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The strongly anisotropic properties of phosphorene makes it an attractive material for applications in deciding the specific direction for different purposes. Here we have particularly reported the competition between strain and electric field stimuli in evaluating the band gap and electron energy loss spectrum (EELS) of single-layer black phosphorus using the tight-binding method and the Kubo conductivity. We construct possible configurations for this competition and evaluate the interband optical excitations considering the corresponding band gap variations. The band gap increases with the individual electric field, while it increases (decreases) with tensile (compressive) uniaxial in-plane strain. Contrary to the in-plane strains, the uniaxial out-of-plane strain shows a critical strain at which the system suffers from a phase transition. Furthermore, the presence of these stimuli simultaneously results in an extraordinary band gap engineering. Based on the EELS response in the electromagnetic spectrum, the armchair (zigzag) direction is classified into the infrared and visible (ultraviolet) region. We report that the electric field gives rise to the blue shift in the interband optical transitions along the armchair direction, while the compressive/tensile (tensile/compressive) in-plane/out-of-plane strain provides a red (blue) shift. Moreover, we observe an inverse behavior of EELS response to the individual and combined effects of electric field and strains compared to the band gap behavior except at critical out-of-plane strain for which the physical theory of interband excitation is simply violated. Our results provide a new perspective on the applicability of phosphorene in stimulated optical applications.

We theoretically address the electronic structure of mono- and simple bi-layer armchair graphene nanoribbons (AGNRs) when they are infected by extrinsic charged dilute impurity. This is done with the aid of the modified tight-binding method considering the edge effects and the Green’s function approach. Also, the interplay of host and guest electrons are studied within the full self-consistent Born approximation. Given that the main basic electronic features can be captured from the electronic density of states (DOS), we focus on the perturbed DOS of lattices corresponding to the different widths. The modified model says that there is no metallic phase due to the edge states. We found that the impurity effects lead to the emergence of midgap states in DOS of both systems so that a semiconductor-to-semimetal phase transition occurs at strong enough impurity concentrations and/or impurity scattering potentials. The intensity of semiconductor-to-semimetal phase transition in monolayer (bilayer) ultra-narrow (realistic) ribbons is sharper than bilayers (monolayers). In both lattices, electron-hole symmetry breaks down as a result of induced-impurity states. The findings of this research would provide a base for future experimental studies and improve the applications of AGNRs in logic semiconductor devices in industry.

In the present paper, we study the impacts of possible uni-, bi-, and tri-axial strains on the electronic band gap of phosphorene using the density of states (DOS) quantity through the Harrison relation. To reach this goal, we use a tight-binding Hamiltonian model and the Green’s function method. The findings report that the electronic phase of phosphorene can be adjustable in the presence of strain. The band gap increases (decreases) when applying the tensile uniaxial strains along the { x , y } (z) direction, while it decreases (increases) when the compressive uniaxial ones are applied. Interestingly, due to the inherent highly anisotropic structure of phosphorene, there is a semiconductor-to-metal phase transition along the z direction as the tensile strain is increased. Furthermore, we found that among all possible configurations for uniform biaxial strains, a flat band emerges at e x = e y = - 15 % , which is a quite new outcome. In addition, phosphorene supports different phase transitions when the triaxial strains are applied, introducing new electro-optical features. Hence, these findings can provide insights into the future experimental research and improve the applications of phosphorene in the real industry such as field-effect transistors.

The continuum model, the Born approximation , and the Green's function approach are used to investigate thermoelectric properties of impurity-infected phosphorene subjected to a Zeeman magnetic field. We found that both disordered electrical and electronic thermal conductivities decrease (increase) with the magnetic field with (without) a threshold value in x−(y −)direction, stemming from the distributed charge with new potential. Furthermore, we found an anisotropic Drude weight at fairly low temperatures in the electrical conductivity . The opposite sign for low-temperature thermopower is captured by strong magnetic fields. Moreover, this information for the thermoelectric efficiency of disordered phosphorene tells us that the efficiency of the disordered phosphorene oscillates (increases) with the magnetic field in x−(y−)direction. Investigation of impurity configuration effects results in the significant differences. The efficiency regardless of the direction oscillates with high impurity concentrations and vanishes at low temperatures, whereas it does not change (oscillates) in x−(y−)direction with impurity scattering potential.

We study the magnon-charged impurity interaction effects on the magnonic heat capacity (MHC) and the Pauli magnetic susceptibility (PMS) of spin-1/2 and spin-3/2 Lieb lattice within the J 1 - J 2 Heisenberg model in the presence of the Dzyaloshinskii-Moriya interaction (DMI) and the Zeeman magnetic field. In so doing, we use the Green’s function technique and the full self-consistent Born approximation. We found that in the case of J 1 Heisenberg model, the MHC and PMS increase (decrease) with impurity concentration for spin-1/2 (spin-3/2), while oscillate (decrease) when coupling constants next-nearest-neighbor (NNN), DMI, and the Zeeman field are taken into account. We have also investigated perturbed MHC and PMS by ascertained impurity for different coupling strengths, resulting in the oscillating (decreasing) behavior with NNN for spin-1/2 (spin-3/2) and a decreasing behavior with DMI and Zeeman magnetic field for both spin-1/2 and spin-3/2. Moreover, magnetic phase transitions in the case of spin-3/2 are established when tuning coupling constants on the non-zero, whilst no magnetic phase transition is observed for spin-1/2 Lieb lattice. Our findings are useful to tune the physical properties of optical lattices subjected to the electric field, magnetic field, and charged impurity.

We address perpendicular magnetic field effects on the electrical conductivity, electronic thermal conductivity, thermopower, and figure of merit in phosphorene. By means of continuum Hamiltonian model and the Green’s function approach, we report a significant difference in the magneto-thermoelectric properties. Results showcase an irregular (regular) behavior in AC (ZZ) direction for electrical conductivity in the presence of a magnetic field. Furthermore, we found an oscillating (increasing) treatment for the Drude weight with increasing magnetic field in AC (ZZ) direction at low temperatures. Also, the electronic thermal conductivity findings introduce the ZZ direction as the right direction in thermoelectric real applications of phosphorene because electronic thermal conductivity in ZZ direction decreases with the magnetic field. In contrast to the electrical conductivity and electronic thermal conductivity, thermopower in AC direction satisfies the requirements of efficient thermoelectric properties at strong magnetic fields. These dependency aspects of the electronic transport coefficients provide insights to modulate the real applications in nanoelectronics.

Using the synergy between the tight-binding model and the Green's function approach, this paper deals with the linear magneto-electron-light interaction in ultranarrow armchair graphene and boronitrene nanoribbons theoretically. In particular, we study the effect of ribbon width and the magnetic field on the refractive index, the absorption coefficient, and the extinction coefficient of abovementioned structures. Here, we reveal that there exists an increasing behavior for static refractive index of both structures when the ribbon width and the magnetic field are increased, whereas the dynamical refractive index increases (oscillates) in average with the ribbon width (magnetic field) at different light frequencies. We further show that the effect of ribbon width (magnetic field) on the absorption coefficient is generally increasing (oscillating) when the photon energy is altered. No refraction and absorption is reported at high enough photon frequencies independent of the ribbon width and the magnetic field strength. In addition, we find that the scattering process in the presence of the magnetic field manifests itself in the extinction coefficient. This information provides insights into the engineers in determining which ribbon width and magnetic field strength to use in the solar cell designs.

With the aid of the two-dimensional tight-binding effective model in combination with effective models for perpendicular electric and magnetic fields, we describe the dynamics of carriers in Bernal bilayer phosphorene. Both electric and magnetic fields are in reliable ranges. We study the electronic phase transition beyond the continuum approximation focusing on the electronic band structure and density of states. In the present work, it is shown that the system subjected to the perpendicular electric field takes a semiconductor-semimetal phase transition with linear- and quadra-like bands along the Γ − X and Γ − Y direction, respectively when the magnetic field is absent. However, turning on the magnetic field shows again the semiconductor-semimetal phase transition for electric field-induced bilayer phosphorene but with only a linear-like band along the Γ − Y direction. Furthermore, we found that by varying the magnetic field, the Landau levels exhibit different treatments in the absence and presence of the electric field. Concisely, we argue that these tunable phase transitions with new rich features give proper motivations to experimentalists to tune the electro-optical properties of bulk phosphorus.

The bosonic phase of the three-band [one flat band and two dispersive bands] Lieb lattice can be significantly influenced by the impurity . Taking into account the combined Zeeman magnetic field , the next-nearest-neighbor (NNN) coupling, and the Dzyaloshinsky-Moriya interaction (DMI), we demonstrate this by studying the Heisenberg model on the spin S = 1∕2 Lieb lattice. We perform numerical calculations of the density of states (DOS) within the Green's function method and the full-consistent Born approximation . The flat band becomes unstable upon the inclusion of the impurity and new degenerate- and mid-states appear. Our numerical results show that the energy of these states depends strongly on the NNN coupling, DMI, and the Zeeman splitting effect. Furthermore, we found different phase transitions depending on the strengths of potentials above. Also, the number of van Hove singularities provided by DOS have a high susceptibility to the Hamiltonian model terms, which provide an intriguing possibility to control the finite impurity doping.

In this paper, the possible electronic phase transitions of β12-borophene crystal are examined using a five-band tight-binding calculation. For different tight-binding models, the Green's function technique is employed for the electronic density of states (DOS). We focus on the modulation of the DOS around the Fermi level with a perpendicular electric field and the dilute charged impurity. The steps to incorporate the effects of external electric field and charged impurity are also detailed with the local Hamiltonian model and the Born approximation, respectively. Our calculations show that the inversion symmetric model is the proper model to discuss the metallic phase of the system, entailing different results compared to the homogeneous model. We find that the electric field opens a tunable band gap and a metal-to-p-doped semiconductor phase transition emerges at the strong perpendicular electric field. The influence of impurity scattering potential on the electronic phase of β12-borophene is much larger than the impurity concentration, in which a metal-to-n-doped semiconductor (metal-to-semimetal) transition takes place at high scattering potentials for the homogeneous (inversion symmetric) model, whereas there is no transition when the impurity concentration is changed. Thereby, producing semimetallic/semiconducting properties by applying an appropriate external electric field and dilute charged impurities paves the way for the realization of β12-borophene-based nano-optoelectronic devices.

In this article we report the effect of charged impurity doping on the thermal spin-wave excitations of 2D Lieb lattice using the longitudinal/transverse and in-plane magnonic heat transport properties in a systematically Green’s function approach and the Born approximation. We present numerical results for XY Heisenberg model. Accordingly, the flow of magnonic current is enhanced depending on the impurity concentration, in which (i) at very low and very high concentrations, in-plane magnon conductivity becomes negative, (ii) the magnon Seebeck coefficient decreases (increases) before (after) a critical impurity concentration for both longitudinal/transverse and in-plane components, (iii) the longitudinal/transverse magnon thermal conductivity increases slightly with impurity concentration, while the in-plane one becomes flat, and (iv) both longitudinal/transverse and in-plane magnonic heat transport efficiencies decrease (increase) at low (high) temperatures with low impurity concentrations, whereas become flat (decrease) at high impurity concentrations. Our results are useful for the implementation of a new spin energy conversion and capable spin-filter devices for the spintronic community.

Optical activity is one of the most fascinating fields in current physics. The strong anisotropic feature in monolayer phosphorene leads to the emergence of non-trivial optoelectronic physics. This paper is devoted to a detailed analysis of strain effects on the optical activity of phosphorene ranging from low-optical-field to high-optical-field. To do so, a numerical study of the two-band tight-binding model is accomplished using the Harrison rule and the linear response theory. Although the transparency of phosphorene confirms at all frequencies independent of the strain modulus and direction, on average, from low- to high-optical-field limit, the polarization of the reflected wave at critical strains becomes circular and the ellipse axis tends to a rotation of 180°. It is found that the maximum absorption takes place at high-energy transitions, which quantitatively depends strongly on the strain modulus and direction. Furthermore, a detailed investigation of compressive and tensile strains results in the dominant contribution of the in-plane compressive and out-of-plane tensile strains to the reflected/transmitted light for low- and intermediate-optical-field ranges, whilst both contribute for the high-optical-field limit. However, overall, in-plane compressive and out-of-plane tensile strains come in to play a role in the absorption spectra. Thereby, the quality of the determined reflection, transmission and absorption waves depends on the regarded regime of the optical field, strain modulus, and strain orientation. These findings if sufficient can be performed and/or tuned experimentally, and a vast number of phosphorene-based optoelectronic devices can be achieved.

In this paper, we have concentrated on the orbital and hybridization effects induced by applied triaxial strain on the interband optical conductivity (IOC) of phosphorene using a two-band Hamiltonian model, linear response theory and the Kubo formula. In particular, we study the dependence of the electronic band structure and of the IOC of a phosphorene single layer on the modulus and direction of the applied triaxial strain. The triaxial strain is included in a model through the introduction of strain-dependent hopping parameters using the Harrison rule. Among the various configurations for applying the triaxial strain, considerable findings are presented here in three classes: (i) uniform, (ii) in-plane uniform and (iii) non-uniform triaxial strain. The main consequence of applying triaxial strain is that of increasing and decreasing the band gap depending on the considered class of study, resulting in a blue shift and red shift of the interband optical transitions, respectively. Our results show that a pure blue shift independent of the strain modulus as well as strain sign (tensile or compressive) emerges when applying non-uniform triaxial strain. The overall feature of our outcomes is tailoring the edge-dependent optical responses of phosphorene in the presence of triaxial strain, which provides the required conditions of tuning the optical properties of phosphorene for future experimental research.

Being able to tune the anisotropic interband transitions in phosphorene at finite temperature offers an enormous amount of possibilities in finding new insights in the optoelectronic community. To contribute to this goal we propose a Zeeman spin-splitting field aiming at absorbing various frequencies of the incident light. Employing the tight-binding Hamiltonian to describe the carrier dynamics and the Kubo formalism to formulate the orientation-dependent interband optical conductivity (IOC) and optical activity of phosphorene we investigate the absorption and scattering mechanisms in phosphorene depending on the Zeeman field strength and optical energy parameters. The optical activity features are characterized by exploring the eccentricity and shift phase of reflected and transmitted electromagnetic waves of the incident light. Different electronic phases in the absence and presence of Zeeman field ultimate different types of interband transitions of which in all cases the IOC along the armchair direction is larger than the zigzag one. However, we observed an irregular (regular) process for IOC with the Zeeman field along the armchair (zigzag) direction, resulting in irregular (regular) absorption and scattering mechanisms. Additionally, a little to no effects for temperature-dependent IOC are provided with the Zeeman field in undoped phosphorene. Further, almost linearly and elliptically polarizations are reported for the transmitted and reflected waves, respectively, indicating that the phosphorene is almost transparent. The emergence of Zeeman spin-splitting effects in optoelectronic properties of phosphorene is pleasant to make it a great potential candidate for logic applications.

We address the electronic phase engineering in the impurity-infected functionalized bilayer graphene with hydrogen atoms (H-BLG) subjected to a uniform Zeeman magnetic field, employing the tight-binding model, the Green's function technique, and the Born approximation. In particular, the key point of the present work is focused on the electronic density of states (DOS) in the vicinity of the Fermi energy. By exploiting the perturbative picture, we figure out that how the interaction and/or competition between host electrons, guest electrons, and the magnetic field potential can lead to the phase transition in H-BLG. Furthermore, different configurations of hydrogenation, namely reduced table-like and reduced chair-like, are also considered when impurities are the same and/or different. A comprehensive information on the various configurations provides the semimetallic and gapless semiconducting behaviors for unfunctionalized bilayer graphene and H-BLGs, respectively. Further numerical calculations propose a semimetal-to-metal and gapless semiconductor-to-semimetal phase transition, respectively, when only turning on the magnetic field. Interestingly, the results indicate that the impurity doping alone affects the systems as well, leading to semimetal-to-metal and no phase transition in the pristine system and hydrogenated ones, respectively. However, the combined effect of charged impurity and magnetic field shows that the pristine bilayer graphene is not influenced much as the functionalized ones and phase back transitions appear. Tuning of the electronic phase of H-BLG by using both types of electronic and magnetic perturbations play a decisive role in optical responses.

A detailed physical meaning of the electronic phase transition in monolayer black phosphorus (BP) has been addressed in the presence of local gate voltage and Zeeman magnetic field. The main features of this transition characterize through the electronic density of states (DOS) in the vicinity of the Fermi level. The numerical calculations have been performed within the continuum approximation of tight-binding model and the Green’s function method. The anisotropy crystal structure of BP causes different behaviors in each component of DOS. First, we have confirmed the Zeeman effect, i.e. the splitting of Van Hove singularities. Then, our results show that the electronic band gap of phosphorene along the x-direction decreases with weak magnetic fields when the gate voltage is absent and the system transits to the semimetallic phase at strong regimes, whereas there is no phase transition along the y-direction. Interestingly, turning on the gate, phase transition independent of the direction does not occur at both weak and strong magnetic fields. Another remarkable point refers to the increase of the band gap with gate voltage at both directions, leading to the semimetallic-semiconductor transition along the x-direction at strong magnetic fields. Tuning the band gap by the gate voltage and magnetic field are useful for future applications of BP.

In this paper, we study theoretically the magneto-topological property and gap engineering of two-dimensional ferromagnetic Lieb lattice by taking into account the next-nearest-neighbors (NNN) coupling and the important Dzyaloshinsky-Moriya interaction (DMI). In particular, the density of states and dispersion energy of the system in terms of various NNN and DMI in the presence of Zeeman field produce the main features in the context of Heisenberg model, Holstein-Primakoff transformation, and Green’s function approach. It is found that the inclusion of NNN coupling opens a gap, leading to the metal-semiconductor-insulator transition depending on its intensity. Furthermore, DMI introduces two extra degenerate bands in the vicinity of Fermi-level due to the Stark effect. The magneto-topological property of the gap in our model is determined by the tunneling probability of particles at both weak and strong NNN and DMI regimes. Finally, we discuss the extension of nuclear spins to arbitrary values.

Combining the frame of the continuum Hamiltonian model, the Born approximation, and the Green's function technique, we systematically investigate the magneto-magnetic susceptibility (MS) of phosphorene in both armchair (AC) and zigzag (ZZ) directions in the presence of impurity doping, electronic dopant, and bias voltage. In particular, we give comparisons between different spin configurations in impurity-infected doped biased phosphorene. Depending on the impurity concentration/scattering potential strength and independent of the direction, there is an antiferromagnetic-ferromagnetic-paramagnetic phase transition. We found that there is no phase transition in the perturbed system along ZZ direction with the magnetic field, whereas there is a magnetic phase transition in AC one. Furthermore, the influence of chemical potential and bias voltage on MS of impurity-infected phosphorene lead to the antiferromagnetic-paramagnetic-ferromagnetic and paramagnetic-ferromagnetic-antiferromagnetic phase transition in AC and ZZ direction, respectively. These dependence aspects of MS provide insights to modulate the inherent electronic and magnetic perturbations effects in spintronic applications of phosphorene.

We theoretically investigate the electron-impurity interaction effects on the direction-dependent electronic heat capacity (EHC) and electrical conductivity (EC) of single-layer biased black phosphorus (BP) as a function of temperature. By means of the Born approximation besides the continuum approximation of the tight-binding Hamiltonian and the Green's function approach, we obtain a significant difference in the x− and y−direction EHC and EC. Inherent higher value of the y−direction density of states around the Fermi energy in the valence band exhibits interesting anisotropic EHC and EC results. The Schottky anomaly in EHC quantity decreases slightly with impurity in x−direction of unbiased BP, whereas there is no change in y−direction. On the other hand, the probability of transition between energy levels provided by the decreased critical temperature decreases with impurity in x−direction EC and similarly there is no alteration in y−direction EC. We show that the Drude weight in EC appears with impurity in both directions. Furthermore, we study the influence of bias voltage on the EHC and EC of impurity-infected BP. Our findings showcase the increased (decreased) EHC (EC) of disordered BP and disappeared the Drude weight of EC in the presence of bias voltage.

By using the Green's function and the tight-binding model within the Born approximation , we obtain the electronic thermal conductivity (TC) and Seebeck coefficient [or thermopower (TP)] of monolayer black phosphorus (BP) under charged impurity doping. The results show that the TC and TP along the y−direction are larger than the x−direction in clean BP due to the inherent higher value of the electronic density of states in y−direction. Furthermore, the TC linearly decreases with low impurity concentrations in both directions, whereas it decreases and does not change with low impurity scattering potentials in x− and y−direction, respectively. On the other hand, TP increases (decreases) with very dilute impurity concentrations and scattering potentials in x−(y−)direction. To be a complete work, we also studied these properties in impurity-infected biased BP. We found that the TC of BP in the presence of impurity gradually increases with bias voltage in both directions and TP has a decreasing (increasing) treatment in x−(y −)direction. Our findings move the thermoelectric (TE) applications of BP into promising TE materials.

Taking into account the electron-impurity interaction within the continuum approximation of tight-binding model, the Born approximation, and the Green's function method, the main features of anisotropic electronic phase transition are investigated in monolayer biased black phosphorus (BP). To this end, we concentrated on the disordered electronic density of states (DOS), which gives useful information for electro-optical devices. Increasing the impurity concentration in both unbiased and biased impurity-infected single-layer BP, in addition to the decrease of the band gap, independent of the direction, leads to the midgap states and an extra Van Hove singularity inside and outside of the band gap, respectively. Furthermore, strong impurity scattering potentials lead to a semiconductor-semimetal transition and one more Van Hove singularity in x-direction of unbiased BP and surprisingly, this transition does not occur in biased BP. We found that there is no phase transition in y-direction. Since real applications require structures with modulated band gaps, we have studied the influence of different bias voltages on the disordered DOS in both directions, resulting in the increase of the band gap.

Motivated by the growing interest in solving the obstacles of spintronics applications, we study the Ruderman-Kittel-Kasuya-Yosida (RKKY) effective pairwise interaction between magnetic impurities interacting through the π -electrons embedded in both electronically doped-semiconducting and metallic armchair graphene nanoribbons. In terms of the Green’s function formalism, treated in a tight-binding approximation with hopping beyond Dirac cone approximation, the RKKY coupling is an attraction or a repulsion depending on the magnetic impurities distances. Our results show that the RKKY coupling in semiconducting nanoribbons is much more affected by doping than metallic ones. Furthermore, we found that the RKKY coupling increases with ribbon width, while there exist some critical electronic concentrations in RKKY interaction oscillations. On the other hand, we find an unusual incoming wave-vector direction for electrons which describes more clearly the ferro- and antiferromagnetic spin configurations in such system. Also, the RKKY coupling at low and high-temperature regions has been addressed for both ferro- and antiferromagnetic spin arrangements.

The spin-dependent electrical conductivity of counterparts of graphene, transition-metal dichalcogenides (TMDs) and group-IV nanosheets, have investigated by a magnetic exchange field (MEF)-induction to gain the electronic transport properties of charge carriers. We have implemented a k.p Hamiltonian model through the Kubo-Greenwood formalism in order to address the dynamical behavior of correlated Dirac fermions. Tuning the MEF enables one to control the effective mass of carriers in group-IV and TMDs, differently. We have found the Dirac-like points in a new quantum anomalous Hall (QAH) state at strong MEFs for both structures. For both cases, a broad peak in electrical conductivity originated from the scattering rate and entropy is observed. Spin degeneracy at some critical MEFs is another remarkable point. We have found that in the limit of zero or uniform MEFs with respect to the spin-orbit interaction, the large resulting electrical conductivity depends on the spin sub-bands in group-IV and MLDs. Featuring spin-dependent electronic transport properties, one can provide a new scenario for future possible applications.

Based on theoretically tight-binding calculations considering nearest neighbors and Green’s function technique, we show that the magnetic phase transition in both semiconducting and metallic armchair graphene nanoribbons with width ranging from 9.83 A to 69.3 A would be observed in the presence of injecting electrons by doping. This transition is explained by the temperature-dependent static charge susceptibility through calculation of the correlation function of charge density operators. This work showed that charge concentration of dopants in such system plays a crucial role in determining the magnetic phase. A variety of multicritical points such as transition temperatures and maximum susceptibility are compared in undoped and doped cases. Our findings show that there exist two different transition temperatures and maximum susceptibility depending on the ribbon width in doped structures. Another remarkable point refers to the invalidity (validity) of the Fermi liquid theory in nanoribbons-based systems at weak (strong) concentration of dopants. The obtained interesting results of magnetic phase transition in such system create a new potential for magnetic graphene nanoribbon-based devices.

We address control of electronic phase transition in charged impurity-infected armchair-edged boron-nitride nanoribbons (ABNNRs) with the local variation of Fermi energy. In particular, the density of states of disordered ribbons produces the main features in the context of pretty simple tight-binding model and Green's functions approach. To this end, the Born approximation has been implemented to find the effect of π-band electron-impurity interactions. A modulation of the π-band depending on the impurity concentrations and scattering potentials leads to the phase transition from insulator to semimetallic. We present here a detailed physical meaning of this transition by studying the treatment of massive Dirac fermions. From our findings, it is found that the ribbon width plays a crucial role in determining the electronic phase of disordered ABNNRs. The obtained results in controllable gap engineering are useful for future experiments. Also, the observations in this study have also fueled interest in the electronic properties of other 2D materials.

In this work, we show that the magnetic phase transition in both semiconducting and metallic armchair graphene nanoribbons would be observed in the presence of electronic dopant. However, the mutual interactions between electrons are also considered based on theoretically tight-binding and Hubbard model calculations considering nearest neighbors within the framework of Green's function technique. This work showed that charge concentration of dopant in such system depending on the weak and strong mutual repulsions plays a crucial role in determining the magnetic phase. It follows from the obtained results that the ground state turns paramagnetic in a range of carrier concentrations by neglecting the electronic correlations. The inclusion of a Coulombic repulsion between electrons stops the phase transition and system remains in its ground state antiferromagnetic phase. Furthermore, we concluded that magnetic phases are insensitive to the electron-electron interaction at all weak and strong concentrations of dopant. In addition, this paper provides a controllable gap engineering by doping and inclusion of electron-electron repulsions for further studies on such system as a new potential nanomaterial for magnetic graphene nanoribbon-based applications.

In this work, based on the Kubo-Greenwood formalism and the k . p Hamiltonian model, the impact of Rashba spin-orbit coupling on electronic band structure and electrical conductivity of spin-up and spin-down subbands in counterparts of graphene, including silicene, stanene, and germanene nanosheets has been studied. When Rashba coupling is considered, the effective mass of Dirac fermions decreases significantly and no significant change is caused by this coupling for the subband gaps. All these nanosheets are found to be in topological insulator quantum phase at low staggered on-site potentials due to the applied perpendicular external electric field. We point out that the electrical conductivity of germanene increases gradually with Rashab coupling, while silicene and stanene have some fluctuations due to their smaller Fermi velocity. Furthermore, some critical temperatures with the same electrical conductivity values for jumping to the higher energy levels are observed at various Rashba coupling strengths. For all structures, a broad peak appears at low temperatures in electrical conductivity curves corresponding to the large entropy of systems when the thermal energy reaches to the difference between the energy states. Finally, we have reported that silicene has the larger has the larger electrical conductivity than two others.

Motivated by the recent nanophotonic community, in this work, we address the behavior of quantized charge-density fluctuations of doped and gated semiconductor armchair-type graphene nanoribbons within the tight-binding model and the Green’s function technique. In particular, we study the behavior of frequency-dependent susceptibility, when the system is exposed to photons or electrons. Injecting electrons by doping or ejecting ones by gating lead to different treatments in response function. Doping offers new collective modes due to added states between the valence and conduction bands (provided by the density of states) corresponding to intraband transitions, while gating distributes intraband modes. The results show that both ribbon width and doping concentrations affect the intraband transitions in electro-optical devices. Another remarkable point is the strong sensitivity of intraband plasmons to the direction of incoming photons or electrons. We found that the susceptibility of doped nanoribbons vanishes at perpendicular angles due to the distribution of intraband modes.

With the help of the simple tight-binding Hamiltonian and Green's function technique, we study how intraband and interband plasmon modes of both semiconducting and metallic armchair graphene nanoribbons are influenced by the width, chemical doping, and incident momentum direction. In particular, we investigate the behavior of the frequency-dependent susceptibility when the system is exposed to photons or electrons. Injecting electrons by doping creates a new collective mode due to new states between the valence and conduction bands corresponding to intraband transition for which the effect of ribbon width on these transitions in the semiconducting case is much more sensitive than metallic ones. Furthermore, some critical chemical potential and momentum values for both intraband and interband modes lead to different behaviors for resonant peaks. Another remarkable point is the high sensitivity of intraband plasmons to the direction of incident momentum. In particular, the susceptibility of doped nanoribbons vanishes at perpendicular directions, i.e., the intraband plasmons disappear.

In this paper, we analytically investigate the electronic density of states (DOS) of bilayer Bernal black phosphorus (BBP) in order to study the anisotropic electronic phase transition. We employ the Green's function approach, the Born approximation, and the tight-binding Hamiltonian model including ten intra-layer and four inter-layer hopping energies. BBP consisting of two coupled layers of black phosphorus is a suitable candidate for studying the layer-dependent electronic properties of few-layer black phosphorus. We examine the electronic properties of BBP under conditions in which only one layer and both layers are subjected to a dilute charged impurity and a perpendicular electric field. Our findings show that there is no phase transition when the impurity is doped on only one layer, whereas in the case of both layers, BBP suffers a phase transition from semiconductor to semimetal at strong impurity scattering potentials. Also, applying the electric field on one layer of BBP leads to an increase in the band gap, whereas in the case of both layers, the band gap decreases with the electric field and eventually, a phase transition appears at a bias voltage of more than 1.8 eV. Consequently, the band gap of BBP can be tuned by applying an electric field and a charged impurity, and thereby these findings provide insights for future experimental research on black phosphorus.

We studied how the strain, induced exchange field and extrinsic Rashba spin-orbit coupling (RSOC) enhance the electronic band structure (EBS) and electronic heat capacity (EHC) of ferromagnetic silicene in presence of external electric field (EF) by using the Kane-Mele Hamiltonian, Dirac cone approximation and the Green’s function approach. Particular attention is paid to investigate the EHC of spin-up and spin-down bands at Dirac K and K ′ points. We have varied the EF, strain, exchange field and RSOC to tune the energy of inter-band transitions and consequently EHC, leading to very promising features for future applications. Evaluation of EF exhibits three phases: Topological insulator (TI), valley-spin polarized metal (VSPM) and band insulator (BI) at given aforementioned parameters. As a new finding, we have found a quantum anomalous Hall phase in BI regime at strong RSOCs. Interestingly, the effective mass of carriers changes with strain, resulting in EHC behaviors. Here, exchange field has the same behavior with EF. Finally, we have confirmed the reported and expected symmetry results for both Dirac points and spins with the study of valley-dependent EHC.

Thermodynamic properties of gapped graphene-like structures by considering the effects of interaction between electrons and Holstein phonons have been studied. Particularly, we study the heat capacity and paramagnetic susceptibility of structures as a function of temperature within the Green’s function method with the help of Holstein model. The paramagnetic susceptibility and heat capacity can be derived by using density of states based on the Kubo formula. We have found the energy dependence of density of states for various values of gap in the presence of Holstein phonons. Finally, the temperature behaviors of specific heat and spin susceptibility of gapped graphene structure due to electron-phonon coupling have been investigated. Our results show the electron-phonon interaction leads to the appearance of a double van Hov singularity for each value of gap parameter. Also, electron-phonon coupling affects the value of heat capacity and magnetic susceptibility at low temperatures.

The k.p Hamiltonian model at low-energy regime and Green’s function technique are carried out to study the effect of Rashba spin-orbit coupling (RSOC) on the spin-dependent thermodynamic properties in monolayer molybdenum disulphide. In particular, we have studied electronic band structure (EBS), electronic heat capacity (EHC), and magnetic susceptibility (MS) of spin-up and spin-down subbands by variation of RSOC strength. An enhancement of the spin splitting can be achieved by increasing the Rashba coupling strength. Electronic subband structure shows that the effective mass of carriers decreases with Rashba coupling while the spin-up and spin-down subband gaps remain constant. The reduced effective mass leads to the increase of both EHC and MS as reasonable results. Spin splitting can be also obtained from the EHC and MS findings. At very strong Rashba couplings, two Dirac-like points appear for spin-up subbands which we can claim that the system gets a new quantum anomalous Hall phase.

Using Landauer-Buttiker formalism, thin barrier approximation, and Klein tunneling theory, the effects of external electric field and induced exchange magnetic gap on the conductance and specifically magnetoresistance (MR) of a topological insulator-based ferromagnet/insulator/ferromagnet (TI-based F/I/F) junction have investigated. Conductance and MR for parallel (P) and antiparallel (AP) magnetic gap configurations are obtained and discussed. The results of MR reveal a significant negative value for maximum magnetic gap at z = (n + 3/4)π (n = 0, 1...), i.e., in smaller insulator barrier strengths in the presence of an external electric field, while it appears at z = n π (n = 1, 2...) without electric field in Ref. Linder et al., Phys. Rev. Lett., 104:067001, 2010 (z is the barrier strength). Also, we have found that maximum transmission probability (TP) increases with magnetic gap in the presence of electric field which is one for P case in z = (n + 1/6)π and approaches to one for induced magnetic gap near the Fermi energy for AP case in the absence of electric field and z = (n + 1/2)π. Finally, the conductance and MR oscillate with the magnetic gap and barrier strength.

By applying the full π-band Holstein model, we analytically have calculated the electronic density of states (DOS) and optical conductivity of doped gapped graphene-like structures including silicon carbide (SiC), boron nitride (BN), and beryllium monoxide (BeO) beyond the Dirac cone approximation. We have implemented the Kubo linear response formalism which is established to get the retarded self-energy here. For strong electron-phonon (e-ph) coupling strengths, an addition peak in the optical conductivity has been found, associated with transitions between the midgap states and the Van Hove singularities of the main π-bands. Optical conductivity (optical absorption) decreases (increases) with the gap which is useful in the fabrication of low-dimensional-based solar cells. At large gaps, a clean sheet of doped graphene has a zero optical conductivity at low energies because of the optically interband excitations through the Dirac points. Also, the Drude weight remains unchanged for all cases at low energy regimes. Consequently, DOS and optical conductivity remain constant with temperature at low e-ph interaction strengths.

Dirac theory and Green's function technique are carried out to compute the spin dependent band structures and corresponding electronic heat capacity (EHC) of monolayer (ML) and AB-stacked bilayer (BL) molybdenum disulfide ( MoS 2 ) two-dimensional (2D) crystals. We report the influence of induced exchange magnetic field (EMF) by magnetic insulator substrates on these quantities for both structures. The spin-up (down) subband gaps are shifted with EMF from conduction (valence) band to valence (conduction) band at both Dirac points in the ML because of the spin-orbit coupling (SOC) which leads to a critical EMF in the K point and EHC returns to its initial states for both spins. In the BL case, EMF results split states and the decrease (increase) behavior of spin-up (down) subband gaps has been observed at both K and K ′ valleys which is due to the combined effect of SOC and interlayer coupling. For low and high EMFs, EHC of BL MoS 2 does not change for spin-up subbands while increases for spin-down subbands.

In the current work, we theoretically study the electronic band structure (EBS), electronic heat capacity (EHC) and magnetic susceptibility (MS) of three structures including monolayer, AA-stacked and AB-stacked bilayer silicene based on the Kane–Mele Hamiltonian model and Green's function method. The particular attention of this study is paid to the effect of external electric field on the aforementioned physical properties. By variation of the electric field, three phases are found: Topological insulator (TI), valley–spin polarized metal (VSPM) and band insulator (BI). Marvellously, its electronic minibands show that the spin-up contribution of charge carriers with lowest energy bands behaves like relativistic Dirac fermions with linear (parabolic) energy dispersions in monolayer (bilayer) case near the Dirac points. An insightful analysis shows that the maximum and minimum value of EHC peak appear for (AA) AB-stacked bilayer and monolayer silicene in TI (BI) regime while in MS curves appear for (AB) AA-stacked bilayer and monolayer lattices in TI (BI) regime, respectively. Moreover, we have observed a phase transition from antiferromagnetic to ferromagnetic and paramagnetic in the monolayer and bilayer structures in the VSPM regime based on the MS findings, respectively.

In this work, we have studied the influences of electron-phonon (e-ph) coupling and chemical potential on the boron nitride graphene-like sheet. In particular, by starting the Green's function technique and Holstein model, the electronic density of states (DOS), electronic heat capacity (EHC) and magnetic susceptibility (MS) of this system have been investigated in the context of self-consistent second order perturbation theory which has been implemented to find the electronic self-energy. Our findings show that the band gap size decreases (increases) with e-ph coupling (chemical potential) parameters. The Schottky anomaly (crossover) decreases in EHC (MS) as soon as e-ph coupling increases. Also, the corresponding temperature with Schottky anomaly is considerably affected by e-ph coupling.

In this paper, we have investigated the effects of strain on DC transverse and spin-valley Hall conductivity (SHC-VHC) of two-dimensional buckled materials ferromagnetic graphene's analog, MoS2 and silicene due to their spin–orbit coupling. The Kubo formalism has been used to investigate the dynamics of carriers under strain along the armchair (AC) direction of systems in the context of the Kane–Mele Hamiltonian and the Dirac cone approximation. The effective mass of carriers increases with strain and this leads to the reduction of their transport. We have found that SHC-VHC changes symmetrically with respect to a critical strain around 13% and 45% for MoS2 and silicene, respectively. Furthermore, the reflection symmetry of silicene has been broken with electric field and a phase transition to topological insulator for strained ferromagnetic silicene has been seen.

By applying the Green’s function technique and using the tight-binding Hamiltonian model, thermodynamic properties of gapped graphene-like structures, including silicon carbide (SiC), boron nitride (BN) and beryllium monooxide (BeO) in the presence of a transverse magnetic field are investigated. In fact, we have studied electronic density of states (DOS), electronic heat capacity (EHC) and magnetic susceptibility (MS) in order to investigate the dynamics of Dirac fermions. At an applied certain value of magnetic field, the band gap width increases for SiC, BN and BeO structures with respect to the gapless graphene and a double peak appears in DOS with increasing of quantum states. On the other hand, the band gap size decreases with magnetic field. We have found that EHC and MS increase slightly at low temperatures with gap and magnetic field. Also, EHC and MS reach to their maximum value at a critical temperature point while an increase behavior has been observed for high temperatures significantly.

In this work, the spin-dependent electronic band structure and magnetic susceptibility (MS) of two graphene’s analogue, monolayer and AB-stacked bilayer MoS 2, are investigated in the presence of induced magnetization by magnetic insulators. Here, we have assumed that the type of magnetic insulators can change the magnetization strength. By focusing on the Dirac points of Brillouin zone and using the Dirac theory within the Green’s function approach, band structures and MSs of two systems are compared. In the bilayer case, gate voltage produces an electric field in order to have spin-orbit coupling (SOC) for spintronic applications. Our results represent the Stark and Zeeman effects because of the spin splitting states by electric and magnetic fields. We have found that the MS has marvelous behaviors before and after a critical magnetization in the monolayer MoS 2 while it increases slightly in bilayer lattice. Up (down) spin band gaps in bilayer MoS 2 decreases (increases) at both Dirac points due to the combined SOC and interlayer hopping potentials.

The Kane-Mele model and Green’s function technique help us to describe and survey the influences of electric and exchange magnetic fields (EF and EMF) on the dynamics of Dirac fermions in the monolayer silicene by studying the energy dispersion (ED), electronic heat capacity (EHC) and Pauli magnetic susceptibility (MS). Spin band gaps show three phases including topological insulator (TI), valley-spin polarized metal (VSPM) and band insulator (BI) in the presence of external EF. The maximum value for EHC and MS have been observed in the VSPM phase. Also, ED evolution with EMF shows that the Fermi level decreases. Interestingly, two critical temperatures have been found in the VSPM regime of 0.5 and 2 in terms of spin-orbit coupling (SOC) strength in the EHC and MS curves, respectively. In addition, remarkable point in the VSPM regime is the observed phase transition from antiferromagnetic to paramagnetic in MS curves. Competition between thermal and quantum effects is the main reason of these critical points which these findings can control the thermal properties of silicene-based devices.

Through restructuring of the electronic spectrum of two-dimensional massive fermions in buckled silicene under an applied electric field and induced exchange field, we have studied how Rashba spin–orbit coupling enhances the electronic band structure and electronic heat capacity of the system. Special attention was given to investigate the spin- and valley-dependent electronic heat capacity. By variation of the electric field, system transitions occurred from the topological insulator phase to the band insulator phase. The Kane–Mele Hamiltonian model and the Green’s function technique were used in this work. The first remarkable point is the unchanged (changed) subband gap size (effective mass of fermions) with Rashba coupling for all phases. We have found a critical Dirac-like point which affects the effective mass of the carriers in the band insulator phase. And finally, we found that variation in the Hall conductivity with Rashba coupling leads to quantized Hall conductivity, which was the main result of the current study: a new quantum anomalous Hall effect at large Rashba coupling strengths. The presented methodology may be extended to other two-dimensional materials, like germanene and stanene.

The electronic heat capacity (EHC) and magnetic susceptibility (MS) of the two-dimensional material ferromagnetic graphene's silicon analog, silicene, are investigated by the strain-induced and the applied electric field within the Green's function technique and the Kane-Mele Hamiltonian. Dirac cone approximation has been performed to investigate the system under strain along the zigzag (ZZ) direction. The main attention is focused on the effects of external static electric field in the presence of strain on EHC and MS of a ferromagnetic silicene sheet. In the presence of strain, carriers have a larger effective mass and transport decreases. As a result, the temperature dependence of EHC and MS gives a critical strain around 10%. Furthermore, electric field breaks the reflection symmetry of the structure and a transition to the topological insulator for strained ferromagnetic silicene has occurred when the electric field is increased. In this phase, EHC and MS have weird behaviors with temperature.

Using the Harrison model and Green's function technique, impurity doping effects on the orbital density of states (DOS), electronic heat capacity (EHC) and magnetic susceptibility (MS) of a monolayer hydrogenated graphene, chair-like graphane, are investigated. The effect of scattering between electrons and dilute charged impurities is discussed in terms of the self-consistent Born approximation. Our results show that the graphane is a semiconductor and its band gap decreases with impurity. As a remarkable point, comparatively EHC reaches almost linearly to Schottky anomaly and does not change at low temperatures in the presence of impurity. Generally, EHC and MS increases with impurity doping. Surprisingly, impurity doping only affects the salient behavior of py orbital contribution of carbon atoms due to the symmetry breaking.

An optical absroption in the ferromagnetic insulator with the Dzyaloshinskii–Moriya interaction (DMI), theoretically and numerically is investigated. In other words, we investigate the optical absroption, σ ( ω ) , of the spin Lieb nanolattice (SLN) a face-centered square lattice, subjected to a light spectrum with frequency ω. Using linear response theory and Green's function approach, the frequency dependence of optical conductivity (OC) has been obtained in the context of Heisenberg Hamiltonian. At low frequencies, the OC is found to be monotonically increasing with frequency for next-nearest neighbor (NNN) coupling, DMI strength (DMIS) and temperature. Also we have found that OC has a peak (absorption). Eventually, we are investigated the effect of magnetic field on the OC of this system.

Alongside a vast number of two-dimensional (2D) materials, $g\text{\ensuremath{-}}{\mathrm{SiC}}_{7}$ siligraphene has already been introduced to the 2D nanomaterial family with extraordinary sunlight absorption. In this work, we exploit the concept of the in-plane biaxial strain-induced electro-optical engineering in siligraphene from ab initio many-body perturbation theory calculations (GW + Bethe-Salpeter equation). While the semiconductor nature of the system is robust for both tensile and compressive strains, the direct GW band gap of 1.35 eV is decreased (increased) linearly with tensile (compressive) strain. This is accompanied by a variation in the optical transitions. The tensile strain in $g\text{\ensuremath{-}}{\mathrm{SiC}}_{7}$ shifts the optical absorption peaks to the lower photon energies, the so-called red shift, whereas the compressive strain causes a shift to the higher frequencies, the so-called blue shift. Eventually, the exciton binding energy trend and the real-space exciton distribution for the bright bound state is studied with strain, resulting in a alteration of the spatial distribution radius of exciton wave functions from tensile to compressive. Our findings extend the logic applications of siligraphene in photovoltaic devices, especially in tailoring the power conversion efficiency of excitonic solar cells.

We theoretically address the perpendicular magnetic field effects on the electronic phase of Bernal bilayer graphene and hexagonal boron-nitride (h-BN) taking into account the total and orbital-projected electronic bands using the tight-binding parameters in the Harrison model, followed by the Green's function method. First, we confirm that our model is computationally efficient and accurate for calculating the magneto-orbital electronic phase transition by reproducing the semimetallic and insulating treatments of pristine Bernal bilayer graphene and h-BN, respectively. In our model, the magnetic field couples only to the electron spin degrees of freedom (with the same contributions for spin-up and spin-down) due to the low dimension of the systems. Here, the main features of the phase transitions are characterized by the electronic density of states (DOS). We found that sp2-hybridization is destroyed when the systems are immersed in the magnetic field, leading to a phase transition to metal for both systems at strong magnetic fields. While there is no phase transition for bilayer graphene at weak magnetic fields, for the case of bilayer h-BN, an insulator to semiconductor phase transition can be viewed, making h-BN more applicable in industry. In bilayer graphene, the anisotropic phase transition appears as insulator–semiconductor, insulator–metal, and semimetal–metal for s-, {px + py}-, and pz-orbitals, respectively, whereas in the case of bilayer h-BN, one observes the same transitions for {s,pz}-orbitals but insulator–semiconductor for {px + py} orbitals. Generically, our findings highlight that the applied magnetic field manipulates the band structure of bilayer graphene and h-BN, and gives ideas to experimentalists for tuning the electro-optical properties of these materials.

In this paper, the effects of dilute charged impurity doping on electronic heat capacity (EHC) and magnetic susceptibility (MS) of a two-dimensional material ferromagnetic gapped graphene-like, MoS2, are investigated within the Green’s function approach by using the Kane-Mele Hamiltonian and self-consistent Born approximation (SCBA) at Dirac points. Our findings show that there is a critical impurity concentration (IC) and scattering strength (ISS) for each valley in EHC and MS curves. Also, we have found that the spin band gap decreases with impurity only for valley K, ↑ and $K^{\prime }, \downarrow $ due to the existence of inversion symmetry between valleys. On the other hand, a magnetic phase transition from ferromagnetic to antiferromagnetic and paramagnetic has been observed. The increase of scattering rate of carriers in the presence of impurity is the main reason of these behaviors.

Using linear response theory, Heisenberg model Hamiltonian and Green's function technique, the influences of Dzyaloshinskii–Moriya interaction (DMI), external magnetic field and next-nearest-neighbor (NNN) coupling on the density of magnon modes (DMM), the magnetic susceptibility (MS) and the magnon heat capacity (MHC) of a spin Lieb lattice, a face-centered square lattice, are investigated. The results reveal a band gap in the DMM and we witness an extension in the bandwidth and an increase in the number of van-Hove singularities as well. As a notable point, besides the magnetic nature which includes ferromagnetism in spin Lieb-based nanosystems, MS is investigated. Further, we report a Schottky anomaly in the MHC. The results show that the effects of the magnetic field on the MHC and MS have different behaviors in two temperature regions. In the low temperature region, MHC and MS increase when the magnetic field strength increases. On the other hand, the MHC and MS reduce with increasing the magnetic field strength in the high temperature region. Also comprehensive numerical modelling of the DMM, the MS and the MHC of a spin Lieb lattice yields excellent qualitative agreement with the experimental data.