductor materials emerged in ensuing years. In particular, InAs/GaSb quantum wells (QWs) exhibit unique inverted band structure with finite overlap of the conduction and valance bands, allowing the coexistence of spatially separated electrons and holes without photoexcitation, which offers a natural setting for equilibrium excitons and consequent formation of condensates including the EI phase^{9,14,15}. Electrons are located in the InAs QW and holes are located in the GaSb QW, and thus, they are spatially separated in real space. The average separation can be defined as one half of the thickness of the doubleQW structure. In this system, signatures of magnetoexcitons were previously reported^{16}. Recently, the quantum spin Hall (QSH) effect was explored in such QWs^{17,18,19,20}. Robust quantized edge transport was observed^{19}, persisting even at strong magnetic fields and high temperatures; it was theoretically suggested^{20} that the formation of a topological EI^{21} state may account for these unexpected properties. A topological EI in 2D has helical edge states propagating on the perimeters of a bulk EI state.
Here, we report optical spectroscopic and electronic transport evidence for the appearance of a BCSlike excitonic insulator gap in gated InAs/GaSb QW devices (Fig. 1f) at low temperatures with a low intrinsic e–h pair density, n _{0} ~ p _{0} ~ 5.5 × 10^{10} cm^{−2} (we use n and p to denote, respectively, the band electron and hole densities, and in particular, n _{0} and p _{0} denote the electron and hole densities at the chargeneutrality point). Our temperature and magnetic fielddependent terahertz (THz) transmission spectroscopy data can be quantitatively explained through our calculated pairbreaking excitation dispersion, E(k), of the presumed EI state formed in the system. Complementarily, our lowtemperature electronic transport measurements also suggested the existence of a gap, determining the gap energy to be ~2 meV (or ~25 K). We found the gap value to be roughly independent of the strength of an applied inplane magnetic field but close quickly with increasing n _{0} or temperature. Together with theoretical analysis, we interpret these results as the observation of EI gap opening in this 2D equilibrium e–h gas, suggesting that a 2D EI phase is realized in our QWs. Moreover, the system exhibited helical edge transport behavior, which supports the notion that the observed lowtemperature e–h phase is a topological EI.
Results
Devices
Our devices were made from inverted InAs/GaSb QWs grown by molecular beam epitaxy^{18,19} (also see Methods and Supplementary Figs. 1 and 2). Transport measurements were performed on wafer A and C, in which a conducting GaAs or GaSb substrate serves as a backgate. In this structure (Fig. 1f), the backgate bias voltage (V _{b}) was limited to the negative range, V _{b} ≤ 0; starting with V _{b} = 0 and with increasing negative bias, we were able to introduce more holes (p) in GaSb QW. Correspondingly, we could sweep the frontgate bias V _{f} (for electrons in InAs QW) and reach the chargeneutrality point (CNP), i.e., n _{0} = p _{0}. The lowest n _{0} in our devices was ~5.5 × 10^{10} cm^{−2}, which was achieved at V _{b} = 0. Figure 1g shows the band structure of InAs/GaSb QWs calculated using the 8band k⋅p method for low n _{0} at V _{b} = 0. The average interexciton inplane distance, 2r _{avg}, corresponding to n _{0} ~ 5.5 × 10^{10} cm^{−2}, defined through 1 = n _{0}·πr _{avg} ^{2}, is ~48 nm. This value should be compared with the effective Bohr radius, a _{B,} which is estimated to be ~30 nm within a simple effectivemass approximation^{15} using the following parameters^{22}: electron effective mass \(m_{\mathrm{e}}^*\) ~ 0.032m _{0} (m _{0} = 9.11 × 10^{−31} kg), hole effective mass \(m_{\mathrm{h}}^*\) ~ 0.136 m _{0}, dielectric constant ε ~ 15, and interlayer distance between the centers of the electron and hole wells d ~ 10 nm. Therefore, we have r _{avg}/a _{B} ~ 0.8, indicating strong wavefunction overlap between excitons, a situation reminiscent of Cooper pairs.
Theoretical model
The EI state is expected to have distinctly different optical and transport properties from an exciton BEC. Our system is described by the Hamiltonian^{3,7,8,9}:
where \({\it{E}}_{\mathrm{k}}^{{\mathrm{e,h}}}\) are the singleparticle electron and hole energies, k is the inplane momentum, \(a_{\mathrm{k}}^{\mathrm{\dagger }}\left( {a_{\mathrm{k}}} \right)\) and \(b_{\mathrm{k}}^{\mathrm{\dagger }}\left( {b_{\mathrm{k}}} \right)\) are the creation (annihilation) operators for electrons in the conduction and holes in the valence bands, respectively, \(V_{\mathrm{q}}^{{\mathrm{ee}}} = V_{\mathrm{q}}^{{\mathrm{hh}}} = \frac{{e^2}}{{2\varepsilon \left {\mathrm{q}} \right}}{\mathrm{,}}\)and \(V_{\mathrm{q}}^{{\mathrm{eh}}} = V_{\mathrm{q}}^{{\mathrm{hh}}} = \frac{{e^2}}{{2\varepsilon {\mathrm{q}}}}{\mathrm{e}}^{  {\mathrm{q}} \cdot {\mathrm{d}}}{\mathrm{,}}\) where ε is the dielectric constant.
To solve this manybody interaction problem, we used a meanfield treatment^{3,7,8,9} to calculate the exciton dispersion, E(k), and the gap function Δ(k) (Fig. 2a). Similarly to the case of Cooper pairs, the e–h Coulomb correlations in the EI phase lead to an unstable Fermi surface, spontaneously opening a BCSlike gap (EI gap) Δ _{max} near the CNP reached by the Fermi level; this is the gap probed by transport measurements. On the other hand, as a ground state, the EI state absorbs incident photons, with individual electrons and holes in the final state. E(k) is the pairbreaking excitation spectrum: the energy cost of taking one loosely bound exciton out of the condensate and placing a pair of an individual electron and an individual hole inplanewave states of momentum k. Figure 2b shows the joint density of states (JDOS) we calculated, which predicts two singularity peaks: one peak near E _{min} ~ 1.5 meV (or ~18 K), and the other near E(k) ~ 7 meV (or ~80 K) with k ~ 0. There is fine structure in the 7 meV peak due to the existence of two nearby peaks. Note that the E(k) and Δ(k) curves in Fig. 2a are general EI characteristics, independent of material details^{3,7,8,9}, making an EI distinguishable from an exciton BEC (or an exciton gas). For an exciton BEC/exciton gas, the two peaks merge at k = 0, corresponding to the exciton binding energy; furthermore, there is no spontaneous gap opening near the Fermi surface^{7}. In the following, from both optics and transport perspectives, we present key findings that support these calculation results, thereby evidencing the existence of an EI in our system.
Pairbreaking excitation measurement of excitonic insulator by terahertz spectroscopy
We performed lowtemperature THz transmission spectroscopy experiments^{16,23,24} (see Methods) on a device covered by a 5 mm × 5 mm semitransparent gate, in a frequency (energy) range of 0.25–2.4 THz (~1–10 meV). Note for THz experiment the wafer B is used which was prepared on a semiinsulating GaAs substrate. A transmittance spectrum when the system is at the CNP at 1.4 K is shown in Fig. 2c. Here, two absorption lines (or transmission dips) are present in the spectrum, line A at ~2 meV, and line B at 7.3 meV, consistent with what we expect from Fig. 2a and b. We interpret line A as coming from pairbreaking excitation near the Fermi level and line B corresponding to excitation near k = 0. Mention that what we are measuring is the joint density of states of the pairbreaking excitation process, which is characterized by the spectrum E(k), not a direct extraction of the gap. It should be noted that these features are observed only when n _{0} is low (~5.5 × 10^{10} cm^{−2}); they disappear in an electron or holedominating regime, or in a high n _{0} (~>10^{11} cm^{−2}) case. In the case of line B, in addition to the fine structure in JDOS (Fig. 2b), inhomogeneous broadening due to random potential fluctuations would contribute to the linewidth^{25}. Moreover disorder has stronger effect to low k states that have lower energy, contributing to a broader line. Also, random potential fluctuations are expected to enhance the exciton density at low k ^{10}.
The interpretation of the lines as pairbreaking excitations across the BCSlike gap of an EI state is also supported by their temperature dependence (Fig. 2d). At 5 K, the unique twoline structure is still present, which cannot be explained by any singleparticle gap in this system. The absorption amplitude of the lines, which directly reflects the JDOS of the excitations, significantly decreases when the temperature is raised to 10 K. At higher temperatures, the twoline structure is absent. A marked decrease of intensity of both lines occurs at a temperature (10 K) that is much less than the line energies (25 and 90 K, respectively). These results cannot be explained by a singleparticle gap but can be interpreted as the transition from an EI state into a metallic state with a critical temperature T _{c} ~ 10 K, which also agrees with transport results as shown below.
When a magnetic field perpendicular to the QWs, B _{⊥}, is applied at 1.4 K, the amplitudes of the absorption lines increase and line B slightly blueshifts with the magnetic field, as shown in Fig. 2e. This B _{⊥}induced enhancement of the absorption lines is particularly marked at 20 K, where the lines are initially absent at 0 T, but the B _{⊥} causes them to reemerge (Fig. 2f). The strengthened absorption under B _{⊥} is likely due to the fact that a B _{⊥} makes e–h pairs more tightly bound^{7,26}, hence more stable against dissociation. Furthermore, it has theoretically been predicted that a magnetic field tends to stabilize the EI phase^{27,28}, which is in stark contrast to BCS superconductors involving Cooper pairs (which are destroyed by a magnetic field).
Although the two lines (lines A and B) have different energies, they disappear at the same T _{c} and reappear simultaneously in B _{⊥}. Such correlation can be quantitatively understood: both lines are associated with transitions from the EI state to distinct final states with different k values, as shown in the calculated spectrum (Fig. 2a). It not only confirms that the lines have the same origin but also provides optical spectroscopic evidence for the spontaneous formation of an EI gap through the Coulomb attraction between spatially separated electrons and holes.
Excitonic insulator gap by Corbino measurement
To access the gap function (order parameter) of the EI state, bulk conductance measurements (presented in Figs. 3 and 4) are necessary, and hence, we utilized a Corbino device (Fig. 3a) to exclusively measure bulk properties^{19}. The red trace in Fig. 4a shows that the bulk conductivity drops fast with a conductance dip coming to zero at the CNP. Red lines from Fig. 4a (low n _{0} case, V _{b} = 0) to Fig. 4e (high n _{0} case, V _{b} = −6 V), the width of the zeroconductance region decreases; the σ _{ xx } dip around the CNP is lifted from zero, indicating that the EI state is weakened by increasing n _{0}. The opening of a hard gap in the low n _{0} case can be confirmed quantitatively via thermal activation measurements. The σ _{ xx } at CNP vs. 1/T can be fit over one order of magnitude in conductivity with an Arrhenius function \(\sigma_{xx} \propto\) exp(−Δ/2k _{b} T), with activation energy Δ ~ 2 meV (or ~ 25 K) (black dashed line in Fig. 3b, also see Supplementary Fig. 3 and Supplementary Note 1). Our calculated gap energy (blue dashed line in Fig. 2a) agrees with this energy value well.
Next, we measured the gap as a function of n _{0}. The data taken at the CNP follows the relation \(\sigma_{xx}\propto\) exp(−Δ/2k _{b} T) well, yielding a set of Δ values as plotted in Fig. 3c. The gap energy diminishes steeply to nearly zero as n _{0} increases (Fig. 3c). Hence, the gap energy is strongly correlated with 1/n _{0}, suggesting that a low n _{0} is crucial for the formation of an EI gap. These results can be understood as a consequence of weakened e–h binding through increased screening as the n _{0} increases.
Temperature dependence of excitonic insulator gap by capacitance measurement
On the other hand, as the temperature increases, the EI state becomes unstable and the gap eventually closes. We studied the behavior of the gap as a function of temperature through capacitance spectroscopy experiments (Fig. 3d) (see Methods and Supplementary Fig. 4). c _{m} = 1/(1/c _{g} + 1/c _{q}) is the capacitance measured per unit area, where c _{g} is the geometry capacitance per unit area, \({\it{c}}_{\mathrm{q}} = {\it{e}}^{\mathrm{2}}{\it{D}}\) is the quantum capacitance per unit area, and D is the density of states (DOS). We begin with an analysis of the quantum capacitance in different regimes. At T _{c} ~ 10 K, the gap vanishes and the D is dominated by electrons and holes; we take 1/c _{q} = 0 as a reference point. At T ~ 0, the D is zero since it would take an energy cost of Δ(T) to excite a pair of electron and hole. At an intermediate temperature, D is proportional to (1/k _{b} T)exp(−Δ/2k _{b} T); measurement of c _{q} is yielding a semiquantitative estimation of the gap function Δ(T) as shown in Fig. 3e. We can see that Δ starts decreasing at 5 K and finally diminishes at 10 K. This behavior can be well described by the gap function predicted for EI (see Supplementary Fig. 5 and Supplementary Note 2), but is inconsistent with alternative interpretations such as thermal excitations over a singleparticle band gap.
Nonlocal measurement of edge states in topological excitonic insulator
In the low n _{0} regime comparable to the case here, the quantization plateau of helical edge states has been previously observed^{19} and taken as the evidence of the QSH effect. Moreover, the plateaus were found to be quite robust under the variance of external parameters such as inplane magnetic field or temperature. Such a robustness of helical edge states cannot be explained by existing singleparticle theory concerning 2D topological insulators. Pikulin and Hyart subsequently proposed^{20} the emergence of an unconventional EI ground state in the inverted InAs/GaSb QWs, providing a plausible explanation for these experimental observations. In their model, the interplay between an excitonic ground state and interlayer tunneling, which is naturally existing in the present InAs/GaSb structure (i.e., without a tunneling barrier between the two layers) can lead to a pwave EI with topologically protected edge states^{19}. The present work has provided convincing experimental evidences for the existence of EI gap in this system when it is tuned into low densities.
To explore the nontrivial topological properties of the condensate, we performed nonlocal transport measurements in a mesoscopic Hbar device at low n _{0}. According to the Landauer–Büttiker formula (see Supplementary Note 3), for helical edge transport, R _{12,34} = V _{12} /I _{43} should measure quantized resistance of h/4e ^{2} ~ 6.45 kΩ. As shown in Fig. 3f, we indeed observed a quantized plateau close to this value, which confirms that helical edge transport^{29} is indeed realized, whereas the bulk is insulating (also see Supplementary Notes 4 and 5, and Supplementary Figs. 6 and 7). Together with the abovepresented optical and transport evidence, these results suggest that an EI spontaneously emerges in the bulk of InAs/GaSb QWs with helical edge modes propagating along the perimeters, consistent with the observed unusual QSH properties in InAs/GaSb^{19} within the picture of the proposed topological EI^{20}.
Measurement of densitydependent hybridization gap under inplane magnetic fields
In InAs/GaSb QWs without a middle barrier layer, conduction band/valence band hybridization is a potential mechanism for the opening of a bulk gap. However, its contribution to the gap can be distinguished by the application of an inplane magnetic field B _{//} ^{30}. Within the singleparticle picture, electrons and holes with the same momentum tunnel between QWs, forming a hybridization gap. An applied B _{//} induces a relative shift of band dispersions by the amount eB _{//} d/h. Consequently, the interwell tunneling is suppressed due to momentum mismatch, or in other words, B _{//} creates an effective barrier for tunneling. As shown in our 8band selfconsistent calculations (Fig. 4f) in the tunneling regime (see Supplementary Notes 6 and 7, and Supplementary Fig. 8), as B _{//} increases beyond 18 T, the two bands will separate in k space, and the system becomes a semimetal. Here, we applied a B _{//} of 35 T to the Corbino device (also see Supplementary Note 8 and Supplementary Fig. 9) and took the conductance increment (σ _{ xx }(35 T) − σ _{ xx }(0 T)) at the CNP as a qualitative measure of the contribution from singleparticle hybridization to the gap. For V _{b} = −6 V (the highest n _{0} ~ 1 × 10^{11} cm^{−2}), σ _{ xx }(35 T) increases from the B = 0 value by four times and the dip vanish at 35 T, indicating that the hybridization effect is dominant. As n _{0} decreases, σ _{ xx }(35 T) still deviates from σ _{ xx }(0 T) but has a dip at the CNP, showing that the hybridization has a weaker role and the EI gap gradually appears. At V _{b} = 0 V (lowest n _{0} ~5.5 × 10^{10} cm^{−2}), σ _{ xx } is characteristically the same as that at B = 0, which is further demonstrated by a plot of σ _{ xx } at various B _{//} with a broad zeroconductance region seen from B _{//} = 0 to 35 T (Fig. 4g). In the presence of a B _{//}, σ _{xx} markedly depends on n _{0}, demonstrating that the EI phase is more stable at a lower n _{0}. At the lowest n _{0}, the gap remains open in spite of strong inplane magnetic fields, confirmed by activation measurements under different B _{//} up to 35 T in Fig. 4h and i. The observed systematic responses of the gap conductance to the B _{//} support the notion that the EI phase, not the hybridization effect, is responsible for the appearance of the gap in such a low density, equilibrium electron–hole gas.
Discussion
The realization of an excitonic insulator state in our InAs/GaSb system paves the way for further studying manyexciton physics in great detail as well as depth (e.g., BCSBEC crossover, and BEC exciton) in an equilibrium electron–hole system without optical pumping. Due to the weak binding nature of the EI and a lack of middle barrier, counterflow experiments on the current system would not be as effective as in the BEC case^{12}. With an additional AlGaSb barrier between the electron and hole layers, counterflow studies would be enabled to explore BCS to BEC crossover. Moreover, the inverted band structure of the current system brings in topological nature to the EI state, which will allow one to study 2D interacting topological insulators in a highly controllable manner. For example, with a thin AlGaSb middle barrier or utilizing strainedlayer InAs/GaInSb QWs, the symmetry of the order parameter can be controlled by tuning the interplay of interlayer interactions and tunneling.
Methods
Characterization and transport measurements
Al_{0.8}Ga_{0.2}Sb–InAs/GaSb–Al_{0.8}Ga_{0.2}Sb wafers were prepared by molecular beam epitaxy (MBE) on (001) substrate with a 1 μm thick buffer layer, with the following nominal parameters: Wafer A, on N + GaAs substrate, with 12.5 nm InAs/10 nm GaSb QWs; Wafer B, on SI GaAs substrate, with 11.5 nm InAs/8 nm GaSb QWs; Wafer C, on nGaSb substrate, with 11 nm InAs/7 nm GaSb QWs. In Wafer A and B, the interface between the GaSb and InAs QWs was doped with a dilute sheet of Si with a concentration of ~1 × 10^{11}cm^{−2}, whereas there was no doping in Wafer C.
Characterization of wafer A can be found in ref. ^{19}. In wafer A, Supplementary Fig. 2a and b shows B/eR _{ xy } vs. ΔV _{f} for a Hall bar device at V _{b} = −6 V and 0 V, respectively; B is the perpendicular magnetic field and ΔV _{f} is the frontgate bias increment from the CNP. For V _{b} = −6 V, at the high electron density (regime I), B/eR _{ xy } is consistent with the density obtained from Subinikov de Haas oscillations, linear with V _{f}. As the top of the hole band is reached by the Fermi level, holes are introduced and twocarrier transport dominates, with R _{ xy } traces divergent and the system reaching the e–h hybridized regime (regime II). Owing to the absence of a hard gap, B/eR _{ xy } is still dominated by electron–hole residual carriers even in the hybridization gap (regime III). As the Fermi level is below the bottom of the electron band, holes dominate and R _{ xy } becomes linear again (regime IV). For V _{b} = 0 V, where less holes are introduced by V _{b}, as the CNP is approached, similarly the system goes from regime i to regime II. However, within regime II, a plateaulike feature is observed, indicating the formation of an EI gap (regime V).
In Wafer C, lattice matched epitaxial layers were grown on a GaSb substrate, which should result in enhanced carrier mobility. The carrier mobility of Wafer C measured at 300 mK indeed showed significant improvement. The typical electron mobility was 90,000 cm^{2} V^{−1} s^{−1} for a density of 5 × 10^{11} cm^{−2}, and the hole mobility was about one order of magnitude lower at similar densities. Magnetotransport experiments showed wellresolved quantum oscillations, as shown in Supplementary Fig. 1a. In a perpendicular magnetic field, the device can be tuned by the frontgate from the electron dominating regime to the holedominating regime in an asymmetrical Hall bar, as shown in Supplementary Fig. 1b. As the electron density decreases, holes emerge, which bends the trace. When the hole and electron densities become comparable, the trace drops again with an EI gap emerging and helical edges dominating. The equilibrium density n _{o} was ~5 × 10^{10} cm^{−2}.
The V _{f} dependence of the conductance in a Corbino device C2 (made from Wafer C) is shown in Supplementary Fig. 1c. Similar to Wafer A, there is a conductance dip at the CNP with the resistivity equal to ~1 MΩ/square. As we apply B _{//} up to 35 T, the dip conductance stays nearly the same, which confirms that this gap is not from hybridization, as shown in Supplementary Fig. 1d. Furthermore, the constant dip conductance also shows that the gap does not close between 0 T and 35 T. These findings are very similar to those from Wafer A, suggesting that Si doping is not essential for the emergence of the EI gap.
Terahertz measurements
In optical spectroscopy experiments, we measured the transmission of a terahertz (THz) beam through the device made from Wafer B, which was prepared on a semiinsulating GaAs substrate. The device had a semitransparent frontgate, and the gated area was 5 mm × 5 mm for maximizing the THz transmission. Through the semitransparent frontgate, we modulated the system between the CNP and the low mobility electron regime. The electron regime we selected had low mobility and low density, so it had no feature in the 0.25–2.4 THz frequency range, working as a good reference for the excitoninduced transmission. The transmittance was defined as the intensity of the THz beam transmitted through the sample normalized by the reference signal.
Capacitance–voltage measurements
We performed capacitance–voltage (CV) measurements^{30} under different V _{b} in Corbino devices C1 (Wafer A) and C2 (Wafer C) (Supplementary Fig. 4a). In addition to the dc V _{f} and V _{b}, a small ac (f = 100 Hz) voltage was applied to the frontgate, and the capacitance between the frontgate and the QWs was measured. In CV measurements^{30}, the geometry capacitance c _{q} and quantum capacitance c _{q} = e ^{2} D contribute to the measured capacitance per unit area c _{m} as c _{m} = 1/(1/c _{g} + 1/c _{q}), where D is the density of states (DOS). In the electron or holedominating regime, c _{g} is much smaller than the quantum capacitance, so c _{g} dominates in c _{m}. When the DOS decreases and quantum capacitance is smaller than c _{g}, c _{m} drops and the quantum capacitance dominates.
The outofphase signal was one order of magnitude less than the inphase signal. In this case, for devices from both Wafers A and C, the observed capacitance drop near the CNP represented the reduced DOS. For V _{b} = −6 V, as illustrated in the blue trace of Supplementary Fig. 4a, the capacitance near the CNP was nearly constant, agreeing with previous CV results^{30} about the hybridized gap (the tunneling is too weak to form a hard gap). The DOS in this gap can be treated as the mixture of electron’s and hole’s. For V _{b} = 0 V, the capacitance at the CNP drops to 10% of c _{g}, which indicates that the hard gap (EI gap) forms with a reduced DOS. CV measurements performed on Corbino device C2 with V _{b} = 0 V also showed that there is a capacitance drop near the CNP (Supplementary Fig. 4b), confirming that the EI gap exists and the gap is not from Si doping. It should be mentioned that in Wafer C there was no doping and thus, Anderson localization (AL) was suppressed. Moreover, AL could not reduce the DOS. Therefore, the CV results confirm that the hard gap does not originate from AL. A large capacitance drop shows that the coexisting electrons and holes can form neutral particles (reduced DOS) with the formation of the EI gap.
Next we increased the temperature. Supplementary Fig. 4c shows CV traces at V _{b} = 0 V for different temperatures. We find that the EI gap starts collapsing at 6 K and disappears at 10 K. This confirms that the EI gap dominates in the lowtemperature regime, and also provides the temperature window for thermal activation energy measurements.
Determination of equilibrium density of electron and hole
Ideally, the electrons are tuned by the frontgate, whereas the holes are tuned by the backgate. Owing to the imperfect screening from the electron layer, the frontgate could simultaneously modulate the electrons and holes when holes appear.
Supplementary Fig. 12a shows that, for ΔV _{f} sweeping from 2 V, the electron density is high and linear with ΔV _{f}, so we can extract the electron density increment per ΔV _{f}. Supplementary Fig. 12b shows the electron density rate under different V _{b}. The rate is constant for V _{b} < 3 V. When V _{b} increases and holes appear, the rate decreases until V _{b} = −7.5 V, where the rate saturates.
From CV measurements, we know that the capacitance stays constant until the EI gap appears. The capacitance per unit area c _{m} means the absolute density (n + p) increment per ΔV _{f}. So we can integrate the capacitance over ΔV _{f} to obtain the absolute density, as shown in Supplementary Fig. 12a. It should be mentioned that, for the high electron density, the hole density can be neglected, so the absolute density agrees with the electron density. The absolute density decreases with a constant rate until it gets to nearly zero, as shown in the blue dashed line, where the CNP is approached. In this case, we can obtain the voltage where the CNP is reached (vertical dashed black line).
In the magnetotransport trace (red dotted line in Supplementary Fig. 12a), as the holes emerge, the trace bends up and it is not straight forward to extract the electron density. However it is reasonable to make an approximation that, when holes appear, the electrons change in the saturated rate (green dashed line) until the CNP is reached, as illustrated in Supplementary Fig. 12a. This way we can estimate the equilibrium density n _{0} (marked by the circle). The density uncertainty is 5 × 10^{9} cm^{−2}. Similarly, we obtain n _{0} under different V _{b}, as shown in Supplementary Fig. 13.
Data availability
The authors declare that the data supporting the findings of this study are available within the paper and its Supplementation Information files or from the corresponding author upon reasonable request.
References
 1.
Mott, N. F. The transition to the metallic state. Phil. Mag. 6, 287–309 (1961).
 2.
Knox, R. S. Theory of excitons. Solid State Phys. Suppl. 5, 100 (1963).
 3.
Keldysh, L. V. K. & Kopaev, Y. V. Possible instability of the semimetallic state toward coulomb interaction. Fiz. Tverd. Tela 6, 2791 (1964).
 4.
Jérome, D., Rice, T. M. & Kohn, W. Excitonic insulator. Phys. Rev. 158, 462–475 (1967).
 5.
Halperin, B. I. & Rice, T. M. Possible anomalies at a semimetalsemiconductor transition. Rev. Mod. Phys. 40, 755–766 (1968).
 6.
Lozovik, Yu. E. & Yudson, Y. I. Feasibility of superfluidity of paired spatially separated electrons and holes; a new superconductivity mechanism. JETP Lett. 22, 274 (1975).
 7.
Littlewood, P. B. & Zhu, X. Possibilities for exciton condensation in semiconductor quantumwell structures. Phys. Scripta T68, 56–67 (1996).
 8.
Zhu, X., Littlewood, P. B., Hybertsen, M. S. & Rice, T. M. Exciton condensate in semiconductor quantum well structures. Phys. Rev. Lett. 74, 1633–1636 (1995).
 9.
Naveh, Y. & Laikhtman, B. Excitonic instability and electricfieldinduced phase transition towards a twodimensional exciton condensate. Phys. Rev. Lett. 77, 900–903 (1996).
 10.
Butov, L. V., Gossard, A. C. & Chemla, D. S. Towards Bose–Einstein condensation of excitons in potential traps. Nature 414, 47–52 (2002).
 11.
Snoke, D. Spontaneous Bose coherence of excitons and polaritons. Science 298, 1368–1372 (2002).
 12.
Eisenstein, J. P. & MacDonald, A. H. Bose–Einstein condensation of excitons in bilayer electron systems. Nature 432, 691–694 (2004).
 13.
Luin, S. et al. Observation of soft magnetorotons in bilayer quantum Hall ferromagnets. Phys. Rev. Lett. 90, 236802 (2003).
 14.
Datta, S., Melloch, M. R. & Gunshor, R. L. Possibility of an excitonic ground state in quantum wells. Phys. Rev. B 32, 2607–2609 (1985).
 15.
Xia, X., Chen, X. M. & Quinn, J. J. Magnetoexcitonsin a GaSbAlSbInAs quantumwell structure. Phys. Rev. B 46, 7212–7215 (1992).
 16.
Kono, J. et al. Farinfrared magnetooptical study of twodimensional electrons and holes in InAs/AlGaSb quantum wells. Phys. Rev. B 55, 1617–1636 (1997).
 17.
Liu, C. X., Hughes, T. L., Qi, X. L., Wang, K. & Zhang, S. C. Quantum spin Hall effect in inverted typeII semiconductors. Phys. Rev. Lett. 100, 236601 (2008).
 18.
Knez, I., Du, R. R. & Sullivan, G. Evidence for helical edge modes in inverted InAs/GaSb quantum wells. Phys. Rev. Lett. 107, 136603 (2011).
 19.
Du, L. J., Knez, I., Sullivan, G. & Du, R. R. Robust helical edge transport in gated InAs/GaSb bilayers. Phys. Rev. Lett. 114, 096802 (2015).
 20.
Pikulin, D. I. & Hyart, T. Interplay of exciton condensation and the quantum spin Hall effect in InAs/GaSb bilayers. Phys. Rev. Lett. 112, 176403 (2014).
 21.
Seradjeh, B., Moore, J. E. & Franz, M. Exciton condensation and charge fractionalization in a topological insulator film. Phys. Rev. Lett. 103, 066402 (2009).
 22.
Mu, X., Sullivan, G. & Du, R. R. Effective gfactors of carriers in inverted InAs/GaSb bilayers. Appl. Phys. Lett. 108, 012101 (2016).
 23.
Wang, X., Belyanin, A. A., Crooker, S. A., Mittleman, D. M. & Kono, J. Interferenceinduced terahertz transparency in a semiconductor magnetoplasma. Nat. Phys. 6, 126–130 (2010).
 24.
Zhang, Q. et al. Collective nonperturbative coupling of 2D electrons with highqualityfactor terahertz cavity photons. Nat. Phys. 12, 1005–1011 (2016).
 25.
Castella, H. & Wilkins, J. W. Splitting of the excitonic peak in quantum wells with interfacial roughness. Phys. Rev. B 58, 16186–16193 (1998).
 26.
Zhang, Q. et al. Stability of highdensity twodimensional excitons against a Mott transition in high magnetic fields probed by coherent terahertz spectroscopy. Phys. Rev. Lett. 117, 207402 (2016).
 27.
Fenton, E. W. Excitonic insulator in a magnetic field. Phys. Rev. 170, 816–821 (1968).
 28.
Kuramoto, Y. & Horie, C. Twodimensional excitonic phase in strong magnetic fields. Solid State Commun. 25, 713–716 (1978).
 29.
Roth, A. et al. Nonlocal transport in the quantum spin Hall state. Science 325, 294–297 (2009).
 30.
Yang, M. J., Yang, C. H., Bennett, B. R. & Shanabrook, B. V. Evidence of a hybridization gap in “semimetallic” InAs/GaSb systems. Phys. Rev. Lett. 78, 4613–4616 (1997).
Acknowledgements
We acknowledge helpful conversations with A.H. MacDonald, B.I. Halperin, and D.I. Pikulin. Work at Rice University was supported by NSF Grants No. DMR1207562 and No. DMR1508644 (R.R.D.), and by DOE Grant No.DEFG0206ER46274 (L.D.). W.L. and K.C. were supported by NSFC (No. 11434010). X.L. and J.K. acknowledge support from NSF (Grant No. DMR1310138). A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by NSF Cooperative Agreement under No. DMR1157490, and by the State of Florida.
Author information
Affiliations
Contributions
L.D. fabricated the devices and performed transport experiments. L.D. and X.L. performed the THz experiments under the supervision of J.K., L.D., X.L., J.K., and R.R.D. analyzed the data. W.L. and K.C. developed the theoretical model and performed the numerical calculations. G.S. prepared the MBE semiconductor materials. L.D., J.K., and R.R.D. cowrote the manuscript with input from the other authors. L. D. and R.R.D. conceived the project. R.R.D. provided overall supervision and coordination of the project.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Additional information
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Du, L., Li, X., Lou, W. et al. Evidence for a topological excitonic insulator in InAs/GaSb bilayers. Nat Commun 8, 1971 (2017). https://doi.org/10.1038/s41467017019881
Received:
Accepted:
Published:
Further reading

A monolayer transitionmetal dichalcogenide as a topological excitonic insulator
Nature Nanotechnology (2020)

Scattering in InAs/GaSb coupled quantum wells as a probe of higher order subband hybridization
Physical Review B (2020)

SpinTriplet Excitonic Insulator: The Case of Semihydrogenated Graphene
Physical Review Letters (2020)

Experimental conditions for the observation of electronhole superfluidity in GaAs heterostructures
Physical Review B (2020)

Frictional Drag Effect between Massless and Massive Fermions in SingleLayer/Bilayer Graphene Heterostructures
Nano Letters (2020)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.