Advances and key technologies in magnetoresistive sensors with high thermal stabilities and low field detectivities

The magnetoresistive (MR) effect in magnetic materials was published for the first time in 1857 based on the electrical resistance changes in bulk Ni and Fe. The discovery of anisotropic magnetoresistance (AMR)¹ was followed by the discovery of anomalous Hall magnetoresistance (AHMR) in 1930.² Later, MR sensing technology was developed via the revolutionary discoveries of pseudo-Hall magnetoresistance and planar Hall magnetoresistance (PHMR) in thin films,^3,4 as well as giant magnetoresistance (GMR), and tunnel magnetoresistance (TMR) in multilayered materials.^5–8 Even though micro-magnetic sensors based on the semiconductor Hall effect dominate the market due to their low cost, newly discovered micro-GMR/TMR sensors have become attractive because of their ease of integration with electronic devices and their higher field resolutions (better than 10 nT). These magnetic field sensors have various applications in different industrial sectors. However, they dominate the market for 3D electronic compasses in modern mobile devices. In addition, recent developments in magnetic sensors and emerging technologies demand a miniaturized MR sensor with high thermal stability and low power consumption. In addition, they require a field resolution that can reach a few pT. The resulting device must be cost-effective. To this end, a recent high-resolution MR-based bio-chip demonstrated a resolution of ∼10 fM.⁹ This creates a new horizon in modern biosensing. Furthermore, recent studies on novel MR-based magnetometers¹⁰ and automotive current controllers (which require high thermal stability) strengthen its potential value in the modern magnetic sensing markets.

One particular type of MR sensor, the PHMR sensor, has recently exhibited promising potential in various applications due to its unique characteristics, such as a zero-base line, tunable sensitivity, low Hooge parameters, and ease of fabrication. Generally, PHMR has the common physical mechanism as AMR and AHMR¹¹ of spin–orbital coupling between transporting electron spins and its orbital angular momentum. However, electrode configurations for external field measurement are different. Here, planar-Hall refers to the conventional Hall electrode configuration in the presence of an in-plane field instead of out of the plane field, where Hall first discovered the Hall effect in conductors in 1879 (see Fig. 1). Thus, the electrical voltage is measured via an electrode arranged transverse to the sensing current. The effect is referred to as the planar Hall effect (PHE) and was discovered for the first time by Goldberg and Davis in 1954.¹² The researchers found that the PHE arises from the anisotropic behaviors of p- and n-type germanium crystals via a sample magnetoresistance study. Later, Ky⁴ found that, unlike the conventional Hall effect, PHE does not depend on scattering and reflection of electrons on film surfaces. Rather, it is associated with the spin–orbit interaction in ferromagnetic (FM) materials, such as Ni, Co, Fe, and NiFe.^4,13,14 For more than 30 years, subsequent developments in PHMR technology occurred slowly. This period ended in the late 20th century. Subsequently, a breakthrough in the evaluation of PHMR characteristics occurred when Schuhl et al. enhanced the field resolution of such a device down to a few nT using the same material in 1995.¹⁵ Their results make it possible to detect field variations on the order of nT at room temperature. However, the hysteretic behavior of the MR curve during a cyclic field is a major drawback to performing absolute field measurements. The hysteretic MR-curve behavior originates primarily from incoherent magnetization over the sensor element. This occurs even when the sensor element is μm-sized.

Incoherent magnetization near the edge of the sensor element is inevitable. However, coherent magnetization can be achieved across most of the sensing area via domain rotation in a single domain or effective single domain structure. Intrinsically, coherent magnetization has been revealed by introducing unidirectional anisotropy via exchange coupling between antiferromagnetic (AFM) and ferromagnetic (FM) materials. The PHMR effect was first reported in an exchange-coupled system by Nemoto et al. in 1999¹⁶ although the sensor element was not hysteresis-free due to the presence of magnetization reversal in the AFM layer under the cyclic field. Subsequently, Kim et al. observed a non-hysteretic PHMR curve in 2000.¹⁷ It exhibited good linearity and an antisymmetric profile in the bidirectional field interval that included negative and positive signs. The major advantage of the exchange-coupled sensor was customized tailoring of the field sensitivity, which could be achieved by adjusting the exchange-coupling field (H_ex) by using an ultra-thin nonmagnetic layer (spacer layer) between the FM and AFM layers. Alternatively, hysteresis-free PHMR sensor behavior can be observed in uniaxially anisotropic materials within a field range smaller than the coercive fields of FM materials (H_C). In this approach, mm-sized sensors can exhibit effective single-domain behavior using sensor-shape anisotropy after one adjusts the demagnetizing factor to account for the major and minor axes of an elliptically shaped single-layer NiFe film.¹⁸ Anisotropy can be induced by a deposition field,¹⁹ a single-crystal substrate,²⁰ or a buffer layer.^21,22

Typically, the PHMR is studied in simple cross structures and the MR effect is related to the off-diagonal components of the 2D resistance tensor. However, it has recently been observed that even in bridge-type hybrid sensor architectures using AMR and PHMR effects simultaneously, an antisymmetric MR profile can be observed in which the MR is mainly related to the off-diagonal components of the resistance tensor for flowing current. In these multi-ring^23–25 and diamond^26–28 sensor geometries, the sensor field sensitivity can be improved by a factor of 1000 (proportional to its length of the sensing element) compared to the cross-type sensor. In particular, because of the self-balancing nature of the multi-ring bridge-type sensors, which is related to its resistive arm manipulation, the temperature effect on the electrical resistance can be eliminated effectively as it cancels out the field-independent resistance component in the Wheatstone-bridge configuration. However, the field-dependent component incorporates an additive MR effect related to the four bridge arms, which eventually favors higher field sensitivity. Apart from this, all characteristics are similar to those of a cross-type element. Therefore, in this paper, hybrid-type sensors, such as ring-type or diamond-type sensors, are also classified as a type of the PHMR sensor using off-diagonal components, such as simple cross-type sensors.

In PHMR sensors, low-frequency noise spectra are primarily governed by intrinsic, extrinsic, and intermixing noise.²⁹ The major noise components of these primary noise sources are 1/f, thermal, magnetic Barkhausen, and system electronic noise. We note that the effective Hooge’s parameter of a PHMR sensor is ∼10⁻² times lower than those of other MR sensors (GMR and TMR).³⁰ This enables the device to exhibit low 1/f noise, whereas the contribution of the 1/f noise is dominant and unavoidable among GMR and TMR sensors. Moreover, as discussed earlier, intermixing noise related to the sensor offset voltage noise can be diminished to nearly zero via self-balancing, which is achieved by changing the resistances of the electrodes. Remarkably, this reduces the noise such that it is two orders of magnitude smaller than that encountered in other types of micro-MR sensors. Generally, sensor noise is inversely proportional to the square root of the sensor area.³¹ Thus, it is possible to reduce the noise by increasing the sensing area. Even when the sensor noise is low enough, the overall detectivity can be dominated by the electronic noise from the integrated chip. Thus far, sub-mm-sized PHMR sensors have exhibited a few pT field detectivities.³² 

In recent studies, PHMR sensors have exhibited several advantages over other sensors in this category. In particular, their high field sensitivities and good thermal stability characteristics have led to novel applications in robust environments and research areas, such as bio-chip detection of superparamagnetically labeled biomarkers, low-field detection down to sub-pT field resolutions, micro-magnetometers with resolutions of 100 pT, and magnetic-flux leakage detection for non-destructive testing. Moreover, they are easy to integrate with flexible modules for physico-mechanical signal monitoring of the human body, etc. The objectives of this review are to describe the PHE and revisit the operating principles of important existing and emerging PHMR sensing technologies. For each sensor type, the operating principle is described and the characteristic studies are discussed to highlight its suitability for various applications.

As shown in Fig. 2, there are many different PHMR sensor applications in development.^15–119 Parallel, fundamental studies of PHMR sensors are quite important because of the rich physics that underlie their thermal stability. Most PHMR principles have been known for decades. In order to meet the demands of future magnetic sensing technologies, there is extensive, continuous research on improving the sensitivity, detectivity, selectivity, reliability, and applicability to bio-sensing and medicine. This work occurs in parallel with research on reducing the sensor size, cost, and power consumption. In this context, worldwide PHMR research activities in various sectors can be categorized as follows: (i) magnetometers,^{26,27,32,39–41} (ii) field sensing,^{15,19,25,28,32,42,69} (iii) noise analysis,^{32,39,41,57,70,71} (iv) magnetic nanoparticle (MNP) detection,^{28,40–42,51,53,72,81} (v) biochips,^{50,79,81–84} (vi) current sensors,³⁷ (vii) flux-leakage measurements,^85,86 (viii) micro-compasses,⁵⁴ and (ix) fundamental studies,^87–114 including studies of topological insulators.^19,115 Moreover, PHMR sensors can prevail over modern technology and explore new horizons via their flexible substrates.^36,116–119 Their sensitivity enhancement characteristics, high thermal stabilities, and high output voltage levels enable PHMR sensors to be integrated easily with consumer electronic components. Furthermore, a monotonically increasing interest in R&D and escalating commercialization of PHMR sensors can be observed due to the progressive emergence of new and widespread applications.

In general, for a single-domain metallic ferromagnetic material in a completely isothermal condition with an applied electric field (⁠$E⃗$⁠), in-plane magnetic field (⁠$H⃗$⁠), and current density (⁠$J⃗$⁠), Ohm’s law can be described in rectangular coordinates using¹²⁰ the following equation:

Here, i and j denote spatial dimensions and ρ_ij, E_i, and J_j represent, respectively, the resistivity tensor, electric field, and current density vector used for 3D magnetization of the sample. Based on Onsager’s theory and the symmetry of resistivity with respect to saturated magnetization in a thin film, $ρijM⃗=ρij−M⃗$⁠. Thus, Ohm’s law can be written as

In a single-domain magnetic material, the resistivity tensor can be simplified in the direction of magnetization instead of magnetization. Here, $α⃗$ is the direction of magnetization (=$M⃗s/Ms$⁠), ρ_∥ and ρ_⊥ are the longitudinal (⁠$M⃗∥J⃗$⁠) and transverse (⁠$M⃗⊥J⃗$⁠) resistivities, Δρ represents ρ_∥ − ρ_⊥, and ρ_H is the Hall coefficient.

For a moderate film thickness, the sample magnetization and resistivity are confined in the 2D plane with θ = 90°, as shown in Fig. 3(a). In this condition, the magnetization direction can be described using an angle ϕ, and the sample resistivity tensor can be expressed in the (2 × 2) matrix form. Thus, Ohm’s law can be expressed in the following form:

It is noted that the electric field components depend on the bias current direction. The induced voltage in the in-plane electrodes is governed by the relative angles of magnetization and the current, ϕ.

The most conventional 2D planar-Hall magnetic sensor designs use the cross-type junction shown in Fig. 3(b). Here, current passes through the electrodes along the x axis and the output voltages are induced at the electrodes along both the x and y axes. For cross-type junction dimensions (l, w) in the x and y axes, the effective induced voltages can be expressed as

Here, V_x is associated with the conventional anisotropic magnetoresistance (AMR) through the diagonal resistivity component of ρ_xx and V_y is the planar Hall magnetoresistance, which is caused by the off-diagonal resistivity component of ρ_xy.

In order to examine the role of anisotropy in the sensor signal profile, the sensor junction is usually fabricated under an external deposition magnetic field (⁠$H⃗$⁠). Thus, multiple domains are involved in its magnetization. Eventually, this drives the hysteretic characteristics that result in uniaxial magnetic anisotropy. The hysteresis during magnetization induce the hysteretic output voltage response in the sensor. The prerequisite for a high-performance PHMR sensor is the presence of an effective single domain where magnetization can rotate coherently, predictably, and reversibly under $H⃗$ in the sensing layer. Thus, it is important to select proper anisotropic magnetic materials in order to achieve superior sensor performance.

Over the past three decades, several methods of achieving sustainable uniaxial anisotropy in soft FM materials have been proposed in support of the development of highly sensitive PHMR sensors. The first approach induces anisotropy in the FM material by applying $H⃗$ during thin-film growth.^80,121–124 In the second approach, the easy magnetization direction is defined using specific crystal substrates or a Fe/Pd buffer layer.^54,96 In addition to these two conventional approaches, researchers use the shape-induced anisotropy approach, in which the sensor is elliptical in shape. In this case, stable FM material anisotropy is induced by implementing a custom-designed elliptical shape.^18,125,126 Here, H_C of the FM material can be controlled by manipulating the minor and major axis lengths of the ellipse.

The use of a uniaxial ferromagnetic layer for PHMR sensing offers several advantages: (i) It is possible to fabricate a thick sensing layer, which eventually reduces the thermal noise in the sensor itself. (ii) Only one single magnetic layer is used as a sensing material through which a high sensing current flows. In addition to these advantages, this structure exhibits several drawbacks. (i) Linear behavior and good reproducibility exist only in a small field range (normally, <H_C). (ii) The magnetization tended to change the direction if the exposed strength field H is higher than H_C.

In addition, after the development of unidirectional anisotropy in the spin-valve GMR structure, several studies have discussed the development of PHMR sensor materials based on interfacial coupling between the AFM and FM layers^{35,55,121,127–131} and interlayer coupling between pinned and free FM layers in the AFM/FM/Spacer/FM spin-valve structure.^{36,59,69,132–134}

Exchange-coupled magnetic sensors offer several advantages. (i) They are easy to fabricate, and their designs are flexible enough to enable increases in sensitivity; thus, it is possible to produce a sensor output voltage in the mV range by manipulating the sensor design. (ii) They have highly stable anisotropic fields from a few mT to a few tens of mT, which eventually produces a highly stable easy magnetization axis for high measurement reproducibility. (iii) Exchange coupling can be optimized for a desired magnetic field measurement range. However, a minor drawback of the exchange-coupled structure used in PHMR sensors is that there is a small shunt current through the pinning and seed layers in the exchange-coupled structure, which can lead to wasting a small portion of the bias power applied to the sensor. Key parameters used to control exchange coupling in thin-film structures are (i) the thickness dependences of the FM and AFM layers,^{66,82,135–139} (ii) the choice of AFM and FM materials,^{17,25,109,124,130,140–143} and (iii) the role of spacer layers at the interface.^{25,64,144–150} The magnetic energy of the FM sensing layer in these unidirectionally anisotropic structures in the presence of an external magnetic field is given by

Here, the angle ϕ represents the angle between the easy axis and the film magnetization vector. The parameters M_FM and t_FM represent the saturation magnetization and thickness of the ferromagnetic layer, respectively. H_k is the anisotropy field, and β is the angle between the easy axis and the applied magnetic field direction [see Fig. 3(b)]. In this approach, a theoretical estimate of the MR voltage response can be obtained as a function of H when the other parameters such as H_k, β, and the exchange-coupling field, H_ex, are known.¹⁵¹ 

A schematic comparison of the atomic magnetic orientations and PHMR signals associated with uniaxial and dominant unidirectional anisotropy are shown in Fig. 4. Here, Figs. 4(a) and 4(b) provide schematic representations of the magnetic spin models of the FM layer and FM/AFM bilayer PHMR structures. Figures 4(c) and 4(d) exhibit the easy axis magnetization curves of the single FM layer and the FM/AFM bilayer, whereas Figs. 4(e) and 4(f) represent the measured magnetization profile of the same samples along the hard axis, respectively. It can be seen that H_ex is induced in the FM/AFM bilayer in the easy-axis magnetization profile.^152,153 Comparing the magnetization profiles of the hard axis and the easy axis, it is seen that the single FM layer demonstrates a uniaxial anisotropy and FM/AFM bi-layered structure is dominated by unidirectional anisotropy. In the FM layer, magnetization hysteresis is found for both axes’ measurements due to the incoherent magnetization reversal of the internal spins. On the other hand, a unidirectionally anisotropic FM/AFM bilayer shows no hysteresis because of the coherent magnetization reversal of internal spins associated with respect to the hard axis.

Figures 4(g) and 4(h) represent the measured PHMR profiles in the hard axis direction for the FM single-layer and FM/AFM layers, respectively. Here, the PHMR profiles exhibit a double hysteresis loop with recurrence points at magnetic fields near ±0.2 mT due to multi-domain magnetization^{82,151,154,155}. On the other hand, an antisymmetric PHMR profile is revealed in the FM/AFM bilayer structure.

The estimated coercive fields from the measured hysteresis loops are around H_c = 0.27 mT on the easy axis and H_c = 0.23 mT on the hard axis for the single-layer film. For the bilayer structures, the estimated H_ex is around 17.4 mT and H_c = 1.06 mT for the easy axis. Any signature of hysteresis or exchange bias mechanism is not observed along the hard-axis measurement [Fig. 4(f)]. Since the minimum energy condition for a single layer is H_ex = 0 and that for exchange-bias multilayers is H_ex ≠ 0, the equilibrium magnetization angle, ϕ, can be determined using Eq. (5). Here, Figs. 4(i) and 4(j) show the calculated PHMR curves V_y(H) under the H-field, which has been obtained by Eq. (4) using extracted ϕ values. It is evident from Figs. 4(i) and 4(j) that it exhibits a good agreement between measured and calculated PHMR profiles for uniaxial anisotropy of FM and the dominant unidirectional anisotropy of the FM/AFM bilayer structure. These four MR peaks in the FM layer eventually refer to the midpoint of the coercive field and the saturation points. The signatory behavior of PHMR confirms that the exchange-coupled sensors exhibit effective single-domain behavior with a wide dynamic range.⁸² Moreover, the exchange-coupled structures exhibit better operational stability, such as high linearity, near zero field offsets, and polarity for ± field regions. In addition, the exchange-coupling field affords a wider dynamic field range.

It is useful to generalize the MR voltage for any arbitrary current angle, γ, of an arbitrary geometric shape using a unidirectional anisotropy sensor junction (see Fig. 5). Taking into account the sensing element for the current angle γ from unidirectional anisotropy H_ex (x axis),^156–161 the electric field components are expressed as follows:

The easy axis magnetization angle is ϕ. The electric fields along the current and perpendicular directions, E_γ∥ and E_γ⊥, are, respectively, re-written as follows:

Here, the electrodes for the bias current and induced voltages are the same; thus, the induced voltage along the junction length is given as follows:

From this equation, it is noted that the voltage induced in the electrode configuration used for conventional AMR depends on the current and easy axis angles. First, if γ = nπ (n = 0, 1, 2, 3…), then cos γ = 1 and sin γ = 0. If γ = (2n + 1) π/2 (n = 0, 1, 2, 3…), then cos γ = 0 and sin γ = 1. In these two cases, only the diagonal component contributes to the magnetoresistive voltage. Second, if γ = (2n + 1) π/4, the diagonal component becomes a constant and only the off-diagonal component contributes to the magnetoresistive voltage. Finally, if γ ≠ nπ/4 (n = 0, 1, 2, 3…), both the diagonal and off-diagonal components contribute to the effective magnetoresistive voltage.

Figure 6 shows the contributions of the diagonal and off-diagonal components for a bar-type junction (see Fig. 5) for γ = 0, π/4, π/2, π/12, π/3, and 5π/12, respectively. The magnetic field, H, is applied perpendicular to the easy axis. The measured datasets for different γ are fitted using Eq. (9) by determining the angle φ and finding the minimum energy using Eq. (5). In each case, the measured data depict good agreement with the estimated fit lines. Figure 5 represents a typical AMR electrode configuration. However, an antisymmetric profile is shown for γ = π/4 because of the contribution of the off-diagonal resistivity component. This is the same as is noted for a typical cross-type PHMR junction. We refer to an apparent PHMR due to off-diagonal resistivity contributions. This enables the design of sensors with any shape that exhibits antisymmetric voltage profiles.

To introduce the exchange-coupling field, one must investigate the spacer layer effect. Figures 7(a) and 7(b) show the hysteresis loops and MR voltage profiles of bilayer, spin-valve, and trilayer structures. The exchange-coupling field of a bilayer structure is ∼15 mT, and the resulting antisymmetric MR profile is in the same range as the exchange field. In a spin-valve GMR structure, exchange coupling shifts the hysteresis loop of the FM sensing layer to one established direction, and an antisymmetric MR profile with a reduced MR voltage appears due to the shunt current through the spacer and the pinned layers. Trilayer structures receive more attention than spin-valve structures in the context of MR sensing materials due to their small shunt currents and their ability to control interfacial coupling by introducing a spacer layer at the FM/AFM interface. This is the so-called tailored exchange-coupled structure.^{25,64,108,145,146} Importantly, the introduction of a thin spacer layer allows the exchange-coupling field of a trilayer structure to be optimized because of its advantages of a small shunt current and high field sensitivity.

In the most general cases, the PHMR sensor is of the cross-junction form, where the junction size is between 3 and 50 μm.^141,162,163 These cross-junction sensors exhibit small output voltages and low field sensitivities and, therefore, are not sufficient for practical sensing applications. Therefore, it is important to enhance the sensitivity of the PHMR sensor and, thus, enable it to be used more broadly in industry and biomedicine. In order to address this issue, a few special designs for hybrid MR sensors have been proposed, considering the planar-Hall effect and the AMR effect in the same framework.¹⁶⁴ Relying on this model, Wheatstone-bridge structured hybrid sensors, such as ring-type^23,24,165 and diamond-type^160,167 sensor elements, were proposed to improve the sensitivity further (see Fig. 8); the first study was reported by Oh et al. in 2011. In these bridge architectures, the MR voltage exhibits a geometric dependence based on its operational parameters and can be improved further by tuning various parameters.

Later, several architectures including hybrid architectures^167,168 were proposed to investigate the role of the output signal in each arm. Ring-type bridge structures amplify the MR signal and simultaneously eliminate the sensor offset. To explain the resistance, a change in each bridge arm of these sensors, Eq. (7), can be formulated as^23,24,162

where r, w, and t_FM are the radius, width, and ferromagnetic layer thickness. We note that ∆ρ is the difference in anisotropic resistivity (ρ_∥ − ρ_⊥) and ϕ is the angle between the induced magnetization and the sensing current. In this case, n is an integer that represents the number of arms in Fig. 8, i.e., n = 1–4. This aids us in understanding the role of the sensing current (I_x) in each bridge arm, which eventually causes a significant change in the output voltage in the bridge architecture. After simplifying Eq. (10), we can express the effective MR voltage of this structure,¹⁶⁸ 

Here, the offset voltage is attributed to the unbalancing of each arm, and the output voltage is proportional to the radius, r. In the case of GMR and TMR sensors in a bridge configuration, such as the hybrid PHMR one, we get the same resistance change in all bridge arms for $ΔRnϕ$⁠. This causes the cancellation of the voltage regardless of the change in the external magnetic field. Therefore, GMR or TMR bridge sensors and additional dummy sensors must be implemented to avoid this drawback. However, in hybrid PHMR structures, the change in the output PHMR response is additive and does not require any dummy sensor.

In this regard, the hybrid PHMR sensor exhibits a self-balancing feature that can cancel out the offset component of each bridge arm and amplifies the MR component. It also exhibits promising potential for sensor sensitivity improvement. In a similar fashion, a bridge-type, diamond-shaped hybrid PHMR sensor was proposed by Østerberg et al.¹⁶⁰ This sensor maintains a constant current density in each branch arm. The MR voltage of a diamond-shaped Hybrid PHMR sensor can be obtained by replacing the radius r in Eq. (10) with (n · l).^28,160 In fact, the two architectures comply the same concept, i.e., cancellation of the sensor signal offset and output signal additivity from the branches, except their geometric coefficients.

Clearly, the MR voltage of the sensor and its sensitivity increase with the number and length of the arms. To explain the advantage of these bridge architectures more explicitly, a schematic diagram of a multi-ring hybrid PHMR sensor is shown in Fig. 9(a). In this structure, the angle changes with the current directions of two consecutive rings from ϕ to π + ϕ. As indicated in the diagram, the MR output at the ring junction varies with sin 2 ϕ for odd-numbered sensing elements and with sin 2(π + ϕ) for even-numbered elements. Therefore, the effective MR voltage contribution at the ring junction due to these two consecutive arms is additive. Thus, if we consider n ring elements in a hybrid PHMR sensor, the effective MR voltage is enhanced by a multiplication factor of n. Hence, the sensitivity increases with the number of rings.

Figure 9(b) experimentally shows how the field sensitivity of a hybrid PHMR sensor increases with the number of ring elements. The study was performed using bilayer and trilayer hybrid PHMR sensors and a sensing current of 1 mA. The sensitivity enhancement for a bilayer structure with multiple ring elements (up to 17) is ∼12 times. For the trilayer structure, the enhancement is ∼10 times. However, the sensitivity is much higher in the trilayer structure than in the bilayer structure.

In Sec. III, we discussed the advantage that the exchange-coupled structures offer with regard to improving the dynamic operational field range and sensor sensitivity. Despite this advantage, some of these exchange-coupled structures use expensive sensing materials. This hinders their usefulness in producing commercially viable magnetic sensors. In this context, most reports demonstrate simplicity in sensing material synthesis and deposition. Basic development of magnetic sensing materials for uniaxial FM single-layered and unidirectional FM multi-layered thin-film sensors is discussed in Sec. III A 1. This diverse range of sensing materials has been developed for potential use in sensor fabrication to support fundamental studies and industrial applications. Tuning H_ex allows one to enhance the sensitivity, as the sensitivity is inversely proportional to H_ex. Therefore, it is essential to select AFM, FM, and spacer materials to optimize the sensitivity and dynamic operational field range of the PHMR sensors.

The choice of AFM materials plays an important role in fabricating stable, highly sensitive exchange-coupled unidirectional PHMR sensors. From this perspective, physical properties such as the Néel temperature (T_N), high-blocking temperature (T_B), chemical stability, and antiferromagnetic states are the primary determinants of the exchange coupling strength of the device and enable the device to be used across a wide temperature range. Thus, it is essential to categorize the AFM materials based on their physical properties. In this section, we discuss the most common AFM materials, which have been used in MR sensing applications for years and are summarized alongside their physical properties in Table I. The listed materials do not offer ideal combinations of material properties such as good exchange-coupling fields, thermal stability, a high blocking temperature, thickness effects, and corrosion resistance. Of these AFM materials, NiMn is promising but requires high-temperature annealing (≥530 K) for a long time period of ∼5–10 h.^{130,142,169,170}

Recent PHMR material trends^{25,83,110,131,143,164,171–179} indicate that IrMn is the most common AFM material as it offers a good combination of coupling properties and does not require heat treatment. In its disordered fcc (ɣ) phase from ∼10 to 30 at. % Mn, T_N can be increased from ∼600–750 K.¹⁸⁰ The highest exchange coupling is achieved when 20 at. % Ir is used.^181,182 Recently, Vas’kovskiy et al.¹⁸³ showed that NiMn can exhibit exchange coupling via an auxiliary NiFe layer up to a relatively high temperature. In the case of FeMn, the structural phase generally extends to about 30–55 at. % Mn at room temperature and T_N increases from 425 to 525 K as the Mn concentration increases. To address this issue, most investigations have used alloys with 50% or more Mn to meet high T_N (and, hence, T_B) requirements.¹⁸⁴ However, high-temperature annealing is required. In contrast, PtMn reaches its maximum H_ex at 50 at. % Pt, but only after annealing (∼600 K for 3 h). This eventually produces an ordered L1₀ phase (T_N ∼ 975 K).¹⁸⁵ In some cases, it is found that the PtMn annealing process is not favorable because of its ordering phase transformation. Bulk NiO has a rhombohedral structure and is AFM below 525 K, but has a cubic structure and is paramagnetic above this temperature. NiO has the disadvantage of low H_ex values (20 mT). In addition, precise control of the partial background gas pressure is required during sputter deposition for growth to occur in the correct (111) orientation.^16,17,112

Undoubtedly, the most favorable sensing materials are Permalloy, Py (Ni_xFe_(1−x)), and its related composites because of their high permeabilities, low coercivities, and near-zero magnetostriction characteristics.¹⁹² In this material, a small change in the composition of the Py thin film produces a substantial effect on the MR coefficient and saturation magnetization, which eventually favors for sensing voltage. Other soft FM materials such as CoFe,¹⁹³ YFeO,¹⁹⁴ CoFeB,¹⁹⁵ NiCo,¹⁹⁶ NiCr,¹⁶²NiFeCr,^45,162 and NiFeMo⁴³ have been used as sensing FM materials although the choice was very much specified. Figure 10(a) shows the exchange-coupling field dependence on the thicknesses of NiFe and NiFeMo materials. It reveals that, in these two FM families, H_ex decreases with the FM material thickness (t_FM) according to the function $Hexo$/t_FM. $Hexo$ is 135 and 94 mT for NiFe and NiFeMo, respectively.

The spacer layer in an exchange-coupled senor reduces the exchange coupling strength, ΔE, at the FM/AFM interface. The exchange-coupling field decreases exponentially as the spacer layer becomes thicker. Gökemeijer et al.¹⁶³ first reported this in 1997 and modeled the relationship as follows:

where t_spacer is the spacer-layer thickness, L is the long-range coupling constant, H_ex is the effective exchange coupling field, and $Hexo$ is the exchange-coupling field of a bilayer system. Interestingly, FM/spacer/AFM interlayer coupling is in the form of a long-range exponentially decaying coupling system with non-oscillatory behavior. This is completely different from the oscillatory interlayer exchange coupling that one finds in an FM/spacer/FM system. This reduction in H_ex continues up to a critical thickness t_c, beyond which it disappears. For Ag, Au, and Cu spacer layers, the values of L are 1.73, 0.92, and 0.41 nm, respectively.¹⁶³ However, these values depend on other parameters such as the type of magnetic materials and thicknesses of the FM/AFM materials, spacer materials, and fabrication conditions. This finding has attracted enormous attention with regard to the development of PHMR sensing materials, as it indicates the possibility of controlling the linear response range and sensitivity of the sensor. Later, Hung et al.³⁵ proposed a NiFe/Cu/IrMn sensor stack that can enhance the sensitivity of a PHMR sensor. The trilayer structure sensitivity was one order of magnitude higher than that of its bilayer counterpart, NiFe/IrMn. To develop other trilayer sensors, various spacer materials, such as Ta,45^,197 Au,^45,46 Pt,²⁵ and Ag,⁴⁵ have also been proposed.

A typical experimental result is shown in Fig. 10(b). Here, two fabrication conditions produce two different sets of $Hexo$ and L, even with the same NiFe/Cu/IrMn structure, which are $Hexo$ = 12.5 mT, L = 0.058 nm³⁵ and $Hexo$ = 12.5 mT, L = 0.325 nm,⁴³ respectively.

PHMR sensors exhibit few unique features compared to other MR sensors, such as small temperature coefficients, high thermal stabilities, minimal offset voltages, and ultra-low field limits of detection (LODs), which makes it more advantageous in the applied field. In this section, we review these specific features, which are commercially attractive, along with other general advantages that PHMR sensors offer and compare them to other magnetic-field sensors.

For accurate field measurements in a robust environment, the PHMR signal must not be sensitive to the temperature. Here, the thermo-parameters that obstruct the field value can be classified as follows:

Here, α_MR, α_sensitivity, and α_drift are the MR signal temperature coefficient, field sensitivity, and offset voltage, respectively.

There has been recent interest in demonstrating improved thermal stability and low-temperature coefficients, particularly with regard to the recent emergence of bridge PHMR sensors. We have already shown that the bridge configuration exhibits a fascinating self-balancing feature, as it is possible to eliminate or reduce the sensor-offset voltage.²⁹ Despite this advantage, it is not possible to eliminate fabrication-driven imperfections that result in nominal changes in the baseline drift of the PHMR signal. We point here that the base resistances of AMR, GMR, and TMR sensors are 10–100 times larger than their MR signals. Thus, the changes in their resistances are significant. This causes a substantial drift in the output signal and large temperature coefficients.

Thus, in 1995, Schuhl et al.¹⁵ were the first to measure the temperature dependence of a cross PHMR sensor. They reported a value of α_MR ∼ 0.5 × 10⁻² % K⁻¹, which is much lower than the standard longitudinal magnetoresistivity (AMR) (α_MR = 50% K⁻¹). Recently, Jeon et al.^44,198 also investigated the thermal drift among PHMR sensors for three different geometries: the cross, the balanced ring, and the unbalanced ring, between 298 and 363 K. In addition, they also studied the thermal dependence of these architectures in the AMR sensing configuration (see Fig. 11).

The temperature coefficients of the offset voltage and the voltage signal are presented for both configurations: α_drift = −9.8 × 10⁻³ % K⁻¹ and α_MR = −2.1 × 10⁻¹ % K⁻¹, respectively. These reported values are much lower than those of other MR sensors and can be improved to less than 10⁻³ % K⁻¹. The thermal drift among bridge PHMR sensors and AMR devices is compared in Fig. 11.

The thermo-sensitivity coefficient affects the accuracy of the transfer function that converts a voltage into the magnetic field of a sensor. The drift voltage largely affects the DC field measurement; however, it can be filtered from an AC field measurement. Thermo-sensitivity cannot be filtered out using AC measurement technologies. Therefore, a PHMR sensor with a sensitivity that is highly thermally stable is desirable.

In order to investigate PHMR thermal stability performance, researchers studied the temperature dependence of the sensor sensitivity, which is an intrinsic property of the sensor. In 2009, Damsgaard et al. investigated the temperature dependence of a bilayer exchange-coupled PHMR sensor between 263 and 343 K⁷⁰ and found that the temperature coefficient of the sensitivity α_sensitivity was ∼0.32% K⁻¹. Thus, the sensitivity varies substantially with the temperature. In order to minimize this variation, the authors proposed a real time experimental technique that uses an additional sensor as a reference. They demonstrated that the thermal sensitivity of the PHMR sensor can be reduced by a factor of 40 using a differential method. In another work, Li et al. proposed another approach to increase the thermal stability of the sensor sensitivity by using an ultra-thin layer of Au as a nonmagnetic layer between the ferromagnetic and the antiferromagnetic layers. The temperature coefficient of the sensitivity corresponding to this structure is 0.25% K⁻¹ in the temperature range from 253 to 333 K.¹⁰⁹ Mahfoud et al. achieved high thermal stability of the sensor sensitivity by controlling the interplay between anisotropic, Zeeman, and exchange bias energies.⁴⁸ Their thermal stability results for a wide temperature range from 110 to 390 K are shown in Figs. 12(a) and 12(b), respectively. It was observed that (i) in the field range of −2 to +2 mT, S increases with the temperature; (ii) outside of this field interval, S decreases with the temperature; and (iii) at a bias field of ±2 mT, S is independent of the temperature. At the optimized condition, the temperature coefficient of the sensitivity is ∼0.02% K⁻¹.⁴⁸ A high-temperature dependence study of a cross PHMR sensor also revealed that major physical properties such as the exchange coupling, coercive field, and anisotropic resistivity decrease as the temperature increases. This drives high thermal stability with respect to the sensor sensitivity.

In order to support a comparative study of the different MR sensors, Table II summarizes the temperature coefficient of the MR output voltage (∼% K⁻¹), the temperature coefficient of the sensitivity (% K⁻¹), and temperature coefficient of the offset voltage α_drift. It is clear from Table II that the temperature coefficient of the PHMR sensor sensitivity is one order magnitude smaller than those of AMR, GMR, and Hall sensors and comparable to the TMR sensor. The baseline drift relative to the sensor signal variation is important for a precise determination of the field variation. Thus, we tabulate the baseline drift characteristics of all of the magnetic sensors in Table I and found that the PHMR sensor exhibits a better thermal drift than the other magnetic sensors.

Low-frequency noise measurements and their decomposition always play important roles in sensor characterization in various environments. The total noise power spectral densities of PHMR sensors are composed of three primary noise contribution sources: (i) intrinsic noise, (ii) external noise, and (iii) intermixing noise components. A detailed discussion in this regard was reported recently by Jeon et al. for three different PHMR sensor structures.¹⁹⁸ In brief, intrinsic sensor noise generally includes contributions from 1/f-noise, white noise, and magnetic Barkhausen noise (⁠$MBN$ sources. In contrast, external noise comprises the system noise and environmental noise. In addition, the sensor exhibits offset noise, which is intrinsic in nature and is the primary contributor to intermixing noise. Therefore, the total noise of a PHMR sensor (⁠$Vnoisetotal$ can be modeled as^29,198

with

where V_int, V_ext, and V_mix represent the contributions of intrinsic, extrinsic, and intermixing noise, respectively. The first part of the sensor noise expression refers to the 1/f-noise contribution, which can have both electrical and magnetic origins, as demonstrated by Hooge in 1969.^208,209 It originates from energy fluctuations around equilibrium. Here, δ_H is the Hooge constant,^210,211n_C is the “free” electron density and is equal to 1.7 × 10²⁹ (1/m³) for Ni₈₀Fe₂₀ Py, f is the frequency, α is a parametric constant (∼1), and Vol is the effective volume within which the electrons contribute to conduction in a homogeneous sample.^125,212,213 In highly sensitive PHMR sensors, δ_H reduces to ≤0.01. The second term in Eq. (15) is the white noise component (∼4 k_B T R_yy), which originates from random thermal motion of electrons.²¹⁴ Here, the Boltzmann constant k_B = 1.3806 × 10⁻²³ J K⁻¹, T is the temperature, and R_yy is the y-terminal resistance for the current direction in the x axis. The remainder of the noise component is MBN, which appears due to domain wall (DW) movements and generally varies with the sensor magnetization in proportion to $∂M∂H$^31,198 (see Sec. IV B 2 for details).

$Vnoise2env$ is the environmental noise, which originates due to the harsh environmental condition. It includes electromagnetic noise, vibration, and thermal noise and can be reduced using several noise-rejection techniques. Examples include shields and adjustable electronic circuits.^215,216 The other type of noise, which is referred to as system or amplifier noise, mainly originates from the preamplifier and current and voltage sources. In a recent report, Grosz et al. found that the thermal and preamplifier noise components can degrade the sensor optimized equivalent magnetic noise by 40% at 0.1 Hz, 300% at 1 Hz, and >1000% at 10 Hz.⁷¹ Furthermore, they designed a transformer-matched operational amplifier (TMA) to address this issue and successfully achieved <200 pV/√Hz voltage-noise at 10 Hz.

Intermixing noise is correlated primarily with system voltage offsets. A recent report showed that the offset voltage noise can be minimized using a specially designed resistance compensator. Here, δI_n is the Nyquist noise of the operating current from the power source. Note that this specially designed sensor structure acts as a self-balancing bridge compensator and improves the sensor detectivity by at least one order of magnitude.²⁹ The noise modeling demonstrated above reveals that the total spectral noise density (TSND) of the PHMR sensor is highly dependent on the sensor shape, size, and magnetic material.⁷¹ 

Recent PHMR sensor data¹⁹⁸ show that thermal noise dominates other noise sources at high frequencies (∼100 Hz). At room temperature, the thermal noise contributes 79.1% of the overall white noise, but the least dominant noise source, intermixing noise, comprises only 3.5% of the total white noise. At low frequencies (below 50 Hz), the 1/f noise starts to dominate the white noise component.

In order to demonstrate the reference low-noise power spectral density of an MR sensor, the voltage noise levels of a standard hybrid PHMR sensor are shown in Fig. 13(a). The noise measurement was performed using a standard DC-excitation current method via a customized low-noise amplifier (LNA) without any shielding in the constant voltage (CV) mode. The bias voltage was varied between 2.3 and 16.2 V using a Keithley 2400 sourcemeter. The overall noise is dominated by the 1/f noise at frequencies <10 Hz but is dominated by white noise above 100 Hz. A similar trend is observed for all bias voltages, although the amplitudes of the average noise density levels are different. The measured amplitude of the white noise level is estimated to be ≤7 × 10⁻⁹ V/√Hz for 2.3 V biasing. At the highest bias voltage (∼16.2 V), the average noise level appears to be ∼30 × 10⁻⁹ V/√Hz at 100 Hz. The average noise level (V_noise) and sensor sensitivity (S) increase linearly with the bias voltage, although the observed enhancement in sensitivity (more than six times) is larger than the increase in the total voltage noise (three times) from their corresponding base values [see Figs. 13(b) and 13(c), respectively].

As discussed above, the magnetic noise of the sensor is related to the stochastic behavior of the magnetization fluctuation due to thermal agitation and external noise.²¹⁷ Typically, this is accompanied by irreversible magnetic domain wall (DW) movement.^217–219 Recently, the manifestation of MBN was analyzed by observing magnetic domains applied to parallel and transverse fields using Magneto-Optical Kerr Microscopy (MOKE).¹⁹⁸ 

Figure 14 shows how the magnetic domains and magnetization reversal loop evolve in bilayer structures. When the external magnetic field is applied along the easy axis (black curve in Fig. 14), a square hysteresis loop is observed. After saturating the sample with a negative field (state 1 in Fig. 14), application of a small reversal field results in the nucleation of magnetic domains with opposite magnetization (state 2). This domain nucleation is followed by 180° magnetization domain wall motion as the magnetic field increases until the entire area is switched (states 3–5). Such behavior suggests the existence of MBN even in exchange coupled PHMR sensors (along the easy axis). In most cases, the contribution of the MBN to the total sensor noise in the magnetic sensor varies between ∼$pVHz$ and a few $nVHz$ depending on the sensor structures and the sensor mechanism.²¹³ 

On the other hand, when the external magnetic field is applied along the hard axis (red curve in Fig. 14), coherent magnetization rotation is observed instead of domain nucleation and domain wall movement. The image gray level gradually changes during the field sweep from negative to positive saturation states (states 6–7). Thus, no MBN is expected in the sensor signal. A similar behavior has been observed in trilayer structures, i.e., coherent rotation of magnetization is revealed when the field is applied along the hard-magnetic axis. Consequently, the coherent magnetization MBN noise value of ∼$15nVHz$ is 1/10 times smaller than incoherent magnetization ∼$215nVHz$ at 1 Hz. The MBN contributions in the hybrid PHMR sensors in both parallel (incoherent) and transverse (coherent) directions to the field are shown in Fig. 15.

The Hooge constant, δ_H, is generally used as a comparative reference parameter to denote the 1/f-noise of different magnetic sensors. Its order of magnitude in a standard metallic system is usually 10⁻²–10⁻³ although its value can be much higher in the presence of more magnetic fluctuations in the system.²²⁰ In addition to δ_H, other factors such as the sensing volume, n_C, and $Voffset2$ significantly contribute to the low-frequency 1/f-noise components of PHMR sensors, as indicated in Eq. (15). The collective results of these parameters eventually determine the nature of the effective low-frequency noise contributions in MR sensors. For a specific sensor, it is straightforward to introduce the effective Hooge’s parameter, $(δH)eff$⁠, to compare the low-frequency characteristics of the 1/f-noise. The effective Hooge’s parameter can be expressed as $(δH)eff=Voffset2⋅δHnC⋅Vol$⁠.

Usually, in AMR, GMR, and TMR sensors, the base voltage is at least a few times larger (or higher by a few percentages) than their corresponding MR signal and is inevitable. Thus, in the low-frequency regime below 100 Hz, $(δH)eff$ is usually quite high in TMR and GMR sensors themselves. For PHMR sensors, the obtained baseline voltage is adjusted and exhibits a low value. Because of this fact, PHMR sensors are more advantageous than TMR and GMR sensors below 100 Hz. To demonstrate the dominance of the 1/f noise in the low-frequency regime, recent results from four different MR sensors are summarized in Table III.

It is evident from Table III that the low-frequency noise amplitudes are much larger in TMR and GMR sensors than in PHMR sensors. PHMR sensors exhibit low noise levels and demonstrate a moderate noise level enhancement of <61% in the 1–100 Hz frequency range. Figure 16 compares low-noise characteristics $(δH)eff$ by assuming just 1/f dependence. Based on the reported values, $(δH)eff$ of a PHMR sensor is ≤5. For GMR and TMR sensors, $(δH)eff$ is ≥100. All other noise contributions that originate from thermal, external, and intermixing noise sources are observed within the range of ∼6 nV/√Hz.

As discussed in Sec. IV B 3, the rings or arms in a sensor enhance the field sensitivity substantially. For bridge- and ring-type sensors, the sensitivity also strongly depends on the FM thickness, t_FM.^28,50,72 Other factors such as the selection of thin film materials^{41,64,68,97,222} and sensor fabrication process^137,164 have substantial roles in determining the sensor sensitivity. Thus, soft ferromagnetic materials with low-saturation fields are always preferred for sensitivity improvement.²²³ Furthermore, magnetic flux concentrators (MFCs) can be integrated on the sensor device to enhance the sensor sensitivity.^{32,52,224,225}

A timeline of the sensitivity performance development among PHMR sensors is shown in Fig. 17. As shown in Fig. 17, extensive progress has been made in the development of high-sensitivity in PHMR sensors since 2004. In Fig. 17, we consider only sensors with field sensitivities >1000 V/(TA). Figure 17 reveals that most of the sensors that exhibit the highest sensitivities are hybrid PHMR sensors, as reported by Oh et al. (S = 27 900 V/[TA]),³⁶ Kamara et al. (S = 15 000 V/[TA]),⁴¹ Hung et al. (S = 12 000 V/[TA]),¹⁶⁶ and Qejvanaj et al. (S = 11 000 V/[TA]).⁴⁹ For cross PHMR sensors, the best reported sensitivities are 1400 V/[TA],¹¹⁰ 1260 V/[TA],²²² and 1200 V/[TA].⁹⁷ Multi-fold enhancements in field sensitivity eventually improve the magnetic moment detection limit from 10⁻⁸ Am²⁷⁹ to 3.1 × 10⁻⁹ Am².²²⁶ 

The detectivity (D) or equivalent magnetic noise (EMN) of the sensor is defined as^26,39,71,198

where $Vnoisetotal$ is estimated from the measured spectrum and S is derived from the slope of the PHMR voltage vs μ₀H_y plot, as discussed above. Equation (18) shows that the detectivity can be improved in two ways: (i) by increasing the sensor sensitivity and (ii) by reducing the total noise level of the sensor. It is also possible to predict the achievable detectivity of the sensor based on its sensitivity enhancement and reduction of the total effective noise level. Figure 18(a) shows S–V_noise correlations for three different EMN levels: 10 nT/√Hz, 1 nT/√Hz, and 100 pT/√Hz. Given the same sensitivity, different noise levels can be reached by tuning the TSND of the system. For example, the 1 V/T sensitivity (depicted by a blue dashed line) intersects the three depicted EMN levels 100 pT/√Hz, 1 nT/√Hz, and 10 nT/√Hz. This demonstrates its importance with regard to sensor device TSND reduction.

Note that addition and integration/implementation of flux concentrators into the close vicinity of the sensing area can also improve the sensor detectivity by converging more magnetic field lines inside the sensing area.³² MFCs increase the overall sensitivity of the sensor by an additional gain factor, G, due to field line concentrations. Thus, with MFCs, the effective S can be expressed as S_eff = S · G. Recent studies estimate that the development of high-gain (>1000) MFCs can reduce the PHMR sensor detectivity level to the sub-pT/√Hz level at ∼100 Hz⁵⁶ and room temperature. In addition, a hard-axis bias field^228,229 or orthogonal soft-pinning of the sensing layer^230,231 can stabilize magnetization of the sensing layer for exchange-coupled MR sensors. In an unshielded environment, it is quite difficult to achieve ultra-low detection limits (sub-pT or fT level) in the low-frequency range due to the presence of background-field noises.^232–237 External background field noise always affects the low-field detectivity and eventually reduces sensor functionality at low frequencies. In this context, Silva et al.²²¹ and Jeon et al.¹⁹⁸ recently demonstrated sub-nT detectivity in MR sensors in the low-frequency range in unshielded environments. The sub-nT/√Hz detectivity achieved in an unshielded environment at room temperature is the most promising thus far for biomagnetic and biomedical applications.

Most of the significant improvements in the environmental noise have been achieved using magnetic shielding to reduce the overall improvement in noise measurements. A three-layer cylindrical magnetic shielding apparatus can increase the total shielding factor²³⁸ by a factor of 10⁷.⁵⁶ Unremitting efforts have led researchers to achieve detection limits of a few pT or less than one pT at ≤ 100 Hz^32,56,57,138 using elliptical or bridge (meander) structured sensors. In this context, historical development of the D values of various state-of-the-art PHMR sensors is summarized in Table IV. In Table IV, the low-frequency noise performance characteristics of PHMR sensors are summarized to compare the sensor structures and sensing areas. The best detectivity thus far has been achieved by integrating MFCs into ellipse-shaped cross PHMR sensors. The corresponding detectivity values are ∼7.4 pT/√Hz at 1 Hz, 5 pT/√Hz at 10 Hz, and 100 pT/√Hz at 0.01 Hz, as demonstrated by Nhalil et al.³² These values were measured inside a multi-layer shield environment. Das et al.⁵⁶ extended this study by reducing the measurement bandwidths (BWs) using a similar MFC integrated cross PHMR sensor and achieved sensor resolutions (lowest critical field limits, B_min) of 700 fT, 910 fT, and ∼1 pT at 10, 1, and 0.1 Hz, respectively, using 0.0025 Hz BW at room temperature. These reported field resolutions conclusively indicate significant improvement in the detectivities of PHMR sensors relative to previous reports available in the nT range^{15,55,116,131} and create new opportunities for the use of PHMR sensors at room temperature in applications that require the detection of pT or sub-pT scale magnetic fields. Thus, a typical example of the relationship between the sensor detectivity and the bias voltage for a trilayered hybrid PHMR sensor is shown in Fig. 18(b). It is evident from Fig. 18(b) that the sensor detectivity enhanced with the increase of bias voltage.

As discussed in Sec. IV C 2, the benefits of PHMR magnetic sensors include high thermal stability, high sensitivity, and linear responses to low-magnetic fields. Therefore, PHMR sensors can be deployed in many applications, including those in harsh environments and those that require high-accuracy measurements. They can detect small magnetic fields in both near- and far-field configurations. These PHMR sensor applications are summarized in Sec. V A.

An important prerequisite to robotic and biomedical applications is the development of flexible magnetic sensors that have high sensitivities and good mechanical performance levels. PHMR sensors based on flexible substrates have recently attracted extensive attention due to their high sensitivities and their good mechanical stability characteristics. In 2013, Oh et al. were the first to demonstrate a flexible PHMR sensor based on a spin valve structure. The sensor was grown on a polyethylene naphthalate (PEN) substrate.⁷³ The flexible PHMR sensor exhibits a sensitivity of 0.095 V/[TA] and is able to detect magnetospirillum magneticum AMB-1 bacteria at a low concentration of 1.3 × 10⁸ cells/ml. The same group studied the effect of bending on the sensitivity of a planar Hall ring sensor bilayer structure grown on a similar flexible substrate.¹³³ The sensor sensitivity decreases from 418.5 V/[TA] to 314.8 V/[TA] when the bending angle changes from 0 to π/4.

Moreover, Ozer et al. proposed a shapeable PHMR sensor grown on a Kapton/polydimethylsiloxane (PDMS) substrate with a sensitivity of 7.4 V/[TA].¹¹⁷ The sensor shows a wide range of linear responses of ±14 mT. Granell et al. also demonstrated a single Py layered PHMR sensor with a 20 nT LOD that was grown on polyethylenterephthalate (PET)¹¹⁶ (Fig. 19). Despite its narrow range of linear responses, it exhibits nominal mechanical deformation of only up to 0.3% even after more than 150 cycles of bending between bending radii of 4 and 2.4 mm. In this context, Kim et al.¹¹⁹ also demonstrated that sensors grown on polydimethylsiloxane (PDMS) and parylene C polymer exhibit high sensitivities, linear responses at under weak magnetic fields, and small hystereses. Detailed descriptions of these PHMR sensors, which are grown on flexible substrates, are provided in Table V.

PHMR sensors have been integrated with consumer electronics to perform various tasks. For instance, integration at the printed circuit board (PCB) level is used to detect magnetic flux leakage in pipelines and integration with an application-specific integrated circuit (ASIC) is used for high-accuracy SoC current sensors.

Detection of pipeline leakage and defects requires good temperature stability at harsh temperatures and measurement stability in a wide range of magnetic fields. From this perspective, PHMR sensors exhibit high thermal stability during extended operation. These characteristics make them effective for this type of application. Therefore, Pham et al.⁸⁵ proposed a highly sensitive magnetic flux leakage detection device that uses a PHMR sensor. They used a trilayer exchange coupling hybrid PHMR sensor based on NiFe/Cu/IrMn layers to investigate the effects of the defect depth in the sample and the distance between the sensor and the test sample. Successful demonstration of this work suggests the ability of these sensors to detect shallow defects as small as a few percent of the wall thickness of a pipe. Such defects can be located at various positions on the pipe wall, such as the outer surface, the inner surface, and inside the pipe wall.

High-precision current sensors are needed urgently for many automotive applications in this modern era of smart grids and smart cities.^239–246 Thus, high-accuracy shunt resistors,^{240,242–250} closed-loop ferromagnetic transducers, and optical current sensing methods^251–253 have been developed. These sensors have current measurement precision limitations due to their voltage and size limitations. They also experience signal distortion as temperatures rise. In contrast, open-type magnetic sensors measure the magnetic field produced by the current directly, are miniaturized, and are inexpensive. In this method, the measured current range can be adjusted via distance adjustments. However, magnetic sensors are susceptible to interference from external magnetic fields or nearby currents^254,255 and are sensitive to temperature changes. Because of this, it is difficult to measure small sensing currents.^256–259 

From this perspective, PHMR sensors can be a good choice as they offer high sensitivity and high temperature stability and thus enable high-accuracy current detection. The first use of a PHMR sensor as an electrical current sensor was introduced by Kim et al. in 2014.²⁶⁰ The investigation studied the performance characteristics of PHMR current sensors using two sensor structures: bilayer Ta(3 nm)/NiFe(10 nm)/IrMn(10 nm)/Ta(3 nm) and trilayer Ta(3 nm)/NiFe(10 nm)/Cu(1.2 nm)/IrMn(10 nm)/Ta(3 nm). Using these structures, the smallest current amplitudes that could be measured were 0.51 μA and 54 nA for the bilayer and the trilayer structures, respectively.

In order to develop high-accuracy current sensors at the chip level, PHMR sensors have been integrated into readout integrated circuits (ROICs) in the differential configuration. The differential PHMR configuration used for current sensors proposed in this work has three advantages: (i) removal of magnetic fields from the earth field and external field sources, (ii) doubling of the sensitivity via a current pad design that allows the current to pass the two sensors in opposite directions such that the differentials of the two opposite signals are added, and (iii) detection of small signals by reducing the distance between the sensor and the current line.

The multi-chip structure of the designed current sensor is shown in Fig. 20. The developed SoC current sensor has two identical signal processing blocks that consist of a low-noise amplifier (LNA), a low-pass filter, and a 12-bit analog to digital converter (ADC). A maximum voltage gain of 30 dB is reported, and the simulated input referred noise is 24 nV/√Hz. The current path is placed on the bottom side of the signal processing unit, as shown in Fig. 20(a), and the PHMR sensor is deposited directly on the top side of the signal processing unit. The final size of the PHMR single-chip current sensor is 4 × 4 mm², as shown in Fig. 20(b). The developed SoC solution is designed to measure 1 A of current with an accuracy of 0.5%.

The PHMR magnetic sensor, which is characterized by its small mass, small size, high sensitivity, and outstanding temperature stability, is often used to detect ultra-small magnetic fields, such as biomagnetic fields. Generally, when magnetically labeled biomarkers are hybridized on the surface of a magnetic biosensor, the sensor output voltage changes by an amount ΔV. This voltage change is proportional to the sensitivity (S) and the magnetizing field of the particles⁸⁴ and can be expressed as

Here, ΔV is a function of the sensor’s sensitivity S, N denotes the number of particles, v denotes the bead volume, χ is the susceptibility of the particles, z is the normal distance between the center of the magnetic bead and the sensor surface, k is the active coefficient depending on the fraction of the sensor area influenced by the bead area, and H is the applied magnetic field. These parameters can be optimized to enhance ΔV. k is a constant that depends on the sensor geometry and particles. ΔV varies in proportion with k. For this reason, understanding the interactions between the magnetic particles and the sensor surface in various geometries is important.

In a homogeneous applied magnetic field, the stray field produced by a spherical bead and observed in position $r⃗$ = (x, y, 0) on the sensor can be estimated using the following dipole approximation:

where $M⃗$ is the magnetization vector, R is the bead radius, and $r⃗0$ = (0, 0, z) is the position vector of the bead, as shown in Fig. 21(a). When a homogeneous external magnetic field is applied along the x axis (perpendicular to the easy axis and the applied current), only the H_stray−x component needs to be considered.

To understand the effect of the sensor architecture on the stray field produced by magnetic particles, the H_stray fields produced by individual beads located at the centers of various geometrical shapes can be calculated. To investigate the geometrical shape effects, five different geometrical shapes were considered: rectangles, disks, squares, ellipses, and rings. Even though they are different, if these architectures possess the same sensor area (A), the average stray field created in these sensors can be calculated via the following common relation:

Importantly, the average stray field [see Eq. (21)] does not have any dependence on geometrical shapes. The variation in the average magnetic field as a function of the area ratio ρ between the magnetic particles (ρ = 2R/A) is calculated and shown in Fig. 21(b). In these calculations, the distance between the particle and the sensor surface is assumed small compared to the dimension of the sensor, which means that we neglected the passivation layer of the sensor. Figure 21(b) shows that when ρ is small, the ring sensor produces the largest 〈H_stray〉_(0,0,z). When the area ratio is high, the average magnetic fields created on the rectangular, elliptical, square, and disk sensors are larger than that created on the ring sensor. These theoretical studies indicate that the ring sensor is among the best choices for the detection of small, sub-micron, and nanometer sized particles and, therefore, for biomagnetic applications, such as the detection of magnetic particles used in labeling technology. Moreover, other sensor shapes, such as rectangles, ellipses, squares, and disks, might be effective for the detection of weak magnetic objects, such as nanoparticles, bioentities (e.g., proteins), cells, and tissues at the μm scale.

In the following, we discuss the use of the PHMR sensor to biochip applications using nanoparticle labels.^261,262 Assuming that the physical parameters of the sensor and particles do not change, the interplay between the sensor sensitivity, S, and the stray field from the particles, H_stray, as a function of the magnetic field can enable detection optimization. For example, in Fig. 22(c), the change in output voltage, ΔV, shows several extrema in one field direction, for example, at the positive pole, the peaks are observed at H₁ and H₂ where the particle detection is maximized at these two magnetic fields.

An example of selection of a magnetic field that is optimized for the detection of magnetic labels using a PHMR biosensor is demonstrated in Fig. 22.²⁶² Figure 22(c) shows the voltage change of a biosensor when it detects a magnetic particle as a function of the magnetic field. As revealed in Eq. (19), this voltage change is a product of the field sensitivity of the PHMR sensor [Fig. 22(a)] and the demagnetizing field of the particle [Fig. 22(b)]. Figure 22(c) clearly shows that the field is optimized for the detection of magnetic particles at H₁ and H₂. Interestingly, this calculation method can be applied to all types of MR sensors. Using the simple DC measurement technique, several works have reported the ability to detect single magnetic beads or ensembles of magnetic nanoparticles.^84,174 Due to the simple measurement techniques and high sensitivity, which are essential requirements to detect the magnetic labels, this DC field technique could be useful in many applications where the detection of large and medium magnetic label volumes is required.

A new prototype PHMR sensor has been implemented to create a superconducting quantum interference device (SQUID) magnetometer for the indirect detection of room-temperature switching in an spin crossover (SCO) nanoparticle. An ensemble of [Fe(hptrz)₃](OTs)₂ nanoparticles with a volume of ∼3 × 10⁻³ mm³ gives rise to a voltage signal of about 5.5 μV [0.55 mV after amplification, as depicted in Fig. 23(b)]. In this study, in order to obtain the selective field component for the detecting object, a magnetic field generated from a sensor junction is used to magnetize the magnetic labels.^134,257,258 Thus, the magnetic biochip does not require an external magnetic field. Assuming that the current I is driven to the sensor at the frequency ω, the self-field generated from the sensor junction is given by

where c is a constant that depends on the geometry, size, and self-field strength of the sensor junction. The voltage change in the sensor is given by

Here, N is the number of magnetic labels that contribute to the sensor signal change, V is the volume of the magnetic labels, and t is the time. Simplification of Eqs. (22) and (23) shows that if a current is driven at a frequency of ω, the biosensor voltage change caused by the nanoparticles should be measured at double the frequency (2ω). Several high-sensitivity PHMR sensor magnetic nanoparticle measurements were validated based on this concept.^74,171 Measurements performed to detect spin switching of spin-crossover nanoparticles were validated as well.^166,263 Furthermore, several sensor stack²⁶¹ and sensor architecture¹⁵ developments targeted toward self-field detection were proposed.

This self-field method was used to detect spin switching of spin-crossover nanoparticles.^166,263 This work serves as a proof of concept for this novel micro-magnetometry approach. The substantial and significant benefits of this approach over conventional SQUID (significantly increased sensitivity for detection of small quantities of nanoparticles) and nano-SQUID (room temperature operation) techniques are highlighted.

Various magnetic sensors have been developed to measure the AC susceptibilities, velocities, sizes, and flow rates of magnetic entities, including via low magnetic moment detection.^{171,261,264–273} Researchers typically prefer to measure the M(H) magnetization behavior in order to evaluate the magnetic state and other magnetization parameters, such as the saturation field, saturation magnetization, and magnetization volume. From this perspective, the vibrating sample magnetometer (VSM) and the SQUID magnetometer both exhibit excellent sensitivity in the order of 10⁻⁹–10⁻¹¹ Am². In these measurement techniques, both magnetometers are used to measure the dipole fields of the magnetic samples. However, since the dipole field of the magnetic moment has a 1/d³ (d: distance) dependence, VSM and SQUID cannot measure on-chip samples with magnetic moments smaller than 10⁻¹³ Am² or volumes of tens of pL.^273,274

An on-chip magnetometer can measure the magnetic properties of samples with volumes of tens of pL if the distance “d” is reduced to a few μm or less. In particular, PHMR sensors are known for their high magnetic field sensitivities, high signal-to-noise ratios (SNRs), and low offset voltages. Therefore, PHMR sensors provide the performance necessary for the development of high-sensitivity, room-temperature magnetometers.

A PHMR sensor-integrated microfluidic-channel magnetometer may be among the best choices for the measurement of the small magnetic moments associated with small volumes of any magnetic entities. It enables high-throughput measurement of sample magnetization characteristics²⁷⁵ by measuring liquid sample volumes dynamically. Kim et al. fabricated an “on-chip” ferrofluid droplet (35 pL) magnetometer with an integrated PHMR sensor [Fig. 24(a)] and measured the magnetization of superparamagnetic fluid droplets in dynamic conditions in oscillation and flow modes.⁴⁰ The on-chip magnetometer is fabricated by integrating a PHMR sensor with a microfluidic channel. The in-plane field sensitivity of the PHMR sensor is 95, 91, 85, 75, and 66 mV/T for z-direction magnetizing fields of −40, −20, 0, +20, and +40 mT, respectively. The PHMR signals monitored during the oscillation of a 35 pL MNP droplet indicate the presence of superparamagnetic properties [see Fig. 24(b)]. When similar measurements are performed on the same sample volume using a SQUID magnetometer, magnetic hysteresis indicates the diamagnetic behavior of the sample. In this context, recently, Schütt et al.²⁷⁶ also showed a nice demonstration for the detection of magnetic droplets in a lower magnetic field using cross PHMR sensors. In this study, they succeeded to detect 100 nl magnetic particle droplets of 0.04 mg/cm³ concentration with the applied magnetic field of 5 mT. However, in the presence of only earth's magnetic field (50 µT), they succeeded to detect the magnetic droplets down to 0.58 mg/cm³.

A magnetic bead placed in an external applied magnetic field will align its magnetic moment to the field via either internal flipping of the magnetic moment or physical rotation of the entire bead.²⁷⁷ Internal flipping of the magnetic moment is called Néel relaxation,²⁷⁸ whereas physical rotation is called Brownian relaxation.²⁷⁹ Therefore, the effective relaxation time is given by $1τeff=1τB+1τN$⁠, where τ_B and τ_N indicate the Brownian and Néel relaxation times, respectively. Brownian relaxation usually dominates for larger beads in the diametric range of 80 nm and above. Thus, we can neglect Néel relaxation in such situations. Relaxation of a nanomagnetic bead (NMB) can be measured in both the time and frequency domains. The governing equations for both domains are described briefly.

The frequency dependence of NMB relaxation in the presence of a magnetic field can be described using the complex magnetic susceptibility technique,

where χ′ and χ″ represent the in-phase (real part) and out-of-phase (imaginary part) components of fundamental susceptibility, respectively. For an ensemble of polydisperse particles, Debye’s monodisperse non-interacting particle model²⁸⁰ can be modified to the polydisperse Cole–Cole model, as demonstrated by the following empirical equation:²⁸¹ 

where χ₀ and χ_∞ are the magnetic susceptibilities of the sample at frequencies of zero and infinity and f is the frequency of the applied field. The equation depends on the magnetic field. The fitting parameter α is a measure of sample polydispersity and lies between 0 and 1. In the low frequency regime (f ≪ f_B), the NMBs rotate in phase with the field. At high frequencies, f ≫ f_B, the NMBs cannot follow the field because the field oscillates much faster the NMBs can respond. Interestingly, when f = f_B, the component that represents lagging of the bead moment behind the applied field is maximized, resulting in peak out-of-phase magnetic susceptibility.¹⁶⁰ Therefore, when an NMB is placed in an oscillating magnetic field, the Brownian relaxation measurement proposed for biosensing²⁷⁷ is characterized by the Brownian relaxation frequency, f_B,²⁷⁸