Feodor V. Kusmartsev et al- Transformation of strings into an inhomogeneous phase of stripes and...

8/3/2019 Feodor V. Kusmartsev et al- Transformation of strings into an inhomogeneous phase of stripes and itinerant carriers

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2 October 2000

Ž .Physics Letters A 275 2000 118–123

www.elsevier.nlrlocaterpla

Transformation of strings into an inhomogeneous phase of stripesand itinerant carriers

Feodor V. Kusmartsev a,b, Daniele Di Castro c, Ginestra Bianconi d,Antonio Bianconi e

a Landau Institute, Moscow, Russia

bPhysics Department, Loughborough UniÕersity, Leicestershire LE 113 TU, UK

c Dipartimento di Fisica, UniÕersita di Roma La Sapienza, P. Aldo Moro 2, 00185 Roma, Italy`

d Department of Physics, Notre Dame UniÕersity, 46556 Indiana, USAe

Unita INFM, Dipartimento di Fisica, UniÕersita di Roma La Sapienza, P. Aldo Moro 2, 00185 Roma, Italy` `

Received 31 July 2000; accepted 7 August 2000

Communicated by V.M. Agranovich

Abstract

We discuss the transformation of a network of strings consisting of charges self-trapped by linear cooperative local lattice

distortions into an inhomogeneous phase of stripes and itinerant carriers in cuprate superconductors. This scenario is

observed by X-ray diffraction in oxygen doped La CuO where the doped charges at the critical doping 1r8 are self trapped2 4

into a crystal of ordered strings of finite length. Above this critical density in the superconducting phase the stripes co-existwith itinerant carriers. q 2000 Published by Elsevier Science B.V.

Doping drives a high correlated electronic system

toward a microscopic electronic phase separation as

it was first shown in doped magnetic semiconductorsw x1,2 . It has been recently shown that in the presence

of a strong electron lattice interaction the doped

charges have a tendency to create electronic stringsw x3–7 . Each string consists of M charged particles

that are self-trapped by local lattice deformation and

polarization in a linear array of N sites. The electron

correlation and in particular the antiferromagnetic

Ž . E-mail address: [email protected] F.V. Kusmartsev .

spin–spin interaction strongly enhance the tendency

to phase separation and to string formation. This

string is a generalization of the idea of the isolated

polaron to a system of many particles and high

density. In the range of physical parameters relevant

to the doped perovskites these strings have lowerw xenergy than isolated polarons 3– 5 .

The doped cuprate perovskites are strongly corre-

lated materials where the on site Coulomb repulsion

U ;6 eV forbids double hole occupancy on the

same Cu ions and holes are doped into oxygenw xorbitals 6,7 . At very low doping the doped holes

segregate into strings of charges that play the role of

domain walls between anti ferromagnetic domains

0375-9601r00r$ - see front matter q 2000 Published by Elsevier Science B.V.Ž .P I I : S 0 3 7 5 - 9 6 0 1 0 0 0 0 5 5 5 - 7



( )F.V. KusmartseÕ et al.r Physics Letters A 275 2000 118–123 119

w x8 . At low doping the strings form a glassy phase.

At a critical doping d s 1r8 the doped holes form

an insulating crystal of strings where the charges arew xtrapped into mesoscopic linear domains 9,10 by the

w xpseudo Jahn–Teller electron lattice coupling 11 . At

higher doping the charges are self-trapped into aw xsuperlattice of stripes of a distorted lattice 12,13

that co-exist with 2D free charge carriers that give aw xquasi 2D Fermi surface 3,14 . In this scenario the

superconducting phase occur in a superlattice of w xquantum stripes 9,10,15– 19 .

It has been recently shown that the Jahn–Teller

electron–phonon coupling drives the formation of

strings where the doped charges are trapped by thew xcooperative local lattice distortions 20,21 also in

manganites. In the very low density limit we can

consider a system of non interacting strings where in

each string the elastic deformation Q is proportional

to the number of trapped charges M . The elasticenergy of the lattice is proportional to Q2

; M 2. The

electron kinetic energy of the self trapped particles

and therefore the lattice adiabatic potential of the

string state decreases as ; E M 2 where E is thep p

energy for trapping a single charged particle. This

localization energy is opposed by the Coulomb re-

pulsion between particles trapped by the string po-

tential well which energy has an additional factor log

M , so that the Coulomb energy is approximately

equal ;VM 2 log M where V is a constant of the

inter-site inter-electron Coulomb repulsion. A bal-ance between these energies which is determined by

a minimization of the total energy gives a stationary

many particles self trapped string state where thew xstring length is equal to 3,21 :

E 1p N ; M ;exp y . 1Ž .ž /V 2

Let us estimate the length of these strings in the

perovskite materials. We take the elastic modulas of

the order of c s 11=1010 ergrcm3, the inter-11

˚atomic distance is a s 4 A, then we get K ;4.1 eV.

The deformation potential may be approximated as

D;e2ra s 3.4 eV, then for the electron–phonon

coupling we obtain E s D2 K s 2.5 eV. Taking ap

dielectric constant of ´ s 5 for a doped system, we

get V s e2 ´ as 0.68 eV. This gives estimation for

the length of the string of the order of 10 inter-atomic

distances.

Increasing the doping the system of strings under-

goes into a kind of nematic liquid phase consisting

of these oriented ordered cigar shaped electronic

molecular strings that can constitute the underdoped

phase of cuprate perovskites. Each of these molecules

is a charged object and therefore by further increas-

ing the doping the long range inter-string Coulomb

repulsion increases and it is expected to stabilize an

ordered phase made of a crystal of strings that isw xexpected near the doping d s 1r8 12,13 .0

We have studied here the formation of this type of

liquid crystal of strings in oxygen doped La CuO at2 4

hole doping d near 1r8. We have focused our

interest to the superconducting L CuO system2 4q y

w x22–41 . The single crystal of La CuO was grown2 4q y

first as La CuO single crystal by flux method and2 4

then doped by electrochemical oxidation up to reachand average oxygen concentration y s 0.1 deter-

mined by the increase of molecular weight. Thew xoxygen ions are mobile above 200 K 36 and the

oxygen distribution al low temperature is known to

be non-homogeneous. Our crystal shows a spinoidal

macroscopic decomposition into two domains withŽ .about equal probability: phase 1 made of macro-

scopic domains with static antiferromagnetic orderw x Ž .below 40 K 41 and a second phase 2 made of

superconducting domains with a single supercon-

ducting transition T s 40 K where the hole densityc

is the same as in optimum doping in Sr doped

La CuO superconductors with d s 0.16 holes per2 4

w xCu site 23–25,38 .

Neutron diffraction data of our crystal show the

presence of two different sites for interstitial oxy-

gens. The first one are interstitial sites O4 at theŽ .1r4,1r4,1r4 lattice position within the LaO layers

Ž . w xwith probability y s 0.064 4 22 . The second site1

for the interstitial doped oxygen is in the LaO planeŽ Ž . Ž . Ž ..at the site O3 0.031 3 ,0.135 3 ,0.176 1 near the

Ž Ž ..apical oxygen at O1 0,0,0.1822 1 forming about

0.08 pairs of apical oxygens.

Therefore the stoichiometry of our crystal givenw xby neutron diffraction is O4 La O1 O3 -0.064 2 1.85 0.232

CuO .2

The first set of interstitial oxygen sites O4 are

associated with the insulating antiferromagnetic do-

mains with about d ;0.125 holes per Cu sites i.e., at



( )F.V. KusmartseÕ et al.r Physics Letters A 275 2000 118–123120

the critical doping 1r8. The occupation of interstitial

sites O3 are associated with the formation of the

second metallic and superconducting domains with

hole doping d s 0.16 in the CuO plane. Therefore2

in the same crystal we can compare the structure of

insulating charge ordered phase and the supercon-

ducting phase at higher doping.

The charge ordering has been studied by tempera-

ture dependent diffraction data collected on the crys-

tallography beam-line at the synchrotron radiation

facility Elettra at Trieste. The X-ray beam emitted by

the wiggler source on the 2 GeV electron storageŽ .ring was monochromatized by a Si 111 double crys-

tal monochromator, and focused on the sample. The

temperature of the crystal was monitored with an

accuracy of "1 K. We have collected the data with˚a photon energy of 12.4 keV, wavelength l s 1 A,

using an imaging plate as a 2D detector. The sample

oscillation around the b-axis was in a range 0-f -308, where f is the angle between the direction of

the photon beam and the a-axis. We have investi-˚ y1gated a portion of the reciprocal space up to 0.6 A

momentum transfer i.e., recording the diffraction

spots up to the maximum indexes 3, 3, 19 in the a),

b), c) direction respectively. The orthorhombic lat-˚tice parameters of our crystal are a s 5.351 A, b s

˚ ˚5.418 A, c s 13.171 A. Thanks to synchrotron radia-

tion it has been possible to record a large number of

weak superstructure spots due to charge ordering

around the main peaks of the average structure. Theindexing of the superstructure has been conducted

taking into account the twinning of the crystal. The

oxygen ordering has been found to occur in the

temperature range 270–330 K. Charge ordering de-

velop below 190 K. At 100 K we have found the

competition and coexistence of two charge orderedŽ .modulation see Fig. 1 . The first one is character-

ized by 3D long range charge ordering and narrowŽ .resolution limited diffraction peaks with wavevec-

tor

q s 0.089 "0.0031 a) , 0 .244 "0.0024 b) ,Ž . Ž .1

0.495 "0.0046 c) .Ž .

This charge ordered phase shows the formation of a

crystal made of charge strings of finite length 11 a˚Ž . Žwhere a s 5.35 A , separated by R s 4 b where

Ž .Fig. 1. Scans along the Qs 0,k ,6q0.29 due to the diffuseŽ .scattering peaks of stage 3.5 superstructure 2 , squares, and along

Ž . Ž .the Qs 0,k ,6q0.5 squares due to the superstructure 1 .

˚ .b s 5.41 A and doubling of the unit cell along the

c-axis.The metallic superconducting phase shows a pat-

tern of diffuse spots due to a second superstructure,˚with a coherence length of about 350 A with

wavevector

q s 0.2080 "0.0016 b) , 0 .290 "0.0055 c) .Ž . Ž .2

This second wavevector q is associated to in plane2

superconducting stripe ordering. The length of the

stripes in the a direction becomes infinite or longer˚than 500 A. The transverse modulation of the super-

lattice of stripes in plane along the b-axis is the sameas the one found in superconducting Bi2212 and

characterized by diagonal stripes with wavevector of

0.208 b) indicating a separation between the stripes

of R;4.8 b in the b-axis direction. The period of of

the superlattice is 3.5 c in the c-axis direction.

The structure of the crystal of strings that we have

determined by X-ray diffraction in oxygen doped

L CuO at hole doping close to d s 1r8 is shown2 4q y

in Fig. 2.

At higher doping the crystal of strings shown in

Fig. 2 is unstable for the formation of an inhomoge-

neous phase where delocalized charge carriers co-ex-

ist with charges trapped into stripes as shown in Fig.

3. This phase is characterized by the diffuse short

range stripe fluctuations associated with the

wavevector q .2

Now we discuss the transformation of the crystal

of strings into an inhomogeneous phase where delo-




Fig. 2. Pictorial view of the striped charge ordered pattern at

doping d s1r8 in the CuO plane of oxygen doped La CuO2 2 4

forming a crystal of strings along the a-direction. The unit cell of

the crystal is orthorhombic with the a-axis shorter than the b-axis.

The strings of N ;11 sites are occupied by M ;10 charges run

along the a-axis. Each string is at distance R from its nearest

neighbor in the b-axis direction. The dashed line indicate thew xelastic antiferromagnetic scattering wavevector 41 .

calized charge carriers co-exist with charge trapped

into stripes at higher doping as shown in Fig. 3.

The total free energy of the crystal of stringsŽshown in Fig. 2 may be described as see, also for a

w x.comparison, in Ref. 1,2

F s F q F s F single inter single

1 n r y r n r X

y r Ž . Ž .Ž . Ž .X

q d r d r , 2Ž .H X2 ´ r y r

where F is the free energy per particle neededsingle

for a single string formation while the second term

F describes a contribution from inter-stringinter

Ž .Coulomb repulsion forces. The value n r describes

an inhomogeneous charge distribution associated with

the formation of liquid crystal of strings while the

value r is an average electron charge density associ-

ated with the doping.

We propose a very simple description of the free

energy per particle based on the assumption that the

electronic molecule is preserved at strong doping

while the molecule charge remains the same. At

doping n;1r8 such linear string molecules will be

ordered in the form presented in Fig. 2. Here dis-

tance between molecules R is determined from the

charge neutrality condition

Me y N r R q 1 e s 0, 3Ž . Ž .

whence from this equation we have that R s n r y 1

and we have used the definition that n s M N . The

inter-string Coulomb energy may be estimated withthe use of the electrostatics force law and it is equal

to

Q2

F s C , 4Ž .inter 1´ aR

where the string charge is equal to Q s Me y N r e

and the numerical factor C is related to a number of 1

strings which are interacting with each other within a

Debye screening radius. In other words, we have to

take into account an effective screening length j for

the inter-string Coulomb interaction. So, for exam-ple, for strings localized in a single CuO plane and

interacting via nearest neighbors, that is, with the

screening length j equal to an inter-string distance

R the value C s 1. When the next nearest neighbors1

are taken into account, i.e., when the screening

length j is equal to twice the inter-string distance,

2 R, the value C s 3r2. Analogously for the next1

Fig. 3. A crystal of charged strings of finite length is formed at

doping d ;1r8 in the phase diagram of the cuprate supercon-0

ductors while an inhomogeneous phase where a superlattice of

stripes coexists with the free carriers is formed by increasing the

density.



( )F.V. KusmartseÕ et al.r Physics Letters A 275 2000 118–123122

nearest neighbors taken into account C s 11r6, and1

so on. In general this numerical factor may be esti-

mated with the use of the simple equation:

j 1C s .Ý1

k k s1

Using this expression for the free energy per particleassociated with the Coulomb inter-string interaction,

Ž .Eq. 4 we get the equation for the total free energy

equal to

F r Me2r n y r Ž . Ž .s y E q C , 5Ž .p 12 M ´ an

Žwhere E is the energy shift per particle or ap

.condensation energy associated with the formation

of individual strings. With the doping this value

changes slightly and it is of the order of a single

particle polaron shift. Therefore for the next consid-Ž .eration we put it as equal to a single particle polaron

shift. We also count the energy from the bottom of

the conduction band that we put equal to E s 0.c

From the obtained expression we may immedi-

ately determine the critical doping at which 2D free

charge carriers may appear and where they can

co-exist with localized string charges. Since in this

case at the value r )r the value of the free energyc

Ž .becomes positive F r )0, the critical value of the

doping may be found as equal to

E pr s . 6Ž .cVNC 1

Here we assume that the value r <n. Using theŽ .same parameters as in Eq. 1 , E s 2.5 eV andp

V s 0.68 eV, the number N s 11 obtained in ourŽ .experiment and also with the use of Eq. 1 and the

numerical factor 2.5-C -3 we apply the formula1

Ž . Ž .6 . Then with the use this formula 6 we obtain the

value for the critical hole doping as equal close to

0.1–0.12, which is in a remarkable agreement with

the present experimental findings.

Thus at the doping r )r the second subsystemc

associated with free charge carriers coexists with

charges trapped into stripes. The concentration of

these free particles is determined via a balance con-

trolled by the chemical potential m. For free fermions

the chemical potential is equal to the Fermi energy

m s E . Let the concentration of the free particles inF

the CuO plane be n , then the Fermi energy is2 F

equal to m s"2 n 4p ma2. Due to the conserva-F F

tion of charge the value of doping r s r q n ,s F

where the value of r is the part of the doping whichs

contributes to the formation of stripes. Thus theŽbalance between these two subsystems correlated

.Fermi liquid and stripes may be presented in the

form

"2 n M r n y r Ž .F s s

m s m s s y E q VC .F p 12 24p ma n

Since n s r y r then with the use of this equationF s

we obtain finally the equation to determine the con-

centration of free particles n or r in the form:F s

"2 n VC M F 1

s y E q r y n n q n y r .Ž . Ž .p F F2 24p ma n

7Ž .

The solution of this equation gives

2n a n a a r n s r y y q q y y r ,(F cž /2 2 b 2 2 b b

8Ž .

where a s"2 2 mp a2 and b s Me2 C ´ an2. From1

this equation one may see that there is a second

critical value for the doping r at which this chemi-c2

cal equilibrium can not be maintained. This second

critical value is equal to

2n a br s q y r . 9Ž .c2 cž /2 2 b a

Probably that at the higher densities r )r thec2

stripes disappear and only a correlated Fermi liquidŽ .or, more precisely, a system of correlated fermions

arises.

In conclusion this work provides a direct measure

of the instability of a crystal of strings at doping 1r8

into a superlattice of superconducting stripes in the

high T superconducting phase. The physical mecha-cnism driving the transformation of a phase made of

strings of finite length into the metallic striped phase

that gives high T superconductivity has been de-c

scribed. We show the presence of a phase for the

co-existence of free carriers and a superlattice of

stripes and we provide a new scenario for the forma-

tion of the high T superconducting phase.c




Acknowledgements

Thanks are due the A. Valletta and P. Radaelli for

help in the early stage of this work, to C. Chaillout

for neutron diffraction sample characterization and to

M. Colapietro for experimental help at the Elettra

X-ray diffraction beam line. This research has been

supported by Istituto Nazionale di Fisica della Mate-Ž .ria INFM in the frame of the progetto PAIS

AStripesB, the Ministero dell’Universita e della`Ž .Ricerca Scientifica MURST Programmi di Ricerca

Scientifica di Rilevante Interesse Nazionale coordi-

nated by R. Ferro, e Progetto 5% SuperconduttvitaŽ .del Consiglio Nazionale delle Ricerche CNR .

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