Expanding the Lorentz concept in magnetism

In 1878, the Dutch physicist Hendrik Antoon Lorentz first addressed the calculation of the local electric field at an atomic site in a ferroelectric material, generated by all the other electric dipoles within the sample. This calculation, which applies equally well to ferromagnets, is taught in Universities around the World. Here we demonstrate that the Lorentz concept can be used to speed up calculations of the local dipolar field in square, circular, and elliptical shaped monolayers and thin films, not only at the center of the film, but across the sample. Calculations show that long elliptical and rectangular films should exhibit the narrowest ferromagnetic resonance (FMR) linewidth. In addition, discrete dipole calculations show that the Lorentz cavity field μ 0 M / 3 does not hold in tetragonal films. Depending on the ratio (b/a), the local field can be either less/greater than μ 0 M / 3 : an observation that has implications for FMR. 3D simple cubic (SC) systems are also examined. For example, while most texts discuss the Lorentz cavity field in terms of a Lorentz sphere, the Lorentz cavity field still holds when a Lorentz sphere is replaced by a the Lorentz cube, but only in cubic SC, FCC and BCC systems. Finally, while the primary emphasis is on the discrete dipole–dipole interaction, contact is made with the continuum model. For example, in the continuous SC dipole model, just one monolayer is required to generate the Lorentz cavity field. This is in marked contrast to the discrete dipole model, where a minimum of five adjacent monolayers is required.


Introduction
In recent years there has been growing interest into incorporating thin magnetic films (100 μm-100 nm) into metamolecules [1][2][3][4] and magnonic devices [5,6]. The properties of thin patterned magnetic films are therefore of importance. In general, such devices rely on exciting ferromagnetic resonance (FMR), which is dependent on the local dipolar field  [7,8]). Here, we first review the calculation of the local field, as laid down by Lorentz [9] 5 and Brown [10]. Secondly, we adapt the Lorentz method to monolayers and thin films. Thirdly, an extension of the Lorentz method is used to speed up calculation of local dipolar fields, as a function of position within a given thin film. Fourthly, we compare local dipolar fields in square, rectangular, circular, and elliptical shapes, paying particular attention to the local dipolar fields at the edges. In essence, the work presented here can be seen as an attempt to identify that shape of film (pattern) which should yield the narrowest FMR linewidth. Such experiments should have relevance in magnetic hybrid split ring resonator (SRR) metamolecules, where strong photon-magnon coupling occurs [2][3][4]. In the latter experiments, the permalloy and other magnetic films were simply patterned into small circular disks of ∼100 μm diameter, primarily to suppress eddy currents. Little regard was paid as to how to strengthen the FMR/SRR (magnon/ photon) interaction. In this paper, it is argued that the use of long rectangular or elliptical shapes should lead to sharper FMR resonances, and hence stronger photon-magnon coupling. However, while the above provides some motivation for the present work, the results presented here, advance the Lorentz concept in basic magnetism, particularly in monolayers and thin films.

Brief review of magnetostatics
Using a uniformly magnetized ferromagnetic slab as an example, the local dipolar field at a target site r i can be expressed in the form: where the remaining symbols take on their usual meanings. So in addition to the applied field B , app the actual local dipolar field ( ) B r i loc at the target atom consists of three terms. One, a contribution generated by all the dipoles within the Lorentz sphere: Two, the Lorentz field B Lor generated by the free poles at the edges of the spherical hole (often referred to as the Lorentz cavity or remagnetizing field m / M 3 0 ). Three, a demagnetization field B , demag generated by uncompensated magnetic poles at the extreme left and right of the slab. The contribution from all other dipoles, the vast majority, amounts to zero by a fundamental theorem of electrostatics (see appendix A). This theorem rests on two pillars. One, the magnetization in the sample with the Lorentz sphere removed, can be considered as continuous, as viewed from the center of the Lorentz sphere. Two, the divergence of the magnetization = M div 0, i.e. uniform magnetization regardless of shape. In the following sections, we apply the Lorentz method to thin films and monolayers, where the concept of a Lorentz sphere is now meaningless.

Monolayer magnetostatics
For thin films, the dipole-dipole contribution to the local field ( ) B r loc at a given site can be determined by summing over all the fields generated by dipoles, within individual monolayers, and subsequently adding all the monolayer contributions together. This method works in thin films because, in general, only five monolayers are required [11].
Specifically, if all the magnetic moments within a given monolayer are parallel, the local field at the origin (0, 0, 0) in a simple cubic (SC) compound can be written in compact matrix form:  Here (i) D = k 0 refers to that monolayer which contains the target atom = ( ) r 0, 0, 0 , while D =   k 1, 2 refers to the monolayers above and below the target monolayer, etc, (ii) the dimensionless term  D xx k is a shorthand notation for the planewise summations ; . 5 (viii) the matrix is traceless, and (ix) all other symbols possess their usual meanings. Also, note that (i) this method only works, if all the magnetic moments m k within a given monolayer are parallel to each other, and (ii) in the Lorentz limit, the dipolar sums  D xx k will give rise to the local field ( ) B r , loc as described by equations (1)- (2).
In [11], it was shown that for a SC square monolayer film with converge asymptotically according to: This is exactly the same cavity or remagnetizing Lorentz field as that found at the center of the Lorentz sphere (see figure 1). Finally, before leaving this section, we state that in the main we shall concentrate primarily on the properties of the dimensionless dipolar terms  D ( ) m .

xx k
In practice, such terms can be rapidly converted into dipolar fields using equations such as equation (9). However, this involves additional prefactors that reduce clarity of presentation. In addition, we shall also concentrate primarily on the properties of  D ( ) m xx k with D = k 0, since it is this monolayer which contributes the largest contribution to the local dipolar field.

The Lorentz method in monolayers
In [11] it was shown that the slowly converging term /m 2 2 in equation (6) originates from demagnetizing fields at the edges of the film. This result allows us to adapt the Lorentz method to a square monolayer of spins, as illustrated schematically in figure 2.
Here, the Lorentz square, defined by Lor Lor spins, is located in the center of the square film > ( ) m m .
Lor Thus for the target atom at the center of the film:

Lor Lor
The first term  D ( ) m xx k Lor is the discrete contribution from all the dipoles within the Lorentz square, the second term /m 2 2 Lor is the monolayer equivalent of the remagnetizing Lorentz field, while the third term is the layerequivalent demagnetizing contribution, arising from the free poles on the edges of theḿ m square film. All contributions from other dipoles (the vast majority) amount to zero, as in the Lorentz method (see appendix A). Equivalently, in terms of magnetic field, for a uniformly magnetized monolayer, magnetized along the x-axis: However, there is one big difference between the original Lorentz method and that of the monolayer method presented here. In the original Lorentz model, the sum over all the dipoles enclosed by the Lorentz sphere is zero: for a SC lattice. However, this is not the case for the sum over the spins in a SC Lorentz square. Indeed, this term is the major contributor to the value of   ¥ D ( ) m :     Indeed, this observation allows us to examine the variation of the local dipolar field across a given monolayer or thin film, as discussed in the next section.

Local dipolar field spatial variation in a uniformly magnetized square thin film
So far, we have concentrated on calculating the local dipolar field at the center of say a sphere or thin film. Here, we show that the Lorentz method can be used to speed up the calculation of the local dipolar-field variations across a uniformly magnetized thin film. Instead of fixing the position of the Lorentz square in the center of the film, we now allow the Lorentz square to travel (metaphorically) in both the x and y directions. This situation is illustrated schematically in figure 3.
Here we can think of the Lorentz square as a Lorentz probe which we move around the film. Next, we use the surface charge analogy of demagnetization fields, to work out the value of the demagnetizing field at a given position within the film, arising from the free poles on the edges of the film. Note that a free north (south) pole, corresponds to a negative (positive) charge, respectively. Thus the demagnetizing field is in the same direction as the equivalent electric field.
If we set σ equal to the charge per unit length the charge dq at y in a small region dy is given by: This charge will give rise to an electric field dE at the target atom, located in the center of the Lorentz square (red) Next, we observe that: 14 y x Thus: . 16 x Consequently, on combining equations (12)-(16) we find: 1 7 x Thus the total electric field, arising from the positive charges on the top of the film, is given by: Here the angle θ is swept from q 1 to q -, 2 as indicated by the dotted red lines in figure 3, specifically: Likewise from the negative charges at the bottom of the film: where: Thus: in accord with [11]. However it does allow us to approach the edge of the film to within Δm =−50. The results are detailed in figure 4(a), where it will be observed that most of the film is characterized by a constant value, close to 4.516, except at the film edges where it falls sharply. We shall refer to these two sharp falls as the cliff edges at the North and South poles of the film.
Concomitantly, in addition to the changes in the magnitude of  , xx 0 transverse fields (y-axis) are also generated by the demagnetization fields. Their amplitudes, which we shall denote by the cross term  , xy 0 are summarized in figure 4(b). Note that the amplitude of the  xy 0 is close to zero for most of the film, except at the corners where it changes quite rapidly.
Finally, the local dipolar field, across the sample, can be determined using the matrix formulation: Here it is understood that (i) the  D From an examination of figure 5(a) it will be seen that the variation of  xx 0 parallel to the magnetization M, changes dramatically (4.51→1) at the North and South pole cliff edges of the film. However, perhaps surprisingly,  xx 0 perpendicular to the magnetization M, is relatively constant, only changing modestly right at the very edge of the film (4.51→4.66). These observations therefore, suggest that if we wish to maximize the number of spins with the same local dipolar field, it is best to choose elongated rectangular films with the long axis parallel to the magnetization M. In fact, it is possible to put this statement on a semi-quantitative footing, by calculating the mean value and variance of say  xx 0 across the sample. The results can be seen in figures 6(a), (b), for ever increasing rectangular shapes.
From the k=0 FMR point of view, the actual situation is, of course, more complicated than that implied by mean values and standard deviations. In practice, both the North and South pole cliff edges, will almost certainly lead to pinning at these edges of the film (different FMR frequency). This situation is examined in more detail in appendix B, where it is argued that (i) if the spin wave mode p = / k lis excited, where l is the length of the film in the direction of the magnetization, and (ii) the shift in resonance is say 1/10 th of the inhomogeneous linewidth, then Kittel's FMR equations are still, essentially, valid. Of course, both the North and South pole cliff-edges can be avoided entirely, if a long rectangular strip is turned into a flattened ring. However, such 3D geometries will limit applications.
In the next section, we examine the local dipolar field across elliptically shaped monolayer or thin films. In practice, there are significant differences between rectangular and elliptical thin films.

Local dipolar fields in a uniformly magnetized elliptical disk
First, we obtain an expression for the demagnetization factor for a target atom at the geometric center of a monolayer in the form of an elliptical disk.
The equation of an ellipse takes the form:   for rectangular films. The mean value for  xy 0 is always zero (see figure 4(b)).
Here, we observe the usual convention that the axes (a, b) are associated with the (long, short) axes, respectively, i.e. the ellipticity  e = / b a 1. Unfortunately, this symbolism clashes with our earlier definition of a as the unit length of the SC lattice. So ultimately, it will be necessary to replace the elliptical length a by m a a, where m a is an integer.
In practice, it is convenient to use the parametric representation of an ellipse: sin , 25 where f is measured from the long axis of the ellipse, and  a b. The situation is illustrated in figure 7. First, we need an expression for the charge distribution around the edges of the ellipse. In the parametric representation, the tangent to the ellipse at a general point (x, y) takes the form: In practice, analytic forms can be obtained for the above integrals, if the ellipticity e can be expressed in the form of a fraction:  We find: where K and E are elliptic integrals. Alternatively, in terms of α and β exponents, we have: where the exponent b a is given by:  i.e. that for a circle. Also if e  0, i.e.  a b, then b  0, a which is to be expected. As the ellipse becomes more and more 'rod-like', the charge is now concentrated at the ends of the rod.
Next, some numerical calculations for an ellipse with e = / 1 2, are summarized in table 2. Once again, as with square monolayers, it is clear that the value of   ¥ D ( ) m xx k can be obtained using quite small elliptical disks. As with square monolayers discussed earlier, small elliptical Lorentz disks can be used to probe the variation of magnetic fields across larger elliptical monolayers.
Finally, before leaving this section we state that  ( ) In summary, we have obtained expressions for the local dipolar field at the center of an ellipse. In the following section, we use our extension of the Lorentz method to examine the local dipolar field at points other than that at the center of the ellipse. (see [12,14]). It will also be observed that even for m a as small as 50, the error in Thus the electric field at r 1 due to the surface charge at r is given by: Proceeding in this manner, we find:ò d 45 x 0 2 2 as expected [11]. However, in general it will be necessary to integrate equation ( However, for ellipses, the cliff edge extends into the sides of the film. This is also true for the  xy 0 term, where the spread in magnitude is even more apparent. In summary therefore, even a visual comparison between elliptical and rectangular films, with similar ( ) m m , a b and ( ) m m , x y values, suggests that the rectangular films will possess a greater number of spins experiencing the same local dipolar field.

The local field in a simple tetragonal system
Next, we examine the simplest tetragonal system, with unit-cell volume a 2 b and ask the question: 'Is the local field equal to the Lorentz cavity field in a tetragonal system?' If we align the z-axis along the b-axis, previous calculations for  D = xx k 0 still hold ( º z 0) (only x-y coordinates are involved). However, there will be differences in the values of  D =  So the local field is some 14% larger than the Lorentz field. This observation is not difficult to understand. The largest contribution to the local field B loc is positive, coming from all the in-plane dipoles D = ( ) k 0 . All other planes contribute smaller negative contributions, which decrease very rapidly as Dk increases. Thus in the case of a simple tetragonal system, with the longer b-axis perpendicular to the plane, an increase in local field is to be expected.
Such observations have consequences for FMR. For a thin infinite tetragonal film uniformly magnetized along the x-axis we may write: Alternatively, as in a matrix equation: In practice, differences between this formula and the normal Kittel solution (with h = 1) may be difficult to spot. Furthermore, tetragonal systems will almost certainly exhibit anisotropy, which again will change the resonance condition [12]. Also, sometimes very thin layers appear to be characterized by demagnetization factors of other than −1 perpendicular to the film [12]. Nevertheless, the above discussion highlights the simple fact that only cubic systems are properly characterized by the Lorentz cavity field of m ( ) / M 3 . 0

Lorentz demagnetization factor for a cube
In the bulk of this paper, we have been concerned primarily with the properties of thin films. However, it is of some interest to examine the situation in three dimensions. As we shall see, in section 10, this also has relevance to 2D monolayers with continuous magnetization.
First, imagine a ferromagnetic cube uniformly magnetized along say the x-axis. Second, imagine that a Lorentz box of magnetic moments is cut out and lifted out of the cube (metaphorically speaking), as illustrated schematically in figure 9 below. We set the dimensions of the Lorentz box equal to (2i 0 +1) 3 where i o is an integer.
We can now make the following familiar statements. One, the contribution of the dipole moments to the field at the center of the Lorentz box is zero: this is well known. So the field at the center of the Lorentz box is governed by the Lorentz remagnetizing field, arising from (i) the free poles on the insides of the empty box (which originally contained the Lorentz box of dipoles), and (ii) the demagnetization field, arising from the free poles on the outside edges of the overall cube. Once again, the surface charge analogy can be used to calculate the positive Lorentz remagnetizing field. We find: In summary therefore, for both spheres and cubes, the Lorentz remagnetizing factor of p/ 4 3 is obtained. Nevertheless, even for i 0 as small as ∼50, the error in the Lorentz remagnetizing field is only ∼1.2%. So, in principle, similar methods to those advanced in sections 5 and 7, can be used to examine demagnetizing fields in uniformly magnetized 3D samples.
Next, we note that if the surface charge s is spread uniformly over the sides of the cube, as in the continuum model, the double summation in equation (52) is readily converted into a double integral. We find: 5 5  Lorentz probe is moved away from the center of the ferromagnetic box. Here, the remagnetizing field, associated with the Lorentz probe stays the same, while the demagnetizing field associated with the distant surface poles/ charges will change. Once again, the latter can be easily calculated using simple integrals, thereby allowing a fast determination of the variation of the local field across the bulk of say a ferromagnetic cube.
First, we examine the case of moving the Lorentz box across the ferromagnetic cube in the x-direction, with y=z=0. Here we set the displacement along the x-axis as x is obtained. However, away from the center of the cube the strength of demagnetizing field rapidly increases, almost doubling as it reaches the faces of the Lorentz box. This result is in marked contrast to the monolayer results presented earlier in section 5. Clearly, the demagnetizing field associated with cube/rectangular boxes is highly non-uniform, in marked contrast to the situation encountered in uniformly magnetized spheres, discussed in the next section.

Local dipolar fields inside a uniformly magnetized sphere
Given the highly non-uniform demagnetization fields inside a ferromagnetic cube, it is of some interest to examine the case of a uniformly magnetized 3D sphere. Here it is well-known that the demagnetizing field is always m -/ M 3, 0 anywhere within the sphere. Thus if we use a smaller 3D sphere, as our Lorentz probe, it is easy to show that the local dipolar field is zero, at least up to the edge of the big sphere. Here, the Lorentz remagnetizing field from the small Lorentz spherical probe exactly cancels the demagnetizing field arising from the free poles on the edge of the big sphere. Of course, from a discrete point of view differences may be expected.
In figure 12, a discrete local field calculation can be seen for a sphere of radius 200 units. Note that the value of  xx 0 is shown right up to and beyond the edge of the sphere. From an examination of figure 12, it is clear that well inside the sphere  xx 0 is close to zero, in accord with expectations. However as we approach the inside edge of the sphere,  xx 0 dips down before turning sharply upwards to a value of  = 8.248 xx 0 right at the edge (n=200). We conclude therefore that the local dipolar field is essentially equal to the applied field, except within a range of 10 units of the sphere radius. So despite classical expectations, the local dipolar field at or near the edge of the sphere is not equal to that at the center. Consequently, even for a sphere, some pinning of the FMR modes, at the edges, can be anticipated. Finally, we note that for x-axis trajectory taken in figure 12, the local dipolar field is given by: Consequently, for an x-axis direction of magnetization:

Continuous magnetization model
It is also of some interest to examine the case of 'continuous magnetization', widely used by the magnetic modeling community. Here we assume that every magnetic dipole is spread out uniformly over its unit-cell (a 3 ), thereby allowing discrete dipole summations to be replaced by simple integrals. For example, consider a single monolayer, containing the target atom at the origin, as illustrated schematically in figure 13. In order to calculate the local field at the origin, we first empty the cubic cell ( ) a 3 at the origin, and subsequently calculate the local field at the origin of the empty cell, generated by all the other dipoles in the monolayer. For an infinite film, we find: Finally, we note that the predicted FMR frequency of a single monolayer is equal to that of the discrete model, provided we make the following assumptions. One, we empty out the unit cell, as shown in figure 13. Two, place a single magnetic point-dipole at the center of the empty cell. Three, calculate the local field at the single point-dipole arising from all other dipoles in the monolayer, now assumed to be continuous.

Conclusions and discussions
In this paper, an extension of the Lorentz method has been discussed which is subsequently used to study the variation of local dipolar fields across thin films/monolayers, and in some 3D systems. In particular, we have concentrated on determining demagnetizing fields arising from uncompensated magnetic poles at the edges of rectangular and elliptical films, using the electrostatic analogy. Initially, it was hoped that a particular shape of thin film could be identified which offered a constant demagnetization field, inside the sample. In essence, a 2D equivalent of 3D spheres and ellipses, which both offer a constant demagnetization field inside the sample, regardless of position. Sadly, no such 2D shape has been found. However, long thin rectangular shapes, magnetized along the long axis, offer the best compromise, and should therefore be the choice of sample for FMR applications, and possible incorporation into hybrid magnetic metamolecules.
Finally, some readers may question the usefulness or otherwise of planar monolayer studies. We answer as follows. Firstly, it has been shown that a minimum of five monolayers is required to generate the expected Lorentz cavity field in thin films (see section 3 of this paper, and [11]). Secondly, recently, 2D-magnetism has been observed in a single monolayer of CrI 3 [13,14]. Here, magnetocrystalline anisotropy at the Cr sites negates the famous Mermin-Wagner theorem, which states that no magnetic order can exist in isotropic 1D or 2D systems [15]. Thirdly, the properties of hyper-thin film dielectrics are of commercial importance [16]. Here, the Lorentz field factor plays a crucial role in the determination of the local electric field, and hence possible electrical breakdown. More dielectric applications of the Lorentz method, and the subsequent extension to the Clausius-Mossiti relationship, can be found in [17,18].
However, if the polarization P is uniform, = P div 0 and so: Finally, on invoking the Gauss theorem, the volume integral in equation (4) is transformed into a surface integral: this failure in principle is probably of little relevance. However for larger turn angles, of say of up to 90°, it would constitute a serious issue.