# Spin Contraction at T = 0 and Transition Temperature

It is known that in antiferromagnets, quantum fluctuations give rise to a contraction of the spin length at zero temperature (see Chapter 3 and Ref. [87]). We will see here that a spin under a stronger antiferromagnetic interaction has a stronger zero-point spin contraction. The spins near the surface serve for such a test. In the case of the film considered above, spins in the first and in the second layers have only one antiferromagnetic NNN while interior spins have two NNN, so the contraction at a given *J**2**/J**1 *is expected to be stronger for interior spins. This is verified with the results shown in Fig. 11.4. When I/2I//1 increases, namely the antiferromagnetic interaction becomes stronger, we observe stronger contractions. Note that the contraction tends to zero when

Figure 11.4 (Color online) Spin lengths of the first four layers at *T* = 0 for several values of *p = J**2**/J**1* with *d =* 0.1, *N _{z}* = 8. Black circles, void circles, black squares and void squares are for first, second, third and fourth layers, respectively. See text for comments.

the spin configuration becomes ferromagnetic, namely *] _{2}* tends to -1.

# Layer Magnetizations

Let us show two examples of the magnetization, layer by layer, from the film surface in Figs. 11.5 and 11.6, for the case where У2//1 = —1.4 and — 2 in a *N _{z}* = 8 film. Let us comment on the case

*J*

*2*

*/J*

*1*= —1.4:

- (i) the shown result is obtained with a convergence of 1%. For temperatures closer to the transition temperature
*T*we have to lower the precision to a few percents which reduces the clarity because of their close values (not shown)._{c}, - (ii) the surface magnetization, which has a large value at
*T*= 0 as seen in Fig. 11.4, crosses the interior layer magnetizations at*T*~ 0.42 to become much smaller than interior magnetizations at higher temperatures. This crossover phenomenon is due to the competition between quantum fluctuations, which dominate low-Г behavior, and the low-lying surface spin wave modes which strongly diminish the surface magnetization at higher*T.*Note that the second-layer magnetization makes also

**Figure 11.5 **(Color online) Layer magnetizations as functions of *T* for }_{2}//i = —1.4 with *d* = 0.1, *N _{z} =* 8 (top). Zoom of the region at low

*T*to show crossover (bottom). Black circles, blue void squares, magenta squares and red void circles are for first, second, third and fourth layers, respectively. See text.

a crossover at Г ~ 1.3. Similar crossovers have been observed in quantum antiferromagnetic films [79] and quantum superlattices [80].

Similar remarks can be also made for the case *J**2**/J**1* = —2.

Note that although the layer magnetizations are different at low temperatures, they will tend to zero at a unique transition temperature as seen below. The reason is that as long as an interior layer magnetization is not zero, it will act on the surface spins as an external field, preventing them to become zero.

The temperature where layer magnetizations tend to zero is calculated by Eq. (11.36). Since all layer magnetizations tend to

Figure 11.6 (Color online) Layer magnetizations as functions of *T* for }_{2}//i = —2 with *d* = 0.1, *N _{z}* = 8 (top). Zoom of the region at low

*T*to show crossover (bottom). Black circles, blue void squares, magenta squares and red void circles are for first, second, third and fourth layers, respectively. See text.

zero from different values, we have to solve self-consistently *N _{z }*equations (11.36) to obtain the transition temperature

*T*One way to do it is to use the self-consistent layer magnetizations obtained as described above at a temperature as close as possible to

_{c}.*T*as input for Eq. (11.36). As long as the

_{c}*T*is far from

*T*the convergence is not reached: We have four 'pseudo-transition temperatures’

_{c}*T*as seen in Fig. 11.7, one for each layer. The convergence of these

_{cs}*T*can be obtained by a short extrapolation from temperatures when they are rather close to each other.

_{cs }*T*is thus obtained with a very small extrapolation error as seen in Fig. 11.7 for

_{c}*p = J*

*2*

*/J*

*1*= —1.4:

**Figure 11.7 **(Color online) Top: Transition temperature is calculated at *p = J*_{2}//i = —1.4 for *d =* 0.1, *N _{z} =* 8: At each temperature, using the self- consistent values of layer magnetizations at Г <

*T*Eq. (11.36) is solved to obtain

_{c},*T*for each layer. The convergence is reached when

_{cs}*T*tend to a single value

_{cs}*T*One has

_{c}.*T*~ 2.313 ± 0.010. Red circles, black void circles, blue squares and blue void squares are

_{c}*T*obtained from Eq. (11.36) for first, second, third and fourth layers, respectively, at different temperatures. Bottom: Extrapolation by lines to obtain

_{cs}*T*is shown for surface parameter

_{c}*J*

*1*

*/J*

*1*

*=*0.7. The precision for self-consistent convergence is 1% for layer magnetizations. See text for comments.

*T _{c}* ~ 2.313 ± 0.010. The results for several

*p*=

*J*

_{2}*/J*1 are shown in Fig. 11.8.

# Effect of Anisotropy and Surface Parameters

The results shown above have been calculated with an in-plane anisotropy interaction *d* = 0.1. Let us show now the effect of *d. *Stronger *d* will enhance all the layer magnetizations and increase *T _{c}. *Figure 11.9 shows the surface magnetization versus

*T*for several

**Figure 11.8 **(Color online) Transition temperature versus *p = J**2**/J**1* for an 8-layer film with d = 0.1. See text for comments.

**Figure 11.9 **Surface magnetization versus *T* ford = 0.05 (circles), 0.1 (void circles), 0.2 (squares), 0.3 (void squares) and 0.4 (triangles), with /2//1 = -1.4, *N _{z} =* 16.

values of *d* (other layer magnetizations are not shown to preserve the figure clarity). The transition temperatures are 2.091 ± 0.010, 2.313 ± 0.010, 2.752 ± 0.010,3.195 ± 0.010 and 3.630 ± 0.010 for *d*** = **0.05, 0.1, 0.2, 0.3 and 0.4, respectively. These values versus *d* lie on a remarkable straight line.

Let us examine the effects of the surface anisotropy and exchange parameters *d _{s}* and

*][.*As seen above, even in the case where the surface interaction parameters are the same as those in the bulk the surface spin wave modes exist in the spectrum. These localized modes cause a low surface magnetization observed in Figs. 11.5 and 11.6. Here, we show that with a weaker NN exchange interaction between surface spins and the second-layer ones, namely

*){ < ]*

_{b }the surface magnetization becomes even much smaller with respect to the magnetizations of interior layers. This is shown in Fig. 11.10 for several values of

*J*j. We observe again here the crossover of layer magnetizations at low

*T*due to quantum fluctuations as discussed earlier. The transition temperature strongly decreases with

*][*We have

*T*2.103±0.010,1.951±0.010,1.880±0.010 and 1.841± 0.010 for

_{c}=*J[*= 1,0.7,0.5 and 0.3, respectively

*[N*= -2,

_{z}= 16,]_{2}/J i*d = d*= 0.1). Note that the value

_{s}*J(*= 0.5 is a very particular value: the GS configuration is a uniform configuration with all angles equal

Figure 11.10 (Color online) Layer magnetizations as functions of *T* for the surface interaction *J[* = 0.3 (top, left), 0.5 (top, right) and 0.7 (bottom) with *Ji/j* = —2, *d =* 0.1 and *N _{z} =* 16. Black circles, blue void squares, magenta squares and red void circles are for first, second, third and fourth layers, respectively.

**Figure **11.11 Left: Surface magnetization versus *T* for *d _{s}* = 0.01 (circles), 0.1 (void circles), 0.2 (squares) and 0.3 (void squares), with

*Jf =*l,/

_{2}//i = —2 and

*N*16. Right: Transition temperature versus

_{z}=*d*for / f=0.7, 0.5 and 0.3 (curves from up to down), with /2//1 = —2,

_{s}*d =*0.1 and

*N*= 16.

_{z}to 60°, namely there is no surface spin rearrangement. This can be explained if we look at the local field acting on the surface spins: the lack of neighbors is compensated by this weak positive value of *J[* so that their local field is equal to that of a bulk spin. There is thus no surface reconstruction. Nevertheless, as *T* increases, thermal effects will strongly diminish the surface magnetization as seen in Fig. 11.10 (middle). As for the surface anisotropy parameter *d _{s},* it affects strongly the layer magnetizations and the transition temperature: We show in Fig. 11.11 the surface magnetizations and the transition temperature for several values of

*d*

_{s}.# Effect of the Film Thickness

We have performed calculations for *N _{z}* = 8, 12 and 16. The results show that the effect of the thickness at these values is not significant: The difference lies within convergence errors. Note that the classical ground state of the first four layers is almost the same: For example, here are the values of cosinus of the angles of the film first half for

*N*= 16 which are to be compared with the values for

_{z}*N*= 8 given in Table 11.1, for

_{z}*p*=

*J*

*2*

*/J*

*1*

*=*-2 (in parentheses are angles in degree):

0.86737967 (29.844446), 0.41125694 (65.716179), 0.52374715 (58.416061), 0.49363765 (60.420044), 0.50170541 (59.887100),

0.49954081 (60.030373), 0.50013113 (59.991325), 0.49993441 (60.004330).

From the 4th layer, the angle is almost equal to the bulk value (60°).

At *p = J**2**/J**1* = -2, the transition temperature is 2.090 ± 0.010 for *N _{z} =* 8, 2.093 ± 0.010 for

*N*= 12 and 2.103 ± 0.010 for

_{z}*N*16. These are the same within errors. At smaller thicknesses, the difference can be seen. However, for helimagnets in the

_{z}=*z*direction, thicknesses smaller than 8 do not allow to see fully the surface helical reconstruction which covers the first four layers (see Section 11.2).

At this stage, it is interesting to note that our result is in excellent agreement with experiments: It has been experimentally observed that the transition temperature does not vary significantly in MnSi films in the thickness range of 11-40 nm [176]. One possible explanation is that the helical structure is very stable as seen above: The surface perturbs the bulk helical configuration only at the first four layers, so the bulk 'rigidity' dominates the transition. This has been experimentally seen in holmium films [206].

# Classical Helimagnetic Films: Monte Carlo Simulation

To appreciate quantum effects causing crossovers of layer magnetizations presented above at low temperatures, we show here some results of the classical counterpart model: Spins are classical XY spins of amplitude S = 1. We take the XY spins rather than the Heisenberg spins for comparison with the quantum case because in the latter case we have used an in-plane Ising-like anisotropy interaction *d.* Monte Carlo simulations have been carried out over film samples of 100 x 100 x 16. Periodic boundary conditions are applied in the *xy* plane. One million of MC steps are discarded to equilibrate the system and another million of MC steps are used for averaging. The layer magnetizations versus *T* are shown in Fig. 11.12 for the case where surface interaction *)[ = l* (top) and 0.3 (bottom) with *J**2**/J**1* = -2 and *N _{z}* = 16. One sees that (i) by extrapolation there is no spin contraction at

*T*= 0 and

**Figure 11.12 **(Color online) Monte Carlo results: Layer magnetizations as functions of *T* for the surface interaction /f = 1 (left) and 0.3 (right) with *J _{2}/J* i = —2 and

*N*= 16. Black circles, blue void squares, cyan squares and red void circles are for first, second, third and fourth layers, respectively.

_{z}there is no crossover of layer magnetizations at low temperatures, (ii) from the intermediate temperature region up to the transition the relative values of layer magnetizations are not always the same as in the quantum case: For example, at *T =* 1.2, one has *Mi <* M_{3} < M_{4} < *M** _{2}* in Fig. 11.12 (top) and Mj <

*M*

_{2}*< M*

_{4}*< M*

*in Fig. 11.12 (bottom) which are not the same as in the quantum case shown in Fig. 11.6 (top) and Fig. 11.10 (top). Our conclusion is that even at temperatures close to the transition, helimagnets may have slightly different behaviors according to their quantum or classical nature. Extensive MC simulations with size effects and detection of the order of the phase transition do not fall within the scope of this present chapter.*

_{3}