Physica E-low-dimensional Systems & Nanostructures最新文献

{"title":"Robust structural superlubricity of twisted graphene bilayer and domain walls between commensurate moiré pattern domains from first-principles calculations","authors":"Irina V. Lebedeva ,&nbsp;Andrey M. Popov ,&nbsp;Yulia G. Polynskaya ,&nbsp;Andrey A. Knizhnik ,&nbsp;Sergey A. Vyrko ,&nbsp;Nikolai A. Poklonski","doi":"10.1016/j.physe.2025.116399","DOIUrl":"10.1016/j.physe.2025.116399","url":null,"abstract":"<div><div>Twisted graphene layers exhibit extremely low friction for relative sliding. Nevertheless, previous studies suggest that the area contribution to friction for commensurate moiré systems is finite and might restrict macroscopic superlubricity for large layer overlaps. In this paper, we investigate the potential energy surface (PES) for relative displacement of the layers forming moiré patterns (2,1) and (3,1) by accurate density functional theory calculations using the vdW-DF3 functional. The amplitudes of PES corrugations on the order of 0.4 and 0.03 <span><math><mrow><mi>μ</mi><mi>eV</mi></mrow></math></span> per atom of one layer, respectively, are obtained. The account of structural relaxation doubles this value for the (2,1) pattern, while causing only minimal changes for the (3,1) pattern. We show that different from aligned graphene layers, for moiré patterns, PES minima and maxima can switch their positions upon changing the interlayer distance. The PES shape is closely described by the first spatial Fourier harmonics both with and without account of structural relaxation. A barrier for relative rotation of the layers to an incommensurate state that can make superlubricity robust is estimated based on the approximated PES. We also derive a set of measurable physical properties related to interlayer interaction including shear mode frequency, shear modulus and static friction force. Furthermore, we predict that it should be possible to observe domain walls separating commensurate domains, each comprising a large number of moiré pattern unit cells, and provide estimates of their characteristics.</div></div>","PeriodicalId":20181,"journal":{"name":"Physica E-low-dimensional Systems & Nanostructures","volume":"175 ","pages":"Article 116399"},"PeriodicalIF":2.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Charge polarization-driven type-II band alignment and enhanced piezoelectricity in tin nitride halide heterostructures","authors":"Arneet Kaur ,&nbsp;Pradip Nandi ,&nbsp;Abir De Sarkar","doi":"10.1016/j.physe.2025.116358","DOIUrl":"10.1016/j.physe.2025.116358","url":null,"abstract":"<div><div>In the quest for efficient energy conversion materials, we investigate piezoelectric properties of tin nitride halide (SnNX) through strategic design of vertical and lateral SnNCl/SnNBr heterostructures, using first-principles calculations. Two vertical configurations (HB-I and HB-II), based on the choice of the basal atomic layer (SnNBr or SnNCl), are studied with six stacking sequences featuring parallel and antiparallel orientations. The interlayer registry index highlights the dominant role of interface interactions in determining the energy landscape. HB-II configuration exhibits a type-II band alignment for all stacking orders (AA, AB, AC). While only the AC stacking order of HB-I displays a type-II band alignment, which correlates with the reversal in the direction of charge polarization. Lateral heterostructures composed of eight-unit cells of SnNCl and SnNBr [(SnNCl)₈/(SnNBr)₈] are also constructed along armchair and zigzag directions, revealing mixed band alignment at the interfaces. A comprehensive analysis indicates that interfacial charge polarization critically determines the piezoelectric response. The out-of-plane piezoelectric strain coefficient, <span><math><mrow><msub><mi>d</mi><mn>33</mn></msub></mrow></math></span> reaches 90 p.m./V in the vertical heterostructure, comparable to leading bulk perovskites. Our findings provide a deeper understanding of band alignment and piezoelectricity in SnNCl/SnNBr heterostructures, paving the way for future experimental efforts to design advanced 2D energy conversion materials with tailored properties.</div></div>","PeriodicalId":20181,"journal":{"name":"Physica E-low-dimensional Systems & Nanostructures","volume":"175 ","pages":"Article 116358"},"PeriodicalIF":2.9,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145061261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Electron-optical phonon scattering in quantum wells in a tilted quantizing magnetic field","authors":"M.P. Telenkov,&nbsp;Yu.A. Mityagin,&nbsp;D.S. Korchagin","doi":"10.1016/j.physe.2025.116351","DOIUrl":"10.1016/j.physe.2025.116351","url":null,"abstract":"<div><div>Electron scattering with longitudinal polar optical phonons in a quantizing magnetic field tilted to the plane of quantum well layers is studied. The expressions are derived for the electron scattering rate due to longitudinal polar optical phonon emission in a tilted quantizing magnetic field. The effect of variations in both the magnitude and orientation of the magnetic field on the electron-phonon scattering processes is studied. The different relation between the inter-subband spacing and the optical phonon energy are considered – when the inter-subband spacing is greater than, equal to, and lower than the optical phonon energy. Principal transformation of the inter-subband scattering rate dependence on the quantizing component of the magnetic field – from oscillatory to monotonic – was found to occur when inter-subband spacing comes closer to optical phonon frequency.</div></div>","PeriodicalId":20181,"journal":{"name":"Physica E-low-dimensional Systems & Nanostructures","volume":"174 ","pages":"Article 116351"},"PeriodicalIF":2.9,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144878252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Effect of electric field on excitons in wide quantum wells","authors":"Shiming Zheng,&nbsp;E.S. Khramtsov,&nbsp;I.V. Ignatiev","doi":"10.1016/j.physe.2025.116333","DOIUrl":"10.1016/j.physe.2025.116333","url":null,"abstract":"<div><div>A microscopic model of a heterostructure with a quantum well (QW) is proposed to study exciton properties in an external electric field. The effect of an electric field ranging from 0 to 6 kV/cm applied to a GaAs/AlGaAs QW structure in the growth direction is studied for several QWs of various widths up to 100 nm. The three-dimensional Schrödinger equation (SE) for the exciton is numerically solved using the finite difference method. Wave functions and energies of several states of the heavy-hole and light-hole excitons are calculated. The dependencies of exciton state energy, binding energy, radiative broadening, and static dipole moment on the applied electric field are determined. Additionally, the threshold for exciton dissociation in the 100-nm QW is established. Furthermore, we calculate an electric-field-induced shift in the center of mass of heavy-hole and light-hole excitons in the QWs. Finally, we simulate the reflection spectra of heterostructures with GaAs/AlGaAs QWs under an electric field using the calculated energies, radiative broadenings, and phases of exciton resonances.</div></div>","PeriodicalId":20181,"journal":{"name":"Physica E-low-dimensional Systems & Nanostructures","volume":"174 ","pages":"Article 116333"},"PeriodicalIF":2.9,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144781827","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Structural and optical characterization of electron Beam–Deposited Dy2O3 thin films on n-GaAs(111) substrate for photonic applications","authors":"Hayet Saghrouni ,&nbsp;Lotfi Beji","doi":"10.1016/j.physe.2025.116350","DOIUrl":"10.1016/j.physe.2025.116350","url":null,"abstract":"<div><div>In this study, dysprosium oxide thin films with a thickness of 15 nm were deposited onto an n-type gallium arsenide (111) substrate using electron beam evaporation under ultra-high vacuum conditions. The deposition process, carried out at 250 °C and followed by thermal annealing at 400 °C, yielded uniformly smooth films with good structural quality and interfacial integrity. Structural analysis through scanning electron microscopy and X-ray diffraction revealed uniform triangular surface features and confirmed the cubic phase of dysprosium oxide, with crystallographic alignment to the gallium arsenide substrate. Fourier-transform infrared spectroscopy validated the formation of dysprosium–oxygen bonds. The optical properties were extracted using spectroscopic ellipsometry. The refractive index exhibited normal dispersion, and the optical band gap was determined to be 4.09 ± 0.03 eV. Wemple–DiDomenico modeling provided the oscillator energy (5.02 ± 0.05 eV) and dispersion energy (9.80 ± 0.07 eV), while dielectric function analysis identified sharp electronic transitions attributed to intra-4f states of trivalent dysprosium ions. In the infrared region, classical dispersion relations enabled the extraction of key parameters, including the high-frequency dielectric constant (3.28 ± 0.05), plasma frequency (1.81 ± 0.04) × 10¹⁴ Hz, carrier relaxation time (2.26 ± 0.03) × 10⁻¹⁵ s, and carrier concentration to effective mass ratio ((3.69 ± 0.08) × 10⁴⁷ g⁻¹ cm⁻³). Energy loss analysis through surface and volume energy loss functions revealed distinct plasmonic and interband excitation features. Optical conductivity measurements highlighted excitonic and bulk plasmon activity in the 2.7–3.8 eV range. Nonlinear optical behavior was evaluated using Miller's rule, yielding a linear susceptibility of 0.155 ± 0.004 esu, a third-order susceptibility of (9.74 ± 0.21) × 10⁻¹⁴ esu, and a nonlinear refractive index of (2.40 ± 0.06) × 10⁻¹² esu. Overall, the dysprosium oxide thin films demonstrated excellent crystallinity, high optical quality, and strong third-order nonlinear response, positioning them as promising candidates for advanced optoelectronic and photonic applications, including ultraviolet photodetectors, nonlinear optical modulators, and integrated optical devices.</div></div>","PeriodicalId":20181,"journal":{"name":"Physica E-low-dimensional Systems & Nanostructures","volume":"174 ","pages":"Article 116350"},"PeriodicalIF":2.9,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144810454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Single transition-metal atom doping enhances NO2 adsorption on 1T-PtSe2 monolayer","authors":"Jiaqi Zhang ,&nbsp;Fangfang Li ,&nbsp;Jiakang Li ,&nbsp;Wenhui Han ,&nbsp;Guangwei Wang ,&nbsp;Kaixing Zhu ,&nbsp;Yan Xu ,&nbsp;Peng Wang","doi":"10.1016/j.physe.2025.116341","DOIUrl":"10.1016/j.physe.2025.116341","url":null,"abstract":"<div><div>The structural and electronic properties of monolayer 1T-PtSe₂ doped with various single transition metal (TM) atoms (Co, Ni, Cu, Mo, Rh, Pd, Ir, and Au) at selenium vacancy sites, along with their interaction with NO₂ gas molecules, have been systematically investigated using density functional theory (DFT). TM doping notably enhances the chemical reactivity and alters the electronic structure of PtSe₂, as reflected in stronger NO₂ adsorption, increased charge transfer, and the emergence of impurity states near the Fermi level. These effects are primarily attributed to the introduction of TM nd-orbitals, which facilitate molecular activation. Among the doped systems, Pd-PtSe₂ demonstrates superior gas sensing performance with a short recovery time in the 400–500 K range. These findings provide valuable theoretical guidance for designing high-efficiency gas sensors based on TM-doped PtSe₂ monolayers.</div></div>","PeriodicalId":20181,"journal":{"name":"Physica E-low-dimensional Systems & Nanostructures","volume":"174 ","pages":"Article 116341"},"PeriodicalIF":2.9,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144771003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Solid state synthesis of silver doped graphitic carbon nitride and its efficacy in removing textile dyes","authors":"K. Chauhan ,&nbsp;D. Banerjee,&nbsp;V.P. Shrivastava","doi":"10.1016/j.physe.2025.116353","DOIUrl":"10.1016/j.physe.2025.116353","url":null,"abstract":"<div><div>The work reports the synthesis of silver doped graphitic carbon nitride (GCN) using urea as precursor. Both the pure and doped samples were characterized by X-ray diffraction (XRD), Field emission scanning electron microscope (FESEM), Energy-dispersive X-ray spectroscopy (EDX) and Fourier transformed infrared (FTIR) spectroscopy analysis.</div><div>XRD confirmed the proper phase formation of the samples with preferential growth along (002) direction whereas FESEM shows the growth of chip-like morphologies with high yield, EDX analysed the stoichiometric ratio of sample and FTIR revealed the different vibrational energy level present.</div><div>The efficacies of the samples in degrading Bengal Rose under UV light irradiation was studied. The results demonstrated that the doped samples exhibited excellent dye removal performance, achieving an efficiency of more than 85 % within just 40 min. Not only that, here the electrical energy per order as well as degradation turnover of both samples were calculated and shown that the doped sample has much higher promises compared to pure GCN. Especially, degradation turnover came out to be over 95 % for the doped sample with an exposure time of just 10 min.</div><div>When reaction kinetics was studied it is seen that the reaction mainly followed 1st order kinetics with regression coefficient almost unity. Here also doped sample showed faster kinetics with reaction constant value 0.050/minute.</div><div>It has been concluded that dopant create intermediate energy states where charge carrier can rest prolonging electron hole pair recombination which in turn facilitate the interaction with dyes and thus enhancing its removal performance.</div></div>","PeriodicalId":20181,"journal":{"name":"Physica E-low-dimensional Systems & Nanostructures","volume":"174 ","pages":"Article 116353"},"PeriodicalIF":2.9,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144864403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Chern insulator phase transition in valley-dependent altermagnet","authors":"Yajun Wei,&nbsp;J. Wang","doi":"10.1016/j.physe.2025.116352","DOIUrl":"10.1016/j.physe.2025.116352","url":null,"abstract":"<div><div>We investigate the possibility of a Chern insulator phase transition in two-dimensional altermagnetic (AM) metals by incorporating the effects of Rashba spin–orbit coupling (RSOC) and an additional exchange field. We present a phase diagram of the system as a function of the AM and exchange field strengths, and demonstrate that when the AM exchange strength exceeds a critical value of the exchange field, the system shows a nonzero Chern number. Moreover, we find that not only do the RSOC and exchange field significantly affect the topological properties, but the AM exchange energy and band energy also play essential roles. Specifically, when the AM exchange energy exceeds the AM band energy and the spin-momentum coupling exhibits clear valley separation, a bulk energy gap opens, and the system transitions into the quantum anomalous Hall insulator regime. Conversely, when the AM exhibits only anisotropic spin-momentum coupling, the system retains the same Chern number but lacks corresponding chiral edge states. We further compute the transport properties of a valley-dependent AM ribbon structure, confirming the existence of possible chiral edge states and their robustness.</div></div>","PeriodicalId":20181,"journal":{"name":"Physica E-low-dimensional Systems & Nanostructures","volume":"174 ","pages":"Article 116352"},"PeriodicalIF":2.9,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144922305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Terahertz multi-resonator refractive index sensor with graphene and MXene integration for cancer biomarker analysis","authors":"Jacob Wekalao ,&nbsp;Mahmood Basil A. AL-Rawi ,&nbsp;Ahmed Zohier Ahmed Elhendi ,&nbsp;Ahmed Mehaney ,&nbsp;Hussein A. Elsayed ,&nbsp;Mostafa R. Abukhadra ,&nbsp;Haifa A. Alqhtani ,&nbsp;Amuthakkannan Rajakannu","doi":"10.1016/j.physe.2025.116357","DOIUrl":"10.1016/j.physe.2025.116357","url":null,"abstract":"<div><div>In this study, we propose an innovative biosensor design that incorporates multiple resonators specifically engineered for detecting carcinoembryonic antigen (CEA). The biosensor integrates a unique combination of materials, including MXene, black phosphorus, and graphene, arranged in a hybrid configuration. The design consists of a centrally positioned circular MXene resonator encircled by a square ring of black phosphorus, complemented by four gold circular resonators. These components are assembled on a silicon dioxide substrate. A comprehensive performance evaluation was conducted across the terahertz spectrum (0.1–1.0 THz) using finite element method modeling in COMSOL Multiphysics 6.2. The biosensor demonstrated impressive metrics, including a maximum sensitivity of 811 GHz RIU⁻¹, a figure of merit of 14.479 RIU⁻¹, and a quality factor of 6.946. When tested with varying CEA concentrations ranging from 0 to 5 ng/mL, the device maintained stable and reliable operation. The transmission characteristics revealed systematic frequency variations from 0.389 THz to 0.382 THz. Additionally, a machine learning approach based on stacking ensemble regression was implemented to optimize sensor parameters. This computational strategy delivered outstanding predictive performance, achieving near-perfect accuracy across most operational variables. The biosensor's combination of high sensitivity, compact design, and reliable functionality positions it as a promising technology for early cancer screening and patient monitoring applications.</div></div>","PeriodicalId":20181,"journal":{"name":"Physica E-low-dimensional Systems & Nanostructures","volume":"174 ","pages":"Article 116357"},"PeriodicalIF":2.9,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144922306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Controllable spin Nernst effect of surface states in three-dimensional topological insulator film","authors":"Xin-Ning Li,&nbsp;Ning-Xuan Yang,&nbsp;Rui Wang,&nbsp;Chun-Yan Song,&nbsp;Hui Liao,&nbsp;Ting-Ting Song,&nbsp;Xue-Yan Cheng,&nbsp;Jiu-Ming Wang","doi":"10.1016/j.physe.2025.116331","DOIUrl":"10.1016/j.physe.2025.116331","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We systematically investigate the spin Nernst effect (SNE) of surface states in three-dimensional topological insulator film under a perpendicular magnetic field. The spin Nernst coefficient &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; of surface states, which are lie in the quantum spin Hall regime (QSH), quantum anomalous Hall regime (QAH), and quantum pseudospin Hall regime (QPH), is theoretically calculated by using the Non-equilibrium Green’s function method combined with the square lattice model. Regardless of the presence of a magnetic field, SNE will occur in the system because it primarily arises from spin–orbit coupling. When the Fermi energy &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; crosses the discrete transverse channels, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; exhibits a pronounced peak. The height of peak strongly depends on the temperature, decreasing with increasing temperature. In the QPH regime, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is an even function of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; without applying a magnetic field. However, this the symmetrical property is destroyed when a magnetic field is applied. In the QSH and QAH regimes, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is also an even function of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; in magnetic field &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mi&gt;ϕ&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. When a magnetic field is applied, &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; of QSH regime retains this symmetrical property due to protection from time-reversal symmetry. But the symmetrical property &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is broken in the QAH regime. This is because that the combined influence of the exchange field and magnetic field breaks time-reversal symmetry, leading to the peak structure of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; to reverse around &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub","PeriodicalId":20181,"journal":{"name":"Physica E-low-dimensional Systems & Nanostructures","volume":"174 ","pages":"Article 116331"},"PeriodicalIF":2.9,"publicationDate":"2025-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144724030","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}