On the broadening of low dimension van Hove singularities in oxide superconductors

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On the broadening of low dimension van Hove singularities in oxide superconductors

J. Friedel

To cite this version:

J. Friedel. On the broadening of low dimension van Hove singularities in oxide superconductors.

Journal de Physique, 1988, 49 (8), pp.1435-1441. �10.1051/jphys:019880049080143500�. �jpa-00210823�

On the broadening of low dimension

^van

Hove singularities ^{in oxide} superconductors

J. Friedel

Physique ^des Solides, Université Paris Sud, Laboratoire Associe au CNRS, 91405 Orsay, ^France

Institute of Theoretical Physics, University ^of California, ^Santa Barbara, California, ^U.S.A.

(Reçu le 20 avril 1988, accepté ^{le 1er} juin 1988)

Résumé. ²⁰¹⁴ On remarque que les couplages transverses, et, accessoirement, la diffusion longitudinale ^{due aux} impuretés ^{et aux} phonons, élargissent suffisamment les singularités ^{de van Hove des} plans ^Cu O2 pour

empêcher leur dédoublement Jahn Teller spontané. ^Les changements ^de phase tétragonale-orthorhombique

observés dans La2-x Srx ^Cu

O4-y

^{et Y Ba2} Cu3 O7-03B4 ^{ont donc des} origines ^{et des natures} différentes. Dans de bons cristaux, ^cet élargissement n’est pas suffisant pour détruire les instabilités antiferromagnétique ^ou supraconductrice, contrairement à la diffusion par les lacunes le long des chaînes 1d Cu O dans les

YBa2Cu3O7-03B4 sousst0153chiométriques. ^Une analyse précédente ^{de la} supraconductivité ^à partir ^{de la} quasibasse dimensionalité et d’un couplage ^{faible est} corrigée pour tenir compte ^{de cet} élargissement.

Abstract. ²⁰¹⁴ It is pointed ^{out that the} transverse couplings ^and, accessorily, ^the longitudinal scattering by doping ^and phonons ^broaden enough ^{the 2d van Hove} singularities ^{of Cu} O2 planes ^to prevent ^their spontaneous Jahn Teller splitting. ^The tetragonal-orthorhombic phase changes observed in La2-x Srx ^Cu

O4-y

and Y Ba2 Cu3 O7-03B4 have then different origins ^{and nature. In} good crystals, ^this broadening ^{is not sufficient}

to destroy ^the antiferromagnetic ^or superconductive instabilities, contrary ^{to the} scattering by ^vacancies along

the 1d Cu O chains in underst0153chiometric Y Ba2 Cu3 O7-03B4. ^A previous analysis ^of superconductivity, starting

from quasilowdimensionality ^{and weak} coupling, is revised to take this broadening ^{into account.}

Classification

Physics ^Abstracts

71.00 ^- 74.00

Introduction.

A previous analysis ^{of the new oxide} superconduc-

tors

[1, 2]

stressed their

quasilowdimensionality :

their electrons are able to move much more rapidly along ^{the Cu}

02

planes ^{or Cu 0} chains than between these units, ^a point clearly proved

subsequently

by

the strong anisotropy ^{of their} transport

properties

and penetration depth

[3,

^4,

5].

It also stressed that the band structure compu- tations and X rays and optical ^{data were coherent}

with a weak coupling ^{limit for} superconductivity ^as

well as magnetism.

Using ^{these two} points, ^one necessarily ^{starts from}

an independent ^electron picture which, for isolated Cu

02

planes ^{as well as Cu 0} chains, possesses infinite van Hove

singularities

^{of the} density ^{of states}

[6].

^It is then natural to try ^and analyse in this way the interplay ^{between the various} instabilities ob- served : tetragonal-orthorhombic phase

change,

^anti- ferromagnetism ^and

superconductivity [7-10].

More

precisely,

ⁱⁿ undoped La2 ^Cu

04,

^{the Fermi}

level should just fall, ^{in the} tetragonal phase, ^{on the}

van Hove

singularity,

^{if a} tight binding description ^is

used for the

corresponding

^{Cu 3d and 0} 2p ^{states. It}

was then remarked that an orthorhombic distortion

e which would shorthen the Cu 0 Cu bonds

along

the a direction and lengthen ^them along ^{the b}

direction would lift the degenerary ^{of the van Hove} singularities ^at

this lowers the total electronic energy by ^{a term in} E2 In E which would necessarily dominate any elastic reaction in E2 of the lattice

[1].

The maximum of

Tc

^{for x -} 2 y = 0.10 in the orthorhombic phase ^was

then attributed to the Fermi level u reaching ^the position Ei ^{of the van Hove} singularity ^{with lower}

energy.

This interpretation ^{of the role of the} orthorhombic distortion relies however on a wrong interpretation

of the nature of the distortion, ^{and therefore cannot}

hold.

In fact, ^{as stressed} recently by several authors

[11,

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049080143500

1436

12,

13],

^the orthorhombic distortion is essentially

due to a rotation of the Cu

06

hexahedra, ^which

preserves the equality ^of length ^of the various Cu 0 Cu bonds. This rotation and a small

change

ⁱⁿ length of these bonds change slightly ^{the relative}

energies

^{of 0} 2p ^{and Cu 3d states} and the transfer

integrals td/3 between Cu and 0, ^{but in a} symmetrical

way which does ^not alter the degeneracy ^{of the van}

Hove singularity ^{of the} density ^{of states.}

One is then lead to think

[12, 13]

^{that the} tetragonal-orthorhombic phase

change

^of

La2 - x Srx

^Cu

04 - y

^{has no}

important bearing

^{on the}

electronic structure of Fermi electrons. This leads to two

questions.

Why ^{does the} predicted spontaneous Jahn Teller

splitting

^{of the van Hove}

singularity

^{not occur for}

La2

^Cu 04 ? ^{A reason}

might

be that the van Hove

singularity

^{is broadened in some} way by ^{factors not}

taken into account in the

preceding simplified

analysis

[13].

Why ^{then does not such a} broadening ^also prevent the other observed instabilities, ^i.e. antiferromagnet-

ism and

superconductibility ?

We examine these two points ^{in turn for}

La2-x Srx Cu °4-y,

^{and comment in fine on}

YBa2Cu30y_g.

1. Broadening ^{of van Hove} singularity prevents Jahn Teller splitting ⁱⁿ

La2 - x Sr x

^Cu 04 - y.

We want to stress that a modest broadening ^{of the 2d}

van Hove

singularity

^{of the Cu}

02

planes ^{is sufficient}

to prevent ^a spontaneous ^{Jahn Teller} splitting.

There are a number of reasons for such a broaden-

ing, ^if only ^the transverse coupling between Cu

02

planes.

We shall ^use here, ^{for an order of} magnitude estimate, ^{the most}

simplified

description

[1]

^{of the}

Cu 3d-0 2p antibonding band, ^where

Ed - Ep

^>

I tdp I

^{. But the same} conclusions are easily ^{seen to be}

valid for the more general ^case

[8, 9],

^where

Ed

>

Ep.

Thus, ^{for an} isolated and

perfect

^Cu

02 plane,

in the tetragonal phase, ^with

From

(1),

^{one deduces a}

density

^{of states which} reads, per Cu ^atom and per spin

if the origin ^of energies ^{is taken at mid} band, ^i.e.

Ed

^{+ 4 t = 0. In a small} distortion such that

the

logarithmic singularity

^of

(3)

splits ^{into two of}

half amplitude, ^{centered at} Ei ^and E2 ^{such that}

if

t (a)

^{oc e qo a.} The energy gained by ^{such a Jahn}

Teller distortion is then

where the total number of carriers per Cu ^atom is

given by

One deduces easily ^that

This becomes infinite for a half filled band

(N =1 )

where

’ The half filled band situation is, ^as expected, ^the

most unstable one. The Jahn Teller distortion should then always ^{win over} the elastic reaction of the

lattice, ^{which is} proportional ^to £2.

Let us now assume that the van Hove singularity ^is

broadened for some reason that we shall discuss below. Then

n (E)

^as given by

(3)

^{should be} replaced by

for the tetragonal phase, ^{and a similar} expression ^for

the Jahn Teller distorted phase. ^{A measures the} broadening.

For a half filled band, the energy gained by ^the

Jahn Teller distortion now reads

where E2 ^{= -} El ^{is still} given

(5).

^A simple compu- tation

gives

where

For I Ell

^{A, this gives}

As expected, ^the broadening ^A provides ^{a cut off}

in the effect of the singularity ^which

changes

^the

nature of the Jahn Teller

instability.

^The

driving

term

(13)

^{is now} only ⁱⁿ E2

Instability

^{should not}

occur if its

multiplicative

^{constant is smaller than}

that B of the elastic reaction :

Taking ^{as a} very rough ^estimate

this gives

A broadening ^{of this}

magnitude

^{is indeed} expected

in

La2-x Srx Cu °4-y

^{from the} transverse coupling

between planes

[1].

^As already stressed, ^{there are}

two possible regimes ^of coupling, either coherent or

incoherent, depending ^{on the mean free} path f along ^{the Cu} 02 planes. ^{But in both cases, an}

estimate is

As also stressed in

(1),

^the transport

properties

suggest

Condition

(15)

^{is then fulfilled.}

Another origin ^of broadening ^{is of course the}

mean free path ^{f in the Cu}

02

planes. ^This gives ^a supplementary contribution to A which should add to the first. For most of the Fermi surface, ^{one can}

write

But in the van Hove singularity, where the electron

velocity ^is especially low, ^{the real} broadening ^is

much less. Resistivity measurements show that

f can be large ^{in the metallic} range. In the best

samples, ^and

taking

^{into account the} anisotropy ^of resistivity, ^{one deduces}

where the upper ^{limit is} obtained at low temperature, where impurity scattering ^dominates.

In conclusion, phonon scattering probably ^des- troys sufficiently ^{the van Hove 2d} anomaly ^at

high

temperatures ^to completely prevent ^a spontaneous Jahn Teller distortion. At low temperatures, ^trans-

verse couplings

(and impurity

scattering ^{in the}

large

concentration

range)

contribute to a total broadening

A such that

which seems also sufficient to prevent ^{the Jahn} Teller distortions.

2. Broadening ^versus antiferromagnetism ^and super-

conductivity Laz-x Srx ^Cu

04 - y-

We now want to show that the broadening ^{due to}

transverse coupling ^and impurity scattering ^{is not} necessarily strong enough ^to kill these low tempera-

ture instabilities, when described in the same weak

coupling ^limit.

Taking again ^{the zero} of energy ^at Ed ⁺ 4 t,

antiferromagnetism

will be stable at 0 K if

where

for an isolated and perfect ^Cu

02

plane ^{in the} tetragonal phase

[1].

This is to be replaced, ^{for a broadened} density ^of

states, by

with A 0 given again by

(7).

An integration by parts ^shows that, ^{for a} nearly

half filled band where JLo -+ 0, ^condition

(21)

^reads

This is always fullfilled for uo --> 0 : the broadening

does not kill the antiferromagnetism ^{for half filled}

bands.

1438

If, ^on the other

hand, I ILo 1

> A, condition

(21)

reads

Now one knows that antiferromagnetism

disappears

at 0 K for x + 2 y ⁼ 2 %. From

(10),

^{one deduces in} general

From the previous discussion, it is reasonable to take A 4 x 10- 2 t ^at low temperatures. Conditions

(23)

and

(24)

^{then lead}

to | Uo| | = 4 x 410- 2 t,

^{which is}

indeed larger than A, and U - 2 eV. This is a very

acceptable ^{value in the weak} coupling ^limit

[1, 14].

The mean field temperature T2 ^{for the} supercon-

ductivity ^{of Cu} 02 planes ^is given, ^{in the same} way and within the simplest ^BCS expression, by

where V is the effective pair interaction. With n broadened into n * as given by

(10),

^the correspond- ing

T2*

^satisfies

where

A simple discussion can be obtained if, ^{as in}

[1],

one replaces ^{th x} by x

for I x I

^{1 and} by ^{± 1 for}

I x I :::. 1.

^The leading ^{terms is} then obtain for

I u I >

^{a *,} if uo is ^near enough ^{to the van Hove}

singularity.

Furthermore, ^as long ^as

the limits of integration ^± UD play ^{no role.}

The maximum of

T2

^is then obtained for zero

doping

(uo=0).

^If

it is given by

For large enough dopings ^{such that}

T2

decreases rather slowly ^as

To explain the observed variation of

7c

^with doping

[15],

^one is lead in this model to think that the antiferromagnetism, ^{which dominates at low} dopings, destroys superconductivity there. The maxi-

mum of Tc ^and therefore of

T2,

^{observed at}

x - 2 y ⁼ 0.10, would then be due to the disappear-

ance at these large dopings ^{of sizeable antifer-} romagnetic fluctuations. This seems coherent with the fact

[16]

that these fluctuations are already ^weak

and short range for x ⁼ 0.03.

For x - 2 y ⁼ 0.10, the observed

Tc

^{and the} likely

relation

tll It 1.

^{=102 lead}

^[1]

^to

Tz = 60 K.

^From

(24),

^we

deduce JLo/4 t = 0.13.

^Condition

(32)

^is

fullfilled if we assume as before A 4 x 10- 2 t.

Equation (33)

^is then satisfied if

V /t

=1.7. This is indeed in the weak coupling ^limit.

Finally ^{the weak} isotope effect observed

[17]

ⁱⁿ

Laz - x Sr x Cu 04 - y

^suggests that V is due to a

phonon ^mediated interaction in agreement ^{with the} negative ^role attributed to the

antiferromagnetic

fluctuations. The weak isotope ^effect is coherent with

(26) ;

^it suggests ^{that condition}

(30)

^{is not} sufficiently fullfilled for the large

dopings

^considered

for the terms in UD ^to be completely neglected. ^But

the fact that the isotope ^effect is small would show that

(33)

^{is not a bad} approximation.

A full discussion should then treat together ^the

various types ^of possible instabilities

[35].

3. Broadening effects in Y Ba2 CU3 07 - 8.

In the case of Y Ba2

Cu3 07 -

^{s, one} expects ^a very similar weak broadening ^{of the 2d} singularity ^{of the}

Cu 02 planes ; ^the strength ^of transverse coupling

and impurity scattering ^{are similar to the case of}

Laz-x Srx Cu °4-y.

The presence of 0 vacancies along ^{the Cu 0}

chains act, ^on the contrary, ^as strong ^{cuts. This} should produce ^a very sizeable broadening ^{even for} relatively ^{small 0} vacancies concentrations

[18].

Also the presence of ^a sufficient number of Cu 0 chains produces ^an orthorhombic assymetry ^{of the} Jahn Teller type ^{in the Cu}

02

planes.

One then expects and indeed observes two differ- ent ranges

[16] :

(1)

^{for 0.5} d 1, the Cu 0 chains have lost all

physical meaning. The oxygens ^on the planes ^{of the}

Cu 0 ^« chains ^» are few and more or less random. In

an ionic

description

that would neglect dp covalency,

isolated copper ions would be Cu+ , ^{those near to}

one 0 would be Cu+ + , ^and the few that are near to two oxygens, Cu+ + + . The corresponding dp ^states

of these very broken ^« chains ^» would be essentially

localized. They ^{would not take} part directly ⁱⁿ transport

properties.

The Cu

02

planes ^{are then} expected ^{to be with a}

half full antibonding dp band, ^{thus to} correspond ^to Cu+ + in the purely ^ionic

description.

^{The small} broadening of the Cu

02

planes ^{due to transverse} couplings ^to the Cu 0 chains and to other Cu

02

planes ^is

expected

^{to be} sufficient to prevent ^a spontaneous Jahn Teller effect of the Cu

02

planes.

Indeed the low temperature phase ^is

tetragonal.

Furthermore, because of the

halffilling,

^on expects

[1]

and indeed observes

[16]

all this range ^to be

antiferromagnetic, and therefore an insulator. There is therefore a very close analogy, for the Cu

02

planes of this range, with those of pure La2 ^Cu 04, except ^{for the} orthorhombic distortion of the latter

by rotation of the Cu

06

hexahedra, ^{which does not}

alter deeply ^{the band structure of the Cu}

02

^anti- bonding dp ^band;

(2)

^{for 0 6 0.5, some at least of the Cu 0}

chains have a meaning, especially ⁱⁿ phases ^where

the 0 vacancies are ordered on some chains or

completely

precipitate

^{on some} chains. These paral-

lel chains impose ^a Jahn Teller distortion, ^{so that the} orthorhombic phase is of the type discussed pre-

viously

[1].

The continuous Cu 0 chains have van Hove

singularities

^{at the} edges ^{of their} dp bands. Their

antibonding dp ^{band is}

expected

^{to be}

nearly

empty, with a small number ^e of electrons per Cu ^atom.

These electrons are taken from the Cu

02

planes,

with have then a

slightly

^{less than}

halffilled

^antibond- ing dp band ; ^{in ionic} terms, they ^{should have} nearly Cu+ + character, ^{as indeed observed}

[19].

^{This is}

what an electrostatic estimate

predicts (cf.

appen- dix

A).

Such conditions should be rather favourable for high

Tc’s [1, 2J,

^and

defavorable for anti ferromagnet-

ism. Thus the Fermi level could be ^at the same time

near to the lower

singularity

^of the orthorhombic Cu

02

planes ^{and near the}

optimum

^{situation for} superconductive couplings ^{in the Cu 0} chains. The transverse couplings between Cu

02

^{and Cu 0 in the}

Cu

02

^{Cu 0 Cu}

0 2

sandwiches should be weaker than those between Cu

02

planes ^{across the Y} planes ; ^{one then} expects ^{each Cu}

02

pair ^of planes

to have a large ^{effective mean field} superconductive temperature T2, leading ^{to a} large

Tc.

^{Also the} proximity of the Fermi level to the 2d and 1d

singularities is coherent with a weaker

isotope

^effect

[20],

^{even if} coupling through phonons prevails, ^as again ^seems likely.

The exact variation of Tc with 5 and heat treat- ments would

require

further studies. It seems at

present ^that chains and planes ^{each have their} weakly coupled ^{mean field} temperatures Tl ^and T2 ^as suggested by ^{some NQR} measurements

[21,

22, 23,

24].

It is also clear that, ⁱⁿ this range, the concentration of 0 vacancies and their state of

ordering ^or precipitation play ^some role. Low con-

centrations of disordered vacancies broaden the van

Hove singularities ^{of the} chains, and decrease

Tl ^and

T2,

^thus Tc. Larger concentrations of 0

vacancies, if ordered or precipitated, ^{decrease the}

proportion

^of superconductive chains, thus their contribution to Tl, ^without changing

T2 [27-35].

Conclusion.

The broadening ^{due to} transverse coupling ^and

imperfection

scattering ^is probably ^small enough ⁱⁿ La2 -x Srx ^Cu

°4-y

^to preserve the

antiferromagnet-

ism and superconductive instabilities while prevent- ing ^a spontaneous Jahn Teller distortion. A coherent

picture ^{of the} phase diagram ^{can therefore be}

obtained in the weak

coupling

limit, ^if the supercon- ductive coupling is assumed through phonons.

The same broadening ^would explain the presence of the tetragonal phase ^{in Y}

Ba2 Cu3 07 -

^{s with} 0.561, ^while allowing again the antifer-

romagnetic instability ^to arise. The Jahn Teller distortion imposed by ^the Cu 0 chains for 0 , 5 0.5, together ^{with a} small transfer of elec- trons from the halffilled

antibonding

dp ^{band of}

Cu

02

^{to the} empty antibonding dp band of Cu 0,

would be especially favourable for high

T,’s.

^The broadening ^{of the} singularity ^{of the Cu 0 chains due}

to 0 vacancies is on the other hand a strong effect,

which puts ^{the affected chains out of the} picture ^for already ^{modest 0} vacancy concentrations. The ^state of ordering ^or

precipitation

of 0 vacancies is there- fore of importance ⁱⁿ this range.

Acknowledgments.

The author wishes to thank S. Barisic, ^C. Noguera

and J. P. Pouget ^for pointing ^{out his error in} interpretation of the orthorhombic distortion of

La2 -.,, Srx

^Cu

04 _ y

^and for fruitful discussions. After the completion ^{of this work, the author received a}

paper by ^{J. Labbe} and R. Combescot which express-

es on a number of

points

very similar views.

This work was realised at the ITP of Santa Barbara and was

supported

ⁱⁿ part by ^{the National}

Science Foundation under grant ^n° PHY82 17853,

supplemented

by funds from the National Aeronau- tics and Space Administration.

Appendix ^{A. I}

ELECTRON TRANSFER FORM Cu

02

^{PLANES TO Cu 0}

CHAINS IN Y Ba2

CU3

07- ^{- In the} simplest tight binding

description

^{of the} compound, ^Cu

02

planes

1440

have

antibonding

dp bands with energy

while Cu 0 chains have antibonding dp ^{bands with}

energy

With tx - ty,

^{the two} bands should have nearly ^the

same center of gravity ^if

Eal = Ed2.

The presence of the other ions produces ^an electrostatic potential

such that Eal =

Ea2 if

^{the Cu}

02

^{have a} halffilled dp

band and if the Cu 0 have an empty

antibonding

dp

band.

This

description

^is however inconsistent : from

(A.1), (A.2),

the kinetic energy ^terms would favour

a transfer of electrons from planes ^{B to chains A}

such that the Fermi levels would become equal ^{in the}

two systems. ^{Thus the} change ⁱⁿ

Madelung

potential

energy

VA -

VB ^{due to} the transfer of electrons should essentially compensate the initial energy difference :

if e is the amount of electron transfer per Cu ^atom

on the chains. The corresponding ^{amount on the} plane will be - 1/2 E.

It is easily ^seen that, ^{if one treats} separately ^the

central atomic cell,

where d is the distance between Cu 0 chains and Cu

02

planes, a, ^{b, c the} elementary ^lattice periods along ^the tetragonal ^{axes and K the} periods ^{of the} reciprocal ^lattice.

Thus

A development ⁱⁿ K a, K b, Kc shows that, ^because Kc >> K a = K b, the first ^term dominates :

With

K, = 2,7r Ic, a =- 1/3 c = 3.85 A,

^d=4.3A

and U = 2 eV, ^this gives ^{in eV}

where the on the site contribution dominates.

Condition

(A.3)

^{with t = 0.8 eV} gives

A

previous

analysis

[1]

^{made for} La2 _ x

Srx

^Cu

04 _ y

^{can be applied to predict that the}

observed orthorhombic distortion

[27]

^for

Y Ba2

Cu3 07,

equal ^{to 1.65 %, would}

give

^{a lower}

2d singularity ^at Fermi level for E1= 2 ^x 0.165 ⁼ 0.33. In other words, the very rough ^estimate

(A.4)

would give ^a Fermi level not much above the lower 2d singularity ^{of the Cu} 02 planes.

For chains, ^a previous analysis

[1]

shows that the

optimum doping ^is given by :

with

Thus

If we take the same coupling ^{constant V for Cu 0 as}

for Cu

O2,

^we find Eo = 0.4, ^somewhat larger ^than

the estimate

(A.4).

There is however no a priori

reason for taking ^{the same} value of V. A more

elaborate study should also take into account the

broadening ^{of the}

density

^{of states} of the Cu 0 chains due to transverse coupling ^and

imperfection

scattering. ^This analysis merely shows that the

predicted

transfer e given by

(A.4)

^is roughly ^{of the} right ^{order of}

magnitude

^for having simultaneously large ^{values of}

Tl

^and T2.

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