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Quantum-mechanical calculations of electron magnetotransport in graphene Fabry-P\'{e}rot interferometers are presented with a focus on the role of spatial structure of edge channels. For an interferometer that is made by removing carbon atoms, which is typically realized in nanolithography experiments, the constrictions are shown to cause strong inter-channel scattering that establishes local equilibrium and makes the electron transport non-adiabatic. Nevertheless, two-terminal conductance reveals a common Aharonov-Bohm oscillation pattern, independent of crystallographic orientation, which is accompanied by single-particle states that sweep through the Fermi energy for the edge channels circulating along the physical boundary of the device. The interferometer constrictions host the localized states that might shorten the device or disrupt the oscillation pattern. For an interferometer that is created by electrostatic confinement, which is typically done in the split-gate experiments, electron transport is shown to be adiabatic if the staggered potential is introduced additionally into the model. Interference visibility decays exponentially with temperature with a weaker dependence at low temperature., Comment: Electrostatic confinement with staggered potential

Quantum-mechanical calculations of electron magneto-transport in ideal graphene nanoribbons are presented. In noninteracting theory, it is predicted that an ideal ribbon that is attached to wide leads should reveal Fabry-Perot conductance oscillations in magnetic field. In the theory with Coulomb interaction taken into account, the oscillation pattern should rather be determined by the Aharonov-Bohm interference effect. Both of these theories predict the formation of quasi-bound states, albeit of different structures, inside the ribbon because of strong electron scattering on the interfaces between the connecting ribbon and the leads. Conductance oscillations are a result of resonant backscattering via these quasi-bound states., Comment: Recalculated Hartree DOS, minor updates

A model of charge hopping transport that accounts for anisotropy of localized states and Coulomb interaction between charges is proposed. For the anisotropic localized states the degree of orientation relates exponentially to the ratio of conductivities in parallel and perpendicular directions, while the ratio of Seebeck coefficients stays nearly unaffected. However, the ratio of Seebeck coefficients increases if Coulomb interaction is screened stronger in a direction parallel to the predominant orientation of the localized states. This implies two different physical mechanisms responsible for the anisotropy of thermoelectric properties in the hopping regime: electronic state localization for conductivities, and screening for Seebeck coefficients. This provides explanation for recent experimental findings on tensile drawn and ribbed polymer films., Comment: 7 pages, 4 figures

Numerical calculations of anisotropic hopping transport based on the resistor network model are presented. Conductivity is shown to follow the stretched exponential dependence on temperature with exponents changing from 1/4 to 1 as the wave functions become anisotropic and their localization length in the direction of charge transport decreases. For sufficiently strong anisotropy, this results in nearest-neighbor hopping at low temperatures due to the formation of a single conduction path, which adjusts in the planes where the wave functions overlap strongly. In the perpendicular direction, charge transport follows variable-range hopping, a behavior that agrees with experimental data on organic semiconductors., Comment: Published version

Electronic, transport, and spin properties of grain boundaries (GBs) are investigated in electrostatically doped graphene at finite electron densities within the Hartree and Hubbard approximations. We demonstrate that depending on the character of the GBs, the states residing on them can have a metallic character with a zero group velocity or can be fully populated losing the ability to carry a current. These states show qualitatively different features in charge accumulation and spin polarization. We also demonstrate that the semiclassical Thomas-Fermi approach provides a satisfactory approximation to the calculated self-consistent potential. The conductance of GBs is reduced due to enhanced backscattering from this potential., Comment: 7 pages, 6 figures

We present numerical studies of conduction in graphene nanoribbons with reconstructed edges based on the standard tight-binding model of the graphene and the extended Huckel model of the reconstructed defects. We performed atomic geometry relaxation of individual defects using density functional theory and then explicitly calculated the tight-binding parameters used to model electron transport in graphene with reconstructed edges. The calculated conductances reveal strong backscattering and electron-hole asymmetry depending on the edge and defect type. This is related to an additional defect-induced band whose wave function is poorly matched to the propagating states of the pristine ribbon. We find a transport gap to open near the Dirac point and to scale inversely with the ribbon width, similarly to what has been observed in experiments. We predict the largest transport gap to occur for armchair edges with Stone-Wales defects, while heptagon and pentagon defects cause about equal backscattering for electrons and holes, respectively. Choosing the heptagon defect as an example, we show that although electron interactions in the Hartree approximation cause accumulation of charge carriers on the defects, surprisingly, their effect on transport is to reduce carrier backscattering by the defects and thus to enhance the conductance., Comment: 8 pages, 4 figures

We provide a systematic quantitative description of spin polarization in armchair and zigzag graphene nanoribbons in a perpendicular magnetic field. We first address spinless electrons within the Hartree approximation studying the evolution of the magnetoband structure and formation of the compressible strips. We discuss the potential profile and the density distribution near the edges and the difference and similarities between armchair and zigzag edges. Accounting for the Zeeman interaction and describing the spin effects via the Hubbard term we study the spin-resolved subband structure and relate the spin polarization of the system at hand to the formation of the compressible strips for the case of spinless electrons. At high magnetic field the calculated effective g-factor varies around a value of ~2.25 for armchair nanoribbons and ~3 for zigzag nanoribbons. An important finding is that in zigzag nanoribbons the zero-energy mode remains pinned to the Fermi-energy and becomes fully spin-polarized for all magnetic fields, which, in turn, leads to a strong spin polarization of the electron density near the zigzag edge., Comment: 9 pages, 4 figures

We present self-consistent calculations of electron transport in graphene nanoconstrictions within the Hartree approximation. We consider suspended armchair ribbons with V-shaped constrictions having perfect armchair or zigzag edges as well as mesoscopically smooth but atomically stepped constrictions with cosine profiles. Our calculations are based on a tight-binding model of the graphene and account for electron-electron interactions in both the constriction and the semi-infinite leads explicitly. We find that electron interactions result in (i) Electrons accumulating along edges of the uniform ribbon and along zigzag and cosine constriction edges but not along armchair constriction edges. (ii) The first subband showing almost perfect transmittance due to localization at the uniform graphene boundary except at low energies for the cases of zigzag and cosine constrictions where Bloch stop-bands form in related periodic structures. (iii) The second subband being almost perfectly blocked by the constriction. (iv) Electron interactions favor intra-subband scattering while the non-interacting electron theory predicts predominance of inter-subband scattering. (v) Conductance quantization for the first few conductance steps being more pronounced for armchair constrictions but less so for zigzag constrictions. (vi) A much more prominent 2e^2/h conductance plateau for the cosine constriction than is found in the absence of electron interactions. Possible implications for recent experiments are briefly discussed., Comment: 8 pages, 7 figures

We present the results of million atom electronic quantum transport calculations for graphene nanoconstrictions with edges that are smooth apart from atomic scale steps. We find conductances quantized in integer multiples of 2e2/h and a plateau at ~0.5*2e2/h as in recent experiments [Tombros et al., Nature Physics 7, 697 (2011)]. We demonstrate that, surprisingly, conductances quantized in integer multiples of 2e2/h occur even for strongly non-adiabatic electron backscattering at the stepped edges that lowers the conductance by one or more conductance quanta below the adiabatic value. We also show that conductance plateaus near 0.5*2e2/h can occur as a result of electron backscattering at stepped edges even in the absence of electron-electron interactions., Comment: 5 pages, 4 figures

The performance potential for simulating quantum electron transport on graphical processing units (GPUs) is studied. Using graphene ribbons of realistic sizes as an example it is shown that GPUs provide significant speed-ups in comparison to central processing units as the transverse dimension of the ribbon grows. The recursive Green's function algorithm is employed and implementation details on GPUs are discussed. Calculated conductances were found to accumulate significant numerical error due to single-precision floating-point arithmetic at energies close to the charge neutrality point of the graphene., Comment: published version with corrections

We present numerical studies of non-linear conduction in graphene nanoribbons when a bias potential is applied between the source and drain electrodes. We find that the conductance quantization plateaus show asymmetry between the electron and hole branches if the potential in the ribbon equals the source or drain electrode potential and strong electron (hole) scattering occurs. The scattering may be at the ends of a uniform ballistic ribbon connecting wider regions of graphene or may be due to defects in the ribbon. We argue that, in ribbons with strong defect scattering, the ribbon potential is pinned to that of the drain (source) for electron (hole) transport. In this case symmetry between electron and hole transport is restored and our calculations explain the upward shift of the conductance plateaus with increasing bias that was observed experimentally by Lin et al. [Phys. Rev. B 78, 161409 (2008)]., Comment: 6 pages, 3 figures

Electron transport properties of few-electron open quantum dots within the spin-restricted Hartree-Fock approximation are studied. The self-consistent numerical calculations were performed for a whole device, including the semi-infinite leads, without employing any phenomenological or adjustable parameters. Inclusion of the non-local Fock potential brings qualitatively new physics in comparison to the Hartree approach: electron screening decreases, resonant energy levels become less pinned to the Fermi energy and clearly correlate with conductance peaks. When coupling between the dot and leads decreases the number of electrons inside the dot becomes quantized and the model predicts the Coulomb blockade of electron transport. This is confirmed by comparison with the master equation approach for an equivalent quantum dot., Comment: v2 with several improvements

We present calculations of electronic quantum transport in graphene nanoribbons with adsorbed H, F, OH and O, based on a tight binding model derived from extended Huckel theory. The relaxed atomic geometries of the adsorbates and graphene are calculated using density functional theory. Our model includes the effects of the local rehybridization of the graphene from the sp2 to sp3 electronic structure that occurs when H, F, OH or O bonds covalently to the graphene. It captures the physics of the scattering resonances that are induced in the graphene near the Dirac point by the presence of these adsorbates. We find these Dirac point resonances to play a dominant role in quantum transport in ribbons with these adsorbates: Even at low adsorbate concentrations the conductance of the ribbon is strongly suppressed and a transport gap develops for electron Fermi energies near the resonance. The transport gap is centered very near the Dirac point energy of for H, below it for F and OH and above it for O. We predict ribbons with these adsorbed species under appropriate conditions to exhibit quantized conductance steps of equal height similar to those that have been observed by Lin et al. [Phys. Rev. B 78, 161409 (2008)] at moderately low temperatures, even for ribbons with conductances a few orders of magnitude smaller than 2e2/h., Comment: This paper has been withdrawn by the authors. This paper has been combined with 1008.4110 and published as Phys. Rev. B 83, 245442 (2011)

We present a tight binding theory of the Dirac point resonances due to adsorbed atoms and molecules on an infinite 2D graphene sheet based on the standard tight binding model of the graphene p-band electronic structure and the extended Huckel model of the adsorbate and nearby graphene carbon atoms. The relaxed atomic geometries of the adsorbates and graphene are calculated using density functional theory. Our model includes the effects of the local rehybridization of the graphene from the sp^2 to sp^3 electronic structure that occurs when adsorbed atoms or molecules bond covalently to the graphene. Unlike in previous tight-binding models of Dirac point resonances, adsorbed species with multiple extended molecular orbitals and bonding to more than one graphene carbon atom are treated. More accurate and more general analytic expressions for the Green's function matrix elements that enter the T-matrix theory of Dirac point resonances than have been available previously are obtained. We study H, F, OH and O adsorbates on graphene and for each we find a strong scattering resonance (two resonances for O) near the Dirac point of graphene, by far the strongest and closest to the Dirac point being the resonance for H. We extract a minimal set of tight binding parameters that can be used to model resonant electron scattering and electron transport in graphene and graphene nanostructures with adsorbed H, F, OH and O accurately and efficiently. We also compare our results for the properties of Dirac point resonances due to adsorbates on graphene with those obtained by others using density functional theory-based electronic structure calculations, and discuss their relative merits. We then present calculations of electronic quantum transport in graphene nanoribbons with these adsorbed species..., Comment: 21 pages, 9 figures

We present numerical studies of conduction in graphene nanoribbons with different types of disorder. We find that even when defect scattering depresses the conductance to values two orders of magnitude lower than 2e^2/h, equally spaced conductance plateaus occur at moderately low temperatures due to enhanced electron backscattering near subband edge energies if bulk vacancies are present in the ribbon. This work accounts quantitatively for the surprising conductance quantization observed by Lin et al. [Phys. Rev. B 78, 161409 (2008)] in ribbons with such low conductances., Comment: 4 pages, 3 figures

We calculate the band structure and the conductance of periodic corrugated graphene nanoribbons within the framework of the tight-binding $p$-orbital model. We consider corrugated structures based on host ribbons with armchair and zigzag edges and three different types of corrugations (armchair edges, zigzag edges as well as a rectangular corrugation). We demonstrate that for armchair host ribbons, depending on the type of corrugation, a band gap or low-velocity minibands appear near the charge neutrality point. For higher energies the allowed Bloch state bands become separated by mini-stopbands. By contrast, for corrugated ribbons with the zigzag host, the corrugations introduce neither band gaps nor stopbands (except for the case of the rectangular corrugations). The conductances of finite corrugated ribbons are analyzed on the basis of the corresponding band structures. For a sufficiently large number of corrugations the conductance follows the number of the corresponding propagating Bloch states and shows pronounced oscillations due to the Fabry-Perot interference within the corrugated segments. Finally we demonstrate that edge disorder strongly affects the conductances of corrugated ribbons. Our results indicate that observation of miniband formation in corrugated ribbons would require clean, edge-disorder free samples, especially for the case of the armchair host lattice., Comment: 7 pages, 7 figures

We present a microscopic picture of quantum transport in quantum antidots in the quantum Hall regime taking electron interactions into account. We discuss the edge state structure, energy level evolution, charge quantization and linear-response conductance as the magnetic field or gate voltage is varied. Particular attention is given to the conductance oscillations due to Aharonov-Bohm interference and their unexpected periodicity. To explain the latter we propose the mechanisms of scattering by point defects and Coulomb blockade tunneling. They are supported by self-consistent calculations in the Hartree approximation, which indicate pinning and correlation of the single-particle states at the Fermi energy as well as charge oscillation when antidot-bound states depopulate. We have also found interesting phenomena of anti-resonance reflection of the Fano type., Comment: 12 pages, 8 figures

We calculate the non-linear conductance of a quantum point contact using the non-equilibrium Greens function technique within the Hartree approximation of spinless electrons. We quantitative reproduce the 0.25-anomaly in the differential conductance (i.e. the lowest plateau at 0.25-0.3*2e^2/h) as well as an upward bending of higher conductance half-integer plateaus seen in the experiments, and relate these features to the non-linear screening and pinning effects., Comment: 6 pages, 4 figures

We present a microscopic picture of quantum transport in the Aharonov-Bohm (AB) interferometer taking into account electron interaction within the Hartree and the spin density functional theory approximations. We discuss the structure of the edge states for different number of the Landau levels in the leads, their coupling to the states in the central island and the formation of compressible/incompressible strips in the interferometer. Based on our results we discuss the existing theories of the unexpected AB periodicity, which essentially rely on specific phenomenological models of the states and their coupling in the interferometer. Our work provides a basis for such the theories, giving a detailed microscopic description of the propagating states and the global electrostatics in the system at hand., Comment: 10 pages, 7 figures

We study spin polarization in a split-gate quantum wire focussing on the effect of a realistic smooth potential due to remote donors. Electron interaction and spin effects are included within the density functional theory in the local spin density approximation. We find that depending on the electron density, the spin polarization exhibits qualitatively different features. For the case of relatively high electron density, when the Fermi energy $E_{F}$ exceeds a characteristic strength of a long-range impurity potential $V_{donors}$, the density spin polarization inside the wire is practically negligible and the wire conductance is spin-degenerate. When the density is decreased such that $E_{F}$ approaches $V_{donors}$, the electron density and conductance quickly become spin polarized. With further decrease of the density the electrons are trapped inside the lakes (droplets) formed by the impurity potential and the wire conductance approaches the pinch-off regime. We discuss the limitations of DFT-LSDA in this regime and compare the obtained results with available experimental data., Comment: 7 pages, 5 figures

We present a detailed comparison of the self-consistent calculations based on the Hartree-Fock and the spin density functional theory for a spit-gate quantum wire in the IQH regime. We demonstrate that both approaches provide qualitatively (and in most cases quantitatively) similar results for the spin-resolved electron density, spin polarization, spatial spin separation at the edges and the effective $g$ factor. The both approach give the same values of the magnetic fields corresponding to the successive subband depopulation and qualitatively similar evolution of the magnetosubbands. Quantitatively, however, the HF and the DFT subbands are different (even though the corresponding total electron densities are practically the same). In contrast to the HF approach, the DFT calculations predict much larger spatial spin separation near the wire edge for the low magnetic fields (when the compressible strips for spinless electrons are not formed yet). In the opposite limit of the large fields, the Hatree-Fock and the DFT approaches give very similar values for the spatial spin separation., Comment: 5 pages, 3 figures

We perform self-consistent quantum transport calculations in open quantum dots taking into account the effect of electron interaction. We demonstrative that in the regime of the ultralow temperatures $2\pi k_BT\lesssim\Delta$ ($\Delta $ being the mean level spacing), the electron interaction strongly affects the conductance oscillations and their statistics leading to a drastic deviation from the corresponding predictions for noninteracting electrons. In particular, it causes smearing of conductance oscillations, which is similar to the effect of temperature or inelastic scattering. For $2\pi k_BT\gtrsim\Delta$ the influence of electron interaction on the conductance becomes strongly diminished. Our calculations (that are free from phenomenological parameters of the theory) are in good quantitative agreement with the observed ultralow temperature statistics (Huibers \textit{et al.}, Phys. Rev. Lett. \textbf{81}, 1917 (1998)). Our findings question a conventional interpretation of the ultralow temperature saturation of the coherence time in open dots which is based on the noninteracting theories where the electron interaction is neglected and the agrement with the experiment is achieved by introducing additional phenomenological channels of dephasing.

We present full quantum mechanical conductance calculations of a quantum point contact (QPC) performed in the framework of the density functional theory (DFT) in the local spin-density approximation (LDA). We show that a spin-degeneracy of the conductance channels is lifted and the total conductance exhibits a broad plateau-like feature at 0.5*2e^{2}/h. The lifting of the spin-degeneracy is a generic feature of all studied QPC structures (both very short and very long ones; with the lengths in the range 40

The origin of the 0.7-conductance anomaly in quantum point contacts (QPCs) has been a subject of lively discussions since its discovery more than 10 years ago. In a recent letter, based on spin density functional theory (DFT), Rejec and Meir explained the origin of the 0.7-anomaly as being due to the formation of a magnetic impurity in the QPCs [T. Rejec and Y. Meir, Nature 442, 900 (2006)]. They did not, however, perform transport conductance calculations so the central question whether the 0.7-anomaly is indeed related to the formation of a magnetic impurity has remained unanswered. In this communication, we perform spin DFT transport calculations for the structures considered by Rejec and Meir. While we recover the findings reported by Rejec and Meir concerning the formation of localized spin-degenerate quasi-bound states, our transport calculations do not contain the 0.7 anomaly in the conductance. We suggest that the inability of the DFT to reproduce quantitatively the 0.7 anomaly may be due to the uncorrected self-interaction errors in the DFT calculation of electronic transport for the case when localization of charge is expected to occur, so that the magnetic impurity formation may be an artefact of DFT due to the spurious self-interaction., Comment: comment on T. Rejec and Y. Meir, "Magnetic impurity formation in quantum point contacts", Nature 442, 900-903 (2006)

The conductance of a quantum wire containing a single magnetic barrier is studied numerically by means of the recursive Greens function technique. For sufficiently strong and localized barriers, Fano - type reflection resonances are observed close to the pinch-off regime. They are attributed to a magnetoelectric vortex-type quasibound state inside the magnetic barrier that interferes with an extended mode outside. We furthermore show that disorder can substantially modify the residual conductance around the pinch-off regime., Comment: 7 pages, 5 figures

The role of electron-electron interaction in transport properties of open quantum dots is studied. The self-consistent full quantum mechanical magnetotransport calculations within the Hartree, Density Functional Theory and Thomas-Fermi approximations were performed where a whole device, including the semi-infinitive leads, is treated on the same footing (i.e. the electron-electron interaction is accounted for both in the leads as well as in the dot region). The main finding of the present paper is the effect of pinning of the resonant levels to the Fermi energy due to the enhanced screening. Our results represent a significant departure from a conventional picture where a variation of external parameters (such as a gate voltage, magnetic field, etc.) causes the successive dot states to sweep past the Fermi level in a linear fashion. We instead demonstrate highly nonlinear behavior of the resonant levels in the vicinity of the Fermi energy. The pinning of the resonant levels in open quantum dots leads to the broadening of the conduction oscillations in comparison to the one electron picture. The effect of pinning becomes much more pronounced in the presence of the perpendicular magnetic field. This can be attributed to the enhanced screening efficiency because of the increased localization of the wave function. The strong pinning of the resonant energy levels in the presence of magnetic field can have a profound effect on transport properties of various devices operating in the edge state transport regime. We also critically examine an approximation often used in transport calculations where an inherently open system is replaced by a corresponding closed one., Comment: 10 pages, 7 figures, submitted to Phys. Rev. B

We demonstrate that a split-gate quantum wire in the integer quantum Hall regime can exhibit electronic transport hysteresis for up- and down-sweeps of a magnetic field. This behavior is shown to be due to phase spin transitions between two different ground states with and without spatial spin polarization in the vicinity of the wire boundary. The observed effect has a many-body origin arising from an interplay between a confining potential, Coulomb interactions and the exchange interaction. We also demonstrate and explain why the hysteretic behavior is absent for steep and smooth confining potentials and is present only for a limited range of intermediate confinement slopes., Comment: submitted to PRL

We provide a systematic quantitative description of the edge state structure around a quantum antidot in the integer quantum Hall regime. The calculations for spinless electrons within the Hartree approximation reveal that the widely used Chklovskii et al. electrostatic description greatly overestimates the widths of the compressible strips; the difference between these approaches diminishes as the size of the antidot increases. By including spin effects within density functional theory in the local spin-density approximation, we demonstrate that the exchange interaction can suppress the formation of compressible strips and lead to a spatial separation between the spin-up and spin-down states. As the magnetic field increases, the outermost compressible strip, related to spin-down states starts to form. However, in striking contrast to quantum wires, the innermost compressible strip (due to spin-up states) never develops for antidots., Comment: submitted to Phys. Rev. Lett

We provide a systematic quantitative description of the structure of edge states and magnetosubband evolution in hard wall quantum wires in the integer quantum Hall regime. Our calculations are based on the self-consistent Green's function technique where the electron- and spin interactions are included within the density functional theory in the local spin density approximation. We analyze the evolution of the magnetosubband structure as magnetic field varies and show that it exhibits different features as compared to the case of a smooth confinement. In particularly, in the hard-wall wire a deep and narrow triangular potential well (of the width of magnetic length $l_B$) is formed in the vicinity of the wire boundary. The wave functions are strongly localized in this well which leads to the increase of the electron density near the edges. Because of the presence of this well, the subbands start to depopulate from the central region of the wire and remain pinned in the well region until they are eventually pushed up by increasing magnetic field. We also demonstrate that the spin polarization of electron density as a function of magnetic field shows a pronounced double-loop pattern that can be related to the successive depopulation of the magnetosubbands. In contrast to the case of a smooth confinement, in hard-wall wires the compressible strips do not form in the vicinity of wire boundaries and spatial spin separation between spin-up and spin-down states near edges is absent., Comment: 9 pages, submitted to Phys. Rev. B

We perform systematic numerical studies of the structure of spin-resolved compressible strips in split-gate quantum wires taking into account the exchange and correlation interactions within the density functional theory in the local spin-density approximation. We find that for realistic parameters of the wire the exchange interaction can completely suppress the formation of the compressible strips. As the depletion length or magnetic field are increased, the compressible strips starts to form first for the spin-down and then for spin-up edge channels. We demonstrate that the widths of these strips plus the spatial separation between them caused by the exchange interaction are equal to the width of the compressible strip calculated in the Hartree approximation for spinless electrons. We also discuss the effect of electron density on the suppression of the compressible strips in quantum wires., Comment: 5 pages, 4 figures, submitted to Phys. Rev. B

We provide a quantitative description of the structure of edge states in split-gate quantum wires in the integer quantum Hall regime. We develop an effective numerical approach based on the Green's function technique for the self-consistent solution of Schrodinger equation where electron- and spin interactions are included within the density functional theory in the local spin density approximation. The major advantage of this technique is that it can be directly incorporated into magnetotransport calculations, because it provides the self-consistent eigenstates and wave vectors at a given energy, not at a given wavevector (as conventional methods do). We use the developed method to calculate the subband structure and propagating states in the quantum wires in perpendicular magnetic field starting with a geometrical layout of the wire. We discuss how the spin-resolved subband structure, the current densities, the confining potentials, as well as the spin polarization of the electron and current densities evolve when an applied magnetic field varies. We demonstrate that the exchange and correlation interactions dramatically affect the magnetosubbands in quantum wires bringing qualitatively new features in comparison to a widely used model of spinless electrons in Hartree approximation., Comment: 15 pages and 10 figures, submitted to Phys. Rev. B