Introduction

In the last couple of decades, many candidate systems for performing quantum simulations have been investigated including ultracold (Rydberg) atoms in optical lattices or cavities, trapped ions or electrons, and superconducting circuits1,2,3. A key advantage of Rydberg atom-mediated quantum simulation is that the long-range dipole-dipole interaction strength scales favourably with their principal quantum numbers4, enabling access to the strongly interacting, hence, strongly correlated regime. While there has been significant progress in Rydberg atom quantum technologies5, a solid-state Rydberg platform with reduced technical overhead and better integration capability is more desirable. Solid state systems provide a robust and miniaturised alternative, where the samples are usually in sub-mm dimensions and can fit inside a very compact experimental setup. This allows for easy tuning and control of the individual excitons and can also be easily scaled. An additional feature of solid-state platforms is their driven-dissipative nature that enables easy access to non-equilibrium physics6.

Only recently, high principal quantum number Rydberg excitons up to n = 30 have been demonstrated in a Cu2O crystal7,8,9 with exponentially scaling properties10. This platform provides an excellent opportunity to explore the feasibility of solid-state quantum simulators based on Rydberg excitons as they exceed micron dimensions. Two interactions in Rydberg excitons of Cu2O are the Rydberg blockade7,11 and plasma blockade12 effects. One requirement in this direction is the ability to confine two or more Rydberg excitons within their blockade volume and modulate their interaction with each other. Non-linear interactions in excitons can also be mediated through strong light-matter coupling13 leading to single particle quantum interaction14,15. Low n Rydberg exciton-polaritons have been demonstrated with perovskites16, TMDCs17, and in Cu2O up to n = 618. The confinement of the excitons can be demonstrated by two means- optically inducing a potential19,20 to trap the excitons, or by means of spatial confinement21,22 by creating micro-/nano-structures. While working with optical confinement is experimentally simpler, it lacks resolution when it comes to creating the potential as it cannot go down to submicron levels. On top of that, the potential in itself is quite weak, leading to reduced coherence time for interaction. Creating spatial confinement circumvents both issues by going below the sub-wavelength regime using a focused ion beam (FIB) etching technique23. While cavity polaritons are better coupled to light for optical transmission, we focus our attention to tuning the properties of Rydberg excitons through spatial confinement. There are two major methods for microfabrication, viz., bottom-up and top-down approaches. Under bottom-up methods, Cu2O crystals have been grown by melting24, electrodeposition25, sputtering26,27, thermal oxidation28, and e-beam evaporation29. Despite the tremendous progress, synthetic Cu2O has been found to demonstrate a high density of defects30 and impurities that reduce the visibility of a high quantum number Rydberg excitons8. In our top-down approach, we start with a high-quality natural Cu2O crystal, thin and polish it down, and etch out microstructures using the FIB technique. Other methods of etching semiconductors include: metal-assisted chemical etching31 and focused electron beam induced etching32,33.

In this work, we fabricate square micropillars that are 70 microns deep with different lateral cross-sectional areas using FIB etching technique. This allows us to investigate the properties of the yellow 1s ortho-excitons and the yellow np excitons through photoluminescence (PL) studies. While PL of excitons in Cu2O has been widely investigated, properties of bulk Cu2O can be significantly different from that of nanostructures. Other studies on Cu2O nano- and microparticles only investigate the average properties of an ensemble29,34. Here, for the first time, we demonstrate a size-dependent tuning of exciton properties within one microstructure at a time.

Results and discussions

Fabrication of micropillars

FIB etching has been used to micromachine nanostructures on metallic contacts and silicon in the manufacturing of semiconductor chips, as well as the preparation of samples for transmission electron microscopy. Gallium ions are the standard source material for this technique due to their easy melting and focusing using a high-voltage electrostatically charged tip. Gallium atoms are also heavy and enable faster sputtering rates of the target materials using elastic and inelastic collisions.

Here, we use this technique to etch pillars on a bulk crystal of Cu2O (see “Methods” for details on crystal preparation). With incident gallium ions accelerated through 30 kV, the etching rate is about 200 μm3 per minute at 3 nA current. In order to have a 2.5 μm separation around a 5 μm × 5 μm pillar, a lateral area of 75 μm2 would need to be etched. Setting the depth to be 70 μm, it takes about half an hour to etch out this required region. While attempting pillars of smaller lateral area, it was found that as the ion beam etches deep into the sample, the conical shape of the beam also results in etching away the top surface of the pillar resulting in a tapered pillar. While the size of the cone can be reduced by etching at a lower ion current of 300 pA, this would require an etching time of 5 h for one nanopillar. Therefore, a 3 nA was chosen as the optimum current to etch a sufficient number of pillars.

Four different pillars are etched by leaving different exclusion zones at the centre of a 10 μm × 10 μm × 70 μm volume. The exclusion zones for pillars A, B, C, and D are squares of 5 μm × 5 μm, 4 μm × 4 μm, 3 μm × 3 μm, and 2 μm × 2 μm, respectively. Etching of the region around the exclusion zone is performed in steps of four of rectangular cross-sectional regions, one for each side of the pillar. Each cross-section is etched using a raster multi-pass scan such that the etching would end at the edge closest to the pillar during each scan (see Fig. 1a). This ensures that any material redeposited close to the pillars during the sputtering process will be removed at the end of each pass. As can be seen in the electron microscopy images (Fig. 1b), while the dimension at the bottom of the pillars is close to the designed ones, the pillars taper at the top with pillar D resulting in a very small tip of 1.6 × 1.0 μm2 area. In the remaining part of this work, we investigate the PL from these four pillars at 4 K temperature.

Fig. 1: Fabrication of the pillars and their PL spectra at low laser power.
figure 1

a Steps taken in FIB fabrication of pillars to minimise redeposition (shown in white) on pillars. b SEM images of the four micropillars constructed using FIB technique. The actual dimensions of the pillars are: A: 4.4 × 4.0 μm2, B: 3.5 × 4.3 μm2, C: 2.7 × 2.0 μm2, and D: 1.6 × 1.0 μm2, (c) PL spectra of the sample for pillar B (green) and a bulk position on the crystal (red) showing Rydberg excitons up to n = 6 at low optical excitation power of 1.68 mW and the cryostat temperature T = 4 K.

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PL at low powers

Here, we study the lowest energy excitons in Cu2O (the so-called yellow series). They originate from the \({\Gamma }_{7}^{+}\to {\Gamma }_{6}^{+}\) transition35 with a bandgap energy of 2.17 eV. The brightest transition is the quadrupole 1s ortho-exciton transition at 2.03 eV and a broad phonon replica at 2.02 eV. The p-exciton Rydberg series starts with the principal quantum number n = 2 at 2.14 eV. PL for np excitons has been observed up to n = 10 previously36. Here, we observe PL (see Fig. 1c) from 1s and np excitons up to n = 6 both in the bulk crystal region and in the micropillars (see “Methods” for more details). This observation on its own is important because it shows that, at least for PL, the bulk of the crystal has stayed intact during the etching process.

Laser power dependence

As the laser power increased, we observed two distinct trends (see Fig. 2). Firstly, both np Rydberg and 1s ortho-excitons exhibit a clear redshift of PL energy with an increase in incident laser power for all pillars. The redshift becomes more dramatic as the dimensions of the pillars become smaller. This energy shift with power for the smallest pillar is ~ 14 times larger than the bulk. This redshift is due to the temperature increase of the crystal by the high absorption of laser power resulting in shifting of the energy bandgap37. Another feature is the splitting in the PL spectra at higher powers, strikingly similar to what has been reported due to mechanical stress38. While one may attribute the splitting to local strain from the large temperature gradient along the long axis of the pillar, we investigate this source using simulations that estimate the temperature gradient and the resultant strain.

Fig. 2: PL spectra of all the pillars and off-pillar data for np excitons and 1s excitons with increasing laser power and decreasing dimensions of the pillars.
figure 2

Plots (b) to (e) show the trend in the np excitons as the size of the pillars decreases, while plots (g) to (j) show the trend for 1s ortho-excitons. In both cases, the ‘splitting’ becomes more prominent with the decreasing size of the pillars, and starts to appear with lower laser powers as well. Red/blue areas mark estimated positions and line widths of PL peaks originating from hot/cold parts of the sample. In the case of the off-pillar measurements as seen in plots (a) and (f), no splitting is visible since all the power is absorbed in the hot region. All measurements were performed at the cryostat temperature T = 4 K.

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Estimation of crystal temperature

We make some preliminary estimations on the expected pillar temperature and compare it with the absorbed laser power. For pillar A, we have the following characteristics: (i) volume: 4.3 × 4.8 × 70 μm3, (ii) cross-section/top area: 20.6 μm2 (note that pillar is slightly tapered towards the top, which we neglect here since we don’t have a good measurement of the bottom cross-section), (iii) side area: 640 μm2 (again, due to the fact that it is getting wider towards the bottom, this value is underestimated), (iv) top temperature: T = 100 K at 8 mW (approximate value derived from the observed band gap shift).

For Cu2O, the relative permittivity is εr = 7.5, so the refractive index is n = 2.74, and transmission at normal incidence is \(T=1-{(\frac{n-1}{n+1})}^{2}\simeq 0.78\). For an 8 mW incident beam, the absorbed power is P ≈ 6.3 mW. On including the total transmission coefficient of the microscope objective and cryostat window of 0.7, and the curved surface of the pillar top increasing the reflection coefficient, we estimate the actual absorbed power is of the order of ~1 mW. We now consider the two primary ways of dissipating the absorbed energy.

Thermal radiation

Using the Stefan-Boltzmann law the energy flux is j = εσT 4 = 2.835 Wm−2, where σ = 5.670 × 10−8 Wm−2K−4 is the Stefan-Boltzmann constant, and we obtained an emissivity ε ~ 0.5. Overall, 0 < ε < 1, and as we are only interested in estimating the order of magnitude, the thermal radiation is negligible even for ε = 1. From the entire pillar surface area, we obtain a radiated power of PR = 1.86 × 10−6 mW. This is negligible compared to the power absorbed by the laser.

Thermal conduction

As an approximation, we take the pillar to be of a constant cross-section through its length and that the bulk material in contact with its base has nearly the same temperature as the thermostat, e.g. T < 10 K. Across the pillar, there is a temperature gradient of ΔT/L = 1.4 × 106 Km−1. Using the thermal conductivity of Cu2O as approximately the same as CuO39K = 30 Wm−1K−1 the heat flux is j = KΔT/L = 42.8 × 106 Wm−2. For a cross-sectional area of S = 20.6 × 10−12 m2 we obtain a power of P ≃ 0.9 mW. This is close to the absorbed power of 1 mW. Our assumption of the temperature gradient to be parallel to the axis of the pillar is justified by the smaller diameter of the pillar compared to its length, and a laser spot size (3 μm) which covers most of the pillar top surface. Therefore, we can expect a linear gradient of the temperature along the pillar length. To verify this, we have performed finite element analysis of the thermal conduction (see Fig. 3) and find that the variance of temperature across the top face is within 10 K and the heating at the base of the pillar is negligible.

Fig. 3: Temperature and stress distribution simulation for pillar A.
figure 3

a FEM analysis results for Pillar A, assuming the top temperature of 100 K. b Stress distribution of the pillar due to thermal expansion. c FDTD simulation result of beam propagation; time-averaged energy density ρ = ϵE2 is calculated.

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On strain due to temperature gradient

To obtain temperature distribution, we use thermal expansion and Young’s modulus of Cu2O, E = 100 GPa40 to estimate stress in the pillars. Using the finite element modelling, we obtain the stress distribution shown in Fig. 3b.

As expected, the highest stress occurs at the corners of the base, which are stress concentration points. The peak value obtained in simulation is about 530 bar (53 MPa), although it should be mentioned that this value depends on the radius of curvature of the base which is not known. The top face is mostly stress-free, with a maximum value of ~ 30 bar. As shown in previous results38, a pressure of 2000 bar would be needed to be applied in order to observe a 10 meV shift. With only 30 bar temperature-related stress at the top of the pillar, the shift in the spectra would be on the order of μeV. This value is smaller than the resolution of our spectrometer. To ensure that, we can analyse the fitting error of the formula (1) to the pillar and off-pillar data; if any stress-related effects are present, they should be different in these two cases. The results show no difference (see Supplementary Fig. S1). Thus we conclude that the stress levels are not sufficient to produce a noticeable shift in exciton energy.

Origin of splitting in PL spectra

Using the above heating model we estimate the source of ‘hot’ and ‘cold’ bands in the PL spectrum (see Fig. 2). We rule out the possibility of having cold regions of significant size on the top face of the pillar that would result in a detectable PL level. The observed lines originating from cold regions could be emitted either from pillar sides or its surrounding implying that a significant fraction of illuminating laser power is not absorbed by the top face. To investigate this issue, we use the FDTD method to simulate wave propagation in the system. Since we are only interested in some general characteristics of the system, a two-dimensional simulation is used where a cross-section of the geometry is investigated. Since the pillar size (3 μm) is approximately 5 times larger than the wavelength, the assumption that the system is much larger than the wavelength in the direction perpendicular to the cross-section plane is reasonable.

The FDTD simulation shows several interesting features in the calculated energy density distribution (see Fig. 3c). The vertical beam is reflected from the pillar top, which can be seen as two small “wings” on the sides of the beam. Their cutoff near y = 75 μm is a result of absorbing layers placed in the simulation to avoid stray reflections. Due to the limited simulation space, the radiation source is placed close to the pillar top at y = 80 μm; reflections between the source and pillar top result in a standing wave pattern across the beam. A key result is the diffraction of the incident beam which results in two side lobes propagating at an angle of approximately 30 degrees to the pillar and hitting the trench walls. Since this power density is approximately 10-20 times smaller than in the beam centre at the top face, they do not result in any appreciable heating. However, these side beams illuminating side walls and the bottom comprise a significantly larger area than the pillar top. This results in a non-negligible PL signal from these cold regions. The exact shape of cold emission regions is not well known, as it depends on the precise angle of the trench and pillar walls, which could be only roughly estimated from SEM images. Another interesting feature is the fact that the rounded pillar top seems to act as a lens, focusing the beam to a spot a few μm under the surface. This would suggest that with sufficient sensitivity, one could see a heavily red-shifted PL band originating from the small but hot spot at the focus.

Exciton energy tuning with laser power

To further study the effect of temperature and to validate our theoretical model of band gap energy shift, we have performed a temperature sweep scan where a series of PL spectra have been measured at constant laser power while varying the cryostat temperature. For the theoretical Eg(T) dependence, we use the so-called Varshni formula41:

$${E}_{g}={E}_{{g}_{0}}-\frac{\alpha {T}^{2}}{\beta +T},$$
(1)

where α, β are fitting parameters of a particular material. Based on data by Snoke et al.42, we obtain values for α = 4.8 × 10−4 eVK−1 and β = 275 K. By fitting the above model to the recent experimental data by Malerba et al. values for43, we obtain α = 6.125 × 10−4 eVK−1 and β = 375.57 K. Despite apparent large disagreement between values, the difference between models is only 0.08 meV at T = 100 K and scales approximately as T2. A laser power P = 1 mW results in approximately 15 K of heating from the cryostat temperature according to the observed energy shift. With this correction, we get an excellent agreement with theoretical relation (see Supplementary Fig. S1).

Another temperature-dependent parameter is spectral linewidth. We use the theoretical relation44

$$\Gamma (n)={\Gamma }_{0}(n)+{\gamma }_{{{{{{{{\rm{AC}}}}}}}}}T+{\gamma }_{{{{{{{{\rm{LO}}}}}}}}}{\left({e}^{{\omega }_{{{{{{{{\rm{LO}}}}}}}}}/kT}-1\right)}^{-1},$$
(2)

with constants γAC, γLO representing acoustic phonons and LO phonons. The value Γ0(n) is the linewidth of the excitonic state with principal number n, in the limit of low temperature, taken from fitting to the experimental data7. The comparison of measured line widths with the theoretical prediction is shown in Supplementary Fig. S2.

There is an excellent match with the data from Kang et al.37. In our results, there is a constant broadening of about 0.3 meV. This broadening could be due to the lower quality of our sample as well as the overlap of multiple lines corresponding to confinement states45.

Using the relation between the bandgap shift with crystal temperature, we can estimate the hottest temperature of the pillars for a given laser power. The power dependence of the energy shift of 2p excitons in pillars shows a quadratic dependence with laser power (see Fig. 4a). Since the bandgap redshift has a quadratic relation with temperature, it follows that the temperature of the pillar varies linearly with incident laser power (see Fig. 4b). The effective temperature of the exciton can be determined by fitting the 1s ortho-exciton phonon replica (see Supplementary Fig. S3). The resulting effective temperature trend is consistent with the temperatures extracted using the Varshni formula, except for a noticeable offset. This discrepancy is attributed to the sample’s elevated temperature relative to the coldfinger (see Supplementary Fig. S4). The PL intensity per unit area of the pillar tips for 2p exciton for all pillars follows a super-linear fit aP1.5, with the excitation laser power P (Fig. 5). The fall-offs from the trend are consistent with the thermal ionization of the excitons, which leads to the energy being redshifted, as seen in Fig. 4a. The effect is more discernible in smaller pillars, as they heat more with the same laser power. The trend is similar for 3p excitons, but with an earlier fall-off from the super-linear trend as they are closer to the bandgap and their binding energy is smaller.

Fig. 4: Energy shifts of 2p excitons and their corresponding temperature estimates.
figure 4

a Energy shift of 2p excitons for all the pillars and a bulk position on the sample. They follow a quadratic fit ΔE = aP2 with power P, where a = [0.1, 0.25, 0.2, 1.15, 1.35] eV K−1 for the bulk, pillar A, B, C, D respectively. b The temperature estimates are plotted based on the energy shift. They follow a linear fit T = bP with power, where b = [9, 13, 11.5, 29, 32] K mW−1 for the bulk, pillar A, B, C, D.

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Fig. 5: Laser power dependence of pillars’ PL intensity.
figure 5

a Power dependence of the PL intensity per unit area of the pillar tips for 2p exciton (a) and 3p excitons (b) follows a superlinear fit followed by a sudden fall off.

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The capability to locally tune the energies of the excitonic states by changing the bandgap energy using optical means is a simple yet powerful tool for lattices made from these pillars21. This simple and versatile technique has significant advantages over the previously existing methods like external electric field46, magnetic field47, or strain38. Local tuning of energy is possible with electric fields and strain. Achieving strain in a micron scale is challenging and tunability with electric fields is too small. Both methods inhibit higher-lying excitons due to the broadening of transitions and ionization10. None of these three methods have shown local tuning of energies to the order of 10 meV as demonstrated here.

Conclusion

In conclusion, we have successfully fabricated microstructures from the bulk Cu2O, whilst preserving their excitonic properties. The micropillars show PL for 1s ortho-excitons and np Rydberg excitons up to n = 6. We can locally tune the energy of the excitons in the pillar by changing the power of the laser. The non-resonant absorption of the laser power induces a local temperature change, which locally shifts the band gap of the crystal. The microstructures demonstrated here could be used for studying single-photon optical nonlinearities 48,49. The local tuning of exciton energies is a useful tool to circumvent disorder for the realisation of strongly correlated lattices of Rydberg polaritons50.

Methods

Crystal preparation

We first start with a naturally mined crystal of Cu2O. We cut the crystal along one of the crystallographic planes, and manually thin down and polish both surfaces using sheets of different grit sizes. The thinned sample (~70 μm thick) is then mounted on a CaF2 substrate using a UV-cured epoxy owing to its excellent cryogenic thermal conductivity.

FIB fabrication

The micropillars were etched using an FEI Scios Dual-beam system. In order to etch microstructures using FIB, it is necessary to electrically ground the surface of the material in order to discharge the implanted beam of ions. As neither Cu2O nor CaF2 substrate is a good electrical conductor, a thin line of conductive silver paste is traced from the sample surface to the aluminium chuck at the bottom. The sample is evacuated to low pressures to avoid the formation of a charged atmosphere that could deflect incoming ions. The instrument consists of a vertical electron beam for microscopy and an ion beam at 52 degrees for etching. The sample is elevated to the eucentric point of the dual column beams and then tilted to face the ion beams head-on. A voltage of 30 kV and a low current of 30 pA is chosen to optimise the focus and astigmatism of the ion beams while incident on the sample surface at a high magnification of 5000×. It is apparent that even a few seconds of exposing the sample during the alignment to the ion beam is enough to start etching of the sample surface leading to roughening of the entire sample area in view. Therefore, a new surface is chosen for etching our microstructures after the alignment of the electron and ion beams. Since Cu2O is softer than silicon, the rate of etching a volume is subsequently about 7 times higher.

PL measurements

The sample is mounted in a continuous-flow cryostat and is cooled down to 4 K. We perform PL spectroscopy on individual micropillars using a green continuous wave laser of 520 nm wavelength (Thorlabs PL203) in conjunction with a 550 nm shortpass filter. The laser beam focuses onto a 2 μm diameter spot after passing through a Mitutoyo 20x objective lens (NA = 0.42). Filtered emission from the micropillar(s) using a 550 nm longpass filter is collected through the same objective, distributed, and analyzed using a spectrometer (Andor Shamrock 750) coupled to an EMCCD camera (Andor Newton). A CMOS camera (Thorlabs Quantalux) provides a live image of the sample to align the optics and create excitation directly on the micropillars.

FDTD simulation

The performed simulation is based on a standard Yee algorithm51, where a set of field evolution equations is derived from Maxwell’s equations and used to update the electric and magnetic field values within some defined computation domain. The domain is divided by a rectangular grid with a single cell size Δx = 50 nm, which is a compromise between calculation accuracy and memory demand. For the same reason, a two-dimensional representation of the system with TM field configuration is chosen; the electric field has two components in the plane of the propagation E = [Ex, 0, Ez], and the magnetic field has a single component perpendicular to the xz plane H = [0, Hy, 0]. Such a representation is valid as long as the size of the system in z axis is much larger than wavelength. In our case, the pillar top size is on the order of 3 μm, as compared to λ ~ 570 nm. This means that the conditions for accurate simulation are approximately met; we stress that the goal of the simulation is only to get some qualitative insight into EM field distribution.

The full set of equations solved is as follows

$$\frac{\partial {H}_{y}}{\partial t}=\frac{1}{{\mu }_{0}}\left(\frac{\partial {E}_{x}}{\partial z}-\frac{\partial {E}_{z}}{\partial x}\right),$$
(3)
$$\frac{\partial {E}_{x}}{\partial t}=\frac{1}{{\epsilon }_{0}\epsilon }\left(\frac{\partial {H}_{y}}{\partial z}+{j}_{x}\right),$$
(4)
$$\frac{\partial {E}_{z}}{\partial t}=\frac{1}{{\epsilon }_{0}\epsilon }\left(\frac{\partial {E}_{x}}{\partial z}+{j}_{z}\right),$$
(5)

where μ0, ϵ0 are the vacuum permittivity and permeability, and jx,jz are the components of current density used to introduce a radiation source to the system. We use ϵ = 1 for vacuum and ϵ = 7.5 for Cu2O.

The equations above are rearranged to obtain evolution equations for Ex, Ez, Hy fields, allowing one to calculate the next field value based on the current one, with some fixed time step Δt.