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Magnetic Polarons, Charge Ordering and Stripes 24.10.2002 1
Magnetic Polarons, Charge Ordering and Stripes
There is intensive research going on worldwide in order to unravel the mechanisms
responsible for the remarkable properties of a whole family of complex materials. In thesematerials, competing interactions lead to the spontaneous formation of nano-sized regions of
a different phase. In the case of magnetoresistive manganites, these might be metallic or
polaronic in nature, their size may depend sensitively on temperature and magnetic field. High-Tc superconductors may present charge-ordered stripes that are superconducting,
separated by antiferromagnetic regions which act as Josephson junctions by a proximityeffect. In relaxor ferroelectric also, there are nano-sized polar domains.
The issues confronted by current researchers working with strongly correlated systems areenormous, intricate and complex. The challenge both to experimental and theoretical physics
stems from the fact that the relevant physical mechanisms and the material science aspectscover a very wide range of properties, such as the interplay of charge, spin, orbital degrees of
freedom.
The purpose of this chapter is to bring out some interesting phenomena that one might expect
to observe in magnetic semiconductors and transition metal oxides. The descriptions aremeant to develop in the reader an intuitive idea of what might be happening in the
prototypical systems considered in these notes. Hopefully, these simple heuristic arguments
may help give a sense that some heterogeneity can be expected, which is intrinsic to the physics that drives their properties. I am not considering here any kind of heterogeneity due
to crystalline imperfections.
Magnetic Polarons
Consider an antiferromagnetic semiconductor, doped with electron donors. Assume further
that the extra electrons are confronted to a large on-site exchange repulsion. Because of it, you
could assume the simplistic senario below left (shown for one electron).
However, this constitutes a severe confinement of this electron. The kinetic energy of an
electron tends to increase with confinement. We expect a kinetic energy of the form2 2
2
k E
m=h
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with some quantization rule for k. If the electron is confined to a sphere of radius R, then k
will be of the order of
2
R
π
This entity is referred to as a “magnetic polaron”.
Let’s call J the exchange coupling of an s electron on the ion with the core electron, typically
an f electron, of the same ion. A typical value for J is a tenth of an eV. Now, when the
electron is spread (figure above, right), the exchange is reduced. Call Jeff the exchange
energy. Since the exchange integral contains an integrand of the order of 2
sϕ , the effective
coupling is scaled by the ratio :
3
3
4
34
3
R
a
π
γ π
= that is : eff
J J
γ =
in which R is the radius of the presumed polaron and a is the radius of the ion carrying the
magnetic moment. At each site, the exchange interaction is of the order of r
eff J S s⋅r
The electron so partially localized interacts with a number of sites (carrying a magnetic
moment) equal to γ . Hence the magnetic energy due to the coupling of the partially
delocalized spin is :3
mag eff z z E J S J γ ∆ ≈ − = − S
s
where Sz is meant to be the average spin of the magnetic moment at one site. Now we just
need to work out what Sz might be under these circumstances. We follow the usual mean fieldtreatment, stating that the spin Sz is the Curie susceptibility times the magnetic field
composed of the field due to the exchange coupling to the electron (strength Jz) and the
coupling among the local moments (strength ). Generally, if an interaction has the form : ferro J r
int H JS =r
we can think of it as the spin coupled to the fieldS r
/ B Js g µ r
. So we have :
62
eff
z Curie ferro z
J S J χ S
= +
From this, we deduce a Curie-Weiss susceptibility χ , that is the susceptibility of the
ferromagnet, and
32 2
eff
z
J J S χ χ
γ = =
With this result the magnetic coupling amounts to2
3
J E
χ
γ ∆ ≈ −
Recall that the J here is describes the s-f coupling and the susceptibility has the form :
0
cT T
χ χ =
−
The total energy has the form :
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2 3tot
A B E
γ γ = −
We cannot assume freely that 0γ →
in the above, because this would make S
r
diverge,which contradicts the fact that S is limited to a finite value (e.g. 7/2 for Eu in EuO), and
contradicts also our assumption of linear response. We can consider that solution may arise if
the general form of the dependence of leaves a region of negative values inside the region
where Sz is reasonable.
tot E
0 1
Sz < S
We see from these considerations however that it may be possible to have a gain in energy in
such a process. The gain is more likely when the susceptibility is large, hence near Tc. As the
radius is to be taken rather small, our expression32 2
eff
z
J J S χ χ
γ = = implies that Sz is
saturated to its maximum value S. This is a characteristic feature used e.g. in an NMR study
as evidence for magnetic polaron formation : while the magnetization decreases with
temperature, the NMR frequency, that is, the hyperfine field, remains unchanged! 1
We have not considered above the possibility that the electron is bound near a donor by
electrostatic interactions. This is the case for example when EuO is doped with Gd. Eu is
divalent, Gd is trivalent, hence gives out an extra electron. It is known that Gd substitutes for
Eu in EuO. It was found that this doping increases Tc enormously. This can be taken as an
indication of enhance ferromagnetic coupling in the doped system compared to the
stoichiometric one.
1Coey, Viret et al, J. Physics C : Cond. Mat.
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Figure 4.120 of “Electronic conduction in oxides”, Springer Solid State Physics Series, vol. 94. Temperaturedependence of the magnetitzation of EuO, substituted by 2% Gd(+++) and a thin film with excess Eu(…). The
electrical resistivity of the latter specimen is about 5x10-3
Ohm.m The solid curve in the figure is thethermomagnetic curve of stoichiometric EuO.
One dimensional model for polaronsand introduction to double exchange
From S. Pathak and S. Satpathy, Columbia, Missouri, PRB 63, 214413 (2001)
We consider the fate of a conduction electron in an antiferromagnetic one-dimensional chain
of localized moments. This electron experiences an exchange coupling with the local
moments.
The author refers to the paper of Anderson and Hasegawa on double exchange to state that the
kinetic energy of the electron comprises a term of the form :
( )cos / 2t χ −
where t is the transfer integral. See my notes on exchange where this exchange intergral
appears. χ is the angle between the local moments on the two adjacent sites for which the
transfer integral is calculated. This term can be thought of as follows. It is known in quantum
chemistry (see e.g. Coulson’s “Valence”) that the estimate of the energy of the ground state of
a diatomic hydrogen molecule is enhanced if one includes the possibility for both electrons to
be on the same atom. This leads to the transfer integral term –t in the energy. Now, Andersonand Hasegawa consider a diatomic molecule that carry each a core electron magnetic moment.
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The work out the passage from discrete terms to continuous terms by a Taylor development of
ψ . The schematics below shows that the first order terms cancel out in the summation.
( )d
x adx
ψ ψ − ( ) xψ ( )
d x a
dx
ψ ψ +
The c.c term doubles the contribution, so the energy in the continuous limit is given by :
The authors find the minimum either by an exact method, or using a function of their choice,
which they justify in their text. Their calculation demonstrates that for any value of the
transfer integral, the formation of polarons is favored. That is, the electron is trapped
somewhere in space and the antiferromagnetic lattice at this point is transformed into a
ferromagnetic region (see schematics).
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This calculation yields a size of the polaron of the order of a few lattice constants. The angle
between adjacent local moments jumps from parallel to antiparallel in about one latticespacing. The paper refers to the model of Mott (Metal-Insulator Tranistions, N. Mott, Taylor
& Francis, 1990) who assumed a jump from ferro to antiferromagnetism at one lattice
spacing. Overall, the proposed calculation does not improve much over the Mott model.
One usuful outcome of this paper is that ,based on a simple idea, it shows how carrier
delocalization can lead to ferromagnetism. The authors calculated the size of the magnetic
polaron as a function of the parameter 2t JS
α = , the ratio of the transfer integral and the
antiferromagnetic coupling of the localized moments.
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Type II polarons are saturated (region with 0 χ = ) whereas so-called type I have the moment
fully saturated. Hence, this model finds that when the carriers are delocalized, the magnetic
moments align. This notion was first introduced by Zener (Phys. Rev 82, 403 (1951) as the
double exchange mechanism. It is supposed to be one of the leading mechanism of the
ferromagnetism in manganites, the materials that exhibit colossal magnetoresistance.
We can well imagine at this point that one of the questions that one might want to address isthe following. What happens as the electron doping is increased and one gets to have a dense
system of self-trapped magnetic polarons. This question was addressed recently in the
literature. (M. Umehara, PRB 63, 134405 (2001) )
Bound magnetic polarons
The magnetic transition metal oxides, in particular the manganites, and related systems such
as EuO, EuTe, EuS, are prone to defects, excess metal ions, oxygen vacancies etc… Hence,
the polaron might form around such a defect and the electrostatic interaction might be the oneresponsible for the trapping of the polaron. Furthermore, polarons in dilute magnetic
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semiconductors, like Mn doped CdTe have clearly charged defects at the Mn site. The
literature on bound magnetic polarons is abundant.
If the polaron is not bound, then it contributes to the conductivity of the material. Hence the
question of the trapping of magnetic polaron is relevant to the modeling of the colossal
magnetoresistance of materials such as EuO. Here we consider a theoretical description madein terms of the linear response of an electron gas coupled by exchange to an array of localized
magnetic moments. This is the work of P. Leroux-Hugnon (PRL 29(14), 939 (1972), who
was then at CNRS Meudon, and is now at Paris VII. (His recent work might be of interest :
PRB65, 125210 (2002) “Application of a self-consistent LSDA-CPA method to the Mott-
Anderson transition in doped semiconductors”, see also PRB 28(7), 3929 (983)“Functional-
integral approach to the linear responses of the Hubbard model : role of exchange-field
fluctuations”, for functional integral, see Philippe Martin, PPUR, EPFL, Initiation à
l’intégrale fonctionnelle)
Charge Ordering
I continue the exploration of reasons why intrinsic heterogeneities can be present in transition
metal oxides with an argument put forth as early as 1955. Koehler and Wollan of Oak Ridge
National Laboratory used neutron scattering and demonstrated the coexistence of
antiferromagnetic and ferromagnetic regions in1-x x 3
La Ca nO . (Phys. Rev. 100(2), 545
(1955) ).
3LaMnO has Mn in a state. Oxygen is –2, La is La 3+, so Mn has to be . Now Ca
is divalent, so the addition of Ca constitutes a hole doping. As a consequence, the system
presents a mixed valence of Mn and . The exchange coupling is known to be :
3+Mn 3+Mn
4+ 3+Mn
- strongly ferromagnetic between and ,4+Mn 3+Mn
- antiferromagnetic between Mn ions,4+
- either ferro- or antiferromagnetic between ions, depending on their separation.3+Mn
Consider now a doping at 25% Mn . Then you can imagine a charge ordering of the crystal
which favours the ferromagnetic interactions between Mn and . Indeed, a unit cell can
be as shown below.
4+
4+ 3+Mn
+4+3
+3+3
+3+3
+3+4
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In this unit cell, every is surrounded by ions. Hence we perceive the possibility
for ferromagnetism with this charge ordering. Indeed, the onset of ferromagnetism and
metallic behaviour is at 25%, as shown on the phase diagram below.
4+Mn 3+Mn
Jaime and Salamon, Cond-Mat/9902284 (1999)
If the doping goes to 50%, then we might have the following charge-ordered unit cell :
+3+4
+3+4
+4+3
+4+3
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Again, this structure favors ferromagnetism. At intermediate doping, one could expect an
ordered array of alternating structures. Such a structure would have produced superlattice
lines in the neutron diffraction pattern. They were not observed by Koehler and Wollan. But
wait a few decades an read below about “stripes” !
Now, what can happen if the doping is below 25% ? It can happen (as Kohler and Wollan
showed in 1955) that the cluster in regions where this ratio of 2 for 6 is
satisfied so as to “take advantage” of the strong - interaction.
4+Mn 4+Mn 3+Mn3+Mn 4+Mn
StripesThe idea about stripes started, as far as I can tell, from a theoretical paper by Haanen and
Gunnarsson of the Max Plank Institut, Stuttgart (Phys. Rev. B40(10) 7391 (1989) ) Theseauthors considered the antiferromagnetism of high-Tc oxide superconductors.
They considered a strong on-site Coulomb repulsion of d electrons, U. They considered a
hybridisation of p and d orbitals assumed to require an energy V small compared to U. Finally
they considered charge hopping in which a d9 atom becomes a d10 atom, leaving a p hole.
This process is assumed to require an energy D, also somewhat small compared to U. The
main result of the numerical calculation based on a two-band Hamiltonian is shown in the
figure below.
This is supposed to represent a copper - oxygen plane in perovskites. The density of excess
holes on the oxygen ions is proportional to the radius of the circles and the spins on the Cu
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lattice are represented by the arrows. Case a) is a 9x10 array, case b a 10x10 array. Periodic
boundary conditions force the formation of a loop in case b).
A so-called Néel line forms. The periodicity of spins has a discontinuity on the line with spin
zero on the line :
up down up down zero up down up down up
as opposed to the regular
up down up down zero down up down up down
It turns out that the stripes that form in doped antiferromagnets can be either insulating or
conducting. Current theoretical research points to the possibility that stripes could form some
sort of electronic quantum liquid crystal which would constitute a “new state of matter”.
(Nature, S.A. Kivelson, E. Fradkin, V.J. Emery, “Electronic liquid-crystal phases of a dopedMott insulator”, vol. 393 1998, p. 550)
Clear evidence for stripes was reported in manganites thin films. (Nature, S. Mori, C.H. Chen,
S.W. Cheong, vol. 392, 473(1998) and Physics Today June 1998 page 19). High resolution
electron microscopy carried out on manganite thin films showed that stripes form stable pairs
which repeat periodically. Because these pervoskites are three-dimensional, the stripes here
are actually planes. The sharing of electrons between 3n + and oxygen causes a distortion of
lattice which produced the contrast in the image. The regions around 4n + were less
distorted and yielded less contrast. Increasing the doping level simply separated the pairs from
one another.