- Email: [email protected]
The damping of helicon resonances in pure type II superconductors
Solid State Communications, Vol. 5, pp. 585-589, 1967. Pergamon Press Ltd. Printed in Great Britain
THE DAMPING OF HELICON RESONANCES IN PURE TYPE II SUPERCONDUCTORS * B.W. Maxfield Laboratory of Atomic and Solid State Physics and Department of Physics, Cornell University, Ithaca, New York, U. S. A. (Received 19 May 1967 by J. A. Krnmhansl)
The field dependence of the helicon resonance parameters in the mixed state of pure niobium have been measured. We find the quality factor, Q, of the observed resonance to be field independent in the mixed state and field dependent in the normal state. From this observation we infer a field independent Hall angle in the mixed state.
HELICON resonances have been observed in niobium having residual resistance ratios (RRR)
where w~is the cyclotron frequency at the upper critical field, H02 7 is the electron-lattice col-
of 16001 and 41002. In this letter we report measurements of helicon resonances in a number largeofas niobium 11, 000.samples We observe with resistance a field dependent ratios as Q for the helicon resonance in the normal state but a field independent Q in the mixed state. We interpret this observation to mean that the ~ angle in the mixed state is field independent, Even in specimens of comparable purity, our experimental results do not agree with those of Druyvesteyn et al.2 who find a field dependent Hall angle in1E~mtxedstate using a similar helicon resonance method. our interpretation of a field independent Hall angle does not agree with conventional d. c. Hall effect measurements3 which yield a field and current dependent Hall angle but it is possible that the a. c. and d. experiments measure different properties of the mixed state.
lision time and H is the field In the vortex core. A qutte due toOtherwse, Nozieres and 5 different gives tan model 8H = W~T. these two Vinen models give similar predictions.
Both the above models predict that helicon-like oscillations of the flux vortices should be possible. The heltcon resonance parameters should depend in the usual way on the Hall angle, so measurements of the helicon resonance parameters may distinguish between these two models. We have determined the resonance parameters for mixed state helicon standing wave resonances in three flat plate samples of niobium8 with resistance ratios between 2000 and 11, 000 using a standard crossed-coil technique. Both fundamental and higher order standing waves have been seen but only the fundamental resonance was studied in detail. A static magnetic field, H 0, is applied perpendicular to the flat plate. The There are two models of flux motion in static magnetic field, an exciting magnetic field the mixed state which take into account the Hall h0 sin w t and the axis of the pickup coil are mueffect of the electrons in the normal vortex core. tually orthogonal. The pickup voltage is detected Each model predicts that the observed Hall angle, phase-sensitively with a product detector at coneH, should be the Hall angle in the normal vortex stant static magnetic field as a function of the core. The models differ in the assumed depenfrequency of the exciting magnetic field. dence of the field in the core upon the applied 4 In no case was a minimum h field. A model due to Bardeen and Stephen 0 required yields a Hall angle given by tan ~jj = W027 H /H,2 to observe the resonances but the pickup voltage *This work is supported mainly by the U.S. Atomic Energy Commission Contract No. AT(30-1) -2150, NYO-2150-29 and to a lesser degree by Advanced Research Projects Agency, MSC Report No. 690. 585 ~.
586
DAMPING OF HELICON RESONANCES
Vol. 5, No. 8
T~4.2°K h 0. 5 Oe 0 Nb-4a 1.5
-
•
Nb-So
0
______.—~--~-~
FIG. 1
0.5
I
0
0.5
I
H /H
1.0 c2
was only linear in the exciting field amplitude above some small h 0 that varied from sample to sample but was never larger than I Oe at a frequency of 400 Hz at fields above H02/2. This exciting field corresponds 2. Fromtothis an induced we conclude current that pinning the flux vortices should be negliof about 100 of A/cm gible for h 0 > 50e, a typical exciting field value, At fixed magnetic fields, we have determined the resonant frequency and Q of the mixed state helicon resonance. Figure 1 shows the dependence of the Q upon H0 for the two2-1 be~t )a instead Q, whereActually, u = tan e~,As samplesofstudied. we plot expected, u = (4Q u is field-dependent in the normal state above H,2 but notice that u remains essentially constant in the mixed state below H,2. The error bars show the uncertainty in an individual measurement due primarily to a small fraction of the pickup voltage (<5%) that does not change sign when the magnetic field is reversed. Meaningful values of u could not be obtained from the Q for sample Nb-3 having a RRR = 2000. In Fig. 2 we show the field dependence of the resonant frequency, V~ Only strain free samples gave data .
1.5
2-lI~ where The field Q isdependence determined of directly u=(4Q from the shape of the resonance curve. Sample Nb-4a has a HER = 6300 and sample Nb-5a has a RRR = 11, 000. Both samples are 0.31 mm thick. Extrapolated normal state curves do not pass through the origin because of a large normal state magnetoresistance.
similar to that shown in Fig. 2. Samples that had been spark cut or mechanically worked all showed a strong field dependence of v,. in the neighborhood of H,2. This was observed in the been sample spark reported cut. The in Ref. helicon 1, and resonant that sample frequency had for the first standing wave resonance in a plate of thickness d is related the 2. average Since Qresistivity, Is field independent, p, and Q by the v,. field = pQu/i..tod dependence of p is Just the field dependence of ~ Figure 2 shows that p is nearly linear in H 0 which is just the behavior expected if p is the flux-flow resistivity, p, = ~N HO/H, 2. .
There are however, two important differences between the resistivity that we determine and the usual flux-flow resistivity. First, at finite temperatures we do not see a sharp change inthe resistivity at H,2 which has been observed in all flux-flow studies to date. A second important difference is that v~ does not extrapolate to zero at B = 0 but has a finite value. This may indicate the presence of a large, field independent damping at fintte temperature.
Vol. 5, No. 8
DAMPING OF BELICON RESONANCES
I
•
4.2°K Ij.
1.5
0 • 6
Nb-3 Nb—4a Nb-So
587
(Ha)
800Hz 588 Hz 691 Hz
Z’r(Ho) J4(H~ 2)
0__o— 1.0
0.5•
FIG. 2
0
1.0
H,/ H82
The resonant amplitude, A~,should give information about the damping of the mixed state resonance as it does for the normal state helicon resonance. We expect that each oscillating vortex contributes about equally to the pickup voltage since each vortex contains one flux quantum. Thus A,. should be proportional to the magnetic induction, B (if all vortices are oscillating) and the average damping experienced by the oscillating flux vortices. We observe afield independent damping and therefore we would expect to observe A,. proportional to B if all the vortices are oscillating. Fairly linear behavior is observed for H~ > H,2/2 (except for a small dip at H,2 in some specimens) but there are marked deviations from linearity at lower fields. The resonant amplitudes go rapidly to zero for a finite magnetic induction which says that not all the flux in the sample is responding to the a. c. fields for H~,< H,2/2. Magnetoresistance lit the normal cores will also cause A,. to deviate from a linear behavior. Although the field we dependence do not understand of A,. is qualitatively the details ofunderstood, the field dependence.
15
The reduce resonant frequency v~(H~)/v, (H,2) as a function of H~/11,2. Specimen Nb- 3 has aRRR= 2000.
Our observation of a field independent Q in the mixed state does not agree with the expertments of Druyvesteyn et al. 2 We find this surprising since the Hail ii~1~ in their material is comparable to ours and their material is also Union Carbide electrolytic niobium. The samples that we studied were all polycrystalline, consisting of crystallites about 1 mm on a side. The magnetoresistance varied considerably from sample to sample, presumably because the nwnber of crystallttes with open orbits near the plane of the sample varied. The largest Hall angle at 11,2 (about 50°)was a specimen having a resistance ratio of 6300 whereas specimens having resistance ratios as large as 11, 000 were studied. The highest resistance ratio specimens however, did have the lowest critical currents. We can make a few comments on the results of Druyvesteyn et al. 2 At the frequencies that they must have u~~’anexciting field of 1 Oe corresponds to2,anso induced it is certainly current density possible of about that all non-linear 2000 A/cmamplitude effects were absent. In our previous work1 the largest induced current
588
DAMPING OF HELICON RESONANCES
Vol. 5, No. 8
density used was about 1500 A/cm2 and the largest induced current densities used in the expertments reported in this letter are 500 A/cm2 at the resonant frequency. Druyvesteyn et al.2 find a field dependent Hall angle in the~E’ed state. If we had tried to determine u directly from the amplitude, then our results would agree qualitatively with those of Druyvesteyn et al.2 However, they say that u is determined ?i~i the Q which is the method that we used. It is worth noting that we could not get meaningful u values from the resonance curves in sample Nb-3 having a ERR = 2000, however, a “value” of u was easily determined from Ar. Druyvesteyn et al.2 give Hall effect data for a sample havjii:IIRR = 2500.
a. c. experiments where it is not necessary for flux to cross the metal-vacuum boundary. It is also possible that the superconductor is damaged In the area of the contacts. The presence of probes has affected Hail effect and resistivity measurements In the alkali metals but to the best of our knowledge, not in materials like nbbium. An a. C. Hall effect experiment where the magnetic field Is kept constant and the current through the sample modulated at a frequency well away from the helicon resonant frequency might help clarify the difference between the helicon and d. c. experiments. The superposition of a direct current might give clues of some real difference between what the a. c. and d. c. expertments actually measure.
A frequency dependent background (which is sometimes, but not always, present) distorts the shape of the resonance curve and hence determining the Q when a large background is present can lead to large errors. To eliminate most of these errors (when present) we always take resonance curves for both positive and negative field directions. We have also fit our curves to the Chambers and Jones8 expression fdr normal helicons and find excellent agreement between the observed and predicted frequency response.
In summary, we find helicon resonances in the mixed state of pure niobium having a large Hail angle. At a fixed magnetic field, the frequency dependence of the mutual inductance between the crossed-coils used in the experiment is given to within a few percent by the expression derived by Chambers and Jones8 for normal state helicons. The niobium is pure enough to determine the hail angle with an accuracy of ± 10% directly from the Q of the resonance. We find that the Hall angle in the mixed state Is independent of the applied field and equal to the Hall angle at 11,2. Although the resistivity In the mixed state is nearly the flux-flow resistivity there appear to be important differences in the case of pure superconductors that need further study. Our experimental results agree with the field independent Hail angle predicted by Nozieres and Vinen5 as opposed to the field dependent Hall angle predicted by Bardeen and Stephen.4
Our observation of a field independent Hall angle also does not agree with the field dependent Hall angle found by Reed et al.3 using a conventional d. c. method. The~1~vo expertments, however, may measure different properties of the superconductors. In both cases the Hall effect of moving fluxotd.s is observed with comparable current densities. However, in the d. c. experiments flux must cross a metalvacuum boundary. Since the superconducting properties in the vicinity of a boundary are different from the bulk, It is conceivable that d. c. experiments could give results different from
Acknowledgments We are extremely grateful to Dr. R. W. Meyerhoff and the Union Carbide Corporation for supplying the samples and thereby making this investigation possible. Discussions with J. W. Wilkins have been very helpful. -
References 1.
MAXFIELD B.W. and JOHNSON E. F., Phys. Rev. Left. 16, 652 (1966).
2.
DRUYVESTEYN W. F., VAN GURP G.J. and GREEBE C .A.A.J. Phys. Left. ~,
3.
REED W.A., FAWCETT E. and KIM Y.B., Phys. Rev. Left. 14, 790 (1965).
4.
BARDEEN JOH}~and STEPHEN M.J.,
5.
NOZIERES P. and VINEN W. F., Phil. Mag. 14, 667 (1966).
Phys. Rev. 140, A1197 (1965).
248 (1966).
Vol. 5, No. 8
DAMPING OF HELICON RESONANCES
6.
Dr. R. W. Meyerhoff of Union Carbide Corporation, Indianapolis, Indiana, prepared these specimens from electrolytic niobium subsequently outgassed at 10’° torr.
7.
KIM Y.B., HEMW~EADC.F. and STRNAD A.R., Phys. Rev. 139, A1163 (1965).
8.
CHAMBERS R.G. and JONES LK., Proc. Roy. Soc. A270, 417 (1962).
Die Feldabheingtgkeit der Parameter der Heliconresonant von reinem Niob im gemischten Zustand wurde gemessen. Die Gilte Q erwies sich hierbei als feldunabNângig im gemisditen Zustand und feldabhängig im normalleitenden Zustand, woraus wir scblierssen, darsa der Hallwinkel tm gemischten Zustand feldunabhâ~ngig1st.
589
THE DAMPING OF HELICON RESONANCES IN PURE TYPE II SUPERCONDUCTORS * B.W. Maxfield Laboratory of Atomic and Solid State Physics and Department of Physics, Cornell University, Ithaca, New York, U. S. A. (Received 19 May 1967 by J. A. Krnmhansl)
The field dependence of the helicon resonance parameters in the mixed state of pure niobium have been measured. We find the quality factor, Q, of the observed resonance to be field independent in the mixed state and field dependent in the normal state. From this observation we infer a field independent Hall angle in the mixed state.
HELICON resonances have been observed in niobium having residual resistance ratios (RRR)
where w~is the cyclotron frequency at the upper critical field, H02 7 is the electron-lattice col-
of 16001 and 41002. In this letter we report measurements of helicon resonances in a number largeofas niobium 11, 000.samples We observe with resistance a field dependent ratios as Q for the helicon resonance in the normal state but a field independent Q in the mixed state. We interpret this observation to mean that the ~ angle in the mixed state is field independent, Even in specimens of comparable purity, our experimental results do not agree with those of Druyvesteyn et al.2 who find a field dependent Hall angle in1E~mtxedstate using a similar helicon resonance method. our interpretation of a field independent Hall angle does not agree with conventional d. c. Hall effect measurements3 which yield a field and current dependent Hall angle but it is possible that the a. c. and d. experiments measure different properties of the mixed state.
lision time and H is the field In the vortex core. A qutte due toOtherwse, Nozieres and 5 different gives tan model 8H = W~T. these two Vinen models give similar predictions.
Both the above models predict that helicon-like oscillations of the flux vortices should be possible. The heltcon resonance parameters should depend in the usual way on the Hall angle, so measurements of the helicon resonance parameters may distinguish between these two models. We have determined the resonance parameters for mixed state helicon standing wave resonances in three flat plate samples of niobium8 with resistance ratios between 2000 and 11, 000 using a standard crossed-coil technique. Both fundamental and higher order standing waves have been seen but only the fundamental resonance was studied in detail. A static magnetic field, H 0, is applied perpendicular to the flat plate. The There are two models of flux motion in static magnetic field, an exciting magnetic field the mixed state which take into account the Hall h0 sin w t and the axis of the pickup coil are mueffect of the electrons in the normal vortex core. tually orthogonal. The pickup voltage is detected Each model predicts that the observed Hall angle, phase-sensitively with a product detector at coneH, should be the Hall angle in the normal vortex stant static magnetic field as a function of the core. The models differ in the assumed depenfrequency of the exciting magnetic field. dence of the field in the core upon the applied 4 In no case was a minimum h field. A model due to Bardeen and Stephen 0 required yields a Hall angle given by tan ~jj = W027 H /H,2 to observe the resonances but the pickup voltage *This work is supported mainly by the U.S. Atomic Energy Commission Contract No. AT(30-1) -2150, NYO-2150-29 and to a lesser degree by Advanced Research Projects Agency, MSC Report No. 690. 585 ~.
586
DAMPING OF HELICON RESONANCES
Vol. 5, No. 8
T~4.2°K h 0. 5 Oe 0 Nb-4a 1.5
-
•
Nb-So
0
______.—~--~-~
FIG. 1
0.5
I
0
0.5
I
H /H
1.0 c2
was only linear in the exciting field amplitude above some small h 0 that varied from sample to sample but was never larger than I Oe at a frequency of 400 Hz at fields above H02/2. This exciting field corresponds 2. Fromtothis an induced we conclude current that pinning the flux vortices should be negliof about 100 of A/cm gible for h 0 > 50e, a typical exciting field value, At fixed magnetic fields, we have determined the resonant frequency and Q of the mixed state helicon resonance. Figure 1 shows the dependence of the Q upon H0 for the two2-1 be~t )a instead Q, whereActually, u = tan e~,As samplesofstudied. we plot expected, u = (4Q u is field-dependent in the normal state above H,2 but notice that u remains essentially constant in the mixed state below H,2. The error bars show the uncertainty in an individual measurement due primarily to a small fraction of the pickup voltage (<5%) that does not change sign when the magnetic field is reversed. Meaningful values of u could not be obtained from the Q for sample Nb-3 having a RRR = 2000. In Fig. 2 we show the field dependence of the resonant frequency, V~ Only strain free samples gave data .
1.5
2-lI~ where The field Q isdependence determined of directly u=(4Q from the shape of the resonance curve. Sample Nb-4a has a HER = 6300 and sample Nb-5a has a RRR = 11, 000. Both samples are 0.31 mm thick. Extrapolated normal state curves do not pass through the origin because of a large normal state magnetoresistance.
similar to that shown in Fig. 2. Samples that had been spark cut or mechanically worked all showed a strong field dependence of v,. in the neighborhood of H,2. This was observed in the been sample spark reported cut. The in Ref. helicon 1, and resonant that sample frequency had for the first standing wave resonance in a plate of thickness d is related the 2. average Since Qresistivity, Is field independent, p, and Q by the v,. field = pQu/i..tod dependence of p is Just the field dependence of ~ Figure 2 shows that p is nearly linear in H 0 which is just the behavior expected if p is the flux-flow resistivity, p, = ~N HO/H, 2. .
There are however, two important differences between the resistivity that we determine and the usual flux-flow resistivity. First, at finite temperatures we do not see a sharp change inthe resistivity at H,2 which has been observed in all flux-flow studies to date. A second important difference is that v~ does not extrapolate to zero at B = 0 but has a finite value. This may indicate the presence of a large, field independent damping at fintte temperature.
Vol. 5, No. 8
DAMPING OF BELICON RESONANCES
I
•
4.2°K Ij.
1.5
0 • 6
Nb-3 Nb—4a Nb-So
587
(Ha)
800Hz 588 Hz 691 Hz
Z’r(Ho) J4(H~ 2)
0__o— 1.0
0.5•
FIG. 2
0
1.0
H,/ H82
The resonant amplitude, A~,should give information about the damping of the mixed state resonance as it does for the normal state helicon resonance. We expect that each oscillating vortex contributes about equally to the pickup voltage since each vortex contains one flux quantum. Thus A,. should be proportional to the magnetic induction, B (if all vortices are oscillating) and the average damping experienced by the oscillating flux vortices. We observe afield independent damping and therefore we would expect to observe A,. proportional to B if all the vortices are oscillating. Fairly linear behavior is observed for H~ > H,2/2 (except for a small dip at H,2 in some specimens) but there are marked deviations from linearity at lower fields. The resonant amplitudes go rapidly to zero for a finite magnetic induction which says that not all the flux in the sample is responding to the a. c. fields for H~,< H,2/2. Magnetoresistance lit the normal cores will also cause A,. to deviate from a linear behavior. Although the field we dependence do not understand of A,. is qualitatively the details ofunderstood, the field dependence.
15
The reduce resonant frequency v~(H~)/v, (H,2) as a function of H~/11,2. Specimen Nb- 3 has aRRR= 2000.
Our observation of a field independent Q in the mixed state does not agree with the expertments of Druyvesteyn et al. 2 We find this surprising since the Hail ii~1~ in their material is comparable to ours and their material is also Union Carbide electrolytic niobium. The samples that we studied were all polycrystalline, consisting of crystallites about 1 mm on a side. The magnetoresistance varied considerably from sample to sample, presumably because the nwnber of crystallttes with open orbits near the plane of the sample varied. The largest Hall angle at 11,2 (about 50°)was a specimen having a resistance ratio of 6300 whereas specimens having resistance ratios as large as 11, 000 were studied. The highest resistance ratio specimens however, did have the lowest critical currents. We can make a few comments on the results of Druyvesteyn et al. 2 At the frequencies that they must have u~~’anexciting field of 1 Oe corresponds to2,anso induced it is certainly current density possible of about that all non-linear 2000 A/cmamplitude effects were absent. In our previous work1 the largest induced current
588
DAMPING OF HELICON RESONANCES
Vol. 5, No. 8
density used was about 1500 A/cm2 and the largest induced current densities used in the expertments reported in this letter are 500 A/cm2 at the resonant frequency. Druyvesteyn et al.2 find a field dependent Hall angle in the~E’ed state. If we had tried to determine u directly from the amplitude, then our results would agree qualitatively with those of Druyvesteyn et al.2 However, they say that u is determined ?i~i the Q which is the method that we used. It is worth noting that we could not get meaningful u values from the resonance curves in sample Nb-3 having a ERR = 2000, however, a “value” of u was easily determined from Ar. Druyvesteyn et al.2 give Hall effect data for a sample havjii:IIRR = 2500.
a. c. experiments where it is not necessary for flux to cross the metal-vacuum boundary. It is also possible that the superconductor is damaged In the area of the contacts. The presence of probes has affected Hail effect and resistivity measurements In the alkali metals but to the best of our knowledge, not in materials like nbbium. An a. C. Hall effect experiment where the magnetic field Is kept constant and the current through the sample modulated at a frequency well away from the helicon resonant frequency might help clarify the difference between the helicon and d. c. experiments. The superposition of a direct current might give clues of some real difference between what the a. c. and d. c. expertments actually measure.
A frequency dependent background (which is sometimes, but not always, present) distorts the shape of the resonance curve and hence determining the Q when a large background is present can lead to large errors. To eliminate most of these errors (when present) we always take resonance curves for both positive and negative field directions. We have also fit our curves to the Chambers and Jones8 expression fdr normal helicons and find excellent agreement between the observed and predicted frequency response.
In summary, we find helicon resonances in the mixed state of pure niobium having a large Hail angle. At a fixed magnetic field, the frequency dependence of the mutual inductance between the crossed-coils used in the experiment is given to within a few percent by the expression derived by Chambers and Jones8 for normal state helicons. The niobium is pure enough to determine the hail angle with an accuracy of ± 10% directly from the Q of the resonance. We find that the Hall angle in the mixed state Is independent of the applied field and equal to the Hall angle at 11,2. Although the resistivity In the mixed state is nearly the flux-flow resistivity there appear to be important differences in the case of pure superconductors that need further study. Our experimental results agree with the field independent Hail angle predicted by Nozieres and Vinen5 as opposed to the field dependent Hall angle predicted by Bardeen and Stephen.4
Our observation of a field independent Hall angle also does not agree with the field dependent Hall angle found by Reed et al.3 using a conventional d. c. method. The~1~vo expertments, however, may measure different properties of the superconductors. In both cases the Hall effect of moving fluxotd.s is observed with comparable current densities. However, in the d. c. experiments flux must cross a metalvacuum boundary. Since the superconducting properties in the vicinity of a boundary are different from the bulk, It is conceivable that d. c. experiments could give results different from
Acknowledgments We are extremely grateful to Dr. R. W. Meyerhoff and the Union Carbide Corporation for supplying the samples and thereby making this investigation possible. Discussions with J. W. Wilkins have been very helpful. -
References 1.
MAXFIELD B.W. and JOHNSON E. F., Phys. Rev. Left. 16, 652 (1966).
2.
DRUYVESTEYN W. F., VAN GURP G.J. and GREEBE C .A.A.J. Phys. Left. ~,
3.
REED W.A., FAWCETT E. and KIM Y.B., Phys. Rev. Left. 14, 790 (1965).
4.
BARDEEN JOH}~and STEPHEN M.J.,
5.
NOZIERES P. and VINEN W. F., Phil. Mag. 14, 667 (1966).
Phys. Rev. 140, A1197 (1965).
248 (1966).
Vol. 5, No. 8
DAMPING OF HELICON RESONANCES
6.
Dr. R. W. Meyerhoff of Union Carbide Corporation, Indianapolis, Indiana, prepared these specimens from electrolytic niobium subsequently outgassed at 10’° torr.
7.
KIM Y.B., HEMW~EADC.F. and STRNAD A.R., Phys. Rev. 139, A1163 (1965).
8.
CHAMBERS R.G. and JONES LK., Proc. Roy. Soc. A270, 417 (1962).
Die Feldabheingtgkeit der Parameter der Heliconresonant von reinem Niob im gemischten Zustand wurde gemessen. Die Gilte Q erwies sich hierbei als feldunabNângig im gemisditen Zustand und feldabhängig im normalleitenden Zustand, woraus wir scblierssen, darsa der Hallwinkel tm gemischten Zustand feldunabhâ~ngig1st.
589