PHYSICAL REVIEW B VOLUME 56, NUMBER 5 1 AUGUST 1997-I Heat capacity as a probe of surface phase transitions in thin antiferromagnetic films R. E. Camley Department of Physics, University of Colorado at Colorado Springs, Colorado Springs, Colorado 80933-7150 Received 10 February 1997 External magnetic fields can produce a phase transition from an antiferromagnetic state to a spin-flop state in antiferromagnets. In ultrathin antiferromagnets, the critical magnetic field is sensitive to whether the net number of magnetic layers is odd or even. We show that this transition will produce a distinct feature in the heat capacity as a function of applied magnetic field. The feature is on the order of 5­10 % of the magnetic heat capacity and should be measurable. S0163-1829 97 06030-X The properties of ultrathin magnetic films have been a 100 planes. This is in sharp contrast to the known bulk struc- subject of considerable interest in the last decade. Most of ture where the magnetic moments lie parallel to 111 planes. this work has focused on metallic ferromagnetic films, pri- One consequence of having such high-quality films is that marily because of the interesting coupling of ferromagnetic one may examine the effect finite-size effects have on mag- films across a nonmagnetic spacer1,2 and because of the giant netic phase transitions.7,9,10 This is particularly interesting magnetoresistance effect.3,4 because in antiferromagnets some of the finite-size effects In the last few years, ultrathin antiferromagnetic depend on whether the number of magnetic layers is even or multilayers5,6 and thin films7,8 have also been constructed odd. This is in distinct contrast to ferromagnets. For ex- and studied. Such materials are interesting both for funda- ample, a sufficiently large external magnetic field will cause mental and technological reasons. In contrast to the metallic a phase transition from the antiferromagnetic state to a spin- ferromagnetic systems, the antiferromagnets are generally in- flop or canted structure. In thin antiferromagnetic films the sulators and thus the simpler localized spin models should be critical field necessary strongly depends on whether there is more appropriate. Furthermore, the antiferromagnetic system an uncompensated layer of spins or not. has many fundamental differences with ferromagnets. The The purpose of this paper is to show theoretically that the magnetic structure is clearly more complex and allows a different kinds of phase transitions may be observed by mak- larger number of stable configurations. Also, the anisotropy ing heat capacity measurements as a function of applied field in antiferromagnets can be quite large, on the order of magnetic field. We note that heat capacity measurements or several hundred kG, compared to the 1­2 kG typically found related measurements such as thermal expansion have been in the ferromagnetic metals. In terms of technological use, used frequently in characterizing antiferromagnetic thin films antiferromagnets and their coupling to ferromagnets are fun- and multilayers.5,8 Our results indicate that the phase transi- damental to the giant magnetoresistance spin-valve struc- tions should show a distinct feature in the heat capacity. tures which will be vital to the next generation of reading Furthermore, the feature is on the order of about 6­10 % of heads in hard-disk systems. the magnetic heat capacity itself and thus should be able to Since antiferromagnets have a magnetic structure which be measured readily. can change significantly from layer to layer, high-quality We use the self-consistent local field method developed samples are of particular importance. For example, one can earlier and applied to a number of different magnetic super- consider an antiferromagnet with all the spins in one layer, lattices and antiferromagnetic films.11 In this method one parallel to the surface, pointing in the same direction, but simply calculates the total effective field the sum of the where the spins in the neighboring layers above and below exchange field, the anisotropy field, and the external field all point in the opposite direction. Such a structure has very acting on an individual spin. This spin is then rotated in such different properties depending on whether the total number a way as to minimize its energy. This is done for each spin in of magnetic layers is odd or even. For an even number of the structure and repeated until a stable configuration ap- layers there is no net magnetic moment, while for an odd pears. To take into account thermal effects, the Brillouin number of layers one of the spin sheets is uncompensated function is used to give the thermally averaged magnitude of and a net magnetic moment results. the spin in a total effective field at a given temperature T. A particularly exciting experimental development is the The heat capacity can be easily calculated once the equilib- recent observation of clear oscillations in the magnetization rium structure is known. One simply evaluates the average of thin antiferromagnetic films as a function of the thickness energy at a given temperature and then numerically takes the of the film.7 These experiments provide evidence that high- derivative CV ( U/ T)V by taking the difference of the quality antiferromagnetic films, with a well-defined number energies at nearby temperatures. of magnetic layers, can now be grown and studied. Further- We consider a thin antiferromagnetic film, with param- more, these experiments indicate that the magnetic structure eters generally characteristic of MnF2 with the easy axis par- of ultrathin antiferromagnets may differ considerably from allel to the surface. An external magnetic field H0 is also that of the bulk. For example, in Ref. 7 it was reported that oriented parallel to the surface and along the easy axis. We the magnetic moments in a thin film of CoO were parallel to take the magnetic structure to consist of sheets of spins, ori- 0163-1829/97/56 5 /2336 4 /$10.00 56 2336 © 1997 The American Physical Society 56 BRIEF REPORTS 2337 FIG. 1. Normalized heat capacity as a function of temperature for different numbers of magnetic layers in a thin antiferromagnetic film. The different transition temperatures for different N are due to finite-size effects. ented parallel to the surface, with the spins on one plane antiparallel to those on the neighboring planes. We consider FIG. 2. Normalized heat capacity as a function of magnetic field systems with the number of magnetic planes, N, being both for different numbers of magnetic layers. Thicker films are shown odd and even. The parameters are He 46.7 T for the ex- in a and thinner ones in b . The phase transition of films with change field and Ha 0.7 T for the anisotropy field. We will N even are much lower than for those with N odd. also look at a MnF2-like material where all the parameters are the same as in MnF2 except that the anisotropy field has a function of applied field for a temperature of 50 K for been increased by a factor of 2. thicker films N 8 and N 9 , while in Fig. 2 b we exam- In Fig. 1 we present the normalized heat capacity ine the field dependence for thinner films. The system with (CV /N) as a function of temperature for films with different an even number of layers shows a rapid change in heat ca- numbers of layers. The most prominent feature is that the pacity at a low field, while the system with an odd number of transition temperature is strongly dependent on the number layers shows a similar change at much higher fields. For the of layers and decreases as the number of layers is decreased. thicker films the relative change in CV is on the order of 4%. This is a clear result of finite-size effects; i.e., the reduced For the thinner films the relative change is on the order of coordination of the spins at the surfaces results in a smaller 6­10 %. effective field for those spins. This in turn means that the The origin of this behavior is due to a phase transition surface spins are more susceptible to thermal agitation and from the antiferromagnetic state to a spin-flop state where the thermal averaged magnitude of these spins is reduced the spin moments are canted with respect to the applied field from what is expected in the bulk. For very thin films, this as shown in Fig. 3. The magnetic structure, of course, is reduction propagates through the entire structure and the determined by finding the state which minimizes the sum of transition temperature is also reduced. The change in the the Zeeman energy, the anisotropy energy, and the exchange critical temperature as a function of the number of magnetic energy. In the antiferromagnetic state and with N even, the layers is in good qualitative agreement with recent experi- net Zeeman energy is zero, with as many spins pointing op- ments on CoO.8 posite to the field as there are spins pointing parallel to the When the thickness of the film reaches N 8 and N 9, field. As the external magnetic field is increased, the struc- the heat capacity curves are nearly the same. There is a near- ture can lower its energy by changing to the canted configu- linear increase in heat capacity as T increases and a leveling ration seen in Fig. 3 b , where each magnetic moment has a off and a rapid drop at the transition temperature. We note component parallel to the applied field. This change in con- that for a bulk sample in the mean-field approximation the figuration comes with an increase in exchange and anisot- linear region extends directly to the phase transition. The ropy energies, but this is overcome by the lowering of the leveling off, which can be seen in experimental data, occurs Zeeman energy. here because of finite-size effects. The only difference be- It has been known for many years that the magnetic-field- tween the N 8 case and the N 9 case is that the transition induced transition at T 0 to a spin-flop state in a large temperature is slightly higher for N 9 since finite-size ef- bulk structure occurs at a critical field given by fects play a smaller role for larger systems. In contrast to the results above, the heat capacity as a H 2 bulk 2HeHa Ha. 1 function of applied field shows a striking difference between a structure with an odd number of layers and one with an In addition to the bulk spin-flop transition, surface-induced even number of layers. In Fig. 2 a we show heat capacity as spin-flop transitions have also been predicted.12 In that work 2338 BRIEF REPORTS 56 FIG. 4. Normalized heat capacity as a function of field for N 8 and N 9. The solid lines indicate the results for MnF2, and the dashed lines show the results for a material with MnF2 param- eters except that the anisotropy field has been doubled. FIG. 3. Illustration of the antiferromagnetic state and spin-flop This behavior helps to explain a very interesting result to state geometry. The applied field is applied parallel to the easy axis. be found in Fig. 2. We note that the jump in the heat capacity occurs at about the same critical field when the number of it was shown that the antiferromagnetic phase would become layers is even. When the number of layers is odd, the critical unstable at a field which was considerably lower than that of field is much larger for the thinner antiferromagnetic film.16 the bulk critical field. The critical field for this surface tran- The key issue is that the Zeeman energy in the antiferromag- sition at T 0 is given by netic state always comes from only one uncompensated plane of spins for any structure where N is odd. In contrast, all the other energies exchange and anisotropy scale lin- Hsurf Hbulk / 2. 2 early with the number of layers. Thus the Zeeman energy associated with the uncompensated layer and it is this which Keffer and Chow later showed that the surface flop would inhibits the phase transition becomes less important as N evolve into a bulk spin-flop state as the magnetic field was increases. As a result, the critical field for the phase transi- increased.13 Although the precise validity of these results has tion is reduced as N is increased. been challenged recently,14 experimental results on metallic We comment briefly on the numerical values in Fig. 2. multilayers with ferromagnetic films coupled antiferromag- With the parameters for MnF netically through a nonmagnetic spacer film15 have indeed 2 , one expects the bulk spin- shown that a surface spin flop takes place at approximately flop transition at T 0 to occur at an external field of H0 the value given by Eq. 2 . 8.1 T. The transitions here, for both N even and N odd, at In thin films the critical field is further modified by sur- T 50 K lie above that value. This is consistent with recent face effects.16 When the number of magnetic layers is odd, theoretical calculations exploring the temperature-field phase there is a net magnetic moment for the antiferromagnetic thin diagram for thin antiferromagnetic films which showed that film. If we put an applied field along the easy axis, this the critical field for a transition to the canted state rises as the moment will be parallel to the field, and thus the Zeeman temperature rises.16 energy will not be zero in the antiferromagnetic state. As a We have also calculated the heat capacity at lower tem- result, the transition from the antiferromagnetic state to the peratures. The percentage shift in the heat capacity at the spin-flop phase, which in the bulk material is associated with phase transition decreases as T decreases, except for the case a change in Zeeman energy from zero to some negative of very low temperatures and N even. Since the heat capacity value, is inhibited since the antiferromagnetic state already is significantly lower at lower T as seen in Fig. 1 , this is has a negative energy contribution from the Zeeman energy. likely to be a more difficult measurement. In contrast, in a thin film in the antiferromagnetic state with It is of interest to see how the size of the jump in the heat an even number of layers has no Zeeman energy when an capacity depends on the characteristic material parameters. external magnetic field is applied along the easy axis. As a In Fig. 4 we explore the spin-flop transition for a material result, the system can more easily take advantage of the low- with parameters closer to that of MnF2, but where the anisot- ering of the energy produced by the spin-flop transition and ropy field has been increased by a factor of 2. We see that for the phase transition occurs at a much lower applied field. both an even and odd number of layers the phase transition 56 BRIEF REPORTS 2339 occurs at a higher field as might be expected from Eqs. 1 In real films, of course, the number of layers will not be and 2 . In addition, the shift in the heat capacity is larger in constant over an entire film. In that case a heat capacity both cases. measurement may be a way of characterizing the magnetic In summary, we have explored whether the spin-flop structural quality. For example, if the regions where a well- phase transition in thin antiferromagnetic films can be ob- defined number of layers exist are reasonably large, one served by heat capacity measurements. It was shown that the might find two jumps in heat capacity as a function of field, spin-flop transition is accompanied by a jump in heat capac- one at higher field corresponding to an odd number of layers ity which was on the order of 6% of the heat capacity itself. and one at lower field corresponding to an even number of In contrast to heat capacity measurements made in zero field, layers. If the spatial regions of a well defined number of films with an even and an odd number of layers behave very layers are small, then presumably one sees only one phase differently. Films with an even number of layers have tran- transition where the entire structure changes together at some sitions which occur at fields below that of the bulk phase average field. transition. 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