Journal of Low Temperature Physics, Vol. 94, Nos. 1/2, 1994

Magnetic Flux Noise in Copper Oxide Superconductors M. J. Ferrari,* Mark Johnson, t F. C. Wellstood,** J. J. Kingston, ~ T. J. Shaw, and John Clarke Department of Physics, Universityof California, Berkeley, California94720, and Centerfor AdvancedMaterials, MaterialsSciencesDivision, Lawrence Berkeley Laboratory, Berkeley, California 94720 (Received June 23, 1993; revised August 6, 1993)

We report on the magnetic flux noise in thin films of YBa2Cu307_~ ( YBCO), Tl2Ca2Ba2Cu30 x, and TlCa2Ba2Cu30 x and in crystals of YBCO and Bi2Sr2CaCu208+ x, measured with a Superconducting QUantum Interference Device (SQUID). We ascribe the noise to the motion of flux vortices. In the low magnetic fields in which the experiments are performed the average vortex spacing always exceeds the superconducting penetration depth. The spectral density of the noise usually scales as 1If (f is frequency) from 1 Hz to 1 kHz and increases with temperature to a peak which is of the same magnitude in all samples, at the transition temperature. Furthermore, the noise power increases with the magnitude of the magnetic field in which the sample is cooled, with a power-law dependence over several decades, whereas a supercurrent well below the critical current density applied to YBCO films suppresses the noise power by an order of magnitude. Most of the measurements were made on YBCO fiIms, and for this set of samples the noise decreases dramatically as the crystalline quality is improved. A model of thermally activated vortex motion is developed which explains the dependence of the noise on frequency, temperature, magnetic field, and current. The pinning potential is idealized as an ensemble of symmetrical double wells, each with a different activation energy separating the two states. From the noise measurements, this model yields the distribution of pinning energies, the vortex hopping distance, the number density of mobile vortices, and the restoring force on a vortex at a typical pinning site. The *Present address: AT&T Bell Laboratories, 600 Mountain Avenue, Murray Hill, New Jersey 07974-0636. tPresent address: Bellcore, 331 Newman Springs Road, Red Bank, New Jersey 07701-7030. **Present address: Center for Superconductivity Research, Department of Physics, University of Maryland, College Park, Maryland 20742-4111. ;Present address: Xerox PARC, 3333 Coyote Hill Road, Palo Alto, California 94304. 15 0022-2291/94/0100-0015507.00/0 ~ 1994 Plenum Publishing Corporation

16

M.J. Ferrari et al.

distribution of pinning energies in YBa2Cu3 0 7 x shows a broad peak below 0.1 eV. Over narrow temperature intervals, most samples exhibit random telegraph signals in which the flux switches between two discrete levels, with activation energies and hopping distances much greater than those deduced from the 1If noise measurements.

1. INTRODUCTION The dynamics of Abrikosov vortices in high transition temperature (To) superconductors has attracted considerable experimental and theoretical attention. In these extremely anisotropic materials T c is an order of magnitude higher and vortex pinning energies Uo(T) are an order of magnitude lower than in the low-To superconductors, opening regimes that were previously inaccessible experimentally. Phenomena of interest include giant flux creep, 1 thermally activated resistivity, 2'3 a Kosterlitz-Thouless unbinding transition, 4 dissipation in samples oscillating in a magnetic field, 5'6 and novel vortex phases. 7 Many of these experiments enable one to infer pinning energies, which have important technological implications because they determine the critical current density. However, these measurements are generally performed in high magnetic fields, often with strong driving forces applied by currents, and thus provide information about the pinning of the vortex lattice. In contrast, the present work is principally concerned with magnetic flux noise generated by the dynamics of vortices in films cooled in magnetic fields below 10 -6 T, so that the vortices are, on the average, far apart and interact only very weakly. Thus, one is able to extract information on the interaction of individual vortices with pinning sites at or close to thermal equilibrium. The original motivation for this work was the study of flux noise in high-To Superconducting QUantum Interference Devices (SQUIDs), ~ and it was demonstrated that the magnitude of this noise decreases dramatically as the crystalline quality of thin films of YBa2Cu307_ x (YBCO) films was improved. 9 We subsequently realized the usefulness of these measurements for studying 1If noise ( f is frequency) in a system where the microscopic fluctuators, moving vortices, could be modeled theoretically. We begin, in Sec. II, with a brief description of the experimental apparatus. Section III describes measurements of random telegraph signals that are generally observed only over very narrow ranges of temperature. Section IV is concerned with the much more common situation in which the power spectrum of the flux noise scales as l/f, and presents results on the inferred distribution of pinning energies. The application of a current, described in Sec. V, reduces the magnitude of the 1If noise and enables one

Magnetic Flux Noise

17

to deduce the pinning force Constant, the average vortex hopping distance, and the number density of vortices. Section VI shows that the magnitude of the 1If noise increases with the magnetic field in which the sample is cooled. Section VII is a concluding summary. Portions of this work have been published previously.9-13 We note that, recently, Rogers et aL 14 have explored vortex diffusion in very thin B i z S r z C a C u 2 0 8 films using a technique similar to ours. 2. EXPERIMENTAL TECHNIQUES

2.1. Apparatus In our experiments we measured the fluctuating magnetization of superconducting samples using a low-To thin-film Superconducting QUantum Interference Device (SQUID) operated in a flux-locked loop. 15'16 The dc SQUID had a square washer geometry with inner and outer dimensions of 200/~m and 1000 #m, respectively, and an inductance L = 0.4 nil. We designed the experiment to allow tight inductive coupling between the SQUID, maintained at or near 4.2 K, and the sample, the temperature of which we could raise to above Tco This was accomplished by attaching the SQUID with vacuum grease to a sapphire plate attached to one end of a vacuum can immersed in liquid 4He (Fig. 1). The sample was similarly attached to a Si chip on the reverse side of which we mounted a Pt resistance thermometer and a 1 kf2 metal film resistor to serve as a heater. The Si chip was attached to a fiberglass mount which was separated from

G

F

E

(a) Sampl SQUIE

I H J

S Fig. 1. Apparatus used to measure flux noise. (a) Schematic of sample supported parallel to SQUID. (b) Front view: A: SQUID substrate, B: sapphire plate, C: modulation and feedback coil, D: sample substrate, E: Si plate, F: heater, G: thermometer, H: spacer, I: bristle, J and K: magnetic field coils. Mechanical support by fiberglass components (shaded).

18

M . J . Ferrari et al.

the sapphire plate by spacers and bristles from a brush that provided the necessary low thermal conductance. The gap between the SQUID and the sample was roughly 0.1 mm. Minimizing the volume of normal metal objects within 10mm of the SQUID rendered their Johnson noise negligible. A small, 10-turn Nb coil was mounted on the reverse side of the sapphire plate to allow us to operate the SQUID in a flux-locked loop. Two larger Nb coils, one of which could store a persistent current, enabled us to apply magnetic fields perpendicular to the plane of the samples. The assembly shown in Fig. 1 was enclosed in a Pb shield inside the vacuum can. The dewar was surrounded by a mu-metal shield which reduced the earth's field to no more than 0.3 #T normal to the sample surface. Under typical conditions the residual 60 Hz field in the shield was about 10 nT rms.

2.2. Samples Table I lists the samples on which we made measurements. 17 26 The YBa2Cu3Ov_x (YBCO) samples are thin films except for the single crystal YBCO(10); samples TO, T1 and T2 are flux transformers containing two YBCO films with an intermediate SrTiO3 layer; TCBCO(1) and (2) are thin films of T12Ca2Ba2Cu3Ox and T1Ca2Ba2Cu30~, respectively; the two BizSr2CaCu2Os+ x (BSCCO) samples are single crystals with possible intergrowths of second phases (Sec. III). The second column of Table I(a) states the method of preparation: cosputter, three sputter sources (Y, Ba, Cu); sputter, single sputter source; laser, laser ablation of a stolchiometric target; coevaporate, e-beam evaporation from three sources; melt, single crystal grown from melt. The third column lists the substrate for the thin films (YSZ is yttrium-stabilized zirconia). The thicknesses of the films were estimated from the deposition parameters rather than measured directly; for the flux transformers, the thickness is of a single YBCO layer. In the fifth column, "film" denotes an unpatterned film, "ring" a thin film patterned by photolithography and an acid etch into a square washer mimicking the dimensions of the SQUID, "FT" is a flux transformer, and "flake" is a cleaved single crystal, typically a few square millimeters in area. Table I(b) summarizes the characterization of the samples. The crystalline orientation, determined with x-ray diffraction, is specified along the normal to the plane of the film or flake. We did not perform x-ray diffraction on several of the YBCO samples, but the in situ growth conditions are known to yield c-axis films. We determined Tc inductively (Sec. II.B) and the critical current Jc of most samples from the width of the magnetic hysteresis loops using the Bean model. 27 However, Jc for the flux transfor-

Magnetic Flux Noise

19

TABLE I(a) Parameters of Samples: Fabrication Technique, Substrate, Thickness (d,), Configuration, Reference Sample

Deposition

Substrate

ds (#m)

Pattern

Ref

YBCO(1 ) YBCO (2) YBCO(3) YBCO(4) YBCO(5) YBCO(6) YBCO(7) YBCO(8) YBCO(9) YBCO(10) YBCO(11 )

Cosputter Sputter Laser Coevaporate Coevaporate Sputter Laser Laser Laser Melt Laser

SrTiO 3 MgO LaA103 SrTiO 3 SrTiO3 SrTiO 3 MgO MgO MgO None SrTiO 3

0.2 0.4 0.3 0.2 0.5 0.4 I 0.4 0.4 35 0.3

Ring Film Film Ring Ring Film Film Film Film Flake Film

17 18 19 20 20 21 22 22 22 26 22

TO T1 T2

Laser Laser Laser

MgO MgO MgO

0.3 0.3 0.3

FT FT FT

23 23 23

BSCCO(1 ) BSCCO(2)

Melt Melt

None None

40 150

Flake Flake

24 24

TCBCO(1) TCBCO(2)

Sputter Sputter

YSZ YSZ

2 2

Film Film

25 25

TABLE I(b) Measured Properties of Samples: Orientation, Critical Current Density, Flux Noise Spectral Density Sample

Orientation

Tc (K)

YBCO(1) YBCO(2) YBCO(3) YBCO(4) YBCO(5) YBCO(6) YBCO(7) YBCO(8) YBCO(9) YBCO(10) YBCO(ll)

>90% c c c a and c polycrystalline >90% a

85 84.4 89.8 85 47 81

Jc(4.2 K) (A/cm 2)

--

8 5 (a)

--c --

88 (a) 82 89 90

--3 • 106 --

2• 10 .8 2 • 10 -9 < 10-9 1 x 10 7 3 x 10 -4 4 x 10 -7 6 X 10 .8 ~ 10--9 (b) 2 X 10 .6 2 x 10 9(~) 3 • 10 -lO(d)

TO T1 T2

----

71 77 59

2 x 104 3 x 105 3 x 104

< 10 -6 1 x 10-6 2 x 10 -6

BSCCO(1) BSCCO(2)

c c

90 93

~ 106 ~ 106

1 x 10 -7 6 x 10 -7

TCBCO(1 ) TCBCO(2)

e c

119 102

~ 10 s ~ 105

4 x 10 -8 4 x 10-7

(a)Transport (b)Measured (r (~)Measured

measurement. at 4.2 K. at 75 K. at 77 K.

5• 106 4 x 107 2 • 107 (a) 2 x 104 -1 • 10 6

S . ( 1 Hz, 40 K) (qb02/Hz)

--

20

M.J. Ferrari et al.

mers was obtained by dividing the critical current of the entire device by the cross section of the narrowest segment (input coil). It is likely that the critical current was limited by grain boundaries where the upper YBCO film climbs the edge of a lower film, and that the values of Jc in the planar regions of the films were significantly higher, The last column in Table I(b) gives the flux noise power, referred to the sample, measured at 1 Hz and 40 K [except for YBCO(8), (10), and (11)]. Noise is tabulated only when the power spectrum scales as 1If (Sec. IV); larger values may occur in the presence of random telegraph signals (Sec. III). 2.3. Experimental Procedures

After the initial cooldown of the apparatus and after each transfer of l i q u i d 4He, we raise the temperature of the sample to well above To, open

and close the heat switch on the field coil to release any trapped flux, and set a heater current corresponding to the temperature at which we wish to make a noise measurement. Thus the sample is cooled through T C in a magnetic field of less than 1 pT. Figure 2 shows that the mutual inductance M between the modulation feedback coil and the SQUID decreases dramatically when the sample cools through Tc and begins to shield the

1.00

-o- o

"~" 075 I III

0.50 --ZZ

Ill

I,i

9 9 9 9

D~

L

100

75 9

oo

50

I

0

I

9 b

I

L

i

J

i

i

I

,

5O 100 Temperature (K)

I

50

Fig. 2. Temperature dependence of SQUID parameters for YBCO(1). (a) normalized modulation amplitude AIc(T)/AIc(4.2K); (b) mutual inductance M between SQUID and feedback coil.

Magnetic Flux Noise

21

SQUID. We use M to measure Tc, which we define as the highest temperature at which is M is depressed below its constant value for T > To. The small jump in M near 60 K in Fig. 2 is reproducible, and probably results from a small amount of the oxygen-deficient phase of YBCO with Tc ~ 60 K. The sharpest inductive transition we observed was in sample YBCO(3), with a width of less than 1 K [-Fig. 10(a)]. For the samples patterned into rings with inductance L~, we determined the mutual inductance M,=~(LL~) ~/2 to the SQUID from the modulation depth AIo of the critical current Ic of the SQUID; c~s is the coupling coefficient and AIo is the difference between the maximum and minimum values of I c as the flux through the SQUID is varied. Above Tc of the sample, we estimate fl=-2LIo/~o~2. Below Tc the SQUID inductance is reduced to L' = L ( l - c ~ ) by the diamagnetic screening of the sample, and analogously fl'= f l ( 1 - C~s2).From the observed increase in AIc with decreasing temperature illustrated in Fig. 2, we can estimate fi' according to Fig. 4 of Ref. 28, and thereby deduce L' and ~s. The coupling coefficient ~ ranged from 0.6 to 0.7 for the "ring" samples. In fact, the above procedure is also reasonable for an unpatterned film (Sec. III). In this paper, S . ( f ) refers to the spectral density of flux noise in the SQUID divided by ~ to refer the noise to the sample, unless stated otherwise. 3. RANDOM TELEGRAPH SIGNALS The noise generated by our samples can be divided into two types:

I/f noise and random telegraph signals (RTSs). Although at most temperatures the samples produced only I/f noise, we analyze RTSs first because we model l / f noise in Sec. IV as a superposition of RTSs. Random telegraph signals in YBCO have been observed in both the voltage across current biased samples 29 and in the flux. 3~ Figures 3(a) and (b) illustrate a typical RTS in which the flux linking the SQUID switched back and forth between two discrete levels at a rate that increased markedly with temperature. The amplitude Aqb= 1.3 x 10-3~0 of this process remains constant over observational times of hours and also as the temperature is varied. The simplest explanation for these data is that one vortex hops between two pinning sites, although it is not impossible that a bundle of several vortices is involved. We detected similar processes in 13 of the 18 samples in Table I: two other examples are shown in Figs. 3(c) and (d). The latter RTS occurred in BSCCO(2), which had a magnetically determined transition temperature of 93 K. The existence of large levels of flux noise at higher temperatures suggests the presence of a second, higher temperature superconducting phasefl I In the temperature range from 95 K to 101 K, the flux switches

22

M . J . Ferrari et aL

(b)

88.5 K

e X

_= ._o cO')

Bsooo, 06,.

Time Fig. 3. Flux through SQUID vs. time generated by (a) laser-deposited YBCO(3) at 88.4 K, (b) same film at 88.5 K, (c) sputtered YBCO(2) at 81.6 K, and (d) BSCCO(2) flake at 96.5 K. Zeros of axes are arbitrary.

among a number of values, approximately equally spaced by Aq~ = 6.5 x 10 _3 ~o. A likely explanation for this behavior is that a fixed number of flux quanta occupy two pinning sites, and the various flux levels correspond to differing occupancies of these sites. We will adopt this interpretation, and assume uncorrelated hopping, in the analysis which follows. Another initially plausible explanation is that the sample contains a superconducting portion which is multiply connected and traps a flux ~x, so that the switching in Fig. 3(d) corresponds to thermally activated transitions among the states ~ x = 0, +q~o, _+2~o, etc. However, the variance of ~ x from its mean would provide a measure of the self inductance Lx of the hypothetical multiply connected structure according to (rbZ)/Lx~kBT, which yields the unreasonably large value Lx ~ 6 nil.

Magnetic Flux Noise

23

3.1. Direct Noise Mechanism We now consider the mechanism (inset to Fig. 4) by which the motion of one vortex in the sample produces an RTS. We call this mechanism direct noise because the flux from the moving vortex directly links the SQUID. Since the magnetic field lines from a vortex must close around the sample, and the gap between the sample and the SQUID is small on the scale of the sensing area of the SQUID, 32 a vortex near the center of the sample couples a large fraction of a flux quantum q~ ~ ccs~o into the SQUID, where ~,~0.7 (Sec. II). Note that if a hole were opened in the sample above the hole in the SQUID (thereby converting from "film" to "ring" geometry in Table I), flux quantization implies that the field lines in the inset to Fig. 4 would be only slightly distorted. We assume that

J

88.5 K 10-8

"7 U

-r 0,10

88.6 K , 4 ~

e

v

10 -9 e

~--w~

10-1~ 10-1

I 10 0

I 101

L I 10 2

10 3

Frequency (Hz) Fig. 4. Spectral densities of RTSs from YBCO(3) at two temperatures. Light curves are power spectra of noise measured in SQUID. Heavy curves are Eq. (3.3) with Aq~ = 1.3 x 10 -3 qb0 and ,/=4.5, (determined from time domain data) and 31 =48 and 7.3 ms, respectively. Inset: Crosssection of sample and SQUID. Vortex pinned at site A couples flux q ) ~ s q ) 0 (dotted field lines) into the SQUID, while one at site B couples negligible flux (solid field lines).

24

M. ,1. Ferrari et aL

decreases linearly with distance as the vortex is moved the entire distance w from the inner edge of the S Q U I D to the outer. We have given a better approximation elsewhere, 33 and Clem has calculated the coupling exactly for specific geometries. 34 A vortex which hops a distance l at an angle 0 from the radial direction causes the flux through the S Q U I D to change by zX~ = (~s~o/w)l cos 0 = ~ t l cos 0.

(3.1)

Thus l/> w A~/c~s~ o. We list our inferred lower limits on l in Table II for seven samples. The process observed in BSCCO(2) in the range 95 K < T < 101 K occurred when most of the sample was normal and no longer provided a superconducting groundplane, so the inferred hopping distance may considerably underestimate the actual value. Changing the temperature of the sample by a fraction of a degree often caused the RTS to vanish, but when we could observe the same process over a range of temperatures we saw no variation in l, implying that the vortex continues to hop between the same two sites. A hopping distance of micrometers, observed in several samples (see Table II), is quite long, suggesting that the vortex is moving along a path of weak superconductivity, such as a twin or grain boundary. In Sec. V, we demonstrate that the hopping distances of vortices involved in 1If noise are orders of magnitude smaller than those producing observable RTSs. TABLE II Lower Limits of Vortex Hopping Distances (l) for RTSs Oberved in 7 Samples at Temperature T Sample

T (K)

l (/tm)

YBCO(2)

78.4 81.1

0.80 2.7

YBCO(3)

85.6 88.4 ~a)

0.16 0.76

YBCO(4)

50.9

5.8

T2

45.3

6 ~b)

BSCCO(1)

89.9 90.5 91.1

BSCCO(2)

35.3 50.6 95.1 ~")

TCBCO(1)

4.2 78.1

(a)Process observed over a range of temperatures. (b)Assuming indirect coupling mechanism with wj = 20/tin. (C)More than one process observed at this temperature.

26 32 10 5.1 5.2 3.7 3.5 5-10 ~)

Magnetic Flux Noise 10 2

....

I ....

I

'

'

25

I

I

I

~

I

(a) 101

I

(b)

F

el.)

E dJ

lOOl{~

!I " 9 i~

N

10 -1 o z 10 - 2 A 1:1/1 s

I

10 -3 , , 88.4

, i , , ~ , , , 94 88.5 88.6

I

I

]

I

96

98

I

I

1 O0

Temperature(K)

10 -2 -

-N

1 0-4

~-

10-6

m+

10-8

(c)~. / .

BSCC0(2)

F

10-10 10-3

Y

i

i

i

J

10-2

10-1

10 0

101

~l(s) Fig. 5. Lifetimes vs. temperature for (a) YBCO(3) and (b) BSCCO(2). Solid lines are least-squares fits of Eq. (3.2) to rl(T) assuming U(T) = U0(1 -- (T/T~)4) and zA1=I 0 - u s . Fitting parameters are U0=4.1eV, T,.=89.5 K for YBCO(3), and Uo=0.48eV, T c = l l 2 K for BSCCO(2). (c) Noise power (referred to SQUID) on low-frequency plateau S,v(f-+ 0) vs. vl for YBCO(3) (triangles) and BSCCO(2) (circles). Lines are least-squares fits with unit slope.

26

M.J.

Ferrari

et

al.

3.2. Activation Energies for Vortex Motion If the RTSs are thermally activated, the lifetimes each of the two potential wells are given by

ri(T)=rAiexp[g(i)(T)/k~T],

zl(T) and "c2(T ) in

i = 1, 2.

(3.2)

Here the states are labeled by i = 1 and 2, z~i I is the attempt frequency for the ith state, and U(~ is the temperature-dependent energy barrier for hopping out of that state. The ratio of lifetimes 7 =zl/% is exponentially sensitive to the difference in activation energies U(t)(T)-U(2)(T). In Fig. 5(a) we plot the temperature dependence of ? obtained from the time traces of the process in YBCO(3). Note that the measured values of ? are consistent with a constant value of 4.5 + 1.0, even though r~ and z2 change by two orders of magnitude over the same temperature range. It is sometimes assumed 35 that the observability of an RTS implies a small difference in activation energies, U(1)(T) - U(2)(T)~2kBT, on the grounds that, according to Eq. (3.2), a large energy difference would make it impossible for both z~-~ and zy 1 to be in the measured bandwidth at the same temperature. However, this statement is true only if the attempt times zAi are comparable, so there is no reason a priori for the measured values of 7 to be temperature independent. The observed constancy of 7 allows us to take the activation energies in the two states of the RTS to be exactly equal in order to simplify the following discussion, and to attribute the asymmetry in the lifetimes to different attempt frequencies Z~l~ and ZA~. However, the data in Fig. 5(a) are fitted nearly as well by the assumption of equal attempt frequencies and slightly different activation energies, U(1)(T ) - U(2)(T)~ 11 meV. The power spectrum of a random telegraph signal is a Lorentzian, 36 4(Ao) 2

S ~ ( f , T ) - (Zl '~-z2)[-('fl 1 "~-"c21) 2"~- (27Ef)2] '

(3.3)

and in Fig. 4 we plot S.(f) for YBCO(3) at two temperatures. The solid curves are obtained from Eq. (3.3) with fixed values of ? and Aqb, as determined from the time traces; Zl is the only fitting parameter. From fits of this kind, we obtain zl vs. T, shown in Fig. 5(a). The data for BSCCO(2) imply 7 ~ 1, since 9 is distributed symmetrically about its mean, and a similar analysis yields the values of zl plotted in Fig. 5(b). Equation (3.3) implies that the low frequency noise plateau S . ( f ~ O) should be proportional to (A(I)) 2 "y(y q-1)-3r 1 . For the switching processes in YBCO(3) and BSCCO(2), the slope ofln S.(f--* 0) vs. In z~ is very close to unity, as shown in Fig. 5(c). This implies that the temperature

Magnetic Flux Noise

27

dependence of the noise power spectrum can be attributed solely to rl(T). Other parameters, such as 7, l, cq, and the number of vortices involved in the process, do not change significantly over this temperature range. We can extract the zero-temperature vortex pinning energy U0 from the temperature dependence of rl(T). Since the measurements were made within a few Kelvin of Tc, we require some estimate of the functional form of U(T), but the temperature intervals over which values of zI(T) exist are too small to enable us to determine this unambiguously. If we were to assume that U(T) is well approximated by a temperatureindependent Uo (for example, Ref. 2) a least-squares fit of Eq. (3.2) to the data from YBCO(3) would yield Uo = 16eV and z~] ~ 109~176 Since this attempt frequency is absurdly high, we can rule out a temperatureindependent pinning energy. Most types of pinriing interactions for isolated vortices 37 give energies proportional t o the vortex self-energy38, e t ( T ) = (p21ntc/4rCpo[2(T)]2, where 2(T) is the penetration depth. Using the empirical two-fluid temperature dependence 2(T)/2(0) = [ 1 - (T/T~.) 4] -1/2, which agrees well with the theoretical result for a strong-coupled BCS superconductor 39 and with the values measured for YBCO, 4~ we find

U(T) = U0[1 - (T/Tc)4].

(3.4)

Hagen and Griessen have found this temperature dependence to be consistent with flux creep data analyzed according to their model. 41 We regard To. as a fitting parameter, since the small region in which the vortex moves may have a transition temperature different from that measured for the sample as a whole. Adopting ZA1 = 10-11 S in accord with several measurements 2'41'43 which yielded attempt times of that order, and fitting Eq. (3.2) to our measured values of zl(T), we obtain U0=4.1 eV and Tc = 89.5 K for YBCO(3) and Uo = 0.48 eV and Tc = 112 K for BSCCO(2). Note that the transition temperatures agree well with the expected values. Activation energies for models of the temperature dependence other than Eq. (3.4), and for other values of r m , are tabulated in Ref. 12. 3.3. Noisy and Quiet Metastable States So far we have considered processes in which the noise power is dominated by switching between readily identified metastable states, and have neglected the fluctuations about the mean value in each state. Figure 3(c) depicts one event from a more complex process in which the flux switches between two states separated by 2.8 x 10 3 (ID0" In contrast to the processes in Figs. 3(a), (b), and (d), the noise in state 1 (the lower level) is much higher than in state 2. We interpret this event as the hopping of a vortex over a barrier from a region where it is relatively mobile

28

M . J . Ferrari et al.

l(a)

TCBCO(1) 78.1 K ]I 0 m O O ls

(b)

TCBCO(1) 4.2 K 5 mOo 1

ls

§

0 t-'-

(d)

TCBCO(1) 4.2 K

(e)

BSCCO(2) 50.6 K

T5m% ~11'2oms

IlOm% ls

(f)

Time

Magnetic Flux Noise

29

(state 1), producing a high noise level, to another site where it is much more restricted spatially. A more dramatic example of such a process appears in Fig. 6(a). These data, obtained from sample TCBCO(1), indicate three metastable states for the vortex, each producing a different level of noise. This process persisted for many minutes, with hopping among the same three states occurring every few seconds. Noise processes such as these, in which the state of one fluctuator influences the noise power of another, have been observed in other systems such as metal-oxide-semiconductor diodes 43 and metallic constrictions. 44 Those data have generally been interpreted as the result of interactions ' among several reconfiguring defects, rather than as the motion of a single degree of freedom (analogous to vortex position) in a complicated potential. In the absence of microscopic information about the fluctuators, both explanations are tenable, but in the case of flux noise the low vortex density makes the single-vortex interpretation more plausible, However, vortices in the sample need not be uniformly distributed, and we can not rule out the possibility that the processes in Figs. 3(c) and 6(a) represent the interaction of one slowly hopping vortex with one or more rapidly moving vortices. We have found that under certain circumstances it is possible to activate and deactivate single RTSs. Closing the SQUID feedback loop applies a small magnetic field step (approximately 0.8/~T) of known sign to the S Q U I D and sample. The effect of this step is shown in Fig. 6 for TCBCO(1). After closing the feedback loop with a negative-going field step, we observed the relatively featureless flux noise in Fig. 6(b). However, when we caused a positive-going field step, two dramatic changes occurred. First, an RTS appeared in the time domain, evident as spikes in Fig. 6(c); both states in the process are clearly resolved on the expanded time scale in Fig. 6(d). Second, the time-averaged flux (q~) was larger by 0.1 q)o when the RTS was active compared to its value in the absence of the RTS. The amplitude of the RTS is too small to explain this increase, which we believe corresponds to a change in the magnetization of the sample caused by the trapping or displacement of a single vortex. We observed similar behavior in BSCCO(2) at 50.6 K. After a positive field step, we observed the noise in Fig. 6(e), while Fig. 6(f) shows addiFig. 6. Flux through SQUID generated by RTSs vs. time (a) TCBCO(1) at 78.1K with three metastable states. Several seconds of data have been omitted (broken segment) during which time the fluctuator remained in the middle state. (b) and (c) TCBCO(1) at 4.2 K after negative and positive magnetic field steps, respectively. (d) Single event from (c) with time resolution increased by a factor of 40. High-frequencynoise is also increased because of larger bandwidth. (e) and (f) BSCCO(2) at 50.6K after positive and negative field steps, respectively. Zeros of axes are arbitrary.

30

M . J . Ferrari et aL

tional noise from an RTS after a negative field step. The average flux (qb) is 0.02 ~o less when the RTS is present than when it is not. The noise power spectra under these two conditions are plotted in Fig. 7(a). In the absence of the RTS, the noise exhibits a featureless 1If spectrum, of the type discussed in Sec. IV. The negative field step causes a weak knee to appear in the frequency domain near 10Hz. As shown, the difference spectrum in Fig. 7(b) is well described by Eq. (3.3), indicating that the effect of the field step is to superpose a single RTS on 1If noise from another source, presumably the motion of other vortices not involved in the RTS. We found that the RTSs in Figs. 6(c) and (f) could be reproducibly activated by positive and by negative field steps, respectively, and deactivated by a step of the opposite polarity. It seems likely that the shielding currents induced by the changes in the applied magnetic field exert Lorentz forces which, depending on the sign, force a vortex into or out of a noisy configuration.

l

--

I

-I

- - I

10-6

U

1-

%o 10-8

I 0-lc

(b)

(a) 1

I

10

I

__i

10 2 1 10 Frequency (Hz)

I

10 2

Fig. 7. Spectral densities of noise from BSCCO(2) at 50.6 K. (a) After positive-going (lower) and negative-going (upper) magnetic field step. (b) Heavy curve, difference of spectra in (a), light curve Eq. (3.3) fitted with z = ('ri-1 + T21)-1 = 18 ms and (2"cA~)2/('Cl + "c2) = 2 x 10 -6 ~2/Hz.

Magnetic Flux Noise

31

4. 1/f NOISE The observed RTSs in Sec. III occur in the unusual circumstance that a single hopping process dominates the noise. More typically, a large number of mobile vortices produces a 1If power spectrum as shown in Fig. 8 for YBCO(3), the least noisy sample we measured. 1~ At 4.2 K the power spectrum is dominated by noise in the SQUID, but at 77.4 K the noise at 1 Hz has increased by a factor of approximately four. At higher temperatures the spectra clearly exceed the SQUID background and scale nearly as 1/f Figures 9 and 10(b) show that S~, (1 Hz) increases slowly as temperature is raised from 4.2 K to 87 K, then peaks sharply at To. Figure ! 1 plots the spectral density of the noise at 1 Hz as a function of temperature for the three samples which were patterned into rings 9 as described in Sec. II. As for YBCO(3), the noise power generally increases with temperature, rising steeply just below To. Just above To, the noise amplitude drops very quickly with increasing temperature, and the slope of its power spectrum decreases from unity [Figs. 10(c) and 11]. Rogers et al.14 have also observed significant departures from 1If scaling in the flux noise produced by thin BSCCO films near T,., and attribute this to vortex diffusion, rather than to a hopping process. From the data in Figs. 10 and tl it is evident that noise decreases as sample quality improves, and that the noise well below Tc is less temperature dependent for the quieter films. As Table I(a) indicates, the parameters used to measure sample quality generally vary together, so that low-noise films tend to have good epitaxial orientation, high transition temperatures, and high critical currents. Since we do not vary these parameters independently, we cannot precisely identify which control the noise. The hopping processes in a quieter sample may have higher activation energies U0, shorter hopping distances/, or a combination of the two. Alternatively, the sample may simply contain fewer vortices; the data of Secs. V and VI suggest that their number does not always correspond to the applied field, and may be sample-dependent. 4.1 Extended Dutta-Dimon-Horn Model

In analyzing the 1If noise we use the Dutta-Dimon-Horn (DDH) model, 45 which consolidates early insights by other workers. 46 The central assumption is that 1/f-like noise arises from the incoherent superposition of many thermally activated switching processes, similar to those analyzed in Sec. III. An attempt has been made to explain our measurements in terms of universal conductance fluctuations, 47 but this mechanism seems unlikely. 13'48 We make two principal extensions to the D D H model: a

~ . ,I, Ferl"a~i r at,

32

10-2

10-4

89.3 K

T""

I

N

:E

89.7 K

10-6

cJO

4) o3 77.4K I 10-10 t 10

10 2

10 3

Frequency (Hz) Fig, 8. Spectral density S,p(f) of ftux noise in YBCO(3) at five temperatures. At 4,2 K, observed noise is due to SQUID; this contributio~ is also significant at 77.4 K, Spikes are from 66 Hz pickup and microphonics,

Magnetic Flux Noise

10 -2

i

r

I

i

]

p

33

i

i

i

10 -4 I

N "1oJo

g

10-6

-1-

e

CO

10-8 9

10-10

9

i

9

~

9

i

0

J

J

i

i

i

50 Temperature (K)

J

100

Fig. 9. Flux noise S~(1 Hz) vs. temperature for YBCO(3). Values

below 8xl0-1~ include large contributions from SQUID. A reproducible peak occurs near 75 K (arrow). generalization to temperature-dependent activation energies, H and (in Sec. V) in inclusion of a driving force which acts on the vortices. 13 The D D H model postulates an ensemble of two-state fluctuators, or hopping processes, with a distribution of pinning energies. In the absence of detailed knowledge of the pinning potential, we assume the two wells to be symmetrical, 35 so that U~(T)= U(2)(T)=- U(T) and ZA~ = r ~ 2 - 2%; we justify this symmetry in Sec. V. From Eq. (3.2) we write 1

~(T)=~I + z21 ~o exp[Uofl(T)],

(4.1)

where U o = U ( 0 ) and fl(T)=--U(T)/Uok~T. The spectral density of the noise from this single process is, from Eq. (3.3),

(A(I))2 "C

S~(f,

T) = 1 + (2zcfz) 2'

(4.2)

where A~ is given by Eq. (3.1). Assuming that the noise processes are incoherent (see Sec. VI) we can combine them to find

ZS~,(f, T) = ~ffo dUo D(Uo) 1 +((I)l/)2 (2~fz) 2'

(4.3)

34

M . J . Ferrari et aL

i

i

1.0

(a)

,4

v O

m

0.5

1.ol,.,.,.:._.. .... 1 %

v O

rn

.... s,,,,i,0,,t 0

I

I

h

..o" IN -r10-4

(b) ~

Q

o

t~lO

e

~

10 -6!

N

-1-

o

~ o

e 10-8!

O9

o

o

oo ~

10-10

I

I

1.3

E o

(c)

1.0 0.7 0.4 87

1

89

__1

Temperature (K)

91

Fig. 10. YBCO(3) near Tc=89.8 K. (a) Normalized applied magnetic field B 0 required to generate one flux quantum in the SQUID vs. temperature. Inset shows B0 over wider temperature range. (b) Noise power at 1 Hz vs. temperature. Solid symbols. 1If spectra; open symbols, Lorentzian spectra from RTSs; downward arrow, upper limit on noise. (c) Slope m of noise spectrum vs. temperature.

MagneficFluxNoise

10

47K

-2

1-,

35

85K .

.

,~

,

"rs 10-4 Z: V

-

10 -6 _

r

10 -IO 0

'l

50

100

Temperature (K) Fig. 11. Spectral density of flux noise at 1 Hz vs. temperature for: YBCO(5), squares; YBCO(4), triangles; YBCO(1), circles. SQUID noise has been subtracted. Solid symbols imply that the spectral density is 1If at 1 Hz, open symbols that it is white or nearly white. Dashed lines indicate temperature ranges where the noise prevented flux-locked operation of the SQUID. Downward arrows above Tc indicate upper limits on the noise from samples. (Small errors in the calibration of the thermometer and the subtraction of background noise have been corrected since the original appearance of this figure in Ref. 9.)

where D(Uo)dUo is the number of processes with zero-temperature activation energies between U0 and Uo + dUo. We have removed the angular dependence from Eq. (4.2) by averaging over 0, which we assume to be uncorrelated with Uo. We assume that %, fl(T), and l are the same for all processes. Of these, fl(T) is the least likely to vary from process to process, since it is notoriously difficult to distinguish core pinning interactions from magnetic interactions, for example, on the basis of their temperature dependence. 37 We discuss the consequences of a distribution of r0 at the end of this section, and of an energy-dependent hopping distance l(Uo) in Sec. V. The time 9 is an exponential function of U0, causing the kernel in Eq. (4.3) to be sharply peaked at a characteristic energy

C7o(f T) = ln(1/2nfro)/fl( T).

(4.4)

M . J . Ferrari et aL

36

Assuming the distribution of activation energies D(Uo) varies slowly across the width of the kernel, we can take D(Uo) outside the integral to find D[Uo(f, T)]

8r

....

=(--~tl)aj~,tj, T).

(4.5)

This equation summarizes the physical content of the D D H model. By measuring S~,(f, T) and taking an appropriate functional form for/}(T), we can determine D(Uo). Because l is unknown, D(Uo) contains an unknown scale factor, a deficiency we resolve in Sec. V. The model makes an additional prediction, which serves as a consistency check. From Eqs. (4.4) and (4.5) it can be shown that the slope m = - ~ in S~,fi?In f obeys ln(1/Z~fzo) 1 ( 1 + 0--'i-n~n ~ in S./~? In 17~ T/~-lnn T)"

m(f, T ) = 1

(4.6)

Equation (4.6) shows the D D H model to be applicable only if the power spectrum is steep in temperature ranges over which the noise is increasing, and has a shallower slope when the noise is decreasing. The 1/f noise measurements from which we extract D(Uo) are plotted in Fig. 12. We exclude the region near Tc where RTSs are observed

10-2

~"

10-4

I

c~lo

10_6

"~ cO

!

10-8 10-10

0

i

k

I

50 Temperature (K)

L

i

100

Fig. 12. Spectral density of flux noise at 1 Hz vs. temperature for YBCO(1), squares; YBCO(2), triangles; BSCCO(1), circles. SQUID noise has been subtracted. All data represent 1/f-like noise power spectra. Curves are piecewise cubic interpolants used to calculate rn(T) (solid and dashed segments) and D(Uo) (solid segment only).

Magnetic Flux Noise

1.8

37

la'YBCO/ t 'I' 1

1.4 1.(3 E 0.6 (1) 00

I

I

I

1.2

I

I

I

I

I

I

t

E

(b) YBCO(2)

1.0 0.8 1.4

(c) B S C C O

^

1.2 1.0 0.8 0

i

~

i

i

I

i

I

50 Temperature (K)

100

Fig. 13. Slope m(T) of noise power s p e c t r u m vs. temperature for: (a) Y B C O ( 1 ) , (b) Y B C O ( 2 ) , (c)

BSCCO(1). Points are experimental data. Curves are predictions of Eq. (4.6) from fits to noise in Fig. 12, with ~0= 10-H s and/~(T) = [1 - ( T / T c ) 4 ] / k B T . (Table II) and where our analysis depends sensitively on/~(T). We need to specify two parameters to obtain quantitative results from the D D H model (see Sec. III): the prefactor Co = 10-11s (ignoring an insignificant factor of two) and the temperature dependence/~(T) = [ 1 - (T/Tc)4]/k~T, for which we use values of Tc from Table I(b). Figure 13 shows the experimental values of m(T) for each sample together with the valuies calculated from Eq. (4.6). Although the measured and predicted values of m can differ by as much as 10 %, the major trends are reproduced, making our application of the D D H model credible. 4.2. Distribution of Activation Energies

Figure 14 shows D(Uo) obtained from Eq. (4.5) with the fit to S~(T) in Fig. 12. Both YBCO films show a peak in D(Uo) below 0.1 eV while the distribution of activation energies for BSCCO(1) is broader. In addition to

M . J . Ferrari et aL

38

l~: U1/2

I constant

2O

'

I

'

~(a) YBCO(1)

I

3.0

(d)

r :3

1.5

~10

~., ~.u

~

(b) YBCO(2)

(e)

0.5

LLI ~3

.c_ ._c 13_

I

i

(f)

(c) BSCCO

60

'5 E

._o

*5 30

10

.13

i

I

0.2

i

014

0 Energy (eV)

0,2

0.4

Fig. 14. D(Uo) vs. energy U o for three samples: (a) and (d), YBCO(1); (b) and (e), YBCO(2); (c) and (f), BSCCO(1). Two models of the hopping distance are shown: left column, constant l [Eqs. (4.4) and (4.5)]; right column, l oc Ulo/z [Eqs. (5.8) and (5.9)]. Arbitrary units on vertical scale are same within each column. Fitting parameters: % = 10 -11 s and fl(T)= [1 - (T/Ty]/kBT.

the low-energy peak, YBCO(1) shows pronounced structure near 0.35 eV produced by the noise peak at 73 K. This structure is absent in the other samples we have measured, with the possible exception of YBCO(3) (Fig. 9). Note that according to Eq. (4.5) an increase in noise can correspond to a peak in D(Uo) or to an increase in the hopping distance l. The curve for YBCO(1) in Fig. 14(a) assumes the former explanation, but the noise near 73 K in this sample may simply be dominated by a few processes with unusually large hopping distances. For several reasons, we are reluctant to apply the D D H model to the very large peak in the noise near T c. Calculation of D(Uo) in this tern-

Magnetic Flux Noise

39

perature range is problematic because Eqs. (4.4) and (4.5) become sensitive to the temperature dependence of//(T), which we do not know exactly. Furthermore, the noise generation mechanism close to Tc may differ from that at lower temperatures. Possible additional mechanisms include the creation of vortex-antivortex pairs when their energy approaches k B T , time-dependent shielding as regions of the sample fluctuate across the normal-superconducting phase boundary, a vanishingly small barrier to thermally activated phase slips in the order parameter, or the appearance of cooperative vortex motion as 2(T) diverges. There are two indications that the noise near T c may be intrinsic to YBCO and independent of microstructure, thereby adding doubt to the validity of the D D H model near To. First, the peak S . (1 Hz) is approximately 10 -3 (I)2/Hz for all our samples, and also for much thinner (d~= 10nm) samples measured by Missert. 49 Second, the maximum noise occurs at precisely the temperature where the static diamagnetic susceptibility of the sample vanishes, as shown in Fig. 10. Further investigation of this peak would be of considerable interest. When comparing our measured activation energies to the work of others, one should bear in mind that the noise measurements are insensitive to processes with larger values of Uo than those plotted in Fig. 14. A vortex pinned at a high-energy "silent site" would be extremely unlikely to move during the time of the measurement. These silent sites might be of great importance, however, in an experiment performed far from equilibrium, such as flux creep near Jc or thermally activated resistivity near To. Nevertheless, there is remarkable agreement between the low-energy peak in D ( U o ) observed in our YBCO samples and the distributions derived by Hagen and Griessen 41 from flux creep measurements. Zero-field critical current measurements on YBCO by Mannhart et al. 5~ and Tahara et al., 5~ and current-voltage characteristics measured by Tahara et al., 52 have also yielded pinning energies of 0.02 to 0.2 eV.

4.3. An a-Axis Sample

Figure 15(a) shows the noise measured in YBCO(6), a thin film with the a - b plane oriented perpendicular to the substrate in more than 90 % of its grains. Although this sample is not very different in transition temperature, critical current density, and degree of orientation from YBCO(1 ) [Table I(b)], a comparison of Figs. 12 and 15(a) shows YBCO(6) to be much noisier than YBCO(1) at all temperatures. This observation is in accord with theoretical 53 and experimental 54 evidence that vortices parallel to the a - b plane move along the plane relatively freely, so that the

40

M.J. Ferrari et al.

10- 2 '

'

'

'

I

'

'

'

10-4 Iu

v

~- 10-6

% co 10-8 (a) 10-10 E " 0

I

I

I

I

I

I

I

I

b

I

I

i

I

1.2 1.0

---1F- m

m m

9

co

(b)

0.8 I

50 Temperature (K)

I

IO0

Fig. 15. Flux noise from a-axis sample YBCO(6). (a) Spectral density of i/f-like noise at 1 Hz vs. temperature. Curve is cubic spline fit to data. (b) Slope m(T) of noise power spectrum vs. temperature. Points are experimental data; statistical error does not exceed height of symbols. Curve is prediction of Eq. (4.6) from fitted in (a), with %= 10-Us and fl(Z)

= [1 --

(Z/Zc)4]/kBZ.

magnitude of the noise is the result of longer hopping distances, lower pinning energies, or both. The slopes m ( T ) measured in YBCO(6) and calculated from Eq. (4.6) are plotted in Fig. 15(b). The measured slope is remarkably independent of temperature, in contrast to that in Fig. 13, except for a small dip between 70 K and 80 K which matches the prediction of the D D H model. This temperature independence reinforces a trend suggested by Fig. 13, namely that the predicted slope consistently exceeds the measured value by 0.05 to 0.1. We have implicitly assumed in the derivation of Eq. (4.6) that the vortex hopping distance l is temperature independent, so that the temperature dependence of the noise from each process rEq. (4.2)] is due solely to z(T). As plotted in Fig. 5(c), the RTSs we observed behave in accordance with this assumption, but for narrow ranges of temperature. Nevertheless, if we

Magnetic Flux Noise

2000

i

I

41

t

I

(a)

1000

"E -m

l constant (b)

"-S v

100

tm

50

l~ 6

i

,1/2 u0

I i 0.2 E n e r g y (eV)

I 0.4

Fig. 16. b(Uo) vs. U 0 for a-axis sample YBCO(6): (a) Constant l, [Eqs. (4.4) and (4.5)], (b) l oc g 1/2 [gqs. (5.8) and (5.9)]. In each case ro = 10 -H s, fl(T)= [1 -(T/Tc)4]/kBT, and D(Uo) is in same arbitrary units as Fig. 14.

were to postulate that l increases with temperature, the curves in Figs. 13 and 15(b) would be displaced downward, improving their agreement with the experimental data. More generally, the fact that Eq. (4.6) overestimates m indicates an additional temperature dependence, unaccounted for in the model, which causes the noise to increase with temperature. We observe in Fig. 16(a) that D(Uo) for YBCO(6) increases steadily with energy, with no evidence of the low energy peak displayed by the c-axis films.

4.4. Why Not a Distribution of Attempt Times? To produce a 1/f spectrum from an ensemble of Lorentzians, one requires a distribution in z. This can arise from a distribution of U0, as we

M . J . Ferrari et aL

42

have assumed up to now, but one could alternatively postulate a distribution of %. Clearly these are two limiting cases of a more general model in which both Uo a n d ~o are distributed, but such a theory contains too many free parameters to have much predictive power. If we assume z is still given by Eq. (4.1), but that Uo is constant while ~o was a distribution D(z0), the analog of Eq. (4.6) becomes

m(f,T)=l

1

O_ln_S./O__In fl

(4.7)

Uo~(T) OlnT/OlnT"

Note that /~(T) vanishes as T ~ Tc, so that Eq. (4.7) predicts that the slope diverges near the transition temperature, in contrast to the result of Eq. (4.6) and contrary to the experimental data. Even if we avoid this problem by applying the model only well below To, the activation energy required for Eq. (4.7) to reproduce approximately the curve in Fig. 13(a) is Uo=0.4eV. This value implies that the processes contributing to the noise measured at T = 4 . 2 K and f = l Hz are those for which o 1 = 27rf exp [ Uo/~(T) ] ~ 10 480 Hz, which is unreasonable to say the least. The assumption that the observed 1If noise arises from a distribution in Zo leads to difficulties at both ends of the experimental temperature range, while a distribution in U0 is consistent with all of our measurements. It is interesting to note that the temperature dependence of U(T) enables us to make a meaningful test of the distribution of %, one which is not available in most systems. Although we cannot rule out some distribution in Zo, our data imply that the distribution in Uo dominates.

5. C U R R E N T D E P E N D E N C E

5.1. Experimental Results The availability 22'23'55 of flux transformers fabricated from YBCO enables us to study the response of noisy vortices to a driving force. A schematic of the transformer on which we made noise measurements appears in Fig. 17. It was attached to the hot side of the apparatus with its input coil of self-inductance Li centered above the SQUID and coupled to it via mutual inductance Mi = ei(LLi) m. We estimate Li ~ 75 nil, ~e ~ 0.5, Mi~3nH; the geometrical self inductance of the pickup loop is Lp ~ 20 nil. We measured the noise in such transformers in the presence of a circulating current /, which we induced by applying a static magnetic field ( < 10 #T) from a persistent current coil below T c. For I < 3 mA, the current density in the input coil does not exceed 5 x 104Acm -z. As showing in Fig. 18(a), a current of I = 2 mA in transformer T1 reduces

Magnetic Flux Noise

43

(a)

|-|

(b)

m

100pm (c) coil . . . . . . . . . . . . . . . . . . SQUID

2a

-~

4

2c

100gm

Fig. 17. YBCO flux transformer. (a) Pickup loop (wide lines), input coil centered on bottom edge, insulating SrTiO3 layer indicated by dashed box. (b) Enlargment of input coil: SrTiO3 is shaded region, crossunder is vertical strip connected to center of coil. (c) Cross section of input coil a distance d from SQUID. S~,(1 Hz) by an order of magnitude from its I = 0 value; S.(f) returns to the latter value when the current is removed. This reversibility suggests that the noise reduction can not be explained by driving the vortices from shallow, noisy pinning sites into deeper, quieter ones, since they would then remain in the deeper sites after the removal of the current. The highfrequency regions of the spectra in Fig. 18(a) are dominated by white noise from normal metal components of the hot side, but the difference spectra in Fig. 18(b) scale as I/f, demonstrating that the effect of the current is to remove a source of 1If noise. We have shown 33 that noise in our transformers is dominated by fluctuations in the circulating supercurrent,

44

M . J . Ferrari et aL

,

--F

I=0

tm-~

10-6 ~ ,,~...

~

Iu

1-11 1

Distance ~

- ['>",:,

'N -r"

-

m

I

0

0

;

0.1 Energy (eV)

I~ '~-'--, ',2,..

'%;,0,.<~mA

2 mA "-,

I

/

if

1

]-

"'~,~

...... .,'k.j"" ..... ,

\~

0.2

"

.,

-

'%"k,~,(

(b) 10-10

(a)

1

, 10

,

I

10 102 Frequency (Hz)

I

102

Fig. 18. (a) Total noise power (referred to SQUID) vs. frequency for transformer TI at 39 K. Upper solid curve, initial spectrum (I= 0), lower solid curve, l = 2 m A , dashed curve, subsequent return to I=0. Inset: Schematic pinning potential for I = 0 and I > 0. (b) Spectra for I = 0 with spectrum for I = 2 mA subtracted. Note 1/f scaling. Inset: Solid curve, D(Uo) determined from noise measurements according to Eqs. (5.8) and (5.9). Peak near 0.05 eV is significant, weak 0.1 eV feature is not. Dashed curve, theoretical distribution from Eq. (5.7) with k0=6 x 10 -5 N/m, Np=4x 106, and ns = 600 #m -2. caused by vortex m o t i o n in the i n p u t coil. W e call this m e c h a n i s m indirect noise because the m a g n e t i c field a s s o c i a t e d with a vortex does n o t link the S Q U I D directly, in c o n t r a s t to the direct noise m e c h a n i s m of Sec. III. F u r t h e r m o r e , a sufficiently large c u r r e n t suppresses indirect noise, so t h a t the residual noise at I = 2 m A is direct noise from the crossunder. 33 S u b t r a c t i n g the l a r g e - c u r r e n t b a c k g r o u n d from the noise m e a s u r e m e n t s at each t e m p e r a t u r e , we o b t a i n the c u r r e n t d e p e n d e n c e of the indirect noise p o w e r S . ( 1 H z ) which is p l o t t e d in Fig. 19. T h e noise p e a k s s y m m e t r i c a l l y a b o u t I = 0 a n d the width of the p e a k s increases slightly with t e m p e r a t u r e .

5.2. D u t t a - D i m o n - H o r n Model with Lorentz Force Terms W e e x p l a i n t3 the c u r r e n t s u p p r e s s i o n of noise within the D D H m o d e l by considering the effect of a L o r e n t z force on a single switching process

i

i

]

1

I

I

I

I

I

I

[

I

I

4.2K 10- 8

10 - I

[

13.7 K 10 - 8

I

10-1 o!

IN "T

.co

I

2 4 . 5 ~~ K

= ----~

10 - 7

-110 -9

~

39.2 K

,

I

!o-71 / 10-8| ~---aJ, 45.7 K 10-7 v _ ~

J

,~

i ,'-~--, /~ - ~ ~

I(mA)

Fig. 19. Noisepower So(1 Hz) at SQUID vs. I in transformer T1 for six temperatures. Points are experimental data, from which least noise measured at each temperature has been subtracted. Curves are prediction of Ec1. (5.2) with indirect noise transfer function q)z. Central peak is due to input coil, shoulders to pickup loop.

46

M . J . Ferrari

et aL

as illustrated in the inset to Fig. 18(a). A vortex hops between pinning sites 1 and 2, separated by a distance /. As in Sec. IV, the pinning potential is assumed to be symmetric in the absence of a supercurrent, with zero-temperature activation energy Uo. We assume that I is uniformly distributed across the line of width wj, so that it exerts a Lorentz force F = IOo/Ws on each vortex. A more realistic current distribution 56 improves the agreement between theory and experiment in Fig. 19 only slightlty while making the calculation more difficult and leaving the quantitative conclusions of this section substantially unchanged. The force introduces a misalignment A U = Fl cos 0 between the minima of the pinning potential [-Fig. 18(a)], where 0 is the angle between the Lorentz force and the vortex trajectory. An increase in the current reduces the noise from this process because the vortex is less likely to be activated out of site 2 when the pinning potential is tipped. This simple observation provides the physical basis for the suppression of indirect noise by current, and also reveals why direct noise is relatively unaffected in our geometry. The noisiest indirect processes (largest AO) are those involving vortices in the input coil hopping across the line, in the direction of the Lorentz force. However, direct noise from the crossunder arises primarily from vortices hopping radially with respect to the SQUID, perpendicularly to the Lorentz force, which thus has little effect. In addition, the linewidth of the crossunder is five times that of the input coil, significantly reducing the local Lorentz force. In the presence of a supercurrent Eq. (3.2) becomes Zl(T, I) = 2% exp[ Uo~(T) - 3]

(5.1a)

zz(T, I) = 2% exp[Uo/~(T) + 6]

(5.1b)

where 6 = I O o l c o s O / 2 w j k B T . The power spectrum S~(f, T , I ) of each hopping process is a Lorentzian as in Eq. (3.3) with Aq~ = Oil cos 0 and the indirect noise coefficient 33 Or= OoMi/ws(Li + Lp). The total noise becomes

Sr

T, I) = ~ -dO 7C

dUo D(Uo) s | i

T, ~r).

(5.2)

Unlike Eq. (4.3), Eq. (5.2) includes an explicit angular integral because processes with different values of 0 have different current dependences. Furthermore, S~ differs among the segments of the transformer because and 9 t depend on wj, an effect we take into account by assuming that processes are randomly distributed with uniform areal density.

Magnetic Flux Noise

47

5.3. Two-Dimensional Pinning Model

In Sec. IV we assumed for simplicity that l is the same for all processes in the ensemble. The data in Fig. 19 allow us to test this assumption, and in fact reveal that l increases with Uo. Note that the width of the peak is only weakly dependent on temperature, although Do is roughly proportional to T. If l and therefore A U were independent of temperature, the fractional misalignment AU/Uo would decrease with increasing temperature, the current would suppress the noise less effectively, and the peaks would broaden. The energy dependence of 1 corresponds to one's intuition that the barrier for a vortex to hop a large distance should exceed that to hop a short distance. Since the data do not constrain a particular functional form for l(Uo), we propose a model with the following assumptions: (i) We reduce the problem to two dimensions. Certainly our measurement is sensitive only to the configuration of points at which the vortices emerge from the surface of the sample facing the SQUID; motion of internal segments does not generate noise. (ii) Pinning sites are randomly distributed with areal density ns, and are not extended in the sense of twin or grain boundaries, that is, we assume that the pinning which underlies 1If noise is provided by localized features such as point defects 57'58 or the cores of screw dislocations. 59 (iii) Vortices do not interact with each other. This is a good approximation in our experiment because n~1/2>>Z, where nv is the areal density of vortices. (iv)

Each vortex interacts with only the nearest pinning site.

(v) The potential energy u(lr~-rsl, T) of a vortex at position r~ relative to a pinning site at r s is the same for all pinning sites. To reconcile this model with the ensemble of bistable hopping processes described earlier, we associate each hopping process in the ensemble with a pair of pinning sites in the two-dimensional model. This is a valid identification at low enough temperatures where IE~'o(T)] "~ n7 ~/2.In this limit, most vortices can not escape from their pinning sites on the time scale of the experiment, and the only ones which produce a measurable amount of noise are those which happen to be located at one of a pair of sites which are unusually closely spaced. The association is certainly invalid when I[Uo(T)] >> n s 1/2, since hopping events will be fast compared to the experimental bandwidth and the slope of the noise power spectrum will decrease from unity, as we observe just above Tc (see Figs. 10 and 11). Data presented later in this section imply that transformer T1 is in the

M . J . Ferrari et aL

48

intermediate regime, /[U0(T)] " ~ t t s 1/2. I n this case we can not rigorously justify the identification of a process with each pair of pinning sites in the two-dimensional model, but the issue may be amenable to further theoretical analysis or to numerical simulation.

5.4. Multiplicity of Processes and Distribution of Activation Energies Although we have assumed that the pinning interaction is the same at each site, the activation energies remain distributed, because the distances between pinning sites are random. Each vortex may hop to any of several sites and is thus involved in some number P~ of processes which can be calculated exactly. A pair of sites a distance l apart constitutes a process provided that a vortex moving along the line connecting them is never more strongly attracted to a third site than it is to the nearer of the pair. Thus, the two sites in a process are located at opposite ends of a diameter of a circle of area ~12/4 which contains no other sites. In a sample of total area A, there will be n ~ A - 2 other sites, and, in the limit A>>I 2, n j 1 the probability that none of them falls within the circle is p ( l ) = exp [-~n~12/4]. Integrating over all possible hopping distances, we find

P~ =

d127zlnsp(l ) =4.

(5.3)

This result depends on the dimensionality of the problem and on our definition of what constitutes a process, but not on n~ or the functional form of the pinning potential. To calculate the distribution of activation energies, we consider the number of available processes with hopping distances in the range l, l + dl,

D(1) dl = (nsA/2) 2~tl dl nsp(l),

(5.4)

where the factor 1/2 eliminates double counting. The distribution of activation energies is then D(Uo) ~ D(1)dl/dUo, and is a proportionality, rather than an equality, because not all of the available processes will be occupied by a vortex. The constant of proportionality can be determined from the normalization condition

I •

dUo D(Uo) = PvnvA -

(5.5)

where Np is the number of processes in the ensemble. To obtain an explicit formula for D(Uo) we have to assume a functional form for u(r, T). In the absence of detailed knowledge of the physical

Magnetic Flux Noise

49

origin of the pinning, we adopt the common assumption 6~ u(r, T ) = k(T)rS/2, which is, of course, no more than the leading term of the Taylor series expansion of a general potential. We assume all pinning sites to be identical, so that no additive constant is necessary. This is a crucial assumption; it insures that when we extract a D D H ensemble from this two-dimensional model, each process will be symmetrical as shown in the inset to Fig. 18(a) for I = 0, and therefore the effect of a current will be to suppress the noise. We will return to this point in greater detail below. Note that the temperature dependence of k(T) is the same as that of U(T) [Eq. (3.4)]. Defining k o = k(0), we find

(su~ \ koJ

,

(5.6)

and the distribution of activation energies is then 27~/7s

D(Uo) = Np ~

exp[ -2un~ Uo/ko].

(5.7)

Our simple model thus provides a possible physical basis for the exponential distribution assumed in the theories of Inui et al.57 and Martin and Hebard. 61 We wish to extract D(Uo) from our noise measurements, both for comparison with the distributions obained in Sec. IV and to test the theoretical prediction of Eq. (5.7). For I = 0 we can evaluate the integrals in Eq. (5.2) analytically, subject to the usual condition that D(Uo) is slowly varying, to find n [ U o ( f , T)] = ~(,,)ln(1/2uf%) \ - ~ 0 /

fS|

T, 0),

(5.8)

where ~(i,)~ 1.2 is a geometrical constant, 33 S.(f, T, I) is the spectral density of the noise in the presence of a current/, and the characteristic ~'o satisfies Uo(f, T ) =

ln(1/2~f%) . //Uo(f, fl-~ + 2fi--~)T)In . -

+

T) fl(T) i)" \Go( f, T) fl( T) -

(5.9)

These results look quite different from those obtained in Sec. IV because the hopping distance l is no longer the same for all processes, but assumed to increase with energy according to Eq. (5.6). For f % = 10 -11, however, the second term on the right side of Eq. (5.9) introduces only a 0.2%

50

M . J . Ferrari et al.

correction to Uo; otherwise the result is the same as Eq. (4.4). The only qualitative difference between Eqs. (4.5) and (5.8) is the additional factor of fl(T) in the latter, which causes the distribution of activation energies to roll off more markedly with increasing energy, as is apparent in Figs. 14 and 16. In the inset to Fig. 18(b) we plot D(Uo) for T1 obtained from the measured values of S.(1 Hz, T, 0); D(Uo) shows a broad peak near 0.05 eV similar to those in Figs. 14(a), (b), (d) and (e), regardless of whether l is taken to be constant or proportional to U~/2.There is also a strong upturn at lower energies; note the remarkable similarity to Fig. 3 of Ref. 62. Neglecting the second term on the right side of Eq. (5.9), one can easily show that the consistency relation for the slope m of the noise power spectrum in the absence of a supercurrent is unchanged from the constant-/ result in Eq. (4.6). Neither the predicted nor the measured values of rn for T1 shows any significant departure from unity.

5.5. Hopping Distances and Restoring Forces Having obtained the form of D(Uo), we can solve for the noise in the presence of a supercurrent by integrating Eq. (5.2) numerically. The resulting curves are shown in Fig. 19. The only fitting parameter is the zero-temperature spring constant ko = 6 • 10 -5 N/re. The measured and calculated noise in Fig. 19 agree trivially at I=0 because D(Uo) was obtained from these data; the shape of the peaks for non-zero current depends only weakly on the form of D(U0), but is sensitive to k o (peak width) and to the geometry of the transformer (high-current shoulders). Knowing k 0 allows us to calculate l from Eq. (5.6). Since Uo increases from 8.5 meV to 190 meV over the temperature range from 4.2 K to 60 K, the hopping distance of processes which contribute to the measured noise ranges from 13 nm to 64nm. These values are one to two orders of magnitude smaller than those given in Table II for the random telegraph signals in YBCO, a finding consistent with our assertion that each RTS is a single long-range process, while 1/f noise results from many short-range hopping events. As discussed in Sec. IV, the DDH model assumes a symmetrical pinning potential, so that both sites in the inset to Fig. 18(a) have the same activation energy for I= O. One explanation for this is that the pinning sites arise from identical defects. The suppression of noise provides strong evidence for this symmetry. If U~ and U(2)(T) were completely uncorrelated, tipping the double well would bring as many processes into alignment as it removed, and the noise would not depend on current. The sharpness of the peaks in Fig. 19 requires that the processes in the ensemble be approximately symmetrical, although a small random asymmetry could

Magnetic Flux Noise

51

explain the discrepancy between theory and experiment at 4.2 K. This discrepancy might also be the result of a hopping distance smaller than that given by Eq. (5.6), or of vortex motion by configurational tunneling. 6 The suppression of noise in each bistable process requires that there be no pinning site to the right of site 2 in the inset to Fig. 18(a) into which the vortex can hop. In terms of the two-dimensional model, this means that the next site in that direction is more distant than site I, so that the barrier is relatively large. Even if this were not the case, however, the vortex would simply move under the influence of the Lorentz force until it encountered such a barrier. The idea of a distribution of activation energies is given powerful support by the data in Fig. 19, since there must exist both small barriers over which vortex hopping occurs ("hoppers") and large barriers ("stoppers") over which the Lorentz force is unlikely to drive a vortex. Thus we believe that our noise measurements ,probe the low-energy region of D(Uo). Experiments such as a flux creep 1 employing current densities orders of magnitude larger than in our experiment or thermally activated resistivity2'3 measured closer to Tc [where U(T)~ Uo] very likely involve vortex motion over the "stoppers." Experiments such as these are also generally performed in magnetic fields 105 times larger than the residual field in our experiment, and the resulting strong vortex-vortex repulsion may compete with the pinning forces. 5.6. Excess Vortices

To calculate D(Uo) from Eq.(5.7) we require two additional parameters, Np and ns. We require Np = 4 x 10 6 t o make the integral of the distribution over the experimental energy range (8.5 meV to 190meV) equal to that of the measured distribution. We then adjust the pinning site density to give the best fit, and find ns = 600 #m -2. This value is within a factor of two of the areal density of Y203 inclusions is sputtered YBCO films observed by Catana et aL, 58 and is consistent with the limit set by Dolan et al., 63 n~ > 100/.tm -2, for an unidentified but uniformly spatially distributed pinning mechanism inferred from images of decorated flux lattices in YBCO. The theoretical distribution of activation energies is plotted in the inset to Fig. 18(b). The qualitative agreement with the measured distribution is reasonable, although the simple exponential in Eq. (5.7) does not reproduce the measured peak at Uo, peak~ 0.05 eV. This peak may indicate that the defects which cause the pinning have a preferred intersite distance of (8Uo, peak/ko)l/2,~ 30 rim, or it may indicate a pinning mechanism beyond the scope of the two-dimensional model. The number density of vortices in transformer T1 is given by n~=Np/P~A =0.03 #m -2, indicating that only one pinning site out of

52

M.J. Ferrari et

al.

n~/n, ~ 2 x

l 0 4 is occupied by a vortex. Integrating D(Uo) from 8.5 meV to 190meV, we find that almost 80% of the processes in the ensemble fall within the experimentally accessible energy range. Thus there are few silent sites, provided that the form of Eq. (5.7) is valid for Uo> 190meV. However, we have no evidence that our two-dimensional model applies to the entire sample, but only to the regions producing the noise, and much larger barriers may well exist. For example, the pinning sites could be nonuniformly distributed, with hoppers in the regions where they cluster and large site-free areas acting as stoppers. The vortex density can be expressed as an effective field Besf= n ~ o = 60 #T, which is much larger than the field in which the sample was cooled. The sample may have nearly zero magnetization if it contains approximately equal numbers of vortices and antivortices, but the absolute number of vortices determined from our model is too large to be created by the ambient field. Some other mechanism could be generating vortices, such as the freezing in of vortex-antivortex pairs as the film is cooled through T C. We return to this issue in Sec. VI.

5.7. Current Dependence of a Random Telegraph Signal According to our model, a supercurrent suppresses the total indirect noise because it reduces 9 for each process in the ensemble; thus we might hope to observe such behavior in a single RTS. Although we often found knee-like features in the power spectra of the transformer which appeared and disappeared reproducibly as a function of I, only one RTS was both clearly resolved in the time domain and sufficiently stable to allow us to investigate its current dependence. It appeared in transformer T2 at 45 K, as shown in Fig. 20. The flux through the SQUID for I = 0 switches between two levels separated by A~ = 9 x 10 3 ~o- Additional noise is present from the motion of other vortices in the film, including a second slow RTS which executes a single cycle in Fig. 20(b). As we increase I, Aq~ remains constant but the switching rate of the dominant RTS increases; this change is approximately reversible when we decrease I to zero. This observation is inconsistent with a symmetrical double well, which would imply a decrease in v. The data cannot be explained by an inherent misalignment in the pinning potential because, according to Eq. (5.1), the product ~1r2 should be constant, whereas both lifetimes are observed to decrease with increasing I. The RTS observed in transformer T2 is not as well-behaved as one might infer from Fig. 20. The ratio 7 at I = 0 varied apparently randomly among three separate measurements, each immediately following a noise measurement at higher current. The fluctuator also tended to alternate

Magnetic Flux Noise

53

0 mA

[ (a)

~

!lOm~ o ~r - - f l

(b)

lOs 0,09 mA

(c)

0.13 mA

(d)

0.18 mA /

X II O ..i.-, (D O3

Time Fig. 20. Flux through SQUID vs. time generated by transformer T2 at 45 K for four different values of I: (a) 0mA, (b) 0.09 mA, (c) 0.13 mA, (d) 0.18 mA. Zeros of axes are arbitrary, traces are displaced for clarity.

several seconds of inactivity with short bursts of switching in a decidedly non-Poisson manner. It vanished entirely before we could investigate its temperature dependence. Nevertheless, it is obvious that this RTS is not representative of the processes which produce l / f noise presumably because the extended defects which produce RTSs differ from the localized defects relevant to 1If noise, yielding longer hopping distances and different current dependences. The data are consistent with an approximately symmetric process with a current-dependent barrier U(T, I) that, according to Eq. (5.1) with % = 1 0 - 1 1 s , decreases linearly from 0.11eV at I = 0 to 0.09 eV at I = 0 . 1 8 mA. A possible extended defect is a high-angle grain boundary acting as a Josephson junction with its critical current I o ,,~ eU(T, O)/h = 27#A reduced by an applied current.

54

M.J. Ferrari et aL

6. MAGNETIC FIELD DEPENDENCE All the measurements described thus far were made on samples cooled through T c in the ambient magnetic field ( < 1 #T). We investigate the effect of larger fields by raising the temperature of the sample to above To, storing a persistent current in a Nb coil which produces a field B parallel to the c axis, and then cooling the sample to the temperature of interest. For a thin-film sample in thermal equilibrium, this produces a vortex density n v = B / d P o proportional to the applied field, which remains constant during the noise measurement. As shown in Fig. 21, the noise power in samples YBCO(10) and YBCO(11) increases linearly with B over several decades of magnetic field. This linearity provides experimental justification for the neglect of correlations among the processes summed in Eq. (4.3). Since flux creep 64 and vortex lattices 65 exist even when the average vortex spacing I

I

I

I

10 -6

v

iN

10 -8

&A

, f 10-~~"

I 10 -3

I

I 10 -1 g(mT)

I 01

Fig. 21. Noise power Sr Hz) vs. magnetic field during cooling for three samples: [], BSCCO(1)at 60K; V and ~ YBCO(10)at 75 K and 60K; 9 YBCO(ll) at 77 K. Open and closed symbols represent opposite signs of the field. Fitted lines have slope 1 for YBCO and 1/2 for BSCCO.

Magnetic Flux Noise

55

exceeds the penetration depth, we do expect non-zero correlation among the vortices producing the noise in our experiment, but they must exhibit correlations of random sign which cancel when summed over the ensemble. The field dependence exhibited by BSCCO(1) in Fig. 21 is less well understood. The noise power increases a s B 1/2 with increasing field over three decades up to B ~ 2 mT, but scales linearly in B for higher fields. The crossover field is comparable to that observed 66 for an ordering transition in BSCCO; alternatively, the random-correlation argument may break down, or the vortex hopping distance diverge, below a critical vortex density in this strongly anisotropic material. The scaling of Sr with B observed in YBCO(10) and (11) is consistent with our model of flux noise provided the noise flattens out below an effective field Bert, where Befr/r is the vortex density that remains after the sample has been cooled through Tc in zero field. In Sec. V we deduced Berf~60pT for T1; in Fig. 21 we note that YBCO(ll) has at least a weakened field dependence below 10/~T, and YBCO(9) shows no field dependence in the range 0 ~< [BI ~<1 mT. On the other hand, the noise in the single crystal YBCO(10) at 75 K is linear down to 1 #T. We can account for these variations in B~rf for YBCO in terms of differing microstructure. Both YBCO(9) and T1 had significantly higher noise levels than YBCO(10) and (11) when cooled in nominally zero field. These high noise levels could be explained by a relatively high density of pinning sites, or by a high pinning energy per site, so that a significant fraction of the vortices generated at Tc would be trapped; consequently, B~frwould also be higher in YBCO(9) and T1 than in YBCO(10) and YBCO(ll). Furthermore, pinning in the single crystal YBCO(10) is expected to be much weaker than that in the films. Although Befr in this crystal is below our threshold of detection, the measured noise exceeds that in YBCO(ll) because the higher vortex mobility produces a much higher noise level per vortex. We note that Glyantsev et al. 67 have found that the 1If magnetic flux noise generated near Tc by bulk samples of YBCO is independent of field over the range 10-1~ < B < 10 -5 T. Although they claim that this result rules out vortex motion as the source of the noise, it is also consistent with an intrinsic vortex density comparable to those discussed in this paper. We have also performed flux creep measurements, in which there is net motion of vortices across the sample, driven by their density gradient. As shown in Fig. 22, the metastable configurations the vortices assume are not equally noisy. The spectra were taken with sample TCBCO(1) at 77 K, after cooling through Tc in B = 0.linT. The field was removed at t = 0 and the noise mesured subsequently with an averaging time of 40 s to 120 s, during which the time-averaged flux through the SQUID drifted by d(~}/dt<6xlO-3Oo/s as a result of flux creep, a rate too small to

M . J . Ferrari et aL

56

10 -4

,

,

",... l\.~ ~ 10- 6

:~ -.:..... ,,.,, 1i~ '.........

L,\ /

/

&

\~

,,~

~'v

,~,,~ ........."..8000 \ /~\1 ~ ~r

~

Hooos

s

......

"... 10 -8

I 1

I

10 10 2 F r e q u e n c y (Hz)

03

Fig. 22. Noise spectral density S,r,(f) vs. frequency for TCBCO(1) at three different times t after cooling to 77 K in B = 0.1 mT. Measurements made after field was removed at t = 0 : solid spectrum, t = 4 0 0 s ; dashed spectrum, t = 1000 s; dotted spectrum, t = 8000 s. Increasing smoothness of spectra with increasing t is artifact caused by longer averaging times.

contribute measurably to the noise. It is conventional in theories of flux creep to neglect hopping in the direction oppositve the driving force because most experiments are insensitive to it, but a noise measurement reveals vortex motion in both direction. F r o m 1 Hz to 1 k H z the noise power spectra in Fig. 22 all scale roughly as l/f, but on a finer scale the slope shows significant variations. These features are reminiscent of those seen in the voltage noise of metalinsulator-metal tunnel junctions 68 when a small number of traps are active in the barrier, producing a few superposed Lorentzians in the experimental bandwidth. It seems reasonable that a few vortex hopping processes dominate the noise at a given time. Subsequent spectra exhibit different features; vortices may have moved out of some processes and into others, or enduring processes may be shifted as a result of interactions with

Magnetic Flux Noise

57

vortices that have moved. The noise power at a given frequency evolves apparently at random, with no discernable trend over more than a decade in time, 400 s ~
58

M . J . Ferrari et aL

vortex motion, and issues of bundle size and collective pinning complicate their analysis. Thus our pinning energies represent intrinsic materials properties rather than the properties of a vortex lattice. When we apply a driving force, it suppresses the noise in a manner that can be understood as a perturbation on the original model. The suppression would not occur if the activation energies were not distributed; given that, it still would not occur if the double wells were not symmetrical. From this phenomenon we demonstrate that the vortex hopping distance increases with energy, and determine the pinning spring constant and the areal densities of vortices and pinning sites. We hypothesize that the number of vortices pinned in some samples exceeds the density corresponding to the ambient field in our experiment as a result of the freezing-in of vortices generated at T~. This hypothesis is supported by integrating D(Uo) for T1, and by the weakened field dependence of the noise as the cooling field is reduced in some noisy samples, but we have not observed this phenomenon in high-quality YBCO films and crystals. The linearity of S . in B in these latter samples provides strong support for our analysis of 1If noise into an ensemble of bistable hopping processes. However, the clear B 1/z dependence of the low-field noise in BSCCO remains unexplained. Three further issues remain open: first, the DDH slope equation consistently overestimates the measured slope of the noise power spectrum; this implies an overlooked effect which causes the noise to increase with temperature, such as an increase in the number of mobile vortices, or individual processes with temperature dependent hopping distances. Second, the magnitude of the noise peak at Tc appears to be independent of materials parameters, and deserves a theoretical explanation. Finally, the distribution of activation energies in the a-axis sample YBCO(6) is so featureless that it begs to be explained by some simple model derived from the highly anisotropic pinning potential present in a-axis films. ACKNOWLEDGMENTS We are indebted to K. Char and T. Hylton for YBCO(1); C.-B. Eom for YBCO(2) and YBCO(6); A. Inam, L. Nazar, T. Venkatesan, and X. D. Wu for YBCO(3); P. Rosenthal and R. Hammond for YBCO(4) and YBCO(5); L. F. Schneemeyer, R. B. Van Dover, and J. V. Waszeak for YBCO(10); E. Dantsker, D. Koelle and D. T. Nemeth for YBCO(ll); D. Mitzi for the BSCCO crystals; and W. Lee and S. S. P. Parkin for the TCBCO films. We thank M. R. Beasly, T. H. Geballe, and A. Kapitulnik for assistance and stimulating discussions during much of this work. The two-dimensional pinning model introduced in Sec. V! was inspired by

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