Showing papers on "Charge density published in 2011"
TL;DR: An efficient, accurate method to integrate the basins of attraction of a smooth function defined on a general discrete grid and apply it to the Bader charge partitioning for the electron charge density is proposed.
Abstract: We propose an efficient, accurate method to integrate the basins of attraction of a smooth function defined on a general discrete grid and apply it to the Bader charge partitioning for the electron charge density. Starting with the evolution of trajectories in space following the gradient of charge density, we derive an expression for the fraction of space neighboring each grid point that flows to its neighbors. This serves as the basis to compute the fraction of each grid volume that belongs to a basin (Bader volume) and as a weight for the discrete integration of functions over the Bader volume. Compared with other grid-based algorithms, our approach is robust, more computationally efficient with linear computational effort, accurate, and has quadratic convergence. Moreover, it is straightforward to extend to nonuniform grids, such as from a mesh-refinement approach, and can be used to both identify basins of attraction of fixed points and integrate functions over the basins.
1,273 citations
TL;DR: The present study combines earlier extensions and improves them consistently with, first, an improved Coulomb interaction between atomic partial charges, and second, the complete third-order expansion of the DFT total energy, leading to the next generation of theDFTB methodology called DFTB3, which substantially improves the description of charged systems containing elements C, H, N, O, and P.
Abstract: The self-consistent-charge density-functional tight-binding method (SCC-DFTB) is an approximate quantum chemical method derived from density functional theory (DFT) based on a second-order expansion of the DFT total energy around a reference density. In the present study we combine earlier extensions and improve them consistently with, first, an improved Coulomb interaction between atomic partial charges, and second, the complete third-order expansion of the DFT total energy. These modifications lead us to the next generation of the DFTB methodology called DFTB3, which substantially improves the description of charged systems containing elements C, H, N, O, and P, especially regarding hydrogen binding energies and proton affinities. As a result, DFTB3 is particularly applicable to biomolecular systems. Remaining challenges and possible solutions are also briefly discussed.
816 citations
TL;DR: Nuclear magnetic resonance measurements are reported showing that high magnetic fields actually induce charge order, without spin order, in the CuO2 planes of YBa2Cu3Oy, and it is argued that it is most probably the same 4a-periodic modulation as in stripe-ordered copper oxides.
Abstract: Nuclear magnetic resonance measurements of the model high-temperature copper oxide superconductor YBa2Cu3Oy demonstrate that high magnetic fields induce charge order, without spin order, within the material's CuO2 planes. The observed charge order has characteristics similar to those of stripe-ordered copper oxides, in which electronic charges spontaneously organize themselves into 'stripes'. The charge order develops only when superconductivity fades away. This work suggests that stripes are more common objects in the cuprates than was thought. They seem to compete with superconductivity, although the tendency to form stripes may be a necessary ingredient of high temperature superconductivity. Electronic charges introduced in copper-oxide (CuO2) planes generate high-transition-temperature (Tc) superconductivity but, under special circumstances, they can also order into filaments called stripes1. Whether an underlying tendency towards charge order is present in all copper oxides and whether this has any relationship with superconductivity are, however, two highly controversial issues2,3. To uncover underlying electronic order, magnetic fields strong enough to destabilize superconductivity can be used. Such experiments, including quantum oscillations4,5,6 in YBa2Cu3Oy (an extremely clean copper oxide in which charge order has not until now been observed) have suggested that superconductivity competes with spin, rather than charge, order7,8,9. Here we report nuclear magnetic resonance measurements showing that high magnetic fields actually induce charge order, without spin order, in the CuO2 planes of YBa2Cu3Oy. The observed static, unidirectional, modulation of the charge density breaks translational symmetry, thus explaining quantum oscillation results, and we argue that it is most probably the same 4a-periodic modulation as in stripe-ordered copper oxides1. That it develops only when superconductivity fades away and near the same 1/8 hole doping as in La2−xBaxCuO4 (ref. 1) suggests that charge order, although visibly pinned by CuO chains in YBa2Cu3Oy, is an intrinsic propensity of the superconducting planes of high-Tc copper oxides.
639 citations
TL;DR: It is pointed out that this electric quadrupole deformation of the quark-gluon plasma produced in heavy ion collisions lifts the degeneracy between the elliptic flows of positive and negative pions leading to v(2)(π(+))
Abstract: Chiral Magnetic Wave (CMW) is a gapless collective excitation of quark-gluon plasma in the presence of external magnetic field that stems from the interplay of Chiral Magnetic (CME) and Chiral Separation Effects (CSE); it is composed by the waves of the electric and chiral charge densities coupled by the axial anomaly. We consider CMW at finite baryon density and find that it induces the electric quadrupole moment of the quark-gluon plasma produced in heavy ion collisions: the ”poles” of the produced fireball (pointing outside of the reaction plane) acquire additional positive electric charge, and the ”equator” acquires additional negative charge. We point out that this electric quadrupole deformation lifts the degeneracy between the elliptic flows of positive and negative pions leading to v2(� + ) < v2(� ), and estimate the magnitude of the effect.
238 citations
TL;DR: The spin-relaxation time τ(s) scales inversely with the mobility μ of BLG samples both at room temperature (RT) and at low temperature (LT), indicating the importance of D'yakonov-Perel' spin scattering in BLG.
Abstract: We report on the first systematic study of spin transport in bilayer graphene (BLG) as a function of mobility, minimum conductivity, charge density, and temperature. The spin-relaxation time τ(s) scales inversely with the mobility μ of BLG samples both at room temperature (RT) and at low temperature (LT). This indicates the importance of D'yakonov-Perel' spin scattering in BLG. Spin-relaxation times of up to 2 ns at RT are observed in samples with the lowest mobility. These times are an order of magnitude longer than any values previously reported for single-layer graphene (SLG). We discuss the role of intrinsic and extrinsic factors that could lead to the dominance of D'yakonov-Perel' spin scattering in BLG. In comparison to SLG, significant changes in the carrier density dependence of τ(s) are observed as a function of temperature.
227 citations
TL;DR: In this paper, the sheet carrier densities in all structures containing more than one unit cell of SrTiO3 are independent of layer thicknesses and growth sequences, indicating that the mobile carriers are in a high concentration, two-dimensional electron gas bound to the interface.
Abstract: Heterostructures and superlattices consisting of a prototype Mott insulator, GdTiO3, and the band insulator SrTiO3 are grown by molecular beam epitaxy and show intrinsic electronic reconstruction, approximately ½ electron per surface unit cell at each GdTiO3/SrTiO3 interface. The sheet carrier densities in all structures containing more than one unit cell of SrTiO3 are independent of layer thicknesses and growth sequences, indicating that the mobile carriers are in a high concentration, two-dimensional electron gas bound to the interface. These carrier densities closely meet the electrostatic requirements for compensating the fixed charge at these polar interfaces. Based on the experimental results, insights into interfacial band alignments, charge distribution, and the influence of different electrostatic boundary conditions are obtained.
199 citations
TL;DR: In this article, the authors compare the layer thickness dependency of the measured surface potential with ab initio density functional theory calculations of the work function for substrate-doped graphene and independently find an interlayer screening length in the order of four to five layers.
Abstract: substrates. We compare the layer thickness dependency of the measured surface potential with ab initio density functional theory calculations of the work function for substrate-doped graphene. The ab initio calculations show that the work function of single- and bilayer graphene is mainly given by a variation of the Fermi energy with respect to the Dirac point energy as a function of doping, and that electrostatic interlayer screening only becomes relevant for thicker multilayer graphene. From the Raman G-line shift and the comparison of the Kelvin probe data with the ab initio calculations, we independently find an interlayer screening length in the order of four to five layers. Furthermore, we describe in-plane variations of the work function, which can be attributed to partial screening of charge impurities in the substrate, and result in a nonuniform charge density in single-layer graphene.
190 citations
TL;DR: In this paper, the effect of oxygen on the degradation of inverted bulk heterojunction solar cells based on poly(3-hexylthiophene):[6,6]-phenyl-C61-butyric acid methyl ester (P3HT:PCBM) blends has been investigated by monitoring current-voltage (jV)-curves, impedance spectra and charge extraction by linearly increasing voltage (CELIV) traces during the degradation process.
186 citations
TL;DR: It is shown that the electrostatic surface potential from a quantum mechanical charge distribution compares well to high-energy electron diffraction and electron holography measurements, as opposed to the comparison with electrochemical measurements.
Abstract: We have resolved the inconsistency in quantifying the surface potential at the liquid−vapor interface when using explicit ab initio electronic charge density and effective atomic partial charge models of liquid water. This is related, in part, to the fact that the resulting electric potentials from partial-charge models and ab initio charge distributions are quite different except for those regions of space between the molecules. We show that the electrostatic surface potential from a quantum mechanical charge distribution compares well to high-energy electron diffraction and electron holography measurements, as opposed to the comparison with electrochemical measurements. We suggest that certain regions of space be excluded when comparing computed surface potentials with electrochemical measurements. This work describes a novel interpretation of ab initio computed surface potentials through high-energy electron holography measurements as useful benchmarks toward a better understanding of electrochemistry.
163 citations
TL;DR: In this article, the conduction band offset and interface charge density of the alumina/GaN interface were determined by analyzing the capacitance-voltage characteristics of atomic layer deposited Al2O3 films on GaN substrates.
Abstract: We report on our investigation of the electrical properties of metal/Al2O3/GaN metal-insulator-semiconductor capacitors. We determined the conduction band offset and interface charge density of the alumina/GaN interface by analyzing the capacitance-voltage characteristics of atomic layer deposited Al2O3 films on GaN substrates. The conduction band offset at the Al2O3/GaN interface was calculated to be 2.13 eV, in agreement with theoretical predications. A non-zero field of 0.93 MV/cm in the oxide under flat-band conditions in the GaN was inferred, which we attribute to a fixed net positive charge density of magnitude 4.60 × 1012 cm−2 at the Al2O3/GaN interface. We provide hypotheses to explain the origin of this charge by analyzing the energy band line-up.
156 citations
TL;DR: In this article, the authors introduce an appropriate energy term into the total energy that represents the hard sphere repulsion of ions and derive a boundary value problem that includes contributions from the repulsive term with a no flux boundary condition for charge density which is a consequence of the variational approach and physically implies charge conservation.
Abstract: We introduce a mathematical model for the finite size (repulsive) effects in ionic solutions. We first introduce an appropriate energy term into the total energy that represents the hard sphere repulsion of ions. The total energy then consists of the entropic energy, electrostatic potential energy, and the repulsive potential energy. The energetic variational approach derives a boundary value problem that includes contributions from the repulsive term with a no flux boundary condition for charge density which is a consequence of the variational approach, and physically implies charge conservation. The resulting system of partial differential equations is a modification of the Poisson-Nernst-Planck (PNP) equations widely if not universally used to describe the drift-diffusion of electrons and holes in semiconductors, and the movement of ions in solutions and protein channels. The modified PNP equations include the effects of the finite size of ions that are so important in the concentrated solutions near electrodes, active sites of enzymes, and selectivity filters of proteins. Finally, we do some numerical experiments using finite element methods, and present their results as a verification of the utility of the modified system.
TL;DR: The existence of local orbital angular momentum on the surfaces of high-Z materials plays a crucial role in the formation of Rashba-type surface band splitting and the band splitting originates from electric dipole interaction, not from the relativistic Zeeman splitting as proposed in the original Rashba picture.
Abstract: We propose that the existence of local orbital angular momentum (OAM) on the surfaces of high-Z materials plays a crucial role in the formation of Rashba-type surface band splitting. Local OAM state in a Bloch wave function produces an asymmetric charge distribution (electric dipole). The surface-normal electric field then aligns the electric dipole and results in chiral OAM states and the relevant Rashba-type splitting. Therefore, the band splitting originates from electric dipole interaction, not from the relativistic Zeeman splitting as proposed in the original Rashba picture. The characteristic spin chiral structure of Rashba states is formed through the spin-orbit coupling and thus is a secondary effect to the chiral OAM. Results from first-principles calculations on a single Bi layer under an external electric field verify the key predictions of the new model.
TL;DR: In this article, a model of a negative corona discharge (without dielectric barriers) in a needle-plane geometry is analyzed, where a set of continuity equations accounting for the movement, generation and loss of charge carriers (electrons, positive and negative ions) is coupled with Poisson's equation to take into account the effect of space and surface charges on the electric field.
Abstract: Axisymmetric finite element models have been developed for the simulation of negative discharges in air without and with the presence of dielectrics. The models are based on the hydrodynamic drift-diffusion approximation. A set of continuity equations accounting for the movement, generation and loss of charge carriers (electrons, positive and negative ions) is coupled with Poisson's equation to take into account the effect of space and surface charges on the electric field. The model of a negative corona discharge (without dielectric barriers) in a needle-plane geometry is analysed first. The results obtained show good agreement with experimental observations for various Trichel pulse characteristics. With dielectric barriers introduced into the discharge system, the surface discharge exhibits some similarities and differences to the corona case. The model studies the dynamics of volume charge generation, electric field variations and charge accumulation over the dielectric surface. The predicted surface charge density is consistent with experimental results obtained from the Pockels experiment in terms of distribution form and magnitude.
TL;DR: In this article, a holographic description of a system of strongly coupled fermions in 2 + 1 dimensions based on a D7-brane probe in the background of D3-branes is considered.
Abstract: We consider a holographic description of a system of strongly-coupled fermions in 2 + 1 dimensions based on a D7-brane probe in the background of D3-branes. The black hole embedding represents a Fermi-like liquid. We study the excitations of the Fermi liquid system. Above a critical density which depends on the temperature, the system becomes unstable towards an inhomogeneous modulated phase which is similar to a charge density and spin wave state. The essence of this instability can be effectively described by a Maxwell-axion theory with a background electric field. We also consider the fate of zero sound at non-zero temperature.
TL;DR: In this paper, the authors present governing equations, scaling analyses and numerical simulations that describe the motion of bimetallic rod-shaped motors in hydrogen peroxide solutions due to reaction-induced charge auto-electrophoresis.
Abstract: Mitchell originally proposed that an asymmetric ion flux across an organism's membrane could generate electric fields that drive locomotion. Although this locomotion mechanism was later rejected for some species of bacteria, engineered Janus particles have been realized that can swim due to ion fluxes generated by asymmetric electrochemical reactions. Here we present governing equations, scaling analyses and numerical simulations that describe the motion of bimetallic rod-shaped motors in hydrogen peroxide solutions due to reaction-induced charge auto-electrophoresis. The coupled Poisson–Nernst–Planck–Stokes equations are numerically solved using Frumkin-corrected Butler–Volmer equations to represent electrochemical reactions at the rod surface. Our simulations show strong agreement with the scaling analysis and experiments. The analysis shows that electrokinetic locomotion results from electro-osmotic fluid slip around the nanomotor surface. The electroviscous flow is driven by electrical body forces which are generated from a coupling of a reaction-induced dipolar charge density distribution and the electric field it creates. The magnitude of the electroviscous velocity increases quadratically with the surface reaction rate for an uncharged motor, and linearly when the motor supports a finite surface charge.
TL;DR: In this paper, the differential capacitance of electric double layers in ionic liquids and its correlation with the surface charge density, ion size and concentration are studied within the framework of the classical density functional theory (DFT).
TL;DR: It is observed, that the structure of the butyl chain is mostly unaffected by the choice of the charge set, an indirect proof for separation into ionic parts and nonpolar domains.
Abstract: We carried out classical molecular dynamics simulations with a standard and two quantum chemistry based charge sets to study the ionic liquid 1-n-butyl-3-methylimidazolium bromide, [C4C1im][Br]. We split the cation up into different charge groups and found that the total charge and the charge distribution in the imidazolium ring are completely different in the three systems while the total charge of the butyl chain is much better conserved between the methods. For comparison, the spatial distribution functions and the radial distribution functions as well as different time correlation functions were calculated. For the structural properties we obtained a good agreement between the standard and one of the two quantum chemistry based sets, while the results from the second quantum chemistry based set led to a completely different picture. The opposite was observed for the dynamic properties, which agree well between the standard set and the second quantum chemistry based set, whereas the dynamics in the fir...
TL;DR: In this paper, the energy density of supercapacitors based on a single-sheet graphene electrode was studied via molecular dynamics (MD) computer simulations, and two electrolytes of different types, pure 1-ethyl-3-methylimidazolium tetrafluoroborate (EMI+BF4-) and an 1.1 M solution of EMI+BF 4 in acetonitrile, were considered as a prototypical room-temperature ionic liquid (RTIL) and organic electrolyte, respectively.
Abstract: Energy density of supercapacitors based on a single-sheet graphene electrode is studied via molecular dynamics (MD) computer simulations. Two electrolytes of different types, pure 1-ethyl-3-methylimidazolium tetrafluoroborate (EMI+BF4–) and an 1.1 M solution of EMI+BF4– in acetonitrile, are considered as a prototypical room-temperature ionic liquid (RTIL) and organic electrolyte, respectively. Structure of ions near the electrode surface varies significantly with its charge density, especially in pure RTIL. Specific capacitance normalized to the electrode surface area is found to be higher in EMI+BF4– than in acetonitrile solution by 55–60%. This is due to strong screening of the electrode charge by RTIL ions in the former. The RTIL screening behavior is found to be rather insensitive to temperature T. As a result, the capacitance of supercapacitors based on pure EMI+BF4– decreases by less than 5% as T increases from 350 to 450 K. The difference in size and shape between cations and anions and the resulti...
TL;DR: In this article, the authors investigated dilute Mg-X alloys, where X denotes Al, Zn, Sn and Ca impurities, with first-principles density functional theory in the local density approximation.
TL;DR: In this paper, the authors compute the field theory Green's function of the fermionic operators making up the star, which contains a large number of closely spaced Fermi surfaces, the volumes of which add up to the total charge density in accordance with the Luttinger count.
Abstract: Electron stars are fluids of charged fermions in Anti-de Sitter spacetime. They are candidate holographic duals for gauge theories at finite charge density and exhibit emergent Lifshitz scaling at low energies. This paper computes in detail the field theory Green’s function G
R(ω,k) of the gauge-invariant fermionic operators making up the star. The Green’s function contains a large number of closely spaced Fermi surfaces, the volumes of which add up to the total charge density in accordance with the Luttinger count. Excitations of the Fermi surfaces are long lived for ω ≲ k
z. Beyond ω ∼ k
z the fermionic quasiparticles dissipate strongly into the critical Lifshitz sector. Fermions near this critical dispersion relation give interesting contributions to the optical conductivity.
TL;DR: It is found that the energy density required to melt the chargedensity wave nonthermally is substantially lower than that required for thermal suppression and is comparable to the charge density wave condensation energy.
Abstract: We use time-resolved optical reflectivity and x-ray diffraction with femtosecond resolution to study the dynamics of the structural order parameter of the charge density wave phase in TiSe2. We find that the energy density required to melt the charge density wave nonthermally is substantially lower than that required for thermal suppression and is comparable to the charge density wave condensation energy. This observation, together with the fact that the structural dynamics take place on an extremely fast time scale, supports the exciton condensation mechanism for the charge density wave in TiSe2.
TL;DR: In this paper, a Monte Carlo simulation of surface-ions interactions is presented to predict surface charge density, electrokinetic potential, and ions adsorption of mineral surfaces in equilibrium with a given electrolyte solution.
TL;DR: In this article, an all-electron calculation of the hyperfine parameters for conduction electrons in Si was presented, showing that all parameters scale linearly with the spin density at a $−29$Si site.
Abstract: We present an all-electron calculation of the hyperfine parameters for conduction electrons in Si, showing that: (i) all parameters scale linearly with the spin density at a $^{29}$Si site; (ii) the isotropic term is over 30 times larger than the anisotropic part; (iii) conduction electron charge density at a Si nucleus is consistent with experimental estimates; (iv) Overhauser fields in natural Si quantum dots (QDs) are two orders of magnitude smaller than in GaAs QDs. This reinforces the outstanding performance of Si in keeping spin coherence and opens access to reliable quantitative information aiming at spintronic applications.
TL;DR: In this paper, the authors report and discuss the effects of geometry, charge distribution and heteroatom substitution on the migration of epoxy oxygen on the basal plane: both the driving force and the ease of surface hopping are very sensitive to their variations.
TL;DR: Studying the ionic conductance through Al₂O₃ nanopore transistors to probe the surface charge density and its dependence on the applied gate field revealed how reactive surface groups dominate the response of nanofluidic field-effect devices via a chemical effect called charge regulation.
Abstract: We studied the ionic conductance through Al${}_{2}$O${}_{3}$ nanopore transistors to probe the surface charge density and its dependence on the applied gate field. The observed conductance modulations are entirely attributable to the electrostatic field effect, and their dependence on pH, ionic strength, and gate voltage is described by a quantitative model. Importantly, these experiments revealed how reactive surface groups dominate the response of nanofluidic field-effect devices via a chemical effect called charge regulation. A quantitative understanding of this effect enables the development of new nanofluidic technologies.
TL;DR: In this paper, the structural properties of aqueous electrolytes confined within graphene pores were investigated using molecular dynamics simulations, and the effects of pore size and surface charge density were quantified by calculating ionic density profiles within the pores and pore-bulk partition coefficients.
Abstract: Molecular dynamics simulations have been employed to study the structural properties of aqueous electrolytes confined within graphene pores. The effects of pore size and graphene surface charge density were quantified by calculating ionic density profiles within the pores and pore-bulk partition coefficients. Carbon-slit pores of width 0.9, 1.2, and 1.6 nm were considered. The graphene surfaces were charged with densities ranging from 0 (neutral pore), 20, 30, and 40 μC/cm2, simulating various applied voltages. Aqueous solutions of NaCl at 1.5–1.6 M concentrations were considered at ambient conditions. When the graphene sheets are neutral, most electrolytes remain outside of the pores. The few sodium and chloride ions that are found within the pores remain preferentially at the center of pores, where they can be hydrated. As the graphene surface charge density increases, more Na+ and Cl– enter the pores. At the maximum graphene surface charge density considered (40 μC/cm2) the ionic concentration within t...
TL;DR: In this paper, the influence of the internal electric field on the qualitative behaviors of solutions is analyzed and it is shown that the electric field leads to the rotating phenomena in charge transport and reduces the speed of fluid motion, but it does not influence the transport of charge density and the heat diffusion.
TL;DR: In this article, a microporous polypropylene membrane (MPPM) was endowed with antibacterial property by grafting polycation and the action mechanism was studied from the viewpoint of the mobility of polycation chains.
TL;DR: In this article, an approach is described that employs cation exchange experiments to develop data that can be interpreted using newly derived set of theoretically-based equations, which can be directly applied to any other materials determination for a given electrolyte system.
Abstract: Specific surface area and surface charge properties such as surface potential, surface charge density, electrostatic field strength at surface, and surface charge number are important characteristics of charged particles in soil and the environment. Currently there is not a theory or method that could give a combined determination for those parameters from a single experiment. In this paper, an approach is described that employs cation exchange experiments to develop data that can be interpreted using newly a derived set of theoretically-based equations. First we give two modifications, respectively, for the nonlinear Poisson-Boltzmann equation and for the cation exchange equilibrium equation. Second, from the modified Poisson-Boltzmann equation, new equations for describing cation exchange equilibrium as considering the cationic hydration effect were obtained. Finally, the theory for the combined determination of the five surface properties through a single experiment of cation exchange equilibrium was established. In the experimental study, the three constant parameters in the theory have been calibrated through a standard sample, illite, and the calibrated variables can be directly applied to any other materials determination for a given electrolyte system. In addition, Na/Ca exchange equilibrium for three different soils were determined, and then the five surface properties were calculated. The application of the new theory showed that this new method could give reliable determination results of the five surface properties for both permanently and variably charged materials. For the specific surface area determination, the new method could be used in both swelling and nonswelling materials. For the surface potential determination, this method could give the surface potential values under different electrolyte composition and concentration conditions.
TL;DR: In this paper, a lattice Hartree-Fock model was used to explore the physics which controls whether a type of broken inversion symmetry state, which can be viewed as a pseudospin ferromagnet, occurs in nature.
Abstract: In mean-field-theory bilayer graphene's massive Dirac fermion model has a family of broken inversion symmetry ground states with charge gaps and flavor dependent spontaneous inter layer charge transfers. We use a lattice Hartree-Fock model to explore some of the physics which controls whether or not this type of broken symmetry state, which can be viewed as a pseudospin ferromagnet, occurs in nature. We find that inversion symmetry is still broken in the lattice model and estimate that transferred areal densities are $\sim 10^{-5}$ electrons per carbon atom, that the associated energy gaps are $\sim 10^{-2} {\rm eV}$, that the ordering condensation energies are $\sim 10^{-7} eV$ per carbon atom, and that the energy differences between competing orders at the neutrality point are $\sim 10^{-9} eV$ per carbon atom. We explore the quantum phase transitions induced by external magnetic fields and by externally controlled electric potential differences between the layers. We find, in particular, that in an external magnetic field coupling to spontaneous orbital moments favors broken time-reversal-symmetry states that have spontaneous quantized anomalous Hall effects. Our theory predicts a non monotonic behavior of the band gap at neutrality as a function of interlayer potential difference in qualitative agreement with recent experiments.