Photon condensation: A new paradigm for Bose–Einstein condensation

Abstract

Bose–Einstein condensation is a state of matter known to be responsible for peculiar properties exhibited by superfluid Helium-4 and superconductors. Bose–Einstein condensate (BEC) in its pure form is realizable with alkali atoms under ultra-cold temperatures. In this paper, we review the experimental scheme that demonstrates the atomic Bose–Einstein condensate. We also elaborate on the theoretical framework for atomic Bose–Einstein condensation, which includes statistical mechanics and the Gross–Pitaevskii equation. As an extension, we discuss Bose–Einstein condensation of photons realized in a fluorescent dye filled optical microcavity. We analyze this phenomenon based on the generalized Planck’s law in statistical mechanics. Further, a comparison is made between photon condensate and laser. We describe how photon condensate may be a possible alternative for lasers since it does not require an energy consuming population inversion process.

Keywords

Bose–Einstein condensation / photon condensation / magneto-optical trap / Gross–Pitaevskii equation / Planck’s radiation law

1 1 Introduction

Two-dimensional (2D) semiconductor and related van der Waals heterostructures (vdWHs) have attracted tremendous attention because they possess excellent characteristics [1-5]. Also, the 2D vdWHs can form different band alignments, such as type-I, II and III, which is useful to diverse device applications [6-9]. Among them, type-II band alignment plays a critical role in facilitating the electron and hole separation, resulting in long photogenerated carrier lifetimes and tunable interlayer coupling [10-13]. However, the type-II band alignments of most 2D vdWHs are sensitive to external conditions [14-16], which brings a challenge for nanodevices to achieve stable properties. Therefore, it is valuable to find 2D vdWHs with robust type-II band alignment.

More recently, MoSi₂N₄ monolayer, an ultrathin transition metal nitride, is in the spotlight for its successful synthesis via passivating MoN₂ monolayer with silicon [17]. The MoSi₂N₄ monolayer is a seven-layer nanosheet consisting of one MoN₂ inner layer and two Si-N outer layers [18]. Furthermore, MoSi₂N₄ is reported to show great potential application due to its superior performance [19-31], including the impressive ambient stability [27], large theoretical electron/hole mobility (~270/1200 cm²·V^–1·s^–1) [17], excellent thermoelectric property [28,29], and remarkable optical absorption in the visible light range [30,31]. Stimulated by the excellent performance, the 2D vdWHs based on MoSi₂N₄ have become an emerging field [32-41]. For instance, C₃N₄/MoSi₂N₄ vdWHs have achieved a tunable type-II band alignment through interlayer coupling and external electric field [38]. The MoSi₂N₄/NbS₂ vdWHs have robust ultra-low p-type Schottky barrier heights (SBH), while the SBH of MoSi₂N₄/graphene can be modulated via the interlayer distance and electric fields [32]. Compared to isolated transition metal dichalogens (TMDs), the MoSi₂N₄/TMDs vdWHs possess higher mobility, improved optical absorption, and tunable band structures [37, 40, 41]. Additionally, Jin et al. [42] have found that the vdWHs based on MoSi₂N₄ and wide-band gap semiconductors, such as ZnO and GaN, have retained the type-II band alignment when the electric field is smaller, but there is a transition from type-II to type-I band alignment with the application of out-of-plane strain. Among the wide-band gap semiconductors, monolayer GeO₂ features an excitonic gap in the deep ultraviolet (UV) region with an energy of 6.24 eV [43]. It is found that the GeO₂ has a relatively high work function, which is higher than that of reported results for the thin films of transition metal oxides [44] and traditional monolayers of TMDs [45], suggesting that GeO₂ is advisable to be integrated into optical and electronic devices as charge extraction and injection layers that provide improvements in the device ability. Also, the 2D GeO₂ material is fabricated by strictly controlling the degree of oxidation at the metal-gas interface [46].

Considering better lattice mismatch and interesting performance between the GeO₂ and MoSi₂N₄ monolayers, and there are few works on GeO₂-based vdWHs, so we select GeO₂ as representative one of wide-band gap semiconductors to construct the heterojunction with MoSi₂N₄ and study the corresponding performance. In this work, the most stable structures are obtained by the global structure search method of passing through 12 × 12 displacement grids. Additionally, the robust type-II band arrangement of GeO₂/MoSi₂N₄ vdWHs is insensitive to the regulation of biaxial strain, interlayer distance and external electric field, which opens up more possibilities for developing the stable optoelectronic devices.

2 2 Computational details

All the calculations are gotten via the Vienna ab initio simulation package (VASP) based on DFT [47]. With the framework of generalized gradient approximation (GGA), the Perdew−Burke−Ernzerhof (PBE) [48] functional is employed to deal with the exchange-correlation energy, and the Heyd−Scuseria−Ernzerhof (HSE06) method is utilized to gain more precise band gap value [49]. In addition, the projector augmented wave (PAW) pseudopotentials are adopted to simulate the electron-ion potential [50]. The vdW interlayer interaction in the GeO₂/MoSi₂N₄ heterostructure is corrected by DFT-D3 correction method. Besides, the out-of-plane vacuum region, about 20 Å, is used, and the plane-wave cutoff is employed as 500 eV. A 11×11×1 k-point mesh is adopted for the sampling in reciprocal space. The structures are relaxed until the Hellmann−Feynman forces and the total energy converge to 0.01 eV/Å and 10⁻⁵ eV, respectively. Moreover, we would like to point out that the dipole correction is utilized in our calculations.

3 3 Results and discussion

3.1 3.1 Crystal structure and stability of isolated layer and heterojunctions

The geometric structures, band structures and phonon dispersion curves of GeO₂ and MoSi₂N₄ monolayers are listed in Fig.1. Both GeO₂ and MoSi₂N₄ monolayers have hexagonal structures, as shown in Fig.1(a) and (d), and Tab.1 demonstrates that the optimized lattice parameters are a = b = 2.909 Å and a = b = 2.911 Å, respectively. All these results are consistent with previous reports [22,43]. The bond lengths of monolayers are 1.95 Å for Ge-O, 2.09 Å for Mo-N, and 1.75 Å for Si-N. Fig.1(b) and (e) indicate that the GeO₂ and MoSi₂N₄ monolayers are semiconductors with indirect band structure, and the band gap of monolayer GeO₂ and MoSi₂N₄ is 3.56 (5.32) and 1.79 (2.35) eV for PBE (HSE06) calculation, respectively. In addition, Fig.1(c) and (f) depict the phonon dispersion curves of GeO₂ and MoSi₂N₄ monolayers. Obviously, there are no imaginary phonon modes, which confirms the dynamical stability of the two kinds of 2D materials.

Fig.1 (a, d) The atomic structures, (b, e) band structures, (c, f) phonon dispersion curves of GeO2 and MoSi2N4 monolayers, respectively. The Fermi level is defined as zero. In the diagram of the atomic structures, different colored balls represent different atoms.

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Tab.1 The structural and electronic parameters. The lattice constant (a), interlayer distance (d), total energy (Etotal), and band gap of PBE (EgPBE) and HSE06 (EgHSE06).

	a (Å)	d (Å)	Etotal (eV)	EgPBE (eV)	EgHSE06 (eV)
MoSi2N4	2.911	−	−61.603	1.79	2.35
GeO2	2.909	−	−18.88	3.56	5.32
GeO2/MoSi2N4	2.909	2.78	−82.816	0.275	0.946

In order to obtain the credible structures of 2D GeO₂/MoSi₂N₄ vdWHs, a global structure search method is implemented. The total energy as a function of the in-plane shift is presented in Fig.2(a). We consider 144 stacking structures A_m,n passing through a 12 × 12 displacement grids. The A_m,n means the displacement of the GeO₂ layer relative to the MoSi₂N₄ layer is $\left(\frac{m}{12}\right)\mathit{a}+\left(\frac{n}{12}\right)\mathit{b}$, where $\mathit{a}$ and $\mathit{b}$ are the lattice vectors of the vdWHs. Among them, the A_3,6 stacking pattern is the most stable structure with the interlayer distance between GeO₂ and MoSi₂N₄ layer of 2.78 Å. Also, Fig.2(b) shows the side view of the most stable vdWHs structure consisting of GeO₂ and MoSi₂N₄ layer, in which the lattice mismatch is 0.07%. From the bright red region in Fig.2(a), we find that there are four metastable structures around the A_3,6 stacking pattern, which feature that the total energy is close to that of the most stable structure. The total energy of these four metastable structures is shown in Table S1. In addition, Fig. S1 presents the projected band structures of the metastable and most stable structures calculated by the PBE and HSE06 methods. Similar total energy and electronic properties are given for the considered five configurations, and thus only the most stable structure is discussed in the following section. The HSE06 method is adopted for all the next calculations. Fig.2(c) indicates that GeO₂ and MoSi₂N₄ can form a type-II heterojunction with direct band gap of 0.946 eV. To further determine the dynamical stability of the structure with the lowest energy, Fig.2(d) shows the phonon dispersion curves of the vdWHs, and all the phonon frequencies are positive, illustrating that the most stable GeO₂/MoSi₂N₄ vdWHs is also dynamically stable.

Fig.2 (a) Color contour plot of total energy versus in-plane shift for GeO2/MoSi2N4 vdWHs, considering the 12 × 12 grids. (b) The geometric structure, (c) band structure and (d) phonon dispersion curves of the most stable GeO2/MoSi2N4 vdWHs. The Fermi level is defined as zero.

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To further explore the electronic characteristics of the most stable structure GeO₂/MoSi₂N₄ vdWHs, Fig.3 draws the orbital-projected electronic band structures, partial charge densities, plane-averaged differential charge density and electrostatic potential. As illustrated in Fig.3(a) and (b), the CBM is mostly contributed by the Ge_s and O_p orbitals of the GeO₂ layer, while the VBM mainly comes from the N_p and Mo_d orbitals in the MoSi₂N₄ layer. Besides, the band shapes of GeO₂ and MoSi₂N₄ are well preserved after forming the heterojunction due to the vdWHs interaction. The charge distribution of Fig.3(c) shows the CBM is mainly contributed by Ge and O atoms, and the VBM comes from Mo and N atoms, which coincides with the above discussion.

Fig.3 The orbital-projected band structures of GeO2/MoSi2N4 vdWHs on GeO2 (a) and MoSi2N4 (b) layer. Partial charge density (c) and plane-averaged charge density difference (d) of GeO2/MoSi2N4 vdWHs. Yellow (green) colored isosurfaces represent the charge accumulation (depletion), respectively. (e) Electrostatic potential of GeO2/MoSi2N4 vdWHs. The isosurface is set to be 0.008 e/bohr3 for (c) and 0.0001 e/ Å3 for (d).

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The plane-averaged charge density difference is calculated to assess the charge redistribution in the GeO₂/MoSi₂N₄ vdWHs, which can be calculated as $\mathrm{\Delta }\rho =$ ${\rho }_{\text{vdWHs}}-{\rho }_{\text{Ge}{\text{O}}_{\text{2}}}-{\rho }_{\text{MoS}{\text{i}}_{\text{2}}{\text{N}}_{\text{4}}}$, where the ${\rho }_{\text{vdWHs}}$, ${\rho }_{\text{Ge}{\text{O}}_{\text{2}}}$, and ${\rho }_{\text{MoS}{\text{i}}_{\text{2}}{\text{N}}_{\text{4}}}$ represent the charge density of the vdWHs, isolated GeO₂ and MoSi₂N₄ monolayers, respectively. In the inset of Fig.3(d), the charge transfer mainly occurs at the interface of the vdWHs. The yellow and green region corresponds to the ρ > 0 and ρ < 0, indicating the accumulation and depletion of charge, respectively. Meanwhile, the charge is depleted near the MoSi₂N₄ layer and accumulated in the vicinity of the GeO₂ layer, which implies that the charge can be transferred from the MoSi₂N₄ layer to the GeO₂ layer.

In addition, the electrostatic potential along the perpendicular direction of the vdWHs is explored in Fig.3(e). The potential drop (ΔV) across the two isolated layers is 10.51 eV, which indicates the presence of a strong built-in field across the interface. The electric field strength of GeO₂/MoSi₂N₄ vdWHs exceeds that of other MoSi₂N₄-based heterostructures, such as the C₃N₄/MoSi₂N₄ (6.28 eV) [38] and MoSe₂/MoSi₂N₄ (4.03 eV) [40]. The MoSi₂N₄ possesses a much deeper potential than GeO₂, owing to the stronger electronegativity of the MoSi₂N₄ monolayer. Thus, the deeper potential may facilitate the separation of electron-hole pairs, which is beneficial for photoelectric detection applications based on the GeO₂/MoSi₂N₄ vdWHs.

3.2 3.2 Biaxial strain tunable electronic properties of the vdWHs

To understand the strain influence on the GeO₂/MoSi₂N₄ vdWHs, we define the biaxial strain as ε = (a − a₀)/a₀ on GeO₂/MoSi₂N₄ vdWHs, where a₀ and a represent the lattice parameter of heterostructures without and with biaxial strain, respectively. In Fig.4, the negative and positive values are used to denote the compressive and tensile strain, respectively. Fig.4(a−c) display the variation of the total energy, band gap, and interlayer charge transfer of GeO₂/MoSi₂N₄ vdWHs under the biaxial strains. It is obvious from Fig.4(a) that the total energy reaches the minimum value under the pristine condition. The band gap decreases for GeO₂/MoSi₂N₄ vdWHs with increasing tensile, which can be seen in Fig.4(b). On the contrary, as the compressive strain is applied, the band gap value tends to increase. When the tensile strain increases up to 6%, the band gap of the vdWHs decreases from 0.946 eV to 0.124 eV. And the variation trend of the band gap gradually slows down and approaches saturation at 4% strain. Additionally, when the compressive strain range is 0 ~ −6%, the band gap increases linearly from 0.946 eV to 2.806 eV. Moreover, it can realize a transition from direct to indirect band gap when the compressive strain is greater than -3%. As shown in Tab.2, we can see that with increasing the tensile strain for the GeO₂/MoSi₂N₄ vdWHs, the atomic bond lengths of Ge-O, Mo-N and Si-N can be raised, while the bond angles exhibit the reduced trend. In contrast to the case of tensile strain, the increasing compressive strain can make the Ge-O, Mo-N and Si-N bond lengths decrease, and the bond angles consequently enhance. However, it is worth noting that the type-II band arrangement is well preserved in the GeO₂/MoSi₂N₄ vdWHs under all the studied biaxial strain cases.

Tab.2 The bond lengths as a function of biaxial strain for GeO2/MoSi2N4 vdWHs.

Strain	dGe-O (Å)	dMo-N (Å)	dSi-N (Å)
−6%	1.90	2.05	1.68
−5%	1.91	2.06	1.70
−4%	1.91	2.06	1.71
−3%	1.92	2.07	1.72
−2%	1.93	2.08	1.73
−1%	1.94	2.08	1.74
0	1.95	2.09	1.75
1%	1.96	2.10	1.76
2%	1.96	2.11	1.78
3%	1.97	2.12	1.79
4%	1.99	2.13	1.81
5%	2.00	2.14	1.82
6%	2.01	2.15	1.83

Fig.4 The variation of the total energy (a), band gap (b), and interlayer charge transfer (c) of GeO2/MoSi2N4 vdWHs as a function of biaxial strain, respectively. (d−g) Projected band structures of GeO2/MoSi2N4 vdWHs with the biaxial strains of −3%, −2%, 2% and 3%.

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Fig.4(c) exhibits the evolution of interlayer charge transfer as a function of biaxial strain. The results show that the interlayer charge transfer of GeO₂/MoSi₂N₄ vdWHs decreases with increasing tensile and compressive strains. Under the case of tensile and compressive strain, the interlayer charge transfer declines by 0.018 e and 0.005 e, respectively. The charge transfer tends to diminish when the compressive strain is larger than −5%. Furthermore, it reaches a maximum value of 0.031 e under the 1% tensile strain.

To better understand the above-mentioned behavior, Fig.4(d)−(g) present the projected band structures of GeO₂/MoSi₂N₄ vdWHs under different biaxial strains. When tensile strain is applied, the conduction band edge moves to the Fermi level significantly, while the valence band edge barely moves, which is consistent with the decrease in the band gap. On the other side, the compressive strain causes the conduction band edge to move away from the Fermi level substantially, corresponding to the increase of the band gap. As the −2% compressive strain is applied, both CBM and VBM remain at Γ point. However, when the compressive strain is greater than −3%, the CBM of GeO₂/MoSi₂N₄ vdWHs is at Γ point, while the VBM is at K point, thus an indirect band gap can be observed. In particular, for all the studied strain cases, the CBM and VBM of GeO₂/MoSi₂N₄ vdWHs are always formed by the GeO₂ and MoSi₂N₄ layers, respectively, confirming that it always maintains type-II band alignment characteristics. In addition, we notice that the application of strain can induce the band alignment transition from type-II to type-I or type-III in many other 2D vdWHs, such as GaSe/SnX₂ (X = S, Se) [14], MoSSe/MBP [51] and BP/β-AsP vdWHs [52].

3.3 3.3 Interlayer distance tunable electronic properties of the vdWHs

The interlayer interaction has obvious effects on the electronic structures of the vdWHs. Thus, Fig.5 plots the interlayer distance change on the electronic properties of the GeO₂/MoSi₂N₄ vdWHs. We can see from Fig.5(a) that the curve of the total energy has the minimum energy value when the interlayer distance is 2.78 Å. From Fig.5(b), the band gap increases for GeO₂/MoSi₂N₄ vdWHs with increasing interlayer distance. Also, the band gap changes slightly when the interlayer distance is larger than 3.28 Å. In addition, Fig.5(c) shows that the interlayer charge transfer declines gradually with the increase of the interlayer distance. To reveal the related physical mechanism, in Fig.5(d−g), we give the interlayer distance effects on the projected band structure of GeO₂/MoSi₂N₄ vdWHs. For different distances, the CBM and VBM contributions to GeO₂/MoSi₂N₄ vdWHs are provided by the GeO₂ and MoSi₂N₄ layers separately, showing that this type-II band alignment is insensitive to the interlayer interactions. Furthermore, the variation of the band gap is mainly due to the up and down movement of CBM.

Fig.5 The total energy (a) and band gap (b), interlayer charge transfer (c) and projected band structures (d−g) of GeO2/MoSi2N4 vdWHs at different interlayer distances, respectively.

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3.4 3.4 Electric field tunable electronic properties of the vdWHs

The external vertical electric field provides another effective strategy for controlling the electronic structure. In the following, the variation of the band gap for GeO₂/MoSi₂N₄ vdWHs as a function of the electric field is shown in Fig.6 (a). It should be noted that the electric field is perpendicular to the surface of the GeO₂/MoSi₂N₄ vdWHs, and the positive direction is defined from MoSi₂N₄ layer to GeO₂ layer, as illustrated in Fig. S2. When the electric field increases from −0.5 to 0.5 V/Å, the band gap decreases linearly, maintaining the type-II band alignment and direct band structures. Here, the band edge and projected band structures are also shown, and the reasons for the band alignment transition are discussed. In Fig.6(b), with the increase of the external electric field, the CBM of GeO₂/MoSi₂N₄ vdWHs decreases significantly, while the VBM shifts upward slightly. Moreover, when the positive electric field reaches 0.4 V/Å, the VBM decreases more slightly, which causes the value of band gap to decrease slowly as the electric field changes from 0.4 V/Å to 0.5 V/Å. As can be seen from Fig.6(c−f), with the increase of the positive electric field, the CBM of GeO₂ layer moves down gradually, while the CBM and VBM of MoSi₂N₄ keep almost unchanged. Compared with the 0.2 V/Å, the CBM of GeO₂ layer moves towards the Fermi level under the electric field of 0.4 V/Å. On the other hand, Fig.6(f) indicates that with increasing the negative electric field, the CBM of GeO₂ layer moves away from the Fermi level when the electric field reaches −0.4 V/Å. However, further increased electric field does not change the type of band arrangement. Thus, the external electric field can change the band gap, while the band alignment remains type-II for the GeO₂/MoSi₂N₄ vdWHs. What’s more, for many other vdWHs, the band alignment is sensitive to the external electric field [16, 51, 53-56]. For example, when the positive electric field is larger than 0.2 V/Å, the band alignment of the AlN/VSe₂ vdWHs changes from type-II to type-I [16]. The MoXY/WXY (X, Y = S, Se, Te; X ≠ Y) vdWHs maintain robust type-II band alignment under positive electric field, while a reverse electric field can make the band alignment change [53].

Fig.6 The band gap (a) and band alignment (b) of GeO2/MoSi2N4 vdWHs as a function of external electric field, respectively. (c−f) The projected band structures of GeO2/MoSi2N4 vdWHs with different external electric fields. The red (green) line indicates the contribution of the GeO2 (MoSi2N4) layer.

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4 4 Conclusions

In conclusion, by using a global structure search approach, we design the GeO₂/MoSi₂N₄ vdWHs to exactly explore the band structures, and the corresponding modulations of biaxial strain, electric field, and interlayer distance are considered to improve the performance of GeO₂/MoSi₂N₄ vdWHs. The results show that the monolayer GeO₂, MoSi₂N₄ and GeO₂/MoSi₂N₄ vdWHs are dynamically stable at room temperature. The most stable GeO₂/MoSi₂N₄ vdWHs possess direct band gap of 0.946 eV and type-II band alignment with the CBM and VBM from the GeO₂ and MoSi₂N₄ layers. When the biaxial strain is applied, the direct-to-indirect band gap transition can be achieved. More interestingly, the robust type-II band arrangement of GeO₂/MoSi₂N₄ vdWHs remains unchanged when the external electric field, interlayer distance, and biaxial strain are applied. These results reveal that the GeO₂/MoSi₂N₄ vdWHs can be in promising optoelectronic devices with stable characteristics due to their robust type-II band arrangement.

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