Back to the A15 instability problems

A.F. Khoder, J. Labbe, M. Couach, J.P. Senateur

To cite this version:

A.F. Khoder, J. Labbe, M. Couach, J.P. Senateur. Back to the A15 instability problems. Jour-nal de Physique, 1986, 47 (7), pp.1233-1238. <10.1051/jphys:019860047070123300>. <jpa-00210311>

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Submitted on 1 Jan 1986

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Back to the A15 instability problems

A. F. Khoder (* +), J. Labbé (**), M. Couach (*) and J. P. Sénateur (***)

(*) Service des Basses Températures, CEN Grenoble, 85 X, 38041 Grenoble Cedex, France(**) Ecole Normale Supérieure, GPS, 24, rue Lhomond, 75231 Paris Cedex 05, France(***) ENSIEG-ER 155, B.P. 46, 38042 St-Martin-d’Hères, France

(Reçu le 30 septembre 1985, accepté sous forme définitive le 13 mars 1986)

Résumé. 2014 De récents résultats expérimentaux sur les composés A15 permettent une meilleure compréhensionde la relation entre la supraconductivité et l’instabilité de réseau dans ce système. L’effet de la déformation struc-turale est de réduire la température de transition supraconductrice Tc. Une augmentation de Tc, dans le cas deV3 Si, est obtenue lorsque la température de la transition structurale est abaissée (par effet de pression ou de défauts).11 est possible, d’un point de vue théorique, de comprendre et de prévoir ces effets dans le cadre des modèles Jahn-Teller de bandes, ce qui donne des arguments supplémentaires en faveur de ce mécanisme.

Abstract 2014 Recent experimental results obtained on A15 compounds allow a better understanding of the cor-relation between superconductivity and lattice instability in this system. The effect of the lattice distortion is shownto reduce the superconducting transition Tc. An enhancement of Tc in the case of V3Si is obtained when the tem-perature of the martensitic transition is lowered (by pressure or defects). Theoretically, these effects are well explain-ed and predictable within the frame work of the band Jahn-Teller « models »; then also give more arguments infavour of these models.

J. Physique 47 (1986) 1233-1238 JUILLET 1986,

Classification

Physics Abstracts74.10 - 74.70 - 63.20D

1. Introduction

The anomalous physical properties of the A15 com-pounds have been known for a long time to be cor-related with the high superconducting transition

temperatures found in this class of compounds.Lattice instabilities, such as phonon softening andstructural transition, strong temperature dependenceof the susceptibility and the Knight shift are exhibitedby the compounds V3Si and Nb3Sn which are themost studied compounds of the A15 family [1].Moreover, the resistivity « saturation &#x3E;&#x3E; with increasingtemperature exhibited by the A15 [2] seems to be ageneral behaviour encountered in most of the d andf-band compounds.

Early in the sixties and seventies, models with adegenerate peak in the density of states at the Fermilevel were proposed [3-5] and succeeded in accountingfor the anomalous physical properties.The origin of the degenerate peak of the density of

states is admitted$ be due to the particular dispo-sition of transition ‘atoms (V or Nb) along threeorthogonal sets of linear chains in the A15 structure,

(+) Permanent address : Universite Libanaise, Facultedes Sciences, Tripoli, Liban.

whatever the coupling between them. As far as thelattice instabilities are concerned, the band Jahn-Teller mechanism is the underlying one in the differentproposed models which, in fact, should be consideredas different versions (different N(E) shapes) of theband Jahn-Teller model as the latter, to be applicable,needs exactly what these versions share in common :

a. degeneracy at the Fermi level EF,b. high density of states N(EF).The Peierls instabilities model of Gor’kov [5] has

also degeneracy (electron-hole) at EF and singulardensity of states and, analytically, cannot be differen-tiated from the other models as has been shown indetail [6]. Although these different versions (or models)are not based on strong microscopic arguments(the anisotropy is too weak to justify a one-dimensionalapproximation of the band structure), the locationof EF close to a singularity of the density of states isshown to be a fact experimentally well establishedActually, the temperature of the structural transitionTm and the sign of the tetragonal distortion s areextremely sensitive to a small variation of the numberof electrons in the band whatever the way of obtainingsuch a variation (pressure, irradiation with low level ofdefects, chemical substitution, etc.).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019860047070123300

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The clearest evidence of this sensitivity is given in arecently published paper [7] on the effect of experi-mental substitution of Sn by Sb in Nb3Sn. A lowtemperature phase diagram of Nb3Sn1 _xSbx wasobtained by means of X-rays and it was possible todeduce the main characteristics of this experimentalresults theoretically by using the Labbe-Friedel versionof the Jahn-Teller band model [8].

In this paper we will give new arguments supportingthe Jahn-Teller band model by studying the connectionbetween superconductivity and tetragonality and

introducing the modifications of the band structuredue to the lattice distortion 8 in the calculation of T cof the tetragonal phase.

Before going on, we recall the limitations of theJahn-Teller band model in the case of A 15.

First, this model does not prejudge of the strengthof the electron-phonon interaction which, if strongenough, can provide a localisation of some phononmodes. This was recently pointed out by Yu andAnderson [9] and seems to explain the behaviour ofresistivity and susceptibility at high temperature.

Second, the fine structure of N(E) is washed outby a strong or even moderate level of disorder. Then,the model is no longer adequate to study the effectof strong disorder on the physical properties of the Al 5and especially the superconductivity which, for disor-dered systems, is mainly limited by the increase ofthe Coulomb disorder renormalized p* parameter,as stated and pointed out by Anderson et al. [10]for strong disorder. But, for weak disorder, as we willshow further, the behaviour of Tc exhibits some

particularities which are well accounted for by theJahn-Teller band model.

2. Experimental evidence of the T,,-structural trans-formation relationship.

Experimental investigations of the effect of the latticetransformation on Tc for A15 were mainly motivatedby the theoretical ideas concerning the favourablerole of soft phonons in high Tc superconductivity [11].Such ideas cannot be easily confirmed from an

experimental point of view. In fact, V3Si and Nb3Sn,and other A15 compounds which show lattice insta-bilities, undergo a structural transition at a tempera-ture Tm higher than Tc (Tm - 21 K for V3Si andTm ~ 45 K for Nb3Sn) and thus the phonon softeningis more or less reduced. The high temperature softmodes stiffen in the low temperature phase before theoccurrence of the superconducting transition

(Tc -- 17 K for V3Si and Tc - 18 K for Nb3Sn).A confusing situation arises from the extreme sensi-tivity of the lattice transformation to defects. It

appeared earlier that the occurrence of the latticetransition is correlated with the defect concentrationin the sample. In the case of V3Si, only the sampleswith a residual resistivity ratio (RRR) higher than20 undergo this transformation and it is usual, in theAl 5’s literature, to classify the samples in transforming,

non-transforming and even potentially transformingsamples.The superconducting transition temperature Tc

of some non-transforming V3 Si samples has beenshown to be higher than the Tc’s of the transformingones [12]. If this fact gives a strong indication of theincompatibility between the lattice and the super-conducting transitions, it also points out the needfor more experimental investigations to clarify theway by which a good transforming sample can moveto the non-transforming state. Moreover, the problemof the inhomogeneity of samples which obscures thedefinition of a precise critical temperature and theextent of the structural transformation has to be

carefully considered.The literature exhibits an important tc spread

which cannot always be explained by the samplequality (e.g. homogeneity) or measurement techniques.

It is a fact that inhomogeneity can often arise insingle crystals which behave as an assembly of macro-scopic subgrains with small differences in the physicalproperties. Also in order to investigate the particularproperties of each subgrain and to achieve a T,spectroscopy of the studied crystal we carried out ACsusceptibility measurements which allow the differentTc regions of the sample to be seen by recording theimaginary part (x") of the susceptibility as a functionof temperature in the vicinity of the superconductingtransition [13].The x" peak is associated with the occurrence of

bulk superconductivity at scales larger than the

penetration depth A [14]. The temperature corres-ponding to the x" peak is taken as Tc ( 1 ).By means of this technique, many V3 Si samples,

single crystals as well polycrystalline samples, werestudied.

In the case of V3Si, a complicated structure of x"peaks on a narrow temperature scale was found

especially with single crystals (Fig. 1). The mosaicnature of such single crystals can be studied care-fully by y-ray diffraction [6,15] which clearly showsthe existence of many diffraction peaks, each cor-responding to a subgrain (Fig. 2). Their number canbe correlated to the number of x" peaks.By applying a d.c. magnetic field, we could associate

a critical field curve to each x" peak, thus charac-terizing the different grains of the samples. This

analysis, in the case of V3Si, is essential to clarifythe origin of misunderstood anomalies at Tc observedby other measurement techniques such as the specificheat double-superconducting transition in V3 Si report-ed by Dayan et al. [16].

Figure 3 gives the results obtained for many samplesof V3Si, by plotting Tc as a function of the critical field

(1) We may suggest the use of the x" peak as an unambi-guous physical criterion of T c determination to be adoptedby physicists.

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Fig. 1. - Result of an AC susceptibility measurement(frequency = 1 033 Hz, excitation field = 1 Oe, staticfield = 0) on a single crystal of V3Si(SI Cl) which shows atleast four x" peaks. Peaks 1 and 2 correspond to the tetra-gonal phase, peaks 3 and 4 to the cubic phase, as shown bycritical field slope measurements.

slope - . This diagram clearly shows thedT Tc

g y

effect of tetragonality on T c.

The plot of Tc versus - dT2 displays a maximumdT Tcwhich separates transforming (left side of the curve)from non-transforming V3Si samples (right side of thecurve). The main effect of the tetragonal distortion is areduction of Tc and we estimate from figure 3 areduction (å T c) of 0.8 K for the purest V3Si sample.

A Tc depends strongly on the value of the distortionBc at Tc. This strong dependence åTc(8c) is displayed

by the increase of Tc with the increase of - d:i21 TcdT Tcwithin the tetragonal domain. In fact, the increase ofthe critical field slope is mainly due to an increase ofthe defect concentration which reduces Tm and thusthe value Bc of the tetragonal distortion at T c’We should mention that this first increase of T c

with the defect concentration has been noticed bymany authors studying the effect of irradiations onV3Si [17]. A Similar effect in Nb3Sn and V3Ga hasalso been pointed out by Kamezos and Weinstock

Fig. 2. - y-ray diffraction rocking-curve at room tempera-ture which reveals the mosaic nature of the same singlecrystal of V 3Si(SsC1) as shown in figure 1. Each peakcorresponds to a subgrain. Only the narrowest peaks showa broadening for T 20 K indicating the occurrence of thetetragonality in the corresponding subgrains.

Fig. 3. - The relationship between Tr and - dH,, TcdT Tc

resulting from the study of different V3Si samples, singlecrystals as well as polycrystals. Note that the same singlecrystal can contribute to both the tetragonal region and thecubic region.

[18]. We studied this effect extensively [6] and similarresults have been obtained recently by Khlopkin [19]although the height of the maximum of T, is less

important than that obtained in figure 2. An interestingresult of Khlopkin [19] is the observation of thestructural transformation at Tm below T c when thesuperconducting transition is inhibited by a strongmagnetic field

3. The interaction between superconductivity and latticetransformation analysed by the Jahn-Teller band modeL

The removal of the degeneracy of N(EF) by thelattice distortion (Jahn-Teller band mechanism) andthe new symmetry of the low temperature phase causemodifications of the Fermi surface shape.

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The impact of the band structure modifications onT, will depend on the strength of these modificationsi.e. of the amplitude of the lattice distortion I 8c I at T C’Since the lattice distortion (E) increases as the tempe-rature decreases, the value of I 8c will depend on theinterval Tm - T c’ Changing this interval (by changingTm) will change Ec , and thus the effect of tetragonalityon Tr.

Obviously, if the interval Tm - T c is larger thanTm/2, as in the case of Nb3Sn, 8c is quite temperatureindependent and we can anticipate that a few Kelvinchange of Tm will have no effect on T C’

Let us consider the Labb6-Friedel version of theJahn-Teller band model of the A 15 [4], where T, ismainly related to, Q, the number of carriers in the peakof N(E). For the cubic phase (c = 0) we have the twocoupled equations [20] :

where B is a constant related to the shape of N(E) andV is the electron-phonon interaction constant.L and K are given by :

with

Çm being the interval between the Fermi level and thesingularity (the bottom of the band). The relationTc(Q) is obtained by eliminating p.The corresponding set of equations for T’ c of the

distorted phase (s 0 0) can be easily obtained [6] :

where

and for

and

and for

L1 and K, have the same expressions as L and K.L2 and K2 are given by the same integrals with preplaced by :

where a is the distance between transition atoms on the

chain, q the Slater coefficient which governs the expo-

nential decrease of the d-orbital function and W is aband scale parameter.

In the derivation of the last equations for Tf and Q,we kept the same V as in the cubic phase. This assump-tion is justified by Anderson’s Theorem [21] for lowfrequency phonons (seen as static defects) and sup-ported by many recent theoretical papers which

pointed out the negligible effect of the phonon softeningon the enhancement of Tc [22].

Eliminating p now gives T’ c as a function of I 8 fordifferent Q. This result is drawn in figure 4 where thereduction of Tc is shown to be proportional to 82 andquite negligible for high values of Q (i.e. EF far fromthe peak of the density of states).

In the case of V3Si and Nb3Sn and with the parame-ters of the Labb6-Friedel model [4], the reductionof Tc is estimated to be 0.7 to 1.6 K (2.5 x 10- 3 I s I 4 x 10 - 3). This estimation scales with the expe-rimental result of figure 3. Modifications of the

density of states N(EF) can be produced by differentways : disappearance of the fine structure of N(EF)by introducing defects, partial or full removal of

degeneracy by uniaxial stress, moving the Fermi levelposition by hydrostatic pressure or by a substitutionelement, etc. The parameter p which is used to modifythe band structure will have a « direct » effect on T.. Tcwill be directly affected by the band structure modifica-tions due to p and indirectly by the modification of Tm(i.e. of the distortion amplitude e I at Tc). We willshow that the indirect effect is by far the most impor-tant. If we accept for I 8 1 the expression given by

a mean field phase transition theory : 1 -1/2

T) T 1/2 for temperatures not far from Tm we will haveTm/

Fig. 4. - Effect of the tetragonality (distortion e) on thecritical temperature T c’ This diagram shows that Tc, behavesas s2 and that this dependence smooths as the electroncontent of the d-band increases.

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where a is a positive constant to be determined and Tcis the superconducting transition temperature of avirtual cubic phase.The last expression is convenient in the case of V3Si

because Tc is close to Tm and the expression used fore is still valid

If we consider the known values of Tc, Tm and theestimated value of the Tc reduction (- 1 K) we finda - 5K.The derivation of the last expresion with respect to

the parameter p gives :

As far as Tm is not strongly depressed, i.e. the Fermisurface is weakly perturbed by p, we can neglectaTOT ’ in the right side of the equation, because TcP

is far less sensitive to the Fermi surface perturbation,as was shown by Labbé [4].

The « direct » term OT becomes important in theap case of strong modifications of the band structure :high pressure, high level of defects, etc.The simplified equation

concerns the indirect effect of p on Tt c due to the latticedistortion and shows an initial increase of Tc with thedecrease of Tm before the direct term starts to interferein a consequent way.

This equation explains the behaviour of Tc shownin figure 3 and the initial increase of 7c for smallirradiation doses [17,18].Another related observation of this effect, in the case

of V3Si, is the dependence of 7c and Tm on hydrostatic

pressure studied by Chu and Testardi [23], whoobtained

while for this ratio the above equation yields the valueof - 0.27, surprisingly close to the experimental one.

Finally, the observed insensitivity of Tc to lowfluences of neutron or a-particle irradiation of V3Si,Nb3Sn and other A15 [24] which suggest the existenceof a threshold fluence, apart from the experimentaluncertainties about the determination of T c should beconsidered as another argument in favour of theabove considerations. The observed insensitivity is

only due to the competition between the direct effectof irradiation which tends to decrease T c and theindirect effect, a consequence of the tetragonal distor-tion which, on the contrary, tends to increase Tr.

4. Conclusion.

Although the microscopic origin of the singular densityof states N(E) at the Fermi level is still controversial,many experimental results support the existence of adegenerate peak of N(E) as we pointed out above.The observed incompatibility between superconduc-

tivity and lattice distortion in A15’s as evidenced bythe first increase of T, with disorder in V3Si, is wellaccounted for by the band Jahn-Teller model consi-dered in this paper.We should insist on the fact that such a model is no

longer adequate for the strongly disordered A15.In this case, as shown by Anderson et al. [10], thedecrease of Tc is governed by another physical mecha-nism based on the increase of the Coulomb parameterJl* with disorder.

Acknowledgments

The authors are grateful to J. Doulat for his criticalreading of the manuscript and to Dr A. Freund fromILL for the successful y experiment. We also thankF. Monnier, Y. Nahajczuk from SBT and A. Escoffierfrom ILL for technical assistance.

References

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