Spintronika
Spindiffúziós hossz
Egy makroszkopikus vezetőben az elektronok számtalanszor szóródnak miközben eljutnak eaz egyik elektródából a másikba. Ahogy a nanovezetékek tárgyalásának bevezetésekor láttuk a szennyezőkön és rácshibákon történő rugalmas szórás az elektronok elektromos tértől nyert impulzusának elvesztéséhez vezet. Ennek a folyamatnak a karakterisztikus skáláját az átlagos momentumrelaxációs szabadúthossz, 
jellemzi. Rugalmatlan szórások esetén (pl. kölcsönhatás rácsrezgésekkel) az elektronok energiája megváltozik, és így elveszik a a fázisinformáció, azaz megszűnik az interferenciaképesség. Ennek a folyamatnak a karakterisztikus skálája az ún. fázisdiffúziós szabadúthossz, 
. Megfelelő távolságon belül az elektronok a mágneses momentumuk, azaz a spinjükhöz tartozó információt is elvesztik, amit a spindiffúziós hosszal, 
jellemezhetünk.
Egy spindiffúziós hossznál kisebb, mágnesesen rendezett építőelemeket tartalmazó nanoszerkezetben azonban számos érdekes, az elektronok spin szerinti polarizáltságához kötődő jelenséggel találkozhatunk.
Spinpolarizáció ideális nanovezetékekben
A korábbiakban láttuk, hogy ha két elektróda közé egy ideális, szórásmentes nanovezetéket helyezünk, akkor abban keresztirányban állóhullámok, hosszirányban pedig síkhullám terjedés alakul ki, a vezetőképesség pedig 
, ahol M a nyitott vezetési csatornák száma, azaz azon különböző keresztmódusokhoz tartozó egydimenziós parabolikus diszperziók száma, melyek metszik a Fermi-energiát. A kettes szorzó a spin szerinti degenerációból adódott.
Egy ferromágneses nanovezetékben azonban különbséget kell tenni a fel és a le spinű elektronok között.
A vezeték mágnesezettségét legegyszerűbben Stoner-képben vehetjük figyelembe, azaz az energidiszperziókhoz hozzáadjuk a kicserőlédési energiát, mely 
-el különbözik a fel illetve le spinű (
) elektronokra:
![\[\varepsilon_n^{\sigma}(k)=\varepsilon(k)+\varepsilon_n-\sigma\varepsilon_{\mathrm{ex}}.\]](/images/math/1/e/c/1ec90d32189d7f827dd5dcf27b159654.png)
Emlékeztetőül: 
a hosszirányú terjedést, 
pedig a keresztirányú állóhullámok energiáját írja le. Fontos megjegyezni, hogy lehet tetszőleges Bloch-állapot diszperziója, nem kell feltétlenül szabad elektronoknak megfelelő parabolikus diszperziót feltételezni.
Az 1. ábra parabolikus szabad elektron diszperzió esetén szemlélteti az energiaviszonyokat. A kék görbék a fel, a piros görbék pedig a le spinű elektronokhoz tartoznak, azaz a piros parabolák minig energiával a megfelelő kék parabola felett helyezkednek el. A korábbiaknak megfelelően a 
állapotok a bal oldali eletródából származnak, így annak a 
kémiai potenciákjáig vannak betöltve, míg a 
állapotok a jobb oldali, 
kémiai potenciálú eletródából származnak.
 |
| 1. ábra |
Értelemszerűen egy ferromágneses nanovezetékben a nyitott csatornák száma különbözhet a különböző spinű elektronokra (2. ábra), így az áram
![\[ I=\sum_{\sigma =\frac{1}{2},-\frac{1}{2}} \frac{e^2}{h}M^{\sigma}V \]](/images/math/2/b/7/2b7d2e5ba1e46a9261434fa189e97c4c.png)
Alakban írható, ahol 
(illetve máshogy jelölve 
és 
) a fel és le spinű elektronokhoz tartozó nyitott csatornák száma.
 |
| 2. ábra. Diszperziós görbék ferromágneses nanovezetékben a Fermi-energia körül. Fel spinű elektronoknak (kék) több nyitott vezetési csatorna áll rendelkezésre mint le spinű elektronoknak (piros). |
Fontos megjegyezni, hogy itt figyelembe vettük, hogy a nanovezetéken belül az elektronok spinállapota nem változik, ennek köszönhető hogy a fel és le spinű elektronokat egymástól független csatornaként kezelhetjük az áram felírásakor.
. Látjuk, hogy ferromágneses ideális nanovezetékben a vezetőképesség 
szerint kvantált.
Az elektronok spin szerinti polarizáltságának fokát
![\[ P_c=\frac{I^{\uparrow}-I^{\downarrow}}{I^{\uparrow}+I^{\downarrow}} \]](/images/math/0/e/5/0e5488e876f0c988e1bdb1710aabad98.png)
képlettel definiálhatjuk, ami ideális nanovezetékben 
formában is írható. Ez a spinpolarizáció 
és 
közötti értékeket vehet föl, tökéletes spinpolarizáció (
) félfémben érhető el, amikor csak fel spinű elektronok találhatók a Fermi-energia környékén. Fontos megjegyezni, hogy mindig a fel spint vesszük többségi spinorientációnak, mely a ferromágneses tartomány mágnesezettség-irányának felel meg. Később látni fogjuk, hogy a spinpolarizáció lehet a mágnesezettséggel ellentétes előjelű.
Landauer formula spinpolarizált esetben
Egy mágneses nanovezeték valósághűbb leírását kapjuk, ha szórást is megengedünk a vezetéken belül. Ezt a legegyszerűbben a korábban megismert Landauer formalizmus keretében tehetjük meg. A Landauer képben az elektrontranszportot 
transzmisszós valószínűségekkel írjuk le, melyek a bal oldali 
-edik vezetési csatornából a jobb oldali 
-edik csatornábó történő szóródás valószínűségét adják meg. Míg 
a fent tárgyalt ideális kvantumvezetéknek felel meg. Ha a vezetéken belül az elektronok rácshibákon, illetve szennyezőkön szóródnak, akkor 
transzmissziós valószínűségeket kapunk. Fontos megjegyezni, hogy a Landauer-kép inelasztikus szórásokat (pl. elektron-fonon kölcsönhatás) és spinszórásokat nem tud figyelembe venni.
Vizsgáljuk me a 3. ábrán szemléltetett rendszer vezetési tulajdonságait: a nanovezeték két nemmágneses ideális kvantumvezeték között egy ferromágneses tartományt tartalmaz, melyben szóródhatnak az elektronok.
 |
| 3. ábra |
Mivel spinszórás nem történik, az áramot továbbra is egymástól függetlenül számolhatjuk a két spincsatornára a transzmissziós valószínűségek figyelembe vételével:
![\[ I^\sigma=\frac{e^2}{h}\sum_{n,m=1}^{M^\sigma}T_{nm}^\sigma V. \]](/images/math/c/5/d/c5df7038656a710a49bbe4b0a8bff651.png)
A 
értékek függnek az elektronok Fermi-hullámhosszától, ami egy adott -edik csatorna esetén jelentősen eltérhet a két spin-orientációra a kicserélődési felhasadás miatt. Ez alapján spinszórás (pl. spin-pálya kölcsönhatás) nélkül is a transzmisszós valószínűségek különböznek a két spincsatornára, ezt vesszük figyelembe a 
indexszel.
A spinfüggő áramot átírhatjuk
![\[ I^\sigma=\frac{e^2}{h}M^\sigma \bar{T}^\sigma V \]](/images/math/d/2/d/d2d35eeca080566f9f4c66f8bf6956c4.png)
formába, ahol 
az átlagos transzmissziós valószínűség a spin-orientációhot tartozó összes nyitott vezetési csatornára. Ezzel a jelöléssel az áram spin szerinti polarizáltságát
![\[ P_c=\frac{I^{\uparrow}-I^{\downarrow}}{I^{\uparrow}+I^{\downarrow}}=\frac{M^{\uparrow}\bar{T}^\uparrow-M^{\downarrow}\bar{T}^\downarrow}{M^{\uparrow}\bar{T}^\uparrow+M^{\downarrow}\bar{T}^\downarrow}. \]](/images/math/e/e/f/eef78545d0cb4c40985d9947b3b4657e.png)
formában írhatjuk.
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 
and
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}), 
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, 
can be assumed. Accordingly the key contribution to the spin
polarization is given solely by the difference of 
and 
. In this limit, Eq.~\ref{Pc}
gives a reasonable approximation for the spin polarization, even for non-ideal transmission probabilities
(
).
\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}
According to Eq.~\ref{vn} the number of conductance channels can be formally written as:
![\[ M^\sigma=\frac{2\pi\hbar}{L}\sum_{n=1}^{M^\sigma}v_n^{\sigma}g_n^{\sigma}. \]](/images/math/1/4/4/144e6325977480a78786c18fdfefa79f.png)
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,
![\[ \bar{v}_F^\sigma=\frac{\sum_{n} g_n^\sigma v_n^\sigma}{\sum_{n} g_n^\sigma}, \]](/images/math/a/7/f/a7f2fe684a4828f81cc678f1dcc37766.png)
and noting that 
is the total density of states at the Fermi level, the formula for the current spin polarization can be rewritten as:
![\[ P_c\approx\frac{M^{\uparrow}-M^{\downarrow}}{M^{\uparrow}+M^{\downarrow}}=\frac{g_F^{\uparrow}\bar{v}_F^{\uparrow}-g_F^{\downarrow}\bar{v}_F^{\downarrow}}{g_F^{\uparrow}\bar{v}_F^{\uparrow}+g_F^{\downarrow}\bar{v}_F^{\downarrow}}. \]](/images/math/7/f/8/7f84d7e5b0af9c5d7c274bf6cad4e433.png)
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.
 |
| 4. ábra |
\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 
(or 
)
bands. Panel (a) demonstrates the DOS in Fe: for minority spins (left side) the
band is above the Fermi level, whereas for majority spins (right side) the
Fermi level lies inside the 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 
is above the band for majority
spins, whereas it lies inside the 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 (or ) 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{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.
 |
| 5. ábra |
\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 
magnetic orientation), whereas panel (d) demonstrates the antiparallel
orientation ( in the first layer and 
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 , whereas for minority
spins (antiparallel spin direction to the magnetization direction) the number of
open channels is 
. (Note, that here the superscript
denotes majority orientation instead of the real spin direction,
and the number of open channels is also 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 , and so the conductance for both spin
channels is 
, i.e. the total conductance is
. For parallel orientation of the magnetic
layers the conductance is 
if the spin direction of the
carriers is the same as the magnetization, and it is if
the magnetization has opposite orientation to the spin orientation of the
carriers. The net conductance is
. The magnitude of the
magnetoresistance effect can be characterized by 
, where
. 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,
![\[ \frac{\Delta G}{G^\mathrm{P}}=\frac{M^\uparrow-M^\downarrow}{M^\uparrow+M^\downarrow}=P_c. \]](/images/math/8/a/a/8aae9d2087178d7af06b43665d07c925.png)
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,
the scattering in the two layers are added coherently, and accordingly
the total resistance relies on the fine details of quantum interference
phenomena.
 |
| 6. ábra |
\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
magnetization direction), whereas the bottom panel demonstrates the
antiparallel alignment (the left layer has and the right layer has
magnetization direction). 
and 
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 (
), but the spin information is still preserved
(
), 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 , whereas for minority spins it is .
For antiparallel oriented magnetic layers the total conductance is
, whereas for
parallel oriented layers it is 
. From these
![\[ \frac{\Delta G}{G^\mathrm{P}}=\left(\frac{G^\uparrow-G^\downarrow}{G^\uparrow+G^\downarrow}\right)^2=P_c^2 \]](/images/math/1/9/3/193c43bfdda2554c0ee0f1ba2ac691a3.png)
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 
is positive, i.e. the
parallel orientation has higher conductance than the antiparallel one. Depending
on the model, the relative conductance change ranges between 
and 
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
, 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.
