**5. Physical realization: simulation results based on Ising model**

We now discuss some of the simulation results obtained using Ising model. **Figure 9** shows spontaneous magnetization for a simple cubic crystal (i.e., scc lattice). As the strength of exchange coupling between spin-up and spin-down (JAB) decreases, the critical temperature lowers down. Lower values of JAB weaken the spin flip-flop mechanism; henceforth the system requires further cooling, so that the spin-spin correlation overcomes the fluctuations. Spontaneous magnetization occurs in the absence of external magnetic field [28]. The confirmation of spontaneous process is further confirmed in **Figure 10**. **Figure 10** is plotted for spin

**Figure 9.** *Spontaneous magnetization in two-dimensional thin film (this figure is reproduced with permission from Singh [28]).*

#### **Figure 10.**

*Correlation function vs. temperature for a two-dimensional thin film. Spontaneous magnetization is marked by discontinuity in it (this figure is reproduced with permission from Singh [28]).*

### *The Ising Model: Brief Introduction and Its Application DOI: http://dx.doi.org/10.5772/intechopen.90875*

length transformation or a change of scale, Wilson introduced the concept of renormalization group theory after removing certain deficiencies in Kadanoff's scaling hypothesis. A greater detail of this is omitted here, because that is beyond

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

**5. Physical realization: simulation results based on Ising model**

We now discuss some of the simulation results obtained using Ising model. **Figure 9** shows spontaneous magnetization for a simple cubic crystal (i.e., scc lattice). As the strength of exchange coupling between spin-up and spin-down (JAB) decreases, the critical temperature lowers down. Lower values of JAB weaken the spin flip-flop mechanism; henceforth the system requires further cooling, so that the spin-spin correlation overcomes the fluctuations. Spontaneous magnetization occurs in the absence of external magnetic field [28]. The confirmation of spontaneous process is further confirmed in **Figure 10**. **Figure 10** is plotted for spin

*Spontaneous magnetization in two-dimensional thin film (this figure is reproduced with permission from Singh [28]).*

*Correlation function vs. temperature for a two-dimensional thin film. Spontaneous magnetization is marked by*

*discontinuity in it (this figure is reproduced with permission from Singh [28]).*

the scope of the chapter.

**Figure 9.**

**Figure 10.**

**126**

correlation function vs. temperature of the system [28]. The critical temperature is marked by the presence of discontinuity in it. Above critical temperature, the magnetization abruptly falls to zero, which is an indication of paramagnetic state. The critical temperature in ferromagnetic thin film is known as Curie temperature. We observe similar kind of behavior with antiferromagnetic films, though below critical point (also known as Neel temperature), the net average magnetization becomes zero, because opposite spins are energetically favored in this case. The schematic diagram is shown in **Figure 11** [28]. Magnetization vs. external magnetic field curves are plotted in **Figure 12(a)**–**(d)** for different sets of parameters [28].

Simulation results obtained for a magnetically striped system as schematically shown in **Figure 13** are reported in **Figures 14–17** [29]. One or two alternate rectangular regions are created, using external field. **Figure 14** shows the gradual transition at the interface, where a definite value of external field suddenly gets zero. The spin polarizations in two regions show sharp boundary. The magnetized film, in presence of magnetic field, induces the magnetic zones in proximity where its close external field is zero. Micrograph also indicates for spin-spin phase separation. The corresponding average magnetization vs. temperature and spin correlation function vs. temperature are also plotted in **Figures 15** and **16**, respectively, but these studies are done using Monte Carlo simulation with semi-infinite free boundary conditions. It has been observed that these systems have relatively high critical transition temperatures. **Figure 17** shows the magnetization process with two alternate magnetized zones [29].

Low-dimensional magnetic heterostructures play vital role in spinotronics. Ferromagnets can induce magnetic ordering through a 40-nm-thick amorphous paramagnetic layer, when placed in its close proximity. One has to reconcile with long-range magnetic interaction to correctly measure the extent of induced magnetization. Readers may go through the *Nature Communications* article of F. Magnus et al. published in the year 2016 [17]. The magnetic properties of ferromagnetic materials with reduced dimensions get altered; when the thickness of a film is

**Figure 11.** *Schematic representation of ferromagnetic to paramagnetic and antiferromagnetic to paramagnetic transitions.*

#### **Figure 12.**

*(a) Magnetization vs. external fields at different temperature T = 0.50, 1.0, 1.5, and 2.0. (b) Magnetization vs. external fields for different exchange couplings J = 0.0, 0.25, 0.50, 0.75, and 1.0. These cases are for ferromagnetic thin films. (c) Magnetization vs. external fields at different temperature T = 0.50, 1.0, 1.5, and 2.0. (d) Magnetization vs. external fields for different exchange couplings J = 0.0, 0.25, 0.50, 0.75, and 1.0. These cases are for antiferromagnetic thin films (this figure is reproduced with permission from Singh [28]).*

**Figure 14.**

**Figure 15.**

**Figure 16.**

**129**

*the presence of depletion layer near the barrier.*

*The Ising Model: Brief Introduction and Its Application DOI: http://dx.doi.org/10.5772/intechopen.90875*

*The micrograph of the coexisting phases in the regions of close proximity of the magnetic barrier indicating for*

*Magnetization vs. temperature for magnetically striped system. Only one region experiences the presence of external magnetic field as illustrated in Figure 12(a). This simulation is done for simple cubic lattice with semi-infinite free boundary conditions (the figure is reproduced with permission from Singh [29]).*

*Spin correlation function vs. temperature for magnetically striped system. Only one region experiences the presence of external magnetic field as illustrated in Figure 13(a). This simulation is done for simple cubic lattice with semi-infinite free boundary conditions (the figure is reproduced with permission from Singh [29]).*

#### **Figure 13.**

*(a) The system with one slab of size nx ny nz = 50 100 100 exposed to an external magnetic field. (b) The system with two alternate slabs of size nx ny nz = 50 100 100 exposed to an external magnetic field.*

reduced below a critical value, the ferromagnetic to paramagnetic transition disappears [18]. Finite-size effects may also weaken or enhance magnetic interactions at the boundaries, as well as restrict the evolution of spin-spin correlation length. Extension of these ideas to model magnetic heterostructures, comprising of multiple magnetic and/or nonmagnetic layers, gives insight into interfacial phenomena. Many current and emerging technologies are based on this central problem. This may be very useful in understanding and exploring problems as metalinsulator transition, which is at the core of many state-of-the-art technologies. Henceforth, computational techniques, especially Ising model, can now be extended to develop and enrich science, for making new technologies. Though, its use can be said at the nascent stage, but with the advancement in computer hardware and efficient algorithms, it's applications in areas related to spinotronics appears to be bright.

*The Ising Model: Brief Introduction and Its Application DOI: http://dx.doi.org/10.5772/intechopen.90875*

#### **Figure 14.**

*The micrograph of the coexisting phases in the regions of close proximity of the magnetic barrier indicating for the presence of depletion layer near the barrier.*

#### **Figure 15.**

reduced below a critical value, the ferromagnetic to paramagnetic transition disappears [18]. Finite-size effects may also weaken or enhance magnetic interactions at the boundaries, as well as restrict the evolution of spin-spin correlation length. Extension of these ideas to model magnetic heterostructures, comprising

*(a) The system with one slab of size nx ny nz = 50 100 100 exposed to an external magnetic field. (b) The system with two alternate slabs of size nx ny nz = 50 100 100 exposed to an external*

*(a) Magnetization vs. external fields at different temperature T = 0.50, 1.0, 1.5, and 2.0. (b) Magnetization vs. external fields for different exchange couplings J = 0.0, 0.25, 0.50, 0.75, and 1.0. These cases are for ferromagnetic thin films. (c) Magnetization vs. external fields at different temperature T = 0.50, 1.0, 1.5, and 2.0. (d) Magnetization vs. external fields for different exchange couplings J = 0.0, 0.25, 0.50, 0.75, and 1.0. These cases are for antiferromagnetic thin films (this figure is reproduced with permission from Singh [28]).*

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

of multiple magnetic and/or nonmagnetic layers, gives insight into interfacial phenomena. Many current and emerging technologies are based on this central problem. This may be very useful in understanding and exploring problems as metalinsulator transition, which is at the core of many state-of-the-art technologies. Henceforth, computational techniques, especially Ising model, can now be extended to develop and enrich science, for making new technologies. Though, its use can be said at the nascent stage, but with the advancement in computer hardware and efficient algorithms, it's applications in areas related to spinotronics

appears to be bright.

**128**

**Figure 13.**

**Figure 12.**

*magnetic field.*

*Magnetization vs. temperature for magnetically striped system. Only one region experiences the presence of external magnetic field as illustrated in Figure 12(a). This simulation is done for simple cubic lattice with semi-infinite free boundary conditions (the figure is reproduced with permission from Singh [29]).*

#### **Figure 16.**

*Spin correlation function vs. temperature for magnetically striped system. Only one region experiences the presence of external magnetic field as illustrated in Figure 13(a). This simulation is done for simple cubic lattice with semi-infinite free boundary conditions (the figure is reproduced with permission from Singh [29]).*

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*The Ising Model: Brief Introduction and Its Application DOI: http://dx.doi.org/10.5772/intechopen.90875*

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**Figure 17.**

*Magnetization vs. temperature for magnetically striped system. Two alternate regions experience the presence of external magnetic field as illustrated in Figure 12(b). This simulation is done for simple cubic lattice with semi-infinite free boundary conditions (the figure is reproduced with permission from Singh [29]).*
