Introduction

Many-body effects play important roles in low-dimensional electron systems and attract great deal of attention in graphene physics due to the interplay with the honeycomb topology1,2,3,4,5,6,7,8. While theoretical treatments of interaction effects in graphene frequently assume that the systems are clean and controllable, these preconditions are difficult to satisfy in the natural material1,2,3. For instance, the theoretical prediction of chiral superconductivity requires tuning the Fermi energy to the M saddle-points with flatband characteristic2, which is difficult in graphene given the large energy gap (~5 eV, a consequence of the small lattice spacing) at the Brillouin-zone (BZ) M-point. Two-dimensional (2D) saddle-point excitons4,5,6,7 in graphene, redshifted from the saddle-point singularity at the M-point, are subject to prominent many-body effects, but their optical responses are at a large energy of 4.6 eV, far from the relevant energy ranges for device applications.

Artificial graphene (AG)9,10,11,12,13,14 is a controllable platform for simulation of quantum behavior in the physics of 2D crystals15,16,17,18,19. Linearly dispersing Dirac bands have been reported in several AG systems, including molecular assemblies on copper11, fermionic atoms trapped in optical lattices12, photonic systems13, and nanopatterned GaAs quantum wells (QWs)14. AG systems with tunable honeycomb lattices should be suitable for explorations of quantum regimes of many-body effects in graphene-like band structures. Thus far, however, electron–electron interactions in solid-state AG have not been reported, and, particularly in the case of GaAs-based AG, the low-disorder conditions for observing such effects are difficult to achieve.

Here, we report the realization of low-disorder semiconductor AG on a high-mobility GaAs QW. We fabricate small period triangular antidot lattices that significantly suppress the impact of processing disorder on electrons, and thus preserve the high-quality of states in as-grown QWs. The achievement enables observations of collective saddle-point spin excitons that are subject to exchange Coulomb interactions. For AG lattice periods in the range of tens nanometers the energy of the observed saddle-point excitons is in a much sought after terahertz range. Relatively large exchange Coulomb interactions, which have energies comparable to the gap of Dirac bands at the M-point of the BZ, could be modulated by tuning the carrier density. The capability of observing collective saddle-point spin excitons and the emergence of relatively large Coulomb interactions in the low-disorder AG lattices demonstrate access to a regime in AG that is dominated by electron–electron interaction effects, allowing the exploration of intriguing many-body effects that are inaccessible in graphene. Combining Coulomb interactions in a clean electron environment with a meV-energy gap at the M-point provides opportunities for experimental studies of many-body effects in honeycomb lattices such as chiral superconductivity2.

Results

Triangular antidot lattice

We employ state-of-the-art nanofabrication to realize small period triangular antidot lattices in high-mobility GaAs QWs. The antidots are created by deep etching of circular holes in the semiconductor in a carefully controlled triangular lattice pattern with period b. Figure 1a shows that the triangular antidot lattice supports a honeycomb lattice of dots (the darker areas in Fig. 1a) with period a. The main advantage of this approach is twofold. First, the effective lattice period is smaller than the patterned period by a factor of \(\sqrt 3\) (a = b/\(\sqrt 3\)). Second, electrons in the antidot lattice avoid the areas under the etched features due to a large repulsive potential, resulting in a greatly reduced impact of process-induced disorder. This is confirmed in the evaluation of electron density reported in Fig. 1b, which is a characteristic feature of the antidot triangular pattern that results in a 2D electron gas with physically connected nearest-neighbor honeycomb lattice sites in the unetched areas of the pattern (Fig. 1a, b). Figure 1c displays the calculated dispersion of the two lowest Dirac bands of AG for a typical triangular antidot lattice imprinted on a GaAs QW. The Dirac bands shown in Fig. 1c are gapless with linear momentum dispersion near the K (K’) points of the BZ and with a gap (EM) at the M-point that is comparable to the Fermi energy (~1 meV), typical of semiconductor AG.

Fig. 1
figure 1

Low-disorder artificial graphene in triangular antidot lattices. a Schematic of a triangular antidot lattice with period b that is imprinted on a GaAs QW. The white circles represent etched antidots of radius r. The black and gray areas represent the unetched areas, and the black circles indicate the effective dots arranged in a honeycomb pattern of period a = b /\(\sqrt 3\). In the evaluation of AG band structure, we used a muffin–tin potential (red dashed line) with Fermi energy EF in the single-particle approximation9,10,14. V0 is the potential depth in the etched area. Each antidot produces an effectively repulsive potential V0. b Evaluations of electron density in a triangular antidot lattice with V0 = 6 meV, b = 70 nm, a = 40.4 nm, and r = 20 nm. Yellow arrows highlight strengthened electron coupling between nearest-neighbor dots. The color bar indicates the electron density. c The two lowest Dirac bands of AG with the parameters in b. EM indicates the value of the gap near the M-point

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Device fabrication

The nanofabrication of small period (60–70 nm) triangular antidot lattices, with periods much smaller than in previous works (200 nm)20, requires precise control of the lattice period, the antidot radius and the deep etching profile. Figure 2a–d describes schematically the processing steps involved in the creation of the antidot lattices (see Methods). Electron-beam lithography is employed to define the antidot pattern using a two-step pattern transfer process, that is, based on a Zep 520 electron-beam resist, which possesses both high-resolution capability and good resistance to etching. The e-beam writer is operated at an accelerating voltage of 80 kV to mitigate proximity effects that are relatively strong in GaAs21,22,23. A temperature-controlled, modified developing process with ultrasonication is applied (see Methods), followed by a low voltage (~3 kV) blanket electron-beam exposure to harden the resist24. This is followed by a BCl3-based inductively coupled plasma reactive-ion etching step designed to transfer the triangular antidot array to the QW. Arrays with lattice constants b as small as 60 nm are fabricated over an area of 200 × 200 µm2. Figure 2e shows top and side views of samples having patterns with b = 60 nm (a = 34.6 nm) and b = 70 nm (a = 40.4 nm) (sample parameters are listed in Supplementary Table 1). The structures are characterized by photoluminescence to determine the Fermi energy (Supplementary Figure 1). Photoluminescence outside the AG pattern where no mask is present shows that the etching depth in the sample with a = 40.4 nm (sample I) leads to a full depletion of electrons, suggesting that electron wavefunctions are strongly suppressed under the antidots. The estimated potential barrier is V0 = 6 meV (see Supplementary Note 1).

Fig. 2
figure 2

Creation of artificial graphene in triangular antidot lattices. Processing steps: a a Zep 520 resist is exposed by an electron beam at 80 kV accelerating voltage. After exposure, the development of the resist produces triangular antidots pattern. b The resist with pattern is exposed by an electron flood at 3 kV accelerating voltage, to enhance the chemical stability of the resist. c The BCl3-based dry etch is performed to transfer the pattern to the substrate with depth control. The final device after the removal of the residue resist is shown in d. The QW is positioned 80 nm below the surface and 30 nm below the Si δ-doping layer. The as-grown electron density is 2.1 × 1011 cm−2, with a Fermi energy EF of 7.5 meV and a typical low-temperature mobility of 106 cm2 V−1 s−1. e Scanning electron microscopy micrographs of AG lattices with different periods in 0° tilt (upper panel, top view) and 40° tilt (lower panel, side view). The white dashed lines define the etching depth. The yellow scale bar is 50 nm for all panels in e. f The comparison between the RILS spectrum of intersubband excitations of AG sample I (with incident photon energy of 1554.36 meV, blue trace) and intersubband excitations of the as-grown GaAs QW (with incident photon energy of 1550.92 meV, black trace). The peak width of intersubband excitations of sample I is nearly constant for different incident photon energies (see Supplementary Note 5 and Supplementary Figure 8). The spectra were taken at 5 K under crossed polarization

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Low-disorder AG

The impact of disorder resulting from nanofabrication processing is assessed by resonant inelastic light scattering (RILS)14,19 spectra of intersubband collective excitations25, in which the only electron degree of freedom that changes is the confinement state in the QW. In Fig. 2f, a comparison between intersubband excitations of sample I and the as-grown QW under crossed incident and scattered photon polarizations shows that the peak in AG has a full width at half maximum (FWHM) of 0.3 meV, which is extremely close to the FWHM ~0.25 meV of the mode in the as-grown QW (the FWHM reflects the degree of disorder in the device, which is typically due to ionized defects that scatter electrons26). The slight increase of FWHM could be attributed to the reduced screening of potential of ionized defects due to the greatly reduced Fermi energy (reduced charge density) in the AG sample (see Supplementary Table 1 and Supplementary Note 1). The nearly identical width of the modes in AG to that in the as-grown QW indicates the exceptional low-disorder environment for charge carriers in this small period triangular antidot lattice. This is in contrast with results from a honeycomb dot lattice with optimized fabrication processing and similar period under crossed polarization, where the FWHM increases by a factor of three from the as-grown QW value25.

Low-energy Dirac-band transitions

The AG band structure in sample I is probed by RILS measurements of low-energy excitations from Dirac-band transitions shown in Fig. 3a. RILS spectra under crossed polarization such as those in Fig. 3b, c access excitations in the spin degree of freedom27. Spin-density excitations involving changes in the spin degrees of freedom are redshifted from single-particle transitions due to exchange Coulomb interactions. This is an excitonic shift due to Coulomb coupling between the excited electrons and the holes in the lower state28,29. Charge density excitations that preserve spin orientation are active in parallel polarization where the laser tail becomes strong and obscures RILS signals in low-energy excitations. Spectra of low-energy excitations presented here are under cross polarization.

Fig. 3
figure 3

Dirac-band excitations in low-disorder AG in a triangular antidot structure. a AG energy band calculation with parameters in Supplementary Table 1 and diagram of Dirac-band transitions for sample I. Transitions in different regimes are illustrated by different colors. b The lowest plot is the joint density of states (JDOS) for the transitions in a. The colored areas show transitions in different regimes. The red trace is the JDOS under Gaussian broadening with width γ = 0.1 meV (this γ is chosen to fit the spectra line shape near EM). The top yellow trace is the RILS spectrum of Dirac-band excitations in sample I obtained with incident photon energy ħωi of 1522.88 meV. c RILS spectra of Dirac-band excitations. A non-RILS background has been removed (subtraction details are reported in Supplementary Figures 4–6, and in Supplementary Note 3). ħωi in meV are indicated. The vertical dashed lines indicate the positions of EM and EX. The inset shows a level diagram for EM and EX. In the inset, black circles are for holes in c00 band and red circles are for electrons in c01 band. d Top shows a RILS spectrum (purple trace) with ħωi of 1523.06 meV, and the JDOS (red dashed line, γ = 0.1 meV) that is proportional to scattering intensity due to the single-particle transitions. Bottom: upper traces reveal the spectrum after subtraction of the single-particle JDOS intensities with ħωi of 1523.06 meV (purple), 1523.01 meV (blue), and 1522.97 meV (green). A schematic plot shows the spin-exciton energy and the gap energy at the M-point. Due to Landau damping, the width of the spin-exciton mode increases

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In the absence of many-body interaction effects the RILS spectra are proportional to the joint density of states (JDOS) for the transitions14. The two lower curves in Fig. 3b are the calculated JDOS from the parameters in Fig. 3a for sample I with γ = 0 meV and γ = 0.1 meV (γ is Gaussian broadening width). The evaluations show that the JDOS has a maximum at EM = 0.9 meV (0.2 THz) arising from the saddle-point singularity for transitions at the M-point of the BZ (cyan areas in Fig. 3a, b). The rapid collapse of the RILS intensity at energies below EM, seen in the top curve in Fig. 3b, is a signature of the formation of linearly dispersing Dirac cones14, thus confirming the presence of a Dirac-band structure in the triangular antidot lattice in sample I.

The RILS spectrum in Fig. 3b also uncovers departures from the calculated single-particle JDOS at energies just below EM. The resonance behavior of RILS spectra shown in Fig. 3c reveals that the spectral lineshapes of low-energy Dirac-band excitations have a strong dependence on incident photon energy ħωi (see Supplementary Note 7). Major departures of the RILS spectral lineshapes from the JDOS predictions for energies below EM can be seen in spectra in Fig. 3d, where the single-particle JDOS has been subtracted from the spectra in Fig. 3c, revealing a relatively broad intensity maximum at EX. The redshift from EM is interpreted as arising from many-body electron interactions that emerge in the low-disorder AG. For spin excitations in Fig. 3 the relevant many-body effects emanate from exchange Coulomb interactions. Then, it is assigned to collective spin excitations associated with the saddle-point singularity for Dirac-band transitions. An alternative interpretation for the EX mode is discussed in Supplementary Note 6 and Supplementary Figure 9. The energy of saddle-point excitons EX associated with transitions at the M-point singularity would be at an energy of about 0.45 meV (0.1 THz), and tunable (e.g., by changing the lattice constant).

The significant broadening of the EX maximum (in the spectra of Fig. 3d) is due to overlap with the continuum of single-particle excitations, which causes the decay of the spin-exciton mode into electron–hole pairs (Landau damping29, see Fig. 3d); this is also observed at the saddle-point exciton in natural graphene5,6,7. An exchange Coulomb energy in the saddle-point exciton here can be estimated as in the case of intersubband excitations:29 \((E_M^2 - E_X^2)/E_{\mathrm{M}}\) = 0.6 meV. This is a relatively large exchange interaction strength that is about half of the M-point gap EM. As shown in ref. 29, exchange Coulomb energies are equal to Eex = 2nβ, where n is the carrier density and β is the exchange interaction matrix element for the transition. Here, the carrier density in the c00 band is linked to the exchange energy in low-energy excitations (c00 → c01). Exchange energies in the same range could be expected in other transitions associated to the carrier density in the c00 band.

Spin-density intersubband excitations

A large exchange interaction is observed in the spin-density intersubband excitation (SDE) shown in Fig. 4, in which there is no change in the states of the AG potential (intersubband excitations under parallel polarization are presented in Supplementary Figures 7 and 8, and in Supplementary Note 4). The exchange energy associated with SDE can be written as Eex = \((E_{{\mathrm{10}}}^2 - E_{{\mathrm{SDE}}}^2)/E_{10}\)29, where ESDE is the peak position of SDE and E10 = 20.0 meV is the QW subband spacing (see Supplementary Note 2, Supplementary Figures 2 and 3). Figure 4a displays RILS processes for intersubband transitions that are from c00 (c01) band to c10 (c11) band in the second QW subband. As shown in Fig. 4a, c00 → c10 transitions have a resonance enhancement at incident photon energies ħωi that are lower than the resonance for c01 → c11 transitions. Figure 4b shows a color plot of the resonance enhancement for SDE modes at around 20 meV that reveals a remarkable dependence of peak positions on ħωi. This can be explained by the different resonance enhancements in RILS by excitations associated with different transitions. At lower ħωi the SDE spectra come from excitations linked to c00 → c10 transitions, whilst at higher ħωi the spectra are largely due to excitations from c01 → c11 transitions. The changes in the position of SDE peaks as a function of ħωi in Fig. 4b are attributed to the different exchange interactions in those two transitions. For a low ħωi = 1552.5 meV the SDE mode is at ESDE = 19.8 meV and the exchange energy is Eex = 0.43 ± 0.05 meV for c00 → c10 transitions. The large exchange interaction confirms the results obtained in the saddle-point spin-exciton mode. For a high ħωi = 1554.0 meV the SDE mode is at ESDE = 19.9 meV and the exchange energy is Eex = 0.17 ± 0.05 meV for c01 → c11 transitions. The dependence of Eex on ħωi is shown in Fig. 4c.

Fig. 4
figure 4

Coulomb exchange energy probed in intersubband excitations. a Optical transitions in RILS of intersubband spin-density excitations (SDE) are indicated. In transitions with incident photon energy ħωi, the incident photon excites an electron from the valence band (VB) to the c11/c10 state leaving behind a hole. In transitions with scattered photon energy ħωs, an electron in the c01/c00 state recombines with the hole and emits the scattered photon. The incident photon energy ħωi and scattered photon energies ħωs are linked by energy conservation E = ħωi−ħωs, where E is the energy of AG excitations being studied. In SDE, the electrons are excited from a lower populated AG subband to the next higher subband without the change in AG band index. Red (black) line defines SDE exciting electron states with spin-flip processing from c01 and c11 (from c00 and c10) under higher (lower) ħωi. While the c01 band is above the Fermi energy it is populated because of the thermal excitation of charge carriers at finite temperature of our experiments. b Color plot of RILS spectra as a function of ħωi measured at temperature T = 5 K with crossed polarization. The color bar indicates intensities of scattered light. The dashed lines marked with c01 → c10 and c00 → c11 indicate the energies of combined intersubband transitions with the change in both quantum well subband and AG band index (see ref. 25 and Supplementary Note 2). The dashed line marked with E10 indicates the position of quantum well subband spacing (see Supplementary Note 2). The yellow dashed lines indicate the position of SDE. There is a weak photoluminescence that overlaps the c01 → c10 and c00 → c11 transitions, as well as SDE. c The dependence of exchange energies extracted from (b) on ħωi. The vertical error bars represent estimated uncertainties in determination of E10 and ESDE from the measured spectra

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The much smaller exchange energy for c01 → c11 transitions is attributed to lower carrier population in the c01 band of AG, since exchange Coulomb energies are proportional to carrier densities. While the c01 band is above the Fermi energy it is populated because of the thermal excitation of charge carriers at finite temperature of our experiments. Data points in Fig. 4c suggest Eex (c00 → c10)/Eex (c01 → c11) ~ 3. According to the Fermi-Dirac distribution at the temperature 5 K, we estimate nc00/nc01 ~ 3, where nc00 and nc01 are carrier densities in the c00 and c01 bands, showing Eex (c00 → c10)/Eex (c01 → c11) ~  nc00/nc01 ~ 3. Furthermore, as shown in Supplementary Note 8 and in Supplementary Figures 11–13, sample III has a higher Fermi energy, so that nc00 and nc01 are very close. Then, the determined exchange energies for c01 → c11 and c00 → c10 transitions, with similar densities, are comparable. The tuning of Eex by the electron density provides an effective way to control saddle-point spin-exciton modes and for future studies of excitonic ground states.

Exchange interactions in low-energy excitations (c00 → c01) and SDE (c00 → c10) are linked to the carrier density in the c00 band, determining similar exchange energies in these excitations. On the other hand, transitions in c00 → c01 and c00 → c10 have different values of matrix elements β, which would result the difference between the exchange energies (0.6 meV in c00 → c01 transitions and 0.45 meV in c00 → c10 transitions). At a carrier density of 0.4 × 1011 cm−2 in the c00 band, we have β ≈ 0.7 × 10−11 cm2 meV−1 for the low-energy excitations based on c00 → c01 transitions, and β ≈ 0.5 × 10−11 cm2 meV−1 for the SDE based on c00 → c01 transitions. It is known that exchange Coulomb interactions in as-grown GaAs QW are strong due to large values of β29. Values of β we obtained in the AG sample are in the same range as the as-grown samples (0.8 × 10−11 cm2 meV−1 measured in ref. 29), resulting in large exchange interactions.

Discussion

The realization of low-disorder AG in small period triangular antidot lattices, which reveals the presence of collective terahertz saddle-point spin excitons at the M-point singularity in the density of states. We observe relatively large exchange Coulomb interactions, which have energies comparable to the gap of Dirac bands at the M-point. Low-disorder AG is thus a highly tunable condensed-matter system for the investigation of graphene-like physics in a regime where the interplay between electron interactions and the hexagonal topology of AG lattices becomes predominant. The large exchange coupling to the honeycomb lattice with tunable parameters makes low-disorder AG lattice potentially suitable for further studies of exciton liquids and spintronics30. To realize possible exciton liquids31,32, the suppression of Landau damping is required, which would be achieved by opening a gap at K (K’) points (through removing inversion symmetry of the AG lattice). Our findings may open approaches for the investigation of strongly correlated quantum phases in condensed-matter systems such as ferromagnetism and unconventional superconductivity2,3, expanding the tool box for quantum simulation. The low-disorder processing approach can be applied to artificial lattices with intriguing topologies such as the Lieb33,34 and Kagome lattices. Given that the saddle-point excitons are in the terahertz frequency range and subject to tunable parameters, low-disorder AG may provide a semiconductor platform for optoelectronic device applications.

Methods

Fabrication of the AG nanopattern

The semiconductor system we used is 25-nm-wide one-side modulation-doped Al0.1Ga0.9As/GaAs QW. The QW is positioned 80 nm below the surface and 30 nm below the Si δ-doping layer. The as-grown electron density is 2.1 × 1011 cm−2, with a Fermi energy EF of 7.5 meV and a typical low-temperature mobility of 106 cm2 V−1 s−1. Electron-beam lithography at 80 kV and beam current 400 pA (e-beam writer nB4, NanoBeam Ltd.) was employed to pattern 200 × 200 µm2 triangular arrays of circles. We used a single layer resist of Zep 520 as a hard mask, which was developed in a solution of n-Amyl acetate: isopropanol in composition 1:3 with ultrasound sonication. In the development, the temperature is always kept at 0°. After development, a 3 kV electron beam was flood exposed on the pattern to harden the developed resist. An Oxford PlasmaPro System 100 was used to perform the inductively coupled plasma reactive-ion etching. The gas employed was a mixture of 50 sccm Ar and 10 sccm BCl3, shown to etch AlGaAs/GaAs heterostructures with high anisotropy with the etching time 60 s. The reactive-ion etching power and ion coupled plasma power were 45 and 200 W with a pressure of 8 mtorr. The residual resist was removed with dimethylacetamide. The etch recipe was optimized to achieve a high degree of physical bombardment and reduced chemical etching.

Optical measurements

The sample was mounted in an optical cryostat with a base temperature of 4 K. The RILS experiments were performed in a back-scattering configuration, with incident laser beam almost perpendicular (within 10°) to the sample. A tunable Ti:sapphire laser with spot diameter of around 200 µm and power as low as 20 μW was focused on the AG nanopattern. Spectra were collected using a liquid nitrogen cooled CCD through a double grating spectrometer (Spex 1404).

Data availability

The authors declare that the data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.