Atom-field dynamics in curved spacetime

Abstract

Some aspects of atom-field interactions in curved spacetime are reviewed. Of great interest are quantum radiative and entanglement processes arising out of Rindler and black hole spacetimes, which involve the role of Hawking−Unruh and dynamical Casimir effects. Most of the discussion surrounds the radiative part of interactions. For this, we specifically reassess the conventional understandings of atomic radiative transitions and energy level shifts in curved spacetime. We also briefly outline the status quo of entanglement dynamics study in curved spacetime, and highlight literature related to some novel insights, like entanglement harvesting. On one hand, the study of the role played by spacetime curvature in quantum radiative and informational phenomena has implications for fundamental physics, notably the gravity-quantum interface. In particular, one examines the viability of the Equivalence Principle, which is at the heart of Einstein’s general theory of relativity. On the other hand, it can be instructive for manipulating quantum information and light propagation in arbitrary geometries. Some issues related to nonthermal effects of acceleration are also discussed.

Graphical abstract

Keywords

atom-field interactions / general relativity / Minkowski and curved spacetime / quantum field theory in curved spacetime / light−matter interactions / spontaneous excitations

1 Introduction

Ferroelectric materials are a type of functional material that, within a certain temperature range, exhibits spontaneous electric polarization even in the absence of an external electric field [1, 2]. Spontaneous polarization is the fundamental building block for a wide range of technological applications, such as ferroelectric field-effect transistors, non-volatile memory devices, and active elements in electro-mechanical and electro-optical systems. In response to the introduction of an external electric field that breaks the degeneracy, it can be repeatedly switched between two or more energetically equivalent states. The ability to alter electric polarization is a critical component of contemporary nanotechnology. Nowadays, the study of ferroelectric properties at the nanoscale has emerged as a hot topic in the research field of novel functional materials [3, 4], in line with the rapid development of micro-nano technology and the tendency toward miniaturization, integration, and multi-functionalization of electronic components. One of the objectives being pursued by researchers is to figure out how to incorporate ferroelectricity in 2D materials, and ultimately to fabricate multi-state and multi-functional nanoelectronic devices. Through theoretical and experimental investigations, scientists have made significant advancements in recent years. In these experimentally demonstrated 2D ferroelectric materials, such as CuInP₂S₆ [5-7] and α-In₂Se₃ [8-10], the spontaneous polarization generates as a result of the relative displacement of atoms, which is strictly constrained by lattice symmetry and structural stability. Until now, only a few 2D layered materials have had their intrinsic ferroelectricity experimentally verified.

To expand the family of 2D ferroelectrics, researchers have proposed that interlayer sliding can generate ferroelectricity in bilayer or few-layer van der Waals materials via the utilization of the unique layer degrees of freedom. A large number of 2D layered materials can exhibit the special feature of “sliding ferroelectricity” [11, 12], even though their bulk-phase structure is not polarized. Experimental evidence for interlayer sliding-induced 2D ferroelectricity has been reported in metallic WTe₂ [13-15], AB-stacked hBN bilayers [16-19], InSe [20, 21], transition metal dichalcogenides (TMDs) bilayers [22, 23], and an amphidynamic single crystal (15-crown-5)Cd₃Cl₆ [24]. These ground-breaking scientific developments have made it possible to design and manufacture 2D ferroelectrics from non-polar materials, and have also given rise to a brand-new area of research called slidetronics. 2D semiconducting materials with modest bandgaps are preferred for further slidetronics research. The materials are more promising due to their semiconducting characteristics such as efficient gate tunability and capacity in direct integration into optoelectronics. In contrast, it is challenging to implement these advantages in a metallic or insulating system. Itinerant electrons, for instance, may shield the spontaneous vertical polarization of multilayer WTe₂ due to its metallic characteristics, reducing its significance in practical applications. In this sense, the search for sliding ferroelectricity in semiconductors not only benefits the investigation of performance modulation strategies but also facilitates potential technological applications.

Here, we report the experimental evidence of robust vertical ferroelectricity in ReSe₂, a semiconducting counterpart of isostructural semimetal TMDs such as MoTe₂ and WTe₂, based on a combined study of micro-zone scanning probe microscopy techniques and non-linear optical effects. In contrast to 1L ReSe₂, which shows no discernible ferroelectric signal, 2L ReSe₂ exhibits vertical ferroelectricity in an ambient environment. Our density-functional theory (DFT) calculations further demonstrate that the out-of-plane polarization of ReSe₂ may be switched through an interlayer shear motion, in stark contrast to the conventional ferroelectric switching mechanism mediated by ion displacement. The semiconducting characteristics and low switching barrier of ReSe₂ make it an alluring candidate for use in functional nanoelectronics.

2 Experimental section

Sample preparation. Using the elements in the proper stoichiometric ratios and I₂ as the transport agent, high-quality single crystals of ReSe₂ were prepared through a chemical transport reaction. Solid-state precursors were packed together in a quartz ampule (10 mm outer diameter, 200 mm length). The ampule was evacuated to a pressure of around 10⁻⁴ torr, sealed, and then placed in a two-zone furnace with the source and growth zones set at 1060 and 1000°C, respectively.

Property characterization. TEM (Tecnai F30, FEI), XRD (X’ Pert Pro, PANalytical B.V.), and XPS (PHI Quantera II, Ulvac-Phi) were used to analyze the elemental composition and crystalline structure of ReSe₂. Optical microscopy (Axio Imager A2m, Zeiss) and AFM (Asylum Research, Cypher S) were used to determine the sample thickness. Based on a commercial confocal optical system (Alfa300R, WITec) equipped with 2.33 eV (532 nm) and 1.96 eV (633 nm) lasers and dispersed by a spectrometer supplied with a diffraction grating of 1800 grooves/mm, Raman spectra of ReSe₂ were acquired. The excitation beam was focused onto the sample surface, and the scattering lights were concurrently collected using a 100× objective lens.

PFM measurement. The AFM system with the PFM module was used to probe piezoresponse signals from ReSe₂. The rectangular-shaped ASYELEC.01-R2 conductive tips utilized in our PFM experiments have an Ir/Ti (20/5 nm) coating, a spring of 2.8 N/m, and a basis resonant frequency of about 90 kHz. By delivering an AC signal of 200−800 mV beneath the tip-sample contact third-order resonant frequency (270 kHz) and superimposing it on a series of DC triangle saw-tooth waveform voltages (0.1−0.2 Hz), the local switching spectroscopic hysteresis loops were monitored in the resonance-enhanced PFM mode.

SHG measurement. To produce ultrafast light of femtosecond magnitude, a Ti-sapphire lock-in laser is employed as the excitation source with optical pulses of about 800 nm and a repetition frequency of 76 MHz. An optical parameter oscillator (OPO) may then generate lasers with tunable wavelengths between 500 and 1600 nm, which can subsequently be converted to polarized light using a polarizer and a half-wave plate. The light is focused onto the sample by the objective lenses (100 ×, N.A. = 0.95), with a spot size of around 2.0 μm. The same lenses are then used to gather the SHG signals produced by the non-centrosymmetric sample, and a dichroic beam splitter is used to separate them. The excitation components are filtered by placing a short-wave pass filter in front of the spectrometer.

3 Calculational section

Our first-principles calculations based on the DFT were conducted by VASP, where the PAW method and the PBE exchange-correlation potential function were performed. The optimization of the geometry structure was employed by the conjugated gradient minimization scheme with full-relaxed atomic positions and crystal lattice constants. The convergence criteria for ionic and electronic relaxations were set to be 0.001 eV/Å and 10⁻⁵ eV/atom, respectively. And, the vacuum space was set to be above 20 Å, large enough to prevent interlayer interactions. In the 2D Brillouin integration, a 10 × 10 × 1 Monkhorst-Pack grid was involved. The weak long-range vdWs interactions were semi-empirically described by the DFT-D3 method. The electric polarization was calculated by the evaluation of the Berry phase expressions of polarization in modern theory.

4 Results and discussion

Similar to other TMDs, ReSe₂ crystals are layered materials composed of covalently bound monolayers stacked vertically by van der Waals interactions. Three atomic planes (Se−Re−Se) make up each ReSe₂ monolayer. Nevertheless, ReSe₂ does not crystallize in the normal 2H or 1T phases found in other TMDs materials such as MoS₂ and MoTe₂. Instead, it does so in a unique deformed triclinic 1T′ structure belonging to the low symmetry space group of $P_{\overline{1}}$. As schematically depicted in the crystal structure [Fig.1(a)], the formation of the in-plane Re₄ “diamond” chains is considered a distortion of Re atoms from their ideal octahedral sites. The unique behavior of Re atoms caused by the $5d^3$ electronic configuration results in a weaker interlayer coupling than other TMDs and a striking in-plane anisotropy.

Fig.1 (a) Top and side views of 1T′-ReSe2, where the “diamond” chains of Re4 clusters have been highlighted in red. (b) XRD pattern of the prepared ReSe2 sample and the standard card of triclinic ReSe2 (JCPDS No. 50-0537). Inset: The ReSe2 bulk-form crystals prepared by the CVT method in the sealed ampoule. (c) High-resolution XPS spectra collected from the prepared ReSe2 sample.

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In our experiments, 1T′ -phase ReSe₂ bulk crystals were prepared in large quantities using a chemical vapor transport (CVT) method, as seen in the inset of Fig.1(b) and Fig. S1 of the Electronic Supplemental Material (ESM). X-ray diffraction (XRD) technique was employed to examine the crystal structures of the bulk samples of CVT method-prepared ReSe₂ samples. Fig.1(b) indicates that the reflection peaks at 13.9°, 28.0°, 42.5°, and 57.7°, which correspond to the lattice planes of (001), (002), (003), and (004) for ReSe₂, revealing the nature of the triclinic 1T′ phase (JCPDS Nos. 50-0537, 82-1379 and 89-0340) [25, 26]. The preferred orientation effect existent in the growth process of the layered ReSe₂ crystal is indicated by the (001) peak’s strongest intensity. The higher crystallization quality is indicated by the modest full width at half maximum for the (001) plane of around 0.099° (Fig. S2 of the ESM). Moreover, X-ray photoelectron spectroscopy (XPS) was used to examine the elementary composition and bonding configuration of the prepared ReSe₂. The high-resolution XPS spectrum for Re 4f states as plotted in the left panel of Fig.1(c) was used to study the bonding states of Re. Two characteristic peaks can be seen at energies of 40.3 eV and 39.3 eV, respectively, which correspond to the core $4f^{5/2}$ and $4f^{7/2}$ level peaks of Re⁴⁺. Two distinguishing peaks for the bonding states of Se are found at 53.3 eV and 52.3 eV, respectively, and correspond to the core $3d^{3/2}$ and $3d^{5/2}$ level peaks of Se²⁻, as depicted in the right panel of Fig.1(c). The reported statistics for ReSe₂ [27, 28] are consistent with our measured XPS results. Moreover, the stoichiometric value of ideal ReSe₂ is 1:2, which matches the Re and Se ratio obtained by the XPS results.

To examine the crystal structure of the CVT method-prepared samples, we thinned ReSe₂ by mechanical exfoliation [Fig.2(a)] and transferred it onto a carbon membrane-coated copper microgrid [Fig.2(b)]. As already mentioned, in contrast to the 1T phase, the 1T′ phase displays covalent bonding between metal atoms (here Re). Re atoms that are covalently connected to one another arrange to form chains that resemble diamonds. More often than not, ReSe₂ crystals cleave along these chains. As seen in the optical microscope observation results in Fig.2(a), the Re chains are currently discovered to correlate with one of the cleaved edges of the few-layer ReSe₂ films, providing a first clue of the crystal orientation. Fig.2(c) presents the captured atomically resolved, high-resolution TEM image of the few-layered ReSe₂ flakes. The formation of the one-dimensional rhenium “diamond” chains was well suited to the 1T′ structure. It was discovered that the orientations of a[100] and b[010] were 120° apart. Additionally, we captured selected area electron diffraction (SAED) patterns from the ReSe₂ flakes, along with the well-ordered diffraction spots revealing that the prepared ReSe₂ is single-crystalline [Fig.2(d)]. We subsequently investigated the structural anisotropy of the prepared ReSe₂ flakes using angle-resolved polarized Raman spectroscopy, a powerful technique for empirically determining the crystal orientations of anisotropic materials. In order to gather the polarization-dependent Raman spectra on ReSe₂ thin flakes, we used 532 nm and 633 nm linearly polarized lasers. The measured ReSe₂ sample region has been marked by a red cross-shaped star shown in Fig.2(a). Fig.2(e) and (f) display the evolution of Raman spectra obtained by the excitation wavelengths of 532 nm and 633 nm, respectively, for ReSe₂, recorded as a function of the angle between the linearly polarized excitation laser and the Re chains in the parallel configuration. The unpolarized Raman spectra of ReSe₂ and the polar coordinate representation for polarization-dependent Raman spectra are provided in Fig. S3 of the ESM. The Raman characteristic peaks of ReSe₂ display a clear variation of intensity with the angle for all modes, which are clear indications of in-plane anisotropy [29].

Fig.2 (a) Optical microscopy image of a mechanically exfoliated ReSe2 supported atop SiO2/Si wafer. (b, c) TEM images of ReSe2. (d) SAED patterns of ReSe2. (e, f) A series of Raman spectra obtained from a ReSe2 sample, recorded as a function of the angle between the linearly polarized excitation and the Re chains at the excitation wavelength of 532 nm (e) and 633 nm (f) under the parallel configuration. The raw spectra are collected from a few-layer ReSe2 region as marked by a red cross-shaped star in (a) and vertically offset for clarity.

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Atomic force microscopy (AFM) presents a workable approach to directly study the surface morphology and to precisely estimate the sample terrace height. In addition, optical contrast provides a more useful, non-destructive method to calculate the number of layers. Both two approaches provide strong support for further characterization of 1L, 2L, and few-layer ReSe₂ samples. The ultrathin ReSe₂ samples were transferred onto a flexible polydimethylsiloxane (PDMS) substrate using mechanical exfoliation. We initially identified the ultrathin regions of the target ReSe₂ sample with an optical microscope and captured an image of it [Fig.3(a)]. The normalized optical contrast (green channel gray value) along the red dotted line depicted in the green channel (G) image was then plotted using the ImageJ software after the G image [inset, Fig.3(b)] was acquired. These variables make it simple to calculate the layer thickness and enable us to look up ultrathin ReSe₂ sample region on a transparent substrate. According to the data, the optical contrasts at the highlighted sites, which correspond to 1L, 2L, and 4L ReSe₂, are 5.1%, 12.5%, and 33.3%, respectively. Previous reports on sister materials of ReSe₂, including grapheme [30], MoS₂ [31], and ReS₂ [32], have all demonstrated the accuracy of this technique for thickness identification. The method can be easily implemented using only an optical microscope.

Fig.3 (a) Optical microscopy image of the ultrathin ReSe2 samples with 1L and 2L regions on PDMS. (b) Contrast distribution and gray value of the red dotted line shown in the inset of (b). Inset: The green channel image of (a) obtained by the ImageJ software.

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A ferroelectric material, by definition, should possess switchable spontaneous polarization. It is feasible to demonstrate that ultrathin samples contain switchable ferroelectric polarization using piezoresponse force microscopy (PFM) [5, 8, 15, 23]. PFM tests need to be performed on conductive substrates. However, the optical contrast of ReSe₂ atop conducting substrates is relatively poor. Therefore, we would like to obtain ReSe₂ fewlayers by mechanical exfoliation from the bulk crystal and transfer them onto PDMS. ReSe₂ samples will exhibit thickness-dependent intensities of reflected light on transparent PDMS substrate. We can then conveniently look up ultrathin ReSe₂ samples under an optical microscope. By optical contrast analysis, we can find a relatively thin sample region, mark the location, and then transfer the targeted sample region from PDMS onto the conducting substrate via a three-axis micromanipulator, which saves time compared with the search for ReSe₂ samples on conducting substrates with poor visibility. The conducting substrates used our experiments for the thinned ReSe₂ nanoflakes with 1L (0.76 nm, region I) and 2L (1.70 nm, region II) areas was the SiO₂/Si wafer covered in a Pt (90 nm)/Ni (10 nm) coating [Fig.4(a) and (b)].

Fig.4 (a) Optical microscopy image of a ReSe2 flake, transferred onto Pt/Ni/SiO2/Si substrate. (b) AFM image for the region denoted by a white dashed square in (a). (c, d) Butterfly-shaped amplitude-voltage loops (c) and phase-voltage hysteretic curves (d) for 1L (region I) and 2L (region II) ReSe2, obtained by the switching spectroscopic PFM technique. (e, f) AFM images of the areas before (e) and after (f) ferroelectric switching. No obvious morphology damage occurred. (g) PFM amplitude image for a 2L ReSe2 with a written box-in-box pattern obtained under reverse DC voltages. (h) PFM phase image corresponding to (g).

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To demonstrate whether the ferroelectric polarization exists in ReSe₂, spectroscopic PFM studies were conducted. The piezoresponse signal from 2L ReSe₂ exhibits switchable hysteretic behavior as a function of applied bias, similar to conventional ferroelectric materials [bottom panels, Fig.4(c) and (d)]. The PFM amplitude signal displays a distinctive “butterfly” shape, and the PFM phase signal exhibits a characteristic phase inversion (i.e., a phase value change of around 180°) at the amplitude response minima. Nevertheless, 1L ReSe₂ displays very little phase or amplitude contrast, as evidenced in the spectroscopic PFM loops randomly recorded from the monolayer region of this material [top panels, Fig.4(c) and (d)], proving the lack of switchable polarization. We used switching bias pulses with the following capture of PFM pictures to record the residual state in order to directly observe these switchable polarization states. The residual polarization that has been bias-driven and oppositely oriented is apparent clearly in the PFM images of different color contrasts [Fig.4(g) and (h)]. The initially nearly uniform contrasts of both the PFM amplitude and phase images are inverted by a negative DC bias and result in opposite phase contrast. By inverting the polarity of the applied writing bias, this bias-induced crossover between anti-parallel equivalent polarization states of the ReSe₂ may be accomplished irreversibly. These tests demonstrate the switchability of polarization in 2L ReSe₂ in the presence of an external bias. The surface topography of the ReSe₂ nanoflake has not changed after domain writing, according to the AFM images.

Second harmonic generation (SHG) is a non-linear optical method that we utilized to examine the structural changes brought on by the transition from the ferroelectric phase to the paraelectric one. In the investigation of ferroelectric order, the SHG method acts as a sensitive probe for broken inversion symmetry [5, 23]. Using polarization-resolved SHG at room temperature with normal incidence, we investigated the structural symmetry of ReSe₂. Fig.5(a) illustrates how the obtained SHG intensity varies with excitation polarization orientation for fixed detection in the horizontal ($I_{//}$) and vertical ($I_{\mathrm{\perp }}$) directions using values from a 2L ReSe₂ nanoflake. The presence of room-temperature ferroelectricity in 2L ReSe₂ is further verified by the non-zero SHG signal. Additionally, we investigated the impact of temperature on the SHG intensity for 2L ReSe₂. For comparison, a bulk-form sample has also been evaluated. Fig.5(b) plots the evolution of SHG intensity with increasing temperature. Both the bulk-form and 2L ReSe₂ samples exhibit a similar trend: there is an obvious SHG signal below T_c, but the SHG intensity gradually decreases as the temperature increases, and finally, it nearly vanishes at higher temperatures. This strongly implies that for 2L ReSe₂, the phase transition from ferroelectric to paraelectric occurs at T_c of around 432 K, which is higher than ambient temperature (see details in Fig. S4 of the ESM). The detailed comparison of the ferroelectric properties of ReSe₂ and its sister material, ReS₂, is provided in Fig. S5 of the ESM. Semiconducting ReSe₂ with ferroelectricity offers tantalizing opportunities for practical applications due to its high transition temperature.

Fig.5 (a) Room-temperature SHG intensity obtained from 2L ReSe2 as a function of the linear excitation polarization angle, along the horizontal ($I_{//}$) and vertical ($I_{\mathrm{\perp }}$) directions in laboratory coordinates. (b) Temperature dependence of SHG intensity for 2L and bulk-form ReSe2.

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The Vienna ab initio simulation software (VASP) [33, 34], which incorporates the projected augmented wave (PAW) approach [35] and the Perdew−Burke−Ernzerhof (PBE) [36] exchange-correlation potential function, was used to perform our first-principles DFT calculations. Each unit cell in the monolayer ReSe₂ crystal structure has four formula units. The optimized lattice constant of monolayer ReSe₂ is 6.65 Å. Monolayer ReSe₂ favors a 1T′ structure with four neighboring Re atoms bonding together to generate diamond-shaped Re chains due to Peierl’s distortion. The inversion symmetry in monolayer ReSe₂ prohibits the emergence of polarization. For bilayer ReSe₂, we move the upper layer along the a and b axes by a sliding distance of (l_a, l_b), respectively, beginning from the AA stacking configuration with xy-plane mirror symmetry. The Berry-phase calculations show that spontaneous polarization occurs in the out-of-plane direction after deviation from AA stacking. The contour of the polarization value is presented in Fig.6(a). There are two structures with maximum polarization values and opposite polarizing directions, which are marked as states A and A′, respectively. According to calculations, the spontaneous polarization at zero temperature is 0.12 pC/m. The upper layer can be moved by sliding distances of (2.2 Å, 4.6 Å) and (4.4 Å, 2.0 Å), respectively, to obtain the structures of states A and A′, which further reduces the symmetry operation. The vertical polarization originates from an uncompensated charge transfer between two non-equivalent layers in states A and A′. The two structures are energy-degenerate, and state A′ could be created by mirroring state A with regard to the horizontal plane in the center. This results in a reversed vertical polarization, which proves the presence of ferroelectricity. According to the findings, vertical ferroelectricity is strongly correlated with interlayer translation and is switchable upon interlayer sliding. The energy pathway of the ferroelectric transition from state A to A′ is given in Fig.6. Only roughly 4.6 meV per unit cell serves as the energy barrier for this slippage-induced polarization shift. The semiconducting nature and low switching barrier of ReSe₂ offer enticing possibilities for functional nanoelectronics applications.

Fig.6 (a) The polarization value contour plot of 2L ReS2 as a function of sliding distance (la, lb). The states A and A′ are marked. The white line represents the path for ferroelectric transition. (b) Side view of the crystal structure of the two energy-degenerate ferroelectric states (A and A′) of 2L ReSe2. The red arrows denote the direction of electric polarization. (c) The energy pathway for ferroelectric transition.

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5 Conclusions

In conclusion, by regulating the growth parameters and temperature gradient in the CVT process, semiconducting 1T′-ReSe₂ crystals were prepared and then thinned down to an ultrathin thickness using the standard mechanical exfoliation method. Based on high-resolution TEM characterizations and polarization-dependent Raman scattering measurements, the lack of centrosymmetric and the significant anisotropy resulting from the puckered structure are confirmed. Remarkably, room-temperature robust vertical ferroelectricity was identified in 2L ReSe₂ via the experimental investigation of PFM and SHG. Combined with the DFT calculations, we hypothesize that the ferroelectric nature of ReSe₂ originates from the uncompensated interlayer charge transfer generated by interlayer sliding. Our findings open up new avenues for the quantum state regulation of 2D materials, laying the theoretical and experimental basis for the design and development of new-principle ferroelectric components.

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Declarations

The authors declare that they have no competing interests and there are no conflicts.

Acknowledgements

This research was supported by the National Natural Science Foundation of China (NSFC) (Grant No. 11974309). SMASB acknowledges financial support from China Scholarship Council at Zhejiang University.

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