(Adapted from the
Mr. Squid Manual (Star Cryotronics) by Jerry Gollub, with updates by Suzanne
Amador Kane 5/2005)
We are used to thinking of quantum mechanics as a way of
understanding the properties of small things:
atoms, nuclei, etc. By contrast,
the macroscopic world often follows the classical laws of Newtonian
mechanics. The reason for this classical
behavior on the macroscopic scale is that the quantum phases of the complex
wavefunctions of different atoms are unrelated and can vary in
complex ways
from moment to moment. We say that there
is no quantum phase coherence for macroscopic objects. For example, electrons that travel by
different paths in a piece of aluminum cannot interfere with each other. However, something amazing happens if the
aluminum is cooled to below 1K: it
changes from a normal metal to become a superconductor, a phase in which
a large number of electrons condense into a single quantum state. It then does
have phase coherence on a macroscopic scale, and hence can show interference
between electrons that travel by quite different paths. The superconducting state exhibits some
remarkable effects by virtue of its quantum mechanical coherence: it can conduct electrical current with zero
resistance, it can completely exclude magnetic flux within the superconductor
(the Meissner effect), and it enables the construction of novel
electronic devices. We will study “macroscopic quantum interference” in this
experiment, using “high temperature superconductors” discovered in the 1980’s
that become superconducting above the temperature of liquid nitrogen (77
K). The difference in behavior of
electrons in a metal below and above the superconducting transition temperature
is analogous to that between coherent laser light and ordinary incoherent
sunlight or light from a lightbulb. In
both cases, a well-defined phase that is coherent over large distances allows
interference effects to be seen on a macroscopic scale.
Superconductivity was first discovered in 1911 in a sample
of mercury metal that lost its resistance just four degrees above absolute
zero. The phenomenon of
superconductivity has been a major focus of research and applications even
since, although until recently efforts were limited by the low temperatures
required. The first experiments only
revealed the zero resistance property of superconductors, and more than twenty
years passed before discovery of the Meissner effect. Magnetic flux quantization – the key to SQUID
operation – was predicted theoretically only in 1950 and was finally observed
in 1961. The Josephson effects were
predicted and experimentally verified a few years after that by Brian
Josephson, who won a Nobel Prize for his work while still in his 20’s.
SQUIDs (superconducting quantum interference
devices) were first studied in the mid-1960s, soon after the first
Josephson junctions were made. These
sensitive magnetic field detectors are also important tools in quantum
computing experiments and tests of the fundamental properties of quantum
mechanics. Practical superconducting
wire for use in moving machines and magnets also became available in the
1960's. For the next twenty years, the
field of superconductivity slowly progressed toward practical applications and
to more profound understanding of the underlying phenomena. A great revolution in superconductivity came
in 1986 when the era of high-temperature superconductivity began. The existence of superconductivity at liquid
nitrogen temperatures has opened the door to applications that are simpler and
more convenient than were ever possible before.
Nevertheless, the product you have in your hands today was made possible
by many aspects of the 80 years of discovery that preceded it. You can read about current applications of
high temperature superconductors in the article “High Temperature Cuprate
Superconductors get to work”, Physics
Today, pp.41-47, April 2005, Vol 58
(4), available in the lab binder.
Introduction to Superconductivity and SQUIDS
There are certain materials – actually, many thousands of
them by now – that exhibit a remarkable transition in their ability to pass
electrical currents: when they are
cooled down to a sufficiently low temperature, their electrical resistance
vanishes completely. How this behavior
comes about was a mystery that occupied the minds of theoretical physicists
for nearly half a century after it was first observed. The answer turned out to be tied to the
quantum-mechanical nature of solids, in particular, to the tendency of
electrons to form organized collective quantum mechanical states. One instance of this is Cooper pairs of electrons that behave cooperatively
in certain materials and form a single quantum-mechanical state. While it is hard to see why two particles of
like charge would form a pair, the reason why is actually easy to understand
intuitively. Phonons (the quantum
mechanical version of sound waves in solids) can promote the pairing of
electrons by creating a local concentration of positive electrical charge when
two nuclei move closer together as a sound wave move past them; a nearby
electron can both be attracted to this local charge concentration and further
concentrate it. The resulting even
greater positive charge imbalance can attract and bind the second Cooper pair
electron. This process has been
described as similar to how a depression in a bed will encourage a person to
roll into the dent, thus creating an even deeper depression and encouraging a
second person in the bed to roll toward the first. This explanation also helps one understand
why electrical currents can flow without resistance in a superconductor. This
is because in normal conductors resistance to electrical current is caused by electrons
scattering off of vibrating atomic nuclei (in other words, off of phonons);
this scattering changes the electron’s direction and transfers energy from the
electrical current to sound waves. In
superconductors, electrons also interact with phonons, but instead of
scattering off of them and diminishing the flow of electrical current, the
phonons actually promote the formation and movement of Cooper pairs.
In the following discussions, we can only explain these
concepts briefly and without theoretical rigor. Fortunately, many such books exist and we
refer you to some in the References in Section 10 (of the full
documentation). What this guide will try
to do is give you some idea of the underlying physical principles behind Mr.
SQUID®.
The Superconducting State
A fundamental aspect of physical systems is that they
naturally seek a state of lowest free energy.
An example of this is that a ball will roll to the lowest spot on an
uneven surface (the lowest potential energy).
An external source of energy (such as kicking the ball) is required to
raise it to a higher spot (energy level).
Similarly, systems of particles, such as the electrons in a metal, will
occupy a lowest-energy state known as the ground
state, unless they are excited by some external source of energy. In certain materials, it is possible for
electrons to achieve a ground state with lower energy than otherwise available
by entering the superconducting state.
In this ground state, all the electrons are described by the same
wavefunction. What does this mean?
In quantum mechanics, physical entities such as electrons
are described mathematically by wavefunctions. Like ordinary waves in water or
electromagnetic waves such as light waves, quantum mechanical wavefunctions are
described by an amplitude (the height of the wave) and a phase (whether it is
at a crest or a trough or somewhere in between). When you are describing waves of any kind,
these two parameters are all that is necessary to specify what part of the wave
you are discussing and how large it is.
Moving waves oscillate both in time and in space. If we sit at one point in space, the
wavefront will oscillate in time. If we
look at one moment in time, the wavefront undulates in space. The quantum mechanical wavefunction is a
mathematical entity that describes the behavior of physical systems such as
electrons and light waves.
So far, you have probably used a wavefunction required to
describe each particle in a physical system.
In the usual or “normal” state, the wavefunctions describing the electrons
in a material are unrelated to one another.
In a superconductor, on the other hand, a single wavefunction
describes the entire population of superconducting electron pairs. That wave function may differ in phase from
one place to another within the superconductor, but knowing the function in one
place determines it in another.
Physicists call such a wavefunction a “many-body wavefunction.” Strictly speaking, electrons are
indistinguishable particles; there is no way to keep track of an individual
electron in the population and it in fact has no meaning to speak of one. The pairs of electrons that inhabit the
superconducting state are constantly forming, breaking, and reforming such that
the wavefunction that describes the superconducting state remains the same. As we will see, it is the existence of this
coherent wavefunction that accounts for the phenomena associated with
superconductivity.
The Quantum of Flux
In quantum mechanics, it is common to find that, for
example, the energy of a bound particle or the energy of a photon of light can
only occur in discrete amounts called quanta rather than being
continuously divisible. We also see
quantization occurring in nature with respect to electrical charge, for
example. To our knowledge, all
electrical charge (excluding quarks and some exotic solid state systems for the
moment) occurs in units of
e = 1.6´10-19 C, the charge on a single electron. Therefore, nothing in nature has 1.5 times this charge, for example. Electromagnetic energy is quantized in units called photons, whose magnitude is set by the product of the frequency of the radiation times Planck's constant. In general, quantization is only readily apparent when we are dealing with microscopic objects. Macroscopic objects are composed of enormous numbers of elementary particles whose energies, complicated interactions and chaotic thermal motion completely masks the discrete nature of the microscopic world.
e = 1.6´10-19 C, the charge on a single electron. Therefore, nothing in nature has 1.5 times this charge, for example. Electromagnetic energy is quantized in units called photons, whose magnitude is set by the product of the frequency of the radiation times Planck's constant. In general, quantization is only readily apparent when we are dealing with microscopic objects. Macroscopic objects are composed of enormous numbers of elementary particles whose energies, complicated interactions and chaotic thermal motion completely masks the discrete nature of the microscopic world.
Superconductivity offers a unique opportunity to observe the
quantization of a physical quantity in a macroscopic, readily observable
system. The
key quantum property to be studied in this experiment is the quantization of the magnetic flux
(field times area), F, penetrating a
hole in a superconducting ring (or penetrating any superconducting closed
circuit). Recall that flux is defined by the integral:
Eq.1
over some loop enclosing an area a. The vector 
points
everywhere normal to the loop’s surface
The unit of quantization (or flux quantum) is h/2e = 2.07 x 10-15 Wb or weber, where h is Planck’s constant. (Note
that the SI unit of flux, is 1 weber = 1 Wb = 1 Tesla-m2.) If the flux through the hole changes (e.g.
as a result of a change in a small magnetic field in the hole), circulating
currents will arise in the ring that will precisely cancel the change. These circulating currents can be detected
using a phenomenon called the Josephson effect, and hence, the small
magnetic field can be measured; this is
the basis of SQUID operation. The
sensitivity to changes in flux is about 0.001 of one flux quantum, good enough
to measure magnetic fields generated outside your skull by the process of thinking
(e.g. thinking about this write-up). For
example, the magnetic field of the earth passing through the area of a typical
SQUID sensor corresponds to over 100 flux quanta.
points
everywhere normal to the loop’s surface
The unit of quantization (or flux quantum) is h/2e = 2.07 x 10-15 Wb or weber, where h is Planck’s constant. (Note
that the SI unit of flux, is 1 weber = 1 Wb = 1 Tesla-m2.) If the flux through the hole changes (e.g.
as a result of a change in a small magnetic field in the hole), circulating
currents will arise in the ring that will precisely cancel the change. These circulating currents can be detected
using a phenomenon called the Josephson effect, and hence, the small
magnetic field can be measured; this is
the basis of SQUID operation. The
sensitivity to changes in flux is about 0.001 of one flux quantum, good enough
to measure magnetic fields generated outside your skull by the process of thinking
(e.g. thinking about this write-up). For
example, the magnetic field of the earth passing through the area of a typical
SQUID sensor corresponds to over 100 flux quanta. Superconducting Rings
We will now explore why superconducting rings exhibit this
flux quantization. Consider the
following experiment. We cool a ring of
superconductor in a small magnetic field that corresponds to one flux quantum
threading the ring. We now have a
superconducting ring threaded by a single flux quantum. Suppose we now turn off the applied
field. According to Faraday's Law of
Induction, the moment that we change the field lines that thread the ring, a
current flows in the ring. The current
induced tries to oppose the change in magnetic field by generating a field to
replace the field we removed. In an
ordinary material, that current would rapidly decay away. In the superconductor, something entirely
different happens. If the induced
current decreased just a little bit in the ring, then the flux threading the ring
would be a little less than a flux quantum.
This is not allowed. The next
allowable value of flux would be zero flux.
Therefore, the current would have to abruptly cease rather than decay
away. Because the superconducting state
is composed of an enormous number of electrons that are paired up and occupying
the same quantum state, a current reduction of the sort needed would require
all the electrons to jump into another state simultaneously. This is an extraordinarily unlikely
event. Practically speaking, it will
never happen. As a result, the current
induced in a superconducting ring will flow indefinitely, and it prevents the
magnetic flux through the loop from changing from its initial value. People have actually tried this experiment
for years on end. As long as the ring is
kept cold, the current flows without resistance.
But why is the flux quantized rather than simply constant at
its initial value? The answer lies in
the long-range coherence of the superconducting wavefunction. As we said before, the value of the
wavefunction in one place in a superconductor is related to the value at any
other place by a simple phase change.
The case of a superconducting ring places special restrictions on the
superconducting wavefunction. The
wavefunction at the point marked by the black dot on the ring in Fig. 1 must
have the same value as the wavefunction obtained by traveling around the ring
one full circuit – it is the same spot.
The phase change for this trip must be 2π in order for the wavefunction
to have a single value at a given point in space. In ordinary wave language, if the wave was at
a crest at the starting point, it must be at a crest 360° (2π) around the
circle.
This condition of “single-valuedness of the wavefunction”
constrains the flux to quantized values or fluxons. According to electromagnetic theory, applying
a magnetic field to a superconductor induces a change in the phase of the
wavefunction. In quantum mechanics, you learned
that a free electron wavefunction can be written as Y(x)= A e i p x / ħ , where p is the linear
momentum along the x-direction. Now,
advanced electricity and magnetism and quantum mechanics explain that you
really should replace the linear momentum, p, in this relationship with
what is called the canonical momentum, p + qA, where p is the mechanical momentum, q
is charge (-2e for Cooper pairs) ,
and A is the magnetic vector potential.
(The magnetic vector potential can be used to generate the magnetic
field using the equation B = Ñ ´ A.) This equation reveals why a given amount of
magnetic field creates a specific phase change in the wavefunction, since now
the phase depends upon the magnetic field through A:
Y(x) = eikx = eipx/ħ =
eipcanonical x/ħ Eq. 2
This means that there is a term in the phase equal to not
just (px/ħ) but (px/ ħ - 2eAx/ ħ). Now
we again apply the single-valuedness condition for the phase to this new
expression. When the particle goes
around a loop, the second phase term becomes a line integral of 2eA/ ħ around
the loop, but that can be transformed into an area integral of the curl of A
(same as the magnetic field) over the area enclosed by the loop, a quantity
equal to the magnetic flux. This means
that the change in phase, Df, as a
particle goes around a loop as in Fig. 1 is :
Eq. 3
So, if we consider a loop within a superconductor, or any
other quantum mechanical system containing magnetic flux, the phase includes a term
proportional to the magnetic flux. (This
is called the Aharonov-Bohm effect
and has been measured in many different experiments.) Since the phase change going completely
around the ring must be some multiple of 2π in order to maintain the
single-valuedness of the wavefunction, the amount of flux contained within in
the ring can only assume certain discrete values: Df = 2eF/ ħ = 2 π . From this relationship, we derive the flux quantum as: F0 = ħ 2 π /2e = h/2e. This quantum mechanical property is the
origin of flux quantization.
The critical current Ic and critical magnetic field, Bc
Superconductors can only remain superconducting if their
currents remain below a critical current, and if they are surrounded by
magnetic fields less than a critical magnetic field magnitude. The origin of both effects is related to the
Meissner effect—that is, the exclusion of magnetic fields from inside the
superconductor. Now, recall from electricity
and magnetism that the energy due to the presence of a magnetic field of
magnitude B is:
U = V B2/8p Eq. 4
where V = volume
of the material. Now, to exclude
the magnetic field a region of space costs a corresponding amount of
energy. The magnetic field is excluded
from the material when it is converted from a normal metal to a
superconductor. This energy cost
is offset by an overall lowering of the free energy of the system, which
changes from the normal state free energy Gn
to that of the superconducting state with free energy Gs. (Why free energy
rather than just plain energy? There is
an entropic change in going from one phase to another as well as an energy
difference. The free energy takes into
account both changes.) In the absence
of a magnetic field and at sufficiently low temperatures, the material lowers
its internal energy by going superconducting:
in other words Gs <
Gn . The
balance between the energy cost of excluding magnetic field (V B2/8p > 0) and energy
lowering by going from normal to superconducting (Gs – Gn < 0) is expressed as:
DU =
(Gs – Gn) + V B2/8p
Eq. 5
Now, for B = 0,
clearly the superconducting state is advantageous energetically. As either T
(temperature) or B increases, the
energy difference, DU reaches zero, then becomes positive
and the superconducting state is no longer favorable. The material becomes a normal conductor
again. The threshold value of B at which this occurs is the critical
field, Bc, and it is
defined by:
DU =
(Gs – Gn) + V Bc2/8p = 0 Eq. 6
So, Bc
is the highest magnetic field at which the superconducting state is
energetically favorable. Higher field
values cause the superconductor to “go normal”.
Now we can understand why there is also a critical current, Ic. Associated with a current in any wire is a
magnetic field. For a simple cylindrical
wire of radius r, the highest current is at the surface of the wire, and
its value is:
B = 2 I/(c r) Eq. 7
Here c is the speed of light. However, this magnetic field cannot exceed Bc, so we also get a limit on
the maximum current that the superconductor can support before it goes normal:
Ic = Bc c r /2 Eq. 8
However, if only the surface layer goes normal, then the
current is just pinched down to smaller radius within the wire, giving an even
higher magnetic field. By this
reasoning, you can see that the entire wire must go normal, not just the region
at the surface. The critical current
limits current throughout the superconductor as a result.
Josephson Junctions
The Josephson effect is yet another manifestation of what we
call the long-range quantum coherence of superconductors. It also takes advantage of the critical
current idea introduced above. The
simplest picture of this effect is shown in Fig. 2.
Figure
2 Schematic diagram of two superconducting
regions separated by a thin gap, in which the phase, f, differs between the
two regions.
Two regions of superconductor are placed very close to one
another, but separated by a thin region of a normal conductor. The quantum mechanical phase on the left is f1 and the
phase on the right is f2. In an ordinary material, the phases at two
different spots are unrelated. In a
single piece of superconductor, the phases at two different places have a
specific relationship to one another.
This arrangement assures the superconducting lower energy ground
state. In the picture above, what will
the phases do? The surprising answer is
that if the two regions of superconductor are close enough together, their
phases will also be related, even though the intervening material is not
superconducting. In other words, they
will act like a single superconductor.
(The origin of this effect isn’t really so mysterious. The superconducting wavefunction always
extends a very short distance into a surrounding normal metal, in a fashion
similar to single particle wavefunctions undergoing exponential decay in a
classically forbidden region. The Josephson
junction (or weak link) is just formed by placing two such regions so close
together that the wavefunctions haven’t decayed to zero on either side before
they meet in the center of the junction.)
Functionally speaking, electrical currents can flow between the two regions
with zero resistance. Such currents are
called Josephson currents and
physical systems composed of two regions of superconductor that exhibit this
property are called Josephson
junctions.
How are such structures formed? You might not know it, but most metals are
actually crystals. That is, they are
composed of atoms arranged in a regular lattice, as shown Fig. 3(a). Most metal wires are polycrystalline—they are
made of multiple, randomly oriented tiny crystals called crystallites or
domains. Within each domain, the atoms
are ordered regularly. Domains share
walls between each region of local crystalline order. One way these walls can arrange themselves is
so as to form part of each domain on either side. The resulting wall is called a grain boundary,
as shown in Fig. 3(b). Electrical
resistance can be thought of as due to scattering of electrons from vibrating
atomic nuclei in the crystalline lattice.
The extra disorder induced by a grain boundary introduces more
scattering and hence greater electrical resistance. In addition, the disorder
caused by grain boundaries can destabilize superconductivity at these points.
(a)
(b)
Figure 3: (a) Schematic
diagram of a crystal lattice. (b) Electron micrograph of strontium-titanate
showing the two mismatched crystal lattices (top and bottom) meeting at a grain
boundary (center line) Image
sources: (a) http://www.localhistory.scit.wlv.ac.uk/Museum/Engineering/Electronics/history/Transistordetails.htm
(b) http://pruffle.mit.edu/~ccarter/NANOAM/images/
Any weak electrical coupling between two regions of
superconductor – such as tiny constrictions, microscopic point contacts, weakly
conducting layers, or certain crystallographic grain boundaries – exhibit the
Josephson effect. This can be exploited
to make Josephson junctions. The Josephson junctions in Mr. SQUID®
are grain boundary weak-link junctions, as shown in Fig. 4. As a result, they have a lower critical
current than the rest of the superconductiving ring. Josephson junctions are the essential active
devices of superconductive electronics, much as the transistor is the essential
active device of semiconductor electronics.
Junctions can be used in a variety of electronic circuits as switching
devices, as sensors, as variable inductors, as oscillators (because of the ac
Josephson effect), and for other applications.
People have built Josephson electronic circuits that contain up to tens
of thousands of junctions. At the
opposite extreme, one of the most useful circuits made from Josephson junctions
is the dc SQUID, which contains only two junctions.
Figure 4: Schematic diagram
of Mr. SQUID™ showing the two grain boundaries (black lines) that constitute
the Josephson junctions. The
superconducting loop is the center circuit, while two outer coils (coil 1 and
coil 2) are used for applying magnetic fields to the device. (Star Cryotronics)
The dc SQUID
Now we will describe how SQUIDs work. A dc SQUID is a superconducting loop with two
Josephson junctions in it. (Fig. 4) Suppose we pass a constant current, known as
a bias current, through the SQUID. If
the SQUID is symmetrical and the junctions are identical, the bias current will
split equally, half on each side. A dc
SQUID is generally represented schematically as shown in Fig. 5.
Figure 5: A schematic
representation of the dc SQUID. (Star Cryotronics)
A supercurrent (superconducting current) will flow through
the SQUID, as long as the total current flowing through it does not exceed the
critical current of the Josephson junctions.
Since the critical current is the maximum current the SQUID can carry and
remain superconducting, it is also the current at which a voltage across the
SQUID develops. You could measure the
critical current of a SQUID by increasing the bias current up slowly from zero
until a voltage appears, then reading the current with an ammeter. The value of current determined in this way
is the critical current of the SQUID.
When the two junctions in the SQUID are identical, the loop
is symmetrical, and the applied field is zero, both junctions will develop a
voltage at the same time. So the
critical current of the SQUID is simply twice the critical current of one of
its junctions. If the critical current
of each junction is 5 microamps, for example, then the critical current of the
SQUID is 10 microamps. The
voltage-current characteristic, or V-I curve, of a SQUID, looks very much like
the V-I curve of a bulk
superconductor, except the value of the critical current is smaller. A typical V-I
characteristic for Mr. SQUID® is shown in Fig. 6, where bias current
is along the x-axis and voltage is along the y-axis.
Figure 6: A typical Mr. SQUID®
V-I characteristic. (Star
Cryotronics)
Now imagine what happens if a magnetic field is applied to
the SQUID. First, let’s bias the SQUID
with a current well below its critical current.
Then, if we apply a tiny magnetic field to the SQUID, the magnetic field
wants to change the superconducting wave function. But the superconducting wavefunction doesn’t
want to change — as discussed earlier, it must maintain an integral number of
wavefunction cycles around the loop. So
the superconducting loop does what you would expect: it opposes the applied magnetic field by
generating a screening current Is,
that flows around the loop (Fig. 7(a)).
The screening current creates a magnetic field equal but opposite to the
applied field, effectively canceling out the net flux in the ring.
(a)
(b)
Figure
7: (a) A dc SQUID in the presence of an applied
magnetic field. (b) The screening current Is has reversed its direction after one additional flux
quantum enters the loop.
The applied magnetic field has lowered the critical current
of the SQUID — in other words, it has reduced the amount of bias current we can
pass through the SQUID without generating a voltage. The reason is that the screening current
superimposes itself on top of the bias current, but we still must have Ibias
+ Iscreening < Ic for the junction to stay
superconducting. Suppose the critical
current of each junction is 5 microamps and the screening current is 1
microamp. Since the bottom junction has
to carry 1 microamp of screening current, it can now carry only 4 microamps of
bias current before it becomes resistive.
When it carries a total of 5 microamps, it becomes normal and
resistive. When the junction on the
bottom goes normal, all the current goes through the junction on the top, which
makes it go normal. That means both
paths are now resistive, so a voltmeter will register a voltage across the
SQUID.
As we increase the applied magnetic flux, the screening
current increases. But when the applied
magnetic flux reaches half a flux quantum, something interesting happens. Just the junctions momentarily go
normal. The continuity of the
superconducting loop is destroyed long enough for one quantum of magnetic flux
to enter the loop. Then
superconductivity around the loop is restored. Consider what happens to the
screening current to see why this is also energetically favorable. Rather than generating enough screening
current to keep 0.51 flux quanta out, now all the SQUID has to do is generate
enough screening current to keep 0.49 flux quanta in, which is, of course, a
little easier (that is, lower in energy.)
Of course, the screening current has to change direction, as shown in
Fig. 7(b).
Thus, the junctions serve as gates that allow magnetic flux
to enter (or leave) the loop. The voltage read with an oscilloscope is the
average voltage across the SQUID. Although the experimenter observes a non-zero
dc voltage (hence the SQUID appears “resistive” just above Ic as shown in Figure
and the left side of) the instantaneous
voltage across the SQUID and the circulating current are actually oscillating
at high frequencies in the microwave range in response to an applied magnetic
field.[i]
Now we will see why the critical current (and hence the
voltage measured across the SQUID) is periodic in magnetic flux. In the discussion above, we have assumed that
an external magnetic field Bext
is generating the magnetic flux in this discussion. (This could be caused by currents flowing
within coil 1 or coil 2 in your Mr. SQUID™, as shown in Fig. 4.) There is also a magnetic field generated by
the total current, I, flowing in the SQUID. The total magnetic field Btotal
inside the superconductor is thus:
Btotal = Bext + 2
I/(cr) Eq.
9
However, this means that it is the total magnetic field, Btotal,
that has to be below the critical field for superconductivity to exist:
Btotal = Bext + 2
I/(cr) £ Bc Eq.
10
This also means that when the total magnetic field equals
the critical magnetic field, the resulting critical current is lower than
before. As a matter of fact, it’s equal
to:
Ic = (rc/2)(Bc – Bext)
Eq.11
This is why the V-I
curve below exhibits a lower Ic
values for greater magnetic fluxes—the critical current decreases as Bext increases. (Fig. 8)
The critical current (and hence the V-I curve) are periodic because the
entire cycle repeats every time one more flux quantum enters the SQUID ring.
Figure 8: Schematic
depiction of how the V-I curve of the SQUID varies as the applied magnetic flux
increases. Different V-I curves
correspond to different values of flux.
The greatest value of critical current occurs when the applied flux
equals an integral number of flux quanta and no screening current is required
Fig. 8 also illustrates why the SQUID’s voltage is periodic
in the applied magnetic flux. The
critical current goes through a maximum when the applied magnetic flux is an
integer multiple of the flux quantum, because that’s when the screening current
is zero. It goes through minima when the
applied magnetic flux is an integer of the flux quantum plus one half, because
that’s when the screening current is largest.
For fixed bias current through the device, the corresponding SQUID
voltage depends upon flux. Because the critical
current of the SQUID is periodic in the way you just saw, the V-I curve of a SQUID oscillates
periodically between two extremes as shown in Fig. 9
Figure 9:
Periodic dependence of the SQUID voltage on applied flux. Note that the slope of each V-I curve is RN
in the normal region. As a result, the
maximum value of V (twice the amplitude of the V vs. F curve) vs. F is equal to IC RN.
To make a magnetic field detector, we operate the SQUID with
a constant bias current slightly greater than the critical current, so the
SQUID junctions are always resistive.
Under these conditions, there is a periodic relationship between the
voltage across the SQUID and the applied magnetic flux, with a period of one
flux quantum (Fig. 9). This flux vs. voltage curve will appear on your
oscilloscope during your experiments with Mr. SQUID®, and it serves
as the basis for magnetic flux measurements using SQUID’s. While it would be possible to obtain quite a
sensitive measure of a magnetic signal simply by counting flux quanta,
practical SQUID systems involve control electronics that interpolates between
whole numbers of flux quanta and greatly enhances their ultimate
sensitivity. SQUID sensitivity is
finally limited by the intrinsic noise in the device, which in
4.2 K niobium dc SQUIDs, for example, typically approaches a millionth of a flux quantum (corresponding to a few billionths of the earth's field passing through a 100-µm diameter SQUID).
4.2 K niobium dc SQUIDs, for example, typically approaches a millionth of a flux quantum (corresponding to a few billionths of the earth's field passing through a 100-µm diameter SQUID).
The inherent periodicity of the SQUID implies that it cannot
distinguish between zero applied field and any other field that generates an
integral number of flux quanta. This
allows the dynamic range of the SQUID to be extended almost indefinitely by
re-zeroing the SQUID in a controlled way.
It also means that in order to measure the absolute value of an applied
field, it is necessary to reset the SQUID in a known field, or to rotate the
SQUID with respect to the field.
Nevertheless, it is very often the case that only changes in field are
of interest; in which case no special measures are necessary.
Although the model of SQUIDs we just discussed is not
rigorous, it is a good approximation.
This model doesn’t really connect quantum interference directly to the
SQUID’s operation, though. A more
complete explanation would describe how the curve showing the critical current
of the SQUID varies with applied flux is an interference pattern, analogous to
an optical interference pattern. If you
pass a current through a dc SQUID, you see maxima and minima of critical
current as you raise or lower the applied flux, because the macroscopic
quantum wave functions at the two junctions interfere with each other.
Experiment A. Magnetic susceptibility vs. Temperature of the YBCO SQUID (the Meissner Effect)
We
own superconducting samples in which the high-T superconductor has been formed
into a pellet embedded in a plastic casing.
The compound used in your sample is called YBCO for
yttrium-barium-copper-oxide. Wrapped
around the superconductor is a wire coil, or inductor; a thermocouple has also
been included in the package to monitor temperature. Now, if the entire inside
of the inductor were filled with superconductor, then no magnetic field could
pass through that region due to the Meissner effect; in other words, it would
have zero inductance. Your wire coil
actually has a diameter of 1.36 cm, while the superconductor pellet inside it
has diameter 1.26cm, so it only partially fills the inductor coil. That means the coil’s inductance, L,
is actually given by:
L @ Lo
(1 + f c)
Eq.12
Where
f = the fraction of the coil volume occupied by the sample, Lo
is the coil inductance in vacuum,
and c is the called the
magnetic susceptibility of the superconductor.
As a result, if you measure the coil’s inductance, you can measure the
magnetic properties of the superconductor inside it! A sudden drop in the inductance at the
superconducting transition is a direct measure of the Meissner effect. (You can also watch the effect of flux
exclusion by levitating a small high field magnet above a block of
superconductor. If you have never seen
this done, ask your instructor for help in doing this.)
Here
is the circuit we will use to measure coil inductance:
Figure
10: Schematic of the induction
measurement.
Table 1: Conversion from thermocouple voltage to
Kelvin.
The
AC voltage will be a f = 1 kiloHz sinewave taken from the low voltage output of
a function generator. From intro physics
you learned that:
Eq.13
Where
I is current through the inductor (and resistor), RL is inductor resistance, L is inductance, and w = 2pf.
STEP 1: Your thermocouple measured temperature by
generating a DC voltage (in the millivolts range) that is depending upon
temperature. Connect up a DMM so it
again reads DC millivolts from your thermocouple (red and blue wires). Make sure you are not on AC mode. You can convert between voltage and
temperature using either Table 1 or by using the following calibration curve on
the calibration curve (determined by Lindsay Subers and Amanda Dillon in
2005):
where V is in milliVolts. (Note from the table above that the sign of
the thermocouple voltage changes close to room temperature, so do not be
surprised if your thermocouple voltage goes negative—just change the polarity
of the leads.)
STEP 2: Connect up a second DMM so it reads AC volts
across the 50 Ohm power resistor. You
will use this to measure current through the entire circuit, I: I = V/50 Ohms. (Measure your power resistor’s actual
resistance first.)
STEP 3: Make a quick reading of your inductor’s
resistance, RL, at room temperature to use later; do this using a 4-point Ohm measurement. (See Appendix A.) Ideally you would measure this as a function
of temperature, but that’s not feasible here.
Instead, you should correct for temperature dependence using the
temperature coefficient of copper, which is 0.393 % per degree C. In other words, if the temperature increases
by 1 degrees C, the resistance will increase by 0.393%.
STEP 4: Connect up a DMM so it reads AC volts across
your inductor. This is VL.
STEP 5: Carefully place your sample in the “sand
cryostat” (foam container filled with
sand) and pour sand around it until the sample is at the center of the
container. Get liquid nitrogen in a transfer
Dewar, being careful to wear eye goggles and gloves to shield your hands from
spills or Dewar glass breakage. Your
instructor can explain where the nearest liquid nitrogen supply is
located. Cool the sample down to 77 K by
pouring liquid nitrogen in a little at a time and allowing it to boil off while
cooling down the sand a little at a time.
You will need to keep adding liquid nitrogen as it boils off; you should
try to keep the top of the sand just covered, while avoiding vigorous boiling
that spills the sand over the table.
Watch the thermocouple voltage to see when it stabilizes at the
temperature of liquid nitrogen (77 K). Record a low temperature inductor resistance
reading once it has approximately stabilized.
Next, allow the sample to warm up slowly while recording temperature,
current (resistor voltage) and inductor voltage. Do this by simply stopping adding the liquid
nitrogen and allowing the sand cryostat + sample to come back to room
temperature. (This will occur more
rapidly if you do not have a large puddle of liquid nitrogen on top of the
sand.)
STEP 6: Make a clearly marked table to keep track of
all these different voltages! Be
especially careful to record data near the transition! Remember to use the inductor resistance that
varies with temperature, interpolating between your room temperature and liquid
nitrogen temperature readings. You will
see a dramatic increase in L (and hence inductor voltage) at the
transition. Take data for a few degrees
above the transition, but you do not have to take data all the way up to room
temperature. Plot up your data using a
computer so as to display inductance L
vs. T[K]. Indicate and comment on the
important features on your plot.
In your lab report, be sure to explain how
the Meissner effect and the magnetic properties of superconductors can explain
your sample’s inductance vs. Temperature curve. What would happen to your curve
if the superconductor entirely filled the coil?
Experiment B: Interpreting Mr. SQUID® ‘s V-I curve
The Mr. SQUID® chip contains a dc SQUID made with
thin film high temperature YBCO superconductor.
The dc SQUID is a simple circuit that can be schematically represented by
our earlier Josephson junction cartoon.
(Fig. 11).
Figure 11: Schematic representation of the dc SQUID in
Mr. SQUID and actual Mr. SQUID.
The Mr. SQUID™ dc SQUID is simply a ring of superconductor
containing two Josephson junctions, which are marked by the X's labeled J1
and J2 in the circuit diagram above.
The actual device is mounted on a post that allows you to insert it into
a bath of liquid nitrogen to cool it to superconducting temperatures. The SQUID is also surrounded by a cylindrical
mu-metal shield that partially screens it from outside magnetic fields. The small fraction of the earth's field that
is not attenuated by the magnetic shield on the Mr. SQUID® probe is
sufficient to shift the V-F curve by several flux quanta!
The simplest experiment to perform with this circuit is to
pass current I from left to right
across the ring and to measure the voltage V
that appears across the ring. If the two
junctions in the SQUID are identical, in the absence of any magnetic field, the
current will divide evenly and half of it will pass through each junction
before recombining at the right side.
The Mr. SQUID® control box allows you to perform this
experiment quite easily. The
instructions in the next section explain how to set up your device so you are
ready to try this voltage-current experiment.
Setting Up the Output Device & measuring V-I for an ordinary resistor
We assume that you are familiar with the operation of your
oscilloscope. You will be using your
oscilloscope in x-y mode. For initial
settings, try an x sensitivity of 0.2 volts per division (or whatever value is
closest to this on your instrument) and a y sensitivity of around 50 millivolts
per division. The Autoscale feature
should work once you have a signal. BE SURE YOU HAVE YOUR OSCILLOSCOPE CHANNELS
BOTH SET TO DC COUPLING! Otherwise, your
V-I curves will not properly retrace, an effect known as hysteresis. You may want to adjust these values as needed
later. Now refer to the diagram of the
Mr. SQUID® control box (Fig. 12).
STEP 1: Since you
will be using an oscilloscope in the X-Y mode, set the oscillator frequency
toggle switch (7) to the OSC (high-speed) position. Once again check that both ‘scope channels
are set for DC coupling (NOT AC coupling).
STEP 2: Before turning on the power, make sure the amplitude
control (4) is at its minimum (fully counter-clockwise) position and set the
flux bias control (2) and the current bias control (3) to their 12-o'clock
(straight up) positions.
Figure 12: Front panel of the Mr. SQUID® electronics
box.
STEP 3:Make sure the function switch (1) is in the V-I
position. In this position, the BNC
cable connected from the X output on the front panel (5) to the oscilloscope
“X” input reads the current being fed through the SQUID. The cable connected from the Y output (6) to
the oscilloscope “Y” input reads the voltage across the SQUID. Set the power switch (9) to the ON
position. On the oscilloscope, there
should be a small bright spot on the center of the screen. You may have to adjust the offset controls
to achieve this.
STEP 4: For
starters, rather than looking at the complicated SQUID outputs, we will just
connect a simple resistor to your circuit and study its V-I behavior. Find the DB-9 connect with the resistor
soldered to it, and measure the resistance using a DMM. Connect this to the end of your Mr. SQUID™
box’s cable in place of Mr. SQUID™ before proceeding.
Understanding the current and amplitude controls in VI mode
First, we will understand what is necessary to measure a V-I
curve. The current bias control (3)
varies the current being sent through the SQUID. Slowly turn this knob in either
direction. The spot on the oscilloscope
screen will move in response to the changing current. As you turn the knob back and forth, you will
trace out a V-I curve representing the relationship between the current fed
through the attached device and the voltage.
This curve is called the V-I curve.
Since you have attached a simple resistor for now, you will see this as
a straight line. You ideally would like
to automatically collect a V-I curve without sweeping the current bias knob by
hand. To do so, return the current bias
control to the 12 o'clock position and now slowly turn the amplitude control
(4) in the clockwise direction. This function
sweeps the current high and low between preset values, thus automating the
procedure you performed by hand using the current bias control. A solid curve should now appear on the
oscilloscope screen. (see Fig. 13) This
is basically the V-I curve for Ohm’s law for a resistor; the line’s slope is
proportional to resistance.
Figure 13: A linear (Ohmic) V-I characteristic.
If you now turn the current bias control, the center point
of the curve being traced on the screen will move. The current bias control sets a single value
of current being passed through the SQUID and the amplitude control sweeps the
current back and forth about that set value.
Interpreting your V-I curve
Your output device acts like a voltmeter. The sensitivity settings on it determine how
much voltage corresponds to a division on the screen. The current output on the Mr. SQUID®
box (5) actually represents the voltage across a 10,000 Ω resistor in the
electronics box. According to Ohm's Law
(I = V/R), the current flowing
through the resistor is therefore equal to the voltage across it divided by
10,000 Ω.
The typical voltage levels from the SQUID are small enough
that the Mr. SQUID® control box amplifies the small signals. Thus, to calculate the actual voltages, the
measured value on the oscilloscope should also be divided by 10,000. You can determine resistance by measuring the
slope of the V-I curve. To obtain a
resistance, you must convert the x-axis value into amperes from volts and take
into account the amplification of the y-axis signal. This
is all very simple since the conversion factor for each axis is the same value
of 10,000, so these factors cancel—resistance is just the slope of the measured
curve on your oscilloscope. Measure
resistance for your simple resistor, and check that you are getting the same
value as the resistance you measure direction using a DMM. Be sure to check with your instructor if you
have any problems at this point!
Now, disconnect the resistor and connect the cable to your
SQUID device. It is at room temperature,
so the entire SQUID is normal and hence resistive. Look at the SQUID”s room temperature V-I
curve and measure its resistance.
Cooling down Mr. SQUID
Make sure the mu-metal magnetic shield is in place around
your SQUID, and that you have the Lakeshore temperature probe connected to the
SQUID (This allows you to measure your sample’s temperature later.) Connect your temperature probe to the display
and make sure it reads room temperature (in Kelvin!) Get a dewar of liquid nitrogen (your
instructor can help you) and prepare to cool down your device. First, make sure you have turned off all
electronic devices around the SQUID, including your oscilloscope, the
temperature display and the Mr. SQUID control box itself. (This eliminates the major sources of stray
magnetic flux near your SQUID, to help avoid trapping so much flux you kill its
superconductivity.) Next, gradually
lower your SQUID into the liquid nitrogen a small section at a time. We have found a lowering rate of about 0.5 cm
per 30 seconds is reasonable to avoid temperature shocks that can damage the
device. Once your SQUID is immersed in
liquid nitrogen, you can lower the device more rapidly until it is close to the
bottom of the dewar. Once the
temperature sensor has stabilized at 77K, proceed to turn on your electronics, reconnect
the SQUID cable to the SQUID and try to get a V-I curve for the SQUID
itself. If the settings on the Mr. SQUID®
box and your output device are correct, and if the SQUID is behaving properly,
you will see a curve that looks more-or-less like Fig. 14 on the screen.
Figure 14: Typical Mr. SQUID® V-I
characteristic. Compare with Fig. 6. The
central region has low resistance because the current is less than the Josephson
junctions’ critical current there. For
currents greater than the critical current, the SQUID is Ohmic, with a normal
state resistance given by the slope.
It is important that there is a flat, low resistance region
in the center of the curve as shown above, although its width may vary from
device to device. (Your SQUID may have a
noticeable slope in the central section, even though superconductors really do
have zero resistance. This is due to a nonzero contact resistance between the
superconductor and its leads.) If what
you see looks like a straight line, as in Fig. 13, then either you don't have enough
liquid nitrogen in the dewar, you have not let the probe get cold, or there may
be trapped magnetic flux in the SQUID.
The latter is a very common occurrence because the SQUID is very
sensitive to external magnetic fields.
You may need to bring the SQUID back to normal and recool it. Refer to the discussion on trapped flux in
Section 6.3 of the full documentation, Magnetic Flux Trapping in SQUIDs, if
this appears to be the problem. Assuming
you see the proper curve, how can we understand its shape?
What you are looking at is the V-I characteristic of two
Josephson junctions connected in parallel with one another. Assuming they are identical junctions (in
practice, they are at least very similar), the V-I characteristic you
see is the same as for a single junction.
The most important feature of the curve is the flat spot in the middle,
corresponding to currents less than the critical current. In this region, there is current flowing with
no voltage – it is a supercurrent. This
is the dc Josephson effect: a
resistanceless current that flows through a superconductor Josephson junction.
At this point, you might want to adjust the flux bias
control (2) on the control box. This
control feeds current into a small gold coil placed above the SQUID and this
current applies a magnetic field upon the loop the SQUID. As explained earlier, a magnetic field will
modulate the critical current in the SQUID in a very specific manner. By turning the flux bias control knob, the
critical current in the junction will change visibly on the oscilloscope screen
or on your plotter page. At this point,
try to adjust the flux bias current such that the flat region of the V-I
curve is widest. The response to these
changes may be quite sensitive; it may take some practice to tune the critical
current to its maximum value. This
procedure may be especially useful if some small amount of magnetic field was
already present in the SQUID loop. A zero
applied field will yield the largest critical current through the junction.
Determining the critical current of your device
You can determine the critical current of the
junctions in Mr. SQUID® by measuring the width (in volts) of the
flat part of the V-I curve and dividing that number by
10,000 Ω to convert your answer into amperes of current. (See Fig. 15.) If the flat region is 1 volt wide, for
example, then the corresponding current is 100 microamps. However, you are looking at the current
through both of the junctions in the SQUID, not just one. Therefore, assuming the junctions are
identical, the current through one junction is half the value you are measuring
(in our example, 50 microamps). Is this
the critical current of the junction?
Not quite. The curve you are
looking at drives the current symmetrically about zero (marked in Fig.
15—compare Fig. 6). The current you have
measured is actually composed of a contribution in the positive direction and a
contribution in the negative direction.
If the curve is symmetric, they are equal. As a result, the real critical current of the
junction is half the value you infer from the measurement. Thus, in our example, the critical current is
25 microamps, 1/4 of the original measurement.
The flat region on the curve corresponds to 4 times the junction
critical current.
There is also a parameter of the junction known as the normal–state
resistance or RN. You can determine it by measuring the slope
of the V-I curve out at the ends where it is essentially a straight line. To obtain a resistance, you must convert the
x-axis value into amperes from volts and take into account the amplification of
the y-axis signal. This is all very
simple since the conversion factor for each axis is the same value of
10,000. Remember that the dc SQUID
contains two junctions in parallel, so that the measured resistance corresponds
to half the resistance of a single junction (assuming they are identical.) Thus, simply taking the slope numbers in
volts off your oscilloscope will give you one-half the normal–state resistance
in ohms. The product of the critical
current and the normal state resistance (IcRN)
is a voltage that is an important parameter for the operation of a SQUID. Make a note of it now for your report. This value sets the maximum voltage change in
the SQUID by an individual magnetic flux quantum, and is discussed later in
this section; to see why this is so,
look at Figs. 8 & 9. Note that IcRN for one of
the junctions has the same value for the SQUID, which has two junctions in
parallel. For the junctions in Mr. SQUID®
operating in liquid nitrogen, you will probably obtain a value between 10 and
100 microvolts; this is just a ball park
number that can vary from device to device.
Figure 15
Determining the junction parameters from the V-I characteristic.
Experiment C: Observing the V-F curve of Mr. SQUID®
Up to this point, we have been looking at the properties of
Josephson junctions. Now we will turn
our attention to the flux-measuring of the dc SQUID itself. The
Mr. SQUID® control box will allow you to observe the periodic
oscillations in the V-F curve on your oscilloscope and hence
measure the flux quantum. You have
already observed the effects of a magnetic field on the V-I characteristics of the SQUID by adjusting the flux bias control
current. The V-F characteristics are
basically an automatically plotted version of this behavior.
As we saw before, in the V-I
operating mode one can apply a magnetic field to the SQUID using the flux bias
control. This dial controls a current
that is applied to the “internal” modulation coil, simply a 3/4 turn gold or
silver coil that creates a magnetic field inside the loop of the SQUID, as
shown in Fig. 16. (The second coil shown
in the photograph is the “external” modulation coil that allows you to couple
other sources of current to the SQUID.) In this experiment, you will use your
SQUID control box to send a time-varying current through this internal
coil. That in turn exposes your SQUID to
a time-varying magnetic flux. Your
control box fixes the bias current flowing through the SQUID at a set fixed value, and measures
the behavior of SQUID voltage, V, vs. magnetic flux, F.
If you slowly turn the flux bias control knob, you will see
the change in the critical current and the changing V-I curve that occurs as the magnetic flux threading the loop of
the SQUID is varied. If you experiment
carefully with the flux bias control, you will see that the critical current of
the SQUID oscillates between a maximum value (the flat region of the V-I curve is at its widest) and a
minimum value at which point the
V-I curve may be more-or-less linear.
V-I curve may be more-or-less linear.
The Mr. SQUID® control box allows you to view this
periodic behavior of the SQUID’s V curve as a function of magnetic flux in a
convenient, automated way. To obtain the
V-F
plot, the bias current is set so that the SQUID voltage is most sensitive to
changes in applied magnetic field. This
occurs at the "knee" of the V-I
curve, the area highlighted in Figure below.
Figure 16: Photograph of the Mr. SQUID®
version 4 chip showing the two modulation coils. See full documentation for Mr.
SQUID® dimensions and specifications.
To find this point, first adjust the flux bias control so
that the critical current is at its largest value. Then, turn down the amplitude control all the
way so that only a point is visible on the oscilloscope screen or so that the
pen is stationary on the plotter page.
In this mode, you can then sweep the point up and down the V-I
curve by adjusting the current bias control, as you did initially. Set the current bias level so that the point
rests at either knee in the V-I
curve. Now turn the flux bias control,
which controls the amount of magnetic flux through the hole in the SQUID
loop. You should see the point move back
and forth vertically on the screen or page.
This periodic motion arises because the screening currents in the SQUID
body depend on the applied magnetic flux in a periodic manner. The period is
determined by the magnetic flux quantum (F0). This phenomenon is a manifestation of the
macroscopic quantum nature of superconductivity.
Figure 17: The “knee” in the V-I characteristic.
To automate this procedure, switch the function switch from V-I to V-F. In this new mode, all of the controls on the
Mr. SQUID® control box work the same way as before, except for the
amplitude control. Instead of
controlling the sweep of the drive current through the SQUID, it now controls
the amplitude of the current through the
modulation coil not through the SQUID itself, which is linearly related to
the magnetic field applied to the SQUID.
Thus, amplitude now sweeps magnetic flux, F. To view as many V-F periods as possible,
turn the amplitude control in the clockwise direction. Your oscilloscope’s Autoscale feature may be
able to find this curve for you. If not,
you can use your previously computed value of IcRN to judge how large DV should be, using the amplification factor explained below. The
voltage amplification provided by the Mr. SQUID® box is unchanged
from before. To determine the actual
voltage across the SQUID, you must divide by 10,000. However, since the SQUID modulation signals
are smaller than the junction voltages you examined earlier, you will need to
increase the sensitivity of the vertical scale on your display device. The maximum peak-to-peak voltage swing of the
SQUID modulation in Mr. SQUID® (∆V)
typically is between 10 and 30 µV (or 100 to 300 mV output from the front panel
of the Mr. SQUID® control box).
Depending on the setting of the amplitude control, you may wish to
change the x sensitivity to more conveniently view the oscillations of the
SQUID.
In addition, it may be necessary to use the position knob on
your output device to center the curve.
If your oscilloscope can be operated in an AC-coupled mode on the
vertical channel, you can use this mode for more convenient viewing of the V-F
curve (never use the AC mode to look at the V-I
curve, or it will be completely distorted on the oscilloscope screen.) You should see a curve that resembles the one
shown in Fig. 18. Depending on the
particular device and on the settings on the control box, you may see many more
periods than shown here. At this point,
you can try to maximize the signal by fine-tuning the current bias control (3)
to the most sensitive part of the V-I
curve (just keep your eye on the V-F curve as you "tweak" the current bias
control until you get the maximum modulation).
There will be some setting of the current bias control that gives the
largest modulation amplitude.
Notice the effect that the flux bias control has on the V-F
curve. It allows you to set the central
value of applied flux about which the amplitude control sweep varies. In other words, this control allows you to
apply a static magnetic field to the SQUID on top of the oscillating field
applied with the amplitude control.
Turning the flux bias control will therefore allow you to move left and
right along the V-F curve and thereby explore more of it than
the amplitude sweep permits.
The voltage change that occurs due to the influence of the
magnetic field appears now on the vertical axis of your display device. Sketch
your voltage-flux curve and measure DV; compare to your measured IcRN.
Now measure the flux quantum using your voltage-flux
curve. The SQUID control box is
sending current through the internal coil surrounding the SQUID. If you flow electrical current, Icoil, through the internal coil,
you create magnetic flux through your device according to the equation:
Figure 18: A typical Mr. SQUID® V-F
characteristic. The maximum ∆V may
range (approximately) from 10 to 30 µV, depending upon your value of IcRN. The voltage V is displayed on the y-axis,
flux on the x-axis.
F = M Icoil Eq. 15
where M is the mutual inductance of the internal coil and SQUID. (Computing this value is pretty complicated
because of the many contributions from the device geometry. The SQUID manufacturer’s manual quotes the
mutual inductance of the internal coil as 37 pH (that’s pico-henry, the unit of
inductance) on page 19.) This allows you
to compute the change in magnetic flux that corresponds to one cycle of voltage
change in the measured V-F curve in
Fig. 18. First, however, you need to
know how to interpret the F axis
(x-axis on your display and their output).
The output in the V-F mode has
the x-axis displaying the current, DI,
through the internal coil. The
conversion between voltage out and current is:
10,000 V/A for this measurement.
So, you can compute the change in magnetic flux through the device using
this conversion and the mutual inductance.
Use this to compute the flux quantum for your device to undergo one
voltage cycle.
Experiment D. Resistance vs. Temperature of Mr. SQUID (OPTIONAL: DO ONLY IF THE SQUID IS NOT WORKING!)
This experiment will allow you to observe the
superconducting transition of the YBCO film that forms the SQUID in Mr. SQUID®. By tracking the resistance of the SQUID while
measuring its temperature, you can watch the YBCO undergo its superconducting
transition. You will start at the
temperature of liquid nitrogen, 77K, and allow your SQUID to slowly warm up to
room temperature while recording both T and R, the resistance in the
superconducting region. The transition
temperature for YBCO superconductors is very near 77K, so take as much data as
possible as the sample begins to warm up.
You do not have to continue taking data once your sample has heated up
to several degrees above its transition temperature. You should see a very noticeable increase in
resistance once you hit the transition temperature (since your sample has
nonzero resistance at the contacts between the ceramic high-T superconductor
and its metal leads, your lowest temperature resistance won’t be truly zero
here.) Also, given the nature of the V-I
curve, you will have trouble measuring the resistance to high accuracy. Once your sample has gone normal (that is,
the entire V-I curve becomes Ohmic) you may use a computer to plot up your data
as a graph of R vs. T, indicating the superconducting transition temperature. This graph and an explanation of your
experiment should form part of your final report.
[i]An
article which describes this is Ryhänen et
al, "SQUID Magnetometers for Low-Frequency Applications," Journal of Low Temperature Physics 76, 287 (1989).
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