Enhancing the sensitivity of magnetic sensors by 3D metamaterial shells

Magnetic sensors are key elements in our interconnected smart society. Their sensitivity becomes essential for many applications in fields such as biomedicine, computer memories, geophysics, or space exploration. Here we present a universal way of increasing the sensitivity of magnetic sensors by surrounding them with a spherical metamaterial shell with specially designed anisotropic magnetic properties. We analytically demonstrate that the magnetic field in the sensing area is enhanced by our metamaterial shell by a known factor that depends on the shell radii ratio. When the applied field is non-uniform, as for dipolar magnetic field sources, field gradient is increased as well. A proof-of-concept experimental realization confirms the theoretical predictions. The metamaterial shell is also shown to concentrate time-dependent magnetic fields upto frequencies of 100 kHz.


Supplementary
: Energy distribution. Normalized energy inside a sphere of radius R as a function of the radius R normalized to R 2 for the cases: (black dashed) when no shell is present, (blue) when (µ r , µ θ ) = (∞, 0) and (red) when (µ r , µ θ ) = (∞, 1/2). The radii ratio of the shell is γ = R 1 /R 2 = 0.5. Figure S2: Measurements for different applied field directions. (a) Angle of the magnetic induction B at the center of the shell (obtained from measurements of B z and B x , see below), θ INT , as a function of the angle of the applied magnetic field, θ EXT (black symbols). Black line is a guide to the eye. Red line indicates the relation θ INT = θ EXT as a reference. (b) Measured B z (black circles) and B x (blue triangles) at the center of the shell as a function of the angle of the applied magnetic field θ EXT . B z follows a cosine dependence (green line) larger than the values of the z-component of the applied field B 0,z (blue line) for all angles, whereas B x is lower than the x-component of the applied field B 0,x (magenta line). All angles are defined with respect to the z-axis.

Supplementary
Supplementary Figure S3: Phase difference between the measured and the applied field. Measured phase difference between the measured and applied field as a function of frequency corresponding to the measurements in Fig. 4b of the article. Line is a guide to the eye.
Supplementary Discussion 1: Magnetic field spherical concentrator in a uniform applied magnetic field In order to study the magnetic concentration properties of a spherical shell in a uniform applied magnetic field, we derive the analytic solutions of the magnetostatic Maxwell equations. Consider a spherical shell of inner radius R 1 and outer radius R 2 , centered at r = 0, made of a linear, homogeneous and anisotropic magnetic material. The shell is characterized by its polar, azimuthal and radial relative permeabilities, µ θ , µ ϕ and µ r , such that B θ = µ θ µ 0 H θ , B ϕ = µ ϕ µ 0 H ϕ and B r = µ r µ 0 H r , being B r,θ,ϕ and H r,θ,ϕ the radial, polar and azimuthal components of the magnetic induction B and the magnetic field H, respectively, and µ 0 the permeability of free space. We choose µ ϕ = µ θ for simplicity. A uniform magnetic field H 0 is applied in the z direction. Since there are not free currents in our system, ∇ ∧ H = 0, the magnetic field can be written in terms of a magnetic scalar potential φ as H = −∇φ everywhere in space. In the shell itself (SHE: R 1 < r < R 2 ), the scalar potential must fulfill the equation while in the interior hole region (INT: r ≤ R 1 ) and in the external region (EXT: r ≥ R 2 ) the scalar potential should satisfy, The general solutions of these equations, taking into account that φ must be finite when r → 0 and tend to −H 0 rcosθ when r → ∞, can be written as where F = −4 − µ r − 4µ r µ θ + 3µ r α, G = 4 + µ r + 4µ r µ θ + 3µ r α, and α 2 = 1 + 8µ θ /µ r . From these equations two important results arise: (i) the field inside the spherical hole is a uniform field in the direction of the applied magnetic field with magnitude H INT = −a INT 1 and (ii) the field created by the sphere at the exterior region corresponds to the field of a dipole centered at the origin of coordinates with magnetic moment m = 4πb EXT 1ẑ .

A. Energy analysis
It is demonstrated in the main text that the maximum concentration of magnetic field inside the hole of a shell is achieved when using a spherical shell with permeabilities µ r → ∞ and µ θ → 0. Since the coefficient b EXT 1 is not zero for these permeabilities, Eq. (S9), this shell distorts the external magnetic field. Another interesting spherical shell, which concentrates the magnetic field inside its hole while being magnetically undetectable is obtained when considering the permeabilities µ r → ∞ and µ θ → 1/2.
In this section we compare the energy in a sphere of radius R for a shell with (µ r , µ θ ) = (∞, 0), E max , to the energy in the same region for a shell with (µ r , µ θ ) = (∞, 1/2), E nd . These energies are normalized by E 0 , which is the energy that there would be in the region r < R if no shell was present. The analytic expressions for these energies are In Supplementary Figure S1, E max (R)/E 0 (R) and E nd (R)/E 0 (R) are plotted as a function of the radius of the sphere normalized by R 2 . It can be observed that E max (R) is larger than E nd (R) in the hole, and also that Supplementary Discussion 2: Field and gradient enhancement for a dipolar source In this section the analytic solutions of Maxwell equations for a dipolar source are derived to analyse how a spherical shell can concentrate a non-uniform magnetic field. Consider a dipole placed at −dẑ. The scalar magnetic potential in terms of the spherical coordinates centered at the origin r, θ, and ϕ, is where θ and ϕ are the spherical angles of the magnetic moment of the dipole m.
A. Dipole pointing towards the center of a spherical shell We consider a spherical shell of external radius R 2 and internal radius R 1 . Its magnetic permeability is homogeneous, linear and anisotropic, being µ r in the radial direction, µ ϕ in the azimuthal direction and µ θ in the polar direction. For simplicity, we choose µ ϕ = µ θ .
When a spherical shell is placed in a uniform applied magnetic field, it creates a dipolar field outside its external surface and a uniform field inside its hole. Interestingly, from Eqs. (S13) -(S15) and Eqs. (S16) -(S19), we see that the response of the spherical shell to a dipolar field is more complex, since the magnetic potential it creates is constituted by infinite terms. The response of a cylindrical shell to a dipolar field could also be understood as a sum of infinite terms. However, that sum was equivalent to that of a non-centered dipole. When considering spherical shells this correspondence is not found.
We would like to see if there is a spherical shell that does not distort the magnetic field created by an external dipole. This would require d n = 0 for all n because, as seen in Eq. (S15), this coefficient indicates how the spherical shell distorts an external magnetic field. For a given value of n, d n = 0 is obtained only when which depends on n. Therefore, a shell with angular and radial permeabilities fulfilling Eq. (S20) for a given n can cancel at most a single n term of the magnetic scalar potential. Consequently, all spherical shells distort the magnetic field created by a dipole in its outer region.

Extreme spherical concentrator
If we consider the extreme spherical concentrator, (µ r , µ θ ) = (∞, 0), the coefficients a n from Eq. (S16), which give the magnetic field inside the shell's hole, are simplified and can be written as a n = m 4πd 2 (−1) n (1 + n)(1 + 2n)R −n 1 R 2 (R 2 /d) From Eqs. (S13) and (S21), one obtains that the magnetic field at r = 0 and the derivative of its z component with respect to z when a spherical concentrator is used are The magnetic field and gradient at r = 0 created by a dipole pointing radially (without the presence of a spherical shell) can be obtained from Eq. (S11) and are H D (r = 0) = m/(2πd 3 )ẑ and (∂H Dz /∂z)| r=0 = −3m/(2πd 4 ), respectively. Hence, using an extreme spherical concentrator the magnetic field and gradient at r = 0 are increased by the factors, which can be increased by designing a spherical concentrator with larger radii ratio R 2 /R 1 .
B. Dipole pointing towards the non-radial plane As in section A, we consider a spherical shell of external radius R 2 and internal radius R 1 . Its magnetic permeability is homogeneous, linear and anisotropic, being µ r in the radial direction, µ ϕ in the azimuthal direction and µ θ in the polar direction. We choose µ ϕ = µ θ to simplify.
Considering a magnetic dipole placed outside the spherical shell, at −dẑ (d > R 2 ), with magnetic moment m perpendicular to the z direction. In this situation, θ = π/2 and Eq. (S10) becomes One can choose the origin of the x and y axis so that ϕ = π/2 (m = mŷ). Then, cos(ϕ − ϕ ) = sinϕ and the magnetic potential of Eq. (S11) can be written as where P 1 l (cosθ) is the first derivative of the Legendre Polynomials P l (cosθ) with respect to θ. Since there are not free currents in the system, ∇ ∧ H = 0, the magnetic field can be written in terms of a magnetic scalar potential φ in the whole space, H = −∇φ. Using ∇ · B = 0, B r = µ r H r , B ϕ = µ ϕ H ϕ and B θ = µ θ H θ one can obtain the equations that the scalar potential should satisfy in the whole space. In the shell itself (SHE: R 1 < r < R 2 ), the scalar potential must fulfill the equation while in the interior hole region (INT: r ≤ R 1 ) and in the external region (EXT: r ≥ R 2 ) the scalar potential should satisfy where we have assumed that the scalar potential should be finite when r → 0 and tend to the applied potential, φ D , Eq. (S27), when r → ∞. By imposing magnetic boundary conditions we obtain, a m l = b m l = c m l = d m l = 0 ∀m = 1, and a 1 l = mµ r (−1) l (2l + 1)qR (q+1)/2 2