Method of imaging magnetic fields at microscopic scales
In condensed matter physics, scanning SQUID microscopy is a technique where a superconducting quantum interference device (SQUID) is used to image surface magnetic field strength with micrometre-scale resolution. A tiny SQUID is mounted onto a tip which is then rastered near the surface of the sample to be measured. As the SQUID is the most sensitive detector of magnetic fields available and can be constructed at submicrometre widths via lithography, the scanning SQUID microscope allows magnetic fields to be measured with unparalleled resolution and sensitivity. The first scanning SQUID microscope was built in 1992 by Black "et al." Since then the technique has been used to confirm unconventional superconductivity in several high-temperature superconductors including YBCO and BSCCO compounds.
Operating principles.
The scanning SQUID microscope is based upon the thin-film DC SQUID. A DC SQUID consists of superconducting electrodes in a ring pattern connected by two weak-link Josephson junctions (see figure). Above the critical current of the Josephson junctions, the idealized difference in voltage between the electrodes is given by
formula_1
where "R" is the resistance between the electrodes, "I" is the current, "I"0 is the maximum supercurrent, "Ic" is the critical current of the Josephson junctions, Φ is the total magnetic flux through the ring, and Φ0 is the magnetic flux quantum.
Hence, a DC SQUID can be used as a flux-to-voltage transducer. However, as noted by the figure, the voltage across the electrodes oscillates sinusoidally with respect to the amount of magnetic flux passing through the device. As a result, alone a SQUID can only be used to measure the change in magnetic field from some known value, unless the magnetic field or device size is very small such that Φ  ; Φ0. To use the DC SQUID to measure standard magnetic fields, one must either count the number of oscillations in the voltage as the field is changed, which is very difficult in practice, or use a separate DC bias magnetic field parallel to the device to maintain a constant voltage and consequently constant magnetic flux through the loop. The strength of the field being measured will then be equal to the strength of the bias magnetic field passing through the SQUID.
Although it is possible to read the DC voltage between the two terminals of the SQUID directly, because noise tends to be a problem in DC measurements, an alternating current technique is used. In addition to the DC bias magnetic field, an AC magnetic field of constant amplitude, with field strength generating Φ « Φ0, is also emitted in the bias coil. This AC field produces an AC voltage with amplitude proportional to the DC component in the SQUID. The advantage of this technique is that the frequency of the voltage signal can be chosen to be far away from that of any potential noise sources. By using a lock-in amplifier the device can read only the frequency corresponding to the magnetic field, ignoring many other sources of noise.
Instrumentation.
A Scanning SQUID Microscope is a sensitive near-field imaging system for the measurement of weak magnetic fields by moving a Superconducting Quantum Interference Device (SQUID) across an area. The microscope can map out buried current-carrying wires by measuring the magnetic fields produced by the currents, or can be used to image fields produced by magnetic materials. By mapping out the current in an integrated circuit or a package, short circuits can be localized and chip designs can be verified to see that current is flowing where expected.
As the SQUID material must be superconducting, measurements must be performed at low temperatures. Typically, experiments are carried out below liquid helium temperature (4.2 K) in a helium-3 refrigerator or dilution refrigerator. However, advances in high-temperature superconductor thin-film growth have allowed relatively inexpensive liquid nitrogen cooling to instead be used. It is even possible to measure room-temperature samples by only cooling a high "Tc" squid and maintaining thermal separation with the sample. In either case, due to the extreme sensitivity of the SQUID probe to stray magnetic fields, in general some form of magnetic shielding is used. Most common is a shield made of mu-metal, possibly in combination with a superconducting "can" (all superconductors repel magnetic fields via the Meissner effect).
The actual SQUID probe is generally made via thin-film deposition with the SQUID area outlined via lithography. A wide variety of superconducting materials can be used, but the two most common are Niobium, due to its relatively good resistance to damage from thermal cycling, and YBCO, for its high "Tc"  ; 77 K and relative ease of deposition compared to other high "Tc" superconductors. In either case, a superconductor with critical temperature higher than that of the operating temperature should be chosen. The SQUID itself can be used as the pickup coil for measuring the magnetic field, in which case the resolution of the device is proportional to the size of the SQUID. However, currents in or near the SQUID generate magnetic fields which are then registered in the coil and can be a source of noise. To reduce this effect it is also possible to make the size of the SQUID itself very small, but attach the device to a larger external superconducting loop located far from the SQUID. The flux through the loop will then be detected and measured, inducing a voltage in the SQUID.
The resolution and sensitivity of the device are both proportional to the size of the SQUID. A smaller device will have greater resolution but less sensitivity. The change in voltage induced is proportional to the inductance of the device, and limitations in the control of the bias magnetic field as well as electronics issues prevent a perfectly constant voltage from being maintained at all times. However, in practice, the sensitivity in most scanning SQUID microscopes is sufficient for almost any SQUID size for many applications, and therefore the tendency is to make the SQUID as small as possible to enhance resolution. Via e-beam lithography techniques it is possible to fabricate devices with total area of 1–10 μm2, although devices in the tens to hundreds of square micrometres are more common.
The SQUID itself is mounted onto a cantilever and operated either in direct contact with or just above the sample surface. The position of the SQUID is usually controlled by some form of electric stepping motor. Depending on the particular application, different levels of precision may be required in the height of the apparatus. Operating at lower-tip sample distances increases the sensitivity and resolution of the device, but requires more advanced mechanisms in controlling the height of the probe. In addition such devices require extensive vibration dampening if precise height control is to be maintained.
High temperature scanning SQUID microscope.
A high temperature Scanning SQUID Microscope using a YBCO SQUID is capable of measuring magnetic fields as small as 20 pT (about 2 million times weaker than the Earth's magnetic field). The SQUID sensor is sensitive enough that it can detect a wire even if it is carrying only 10 nA of current at a distance of 100 μm from the SQUID sensor with 1 second averaging. The microscope uses a patented design to allow the sample under investigation to be at room temperature and in air while the SQUID sensor is under vacuum and cooled to less than 80 K using a cryo cooler. No Liquid Nitrogen is used. During non-contact, non-destructive imaging of room temperature samples in air, the system achieves a raw, unprocessed spatial resolution equal to the distance separating the sensor from the current or the effective size of the sensor, whichever is larger. To best locate a wire short in a buried layer, however, a Fast Fourier Transform (FFT) back-evolution technique can be used to transform the magnetic field image into an equivalent map of the current in an integrated circuit or printed wiring board. The resulting current map can then be compared to a circuit diagram to determine the fault location. With this post-processing of a magnetic image and the low noise present in SQUID images, it is possible to enhance the spatial resolution by factors of 5 or more over the near-field limited magnetic image. The system's output is displayed as a false-color image of magnetic field strength or current magnitude (after processing) versus position on the sample. After processing to obtain current magnitude, this microscope has been successful at locating shorts in conductors to within ±16 μm at a sensor-current distance of 150 μm.
Operation.
Operation of a scanning SQUID microscope consists of simply cooling down the probe and sample, and rastering the tip across the area where measurements are desired. As the change in voltage corresponding to the measured magnetic field is quite rapid, the strength of the bias magnetic field is typically controlled by feedback electronics. This field strength is then recorded by a computer system that also keeps track of the position of the probe. An optical camera can also be used to track the position of the SQUID with respect to the sample.
As the name implies, SQUIDs are made from superconducting material. As a result, they need to be cooled to cryogenic temperatures of less than 90 K (liquid nitrogen temperatures) for high temperature SQUIDs and less than 9 K (liquid helium temperatures) for low temperature SQUIDs. For magnetic current imaging systems, a small (about 30 μm wide) high temperature SQUID is used. This system has been designed to keep a high temperature SQUID, made from YBa2Cu3O7, cooled below 80K and in vacuum while the device under test is at room temperature and in air. A SQUID consists of two Josephson tunnel junctions that are connected together in a superconducting loop (see Figure 1). A Josephson junction is formed by two superconducting regions that are separated by a thin insulating barrier. Current exists in the junction without any voltage drop, up to a maximum value, called the critical current, Io. When the SQUID is biased with a constant current that exceeds the critical current of the junction, then changes in the magnetic flux, Φ, threading the SQUID loop produce changes in the voltage drop across the SQUID (see Figure 1). Figure 2(a) shows the I-V characteristic of a SQUID where ∆V is the modulation depth of the SQUID due to external magnetic fields. The voltage across a SQUID is a nonlinear periodic function of the applied magnetic field, with a periodicity of one flux quantum, Φ0=2.07×10−15 Tm2 (see Figure 2(b)). In order to convert this nonlinear response to a linear response, a negative feedback circuit is used to apply a feedback flux to the SQUID so as to keep the total flux through the SQUID constant. In such a flux locked loop, the magnitude of this feedback flux is proportional to the external magnetic field applied to the SQUID. Further description of the physics of SQUIDs and SQUID microscopy can be found elsewhere.
Magnetic field detection using SQUID.
Magnetic current imaging uses the magnetic fields produced by currents in electronic devices to obtain images of those currents. This is accomplished through the fundamental physics relationship between magnetic fields and current, the Biot-Savart Law:
formula_2
B is the magnetic induction, Idℓ is an element of the current, the constant μ0 is the permeability of free space, and r is the distance between the current and the sensor.
As a result, the current can be directly calculated from the magnetic field knowing only the separation between the current and the magnetic field sensor. The details of this mathematical calculation can be found elsewhere, but what is important to know here is that this is a direct calculation that is not influenced by other materials or effects, and that through the use of Fast Fourier Transforms these calculations can be performed very quickly. A magnetic field image can be converted to a current density image in about 1 or 2 seconds.
Applications.
The scanning SQUID microscope was originally developed for an experiment to test the pairing symmetry of the high-temperature cuprate superconductor YBCO. Standard superconductors are isotropic with respect to their superconducting properties, that is, for any direction of electron momentum formula_3 in the superconductor, the magnitude of the order parameter and consequently the superconducting energy gap will be the same. However, in the high-temperature cuprate superconductors, the order parameter instead follows the equation formula_4, meaning that when crossing over any of the [110] directions in momentum space one will observe a sign change in the order