Thin Film Magnetoresistive Sensors

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Thin Film Magnetoresistive Sensors

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Series in Sensors Senior Series Editor: B E Jones Series Co-Editor: W B Spillman, Jr Solid State Gas Sensors Edited by P T Moseley and B C Tofield Techniques and Mechanisms in Gas Sensing Edited by P T Moseley, J O W Norris and D E Williams Half Effect Devices R S Popovi´c Thin Film Resistive Sensors Edited by P Ciureanu and S Middelhoek Biosensors: Microelectrochemical Devices M Lambrechts and W Sansen Sensors VI: Technology, Systems and Applications Edited by K T V Gratten and A T Augousti Automotive Sensors M H Westbrook and J D Turner Sensors and their Applications VII Edited by A T Augousti Advances in Actuators Edited by A P Dorey and J H Moore Intelligent Sensor Systems, Revised Edition J E Brignell and N M White Sensor Materials P T Moseley and J Crocker Ultrasonic Sensors R C Asher Sensors and their Applications VIII Edited by A T Augousti and N M White Eurosensors XII Edited by N M White Sensors and their Applications X Edited by N M White and A T Augousti

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Thin Film Magnetoresistive Sensors S Tumanski Warsaw University of Technology, Poland

Institute of Physics Publishing Bristol and Philadelphia

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© IOP Publishing Ltd 2001 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with the Committee of Vice-Chancellors and Principals. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0 7503 0702 1 Library of Congress Cataloguing-in-Publication Data are available

Senior Series Editor: Professor B E Jones, Brunel University Series Co-Editor: Dr W B Spillman, Jr, Virginia Tech Production Editor: Simon Laurenson Production Control: Sarah Plenty Cover Design: Victoria Le Billon Marketing Executive: Colin Fenton Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 1035, 150 South Independence Mall West, Philadelphia, PA 19106, USA Typeset by Mackreth Media Services, Hemel Hempstead, Herts. Printed in the UK by MPG Books Ltd, Bodmin

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Contents

Preface 1

AMR SENSORS 1.1 Anisotropic magnetoresistive effect in thin ferromagnetic films 1.1.1 Essential information about magnetoresistive effects 1.1.2 The origin of anisotropic magnetoresistance in ferromagnetic metals 1.1.3 The theory of magnetoresistance in ferromagnetic metals 1.1.4 Thin ferromagnetic film as the magnetic field sensor 1.1.5 Applying the Stoner–Wohlfarth model for analysis of magnetoresistive effects 1.2 Biasing and stabilizing techniques 1.2.1 The biasing and stabilizing fields in ferromagnetic magnetoresistors 1.2.2 The real thin film – the multidomain structure and the dispersion of anisotropy 1.2.3 The sensors biased by the hard magnetic layer 1.2.4 The sensors biased by the current conducting layer 1.2.5 Biasing by exchange coupled antiferromagnetic layer 1.2.6 The soft adjacent layers (SAL) biasing technique 1.2.7 Dual element sensors 1.2.8 AC biasing techniques 1.2.9 The reverse mode of biasing 1.3 Design and performances of AMR sensors 1.3.1 The influence of the sample geometry on the MR effect 1.3.2 The influence of magnetization non-uniformity on the MR effect 1.3.3 Technological factors affecting the performances of AMR sensors 1.3.4 The design and construction of AMR sensors 1.3.5 The performances of AMR sensors References

ix

1 1 8 12 17 22 27 27 33 45 52 58 63 70 76 81 83 83 89 97 115 132 146

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3

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CONTENTS GMR SENSORS 2.1 Giant magnetoresistive effects 2.1.1 An historical review and the main terms 2.1.2 Oscillatory exchange coupling in the magnetic multilayers 2.1.3 Other coupling effects in multilayer structures 2.1.4 Theoretical models of giant magnetoresistance 2.1.5 Ferromagnetic multilayers as magnetic field sensors 2.2 Various types of GMR structures 2.2.1 Structures with antiferromagnetic coupling 2.2.2 Granular GMR structures 2.2.3 Spin-valve structures with asymmetric magnetic layers (uncoupled structures) 2.2.4 Spin valve structures with exchange-biased layer 2.2.5 Current perpendicular to plane (CPP) structures 2.2.6 Magnetic tunnel junction (MTJ) structures 2.2.7 Colossal magnetoresistance (CMR) thin film structures 2.2.8 Giant magnetoimpedance (GMI) structures 2.3 Preparation, design and properties of GMR sensors 2.3.1 The deposition of thin film GMR structures 2.3.2 Technological factors affecting the performances of GMR sensors 2.3.3 The shape effects in thin film GMR devices 2.3.4 Design and construction of GMR sensors References

266 276 286 299

APPLICATIONS OF MAGNETORESISTIVE SENSORS 3.1 Magnetic measurements 3.1.1 Magnetometers and compasses 3.1.2 Gradiometers, magnetic anomaly detection 3.2 Electrical measurements 3.2.1 Current transducers 3.2.2 Electrical transducers, switching and logic elements 3.3 Magnetoresistive elements in data storage applications 3.3.1 Magnetic Random Access Memory (MRAM) devices 3.3.2 Magnetic card readers 3.3.3 Magnetoresistive heads for tape and disk applications 3.3.4 Unshielded magnetoresistive reading heads 3.3.5 Shielded magnetoresistive heads 3.3.6 Yoke-type magnetoresistive heads 3.4 Transducers of mechanical values 3.4.1 Transducers of linear displacement 3.4.2 Transducers of angular position 3.4.3 Measurements of rotational speed

325 325 336 341 341 347 353 353 357 361 372 376 382 384 384 391 396

165 165 173 185 191 199 209 209 217 222 229 241 247 251 256 260 260

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vii

3.5 Material testing and magnetic field imaging 3.5.1 Material testing by means of thin film magnetoresistors 3.5.2 Magnetic imaging systems References

400 400 406 417

List of Symbols List of Abbreviations Index

431 433 435

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Preface

Thin film magnetoresistive sensors (MR sensors) are widely used in various applications. The most important are their applications as read heads. MR sensors are also used as mechanical transducers and of course as magnetic field sensors, for example in compasses. There is a large variety of MR sensors on offer in the marketplace. However, market availability of MR sensors does not mean that the research in this field is less intensive. At scientific conferences devoted to magnetic sensors or to magnetism in general the subject of ‘magnetoresistance’ is covered in a large number of papers. Every year, each conference brings revolutionary news. The progress in thin film MR sensors is possible due to the large scientific and business interest. It has always been to the benefit of MR sensors that thin film technology and magnetoresistance were the objects of interest of other often better financial supported activities. For fundamental physics the investigations of phenomena in thin magnetic films were always valuable because it was easier to explain and to understand the magnetism in this specific form of magnetic material (in comparison with the bulk form). The most significant support for investigations of magnetoresistance in thin film arises from the computer industry because it is obvious that the next generation of high-density recording heads will be based on magnetoresistive effects. That is why the subject ‘thin film magnetoresistive sensors’ is broadly described in this book. Not only are classical measuring transducers described, but also other magnetoresistive devices, such as switching and logical elements, multipliers and insulators, memory devices and most importantly reading heads. Also the physics of thin magnetic films and thin film technology are presented comprehensively. The technology of GMR sensors is very sophisticated and refined and therefore it is described as an example of thin film technology. Thus this book describes not only one class of sensors, the magnetoresistive sensors, but also discusses many more universal subjects, such as magnetism of thin films, thin film technology and magnetic measurements. Therefore it should be useful for various kinds of readers. For high quality specialists it should be interesting as a comprehensive review of all available knowledge to date. The mathematical description is limited to only the most important definitions, therefore the level is also applicable to students and practising engineers. The book is especially addressed to engineers. The author is an engineer who has been active for more than thirty years in the field of industrial and magnetic measurements. That is why there is much practical information about the

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performances of sensors and the author’s experiences in design and construction of transducers. Usually MR sensors are divided into anisotropic MR sensors (AMR) and giant MR sensors (GMR). This classification results from the different mechanisms and features of these effects. The giant magnetoresistance is recently in vogue partially due to the very marketing name. Simultaneously the AMR sensors are treated as slightly old-fashioned. Meanwhile the AMR sensors still exhibit better sensitivity and are much simpler in technology. In the opinion of the author both types of MR sensors have their unique performances and both are more complimentary rather than competitive. That is why in this book AMR sensors and GMR sensors are presented equally. In fact knowledge about AMR sensors is more stabilized and therefore their properties can be discussed in detail. Research into GMR sensors is largely still in progress, and the most upto-date information is presented here. All three parts of the book: AMR Sensors, GMR Sensors and Applications are written as separate sections (with independent references and independent logical concept). So it is not necessary to read the book ‘from the beginning’. If, for example, readers are not interested in the physics of MR effects, and only want information about applications or performances of the sensors then it should be sufficient to read only the separate chapters. In both main parts the structure is the same – first the theory, then a description of the various types of sensors, and finally technology, design and performances. Thin film magnetoresistive sensors exhibit many unique features. They can be extremely small, with dimensions of parts of micrometres. They can detect a wide range of magnetic fields, from picotesla to several tesla. The measurement of DC and AC magnetic fields is possible. Because the sensors detect the field in the film plane they can be placed very near the investigated area, as tangential sensors. Some sensors available in the marketplace may cost only several dollars. MR sensors exist as well as sophisticated transducers, for example three-axis compasses. Therefore it is worth reading this book to gain a better understanding of the features of MR sensors and hopefully to use them more effectively. However, the author also recommends reading this book as a story of one of the most fascinating events of high technology. The sensors fabricated in nanometric atomic scale exhibit extraordinary, not fully understood phenomena. The expected future applications of these sensors, especially in data storage systems, open new perspectives for the whole of science. Slawomir Tumanski May 2000

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1 AMR Sensors

1.1 ANISOTROPIC MAGNETORESISTIVE (AMR) EFFECT IN THIN FERROMAGNETIC FILMS 1.1.1 Essential information about magnetoresistive effects The resistance of a material may depend on its state of magnetization. This phenomenon is called the magnetoresistive effect. We can change the magnetization of the material by applying an external magnetic field. Thus, the magnetoresistive effect may be used to construct magnetic field sensors. The magnetic field may influence the material resistivity in various ways (Heremans 1993, Lenz 1990, Caruso et al 1998, Popovi´c et al 1996). For example the magnetoresistive effect may be a consequence of the Lorentz force – similarly as in the Hall effect. The Lorentz force F is acting on particle with charge q moving with velocity v in a magnetic field B F
q(v  B).

(1.1)

As a result the Lorentz force deflects the path of the moving free charge carrier (figure 1.1(a)). It increases the path length causing a change in the material resistivity. Figures 1.1(b) and (c) present the current paths and

Figure 1.1 The origin of the magnetoresistive effect in semiconductors: (a) the influence of the Lorentz force on the current path length; (b) the current paths and equipotential lines in a conducting rectangular plate in the absence of the external field; (c) the same lines after application of the external magnetic field.

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equipotential lines in a rectangular plate of the conducting material. The resistance R(B) is proportional to the square of the magnetic field induction B perpendicular to the plate: B R(B)
Ro –— (1  2CB2) o

(1.2)

where: R0 is the resistance of the material in the magnetic field B
0, B /0 is the magnetoresistivity coefficient,  is the carrier mobility, and C is the coefficient depending on the geometry of the sample. The carrier mobility is especially large in semiconductors InSb and InAs and therefore these materials are most frequently used to prepare the magnetoresistors (
7.7 104 cm2/Vs for InSb and 
3 104 cm2/Vs for InAs). The coefficient C has been calculated by Lippmann and Kuhrt (1958). For l/w 0.35 (l – length of the plate, w – width of the plate) coefficient C is given by the expression l C
1  0.54 —. w

(1.3)

The l/w ratio should be as small as possible because in the long plate (large l/w ratio) the current path runs parallel to the edges and the Hall electrical field reduces the magnetoresistive effect. On the other hand, to obtain large resistances of the magnetoresistor it is required to elongate the magnetoresistor stripe. Various techniques are adopted to fabricate magnetoresistors with a long current stripe without sacrificing the magnetoresistivity (Popovi´c and Heidenreich 1989). For example shorting bars from a highly conductive metal are introduced into the stripe (figure 1.2(a)). The magnetoresistor consists then of many elementary magnetoresistors connected in series with a small l/w ratio.

Figure 1.2 Various design of semiconductor magnetoresistors: (a) magnetoresistor with shorting bars; (b) magnetoresistor with NiSb needles; (c) Corbino disk (Weiss 1966).

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3

A special kind of semiconductor magnetoresistors with NiSb needles has been developed by Weiss (1966). This magnetoresistor is called the “Feldplatte”. The NiSb needles are built into the polycrystalline InSb. Due to the much greater conductivity of NiSb the current path is larger and the resistance of the sensor is greater than for the simple InSb material (figure 1.2(b)). The magnetoresistive “Feldplatte” sensors are actually manufactured by Siemens AG. In a special geometrical form of magnetoresistor, known as the Corbino disk (Corbino 1911) (figure 1.2(c)), it is possible to obtain a pure magnetoresistive effect (without the Hall effect). The Corbino disk is equivalent to the plate with l/w  0 (the current lines are curved along logarithmic spirals). Unfortunately the Corbino disk structure does not ensure a large enough resistance of the sensor. Typical dependence of the relative change of resistance Rx /Ro on the external magnetic field for the InSb Corbino disk and the Feldplatte magnetoresistor are presented in figure 1.3.

Figure 1.3 Dependence of the relative change of resistance Rx /Ro versus the magnetic field B of the semiconductor magnetoresistors.

All metals exhibit the magnetoresistive effect (Jan 1957). In some semimetals, for example bismuth, this effect can be quite large. The bismuth spiral was the earliest material used for preparing magnetoresistors (Bublitz 1937). The maximal value of the relative change of resistivity x /o for pure bismuth is at room temperature equal to about 2 (for magnetic induction change from 0 to 2 T ). This change of resistivity is about 40 at liquid nitrogen temperature. The appearance of magnetoresistance in non-magnetic metals requires the application of a large external magnetic field. The change of resistivity is described by the Kohler rule (Kohler 1949) in the form of the expression:

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THIN FILM MAGNETORESISTIVE SENSORS B,T 273K ———
1  f B ——— B
0,T B
0,T

(

)

(1.4)

where: B,T – resistivity in magnetic field B and temperature T. This rule is usually presented as the Kohler diagram. The Kohler diagram is the relation between /(0) and B/red. The reduced resistivity red is (T)/( ), where is the Debye temperature. An example of this plot is presented in figure 1.4.

Figure 1.4 The reduced Kohler diagram (B in Tesla).

Apart from the ordinary magnetoresistive effect in metals previously discussed, an anomalous magnetoresistance effect appears in ferromagnetic materials (Mott 1936, Smit 1951, Dorleijn 1976). This anomalous effect in transition metals is explained as the spin-orbit interaction. In this effect the resistivity depends on the orientation of magnetization with respect to the direction of the electric current. Therefore this effect is often called the anisotropic magnetoresistance (AMR) (McGuire and Potter 1975, Mapps 1994, Kwiatkowski and Tumanski 1986, Ciureanu 1992, Dibbern 1989). In comparison with the magnetoresistive effect in semiconductors and nonmagnetic metals anisotropic magnetoresistance occurs for much smaller magnetic fields. The anisotropic magnetoresistance was discovered in 1857 by William Thomson (Lord Kelvin) (Thomson 1857). In a paper entitled “Effects of magnetization on the electric conductivity of nickel and of iron” he described the results of an experiment on a small rectangular piece of ferromagnetic material “subjected to magnetic force”. He also mentioned the anisotropic nature of this effect stating that “iron acquires an increase of resistance to the conduction of electricity along, and a diminution of resistance to the conduction of electricity across, the lines of magnetization”.

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5

Figure 1.5 presents the change of the resistivity of NiCo alloy versus the magnetic field induction. The longitudinal resistivity l of the material magnetized parallel to the current direction is larger than the transverse resistivity t of the material magnetized perpendicularly to the current direction. The magnetoresistivity coefficient / is defined as follows   l  t —–
—–
————–  av 1 2 – l  – t 3 3

(1.5)

where: av – the average resistivity (resistivity of the demagnetized specimen).

Figure 1.5 The change of resistivity of NiCo alloy versus the external magnetic induction (McGuire and Potter 1975).

As a material for preparing AMR sensors the thin film of the NiFe alloy (permalloy) is most frequently used. Figure 1.6 presents as an example the R/R
f(B) dependence for a thin film permalloy magnetoresistor. The thin film permalloy exhibits a magnetoresistivity coefficient equal to about 2.5%. The thin film in comparison to the bulk material offers several advantages. The magnetization process of thin film is relatively simple and fast – approximate to a single domain model. For full magnetization, a relatively small (about 1 mT), external magnetic field is needed. Thin film in the form of the path enables to obtain large resistance of the magnetoresistor. It is convenient that these films are fabricated using standard semiconductor technology. In 1988 Baibich and co-workers discovered another magnetoresistive effect utilising thin ferromagnetic films (Baibich et al 1988). Because the obtained magnetoresistive coefficient was much larger than the AMR effect ( R/R larger than 150% at T
4 K has been reported), this effect was called the giant

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Figure 1.6 The typical transfer characteristic of an AMR permalloy thin film magnetoresistor.

magnetoresistive (GMR) effect (Fert and Bruno 1996, Parkin 1996, Levy 1994, Daughton 1999, Smith and Schneider 1998, Grünberg 1999, Coehoorn 1999). The GMR magnetoresistor consists of magnetic layers separated by a very thin non-magnetic layer. The GMR effect may be explained as the scattering effect of electrons through the non-magnetic interface – spin-dependent scattering. Figure 1.7(a) presents an example of the transfer characteristic determined for the Fe/Cr multilayer structure by Baibich and co-workers. The GMR magnetoresistance in multilayers at room temperature is about 30–40% and needs a much larger magnetic field than AMR. The most promising and recently most frequently used is another GMR thin film structure developed by Dieny and co-workers (Dieny 1994). It is called the spin valve (SV) (Dieny 1994, Daughton 1996, Kools 1996). Currently spin valve magnetoresistive sensors with magnetoresistive coefficient R/R
5–15% and saturation field

Figure 1.7 Examples of transfer characteristics of GMR sensors: (a) Fe/Cr multilayer T
4 K); (b) spin valve sensor at room temperature.

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7

B
2.5 mT are available. Figure 1.7(b) presents transfer characteristics determined for the spin valve structure. Lately promising results have been reported in another multilayer structure – in magnetic layers separated by a very thin insulator layer (Moodera et al 1995, Gallagher et al 1997). It is expected that this magnetic tunnel junction (MTJ) structure could be the most sensitive GMR sensor. The magnetoresistive effect has been recently reported in magnetic oxides (for example in doped manganate perovskites La–Ca–Mn–O thin films) (Jin et al 1994, Sun 1998). This has been called the colossal magnetoresistance (CMR) effect because the change of resistance is very large. Magnetoresistivity as large as 1010 (for T
60 K and B
7 T) has been achieved (Chen et al 1996). Magnetoresistance is not only reported in thin film structures. Permalloy sensing elements have been obtained in micro-fiber shape by rapid solidification (Ciureanu et al 1993, Ciureanu et al 1994). Magnetoresistive nanowires have also been prepared by the electroplating method (Piraux et al 1994, Blondel et al 1994). The giant magnetoimpedance effect (GMI) as some kind of magnetoresistance may also be considered. The GMI effect occurs in wires and thin film structures supplied by high frequency voltages and obtains very sensitive magnetic field sensors (Panina 1995, Mohri et al 1997). Figure 1.8 presents a comparison of a range of magnetic fields detected by various measuring methods. Using magnetoresistive sensors it is possible to detect magnetic fields ranging from picotesla to several tesla. Figure 1.9 presents a comparison of the sensitivity of various magnetoresistive sensors. For large magnetic fields (one tesla or more) semiconductor magnetoresistors or GMR multilayers are recommended (in this range of magnetic field Hall sensors are used most widely). For detecting small magnetic fields the permalloy magnetoresistive sensors are the most suitable. Detectable Field Range

Figure 1.8 The typical field range of various magnetic field sensors (Caruso 1998).

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Figure 1.9 The typical field range of various magnetoresistive sensors – AMR: anisotropic magnetoresistance; SV: spin valve sensors; InSb: semiconductor magnetoresistors; GMR: multilayer GMR sensors; CMR: colossal magnetoresistance (Heremans 1993).

Spin-valve sensors obtain larger output signals than AMR sensors (but for larger magnetic fields). Therefore they are useful for reading information from magnetic tapes or disks (in magnetoresistive heads) where small dimensions of the sensor are the most important parameter. Due to large magnetoresistivity GMR heads allow reading of information with the highest density reported to date. The most frequently used are AMR and GMR/SV sensors – they are also available as commercial products (Philips, Siemens, Honeywell, Nonvolatile Electronics). The semiconductor magnetoresistors are of smaller technical importance due to rather strong temperature dependence and high non-linearity (Popovi´c et al 1996). The thin film magnetoresistive effects and sensors presented above (with the exception of semiconductors described elsewhere (Popovi´c and Heidenreich 1989, Metschl 1987, Wieder 1971)) will be described in more detail in the next chapters.

1.1.2 The origin of anisotropic magnetoresistance in ferromagnetic metals The microscopic origin of electrical transport in ferromagnetic metals was described in detail in many papers in the 1960s and the 1970s. As a basis for these papers Smit’s results of investigations of magnetoresistance in ferromagnetics (Smit 1951) have been used. Smit had explained the phenomenon of anisotropic magnetoresistivity using Mott’s two-current model of conduction in transition metals (Mott 1936). Jan (1957) and McGuire and Potter (1975) summarized the knowledge about anisotropic magnetoresistance.

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9

It is convenient to explain the anisotropy of magnetoresistance by taking into account the existence of spin-orbit coupling and the anisotropic scattering mechanism of s and d electrons (Kondo 1962, Berger 1965, Vu Dinh Ky 1967). Figure 1.10(a) shows density of states curves for ferromagnetic nickel determined by Langlinais and Callaway (1972) and figure 1.10(b) shows the same curves in a simplified schematic form. This band structure is split into two different sub-bands representing the different orientations of the electron spins (with magnetic moments parallel or antiparallel to the total magnetization). When the 3d band is not fully filled, scattering of 4s electrons to the 3d band is probable. The current of 4s electrons with small effective mass m* s (comparable with free electron mass) is predominant. The 3d electrons with large effective mass m*d are low mobile (for iron m*d  30 m* s ). Due to magnetic ordering the 3d sub-bands are not equally filled. Therefore there is another probability of scattering from the 4s state to the 3d state than from the 4s state to the 3d state. For ferromagnetic nickel only scattering from the 4s state to the 3d state is probable, because the 3d sub-band is below the Fermi level.

Figure 1.10 Density of states curve of ferromagnetic nickel: determined by Langlinais and Callaway (1972a) and in simplified form (Miedema and Dorleijn 1975b).

In Smit’s model it is assumed that the influence of the magnetic field and spin-orbit interaction cause mixing of d and d states. It can be demonstrated that this mixing is anisotropic and the probability of sd scattering is larger for electrons travelling parallel to the magnetization. Thus resistivity l is larger than t, as shown in figure 1.5. Figure 1.11 shows the main idea of Mott’s two-current model of conductivity in ferromagnetic metals. The current can be divided into two parts, one current of s- electrons with parallel spin direction, and the other with the opposite spin. Then the resistivity can be described with the following formula 1 1 1 —
—————  —————.  ss  sd ss  sd

(1.6)

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Figure 1.11 Two-current model of the conduction in ferromagnetic metals.

Further developing this two-current model, Marsocci (1965) and Thomas and co-workers (1969) determined the spin-orbit interaction in nickel–iron thin films with respect to the crystalline axes. Let us consider a ferromagnetic fcc crystal with x,y,z axes corresponding to the main crystal axes. The current is in [100] direction (along the x axis) and the magnetic field is rotated in the (100) plane (y–z plane) with the angle  to current direction. The atomic wave functions for the d electrons in XYZ space are 1
yzf (r) 2
zxf (r) 3
xyf (r) 1 4
 (x2 – y2) f (r) 2 1 5
——– — (r2  3z2) f (r). 2√ 3

(1.7)

The spin-orbit interaction operator is of the form

[

]

1 1 K (L · S)
K Lz Sz  – (Lx  iLy)(Sx  iSy )  – (Lx  iLy )(Sx  iSy ) 2 2

(1.8)

where: L is the orbital angular momentum operator, Sx Sy Sz are the Pauli matrices and K is the spin-orbit interaction coefficient. The spin operators for assumed directions can be determined by

( )

1 0 1 Sx
– , 2 1 0 1  sin   i cos  Sy
– , 2 i cos  sin 

( (

1 cos  Sz
– 2 i sin 

 i sin   cos 

) )

.

(1.9)

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11

Taking into account equations 1.7–1.9 and using first-order perturbation theory, the perturbed wave functions can be described as (Thomas et al 1969) K — — —  1
1  ——  2 i sin   3 i cos   4  √ 3 5 , 2

(

)

K — — —  2
2  —— 1 i sin   3  4 i cos   √ 3 5 i cos  , 2

(

)

K — —  3
3  —— 1 i cos   2  2 4 i sin  , 2

(

)

(1.10)

K — —  4
4  ——  1  2 i cos   2 3 i sin  , 2

(

)

— K — — —  5
5  —— √ 3 1  √ 3 2 i cos  , 2

(

)

where  is the exchange energy and / exponents of wave functions indicating the directions of spin. The spin-orbit interaction causes non-symmetrical mixing of wave functions and the resultant perturbed wave function is a complex dependence on the angle . Knowing wave functions it is possible to determine respectively: the scattering potentials V, the transition probabilities Psd, relaxation times sd and conductivities l and t. All these parameters are the complex function of the angle  (Thomas et al 1969). Potter (1974) demonstrated calculations of the magnetoresistance conductivity of ferromagnetic NiCu alloys. After sophisticated calculations the following results have been reported:

{

[( ) (

Ns K 2 N  2.6 ——ln 0.06 —  ——s Nd  Nd

)]

N K 2 K l
o 10.007 —–d —–  0.4 ——— Ns 2 2

2

[ ()

]}

(1.11) 2 Nd K 2 K Ns K 2 N t
o 10.007 —– 1.2 —–  1.76 ———  2.6 ——ln 0.03 —  ——s Ns 2 2 Nd  Nd

{

[ ( ) ( )]

[ ()

]}

where:  is splitting between the uppermost two d bands, 2 is exchange splitting, K is the spin-orbit coupling parameter, Ns, Nd – density of s, d states respectively, (Ns /Nd) is a term due to isotropic ss scattering and ns e2ss 0
———–. ms

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The coefficient of magnetoresistivity according to equation (1.5) can be calculated as  l  t 3(t  l ) —–
———–——
————–. av 1 2 t  2l – l  – t 3 3

(1.12)

For NiCu 70/30 the parameters in equation (1.11) have been estimated as: Nd /Ns
14, K
0.1 eV, K/
0.33, K/2
0.125, K/(2)
0.09 and /av
0.44%. Figure 1.12 shows the conductivities for majority and minority spins versus the Nd /Ns parameter (Nd /Ns
14 for pure Ni and Nd /Ns
21 for Ni71Cu29). The anisotropic magnetoresistance is mainly influenced by the conductivity of minority spins, as only minority spin s electrons can scatter into d states (see figure 1.10).

Figure 1.12 The conductivity of minority and majority spin electrons determined by Potter (1974) for NiCu alloy.

1.1.3 The theory of anisotropic magnetoresistance in ferromagnetic metals Under isothermal conditions the Ohm law for a chemically homogeneous metal is described by the following expression 3

Ei
∑ij Jj
ij Jj

(1.13)

j
1

where: J (J1 J2 J3) is the current density, E (E1 E2 E3) is the electric field and ij is the tensor of resistivity. If this metal is in the external magnetic field H the resistivity tensor depends on the magnetic field strength or on the magnetization M

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AMR SENSORS Ei
ij (H)Jj
ij (M)Jj .

13 (1.14)

If the metal is a saturated single crystal the magnetization may be described by the direction cosines of magnetization with respect to the crystallographic axes M
(M cos 1, M cos 2, M cos 3,)

(1.15)

and the resistivity tensor in equation (1.13) depends only on  Ei
ij (a)Jj .

(1.16)

Taking into account the crystal symmetry (Birss 1964, Bhagavantam 1966) the resistivity tensor can be simplified. After expansion using MacLaurin’s series this tensor is described by the expression ij 1 2 ij ij ()
ij (0)  k ——  – k l ———  ... k 2 k l
aij  akij cos k  klij cos k cos l  ... .

(1.17)

The resistivity tensor can be divided into a symmetric part  ijs and an asymmetric part ija ij  ji ijs
———— ; 2

ij  ji aij
———— . 2

(1.18)

Taking into consideration Onsager’s theory of irreversible thermodynamics in the form ij ()
ij ()

(1.19)

it is assumed that ijs ()
ijs ()
jis () aij ()
aij ()
aji ().

(1.20)

An asymmetric part of the expression (1.17) aij
akij cos ak  mlkij cos m cos k cos l  ...

(1.21)

represents the Hall effect and does not take part in Joule heating of the material. The magnetoresistivity is represented by a symmetric part ijs
aij  aklij cos k cos l  ... .

(1.22)

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THIN FILM MAGNETORESISTIVE SENSORS

According to Neumann’s principle, any type of symmetry, which is exhibited by the point group of the crystal, is possessed by every physical property of the crystal. For the m3m group, representing the ferromagnetic crystals with cubic symmetry, Birrs (1964) determined the following condition aij
0

except

a11
a22
a33

aklij
0

except

a1111
a2222
a3333

aklij
0

except

a1122
a1133
a2233
a3322
a3311
a2211.

(1.23)

The magnetoresistivity tensor s through fourth order can be described as co  c1 cos2 1 c2 cos 1 cos 2 c2 cos 1 cos 3 s
c2 cos 1 cos 2 co  c1 cos2 1 c2 cos 2 cos 3 c2 cos 1 cos 2 c2 cos 2 cos 3 co  c1 cos 1

[

]

(1.24)

where: co
a11  a1122; c1
a1111  a1122, c2
a2323. The magnetoresistivity tensor for a rectangular prism declined on the angle  with respect to crystallographic axes is described as J·E (,)
———
ijs () cos i cos j . J2

(1.25)

Taking into account equations (1.22–1.24) the expression (1.25) can be rewritten as

(∑

(,)
co  c1

3

)

2

∑cos 

cos 1 cos 1  c2

i
1

i

cos j cos i cos j . (1.26)

ij

Equation (1.26) is similar to the Döring formula (Döring 1938) describing the even effects in cubic crystals

(

)

1 (,)
1 cos2 1 cos2 1  cos2 2 cos2 2  cos2 3 cos2 3  – 3

 22 (cos 1 cos 2 cos 1 cos 2  cos 2 cos 3 cos 2 cos 3  cos 3 cos 1 cos 3 cos 1)

(1.27)

3 3 where: 1
— a11[100] , 2
— a11[111]; 2 2 and a11[100], a11[111] are the resistivities measured along an edge and a diagonal of the cubic prism. Taking into account the symmetry conditions, the asymmetric component in equation (1.16) can be described as axial vector a123 J  , and the symmetric

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15

component as (a11a1122)J  2a2323(J·). Equation (1.16) can be written in vector form E
t J   · ( · J)  H (  J)

(1.28)

where H is the Hall coefficient and 
l – t. According to equation (1.26) the resistivity tensor is

[

t   cos2 1

 cos 1 cos 2  H cos 3

 cos 1 cos 2  H cos 3

t   cos2 2

 cos 1 cos 3  H cos 2

 cos 2 cos 3  H cos 1

 cos 1 cos 3  H cos 2

]

 cos 2 cos 3  H cos 1 . t   cos2 3

(1.29) For a simplified case when the magnetic field is in the film plane (cos 3
0, cos 2
sin 1) matrix (1.29) is

[

t sin2   l cos2 

1 –  sin 2 2

H sin 

1 –  sin 2 2 H sin 

t cos2   l sin2 

H cos 

H cos 

t

]

. (1.30)

Thus the dependence of resistivity on the orientation of the magnetization vector can be described as ()
t sin2   l cos2 .

(1.31)

The dependence (1.31) is more often presented as the so-called Voigt–Thomson formula, where it is assumed that the current is in one axis and resistivity is a dependence of the angle  ()
t sin2   l cos2 
t   cos2 

(1.32)

where the angle  is an angle describing the orientation of the magnetization with respect to the current direction. Usually in the initial state the magnetoresistor is magnetized along the current direction (
0) and R(H
0)
Rl.

(1.33)

Therefore it is convenient to present expression (1.32) with respect to the l value as: ()
l   · sin2 .

(1.34)

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Page 16

THIN FILM MAGNETORESISTIVE SENSORS The relative change of resistance can be described in the form Rx  ——
 —— sin2  Rx 

(1.35)1

where magnetoresistivity coefficient is  l  t ——
———— .  l In case of polycrystalline sample it is necessary to determine the average resistivity  of a large number of randomly oriented crystallites. When the orientation of the current density vector J is described by the polar coordinates

and and the orientation of the vector of magnetization M by the angle  (for the fixed value of the angle ), then the resistivity of polycrystalline film is possible to calculate by integrating (,) 1 2  2 
—— ∫d ∫ d ∫ d (,). 82 0 0 0

(1.36)

This integration has been calculated by Birss (1960). The following results have been reported: 
t  (l  t ) cos2 

(1.37)

where: 1 1 1 3 1 t
ko  – k1  — k2  – k3  — k4  — k5 5 10 5 35 70 3 2 1 3 1 l
ko  – k1  — k2  – k3  — k4  — k5 5 10 5 7 35 and ko
a11  a1122  a111122 k1
a1111  a1122  a112211  a112233 k3
a112233  2a111122 k4
a111111  a111122  a112211  a112233 k5
2a112323  2a111212. 1 The ‘minus’ sign in equation 1.35 indicates that the resistance of the sensor is largest when the angle between the current and magnetization is zero (Rl ). After applying the external field resistance decreases.

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17

The resistivity of the demagnetized polycrystalline sample is (Birss 1960) 1 1 1 2 dem  ko  – k1  – k4
– l  – t . 3 3 3 3

(1.38)

The formula (1.37) is similar to the Voigt–Thomson formula (1.32). From the above analysis it appears that the change of resistivity can be determined from the materials coefficients (for example a11[100] and a11[111] from equation (1.25)) as a dependence on the angles  and . In the special case of thin ferromagnetic film it is possible to describe this change of resistivity as a dependence only on the angle  (knowing the magnetoresistivity coefficient /). As it will be proved in the next chapter, in the case of thin ferromagnetic film, there exists a relatively simple dependence between the angle  and external magnetic field strength H x.

1.1.4 Thin ferromagnetic film as the magnetic field sensor For users of magnetoresistive sensors the most important aspect is the dependence R/R
f(H). Thin film is usually anisotropic with uniaxial anisotropy in the film plane. When the thin film structure is anisotropic then in the absence of external magnetic field H the resultant vector of magnetization M is directed along the anisotropy axis (the so-called easy axis of magnetization L – see figure 1.13). To magnetize the thin film (to rotate the vector of magnetization) the magnetic field should be directed perpendicular to the anisotropy axis (and in the film plane). Therefore it is usually assumed that the measured field component Hx is perpendicular to the resultant easy axis of magnetization (for convenience it will be assumed that the y axis is the anisotropy axis as in figure 1.13). To determine the desired dependence Rx/Rx
f(Hx) it will be assumed under simplified conditions that the thin film is magnetized uniformly and by coherent rotation of magnetization. This assumption is valid for the single domain state of the thin film. As will be shown in the next chapters in real thin film structures it is possible to ensure conditions very near to this assumption. The direction of the magnetization vector M can be calculated by minimizing the expression of the free energy of the system. This energy W is described as W
H · M  Ku sin2

(1.39)

where is the angle of magnetization direction with respect to the anisotropy

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THIN FILM MAGNETORESISTIVE SENSORS

Figure 1.13 The thin film ferromagnetic path – the simplest ferromagnetic magnetoresistive sensor (L – anisotropy axis).

axis (figure 1.13) and Ku is the anisotropy coefficient. The first term in equation (1.37) represents the magnetostatic energy and the second one is the anisotropy energy. Equation (1.37) can be rewritten as 1 W
o Ms Hx sin  o Ms Hy cos  – o Ms Hk sin2 . 2

(1.40)

The external magnetic field H is represented by two components Hx and Hy and the magnetization M is represented by the angle and the saturation magnetization Ms. The anisotropy is described by the anisotropy field Hk: 2K Hk
——u . Ms

(1.41)

The minimization of the energy with respect to the magnetization angle leads to W ——
o Ms Hx cos  o Ms Hy sin  o Ms Hk sin cos
0. (1.42)  For a small value of the angle we can assume that cos  0 and sin ≈ tg and then from (1.42) the requested dependence
f(H) is expressed as Hx sin
———— . Hk  Hy

(1.43)

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When the magnetoresistor path is situated along the anisotropy axis (
0) the angle  between the magnetization direction and current direction is equal to the angle


(1.44)

and it is possible to determine the angle  as a function of measured field Hx: Hx sin 
———— . Hk  Hy

(1.45)

Taking into account equations (1.32) and (1.33) the resistance of the sensor with its path along the anisotropy axis can be expressed as 1 R(H)
Rl  R —–——— H x2 (Hy  Hk )2

(1.46)

and the change of resistance is described by the following expression R  1 ——–x
—– ——–—— H x2. Rx  (Hy  Hk )2

(1.47)

Figure 1.14 presents the calculated and experimentally determined transfer characteristics of a magnetoresistive magnetic field sensor with its path along the anisotropy axis. This kind of sensor is non-linear with the parabolic characteristic Rx /Rx
f(Hx). In the general case of the path situated in an arbitrary direction with respect to the anisotropy axis the angle  is expressed as 
  .

(1.48)

In this case the change of resistance is described by the following expression

[

Rx  H x2 ——–
—– cos2   cos 2 ——–—— Rx  (Hy  Hk )2 ——————— 2 Hx 1  ———— . (Hy  Hk )

√ (

Hx  sin 2 ———— Hx  Hk

)]

(1.49)

The change of resistance Rx /Rx
f(Hx) is the sum of the square and quasilinear components. Figure 1.15 presents an example of the calculated transfer characteristics for various locations of the path with respect to the anisotropy axis.

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THIN FILM MAGNETORESISTIVE SENSORS

Figure 1.14 The change of the resistance of a magnetoresistor with the path along the anisotropy axis: (a) Hy
0; (b) Hy
0.5 Hk ; (c) Hy
Hk (grey lines – experimental results).

Figure 1.15 The change of the resistance of a magnetoresistor with the path inclined of the angle  with respect to the anisotropy axis (Hy
0).

The pairs of the magnetoresistors with the angles 1
0 and 2
0 90° (for example 1
0° and 2
90° or 1
45° and 2
45°) are complementary (differential) and can be connected in the neighbouring arm of the bridge circuit. It is an important behaviour of these magnetoresistors because in this way it is possible to compensate the temperature error. Especially interesting is the case of the magnetoresistor with a path located on the angle 
45° with respect to the anisotropy axis. Its change of the resistance is described by following expression;

[

——————— 2 Hx 1  ———— (Hy  Hk )

√ (

R  1 Hx ——–x
—– –  ———— Rx  2 Hx  Hk

)]

(1.50)

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21

and for a small value of the measured magnetic field (Hx0.5 Hk) R  1 Hx ——–x  —– –  ———— .  2 Hk  Hy Rx

(

)

(1.51)

When four differential magnetoresistors are connected into the bridge circuit supplied by the voltage Uo the constant component is eliminated and the output signal Uout is described by the expression:  1 Uout  —– Uo ———— Hx .  Hk  Hy

(1.52)

If the orthogonal component Hy is very small (Hy