The Effect of Geometrical Overlap between Giant Magnetoresistance Sensor and Magnetic Flux Concentrators: A Novel Comb-Shaped Sensor for Improved Sensitivity

Abstract

The combination of magnetoresistive (MR) element and magnetic flux concentrators (MFCs) offers highly sensitive magnetic field sensors. To maximize the effect of MFC, the geometrical design between the MR element and MFCs is critical. In this paper, we present simulation and experimental studies on the effect of the geometrical relationship between current-in-plane giant magnetoresistive (GMR) element and MFCs made of a NiFeCuMo film. Finite element method (FEM) simulations showed that although an overlap between the MFCs and GMR element enhances their magneto-static coupling, it can lead to a loss of magnetoresistance ratio due to a magnetic shielding effect by the MFCs. Therefore, we propose a comb-shaped GMR element with alternate notches and fins. The FEM simulations showed that the fins of the comb-shaped GMR element provide a strong magneto-static coupling with the MFCs, whereas the electric current is confined within the main body of the comb-shaped GMR element, resulting in improved sensitivity. We experimentally demonstrated a higher sensitivity of the comb-shaped GMR sensor (36.5 %/mT) than that of a conventional rectangular GMR sensor (28 %/mT).

1. Introduction

Giant magnetoresistance (GMR) devices with current-in-plane geometry provide magnetic sensors for various applications such as gradiometers and magnetic encoders owing to their large output voltage, controllable field range and reliability [1,2,3,4]. The GMR sensors are promising not only for general purpose magnetic sensors but also for high-sensitive magnetic sensors to detect small magnetic fields, e.g., biomagnetic fields or stray fields from magnetic nanoparticles for biomedical applications, in combination with magnetic flux concentrators (MFCs) [5,6]. MFCs are made of soft-magnetic film/foil/bulk materials that concentrate magnetic flux into magnetic sensor elements, e.g., Hall [7,8], anisotropic magnetoresistance [9], GMR [3,10,11], and tunnel magnetoresistance [12,13] sensors. Typically, larger MFCs in size provide more effective magnetic flux concentration; however, the large footprint of MFC is a tradeoff for sensor applications. In magnetic sensors with MR element, generally the size of the MFC is larger than the sensing element and acts as a limiting factor to the spatial resolution of a sensor. Therefore, it is important to develop the sensors with smaller MFCs. Various methods have been implemented to improve the sensitivity of magnetic sensor with an on-chip MFC and MR element. In 2004, Pannetier et al. proposed a GMR sensor with a superconducting MFC loop for a femto-tesla field measurement at low temperature [14]. Other reports show that the detectivity of MR sensor can be improved by modulating the magnetic field to higher frequency using MFCs or MR elements with microelectromechanical systems-based (MEMS) mechanisms [15,16,17]. Recently, Kikitsu et al. [18] have implemented highly sensitive GMR sensors (detectivity~13 pT/Hz^1/2 at 100 Hz) with on-chip MFCs and an external ac modulation field in a magnetic field microscope and observed a better spatial resolution in comparison with magneto-impedance sensors.

Figure 1 shows a schematic configuration of a GMR sensor with MFCs. The GMR element is typically a rectangular-shaped spin-valve (SV) film with a width of a few micrometers to a few tens micrometers patterned by the microfabrication technique. A pair of MFCs, typically made of permalloy, are placed next to the GMR element. In principle, a larger relative permeability (μ_r) of the MFC material and a narrower gap between the MFCs create a lower magnetic reluctance path for magnetic flux. Hence, MFCs concentrate external magnetic flux into the sensor element placed between the gap of the MFC. The efficiency of an MFC is often described by MFC gain, which is defined as $B_{\mathrm{sen}}^{ \mathrm{MFC}}{\mathbf{/}B}_{\mathrm{sen}}^{ \mathrm{noMFC}}$, where $B_{\mathrm{sen}}^{ \mathrm{MFC}}$ and $B_{\mathrm{sen}}^{ \mathrm{noMFC}}$ are the magnetic flux densities inside the sensor element with and without MFC, respectively. However, for magnetoresistive (MR) sensors consisting of MFCs, the net magnetic reluctance of the whole sensor system depends not only on the μ_r of the MFCs but also on that of the sensor element. Therefore, it is important to design the magnetic circuit of sensors with MFCs considering the magnetic properties and geometrical parameters (e.g., shape, gap, width, and length) for both MFC and magnetic free layer (FL) of the MR element. Although there are studies on the effects of the magnetic properties and geometrical parameters of MFCs on the MFC gain, [13,19,20,21,22,23,24] discussions including the magnetostatic coupling between MFCs and FL are limited.

In this work, we studied the effect of the geometrical configuration of the GMR element and MFCs on the sensitivity of the GMR sensor. We simulated the distribution of magnetic flux in the GMR elements with two kinds of shape: a conventional rectangular and a comb shape. We also studied the sensitivity of these sensors experimentally. We found that the geometrical overlap between the rectangular GMR element and MFCs can give either enhancement or reduction of the GMR ratio depending on the degree of overlap. For the comb-shaped GMR elements, however, we obtained an improved sensitivity compared to the case with the rectangular GMR elements.

2. Magnetic Circuit in the MFC–FL System

Since MR sensor elements consist of ferromagnetic FLs with relatively large μ_r, the magneto-static coupling between the FL and MFCs influences the magnetic flux density (B) in the FL. This is qualitatively explained in Figure 2. As shown in Figure 2a, when only a pair of MFCs presents without a MR element in the gap, the magnetic flux (Φ₁) through the MFCs is determined by the magnetic reluctances of the MFCs and the air gap (R_air). When a FL with a magnetic reluctance of R_FL is placed in the gap (Figure 2b), the magnetic reluctance between the MFCs is reduced compared to R_air because of the lower magnetic reluctance of R₁ + R_FL + R₂ than R_air, where R₁ and R₂ are the magnetic reluctances of the spacing between the FL and MFC. The FL provides a low magnetic reluctance path due to its large μ_r ($\gg$1). Therefore, the presence of FL in the MFC gap not only increases the magnetic flux flowing through itself, but also increases the total magnetic flux flowing through the whole sensor (i.e., Φ₂ > Φ₁ in Figure 2). Therefore, by increasing the magneto-static coupling between MFC and FL, the effect of the MFC, i.e., the sensitivity of the sensor, improves. As shown in Figure 2c, by introducing an overlap between the FL and MFCs with a small vertical spacing for electrical separation between them, the magneto-static coupling between the FL and MFCs is further enhanced.

3. Methods

3.1. Finite Element Method Simulations

We simulated the distributions of B and electric current (I) in sensor systems composed of a GMR element and a pair of MFCs using the finite element method (FEM) with the AC/DC module of COMSOL Multiphysics^® software [25]. The region surrounding the GMR element and MFCs was defined as “air” (i.e., magnetic permeability of μ₀) with boundary condition at infinity. The magnetic field generated from an electric current I flowing through the GMR element was not considered in the analysis. For the GMR element, only the FL was considered because the magnetizations of the other ferromagnetic layers (i.e., pinned layer (PL) and reference layer (RL)) are fixed. Figure 3 shows the three-dimensional (3D) and two-dimensional (2D) (in the xy-plane) schematics of the simulated structures consisting of the MFCs and FL. We simulated two types of the shape of the FL; a conventional rectangular (Figure 3a) and a comb shape (Figure 3b). Both the rectangular and comb-shaped FLs are located in the gap between the MFCs with a spacing of 0.1 μm in the z-direction, which is for the electrical separation between them. In the comb-shaped FL, the “fins” are separated by the “notches” and connected to the central “spine” of the FL. The lengths of the fins and notches in the y-direction were defined as L_f and L_n, respectively. I was flown in the y-direction.

For both the rectangular and comb-shaped FLs, the shape and size of the MFC were fixed to a cuboid with a length of L_MFC = 1000 μm, a width of W_MFC = 100 μm, and a thickness of t_MFC = 1 μm. Typically, a large (MFC length/MFC gap) ratio (greater than 100) is desirable for a higher MFC gain. At the same time, if the MFC gap is comparable to or less than t_MFC, it may increase the flux leakage between the two MFCs. Considering these aspects, the gap between MFCs was set to W_G = 5 μm in the present study. Note that MFCs do not necessarily have to be cuboids but can assume various shapes, such as trapezoids, and taper in thickness [23]. However, in this work we simulated only for cuboidal MFCs for simplicity. The thickness of the FL was fixed to 10 nm for both the rectangular and comb-shaped FLs. Although the actual thickness of the FL in the experiments was 5.5 nm, the choice of 10 nm for the FL thickness was limited by the finite memory size of the computer used for the FEM simulations. For the rectangular FLs, the MFC and FL overlap (hereafter, MFC–FL overlap) when the width of the FL (W_FL) was larger than W_G. For the comb-shaped FLs, the width of the spine (W_S) was kept the same as W_G (=5 μm). Therefore, the fins of the comb-shaped FL provided an MFC–FL overlap. The total width of the comb-shaped FL including the spine and fins was defined as W_L. The relative permeabilities of the FL (${\mu }_r^{FL}$) and MFC (${\mu }_r^{MFC}$) were assumed to be 500 and 2000, respectively, based on the experiments described below. The effects of W_FL (for the rectangular FL), W_L, L_f, and L_n (for the comb-shaped FL) on B inside the FLs were simulated under the application of a uniform external B (B_ex) in the x-direction.

3.2. Experimental Procedures

We deposited the SV structure on a 2 cm × 2 cm substrate with a confocal sputtering system with a rotating substrate. Then, the SV devices and corresponding MFCs were fabricated on the same piece of substrate. This allowed us to keep the fabrication conditions the same for all devices, which included (i) the temperature, field, and time of magnetic annealing; (ii) the microfabrication process (photolithography, ion-milling etc.); and (iii) the misalignment between MFC and GMR patterns.

First, a bottom-pinned SV with the structure of Ta (2)/Ru (2)/IrMn (6)/Co₅₀Fe₅₀ (2.5)/Ru (0.8)/Co (3)/Cu (3)/Co (0.5)/Ni₈₀Fe₂₀ (5)/Ru (4) (thickness in nm) was deposited on a thermally oxidized Si substrate by a DC magnetron sputtering at room temperature. The Co₅₀Fe₅₀ (2.5), Co (3), and Co (0.5)/Ni₈₀Fe₂₀ (5) layers acted as PL, RL, and FL, respectively. The film was annealed at 250 °C for 1 h under a magnetic field of 0.7 T. The unpatterned FL showed a saturation magnetic flux density (B_s) of 1.1 T, a magnetic coercivity (μ₀H_c) of 0.2 mT, and a saturation field (μ₀H_s) of 2.15 mT along the hard axis perpendicular to the pinning direction; thus, μ_r = 512. The microfabrication process of the GMR sensors with MFCs is shown in Figure 4a. First, the GMR elements with rectangular and comb shapes were patterned by a combination of photolithography and Ar ion milling. The GMR elements were patterned in such a way that their widths (i.e., W_FL and W_L) were aligned perpendicular to the pinning direction (μ₀H_pin) as indicated in Figure 4b. Before photoresist was stripped for the milling of the GMR elements, a SiO₂ insulation layer of the same thickness as the total GMR SV stack (~30 nm) was deposited, by which the surface of the sample remained flat after the photoresist was stripped. Then, another SiO₂ layer with a thickness of 100 nm was deposited on the whole surface of the substrate to electrically insulate the GMR elements from the MFCs.

Next, the MFCs with a rectangular shape were fabricated by a lift-off process using a 3 μm-thick negative photoresist (ZEON Corporation, ZPN 1150) with an undercut (readers may refer to ref. [26] for general information on the use of a negative photoresist in the lift-off process). The size of the MFCs was 2000 μm × 500 μm as shown in Figure 4b. The MFCs were sputter-deposited as a multilayer of Conetic alloy (Ni₇₇Fe₁₄Cu₅Mo₄ (at. %)) and Ta: [Ta (5 nm)/Ni₇₇Fe₁₄Cu₅Mo₄ (50 nm)]_×20/Ta (5 nm) under a magnetic field (μ₀H_depo) of 10–20 mT to induce uniaxial magnetic anisotropy. μ₀H_depo was kept parallel to μ₀H_pin (i.e., y-axis). The multilayering of the Ni₇₇Fe₁₄Cu₅Mo₄ films with the Ta intermediate layers resulted in a soft-magnetic MFC film with B_s = 0.6 T, an anisotropy field (μ₀H_k) of 0.28 mT, a hard axis μ₀H_c of 0.01 mT; thus, μ_r = 2140. Figure 4c shows the cross-sectional scanning electron microscopy (SEM) image of the MFCs with a rectangular GMR element, exhibiting tapered thickness of the MFCs, which naturally formed by the shadow of the photoresist. The tapered shape is reported to be effective in concentrating the magnetic flux toward an MR element [22,24]. ΔR/R_min measurements were carried out at room temperature by connecting the electric current (I) and voltage (V) probes, as shown in Figure 4b.

4. Results and Discussion: FEM Simulations

4.1. Rectangular FL

When B_ex is applied to the GMR sensor with MFCs, the magnetic flux density inside the FL (B_FL) is larger than B_ex due to the lower magnetic reluctance through the MFCs than that in the air. Thus, the ratio of B_FL/B_ex represents the figure-of-merit of the MFCs. Therefore, the optimal sensor design is the one which maximizes B_FL/B_ex.

In this section, we discuss the effect of the MFC–FL overlap on B_FL/B_ex. We simulated the distribution of magnetic flux in the FL for different W_FL (from 1 μm to 20 μm) while keeping the other parameters unchanged. Figure 5a shows the profiles of B_FL/B_ex along the line parallel to the x-direction and passing through the center of the FL. The maximum B_FL/B_ex was at the center of the FL (represented at x = 0) for W_FL ≤ W_G (= 5 μm). On the other hand, for W_FL > W_G, the maximum B_FL/B_ex was observed near the edges of the MFCs. In addition, for W_FL > W_G, B_FL/B_ex rapidly decreased in the region of the FL overlapping with the MFCs (i.e., at x > 2.5 μm and x < −2.5 μm). This is because the regions of FL under the MFCs are magnetically shielded by the MFCs; thus, only a small magnetic flux flows in those regions. Figure 5b shows B_FL/B_ex at the center of the FL (x = 0) for different W_FL. The inset of Figure 5b shows the B_FL/B_ex values in the absence of MFCs, which are lower and reaching toward ${\mu }_{\mathrm{r}}^{ \mathit{FL} }=500$ with an increase in W_FL. This is due to the demagnetization field (H_d) of the patterned FL, which reduces B_FL as B_FL = ${\mu }^{ \mathit{FL}}{(H}_{\mathrm{ex}}{ - H}_{\mathrm{d}})$. Here, H_ex (=B_ex/μ₀) is the external magnetic field applied to the sensor. From a practical point of view, we can tell that H_d in the patterned FL reduces the effective value of ${\mu }_{\mathrm{r}}^{ \mathit{FL}}$ from the value in the unpatterned film (in the present case, ${\mu }_{\mathrm{r}}^{ \mathit{FL}}=500$).

The value of B_FL/B_ex in Figure 5b increases as W_FL increases from 1 μm to 8 μm and reaches the saturation value of 21,900 for W_FL ≥ 8 μm. This indicates that the MFC–FL overlap enhances B_FL/B_ex in the middle of the FL because the magnetic reluctance of the sensor is reduced by the MFC–FL overlap as described schematically in Figure 2c. Figure 5c shows the B_MFC/B_ex vs. W_FL curve, where B_MFC is the average magnetic flux density in the MFCs. B_MFC/B_ex increases with the increase in W_FL because of the reduction of the overall magnetic reluctance of the system and attains to its maximum value for W_FL ≥ 8 μm.

From Figure 5a,b, we observe that the MFC–FM overlap enhances B_FL in the region of the FL under the MFC gap but reduces B_FL in the other region of the FL. Since the FL in the MFC–FL overlap region receives a much smaller B, we lose the output voltage from the GMR element in the region if the length of the MFC–FL overlap region is longer than the optimal value. To understand the gain and loss factors of the MFC–FL overlap, we introduce average magnetic flux density as described below.

The output voltage of a GMR sensor is expressed as ${V=\mathrm{d}R/\mathrm{d}B}_{\mathrm{x}}·B_{\mathrm{FL}}·I$, where ${\mathrm{d}R/\mathrm{d}B}_{\mathrm{x}}$ is the intrinsic sensitivity of the GMR element in the sensing axis (x) determined by the MR ratio and the anisotropy field in the FL, B_FL is the magnetic flux density inside the FL, and I is the bias current. ${\mathrm{d}R/\mathrm{d}B}_{\mathrm{X}}$ and the bias current density (J in A/m²) can be assumed to be uniform in rectangular GMR elements. However, B_FL is not uniform because of the magneto-static coupling between the FL and MFC and the magnetic shielding effect of the MFC, as shown in Figure 5a. In this case, the output voltage V is proportional to the average of B_FL weighted by the bias current density (J). Therefore, we define the average magnetic flux density inside FL, $\overline{B}$_FL as follows:

$${\overline{B}}_{\mathrm{FL}}=\frac{ʃʃʃ J\left(x,y,z\right)·B\left(x,y,z\right) dx dy dz }{ʃʃʃ J\left(x,y,z\right) dx dy dz }$$

(1)

where, the integrations are taken over the whole FL, and J(x,y,z) and B(x,y,z) represent the bias current density and magnetic flux density at each point of the FL, respectively, with coordinates (x, y, z). Figure 5d shows $\overline{B}$_FL/B_ex plotted for W_FL. The maximum $\overline{B}$_FL/B_ex of 19,000 was obtained at W_FL = 6 μm, indicating that the small MFC–FL overlap of only 0.5 μm on both sides of the FL is optimal for the MFC gap W_G = 5 μm. When W_FL ≥ 7 μm, $\overline{B}$_FL/B_ex rapidly decreased due to the magnetic shielding effect of the MFCs.

Therefore, for rectangular GMR elements with a uniform width W_FL, we found that the MFC–FL overlap (1) enhances B_FL because the overlapped FL region works as a guide for magnetic flux but (2) loses the output voltage if the MFC–FL overlap is large due to the magnetic shielding effect of the MFCs.

4.2. Comb-Shaped FL

Figure 6a shows the distribution of J across the comb-shaped GMR element with W_S = 5 μm, W_L = 20 μm, and L_f = L_n = 1 μm (a portion of the FL with a length of spine = 30 μm is shown). The electric current is mostly confined in the spine region of the comb-shaped GMR element. Through this effect, the spine region acts as an active GMR element, whereas the fins provide an MFC–FL overlap which helps to increase the magnetic flux flowing into the FL. Figure 6b shows $\overline{B}$_FL/B_ex vs. W_L for the comb-shaped FL with different L_f/L_n combinations of 0.1/0.1, 1/1, and 4/2 (in μm). The $\overline{B}$_FL/B_ex vs. W_FL values for the rectangular FL are also plotted for comparison. The inset of Figure 6b shows the B_FL/B_ex value at the center of the FL for different W_L in the comb-shaped FL with L_f = L_n = 1 μm. The B_FL/B_ex vs. W_L behavior is very similar to that in the rectangular FL (Figure 5b) and has the saturation value of 20,740 for W_FL ≥ 8 μm. The small reduction in the saturation value of B_FL/B_ex in the comb-shaped FL was due to the comparatively smaller area of overlap with the MFCs. However, the behavior of $\overline{B}$_FL/B_ex with an increase in W_FL (i.e., MFC–FL overlap) is different between the comb-shaped FL and the rectangular FL. As mentioned in the previous section, $\overline{B}$_FL/B_ex decreased with an increase in the MFC–FL overlap in the rectangular FLs (W_FL > 6 μm). However, the MFC–FL overlap for the comb-shaped FLs did not cause a reduction in $\overline{B}$_FL/B_ex. We observed an increase in $\overline{B}$_FL/B_ex with W_L until it attained a saturation value at W_L = 8 μm, and it retained the same value for a further increase in W_L.

The maximum value of $\overline{B}$_FL/B_ex was larger with finer comb shapes, i.e., smaller L_f and L_n values. This is because of the more effective current confinement in the spine region for the smaller L_f and L_n values. The degree of current confinement and the value of $\overline{B}$_FL/B_ex for various shapes of FL are summarized in

Table 1. Degree of confinement of the electric current, $\overline{B}$_FL/B_ex with and without MFC, and MFC gain in different FLs with a constant W_L = 15 μm and different L_f and L_n. The values for the rectangular FL with W_FL = 5 and 15 μm are also given for comparison. For simulations, the MFC gap was fixed to W_G = 5 μm.

FL Shape (in μm)	Degree of Confinement of Electric Current (Ispine/Itotal)	${\overline{\mathit{B}}}_{\mathit{F}\mathit{L}}/{\mathit{B}}_{\mathbf{e}\mathbf{x}}$ with MFC	${\overline{\mathit{B}}}_{\mathit{F}\mathit{L}}/{\mathit{B}}_{\mathbf{e}\mathbf{x}}$ without MFC	MFC Gain
Comb shape, WL = 15 Lf = Ln= 0.1	0.992	21,550	340	63
Comb shape, WL = 15 Lf = Ln = 1.0	0.952	20,200	338	60
Comb shape, WL = 15 Lf = 4, Ln = 2	0.787	19,050	359	53
Rectangular WFL = 5	(Igap/Itotal =) 1.0	16,300	218	75
Rectangular WFL = 15	(Igap/Itotal =) 0.333	9470	345	27

. For the comb-shaped FL, various L_f and L_n values with a constant W_L = 15 μm were considered. The data of the rectangular FLs with W_FL of 5 μm and 15 μm are shown for comparison. The degree of current confinement in the comb-shaped FL is defined as I_spine/I_total, where I_spine is the current in the spine region and I_total is the total current flowing in the whole FL. For the rectangular FLs, the current in the region of the FL under the MFC gap (I_gap) is used instead of I_spine. Table 1 also shows the values of $\overline{B}$_FL/B_ex without MFC and the MFC gain. Here, the MFC gain is defined as the ratio of the $\overline{B}$_FL/B_ex with MFCs to that without MFCs.

The maximum MFC gain of 75 was obtained for the rectangular FL with W_FL = 5 μm. This is because of the lowest $\overline{B}$_FL/B_ex without MFCs for this FL, which is caused by the strong demagnetization field for a narrow W_FL. Despite the largest MFC gain for this sensor, the value of $\overline{B}$_FL/B_ex, which represents the sensitivity of the sensor, was not the highest for this sensor. The value of $\overline{B}$_FL/B_ex with MFCs was larger for the comb-shaped FLs than for the rectangular FLs. In addition, for comb-shaped FLs, higher $\overline{B}$_FL/B_ex was obtained for higher I_spine/I_total. For example, I_spine/I_total of 0.992 was obtained for the comb-shaped FL with L_f = L_n = 0.1 μm, indicating that most of the electric current was confined to the spine region of the FL. For this sensor, we obtained the highest $\overline{B}$_FL/B_ex of 21,550 with a MFC gain of 63. On the other hand, for the rectangular FL with W_L = 15 μm, we obtained the lowest $\overline{B}$_FL/B_ex of 9470. The data in Table 1 indicate that a maximum MFC gain may not provide a maximum value of $\overline{B}$_FL/B_ex. This is because MFC gain represents only the change in sensitivity ($\overline{B}$_FL/B_ex) by an addition of MFCs but does not represent the absolute value of $\overline{B}$_FL/B_ex. Moreover, the sensitivity of MR sensors with the MFCs depends on factors such as magnetic reluctance, nonuniformity of the magnetic flux density, and the demagnetizing field in the MR element.

Note that the FEM simulation study presented here did not consider magnetization domains in the FLs and MFCs. See the Supplementary Materials for micromagnetic simulations for the representative rectangular and comb-shaped FLs.

5. Results and Discussion: Experiments

To validate the above findings with the FEM simulations, we fabricated the GMR SV sensors with rectangular and comb-shaped FLs in combination with MFCs. The rectangular FL had dimensions of W_FL = 5 and 36 μm and those of the comb-shaped FL were W_S = 5 μm and W_L = 36 μm. We compared the GMR ratio (ΔR/R_min) and sensitivity, defined as (dR/dH)/R, between the rectangular and comb-shaped FL sensors.

Figure 7a shows the bright field (BF) and dark field (DF) optical microscope images of these sensors. Although the simulation results showed that smaller L_f and L_g gave larger $\overline{B}$_FL/B_ex, we made a comb-shaped FL with L_f = 4 μm and L_g = 2 μm due to the resolution limit of our photolithography system. The magnetizations of the PL and RL of the SV were pinned in the $\mp$y-directions, respectively. Figure 7b,c show ΔR/R_min and the sensitivity, defined by (dR/dH)/R, of the GMR elements without the MFCs for an external magnetic field (μ₀H_ex) in the x-direction, i.e., perpendicular to the pinning direction. The inset of Figure 7b shows the ΔR/R_min curve of a rectangular SV with W_FL = 5 μm for an external magnetic field μ₀H_y in the y-direction, i.e., parallel to the pinning direction. ΔR/R_min vs. μ₀H_y shows ΔR/R_min~6.9% for a complete rotation of the FL from the +y to −y direction. Moreover, a shift of 1.1 mT in ΔR/R_min vs. μ₀H_y is observed from μ₀H_y = 0, giving ΔR/R_min = 0 at μ₀H_y = 0. This indicates that the FL magnetization is weakly stabilized in the +y-direction, probably through an orange-peel ferromagnetic coupling between RL and FL. Due to this ferromagnetic coupling, the magnetization of FL remains parallel to that of RL in the absence of an external magnetic field. Therefore, this configuration gave even-function type R-H curves in x-direction with minimum resistance at zero field. These devices with the even-function R-H response were intended for use with the AC magnetic field modulation for reduction of 1/f noise [18,27]. At μ₀H = 10 mT, we obtained ΔR/R_min~2.3%, where the FL magnetization was nearly saturated in the x-direction. The asymmetries of the ΔR/R_min and (dR/dH)/R curves with respect to μ₀H = 0 were derived from a misalignment of the direction of the magnetic field during annealing from the y-direction.

For the both comb-shaped SVs and rectangular SV with W_FL = 36 μm, the maximum value of sensitivity (dR/dH)/R without MFCs was 0.7 %/mT and 1 %/mT for negative and positive H, respectively, resulting in the average maximum sensitivity (S_max) of 0.85 %/mT. On the other hand, the rectangular SV with a smaller W_FL = 5 μm showed a S_max~0.6 %/mT. The smaller sensitivity for the smaller W_FL is due to the larger demagnetizing field in the x-direction.

Figure 7d,e show ΔR/R_min and (dR/dH)/R, respectively, of the same sensors after fabrication of the MFCs. All sensors showed μ₀H_c of 0.02 mT, which might be derived from that of the MFCs. The rectangular sensor with W_FL = 36 μm showed the lowest ΔR/R_min due to the loss of the GMR by the MFC–FL overlap, as discussed in Section 4.1, resulting in S_max~6 %/mT. On the other hand, the comb-shaped SV and the rectangular SV with W_FL = 5 μm maintained ΔR/R_min above 2.0% after fabrication of the MFCs. However, the sensitivity was higher for the comb-shaped SV: S_max~36.5 %/mT (31 %/mT and 42 %/mT for negative and positive H, respectively). On the other hand, the rectangular SV with W_FL = 5 μm exhibited S_max~28 %/mT. The S_max and MFC gains of these sensors with and without MFCs are summarized in

Table 2. Summary of the maximum sensitivity values and MFC gains of the rectangular and comb-shaped GMR sensors with and without MFCs. −S_max and +S_max indicate the maximum sensitivities measured for negative and positive H, respectively (see Figure 7).

Device	Max Sensitivity without MFC (%/mT)			Max Sensitivity with MFC (%/mT)			MFC Gain
	−Smax	+Smax	Avg. Smax	−Smax	+Smax	Avg. Smax
Rectangular WFL = 5	0.6	0.6	0.6	24	32	28	47
Rectangular WFL = 36	0.7	1	0.85	6	6	6	7
Comb shape WL = 36, WS = 5	0.7	1	0.85	31	42	36.5	43

. The MFC gains of the rectangular SV with W_FL = 5 μm and W_FL = 36 μm and the comb-shaped SV were 47, 7, and 43, respectively. The highest MFC gain for the rectangular SV with W_FL = 5 μm was a result of its lowest sensitivity (~0.6 %/mT) without the MFCs owing to a stronger demagnetization field for a smaller W_FL. Due to this, the effectiveness of the MFCs in reducing H_d was greater in the rectangular SV with W_FL = 5 μm. However, as mentioned in the previous section, the maximum MFC gain does not necessarily give the maximum sensitivity. The sensitivity was the highest in the comb-shaped SV owing to the low magnetic reluctance caused by the magneto-static coupling between the fins and MFCs as shown by the FEM simulations.

These results demonstrate that MFC gain is not a sufficient parameter to describe the effect of the MFC. For the appropriate design of MR sensors combined with MFCs, the magnetic reluctance of the whole sensor system (i.e., the MR element and MFCs) and the nonuniformity of the magnetic flux density in the MR element should be considered.

6. Conclusions

We studied the effect of the geometrical overlap between the MFCs and FL (MFC–FL overlap) of GMR sensors using FEM simulations and experiments. We observed that the MFC–FL overlap enhances the magnetic flux density in the center of the FL, i.e., the region of the FL under the MFC gap. However, the magnetic flux density in the region of the FL overlapped with the MFC decreases due to the magnetic shielding effect of the MFC, by which the MR ratio of the GMR sensor was decreased in the conventional rectangular-shaped GMR. To quantify the effect of the MFC–FL overlap on the sensitivity of the sensor, we defined the effective magnetic flux density, $\overline{B}$_FL, which incorporates the effect of nonuniformity in the magnetic flux density on the output voltage of the GMR sensor.

To overcome the reduction of output voltage in the case of the rectangular FLs overlapped with the MFCs, we propose a comb-shaped GMR sensor with alternate fins and notches. The notches confine the electric current in the spine region of the comb-shaped GMR sensor, and the fins provide the MFC–FL overlap. Using this shape, we obtained a strong magneto-static coupling between the MFC and GMR sensor without losing its ΔR/R_min. The advantage of the comb shape in comparison with the conventional rectangular GMR sensor has been demonstrated experimentally. We obtained a sensitivity of 36.5 %/mT for the comb-shaped GMR sensor as compared to 28 %/mT for the rectangular GMR sensor. This study shows that the magnetic reluctance and the nonuniformity of the magnetic flux density in MR elements should be considered when designing MR sensors with MFCs.