Ultrasonic attenuation in a type II superconducting alloy

J. Phys. Chem. Solids

PergamonPress 1970.Vol. 3I,

pp. 1687-I69I. Printedin Great

ULTRASONIC ATTENUATION SUPERCONDUCTING

Britain.

IN A TYPE ALLOY

II

B. R. TITTMANN Science Center, North American Rockwell Corporation, Thousand Oaks, Calif. 9 1360, U.S.A. (Received

5 September

1969: in revisedform

I November

1969)

Abstract- Ultrasonic attenuation measurements have now been obtained on a magnetically reversible type-11 superconductor with both a high Ginzburg-Landau parameter (Q > I) and with BCS coherence length > electronic mean free path. For applied fields close to H,,, the data appear to agree with dirty-limit (&,/1* I) theory which predicti a linear dependence of a on H. At fields farther from H,,, a(H) deviates from linearity and appears to fit empirically to a parabolic function of H. At very low fields, H 2 H,, the H- and T-dependence is in apparent agreement with another theory. Physical insight into the behavior in this region could be obtained with a model in which vortices are replaced by cylinders of completely normal material imbedded in a superconducting matrix. 1. INTRODUCTION THEORETICAL

and experimental work has been carried out on the ultrasonic attenuation in clean type II superconductors characterized by having an electronic mean free path greater than the BCS coherence length, 1> &, and having a small Ginzburg-Landau parameter, KG = 1. Although there have been theoretical[ l-51 studies of the dirty regime characterized by I< eu and KG > 1 only few experimental investigations [6,7] have been reported and discussed in the light of recent theory [ l-51. One [6] of these is confined to high-field superconducting alloys, and does not study electron-phonon interaction which is basic to the present investigation. The other experiment [7], performed on the alloy Mo25 per cent Re (KG = 4), yielded a linear dependence of the electronic attenuation on magnetic field near the upper critical field H,, in qualitative agreement with the dirty limit theory[l-51. However, the measured slope was in disagreement with the predicted value and did not show the temperature dependence expected from the theory [ l-51 For H Q H,, no mention was made of the degree of magnetic reversibility and no interpretation was given for the ultrasonic data. The present experiments were undertaken to see if a more quantitative comparison with the theory [l-5] was feasible and in particular MUCH

to explore the region in which H 4 H,, where few data for the electronic attenuation in the dirty regime are available. For this purpose a well characterized magnetically reversible crystal with reasonable T, and Zf,, values and moderately high KG and to/1 values was necessary. These requirements could be fulfilled with the solid solution alloy V-5.6 at% Ta and single, oriented crystals of this composition were grown by H. Nadler of this laboratory. Previous caloric [8] and magnetization[9] studies and the present resistivity and ultrasonic data combine to render the alloy a wellcharacterized metal with KG = 5 and to/1 = 5. The ultrasonic measurements [ lo] display a substantial electron-phonon interaction, a high degree of magnetic reversibility, theoretically [ 1l- 131 predicted upper critical fields H,, and a behavior consistent with previous studies [8,9] so that a quantitative comparison with theory is possible. The lack of hysteresis (except near the field of first flux penetration If& means that the attenuation data in the range Hfp < H 4 H,, is physically significant and, therefore, its magnitude and temperature dependence should be amenable to detailed analysis and quantitative interpretation. 2. EXPERIMENTAL TECHNIQUE

The apparatus and measurement techniques used in this experiment are essentially the same as those already described elsewhere

1687

B. R. TI~MANN

1688

[ 141, except that the experimental arrangement allowed measurement on specimens with relatively low demagnetizing coefficient. Sample shape is well known to influence the initial flux penetration field HfP and the slope [ 15,161 of the magnetization curve near H,, in type-II superconductors, the latter especially when the condition~l6J (2Kc2 - 1)p @ I is not satisfied. Although in the present experiment KG = 5 so that the above condition is satisfied, it was possible to reduce the effective demagnetization factor[l7] to IZ~~= 0.015 by placing the cylindrical sample (length L. = 3.8cm, dia. D = 1.3 cm) in between tapered cylindrical conductors machined from polycrystalline rods of nominally V-56 at% Ta com~sition. The attenuation was measured by the pulse method at a frequency of 0*54 GHz for iongitudinal waves along the (110) direction and parallel to the applied magnetic jield. The magnetic fields of initial flux penetration HfP and the upper critical fields H, are clearly defined by the attenuation data both for increasing and decreasing fields. The only hysteresis was found near H = 0 where some flux trapping can be expected. 3. RESULTS

For dirty limit &,/l S 1 superconductors no simple theoretical expression is available describing the attenuation over the whole extent of its magnetic field dependence but theoretical predictions have been made for two limits; the region121 for fields close to the flux penetration field HfP and the region [ 1,3-51 for fields close to H, corresponding respectively, to the limits of isolated, weakly interacting vortex lines, or D/S < 1, and to the case of many strongly interacting vortex lines forming a very dense lattice, or D/S = 1 (where D is the effective core diameter and S is the spacing between vortex lines). ( 1) Ultrasonic atfenuatio~

forH < H,,

A theoretical expression for the electronic attenuation of longitudinal soundwaves

rr(H, T)la, as a function of applied magnetic field H near H,, is given [ 1,3-51 for the limit [o/l * 1 by cw(H, T)/a, = I--

w,c 8’,7’kgT,

Ha--H [2Kz2(t)

-

‘]/?‘(:),)

where C,(t) is a universal function[ 1] of the reduced temperature t = TIT,, e is the electronic charge, kB is Boltzmann’s constant, c is the speed of light, /3 = l-16, and pn is the normal state electronic resistivity. Equation (1) predicts a linear dependence of the attenuation on the applied field near H,,. Figure 1, which shows the measured attenuation as a function of the field for three different temperatures, exhibits indeed such a linear dependence near Hc,. In order to verify equation ( 1), the magnetization parameter K2 was calculated from the data and compared with values obtained in previous& 91 studies. In terms of the slope at H,, of the field dependent attanuation curve the parameter Kp may be written as

From the measured values of the resistivity temperature T,, the numerical values[l] of C,(t), the observed slopes and equation (2) the parameter K2 was calculated. Figure 2 displays the ultrasonic values of Q in rough agreement with previous values obtained directly from caloric [8] and magnetization[9] data. For T = T, the accuracy of (&x/aHi) at H = H,, is limited because of the limited region for which the curve is linear. The extrapolation to T = T, of the values of K~(T) yields K2(T = T,) = KG = 4.7. A comparison with theory [ 1 l- 131 of the temperature dependence of u2 shows no disagreement for &Jl = 5 without p-wave scattering. The theory [ 1,3-57 does not predict the iield H < H,, down to which the linear solution holds, nor how the length of the linear segment changes with temperature. However, an estimate has been made[ 181 that the linear dependence

pn, the transition

ULTRASONIC

ATTENUATION

\\\ 11 T= 4.3

o-

IN A TYPE

K

FNormol

II SUPERCONDUCTING T= 1.7K

T=3*0K

Siote

-Fe

-

f

Moki

1689

ALLOY

E,/Pw>

-

--F

I

\

0.2

\

k-1

ii

\

of \ _ 0.4 E 0 1\

\

i

\

,

Empirical

J

-

Vortex spacing equals penetration depth

I I.0

t

“fP

I

0

_1

6.0

4.0

2.0

IWO

u.0

H(kG)

Fig. 1. Ultrasonic attenuation a, relative to the attenuation Q, in the normal state as a function of magnetic field H for three representative temperatures at a frequencyf = 0.54 GHz in a sample of V-5.6 at% Ta (with transition temperature T, = 4.7”K, and normal state resistivity pn = 4.1 X 10-“&m). IO-0

should hold in the regime (H,,-_)/H,, 4 [(&,/l)( 1 - T/T,)IZ. The observed temperature dependence is in reasonable agreement with this criterion for the to/l = 5 purity value of the V-56 at% Ta specimen.

9.0 6.0 7-o

t

6.0

(2) Ultrasonic attenuation for H >, HrP For weak magnetic fields such that BeX2 Q 1 the electronic attenuation &(H, T)/a, as a function of the applied magnetic field B is given [23 by

3-o 2.0

.

Ultrasonic

n

Calorimetric

I.0

0

i I

(Tc ~4-7~K)

I

I

I.0

2.0

+k(T)$

(Tc=4_3OK)

Mognetization:O(Tc A (Tc n4.S°K) +(Tc I

3.0 T(OK)

unknown) 5 4*S°K) I

4.0

cz

1

tGO

Fig. 2. Magnetization parameter I+ as a function of temperature. The values obtained with the help of equation ( 1) from ultrasonic data on a single crystal oriented along the (110) direction are compared with values obtained directly from previous caloric[l] and magnetization[9] data on polycrystalline samples of nominally V-5 at% Ta composition.

k(T) =AKG"~-

A0 2k,T

A0

cosh2 2k,T

(2)

where A0 is the temperature dependent superconducting energy gap as given by the BCS theory [ 193, A is the penetration depth and A is a proportionality constant. Equation (2) predicts a linear dependence of the attenuation

1690

B. R. TITTMANN

on the applied field[20] H near HfP in agreement with the data on Fig. 1. From the ultrasonic data for H,,, KG, and T, plus the tabulated values [21] for the energy gap it was possible to calculate the slopes of the attenuation as a function of temperature. As shown in Fig. 3, the theoretical slopes when fitted to the data near T = T, (A -L 0.31) are in reasonable

I

and matrix have, nearly the same moduli, the total attenuation is given by a(H, UH,

T) = (as+ Q) [lT)

material assumed to be made up of N cores each of diameter D and unit length. With the substitution of %I% = 2/(exp A,IkBT + 1) from the BCS theory[l9] and taking N = H/&,, the slopes are given by

%,H

Vortex

core

model

I.0

C

0.2

0.6

o-4

A0 ?r?tanh_ 4 40

2k,T

(4)

where &, = 2.1 X lo-’ G cm2 (the flux quantum). The predicted [22] Ginzburg-Landau coherence distance for the dirty limit &,/l % I is .$G2= 0.723 &r)/( I- TIT,).

2.0

‘.l 0 * ‘E .’ .”

(3)

where CY,= a(H = 0, T), a, = a(H = H,., T), aB is the non-electronic background attenuation, and V(T) is the total volume of normal

a(H, 7’) ---a, = : -’ ‘0 ;

J’(H, T)] + (a,+ a~)

0.6

I.0

T/Tc

Fig. 3. Electronic attenuation per umt field [a(H. T) a(H = 0. T)l/H = Cn,, - cu,)/a,,H as a function of temperature in the low magnetic field region H,,, < H =S H,,. The ultrasonic data is compared with theory [2] and the results of a simple vortex core model.

agreement with those obtained from the ultrasonic data except in the region T + 0°K. In an effort to gain physical insight into the behavior of the attenuation in this region a model was constructed in which vortices were replaced by cylinders for which the energy gap parameter is nearly zero. The matrix surrounding the cylinders acts with a BCS zero field energy gap. If the normal core

A reasonable fit of the calculated slopes to the experimental values in the temperature regime near T = T, could be obtained when D = 2f,. Although the agreement near T = O”K is poor, it is interesting to find approximate agreement with experiment in the magnitude and temperature dependence of the attenuation for such a simple model. One might expect the above considerations to break down when the vortices are made to approach (by increasing the field H) one another to within the critical distance given [22] by the penetration depth A. This is born out by the data such as shown in Fig. 1 where the limits of linear dependence of attenuation on field approximately coincide with field values H,,i, such that the critical spacing S, = (H,,,&,)-I’” = A. (3) Ultrasonic

attenuationfor Hf, < H < H,, As seen in Fig. 1 the data exhibits a substantial transition region lying between the two limits H = Hf, and H = H,,. Apparently no

ULTRASONIC e--t

I-

H /

ATTENUATION HC2)

l/2

o-4

O-6

1N A TYPE

0.2

0

IL SUPERCONDUCTING

)a(i-h't"2

Fig. 4. Electronic attenuation (1 -a’) as a function of field ( 1- H/H,,)“Z for several temperatures where (Y’= a(H, T)/a(H = H,,, T). Note that far away from H,, the data agree well with f 1- a’) CQ t I- h’)*‘* where h’ = H/H,,.

theoretical treatment covers this region. Figure 4 shows the field dependent attenuation below H,, down to about H/H,, = O-3 plotted against (1 -H/HC2)1’2. The resulting linear plots suggest that (l-~~‘)~(l-/z’)“~ where N’= a(H, Q&H= HCt, T) and h’= HjH, is a possible empirical fit. 4. CONCLUSION

For magnetic field values sufficiently close to the upper critical field H,,, the field and temperature dependence of the attenuation is not in disagreement with the dirty-limit theory. The only hysteresis observed in the ultrasonic measurements is near H = 0 where some flux trapping can be expected. This magnetic reversibility makes it possible to interpret the data in the range H 6 IS,,. The field and temperature dependence appears to be in agreement with the theory. Physical insight into the behavior in this regime could be obtained by a model which replaces vortices by cylinders in which the energy gap parameter is nearly zero. An empirical fit to the data was possible in the region of intermediate field values yielding a

1691

parabolic field dependence of the attenuation. Although most of the field dependence of the attenuation thus appears to be described by three different functions, it is interesting to note that the data behaves as if it might just as well be described by a single smoothly varying function if such were available. REFERENCES K., Sl{~erc~ndi~~t~~ity (Edited by R. D. Parks), Vol. 2, p. 1035. Marcel Dekker, New York, [ 1969). {The factor fs’ in equation (224) is in error and should read &r*; this has been confirmed by K. Maki, private communication.) 2. GALAIKO V. P. and FALKO 1. I., J. exp. theor. Phys. 52, 976 (1967); English transi: Soviet Phys. JETP 25,646 61967). MAKI K. and FU LDE P., Solid St. e~~~~?~~. 5.2 1 t 1967). MCLEAN F. B. and HOUGHTON A., Ph_vs. Rev. 157,350 C1967). MAKi K., Phys. Rev. 148,370 (1968). SHAPIRA Y. and NEURINGER L. J., Phys. Rec. 154, 375 (1967): also NEURINGER L. J. and SHAPiRA Y., Phys. Rar. Left. 17,81 (1966). 7. GOTTLIEB M.,JONES C. K. and GARBUNY M., Phys. Lerr. 25A, 107 t 1967). 8. HAKE R. R. and BRAMMER W. G., Phys. Rev. 133,A71911964). 9. CAPE J. A., unpublished data for polycrystaiiine V-5 at% Ta; also BARNES L. J. and HAKE R. R., unpublished. 10. TITTMANN B. R. and BGMMEL H. E., Phys. Lett. 28A. 396 f i 968). E. and WERTHAMER N. R.. Phys. 11. HELFAND Rev. Let?. 13,686 t 1968); Phys. Rec. 147.288 11966). N. R., HELFAND E. and 12. WERTHAMER HOHENBERG P. C., Phys. Rec. 147,295 (1966). 13. El LENBERGER G., Phys. Rec. 153,584 f 1967). B. R. and BGMMEL H. E., Rev. 14. TITTMANN scient In~r~~t~. 39, 614 f 1968); Rev. scient. Inslrff~. 38,1491 (i967). 15. KULIK I. O., JETP PIS’MA v Redaktisyu 3, 395 ( 1966); English Transl: JETP lett. 3,259 (1966). J. M., Phys. Rev. 16. CAPE J. A. and ZIMMERMAN 153,416(1967). 17. STONER E. G.. Phil. Mug. 36,803 (1945). 18. MAKI K., Private communication. 19. BARDEEN J.. COOPER L. and SCHRIEFFER J.. Ph_vs. Rec. 108, 1175 (1957). 20. Since K,; = 5 for this material, /4nMi Q H in the range 0.2 < H/H,, < 1.0 (e.g. at H = 0.5 H,,, I4&‘i = fH -N,,)/(~K,‘l,p = 0.009 H,, < H) so that B = H. 21. MUHLSCHLEGEL B., 2. Phys. 155,313 (1959). 22. See, eg. De Gennes P. G., S~perconductjvi~ of Metals and Alloys (Translated by P. A. Pincus). Benjamin, New York (1966). I. MAKI

(l-a’

ALLOY