Gross R., Sidorenko A., Tagirov L. Nanoscale Devices

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54 Rudolf Gross

Fig. 1. Schematic illustration of the tunneling in a FM/I/FM tunnel junction

according to the Jullière model for parallel (a) and anti-parallel (b) magnetization

orientation. In the lower part the spin-resolved density of states of a ferromagnetic

metal is shown, which is exchange split by E

. The dotted arrows mark the spin

conserving tunneling of the spin-up and spin-down electrons.

JMR

TMR

+−

(3)

Here, the spin polarization is expressed in terms of the spin-resolved

density of states N

↑

and N

↓

, the majority and minority spin in the

ferromagnetic electrodes, as

12=)(1= −−−

−

≡

↓↑

aaa

(4)

with

)/2(1=)/(

iii

PNNNa

+≡

↓↑↑

being the fraction of majority charge

carriers and

)/(1

↓↑↓

≡−

iii

NNNa

the fraction of minority charge carriers

at the Fermi level in the two (

1,2=i ) junction electrodes. Conductance in

eqs.(1) and (3) can be expressed as

,2,1,2,1 ↓↓↑↑

∝

NNNNG

and

,2,1,2,1 ↑↓↓↑

+∝ NNNNG

(compare Fig. 1) to give eq. (3). Evidently, with

the assumptions made the Jullière's model predicts that the spin polarization

of the tunneling current is determined solely by the spin polarization of the

density of states (DOS) at the Fermi level. Jullière's result can be obtained

from a much more general Kubo/Landauer approach by assuming that the

wavevector parallel to the tunneling barrier is not conserved (incoherent

Magnetic Tunnel Junctions Based on Half-Metallic Oxides 55

tunneling) [73]. The loss of coherence can be attributed to the amorphous

tunneling barrier (e.g. Al

) commonly used in magnetic tunnel junctions.

Of course, Jullière's assumption that the tunneling conductance is

proportional to the product of the effective DOS is oversimplifying. In fact

the tunneling conductance not only depends on the number of the electrons

at the Fermi level, but also on their tunneling probability, which may be

different for different electronic states. This is in particular important for

the 3d transition metals, where the electronic structure is characterized by

dispersive s-bands which are hybridized with more localized d-band. These

features were taken into account by Stearns [74], who pointed out that the

tunneling probability depends on the effective mass, which is different for

different bands. Since the effective mass of the dispersive s-like electrons is

much smaller, the tunneling probability of these electrons is much larger

and therefore they essentially carry most of the tunneling current. Stearns

also pointed out that in many cases the dispersive bands dominating the

tunneling current are close to free-electron bands and therefore the DOS of

these bands at the Fermi level is proportional to the Fermi wave vector.

Then, assuming that the tunneling conductance is proportional to the DOS

of these itinerant electrons we can rewrite eq.(4) as

↓↑

−

≡

(5)

where

↑

k and

↓

k are the Fermi wave vectors for the spin-up and spin-

down electrons. Using band structure calculations, a spin polarization of

45% and 10% were found for Fe and Ni, respectively, in good agreement

with experiment. We see that in the model by Stearns the relevant DOS is

identified by the Fermi wave vectors of the itinerant electrons. This was an

early hint for the fact that the spin dependent tunneling sensitively depends

on the electronic structure of the electrode material.

Model of Stearns

56 Rudolf Gross

Fig. 2. Effective spin polarization

plotted versus

↑

for different values of

))/((=

↓↑↓↑

+− NNNNP

is calculated according to eq. (6).

Although the models by Jullière and Stearns have been quite successful for

the interpretation of some experiments, they have several drawbacks that

will be discussed in more detail below. The first more accurate theoretical

description of MTJs taking into account the effect of the tunneling barrier

was made by Slonczewski [75]. He considered the tunneling through a

rectangular potential barrier of height V

modeling the ferromagnetic

electrodes by two parabolic bands shifted rigidly against each other by the

exchange splitting. By solving Schrödinger's equation for the left and the

right electrode as well as for the barrier region for the parallel and

antiparallel magnetization orientation and matching the solutions at the

interfaces (doing so coherent tunneling, i.e. k

║

conservation was assumed),

he recovered the result (3), however with the effective spin polarization

↓↑

−

≡

(6)

Here,

))(/(2=

EVm −h

is the decay constant of the electron wave

function in the barrier region. As seen from Fig. 2,

depends on the

Model of Slonczewski

Magnetic Tunnel Junctions Based on Half-Metallic Oxides 57

barrier height. For high barriers (large

↑

), the Jullière result is

recovered:

PP ≅

. However, for low barriers (small

↑

), the effective

spin polarization decreases with decreasing V

and even changes sign. This

result clearly shows that the measured spin polarization may not be

characteristic for the electronic structure of the electrode material alone but

also reflects properties of the tunneling barrier. Since coherent tunneling is

assumed in Slonczewski's model, it is appropriate for the description of

junctions with epitaxially grown barriers [67].

More advanced models

Recently, it was found that the sign of the spin polarization derived from

the measured TMR effect using Jullière's model depends on the material

used for the tunneling barrier. Whereas for Co a negative spin polarization

was found for MTJs using a SrTiO

barrier [76, 77], in experiments with

barriers for Co and all other transition metals a positive spin

polarization is derived [44]. This clearly demonstrated the relevance of the

nature of the tunneling barrier. The energy, symmetry and orientation of the

unoccupied orbitals in the insulating barrier have a significant influence on

the tunneling probability and therefore the observed TMR. Actually, it has

been predicted that crystalline tunneling barriers may give rise to much

higher TMR values due to a highly spin-dependent decay of certain wave

functions with specific transverse momentum in the tunneling barrier [66-

78]. This clearly shows that the early models such as Jullière model are

much to simple, since they do neither include effects due to imperfect

barriers nor the detailed electronic structure at the interface between the

barrier and the junction electrodes [79-81]. Recent theoretical and

experimental work (for reviews see [20, 21, 82, 83]) made quite clear that a

quantitative description of spin polarized tunneling is a demanding task, if

one has to include all the details on structural and electronic properties of

the junction electrodes and the barrier material as well as specific properties

of the ferromagnet/insulator interfaces. A discussion of further aspects is

Decay of TMR with temperature and bias voltage

In most experiments a significant decrease of TMR with increasing

temperature is observed. This effect is not predicted by the majority of

theoretical

models and can be attributed to various mechanisms. First, there

given in the section on the tunneling definition of the spin polarization.

58 Rudolf Gross

may be an increase of inelastic tunneling processes with increasing

temperature which are not spin conserving. These processes can be caused

by tunneling via magnetic impurity states in the barrier [84-91]. A further

process that results in a decrease of TMR with increasing temperature is

magnon scattering [90, 92]. Tunneling including the excitation of a magnon

can be viewed as an inelastic tunneling process accompanied by spin-flip

scattering. Finally, the decrease of the surface magnetization with

increasing temperature obviously leads to a reduction of TMR [93-95]. For

example,

3/2

)(

−

∝TTM

is expected according to Bloch's law due to thermal

magnon excitation.

The decrease of TMR with increasing voltage can have similar reasons

as that due to an increasing temperature. Both for increasing k

T or eV there

may be additional inelastic tunneling channels or the excitation of magnons

resulting in additional tunneling channels which are not spin conserving

[85, 86]. For sufficiently small temperatures and voltages, the contribution

of inelastic tunneling processes due to magnon excitations is expected to

,)()(

3/2

VVGandTTG

∝∝

(7)

whereas the inelastic tunneling current due to multi-step inelastic tunneling

via impurity states is expected to follow [86, 87]

TkeVforTaTaTG

>>++ K

5/2

4/3

=)(

(8)

.=)(

5/2

4/3

eVTkforVbVbVG

>>++ K

(9)

Here, a

and b

are constants depending on the barrier thickness and the

We finally note that a variation of the TMR with varying bias voltage

also may be caused by the energy dependence of the density of states. In

this case even a sign change of the TMR with varying bias voltage may be

obtained as already discussed above [76, 77].

Form the application point of view MTJs with high TMR effect are

desirable. Based on the most simple models this can be achieved only with

density of defect states.

Magnetic tunnel junctions based on half-metals

materials having a large spin polarization (the more detailed discussion in the

section dealing with the spin polarization will show that actually the so-called

Magnetic Tunnel Junctions Based on Half-Metallic Oxides 59

That is, in half-metals only half of the electrons are conducting. With only

one spin band present at the Fermi level, half-metals are 100% spin-

polarized. Several classes of potentially half-metallic materials such as the

doped manganites, the double perovskites, the Heusler compounds,

magnetite, CrO

or diluted magnetic semiconductors have been proposed as

electrode materials for MTJs or as materials for spin injectors [97]. Some of

them already have been successfully used for MTJs and TMR values well

above 100% have been reported [98]. However, whereas the fabrication

techniques for MTJs based on ferromagnetic transition metals is well

developed and the magnetic properties of these materials are well

understood, this is often not the case for the half-metallic materials listed

above.

In this book chapter we will discuss the status of the fabrication and the

understanding of MTJs based on oxide materials with large spin

polarization. We try to address the key factors affecting the

magnetotransport properties of such MTJs, in particular the magnitude of

the measured TMR effect. In order to do so we first give different

definitions of the quantity "spin polarization'', which are relevant for

different kinds of experiments, and briefly address methods to measure the

spin polarization. We then introduce the relevant oxide materials and

present some selected experimental results on MTJs based on these

materials. Here, we will in particular address the requirement of nano-

engineering of interfaces in multilayer structures used for the

implementation of MTJs.

Spin Polarization

Discussing spin dependent transport the term spin polarization is often

used in a different context. In particular, the degree of spin polarization

determined

by different experimental methods such as spin-polarized

photo

emission, spin-dependent tunneling, or Andreev reflection does not

tunneling spin polarization is

the relevant quantity). Until now, MTJs are

mostly based on ferromagnetic transition metals and alloys. However, the

only partial spin polarization of the charge carriers at the Fermi level in these

materials sets an upper limit for the maximum TMR (about 60% at room

of MTJs, materials with a full spin polarization of the charge carries, so called

half-metals, are

temperature according to Jullière's model). In order to further improve the TMR

level, whereas there is a gap in the density of states for the other spin direction.

structure in which only states of one spin direction are present at the Fermi

desired. Half-metals are ferromagnets with an unusual band

60 Rudolf Gross

necessarily coincide with the definition (4) of the spin polarization. That is,

one has to analyze in detail how to extract the spin polarization from a

specific experiment. For example, the information obtained from a spin-

dependent tunneling experiment is giving useful information on the degree

of spin polarization only if the experimental data can be compared to

suitable model predictions. Therefore, in the following subsections we will

deepen our knowledge on the spin polarization in ferromagnets. We

introduce different definitions of this quantity and discuss, which quantity

is actually measured in which experiment.

DOS definition of spin polarization

The most natural and most often used definition of the spin polarization is

the DOS definition P

DOS

which coincides with eq. (4). In this definition, N

↑

and N

↓

are the densities of states of the spin-up and spin-down electrons at

the Fermi level given by

)(2

)(

)(2

σα

kdEEN

∫

∑

∫

∑

−

(10)

Here, E

k,α,σ

and v

k,α,σ

are the energy and velocity of an electron in band α

with spin σ = ↑,↓ and wave vector k. In eq. (10) we have replaced the

volume integral in k-space (d³k) by integrals on surfaces of constant energy

(dS

dE) by using the relation d³k = dS

┴

v(k)h

∇

Although P

DOS

can be measured by spin-polarized photo electron

spectroscopy [103-105], this definition is of limited meaning for transport

experiments, since transport properties are usually not determined by the

DOS of the spin-up and spin-down electrons alone. This is of particular

importance for materials such as transition metals with "heavy'' (e.g. d-

electrons) and "light'' (e.g. s-electrons) electrons at the Fermi level. While

the DOS often is dominated by the d-electrons, the transport properties are

dominated by the s-electrons due to their smaller effective mass [102].

Magnetic Tunnel Junctions Based on Half-Metallic Oxides 61

Transport definition of the spin polarization - the diffusive case

We now derive a transport definition of the spin polarization which is

more relevant for magnetotransport studies and spintronic devices. We start

with the diffusive transport in a metal. Classical Boltzmann transport theory

allows us to formally distinguish the current J

↑

und J

↓

of the spin-up and

spin-down electrons and therefore to give a transport definition of the spin

polarization as

↓↑

−

≡

(11)

By using the Boltzmann expression for the current density (diffusive

transport) [106]

)()(2

Ε⋅

∫

∑

dSe

v(k)v(k)

σα

(12)

where

Ε is the electrical field and we take into account that several bands

with index α can contribute to transport, we obtain for the current density

of the two spin directions

σσσ

〉〈∝

NveJ

(13)

with

)(2

||)(2

dSv

dSNv

ασ

ασασ

∫

∑

∫

∑

〉〈

(14)

Note that the second equality only holds for an isotropic material. With this

expression we can express the electrical conductivity as

2222

↓↓↑↑

〉〈+〉〈

ττσ

eNveNv

(15)

The comparison with the simple Drude expression

shows, that the

expression

〉〈

corresponds to

)*/( mn

, that is, to the effective density

of electrons of a specific spin direction weighted by 1/m*, i.e. their

effective contribution to transport.

62 Rudolf Gross

Assuming that the scattering time

is spin independent, we can give

the following simplified definition of the transport spin polarization in the

diffusive limit [108, 109]:

↓↑

〉〈+〉〈

〉〈−〉〈

≡

NvNv

iff

(16)

If, however, the scattering time is spin dependent and/or if there is spin-flip

scattering, the expression for the spin polarization becomes much more

complicated, since the current density in each spin channel then depends on

the properties of both spin channels.

We see that in contrast to the DOS definition of the spin polarization,

where only N

enters, the transport definition of the spin polarization is

determined by

〉〈

Nv . This is expected, since transport is not only

affected by the DOS, i.e. the number of charge carriers, but also by their

velocity

*1/mv

∝

The definition (16), although interesting from the pedagogical point of

view, is however of minor practical relevance, since usually the currents J

↑

and J

↓

cannot be measured separately in a specific material. In typical

experiments the spin dependent transport between a ferromagnetic and a

non-magnetic material (e.g. a superconductor, semiconductor or normal

metal) is measured. Therefore, we have to discuss the transport across the

interface between such materials. In the next subsection, we start with the

purely ballistic case, where interface scattering and the mismatch between

the Fermi velocities is negligible. Then, this case is extended to more

general situations.

Transport definition of the spin polarization - the ballistic case

We now derive a definition of the spin polarization for the transport across

a contact between a ferromagnetic and non-magnetic material. We denote

this as the contact definition of spin polarization. Our discussion follows

the original work of Sharvin [110], which has been extended by Mazin et

al. to allow for an arbitrary Fermi surface [108, 109]. We assume that an

electron passing the contact area experiences an acceleration through the

electrical field so that its energy has increased by eV after passing the

contact. Here, V is the voltage drop across the contact. If in this process the

quasi-momentum of the electrons is changed from ħk to ħk', we can

immediately state the phase space available for this process at T=0. Since

only initial states below E

are occupied, i.e. ε

= E

- E

≤ 0 and there are

Magnetic Tunnel Junctions Based on Half-Metallic Oxides 63

free final states only above E

, i.e. ε

k´

≥ 0, we can write the available phase

space by using the Heavyside function

.)(=)()(=)()(

kkkk'

eVeV

−

(17)

Following the discussion in [108, 109] we now consider the fraction of

electrons with a given k, which can reach the contact area per unit time. If

the contact plane is perpendicular to the z-direction, this fraction is just v

with A the contact area and v

the z-component of the electron velocity. The

total single spin current density is given by the product of the charge unit,

velocity and the number of available states, that is, by

,||=)(

)(2

σασασ

εδ

〉〈

∫

∑

vNVekdeVv

(18)

where we have defined

〉〈

Nv completely analogous to eq. (14). Using

eq. (10) we can rewrite this expression as

)(2

edS

vVeJ

ππ

ασ

∫

∑

(19)

where S

F,z

is the projection of the Fermi surface on the contact plane, which

we have assumed perpendicular to the z-direction. Note that for a spherical

Fermi surface we just recover the well known Sharvin result.

We can compare eq. (19) with the well-known Landauer expression

saying that the conductance G = J·A/V of a single conduction channel is

just G

= e²/h. The total conductance is then given by G

times the number

of conduction channels N

, which is given by the number of electrons that

can pass the contact area. If translational symmetry within the contact plane

is not violated, the parallel component k

║

of the wave vector is conserved

and N

is given by the product of the contact area A and the two-

dimensional momentum density. The latter is just given by S

F,z

/(2π)², that

is, by the product of the projection S

F,z

of the Fermi surface on the contact

plane and the two-dimensional density of states 1/(2π)² in k-space. We

obtain for the single spin conductance

.||=

)(2

AvNeA

σσσ

〉〈

(20)

Gross R., Sidorenko A., Tagirov L. Nanoscale Devices - Fundamentals and Applications

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