The material Gd_{5}(Si _{x}Ge_{1−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 Gd_{5}Si_{1.5}Ge_{2.5}, 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 “Gd_{5}Si _{4}” structure. Because of the change in the crystal structure, the phase transition is accompanied by a colossal strain (λ_{a} ≈ 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 _{5}Ge_{4}” structure to a lower temperature ferromagnetic orthorhombic “Gd_{5}Si_{4}” 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 Gd_{5}(Si_{x}Ge _{1−x})_{4}. Experimental data were taken on single crystal Gd_{5}Si _{1.5}Ge_{2.5} 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 Gd_{5}Si_{1.8}Ge_{2.2} 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 Gd_{5}Si _{1.5}Ge_{2.5} 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.

When Gd_{5}(Si_{x}Ge_{1−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]
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$$ \rho = \rho _{{\rm Lattice}} + \rho _{{\rm Dislocations}} + \rho _{{\rm Microstructure}}. \eqno{\hbox{(1)}} $$

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 Gd_{5}(Si_{x}Ge_{1−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)
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$$ \rho _{{\rm Dislocations}}= kN \eqno{\hbox{(2)}} $$and hence
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$$ \rho = \rho _{{\rm Lattice}} + kN + \rho_{{\rm Microstructure}} . \eqno{\hbox{(3)}} $$

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*/*k*_{B} *T*) where *T* is the thermodynamic temperature and *k*_{B} 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
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$$ P = s\exp \left({{{ - \Delta E}\over {k_{\rm B} T}}}\right). \eqno{\hbox{(4)}} $$

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)
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$$ {{dN}\over {dt}} =- NP. \eqno{\hbox{(5)}} $$

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
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$$\eqalignno{\int_{N_0 }^N { - {1\over N}dN} &= \int_0^t {Pdt} &{\hbox{(6)}}\cr - \log _e \left({{N\over {N_0 }}} \right) &= \int_0^t {Pdt} &{\hbox{(7)}}}$$and hence, for fixed temperature *T*, the dislocation density *N* decreases exponentially with time *t*
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$$ N = N_0 \exp \left({- Pt} \right). \eqno{\hbox{(8)}} $$

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*
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$$ N = N_0 \exp \left({ - s\exp \left({{{ -\Delta E}\over {k_{\rm B} T}}} \right)t} \right). \eqno{\hbox{(9)}} $$

The expected dependence of resistivity ρ on annealing temperature *T* and time *t*, therefore, can be expressed as
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$$\rho = \rho _{{\rm Lattice}} + kN_0 \exp \left({ - s\exp \left({{{ - \Delta E}\over {k_{\rm B} T}}} \right)t} \right) + \rho _{{\rm Microstructure}}\eqno{\hbox{(10)}} $$typically *N*_{0} = 10^{8} 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].

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.

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.