Home Mathematics



Spin Contraction at T = 0 and Transition TemperatureTable of Contents:
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 zeropoint 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 J2/J1 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 = J2/J1 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 MagnetizationsLet 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 J2/J1 = —1.4:
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 J2/J1 = —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 selfconsistently N_{z }equations (11.36) to obtain the transition temperature T_{c}. One way to do it is to use the selfconsistent layer magnetizations obtained as described above at a temperature as close as possible to T_{c} as input for Eq. (11.36). As long as the T is far from T_{c} the convergence is not reached: We have four 'pseudotransition temperatures’ T_{cs} as seen in Fig. 11.7, one for each layer. The convergence of these T_{cs }can be obtained by a short extrapolation from temperatures when they are rather close to each other. T_{c} is thus obtained with a very small extrapolation error as seen in Fig. 11.7 for p = J2/J1 = —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_{c}, Eq. (11.36) is solved to obtain T_{cs} for each layer. The convergence is reached when T_{cs} tend to a single value T_{c}. One has T_{c} ~ 2.313 ± 0.010. Red circles, black void circles, blue squares and blue void squares are T_{cs} obtained from Eq. (11.36) for first, second, third and fourth layers, respectively, at different temperatures. Bottom: Extrapolation by lines to obtain T_{c} is shown for surface parameter J1/J1 = 0.7. The precision for selfconsistent convergence is 1% for layer magnetizations. See text for comments. T_{c} ~ 2.313 ± 0.010. The results for several p = J_{2}/J1 are shown in Fig. 11.8. Effect of Anisotropy and Surface ParametersThe results shown above have been calculated with an inplane 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 = J2/J1 for an 8layer 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 secondlayer 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_{c} = 2.103±0.010,1.951±0.010,1.880±0.010 and 1.841± 0.010 for J[ = 1,0.7,0.5 and 0.3, respectively [N_{z} = 16,]_{2}/J i = 2, d = d_{s} = 0.1). Note that the value 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_{z} = 16. Right: Transition temperature versus d_{s} for / f=0.7, 0.5 and 0.3 (curves from up to down), with /2//1 = —2, d = 0.1 and N_{z} = 16. 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 ThicknessWe 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_{z} = 16 which are to be compared with the values for N_{z} = 8 given in Table 11.1, for p = J2/J1 = 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 = J2/J1 = 2, the transition temperature is 2.090 ± 0.010 for N_{z} = 8, 2.093 ± 0.010 for N_{z} = 12 and 2.103 ± 0.010 for N_{z} = 16. These are the same within errors. At smaller thicknesses, the difference can be seen. However, for helimagnets in the 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 1140 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 SimulationTo 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 inplane Isinglike 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 J2/J1 = 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_{z} = 16. Black circles, blue void squares, cyan squares and red void circles are for first, second, third and fourth layers, respectively. 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_{3} 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. 
<<  CONTENTS  >> 

Related topics 