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AMR-Effekt
[Bearbeiten | Quelltext bearbeiten]Englischer Text aus meiner Diplomarbeit (unformatiert)
[Bearbeiten | Quelltext bearbeiten]In many ferromagnetic materials and alloys such as iron, nickel, or a mixture of those like permalloy (Ni80Fe20) the anisotropic magnetoresistance (AMR) 1Synonyms are the spontaneous magnetoresistance anisotropy (SMA) or the ferromagnetic resistivity anisotropy. 4 2. THEORY appears as the change of few percent in electrical resistivity �(#) depending on the angle # = \(~j, ~M ) between the direction of the electrical current ~j(~r) and the orientation of the samples magnetisation ~M (~r). Extreme cases are collinear and perpendicular orientation with maximal �k or minimal resistivity �?. A cos2-term can well describe the variation between those extremes [8]. �(#) = �? + (�k − �?) cos2 # = �? + �� cos2 #, = �k + (�? − �k) sin2 #, (2.1) = �? cos2 # + �k sin2 #. Resistivity r [W] -180 -90 0 90 180 Angle q between current and magnetisation [°] r∥ r⊥ -180° -90° 0° 90° 180°
Figure 2.1: Electrical resistivity �(#) as a function of the angle
- = \(~j, ~M ) between the direction of the electrical current ~j(~r) and the
orientation of the samples magnetisation ~M (~r). 2.1.1 Formalism of the AMR The AMR ratio �AMR is defined as the normalized variation in resistivity and is a good measure for the size of the effect2. It can be obtained directly 2In many publications the AMR ratio is also given as ��/�ave where �ave = 1 3�k + 2 3�? is the average value for truly demagnetized polycrystalline bulk material. In this case 2.1. Anisotropic magnetoresistance 5 from experiment by measuring the resistance parallel and perpendicular to the magnetisation. �AMR = (Rk − R?) Rk = �R Rk , = (�k − �?) �k = �� �k . (2.2) Taking the z-axis along the direction of magnetisation, the resistivity can be written in form of a tensor as ˆ�ik = 0@ �? −�H 0 �H �? 0 0 0 �k 1A . (2.3) With ~uM being the unit vector in the direction of the magnetisation, Ohm’s law can be given in the form of ~E = ˆ�ik ~j = �? ~j + �� (~uM ·~j)~uM + �H ~uM×~j (2.4) The diagonal elements are the resistivities along or perpendicular to the magnetisation while the off-diagonal elements ±�H represent the spontaneous or anomalous Hall effect, which is small for permalloy and shall not be discussed here. 2.1.2 Fundamentals of the AMR An exhaustive quantum mechanical description of the effect is rather lengthly. Solely an overview on the microscopic origin of the anisotropic magnetoresistance as well as a short introduction to the basic principles of ferromagnetism in iron and nickel will be given. This should give a qualitative understanding of the nature of the AMR (see also [11], [12] and [13]). A more extensive description can be found in [8]. Ferromangetism in transition metals Exchange interaction In a model assuming localized electrons one can describe the interaction between the electron spins with the Hamiltonian introthe magnetisation of the domains is randomly orientated throughout the three dimensions. In thin films where the magnetisation is only in-plane the expression changes to �ave = 1 2�k + 1 2�?. 6 2. THEORY duced by Heisenberg [14]: HHeisenberg = −2A �1�2. (2.5) For a positive exchange constant (A > 0) the two spins �1 and �2 will energetically prefer a parallel orientation, which leads to a ferromagnetic spin lattice. A negative exchange constant (A < 0) will cause the spins to orientate antiparallel and promote an antiferromagnetic spin lattice. This exchange interaction can issue from divers sources: either directly from a relevant overlap of the electron orbits or indirectly via interaction with electrons of diamagnetic atoms situated between the atoms of the (anti-) ferromagnetic lattice. This is a good model for antiferromagnetic manganous oxide (Mn2+O2−). Additionnaly the conduction electrons can act as mediator between the spins as proposed by M. A. Rudermann, C. Kittel, T. Kasuya and K. Yosida. RKKY-interaction can play a relevant role in the ferromagnetism of rare earths [15]. In transition metals the spins of the 3d-like electrons are responsible for the ferromagnetism. In those materials these electrons cannot be considered as localized since they hybridize with the 4s-like electrons and form a half filled conduction band. Here the exchange interaction between the quasi free electrons lowers the energy of the majority electrons (") and rises that of the minority electrons (#) as shown in Fig. 2.2. a) 10 20 30 10 20 30 0 EF 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Energy (Ry) D(E) [103/Ry]
- "
b) 0.4 0.3 0.2 0.1 0.5 0.6 0.7 0.8 0.9 Energy (Ry) 0 10 20 30 10 20 30 EF s,p-band d-band D(E) [103/Ry]
- "
Figure 2.2: Density of states in nickel divided into majority (") and minority (#) spin states according to J. Callaway and C. S. Wang [16] a) From self consistent band structure calculations b) Schematic illustration of the s,p- and d-bands 2.1. Anisotropic magnetoresistance 7 Domain patterns In a macroscopic ferromagnetic structure the magnetisation will generally not point homogeneously in one direction, but divide into domains in which the magnetisation is parallel, divided by domain walls, where the magnetisation changes direction. The average over many domains eventually sums up to zero, so that an iron nail may seem without magnetisation. This is due to a series of energy terms, that oppose and balance: Etotal = Eexchange + Estray-field + Eanisotropy + EZeeman + Eother. (2.6) The exchange energy, as described by the Heisenberg Hamiltonian, tends to parallelize the spins and therefore prefers a homogeneous magnetisation. This is in contradiction to the second term, that represents the energy of the stray field, which would maximize for a homogeneous magnetisation. The third term denotes the symmetry of the lattice, that makes certain directions of magnetisation more favourable than others. The face-centred cubic lattice of nickel, for example, prefers magnetisations in one of the four [111] directions, i. e. the space diagonals. The Zeeman energy arises from the interaction with the applied magnetic field and tends to align the magnetisation with ~B ext. The last term contains other energies like the magnetostrictive energy, which can be neglegted here. As a result, a magnetic object may have a magnetisation as in Fig. 2.3, where the stray field is minimized and seven domains result with a near zero average magnetisation. a) b) c) Figure 2.3: Possible domain pattern in a 2×4 µm2 permalloy element of 25 nm thickness: a) Measured magnetisation with a magnetic force microscope (MFM). b) Calcutated MFM image from simulation. c) Micromagnetic computer simulation of the magnetisation. 8 2. THEORY Origin of the AMR Spin-orbit coupling The dependence of the resistivity on the angle between the magnetisation and the current as illustrated in Fig. 2.4 is due to electron scattering from the 4s,p-band to the 3d-band connected by a spin flip. This additional scattering channel is opened by the spin-orbit interaction which contributes to the Hamiltonian in the form of Hspin-orbit = K L S = K Lz Sz + K2 L+ S− + K2 L− S+. (2.7) Here the generators and annihilators L± = Lx ± iLy and S± = Sx ± iSy, if applied to an electron wave function, can increase or decrease the orbital quantum number or flip the spin. The operator product L+ S− turns a majority spin p"-electron wave function into that of a d#-electron. Unlike the rather isotropically widely spread s,p-states, the more localized d-states have a strong orbital anisotropy. This accounts for an anisotropic scattering cross section for interband scattering with spin flip, that is bigger for s,p-electrons with a momentum parallel to the orbit of the empty 3d-state and therefore, due to spin-orbit coupling, bigger if parallel to the magnetisation [4]. a) b) Figure 2.4: Illustration of the origin of the anisotropic magnetoresistance. The direction of the sample’s magnetisation is connected to the spin of the 3d-electrons. The anisotropic 3d-orbits have a bigger cross section for an electron current in the direction of the magnetisation. a) If the magnetisation is perpendicular to the current direction ( ~M ?~j), then the resistivity is low. b) For parallel orientation (Mk~j) the resistivity is higher. Since the s,p-band has a few times higher group velocity vg(EFermi) = 1 ~ @E(k) @k and a smaller effective mass m? = ~2( @2E(k) @k2 )−1 at the Fermi energy than the d-band, as one can see from band structure 2.1. Anisotropic magnetoresistance 9 calculations and photoelectron spectroscopy [18, 16, 19], mostly s,p-like electrons contribute to the conductivity in permalloy. It is assumed, that the high density of states D#d(EFermi) of the minority spin d-electrons at the Fermi energy is responsible for the short mean free path of the minority s,p-conduction electrons, which results in a high resistivity for the minority s,p-subband. Thus mostly majority spin s,p-electrons contibute to the conductivity and therefore the spin-orbit interaction plays a relevant role, since it scatters the main conduction electrons into the biggest reservoir of empty states at the Fermi energy. r (B) r^ r | | | | ^ Bext[mT] Figure 2.5: Resistivity of a multi-domain ferromagnetic structure in dependence of an applied magnetic field ~Bext. The red curve shows the behaviour with the magnetic field in the direction of the current (k). The blue curve corresponds to the perpendicular case (?). Applying magnetic fields As described above, a macroscopic ferromagnetic structure consists of domains with different directions of magnetisation. These domains can be aligned by an applied magnetic field ~Bext. The strength of the magnetic field ~Bsat, that is needed to saturate the alignment varies with material, form, direction, and temperature of the ferromagnetic structure. Although the alignment of the domains is not brought about by simultaneously rotating the magnetisation in each domain, a quasi continuous alignment process of the magnetisation of the sample is possible. The domain walls are shifting in such a way, that the domains with a magnetisation in direction of ~Bext grow on cost of those pointing in other directions. 10 2. THEORY A typical AMR signature is depicted in Fig. 2.5. Starting from a demagnetized multi-domain state at zero field, the magnetisation becomes aligned as the field strength rises. An external magnetic field ~Bext either parallel or perpendicular to the current results in a rise or fall of the resistance, respectively. In ferromagnetic microstructures certain domain configurations like that in Fig. 2.3 are energetically favourable. In these cases, the domain configuration can switch from one state to another, so that the magnetisation process is a series of quasi continuous (reversible) domain-wall movements separated by (irreversible) configuration changes. The shape of the ferromagnetic microstructure can induce hard and easy axes of magnetisation, e.g. directions in which the structure can easily be saturated and othes in which the magnetic field strength for saturation Bsat is much higher. In cases of extreme shape anisotropy quasi single-domain microstructures can be produced. In thin permalloy films with a thickness of about or less than t . 100 nm the magnetisation is preferably in-plane and the domain walls are N´eel walls, where the magnetisation between the domains is rotating in the plane of magnetisation. 2.1.3 Magnitude of the AMR a) 1 2 3 4 5 60 70 80 90 50 Concentration of nickel in iron x [%] b) 1 2 3 4 5 40 60 80 20 Film thickness t [nm] c) 2 100 200 4 6 8 Temperature T [K] Figure 2.6: Magnitude of the AMR-Ratio in NixFe1−x alloys: a) Depending on the concentration x of nickel at room temperature [2]. b) Depending on the film thickness in Ni80Fe20 at 4 K [20]. c) Depending on the temperature in Ni80Fe20 [4]. The size of the effect usually given in percent as �AMR = �R R changes with material, temperature, shape, and many other parameters. Especially 2.2. Magnetic-force microscopy 11 in thin films the resistance R is influenced by grain size, surface quality, and film thickness and therefore varies with deposition rate, groth temperature, heat treatment, and vacuum quality. In Fig. 2.6 the variation with three main parameters is shown. For room temperature measurements Ni80Fe20 permalloy films with a thickness of t & 20 nm still show a reasonable effect.