Energy Levels Of States Research Articles

Topological states in finite graphene nanoribbons tuned by electric fields

In this comprehensive study, we conduct a theoretical investigation into the Stark shift of topological states (TSs) in finite armchair graphene nanoribbons (AGNRs) and heterostructures under transverse electric fields. Our focus centers on the multiple end zigzag edge states of AGNRs and the interface states of9--7--9AGNR heterostructures. For the formal TSs, we observe a distinctive blue Stark shift in energy levels relative to the electric field within a range where the energy levels of TSs do not merge into the energy levels of bulk states. Conversely, for the latter TSs, we identify an oscillatory Stark shift in energy levels around the Fermi level. Simultaneously, we reveal the impact of the Stark effect on the transmission coefficients for both types of TSs. Notably, we uncover intriguing spectra in the multiple end zigzag edge states. In the case of finite9--7--9AGNR heterostructures, the spectra of transmission coefficient reveal that the coupling strength between the topological interface states can be well controlled by the transverse electric fields. The outcomes of this research not only contribute to a deeper understanding of the electronic property in graphene-based materials but also pave the way for innovations in next-generation electronic devices and quantum technologies.

A new search for the variation of fundamental constants using the rovibrational levels and isotope effects of magnesium fluoride molecule

Abstract The recently demonstrated methods for cooling and trapping diatomic molecules offer new possibilities for precision searches in fundamental physical theories. Here, we propose to study the variations of the fine-structure constant (α=e 2/ħc) and the proton-to-electron mass ratio (μ = mp/me) with time - by taking advantage of the nearly degenerate rovibrational levels in the electronic states of the magnesium fluoride (MgF) molecule. Specifically, due to the cancellation between the fine structure splitting and the rovibrational intervals in the different MgF natural isotopes, a degeneracy occurs for the A 2Π(3/2) (v'=0,J'=18.5,-) and the A2Π(1/2) (v''=0,J''=20.5,-). We find that using the nearly degenerate energy level of such states can be 104 times more sensitive than using a pure rotational transition to measure the variations of α and μ. To quantify the small gap between the A2Π(3/2) (v'=0,J'=18.5,-) and the A2Π(1/2) (v''=0,J''=20.5,-), the special transitions of choice are feasible: the X 2Σ(1/2) + (v=0,J=19.5,+) to the A2Π(3/2) (v'=0,J'=18.5,-) and the X 2Σ(1/2) + (v=0,J=19.5,+) to the A 2Π(1/2) (v''=0,J''=20.5,-). In addition, we estimate the frequency uncertainties caused by the narrow linewidth, Zeeman shift, the Stark shift, Doppler boarding and the Blackbody radiation.

Benchmarking the optimization of optical machines with the planted solutions

This research focuses on developing effective benchmarks for quadratic unconstrained binary optimization instances, crucial for evaluating the performance of Ising hardware and solvers. Currently, the field lacks accessible and reproducible models for systematically testing such systems, particularly in terms of detailed phase space characterization. Here, we introduce universal generative models based on an extension of Hebb’s rule of associative memory with asymmetric pattern weights. We conduct comprehensive calculations across different scales and dynamical equations, examining outcomes like the probabilities of reaching the ground state, planted state, spurious state, or other energy levels. Additionally, the generated problems reveal properties such as the easy-hard-easy complexity transition and complex solution cluster structures. This method offers a promising platform for analyzing and understanding the behavior of physical hardware and its simulations, contributing to future advancements in optimization technologies.

Highly Efficient Blue Organic Light‐Emitting Devices Based on “Cross”‐Shaped Hot Exciton Emitters

AbstractThe development of blue electroluminescent (EL) materials remains a significant challenge in organic light‐emitting diode (OLED) technology. In this study, a novel design strategy is proposed for blue hot exciton (HE) materials, which involves utilizing a “cross” shaped molecular structure characterized by substantial steric hindrance and a highly twisted conformation. The unique cross‐shaped molecular architecture with distinct “arms” enables flexible control over the excited state properties of the molecule, thereby facilitating precise modulation of high‐lying triplet and low‐lying singlet excited state energy levels. Furthermore, the 3D spatial configuration of the molecule effectively reduces close molecular packing, thereby minimizing the risk of material concentration quenching. The proof‐of‐concept HE emitters CN‐PI and TP‐PI exhibit non‐π‐π stacking configurations in single crystals, achieving high photoluminescence quantum yield (PLQY) values up to 51.3% and 46.5% in non‐doped thin films, respectively, along with rapid radiation decay rates and reasonable distribution of Tm (m ≤ 5) and S1 states. Non‐doped OLEDs incorporating these emitters demonstrate exceptional external quantum efficiencies (EQE), reaching 7.3% and 6.4%, respectively, while exhibiting minimal efficiency roll‐off at high luminance. This research introduces a promising approach for developing high‐performance blue HE emitters.

The characterization of electronic structures and spin-orbit couplings in the diatomic sodium arsenide cation

A high-level ab initio computation has been performed in order to determine the molecular structures, electronic characters and transition properties of a hitherto unknown diatomic cation of NaAs+. The electronic states associated with five dissociation channels, including both the Na++As and Na+As+ limits, are subjected to detailed analysis. The results of the potential energy functions, spectroscopic constants and vibrational energy levels of the relevant states are presented herewith. The impact of the spin-orbit coupling effect on the low-lying electronic states is evaluated as well. Additionally, predictions of transition features, such as Einstein coefficients, Frank-Condon factors, and radiative lifetimes, are considered. It is concluded that the direct laser cooling of NaAs+ is infeasible.

Toward Efficient Modeling of Nonradiative Decay in Extended INVEST: Overcoming Computational Challenges in Quantum Dynamics Simulations.

In recent years, an increasing number of fully organic molecules capable of thermally activated delayed fluorescence (TADF) have been reported, often with very small or even inverted singlet-triplet (INVEST) energy gaps. These molecules typically exhibit complex photophysics due to the close energy levels of multiple singlet and triplet states, which create various transition pathways toward emission. A predictive model for the rates of these transitions is thus essential for assessing the suitability of new materials for light-emitting devices. Quantum Dynamics (QD) calculations are ideal for this purpose, as they include quantum effects, without the limitations of first-order perturbative approaches, also allowing taking into account more than two electronic states at once. However, the huge computational demands of QD methodologies, especially for large molecules, currently limit their use as a standard tool. To address this problem, we here employ a strategy that allows us to include almost the whole set of the vibrational coordinates by selecting the key elements of the Hilbert space that significantly impact dynamics, thereby hugely reducing the computational burden. Application of this protocol to two relatively large INVEST molecules reveals that internal conversion in these systems is very fast, making indirect emissive pathways a possible channel for the population of the S1 state. More importantly, this study demonstrates that the dynamics can be accurately described even with a significantly reduced vibrational space, thus allowing quantum dynamics calculations that yield accurate transition rates in a few minutes of computational time.

Absence of Additional Stretching-Induced Electron Scattering in Highly Conductive Cross-linked Nanocomposites with Negligible Tunneling Barrier Height and Width.

The intrinsic resistance of stretchable materials is dependent on strain, following Ohm's law. Here the invariable resistance of highly conductive cross-linked nanocomposites over 53% strain is reported, where additional electron scattering is absent with stretching. The in situ generated uniformly dispersed small silver nanosatellite particles (diameter=3.6nm) realize a short tunneling barrier width of 4.1nm in cross-linked silicone rubber matrix. Furthermore, the barrier height can be precisely controlled by the gap state energy level modulation in silicone rubber using cross-linkers. The negligible barrier height (0.01eV) and short barrier width, achieved by the silver nanosatellite particles in cross-linked silicone rubber, dramatically increase the electrical conductivity (51710Scm-1) by more than 4 orders of magnitude. The high conductance is also maintained over 53% strain. The quantum tunneling behavior is observed when the barrier height is increased, following the Simmons approximation theory. The transport becomes diffusive, following Ohm's law, when the barrier width is increased beyond 10.3nm. This study provides a novel strain-invariant resistance mechanism in highly conductive cross-linked nanocomposites.

Analysis of Chemical Kinetics of Multistep Reactions by Mean Reaction Time.

Although methods to derive the rate equation from a kinetic model have been known for over a century, it remains mathematically challenging to derive the rate equation for complex reactions involving multiple steps, as the derivation requires a solution for simultaneous differential equations. Furthermore, the derived kinetic equations are often difficult to intuitively understand. Here, we report a radically different approach to analyze chemical kinetics using the mean reaction time, the average of the time required for the completion of a chemical reaction. This mathematical description of chemical kinetics provides an intuitive understanding of how individual steps in a multistep reaction contributes to the overall kinetics. We demonstrate here the step-by-step approach to derive the formula for the mean reaction time from kinetic models. Surprisingly, one can derive the formula for the mean reaction time without solving simultaneous differential equations or using a steady-state approximation. Being an ensemble-averaged value, the mean reaction time cannot describe the distribution of the reaction time of the individual chemical entities. However, the mean reaction time reveals invaluable insights on how the energy levels of the ground states and the transition states affect the kinetics of the multistep reaction. As proof of principle, we apply the mean reaction time to enzyme kinetics and demonstrate that one can readily derive the expressions of the kinetic parameters (k cat/K M and k cat) even without deriving the Michaelis-Menten equation.

Intracage vibrations and Zeeman effect in Y3N@C80 single-molecule transistors

Clusterfullerenes, which contain a cluster rather than single-atom inclusions, exhibit more complex internal structures and greater degrees of freedom for motion. Trimetallic nitride clusterfullerenes have attracted significant attention due to their diversity and potential applications, among which Y3N@C80 stands out for its charge-transfer characteristics in electronic excitations, owing to the unique distribution of molecular orbitals near the Fermi level. Here, we have fabricated single-molecule transistor devices using Y3N@C80. Transport measurements at liquid helium temperature revealed a series of excited state energy levels, which were matched to corresponding vibrational modes through comparison with Raman spectra and density functional theory calculations. Additionally, we measured charge-state-dependent magnetic responses, revealing the electron and spin-filling patterns of the molecular orbitals in Y3N@C80. These results enhance our understanding of the dynamics and molecular spin–orbit characteristics of clusterfullerenes, indicating their potential for multifunctional applications.

Optical thermometry using efficient upconversion luminescence in Er3+ self-sensitized NaYS2 under multi-wavelength excitation

Herein, a series of Er3+ self-sensitized NaYS2 phosphors are synthesized by the solid-gas reaction method for a novel upconversion luminescence thermometer. Er3+ possesses abundant excited state energy levels in the near-infrared region, enabling efficient upconversion luminescence by absorbing different near-infrared wavelength light. Compared to 980 nm excitation, the emission intensity is enhanced by nearly an order of magnitude under 1532 nm excitation, which can be attributed to the larger absorption cross-section of 4I13/2 and stronger absorption efficiency for Er3+. Based on the luminescence intensity ratio technique, the optical thermometry behaviors of NaYS2:Er3+ under different wavelength excitation are evaluated by employing the thermally coupled energy levels of 2H11/2/4S3/2. It can be deduced that the excitation wavelength has no significant effect on the temperature sensing parameters. Compared to other typical upconversion luminescence thermometers, NaYS2:Er3+ thermometer exhibits not only excellent sensitivity performance but also high upconversion luminescence efficiency, which is expected to be applied in wide-temperature-range and highly-sensitive temperature sensing.

Enhancement mechanism of copper ions doped TiO2 substrates based on surface-enhanced Raman scattering

Surface-enhanced Raman scattering (SERS) is considered as one of the most attractive analytical tools, and it is necessary to develop more semiconductor substrates. Herein, copper ions-doped TiO2 (Cu-TiO2) nanoparticles (NPs) with different doping ratios were synthesized by a sol-hydrothermal method and used as semiconductor SERS substrates to explore the possible enhancement mechanisms. The synthesized Cu-TiO2 NPs show a cubic crystalline structure of anatase with a small particle size at 8 nm. We discussed the SERS enhancement ability of Cu-TiO2 substrates using crystal violet (CV) used as a probe. It can be found that there is a charge transfer (CT) effect between CV molecules and Cu-TiO2 substrates. In addition, Cu-TiO2 substrates exhibit the most SERS enhancement with an enhancement factor (EF) up to 2.4 × 104 at the Cu2+ doping ratio of 3 %, with a detection limit of 5 × 10–7 M. The experiments demonstrated that Cu-TiO2 NPs can be used as SERS substrates because of the CT route among the valence band, the surface state energy level, and the lowest unoccupied molecular orbital, which facilitates the development of semiconductor SERS substrates and provides a new perspective in exploring SERS enhancement mechanisms.

Benzimidazole based electron-transport hosts for efficient pure blue narrowband OLED device

Two electron-transport host materials are synthesized by linking anthracene with benzimidazole derivatives through a unit of methyl substituted phenyl. The methyl substitution generates a large dihedral angle between the benzimidazole plane and bridging phenyl, keeping the excited state distribution and energy levels of T1 and S1 unchanged. The electron-only devices reveal that the two hosts have obviously enhanced electron transport ability than that of Ph-AN-MBP without benzimidazole. As a result, the devices with Ph-AN-BIZ and N1-AN-BIZ as hosts and a blue multiple resonance material of tBuCz-DABNA as emitter exhibit high external quantum efficiency of 10.1 % and 9.1 %, respectively, and low turn-on voltage of 2.7 V, which is better than those of Ph-AN-MBP based device (7.9 %, 3.4 V). And their electroluminescence spectra have a very small full-width at half-maximum of 16.2 nm and pure blue color with CIE coordinates of (0.12, 0.11).

Regulating the redshift of the charge transfer band edge and antithermal quenching by ion substitution strategy in YP1-xVxO4:Eu3+ phosphor

The redshift of the charge transfer band (CTB) edge of phosphor induced by temperature rise is pervasively accompanied by antithermal quenching. Accordingly, exploring the mechanism of the CTB redshift is thus not only propitious for optimizing the sensitivity of optical thermometry but also tenders a new idea for the design of antithermal quenching phosphors. Herein, a family of YP1-xVxO4:Eu3+ phosphors were designed via the V5+/P5+ ion substitution strategy. With the increase of the V5+ ratio, the relative integral intensity of the 5D0→7F2 transition multiplies from 1.25 to 28.67 times under CTB edge excitation. Moreover, by Gaussian fitting, the reduction degree of vanadate band gap energy 1E (1T1)→1B2 (1T2), 1A2 (1T1)→1B2 (1T2) increased from 0.122 eV and 0.065 eV to0.162 eV and 0.083 eV, respectively, in the range of 303–523 K. Both reveal an enhance in redshift of the excitation spectra. The decrease of charge transfer state (CTS) energy level and the alteration of photoelectron thermal population are two mechanisms that induce CTB redshift. Specifically, the decrease of the CTS level is dominant in the YPO4:Eu3+ sample, and the thermal population of the higher vibrational levels of the VO43−ground state is dominant in the presence of a tiny amount of V5+ doping. In addition, according to the experimental phenomenon, a dual-mode temperature sensing scheme based on excitation intensity ratio (EIR) and single band ratiometric (SBR) is proposed. Y0.9P0.1V0.9O4:0.1Eu3+ achieves a maximum sensitivity of Sr-max = 4.040 % K−1 and a minimum temperature determination uncertainty of δT = 0.124 K at 323 K, which excellent performance indicates that this is a promising non-contact optical thermometer.

Activation/ Deactivation of Surface States of Heterostructure (Anatese-TiO2/Au) Via Addition of Materials with High Band Gap, Rutile TiO2 (R-TiO2) or Low Band Gap Silicon (Si), and Fabrication Strategy

The addition of high band gap or low band gap material not only modulates the optical absorption cross-section but also affects the surface state reactivity of heterostructure towards the photoelectrochemical (PEC) water splitting. Further, the deposition strategy to assemble the heterostructure also affects the effectiveness of the heterostructure. To explore such issues, we fabricated the two sets of heterostructures containing (A-TiO2, R-TiO2) and (Si, A-TiO2) via the click chemistry approach through two different strategies, i.e., layer-by-layer heterostructure (LLH), and randomly deposited heterostructure. In LLH strategy, only the addition of high band gap material R-TiO2 enhances the reactivity of the surface states. On the other hand, the addition of the low band gap material Silicon (Si) hampers the reactivity of the surface states of the heterostructure deposited via the LLH. Further, by changing the deposition strategy to RDH, the reactivity of surface states diminishes by adding either the high band gap or the low band gap material. These observations were rationalized from the framework of band structure/band-alignment in the heterostructure and also via electrochemical impedance spectroscopy (EIS).In LLH, the addition of the A-TiO2 quenches electrons from the conduction band (CB) of R-TiO2, and the holes of A-TiO2 are transferred to R-TiO2. The holes from R-TiO2 participate in activating the surface states present at the surface. On the other hand, the addition of silicon to LLH, having CB above the CB of A-TiO2, does not allow the electrons to transfer from A-TiO2 to Si; hence, the CB electrons from A-TiO2 transfer to surface states, which enhances the energy level of surface states, making it more negative than the VB of Si. The negative shift of surface states of A-TiO2 makes it the recombination center. Interestingly, changing the fabrication strategy from LLH to RDH makes the nature of surface states from reactive to recombination states due to the interfacing of all materials with the underlying substrates.EIS was utilized to measure and analyze the low-frequency capacitance due to the reactive surface state. The low-frequency capacitance of LLH with A-TiO2 shows the maxima at onset potential, confirming the role of reactive surface states. On the contrary, low-frequency capacitance increases monotonically for LLH with Si addition, showing the effect of recombination surface states. Overall, the study provides pointers on the interplay between band positions and electron-hole separation in heterostructures and how they can be effectively used to enhance surface reactivity in PEC water splitting.

(Invited) Electronic Trap Characterization in QD Devices Using Impedance Analysis Techniques

Colloidal quantum dots (QDs) are semiconductor nanocrystals with dimensions in the range of tens of nanometers. They exhibit unique optical and electronic properties, such as a size-dependent band gap and high absorption cross-section. These properties make them attractive for optoelectronic applications such as photovoltaics, light-emitting diodes (LEDs), and photodetectors. However, QD optoelectronics pose significant challenges due to trap states, which mainly originate from the surface states of nanocrystals. Therefore, it is necessary to analyze the trap states of QDs to understand their electronic properties and optimize their performance in optoelectronic applications. With this understanding, we want to correlate the cause of the trap states with the chemical defect to design better passivation strategies.In this discussion, we will explore the principles of electronic trap analysis in colloidal quantum dots based on impedance analysis techniques as a complementary method to spectroscopy. We will provide examples of electronic trap analysis in PbS QD photovoltaics and InP QD LEDs to demonstrate the effectiveness of this approach. We can electrically probe the QD layers in the device using an impedance analyzer to analyze the energy level and density of trap states. By combining chemical analysis and a theoretical approach with this technique, we attempted to correlate the trap states and chemical defects.

Multi‐Resonance Thermally Activated Delayed Fluorescence Molecules for Triplet‐Triplet Annihilation Upconversion

AbstractTriplet‐triplet annihilation upconversion (TTA‐UC) has made significant progress in recent years in several key applications, including solar energy harvesting, photocatalysis, stereoscopic 3D printing, and disease therapeutics. In TTA‐UC research, photosensitizers serve the vital function of harvesting low‐energy photons. The photophysical characteristics of photosensitizers, including absorbance, triplet state quantum yield, triplet state energy level, triplet state lifetime, etc., determine the performance of TTA‐UC. Thus, the study of photosensitizers has been a key aspect of TTA‐UC. In recent years, multi‐resonance thermally activated delayed fluorescence (MR‐TADF) molecules have received extensive attention due to their excellent photophysical properties and electroluminescent device performance. MR‐TADF molecules not only present a narrow energy gap between the singlet and triplet excited states, but also have stronger absorption and better wavelength regulation than conventional TADF molecules. Nowadays, the preliminary attempts in TTA‐UC using MR‐TADF molecules as photosensitizers have resulted in the development of green to ultraviolet, blue to ultraviolet, and even near‐infrared to blue emission. This concept will summarize the research progress of MR‐TADF molecules as photosensitizers in TTA‐UC, analyzing the challenges and giving possible solutions. Finally, we prospect the future development of MR‐TADF molecules as photosensitizers, including the molecular design as well as the possible application areas.

Impact of electronic correlation on strong laser-induced bound-state transitions.

Electron correlation (EC) plays a crucial role in all multi-electron systems and dynamic processes. In this work, we focus on strong laser-induced bound-bound transitions (BBT), which are fundamental to optical absorption measurements. We use the helium atom, the simplest two-electron system, as our test case, utilizing the ab initio code package HeTDSE. We examined the bound state energy levels, transition dipole moments (TDMs), and the dynamics of strong laser-induced BBT, both with and without considering EC. Our results indicate that EC significantly impacts the energy levels of the bound states and the TDMs. These effects collectively influence the transition dynamics of the excited states. Although EC does not alter the quantum transition pathways between resonance states, it generally increases the probability of resonance transitions in most cases. Our findings provide a quantitative description of EC in laser-induced BBT.

Theoretical Investigation of Transport Properties of X-doped (X = Cl, P, N) Graphene Quantum Dots Layer Adsorbed with Ammonia Molecule Using Non-Equilibrium Green’s Function

This study investigates the various transport properties, including Density of States (DOS), Transmission Coefficient, Current-Voltage Characteristics (I-V), and HOMO-LUMO energy levels, of X-doped (X = Cl, P, N) graphene quantum dot (GQD) layers with adsorbed ammonia molecules using density functional theory and non-equilibrium Green's function techniques. Our findings revealed the profound influence of different doping and adsorption scenarios on the transport properties of GQD layers. Specifically, chlorine doping leads to notably stronger resonance peaks, resulting in enhanced conductivity of the GQD layer. This effect is less pronounced for P-doped and N-doped GQDs, which exhibit minimal changes in conductance capability. Importantly, we observed a prominent interaction between ammonia molecules and chlorine-doped GQD layers. The HOMO and LUMO energy levels exhibit double degeneracy, significantly impacting edge passivation and the finite dimensions of GQDs, consequently reducing the energy gap. As a result, chlorine-doped GQD layers with adsorbed ammonia molecules exhibit increased energy levels and stronger current characteristics compared to P-GQDs and N-GQDs. HOMO and LUMO are doubly degenerated which has a clear effect on edge passivation as well as the finite size of GQDs and reduces energy gap. Therefore, more energy level and stronger current is available for chlorine doped graphene quantum dots layer with adsorbed ammonia molecule than P-GQDs and N-GQDs. Our theoretical finding highlighted the significant features of edge passivation, doping effect and adsorption of ammonia molecule onto graphene quantum dots layer. Our theoretical analysis highlighted key features encompassing edge passivation, doping effects, and ammonia molecule adsorption on GQD layers. These insights not only develop our fundamental understanding of these systems but also hold promise for novel applications that harness their distinct electronic properties.

White Light Emissive Eu(III) Complexes through Ligand Engineering and their Applications in Cool Near Ultraviolet White Light Emitting Diodes and Thermometer.

In pursuit of enhancing white light quality for solid-state lighting (SSL) applications, an attempt has been made to design novel imidazo-bipyridyl ligands as an ancillary ligand to obtain multiple emissions (mimic sunlight) in the Eu-complex. By strategically modifying the phenanthroline core with imidazo-bipyridyl incorporation with 1 or 2-Napthyl groups at the C1 position, the excitation spectral line is successfully shifted from Ultraviolet (UV) to near UV/visible spectrum (where the LED emission occurs). The ligands showed greenish blue emission in solid and solution. Density Functional Theory (DFT) calculations were utilized to understand the energy transfer processes from ligand to Eu ion in the Eu complexes. The analysis revealed that the energy transfer is incomplete, primarily attributed to the proximity of triplet state energy levels to the resonance level of Eu(III) ions as reflected in solvatochromism. These complexes exhibit a unique dual emissive behavior (emitting multi-color) including white light across various solvents. These complexes hold great promise as single-component white light-emissive materials, with potential applications in white light-emitting diodes (WLED). The fabricated white LED showed an excellent color rendering index (CRI ~93 %). Beyond lighting, this distinctive property opens avenues for temperature sensing ([Eu(DBM)31-Naph] shows the highest sensitivity of Sr=10.97 %, and [Eu(DBM)32-Naph] shows the highest sensitivity of Sr=5.5 % at 333 K) and vapoluminescent (acid-base on-off-on luminescence) studies. This research pioneers the development of these complexes as potential single-component materials for superior white LEDs, underlining their multifaceted utility in cutting-edge lighting and sensing technologies.

Analytical solution and spectral structure of the two-photon anisotropic Rabi-Stark model

Since the realization of the strong coupling between light and matter in experimental setups, the quantum Rabi model and its generalized models describing the interaction between the boson field and the two-level system have attracted extensive interest again. The study of anisotropic generalized Rabi models enables us to better understand the novel physical properties of the interaction between light and matter in the ultra-strong and deep-strong coupling regions. In this work, the two-photon anisotropic Rabi-Stark model (tpARSM) is analytically solved by using the Bogoliubov operator approach and the su(1, 1) Lie algebra. We derive the G-function, whose zeros give the regular spectrum of the system. By studying the pole structure of the G-function and the coefficients in the function, exceptional solutions, including the first-order quantum phase transition points, doubly degenerate exceptional solutions and nondegenerate exceptional solutions, are obtained. By discussing the spectral structure, we give the conditions for the first-order quantum phase transition of tpARSM. Furthermore, we find that the property that all of the lowest doubly degenerate crossing points in the two-photon Rabi-Stark model have the same energy only holds for the special case of the tpARSM in which the anisotropy parameter is equal to 1. Finally, from the perspective of first-order quantum phase transitions, concise conditions for the ground state energy level to collapse to or escape from the collapse point for the tpARSM are presented. A good understanding of the tpARSM will lay a good foundation for studying the extended two-photon systems involving multiple levels and multiple bosonic modes, and even the relevant open quantum systems.