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Thermal activation in the quantum regime and macroscopic quantum tunnelling in the thermal regime in a metabistable system consisting of a superconducting ring interrupted by a weak junction
Physica 133B (1985) 196-209 North-Holland, Amsterdam
THERMAL ACTIVATION IN THE QUANTUM REGIME AND MACROSCOPIC QUANTUM TUNNELLING IN THE THERMAL REGIME IN A METABISTABLE SYSTEM CONSISTING OF A SUPERCONDUCTING RING INTERRUPTED BY A WEAK JUNCTION PART I: THERMAL
ACTIVATION
IN THE QUANTUM REGIME
D.W. BOL, J.J.F. SCHEFFER, Kamerlingh
Onnes Laboratorium
W.T. GIELE
and R. DE BRUYN OUBOTER
der Rijksuniversiteit Leiden,
Postbus 9506, 2300 RA Leiden,
The Netherlands
Received 16 July 1985
Under certain circumstances a superconducting ring containing a weak superconducting junction, a SQUID, has two metastable magnetic flux states separated by a potential barrier AU. If the junction has a small capacitance stochastic magnetic flux transitions are observed even at very low temperatures. This article mainly concerns the intrinsic thermal activation in the quantum regime, where the mean energy of the oscillator, corresponding to the metastable potential wells, is different from the equipartition value kT due to the high frequencies and low temperatures involved in our experiments. We have found strong deviations from the classical thermodynamic activation theory of Kramers where the transition rate is proportional to exp(-AUikT). We are able to explain our experimental findings by replacing the classical equipartition value kT for the mean energy in the exponent with the value for the mean excitation energy of the Brownian motion of a quantum oscillator, accounting for the appropriate damping.
1. Introduction
Under certain circumstances a superconducting ring containing a weak superconducting junction (a SQUID) has two metastable magnetic flux states separated by a potential barrier if an external magnetic field is applied of appropriate strength. If the junction has a small capacitance magnetic-flux transitions are observed, even at low temperatures where kT is much smaller than the barrier height AU separating the two metastable flux states. These flux transitions from one flux state into the other and vice versa correspond to the stochastically switching of the weak “persistent supercurrent” from one direction into the opposite. Our earlier investigations [l-5] have been extended in two regimes: (1) thermal activation over the barrier in the quantum regime, where kT = hw,, w,, being the free oscillation frequency corresponding to the metastable potential wells and (2) macroscopic quantum tunnelling through the barrier in the thermal regime where kT < hw,. This article mainly concerns the intrin-
sic thermal activation in the quantum regime, where the mean energy of the oscillator is different from the equipartition value kT due to the high frequencies and low temperatures involved in our experiments. We have found strong deviations from the classical thermal activation theory of Kramers [6,7] for the shuttling action of a Brownian particle which is subject to the irregular forces of the surrounding medium in temperature equilibrium and which can therefore escape over the potential barrier. In the classical thermal activation theory the transition rate, caused by intrinsic thermal fluctuations over the barrier, is given by l/(7-,.) = (w,/27r) exp(-AUlkT), in which (T=) is the mean lifetime of the metastable state and w, the attempt frequency. We are able to explain our experimental findings by replacing in the exponent the classical equipartition value kT for the mean energy with the value for the mean excitation energy of the Brownian motion of a quantum oscillator as calculated by Ford et al. [8]. Essential for our analysis of the thermal activation data is that we have used a fluctuating
0378-4363/85/$03.30 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
D.W. Bol et al. I Thermal activation in the quantum regime
quantum force of the Planck type without zero point fluctuations [8] in the Langevin equation. Zero point energy cannot be extracted from the environment and is therefore not included in the thermal excitation energy. Just as in the case of macroscopic quantum tunnelling [9], the exponent in the expression for the transition rate for the case of thermal activation in the quantum regime contains a damping term [8] contrary to the case of classical activation [6]. We can define a characteristic temperature Tq below which quantum effects become important by means of the relation [lo]: kT, = hwi = fiwh{d(l+ 7”) - r}, where y = 1 I2RC4 is the damping parameter (R the resistance and C the capacitance of the weak junction) and WA= 1 Iv(K)*. Dissipation lowers the characteristic temperature T, so that quantum effects are diminished by the damping [lo]. Although this article mainly discusses thermal activation in the quantum regime we also present some of our results for macroscopic quantum tunnelling in the thermal regime. Now a fluctuating quantum force of the Schrtidinger type with zero point fluctuations included plays an important role in the Langevin equation [ll, 21. At weak damping the tunnelling itself can be considered to be energy-conserving. In the crossover region from thermal activation to quantum tunnelling we suggest that the total transition rate 1 /(T) is equal to the sum of the thermal transition rate l/( rT) and the transition rate l/ (T,) due to macroscopic quantum tunnelling, hence l/(7)= l/(7,)+ l/(~~). Macroscopic quantum tunnelling will be favoured by a very small capacitance of the weak junction [l] and has already been observed at moderately low temperatures in a superconducting ring interrupted by an adjustable superconducting point contact with a very small intrinsic capacitance of about lo-l5 F [l]. Later on, Voss and Webb [12] studied the influence of damping on the rate of tunnelling in a single low critical current junction driven by an external applied current, confirming qualitatively the theoretical findings of Caldeira and Leggett [9] that dissip*WA insteadof usualpractice in theoretical literature is defined by eqs. (11) and (12).
197
ation reduces the tunnelling rate. We extended our study on macroscopic quantum tunnelling as reported in section 7 of ref. 5 by investigating evaporated Nb-Nb tunnel junctions and made a comparison between both types of devices: Nb-point contacts and evaporated NbNb tunnel junctions with different device characteristics. Since the capacitance of an evaporated junction ( lo-l4 F) is larger than that of a point contact, macroscopic quantum tunnelling was now observed for smaller barrier heights and consequently at lower temperatures. The temperature dependence of the observed mean lifetime (T,) might possibly be due to a contribution of macroscopic quantum tunnelling from the thermal activated first excited energy level in the potential well (macroscopic quantum tunnelling in the thermal regime). In Part II macroscopic quantum tunnelling will be discussed in greater detail. The emphasis will be on the behaviour of our evaporated Nb-Nb tunnel junctions at very low temperatures in the millikelvin range in order to make a comparison between experiment and the recent theoretical predictions on the temperature dependence of the quantum-tunnelling rates for symmetric and asymmetric double-well systems with ohmic dissipation [ 161.
2. Metabistability in a SQUID Through a superconducting ring containing a weak junction of the Josephson type [l], see fig. 1, a variable magnetic flux B, 0 is applied (B, is the external magnetic field applied perpendicular to the embraced area 0). in the presence of the external applied magnetic field a circulating current ifirc will be induced, so that the net magnetic flux inside the ring will be @ = B,O + Licirc ,
(1)
L is the self-inductance of the ring. The circulating current cannot provide complete shielding even at the lowest value of B,O due to the presence of the weak junction. The steady-state current through the junction is related to the
D.W. Bol et al. I Thermal activation in the quantum regime
198
ic,,,=-i~sin(~~,-i~j,R-~~,
(6)
with R and C the normal resistance and the intrinsic capacitance of the junction. Combining eq. (1) with eq. (6) leads, after rearrangement of the terms, to a “classical equation of motion” for @(t) of the following form [l, 71: Fig. 1. A superconducting ring with inductance L and containing a weak junction 6 with intrinsic capacitance C. The direction of the applied magnetic field B I is indicated. 0 is the embraced area.
gauge-invariant phase difference of the order parameter A$J *, across the junction, by the first Josephson relation [ 131
(7) C plays the role of mass and (1 lR) is the friction coefficient. The corresponding potential energy U( @, B, 0) is given by U = --(&ii,
icirc
= i, sin(-A$*),
(2)
where i, is the maximum supercurrent through the junction. For the gauge-invariant phase difference one finds AC+*=
$ @+n2m,
n = 0, +l, *2,.
..,
(3)
by integrating the Aharonov-Bohm relations for the electromagnetic potentials around the ring and using the fact that the wave function of the superfluid condensate due to the formation of Cooper pairs is always single valued. In the steady state one obtains the well-known relation for the total embraced magnetic flux as a function of the applied magnetic field:
@(B,) = B,O - Li, sin ($
@(B,)) .
During the approach to the steady-state eqs. (1) and (3) should still hold, but a voltage difference AV will be present across the junction [13]:
AVz_!!$-h+f. 2e
This voltage difference gives rise to a normal current, AVIR, as well as a displacement current, CaAVldt, through the junction so that instead of eq. (2), i,,,, becomes
. (8)
The total embraced flux @ is here regarded as a dynamic variable. The first term on the right-hand side of eq. (8) can be identified as the Josephson coupling energy [13] and the second term as the magnetic energy of the system. In the steady state (a U/a @ = 0) one obtains again the well-known relation (4) for the flux as a function of the applied magnetic field. When the dimensionless coupling constant 2 E FLi,
(9)
is larger than one and for suitable values of the external magnetic field B, the potential energy U(@) has one or more metastable local minima separated by an energy-barrier AU (see figs. 2a and 2b), resulting in the multi valuedness of the flux-applied magnetic field relation which is shown in fig. 3. In this way metabistability arises in a SQUID. In the case of zero damping the frequency w,, of free oscillations in the metastable local minima is given by 2 w()=
at
cos( $f@) + ((a - B,O)*/2L
c
a2U 2
)
= ,la:~::;[email protected],,L,.
(10)
For later convenience this expression for the frequency w0 of free oscillations can be written as
D.W.
Bol et al. I Thermal activation
I
0
in the quantum
199
regime
I
I
I
h12Q
0
h12Q
;P
9
a
b
Fig. 2. (a) The potential energy U as a function of the totally embraced magnetic flux Q,at constant applied magnetic flux B,O. The corresponding constant magnetic field value and the minimum and maximum are indicated on the @(B,) curve of fig. 3. The illustrated case is for 9 = rr. (b) The potential energy U as a function of the totally embraced magnetic flux @ at constant applied magnetic flux B, 0 = 4 (h/2e).
@Jo =
t+oT
(11)
)
in which 0; = 1 l(Lc)“2
)
(12)
If we introduce the embraced magnetic flux and the applied magnetic flux in dimensionless quantities: y = @l(h/2e)
and
x = B,Ol(h/2e)
we are able to write the eqs. (7) and (8) in reduced flux quantities:
and (13)
h
Lc a2ylat2 + (LIR)aylat
= -aday,
(W
u( y, x) = $( y - x)” - (Y/4T2)
cos 297y
= { Ll(h/2e)2}
,
U(@, B,O)
(84
The expression for the energy barrier AU for a symmetric potential, i.e. x = &or B,O = ih/2e in reduced quantities becomes Au = (Z/47r2)[1 + cos 2ry,,,J = { Ll(h/2e)2}AU 810
Fig. 3. The totally embraced magnetic flux @ vs. the applied flux B,O in the steady state P’= r. The local potential minimum at A is separated by a potential maximum at C from a lower potential minimum at B. In fig. 2a these points are indicated in the U(D) curve. In A and B the energv barrier is equal to the shaded area a respectively b dividerby L.
.
- 4 ( ymin - 1)’ WI
The energy I_ barriers AU,u , AU,Y for an asymmetric situation are indicated in fig. 2a and are equal to the shaded area a respectively b in fig. 3 divided by the self-inductance L of the ring. In fig. 2b the symmetric potential barrier is drawn when AU, = AU,= AU [l].
200
D. W. Bol et al. / Thermal activation
3. The Langevin equation and the fluctuating quantum forces
in the quantum
(Et) =
gw2“i)’ wi + (oIRC)* -
0
In the quantum regime, i.e. at high frequencies or low temperatures, the mean energy of the Brownian motion of a harmonic oscillator is no longer equal to the equipartition value kT as used by Kramers in the classical activation theory. Therefore we like to derive an expression for the mean energy (E) of the Brownian motion of a harmonic quantum oscillator. We start from the Langevin equation [8, 11, 21 by adding to the equation of motion for Q, with “mass” C, in a potential well with a characteristic frequency wO and a friction coefficient R-r, a fluctuating random noise “force” F associated to the normal current noise and obtain [2]
regime
hw
dw .
x 2RCcoth
(18)
In the limit no G kT both expressions (.+) and (et) for the mean energy of a Brownian particle approach kT, which is the classical equipartition result. Another simple limit is the weak coupling limit for the mean thermal excitation energy H.A.Kromers,bO
a We distinguish two expressions [8] for the fluctuating random noise “force” F: the noise “force” FT due to the thermal fluctuations of the Planck type and the total noise “force” F, of the Schriidinger type including both thermal and zero-point fluctuations. Their spectral densities are related to (Fc),, (F:)
kT/hw,,
H.A.Kromers.bO
o+dw = o, w+dw= &
coth&T
= %{i
. Aw
+ ll(efi”‘kT-
l)}Aw
. (16)
For the mean thermal excitation energy (s,) find [8, 141
we
b iiwIRC
x e
fiolkT
-1
kT/hw,
dw ’
(17)
and for the mean total energy (Ed) (zero point energy plus thermal excitation energy) we find 1141
Fig. 4a, b. The mean thermal excitation energy ( .Y,) of a Brownian harmonic quantum oscillator with a free oscillation frequency 00. (m)IkT is plotted vs. kT/hm for different damping parameters y = 1/2RCw,. The classical equipartition result fi-+0 and the Planck result y = O.Ol+ 0 are also indicated.
D.W. Bol et al. I Thermal activation in the quantum regime la-
4. The attempt frequency
t 16
201
t,kT
,kT
Fig. 5. The mean total energy (Ed) and the mean thermal excitation energy (E,) of a Brownian harmonic quantum oscillator with a free oscillation frequency wO. (E,) lkT and (.e,)lkT are plotted vs. fiw,lkT for different damping parameters y. (Ed), in which the coupling of the Brownian particle to the environment is weak, i.e. when w0 % 1IRC. In this limit the resonance denominator in (17) becomes sharply peaked at w = w,, with a width equal to 1IRC. We find (c,) = hw,,l(exp(ho,JkT) - l), which is the wellknown Planck result for the mean thermal energy of an independent oscillator of frequency o0 at temperature T. We solved eqs. (17) and (18) by numerical calculation for different damping parameters y = 1 I2RG.0,. In figs. 4a and b (c,) lkT as a function of kT/hw, is plotted for different damping parameters y. The classical equipartition result (h = 0) and the Planck result ( y = 0) are also indicated. We see in fig. 4 that the characteristic temperature Tq below which quantum effects become important is lowered when the damping increases. In fig. 5 (c,) lkT and (~,)lkT are plotted as a function of hw,,lkT. Ford et al. [8] studied the Brownian motion of a quantum oscillator in a model of a Brownian particle coupled to a system of harmonic oscillators acing as a heat bath. Eq. (17) was one of their results. As already mentioned in the Introduction the results obtained from eq. (17) and presented in fig. 4 will be used later for interpreting our thermal activation data in the quantum regime.
The frequency appearing in the prefactor of the expressions for the mean lifetime is named the attempt frequency w, and is interpreted as the frequency at which the flux coordinate strikes the potential barrier AU. In the classical thermal activation theory of Kramers [6, 71, valid in the region hw, 4 kT @ AU, the attempt frequency is defined by the relation
In this theory the noise activated escape over the potential barrier was discussed in terms of the properties of the motion of a Brownian particle in a field of force. Kramers discusses three separate approximations of the case of large viscosity, depending on the degree of damping. Here we mention the three regimes of our interest: (a) The weakly damped regime where y < 1 leading to the Arrhenius transition state theory:
(20)
w, = w, .
(b) The heavily damped regime, where y S 1: w, = w,lw,]RC = 21w&y ,
(21)
in which 1wb[ = q/(.2 - 1) IVLC = whd(2 - 1) is the absolute value of the imaginary frequency associated with the unstable potential curve at the top of the barrier. (c) The regime of moderate damping between the cases (a) and (b):
(24 We will not discuss the extremely weakly damped situation (y + 0) which follows from Kramers theory for the case of small viscosity [6]. Eq. (22) can be written as wJwO = w:/w;, = (1 + ,,)l’*
- p ) (224
with p = wgeyl(Z - 1)1’2 .
202
D.W. Bol et al. I Thermal activation in the quantum regime 0.46 +/
In the case of macroscopic quantum tunnelling without damping one finds, in WKB approximation, for the mean lifetime [9]
mm /t
(23) where CY= 7.15 for the shape of our nearly symmetrical double potential well [9] (B,O = hh/2e) and for the attempt frequency [12, 151 &--
(24) Following Caldeira and Leggett [9] and Voss and Webb [12] one uses the following expression in case of moderate damping:
(7,) = Feb,
(25)
a
where CYAU + a(A@)*
b=Zlo
~
0
TLR ’
(26)
in which A@ is the tunnel distance under the barrier and a a numerical factor of order unity (a = 0.47). The attempt frequency is now equal to
[=I (27) Again we see that dissipation plays an important role [9, 2, 121. We have to realize that these expressions (23) and (25) of course cannot be applied in the purely symmetric potential case, but are derived for an escape into a “continuum”
[W 5. The experiment The superconducting ring interrupted by a Josephson junction consists of an evaporated planar Nb ring structure in which a Nb-Nb junction is incorporated (fig. 6). The ring structures were made six at a time on a flat silicon
5mm a
+i b
Fig. 6. (a) The evaporated planar Nb ring structure interrupted with a Nb-Nb junction (L = 9.2 x lo-“‘H, C= lo-l4 F). (b) Detail of the Nb ring structure with the Nb-Nb junction formed by the overlap of the two 10 km banks.
substrate using photoresist techniques. The NbNb junction with a silicon oxide barrier between it has an overlap area smaller than 1 urn* and an estimated capacitance of lo-l4 F [17]. Due to the special geometry chosen for the ring the inductance can be calculated to be 7.2 x lo-” H [18]. The 10 km wide banks leading to the junction have an estimated inductance of 2 x lo- lo H [ 171. So the total inductance of the ring is approximately 9.2 X lo-” H. All the samples have the same inductance and capacitance. The advantages of this kind of ring structure are its well-defined geometry, the resulting knowledge of the experimental parameters [17] and the stability of the junction with respect to thermal cycling. Storage over the period of a year at room temperature resulted in a decrease of the critical current with a few percent. This reduction can be avoided by storage at liquid nitrogen temperature. The magnetic flux in the ring was measured with a commercial rf SQUID system [19]. By means of a Nb wire, wound as a spiral pick-up coil, a weak but sufficient coupling to the ring was made. The ring was placed in an Araldite sample holder and shielded from the environment by two copper-lead shields at liquid helium temperature and a p-metal and copper shield which surrounded the cryostat. Except for the leads to the external field coil which are carefully filtered, no electrical connections were made to the inner
203
D.W. Bol et al. I Thermal activation in the quantum regime
cryostat to prevent external disturbances during the measurements. The coupling constant Ywhich according to eq. (8b) determines the barrier height AU [l] together with the inductance L, can be measured in two ways. First by comparing the measured magnetic flux @ vs. the applied flux B,O-curve with that from an identical but short-cutted ring so that the magnetic flux picked up directly from the external field coil can be subtracted from the SQUID output. Then 3 follows from the slope of the @ - B,O curve at an applied magnetic flux equal to an integral number of magnetic flux quanta. The second way is by fitting the relation (see (4)) @ = B,O - $-gsin[L?+r&/(
d))
the coupling strength of the ring to the pick-up coil is reflected in the measured amplitude of the @ - B, 0 curve. From the coupling strength thus determined and again the slope of the @ - B,O curve at an integral number of applied magnetic flux quanta, the coupling constant 3follows when it is larger than one at lower temperatures. In these two ways P’is known within 5% of its absolute value. Changes in 3, for instance due to temperature variations, are known within 1%. For 3 > 1 and B,O = $ (M2e) the potential energy as a function of @ is symmetric and has at least two metastable minima. Switching between these two flux states can be observed by measuring the magnetic flux as a function of time. The SQUID output is recorded by a transient recorder which is connected to a microcomputer for storing and analyzing the signal. In this way it is possible to measure the mean lifetimes (T) of the magnetic flux states in a “window” which extends in time from 0.5 ms up to 1 s. The lower bound is set by
(28)
to the measured @ - B,O curve for 3 < 1 when this relation is single valued. When 3 is known,
/ 1o-4
4
I
6I
/
/ 1
8I
,.l.“’
/ I
I
10
I
I
12
I
I
14
I
I
I
16
LAU/kT (lo-‘HI Fig. 7. The measured mean lifetimes (T) vs. LAUIkT for the samples (a) up to (f). The thermal activation data of the samples (a) up to (d) are represented by straight lines. The classical thermal activation result according to the theory of Kramers [6] is also indicated (w: = o,/w~ = 10” s-r). In the corresponding fig. 8 is indicated for the devices (a) up to (d) how the quantities T, JZaand oP vary as a function of LAUIkT in the “measuring window”. For device (f) the measurements go down in temperature to 0.4 K.
204
D.W. Bol et al. I Thermal activation
in the quantum
regime
12 LAUlkT e
d
C
b
Fig. 8. For the devices (a) up to (e) is indicated how the quantities T, 3’ window”.
the intrinsic noise of the rf SQUID and beyond the upper bound statistical interpretation of the magnetic flux transitions becomes very difficult. In the thermal regime the mean lifetime (TV) of the magnetic flux state will inevitably get out of the measuring window as we vary the temperature (fig. 7). To investigate thermal fluctuations over the whole temperature range we needed several rings (the devices (a) up to (f)) with suitable coupling constants 9 for the various temperatures (fig. 8). The variation of the critical current of the rings, even in one evaporation run, was large enough to make this possible. For example, 4 out of 24 rings which were made were suitable for investigation of thermal activation in the quantum regime and only 2 for investigation of macroscopic quantum tunnelling. In order to study the thermal activation over the barrier we determined the temperature dependence of the mean lifetime (r) of a magnetic flux state in a symmetric potential, i.e. B,O =
14
16
18
(lo-‘H)
and wr vary
as a function of LAUIkT
a
in the “measuring
(n + i)(h/2e). Due to the temperature dependence of the critical current i, of the junction or the coupling constant 55 of the ring, the barrier height AU changed as we followed ( C-) within its measuring window. This change of AU is taken into account by plotting the mean lifetime measurements for the various rings logarithmically as a function of LAUIkT as done in fig. 7. LAU only depends on the coupling constant 3, see eq. (8b), and 3 was measured at all temperatures together with the mean lifetimes. The change of T, Z’and ox within the measuring window of the mean lifetime is plotted in fig. 8. The mean values of T, 2, o, and kTlfiq, which are indicated by a bar above the symbol are given in table I. We have plotted (T) logarithmically vs. LAUIkT, to compare our results with the classical thermal activation theory of Kramers [6]. The classical result is represented in fig. 7 by the dotted line on the right with slope l/L and intersection (7) = 27r/wl at LAUIkT = 0. Our
205
D.W. Bol et al. I Thermal activation in the quantum regime Table I
L = 9.2 x lo-”
H, C = IO-l4 F, 06 = 3.3 x 10” s-l, w: = 10’” s-’ (b)
(c)
(d)
(e)
4.76
4.3
4.2
4.0
5.3
3.3
2.9
2.0
3.69
2.56
1.84
1.63
1.17
5.38
3.90
2.89
2.61
1.92
2.4
2.0
1.7
1.6
1.3
1.15
1.17
1.24
1.26
1.36
0.84
0.73
0.55
0.49
0.26
0.61
0.51
0.43
0.41
0.36
4.8
2.4
1.0
0.78
0.16
11.5
4.80
1.70
1.25
0.208
0.40
0.75
0.88
0.90
0.96
0.567
0.236
0.0837
0.0615
0.0102
0.566
0.202
0.0878
0.0651
0.0151
(a)
4.76 13.5
‘VT,) _=R(T)
Wrpy
rr,
R(T,) R(T) [eq. (30),theor.]
devices (a), (b) (c) and (d) show an exponential temperature dependence of the mean lifetime (r) above -1.9 K, but the exponent differs from the classical result. This difference decreases at higher temperature. A discussion will be given in the next section. Below -1.9 K our devices (c), (d), (e) and (f) show a deviation from the exponential temperature dependence which gets more significant at lower temperatures. This deviation is due to the crossover from the mechanism of thermal activation over the potential barrier to macroscopic quantum tunnelling through the potential barrier and is briefly discussed in the last section.
6. Thermal activation in the quantum
regime
The thermal activations of the samples (a), (b),
(c) and (d) are represented by straight lines in fig. 7. If the data could be interpreted by the classical thermal activation theory of Kramers [6] (eq. (19)) all these lines should nearly coincide (all the samples have nearly the same CO:= w,/w~ and we have only small changes in We), with the classical activation result indicated by the dotted line. We observe, however, strong deviations from the classical result of Kramers which depend strongly on the mean value k~lhZ&,. The smaller this quantity the greater the slope and the shift of the straight lines to the left of the classical line. These deviations can also be seen in the fact that magnetic flux transitions due to thermal activation are observed in the same “measuring window” for the devices (a) up to (e) for AUIkT values in the range between 13 and 18.5 for device (a), between 11.2 and 15.5 for (b), between 8.5 and 9.8 for (c), between 7.7 and 8.6 for (d) and
206
D.W. Bol et al. I Thermal activation
between 3.8 and 5 for (e). As discussed in the Introduction we can interpret our experimental data on thermal activation over the barrier in the quantum regime on the basis of the following relation:
in the quantum
regime
‘E o=o q =b
t OlL
Arc
x=d
t
;;
where the equipartition value kT in the classical thermal activation theory is replaced by the mean thermal excitation energy (E,) which is given by eq. (17). (E,) lkT is calculated and plotted vs. kTlhw, in figs. 4a, b for different damping parameters y. In the limit fro < kT, eq. (29) reduces to the classical activation result of Kramers (eq. (19)). The straight lines for the samples (a), (b), (c), (d) and (e) in the log r vs. LAUIkT plot of fig. 7 are drawn in agreement with eq. (29), using only one value w: = o,lwY (=lO’O s-‘) for all the samples. This result does not depend very critically on possible small individual changes in wi. According to eq. (29) the slope of the straight lines gives kT/ (c,) L. With L = 9.2 X lo-‘” H the mean value (q) /kT for the samples is determined and given in table I. With the use of the figs. 4a, b the mean damping parameter 7 for the samples follows from the (E,)lkT va 1ue found in this way and the corresponding value of kTlfiZ&. The values for 7 are given again in table I and will be discussed later on. Finally, the logarithm of oz rl. vs. LAUl ( Ed) is plotted in fig. 9 to illustrate the analysis of our experimental data on the basis of eq. (29). We see that all our thermal activation data for all the different samples fall on one straight line with wi = lOlo s-‘. The analysis shows that the thermal activation data are in agreement with eq. (29) if the temperature dependence which is found for the damping parameter y(T) = 1/2R( T) Cw, or for the quasi-particle resistance R(T) can be understood. This will be done in the next section. We like to remark that we were unable to interpret our thermal activation data in the quantum regime by using, instead of the mean thermal excitation energy (Ed), the mean total energy (E,) as given by eq. (18) and presented in fig. 5 as an (Et) lkT vs. tiw,lkT plot, since (et) > kT. On
‘;
0.01
3’
0.001
0.0001’
;*
’
I 14
I
I 16
/ 18
/
I 20
LAU/(lOegH)
Fig. 9. CUT(V) vs. LAU/(ET) for the devices (a) up to (d). The straight line is represented by the equation W~P(TT) = (2?r/w:) exp[AU/ (&,)I, (29a) with W: = 10’” sC1and describes all our thermal activation data.
the other hand, for the mean thermal excitation energy we find (E,) < kT.
7. The damping parameter and the quasi-particle resistance As mentioned in the preceding section the mean damping parameters 7 (7 = 1/2RCG&J of our devices are determined from the experimentally found values of (q) / kT and are given in table I. From the obtained results we see immediately that, when lowering the temperature T, the damping 7 decreases. We interpret the data by making the assumption that in the resistively shunted junction model, which is implicitly used in our calculations, the normal current may be identified with the normal quasi-particle current which tunnels through the (oxide) barrier of the junction. The supercurrent and the normal quasiparticle current flow in parallel through the junction. Let us use the following expression [20] for the ratio of the resistance in the complete normal state R(T,) and the quasi-particle resistance
207
D. W. Bol et al. I Thermal activation in the quantum regime
R,,(T)
in the limit of zero voltage:
(30) where K, is a modified Bessel function of the second kind. This ratio is only a function of T and of the BCS gap A(T). At very low temperatures (kT 6 A(T)) the normal quasi-particle resistance in the limit of zero voltage becomes exponentially large and eq. (30) gives (m = 1): E$.)!
= ( 2rfr))“2
exp[ ?!!!$I
(31)
.
in fig. 10a the drawn line gives the result of a numerical calculation of R(T,)IR(T) vs. the reduced temperature T/T,, using eq. (30) and the relation A( T = 0) = 1.764kT, and the tabulated values of Miihlschlegel [21] for A (T)/A (T = 0). The experimental determined values of W,y (= 1/2R(T)Cw9 for our devices are given in table I. In fig. lob these values wYy are plotted vs. 1.4-
the theoretically calculated value of R( T,) / R( T) at 7. Obviously the decrease of (oZy),,r at lower temperatures ?? corresponds with the temperature dependence of R( T,) / R( T) according to eq. (30), assuming a constant R( T,). From the slope of the drawn straight line yr, is determined, this gives R( T,) = 1/2y,~o,‘,C. We found yT, = 20.3 and Now we can calculate R( T,) = 7.46 R. R( T,) I R(T) for our devices. The results obtained in this way are plotted in fig. 10a. It appears that the damping parameter y for all our devices can be described with a single yr,, and that they have the same WI in the limit .Y+O. The experimentally determined value for R( T,) / R(T) = W,y/20.3 and the theoretically calculated value for R( TJl R(T) are given in table I. Finally we remark that from eq. (22a) follows that ~1 can be calculated by the assumption of using the normal resistance value R, = R(T,). In that case, from eq. (22a) follows that for the devices (a)-(e) /? is equal to, respectively, 23.3, 23.8, 25.1, 25.6 and 27.5; w: is equal to, respectively, 7.0 X 104 7.0 X lo’, 6.7 x 10, 6.7 x 10’ and 6 x 10’s_‘. These values are not far different from 10”’ s-i, showing the consistency of the analysis. 14r
1.2 1 .o $ .
0.8 -
t” 0.6 lx
cl / Cl
0.4 ,/,,,
0.2 1
0L-L -do 0 0.2
b.
d;o 0.4 0.6 TIT, a
0.8
1.0 R CT, )/R(T) I-
Fig. 10. (a) The drawn line gives the result of a numerical calculation of R( T,)IR(T) vs. the reduced temperature TIT,. R(T) is the quasi-particle resistance in the limit of zero voltage. R( T,) lR( T) is also determined from the measured damping parameter for the devices (a) up to (e) and indicated in the figure. (b) The experimentally determined value of cy vs. the calculated value of R( T,)IR( T) for the devices (a) up to (e) (see fig. lOa). From the slope are determined y,, and R( T,) = 1/2yrcwAC ( yr,= 20.3 and R( T,) = 7.46 a).
208
D. W. Bol et al. I Thermal activation in the quantum regime
We should like to remark that the value of R, = R(7'Jin this way obtained is about a factor ten smaller than the directly measured resistance. We have found up to now no satisfactory explanation. This will be investigated in more detail. Finally we hke to make some remarks about the experiments of Voss and Webb [12] where a single low critical current Josephson junction was driven by an external current source and the transition probability was measured for switching out of a metastable superconducting state to a finite voltage state as a function of temperature down to the millikelvin range. At high temperatures the transition rate out of the metastable state was dominated by thermal activation over the barrier and at very low temperatures it was observed that the distribution widths of the transition probabilities became temperature independent and in qualitative agreement with the predictions for macroscopic quantum tunnelling. But feature of thermal activation in the quantum regime are also hidden in their measurements. For instance from fig. 2a of their publication [12] we got data in order to calculate the energy barrier AU and w0 for the current (i) driven junction: AU(ili,)
= (fi/2e)2i1[{l
- (ili,)2}1’2
- (i/i,) cosml(ili,)] w. = WI(l - (ili,)2)1’4;
,
w: = (2elti)(i,lC).
Again, by making a plot of log rr(1 - (ili,)2)1’4 vs. A UIkT we found a nearly exponential temperture dependence for their thermal activation data with the typical feature of an “apparent” attempt frequency which is decreasing rapidly with decreasing temperature. For instance, from their measurements at 0.740, 0.558 and 0.414K (with w0 equal to, respectively, 1.94 x 10”; 1.83 X 10” and 1.76 x 1011s-l and with kTlG& equal to, respectively, 0.50, 0.40 and 0.24) we find an “apparent” attempt frequency of, respectively, 1.08 x loll, 2.62 x 10” and 8.31 x lo9 s-‘. At lower temperatures we come in the region of macroscopic quantum tunnelling, for instance at 0.005 K (with T3,= 1.34 x 10” s-l) we find an
attempt frequency of 2.45 X 10” SC’ in agreement with eqs. (24) and (27) (see next section).
8. Macroscopic regime
quantum tunnelling
in the thermal
As already reported in fig. 8 of ref. 5 and mentioned in the Introduction, macroscopic quantum tunnelling is observed below 0.5 K in our Nb-Nb evaporated tunnel junctions with capacitances of ~1O-l~ F and for barrier heights corresponding to 2~2, whereas in ring devices interrupted by a Nb-point contact with capacitances of ~10-15 F this phenomenon is already observed [l] below 1.5 K for barrier heights corresponding to a value of the coupling constant L? between 3 and 4. Just as in the case of thermal activation it is only possible to measure mean lifetimes TVfor tunnelling in the region between 0.5 ms and 1 s. Our experiments on macroscopic quantum tunnelling in the thermal regime at very low temperatures in the millikelvin range will be extensively discussed in Part II. Here we only make some rough calculations with respect to the results as presented in fig. 7. We take as example the low temperature value for (7,) of the curve (e) in fig. 7. We first calculate the mean lifetime (T:) for tunnelling in the undamped case, using eqs. (23), (24), (ll), (12) and (13): (7:) = (27r/w, (aAU127rfiwo)1’2) exp(cuAulhw,) with the constant (Y in WKB approximation for our potential equal to 7.15. We find for (7:) = 8.3 x lop5 s and for the effective undamped attempt frequency wi = 6.7 x 10” SC’ using eq. (24). Use has been made of the data of table I. In the case of damping (for sample (e) R = 728 R in the low temperature region) eqs. (25), (26) and (27) are used with a = 0.47 and (T,) = 0.85 s is not far from the experimental values shown in fig. 7 for curve (e). For the samples corresponding to the curves (d) and (c) of fig. 7 we find for (7:)) respectively 400 s and 3.7 x lo5 s. Again we have to realize that the expressions, which we have used in the calculation, cannot be applied in the purely symmetric potential case [16]. For device (f) the measurements shown in fig. 7 go down in temperature to 0.4K. We should like to remark
D. W. Bol et al. 1 thermal activation in the quantum regime
that in this temperature range the resistance values R(T) are still smaller than the critical resistance value (A@)2/di as discussed in ref. 16, [(A@92/27rfiR(T) > $1.
Acknowledgements It is a pleasure to thank Dr. H. Dekker (F.E.L.-T.N.O. - Den Haag) for his stimulating interest and advice in this work. The evaporated devices were made by Mr. J. Kortlandt of the research group of Prof. J.E. Mooij, University of Technology, Delft, and the masks used for their fabrication by the Delft Centre for Submicron Technology. The advice of Dr. T.M. Klapwijk with regard to these devices and their characteristics is greatly acknowledged. Mr. G. Vis and Mr. P.J.M. Vreeburg delivered the technical support for the experiments. This investigation is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)“, which is financially supported by the “Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (Z.W.O.)“. References PI W. den Boer and R. de Bruyn Ouboter, Physica 98B (1980) 185. PI R. de Bruyn Ouboter and D.W. Bol, Physica 108B (1981) 965; 112B (1982) 15. 131 D.W. Bol, R. van Weelderen and R. de Bruyn Ouboter, Physica 122B (1983) 1. 141 R. de Bruyn Ouboter, Proc. Int. Symp. Foundations of Quantum Mechanics, Tokyo (1983) p. 83. PI R. de Bruyn Ouboter, Physica 126B (1984) 423. PI H.A. Kramers, Physica 7 (1940) 284. [71 J. Kurkijarvi, Phys. Rev. B6 (1972) 832. M. Biittiker, E.P. Harris and R. Landauer, Phys. Rev. B28 (1983)
209
1268. H. Dekker, to be published (1985). Experiments by L.D. Jackel, W.W. Webb, J.E. Lukens and S. Pei, Phys. Rev. B9 (1974) 115. PI G.W. Ford, M. Kac and P. Mazur, J. Math. Phys. 6 (1965) 504; p. 513, eqs. (82), (84). See also, D.K.C. MacDonald, Physica 28 (1962) 409. [91 A.O. Caldeira and A.J. Leggett, Phys. Rev. Lett. 46 (1981) 211; Ann. Phys. 149 (1983) 374; A.J. Leggett, Macroscopic quantum tunnelling, in: Percolation, Localization, and Superconductivity, eds. A.M. Goldman and S.A. Wolf (Plenum, New York, 1984). [lOI See also, H. Grabert and U. Weiss, Phys. Rev. Lett. 53 (1984) 1787. Pll H.B. Callen and T.A. Welton, Phys. Rev. 83 (1951) 34. L.D. Landau and I.M. Lifshitz, Statistical Physics (Pergamon, London, 1958). R.K. Pathria, Statistical Mechanics (Pergamon, London, 1972) p. 77. WI R.F. Voss and R.A. Webb, Phys. Rev. Lett. 47 (1981) 265. 1131 B.D. Josephson, Rev. Mod. Phys. 36 (1964) 216; Adv. Phys. 14 (1965) 417. 1141 H. Dekker, Physics Reports80 (1981)~~. 1-112; p. 39, eq. (577). WI E. Merzbacher, Quantum Mechanics (John Wiley, New York, 1961). [161 S. Chakravarty and A.J. Leggett, Phys. Rev. Lett. 52 (1984) 5. H. Grabert and U. Weiss, Phys. Rev. Lett. 54 (1985) 1605. M.P.A. Fisher and A.T. Dorsey, Phys. Rev. Lett. 54 (1985) 1609. H. Dekker, to be published: here the relation l/(7)= l/(7,)+ l/(rr) will be discussed. (171 V.J. de Waal, Ph.D. Thesis, University of Technology, Delft (1983). V.J. de Waal, P. van den Hamer and T.M. Klapwijk, Appl. Phys. Lett. 42 (1983) 389. P81 J.M. Jaycox and M.B. Ketchen, IEE Trans. Magn. 17 (1981) 400. P91 rf SQUID system 330, S.H.E. Corporation. WI AI. Larkin and Yu N. Qvchinnikov, Sov. Phys. JETP 24 (1967) 1035; Phys. Rev. 28B (1983) 6281. Yu N. Ovchinnikov, R. Cristiano and A. Barone, J. Appl. Phys. 56 (1984) 1473. D.H. Douglass Jr. and L.M. Falicov, in: Progr. in Low Temp. Phys., ed. C.J. Gorter (NorthHolland, Amsterdam, 1964) Vol. 4, chap. III, p. 144, eq. (4.15), fig. 4.3. R. Meservey and B.B. Schwartz, in: Superconductivity, ed. R.D. Parks (Dekker, New York, 1969) p. 135, eqs. (18b) and (18~); p. 137, eq. (22) [21] B. Miihlschlegel, Z. Phys. 155 (1959) 313, p. 322.
THERMAL ACTIVATION IN THE QUANTUM REGIME AND MACROSCOPIC QUANTUM TUNNELLING IN THE THERMAL REGIME IN A METABISTABLE SYSTEM CONSISTING OF A SUPERCONDUCTING RING INTERRUPTED BY A WEAK JUNCTION PART I: THERMAL
ACTIVATION
IN THE QUANTUM REGIME
D.W. BOL, J.J.F. SCHEFFER, Kamerlingh
Onnes Laboratorium
W.T. GIELE
and R. DE BRUYN OUBOTER
der Rijksuniversiteit Leiden,
Postbus 9506, 2300 RA Leiden,
The Netherlands
Received 16 July 1985
Under certain circumstances a superconducting ring containing a weak superconducting junction, a SQUID, has two metastable magnetic flux states separated by a potential barrier AU. If the junction has a small capacitance stochastic magnetic flux transitions are observed even at very low temperatures. This article mainly concerns the intrinsic thermal activation in the quantum regime, where the mean energy of the oscillator, corresponding to the metastable potential wells, is different from the equipartition value kT due to the high frequencies and low temperatures involved in our experiments. We have found strong deviations from the classical thermodynamic activation theory of Kramers where the transition rate is proportional to exp(-AUikT). We are able to explain our experimental findings by replacing the classical equipartition value kT for the mean energy in the exponent with the value for the mean excitation energy of the Brownian motion of a quantum oscillator, accounting for the appropriate damping.
1. Introduction
Under certain circumstances a superconducting ring containing a weak superconducting junction (a SQUID) has two metastable magnetic flux states separated by a potential barrier if an external magnetic field is applied of appropriate strength. If the junction has a small capacitance magnetic-flux transitions are observed, even at low temperatures where kT is much smaller than the barrier height AU separating the two metastable flux states. These flux transitions from one flux state into the other and vice versa correspond to the stochastically switching of the weak “persistent supercurrent” from one direction into the opposite. Our earlier investigations [l-5] have been extended in two regimes: (1) thermal activation over the barrier in the quantum regime, where kT = hw,, w,, being the free oscillation frequency corresponding to the metastable potential wells and (2) macroscopic quantum tunnelling through the barrier in the thermal regime where kT < hw,. This article mainly concerns the intrin-
sic thermal activation in the quantum regime, where the mean energy of the oscillator is different from the equipartition value kT due to the high frequencies and low temperatures involved in our experiments. We have found strong deviations from the classical thermal activation theory of Kramers [6,7] for the shuttling action of a Brownian particle which is subject to the irregular forces of the surrounding medium in temperature equilibrium and which can therefore escape over the potential barrier. In the classical thermal activation theory the transition rate, caused by intrinsic thermal fluctuations over the barrier, is given by l/(7-,.) = (w,/27r) exp(-AUlkT), in which (T=) is the mean lifetime of the metastable state and w, the attempt frequency. We are able to explain our experimental findings by replacing in the exponent the classical equipartition value kT for the mean energy with the value for the mean excitation energy of the Brownian motion of a quantum oscillator as calculated by Ford et al. [8]. Essential for our analysis of the thermal activation data is that we have used a fluctuating
0378-4363/85/$03.30 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
D.W. Bol et al. I Thermal activation in the quantum regime
quantum force of the Planck type without zero point fluctuations [8] in the Langevin equation. Zero point energy cannot be extracted from the environment and is therefore not included in the thermal excitation energy. Just as in the case of macroscopic quantum tunnelling [9], the exponent in the expression for the transition rate for the case of thermal activation in the quantum regime contains a damping term [8] contrary to the case of classical activation [6]. We can define a characteristic temperature Tq below which quantum effects become important by means of the relation [lo]: kT, = hwi = fiwh{d(l+ 7”) - r}, where y = 1 I2RC4 is the damping parameter (R the resistance and C the capacitance of the weak junction) and WA= 1 Iv(K)*. Dissipation lowers the characteristic temperature T, so that quantum effects are diminished by the damping [lo]. Although this article mainly discusses thermal activation in the quantum regime we also present some of our results for macroscopic quantum tunnelling in the thermal regime. Now a fluctuating quantum force of the Schrtidinger type with zero point fluctuations included plays an important role in the Langevin equation [ll, 21. At weak damping the tunnelling itself can be considered to be energy-conserving. In the crossover region from thermal activation to quantum tunnelling we suggest that the total transition rate 1 /(T) is equal to the sum of the thermal transition rate l/( rT) and the transition rate l/ (T,) due to macroscopic quantum tunnelling, hence l/(7)= l/(7,)+ l/(~~). Macroscopic quantum tunnelling will be favoured by a very small capacitance of the weak junction [l] and has already been observed at moderately low temperatures in a superconducting ring interrupted by an adjustable superconducting point contact with a very small intrinsic capacitance of about lo-l5 F [l]. Later on, Voss and Webb [12] studied the influence of damping on the rate of tunnelling in a single low critical current junction driven by an external applied current, confirming qualitatively the theoretical findings of Caldeira and Leggett [9] that dissip*WA insteadof usualpractice in theoretical literature is defined by eqs. (11) and (12).
197
ation reduces the tunnelling rate. We extended our study on macroscopic quantum tunnelling as reported in section 7 of ref. 5 by investigating evaporated Nb-Nb tunnel junctions and made a comparison between both types of devices: Nb-point contacts and evaporated NbNb tunnel junctions with different device characteristics. Since the capacitance of an evaporated junction ( lo-l4 F) is larger than that of a point contact, macroscopic quantum tunnelling was now observed for smaller barrier heights and consequently at lower temperatures. The temperature dependence of the observed mean lifetime (T,) might possibly be due to a contribution of macroscopic quantum tunnelling from the thermal activated first excited energy level in the potential well (macroscopic quantum tunnelling in the thermal regime). In Part II macroscopic quantum tunnelling will be discussed in greater detail. The emphasis will be on the behaviour of our evaporated Nb-Nb tunnel junctions at very low temperatures in the millikelvin range in order to make a comparison between experiment and the recent theoretical predictions on the temperature dependence of the quantum-tunnelling rates for symmetric and asymmetric double-well systems with ohmic dissipation [ 161.
2. Metabistability in a SQUID Through a superconducting ring containing a weak junction of the Josephson type [l], see fig. 1, a variable magnetic flux B, 0 is applied (B, is the external magnetic field applied perpendicular to the embraced area 0). in the presence of the external applied magnetic field a circulating current ifirc will be induced, so that the net magnetic flux inside the ring will be @ = B,O + Licirc ,
(1)
L is the self-inductance of the ring. The circulating current cannot provide complete shielding even at the lowest value of B,O due to the presence of the weak junction. The steady-state current through the junction is related to the
D.W. Bol et al. I Thermal activation in the quantum regime
198
ic,,,=-i~sin(~~,-i~j,R-~~,
(6)
with R and C the normal resistance and the intrinsic capacitance of the junction. Combining eq. (1) with eq. (6) leads, after rearrangement of the terms, to a “classical equation of motion” for @(t) of the following form [l, 71: Fig. 1. A superconducting ring with inductance L and containing a weak junction 6 with intrinsic capacitance C. The direction of the applied magnetic field B I is indicated. 0 is the embraced area.
gauge-invariant phase difference of the order parameter A$J *, across the junction, by the first Josephson relation [ 131
(7) C plays the role of mass and (1 lR) is the friction coefficient. The corresponding potential energy U( @, B, 0) is given by U = --(&ii,
icirc
= i, sin(-A$*),
(2)
where i, is the maximum supercurrent through the junction. For the gauge-invariant phase difference one finds AC+*=
$ @+n2m,
n = 0, +l, *2,.
..,
(3)
by integrating the Aharonov-Bohm relations for the electromagnetic potentials around the ring and using the fact that the wave function of the superfluid condensate due to the formation of Cooper pairs is always single valued. In the steady state one obtains the well-known relation for the total embraced magnetic flux as a function of the applied magnetic field:
@(B,) = B,O - Li, sin ($
@(B,)) .
During the approach to the steady-state eqs. (1) and (3) should still hold, but a voltage difference AV will be present across the junction [13]:
AVz_!!$-h+f. 2e
This voltage difference gives rise to a normal current, AVIR, as well as a displacement current, CaAVldt, through the junction so that instead of eq. (2), i,,,, becomes
. (8)
The total embraced flux @ is here regarded as a dynamic variable. The first term on the right-hand side of eq. (8) can be identified as the Josephson coupling energy [13] and the second term as the magnetic energy of the system. In the steady state (a U/a @ = 0) one obtains again the well-known relation (4) for the flux as a function of the applied magnetic field. When the dimensionless coupling constant 2 E FLi,
(9)
is larger than one and for suitable values of the external magnetic field B, the potential energy U(@) has one or more metastable local minima separated by an energy-barrier AU (see figs. 2a and 2b), resulting in the multi valuedness of the flux-applied magnetic field relation which is shown in fig. 3. In this way metabistability arises in a SQUID. In the case of zero damping the frequency w,, of free oscillations in the metastable local minima is given by 2 w()=
at
cos( $f@) + ((a - B,O)*/2L
c
a2U 2
)
= ,la:~::;[email protected],,L,.
(10)
For later convenience this expression for the frequency w0 of free oscillations can be written as
D.W.
Bol et al. I Thermal activation
I
0
in the quantum
199
regime
I
I
I
h12Q
0
h12Q
;P
9
a
b
Fig. 2. (a) The potential energy U as a function of the totally embraced magnetic flux Q,at constant applied magnetic flux B,O. The corresponding constant magnetic field value and the minimum and maximum are indicated on the @(B,) curve of fig. 3. The illustrated case is for 9 = rr. (b) The potential energy U as a function of the totally embraced magnetic flux @ at constant applied magnetic flux B, 0 = 4 (h/2e).
@Jo =
t+oT
(11)
)
in which 0; = 1 l(Lc)“2
)
(12)
If we introduce the embraced magnetic flux and the applied magnetic flux in dimensionless quantities: y = @l(h/2e)
and
x = B,Ol(h/2e)
we are able to write the eqs. (7) and (8) in reduced flux quantities:
and (13)
h
Lc a2ylat2 + (LIR)aylat
= -aday,
(W
u( y, x) = $( y - x)” - (Y/4T2)
cos 297y
= { Ll(h/2e)2}
,
U(@, B,O)
(84
The expression for the energy barrier AU for a symmetric potential, i.e. x = &or B,O = ih/2e in reduced quantities becomes Au = (Z/47r2)[1 + cos 2ry,,,J = { Ll(h/2e)2}AU 810
Fig. 3. The totally embraced magnetic flux @ vs. the applied flux B,O in the steady state P’= r. The local potential minimum at A is separated by a potential maximum at C from a lower potential minimum at B. In fig. 2a these points are indicated in the U(D) curve. In A and B the energv barrier is equal to the shaded area a respectively b dividerby L.
.
- 4 ( ymin - 1)’ WI
The energy I_ barriers AU,u , AU,Y for an asymmetric situation are indicated in fig. 2a and are equal to the shaded area a respectively b in fig. 3 divided by the self-inductance L of the ring. In fig. 2b the symmetric potential barrier is drawn when AU, = AU,= AU [l].
200
D. W. Bol et al. / Thermal activation
3. The Langevin equation and the fluctuating quantum forces
in the quantum
(Et) =
gw2“i)’ wi + (oIRC)* -
0
In the quantum regime, i.e. at high frequencies or low temperatures, the mean energy of the Brownian motion of a harmonic oscillator is no longer equal to the equipartition value kT as used by Kramers in the classical activation theory. Therefore we like to derive an expression for the mean energy (E) of the Brownian motion of a harmonic quantum oscillator. We start from the Langevin equation [8, 11, 21 by adding to the equation of motion for Q, with “mass” C, in a potential well with a characteristic frequency wO and a friction coefficient R-r, a fluctuating random noise “force” F associated to the normal current noise and obtain [2]
regime
hw
dw .
x 2RCcoth
(18)
In the limit no G kT both expressions (.+) and (et) for the mean energy of a Brownian particle approach kT, which is the classical equipartition result. Another simple limit is the weak coupling limit for the mean thermal excitation energy H.A.Kromers,bO
a We distinguish two expressions [8] for the fluctuating random noise “force” F: the noise “force” FT due to the thermal fluctuations of the Planck type and the total noise “force” F, of the Schriidinger type including both thermal and zero-point fluctuations. Their spectral densities are related to (Fc),, (F:)
kT/hw,,
H.A.Kromers.bO
o+dw = o, w+dw= &
coth&T
= %{i
. Aw
+ ll(efi”‘kT-
l)}Aw
. (16)
For the mean thermal excitation energy (s,) find [8, 141
we
b iiwIRC
x e
fiolkT
-1
kT/hw,
dw ’
(17)
and for the mean total energy (Ed) (zero point energy plus thermal excitation energy) we find 1141
Fig. 4a, b. The mean thermal excitation energy ( .Y,) of a Brownian harmonic quantum oscillator with a free oscillation frequency 00. (m)IkT is plotted vs. kT/hm for different damping parameters y = 1/2RCw,. The classical equipartition result fi-+0 and the Planck result y = O.Ol+ 0 are also indicated.
D.W. Bol et al. I Thermal activation in the quantum regime la-
4. The attempt frequency
t 16
201
t
Fig. 5. The mean total energy (Ed) and the mean thermal excitation energy (E,) of a Brownian harmonic quantum oscillator with a free oscillation frequency wO. (E,) lkT and (.e,)lkT are plotted vs. fiw,lkT for different damping parameters y. (Ed), in which the coupling of the Brownian particle to the environment is weak, i.e. when w0 % 1IRC. In this limit the resonance denominator in (17) becomes sharply peaked at w = w,, with a width equal to 1IRC. We find (c,) = hw,,l(exp(ho,JkT) - l), which is the wellknown Planck result for the mean thermal energy of an independent oscillator of frequency o0 at temperature T. We solved eqs. (17) and (18) by numerical calculation for different damping parameters y = 1 I2RG.0,. In figs. 4a and b (c,) lkT as a function of kT/hw, is plotted for different damping parameters y. The classical equipartition result (h = 0) and the Planck result ( y = 0) are also indicated. We see in fig. 4 that the characteristic temperature Tq below which quantum effects become important is lowered when the damping increases. In fig. 5 (c,) lkT and (~,)lkT are plotted as a function of hw,,lkT. Ford et al. [8] studied the Brownian motion of a quantum oscillator in a model of a Brownian particle coupled to a system of harmonic oscillators acing as a heat bath. Eq. (17) was one of their results. As already mentioned in the Introduction the results obtained from eq. (17) and presented in fig. 4 will be used later for interpreting our thermal activation data in the quantum regime.
The frequency appearing in the prefactor of the expressions for the mean lifetime is named the attempt frequency w, and is interpreted as the frequency at which the flux coordinate strikes the potential barrier AU. In the classical thermal activation theory of Kramers [6, 71, valid in the region hw, 4 kT @ AU, the attempt frequency is defined by the relation
In this theory the noise activated escape over the potential barrier was discussed in terms of the properties of the motion of a Brownian particle in a field of force. Kramers discusses three separate approximations of the case of large viscosity, depending on the degree of damping. Here we mention the three regimes of our interest: (a) The weakly damped regime where y < 1 leading to the Arrhenius transition state theory:
(20)
w, = w, .
(b) The heavily damped regime, where y S 1: w, = w,lw,]RC = 21w&y ,
(21)
in which 1wb[ = q/(.2 - 1) IVLC = whd(2 - 1) is the absolute value of the imaginary frequency associated with the unstable potential curve at the top of the barrier. (c) The regime of moderate damping between the cases (a) and (b):
(24 We will not discuss the extremely weakly damped situation (y + 0) which follows from Kramers theory for the case of small viscosity [6]. Eq. (22) can be written as wJwO = w:/w;, = (1 + ,,)l’*
- p ) (224
with p = wgeyl(Z - 1)1’2 .
202
D.W. Bol et al. I Thermal activation in the quantum regime 0.46 +/
In the case of macroscopic quantum tunnelling without damping one finds, in WKB approximation, for the mean lifetime [9]
mm /t
(23) where CY= 7.15 for the shape of our nearly symmetrical double potential well [9] (B,O = hh/2e) and for the attempt frequency [12, 151 &--
(24) Following Caldeira and Leggett [9] and Voss and Webb [12] one uses the following expression in case of moderate damping:
(7,) = Feb,
(25)
a
where CYAU + a(A@)*
b=Zlo
~
0
TLR ’
(26)
in which A@ is the tunnel distance under the barrier and a a numerical factor of order unity (a = 0.47). The attempt frequency is now equal to
[=I (27) Again we see that dissipation plays an important role [9, 2, 121. We have to realize that these expressions (23) and (25) of course cannot be applied in the purely symmetric potential case, but are derived for an escape into a “continuum”
[W 5. The experiment The superconducting ring interrupted by a Josephson junction consists of an evaporated planar Nb ring structure in which a Nb-Nb junction is incorporated (fig. 6). The ring structures were made six at a time on a flat silicon
5mm a
+i b
Fig. 6. (a) The evaporated planar Nb ring structure interrupted with a Nb-Nb junction (L = 9.2 x lo-“‘H, C= lo-l4 F). (b) Detail of the Nb ring structure with the Nb-Nb junction formed by the overlap of the two 10 km banks.
substrate using photoresist techniques. The NbNb junction with a silicon oxide barrier between it has an overlap area smaller than 1 urn* and an estimated capacitance of lo-l4 F [17]. Due to the special geometry chosen for the ring the inductance can be calculated to be 7.2 x lo-” H [18]. The 10 km wide banks leading to the junction have an estimated inductance of 2 x lo- lo H [ 171. So the total inductance of the ring is approximately 9.2 X lo-” H. All the samples have the same inductance and capacitance. The advantages of this kind of ring structure are its well-defined geometry, the resulting knowledge of the experimental parameters [17] and the stability of the junction with respect to thermal cycling. Storage over the period of a year at room temperature resulted in a decrease of the critical current with a few percent. This reduction can be avoided by storage at liquid nitrogen temperature. The magnetic flux in the ring was measured with a commercial rf SQUID system [19]. By means of a Nb wire, wound as a spiral pick-up coil, a weak but sufficient coupling to the ring was made. The ring was placed in an Araldite sample holder and shielded from the environment by two copper-lead shields at liquid helium temperature and a p-metal and copper shield which surrounded the cryostat. Except for the leads to the external field coil which are carefully filtered, no electrical connections were made to the inner
203
D.W. Bol et al. I Thermal activation in the quantum regime
cryostat to prevent external disturbances during the measurements. The coupling constant Ywhich according to eq. (8b) determines the barrier height AU [l] together with the inductance L, can be measured in two ways. First by comparing the measured magnetic flux @ vs. the applied flux B,O-curve with that from an identical but short-cutted ring so that the magnetic flux picked up directly from the external field coil can be subtracted from the SQUID output. Then 3 follows from the slope of the @ - B,O curve at an applied magnetic flux equal to an integral number of magnetic flux quanta. The second way is by fitting the relation (see (4)) @ = B,O - $-gsin[L?+r&/(
d))
the coupling strength of the ring to the pick-up coil is reflected in the measured amplitude of the @ - B, 0 curve. From the coupling strength thus determined and again the slope of the @ - B,O curve at an integral number of applied magnetic flux quanta, the coupling constant 3follows when it is larger than one at lower temperatures. In these two ways P’is known within 5% of its absolute value. Changes in 3, for instance due to temperature variations, are known within 1%. For 3 > 1 and B,O = $ (M2e) the potential energy as a function of @ is symmetric and has at least two metastable minima. Switching between these two flux states can be observed by measuring the magnetic flux as a function of time. The SQUID output is recorded by a transient recorder which is connected to a microcomputer for storing and analyzing the signal. In this way it is possible to measure the mean lifetimes (T) of the magnetic flux states in a “window” which extends in time from 0.5 ms up to 1 s. The lower bound is set by
(28)
to the measured @ - B,O curve for 3 < 1 when this relation is single valued. When 3 is known,
/ 1o-4
4
I
6I
/
/ 1
8I
,.l.“’
/ I
I
10
I
I
12
I
I
14
I
I
I
16
LAU/kT (lo-‘HI Fig. 7. The measured mean lifetimes (T) vs. LAUIkT for the samples (a) up to (f). The thermal activation data of the samples (a) up to (d) are represented by straight lines. The classical thermal activation result according to the theory of Kramers [6] is also indicated (w: = o,/w~ = 10” s-r). In the corresponding fig. 8 is indicated for the devices (a) up to (d) how the quantities T, JZaand oP vary as a function of LAUIkT in the “measuring window”. For device (f) the measurements go down in temperature to 0.4 K.
204
D.W. Bol et al. I Thermal activation
in the quantum
regime
12 LAUlkT e
d
C
b
Fig. 8. For the devices (a) up to (e) is indicated how the quantities T, 3’ window”.
the intrinsic noise of the rf SQUID and beyond the upper bound statistical interpretation of the magnetic flux transitions becomes very difficult. In the thermal regime the mean lifetime (TV) of the magnetic flux state will inevitably get out of the measuring window as we vary the temperature (fig. 7). To investigate thermal fluctuations over the whole temperature range we needed several rings (the devices (a) up to (f)) with suitable coupling constants 9 for the various temperatures (fig. 8). The variation of the critical current of the rings, even in one evaporation run, was large enough to make this possible. For example, 4 out of 24 rings which were made were suitable for investigation of thermal activation in the quantum regime and only 2 for investigation of macroscopic quantum tunnelling. In order to study the thermal activation over the barrier we determined the temperature dependence of the mean lifetime (r) of a magnetic flux state in a symmetric potential, i.e. B,O =
14
16
18
(lo-‘H)
and wr vary
as a function of LAUIkT
a
in the “measuring
(n + i)(h/2e). Due to the temperature dependence of the critical current i, of the junction or the coupling constant 55 of the ring, the barrier height AU changed as we followed ( C-) within its measuring window. This change of AU is taken into account by plotting the mean lifetime measurements for the various rings logarithmically as a function of LAUIkT as done in fig. 7. LAU only depends on the coupling constant 3, see eq. (8b), and 3 was measured at all temperatures together with the mean lifetimes. The change of T, Z’and ox within the measuring window of the mean lifetime is plotted in fig. 8. The mean values of T, 2, o, and kTlfiq, which are indicated by a bar above the symbol are given in table I. We have plotted (T) logarithmically vs. LAUIkT, to compare our results with the classical thermal activation theory of Kramers [6]. The classical result is represented in fig. 7 by the dotted line on the right with slope l/L and intersection (7) = 27r/wl at LAUIkT = 0. Our
205
D.W. Bol et al. I Thermal activation in the quantum regime Table I
L = 9.2 x lo-”
H, C = IO-l4 F, 06 = 3.3 x 10” s-l, w: = 10’” s-’ (b)
(c)
(d)
(e)
4.76
4.3
4.2
4.0
5.3
3.3
2.9
2.0
3.69
2.56
1.84
1.63
1.17
5.38
3.90
2.89
2.61
1.92
2.4
2.0
1.7
1.6
1.3
1.15
1.17
1.24
1.26
1.36
0.84
0.73
0.55
0.49
0.26
0.61
0.51
0.43
0.41
0.36
4.8
2.4
1.0
0.78
0.16
11.5
4.80
1.70
1.25
0.208
0.40
0.75
0.88
0.90
0.96
0.567
0.236
0.0837
0.0615
0.0102
0.566
0.202
0.0878
0.0651
0.0151
(a)
4.76 13.5
‘VT,) _=R(T)
Wrpy
rr,
R(T,) R(T) [eq. (30),theor.]
devices (a), (b) (c) and (d) show an exponential temperature dependence of the mean lifetime (r) above -1.9 K, but the exponent differs from the classical result. This difference decreases at higher temperature. A discussion will be given in the next section. Below -1.9 K our devices (c), (d), (e) and (f) show a deviation from the exponential temperature dependence which gets more significant at lower temperatures. This deviation is due to the crossover from the mechanism of thermal activation over the potential barrier to macroscopic quantum tunnelling through the potential barrier and is briefly discussed in the last section.
6. Thermal activation in the quantum
regime
The thermal activations of the samples (a), (b),
(c) and (d) are represented by straight lines in fig. 7. If the data could be interpreted by the classical thermal activation theory of Kramers [6] (eq. (19)) all these lines should nearly coincide (all the samples have nearly the same CO:= w,/w~ and we have only small changes in We), with the classical activation result indicated by the dotted line. We observe, however, strong deviations from the classical result of Kramers which depend strongly on the mean value k~lhZ&,. The smaller this quantity the greater the slope and the shift of the straight lines to the left of the classical line. These deviations can also be seen in the fact that magnetic flux transitions due to thermal activation are observed in the same “measuring window” for the devices (a) up to (e) for AUIkT values in the range between 13 and 18.5 for device (a), between 11.2 and 15.5 for (b), between 8.5 and 9.8 for (c), between 7.7 and 8.6 for (d) and
206
D.W. Bol et al. I Thermal activation
between 3.8 and 5 for (e). As discussed in the Introduction we can interpret our experimental data on thermal activation over the barrier in the quantum regime on the basis of the following relation:
in the quantum
regime
‘E o=o q =b
t OlL
Arc
x=d
t
;;
where the equipartition value kT in the classical thermal activation theory is replaced by the mean thermal excitation energy (E,) which is given by eq. (17). (E,) lkT is calculated and plotted vs. kTlhw, in figs. 4a, b for different damping parameters y. In the limit fro < kT, eq. (29) reduces to the classical activation result of Kramers (eq. (19)). The straight lines for the samples (a), (b), (c), (d) and (e) in the log r vs. LAUIkT plot of fig. 7 are drawn in agreement with eq. (29), using only one value w: = o,lwY (=lO’O s-‘) for all the samples. This result does not depend very critically on possible small individual changes in wi. According to eq. (29) the slope of the straight lines gives kT/ (c,) L. With L = 9.2 X lo-‘” H the mean value (q) /kT for the samples is determined and given in table I. With the use of the figs. 4a, b the mean damping parameter 7 for the samples follows from the (E,)lkT va 1ue found in this way and the corresponding value of kTlfiZ&. The values for 7 are given again in table I and will be discussed later on. Finally, the logarithm of oz rl. vs. LAUl ( Ed) is plotted in fig. 9 to illustrate the analysis of our experimental data on the basis of eq. (29). We see that all our thermal activation data for all the different samples fall on one straight line with wi = lOlo s-‘. The analysis shows that the thermal activation data are in agreement with eq. (29) if the temperature dependence which is found for the damping parameter y(T) = 1/2R( T) Cw, or for the quasi-particle resistance R(T) can be understood. This will be done in the next section. We like to remark that we were unable to interpret our thermal activation data in the quantum regime by using, instead of the mean thermal excitation energy (Ed), the mean total energy (E,) as given by eq. (18) and presented in fig. 5 as an (Et) lkT vs. tiw,lkT plot, since (et) > kT. On
‘;
0.01
3’
0.001
0.0001’
;*
’
I 14
I
I 16
/ 18
/
I 20
LAU/
Fig. 9. CUT(V) vs. LAU/(ET) for the devices (a) up to (d). The straight line is represented by the equation W~P(TT) = (2?r/w:) exp[AU/ (&,)I, (29a) with W: = 10’” sC1and describes all our thermal activation data.
the other hand, for the mean thermal excitation energy we find (E,) < kT.
7. The damping parameter and the quasi-particle resistance As mentioned in the preceding section the mean damping parameters 7 (7 = 1/2RCG&J of our devices are determined from the experimentally found values of (q) / kT and are given in table I. From the obtained results we see immediately that, when lowering the temperature T, the damping 7 decreases. We interpret the data by making the assumption that in the resistively shunted junction model, which is implicitly used in our calculations, the normal current may be identified with the normal quasi-particle current which tunnels through the (oxide) barrier of the junction. The supercurrent and the normal quasiparticle current flow in parallel through the junction. Let us use the following expression [20] for the ratio of the resistance in the complete normal state R(T,) and the quasi-particle resistance
207
D. W. Bol et al. I Thermal activation in the quantum regime
R,,(T)
in the limit of zero voltage:
(30) where K, is a modified Bessel function of the second kind. This ratio is only a function of T and of the BCS gap A(T). At very low temperatures (kT 6 A(T)) the normal quasi-particle resistance in the limit of zero voltage becomes exponentially large and eq. (30) gives (m = 1): E$.)!
= ( 2rfr))“2
exp[ ?!!!$I
(31)
.
in fig. 10a the drawn line gives the result of a numerical calculation of R(T,)IR(T) vs. the reduced temperature T/T,, using eq. (30) and the relation A( T = 0) = 1.764kT, and the tabulated values of Miihlschlegel [21] for A (T)/A (T = 0). The experimental determined values of W,y (= 1/2R(T)Cw9 for our devices are given in table I. In fig. lob these values wYy are plotted vs. 1.4-
the theoretically calculated value of R( T,) / R( T) at 7. Obviously the decrease of (oZy),,r at lower temperatures ?? corresponds with the temperature dependence of R( T,) / R( T) according to eq. (30), assuming a constant R( T,). From the slope of the drawn straight line yr, is determined, this gives R( T,) = 1/2y,~o,‘,C. We found yT, = 20.3 and Now we can calculate R( T,) = 7.46 R. R( T,) I R(T) for our devices. The results obtained in this way are plotted in fig. 10a. It appears that the damping parameter y for all our devices can be described with a single yr,, and that they have the same WI in the limit .Y+O. The experimentally determined value for R( T,) / R(T) = W,y/20.3 and the theoretically calculated value for R( TJl R(T) are given in table I. Finally we remark that from eq. (22a) follows that ~1 can be calculated by the assumption of using the normal resistance value R, = R(T,). In that case, from eq. (22a) follows that for the devices (a)-(e) /? is equal to, respectively, 23.3, 23.8, 25.1, 25.6 and 27.5; w: is equal to, respectively, 7.0 X 104 7.0 X lo’, 6.7 x 10, 6.7 x 10’ and 6 x 10’s_‘. These values are not far different from 10”’ s-i, showing the consistency of the analysis. 14r
1.2 1 .o $ .
0.8 -
t” 0.6 lx
cl / Cl
0.4 ,/,,,
0.2 1
0L-L -do 0 0.2
b.
d;o 0.4 0.6 TIT, a
0.8
1.0 R CT, )/R(T) I-
Fig. 10. (a) The drawn line gives the result of a numerical calculation of R( T,)IR(T) vs. the reduced temperature TIT,. R(T) is the quasi-particle resistance in the limit of zero voltage. R( T,) lR( T) is also determined from the measured damping parameter for the devices (a) up to (e) and indicated in the figure. (b) The experimentally determined value of cy vs. the calculated value of R( T,)IR( T) for the devices (a) up to (e) (see fig. lOa). From the slope are determined y,, and R( T,) = 1/2yrcwAC ( yr,= 20.3 and R( T,) = 7.46 a).
208
D. W. Bol et al. I Thermal activation in the quantum regime
We should like to remark that the value of R, = R(7'Jin this way obtained is about a factor ten smaller than the directly measured resistance. We have found up to now no satisfactory explanation. This will be investigated in more detail. Finally we hke to make some remarks about the experiments of Voss and Webb [12] where a single low critical current Josephson junction was driven by an external current source and the transition probability was measured for switching out of a metastable superconducting state to a finite voltage state as a function of temperature down to the millikelvin range. At high temperatures the transition rate out of the metastable state was dominated by thermal activation over the barrier and at very low temperatures it was observed that the distribution widths of the transition probabilities became temperature independent and in qualitative agreement with the predictions for macroscopic quantum tunnelling. But feature of thermal activation in the quantum regime are also hidden in their measurements. For instance from fig. 2a of their publication [12] we got data in order to calculate the energy barrier AU and w0 for the current (i) driven junction: AU(ili,)
= (fi/2e)2i1[{l
- (ili,)2}1’2
- (i/i,) cosml(ili,)] w. = WI(l - (ili,)2)1’4;
,
w: = (2elti)(i,lC).
Again, by making a plot of log rr(1 - (ili,)2)1’4 vs. A UIkT we found a nearly exponential temperture dependence for their thermal activation data with the typical feature of an “apparent” attempt frequency which is decreasing rapidly with decreasing temperature. For instance, from their measurements at 0.740, 0.558 and 0.414K (with w0 equal to, respectively, 1.94 x 10”; 1.83 X 10” and 1.76 x 1011s-l and with kTlG& equal to, respectively, 0.50, 0.40 and 0.24) we find an “apparent” attempt frequency of, respectively, 1.08 x loll, 2.62 x 10” and 8.31 x lo9 s-‘. At lower temperatures we come in the region of macroscopic quantum tunnelling, for instance at 0.005 K (with T3,= 1.34 x 10” s-l) we find an
attempt frequency of 2.45 X 10” SC’ in agreement with eqs. (24) and (27) (see next section).
8. Macroscopic regime
quantum tunnelling
in the thermal
As already reported in fig. 8 of ref. 5 and mentioned in the Introduction, macroscopic quantum tunnelling is observed below 0.5 K in our Nb-Nb evaporated tunnel junctions with capacitances of ~1O-l~ F and for barrier heights corresponding to 2~2, whereas in ring devices interrupted by a Nb-point contact with capacitances of ~10-15 F this phenomenon is already observed [l] below 1.5 K for barrier heights corresponding to a value of the coupling constant L? between 3 and 4. Just as in the case of thermal activation it is only possible to measure mean lifetimes TVfor tunnelling in the region between 0.5 ms and 1 s. Our experiments on macroscopic quantum tunnelling in the thermal regime at very low temperatures in the millikelvin range will be extensively discussed in Part II. Here we only make some rough calculations with respect to the results as presented in fig. 7. We take as example the low temperature value for (7,) of the curve (e) in fig. 7. We first calculate the mean lifetime (T:) for tunnelling in the undamped case, using eqs. (23), (24), (ll), (12) and (13): (7:) = (27r/w, (aAU127rfiwo)1’2) exp(cuAulhw,) with the constant (Y in WKB approximation for our potential equal to 7.15. We find for (7:) = 8.3 x lop5 s and for the effective undamped attempt frequency wi = 6.7 x 10” SC’ using eq. (24). Use has been made of the data of table I. In the case of damping (for sample (e) R = 728 R in the low temperature region) eqs. (25), (26) and (27) are used with a = 0.47 and (T,) = 0.85 s is not far from the experimental values shown in fig. 7 for curve (e). For the samples corresponding to the curves (d) and (c) of fig. 7 we find for (7:)) respectively 400 s and 3.7 x lo5 s. Again we have to realize that the expressions, which we have used in the calculation, cannot be applied in the purely symmetric potential case [16]. For device (f) the measurements shown in fig. 7 go down in temperature to 0.4K. We should like to remark
D. W. Bol et al. 1 thermal activation in the quantum regime
that in this temperature range the resistance values R(T) are still smaller than the critical resistance value (A@)2/di as discussed in ref. 16, [(A@92/27rfiR(T) > $1.
Acknowledgements It is a pleasure to thank Dr. H. Dekker (F.E.L.-T.N.O. - Den Haag) for his stimulating interest and advice in this work. The evaporated devices were made by Mr. J. Kortlandt of the research group of Prof. J.E. Mooij, University of Technology, Delft, and the masks used for their fabrication by the Delft Centre for Submicron Technology. The advice of Dr. T.M. Klapwijk with regard to these devices and their characteristics is greatly acknowledged. Mr. G. Vis and Mr. P.J.M. Vreeburg delivered the technical support for the experiments. This investigation is part of the research program of the “Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.)“, which is financially supported by the “Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek (Z.W.O.)“. References PI W. den Boer and R. de Bruyn Ouboter, Physica 98B (1980) 185. PI R. de Bruyn Ouboter and D.W. Bol, Physica 108B (1981) 965; 112B (1982) 15. 131 D.W. Bol, R. van Weelderen and R. de Bruyn Ouboter, Physica 122B (1983) 1. 141 R. de Bruyn Ouboter, Proc. Int. Symp. Foundations of Quantum Mechanics, Tokyo (1983) p. 83. PI R. de Bruyn Ouboter, Physica 126B (1984) 423. PI H.A. Kramers, Physica 7 (1940) 284. [71 J. Kurkijarvi, Phys. Rev. B6 (1972) 832. M. Biittiker, E.P. Harris and R. Landauer, Phys. Rev. B28 (1983)
209
1268. H. Dekker, to be published (1985). Experiments by L.D. Jackel, W.W. Webb, J.E. Lukens and S. Pei, Phys. Rev. B9 (1974) 115. PI G.W. Ford, M. Kac and P. Mazur, J. Math. Phys. 6 (1965) 504; p. 513, eqs. (82), (84). See also, D.K.C. MacDonald, Physica 28 (1962) 409. [91 A.O. Caldeira and A.J. Leggett, Phys. Rev. Lett. 46 (1981) 211; Ann. Phys. 149 (1983) 374; A.J. Leggett, Macroscopic quantum tunnelling, in: Percolation, Localization, and Superconductivity, eds. A.M. Goldman and S.A. Wolf (Plenum, New York, 1984). [lOI See also, H. Grabert and U. Weiss, Phys. Rev. Lett. 53 (1984) 1787. Pll H.B. Callen and T.A. Welton, Phys. Rev. 83 (1951) 34. L.D. Landau and I.M. Lifshitz, Statistical Physics (Pergamon, London, 1958). R.K. Pathria, Statistical Mechanics (Pergamon, London, 1972) p. 77. WI R.F. Voss and R.A. Webb, Phys. Rev. Lett. 47 (1981) 265. 1131 B.D. Josephson, Rev. Mod. Phys. 36 (1964) 216; Adv. Phys. 14 (1965) 417. 1141 H. Dekker, Physics Reports80 (1981)~~. 1-112; p. 39, eq. (577). WI E. Merzbacher, Quantum Mechanics (John Wiley, New York, 1961). [161 S. Chakravarty and A.J. Leggett, Phys. Rev. Lett. 52 (1984) 5. H. Grabert and U. Weiss, Phys. Rev. Lett. 54 (1985) 1605. M.P.A. Fisher and A.T. Dorsey, Phys. Rev. Lett. 54 (1985) 1609. H. Dekker, to be published: here the relation l/(7)= l/(7,)+ l/(rr) will be discussed. (171 V.J. de Waal, Ph.D. Thesis, University of Technology, Delft (1983). V.J. de Waal, P. van den Hamer and T.M. Klapwijk, Appl. Phys. Lett. 42 (1983) 389. P81 J.M. Jaycox and M.B. Ketchen, IEE Trans. Magn. 17 (1981) 400. P91 rf SQUID system 330, S.H.E. Corporation. WI AI. Larkin and Yu N. Qvchinnikov, Sov. Phys. JETP 24 (1967) 1035; Phys. Rev. 28B (1983) 6281. Yu N. Ovchinnikov, R. Cristiano and A. Barone, J. Appl. Phys. 56 (1984) 1473. D.H. Douglass Jr. and L.M. Falicov, in: Progr. in Low Temp. Phys., ed. C.J. Gorter (NorthHolland, Amsterdam, 1964) Vol. 4, chap. III, p. 144, eq. (4.15), fig. 4.3. R. Meservey and B.B. Schwartz, in: Superconductivity, ed. R.D. Parks (Dekker, New York, 1969) p. 135, eqs. (18b) and (18~); p. 137, eq. (22) [21] B. Miihlschlegel, Z. Phys. 155 (1959) 313, p. 322.