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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June 1961 (unpublished); T. H Geballe, B. T. Matthias,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 LowTemErature
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 freeelectron
model for the simple metals and a twoband 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 dband
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 zeroorder approximation the
simple two—squarewell 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 squarewell 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 freeelectron 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—squarewell 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 squarewell
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 electronphonon 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 finitetem—
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 strongcou—
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—squarewell model of Tolmachev
(see reference 9). 8This result follows from the relative importance of
small—q matrix elements and the fact that smallq 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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