Magnetic monopole detection in magnetoelectric materials

The magnetic monopole has often been a perplexing concept for physicists due to its illusive nature. As a result of the similarities between electricity and magnetism, the presence of a single polarity magnetic particle, which is the magnetic counterpart of the electron or positron, seems inevitable. However, the detection of such particles has never been achieved. This work discusses the properties of magnetic monopoles and considers the possible methods of detecting such particles. It is shown that magnetic textures within solid-state materials can display properties similar to magnetic monopoles. These textures behave like magnetic monopoles, exhibiting a $\frac{1}{r}$ and $\frac{1}{r^2}$ dependence of magnetic potential and magnetic field, respectively, as well as allowing for the isolation of positive and negative magnetic charge. The formation of these particles within Spin-Ice materials is discussed through the flip of dipole moments, causing lattice tetrahedra with a negative or positive net charge.

As a result of a strong coupling between magnetic and electric properties within magnetoelectrics, these materials can display a magnetic response comparable to that of a magnetic monopole. The formation of magnetoelectric states is discussed, with the formation of monopole-like magnetic textures within these materials. Consideration is taken for a magnetoelectric that can exhibit monopole-like behaviour in the form of Chromium (III) Oxide (Cr\textsubscript{2}O\textsubscript{3}) as a result of a diagonal magnetoelectric tensor. As a result of the coupling between the magnetic and electric properties, the characteristic magnetic monopole response to the free electric charge within these materials is outlined. The framework for determining the external magnetic response is outlined by considering boundary conditions upon the surface of magnetoelectric samples.

Using Legendre functions, the magnetic response of a semi-infinite uniaxial magnetoelectric is numerically calculated in the presence of a charged electric tip. The response above the surface of the magnetoelectric was shown to be perfectly radial, as expected for a magnetic monopole within the magnetoelectric.

Using Green's functions, a new method for determining the magnetic response of finite-thickness isotropic magnetoelectrics is derived. The magnetic response of these systems is derived and shown that it can be represented as a series of magnetic image monopoles. An electric charge $q_n$ is placed at a distance $z_n$ above a magnetoelectric of finite-thickness $h$, inducing a B-field above the sample, which is representative of a monopole charge at $-z_n$ along with a series of image monopoles at $-z_{nm}$ where $z_{nm} = z_n + 2(m+1)h$. The magnetic response below the sample represents a magnetic monopole at $z_n$ and a series of monopoles at $+z_{nm}$. It is calculated that the potential inside the slab is a result of two series of image dyons $\mathbf{\mathcal{D}^+}$ and $\mathbf{\mathcal{D}^-}$ situated above and below the slab possessing both magnetic and electric charge. This is advanced further by approximating an electric tip as a series of physical electric point charges to calculate the overall magnetic response of a finite-thickness magnetoelectric to a charge configuration which matches experimental techniques. It is shown that the magnetic response is radial and has a $1/r^2$ dependence, which is indicative of magnetic monopole behaviour.

The model is extended to treat uniaxial anisotropic magnetoelectrics which are representative of physical magnetoelectric materials. Through use of a modified Green's functions, formed by determining the eigenvalues of the governing equations of the magnetoelectric by imposing boundary conditions. The single eigenvalue pair system is considered an analytical tool for understanding the impact of anisotropy. The two-eigenvalue pair system is representative of physical magnetoelectric samples, and a numerical model of the magnetic response of a finite-thickness uniaxial anisotropic magnetoelectric is presented. It is shown that an electric charge $q$ above a magnetoelectric of thickness $h$ produces a radial B-field above and below the magnetoelectric sample. Logarithm plots show that the field above and below has a $1/r^2$ dependence, indicating magnetic monopole behaviour.

Current attempts at measuring the monopole-like magnetic response of magnetoelectrics are discussed. A new method in the form of SQUID magnetometry is outlined, along with the modelling of the expected response. This experimental method compares the expected readings for a dipole and monopole. It is shown that a discontinuous derivative of the magnetic flux across each pickup coil can be an indicator of magnetic monopole behaviour. A $(R_C^2 + z^2)^{-\frac{3}{2}}$ behaviour of the derivative of the magnetic flux above and below the pickup coils can act as a confirmation of the $1/r^2$ distance dependence of these systems compared to the $1/r^3$ seen within dipoles, where $R_C$ is the radius of the pickup coil. A novel method of fitting a magnetic dipole moment to the magnetic flux data and measuring it as a function of charge distance to show magnetic monopole behaviour is also outlined. Additional analytical methods are discussed to confirm the magnetic monopole-like properties of these systems.