![8.2
Voltmeters
One of the earliest applications of the dc SQUID was as a voltmeter. The sensor
was in fact a SLUG (superconducting low-inductance undulatory galvanometer)
[1] described briefly in Chapter 5. In essence, the SLUG consists of a bead of
PbSn solder frozen around a length of Nb wire. The critical current measured be-
tween the two superconductors is periodic (often multiply periodic) in the super-
current passed along the Nb wire. In the early days of this device, it was possible
to measure changes in this current of about 1 lA Hz–1/2. The fact that the input
circuit had a low inductance – a few nanohenries – enabled one to measure volt-
ages developed by much smaller resistances than had been previously possible
since the time constant of the measurement could be kept to below one second.
Figure 8.1 shows the original voltmeter circuit used with a SLUG. The voltage
source Vs was connected in series with a standard resistor rs and the Nb wire of
the SLUG. The SLUG was operated in a flux-locked loop (Section 4.2) that fed a
current is into rs to maintain a null current in the Nb wire: evidently the value of
Vs is given by isrs. With a SLUG current resolution of 1 lA Hz–1/2 determined by
the readout electronics, the voltage resolution for rs = 10–8 X was 10–14 VHz–1/2.
This represented a five orders of magnitude improvement over the resolution of
semiconductor amplifiers. Since the Nyquist voltage noise across a 10–8 X resistance
at 4.2 K is about 1.5 fV Hz–1/2, these early measurements were not Nyquist noise
limited. Nonetheless, the SLUG voltmeter was used successfully to make mea-
surements of the characteristics of superconductor–normal metal–superconduc-
tor (SNS) Josephson junctions [2] and of thermoelectric voltages [3]. Subsequently,
it was used in studies of the resistance of the SN interface [4] and to make the first
measurements of quasiparticle charge imbalance in superconductors [5].
4 8 SQUID Voltmeters and Amplifiers
5 mm
Copper wire
Niobium wire
Solder
Vs
rs
V
I
I
is is
From output of
flux-locked loop
Fig. 8.1 The SLUG. The configuration of a voltmeter measuring
a voltage source Vs has been superimposed on a photograph.](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-26-320.jpg)
![8.3 The SQUID as a Radiofrequency Amplifier
The development of much lower noise SQUIDs with multiturn input coils,
notably the Ketchen Jaycox square-washer design [6], has greatly reduced the
equivalent current noise. For example, for a low-Tc SQUID with a flux noise of
2 10–6U0 Hz–1/2 at frequencies above the l/f knee (f is frequency) of typically
1 Hz, coupled to an input coil with a mutual inductance of 5 nH, the current
noise S
1=2
I (f ) » 1 pA Hz–1/2. At 4.2 K, this resolution enables one to make Nyquist-
noise-limited measurements in resistors
r 4kBT/SI(f ) » 200 X . (8.1)
In making this estimate, we have neglected the effects of current noise in the
SQUID loop which induces noise voltages into the input circuit. This subject is
discussed at length in Section 8.3. These devices are generally used with current
feedback to the standard resistor to obtain a null balancing voltmeter [7].
Voltmeters have also been based on high-Tc SQUIDs operating at 77 K [8–10].
The unavailability of flexible, bondable wire made from a high-Tc superconductor
means that normal wire must be used to connect the components in the input
circuit. Contact resistance between this wire and the YBa2Cu3O7–x (YBCO) input
coil adds to the total resistance. As a result, the voltage resolution is limited to
roughly 1 pV Hz–1/2.
SQUID packages suitable for use as voltmeters are available commercially from
several companies. The SQUID is enclosed in a niobium can to shield it from
ambient magnetic noise. The two ends of the input coil are connected to niobium
pads to which external niobium wires can be clamped with screws to produce
superconducting contacts. Thus, the user can readily couple any desired external
circuit to the SQUID.
8.3
The SQUID as a Radiofrequency Amplifier
8.3.1
Introduction
This section is concerned with the use of the dc SQUID as a radiofrequency (rf)
amplifier. We confine our attention to the situation in which the SQUID is oper-
ated open loop, biased near (2n + 1)U0/4 to maximize VU. A thorough discussion
of such amplifiers is quite complicated. Although these issues are often ignored
in the design of SQUID input circuits, the coupling of a circuit to a SQUID may
modify its properties significantly, while at the same time the SQUID reflects
both a nonlinear impedance and a voltage noise source into the input circuit. The
modification of the SQUID by a coupled inductance was pointed out by Zimmer-
man [11], and studied extensively in a series of papers by Clarke and coworkers
[12–14]. The fact that the SQUID loop contains a noise current that is partially
correlated with the voltage noise [15] across the SQUID was computed by Tesche
5](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-27-320.jpg)
![8 SQUID Voltmeters and Amplifiers
and Clarke [16], and subsequently used by various authors to calculate the noise
temperature of amplifiers [13, 14, 17–20]. A complete treatment of these issues
would make this chapter unwieldy, and we limit ourselves to summarizing the
key theoretical results and to describing some experimental amplifiers.
8.3.2
Mutual Interaction of SQUID and Input Circuit
Consider a SQUID with loop inductance L and two identical Josephson junctions
each with critical current I0, self capacitance C and shunt resistance R. For a typi-
cal SQUID in the 4He temperature range, the noise parameter C ” 2pkBT/ I0U0 ~
0.05. The noise energy e(f ) ” SU(f )/2L is optimized [15] when bL ” 2LI0/U0 = 1.
The Stewart–McCumber parameter [21, 22] bC ” 2pI0R2C/U0 should be somewhat
less than unity to avoid hysteresis in the current voltage (I–V) characteristic (see
Chapters 1 and 2). Under these conditions, one finds the following results [15].
The maximum flux-to-voltage transfer coefficient is
VU ” ¶V
j =¶UjIB
» R=L (8.2)
where IB is the value of the bias current that maximizes VU, and the flux in the
SQUID is near (2n + l)U0/4 (n is an integer). When VU is maximized, the spectral
density of the voltage noise across the SQUID, which is assumed to arise from
Nyquist noise in the shunt resistors, is [15]
SV ðf Þ » 16 kBTR . (8.3)
The current noise in the SQUID loop has a spectral density [16]
SJðf Þ » 11 kBT=R (8.4)
and is partially correlated with the voltage noise with the cross spectral density
[16]
SVJðf Þ » 12 kBT . (8.5)
Figure 8.2(a) shows an input circuit consisting of a voltage source Vi in series with
a resistance Ri, the inductance Lp of a pickup coil, a stray inductance Ls, a capaci-
tor Ci and the input inductance Li of the SQUID. Depending on the application,
some of the components may be omitted. The mutual inductance to the SQUID
is Mi = ki(LLi)1/2, where ki £ 1 is the coupling coefficient. The SQUID reflects a
complex impedance into the input circuit which is derived from the dynamic
input impedance Z of the SQUID; in turn, Z can be related to the flux-to-current
transfer function JU ” (¶J/¶U)IB
by the equation [18]
–JU = jx/Z = 1/L + jx/R . (8.6)
6](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-28-320.jpg)
![8.3 The SQUID as a Radiofrequency Amplifier
The parameters Z, L and R refer to currents flowing around the SQUID loop. At
x = 0, –JU reduces to the inverse of the dynamic input inductance L, while for
x 0 there are resistive losses, represented by the dynamic input resistance R.
Figure 2(b) shows a schematic representation of L and R, which define the
response of the SQUID to an applied flux U.
Figure 8.3 shows the variation of L/L and R/R with applied flux [13] for four
values of bias current. Typically, SQUIDs are operated with IB » 2I0. We observe
that both parameters depend strongly on U, with L/L becoming negative in some
regions.
We next discuss the effect of the input circuit on the SQUID parameters.
Throughout this discussion we assume that the SQUID is operated open-loop,
with its current and flux biases adjusted to maximize VU. We also assume that the
loading of the readout amplifier on the SQUID is negligible. To illustrate the
point, consider a superconducting pickup inductance Lp in series with a stray
inductance Ls connected across the input inductance Li, as in a magnetometer. We
assume that the SQUID is current-biased at a voltage corresponding to a Joseph-
son angular frequency xJ. In the absence of parasitic capacitance, currents in the
7
o
Fig. 8.2 (a) Schematic of a generic tuned
amplifier. The voltage source Vi is connected
in series with a pickup loop of inductance Lp,
a stray inductance Ls, a capacitor Ci, a resis-
tance Ri and the input inductance Li of the
SQUID. (b) Dynamic input impedance of the
SQUID represented by an inductance L and
resistance R. In both figures, J is the current
induced in the SQUID loop by signal and
noise sources in the input circuit. (Repro-
duced with permission from ref. [13].)
Fig. 8.3 Simulated values of L/L and R/R. versus
reduced flux U/U0 for a bare SQUID versus flux U
for four values of bias current. SQUID parameters
were bL = 1.0, bC = 0.2 and C » 0.06 (Reproduced
with permission from ref. [13].)](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-29-320.jpg)
![8 SQUID Voltmeters and Amplifiers
SQUID loop at xJ and its harmonics will induce currents into the input circuit. It
is easy to show that the SQUID loop inductance will be reduced by the presence of
the input circuit to a value
Lr ¼ ð1 k2
ieÞL , (8.7)
where
kie ¼ ki½ðLi þ Lp þ LsÞ=Li1=2
(8.8)
is the effective coupling coefficient between the SQUID and the total inductance
of the input circuit. Other parameters of the SQUID take the reduced values Vr
U,
Jr
U, Zr and Rr corresponding to a SQUID with loop inductance Lr. In practice,
things may be not so simple: parasitic capacitance between the coil and the
SQUID washer modifies the coupling between them at the Josephson frequency
and its harmonics. In the limiting case where this parasitic capacitance prevents
any high-frequency currents from flowing in the input circuit, the SQUID param-
eters are unaffected by the input circuit [12]. In a real system, the result is likely to
be somewhere between the two extremes; it will also depend, for example, on the
number of turns in the input coil which determines the parasitic capacitance. By
studying a series of SQUIDs with 20-turn input coils, Hilbert and Clarke [13]
found that VU was increased by roughly the expected amount (corresponding to
the reduced loop inductance) when the previously open coil was shorted.
We are now in a position to consider the modification of the input circuit by the
SQUID impedance reflected into it. A productive way of writing the result is in
terms of the output voltage across the SQUID in the presence of a signal applied
to the input circuit shown in Figure 8.2(a). After some calculation, one finds [12]
VðxÞ ¼ Vr
NðxÞ þ MiVr
U
ViðxÞ þ Mi Jr
NðxÞðRi þ l=jxCiÞ=LT
ZTðxÞ Jr
U M2
i ðRi þ l=jxCiÞ=LT
. (8.9)
Here, Vr
N(x) and Jr
N(x) are the reduced voltage and current noises of the SQUID,
and Vi(x) is the input voltage applied to the resistance Ri and capacitance Ci in
series with the total inductance of the input circuit LT = Li + Lp + Ls. The total
impedance of the (uncoupled) input circuit is
ZTðxÞ ¼ Ri þ jxðLi þ Lp þ LsÞ þ l=jxCi . (8.10)
The denominator of Eq. (8.9) contains the term Jr
UM2
i ðRi þ l=jxCiÞ=LT that rep-
resents the impedance reflected into the input circuit from the SQUID. The term
involving Jr
NðxÞ in the square brackets is the noise current generated in the input
circuit by the SQUID. We can readily derive the voltage gain for a SQUID ampli-
fier from Eq. (8.9):
Gv ¼
MiVr
U
ZT k2
ieLJr
UðRi þ l=jxCiÞ
”
MiVr
U
Z*
T
. (8.11)
8](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-30-320.jpg)
![8.3 The SQUID as a Radiofrequency Amplifier
Using Eq. (8.6), we see that the impedance can be written in the form
Z*
T ¼ Ri 1 þ
k2
ieL
Lr
þ
k2
ieL
RrCi
þ jx Li þ a2
eL
Ri
Rr
1
x2CiLr
þ Lp þ Ls
þ
1
jxCi
:
(8.12)
We note that Rr and Lr contribute to both the real and imaginary parts of Z*
T.
Hilbert and Clarke [13] made extensive measurements of Jr
U as a function of U
by connecting a capacitor Ci across the input coil of both a 4-turn and a 20-turn
SQUID. The resonant frequency f 0
0 and full width at half maximum (FWHM) of
this tank circuit were measured directly with the SQUID biased with a large cur-
rent ( 2I0) where the SQUID has negligible inductive screening and a dynamic
input impedance of approximately 2Rr. The SQUID was then biased at its usual
operating point, and the Nyquist noise power P(f0) generated by the tank circuit
was measured at the output of the SQUID with a spectrum analyzer. The value of
Vr
U was determined from the height of the peak. The frequency f0 at which the
noise power peaked generally differed from f 0
0 , and yielded the inductance change
DLi reflected into the tank circuit. Similarly, the value of the FWHM, Df, yielded
the resistance change DRi. From these values, it was straightforward to infer the
values of L/Lr and R/Rr from Eq. (8.12) inserted into Eq. (8.11).
The results for a 20-turn SQUID are shown in Figure 8.4 for three values of bias
current. The behavior of jVr
Uj is much as expected. However, the magnitudes of
the curves suggest that the inductive screening of the SQUID is more effective at
the lowest bias current than at the highest bias current; this result is consistent
with a reduction in screening by the parasitic capacitance as the Josephson fre-
quency increases. The measured values of L/Lr follow the trends in the simula-
tions quite well. For example, at the lowest bias current there is a broad maximum
at U = 0 and a negative region around U = –U0/2. The overall magnitude of L/Lr
9
Φ/Φ0
Fig. 8.4 Measured values of L/ L r, R/R r and |Vr
U|L/R versus reduced flux U/U0
for a SQUID with L » 400 pH, 2I0 » 6 – 1 lA, C » 0.5 pF and R » 8 X, corresponding
to bL » 1 and bC » 0.2. The temperature was 4.2 K corresponding to C » 0.06. Bias
current: (a) 4.0 lA, (b) 5.0 lA, (c) 6.0 lA. (Reproduced with permission from ref. [13].)](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-31-320.jpg)
![8 SQUID Voltmeters and Amplifiers
is generally in fair agreement with the simulations. On the other hand, the mea-
sured values of R/Rr, which vary between +30 and –5, are in sharp disagreement
with the simulated values, which are always positive, with a maximum of about 2.
Thus, DRi is evidently dominated by a mechanism other than the resistance
reflected from the SQUID.
Further investigation showed that the change in resistance in the tank circuit
was dominated by feedback from the output of the SQUID via the parasitic capac-
itance between the washer and the input coil. Approximating the distributed ca-
pacitance with a lumped capacitor, Hilbert and Clarke [13] were able to account
for the observed change in resistance in the input circuit to within a factor of 2.
The reader is referred to the original paper for details.
This concludes our discussion of the input impedance of the dc SQUID and of
the mutual loading of the SQUID and input circuit. We next apply these ideas to
the design of SQUID amplifiers.
8.3.3
Tuned Amplifier: Theory
To simplify our initial discussion we first neglect capacitive feedback, and later
return briefly to this issue. The circuit is shown schematically in Figure 8.2(a); we
interpret Ri as the impedance of the voltage source Vi. For Vi = 0, the noise voltage
at the SQUID output can be written from Eqs. (8.8) and (8.9) as
VNðxÞ ¼ Vr
NðxÞ þ k2
ieLVr
UðRi þ l=jxCiÞJr
NðxÞ=Z*
TðxÞ (8.13)
where Z*
TðxÞ is given by Eq. (8.12). We now assume that the amplifier is operated
at the resonant frequency f0 = x0/2p at which the imaginary terms in Z*
T tune to
zero:
x0 ¼ ½ðLi þ Ls þ k2
ieLRi=Rr
ÞCi=ð1 þ k2
ieL=Lr
Þ1=2
. (8.14)
Thus, at the resonant frequency, Eq. (8.13) reduces to
VNðx0Þ ¼ Vr
Nðx0Þ þ
k2
ieL Ri þ 1=jx0Ci
ð ÞJr
N x0
ð ÞVr
U
Ri þ DRi
(8.15)
where
DRi ¼ k2
ieLðRi=Lr
þ 1=Rr
CiÞ . (8.16)
To simplify matters, we now assume that Q is high so that Ri 1/jxCi. As we
shall see later, optimization of the noise temperature implies that k2
ieQ » 1, so that
k2
ie 1. Thus, to an excellent approximation the SQUID parameters assume their
bare values, the resonant frequency becomes x0 » [(Li + Ls)Ci]–1/2 and Eq. (8.15)
reduces to
10](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-32-320.jpg)
![8.3 The SQUID as a Radiofrequency Amplifier
VNðx0Þ » V0
Nðx0Þ jx0M2
i VUJNðx0Þ=ðRi þ DRiÞ . (8.17)
Here, V0
Nðx0Þ is the voltage noise across the bare SQUID. We have retained the
term DRi for later discussion, since in practice it is dominated by capacitive feed-
back. Neglecting DRi for the moment, we can interpret Eq. (8.17) quite simply.
The current noise JNðxÞ produces a voltage noise jxMiJNðxÞ into the input cir-
cuit which, on resonance, yields a current noise jxMiJNðxÞ=Ri and hence a flux
noise jxM2
i JNðxÞ=Ri in the SQUID. Multiplying this term by VU yields the sec-
ond term in Eq. (8.17).
We now introduce the noise temperature TN(f0) of the SQUID amplifier on res-
onance through the definition
4kBTNðf0ÞRiG2
vðf0Þ ¼ SV ðf0Þ . (8.18)
Here, SV(f0) is the spectral density of the voltage noise at the output of the SQUID
amplifier, which we can readily calculate from Eq. (8.17). We find the value of Ri
that optimizes TN(f0) by calculating ¶TNðf0Þ=¶Ri ¼ 0 from Eq. (8.18):
R
opt
i ¼ ½ðR
opt
i0 Þ2
þ ðDRiÞ2
1=2
. (8.19)
Here,
R
opt
i0 ¼ ðSJ=SV Þ1=2
x0M2
i VU . (8.20)
This result can also be obtained from the usual [23] treatment of an amplifier with
a voltage noise source –jxMiJN(x) and a current noise source VN(x)/MiVU placed
at its input terminals. With DRi R
opt
i0 , the corresponding optimized noise tem-
perature is
T
opt
N ðf0Þ=T ¼ 8RR
opt
i =M2
i V2
U . (8.21)
Finally, setting VU » R/L, we find
R
opt
i0 » k2
i x0Li (8.22)
and
T
opt
N » ðSV SJÞ1=2
x0=2kBVU » 7Tx0=VU . (8.23)
We note that T
opt
N scales as the ratio x0/VU.
Furthermore, neglecting DRi we can readily show from Eq. (8.22) that the opti-
mum value of Q » x0(Li + Ls)/Ri is Qopt » ð1 þ Ls=LiÞ=k2
i , or
Qoptk2
ie » 1 . (8.24)
In general, for reasonably high values of Q this result implies that if necessary
one should add inductance to the input circuit to reduce kei. The resulting weak
11](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-33-320.jpg)
![8 SQUID Voltmeters and Amplifiers
coupling justifies the use of the bare SQUID parameters. Finally, the optimized
power gain at resonance, |Vo/Vi|2Ri/Rdyn (Vo is the output signal voltage), is found
from the square of Eq. (8.11) to be
G
opt
p » M2
i V2
U=RiRdyn » VU=x0 (8.25)
where we have set the dynamic resistance of the SQUID Rdyn » R. Combining
Eqs. (8.23) and (8.25), we find the gain noise temperature product
G
opt
p T
opt
N » 7T . (8.26)
Thus, high gain is synonymous with low noise temperature.
The results given in Eqs. (8.15)–(8.26) are at the resonant frequency. We observe
that the cross spectral density SVJ(f ) does not enter the noise temperature, for the
following reason. The noise voltage induced into the input circuit is in quadrature
with the current noise JN(t) in the SQUID that generates it. On resonance the
input circuit has a real impedance, so that the noise produced at the SQUID out-
put by this voltage noise is also in quadrature with that component of the output
noise produced by the circulating current. As a result, the cross-correlation term
vanishes. However, in principle one can obtain a lower noise temperature [17] by
operating the amplifier off resonance, so that the noise at the SQUID output pro-
duced by the voltage noise in the input circuit partially cancels the component
due to the current noise in the SQUID. It is straightforward to show that the mini-
mum noise temperature so obtained is given by replacing (SVSJ)1/2 in Eq. (8.23) by
ðSV SJ S2
VJÞ1=2
. However, in practice the gain of the amplifier may well be too
low to make operation off resonance realistic.
To give an example of the predicted performance of a tuned amplifier on reso-
nance, we assume L » 400 pH, R » 8 X, Li » 160 nH, k2
i = 0.7, bL = 1, VU » R/L »
2 1010 s–1, SV » 16 kBTR, SJ » 11 kBT/R and f0 = 100 MHz to find R
opt
i » 70 X,
G
opt
p » 15 dB and T
opt
N » T/5.
To conclude the discussion of the tuned amplifier, we note that the resistance
induced into the input circuit in practical devices is dominated by capacitive feed-
back. This effect modifies both the resonant frequency and the Q of the amplifier
[14]. This frequency shift is of the order of 1/ Q and thus is small for high values
of Q. The effect on Q, on the other hand, may be much larger, and could be a
factor of 2 for the numerical example considered above [14].
8.3.4
Untuned Amplifier: Theory
To represent an untuned amplifier, we set Lp = 1/jxCi = 0. The analysis proceeds
along the same lines as that for the tuned case. Thus, we set 1/jxCi = 0 in Eq.
(8.13) for V(x) and l/jxCi = Lp = 0 in Eq. (8.12) for Z*
TðxÞ. After some calculation,
we find the following expression for the noise temperature:
12](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-34-320.jpg)
![8.3 The SQUID as a Radiofrequency Amplifier
TNðf Þ =
Sr
V Z*
T
2
þ2Sr
VJk2
ieLRiVr
U ReðZ*
TÞ þ Sr
Jðk2
ieVr
ULRiÞ2
4kBRiM2
i ðVr
UÞ2 . (8.27)
To make progress, we neglect the terms k2
ieL/Lr and k2
ieLRi/LiRr in Z*
TðxÞ. These
are reasonable approximations since L/Lr £ 1/10 for U near U0/4 or 3U0/4, and
LRi/LiRr ~xL/R 1. We can now optimize TN(f ) with respect to Ri to find
R
opt
i ðf Þ » 2pf ðLi þ LsÞ 1 þ
2k2
ieLVr
USr
VJ
Sr
V
þ ðk2
ieLVr
UÞ2
Sr
J
Sr
V
#1=2
. (8.28)
The corresponding optimized noise temperature is
T
opt
N ðf Þ » 4p2f 2ðLi þ LsÞSr
V =2kBk2
ieLðVr
UÞ2
R
opt
i ðf Þ . (8.29)
In contrast to the tuned case, where ke 1 and we could replace the reduced
SQUID parameters with their corresponding bare parameters, we now need to
estimate the reduced parameters. We assume the same bare SQUID parameters
used for the example of a tuned amplifier in Section 8.3.3, and take Ls = 20 nH,
corresponding to the case of the particular amplifier discussed in Section 8.3.5.
We find k2
ie » 0.6 and hence br
L ¼ ð1 k2
ieÞbL » 0:4. From simulations we find
Vr
U » 2.5 1010 s–1, Sr
V » 18 kBTR, Sr
J » 12 kBT/R and Sr
VJ » 12 kBT [15, 16]. Insert-
ing these values into Eq. (8.28) we find R
opt
i » 0.7x(Li + Ls), which differs little
from the value for the tuned case provided Ls Li. From Eq. (8.29) we find T
opt
N »
0.6T, three times greater than the value for the tuned amplifier.
As a final remark, we note that the reduced SQUID parameters, even for a value
of k2
ie as large as 0.6, differ little from the bare parameters. Given that capacitive
feedback, which we have neglected, also modifies the gain and noise temperature
to some extent [14], it appears that the trouble of calculating the reduced parame-
ters is not really justified unless k2
ie becomes rather close to unity.
8.3.5
Tuned and Untuned Amplifiers: Experiment
Hilbert and Clarke [14] measured the gain and noise temperature of a SQUID
used as a tuned and an untuned amplifier. The configurations used to measure
the power gain Gp and noise temperature TN are shown in Figure 8.5. The output
of the SQUID, which was enclosed in a superconducting shield, was coupled to a
low-noise, room-temperature amplifier followed by a spectrum analyzer and a
power meter. To measure the gain, a calibrated signal was coupled to a cold
attenuator that presented an impedance Ri to the input coil. To measure the noise
temperature, the input coil was connected to a resistor Ri, the temperature Ti of
which could be raised above the bath temperature with a heater. The total noise
temperature referred to the input of the amplifier is
TT
N ¼ TN þ Ti þ TP
N=GP (8.30)
13](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-35-320.jpg)
![8 SQUID Voltmeters and Amplifiers
where TP
N is the noise temperature of the postamplifier. By measuring the output
noise power as a function of Ti one can obtain TN.
In the case of the tuned amplifier, the four-turn input coil with Li » 5.6 nH was
connected in series with a capacitor Ci » 20 pF and the source resistance Ri. The
measured resonant frequency was about 93 MHz and Q was about 45. The corre-
sponding optimized value of Ri (Eq. (8.22)) is about 2 X. The resonant frequency
implies a large stray inductance, Ls » 140 pH. Thus, from Eq. (8.8) we find k2
ie »
0.023, so that Qk2
ie » 1, as required by Eq. (8.24). The measured gain of 18.6 dB
was in quite good agreement with the predicted value (see Table 8.1). The mea-
sured output noise power as a function of Ti is shown in Figure 8.6, and leads to
TN = 1.7 – 0.5 K, slightly above the predicted value. The shift in resonant fre-
quency with flux bias was less than 1%, as expected given the low value of k2
ie. The
change in Q when the flux bias was changed from (n + 1/4)U0 to (n + 1/2)U0 was
substantially higher, about 25%, and demonstrated that the additional resistance
in the input circuit due to the SQUID was dominated by capacitive feedback.
14
Fig. 8.5 Circuit used to measure (a) the power gain and (b) the noise
temperature of an untuned SQUID amplifier. Components in the dashed
boxes are immersed in liquid helium. (Reproduced with permission
from ref. [14].)
Fig. 8.6 Total output noise power (arbitrary units) versus Ti at 93 MHz for
a tuned SQUID amplifier at 4.2 K. (Reproduced with permission from ref. [14].)](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-36-320.jpg)
![8.3 The SQUID as a Radiofrequency Amplifier
Table 8.1 Measured and predicted power gain Gp and noise
temperature TN for a dc SQUID radiofrequency amplifier.
Frequency
(MHz)
Gp (dB) TN (K)
Measured Predicted Measured Predicted
T = 4.2 K (tuned) 93 18.6 – 0.5 17 1.7 – 0.5 1.1
T = 1.5 K (untuned) 60 24.0 – 0.5 – 1.2 – 0.3 –
80 21.5 – 0.5 – 0.9 – 0.3 –
100 19.5 – 0.5 18.5 1.0 – 0.4 0.9
T = 4.2 K (untuned) 60 20.5 – 0.5 – 4.5 – 0.6 –
80 18.0 – 0.5 – 4.1 – 0.7 –
100 16.5 – 0.5 16.5 3.8 – 0.9 2.5
For the untuned amplifier, the SQUID had a 20-turn input coil, with Li » 120 nH
and Mi » 6 nH. The estimated value of k2
ie was about 0.6. At 80 MHz, the total
inductance Li + Ls » 140 nH was approximately optimized to a 50-X source impe-
dance. Figure 8.7 shows the gain versus frequency for positive and negative values
of VU. In each case, the gain drops by about 5 dB as the frequency is increased
from 10 to 100 MHz as a result of the increasing impedance of the input coil.
There is a resonance around 150 MHz that produces a dip in the gain for negative
VU and a peak for positive VU. This resonance corresponds to the self-resonance
of the input coil, and corresponds to a parasitic capacitance of about 8 pF. The
measured values of gain and noise temperature at three frequencies and two
temperatures are listed in Table 8.1, and are in quite good agreement with predic-
tions.
15
Fig. 8.7 Gain versus frequency for an untuned SQUID
amplifier for (a) negative and (b) positive VU. (Reproduced
with permission from ref. [14].)](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-37-320.jpg)
![8.3 The SQUID as a Radiofrequency Amplifier
of the SQUID is not too far from its value in the absence of the input circuit and
we may neglect the distinction between VU and Vr
U. These approximations enable
us to find the minimum detectable signal energy simply by setting Eq. (8.33)
equal to the mean square flux noise of the bare SQUID, 16kBTRdf/V2
U, to find
Euntuned
min » 32kBTdf =k2
i VU . (8.34)
We have set VU = R/L.
From Eqs. (8.32) and (8.34), we see that it is advantageous to use a tuned circuit
provided Etuned
min Euntuned
min , that is, with ki ~ 1,
f0Q VU=16p . (8.35)
For VU ~2 1010 s–1, this result implies that it is desirable to use a tuned circuit
only if f0Q 400 MHz. For frequencies of a few megahertz, this implies a tuned
circuit with Q of a few hundred, which is generally realistic. On the other hand,
for a frequency of (say) 10 kHz, it would be quite impracticable to tune – the re-
quired Q ~ 4 104 would imply a bandwidth of 0.25 Hz – and one should clearly
use an untuned circuit. The crossover from untuned to tuned circuits is likely to
occur around 1 MHz, at which frequency a more careful evaluation of the
untuned case would be warranted.
8.3.7
SQUID Series Array Amplifier
A very useful extension of the basic design of the dc SQUID is the SQUID series
array [24]. The array consists of Ns SQUIDs with their current leads connected in
series; Ns is typically 100. The SQUIDs are biased with a single current and the
voltage is measured across the entire array. The input coils to the SQUIDs are
similarly connected in series. Provided the flux in each SQUID is the same, the
value of VU is increased to NsVU, which is thus typically 5 mV/U0. Since the noise
voltages VN(f ) across the SQUIDs are incoherent, the voltage noise is increased
by the factor Ns
1/2 to Ns
1/2VN(f ). The large flux-to-voltage transfer coefficient
enables one to connect the array directly to a low-noise, room-temperature ampli-
fier while having the SQUID noise dominate over the amplifier noise.
The SQUID array can readily be used as an amplifier. Following the discussion
in Section 8.3.3, we can write the voltage noise induced into the input circuit as
–jxMiNs
1/2JN(x), where Ns
1/2JN(x) is the contribution of the Ns incoherent current
noise sources in the SQUIDs. The equivalent current noise in the input circuit
due to the voltage noise across the array is Ns
1/2VN(x)/Mi Ns VU. We observe that
the product of the current and noise sources in the input circuit is independent of
Ns. Consequently, for an array SQUID used as an amplifier with a tuned input
circuit, one expects the optimized noise temperature to be the same as for a single
SQUID, Eq. (8.23).
17](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-39-320.jpg)
![8 SQUID Voltmeters and Amplifiers
Huber et al. [25] fabricated SQUID arrays with 10, 30 and 100 SQUIDs. A novel
feature of their design was that the flux focusing washer of each SQUID was elec-
trically isolated from the SQUID itself. The relatively small capacitance between
the SQUID and washer was substantially less than that between the input coil
and the washer. Since these two capacitances are in series, the effective parasitic
capacitance between the input coil and the SQUID was substantially reduced,
largely eliminating resonances induced on the current–voltage characteristic. The
3-dB point of a 100-SQUID array was 120 MHz. The fact that no coupling network
is required between the array and the room-temperature amplifier makes this con-
figuration particularly appealing for radiofrequency applications.
8.3.8
The Quantum Limit
At signal frequencies greater than kBT/h, one expects quantum effects to become
important. For a device cooled in a dilution refrigerator to 20 mK, the correspond-
ing frequency is about 0.4 GHz so that the quantum limit is likely to be relevant
only to the microstrip SQUID amplifier (Section 8.4). We briefly discuss quantum
effects in Josephson junctions and SQUIDs.
In the case of a single resistively shunted junction (RSJ) (Section 2.1.1), quan-
tum effects become significant when hfJ kBT, where fJ = 2eV/h is the Josephson
frequency and V is the time-averaged voltage across the junction. In this limit,
one replaces the spectral density of the current noise in the shunt resistor with
(2hfJ/R) coth(hfJ/2kBT) [26, 27]. In the limit hfJ kBT this spectral density reduces
to 2hfJ/R, representing the zero-point fluctuations of an ensemble of harmonic
oscillators with random phases. This term can be observed by measuring the volt-
age noise across a junction at a frequency fm much less than fJ. This noise is pre-
dicted to have a spectral density
SV(fm) = [4kBT +
1
2
(I/I0)2 2eV coth (eV/kBT)] R2
dyn (8.36)
where I I0. The first term in brackets arises from noise generated at frequency
fm. The second term arises from noise generated at the Josephson frequency
2eV/h and mixed down to the measurement frequency by the nonlinearity of the
junction; the mixing coefficient is 1
2(I0/I)2. Equation (8.36) represents the solution
of a quantum Langevin equation in which one uses zero-point fluctuations in the
classical equation of motion for the RSJ (Eq. (2.22)). Koch et al. [28] measured the
voltage noise at a frequency of about 100 kHz for Josephson frequencies up to
about 500 GHz, and found good agreement with Eq. (8.36).
At sufficiently low temperatures and high signal frequencies, one would expect
quantum effects to become important in the dc SQUID. Unfortunately, since
SQUIDs are typically operated at a bias current rather close to the (non-noise-
rounded) value of their critical current, the quantum Langevin equation is of
dubious validity and one should instead undertake a full quantum mechanical
treatment of the SQUID as an amplifier. This problem turns out to be remarkably
18](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-40-320.jpg)
![8.3 The SQUID as a Radiofrequency Amplifier
challenging, and, to date, remains unsolved. Despite the questionable validity of
the quantum Langevin approach, a quarter of a century ago Koch et al. [29] carried
out a series of simulations for noise in a SQUID coupled to a tuned input circuit
to make an amplifier (as in Figure 8.2 with Lp = Ls = 0). They computed the spec-
tral densities SV(f ), SJ(f ) and SVJ(f ) at T = 0 assuming that the noise arose from
uncorrelated zero-point fluctuations in the two shunt resistors. As a figure of
merit they defined the quality n(i) (f )hf = S
ðiÞ
V (f )/4Ri as the mean photon energy in
the input circuit due to intrinsic SQUID noise; S
ðiÞ
V (f ) is the spectral density of the
voltage noise referred to the input terminals of the amplifier. When the amplifier
is operated off resonance (Section 8.3.3) to minimize the noise temperature, one
finds
n(i) = p[SV(f ) SJ(f ) – S2
VJ(f )] h/VU . (8.37)
For an optimized SQUID, Koch et al. [29] found n(i) » 1
2, that is to say, on the aver-
age the SQUID adds one half photon of noise to the input circuit. When this ener-
gy is added to the zero-point energy 1
2hf of the resonant circuit, the total noise of
the amplifier becomes hf, the result for any quantum-limited amplifier.
Since, as we shall see in Section 8.4.4, the microstrip SQUID amplifier is within
a factor of two of this quantum limit, it would be of great interest to perform a full
quantum mechanical treatment of the SQUID amplifier to find out whether or
not it is indeed strictly quantum-limited and to examine the validity of the quan-
tum Langevin approach.
8.3.9
Future Outlook
The theory and operation of single dc SQUIDs used as radiofrequency amplifiers
at frequencies up to about 100 MHz are well in hand. The measured gain and
noise temperature are in quite good agreement with predictions. However, a com-
plication in the understanding of these amplifiers is the presence of parasitic ca-
pacitance between the input coil and the SQUID washer. This capacitance not
only introduces resonance on the current–voltage characteristic but also partially
screens currents at the Josephson frequency from the input circuit, making it dif-
ficult to calculate the effects of mutual coupling between the SQUID and input
circuit with any precision (Section 8.3.2). The use of a design in which the flux-
focusing washer is electrically isolated from the SQUID [25] can substantially
reduce the parasitic capacitance between the SQUID and the input coil (Section
8.3.7). This decreases the magnitude of resonances on the current–voltage charac-
teristic and at the same time largely eliminates the interaction between the
SQUID and input circuit at the Josephson frequency. As a result, the effect of the
input circuit on the SQUID parameters may become relatively unimportant.
These effects are well worth exploring experimentally. The use of SQUID series
array amplifiers, with their substantially enhanced output signal voltage, allows
one to use a low-noise amplifier at room temperature without the need of an
19](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-41-320.jpg)
![8 SQUID Voltmeters and Amplifiers
impedance-matching network between them. This simplification is very appealing
for radiofrequency applications.
Although the theory of the SQUID amplifier in the classical regime is well
understood, the same cannot be said for the quantum limit. A theory that fully
accounts for the quantum mechanical nature of the SQUID is very much needed,
particularly to understand whether or not the SQUID is truly a quantum-limited
amplifier.
8.4
Microstrip SQUID Amplifier
8.4.1
Introduction
As we have seen in Section 8.3, the conventional square-washer SQUID config-
uration [6] can be operated as a low-noise amplifier at frequencies up to about
100 MHz. As the frequency is increased, however, parasitic capacitance between
the input coil and the square washer causes the gain to fall off to levels that are no
longer useful. This problem was addressed by Tarasov et al. [30] who made a four-
loop SQUID with input coils in series deposited inside the loops rather than on
top of the superconducting washer. As a result, the parasitic capacitance is
reduced, and the operating frequency range is substantially extended. For exam-
ple, in a tuned amplifier configuration a gain of nearly 20 dB was achieved at
420 MHz. The development of this amplifier was described in a subsequent series
of publications [31–35], and its frequency range increased by reducing the number
of SQUID loops to two. For an operating temperature of 4.2 K and a frequency of
3.65 GHz, a gain of (11 – 1) dB and a noise temperature of (4 – 1) K were achieved
[34]. A two-stage amplifier at the same frequency achieved a gain of (17.5 – 1) dB
[33]. This research is currently focused on developing an intermediate-frequency
amplifier to follow an SIS (superconductor–insulator–superconductor) mixer for
radio astronomy.
In an alternative approach, one makes a virtue of the capacitance between the
coil and the washer by using it to form a resonant microstrip [36]. The signal to be
amplified is applied between one end of the coil and the washer, while the other
end of the coil is left open. Provided that the source impedance is greater than the
characteristic impedance of the microstrip, there is a peak in the gain when a half
wavelength of the standing wave is approximately (but not exactly) equal to the
length of the coil. Gains of well over 20 dB and noise temperatures well below the
bath temperature can be achieved. However, as we shall see, the actual behavior of
the device differs markedly from that of a simple microstrip because the induc-
tance coupled into the input coil from the SQUID is generally substantially
greater than the intrinsic microstrip inductance.
20](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-42-320.jpg)
![8.4 Microstrip SQUID Amplifier
8.4.2
The Microstrip
A microstrip consists of a superconducting strip of width w separated from an
infinite superconducting sheet by an insulator with dielectric constant e and thick-
ness d. We assume that the thicknesses of the two superconductors are much
greater than the superconducting penetration depth k, and that w d. The capac-
itance and inductance per unit length of the microstrip are given by C0 = e e0w/d
(Fm–1) and L0 = (l0d/w)(1 + 2k/d) (H m–1) [37]. Here, e0 = 8.85 10–12 Fm–1 and
l0 = 4p 10–7 H m–1 are the permittivity and permeability of free space, and c =
1/(e0l0)1/2 = 3 108 m s–1 is the velocity of light in vacuum. The factor (1 + 2k/d)
accounts for the penetration of the magnetic field into the (identical) supercon-
ductors. The velocity of an electromagnetic wave on the microstrip is thus given
by
c
c ¼ c=½eð1 þ 2 =dÞ1=2
, and its characteristic impedance by
Z0 ¼
L0
C0
1=2
¼
d
w
l0ð1 þ 2k=dÞ
ee0
1=2
. (8.38)
The microstrip represents an electromagnetic resonator. For a microstrip of length
‘ with its two ends either open or terminated with resistances greater than Z0, the
fundamental frequency occurs when ‘ is equal to a half wavelength [37],
f0(L0) = c/2 ‘[e(1 + 2k/d)]1/2 . (8.39)
In this mode, the resonator is analogous to a parallel tuned circuit, and, for a resis-
tance Ri Z0 connected to one end, the quality factor of the resonance is given by
Q = pRi/2Z0.
8.4.3
The Microstrip SQUID Amplifier: Gain
The microstrip SQUID has the conventional square-washer configuration (Figure
8.8). However, in contrast to the conventional input scheme in which the signal is
connected to the two ends of the coil, the signal is instead coupled between one
end of the coil and the square washer, which provides the groundplane for the
microstrip. In an early set of experiments [36], the square washer had inner and
outer dimensions of 0.2 0.2 mm2 and 1 1 mm2, and the input coil had n = 31
turns, a width w = 5 lm and a length ‘ = 71 mm. Estimated parameters were
L » 320 pH, L i » 300 nH and Mi » 10 nH. The critical current and shunt resis-
tance per junction were typically 5 lA and 10 X, and the maximum value of VU
was about 60 lVU1
0 . At 4.2 K, the white flux noise measured in a flux-locked loop
at low frequencies was typically 2–3 lU0 Hz–1/2. We note that virtually the entire
length of the coil overlays the washer, which is at a uniform potential. The SQUID
chip was mounted on a printed circuit board, and each pad was wire bonded to a
50-X trace patterned on the board. The grounds of the input and output SMA con-
21](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-43-320.jpg)
![8 SQUID Voltmeters and Amplifiers
nectors were soldered to a groundplane on the reverse side of the board, and their
center conductors were soldered to the traces coupled to the input of the micro-
strip and to the output from the SQUID, respectively. The assembly was rigidly
mounted inside a superconducting box that eliminated fluctuations in the ambi-
ent magnetic field. Most measurements were made with the package immersed
in liquid 4He.
The circuit shown in Figure 8.9 was used to measure the gain; the input coil
over the square washer is shown as a distributed line. The current bias was sup-
plied by batteries. In later versions of the device, the flux bias was provided by a
directly coupled flux-locked loop that maintains the SQUID near its maximum
gain [38]. The loop rolls off at frequencies above a few kilohertz, and has no effect
on the high-frequency performance. A sweep oscillator was coupled to the micro-
strip via a room-temperature 100-dB attenuator and a cold 20-dB attenuator that
prevented noise produced by the generator from saturating the SQUID. The cold
attenuator also presented an impedance of 50 X to both the input coaxial line and
the microstrip. A second cold, 4-dB attenuator coupled the output of the SQUID
to a room-temperature postamplifier. The gain of the system excluding the
SQUID was calibrated by disconnecting the SQUID and connecting together the
input and output attenuators. All measurements of the gain of the SQUID ampli-
fier were referred to the baseline so obtained. Because the washer SQUID is an
asymmetric device – the two Josephson junctions are situated close together
rather than on opposite sides of the SQUID loop – one can either ground the
washer or ground the counter-electrode close to the Josephson junctions. Since
the washer acts as a groundplane for the input coil, at first sight it might seem
plausible to ground the washer. However, it is also possible to ground the counter-
electrode and have the washer at output potential. In the latter case, there is feed-
back from the output voltage generated on the washer to the input coil, via the
capacitance between them. If the sign of VU is such that the output voltage has
22
Fig. 8.8 Configuration of microstrip SQUID amplifier. The
input signal is connected between one end of the coil and the
square washer.](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-44-320.jpg)
![8.4 Microstrip SQUID Amplifier
the same sign as the input voltage, the feedback is positive; if the signs are oppo-
site, the feedback is negative. We designate the flux-to-voltage transfer coefficients
as Vþ
U and V
U , respectively.
Figure 8.10 shows the gain as a function of frequency for six devices with coil
lengths ranging from 98 mm to 3 mm. The two SQUID configurations are shown
in the figure: for the four longer coils the hole in the washer was 200 200 lm2
and the estimated inductance was 350 pH; for the two shorter coils, the hole was
10 200 lm2 and the estimated inductance was 90 pH. These inductances, the
number of turns on the coils and the length of the coils are listed in Table 8.2. The
peak gains achieved with SQUIDs 5 and 6 are lower than for the other four
devices because of the reduced mutual inductance between the input coil and the
washer.
Table 8.2 Measured and calculated frequency of the fundamental
resonance (half–wavelength) for six microstrip SQUIDs with
inductance L and coils of n turns and length ła.
SQUID L
(pH)
n ł
(mm)
f0
(MHz)
f calc
0 ðL0Þ
(MHz)
f calc
0 ðn2LÞ
(MHz)
1 350 40 98 105 500 91
2 350 15 23 370 2100 500
3 350 11 16 590 3000 820
4 350 6 8 1200 6200 2200
5 90 8 4 2200 12330 4400
6 90 7 3 2650 15000 5300
a The measured values f0 were obtained with a reverse gain techni-
que. The calculated value f calc
0 ðL0Þ was obtained from Eq. (8.39)
with e= 5.5, k = 0.15 lm and d = 0.4 lm. The calculated value
f calc
0 ðn2LÞ was obtained from Eq. (8.40) with C0 = 0.6 nFm–1.
23
B
Φ
Fig. 8.9 Circuit used to determine the gain of the microstrip SQUID, which is shown
with counter-electrode grounded. IB is the current bias, and IU provides the flux bias.
The input and output p-networks have attenuations of 20 dB and 5 dB, respectively.
(Reproduced with permission from ref. [36].)](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-45-320.jpg)
![8 SQUID Voltmeters and Amplifiers
In a more recent set of devices [39], the peak in the gain was moved to high
frequency by scaling down the hole in the SQUID to 10 200 lm2, and reducing
the dimensions of the coils. The gains achieved ranged from 12 – 1 dB at 2.2 GHz
to 6 – 1 dB at 7.4 GHz. The reduction in gain compared with devices operating at
lower frequencies was due largely to the lower mutual inductance between the
coil and the SQUID which, in turn, arose from the smaller SQUID inductance.
Nonetheless, it is encouraging that useful levels of gain can be achieved at fre-
quencies well into the gigahertz range.
It is important to realize that the peak in the gain is shifted to a frequency high-
er than the resonant frequency of the coil – typically by 30% – by positive feedback
[40]. The true resonant frequency f0 can be determined by a reverse gain technique
[40] as follows. With the SQUID washer grounded, a signal source is connected
across the SQUID and the signal transmitted through the SQUID into the coil
resonator is measured with a spectrum analyzer. The current flux biases are
adjusted to their usual operating values, that is, to produce maximum forward
gain. At the resonant frequency of the resonator, where ‘ = k/2, there is a mini-
mum in the observed power. This minimum arises from the asymmetric voltage
distribution on the resonator, which induces a positive current in one half of the
resonator and an equal, negative current in the other half. The measured values of
f0 are listed in Table 8.2.
Also listed in Table 8.2 are the resonant frequencies f0(L0) predicted by the
microstrip formula, Eq. (8.39), using e = 5.5, d = 0.4 lm and k = 0.15 lm. We see
that in all cases these exceed the measured value by a factor of roughly 5. Thus,
the input coil does not behave as a simple microstrip resonator. A much better
agreement between the measured and predicted resonant frequencies is obtained
by taking n2L/‘ as the inductance per unit length, instead of L0; here, we have
used n2L as the inductance of the input coil (Chapter 5). This assumption leads to
the predicted resonant frequency [40]
24
Fig. 8.10 Gain versus frequency for six microstrip SQUID
amplifiers in two different configurations with counter-
electrode grounded and Vþ
U . Numbers refer to coil lengths in
millimeters.](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-46-320.jpg)
![8.4 Microstrip SQUID Amplifier
f0(n2L) = 1/2n(‘LC0)1/2 . (8.40)
The resonant frequencies predicted by this model are also listed in Table 8.2. The
agreement with the measured values is acceptable for the longer coils, but
becomes progressively worse as the coil length is reduced. This trend may possi-
bly be explained by the fact that the parasitic inductance of the wiring to the chip
becomes progressively more important as the length of the coil is shortened.
The electromagnetic behavior of the microstrip SQUID amplifier has been stud-
ied using both an analog model and numerical calculations [40]. The analog
model confirmed the validity of Eq. (8.40) for the resonant frequency. The numer-
ical simulations, using both lumped circuit and distributed element models, were
used to study the effects of positive and negative feedback, corresponding to a
grounded electrode with transfer functions Vþ
U and V
U . These simulations agree
well with the observation that the peak in the gain occurs at a frequency above
and below the resonance, respectively. The same model was used to investigate
the input impedance of the microstrip, which is dominated by the complex impe-
dance of the SQUID. For Vþ
U the input resistance becomes negative below the k/2
resonance in the frequency range where the gain is high. Correspondingly, for V
U
the input resistance is negative above the k/2 resonance, where the gain is also
high. For low gains (above the resonance for V
U and below the resonance for Vþ
U ),
the input resistance is positive and the return loss is high. These simulations fol-
lowed the trends in measured values of the input impedance remarkably well.
In a further series of experiments, the harmonic distortion and intermodulation
distortion were investigated [41]. Biased for maximum gain, the microstrip
SQUID amplifier generates third harmonic signals with an amplitude in good
agreement with a model based on a sinusoidal flux-to-voltage transfer function.
The amplitude of the third harmonic is less than 1% of the fundamental for a flux
amplitude of 0.1 U0. However, under the same bias conditions, departures from a
sinusoidal transfer function produce a second harmonic signal. This signal can be
reduced or even eliminated by adjusting the flux bias empirically away from the
point of maximum gain. Similarly, the third- and fifth-order intermodulation
products can be non-negligible with the SQUID biased for maximum gain, but
one or the other can be reduced if the flux bias is adjusted appropriately.
We have seen that substantial levels of gain can be achieved with the microstrip
SQUID amplifier. However, the frequency at which the gain peaks is evidently
fixed by the length of the microstrip, whereas some applications demand tuneabil-
ity. Fortunately, one can tune the frequency quite simply by connecting a varactor
diode between the otherwise open end of the input coil and the washer [42]. The
capacitance of the diode can be varied by changing the value of the reverse bias
voltage. Changing the capacitance modifies the phase shift of the electromagnetic
wave when it is reflected, thereby increasing or decreasing the effective length of
the microstrip and lowering or raising the peak frequency.
Experiments were carried out with a GaAs varactor diode, the capacitance of
which could be varied from 1 to 10 pF by changing the bias voltage from 1 V to
–22 V. Two diodes in parallel were used to increase the tuning range. The diodes,
25](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-47-320.jpg)
![8 SQUID Voltmeters and Amplifiers
in series with a capacitor, were connected between the washer and the end of the
input coil not connected to the signal source. The gain for optimized current and
flux biases for a SQUID with 31 turns is shown in Figure 8.11 for 9 values of the
capacitance of the two diodes. We see that the peak frequency is progressively low-
ered, from 195 MHz to 117 MHz, as the capacitance is increased. The maximum
gain is constant to within 1 dB over this range. In the absence of the varactor, the
peak frequency is about 200 MHz. In fact, the presence of the varactors increases
the gain, most likely by increasing the degree of positive feedback. The depen-
dence of the peak frequency on the varactor capacitance is in reasonable agree-
ment with a simple model [42].
A potential concern is whether the varactor diode can introduce additional noise
into the amplifier. Estimates of the contributions of the Nyquist noise and shot
noise of the diode and of the noise on the bias voltage indicate that they should
not be significant. The measured noise temperatures of a particular device with
and without the varactor diodes were identical to within the uncertainties.
8.4.4
The Microstrip SQUID Amplifier: Noise Temperature
We turn now to the central issue, the noise temperature. An accurate way of mea-
suring TN is to increase the temperature T of the input load by means of a heater,
so that the resistor provides a well-defined source of Nyquist noise power. The
noise power at the output of the postamplifier is given by
PN(f) = kB(T + TN)RiGGP + kBTPRiGP , (8.41)
where G and GP are the (power) gains of the SQUID amplifier and postamplifier,
and TP is the noise temperature of the postamplifier. By measuring the output
26
Fig. 8.11 Tuning the microstrip SQUID amplifier. Gain versus frequency
for a 31-turn SQUID at 4.2 K for 9 values of reverse bias voltage applied to
the varactor diodes connected between the open end of the coil and the
washer. (Reproduced with permission from ref. [42].)](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-48-320.jpg)
![8.4 Microstrip SQUID Amplifier
noise power with a spectrum analyzer for several values of T, one can infer TN +
TP/G. Measuring TP separately using a similar method, one can deduce TN. This
discussion also makes it clear that one requires TP/G TN to ensure that the post-
amplifier noise does not contribute significantly to the system noise temperature.
In preliminary experiments, the postamplifier was at room temperature and
had a noise temperature of abut 80 K. Thus, with a typical SQUID power gain of
200 at the peak, the postamplifier contributed a noise temperature of about 0.4 K.
Subsequent experiments made use of a single-stage postamplifier using a hetero-
junction field effect transistor (HFET – Fujitsu FHX 13LG) operated in the 4He
bath. This postamplifier had a noise temperature of 10 – 1.5 K. Figure 8.12 shows
the system noise temperature versus frequency for a device with a peak frequency
near 365 MHz, cooled to 1.8 K. The peak gain of the microstrip SQUID was 24.5
– 0.5 dB. The noise temperature increases rapidly as the frequency moves away
from the resonance; this effect is due largely to the resonant network used to cou-
ple the SQUID to the HFET. The minimum system noise temperature is 0.28 –
0.06 K, to which the postamplifier contributes 0.09 – 0.02 K. Thus, the intrinsic
noise temperature is 0.19 – 0.06 K, an order of magnitude lower than the bath
temperature. The noise temperature is also an order of magnitude lower than that
of state-of-the-art, cooled semiconductor amplifiers [43].
An alternative way to determine TN is to couple the input of the microstrip to a
tuned circuit, as shown in Figure 8.13 [44]. The tuned circuit consisted of a 1 pF
capacitor and a four-turn copper coil, about 4 mm in diameter, inductively coupled
to the microstrip by means of a loop of wire. The resonant frequency was about
438 MHz. In a separate experiment, the loop was connected to a 50-X cable, and
the distance between the coil and the loop was adjusted to produce a 2-dB loss,
thus reducing Q by a factor of about two. To determine the gain and Q, a second
loop, with a coupling loss of 10 dB to the tuned circuit, was connected to a signal
generator via a cold 20-dB attenuator and a stainless steel cable with a loss of 3 dB.
The measured transmitted power at 4.2 K is shown in Figure 8.14 (upper trace).
27
Fig. 8.12 Noise temperature for a microstrip SQUID amplifier with 29-turn
spiral input coil. The device and its HFET postamplifier were cooled to 1.8 K.
(Reproduced with permission from ref. [44].)](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-49-320.jpg)
![8 SQUID Voltmeters and Amplifiers
The Q is about 270 and the maximum gain is 22 dB. The gain is referred to the
input coupling loop of the resonant circuit, and includes an added 12 dB to
account for the coupling losses.
The noise generated by the resonant circuit, with the generator disconnected, is
shown in the lower trace of Figure 8.14. The measured peak is 4.7 dB above the
(nearly white) noise at 432 MHz. This peak contains contributions from the
Nyquist noise of the resonant circuit and from the system noise of the SQUID
amplifier. Although it is not entirely straightforward to separate these contribu-
tions, using the following approximate argument one can show that this result is
consistent with the measured noise temperature. On resonance, since the micro-
strip reduces Q of the resonant circuit to approximately half of its unloaded value,
the source impedance presented to the microstrip is roughly equal to the charac-
teristic impedance of the microstrip. At frequencies well below resonance, the
magnitude of the source impedance is approximately xL‘, where L‘ is the induc-
tance of the coupling loop. Estimating L‘ ~ 10 nH, we find xL‘ ~25 X at 400 MHz.
Since TN T, to a first approximation one can ignore any variation in the noise
power with source impedance. Thus, referred to the input of the preamplifier, the
28
Fig. 8.13 Configuration of circuit used to detect Nyquist noise in a
resonant circuit inductively coupled to the input of the microstrip
amplifier. (Reproduced with permission from ref. [44].)
Fig. 8.14 Microstrip SQUID amplifier coupled to the resonant circuit of
Figure 8.13. Transmitted power (referred to the loop that couples signal into
the resonant circuit) and relative noise power with no input signal at 4.2 K.
(Reproduced with permission from ref. [44].)](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-50-320.jpg)
![8.4 Microstrip SQUID Amplifier
total noise on resonance can be characterized by the temperature [G¢(T/1.58) +
G¢TN + TP]; off-resonance, where we assume the noise from the resonant circuit to
be negligible, the corresponding temperature is (G¢TN + TP). Here G¢ = 16 dB is
the gain of the microstrip SQUID amplifier reduced by the 4-dB loss in the
attenuator coupling it to the HFET, and Tp = 15 K is the measured noise tempera-
ture of the HFET of 438 MHz. The factor of 1/1.58 accounts for the 2-dB loss be-
tween the resonant circuit and the input to the microstrip SQUID amplifier. We
can deduce TS = TN + TP/G¢ from the relation 10 log10 {[G¢(T/1.58) + G¢TS]/ G¢TS}
= 4.7, and find TS = 1.4 K. For the given values of TP and G¢ we calculate TN = 1 K.
It is evident from Eq. (8.23) that TN should scale with the bath temperature.
Thus, even lower noise temperatures should be possible with the device cooled to
millikelvin temperatures in a dilution refrigerator. However, the contribution of
the HFET amplifier, about 0.1 K, would then become dominant. To circumvent
this problem, a second microstrip SQUID amplifier was used as a postamplifier
[44]. If necessary, the peak frequency of the second SQUID could be tuned with a
varactor diode to coincide with that of the first. To prevent the two SQUIDs from
interacting with each other, it was necessary to separate them with an attenuating
network. For a particular pair of SQUIDs at 4.2 K, the maximum gain was 33.5 –
1 dB.
Such a system was cooled in a dilution refrigerator [45]. The two peak frequen-
cies were made to coincide by means of small changes in the bias fluxes. All leads
connected directly to the SQUIDs were very heavily filtered over a wide frequency
range using a combination of lumped circuit and copper powder filters, and a
superconducting shield surrounding the SQUIDs eliminated ambient magnetic
field fluctuations. The overall gain of the two SQUIDs at 538 MHz was 30 – 1 dB
and 32 – 1 dB at 4.2 K and 100 mK, respectively. A third stage of amplification
was provided by an HFET with a resonant input circuit, cooled to 4.2 K and con-
nected to the SQUID postamplifier via a cryogenic cable with a loss of 6 dB. At
550 MHz, the gain of the HFET was 22 – 1 dB, and its noise temperature TP about
6 K. At the lowest temperatures, the cable loss reduced the effective gain G¢ of the
two SQUIDs to 26 – 1 dB so that the HFET contributed a noise temperature TP/G¢
» 15 mK referred to the input of the first SQUID. Two different input circuits
were used to measure the noise: one involved measuring the signal-to-noise ratio
in the presence of an accurately known signal, and the other measuring the noise
from a resonant circuit. The two methods gave nearly identical results, and we
describe only the second.
During the measurement, a small (about –140 dBm) signal was applied to the
tuned circuit, via the coupling loop, at a frequency about 2 MHz above its
519 MHz resonant frequency. This signal was used to optimize the bias currents
and fluxes of the two SQUIDs for maximum gain and optimum signal-to-noise
ratio. The peak was monitored throughout the measurements to verify that the
gain did not drift. The inset in Figure 8.15 shows this peak, together with the
noise peak from the resonant circuit. The value of TN is extracted from the peak
using the method described above, with the Nyquist noise power replaced by
(hf/2kB)coth(hf/2kBT). Figure 8.15 shows the inferred values of TN versus T. The
29](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-51-320.jpg)
![8 SQUID Voltmeters and Amplifiers
error bars are determined solely by the uncertainty in the spectrum analyzer mea-
surement. We see that TN scales as T above about 150 mK, and flattens off at tem-
peratures below about 70 mK to 47 – 10 mK; by comparison, the quantum-limited
noise temperature, TQ » hf/kB, is about 25 mK.
A potential source of the low-temperature saturation of TN is hot electrons pro-
duced in the resistive shunts by bias current heating. Wellstood et al. [46] obtained
remarkably similar results for the noise spectral density measured at frequencies
below about 50 kHz in SQUIDs cooled to around 20 mK, and found good agree-
ment with a model in which the temperature of the electrons is determined by
their coupling to the phonons. To investigate whether hot electrons were indeed
responsible for the saturation in Figure 8.15, Mck et al. [45] remeasured the noise
of the same SQUID at 140 kHz, where TQ 10 lK. For this experiment, the out-
put of the first SQUID was coupled to the two ends of the input coil of a second
SQUID, via a superconducting transformer with a current gain of about 3. The
output of the second SQUID was coupled to a room-temperature amplifier with a
noise temperature of about 3 K. The temperature dependence of the noise energy
at 140 kHz mimicked that plotted in Figure 8.15 quite remarkably, leveling off to
7.5 – 1.3 at low temperatures, thus providing strong evidence that the low-tem-
perature saturation of TN indeed arose from hot electrons.
30
Fig. 8.15 Noise temperature of input micro-
strip SQUID at 519 MHz versus temperature
measured with a resonant source. The dashed
line through the data corresponds to TN T,
and the dot-dash line indicates TQ = hf/kB
» 25 mK. The inset shows the noise peak pro-
duced by an LC-tuned circuit at 20 mK. The
peak at 520.4 MHz is a calibrating signal.
(Reproduced with permission from ref. [45].)](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-52-320.jpg)
![8.4 Microstrip SQUID Amplifier
8.4.5
High-Tc Microstrip SQUID Amplifier
Tarasov and coworkers [47–50] fabricated and tested microstrip SQUID amplifiers
involving YBCO SQUIDs with grain boundary junctions. The spiral microstrip
was made either of YBCO, deposited on a separate substrate and coupled to the
SQUID in a flip-chip arrangement, or of Au, deposited on the SQUID washer
with an intervening layer of insulator. The maximum gain of these devices to date
is a few decibels at frequencies of around 1 GHz. However, it is possible that such
amplifiers operated at low temperature could achieve very high levels of gain. For
example, Tarasov and coworkers [49, 50] fabricated YBCO devices with 0.5 lm
grain boundary junctions, and achieved values of I0Rn as high as 8 mV and values
of VU as high as 1 mVU0
–1 at 20 K. These values are substantially higher than
those achieved with low-Tc devices, where the external resistive shunts drastically
reduce the values of I0R that can be achieved. Since the relatively high levels of
l/f noise observed in high-Tc SQUIDs (Chapter 5) are not an issue at high frequen-
cies, the high-Tc microstrip amplifier is worthy of further investigation for use at
or below 4He temperatures.
8.4.6
Future Outlook
Several challenges remain for the microstrip SQUID amplifier. Although substan-
tial levels of gain have been achieved at frequencies up to 3 or 4 GHz [39], one
would like to extend this frequency range up to (say) 10 GHz. Another interesting
endeavor would be the development of high-Tc SQUIDs with low critical current
– and correspondingly high resistance – at 4.2 K. Such SQUIDs might well have
substantially higher values of I0Rn than resistively shunted low-Tc SQUIDs, and
thus higher levels of gain as microstrip amplifiers. Perhaps the biggest challenge,
however, is to achieve the quantum-limited noise temperature at frequencies
of 0.5–1 GHz. The lowest noise temperature yet achieved [45], 47 – 10 mK at
0.519 GHz, was in fact limited by hot electrons in the resistive shunts [46]. It
would thus be of great interest to reduce the hot electron temperature by means
of cooling fins to attempt to attain the quantum limit.
The current understanding of the operation of the microstrip SQUID amplifier
[40] is largely empirical. Although this model is adequate for the design of new
devices, it would be a significant advance if a first-principles theory were to be de-
veloped that would enable one to calculate with reasonable accuracy the gain, fre-
quency response and input impedance.
The original motivation for the development of the microstrip SQUID amplifier
was to improve the performance of an axion detector (Section 8.7). However,
another intriguing application is as a postamplifier for the radiofrequency single-
electron transistor (RFSET) [51]. This device is a charge detector typically operat-
ing at frequencies of several hundred megahertz. Although potentially very sensi-
tive, the RFSET has relatively low gain and its noise temperature is generally lim-
31](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-53-320.jpg)
![8 SQUID Voltmeters and Amplifiers
ited by noise in the HFET to which its output is coupled. It appears likely that the
use of a microstrip SQUID as a postamplifier will enable the RFSET to reach the
quantum limit of charge detection [52], and several groups are working towards
this goal.
8.5
SQUID Readout of Thermal Detectors
8.5.1
Introduction
The detection of electromagnetic waves plays an important role in science, tech-
nology and everyday life. Several types of cooled sensor for the detection of milli-
meter-wave to X-ray radiation are being developed which use a SQUID readout.
The development of these sensors is strongly motivated by the requirements for
astrophysical observations. A particularly promising development is the use of the
voltage-biased superconducting transition-edge sensor (TES) for thermal detectors
[53]. Several types of readout multiplexer for TESs are under development with
the near-term goal of producing imaging arrays with 104 elements. In the follow-
ing sections, we discuss the operation of the voltage-biased TES and the require-
ments for the SQUID readout, including SQUID output multiplexers. We then
describe the application, design, and performance of far-infrared to millimeter-
wave bolometers and X-ray calorimeters. These devices have been the focus of
intensive recent development. Finally, we briefly describe several other types of
sensor that use SQUID readouts, including magnetic calorimeters, SIS tunnel
junction sensors, normal–insulator–superconductor (NIS) tunnel junction sen-
sors and kinetic inductance sensors.
In a thermal detector, a photon absorbing layer is attached to a low-temperature
heat sink by a weak thermal conductance. A thermistor, such as a TES, is attached
to the absorber which senses the energy released by one (or more) absorbed
photon(s). At far-infrared to millimeter wavelengths, TES thermal detectors are
used to measure the rate of arrival of photons. In this mode, the characteristic
time for energy deposition is long compared with the thermal relaxation time, so
that the temperature rise is proportional to the absorbed power. Thermal detectors
operated in this mode are called bolometers. Arrays of TES bolometers are being
developed for ground-based, airborne, balloon-borne and space telescopes for use
at millimeter to far-infrared wavelengths.
From near-infrared to X-ray wavelengths, the temperature pulse from a single
absorbed photon is detected. The characteristic time for energy deposition is short
compared with the thermal relaxation time of the device, so that the height of the
temperature pulse is proportional to the total energy of the absorbed photon.
Thermal detectors operated in this mode are called calorimeters. They are used to
measure the spectrum of a source as well as its flux. Individual TES calorimeters
are used for laboratory X-ray spectroscopy. Arrays of TES calorimeters are being
32](https://image.slidesharecdn.com/2156168-250607115511-eccfc3ab/85/The-Squid-Handbook-Applications-Of-Squids-And-Squid-Systems-Volume-Ii-John-Clarke-54-320.jpg)
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The Squid Handbook Applications Of Squids And Squid Systems Volume Ii John Clarke
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- 5. J. Clarke, A. I. Braginski (Eds.) The SQUID Handbook. Vol. II: Applications of SQUIDs and SQUID Systems. John Clarke and Alex I. Braginski (Eds.) Copyright 2006 WILEY-VCH Verlag GmbH Co. KGaA, Weinheim ISBN: 3-527-40408-2 The SQUID Handbook Vol. II
- 6. Related Titles Buckel, W., Kleiner, R. Superconductivity Fundamentals and Applications Second Edition 475 pages with approx. 247 figures 2004 Hardcover ISBN 3-527-40349-3 Andr, W., Nowak, H. (eds.) Magnetism in Medicine A Handbook Second Edition 550 pages with 155 figures and 11 tables 2006 Hardcover ISBN 3-527-40558-5
- 7. John Clarke, Alex I. Braginski (Eds.) The SQUID Handbook Vol. II Applications of SQUIDs and SQUID Systems
- 8. The Editors Prof. John Clarke Department of Physics 366 LeConte Hall University of California Berkeley, CA 94720-7300 USA and Materials Science Division Lawrence Berkeley National Laboratory One Cyclotron Road Berkeley, CA 94720 jclarke@berkeley.edu Prof. Dr. Alex I. Braginski Research Center Jlich IBN-2 D-52425 Jlich Germany a.braginski@fz-juelich.de All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de. 2006 WILEY-VCH Verlag GmbH Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Typesetting Khn Weyh, Satz und Medien, Freiburg Printing Strauss GmbH, Mrlenbach Bookbinding Litges Dopf Buchbinderei GmbH, Heppenheim Printed in the Federal Republic of Germany. Printed on acid-free paper. ISBN-13: 978-3-527-40408-7 ISBN-10: 3-527-40408-2
- 9. This Handbook is dedicated to the memory of Robin P. Giffard, Christoph Heiden and James E. Zimmerman.
- 11. VII Volume I Preface XI 1 Introduction 1 1.1 The Beginning 2 1.2 Subsequent Developments 5 1.3 The dc SQUID: A First Look 7 1.4 The rf SQUID: A First Look 12 1.5 Cryogenics and Systems 16 1.6 Instruments: Amplifiers, Magnetometers and Gradiometers 17 1.7 Applications 21 1.8 Challenges and Perspectives 24 1.9 Acknowledgment 26 2 SQUID Theory 29 2.1 Josephson Junctions 30 2.2 Theory of the dc SQUID 43 2.3 Theory of the rf SQUID 70 3 SQUID Fabrication Technology 93 3.1 Junction Electrode Materials and Tunnel Barriers 94 3.2 Low-temperature SQUID Devices 96 3.3 High-temperature SQUID Devices 107 3.4 Future Trends 118 4 SQUID Electronics 127 4.1 General 128 4.2 Basic Principle of a Flux-locked Loop 128 4.3 The dc SQUID Readout 137 4.4 The rf SQUID Readout 155 4.5 Trends in SQUID Electronics 165 Contents The SQUID Handbook. Vol. II: Applications of SQUIDs and SQUID Systems. John Clarke and Alex I. Braginski (Eds.) Copyright 2006 WILEY-VCH Verlag GmbH Co. KGaA, Weinheim ISBN: 3-527-40408-2
- 12. VIII 5 Practical DC SQUIDS: Configuration and Performance 171 5.1 Introduction 172 5.2 Basic dc SQUID Design 175 5.3 Magnetometers 186 5.4 Gradiometers 193 5.5 1/f Noise and Operation in Ambient Field 200 5.6 Other Performance Degrading Effects 208 6 Practical RF SQUIDs: Configuration and Performance 219 6.1 Introduction 220 6.2 Rf SQUID Magnetometers 220 6.3 Rf SQUID Gradiometers 236 6.4 Low-Frequency Excess Noise in rf SQUIDs 237 6.5 Response of rf SQUIDs to High-frequency Electromagnetic Interference 239 6.6 Characterization and Adjustment of rf SQUIDs 241 6.7 The rf SQUID versus the dc SQUID 244 6.8 Concluding Remarks and Outlook 246 7 SQUID System Issues 251 7.1 Introduction 254 7.2 Cryogenics 255 7.3 Cabling and Electronics 272 7.4 Data Acquisition and Rudimentary Signal Processing 289 7.5 Characterization, Calibration and Testing 292 7.6 Conditions Imposed on SQUID Systems by the Environment and Applications 309 7.7 Noise Suppression 315 7.8 Signal and Noise Implications for the SQUID System Design 335 7.9 Concluding Remarks and System Trends 344 Appendix 1 357 Basic Properties of Superconductivity Appendix 2 367 Abbreviations, Constants and Symbols Index 383 Contents
- 13. IX Volume II Preface XI List of Contributors XV 8 SQUID Voltmeters and Amplifiers 1 J. Clarke, A. T. Lee, M. Mck and P. L. Richards 8.1 Introduction 3 8.2 Voltmeters 4 8.3 The SQUID as a Radiofrequency Amplifier 5 8.4 Microstrip SQUID Amplifier 20 8.5 SQUID Readout of Thermal Detectors 32 8.6 Nuclear Magnetic and Quadrupole Resonance and Magnetic Resonance Imaging 56 8.7 The Axion Detector 81 9 SQUIDs for Standards and Metrology 95 J. Gallop and F. Piquemal 9.1 Introduction 96 9.2 SQUIDs in Voltage Metrology 97 9.3 Cryogenic Current Comparator (CCC) 101 9.4 Other Current Metrological Applications of SQUIDs 123 9.5 Future Trends and Conclusion 129 10 The Magnetic Inverse Problem 139 E. A. Lima, A. Irimia and J. P. Wikswo 10.1 The Peculiarities of the Magnetic Inverse Problem 141 10.2 The Magnetic Forward Problem 145 10.3 The Magnetic Inverse Problem 168 10.4 Conclusions 254 11 Biomagnetism 269 J. Vrba, J. Nenonen and L. Trahms 11.1 Introduction 271 11.2 Magnetoencephalography 274 11.3 Magnetocardiography 321 11.4 Quasistatic Field Magnetometry 342 11.5 Magnetoneurography 346 11.6 Liver Susceptometry 351 11.7 Gastromagnetometry 356 11.8 Magnetic Relaxation Immunoassays 360 Contents
- 14. 12 Measurements of Magnetism and Magnetic Properties of Matter 391 R. C. Black and F. C. Wellstood 12.1 Introduction 392 12.2 The SQUID Magnetometer–Susceptometer 392 12.3 Scanning SQUID Microscopy 409 13 Nondestructive Evaluation of Materials and Structures using SQUIDs 441 H.-J. Krause and G. Donaldson 13.1 Introduction 442 13.2 Detection of Magnetic Moments 445 13.3 Magnetic Flux Leakage Technique 448 13.4 Static Current Distribution Mapping 452 13.5 Eddy Current Technique 453 13.6 Alternative Excitation Techniques 467 13.7 Conclusion and Prospects 472 14 SQUIDs for Geophysical Survey and Magnetic Anomaly Detection 481 T. R. Clem, C. P. Foley, M. N. Keene 14.1 Introduction 483 14.2 Magnetic Measurements in the Earth’s Field 484 14.3 Operation of SQUIDs in Real World Environments 494 14.4 Data Acquisition and Signal Processing 499 14.5 Geophysical Applications of SQUIDs 504 14.6 Magnetic Anomaly Detection Systems using SQUIDs 527 14.7 Future Prospects 536 15 Gravity and Motion Sensors 545 Ho J. Paik 15.1 Introduction 546 15.2 The Superconducting Accelerometer 547 15.3 Superconducting Transducer for Gravitational-Wave Detectors 548 15.4 Superconducting Gravity Gradiometers (SGGs) 554 15.5 Applications of the SGG Technology 563 15.6 Outlook 575 Appendix 581 Physical Constants, Abbreviations and Symbols Index 617 Contents X
- 15. XI We hope that this two-volume Handbook will provide an in-depth, systematic treatment of Superconducting QUantum Interference Devices (SQUIDs) and their many applications. Our intent is to offer the reader a reasonably complete, balanced and up-to-date presentation of the entire field, with as few omissions and duplications as possible. Although our publisher initially suggested that one or two of us write the Handbook, we pointed out that the field had become so large and diverse that this would be an almost impossible undertaking. Many aspects of SQUIDs, especially applications, have become so specialized that no single person can realistically provide adequate coverage. Consequently, we invited various colleagues collectively to write a comprehensive treatise. Fortu- nately, virtually everyone we asked graciously agreed to participate. The first volume of the Handbook, published in 2004, contained seven chapters devoted to the fundamental science, fabrication and operation of low-Tc and high- Tc, dc and rf SQUIDs. After an introductory overview, subsequent chapters were entitled SQUID Theory, SQUID Fabrication Technology, SQUID Electronics, Practical DC SQUIDs: Configuration and Performance, Practical RF SQUIDs: Configuration and Performance, and SQUID System Issues. Appendix 1 briefly described the Basic Properties of Superconductivity and Appendix 2 listed the acronyms and symbols used in the Handbook. Volume II contains eight chapters concerned with applications using SQUIDs as sensors and readout devices. In Chapter 8, Clarke, Lee, Mck and Richards describe the theory and imple- mentation of SQUID voltmeters and amplifiers. The first sections describe mea- surements of quasistatic voltages, the use of the dc SQUID as a radiofrequency amplifier, and the extension of the frequency range into the microwave regime by means of a microstrip input circuit. Subsequently, the application of SQUIDs to read out thermal detectors and their multiplexing in the time- and frequency- domains are discussed. SQUID amplifiers for nuclear magnetic resonance and magnetic resonance imaging are reviewed, and various examples are presented. The chapter concludes with a brief discussion of the implementation of a near- quantum-limited SQUID amplifier on a detector to search for the axion, a candi- date for cold dark matter. Preface The SQUID Handbook. Vol. II: Applications of SQUIDs and SQUID Systems. John Clarke and Alex I. Braginski (Eds.) Copyright 2006 WILEY-VCH Verlag GmbH Co. KGaA, Weinheim ISBN: 3-527-40408-2
- 16. XII In Chapter 9, Gallop and Piquemal describe the role of SQUIDs in standards and metrology. After a brief discussion of highly accurate voltage measurement, the authors focus on the principles and accuracy limits of the cryogenic current comparator (CCC). Among its applications are measurements of resistance ratios, very low currents from superconducting electron transistors, and currents in beams of charged particles. Other metrology applications include secondary ther- mometers based on magnetic susceptibility and resistance, and a primary thermo- meter based on Nyquist noise. In Chapter 10, Lima, Irimia and Wikswo tackle the magnetic inverse problem that is central to interpreting measurements in biomagnetism, geophysics and nondestructive evaluation. They first describe the forward problem – the determi- nation of magnetic fields produced by distributions of magnetization and current and by multipoles. They begin their discussion of the inverse problem with the law of Biot and Savart, and go on to discuss the imaging of distributions of mag- netization. An important aspect of the inverse problem is “silent sources” – for example, source configurations that produce either an electric or a magnetic field but not both. They conclude with a treatment of the three-dimensional inverse problem – which, in general, has no unique solution – that highlights some of the most widely used algorithms. In Chapter 11, Vrba, Nenonen and Trahms address biomagnetism, unquestion- ably the largest single consumer of SQUIDs. They begin with magnetoencephalo- graphy (MEG) – magnetic signals from the brain – and describe whole cortex sys- tems, types of sensors, fetal MEG, and data analysis with clinical examples. They continue with magnetocardiography, describing the kinds of instrumentation, types of sensors, and clinical applications. There follows a miscellany of topics in biomagnetism, including the measurement of static fields from the body, detect- ing signals propagating along nerves, the susceptibility of the liver as a diagnostic tool, gastro-magnetometery, and immunoassay using magnetic labeling of cells. In Chapter 12, Black and Wellstood describe measurements of magnetism and magnetic properties of matter. The first part describes the history, development and operation of the most widely used SQUID system, namely a commercially available magnetometer and susceptometer. Issues of accuracy and sensitivity are discussed. The second part of the chapter is concerned with the scanning SQUID microscope. The authors outline the special requirements for the SQUIDs and cryogenics, describe the techniques for scanning and image processing, and dis- cuss issues of spatial resolution. They conclude with a review of current and potential applications. In Chapter 13, Krause and Donaldson give an overview of methods for nondes- tructive evaluation. These include the detection of static magnetic moments, the magnetic flux leakage technique, static current distribution mapping, and the eddy current technique. A number of examples is presented. The chapter con- cludes with a brief discussion of alternative ways of exciting a magnetic response. In Chapter 14, Clem, Foley and Keene describe the application of SQUIDs to geophysical survey and magnetic anomaly detection. They begin with issues of magnetic measurements in the presence of the Earth’s field and operating Preface
- 17. XIII SQUIDs in harsh environments, and continue with data acquisition and signal processing. A major portion of the chapter is concerned with geophysical applica- tions, ranging from rock magnetometry to a variety of prospecting and surveying methods. They conclude with an overview of the detection of magnetic anomalies, for example, buried ordnance. Finally, in Chapter 15, Paik addresses gravity and motion sensors. He describes in turn a superconducting accelerometer, a superconducting transducer for gravi- tational-wave detectors, and the superconducting gravity gradiometer (SGG). Ap- plications of the SGG include precision tests of the laws of gravity, searching for new weak forces, gravity mapping and mass detection, and inertial navigation and survey. In the Appendix, we duplicate Appendix 2 of Volume I and provide a list of addi- tional acronyms and symbols for each chapter of Volume II. This very brief survey illustrates the remarkable diversity of the SQUID, which finds applications to physics, astrophysics, cosmology, chemistry, materials science, standards, biology and medicine. We would like to believe that the Hand- book will be of use not only to practitioners of the art of SQUIDs but also to stu- dents and professionals working in these fields. In conclusion, we express our heartfelt thanks to the authors of both volumes of the Handbook for their hard work, their attention to quality and accuracy and not least for their patience and perseverance during our editing of their manuscripts. One of us (JC) expresses his grateful thanks to his assistant, Barbara Salisbury, for her unflagging help with all the manuscripts for both volumes. We owe an enor- mous debt of gratitude to the staff at Wiley-VCH, particularly to Dr. Michael Br, who first asked us to co-write the Handbook, and to Mrs. Vera Palmer and Mrs. Ulrike Werner without whose expert guidance and extraordinary patience the Handbook would never have seen the light of day. Finally, we thank our wives Maria Teresa and Grethe for their patience and understanding during our editing of both volumes of the Handbook, which took much of our time away from them. Alex Braginski and John Clarke Preface
- 19. XV Volume I Alex I. Braginski (Chapters 1 and 6) Research Centre Jlich, ISG-2, D-52425 Jlich, Germany, (retired), and Physics Department, University of Wuppertal, 42097 Wuppertal, Germany A.Braginski@fz-juelich.de Robin Cantor (Chapters 3 and 5) STAR Cryoelectronics, 25-A Bisbee Court, NM 87508 Santa Fe, USA rcantor@starcryo.com Boris Chesca (Chapter 2) Institute of Physics, University of Tbingen, Auf der Morgenstelle 14, 72076 Tbingen, Germany boris.chesca@uni-tuebingen.de John Clarke (Chapter 1) Department of Physics, 366 LeConte Hall, University of California, Berkeley CA 94720-7300, USA, and Materials Sciences Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley CA 94720, USA jclarke@berkeley.edu Dietmar Drung (Chapter 4, Appendix 2) Physikalisch-Technische Bundesanstalt, Abbestrasse 2–12, 10587 Berlin, Germany Dietmar.Drung@ptb.de Catherine P. Foley (Chapter 7) CSIRO Industrial Physics, P.O. Box 218, Lindfield, NSW 2070 Australia Cathy.Foley@csiro.au Mark N. Keene (Chapter 7) QinetiQ Ltd., St. Andrews Road, Malvern, Worcestershire WR14 3PS, UK mnkeene@qinetiq.com Reinhold Kleiner (Chapter 2, Appendix 1) Institute of Physics, University of Tbingen, Auf der Morgenstelle 14, 72076 Tbingen, Germany kleiner@uni-tuebingen.de List of Contributors The SQUID Handbook. Vol. II: Applications of SQUIDs and SQUID Systems. John Clarke and Alex I. Braginski (Eds.) Copyright 2006 WILEY-VCH Verlag GmbH Co. KGaA, Weinheim ISBN: 3-527-40408-2
- 20. XVI List of Contributors Dieter Koelle (Chapter 2 and 5 and Appendices 1 and 2) Institute of Physics, University of Tbingen, Auf der Morgenstelle 14, 72076 Tbingen, Germany koelle@uni-tuebingen.de Frank Ludwig (Chapter 3) Institute of Electrical Metrology and Electrical Engineering, Technical University of Braunschweig, 38092 Braunschweig, Germany f.ludwig@tu-bs.de Michael Mck (Chapter 4) Institute of Applied Physics, University of Giessen, Heinrich-Buff-Ring 16, 35392 Giessen, Germany Michael.Mueck@ap.physik.uni- giessen.de H. J. M. ter Brake (Chapter 7) Department of Applied Physics, Twente University of Technology, P.O. Box 217, 7500AE Enschede, The Netherlands H.J.M.terBrake@tn.utwente.nl Jiri Vrba (Chapter 7) VSM MedTech Ltd, 9 Burbidge Street, Coquitlam, B.C., Canada jvrba@vsmmedtech.com Yi Zhang (Chapter 6) Research Centre Jlich, ISG-2, 52425 Jlich, Germany y.zhang@fz-juelich.de Volume II Randall C. Black (Chapter 12) Quantum Design, Inc., 6325 Lusk Blvd., San Diego CA 92121, USA randy@blacksdesign.com John Clarke (Chapter 8) Department of Physics, 366 LeConte Hall, University of California, Berkeley CA 94720-7300, USA, and Materials Sciences Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley CA 94720, USA jclarke@berkeley.edu Ted R. Clem (Chapter 14) Naval Surface Warfare Center Panama City, 110 Vernon Avenue, Panama City FL 32407-7001, USA ted.clem@navy.mil Gordon B. Donaldson (Chapter 13) Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK g.b.donaldson@strath.ac.uk Catherine P. Foley (Chapter 14) CSIRO Industrial Physics, P.O. Box 218, Lindfield, NSW 2070 Australia Cathy.Foley@csiro.au
- 21. XVII John Gallop (Chapter 9) National Physical Laboratory, Hampton Rd., Teddington TW11 0LW, UK John.Gallop@npl.co.uk Andrei Irimia (Chapter 10) Department of Physics and Astronomy, Vanderbilt University, VU Station B 351807, Nashville TN 37235, USA andrei.irimia@vanderbilt.edu Mark N. Keene (Chapter 14) QinetiQ Ltd., St. Andrews Road, Malvern, Worcestershire WR14 3PS, UK mnkeene@qinetiq.com Hans-Joachim Krause (Chapter 13) Institute of Thin Films and Interfaces, Research Center Jlich, 52425 Jlich, Germany h.-j.krause@fz-juelich.de Adrian T. Lee (Chapter 8) Department of Physics, University of California, 363 LeConte Hall, Berkeley CA 94720-7300, USA atl@physics7.berkeley.edu Eduardo Andrade Lima (Chapter 10) Department of Biomedical Engineering, Vanderbilt University, VU Station B 351807, Nashville TN 37235, USA eduardo.a.lima@vanderbilt.edu Michael Mck (Chapter 8) Institute of Applied Physics, University of Giessen, Heinrich-Buff-Ring 16, 35392 Giessen, Germany Michael.Mueck@ap.physik.uni- giessen.de Jukka Nenonen (Chapter 11) Laboratory of Biomedical Engineering, Helsinki University of Technology, Espoo, Finland Jukka.Nenonen@neuromag.fi Ho Jung Paik (Chapter 15) Department of Physics, University of Maryland, College Park MD 20742, USA hpaik@physics.umd.edu Franois Piquemal (Chapter 9) Bureau National de Mtrologie, LNE: Laboratoire National de Mtrologie et d’Essais, Avenue Roger Hennequin 29, 78197 Trappes cedex, France francois.piquemal@lne.fr Paul L. Richards (Chapter 8) Department of Physics, University of California, 363 LeConte Hall, Berkeley CA 94720-7300, USA richards@physics.berkeley.edu Lutz Trahms (Chapter 11) Department of Bioelectricity and Biomagnetism, Physikalisch- Technische Bundesanstalt, Abbestr. 2–12, 10587 Berlin, Germany lutz.trahms@ptb.de List of Contributors
- 22. Jiri Vrba (Chapter 11) VSM MedTech Ltd., 9 Burbidge Street, Coquitlam, B.C., Canada jvrba@vsmmedtech.com Frederick C. Wellstood (Chapter 12) Center for Superconductivity Research, Department of Physics, University of Maryland, College Park MD 20742- 4111, USA well@squid.umd.edu John P. Wikswo (Chapter 10) Departments of Biomedical Engineering, Physics and Astronomy, Molecular Physiology and Biophysics, Vanderbilt University, VU Station B 351807, Nashville TN 37235, USA john.wikswo@vanderbilt.edu List of Contributors XVIII
- 23. 1 8 SQUID Voltmeters and Amplifiers John Clarke, Adrian T. Lee, Michael Mck and Paul L. Richards 8.1 Introduction 3 8.2 Voltmeters 4 8.3 The SQUID as a Radiofrequency Amplifier 5 8.3.1 Introduction 5 8.3.2 Mutual Interaction of SQUID and Input Circuit 6 8.3.3 Tuned Amplifier: Theory 10 8.3.4 Untuned Amplifier: Theory 12 8.3.5 Tuned and Untuned Amplifiers: Experiment 13 8.3.6 To Tune or Not to Tune? 16 8.3.7 SQUID Series Array Amplifier 17 8.3.8 The Quantum Limit 18 8.3.9 Future Outlook 19 8.4 Microstrip SQUID Amplifier 20 8.4.1 Introduction 20 8.4.2 The Microstrip 21 8.4.3 The Microstrip SQUID Amplifier: Gain 21 8.4.4 The Microstrip SQUID Amplifier: Noise Temperature 26 8.4.5 High-Tc Microstrip SQUID Amplifier 31 8.4.6 Future Outlook 31 8.5 SQUID Readout of Thermal Detectors 32 8.5.1 Introduction 32 8.5.2 Transition-Edge Sensors 33 8.5.3 SQUID Multiplexers 35 8.5.3.1 Time-Domain Multiplexing 35 8.5.3.2 Frequency-Domain Multiplexing 39 8.5.4 TES Bolometers 45 8.5.4.1 TES Bolometer Designs 46 8.5.4.2 Bolometer Performance 49 8.5.5 TES Calorimeters and Nonequilibrium Detectors 50 8.5.5.1 Calorimeter Designs 51 8.5.5.2 Calorimeter Noise Performance 52 The SQUID Handbook. Vol. II: Applications of SQUIDs and SQUID Systems. John Clarke and Alex I. Braginski (Eds.) Copyright 2006 WILEY-VCH Verlag GmbH Co. KGaA, Weinheim ISBN: 3-527-40408-2
- 24. 8.5.6 SQUID Readout of Non-TES Detectors 53 8.5.6.1 Magnetic Calorimeter 53 8.5.6.2 SIS Tunnel Junction 54 8.5.6.3 NIS Junctions 55 8.5.6.4 Kinetic-Inductance Thermometer 55 8.5.7 Future Outlook 56 8.6 Nuclear Magnetic and Quadrupole Resonance and Magnetic Resonance Imaging 56 8.6.1 Introduction 56 8.6.2 Principles of NMR and NQR 57 8.6.3 SQUID-Detected NMR and NQR 61 8.6.3.1 NQR of 14N 61 8.6.3.2 Spin Noise 64 8.6.3.3 NMR of Hyperpolarized 129Xe 67 8.6.3.4 Liquid-State Proton NMR and MRI 69 8.6.4 Future Outlook 80 8.7 The Axion Detector 81 8 SQUID Voltmeters and Amplifiers 2
- 25. 8.1 Introduction Volume I of this handbook is concerned with the theory, fabrication and perfor- mance of dc and rf SQUIDs, and with the implementation of SQUIDs as magnet- ometers and gradiometers using appropriate superconducting input circuits. Most of these devices are used at frequencies ranging from zero to a few kilohertz, for example for quasistatic measurements of susceptibility, for geophysical appli- cations and for biomagnetism. In this chapter we are concerned with the use of SQUIDs as voltmeters and amplifiers. Since rf SQUIDs are almost never used for such purposes, we confine ourselves to dc SQUIDs. Broadly speaking, we can divide these applications into three frequency ranges. The first is the measurement of quasistatic voltages – for example, thermoelectric voltage and the voltage generated by quasiparticle charge imbalance in a super- conductor. These voltmeters are described briefly in Section 8.2. The second fre- quency range extends from a few tens or hundreds of hertz to perhaps 100 MHz, and is discussed in Section 8.3. Major applications include readout schemes for bolometers and calorimeters for particle detectors, discussed in Section 8.5, and nuclear magnetic resonance (NMR) , nuclear quadrupole resonance (NQR) and magnetic resonance imaging (MRI), discussed in Section 8.6. In the frequency range up to a few megahertz, the SQUID is generally operated in a flux–locked loop, while at higher frequencies it is operated open–loop, with an applied flux near (2n + 1)U0/4 (U0 = h/2e » 2.07 10–15 Wb is the flux quantum and n is an integer) chosen to maximize the flux–to–voltage transfer coefficient (¶V/¶U)I ” VU. At frequencies up to, say, 100 MHz, the conventional square–washer SQUID design described in Chapter 5 is entirely adequate. In the third range of frequen- cies, a few hundred megahertz to a few gigahertz, however, the parasitic capaci- tance between the input coil and the SQUID washer can substantially reduce the gain of the conventional design. An alternative option is the so–called microstrip SQUID amplifier, in which the input coil is used as a resonant microstrip. This device is described in Section 8.4. Applications of the microstrip amplifier include the axion detector described in Section 8.7 and a postamplifier for the radiofre- quency single–electron transistor (RFSET). 8.1 Introduction 3
- 26. 8.2 Voltmeters One of the earliest applications of the dc SQUID was as a voltmeter. The sensor was in fact a SLUG (superconducting low-inductance undulatory galvanometer) [1] described briefly in Chapter 5. In essence, the SLUG consists of a bead of PbSn solder frozen around a length of Nb wire. The critical current measured be- tween the two superconductors is periodic (often multiply periodic) in the super- current passed along the Nb wire. In the early days of this device, it was possible to measure changes in this current of about 1 lA Hz–1/2. The fact that the input circuit had a low inductance – a few nanohenries – enabled one to measure volt- ages developed by much smaller resistances than had been previously possible since the time constant of the measurement could be kept to below one second. Figure 8.1 shows the original voltmeter circuit used with a SLUG. The voltage source Vs was connected in series with a standard resistor rs and the Nb wire of the SLUG. The SLUG was operated in a flux-locked loop (Section 4.2) that fed a current is into rs to maintain a null current in the Nb wire: evidently the value of Vs is given by isrs. With a SLUG current resolution of 1 lA Hz–1/2 determined by the readout electronics, the voltage resolution for rs = 10–8 X was 10–14 VHz–1/2. This represented a five orders of magnitude improvement over the resolution of semiconductor amplifiers. Since the Nyquist voltage noise across a 10–8 X resistance at 4.2 K is about 1.5 fV Hz–1/2, these early measurements were not Nyquist noise limited. Nonetheless, the SLUG voltmeter was used successfully to make mea- surements of the characteristics of superconductor–normal metal–superconduc- tor (SNS) Josephson junctions [2] and of thermoelectric voltages [3]. Subsequently, it was used in studies of the resistance of the SN interface [4] and to make the first measurements of quasiparticle charge imbalance in superconductors [5]. 4 8 SQUID Voltmeters and Amplifiers 5 mm Copper wire Niobium wire Solder Vs rs V I I is is From output of flux-locked loop Fig. 8.1 The SLUG. The configuration of a voltmeter measuring a voltage source Vs has been superimposed on a photograph.
- 27. 8.3 The SQUID as a Radiofrequency Amplifier The development of much lower noise SQUIDs with multiturn input coils, notably the Ketchen Jaycox square-washer design [6], has greatly reduced the equivalent current noise. For example, for a low-Tc SQUID with a flux noise of 2 10–6U0 Hz–1/2 at frequencies above the l/f knee (f is frequency) of typically 1 Hz, coupled to an input coil with a mutual inductance of 5 nH, the current noise S 1=2 I (f ) » 1 pA Hz–1/2. At 4.2 K, this resolution enables one to make Nyquist- noise-limited measurements in resistors r 4kBT/SI(f ) » 200 X . (8.1) In making this estimate, we have neglected the effects of current noise in the SQUID loop which induces noise voltages into the input circuit. This subject is discussed at length in Section 8.3. These devices are generally used with current feedback to the standard resistor to obtain a null balancing voltmeter [7]. Voltmeters have also been based on high-Tc SQUIDs operating at 77 K [8–10]. The unavailability of flexible, bondable wire made from a high-Tc superconductor means that normal wire must be used to connect the components in the input circuit. Contact resistance between this wire and the YBa2Cu3O7–x (YBCO) input coil adds to the total resistance. As a result, the voltage resolution is limited to roughly 1 pV Hz–1/2. SQUID packages suitable for use as voltmeters are available commercially from several companies. The SQUID is enclosed in a niobium can to shield it from ambient magnetic noise. The two ends of the input coil are connected to niobium pads to which external niobium wires can be clamped with screws to produce superconducting contacts. Thus, the user can readily couple any desired external circuit to the SQUID. 8.3 The SQUID as a Radiofrequency Amplifier 8.3.1 Introduction This section is concerned with the use of the dc SQUID as a radiofrequency (rf) amplifier. We confine our attention to the situation in which the SQUID is oper- ated open loop, biased near (2n + 1)U0/4 to maximize VU. A thorough discussion of such amplifiers is quite complicated. Although these issues are often ignored in the design of SQUID input circuits, the coupling of a circuit to a SQUID may modify its properties significantly, while at the same time the SQUID reflects both a nonlinear impedance and a voltage noise source into the input circuit. The modification of the SQUID by a coupled inductance was pointed out by Zimmer- man [11], and studied extensively in a series of papers by Clarke and coworkers [12–14]. The fact that the SQUID loop contains a noise current that is partially correlated with the voltage noise [15] across the SQUID was computed by Tesche 5
- 28. 8 SQUID Voltmeters and Amplifiers and Clarke [16], and subsequently used by various authors to calculate the noise temperature of amplifiers [13, 14, 17–20]. A complete treatment of these issues would make this chapter unwieldy, and we limit ourselves to summarizing the key theoretical results and to describing some experimental amplifiers. 8.3.2 Mutual Interaction of SQUID and Input Circuit Consider a SQUID with loop inductance L and two identical Josephson junctions each with critical current I0, self capacitance C and shunt resistance R. For a typi- cal SQUID in the 4He temperature range, the noise parameter C ” 2pkBT/ I0U0 ~ 0.05. The noise energy e(f ) ” SU(f )/2L is optimized [15] when bL ” 2LI0/U0 = 1. The Stewart–McCumber parameter [21, 22] bC ” 2pI0R2C/U0 should be somewhat less than unity to avoid hysteresis in the current voltage (I–V) characteristic (see Chapters 1 and 2). Under these conditions, one finds the following results [15]. The maximum flux-to-voltage transfer coefficient is VU ” ¶V j =¶UjIB » R=L (8.2) where IB is the value of the bias current that maximizes VU, and the flux in the SQUID is near (2n + l)U0/4 (n is an integer). When VU is maximized, the spectral density of the voltage noise across the SQUID, which is assumed to arise from Nyquist noise in the shunt resistors, is [15] SV ðf Þ » 16 kBTR . (8.3) The current noise in the SQUID loop has a spectral density [16] SJðf Þ » 11 kBT=R (8.4) and is partially correlated with the voltage noise with the cross spectral density [16] SVJðf Þ » 12 kBT . (8.5) Figure 8.2(a) shows an input circuit consisting of a voltage source Vi in series with a resistance Ri, the inductance Lp of a pickup coil, a stray inductance Ls, a capaci- tor Ci and the input inductance Li of the SQUID. Depending on the application, some of the components may be omitted. The mutual inductance to the SQUID is Mi = ki(LLi)1/2, where ki £ 1 is the coupling coefficient. The SQUID reflects a complex impedance into the input circuit which is derived from the dynamic input impedance Z of the SQUID; in turn, Z can be related to the flux-to-current transfer function JU ” (¶J/¶U)IB by the equation [18] –JU = jx/Z = 1/L + jx/R . (8.6) 6
- 29. 8.3 The SQUID as a Radiofrequency Amplifier The parameters Z, L and R refer to currents flowing around the SQUID loop. At x = 0, –JU reduces to the inverse of the dynamic input inductance L, while for x 0 there are resistive losses, represented by the dynamic input resistance R. Figure 2(b) shows a schematic representation of L and R, which define the response of the SQUID to an applied flux U. Figure 8.3 shows the variation of L/L and R/R with applied flux [13] for four values of bias current. Typically, SQUIDs are operated with IB » 2I0. We observe that both parameters depend strongly on U, with L/L becoming negative in some regions. We next discuss the effect of the input circuit on the SQUID parameters. Throughout this discussion we assume that the SQUID is operated open-loop, with its current and flux biases adjusted to maximize VU. We also assume that the loading of the readout amplifier on the SQUID is negligible. To illustrate the point, consider a superconducting pickup inductance Lp in series with a stray inductance Ls connected across the input inductance Li, as in a magnetometer. We assume that the SQUID is current-biased at a voltage corresponding to a Joseph- son angular frequency xJ. In the absence of parasitic capacitance, currents in the 7 o Fig. 8.2 (a) Schematic of a generic tuned amplifier. The voltage source Vi is connected in series with a pickup loop of inductance Lp, a stray inductance Ls, a capacitor Ci, a resis- tance Ri and the input inductance Li of the SQUID. (b) Dynamic input impedance of the SQUID represented by an inductance L and resistance R. In both figures, J is the current induced in the SQUID loop by signal and noise sources in the input circuit. (Repro- duced with permission from ref. [13].) Fig. 8.3 Simulated values of L/L and R/R. versus reduced flux U/U0 for a bare SQUID versus flux U for four values of bias current. SQUID parameters were bL = 1.0, bC = 0.2 and C » 0.06 (Reproduced with permission from ref. [13].)
- 30. 8 SQUID Voltmeters and Amplifiers SQUID loop at xJ and its harmonics will induce currents into the input circuit. It is easy to show that the SQUID loop inductance will be reduced by the presence of the input circuit to a value Lr ¼ ð1 k2 ieÞL , (8.7) where kie ¼ ki½ðLi þ Lp þ LsÞ=Li1=2 (8.8) is the effective coupling coefficient between the SQUID and the total inductance of the input circuit. Other parameters of the SQUID take the reduced values Vr U, Jr U, Zr and Rr corresponding to a SQUID with loop inductance Lr. In practice, things may be not so simple: parasitic capacitance between the coil and the SQUID washer modifies the coupling between them at the Josephson frequency and its harmonics. In the limiting case where this parasitic capacitance prevents any high-frequency currents from flowing in the input circuit, the SQUID param- eters are unaffected by the input circuit [12]. In a real system, the result is likely to be somewhere between the two extremes; it will also depend, for example, on the number of turns in the input coil which determines the parasitic capacitance. By studying a series of SQUIDs with 20-turn input coils, Hilbert and Clarke [13] found that VU was increased by roughly the expected amount (corresponding to the reduced loop inductance) when the previously open coil was shorted. We are now in a position to consider the modification of the input circuit by the SQUID impedance reflected into it. A productive way of writing the result is in terms of the output voltage across the SQUID in the presence of a signal applied to the input circuit shown in Figure 8.2(a). After some calculation, one finds [12] VðxÞ ¼ Vr NðxÞ þ MiVr U ViðxÞ þ Mi Jr NðxÞðRi þ l=jxCiÞ=LT ZTðxÞ Jr U M2 i ðRi þ l=jxCiÞ=LT . (8.9) Here, Vr N(x) and Jr N(x) are the reduced voltage and current noises of the SQUID, and Vi(x) is the input voltage applied to the resistance Ri and capacitance Ci in series with the total inductance of the input circuit LT = Li + Lp + Ls. The total impedance of the (uncoupled) input circuit is ZTðxÞ ¼ Ri þ jxðLi þ Lp þ LsÞ þ l=jxCi . (8.10) The denominator of Eq. (8.9) contains the term Jr UM2 i ðRi þ l=jxCiÞ=LT that rep- resents the impedance reflected into the input circuit from the SQUID. The term involving Jr NðxÞ in the square brackets is the noise current generated in the input circuit by the SQUID. We can readily derive the voltage gain for a SQUID ampli- fier from Eq. (8.9): Gv ¼ MiVr U ZT k2 ieLJr UðRi þ l=jxCiÞ ” MiVr U Z* T . (8.11) 8
- 31. 8.3 The SQUID as a Radiofrequency Amplifier Using Eq. (8.6), we see that the impedance can be written in the form Z* T ¼ Ri 1 þ k2 ieL Lr þ k2 ieL RrCi þ jx Li þ a2 eL Ri Rr 1 x2CiLr þ Lp þ Ls þ 1 jxCi : (8.12) We note that Rr and Lr contribute to both the real and imaginary parts of Z* T. Hilbert and Clarke [13] made extensive measurements of Jr U as a function of U by connecting a capacitor Ci across the input coil of both a 4-turn and a 20-turn SQUID. The resonant frequency f 0 0 and full width at half maximum (FWHM) of this tank circuit were measured directly with the SQUID biased with a large cur- rent ( 2I0) where the SQUID has negligible inductive screening and a dynamic input impedance of approximately 2Rr. The SQUID was then biased at its usual operating point, and the Nyquist noise power P(f0) generated by the tank circuit was measured at the output of the SQUID with a spectrum analyzer. The value of Vr U was determined from the height of the peak. The frequency f0 at which the noise power peaked generally differed from f 0 0 , and yielded the inductance change DLi reflected into the tank circuit. Similarly, the value of the FWHM, Df, yielded the resistance change DRi. From these values, it was straightforward to infer the values of L/Lr and R/Rr from Eq. (8.12) inserted into Eq. (8.11). The results for a 20-turn SQUID are shown in Figure 8.4 for three values of bias current. The behavior of jVr Uj is much as expected. However, the magnitudes of the curves suggest that the inductive screening of the SQUID is more effective at the lowest bias current than at the highest bias current; this result is consistent with a reduction in screening by the parasitic capacitance as the Josephson fre- quency increases. The measured values of L/Lr follow the trends in the simula- tions quite well. For example, at the lowest bias current there is a broad maximum at U = 0 and a negative region around U = –U0/2. The overall magnitude of L/Lr 9 Φ/Φ0 Fig. 8.4 Measured values of L/ L r, R/R r and |Vr U|L/R versus reduced flux U/U0 for a SQUID with L » 400 pH, 2I0 » 6 – 1 lA, C » 0.5 pF and R » 8 X, corresponding to bL » 1 and bC » 0.2. The temperature was 4.2 K corresponding to C » 0.06. Bias current: (a) 4.0 lA, (b) 5.0 lA, (c) 6.0 lA. (Reproduced with permission from ref. [13].)
- 32. 8 SQUID Voltmeters and Amplifiers is generally in fair agreement with the simulations. On the other hand, the mea- sured values of R/Rr, which vary between +30 and –5, are in sharp disagreement with the simulated values, which are always positive, with a maximum of about 2. Thus, DRi is evidently dominated by a mechanism other than the resistance reflected from the SQUID. Further investigation showed that the change in resistance in the tank circuit was dominated by feedback from the output of the SQUID via the parasitic capac- itance between the washer and the input coil. Approximating the distributed ca- pacitance with a lumped capacitor, Hilbert and Clarke [13] were able to account for the observed change in resistance in the input circuit to within a factor of 2. The reader is referred to the original paper for details. This concludes our discussion of the input impedance of the dc SQUID and of the mutual loading of the SQUID and input circuit. We next apply these ideas to the design of SQUID amplifiers. 8.3.3 Tuned Amplifier: Theory To simplify our initial discussion we first neglect capacitive feedback, and later return briefly to this issue. The circuit is shown schematically in Figure 8.2(a); we interpret Ri as the impedance of the voltage source Vi. For Vi = 0, the noise voltage at the SQUID output can be written from Eqs. (8.8) and (8.9) as VNðxÞ ¼ Vr NðxÞ þ k2 ieLVr UðRi þ l=jxCiÞJr NðxÞ=Z* TðxÞ (8.13) where Z* TðxÞ is given by Eq. (8.12). We now assume that the amplifier is operated at the resonant frequency f0 = x0/2p at which the imaginary terms in Z* T tune to zero: x0 ¼ ½ðLi þ Ls þ k2 ieLRi=Rr ÞCi=ð1 þ k2 ieL=Lr Þ1=2 . (8.14) Thus, at the resonant frequency, Eq. (8.13) reduces to VNðx0Þ ¼ Vr Nðx0Þ þ k2 ieL Ri þ 1=jx0Ci ð ÞJr N x0 ð ÞVr U Ri þ DRi (8.15) where DRi ¼ k2 ieLðRi=Lr þ 1=Rr CiÞ . (8.16) To simplify matters, we now assume that Q is high so that Ri 1/jxCi. As we shall see later, optimization of the noise temperature implies that k2 ieQ » 1, so that k2 ie 1. Thus, to an excellent approximation the SQUID parameters assume their bare values, the resonant frequency becomes x0 » [(Li + Ls)Ci]–1/2 and Eq. (8.15) reduces to 10
- 33. 8.3 The SQUID as a Radiofrequency Amplifier VNðx0Þ » V0 Nðx0Þ jx0M2 i VUJNðx0Þ=ðRi þ DRiÞ . (8.17) Here, V0 Nðx0Þ is the voltage noise across the bare SQUID. We have retained the term DRi for later discussion, since in practice it is dominated by capacitive feed- back. Neglecting DRi for the moment, we can interpret Eq. (8.17) quite simply. The current noise JNðxÞ produces a voltage noise jxMiJNðxÞ into the input cir- cuit which, on resonance, yields a current noise jxMiJNðxÞ=Ri and hence a flux noise jxM2 i JNðxÞ=Ri in the SQUID. Multiplying this term by VU yields the sec- ond term in Eq. (8.17). We now introduce the noise temperature TN(f0) of the SQUID amplifier on res- onance through the definition 4kBTNðf0ÞRiG2 vðf0Þ ¼ SV ðf0Þ . (8.18) Here, SV(f0) is the spectral density of the voltage noise at the output of the SQUID amplifier, which we can readily calculate from Eq. (8.17). We find the value of Ri that optimizes TN(f0) by calculating ¶TNðf0Þ=¶Ri ¼ 0 from Eq. (8.18): R opt i ¼ ½ðR opt i0 Þ2 þ ðDRiÞ2 1=2 . (8.19) Here, R opt i0 ¼ ðSJ=SV Þ1=2 x0M2 i VU . (8.20) This result can also be obtained from the usual [23] treatment of an amplifier with a voltage noise source –jxMiJN(x) and a current noise source VN(x)/MiVU placed at its input terminals. With DRi R opt i0 , the corresponding optimized noise tem- perature is T opt N ðf0Þ=T ¼ 8RR opt i =M2 i V2 U . (8.21) Finally, setting VU » R/L, we find R opt i0 » k2 i x0Li (8.22) and T opt N » ðSV SJÞ1=2 x0=2kBVU » 7Tx0=VU . (8.23) We note that T opt N scales as the ratio x0/VU. Furthermore, neglecting DRi we can readily show from Eq. (8.22) that the opti- mum value of Q » x0(Li + Ls)/Ri is Qopt » ð1 þ Ls=LiÞ=k2 i , or Qoptk2 ie » 1 . (8.24) In general, for reasonably high values of Q this result implies that if necessary one should add inductance to the input circuit to reduce kei. The resulting weak 11
- 34. 8 SQUID Voltmeters and Amplifiers coupling justifies the use of the bare SQUID parameters. Finally, the optimized power gain at resonance, |Vo/Vi|2Ri/Rdyn (Vo is the output signal voltage), is found from the square of Eq. (8.11) to be G opt p » M2 i V2 U=RiRdyn » VU=x0 (8.25) where we have set the dynamic resistance of the SQUID Rdyn » R. Combining Eqs. (8.23) and (8.25), we find the gain noise temperature product G opt p T opt N » 7T . (8.26) Thus, high gain is synonymous with low noise temperature. The results given in Eqs. (8.15)–(8.26) are at the resonant frequency. We observe that the cross spectral density SVJ(f ) does not enter the noise temperature, for the following reason. The noise voltage induced into the input circuit is in quadrature with the current noise JN(t) in the SQUID that generates it. On resonance the input circuit has a real impedance, so that the noise produced at the SQUID out- put by this voltage noise is also in quadrature with that component of the output noise produced by the circulating current. As a result, the cross-correlation term vanishes. However, in principle one can obtain a lower noise temperature [17] by operating the amplifier off resonance, so that the noise at the SQUID output pro- duced by the voltage noise in the input circuit partially cancels the component due to the current noise in the SQUID. It is straightforward to show that the mini- mum noise temperature so obtained is given by replacing (SVSJ)1/2 in Eq. (8.23) by ðSV SJ S2 VJÞ1=2 . However, in practice the gain of the amplifier may well be too low to make operation off resonance realistic. To give an example of the predicted performance of a tuned amplifier on reso- nance, we assume L » 400 pH, R » 8 X, Li » 160 nH, k2 i = 0.7, bL = 1, VU » R/L » 2 1010 s–1, SV » 16 kBTR, SJ » 11 kBT/R and f0 = 100 MHz to find R opt i » 70 X, G opt p » 15 dB and T opt N » T/5. To conclude the discussion of the tuned amplifier, we note that the resistance induced into the input circuit in practical devices is dominated by capacitive feed- back. This effect modifies both the resonant frequency and the Q of the amplifier [14]. This frequency shift is of the order of 1/ Q and thus is small for high values of Q. The effect on Q, on the other hand, may be much larger, and could be a factor of 2 for the numerical example considered above [14]. 8.3.4 Untuned Amplifier: Theory To represent an untuned amplifier, we set Lp = 1/jxCi = 0. The analysis proceeds along the same lines as that for the tuned case. Thus, we set 1/jxCi = 0 in Eq. (8.13) for V(x) and l/jxCi = Lp = 0 in Eq. (8.12) for Z* TðxÞ. After some calculation, we find the following expression for the noise temperature: 12
- 35. 8.3 The SQUID as a Radiofrequency Amplifier TNðf Þ = Sr V Z* T 2 þ2Sr VJk2 ieLRiVr U ReðZ* TÞ þ Sr Jðk2 ieVr ULRiÞ2 4kBRiM2 i ðVr UÞ2 . (8.27) To make progress, we neglect the terms k2 ieL/Lr and k2 ieLRi/LiRr in Z* TðxÞ. These are reasonable approximations since L/Lr £ 1/10 for U near U0/4 or 3U0/4, and LRi/LiRr ~xL/R 1. We can now optimize TN(f ) with respect to Ri to find R opt i ðf Þ » 2pf ðLi þ LsÞ 1 þ 2k2 ieLVr USr VJ Sr V þ ðk2 ieLVr UÞ2 Sr J Sr V #1=2 . (8.28) The corresponding optimized noise temperature is T opt N ðf Þ » 4p2f 2ðLi þ LsÞSr V =2kBk2 ieLðVr UÞ2 R opt i ðf Þ . (8.29) In contrast to the tuned case, where ke 1 and we could replace the reduced SQUID parameters with their corresponding bare parameters, we now need to estimate the reduced parameters. We assume the same bare SQUID parameters used for the example of a tuned amplifier in Section 8.3.3, and take Ls = 20 nH, corresponding to the case of the particular amplifier discussed in Section 8.3.5. We find k2 ie » 0.6 and hence br L ¼ ð1 k2 ieÞbL » 0:4. From simulations we find Vr U » 2.5 1010 s–1, Sr V » 18 kBTR, Sr J » 12 kBT/R and Sr VJ » 12 kBT [15, 16]. Insert- ing these values into Eq. (8.28) we find R opt i » 0.7x(Li + Ls), which differs little from the value for the tuned case provided Ls Li. From Eq. (8.29) we find T opt N » 0.6T, three times greater than the value for the tuned amplifier. As a final remark, we note that the reduced SQUID parameters, even for a value of k2 ie as large as 0.6, differ little from the bare parameters. Given that capacitive feedback, which we have neglected, also modifies the gain and noise temperature to some extent [14], it appears that the trouble of calculating the reduced parame- ters is not really justified unless k2 ie becomes rather close to unity. 8.3.5 Tuned and Untuned Amplifiers: Experiment Hilbert and Clarke [14] measured the gain and noise temperature of a SQUID used as a tuned and an untuned amplifier. The configurations used to measure the power gain Gp and noise temperature TN are shown in Figure 8.5. The output of the SQUID, which was enclosed in a superconducting shield, was coupled to a low-noise, room-temperature amplifier followed by a spectrum analyzer and a power meter. To measure the gain, a calibrated signal was coupled to a cold attenuator that presented an impedance Ri to the input coil. To measure the noise temperature, the input coil was connected to a resistor Ri, the temperature Ti of which could be raised above the bath temperature with a heater. The total noise temperature referred to the input of the amplifier is TT N ¼ TN þ Ti þ TP N=GP (8.30) 13
- 36. 8 SQUID Voltmeters and Amplifiers where TP N is the noise temperature of the postamplifier. By measuring the output noise power as a function of Ti one can obtain TN. In the case of the tuned amplifier, the four-turn input coil with Li » 5.6 nH was connected in series with a capacitor Ci » 20 pF and the source resistance Ri. The measured resonant frequency was about 93 MHz and Q was about 45. The corre- sponding optimized value of Ri (Eq. (8.22)) is about 2 X. The resonant frequency implies a large stray inductance, Ls » 140 pH. Thus, from Eq. (8.8) we find k2 ie » 0.023, so that Qk2 ie » 1, as required by Eq. (8.24). The measured gain of 18.6 dB was in quite good agreement with the predicted value (see Table 8.1). The mea- sured output noise power as a function of Ti is shown in Figure 8.6, and leads to TN = 1.7 – 0.5 K, slightly above the predicted value. The shift in resonant fre- quency with flux bias was less than 1%, as expected given the low value of k2 ie. The change in Q when the flux bias was changed from (n + 1/4)U0 to (n + 1/2)U0 was substantially higher, about 25%, and demonstrated that the additional resistance in the input circuit due to the SQUID was dominated by capacitive feedback. 14 Fig. 8.5 Circuit used to measure (a) the power gain and (b) the noise temperature of an untuned SQUID amplifier. Components in the dashed boxes are immersed in liquid helium. (Reproduced with permission from ref. [14].) Fig. 8.6 Total output noise power (arbitrary units) versus Ti at 93 MHz for a tuned SQUID amplifier at 4.2 K. (Reproduced with permission from ref. [14].)
- 37. 8.3 The SQUID as a Radiofrequency Amplifier Table 8.1 Measured and predicted power gain Gp and noise temperature TN for a dc SQUID radiofrequency amplifier. Frequency (MHz) Gp (dB) TN (K) Measured Predicted Measured Predicted T = 4.2 K (tuned) 93 18.6 – 0.5 17 1.7 – 0.5 1.1 T = 1.5 K (untuned) 60 24.0 – 0.5 – 1.2 – 0.3 – 80 21.5 – 0.5 – 0.9 – 0.3 – 100 19.5 – 0.5 18.5 1.0 – 0.4 0.9 T = 4.2 K (untuned) 60 20.5 – 0.5 – 4.5 – 0.6 – 80 18.0 – 0.5 – 4.1 – 0.7 – 100 16.5 – 0.5 16.5 3.8 – 0.9 2.5 For the untuned amplifier, the SQUID had a 20-turn input coil, with Li » 120 nH and Mi » 6 nH. The estimated value of k2 ie was about 0.6. At 80 MHz, the total inductance Li + Ls » 140 nH was approximately optimized to a 50-X source impe- dance. Figure 8.7 shows the gain versus frequency for positive and negative values of VU. In each case, the gain drops by about 5 dB as the frequency is increased from 10 to 100 MHz as a result of the increasing impedance of the input coil. There is a resonance around 150 MHz that produces a dip in the gain for negative VU and a peak for positive VU. This resonance corresponds to the self-resonance of the input coil, and corresponds to a parasitic capacitance of about 8 pF. The measured values of gain and noise temperature at three frequencies and two temperatures are listed in Table 8.1, and are in quite good agreement with predic- tions. 15 Fig. 8.7 Gain versus frequency for an untuned SQUID amplifier for (a) negative and (b) positive VU. (Reproduced with permission from ref. [14].)
- 38. 8 SQUID Voltmeters and Amplifiers 8.3.6 To Tune or Not to Tune? Given the preceding discussion of tuned and untuned amplifiers, it is natural to ask which type one should use under a given set of conditions. This question becomes particularly relevant for the case of NMR discussed in Section 8.6, and it is convenient to discuss it at this juncture. We again consider the configuration of Figure 8.2, but now assume that the voltage source Vi is generated by an oscillat- ing magnetic flux in the pickup loop which has an inductance Lp. The untuned amplifier is obtained by setting Ri = 1/Ci = 0. The issue is the following. On reso- nance, the tuned input circuit enhances the signal amplitude by a factor Q, but the Nyquist noise in the input circuit also peaks at the resonance frequency. This noise may well overwhelm the intrinsic noise of the SQUID. In the untuned case, there is no resonant enhancement of the signal, but the input circuit, at least at low frequencies, is noiseless, and the SQUID noise determines the noise in the input circuit. Which of the two cases offers the higher signal-to-noise ratio? To dis- cuss this question, it is convenient to focus on the signal energy in the pickup loop E = ÆV2 i æ=2x2LP (8.31) where ÆV2 i æ is the mean square voltage induced by the oscillating flux. We first consider the tuned amplifier, operated at the resonant frequency x0. If we assume an optimized input circuit at 4.2 K, the situation is very simple at fre- quencies below 100 MHz where T opt N T (Section 8.3.3). The minimum detect- able signal energy in a bandwidth df f 0/Q is found by setting ÆV2 i æ = 4kBTRidf in Eq. (8.31) to find Etuned min ¼ 2kBTRidf =4p2f 2Lp » kBTðLi þ Lp þ LsÞdf =pfQLp . (8.32) We have neglected the impedance reflected into the input circuit by the SQUID, which is a reasonable approximation for Q 1 so that k2 ie 1 (Eq. (8.24)), and we thus set Q = x0(Li + Lp + Ls)/Ri. For the untuned case, the situation is more complicated. Strictly speaking, we should optimize the parameters of the input circuit in a manner analogous to that for the untuned amplifier in Section 8.3.4. However, since we are concerned with finding only an order of magnitude result, we shall resort to the following approx- imate, but much simpler, estimate. The mean square flux induced in the SQUID is given approximately by M2 i ÆV2 i æ x2 Li þ Lp þ Ls 2 ¼ 2k2 i LiLLpE Li þ Lp þ Ls 2 (8.33) where we have neglected the impedance reflected into the input circuit from the SQUID. We now assume Ls Li + Lp and Lp ~ Li so that the output voltage noise 16
- 39. 8.3 The SQUID as a Radiofrequency Amplifier of the SQUID is not too far from its value in the absence of the input circuit and we may neglect the distinction between VU and Vr U. These approximations enable us to find the minimum detectable signal energy simply by setting Eq. (8.33) equal to the mean square flux noise of the bare SQUID, 16kBTRdf/V2 U, to find Euntuned min » 32kBTdf =k2 i VU . (8.34) We have set VU = R/L. From Eqs. (8.32) and (8.34), we see that it is advantageous to use a tuned circuit provided Etuned min Euntuned min , that is, with ki ~ 1, f0Q VU=16p . (8.35) For VU ~2 1010 s–1, this result implies that it is desirable to use a tuned circuit only if f0Q 400 MHz. For frequencies of a few megahertz, this implies a tuned circuit with Q of a few hundred, which is generally realistic. On the other hand, for a frequency of (say) 10 kHz, it would be quite impracticable to tune – the re- quired Q ~ 4 104 would imply a bandwidth of 0.25 Hz – and one should clearly use an untuned circuit. The crossover from untuned to tuned circuits is likely to occur around 1 MHz, at which frequency a more careful evaluation of the untuned case would be warranted. 8.3.7 SQUID Series Array Amplifier A very useful extension of the basic design of the dc SQUID is the SQUID series array [24]. The array consists of Ns SQUIDs with their current leads connected in series; Ns is typically 100. The SQUIDs are biased with a single current and the voltage is measured across the entire array. The input coils to the SQUIDs are similarly connected in series. Provided the flux in each SQUID is the same, the value of VU is increased to NsVU, which is thus typically 5 mV/U0. Since the noise voltages VN(f ) across the SQUIDs are incoherent, the voltage noise is increased by the factor Ns 1/2 to Ns 1/2VN(f ). The large flux-to-voltage transfer coefficient enables one to connect the array directly to a low-noise, room-temperature ampli- fier while having the SQUID noise dominate over the amplifier noise. The SQUID array can readily be used as an amplifier. Following the discussion in Section 8.3.3, we can write the voltage noise induced into the input circuit as –jxMiNs 1/2JN(x), where Ns 1/2JN(x) is the contribution of the Ns incoherent current noise sources in the SQUIDs. The equivalent current noise in the input circuit due to the voltage noise across the array is Ns 1/2VN(x)/Mi Ns VU. We observe that the product of the current and noise sources in the input circuit is independent of Ns. Consequently, for an array SQUID used as an amplifier with a tuned input circuit, one expects the optimized noise temperature to be the same as for a single SQUID, Eq. (8.23). 17
- 40. 8 SQUID Voltmeters and Amplifiers Huber et al. [25] fabricated SQUID arrays with 10, 30 and 100 SQUIDs. A novel feature of their design was that the flux focusing washer of each SQUID was elec- trically isolated from the SQUID itself. The relatively small capacitance between the SQUID and washer was substantially less than that between the input coil and the washer. Since these two capacitances are in series, the effective parasitic capacitance between the input coil and the SQUID was substantially reduced, largely eliminating resonances induced on the current–voltage characteristic. The 3-dB point of a 100-SQUID array was 120 MHz. The fact that no coupling network is required between the array and the room-temperature amplifier makes this con- figuration particularly appealing for radiofrequency applications. 8.3.8 The Quantum Limit At signal frequencies greater than kBT/h, one expects quantum effects to become important. For a device cooled in a dilution refrigerator to 20 mK, the correspond- ing frequency is about 0.4 GHz so that the quantum limit is likely to be relevant only to the microstrip SQUID amplifier (Section 8.4). We briefly discuss quantum effects in Josephson junctions and SQUIDs. In the case of a single resistively shunted junction (RSJ) (Section 2.1.1), quan- tum effects become significant when hfJ kBT, where fJ = 2eV/h is the Josephson frequency and V is the time-averaged voltage across the junction. In this limit, one replaces the spectral density of the current noise in the shunt resistor with (2hfJ/R) coth(hfJ/2kBT) [26, 27]. In the limit hfJ kBT this spectral density reduces to 2hfJ/R, representing the zero-point fluctuations of an ensemble of harmonic oscillators with random phases. This term can be observed by measuring the volt- age noise across a junction at a frequency fm much less than fJ. This noise is pre- dicted to have a spectral density SV(fm) = [4kBT + 1 2 (I/I0)2 2eV coth (eV/kBT)] R2 dyn (8.36) where I I0. The first term in brackets arises from noise generated at frequency fm. The second term arises from noise generated at the Josephson frequency 2eV/h and mixed down to the measurement frequency by the nonlinearity of the junction; the mixing coefficient is 1 2(I0/I)2. Equation (8.36) represents the solution of a quantum Langevin equation in which one uses zero-point fluctuations in the classical equation of motion for the RSJ (Eq. (2.22)). Koch et al. [28] measured the voltage noise at a frequency of about 100 kHz for Josephson frequencies up to about 500 GHz, and found good agreement with Eq. (8.36). At sufficiently low temperatures and high signal frequencies, one would expect quantum effects to become important in the dc SQUID. Unfortunately, since SQUIDs are typically operated at a bias current rather close to the (non-noise- rounded) value of their critical current, the quantum Langevin equation is of dubious validity and one should instead undertake a full quantum mechanical treatment of the SQUID as an amplifier. This problem turns out to be remarkably 18
- 41. 8.3 The SQUID as a Radiofrequency Amplifier challenging, and, to date, remains unsolved. Despite the questionable validity of the quantum Langevin approach, a quarter of a century ago Koch et al. [29] carried out a series of simulations for noise in a SQUID coupled to a tuned input circuit to make an amplifier (as in Figure 8.2 with Lp = Ls = 0). They computed the spec- tral densities SV(f ), SJ(f ) and SVJ(f ) at T = 0 assuming that the noise arose from uncorrelated zero-point fluctuations in the two shunt resistors. As a figure of merit they defined the quality n(i) (f )hf = S ðiÞ V (f )/4Ri as the mean photon energy in the input circuit due to intrinsic SQUID noise; S ðiÞ V (f ) is the spectral density of the voltage noise referred to the input terminals of the amplifier. When the amplifier is operated off resonance (Section 8.3.3) to minimize the noise temperature, one finds n(i) = p[SV(f ) SJ(f ) – S2 VJ(f )] h/VU . (8.37) For an optimized SQUID, Koch et al. [29] found n(i) » 1 2, that is to say, on the aver- age the SQUID adds one half photon of noise to the input circuit. When this ener- gy is added to the zero-point energy 1 2hf of the resonant circuit, the total noise of the amplifier becomes hf, the result for any quantum-limited amplifier. Since, as we shall see in Section 8.4.4, the microstrip SQUID amplifier is within a factor of two of this quantum limit, it would be of great interest to perform a full quantum mechanical treatment of the SQUID amplifier to find out whether or not it is indeed strictly quantum-limited and to examine the validity of the quan- tum Langevin approach. 8.3.9 Future Outlook The theory and operation of single dc SQUIDs used as radiofrequency amplifiers at frequencies up to about 100 MHz are well in hand. The measured gain and noise temperature are in quite good agreement with predictions. However, a com- plication in the understanding of these amplifiers is the presence of parasitic ca- pacitance between the input coil and the SQUID washer. This capacitance not only introduces resonance on the current–voltage characteristic but also partially screens currents at the Josephson frequency from the input circuit, making it dif- ficult to calculate the effects of mutual coupling between the SQUID and input circuit with any precision (Section 8.3.2). The use of a design in which the flux- focusing washer is electrically isolated from the SQUID [25] can substantially reduce the parasitic capacitance between the SQUID and the input coil (Section 8.3.7). This decreases the magnitude of resonances on the current–voltage charac- teristic and at the same time largely eliminates the interaction between the SQUID and input circuit at the Josephson frequency. As a result, the effect of the input circuit on the SQUID parameters may become relatively unimportant. These effects are well worth exploring experimentally. The use of SQUID series array amplifiers, with their substantially enhanced output signal voltage, allows one to use a low-noise amplifier at room temperature without the need of an 19
- 42. 8 SQUID Voltmeters and Amplifiers impedance-matching network between them. This simplification is very appealing for radiofrequency applications. Although the theory of the SQUID amplifier in the classical regime is well understood, the same cannot be said for the quantum limit. A theory that fully accounts for the quantum mechanical nature of the SQUID is very much needed, particularly to understand whether or not the SQUID is truly a quantum-limited amplifier. 8.4 Microstrip SQUID Amplifier 8.4.1 Introduction As we have seen in Section 8.3, the conventional square-washer SQUID config- uration [6] can be operated as a low-noise amplifier at frequencies up to about 100 MHz. As the frequency is increased, however, parasitic capacitance between the input coil and the square washer causes the gain to fall off to levels that are no longer useful. This problem was addressed by Tarasov et al. [30] who made a four- loop SQUID with input coils in series deposited inside the loops rather than on top of the superconducting washer. As a result, the parasitic capacitance is reduced, and the operating frequency range is substantially extended. For exam- ple, in a tuned amplifier configuration a gain of nearly 20 dB was achieved at 420 MHz. The development of this amplifier was described in a subsequent series of publications [31–35], and its frequency range increased by reducing the number of SQUID loops to two. For an operating temperature of 4.2 K and a frequency of 3.65 GHz, a gain of (11 – 1) dB and a noise temperature of (4 – 1) K were achieved [34]. A two-stage amplifier at the same frequency achieved a gain of (17.5 – 1) dB [33]. This research is currently focused on developing an intermediate-frequency amplifier to follow an SIS (superconductor–insulator–superconductor) mixer for radio astronomy. In an alternative approach, one makes a virtue of the capacitance between the coil and the washer by using it to form a resonant microstrip [36]. The signal to be amplified is applied between one end of the coil and the washer, while the other end of the coil is left open. Provided that the source impedance is greater than the characteristic impedance of the microstrip, there is a peak in the gain when a half wavelength of the standing wave is approximately (but not exactly) equal to the length of the coil. Gains of well over 20 dB and noise temperatures well below the bath temperature can be achieved. However, as we shall see, the actual behavior of the device differs markedly from that of a simple microstrip because the induc- tance coupled into the input coil from the SQUID is generally substantially greater than the intrinsic microstrip inductance. 20
- 43. 8.4 Microstrip SQUID Amplifier 8.4.2 The Microstrip A microstrip consists of a superconducting strip of width w separated from an infinite superconducting sheet by an insulator with dielectric constant e and thick- ness d. We assume that the thicknesses of the two superconductors are much greater than the superconducting penetration depth k, and that w d. The capac- itance and inductance per unit length of the microstrip are given by C0 = e e0w/d (Fm–1) and L0 = (l0d/w)(1 + 2k/d) (H m–1) [37]. Here, e0 = 8.85 10–12 Fm–1 and l0 = 4p 10–7 H m–1 are the permittivity and permeability of free space, and c = 1/(e0l0)1/2 = 3 108 m s–1 is the velocity of light in vacuum. The factor (1 + 2k/d) accounts for the penetration of the magnetic field into the (identical) supercon- ductors. The velocity of an electromagnetic wave on the microstrip is thus given by c c ¼ c=½eð1 þ 2 =dÞ1=2 , and its characteristic impedance by Z0 ¼ L0 C0 1=2 ¼ d w l0ð1 þ 2k=dÞ ee0 1=2 . (8.38) The microstrip represents an electromagnetic resonator. For a microstrip of length ‘ with its two ends either open or terminated with resistances greater than Z0, the fundamental frequency occurs when ‘ is equal to a half wavelength [37], f0(L0) = c/2 ‘[e(1 + 2k/d)]1/2 . (8.39) In this mode, the resonator is analogous to a parallel tuned circuit, and, for a resis- tance Ri Z0 connected to one end, the quality factor of the resonance is given by Q = pRi/2Z0. 8.4.3 The Microstrip SQUID Amplifier: Gain The microstrip SQUID has the conventional square-washer configuration (Figure 8.8). However, in contrast to the conventional input scheme in which the signal is connected to the two ends of the coil, the signal is instead coupled between one end of the coil and the square washer, which provides the groundplane for the microstrip. In an early set of experiments [36], the square washer had inner and outer dimensions of 0.2 0.2 mm2 and 1 1 mm2, and the input coil had n = 31 turns, a width w = 5 lm and a length ‘ = 71 mm. Estimated parameters were L » 320 pH, L i » 300 nH and Mi » 10 nH. The critical current and shunt resis- tance per junction were typically 5 lA and 10 X, and the maximum value of VU was about 60 lVU1 0 . At 4.2 K, the white flux noise measured in a flux-locked loop at low frequencies was typically 2–3 lU0 Hz–1/2. We note that virtually the entire length of the coil overlays the washer, which is at a uniform potential. The SQUID chip was mounted on a printed circuit board, and each pad was wire bonded to a 50-X trace patterned on the board. The grounds of the input and output SMA con- 21
- 44. 8 SQUID Voltmeters and Amplifiers nectors were soldered to a groundplane on the reverse side of the board, and their center conductors were soldered to the traces coupled to the input of the micro- strip and to the output from the SQUID, respectively. The assembly was rigidly mounted inside a superconducting box that eliminated fluctuations in the ambi- ent magnetic field. Most measurements were made with the package immersed in liquid 4He. The circuit shown in Figure 8.9 was used to measure the gain; the input coil over the square washer is shown as a distributed line. The current bias was sup- plied by batteries. In later versions of the device, the flux bias was provided by a directly coupled flux-locked loop that maintains the SQUID near its maximum gain [38]. The loop rolls off at frequencies above a few kilohertz, and has no effect on the high-frequency performance. A sweep oscillator was coupled to the micro- strip via a room-temperature 100-dB attenuator and a cold 20-dB attenuator that prevented noise produced by the generator from saturating the SQUID. The cold attenuator also presented an impedance of 50 X to both the input coaxial line and the microstrip. A second cold, 4-dB attenuator coupled the output of the SQUID to a room-temperature postamplifier. The gain of the system excluding the SQUID was calibrated by disconnecting the SQUID and connecting together the input and output attenuators. All measurements of the gain of the SQUID ampli- fier were referred to the baseline so obtained. Because the washer SQUID is an asymmetric device – the two Josephson junctions are situated close together rather than on opposite sides of the SQUID loop – one can either ground the washer or ground the counter-electrode close to the Josephson junctions. Since the washer acts as a groundplane for the input coil, at first sight it might seem plausible to ground the washer. However, it is also possible to ground the counter- electrode and have the washer at output potential. In the latter case, there is feed- back from the output voltage generated on the washer to the input coil, via the capacitance between them. If the sign of VU is such that the output voltage has 22 Fig. 8.8 Configuration of microstrip SQUID amplifier. The input signal is connected between one end of the coil and the square washer.
- 45. 8.4 Microstrip SQUID Amplifier the same sign as the input voltage, the feedback is positive; if the signs are oppo- site, the feedback is negative. We designate the flux-to-voltage transfer coefficients as Vþ U and V U , respectively. Figure 8.10 shows the gain as a function of frequency for six devices with coil lengths ranging from 98 mm to 3 mm. The two SQUID configurations are shown in the figure: for the four longer coils the hole in the washer was 200 200 lm2 and the estimated inductance was 350 pH; for the two shorter coils, the hole was 10 200 lm2 and the estimated inductance was 90 pH. These inductances, the number of turns on the coils and the length of the coils are listed in Table 8.2. The peak gains achieved with SQUIDs 5 and 6 are lower than for the other four devices because of the reduced mutual inductance between the input coil and the washer. Table 8.2 Measured and calculated frequency of the fundamental resonance (half–wavelength) for six microstrip SQUIDs with inductance L and coils of n turns and length ła. SQUID L (pH) n ł (mm) f0 (MHz) f calc 0 ðL0Þ (MHz) f calc 0 ðn2LÞ (MHz) 1 350 40 98 105 500 91 2 350 15 23 370 2100 500 3 350 11 16 590 3000 820 4 350 6 8 1200 6200 2200 5 90 8 4 2200 12330 4400 6 90 7 3 2650 15000 5300 a The measured values f0 were obtained with a reverse gain techni- que. The calculated value f calc 0 ðL0Þ was obtained from Eq. (8.39) with e= 5.5, k = 0.15 lm and d = 0.4 lm. The calculated value f calc 0 ðn2LÞ was obtained from Eq. (8.40) with C0 = 0.6 nFm–1. 23 B Φ Fig. 8.9 Circuit used to determine the gain of the microstrip SQUID, which is shown with counter-electrode grounded. IB is the current bias, and IU provides the flux bias. The input and output p-networks have attenuations of 20 dB and 5 dB, respectively. (Reproduced with permission from ref. [36].)
- 46. 8 SQUID Voltmeters and Amplifiers In a more recent set of devices [39], the peak in the gain was moved to high frequency by scaling down the hole in the SQUID to 10 200 lm2, and reducing the dimensions of the coils. The gains achieved ranged from 12 – 1 dB at 2.2 GHz to 6 – 1 dB at 7.4 GHz. The reduction in gain compared with devices operating at lower frequencies was due largely to the lower mutual inductance between the coil and the SQUID which, in turn, arose from the smaller SQUID inductance. Nonetheless, it is encouraging that useful levels of gain can be achieved at fre- quencies well into the gigahertz range. It is important to realize that the peak in the gain is shifted to a frequency high- er than the resonant frequency of the coil – typically by 30% – by positive feedback [40]. The true resonant frequency f0 can be determined by a reverse gain technique [40] as follows. With the SQUID washer grounded, a signal source is connected across the SQUID and the signal transmitted through the SQUID into the coil resonator is measured with a spectrum analyzer. The current flux biases are adjusted to their usual operating values, that is, to produce maximum forward gain. At the resonant frequency of the resonator, where ‘ = k/2, there is a mini- mum in the observed power. This minimum arises from the asymmetric voltage distribution on the resonator, which induces a positive current in one half of the resonator and an equal, negative current in the other half. The measured values of f0 are listed in Table 8.2. Also listed in Table 8.2 are the resonant frequencies f0(L0) predicted by the microstrip formula, Eq. (8.39), using e = 5.5, d = 0.4 lm and k = 0.15 lm. We see that in all cases these exceed the measured value by a factor of roughly 5. Thus, the input coil does not behave as a simple microstrip resonator. A much better agreement between the measured and predicted resonant frequencies is obtained by taking n2L/‘ as the inductance per unit length, instead of L0; here, we have used n2L as the inductance of the input coil (Chapter 5). This assumption leads to the predicted resonant frequency [40] 24 Fig. 8.10 Gain versus frequency for six microstrip SQUID amplifiers in two different configurations with counter- electrode grounded and Vþ U . Numbers refer to coil lengths in millimeters.
- 47. 8.4 Microstrip SQUID Amplifier f0(n2L) = 1/2n(‘LC0)1/2 . (8.40) The resonant frequencies predicted by this model are also listed in Table 8.2. The agreement with the measured values is acceptable for the longer coils, but becomes progressively worse as the coil length is reduced. This trend may possi- bly be explained by the fact that the parasitic inductance of the wiring to the chip becomes progressively more important as the length of the coil is shortened. The electromagnetic behavior of the microstrip SQUID amplifier has been stud- ied using both an analog model and numerical calculations [40]. The analog model confirmed the validity of Eq. (8.40) for the resonant frequency. The numer- ical simulations, using both lumped circuit and distributed element models, were used to study the effects of positive and negative feedback, corresponding to a grounded electrode with transfer functions Vþ U and V U . These simulations agree well with the observation that the peak in the gain occurs at a frequency above and below the resonance, respectively. The same model was used to investigate the input impedance of the microstrip, which is dominated by the complex impe- dance of the SQUID. For Vþ U the input resistance becomes negative below the k/2 resonance in the frequency range where the gain is high. Correspondingly, for V U the input resistance is negative above the k/2 resonance, where the gain is also high. For low gains (above the resonance for V U and below the resonance for Vþ U ), the input resistance is positive and the return loss is high. These simulations fol- lowed the trends in measured values of the input impedance remarkably well. In a further series of experiments, the harmonic distortion and intermodulation distortion were investigated [41]. Biased for maximum gain, the microstrip SQUID amplifier generates third harmonic signals with an amplitude in good agreement with a model based on a sinusoidal flux-to-voltage transfer function. The amplitude of the third harmonic is less than 1% of the fundamental for a flux amplitude of 0.1 U0. However, under the same bias conditions, departures from a sinusoidal transfer function produce a second harmonic signal. This signal can be reduced or even eliminated by adjusting the flux bias empirically away from the point of maximum gain. Similarly, the third- and fifth-order intermodulation products can be non-negligible with the SQUID biased for maximum gain, but one or the other can be reduced if the flux bias is adjusted appropriately. We have seen that substantial levels of gain can be achieved with the microstrip SQUID amplifier. However, the frequency at which the gain peaks is evidently fixed by the length of the microstrip, whereas some applications demand tuneabil- ity. Fortunately, one can tune the frequency quite simply by connecting a varactor diode between the otherwise open end of the input coil and the washer [42]. The capacitance of the diode can be varied by changing the value of the reverse bias voltage. Changing the capacitance modifies the phase shift of the electromagnetic wave when it is reflected, thereby increasing or decreasing the effective length of the microstrip and lowering or raising the peak frequency. Experiments were carried out with a GaAs varactor diode, the capacitance of which could be varied from 1 to 10 pF by changing the bias voltage from 1 V to –22 V. Two diodes in parallel were used to increase the tuning range. The diodes, 25
- 48. 8 SQUID Voltmeters and Amplifiers in series with a capacitor, were connected between the washer and the end of the input coil not connected to the signal source. The gain for optimized current and flux biases for a SQUID with 31 turns is shown in Figure 8.11 for 9 values of the capacitance of the two diodes. We see that the peak frequency is progressively low- ered, from 195 MHz to 117 MHz, as the capacitance is increased. The maximum gain is constant to within 1 dB over this range. In the absence of the varactor, the peak frequency is about 200 MHz. In fact, the presence of the varactors increases the gain, most likely by increasing the degree of positive feedback. The depen- dence of the peak frequency on the varactor capacitance is in reasonable agree- ment with a simple model [42]. A potential concern is whether the varactor diode can introduce additional noise into the amplifier. Estimates of the contributions of the Nyquist noise and shot noise of the diode and of the noise on the bias voltage indicate that they should not be significant. The measured noise temperatures of a particular device with and without the varactor diodes were identical to within the uncertainties. 8.4.4 The Microstrip SQUID Amplifier: Noise Temperature We turn now to the central issue, the noise temperature. An accurate way of mea- suring TN is to increase the temperature T of the input load by means of a heater, so that the resistor provides a well-defined source of Nyquist noise power. The noise power at the output of the postamplifier is given by PN(f) = kB(T + TN)RiGGP + kBTPRiGP , (8.41) where G and GP are the (power) gains of the SQUID amplifier and postamplifier, and TP is the noise temperature of the postamplifier. By measuring the output 26 Fig. 8.11 Tuning the microstrip SQUID amplifier. Gain versus frequency for a 31-turn SQUID at 4.2 K for 9 values of reverse bias voltage applied to the varactor diodes connected between the open end of the coil and the washer. (Reproduced with permission from ref. [42].)
- 49. 8.4 Microstrip SQUID Amplifier noise power with a spectrum analyzer for several values of T, one can infer TN + TP/G. Measuring TP separately using a similar method, one can deduce TN. This discussion also makes it clear that one requires TP/G TN to ensure that the post- amplifier noise does not contribute significantly to the system noise temperature. In preliminary experiments, the postamplifier was at room temperature and had a noise temperature of abut 80 K. Thus, with a typical SQUID power gain of 200 at the peak, the postamplifier contributed a noise temperature of about 0.4 K. Subsequent experiments made use of a single-stage postamplifier using a hetero- junction field effect transistor (HFET – Fujitsu FHX 13LG) operated in the 4He bath. This postamplifier had a noise temperature of 10 – 1.5 K. Figure 8.12 shows the system noise temperature versus frequency for a device with a peak frequency near 365 MHz, cooled to 1.8 K. The peak gain of the microstrip SQUID was 24.5 – 0.5 dB. The noise temperature increases rapidly as the frequency moves away from the resonance; this effect is due largely to the resonant network used to cou- ple the SQUID to the HFET. The minimum system noise temperature is 0.28 – 0.06 K, to which the postamplifier contributes 0.09 – 0.02 K. Thus, the intrinsic noise temperature is 0.19 – 0.06 K, an order of magnitude lower than the bath temperature. The noise temperature is also an order of magnitude lower than that of state-of-the-art, cooled semiconductor amplifiers [43]. An alternative way to determine TN is to couple the input of the microstrip to a tuned circuit, as shown in Figure 8.13 [44]. The tuned circuit consisted of a 1 pF capacitor and a four-turn copper coil, about 4 mm in diameter, inductively coupled to the microstrip by means of a loop of wire. The resonant frequency was about 438 MHz. In a separate experiment, the loop was connected to a 50-X cable, and the distance between the coil and the loop was adjusted to produce a 2-dB loss, thus reducing Q by a factor of about two. To determine the gain and Q, a second loop, with a coupling loss of 10 dB to the tuned circuit, was connected to a signal generator via a cold 20-dB attenuator and a stainless steel cable with a loss of 3 dB. The measured transmitted power at 4.2 K is shown in Figure 8.14 (upper trace). 27 Fig. 8.12 Noise temperature for a microstrip SQUID amplifier with 29-turn spiral input coil. The device and its HFET postamplifier were cooled to 1.8 K. (Reproduced with permission from ref. [44].)
- 50. 8 SQUID Voltmeters and Amplifiers The Q is about 270 and the maximum gain is 22 dB. The gain is referred to the input coupling loop of the resonant circuit, and includes an added 12 dB to account for the coupling losses. The noise generated by the resonant circuit, with the generator disconnected, is shown in the lower trace of Figure 8.14. The measured peak is 4.7 dB above the (nearly white) noise at 432 MHz. This peak contains contributions from the Nyquist noise of the resonant circuit and from the system noise of the SQUID amplifier. Although it is not entirely straightforward to separate these contribu- tions, using the following approximate argument one can show that this result is consistent with the measured noise temperature. On resonance, since the micro- strip reduces Q of the resonant circuit to approximately half of its unloaded value, the source impedance presented to the microstrip is roughly equal to the charac- teristic impedance of the microstrip. At frequencies well below resonance, the magnitude of the source impedance is approximately xL‘, where L‘ is the induc- tance of the coupling loop. Estimating L‘ ~ 10 nH, we find xL‘ ~25 X at 400 MHz. Since TN T, to a first approximation one can ignore any variation in the noise power with source impedance. Thus, referred to the input of the preamplifier, the 28 Fig. 8.13 Configuration of circuit used to detect Nyquist noise in a resonant circuit inductively coupled to the input of the microstrip amplifier. (Reproduced with permission from ref. [44].) Fig. 8.14 Microstrip SQUID amplifier coupled to the resonant circuit of Figure 8.13. Transmitted power (referred to the loop that couples signal into the resonant circuit) and relative noise power with no input signal at 4.2 K. (Reproduced with permission from ref. [44].)
- 51. 8.4 Microstrip SQUID Amplifier total noise on resonance can be characterized by the temperature [G¢(T/1.58) + G¢TN + TP]; off-resonance, where we assume the noise from the resonant circuit to be negligible, the corresponding temperature is (G¢TN + TP). Here G¢ = 16 dB is the gain of the microstrip SQUID amplifier reduced by the 4-dB loss in the attenuator coupling it to the HFET, and Tp = 15 K is the measured noise tempera- ture of the HFET of 438 MHz. The factor of 1/1.58 accounts for the 2-dB loss be- tween the resonant circuit and the input to the microstrip SQUID amplifier. We can deduce TS = TN + TP/G¢ from the relation 10 log10 {[G¢(T/1.58) + G¢TS]/ G¢TS} = 4.7, and find TS = 1.4 K. For the given values of TP and G¢ we calculate TN = 1 K. It is evident from Eq. (8.23) that TN should scale with the bath temperature. Thus, even lower noise temperatures should be possible with the device cooled to millikelvin temperatures in a dilution refrigerator. However, the contribution of the HFET amplifier, about 0.1 K, would then become dominant. To circumvent this problem, a second microstrip SQUID amplifier was used as a postamplifier [44]. If necessary, the peak frequency of the second SQUID could be tuned with a varactor diode to coincide with that of the first. To prevent the two SQUIDs from interacting with each other, it was necessary to separate them with an attenuating network. For a particular pair of SQUIDs at 4.2 K, the maximum gain was 33.5 – 1 dB. Such a system was cooled in a dilution refrigerator [45]. The two peak frequen- cies were made to coincide by means of small changes in the bias fluxes. All leads connected directly to the SQUIDs were very heavily filtered over a wide frequency range using a combination of lumped circuit and copper powder filters, and a superconducting shield surrounding the SQUIDs eliminated ambient magnetic field fluctuations. The overall gain of the two SQUIDs at 538 MHz was 30 – 1 dB and 32 – 1 dB at 4.2 K and 100 mK, respectively. A third stage of amplification was provided by an HFET with a resonant input circuit, cooled to 4.2 K and con- nected to the SQUID postamplifier via a cryogenic cable with a loss of 6 dB. At 550 MHz, the gain of the HFET was 22 – 1 dB, and its noise temperature TP about 6 K. At the lowest temperatures, the cable loss reduced the effective gain G¢ of the two SQUIDs to 26 – 1 dB so that the HFET contributed a noise temperature TP/G¢ » 15 mK referred to the input of the first SQUID. Two different input circuits were used to measure the noise: one involved measuring the signal-to-noise ratio in the presence of an accurately known signal, and the other measuring the noise from a resonant circuit. The two methods gave nearly identical results, and we describe only the second. During the measurement, a small (about –140 dBm) signal was applied to the tuned circuit, via the coupling loop, at a frequency about 2 MHz above its 519 MHz resonant frequency. This signal was used to optimize the bias currents and fluxes of the two SQUIDs for maximum gain and optimum signal-to-noise ratio. The peak was monitored throughout the measurements to verify that the gain did not drift. The inset in Figure 8.15 shows this peak, together with the noise peak from the resonant circuit. The value of TN is extracted from the peak using the method described above, with the Nyquist noise power replaced by (hf/2kB)coth(hf/2kBT). Figure 8.15 shows the inferred values of TN versus T. The 29
- 52. 8 SQUID Voltmeters and Amplifiers error bars are determined solely by the uncertainty in the spectrum analyzer mea- surement. We see that TN scales as T above about 150 mK, and flattens off at tem- peratures below about 70 mK to 47 – 10 mK; by comparison, the quantum-limited noise temperature, TQ » hf/kB, is about 25 mK. A potential source of the low-temperature saturation of TN is hot electrons pro- duced in the resistive shunts by bias current heating. Wellstood et al. [46] obtained remarkably similar results for the noise spectral density measured at frequencies below about 50 kHz in SQUIDs cooled to around 20 mK, and found good agree- ment with a model in which the temperature of the electrons is determined by their coupling to the phonons. To investigate whether hot electrons were indeed responsible for the saturation in Figure 8.15, Mck et al. [45] remeasured the noise of the same SQUID at 140 kHz, where TQ 10 lK. For this experiment, the out- put of the first SQUID was coupled to the two ends of the input coil of a second SQUID, via a superconducting transformer with a current gain of about 3. The output of the second SQUID was coupled to a room-temperature amplifier with a noise temperature of about 3 K. The temperature dependence of the noise energy at 140 kHz mimicked that plotted in Figure 8.15 quite remarkably, leveling off to 7.5 – 1.3 at low temperatures, thus providing strong evidence that the low-tem- perature saturation of TN indeed arose from hot electrons. 30 Fig. 8.15 Noise temperature of input micro- strip SQUID at 519 MHz versus temperature measured with a resonant source. The dashed line through the data corresponds to TN T, and the dot-dash line indicates TQ = hf/kB » 25 mK. The inset shows the noise peak pro- duced by an LC-tuned circuit at 20 mK. The peak at 520.4 MHz is a calibrating signal. (Reproduced with permission from ref. [45].)
- 53. 8.4 Microstrip SQUID Amplifier 8.4.5 High-Tc Microstrip SQUID Amplifier Tarasov and coworkers [47–50] fabricated and tested microstrip SQUID amplifiers involving YBCO SQUIDs with grain boundary junctions. The spiral microstrip was made either of YBCO, deposited on a separate substrate and coupled to the SQUID in a flip-chip arrangement, or of Au, deposited on the SQUID washer with an intervening layer of insulator. The maximum gain of these devices to date is a few decibels at frequencies of around 1 GHz. However, it is possible that such amplifiers operated at low temperature could achieve very high levels of gain. For example, Tarasov and coworkers [49, 50] fabricated YBCO devices with 0.5 lm grain boundary junctions, and achieved values of I0Rn as high as 8 mV and values of VU as high as 1 mVU0 –1 at 20 K. These values are substantially higher than those achieved with low-Tc devices, where the external resistive shunts drastically reduce the values of I0R that can be achieved. Since the relatively high levels of l/f noise observed in high-Tc SQUIDs (Chapter 5) are not an issue at high frequen- cies, the high-Tc microstrip amplifier is worthy of further investigation for use at or below 4He temperatures. 8.4.6 Future Outlook Several challenges remain for the microstrip SQUID amplifier. Although substan- tial levels of gain have been achieved at frequencies up to 3 or 4 GHz [39], one would like to extend this frequency range up to (say) 10 GHz. Another interesting endeavor would be the development of high-Tc SQUIDs with low critical current – and correspondingly high resistance – at 4.2 K. Such SQUIDs might well have substantially higher values of I0Rn than resistively shunted low-Tc SQUIDs, and thus higher levels of gain as microstrip amplifiers. Perhaps the biggest challenge, however, is to achieve the quantum-limited noise temperature at frequencies of 0.5–1 GHz. The lowest noise temperature yet achieved [45], 47 – 10 mK at 0.519 GHz, was in fact limited by hot electrons in the resistive shunts [46]. It would thus be of great interest to reduce the hot electron temperature by means of cooling fins to attempt to attain the quantum limit. The current understanding of the operation of the microstrip SQUID amplifier [40] is largely empirical. Although this model is adequate for the design of new devices, it would be a significant advance if a first-principles theory were to be de- veloped that would enable one to calculate with reasonable accuracy the gain, fre- quency response and input impedance. The original motivation for the development of the microstrip SQUID amplifier was to improve the performance of an axion detector (Section 8.7). However, another intriguing application is as a postamplifier for the radiofrequency single- electron transistor (RFSET) [51]. This device is a charge detector typically operat- ing at frequencies of several hundred megahertz. Although potentially very sensi- tive, the RFSET has relatively low gain and its noise temperature is generally lim- 31
- 54. 8 SQUID Voltmeters and Amplifiers ited by noise in the HFET to which its output is coupled. It appears likely that the use of a microstrip SQUID as a postamplifier will enable the RFSET to reach the quantum limit of charge detection [52], and several groups are working towards this goal. 8.5 SQUID Readout of Thermal Detectors 8.5.1 Introduction The detection of electromagnetic waves plays an important role in science, tech- nology and everyday life. Several types of cooled sensor for the detection of milli- meter-wave to X-ray radiation are being developed which use a SQUID readout. The development of these sensors is strongly motivated by the requirements for astrophysical observations. A particularly promising development is the use of the voltage-biased superconducting transition-edge sensor (TES) for thermal detectors [53]. Several types of readout multiplexer for TESs are under development with the near-term goal of producing imaging arrays with 104 elements. In the follow- ing sections, we discuss the operation of the voltage-biased TES and the require- ments for the SQUID readout, including SQUID output multiplexers. We then describe the application, design, and performance of far-infrared to millimeter- wave bolometers and X-ray calorimeters. These devices have been the focus of intensive recent development. Finally, we briefly describe several other types of sensor that use SQUID readouts, including magnetic calorimeters, SIS tunnel junction sensors, normal–insulator–superconductor (NIS) tunnel junction sen- sors and kinetic inductance sensors. In a thermal detector, a photon absorbing layer is attached to a low-temperature heat sink by a weak thermal conductance. A thermistor, such as a TES, is attached to the absorber which senses the energy released by one (or more) absorbed photon(s). At far-infrared to millimeter wavelengths, TES thermal detectors are used to measure the rate of arrival of photons. In this mode, the characteristic time for energy deposition is long compared with the thermal relaxation time, so that the temperature rise is proportional to the absorbed power. Thermal detectors operated in this mode are called bolometers. Arrays of TES bolometers are being developed for ground-based, airborne, balloon-borne and space telescopes for use at millimeter to far-infrared wavelengths. From near-infrared to X-ray wavelengths, the temperature pulse from a single absorbed photon is detected. The characteristic time for energy deposition is short compared with the thermal relaxation time of the device, so that the height of the temperature pulse is proportional to the total energy of the absorbed photon. Thermal detectors operated in this mode are called calorimeters. They are used to measure the spectrum of a source as well as its flux. Individual TES calorimeters are used for laboratory X-ray spectroscopy. Arrays of TES calorimeters are being 32
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- 59. The Project Gutenberg eBook of Our Little Siamese Cousin
- 60. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: Our Little Siamese Cousin Author: Mary Hazelton Blanchard Wade Illustrator: L. J. Bridgman Release date: October 8, 2013 [eBook #43908] Most recently updated: October 23, 2024 Language: English Credits: Produced by Emmy, Beth Baran and the Online Distributed Proofreading Team at http://www.pgdp.net (This book was produced from images made available by the HathiTrust Digital Library.) *** START OF THE PROJECT GUTENBERG EBOOK OUR LITTLE SIAMESE COUSIN ***
- 61. Our Little Siamese Cousin THE Little Cousin Series (TRADE MARK) Each volume illustrated with six or more full-page plates in tint. Cloth, 12mo, with decorative cover, per volume, 60 cents LIST OF TITLES By Mary Hazelton Wade (unless indicated otherwise) Our Little African Cousin Our Little Alaskan Cousin By Mary F. Nixon-Roulet Our Little Arabian Cousin By Blanche McManus Our Little Argentine Cousin By Eva Cannon Brooks
- 62. Our Little Armenian Cousin Our Little Australian Cousin By Mary F. Nixon-Roulet Our Little Belgian Cousin By Blanche McManus Our Little Bohemian Cousin By Clara V. Winlow Our Little Brazilian Cousin By Mary F. Nixon-Roulet Our Little Canadian Cousin By Elizabeth R. MacDonald Our Little Chinese Cousin By Isaac Taylor Headland Our Little Cuban Cousin Our Little Danish Cousin By Luna May Innes Our Little Dutch Cousin By Blanche McManus Our Little Egyptian Cousin By Blanche McManus Our Little English Cousin By Blanche McManus Our Little Eskimo Cousin Our Little French Cousin By Blanche McManus Our Little German Cousin Our Little Grecian Cousin By Mary F. Nixon-Roulet Our Little Hawaiian Cousin Our Little Hindu Cousin By Blanche McManus Our Little Hungarian Cousin By Mary F. Nixon-Roulet
- 63. Our Little Indian Cousin Our Little Irish Cousin Our Little Italian Cousin Our Little Japanese Cousin Our Little Jewish Cousin Our Little Korean Cousin By H. Lee M. Pike Our Little Malayan (Brown) Cousin Our Little Mexican Cousin By Edward C. Butler Our Little Norwegian Cousin Our Little Panama Cousin By H. Lee M. Pike Our Little Persian Cousin By E. C. Shedd Our Little Philippine Cousin Our Little Polish Cousin By Florence E. Mendel Our Little Porto Rican Cousin Our Little Portuguese Cousin By Edith A. Sawyer Our Little Russian Cousin Our Little Scotch Cousin By Blanche McManus Our Little Siamese Cousin Our Little Spanish Cousin By Mary F. Nixon-Roulet Our Little Swedish Cousin By Claire M. Coburn Our Little Swiss Cousin Our Little Turkish Cousin
- 64. L. C PAGE COMPANY 53 Beacon Street, Boston, Mass. CHIN.
- 65. Our Little Siamese Cousin By Mary Hazelton Wade Illustrated, by L. J. Bridgman Boston L. C. Page Company PUBLISHERS Copyright, 1903 By L. C. Page Company (INCORPORATED) All rights reserved Published June, 1903
- 66. Fourth Impression, June, 1909 Fifth Impression, November, 1912
- 67. Preface Many years ago there came to America two young men who were looked upon as the greatest curiosities ever seen in this country. They belonged to another race than ours. In fact, they were of two races, for one of their parents was a Chinese, and therefore of the Yellow Race, while the other was a Siamese, belonging to the Brown Race. These two young men left their home in far-away Siam and crossed the great ocean for the purpose of exhibiting the strange way in which nature had joined them together. A small band of flesh united them from side to side. Thus it was that from the moment they were born to the day of their death the twin brothers played and worked, ate and slept, walked and rode, at the same time. Thousands of people became interested in seeing and hearing about these two men. Not only this, but they turned their attention to the home of the brothers, the wonderful land of Siam, with its sacred white elephants and beautiful temples, its curious customs and strange beliefs. Last year the young prince of that country, wishing to learn more of the life of the white people, paid a visit to America. He was much interested in all he saw and heard while he was here. Now let us, in thought, return his visit, and take part in the games and sports of the children of Siam. We will attend some of their festivals, take a peep into the royal palace, enter the temples, and learn something about the ways and habits of that far-away eastern country.
- 69. Contents CHAPTER PAGE I.The First Birthday 9 II.Little Chie Lo 25 III.Night on the River 36 IV.Work and Play 47 V.New Year's 55 VI.White Elephants 61 VII.In the Temple 67 VIII.The Legend of the Peace-Offering 78 IX.Queer Sights 87 X.The Queen's City 98 XI.The Monsoon 104
- 70. List of Illustrations PAGE Chin Frontispiece Chin's Home 29 The Great Temple at Bangkok 40 They carried some of their flowers to the statue of Buddha 57 'They would pick up the logs with their trunks' 63 Siamese Actors 92 Our Little Siamese Cousin
- 71. CHAPTER I. THE FIRST BIRTHDAY If you had seen Chin when he was born, you would have thought his skin yellow enough to suit anybody. But his mother wasn't satisfied, for the baby's nurse was told to rub him with a queer sort of paste from top to toe. This paste was made with saffron and oil, and had a pleasant odour. It made Chin's skin yellower and darker than ever. It did not seem to trouble him, however, for he closed his big brown eyes and went to sleep before the nurse had finished her work. After this important thing had been done, the tiny baby was laid in his cradle and covered over. This does not appear very strange until you learn that he was entirely covered. Not even the flat little nose was left so the boy could draw in a breath of fresh air. It is a wonder that he lived, for his home is very near the equator and the weather is extremely warm there all the time. But he did live, and grew stronger and healthier every day. Each morning he was rubbed afresh and stowed away under the covers of his crib. He had one comfort, although he did not realize it. The mosquitoes could not reach him, and that was a greater blessing than you can, perhaps, imagine. There are millions of these insects in Siam,—yes, billions, trillions,—and the people of that country are not willing to kill one of them! Destroy the life of a living creature! It is a dreadful idea, Chin's mother would exclaim. Why, it is against the laws of our religion. I
- 72. could never think of doing such a thing, even if my darling boy's face were covered with bites. If she were to see one of Chin's American cousins killing a fly or a spider, she would have a very sad opinion of him. She was only fourteen years old when Chin was born. People in our country might still call her a little girl, yet she kept house for her husband, and cooked and sewed and spun, and watched over her new baby with the most loving care. The father was only a little older than the mother. He was so glad that his first baby was a boy that he hardly knew what to do. He was quite poor and had very little money, but he said: I am going to celebrate as well as I can. Rich people have grand parties and entertainments at such times. I will hire some actors to give a little show, at any rate. He invited his friends, who were hardly more than boys themselves, to come to the show. The actors dressed themselves up in queer costumes, and went through with a play that was quite clever and witty. Every one laughed a great deal, and when it was over the guests told the new father they had enjoyed themselves very much. After a few months, Chin had grown strong enough to walk alone. He did not need to be covered and hidden away any longer. His straight black hair was shaved off, with the exception of a round spot on the top of his head, and he was allowed to do as he pleased after his morning bath in the river was over. The bath did not last long, and was very pleasant and comfortable. There was no rubbing afterward with towels, for the hot sunshine did the drying in a few moments. Nor was there any dressing to be done, for the brown baby was left to toddle about in the suit Dame Nature had given him. It was
- 73. all he could possibly desire, for clothing is never needed in Siam to keep one from catching cold. Chin's mother herself wears only a wide strip of printed cloth fastened around her waist and hanging down to the knees. Sometimes, but not always, she has a long scarf draped across her breast and over one shoulder. There are no shoes on her little feet, nor is there a hat on her head except in the hottest sunshine. There are many ornaments shining on her dark skin, even though she is not rich; and baby Chin did not have his toilet made till a silver bracelet had been fastened on his arms, and rings placed on his fingers. After a year or two the boy's ears were bored so that gilt, pear- shaped earrings could be worn there. Soon after that a kind relative made him a present of silver anklets, and then he felt very much dressed indeed. Few boys as poor as he could boast of as much jewelry. Chin was born on the river Meinam in a house-boat. There was nothing strange about that, for the neighbours and friends of the family had homes like his. It was cool and pleasant to live on the water. It was convenient when one wished to take a bath, and it was easy for the children to learn to swim so near home. Yes, there were many reasons why Chin's parents preferred to make their home on the water. Perhaps the strongest one of all was that they did not have to pay any rent for the space taken up by the boat. A piece of land would have cost money. Then, again, if they should not like their neighbours, they could very easily move to a new place on the river. Chin's father built the house, or the boat, just before he was married. He had some help from his friends, but it was not such hard work that he could not have done it all alone.
- 74. A big raft of bamboo was first made. This served as the floating platform on which the house should stand. The framework of the little home was also made of bamboo, which could be got from the woods not far away, and was very light and easy to handle. How should the roof be protected from the heavy rains that fell during a portion of the year? That could be easily managed by getting quantities of the leaves of the atap palm-tree for thatching. These would make a thick, close covering, and would keep out the storms for a long time if they were carefully cemented with mud. The broad, overhanging eaves would give shade to baby Chin when he was old enough to play in the outdoor air, and yet not strong enough to bear the burning sunshine. Of course, there were many windows in the little house, you would think. There were openings in the walls in the shape of windows, certainly, but they were openings only, for they were not filled with glass, nor any other transparent substance. Chin's father would say: We must have all the air we can get. At night-time, when the rain falls heavily, we can have shutters on the windows. They are easily taken down whenever we wish. Why, the whole front of the house was made so it could be opened up to the air and sunshine, as well as the view of passers-by. The family have few secrets, and do not mind letting others see how they keep house. At this very moment, perhaps, Chin's mother is sitting on the edge of the bamboo platform, washing her feet in the river; his grandmother may be there preparing the vegetables for dinner; or, possibly, Chin himself is cleaning his teeth with a stick of some soft wood. The boy's mother has taught him to be very careful of his teeth. It is a mark of beauty with her people to have them well blacked.
- 75. They will tell you, Any dog can have white teeth. But there is nothing they admire more than bright red gums showing plainly with two rows of even, dark-coloured teeth. How do they make their gums such a fiery red? It is caused by chewing a substance called betel, obtained from a beautiful kind of palm-tree very common in Siam. Many of Chin's brown cousins chew betel, as well as the people of his own land. It is even put in the mouths of babies. Betel-chewing grows to be such a habit with them that they become unhappy and uncomfortable if long without it. Even now, although Chin is only ten years old, he would say: I can go without food for a long time, if need be, but I must have my betel. Let us go back to the boy's home. If we should count the windows, we should find their number to be uneven. The Siamese believe something terrible would be sure to happen if this were not so. They seem to think There is luck in odd numbers, for not only the steps leading to the houses, but the stairs leading from one floor to another must be carefully counted and made uneven. There are three rooms in Chin's home. First, there is the sitting- room, where friends are received, although there is much less visiting done in Siam than in many other countries. It took little time and money to furnish the room. There are no pictures or ornaments here. There are two or three mats on which one may sit, and there is a tray filled with betel from which every one is invited to help himself. If callers should arrive and the betel were not offered to them, they would feel insulted and would go away with the intention of never coming to that house again.
- 76. The second room is that set apart for sleeping. Very little furniture is found here, as well, for all that Chin's father had to prepare was a number of long, narrow mattresses, stuffed with tree-cotton. Some pillows were made in the shape of huge bricks. They were also packed full of tree-cotton, and were stiff, uncomfortable-looking things; but Chin and his parents like them, so we should certainly not find fault. You remember there are great numbers of mosquitoes in the country. How do they manage to sleep when the air around them is filled with the buzzing, troublesome creatures? Coarse cotton curtains hang from the roof down over the beds. While these keep the mosquitoes away from the sleepers, they also keep out the air, so it is really a wonder that one can rest in any comfort. When Chin is in the house during the day, he spends most of his time in the kitchen, which is also the eating-room. But, dear me! it is a smoky place, for the boy's father never thought of building a chimney. The cooking is done over a little charcoal stove and, as the flames rise, the smoke rises, too, and settles on the ceiling and walls. Chin has had many good meals cooked over the little fire, and eaten as the family squatted around the tiny table. Just think! It stands only four inches above the floor, and is not large enough to hold many dishes. That does not matter, for each one has his own rice-bowl on the floor in front of him. Chin has been brought up so that he is satisfied with one or two things at a time. The little table is quite large enough to hold the dish of curried fish or meat from which each one helps himself. Chin is a very nice boy, yet I shall have to confess that he usually eats with his fingers! Yes, not only he, but his father and mother and sister, and even grandmother, do the same thing. One after another helps himself from the same dish and thinks nothing of it.
- 77. People who are a little richer use pretty spoons of mother-of- pearl; Chin's mother owns one of these useful articles herself, but of course, that won't serve for five persons, so it is seldom seen on the table. As for knives and forks, she never even saw any. One of her friends once watched a stranger from across the great ocean eating with these strange things. She laughed quietly when she told of it, and said: It must take a long, long time before one can get used to them. They are very clumsy. As Chin squats at his dinner he can look down through the split bamboos and see the water of the river beneath the house. It does not matter if he drops some crumbs or grains of rice. They can be easily pushed through the cracks, when down they will fall into the water to be seized by some waiting fish. The good woman doesn't even own a broom. Her house-cleaning is done in the easiest way possible. Anything that is no longer useful is thrown into the river, while the dirt is simply pushed between the wide cracks of the floor. The dish-washing is a simple matter, too. Each one has his own rice-bowl, and after the meal is over it is his duty to clean it and then turn it upside down in some corner of the kitchen. It is left there to drain until it is needed again. Chin's mother cooks such delicious rice that he wonders any one can live without it. He needs no bread when he can have that, for it is a feast in itself. When poured out, it looks like a mountain of snow; each grain is whole and separate from the others. It is cooked in an earthen pot with the greatest care, and, when it is done, never fails to look beautiful and delicate. Chin's mother would think herself a very poor housekeeper if she should make a mistake in preparing the rice.
- 78. When a dish of rat or bat stew is added to the meal, Chin feels that there is nothing more in the world that he could wish. He knows that the rich people in the city often have feasts where twenty or thirty different dainties are served. But he does not envy them. A person can taste only one thing at a time, and nothing can be better than a stew with plenty of curry and vegetables to flavour it. We don't need to think of the rats and bats if it is an unpleasant idea. As for Chin, if he had seen you shudder when they were spoken of, he could not have imagined what was the matter.
- 79. CHAPTER II. LITTLE CHIE LO Chie Lo! Chie Lo! come out quickly, or you won't see it before it passes, called Chin to his sister. She was playing with her dolls in the sitting-room, but when she heard Chin calling she put them down and came out on the platform where her brother sat dangling his feet in the water and holding his pet parrot. Chie Lo! Chie Lo! screamed the parrot, when she appeared. He was a bright-looking bird with a shining coat of green feathers and a red tuft on his head. He must have loved Chie Lo, for he reached up for her to pat him as she squatted beside her brother. Look, look, said Chin, isn't that grand? The boy pointed to a beautiful boat moving rapidly down the river. It is the king's, you know, he whispered. Do you see him there under the canopy, with his children around him? Yes, yes, Chin, but don't talk; I just want to look. It was no wonder that Chie Lo wished to keep still, for it was a wonderful sight. The boat was shaped like a huge dragon, whose carved head, with its fierce eyes, could be seen reaching out from the high bow. The stern was made in the shape of the monster's tail. The sides of the royal barge were covered with gilded scales, inlaid with pearls, and these scales shone and sparkled in the sunlight. A hundred men dressed in red were rowing the splendid boat, and they must have had great training, for they kept together in
- 80. perfect time. Isn't the canopy over the king the loveliest thing you ever saw? said Chin, who could not keep still. It is made of cloth-of-gold, and so are the curtains. Look at the gold embroidery on the king's coat. Oh, Chie Lo, it doesn't seem as though he could be like us at all. I feel as though he must be a god. The young prince who took the long journey across the ocean last year is there with him, Chin went on. Father told me that he visited strange lands where all the people have skins as white as pearls, and that he has seen many wonderful sights. But, Chie Lo, there is nothing in the world grander than our king and his royal boat, I'm sure. As the barge drew nearer, the children threw themselves face downward on the platform until it had passed down the river. It was their way of showing honour to the ruler of the land. In the olden times all who came into the presence of the king, did so in one way only. They crawled. Even his own little children were obliged to do this. No one dared to stand in his presence. But such things have been changed now. The king loves his people and has grown wiser since he has learned the ways of other countries. When he was a little boy, an English lady was his teacher for a long time, and she taught him much that other Kings of Siam had never known. It is partly because of this that he is the best ruler Chin's people have ever had. The royal barge was decorated with beautiful white and yellow umbrellas, many stories high. There was also a huge jewelled fan, such as no boat was allowed to carry except the king's. Other dragon-shaped boats followed the royal barge, but they were smaller and less beautiful. They were the king's guard-boats, and moved along in pairs.
- 81. CHIN'S HOME Many other interesting sights could be seen on the river this morning. Vessels were just arriving from distant lands, while here and there Chinese junks were scattered along the shores. Chin and his sister can always tell such boats from any others. An eye is always painted on the bow. A Chinaman who was once asked why he had the eye there, answered, If no have eye, how can see? It is so much pleasanter outside, it is no wonder that Chin and his sister do not spend much time indoors. After the royal procession had passed out of sight, Chie Lo went into the house and brought out her family of dolls. Of course they did not look like American dolls; you wouldn't expect it. Some of them were of baked mud and wore no clothes. Others were of stuffed cotton and made one think of the rag dolls of Chie Lo's white cousins. The father and mother dolls were dressed in strips of cloth wound around their bodies, just like the real grown-up people of Siam, but the baby dolls had no more clothes than the children of the country. Chie Lo talked to her dolls and sang queer little songs to them. She made believe they were eating, just as other little girls play,
- 82. far away across the great ocean. Then she kissed them and put them to bed on tiny mattresses under the shady eaves of the house. Perhaps you wouldn't have known that Chie Lo was kissing them, however, for the fashions of Siam are quite different from those of our country. She simply touched the dolls' noses with her own little flat one and drew in a long breath each time she did so. That was her way of showing her love,—gentle little Chie Lo. Chin didn't laugh, of course. He was used to seeing his sister playing with her dolls, and as for the kissing, that was the only way of doing it that he knew himself. Chie Lo, I saw some beautiful dolls in a store yesterday, he said, as he stopped working for a minute. He was making a new shuttlecock for a game with his boy friends the next day. What kind were they, Chin? asked his sister. They were lovely wooden ones. Only rich children could buy them, for they cost a great deal. I wish I could get one for you, Chie Lo, but you know I haven't any money. What else did you see, Chin? There were doll-temples in the store, and boats filled with sailors, and lovely ivory furniture for the doll-houses. You must see the things yourself. Chie Lo went on with her play. She finished putting her own toy house in order. It was one Chin had made for her. It looked like her own home,—it stood on a bamboo platform, it had a high, slanting roof, covered with palm leaves, and there were three rooms inside. Chin was a good boy to make it. All brothers were not as kind as he. Yes, I should like to see all those things, Chie Lo answered, after awhile. But I am happy here with my own toys. I must row up the river to-morrow and sell some fruit for father. I won't have any time for play then.
- 83. Come to dinner, children, called their mother. Chin, take this jug and get some fresh water before you come in. She handed a copper jug to Chin. He quickly filled it by reaching over the platform, and followed his sister into the kitchen a moment later. Every one was thirsty, and the jug was passed from one to another for each to help himself. There were no tumblers nor cups. Chin had made small dishes for his mother by cutting cocoanuts in halves and scooping out the delicious cream from the inside; but they did not use them for drinking the water. Nor did they put their lips to the jug. Each one cleverly twisted a palm leaf into the shape of a funnel and received the water through this. It was done more quickly than I can tell you about it. Chin and his sister thought it was a fine dinner. The evening dews were falling, and a gentle breeze came floating down the river. The terrible heat of the day was over and it was the very time to enjoy eating. In the first place, there was the dish of steaming rice. There was also a sort of stew made of meat chopped very fine and seasoned with red pepper. If you had tasted it, you would probably have cried: Oh dear, my mouth is burnt; give me a drink of water at once. But Chin and Chie Lo thought it very nice indeed, and not a bit too hot. Isn't this pickled turnip fine? said Chin's mother. I bought it this morning from a passing store. What could she mean by these words? It was a very common thing for these little brown cousins to see not only houses but stores moving past them down the river. The storekeepers were always ready to stop and sell their goods to any one who wished them.
- 84. Chin's mother never made bread, nor pies, nor cake, nor puddings. She bought most of the vegetables already cooked from the floating stores, so you can see she had quite an easy time in preparing her meals. But to-day, after the rice and stew had been cooked, she laid bananas to roast in the hot coals, and these were now taken out and handed to her family as they squatted on the mats around the table. If the children had no bread with their dinner, they ought to have had milk, you think. But they never drink it. The cows of Siam are not milked at all, and so the rich children of the country are brought up in the same way as Chin and his sister. When the meal was finished, Chie Lo did not forget that her dear pussy must still be fed. It was an odd-looking little creature. Although it was a grown-up cat, yet its eyes were as blue as those of a week-old American kitten. It had a funny little tail twisted up into a knot. It was better off than many other cats of Siam, however, who go about with none at all.
- 85. CHAPTER III. NIGHT ON THE RIVER After Chie Lo had watched her pussy eat all the fish she could possibly wish, the children went outdoors again to sit in the cool evening air. The night was already pitch-dark, for there was no moon, and there is no long twilight in the tropics at any season of the year. But what a beautiful sight now met the children's eyes! It seemed almost like fairy-land, there were so many lights to be seen in every direction. Their home stood just below the great city of Bangkok, and along the shores of the river the houses and palaces and temples could be seen almost as plainly as in the daytime. Floating theatres were passing by, each one lighted with numbers of coloured paper lanterns. Look! look! cried Chin. There are some actors giving a show outside. They want to tempt people to stop and come in to the play. See the beautiful pointed finger-nails on that one. What fine care he must take of them! It is no wonder Chin noticed the man's finger-nails, for they were at least five inches long. See the wings on the other actor, Chin, said his sister. I suppose he represents some strange being who does wonderful deeds. I should like to go to the play. Look! there is a party of people who are going on board of the theatre.
- 86. The children now turned their eyes toward the small boat of a Chinaman who was calling aloud to the passers-by: Come here and buy chouchou; it is a fine dish, indeed. A moment afterward he was kept so busy that he had no time to call. His canoe was fairly surrounded by other boats, for many people were eager to taste the delicious soup he served from an odd little stove in front of him. It is hard to tell how chouchou is made. Many kinds of meat and all sorts of vegetables are boiled down to jelly and seasoned with salt and pepper. He must have had a good recipe, for every one that tasted his chouchou seemed to like it and want more. Listen to the music, Chie Lo, said her brother, as he turned longingly away from the chouchou seller. It seemed more like noise than music. Two men stood on a bamboo raft causing loud, wailing sounds to come from some queer reed instruments. A third player was making the loudest noise of all. He sat in the middle of a musical wheel, as it is called. This wheel is made of metal cups of different sizes placed next each other in a circle. It seems strange that Chin and his sister should enjoy such music, and stranger still that the grown-up people should also like it; but they seemed to do so. Were they doing it for their own pleasure? Oh no, they had dainties to sell as well as the chouchou maker, and this was their way of attracting attention. New sights could be seen constantly. Here were the beautifully- trimmed boats of the rich people taking a ride for pleasure after the heat of the day. There were the canoes of the poor, who were also out to enjoy the sights, for Bangkok is a city built upon the water. The river Meinam flows through its very centre. The name of the river means Mother of Waters, just as the name of our own
- 87. THE GREAT TEMPLE AT BANGKOK. Mississippi means The Father of Waters. It is well named, for many canals reach out from it in different directions. If a person is going to a temple to worship, if he has shopping to do, or a visit to make, he does not take a car or carriage, nor does he often walk. He steps into a boat, and after a pleasant sail or row, he finds himself at his journey's end. Let's go down the river before we go to bed, said Chin, who had grown tired of sitting still. He stepped from the platform into his own little canoe and Chie Lo followed him. The children looked very much alike. Their faces were of the same shape, their eyes were of the same colour, and the two little round heads were shaved in exactly the same way. A tuft of hair had been left on the top of each and was coiled into a knot. When Chin grew a little older there would be a great celebration over the shaving of his tuft. It would mark his coming of age, but that would not be for two or three years yet. He was only eleven years old now and was left to do much as he pleased. The little canoe made its way in and out among the big boats and soon left the city behind. Tall palm-trees lined the banks of the river
- 88. and waved gently in the evening breeze. Suddenly there was a loud sound, like a big drum, in the water directly under the boat. Tom, tom! Tom, tom! It startled Chie Lo, and she exclaimed: What is it, Chin? What is it? It must be a drum-fish, Chie Lo. Nothing else could make a sound like that. Of course, Chin. It was all so quiet, and then the sound was so sudden, I didn't think for a moment what it could be. They had often seen this ugly-looking fish, which is never eaten by the people of their country. It is able to make a loud noise by means of a sort of bladder under its throat, and it is well called the drum-fish. The children still went onward, keeping time with their sculls. Suddenly the air around them blazed with countless lights, and a moment afterward the darkness seemed blacker than ever. Then, again the lights appeared, only to be lost as suddenly, while Chin and his sister held their oars and watched. Aren't they lovely? said Chie Lo. I never get tired of looking at the fireflies. It is no wonder she thought so. The fireflies of Siam are not only very large and brilliant, but they are found in great numbers. And, strange to say, they seem fond of gathering together on certain kinds of trees only. There they send forth their light and again withdraw it at exactly the same moment. It seems as though they must be under the orders of some leader. How else do they keep together? I can hear the trumpeter beetle calling along the shore, said Chin, as the boat floated about. He makes a big noise for his size, and takes his part in the song of the night. There must be hundreds
- 89. of lizards singing up there among the bushes, too, and I don't know what else. I suppose the parrots are asleep in the tree-tops by this time, as well as the monkeys. Don't you love to go about in the woods, Chin? It is almost the best fun in the world, I think. Oh, Chie Lo, I saw something the other day I didn't tell you about. You made me think of it when you spoke of the monkeys. Father and I had gone a long way up the river in the canoe to get wild bananas. We had just turned to come home when I saw a crocodile ahead of us, lying close to the shore. His wicked mouth was wide open and his eyes were glittering. All at once I saw what was the matter. A chain of monkeys was hanging from a tree-top above him. They were having sport with the monster. The lowest monkey would suddenly strike out with his paw and touch the crocodile's head when he was off his guard. Then the whole chain of monkeys would swing away as quick as a flash, and the crocodile would snap too late. Oh, he did get so angry after awhile, it made me laugh, Chie Lo. The monkeys grew bolder after awhile, and chattered more and more loudly. Then the crocodile began to play a game himself. He shut his eyes and pretended to be asleep. Down swung the monkeys, straight over his head. His jaws opened suddenly in time to seize the little fellow who had been teasing him. That was the last of the silly little monkey, whose brothers and sisters fled up into the tree-tops as fast as they could go. I didn't see them again, but we could hear them crying and wailing as long as we stayed near the place. I wish I had been there, sighed Chie Lo. It must have made you laugh to watch the monkeys before they were caught. But they are easily scared. I shouldn't be afraid of monkeys anywhere.
- 90. Chin smiled when his sister said these words. If there were enough monkeys together, Chie Lo, and if they were all angry and chasing you, I don't think you would exactly enjoy it. Father told me of a time when he was off with a party of men in a deep forest. They caught a baby monkey, and one of the men was going to bring it home. It made the mother wild to have her child taken from her. She raised a loud cry and started after the men. Her friends and relatives joined her, crying and screaming. But this was not all, for every other monkey in the forest seemed to get the idea of battle. On they came by the hundreds and the thousands. Do you think those men weren't scared? They hurried along as fast as they could, stumbling over bushes and floundering in the mud. They were only too glad to reach the bank of the river, where they jumped into the canoes and paddled quickly away. The monkeys crowded on the shore and screamed at them. I wish I could have seen them. Chin lay back and laughed as he finished the story. We mustn't stop to talk any more, for it is getting late, said Chie Lo. But I love to hear you tell these stories, Chin. I hope you will remember some more to-morrow night. Now we must paddle home as fast as we can go.
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