rus
Issue #3-4/2020
V.V.Amelichev, D.V.Kostyuk, D.A.Zhukov, A.B.Shevchenko, S.I.Kasatkin, O.P.Polyakov, V.S.Shevtsov, P.A.Polyakov
Calculation of the transfer characteristic of the anisotropic magnetoresistive magnetic field transducer
Calculation of the transfer characteristic of the anisotropic magnetoresistive magnetic field transducer
DOI: 10.22184/1993-8578.2020.13.3-4.230.238
The developed theoretical model of one-dimensional heterogeneity (MODH) is presented. Described are the main factors affecting distribution of a thin magnetoresistive element magnetization vector of an anisotropic magnetoresistive transducer (AMRT). The calculated results of AMRT volt-oersted characteristics obtained using MODH are consistent with the experimental data.
The developed theoretical model of one-dimensional heterogeneity (MODH) is presented. Described are the main factors affecting distribution of a thin magnetoresistive element magnetization vector of an anisotropic magnetoresistive transducer (AMRT). The calculated results of AMRT volt-oersted characteristics obtained using MODH are consistent with the experimental data.
Теги: anisotropic magnetoresistive effect magnetic field transducer magnetoresistive nanostructure theory of micromagnetism анизотропный магниторезистивный эффект магниторезистивная наноструктура преобразователь магнитного поля теория микромагнетизма
Calculation of the transfer characteristic of the anisotropic magnetoresistive magnetic field transducer
INTRODUCTION
AMRTs of a magnetic field are quite widely demanded by consumers in the civil, space and military fields. Examples of such applications are: highly sensitive magnetic field and electric current sensors, gradient read heads, galvanic isolation, biosensor devices and electronic compasses. AMRT is chosen for these applications mainly due to a good combination of consumer properties, such as high sensitivity in the range of weak magnetic fields (sensitivity threshold is units of nT), small weight and size parameters and reasonable cost.
These advantages come from the used ferromagnetic nanostructures, the AMRT design features and integrated methods of their manufacture. Due to a number of factors, including the odd form of the transfer characteristic, these transducers successfully withstand competition with the magnetic field transducers that appeared much later on the basis of the giant magnetoresistive effect. The main functional part of the magnetically sensitive element in AMRT is a Ti(Ta)-FeNiCo-Ti(Ta) strip of metal ferromagnetic nanostructure with a thickness of about (25–35) nm. AMRTs have an axis of sensitivity parallel to the plane of the crystal and perpendicular to the axis of easy magnetization (EMA) of the ferromagnetic nanostructure. EMA is formed during deposition of a ferromagnetic nanostructure in a constant uniform magnetic field.
The development and manufacture of instruments and devices based on AMRT is one of the urgent tasks of modern microelectronics of the Russian Federation. As a result of fundamental, experimental and technological research, new opportunities are opening up for the creation of promising products with improved characteristics and new functions. Magnetic straintronics and spintronics present examples of this new developing areas. In new devices, for example, magnetostrictive and magnetoresistive properties can be combined [1], which will expand their functionality.
RESEARCH METHODS
In SMC "Technological Centre", together with V.A.Trapeznikov Institute of Control Sciences of RAS, AMRT devices designed to measure the magnitude of magnetic field and electric current were developed and patented. AMRT is a bridge circuit, each arm of which consists of several AMR strips connected in series by low-resistance non-magnetic jumpers, and has an even or odd transfer volt-oersted characteristic (VOC), the parameters of which are determined by the design of the magnetically sensitive element and the type of nanostructure. Above the AMR strips are three layers of planar low-resistance conductors, separated by layers of a dielectric, usually SiO2. In the first layer, the conductors combine AMR strips into a bridge circuit. The conductor formed in the second metallization level is a planar offset coil, which creates a magnetic field directed parallel to the sensitivity axis of the transducer to minimize the initial unbalance of the bridge circuit. The set/reset conductor, made in the third level of metallization in the form of a meander, is necessary to set the initial direction of the magnetization vectors of the AMR strips along their EMA and to reduce the hysteresis of the AMRP VOC.
For experimental studies of AMRT, galvanomagnetic methods were used, which make it possible to measure the VOC of AMRT for various parameters of the device and its operating modes.
For theoretical studies, the developed models of interaction in the two-dimensional distribution of magnetization vectors in ferromagnetic strips are used to calculate their configuration and AMRP VOC based on the theory of micromagnetism.
THEORETICAL MODEL
When developing a theoretical model, the following factors were identified as the main factors affecting distribution of the magnetization vector M→ in the studied sample: the effective uniaxial anisotropy field Ha, the magnetostatic demagnetization field H→mand the external magnetic field H→0. For theoretical calculation of the AMR characteristics of a sample, it is necessary to determine the resulting distribution of the magnetization vector M→ in the sample. This became possible due to finding an effective algorithm for solving the variational problem of finding the minimum of the total magnetic energy of a ferromagnet [2]:
(1)
Here wa is density of magnetic anisotropy energy [3]:
, (2)
where K – uniaxial anisotropy constant, n→ – unit vector directed along the coordinate axis z, m→ –unit vector in the M→ direction.
Value wm is density of magnetostatic energy defined by the expression
, (3)
where H→m – a magnetic field created by internal and surface magnetic "charges" and called a magnetostatic or demagnetizing field. The demagnetizing field H→m is determined by the equations of magnetostatics:
rot H→m = 0, (4)
(5)
The last term in (1) is the density of Zeeman energy:
(6)
The developed model of one-dimensional heterogeneity takes into account the features of the geometric parameters of the magnetoresistive (MR) strip. In case of the test sample, the strip length is an order of magnitude greater than the width. Thus, we can assume that the heterogeneity of the distribution of magnetization at the ends of the strip is negligible.
To solve the variational problem (1) numerically, taking into account (2)–(6), we divide the integration region into N narrow strips, within which the distribution of the magnetization vector M→i can be considered homogeneous (see Fig.1). Then the continuous equation (1) can be replaced by a discrete (grid) equation for N variable projections of the magnetization vector Miy = |M→| sinθi.
The magnetic field created by uniformly magnetized rectangular stripes is determined by analytical expressions [4–5]. With this in mind, solving variational problem (1) reduces to solving a system of N Lagrange equations, which, in this case, reduces to a system of linear equations with respect to sinθi:
, (7)
where (look at form. 8).
Having obtained the solution of the system of equations (7), we can calculate the change in resistance in each MR strip caused by an external magnetic field H→0, which is applied perpendicular to the EMA (see Fig.1), in accordance with the formula for the AMR effect [6]:
, (9)
where Ri⊥ is the resistance of the strip at the perpendicular orientation of the magnetization vector with respect to the EMA (along the y axis), Δρ/ρ is the coefficient of the magnetoresistive effect. For the material studied in our case, Δρ/ρ = 0,02. Knowing the resistance of each strip, it is easy to calculate the resulting resistance R of the original strip according to the formula:
, (10)
here R⊥ = Ri⊥ / N.
RESULTS
Permalloy (FeNi) AMP strips, as well as strips based on FeNiCo alloy, with 6 and 20 percent cobalt content are used in the AMRT sensitive element. The use of permalloy in AMRT allows to achieve maximum sensitivity values. But, at present, despite the high sensitivity of permalloy based converters, this material is less commonly used in the AMRT design, due to increased values of hysteresis, magnetic noise, and temperature coefficient of resistance (TCR). The use of FeNiCo alloy increases the magnetic anisotropy field, expands the range of the measured magnetic field and increases the magnitude of the AMR effect to 2.0–2.5%. In this case, sensitivity of the transducers somewhat decreases but their magnetic noise, hysteresis, and TCR decrease; during magnetization reversal of AMR strips, the hysteresis-free rotation of the magnetization vectors prevails.
In [7], the results of experimental studies of AMRT with an odd transfer characteristic of MR with stripes at angles of ± 45° relative to the EMA were considered (see Fig.2). This means that to eliminate the influence of hysteresis before measuring the magnetic field through the set/reset conductor, it is necessary to pass current pulses from 0.5 to 2.5 A with duration of at least 2 μs, as a result of which the MR strips in the adjacent shoulders of the bridge circuit will be magnetized in opposite sides. The location of the magnetization vectors in the AMRT bridge circuit, established using the set/reset current pulses, in combination with the location of the strips themselves relative to the EMA, creates an odd VOC with a linear section.
Structurally, such an AMRT of a magnetic field is a substrate with a dielectric layer on which four rows of MR strips with Ti (Ta)-FeNi(FeNiCo)-Ti(Ta) metal ferromagnetic nanostructure are located. Above the MP strips, along each row, a planar offset coil with a second protective layer can be formed. The coil is designed to eliminate technological imbalance in the measurement of a constant magnetic field by applying a direct current to it, creating a magnetic field along the direction of the measured field. The set/reset conductor is located above the upper insulating layer (see Fig.2). A protective SiO2 layer with windows for contact pads is formed on top.
When a current pulse is applied to the set/reset conductor, the magnetic field created by it acts along the EMA on two rows of MR strips in one direction, and on the other two rows in the opposite direction. Under the action of the magnetic field created by the current pulse in the set/reset conductor, the magnetization vectors in two pairs of rows of MR strips will turn in opposite directions.
Under the action of an external magnetic field directed across the EMA, all the magnetization vectors of the rows of MR strips rotate in its direction, and in two rows of MR strips, the angle between the magnetization vector and OLS increases, and decreases in the other two. Consequently, the angles between the magnetization vectors and the current directions in the strips increase and decrease. Thus, the bridge circuit will be unbalanced, and an output signal appears at the AMRT output of the magnetic field which polarity depends on the direction of the measured magnetic field, and an odd VOC of the converter is formed.
A theoretical study of the samples described above is hindered, first of all, by the fact that the structure of the arising magnetic inhomogeneities can be very diverse [8, 9]. Various types of magnetic domains may appear in the sample, separated by domain boundaries [10], which may contain subdomain structures (Bloch lines, points) [11]. The magnetization reversal mechanism of such structures under the action of an external magnetic field is complicated for theoretical presentation.
The research team, within the framework of the mathematical package for micromagnetic simulation OOMMF [12] for the structures described above and comparison with experimental data, revealed the following regularity. In the central part of the strip, complex two-dimensional micromagnetic structures and domains are not observed (see Fig.3), and inhomogeneities at the edges practically do not affect the resulting magnetoresistance, since the length of the strip is an order of magnitude greater than its width. It can also be noted that the heterogeneity of the distribution of magnetization in the central part is one-dimensional, which allows us to use the MODH developed by us and described above.
Figures 4–5 show comparisons of the experiment with theoretical calculations using the OOMMF software package, and with use of the MODN. The linear nature of the experimental dependences of the output voltage on the magnitude of the external magnetic field additionally indicates the absence of complex magnetic structures in the sample.
CONCLUSIONS
An experimental study of AMRT of a magnetic field with a linear odd VOC and various cobalt content in inclined stripes shows that the converter has state-of-the-art technical characteristics. The developed theory of one-dimensional inhomogeneity is consistent with experimental data and leads to good agreement between the calculated and experimental curves of the VOC. The described theory of the distribution of magnetization vectors will also be valid for the case of nanostructures with a giant MR effect, which are being actively developed at present.
The study of AMRT parameters was carried out using specialized equipment of the central control center "Functional control and diagnostics of micro- and nanosystem equipment" on the basis of SMC "Technological Centre". ■
INTRODUCTION
AMRTs of a magnetic field are quite widely demanded by consumers in the civil, space and military fields. Examples of such applications are: highly sensitive magnetic field and electric current sensors, gradient read heads, galvanic isolation, biosensor devices and electronic compasses. AMRT is chosen for these applications mainly due to a good combination of consumer properties, such as high sensitivity in the range of weak magnetic fields (sensitivity threshold is units of nT), small weight and size parameters and reasonable cost.
These advantages come from the used ferromagnetic nanostructures, the AMRT design features and integrated methods of their manufacture. Due to a number of factors, including the odd form of the transfer characteristic, these transducers successfully withstand competition with the magnetic field transducers that appeared much later on the basis of the giant magnetoresistive effect. The main functional part of the magnetically sensitive element in AMRT is a Ti(Ta)-FeNiCo-Ti(Ta) strip of metal ferromagnetic nanostructure with a thickness of about (25–35) nm. AMRTs have an axis of sensitivity parallel to the plane of the crystal and perpendicular to the axis of easy magnetization (EMA) of the ferromagnetic nanostructure. EMA is formed during deposition of a ferromagnetic nanostructure in a constant uniform magnetic field.
The development and manufacture of instruments and devices based on AMRT is one of the urgent tasks of modern microelectronics of the Russian Federation. As a result of fundamental, experimental and technological research, new opportunities are opening up for the creation of promising products with improved characteristics and new functions. Magnetic straintronics and spintronics present examples of this new developing areas. In new devices, for example, magnetostrictive and magnetoresistive properties can be combined [1], which will expand their functionality.
RESEARCH METHODS
In SMC "Technological Centre", together with V.A.Trapeznikov Institute of Control Sciences of RAS, AMRT devices designed to measure the magnitude of magnetic field and electric current were developed and patented. AMRT is a bridge circuit, each arm of which consists of several AMR strips connected in series by low-resistance non-magnetic jumpers, and has an even or odd transfer volt-oersted characteristic (VOC), the parameters of which are determined by the design of the magnetically sensitive element and the type of nanostructure. Above the AMR strips are three layers of planar low-resistance conductors, separated by layers of a dielectric, usually SiO2. In the first layer, the conductors combine AMR strips into a bridge circuit. The conductor formed in the second metallization level is a planar offset coil, which creates a magnetic field directed parallel to the sensitivity axis of the transducer to minimize the initial unbalance of the bridge circuit. The set/reset conductor, made in the third level of metallization in the form of a meander, is necessary to set the initial direction of the magnetization vectors of the AMR strips along their EMA and to reduce the hysteresis of the AMRP VOC.
For experimental studies of AMRT, galvanomagnetic methods were used, which make it possible to measure the VOC of AMRT for various parameters of the device and its operating modes.
For theoretical studies, the developed models of interaction in the two-dimensional distribution of magnetization vectors in ferromagnetic strips are used to calculate their configuration and AMRP VOC based on the theory of micromagnetism.
THEORETICAL MODEL
When developing a theoretical model, the following factors were identified as the main factors affecting distribution of the magnetization vector M→ in the studied sample: the effective uniaxial anisotropy field Ha, the magnetostatic demagnetization field H→mand the external magnetic field H→0. For theoretical calculation of the AMR characteristics of a sample, it is necessary to determine the resulting distribution of the magnetization vector M→ in the sample. This became possible due to finding an effective algorithm for solving the variational problem of finding the minimum of the total magnetic energy of a ferromagnet [2]:
(1)
Here wa is density of magnetic anisotropy energy [3]:
, (2)
where K – uniaxial anisotropy constant, n→ – unit vector directed along the coordinate axis z, m→ –unit vector in the M→ direction.
Value wm is density of magnetostatic energy defined by the expression
, (3)
where H→m – a magnetic field created by internal and surface magnetic "charges" and called a magnetostatic or demagnetizing field. The demagnetizing field H→m is determined by the equations of magnetostatics:
rot H→m = 0, (4)
(5)
The last term in (1) is the density of Zeeman energy:
(6)
The developed model of one-dimensional heterogeneity takes into account the features of the geometric parameters of the magnetoresistive (MR) strip. In case of the test sample, the strip length is an order of magnitude greater than the width. Thus, we can assume that the heterogeneity of the distribution of magnetization at the ends of the strip is negligible.
To solve the variational problem (1) numerically, taking into account (2)–(6), we divide the integration region into N narrow strips, within which the distribution of the magnetization vector M→i can be considered homogeneous (see Fig.1). Then the continuous equation (1) can be replaced by a discrete (grid) equation for N variable projections of the magnetization vector Miy = |M→| sinθi.
The magnetic field created by uniformly magnetized rectangular stripes is determined by analytical expressions [4–5]. With this in mind, solving variational problem (1) reduces to solving a system of N Lagrange equations, which, in this case, reduces to a system of linear equations with respect to sinθi:
, (7)
where (look at form. 8).
Having obtained the solution of the system of equations (7), we can calculate the change in resistance in each MR strip caused by an external magnetic field H→0, which is applied perpendicular to the EMA (see Fig.1), in accordance with the formula for the AMR effect [6]:
, (9)
where Ri⊥ is the resistance of the strip at the perpendicular orientation of the magnetization vector with respect to the EMA (along the y axis), Δρ/ρ is the coefficient of the magnetoresistive effect. For the material studied in our case, Δρ/ρ = 0,02. Knowing the resistance of each strip, it is easy to calculate the resulting resistance R of the original strip according to the formula:
, (10)
here R⊥ = Ri⊥ / N.
RESULTS
Permalloy (FeNi) AMP strips, as well as strips based on FeNiCo alloy, with 6 and 20 percent cobalt content are used in the AMRT sensitive element. The use of permalloy in AMRT allows to achieve maximum sensitivity values. But, at present, despite the high sensitivity of permalloy based converters, this material is less commonly used in the AMRT design, due to increased values of hysteresis, magnetic noise, and temperature coefficient of resistance (TCR). The use of FeNiCo alloy increases the magnetic anisotropy field, expands the range of the measured magnetic field and increases the magnitude of the AMR effect to 2.0–2.5%. In this case, sensitivity of the transducers somewhat decreases but their magnetic noise, hysteresis, and TCR decrease; during magnetization reversal of AMR strips, the hysteresis-free rotation of the magnetization vectors prevails.
In [7], the results of experimental studies of AMRT with an odd transfer characteristic of MR with stripes at angles of ± 45° relative to the EMA were considered (see Fig.2). This means that to eliminate the influence of hysteresis before measuring the magnetic field through the set/reset conductor, it is necessary to pass current pulses from 0.5 to 2.5 A with duration of at least 2 μs, as a result of which the MR strips in the adjacent shoulders of the bridge circuit will be magnetized in opposite sides. The location of the magnetization vectors in the AMRT bridge circuit, established using the set/reset current pulses, in combination with the location of the strips themselves relative to the EMA, creates an odd VOC with a linear section.
Structurally, such an AMRT of a magnetic field is a substrate with a dielectric layer on which four rows of MR strips with Ti (Ta)-FeNi(FeNiCo)-Ti(Ta) metal ferromagnetic nanostructure are located. Above the MP strips, along each row, a planar offset coil with a second protective layer can be formed. The coil is designed to eliminate technological imbalance in the measurement of a constant magnetic field by applying a direct current to it, creating a magnetic field along the direction of the measured field. The set/reset conductor is located above the upper insulating layer (see Fig.2). A protective SiO2 layer with windows for contact pads is formed on top.
When a current pulse is applied to the set/reset conductor, the magnetic field created by it acts along the EMA on two rows of MR strips in one direction, and on the other two rows in the opposite direction. Under the action of the magnetic field created by the current pulse in the set/reset conductor, the magnetization vectors in two pairs of rows of MR strips will turn in opposite directions.
Under the action of an external magnetic field directed across the EMA, all the magnetization vectors of the rows of MR strips rotate in its direction, and in two rows of MR strips, the angle between the magnetization vector and OLS increases, and decreases in the other two. Consequently, the angles between the magnetization vectors and the current directions in the strips increase and decrease. Thus, the bridge circuit will be unbalanced, and an output signal appears at the AMRT output of the magnetic field which polarity depends on the direction of the measured magnetic field, and an odd VOC of the converter is formed.
A theoretical study of the samples described above is hindered, first of all, by the fact that the structure of the arising magnetic inhomogeneities can be very diverse [8, 9]. Various types of magnetic domains may appear in the sample, separated by domain boundaries [10], which may contain subdomain structures (Bloch lines, points) [11]. The magnetization reversal mechanism of such structures under the action of an external magnetic field is complicated for theoretical presentation.
The research team, within the framework of the mathematical package for micromagnetic simulation OOMMF [12] for the structures described above and comparison with experimental data, revealed the following regularity. In the central part of the strip, complex two-dimensional micromagnetic structures and domains are not observed (see Fig.3), and inhomogeneities at the edges practically do not affect the resulting magnetoresistance, since the length of the strip is an order of magnitude greater than its width. It can also be noted that the heterogeneity of the distribution of magnetization in the central part is one-dimensional, which allows us to use the MODH developed by us and described above.
Figures 4–5 show comparisons of the experiment with theoretical calculations using the OOMMF software package, and with use of the MODN. The linear nature of the experimental dependences of the output voltage on the magnitude of the external magnetic field additionally indicates the absence of complex magnetic structures in the sample.
CONCLUSIONS
An experimental study of AMRT of a magnetic field with a linear odd VOC and various cobalt content in inclined stripes shows that the converter has state-of-the-art technical characteristics. The developed theory of one-dimensional inhomogeneity is consistent with experimental data and leads to good agreement between the calculated and experimental curves of the VOC. The described theory of the distribution of magnetization vectors will also be valid for the case of nanostructures with a giant MR effect, which are being actively developed at present.
The study of AMRT parameters was carried out using specialized equipment of the central control center "Functional control and diagnostics of micro- and nanosystem equipment" on the basis of SMC "Technological Centre". ■
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