The Direct Conversion of Heat to Electricity Using Multiferroic Alloys


  • Vijay Srivastava,

    1. Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA
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  • Yintao Song,

    1. Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA
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  • Kanwal Bhatti,

    1. Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA
    2. Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
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  • R. D. James

    Corresponding author
    1. Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA
    • Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA.
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We demonstrate a new method for the direct conversion of heat to electricity using the recently discovered multiferroic alloy, Ni45Co5Mn40Sn101. This alloy undergoes a low hysteresis, reversible martensitic phase transformation from a nonmagnetic martensite phase to a strongly ferromagnetic austenite phase upon heating. When biased by a suitably placed permanent magnet, heating through the phase transformation causes a sudden increase of the magnetic moment to a large value. As a consequence of Faraday’s law of induction, this drives a current in a surrounding circuit. Theory predicts that under optimal conditions the performance compares favorably with the best thermoelectrics. Because of the low hysteresis of the alloy, a promising area of application of this concept appears to be energy conversion at small ΔT, suggesting a possible route to the conversion of the vast amounts of energy stored on earth at small temperature difference. We postulate other new methods for the direct conversion of heat to electricity suggested by the underlying theory.

1. Introduction

While the goal of discovering a material that is both strongly ferroelectric and strongly ferromagnetic remains elusive, numerous other interesting multiferroic materials are emerging.1–9 A particularly fertile area for discovery are crystalline materials having a big first order phase transformation, but with no diffusion.10 Multiferroic properties like polarization or magnetization are generally sensitive to changes of lattice parameters.2, 8, 11 In a material that undergoes a phase transformation having an abrupt change of lattice parameters, one can expect the two phases to exhibit diverse electromagnetic properties. The alloy Ni45Co5Mn40Sn10, that is used in energy conversion device described here, is of this type. It has a strongly ferromagnetic austenite phase, with magnetization approaching that of iron, and a nonferromagnetic martensite phase.

Alloys that undergo such martensitic phase transformations are also attractive for energy storage and conversion. Fundamentally, this is owing to the absence of diffusion and other relaxation processes and the presence of a low energy mode of transformation between crystalline lattices, i.e., the passage of many austenite-martensite interfaces. Martensitic transformations are typically quite fast, with speeds of interfaces an appreciable fraction of the speed of sound in the material.12 With a large change in the lattice parameters, large free energy changes are typical. Partly for these reasons, the highest energy density actuator, that converts heat to mechanical work, is the shape memory alloy NiTi,13 which undergoes a reversible martensitic transformation from a cubic to a monoclinic phase. Numerous emerging devices convert mechanical vibrations into electricity using piezoelectric, ferroelectric, magnetostrictive or ferromagnetic shape memory effects. These rely on quite different physical processes so we do not review them here.

With its low hysteresis and its large and abrupt change in magnetization at transformation, we conjecture that the natural application area for our device is energy conversion at small temperature difference. Cyclic heat engines of this type are necessarily of low efficiency, because the efficiency of all such devices is bounded by the Carnot efficiency 1 − Tmin/Tmax, Tmin (resp., Tmax) being the lowest (resp., highest) temperature experienced by any part of the device during the cycle, according to a fundamental theorem of thermodynamics. However, the definition of efficiency, η = (net work done)/(total heat absorbed) during the cycle, expressed in purely economic terms, is typically (energy sold)/(fuel bought). Given that all internal combustion engines waste heat at small temperature difference through cooling, it may be that the standard definition of efficiency is irrelevant. More generally, there are enormous reserves of energy on earth stored at small temperature difference, notably, the ∼20 °C difference between surface ocean temperatures and temperatures just below the thermocline in mid-latitude waters. What is needed in such cases is a more suitable measure of efficiency, consistent with thermodynamics at small temperature difference.

Multiferroic alloys that could be utilized in energy conversion devices must have highly reversible phase transformations.14 A change of lattice parameters typically implies the presence of stress, notably in the transition layer between phases at the austenite/martensite interface.15 If this stress is sufficiently large, it can drive the formation of dislocations and other defects that, after sufficiently many cycles of transformation, may lead to widening thermal hysteresis, migration of transformation temperatures, or failure.16–18 At a macroscopic level such processes are often seen as the competition between phase transformation and crystal plasticity.

The particular alloy Ni45Co5Mn40Sn10 used here was found by implementing a new strategy for improving the reversibility of phase transformations. This can be expressed11, 15, 19 as the condition λ2 = 1, where λ2 is the middle eigenvalue of the transformation stretch matrix. When λ2 = 1, the stressed transition layer in the austenite/martensite interface disappears.20 In recent alloy development programs done by combinatorial synthesis methods,21, 22 the thermal hysteresis of phase transformations in NiTiX (X = Pd, Pt, Au) has been shown to drop precipitously from more than 30° to less than 6 °C as λ2 → 1. A particularly dramatic example of Zarnetta et al.22 in NiTiCuPd alloys shows a hysteresis near zero23 and corresponding improvement of the reversibility of transformation, quantified by the migration of hysteresis loops under repeated transformation, as λ2 → 1. The alloy Ni45Co5Mn40Sn10 (λ2 = 1.0032) used here was discovered by this kind of tuning of lattice parameters by systematic change of composition.1 Having λ2 = 1 does not contradict an otherwise large change of lattice parameters, i.e., the other eigenvalues of the stretch matrix can remain far from 1, so the aforementioned abrupt change of electromagnetic properties is possible. In this general alloy system we found a correlation between the large magnetization of austenite and λ2, with the maximum magnetization occurring near λ2 = 1. An understanding of the the origins of this correlation is currently unavailable.

2. Characterization of Ni45Co5Mn40Sn10 for Energy Conversion

The particular properties of this alloy Ni45Co5Mn40Sn10 influenced the design of the demonstration. Figure1 shows the magnetization vs. temperature measured in this alloy. At the applied field of 1T the phase transformation on heating causes an increase of the magnetization from less than 10 emu/cm3 in the martensite phase to more than 1100 emu/cm3 in the austenite phase. The austenite is nevertheless magnetically very soft with a susceptibility of equation image measured from the data of Figure 1d near H = 0. Magnetization vs. temperature at 0.5 T is nearly the same as shown in Figure 1a. Figure 1c suggests an inhomogeneous structure containing a small volume fraction of ferromagnetic particles in an antiferromagnetic matrix. Notice that this high magnetization of austenite occurs even though the transformation temperature equation image measured23 at low field is quite close to the approximate Curie temperature of 167 °C. These temperatures are somewhat high for the design of actual devices, but in this family of alloys, and in martensitic transformations in general, transformation temperatures are extremely sensitive to composition. Further alloy development would be needed to tune both λ2 and θc. The thermal hysteresis measured by calorimetry (Figure 1b) is equation image and the latent heat is 16.5 J/g. The magnetization curves show zero remanence and coercivity within experimental error. This implies that changes of magnetization contributed negligible loss to the device. This also necessitated the use of a permanent magnet to bias the material, so that, upon heating to austenite, a uniformly magnetized sample was achieved. The loss due to the phase transformation is indicated by the hysteresis loops in several of the measurements.

Figure 1.

Relevant properties of Ni45Co5Mn40Sn10. a) Magnetization vs. temperature under 1 T field. b) Heat flow vs. temperature. Magnetization vs. applied field in martensite (c) and in austenite (d) (Note the different scales). Reproduced with permission.1 Copyright 2010, American Institute of Physics.

3. Energy Conversion Device and Voltage Measurements

We show a schematic view of our demonstration in Figure2. The specimen is fixed (to prevent movement) near a pole of a permanent bar magnet and is surrounded by a coil. Heat is supplied by an external heat gun. The basic idea is that, upon transformation on heating, the average magnetization M is increased in the specimen, giving a substantial component of dM/dt. Demagnetization is prevented by the presence of the nearby permanent magnet. The fundamental dipolar relationship between the magnetizationM, the magnetic inductionB and magnetic fieldH,

equation image((1))

then gives a nonzero dB/dt. The precise way in which the magnetization is partitioned between H and B is governed by micromagnetic considerations. A consequence of the design is that equation image is induced in by the phase transformation. According to Faraday’s law

equation image((2))

an electric field is created during transformation. The design of the coil is to give a maximal component of E parallel to the wire, thereby driving a current. A potential difference across the coil of opposite polarity is obtained on the reverse phase transformation upon cooling.

Figure 2.

Schematic (a) and actual (b) views of the demonstration. C, coil; R, heat source; S, specimen of Ni45Co5Mn40Sn10; M, permanent magnet with direction of magnetization indicated; T, thermocouple; V, voltmeter.

Besides the micromagnetic considerations mentioned above, this analysis is too simple in another respect. The current created in the coil itself places a back field on the material, decreasing the overall rate of change of B. A more accurate analysis accounting for this effect leads to a differential equation for the current described below.

We show a picture of the actual device in Figure 2b. The specimen of Ni45Co5Mn40Sn10 was arc-melted from elemental materials and, to retain as much mass as possible, it was not cut down to a convenient shape. The surrounding coil consisted of 2000 turns of fine insulated copper wire. A large resistance of equation image, placed between the ends of the coil, gave a potential difference between its ends during heating. Figure3 shows the potential difference across the coil vs. time, and the temperature vs. time (inset of 3b), measured simultaneously upon heating. The temperature measurement was done with a thermocouple placed below the specimen and therefore must be considered an approximate representation of the temperature field on the specimen. The measured temperature vs. time is approximately monotone, so in Figure 3b we eliminate time between these plots and get the graph of voltage vs. temperature. We interpret the plateau in the inset as being due the absorption of latent heat: the lack of strict monotonicity here causes a slight reversal on voltage vs. temperature. By comparison with magnetization vs. temperature shown in Figure 1a, we interpret the first rapid rise as due to the phase transformation. We interpret the subsequent sharp decrease as due to the rapid fall of magnetization vs. temperature close to the Curie temperature, as seen in Figure 1a. The cooling curves confirm this interpretation.

Figure 3.

Measurements of a) voltage vs. time, and b) voltage vs. temperature. The graph (b) is obtained from (a) by elimination of the time using the inset. The open circles give the corresponding measurements done with the specimen absent.

We have written down Faraday’s law in a form appropriate to the assumed geometry, including the back field of the induced current. The details will be presented elsewhere.24 We treat the shape effect using an axial demagnetization factor equation image. For an assumed time-dependent susceptibility χ(t) we get the following differential equation for the current induced in the coil:

equation image((3))

Here, equation image is the cross-sectional area of the specimen and the other quantities are defined in Table1. Leff is given by a standard formula for the effective length of the coil of inner/outer radii r1, r2 and length L, accounting for flux leakage at the end:

equation image((4))

In the special case of a susceptibility that is linear in time,

equation image((5))

(3) becomes

equation image((6))


equation image((7))

Here, equation image is the susceptibility of martensite. The special case (6) is particularly useful for parameter studies because the solution can be written down explicitly. The general solution of (6) is

equation image((8))
Table 1.. Values of parameters used in the evaluation.
cspeed of light3 × 1010cm s−1
Ddiameter of the specimen0.75cm
δdemagnetization factor2.5 π 
Nnumber of turns in the coil2000 
r1inside radius of the coil0.95cm
r2outside radius of the coil1.9cm
rcaverage radius of the coil1.425cm
Llength of the coil0.8cm
ddiameter of the wire0.013cm
σelectrical conductivity of the wire (Cu)5.4 × 1017s−1
Rexternal resistance10 k ohm = 1013/c2s cm−1
h0magnitude of the applied magnetic field2000Oe
χaaverage susceptibility of the austenite0.1emu cm−3Oe−1
Q|latent heat of the specimen13.2 × 108erg cm−3
t1transformation time10s

This solution has a short-lived transient followed by saturation to the value equation image. This current gives a constant back field that opposes the field due to the permanent magnet. Rapid heating through the phase transformation strongly favors a big back field, and, together with the large magnetization, gives a higher work output.

4. Comparison Between Theory and Experiment and Optimization

In this section we give two kinds of predictions of the model. In the first comparison we use the actual parameters estimated from the experiment. We also examine the effect of certain minor changes of the experiment that can lead to improvements. In the second comparison we seek a geometry and parameter regime that is expected to be realizable, and which leads to substantial improvements. We call the latter an optimized demonstration.

The parameters corresponding to the actual demonstration are given in Table 1. Most of values are accurately known or straightforward to estimate. There is difficulty with equation image, because the precise location of the specimen while heating was not known. But the main difficulty is equation image, because the specimen contained a temperature gradient while being heated and therefore experienced a range of susceptibilities (cf., Figure 1a) at any given time. The value equation image was obtained by averaging the graph of Figure 1a over an estimated temperature interval. The value of equation image is taken to be 0. The value of the demagnetization factor was also approximate, since the shape of the active volume was not known (The specimen contained several cracks.) The time t1 is taken to be the approximate duration of the first peak in Figure 3a. The applied field was measured directly in the device with the specimen absent, and equation image is measured directly from the area under the curve in Figure 1b. With these values the transient term in (8) decays very quickly in a small fraction of t1. Inserting these values into (8) and multiplying by R we predict an output voltage of 0.81 mV. The measured peak voltage is about 0.6 mV (cf., Figure 3a).

To give a more realistic description of the transformation process, we now use a time-dependent equation image but leave the rest of the Table 1 the same. Referring to Figure4, we now discard the assumption (5) and use (3), choosing equation image that gives an approximate representation of the variation of susceptibility during the full heating cycle seen in Figure 3a. This can only be done semi-quantitatively, because some details (precise shape, thermal conductivity) needed for the heat transfer problem are inexactly known and the kinetics of this phase transformation is completely unknown. In this case the solution (8) is no longer valid, and we did a direct numerical integration of (6) with the time-dependent equation image shown in Figure 4a. The result is seen in Figure 4b. This shows qualitative agreement with Figure 3a. As above, the peak values are a little overestimated: this is likely due to overestimation25 of the active volume of the specimen.

Figure 4.

Numerical solution of (3) with assumed susceptibility χ(t) modeling the demonstration.

Better performance would be expected with a specimen that fills the region inside the coil. If we fill the volume inside the coil with the specimen (D = 1.9 cm), use an appropriate demagnetization factor (equation image, based on the aspect ratio of 1.9/1 of the coil used) and use the peak susceptibility measured above (equation image), leaving the other values in Table 1 the same, we predict a voltage several orders of magnitude bigger, 156 mV.

Now we use our measured properties of Ni45Co5Mn40Sn10 but consider the effect of optimizing the geometry and external resistance. Further improvements are expected by taking advantage of the natural scaling of (8). The role of Leff in (8) suggests that platelike specimens are advantageous. The scaling of the heat equation is such that if the specimen thickness is scaled by k, then the time for the center of the plate to change temperature by a fixed amount is scaled by equation image. Assuming again that the specimen fills the region inside the coil, we replace L by equation image in the formula for Leff and t1 by equation image in (8), and we examine what happens when k is increased. Since this also changes the volume of the specimen, we divide by the volume in comparisons below.

This scaling also however affects other material constants in Table 1. In particular, we replace N by equation image to ensure that the coil always has the length of the specimen. In addition, demagnetization factors degrade as the specimen gets thinner. For this we use the standard formula (see ref. 26, eq. (2.24)) for the demagnetization factor of an oblate ellipsoid with aspect ratio k,

equation image((9))

At equation image, equation image (sphere) and equation image as equation image (i.e., the standard formula for a thin plate). We also choose an optimized external resistance. Except where noted, we choose the external resistance R to be the unique value that maximizes the power output for each choice of parameters given below.

Modest increases in k (thinner plate, shorter heating time) have a big effect on voltage and power density. For example, if we begin with the constants for the improved experiment above (Table 1, but with equation image, and equation image scaled with k as described above) we find that the power density increases rapidly with k, then linearly. At the modest value equation image we already have a power density of equation image, an energy density per (half) cycle of equation image, and a predicted voltage of 502 mV with equation image. Greater improvements (per volume) are suggested for thin films with patterned coils.

The behavior of demagnetization factors with k given in (9) suggests another interesting scaling. The demagnetization factor behaves much better for a thin rod, while the advantages of the thermal scaling persist. But the design of the experiment has to change to permit the rapid heating of one or more rods. We consider several alternatives below, in which k is now the aspect ratio (length/diameter) of the rod:

  • (1)Thin rods with a tightly wound coil of one layer deployed on a surface. This is intended to allow rapid heating and to accommodate a flux path below the surface. We now let equation image be the aspect ratio, so again we are interested in large values of k. In this case, to a very good approximation for equation image (particularly with a flux path), equation image. Also, for a single layer coil we have equation image in the formula for Leff in which case equation image. Other constants are modified as follows: equation image, equation image, equation image, equation image, equation image, equation image, equation image (peak susceptibility), equation image. In this case again the power density increases rapidly with k and then linearly, while the energy density per (half) cycle saturates, but more slowly than in the case of a thin plate. At equation image the power density is equation image and the energy density per (half) cycle is equation image. This high power density is only produced at a voltage of equation image.
  • (2)Hollow rods, or rods separated from the coil by a small space, heated/cooled by a flowing fluid. We now envision a long thin specimen (equation image) but heated/cooled through its hollow core, or between itself and the coil. Obviously, the temperature changes cannot be as rapid in this case (in practice, k is limited). With the constants and scaling of the preceding item except with equation image, the power and energy densities are similar to those in Item 1, but this is produced at the higher voltage of equation image.

5. Comparison with Thermoelectric Materials

It is illustrative to compare our results with the most promising other device for the direct conversion of heat to electricity, thermoelectrics.27 As explained below, even an optimized version of our demonstration has low efficiency. However, the predicted voltage output for an optimized device is comparable to that of good thermoelectric materials. The predicted power density of an optimized device is more than an order of magnitude greater than thermoelectrics with the highest known equation image value. Our predictions involve optimizing the geometry, but otherwise only use the measured data of Ni45Co5Mn40Sn10 shown in Figure 1.

The thermoelectric effect refers to the direct production of electricity from a temperature gradient using the Seebeck effect, and its converse. Normally thermoelectrics use a larger temperature difference than that for which the present demonstration seems best suited. However, thermoelectrics and our demonstration are both capable of converting heat to electricity with no moving parts, and it is natural to compare them. It should be born in mind that the Seebeck effect was discovered in 1821, and thermoelectrics have been the subject of intense study and optimization, especially during the past 53 years.

In all the comparisons below we use the temperature interval of equation image which includes the support of the first peak in Figure 3b, which has been interpreted as the phase transformation.

We re-emphasize that the present materials seem best suited to conversion of energy at small temperature difference in which efficiency may not be the best measure of performance. Nevertheless, this measure is often applied to thermoelectrics. The efficiency of thermoelectric materials is measured by equation image where equation image, equation image is electrical conductivity, equation image is thermal conductivity, and S is the temperature dependent Seebeck coefficient. Peak values of equation image for the the material of best thermoelectric devices is near 1. The whole device efficiency is often measured by equation image, also of order 1 in the best devices,28 and the efficiency is given by

equation image((10))

where equation image and Tmax are the temperature difference and maximum temperature in a thermal cycle of the device.27 With equation image, equation image, and equation image, we get an efficiency of 0.57%. If we consider our optimized demonstration of the preceding section, we have an energy density per cycle of equation image. As a reasonable estimate of the heat added we take the latent heat, measured to be equation image. This gives a rather low efficiency of equation image. In our present designs a lot of the latent heat is not being used to make electricity.

Other measures of performance are more favorable. The voltage output of of a thermoelectric material is given directly by its Seebeck coefficient. The thermoelectric material Bi2Te3 has a large Seebeck coefficient of equation image. The standard thermocouple with the highest Seebeck coefficient is Chromel-Constantan at equation image. At equation image these values give respectively 2.3 mV and 0.6 mV. We have measured 0.6 mV in our device. Under reasonable hypotheses in an optimized device we predict 502 mV. Thermoelectric devices arrange many individual thermoelectric elements in series to boost the voltage. Apparently, we could do the same, so we have compared only materials.

An important comparison is power density. A commercial thermoelectric generator (TG12-8, Marlow Industries29) has equation image and a power density at equation image of equation image. Scaling this to 10 °C gives a power density of equation image. A recent high power thermoelectric device28 exhibited a measured equation image power density (not including fluid manifolds), achieved at a temperature difference of 180 °C. Scaling proportionately to 10 °C, this is equation image. Our predicted values of equation image for a thin plate and equation image for a thin rod (excluding the coil and permanent magnet in both cases) compare favorably.

6. Related Methods of Conversion of Heat to Electricity using Multiferroic Materials

The device presented here represents one of several ways multiferroics can be used as new kinds of energy conversion devices. For the purpose of this report we confine attention to what appear to be the most promising: devices that convert heat directly to electricity, realized, say, as current flow through a resistor, and that involve a phase transformation without diffusion.

We present a summary in Table2. To our knowledge, only rows 6 and 7 have been previously realized. Although the design of the device would be affected, it is otherwise unimportant whether phase 1 is the high temperature or low temperature phase. For all these cases one must pay due attention to reversibility and low hysteresis, so these terms are omitted from the last column. Some of these cases are obvious. Others are more subtle. While the present alloy uses a big difference in magnetization of the two phases, this is not essential. One could have a transformation between phases having nearly the same magnetization, but quite different magnetic anisotropy, as in the prototype alloy Ni2MnGa.30 In that case a carefully tuned applied field would easily rotate the magnetization in one phase (austenite, in Ni2MnGa) but not in the other, leading to an effective large change of the magnetic moment during transformation. Ferroelectrics can be used like ferromagnetics in many ways, Table 2, but have the additional flexibility of being able to directly move free charges. An interesting possibility revealed by the presence of the second peak Figure 3a is that the quite sharp drop at the Curie temperature can also induce a significant voltage. In fact, this possibility was noticed long ago by Tesla.31 We have concentrated on simple mechanisms in Table 2. Given a suitable material exhibiting two or more effects, more sophisticated devices are possible.

Table 2.. Methods of conversion of heat to electricity using phase transformations in multiferroics. In all cases the transformation is induced by direct heating and cooling. Ms is the saturation magnetization. Ps the saturation polarization, and Tc the Curie temperature (magnetic or ferroelectric, resp.).
 Phase 1Phase 2PhysicsNotes
1.FerromagneticNonmagneticFaraday’s lawBiasing by a permanent magnet; external coil
2.FerroelectricNonferroelectricOhm’s lawBiasing by a capacitor; polarization-induced current
3.Ferromagnetic; high anisotropyFerromagnetic; low anisotropyFaraday’s lawBiasing by a permanent magnet, intermediate magnetic field; external coil
4.Ferroelectric; high permittivityFerroelectric; low permittivityOhm’s lawBiasing by a capacitor, intermediate electric field; polarization-induced current
5.Ferroelectric; large Ps near Tc (large pyroelectric coefficient)NonpolarOhm’s lawSecond-order transition; biasing by a capacitor
6.Ferromagnetic; large Ms near Tc (small critical exponent)NonmagneticFaraday’s lawSecond-order transition; biasing by a permanent magnet31
7.Nonpolar; nonmagneticNonpolar; nonmagneticStress-induced transformation; Faraday’s lawShape-memory engine driving generator; biased by stress32

7. Experimental Section

The active material for both the property measurements and the demonstration was obtained from a polycrystalline button (3equation image) of Ni45Co5Mn40Sn10 prepared by arc melting the elemental materials Ni (99.999%), Mn (99.98%), Co (99.99%) and Sn (99.99%) under positive pressure of argon. The arc melting furnace was purged three times and a Ti getter was melted prior to melting each sample. To promote homogeneity, the ingot was melted and turned six times. Conversion from emu/g to emu/cm3 was done with a density 8.0 g/cm3. All samples were weighed before and after melting and lost less than 1% by mass. The resulting buttons were homogenized in an evacuated and sealed quartz tube at 900 °C for 24 h, and quenched in water.

Magnetic measurements reported in Figure 1a,c,d were done on a Princeton Measurements micro vibrating sample magnetometer (VSM), calibrated using magnetite, at a heating/cooling rate of 6 °C/min. The magnetic measurements reported in this paper were checked by cutting another sample from the same button and repeating the measurements. Differential scanning calorimetry (DSC) measurements were done on a Thermal Analyst, calibrated with indium, at a heating and cooling rate of equation image. For the DSC measurements each sample was thinned and finely polished to ensure good thermal contact with the pan.

The experimental setup for the measurement of the induced emf is as shown in Figure 2. The permanent magnet having 6.35 cm diameter was used. The field at the pole is equation image and, near the place later occupied by the center of the sample, it is 0.2equation image. The coil has 2000 turns of copper wire having 0.13 mm diameter. The inner diameter of the coil is 1.9 cm and the outer diameter is 3.8 cm. A 10 k ohm resistance was connected in series with the device as indicated in Figure 2. A Keithley 2182 nanovoltmeter having an internal resistance of 10 G ohm was used to measure the voltage across the 10 k ohm resistance. The temperature was measured using a chromel alumel thermocouple, whose signal was measured in the second channel of the voltmeter. The thermocouple was placed just on the surface and below the sample, away from the direct flow of the heat gun. The nanovoltmeter was interfaced with the computer and the data were recorded at time intervals of 0.05 s. The sample was heated to above the transition temperature using a heat gun. The cooling occurred by natural conduction and convection.


We wish to thank Dave Hultman for numerous helpful suggestions.

This work was supported by MURI W911NF-07-1-0410 administered by ARO. It also benefitted from the support of AFOSR GameChanger program GRT00008581/RF60012388 and MURI N000140610530 ONR. We thank the Institute for Rock Magnetism, supported by NSF and UMN, for the the use of its facilities for magnetic measurements.