Doped antiferromagnetic insulators : a model for high
temperature superconductivity
N.F. Mott
To cite this version:
N.F. Mott. Doped antiferromagnetic insulators : a model for high temperature superconductivity. Journal de Physique, 1989, 50 (18), pp.2811-2822. <10.1051/jphys:0198900500180281100>.
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J.
Phys.
France 50
(1989)
15
2811-2822
SEPTEMBRE
1989,
2811
Classification
Physics Abstracts
74.65
-
74.70H
-
71.30
Doped antiferromagnetic insulators :
temperature superconductivity
a
model for
high
N. F. Mott
Cavendish
(Reçu
le 6
Laboratory, Cambridge,
mars
1989, accepté
sous
G.B.
forme définitive
le 29 mai
1989)
Une esquisse de la nature de la transition métal-isolant se produisant lorsque les
composés antiferromagnétiques non métalliques LaVO3 et La2CuO4 sont dopés avec du strontium
ou du baryum est donnée. Il est suggéré, en accord avec d’autres auteurs, qu’un gaz dégénéré de
polarons de spin est formé. Ce gaz, particulièrement dans des structures bidimensionnelles, peut
conduire à la formation de bipolarons, constituant des paires de bosons donnant naissance à la
supraconductivité à haute température dans ce composé et d’autres oxydes du même type.
Résumé.
2014
An outline is given of the nature of the metal-insulator transition when the
antiferromagnetic non-metals LaVO3 or La2CuO4 are doped with strontium or barium. It is
suggested, following other authors, that a degenerate gas of spin polarons is formed, which
particularly in two-dimensional structures, can form bipolarons, and that these could be the boson
pairs which in the latter and similar oxides give rise to high temperature superconductivity.
Abstract.
2014
1. Introduction.
The first of the high-temperature superconductors to be discovered was La2Cu04 doped with
strontium or barium (Bednorz and Müller [1]). This material is now known to be an
antiferromagnetic insulator with the magnetic Cu2 + ions arranged antiferromagnetically on
the CU02 planes of the layer structure and a Néel temperature of c. 200 K. On doping with
one of the divalent metals the ordered state of the moments rapidly disappears, the Néel
temperature dropping to zero, and the material has the properties of a spin glass [2], the
moments having fixed or slowly varying orientations. Further doping leads to a metalinsulator transition, and in the metallic state the material is superconducting.
The aim of this paper is to examine the nature of the metal-insulator transition in doped
antiferromagnetic oxides, and to compare the properties of the superconductors with those of
other oxides such as La, -.,Sr,,V03 which also show a metal-insulator transition. This material
is not, as far as is known, a superconductor, and does not have a layer structure. We examine
the possibility that the carriers in all these materials are spin polarons, and that the concept of
a degenerate gas of spin polarons is a useful approximation with which to describe their
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500180281100
2812
Several authors (1) have suggested that polarons can combine to form bipolarons,
and that these are the boson pairs needed for superconductivity ; we shall examine this
hypothesis for spin polarons.
properties.
2. Metal-insulator transitions.
Two kinds of metal-insulator transition are relevant to this discussion. The first 1 call a Mott
transition, the second an Anderson transition.
1 define a Mott transition as one, in a crystalline system, in which a change of volume under
pressure, a change of temperature or of composition in an alloy leads to a transition from an
antiferromagnetic or possibly RVB insulator to a metal, which may be antiferromagnetic or
may be « highly correlated » in the sense of Brinkman and Rice [7]. Such transitions are first
order, and are necessarily accompanied by a change in volume and sometimes by a change of
structure. The treatment given by Mott [8] in 1949 is no longer relevant and following
Hubbard [9] we suppose that the transition occurs when two « Hubbard bands » overlap, so
that if their widths are Bl, B2, and a tight binding treatment is valid, the criterion should be
if the influence of any change of structure or volume is neglected. Here Bl, B2 can be equated
to 2 Ztl, 2 Zt2 where Z is the coordination number and tl, t2 are the transfer integrals for the
two Hubbard bands. In general t2:&#x3E; t1. U is the Hubbard intra-atomic energy
Even so, equation (1) is not exact, since when long-range interaction is taken into account, all
transitions of this kind are shown to be first order (Brinkman and Rice [10]). Typical
examples of these transitions are observed in V203 under pressure, temperature or on
addition of Ti203, and in the series Ni(Sl-xSex)2 (Wilson [11]).
The influence of disorder on Mott transitions has been widely discussed ; it was originally
thought that the transition in doped silicon was of this kind, if the material is not
compensated. However, following calculations by Bhatt and Rice [12] and certain experimental data, it now appears (Mott [13]) that in the conduction band of many-valley materials the
disorder induces a kind of self-compensation, so the transition is of Anderson type. Whether
this is so for p-type materials or single-valley semi-conductors is not known.
Turning now to Anderson transitions, Anderson’s [14] paper of 1958 and research based on
it showed that, in the approximation of non-interacting electrons, a degenerate electron gas in
a non-periodic field would undergo a second order transition to a non-conducting state as the
ratio of a disorder parameter Vo to the original band-width B is increased or as the mobility
edge Ec passes through the Fermi energy EF. The scaling theory [15] of 1979 and much
experimental evidence showed that the zero-temperature conductivity increases linearly from
zero as Vo/B decreases from the critical value, there being no minimum metallic conductivity.
So if a is the conductivity and x the concentration,
(e2/Kr12&#x3E;.
Many-valley uncompensated semiconductors seem to be exceptions for n-type
conductors ; for these the effects of long-range interactions are very important, and have been
with v = 1.
(1)
Edwards
Jongh [63].
[3], Cyrot [4],
Kamimura
[5],
Su and Chen
[6],
Emin
[58],
Kamimura et al.
[59],
de
2813
studied
extensively
behaviour with v
in the present decade ;
they
are
probably responsible [16]
for the
= 21
.
However calculations by Schreiber et al. [65] suggest that, for non-interacting electrons,
1.6.
The theory however is most directly applicable to compensated semiconductors, where the
lower Hubbard band is only partly filled. Work in Professor Friedel’s laboratory [17] has been
instrumental in showing that the transition takes place in an impurity band, to which a tightbinding model and the Anderson localization theory can be directly applied. Calculated
values of the electron concentration at which the transition occurs, using the Anderson model
or that based on the Hubbard U, give very similar values for the critical concentration
v
here aH is the hydrogen radius of a donor. The reason is that the one depends on the ratio of
B to U, the other of B to Vo, but both occur in the logarithm of a fairly large number. This is
probably why the well-known plot, reproduced in figure 1, given by Edwards and Sienko [18]
shows a constant value of the quantity (3), namely 0.26, though the transitions sampled may
well be of both kinds.
Fig.
1.
-
Plot of effective radius in
equation (3) against log ne (Edwards
and Sienko
[18]).
2814
As first pointed out by Alexander and Holcomb [19], for a value of n about three time
nc the impurity band, in which the upper and lower Hubbard bands are already merged,
merges in its turn with the conduction band.
The non-conducting state for n .-- nc has also been extensively investigated, particularly in
the « intermediate » range below the transition where the density of states at the Fermi energy
remains finite and conduction at low enough temperatures is by variable-range hopping. The
material is not an amorphous ferromagnet, but rather similar to a spin-glass, the moments
being frozen in random directions. It was suggested first by de Gennes [20] and later by the
present author [21] that an electron with energy at the mobility edge may effect the
orientation of the electrons on neighbouring sites, forming what is called a spin polaron ; if
the material were antiferromagnetic this might be necessary to give sufficient mobility
because an electron could not move to a nearest neighbour site without changing its spin
directions. For a spin glass, on the other hand, a transfer integral I involving the whole
systems will, on each atomic site, include wave functions for both spin directions, though if
the number of sites within the polaron is large, I is likely to be small, giving a high effective
mass.
3. Metal-insulator transitions in
antiferromagnetic
insulators.
We have no reason to believe that a metal-insulator transition in LaV03 or La2Cu04 doped
with Sr or Ba is of Mott type. The former (see Fig. 2) has been extensively discussed by the
present author [21] and by Doumerc et al. [22] as an Anderson transition but perhaps with
insufficient attention to the possible formation of spin polarons. By a spin polaron we mean a
situation in which a carrier is surrounded by a region in which the antiferromagnetic order is
broken down ; this may merge gradually into an ordered region. First, then, we give evidence
for the existence in at least one system of spin polarons. The clearest comes from the work of
Fig. 2.
[57]).
-
Variation of the electrical
conductivity of La, -,,SrV03 with 1 / T (Dougier
and
Hagenmuller
2815
Von Molnar and
Penney [23] on Gd3 _ xVxS4. Here V stands for a gadolinium vacancy in the
cubic structure, the concentration of vacancies being high, and through their random
positions producing Anderson localized states in the conduction band. Charge neutrality
ensures that x electrons per Gd atom are in the conduction band, forming a degenerate
electron gas ; at low temperatures the conductivity is low, tending to zero with temperature.
In a magnetic field, however, a transition to metallic behaviour takes place, which seems of
standard Anderson type ; this is shown in figure 3. Magnetic fields can cause transitions in
doped semiconductors from metal to insulator or vice-versa, as first pointed out by Shapiro
[24], by suppressing the quantum interference effect and thus increasing 0-, or by shrinkage or
orbits in doped semiconductors and decreasing u. Here, however, the accepted explanation is
that the carriers form antiferromagnetic spin polarons with the moments on the Gd ions, and
this increases their effective mass and so allows Anderson localization. The system then is
what Anderson has called a « Fermi glass »
that is a Fermi distribution of states described
by localized wave functions. In the presence of a strong field, however, the Gd moments are
oriented parallel to it, no magnetic polarons can form and so the effective mass drops ; the
mobility edge, initially deep down in the occupied states, rises through EF, leading to an
Anderson transition to metallic values of the conductivity.
-
Fig.
and
3.
Dependence of the conductivity of Gd3 _ xVxS4 in a magnetic field at
Penney [23]).
-
This
work, then, shows that
acceptable approximation
a
«
to the behaviour of a
ask whether the same can be said for
author [25] has proposed that the
=
300 mK
(Von Molnar
spin polarons can exist, and can be an
degenerate electron gas. It is interesting to
a degenerate gas of dielectric polarons. The present
slightly reduced crystalline materials SrTi03 and
degenerate
gas of
T
»
2816
KTa03 must be described in this way. The former shows metallic conductivity if the density of
carriers is greater than 3 x10 18 CM- 3. According to equation (3), this implies a very large
value of the hydrogen radius.
The materials have very high static dielectric constants, so we deduce that K in (4) must be the
static dielectric constant. This can only be so if both the positive defects and the carriers are
able to polarize the medium, in other words that the carrier forms a (dielectric) polaron. Thus
the force between them will be
In real space the polaron must be small, to ensure that
the force attracting it to a positive charge is
not e2/ K o r2, where K () is the high
frequency and K the static dielectric constant. At the same time the mass enhancement must
not be too great ; Eagles [26] has estimated it as 7 - 10 me, so the polaron should not be too
small.
We now look further at the substance Lal _ xSrxV03. The strontium produces a hole, which
has been assumed until now to move in the vanadium d-band, and is thus an electronic
configuration 3d1 moving through the antiferromagnetic lattice of 3d2 states. If this is what is
formed, it seems to us highly likely that a spin polaron will result ; otherwise, as argued
above, the carrier would need to move to next nearest neighbours, giving a very low mobility.
There remains the possibility however, as seems to be the case for the superconductors, that
the carrier is a hole in the oxygen p-band (see discussion below), in which case this argument
would not be so strong. However, in either case the formation of a spin polaron seems
possible with relatively large effective mass, of order 10 me.
For low concentrations of Sr, the carrier will be trapped by the negative charge produced by
the substitution for a Sr2 + for a La3 + ion.
As the concentration increases, the trapping will be of Anderson type ; EF will increase,
Ec - EF diminish and a conventional Anderson transition occurs.
We have to ask, however, whether the transition takes place in an impurity band, or
whether the impurity and conduction band have merged. We think it unlikely that there is any
self-compensation here, as there appears to be in many-valley conduction bands. The
transition, if it took place in an impurity band, would therefore be of Mott type, taking place
when U - B. If U is too large to allow this, B for an impurity band being small, it cannot occur
until the bands have merged. It will then occur in the merged band when EF
Ec. As the
number of carriers increases, EF moves away from the band edge as does the mobility edge
Ec, which results from increasing disorder. It is supposed that EF catches up with
Ec. If spin polarons are formed, EF and Ec refer to the degenerate gas of spin polarons.
There is an interesting difference between Lal _ xSrxV03 and the superconductor
La2-,,Sr,,CU04. In the former [22], on adding Sr, the Néel temperature does not drop much
until the transition, and then antiferromagnetic order disappears. That Ec EF goes
continously to zero suggests an Anderson transition. The different behaviour of the Néel
temperatures is doubtless a consequence of the two dimensional nature of the latter material.
Also in the vanadate at the transition the thermopower changes sign and becomes negative,
suggesting a normal, not a p-type metal. In the insulating state the activation energy for
conduction varies as (xc - x )1.8, which is near the index expected for classical percolation
theory rather than the normal index v 1. Perhaps a rapidly changing effective mass due to
spin polarons could be responsible, if as we suppose these are formed. There are several
unexplained features here, perhaps also related to the merging of an impurity bad into a
conduction band ; the rate of delocalization could be increased as Meff decreases.
In La2 - xSrxCu04, on the other hand, TN drops rapidly with increasing x with the formation
of a spin glass. This seems to me by no means surprising ; the spins are under the influence of
e2/ K r2.
e2/Kr2,
=
=
2817
neighbours and the trapped carriers, and TN should drop. When it disappears a random
spin glass (localized random spins) remains (2) . But metallic behaviour must await the
merging of impurity with the valence bands for the spin polarons ; when this happens there
their
will be many sites for each carrier.
Many of the other high temperature
superconductors can, we suggest, be described in the
same way. Thus YBa2Cu3O7 - x should be insulating when x
while with x &#x3E; 1/2 holes are
2
2
introduced into the oxygen p band, so forming spin polarons with copper ions, and a metalinsulator transition can take place.
The interesting case of Bal - xKxBi03 apparently shows no magnetic order [28] ; it is a
superconductor when x &#x3E; 0.25. We suggest that the moments are Bi4 + (it is interesting that
they form Bi3 + and Bi5 + for x 0.25), and that for the insulating composition they form a
=1/2
RVB state.
Anderson’s proposal [29] that insulating RVB states exist is verified by materials such as
TiBr3 which are insulators but show no antiferromagnetic order but a small paramagnetism
nearly independent of temperature (Wilson et al. [30], Maule et al. [31]). Also calculations
such as those of Kohmoto and Friedel [32] and Gros [33] suggest that the RVB state can be
stable. In the metallic state, however, its significance is less clear (see § 5).
4.
Bipolarons.
As already stated, photoemission observations suggest that in La2 - xSrxCu04 the carriers are
holes in the oxygen valence band, or at any rate in a band with strong oxygen 2p content, as
shown by X-ray absorption [34, 35]. It seems to the author improbable that there will be
strong hybridisation for the states occupied by these holes. We consider hybridisation
between a 3d e. state in the Cu ions and the 2p oxygen, before the Hubbard
U is introduced for the former ; this should be as in figure 4 ; the dotted lines represent the
hybridised states and N(E) is finite at the Fermi energy. The Hubbard U will produce
antiferromagnetism and open up a gap in the Cu d-band, and the wave functions here may
have considerable p admixture. But the holes introduced by doping will, after U is introduced,
lie at P, where the 3d component is small. So in our view a relative position of the two bands,
which leads to strong hybrisation for the d functions, necessarily locates the holes in a part of
the p-band where hybridisation is small. These we suggest form spin polarons with the Cu 3d
moments, which combine into bipolarons which are the pairs needed for superconductivity.
Fig.
4.
copper
Showing hybridisation between the 2p oxygen band and that for motion of 3d8 through the
3d9 states.
-
(2) See Maekawa et al. [60] « Motion of holes in magnetic insulators » who maintain that each hole
disturb
spins, that t &#x3E; J and so as little as one per cent of holes can disorder the spins.
’TT’ 2 t / J
2818
We review first the evidence for the existence of dielectric bipolarons in certain other
materials. One asks first whether under any circumstance two such polarons can combine,
because at large distances they must repel each other. Of course in a sense Cooper pairs are
bipolarons, but the correlation length is so large that any repulsion is screened out. The
clearest evidence for dielectric bipolarons in the literature comes from the work of Lakkis
et al. [36] on Ti407 ; here every other Ti ion carries an electron, so every Ti3 + must have
neighbours with the same charge. Repulsion cannot prevent the formation of bipolarons
which give over a range of temperature an activated conductivity but no paramagnetism. The
binding energy which forms these bipolarons is probably of homopolar type, so in a sense it is
a spin-dependent type, caused by the exchange of antiparallel spins. Other evidence does
however suggest that isolated bipolarons can form ; if so, the interaction energy must be of
the form shown in figure 5, with the attractive force of homopolar type. In this case, as with
any alternative force localization will clearly be more probable in two dimensions than in
three.
Fig.
5.
-
Showing the potential energy between two polarons (dielectric or spin) if bipolarons can form.
We have however no direct evidence for the formation of spin bipolarons apart from the
existence of superconductivity, with in the planes a small coherence length (-20 Â), and
even smaller across in planes (3).
Su and Chen [6] consider the formation of bipolarons from carriers with parallel and
antiparallel spins, finding that both can be stable, particularly in two dimensional structures,
because localization is always easier in one or two dimensions. Essential to ensure the validity
of such a model, it has to be shown that the spin interaction can overcome the Coulomb
repulsion, as in figure 5. This must occur, whether or not our polaron model is a good one,
given that superconduction occurs with the observed small correlation length. The calculation
of Su and Chen, suggesting a cigar shaped form for the polaron in the CU02 planes, makes the
problem of the bipolaron almost one-dimensional and thus favours its stability even more
strongly than another form would do.
Islam et al. [39] have attempted to calculate whether a spin bipolaron can form ; though
they cannot obtain a positive binding energy, they consider it not impossible.
We next ask whether all the spin polarons, that it all the carriers, form bipolarons, or only a
fraction, that is those in the upper part of the Fermi distribution. Another question is,
(3) Worthington
et al.
[37], Gallagher [38].
2819
whether a bipolaron, moving through the 3d9 states, will suppress the antiferromagnetism,
since the influence on the spins round a bipolaron will be weaker than that round a polaron.
This point needs further investigation. Anderson’s model does involve, as well as bosons,
fermions though his, unlike ours, carry no charge. We think however it is more likely that the
transition temperature is that at which the Bose gas becomes non-degenerate, and if this is so,
according to Prelovsek et al. [64], the effective mass is in the range 20-40 me.
In the superconducting region the transition temperature Tc first rises with increasing
x and then drops. If the transition temperature is that at which the bipolaron dissociates, then
for concentrations near the Anderson transition the wave functions of any carrier according to
Mott [40] fluctuate strongly within a length e, which tends to infinity (4) as 1/(EF - E,,). It is
likely that such fluctuations impede pair formation, and also because below EF the wave
functions remain localized. For high values of x, doubtless the pairs can impede each other’s
formation.
5.
Comparison
with other models.
A degenerate gas of spin polarons is a theoretical model very similar to the highly correlated
electron gas introduced by Brinkman and Rice [10] ; in both models a small number of
carriers moves, causing the moments to resonate between their possible orientation. In both
there is a mass enhancement. The quantity X (q, w ), of importance for neutron diffraction,
should we believe vary with q only when - hw - EF, where EF is the width of the band of
occupied states reduced by this interaction.
The model would suggest a smaller correlation length than actually observed. A detailed
examination by Liang [41] in which the spin polaron is envisaged as causing local strains,
which in their turn are responsible for the scattering, suggests that coupling should be
between more distant sites. Liang describes his bipolarons as spin-phonon induced, and some
small isotope effect is to be expected. Here the spin produces a distortion, which is
responsible for part of the cohesion. Fujimari [62] gives a discussion of various ways in which
a spin polaron can give cohesion. Calculations are needed to give quantitative values. The
binding energy may be much higher than kTc, so that above Tc the carriers are still bipolarons.
There is much evidence, to be reviewed elsewhere [65], that above Tc the carriers are very
heavy, as spin bipolarons might be.
6. Some relevant
expérimental
evidence.
This paper has taken the concept of a doped « Mott » insulator, and asked whether it can
produce superconductivity, and found that Cooper pairs with high binding energy, of order
kTN, are possible. This final section summarises some of the relevant experimental evidence.
It seems certain that pairs (Cooper pairs, holons, bipolarons ?) exist with charge
2 e ; the experimental evidence is summarized in the review by Bednorz and Müller [42].
Experiments on neutron scattering and muon spin rotation is stated to show that magnetism
plays an essential role in the Cu02-based superconductors. Some of the evidence is given by
Birgeneau et al. [43], who state that for L2-xSrxCu04 the Cu 2+ moment is independent of
x but the Cu 2+ spin-spin correlation function exhibits a dramatic x-dependence, being of the
order of the distance between the holes. This is in accord with the ideas expressed in this
paper. Greene et al. [44] show that the susceptibility above 7c is enhanced by electron
correlations, as we expect. A small isotope effect is observed [45] which must in our view
(4)
such
A parameter which tends to infinity at a second-order transition appears to be
transition, as at critical or Curie points.
a
general feature of
2820
accompany the formation of a spin polaron sharply localized in space. In the non-metallic
state, in crystals of La2 - ySryCu1-xLi204 - 3 variable range hopping is observed [46] ; thus
N (EF) must be finite, with Anderson localization at EF. Endo et al. [47], in work on neutron
scattering, say that the large spin fluctuations give credence to the models in which the pairing
is magnetic in origin, but according to Birgeneau et al. [48] there is no significant difference
between the superconducting and normal states. Mawdsley et al. [49] consider that the
thermopower of YBa2CU307or- is much enhanced ; they suggest an enhancement of the
effective mass by phonons but in our view spins may be equally responsible, as in our model of
spin polarons.
Torrance et al. [50] have found that for La2-,,Sr,,CuO4 annealing in oxygen to remove
oxygen vacancies leads to a constant value of Tc(36K) for x between 0.15 and 0.25 ;
superconductivity disappears at x 0.34 though holes are formed till x 0.4. Without
annealing they claim that increasing x beyond 0.15 just adds oxygen vacancies rather than
holes.
The material [51] YBa2CU307-y seems to have no magnetic system, if the copper is
envisaged as a metallic system made up of two Cu2 + and one Cu3 + , as might be the case for
highly compressed Fe304(Wilson [52]). It seems to us, however, that the Cu3 + may be absent,
being replaced by holes in the oxygen valance band. If so, this material may be classed with
the others described here.
The newly discovered n-type superconductor [53, 54] La2 - xCexCu04 might fall into the
same pattern with of course no holes in the valence band and the stable [52] Cu ion
(3d )1° forming a spin polaron with the surrounding 3d9.
From the Zurich school, Takahashi and Zhang [55] show, in agreement with our postulate,
that a small concentration of holes in the Cu3d9 states on the quadratic lattice can destroy the
antiferromagnetic order. On the other hand Zhang and Rice [56] consider that a hole in the
oxygen p-band can combine with a Cu 3d9 ion to form a singlet, which is a kind of spin
polaron. We think that, whether a singlet or triplet is formed, it will disturb the spin of its
neighbours, forming a spin polaron and breaking down the AF order.
=
=
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