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Submitted on 1 Jan 1978
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ANOMALOUS BEHAVIOR OF THE RF-SQUID IN
THE NON-HYSTERETIC CASE : DC-FLUX
DEPENDENCE OF THE OUTPUT SIGNAL WITH
PERIOD ø0/2
S. Erné, H. Luther
To cite this version:
ANOMALOUS BEHAVIOR OF THE RF-SQUID IN THE NON-HYSTERETIC CASE :
DC-FLUX DEPENDENCE OF THE OUTPUT SIGNAL WITH PERIOD <j>
0/2
Abstract.- It is shown that the dependence of the RF-SQUIDoutput signal is not always periodical in 4> but, for special parameter configurations the period is <f 12. Experimental and theoretical results are in agreement. From the general theory for the non-hysteretic SQUID conditions have been develo-ped for this anomalous behaviour.
Usually it is assumed that the amplitude of the tank circuit voltage U of an RF-biased SQUID is a periodical function of the applied dc-magnetic flux <|), with a period of the flux quantum <J> . This is the reason for several authors /1-4/ to define a modulation depth U„ as the difference of the values Um for <t>, = 0 and (j>, = <J> /2. As the dependence of
T dc dc o r
U on the bias-frequency u>/2ir strongly is influen-ced by the SQUID-parameters some attempts /1-3/ have been done to determine the SQUID-parameters by
ana-lysing measurements of the off-tune-characteristic UM(OJ) . However, a detailed numerical treatment of the theory for the RF-SQUID given in /3/ shows that the assumption mentioned above for the dependence of U on <f> is not always satisfied. We here report some theoretical and first experimental results showing a periodical dependence of the output signal on the dc-flux with period <!> /2.
Quantitative expressions for the output si-gnal U can be attained from the theory /3/ only by numerical calculations. In figure 1 we present such numerical results in comparison with experimental ones, measured for a toroidal SQUID with a Nb-NbTi point contact driven at about 300 MHz and 4.2 K. The values of the Q-factor and the coupling coefficient k were Q = 100 and k = 0.2 the resonant frequency of "the tank circuit «T/2'IT = 332.6 MHz. The amplitu-de of the RF-bias-current I corresponds to that value which leads to a maximum dc-flux sensitivity at 6 = 2 Q(w - w )/w = 1. Calculated and measured values of U_ are normalized to the amplitude of the signal voltage at 5 = 1. Further SQUID parameters (as they are defined in /3/) 2Trg, OOT and a have
been fitted. Experiments and theory exhibit the ap-pearence of a dc-flux dependence with period <|> /2.
We now want to analyse these results on the basis of a simplified model to obtain analytical expressions for the Fourier coefficients of the output-voltage VT(<t>. ) • Under the conditions 2irg « 1, MI /<(> « 1, u/w « 1 the equations
cal-T O S
Fig. 1 : <(>, -dependence of the tank voltage amplitu-de fl_, 000 = measured, = calculated for 2ng A 0.7, a = - 0.9, U T = 0.1
q
JOURNAL DE PHYSIQUE
Colloque
C6,
supplément au n°
8,
Tome
39,
août
1978,
page
C6-1208
S.N. Erné and H. Luther
Physikaliseh-Teehnisahe Bundesanstalt - Institut Berlin - Abbestr. 2-12, D-1000 Berlin 10, R.F.A.
Résumé.- Il est démontré que la dépendance du signal de sortie du SQUID-RF n'est pas toujours pério-dique en 4> , mais que pour certaines configurations des paramètres, la période est <f>„/2. Les résul-tats expérimentaux confirment ce comportement. Les conditions de ce comportement anormal ont été ex-primées à partir de la théorie générale du SQUID-non-hystérétique.
led the characteristic ones in /3/ are linearized :
ST
k25;
wr IRF sin 6 = I { -- -
q TQ
r S k " ;TRF
cos 6 = IT (1-
S2
T+
-
r S 2aB cos (- 2r 0''dc)l
I
The amplitude of the output voltage can be approximated as follows A 'dc cos (2a -) ' 0
-
-
2Yo-
Since y and y are a linear respectively a qua- dratic function of 2 ~ 6 Q k2, this effect becomes im- portant for large values of B, high Q and strong coupling between SQUID and resonant tank circuit.From this anomalous behaviour important con- sequences arise if one intends to usetheSQUID con- figuration for junction diagnostic purpose. In this case the comparison between experiments and theory on the basis of the old "modulation depth" defini- tion described above could lead to severe difficul- ties. We therefore propose the introduction of a new definition of modulation depth Urn twice the Fourier coefficient of U (@ ) corresponding to the period
T dc
Qo. This definition is suitable over the whole range of each parameter.
References
/I/ Hansma, P.K., Phys. Rev. B
2
(1975) 1707where we get for yo, y l and
Y,
using equation (I) 121 Nisenoff, M. and Wolf, S., Phys. Rev. B12
(1975)and (2) : 1712
/3/ Erne, S.N., Hahlbohm, H.-D. and Liibbig, H., J.
'
[l+
(9
K2ST
@-cq)'
Q
K'ST
7 Appl. Phys.47
(1976) 5440Yo =
-
+ 2Q~
I + w2-c2
I + u2 r2 /4/ Soerensen, O.H., SQUID, Superconducting Quantum9 P Interference Devices and their Applications, ( 1 - 6 w r ) + 6 g + y 2 9 ( 4 ) De Gruyter Berlin (1977) 365
Some essential features concerning the func- tion follow from equation 13)
-
the function ($ ) is a superposition of two T dcterms with period $ and (Po/2
0
-
the amplitude of the $I -component vanishes at a0
bias frequency given by
(7) in agreement with definitions (4) and (5), (yo is a quadratic, y l is a linear function of
