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In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is said to be polarized. The electric dipole moment induced per unit volume of the dielectric material is called the electric polarization of the dielectric.^[1][2]

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 to magnetization, which is the measure of the corresponding response of a material to a magnetic field in magnetism. The SI unit of measure is coulombs per square meter, and polarization density is represented by a vector P.^[2]

Definition

An external electric field that is applied to a dielectric material, causes a displacement of bound charged elements. These are elements which are bound to molecules and are not free to move around the material. 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 element ΔVdisplaystyle Delta V in the material, which carries a dipole moment Δpdisplaystyle Delta mathbf p , we define the polarization density P:

P=ΔpΔVdisplaystyle mathbf P =frac Delta mathbf p Delta V

In general, the dipole moment Δpdisplaystyle Delta mathbf p changes from point to point within the dielectric. Hence, the polarization density P of a dielectric inside an infinitesimal volume dV with an infinitesimal dipole moment dp is:

P=dpdV(1)displaystyle mathbf P =mathrm d mathbf p over mathrm d Vqquad (1)

The net charge appearing as a result of polarization is called bound charge and denoted Qbdisplaystyle Q_b.

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

Other expressions

Let a volume dV be isolated inside the dielectric. Due to polarization the positive bound charge dqb+displaystyle mathrm d q_b^+ will be displaced a distance ddisplaystyle mathbf d relative to the negative bound charge dqb−displaystyle mathrm d q_b^-, giving rise to a dipole moment dp=dqbddisplaystyle mathrm d mathbf p =mathrm d q_bmathbf d . Substitution of this expression in (1) yields

P=dqbdVddisplaystyle mathbf P =mathrm d q_b over mathrm d Vmathbf d

Since the charge dqbdisplaystyle mathrm d q_b bounded in the volume dV is equal to ρbdVdisplaystyle rho _bmathrm d V the equation for P becomes:^[3]

P=ρbd(2)displaystyle mathbf P =rho _bmathbf d qquad (2)

where ρbdisplaystyle rho _b is the density of the bound charge in the volume under consideration.

Gauss's law for the field of P

For a given volume V enclosed by a surface S, the bound charge Qbdisplaystyle Q_b inside it is equal to the flux of P through S taken with the negative sign, or

−Qb=displaystyle -Q_b= Sdisplaystyle scriptstyle S P⋅dA(3)displaystyle mathbf P cdot mathrm d mathbf A qquad (3)

Proof:

Let a surface area S envelope part of a dielectric. Upon polarization negative and positive bound charges will be displaced. Let d1 and d2 be the distances of the bound charges dqb−displaystyle mathrm d q_b^- and dqb+displaystyle mathrm d q_b^+, respectively, from the plane formed by the element of area dA after the polarization. And let dV1 and dV2 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 charge dqb−=ρb− dV1=ρb−d1 dAdisplaystyle 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 charge dqb+=ρb dV2=ρbd2 dAdisplaystyle mathrm d q_b^+=rho _b mathrm d V_2=rho _bd_2 mathrm d A moved from the inner part of the surface outwards. By the law of conservation of charge the total bound charge dQbdisplaystyle mathrm d Q_b left inside the volume dVdisplaystyle mathrm d V after polarization is: dQb=dqin−dqout=dqb−−dqb+=ρb−d1 dA−ρbd2 dAdisplaystyle beginalignedmathrm d Q_b&=mathrm d q_textin-mathrm d q_textout\&=mathrm d q_b^--mathrm d q_b^+\&=rho _b^-d_1 mathrm d A-rho _bd_2 mathrm d Aendaligned Since ρb−=−ρbdisplaystyle rho _b^-=-rho _b and (see image to the right) d1=(d−a)cos⁡(θ)d2=acos⁡(θ)displaystyle beginalignedd_1&=(d-a)cos(theta )\d_2&=acos(theta )endaligned The above equation becomes dQb=−ρb(d−a)cos⁡(θ) dA−ρbacos⁡(θ) dA=−ρbd dAcos⁡(θ)displaystyle beginalignedmathrm d Q_b&=-rho _b(d-a)cos(theta ) mathrm d A-rho _bacos(theta ) mathrm d A\&=-rho _bd mathrm d Acos(theta )endaligned By (2) it follows that ρbd=Pdisplaystyle rho _bd=P, so we get: dQb=−P dAcos⁡(θ)−dQb=P⋅dAdisplaystyle beginalignedmathrm d Q_b&=-P mathrm d Acos(theta )\-mathrm d Q_b&=mathbf P cdot mathrm d mathbf A endaligned And by integrating this equation over the entire closed surface S we find that −Qb=displaystyle -Q_b= Sdisplaystyle scriptstyle S P⋅dAdisplaystyle mathbf P cdot mathrm d mathbf A which completes the proof.

Differential form

By the divergence theorem, Gauss's law for the field P can be stated in differential form as:

−ρb=∇⋅Pdisplaystyle -rho _b=nabla cdot mathbf P ,

where ∇ · P is the divergence of the field P through a given surface containing the bound charge density ρbdisplaystyle rho _b.

Proof:

By the divergence theorem we have that −Qb=∭V∇⋅P dVdisplaystyle -Q_b=iiint limits _Vnabla cdot mathbf P mathrm d V, for the volume V containing the bound charge Qbdisplaystyle Q_b. And since Qbdisplaystyle Q_b is the integral of the bound charge density ρbdisplaystyle rho _b taken over the entire volume V enclosed by S, the above equation yields −∭Vρb dV=∭V∇⋅P dVdisplaystyle -iiint limits _Vrho _b mathrm d V=iiint limits _Vnabla cdot mathbf P mathrm d V, which is true if and only if −ρb=∇⋅Pdisplaystyle -rho _b=nabla cdot mathbf P

Relationship between the fields of P and E

Homogeneous, isotropic dielectrics

Field lines of the D-field in a dielectric sphere with greater susceptibility than its surroundings, placed in a previously-uniform field.^[6] The field lines of the E-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 a homogeneous, linear and isotropic dielectric medium, the polarization is aligned with and proportional to the electric field E:^[7]

P=χε0E,displaystyle mathbf P =chi varepsilon _0mathbf E ,

where ε₀ is the electric constant, and χ is the electric susceptibility of the medium. Note that in this case χ simplifies to a scalar, although more generally it is a tensor. This is a particular case due to the isotropy of the dielectric.

Taking into account this relation between P and E, equation (3) becomes:^[3]

−Qb=χε0 displaystyle -Q_b=chi varepsilon _0 Sdisplaystyle scriptstyle S E⋅dAdisplaystyle mathbf E cdot mathrm d mathbf A

The expression in the integral is Gauss's law for the field E which yields the total charge, both free (Qf)displaystyle (Q_f) and bound (Qb)displaystyle (Q_b), in the volume V enclosed by S.^[3] Therefore,

−Qb=χQtotal=χ(Qf+Qb)⇒Qb=−χ1+χQf,displaystyle beginaligned-Q_b&=chi Q_texttotal\&=chi left(Q_f+Q_bright)\[3pt]Rightarrow Q_b&=-frac chi 1+chi Q_f,endaligned

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+χρfdisplaystyle 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 (denoted σbdisplaystyle sigma _b to avoid ambiguity with the volume bound charge density ρbdisplaystyle rho _b).^[3]

σbdisplaystyle sigma _b may be related to P by the following equation:^[8]

σb=n^out⋅Pdisplaystyle sigma _b=mathbf hat n _textoutcdot mathbf P

where n^outdisplaystyle mathbf hat n _textout is the normal vector to the surface S pointing outwards. (see charge density for the rigorous proof)

Anisotropic dielectrics

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

In such materials, the i^th component of the polarization is related to the j^th component of the electric field according to:^[7]

Pi=∑jϵ0χijEj,displaystyle P_i=sum _jepsilon _0chi _ijE_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 of crystal 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. The polarizability of individual particles in the medium can be related to the average susceptibility and polarization density by the Clausius-Mossotti relation.

In general, the susceptibility is a function of the frequency ω of the applied field. When the field is an arbitrary function of time t, the polarization is a convolution of the Fourier transform of χ(ω) with the E(t). This reflects the fact that the dipoles in the material cannot respond instantaneously to the applied field, and causality considerations lead to the Kramers–Kronig relations.

If the polarization P is not linearly proportional to the electric field E, the medium is termed nonlinear and is described by the field of nonlinear optics. To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), P is usually given by a Taylor series in E whose coefficients are the nonlinear susceptibilities:

Piϵ0=∑jχij(1)Ej+∑jkχijk(2)EjEk+∑jkℓχijkℓ(3)EjEkEℓ+⋯displaystyle frac P_iepsilon _0=sum _jchi _ij^(1)E_j+sum _jkchi _ijk^(2)E_jE_k+sum _jkell chi _ijkell ^(3)E_jE_kE_ell +cdots !

where χ(1)displaystyle chi ^(1) is the linear susceptibility, χ(2)displaystyle chi ^(2) is the second-order susceptibility (describing phenomena such as the Pockels effect, optical rectification and second-harmonic generation), and χ(3)displaystyle chi ^(3) is the third-order susceptibility (describing third-order effects such as the Kerr effect and electric field-induced optical rectification).

In ferroelectric materials, there is no one-to-one correspondence between P and E at all because of hysteresis.

Polarization density in Maxwell's equations

The behavior of electric fields (E and D), magnetic fields (B, H), charge density (ρ) and current density (J) are described by Maxwell's equations in matter.

Relations between E, D and P

In terms of volume charge densities, the free charge density ρfdisplaystyle rho _f is given by

ρf=ρ−ρbdisplaystyle rho _f=rho -rho _b

where ρdisplaystyle 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 the electric displacement field D, E and P in that order), this can be written as:^[9]

D=ε0E+P.displaystyle mathbf D =varepsilon _0mathbf E +mathbf P .

This is known as the constitutive equation for electric fields. Here ε₀ is the electric 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 field E, whereas D is the field due to the remaining charges, known as "free" charges.^[5][10]

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

Therefore, a polarized medium, by way of Green's Theorem can be split into four components.

• The bound volumetric charge density: ρb=−∇⋅Pdisplaystyle rho _b=-nabla cdot mathbf P
  


• The bound surface charge density: σb=n^out⋅Pdisplaystyle sigma _b=mathbf hat n _textoutcdot mathbf P
  


• The free volumetric charge density: ρf=∇⋅Ddisplaystyle rho _f=nabla cdot mathbf D
  


• The free surface charge density: σf=n^out⋅Ddisplaystyle sigma _f=mathbf hat n _textoutcdot mathbf D
  

Time-varying polarization density

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

Jp=∂P∂tdisplaystyle 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∂tdisplaystyle mathbf J =mathbf J _f+nabla times mathbf M +frac partial mathbf P partial t

where J_f is the free-charge current density, and the second term is the magnetization current density (also called the bound current density), a contribution from atomic-scale magnetic dipoles (when they are present).

Polarization ambiguity

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.

The polarization inside a solid is not, in general, uniquely defined: It depends on which electrons are paired up with which nuclei.^[11] (See figure.) In other words, two people, Alice and Bob, looking at the same solid, may calculate different values of P, and neither of them will be wrong. Alice and Bob will agree on the microscopic electric field E in the solid, but disagree on the value of the displacement field D=ε0E+Pdisplaystyle mathbf D =varepsilon _0mathbf E +mathbf P . They will both find that Gauss's law is correct (∇⋅D=ρfdisplaystyle nabla cdot mathbf D =rho _f), but they will disagree on the value of ρfdisplaystyle rho _f at the surfaces of the crystal. For example, if Alice interprets the bulk solid to consist of dipoles with positive ions above and negative ions below, but the real crystal has negative ions as the topmost surface, then Alice will say that there is a negative free charge at the topmost surface. (She might view this as a type of surface reconstruction).

On the other hand, even though the value of P is not uniquely defined in a bulk solid, variations in P are 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 in P. The current can be calculated in computer simulations (such as density functional theory); the formula for the integrated current turns out to be a type of Berry's phase.^[11]

The non-uniqueness of P is not problematic, because every measurable consequence of P is in fact a consequence of a continuous change in P.^[11] For example, when a material is put in an electric field E, which ramps up from zero to a finite value, the material's electronic and ionic positions slightly shift. This changes P, and the result is electric susceptibility (and hence permittivity). As another example, when some crystals are heated, their electronic and ionic positions slightly shift, changing P. The result is pyroelectricity. In all cases, the properties of interest are associated with a change in P.

Even though the polarization is in principle non-unique, in practice it is often (not always) defined by convention in a specific, unique way. For example, in a perfectly centrosymmetric crystal, P is usually defined by convention to be exactly zero. As another example, in a ferroelectric crystal, there is typically a centrosymmetric configuration above the Curie temperature, and P is defined there by convention to be zero. As the crystal is cooled below the Curie temperature, it shifts gradually into a more and more non-centrosymmetric configuration. Since gradual changes in P are uniquely defined, this convention gives a unique value of P for the ferroelectric crystal, even below its Curie temperature.

Another problem in the definition of P is related to the arbitrary choice of the "unit volume", or more precisely to the system's scale.^[5] For example, at microscopic scale a plasma can be regarded as a gas of free charges, thus P should be zero. On the contrary, at a macroscopic scale the same plasma can be described as a continuous medium, exhibiting a permittivity ε(ω)≠1displaystyle scriptstyle varepsilon (omega )neq 1 and thus a net polarization P≠0displaystyle scriptstyle neq mathbf 0 .

See also

• Crystal structure


• Electret


• Polarization (disambiguation)

References and notes