# Giant magnetoresistance

Albert Fert (2011), Scholarpedia, 6(2):6982. | doi:10.4249/scholarpedia.6982 | revision #91316 [link to/cite this article] |

The **Giant Magnetoresistance** (GMR) is the large change in the electrical resistance which is induced by the application of a magnetic field to thin films composed of alternating ferromagnetic and nonmagnetic layers. This change in resistance, in general a reduction, is related to the field-induced alignment of the magnetizations of the magnetic layers. In the first experiments, the film was composed of layers of Fe (ferromagnetic) and Cr (nonmagnetic) with typical thicknesses of a few nm and the current was in the plane of the film. GMR effects can also be obtained with the current perpendicular to the layers. The origin of the GMR is the dependence of the electrical conduction in ferromagnetic materials on the spin state of the carriers (electrons).

## Discovery and first experiments

The GMR was discovered in 1988 by the team of Albert Fert (Baibich 1988) in France on Fe/Cr(001) multilayers and, independently, by Peter Grünberg (Binash 1989) and coworkers in Germany on Fe/Cr/Fe(001) trilayers, in both cases on samples grown by Molecular Beam Epitaxy (MBE). We show in Figure 1 some of the first experimental curves. The GMR can be described as the reduction of the resistance of the magnetic multilayers due to the alignment of the magnetizations of the Fe layers by the applied magnetic field.

The discovery of 1988 had been preceded in 1986 by Grünberg’s Brillouin scattering experiments (Grünberg 1986) showing that two layers of Fe separated by an ultra-thin (« 1 nm) layer of Cr were antiferromagnetically coupled. Consequently, in zero applied field, the magnetizations of the Fe layers are antiparallel (in opposite direction) but can be aligned by an applied field. The field-induced decrease of the resistance in the experiments of the discovery of the GMR (Baibich 1988, Binash 1989) is thus the resistance variation between the resistance R_{AP} and R_{P} of the AP (antiparallel) and P (parallel) magnetic configurations. The amplitude of the GMR is generally characterized by the ratio \((R_{AP}- R_{P})/ R_{P}\ .\) For example, this ratio reaches 80% for the multilayer with 0.9nm thick Cr layers in the results of Figure 1 a. The GMR decreases rapidly as the layer thickness increases. The interpretation of the GMR in terms of spin dependent scattering, proposed by Fert’s group (Baibich 1988), and the corresponding theoretical models are presented in Section 3.

The publications reporting the discovery of GMR rapidly attracted attention for its fundamental interest as well as for the many possibilities of applications, and the research on magnetic multilayers became very active. In 1990 Parkin and coworkers (Parkin 1990) demonstrated the existence of GMR in multilayers (Fe/Cr, Co/Ru and Co/Cr) made by the simpler and faster technique of sputtering. They could explore very broad thickness ranges and found the oscillatory variation of the magnetoresistance which reflects the oscillations of the interlayer exchange coupling as a function of the spacer thickness. GMR effects exist in the thickness ranges where the coupling is antiferromagnetic (AF) and vanishes when the coupling is ferromagnetic, as shown in Figure 2. The oscillations are modulated by the general decrease of the GMR with the thickness. The oscillatory behavior disappears and only a continuous decrease subsists in the thickness range where the exchange coupling becomes weaker than the coercive field.

Other important advances were obtained at the beginning of the nineties. In 1990 Shinjo and Yamamoto (1990), as well as Dupas et al (1990), demonstrated that GMR effects can be found in multilayers without antiferromagnetic interlayer coupling but composed of magnetic layers having different coercivities. The first results on trilayers of “spin valve” type in which the magnetization of one of the magnetic layer is pinned by interaction with an antiferromagnetic layer were also obtained in 1991. We will present the spin valves and their applications in Section 4. Another important result in 1991, illustrated by Figure 2 b, was also the observation of large and oscillatory GMR effects in Co/Cu, which became an archetypical GMR system. The first observations were obtained at Orsay (Mosca 1991) with multilayers prepared by sputtering at Michigan State University and at about the same time at IBM (Parkin 1991). Finally it can be noted that the highest GMR ratio, 220%, was obtained by Schad et al. (1994) on Fe/Cr multilayers . All these results are for the GMR with the Current In the layer Planes (CIP-GMR). After 1991, measurements have been also performed with the Current Perpendicular to the layer Planes (CPP-GMR), which leads to the different properties described in Section 5.

## Spin dependent conduction in a ferromagnetic conductor

The origin of the GMR is the dependence of the electrical conduction in ferromagnetic materials on the spin state of the carriers (electrons). This is a consequence of the spin spitting of the energy bands in the ferromagnetic state illustrated in Figure 3 a.

The spin dependence of the conduction in ferromagnetic metals or alloys has been first suggested by Mott (1936) before being experimentally demonstrated and quantitatively described at the end of the sixties by Fert and Campbell (Fert 1968) for series of iron- and nickel-based alloys. Similar results could be rapidly found in several other systems (Loegel 1971, Dorleijn 1977, Fert 1976). The experimental results can be accounted for in the “two current model” of the conduction in ferromagnetic metals (Fert 1968, 1976). In this model the resistivity of a ferromagnetic conductor is expressed as \[\tag{1} \rho = {{\rho _ \uparrow \rho _ \downarrow + \rho _{ \uparrow \downarrow } (\rho _ \uparrow + \rho _ \downarrow )} \over {\rho _ \uparrow + \rho _ \downarrow + 4\rho _{ \uparrow \downarrow } }} \]

\(\rho_\uparrow\) and \(\rho_\downarrow\) are the resistivities of the spin\(\uparrow\) (majority spin direction) and spin\(\downarrow\) (minority spin direction) channels. \(\rho_{\uparrow \downarrow}\) is the spin mixing resistivity term expressing the transfer of momentum between the two channels by spin-flip scattering. In the low temperature limit (T<<T_{C}) of the “two current model”, when the spin flip scattering of the conduction electrons by magnons is frozen out, the spin mixing rate is much smaller than the momentum relaxation rate and the resistivity of the ferromagnet is expressed as
\[\tag{2}
\rho = {{\rho _ \uparrow \rho _ \downarrow } \over {\rho _
\uparrow + \rho _ \downarrow }} \]

This correspond to the situation with conduction in parallel by two independent channels illustrated in Figure 3 b. Surprisingly this simplified version of the two current model has been generally adopted in the theories of the GMR and only few models take into account the spin mixing term. The asymmetry between the two channels is characterized by spin asymmetry coefficients: \[\tag{3} \alpha = {{\rho _ \downarrow } \over {\rho _ \uparrow }} \]

or alternatively, \[\tag{4} \beta = {{(\rho _ \downarrow - \rho _ \uparrow )} \over {(\rho _ \downarrow + \rho _ \uparrow )}} = {{\alpha - 1} \over {\alpha + 1}}\]

There are several origins for the difference between \(\rho_\uparrow\) and \(\rho_\downarrow\ .\) Schematically, the resistivity \(\rho_\sigma\) can be written as a function of the number \(n_\sigma\ ,\) effective mass \(m_\sigma\ ,\) relaxation time \(\tau_\sigma\) and density of states (DOS) at the Fermi level \(n_\sigma(E_F)\) of spin \(\sigma\) electrons in the following way \[\tag{5} \rho _\sigma = {{m_\sigma } \over {n_\sigma e^2 \tau _\sigma }} \]

with, for a given type of scattering potential (without spin-flip) characterized by its matrix elements \(V_\sigma\) and in the Born approximation \[\tag{6} {\tau _\sigma}^{-1} \sim {\mid {V_\sigma} \mid}^2 n_\sigma (E_F) \]

There are intrinsic origins of the spin dependence of \(\rho_\sigma\) that are related to the spin dependence of \(n_\sigma\ ,\) \(m_\sigma\ ,\) or \(n_\sigma (E_F) \ .\) In transition metals, the most important of these intrinsic origins comes from the proportionality of the relaxation rate to the DOS, \(n_\sigma (E_F) \ ,\) in Eq. (6). In first approximation, it can be said that a major part of the current is carried by light electrons of s character and that these electrons are more strongly scattered when they can be scattered into heavy states of the d band for which the DOS is large. In Ni, Co and alloys like NiFe or CoFe, see Figure 3 a, the \(d_\uparrow\) band is below the Fermi level and \( n_{d\uparrow}(E_F)=0 \ .\) From Eq. (6) with \( n_{d\sigma}(E_F)\neq 0 \) only for spin\(\downarrow\) direction, s-d scattering exists only for the spin\(\downarrow\) s electrons, so that there is a general intrinsic tendency for stronger scattering and larger resistivity in the spin\(\downarrow\) channel. However the largest asymmetries between \(\rho _ \uparrow\) and \(\rho _ \downarrow\) can be induced by extrinsic effects, in particular by doping with impurities presenting a strongly spin dependent scattering cross section (Fert 1968, Loegel 1971, Dorleijn 1977, Fert 1976). In Figure 3 c, we show the examples of the spin\(\uparrow\) and spin\(\downarrow\) resistivities induced by 1% of several magnetic types in Ni. For Co and Fe, the ratio \(\alpha\) of \(\rho _ \uparrow\) to \(\rho _ \downarrow\) is larger than 1 (\( \beta > 0 \)) and can be as large as 20. In contrast, for Cr or V, \(\alpha\) is smaller than 1 (\( \beta < 0 \)). This can be explained by the electronic structure of these different impurities in Ni (Fert 1976). In multilayers, similar arguments can also explain the opposite spin asymmetries of the scattering at, for example, Co/Cu and Co/Cr interfaces.

## Physics and theoretical models of the GMR with Current In the layer Planes (CIP-GMR)

The physical mechanism of the GMR proposed by Fert and coworkers in their first publication on GMR (Baibich 1988) is illustrated in a simple free electron picture by Figure 1 c. Let us suppose that the majority and minority spin electrons are very differently scattered when they go through a magnetic layer. If the magnetizations of all the magnetic layers are parallel, one of the spin directions will be weakly scattered by all of them and there will be shorting of the conduction by the corresponding channel (bottom sketch). On the contrary, if the magnetizations are alternating (top sketch), the electrons of both channels will be strongly scattered by every second magnetic layer, the short-circuit by one of the channel disappears and the resistance is much higher. However this simple picture assumes that the electrons average the scatterings occurring in different magnetic layers, what holds only when the distance between these layers, i.e. the nonmagnetic layer thickness, is much smaller than the electron mean free path (MFP). Also a part of the scatterings inside the magnetic layers becomes inactive when their thickness exceeds the MFP. This explains that large GMR effects can be observed only for layers thinner than the MFP, practically for thicknesses in the nm range. For a simple estimate of the GMR ratio, let us call \(r_{+}^{P}\) and \(r_{-}^{P}\) the resistances of the spin+ and spin- channels in the P configuration (throughout the paper, our notation is + and – for the spin directions corresponding to \(s_z =+1/2\) and \(s_z =-1/2\ ,\) z being an absolute axis, and \(\uparrow\) and \(\downarrow\) for the majority and minority spin directions inside a ferromagnet). If for example, \(r_{+}^{P}\) is much smaller than \(r_{-}^{P}\ ,\) the current is shorted by the spin + channel and the final resistance is small \( \approx r_{+}^{P} \ .\) In the antiparallel (AP) configuration, the electrons are alternatively majority and minority spin electrons, so that, at least in the limit of thicknesses much smaller than the “mean free path”, the resistance of both channels can be written as \[\tag{7} r_{+}^{AP} = r_{-}^{AP} = \frac{ r_{+}^{P} + r_{-}^{P} }{2}\]

For the resistance of the two-channel system, this leads to \[\tag{8} {\rm{r}}_{{\rm{AP}}} = {{{\rm{r}}_ {+}^{P} + {\rm{r}}_ {-} ^{P} } \over {\rm{4}}} > {\rm{r}}_{\rm{P}}= \frac{{\rm{r}}_ {+}^{P}{\rm{r}}_ {-} ^{P}}{{\rm{r}}_ {+}^{P}+{\rm{r}}_ {-} ^{P}} \]

and to the GMR ratio
\[\tag{9}
{\rm{GMR}} = {{{\rm{r}}_{{\rm{AP}}} - {\rm{r}}_{\rm{P}} } \over
{{\rm{r}}_{\rm{P}} }} = {{{\rm{(r}}_ -^{P} - {\rm{r}}_ +^{P}
{\rm{)}}^{\rm{2}} } \over {{\rm{4r}}_ +^{P} {\rm{r}}_ -^{P} }}\]

For a more realistic picture, one must take into account that the layers are not much thinner than the MFP, consider the physical origin of the spin dependent scattering and also go beyond free electron models. Figure 4 represents schematically the potential landscape seen by the electrons. Figure 4 (a) and (b) are for the spin + and spin – electrons in the parallel (P) configuration and (c) is for any spin directions in the antiparallel (AP) configuration. The potential can be separated into, (a) the intrinsic potential of the perfect multilayered structure (superlattice), which determines the wave functions of the electrons carrying the current, and (b) the scattering (extrinsic) potentials due to defects (atomic disorder, impurities, interface roughness) and represented by spikes. In a magnetic multilayer both the intrinsic and extrinsic potentials are spin dependent.

(a) Let us first consider the role of the intrinsic potential represented in Figure 4 by steps of Kronig-Penney potentials. The spin dependence of these steps are related to the exchange splitting of the energy bands in a ferromagnetic metal. For a perfectly ordered structure the interferences between Bragg-like specular reflections at the interfaces would build the Bloch functions of an artificial superlattice. However the superlattice approach is valid for coherent interferences between the specular reflections, that only applies if the MFP is much longer than the multilayer period. For real multilayers, with bulk scattering and also a significant probability of scattering by roughness defects at each interface, the MFP cannot be many times longer than the layer thickness. Consequently a superlattice approach is rarely appropriate, which is confirmed by the absence of most superlattice effects, oscillations of the conductance as a function of the layer thickness for example. A more realistic approach is the so-called “layer by layer” approach, in which one considers the specular reflections of the wave functions in each layer but not the interference between the reflections at successive interfaces. Nevertheless the specular reflections play an important role for the GMR as they can channel the current in some of the layers and lead to different efficiencies for the spin dependent scatterings in different layers.

(b) The extrinsic potentials, represented by spin dependent spikes in Figure 4, are associated with bulk or interface scatterings. Both are spin dependent and contribute to the GMR. Their respective contributions depend on the density of interfaces (i.e. on the thicknesses) and also, as described a few lines above, on the different channelling of the electrons in the different layers. More quantitative data on the respective importance of bulk and interface effects can be derived from the analysis of the CPP-GMR (see Section 5). It turns out that the interface contribution is generally predominant for a few nm thick layers, the bulk contributions becoming larger for thicknesses exceeding the 5-10 nm range.

To sum up, a realistic description of the GMR demands to treat the bulk and interface spin dependent scatterings of wave functions more or less channelled inside the layers by interface specular reflections. The MFP fixes the range in which the different spin dependent scatterings must be averaged. The main difficulty for quantitative theoretical predictions is the limited information we have on the defects at the origin of the bulk and interface spin scattering potentials and on their spin dependence. However the theoretical models can describe qualitatively all the main features, as it turns out from the review of models in the next lines.

The first model of GMR was the semi-classical free electron model of Camley and Barnas (1989). This is a free electron model in which the GMR is calculated from bulk scattering probabilities and interface scattering, reflection and transmission coefficients. The scattering probabilities inside the magnetic layers and the interface coefficients are spin dependent. The major success of this model was to predict the thickness dependence of the GMR. In the limit of thick nonmagnetic layers the GMR decreases exponentially as a function of the ratio of the thickness to the MFP. As a function of the thickness of the magnetic layers, the GMR vanishes as the inverse of the thickness.

The first quantum mechanical model of the GMR was introduced by Levy and coworkers (1990) who used the Kubo formalism to calculate the conductivity of free electrons scattered by a distribution of spin dependent potentials. The model, as the one of Camley of Barnas (1989), explains the thickness dependence of the GMR. However, for a realistic comparison with experimental results, it is necessary to replace the free electron picture by an accurate description of the spin-polarized band structure. This has been done first using tight-binding models and then in several types of ab initio models based on the Local Spin Density Approximation (LSDA). One of the important results of such ab initio calculations is the concept of quantum channelling discussed above (Zahn 1998). For quantitative predictions, the scattering potentials of defects, interface roughness or impurities must be introduced into the models but little is know on these imperfections of the multilayers. Consequently the theory of the GMR cannot be really predictive. For an extensive review of the theoretical models, we refer to a review article of Levy and Mertig (2002).

## Spin-valves, applications

GMR requires that an antiparallel configuration of the magnetizations in the multilayers can be switched into a parallel one by applying a magnetic field. In the first GMR experiments the AP configuration was induced by antiferromagnetic interlayer exchange but this is not the only way to obtain an antiparallel configuration. For example, in multilayers combining hard and soft magnetic layers, the GMR effects can be obtained by switching only the soft layer (Shinjo 1990, Dupas 1990). The best known structure in which interlayer exchange is not used to obtain an AF configuration and GMR effects, is the spin valve structure, introduced in 1991 by Dieny et al (1991) and now used in most applications of GMR.

A spin valve structure, in its simplest form shown in Figure 5 a, consists of a magnetically soft layer separated by a nonmagnetic layer from a second magnetic layer which has its magnetization pinned by an exchange biasing interaction with an antiferromagnetic (FeMn) or ferrimagnetic layer. The operation of the spin valve can be understood from the magnetization and magnetoresistance curves shown in Figure 5 b. One of the permalloy layers has its magnetization pinned by the FeMn in the negative direction. When the magnetic field is increased from negative to positive values, the magnetization of the free layer reverses in a small field range close to H=0, whereas the magnetization of the pinned layer remains fixed in the negative direction. Consequently, the resistance increases steeply in this small field range. Magnetic multilayers of the spin valve type are used in most applications of GMR, in particular the read heads of hard discs, see Figure 6. More details about the applications of GMR can be found in review articles (Parkin 2002, Chappert 2007).

## GMR with the Current Perpendicular to the layer Planes (CPP-GMR), spin accumulation effects

During the first years of the research on GMR, the experiments were performed only in the CIP geometry, that is with currents flowing along the layer planes. It is only in 1991 that experiments of GMR with the Current Perpendicular to the layer Planes (CPP-GMR) begun to be performed. This was done first by sandwiching a magnetic multilayer between superconducting electrodes (Pratt 1991, Bass 1999), then by electrodepositing multilayers into the pores of a polycarbonate membrane (Piraux 1994, Fert 1999) and, more recently, in vertical nanostructures (pillars) fabricated by e-beam lithographic techniques (Albert 2000). In the CPP-geometry, the GMR is not only definitely higher than in CIP but also subsists in multilayers with relatively thick layers, up to the micron range in Figure 7 a for example. Actually, as explained in the Valet-Fert model of the CPP-GMR (Valet 1993), spin-polarized currents flowing perpendicularly to the layers induce spin accumulation effects and the final result is that the length scale governing the thickness dependence becomes the “long” spin diffusion length (related to the spin relaxation) in place of the “short” mean free path in the CIP-geometry. Similar effects for single interfaces had already been described by Johnson and Sisbee (1987).

The physics of the spin-accumulation occurring when an electron flux crosses an interface between a ferromagnetic (F) and a nonmagnetic (N) material is explained in Figure 8 for a simple situation (single interface, no interface resistance, no band bending, single polarity). In Figure 8 a, the incoming electron flux is predominantly carried by the spin up direction whereas the outgoing flux is carried equally by both spins. Consequently there is accumulation of spin up electrons at the interface and this accumulation diffuses on both sides of a F/N interface to a distance of the order of the spin diffusion length. In terms of electron distribution the spin accumulation is described as a splitting of the spin up and spin down Fermi energies (chemical potentials), as shown in Figure 8 b. The spin-flips generated by this out of equilibrium electron distribution in the spin accumulation zone provide the mechanism of the adjustment between the incoming and outgoing spin currents. To sum up, the spin polarization of the current decreases progressively as it goes through this broad spin accumulation zone. In a similar way, for the current in the opposite direction, a similar mechanism progressively polarizes the current. In both cases, the current spin-polarization just at the interface depends on the proportion of the depolarizing (or polarizing, depending on the direction of the current), spin-flips induced by the spin accumulation in F and N.

In the multi-interface structure of a CPP-GMR experiment, there is an interplay between the spin accumulation effects at successive interfaces. The spin accumulation in a non-magnetic layer is larger for an AP magnetic configuration in which the easily injected spin direction is the less easily extracted. The CPP-GMR is related to the difference between the spin accumulation in the P and AP configuration. The GMR ratio vanishes only when the thickness becomes larger than the spin diffusion length, in agreement with the persistence of the GMR up to much thicker layer than in the CIP geometry, see Figure 7.

The physics of spin accumulation can be described by new types of transport equations (Valet 1993), often called drift/diffusion equations, in which the electrical potential is replaced by a spin and position dependent electro-chemical potential. The electro-chemical potentials in different layers are coupled by boundary conditions involving spin dependent interface resistances. These equations have been extensively applied to the interpretation of the experimental results of CPP-GMR (Pratt 1991, Bass 1999, Fert 1999) and an example is shown in Figure 7 b.

More generally, the spin accumulation effects govern the propagation of a spin-polarized current through any succession of magnetic and nonmagnetic materials and play an important role in all most recent developments of spintronics. The diffusion current induced by the accumulation of spins at the magnetic/nonmagnetic interface is the mechanism driving a spin-polarized current at a long distance from the interface, well beyond the ballistic range (i.e. well beyond the mean free path) up to the distance of the spin diffusion length (SDL). The drift-diffusion equations of the CPP-GMR can be applied to understand the spin transport in various types of devices. In particular they have been applied to explain the difficulty of the spin injection from a magnetic metal into a semiconductor, the so-called “conductivity mismatch” problem (Schmidt 2000), and to show how this problem can be solved by the insertion of spin dependent interface resistances (Rashba 2000, Fert 2001).

## Concluding remarks

GMR is best known by the grand public for its application to the hard disc drives and the resulting considerable increase of the disc capacities. However more important, in my opinion, is that the GMR boosted the research of other spin-induced transport effects and, finally, triggered the development of the new field of research and technology called spintronics. An important second stage, after 1995, was the research on the magnetoresistance of the magnetic tunnel junctions (TMR) (Moodera 1995, Miyasaki 1995). The TMR has now replaced the GMR in a majority of hard disc drives and is also applied in the type of memory called M-RAM (Magnetic Random Access Memory). The physics of spin accumulation revealed by the CPP-GMR has also been extended to the situation of spin transport in a lateral nonmagnetic channel between magnetic contacts. The channel can be a metal, a semiconductor or a carbon-based conductor like carbon nanotubes (CNT) or graphene, the lateral geometry introducing the possibility of a gate to manipulate the spin polarization for logic or transistor-like applications. Promising results have been obtained with CNT (Hueso 2007) in which spin-polarized currents turn out to be propagated to distances well above the micron range. Similar spin propagations at long distances can be expected for graphene and probably other carbonic materials, which can lead to new concepts of information processing based on the manipulation of spin currents. Of particular importance is also the concept of spin transfer introduced by Slonclewski (Slonczewski 1996). Spin transfer is the opposite of magnetoresistance effects like GMR or TMR. Whereas, in GMR or TMR, a magnetic configuration is detected by a current, in spin transfer a magnetic configuration is created by a “spin transfusion” from a spin polarized current. Spin transfer will be applied to write MRAM memories or to generate oscillations in the microwave frequency range. Many other fascinating directions of research are emerging today on the road of spintronics: spin photonics, Spin Hall Effect, topological insulators, spin quantum computing, neuromorphic electronics…GMR was only the first step.

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## See also

- Fert, A., Barthélémy, A. and Petroff, F. (2006). Spin transport in Magnetic Multilayers and Tunnel Junctions. In Nanomagnetism: Ultrathin Films and Nanostructures, ed. F. Mills and J. A.C. Bland, 153-226 Amsterdam : Elsevier
- Parkin, S.S.P. (2002). Applications of magnetic nanostructures. In Spin Dependent Transport in Magnetic nanostructures, ed S. Maekawa and T. Shinjo, 237-279. Taylor and Francis.