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Flux-flow noise measurements on type II superconductors
Physica
64 (1973) 587-594
FLUX-FLOW
0
North-Holland
NOISE
Publishing
Co.
MEASUREMENTS
ON TYPE
II
SUPERCONDUCTORS* S.W. SHEN and A. VAN DER ZIEL Department
of Electrical
Engineering,
Minneapolis,
University
Minnesota,
of Minnesotn,
USA
Received 25 January 1972
Synopsis Flux-flow noise in type II superconductors is investigated experimentally. The observed noise turns out to be shot noise as expected from the assumption that the flux-flow voltage is generated by independently drifting flux bundles. Bundle sizes as large as lo2 to IO5 are found in vanadium foils dependent on the bias condition. The experiment shows qualitative agreement with the theory.
I. Introduction. The calculation of the flux-flow noise spectrum in type II superconductors is modified by taking the geometric dependence of the induced noise of drifting fluxoids into account’). Therefore, the noise voltage appearing across potential probes strongly depends on the shape of the superconductor foil and on the path of the moving fluxoids. Flux-bundle effects2v3) explain the magnitude of the flux-flow noise which is several orders of magnitude higher than that predicted by the assumption that the singly quantized fluxoids are independently drifting entities. However, it should be noted that the pinning effect which causes flux bundles, changes only S(0) and leaves the relative shape of spectral intensity S(f)/S(O) unaltered. 2. Theory. Suppose a perpendicular magnetic field is applied on a infinitely long superconducting foil with a finite width w. Fig. I shows the geometry of the foil where two potential probes are placed at x = +x, and x = -x,. Assume FJf) is the Fourier transform of the voltage impulse induced between the probes by a fluxoid drifting along the flow-line x = constant. According to Carson’s theorem4), the total noise spectral intensity due to all the drifting fluxoids should be given by
S,,(f)
= 2
* Work sponsored
if!_tisIFx(f>l’ dx, (%lC) R by ONR contract.
587
588
S.W.
SHEN
AND
A. VAN
DER
ZIEL
where B is flux density, y0 is the fluxoid quantum and v is the drift velocity of the fluxoids. R stands for the flux-flow region where both magnetic field and transport current exist. Since the voltage impulse due to a single moving fluxoid is knownl), once the flux-flow region R is determined by the experimental arrangement, eq. (1) will give the spectral intensity.
Fig.
If a transport eq. (1) yields -W) S(0)
current
1. Geometrical
Z is passed
diagram
through
for eq. (2).
the whole foil in the + .Ydirection,
f f.
= 4 exp
)I’
(2)
where fro = n, n = 0, I, 2, . . .; z. = w/u, fx = v/2x,, z0 is the transit time and v is the drift velocity. Notice that here all the fluxoids are in motion. Although only those drifting between the probes contribute to the d.c. voltage; as far as noise is concerned, the fluxoids drifting outside the probes must also be considered. The above model requires that both the current density and the magnetic field are uniform all over the infinitely long strip, therefore this does not agree with the experimental arrangement. In the experimental investigation the transport current is only applied in the region between the potential probes, i.e. 1x1 < x,. This arrangement will give the same de. voltage as before. But, the spectral intensity becomes
-S(f) = -J exp S(O)
-2~s C
+ 8i:;fFX)
[1 - exp (1 - 4~31.
(3)
1
Both eqs. (2) and (3) are plotted in fig. 2. It shows that both of them have an f- l tail except that eq. (3) has a lower cutoff frequency. According to the flux-creep theory*), because of the presence of pinning, the fluxoids do not move singly but in bundles and this increases the noise. The ex-
FLUX-FLOW
NOISE 1N TYPE II SUPERCONDUCTORS
589
pression for the spectral intensity atf = 0 gives a value (4) where n is the average number of fluxoids in the bundle. Here V,, and S(0) can be found by d.c. and noise measurements, hence n can be determined.
Fig. 2. Power spectra for two different models.
3. Measurements and results. The experimental arrangement is shown in fig. 3. The potential probes are spot-welded on the vanadium foil whereas the current leads are spot-welded on a copper strip first, then the strip is welded on the foil so that the current density can be uniform over (xl < x,. All measurements are
Fig. 3. Geometrical
diagram for eq. (3).
590
S. W. SHEN
AND
A. VAN
DER
ZIEL
conducted at T = 4.2 K. Figs. 4, 5 and 6 show the d.c. characteristics for three samples with different dimensions used in the experiment. The flux-flow resistance approaches the normal resistance as the flux-flow driving force increases. As for the noise measurement, instead of using a step-up transformer3), the flux-flow voltage is fed directly into a low-noise, low-impedance transistor amplifier. The theory contains as a parameter the characteristic frequency f, = ~/2x, where 2x, is the distance between the potential probes and the value of v follows from the d.c. characteristic. Knowing,f, for each particular experiment one can fit the theoretical curve through the measured points. The low-frequency spectral intensity S(0) is determined from such a fit. The fact that such a fit is at all possible, indicates that the results agree reasonably well with the theory. There is usually a discrepancy at high frequencies, however, in that the noise spectrum drops much faster with increasing frequency that the f-’ tail required by the theory. This is because of the fact that the potential probes have appreciable size instead of being point contacts. To avoid flicker noise, both the transport current density and the magnetic field intensity must be kept quite low.
Fig. 4. D.C.
characteristic
for device A.
FLUX-FLOW
591
NOISE IN TYPE II SUPERCONDUCTORS r
I
I
100 Fig. S. D.C. characteristic
for device B.
Fig. 7. Noise spectrum for device A.
Fig. 6. D.C. characteristic
I
(Al-,
IO'
for device C.
Fig. 8. Noise spectrum for device A.
592
S.W. SHEN AND A. VAN DER ZIEL
5 3
Fig. 9. Noise spectrum for device A.
Fig. 11. Noise spectrum for device C.
Fig. 10. Noise spectrum for device B.
Fig. 12. Noise spectrum for device C.
FLUX-FLOW
NOISE IN TYPE II SUPERCONDUCTORS
593
Fig. 7 shows a typical noise spectrum for device A under the bias-condition such that H = 430 Oe and I = 0.20 A. The value of fx is only 20 Hz, therefore the data points shown in the figure correspond to the tail part of eq. (3) and it shows an f -Icharacteristic. In the figure, the solid line represents the theoretical curve. Figs. 8 and 9 give two more spectra for device A at H = 360 Oe. In fig. 8, since it has anJX equal to 16.5 Hz, it shows essentially an f-l spectrum as expected. Extrapolation gives for S(0) a value of 2 x 10-15vz s. In fig. 9 the fxis increased to 46.7 Hz. Device B has a smaller value for 2x,. As a result, for the same transport current density, it would have a larger value for f,. Fig. 10 shows a typical spectrum for device B with fx = 435 Hz and H = 360 Oe. The measurements fit well with the theoretical curve exhibiting a flat low-frequency part because of its large fx. Device C is very wide, therefore its fx would be smaller for the same transportcurrent level as before. It is expected to show an f-' spectrum over the frequency range of 20 to 200 Hz. This is really the case as shown in figs. 11 and 12; at H = 1360Oe each has fx = 6.5 Hz and 14 Hz, respectively. The conclusion to be
“a 360
oc
OEWCE
c
Fig. 13. Flux-bundle
-
size n us. J.
594
S. W. SHEN AND A. VAN DER ZIEL
drawn from all spectral measurements is that the observed noise fits eq. (3) very well. Using eq. (4), the values of flux-bundle size y1 = q/q0 are plotted against transport-current density for three devices in fig. 13. It shows that the bundle size decreases rapidly with increasing transport-current density, it varies as J-” (2 < cx < 3). This part of the work agrees qualitatively with the results found by Van Gurp3). This is not surprising, for the theory given by Van Gurp leads to the same value of S(0) as described by the theory summed up in eq. (3).
REFERENCES 1) Clem, J.R., Time-Dependent Voltage Across Superconductors During Flux Motion. Proceedings ofthernternational Conference on the Science of Superconductivity, Stanford, (1969). 2) Anderson, P. W., Phys. Rev. Letters 9 (1962) 309. 3) Van Ooijen, D. J. and Van Gurp, G. J., Philips Research Reports 21 (1966) 343. 4) Van der Ziel, A., Noise Source, Characterization, Measurement, Prentice Hall (London, 1970).
64 (1973) 587-594
FLUX-FLOW
0
North-Holland
NOISE
Publishing
Co.
MEASUREMENTS
ON TYPE
II
SUPERCONDUCTORS* S.W. SHEN and A. VAN DER ZIEL Department
of Electrical
Engineering,
Minneapolis,
University
Minnesota,
of Minnesotn,
USA
Received 25 January 1972
Synopsis Flux-flow noise in type II superconductors is investigated experimentally. The observed noise turns out to be shot noise as expected from the assumption that the flux-flow voltage is generated by independently drifting flux bundles. Bundle sizes as large as lo2 to IO5 are found in vanadium foils dependent on the bias condition. The experiment shows qualitative agreement with the theory.
I. Introduction. The calculation of the flux-flow noise spectrum in type II superconductors is modified by taking the geometric dependence of the induced noise of drifting fluxoids into account’). Therefore, the noise voltage appearing across potential probes strongly depends on the shape of the superconductor foil and on the path of the moving fluxoids. Flux-bundle effects2v3) explain the magnitude of the flux-flow noise which is several orders of magnitude higher than that predicted by the assumption that the singly quantized fluxoids are independently drifting entities. However, it should be noted that the pinning effect which causes flux bundles, changes only S(0) and leaves the relative shape of spectral intensity S(f)/S(O) unaltered. 2. Theory. Suppose a perpendicular magnetic field is applied on a infinitely long superconducting foil with a finite width w. Fig. I shows the geometry of the foil where two potential probes are placed at x = +x, and x = -x,. Assume FJf) is the Fourier transform of the voltage impulse induced between the probes by a fluxoid drifting along the flow-line x = constant. According to Carson’s theorem4), the total noise spectral intensity due to all the drifting fluxoids should be given by
S,,(f)
= 2
* Work sponsored
if!_tisIFx(f>l’ dx, (%lC) R by ONR contract.
587
588
S.W.
SHEN
AND
A. VAN
DER
ZIEL
where B is flux density, y0 is the fluxoid quantum and v is the drift velocity of the fluxoids. R stands for the flux-flow region where both magnetic field and transport current exist. Since the voltage impulse due to a single moving fluxoid is knownl), once the flux-flow region R is determined by the experimental arrangement, eq. (1) will give the spectral intensity.
Fig.
If a transport eq. (1) yields -W) S(0)
current
1. Geometrical
Z is passed
diagram
through
for eq. (2).
the whole foil in the + .Ydirection,
f f.
= 4 exp
)I’
(2)
where fro = n, n = 0, I, 2, . . .; z. = w/u, fx = v/2x,, z0 is the transit time and v is the drift velocity. Notice that here all the fluxoids are in motion. Although only those drifting between the probes contribute to the d.c. voltage; as far as noise is concerned, the fluxoids drifting outside the probes must also be considered. The above model requires that both the current density and the magnetic field are uniform all over the infinitely long strip, therefore this does not agree with the experimental arrangement. In the experimental investigation the transport current is only applied in the region between the potential probes, i.e. 1x1 < x,. This arrangement will give the same de. voltage as before. But, the spectral intensity becomes
-S(f) = -J exp S(O)
-2~s C
+ 8i:;fFX)
[1 - exp (1 - 4~31.
(3)
1
Both eqs. (2) and (3) are plotted in fig. 2. It shows that both of them have an f- l tail except that eq. (3) has a lower cutoff frequency. According to the flux-creep theory*), because of the presence of pinning, the fluxoids do not move singly but in bundles and this increases the noise. The ex-
FLUX-FLOW
NOISE 1N TYPE II SUPERCONDUCTORS
589
pression for the spectral intensity atf = 0 gives a value (4) where n is the average number of fluxoids in the bundle. Here V,, and S(0) can be found by d.c. and noise measurements, hence n can be determined.
Fig. 2. Power spectra for two different models.
3. Measurements and results. The experimental arrangement is shown in fig. 3. The potential probes are spot-welded on the vanadium foil whereas the current leads are spot-welded on a copper strip first, then the strip is welded on the foil so that the current density can be uniform over (xl < x,. All measurements are
Fig. 3. Geometrical
diagram for eq. (3).
590
S. W. SHEN
AND
A. VAN
DER
ZIEL
conducted at T = 4.2 K. Figs. 4, 5 and 6 show the d.c. characteristics for three samples with different dimensions used in the experiment. The flux-flow resistance approaches the normal resistance as the flux-flow driving force increases. As for the noise measurement, instead of using a step-up transformer3), the flux-flow voltage is fed directly into a low-noise, low-impedance transistor amplifier. The theory contains as a parameter the characteristic frequency f, = ~/2x, where 2x, is the distance between the potential probes and the value of v follows from the d.c. characteristic. Knowing,f, for each particular experiment one can fit the theoretical curve through the measured points. The low-frequency spectral intensity S(0) is determined from such a fit. The fact that such a fit is at all possible, indicates that the results agree reasonably well with the theory. There is usually a discrepancy at high frequencies, however, in that the noise spectrum drops much faster with increasing frequency that the f-’ tail required by the theory. This is because of the fact that the potential probes have appreciable size instead of being point contacts. To avoid flicker noise, both the transport current density and the magnetic field intensity must be kept quite low.
Fig. 4. D.C.
characteristic
for device A.
FLUX-FLOW
591
NOISE IN TYPE II SUPERCONDUCTORS r
I
I
100 Fig. S. D.C. characteristic
for device B.
Fig. 7. Noise spectrum for device A.
Fig. 6. D.C. characteristic
I
(Al-,
IO'
for device C.
Fig. 8. Noise spectrum for device A.
592
S.W. SHEN AND A. VAN DER ZIEL
5 3
Fig. 9. Noise spectrum for device A.
Fig. 11. Noise spectrum for device C.
Fig. 10. Noise spectrum for device B.
Fig. 12. Noise spectrum for device C.
FLUX-FLOW
NOISE IN TYPE II SUPERCONDUCTORS
593
Fig. 7 shows a typical noise spectrum for device A under the bias-condition such that H = 430 Oe and I = 0.20 A. The value of fx is only 20 Hz, therefore the data points shown in the figure correspond to the tail part of eq. (3) and it shows an f -Icharacteristic. In the figure, the solid line represents the theoretical curve. Figs. 8 and 9 give two more spectra for device A at H = 360 Oe. In fig. 8, since it has anJX equal to 16.5 Hz, it shows essentially an f-l spectrum as expected. Extrapolation gives for S(0) a value of 2 x 10-15vz s. In fig. 9 the fxis increased to 46.7 Hz. Device B has a smaller value for 2x,. As a result, for the same transport current density, it would have a larger value for f,. Fig. 10 shows a typical spectrum for device B with fx = 435 Hz and H = 360 Oe. The measurements fit well with the theoretical curve exhibiting a flat low-frequency part because of its large fx. Device C is very wide, therefore its fx would be smaller for the same transportcurrent level as before. It is expected to show an f-' spectrum over the frequency range of 20 to 200 Hz. This is really the case as shown in figs. 11 and 12; at H = 1360Oe each has fx = 6.5 Hz and 14 Hz, respectively. The conclusion to be
“a 360
oc
OEWCE
c
Fig. 13. Flux-bundle
-
size n us. J.
594
S. W. SHEN AND A. VAN DER ZIEL
drawn from all spectral measurements is that the observed noise fits eq. (3) very well. Using eq. (4), the values of flux-bundle size y1 = q/q0 are plotted against transport-current density for three devices in fig. 13. It shows that the bundle size decreases rapidly with increasing transport-current density, it varies as J-” (2 < cx < 3). This part of the work agrees qualitatively with the results found by Van Gurp3). This is not surprising, for the theory given by Van Gurp leads to the same value of S(0) as described by the theory summed up in eq. (3).
REFERENCES 1) Clem, J.R., Time-Dependent Voltage Across Superconductors During Flux Motion. Proceedings ofthernternational Conference on the Science of Superconductivity, Stanford, (1969). 2) Anderson, P. W., Phys. Rev. Letters 9 (1962) 309. 3) Van Ooijen, D. J. and Van Gurp, G. J., Philips Research Reports 21 (1966) 343. 4) Van der Ziel, A., Noise Source, Characterization, Measurement, Prentice Hall (London, 1970).