Large exciton binding energy, superior mechanical flexibility, and ultra-low lattice thermal conductivity in BiI₃ monolayer

The appropriate choice of computational methods is vital to accurately describe the band gap characteristics and optical properties of BiI₃ monolayer. To this end, we first calculated the relevant properties of bulk BiI₃. The corresponding analyses are provided in the supporting information (supplemental material (https://stacks.iop.org/JPCM/34/055302/mmedia)). Calculations without considering SOC reproduced a sound E_b for bulk BiI₃, thereby giving us some confidence in using this approach for the BiI₃ monolayer. Therefore, we calculated the physical properties without including the SOC effect in the following sections. The calculated electronic band gaps of monolayer BiI₃ within the PBE + SOC, PBE, and HSE06 + SOC schemes are 1.635, 2.580, and 2.214 eV respectively, with conduction band minimum (CBM) at Γ point and valence band minimum at the midway of the Γ → K direction, as presented in figure S3, which is perfectly consistent with references [3, 4].

As mentioned previously, we investigated the E_b in the framework of the G₀W₀ + BSE scheme with the Tamm–Dancoff approximation, which accounts for the many-body electron interactions and the electron–hole interactions. For a 2D system, a subtlety arises from the calculated dielectric constant, which includes the contributions from the layer of material and the layer of vacuum with ε₀ (permittivity of vacuum). The size of the vacuum layer should approach to ∞ in principle to eliminate the influence [26]. The inset in figure 2 shows E_b as a function of L_z , indicating that L_z of 30 Å provides a relatively well-converged test result. Figure 2 shows the imaginary part of dielectric constant ε₂ obtained from DFT + G₀W₀ + BSE scheme and the G₀W₀ band structures of BiI₃ monolayer with L_z = 30 Å. The minimum direct band gap occurs at Γ point with a value of 4.196 eV; this value is slightly smaller than the direct excitation at M point (4.272 eV). In figure 2(a), the first peak is observed at the photon energy of 3.176 eV, which corresponds to the optical gap. E_b is calculated by taking the difference between the minimum direct G₀W₀ band gap and the G₀W₀ + BSE optical gap.

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The obtained E_b is 1.02 eV, which is larger than that of bulk (0.224 eV). This condition is due to the reduced dielectric screening in 2D materials. This result is fairly consistent with the theoretical modification ${E}_{\text{b}}^{2\text{D}}=\mathrm{4}{E}_{\text{b}}^{3\text{D}}$ [27] for 2D materials. The large E_b observed in monolayer BiI₃ is comparable with the reported value of 500–1100 meV in the monolayers of molybdenum and tungsten dichalcogenides [28, 29]. This finding indicates that the strongly bound exciton is fairly stable and has an excellent resistance to the thermal disturbance and external electric field [30, 31]. At the same time, a large E_b may indicate a defining role in the optoelectronic processes in this strongly quantum-confined material after photoexcitations [32].

Therefore, the material exhibits great potential for the development of new excitonic lasers [33]. Theoretical calculations revealed that the CBM states mainly originate from the Bi-6p_z orbitals, and the valence band states near the Fermi level is dominated by I-5p states with a large admixture of Bi-6s states [3, 4], as shown in the insets of figure 3. This scenario in monolayer BiI₃ is extremely similar to in its bulk counterpart, indicating that the exciton transitions s → p can be attributed to the transitions in the Bi metal ion [34].

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To obtain preliminary insight into the mechanical property, we calculated the elastic constants and ideal tensile strength of monolayer BiI₃ from the strain–energy and stress–strain relationship. The monolayer BiI₃ is a hexagonal structure, where the independent linear elastic constants reduce to two components of C₁₁ and C₁₂, and the mechanical stability criteria are given as C₁₁ > C₁₂ and C₆₆ = (C₁₁ − C₁₂)/2 > 0 [35]. The elastic constants can be deduced from the strain–energy relation as follows:

$$\begin{equation*}{E}_{\text{s}}=\frac{V}{\mathrm{2}}\sum\limits _{i=1}^{6}\sum\limits _{j=1}^{6}{C}_{i,j}{e}_{i}{e}_{j},\end{equation*}$$

where V is the volume of the undistorted crystal, e_i is an element of strain tensor e = (e₁, e₂, e₃, e₄, e₅, e₆), C_i,j are elastic constants [36, 37]. For the unstained structure, C₁₁ = C₂₂ = 7.216 N m⁻¹, and C₁₂ = C₂₁ = 2.196 N m⁻¹; thus, they satisfy the Born–Huang criteria [35], indicating that the monolayer is mechanically stabile. The Young's modulus (Y) and Poisson's ratio (ν) can be calculated from equations: $Y=({C}_{11}^{2}-{C}_{12}^{2})/{C}_{11}$, and ν = C₁₂/C₁₁. The obtained Y and ν are 6.547 N m⁻¹ and 0.304, respectively. The C_ij values of graphene, monolayer MoS₂, and BiI₃ monolayer with strain ε at 5% are tabulated in table 1. Such small Y and large ν demonstrate that BiI₃ is more flexible and ductile than grapheme (330–348 N m⁻¹) [37, 39] and MoS₂ (122–126 N m⁻¹) [39, 40], implying that it is unlikely to be free-standing. Therefore, a suitable substrate is highly desirable to experimentally support the 2D film. Hexagonal BN is a suitable choice for the epitaxial growth for BiI₃ monolayer due to its low cost, resistance to high temperature and strong acid corrosion, extremely stable chemical properties, ultrawide band gap [41]. The lattice parameters of h-BN were determined to be a = b = 2.50 Å and c = 7.44 Å in our calculation, therefore, in-plane parameters are very close to the experimental value of 2.504 Å [42]. The lattice parameters of BiI₃ monolayer were determined to be a = b = 7.80 Å which means a small mismatch of 1.9% = [7.80 − (7.50 + 7.80)/2]/[(7.50 + 7.80)/2] with the lattice parameter of 2.50 Å of h-BN.

To explore the flexibility and ideal tensile strengths, we investigated the mechanical responses to tensile strain. The strain is defined as ɛ = (a − a₀)/a₀, where a₀ and a denote the lattice parameters of relaxed and strained systems. Primitive unit cell and rectangular cell (figure 1(c)) were adopted to calculate the responses to the biaxial and uniaxial strains, respectively. When a uniaxial strain was applied in x/y direction, the lattice constant in the y/x direction and internal atomic coordinates were fully relaxed by using the minimum energy method [38]. Under a biaxial strain, the internal atomic coordinates fully relaxed with the given lattice unchanged. The in-plane stress must be rescaled by L_z /d₀ because the unit cell includes the vacuum space along the z direction [38, 39]. Here, d₀ is the effective layer thickness of BiI₃ monolayer. Following the approach descripted in reference [38], we take the interlayer spacing 7.68 Å of bulk BiI₃ as d₀ for the monolayer.

When the strain is less than 4% along the x or y direction, we can observe the near degeneracy of energy and stress responses, as shown in figure 3. This condition implies that the monolayer BiI₃ is characterized by an isotropic elastic behavior in linear elastic regime, which is consistent with the theory for 2D system with hexagonal symmetry [37]. With the increase in ε, the calculated energy and stress curves are nonlinear and exhibit difference between the x and y directions. The critical strains along the x and y directions are 37% and 34%, respectively, and the corresponding stresses are 4.09 and 3.09 GPa, respectively. The biaxial stress–strain curve reaches its maximum of 2.60 GPa when the strain reaches up to 30%, which is larger than the previously reported (13%) [3]. The discrepancy stems from the relaxation for the strained structure being done or not. The maximum value of stress at critical strain is referred to as the ideal strength, which is the ultimate stress that a material can bear. The biaxial critical strain and ideal strengths are summarized in table 1. One can see that the monolayer BiI₃ is more flexible and ductile than graphene and MoS₂ by comparison.

Determining whether the monolayer BiI₃ remains stable before approaching the critical strain is necessary because elastic instability or dynamical instability may intrude prior to the maximum stress on the strain path. We calculated the phonon dispersions of the monolayers at various biaxial strains. Figure 4 shows that the BiI₃ monolayer is still dynamically stable at ε = 7%. It becomes dynamically instable when the tensile strain rises up to 8% because imaginary frequencies are identified at the M point. Since the system with strain of 7% is dynamically stable, the system at the strain of 5% is dynamically stable. The elastic constants C₁₁ = C₂₂ = 7.216 N m⁻¹ and C₁₂ = C₂₁ = 2.196 N m⁻¹ at ε = 5% also indicate the elastic stability. Interestingly, the Poisson's ratio (ν = 0.469) approaches the limiting value (0.5). Therefore, tensile strain sharply softens the material and highly controls the band gap and electronic structure of single layer BiI₃ [3, 4].

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We solve the linearized phonon BTE with RTA [25] to obtain the macroscopic thermal conductivity tensor k _L which can be calculated as a sum of contributions from individual phonon modes λ according to [44]:

$$\begin{equation*}{\kappa }_{\text{L}}=\frac{1}{V}\sum\limits _{\lambda }{\;C}_{\lambda }{\mathbf{v}}_{\lambda }\otimes {\mathbf{v}}_{\lambda }{\tau }_{\lambda },\end{equation*}$$

where V is the volume of the unit cell. Suffix λ ≡ q, j represents the mode index, which enumerates phonon wave vector q and branch j. C_λ , v_λ , and τ_λ are the specific heat, group velocity, and phonon lifetime, respectively. The thermal conductivity tensor k _L give the cumulative lattice thermal conductivity of six elements, k_xx , k_yy , k_zz , k_yz , k_xz , k_xy . The total lattice thermal conductivity k_L can be calculated by k_L = (k_xx + k_yy )/2 for 2D materials [45]. Figure 5 shows the temperature-dependent k_L of monolayers. At strains ε = 0% and 5%, the calculated k_L values at 300 K is 0.247 and 0.206 W m⁻¹ K⁻¹, respectively, which is smaller than that of MoS₂ monolayer (31.2 W m⁻¹ K⁻¹) by two orders of magnitude and is extremely lower than that of graphene (3080–5150 W m⁻¹ K⁻¹) [46]. The iterative method as performed in ShengBTE code [47] gives rise to a value of 0.398 W m⁻¹ K⁻¹, as shown in figure S2.

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To explore the origin of ultralow k_L, we examined the phonon dispersion. Figure 6 shows the phonon dispersion spectra along high-symmetry directions in the Brillouin zone. Twenty-four phonon branches, including three acoustical branches and 21 optical branches, are found due to eight atoms in the primitive cell. In the small region of long wavelength limit (λ → ∞), the transverse acoustic (TA) and longitudinal acoustic (LA) phonon modes are linear, whereas the out-of-plane acoustic (ZA) phonon mode is parabolic. In pristine semiconductors and insulators, phonons rather than electrons take the main responsibility for heat transport, where the low-frequency acoustic phonon branches are the dominant contributors to k_L. This condition is consistent with the inset in figure 5. No phonon frequency gap is found between acoustic and optical phonon branches. Therefore, acoustical and optical branches in the range of 0.49 THz to 0.82 THz involve the resistive Umklapp scattering through the channel of acoustic + acoustic → optical [48]. The fitting lines of k_L are inversely proportional to the temperature at high temperature, further verifying that the anharmonic phonon–phonon interaction is a key to the thermal resistivity at high temperature.

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To explain the reason for the low k_L, we analyzed the phonon group velocity v_λ and phonon lifetime τ_λ . Phonon v_λ can be determined from

$$\begin{equation*}{v}_{\lambda ,j}=\frac{\partial {\omega }_{j}(q)}{\partial q},\end{equation*}$$

where ω, j, and q represent the frequency, phonon branch index, and the wave vector, respectively. Figure 6(a) shows the v_λ distributions for each acoustic phonon mode in the Brillouin zone. The calculated v_λ for LA and TA phonons close to the Γ point ranges from 1.34 to 1.78 km s⁻¹, which are on the same magnitude as that (6.61 and 3.90 km s⁻¹) of MoS₂ monolayer, but they are smaller than that of graphene (21–22 km s⁻¹) [43, 49, 50]. Although large v_λ is observed around 2.5 and 3.6 THz for the high-lying optic modes, the corresponding phonon lifetime τ_λ is smaller than of the acoustic branches by two orders of magnitude. Thus, the acoustic branch is responsible for thermal transport. Their small group velocity lowers the thermal conductivity. Such low v_λ originates from the weak interaction between Bi and I atoms and their heavy atomic mass. At the long wavelength limit, the group velocity v_λ reaches the speed of sound, ${v}_{\text{s}}=\sqrt{Y/\rho }\propto \sqrt{Y/M}$ [51], in a crystal. Here, ρ and M are the mass density and mass in a unit cell, respectively, and Y is intimately associated with the bond strength. Thus, the larger the atomic number and the smaller the Young's modulus, the lower the phonon thermal conductivity.

Another underlying cause for the ultralow thermal conductivity is the short τ_λ given that k_L is directly proportional to τ_λ . In the RTA, the τ_λ are calculated as the inverse of the phonon linewidths Γ_λ : τ_λ = 1/(2Γ_λ ) [25]. Figure 6(b) shows that the average τ_λ is less than 200 ps, which is approximately two orders of magnitude lower than that of MoS₂ monolayer [52]. Such short τ_λ results in a low k_L. The scattering between acoustic and optical phonon modes is the main reason for the extremely short phonon relaxation time.

As mentioned previously, the strained monolayer within a strain of less than 7% is a thermodynamically stable structure. Therefore, determining the effect of strain on k_L is important. The corresponding k_L is illustrated in figure 5(a) when the biaxial strain of 5% is applied. The estimated k_L value is 0.206 W m⁻¹ K⁻¹ at 5% strain and 300 K. Compared with the unstrained case, the strain at 5% slightly lowers k_L. The value is lower than that (0.30 W m⁻¹ K⁻¹) of penta-germanene with the lowest k_L in 2D crystal materials [53] and most other 2D materials [54]. The low k_L results from the short length of Bi–I bonds (3.118 Å → 3.144 Å). Similar to the electronic band gap, k_L can be effectively regulated and controlled by applying external strain, which can result in a wider and more flexible range of application.