Spin electronics-ZieseThornton, Notas de estudo de Física

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Contents

Part I Introduction

1 Introduction to Spin Electronics J. F. Gregg........................................................ 3

Part II Basic Concepts

2 An Introduction to the Theory of Normal and Ferromagnetic Metals G. A. Gehring..................................................... 35

3Basic Electron Transport B. J. Hickey, G. J. Morgan and M. A. Howson........................ 52

4 Phenomenological Theory of Giant Magnetoresistance J. Mathon........................................................ 71

5 Electronic Structure, Exchange and Magnetism in Oxides D. Khomskii...................................................... 89

6 Transport Properties of Mixed-Valence Manganites M. Viret.......................................................... 117

7 Spin Dependent Tunneling F. Guinea, M. J. Calder´on and L. Brey............................... 159

8 Basic Semiconductor Physics H. J. Jenniches.................................................... 172

9 Metal–Semiconductor Contacts D. I. Pugh........................................................ 199

10 Micromagnetic Spin Structure R. Skomski........................................................ 204

11 Electronic Noise in Magnetic Materials and Devices B. Raquet......................................................... 232

1 Introduction to Spin Electronics

J. F. Gregg

Clarendon Laboratory, Oxford University, Parks Road, Oxford OX1 3PU, U.K.

1.1 Coey’s Lemma

The driving force behind Spin Electronics is neatly summarized in J. M. D. Coey’s incisive observation [1] that “Conventional Electronics has ignored the spin of the electron”. In every hi-fi and radio set, 50% of the conducting electrons tend to be spin-up and the remainder are spin down (where up and down relate to some locally induced quantisation axis in the relevant wires and devices). Yet, although electron spin was known about for most of the 20th Century, no technical use is made of this fact.

1.2The Two Spin Channel Model

The mechanistic basis for Spin Electronics is almost as old as the concept of electron spin itself. In the mid-thirties, Mott postulated [2] that certain elec- trical transport anomalies in the behaviour of metallic ferromagnets arose from the ability to consider the spin-up and spin-down conduction electrons as two independent families of charge carriers, each with its own distinct transport prop- erties. Mott’s hypothesis essentially is that spin-flip scattering is sufficiently rare on the timescale of all the other scattering processes canonical to the problem that defections from one spin channel to the other may be ignored, hence the relative independence of the two channels [3,4,5].

1.2.1 Spin Asymmetry

The other necessary ingredient of this model is that the two spin families con- tribute very differently to the electrical transport processes. This may be because the number densities of each carrier type are different, or it may because they have different mobilities – in other words that the same momentum or energy scattering mechanisms treat them very differently. In either case, the asymmetry which makes spin-up electrons behave differently to spin-down electrons arises because the ferromagnetic exchange field splits the spin-up and spin-down con- duction bands, leaving different bandstructures evident at the Fermi surface. If the densities of electron states differs at the Fermi surface, then clearly the number of electrons participating in the conduction process is different for each spin channel. However, more subtly, different densities of states for spin-up and spin-down implies that the susceptibility to scattering of the two spin types is different, and this in turn leads to their having different mobilities.

M.J. Thornton and M. Ziese (Eds.): LNP 569, pp. 3–31, 2001. ©c Springer-Verlag Berlin Heidelberg 2001

4J. F. Gregg

1.2.2 Spin Accumulation

Let us consider two spin channels of different mobility (Fig. 1.1). When an electric field is applied to the metal, there is a shift, ∆k, in momentum space of the spin-up and spin-down Fermi surfaces in accordance with the equation:

F = eE = ℏ dk dt

∆k τ

where F is force on carrier, E is electric field, e is electronic charge, τ is electron scattering time given by μ = eτ /m∗, μ being the electron mobility and m∗^ the electron effective mass. Since the channels have different mobilities, this shift is different for the spin-up and spin-down Fermi surfaces as illustrated.

displaced Fermi spheres

Electric Field

Brillouin zone

k = 0

∆k

Fig. 1.1. The shift of the Fermi surface when an electric field is applied to a ferro- magnet is shown. The solid circles represents the Fermi sphere of up and down spin electrons in a field, the dashed circle represents the Fermi sphere in zero external field.

From Fig. 1.1, it is evident that the spin-up electrons are performing the lion’s share of the electrical conducting, and, moreover, that if a current is passed from such a spin-asymmetric material – for example cobalt – into a paramagnet like silver (where there is no asymmetry between spin channels [6]), there is a net influx into the silver of up-spins over down-spins. Thus a surplus of up-spins appears in the silver and with it a small associated magnetic moment per volume. This surplus is known as a “spin accumulation”. Evidently, for constant current flow, the spin accumulation cannot increase indefinitely; this is because as fast as the spins are injected into the silver across the cobalt-silver interface, they are converted into down-spins by the slow spin-flip processes which we have hitherto ignored. This spin-flipping goes on throughout all parts of the silver which have been invaded by the spin accumulation. So now we have a dynamic equilibrium between influx of up-spins and their death by spin-flipping. This in turn defines a

6 J. F. Gregg

other materials. For a mathematically rigorous analysis of the spin-accumulation in terms of the respective electrochemical potential of the spin channels, the reader is referred to Valet and Fert [12] from which it can be seen, numerical factors apart, that the crude “drunken sailor” model gives a remarkably accurate insight into the physics of this problem.

1.2.6 How Large is a Typical Spin Accumulation?

It is also of interest to estimate how large is the spin accumulation for typical current densities. The calculation is done by balancing the net spin injection across the interface: dn dt

Aαj e

with the total decay rate of spins due to spin flipping in the entire volume influenced by the spin accumulation:

A τ↑↓

0

ndx =

n 0 A τ↑↓

0

exp

−x λsd

dx =

An 0 λsd τ↑↓

A is sectional area, j is current density, n is number density of excess spins, x is distance from the interface, α is ferromagnet spin polarization. This in turn gives a spin accumulation just inside the interface of

n 0 =

αjτ↑↓ eλsd

3 αjλsd evFλ

Putting in typical numbers of j = 1000 Amps/cm^2 , α = 1, vF = 10^6 m/s, λ = 5 nm, λsd = 100 nm, gives n 0 = 4 × 1022 m−^3. Thus, given an electron density of 3 × 1028 m−^3 , it is seen that only one part in 10^6 of the electrons are spin polarized. The significance of this will be discussed below. Incidentally the magnetic field B associated with this spin accumulation is:

B = μ 0 M = μ 0 μB n 0 (1.6) = 10−^6 × 10 −^24 × 1022 = 10 nTesla!! (1.7)

This is experimentally very hard to detect, especially considering the magnetic fields caused by the current which generates the spin accumulation in the first place.

1.3 Two Terminal Spin Electronics

The next step in the Spin Electronic story is to make a simple device and this is realized by making a sandwich in which the “bread” is two thin film layers of ferromagnet and the “filling” is a thin film layer of paramagnetic metal (Fig. 1.2). This is the simplest Spin Electronic device possible. It is a two-terminal passive device which in some realizations is known as a “spin valve” and it passes muster in the world of commerce as a Giant Magnetoresistive hard-disk read-head.

1 Introduction to Spin Electronics 7

Ferro Para Ferro

Fig. 1.2. Passive two terminal spin electronic device.

Empirically, the device functions as follows [13]: The electrical resistance is measured between the two terminals and an externally applied magnetic field (supplied for example by the magnetic information bit on the hard disk whose orientation it is required to read) is used to switch the relative magnetic orien- tations of the ferromagnetic layers from parallel to antiparallel. It is observed that the parallel magnetic moment configuration corresponds to a low electrical resistance and the antiparallel state to a high resistance. Changes in electrical resistance of order 100% are possible in quality devices, hence the term gi- ant magnetoresistance, since by comparison with, for example, anisotropic magnetoresistance in ferromagnets, the observed effects are about 2 orders of magnitude bigger.

1.3.1 The Analogy with Polarized Light

There are a variety of different ways – of varying rigour – to consider the op- eration of this spin valve structure. To keep things simple, let us analyse it by analogy with the phenomenon of polarized light. In the limit in which the fer- romagnets are half-metallic, the left hand magnetic element supplies a current consisting of spin-up electrons only which produce a spin accumulation in the central layer. If the physical thickness of the silver layer is comparable with or smaller than the spin diffusion length, this spin accumulation reaches across to the right hand magnetic layer which, on account of its being half-metallic, acts as a spin filter, just as a piece of Polaroid spectacle lens acts as a polarized light filter. The spin accumulation presents different densities of up and down electrons to this spin filter which thus lets through different currents depending on whether its magnetic orientation is parallel or antiparallel to the orientation of the polariser (i.e. the first magnetic layer). The only difference with the case of crossed optical polarisers is that in optics the extinction angle is 90 degrees. In the spin electronic case it is 180 degrees [14].

1 Introduction to Spin Electronics 9

Fig. 1.3. A 10×10 matrix with the memory elements 0.1 microns in size. One of the project goal of the European funded framework 5 network NANOMEM (courtesy of M. Hehn, Universit´e Henri Poincar´e, Nancy, France).

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 1.4. Currently state of the art MRAMs use: (a) semiconductor diodes to prevent current shortcuts. Shown in (b) MIM diodes and (c) TTRAMs with selective polar- isation are being developed to replace the semiconductor diodes and prevent current shortcuts. With (d), (e) & (f) the respective MRAMs in array form (courtesy of M. Hehn, Universit´e Henri Poincar´e, Nancy, France).

results. There is a useful lesson here for later design work: always make sure your spin polarization is injected at the right part of the energy bandstructure.

10 J. F. Gregg

1.3.4 CIP and CPP GMR [19]

In fact there are two configurations in which our simple two terminal device can work – they are respectively described as current in plane (CIP) and cur- rent perpendicular to plane (CPP). Above, we have discussed only the latter in which the critical lengthscale for the magnetic phenomena is the spin diffusion length. The physics involved in CIP operation is rather different and the critical lengthscale here is the mean free path. However we shall leave the discussion of this case since it is not central to the theme of this chapter. The reader is referred to G. Mathon’s chapter for further details.

1.4 Three Terminal Spin Electronics

Electronically, the natural progression is from this two terminal device to a three terminal one, and this step was made by Mark Johnson [20,21,22] who achieved it simply by introducing a third contact to the intermediate paramagnetic base layer to create the Johnson Transistor (Fig. 1.5). Now in the language of bipolar transistors, we can speak of a base, an emitter, and a collector, the last two being

Ferro Ferro

Pump V

Fig. 1.5. Johnson transistor.

the ferromagnetic layers. Just like its bipolar counterpart, the Johnson transistor may be used in various configurations; the one we discuss here is chosen because it gives insight into yet another way to analyse spin filtering and spin accumula- tion. We leave the collector floating and monitor the potential at which it floats using a high impedance voltmeter. Meanwhile a current is pumped round the emitter-base circuit and this causes a spin accumulation in the base layer as before. The potential at which the collector floats now depends on whether its magnetic moment is parallel or antiparallel to the magnetization of the polarizing emitter electrode which causes the spin accumulation. Evidently this potential may be altered by using an external magnetic field to switch the relative orienta- tion of the emitter and collector magnetic moments. To analyse this behaviour,

12 J. F. Gregg

1.5.1 Giant Thermal Magnetoresistance

Magnetic Field (T)

T=10K

Phase (degrees) arbitrary origin

(a)

(b)

Fig. 1.6. Schematic set-up for measurement of the giant thermal magnetoresistance in a GMR mechanical alloy shown in (a). With the thermal GMR effect in a mechanical alloy shown in (b). For comparision the electrical GMR is also shown inverted and superimposed on the lower trace, with the axes arbitary.

As an interesting aside, the Wiedemann Franz Law (WFL) tells us that there is a close relationship between electrical transport and heat transport in most materials. Thermal and electrical conductivities are limited in most regimes by the same scattering processes and the WFL tells us that in these circumstances their quotient is a constant times absolute temperature. Moreover, this close relationship extends to magnetotransport in mesomagnetic systems. Figure 1. shows measurement of the Giant Thermal Magnetoresistance in a giant magne-

1 Introduction to Spin Electronics 13

toresistive mechanical alloy. The analysis is identical to the electrical case. Spin information is encoded onto a thermal current in one part of the device and read off again in a different part of the device: the result is a thermal resistance which varies with applied magnetic field by many percent [23].

1.5.2 The Domain Wall in Spin Electronics

Another example of the intrigue of Mesomagnetism may be seen by considering the geometrical similarity between a spin-valve structure and a ferromagnetic domain wall as illustrated in Fig. 1.7. In both cases, regions of differential

Maj Maj Min Min

Maj Maj Min Min

(a)

(b)

Fig. 1.7. Geometric similarities of (a) FM domain wall and (b) a GMR trilayer.

magnetization are separated by an intermediate zone which takes the form of a thin film of nonmagnetic metal and a region of twisted magnetization in the respective cases. The spin valve functions provided that spin conservation occurs across the intermediate zone. This suggests a model of domain wall resistance [24,25,26] in which the value of the resistance is determined by the degree of spin depolarization in the twisted magnetic structure which forms the heart of the domain wall. The model invokes magnetic resonance in the ferromagnetic exchange field to determine the degree of electron spin mistracking on passing

1 Introduction to Spin Electronics 15

gain without the addition of two extra electrodes and a transformer structure. The underlying design problem with the device is that it is entirely Ohmic in operation since all its constituent parts are metals. Clearly another technology progression is needed and this is the introduction of Hybrid Spin Electronics – the combination of conventional semiconductors with spin-asymmetric conducting materials. At a stroke, this releases to the Spin Electronic designer all the armoury of semiconductor physics such as exploiting diffusion currents, depletion zones and the tunnel effect to create new high- performance spin-devices.

1.6.1 The Monsma Transistor

The first Hybrid Spin Electronic device was the Monsma transistor [28,29,30] produced by the university of Twente which was fabricated by sandwiching a traditional spin valve device between two layers of silicon. Three electrical con- tacts are made to the spin-valve base layer and to the respective silicon layers. The spin valve is more sophisticated than that illustrated in Fig. 1.9a and com- prises multiple magnetic/nonmagnetic bilayers, but its operating principle is the same. Schottky barriers form at the interfaces between the silicon and the metal structure and these absorb the bias voltages applied between pairs of terminals. The collector Schottky barrier is back biased and the emitter Schottky is for- ward biased. This has the effect of injecting (unpolarised) hot electrons from the semiconductor emitter into the metallic base high above its Fermi energy. The question now is whether the hot electrons can travel across the thickness of

Emitter Base Collector

GMR Multilayer

Si Si

e -

(a) (b)

x

Collector barrier

λ 1

n(x)^ λ^2

Fig. 1.9. Monsma transistor: first attempt to integrate ferromagnetic metals with sili- con shown in (a). In (b) the average energy of both spin types plotted as a function of distance. The thick line denotes scattering for both spin types in an antiferromagnet- ically aligned mutilayer (both species experiences strong scattering) and the thin line denotes the scattering when the layers are ferromagnetically aligned (only one species will experiences strong scattering).

16 J. F. Gregg

the base and retain enough energy to surmount the collector Schottky barrier. If not, they remain in the base and get swept out the base connection. By varying the magnetic configuration of the base magnetic multilayer the operator can determine how much energy the hot electrons lose in their pas- sage across the base. If the magnetic layers are antiferromagnetically aligned in the multilayer then both spin types experience heavy scattering in one or other magnetic layer orientation, so the average energy of both spin types as a function of distance into the base follows the thick line exponential decay curve (λ 1 ) of Fig. 1.9b. On the other hand, if the magnetic multilayer is in applied field and its layers are all aligned, one spin class gets scattered heavily in every magnetic layer, whereas the other class has a passport to travel through the structure relatively unscathed and the average energy vs distance of this priv- ileged class follows the thin curve (λ 2 ). It may thus be seen that for parallel magnetic alignment, spins with higher average energy impinge on the collector barrier and the collected current is correspondingly higher. Once again we have a transistor whose electrical characteristics are magnetically tunable. This time, however, the current gain and the magnetic sensitivity are sufficiently large that, with help from some conventional electronics, this is a candidate for a practical working device. It may be seen from comparison of the two traces of Fig. 1.9b that there is a trade-off to be made in determining the optimum base thickness. A thin base al- lows a large collector current harvest but affords little magnetic discrimination. A thick base on the other hand means a large factor between the collector currents corresponding to the two magnetic states of the multilayer but an abysmally small current gain. (The low current gain has always been the Achilles Heel of metal base transistors, and is probably the main reason for their fall from grace as practical devices despite their good high frequency performance owing to the absence of base charge storage.) An interesting feature of the Monsma transistor is that the transmission se- lection at the collector barrier is done on the basis of energy. Thus the scattering processes in the base which determine collected current are the inelastic ones. Elastic collisions which change momentum but not energy are of less significance. This contrasts with the functioning of a spin valve type system in which all mo- mentum changing collision processes have the same status in determining device performance [31].

1.6.2 Spin Transport in Semiconductors

The Monsma transistor represents a very important step in the evolution of Spin Electronics. It is the first combination of spin-selective materials with a semiconductor. However, as yet, the semiconductor is used only to generate barriers and shield the spin-dependent part of the device from electric fields. To release the full potential of Hybrid Spin Electronics we need to make devices which exploit spin-dependent transport in the semiconductor itself.

18 J. F. Gregg

100^ Cobalt

Doped Silicon gap

10

30

-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12

Sweep 1 Sweep 2 Sweep 3

(R

-RH

)/R 0

(%) 0

Field (kOe)

(a)

(b)

Fig. 1.10. Experiment performed to directly spin-inject into a semiconductor and ob- serve the polarization of the current which emerges on the other side is shown in (a). The resulting transport measurements (b) suggests that the spin diffusion length in sil- icon is at least many tens of microns, but the spin injection process at the metal/silicon interface is highly inefficient.

cause) and they are compatible with the observed domain magnetization pro- cesses for the cobalt pads. They appear to correspond to spin transport through the semiconductor, and as such they correlate well with earlier experiments us- ing nickel injectors [34]. Interestingly however the spin transport effects are of order a few percent at best, yet the effect decays only very slightly with silicon channel length and was still well observable for 64 micron channels. The message would seem to be that the spin diffusion length in silicon is many tens of microns at the least, but that the spin injection process at the metal/silicon interface is highly inefficient. This direct injection inefficiency is being widely observed and its cause is still hotly debated. It may arise from spin depolarization by surface states [36], or it may be explainable by the Valet/Fert model in which spin injec- tion is less efficient for materials of very different conductivities [37]. It may also be because the spin injection is not being implemented at the optimum point in the semiconductor bandstructure. From the latter point of view, spin tunnel injection into semiconductors is a more versatile technique, since, for a given injected tunnel-current density, the necessary bias (and hence the point in the band-structure where injection occurs) may be tuned by varying the thickness and/or the tunnel barrier height.

1 Introduction to Spin Electronics 19

A very beautiful direct measurement of semiconductor spin diffusion length has been made by decoupling from the spin injection problem [38,39] and gen- erating the spin polarized carriers in the semiconductor itself (see Fig. 1.11). Gallium Arsenide [40,41] was used as the host which has the property that,

(a)

(b) (c)

Fig. 1.11. Lateral drag of spin coherence in Gallium Arsenide has been measured by Faraday rotation as shown in (a). A new spin population is created every time a pump pulse hits the sample as shown in (b). The electrons in each new population then drift along the electric field. When observed at some time after injection, each population will have drifted an amount proportional to its age as well as experienced an exponential decay in its Faraday signal. A number of field scans can be taken over a range of displacements in order to identify the spatial extent of each spin population and track its movement in time, as shown in (c). Spin transport can be observed at distances exceeding 100 microns (after [39]).

when pumped with circularly polarized light, the selection rules are such as to