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References
- Kruglyak Yu. А., Kostritskаya L. S. Thermoelectric phenomena and devices in the generalized model of electron transport
- Kruglyak Yu. A., Shtefan N. Z. Heat transfer by phonons in the generalized transport model
- Kruglyak Yu.A., Kryzhanovskaya T.V. Non-equillibrium green’s functions method in matrix represenatation. 4. Quantum interference and dephasing
- Kruglyak Yu.A., Remenyak L.V. Thermoelectric coefficients in generalized electron transport model approach
- Kruglyak Yu.A., Remenyak L.V. Electric current and the second law of thermodynamics in the «bottom – up» concept of nanoelectronics
- Kruglyak Yu.A., Tereschenko T.M. Landauer – Datta – Lundstrom generalized electron transport model
- Kruglyak Yu.A., Kryzhanovskaya T.V. Non-equilibrium Green’s functions method in matrix representation. 3. Huckel model of graphene
- Yu.A. Kruglyak, L.V. Remenyak The role of electrostatics and contacts in the «bottom – up» approach of nanoelectronics
- Yu.A. Kruglyak, T.V. Kryzhanovskaya Non-equilibrium Green’s functions method in matrix representation. 2. Model transport problems
- Kruglyak Yu.A., Kryzhanovskaya T.V. Non-Equilibrium Green’s Functions Method in Matrix Presentation. 1. Theoretical Basics
- Kruglyak Yu.A. Basics of Spintronics by «bottom – up» approach
- Kruglyak Yu.A., Kruglyak N.E. Lessons of nanoelectronics. 3. Electronic conductivity and conductivity modes by «bottom – up» approach
Papers
In the tutorial review article intended for researchers, university lecturers and students the thermoelectric Seebeck and Peltier phenomena are considered in the framework of a generalized transport model due to R. Landauer, S. Datta, and M. Lundstrom of modern nanoelectronics within the “bottom – up” approach. The Wiedemann – Franz law and Lorenz numbers as well as the four transport coefficients (specific resistivity, Seebeck and Peltier coefficients, and electronic thermal conductivity) are also qualitatively discussed. Referring to a 3D resistor in the diffusion regime the thermoelectric cooler and energy power generator are analyzed with an account of only electrons as real current carriers as well as with artificial but useful electron/hole conception. Coefficient of performance, power factor and figure of merit for thermoelectric devices are introduced and defined. How transport coefficients depend on the properties of electrotermics are also discussed. Qualitative dependence of the Seebeck coefficient and electronic conductivity from the position of the Fermi level relative to the bottom of the conduction band is demonstrated. Maximization of the power factor near the bottom of the conduction band is shown. As the Fermi level approaches to the bottom of the conduction band and then moves up, the Seebeck coefficient decreases. At the same time, the electronic conductivity increases due to the appearance of an increasing number of conductivity modes. Their product is the power factor, which is maximal in the vicinity of the bottom of the conduction band. The position of the maximum for a specific electrotermics is dependent on the band structure of the conductor and the physics of its scattering centers. It is shown why in practice we try by doping the semiconductor to shift the Fermi level closer to the bottom of the conduction band.
- Kruglyak Yu. S. Nanosystems, Nanomaterials, Nanotechnologies, 2013, vol. 11, no. 3, pp. 519 – 549. (In Russian)
- Kruglyak Yu. S. Nanosystems, Nanomaterials, Nanotechnologies, 2013 vol. 11, pp. 655 – 677. (In Russian)
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On the basis of Landauer – Datta – Lundstrom transport model the generalized model of heat transfer by phonons is formulated. Similarly to the Fermi window for electron conductivity the concept of the Fermi window for phonon conductivity is introduced and used to obtain the general expression for the lattice thermal conductivity with the quantum of thermoconductance appearing at the very beginning. There are emphasized the similarity and differences in the construction of the theory of electron conductivity and the theory of heat conduction. There are discussed the thermal conductivity of the conductors, the physical sense of proportionality between the thermal conductivity and specific heat capacity at constant volume, the relationship between the transmission coefficient and the mean-free-path, the calculation of the number of phonon modes and density of phonon states, the Debye model of heat conductivity and scattering of phonons, the temperature dependence of the lattice thermal conductivity, the difference between the lattice thermal conductivity and electron conduction, and quantization of thermal conductivity.
In the present review it is emphasized that the concepts used to describe the electron transport can be successfully transferred to the phonon transport. And in both cases the Landauer approach, generalized subsequently by Datta and Lundstrom, allows a quantitative description of transport processes in conductors of any dimension and in all modes of transport from diffusive to ballistic. Lattice thermal conductivity and electronic conductivity are described very similar as far as the shape of corresponding equations. There are, however, two significant differences related to the physics of electron and phonon processes which are also emphasized in the review.
- Datta Supriyo. Lessons from Nanoelectronics: A New Perspective on Transport. Hackensack, New Jersey: World Sci-entific Publishing Company, 2012, 473 p. www.nanohub.org/courses/FoN1.
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The following topics are discussed in the frame of the «bottom – up» approach of modern nanoelectronics: a super-brief but hopefully self-containing introduction to the Hamiltonian matrix whose eigenvalues tell us the allowed energy levels in the channel. However, the Hamiltonian describes an isolated channel and we can not talk about the steady-state resistance of an isolated chan-nel without bringing in the contacts and the battery connected across it. Non-equilibrium Green’s functions method in matrix presentation was initially formulated and applied to model transport problems for 1D and 2D conductors using a nearest neighbor orthogonal tight-binding model in the frame of the «bottom – up» approach of modern nanoelectronics. General method to account for electric contacts in Schrödinger equation to solve electron quantum transport problems is given. There are also discussed the elastic and spin dephasing modeling, account for non-coherent processes using Buttiker probe, 1D conductor with two and more scatterers, quantum interference, strong and weak localization, potential drop across scatterers, quantum oscillations in NEGF method without dephasing and with its account under phase and impulse relaxation regimes, destructive and constructive interference effects, four-component description of spin transport with account for dephasing and ending with discussion of quantum nature of classics including spin coherence and pseudo-spin formalism.
- Kruglyak Yu. A. Nanoelectronics “bottom – up”: Non-equilibrium Green function method, model transport prob-lems and quantum interference. Science-Rise, 2015, vol. 9, no. 2 (14), pp. 41–72. (In Russian)
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- Kruglyak Yu. О., Strіkha M. V. Hall effect and measurement of electrochemical potentials within «bottom – up» approach. Sensor Electronics Microsys. Tech., 2014, vol. 11, no. 1, pp. 5–27. (In Ukrainian)
- Datta Supriyo. Lessons from Nanoelectronics: A New Per-spective on Transport. Hackensack, New Jersey: World Sci-entific Publishing Company, 2012. 473 р.
- Kruglyak Yu. O., Kruglyak N. E., Strіkha M. V. Lessons of Nanoelectronics. Spintronics in «Bottom – Up» Approach. Sensor Electronics Microsys. Tech., 2013, vol. 10, no. 2, pp. 5–35. (In Ukrainian)
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- Kruglyak Yu. A. Graphene in the Landauer – Datta – Lund-strom Transport Model. ScienceRise, 2015, vol. 2, no. 2 (7), pp. 93–106. (In Russian)
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General issues of electronic conductivity and the causes for the current flow, role of electro-chemical potentials, Fermi functions, and Fermi window for conduction are discussed, as well as there given detailed description of the Landauer elastic resistor model, different transport regimes from ballistic to diffusion and in between, conductivity modes, and transmission coefficient in the frame of the «bottom – up» approach of modern nanoelectronics. Generalized model of electron transport in the linear response regime developed by R. Landauer, S. Datta, and M. Lundstrom with application to the resistors of any dimension, any size and arbitrary dispersion working in ballistic, quasi-ballistic or diffusion regime is summerized.
In summary, the Landauer equation for the conductivity describes the electron transport in the conductor from the very general positions. The conductivity is proportional to the fundamental constants q and h, which determine the quantum of conductance, associated with contacts. The conductivity depends on the number of modes of conductance and transmission coefficient, representing the probability that an electron with energy E injected by one contact to reach another contact. Conductivity we finally find by integrating the contributions from all modes of conduction. The equations valid for 1D, 2D and 3D conductors for ballistic nanoreactors as well as for massive conductors.
- Kruglyak Yu.O., Kruglyak N.E., Strikha M.V. Lessons of Nanoelectronics: Current generation, Ohm’s Law Formulation and Conduction modes in «Bottom – Up» Approach. Sensor Electronics Microsys. Tech, 2012, vol. 9, no. 4, pp. 5-29 (In Ukrainian)
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Within the «bottom – up» concept of modern nanoelectronics flows of electrons and heat through a conductor taking into account a respective environment are studied and equilibrium thermodynamics of an energized conductor is developed with emphasis of the role of the Fock states in such concept, accumulation of information in a non-equilibrium state is discussed together with a detailed analysis of the model of information-driven battery and its connection with the Landauer principle on minimum amount of energy required to erase one bit of information which was experimentally proved recently. The concept of quantum entropy is introduced and some aspects of its application are discussed, and topicality of integration of spintronics and magnetronics in connection with upcoming transfer to the spin architecture of computing devices is emphasized. Also a question with respect to possibility of electron cloning in a certain state in the form of numerous identical electrons with the same wave functions is discussed. There are no theorems prohibiting such cloning. It is only necessary to inject numerous electrons prepared in the same way and then to carry out competent processing of measured results. All these issues are dealt by rapidly developing quantum informatics. But it is already obvious that the spin of an electron in comparison with its charge opens up qualitatively new perspectives for information processing.
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This article briefly summarizes the Landauer – Datta – Lundstrom electron transport model. Provided a band structure is given, number of conduction modes can be evaluated and, if a model for a mean free pass for backscattering can be established, then the near-equilibrium thermoelectric transport coefficients can be calculated using the final expressions listed below for 1D, 2D, and 3D resistors in ballistic, quasi-ballistic and diffusive linear response regimes when there are differences in both voltage and temperature across the device. The final expressions of thermoelectric transport coefficients through the Fermi – Dirac integrals are collected for 1D, 2D and 3D semiconductors with parabolic band structure and for 2D graphene linear dispersion in ballistic and diffusive regimes with the power law scattering.
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Non-equilibrium Green’s functions method is applied to graphene, using the nearest neighbor orthogonal tight-binding model in the frame of the «bottom – up» approach of modern nanoelectronics. There is also a general method to account for electric contacts in Schrödinger equation in order to solve electron quantum transport problems given.
As an illustrative example of the calculation results for the transmission coefficients and the density of states in Huckel model the graphene ribbons are computed in two boundary configurations – Zigzag and Armchair Graphene Nanoribbons. The calculations were performed for ribbons with width of 53 nm and with resonance integral 2.7 eV. Noteworthy is the high density of states at E = 0 for a zigzag configuration of a graphene ribbon. These are so-called edge local states near the Fermi level that are not available in Armchair configuration. It is worth to note the high plausibility of the results obtained even within such a simple model of graphene as in strong-coupling approximation in orthogonal parametric basis with account of the interaction of neighboring atoms only.
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The theory of electrical conduction is developed in the framework of the “bottom – up” approach without invoking the concept of an external electric field generated by a potential difference applied to the conductor. Within the concept of «bottom – up» approach of modern nanoelectronics the diffusion-drift model of a current on the basis of the Boltzmann transport equation is described. There are also discussed the role of the external electric field beyond the linear response regime, field-effect transistor and saturation current, the role of conductor charging, the point and extended models of a conductor, the role of contacts, the model of p-n junctions, the generation of a current in a conductor with asymmetric contacts.
In summary, we conclude, that when a band structure is given, number of modes can be evaluated and, if a model for the mean-free-pass for backscattering can be chosen, then the near-equilibrium transport coefficients can be evaluated. Next, the new generalized Ohm’s law was formulated and used which provides a quite different view of resistivity in terms of the number of modes per unit area and the mean-free-path. Finally, the transport model given is equally well applied either to nanoresistors or as well to micro- and macroconductors made of any kind of materials.
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Non-equilibrium Green’s functions method is applied to model transport problems for 1D and 2D uniform conductors using the nearest neighbor orthogonal tight-binding approximation in the frame of the «bottom – up» approach of modern nanoelectronics.
First of all we discuss the construction of the contact matrices of self-energies. The basic idea is that infinitely long conductor described by the Hamiltonian [H] is replaced by a conductor of the finite length described by the matrix [H + Σ1 + Σ2] with the open boundary conditions at the ends meaning “good” contacts, which do not create the reflected streams at its ends. Further we discuss 1D ballistic conductor, a 1D conductor with a single scattering center, then 2D conductor is modeling and explanation is given to the steplike dependence of the transmission coefficients over the energy, and finally there is given the representation of 2D/3D conductor in the form of parallel 1D conductors, which is not only physically correct but also extremely useful in interpreting experimental data.
In summary the physical adequacy of the Huckel approach is stated in the framework of the method of nonequilibrium Green’s functions.
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