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Polarization density

From Wikipedia, the free encyclopedia
Vector field describing the density of electric dipole moments in a dielectric material
This article is about the polarization of dielectric media by electric fields. For the polarization of electromagnetc waves themselves, seePolarization (waves). For other uses, seePolarization (disambiguation).
Electromagnetism
Solenoid

Inclassical electromagnetism,polarization density (orelectric polarization, or simplypolarization) is thevector field that expresses the volumetric density of permanent or inducedelectric dipole moments in adielectric material. When a dielectric is placed in an externalelectric field, its molecules gainelectric dipole moment and the dielectric is said to be polarized.

Electric polarization of a given dielectric material sample is defined as the quotient of electric dipole moment (a vector quantity, expressed ascoulombs*meters (C*m) inSI units) to volume (meters cubed).[1][2]Polarization density is denoted mathematically byP;[2] in SI units, it is expressed in coulombs per square meter (C/m2).

Polarization density also describes how a material responds to an applied electric field as well as the way the material changes the electric field, and can be used to calculate the forces that result from those interactions. It can be compared tomagnetization, which is the measure of the corresponding response of a material to amagnetic field inmagnetism.

Similar toferromagnets, which have a non-zero permanent magnetization even if no external magnetic field is applied,ferroelectric materials have a non-zero polarization in the absence of external electric field.

Definition

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An external electric field that is applied to a dielectric material, causes a displacement of bound charged elements.

Abound charge is a charge that is associated with an atom or molecule within a material. It is called "bound" because it is not free to move within the material likefree charges. Positive charged elements are displaced in the direction of the field, and negative charged elements are displaced opposite to the direction of the field. The molecules may remain neutral in charge, yet an electric dipole moment forms.[3][4]

For a certain volume elementΔV{\displaystyle \Delta V} in the material, which carries a dipole momentΔp{\displaystyle \Delta \mathbf {p} }, we define the polarization densityP:P=ΔpΔV{\displaystyle \mathbf {P} ={\frac {\Delta \mathbf {p} }{\Delta V}}}

In general, the dipole momentΔp{\displaystyle \Delta \mathbf {p} } changes from point to point within the dielectric. Hence, the polarization densityP of a dielectric inside an infinitesimal volume dV with an infinitesimal dipole momentdp is:

P=dpdV{\displaystyle \mathbf {P} ={\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} V}}}1

The net charge appearing as a result of polarization is called bound charge and denotedQb{\displaystyle Q_{b}}.

This definition of polarization density as a "dipole moment per unit volume" is widely adopted, though in some cases it can lead to ambiguities and paradoxes.[5]

Other expressions

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Let a volumedV be isolated inside the dielectric. Due to polarization the positive bound chargedqb+{\displaystyle \mathrm {d} q_{b}^{+}} will be displaced a distanced{\displaystyle \mathbf {d} } relative to the negative bound chargedqb−{\displaystyle \mathrm {d} q_{b}^{-}}, giving rise to a dipole momentdp=dqbd{\displaystyle \mathrm {d} \mathbf {p} =\mathrm {d} q_{b}\mathbf {d} }. Substitution of this expression in(1) yieldsP=dqbdVd{\displaystyle \mathbf {P} ={\mathrm {d} q_{b} \over \mathrm {d} V}\mathbf {d} }

Since the chargedqb{\displaystyle \mathrm {d} q_{b}} bounded in the volumedV is equal toρbdV{\displaystyle \rho _{b}\mathrm {d} V} the equation forP becomes:[3]

P=ρbd{\displaystyle \mathbf {P} =\rho _{b}\mathbf {d} }2

whereρb{\displaystyle \rho _{b}} is the density of the bound charge in the volume under consideration. It is clear from the definition above that the dipoles are overall neutral and thusρb{\displaystyle \rho _{b}} is balanced by an equal density of opposite charges within the volume. Charges that are not balanced are part of the free charge discussed below.

Gauss's law for the field ofP

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For a given volumeV enclosed by a surfaceS, the bound chargeQb{\displaystyle Q_{b}} inside it is equal to the flux ofP throughS taken with the negative sign, or

−Qb={\displaystyle -Q_{b}=}\oiintS{\displaystyle {\scriptstyle S}}P⋅dA{\displaystyle \mathbf {P} \cdot \mathrm {d} \mathbf {A} }3
Proof

Let a surface areaS envelope part of a dielectric. Upon polarization negative and positive bound charges will be displaced. Letd1 andd2 be the distances of the bound chargesdqb−{\displaystyle \mathrm {d} q_{b}^{-}} anddqb+{\displaystyle \mathrm {d} q_{b}^{+}}, respectively, from the plane formed by the element of area dA after the polarization. And letdV1 anddV2 be the volumes enclosed below and above the area dA.

Above: an elementary volume dV = dV1+ dV2 (bounded by the element of area dA) so small, that the dipole enclosed by it can be thought as that produce by two elementary opposite charges. Below, a planar view (click in the image to enlarge).

It follows that the negative bound chargedqb−=ρb− dV1=ρb−d1 dA{\displaystyle \mathrm {d} q_{b}^{-}=\rho _{b}^{-}\ \mathrm {d} V_{1}=\rho _{b}^{-}d_{1}\ \mathrm {d} A} moved from the outer part of the surface dA inwards, while the positive bound chargedqb+=ρb dV2=ρbd2 dA{\displaystyle \mathrm {d} q_{b}^{+}=\rho _{b}\ \mathrm {d} V_{2}=\rho _{b}d_{2}\ \mathrm {d} A} moved from the inner part of the surface outwards.

By the law of conservation of charge the total bound chargedQb{\displaystyle \mathrm {d} Q_{b}} left inside the volumedV{\displaystyle \mathrm {d} V} after polarization is:

dQb=dqin−dqout=dqb−−dqb+=ρb−d1 dA−ρbd2 dA{\displaystyle {\begin{aligned}\mathrm {d} Q_{b}&=\mathrm {d} q_{\text{in}}-\mathrm {d} q_{\text{out}}\\&=\mathrm {d} q_{b}^{-}-\mathrm {d} q_{b}^{+}\\&=\rho _{b}^{-}d_{1}\ \mathrm {d} A-\rho _{b}d_{2}\ \mathrm {d} A\end{aligned}}}

Sinceρb−=−ρb{\displaystyle \rho _{b}^{-}=-\rho _{b}} and (see image to the right)d1=(d−a)cos⁡(θ)d2=acos⁡(θ){\displaystyle {\begin{aligned}d_{1}&=(d-a)\cos(\theta )\\d_{2}&=a\cos(\theta )\end{aligned}}}

The above equation becomesdQb=−ρb(d−a)cos⁡(θ) dA−ρbacos⁡(θ) dA=−ρbd dAcos⁡(θ){\displaystyle {\begin{aligned}\mathrm {d} Q_{b}&=-\rho _{b}(d-a)\cos(\theta )\ \mathrm {d} A-\rho _{b}a\cos(\theta )\ \mathrm {d} A\\&=-\rho _{b}d\ \mathrm {d} A\cos(\theta )\end{aligned}}}

By (2) it follows thatρbd=P{\displaystyle \rho _{b}d=P}, so we get:dQb=−P dAcos⁡(θ)−dQb=P⋅dA{\displaystyle {\begin{aligned}\mathrm {d} Q_{b}&=-P\ \mathrm {d} A\cos(\theta )\\-\mathrm {d} Q_{b}&=\mathbf {P} \cdot \mathrm {d} \mathbf {A} \end{aligned}}}

And by integrating this equation over the entire closed surfaceS we find that

−Qb={\displaystyle -Q_{b}=}\oiintS{\displaystyle \scriptstyle {S}}P⋅dA{\displaystyle \mathbf {P} \cdot \mathrm {d} \mathbf {A} }

which completes the proof.

Differential form

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By the divergence theorem, Gauss's law for the fieldP can be stated indifferential form as:−ρb=∇⋅P,{\displaystyle -\rho _{b}=\nabla \cdot \mathbf {P} ,}where∇ ·P is the divergence of the fieldP through a given surface containing the bound charge densityρb{\displaystyle \rho _{b}}.

Proof

By the divergence theorem we have that−Qb=∭V∇⋅P dV,{\displaystyle -Q_{b}=\iiint _{V}\nabla \cdot \mathbf {P} \ \mathrm {d} V,}for the volumeV containing the bound chargeQb{\displaystyle Q_{b}}. And sinceQb{\displaystyle Q_{b}} is the integral of the bound charge densityρb{\displaystyle \rho _{b}} taken over the entire volumeV enclosed byS, the above equation yields−∭Vρb dV=∭V∇⋅P dV,{\displaystyle -\iiint _{V}\rho _{b}\ \mathrm {d} V=\iiint _{V}\nabla \cdot \mathbf {P} \ \mathrm {d} V,}which is true if and only if−ρb=∇⋅P{\displaystyle -\rho _{b}=\nabla \cdot \mathbf {P} }

Relationship between the fields ofP andE

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Homogeneous, isotropic dielectrics

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Field lines of theD-field in a dielectric sphere with greater susceptibility than its surroundings, placed in a previously uniform field.[6] Thefield lines of theE-field are not shown: These point in the same directions, but many field lines start and end on the surface of the sphere, where there is bound charge. As a result, the density of E-field lines is lower inside the sphere than outside, which corresponds to the fact that the E-field is weaker inside the sphere than outside.

In ahomogeneous, linear, non-dispersive andisotropicdielectric medium, thepolarization is aligned with andproportional to the electric fieldE:[7]P=χε0E,{\displaystyle \mathbf {P} =\chi \varepsilon _{0}\mathbf {E} ,}

whereε0 is theelectric constant, andχ is theelectric susceptibility of the medium. Note that in this caseχ simplifies to a scalar, although more generally it is atensor. This is a particular case due to theisotropy of the dielectric.

Taking into account this relation betweenP andE, equation (3) becomes:[3]

−Qb=χε0 {\displaystyle -Q_{b}=\chi \varepsilon _{0}\ }\oiintS{\displaystyle \scriptstyle {S}}E⋅dA{\displaystyle \mathbf {E} \cdot \mathrm {d} \mathbf {A} }

The expression in the integral isGauss's law for the fieldE which yields the total charge, both free(Qf){\displaystyle (Q_{f})} and bound(Qb){\displaystyle (Q_{b})}, in the volumeV enclosed byS.[3] Therefore,

−Qb=χQtotal=χ(Qf+Qb)⇒Qb=−χ1+χQf,{\displaystyle {\begin{aligned}-Q_{b}&=\chi Q_{\text{total}}\\&=\chi \left(Q_{f}+Q_{b}\right)\\[3pt]\Rightarrow Q_{b}&=-{\frac {\chi }{1+\chi }}Q_{f},\end{aligned}}}

which can be written in terms of free charge and bound charge densities (by considering the relationship between the charges, their volume charge densities and the given volume):ρb=−χ1+χρf{\displaystyle \rho _{b}=-{\frac {\chi }{1+\chi }}\rho _{f}}

Since within a homogeneous dielectric there can be no free charges(ρf=0){\displaystyle (\rho _{f}=0)}, by the last equation it follows that there is no bulk bound charge in the material(ρb=0){\displaystyle (\rho _{b}=0)}. And since free charges can get as close to the dielectric as to its topmost surface, it follows that polarization only gives rise to surface bound charge density (denotedσb{\displaystyle \sigma _{b}} to avoid ambiguity with the volume bound charge densityρb{\displaystyle \rho _{b}}).[3]

σb{\displaystyle \sigma _{b}} may be related toP by the following equation:[8]σb=n^out⋅P{\displaystyle \sigma _{b}=\mathbf {\hat {n}} _{\text{out}}\cdot \mathbf {P} }wheren^out{\displaystyle \mathbf {\hat {n}} _{\text{out}}} is thenormal vector to the surfaceS pointing outwards. (seecharge density for the rigorous proof)

Anisotropic dielectrics

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The class of dielectrics where the polarization density and the electric field are not in the same direction are known asanisotropic materials.

In such materials, thei-th component of the polarization is related to thej-th component of the electric field according to:[7]

Pi=∑jε0χijEj,{\displaystyle P_{i}=\sum _{j}\varepsilon _{0}\chi _{ij}E_{j},}

This relation shows, for example, that a material can polarize in the x direction by applying a field in the z direction, and so on. The case of an anisotropic dielectric medium is described by the field ofcrystal optics.

As in most electromagnetism, this relation deals with macroscopic averages of the fields and dipole density, so that one has a continuum approximation of the dielectric materials that neglects atomic-scale behaviors. Thepolarizability of individual particles in the medium can be related to the average susceptibility and polarization density by theClausius–Mossotti relation.

In general, the susceptibility is a function of thefrequencyω of the applied field. When the field is an arbitrary function of timet, the polarization is aconvolution of theFourier transform ofχ(ω) with theE(t). This reflects the fact that the dipoles in the material cannot respond instantaneously to the applied field, andcausality considerations lead to theKramers–Kronig relations.

If the polarizationP is not linearly proportional to the electric fieldE, the medium is termednonlinear and is described by the field ofnonlinear optics. To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present),P is usually given by aTaylor series inE whose coefficients are the nonlinear susceptibilities:

Piε0=∑jχij(1)Ej+∑jkχijk(2)EjEk+∑jkℓχijkℓ(3)EjEkEℓ+⋯{\displaystyle {\frac {P_{i}}{\varepsilon _{0}}}=\sum _{j}\chi _{ij}^{(1)}E_{j}+\sum _{jk}\chi _{ijk}^{(2)}E_{j}E_{k}+\sum _{jk\ell }\chi _{ijk\ell }^{(3)}E_{j}E_{k}E_{\ell }+\cdots }

whereχ(1){\displaystyle \chi ^{(1)}} is the linear susceptibility,χ(2){\displaystyle \chi ^{(2)}} is the second-order susceptibility (describing phenomena such as thePockels effect,optical rectification andsecond-harmonic generation), andχ(3){\displaystyle \chi ^{(3)}} is the third-order susceptibility (describing third-order effects such as theKerr effect and electric field-induced optical rectification).

Inferroelectric materials, there is no one-to-one correspondence betweenP andE at all because ofhysteresis.

Polarization density in Maxwell's equations

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The behavior ofelectric fields (E,D),magnetic fields (B,H),charge density (ρ) andcurrent density (J) are described byMaxwell's equations in matter.

Relations between E, D and P

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In terms of volume charge densities, thefree charge densityρf{\displaystyle \rho _{f}} is given by

ρf=ρ−ρb{\displaystyle \rho _{f}=\rho -\rho _{b}}

whereρ{\displaystyle \rho } is the total charge density. By considering the relationship of each of the terms of the above equation to the divergence of their corresponding fields (of theelectric displacement fieldD,E andP in that order), this can be written as:[9]

D=ε0E+P.{\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} .}

This is known as theconstitutive equation for electric fields. Hereε0 is theelectric permittivity of empty space. In this equation,P is the (negative of the) field induced in the material when the "fixed" charges, the dipoles, shift in response to the total underlying fieldE, whereasD is the field due to the remaining charges, known as "free" charges.[5][10]

In general,P varies as a function ofE depending on the medium, as described later in the article. In many problems, it is more convenient to work withD and the free charges than withE and the total charge.[1]

Therefore, a polarized medium, by way ofGreen's theorem can be split into four components.

Time-varying polarization density

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When the polarization density changes with time, the time-dependent bound-charge density creates apolarizationcurrent density of

Jp=∂P∂t{\displaystyle \mathbf {J} _{p}={\frac {\partial \mathbf {P} }{\partial t}}}

so that the total current density that enters Maxwell's equations is given by

J=Jf+∇×M+∂P∂t{\displaystyle \mathbf {J} =\mathbf {J} _{f}+\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}}

whereJf is the free-charge current density, and the second term is themagnetization current density (also called thebound current density), a contribution from atomic-scalemagnetic dipoles (when they are present).

Polarization ambiguity

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Example of how the polarization density in a bulk crystal is ambiguous. (a) A solid crystal. (b) By pairing the positive and negative charges in a certain way, the crystal appears to have an upward polarization. (c) By pairing the charges differently, the crystal appears to have a downward polarization.

Crystalline materials

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In a simple approach the polarization inside a solid is not, in general, uniquely defined. Because a bulk solid is periodic, one must choose a unit cell in which to compute the polarization (see figure).[11][12] In other words, two people, Alice and Bob, looking at the same solid, may calculate different values ofP, and neither of them will be wrong. For example, if Alice chooses a unit cell with positive ions at the top and Bob chooses the unit cell with negative ions at the top, their computedP vectors will have opposite directions. Alice and Bob will agree on the microscopic electric fieldE in the solid, but disagree on the value of the displacement fieldD=ε0E+P{\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} }.

Even though the value ofP is not uniquely defined in a bulk solid,variations inPare uniquely defined.[11] If the crystal is gradually changed from one structure to another, there will be a current inside each unit cell, due to the motion of nuclei and electrons. This current results in a macroscopic transfer of charge from one side of the crystal to the other, and therefore it can be measured with an ammeter (like any other current) when wires are attached to the opposite sides of the crystal. The time-integral of the current is proportional to the change inP. The current can be calculated in computer simulations (such asdensity functional theory); the formula for the integrated current turns out to be a type ofBerry's phase.[11]

The non-uniqueness ofP is not problematic, because every measurable consequence ofP is in fact a consequence of a continuous change inP.[11] For example, when a material is put in an electric fieldE, which ramps up from zero to a finite value, the material's electronic and ionic positions slightly shift. This changesP, and the result iselectric susceptibility (and hencepermittivity). As another example, when some crystals are heated, their electronic and ionic positions slightly shift, changingP. The result ispyroelectricity. In all cases, the properties of interest are associated with achange inP.

In what is now called themodern theory of polarization, the polarization is defined as a difference. Any structure which has inversion symmetry has zero polarization; there is an identical distribution of positive and negative charges about an inversion center. If the material deforms there can be a polarization due to the charge in the charge distribution.[12]

Amorphous materials

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Another problem in the definition ofP is related to the arbitrary choice of the "unit volume", or more precisely to the system'sscale.[5] For example, atmicroscopic scale a plasma can be regarded as a gas offree charges, thusP should be zero. On the contrary, at amacroscopic scale the same plasma can be described as a continuous medium, exhibiting a permittivityε(ω)≠1{\displaystyle \varepsilon (\omega )\neq 1} and thus a net polarizationP ≠0.

See also

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References and notes

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  1. ^abGriffiths, D.J. (2007).Introduction to Electrodynamics (3rd ed.). Dorling Kindersley: Pearson Education.ISBN 978-81-7758-293-2.
  2. ^abParker, Sybil P., ed. (1994).McGraw Hill Encyclopaedia of Physics (2nd ed.). New York: McGraw-Hill.ISBN 978-0-07-051400-3.
  3. ^abcdeIrodov, I.E. (1986).Basic Laws of Electromagnetism. Mir Publishers, CBS Publishers & Distributors.ISBN 81-239-0306-5.
  4. ^Matveev, A. N. (1986).Electricity and Magnetism. Mir Publishers.
  5. ^abcC.A. Gonano; R.E. Zich; M. Mussetta (2015)."Definition for Polarization P and Magnetization M Fully Consistent with Maxwell's Equations"(PDF).Progress in Electromagnetics Research B.64:83–101.doi:10.2528/PIERB15100606.
  6. ^Based upon equations fromGray, Andrew (1888).The theory and practice of absolute measurements in electricity and magnetism. Macmillan & Co. pp. 126–127., which refers to papers by Sir W. Thomson.
  7. ^abFeynman, R.P.; Leighton, R.B.; Sands, M (1964).Feynman Lectures on Physics: Volume 2. Addison-Wesley.ISBN 0-201-02117-X.{{cite book}}:ISBN / Date incompatibility (help)
  8. ^Grant, I.S.; Phillips, W.R. (2008).Electromagnetism (2nd ed.). Manchester Physics, John Wiley & Sons.ISBN 978-0-471-92712-9.
  9. ^Saleh, B.E.A.; Teich+, M.C. (2007).Fundamentals of Photonics. Hoboken, NJ:Wiley. p. 154.ISBN 978-0-471-35832-9.
  10. ^A. Herczynski (2013)."Bound charges and currents"(PDF).American Journal of Physics.81 (3):202–205.Bibcode:2013AmJPh..81..202H.doi:10.1119/1.4773441.
  11. ^abcdResta, Raffaele (1994)."Macroscopic polarization in crystalline dielectrics: the geometric phase approach"(PDF).Rev. Mod. Phys.66 (3):899–915.Bibcode:1994RvMP...66..899R.doi:10.1103/RevModPhys.66.899. See also:D Vanderbilt,Berry phases and Curvatures in Electronic Structure Theory, an introductory-level powerpoint.
  12. ^abSpaldin, Nicola A. (2012)."A beginner's guide to the modern theory of polarization".Journal of Solid State Chemistry.195:2–10.arXiv:1202.1831.Bibcode:2012JSSCh.195....2S.doi:10.1016/j.jssc.2012.05.010.S2CID 55374298.

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