ELECTRIC FIELD PDF

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Aims. In studying this chapter you should aim to understand the basic concepts of electric charge and field and their connections. Most of the material provides. study of forces, fields and potentials arising from static charges. . by saying that electric charges in motion produce magnetic fields and moving magnets. “stresses” transmitted by electric fields. We use both the “grass seeds” representation and the ”field lines” representation of the electric field of the two charges.


Electric Field Pdf

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Now, consider point PP a distance rr from +Q. 2. An electric field EE exists at PP if a test test charge +q has a force FF at that point. 3. The direction direction of. Worked Examples from Introductory Physics. Vol. IV: Electric Fields. David Murdock. Tenn. Tech. Univ. September 11, Electric Fields from Particular Charge Distributions. • Electric Dipole. An electric dipole is a pair of charges of opposite sign (±q) separated by a distance d .

Because of its transparency, silicon is suitable for low-loss optical waveguides in the fiber-optical communications wavelengths 1. Silicon exhibits loss than 0. This characteristic permit to silicon competes with the inherently faster III-V- based devices and electro-optic materials such as lithium niobate LiNbO3 [1].

During some years the role of silicon in integrated optics was limited as a substrate for a variety of guided wave structures [2,3,4]. However, taking into account that optical waveguides in which the effective index can be modulated, now it is possible to have important applications as electro-optic modulators, and phase-shifters in such areas as optical communications an high-speed signal processing [5,6,7]. Currently, researches are being focused to obtain optimum electro-optical modulators performances.

In this work, we propose and analyze a versatile structure of an electro-optic modulator using polysilicon as rib waveguide. The electro-optic interaction is obtained by the use of a coplanar waveguide CPW structure. Software based on the Finite Element Method FEM , allows an analysis of the penetration of the electric field E within the polysilicon rib waveguide.

This paper has been divided into the followings sections. Section 2 describes the structure and the main parameters of the device analyzed.

In section 3, results of numerical simulations are shown. Finally, in section 4 conclusions are presented.

It consists of a n-type doped crystalline silicon cm-3 , and a p-type doped polysilicon cm-3 rib embedded into a CPW. The region required for light confinement on the polysilicon rib waveguide can be obtained by a proper choice of the geometrical dimensions for to obtain single mode propagation.

Lateral oxide regions on either side of the rib waveguide maintain optical confinement. Vertical optical confinement is due to a buried oxide and an oxide cover.

A thin insulating oxide layer tox separates the rib waveguide of the n-type region. The CPW structure formed by a central conductor strip and ground planes on both sides permits that a DC voltage can be applied.

We consider a doping level of cm-3 for these regions. Width of the doped regions and their distance to the rib sidewalls are referred to as wd 0. Finally, thickness and width of the polysilicon rib waveguide are referred to as H 0.

Righini, Proc. Its application requires division of the structure into sub- regions or cells. The mesh is built so that the points of some of the triangles called nodes fall throughout the border of the solution region.

Within each element, the parameter under study is approximated by a polynomial. Once the series of equations in each element is available, a matrix is generated with all the coefficients that describe the connections between the nodes. The matrix obtained is solved and finally the solution for this region is found.

HC Verma Class 12 Physics Part-2 Solutions for Chapter 29 - Electric Field and Potential

Each triangular elements and nodes are numbered, in such a way that two systems are obtained. The first system relates the nodes to the elements, whereas the second system contains the coordinates x and y of each node.

Once the systems of nodes and elements have been configured, FEM is used to solve the Laplace equation over the region around of central node. As discussed previously, most of the parameters determining EFM phase, except V, should be constant during each measurement. In turn, the local capacitance C should change slightly accordingly. In the following sections, we will show that such asymmetric behavior can be explained at two theoretical levels, both by ab initio simulations with vdW-functionals, as well as a quantum-capacitance-based classical electrostatic model.

Quantum mechanical first-principle simulations To better understand this intriguing phenomenon, we performed two levels of theretical analysis using quantum mechanical ab initio calculations based on density functional theory DFT ; and a classical electrostatic approach using a capacitance model based on charge conservation equation solved variationally see Methods for details.

We used the quantum mechanical model presented in refs. No external parameters apart from the magnitude of the external electric fields were utilized in a self-consistent calculation.

The simulations also took into account vdW dispersion forces, electrostatic interactions, and exchange-correlation potential within DFT at the same footing. This is in remarkable agreement with the experimental results, where an asymmetrical response was recorded only from graphene and few-layer MoS2 heterostructures Fig.

The effect is enhanced, as more MoS2 layers are included into each graphene system. This follows the behavior observed from EFM measurements, which heterostructures involving thicker MoS2 sheets in contact with graphene gave rise to a more asymmetric EFM phase parabola Fig.

On the basis of these results, it becomes clear that the transition metal dichalcogenide layers play a key role on this screening effect. We will analyze in the following the modifications of the electronic structure of the heterostructures at finite electric fields, and elucidate the origin of this asymmetric susceptibility dependence on the external bias.

The number of graphene layers systematically increases in each panel at a fixed number of MoS2 sheets following the labeling shown in a.

The polarization of the field follows the orientation in Fig. Different curves correspond to different number of MoS2 layers on the each vdW-heterostructures. As the number of MoS2 layer is small, little differences are noticed under the reversed electric field, as the dipole moment formed at the interface roughly compensated each other Fig. This effect is enlarged, as thicker MoS2 sheets are included.

This was due to the amount of interfacial charge redistribution, which generated electric dipole moments preferentially aligned along one direction Fig. This means that higher magnitudes of electric field were observed inside thinner heterostructures, rather than thicker ones Fig.

That is, the thicker the MoS2 sheets, the larger the polarization. For negative fields towards MoS2 layers, the second term on the right-hand side in Eq.

This resulted in smaller electric fields inside the heterojunction Fig. A similar effect is observed to fields directed to graphene layers, but higher in magnitudes inside the sheet due to smaller induced polarization. Blue green curves correspond to positive negative fields.

Positive negative fields go towards graphene MoS2 , and vice versa. Geometries for all systems are highlighted at the background of each panel in opacity tone Full size image On the basis of the previous analysis, several implications on the electronic structure of the heterojunction can be foreseen. At negative bias, the induced dipole moments associated to the S—Mo—S bonds displace the charge towards the surface of the MoS2 sheet Fig. This charge rearrangement is smaller for positive fields because of the semiconducting nature of the MoS2 layer with less charge-carriers on its surface, and the semi-metallic character of graphene.

This results in less polarizable field-dependent facet, smaller charge-transfer from MoS2 to graphene, and consequently better screening. Thin systems e. Such asymmetric behavior has been observed when MoS2 layers are used in metal-insulator-semiconductor junctions Carrier doping induced by the electric field was responsible for the variation of the Fermi level or the work function of MoS2, mainly along one direction, which is directly related to the unbalance of charge density between both sides of the semi-metal and the semiconductor interface.

This indicates that the intrinsic character of the electronic structure of each system in vdW heterostructures contributes to the formation of the asymmetric screening observed.

Similar trends are observed for different thicknesses of graphene and MoS2. Several MoS2 states at the conduction band were observed at This modifies their occupation, as some graphene states can become occupied positive bias or unoccupied negative bias according to the field polarization. The insets in Fig. This indicates that for electric fields toward the dichalcogenide layer, the states mainly composed of the conduction band of MoS2 with minor contribution from graphene were responsible for the charge-screening effect and vice versa.

The effect is enhanced, as more MoS 2 layers are included into each graphene system. This follows the behavior observed from EFM measurements, which heterostructures involving thicker MoS 2 sheets in contact with graphene gave rise to a more asymmetric EFM phase parabola Fig. On the basis of these results, it becomes clear that the transition metal dichalcogenide layers play a key role on this screening effect.

We will analyze in the following the modifications of the electronic structure of the heterostructures at finite electric fields, and elucidate the origin of this asymmetric susceptibility dependence on the external bias.

The number of graphene layers systematically increases in each panel at a fixed number of MoS 2 sheets following the labeling shown in a. The polarization of the field follows the orientation in Fig. Different curves correspond to different number of MoS 2 layers on the each vdW-heterostructures.

As the number of MoS 2 layer is small, little differences are noticed under the reversed electric field, as the dipole moment formed at the interface roughly compensated each other Fig. This effect is enlarged, as thicker MoS 2 sheets are included. This was due to the amount of interfacial charge redistribution, which generated electric dipole moments preferentially aligned along one direction Fig.

This means that higher magnitudes of electric field were observed inside thinner heterostructures, rather than thicker ones Fig. That is, the thicker the MoS 2 sheets, the larger the polarization. In electrostatic boundary conditions, where the normal component of the displacement field D has to be preserved into the system 25 , it gives:.

For negative fields towards MoS 2 layers, the second term on the right-hand side in Eq.

This resulted in smaller electric fields inside the heterojunction Fig. A similar effect is observed to fields directed to graphene layers, but higher in magnitudes inside the sheet due to smaller induced polarization. Blue green curves correspond to positive negative fields. Positive negative fields go towards graphene MoS 2 , and vice versa. Geometries for all systems are highlighted at the background of each panel in opacity tone.

On the basis of the previous analysis, several implications on the electronic structure of the heterojunction can be foreseen. At negative bias, the induced dipole moments associated to the S—Mo—S bonds displace the charge towards the surface of the MoS 2 sheet Fig.

This charge rearrangement is smaller for positive fields because of the semiconducting nature of the MoS 2 layer with less charge-carriers on its surface, and the semi-metallic character of graphene. This results in less polarizable field-dependent facet, smaller charge-transfer from MoS 2 to graphene, and consequently better screening.

Thin systems e. Such asymmetric behavior has been observed when MoS 2 layers are used in metal-insulator-semiconductor junctions Carrier doping induced by the electric field was responsible for the variation of the Fermi level or the work function of MoS 2 , mainly along one direction, which is directly related to the unbalance of charge density between both sides of the semi-metal and the semiconductor interface.

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This indicates that the intrinsic character of the electronic structure of each system in vdW heterostructures contributes to the formation of the asymmetric screening observed. Similar trends are observed for different thicknesses of graphene and MoS 2. Several MoS 2 states at the conduction band were observed at This modifies their occupation, as some graphene states can become occupied positive bias or unoccupied negative bias according to the field polarization.

The insets in Fig. This indicates that for electric fields toward the dichalcogenide layer, the states mainly composed of the conduction band of MoS 2 with minor contribution from graphene were responsible for the charge-screening effect and vice versa. This suggests the important role of the interface on the electrical properties of the vdW heterostructures, as the polarity of the electric field can select, which states can screen the system against external bias.

Asymmetric electronic response under external electric fields. The different curves and colors correspond to the different number of graphene and MoS 2 layers.

The same labeling in a for the number of graphene layers and colors for the MoS 2 are used throughout the different plots. Positive negative fields point towards graphene MoS 2 layers as shown in the inset in a.

That is, positive bias induces charge transfer from MoS 2 to graphene, and vice versa. Also notice the relative shifts of graphene and MoS 2 states with the electric bias. Graphene states are highlighted in blue and MoS 2 bands in faint pink. Fermi level is shown by the dashed-line in each panel. An asymmetrical dependence of the electronic properties with the electric field is noted in all calculated quantities.

Apart from the mighty ab initio approach that gives the full picture of electronic states in the vdWHs, it is more desirable that such asymmetric behavior of the vdWHs can be modeled inexpensively using several key electronic properties from the individual 2D material layers.

As the 2D vdWHs are stacked via non-covalent interactions, it is found that the individual properties of 2D materials can still be largely preserved in their stacked layers, which are coupled by the Coulombic interactions For simplicity of the model, we further assume that i the density of states DOS of individual layer is invariable with the stacking order and the external electric field and ii the interlayer distances d i are not affected by the external electric field.

Note that although the transition of band structure is ignored in assumption i , it has been shown that such classical treatment using Coulombic coupling has relatively high consistency with the ab initio simulations The charge and potential distribution in the vdWH is solved by several conservation equations in a self-consistent approach More details about the mean-field model can be found in Methods.

The fermi level of MoS 2 shifts close to its conduction band CB or valence band VB in both regimes, respectively, which is accounted for the charge accumulation in the vdWH. We ascribe such asymmetry to the difference between the electronic structures of graphene and MoS 2: Due to their close Fermi level values, little charge transfer occurs between graphene and MoS 2 under weak electric field, and the degree of charge transfer is mainly determined by the position of the Fermi level with respect to the CB or VB of MoS 2.

The results from the classical model show good consistency compared with the quantum mechanical ab initio calculations of dipole moment Fig.

Note that for thicker graphene layers e. This indicates that the electric field is well screened by multilayer graphene under such conditions, since the DOS of graphene is finite around the intrinsic Fermi level.

The screening in the MoS 2 becomes more important only when the Fermi level reach the band edges, that is, when the DOS increases greatly.

Asymmetric electric field screening in van der Waals heterostructures

Charge accumulation occurs mostly on graphene and the outmost MoS 2 layer, due to the larger DOS of both layers. The regimes of the n-doped and p-doped MoS 2 are highlighted in green and faint red, respectively.

The Fermi level reaches the conduction band CB or valence band VB of MoS 2 when large negative or positive E ext is applied, relatively as shown by the arrows. The curves follow the labeling in c. Positive charges are shown in faint red and negative charges are shown in blue, respectively.

Only band structures near the Fermi level are shown for illustration. Inspired by the equation of EFM response Eq. We find the layer dependency of quantum capacitance is very similar to that of the dipole moment and charge transfer: This is reasonable due to the higher quantum capacitance of MoS 2 than graphene when Fermi level shifts to the band edges Note that the DFT calculations predict a non-zero quantum capacitance of vdWHs with 2L and 3L graphene even without external electric field, as a result of the interlayer coupling, which is not included in the classical model.

The quantum capacitance increase more under negative electric field than positive field. Both classical and DFT calculations show similar trends of relationship between layer numbers and quantum capacitance: In summary, our findings reveal fundamental knowledge of the screening properties of van der Waals heterostructures using widely used two-dimensional materials, such as graphene and MoS 2.

Our ab initio calculations are further unified in a quantum capacitance-based model, showing that the difference between the energy levels and band structures between graphene and MoS 2 is account for the asymmetric screening behavior. After the transfer of MoS 2 on graphene, the screening of either isolated graphene or MoS 2 changes accordingly to the sign of the electric bias utilized becoming polarity-dependent.

The EFM phase spectrum shows an asymmetry with the tip voltage, as the number of MoS 2 layers increases relative to that of the graphene. Such charge rearrangement also polarized the interface inducing the appearance of dipole moments and consequently giving a directional character to the underlying electronic structure. In particular, external fields in such vdW heterostructures can select which electronic states can be used to screen the gate bias, which clearly give an external control on the screening properties according to the stacking order and thickness.

Our computational-experimental framework paves the way to understand and engineer the electronic and dielectric properties of a broad class of 2D materials assembled in heterojunctions for different technological applications, such as optoelectronics and plasmonics. The graphene and MoS 2 nanosheets were mechanically exfoliated by Scotch tape.

The Raman spectra were collected by a Renishaw Raman microscope using The Au coating was produced by a Leica ACE sputter with a crystal balance monitoring the coating thickness in real time.

Projected augmented wave method PAW 44 , 45 for the latter, and norm-conserving NC Troullier-Martins pseudopotentials 46 for the former, have been used in the description of the bonding environment for Mo, S and C. The shape of the numerical atomic orbitals NAOs was automatically determined by the algorithms described in ref.

The self-consistent field SCF convergence was also set to 1.

To model the system studied in the experiments, we created large supercells containing up to atoms to simulate the interface between different number of graphene and MoS 2 layers. We have kept the lattice constant of the MoS 2 at equilibrium, and stretched the one for graphene by that amount. Negligible variations of the graphene electronic properties are observed with the preservation of the Dirac cone for all systems. Such potential is convenient to analyze the response of finite systems e.

In the classical model, the graphene and MoS 2 layers are treated as individual layers, with the the band structures band gap, DOS and intrinsic work functions considered as invariable to external electric field i.

We further consider that the interlayer distance d i between the i-1 and i-th layers is fixed, and taken as the interlayer distance in DFT calculations under zero field.

We consider the interlayer electric field E i to be uniform. We simplify the quantum capacitance of graphene using a linear model: The intrinsic work function of graphene and MoS 2 are set at 4. The data that support the findings of this study are available from the corresponding author upon reasonable request.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Geim, A. Van der Waals heterostructures. Nature , — Jariwala, D. Mixed-dimensional van der Waals heterostructures.

Novoselov, K. Science , — Liu, Y.To compare this phenomenon from the different samples, we normalized all the EFM data, as shown in Fig. This suggests the important role of the interface on the electrical properties of the vdW heterostructures, as the polarity of the electric field can select, which states can screen the system against external bias.

Yu, L. ACS Nano 9 , — Geometries for all systems are highlighted at the background of each panel in opacity tone. Solving the corresponding equation and using the values of potential in each node and the form function. This suggests the important role of the interface on the electrical properties of the vdW heterostructures, as the polarity of the electric field can select, which states can screen the system against external bias. On the other hand, it knows that the bandwidth of modulators is mainly restricted by the velocity mismatch between the traveling-wave electrodes and the microwave connector.

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