Properties of spin–orbit-coupled Bose–Einstein condensates

Abstract

The experimental and theoretical research of spin–orbit-coupled ultracold atomic gases has advanced and expanded rapidly in recent years. Here, we review some of the progress that either was pioneered by our own work, has helped to lay the foundation, or has developed new and relevant techniques. After examining the experimental accessibility of all relevant spin–orbit coupling parameters, we discuss the fundamental properties and general applications of spin–orbit-coupled Bose–Einstein condensates (BECs) over a wide range of physical situations. For the harmonically trapped case, we show that the ground state phase transition is a Dicke-type process and that spin–orbit-coupled BECs provide a unique platform to simulate and study the Dicke model and Dicke phase transitions. For a homogeneous BEC, we discuss the collective excitations, which have been observed experimentally using Bragg spectroscopy. They feature a roton-like minimum, the softening of which provides a potential mechanism to understand the ground state phase transition. On the other hand, if the collective dynamics are excited by a sudden quenching of the spin–orbit coupling parameters, we show that the resulting collective dynamics can be related to the famous Zitterbewegung in the relativistic realm. Finally, we discuss the case of a BEC loaded into a periodic optical potential. Here, the spin–orbit coupling generates isolated flat bands within the lowest Bloch bands whereas the nonlinearity of the system leads to dynamical instabilities of these Bloch waves. The experimental verification of this instability illustrates the lack of Galilean invariance in the system.

Keywords

atomic Bose–Einstein condensate / spin–orbit coupling / collective excitations / optical lattice

1 1 Introduction

Two-dimensional (2D) materials have attracted great attention due to their excellent electronic and optoelectronic properties [1, 2]. So far, the most widely studied 2D semiconductor materials are graphene [3-6], black phosphorus [7-10], and transition metal dichalcogenides such as MX₂ (M = Mo, W; X = S, Se) [11-14]. The in-plane chemical bonds of 2D materials are very strong, and the interaction between layers is relatively weak, generally van der Waals interaction [15]. Different 2D materials can be stacked in a horizontal or vertical manner to form a 2D heterostructure in a certain sequence by artificial means to realize a variety of different functional devices [16, 17]. Whether it is a single 2D material or its heterostructure, its energy band structure and optical properties have significant layer number dependence [18-22].

Gallium selenide (GaSe) is a 2D layered material with a unique four-layer Se−Ga−Ga−Se structure and the thickness of a single layer of GaSe is 0.8 nm [23]. The band gap of GaSe increases as the number of layers decreases. When the number of atomic layers of the GaSe crystal is less than 7, it changes from a direct band gap semiconductor to an indirect band gap semiconductor [24]. It is worth noting that GaSe is a p-type 2D semiconductor [25], which can be combined with other n-type 2D semiconductors to form a p-n junction with unidirectional conductivity. Due to its unique electronic structure, GaSe can achieve power output in the terahertz band, and has a wide range of applications in non-linear optics, optoelectronic devices and other fields [26-28].

So far, many methods have been tried to obtain high-quality 2D GaSe, including mechanical exfoliation [29], pulsed laser deposition [30, 31], molecular beam epitaxy [32-34], electrochemical deposition [35], vapor−liquid−solid [36], and chemical vapor deposition (CVD) method [26, 37-44]. Among all these methods, the CVD method is the most effective method to achieve high-quality 2D GaSe growth. Here, we report a facile method for growing GaSe nanoflakes with different thicknesses via CVD method. The synthesized GaSe nanoflakes showed a triangular morphology with an average size of ~10 μm. Moreover, thickness-dependent optical properties of GaSe nanoflakes were investigated by Raman, photoluminescence (PL) and second harmonic generation (SHG). In addition, GaSe nanoflake-based photodetector was constructed to study the optoelectronic properties of synthesized GaSe nanoflake, which exhibited a stable and fast response under visible light illumination.

2 2 Experimental section

Synthesis of GaSe samples. The growth of GaSe samples was carried out in a single temperature zone tube furnace equipped with a quartz tube. Ga₂Se₃ powder (Alfa, purity 99.99%) and Ga (MACKLIN, purity 99.999%) were placed in the center of the heating zone. Mica substrate were placed at the downstream of the heating zone. Prior to heating, the whole system was purged by Ar for ~30 minutes to remove the air in the tube. The furnace was heated up to 880−920 °C at a rate of 25 °C/min with 35−45 sccm Ar and then maintained for 10 min for the growth of GaSe. After the annealing, the furnace was naturally cooled to room temperature. The pressure for the growth of GaSe was under ambient pressure.

Transfer of CVD-grown GaSe nanoflakes. The as-grown GaSe sample on mica substrate was then transferred onto an SiO₂/Si substrate. First, the GaSe sample on mica substrate was spin-coated with PMMA at 4000 rpm for 1min, and then baked at 155 °C for 5 min in order to enhance the adhesion force between PMMA and GaSe samples. Then, PPC (20 wt%, dissolved in anisole) was spin-coated on the top of PMMA film at 2000 rpm for 1 min, followed by baking at 110 °C for 10 min. Afterwards, the PPC/PMMA polymer carrying GaSe samples was peeled off slowly from the mica substrate by tweezers. Next, the PPC/PMMA polymer was placed on the SiO₂/Si (300 nm) substrate and isopropanol was added dropwise to make it adsorb on the surface of the substrate. The PPC/PMMA polymer was uniformly attached to the surface of the SiO₂/Si substrate by baking at 90 °C for 15 min. The TEM samples were prepared by the same method, except that the SiO₂/Si substrate was replaced by a microgrid.

Characterizations. The as-prepared products were further characterized by optical microscopy (Olympus BX41 microscope), atomic force microscope (Dimension Icon, BRUKER), and transmission electron microscope (TEM, JEM-2100, JEOL). Raman, PL spectra of GaSe samples with different thicknesses were recorded in a confocal Raman spectroscopy (Alpha 300RS+, WITec) using a 532 nm laser as the excitation source. For SHG measurement, we use an Alpha 300RS+ Raman spectroscopy by introducing a femtosecond laser as the excitation source. A mode-locked Ti: sapphire was used to generate a continuously adjustable laser wavelength from 800 nm to 1080 nm with pluse duration of 140 fs and repetition rate of 80 MHZ and filtered into optical parametric oscillator (Chameleon Compact OPO-Vis). The output laser beam was focused on the sample with a spot size of about 1.8 μm by 100× objective. For SHG polarization measurement, the collected parallel-polarized SHG signal was sent through a linear polarized analyzer by rotating the sample with a step of 15 ° relative to the fixed light polarization. More details about the SHG test setup were described in previous work [45]. All experiments were performed at room temperature.

Device fabrication and measurement. The devices were fabricated by transferring GaSe nanoflakes onto SiO₂/Si (300 nm) substrate. The electrodes were patterned on the samples by electron beam lithography (Quanta 650 SEM, FEI and ELPHY Plus, Raith GmbH), and then Cr/Au (10 nm/50 nm) layers were deposited on the GaSe nanoflakes with thermally evaporated deposition (Nexdap, Angstrom Engineering). The photodetector measurements were carried out by a semiconductor characterization system (B1500A, Agilent) under the illumination of 532 nm light pulse with area of light spot of 0.44 cm² in a probe station (CRX-6.5K, Lake Shore). All the measurements were performed in air and at room temperature.

3 3 Results and discussion

In this study, the GaSe samples were grown on mica substrates via a vapor phase growth process in a tube furnace [Fig.1(a)]. Growth kinetics depend on the Ga/Se ratio of the precursor in the local growth zone, but the Ga/Se ratio at different positions is usually not constant due to the weight difference between Ga and Se. Therefore, by deviating the Ga/Se ratio from the theoretical ratio of 1:1 throughout the growth region, a uniform morphology of GaSe nanoflakes can be obtained, which is also manifested in the vapor phase growth of other 2D materials [38]. The growth of GaSe nanoflakes can be carried out at a mass ratio of Ga₂Se₃:Ga = 1:1.1. Fig.1(b) shows the optical images of GaSe nanoflake with size of ~10 μm. The thickness of a typical GaSe nanoflake mica substrate was determined to be ~27 nm by AFM measurements [inset of Fig.1(b)]. Fig.1(c) shows a TEM image of a typical GaSe nanoflake, indicating the standard triangular morphology of as-synthesized GaSe nanoflakes. The high-resolution TEM (HRTEM) image of GaSe nanoflake is shown in Fig.1(d). The lattice spacing of 0.33 nm is in accordance with the (100) interplanar distance of GaSe (PDF#01-078-2499 [46]). The corresponding selected area electron diffraction (SAED) pattern is presented in the inset of Fig.1(d), showing single crystalline of as-synthesized GaSe nanoflakes. From the EDS spectrum of the GaSe nanoflake as shown in Fig.1(e), the signals of Ga and Se can be detected with atomic ratio of Ga/Se of 1.17 [inset of Fig.1(e)], approximately equal to the theoretical stoichiometric value of GaSe.

Fig.1 Synthesis of GaSe nanoflakes by CVD method. (a) Schematic diagram of 2D GaSe nanoflakes growth by a CVD system. (b) Optical image of GaSe nanoflake. Inset: AFM image of GaSe nanoflake. (c) TEM image of GaSe nanoflake. (d) HRTEM image of GaSe nanoflake. Inset: Corresponding SAED pattern. (e) EDX spectrum of GaSe nanoflake. Inset: Atomic ratio of Ga/Se.

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Fig.2(a) shows the Raman spectra of 2D GaSe nanoflakes with different thicknesses with a laser excitation of 532 nm. For a GaSe nanoflakes with thickness of 6 nm, Raman characteristic peaks of GaSe and mica substrate located at 197 and 270 cm⁻¹ can be observed. With the increase of thickness of GaSe nanoflake, the characteristic peak intensity of GaSe increases, while the Raman peak of mica substrate gradually weakens until disappears. Three distinct characteristic Raman peaks located at 132, 213, and 308 cm⁻¹ are clearly observed, assigned to the ${\mathrm{A}}_{1\mathrm{g}}^1$, ${\mathrm{E}}_{2\mathrm{g}}^1$, ${\mathrm{A}}_{1\mathrm{g}}^2$ modes of GaSe, respectively [23]. As the thickness increases, the Raman characteristic peaks of GaSe shift to the lower wavenumber direction. This is because with the increase of the number of layers, the vibrational activity of atoms in the layer is hindered, thereby reducing its vibrational energy, resulting in a red shift of the Raman peak positions corresponding to the three vibrational modes [47]. PL spectroscopy was employed to evaluate the optical properties of GaSe nanoflakes, as seen in Fig.2(b). Under the excitation of 532 nm wavelength, the GaSe nanoflake with thickness of 30 nm exhibits an obvious peak at ~629 nm (1.97 eV), which is similar to the GaSe reported [39], corresponding to the bandgap of GaSe [30]. A weak PL peak at ~606 nm (2.05 eV) can be observed in a thin GaSe flake with thickness of ~4 nm, which may be due to the transition from direct to indirect bandgap when the thickness is reduced to few-layer [24]. As the thickness increases, the PL intensity of GaSe is found to increase and a broad peak from 650 nm to 800 nm assigned to the underneath mica substrate is observed to decrease. Owing to the strong quantum confinement effects [48], a red-shifted variation can be observed as the thickness increases.

Fig.2 (a) Thickness-dependent Raman spectra of GaSe nanoflakes. (b) Thickness-dependent PL spectra of GaSe nanoflakes.

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The non-centrosymmetric structure is the key factor of second harmonic generation (SHG)[49]. Fig.3(a) shows the SHG signal at 400 nm wavelength under excitation at 800 nm wavelength, and its intensity increases with laser power, which shows a linear dependence with a slope of 2.03 by power-law fitting [shown in Fig.3(b)]. Compared with transition metal chalcogenides (monolayer has a non-centrosymmetric structure), GaSe maintains a non-centrosymmetric structure at all thicknesses, so its SHG signal intensity increases with its thickness [shown in Fig.3(c)]. The key that must be considered in nonlinear applications is the spectral range of the material that can generate SHG signals. Fig.3(d) shows the SHG generated by the excitation of GaSe nanoflake within the different wavelength range of 800−1080 nm. In order to study the crystal symmetry of a single GaSe nanoflake, we tested the polarization second harmonic generation by rotating the sample during the test, and collecting parallel polarized light. As shown in Fig.3(e), as the azimuth changes, we can observe that the SHG intensity exhibits a 6-axis symmetry pattern. Fig.3(f) shows the SHG mapping of a single triangular-shaped GaSe nanoflake under excitation of 840 nm. SHG results indicate that the synthesized GaSe exhibits excellent nonlinear optical properties due to its non-centrosymmetric structure.

Fig.3 The second harmonic generation (SHG) test of 2D GaSe nanoflake: (a) SHG intensity of GaSe flake with different excitation power and (b) the corresponding linear fitting. (c) SHG intensity of GaSe nanoflakes with different thicknesses. (d) Wavelength dependence of SHG intensity from 800 to 1080 nm. (e) SHG polarization test of 2D GaSe nanoflake, rotating the sample angle θ at a step of 15o, showing a 6-axis rotating scale; (f) SHG mapping of a single GaSe nanoflake.

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To explore the optoelectronic properties of the GaSe nanoflakes, GaSe-based photodetectors were fabricated on SiO₂/Si substrate. Fig.4(a) shows the schematic image of the photodetector. Fig.4(b) shows the current−voltage (I−V) curve measured of the GaSe-based photodetector under 532 nm laser excitation and in the dark, respectively. It is found that the device exhibited a Schottky contact between the Cr/Au electrodes and the GaSe nanoflake. Under excitation of 532 nm, the optimal photoresponsivity (R_λ) is 2.7 mA/W. The external quantum efficiency (EQE) and detectivity (D^*) of the GaSe-based photodetector at a wavelength of 532 nm is 0.63 % and 8.7 × 10⁷ Jones, respectively. Calculated by the relations of EQE = hcR_λ/eλ and D^* = R_λS^1/2/(2eI_dark)^1/2, where the parameters h, c, λ, e, and I_dark are Plank’s constant, light velocity, excited wavelength, elementary electronic charge and dark current, respectively. Fig.4(c) shows that the GaSe-based photodetector exhibits a repeatable and stable response to incident light. Photo-response time is another critical parameter used to judge device performance. We investigated the time-resolved photo-response of the device by switching the laser on and off. Fig.4(d) displays a single cycle response of laser on and off. The photodetector shows a fast response rate of 6 ms for the rise and 10 ms for the decay times. The trend of the photocurrent changes with time under different illumination intensities of 9.93−41.65 mW/cm² for the GaSe-based photodetector is shown in Fig.4(e). It can be seen from Fig.4(e) that as the optical power density increases, its photocurrent also increases. The relationship between light intensity and photocurrent can be fitted by the Power Law formula of I_ph∝P^θ [50, 51] (where I_ph is the light response current, P is the light power density, and θ is an index associated with the light response at a certain light intensity). As shown in Fig.4(f), the photocurrent increases linearly with the increase of optical power density, and its θ is 0.90, which indicates that the synthesized GaSe-based photodetector has excellent photocurrent capability. Some reported device performance data are summarized in Tab.1.

Tab.1 Comparison of the key parameters of our device to the reported 2D materials and the other structures of GaSe-based photodetectors.

Device	Fabrication methods	Rλ (mA·W−1)	EQE (%)	D* (Jones)	Rise time (ms)	Decay time (ms)	Ref.
Graphene	ME	1.0	6−16	−	−	−	[52]
MoSe2	CVD	13	−	−	~60	~60	[53]
WS2	CVD	7.3 × 103	1814	−	5	5	[13]
ReSe2	CVD	2.98 × 103	458	−	5.47 × 103	8.41 × 103	[54]
HfS2	CVD	2.8	−	−	55	55	[55]
InSe	CVD	1.5 × 103	230	3.1 × 108	500	800	[56]
In2Se3	MBE	3	0.67	109	≈7	≈7	[57]
GaS	CVD	50	23	−	−	−	[58]
GaSe	VPM	17	5.2	−	−	−	[59]
GaSe	CVD	2.7	0.63	8.7 × 107	6	10	This work

Note. ME: Mechanical exfoliation; MBE: Molecular beam epitaxy; VPM: Vapor phase mass transport.

Fig.4 (a) Schematic image of the photodetector. (b) I−V characteristics of the device in the dark and under light illumination with wavelength at 532 nm (Vbias = 5 V). (c) Time-resolved photoresponse of the device at 532 nm (Vbias = 2 V). (d) Rise and decay curve measured under 532 nm excitation at Vbias = 2 V. (e) Photocurrent as a function of illumination intensity at Vbias = 1 V under 532 nm excitation. (f) The corresponding fitting curve of photocurrent versus incident light intensities by the power law.

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4 4 Conclusion

In conclusion, we achieved the synthesis of GaSe nanoflakes with different thicknesses via CVD method. GaSe nanoflakes with triangle morphology can be obtained under different Ga/Se ratios by adjusting the ratio of two source species. A series of characterizations showed that the synthesized GaSe samples crystallized well and exhibited excellent performance. The layer-dependent optical properties of GaSe nanosheets were investigated by Raman, PL and SHG characterization. Simultaneously, SHG characterization shows that the synthesized GaSe samples exhibit excellent nonlinear optical properties due to their non-centrosymmetric structure. Under visible light illumination, the photodetectors based on GaSe nanoflakes exhibit stable and fast photoresponse with a rise time of 6 ms and decay time of 10 ms. This provides a reference for the preparation of 2D materials and the possibility of using GaSe nanoflakes for potential applications in nonlinear optics and (opto)-electronics.

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