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Garland Isotope Effect Paper PRL 114_1

Garland Isotope Effect Paper PRL 114_1 - VOLUM511,NUMBER5...

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Unformatted text preview: VOLUM511,NUMBER5 PHYSICAL REVIEW LETTERS 1AUGUST1963 T. H. Geballe and B. T. Matthias, IBM J. Res. De— velop. 6, 256 (1962); B. T. Matthias, T. H. Geballe, E. Corenzwit, and G. W. Hull, Jr., Phys. Rev. 129, 1025 (1963). 2K. Andres, J. L. Olsen, and H. Rohrer, IBM J. Res. Develop. 6, 84 (1962); E. Bucher and J. L. Ol— sen, Proceedings of the Eighth International Confer— ence on Low—Temperature Physics, London, 1962 (to be published). 3J. K. Hulm, R. D. Blaugher, T. H. Geballe, and B. T. Matthias, Phys. Rev. Letters 1, 302 (1961); A. B. Pippard, IBM Conference on Superconductivity, June 1961 (unpublished); T. H Geballe, B. T. Matthias, V. B. Compton, E. Corenzwit, and G. W. Hull, Jr. (to be published). 4B. T. Matthias, IBM J. Res. Develop. fi, 250 (1962); B. T. Matthias, Suppl. J. Appl. Phys. 91, 23 (1960). 5B. T. Matthias, Proceedings of the Eighth Interna» tional Conference on Low—Temperature Physics, Lon- don, 1962 (to be published); B. T. Matthias, T. H. Ge— balle, and V. B. Compton, Rev. Mod. Phys. §_5_, 1 (1963); T. H. Geballe and B. T. Matthias (to be pub— lished). 6J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 1%, 1175 (1957). 7L. Landau, Zh. Eksperim. iTeor. Fiz. gm, 1058 (1956) [translationz Soviet Phys. —-JETP g, 920 (1956)]; _3_2, 59 (1957) [translatiom Soviet Phys.—JETP 5, 101 (1957)]. 8F. Englert, J. Phys. Chem. Solids 1_1, 78 (1959). 9N. Wiser, Phys. Rev. lg, 62 (1963). 10The dielectric functions in the superconducting state and the normal state have been shown to be approxi— mately equal. R. E. Prange, Phys. Rev. 129, 2495 (1963). 11We define “clean” and “dirty” in accordance with the Anderson theory of dirty superconductors: P. W. An— derson, J. Phys. Chem. Solids 11, 26 (1959). 12This assumption is necessary to our two~gap theory, and may not be justified. However, the theory remains essentially the same even if the d—like area of the Fer— mi surface is partially hybridized or anisotropic so that Ad cannot be written as a function of 5 alone, pro— vided that some section of the Fermi surface is purely 5 —like. 13H. Suhl, B. T. Matthias, and L. R. Walker, Phys. Rev. Letters g, 552 (1959). 1‘Englert first pointed out the possibility of such an attractive Coulomb contribution: F. Englert, Proceed- ings of the International Conference on Semiconductor Physics, Pragye, 1960 (Czechoslovakian Academy of Sciences, Prague, 1961). 15J. Kondo, Progr. Theoret. Phys. (Kyoto) 22, 1 (1963); J. Paretti, Phys. Letters 2, 275 (1962). 16This contribution to the kernel is illustrated by Gar— land: J. W. Garland, Jr., Proceedings of the Eighth International Conference on Low—Temperature Physics, London, 1962 (to be published). ”J. I. Budnick, Phys. Rev. E, 1578 (1960); n. P. Seraphim (unpublished). 18T. H. Geballe (private communication). 19J. W. Garland, Jr., following Letter [Phys. Rev. Letters _1_1_, 114 (1963)]. 20Both (Kph> and KC" are defined more exactly by ref— erence 19. 21This form for Sph follows from the exact self—con- sistent dielectric function remembering that Vph is screened by the square of the dielectric function and neglecting the energy dependence of N(£) for g gfiEd. We have 4m?z , 47w2 “7 Q2 Yn'za'wpmal" M“ Q2 ' where Q is a typical phonon wave number, and the state 71 is on the Fermi surface. 22B. T. Matthias, Progress in Low-TemErature Physics, edited by C. J. Gorter (North—Holland Pub— lishing Company, Amsterdam, 1957), Vol. II, p. 138. ISOTOPE EFFECT IN SUPERCONDUCTIVITY James W. Garland, Jr .* Department of Physics and Institute for the Study of Metals, The University of Chicago, Chicago, Illinois (Received 28 May 1963) The transition temperature To of a superconduc— tor depends upon the average atomic mass M of its constituents: Tc an“ =M'0- 5‘1 ‘ i). (1) Although the existence of this isotope effect has played an important role in the development of the theory of superconductivity, the experimentally observed values of g have never been properly ex— 114 plained. Here we present calculated values of g in agreement with experiment and show the ob- served qualitative difference between the reduced isotope effect (g 2 0.3) of the transition metals and the nearly complete isotope effect (§~O. 1) of the simple1 metals to follow from elementary ideas of band structure. We have assumed an isotropic free-electron model for the simple metals and a two-band mod- el consisting of nearly free electronic states or- VOLUME 11, NUMBER 3 thogonalized to tightly bound d states for the tran— sition metals.2 We have already shown2 that the experimentally observed properties of the transi— tion metals must be described within the frame— work of the Anderson theory3 of dirty supercon— ductors. Thus, we describe the superconductivity of both the transition metals and the simple metals in terms of a single gap Mg) which is a function only of the renormalized normalsstate energy, A“) = 'EdE’ReiAQ'HKu, r) tanh(E'/2kBT) “T75, (2) where g is measured relative to the Fermi level and where 5' =Re{[£'“+A2(g')]”"} (3) is the superconducting quasisparticle energy.‘ We assume the existence of only two contributions to the kernel of Eq. (2), a Coulomb contribution KC and a contribution Kph arising from the virtual exchange of phonons“: m, £’;M) =Kc(g, 5') +Kph(x,xl), (4) where x = g/kBBD is measured in units of the M- dependent Debye energy. We first discuss the general properties of the gap solutions A(§) in order to understand quali- tatively the difference between the observed iso— tope effect in the simple metals and in the transi- tion metals. As the isotope effect arises solely from Kph through the M"’2 dependence of 9D, we clearly expect a significant deviation 5 from the ideal isotope effect if Tc depends strongly on the average magnitude of K (g, g’). In all supercon- ductors, the real part 0? Mg) changes sign at an energy very nearly equal to any energy at which the kernel K(O, g) changes sign.6 Thus, the prod- uct Re[A(g)}K(O, g) is negative for essentially all energies g {assuming Re[A(0)1K(0, 0) <0} , so that the contribution of KC(O, 5) in the region g>>kBBD reduces the effect of KC(O, g<) in the region g< <k36D.7 We find that in the simple metals KC(O, g) re— mains large up to energies g> 10 eV. On the other hand, we expect KC(0, g)/KC(O, 0) to be very small for [£1 greater than half a d-band width (~1 eV) for the case of the transition met- als. Our explicit calculations have shown the ef- fective cutoff energy £6 for KC(0, .5) actually to be of the order of half a d—subband width éEd<1 eV for this caseF:a Therefore, we expect the con— tribution of KC(O’ £>) more nearly to cancel the effect of KC(O, g<) in the simple metals than in the transition metals, leading to a smaller devi- PHYSICAL REVIEW LETTERS 1 AUGUST 1963 ation g from the ideal isotope effect in the sim- ple metals. In order to obtain simple quantitative formulas for g, we must investigate the approximate form of the kernel K(g, g'). We find that Kph(x,x’) de— pends strongly on x —x’ and approaches zero for Ix -x’I >> 1, but that the explicit dependence on x +x’ is weak. We also find that KC(§, g') is very nearly constant over any range in energy of order kBOD (at least for the simple metals). We are thus led to assume the existence of a phonon cut— off parameter xc 2 1 such that Kph(x -x’) may be neglected whenever 1x -x’] >xc and such that both the explicit 5+ g' dependence of Kph(x,x’) and the g and g' dependence of Kc(g, 5’) over any range in energy less than kaBQD may be neglected. We consider as a zero-order approximation the simple two—square-well model of Tolmachev" and of Morel and Anderson”: 0 I = l : Kph (x,x ) Kph(0,0) for both Ix] and Ix l<xc 1, = 0 otherwise; 0 I _ I Kc (g,§)—KC(0,0) for both :5: and lg 1<gc, : 0 otherwise. This model yields the simple formula )2. (5) ._ o_ §-§ —(KC*/Keff The experimentally determined quantity K eff = 1/ [2 ln(Tc/1. 14 9D)] corresponds to the parameter -N(0)V of Bardeen, Cooper, and Schrieffer (BCS),“ and KC* is given by Kc * =KC<0, 0)/[1 +KC(0, 0) 1n(gC/kBeD)1. (6) As Keff may be expressed as the sum of an av— erage phonon contribution (Kph) plus the net Cou- lomb contribution KC *, Eqs. (5)-(7) clearly sup— port our contention that the deviation g from the ideal isotope effect is significant only if the net Coulomb contribution KC* is of the same order as the phonon contribution (Kph). An integral equation for the variation 6<p(g) of the normalized energy gap lim 3(3)? =R <p(£) e} T4 M0) C with respect to small changes in Tc and GD fol— lows immediately from Eqs. (2) and (3) without any square—well approximations, given our as— 115 VOLUME 11, NUMBER 3 PHYSICAL REVIEW LETTERS 1 AUGUST 1963 sumption of a phonon cutoff parameter xc: bets) = 2[I?ph(x) — Kph(0)][51n(rc) - 51n(9D)] +2[ROW —KC(0)]61n(TC) W (“i ) — , tanh -00 a ZkBTC xim, 5')-K(0, film/1(8), (7) where 6D x+x c “f dx’K(x,x’)qo(x’) 4T x -x C c x’B D xsech2<2 T ) 0 However, in order to solve Eqs. (2) and (7) nu— merically and thus calculate g, we have employed the square-well approximation Kph(x,x') =Kph°(x, x’) which we justify after a discussion of our treatment of the Coulomb contribution KC(§, g') to the kernel. Our isotropic free-electron model for the sim- ple metals yields a Coulomb contribution KC(£, g') which may be written in the form KC(O, 0)Kcn(y, y’;E ), where y = g/EF is measured in units of the ermi energy and where the explicit depend- ence of KC” upon EF is very weak. Therefore, we may write g; in the form given by Eqs. (5) and (6) and calculate the ratio of the effective Cou— lomb cutoff energy to the Fermi energy, gc/EF (:4 for all of the superconducting simple metals), simply by calculating KC(§, g') numerically5 and solving Eqs. (2) and ('7) for any given simple metal. Our lack of knowledge of the detailed band structure of the transition metals leads us to choose a simple two—square-well model for Kc(g, g’) in the transition metals: KC(€,£’)=K K(x) =Re =K(x,0). C for :5] and [g’l both<gc’, as = ‘ I I KC for either [5| or lg [>56 and both lg! and It'REC”, =0 otherwise. Here the cutoff gc’ corresponds roughly to half a d—subband width éEd (<1 eV),12 and the cutoff 50” is of the order of 10 eV; KC is the total Cou- 116 lomb contribution to K(0, 0), and KC35 is the small contribution arising from matrix elements of the Coulomb interaction between the s—like parts of the electronic wave functions.2 We again write g in the form given by Eqs. (5) and (6), de- termining the effective Coulomb cutoff energy gc numerically for each transition metal. As the ratio Kcss/KC is much less than unity, 5c is only slightly greater than gc'. Although our lack of knowledge of the band structure of the transi- tion metals introduces a large probable error into our calculation of gc, the weak dependence of KC* upon gc makes this error relatively un— important.“ The largest error in our calcula- tions for the transition metals arises from the uncertainty in our calculation of KC(0, 0), which depends both on the relative importance of um- klapp processes and on the position of the Fermi surface in 12 space. In order to justify the use of the square-well approximation Kph(x,x') =Kph°(x,x’), we derive an equation for g valid for any physical Kph(x,x') and show that the ratio z/g" is approximately unity. From Eq. (7) we derive the equation 3<p(x)/31n(Tc)=(1 +t)[I?(x) 430)], (8) where O<t<1, and see that qa’(x) = 8cp(x)/8 ln(TC) + 8<p(x)/a 1n(9D) is very nearly zero for ix I <xc. Then, the deviation g from the ideal isotope ef- fect is given by <Kph> -I?ph(0) (1 +t)[KC*/2<Kph>] __ + _ Kph(0) +KC * K c - KC *[<Kph> - K ph(0)] §—§° 1+ xcdx x9D Xf ~K(0,x)[1?(x)-I?(0)]tanh(-——) — C x 2T0 _ '— a ([prh> K ph(0)] ) +0 T.— eff where the average phonon contribution is given by , (9) xc dx [x TKph(0,x)qo(x) C x9 D X tanh (ET) C : _ * eff KC ' (Kph) 2 -KeffRe VOLUME 11, NUMBER 3 As numerical calculations indicate that the ine- quality thh(0)/<Kph> — 1! S lKeffl should hold for all metals, we find that Eq. (9) shows our square—well approximation Kph(x,x’) =Kph°(x,x’) to lead to an error of less than or approximately equal to 25% in the calculation of r. We estimate the over—all probable error in our calculation of g to be less than or approxi— mately equal to 30% for the simple metals and 40% for the transition metals. Our calculated values of g and our estimated range of probable error are compared in Table I with the results of Bardeen, Cooper, and Schrief— fer (BCS), Swihart (Swi),14 and Morel and Ander— son (MA), as well as with experiment. ”"23 The great improvement embodied in our results fol— lows from (1) the recognition of the importance of band structure effects in the transition metals, and (2) a more exact treatment of the Coulomb contribution KCQ, 5') than has previously been attempted. For the simple metals, when warran- ted by improved experimental determinations of g, our formalism can be extended through lengthy PHYSICAL REVIEW LETTERS 1 AUGUST 1963 numerical calculations to include the effect of the detailed structure of the kernel K(g, g'). In many recent articles,“ it has been suggested that the superconductivity of the transition met— als arises primarily from some interaction other than the electron-phonon interaction. However, the excellent agreement between our calculated values of g and the few available experimental values clearly demonstrates that one need not infer this notion from the reduced isotope effect of the transition metals. Indeed, the observed vanishing of the isotope effect in ruthenium can even be used as the basis for an argument against the existence of any important attractive interac— tion between electrons in the transition metals other than the interaction Vph(x,x’) arising from the virtual exchange of phonons. Even if we dis— regard our explicit numerical calculations, we see that the large partial isotope effect observed in molybdenum and Moslr (Table 1) implies a ra— tio (Kph)/(KC) nearly as great as that in the sim— ple metals. The assumption that the supercon- ductivity of ruthenium arises from some interac— tion other than the electron—phonon interaction, coupled with its observed zero isotope effect, then implies that the ratio (Kph)/<Kc) is an or- Deviations 5 from the ideal isotope effect. Table I. z r Experiment Material (exp) (exp) reference Zn 0.45i 0.05 0.10: 0.10 17,18,24 Cd 0.50i 0.10 0.0 i0.2 19 Sn 0.47i 0.02 0.06t0.04 20 Hg 0.50i0.03 0.00i 0.06 20 Pb 0.48:9: 0.01 0.04i0.02 21 T1 0.50i 0.10 0.0 i0.2 20,22 Al Ru 0,001 0.05 1.0 40.10 23 Os 0.101010 0.8 i0.2 17 Mo 0.37i 0.07 0.25:1: 0.15 24 Ir Hf V Ti Zr Ta Re Nb3Sn 0.081: 0.02 0.84i0.04 25 Moalr 0.33i 0.03 0,344 0.06 24 V3Ge V3Ga Vgsi i C C C (BCS) (Swi) (MA) (Present calculation) 0.0 0.6 0.35 0.204 0.06 0.0 0.6 0.32 0.274 0.08 0.0 0.43 0.16 0.12: 0.04 0.0 0.2: 0.08 0.074 0.02 0.0 0'23 0.06 0.06i 0.02 0.0 0.43 0.14 0.11: 0.03 0.0 0.6 0.32 0.31:0.09 0.0 1.0 0.5113 1.0 40.4 0.0 0.8a 0.5b 0.8 i0.4 0.0 0.7 0.4b 0.3 :015 0.0 0.6b 1.4 :0.5 0.0 0.5 0.4 40.2 0.0 0.18 0.7 ::0.3 0.0 0.50 0.6 :03 0.0 0.40 0.3 :0.15 0.0 0.1610 0.3 40.15 0.0 0.18b 0.4 $0.2 0.0 0.10b 0.6 40.25 0.0 0.16b 0.35:0.15 0.0 0.20b 0.7 40.3 0.0 0.10b 1.3 40.5 0.0 0.10 1.0 40.4 a lOThese values are calculated from the model used by Swihart,” but were not published by Swihart. These values are calculated from the formulas of Morel and Anderson,10 but were not published by Morel and Anderson. 117 VOLUME 11, NUMBER 3 PHYSICAL REVIEW LETTERS 1 AUGUST 1963 der of magnitude smaller in ruthenium than in molybdenum. This great fluctuation in the ratio (Kph)/(KC) from one transition metal to another seems unreasonable. By contrast, our calcula— tions indicate only a small variation in the ratio (Kph)/(KC) as a function of position in the peri- odic table."’ The pressure dependence of the transition tem- perature Tc is also susceptible to an analysis similar to that given above. It is found experi- mentally”5 that the pressure dependence of Tc in the simple metals is an order of magnitude greater than in the transition metals. This ex- perimentally observed qualitative difference be- tween the pressure dependence of To in the case of the transition metals and in the case of the simple metals is consistent with our calculations.”3 However, we find that in both cases the pressure dependence arises almost entirely from the pres- sure dependence of (K h), which is extremely sensitive to the exact gorm of the pseudopotential in the simple metals and to the effective nuclear charge felt by a d electron at the Fermi surface in the transition metals. This sensitivity was too great in both cases to allow us to claim any- thing more than a fortuitous, qualitative agree- ment with experiment for the pressure depend- ence of the transition temperature. I wish to thank Professor Morrel H. Cohen and Professor L. M. Falicov for many helpful discus- sions and to thank Dr. T. H. Geballe and Dr. R. A. Hein for submitting their results prior to publica- tion. We are grateful to the U. S. Office of Naval Research, the National Science Foundation, and the National Aeronautics and Space Administra— tion, for direct financial support of this work. In addition, the research benefited from general support of the science of materials at the Univer— sity of Chicago by the Atomic Energy Commission and the Advanced Research Projects Agency, as well as support of the Low Temperature Labora— tory of the Institute for the Study of Metals at the University of Chicago by the National Science Foundation. *National Science Foundation Fellow, 1961—1963. 1By simple metals, we mean those metals which are not transition metals, rare earths, or actinides. 2J. W. Garland, Jr., preceding Letter [Phys. Rev. Letters 11, 111 (1963)]. 3P. w. Anderson, J. Phys. Chem. Solids 11, 26 (1959). _ 4As was first pointed out by Eliashberg (G. M. Eliash— berg, Zh. Eksperim. i Teor. Fiz. E, 966 (1960) [trans— 118 lation: Soviet Phys. —JETP 11, 696 (1960)”. Eqs. (2) and (3) are not strictly correct for temperatures below Tc. However, Eqs. (2) and (3) contain lifetime effects and are equivalent in the limit T” TO to a finite-tem— perature generalization of the Green’s function equa— tions of Schrieffer, Scalapino, and Wilkins [J. R. Schrieffer, D. J. Scalapino, and J. W. Wilkins, Phys. Rev. Letters E, 336 (1963)]. Although our calculations apply only to the isotope effect as defined by Eq. (1), not to the isotopic mass dependence of the critical mag- netic field, HCTmMO' 5(1 ' t), it may be shown that I, and t’ are very nearly equal, even in the strong-cou— pling case of lead. 5Formulas for the calculation of KC(§,g’) valid for both the transition metals and the simple metals are given in reference 2. 3This statement is not necessarily true of the s—band gap A509 in the case of a clean transition metal dis— cussed in reference 2, but is true in all cases thus far experimentally observed. 1The reader may easily verify the above arguments for the simple two—square-well model of Tolmachev (see reference 9). 8This result follows from the relative importance of small—q matrix elements and the fact that small-q ma— trix elements between different subbands vanish. 9N. N. Bogoliubov, V. V. Tolmachev, and D. V. Shirkov, New Method in the Theory of Sugrconduc— tivity (Academy of Sciences of U. S. S. R. , Moscow, 1958), Sect. 6.3. 10P. Morel and P. W. Anderson, Phys. Rev. 125, 1263 (1962). 11J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 1_0_8_, 1175 (1957). ”.50' is roughly proportional to the inverse of the den— sity of states mm at the Fermi surface and is only weakly dependent upon valence z. The detailed calcu— lation of EC’ depends upon a complicated model (to be published). 13Of course, an important error would be introduced into our calculations by the occurrence of a subband edge at an en...
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