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Effect of soft phonons on superconductivity: A re-evaluation and a positive case for Nb3Sn
Solid State Communications,
Vol. 14, pp. 937—940, 1974. Pergamon Press.
Printed in Great Britain
EFFECT OF SOFT PHONONS ON SUPERCONDUCTIVITY: A RE-EVALUATION AND A POSITIVE CASE FOR Nb3 Sn* Philip B. Allent
t
Cavendish Laboratory, Madingley Rd, Cambridge CB3 OHE, U.K. (Received 17 December 1973; in revised form 14 January 1974 by R.A. Cowley)
The effect of low freqUency phonons on superconductivity (which was described as detrimental in a previous paper) is re-assessed following the work of Bergrnann and Rainer. An anomalously large positive contribution to T~is quantitatively evaluated for soft phonons in Nb3 Sn using neutron data of Axe and Shirane. Low frequency phonons appear helpful in raising T~,but the coupling to high frequency phonons is of greater ultimate importance.
THIS PAPER has two purposes: (1) to retract some of the conclusions of an earlier paper’ which2have and been (2) to shown incorrect by Bergmann and Rainer; present some new thoughts which arise from this revised point of view. Specifically, it was claimed in reference I that low frequency phonons were ineffective in raising the transition temperature 7’, of superconductors, and could in fact lower T~if the frequencies were too low. These conclusions followed from a general treatment of the Eliashberg equations using a two-square-well approximation for the gap function Bergmann and Rainer2 have made much more accurate calculations of the effects of phonons of various frequencies on T~by numerical solution of the Eliashberg equations for certain specific types of phonon spectrum aF~wQ).In this exact approach, it is clearly shown that while very low frequencies become ineffective, they are never harmful to T~.I have found no fault with the calculations of reference 2, and accept their conclusions as correct. Apparently the ‘repulsive’ effect ascribed to very low frequency
phonons in reference 1 was an artifact of the twosquare-well approximation. is that dispersion which occurs in ~The as apoint function of the w is self-consistently determined so as to minimize the free energy, and thus (plausibly) to increase T~.This effect (which is omitted in a two-square-well model) enables the superconductor to make best use of each phonon. Having accepted this revised point of view, there is grounds for a more positive assessment of the role of soft phonons in raising T~,.Testardi3 has argued that the lattice softening associated with proximity to a lattice instability is beneficial to T,,. Evidence for this can be drawn from the experiments of Chu eta!.4 on V—Ru alloys in the vicinity of a secondorder lattice transition. A more general appreciation of the connection between high T~,and instabilities has been developing for several years and has been summarized by Matthias.5 In the remainder of this paper, a new quantitative argument is given in support of Testardi’s point of view. Specifically, it is shown that soft phonons in Nb 3Sn (TC ~ 1 8°K)have an anomalously large contribution to the electron—phonon coupling constant X and to T~.
~
_____________
*
.
.
.
Work supported in part by National Science Foundation Grant no. GH037925.
t *
.
Alfred P. Sloan Research Fellow.
.
6 at temperatures below 45°KUltraA small tetragonal distortion was found in Nb3Sn sonic measurements by Keller and Hanak7 showed that by Mailfert eta!.
Stony Brook, New York 11790, U.S.A. Permanent address: Department of Physics, SUNY, 937
938
EFFECT OF SOFT PHONONS ON SUPERCONDUCTIVITY
the distortion is connected with softening of the elastic modulus C11—C12 which determines the velocity of transverse acoustic (Ti) phonons propagating in the 8 have demonstrated [1101 direction. Shirane by neutron scattering thatand thisAxe softening extends to large wavevectors in the [110] direction. An assessment of therecent role ofmeasurements these soft TAof phonons on T~,can made using the linewidth ~ be of these phonons by Axe and Shirane,9 and recent theoretical work1°relating ‘y~to T~. The theory of reference 10 allows an explicit evaluation of the importance of each phonon in superconductivity, ~.
=
i~
=
—
2 jfdw —cr2F(w) W
(la)
a2~w) =
(wQ~/~)5(w
~
8.8 states/eV atom]. Before using equation (lc) the mass renormalization factor 1 + A must be divided out of states. about The result is a value = the 9.4 density for this of phonon, 18 times larger than the average value of XQ for Nb 3Sn. 2 have calculated the functional Bergmann and Rainer derivative S T~/6a2F of T~with respect to a2F which is defined by = .1 dw(5T~/5a2F(w))&i2F(w) (3) 0
From this equation and the numerical value of 5 T~/5cx2F we can get a precise measure of how much T~would diminish if a phonon Q were fictitiously uncoupled from electrons. From equation 1 we see that the change ina2Fis ~a2F(Q)
BZ —
~)
(lb)
Vol. 14, No. 10
=
—
~ WQXQ 5(w
—
WQ)
(4)
and the resulting change in T~is
Q
XQ
=
2 27Q/(lrN(O)hwQ)
(lc)
The symbol Q is shorthand for wavevector and branch quantum numbers (Q, v), and )~is the contribution of each phonon to A. Let us normalize so that N is the number of atoms in the crystal. Then an enlarged Brillouin zone (BZ) is needed which contains one state per atom, instead of the usual one which has è state per average valuebranches of XQ is in X/3thebecause in this atom. pictureThe there are three spectrum. According to Shen’s tunneling experiment, the value of A in Nb 3Sn is about 1.6. From Fig. 2 of reference ~ it is possible to learn XQThis for the T1 phonon with Q = (2ir/a) [~0]both and‘YQ=and 0.18. phonon ~‘
~
henceforth be denoted by the label Q*, The value of WQ* at low temperature is 4.0 meV while ~~Q* (half width due to electron—phonon interactions) is given by 2’YQ
~
~~~WQXQ
STC/Scx2F(WQ)
(5)
Bergmann and Rainer2 have calculated S T~/5a2Ffor ten fora which known. callysuperconductors there seems to be nearlya2Fis universal curveEmpiri. with the most effective phonons in a broad maximum near 7T~,the height of this maximum depending mainly on the value of A. For Nb 3Sn it is probably 2Fcalculated for Pb accurate to use the curve 5T~/5a provided WQ is scaled to T~.The maximum value of 5T~/5a2Fis just under 0.2 and occurs at 11 meV. For the specific [~0] phonon consideration, 2Fisunder still 2/3 of its maximum WQ * is 4 meV and S T~/5cx value, giving ~T~(Q*) = —30°K/Nfrom equation (5). This is six times larger than the value ~ for the ‘average’ phonon.
—5°K/N
where r’N and I’~are the experimental full widths in the normal and superconducting states, 1.4 and 0.7 meV respectively; l’~is believed to represent the instrumental resolution. The value of 7Q* is thus 0.6 meV. Finally, the density of states at the Fermi surface N(0) (for one atom and both spin orientations) is needed. Morin and Maita~have measured the linear coefficient of the heat capacity v to be 150 X 10 cal/mole °K2.l’his gives N(0) 6.6 states/eV atom. [Morin and Maita ignore the Sn atoms and obtain
on the instantaneous value of T~.It is perhaps easier to think of starting with no electron—phonon coupling. As the coupling is turned on, T~initially remains zero (aT~/~2F also vanishes) until roughly the point where A exceeds p’~.At this point T~is exp(—oc) and ar~ !a~2F is positive and infinite. The logarithm of T~increases rapidly, but eventually (when A 1) begins to saturate:
—
~—
(2)
=
It is important to exercise some care in interpreting these numbers. For example, the figure —5°K/Nfor the average phonon would apparently yield ~ = —l5°Kif all 3Nphonons were turned off, which tallies well with the actual T~ 1 8°K.However this is purely accidental. The functional derivative aT~/a~2F depends
=
[‘~)~
~T~(Q)
Vol. 14, No. 10
EFFECT OF SOFT PHONONS ON SUPERCONDUCTIVITY
correspondingly a T~/bcx2Fdiminishes rapidly at first, and asymptotically approaches zero at very large A. Qearly there is no single unambiguous way of assessing the importance of a phonon for T~.The contribution A,Q to A is the most significant measure, but in distinguishing between a 15°K superconductor and a 20°K superconductor, the contribution AT~(Q)is a better (and more stringent) measure. It is difficult to estimate how much of 7’~,actually is caused by soft phonons. There are two reasons for this. First the absence of experimental measurements of ‘YQ and (l.)Q except for a few phonons in one branch and one symmetry direction makes it hazardous T~from to the estimate the volume integrated contribution to appropriate of phase space. Second, the distinction between ‘soft’ and ordinary phonons is always ambiguous, and particularly so far away from the point where the susceptibility actually diverges (Q = 0 in Nb 3Sn). A somewhat arbitrary estimate can be made in the 1 following way. The striking feature of Shen’s’ 2F(~Q) is a peak at w~ 8 meV measured spectrum a Nb, and can be tentatively ascribed which is not found in to the soft TA branches at the zone boundary.9 A similar peak13 has been seen in the preliminary density of states F(~Q)measured by incoherent neutron scattering. The peak in F is weaker than that in a2F, implying anomalously strong coupling to this branch, in agreement with our estimates. Let us estimate how much this anomalous peak contributes to A and to T~. The separation between low frequency and ordinary phonons can be taken to be 12 meV, the energy at which the first ‘ordinary’ peak is beginning to rise in Nb and Nb 3Sn. Using Shen’s data, half of A arises from phonons below 12 meV and half from those above 12 meV. However, a further distinction is needed 2F below 12 meV which is ‘normal’ between the which part ofiscr‘anomalous’. The ‘normal’ part and the part is arbitrarily taken to be quadratic in w and normalized to match onto ci’Fat 12 meV. The part above this parabola is then defined as ‘anomalous’, and contains 20% of A. This is probably a conservative estimate of the ‘soft-mode’ contribution. The resulting change in 2Fprocedure is turned is 7’~calculated from the Bergrnann—R.ainer —1 the ‘anomalous’ partalso of cibe used to make off..7°Kwhen The McMilan equation14 can this estimate. Only two parameters are necessary, the coupling A and the scale factor which Dynes’5 has shown to be (c). When the ‘anomalous’ part of~2F is turned off, A decreases by 0.31 and (w> increases by 1.6 meV. The change in A alone would decrease
939
7’~by 3~50 K, but the increase of (s,> offsets some of this, giving AT~= —1 .9°K,in good agreement with the more accurate Bergmann—Rainer procedure. The estimates given here are in severe disagreement with the more optimistic assessment of the importance of soft modes made by Ramakrishnan.16 This disagreement can be ascribed to the semiquantitative nature of the model of reference 16. This work also disagrees with the more pessimistic assessment of Barisic,17 who has recently argued that A is small in Nb 3Sn because of an inconsistency between large negative pressure coefficient (dT~/dp)and large values of A. However the inconsistency from the McMillan 2’ highly arguments which arises are not generally believed‘Ato= be accurate. Further reasons for not trusting the ‘A = 2’ arguments will be published shortly.’8 In summary, low frequency ‘soft’ phonons in Nb3Sn are effect strongly tosoft electrons, and the integrated of coupled the whole T 1 branch may1 8°K. be of order 20% in A which can raise T~,from 16 to Although this is a significant increase, it is by no means a complete explanation of the large value of 7’~.The values of A and T~are both high even if low-frequency modes are discounted. It seems plausible that the large contributions to A and T~,coming from high frequency phonons arise partly because of lattice softening, which leads to large amplitudes of vibration even in modes of comparatively high frequency. Such a point of viewhas been described by Gomersall and Gyorffy’9 for NbN. Finnis and Heine~have recently described a classification of lattice transitions into ‘T-type’ where only a minority of phonons are softened, and ‘R-type’ where all phonons are softened. It seems likely that high‘R’ T~, materials are allitself located neara material an instability of the type. Niobium is such Relative to Mo, the squared phonon energies in Nb have decreased by nearly a factor of two over much of the Brillouin zone. The specific softening of [110] T 1 phonons in Nb3Sn from this point of view is simply the most overt evidence of an overall lattice softening, and provides an extra benefit in raising T~. Acknowledgements I am grateful to J .D. Axe, S.M. Shapiro, G. Shirane, and R. Silberglitt for much help. .~
—
940
EFFECT OF SOFT PHONONS ON SUPERCONDUCTIVITY
Vol. 14, No. 10
REFERENCES 1. 2.
ALLEN P.B., Solid State Commun. 12, 379 (1973). [see also Erratum Solid State Commun. 13, vii (1973)] BERGMANN G. and RAINER D.,Z. Phys. 263, 59(1973).
3. 4.
TESTARDI L.R.,Thys. Rev. B5,4342 (1972). CHU C.W., BUCHER E., COOPER A.S. and MAfIA, J.P., Phys. Rev. B4, 320 (1971).
5.
MATTHIAS B.T.,Thysica 69,54(1973).
6. 7.
MAILFERT R., BATTERMAN B.W. and HANAK J.J.,Phys. Lett. 24A, 315 (1967); Phys. Status Solidi 32, K67 (1969). KELLER KR. and HANAK J.J.,Phys. Lert. 21, 263 (1966);Phys. Rev. 154, 628 (1967).
8.
SHIRANE G. and AXE J.D.,Phys. Rev. Lett. 27, 1803 (1971).
9.
AXE J.D. and SHIRANE G.,Phvs. Rev. Lett. 30, 214 (1973); Phys. Rev. B8, 1965 (1973).
10.
ALLEN P.B., Phys. Rev. B6, 2577 (1972). Errors of a factor of 2 are corrected in the present paper.
11.
SHEN L.Y.L.,Phvs. Rev. Lett. 29, 1082 (1972).
12.
MORIN F.J. and MAlTA J.P.,Phys. Rev. 129,1115(1963).
13. 14.
REICHARDT W., SCHWEISS P., SALGADO J. and GOMPF F., Symp. Superconductivity and Lattice Instabilities, Gatlinburg, Tenesee, 1973 (unpublished). MCMILLANW.L.,Phys. Rev. 167,331 (1968).
15. 16.
DYNES R.C.. Solid State Commun. 10,615 (1972). RAMAKRISHNAN T.V.,J. Phvs. C6, 3041 (1973).
17. 18.
BARISIC S.,Phys. Rev. B8(October. 1973). ALLEN P.B. and DYNES R.C., to be published.
19.
GOMERSALL 1.R. and GYORFFY B.L.,J. Phys. F3, L138 (1973).
20. 21.
FINNIS M.W. and HEINE V.. Cavendish Lab. Preprint no. TCM/35/1973. ALLEN P.B. and COHEN M.L., Phys. Rev. Lert. 29, 1593 (1972); BECHTOLD J. and ALLEN P.B., Internal report no. 73-1, Solid State Theory Group, Dept. of Physics, SUNY Stony Brook (unpublished).
L’effet de phonons de basse frequence sur Ia superconductivité (décrit comme préjudiciable dans un texte précédent) est ré-évalué a la suite des travaux de Bergmann et de Rainer. Une anormalement grande et positive contribution a T~est quantitativement évaluée pour les phonoris doux dans Ic Nb3 Sn en utiisant des données de neutrons d’Axe et de Shirane. Les phonons de basse frequence semblent utiles pour élever T~,mais l’accouplement aux phonons de haute fréquence est d’importance eventuellement plus grande.
Vol. 14, pp. 937—940, 1974. Pergamon Press.
Printed in Great Britain
EFFECT OF SOFT PHONONS ON SUPERCONDUCTIVITY: A RE-EVALUATION AND A POSITIVE CASE FOR Nb3 Sn* Philip B. Allent
t
Cavendish Laboratory, Madingley Rd, Cambridge CB3 OHE, U.K. (Received 17 December 1973; in revised form 14 January 1974 by R.A. Cowley)
The effect of low freqUency phonons on superconductivity (which was described as detrimental in a previous paper) is re-assessed following the work of Bergrnann and Rainer. An anomalously large positive contribution to T~is quantitatively evaluated for soft phonons in Nb3 Sn using neutron data of Axe and Shirane. Low frequency phonons appear helpful in raising T~,but the coupling to high frequency phonons is of greater ultimate importance.
THIS PAPER has two purposes: (1) to retract some of the conclusions of an earlier paper’ which2have and been (2) to shown incorrect by Bergmann and Rainer; present some new thoughts which arise from this revised point of view. Specifically, it was claimed in reference I that low frequency phonons were ineffective in raising the transition temperature 7’, of superconductors, and could in fact lower T~if the frequencies were too low. These conclusions followed from a general treatment of the Eliashberg equations using a two-square-well approximation for the gap function Bergmann and Rainer2 have made much more accurate calculations of the effects of phonons of various frequencies on T~by numerical solution of the Eliashberg equations for certain specific types of phonon spectrum aF~wQ).In this exact approach, it is clearly shown that while very low frequencies become ineffective, they are never harmful to T~.I have found no fault with the calculations of reference 2, and accept their conclusions as correct. Apparently the ‘repulsive’ effect ascribed to very low frequency
phonons in reference 1 was an artifact of the twosquare-well approximation. is that dispersion which occurs in ~The as apoint function of the w is self-consistently determined so as to minimize the free energy, and thus (plausibly) to increase T~.This effect (which is omitted in a two-square-well model) enables the superconductor to make best use of each phonon. Having accepted this revised point of view, there is grounds for a more positive assessment of the role of soft phonons in raising T~,.Testardi3 has argued that the lattice softening associated with proximity to a lattice instability is beneficial to T,,. Evidence for this can be drawn from the experiments of Chu eta!.4 on V—Ru alloys in the vicinity of a secondorder lattice transition. A more general appreciation of the connection between high T~,and instabilities has been developing for several years and has been summarized by Matthias.5 In the remainder of this paper, a new quantitative argument is given in support of Testardi’s point of view. Specifically, it is shown that soft phonons in Nb 3Sn (TC ~ 1 8°K)have an anomalously large contribution to the electron—phonon coupling constant X and to T~.
~
_____________
*
.
.
.
Work supported in part by National Science Foundation Grant no. GH037925.
t *
.
Alfred P. Sloan Research Fellow.
.
6 at temperatures below 45°KUltraA small tetragonal distortion was found in Nb3Sn sonic measurements by Keller and Hanak7 showed that by Mailfert eta!.
Stony Brook, New York 11790, U.S.A. Permanent address: Department of Physics, SUNY, 937
938
EFFECT OF SOFT PHONONS ON SUPERCONDUCTIVITY
the distortion is connected with softening of the elastic modulus C11—C12 which determines the velocity of transverse acoustic (Ti) phonons propagating in the 8 have demonstrated [1101 direction. Shirane by neutron scattering thatand thisAxe softening extends to large wavevectors in the [110] direction. An assessment of therecent role ofmeasurements these soft TAof phonons on T~,can made using the linewidth ~ be of these phonons by Axe and Shirane,9 and recent theoretical work1°relating ‘y~to T~. The theory of reference 10 allows an explicit evaluation of the importance of each phonon in superconductivity, ~.
=
i~
=
—
2 jfdw —cr2F(w) W
(la)
a2~w) =
(wQ~/~)5(w
~
8.8 states/eV atom]. Before using equation (lc) the mass renormalization factor 1 + A must be divided out of states. about The result is a value = the 9.4 density for this of phonon, 18 times larger than the average value of XQ for Nb 3Sn. 2 have calculated the functional Bergmann and Rainer derivative S T~/6a2F of T~with respect to a2F which is defined by = .1 dw(5T~/5a2F(w))&i2F(w) (3) 0
From this equation and the numerical value of 5 T~/5cx2F we can get a precise measure of how much T~would diminish if a phonon Q were fictitiously uncoupled from electrons. From equation 1 we see that the change ina2Fis ~a2F(Q)
BZ —
~)
(lb)
Vol. 14, No. 10
=
—
~ WQXQ 5(w
—
WQ)
(4)
and the resulting change in T~is
Q
XQ
=
2 27Q/(lrN(O)hwQ)
(lc)
The symbol Q is shorthand for wavevector and branch quantum numbers (Q, v), and )~is the contribution of each phonon to A. Let us normalize so that N is the number of atoms in the crystal. Then an enlarged Brillouin zone (BZ) is needed which contains one state per atom, instead of the usual one which has è state per average valuebranches of XQ is in X/3thebecause in this atom. pictureThe there are three spectrum. According to Shen’s tunneling experiment, the value of A in Nb 3Sn is about 1.6. From Fig. 2 of reference ~ it is possible to learn XQThis for the T1 phonon with Q = (2ir/a) [~0]both and‘YQ=and 0.18. phonon ~‘
~
henceforth be denoted by the label Q*, The value of WQ* at low temperature is 4.0 meV while ~~Q* (half width due to electron—phonon interactions) is given by 2’YQ
~
~~~WQXQ
STC/Scx2F(WQ)
(5)
Bergmann and Rainer2 have calculated S T~/5a2Ffor ten fora which known. callysuperconductors there seems to be nearlya2Fis universal curveEmpiri. with the most effective phonons in a broad maximum near 7T~,the height of this maximum depending mainly on the value of A. For Nb 3Sn it is probably 2Fcalculated for Pb accurate to use the curve 5T~/5a provided WQ is scaled to T~.The maximum value of 5T~/5a2Fis just under 0.2 and occurs at 11 meV. For the specific [~0] phonon consideration, 2Fisunder still 2/3 of its maximum WQ * is 4 meV and S T~/5cx value, giving ~T~(Q*) = —30°K/Nfrom equation (5). This is six times larger than the value ~ for the ‘average’ phonon.
—5°K/N
where r’N and I’~are the experimental full widths in the normal and superconducting states, 1.4 and 0.7 meV respectively; l’~is believed to represent the instrumental resolution. The value of 7Q* is thus 0.6 meV. Finally, the density of states at the Fermi surface N(0) (for one atom and both spin orientations) is needed. Morin and Maita~have measured the linear coefficient of the heat capacity v to be 150 X 10 cal/mole °K2.l’his gives N(0) 6.6 states/eV atom. [Morin and Maita ignore the Sn atoms and obtain
on the instantaneous value of T~.It is perhaps easier to think of starting with no electron—phonon coupling. As the coupling is turned on, T~initially remains zero (aT~/~2F also vanishes) until roughly the point where A exceeds p’~.At this point T~is exp(—oc) and ar~ !a~2F is positive and infinite. The logarithm of T~increases rapidly, but eventually (when A 1) begins to saturate:
—
~—
(2)
=
It is important to exercise some care in interpreting these numbers. For example, the figure —5°K/Nfor the average phonon would apparently yield ~ = —l5°Kif all 3Nphonons were turned off, which tallies well with the actual T~ 1 8°K.However this is purely accidental. The functional derivative aT~/a~2F depends
=
[‘~)~
~T~(Q)
Vol. 14, No. 10
EFFECT OF SOFT PHONONS ON SUPERCONDUCTIVITY
correspondingly a T~/bcx2Fdiminishes rapidly at first, and asymptotically approaches zero at very large A. Qearly there is no single unambiguous way of assessing the importance of a phonon for T~.The contribution A,Q to A is the most significant measure, but in distinguishing between a 15°K superconductor and a 20°K superconductor, the contribution AT~(Q)is a better (and more stringent) measure. It is difficult to estimate how much of 7’~,actually is caused by soft phonons. There are two reasons for this. First the absence of experimental measurements of ‘YQ and (l.)Q except for a few phonons in one branch and one symmetry direction makes it hazardous T~from to the estimate the volume integrated contribution to appropriate of phase space. Second, the distinction between ‘soft’ and ordinary phonons is always ambiguous, and particularly so far away from the point where the susceptibility actually diverges (Q = 0 in Nb 3Sn). A somewhat arbitrary estimate can be made in the 1 following way. The striking feature of Shen’s’ 2F(~Q) is a peak at w~ 8 meV measured spectrum a Nb, and can be tentatively ascribed which is not found in to the soft TA branches at the zone boundary.9 A similar peak13 has been seen in the preliminary density of states F(~Q)measured by incoherent neutron scattering. The peak in F is weaker than that in a2F, implying anomalously strong coupling to this branch, in agreement with our estimates. Let us estimate how much this anomalous peak contributes to A and to T~. The separation between low frequency and ordinary phonons can be taken to be 12 meV, the energy at which the first ‘ordinary’ peak is beginning to rise in Nb and Nb 3Sn. Using Shen’s data, half of A arises from phonons below 12 meV and half from those above 12 meV. However, a further distinction is needed 2F below 12 meV which is ‘normal’ between the which part ofiscr‘anomalous’. The ‘normal’ part and the part is arbitrarily taken to be quadratic in w and normalized to match onto ci’Fat 12 meV. The part above this parabola is then defined as ‘anomalous’, and contains 20% of A. This is probably a conservative estimate of the ‘soft-mode’ contribution. The resulting change in 2Fprocedure is turned is 7’~calculated from the Bergrnann—R.ainer —1 the ‘anomalous’ partalso of cibe used to make off..7°Kwhen The McMilan equation14 can this estimate. Only two parameters are necessary, the coupling A and the scale factor which Dynes’5 has shown to be (c). When the ‘anomalous’ part of~2F is turned off, A decreases by 0.31 and (w> increases by 1.6 meV. The change in A alone would decrease
939
7’~by 3~50 K, but the increase of (s,> offsets some of this, giving AT~= —1 .9°K,in good agreement with the more accurate Bergmann—Rainer procedure. The estimates given here are in severe disagreement with the more optimistic assessment of the importance of soft modes made by Ramakrishnan.16 This disagreement can be ascribed to the semiquantitative nature of the model of reference 16. This work also disagrees with the more pessimistic assessment of Barisic,17 who has recently argued that A is small in Nb 3Sn because of an inconsistency between large negative pressure coefficient (dT~/dp)and large values of A. However the inconsistency from the McMillan 2’ highly arguments which arises are not generally believed‘Ato= be accurate. Further reasons for not trusting the ‘A = 2’ arguments will be published shortly.’8 In summary, low frequency ‘soft’ phonons in Nb3Sn are effect strongly tosoft electrons, and the integrated of coupled the whole T 1 branch may1 8°K. be of order 20% in A which can raise T~,from 16 to Although this is a significant increase, it is by no means a complete explanation of the large value of 7’~.The values of A and T~are both high even if low-frequency modes are discounted. It seems plausible that the large contributions to A and T~,coming from high frequency phonons arise partly because of lattice softening, which leads to large amplitudes of vibration even in modes of comparatively high frequency. Such a point of viewhas been described by Gomersall and Gyorffy’9 for NbN. Finnis and Heine~have recently described a classification of lattice transitions into ‘T-type’ where only a minority of phonons are softened, and ‘R-type’ where all phonons are softened. It seems likely that high‘R’ T~, materials are allitself located neara material an instability of the type. Niobium is such Relative to Mo, the squared phonon energies in Nb have decreased by nearly a factor of two over much of the Brillouin zone. The specific softening of [110] T 1 phonons in Nb3Sn from this point of view is simply the most overt evidence of an overall lattice softening, and provides an extra benefit in raising T~. Acknowledgements I am grateful to J .D. Axe, S.M. Shapiro, G. Shirane, and R. Silberglitt for much help. .~
—
940
EFFECT OF SOFT PHONONS ON SUPERCONDUCTIVITY
Vol. 14, No. 10
REFERENCES 1. 2.
ALLEN P.B., Solid State Commun. 12, 379 (1973). [see also Erratum Solid State Commun. 13, vii (1973)] BERGMANN G. and RAINER D.,Z. Phys. 263, 59(1973).
3. 4.
TESTARDI L.R.,Thys. Rev. B5,4342 (1972). CHU C.W., BUCHER E., COOPER A.S. and MAfIA, J.P., Phys. Rev. B4, 320 (1971).
5.
MATTHIAS B.T.,Thysica 69,54(1973).
6. 7.
MAILFERT R., BATTERMAN B.W. and HANAK J.J.,Phys. Lett. 24A, 315 (1967); Phys. Status Solidi 32, K67 (1969). KELLER KR. and HANAK J.J.,Phys. Lert. 21, 263 (1966);Phys. Rev. 154, 628 (1967).
8.
SHIRANE G. and AXE J.D.,Phys. Rev. Lett. 27, 1803 (1971).
9.
AXE J.D. and SHIRANE G.,Phvs. Rev. Lett. 30, 214 (1973); Phys. Rev. B8, 1965 (1973).
10.
ALLEN P.B., Phys. Rev. B6, 2577 (1972). Errors of a factor of 2 are corrected in the present paper.
11.
SHEN L.Y.L.,Phvs. Rev. Lett. 29, 1082 (1972).
12.
MORIN F.J. and MAlTA J.P.,Phys. Rev. 129,1115(1963).
13. 14.
REICHARDT W., SCHWEISS P., SALGADO J. and GOMPF F., Symp. Superconductivity and Lattice Instabilities, Gatlinburg, Tenesee, 1973 (unpublished). MCMILLANW.L.,Phys. Rev. 167,331 (1968).
15. 16.
DYNES R.C.. Solid State Commun. 10,615 (1972). RAMAKRISHNAN T.V.,J. Phvs. C6, 3041 (1973).
17. 18.
BARISIC S.,Phys. Rev. B8(October. 1973). ALLEN P.B. and DYNES R.C., to be published.
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L’effet de phonons de basse frequence sur Ia superconductivité (décrit comme préjudiciable dans un texte précédent) est ré-évalué a la suite des travaux de Bergmann et de Rainer. Une anormalement grande et positive contribution a T~est quantitativement évaluée pour les phonoris doux dans Ic Nb3 Sn en utiisant des données de neutrons d’Axe et de Shirane. Les phonons de basse frequence semblent utiles pour élever T~,mais l’accouplement aux phonons de haute fréquence est d’importance eventuellement plus grande.