Download SDW`s

Presentation Groupmeeting June 3rd, sorry 10th, 2009
by Jacques Klaasse
Spin Density Waves
This talk is based on a book-chapter on antiferromagnetism,
written by Anthony Arrott in Rado-Suhl, Volume IIB, 1966.
Contents:
-
Exchange interactions
Spin Density Waves
Neutron diffraction
Chromium
Conclusions
Exchange interactions
Starting point of the (historical) discussion:
- atoms with intrinsic localised magnetic moment.
- exchange interaction (Heisenberg) because of direct overlap.
- Weiss molecular field.
The interaction can be ferromagnetic or antiferromagnetic, dependent on the
sign of the exchange parameter.
Kramers introduced also “superexchange” mediated by electrons on
intervening non-magnetic atoms (oxygen!).
In 1946 Stoner questioned the picture in case of metals.
He presented the “collective electron picture”: itinerant
(ferro)magnetism by mutual exchange of d-band electrons.
Stoner Criterion: ferromagnetism occurs if D(εF) * IS > 1
where D(εF) is the density of states at the Fermi level,
and IS is the Stoner exchange parameter.
Exchange interactions
The two subbands are shifted in
energy because of the exchange
interaction.
The shift is based on the Hubbard
Hamiltonian Un↑n↓ , which can be
rewritten, with n = n↑ + n↓ , as
(U/4) { n2 – (n↑ - n↓)2 }.
The exchange potential is not a
fixed potential but is governed by
the other electrons.
Moment and potential are both
given by the same medium.
This results in a non-zero ferromagnetic moment even in zero applied field,
as long as the gain in exchange energy is larger than the loss
in kinetic energy.
The Coulomb interaction in metals seems to favour
ferromagnetic coupling!!
Exchange interactions
Let D be the average DOS per spin direction
around εF and let the splitting be ΔE.
Then
(n↑ - n↓) = D.ΔE
Let
s = ( n↑ - n↓ ) / N , and
Let Is be the exchange parameter.
We calculate now the splitting energy.
We do Ekin first.
ΔE/2
Ekin = 0∫
2
2
2
(2ε) D dε = (D/4) ΔE = (D/4) (Ns/D) = (Ns/2) (1/D).
2
For the exchange we found Eexch = -(Is/4) (Ns) , so, for the total splitting
energy we find:
2 2
2 2
Es = (N s /4) { 1/D – Is } = (N s /4D) { 1 – DIs } .
From this formula follows the Stoner Criterion. This shows to work for
ferromagnets like Fe and Ni, but not for much more. Obviously,
a high D (or low density in real space) favours ferromagnetism.
Above Tc the material should be a Pauliparamagnet. This is not seen!
Help screen for calculating Ekin
ε
ε
dN = D dε
n↑-n↓ = D ΔE
=Ns
ΔE/2
ΔE / 2
D(ε)
∫
0
(2ε ) Ddε
Exchange interactions
Another class of materials contain localised moments apart from
conduction electrons.
It is shown by Ruderman, Kittel, Kasuya, and Yosida that the interaction can be
formulated in a way that a Heisenberg-like picture is simulated without direct
overlap (RKKY interaction, published between 1954 and 1957).
This makes the problem similar to the non-metallic problem.
The coupling can be FM as well as AFM, and is strongly oscillating with
distance.
This work followed upon a discussion by Zener on the role of the conduction
electrons in providing (ferro)magnetic interactions.
Zener also revived the old suggestion of Néel that Cr and Mn as metals were
antiferromagnets, where so far antiferromagnetism seemed to be only a
property of non-metals.
What happens here. Is Cr metal an RKKY magnet, or do we have something
special? The moments of Cr++ and Cr+++ are 4.9 and 3.8 Bohrmagnetons
per atom respectively and the Cr++ saturation moment should be 4 μB.
Exchange interactions
Stimulated by this discussion Shull and Wilkinson proved that
these metals are weakly antiferromagnetic.
See Rev. Mod. Phys 25 (1953), p100.
Moreover, the wavelength was not equal to the length
of the cubic unit cell (Corliss et al., PRL 3 (1959), p211) .
These results came on the moment that neutron
diffraction techniques became a suitable tool for
determining magnetic structures.
We come to that later.
Anyhow, there was a problem how to
explain these results.
SDW’s
It was Overhauser (around 1960) who showed that in the onedimensional Hartree-Fock approximation the antiferromagnetic
state could exist and may have lower energy.
The AF periodicity is not given by the lattice but by the wave
vector equal to the diameter of the volume of occupied states in
k-space.
With the help of neutron diffraction techniques, it is shown that
these “Spin Density Waves” indeed exist.
Main conclusions from Overhauser’s work:
a) Spin density waves are allowed states.
b) SDW’s may be ground state.
c) SDW’s with wave vector q = 2kF are most
likely to minimise energy.
SDW’s
Source: A.W. Overhauser,
PRL 3,9 (1959) 414
Source: [Overhauser (1962)]
SDW’s
Here, we will not give all the mathematical details of the HF
procedure.
We only give some flavour of what was going on here, in particular
we will show some pictures to elucidate the situation.
For detailed information we recommend the following paper:
A. W. Overhauser, Phys. Rev. 128 (1962) p 1437 – 1452.
You need a reasonable starting wave function (for fermions this is a
Slater determinant) and a smart trial potential.
Then you have to solve the Schrödinger equation by a variational
method until you find an internally consistent solution where
your potential is stable under continued iterations.
With a proper starting set, the procedure is generally convergent,
but it is not sure your solution is the real ground state.
SDW’s
Some citations on the HF method:
“far from being straightforward”
“coupled integral equations are thoroughly nonlinear
and require an iteration technique for their solution.”
“repeated until a self consistent set of solutions is obtained”
“convergence dependent on initial guess of the
one-particle states.”
Overhauser started his calculations with a helical polarization
(“ this leads to an off-diagonal contribution to the oneelectron exchange potential.”)
SDW’s
Overhauser showed that for spin up
and spin down a gap opens at the
Fermi wavevector, but for the two at
a different sign of kF (=q/2).
The two waves at kF and -kF give
together two charge density waves
at q=2kF with a certain spin polarization,
resulting in a static spin density wave with
constant charge density.
Source: A.W. Overhauser,
PRL 4,9 (1960) 462
It has some resemblance with the opening
of the gap at the Brillouin zone, but
there the potential is fixed, here the
potential is determined by the electron gas,
with largest effect near the Fermi wave
vector.
SDW’s
Source: [Overhauser (1962)]
SDW’s
The spin susceptibility
shows for SDW’s to
diverge at 2kF.
Source: [Overhauser (1962)]
SDW’s
In 3 dimensions the problem
may not yet be solved but here
we give an artists impression
of the situation.
The gap causes a lowering of
the D(εF), and thus of the linear
term, γ, in the specific heat.
From this effect the gapped
surface fraction of the Fermi
Surface can be derived.
In order to conserve entropy, the entropy loss by the lower γ is recovered
at the transition point to the paramagnetic state, resulting over there in a
peak in the observed heat capacity.
In the resistance, a jump should be expected on opening the gap, caused
by a lowering of the number of available carriers, which is proportional to
the opened Fermi Surface fraction.
SDW’s
In order to see whether in reality SDW’s
occur, we have to minimise the total
energy, being the sum of the total
kinetic energy including the effects of
the SDW’s and the total potential
energy including the total exchange
energy.
The algebra necessary to do this, and
calculate the correct parameters, is
considerable even in the one
dimensional case.
SDW’s
In three dimensions, the so called
“nesting vectors” play the role of
the 2kF from the one dimensional
case.
What is told is that SDW vectors
“ should be in directions where the density
of states is high.”
This sounds reasonable.
Source: Wikipedia
Neutron diffraction
It is not surprising that the discovery
of Spin Density Waves (Shull and
Wilkinson, 1953) goes parallel
with the development of neutron
diffraction techniques.
However, also working with neutrons
has its restrictions.
Longitudinal waves cannot be
detected: neutrons see no spins
but only magnetic field.
A longitudinally polarized magnetic
field wave is incompatible with the
Maxwell equations.
Further: domains!!
Domains add ambiguity to the
interpretation of results.
Source: [Shull & Wilkinson (1953)]
( Normal bcc structure: h + k + l = even. )
Neutron diffraction
Arrott p333: “It is not possible to decide
from the diffraction experiments
between the existence of a helical spin
density wave state and the presence
of two types of domains each with
transverse linear spin density waves
but with mutually perpendicular
polarizations.”
A magnetic field may unravel the problem.
Other problems: the intensities of the
SDW reflections are very weak, and you
don’t know where they are.
(needle in hayloft).
With K = 2πG ± q, only the G=0
reflections give sufficient intensity.
So, K is, in reciprocal space, not known
In magnitude, nor in direction.
Source: [Shull & Wilkinson (1953)]
Neutron diffraction
Spin configurations are generally described in terms of
Helical Spin Density Waves:
p(z) = p (excos qz ± eysin qz )
p(z) = p (eycos qz ± ezsin qz )
p(z) = p (ezcos qz ± exsin qz ),
for q // z.
The first are called “normal” helical waves, the last two
lines describe “end-over-end” helical waves.
This set of functions form a complete set to describe any spin
wave with q = ± qez.
The use of this set instead of plane waves gives some profit
in the analysis of neutron diffraction patterns. The
“normal” and the “end-over-end” behave differently
(Arrott, p 300).
Chromium
A review article by E Fawcett, Rev.
Mod. Phys. 60 (1988) p209, gives
75 pages of material on
chromium properties.
Too much to handle here.
There is agreement on the AF low
temperature state with TN=311K
and μ ≈ 0.5μB.
In the picture here you see the
behaviour of the thermal
expansion, resistivity, specific
heat, and thermo electric power.
Chromium
From neutron results it
follows that there is a
second spin-flip
transition at about
115K.
Below 115K the SDW’s are
longitudinal, and above
it the SDW’s are
transverse.
Source: [Fawcett (1988)]
Chromium
My interest is in particular the heat
capacity. This here, on this picture,
looks like a second order transition.
However this is cold rolled material.
The results showed to be strongly
dependent on the strain situation.
In the next slide we show results on
a strain-free single crystal.
Chromium
This is clearly a
First-order
SDW-PM
Transition.
I. S. Williams,
E. S. R. Gopal,
and R. Street,
J. Phys F: Metal
Phys, 9 (1979)
P 431.
Chromium
There is a similarity in the structure of
the SDW HF equations and those of
the BCS model for superconductivity.
The temperature dependence of
the SDW gap, which is
proportional to the amplitude of
the SDW’s is a lookalike of the
BCS curve.
Maybe the similarity is not only
mathematical.
Source: [Overhauser, 1962]
Maybe SDW’s and SC are
two sides of the same coin.
Chromium
The entropy around the transition (obtained from heat capacity
measurements) amounts to about somewhat less than 0.02
J/K.mol. This should be the recovery of the effect of a lower γ .
From γ =1.4 mJ/K.mol we can derive the total entropy of the electron
gas at TN, being about 0.42 J/K.mol. This should point in the
direction that 4.5% of the FS is gapped.
This is in agreement with the resistivity jump of about 5% at TN .
If Cr should have a permanent moment on-site, the entropy in the
peak should be of the order of R.ln(2) = 5.76 J/K.mol.
The observed entropy is about two orders of magnitude lower,
indicating no permanent moment is present on the Cr sites.
Chromium
From the BCS theory follows that the gap is
(3.5 * kBTN ) ≈ 0.1 eV.
For this energy holds roughly ħ2R02/2m*
where R0 is (in reciprocal space!!) the
radius of the truncated part of FS
and m* the effective mass.
If p is the number of truncated faces (here 6), then for the total truncated
2
2
fraction, t, holds t = p R0 / 4 kF .
With m* ≈ 1.5m, a value for R0 can be derived: 0.2*108 cm-1.
From q ≈ 2kF follows kF ≈ 1.1*108 cm-1.
Result: t ≈ 0.05, in good agreement with the earlier estimates.
Conclusions
•
Spin Density Waves are possible solutions for the free
electron state.
•
Overwhelming evidence exists that in Cr this SDW solution is
ground state.
•
In an SDW state a part of the FS is gapped.
•
The wave vector of the SDW is determined
not by the lattice but by a “nesting vector”,
being about 2kF for a simple Fermi sphere.
•
It is not clear whether, in a real situation,
these nesting vectors can be calculated,
or simply follow from experiment.
Thank you for your attention.
Help, a gap,
I’ve to flip over…