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

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64 Rudolf Gross

We see that the conductance in the ballistic limit is proportional to

〉〈 ||

, while the conductivity in the diffusive limit is proportional to

〉〈

in the ballistic limit as

↓↑

〉〈+〉〈

〉

〈

−

〉

〈

≡

all

NvNv

(21)

Again,

all

is determined by

〉

〈

Nv and not by

N alone, since

transport across the contact is not only determined by the DOS in the two

contact electrodes but also by the velocity of the charge carriers. For a

spherical Fermi surface the projection S

F,z

of the Fermi surface on the

contact plane is given by

resulting in

/4=

σσσ

khAkeG ∝

. This

gives

))/((=

2222b

↓↑↓↑

+−

FFFF

all

kkkkP

, in contrast to expressions (5) or (6).

We note that ballistic transport between a ferromagnetic metal and a 2D

electron gas has also been studied in the context of spin filtering effects

[111, 112]. Ballistic point contacts between superconductors and normal

metals resp. ferromagnets have been discussed by de Jong and Beenakker

[113, 114].

Transport definition of the spin polarization - the extended

ballistic case

We now extend our discussion of the transport across an interface between

a ferromagnetic and a non-magnetic material to the case of a non-ideal

interface, where we have an additional barrier and also a mismatch of the

Fermi velocities of the two materials. The non-ideal barrier is modeled by a

δ-type potential U(z) with a strength determined by the dimensionless

parameter Z:

.)(=)(=)(

vZzWzzU h

⋅

(22)

Here, v

and k

are the Fermi velocity and the Fermi wave vector,

respectively. For a vanishing barrier (Z=0) we have an ideal interface

(ballistic limit with transmission probability (T=1), whereas for a strong

barrier (Z>>1) we have T<<1 similar to a tunnel junction. That is, by

introducing the parameter Z we can describe the continuous transition from

an ideal contact to contact with small transparency as a tunnel junction.

. In analogy to eq. (16) we can define the "contact spin polarization''

Magnetic Tunnel Junctions Based on Half-Metallic Oxides 65

In contrast to the case of an ideal interface with Z=0, for the non-ideal

interface we now receive a finite reflection probability. That is, the

transmission probability T through the contact is 0<T<1 and we obtain the

single spin conductance of the contact by using the Landauer-Büttiker

formula [115] as

σα

∑

(23)

where we have to sum up the transmission probabilities of all available

conduction channels. The calculation of the transmission probability

depends on the particular model used for the description of the MTJ. Here,

we use an extended Blonder-Tinkham-Klapwijk model [116].

The transmission probability

ασ

is determined by the barrier strength

Z and the mismatch of the Fermi velocities

in the ferromagnet and

the non-magnetic material. In order to determine the transmission

probability we are solving Schrödinger's equation

ΨΨ













∂

− EzU

=)(

(24)

in the region of the ferromagnet (left) and the non-magnetic material (right)

and match the solutions at the interface taking into account the boundary

conditions. An electron moving from left to right in the ferromagnet will be

reflected/transmitted at the interface to the non-magnetic material with

certain probabilities. For the wave function of a spin-up electron we have

kk ⋅













⋅













↑

−

↑

r e

=)(

(25)

k ⋅













↑

↑↑

t e

=)(

(26)

with equivalent expressions for a spin-down electron. Here,

↑↓

t and

↑↓

r are

the transmission and reflection coefficients for the spin-up and spin-down

electrons at the interface. For Z=0 we have t

↑↓

= 1 and r

↑↓

= 0. The

nomenclature













means that we are dealing with an electron and not with a

hole. If we would use a superconductor instead of a normal metal, the

66 Rudolf Gross

electron can be Andreev reflected as a hole













and in the superconductor

we have quasiparticles













, which represent mixed electron/hole states

[107].

Fig. 3. Sketch of the band structure at the contact between a ferromagnet (left) and

a non-magnetic metal (right). In the ferromagnet the lower conduction band edge of

the minority spin band is shifted upwards by the exchange splitting E

. The

quantity Γ represents the difference of the lower conduction band edges of the

ferromagnet and the non-magnetic metal.

If we assume that only electrons with energies

≅

are contributing

to the electrical transport, we have

nff

EEEE ≅≅≅

↓↑

and we can write

the absolute values of the Fermi wave vectors as

↑

(27)

)(2

exFe

EEm

−

↓

(28)

)(2

Γ−

(29)

Here, Γ is the difference between the lower conduction band edges of the

ferromagnet and the non-magnetic metal (see Fig. 3).

Magnetic Tunnel Junctions Based on Half-Metallic Oxides 67

In order to calculate the transmission probability, we assume that the

interface is perfectly flat so that we can neglect diffusive scattering. In this

case we can assume conservation of k

║

. For an angle of incident

of the

electrons on the interface we then have

/=sin

and we can write for

the wave vectors in z-direction, i.e. perpendicular to the interface (compare

Fig. 4)

φφ

σσ

cos=)(

(30)

.)]([)(=)(

,||

φφ

kkk −

(31)

Applying the boundary conditions at the interface at z=0 [116]

0)=,,(=0)=,,( zyxzyx

σσ

ΨΨ

(32)

=0)=,,(|

+−

∂

Ψ∂

Ψ+

∂

Ψ∂

zyxU

(33)

and using the wave functions (25) and (26) for both spin directions it is

straightforward to calculate the spin dependent transmission and reflection

coefficients (see e.g. [148-150]). One obtains

)/()(

=||=),(

hWvv

σσ

(34)

Here we have assumed parabolic bands, i.e.

m/=

σσ

kv h

and

m/= kv h

. We see that the transmission probability depends on the Fermi

velocities and the parameter Z characterizing the barrier strength. Even for

the case of an ideal interface (Z=0) the transmission probability may be less

than unity due to a possible mismatch of the Fermi velocities. It is evident

from Fig. 4 that for large k

║,σ

the transmission probability will be zero for

the chosen situation, since there is no available k

║

in the non-magnetic

metal. This corresponds to the case of total reflection in optics. Only if the

Fermi velocities are equal in both junction electrodes (E

= Γ = 0), we

obtain the well-known Blonder-Tinkham-Klapwijk result T = 1/(Z²+1) and,

hence, T = 1 for Z = 0 [116].

To calculate the current density across the interface we have to replace

the transmission probability T = 1 in (19) by

1<),( ZT

and obtain

68 Rudolf Gross

),(

)(2

vZTVeJ

ασ

ασσ

∫

∑

(35)

Fig. 4. Wave vectors at an interface between a ferromagnet and a non-magnetic

metal for spin-up (top) and spin-down (bottom) electrons. Note that due to the

conservation of k

║

and different Fermi velocities at both sides of the interface there

is electron diffraction at the interface

If we consider the case of a strong barrier, we can neglect in the expression

for

),( ZT

the first term in the denominator of eq. (34) and obtain

.)(=

)(

)(2

ασ

〉〈∝

∫

∑

(36)

Defining again the spin polarization with this expression for the current

density, we get

Magnetic Tunnel Junctions Based on Half-Metallic Oxides 69

↓↑

〉〈+〉〈

〉〈−〉〈

≡

arrier

NvNv

(37)

We see that in the tunneling limit a definition of the contact spin

polarization is obtained that corresponds to that derived for the diffusive

limit. We finally would like to emphasize that it is difficult to give a simple

expression for the spin polarization for the case of intermediate barrier

strength.

Tunneling definition of the spin polarization

In many experiments spin polarized tunneling through a thin insulating

barrier is used to determine the spin polarization. As already pointed out in

two junction electrodes and the tunneling probability of the spin-up and

spin-down electrons given by the tunneling matrix element |M

↑

|² and |M

↓

|²,

respectively. With ρ

↑

= N

↑

|² and ρ

↓

= N

↓

|² with i= 1,2 for the two

junction electrodes we obtain the tunneling conductance for the parallel and

antiparallel magnetization orientation to

↓↓↑↑

+∝

2121

ρρρρ

(38)

2121

↑↓↓↑

+∝

ρρρρ

(39)

With the definition (1) of the TMR we obtain

,2t,1t

unun

TMR

−

(40)

with

||||

↓↓↑↑

↓↑

−

MNMN

iun

ρρ

(41)

We see that (40) reduces to the well-known Jullière expression (3) for |M

↑

|²

↓

|², that is, for the same tunneling probability of the spin-up and spin-

down electrons, which was explicitly assumed in deriving the Jullière result

(3). In this case the tunneling spin polarization P

tun

is equal to the density of

introduction, the tunneling conductance is determined by both the DOS in the

70 Rudolf Gross

state polarization P

DOS

. However, since the assumption |M

↑

|² = |M

↓

|² is not

valid in most cases, the experiments based on spin dependent tunneling

rather yield the tunneling than the DOS spin polarization.

Although there is a huge amount of experimental data on spin dependent

tunneling, the interpretation and comparison of the experimental data is

complicated, since the measured TMR values are often affected by the

following factors:

- the crystallographic orientation of the ferromagnetic electrodes [117],

- the presence of thin nonmagnetic interface layers [79, 80, 120-126] or

surface layers with reduced or changed magnetic properties [127-130],

- the reduction of TMR due to inelastic tunneling via localized states in

the barrier [85-100] or magnon excitation [92, 93, 118, 119]

- the influence of the barrier material on the tunneling probability

[60, 76, 131-144],

- the influence of the crystallographic orientation of the barrier on the

tunneling probability [61-67].

Hence, an accurate description of spin dependent tunneling taking into

account all the details of the electronic, structural and magnetic properties

of the tunnel junction electrodes and interfaces as well as imperfections of

the tunneling barrier is very demanding.

For pedagogical reasons we briefly discuss the case of a perfect planar

FM/I/FM tunnel junction. Due to periodicity within the plane we have k

║

conservation and hence obtain the tunneling conductance per spin channel

by summing up over all k

║

to (compare (23))

∑

(42)

In order to get G

we have to calculate the transmission coefficient T

k||σ

which depends again on the particular model.

In the most simple case we just can assume a rectangular potential barrier

and free electrons in the junction electrodes (free electron model). Then, the

transmission coefficient is given by [75, 145]

.16=

κκ

−

(43)

Here, d is the thickness of the tunneling barrier,

1/22

||,e

}]))[(/{(2= kEEEmk

Fxi

−−−

σσ

h is the component of the wave

vector perpendicular to the tunneling barrier (i = 1,2 denotes the two

junction electrodes) and

1/22

||0

}))(/{(2= kEVm

+−h

is the decay

Magnetic Tunnel Junctions Based on Half-Metallic Oxides 71

constant of the electron wave function in the barrier region of height V

The exchange splitting of the free electron bands is included via E

,σ

which has opposite sign for the two spin directions. Equation (43) shows

that for a thick enough barrier due to the dominating exponential factor only

those electrons contribute significantly to the tunneling conductance, which

have the smallest κ, i.e. those with k

≈ 0. In this limit we recover the

expression (6) for the spin polarization derived by Slonczewski. For thinner

barriers the spin polarization will depend on the barrier thickness as shown

by MacLaren [145]. Although a free electron description is quite intuitive it

cannot be used for deriving quantitative results, since the calculations are

sensitive on the barrier shape and cannot take into account multi-band

effects [146]. Nevertheless we see that even in the very simple free electron

model the TMR and the derived spin polarization P

tun

are not only

determined by the properties of the ferromagnetic junction electrodes but

also by the barrier properties. Since the transmission coefficient that enters

the Landauer formula (42) is flux conserving, it incorporates a factor of v

the component of the band velocity perpendicular to the interface. This has

been emphasized already above discussing the importance of the band

velocity perpendicular to the interface when interpreting spin-dependent

transport in the ballistic and diffusive limit.

Note that expressions (23) and (42) for the extended ballistic case (strong

barrier) and the tunneling case look quite similar. However, one has to keep

in mind that in the tunneling case the transmission probability is dominated

by the exponential factor exp(-2

d). This results in a selection of states

within a narrow tunneling cone, which have v

≈ v

or equivalently k

≈ 0.

This is important if one has to deal with single crystalline junction

electrodes, since in this case only certain crystallographic directions

contribute to the tunneling current. The derived spin polarization then does

no longer represent a Fermi average. This is different for the extended

diffusive case, where one is averaging over all v

. Nevertheless, in junctions

with polycrystalline electrodes the experimental results for the tunneling

case and the extended diffusive case will be quite similar, since in the

tunneling case the Fermi surface averaging is obtained by the averaging

over many grains with different crystallographic orientations.

An important aspect in spin dependent tunneling is the chemical bonding

at the ferromagnet/insulator interface. It was shown that a change of the

interfacial bonding can reduce the spin polarization and even can change its

sign [131]. The physics behind this is simple. Suppose, for example, that

for a specific interfacial bonding mainly the s states of a 3d transition metal

are coupled to those of the insulator. Then the tunneling current is mainly

72 Rudolf Gross

determined by the s-states and also the resulting TMR and spin

polarization. If in contrast there is a significant coupling of the d-states to

those of the insulator, the d electrons due to their high DOS will mainly

determine the tunneling current and the resulting spin polarization. As a

result, for Co or Ni a different sign of the spin polarization is obtained, if

the s- (positive P) or the d-electrons (negative P) are dominating the

tunneling current.

Today spin dependent tunneling in MTJs is modeled by combining

density functional theory within the local spin density approximation

(LSDA) for the electronic structure and the Landauer-Büttiker formalism

for the conductance [147]. In this approach a multiband description of the

electronic structure is used taking into account the electronic states in the

ferromagnetic junction electrodes, the interfacial states, the variation of the

potential across the tunneling barrier and the evanescent states in the

insulator. Recently, Mavropoulos et al. pointed out the relevance of the

evanescent gap states in the tunneling barrier [78]. They showed that states

of a specific symmetry have a small decay constant κ and therefore will

eventually dominate the tunneling current. Recently, Butler et al. [66]

showed, that due to this effect in Fe/MgO/Fe tunnel junctions for Fe(100)

oriented electrodes the majority spin electrons dominate the conductance

resulting in a very high TMR. This was confirmed by Mathon and Umerski

[67] and by recent experiments [61, 62].

Spin polarization revisited

It is interesting to note that until now the DOS definition of the spin

polarization is still commonly used to define the degree of spin polarization

although this quantity is completely irrelevant for the magnetotransport

properties mostly exploited in devices. Experimental values for P

DOS

are

still derived by interpreting the measured TMR values with the

oversimplifying Jullière's model. A likely explanation for this strange

situation may be the fact that in tunneling experiments usually the

conductance is measured as a function of voltage and then this dependence

is used to derive the DOS. Indeed this can be done for

superconductor/normal metal contacts. However, this is not possible for

ferromagnet/non-magnetic metal junctions, where two different spin

channels are compared.

Discussing the expressions for the spin polarization derived above we

have to make clear that P

DOS

, P

Nv²

and P

tun

may be completely different

in real materials. For example, in transition metals we often can distinguish

parts of the Fermi surface that have overwhelming d- or s-character [102].

Magnetic Tunnel Junctions Based on Half-Metallic Oxides 73

Those parts with d-character are responsible for the major part of the DOS

at the Fermi level given by eq.(10). In contrast, those parts with s-character

have small effective mass and high velocity and, hence, dominate <Nv²>

(compare eq. (14)). In the same way, the tunneling probability is much

larger for s electrons with their small effective mass, since T ~ exp(-2

with

κ ∼

(m*)

1/2

. Hence, it is evident that the differences in P

DOS

, P

Nv²

and P

tun

are the larger the larger the differences in the Fermi velocities of

the d- and s-like electrons. This can be shown by band structure

calculations as shown in Fig. 5, where P

DOS

, P

, and P

Nv²

are plotted as a

function of energy for Fe and Ni [108].

Fig. 5. The different degrees of spin polarization, P

DOS

, P

, and P

Nv²

plotted versus

energy for Fe (a) and Ni (b) (data according to Ref. [108]).

In general, a large value of P

Nv²

or P

tun

can have two different reasons.

First, the magnitude of the Fermi surfaces of the spin-up and spin-down

electrons can be very different while the Fermi velocities (effective masses)

are quite similar. Second, the Fermi surfaces can be quite similar while the

Fermi velocities (effective masses) of the two spin channels are very

different. In the first case the additional factors v and v² do not significantly

change the degree of spin polarization. Furthermore, the tunneling

probability is similar for the spin-up and spin-down electrons. That is, we

expect P

DOS

≈ P

Nv²

≈ P

tun

. We see that in this case the differently

defined values of the spin polarization show only small quantitative

differences. However, qualitative differences such as a sign change are

absent. A typical example for this case is Fe, for which

)(EN〉

〈

and

)(

ENv 〉〈

have a similar shape. In the second case we have electrons with

small and large effective mass at the Fermi level. Then, P

DOS

is dominated

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

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