- Email: [email protected]
Effect of anisotropic phonons on superconductivity
Physica C 171 (1990) 518-522 North-Holland
Effect of anisotropic phonons on superconductivity S.P. T e w a r i a n d P a r a m j e e t K a u r G u m b e r Department of Physics and Astrophysics, University of Delhi, Delhi- 110 O07, India
Received 15 August 1990
Effect of crystal anisotropy which results in the anisotropic phonon frequency distribution function has been studied on superconductivity in a strongly coupled superconductor. The dependence of the superconducting critical temperature, To, on the degree ofanisotropy is quite involved. In highly anisotropic systems superconductivity is suppressed. The expression for Tc is related to the total mean square displacement of the vibrating ion, yielding the result that, for superconductivity, the phonon frequency distribution function from mean square displacement is preferred to that obtained from other studies for consistent physical parameters. Explicit calculations for 67Zn are reported.
It is well known that zinc is a superconductor with a critical temperature Tc=0.85 K [1 ]. A recently measured value o f Tc for 67Zn by Obenhuber et al. [2] is (835_+20) mK. They have also been able to measure the temperature dependent L a m b - M 6 s s bauer recoilless fraction in 67Zn along the c-direction and perpendicular to it at temperatures 4.2, 20.8 and 47 K in spite of a very sharp resonance width F ( = 4 . 8 X 10 -~1 eV) o f its M6ssbauer y-ray radiation (Eo = 93.3 keV). From the observed highly anisotropic recoilless fraction, f the ratio o f f perpendicular and parallel to the c-axis is ( f l / f l ) ~ 23 at 4.2 K and ( f ± / ~ ) ~ 2 1 0 0 at 47 K, [2,3], temperature dependent mean square displacement ( M S D ) of a zinc atom is extracted. It is found that at a given temperature MSD along the c-axis is much larger than the corresponding MSD in the basal plane. Clearly the motion of a zinc atom in its crystal is highly anisotropic and is essentially because o f the large c / a ratio equal to 1.86 in the hexagonal close packed structure o f zinc. From the earlier analysis of superconductivity in zinc one finds that it is a strongly coupled superconductor and can be described rather well using isotropic phonon modes in McMillan's equation [1] which is an outcome of the Eliashberg theory of the strongly coupled superconductor. However, later studies [4,5 ] on the electron-phonon mass enhancement factor, 2, in zinc show that p h o n o n anisotropy
must be considered. Very recently [6] such a study has been made and it turns out that the effect of crystal anisotropy on superconductivity is quite involved and significant. However, in the study a simple phonon frequency distribution function ( F D F ) , i.e. the extended Debye model [3] ( E D M ) , is used. Such a model does not take into account the presence of the planar modes in the basal plane and along the c-direction, expected to be present in such a highly anisotropic hexagonally close packed structure [ 7 10 ]. Very recently [8] a model for phonon F D F in zinc has been suggested, which not only takes into account the presence o f planar modes but is also able to successfully explain ( i ) t h e recently [2] observed MSD of the zinc atom both along the c-axis and in the basal plane in the temperature range 4.2 to 47 K, and (ii) gives gross features of the experimental phonon FDF, obtained from neutron scattering [ 11 ], much better than the FDFs given by other models [2,3,12,13]. We therefore use the suggested anisotropic phonon F D F to study the effect of these anisotropic phonon modes on the superconducting critical temperature in zinc. The superconducting transition temperature, T~, is calculated using the modified McMillan's equation [ 14-17 ], which is known to yield reliable results for 2 < 1 and hence can be confidently used to study superconductivity in a strongly coupled superconductor such as zinc [ 16,17 ] :
0921-4534/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
S.P. Tewari,P.K.Gumber/ Effectof anisotropicphononson superconductivity To- (w) ( 1.04(1+2) _) 1.2 exp - 2 _ / ~ . ( 1 + 0 . 6 2 2 ) , / ,
(1)
where/1" represents the electron-electron coulomb interaction; 2, the electron-phonon mass enhancement factor, is given by 2=2 f
Og2(W)F(w)
'? ) , w - l d w = M(w2
f o~2(w)F(w)wdw (3a)
ol2(w)f(w)w-ldw
structure. For such a highly anisotropic crystal structure, the phonon frequency distribution function along the c-axis ( i = z ) and in the basal plane (i=x, y) can be written in the following form [7-9,22]:
Fi(w)=Ajw 2 O<<,w<~wli =Biw
Wli~W~W2 i
~. 0
W • W2i
(6)
(2)
where r/is a purely electronic factor, M is the ionic mass, and
(w 2) = f
519
where A~ and B, are constants and can be determined using the conditions (i) F~(w) is continuous at w=w~ and (ii) total number of modes in a given direction is equal to N; and are given as N A i-
and
w3i/i
N ,
Bi=
w~i~ i ,
where
f o~2(w)F(w)dw (w) =
.
(3b)
f o~Z(w)F(w)w-ldw
i=x,y,z,
F(w) is the phonon density of states and ot2(w) represents the electron-phonon coupling constant. Making use of the generally accepted approximations (i) the constancy of electron-phonon coupling with phonon frequencies [ 1,6,14-16,18,19 ] and (ii) harmonic approximation, one obtains the following expression for To, when/z*=0: T~= 0.442h
M(U2)o exp
(1.56h(w)') -- q(U2)o
(4)
where 2 = 2~/(U2 )0 3h(w)' and ( u 2) o is the MSD of the vibrating mass of T = 0 K [20,21] and
f ot2(w)F(w)wdw
(w>'= f
o~2(w )F( w)dw
A~---'0.5 [6] - 0 . 3 3 l,
(5)
Using eq. (4) one can obtain an expression of Tc for /t*~0. In order to evaluate (U2)o and ( w ) ' the phonon FDF F ( W ) has to be known. As discussed earlier the observed mean square displacement of a zinc atom in its crystal is highly anisotropic and so is its crystal
8(i=x.~)=6
(~i ~ W 2 i / Wl i ,
and
6~=z)=6'.
In the FDF given by eq. (6) at low frequencies, only the w 2 term contributes so that the variation of the specific heat at low temperatures is T 3 dependent, i.e., the contribution of two-dimensional modes (w dependent) is absent at low frequencies. The relative number of two-dimensional modes to three-dimensional modes is determined by the parameter 6,. When 6~--.~ all the modes in a given direction would essentially be two-dimensional. That is in this case there would not be any coupling between the planes and the crystal would be extremely anisotropic. For other values of 6i, the anisotropy would be less. Thus, 6~ characterizes the degree of anisotropy in a given direction. Making use of the normalization condition i.e. (ii), one can show that the number of modes is not dependent on 6~. Since the number of modes in a given direction is fixed, equal to N, variation in 6i from one value to the other results essentially in the redistribution of the fixed total number of modes. Using eq. (6) and F(w)=2Fx#(w)+Fz(w), in evaluating ( u 2) o and ( w)' which when substituted in eq. (4) give the following expression for To:
S.P. TewarLP.K. Gumber / Effect of anisotropic phonons on superconductivity
520 0.442WlxyW1 ~ rc = X
/
expE--0"347
mWlxyWlz (2W1xyA2-~-- WlzAt2 ~l \2WlzAl + WlxyZ~'l]_]'
(7) where _( ~-0.5 AI - \ 6 2 _ 0 . 3 3 ],
( d ' - 0 . 5 "~ 3'i = \~77_--0-~-. 33j,
( ,3_025
(~3-0.25~
(~= W2xy/ Wlxy,
(~' = W2z/Wlz
W(1,2)xv=W(1,2)x=W(l,2)y
and
/t*=0.
When/1" ~ 0 the expression (7) gets modified. In the case of Debye approximation, i.e. the isotropic phonon frequency distribution function where in contrast to expression (6) Fi(w) oc w 2 in the entire frequency region, expression (7) reduces to the following well known expression: {-0.52mw~'~ Tc -~ 0.2WD exp~ ~ ).
(8)
It is evident from expression (7) that the dependence of Tc on crystal anisotropy is quite involved even though no anisotropic effect in the purely electronic parameter r/has been taken into account. Effect of anisotropic Fermi surface on various superconducting properties of metallic alloys is only marginal (23). Both the cofactor and the exponential term of expression (7) for Tc contain anisotropic effects. If one studies these anisotropy dependent parameters with the degree of anisotropy, i.e. with d~=w2i/wl~ where i( = x , y, z), one finds that with the increase in anisotropy, both the cofactor and the exponential term increase. However, the rate of increase for the exponential term is greater than that for the cofactor. This would result in the suppression of superconductivity in highly anisotropic systems. In the usual analysis of Tc, in the absence of relevant tunneling data, one resorts to a simple Debye approximation for the phonons with the cut off Debye temperature taken from the specific heat study. For example in the case of zinc, 0D is taken to be equal
to 309 K [ 1 ] which reproduces rather well the experimental specific heat [ 12] at very low temperatures, i.e. around 1 K. However, it is well known that in specific heat the low energy phonons are not given as much weightage as in the mean square vibrational displacement of an atom. These low energy elastic modes play a crucial role in superconductivity as is also evident from expression (2). Indeed one may obtain the expression (4) where Tc is directly related to the MSD of the vibrating atom. We therefore suggest that the phonon FDF derived from MSD studies, if available, should be preferred to that given by specific heat to calculate Tc. In fact the value of q(=2.362 eV//~ 2) for zinc which one gets using 0D ( = 309 K) and/z* ( = 0.12 ) [ 1 ], is more than 100% off from the value ( q = 1.164 eV/& 2) using the experimental result of cohesive energy [24]. When the measured value of cohesive energy is used one obtains 0D = 206 K to yield experimental Tc. This value of 0D also gives total MSD of zinc equal to approximately 7.98× 10 -3 ~2 at 4.2 K which is only 3% off from the recently measured value [3 ]. However, as has been emphasized zinc is highly anisotropic and therefore in a proper analysis due consideration should be given to its anisotropicity. Using the anisotropic phonon frequency distribution function given by expression (6) it has been found that 01z= 100 K and 02z = 170 K, 0~xy= 130 K and 02xy= 269 K (0~ 1,2)i= •W(1,2 )i/kB) explains successfully, amongst others [8], measured temperature variation of MSD of a zinc atom in the temperature range 47 to 4.2 K along the c-axis and in the basal plane, respectively. The phonon FDF shown in fig. 1 by solid line ( - - ) also reproduces the gross features of experimental phonon FDF (obtained from neutron scattering) [ 11 ] shown by ( - . - ) much better than that given by other dynamical models: ( - - ) the Debye model with 00=309 K [12], ( - • - ) the extended Debye model [ 3 ] and ( - - - ) the modified axially symmetric model [13], also shown in the figure. All FDFs carry the same normalization. Making use of these characteristic temperatures and modified form of expression (7) with /t* = 0.12 and m = 67 ainu, we have studied the effect of varying the degree ofanisotropy (d, d' ) on critical temperature To. In fig. 2 (a) is plotted the variation of Tc with d, d'. That is, while in one study we keep 6' fixed and vary d, it is vice-versa in the other. For
521
S.P. Tewari, P.K. Gumber / Effect of anisotropic phonons on superconductivity 10 -
200
(2a)
i: iI
i i i
160
i
I
.d
, / !~ : ', / .YIt, ; ',.//I ],",
8O
i °--
//i / i / ./ ;/ , ,
/,0
1.0
2.0
/I
I
,..
olj
,
100
A
/i
~ × 120
,
I ,
10-1
-"-+',J r ,"'4
3.0 z~.O ~) ( 1012 s-1 )
5.0
6.0
,,rt"-t
7.0
2x10- 2 12.0"
I
I
I
1.5
2.0
2.5
~ ~' ~
3.0 (2b)
10.0
Fig. 1. Comparison of the various phonon frequency distribution functions in zinc: (- • -) experimental [ 11 ]; (--) present model; (- - ) Debye model with 0o=309 K [ 12]; (- • - ) extended Debye model [ 3 ]; (- - - ) modified axially symmetric model [ 13 ].
%<
8.0
6.0 x .% 4.o
(u2)xy
. . . . . . . . . . . . . . . . . . . 2. . . . . . . . . . . . . .
{u2}z
x/2. 0
example, the curve m a r k e d I represents the v a r i a t i o n o f T~ with increase in a n i s o t r o p y o f the basal plane, i.e. 5, keeping that along the c-axis fixed, i.e. ~'. The e x p e r i m e n t a l value o f T~ (835 m K ) here corresponds to 6 = 2.07. Similarly the e x p e r i m e n t a l point on curve m a r k e d II is at 6 ' = 1.70. The value o f q = 1.262 e V / A 2 in the present case is less than 9% off from that o b t a i n e d using the m e a s u r e d cohesive energy. As the degree o f a n i s o t r o p y in the basal plane or along the c-axis is increased, T¢ decreases as is shown in fig. 2 ( a ) . While the rate o f decrease o f Tc with 6 ' is m o r e or less constant in the range 1 to 3, it is varying for the other, i.e. with & The rate becomes much faster when the degree o f a n i s o t r o p y is larger. In fig. 2 ( b ) is plotted the v a r i a t i o n o f total MSD, ( u 2) o, with the degree o f a n i s o t r o p y at T = 0 K, which occurs in the expression o f T¢. Also shown in the figure are the values o f M S D in z-direction, ( u 2) z, a n d M S D in the basal plane, 2 ( u 2 ) xy, for the curves I a n d II, respectively. In b o t h the cases total M S D is found to decrease with the increase in degree o f a n i s o t r o p y as shown in fig. 2 ( b ) . As the degree o f anisotropy, say 6, increases, the M S D decreases in an increase in the cofactor but a decrease in the exponential term o f expression ( 4 ) . F u r t h e r ( w ) ' is also found to increase with increase in 6. Both these factors c o m b i n e d together not only arrest the increase in the cofactor but also reverse any increase in T~. At high anisotropy, the increase in the exponential is
0.0 1.0
I 1.5
I 2.0 /
I 2.5
I 3.0
6~6
Fig. 2. (a) Variation of superconducting critical temperature, T¢, with the degree ofanisotropy in 6 7 Z n . Curve I represents the variation when the anisotropy parallel to the c-axis is fixed (i.e. 6' is fixed) while that in the basal plane is varied (i.e. 6 is varied). Curve II represents the same but now 6 is fixed and 6' is varied. ( • ) denote the experimental To. (b) Variation of the total mean square displacement (MSD) of the zinc atom in its crystal with the degree of anisotropy at T= 0 K. While curve I shows the variation of the total MSD for different values of 6 and a fixed value of 6', curve II represents the same for different values of 6' and a fixed value of 6. Also shown are the constant MSD along the zdirection, (u2)=, corresponding to curve I and 2( U2)xy corresponding to the curve II. much faster and therefore the decrease in Tc is much steeper. These results are m o r e or less true for the v a r i a t i o n o f Tc with &' except that the increase in exponential at higher 6' is not as large as in the earlier case. A n d therefore here there is no steeper decrease in T~ with increase in 6'. W h e n one increases the degree o f anisotropy in a given direction, it results in a redistribution o f p h o n o n modes. While the range o f p h o n o n m o d e s increases, the n u m b e r o f low energy phonons decreases, resulting in a decrease o f both the M S D and the critical temperature. F r o m our study we conclude that the effect o f crystal a n i s o t r o p y on superconductivity is fairly involved a n d significant. Hence it should be taken into
522
s.P. Tewari, P.K. Gumber / Effect of anisotropic phonons on superconductivity
a c c o u n t . H i g h l y a n i s o t r o p i c systems are d e t r i m e n t a l to the onset o f s u p e r c o n d u c t i v i t y . F u r t h e r s u p e r c o n d u c t i n g critical t e m p e r a t u r e is f o u n d to be intim a t e l y c o n n e c t e d w i t h t h e M S D o f the v i b r a t i n g ion. T h e r e f o r e , to o b t a i n c o n s i s t e n t results, o n e s h o u l d use the p h o n o n f r e q u e n c y d i s t r i b u t i o n f u n c t i o n obt a i n e d f r o m the M S D studies like M 6 s s b a u e r recoiiles fraction, X - r a y diffraction, etc. at low temperatures.
References [1 ] W.L. McMillan, Phys. Rev. 167 (1968) 331. [2] Th. Obenhuber, W. Adlassing, J. Zenkert, U. Narger, W. Potzel and G.M. Kalvius, Hypertine Interactions 33 (1987) 69. [ 3 ] W. Potzel, W. Adlassing, U. Narger, Th. Obenhuber, K. Riski and G.M. Kalvius, Phys. Rev. B30 (1984) 4980. [4] P.G. Tomlinson and J.C. Swihart, Phys. Rev. BI9 (1979) 1867. [ 5 ] P.T. Truant and J.P. Carbotte, Can. J. Phys. 51 ( 1973 ) 922. [6] S.P. Tewari and Paramjeet Kaur Gumber, Physica C 160 (1989) 545. [7] S.P. Tewari and Poonam Silotia, J. Phys.: Cond. Matter 1 (1989) 7535.
[8 ] S.P. Tewari and Poonam Silotia, J. Phys.: Cond. Matter 1 (1989) 5165. [9] J. Krumhansl and H. Brooks, J. Chem. Phys. 21 (1953) 1663. [ 10 ] G. Boato, P. Cantini, C. Salvo, R. Tatarek and S. Terreni, Surface Science 114 (1982) 485. [ 11 ] I.P. Eremeev, I.P. Sadikov and A.A. Chernyshov, Sov. Phys. Solid State 18 (1976) 960. [ 12] G. Seidel and P.H. Keesom, Phys. Rev. 112 (1958) 1083. [13] L.J. Raubenheimer and G. Gilat, Phys. Rev. 157 (1967) 586. [ 14] S.P. Tewari, Phys. Lett. 98A (1983) 280. [ 15 ] S.P. Tewari and Paramjeet Kaur Gumber, Phys. Rev. B41 (1990) 2619. [ 16 ] S.P. Tewari and Paramjeet Kaur Gumber, J. Phys.: Condens. Matter 1 (1989) 6307. [ 17 ] P.B. Allen and R.C. Dynes, Phys. Rev. B 12 ( 1975 ) 905. [ 18 ] I.R. Gomersell and B.L. Gyorffy, Phys. Rev. Lett. 33 ( 1974 ) 1286. [ 19 ] J.R. Schrieffer, D.J. Scalapino and J.W. Wilkins, Phys. Rev. Len. 10 (1963) 336. [20] S.P. Tewari and Paramjeet Kaur Gumber, Physica C 167 (1990) 307. [21] S.P. Tewari and Paramjeet Kaur Gumber, Physica C 168 (1990) 647. [22]B. Firey, F.W. de Wene, E. De Rouffignac and G.P. Alldredge, Phys. Rev. B28 (1983) 7210. [23] P. Entel and M. Peter, J. Low Temp. Phys. 22 (1976) 613. [24] K.A. Gschnaidner, Solid State Physics 16 (1964) 275.
Effect of anisotropic phonons on superconductivity S.P. T e w a r i a n d P a r a m j e e t K a u r G u m b e r Department of Physics and Astrophysics, University of Delhi, Delhi- 110 O07, India
Received 15 August 1990
Effect of crystal anisotropy which results in the anisotropic phonon frequency distribution function has been studied on superconductivity in a strongly coupled superconductor. The dependence of the superconducting critical temperature, To, on the degree ofanisotropy is quite involved. In highly anisotropic systems superconductivity is suppressed. The expression for Tc is related to the total mean square displacement of the vibrating ion, yielding the result that, for superconductivity, the phonon frequency distribution function from mean square displacement is preferred to that obtained from other studies for consistent physical parameters. Explicit calculations for 67Zn are reported.
It is well known that zinc is a superconductor with a critical temperature Tc=0.85 K [1 ]. A recently measured value o f Tc for 67Zn by Obenhuber et al. [2] is (835_+20) mK. They have also been able to measure the temperature dependent L a m b - M 6 s s bauer recoilless fraction in 67Zn along the c-direction and perpendicular to it at temperatures 4.2, 20.8 and 47 K in spite of a very sharp resonance width F ( = 4 . 8 X 10 -~1 eV) o f its M6ssbauer y-ray radiation (Eo = 93.3 keV). From the observed highly anisotropic recoilless fraction, f the ratio o f f perpendicular and parallel to the c-axis is ( f l / f l ) ~ 23 at 4.2 K and ( f ± / ~ ) ~ 2 1 0 0 at 47 K, [2,3], temperature dependent mean square displacement ( M S D ) of a zinc atom is extracted. It is found that at a given temperature MSD along the c-axis is much larger than the corresponding MSD in the basal plane. Clearly the motion of a zinc atom in its crystal is highly anisotropic and is essentially because o f the large c / a ratio equal to 1.86 in the hexagonal close packed structure o f zinc. From the earlier analysis of superconductivity in zinc one finds that it is a strongly coupled superconductor and can be described rather well using isotropic phonon modes in McMillan's equation [1] which is an outcome of the Eliashberg theory of the strongly coupled superconductor. However, later studies [4,5 ] on the electron-phonon mass enhancement factor, 2, in zinc show that p h o n o n anisotropy
must be considered. Very recently [6] such a study has been made and it turns out that the effect of crystal anisotropy on superconductivity is quite involved and significant. However, in the study a simple phonon frequency distribution function ( F D F ) , i.e. the extended Debye model [3] ( E D M ) , is used. Such a model does not take into account the presence of the planar modes in the basal plane and along the c-direction, expected to be present in such a highly anisotropic hexagonally close packed structure [ 7 10 ]. Very recently [8] a model for phonon F D F in zinc has been suggested, which not only takes into account the presence o f planar modes but is also able to successfully explain ( i ) t h e recently [2] observed MSD of the zinc atom both along the c-axis and in the basal plane in the temperature range 4.2 to 47 K, and (ii) gives gross features of the experimental phonon FDF, obtained from neutron scattering [ 11 ], much better than the FDFs given by other models [2,3,12,13]. We therefore use the suggested anisotropic phonon F D F to study the effect of these anisotropic phonon modes on the superconducting critical temperature in zinc. The superconducting transition temperature, T~, is calculated using the modified McMillan's equation [ 14-17 ], which is known to yield reliable results for 2 < 1 and hence can be confidently used to study superconductivity in a strongly coupled superconductor such as zinc [ 16,17 ] :
0921-4534/90/$03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
S.P. Tewari,P.K.Gumber/ Effectof anisotropicphononson superconductivity To- (w) ( 1.04(1+2) _) 1.2 exp - 2 _ / ~ . ( 1 + 0 . 6 2 2 ) , / ,
(1)
where/1" represents the electron-electron coulomb interaction; 2, the electron-phonon mass enhancement factor, is given by 2=2 f
Og2(W)F(w)
'? ) , w - l d w = M(w2
f o~2(w)F(w)wdw (3a)
ol2(w)f(w)w-ldw
structure. For such a highly anisotropic crystal structure, the phonon frequency distribution function along the c-axis ( i = z ) and in the basal plane (i=x, y) can be written in the following form [7-9,22]:
Fi(w)=Ajw 2 O<<,w<~wli =Biw
Wli~W~W2 i
~. 0
W • W2i
(6)
(2)
where r/is a purely electronic factor, M is the ionic mass, and
(w 2) = f
519
where A~ and B, are constants and can be determined using the conditions (i) F~(w) is continuous at w=w~ and (ii) total number of modes in a given direction is equal to N; and are given as N A i-
and
w3i/i
N ,
Bi=
w~i~ i ,
where
f o~2(w)F(w)dw (w) =
.
(3b)
f o~Z(w)F(w)w-ldw
i=x,y,z,
F(w) is the phonon density of states and ot2(w) represents the electron-phonon coupling constant. Making use of the generally accepted approximations (i) the constancy of electron-phonon coupling with phonon frequencies [ 1,6,14-16,18,19 ] and (ii) harmonic approximation, one obtains the following expression for To, when/z*=0: T~= 0.442h
M(U2)o exp
(1.56h(w)') -- q(U2)o
(4)
where 2 = 2~/(U2 )0 3h(w)' and ( u 2) o is the MSD of the vibrating mass of T = 0 K [20,21] and
f ot2(w)F(w)wdw
(w>'= f
o~2(w )F( w)dw
A~---'0.5 [6] - 0 . 3 3 l,
(5)
Using eq. (4) one can obtain an expression of Tc for /t*~0. In order to evaluate (U2)o and ( w ) ' the phonon FDF F ( W ) has to be known. As discussed earlier the observed mean square displacement of a zinc atom in its crystal is highly anisotropic and so is its crystal
8(i=x.~)=6
(~i ~ W 2 i / Wl i ,
and
6~=z)=6'.
In the FDF given by eq. (6) at low frequencies, only the w 2 term contributes so that the variation of the specific heat at low temperatures is T 3 dependent, i.e., the contribution of two-dimensional modes (w dependent) is absent at low frequencies. The relative number of two-dimensional modes to three-dimensional modes is determined by the parameter 6,. When 6~--.~ all the modes in a given direction would essentially be two-dimensional. That is in this case there would not be any coupling between the planes and the crystal would be extremely anisotropic. For other values of 6i, the anisotropy would be less. Thus, 6~ characterizes the degree of anisotropy in a given direction. Making use of the normalization condition i.e. (ii), one can show that the number of modes is not dependent on 6~. Since the number of modes in a given direction is fixed, equal to N, variation in 6i from one value to the other results essentially in the redistribution of the fixed total number of modes. Using eq. (6) and F(w)=2Fx#(w)+Fz(w), in evaluating ( u 2) o and ( w)' which when substituted in eq. (4) give the following expression for To:
S.P. TewarLP.K. Gumber / Effect of anisotropic phonons on superconductivity
520 0.442WlxyW1 ~ rc = X
/
expE--0"347
mWlxyWlz (2W1xyA2-~-- WlzAt2 ~l \2WlzAl + WlxyZ~'l]_]'
(7) where _( ~-0.5 AI - \ 6 2 _ 0 . 3 3 ],
( d ' - 0 . 5 "~ 3'i = \~77_--0-~-. 33j,
( ,3_025
(~3-0.25~
(~= W2xy/ Wlxy,
(~' = W2z/Wlz
W(1,2)xv=W(1,2)x=W(l,2)y
and
/t*=0.
When/1" ~ 0 the expression (7) gets modified. In the case of Debye approximation, i.e. the isotropic phonon frequency distribution function where in contrast to expression (6) Fi(w) oc w 2 in the entire frequency region, expression (7) reduces to the following well known expression: {-0.52mw~'~ Tc -~ 0.2WD exp~ ~ ).
(8)
It is evident from expression (7) that the dependence of Tc on crystal anisotropy is quite involved even though no anisotropic effect in the purely electronic parameter r/has been taken into account. Effect of anisotropic Fermi surface on various superconducting properties of metallic alloys is only marginal (23). Both the cofactor and the exponential term of expression (7) for Tc contain anisotropic effects. If one studies these anisotropy dependent parameters with the degree of anisotropy, i.e. with d~=w2i/wl~ where i( = x , y, z), one finds that with the increase in anisotropy, both the cofactor and the exponential term increase. However, the rate of increase for the exponential term is greater than that for the cofactor. This would result in the suppression of superconductivity in highly anisotropic systems. In the usual analysis of Tc, in the absence of relevant tunneling data, one resorts to a simple Debye approximation for the phonons with the cut off Debye temperature taken from the specific heat study. For example in the case of zinc, 0D is taken to be equal
to 309 K [ 1 ] which reproduces rather well the experimental specific heat [ 12] at very low temperatures, i.e. around 1 K. However, it is well known that in specific heat the low energy phonons are not given as much weightage as in the mean square vibrational displacement of an atom. These low energy elastic modes play a crucial role in superconductivity as is also evident from expression (2). Indeed one may obtain the expression (4) where Tc is directly related to the MSD of the vibrating atom. We therefore suggest that the phonon FDF derived from MSD studies, if available, should be preferred to that given by specific heat to calculate Tc. In fact the value of q(=2.362 eV//~ 2) for zinc which one gets using 0D ( = 309 K) and/z* ( = 0.12 ) [ 1 ], is more than 100% off from the value ( q = 1.164 eV/& 2) using the experimental result of cohesive energy [24]. When the measured value of cohesive energy is used one obtains 0D = 206 K to yield experimental Tc. This value of 0D also gives total MSD of zinc equal to approximately 7.98× 10 -3 ~2 at 4.2 K which is only 3% off from the recently measured value [3 ]. However, as has been emphasized zinc is highly anisotropic and therefore in a proper analysis due consideration should be given to its anisotropicity. Using the anisotropic phonon frequency distribution function given by expression (6) it has been found that 01z= 100 K and 02z = 170 K, 0~xy= 130 K and 02xy= 269 K (0~ 1,2)i= •W(1,2 )i/kB) explains successfully, amongst others [8], measured temperature variation of MSD of a zinc atom in the temperature range 47 to 4.2 K along the c-axis and in the basal plane, respectively. The phonon FDF shown in fig. 1 by solid line ( - - ) also reproduces the gross features of experimental phonon FDF (obtained from neutron scattering) [ 11 ] shown by ( - . - ) much better than that given by other dynamical models: ( - - ) the Debye model with 00=309 K [12], ( - • - ) the extended Debye model [ 3 ] and ( - - - ) the modified axially symmetric model [13], also shown in the figure. All FDFs carry the same normalization. Making use of these characteristic temperatures and modified form of expression (7) with /t* = 0.12 and m = 67 ainu, we have studied the effect of varying the degree ofanisotropy (d, d' ) on critical temperature To. In fig. 2 (a) is plotted the variation of Tc with d, d'. That is, while in one study we keep 6' fixed and vary d, it is vice-versa in the other. For
521
S.P. Tewari, P.K. Gumber / Effect of anisotropic phonons on superconductivity 10 -
200
(2a)
i: iI
i i i
160
i
I
.d
, / !~ : ', / .YIt, ; ',.//I ],",
8O
i °--
//i / i / ./ ;/ , ,
/,0
1.0
2.0
/I
I
,..
olj
,
100
A
/i
~ × 120
,
I ,
10-1
-"-+',J r ,"'4
3.0 z~.O ~) ( 1012 s-1 )
5.0
6.0
,,rt"-t
7.0
2x10- 2 12.0"
I
I
I
1.5
2.0
2.5
~ ~' ~
3.0 (2b)
10.0
Fig. 1. Comparison of the various phonon frequency distribution functions in zinc: (- • -) experimental [ 11 ]; (--) present model; (- - ) Debye model with 0o=309 K [ 12]; (- • - ) extended Debye model [ 3 ]; (- - - ) modified axially symmetric model [ 13 ].
%<
8.0
6.0 x .% 4.o
(u2)xy
. . . . . . . . . . . . . . . . . . . 2. . . . . . . . . . . . . .
{u2}z
x/2. 0
example, the curve m a r k e d I represents the v a r i a t i o n o f T~ with increase in a n i s o t r o p y o f the basal plane, i.e. 5, keeping that along the c-axis fixed, i.e. ~'. The e x p e r i m e n t a l value o f T~ (835 m K ) here corresponds to 6 = 2.07. Similarly the e x p e r i m e n t a l point on curve m a r k e d II is at 6 ' = 1.70. The value o f q = 1.262 e V / A 2 in the present case is less than 9% off from that o b t a i n e d using the m e a s u r e d cohesive energy. As the degree o f a n i s o t r o p y in the basal plane or along the c-axis is increased, T¢ decreases as is shown in fig. 2 ( a ) . While the rate o f decrease o f Tc with 6 ' is m o r e or less constant in the range 1 to 3, it is varying for the other, i.e. with & The rate becomes much faster when the degree o f a n i s o t r o p y is larger. In fig. 2 ( b ) is plotted the v a r i a t i o n o f total MSD, ( u 2) o, with the degree o f a n i s o t r o p y at T = 0 K, which occurs in the expression o f T¢. Also shown in the figure are the values o f M S D in z-direction, ( u 2) z, a n d M S D in the basal plane, 2 ( u 2 ) xy, for the curves I a n d II, respectively. In b o t h the cases total M S D is found to decrease with the increase in degree o f a n i s o t r o p y as shown in fig. 2 ( b ) . As the degree o f anisotropy, say 6, increases, the M S D decreases in an increase in the cofactor but a decrease in the exponential term o f expression ( 4 ) . F u r t h e r ( w ) ' is also found to increase with increase in 6. Both these factors c o m b i n e d together not only arrest the increase in the cofactor but also reverse any increase in T~. At high anisotropy, the increase in the exponential is
0.0 1.0
I 1.5
I 2.0 /
I 2.5
I 3.0
6~6
Fig. 2. (a) Variation of superconducting critical temperature, T¢, with the degree ofanisotropy in 6 7 Z n . Curve I represents the variation when the anisotropy parallel to the c-axis is fixed (i.e. 6' is fixed) while that in the basal plane is varied (i.e. 6 is varied). Curve II represents the same but now 6 is fixed and 6' is varied. ( • ) denote the experimental To. (b) Variation of the total mean square displacement (MSD) of the zinc atom in its crystal with the degree of anisotropy at T= 0 K. While curve I shows the variation of the total MSD for different values of 6 and a fixed value of 6', curve II represents the same for different values of 6' and a fixed value of 6. Also shown are the constant MSD along the zdirection, (u2)=, corresponding to curve I and 2( U2)xy corresponding to the curve II. much faster and therefore the decrease in Tc is much steeper. These results are m o r e or less true for the v a r i a t i o n o f Tc with &' except that the increase in exponential at higher 6' is not as large as in the earlier case. A n d therefore here there is no steeper decrease in T~ with increase in 6'. W h e n one increases the degree o f anisotropy in a given direction, it results in a redistribution o f p h o n o n modes. While the range o f p h o n o n m o d e s increases, the n u m b e r o f low energy phonons decreases, resulting in a decrease o f both the M S D and the critical temperature. F r o m our study we conclude that the effect o f crystal a n i s o t r o p y on superconductivity is fairly involved a n d significant. Hence it should be taken into
522
s.P. Tewari, P.K. Gumber / Effect of anisotropic phonons on superconductivity
a c c o u n t . H i g h l y a n i s o t r o p i c systems are d e t r i m e n t a l to the onset o f s u p e r c o n d u c t i v i t y . F u r t h e r s u p e r c o n d u c t i n g critical t e m p e r a t u r e is f o u n d to be intim a t e l y c o n n e c t e d w i t h t h e M S D o f the v i b r a t i n g ion. T h e r e f o r e , to o b t a i n c o n s i s t e n t results, o n e s h o u l d use the p h o n o n f r e q u e n c y d i s t r i b u t i o n f u n c t i o n obt a i n e d f r o m the M S D studies like M 6 s s b a u e r recoiiles fraction, X - r a y diffraction, etc. at low temperatures.
References [1 ] W.L. McMillan, Phys. Rev. 167 (1968) 331. [2] Th. Obenhuber, W. Adlassing, J. Zenkert, U. Narger, W. Potzel and G.M. Kalvius, Hypertine Interactions 33 (1987) 69. [ 3 ] W. Potzel, W. Adlassing, U. Narger, Th. Obenhuber, K. Riski and G.M. Kalvius, Phys. Rev. B30 (1984) 4980. [4] P.G. Tomlinson and J.C. Swihart, Phys. Rev. BI9 (1979) 1867. [ 5 ] P.T. Truant and J.P. Carbotte, Can. J. Phys. 51 ( 1973 ) 922. [6] S.P. Tewari and Paramjeet Kaur Gumber, Physica C 160 (1989) 545. [7] S.P. Tewari and Poonam Silotia, J. Phys.: Cond. Matter 1 (1989) 7535.
[8 ] S.P. Tewari and Poonam Silotia, J. Phys.: Cond. Matter 1 (1989) 5165. [9] J. Krumhansl and H. Brooks, J. Chem. Phys. 21 (1953) 1663. [ 10 ] G. Boato, P. Cantini, C. Salvo, R. Tatarek and S. Terreni, Surface Science 114 (1982) 485. [ 11 ] I.P. Eremeev, I.P. Sadikov and A.A. Chernyshov, Sov. Phys. Solid State 18 (1976) 960. [ 12] G. Seidel and P.H. Keesom, Phys. Rev. 112 (1958) 1083. [13] L.J. Raubenheimer and G. Gilat, Phys. Rev. 157 (1967) 586. [ 14] S.P. Tewari, Phys. Lett. 98A (1983) 280. [ 15 ] S.P. Tewari and Paramjeet Kaur Gumber, Phys. Rev. B41 (1990) 2619. [ 16 ] S.P. Tewari and Paramjeet Kaur Gumber, J. Phys.: Condens. Matter 1 (1989) 6307. [ 17 ] P.B. Allen and R.C. Dynes, Phys. Rev. B 12 ( 1975 ) 905. [ 18 ] I.R. Gomersell and B.L. Gyorffy, Phys. Rev. Lett. 33 ( 1974 ) 1286. [ 19 ] J.R. Schrieffer, D.J. Scalapino and J.W. Wilkins, Phys. Rev. Len. 10 (1963) 336. [20] S.P. Tewari and Paramjeet Kaur Gumber, Physica C 167 (1990) 307. [21] S.P. Tewari and Paramjeet Kaur Gumber, Physica C 168 (1990) 647. [22]B. Firey, F.W. de Wene, E. De Rouffignac and G.P. Alldredge, Phys. Rev. B28 (1983) 7210. [23] P. Entel and M. Peter, J. Low Temp. Phys. 22 (1976) 613. [24] K.A. Gschnaidner, Solid State Physics 16 (1964) 275.