Plasticity-Induced Magnetization in Amorphous Magnetic Solids
Amorphous magnetic solids, like metallic glasses, exhibit a novel effect: the growth of magnetic order as a function of mechanical strain under athermal conditions in the presence of a magnetic field. The magnetic moment increases in steps whenever there is a plastic event. Thus plasticity induces the magnetic ordering, acting as the effective noise driving the system towards equilibrium. We present results of atomistic simulations of this effect in a model of a magnetic amorphous solid subjected to pure shear and a magnetic field. To elucidate the dependence on external strain and magnetic field we offer a mean-field theory that provides an adequate qualitative understanding of the observed phenomenon.
The study of the influence of random impurities, disorder and random anisotropy on the magnetic properties of crystalline materials is an often studied and therefore an extremely well established subject of condensed matter physics 73HPZ (); 75IM (); 93Seth (). In contrast, the theoretical study of the consequences of glassy randomness on the interaction between mechanical and magnetic properties in amorphous solids like metallic glasses is still in its infancy. In this paper we study an interesting effect that has been discovered in our group via numerical simulations; an amorphous magnetic solid with random local anisotropy quenched from the liquid in the presence of a magnetic field has an initial magnetic moment (which is zero in the absence of magnetic field), cf. Fig. 1. But when subjected to an external mechanical strain the magnetic moment can increase with the accumulation of plastic events. The magnetization increases to a steady state level that depends on the relative magnitude of the magnetic field compared to the other parameters in the problem. Needless to say, this effect is particular to glassy systems in which the position of particles is free to adjust under mechanical strains and plastic irreversibility; its study requires models that combine glassiness, magnetism and randomness.
In Sect. II we present the model that we use to study the interaction between mechanical and magnetic responses in amorphous solids 12HIP (); 13DHPS (). In Sect. III we present the numerical results of the effect under discussion. We show that when the magnetic field is non-zero, the magnetization increases to a steady-state value that depends on the magnetic field. In Sect. IV we offer a theory for the steady-state value of the magnetization. In Sect. V we present a mean field theory for the actual trajectory of the magnetization as a function of external strain, and present some non-trivial predictions of the theory. Sect. VI contains a summary of the paper and some concluding remarks.
Ii The Model
The model we employ is in the spirit of the Harris, Plischke and Zuckerman (HPZ) Hamiltonian 73HPZ () but with a number of important modifications to conform with the physics of amorphous magnetic solids 12HIP (). First, our particles are not pinned to a lattice. We write the Hamiltonian as
where are the 2-D positions of particles in an area and are spin variables. The mechanical part is chosen to represent a glassy material with a binary mixture of 65% particles A and 35% particles B, with Lennard-Jones potentials having a minimum at positions , and for the corresponding interacting particles 09BSPK (). These values are chosen to guarantee good glass formation and avoidance of crystallization. The energy parameters chosen are , in units for which the Boltzmann constant equals unity. All the potentials are truncated at distance 2.5 with two continuous derivatives. Particles A carry spins ; the B particles are not magnetic. We choose the spins to be classical spins; the orientation of each spin is then given by an angle . We also denote by the local preferred easy axis of anisotropy, and end up with the magnetic contribution to the potential energy in the form 12HIP ():
Here and the sums are only over the A particles that carry spins. For a discussion of the physical significance of each term the reader is referred to Ref. 12HIP (). It is important however to stress that in our model (in contradistinction with the HPZ Hamiltonian 73HPZ () and also with the Random Field Ising Model 93Seth ()), the exchange parameter is a function of a changing inter-particle position (either due to external strain or due to non-affine particle displacements, and see below). Thus randomness in the exchange interaction is coming from the random positions , whereas the function is not random. We choose for concreteness the monotonically decreasing form where with . This choice cuts off at with two smooth derivatives. In our case .
Another important difference is that in our case the local axis of anisotropy is not selected from a pre-determined distribution, but is determined by the local structure: define the matrix :
The matrix has two eigenvalues in 2-dimensions that we denote as and , . The eigenvector that belongs to the larger eigenvalue is denoted by . The easy axis of anisotropy is given by by . Finally the coefficient which now changes from particle to particle is defined as
The parameter determines the strength of this random local anisotropy term compared to other terms in the Hamiltonian. The form given by Eq. (4) ensures that for an isotropic distribution of particles . Due to the glassy random nature of our material the direction is random. In fact we will assume below (as can be easily tested in the numerical simulations) that the angles are distributed randomly in the interval . It is important to note that external straining does NOT change this flat distribution and we will assert that the probability distribution can be simply taken as
The last term in Eq. (2) is the interaction with the external field . We have chosen in the range [-0.08,0.08]. At the two extreme values all the spins are aligned along the direction of .
Iii Plasticity Induced Magnetization
The novel effect that is the subject of this paper is described as follows: we prepare the system described by the Hamiltonian (1) in a fluid state at a high temperature ( in units of where the Boltzmann constant is fixed at unity). In this paper the system is 2-dimensional, containing particles. The system is then quenched to with molecular dynamics, and finally brought to an inherent state using gradient energy minimization 99ML (); 04ML (); 09LP () in the presence of a magnetic field directed in the direction . All the subsequent simulation are performed at . We strain the system using simple shear, such that at each step of strain the particles are first subjected to an affine transformation
where . Subsequent to the affine step the system is again relaxed to equilibrium using energy gradient minimization 99ML (); 04ML (); 09LP (). This procedure of athermal, quasi-static strain (AQS) is continued until we reach the desired values of the strain . The procedure is applied either in the absence of magnetic field () or at any chosen value of which is always directed in the direction.
Immediately after the quench from the liquid at the magnetic moment has a value that depends on the magnetic field , . Denoting the number of spin carrying particles as we write quite generally
The effect of interest here is what happens to as we begin to strain the system in the AQS procedure. The answer is shown in Fig. 1 which exhibits as a function of for for various values of the magnetic field . For (not shown in the figure) the magnetization settles to a steady state value .
Thus for each value of one can associate a steady-state value and an initial value . At this point we turn to a theoretical analysis whose aim is to understand the steady-state value and to derive an equation for the dependence of on for any given value of .
Iv The steady state value of the magnetization
To develop a theory of the effect displayed in Sect. III we start from the obvious remark that in an AQS procedure the force (or torque) on any particle is zero before and after every change in strain, and therefore
Thus this condition gives rise to coupled nonlinear algebraic equations for the spin coordinates given the instantaneous easy axis directions . To proceed we will accept Eq. (5) and in addition will make the mean field approximation that for any finite
Here we used the fact that is in the direction such that for any value of we have for the spins surrounding , . Next we consider the distributions of and , Cf. Fig. 2.
This distributions are well behaved with a very well defined averages. Denoting then the number of nearest neighbors by and the average values of and by and respectively we find the mean field equations
These equations should be solved together with the supplementary equations (5) and (7). The solution should provide the steady state values of as seen in Fig. 3 for given parameters and and for any value of .
iv.1 Numerical Solution
An easy way to solve Eq. (10) numerically is by graphic methods. Since is flatly distributed in the interval we can choose values of and then solve for using a Newton-Raphson method starting with . The procedure provides values for from which we compute a new value of . Solving again, we get a new set of and a new value of . The procedure is stopped when we get back the same value of , see Fig. 3 for an example of this procedure for . The converged value of for this value of is 0.97, compared to 0.9 from the direct numerical simulation.
Considering the mean field approximation that is at the basis of Eq. (10) we find this result very satisfactory.
iv.2 Analytic Solution for large and small
In this subsection we examine the solution of Eqs. (10) when the effect of random anisotropy is large (large ), the exchange is small and the magnetic field small. In this case we expect that will deviate slightly from , . Linearizing Eq. (10) in we find
Finally, since we assumed already that is large and is small, we can rewrite this last equation as
where is the Bessel function of the first kind. As anticipated already from Eq. (10) the relevant parameter that determines the steady-state value is . For a small value of we can use the series expansion of the Bessel function to define and compute the susceptibility ,
Eq. (13) can be also solved numerically, and the predicted value of
can be read from the graphic solution as shown in Fig. 4. For the case for which the condition is valid we get a very good agreement between the value of found here and in the direct numerics ( compared to ). For the large value of the condition is not obeyed; the value of the predicted is not close to the direct numerical value of .
iv.3 General Analytic Solution
The theory derived above is only valid for small magnetic fields and weak magnetization. In the next section we derive a nonlinear relaxation equation Eq. (30) for the magnetization as a function of strain. But in order to integrate this equation we will require expressions for the average magnetization for larger magnetic fields and signification magnetization. Further we will see that our expressions for the magnetization show hysteresis as expected from simulations. We therefore present series solutions in both powers of and powers of that converge respectively in the case of large and small on the one hand and small and large on the other.
iv.3.1 X small
We now proceed to solve the mean field equation by separation of variables in the form
We expect this power series in to converge for (which we estimate below). The coefficients are found from the nonlinear equation
Equating powers of we find
In Fig. 5 we plot the equilibrium position of a spin compared to its local easy axis as is increased. As can be seen, for the spins are essentially pinned along their easy axes, but as increases by, for example, increasing the magnetic field , the spins have a tendency to prefer the magnetic axis. This is especially true for easy axes that by chance are already close to the magnetic axis. As increases further this set increases further, finally only leaving small subsets of spins where the easy axes are oriented close to .
Having introduced a series expansion in we should distinguish between the steady-state value and any intermediate value of that is obtained for . In order to calculate using Eq. (17), we use the series expansion for with the result
Calculating the first few terms of this expansion we find
From Eq. (19) we immediately see that
From the inequality we see that the expansion in powers only exist for . We therefore now consider solutions in powers of .
iv.3.2 Y small
The parameter is small for weak anisotropy and strong magnetic fields. We now proceed to solve the mean field equation for
in a power series in that will converge for from the nonlinear equation
Equating powers of we find
In Fig. 6 we plot , the equilibrium position of a spin, compared to its local easy axis as is increased. As can be seen, for the spins are essentially pointing along the magnetic axis, but as increases by, for example, reducing the magnetic field the spins have a tendency to prefer the local anistropy easy axis. This is especially true for easy axes that by chance are already close to the magnetic axis. As increases further this set increases further, finally only leaving small subsets of spins where the easy axes are oriented close to still preferentially pointing along the magnetic axis.
In order to calculate using Eq. (23), we use the series expansion for with the result
Calculating the first few terms of this expansion we find
From Eq. (25) we immediately see that
From the inequality we see that the expansion in powers can only exist for or .
Thus we see from our analysis the magnetization is hysteretic. For at low magnetic fields only pinned solutions exists. While for at high magnetic fields only the depinned phase exists. Finally for both phases are possible and the chosen solution will depend on initial conditions.
V The magnetization as a function of the strain
In this section we derive an approximate differential equation for as a function of . To this aim we assume that we know the steady state value of as a function of the parameter , . For a value of not in the steady state we write
To proceed, we realize that the change occurs only due to plastic events, and these are localized on a small number of particles , . We will denote the relative number of particles involved in the plastic events as . If were unity, we would expect that the change would be complete, i.e. . Since we estimate
Using this estimate in Eq. (27) we write
Dividing through by we write
To compare these equations to the simulations we need to show that is an intensive parameter, independent of the system size. This is done in the appendix. In addition, we employ a Pade’ approximant form for that captures both the behavior and the behavior . These asymptotic limits are given by the Pade’ approximant
This form is plotted in Fig. 7.
Eq. (30) is a nonlinear relaxation equation for the magnetization describing its approach to steady state as the material is strained. Using the Pade’ approximant form for given by Eq. 31 that captures both the small and large behavior we solved Eq. (30) with the initial condition using the same parameters as in the simulation, namely , . We chose to get a qualitative fit with the direct simulations, and solved for various values of from to . The resultant curves are shown in Fig. 8.
Vi Concluding remarks
In summary, we have presented an interesting effect that is particular to magnetic amorphous solids, showing that at the plastic events can act as an “effective noise” that drives the magnetization from initial conditions to a steady state value. The magnetization is changing in steps that are coincident with the irreversible plastic events. While the effect itself was discovered numerically, we presented above a mean-field theory that provides adequate estimates of both the steady-state values of the magnetization and of the trajectories ( vs. ) to get there. The model employed above appears to offer considerable amount of additional interesting physics that calls for careful study, as will be elaborated in future work.
Appendix A the parameter
To show that is an intensive parameter we rely heavily on the scaling theory of elasto-plastic steady states that is presented in great details in Ref. 10KLP (). Denoting the steady state mean stress as , we note that the value of can be obtained by equating the typical increase in elastic energy in the steady state, i.e. with the typical plastic energy drop where is the plastic energy drop per particle. Writing and we find . Here is the volume per particle. Thus is intensive even though neither nor are intensive. In fact, from Ref. 10KLP () we expect both to scale like . This expectation follows from the scaling behavior and together with the scaling relation 10KLP (). Finally, since the plastic event is associated with a saddle node bifurcation, we know that the barrier to instability scales like . In Ref. 10KLP () it is also argued that the barrier scales like leading finally to . Therefore we also find that and thus the participation ratio also scales like . Accordingly is intensive. This is crucial for the comparison of the theory and the simulations as shown above.
Acknowledgements.This work had been supported by an ERC “ideas” grant STANPAS, the German Israeli Foundation and the Israel Science Foundation.
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