Ellipsoids with negative permeability

To study the properties of a material with negative permeability, we choose the general shape of an ellipsoid with the semiaxesa,bandcinx,yandz, respectively, in a uniform fieldH₀ applied along thec−axis, as sketched in Fig. 4.1. The ellipsoid relative permeabilityµ is assumed to be linear, homogeneous, and isotropic. In this situation, the material magnetization and demagnetizing field are uniform and they are related through the demagnetizing factorN asHd =−NM, where N is a scalar ranging from 0 (long samples,c→ ∞) to 1 (short samples,c→0) [5].

Figure 4.1: Sketch of an ellipsoid with semiaxesa,bandcin a uniform fieldH0 applied along itsc−axis.

4.1 Static negative permeability 59 The behaviour of the ellipsoids can be analysed in terms of the magnetizationM, the magnetic fieldH, and the magnetic inductionB in their volume, which are analytically obtained as [1, 2, 4, 144]

M= µ−1

1 +N(µ−1)H0, H= 1

1 +N(µ−1)H0, B= µµ0

1 +N(µ−1)H0. (4.1) This set of equations shows that all the fieldsM,HandBare uniform and only depend on the demagnetizing factor and on the material permeability. In Fig. 4.2, M, H and Bare plotted as a function ofµfor two particularly interesting ellipsoidal geometries: a sphere (a=b=c, N = 1/3), and a long cylinder in perpendicular field (a=c,b→ ∞, N = 1/2).

Figure 4.2: Normalized magnetization (in red), magnetic field (in green), and magnetic induction (in blue) as a function of the relative permeabilityµwhen a uniform magnetic fieldH0 is applied to (a) a sphere and (b) a long cylinder.

Materials with negative permeability can be analysed as a natural extension of ma-terials with positive permeability. We start by reviewing how mama-terials with µ > 0 interact with uniform applied magnetic fields. Ferromagnetic materials (µ → ∞) ex-hibit the largest magnetization due to the strong field attraction they provide. The magnetization and the magnetic induction in their volume depend on their geometry as 1/N: M(µ→ ∞) = (1/N)H0 and B(µ→ ∞) = (µ₀/N)H0. Since M= (µ−1)H and B =µµ0H, M(µ → ∞) and B(µ → ∞) can only be finite if the magnetic field in the ferromagnetic volume is zero, H(µ→ ∞) →0. As µ is reduced from ∞ to 1, the field attraction decreases and both the modulus ofMandBare reduced. The modulus of the demagnetizing field (H_d =−NM) is reduced due to the decrease in Mand, therefore, the modulus of the magnetic field (H=H₀+H_d) is increased. When reaching the air permeability,µ= 1, materials do not attract (nor expel) magnetic fields;M(µ= 1) = 0, H(µ= 1) =H₀, and B(µ= 1) =µ₀H₀. For positive µ−values lower than 1, materials become diamagnetic and repel the applied magnetic field; M(0< µ < 1) has opposite direction to H0 and, thus, the modulus of B(0 < µ < 1) is lower than µ0H0. In the limit of perfect diamagnetic ellipsoids (µ→0),B(µ→0)→0.

60 Shaping magnetic fields with negative permeability A further decrease of µ leads us to the region of negative permeability [141]. The main feature one appreciates in Fig. 4.2 forµ <0 is that there is a negative permeability value that gives a divergence of the magnetic fields. According to the set of equations (4.1), this permeability isµ_DIV= (N−1)/N;µ_DIV=−2 for the sphere andµ_DIV=−1 for the cylinder. As µ is reduced from 0 to negative values, the fields M, H and B continuously build up (M and B in the opposite and H in the same direction as H0) until they become infinite atµ_DIV. Because materials withµ_DIV< µ <0 are magnetized in the opposite direction toH0, they can be classified as diamagnetic materials. Since the modulus of their magnetization is larger than that of perfect diamagnetic materials, these materials expel magnetic field lines even more strongly than perfect diamagnets (see Figs. 4.3a and c). Forµ-values lower thanµDIVthe ellipsoids change their behaviour completely and the direction of all the fields is inverted. The modulus ofM,Hand B decreases withµ and in the limit µ→ −∞ their magnitude is the same as in the limit µ→ ∞. Because materials withµ < µDIVare magnetized in the direction of the applied field, they can be regarded as paramagnetic materials. The field attraction they provide is even larger than that of ferromagnetic materials (see Figs. 4.3b and d).

To understand how negative-permeability materials behave and why they can pro-duce such strong magnetic responses, it is important to analyse the energy density, E = (1/2)H·B, in their volume. In linear, homogeneous and isotropic materials, the energy density can be written as E = (1/2)µµ₀H², where H is the modulus of the magnetic field. This expression shows that the energy density is positive in the volume occupied by a positive-permeability material and negative in the volume of a material with negative permeability.

Consider an ellipsoid of volumeV in a uniform fieldH0. The total magnetic energy the applied field creates in the region occupied by the ellipsoid is E0 = (1/2)µ₀V H₀². As shown in Figs. 4.3a and b, the positive-permeability ellipsoids exhibiting the largest field attraction (µ→ ∞) and repulsion (µ→ 0) expel all the magnetic energyE0 from their volume (materials withµ→ ∞ because H= 0 and materials withµ→0 because B= 0). This energy is redistributed to the rest of space and is responsible for the dis-tortion of the applied magnetic field. Materials with intermediate positive permeability values also expel some magnetic energy, but their magnetic responses are weaker because they keep part of the energy in their volume. Materials with negative permeability, as shown in Fig. 4.2, show larger magnetic responses than materials with µ → ∞ and µ→ 0. In Fig. 4.3c, a cylinder withµ=−1/2 is shown to expel the applied magnetic induction field lines much further than a cylinder with µ → 0 (Fig. 4.3b) and in Fig.

4.3d a cylinder withµ=−2 is shown to attract more lines than a cylinder withµ→ ∞ (Fig. 4.3a). In terms of energy reorganization, these materials can only achieve these strong magnetic responses if they expel more magnetic energy from their volume than materials with µ→ ∞ and µ→ 0; that is, they expel more energy than the energyE0

that the applied magnetic field provides.

4.1 Static negative permeability 61

Figure 4.3: Numerical calculations of the magnetic induction field lines and (in colors) the normalized energy density E/(µ0H₀²) when a uniform magnetic field H0 is applied along thez-direction to a long cylinder with isotropic relative permeability (a)µ= 10⁴, (b)µ= 10⁻⁴, (c) µ=−1/2, and (d) µ=−2.

In other words, positive-permeability materials can be regarded as materials that redistribute the magnetic energy E0 that the applied magnetic field provides in their volume. The manner in which they redistribute this energy depends on their geometry and permeability. In contrast, negative-permeability materials, do not redistribute the energyE0but a larger amount of energy. Since this energy is not provided by the uniform applied magnetic field, materials with µ <0 can be understood as active because they themselves are responsible for generating the extra amount of magnetic energy they need to expel [145]. In this sense, the negative energy in the µ <0 materials volume can be regarded as energy that the material has to generate in order to behave as a material with negative permeability. Active materials and metamaterials have already been introduced for controlling a wide range of fields, including thermal [146], acoustic [147], electric [148, 149], and electromagnetic waves [150, 151, 152, 153]. A strategy on how to create an active negative-µ material for magnetostatic fields based on electric currents will be presented in Sec. 4.1.3