Marder M.P. Condensed Matter Physics

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Problems

793

Suppose that

Show that

— iPm&

where

2ip

cos(27rmè-K) + ^

i+^m-i = £ipm,

_ ea

B _ a

2TTHC

(25.149)

(25.150)

(25.151)

Quantum electrons in a magnetic field: Work through the energy levels of

an electron in a magnetic field. Beginning with the Hamiltonian

1 - e~

adopt

the

Landau gauge

K=—{P+-A)\

2m c

A = (0,Bx,0).

(25.152)

(25.153)

Show for wave functions proportional to exp[ryfcj,] that Eq. (25.152) becomes

the Hamiltonian for a harmonic oscillator centered about

xo =

rnujr

(25.154)

where

is given by Eq. (25.46). Show that the energy levels are given by

Eq. (25.48).

Careful derivation of Landau diamagnetism: Consider noninteracting elec-

trons confined by a potential

£o/2R

)

where p is the radius in cylindrical

coordinates, with a magnetic field of strength H pointing in the z direction.

Schrödinger's equation is

1 d dip d

ip 1 d

pdp

dp dz

d<p

1 dip mcv

£o i 2 / c,

-j/kj

—

+ [— + -^\p

= Zip.

(a) Define

and

R(p)e>

mui

c 2

d(p

,il<fi+ik

Show that after substituting £ for p one obtains

£R"

k\ £o£ i I

+ T + TT +

2mhio

mRr.uj

4 4£ 2 HLÜ,

(25.155)

(25.156)

(25.157)

R = 0. (25.158)

794

Chapter

25.

Magnetism of Ions and Electrons

(b) Show that R behaves as exp[—v£/2] for large £ and as ^

for small £. Find

the constant v.

exp[-^/2]£l'l/

M0=*(0

(25.159)

and

£=-.

(25.160)

Show that

Cto"-«;'C

+ (|/| + lK +

/k<j

.f 2^

w = 0

(25.161)

(d) The confluent hypergeometric function /

(a, 7; C) solves

QF"

+ (7 - C)/

' - aF = 0

(25.162)

and is well-behaved at infinity only if a is a negative integer. Find the energy

levels of the electron.

(e) Repeat the arguments leading to Eq. (25.66) and show that they are un-

changed in the limit v

— 1

<C 1.

References

Y. Aharonov and D. Böhm (1959), Significance of electromagnetic potentials in quantum theory,

Physical Review, 115,

485-491.

T. Ando, A. B. Fowler, and F. Stern (1982), Electronic properties of two-dimensional systems, Re-

views of Modern Physics, 54, 437-672.

G. Boebinger (1996), Correlated electrons in a million gauss, Physics Today, 49(6), 36-42.

Bohr (1911), Ph.D. thesis, Copenhagen.

Bohr (1922), The Theory of Spectra and Atomic Constitution, Cambridge University Press, Cam-

bridge.

E. A. Boudreaux and L. N. Mulay (1976), Theory and Applications of Molecular Paramagnetism,

John Wiley and Sons, New York.

M. E. Cage (1987), Experimental aspects and metrological applications, in The Quantum Hall Effect,

R. E. Prange and S. M. Girvin, eds., pp. 37-68, Springer-Verlag, New York.

A. M. Chang (1990), Experimental aspects, in The Quantum Hall Effect, R. E. Prange and S. M.

Girvin, eds., pp. 175-232, Springer-Verlag, New York.

R. Hofstadter (1976), Energy levels and wave functions of Bloch electrons in rational and irra-

tional magnetic fields, Physical Review B, 14, 2239-2249.

L. D. Landau (1930), Diamagnetism of metals, Zeitschrift für

Physik,

64, 629-637. In German.

Translated in Landau (1965), pp. 31-38

L. D. Landau (1939), On the de Haas-van Alphen effect, Proceedings of the Royal Society of London,

A170,

363-364. Appendix to paper by Shoenberg. Reprinted in Landau (1965), pp. 268-270.

L. D. Landau (1965), Collected Papers of L D Landau, D. ter Haar, ed., Gordon and Breach, New

York.

References 795

L. D. Landau and E. M. Lifshitz (1977), Quantum Mechanics (Non-relativistic Theory), 3rd ed.,

Pergamon Press, Oxford.

L. D. Landau and E. M. Lifshitz (1980), Statistical

Physics,

Part 1, 3rd ed., Pergamon Press, Oxford.

H. Landolt and R. Bömstein (1959), Numerical Data and Functional Relationships in Science and

Technology, II, Springer-Verlag, Berlin.

R. B. Laughlin

( 1981

Quantized Hall conductivity in two dimensions, Physical Review

23,

5632-

5633.

R. B. Laughlin (1983), Anomalous quantum Hall effect: An incompressible quantum fluid with

fractionally charged excitations, Physical Review Letters, 50, 1395-1398.

A. V. Nurmikko and A. Pinczuk (1993), Optical probes in the quantum Hall regime, Physics Today,

46(6),

24-32.

R. E. Prange and S. M. Girvin, eds. (1987), The Quantum Hall Effect, Springer-Verlag, New York.

L. Saminadayar, D. C. Glattli, Y. Jin, and B. Etienne (1997), Observation of the e/3 fractionally

charged Laughlin quasiparticle, Physical Review Letters, 79, 2526-2529.

S. L. Sondhi, S. M. Girvin, J. P. Carini, and D. Shahar (1997), Continuous quantum phase transitions,

Reviews of Modern Physics, 69, 315-333.

Streda (1982), Theory of quantised Hall conductivity in two dimensions, Journal of Physics C, 15,

L717-L721.

A. Tonomura (1993), Electron Holography, Springer-Verlag, Berlin.

A. Tonomura, N. Osakabe, T. Matsuda, T. Kawasaki, J. Endo, S. Yano, and H. Yamada (1986),

Evidence for the Aharonov-Bohm effect with magnetic field completely shielded from electron

wave, Physical Review Letters, 56, 792-795.

C. Tsui, H. L. Stornier, and A. C. Gossard (1982), Two-dimensional magnetotransport in the

extreme quantum limit, Physical Review Letters, 45, 1559-1562.

H.-J. van Leeuwen

(

1921

Problems in the electronic theory of magnetism, Journal de Physique, pp.

361-377. In French.

J. H. van Vleck (1932), 77ie Theory of Electric and Magnetic Susceptibilities, Oxford University

Press,

London.

K. von Klitzing (1986), The quantized Hall effect, Reviews of Modern Physics, 58,

519-531.

K. von Klitzing, G. Dorda, and M. Pepper (1980), New method for high-accuracy determination of

the fine-structure constant based on quantum Hall resistance, Physical Review Letters, 45, 1545—

1547.

R. Yennie (1987), Integral quantum Hall effect for nonspecialists, Reviews of Modern Physics,

59,781-824.

26.

Quantum Mechanics of Interacting

Magnetic Moments

26.1 Introduction

Permanent magnetic moments could not exist in a world of charges ruled by classi-

cal electromagnetism. Quantum mechanics solves this difficulty through the quan-

tization of orbits. However, there is a second difficulty that quantum mechanics

must also be called upon to solve, and that is the strength with which adjacent

magnetic moments interact.

Nearby magnetic dipoles do interact with each others' magnetic fields. Yet

magnetic fields are so naturally weak that even at distances characteristic of the

atomic scale, magnetic dipoles are too small to explain magnetism. The induction

produced by a magnetic dipole mi is

B = V

_ li 3r(mi-r)-mi

mi-V- = ^ , (26.1)

and so according to Eq. (24.37) the energy of two dipoles separated by fyi and

interacting with each others' fields is

fh\

-/W2 —3(/W2-fi2)(mi 'hi)

(26.2)

which has a maximum magnitude of 2m\m2/r\

when fh\

mi ||

~f\2-

The energy

scale for the dipole interactions is

A tn\ mo /2fln\

0.9- 10 eV • — — —- . a

is the Bohr radius.

Xmun^ /2ao\ (4

0 9.10"

eV .^L^ (—Y

i2 ) a\ ' ßB ßB

m /

(26.3)

Because this energy scale equals kßT for a temperature of

K, magnetic dipole in-

teractions would have difficulty explaining a solid whose ferromagnetic transition

temperature was 10 K, let alone iron for which it is over 1000 K. Apart from their

role in large-scale phenomena such as domains, the magnetic fields produced by

electrons and ions are usually negligible.

26.2 Origin of Ferromagnetism

26.2.1 Heitler-London Calculation

The physical effect that produces magnetic ordering of adjacent magnetic moments

is the same that leads to magnetic ordering within an ion and produces Hund's

797

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

798 Chapter 26. Quantum Mechanics of Interacting Magnetic Moments

rules.

Magnetic fields are irrelevant; what matters is the effort by electrons to

choose a spin state that will minimize the Coulomb repulsion between them once

Fermi statistics are taken into account. Interactions between spins are actually just

a convenient way to record the end result of electrostatic repulsion.

The emergence of interactions between spins from Coulomb forces is most

clearly illustrated by a calculation of Heitler and London (1927) (Figure 26.1).

Consider two adjacent atoms separated by distance Rn = R\

—

Ri, and let (f)\{r)

and </>2(?) be spatial wave functions of the outermost electrons on atoms 1 and

respectively. These wave functions should be determined when the atoms are

infinitely far apart, not interacting, and the Coulomb forces between the two atoms

will be treated as a perturbation.

Figure 26.1. Setting for the calculation of Heitler and London (1927). Two electrons,

possibly of different spin, surround nuclei at locations

and R\. Correlations between the

spins are induced by the overlap of

the

wave functions.

Because the wave functions

4>\

and

4>2

are determined when the atoms are far

separated, they will not be orthogonal when the atoms come close together; the

overlap integral

l=\\

I dr^(r)^

(r)\\

(26.4)

is not zero.

So long as the excited states of the two atoms remain irrelevant, the two-

electron wave function when the two electrons approach one another will consist

of linear combinations of antisymmetric products of

<fi\

and fa. It is not enough,

however, to consider only the spatial dependence of the electron wave functions.

The electrons employ spin degrees of freedom as part of the strategy of lowering

their energy, and spin must be included explicitly.

Spin Singlets and Triplets. The Hamiltonian does not depend upon spin explic-

itly, so all spin operators commute with the Hamiltonian, and the eigenfunctions

can be chosen as eigenfunctions of the commuting operators S

and S

. The spin

eigenfunctions are first

—ß (Xî(<ri)Xi(<72)-X±(<7i)Xî(<72)) , (26.5a)

which has eigenvalues S = S

= 0 and is a spin singlet. Second, there is the triplet

Origin of Ferromagnetism 799

Xî(o"i)Xî(cr

) 5=1;5

^=(XT(^)Xi(^)

Xi(^ikî(^)) S=l;S

= 0 (26.5b)

Xi(^i)Xi(^2) S=l;S

-l.

Spatial Wave Functions. The product of spatial and spin wave functions must

be antisymmetric under exchange of particle numbers. Because the singlet wave

function is odd,

must multiply

spatial wave function that is even, while the

triplet wave functions must multiply something odd. From the two functions </>i

and

fa,

only one even and one odd function can be constructed, and when one

takes into account the overlap integral in Eq. (26.4) the normalized combinations

are

<Mn,

2) = . ^,J(f)\{r\)(p2(r2) + h(r2)Mn)}, ^n, multiply

spin (26.6a)

V2 + 2/

sin

let

4>t{r\,r

) = , J0l(n)(fe('

2)-<?i>l(?2)<fe(n)].

Odd

multiply

spin (26.6b)

y/2 —

triplet and get three

more wave functions.

The Hamiltonian is diagonal in the space spanned by the four wave functions pro-

duced by the allowed products of

Eqs.

(26.5) and (26.6), because the Hamiltonian

is independent

spin, but all the spin functions are orthogonal to one another.

These four wave functions must be eigenfunctions of that subspace, although not

necessarily of the full Hamiltonian. What remains is to calculate the expectation

value of the Hamiltonian in each of them, and to find an approximation for the

wave function of the ground state.

Evaluation of Energies. Let £o be the energies of each atom when the two atoms

are infinitely separated, so that

\r\-R

When the two atoms come together, the full Hamiltonian is

Pi p

,(r

)

0i(ri). (26.7)

Then

2m |n-/?i|

-R

2 2 2

e e e

+TZ

^ + ^ —

—,--

^-7-

(26.8)

\r\-n\

\Ri-R

\ \ri-R

\ k

-/?i

dhdr

tâ(7i)ÏÏ(72)X<h(?i)Mh) (26.9)

2EQ

+ U, (26.10)

where

800 Chapter 26. Quantum Mechanics of Interacting Magnetic Moments

and

df\ df

I0.(n)l

x I02(r



■r

\Ri-R

\7i-R

-R\\

(26.11)

(26.12)

„ 2

Notice that the Coulomb interactions appearing in Eq. (26.14) are

2co/ + v , not the same as those appearing in Eq. (26.12); because ?i and

~?2

(26.1 j)

have been exchanged, the Coulomb terms are grouped differently.

dndr

<#(WÎ(r2Wi(n)02(r2)

dh (t>UnW

(hW

(n)Mh)

with

df\ df

<P\{nW

)

{r\)4>\(r

)

\r\-r

\ \R\-R

\ \r\-Ri\

Finally,

and

F MITTUN o2£o + t/ + 2/

£o +

tf +

£, = (0.

|5t|^) =2 ^ , „,

=2£

2 + 2/

;

= (^|?C|ç!)

) = 2

2/2

=2£

and the difference between triplet and singlet energies is

U -

1 + /

u-v

£r

l-/

-y.

-^2|

(26.14)

(26.15a)

(26.15b)

(26.16)

The sign of

the

triplet-singlet energy difference can vary according to the mag-

nitudes of the three integrals /, U, and V. In the particular case of the helium atom,

Heitler and London (1927) found that Eq. (26.16) is positive, so the singlet is of

lower energy. Spins on the two atoms point in opposite directions, providing a

two-atom example of antiferromagnetism.

Lieb-Mattis Theorem. While Eq. (26.16) is very suggestive of how varying

overlap integrals can lead sometimes to ferromagnetism and other times to antifer-

romagnetism, it results from approximations, and in at least one case where it can

be checked thoroughly it is wrong. Lieb and Mattis (1962) proved that the true

ground state of any two-electron system is always a singlet. Equation (26.16) can

predict that the triplet ground state is lower, but such a prediction must be wrong.

The argument is one earlier used by Feynman (1953) to discuss properties of

the ground state of helium, and it is based upon observing that the ground state

must be free of nodes. In

brief,

the argument proceeds as follows: Consider any

Origin of Ferromagnetism 801

Hamiltonian for two electrons, and suppose one has found the ground-state wave

function </>(n, 7

)- The function </>*(n, r

) must have the same energy, as does

4>*,

so one may as well take

to be real. Using the variational principle of

Appendix B.2,

<f>

minimizes the functional

|_|V

+ |-|V

+ I/(?i, r

)|0(n, r

J dr\dr

|</>(n, r

Suppose that </>(?i, r

) is an odd function of its arguments. In some places it is

positive, in other places it is negative, and somewhere it must pass through zero, so

it has at least one node. Next consider the absolute value of

</>,

\<j>(ri,

)\. Placed

into Eq. (26.17), |</>| gives the same value as

</>;

tiny regions around the nodes of

</>

where derivatives of

\cf>\

might be ambiguous can be excluded from the integral

without changing it appreciably. However, deforming

</>

near the cusp in Figure

26.2 is guaranteed to lower the energy functional (26.17), as shown in Problem 1.

Therefore

cannot be a ground state, and consequently it cannot have nodes. The

singlet is guaranteed to lie lower than the triplet for all potentials U.

Figure

26.2.

The energy of a wave function with a cusp is always lowered by smoothing

out the cusp. If

has a node it has the same energy as

\(j>\,

but the energy of

\4>\

can be

lowered by deforming to the dotted line, so

cannot be a ground state.

Generalizing from this argument, it would be hard to see how ferromagnetism

could occur at all. The conclusions that can be obtained for two electrons are actu-

ally quite particular to problems involving only two electrons. Consider the ground

state of the ion Pr

, which consists essentially of two 4/ electrons sitting outside

a xenon core. If the Lieb-Mattis theorem could be applied, Pr

would be in a

state with S = 0 and nonmagnetic. In fact, Hund's first rule predicts, and experi-

ment confirms, that the ground state is the triplet 5=1. The theorem fails because

the two 4/ electrons must be orthogonal to the inner shells of electrons, and the 4/

electrons are consequently required to have nodes. The theorem based upon avoid-

ance of nodes is irrelevant, and in returning to the Heitler-London approximation,

one finds that it does not do such a bad job after all. Equation (26.16) can be ap-

plied, roughly, to Pr

by allowing R

and R\ to coalesce and letting

cj>\

and

two orthogonal wave functions in the 4/ shell. The overlap integral / vanishes, and

802 Chapter 26. Quantum Mechanics of Interacting Magnetic Moments

Eq. (26.16) predicts that the triplet will lie lower than the singlet by 2U, predicting

ferromagnetic correlations, and correctly recovering Hund's first rule.

26.2.2 Spin Hamiltonian

Because the symmetric and antisymmetric spatial states of two electrons are un-

shakeably correlated with the spin wave functions, Dirac (1926) and Heisenberg

(1926) showed that the original Hamiltonian (26.8), which acts only on spatial

degrees of freedom, can be replaced by a new Hamiltonian acting only upon the

spin degrees of freedom. This new Hamiltonian is constructed to give results iden-

tical to the old one within the subspace of states spanned by the wave functions in

Eqs.

(26.5) and (26.6). It would not, of course, remain correct if excited atomic

states were to become important.

The four electronic states created by the products of Eqs. (26.5) and (26.6)

have two energies. The three triplet states sharing the same energy are related to

each other by symmetry. Their energies are invariant when the spins of the two

electrons, S] and 52, rotate simultaneously by the same amount. This observation

suggests the Hamiltonian

"K = a + bS]

■

amdb are

constants. (26.1 8)

rê+ê- i c+«

See Schiff (1968), pp.

200-201,

= a + b [S\SÏ+-\StSï+StS7\\ . or Landau and Lifshitz (1977), (

6.19)

\ '

2 / P' raising and lowering

operators are defined by

S± =S

±iS>\

When this Hamiltonian acts on the triplet state it gives a + b/A, while when it acts

on the singlet state it gives a

—

3b/A. The difference J between singlet and triplet

energies defined in Eq. (26.16) equals

—b,

and evaluating a gives

Ä = 2£

+ ^^ + (^-5,

-S

)J.

(26.20)

A positive value of J causes the two spins to point in the same direction in the

ground state.

26.3 Heisenberg Model

The Heisenberg model supposes that when a large collection of magnetic ions is

placed in a lattice, Eq. (26.20) remains a guide to the form of the energy, and

Drop additive contributions to the energy, and al-

ar _ \ ^ i , C. . C , low the possibility that J depends in

general fash- (26 21^

' ' " ' ' ion upon the particular pair of spins that is inter- ^ ' '

(//') acting. Carry out the sum over distinct pairs of

spins at / and /'.

Equation (26.21) cannot easily be derived in a controlled fashion, and it is not

necessarily the form that realistic accounts of interactions between electrons will

take.

It has, however, formed the basis for almost all studies of the organization