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Physics 617 Apr. 17, 2017: Magnetism Notes A. General concepts and background: Magnetization and susceptibility: The magnetization (M) is defined as the density of magnetic dipole moments per volume, while the susceptibility (c) is the derivative of M with respect to the magnetic field. Equivalently, M can be defined thermodynamically as a derivative of the Helmholtz free energy, F = E – TS, so that the following relationships determine these quantities:

M =µV

or M = − 1V

∂F∂H T

; χ = ∂M∂H

. (1)

Note, see an advanced thermodynamics/statistical mechanics text such as Pathria & Beale for more discussion of the thermodynamic quantities. Generally M is a vector, pointing in the direction of the local alignment of moments, and accordingly c is defined as a second-rank tensor, for cases where M does not align with H. Causes of such mis-alignment include surface poles due to the sample shape, and microscopic effects (magnetocrystalline anisotropy) associated with spin-orbit interactions. In the discussion below these effects assumed to be weak. Units: Folling the Ashcroft and Mermin text we are using CGS units, in which B = H + 4πM, so one can see that the units on B, H, and M are fundamentally the same. Note that while the general trend in physics is towards SI units, CGS is still encountered very commonly in magnetism. For example commercial systems such as the SQUID magnetometers on our campus provide results in emu units, 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, 1 gauss-cm3 is also the same as 1 erg/G, since 1 G2cm3 = 1 erg [the energy density is B2/8π]. So, for example µB = 9.27 ´ 10-21 erg/G, is a measure of the magnetic moment of the electron. From (1) it can be deduced that c" is a dimensionless number. The dimensionless 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”. Quenching of orbital moment: For transition-metal based materials, the orbital contribution to the magnetization is typically quenched, leaving only the spin part as contributing to the magnetism. In general this occurs because of hybridization (or essentially bonding effects); the degeneracy of different 𝑚ℓ𝓁 sublevels is removed by orientation-dependent interactions, leaving nonmagnetic combinations as the preferred basis states (for example dxy and dx

2-y

2 rather than Y2

±1 states). Thus for most of this handout, S is used for the magnetic moment, except in the discussion of diamagnetism (eqn. 4). However note that for rare-earth materials containing unpaired f orbits, the full unquenched moment must typically be used. For the latter cases, substitute J for S, and use the Landé g-factor, as defined on page 654 of the text. Magnetic order includes ferromagnetism, which means all spins are aligned in one direction, as in the schematic figure below (example: iron (Fe) metal, the archetype ferromagnet, which is ordered below TC = 1043 K.) Antiferromagnetism means spins are aligned in alternating directions, with no net magnetization. Examples: CuO, with a transition temperature TN = 230 K,

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or FeCl3, with TN = 15 K. An antiferromagnet such as IrMn is used in hard-drive read heads to pin the ferromagnet-based sensor via exchange bias. Ferrimagnetism is an intermediate case, which 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 it is not actually a ferromagnet; the Nd moments are anti-aligned. This is a tradeoff; the benefit of the rare earth Nd constituent is that it provides strong anisotropy helping the magnet to maintain its magnetization, even though the Nd moments reduce somewhat the magnitude of M. Magnetocrystalline anisotropy is the term used for the effects leading to orientation-dependent magnetizing energy in these systems (or correspondingly shape anisotropy is for effects due to sample shape). There are also other types of magnetic order, for example in some cases competing interactions can lead to spiral-type magnetic ordering, while in some cases the magnetic moments of disordered alloys can “freeze” into a spin-glass state. A paramagnet is a case where the spins are not aligned, and in general fluctuating rapidly due to competing interactions.

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. Although Fe metal is often pictured as containing localized spin moments in atomic d orbitals, 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 Nd-Fe-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. B. Linear susceptibilities and Hamiltonian: In most materials, the magnetic response is weak and linearly proportional to the applied field. Examples: (a) Diamagnetism, typical of insulators, for example silicon with c�= –3.9 ´ 10-6 cm3/mol. Typical behavior for such cases: very small, temperature-independent c, due to the small orbital magnetic response of such systems. Alternatively, Larmor diamagnetism occurs for metallic electrons, however it is typically hidden by the Pauli paramagnetism of metals defined below. Also this term is difficult to treat analytically, except for the simplest metals. The free-electron Larmor result is

�

χLarmor = − 13 χPauli , although this is not necessarily valid for a real metal.

(b) Paramagnetism, appears in systems with degenerate (or nearly degenerate) ground states. (i) Pauli paramagnetism for metals, example aluminum cPauli�= +16.5 ´ 10-6 cm3/mol.

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Typical behavior: small, T-independent positive c, due to B field-induced polarization of electron spins in metallic energy bands. (ii) Curie paramagnetism, for example FeCl3, c�= +13500 ´ 10-6 cm3/mol at RT. Large, strongly T-dependent c, with Curie law, c"= C/T. This term comes from

paramagnetism of local moments. (Note that FeCl3 really has Néel-type susceptibility,

χ = n CT −TN

since it becomes an antiferromagnet at low temperatures.)

(iii) van Vleck paramagnetism, example KMnO4. T-independent positive c, 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. Hamiltonian: We discussed the Hamiltonian including the vector potential earlier in the semester. The general form is,

H = 12m(p + e

cA)2 +Uo + µB

H ⋅S , (2)

where the last term is the spin interaction, which must be added separately. Internal magnetic interactions such as exchange between spins are also not included here and must also be added later (see part D below). Note that the electron charge is written here as q = –e, where e is considered positive. Squaring the first term on the right yields three terms: (i)

�

p2 / 2m (the H = 0 kinetic energy), (ii) a cross term which is shown below to be equivalent to

�

µB

H ⋅ L , and (iii)

�

e2 A2 / 2mc2 . The last term contains A2 so it is generally small, but it is the only contribution for

diamagnetic materials. Replacing

�

p = −i

∇ , the cross term is

−ie2mc

(∇⋅A +A ⋅∇) . With the case

that

�

H is a uniform field, we can choose

�

A = − 1

2

r ×

H , for which

�

∇ ⋅ A = 0 , so that:

−ie2mc

(∇⋅A +A ⋅∇) = −ie

2mc2A ⋅∇ = ie

2mc(r ×

H ) ⋅∇

= ie2mc

(∇× r ) ⋅

H

= e2mc

L ⋅H

. (3)

The second line is a cyclic equivalence for the triple product, while the third line uses

�

L = r × p

which is the usual angular momentum operator. The factor

�

e2mc

= µB is defined as the Bohr

magneton. Thus the Hamiltonian (2) can be expressed (neglecting interactions):

H =H o + µB

H ⋅(L + go

S)+ e2

2mc2A2 . (4)

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The middle term,

�

µB

H ⋅ ( L + go

S ), generally leads to paramagnetism, while the last term,

�

e2

2mc2 A2 , leads to diamagnetism.

Current density: In QM the current density is derived by expanding the time derivative:

�

∂∂t

ψ2

=ψ* ∂ψ∂t

+ ∂ψ*

∂tψ . (5)

Since

�

ψ2≡ ρ is the quantum probability density, (5) can be placed in a continuity equation,

∂ρ

∂t = − ∇ ⋅ j , where j is a probability current. For the derivatives on the right of (5) we

substitute by using the Schrödinger equation,

∂∂tψ = − i

Hψ = + i

2m∇ + iecA⎛

⎝⎜⎞⎠⎟2

ψ − i(Uo + µB

H ⋅S)ψ (6)

and its complex conjugate. With a little algebra we obtain,

j = − i

2mψ *∇ψ − (

∇ψ *)ψ⎡⎣ ⎤⎦ +

emcAψ *ψ . (7)

The term in (7) proportional to A can be shown to be responsible for diamagnetism. Essentially, 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, which has zero angular momentum, nevertheless 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 will completely exclude applied magnetic fields. 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:

M =µV

= − NV

gµBmSe−gµBmSH /kT

−S

S∑e−gµBmSH /kT

−S

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,

M ≅ n (gµB )2HS(S +1)3kT

, (9)

which gives a positive (paramagnetic) susceptibility,

χ ≡ CT= n

(peffµB )2

3kT, (10)

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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 = ngµBSBS (x) , (11) with

�

x = gµB SH (kT ) [a ratio of magnetic spin energy to thermal energy]. The functions

�

Bs( x) are the Brillouin Functions:

BS =2S +12S

ctnh 2S +12S

x⎛⎝⎜

⎞⎠⎟ −

12Sctnh x

2S⎛⎝⎜

⎞⎠⎟ . (12)

Note: for the special case S = ½, a little manipulation of the hyperbolic functions yields a much simpler form:

�

B1/ 2 = tanh x( ), not given in the text. A plot of the Brillouin functions is given below, with the S = ½ curve at the top:

For large S the curves progressively approach the Langevin Function, P = ctnh x( )− 1

x. The

Langevin function describes the magnetization of classical spins which can assume an arbitrary angle vs. the field; large moment spins are sometimes found in superparamagnetic systems where the large moments are due to small clusters of ferromagnetically-aligned atoms which act like a single fixed spin. Taking the S = ½ 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 (S = ½ case). 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 for all cases. In the opposite limit, the large-x asymptote is BS = 1 for all cases, corresponding to the uniform alignment of the spins by a field sufficiently large to completely overcome the thermal energy.

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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: H = −J Si ⋅Sj

i≠ j∑ + µBg H ⋅Si

i∑ . (13)

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. 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, H = −J (Sz )i (Sz ) j

i≠ j∑ + µBg H ⋅(Sz )i

i∑ . (14)

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): H = − ′J sisj

i≠ j∑ + µB Hsi

i∑ . (15)

The Mean Field approximation simplifies things considerably by replacing each neighbor spin by a mean value,

�

sj . In that case:

H = − ′J si sjj≠i∑ + µB Hsi

i∑ , (16)

where in the case of a ferromagnet,

�

sj is a number which has the same value everywhere in the solid. Another way to write this is,

H = µB si H −′J z sµB

⎛⎝⎜

⎞⎠⎟i

∑ , (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

we note:

�

M = −µB n s , and for high T, the solution of (17) must be a Curie law, M =CHeff

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,

M T − C ′J zµB2

⎛⎝⎜

⎞⎠⎟= nCH , (18)

which is equivalent to,

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M = n CHT −Tc

, (19)

where Tc ≡ nC ′J zµB2 = ′J z

kB is the Curie temperature.

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

is often designated q, (19) becoming M = n CHT −θ

, with Tc reserved for the temperature at which

spontaneous order appears. 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,

M = nµB tanhµBHeff

kT⎛⎝⎜

⎞⎠⎟= nµB tanh

′J zMµBnkT

⎛⎝⎜

⎞⎠⎟

. (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 .

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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. (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.