# Investigation into the Exciton Binding Energy of Carbon Nitrides on Band Structure and Carrier Concentration through the Photoluminescence Effect

^{1}

^{2}

^{*}

## Abstract

**:**

^{+}Cl

^{−}) by employing the photoluminescence spectra and density functional theory (DFT) calculations based on the Wannier-Mott exciton module. The results of self-consistent GW computation were applied. The measurement of photoluminescence spectra, by which exciton binding energies are estimated, is likewise discussed. Generally, compared with the results calculated by GW-BSE, the DFT results based on the Wannier-Mott model are closer to the experimental values. From a materials perspective, on the other hand, the exciton binding energy of the melon is lower than that of PTI/Li

^{+}Cl

^{−}.

## 1. Introduction

^{+}Cl

^{−}) [5,6,7] (Figure S1). During the photoresponse procedure, light absorption is the very first step including the intrinsic absorption of the non-excited state and the exciton absorption. The relatively free motion of interrelated electron–hole pairs can be described as neutral quasiparticles using the effective mass approximation in the Schrodinger equation. The Coulomb binding between them will lower the overall energy of the system, which usually performs as impurity doping of energy levels within the band gap. Therefore, the experimentally measured optical bandgap (denoted as ${E}_{g}^{opt}$) is generally slightly smaller than the electronic band gap (denoted as ${E}_{g}^{el}$):

## 2. Results and Discussion

#### 2.1. Experiment Measurement of ${E}_{b}$

_{B}is the Boltzmann constant, I

_{0}is the peak intensity at 0 K, and $A=\frac{{\tau}_{r}}{{\tau}_{0}}$ is defined as a parameter to describe radiative lifetime, in which ${\tau}_{0}$ and ${\tau}_{r}$ are the radiative lifetime when T → 0 K and the radiative lifetime under usual conditions, respectively. This equation is derived from the fluorescence quantum efficiency:

^{+}Cl

^{−}(~50 meV) is more than twice that of the melon (~22 meV). Although it seems quite outrageous, this observation did not contradict the empirical energy band gap scaling law that is proportional to its energy band gap [12,13]. The reduction of nonradiative lifetime with rising temperature manifests as a reduction in luminescence intensity. No visible shift of peak position is observed as the temperature increases in the PL spectra among all carbon nitrides.

_{0}for fitting data, it has further been confirmed that whatever wavelength is applied to extract fitting data, the reason for the frequent use of the highest peaks may primarily be that the components near the main peak are predominantly bound-state excitons, which can be well described by Equation (2). For example, the ${E}_{b}$ values of PTI/Li

^{+}Cl

^{−}always fluctuate slightly above and below 50 meV, which verifies the validity of the previous assumption in any radiative emission transition. On the other hand, the ${E}_{b}$ values of melon or poly-(heptazine imide) (PHI) float over a wider range (Figure 2) and rise swiftly in the longer wavelength region. The better crystallized the material component is, the more unitary the fluorescence emission mode is, and the more stable is the fitted binding energy it exhibits. The fitted exciton binding energy shows a relatively monotonous change from short to long wavelengths. This observation suggests that the excitons from shallow energy levels are more stable and less prone to dissociation.

^{+}Cl

^{−}emission center is observed at P1 (420 nm, ~2.9 eV), P2 (450 nm, ~2.7 eV), P3 (480 nm, ~2.6 eV) and P4 (509 nm, ~2.43 eV) (Figure 3), which mainly originate from the three different pathways of transition: π*-π, π*-σ and π*-n to non-bonding orbitals (n) [14,15]. The components of the band pairs with an energy gap corresponding to the wavelength of the wave function shapes (Figures S2–S4) as well as the typical emission peaks (Figures S5–S9) are listed. The shape change in the spectrum with the variation in temperature provides information to distinguish free excitons from bound excitons. As shown in the figure, the shoulder at 420 nm becomes progressively weaker with the increase in temperature, which points to the signature of the quenching of bound-state excitons [16,17].

#### 2.2. Calculations of the Dielectric Function

#### 2.3. Calculations of the Effective Mass

#### 2.4. Burstein–Moss Band Shift and Transient Absorption Spectra (TASs)

^{18}cm

^{−3}, while it can rise to as high as ca. 10

^{21}–10

^{22}cm

^{−3}after photoexcitation under AM 1.5G illumination [31]. Thus, the influence of ${\u2206E}_{BM}$ can be observed in transient absorption spectroscopy; the highest band shifts of melon and PTI at a carrier concentration of 10

^{22}cm

^{−3}are about 0.6 eV and 1.2 eV, respectively. Considering the anisotropy of the effective mass, a detailed value of the Burstein–Moss band shift depends on how the carriers fill up the energy state; namely, which energy band and which section of k-space the carriers populate. There might exist some discrepancy in the excitation between the compressed powder sample and electrodes immersed in electrolyte solution.

^{+}Cl

^{−}exhibited a weaker absorption ability and a more rapid bleaching process compared with melon, which is consistent with our knowledge of the size of the conjugate system. The faint peak simultaneously hindered estimation of the band shift value. Generally, the Burstein–Moss band shift takes effect within 1–2 ps; this time range appears in many other semiconductor materials [30,32], indicating a universal response speed of a material against an external field and carrier re-equilibration relaxation time during the photoexcitation process.

#### 2.5. Calculations of the Exciton Binding Energy

#### 2.5.1. Wannier Model

^{+}Cl

^{−}behaves like a 3D exciton. Thus, the exciton binding energy and the exciton radius can be determined by [34,35,36]:

#### 2.5.2. GW-BSE Method

_{0}) and single-shot quasiparticle calculation (G

_{0}W

_{0}) greatly overestimate the bandgap of carbon nitrides alike. Based on the past computation experience of carbon nitride, this metal-free conjugate system has a special compatibility with the PBE functional. PBE did not underestimate the conjugate effect; the undervalued bandgap provided by PBE (~2.5 eV) should be traced to the misunderstanding and misnaming of carbon nitride structures. It is not an exception that GW produces unsatisfactory yield results compared to experimental observations since the computed band gaps often exhibit a substantial overestimation relative to experimental values [37] (Figure 9). As shown in Figure 9b, the calculated results mutually exhibit a distinct blue shift compared with the experiment value.

## 3. Computational and Experimental Details

^{+}Cl

^{−}crystals, DFT calculations were carried out on the basis of the crystal structure proposed by Lin et al. [5].

^{−7}eV and the Hellmann–Feynman forces on each atom were less than 0.01 eV·Å

^{−1}. The regular plane wave cutoff energy was set to 550 eV [44], and the GW cutoff energy was set to 366 eV as the default value of 2/3 of the common cutoff energy.

## 4. Conclusions

^{+}Cl

^{−}; these deviations may not only be attributable to the misalignment of molecule structure, but rather to the backwardness of the description with respect to the excitation model. In the case of GW methods, while it may seem that first-principal methods are employed in describing the energy structure, the fundamental particle remains rooted in a quasi-particle framework. Upon examination of the hydrogen-like model used to calculate binding energy, the formula structure still relies on Columbic interactions, which is essentially a phenomenological approach.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**,

**c**): Temperature-dependent PL spectra of PTI/Li

^{+}Cl

^{−}and melon. (

**b**,

**d**): fitting process of ${E}_{b}$ of melon and PTI applying Equation (2), the blue line is the fitted curve and the scatter points were selected from the peak values of PL data at different temperatures.

**Figure 2.**PL spectra of melon in molten salt treated for different durations. (

**a**) Pristine melon; (

**b**) melon treated in molten salt for 4 h, the most popular processing time to synthesis PHI; (

**c**) melon treated in molten salt for 8 h; (

**d**) melon treated in molten salt over 12 h, widely known as PTI/Li

^{+}Cl

^{−}.

**Figure 3.**The deconvolution of PTI/Li

^{+}Cl

^{−}at (

**a**) 50 K, (

**b**) 200 K and (

**c**) 300 K, the four peaks of similar width from left to right correspond to the P1-P4 labeled in the text. (

**d**) The deconvolution of melon at 300 K, only this temperature is chosen, for no visible shape change with the variation in temperature was observed comparing with PTI/Li

^{+}Cl

^{−}, and an extra peak merged at ~464 nm.

**Figure 4.**Dielectric function of PTI/Li

^{+}Cl

^{−}along major axis of primitive cells. (

**a**–

**c**) The dielectric contribution of the ionic effect. (

**d**–

**f**) The dielectric contribution of electric density.

**Figure 5.**Boxplot of the calculated reduced effective mass of selected bands along different k-point paths. High symmetry k-points were chosen to be the segmentation points of the fitted curve to avoid the cusps introduced by the discontinuity of the paths.

**Figure 6.**Burstein–Moss band shift value calculated by two different effective mass: (

**a**) parabolic-fiited effective mass, (

**b**) optical effective mass calculated by Kane dispersion.In the same was as effective mass, the anisotropy of the resulting band shift values was considered on the same K-path as that used for the band calculation.

**Figure 7.**ΔA vs. the wavelength plot for different times after the light pulse; time zero corresponds to the maximum overlap of the excitation and the probe pulse. (

**a**) Photobleaching process of melon; (

**b**) photobleaching process of PTI/Li

^{+}Cl

^{−}.

**Figure 8.**(

**a**) Binding energy of different transition way, (

**b**) exciton radius of different transition ways. All marked exciton binding energies and corresponding exciton radius values of PTI/Li

^{+}Cl

^{−}within the transition energy range. The circular marks in the heatmap indicate the transition mode that does not satisfy the two screening conditions of TDM and experimental excitation energy, and the square colored blocks indicate the results that pass the screening.

**Figure 9.**(

**a**) Real part of dielectric function obtained by GW; (

**b**) reflection function calculated with the dielectric function in (

**a**) three typical structures. The solid lines are the theoretical value, the dashed lines are from the experimental observation.

**Table 1.**List of dielectric tensors of different representative carbon nitride structures, which fits the real part of the zero points value in Figure 2.

$\mathit{\epsilon}$ | ${\mathit{\epsilon}}_{\mathit{x}\mathit{x}}^{\mathit{e}\mathit{l}\mathit{e}}$ | ${\mathit{\epsilon}}_{\mathit{y}\mathit{y}}^{\mathit{e}\mathit{l}\mathit{e}}$ | ${\mathit{\epsilon}}_{\mathit{z}\mathit{z}}^{\mathit{e}\mathit{l}\mathit{e}}$ | ${\mathit{\epsilon}}_{\mathit{x}\mathit{x}}^{\mathit{i}\mathit{o}\mathit{n}}$ | ${\mathit{\epsilon}}_{\mathit{y}\mathit{y}}^{\mathit{i}\mathit{o}\mathit{n}}$ | ${\mathit{\epsilon}}_{\mathit{z}\mathit{z}}^{\mathit{i}\mathit{o}\mathit{n}}$ | ${\mathit{\epsilon}}_{\mathit{x}\mathit{x}}^{\mathit{r}}$ | ${\mathit{\epsilon}}_{\mathit{y}\mathit{y}}^{\mathit{r}}$ | ${\mathit{\epsilon}}_{\mathit{z}\mathit{z}}^{\mathit{r}}$ |
---|---|---|---|---|---|---|---|---|---|

PTI | 4.88 | 4.87 | 2.44 | 2.03 | 2.25 | 0.876 | 6.91 | 7.12 | 2.90 |

PTI-noion | 3.97 | 3.98 | 1.94 | 1.38 | 1.38 | 0.273 | 5.35 | 5.36 | 1.65 |

PHI | 1.97 | 1.97 | 1.15 | 0.513 | 0.51 | 0.0312 | 2.47 | 2.47 | 0.54 |

PHI(1k) | 4.06 | 3.79 | 1.68 | 1.92 | 1.62 | 0.217 | 5.98 | 5.40 | 1.88 |

melon | 2.38 | 2.23 | 1.21 | 0.72 | 0.45 | 0.0389 | 3.10 | 2.68 | 0.757 |

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**MDPI and ACS Style**

Lin, Z.; Cai, X.; Lin, W.
Investigation into the Exciton Binding Energy of Carbon Nitrides on Band Structure and Carrier Concentration through the Photoluminescence Effect. *Catalysts* **2024**, *14*, 262.
https://doi.org/10.3390/catal14040262

**AMA Style**

Lin Z, Cai X, Lin W.
Investigation into the Exciton Binding Energy of Carbon Nitrides on Band Structure and Carrier Concentration through the Photoluminescence Effect. *Catalysts*. 2024; 14(4):262.
https://doi.org/10.3390/catal14040262

**Chicago/Turabian Style**

Lin, Zhiyou, Xu Cai, and Wei Lin.
2024. "Investigation into the Exciton Binding Energy of Carbon Nitrides on Band Structure and Carrier Concentration through the Photoluminescence Effect" *Catalysts* 14, no. 4: 262.
https://doi.org/10.3390/catal14040262