Magnetic Polarons, Charge Ordering and Stripes

8/3/2019 Magnetic Polarons, Charge Ordering and Stripes

http://slidepdf.com/reader/full/magnetic-polarons-charge-ordering-and-stripes 1/12

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




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 :




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.




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.




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).




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.




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




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




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




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




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.

Magnetic Polarons, Charge Ordering and Stripes

Documents

Transcript of Magnetic Polarons, Charge Ordering and Stripes