Magnetometers have received considerable attention in recent years. Magnetic components offer an alternative methodology to improve the sensitivity. Due to their exceedingly small structural dimensions, metasurfaces exhibit significant competitiveness in field modulation. A magnetic field concentration phenomenon of spheres at the nanoscale is presented in this paper. The sensitivity of a magnetometer is, therefore, improved through the enhanced static or quasistatic magnetic field by the nanosphere concentrator. Magnetic field redistribution due to the assistance of nanospheres is discussed in this paper using the finite element method. The numerical method is verified with classical analytical equations with a single sphere. The simulation results show that the magnetic field concentrates in the near field behind the nanosphere along the direction of the magnetic flux density. The radius, material or permeability exactly, and distribution are critical parameters to the concentration strength. The magnetic gain of a single nanosphere with typical positive permeability of the typical soft magnetic material reaches 3, and thus, the field along the magnetic flux direction concentrates. Furthermore, the amplification factor is more prominent with the nanosphere arrays compared to a single sphere with the same scale of size, and amplification improves with the sphere numbers and distributions in the array arrangement, which provides a novel approach for the designing of the magnetic flux concentrator being monolithically integrated with the magnetometer probe. Our simulation results provide a new degree of freedom by using nanoscale structures to manipulate magnetic fields.
I. INTRODUCTION
Magnetometers with ultrahigh sensitivities have recently garnered scientific attention.1–4 Substantial research efforts have been directed toward the development of novel detection mechanisms that can enhance the sensitivity of magnetic field measurements, such as atomic magnetometers and superconducting quantum interference devices.5–10 To further amplify the sensitivity of these magnetometers, there is a pressing need to identify alternative strategies for efficiently harvesting magnetic fields without additional power supply requirements. An alternative solution is to enhance magnetic field strength, improve energy efficiency, and integrate magnetic separation with other processing techniques.
One promising approach involves increasing the collection efficiency through the utilization of magnetic flux concentrators.11,12 Notably, devices that focus on the magnetic flux density being integrated in front of a magnetometer offer the potential to intensify the magnetic field, thereby increasing the sensitivity of magnetometers, which motivates the development of magnetic flux concentrators or flux intensifiers. Magnetic concentrators are pivotal in various industries, ranging from mining to medical diagnostics, where weak magnetic field detection is required.13 Generally, magnetic flux concentrators focus magnetic flux by harnessing the diamagnetism properties of superconducting materials like niobium–titanium or through the use of high permeability materials shaped as cones or rings.14–16 The magnetic concentrators at the microscale haVE been preliminarily demonstrated.17 However, the application of magnetic lenses using superconductors or materials with ultrahigh permeability is constrained by the critical temperature limitations and the high costs associated with producing these materials. The quest for more efficient, selective, and versatile magnetic concentrators has driven ongoing research and innovation.18
To our knowledge, studies on magnetic field control are primarily devoted to improving material properties.8 Negative permeability metamaterials have enabled the operation of magnetic lenses at radio frequencies.19 Metamaterials or metasurfaces offer a potential to flexibly manipulate electronic and magnetic field distributions.20 By scaling down the size of subcomponents to the subwavelength level, the permeability or permittivity of the materials can be modulated, leading to the exploration of novel phenomena.21,22
For magnetic fields with direct current or low frequency, which lack wave characteristics, spatial modulation of the magnetic field can still be achieved using materials that possess magnetic properties.23–25 When scaling down the dimensions of these structures to the micrometer level or smaller, it becomes worthwhile to study the fundamental physics in magnetic field control.26,27 Previous research has demonstrated that magnetic field harvesting efficiency can be significantly enhanced through the strategic design of structures at the millimeter or microscale levels.28–30 These investigations have primarily concentrated on elucidating the concentration phenomena of magnetic field distribution within materials under low-frequency or static magnetic fields.12 However, the investigations into the redistribution of magnetic fields outside the concentrators, which occurs due to spatial modulation in response to differences in the structure shape and materials, remain a fertile ground for exploration. Investigating the physical insights and potential abnormal modulation effects that may arise when the structure size is further reduced is worthwhile.
In this study, the influences of spheres at the nanoscale with positive permeability to the spatial distribution of static or quasistatic magnetic field are simulated to study the possibilities of designing magnetic concentration with nanometer scale thickness. For the structures with spherical components, magnetic field concentration occurs along the magnetic flux direction. With the designed distribution of the spheres, the magnetic field is enhanced along the flux direction, and, therefore, the magnetometer sensitivity is improved, borrowing the idea of the solid immersion lens.31 Unlike the traditional magnetic flux concentrator, which requires the probe to be placed inside or in-between the concentrator structure, our design provides a novel possibility by placing the probe behind the nano-thickness concentrator, alleviating the limitation imposed by the probe size while capturing the amplified magnetic field. This paper explores the design process of magnetic concentrators, examining fundamental principles and key design considerations. The findings presented offer a promising avenue for future research in improving the sensitivity of magnetometers and other magnetic-related devices. Furthermore, the compressing or enhancing efficiency can be modulated with the distribution and number of nanospheres in the array, thereby increasing the amplification factor compared with a single sphere of the same size.
II. SPHERE CONCENTRATOR MODEL AND SIMULATION METHOD
A schematic of the concentrator composed of nanospheres is plotted in Fig. 1. The sphere concentrator is designed to be fixed in front of a magnetometer probe. A diffraction-like pattern emerges from a curved surface. The field is enhanced along the direction of the magnetic flux, thereby increasing the amplitude passing through the probe. The sphere layout is also discussed to strengthen the concentration efficiency further with the revealed magnetic field concentration phenomenon.
Proposed future work integrating a sphere-array concentration with a magnetometer probe.
Proposed future work integrating a sphere-array concentration with a magnetometer probe.
The magnetic amplification of the spheres is studied considering the interaction between the spheres in groups, such as a row or conical distribution, for which a single domain or dipole assumption is not applicable. The numerical simulation method of finite element analysis is adopted to solve Maxwell's equations in stationary conditions.32 The magnetic field could be obtained with boundary conditions by separating the simulation area into numerous small blocks.33 At the same time, it provides a method to describe a physical process for complicated objects. The numerical method is verified with classical analytical analysis of a single sphere first. Then, spheres in typical structure groups of row and conical shape are studied in detail. Magnetic field redistributions are plotted due to the existence of the nanoscale spheres to study the magnetic gain under a weak magnetic field.
The values of and are contingent upon the geometry and material properties of the nanosphere, respectively. represents the inside magnetic field, which is influenced by both the external magnetic field and the demagnetization of the material. In this paper, the proposed concentrator is designed to focus a weak magnetic field, which may decrease to microtesla or femtotesla.1,4 Therefore, magnetization and permeability variation are disregarded.
In numerical simulation, a homogeneous, linear, and isotropic sphere is placed in a homogeneous medium with a relative permeability of 1 under a weak magnetic field at low frequency or direct current. A schematic of a 3D nanosphere with a geometric center at (0, 0, 0) is plotted in Fig. 2. An incident magnetic flux density along the x-axis is assumed. The magnetic field (B) is set to 50 μT, taking account of the sensitivity limitation of the existing magnetometers, and environmental amplitude in geoscience, materials science, medical imaging, or other fields.44,45 In simulation, a soft magnetic material like ferrite is chosen as the sphere material, considering magnetization and demagnetization.
Schematic of the sphere magnetic flux concentrator, the geometrical parameters, and the magnetic field amplitude in the center of the sphere independence of relative permittivity.
Schematic of the sphere magnetic flux concentrator, the geometrical parameters, and the magnetic field amplitude in the center of the sphere independence of relative permittivity.
The numerical results are compared with the analytical results calculated with Eq. (8). The magnetic amplitude in the center of the sphere undergoes an initial sharp increase and stabilizes subsequently with the relative permeability. Furthermore, the influence of relative permeability turns inconspicuous when larger than 10 000, making it less complicated to find the magnetic flux concentrator materials. Therefore, in our numerical simulation, the relative permeability (μr) of which is set to 2500 without specific mention. The simulation results are influenced by the radius of the structure and the simulation area as well, which leads to an aberration between the numerical and analytical results. However, the variation trend agrees with each other, and the errors between the results are within 0.8%. Therefore, this paper uses the numerical method to illustrate the concentration phenomenon and the changing trend.
The magnetic field in the sphere is intensified compared to the environmental field, and the magnetic flux amplification differences between the spheres at the microscale and nanoscale are less than 0.5%, but the size decreases 1000 times. With the turbulence led by the sphere, the magnetic field is redistributed, forming a concentrated magnetic field compared to the environmental field due to the curved structure, which can be used as a magnetic flux concentrator.
III. SIMULATION RESULTS OF NANOSPHERE METHOD
Other than the amplification inside the spheres, the amplitude outside the sphere increases along the magnetic flux direction are amplified as well. 2D field amplitudes at corresponding positions parallel to xy, xz, and yz planes are plotted in Fig. 3. The figures are calculated with a single sphere with a radius of 100 nm and relative permeability of 2500 under the magnetic field of 50 μT. The magnetic field is concentrated and enhanced along the magnetic flux direction but deconcentrated along the perpendicular direction as plotted at z = 0 in Fig. 3. The focusing spot is smaller than the sphere in its near field as a graph plotted at x = 100 nm and z = 0 nm and decreases sharply with the distance away from the sphere, as illustrated by the field distribution in the yz plane at different x positions. The phenomenon of the focusing-like pattern parallel to the flux direction outside the nanospheres with ultra-thin thickness offers a possibility of enhancing the magnetic flux. The magnetometer probe collects the energy or amplitude along the magnetic flux direction in the near field behind the sphere to improve the sensitivity.
Magnetic field amplitudes at corresponding distances along xy, xz, and yz planes.
Magnetic field amplitudes at corresponding distances along xy, xz, and yz planes.
The redistribution phenomenon disappears hundreds of nanometers away from the single sphere with the radius at the nanoscale, as shown by the image collected in the yz plane at x = 200 nm and xy plane plotted in Fig. 3. The variation trends are similar to those of macroscopic concentrators.46 Meanwhile, the magnetic gain, defined as the ratio between the magnetic field without and with magnetic flux concentrators, is 3 for the single sphere. The magnetic enhancement is less than traditional concentrators but more prominent than those with similar ultra-thin thicknesses, such as petal-shaped structures.18 Meanwhile, the existence of the sphere results in a magnetic field shielding area in the xy and xz plane perpendicular to the magnetic flux direction as well.
The magnetic amplitude distribution along the x, y, and z-axis is plotted in Fig. 4 to observe the amplitude variation in detail. The simulation parameters are the same as those used in the simulation plotted in Fig. 3. The magnetic field inside the sphere is enhanced and magnetified as predicted with the analytical equations. Outside the structure, the magnetic amplitudes parallel to the magnetic flux direction decrease from the level inside the sphere to that of the environments, as illustrated in Fig. 4(a). On the contrary, the ones perpendicular to the flux direction exhibit a sudden decline to approximately zero, followed by an increase in the environmental magnetic field amplitude value. The curves represent the typical magnetic field amplitude variation parallel and perpendicular to the magnetic flux direction along the x-axis and y- or z-axis. The data along the y- or z-axis are in line with each other due to the axially symmetric distribution of the simulation model. Therefore, only the amplitudes along the y-axis are plotted and analyzed in detail.
Magnetic field amplitude distribution comparison along different directions. (a) Magnetic field amplitude along the x or y-axis. (b) Magnetic field amplitude parallel to the x-axis in the xz plane at different y. (c) Magnetic field amplitude along the radius to varying angles in the xy plane.
Magnetic field amplitude distribution comparison along different directions. (a) Magnetic field amplitude along the x or y-axis. (b) Magnetic field amplitude parallel to the x-axis in the xz plane at different y. (c) Magnetic field amplitude along the radius to varying angles in the xy plane.
The decline occurs gradually with the position away from the symmetrical axis as the position shift or angle increases in Figs. 4(b) and 4(c). The curve compares the amplitude distribution with position shift to those at x = 0 and z = 0, while the angle concerns the magnetic flux direction. The decline is due to the redistribution of the magnetic field due to the permeability change, resulting in a concentration phenomenon as shown in Fig. 3.
To analyze the amplitude variation independence of the sphere radius and the distance behind the nanosphere, curves of the magnetic amplitude with the same parameters used in Figs. 3 and 4 are plotted in Fig. 5. The dis is defined from the edge to the point of picking up the field strength, as illustrated in the schematic. The relative permeability remains 2500. The magnetic amplitude increases with radius but decreases with distance. The variation is noticeable at smaller radius and distances. The magnetic amplitude independence on the radius shows that the magnetic gain is influenced not only by the permeability but also by the size of the sphere, which turns unobvious when the radius grows to a micrometer scale, and so does the distance away from the sphere structure.
(a) Magnetic amplitude independence on radius at different distances from the backside of the nanosphere along the environmental magnetic flux direction. (b) Magnetic amplitude independence on distances from the backside of the nanosphere with different radii parallel to the environmental magnetic flux direction. (c) Magnetic amplitude independence on distances from the backside of the nanosphere with different radii perpendicular to the environmental magnetic flux direction.
(a) Magnetic amplitude independence on radius at different distances from the backside of the nanosphere along the environmental magnetic flux direction. (b) Magnetic amplitude independence on distances from the backside of the nanosphere with different radii parallel to the environmental magnetic flux direction. (c) Magnetic amplitude independence on distances from the backside of the nanosphere with different radii perpendicular to the environmental magnetic flux direction.
Furthermore, the magnetic flux amplitude independence on dis is plotted in Fig. 5(b). The decline rate decreases with the increase of the sphere radius, which aligns with the variation trend in Fig. 5(a) that the magnetic field amplified by the sphere increases with the radius at the same dis. Comparing the curves in Figs 5(a) and 5(b), it could be concluded that along the magnetic flux direction, the larger the radius is, the higher the capability of shifting the magnetic field of the radius is, and the slower the amplitude behind the sphere decreases. Meanwhile, the amplitude drops even more when leaving the sphere with increased radius along the direction perpendicular to the magnetic flux direction, illustrating that the concentration capability is related to the radius. We have to admit that the size of the sphere impacts the maximum magnetic field amplitudes after concentration. However, the variation is less than 3% from 146 to 149 μT on the cost of increasing the radius from the nanometer to micrometer scale, which makes it worthwhile to decrease the size of the concentrators.
IV. CONCENTRATION ENHANCEMENT BY NANOSPHERE ARRAYS
The magnetic gain of a single sphere is limited compared to the magnetic flux concentrators at micro- or millimeter scale. Two typical stacking patterns are adopted to improve amplification, taking advantage of both the bar and cone shape concentrator and nanoscale size sphere.
A. Two nanospheres in a row
A simple array model listing two nanospheres in a row is studied first. The simulation results are plotted in Fig. 6. To minimize the condition differences that may make it confusing to compare the concentration function, the radius of the sphere and relativity permeability are set to 100 nm and 2500, as well as those used in the single sphere. The magnetic field redistribution due to the nanosphere influences others nearby, as seen in the field distribution along the x-axis. The schematic in Fig. 6 illustrates D and d as representing different geometrical parameters of the distance between the two nanospheres and the intensity collection point.
Magnetic field redistribution due to a pair of nanospheres in a row along the magnetic flux direction. (a) Magnetic field amplitudes along the x-axis with different distances between the pair spheres. (b) Magnetic field amplitudes independence of the distance between the pair spheres. (c) Magnetic field amplitudes with varying distances along the xy plane at y = 0 nm.
Magnetic field redistribution due to a pair of nanospheres in a row along the magnetic flux direction. (a) Magnetic field amplitudes along the x-axis with different distances between the pair spheres. (b) Magnetic field amplitudes independence of the distance between the pair spheres. (c) Magnetic field amplitudes with varying distances along the xy plane at y = 0 nm.
As the figures plotted in Figs. 6(a) and 6(c), the two spheres in a row along the magnetic flux direction emphasize the magnetic field amplitude amplification. Enhancement is due to the interaction between the closely placed sphere concentrator. The amplifications inside the spheres are significant when the distance is in the range that the enhancements or the magnetic gain are more considerable. However, the enhancement decreases with the distance D between the spheres owing to the sharp decrease behind the sphere concentrator. As a result of the interaction, the average magnetic strength inside the sphere increases, maximum to 1.7 times that of the single sphere with the same radius and permeability. Meanwhile, the field amplitude between the two spheres is more extensive, which is consistent with the magnetic flux concentrators with T-shaped or bar-shaped.28 However, the enhancement by the traditional concentrators limits the magnetometer probe's size.
The amplitudes outside the spheres along the magnetic flux direction increase accordingly. To illustrate the variation trend, the amplitude behind the sphere at a fixed dis with varying sphere distance D is plotted in Fig. 6(b). The strength increases with the decreasing D due to the enhancement of the localized field strength. At dis = 10 nm with r = 100 nm, the magnetic field is increased to 150 μT compared to 124 μT of the single sphere with the same conditions. The intensity outside the sphere along the magnetic flux decreases exponentially with the square distance between the two spheres in a row along the magnetic flux direction, agreeing with the field variation of the single sphere. The magnetic field gain is related to the radius of the sphere, concentrator shape, and surrounding material.
The magnetic gain in the center of each sphere is as high as 5, which is 1.6 times that when only a single sphere is placed in the same environmental magnetic flux. The magnetic gain remains the same when considering the magnetism of the material as a liner in the weak environmental magnetic field. Though the magnetic gain is not as high as those with macro-sizes, the magnetic field concentration efficiency could be further improved based on the variation from a single sphere to double spheres in a row. To illustrate briefly and concisely the design principle of the magnetic flux concentrator in this paper, the spheres with more numbers in a row are not discussed in detail.
B. Four nanospheres in a conical shape
Though a sphere row could improve the magnetic gain, a conical shape composed of nanospheres is further studied to enrich the magnetic gain strategy. The conical shape is chosen as the sphere layout due to the studies that show the conical shape provides a higher magnetic gain.17 The coordinate origin is located at the geometric center of the 4-sphere structure. The relative permeability and the radius are set to 2500 and 100 nm, respectively, the same as the single sphere and sphere row, to facilitate magnetic field enhancement comparison across the spheres in different layouts.
Considering the results obtained in the sphere row when varying the distance between them, the spheres are placed near each other with a D of 0 nm to improve the magnification, as plotted in the schematic in Fig. 7. The redistribution of the magnetic field at different positions corresponding to each picture's title is plotted in Fig. 7. Only the position at the corresponding coordinate is labeled, and the axis is saved for a concise illustration. The magnetic field variation trends are similar to those of the 2-sphere row. The magnetic field is influenced by each sphere, and the magnetic field gain increases. Compared to the row-placed spheres, the magnetic field on the backside of the conical shape concentrator is higher due to the enhancement of the three spheres in the front.
Magnetic field amplitude with the 4-spheres in a conical shape alone yz, xz, and xy plane at different distances.
Magnetic field amplitude with the 4-spheres in a conical shape alone yz, xz, and xy plane at different distances.
The magnetic field amplitude with different radii at different distances away from the spheres in the conical shape is plotted in Fig. 8. Similar to the concentration by single spheres, the amplitudes increase sharply with the radius at the nanoscale and then flatten at the microscale. Compared to a single or double sphere enhancement, the magnetic gain increased to 4 from 3, corresponding to about 150 to nearly 200 μT, at a distance 10 nm away from the sphere along the magnetic flux direction. Increasing the size or area facing the magnetic flux and decreasing the distance between nanospheres increases the concentration efficiency, ignoring abnormal high amplitude points. Compared to cone-shaped or rod-shaped structures, the magnetic gain is not as high as supposed.16,46 However, the thickness decreases to one thousand of the mentioned macrostructures, which offers possibilities to increase the enhancement by cascading layers of the sphere or other structures at the nanoscale.
(a) Magnetic amplitude independence on radius at different distances away from the nanospheres in a conical shape. (b) Magnetic amplitude independence on dis to varying radii of the nanospheres in a conical shape.
(a) Magnetic amplitude independence on radius at different distances away from the nanospheres in a conical shape. (b) Magnetic amplitude independence on dis to varying radii of the nanospheres in a conical shape.
V. CONCLUSIONS AND DISCUSSIONS
This paper studies a magnetic field concentrator composed of spheres at a nanoscale that operates as a magnetic flux concentrator. A magnetic gain of 4 can be achieved with the novel magnetic flux concentrator composed of four spheres with a radius of 100 nm made of soft magnetic material like ferrite. The influence of geometric parameters on the magnetic field distribution and concentration strength is discussed in detail with the numerical model. The single sphere with the soft magnetic material results in a concentration phenomenon for the ultra-weak magnetic field at low or direct frequency. Meanwhile, the nanosphere placed in a row and conical shape is also discussed. The magnetic concentration efficiency increases with the sphere numbers at the nanoscale distance. As such, this work provides a design principle for magnetic field concentration behind the concentrator at a nanoscale along the magnetic flux direction.
In particular, the magnetic field could be strengthened by adding the layer numbers of the sphere group based on these principles. A preliminary result to improve the magnetic gain by adding a layer of six spheres in a triangle layout to the conical shape is plotted in Fig. 9. With the additional added layer to the conical shape sphere, the magnetic gain increases to 4.8. The relationship between the number of spheres used and the magnetic gain has not been discussed in detail in this paper, considering the simulation time and complexity with the numerical method. However, studying the quantifiable relationship between the distribution shape and the magnetic gain might be worthwhile.
Magnetic field amplitude with the 10-spheres in a conical shape alone xy plane at z = 0.
Magnetic field amplitude with the 10-spheres in a conical shape alone xy plane at z = 0.
The simulation is executed under the premise of static or quasistatic magnetic with a nanosized structure. The numerical results are first validated with analytical solutions to demonstrate the accuracy of the numerical solutions. For the experimental verification of the novel sphere magnetic flux concentrator, the spheres could be fabricated with nano-inject or two-photon polymerization directly in front of a typical magnetometer. Alternatively, fixed with materials such as spin-on glass, which is nonmagnetic and could help minimize the impact on magnetic field distribution. The magnetic gain is, therefore, obtained with the comparison between the magnetic field strength measured with and without the sphere concentrator.
However, the assumption used in the analytical solution is idealized and worth studying further to see the impact of magnetization rotation or domain wall movement in detail to study the sphere arrays, which may be done in our future study to refine our theoretical understanding of the permeability modulation due to size scaling down and enhance the predictive capabilities of models pertaining to nanomagnetic devices and sensors.
Furthermore, the nanosphere leads to a shadow area perpendicular to the magnetic flux direction, which could be applied to protect sensitive magnetic micromachines or other relevant amplifications instead of ambient structures.37,47
ACKNOWLEDGMENTS
This research was funded by the R&D Program of Beijing Municipal Education Commission with grant number KM202211232018, the Young Backbone Teacher Support Plan of Beijing Information Science & Technology University with grant number YBT 202412, and the National Natural Science Foundation of China with grant number 62105036.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Huiyu Li: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Lin Zhao: Data curation (equal); Writing – original draft (equal). Guoqing Hu: Funding acquisition (equal); Software (equal); Validation (equal). Zhehai Zhou: Funding acquisition (equal); Investigation (equal); Supervision (equal). Guangwei Chen: Methodology (equal); Project administration (equal); Supervision (equal); Validation (equal); Visualization (equal).
DATA AVAILABILITY
Data is available from the authors upon reasonable request.