What nature demands from us is not a quantum theory or a wave theory; Mathematical models for nanoscience; low dimensional nanostructures and their applications Our interests here lie primarily with low dimensional nanostructures where the motion of electrons can be confined from one, two, and even three spatial dimensions. The respective structures are known as quantum wells, quantum wires, and quantum dots. Such structures will continue transforming our lives as they will be used as parts of environmental, chemical, and biological sensors, in transportation industries, energy saving technologies, biomedicine, communications, and information technology. In order to successfully use them, we have to know their properties which can be understood better with new mathematical models developed for such nanostructures. The area of mathematical modeling for nanoscience & nanotechnology is dotted with many open problems, some of which we have been addressing. This includes the development of mathematical models for more accurate determinations of properties of such structures, accounting for a range of multiscale effects, originating from the influence of mechanical, electrical, thermal, and other fields. Both classical and quantum mechanical models are required in this development. We have also been working on the formulation of correct boundary conditions for multiscale problems arising from modelling low dimensional nanostructures. Properties at the nanoscale are size dependent, allowing to develop new materials with previously unmatched properties by manipulating their nanostructures. We analyze such properties with stateoftheart tools that we have been developing based of the Density Functional Theory (DFT), Green's function and nonequilibrium Molecular Dynamics (MD) approaches, to name just a few. In a number of cases, we have to deal with nonlinear problems and several new models that have allowed us to account for new important effects, previously unaccounted for, have also been developed in this context. In this focus area, our research interests have included the following topics:
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Mathematical models for quantum dots and other lowdimensional structures, quantum information, control, dynamics, coupled problems in quantum mechanics We approach the analysis of low dimensional nanostructures as a multiscale, multiphysics problem. We were the first to point out the importance of deriving the correct boundary conditions for the analysis of quantum dots with wetting layers. The underlying problem is a typical multiscale problem, where despite the smallness of the wetting layer as compared to the quantum dot itself, it can influence substantially the electronic processes in the dot. Such boundary conditions were derived and discussed in detail in [ EJ75 ]. Multiscale systems of quantum dots with wetting layers were analyzed in [ EJ65 ] for different geometries of quantum dots and the results of band structure calculations were systematically compared (also proceedings [ EP25 ] where cylindrical quantum dots with wetting layers were analyzed in detail). In what follows, we briefly describe our results on the development and applications of mathematical models for the analysis of low dimensional nanostructures, focusing first on quantum dots, rods, nanowires, and nanowire superlattices, and then moving to quantum dot arrays. By using the multiband theory, we demonstrated the universality of the phenomenon of inversion, showed its existence for the exited states, and provided a guidelines for the prediction of barrier localization in modulated nanowires. Furthermore, for the first time, on the basis of the multiband theory we demonstrated the existence of critical radii for the inversion of hole states in such nanowires [ EJ61 ]. This phenomenon should be readily observable in both optical spectroscopy and transport. A detailed analysis of the influence of shape, orientation, size and material system on the band structure of quantum wires was carried out in [ EJ62 ] where we also provided results on comparisons between the classical LuttingerKohn Hamiltonian and the BurtForeman correction of this Hamiltonian. Properties of quantum confinement of symmetric and asymmetric nanowire superlattices (NSL) were studied in [ EJ63 ] where it was also demonstrated convincingly that asymmetrical NWSL structures have wellpronounced qualitative and quantitative differences as compared to both symmetrical NWSL structures and infinite periodic NWSL structures. The issue of boundary conditions for the analysis of NWSLs in cylindrical polar coordinates have been analyzed in [ EJ67 ]. In fact, accounting for symmetries, e.g. cylindrical symmetry, whenever possible is an important direction in the computational analysis of low dimensional nanostructures. In [ EJ80 ] we presented a new multiband model wurtzite semiconductor nanostructures with cylindrical symmetry. This model extended the VahalaSercel formulation to the RashbaShekaPikus Hamiltonian for wurtzite semiconductors, without the need for the axial approximation. In [ EJ80 ] we also provided comparisons of our formulation for studying the electronic structure of wurtzite quantum dots with the conventional formulation. The influence of aspect ratio on electronic states of quantum rods was analyzed in [ EJ73 ]. We observed that the nonseparability and multiband nature of the problem leads to a complex groundstate envelope function and also to the phenomenon of level crossing. In [ EJ73 ] we also demonstrated the intrinsic difference between the valence states of a quantum rod and those of a quantum wire. When the multiband theory is applied to the analysis of low dimensional nanostructures, in some cases spurious, nonphysical solutions may arise. On the example of 8band model, in [ EJ98 ] we demonstrated how the spurious solution problem can be resolved. The methodology was applied to studying the electronic properties of InP and InAs freestanding nanowires. We presented band gaps and effective masses as functions of size, shape, and orientation of the nanowires, and compared our results with experimental works and with other calculations. The longstanding problem of spurious solutions in the multiband models has been analyzed in detail in [ EJ133 ]. We analyzed the resulting mathematical model showed that a failure to restrict the Fourier expansion coefficients of the envelope function to small k components would lead to the appearance of nonphysical solutions. We proposed a simple and effective solution to the problem and demonstrated it on an example of a twoband model for both bulk materials and lowdimensional nanostructures. Based on the above requirement of small k, we also derived a model for nanostructures with cylindrical symmetry and applied the developed model to the analysis of quantum dots using an eightband model. We also develop Molecular Dynamics and atomistic based methods for the analysis of nanostructures in those cases when it is necessary. In particular, by using Nonequilibrium Molecular Dynamics (NMD), we explored geometry and temperature dependent properties of materials at different scales, including the analysis of such properties in low dimensional nanostructures. Our interests include structural, thermodynamic, mechanical, transport, and optoelectronic properties, as well as relative and phase stability of materials at the nanoscale. Other tools that we are developing in these studies are based on first principles calculations (applied to nanostructures and intermetallic compounds), nonequilibrium Green's function techniques and density functional theory [EJ123 , EJ125 , EJ134 , EJ139 , EJ140 , EJ144 ] . Nonlinear effects in low dimensional nanostructures are becoming increasingly important. In [ EJ76 ] we focused on the influence of nonlinear strain effects and in [ EJ94 ] on their inclusion in the Hamiltonians for nanostructure modelling. In [ EJ83 ], we proposed a general method of treating Hamiltonians of deformed nanoscale systems is proposed. This method was used to derive a secondorder approximation both for the strong and weak formulations of the eigenvalue problem. The weak formulation was needed in order to allow deformations that have discontinuous first derivatives at interfaces between different materials. It was shown that, as long as the deformation is twice differentiable away from interfaces, the weak formulation is equivalent to the strong formulation with appropriate interface boundary conditions. Transport phenomena were in the focus of our analysis of quantum wires where we applied quantum correction type models [ EJ95 ]. Based on the quantum driftdiffusion model, we analyzed in detail the influence of the metal contact size on the electron dynamics and transport inside such nanowires. We presented in [ EJ95 ] CurrentVoltage (IV) characteristics for the same heterostructure nanowire supplemented with different combinations of the metal contacts of different sizes. It was demonstrated that the size of the metal contacts strongly influences the carrier dynamics and IV characteristics of the heterostructure nanowire and that it is possible to produce and tune Nshape of the IV characteristics by changing the size of the metal contacts. A general framework for the nonlinear analysis of low dimensional nanostructures was laid down in [ EJ110 ] where we demonstrate that the conventional application of linear models to the analysis of optoelectromechanical properties of nanostructures in bandstructure engineering could be inadequate and presented generalizations of the existing models for bandstructure calculations in the context of coupled effects. Coupled effects in low dimensional nanostructures were considered in a series of paper [ EJ90 , EJ135 , EJ136 , EJ138 ]. In [ EJ90 ] the focus was on the dynamic coupling of piezoelectric effects, spontaneous polarization, and strain, while [ EJ136 ] was devoted on coupled electromechanical effects in finite length nanowires. In addition to a comprehensive comparison of two and three dimensional models, accounting for anisotropy and piezoelectricity, we also presented the results on the effects of the finite length of the nanowires on electromechanical field distributions. For the first time, the fully coupled thermoelectromechanical models in conjunction with the band structure analysis of quantum wires and quantum dots have been presented in [ EJ135 ] and [ EJ138 ], respectively. Nonlinear strain models were considered also in the context of quantum dot molecules [ EJ79 , , EJ81 ]. It was pointed out that the influence of strain effects is typically analyzed with simplified linear models based on the minimization of uncoupled, purely elastic energy functionals with respect to displacements . The applicability of such models is limited to the study of isolated idealized quantum dots, and both coupled and nonlinear effects need to be accounted for in the analysis of more realistic structures. In this paper, generalizations of the existing models for bandstructure calculations are discussed in the context of strain effects. Exemplifications are given for hexagonal wurtzite semiconductor nanostructures. Note also that the importance of coupled formulations in the analysis of electromechanical fields (including the context of piezoelectricity) was emphasized in our earlier works on electroelasticity (see details here). Quantum dot molecules (or arrays) mentioned above have a number of promising applications, ranging from the construction of of quantum networks in information processing to biological systems. We analyzed several configurations of coupled quantum dots [ EPQD2010] where we used an external magnetic field as a control variable allowing to taylor the nanostructure properties. Further results in this direction can be found in [ EJ146 ]. Strain, electric, and thermal fields in the fully coupled modelling of low dimensional nanostructures can also serve as control variables. Several special journal issues, where some of the stateoftheart mathematical modelling tools for the analysis of low dimensional nanostructures, their applications and related topics, include [ EJ68 , EJ97 , EJ121 , EJ124 ].
Carbon nanotubes and their arrays in biological imaging and other applications, lowdimensionalnanostructurebased sensors and networks We are interested in the development of mathematical models for biological and chemical sensors (including those can be used in harsh environments for environmental monitoring), as well as for the field emission as an alternative to the conventional Xray radiation. In particular, we proposed a series of new multiscale multiphysics models for the evolution of Carbon Nanotubes (CNTs) on thin films, numerical methods for the solution of the resulting problems, and we reported the efficiency of the proposed models and techniques based on theoretical, computational, and experimental results [ EJ112 ,EJ116 , EJ122 , EJ131 ]. Zincoxide (ZnO) nanowires provide a promising element for efficient chemical sensors as well as for other applications. We have analyzed these materials with density functional theory and nonequilibrium Green's function technique in [ EJ125 , EJ139 ]. Such nanowires have also been used for sensor applications in biodetection and identification. Our interest in this area includes also the development of mathematical models for the analysis of arrays (and ultimately networks) based on low dimensional nanostructures. In addition to CNT arrays described above and quantum dots arrays described in the previous section, other low dimensional nanostructures (e.g., nanowires) can also be used in designing such new structures and networks. Nonlinear problems and low dimensional nanostructures, inverse problems, properties of materials and new phenomena at the nanoscale Nonlinear phenomena and associated mathematical problems are important parts of the analysis of low dimensional nanostructures. Our interests in this area include coupled mathematical models, in particular those that account for both classical and quantum phenomena. Our efforts in the development of nonlinear strain models has been described here with basic summaries of the results found in [ EJ76 , EJ83 , EJ94 , EJ110 ]. Results on nonlinear currentvoltage ( IV) characteristics of quantum nanowires as functions of the size of the metal contacts were reported in [ EJ95 ]. In this work transport phenomena were in the focus of our analysis of quantum wires where we applied quantum correction type models. Many problems in the analysis of low dimensional nanostructures are reducible to eigenvalue PDE problems which subsequently have to be discretized. Nonlinear eigenvalue problems may arise in this context in several different ways and even in the linear case there are many nontrivial problems to be address[ EJ34 ]. Our interests include eigenvalue problems arising in the context of multiband theories and density functional models. Our interest in this field also include size dependent effects, the influence of phase transformations (that may occur under different conditions, e.g. ion irradiation), as well as defects enhanced coupled phenomena and higher order effects (such as electrostriction) on properties of nanostructures [ EP79 , EP80 , EP81 , EPRioAIP2009 , EPASME2010]. Many important problems in nanoscience, nano and bionanotechnological applications can be formulated mathematically as inverse problems. For example, the idea of designing nanostructures via inverse scattering leads to such problems. Subsequently, in some cases they can be reduced to a multiband Riccati model for the envelope function, ultimately allowing to control electronic structure and transport properties of the nanostructures being analyzed. The problems of interest here are not limited to quantum well situations, where the GelfandLevitanMarchenko theory can be applied due to the onedimensionality of the problem, as our interest includes also quantum wires and dots where problems become 2D and 3D, respectively.
Quantum dots as reactive agents, open systems, boundary conditions, foundations of quantum mechanics With new properties of materials at the nanoscale, as compared to bulk materials, and their increasing range of applications, we should understand their properties as reactive agents much better. For example, one of the promising applications of quantum dots includes intracellular imaging where quantum dots are attached to selected molecules. In this and other applications, it is important to know cytotoxicity properties (toxicity to cells) of such quantum dots. Furthermore, key properties of quantum dots, such as quantum confinement, have a direct effect on their catalytic functions due to electron affinity (changes in the ability of quantum dots to accept/donate electron charge). As quantum dots often create reactive oxygen species (ROS), the key mechanisms of toxicity are usually attributed to the oxidation process. We are interested in the development of mathematical models for the description of these and associated processes. This development requires new approaches that can address fundamental problems and challenges that are on the border between classical and quantum mechanics. As nanostructures will play an increasingly important role in bioimaging, visualization of life critical processes in living organisms such as metabolic, and biomedicine, the developed models will be indispensable in this scientific advance. Most problems we are facing in this area are multiscale, multiphysics problems as discussed earlier.
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