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  • Abstract

Irrecoverable and Recoverable Resistivity Resulting From the First Order Magnetic-Structural Phase Transition in Gd5(SixGe1−x)4

An irreversible change in resistivity occurs in Gd5(SixGe1−x)4 at the first-order phase transformation each time the material is cycled through its transition. This results in a progressive increase in resistivity each time the material is cycled through the transition and returned to its initial conditions of temperature and magnetic field. The effect of a first-order magnetic/structural phase transition on the resistivity of a material, and its recovery by annealing have not been reported before, and consequently, there has been no theoretical analysis of the effect until now. We postulate that if the material is held at an elevated temperature, the resistivity recovers and recovery time decreases with temperature. A model has been developed to explain the recovery in resistivity when the sample is held at elevated temperatures over a period of time and this has been verified experimentally.

SECTION I

INTRODUCTION

The material Gd5(Si xGe1−x) 4 exhibits a first-order magnetic-structural phase transition accompanied by extreme changes in properties at the transition [Pecharsky 1997]. This paper reports on results on single crystal Gd5Si1.5Ge2.5, which was thermally cycled through the first-order magnetic-structural phase transition for up to 20 cycles. For 0.41 < x < 0.575, the higher temperature paramagnetic monoclinic structure transforms to a lower temperature ferromagnetic orthorhombic “Gd5Si 4” structure. Because of the change in the crystal structure, the phase transition is accompanied by a colossal strain (λa ≈ 10 000 ppm) [Morellon1998, Han 2003] along the crystallographic “ a” axis of a single crystal sample.This may be induced either by change in magnetic field or by change in temperature.For x <0.41, the structural transition is from a higher temperature paramagnetic orthorhombic “Sm 5Ge4” structure to a lower temperature ferromagnetic orthorhombic “Gd5Si4” structure.

In this paper, we show that there is an irreversible change in the resistivity when the samples pass through the first-order magnetic-structural phase transition, and we present a theoretical model for the change in resistivity of Gd5(SixGe 1−x)4. Experimental data were taken on single crystal Gd5Si 1.5Ge2.5 to test the validity of the model.

The irreversible increase in resistivity was also accompanied by an irreversible increase in the coercivity in single crystal Gd5Si1.8Ge2.2 at 220 K after cycling the sample through the first-order magnetic-structural phase transition [Hadimani 2009].

We have measured the irreversible change in the resistivity for up to20 cycles through the transition. Other investigators have observed that the resistivity exhibited partial recovery of resistivity when held at room temperature over a period of a few years [Zou 2009]. We have also demonstrated partial recovery of the irreversible increase in resistivity of single crystal Gd5Si 1.5Ge2.5 by holding the sample at elevated temperature of 345 K. In this case, the recovery was observed over a period of days instead of years.

SECTION II

EXPERIMENTAL DETAILS

Single crystal Gd5Si1.5Ge2.5 was obtained from Ames Laboratory, U.S. Department of Energy (Materials Preparation Center, www.mpc.ameslab.gov). The sample was prepared by the tri-arc method. The sample was cut into a cuboid of 2.5 × 4.45 × 4.28 mm along a-, b-, and c-axes, respectively. Resistivity was measured with a standard collinear four-probe method using a quantum design physical property measurement system (PPMS). A current of 5 mA was passed through the sample to measure resistivity. The sample was subjected to multiple cycling through the first-order transition temperature, and was then held at a temperature of 345 K to increase the rate of recovery in resistivity compared with that observed at room temperature of 293 K.

SECTION III

RESULTS

Resistivity of the single crystal Gd5Si 1.5Ge2.5 sample was measured as a function of temperature at zero-applied magnetic field over the temperature range 150–230 K. The first-order magnetic-structural phase transition temperature for this composition occurred at 200 K as shown in Fig. 1. At the first-order phase transition temperature, there was a sudden sample cycled through the first-order phase transition change in the resistivity, which increased irreversibly when the temperature. A plot of resistivity at 230K as a function of the number of cycles up to 20 cycles shows a straight line at low numbers of cycles and a tendency to saturate at larger numbers of cycles. To investigate the effects of time and temperature on the resistivity, the sample was held at an elevated temperature of 345 K in situ for more than 50 h and the resistivity was measured periodically. There was a recovery in the resistivity from 160 to 48 μΩ·m as shown in Fig. 2, which amounts to 66% of the irreversible increment in resistivity.

Figure 1
Fig. 1. Irreversible increase in the resistivity of single crystal Gd5Si1.5Ge2.5 for up to 15 thermal cycles through the phase transition at zero applied field.
Figure 2
Fig. 2. Resistivity at 230 K versus number of cycles of a single crystal Gd5Si 1.5Ge2.5 (x = 0.375) for 20 cycles through the first order phase transition.
SECTION IV

DISCUSSION

When Gd5(SixGe1−x)4 undergoes a first-order magnetic-structural phase transition, it exhibits a large volumetric strain ( ∼10 000ppm). Cycling the sample several times through the phase transition with its associated large strain, can cause the resistivity of the sample to increase irreversibly due to the development of dislocations and other microstructural defects including microcracks. The resistivity can be written as a summation of three separate components: lattice, dislocation, and microstructural components as shown in (1), which is analogous to Matthiessen's rule [Jiles 2001, Zurcher 1995] Formula TeX Source $$ \rho = \rho _{{\rm Lattice}} + \rho _{{\rm Dislocations}} + \rho _{{\rm Microstructure}}. \eqno{\hbox{(1)}} $$

The resistivity due to the lattice component, which is caused by phonon scattering of electrons, is expected to be reversible when the first-order phase transition is reversed. This component is, therefore, not expected to contribute to the observed irreversible increase in resistivity, and we have termed this component of resistivity “reversible.”

The resistivity component that is due to the development of dislocations that arise when the sample is cycled through the first-order phase transition will be irreversible over short periods of time because the number of dislocations in the material increases irreversibly, However, this build up of dislocations can be recovered by annealing since the dislocation density can be reduced by annealing. Hence, this component of resistivity we have termed “irreversible and recoverable.”

The microstructural component of the resistivity can increase in Gd5(SixGe1−x)4 when cycled through the first-order phase transition due to the development of features such as microcracks, a phenomenon that was previously observed by Manekar [2006]. The irreversible increase in the microcracks results in an irreversible increase in the resistivity, which cannot be recovered by annealing. Hence, this component of resistivity we have termed “irreversible and irrecoverable.”

The dislocation component of the resistivity should be dependent on the number density N of dislocations present and to a first approximation this should be a linear function of dislocation density N as given in (2) Formula TeX Source $$ \rho _{{\rm Dislocations}}= kN \eqno{\hbox{(2)}} $$and hence Formula TeX Source $$ \rho = \rho _{{\rm Lattice}} + kN + \rho_{{\rm Microstructure}} . \eqno{\hbox{(3)}} $$

The dislocations in the material, which are created by the generation of stress when it passes through the first order phase transition, may be considered to be metastable, being trapped in localized potential wells, from which they can escape by thermal activation. Although in practice there will likely be a range of depths of these localized potential wells, for the purposes of this model we take a typical depth of potential well Δ E as representative of the material, and assume that the material behaves as if all dislocations are trapped in potential wells of the same depth.

The probability P of a dislocation overcoming the energy barrier and escaping from the potential well Δ E is according to statistical thermodynamics, proportional to exp(−Δ E/kB T) where T is the thermodynamic temperature and kB is Boltzmann's constant.The probability of a single dislocation being thermally activated and escaping the potential well in unit time can be expressed in terms of a frequency “s,” the Arrhenius coefficient, which can be thought of as the rate at which the dislocations attempt to escape the potential well Formula TeX Source $$ P = s\exp \left({{{ - \Delta E}\over {k_{\rm B} T}}}\right). \eqno{\hbox{(4)}} $$

The rate of dislocations escaping the potential well, per unit volume per unit time, is then the product of this probability P for a typical single dislocation and the number of dislocations remaining per unit volume N, as given in (5) Formula TeX Source $$ {{dN}\over {dt}} =- NP. \eqno{\hbox{(5)}} $$

The remaining dislocation density N (i.e., the original dislocation density minus the number per unit volume that have escaped over a given time interval t), is given by Formula TeX Source $$\eqalignno{\int_{N_0 }^N { - {1\over N}dN} &= \int_0^t {Pdt} &{\hbox{(6)}}\cr - \log _e \left({{N\over {N_0 }}} \right) &= \int_0^t {Pdt} &{\hbox{(7)}}}$$and hence, for fixed temperature T, the dislocation density N decreases exponentially with time t Formula TeX Source $$ N = N_0 \exp \left({- Pt} \right). \eqno{\hbox{(8)}} $$

Substituting for the probability P from (4), this results in a new equation in which P is no longer a constant but is dependent on temperature and the average depth of the potential wells Δ E Formula TeX Source $$ N = N_0 \exp \left({ - s\exp \left({{{ -\Delta E}\over {k_{\rm B} T}}} \right)t} \right). \eqno{\hbox{(9)}} $$

The expected dependence of resistivity ρ on annealing temperature T and time t, therefore, can be expressed as Formula TeX Source $$\rho = \rho _{{\rm Lattice}} + kN_0 \exp \left({ - s\exp \left({{{ - \Delta E}\over {k_{\rm B} T}}} \right)t} \right) + \rho _{{\rm Microstructure}}\eqno{\hbox{(10)}} $$typically N0 = 108 m −2 [Khanna 1986] and k = 10−6 Ω·m.

The model parameters s and Δ E have been calculated from the experimental data by least squares analysis and the results are shown in Fig. 3 with the results of experimental measurements and theoretical calculations of the variation of resistivity with time. In particular, Δ E ≈ 0.7 eV that is fairly typical of these dislocations [Warner 2009].

Figure 3
Fig. 3. Recovery in the resistivity of single crystal Gd5Si1.5Ge2.5 held at345 K for 50 h after cycling 20 times through the first order phase transition.Values of model parameters of (10) obtained by least squares fitting are s = 0.26 × 106 s−1, Δ E = 1.1 × 10−19 J.

Solutions of the (10) for a variety of different temperatures and times are shown in Fig. 4.The values of the model parameters are the same as those given in Fig. 3.

Figure 4
Fig. 4. Predicted dependence of resistance recovery of single crystal Gd5Si 1.5Ge2.5 on time for various temperatures using s = 0.26 × 106 s−1, Δ E = 1.1 × 10−19 J.

Damask [1963] described the recovery in resistivity of quenched metals such as gold when annealed at temperatures much lower than the quenching temperature. They stated that the resistivity recovery occurred in quenched gold was due to annihilation of defects such as vacancies and dislocations caused by quenching. There are some similarities with the description proposed here. However the effect of dislocations caused by cyclic stress due to a first-order magnetic/structural phase transition of a material on the resistivity recovery by annealing has not been reported and there has been no theoretical analysis of the effect until now.

SECTION V

CONCLUSION

Irreversible changes in resistivity were observed in single crystal Gd5 Si1.5Ge2.5 when the material was cycled through its first-order magnetic/structural phase transition.The irreversible changes in resistivity were measured over 20 cycles and it was found that the material showed a linear increase in resistivity with number of cycles at low numbers of cycles. For higher numbers of cycles, the resistivity showed a tendency to saturate. It was found that 66% of the irreversible increment in resistivity was recovered by holding the sample at an elevated temperature of 345 K for 50 h. An explanation of the dependence of the rate of recovery of resistivity on the annealing temperature has been developed based on statistical thermodynamics using a model consisting of dislocations trapped locally within potential wells with a single (or average) depth of potential well and subjected to thermal activation.

Acknowledgment

This work was supported by the Royal Society of London under a Wolfson Research Merit Award. The authors would like to acknowledge T. A. Lograsso and D. L. Schlagel from the Ames Laboratory, U.S. Department of Energy for sample preparation and Dr. Y. Melikhov and Dr. J. E. Snyder for helpful discussions.

Footnotes

Corresponding author: R. L. Hadimani (Hadimanim@cardiff.ac.uk).

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R. L. Hadimani

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D. C. Jiles

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