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IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, Vol. 17, NO.2, pp. 760 – 763 June 2007
A magnetic-field-fluctuation thermometer for
the mK range based on SQUID-magnetometry
J. Beyer, D. Drung, A. Kirste, J. Engert, A. Netsch, A. Fleischmann, C. Enss
Abstract— We are developing a compact and easy-to-use
thermometer for the temperature range of about 10 mK to 4 K
based on the measurement of magnetic noise above the surface of
a metal body which acts as a temperature sensor. The metal body
is thermally anchored to the temperature to be measured. The
magnetic field fluctuations arise from the thermally agitated
motion of electric charges and can be related to the temperature
of the metal via Nyquist’s relation of the noise in a conductor. We
measure the magnetic field fluctuations with a highly sensitive
low-Tc dc SQUID magnetometer which is at the same
temperature stage as the metal body and in close vicinity to the
metal surface. The temperature to be measured is extracted from
the spectrum of thermal magnetic noise detected by the
magnetometer. The SQUID magnetometer used is a miniature
multiloop magnetometer with maximized field sensitivity and low
power dissipation. The spectrum of thermal magnetic noise
detected by the magnetometer is significantly affected by the
configuration of the metal sensor and the magnetometer. We
discuss considerations regarding the configuration of an
integrated magnetic-field-fluctuation thermometer and present
measurements of its sensitivity and speed.
Index
Terms—Low-temperature
thermometer, SQUID magnetometry
A
thermometry,
noise
I. INTRODUCTION
n absolute thermometer is a thermometer which measures
the thermodynamic temperature T = g ( xi ) without the
need of a second thermometer in order to determine the
parameters xi of its thermometer function g. An absolute
thermometer is considered a primary thermometer when
metrological quality regarding its uncertainty and
reproducibility can be achieved [1]. Thermometers for which
the thermometer function g is known and the variables xi can
be determined at a reference point of known temperature are
called semi-primary thermometers.
Noise thermometry is one of few established approaches to
primary thermometry. It is based on the direct relationship
between the thermal noise, typically referred to as Nyquist or
Johnson noise, generated by the thermally agitated motion of
electric charges in a conductor and the thermodynamic
temperature of that conductor. Noise thermometer
implementations with two main principles of operation have
been developed [2,3]. For the temperature range below 4.2 K
Manuscript received August 24, 2006.
J.Beyer (phone: ++49-30-3481-7379; fax: ++49-30-3481-69-7379; e-mail:
[email protected]), D.Drung, A.Kirste, and J.Engert are with PhysikalischTechnische Bundesanstalt Berlin D-10587 Germany,.
A.Netsch, A.Fleischmann, and C.Enss are with Kirchhoff Institut für
Physik of Universität Heidelberg D-69120 Germany.
most notably the Current Sensing Noise Thermometer (CSNT)
[4], which detects the current noise of a resistor by means of a
SQUID based current sensor, has undergone further
development in recent years in order to improve its
practicality, sensitivity, and speed [5].
We are developing another type of noise thermometer for
the temperature range below 4.2 K which we call the
Magnetic-Field-Fluctuation Thermometer (MFFT). The basic
working principle of the MFFT is the inductive detection of
magnetic field fluctuations in the vicinity of a conductor and
the extraction of the conductor temperature from the spectrum
of this thermal magnetic noise [6]. An inductive approach to
noise thermometry was also used by Varpula et.al [7],
however in a LC-resonant circuit configuration and for a range
of much higher temperatures of 300 K – 1300 K.
II. MAGNETIC-FIELD-FLUCTUATION NOISE THERMOMETER
FOR LOW TEMPERATURES
A. Principle of Operation
The thermal magnetic noise at a point at distance z above
the surface of a conductor at temperature T with electrical
conductivity σ is caused by the superposition of the magnetic
fields arising from the multitude of noise current modes in the
conductor. It has been analyzed theoretically in [8] and [9].
The expressions derived therein for arbitrary conductor
geometries are very complex and were treated either
numerically or approximated for certain conductor shapes. An
expression deduced in [8] approximates the power spectrum
density of the thermal magnetic flux density noise SB at a
distance z above the conductor surface for the case of a large
conducting plate of thickness d with d << z :
µ0 2 k B T σ d
SB =
8 π z2
2f
1+
π fc
2
(1)
with µ0 – permeability of free space and kB – Boltzmann
constant, f – frequency and fc = (4µ0σ zd)-1. It also illustrates
the scaling of SB with frequency for other geometries, for
instance when z becomes comparable or smaller than the
dimensions of the conductor. The spectrum of SB exhibits a
zero-frequency value which is proportional to the conductor
temperature T and a “low-pass-like” frequency fall-off with a
characteristic frequency fc. For a fixed geometry fc is
independent of temperature provided that σ does not vary with
T. This means that for σ = const(T) the thermal magnetic noise
values SB ( f ) at any frequencies are directly proportional to T.
A calibration at one known temperature, therefore, provides
IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, Vol. 17, NO.2, pp. 760 – 763 June 2007
the means for a semi-primary thermometer based on the
measurement of SB ( f ). Following the consideration above,
the resulting thermal magnetic noise spectrum can be
considered as the “superposition” of the spectra of the various
noise current modes which extend over different regions of the
conductor. For frequencies below the characteristic frequency
fc noise current modes extending over larger regions of the
conductor volume contribute more to SB( f ) than for
frequencies above fc. Therefore, fc is a meaningful parameter
for the speed of the Fourier transform based MFFT, similar to
fc = R / 2π L (L – inductance of SQUID input, R – noise resistor)
in the case of the CSNT.
In order to build a MFFT for the low temperature range two
essential components are needed: a temperature sensor made
of a material with an electrical conductivity highly constant in
that temperature range, and a magnetometer sensitive enough
to detect the thermal magnetic field noise. Noble metals are
widely used materials in cryotechnics. They can be prepared
in high purity and typically show both a high and a close to
constant conductivity at low temperatures [10,5].
Regarding the magnetic field sensor for the low temperature
MFFT, a SQUID based device is the natural magnetometer as
it combines highest sensitivity and operability in the relevant
temperature range. A SQUID magnetometer detects integral
magnetic field changes over the (effective) area of its pickup
coil. Therefore, for a fixed configuration of sensor and SQUID
pickup coil a characteristic temperature-independent shape of
the thermal magnetic flux noise spectrum SΦ ( f ) is obtained if
σ = const(T).
A thermometer noise temperature of the MFFT can be
defined analogously to the approach in the CSNT case [4].
The MFFT noise temperature TN is the temperature at which
the thermal magnetic flux noise equals the flux noise of the
SQUID magnetometer. Both the thermal magnetic flux noise
and the SQUID flux noise vary with frequency so that TN
becomes frequency dependent as well. Clearly, the
thermometer noise temperature in the frequency range used
for the MFFT measurement should be much lower than the
minimum operating temperature. The thermal magnetic noise
signal falls off with frequency, and it is common that SQUIDs
show low-frequency excess noise which can even increase at
lower temperature. Hence, it is desirable to obtain a
characteristic frequency fc for the MFFT configuration well
above the onset of the SQUID low-frequency excess noise.
B. Measurements
In a proof-of-principle experiment the operation of a low
temperature MFFT was demonstrated [6]. The setup used
employs a rod of high purity Au as the temperature sensor.
The thermal magnetic noise is picked up by a superconducting
flux transformer consisting of a gradiometric pickup coil
wound around the Au rod connected to the input coil of a
SQUID placed at 1K. Based on a Fourier transform analysis
the thermal magnetic noise spectrum is determined. The
characteristic spectrum obtained for this setup can be
explained by employing the fluctuation-dissipation theorem.
The losses in the superconducting pick up coil due to the
presence of the Au sensor are expressed as the resistive
impedance R ( f ) of the coil. For certain geometries R ( f ) can
be calculated analytically [11], for more complex
configurations a numerical computation is useful. The thermal
magnetic flux noise in the coil is given by [12]
SΦ =
k B T R( f )
.
π2f 2
(2)
With this setup measurements were taken in the temperature
range between 208 mK and 21 mK. A noise temperature of ca.
0.17 mK was obtained in the frequency range used for
determining the temperature. The measured temperatures were
compared with superconducting reference points providing
temperature values according to the Provisional Low
Temperature Scale PLTS - 2000. A direct proportionality
SΦ ∝ T to within 1% was measured, and the characteristic
shape of the spectrum SΦ( f ) was found to be temperature
independent in the observed temperature range.
In order to make the MFFT more compact as well as to
achieve lower noise temperatures we changed the approach of
using a wire-wound superconducting pickup coil and a
SQUID operated at a fixed temperature. Instead, as shown in
Fig.1, the SQUID magnetometer chip is placed directly onto
the temperature sensor. This makes it possible that the SQUID
flux noise reduces with temperature over a significant range of
the MFFT operation temperature. Furthermore, a lithographically fabricated pickup coil can be more easily
configured for MFFT use as well as more precisely defined
with respect to the temperature sensor. The SQUID
magnetometer is a miniaturized multiloop dc SQUID [13] of
type C3WM with 110 pH inductance and with an outer pickup
coil dimension of 2.9 mm. Stripline spokes are used to
improve the magnetic field sensitivity to B / Φ = 2.9 nT/Φ0,
with B - average magnetic flux density, Φ - magnetic flux thru
the SQUID, and Φ0 = 2.07 fT m2. The integrated feedback coil
is led around the pickup coil close to its inner edge. A
minimum of four wires, preferably made from
superconducting material, are needed to read out the SQUIDmagnetometer in a flux-locked loop (FLL) configuration.
superconducting
shield
high purity Cu
thermal
anchoring
3 mm
Fig. 1. Integrated Magnetic-Field-Fluctuation Thermometer: A multiloop
SQUID magnetometer (chip size 3 mm x 3 mm x 0.1 mm) is mounted
directly onto a sheet of high purity (99.999%) Cu of thickness ca. 0.25 mm.
The setup is placed inside a can-shaped superconducting magnetic shield.
IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, Vol. 17, NO.2, pp. 760 – 763 June 2007
S Φ ,FIT ( f ) = S Φ (0 )
p1
Φ
(a)
SΦ,FIT ( f ) = SΦ (0)
2f
1+
π 133 Hz
1.365 1.258
(b)
(c)
p2
!
2 f
1+
π fc
the uncertainty of the fit variable SΦ(0). In Fig.2(b) both TMFFT
and a linear fit are plotted against the reference temperature T.
The relative differences (T - TMFFT) / T are given as percentage
values. The MFFT temperature agrees to better than 1.5 %
with the reference temperature. For the particular SQUID
magnetometer used for these MFFT measurements an upper
bound for the SQUID flux noise level of 4.2 µΦ0/√Hz at
325.5 mK can be estimated. This atypically poor noise
performance was caused by adverse junction parameters. Still,
this SQUID flux noise level corresponds to TN = 0.03 mK.
Hence, we expect TN << T to be fulfilled within the intended
Φ
For the purpose of choosing a suitable shape of the
temperature sensor the SQUID magnetometer can be
considered a system with one degree of freedom which is the
magnetic energy stored in the inductance of the SQUID loop
[12]. Here, we neglect the effect of the Josephson junctions on
the total inductance for different flux values. This is justified
because in MFFT operation the SQUID is run in a FLL. The
actual shape of the normal conducting temperature sensor and
its position relative to the SQUID loop determine the resistive
impedance R ( f ) of the SQUID loop and, according to
equation (2), the frequency spectrum of the thermal magnetic
flux noise threading the loop. Based on simulations and
experiments we chose the shape of the temperature sensor to
be a sheet with ca. 0.25 mm thickness. The lateral dimensions
of the sheet (18 x 22 mm2) are chosen such that the thicker
thermal anchoring mount is several pickup coil dimensions
away from the magnetometer.
The SQUID chip of thickness ca. 100 µm is glued onto the
metal sensor sheet using varnish GE7031, and the setup is
placed inside a can-shaped superconducting shield. In order to
minimize temperature variations of σ the material used for the
metal sensor in the integrated MFFT is high-purity (99.999%)
Cu with < 1 ppm Iron impurities.
In the integrated MFFT configuration, in contrast to
transformer coupled CSNT or MFFT, one is concerned with
heating effects from the power dissipation of the SQUID
magnetometer that is directly placed onto the temperature
sensor. The power dissipation of our SQUID magnetometers is
typically about 100 pW. The volume of the thermal sensor
itself used in our configuration is ca. 1 cm3. Furthermore,
thermal anchoring of the sensor to a larger metal volume - for
instance the base plate of a refrigerator - is straightforward.
Hence, we do not expect significant hot electron effects [14] in
the intended range of operation temperatures down to 10 mK.
The MFFT configuration described above was operated in a
3
He-cryostat. The cryostat temperature was measured with the
MFFT as well as a calibrated Cernox resistance thermometer
as a reference thermometer. This resistance thermometer has a
calibration uncertainty of < 5 mK [15]. In Fig.2(a) the thermal
magnetic flux noise spectra measured for different reference
temperatures T are depicted. The temperatures measured with
the MFFT were extracted from these spectra as follows. The
noise spectrum measured at the calibration temperature
T = 4.544 K was fitted using a function in the style of (1):
(3)
with SΦ(0), fc, p1 and p2 being the fit variables. For the
temperatures lower than T = 4.544 K the spectra were fitted
according to (3), but here using fc = 133 Hz, p1 = 1.365 and
p2 = 1.258 obtained for T = 4.544 K and the zero-frequency
value SΦ(0) being the only fit variable. The temperature TMFFT
was then calculated assuming SΦ ∝ TMFFT. The uncertainty
u(TMFFT) of the MFFT temperature as well as the relative
uncertainty ur(TMFFT ) = u(TMFFT) / TMFFT were calculated from
!
Fig. 2. (a) Thermal magnetic flux noise spectra measured (dots) at different
reference temperatures T, and fit curves (lines) according to SΦ( f ) given. (b)
TMFFT vs. reference temperature T together with a linear fit. The percentage
values are the relative differences (T - TMFFT) / T. (c) TMFFT vs. time tMEAS for
reference temperature T = 325.5 mK. The relative uncertainty of TMFFT is
ur(TMFFT ) = u(TMFFT) / TMFFT . The error bars of TMFFT are deduced from
u(TMFFT) and the deviation of TMFFT from T = 325.5 mK.
IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, Vol. 17, NO.2, pp. 760 – 763 June 2007
range of operation temperatures down to 10 mK.
The measured thermal flux noise shows the expected “lowpass-like” spectral shape. However, for the fit curves used in
Fig.2(a) the exponents p1 and p2 differ from the values 2 and 1
according to expression (1). The reason for this discrepancy is
twofold. Firstly, the integrated MFFT arrangement does not
conform to the above approximation of a single point with
distance z over a conducting plate with thickness d << z.
Secondly, as the SQUID magnetometer is operated in a FLL
the measured thermal magnetic flux noise spectra are effected
by the frequency response of the feedback coil-to-SQUID
mutual inductance MFB. The feedback coil interacts
inductively with the conductor volume as does the pickup coil.
It is clear that for frequencies at which the skin depth of the
conductor is no longer large compared to the distance between
the feedback coil and the conductor, the frequency response of
MFB will be effected by the absolute value of the conductivity.
MFB( f ) can be easily measured in situ. It represents a built-in
ac induction method [11] to check for changes in the
conductivity of the temperature sensor within the range of
operation temperature. For the MFFT measurement depicted
in Fig.2 MFB( f ) was measured in the range f = 5 Hz..100 kHz.
In the frequency range of about 50 Hz to 1000 Hz MFB( f )
decrease with frequency by about 40%. The change in MFB( f ),
however, was found to be temperature independent between
T = 4.544 K and 0.337 K. This indicates that the conductivity
of the high purity Cu temperature sensor did not vary
significantly over the observed temperature range.
Fig.2(c) illustrates the dependence of the TMFFT
measurement on the time tMEAS at a reference temperature
T = 325.5 mK. Time records with 50% overlap were used to
obtain thermal flux noise spectra with bandwidths of 400 Hz.
The error bars of TMFFT are deduced from u(TMFFT) and the
deviation of TMFFT from T = 325.5 mK. They are a measure for
the statistical uncertainty of the measurement. As depicted in
Fig.2(c) (open circles); ur(TMFFT ) reduces with increasing
tMEAS as expected [5]. A measurement time tMEAS ≥ 15 s yields
ur(TMFFT ) < 1.5 %.
III. SUMMARY
We are developing a novel noise thermometer for the
temperature range of about 10 mK to 4 K based on the
measurement of thermal magnetic noise above the surface of a
conductor. The magnetic field fluctuations arise from the
Johnson noise currents in the temperature sensor. The MFFT
has, therefore, various aspects in common with the CSNT.
This includes being a fast, sensitive, linear and easy-to-use
thermometer for a wide temperature range below 4 K. Beyond
that, the MFFT eliminates the need for galvanic contacts to the
metallic temperature sensor. This is a practical advantage and
eliminates potential effects of the contacts, such as non-ohmic
contact resistances and contamination of the noise resistor. In
the MFFT the sensor has a large volume and surface area
compared to a small thin-film resistor typically used in the
case of CSNT. This provides for better thermal anchoring of
the sensor, which is the more important the lower the
operation temperature. In our integrated MFFT approach a
miniaturized multiloop SQUID magnetometer is directly
placed onto the temperature sensor. This allows us to use
adapted SQUID magnetometer designs to detect the magnetic
Johnson noise. We run the integrated MFFT with a
temperature sensor made of high purity Cu, the shape of which
was chosen to obtain both a high characteristic frequency fc
and a sufficiently low thermometer noise temperature TN.
We will further optimize SQUID magnetometers for
integrated MFFT in two directions. Firstly, a multiloop
SQUID gradiometer design is in preparation for a semiprimary MFFT down to about 10 mK with focus on
practicality and speed. The gradiometer design is expected to
eliminate the need of a superconducting magnetic shielding.
Instead, the temperature sensor will be a closed metal
encapsulation around the SQUID gradiometer and at the same
time act as an eddy current magnetic shielding. Secondly, a
multiloop SQUID magnetometer with pickup coil dimensions
of 1.7 mm [13] will be implemented in a MFFT with a
temperature sensor of known conductivity. A narrow-trace and
high symmetry design is chosen for this SQUID magnetometer
in order to be able to calculate the resistive impedance R ( f )
with high accuracy. With this device we will investigate the
potential of MFFT for primary low temperature thermometry.
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http://www.lakeshore.com/pdf_files/Appendices/LSTC_appendixD_l.pdf