Copper(I) sulfide: a two-dimensional semiconductor with superior oxidation resistance and high carrier mobility

Abstract

Two-dimensional (2D) semiconductors with suitable direct band gaps, high carrier mobility, and excellent open-air stability are especially desirable for material applications. Herein, we show theoretical evidence of a new phase of a copper(I) sulfide (Cu₂S) monolayer, denoted δ-Cu₂S, with both novel electronic properties and superior oxidation resistance. We find that both monolayer and bilayer δ-Cu₂S have much lower formation energy than the known β-Cu₂S phase. Given that β-Cu₂S sheets have been recently synthesized in the laboratory (Adv. Mater.2016, 28, 8271), the higher stability of δ-Cu₂S than that of β-Cu₂S sheets suggests a high possibility of experimental realization of δ-Cu₂S. Stability analysis indicates that δ-Cu₂S is dynamically and thermally stable. Notably, δ-Cu₂S exhibits superior oxidation resistance, due to the high activation energy of 1.98 eV for the chemisorption of O₂ on δ-Cu₂S. On its electronic properties, δ-Cu₂S is a semiconductor with a modest direct band gap (1.26 eV) and an ultrahigh electron mobility of up to 6880 cm² V⁻¹ s⁻¹, about 27 times that (246 cm² V⁻¹ s⁻¹) of the β-Cu₂S bilayer. The marked difference between the electron and hole mobilities of δ-Cu₂S suggests easy separation of electrons and holes for solar energy conversion. Combination of these novel properties makes δ-Cu₂S a promising 2D material for future applications in electronics and optoelectronics with high thermal and chemical stability.

---

Conceptual insights

Copper sulfides have been extensively studied owing to their high abundance, easy preparation, and outstanding chemical and physical properties. Here we report a hitherto unreported 2D copper(I) sulfide, namely, a δ-Cu₂S monolayer. The δ-Cu₂S monolayer is a semiconductor with a desirable direct band gap of 1.26 eV, close to that of bulk silicon, as well as a high electron mobility of up to 6880 cm² V⁻¹ s⁻¹, about 27 times that of the β-Cu₂S bilayer (246 cm² V⁻¹ s⁻¹, measured value). Unlike many other 2D modest-band-gap semiconductors which have relatively low chemical stability, notably, the δ-Cu₂S monolayer exhibits superior oxidation resistance, as demonstrated by the high activation energy of 1.98 eV for the chemisorption of O₂ on δ-Cu₂S. Another novel feature of the δ-Cu₂S solid is that it has much lower formation energy than the previously known β-Cu₂S solid, suggesting a high likelihood of achieving a δ-Cu₂S monolayer in the laboratory.

Introduction

Copper sulfide (Cu_2−xS, 0 ≤ x ≤ 1) nanomaterials have been extensively studied owing to their high abundance, easy preparation, and excellent chemical and physical properties.^1–4 In the laboratory, Cu_2−xS can be synthesized using various methods, e.g., the colloidal solution process, the wet chemical method, and the facile solvothermal method.^5–8 Among many Cu_2−xS nanomaterials, Cu₂S (chalcocite) is known for its excellent optical and electronic properties with potential applications in lithium storage,⁹ high-capacity cathode materials in lithium secondary batteries,¹⁰ photocatalysts,¹¹ and absorbers in photovoltaic conversions.^12–14 Bulk Cu₂S solids are usually classified as α (stable above 425 °C), β (high chalcocite; stable between 105 and 425 °C) and γ (low chalcocite; the first solid–liquid hybrid phase, and stable below 105 °C) phases,¹⁵ respectively. Phase transition between the high and low chalcocite phases could be achieved by heating with an electron beam at relatively low temperature.^5,16,17

Two-dimensional (2D) materials have attracted extensive attention due largely to many of their novel properties which are not seen in their bulk counterparts.^18–23 Very recently, 2D β-Cu₂S sheets with thicknesses down to 1.8 nm have been synthesized and proven to exist as a hybrid solid–liquid phase.^8,24,25 The temperature (−15 °C) for high chalcocite to low chalcocite transition is much lower than that of the bulk counterpart (105 °C), rendering 2D β-Cu₂S sheets available for room-temperature applications.⁸ Similarly, the transition temperature from β-Cu₂S to γ-Cu₂S also decreases with decreasing nanoparticle size.²⁶ In addition, Cu₂S sheets exhibit higher photo-activity for solar energy conversion than their bulk counterpart.²⁷ A previous theoretical calculation predicted that a β-Cu₂S bilayer is the thinnest stable layer as the two layers are strongly bonded, evidenced by the large binding energy between the layers (0.93 eV per Cu₂S unit).²⁴ Moreover, the β-Cu₂S bilayer exhibits a direct band gap of 0.9 eV at the high-symmetry point Γ,⁸ implying that the β-Cu₂S bilayer could be a promising candidate for electronic and optical applications.

Although the solid–liquid phase is quite interesting,^8,15,24 it also indicates meta-stability and rapid diffusion of atoms, which may degrade the material's performance. In this article, we predict a new phase of a Cu₂S monolayer, denoted δ-Cu₂S, based on global structural search and first-principles calculations. Our computation indicates that the δ-Cu₂S monolayer and bilayer are semiconductors with modest direct band gaps and ultrahigh electron mobilities of up to 6880 cm² V⁻¹ s⁻¹ and 5786 cm² V⁻¹ s⁻¹, respectively. The latter is more than one order of magnitude higher than that of the bilayer β phase (246 cm² V⁻¹ s⁻¹). Our computation also shows that δ-Cu₂S is dynamically and thermally stable, and chemically inert upon oxidation. In particular, the superior oxidation resistance of δ-Cu₂S, due to the high activation energy (1.98 eV) for the chemisorption of O₂ on this 2D sheet, is the distinct feature compared with many ultrathin 2D materials with poor chemical stability in open air.^28–31 A well-known example is black phosphorus whose 2D counterpart – phosphorene – is hydrophilic and can be easily oxidized in air moisture with a low activation energy of 0.70 eV.³² Oxidation of phosphorene leads to higher contact resistance, lower carrier mobility, and possible mechanical degradation and breakdown.^29,32,33

Results and discussion

After a systematic structural search, hundreds of structures were generated using the CALYPSO code. Four representative low-lying structures are shown in Fig. S1 (ESI†), and their stability analyses are presented in Fig. S2 and Table S1 (ESI†). The δ-Cu₂S monolayer is identified as the most stable 2D structure. Fig. 1a presents the optimized structure of δ-Cu₂S, with a tetragonal structure and lateral isotropy. The unit cell of δ-Cu₂S has four Cu and two S atoms, with each Cu atom being bonded with five Cu atoms and two S atoms, and each S atom being bonded with four Cu atoms. As shown in Fig. 1a, the optimized 2D layered structure has a lattice constant (a) of 5.02 Å and a thickness (h) of 2.55 Å along the z direction, with average Cu–S and Cu–Cu bond lengths of 2.22 Å and 2.57 Å, respectively (see Table 1). To gain insight into the chemical bond of δ-Cu₂S, we calculated the electron localization function (ELF).^34–36 As displayed in Fig. 1b, the electrons are mainly localized around S atoms and absent in the middle region between Cu and S atoms, indicating the distinct feature of ionic bonding. In addition, Bader charge analysis^37,38 shows a substantial number of electrons, i.e., 0.32 e donated from each Cu atom to S atoms (Table 1).

Fig. 1 (a) The atomic structures of δ-Cu₂S from top and side views, respectively. The unit cell is denoted by dashed lines. a, l₁, l₂, θ and h represent the lattice constant, the Cu–Cu and Cu–S bond lengths, the Cu–S–Cu bond angle, and the layer thickness along the z direction, respectively. (b) Electron localization functions (ELFs); ELF = 1 (red) and 0 (blue) indicate the accumulated and vanishing electron densities, respectively. (c) Phonon dispersion of δ-Cu₂S. High-symmetry points of the first Brillouin zone are Y (0, 0.5, 0), Γ (0, 0, 0) and Q (0.5, 0.5, 0), as shown in Fig. 4b. (d) Snapshots of the equilibrium structures of the Cu₂S monolayer at 800 K after 10 ps ab initio molecular dynamics simulation. Cu and S atoms are shown in pink and yellow colors, respectively.

Table 1 Lattice constant (a), Cu–Cu (l₁) and Cu–S (l₂) bond lengths, Cu–S–Cu angle (θ), and thickness (h) of δ-Cu₂S as shown in Fig. 1(a); Bader charge (Q_Cu) transferred from each Cu atom to S atoms; and effective masses (m) of carriers, deformation potential constants (E₁), elastic moduli (C_2D) and carrier mobilities (μ) of the δ-Cu₂S monolayer (δ-monolayer), δ-Cu₂S bilayer (δ-bilayer), and β-Cu₂S bilayer (β-bilayer) along the armchair and zigzag directions

a (Å)	l 1 (Å)	l 2 (Å)	θ (°)	h (Å)	Q Cu (e)
5.02	2.57	2.22	70.76	2.55	0.32

---

Material	Carrier type	m (m0)	E 1 (eV)	C (J m^−2)	μ (cm^2 V^−1 s^−1)
δ-Monolayer	Holes	4.28	1.79	40.59	14.54
	Electrons	0.21	1.71	40.59	6880.17
δ-Bilayer	Holes	0.23	4.28	94.99	2107.74
	Electrons	0.16	3.66	94.99	5786.28
β-Bilayer (armchair)	Holes	0.98	4.79	63.99	79.68
	Electrons	0.49	5.07	63.99	222.09
β-Bilayer (zigzag)	Holes	0.59	5.00	65.78	109.37
	Electrons	0.47	5.35	65.78	246.51

---

To evaluate the energetic stability of δ-Cu₂S, we first calculated the formation energy ΔH defined as

where E_tot is the total energy of δ-Cu₂S, and E_Cu and E_S are the energies per atom of body-centered cubic (bcc) Cu and orthorhombic S₁₂₈ solids, respectively. The factors n₁ and n₂ denote the numbers of Cu atoms and S atoms, while the factor n represents the total number of atoms in the unit cell. The calculated ΔH is −0.21 eV per atom, revealing that the formation of δ-Cu₂S is exothermic. Remarkably, this value is far lower than that of β-Cu₂S (ΔH = −0.01 eV per atom) shown in Fig. S3 (ESI†), implying a high possibility of synthesizing δ-Cu₂S in the laboratory.

Since the experimentally synthesized δ-Cu₂S sheets could be multilayer rather than monolayer, we also considered several high-symmetry stacking configurations for δ-Cu₂S monolayers. We found that the AA stacking in Fig. 4b is the most stable bilayer configuration with an interlayer binding energy of −0.08 eV per atom, comparable with that of graphene and phosphorene (around −0.06 eV per atom) bilayers.^39,40 According to eqn (1), the formation energy (−0.13 eV per atom) for the δ-Cu₂S bilayer is also much lower than that of the β-Cu₂S bilayer, again suggesting the likelihood of the synthesis of the δ-Cu₂S bilayer.

The dynamic stability of δ-Cu₂S is confirmed by the absence of imaginary frequencies in computed phonon dispersion (Fig. 1c). We also performed Born–Oppenheimer molecular dynamics (BOMD) simulations to examine the thermal stability of the 2D structure at elevated temperature. The BOMD simulations show that the δ-Cu₂S monolayer can maintain its structural integrity even up to 800 K for 10 ps (Fig. 1d), indicating its exceeding stability even above room temperature. At a temperature of 900 K, the planar structure is highly distorted within 10 ps of BOMD simulation (Fig. S4d, ESI†). Furthermore, for both Cu and S atoms, we calculated the mean-square-diffusion distance,⁸ defined as , where N is the number of atoms for each element, and Δx, Δy and Δz are the differences of Cartesian coordinates for the same atom at the beginning and end of the BOMD simulation. As shown in Fig. 2, the mean-square-diffusion distances for Cu and S atoms remain almost unchanged with time t. At a temperature of 800 K, the S atoms diffuse a bit more than the Cu atoms since the Cu atoms are covered by the S atoms (see Fig. 1a). The computed time-dependent mean-square-diffusion distances of the Cu and S atoms indicate that the δ-Cu₂S monolayer is still in the solid phase at 800 K, contrary to the hybrid solid–liquid phase of bulk β-Cu₂S at room temperature.⁸ This result further supports that the δ-Cu₂S monolayer exhibits higher structural stability than the β-Cu₂S monolayer at room temperature.

Fig. 2 Average diffusion distance squares as functions of simulated time for δ-Cu₂S at 300 K, 500 K and 800 K from top to bottom panels, respectively.

To examine the chemical stability of the δ-Cu₂S monolayer in an open-air environment, we investigated the physisorption and chemisorption of an O₂ molecule on the surface of a 2D sheet, and computed the dissociative oxidation pathway. The interaction between O₂ and the monolayer is described by the binding energy (E_bind) defined as follows:

E_bind = E_tot − E_δ − E_O2 (2)

where E_tot, E_δ, and E_O2 are the energies of the δ-Cu₂S monolayer with an O₂ molecule adsorbed, the δ-Cu₂S monolayer, and a single O₂ molecule in the triplet spin state, respectively. By definition, a negative E_bind means exothermic adsorption of the O₂ molecule.

Fig. 3 illustrates the adsorption configurations and dissociation process for an O₂ molecule, initially physisorbed on the δ-Cu₂S, turning into two oxygen atoms chemisorbed on the monolayer. In the initial stage of physisorption, the O₂ molecule is located about 3.7 Å above the surface of δ-Cu₂S with a binding energy of −0.09 eV. As the O₂ molecule approaches the 2D sheet, the O–O bond is elongated to 1.65 Å in the transition state. The O₂ molecule undergoes a transition from physisorption to chemisorption by overcoming an activation energy (E^a) of 1.98 eV, suggesting an extremely slow oxidation rate at room temperature. This activation energy is even greater than that calculated for a MoS₂ monolayer (1.59 eV),⁴¹ which already exhibits relatively high resistance to O₂ chemisorption. Thus, the δ-Cu₂S monolayer is expected to be a superior oxidation-resistant material, as the MoS₂ monolayer. Note that we considered several other adsorption sites for O₂ and selected the initial and final structures with the lowest binding energy to simulate the oxidation process of δ-Cu₂S (see S4 (ESI†) for details).

Fig. 3 Reaction pathway for an O₂ molecule to dissociate into two O atoms on the δ-Cu₂S monolayer. The black line segments indicate the energy levels of the initial, transition, and final states, respectively, with the corresponding atomic structures (top and side views) given next to the energy levels. The blue numbers give (from left to right) the binding energy of the initial state, the heat of reaction, the activation energy, and the binding energies of the transition and final states, respectively. The O, S and Cu atoms are painted in red, yellow and pink, respectively.

The bonding energy of O₂ is often overestimated by DFT/PBE calculations. If we adopt the experimental value of O₂ bonding energy (5.16 eV)⁴² instead of the DFT value, the binding energy (E_bind*) could be redefined as:

E_bind* = E_tot − E_δ − 2 × E_O − E_bond (3)

where E_tot, E_δ, E_O, and E_bond are the energies of the δ-Cu₂S monolayer with an O₂ molecule adsorbed, the pristine δ-Cu₂S monolayer, and an O atom, and the experimental O₂ bonding energy, respectively. Based on eqn (3), the binding energies for the initial, transition and final states are −1.74, 0.24 and −1.82 eV, respectively (see Table S2, ESI†), which are lower than those based on using calculated O₂ bonding energy as the reference (see eqn (2)). This indicates that O₂ could bind strongly to the δ-Cu₂S monolayer. However, there is no change with an activation energy of 1.98 eV, which is still hard to overcome.

To characterize the thermal stability of the oxidized products, we defined the heat of reaction as follows:

E^H = E^phys_bind − E^chem_bind (4)

where E^phys_bind and E^chem_bind are the binding energies of O₂ physisorbed and chemisorbed on the δ-Cu₂S monolayer in the corresponding equilibrium state, respectively. As displayed in Fig. 3, E^H obtained for δ-Cu₂S is as low as 0.08 eV, demonstrating weak driving force for an O₂ molecule to dissociate, and to be chemisorbed on the 2D sheet, suggesting again high resistivity to oxidation and superior chemical inertness in ambient air. Thus, the δ-Cu₂S monolayer could offer long-term stability and durability for sustained device performance. To gain additional insight into good chemical stability against O₂ oxidation, we performed BOMD simulation of the δ-Cu₂S monolayer exposed to gas-phase O₂ at room temperature (S6, ESI†). Here, six O₂ molecules (per supercell) are initially placed about 4 Å above the surface of the δ-Cu₂S monolayer. The 5 ps simulation indicates that O₂ molecules tend to either bounce back or move away from the surface, rather than dissociating into oxygen atoms (Movie S1, ESI†). This simulation suggests good stability of the δ-Cu₂S monolayer against O₂ oxidization.

The electronic band structure and local density of states (LDOS) of the δ-Cu₂S monolayer were computed using the HSE06 functional (Fig. 4a). The HSE06 computation suggests that δ-Cu₂S possesses a modest direct band gap of 1.26 eV at the Γ point, 0.24 eV smaller than the HSE06 bandgap of 1.50 eV for bilayer β-Cu₂S (Fig. S3, ESI†). The LDOS analysis reveals that the edges of the valence bands are dominated by the 3d orbitals of the Cu atoms, while the conduction band edges stem mainly from the 3d orbitals of the Cu atoms and partially from the 3p orbitals of the S atoms. Hence, the Cu atoms make the major contribution to both the valence band maximum (VBM) and the conduction band minimum (CBM). Meanwhile, the substantial overlap of the LDOS near the Fermi level implies strong hybridization between the orbitals of the Cu and S atoms. The computed carrier effective masses are 4.28 m₀ for holes and 0.21 m₀ for electrons, respectively. The relatively small effective mass of the electrons suggests that the carriers are rather mobile in the 2D sheet. Moreover, the large difference in effective masses can be exploited to separate the electrons and holes for solar energy conversion.

Fig. 4 (a) Electronic band structure (left panel) and local density of states (LDOS, right panel) of the monolayer δ-Cu₂S sheet. Green and red lines in LDOS represent contributions from S 3p and Cu 3d states, respectively. (b) Atomic structure of the layers of δ-Cu₂S and Brillouin zone of the bulk counterpart. (c) Electronic band structures of bilayer, trilayer and bulk δ-Cu₂S, respectively. E_g is the band gap denoted by a solid arrow. The Fermi levels are set to zero.

Strain engineering offers a useful way to tune the electronic properties and performance of 2D atomic crystals. For example, a linear decrease in the optical band gap of MoS₂ with strain has been experimentally confirmed via photoluminescence.^43,44 To examine the strain effect on the band gap of the δ-Cu₂S monolayer, we computed the electronic band structures of the δ-Cu₂S monolayer under various biaxial strains up to ±3% (see Fig. S7 (ESI†) for details). The results indicate that the band gap of the δ-Cu₂S monolayer linearly increases from 1.07 eV to 1.55 eV under a strain of −3% to 3%, which is similar to the strain effect on MoS₂. The direct character and dispersion (see Fig. S7 and S8, ESI†) are preserved. For the multilayered system, as the layer thickness increases, the band gap decreases monotonically. Consequently, the δ-Cu₂S bilayer and trilayer preserve the semiconducting behavior, with band gaps of 0.52 eV (direct) and 0.28 eV (indirect), while bulk δ-Cu₂S is metallic. Furthermore, the δ-Cu₂S bilayer possesses much smaller carrier effective masses of 0.23 m₀ for holes and 0.16 m₀ for electrons, compared with those of the β-Cu₂S bilayer (0.56–0.98 m₀ for holes and 0.47–0.49 m₀ for electrons), as listed in Table 1, implying easy transport of carriers in the δ-Cu₂S bilayer.

To further characterize the electron transport properties of the δ-Cu₂S monolayer, we calculated their acoustic phonon-limited carrier mobility μ, based on the Takagi model within the deformation potential approximation:^45–47

where e is the electron charge, ℏ is the reduced Planck constant, k_B is the Boltzmann constant, T is the temperature, m is the effective mass along the transport direction, is the average effective mass (m_⊥ is the effective mass perpendicular to the transport direction), and C_2D is the elastic modulus of the 2D sheet, determined by the lattice parameter l along the transport direction via ΔE/S₀ = C_2D(Δl/l)²/2 (ΔE is the energy change of the system under lattice deformation Δl, and S₀ is the area of unit cell). The term E₁ represents the deformation potential constant of the VBM for a hole, or the CBM for an electron along the transport direction, defined by ΔV/(Δl/l) (ΔV is the band edge shift under lattice deformation). All data are calculated using a strain step of 0.5%. The temperature used in the mobility calculations is 300 K. The present carrier mobility calculation has been demonstrated to be physically reasonable and computationally efficient.^48–50

As listed in Table 1, the electron mobility of the δ-Cu₂S monolayer can reach as high as 6880 cm² V⁻¹ s⁻¹, about five times that of phosphorene (1140 cm² V⁻¹ s⁻¹).⁴⁸ Its high carrier mobility is attributed to its small effective mass, large elastic modulus, and small deformation potential constant. The δ-Cu₂S monolayer shows C_2D values up to 40.59 J m⁻², larger than that of phosphorene (28.94 J m⁻² along the armchair direction),⁴⁸ resulting in small-amplitude lattice waves and weak scattering with charge carriers. The deformation potential constant E₁ is another important factor for carrier mobility. It approximately describes the strength of electron–phonon coupling and is determined by the band edge shift under lattice variation due to acoustic phonons. Here, E₁ is as low as 1.71 eV, contributing to the ultra-high carrier mobility. The high electron mobility demonstrates that the δ-Cu₂S monolayer is a good candidate for 2D electronic devices. Furthermore, the carrier mobility of electrons is three orders of magnitude larger than that of holes (14 cm² V⁻¹ s⁻¹). The substantial difference between the transport behavior of electrons and holes could be beneficial to controlling the carrier transport, and separating the carriers to promote the device performance. More importantly, the electron mobility of the δ-Cu₂S bilayer is as high as 5786 cm² V⁻¹ s⁻¹, about 25 times that of the β-Cu₂S bilayer (222 cm² V⁻¹ s⁻¹ along the armchair direction and 246 cm² V⁻¹ s⁻¹ along the zigzag direction). The high mobility of the δ-Cu₂S bilayer also stems from the small carrier effective masses (∼0.16 m₀), relatively small deformation potential constant (∼3.03 eV), and large elastic modulus (∼95 J m⁻²). The hole mobility of the δ-Cu₂S bilayer (up to 2107 cm² V⁻¹ s⁻¹) is notably higher than that of the monolayer counterpart as well as that (79–109 cm² V⁻¹ s⁻¹) of the β-Cu₂S bilayer. Therefore, the δ-Cu₂S monolayer and bilayer exhibit more prominent carrier transport properties than the β-Cu₂S bilayer, making layered δ-Cu₂S a very promising candidate for future applications in electronic devices.

Conclusions

We have predicted a new phase of a Cu₂S monolayer (δ-Cu₂S) and explored its atomic structure, bonding character, structural, thermal and chemical stabilities, and electronic properties. In particular, δ-Cu₂S sheets are energetically more favorable than recently synthesized 2D β-Cu₂S sheets, and exhibit excellent dynamical and thermal stabilities. Importantly, the dissociation and chemisorption of O₂ on δ-Cu₂S are kinetically hampered by the relatively high activation energy of 1.98 eV, suggesting that δ-Cu₂S can better withstand oxidation in ambient air. δ-Cu₂S has a modest direct band gap of 1.26 eV, and an ultrahigh electron mobility of up to 6880 cm² V⁻¹ s⁻¹. The large difference between the electron and hole mobilities can be valuable for the efficient separation of electrons and holes for solar conversion applications. Furthermore, the δ-Cu₂S bilayer possesses lower formation energy and much higher carrier mobility (∼5786 cm² V⁻¹ s⁻¹) than the β-Cu₂S bilayer. Given that the β-Cu₂S sheets have been recently synthesized in the laboratory, it is expected that δ-Cu₂S sheets can be fabricated in the near future. In view of the combination of their novel electronic properties and superior oxidation resistance, δ-Cu₂S sheets may play an even more important role in future 2D electronic devices than β-Cu₂S sheets.

Computational methods

First-principles calculations were performed using the Vienna ab initio simulation package (VASP 5.4),⁵¹ with the plane-wave basis set with an energy cutoff of 500 eV, the projector augmented wave potentials (PAW),^52,53 and the generalized gradient approximation parameterized by Perdew, Burke and Ernzerhof (GGA-PBE) for the exchange–correlation functional.⁵⁴ The convergence criteria for total energy and residual force on each atom were set to 10⁻⁷ eV and 0.01 eV Å⁻¹, respectively. For a unit cell of a 2D δ-Cu₂S system, the Brillouin zone was sampled with a Γ-centered 16 × 16 × 1 Monkhorst−Pack⁵⁵k-point grid. Note that the standard GGA functional tends to underestimate the band gaps; thus a hybrid functional (HSE06)⁵⁶ was also used to compute the electronic band structures of δ-Cu₂S. The climbing-image nudged elastic band (CI-NEB) method⁵⁷ was employed to investigate the adsorption kinetics of the O₂ molecule, and to determine the activation energy for chemisorption. To this end, a 3 × 3 × 1 supercell containing 18 S atoms and 36 Cu atoms was built. Five images were used to calculate the reaction path. The intermediate image of each CI-NEB simulation was relaxed until the perpendicular forces were less than 0.02 eV Å⁻¹. The DFT-D3 method⁵⁸ was used to take into account the long-range van der Waals (vdW) interactions in the O₂ adsorption and layered systems.

To examine the dynamical stability of the δ-Cu₂S monolayer, phonon dispersion was computed using the Phonopy code based on the density functional perturbation theory (DFPT)⁵⁹ as incorporated with VASP. The Born–Oppenheimer molecular dynamics (BOMD) simulations with the PAW method and the PBE functional were carried out to assess the thermal stabilities of the predicted δ-Cu₂S. In the BOMD simulations, the initial configuration of δ-Cu₂S with a 3 × 3 × 1 supercell was annealed at several temperatures. Each BOMD simulation in the NVT ensemble lasted for 10 ps with a time step of 1.0 fs, and the temperature was controlled using the Nosé–Hoover method.⁵⁹

The particle-swarm optimization (PSO) method implemented in the CALYPSO code^60,61 was employed to search for low-energy structures of 2D (Cu₂S)_n (n = 1–4) monolayer sheets. As an unbiased global optimization method, the PSO algorithm has successfully predicted a variety of 2D structures with high stability.^62–67 In our PSO search, both the population size and the number of generation were set to 30. The required structural relaxations were performed using the PBE functional, as implemented in the VASP 5.4 code.⁵¹

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (11574040, 21525311, 21373045, 21773027, 21403005, 21803002), the Fundamental Research Funds for the Central Universities of China (DUT16LAB01, DUT17LAB19), the National Key Research and Development Program of China (No. 2017YFA0204800), and the Jiangsu 333 project (BRA2016353). Y. G. is supported by the China Scholarship Council (CSC, 201706060138). Q. W. is supported by CSC (201606090079) and the Scientific Research Foundation of Graduate School of Southeast University (YBJJ1720). X. C. Z. is supported by the National Science Foundation (NSF) through the Nebraska Materials Research Science and Engineering Center (MRSEC) (Grant No. DMR-1420645). We acknowledge the computing resource from the Supercomputing Center of Dalian University of Technology, the University of Nebraska Holland Computing Center and the National Supercomputing Center in Tianjin.

References

1. Y. Xie, L. Carbone, C. Nobile, V. Grillo, S. D’Agostino, F. Della Sala, C. Giannini, D. Altamura, C. Oelsner, C. Kryschi and P. D. Cozzoli, ACS Nano, 2013, 7, 7352–7369.
2. J. M. Luther, P. K. Jain, T. Ewers and A. P. Alivisatos, Nat. Mater., 2011, 10, 361.
3. S.-W. Hsu, K. On and A. R. Tao, J. Am. Chem. Soc., 2011, 133, 19072–19075.
4. Y. Zhao, H. Pan, Y. Lou, X. Qiu, J. Zhu and C. Burda, J. Am. Chem. Soc., 2009, 131, 4253–4261.
5. H. Zheng, J. B. Rivest, T. A. Miller, B. Sadtler, A. Lindenberg, M. F. Toney, L.-W. Wang, C. Kisielowski and A. P. Alivisatos, Science, 2011, 333, 206.
6. S. He, G.-S. Wang, C. Lu, J. Liu, B. Wen, H. Liu, L. Guo and M.-S. Cao, J. Mater. Chem. A, 2013, 1, 4685–4692.
7. B. Zhao, G. Shao, B. Fan, W. Zhao, Y. Xie and R. Zhang, J. Mater. Chem. A, 2015, 3, 10345–10352.
8. B. Li, L. Huang, G. Zhao, Z. Wei, H. Dong, W. Hu, L.-W. Wang and J. Li, Adv. Mater., 2016, 28, 8271–8276.
9. R. Cai, J. Chen, J. Zhu, C. Xu, W. Zhang, C. Zhang, W. Shi, H. Tan, D. Yang, H. H. Hng, T. M. Lim and Q. Yan, J. Phys. Chem. C, 2012, 116, 12468–12474.
10. F. Han, W.-C. Li, D. Li and A.-H. Lu, ChemElectroChem, 2014, 1, 733–740.
11. Y. Kim, K. Y. Park, D. M. Jang, Y. M. Song, H. S. Kim, Y. J. Cho, Y. Myung and J. Park, J. Phys. Chem. C, 2010, 114, 22141–22146.
12. R. L. N. Chandrakanthi and M. A. Careem, Thin Solid Films, 2002, 417, 51–56.
13. G. Liu, T. Schulmeyer, J. Brötz, A. Klein and W. Jaegermann, Thin Solid Films, 2003, 431-432, 477–482.
14. M. Ashry and S. A. Fayek, Renewable Energy, 2001, 23, 441–450.
15. L.-W. Wang, Phys. Rev. Lett., 2012, 108, 085703.
16. S.-H. Lee, Y. Jung and R. Agarwal, Nat. Nanotechnol., 2007, 2, 626.
17. M. H. R. Lankhorst, B. W. S. M. M. Ketelaars and R. A. M. Wolters, Nat. Mater., 2005, 4, 347.
18. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva and A. A. Firsov, Science, 2004, 306, 666.
19. Y. Liu, N. O. Weiss, X. Duan, H.-C. Cheng, Y. Huang and X. Duan, Nat. Rev. Mater., 2016, 1, 16042.
20. M. Xu, T. Liang, M. Shi and H. Chen, Chem. Rev., 2013, 113, 3766–3798.
21. M. Li, J. Dai and X. C. Zeng, Nanoscale, 2015, 7, 15385–15391.
22. Y. Hong, J. Zhang and X. C. Zeng, Nanoscale, 2018, 10, 4301–4310.
23. L. Ma, M.-G. Ju, J. Dai and X. C. Zeng, Nanoscale, 2018, 10, 11314–11319.
24. F. B. Romdhane, O. Cretu, L. Debbichi, O. Eriksson, S. Lebègue and F. Banhart, Small, 2015, 11, 1253–1257.
25. S. Rauf, K. TaeWan, M. Jihun and K. Sang-Woo, Nanotechnology, 2017, 28, 505601.
26. J. B. Rivest, L.-K. Fong, P. K. Jain, M. F. Toney and A. P. Alivisatos, J. Phys. Chem. Lett., 2011, 2, 2402–2406.
27. K. Anuar, Z. Zainal, M. Z. Hussein, N. Saravanan and I. Haslina, Sol. Energy Mater. Sol. Cells, 2002, 73, 351–365.
28. C.-G. Andres, V. Leonardo, P. Elsa, O. I. Joshua, K. L. Narasimha-Acharya, I. B. Sofya, J. G. Dirk, B. Michele, A. S. Gary, J. V. Alvarez, W. Z. Henny, J. J. Palacios and S. J. V. D. Z. Herre, 2D Mater., 2014, 1, 025001.
29. O. I. Joshua, A. S. Gary, S. J. V. D. Z. Herre and C.-G. Andres, 2D Mater., 2015, 2, 011002.
30. A. Ziletti, A. Carvalho, D. K. Campbell, D. F. Coker and A.H. Castro Neto, Phys. Rev. Lett., 2015, 114, 046801.
31. Y. Guo, S. Zhou, Y. Bai and J. Zhao, J. Chem. Phys., 2017, 147, 104709.
32. Y. Guo, S. Zhou, Y. Bai and J. Zhao, ACS Appl. Mater. Interfaces, 2017, 9, 12013–12020.
33. J.-S. Kim, Y. Liu, W. Zhu, S. Kim, D. Wu, L. Tao, A. Dodabalapur, K. Lai and D. Akinwande, Sci. Rep., 2015, 5, 8989.
34. A. Savin, R. Nesper, S. Wengert and T. F. Fässler, Angew. Chem., Int. Ed., 1997, 36, 1808–1832.
35. A. Savin, O. Jepsen, J. Flad, O. K. Andersen, H. Preuss and H. G. von Schnering, Angew. Chem., Int. Ed., 1992, 31, 187–188.
36. A. D. Becke and K. E. Edgecombe, J. Chem. Phys., 1990, 92, 5397–5403.
37. R. F. W. Bader, Acc. Chem. Res., 1985, 18, 9–15.
38. J. Hernández-Trujillo and R. F. W. Bader, J. Phys. Chem. A, 2000, 104, 1779–1794.
39. G. Gabriella, K. Jiří, F.-A. Felix and M. Angelos, J. Phys.: Condens. Matter, 2012, 24, 424216.
40. Y. Cai, G. Zhang and Y.-W. Zhang, Sci. Rep., 2014, 4, 6677.
41. S. Kc, R. C. Longo, R. M. Wallace and K. Cho, J. Appl. Phys., 2015, 117, 135301.
42. J. G. Speight, Lange's handbook of chemistry, McGraw-Hill, New York, 2005.
43. H. J. Conley, B. Wang, J. I. Ziegler, R. F. Haglund, Jr., S. T. Pantelides and K. I. Bolotin, Nano Lett., 2013, 13, 3626–3630.
44. A. R. Klots, A. K. M. Newaz, B. Wang, D. Prasai, H. Krzyzanowska, J. Lin, D. Caudel, N. J. Ghimire, J. Yan, B. L. Ivanov, K. A. Velizhanin, A. Burger, D. G. Mandrus, N. H. Tolk, S. T. Pantelides and K. I. Bolotin, Sci. Rep., 2014, 4, 6608.
45. G. Fiori and G. Iannaccone, Proc. IEEE, 2013, 101, 1653–1669.
46. S. Bruzzone and G. Fiori, Appl. Phys. Lett., 2011, 99, 222108.
47. S. Takagi, A. Toriumi, M. Iwase and H. Tango, IEEE Trans. Electron Devices, 1994, 41, 2357–2362.
48. J. Qiao, X. Kong, Z.-X. Hu, F. Yang and W. Ji, Nat. Commun., 2014, 5, 4475.
49. G. Yu, Z. Si, Z. Junfeng, B. Yizhen and Z. Jijun, 2D Mater., 2016, 3, 025008.
50. W. Zhang, Z. Huang, W. Zhang and Y. Li, Nano Res., 2014, 7, 1731–1737.
51. G. Kresse and J. Furthmüller, Phys. Rev. B: Condens. Matter Mater. Phys., 1996, 54, 11169–11186.
52. G. Kresse and D. Joubert, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 1758–1775.
53. P. E. Blöchl, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 50, 17953–17979.
54. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865–3868.
55. H. J. Monkhorst and J. D. Pack, Phys. Rev. B: Condens. Matter Mater. Phys., 1976, 13, 5188–5192.
56. J. Heyd, G. E. Scuseria and M. Ernzerhof, J. Phys. Chem. C, 2003, 118, 8207–8215.
57. G. Mills, H. Jónsson and G. K. Schenter, Surf. Sci., 1995, 324, 305–337.
58. L. A. Burns, Á. V. Mayagoitia, B. G. Sumpter and C. D. Sherrill, J. Chem. Phys., 2011, 134, 084107.
59. S. Baroni, S. de Gironcoli, A. Dal Corso and P. Giannozzi, Rev. Mod. Phys., 2001, 73, 515–562.
60. Y. Wang, M. Miao, J. Lv, L. Zhu, K. Yin, H. Liu and Y. Ma, J. Phys. Chem. C, 2012, 137, 224108.
61. Y. Wang, J. Lv, L. Zhu and Y. Ma, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 82, 094116.
62. Q. Wu, W. W. Xu, B. Qu, L. Ma, X. Niu, J. Wang and X. C. Zeng, Mater. Horiz., 2017, 4, 1085–1091.
63. Y. Ma, A. Kuc and T. Heine, J. Am. Chem. Soc., 2017, 139, 11694–11697.
64. Y. Li, Y. Liao and Z. Chen, Angew. Chem., Int. Ed., 2014, 53, 7248–7252.
65. X. Wu, J. Dai, Y. Zhao, Z. Zhuo, J. Yang and X. C. Zeng, ACS Nano, 2012, 6, 7443–7453.
66. X. Luo, J. Yang, H. Liu, X. Wu, Y. Wang, Y. Ma, S.-H. Wei, X. Gong and H. Xiang, J. Am. Chem. Soc., 2011, 133, 16285–16290.
67. F. Ma, Y. Jiao, G. Gao, Y. Gu, A. Bilic, Z. Chen and A. Du, Nano Lett., 2016, 16, 3022–3028.

---

Footnotes

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8nh00216a

‡ Y. G. and Q. W. contributed equally to this work.

---