Marder M.P. Condensed Matter Physics

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Ferromagnetism in Transition Metals 813

which means that the system is unstable to ferromagnetism when

„ •* TWO \ A

This

is the

condition

for

linear

instability

0 =r> — L^Gf) = 4. the

unmagnetized

state. It

predicts

that

the

W net

spin

runs away

from

zero, but

does

not

say

what

the

spin

will

become.

(26.71)

The value of (S) does not necessarily increase until it saturates at 1/2. The details

depend entirely upon the shape of the density of states. If

J£'D(£')£'

2 JEf-At

£F+A

dE' D(E')8.'- ~Jn {Sy

then

D(E

-Ai)

This

condition

results

from

conserving

the num-

ber of

electrons.

ÖA, D(£

+ A

and magnetization tends to increase so long as

öS

A,+A

< — / </£'£>(£').

4« i£

-A,

(26.72)

(26.73)

(26.74)

(26.75)

26.4.2 Calculations Within Band Theory

The magnetic moments of the transition metals can be calculated with almost as-

tonishing reliability by the combination of density functional theory and band cal-

culations. The flavor of how these calculations proceed can be obtained by return-

ing to the Hartree-Fock description of the uniform electron gas. The calculations

of Section 9.2.4 assumed that up- and down-spin states were populated equally, but

it is easy to relax this assumption and see what comes out. Returning to Eq. (9.50),

the energy of

up-spin electrons and

down-spin electrons would be

£ = £

+ £

where

■

n kfr

, and

A-K

identical

expression

holds

for £^.

3 (2vr)

3*fT

(26.76)

(26.77)

(26.78)

If the energy (26.76) can be lowered at all by creating nonzero net spin, then

the energy is minimized by putting all N electrons into a single spin direction. The

energy of such a state is

t-polarized

— Ar

3 h

5 2m

(ÔTT

«)

2/3

4TT

6TT n)

1/3

(26.79)

814 Chapter 26. Quantum Mechanics of Interacting Magnetic Moments

and

must be compared with the unpolarized state

-unpolarized

= N

/, 2 \

Nl/3

3-7T

n e 37T n

5 2m

) ATT V 7

(26.80)

After a bit of algebra, the criterion for the ferromagnetically polarized state to have

lower energy can be put in the form

2-KH

l)<^

(67r

n)-

(26.81)

rw 27T / 1 \ /97r\ '/

—

> -TH + ! 1 ( =

SeeEq

(U-48) for

the

definition (26.82)

ÛQ

5 \

21/^

/ \ 2 / of

nv;

ao is the Bohr radius,

„2

/me

The prediction

(26.82) that ferromagnetism is preferred at low electron den-

sities

not confirmed by any experimental results. Only cesium among the ele-

ments should be ferromagnetic according to this criterion, and cesium is nonmag-

netic.

The validity

the approach

not, however, fairly judged by such

crude

version

the calculation.

more reasonable test

is to

ask what happens when

the full apparatus

density functional theory

set loose on the magnetic prob-

lem, allowing populations of

up-

and down-spin electrons to vary freely and letting

the exchange and correlation functional

from Eq. (9.103) battle kinetic energy

terms over magnetic structures.

The

transition metals present

the

most important test case, because

the

metallic ferromagnets iron, cobalt, and nickel are

this group.

rough terms,

these elements are magnetic because many partly filled

bands are clustered about

the Fermi surface, providing a large density

states and allowing the Stoner crite-

rion

(26.71 )

to be satisfied. Quantitative calculations along these lines are summa-

rized by Moruzzi and Marcus (1993), and some results are presented in Table 26.1.

The predictive power of the band structure calculations is somewhat hampered

by their great sensitivity to changes in lattice structure. Band structure calculations

Table 26.1. Magnetic moment per ion in the ground state for the

3d transition metals in fee and bec minimum energy states

Element:

Sc Ti V Cr Mn Fe Co Ni

Calculated

m/ßß

(bec):

0 0 0

Experimental

m/ßß

(bec):

Calculated m//x

(fee):

0 0 0

Experimental

m/ßß

(fee):

Results are from density functional calculations of

Moruzzi

and Marcus

(1993).

The calculations are not always capable

deciding between

the fee and bec structures, but given the correct structure they correctly

predict the magnetic moment.

0.70

2.15

2.12

1.68

1.56

1.61

0.38

0.60

0.61

Spintronics

815

usually fail to predict experimental lattice constants to better than a few percent,

and they may have trouble correctly deciding which of several candidate lattices is

the true ground state. Equilibrium magnetic properties can vary quickly with small

changes in lattice constant, so if experiment and calculation disagree on what the

lattice constant should be, it is not clear what should be used to find the mag-

netic moment. Some remarkable predictions are nevertheless possible. Iron is bcc

at room temperature, but calculations predict that if it were fee, and if its lattice

constant were expanded by around 5% above equilibrium, it should become ferro-

magnetic with a moment of 2/^g per ion. Pescia et al. (1987) tested this prediction

by depositing a thin epitaxial layer of iron on fee copper, and they found that it was

indeed ferromagnetic.

26.5 Spintronics

Conventional electronics depends upon the manipulation of

charge.

Switches con-

trol the flow of charge currents, and the currents carry information. One hope for

the future of information processing is to move from control of charge to spin. In

one respect, spin has long been used to hold information; domains of aligned spin

form bits in magnetic recording. The vision of spintronics is to move beyond con-

trol of domains with external fields to new devices, where packets of spin travel

through wires, carrying bits as packets of charge do now. The hope is that these

devices will have lower dissipation and faster switching times than ones now used.

26.5.1 Giant Magnetoresistance

In one respect, the hope of technological advance through control of spin has al-

ready been realized. Binasch et al. (1989) and Baibich et al. (1988) discovered

giant magnetoresistance in layered magnetic materials. Figure 26.5(A) shows the

change of in-plane resistivity for three layers of Fe, Cr, and Fe as a function of

magnetic field applied perpendicular to the layers. The effect is around one per-

cent, which is much greater than the change in resistance of crystalline iron alone.

The magnetic coupling between the two iron layers is antiferromagnetic; absent an

external field, they point opposite to each other. Thus the large resistance is due

to forcing electrons through a layered structure where spins point opposite to one

another in the different layers. Theories for the origins of the change in resistance

are reviewed by Zutic et al. (2004).

Combining many iron and chromium layers together, as shown in Figure 26.5(B),

the fractional change in resistivity climbs to 80%. Devices optimized for applica-

tions,

such as spin valves, have an intermediate structural complexity, and might

consist of four layers: an antiferromagnetic top layer, a ferromagnetic layer, a

conducting layer, and a final thin ferromagnetic layer, as described by Wolf et al.

(2001).

816 Chapter 26. Quantum Mechanics of Interacting Magnetic Moments

Figure 26.5. First measurements of giant magnetoresistance in magnetic multilayers. (A)

Change in resistivity as a function of magnetic induction for a three-layer structure of Fe

and Cr. The reference resistivity py is the saturation resistivity B

—>

oo along the easy axis

(001).

The data from the magnetic multilayer are compared with a reference film of pure

iron. [Source: Binasch et al. (1989), p. 4829.] (B) Change in resistivity as a function of

magnetic induction for an 80-layer structure of Fe and Cr at 4.2K. [Source: Baibich et al.

(1988),

p. 2473.]

26.5.2 Spin Torque

Slonczewski (1996) and Berger (1996) observed that since spins carry angular mo-

mentum, currents containing en excess population

one type

spin should

able

flip magnetic domains. Freitas and Berger (1985) had already observed

domain-wall motion triggered by currents

several amperes, but the prediction

was that modest spin current densities in nanometer scale channels could flip do-

mains as well.

A spin current describes the rate

spin transport. Classically,

particles

carrying spin

are moving with velocity

then the current describing component

j of the spin moving in direction

Qij

vtSj. (26.83)

Quantum mechanically, the definition stays essentially the same. The spin current

density at point

carried by state \tp)

Qijij)

Re(lp\Ö(r

R)Sj—\lp) Be careful and work with (H/m)lm(ip* Vip).

(26.84)

Suppose

current

spin

particles

flowing along

with

wave function

of the form

*P(x)

= ^=(a\1)+b\l)).

(26.85)

The spin operators

are given by H/2 times the Pauli matrices of Eq. (26.35),

Eq. (26.84) becomes

Qxx

b + b*a) (26.86a)

2vm

Spintronics 817

Incoming spin current Ferromagnet

f//

Potential seen by spin down electrons

Figure 26.6. A spin current incident upon a ferromagnet creates a spin torque. This

schematic diagram illustrates model calculations.

Qxz =

2Xhn

i(b*a-a*b+)

\a\

-\b\

)

(26.86b)

(26.86c)

Specializing further, suppose that a spin current whose spin axis points at angle

9 away from z in the x

—

z plane impinges upon a ferromagnet where all spins point

along z (Figure 26.6). Use a very simple model of the ferromagnet, in the spirit

of the Stoner model, where the highest occupied spin up states have kinetic energy

2A more than the highest occupied spin down states. Thus assume that down spins

see an energy barrier of height 2A, while up spins pass into the ferromagnet with

kinetic energy unchanged. That is, when spin down electrons with h

/2m > 2A

enter the ferromagnet, their wave vector changes from k to

—

4mA/H

while for spin up &j = k. (26.87)

Computing transmission and reflection coefficients t and r as is customary in bar-

rier penetration problems

k + k

k-k

. a

k + k

For spin down, r = 0 and t = 1.

(26.88)

The main point is that up- and down-spin electrons have different transmission

and reflection coefficients upon entering the ferromagnet; fine details of the coeffi-

cients'

values are less significant.

The result is that outside the ferromagnet there is a wave function

Jkx

^—ikx

(cos 9/2\ Î) + sin 9/2\ [)) + -^

sin 6/2\ |) ffi?^£ÄtSd

Rotation matrices applied to

spin 1/2 particles pr

half the rotation one

expect for a vector.

(26.89)

818 Chapter 26. Quantum Mechanics of Interacting Magnetic Moments

100

(A)

0 0.4

Current (mA)

(B)

Time (ns)

Figure 26.7. Domain flipping and precession induced by spin torques. (A) Magnetiza-

tion flip in room-temperature layered metallic system consisting of 20nm Ni

Fei9/12nm

Cu/4.5nm Ni

Fei

. A constant external magnetic field is used to maintain zero net mag-

netic field on the thinner magnetic layer, which flips and changes the magnetoresistance.

(B) Magnetization precession in a 8nm Ir

oMn

o/4nm Ni

oFe

o/8nm Cu/4nm Ni

o lay-

ered structure at a temperature of

40K.

[Source: Ralph and Stiles (2008), p. 1192.]

and inside the ferromagnet

„ikx

V>"

?-= (cos 0/2| T) +t

e^-V sin e/2\ I)) . (26.90)

Returning to Eqs. (26.86), one can compute the spin currents inside and out. Out-

side,

they are

^ 2Vm

2Vm

Sin u In principle there are terms going as

2lkx

and

, but they oscillate so rapidly they can

be neglected.

cos 0 + r^ sin

sin

6/2)

(26.91a)

(26.91b)

(26.91c)

Inside the ferromagnet, the spin current is

-}v

k + k\

2Vm' 2

Sin 9 COs(k\ - k)x Return to Eq. (26.84); remember (26.92a)

to take real part.

k + ki

~2Xhn

^2~

sin 6 sin(£

—

k±)x

2Vm

Define the current

(k cos

9/2-k^l sin

6/2)

Ahk

h=-r.—e.

V m

(26.92b)

(26.92c)

(26.93)

The components of the spin current are discontinuous at the edge of the ferromag-

net, x = 0, and this discontinuity integrated over the area A of its surface is the

Kondo Effect 819

torque N:

Use

Eq.

(26.88). (26.94a)

(26.94b)

Use?2 + r[ = l. (26.94c)

A qualitative conclusion to draw from Eq. (26.92) is that once spin currents

enter a ferromagnet they will precess in space with a wavelength 2n/{ki —k). A

qualitative conclusion from Eq. (26.94) is that each excess spin-up electron enter-

ing the ferromagnet does bring with it a change in angular momentum on the order

of h/2. A spin polarized current / will thus deliver a torque on the order of IH/e.

Assuming this current arrives in a sample containing N atoms with spin angular

momentum

HN,

the characteristic time to make the sample flip or precess will be

11.

For N ~ 10

and currents on the order of mA, this works out to 10~

Figure 26.7 shows the results of

two

different experiments. In the first, a current

pulse flips a magnetic domain. In the second, the arrival of a polarized spin current

causes the spins in a device to precess.

26.6 Kondo Effect

Resistance Minima. According to Matthiessen's rule, Section 18.2.2, the resis-

tivity of a metal should consist of two additive pieces. The first, due to phonons,

decreases with temperature, dropping as T

at low temperatures. The second, due

to impurities, is temperature-independent. A puzzling exception to this rule was

observed by de Haas et al. (1934), while measuring the resistance of gold. Its re-

sistance dropped to a minimum at a temperature of

K, and then proceeded to rise

at lower temperatures. Some feature of the solid became more effective at scat-

tering electrons as its temperature decreased. It was clear to Wilson (1954) that

small traces of impurities were responsible, because the resistance minimum could

be moved about by varying their concentration, but there was no explanation be-

yond this, and he believed that "some new physical principle seems to be involved."

Some characteristic resistance measurements appear in Figure 26.8.

The explanation lies in the magnetic character of the impurities. At high tem-

peratures, the spin of an isolated magnetic impurity flips about freely, presenting a

small isotropic scattering potential to incoming electrons. At low temperatures, by

pointing in a definite direction, the magnetic impurity becomes more effective in

scattering electrons. Some experimental evidence along these lines was obtained

by Sarachik et al. (1964), and the first theory is due to Kondo (1964). His cal-

culation followed the influence of single magnetic moment upon a collection of

conduction electrons out to third order in perturbation theory. It appeared to ex-

plain the resistance minima observed at temperatures of a few kelvin, but could not

be the full story, because it predicted that at 0 kelvin the resistance should become

infinite, something neither observed experimentally nor believed to be true. An

= ~j sin 6(\-

)

Hz'-

= 0

~ 2 e

-k

fj sin

0/2.

820 Chapter 26. Quantum Mechanics of Interacting Magnetic Moments

Mo.

Nb.

, ß/ß

Mo.

, ß/ßB = 0

Mo.6Nb.4, ß/ßB

—

-3

Mo.

Nb.

, ß/ßB = -6

Mo.

Nb.

, ß/ßB = 1-1

Mo.

Nb.i, ß/ßB = 1-9

0 8 16 24 32 40 48 56

T(K)

Figure 26.8. Resistivity data for Mo^Nbi_

alloys, showing that depth and location of

a minimum of resistivity at low temperatures is correlated with the strength of magnetic

moments in the sample. [Source: Sarachik et

al.

(1964),

A1043.]

intense search for the solution to this problem over the next decade led to many

valuable new ideas, and eventually even exact analytical solutions of Wiegmann

(1980) and Andrei (1982), reviewed by Hewson (1997).

From the host of approaches to this problem it is only possible to choose a small

sample. First it will be demonstrated why a sea of conduction electrons interacting

with an impurity can behave as though interacting with a magnetic moment. The

half-bandwidth W will denote the range of energy states inhabited by the electrons;

their energies will range from —

to W. Next it will be shown that the band of

conduction electrons interacting with a magnetic moment of strength J is indistin-

guishable from an equivalent system in which the bandwidth 2W diminishes a bit,

while J slightly increases. This rescaling of the problem, due to Anderson et al.

(1970);

Anderson (1970) and Wilson (1975), provides a simple introduction to how

ideas from the renormalization group can be employed for systems of interacting

electrons.

Anderson

Model.

To begin, consider a model due to Anderson (1961) which

describes a collection of conduction electrons in contact with a single impurity site

(Figure 26.9). The Hamiltonian is

Ä = £0 [«0Î +

«Oil

+ £/«0î«0i + J2 h

\/ka + ^Aa^ka +

°^

■

)

The first two terms describe the impurity; placing one electron there costs an energy

eo,

but placing two there costs

2EQ

+ U, with U representing the repulsion that

two localized electrons are expected to feel for one another. The remaining terms

1.U4

1.02

1.00

#«£•*$!<

AAA

AAAA

0.98

0.96

<l<]<K3

. I . I . I . I . I . I . I . I  I  I . I

Kondo Effect 821

describe the conduction electrons and provide terms proportional to v^ that allow

them to interact with the impurity. In actual physical contexts, the impurity is likely

to be an atom with an unfilled outer d or f shell, and the shell will usually have

more than one orbital state available. Because the spin degrees of freedom included

in Eq. (26.95) are quite enough to produce many interesting effects, additional

orbital degeneracy can be neglected on a first pass through the theory.

Figure 26.9. A band of conduction electrons, bandwidth 2W, is placed in contact with an

impurity site, where the energy to add a first electron is eo, and the additional energy to add

a second is

€o

+ U. If the first energy lies below the Fermi surface while the second energy

lies above the Fermi surface, it is an excellent approximation to say that the impurity is

singly occupied.

To make the impurity potential in (26.95) start acting like a magnetic moment,

suppose that

• the Fermi level £f of the conduction electrons lies well above eo, so the im-

purity state is almost sure to be occupied,

• but eo + U is much greater than the Fermi level, so double occupation is very

unlikely, and

• the couplings v-

between the impurity and the conduction electrons can be

treated as small.

Rather than simply carrying out perturbation theory to some order in

v-^,

the.

goal is instead to have a general idea how the presence of the impurity affects

properties of the system at low temperature. At low temperatures, only states near

the ground state enter, so the way the impurity affects the whole collection of low-

lying excitations needs to be analyzed. What results is an effective Hamiltonian for

low-lying excitations that results from eliminating high-energy states.

The meaning of the last statement is best clarified by proceeding immediately

to show how it is done. First, break the Hamiltonian up so that it is possible to keep

track of whether the impurity site is unoccupied («o = 0), singly occupied (no = 1)

or doubly occupied («o = 2). Toward this end, define PQ,P\, and

to be projection

operators. Po acting on any quantum state where the impurity is occupied gives

zero,

but if the impurity is unoccupied,

returns the state unchanged. P\ and P2

similarly project out the no = 1 and no = 2 spaces. Formally, one can write

â) = (l-«0i)(l-«0î), (26.96)

822 Chapter 26. Quantum Mechanics of Interacting Magnetic Moments

with similar expressions for the other operators.

Next, letting

\ip)

be some quantum state of many electrons, let

|^o)=A#},

|Vi>=ÂM,andh/>2>=/^>. (26.97)

Also,

define

„

~ ,

For

example,

3<io gives the

, = PiKPv

action of the Hamiltonian on (26.98)

whatever

piece of a wave

function

has the impurity state

empty,

and it selects from the

result

whatever part is singly

occupied.

so 3i\ip) =

£.\ip)

can be rewritten as

The

reason that %%

3<20

that the various terms in the (26.99)

Hamiltonian

can leave the

occupation

the

impurity by 1,

leave it

alone,

but cannot

change

it by 2.

Having broken the Hamiltonian apart to describe its action on different states of the

impurity separately, it is now possible to transform (26.99) so that it is a problem

posed for

\ip\ )

alone, eliminating from the problem the subspaces where

0 or

no = 2. Simply write

|^o)+Äoi|^l)

£|^o) The top equation in

(26.99).

(26.100)

|Vo}

(fi-Äoo)^

Ä01IV1) (26.101)

Similarly from the bottom equation of

and|^2)

(£-Ä

)' Ä

|^>; %^&£^%%® Q6.102)

of (26.99) to obtain

{Äio (£-Äoo)

-1

(Ä

i-£)+Äi2 (£-Ä22)

i}^i)=0.

(26.103)

By eliminating |^o) and |^

), Eq. (26.103) recasts the original problem entirely

in terms of states where the impurity is singly occupied. This formulation is ap-

propriate when one expects on physical grounds that single occupation is to be

expected.

To proceed further, it is necessary to obtain explicit expressions for the entries

in the matrix on the left-hand side of

Eq.

(26.99). Consider

CKi

It needs to act on

a state where the impurity is unoccupied and subsequently produce one where it is

singly occupied. The only piece of

that works that way is

H^la-

(

104

)

However, (26.104) does not annihilate states where the impurity is singly occupied

as it should; one needs to add a projection operator on the right, to get

E¥O/J/O (26.105)