brief nanomagnetism and spintronics -...

1. Brief overview of nanomagnetism and spintronics

What is spin electronics (or “spintronics”)?

4

Electronics: move charge currents

Spin electronics: exploit spin degree of freedom

In ferromagnetic transition metals, currents are naturally spin-polarized(Half-metals: potential for 100% spin polarization)

Magnetism plays an important role in the manifestation of spin-dependent transport

Applications of spintronics already widespreade.g. read-head in hard disk drives based on magnetoresistance (MR) phenomena

Spin electronics as future?

5

The future miniaturization of electronics will be strongly impeded by energy consumption and heat generation in circuits via Joule heating

Spin currents may be a means to circumvent such issues (especially charge free currents)

But how do we generate and detect spin currents? How can these be used effectively?

How much longer can Moore’s law be sustained?

From “charge” to “spin” electronics

High Integration ML Circuits

Magnetic Storage

Magnetic Logic (ML)

HDD, MRAM controlled by

Magnetic field

STT MRAM, DW spin-polarized

charge current control by

Electric field

pure Spin-currents spin transfer and logic with pure spin currents

manipulate magnetization, transport, with gate voltage

phase coherence

2009 CHARGE

SPIN SPIN CHARGE

Moore’s law Beyond CMOS

2016 2020

31 MINATEC Upstream Research, June 24th, 2010

Some Beyond-CMOS initiatives

6

In this set of lectures, we will give an introduction to some spin-dependent transport phenomena related to magnetic metals and explain some of the underlying physics

Combined transport and magnetization dynamics gives clues to how magnetic materials can be used for data processing and storage

Tunneling spin injection into grapheneKawakami group, UC Riverside

Spin-torque MRAM

Ex. 1: Magnetoresistive random access memories (MRAM)

7

Some commercial products already available

Ex. 2: Spin-wave processing circuits

8

A gigahertz-range spin-wave filter composed of width-modulated nanostripmagnonic-crystal waveguides

Sang-Koog Kim,a! Ki-Suk Lee, and Dong-Soo HanResearch Center for Spin Dynamics and Spin-Wave Devices, Nanospinics Laboratory, Research Instituteof Advanced Materials, Department of Materials Science and Engineering, Seoul NationalUniversity, Seoul 151-744, Republic of Korea

!Received 13 May 2009; accepted 1 July 2009; published online 26 August 2009"

We found a robust magnonic-crystal waveguide structure for use as an efficient gigahertz-rangespin-wave filter that passes only spin waves of chosen narrow band frequencies and filters out theother frequencies. The structure consists of the serial combinations of various width modulationswith different periodicities and motifs in planar-patterned thin-film nanostrips composed of a singlesoft magnetic material. The observed magnonic band gaps result from both the translation symmetryof the one-dimensional width modulation and the higher-quantized width-mode spin waves excitedfrom scattering at the periodic edge-steps of the width modulation. This work brings us one stepcloser to practical implementations of spin waves in information transmission and processingdevices. © 2009 American Institute of Physics. #DOI: 10.1063/1.3186782$

Due to the current needs of ultrafast information-processing and ultrahigh-density information-storage de-vices, different types of nanoscale devices have been ex-plored along with the recent developments in advancednanofabrication technologies. A prominent example is spinwaves !called magnons", which are collective excitationsof individual spins in ordered magnets. Through the propa-gation of spin waves having many harmonic frequencies, in-formation signals can be transmitted along patternedwaveguides made of magnetic materials.

Up to now, many previous theoretical, numerical, andexperimental studies have focused mostly on fundamentallyunderstanding the eigenmodes of excited spin waves andtheir dispersion relations including their controllability in in-finite model and micrometer-sized !or larger" real systems.1

These studies have also stimulated technological implemen-tations of spin waves in information transmission and pro-cessing devices, such as logic gates,2,3 logic circuits,4 andfilters.5,6 However, not only the reliable control of propagat-ing spin waves along well-behaved waveguides,3,5–10 butwaveguide miniaturization2,3,11,12 down to less than mi-crometer size, are further necessary for real applications ofsuch spin-wave devices.12

In this letter, as an extension of our earlier study,5 wereport on the discovery of a planar-patterned magnonic-crystal waveguide !MCWG" that can be used for a promisinggigahertz-range broadband spin-wave filter. This filter iscomposed simply of various nanostrips of different widthmodulations serially connected and made only of a singlesoft magnetic material #e.g., permalloy !Py"$. The underlyingphysics of the occurrence of magnonic band gaps and theirrelation to the geometrical width modulations, additionally,were studied by micromagnetic numerical calculations.

In this study, we used 10 nm thick Py thin-film nano-strips for MCWGs. As seen in Fig. 1, the entire MCWGstructure is composed of three parts. One is the width-modulated strip in the middle and the other two are the 30

nm wide strips at both ends. The middle area, acting as aspin-wave filter, is made of a planar-patterned width modu-lation !here, e.g., W1=30 and W2=24 nm", with a periodic-ity P= P1+ P2 and its repeating number N !see the inset ofFig. 1". P1 !P2" corresponds to the segment length of the 30!24" nm wide strip. This proposed MCWG is expected toyield scattering of initially propagating spin waves fromthe periodic edge-steps of the width modulations. The simu-lation code used here is the OOMMF of version 1.2a4!Ref. 13", which employs the Landau–Lifshitz–Gilbertequation of motion.14 Additional descriptions of the presentsimulation are described in Ref. 15. To excite and injectthe lowest-mode spin waves with frequencies, fSW, rangingfrom 0 to 100 GHz, we used a “sinc” function,5,16 Hy!t"=H0%sin#2!"H!t! t0"$& /2!"H!t! t0" with H0=1.0 T !Ref.17" and "H=100 GHz. This field was applied along the yaxis only to the left-end local area !1.5#30#10 nm3" indi-cated by the dark-gray color in Fig. 1.

To explore how the planar-patterned structure affectsmagnonic band structures, frequency spectra and dispersioncurves were plotted #Fig. 2!a"$ for spin waves propagatingalong the x axis, at the pass of y=15 nm, through the indi-cated different structures of MCWGs. They were obtained byconducting fast Fourier transforms !FFTs" of the temporalMz /Ms !the perpendicular component of the magnetizationnormalized by its magnitude" oscillations in the given nano-strip of N=0, 1, 2, 4, or 12. All of the dispersion curves are

a"Author to whom correspondence should be addressed. Electronic mail:[email protected].

10 nm

500 nm30 nmP x N0

P1 P2xyz

P

W2=24nmW1=30nm

FIG. 1. !Color online" Schematic illustration of the model thin-film MCWG.The equilibrium magnetizations point in the !x direction in most areas, dueto the shape anisotropy. The white-colored areas correspond to 30 nm widenanostrips, and the blue-colored area represents the width-modulated part.The left end indicated by the gray color is the local area for spin-waveexcitation. The inset displays the two unit periods of the different widthmodulations.

APPLIED PHYSICS LETTERS 95, 082507 !2009"

0003-6951/2009/95"8!/082507/3/$25.00 © 2009 American Institute of Physics95, 082507-1

nonsymmetric with respect to kx=0, due to the excitationand injection of the lowest-mode spin waves from the leftends of the nanostrips. For N=0, there exists no forbiddenband above 14 GHz, except for the intrinsic forbidden bandbelow the potential barrier of 14 GHz, due to the nanostrip’swidth confinement.8,18,19 In contrast, for N=2 or 4, twoforbidden bands are observed. The higher band gap of !g

h

=50–67 GHz is clearer than the lower band gap of !gl

=26–36 GHz. As the value of N increases further, bothband gaps become more obvious. For example, the value ofN=12 leads to a complex band structure and more distinctband gaps. To understand the role of the periodic widthmodulation in the occurrence of such complex band gaps, itis informative to investigate dispersion spectra obtained fromthe excitation and injection of spin waves from the middle ofthe indicated width-modulated MCWG and two single-widthnanostrips !24 and 30 nm" #Fig. 2!b"$. All of the dispersionspectra show the intrinsic forbidden bands below 14 GHz,due to the strip-width confinements.8,18,19 As seen in the leftpanel of Fig. 2!b", the lower band gaps occur at the positionskx=n" / P !where n=0, #1, #2, . . ., and P=18 nm", i.e., theBrillion zone !BZ" boundaries represented by the bluedashed vertical lines. This indicates that the lower band gapsare associated with the one-dimensional !1D" translationsymmetry of the width modulation along the longitudinaldirection, which yields interference between the initiallypropagating lowest-mode spin waves and their reflectedwaves from the edge-steps. In contrast, the higher band gapsare observed at kx= !2n+1"" / P#$ nm!1, $=0.08 !denotedby the red dotted vertical lines", which are away from the BZboundaries. The variations in W1 /W2 as well as P affect thenumerical value of $ !Ref. 20". This reveals that the higherbands cannot be understood only through the 1D approach.As seen in the dispersion curve of N=1 #Fig. 2!a"$, the ad-

ditional branch above 50 GHz is caused by the 3rd-quantizedwidth mode excited through the scattering of the lowestmode at the edge-steps.5,8,18,19 Consequently, the higher bandgap results from the interaction between the higher-quantizedwidth mode and the initial lowest-mode spin waves.5 Thedifferent physical origins of the occurrences of the higherand lower band gaps are also understood by the distinctstanding wave patterns shown in Fig. 3 for three specificvalues of fSW=31, 58, and 40 GHz, which are selected fromamong the forbidden and allowed bands. For fSW=31 GHz,the nodes of standing waves in the regions before the width-modulated strips are positioned parallel to the width direc-tion. For fSW=58 GHz, by contrast, the nodes are positionedin the transverse direction as well as the longitudinal direc-tion.

According to the above findings, one can readily expectthat magnonic band structures vary by P, N, P1 / P2, andW1 /W2. Figure 4!a" shows some examples of frequencyspectra obtained from MCWGs with the indicated differentperiodic modulations. For the case of P1= P2=7.5 nm, thetwo wide band gaps of !g=31–48 and !g=52–75 GHz ap-pear, and for P1= P2=10.5 nm, the relatively narrow bandgaps of !g=22–30 and !g=48–62 GHz are found. For P1=18 and P2=3 nm, in contrast, four narrow band gaps areobserved. To construct more valuable maps on the variationsin magnonic band gaps by both P and P1 / P !here the ratio ofW1 /W2 is kept constant; see Ref. 20", we conducted addi-tional simulations with different nanostrips of various widthmodulations, i.e., variations in P while retaining P1= P2 andof P1 / P while maintaining P=30 nm. Figure 4!b" shows theresults, where all of the gray regions correspond to the al-lowed bands. The white regions correspond to the forbiddenbands obtained at kx=n" / P, and the blue regions correspondto those observed away from the BZ boundaries. As seen inFig. 4!b", the number of band gaps and the center and widthof the individual gaps change dramatically with P. For theforbidden bands denoted by the white-color-coded region,the center position and width of each band decrease with theincreasing P, whereas the center positions do not change

100

50

0

(a) N = 0

0 0.5 1.0

N = 2

0 0.5 1.0

N = 4

0 0.5 1.0

N = 12

1426365067

0 0.5 1.0 1.50 0.5 1.0

N = 1

( )x mµ 0 0.1FFT Power (a.u.)

f SW(GHz)

f SW(GHz)

-0.3 0 0.3kx(nm-1)

1426365067

0FFT Power (a.u.)

-0.3 0 0.3 -0.3 0 0.3 -0.3 0 0.3 -0.3 0 0.380

100

75

50

25

0-0.4 -0.2 0 0.2 0.4kx (nm-1)

[W1 (nm), W2 (nm)] = [30, 24]

2636

50

67

14

f SW(GHz)

(b)100

50

0

-0.4 -0.2 0 0.2 0.4kx (nm-1)

f SW(GHz)

[24, 24]

[30, 30]

50

0

0FFT Power (a.u.)

200

[W1, W2]

FIG. 2. !Color online" !a" Comparisons of frequency spectra !top" and dis-persion curves !bottom" for a 30 nm wide nanostrip and the width-modulated nanostrips of #P1!nm" , P2!nm"$= #9,9$ and #W1!nm" ,W2!nm"$= #30,24$ but with different N values, as noted. !b" Dispersion curves ofspin-wave excitation at the middle !x=1000 nm" of the indicated nanostripsof N=55.

NfSW (GHz)31

40

1412

1412

1412

58

450 500 550 600 650 7000 0.5 1.0FFT Power (a.u.)

x (nm)

300y

(nm)

58

[P1, P2] = [9 nm, 9 nm][W1, W2] = [30 nm, 24 nm]

FIG. 3. !Color online" Plane-view color-coded images of the standing spin-wave patterns obtained from the FFTs of the spatial distributions of thetemporal Mz /Ms evolution in the width-modulated nanostrips with the indi-cated different N values for the specific frequencies noted.

082507-2 Kim, Lee, and Han Appl. Phys. Lett. 95, 082507 !2009"

Nanostructured materials allows for band-engineering of spin waves

Possible applications in information processing

Ex. 3: RF oscillators

9

Nanoscale radiofrequency oscillators based on sustained magnetization dynamics

2.0

1.0

0.5

1.5

0

7.8

7.5

7.2

6.9

5 6 7 8

7 8 9Frequency (GHz)

Current (mA)

zH/

Vn( e

dutil

pm

A2/

1)

8.5 mA

7.5 mA

6.5 mA

5.5 mA

4.5 mA

4.0 mA

0.23 GHz/mA

kae

Pf

)zH

G(

50 Å Ni80

Fe20

50 Å Cu

200 Å Co90

Fe10

40-nm Au contact

2. Brief review of normal and ferromagnetic metals

Free electron model

11

Conduction electrons are valence electrons that move freely through volume of metal

Conduction electrons do not feel metallic ions, form uniform gas

Subject to Pauli exclusion principle (Fermi statistics)

e.g. 1D model for N electron gas

H�n = � �22m

d2�n

dx2= ⇥n�n

p = �i� d

dxH =

p2

2m ⇤n = A sin

�2⇥

�nx

⇥

�n(0) = �n(L) = 0

In!nite potential energy barriers, standing wave solutions

Free particle Hamiltonian

1D Schrödinger eq.

Box, length L

⇥n =�22m

�n�L

⇥2

Fermi energy

⇥F =�22m

�N�

2L

⇥2

2nF = N

Spin degeneracy

Free electron model

12

H�n = � �22m

⇥2�n = ⇥n�n

p = �i�⇥ H =p2

2m

Simple generalization to 3D

L

kFermi sphere

�F =�22m

k2F k = 2⇥/�

kF =

�3�2N

V

⇥1/3

⇥F =�22m

�3�2N

V

⇥2/3

(kx, ky, kz) (2�/L)3One k state in volume element

Fermi energy de!ned by Fermi sphere

Electrical conductivity, Ohm’s law

13

Momentum of free electron is given by

In an electric field E and a magnetic field B, force acting on charge is

mv = �k

F = mdv

dt= �dk

dt= �e (E+ v ⇥B)

In E field alone, Fermi surface is displaced at a rate of �k = �eE�t/�

Electrical conductivity, Ohm’s law

14

Displacement of Fermi surface can be maintained at a constant value at steady state because of collisions

e.g. scattering with phonons, lattice impurities

For a mean collision time

�k = �eE�t/�

54 B. J. Hickey et al.

f(k)

kx

1

(a)

ky

kx

(b)

Fig. 3.2. Schematic drawing of the distribution function in an applied electric field:(a) change (shaded) in the Fermi function and (b) shift of the projected Fermi sphere.

It is made up of two terms, the first describing the fact that f is being driven awayfrom equilibrium as the electrons are accelerated by the electric field and thesecond describing the relaxation back to equilibrium due to scattering processes.Below we shall show how this simple expression can be used to calculate theconductivity in a simple free electron model, in more complex models and inthin films.

Let’s start by looking at the simple free electron case since this will help usto understand some basic ideas. Thus

df

dt

!

!

!

field=

df

dkx

dkx

dt=

df0

dkx

eE!

; (3.4)

the last step arises as dk/dt is simply related to the rate of change of momentumand so to the accelerating force on the electrons due to the electric field, E . Herewe have allowed the electron charge e to include its sign. In the last step we haveonly kept the derivative of f0 in order to keep terms linear in E . The next stepis to change the derivative with respect to k to one with respect to energy usingthe free electron relationship E = !2k2/2m:

df

dt

!

!

!

field= ! df0

dE!xeE . (3.5)

�

v = �eE�

m

The electric current density then is

j = nqv =ne2�E

mj = �E � =

ne2⇥

m

Define mean-free path as l = vF �

t (fs) @ 273 K vF (x 106 ms-1) l (nm) @ 273 KCu 27 1.57 42Au 30 1.40 56Fe 2.4 1.98 4.8

Ohm’s law � = 1/⇥

Nearly-free electron gas

15

Account for periodic lattice of background metallic ions - periodic potentialElectrons in weak periodic potential are nearly free

Solution to Schrödinger equation are Bloch functions

�k(r) = uk(r) exp(ik · r)

Degeneracies at the Bragg plane are lifted by the perturbations due to potential: band gaps appear

plane wavesdescribes lattice periodicity

Fermi surfaces

16

Fermi surface is surface in k space of constant energy equal to the Fermi energyIt separates filled states from empty states at absolute zero

Electrical properties of metals are determined by shape of Fermi surface, because currents are due to changes in occupancy near this surface

Take into account crystal symmetry (cf band structure, nearly-free electron model)

fcc Brillouin zone fcc free electron bands calculated surface for Cu

Fermi surfaces and conductivity

17

Electrical conductivity is determined by how electrons respond to electric fields

Insulator: allowed energy bands are either empty or full

Metal: One or more bands are partially filled

Metals: Conduction processes occur at Fermi surface

Insulator Metal Semiconductor

Ener

gy

http://www.phys.ufl.edu/fermisurface/

4s 3d10 4s

Potassium Copper Aluminium

3s2 3p

Fermi surface - ferromagnetic metals

18

Different Fermi surface for spin up and spin down electrons

3d6 4s2

bcc iron hcp cobalt

3d7 4s2

maj

ority

min

ority

Conduction not same for spin-up/down electrons

Spin-dependent transport processes

Two-current model

19

Electrical resistance depends on Fermi velocities, density of states @ Fermi surface, etc.

In nonmagnetic metals, Fermi surface is identical for spin-up and spin-down

In ferromagnetic metals, expect resistances for spin-up and spin-down channels to be different

Electrical conduction assumed to take place via two independent spin-channels

R! != R"

↑

↓

bcc iron

j�,⇥ = ��,⇥E ��,⇥ =ne2⇥�,⇥

mTopical Review

Figure 1. Spin splitting of the density of states (!) in a ferromagnetdue to the exchange field.

from a ferromagnet to a paramagnet by tunnelling throughan insulator, this current is also polarized due directly to thedensity of states asymmetry. FM elements may thus be used asspin-polarized current sources in spin-electronic circuits. Mostspin-electronic phenomena are based on either one or both ofthese asymmetries (whose common origin is the band structuresplitting) prevailing in the relevant physical system [1].

2.1. Spin asymmetry: density of states asymmetry versusmobility asymmetry

In fact, the two asymmetries often compete with one anotherin spin electronics. The Fermi surface in most FM materialscontains components which have both s- and d-character.The s-like effective masses are small compared to the d-likemasses and so any current that flows is primarily mediated bys-electrons. However, the d-electrons are significantly split bythe exchange interaction and as a result present very differentdensities of states into which the s-electrons may be scattered.Thus, from figure 1, the down s-channel (whose spin type hasa large d density of states at the Fermi energy) suffers themost scattering and hence has lower mobility than the others-channel: this latter consequently carries most of the current.

Thus in a system with s- and d-like character at the Fermisurface, the tendency is for the current to be carried by theminority carriers (where ‘minority’ is taken to mean thosewith the lower density of states at the Fermi energy, and thisconvention will be used throughout this paper) whereas in ahalf-metallic ferromagnet (see next section), the current mayonly be carried by majority carriers. This conflict between thetwo types of asymmetry is one reason why spin-tunnel devices(section 6) have an advantage over their competitors sincethey exploit only the density of states asymmetry; hence, themobility asymmetry has no chance to compete and reduce theoverall device performance. This has direct relevance to thequestion of spin injection into semiconductors.

2.2. The half-metallic ferromagnet

In the extreme limit of spin asymmetry lies the half-metallicferromagnet [2] in which the band structure splitting is suchthat only one spin channel has available states at the Fermisurface and hence all current must be carried by these so-calledmajority spins. Practical examples include chromium dioxide

Figure 2. Schematics of the difference in the densities of statesbetween a ferromagnet and a half metallic ferromagnet.

Figure 3. Illustration of the spin accumulation at theferromagnet/paramagnet interface.

(CrO2), lanthanum strontium manganite (La0.7Sr0.3MnO3)and some Heusler alloys. In reality, obtaining half-metallicspin-electronic behaviour is fraught with problems mainly todo with the interfaces. Conversely, some materials whosebulk electrical conduction deploys both spin channels may,due to hybridization, form half-metallic interfaces with othermaterials.

2.3. Spin injection across an interface: spin accumulation

Now that we have considered the basic principles behindthe origin of spin asymmetry, we can briefly consider animportant phenomenon which lies at the heart of early spin-electronic devices. Providing one carrier spin type is dominantin the electrical transport of a ferromagnet, when a currentis passed from this ferromagnet to a PM metal such assilver or aluminium, it brings with it a net injection of spinangular momentum and hence also of magnetization [3]. Themagnetization which builds up in the new material is knownas a spin accumulation (figure 3). Its size is determined by theequilibrium between the net spin-injection rate at the interfaceand the spin-flipping rate in the body of the paramagnet. Itfollows that the spin accumulation decays exponentially awayfrom the interface on a length scale called the ‘spin diffusionlength’.

R122

Two-current model

20

Spin-mixing can occur, but rare (at low T) compared with spin-conserving scattering processes

Spin-orbit interaction and electron-magnon scattering can lead to spin flips(Question: Why? Which symmetry principles underlie these processes?)

Hso =�2

2m2c2r

dV

drL · S

Deviations to Matthiessen’s rule in such cases

� =��⇥

�� + �⇥no mixing

with mixing

��

��/2

��/2

��

��/2

��/2

1

�=

1

�1+

1

�2+ · · ·

� =��⇥ + ��⇥(�� + �⇥)

�� + �⇥ + 4��⇥��⇥

3. Localized vs. itinerant models of magnetism

Band model of magnetism

22

Non-integer number of Bohr magnetons per atom

Bandwidths ~ eV

Element Bohr magnetons per formula unit

Fe 2.22Co 1.72Ni 0.606

Cr Mn Fe Co Ni Cu0.0

0.5

1.0

1.5

2.0

2.5

3.0

Ato

mic

mom

ent (+

B)

Electron concentration

pure metals

Fe-V Fe-Cr Fe-Ni Fe-Co Ni-Co Ni-Cu Ni-Zn Ni-V Ni-Cr Ni-Mn Co-Cr Co-Mn

−10 −5 0 5−3

−2

−1

0

1

2

3

DEN

SITY

OF

STA

TES

[sta

tes/e

V.a

tom

.spin

]

−10 −5 0 5

ENERGY (eV)

−3

−2

−1

0

1

2

3

−10 −5 0 5−3

−2

−1

0

1

2

3

Cu(up) Co(up) Fe(up)

Cu(dn) Co(dn) Fe(dn)

Fig. 4.7. Densities of states of copper, cobalt and iron. Broken line denotes the positionof the Fermi level.

Slater-Pauling curve

64 CHAPTER 7. ITINERANT-ELECTRON MAGNETISM

level is high. After the transfer, there will be more spin-up electrons than spin-down electrons, and the magnetic moment, which has arisen, will be equal to

We will first derive a simple band model, which accounts for the existence of ferro-magnetism. The interaction Hamiltonian, following the above definition of can be written as

where and represent the number of electrons per atom for each spin state, and where the total number of 3d electrons per atom equals Because is a positive quantity, Eq. (7.1.2) will lead to the lowest energy if the product is as small as possible. For equally populated subbands, this product has its maximum value and hence the highest energy. Consequently, electron transfer is always favorable for the lowering of the exchange energy, and this electron transfer will come to an end only if one of the two spin subbands is empty or has become completely filled up.

We define N(E) as the density of states per spin subband, and p as the fraction of electrons that has moved from the spin-down band to the spin-up band. This means

Let us assume that the interaction Hamiltonian (Eq. 7.1.2) leads to an increase in the number of spin-up electrons at the cost of the number of spin-down electrons. The corresponding gain in magnetic energy is then

This energy gain is accompanied by an energy loss in the form of the amount of energy needed to fill the states of higher kinetic energy in the band. For a small displacement

(see Fig. 7.1.1b), this kinetic-energy loss can be written as

The total energy variation is then

Ue�N(EF ) > 1

Stoner criterion

Localized moments

23

Magnetic material described by ordered array of local spins

Interaction between nearest-neighbour spins determines magnetic order

Ji,j

Works well for magnetic insulators, rare-earths (well-localized f shells)

Easily generalized to continuum models, micromagnetics simulations

H = ��

�i,j⇥

Ji,jSi · SjHeisenberg interaction

Ji,j > 0

Ji,j < 0

ferromagnetism

antiferromagnetism

Localized moments vs itinerant ferromagnetism

24

Nevertheless, the localized spin picture, which corresponds directly to insulating magnets, produces results that work very well for metals.

What about the 4s electrons, which are very active in conduction processes in transition metals (effective masses for 3d electrons are larger)?

In transition metal ferromagnets, both 3d and 4s electrons play a crucial role, although separating them into two distinct groups is a little questionable ...

In reality, s and d bands are hybridized near the Fermi level

sd model

25

Vonsovsky-Zener / sd model seems like a good compromise (in fact, it is the picture often used in spin electronics today)

mobile 4s electronsconduction only

localized 3d electronsmagnetism only

One supposes that there is an additional exchange interaction that polarizes the conduction electrons

Hsd ! "Jsd!S3d · !s4s

Topical Review













R122

But there are many subtle points ... beyond the scope of these lectures(for a detailed discusion, see Mattis)

4. Semiclassical electron transport

Different transport regimes

27

Full quantum

Semiclassical

Classical

Degr

ee o

f “qu

antu

mne

ss”

Kubo / Keldysh

Boltzmann

Drift-diffusion

!(!r, t)

f(!r,!k, t)

n(!r, t)

Here, we will focus attention on semiclassical Boltzmann theory

Semiclassical : take into account Fermi-Dirac statistics and band structure, but use diffusion equation.

Why do we use it?

Because we get can quick results

Easily adapted to multilayer structures (important for spintronics!)

Captures most of the essential physics

Metals: Remember, transport processes occur only at the Fermi surface.

Boltzmann theory describes time and spatial evolution of distribution function f.

Semiclassical theory

28

f0(!) =1

e!("!µ) + 1

! =1

kBT

Boltzmann equation

29

Let f(k,r) denote the electronic distribution at a point (k,r) in phase space

Evolution of f(k,r,t) given by total time derivative

!f

!t=

!f

!t

!

!

!

!

field

+!f

!t

!

!

!

!

di!usion

+!f

!t

!

!

!

!

scattering

x

kx

dkx

dx

!f

!t

!

!

!

!

field

=e

h"E ·

!f

!"k

!f

!t

!

!

!

!

di!usion

= !"v!k·

!f

!"r

!f

!t

!

!

!

!

scattering

=

"

d"k!{f("k!,"r)[1 ! f("k,"r)]P!k!,!k

! f("k,"r)[1 ! f("k!,"r)]P!k!,!k} out-scattering

in-scattering

drift due to electric !eld

Steady state, relaxation time

30

!f

!t

!

!

!

!

scattering

= !

f(r, k) ! f(r)

"

f(r) =1

4!

!FS

d2k f(r, k)

average over Fermi surface transport relaxation time(one example)

The full Boltzmann equation is very difficult to solve as it is a nonlinear partial integro-differential equationWe can still get insights into the main physics underlying transport with the following approximations:

Assume steady state is reached and neglect transient dynamics

Use a relaxation time approximation to simplify the scattering integral

�f

�t= 0

1

�(k)=

�d3k� Pk,k�(1� k · k�)

Boltzmann equation

31

!(k) =h

2k2

2m

!f

!t

!

!

!

!

field

= e "E · "v!k

!f0

!#

e !E · !v!k

"f0

"#+ !v!k

·

"f

"!r= !

f ! f

$

f(r, k) = f0 !

!f0

!"g(r, k)

f(r) = f0 !!f0

!"

1

4#

!FS

d2k g(r, k)

Example: Let’s apply the Boltzmann treatment for the free electron gas. Recall that

In the steady state and in the relaxation time approximation, we obtain a tractable form of the Boltzmann equation

We can do even better. Let’s linearize about the equilibrium distribution function. Call g(r,k) the nonequilibrium part of the distribution.

Boltzmann equation

32

!v!k·!g(!r,!k) +

g(!r,!k)

"= e!v!k

· !E +µ(!r)

"

µ(r) =1

4!

!FS

d2k g(r, k)

driving terms

Define µ as the non-equilibrium component of the chemical potential

So the Boltzmann equation for g(r,k) becomes

Thus, the electric field E and the chemical potential µ are “driving terms” for the differential equation for g(r,k).

Drude conductivity, revisited

33


f(k)

kx

1

(a)

ky

kx

(b)




df

dt

!

!

!

field=

df

dkx

dkx

dt=

df0

dkx

eE!

; (3.4)


df

dt

!

!

!

field= ! df0

dE!xeE . (3.5)

Trivial example: electrical conduction in an infinite metal

Translational invariance: gradient term and µ → 0.

Suppose E = (E, 0, 0). Then

g(!r,!k) = evxE"

The current is

j = e

!

vg(r, v)d3v =

"

ne2!

m

#

E


f(k)

kx

1

(a)

ky

kx

(b)




df

dt

!

!

!

field=

df

dkx

dkx

dt=

df0

dkx

eE!

; (3.4)


df

dt

!

!

!

field= ! df0

dE!xeE . (3.5)

Drude conductivity

5. Spin-dependent and multilayer transport

Transport in multilayers

35

It becomes harder to solve Boltzmann equation for multilayers because of gradient and chemical potential terms.

But can we even use Boltzmann theory for such thin films??

From experiment, we can estimate the Drude relaxation time from the electrical conductance, from which we can estimate a relaxation length

! =

!

m

ne2

"

"

t (fs) @ 273 K vF (x 106 ms-1) l (nm) @ 273 KCu 27 1.57 42Au 30 1.40 56Fe 2.4 1.98 4.8

l = vF �

Layer-by-layer approach

36


f(k)

kx

1

(a)

ky

kx

(b)




df

dt

!

!

!

field=

df

dkx

dkx

dt=

df0

dkx

eE!

; (3.4)


df

dt

!

!

!

field= ! df0

dE!xeE . (3.5)

Thus, the typical relaxation lengths are larger or equal to the film thicknesses used in metallic multilayers (d ~ 1-5 nm).

Is it justified to use Boltzmann theory?

Recall that all electrons on the Fermi surface participate to conduction, not only those with velocities/wavevectors perpendicular to the interface.

Furthermore, metallic interfaces are rough, which meansany coherences are destroyed across the film.

So a diffusion equation like Boltzmann is OK.


37

The layer-by-layer approach consists of:

(i) Solving the Boltzmann equation for each layer separately

(ii) Joining the solutions together through appropriate interface boundary conditions

1 2 3 4 5

f1(r, k)f2(r, k)f3(r, k)f4(r, k)

f5(r, k)


38

Interface boundary conditions: consider ballistic transport across interface

i.e. Reflection, transmission and diffuse scattering of electron plane-waves

a) b)

1

STSR

1−S

f ( v<1 )

f ( v>1 ) f ( v

<2 )

f ( v>2 )

z = 0

Figure 3.1: Scattering at an interface: a) schematic representation of electron scattering,

b) direction of electron fluxes at an interface.

In the structure consisting of several layers (N), boundary conditions connect distri-

bution function in a layer (Li) with the distribution functions in the adjacent layers (Lj).

Equation (3.4) can then be generalized to describe the chemical potential everywhere in

the system:

µi(z) =N

!

j=1

"

Lj

dz!Kij(z, z!)µj(z

!) + Yi(z). (3.5)

The integral equation (3.5) is a system of Fredholm equations of the second kind,

which can be numerically solved to obtain the chemical potential profile everywhere

in the layered structure. [32, 56] The functions Kij(z, z!) and Yi(z) are defined by theboundary conditions at the interfaces. Numerical procedure of solving the Fredholm

equation of the second kind is presented in the Appendix A.6.

3.2 Boundary conditions at the interface between two

layers

At the interface between two layers, an incoming electron is scattered via reflection and

transmission. There is also diffuse scattering in all directions. In the present work, a

simple model for the diffuse scattering will be used, where a single parameter S rep-

resents an amount of electrons which are not scattered diffusely, so that the specularly

reflected and transmitted electron fluxes are scaled by S. 1!S is the amount of electronsscattered diffusely. Figure 3.1a) represents schematically how an electron is scattered at

an interface.

34

Incident electron wave

Specular re"ection

Specular transmission

Di#use scattering

a) b)

1

STSR

1−S

f ( v<1 )

f ( v>1 ) f ( v

<2 )

f ( v>2 )

z = 0

Figure 3.1: Scattering at an interface: a) schematic representation of electron scattering,

b) direction of electron fluxes at an interface.

In the structure consisting of several layers (N), boundary conditions connect distri-

bution function in a layer (Li) with the distribution functions in the adjacent layers (Lj).

Equation (3.4) can then be generalized to describe the chemical potential everywhere in

the system:

µi(z) =N

!

j=1

"

Lj

dz!Kij(z, z!)µj(z

!) + Yi(z). (3.5)

The integral equation (3.5) is a system of Fredholm equations of the second kind,

which can be numerically solved to obtain the chemical potential profile everywhere

in the layered structure. [32, 56] The functions Kij(z, z!) and Yi(z) are defined by theboundary conditions at the interfaces. Numerical procedure of solving the Fredholm

equation of the second kind is presented in the Appendix A.6.

3.2 Boundary conditions at the interface between two

layers

At the interface between two layers, an incoming electron is scattered via reflection and

transmission. There is also diffuse scattering in all directions. In the present work, a

simple model for the diffuse scattering will be used, where a single parameter S rep-

resents an amount of electrons which are not scattered diffusely, so that the specularly

reflected and transmitted electron fluxes are scaled by S. 1!S is the amount of electronsscattered diffusely. Figure 3.1a) represents schematically how an electron is scattered at

an interface.

34

Left- and right-moving distributions(upstream / downstream)

f(v>

2 , 0) = SR22f(v<

2 , 0) + ST12f(v>

1 , 0) + (1 ! S)F (f)

e.g.


39

−4 −2 0 2 4

0.0

0.2

−0.2

0.4

−0.4

0.6

−0.6

0.8

−0.8

1.0

−1.0

1.2

−1.2

S=1

S=0

S=1

l l

z/l z/l

eEeE1 1 2 2

1 2

1 2µ µ

Figure 4.3: Chemical potential profile in two-layered system for different values of

diffuse scattering S = 0, 0.1, . . ., 1 at the interface, normalized with respect to potentialdrop eEili within a mean-free path li of each metal, so that the plots are anti-symmetricalaround z = 0.

on both sides of the boundary, the chemical potential varies within several mean-free

paths of the interface. Resistance measured far from the interface,R1, is bigger then that

measured right at the interface, R0, for interfaces where diffuse scattering dominates

(S is close to 0). R1 is smaller then R0 for the interfaces where specular scattering

dominates (S is close to 1). Unlike the case of two identical layers, the resistance of thestep-like barrier decreases as the amount of diffuse scattering decreases. [24]

Figure 4.4 demonstrates the effect of diffuse scattering in the bulk of the layers on the

interface resistance. In the case of a completely specular interface, S = 1, the variationof interface resistance as the function of the barrier height V2/V1 is shown in comparison

with that obtained by Barnas and Fert in the system of ballistic layers. [18, 19] In the

presence of diffuse scattering in the bulk of the layers, both R0 and R1 are less than

the interface resistance obtained for purely ballistic transport. In the diffusive layer,

electrons reflected off the barrier can be scattered back on the barrier, and again have a

chance to be transmitted through, making the interface resistance smaller.

Figure 4.5 summarizes the behavior of the interface resistance as the amount of

diffuse scattering at the interface, and the height of the potential barrier experienced by

47

−500

−1.0−1.0

−200 0

−0.5−0.5

−100 500

0.00.0

0

0.50.5

100

1.01.0

200−2.0

−1.5

−1.0

−0.5

0.0

0.5

1.0

−2.0−1.0

−1.0

1.5

−1.5

−0.5

−0.5

2.0

−1.0

0.0

0.0

−0.5

0.5

0.5

0.0

1.0

1.0

0.5

1.0

1.5

2.0

z/l z/l

z/l

µ

µµ

µ

z/l

a) b)

d)c)

eEl

eEleEl

eEl

Figure 4.7: Chemical potential profiles for S = 0.5, normalized with respect to potentialdrop eEl within each layer: a) in the three-layered system with the middle layer thick-ness much larger than the electron mean-free path in the layer, b) in the five-layered

system with the middle layers thicknesses much larger than the electron mean-free path

in the layers, c) in the three-layered systemwith the middle layer thickness much smaller

than the electron mean-free path in the layer, d) in the five-layered system with the mid-

dle layers thicknesses much smaller than the electron mean-free path in the layers.

53

e.g. Calculated chemical potentials from Boltzmann theory for different diffuse scattering parameters

Two-layer system Five-layer system

Discontinuities in chemical potentials = interface resistances

What happens at the interface between a ferromagnet and a non-magnetic metal?

Imagine injecting a spin-polarized current into a nonmagnetic material. Eventually, the current becomes unpolarized far enough from the interface

Spin-flip scattering in the nonmagnetic layer restores this unpolarized state

Spin accumulation appears at the interface. How far does this penetrate into nonmagnetic layer?

Spin accumulation

40

Topical Review













R122

!!"

spin-di"usion length(cf. random walk)

spin-#ip relaxation time

mean free path

Cu: 500 nmCo: 60 nm

l = vF �

F NM

N collisionslsd = l

�N/3 Nl = vF ��⇥

lsd =

�vF ��⇥l

3

total distance travelled

@ T = 4 K

Spin-dependent transport

41

!v!k·!g"(!r,!k) " e!v!k

· !E = "g"(!r,!k) " µ(!r)

"m

"g"(!r,!k) " g!"(!r,!k))

""#

momentumrelaxation

spin-"iprelaxation

We can proceed with a Boltzmann treatment by using spin-dependent distribution functions

This formula shows that the current is unpolarized in the

limit that the leads become infinitely long (L!!).Now introduce spin–flip scattering in the leads. The cur-

rent polarization can vary spatially in this case because only

the sum of the up and down spin currents is conserved. This

is shown as the solid curve in Fig. 2"b# where p!pzz. Note

that the current in the ferromagnet is polarized and the cur-

rent in the leads "far from the interfaces# is unpolarized. Inbetween, pz(x) varies on a scale set by the spin diffusion

length. Therefore, the presence of spin–flip scattering20 al-

lows the system to accommodate as much as possible to the

‘‘polarization desires’’ of both the ferromagnet and the non-

magnet "as determined by their intrinsic conductivities#.Nonzero values of the dashed curve in Fig. 2 identify por-

tions of space where the spin density deviates from its equi-

librium value, i.e., spin accumulation. As mentioned earlier,

the gradient of this quantity contributes to the current polar-

ization.

Returning to the solid curve, the fact that pz(x) is sym-

metrical around the origin tells us that the steady state cur-

rent distribution is equally polarized on both sides of the

ferromagnetic layer. This means that no torque acts on the

magnet. On the other hand, the nonzero gradient of pz(x)

elsewhere tells us that distributed torques act throughout the

leads where spin–flip scattering occurs. These torques are

equal and opposite at points which are symmetrically dis-

posed with respect to the thin film. This means that current

flow in this system with a single ferromagnetic layer induces

a bending stress in the entire structure. In essence, the con-

duction electrons transfer angular momentum from one lead

to the other. This interesting result motivates us to look into

the mechanisms of spin transfer in more detail.

B. Spin transfer by reflection

The fate of a polarized electron incident on a ferromag-

net depends on the angle between the electron spin moment

and the magnetization direction of the magnet. We can en-

code this effect of quantum mechanical exchange most con-

cisely using spin-dependent reflection and transmission coef-

ficients R$ and T$ . This has been discussed qualitatively by

Waintal et al.8 Here, we focus on the scattering state for a

polarized electron in a nonmagnet (x"0) that is incident ona ferromagnet (x#0). If the incident electron spin points inan arbitrary direction (% &) with respect to the permanentmagnetization, we can write its wave function in the form

'!e$i&/2 cos"%/2#!'k"(%ei&/2 sin"%/2#!'k#(. "7#

Here,

!'k"(!"eikx%R"e$ikx#!"( x"0

!T"eik"x!"( x#0

!'k#(!"eikx%R#e$ikx#!#( x"0

!T#eik#x!#( x#0, "8#

are scattering states in a majority/minority basis. Inserting

Eq. "7# into Eq. "3# gives the incident current polarization as

pinc!"sin%cos& ,sin%sin& ,cos%#. "9#

It is straightforward "but tedious# to compute the corre-sponding quantities prefl and ptr from the transmitted and re-

flected waves generated by Eq. "7#. We omit them here and

focus instead on the extreme case where R"!1 and R#!0for an incident electron with a spin pointed in the y direction

"the magnetization in the z direction#. In this case, the inci-dent spin current polarization is pinc!(0,1,0). Only up spinsare reflected, so the reflected spin current polarization is

prefl!(0,0,1). Only down spins are transmitted, so the trans-mitted spin current polarization is ptr!(0,0,$1).

Note first that the z component of pinc is the same as the

z component of prefl%ptr . The numerical value happens to bezero in this case, but the stated equality is a general result.

Nothing very interesting happens to the component of the

electron spin that is parallel to the quantization axis of the

ferromagnet. By contrast, the transverse component of the

spin angular momentum does change. From Newton’s law

"and Ehrenfest’s theorem#, this is possible only if the mag-netization exerts a torque on the conduction electron spins.

For other angles and other reflection amplitudes, the amount

of transferred angular momentum is more complicated, but it

is nonzero in general.

From the sentence below Eq. "2# and using Eq. "3#, thetorque exerted on the permanent magnetization at x!0 dueto this reflection mechanism is

NR!A)

2"pincj inc$ptrj tr$preflj refl#! , "10#

where, A is the cross sectional area of the interface and the

currents j inc , j refl , and j tr are taken to be positive. The sub-

script! reminds us that this vector is transverse to the mag-

netization. We have chosenM!M z, so the torque lies in the

x$y plane specifically, the y direction for our simplified

example.

C. Spin transfer by averaging

The averaging mechanism of spin transfer is also a con-

sequence of the exchange interaction. But, it is completely

distinct from the reflection mechanism. To see this, observe

FIG. 2. Current polarization for a single ferromagnetic layer. "a# A thin Colayer embedded between two semi-infinite Cu leads; "b# Current polarization"solid line# and spin accumulation "dotted line# for a single Co layer embed-ded in Cu. The spin accumulation, defined as a density rather than a mag-

netization, is put in a dimensionless, scaled form by dividing by the ratio of

the current to the Fermi velocity.

6814 J. Appl. Phys., Vol. 91, No. 10, 15 May 2002 M. D. Stiles and A. Zangwill

spin

-cur

rent

Example: Cu/Co/Cu multilayer

spin

accu

mul

atio

n

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