Magnet Field Relaxation

It's possible to qualitatively study the relaxation of the field in the MPMS without looking at the sample moment.  Looking at sample moment as a function of time has significant limitations as the ability to see small changes in field is related to the sample mass and susceptibility, and the ability to see small changes over time is limited to the time it takes to do an extraction through the gradiometer.

In the current study, the sample was removed from the sample tube, and the output of the longitudinal SQUID amplifier was logged every second. The absolute value of the SQUID amplifier is meaningless in this case, since the SQUID input is the flux coupled to the second-derivative gradiometer, and in this study we are not extracting samples through the gradiometer. At the same time, if the magnetic field were to be fixed, then there should be no variation of the SQUID amplifier output.

In this study the magnet was set to 1 T in no-overshoot mode mode and then back to 0 T in no-overshoot mode. This was repeated with the oscillate mode. What is clear is that in no-overshoot mode that the change in the SQUID output has two different regimes. Between zero and about five minutes the magnetic flux trapped on the surfaces of the superconducting magnet is relaxing very quickly, and there is rapid change in the SQUID output. If we were to do a measurement by extracting a sample through the gradiometer in this time period, the extraction profile would be superimposed on this rapidly varying SQUID amplifier background-- the result being increased uncertainty in moment. Above about five to seven minutes there is a regime where the field relaxes very slowly. In this regime one would obtain the best measurement results as the extraction profiles would be superimposed on a gently varying background.

In oscillate mode there is a smooth and steady relaxation of the field trapped in the magnet from the onset. Clearly in this magnet charging mode the measurement could be made almost immediately after the field is set with no impact on measurement accuracy. It should be noted that all of the data sets tend towards this same long-term relaxation behavior.

One immediate conclusion from this is that for optimum measurement accuracy one should use delays of 3-7 minutes when charging the magnet above H1c.

SQUID Drift: Revisited

As discussed in a previous post, type-2 superconductors pin flux above their first critical field, H1c. In practice this is applicable to SQUID magnetometry as the flux in the second-derivative gradiometer changes over time as the trapped flux relaxes. Subsequent gradiometer extraction profiles are thus on a time-dependent background and subject to discontinuities due to jumps in trapped flux. This has been demonstrated in actual hysteresis curves here, where it was shown that the moment uncertainty is greatest as one passes through -H1c and then back through +H1c (or vice versa).

To demonstrate the timescale of this phenomena a Pd calibration standard was subject to -1 T, well above -H1c, and then back to + 1 T where moment was measured as a function of time.  To perform the relaxation study single extractions of 24 points were used to make the measurements as fast as possible. This was performed with the field changes in both no-overshoot mode and oscillate mode. The blue dots show data taken in no-overshoot mode and the red in oscillate mode. It is clear that the the oscillate mode provides less of an initial deviation in average moment, and both curves show the effect of trapped flux relaxation being a non-issue after 400-500 seconds or so.

If one performs the same experiment going just above H1c, the results are much different.  In the second graph the maximum field is only +/- 1000 Oe. It is clear that in the case of the field changes in no-overshoot mode the flux relaxation time is > 1000 s. If one is accustomed to averaging N gradiometer extractions, this will not reduce moment uncertainty because of the time variation in moment as a function of time due to trapped flux relaxation. The extent to which this is catastrophic to one's measurements depends upon the magnitude of differential changes in moment that are relevant to one's experiment. Looking at the vertical scale the effect of flux drift is on the order of micro-EMU.

Finally, if one stays below H1c there is no problem. In the bottom graph we see no time variation in the no-overshoot and oscillate approaches. What we do see is a small offset in moment due to a remnant field in the magnet. If possible, stay at as low of a field as possible and avoid going over H1c if one needs low-noise measurements.

Approach to H1c and Moment Noise

The materials used in the MPMS electromagnet are type-2 superconductors which means that they can be penetrated by magnetic flux. Over the first critical field, H1c, which is around 800-900 Oe for the magnet in the MPMS, magnetic flux begins to penetrate the surfaces of the superconducting wires where it is pinned on various defects in the magnet material.

Flux pinning is of interest when we use the MPMS because it impacts the moment noise of our data. If the trapped flux was stable, it would be of little interest, but it changes dynamically with time. The amount of trapped flux relaxes, and it is also subject to sudden discrete flux jumps. Since this flux is trapped on the surfaces of the MPMS magnet, the dynamics of the trapped flux manifests in the effective applied magnetic field and thus the moment detected by the gradiometer.

In this study the moment of a ferromagnetic sample was measured as a function of field, from +5 kOe to -5 kOe as shown by the blue curve, and then from -5 kOe to +5 kOe as shown by the red curve. The sample was chosen to be in a 10^-5 EMU moment range where the effects of flux relaxation and flux jumps would be noticeable. Notice that in the blue curve there is more noise in the moment as the sample passes through -H1c and on the red curve as it passes through +H1c.  This is seen more clearly in the variance of the measured moment in the second graph.

It can be seen that it's not as simple as the moment noise being the largest at +/-H1c. It is also a function of the approach to the critical field. Approaching +H1c from above has only a minor effect on the moment noise as does approaching -H1c from below. Because of the instability of the trapped flux in the magnet the greatest effect occurs when one goes above +H1c and then below -H1c and vice versa.

To get at why this is so, we can look at the average regression fit. There is clearly a very sharp and sudden drop in the average regression fit as one passes through - H1c from above and +H1c from below.  An examination of the actual SQUID extraction profiles shows discontinuities in some of the extraction profiles-- the result of discrete flux jumps.

The purpose of the short note is to show it's not as simple as "all heck breaks lose at H1c".  It's a function obviously of not only the moment range but the approach to +/-H1c.  At this point the skilled MPMS user would attempt to mitigate the effects of discrete flux jumps by using MultiMeasure, adding significant pauses as one passes through H1c to allow the trapped flux to relax and stabilize, or by logging all extractions and eliminating the statistical outliers.

The Benefits of Multi-Measure

The standard MPMS moment measurement averages a specified number of SQUID profiles. This mode of operation is called DC-measure. In the top graph a 14.4 uEMU moment at 0 Oe is measured using a 4 cm scan defined by Np = 24 points. The number of samples is varied and the error in moment is displayed along with 1-sigma error bars.

A few things are immediately apparent. One, the average moment errors are in the range of 50-60 nEMU which is within the specification for the sensitivity of the instrument at zero applied field. The other observation is that the moment error does not trend with the number of samples. That is peculiar. One would expect the moment error to decrease and approach some limiting value as the number of samples increases. One would also expect the error bars on that moment error to decrease as the number of samples increases.


Implicit in that are some assumptions about the nature of the effects contributing to the moment error. In practice the measurement of the moment is perturbed by low frequency random events. These can be acoustic noise in the laboratory, thunder, etc. that cause the sample rod to vibrate, or random electromagnetic events such as the operation of high power building machinery. The sources of these perturbations is unknown and will remain so. What is important is the sense that they are low frequency random perturbations of the moment measurement.

The multi-measure function allows one to deal with these events. In this case one defines the number of points to define a SQUID extraction profile, N_p, as always, along with the number of samples to be averaged, N_s. In addition one can define a maximum number of measurements to be rejected, N_r, if they exceed a certain number of standard deviations of the mean moment, N_sigma. This allows one to apply statistical criteria for measurement rejection at every data point.

In the bottom graph we are varying the number of samples, N_s, with the number of SQUID profile points fixed at N_p = 24. We have also fixed N_r = 2 and N_sigma = 2. The moment error and 1-sigma error bars are in that error are shown. What is observed is that at small N_s both the moment error and its uncertainty are larger, and that as N_s increases the moment error decreases along with the uncertainty in the moment error. At N_s = 24 we have cut the moment error in the standard DC-measure in one third to 40 nEMU which is the performance specification for the moment uncertainty at zero applied field. Since we can only reject two extreme measurements, the moment error eventually beings to increase as N_s increases to larger and larger values. Also evident is that using multi-measure one can get the same moment error (and uncertainty in that error) using ~ N_s = 8-10 samples as one obtained using DC-measure with N_s = ~ 25 samples.

SQUID Extraction Profile Points

In this graph the uncertainty in moment of a ~ 500 nEMU sample at 30 K and 0 Oe applied field is studied. The extraction distance was set to 4 cm and the moment standard error was measured using the multi-measure function. Multi-measure allows one to place statistical constraints on an ensemble of measurements allowing a certain number to be rejected if they exceed these constraints.

In this study the number of measurements was set to 30 (N_m) with a maximum number of two rejected (N_r) if they exceed two standard deviations from the mean (N_sigma). The solid line is the standard error as a function of the number of points defining the SQUID extraction profile. The dotted lines are 1-sigma error bars. N_m is very large-- impractically large-- to minimize the effects of statistical uncertainty so that the effects of extraction profile points can be isolated.

The result is fairly clear: there is no clear benefit of using more extraction profile points. As little as 12 or 16 points can be used with 4 cm extraction length to produce standard errors in the 10's nEMU range. The most common choice among users seem to be N_p = 24.

Longitudinal Moment Calibration

Calibration of the MPMS SQUID magnetometer is extremely straightforward. There is a calibration factor, the longitudinal calibration factor, that calibrates the moment obtained from the SQUID extraction profile. As long as the SQUID amplifier is appropriately calibrated across its ranges of gain, a single one-point calibration is sufficient.

In practice one uses a high purity paramagnetic sample of known mass and susceptibility. The sample is a cylinder of appropriate size (a few mm in diameter) and shape that it reasonably approximates a point dipole in the gradiometer, eliminating errors due to the presence of higher order multipoles. The standard is also mounted in a way so as to introduce as little diamagnetic background as possible from sample mounting-- the sample mounting is continuous through the extraction length in the gradiometer-- and the standard is not handled with magnetic forceps which might introduce magnetic contamination..

This graph shows M vs. H for a 0.2477 g Pd standard at 298 K where the DC susceptibility is known to be 5.25 x 10^-6. The susceptibility of the standard can be determined from the slope of the M vs. H curve, and the new longitudinal calibration factor determined by: L_new = L_old * (X_meas/X_std). The same data can be used to determine the remnant field in the magnet from the ratio the y-intercept and the slope.

There is also a transverse gradiometer in the CMMP MPMS SQUID magnetometer. This is a bit more complex to calibrate. Because of the design of the transverse gradiometer, the longitudinal moment is not entirely decoupled from the transverse gradiometer. As such, some longitudinal signal "leaks" into the transverse measurement channel. Additionally, paramagnetic standards such as Pd do not produce a transverse moment. Transverse calibration is left as a separate subject.

Temperature Control

The top graph shows a series of temperature changes with increasingly smaller cooling increments. This illustrates several features of the MPMS temperature control on cooling.

The feature between 0 < t < 55 minutes is an under-cooling operation as the temperature is reduced from 300 K to 134 K. Under-cooling allows for more efficient cooling operations when |dT| > 10 K. There is a large temperature gradient between the sample location in the sample tube and the thermometers at the end of the sample tube in the cooling annulus. The MPMS system can exploit knowledge of this gradient to under cool the thermometer end of the sample tube to get the sample portion of the sample tube to the target temperature faster. While it took ~ 50 minutes to stabilize at the target temperature of 134 K, the average cooling rate is around 3 K/min. The amount of time to stabilize the temperature in a large dT cooling operation depends upon not only the magnitude of dT but the temperature range. At lower temperatures stabilization is faster because the heat capacity of the components of the sample tube is lower. It should be noted that the initial cooling portion of this large under cool operation shows a cooling rate of over 40 K/min.

The temperature is then dropped in increments of 10 K, 1 K and 0.1 K. The same general cooling behavior is demonstrated in these cases where under-cooling is not applied because |dT| < 10 K. As the proportional valve opens to increase the flow of cool He vapor, the temperature spikes down, then up and plateaus at a temperature below the temperature set-point. After a period of time the proportional valve closes somewhat as the gas heater heats the He vapor. The temperature increases, slightly overshoots and then approaches the set-point.

The bottom graph shows T vs. t on warming. The temperature is stabilized at 100 K warming from roughly 40 K. The initial part of this operation shows a maximum heating rate of ~ 15 K/min. The rate of warming decreases as the proportional valve starts closing, the temperature overshoots and the set-point is approached in a damped oscillatory fashion. 0.1 K and 1 K increments are followed by 10 K increments. Note that the over-shoots on warming are much less than the under-shoots on cooling: a 10 K warming increment over-shoots by a few 100 mK while a 10 K cooling increment under-shoots by several K's.

Also note the different times scales involved in cooling and heating: a 10 K warming increment takes about 9 minutes while the corresponding 10 K cooling increment about 29 minutes. Of course these numbers depend upon the temperature range one is performing the operation in-- but the moral is that it is easier to put energy into the system than remove it. It's faster and takes less fine control to heat than cool. If at all possible perform all M vs. T measurements while warming. These temperature overshoots and undershoots should not concern one in the performance of M vs. T measurements because what is being measured is the temperature control point of the gas flow annulus-- not the sample temperature.

Very Low Temperature Operation

The boiling point of liquid He4 is 4.2 K at 1 atm. To reach temperatures below 4.2 K one must rely upon vapor at pressures less than 1 atm. This is achieved by filling a reservoir with LHe and pumping on it to achieve supercooled vapor. This supercooled vapor then flows through the cooling annulus to cool the sample as in normal operation.

This graph shows the temperature as a function of time in a sequence of temperature changes that illustrate this process.

The temperature is initially stabilized at T = 5.4 K around t = 2.25 minutes (originally starting at T = 5.0 K). After that the temperature is set to T = 4.4 K. This is below the "low temperature" setting of 4.43 K which is the lowest temperature that can be achieved without resorting to cooling with super-cooled vapor. This value is more related to the heat transfer properties of the cooling annulus than any intrinsic property of LHe4. Any T < T_low = 4.43 K will require filling the reservoir and pumping on it.

What we see after some temperature fluctuations is the temperature stabilizing at T = 4.15 K for a period of ~ 10 min. This is the "fill temperature"-- the temperature of LHe with the small vacuum required to pull vapor through the cooling annulus. This is just below the LHe boiling temperature of 4.2 K at 1 atm. Since the temperature sensor is at the bottom of the sample tube this is indicative of the reservoir filling with LHe. At this point the impedance value is open and the proportional value only slightly open to allow the filling of the reservoir.

After the reservoir is filled the impedance valve is closed ceasing flow of LHe into the reservoir. The vacuum applied on this isolated pool of LHe in the reservoir produces super-cooled He vapor that can be used to cool to temperatures below T_low = 4.43 K. The temperature stabilizes at T = 4.4 K at ~ t = 17 minutes. The temperature is then reduced in 1 K steps to 2.4 K, and then in 0.2 K to the lowest achievable temperature of 1.8 K.

Since this system only has one reservoir, one can work in the range of 1.8 K < T < 4.43 K for as long as the reservoir contains LHe-- approximately 30-45 minutes. In systems that have the CLT or continuous low temperature option one can run continuously in this range. This is generally not a problem for M vs. T measurements that require a few low temperature points, but it does make it difficult to perform a full hysteresis loop at T < 4.43 K. Eventually the reservoir will empty, there will be a huge temperature fluctuation and subsequent reservoir filling period before very low temperature operation can resume.

In practice one can not fill the reservoir if one goes below T_low from high temperatures T > 100K). There is enough residual heat in the sample tube that the reservoir will start filling and immediately vaporize the LHe. It is good practice to stabilize at a temperature just above T_low, such as 5.0 K, for a period of 15-30 minutes or so first. That may seem like a long time, but keep in mind that the reservoir fill time is ~ 10-15 minutes.

SQUID Drift

The SQUID profiles-- the SQUID device output voltage vs. extraction distance-- are generally detrended to remove any background on the SQUID profile. This occurs automatically, but one can inspect the raw SQUID voltages through the diagnostic menus of the MultiVu software. It is not unusual to see the SQUID voltage run through its output range repeatedly as the SQUID amplifier's gain is reset again and again. This is due the effect of SQUID drift.

SQUID drift is a time-dependent drift of the SQUID device output due to field relaxation-- more specifically due to trapped flux on the magnet relaxing. The first critical field, H_c1, of the magnet is around 800-900 Oe. Above this field flux can penetrate the surfaces of the superconducting magnet thus pinning flux vortices. Above H_c1 "flux jumps" can be seen as small discrete changes in field. This effect is seen primarily in the range of 0.8-5.0 kOe. To deal with this effect in practice it is good MPMS practice to include a 120-180 second pause after field changes in this range.

Apart from "flux jumps" the flux trapped on the magnet surface relaxes over a considerable time period. That timescale depends upon the fields applied and the method of magnet charging. The graph above shows M vs. t of time over a 2000 minute period. The blue data is the "baseline"-- the moment of a Pd standard measured at H = 0 Oe. The fact that the moment of this paramagnetic system is negative is a signature that there is magnet hysteresis: there is a negative remnant field due to trapped flux in the magnet. Generally that field is on the order of a few Oe's, but can be as large as 50-100 Oe depending upon the magnet history.

The green data shows M vs. t for the magnet being charged to 5 T and then back to 0 T in "oscillate mode" where the field approaches its target value in a damped oscillatory fashion, while the red data shows M vs. t for the magnet being charged to 5 T and then back to 0 T in "no oscillate mode" where the field approaches its target asymptotically. In the case of the "oscillate mode" approach there is no sign of relaxation: the M vs. t data is as flat as the background data. In the case of the "no oscillate" mode approach, the moment of the paramagnetic sample is changing exponentially as the trapped flux relaxes. The timescale of this relaxation should be noted: it's not fully relaxed after 2000 minutes-- more than one day!

While this effect is very noticeable at low fields where the applied fields are comparable to the remnant field in the magnet due to trapped flux-- it is also detectable at high fields. The bottom graph shows the M vs. t at 5 T as the high field is approached in both "oscillate" and "no oscillate" modes. Approaching the high field from below in "no oscillate" mode causes the field to relax over a period of 5-10 minutes. While it is a small fractional change, it is clearly detectable.

What to do?

In systems with the magnet reset option (the CMMP MPMS has this option but it's currently not active) the magnet can be reset. This essentially drives the magnet normal by heating a small part of the magnet. As the magnet goes through its T_c the trapped flux is lost as it is no longer pinned. Another option is to use a nulling sequence: set the magnet (all in oscillate mode) to + 5 T, then - 2.5 T, then + 1.25 T and so on until one reaches 0 T. The purple data (top graph) shows M vs. t for after this nulling sequence was applied after approaching 0 T from 5 T. There is no sign of relaxation and the remnant field in the magnet due to trapped flux is even closer to zero than the "baseline" data.

A few good MPMS practices:

• Don't go above H_c1 unless you have to. If you are doing low field work, apply an H < H_c1 to center the sample.
• Don't go to high fields unless you have to. If a system saturates at 5 kOe, why go to 50 kOe? This will only lead to increased field relaxation.
• Use oscillate mode whenever possible. The exception: in an M vs. H measurement of a ferromagnetic (or ferrimagnetic) system this is not a good idea as one would be performing a series of small minor hysteresis loops at each point of the measurement.
• Using a nulling sequence to minimize trapped flux and remnant field in the magnet.
• Use pauses after field changes: at high fields ( >= 1 T) 60-90 s; at low fields ( < 0.1 T) 30-60 s; just above H_c1 (0.8-5.0 kOe) 120-180 s; when coming to low field from large fields ( e.g. 5 T to 0 T) as much as a 10 minute initial pause.

Keep in mind that the magnitude of magnet noise due to flux jumps and trapped flux relaxation is not proportional to the moment intensity: small moments may require significantly larger field pauses than larger moments. The above are general guidelines and may have to be modified for smaller moments.

Second Derivative Gradiometer

The superconducting quantum interference device (SQUID) is functionally a magnetic flux to voltage converter. One advantage of SQUID magnetometry is that it directly measures magnetic flux as opposed to torque (torque magnetometer) or current induced by a magnetic field (vibrating sample magnetometer). SQUID magnetometry is also very sensitive measuring moments as small as 80-100 nEMU over a dynamic range of ten orders of magnitude.

A simple loop coupled to a SQUID device would provide some challenges. One, it would detect the flux from the earth's magnetic field, as well as any flux from nearby magnetic systems. To use the SQUID device more effectively, one uses a gradiometer-- a configuration of coils that detects the gradient of the magnetic field. In the case of the Quantum Design MPMS a second derivative gradiometer is used. Such a gradiometer rejects all constant and linearly diverging magnetic flux. Since the sample can be assumed to be a dipole, the geometry of the second derivative gradiometer can be optimized for the detection of dipole moments. Such a gradiometer consists of two coils with the same sense very closely spaced and two coils of the opposite sense spaced symmetrically outside. All four of these coils are tied together. The flux penetrating all four coils is detected by the SQUID device.

In practice the sample is moved or extracted through the gradiometer and the SQUID output measured as a function of position. Based on this profile of voltage (and thus flux) versus position, given the assumption that the signal is from a point dipole, the magnetic moment can be determined. Double-click on the image to see the animated GIF (image borrowed from ICMMO). The gradiometer is interior to the superconducting magnet of the magnetometer. When fields are changed the gradiometer is decoupled from the SQUID and recoupled after field changes are complete. This decoupling is achieved by using a small heater to drive normal a superconducting transformer that couples the gradiometer to the SQUID device.

Proper use of the gradiometer requires not only centering the sample in the gradiometer coils, but mounting samples in such a way that the mounting material provides a continuous signal throughout the gradiometer coils and is thus not detected by the gradiometer.

Superparamagnetism: ZFC & FC Measurements

The MPMS SQUID magnetometer is useful in the study of nanoparticle systems that demonstrate superparamagnetic behavior. For particles that are smaller than the domain size of a magnetic material, the particle size can be small enough that that magnetic domain can flipped by thermal fluctuations. Such systems are characterized by a blocking temperature above which the system is superparamagnetic and below which it is ferromagnetic or ferrimagnetic.

To determine the blocking temperature of a nanoparticle material, one typically performs zero-field cooled and field cooled-- ZFC & FC-- M vs. T measurements. In this case one cools the sample in no applied field from a temperature above the blocking temperature. Then one applies a magnetic field while warming and cooling in the applied field. The presence of temperature hysteresis is characteristic of superparamagnetic behavior with the maximum moment of the ZFC curve corresponding to the blocking temperature

Data shown is for a nanoparticle Prussian blue complex in a porous silica matrix. Thanks to Dr. Al Stiegman, FSU Department of Chemistry.

Lost Samples & Sample Mounting Materials

SQUID samples are generally mounted inside drinking straws using pieces of tape, pieces of straws and capsules to mount the samples themselves. There is an art to this.

This is material retrieved from the sample tube. The block with the relief valve to the right is the top of the sample tube that has been removed from the gas annulus. Lost material includes an entire thin-film sample, scotch tape, kapton tape, and powder material. The long piece of straw was sheared off in the gate valve between the sample transport and the sample tube.

Losing some sample material is acceptable and part of the learning curve-- not telling anyone about it is not! We were able to identify the culprit by performing EDS on the thin-film and powder material. If you lose a sample, or suspect not everything made it back out-- let us know!

5T MPMS SQUID Magnetometer

The CMMP group at the Florida State University Department of Physics has a Quantum Design 5T MPMS SQUID magnetometer that is available to the local reasearch community. A small fee of is assessed for usage of this instrument.

The MPMS allows one to perform measurements of magnetic field as a function of temperature, time, orientation and applied DC field. The maximum applied field is +/- 5.5 T, with a temperature range of 1.8 K to 400 K. Temperatures below ~ 4.8 K require filling and pumping on a small reservoir to achieve temperatures below the LHe boiling point. A reservoir fill lasts about 45 minutes (this MPMS does not have the CLT or continuous low temperature option). The moment sensitivity is 8 x 10^-8 to 1 x 10^-7 EMU. The system has both transverse and longitudinal gradiometers and has the extended range option which allows for the measurement of moments up to 300 EMU. Samples must be fit in a 9 mm diameter bore and be attached to an appropriate probe. Transport measurements can be performed in situ using interfaced test equipment and a specialized probe. Unlike many MPMS systems, it does not have the AC susceptibility option nor the magnet reset option. There are horizontal and vertical rotation options for anisotropy measurements and an oven insert capable of reaching 800 K.

Advantages: sensitivity; two axis measurement; gas management is automatic; in situ transport measurements.

Disadvantages: slow; no magnet reset so one must deal with trapped flux in the magnet when going above the first critical field; no continuous low temperature (CLT) option which limits the time one can stabilize at temperatures below the LHe boiling point.