Physics 617 Nov. 26, 2012: Magnetism Notes
Magnetism and Superconductivity reading:
(I) Magnetism: We will cover diamagnetism and paramagnetism in weak magnetic systems
(chapter 31). Also, time permitting I will cover ferromagnetic order in the mean-field
approximation (chapter 33; chapter 32 provides some background on the interactions leading to
such order). A summary of topics is as follows:
Chapter 31: Basic Hamiltonian and thermodynamics, Curie-law paramagnetism, Larmor
diamagnetism. Time permitting: Pauli paramagnetism and diamagnetism of metals.
Chapter 33: Interacting dipoles, ferromagnetism in mean-field Ising model, second-order
transitions. (Probably won't cover domains or hysteresis.)
(II) Superconductivity: I will discuss the basic phenomena and summarize properties of some
superconducting materials, then discuss the Ginzburg-Landau formalism with a few applications,
followed by a brief introduction to the BCS theory. Read chapter 34.
A. General concepts and physical background:
Magnetization (M): Physically, the magnetization (M) is the overall density of magnetic dipole
moment per volume, while the susceptibility (χ) is the derivative of M with respect to the
magnetic field. Equivalently, M is defined thermodynamically as a derivative of the Helmholtz
free energy, F = E – TS, so that the following relationships determine these quantities:
µ
1 !F
"M
or M = !
;
.
(1)
!=
M=
V !H T
"H
V
Units: We are using CGS units, where B = H + 4πM. While the general trend in physics is
towards SI, or MKS units, that is not so in magnetism. For example commercial magnetometers
such as the SQUIDs on our campus provide measurements in emu, a CGS unit. In CGS, B is
measured in gauss (G), and H in oersteds (Oe), however physically these two units are identical.
The CGS moment (µ) has fundamental units gauss-cm3, which is also the same as erg/G, since 1
G2cm3 = 1 erg [the energy density is B2/8π]. So, for example the Bohr magneton, µB = 9.27 ×
10-21 erg/G, is a measure of the magnetic moment of the electron. From (1) it can thus be
determined that χ is a dimensionless number. The susceptibility thus defined is the volume
susceptibility, distinguished from the molar susceptibility (often given as emu/mol, real units
cm3) and the mass susceptibility (emu/g has units cm3/g). In SI units M is also the moment per
volume (SI moment has units J/T), and the volume susceptibility is also dimensionless, but there
is a conversion factor of 4π from CGS to SI, which is the cause of some confusion. The
dimensionless susceptibility should never be quoted as just as a number, the units need to be
specified, e.g. “CGS dimensionless”.
Magnetic order can include ferromagnetism, which means all spins are aligned in one direction
(Example: Fe metal, the archetype ferromagnet, which is ferromagnetic below TC = 1043 K.)
Antiferromagnetism means spins are aligned in (approximately) alternating directions, with no
net M. Examples: CuO, with a transition temperature TN = 230 K, or FeCl3, with TN = 15 K.
Ferrimagnetism may be observed when there is more than one spin type, pointing in different
directions. Example: Nd2Fe14B is one of the best permanent magnets, but while this material is
often thought of as a ferromagnet, the Nd moments are anti-aligned, which reduces M somewhat
but enforces the strong anisotropy which helps such rare-earth magnets maintain their
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magnetization. There are a number of other types of magnetic order, for example some rare-earth
materials adopt complex spiral-type magnetic ordering, while disordered alloys can “freeze” into
a spin-glass state.
Local-moment vs. band magnetism: The local-moment case features moments occupying
individual ions, often the case in insulator systems such as the CuO and FeCl3 antiferromagnets
mentioned above. Band magnetism is the opposite case, magnetism composed of itinerant
metallic electrons. Fe metal is often pictured as containing localized moments in d orbitals, but it
is closer to the situation of a band magnet.
Hard vs. soft magnets: refers to ferromagnets; soft magnets have little resistance to reversing
spin direction (or, small magnetic anisotropy), while hard magnets resist changes in the
orientation of M. For permanent magnet applications one needs hard magnets (e.g. rare-earth NdFe-B magnets), while transformer cores should be soft, and disk-drive media should be
somewhere in-between since writing involves reversing the magnetization of tracks.
Hamiltonian: We discussed the Hamiltonian including the vector potential (briefly) earlier in the
semester. The general form is,
! !
1
e
(2)
H =
( p + A)2 + Uo + µ B H ! S ,
2m
c
where I absorbed the scalar potential e! into the crystal potential energy U o , and the last term is
the spin interaction, which must be added separately. Internal magnetic interactions such as
exchange between spins are not included here and must also be added separately (see part D
below). Also the electron charge is q = –e, where e is considered positive. Squaring the first term
on the right yields three terms: p 2 / 2m which is the H = 0 kinetic energy, the cross term which is
! !
shown below to be equivalent to µB H ! L , and e2 A2 / 2mc2 . The A2 term is small but it is the only
!
!
contribution for diamagnetic materials. Replacing p = !i"" , the cross term is
!
! !
!
! !
!ie! " " " "
(" # A + A # ") . With H a uniform field, we can choose A = ! 12 r " H , for which ! " A = 0 ,
2mc
so that:
!ie! " " " "
!ie! " " ie! " " "
(" # A + A # ") =
2A #" =
(r $ H )# "
2mc
2mc
2mc
ie! " " "
=
(" $ r )# H
.
(3)
2mc
e! " "
=
L#H
2mc
The second line is a cyclic equivalence for the triple product, while in the third line I used
" " "
e!
= µB
!L = r ! p which is the usual expression for angular momentum operator. The factor
2mc
is defined as the Bohr magneton. Thus the Hamiltonian (2) can be expressed as follows
(neglecting interactions for now):
!
! !
e2
H = H o + µ B H !( L + go S) +
A2 .
(4)
2
2mc
2
! !
!
The middle term, µB H ! ( L + g o S ) , leads to paramagnetism, while the last term,
to diamagnetism.
e2
A2 , leads
2
2mc
Current density: In QM the current density is derived by expanding the time derivative:
! 2
!" *
* !"
(5)
" ="
+
".
!t
!t
!t
2
Since ! " # is the quantum probability density, (5) can be placed in a continuity equation,
! !
!" = #$
% j , where j is a probability current. For the derivatives on the right of (5) we
!t
substitute by using the Schrödinger equation,
2
! !
!
i
i! #
ie &
i
" = ! H! = +
"
+
A
!
!
(U
+
µ
H
) S)!
(6)
%
(
o
B
!t
!
2m $
!c '
!
and its complex conjugate. With a little algebra we obtain,
!
!
!
i!
e ! *
#! *"! ! ("! * )! % +
(7)
j =!
& mc A! ! .
2m $
The relation (7) appears in many QM texts, although sometimes one only sees the first part,
which is the usual kinematic part of the current density.
The last term in (7) can be shown to be responsible for
! diamagnetism (shown in class in a rather
different way). This is because the vector potential A circulates around field lines, and therefore
!
so does the current density. The electric current density circulates the opposite direction as j
defined above, since it is due to negative electrons, and thus it provides a shielding current that
opposes applied magnetic fields. Even an s-orbital, with zero angular momentum, has a net
circulating current density when a field is applied.
The relation (7) also re-appears in the Ginzburg-Landau theory of superconductivity.
Superconductors are perfect diamagnets, and in some cases will completely exclude applied
magnetic fields.
Quenching of orbital moment: For typical materials having magnetic order, the orbital moment
contribution is quenched, since hybridization removes the degeneracy of different orbitals.
Hence for the rest of this handout, S is used for the magnetic moment, with no reference to the L
term in (4). However note that for rare-earth materials containing unpaired f orbits, this is not the
case; the full unquenched moment must be used. In the latter case, substitute J for S, and use the
Landé g-factor, as defined on page 654 of the text.
B. Linear susceptibilities:
(a) Diamagnetism, example silicon with χmol = –3.9 × 10-6 cm3/mol.
Comes from A2 term in (4). Typical behavior: small, T-independent negative χ.
There is also Larmor diamagnetism, the diamagnetism of unbound (metallic) electrons:
it is generally smaller than the Pauli paramagnetism of metallic electrons; hard to
calculate except for the simplest metals, and often the free-electron result is used,
! Larmor = " 13 ! Pauli , though that is not necessarily valid for a typical metal.
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(b) Paramagnetism, appears in systems with degenerate (or nearly degenerate) ground states.
(i) Pauli paramagnetism for metals, example aluminum χmol = +16.5 × 10-6 cm3/mol.
Typical behavior: small, T-independent positive χ. Comes from S-containing term in
eqn. (4): B field produces a small polarization of spins in metallic energy bands.
(ii) Curie paramagnetism, example FeCl3, χmol = +13500 × 10-6 cm3/mol at RT.
Large, T-dependent χ, with Curie law, χ = C/T. Comes from paramagnetism of local
C
moments. (FeCl3 really has Néel-type ! = n
since it is an antiferromagnet as
T ! TN
described above, however it is nearly Curie-law at RT since TN is small.)
(iii) van Vleck paramagnetism, example KMnO4.
T-independent positive χ, typically larger than Pauli term. This is important if there is a
small gap between filled and empty states, and non-degenerate ground state mixes with
the low-lying excited states. Sometimes called “TIP” for temperature independent
paramagnetism, though in a metal there may be a T-independent combination of this
and the Pauli term.
C. Curie Law and local moments:
For the Curie Law we consider the linear response of a density n of non-interacting magnetic
spins. Starting with a Boltzmann distribution for local magnetic spins S, we have for the
magnetization:
gµ m e ! gµBmS H /kT
µ
N
M=
= ! " !S B S
,
S
V
V
" ! S e! gµBmS H /kT
S
(8)
where ms are the integer eigenvalues of S for the different spin eigenstates. The negative sign
comes in because the spin and magnetic moment directions are opposed for electrons. Solving in
the high-T approximation, we find,
(gµ B )2 HS(S +1)
M !n
,
(9)
3kT
which gives a positive (paramagnetic) susceptibility,
( p µ )2
C
! ! = n eff B ,
(10)
T
3kT
where we define peff ! g S(S + 1) , with peff µB called the “effective moment”.
General case: Beyond the high-T limit, the sums in (8) can be solved in the general case:
M = ngµ B SBS (x) ,
(11)
with x = gµB SH (kT ) [a ratio of magnetic spin energy to thermal energy]. The functions Bs ( x)
are the Brillouin Functions:
2S +1
! 2S +1 $ 1
! x $
BS =
ctnh #
x& '
ctnh # & .
(12)
" 2S % 2S
" 2S %
2S
Note: for the special case S = 1/2, a little manipulation of the hyperbolic functions yields a much
simpler form: B1/ 2 = tanh( x ) , not given in the text.
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A plot of the Brillouin functions is given below, with the S = 1/2 curve at the top:
As mentioned above, for the case S = 1/2 the curve is B1/ 2 = tanh x . At the other extreme, for
1
large S the functions all approach a function called the Langevin Function, P = ctnh ( x ) ! . The
x
Langevin function describes the magnetization of classical spins, which can assume an arbitrary
angle vs. the field; this is an example of the correspondence theorem.
Taking the S = 1/2 case for simplicity, in the high-temperature limit (small x), we can use
tanh x ! x , so that eqn. (11) becomes M = ngµB Sx = n(gµB S )2 H / kT . This is identical to the
Curie law for S = 1/2, and in general the low-x portions of all of the curves plotted above are
linear in x, and the Curie law holds in this limit. In the opposite limit, the asymptote for all
curves is BS = 1, corresponding to the uniform alignment of the spins by a field sufficiently large
to completely overcome the thermal energy.
D. Mean Field theory of Local-Moment Ferromagnetism:
Suppose we have an exchange interaction between neighboring spins, which is proportional to
the dot product of the two spins:
(13)
H = !J $ Si " S j + µ B g$ H " Si .
i# j
i
J is a constant parameter which measures the size of the exchange interaction. This is the
nearest-neighbor Heisenberg model. J may be due to exchange in the case that there is overlap of
orbitals between neighboring spins, on the other hand see the description on page 685 where J is
based on a hopping model (the Hubbard model). For insulators normally J is negative, leading to
antiferromagnetism. If J is positive, spins tend to align, leading to ferromagnetism. Here we limit
the sum in (13) to nearest neighbors, otherwise we need to include a more general site-dependent
J(R) in the model.
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Eqn. (13) is the Heisenberg model if the full spin-spin dot product is retained; a simpler model is
the Ising model, where we limit ourselves to S = 1/2 and neglect the x and y parts of the spin-spin
product, leaving,
(14)
H = !J # (Sz )i (Sz ) j + µ B g# H $(Sz )i .
i" j
i
We can also replace gSz with an effective spin s, which has eigenvalues ±1 rather than ±1/2. So
finally we have (with J renormalized to J' = J/4):
(15)
H = ! J " $ si s j + µ B $ Hsi .
i# j
i
The Mean Field approximation simplifies things considerably by replacing each neighbor spin by
a mean value, s j . In that case:
H = ! J " $ si s j + µ B $ Hsi ,
j#i
(16)
i
where in the case of a ferromagnet, s j
solid. Another way to write this is,
#
J "z s
H = µ B ) si % H !
µB
$
i
is a number which has the same value everywhere in the
&
(17)
(' ,
where z is the number of neighbors, from the sum over j. The second term in the bracket acts as a
mean field, as if it were a real atomic-scale magnetic field which supplements the applied field
H.
In the high-T limit this yields the Curie-Weiss law, similar to the Curie law of eqn. (10). First
CH eff
we note: M = !µB n s , and for high T, the solution of (17) must be a Curie law, M =
,
T
exactly the same as eqn. (10), except that Heff is the (round-bracket) total field felt by the spins in
eqn. (17), not the bare applied field. Rearranging, we find,
#
CJ "z &
(18)
M % T ! 2 ( = nCH ,
µB '
$
which is equivalent to,
CH
,
T ! Tc
CJ "z J "z
where Tc ! n 2 =
is the Curie temperature.
µB
kB
M =n
(19)
Note that Tc measured from the high-T susceptibility is the same as the transition temperature in
mean field theory, as shown below. However in general when fluctuations and other factors are
considered, the Curie-Weiss constant may differ from Tc (often substantially). Thus the constant
CH
is often designated θ, (19) becoming M = n
, with Tc reserved for the temperature at which
T !!
spontaneous order appears.
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General solution:
At lower temperatures we expect M to satisfy eqn. (11), except that in the mean field
approximation, every time H appears in (11) we replace it with Heff. Solving for the case of no
applied field (H = 0), and using B1/ 2 = tanh( x ) in eqn, 7 we find,
! J 'zM $
!µ H $
.
M = nµ B tanh # B eff & = nµ B tanh #
" kT %
" µ B nkT &%
(20)
This equation can be solved graphically. The plot below includes an S = 1/2 mean field curve,
which is the solution to this equation, along with some other plots for comparison. There is a
spontaneous magnetization at temperatures below T = Tc .
The transition at Tc is a second-order phase transition. In real materials M will deviate from the
mean-field curve: for example see the data for Fe, and for FeGe2 plotted above. (The FeGe2 data
are data taken from my lab.) Note: the S = 1/2 curve is the upper solid curve; the lower curve for
S = 5/2 is green, which may not show up in your copy. Important points are:
(1) At low temperatures the reason the data drop faster than mean field as T increases is because
of cooperative magnons, or spin waves, which are the important thermal excitations. In the mean
field approximation, non-uniform wave-type excitations were not considered, since we assumed
no deviations from the average spin value. In ferromagnets, magnon terms give the Bloch T3/2
law behavior at low temperatures, for 3-dimensional systems. This is discussed in chapter 33; see
fig. 33.7 for an illustration of a spin wave.
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(2) Near Tc , we must consider random fluctuations of individual spins, not just the mean
value—this is the regime of critical phenomena, which has been a subject of a great deal of
theoretical activity (for instance the work by Wilson and others on the renormalization group,
leading to Wilson’s 1982 Nobel prize). There is a curve in the plot above proportional to (1 –
T/Tc)0.50, which was an attempt to fit to the critical exponent for the phase transition. There is
some discussion of critical exponents on pp. 699-700 of your text, however much work was done
since the text was published. More information is provided in a number of more recent texts;
look for information under the subject “critical phenomena”.
An important consideration for the validity of Mean-field theory is that, if the number of
neighbors is small, replacing them by a mean value may work poorly, while for a large number
of neighbors, the averaging effect of large numbers means that mean field may work reasonably
well as a first approximation. This is closer to the case, for example, for the second order
electronic phase transition in traditional superconductors.
Finally, note that experimental measurements of bulk Magnetization in ferromagnets are also
strongly dependent upon the fact that the magnetization arranges itself into local domains,
something we have not considered here. Thus, measurements of the overall magnetization of a
sample may deviate signifacantly from the intrinsic value of M unless considerable care is taken.
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