Marder M.P. Condensed Matter Physics

Подождите немного. Документ загружается.

Kondo Effect 823

E VL^(1-"0I)(1-"0T)

See Eq.

(26.96).

(26.106)

2_,

^0a^~k(T^ ~

"0,-cr)-

The term

(' -"ou) is superfluous, because(26.107)

if the state with spin a is occupied, cj will

annihilate

it anyway.

Because "K is Hermitian,

Äoi =

Äto

= Ç(l -«

,->f4>- (26.108)

to-

other terms can be worked out similarly, such as

t^];Äoo

A)E^L

and

^2i = ^Î2 = E V(U*>.-- (26-

110

)

From Impurity to Local Moment. With these formal developments as prelude,

it can now be demonstrated that Eq. (26.103) approximately describes conduction

electrons interacting with a magnetic moment at the impurity site. Consider the

first term,

Ä,o(£-Äoo)

Äoi|^i) (26.111)

= Ä

C(l-no,->|4>- (S-IÄn-eo + erj)"' |Vi). (26.112)

UseEq. (26.109) to get

rid of

CKoo,

and add

because in moving

(£

—

9^oo)

to the right

of ct ,

CK«) acts

on an

empty state

that

populated

the left.

As Eq. (26.108) shows, (26.111) is of order

|urj

and very small. To leading order

Eq. (26.103) can be replaced by (Jiw

—

£)|^i) = 0. Up to order

|f^|

—

An can

be replaced by 0 in (26.112). Thus Eq. (26.103) can be interpreted as describing a

situation where the impurity is singly occupied, but makes virtual transitions to the

unoccupied or doubly occupied states and immediately jumps back. Furthermore,

if one restricts attention to excited states k near the Fermi surface, then er can be

' k

replaced whenever it appears by £/?. Equation (26.112) now becomes

5(10

£^*(l-/fo,-

)c| co^i) (26.113)

—

£F - * '

kcr

■ ^ -^cL'^a'(

-"o,-

')(l-«o,-<r)êLêo

|V'i> (26.114)

eo t,f

kk'aa'

The

two terms that have been

f^,fy . i ^ omitted

always

just give 1, unless

^OCT'^7

Cpn-iC0o\lp\)- everything vanishes anyway.

(26.115)

— Co *

Reversing two

the operators

kk'aa'

reverses the sign and produces

spin-independent constant.

824 Chapter 26. Quantum Mechanics of Interacting Magnetic Moments

In the no =

subspace, the impurity has two possible states, | and

which can be

identified with the states of a spin 1/2 particle. Thinking in these terms,

cLcoj.

can

be identified with S

, the raising operator for the spin, while cj, cof = S~. For the

z component of the spin, S

—

(«of

—

"o|)/2, use the identity

not ci er,

+ n

, cî cv,,

This is what shows

(26.116)

*î *'î

k[ k'l

(26.115).

2 ("OT

«Oi ) (c|

c^,

+ cî^

cj,

)

(26.117)

— S

(cl Cr

—

cl,Cr,,) + - Y^ cl Cr, . In the «

subspace, n

T +

«0|

(26.118)

tî *'î k[ k'l> 2 ^ ka k'a

alwaysgivesl

v ;

Therefore, Eq. (26.115) can be rewritten as

^ £F"€0

kk!

§+ê

ï'î

+§

î>i

+§

î'iï

+ 2

£ 4/^

IV'i)-

(26.119)

The first three terms in the bracket of (26.119) represent electron spins interacting

with a spin-1/2 magnetic moment. The last term is a small correction to the ener-

gies of the conduction electrons that does not involve the spin. Treating the third

term of (26.103) in a fashion similar to the first term finally transforms (26.103)

into an effective Hamiltonian, only to be used on states of type ip\. The effective

Hamiltonian is

Äeff = Ä,i +Ç J

, [s+cl^+S-cl^^ +S

]

-c\

lli

kk'

+%E4A„

(

120a

)

with

, * r i . i

- e

U + e

8-F

(26.120b)

Completing the derivation of Eq. (26.120) and finding K^, is the subject of Prob-

lem 6. Under the conditions depicted in Figure 26.9, the conduction electrons are

coupled antiferromagnetically to the local moment, because

J-g,

is positive.

26.6.1 Scaling Theory

The previous section demonstrated that by eliminating states far above and below

the Fermi level from a Hamiltonian, an impurity potential can be revealed as a

magnetic moment. Putting matters this way leads to an additional suggestion: The

states being eliminated might not have to be impurity states. They could also be

conduction electron states. This suggestion is correct. By eliminating states at the

band edges in just the fashion that occupancies of the impurity were eliminated

Kondo Effect

825

before, the coupling between conduction electrons and a local magnetic moment

can be made to grow. A whole family of different Hamiltonians is thereby shown

to be equivalent, some with strong coupling to the spin and a small bandwidth,

others with a weak coupling to the spin and a large bandwidth. The final result is

quite simple, and it appears in Eq. (26.129).

The starting Hamiltonian is a simplified version of the Hamiltonian found in

Eq. (26.120):

J<-

V eve! er +V; Is'cl cp, +S

clc

llt

kfk'î

k'l

kk'

(26.121)

Proceed as in the previous section, except that now Pj is a projection operator

demanding that W

—

\SW\ < ej < W, P\ projects on states with —W+ |<5W| <

e^ < W

—

|<5W|, while

insists that —"W < e^ < —W+ |<5W|, as shown in Figure

26.10.

Figure 26.10. A small band of states, of width |<$W| is eliminated from the upper and

lower band edges. Accounting for the virtual transitions that would have been possible to

these states makes the coupling to the magnetic moment stronger.

Using kork' to denote states obeying —"W+ \SW\ < ej < "W— |<TW|, and using

q or q' to denote states obeying W

—

|<TW| < e^ < W, then

iin = J £ S-c\c

c\c^+S

1=7^5 4'î%+^

4l^'î

\f®-

ïM

q'fk'}

q'ï

k'[

(26.122a)

(26.122b)

k'q'

In the previous section, Ä02 was zero, something that is no longer true. However,

so long as W/J S> 1, processes in which an electron scatters from —W to W are

unimportant. Not all the following discussion is restricted to this limit, so setting

Ä02 = Ä20 = 0 should be regarded as an approximation that is usually, but not

always, well-justified.

826 Chapter 26. Quantum Mechanics of Interacting Magnetic Moments

As in Eq. (26.103), one needs to examine Äi

i —

£ plus two extra terms, one of

which

^(fi-^-'ÄzilVi) (26.123)

«ftÄCE-Ä^-riV-^])-

^!)

(

124

)

particle from the occupied states is destroyed, and one is created in the nar-

row

region near W. The highest-energy states that can be destroyed have energy

around

£f,

and these make the most important contribution to all physical

pro-

cesses;

hence W—

£f.

-£12^21 (-W)->|Vi>- (26.125)

Restrict

attention to low-energy

excitations,

which means

£f,

and measure

all

energies as

distance from

£f.

In acting upon

\tpi),

Ä22 can be taken

vanish,

because the states up near W are empty.

All that remains is the somewhat painful task

multiplying out

Ä12Ä21.

Details

are left to Problem 8, and the result is

Ä„Ä

=j>D<w[-m

E \ E

t^-fe^

n+È+

(26.126)

Eliminating the lower band

states near —W leads to an equal contribution, and

doubles (26.125). Therefore, from

Eqs.

(26.125) and (26.126), the effective Hamil-

tonian produced by Eq. (26.103) has the same form as Eq. (26.121), but with a new

value J + 8J of

+ fiJ

— 2—D(W)S'W,

<5W

negative, because the bandwidth is be- (26.127)

ing reduced.

and the energies of the conduction electrons are altered by addition of a term

^)W?i

;

(26.128)

kk'o

The renormalization of 7 in Eq. (26.127) can be used repeatedly to find how J

changes for large changes of W. One has

-2^(W). (26.129)

The density of states D(W) will of course vary with W, but as W becomes smaller

and smaller, driving

J to

larger and larger values, D(W) must approach Do,

the

density

states at the Fermi level. To see the basic structure of Eq. (26.129),

set

D(W) to this value, so that it can be integrated, giving

exp

constant

ICBTK,

(26.130)

Kondo Effect

827

where

is defined to be the Kondo

temperature.

Two Hamiltonians with different

values of

and J should have the same low-temperature behavior so long as W

and J are related by Eq. (26.130). In addition, as

changes, the Hamiltonian must

acquire an additional term given by Eq. (26.128), but this additional term does not

directly involve the magnetic moment.

Two limits of

Eq.

(26.130) are particularly useful. Imagine fixing

TK,

but vary-

ing W and J. In one limit, J is small and "W is exponentially large. This limit

describes a very large population of electrons interacting very weakly with a mag-

netic moment, and because the interaction is weak, perturbation theory should be

applicable. In the other limit, J is large and W is small. In this limit, a small num-

ber of conduction electrons interacts very strongly with a magnetic moment. It is

natural to assume that the strong interaction will produce a strong antiferromag-

netic correlation between the electrons and the moment at temperatures below the

Kondo temperature, and that this correlation is destroyed as the temperature rises.

Scaling of Resistivity. To obtain physical consequences from the scaling relation

(26.130), assume that physical properties of a system will depend on temperature

only in the form "J(T /TK), and that two systems with the same

are essentially

indistinguishable. Take the particular case of resistivity;

-V

(26.131)

If all else in the problem is fixed, and the coupling / to the magnetic moment

is made very small, the resistivity p should vanish as J

; this conclusion follows

for example from Eq. (18.16), which shows rather generally that scattering from a

potential vanishes as the square of the potential's strength. The way to obtain such

behavior for small J is to guess something like

3"

?(x) =

ln(jc)

>4Dlfi

\+2D

J\n{k

T/W)

(26.132)

(26.133)

J \n(k

T/VJ)) . Catch the leading trends for (26.134)

small J by Taylor

expanding.

The prediction is that the contribution to resistivity from magnetic impurities should

begin to rise in a logarithmic fashion as temperature approaches 7>. The total re-

sistivity is the sum of this magnetic contribution and the contribution from phonons

that according to Bloch's law Eq. (18.20) drops as T

. Taking the density of mag-

netic impurities to be n

, the low-temperature resistivity should behave as

■

- Bn

ln(k

T/W),

-A

and

are constants.

and a minimum in resistivity occurs when

0=>

min

£nmi\

1/5

(26.135)

(26.136)

828 Chapter 26. Quantum Mechanics of Interacting Magnetic Moments

The shape of the resistivity minimum in Figure 26.8 and the movement of the min-

imum with magnetic impurity density

n„à

are well accounted for by Eq. (26.135).

The form guessed in Eq. (26.132) for the resistivity has the unfortunate prop-

erty of diverging when T

—>

7^. The divergence may be eliminated by choosing

instead, for example,

*(£)-

which rises to some finite value as T

—>

0, but produces the proper J

behavior

for small J. Hewson (1997) describes several calculations that make the behavior

indicated by Eq. (26.137) more compelling than a plausible guess.

Specific Heat. The heavy fermion compounds mentioned in Section 6.5.1 are

roughly modeled as lattices of Kondo spins. Compounds such as UPt3 and UBe^

have low-temperature specific heats whose linear term is up to 1000 times larger

than would be anticipated from free-electron theory. If the low-temperature specific

heat is linear in temperature, the scaling theory suggests that

T k

ocn— =n~— exp

While this equation does not allow an immediate connection with experiment, it

does show how small reductions in the density of states Do or coupling constant J

can be expected to produce exponentially large increases in the linear term of the

specific heat.

Figure 26.11 shows the specific heat as a function of temperature for UBen,

one of the heavy fermion compounds. The large values of Cy/T at low temper-

atures are due to the peak at around 0.75 K, which has to decay rapidly in order

to drop to zero at 0 K. If the scaling theory associated with Eq. (26.130) is ap-

plicable, then at very low temperatures conduction electrons have spins pointing

opposite the uranium local moments, while above the Kondo temperature this cor-

relation disappears. Therefore, the entropy S = J dT C-ç/T associated with the

peak should be kß In 2 or 5.8 J mole

-1

. Performing the integral over the peak

in Figure 26.11 actually gives 1.2 J mole

-1

, of the same order of magnitude

as the simple prediction.

26.7 Hubbard Model

Hubbard

(1963,

1964a,b) proposed a seemingly simple extension of the tight-

binding model intended to explore the electronic correlations leading to magnetism.

To the ordinary tight-binding model Hubbard added a term that provides an energy

penalty U for any atomic site occupied by more than one electron. Hubbard argued

that the diagonal piece of the Coulomb interaction has magnitude of around 20 eV,

while the off-diagonal terms are all at least 10 times smaller; therefore he included

cosh

-1

(777k)

(26.137)

(26.138)

Hubbard Model

C/3



 D



- D

D

TJl^

ffD

111

ŒDDCD

irrm

y '2

0 .0

111

DÙ

& D

0.5 1.0

0 1 23456789 10 11 12

Temperature T (K)

Figure

26.11.

Low-temperature specific heat of the heavy fermion compound UBe^. The

very large slope in C\? at low temperatures is due to the peak at around 0.75 K; were it

not for this peak, C\r/T would have a value comparable to that of the free electron metals.

[Source Ott et al.

(1983,

1984).]

only the repulsion between electrons at the same site. The Hubbard Hamiltonian

in second quantized notation is conventionally

i'*+4*

i<?

U^2c

(26.139)

where the sum (//') is taken over distinct nearest-neighbor pairs.

26.7.1 Mean-Field Solution

An exact solution of the Hubbard model is available in one dimension, due to

Lieb and Wu (1968). The ground state has spin zero, with antiferromagnetic

correlations between neighboring spins. These results do not carry over to two or

three dimensions. The only simple thing to do in three dimensions is to employ

mean field theory, which is generally believed to be wrong in most aspects, but

which illustrates the competing effects. To carry out the mean field theory, write

830 Chapter 26. Quantum Mechanics of Interacting Magnetic Moments

the model as

^=Y.-

[

i^ + c\,

cia\ +UY, n/T«U- (26.140)

The difficulty resides in the last term, which is quartic in fermion operators. In the

standard steps of mean field theory, try

"la = n

+ (nia ~ n

)

■

(26.141)

Choosing the average value of n to be the same on all sites prejudices matters in

favor of ferromagnetism, which Hubbard had devised the model to explain. Ex-

panding to quadratic order, one finds that

Ä~5Z

+ c\,

+U^2n

]

-n

]

. (26.142)

Going into k space, the Hamiltonian becomes diagonal, and equals

^ — tel Cj

COS 5-k + U ^2 "jfcî"!

+"î"itj.

— n

l- 5 are nearest-neighbor vectors.

kSa k

(26.143)

The mean field theory is closed by setting the mean occupancy of up and down

electrons equal to nj and n^, respectively. Further progress depends upon choosing

a particular lattice and dimension in which to work. It is easiest to work in one

dimension, where all the algebra is simple. Because the up and down electrons can

have different densities, one must suppose that they are described by two separate

Fermi levels. If a is the lattice spacing and N is the total number of lattice sites,

then

Nn^=Na / — (26.144)

J-k

27T

=>■

7rn| =

akp-[-

(26.145)

Doing the integral in Eq. (26.143), the ground-state energy is

£o =

—

[—2t] [sin nm + sin irn\] +NUmn\. (26.146)

The total number of electrons in the metal is always fixed; assume that the band is

half filled, so that «| +

= 1. Then the ground-state energy becomes

-4tN

= simrn^+NUn^ (1-«

) . (26.147)

Minimizing this expression, one finds that for

^ > —, (26.148)

t 7T

Hubbard Model 831

100

to|-

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

A A A

10,

, 10

Georges et ai. (1996)

Mott Insulator

Coexistence,

Sordi et al. (2010)

Mott Insulator .

Metal

Coexistence

2 0

0.5

1 1.5

F=Ferromagnetic

FC=Short-range ferromagnetic correlations

AF=Antiferromagnetic

AFC=Short-range antiferromagnetic correlations

G=Correlated state of Gutzwiller type

Figure 26.12. Five representative phase diagrams of the two-dimensional Hubbard model,

as a function of on-site repulsion U over hopping parameter t, and filling fraction A, which

tells the fraction of sites that has more than one electron, or chemical potential fi, which

enters by adding (/i

—

U/2) ]T\ n, to Eq. (26.139). Early approximations differed dramat-

ically; more recently, the phase diagrams have started to converge, although the problem

is still not definitively solved. [Results of Kaxiras and Manousakis (1988) Coppersmith

and Yu (1989), Richter et al. (1978), Georges et al. (1996) (hopping energies t made ran-

dom variables here), and Sordi et al. (2010). There are also predictions of superconducting

regions, as in Giamarchi and Lhuillier (1991).]

«1 = 1 and nj, = 0 (or the roles may be reversed), so that the Coulomb repulsion

leads to ferromagnetic ordering. Otherwise, one has

n-j-

n±

= 1/2, and the system

is an ordinary metal.

This calculation illustrates the physics of the Hubbard model in an uncomfort-

able way. By using mean field theory, one obtains simple expressions that illustrate

the competition between kinetic energy and magnetic ordering that the model was

designed to explore. Unfortunately, in one dimension this solution is known to be

qualitatively incorrect. The exact analytical solution has no sharp transition in any

physical quantity as U/t varies. The correlations between neighboring spins are

antiferromagnetic, not ferromagnetic.

832 Chapter 26. Quantum Mechanics of Interacting Magnetic Moments

In two and three dimensions, there is only one point of general agreement. At

half filling, where the number of electrons precisely equals the number of lattice

sites,

and for large enough U, the ground state is antiferromagnetic. Some reasons

for this conclusion are explored in Problem 9. This conclusion has considerable

physical importance, because it provides the simplest way to see how solids like

CuO (Section 23.6.3) that should be metals according to band theory actually turn

out to be antiferromagnetic insulators, called Mott-Hubbard or Mort insulators.

Even this statement needs to be qualified, because charge transfer between oxygen

and copper as discussed in Section 23.6.3 is not included in the Hubbard model,

which is hard enough to solve without it. Away from half filling, the problem is

not completely solved. Figure 26.12 displays five characteristic phase diagrams

describing the ground state of the Hubbard model as a function of U/t and of A

or chemical potential /i; A gives the fraction of electrons per site in excess of half

filling, so there is one electron per site when A = 0, and two electrons per site when

A = 1. The chemical potential ß enters the model by adding (//

—

U/2) J2i

ito

Eq. (26.139).

The fact that the properties of the two-dimensional Hubbard model are not

yet known with certainty is unsatisfactory because the Hubbard model is widely

viewed as the simplest context in which to try to obtain exact results for many inter-

acting electrons; this model has been investigated with a ferocious intensity since

the discovery of high-temperature superconductors. There is no better illustration

of the difficulties involved in progressing systematically beyond the one-electron

pictures of solids.

Problems

Wave functions with cusps: Show that the energy of

\cj)\

in Figure 26.2

can certainly be lowered by smoothing out the cusp. One way to perform

the demonstration is to flatten out the wave function as shown in the figure

throughout some small volume v and then estimate the effect on kinetic and

potential energies.

Ferromagnetic ground state: Consider the Hamiltonian (26.21) with all J

positive, and take S$ to describe spin 1/2 particles.

(a) Show with the aid of the identity in Eq. (26.19) that the state where all spins

are eigenvalues of S

with eigenvalue 1/2 is an eigenfunction of the Hamilto-

nian.

(b) Show that the largest value that (^\S^

•

S$,

|\I/)

can assume for R ^ R' in any

wave function \I> is less than or equal to the largest eigenvalue of S^

■

S^,. By

expanding out the square of S^ + S^,, show that the largest eigenvalue of this

operator is 1/4.

Heisenberg Hamiltonian.