# Phase Transition

We recall that for bulk materials, in spite of their long history, the nature of the phase transition in non-collinear magnets such as stacked triangular XY and Heisenberg antiferromagnets has been elucidated only recently [90, 187, 250, 251]. On the other hand, surface effects in thin films have been intensively studied during the past three decades [36, 87, 374]. Most of theoretical studies were limited to collinear magnetic orderings. Phase transitions in thin films with non-collinear ground states have been only recently studied [89, 102, 229, 248, 249]. MC simulations of a helimagnetic thin film [64] and a few experiments in helimagnets [176,177] have also been carried out. These investigations were motivated by the fact that helical magnets present a great potential of applications in spintronics with spin-dependent electron transport [150,166, 357].

As described in the previous section, the planar helical spin configuration in zero field becomes non-planar in a perpendicular field. In order to interpret the phase transition shown below, let us mention that a layer having a large *z* spin-component parallel to the field cannot have a phase transition because its magnetization will never become zero. This is similar to a ferromagnet in a field. However, layers having large negative *z* spin-components (antiparallel to the field) can undergo a transition due to the magnetization reversal at a higher temperature similarly to an antiferromagnet in a field. In addition, the *xy* spin-components whose *xy* fluctuations are not affected by the perpendicular field can make a transition. Having mentioned these, we expect that some layers will undergo a phase transition, while others will not. This is indeed what we observed in MC simulations shown in the following.

For MC simulations, we use the Metropolis algorithm (see Chapter 6) and a sample size *N* x *N* x *N _{z}* with

*N =*20,40,60,100 for detecting lateral-size effects and

*N*8,12,16 for thickness effects. The equilibrium time is 10

_{z}=^{s}MC steps/spin and the thermal average is performed with the following 10

^{5}MC steps/spin.

**Figure 12.5 **(a) Layer magnetization and (b) layer magnetic susceptibility

versus *T* for *H =* 0, *J*_{2}* =* —1, *N _{z} =* 12. Dark olive green void squares for the first layer, maroon void triangles for the second, red circles for the third, indigo triangles for the fourth, dark blue squares for the fifth, dark green void circles for the sixth layer.

## Results of 12-Layer Film

In order to appreciate the effect of the applied field, let us show first the case where *H =* 0 in Fig. 12.5. We see there that all layers undergo a phase transition within a narrow region of *T.*

In an applied field, as seen earlier, in the GS all layers do not have the same characteristics so one expects different behaviors. Figure 12.6 shows the layer magnetizations and the layer susceptibilities as functions of *T* for *H* = 0.2 with *J*_{2}* = -1, N _{z} =* 12 (only the first six layers are shown, the other six are symmetric). Several remarks are listed below:

**Figure 12.6 **(a) Layer magnetization and (b) layer magnetic susceptibility versus *T* for *H = 0.2, J** _{2}* = —1,

*N*= 12. Dark olive green void squares for the first layer, maroon void triangles for the second, red circles for the third, indigo triangles for the fourth, dark blue squares for the fifth, dark green void circles for the sixth layer.

_{z}- (1) Only layer 3 and layer 5 have a phase transition: Their magnetizations strongly fall down at the transition temperature. This can be understood from what we have anticipated above: These layers have the largest
*xy*components (see Fig. 12.2c). Since the correlation between*xy*components do not depend on the applied field, the temperature destroys the in-plane ferromagnetic ordering causing the transition. It is not the case for the*z*components which are kept non-zero by the field. Of course, symmetric layers 8 and 10 have the same transition (not shown). - (2) Layers with small amplitudes of
*xy*components do not have a strong transverse ordering at finite*T:*The absence of

**Figure 12.7 **(a) *S ^{z}* and (b)

*S***across the film with

*H =*0.7.

pronounced peaks in the susceptibility indicates that they do not make a transition (see Fig. 12.6).

(3) Note that the *xy* spin components of layers 3 and 5 are disordered at *T _{c}* ~ 1.275 indicated by pronounced peaks of the susceptibility.

What we learn from the example shown above is that under an applied magnetic field the film can have a partial transition: Some layers with large *xy* spin components undergo a phase transition (destruction of their transverse *xy* correlation). This picture is confirmed by several simulations for various field strengths. Another example is shown in the case of a strong field *H* =0.7: The GS is shown in Fig. 12.7, where we observe large *xy* spin components

Figure 12.8 (a) Layer magnetization and (b) layer magnetic susceptibility

versus *T* for *H* = 0.7*,J** _{2}* = — 1. = 12. Dark olive green void squares for

the first layer, maroon void triangles for the second, red circles for the third, indigo triangles for the fourth, dark blue squares for the fifth, dark green void circles for the sixth layer.

of layers 3, 4, and 5 (and symmetric layers 7, 8 and 9). We should expect a transition for each of these layers. This is indeed the case: We show these transitions in Fig. 12.8, where sharp peaks of the susceptibilities of these layers are observed.

We close this section by showing some size effects. Figure 12.9 shows the effect of lateral size (xy planes) on the layer susceptibility. As expected in a continuous transition, the peaks of the susceptibilities of the layers undergoing a transition grow strongly with the layer lattice size.

322

*Helimagnetic Thin Films in a Field*

Figure 12.9 Magnetic susceptibility of the third layer versus *T* for *H =* 0.2, *)* _{2} = — 1, *N _{z}* = 12. Dark green void circles, dark blue squares, indigo triangles, red circles are susceptibilities for layer lattice sizes 100x100, 60x60,40x40 and 20x20, respectively.

Figure 12.10 Magnetic susceptibility versus *T* for two thicknesses with *H =* 0.2, *J** _{2}* = —1: (a)

*N*8, (b)

_{z}=*N*16. Dark olive green void squares are for the first layer, maroon void triangles for the second, red circles for the third, indigo triangles for the fourth, dark blue squares for the fifth, dark green void circles for the sixth, black diamonds for the seventh, dark brown void diamonds for the eighth. See text for comments.

_{z}=**Figure 12.11 **Nine-layer film: Spin components across the film in the case where *H* = 0.2. The horizontal axis *Z* represents plane *Z* (Z = 1 is the first plane etc.): (a) *S ^{z};* (b) modulus

*S*of the projection of the spins on the

^{xy}*xy*plane; (c) layer susceptibilities versus

*T:*Dark olive green void squares are for the first layer, maroon void triangles for the second, red circles for the third, indigo triangles for the fourth, dark blue squares for the fifth layer, respectively. Only layers 4 and 1 (also layers 9 and 6, not shown) undergo a transition. See text for comments.

## Effects of the Film Thickness

As for the thickness effects, we note that changing the thickness (odd or even number of layers) will change the GS spin configuration so that the layers with largest *xy* components are not the same. As a consequence, the layers which undergo the transition are not the same for different thicknesses. We show in Fig. 12.10 the layer susceptibilities for *N _{z} =* 8 and 16. For

*N*= 8, the layers which undergo a transition are the first, third and fourth layers with pronounced peaks, while for

_{z}*N =*16, the layers which undergo a transition are the third, fifth, seventh and eighth layers.

Let us show the case of an odd number of layers. Figure 12.11 shows the results for *N _{z}* = 9 with

*H*= 0.2,/2 = — 1. Due to the odd layer number, the center of symmetry is the middle layer (5th layer). As seen, the layers 1 and 4 and their symmetric counterparts (layers 9 and 6) have largest xy spin modulus (Fig. 12.11b). The transition argument shown above predicts that these layers have a transversal phase transition in these

*xy*planes. This is indeed seen in Fig. 12.11c, where the susceptibility of layer 4 has a strong peak at the transition. The first layer, due to the lack of neighbors, has a weaker peak. The other layers do not undergo a transition. They show only a rounded maximum.

# Quantum Fluctuations, Layer Magnetizations and Spin Wave Spectrum

We shall extend here the method used in Chapter 11 and Ref. [89] for zero field to the case where an applied magnetic field is present. The method remains essentially the same except the fact that each spin is defined not only by its angles with the NN in the adjacent layers but also by its azimuthal angle formed with the *c* axis as seen in Section 12.2.

We use in the following the Hamiltonian (12.1) but with quantum Heisenberg spins S, of magnitude 1/2. In addition, it is known that in two dimensions there is no long-range order at finite temperature for isotropic spin models [231] with short-range interaction. Since our films have small thickness, it is useful to add an anisotropic interaction to stabilize the long-range ordering at finite temperatures. Let us use the following anisotropy between S, and S*j *which stabilizes the angle between their local quantization axes *Sf *and SJ:

where /1 is supposed to be positive, small compared to /_{ь} and limited to NN.

The general method has been recently described in details in Refs. [89, 102] and in Chapter 11. To save space, let us give the results for the simple cubic helimagnetic film in a field. We define the following two double-time Green’s functions in the real space:

Writing the equations of motion of these functions and using the Tyablikov decoupling scheme to reduce the higher-order functions, we obtain the general equations for non-collinear magnets [89]. We next introduce the following in-plane Fourier transforms *g, _{hn}>* and of the

*G*and

*F*Green's functions, we finally obtain the following coupled equations:

where *n =* 1, 2,..., *N _{Zl} d_{n}* =

*у*= (cos

*k*+

_{x}a*cosk*The coefficients are given by

_{y}a)/2.*Helimagnetic Thin Films in a Field*

326

*w* is the spin wave frequency, *k _{x}* and

*k*denote the wave vector components in the

_{y}*xy*planes,

*n*is the index of the layer along the c axis with

*n =*1 being the surface layer, /7 = 2 the second layer and so on. The angle is the azimuthal angle formed by a spin in the layer

*n*with the c axis. Note that (i) if

*n*= 1 then there are no

*n*- 1 and /7-2 terms in the matrix coefficients, (ii) if

*n*= 2 then there are no

*n*— 2 terms, (iii) if

*n*=

*N*then there are no

_{z}*n*+1 and

*n + 2*terms, (iv) if /7 =

*N*— 1 then there are no

_{z}*n*+ 2 terms. Besides, we have distinguished the in-plane NN interaction /[' from the inter-plane NN one / j-

^{1}. If we write all equations explicitly for

*n*=

*1,..., N*we can put these equations under a matrix of dimension

_{z }*2N*x 2

_{Z}*N*Solving this matrix equation, one gets the spin wave frequencies

_{z}.*ш*at a given wave vector and a given

*T.*

The layer magnetizations can be calculated at finite temperatures self-consistently. The numerical method to carry out this task has been described in details in Ref. [89]. It is noted that in bulk antiferromagnets and helimagnets the spin length is contracted at *T* = 0 due to quantum fluctuations [87]. Therefore, we also calculate the layer magnetization at *T* = 0 [78, 89]. It is interesting to note that due to the difference of the local field acting on a spin near the surface, the spin contraction is expected to be different for different layers.

We show in Fig. 12.12 the spin length of different layers at *T* = 0 for *N =* 12 and *J** _{2}* = -1 as functions of

*H.*All spin contractions are not sensitive for

*H*lower than 0.4, but rapidly become smaller for further increasing

*H.*They spin lengths are all saturated at the same value for

*H > 2.*Figure 12.13 shows the spin length as a function of

*]*When

_{2}-*J*

*> -0.4, the spin configuration becomes ferromagnetic, and as a consequence the contraction tends to 0. Note that in zero field, the critical value of*

_{2}*J*

*is -0.25. In both Figs. 12.12 and 12.13, the surface layer and the third layer have smaller contractions than the other layers. This can be understood*

_{2}**Figure 12.12 **Spin lengths at *T* = 0 versus applied magnetic field *H.* Dark olive green void squares correspond to the spin length of the first layer, maroon void triangles to that of the second, red circles to the third, indigo triangles to the fourth, dark blue squares to the fifth, dark green void circles to the sixth layer.

**Figure 12.13 **Spin lengths at Г = Oversus *J _{2}-* Dark olive green void squares correspond to the spin length of the first layer, maroon void triangles to that of the second, red circles to the third, indigo triangles to the fourth, dark blue squares to the fifth, dark green void circles to the sixth layer.

by examining the antiferromagnetic contribution to the GS energy of a spin in these layers: They are smaller than those of the other layers.

We show in Fig. 12.14 the layer magnetizations versus *T* for the case where *J** _{2}* = -1 and

*N*12 (top figure). The low-Г region is enlarged in the inset where one observes a crossover between the magnetizations of layers 1, 3 and 6 at Г ~ 0.8: Below this temperature,

_{z}=*M> M*and above they become

_{2}> Me,*Mi < M*

_{3}*< M*

_{6}.**Figure 12.14 **Layer magnetizations versus *T* for several values of *] _{2}* with

*H*= 0.2, and

*N*= 12: (a)

_{z}*J*

*= -1, (b)*

_{2}*J*

*= -0.5, (c)*

_{2}*]*

_{2}*= -2.*Dark olive green void squares correspond to the magnetization of the first layer, maroon void triangles to the second, red circles to the third, indigo triangles to the fourth, dark blue squares to the fifth, dark green void circles to the sixth layer. The inset in the top figure shows an enlarged region at low

*T.*See text for comments.

Figure 12.15 Spin wave spectrum versus *k _{x} = k_{y}* where

*W*stands for spin wave frequency win Eqs. (12.5)-(12.6),at (a)

*T =*0.353 and (b)

*T*= 1.212, with

*H*= 0.2.

This crossover is due to the competition between several complex factors: For example, quantum fluctuations have less effect on the surface magnetization making it larger than magnetizations of interior planes at low *T* as explained above (see Fig. 12.12), while the missing of neighbors for surface spins tends to diminish the surface magnetization at high *T* [79, 89]. The middle figure shows the case where *J*_{2} = -0.5 closer to the ferromagnetic limit. The spin length at *T* = 0 is almost 0.5 (very small contraction) and there is no visible crossover observed in the top figure. The bottom figure shows the case *] _{2}* = —2, which is the case of a strong helical angle. We observe then a crossover at a higher

*T*(—1.2) which is in agreement with the physical picture given above on the competition between quantum and thermal fluctuations. Note that we did not

*Helimagnetic Thin Films in a Field*

330

attempt to get closer to the transition temperature, namely *M <* 0.1 because the convergence of the self-consistency then becomes bad.

Before closing this section, let us discuss about the spin wave spectrum. Let us remind that to solve self-consistently Eqs. (12.5)- (12.6) at each *T,* we use as inputs < *Sf* >, < Sf >>■■••< $n„ > to search for the eigenvalues *w* for each vector *(k _{x}, k_{y})* and then calculate the outputs < Sf >, < Sf >,...,<

*S*>. The self- consistent solution is obtained when the outputs are equal to the inputs at a desired convergence precision fixed at the fifth digit (see other details in Ref. [89]). Figure 12.15 shows the spin wave spectrum in the direction

^{z}Nz*k*=

_{x}*k*of the Brillouin zone at

_{y}*T =*0.353 and Г = 1.212 for comparison. As seen, as

*T*increases the spin wave frequency decreases. Near the transition (not shown), it tends to zero. Figure 12.16 shows the spin wave spectrum at

*T*=0.353 for

Figure 12.16 Spin wave spectrum versus *k _{x} = k_{y}* at

*T =*0.353 where

*W*stands for spin wave frequency

*a>*in Eqs. (12.5)-(12.6), for (a)

*J*—0.5 and (b)

_{2}=*J*—2, with

_{2}=*H*=0.2.

*J _{2} =* 0.5 and у

_{2}= — 1, for comparison. Examining them closely, we see that the distribution of the spin wave modes (positions of the branches in the spectrum) is quite different for the two cases. When summed up for calculating the layer magnetizations, they give rise to the difference observed for the two cases shown in Fig. 12.14.

# Conclusion

In this chapter, we have shown (i) the GS spin configuration of a Heisenberg helimagnetic thin film in a magnetic field applied along the *c* axis perpendicular to the film, (ii) the phase transition occurring in the film at a finite temperature, (iii) quantum effects at low *T* and the temperature dependence of the layer magnetizations as well as the spin wave spectrum.

We emphasize that under the applied magnetic field, the spin configurations of the layers in the GS are different with each other across the film. When the temperature increases, the layers with large *xy* spin-components undergo a phase transition where the transverse (in-plane) xy ordering is destroyed. This "transverse" transition is possible because the *xy* spin-components are perpendicular to the field. Other layers with small *xy* spin- components, namely large *z* components, do not make a transition because the ordering in *S ^{z}* is maintained by the applied field. The transition of a number of layers with large

*xy*spin-components, not all layers, is a new phenomenon discovered here with our present model.

We have also investigated the quantum version of the model by using the Green's function method. The results show that the zero- point spin contraction is different from layer to layer. We also find a crossover of layer magnetizations which depends on /_{2}, namely on the magnitude of helical angles.

Experiments are often performed on materials with helical structures more complicated than the model considered above. However, the clear physical pictures given in our present analysis are believed to be useful in the search for the interpretation of experimental data.