induced superconductivity and gapless state in yba2cu3o7−χ

Physica C 198 (1992) 328-332 North-Holland I l l

Induced superconductivity and gapless state in YBa2fu307_x V l a d i m i r Z. Kres in a a nd Stuar t A. W o l f b

Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720, USA b NavalResearch Laboratory, Washington, DC20375-5000, USA

Received 24 april 1992 Revised manuscript received 7 June 1992

The removal of oxygen from Y B a 2 C u 3 0 7 _ x has a profound impact on various properties of this compound. The model of induced two gap superconductivity that we developed previously allows us to describe these properties. For example, for compo- sitions between 0 < x < 0.15, the energy gap on the chains changes drastically, and in fact, becomes gapless, whereas T¢ remains relatively constant.

1. Introduction

This paper is concerned with several peculiar properties o f the YBa2Cu307_x compound such as zero-bias anomalies, residual microwave losses, etc. In addition, it is known that the properties o f this material are very sensitive to the oxygen content. At the same time, one observes a "plateau" in T~ versus x for 0_<x_<0.2.

This paper is an extension of our theory for induced superconductivity for the chain band [ 1 ]. We will show that based on this model [ 1 ] and its fur- ther development described below, we can explain some peculiar properties of this material and suggest a number of interesting experiments.

2. Two gap induced superconductivity

In this section we review some of the major con- cepts of our model [ 1 ]. YBa2fu307_x is characterized by two conductive subsystems: ~ (Cu-O planes) and 1] ( C u - O chains). Charge transfer between these subsystems along with a short coherence length leads to the appearance of a two gap structure; this means that the superconducting electronic density of states has two distinctive peaks at E~ and ~a where E~ is close to the value of the intrinsic energy gap on the CuO2 planes. The conductive properties of the chains and,

consequently an unique opportunity to observe a two gap spectrum, was considered by us in ref. [2 ] and at present this picture has been confirmed by many experiments (see e.g. refs. [3] and [4]; for a more detailed analysis see ref. [ 1 ] ). The total hamiltonian has the form

I:I=H~ +H~ + ~ gzxqa ~+ a~bq X,tc,q

+ Y. -~-x7""~"" +,, ~ + c . c . _ ~ (1)

H~ and H~ are the terms describing the isolated ct and ~ subsystems; a, a ÷, b are the cartier and phonon amplitudes in the second quantization representa- tion, Z, x, q are the sets o f quantum numbers describing the electronic states for the a and l~ subsystems and the phonons correspondingly; "a gzxq is the e lectron-phonon matrix element for the ct-q3 transition, and Tx,~ is the tunneling matrix element.

The hamiltonian H~ contains the interaction that leads to the pairing. For example H~ contains the term

~ ~+ +a~,bq (2) gzz' az

if the intrinsic superconductivity for the et subsystem is due to the usual phonon superconductivity.

Therefore, the C u - O planes are intrinsically superconducting whereas the superconducting state in the chains is induced by charge transfer. It is ira-

0921-4534/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

V.Z. Kresin, S.A. Wolf~ Induced superconductivity and gapless state 329

portant to realize that there are two charge transfer channels ( 1 ) Intrinsic proximity effect which can be viewed as an ~t~fl tunneling of the Cooper pairs (fourth term ineq. (1) ) . (2) Phonon mediated charge transfer, which is similar to the two band channel [ 5 ], or inelastic tunneling [6] (third term in eq. (1) ) . The amplitude of this process contains the electron-phonon matrix element. More specifically one carrier from the ~t subsystem radiates a phonon and makes a transition to 1~. Another carrier absorbs this phonon and as a result they form a Cooper pair in the fl subsystem. If the et and 1~ subsystems are spatially separated, the transition is described by the same matrix element as inelastic tunneling (see e.g. ref. [6] ). However, because the pairing involves the exchange of virtual (not real) phonons, the process described by this term does not represent a real inelastic process (there is no radiation of real photons). As a result, all ini- tial and final states of the particles are on the Fermi surface so that energy is conserved in the whole process.

An analysis of the hamiltonian (1) based on the method of thermodynamic Green's functions (see e.g. ref. [7] ) allows us to evaluate Tc and the induced energy gap. Equations for the order parameters have the form at Tc

A'~(oo.)Z"(oo.)=nT~., ( 2 . D . . , - f i ) Io9.,-------~ A~(~O.,)

+ ~ )'~I~D-. ' AI3(oo.,_____.~) +F~I3AI3 ~ . , t ~ ., Io~., [ I ( ~ . 1

A t~(~on) Zt~(og~) = nT ~ 2 t ~ D ~ , - ~, Io~,, I

+ ,~ . a~(oJ.) ~ . (3)

Here o9~= (2n+ 1)nT, Dnn, =122 [ (o9~- oJ~,) 2 +I22] -~ is the phonon thermodynamic Green's function, t'2 is the characteristic phonon frequency, F ~ = [T~I2v~, F ~ = I T~12va are the parameters introduced by McMillan [8 ], so that F ~ / F ~ = v ~/ v ~, v ~ and v ~ are the densities of states; 2~, 2~, and 2~ the electron-phonon coupling constants, Z ~ and Z ~ are the renormalization functions. We will not

write out explicit expressions for Z ~ and Za (see ref. [1]) .

The explicit expressions for Tc and e~ depend on the strength of the coupling. We think that the superconducting state in YBa2Cu307 is characterized by strong coupling to soft optical phonon modes. Then we obtain:

rc ~ 0.1821~r212, (4)

2~ + 2 ~ y ~ , +2 ~,y~ (5) 2err= 1 +y,~(1 + y ~ ) _ l + 2 f i ,

and

2 t~ %,~ ~ ( l +2 , )A , (O)+ nTcy~,, (6)

where yl~,= F~" / n T~ The term proportional to y~, in the denominator

of eq. (5) corresponds to the pure proximity effect which depresses To. A very important feature is the appearance of a mixed channel that can be viewed as a two step process a--,13--,a where the first step is a phonon mediated a--,13 transition followed by a 13--,ct proximity tunneling transition. This process enhances Tc which is determined mainly by the in- plane coupling constant 2~. One can see from eq. (6) that both channels make positive contributions to the induced gap %, whereas they make opposite contributions to T¢. We estimate that appropriate parameters for Y - B a - C u - O have the following values (see discussion in ref. [1]) : 2 ~ 3 , 2~ _-__ 0. 5, y~o_-__0.35, 12~280 K, p ~ - 0 . 1 , since v~/v,~-l.7 [9], then 2 , ~ 0 . 9 and y , ~ 0 . 6 . Using eq. (5) we obtain Tc -86 K.

3. Oxygen depletion; gapless state

The picture described above corresponds to a fully oxygenated sample of YBa2Cu307_x, i.e., x = 0, with well developed highly ordered chains. Oxygen depletion leads to a number of drastic changes. As is well known, oxygen is removed almost exclusively from the chains. As a result, instead of a well developed chain structure we have a set of broken chains with Cu-atoms at the end. These Cu atoms form local magnetic states, similar to surface states. These magnetic moments act as a strong pairbreaker

330 KZ. Kresin, S.A. Wolf / Induced superconductivity and gapless state

in the chain band. This leads to the formation of a gapless state [ 1 0 ] on the chains. It remarkable however, that unlike the usual effect of magnetic impurities where Tc is also depressed, in the present case the appearance of an induced gapless state in the chains does not affect in a significant way the value of Tc.

Let us consider this question in more detail. We are dealing with two spatially separated subsystems (planes and chains) with two order parameters A"(03), A~(03). One of these subsystems ([3) which is in the induced superconducting state is affected by magnetic impurities. Let us evaluate the value of the critical concentration of these impurities which leads to a gapless [3-state. The presence of some critical concentration of magnetic impurities leads to the appearance of an imaginary part of the order parameter A~(03) at 03=0; as a result the density of states v~, proportional to Re{03103 2 - (A ~(03))2]- 1/2}, becomes finite at 03 = 0.

The equation for the order parameter A ~ (03) has the form

A 0(09) = Z ~ -~ f d03'A~,D,o,~o, F + (03')

+ F ~ " Z ~ - ' F + ( 0 3 ) . (7)

Here

£2 Dw, w, = -~ [ ( 0 3 - 0 3 ' - Q + i ~ ) - I

-- (03-- 0 3 ' + Q - - i ~ ) - l ]

and F + describes the Cooper pairing. One can write a similar equation for A"(03). Our treatment is similar to ref. [11 ] for usual proximity systems. The presence of magnetic impurities leads to additional terms in the renormalization function Z ~ and the self- energy part Z~=A ~Z ~. After averaging over impur- ity positions [10], we have Z~=F~)[A2~(03)+ 03 2 ] - ~/2, where F ~ ) ~ n~ ; n ~ is the concentration of the magnetic impurities. With the use of eq. (7) and a similar equation for A~(03), one can write the equations for the order parameters at 03=0. The critical values, F~m, corresponds to the appearance of an imaginary part of A~(03) and is equal to:

r~m ~ ~ ~ ~ ( 1 +&)co) + F ~ (8) ).a

Here ~ ~ is the induced energy gap in the absence of any impurities.

4. Discussion

As was noted above, oxygen depletion leads to local magnetic moments appearing on the Cu-O chains in the YBa2Cu307_x compound. Let us estimate the minimum value of x which corresponds to gapless superconductivity on the chains. First of all, the critical concentration can be estimated from the relation.

l c~ .v~r c ~ h v ~ ( F c ) - l ~ . (9)

Here ~t~ =bye~n% is the coherence length for the chains. We can estimate its value as follows. The Fermi velocity v~ is directly related to the density of states at the Fermi level v~, for a quasi-lD band, namely v~= (xhvvaoL)- ~, where ao is the interchain distance and L is the interlayer distance. According to the band structure calculation of ref. [9] v~= rv,, r ~ l . 7 , where v,=rn*(nh2L) -~ is the density of states for the planes. Thus ~ ~- h 2 ( xraom. ¢ o ) - ~. Us- ing the values r~- 1.7, a °~3 .5 A, rn*~5me [2] and %=k~Tc (see ref. [1 ] ) we obtain ~ 2 5 A. There- fore, the coherence length on the chain is larger than that for the planes, but still is small relative to con- ventional materials.

It is now easy to estimate the value of x which corresponds to the appearance of gapless superconductivity in the chains. Since l c ~-~ (see above), and ~ 2 5 A, it is clear that the induced gap % disappears for x~0.15. Therefore for YBa2Cu3Oy with y<0.85, we are dealing with gapless superconductivity on the chain.

The formation of the gapless state on the chain is very important for the spectroscopy of the material. Indeed, the surface resistance, for example, is mainly determined by the smaller gap in the low temperature region. A strong correlation between the oxygen content and the value of the smaller gap, ~, leads to the strong effect of oxygen ordering on spectroscopy. As a result one can observe significant residual losses for a sample with x>0.15. The quantum hw<2 %, where ~ is the gap for the highly ordered sample ( x ~ 0 ) , can be absorbed for x>0.15 where the smaller gap % disappears. On the other hand, the nearly stoichiometric oxygen content in the high

V.Z. Kresin, S.A. Wolf~ Induced superconductivity and gapless state 331

quality samples leads to a significant reduction in the residual losses. Such an effect indeed has been observed recently in ref. [ 12 ].

The appearance of a zero-bias anomaly [ 13 ] is also a manifestation of the gapless state of the chains. An- other consequence of this gapless state is a linear term in the heat capacity that has been observed below Tc near T = 0 K [14].

Of course, the appearance of the gapless state of the chains does not mean that the compound contains superconducting and normal components. The system still contains two superconducting subsystems and both order parameters are not equal to zero, but the 13 subsystem (chains) is in a peculiar gapless superconducting state. Nevertheless, such terminol- ogy as "superconducting" and "normal" components can be used in a qualitative sense because spec- troscopic measurements do not find an energy gap.

Thus the oxygen depletion leads to the disappearance of the induced energy gap E~. However, it does not lead to a noticeable change in the critical temperature. This is a very important property of the model that makes the system with induced superconductivity entirely different form the usual gapless superconductivity [ 10 ]. Qualitatively, this feature is due to the fact that Tc of the system is determined, mainly by the state of the intrinsically superconducting ct subsystem. The validity of this conclusion can be seen directly from eqs. (4,5). Indeed, the effect of magnetic impurities manifests itself as an additional term in the renormalization function Z ~ describing the scattering. In other words, one should make the replacement: 7 ~ Y~ + Y~, where YM = FM/ nTc. An increase in nM leads to an increase in FM, and y c =ixM/nTc. One can see that the additional term yc results in a very small change in 2~ff and, as a result, in the shift ATe << T~.

Indeed if we add the value Y4 _-__0.35 to 2~, we obtain from eq. (5), A21~r 2 -~0.1. This corresponds to a shift in T~; ATc /T~O.06 . Therefore, the oxygen depletion which corresponds to x - 0 . 1 5 leads to a drastic effect on the spectrum of the material, namely to the disappearance of the smaller gap ca, whereas the shift in Tc is small and is of the order of several percent.

Experimentally, one observes a plateau in Tc versus x for 0.2 > x > 0 [ 15,16 ]. From the microscopic point of view, the critical temperature is determined

by three parameters (2~, 2~, y~ - see above). The value of 2.~ is determined by the doping level in the Cu-O planes which is relatively constant in this region [ 15 ]: ~n the other hand, contributions from the other two parameters compensate each other [ l ]. The induced gap is drastically affected by the oxygen depletion, becoming even gapless in this region, but Tc changes very slowly.

It would be interesting to carry out an experimental study of the effect of oxygen content on various properties, especially magnetic susceptibility. Some indication of such effects have been observed (see e.g. ref. [ 17 ] ), but the problem deserves a more detailed study. In addition, the residual losses can be reduced if one can prepare films with a special structure, so ~hat the chains are perpendicular to the elec- tric field vector. The exclusion of the chains would be favorable for a significant decrease in the losses at low temperatures.

Our approach based on ref. [ 1 ] leads to a strong dependence of the spectroscopy of YBaECU307_x on the oxygen content. For x g 0.15 the chains become gapless. This gapless state is responsible for a number of peculiar properties such as large residual losses, zero-bias anomalies and a linear term in the low temperature specific heat. These properties are a clear manifestation of two gap induced superconductivity in the cuprates. However, one should note that these properties are similar in the gapless and the normal states. Nevertheless, it is important to stress that the chains for oxygen content less than 6.85 are in the gapless, not in the normal state. This means that the order parameter describing the pairing is not zero as it is in the normal state. This leads for example, to a finite dissipationless current (see e.g. ref. [ 18 ] ). From the experimental point of view, the difference between the gapless and the normal state can be observed by measuring the electronic density of states by tunneling spectroscopy. The gapless state is still characterized by a peak in the region of voltage ap- proximately equal to e~. This peak would not exist for normal chains. The gapless state is characterized by a tail which continues to zero voltage (zero bias anomaly).

332 V.Z. Kresin, S.A. Wolf~ Induced superconductivity and gapless state

Acknowledgements

The au thors are grateful to B. R a v e a u and D.

K h o m s k i i for f rui t ful discussions. T h e research o f

V Z K was suppor t ed by the U S off ice o f N a v a l Re-

search unde r C o n t r a c t N00014-91 -F0006 . SAW

thanks O N R and D A R P A for suppor t .

References

[ 1 ] V. Kresin, S. Wolf and G. Deutcher, Physica C 191 (1992) 9.

[2] V. Kresin, Solid State Commun. 63 (1987) 725; V. Kresin and S. Wolf, Solid State Commun. 63 (1987) 1141; ibid., in: Novel Superconductivity, eds. S. Wolf and V. Kresin (Plenum, NY, 1987) p. 287; ibid., Phys. Rev. B 41 (1990) 4278.

[3] T. Friedman et al., Phys. Rev. B 42 (1990) 6217; Schlesinger et al., Phys. Rev. Lett. 65 (1990) 801; M. Maghighi et al., Phys. Rev. Lett. 67 ( 1991 ) 382.

[4] S. Barrett et al., Phys. Rev. B 41 (1990) 6283; H. Piel et al., Physica C 153-155 (1988) 1604.

[5] M. Suhl et al., Phys. Rev. Lett. 3 (1959) 552; B. Geilikman et al., Sov. Phys. Solid State 9 (1966) 542; V. Kresin and S. Wolf, Physica C 169 (1990) 476.

[6] M. Belogolovskii et al., Sov. Phys. JETP Lett. 21 (1975) 332.

[ 7 ] A. Abrikosov, L. Gorkov and I. Dzyaloshinskii, Methods of Quantum Field Theory in Statistical Physics (Prentice-Hall, New Jersey, 1963 ).

[8] W. McMillan, Phys. Rev. 175 (1968) 537. [9] W. Pickett, Rev. Mod. Phys. B 44 ( 1991 ) 9164.

[ 10] A. Abrikosov and L. Gorkov, Soy. Phys., JETP 12 ( 1961 ) 1243.

[ 11 ] A. Kaiser and M. Zuckermann, Phys. Rev. B 1 (1970) 229. [ 12] N. Hein et al., High-To Superconductor Thin Films, ed. L.

Correra (Elsevier, Amsterdam, 1992) p. 94. [ 13] M. Gurvitch et al., Phys. Rev. Lett. 63 (1989) 1008. [ 14] N. Philips et al., Phys. Rev. Lett. 65 (1990) 357. [ 15 ] R. Cava et al., Phys. Rev. B 36 ( 1987 ) 5719. [ 16] J. Jorgenson et al., Phys. Rev. B 41 (1989) 1863;

H. Paulsen et al., Phys. Rev. Lett 66 ( 1991 ) 465. [ 17 ] T. Kawagoe et al., J. Phys. Soc. Jpn. 57 (1988) 2272. [ 18] P. DeGennes, Superconductivity of Metals and Alloys

(Benjamin, lqY, 1966).

induced superconductivity and gapless state in yba2cu3o7−χ

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