Spintronika

Bevezetés

The spin dependence of electronic transport is hidden in most metals. However, spin polarized current injected to a nonmagnetic conductor keeps the spin memory within a certain distance, typically in the submicron range. By now, nanoscale devices have been prepared where the size of the components is well below the distance of the spin-memory, and both the control of the spin states and the utilization of the spin information have been demonstrated.\cite{Wolf16112001,Awschalom2007} In most of these electronic applications ferromagnetic components act as spin filter and detector, \cite{RevModPhys.76.323,0022-3727-35-18-201} while a nonmagnetic metal mediates the spin information: this is the simplest spin-valve structure. For the design of such nanodevices a proper understanding of spin dependent transport is necessary, including the knowledge of the carriers' spin-polarization, or the determination of the spin diffusion length.

In the first part of this review we present a simple model for the spin dependent transmission through a nanostructure based on the Landauer formalism widely applied in the field of nanophysics. In the second part a special measurement technique is reviewed, which allows the direct determination of the current spin polarization, and with which even the decay of the spin polarization i.e.~the spin diffusion length can be determined in a nonmagnetic layer. In the last part the nanoscale spin-valve architecture is discussed, demonstrating how the results of basic research find broad technical application and trigger an intensive development of novel memory elements. <\wlatex>

Spinpolarizált vezetés nanoszerkezetekben

In a macroscopic conductor the electrons suffer numerous scattering processes as they move from one electrode towards the other. Due to the scattering on impurities and lattice defects the drift momentum of the electrons gained from the electric field is lost. This process is described by the characteristic

length scale of the momentum relaxation mean free path,

\setbox0\hbox{\setbox0\hbox{$l_m$}%
\message{//depth:\the\dp0//}%
\box0%

}%

\message{//depth:\the\dp0//}% \box0% </latex>. Inelastic scattering processes like the scattering on lattice vibrations lead to a change of energy, which results in the loss of phase coherence for electron waves. The corresponding characteristic length scale is called phase diffusion length,

\setbox0\hbox{\setbox0\hbox{$l_{\phi}$}%
\message{//depth:\the\dp0//}%
\box0%

}%

\message{//depth:\the\dp0//}% \box0% </latex>.\cite{nazarov_book,datta_book} Within a large enough distance the electrons also loose their spin information, as characterized by the spin diffusion length,

\setbox0\hbox{\setbox0\hbox{$l_s$}%
\message{//depth:\the\dp0//}%
\box0%

}%

\message{//depth:\the\dp0//}% \box0% </latex>.\cite{RevModPhys.76.323,fabian:1708} In a nanostructure, however, all these length scales may become comparable to, or even larger than the characteristic size of the structure,

\setbox0\hbox{\setbox0\hbox{$L$}%
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\box0%

}%

\message{//depth:\the\dp0//}% \box0%

</latex>. For instance, in small enough structures (

\setbox0\hbox{\setbox0\hbox{$L}%
\message{//depth:\the\dp0//}%
\box0%
</latex>) the spin information is
preserved, which is a key ingredient for spintronic

applications.\cite{PhysRevB.48.7099} For structures smaller than 
\setbox0\hbox{\setbox0\hbox{$l_{\phi}$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>

coherent quantummechanical features appear, whereas for 
\setbox0\hbox{\setbox0\hbox{$L}%
\message{//depth:\the\dp0//}%
\box0%
</latex> a ballistic
transport is observed, i.e.~the electrons scatter only on the edges of the
structure, but not inside.
Here we give a simple model to demonstrate how spin-polarized current may arise
in ferromagnetic nanostructures, which are smaller than the spin diffusion
length. In order to provide a microscopic insight to the relevant processes we
use the well established approach of the Landauer
formalism,\cite{Landauer_form,datta_book} which has been successfully applied to
describe several nanoscale systems, like atomic-sized
conductors\cite{Agrait200381} or semiconductor
nanostructures.\cite{PhysRevLett.60.848,ihn_book}
\begin{figure} [!h]
\centering
\includegraphics[width=0.7\columnwidth]{fig1.eps}
\caption{\it  Panel (a) demonstrates an ideal 2D quantum wire with parallel
walls and the absence of scattering inside the wire. The standing waves
demonstrate the quantized transverse modes in the wire. Panel (b) demonstrates
the dispersion of the conductance channels inside the wire in a free electron
picture. The blue parabolas correspond to spin up electrons, whereas the red
parabolas stand for the spin down electrons. The two spin subbands are shifted

by an energy 
\setbox0\hbox{\setbox0\hbox{$\varepsilon_{ex}$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>. The thick/thin lines denote the

occupied/unoccupied states. Note, that right going states (
\setbox0\hbox{\setbox0\hbox{$k>0$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>) are occupied

up to an energy higher by 
\setbox0\hbox{\setbox0\hbox{$eV$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex> than the left moving states (
\setbox0\hbox{\setbox0\hbox{$k<0$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>).}
\label{Qwire}
\end{figure}
As a simplified model first we consider an ideal two dimensional wire with
parallel walls (Fig.~\ref{Qwire}(a)). In the ballistic limit no scattering occurs
in the wire, and an electron entering into the wire from one side will propagate
to the other side without changing its energy and spin state. Along

the wire (
\setbox0\hbox{\setbox0\hbox{$x$}%
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}%
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</latex>-direction) the electrons exhibit a free, reflectionless

propagation, whereas perpendicular to the wire (
\setbox0\hbox{\setbox0\hbox{$y$}%
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}%
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</latex>-direction) quantized
transverse modes are formed. In this simple geometry the wavefunction factorizes
to the product of a longitudinal and a transverse wave function, and the
energies corresponding to the transverse standing waves and the longitudinal
propagation are simply added. After including magnetism in a Stoner type picture
the energy dispersion for this system can be written as:
\begin{equation}
\varepsilon_n^{\sigma}(k)=\varepsilon(k)+\varepsilon_n-\sigma\varepsilon_{\mathrm{ex}},
\end{equation}

where 
\setbox0\hbox{\setbox0\hbox{$\varepsilon_n$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> is the energy corresponding to the quantized transverse

modes, 
\setbox0\hbox{\setbox0\hbox{$\varepsilon(k)$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex> is the dispersion of the extended Bloch states in the 
\setbox0\hbox{\setbox0\hbox{$x$}%
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}%
\message{//depth:\the\dp0//}%
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</latex>

direction (
\setbox0\hbox{\setbox0\hbox{$k$}%
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}%
\message{//depth:\the\dp0//}%
\box0%

</latex>=
\setbox0\hbox{\setbox0\hbox{$k_x$}%
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}%
\message{//depth:\the\dp0//}%
\box0%

</latex>), 
\setbox0\hbox{\setbox0\hbox{$\sigma =\pm 1/2$}%
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}%
\message{//depth:\the\dp0//}%
\box0%
</latex> is the spin index, and

\setbox0\hbox{\setbox0\hbox{$\varepsilon_{\mathrm{ex}}$}%
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}%
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</latex> is the exchange energy. The resulting dispersion is a
set of one dimensional dispersion curves, which are vertically shifted by the
transverse energies and the exchange energy (Fig.~\ref{Qwire}(b)). The discrete
one dimensional dispersions are called \emph{conductance
channels}.\cite{PhysRevB.31.6207,datta_book}

In such an ideal wire the states with positive and negative 
\setbox0\hbox{\setbox0\hbox{$k$}%
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}%
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</latex> are well
separated, the former are all coming from the left electrode, while the latter are all
coming from the right electrode. In a symmetric situation no net current flows.
Application of a bias voltage between the two sides of the wire, however, shifts

the chemical potentials of the right and left moving electron states by 
\setbox0\hbox{\setbox0\hbox{$eV$}%
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</latex>
with respect to each other, and this imbalance results in a finite current. For
a selected conductance channel the velocity of the electrons and the density of
states is respectively given as
\begin{equation}
v_n^{\sigma}=\frac{1}{\hbar}\frac{\partial \varepsilon_n^{\sigma}(k)}{\partial k},\ \ \ \ g_n^{\sigma}=\frac{L}{2\pi}\left(\frac{\partial \varepsilon_n^{\sigma}(k)}{\partial k}\right)^{-1},
\label{vn}
\end{equation}

where 
\setbox0\hbox{\setbox0\hbox{$L$}%
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\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> is the length of the nanowire. The spacial density for the electron

states in the 
\setbox0\hbox{\setbox0\hbox{$eV$}%
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}%
\message{//depth:\the\dp0//}%
\box0%
</latex> energy window is obtained as
\begin{equation}
n_n^{\sigma}=eV g_n^{\sigma}/L.
\end{equation}
The current is then calculated as a product of charge, velocity and electron
density, summing over the spins and all the conductance channels:
\begin{equation}
I=\sum_\sigma \sum_{n=1}^{M^{\sigma}}e v_n^{\sigma} n_n^{\sigma} =\sum_\sigma\frac{e^2}{h}M^{\sigma}V.
\end{equation}

Here 
\setbox0\hbox{\setbox0\hbox{$M^{\sigma}$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> is the number of open conductance channels, i.e. the number of
1D dispersion curves crossing the Fermi energy for a given spin orientation.

Note that 
\setbox0\hbox{\setbox0\hbox{$\partial \varepsilon / \partial k$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex> cancels in the product of 
\setbox0\hbox{\setbox0\hbox{$v$}%
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\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> and

\setbox0\hbox{\setbox0\hbox{$n$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>, thus the above result is valid for any shape of the dispersion curve.
Furthermore, the two dimensional model only simplifies the visualization, but
the above arguments also hold for any perfect three dimensional nanowire with

uniform cross section along the 
\setbox0\hbox{\setbox0\hbox{$x$}%
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}%
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</latex> axis.
Here we have taken advantage of the assumption that the spin state is conserved,
i.e. the structure is smaller than the spin diffusion length. This has allowed
the separation of the current to a purely spin up and a purely spin down component
being two independent \emph{routes} of the transport:

\setbox0\hbox{\setbox0\hbox{$I=I^{\uparrow}+I^{\downarrow}=(e^2/h)(M^{\uparrow}+M^{\downarrow})V$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>. The
total current is determined by the number of open conductance channels, and the

conductance (
\setbox0\hbox{\setbox0\hbox{$G=I/V$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex>) is an integer multiple  of 
\setbox0\hbox{\setbox0\hbox{$e^2/h=(25.8\,$}%
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\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex>k
\setbox0\hbox{\setbox0\hbox{$\Omega)^{-1}$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>.
\begin{figure} [!h]
\centering
\includegraphics[width=\columnwidth]{fig2.eps}
\caption{\it The dispersion curves around the Fermi energy
for the different conductance channels. The bottom and top panels
correspond to spin up and spin down electrons, respectively. Due to the exchange
splitting the number of open conductance channels differs for the two spin subbands.}
\label{linear}
\end{figure}

Due to the shifted spin subbands, 
\setbox0\hbox{\setbox0\hbox{$M^{\uparrow}$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> not necessarily equals

\setbox0\hbox{\setbox0\hbox{$M^{\downarrow}$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>. This is indicated by Fig.~\ref{linear}, where only the
linearized part of the 1D dispersions around the Fermi energy is shown
demonstrating that the free electron dispersions in Fig.~\ref{Qwire}(b) are only
illustrations and are generally not considered in this model. The degree of the
resulting spin polarization can be characterized by the ratio of the spin
polarized current and the total current:
\begin{equation}
P_c=\frac{I^{\uparrow}-I^{\downarrow}}{I^{\uparrow}+I^{\downarrow}}=\frac{M^{\uparrow}-M^{\downarrow}}{M^{\uparrow}+M^{\downarrow}}.
\label{Pc}
\end{equation}

This normalized current spin polarization can reach even 
\setbox0\hbox{\setbox0\hbox{$100\%$}%
\message{//depth:\the\dp0//}%
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}%
\message{//depth:\the\dp0//}%
\box0%

</latex> (
\setbox0\hbox{\setbox0\hbox{$P_c=1$}%
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\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>) in
case of a half-metal, where only one type of carrier is present at the
Fermi level.
\begin{figure} [!h]
\centering
\includegraphics[width=0.7\columnwidth]{fig3.eps}
\caption{\it A diffusive ferromagnetic region is sandwiched between two normal
metal ideal quantum wires. The transport is determined by spin dependent
transmission probabilities between the conductance channels at the two sides.}
\label{landauer}
\end{figure}
A more realistic description can be given for the conductance of a piece of
magnetic nanowire inserted between nonmagnetic reservoirs by taking also into
account elastic scatterings (Fig.~\ref{landauer}). According to the Landauer
picture the transport in this system can be be described by the transmission

probabilities 
\setbox0\hbox{\setbox0\hbox{$T_{nm}$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>, i.e. the probability for the transmission of an electron

from channel 
\setbox0\hbox{\setbox0\hbox{$n$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex> on the left side to channel 
\setbox0\hbox{\setbox0\hbox{$m$}%
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\box0%
}%
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\box0%
</latex> on the right side. These
transmission probabilities depend on the ``waveguide parameters of the wire.

While 
\setbox0\hbox{\setbox0\hbox{$T_{nm}=\delta_{nm}$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> corresponds to the previous model of an ideal wire,

\setbox0\hbox{\setbox0\hbox{$T<1$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> represents scatterings in the wire due to defects, impurities, etc.
As no spin mixing occurs between the spin up and spin down states, the current
can still be calculated for each spin state independently taking into account
the finite transmissions of the
channels:\cite{PhysRevB.31.6207,Agrait200381,datta_book}
\begin{equation}
I^\sigma=\frac{e^2}{h}\sum_{n,m=1}^{M^\sigma}T_{nm}^\sigma V.
\end{equation}

The values of 
\setbox0\hbox{\setbox0\hbox{$T_{nm}$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> depend on the Fermi wave numbers of the involved
channels. Due to the spin-dependent shift of the dispersions the spin up and the
spin down channels have different wave numbers at the Fermi energy. Thus, in
general, the transmission probability may become spin dependent even without any
direct spin-interaction, like spin-orbit coupling.




The spin dependent current can be rewritten in the form:
\begin{equation}
I^\sigma=\frac{e^2}{h}M^\sigma \bar{T}^\sigma V,
\end{equation}

where 
\setbox0\hbox{\setbox0\hbox{$\bar{T}^\sigma$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> is the average transmission probability for all the
conductance channels with a given spin direction. With this notation the spin
polarization of the current can be written as:
\begin{equation}
P_c=\frac{I^{\uparrow}-I^{\downarrow}}{I^{\uparrow}+I^{\downarrow}}=\frac{M^{\uparrow}\bar{T}^\uparrow-M^{\downarrow}\bar{T}^\downarrow}{M^{\uparrow}\bar{T}^\uparrow+M^{\downarrow}\bar{T}^\downarrow}.
\end{equation}
It is evident from this formula, that either the difference of the transmission
probabilities for the two spin channels, or the difference in the number of open
channels for spin up and spin down electrons gives contribution to the spin
polarization of the current.

In case of a few conductance channels the difference in 
\setbox0\hbox{\setbox0\hbox{$\bar{T}^\uparrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> and

\setbox0\hbox{\setbox0\hbox{$\bar{T}^\downarrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> may introduce a spin polarized current even for the same
number of open channels. Indeed, for ferromagnetic atomic point contacts, the
transmission probabilities for the two spin channels are expected to differ
significantly.\cite{PhysRevB.77.104409}
However, for larger structures, where

many conductance channels are present (Fig.~\ref{linear}), 
\setbox0\hbox{\setbox0\hbox{$\bar{T}^\sigma$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> is
an average over a broad ensemble of different wavenumbers, and for diffusive
junctions random matrix theory (RMT) predicts a universal value of
\(\bar{T}^{\sigma}=l_m^{\sigma}/L\),\cite{RevModPhys.69.731} where \(L\) is the
length of the diffusive region. As the momentum relaxation

mean free path is expected to be similar for the two spin subbands, 
\setbox0\hbox{\setbox0\hbox{$\bar{T}^\uparrow \approx
\bar{T}^\downarrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> can be assumed. Accordingly the key contribution to the spin

polarization is given solely by the difference of 
\setbox0\hbox{\setbox0\hbox{$M^\uparrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex> and 
\setbox0\hbox{\setbox0\hbox{$M^\downarrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>. In this limit, Eq.~\ref{Pc}
gives a reasonable approximation for the spin polarization, even for non-ideal transmission probabilities

(
\setbox0\hbox{\setbox0\hbox{$\bar{T}^\uparrow\approx\bar{T}^\downarrow<1$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>).
According to Eq.~\ref{vn} the number of conductance channels can be formally
written as:
\begin{equation}
M^\sigma=\frac{2\pi\hbar}{L}\sum_{n=1}^{M^\sigma}v_n^{\sigma}g_n^{\sigma}.
\end{equation}
Introducing a mean Fermi velocity, which is the average of the Fermi velocities for the different conductance channels weighted by the density of states of the channels,
\begin{equation}
\bar{v}_F^\sigma=\frac{\sum_{n} g_n^\sigma v_n^\sigma}{\sum_{n} g_n^\sigma},
\end{equation}

and noting that 
\setbox0\hbox{\setbox0\hbox{$\sum_{n} g_n^\sigma=g_F^\sigma$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> is the total density of states at the Fermi level, the formula for the current spin polarization can be rewritten as:
\begin{equation}
P_c\approx\frac{M^{\uparrow}-M^{\downarrow}}{M^{\uparrow}+M^{\downarrow}}=\frac{g_F^{\uparrow}\bar{v}_F^{\uparrow}-g_F^{\downarrow}\bar{v}_F^{\downarrow}}{g_F^{\uparrow}\bar{v}_F^{\uparrow}+g_F^{\downarrow}\bar{v}_F^{\downarrow}}.
\label{PcDOS}
\end{equation}
This formula supplies a microscopic background for the conventional
formulation of the current spin polarization\cite{PhysRevB.70.054416} expressed
by Fermi surface parameters of the two spin subbands: density of states and Fermi velocity.
\begin{figure} [!h]
\centering
\includegraphics[width=\columnwidth]{fig4.eps}
\caption{\it  Illustration for the density of states in ferromagnets. The

magnetism is related to the exchange splitting of the narrow 
\setbox0\hbox{\setbox0\hbox{$d$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex> (or 
\setbox0\hbox{\setbox0\hbox{$f$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>)
bands. Panel (a) demonstrates the DOS in Fe: for minority spins (left side) the

\setbox0\hbox{\setbox0\hbox{$d$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> band is above the Fermi level, whereas for majority spins (right side) the

Fermi level lies inside the 
\setbox0\hbox{\setbox0\hbox{$d$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> band. Accordingly the DOS is significantly
larger for the majority spins. Panel (b) demonstrates the DOS for Co and Ni type

band filling. In this case 
\setbox0\hbox{\setbox0\hbox{$\varepsilon_F$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex> is above the 
\setbox0\hbox{\setbox0\hbox{$d$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> band for majority

spins, whereas it lies inside the 
\setbox0\hbox{\setbox0\hbox{$d$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> band for minority spins resulting in a
negative Fermi surface spin polarization with respect to the positive
magnetization.}
\label{DOS}
\end{figure}

In real ferromagnets the exchange splitting of narrow 
\setbox0\hbox{\setbox0\hbox{$d$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex> (or 
\setbox0\hbox{\setbox0\hbox{$f$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>) bands and the
filling of the bands determine the magnetic properties (see Fig.~\ref{DOS}). The
integral of the density of states up to the Fermi energy is different for the
two spin orientations, which results in a finite magnetization, conventionally
selecting the up spin as the majority spin orientation. However, the electron
transport is determined by the Fermi surface properties. The spin polarized
transport is not characterized by the magnetization, rather it is described by
the spin polarization of the current as described by Eq.~\ref{PcDOS}. An
interesting example is supplied by the Co or Ni type band fillings: the density
of states at the Fermi level is much larger for the down spin, leading to a
situation where the current spin polarization may take a sign opposite to that
of the magnetization (Fig.~\ref{DOS}b).
\section{Measurement of the current spin polarization and the spin diffusion length}
As shown in the previous section, there is an important difference between
magnetization and current spin polarization. This difference highlights that
magnetization measurements -- for which various sensitive tools are available --
are not an appropriate tool for the study of spin-polarized transport. Somewhat
closer information can be obtained by spin resolved photoemission spectroscopy,

which measures the density of states spin polarization, 
\setbox0\hbox{\setbox0\hbox{$P_{DOS}=(g_F^\uparrow
-g_F^\downarrow )/(g_F^\uparrow+g_F^\downarrow)$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>.\cite{PhysRevLett.50.1623}
Nevertheless, for a close insight to the current spin polarization, direct
transport measurements are necessary. There are several transport measurements
reveling various consequences of spin polarized
currents,\cite{datta:665,Ohno1999,PhysRevLett.83.203,0953-8984-19-18-183201,Valenzuela2006}

but for a direct measurement of 
\setbox0\hbox{\setbox0\hbox{$P_c$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> a few methods are available. In the following we
demonstrate such a method based on point contact Andreev spectroscopy. (An
alternative method is based on superconducting tunneling spectroscopy in
external magnetic field.\cite{PhysRevLett.26.192})
\begin{figure} [!h]
\centering
\includegraphics[width=\columnwidth]{fig5.eps}
\caption{\it  Panel (a): Illustration of Andreev reflection between a
nonmagnetic metal and a superconductor. The incoming spin up electron is
reflected as a spin down hole, and meanwhile a Cooper pair is created in the
superconductor. Panel (b) demonstrates the absence of Andreev reflection
between a fully spin polarized ferromagnet (half metal) and a superconductor.
In this case no spin down states are available, and the Andreev
reflection of the incoming spin up electrons is completely suppressed.}
\label{Andreev}
\end{figure}
The basic idea of Andreev spectroscopy is to measure the Andreev reflection of
spin polarized conduction electrons on a superconductor
surface.\cite{R.J.SoulenJr.10021998,PhysRevLett.81.3247,PhysRevB.69.140502} When
current is injected into the superconductor the charge must be converted to spin singlet Cooper pairs
composed from two electrons with opposite spins. On the normal side an incoming
electron drags an other electron with opposite spin and momentum, which is
equivalent to reflecting a hole with opposite spin and momentum. This process is
called Andreev reflection (Fig.~\ref{Andreev}(a)). Compared to the single particle transfer (which

occurs at voltages above the superconductor gap energy, 
\setbox0\hbox{\setbox0\hbox{$|eV|>\Delta$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>), in the
Andreev reflection double charge quanta are transmitted, and therefore the zero
bias conductance may be even twice the high-bias, normal-state
conductance. However, for a spin polarized metal the spin states available for
the reflected hole are reduced (see Fig.~\ref{Andreev}(b)), and the conductance is
suppressed.
The above idea was successfully used to get a direct experimental measure of the
current spin polarization in various ferromagnetic
metals\cite{R.J.SoulenJr.10021998,PhysRevLett.81.3247,PhysRevLett.86.5585,PhysRevB.63.104510,PhysRevB.66.212403,
PhysRevB.70.054416,PhysRevB.69.140502,PhysRevLett.101.147005,PhysRevLett.105.207001,rajanikanth:022505}
as well as more complex material systems, such as dilute magnetic semiconductors.\cite{PhysRevLett.91.056602,PhysRevB.77.233304} In a real structure, however, a non-perfect barrier should also be considered between the ferromagnet and the
superconductor. Different models have been set up to describe the shape of the
current voltage characteristics of ferromagnet - superconductor nanojunctions as

a function of the current spin polarization (
\setbox0\hbox{\setbox0\hbox{$P_c$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>) and the transparency of the

interface (
\setbox0\hbox{\setbox0\hbox{$T$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>). In the most widely applied method the original
Blonder-Tinkham-Klapwijk (BTK) theory of the Andreev
reflection\cite{PhysRevB.25.4515} is extended for ferromagnetic materials
splitting the current to unpolarized and fully polarized parts, and then
assuming no Andreev reflection for the fully polarized part and applying the BTK
theory for the unpolarized part.\cite{PhysRevB.63.104510} In an other approach
spin-dependent transmission coefficients are considered, and their imbalance

is deduced from the analysis of the 
\setbox0\hbox{\setbox0\hbox{$I-V$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>
characteristics.\cite{PhysRevB.54.7366,PhysRevB.69.140502}
Experimentally the
ferromagnet - superconductor interface is either created in an STM geometry by
pushing a superconducting tip to a magnetic
sample,\cite{R.J.SoulenJr.10021998,PhysRevB.63.104510,PhysRevB.66.212403} or a
nano-hole is fabricated in an isolating membrane, and afterwards the ferromagnetic and the
superconducting layers are evaporated to the two sides of this
hole.\cite{PhysRevLett.81.3247,PhysRevB.69.140502,PhysRevLett.101.147005}
\begin{figure} [!h]
\centering
\includegraphics[width=\columnwidth]{fig6.eps}
\caption{\it Panel (a) demonstrates the differential conductance curve measured
between an Fe sample covered with \(5\,\)nm Au capping layer and a Nb tip. The
red line shows the best fit using the modified BTK theory, yielding a normal

state resistance of 
\setbox0\hbox{\setbox0\hbox{$R_N=890\,\Omega$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex>, a spin polarization of 
\setbox0\hbox{\setbox0\hbox{$P_c=0.62$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex>, a contact transparency of 
\setbox0\hbox{\setbox0\hbox{$T=0.98$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>, a

superconducting gap of 
\setbox0\hbox{\setbox0\hbox{$\Delta=1.50\,$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex>meV at a temperature of 
\setbox0\hbox{\setbox0\hbox{$T=4.2\,$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>K. Note,

that the normalized barrier strength, 
\setbox0\hbox{\setbox0\hbox{$Z$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> in the BTK theory is converted to

transmission probability as 
\setbox0\hbox{\setbox0\hbox{$T=(1+Z^2)^{-1}$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>. The blue and green lines show

fits, where 
\setbox0\hbox{\setbox0\hbox{$P_c$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> is fixed at values intentionally detuned from the best fitting value, and

only 
\setbox0\hbox{\setbox0\hbox{$T$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> is used as a fitting parameter. These curves demonstrate that the

resolution of the method is well below 
\setbox0\hbox{\setbox0\hbox{$\Delta P_c=0.1$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>. Panel (b) demonstrates
the deviation between the best fits based on the modified BTK theory and the
Hamiltonian approach (red line). Although these two models do not give  exactly
the same fit, their deviation is smaller than the deviation of the
experimental curve from the modified BTK fit (black line).}
\label{Pc1}
\end{figure}
To demonstrate the experimental determination of the current spin polarization
Fig.~\ref{Pc1} shows the results of our study on a Fe-Nb nanojunction created in
the tip - sample geometry. In panel (a) the experimental differential
conductance curve is fitted by the modified BTK theory revealing a

current spin polarization of 
\setbox0\hbox{\setbox0\hbox{$P_c=0.62$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>. This value is somewhat higher than the
result of earlier studies,\cite{R.J.SoulenJr.10021998,PhysRevB.63.104510} which
may arise form the different surface preparation: in contrast to previous
studies in our measurements a protective Au surface layer with a thickness of

\setbox0\hbox{\setbox0\hbox{$5\,$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>nm was applied to prevent the contamination of the Fe surface. The figure
also demonstrates the precision of the method: if the spin
polarization is fixed at a value slightly detuned from the optimal fit, no

satisfactory fit is available by varying the barrier transparency (
\setbox0\hbox{\setbox0\hbox{$T$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>).  It
is also found that the two theoretical models lead to essentially
identical spin polarization values, even though they mathematically cannot  be
mapped onto each other.
\begin{figure} [!h]
\centering
\includegraphics[width=\columnwidth]{fig7.eps}
\caption{\it Panel (a) demonstrates the differential conductance curve for a
nanojunction between a Nb tip and a Co sample with a Pt capping layer. The best

fit (red line) yields 
\setbox0\hbox{\setbox0\hbox{$R_N=780\,\Omega$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex> 
\setbox0\hbox{\setbox0\hbox{$P_c=0.39$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex>, 
\setbox0\hbox{\setbox0\hbox{$T=0.87$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex>, 
\setbox0\hbox{\setbox0\hbox{$\Delta=1.62\,$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>meV at a temperature of

\setbox0\hbox{\setbox0\hbox{$1.5\,$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>K. Panel (b) demonstrates the temperature dependence of the fitted gap
values clearly following the BCS prediction.}
\label{Pc2}
\end{figure}
Another experimental example is presented in Fig.~\ref{Pc2}(a), demonstrating a
differential conductance curve on Co-Nb junction. For cobalt the spin
polarization is smaller than for iron: based on numerous measurements on various

junctions 
\setbox0\hbox{\setbox0\hbox{$P_c=0.41\pm0.04$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> is obtained. This value agrees well with previous
measurements performed on nano-hole samples.\cite{PhysRevB.69.140502}
It is worth noting, that the fitting functions contain two additional
parameters: the temperature and the superconducting gap. We have found that for
junctions in the ballistic limit the fitting results in a temperature which
agrees well with the experimental temperature value. The gap also coincides with
the bulk Nb gap, and the BCS temperature dependence of the gap\cite{PhysRev.128.591} is
clearly resolved by fitting the differential conductance curves at various
temperatures (see Fig.~\ref{Pc2}(b)). This consistency of the method is,
however, lost if the contact size is increased and junctions in the diffusive
limit are studied.\cite{PhysRevB.77.233304} In this regime a considerable
smearing of the curves is observed, and good fits are only possible if the fitting temperature is
significantly larger than the experimental value. In extreme cases the fitting
temperature may even grow above the critical temperature of Nb, which is an
obvious contradiction.
The above examples provide an insight to spin polarization measurement based on
Andreev spectroscopy. This method provides a sensitive and

reliable method for the direct determination of 
\setbox0\hbox{\setbox0\hbox{$P_c$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> if small ballistic
junctions are used. For such junctions it is generally found that the results
are not sensitive to the choice of the fitting model. However, for larger
diffusive junctions the quality of the fits is significantly worse and
accordingly the reliability of the method breaks down. Another requirement for
the reliability of the method is the choice of a transparent junction.
As Andreev reflection is a second order process, for low transparency junctions

(
\setbox0\hbox{\setbox0\hbox{$T\ll1$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>) its probability is suppressed and the differential conductance curve is not
sensitive to the degree of spin polarization.
It is worth emphasizing that Andreev spectroscopy acts as a local probe
technique, it is able to probe the spin polarization locally, at the nanometer
scale.\cite{PhysRevB.77.233304} In this sense it can be also applied to detect the
lateral variations of the spin polarization.
Next we show how the spin diffusion length can be determined by Andreev
spectroscopy.\cite{Geresdi2011} For this purpose a nonmagnetic capping layer is
evaporated on the top of the ferromagnetic sample. The degree of spin
polarization is measured at the surface of the capping layer, studying the decay
of the spin polarization as the thickness of the capping layer is systematically
increased. The carrier spin polarization is expected to decay exponentially as a
function of the layer thickness. Fig.~\ref{Spindiff} shows the results for Fe/Au
samples clearly resolving this exponential decay, and yielding a spin diffusion

length of 
\setbox0\hbox{\setbox0\hbox{$53\pm6\,$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>nm. This example further demonstrates that Andreev
spectroscopy is a powerful tool to measure current spin polarization, and it may
find even wider application as a local probe technique to study the lateral

variation of 
\setbox0\hbox{\setbox0\hbox{$P_c$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> at nanometer scale.




\begin{figure} [!h]
\centering
\includegraphics[width=\columnwidth]{fig8.eps}
\caption{\it  The measurement of spin diffusion length by Andreev spectroscopy.
The inset demonstrates the experimental setup: on the top of Fe samples Au
capping layers are evaporated with varying thickness. The current spin
polarization is measured at the top of the Au layer as a function of the layer
thickness. Each point in the figure corresponds to more than 10 independent
junctions, which were created by lateral shifting the sample below the lifted
Nb-tip by a piezo actuator. The error bars represent the scattering of the
calculated polarization values for these independent experiments. The
exponential decay of the spin polarization is clearly resolved as  the Au layer

thickness is increased yielding a spin diffusion length of 
\setbox0\hbox{\setbox0\hbox{$53\pm6\,$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>nm.}
\label{Spindiff}
\end{figure}




\section{The spin valve structure}
The knowledge of current spin polarization and spin diffusion length is
essential for the design of spintronic devices. In the previous part we have
shown an experimental method which allows the determination of both quantities.
In the following we turn to the application of spin polarized transport by
discussing the most widely utilized spintronic device, the nanoscale spin-valve
structure. First we give simple demonstrative models for this structure, and
then we briefly discuss possible applications for next generation data storage
devices.
The basic idea of this structure dates back to the discovery of the giant magnetoresistance (GMR)
effect by Albert Fert \cite{PhysRevLett.61.2472} and Peter Gr\"unberg
\cite{PhysRevB.39.4828}, for which the Nobel price in
physics was awarded in 2007. The architecture is based on two ferromagnetic
layers, which are separated by a paramagnetic layer being thinner than the spin
diffusion length (Fig.~\ref{GMR1}(a)). It was found that the conductance of this device is larger for
the parallel orientation of the two magnetic layers than for the antiparallel
orientation.
\begin{figure} [!h]
\centering
\includegraphics[width=\columnwidth]{fig9.eps}
\caption{\it Panel (a) demonstrates the basic idea of a spin valve structure:
two ferromagnetic regions are separated by a narrow normal region, and this
structure is connected to normal leads at both sides. Panels (b,d) demonstrate
the variation of the exchange energy along the structure. Panel (b) corresponds

to the parallel magnetic orientation of the layers (both layers have 
\setbox0\hbox{\setbox0\hbox{$\uparrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>
magnetic orientation), whereas panel (d) demonstrates the antiparallel

orientation (
\setbox0\hbox{\setbox0\hbox{$\uparrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex> in the first layer and 
\setbox0\hbox{\setbox0\hbox{$\downarrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> in the second one).
Panels (c) and (e) respectively show the equivalent ballistic model, where the
change of the exchange energy is replaced by an effective narrowing or widening
of the channel.}
\label{GMR1}
\end{figure}
To understand this phenomenon, as the simplest model we can consider an ideal
quantum wire, in which the exchange coupling is switched on adiabatically in two
regions (Fig.~\ref{GMR1}(b,d)). The parallel orientation is modeled by a
positive exchange in both regions, whereas the antiparallel orientation
corresponds to opposite signs of the exchange in the two regions. Similarly to
the previous situations the current can be divided to two independent components
corresponding to spin up and spin down electrons. For a given spin orientation
the effect of the exchange energy is very similar to an effective spatial
narrowing or widening of the channel, which would increase/decrease the
transverse quantized energies (Fig.~\ref{GMR1}(c,e)). Therefore the system can
be considered as serial connected quantum point contacts. For electrons with
majority spin direction (spin direction parallel to the magnetization direction)

the number of open channels is denoted by 
\setbox0\hbox{\setbox0\hbox{$M^\uparrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>, whereas for minority
spins (antiparallel spin direction to the magnetization direction) the number of

open channels is 
\setbox0\hbox{\setbox0\hbox{$M^\downarrow(}%
\message{//depth:\the\dp0//}%
\box0%
</latex>. (Note, that here the superscript

\setbox0\hbox{\setbox0\hbox{$\uparrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> denotes majority orientation instead of the real spin direction,

and the number of open channels is also 
\setbox0\hbox{\setbox0\hbox{$M^\uparrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> for down spins in down
oriented magnetization of the magnetic layer.)
In such ballistic quantum point contacts the conductance is determined by the
number of modes at the narrowest cross section. If the two magnetic layers have
antiparallel (AP) orientation either the first or the second layer restricts the

number of channels to 
\setbox0\hbox{\setbox0\hbox{$M^\downarrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>, and so the conductance for both spin

channels is 
\setbox0\hbox{\setbox0\hbox{$(e^2/h)M^\downarrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>, i.e. the total conductance is

\setbox0\hbox{\setbox0\hbox{$G^\mathrm{AP}=2(e^2/h)M^\downarrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>. For parallel orientation of the magnetic

layers the conductance is 
\setbox0\hbox{\setbox0\hbox{$(e^2/h)M^\uparrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> if the spin direction of the

carriers is the same as the magnetization, and it is 
\setbox0\hbox{\setbox0\hbox{$(e^2/h)M^\downarrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> if
the magnetization has opposite orientation to the spin orientation of the
carriers. The net conductance is

\setbox0\hbox{\setbox0\hbox{$G^\mathrm{P}=(e^2/h)(M^\downarrow+M^\uparrow)$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>. The magnitude of the

magnetoresistance effect can be characterized by 
\setbox0\hbox{\setbox0\hbox{$\Delta G/G^\mathrm{P}$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>, where

\setbox0\hbox{\setbox0\hbox{$\Delta G=G^\mathrm{P}-G^\mathrm{AP}$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>. With this notation the relative change of
the conductance turns out to be equal to the current spin polarization
characteristic to the ferromagnetic metal applied,
\begin{equation}
\frac{\Delta G}{G^\mathrm{P}}=\frac{M^\uparrow-M^\downarrow}{M^\uparrow+M^\downarrow}=P_c.
\end{equation}
This simplified model corresponds to the perfectly ballistic limit, where no
backscatterings are considered inside the quantum wire.
In a more realistic model the scattering inside the nanostructure should also be
considered. If the structure is smaller than the phase diffusion length,

\setbox0\hbox{\setbox0\hbox{$l_\phi$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> the scattering in the two layers are added coherently, and accordingly
the total resistance relies on the fine details of quantum interference
phenomena.
\begin{figure} [!h]
\centering
\includegraphics[width=\columnwidth]{fig10.eps}
\caption{\it Resistive model of the GMR effect. The two magnetic layers are
separated by an incoherent nonmagnetic region. The top panel demonstrates the

parallel alignment of the magnetic layers (both layers have 
\setbox0\hbox{\setbox0\hbox{$\uparrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>
magnetization direction), whereas the bottom panel demonstrates the

antiparallel alignment (the left layer has 
\setbox0\hbox{\setbox0\hbox{$\uparrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> and the right layer has

\setbox0\hbox{\setbox0\hbox{$\downarrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex> magnetization direction). 
\setbox0\hbox{\setbox0\hbox{$G^\uparrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex> and 
\setbox0\hbox{\setbox0\hbox{$G^\downarrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex> denote
the conductance of a layer for electrons with majority and minority spin
direction, respectively.}
\label{GMR2}
\end{figure}
For even larger junctions, where phase coherence between the two magnetic layers

is already lost (
\setbox0\hbox{\setbox0\hbox{$L>l_\phi$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>), but the spin information is still preserved

(
\setbox0\hbox{\setbox0\hbox{$L}%
\message{//depth:\the\dp0//}%
\box0%
</latex>), the two scattering regions are connected \emph{incoherently}
(Fig.~\ref{GMR2}), and accordingly their  resistances are simply summed to get
the total resistance.\cite{note1} In this limit the magnetoresistance of the
spin valve structure is also easily calculated. To compare with the previous
ballistic model we calculate with conductances: the conductance for majority

spin states is 
\setbox0\hbox{\setbox0\hbox{$G^\uparrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%

</latex>, whereas for minority spins it is 
\setbox0\hbox{\setbox0\hbox{$G^\downarrow$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>.
For antiparallel oriented magnetic layers the total conductance is

\setbox0\hbox{\setbox0\hbox{$G^{AP}=2G^\uparrow G^\downarrow/(G^\uparrow+G^\downarrow)$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>, whereas for

parallel oriented layers it is 
\setbox0\hbox{\setbox0\hbox{$G^{P}=G^\uparrow/2+G^\downarrow/2$}%
\message{//depth:\the\dp0//}%
\box0%
}%
\message{//depth:\the\dp0//}%
\box0%
</latex>. From these
equations
\begin{equation}
\frac{\Delta G}{G^\mathrm{P}}=\left(\frac{G^\uparrow-G^\downarrow}{G^\uparrow+G^\downarrow}\right)^2=P_c^2
\end{equation}
follows. Note, that this model is equivalent with the common resistor model of the GMR phenomenon.\cite{0022-3727-35-18-201}

Both of the above simple models show that 
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</latex> is positive, i.e. the
parallel orientation has higher conductance than the antiparallel one. Depending

on the model, the relative conductance change ranges between 
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</latex> and 
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</latex>
which indicates that a more complicated model based on the coherent
superposition of two scattering regions would also give a result between the
above two extreme limits. These considerations show that for typical values of

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</latex>, the relative amplitude of the GMR effect is expected to be
significant regardless of the details of the model. Note, however, that this
is only valid if the spin information is fully preserved in the spacer layer. As
the separation of the magnetic layers becomes larger than the spin diffusion length the GMR
effect exponentially decays.\cite{PhysRevB.48.7099,0953-8984-19-18-183201}
The magnetic layers of a spin valve are decoupled by paramagnetic spacer, and
this allows switching between parallel and antiparallel alignments. As the
relative alignment depends on the external magnetic field, the GMR effect can be
used for magnetic sensing. For this purpose the orientation of one layer is
pinned by growing it on top of an antiferromagnet, while the unpinned free layer
can rotate even under the influence of a weak magnetic field. Spin valves are
extensively applied as the read head of hard disks, which utilizes the fast and
reliable reading of the magnetic information by an electric signal.
The spin valve structure can also be used to store magnetic information. This
type of non-volatile memory consists of a large number of spin valves, each of
them being addressed separately in a crossbar wire architecture.  The
orientation of the free layer defines bit ``0 and bit ``1, which can be
changed by the stray field of the nearby crossing wires.  The information written in
this way is red out by the GMR effect. However, the areal density of this type
of magnetoresistive random access memories (MRAM) is limited by the length scale
of the slowly decreasing stray fields.




In the most advanced novel MRAMs no external field (and extra wiring) is
required to write information in a spin valve memory element. In these devices
the orientation of the free layer is manipulated by high density spin polarized
currents, while the magnetic information is obtained by low current GMR
measurements. The writing is based on the spin-flip electron scattering
processes occurring in the free layer, which exert a torque on the
magnetization. Even though such spin-flips are rare events (the spin diffusion
length is much larger than the characteristic size of the nanodomain), at
current densities of about \(10^9 -10^{10}\, \textrm{A/cm}^2\), the induced
torque becomes large enough to reverse the magnetization. The details of the spin transfer torque
phenomenon are reviewed in Ref.~\cite{Ralph20081190}. Here we just point
out the importance of the strongly non-equilibrium state of this nanoscale
device: while the above current densities would lead to melting in any macroscopic metal, in this device the
length scales which determine its transparency (i.e. its resistance) are well
below the inelastic mean free path, and as a consequence the Joule heat is
produced outside the device, in a much larger volume. In spite of the huge
current densities, the nanoscale electronics defines appropriate signal levels
for practical applications (below \(100\,\mu\)A, and around \(1\,\)V).  Finally
it is to be noted that the spin transfer torque magnetoresistive random access memory
(STT-MRAM) is one of the most promising candidate for future spintronic
applications.
\section{Conclusions}
Nanoscale phenomena of spin related transport were discussed both theoretically
and experimentally. Based on the spin dependent band structure of a magnetic
metal, we discussed the propagation of electrons in the ballistic and diffusive
limits, and by applying the Landauer formalism, we supplied a \emph{nanoscopic}
background for the conventional expression of the current spin polarization. As
the most direct experimental method for spin polarization measurements, the
Andreev reflection spectroscopy was introduced, and experimental results on Fe
and Co were shown. The analysis demonstrated the reliability of measurements carried out in
the ballistic limit. The method was also extended for the determination of the
spin diffusion length, one of the most important parameters of metals in
spintronic applications. The operation of spin valves was described in terms of
the Landauer formalism and the magnitude of the GMR effect was determined in
the coherent ballistic and the incoherent diffusive limits. Finally a short
overview on spin valve applications was given.
\section{Acknowledgments}
This work was supported by the Hungarian Research Funds OTKA under Grant Nos.
72916 and 80991. A.~H.~is a grantee of the Bolyai Scholarship.








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