Exciton-plasmon polariton coupling and hot carrier generation in two-dimensional SiB semiconductors: a first-principles study

2.1 Electronic and optical properties calculations

The electronic properties of SiB and H-SiB and the optical properties of H-SiB were computed utilizing first-principles many-body perturbation theory (MBPT) BerkeleyGW (BGW) (BerkeleyGW, Oakland, CA, USA) [34], [35], [36] package with a Perdew, Burke, and Ernzerhof starting point. GW approach and Beth–Salpeter equation (BSE) are described in Sections S3 and S4 in SI, respectively.

2.2 Exciton–plasmon interaction in semiconductors

The main parameters in the calculation of exciton–plasmon interaction are noninteracting χ⁰(q, ω) and interacting χ(q, ω) polarizability. The density–density response function of noninteracting electrons in the reciprocal space [χGG′0(q, ω)] and interacting polarizability χ_GG′ are calculated to understand the exciton–plasmon interactions in H-SiB. The effect of temperature on the polarizability and self-energy is also studied. The details are given in SI (Sections S7–S10). We, then, calculated the diagonal elements of the random phase approximation dielectric function ε_GG′ (see Eq. S30). Then plasmon modes can be obtained when the real part of dielectric function [ε(q, ω)] is set to zero. The electron energy loss (EEL) function is considered as the imaginary part of the inverse of the dielectric function ε(q, ω).

2.3 Light emission

After calculating the scattering process (Section S11 in SI), we used Green’s function and nonequilibrium BSE, taking into account the e–h interactions, to compute the light absorption and emission [37], [38]. The details are given in SI (Section S12).

3.1 Dynamical stability

The graphene-like structure of a silicon boride monolayer is fully relaxed using the quasi-Newton method to determine the equilibrium atomic positions (Figure 1A, B) as this algorithm is very fast and efficient close to local minima. In this method, the atomic forces and the stress tensor are used to determine the equilibrium positions [33]. Also, the lattice constants of this unit cell are optimized using the Birch–Murnaghan equation state [39], [40] The results show that a pure SiB monolayer has a fully flat structure, and the optimized lattice constants, a=b=3.4 Å, are in good agreement with previous results presented by Hansson et al. [8]. The phonon dispersion spectrum for pure SiB monolayer has a negative vibrational mode (Figure S1), establishing that it is dynamically unstable. To reach the goal of stabilizing the structure, surface functionalization using hydrogen atoms is here used in different configurations. The results show that semifunctionalized monolayers, for which the H atoms are adsorbed onto boron atoms, do not reach a stable form after force minimization calculations. Thus, semifunctionalization on silicon sites leads to stability, and we limit force minimization to these configurations (Figure 1C, D).

Figure 1:

Crystallographic structure of SiB and H-SiB monolayers.

(A) Top and (B) side views of pristine honeycomb SiB monolayer. (C) Top and (D) side views of the optimized structure of H-SiB monolayer. The hydrogen atoms are adsorbed onto the Si atomic sites. Red, cyan, and blue balls show silicon, boron, and hydrogen atoms, respectively.

3.2 Electronic properties

The band diagrams of H-SiB monolayer are illustrated in Figure 2. Our results show that SiB monolayer is a nonmagnetic metal with partially occupied states around the Fermi level. As silicon and boron atoms have three sp² bonds in a honeycomb configuration, the σ bands are fully occupied and are located below the Fermi level. On the other hand, the p_z orbitals of Si hybridize with p_z orbitals of boron, and the hybridized π orbitals contribute symmetrically to both sides of the SiB monolayer. These π orbitals are partly filled and create a metallic band around the Fermi level (Figure 3A). As can be seen in this figure, pure SiB monolayer, the Fermi level lies in the π band of the hybridized B-2p_z and Si-3p_z orbitals. So, when a hydrogen atom bonds with a Si atom, it just changes the 3p_z configuration of Si atoms and forms a σ bond between hydrogen and Si-3p_z (σ_Si-H). As a result, the empty 2p_z orbitals of boron do not hybridize with Si-3p_z and locate above the Fermi level. Finally, this mechanism leads to creation of an energy gap between bonding σ_Si-B and nonbonding 2p_z bands (Figure 3B).

Figure 2:

Calculated band structure with GW for pure SiB monolayer.

The band structure is plotted along the high symmetry k-path, K−Γ−M−K. The black dashed line denotes the Fermi level at zero energy.

Figure 3:

Band diagram for pure (A) and (B) semihydrogenated SiB.

Partial density of states (PDOS) calculations indicates that σ (p_x+p_y) and π (p_z) bands are fully and partly filled, respectively, for a pure SiB monolayer. Figure 4A and C exhibit the PDOS of p_z and p_x+p_y orbitals, of silicon and boron atoms in the pure SiB monolayer, respectively. Both bonding p_x+p_y orbitals are fully occupied, while the p_z orbitals cross the Fermi level. The PDOS for silicon and boron atoms in H-SiB (Figure 4B, D) confirms our inferred band diagram of H-SiB structure. Therefore, the SiB monolayer experiences a transition from metal to semiconductor after semihydrogenation.

Figure 4:

Total density of states for (A) pure and (B) semihydrogenated SiB monolayer and partial density of states of p_z and p_x+p_y suborbitals for silicon and boron atoms in (C) pure and (D) semihydrogenated SiB monolayer.

3.3 Strain engineering the electronic and optical properties

The unstrained H-SiB monolayer is a direct semiconductor along the Γ direction and remains of direct type under strains in the range −6% to +6%, where the negative strain is compressive and the positive is tensile (Figure 5A–G).

The energy gap region is highlighted in yellow for all cases. Figure 5A–G show that H-SiB experiences a bandgap reduction under compressive strain and an expansion under tensile strain. We developed an empirical equation to correlate the bandgap energy of the material to the applied external strain (Figure 5H):

Figure 5:

Electronic band structure of unstrained and strained H-SiB under strain ranging from −6% to 6%.

The gap direction remains direct, Γ to Γ, over the whole of applied strain range, −6% to +6%. The gap region is highlighted in yellow.

where E_g is the bandgap energy, and ε is the applied external strain. As can be seen in Figure 5H, this relationship is linear. As the behavior of the material in the domain of applied strain (−6% to +6%), is linear (Figure S2), the gap energy of the material increases linearly when the applied external increases from −6% to +6% (Figure 5).

Optical properties of unstrained and strained H-SiB were computed based on first-principles MBPT as implemented in BGW (see Methods). We compared the optical properties of H-SiB monolayer without considering e–h interactions and with e–h (excitonic) interactions using BSE. Previous studies revealed that excitonic effects modify the optical properties of bulk [41], [42] and nanopore Si [43] and even more enhance the exciton binding energy of Si nanowires [44], [45].

Both real [ε₁(ω)] and the imaginary [ε₂(ω)] parts of the dielectric function are calculated using BGW and demonstrated in Figure 6. In Figure 6, the imaginary and real parts of dielectric function and absorption and reflectivity of SiB and H-SiB with and without e–h (exciton) interactions are indicated. Strong interactions of particle– antiparticle (e–h) result in pair excitonic effects, which drastically affect the photoresponse of nanoscale devices. These graphs confirm that considering the e–h interactions enhances optical absorption spectrum in visible range.

Figure 6:

(A) Imaginary (ϵ₂) and (B) real part (ϵ₁) of dielectric function of unstrained and strained H-SiB. (C) Absorption (α) and (D) Reflectivity coefficients of unstrained and strained H-SiB with and without electron–hole (e–h) interaction. Joint density of states (JDOS) of (E) unstrained and (F) strained H-SiB with (solid lines) and without (dashed lines) e–h (exciton) interactions.

The strain field modifies the absorption spectra (Figure 6C) and the optical gap of the H-SiB monolayer, both of which are crucial physical quantity for the device operation. As displayed in Figure 6C, compressive strains shift the absorption spectrum to a lower energy (red shift), whereas tensile strains shift the absorption spectrum to an upper energy (blue shift). The compressive strain deforms covalent bonds toward reduction of screening effect; therefore, the photoexcited QP in the H-SiB monolayer experiences stronger Coulomb interactions. Consequently, the compressive strained H-SiB emitted lower photon energy in comparison to the tensile strained nanostructures.

3.4 Surface plasmons

Surface plasmons (SPs) are collective longitudinal oscillations of valence electrons in metals or semiconductors that enable control of electromagnetic energy and confinement at subwavelength scale [46]. In general, SPs can exist in any material with mobile charge carriers, which show reactive response to the electric field [46]. The EEL spectroscopy is a main tool for detecting collective excited electrons due to SPs. Figure 7 shows the EEL function for different momentum transfer vectors, q in unstrained and strained H-SiB monolayers with and without considering e–h interactions. This figure suggests that SP exists and shows a blue shift for different momentum transfer vectors as a result of interband transition. Due to the structural deformation in the strained H-SiB monolayers, their EEL function shows three peaks, which confirms the strong inelastic scattering in the strained nanostructures. To compare the dispersion of SP for unstrained and strained H-SiB, we calculated [ω (q)] as a dispersion function, obtained by GW calculations, as can be seen in Figure 8. Importantly, the dispersion modes of SP improve for strained H-SiB, supporting previous EEL results.

Figure 7:

Electron-energy loss (EEL) function for different momentum transfers; for (A) unstrained, (B) 6% compressive strained, and (C) 6% tensile strained H-SiB with/without e–h interactions; (D) 3D plot of EEL as a function of strain and photon energy.

Figure 8:

Plasmon modes dispersion [ω (q)] for unstrained and strained H-SiB.

3.5 Light emission

Surface plasmons enhance photoluminescence in metal nanomaterials [27]. To apply this concept for 2D semiconductors and acquire all light spectral features of H-SiB monolayers including thermal parameters on the light emission, we further compute the polarizability matrix. The polarizability of H-SiB monolayer is reported in Figure 9A. Strain (displacement of atoms) and polarization are correlated due to the fact that applying strain on dielectric crystal creates a net dipole moment, which induces additional polarization to the system [47]. Therefore, strained structures possess larger polarizability. Accordingly, in the current research, the strained H-SiB structures show larger polarizability than unstrained H-SiB (Figure 9A). Figure 9B and D present the temperature evolution of polarizability in both unstrained and strained structures. As can be seen in Figure 9, the polarization of H-SiB monolayers decreases with increasing the temperature, which can be correlated to the screening properties of 2D materials. We note that the screening effect of 2D materials increases by temperature due to increasing the scattering rate. The temperature polarizability function χ(q) is defined from the dielectric screening function ε(q) as follows: ε(q)=1+ν(q)χ(q), where ν(q) is the 2D screening Coulomb interaction, and χ(q)= gsgν∑kf(εk)−f(ε|k+q|)εk−ε|k+q|, where f(ε_k) is the Fermi distribution function [48]. This picture can be linked to the temperature dependency of H-SiB polarizability.

Figure 9:

Dipole matrix elements or polarization χ(ω, q) for unstrained and strained H-SiB (A–D).

As HCs in semiconductors are created due to strain and temperature, we calculated the luminescence spectra at finite temperatures and in the presence and absence of strain to understand the effect of HCs on the light emission in H-SiB (Figures 10 and 11). We used Eqs. S42–S44 to compute the phonon assisted luminescence spectrum (Figure 10). All spectra showed a red shift with increasing temperature. This resulted from the enhanced inelastic scattering (loss of energy) as discussed in the previous section and strong e–ph coupling and interaction during the temperature increase. Zhang et al. [49] experimentally reported temperature-dependent photoluminescence of strained and unstrained GaSe nanosheets and found that red shift happened. In Figure 10A, there are two main peaks at 1.13 and 5.16 eV, which can be seen for the unstrained structure. The main peak is shifted from 1.13 eV to 0.74, 0.62, and 0.53 eV, respectively, when the temperature increased from 0 K to 100, 200, and 300 K (Figure 10A). In detail, Figure 11B highlights that the main peaks (peaks I, II) of unstrained H-SiB move toward invisible light region (IR) by increasing temperature. As discussed earlier in Figure 9B, the carrier energy for unstrained nanostructure decreases with increasing temperature. Therefore, the emitted light placed in the IR region has lower energy in comparison to the visible region. In Figure 10B and C, two main peaks at 2.19 and 3.07 eV are for the structure under 6% compressive, and 2.94 and 3.82 are for the structure under 6% tensile strain, respectively. The luminescence spectrum of the strained H-SiB structures under 6% compression deformation follows a similar pattern of unstrained structure, while here a red shift from 3.07 eV to 2.9, 2.81, and 2.64 eV can be, respectively, seen for the main peaks (peak III) when the temperature increases from 0 K to 100, 200, and 300 K (Figure 11C). For the tensile structure at 6% deformation, the same pattern can be observed in Figure 11C, where the main peak (peak III) is red shifted from 3.82 eV to 3.67, 3.54, and 3.38 eV (Figure 11C). A good explanation for the temperature effect on photoluminescence spectrum of strained structures can be attributed to decrease in polarizability of these structures with temperature (Figure 9C, D). Therefore, by increasing the temperature, the carrier energy decreases, and red shift (low energy) happens in the photoluminescence spectrum. To better understand the contributions of the mechanisms responsible for thermal effect, we considered a steady-state solution for the temperature-dependent intensity I for the exciton recombination, which is given by Eq. 2:

Figure 10:

(A–C) Luminescence spectra for unstrained and strained H-SiB by considering electron–electron (e–e) and electron–phonon (e–ph) interaction. (D, E) Temperature-dependent valley-polarized exciton–polariton emission of upper polariton (UP) and lower polariton (LP) quasiparticle eigenstates in the strained H-SiB. The band structure with zooming in on the conduction band (CB) of (F) compressive strained (−6%) and (G) tensile strained (+6%) H-SiB.

Figure 11:

(A) Three-dimensional plot of luminescence function vs. strain and photon energy. Temperature-dependent of luminescence of peaks I, II, and III in (B) unstrained and (C) strained H-SiB. Solid and dashed lines are representatives of compressive (−6%) and tensile strain (6%), respectively.

where E_A is the thermal activation energy [50]. The application of temperature can lead to a change in activation energy with increasing the potential depth.

Cai et al. [27] showed an enhancement in the photoluminescence of gold nanoparticle due to the generation of HCs. Here, we note that HCs improve interlayer e–h interaction in the 2D semiconductors. Figure 10B and C support the idea that HC exciton created by SPP enhances the luminescence intensity (photon density of states) arising from thermal exciton and thermal SP. Figure 10D and E indicate the temperature-dependent valley-polarized exciton–polariton emission of upper polariton and lower polariton QP eigenstates in the strained (−6% and +6%) H-SiB monolayer. The exciton energies are calculated by GW approach and BSE [see SI (Sections S3, and S4) for more details]. The temperature- and strain-dependent photoluminescence suggests that we can tune the light emission in the visible light range for photoelectronic devices made of 2D semiconductors.

3.6 Decay rate and HC generation

To study SPP–electron coupling in the semiconductors, we developed a framework of perturbation theory and Feynman diagram for 2D semiconductors [see more computation details in the SI (Section S13)]. Figure 12A presents a schematic illumination of H-SiB monolayer, in which SPs are polarized and generate SPPs. Scattering of collective electrons of generated SPPs leads to their decay and results in the creation of excitons (pair of hot electron and hole). Figure 12B–D represent the hot hole and electron distribution for unstrained and strained H-SiB at different temperatures within windows of −4… 4 eV Fermi energy. Figure 12 reveals that at finite temperatures the scattering and decay rates of collective electrons of SPPs in H-SiB become more significant at distances far away Fermi level, which results in the generation of hot electrons and holes in those distances.

Figure 12:

(A) Schematic of an H-SiB semiconductor under illumination, showing a polarization of SP and excited SPP and the decay of an SPP to an exciton (pair of hot electron and hole. (B) The HC distribution (red color represents electron, and blue color represents hole) of unstrained and strained H-SiB at different temperatures.

Figure 13 shows mean free path (MFP) of HCs in both unstrained and strained H-SiB monolayers at ground state. To compute MFPs, the HC velocity (the slope of GW band structure) is multiplied into the total relaxation times of different temperatures [51]. Our results show a large spread of MFPs near the Fermi energy (Figure 13A). The total MFP for unstrained 6% compressive strained and 6% tensile strained structures is obtained as 8, 6, and 5 nm, respectively. By increasing the number of generated HCs in the structure, their MFP decreases. Here, as the maximum HCs are generated in the 6% tensile strained structure, the MFP of HCs is the shortest distance in comparison to the other two structures (unstrained and 6% compressive strained). Additionally, a comparison of our calculated HC-MFP for 2D semiconductors (H-SiB) with the reported HC-MFP for metals (Au and Ag) [51] shows that MFPs in metals can range in the order 10 to 40 nm [51], while MFPs in H-SiB is in the order of 3 to 8 nm (Figure 13A). As can be seen in this figure, the MFP for unstrained H-SiB is the most significant one in comparison to both strained structures.

Figure 13:

(A) The mean free path (MFP) of HCs and (B) mechanism of HC generation in unstrained and strained H-SiB. Red points are representatives of HCs.

With regard to HC generation and its MFP at ground state, it should be noticed that polarizability of both strained H-SiB is greater than unstrained as discussed in Section 3.5 and Figure 9. This polarizability creates a built-in electric field in the system, which affects the carrier energy with ascending rate in the strained nanostructures. Therefore, the scattering rate of SPP on the strained H-SiB nanostructures enhances, which leads to the generation of a larger amount of HCs with smaller MFPs in comparison to the unstrained structures as shown in Figure 13.

As we only have the band structure for ground state, we have calculated only the MFP for the ground state. However, we believe that we will have the similar behavior at finite temperatures (at any finite temperature, unstrained and 6% tensile strained structure will have the maximum and minimum HC-MFP, respectively). Additionally, by increasing temperature, the mobility and scattering rate of carriers increase. This makes them lose their collective behavior. Then, the rate of MFPs for generated HC decreases with increasing temperature. Therefore, we expect to have the minimum HC-MFP for the 6% tensile strained structure at 300 K.