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Thickness dependent Curie temperatures of ferromagnetic Heisenberg films 9 9 9 1 R. Schiller and W. Nolting n a Humboldt-Universita¨t zu Berlin, Institut fu¨r Physik, Invalidenstraße 110, J D-10115 Berlin, Germany 6 ] l e - r Abstract t s . t We develop a procedure for calculating the magnetic properties of a ferromagnetic a m Heisenberg film with single-ion anisotropy which is valid for arbitrary spin and film thickness. Applied to sc(100) and fcc(100) films with spin S = 7 the theory - 2 d yieldsthelayerdependentmagnetizationsandCurietemperaturesoffilmsofvarious n o thicknesses making it possible to investigate magnetic properties of films at the c interesting 2D-3D transition. [ 1 v Key words: A. magnetically ordered materials, A. thin films, D. phase transitions 6 3 0 1 0 In the past the Heisenberg model in thin films and superlattices has been 9 subject to intense theoretical work. Haubenreisser et al. [1] obtained good re- 9 / sults for the Curie temperatures of thin films [2] introducing an anisotropic t a exchangeinteraction(2).ShiandYang[3]calculatedthelayer-dependent mag- m netizations of ultra-thin n-layer films with single-ion anisotropy (3) for thick- - d nesses n ≤ 6. Other recent works are aimed at the question of reorientation n transitions in ferromagnetic films [4] or low-dimensional quantum Heisenberg o c ferromagnets [5]. : v i When investigating the temperature dependent magnetic and electronic prop- X erties of thin local-moment films or at surfaces of real substances it becomes r a desirable to be able to calculate the magnetic properties of the underlying Heisenberg model with no restrictions to neither the film thickness n nor the spin S of the localized moments. We develop a straigthforward analytical approach for the case of Heisenberg film with single-ion anisotropy. Considering the Heisenberg model, H = J S ·S = J S+S− +SzSz , (1) f ij i j ij i j i j Xij Xij (cid:16) (cid:17) Preprint submitted to Elsevier Preprint 1 February 2008 in a system with film geometry one comes to the conclusion that due to the Mermin-Wagner theorem [6] the problem cannot have a solution showing col- lective magnetic order at finite temperatures T > 0. To steer clear of this obstacle there are two possibilities. First, one can apply a decoupling scheme to the Hamiltonian (1) which breaks the Mermin-Wagner theorem. The most common example in the case of the Heisenberg model would bea mean-field decoupling. Forus, the maindrawback ofthe mean-field decoupling is its incapability of describing physical properties at the 2D-3D transition. WhenchoosingabetterdecouplingapproximationtofulfilltheMermin-Wagner theorem, the original Heisenberg Hamiltonian (1) has to be extended to break the directional symmetry. The most common extensions are the introduction of an anisotropic exchange interaction, −D SzSz −D SzSz, (2) i j s i j ij i,j∈surf X X and/or the single-ion anisotropy, −D (Sz)2 −D (Sz)2. (3) 0 i 0,s i i i∈surf X X In (2) and (3) the first sums run over all lattice sites of the film whereas in the second optionalterms the summations include positions within the surface layers of the film, only, according to a possible variation of the anisotropy in the vicinity of the surface. Extending the original Heisenberg Hamiltonian (1) by (2) or (3) one can now calculate the magnetic properties of films at finite temperatures within a non- trivial decoupling scheme. For the following we have choosen a single-ion anisotropy which is uniform within the whole film leaving us with the total Hamiltonian: H = H +H = Jαβ S+S− +Sz Sz +D (Sz )2, (4) f A ij iα jβ iα jβ 0 iα iXjαβ (cid:16) (cid:17) Xiα where we have considered the case of a film built up by n layers parallel to two infinitely extended surfaces. Here, as in the following, greek letters α, β, ..., indicate the layers of the film, while latin letters i, j, ..., number the sites within a given layer. Each layer possesses two-dimensional translational symmetry. Hence, the thermodynamic average of any site dependent operator 2 A depends only on the layer index α: iα hA i ≡ hA i. (5) iα α To derive the layer-dependent magnetizations hSzi for arbitrary values of the α spin S of the localized moments we introduce the so-called retarded Callen Green function [7]: Gαβ (E) ≡ S+;B(a) = S+;eaSjzβS− . (6) ij(a) iα jβ E iα jβ E DD EE DD EE For the equation of motion of the Callen Green function, E Gαβ (E) = ℏ S+,B(a) + S+,H ;B(a) (7) ij(a) iα jβ − iα − jβ (cid:28)h i (cid:29) (cid:28)(cid:28)h i (cid:29)(cid:29)E one needs the inhomogenity, S+,B(a) = η(a)δ δ , (8) iα jβ α αβ ij − (cid:28)h i (cid:29) and the commutators, S+,H =−2ℏ Jαγ Sz S+ −Sz S+ , (9) iα f ik iα kγ kγ iα − h i Xkγ (cid:16) (cid:17) S+,H =D ℏ S+Sz +Sz S+ . (10) iα A 0 iα iα iα iα − h i (cid:16) (cid:17) For the higher Green function on the right hand side of the equation of motion (7) resulting from the commutator relationship (9) one can apply the Random Phase Approximation (RPA) which has proved to yield reasonable results throughout the entire temperature range: Sz S+;B(a) −→hSzi S+;B(a) , (11) iα kγ jβ α kγ jβ E E DDSz S+;B(a)EE −→ Sz DDS+;B(a)EE . (12) kγ iα jβ γ iα jβ E E DD EE D EDD EE For the higher Green functions resulting from the commutator (10) this is not possible due to the strong on-site correlation of the corresponding operators. However, one can look for an acceptable decoupling of the form S+Sz +Sz S+;B(a) = Φ S+;B(a) . (13) iα iα iα iα jβ iα iα jβ E E DD EE DD EE 3 As was shown by Lines [8] an appropriate coefficient Φ = Φ can be found iα α for any given function B(a) = f S− , which is all we need to know for the jβ jβ moment. We will come back to th(cid:16)e ex(cid:17)plicit calculation of the Φα later. Using therelations(8)–(13)andapplying atwo-dimensional Fourier transform introducing the in-plane wavevector k the equation of motion (7) becomes (E −ℏD Φ )Gαβ =ℏη(a)δ 0 α k(a) α αβ +2ℏ Jαγ Sz Gαβ −Jαγ hSziGγβ . (14) 0 γ k(a) k α k(a) γ X(cid:16) D E (cid:17) Writing equation (14) in matrix form one immediately gets the solution by simple matrix inversion: (a) η 0 1 Gαk(βa)(E) = ℏ ... ·(EI−M)−1, (15) 0 η(a)  n    where I represents the n×n identity matrix and (M)αβ = D Φ +2 Jαγ Sz δ −2JαβhSzi. (16) ℏ 0 α 0 γ αβ k α γ ! X D E The local, i.e. layer-dependent, spectral density, Sα = −1ImGαα , can then k(a) π k(a) be written as a sum of δ-functions and with (15) one gets: Sα = ℏη(a) χ (k)δ(E −E (k)), (17) k(a) α ααγ γ γ X where E (k) are the poles of the Green function (15) and χ (k) are the γ ααγ weights of these poles in the diagonal elements of the Green function, Gαα . k(a) Both, the poles and the weigths can be calculated e.g. numerically. Extending the procedure by Callen [7] from 3D to film structures1 one finds an analytical expression for the layer-dependent magnetizations, (1+ϕ )2S+1(S −ϕ )+ϕ2S+1(S +1+ϕ ) hSzi = ℏ α α α α , (18) α ϕ2S+1 −(1+ϕ )2S+1 α α 1 The only pre-condition for the extension is that the spectral density has the multipole structure (17) 4 where 1 χ (k) ααγ ϕ = . (19) α N eβEγ(k) −1 s k γ XX Here, N is the number atoms in a layer and β = 1 . The poles and weigths s kBT in (19) have to be calculated for the special Green function Gαα with a = 02. k(a) In this case the Callen Green function (6) simply becomes: Gαβ = Gαβ = S+;B(0) = S+;S− , (20) ij(0) ij iα jβ iα jβ DD EE DD EE and, according to (8), η(0) = η = 2ℏhSzi. (21) α α α Having solved the problem formally we are left with explicitly calculating the coefficients Φ of equation (13). Applying the spectral theorem to (13) for the α special case of a = 0 one gets, using elementary commutator relations: S−S+(2Sz +ℏ) = Φ S−S+ . (22) jβ iα iα iα jβ iα E D E D E We now define the Green function Dβα = S−;C , (23) ji jβ iα E DD EE where C isa functionof thelattice site. Writing down theequation of motion iα βα of D for the limit D → 0, ji 0 E Dβα(E) = ℏ S−,C + S−,H ;C , (24) ji jβ iα jβ f iα − − (cid:28)h i (cid:29) (cid:28)(cid:28)h i (cid:29)(cid:29)E and decoupling all the higher Green functions using the RPA one arrives after transformation into the two-dimensional k-space at: S−,C 0 1 1 − Dkβα = ℏ(cid:28)h i (cid:29) ... ·(EI−A)−1, (25)  0 [Sn−,Cn]−   D E 2 The parameter a had been introduced to derive (19) for arbitrary spin S. 5 where A is a matrix which is independent on the choice of C . Now putting iα C in (23) in turn equal to S+ and to S+(2Sz +ℏ) and applying the spectral iα iα iα iα theorem to equation (25) one eventually gets the relation: S−S+ S−S+(2Sz +ℏ) jβ iα jβ iα iα = . (26) D E D E S−,S+ S−,S+(2Sz +ℏ) iα iα iα iα iα − − (cid:28)h i (cid:29) (cid:28)h i (cid:29) The coefficients Φ are then with (22) given by iα Si−α,Si+α(2Sizα +ℏ) 2h(Sz )2i−ℏ2S(S +1) Φiα = (cid:28)h S−,S+ i−(cid:29) = iα hSizαi , (27) iα iα − (cid:28)h i (cid:29) where, along with commutator relations, the identity S±S∓ = ℏ2S(S +1)±ℏSz −(Sz )2 (28) iα iα iα iα has been used. To avoid the unknown expectation value h(Sz )2i we apply the iα spectral theorem to the spectral density (17) with a = 0 and get using (21): 1 χ (k) S−S+ = 2ℏhSzi ααγ (=19) 2ℏhSziϕ . (29) α α α Ns k γ eβEγ(k) −1 α α D E XX Hence, with (28) and (29), we get (Sz)2 = ℏ2S(S +1)−ℏhSzi(1+2ϕ ), (30) α α α D E and the coeffcients Φ can be written in the convienient form α 2ℏ2S(S +1)−3ℏhSzi(1+2ϕ ) Φ = α α . (31) α hSzi α Together with (31), the equations (15), (16), (18), and (19) represent a closed system of equations, which can be solved numerically. All the following calculations have been performed for spin S = 7, applicable 2 to a wide range of interesting rare-earth compounds, and for an exchange interaction in tight-binding approximation J = 0.01eV which is uniform within the whole film. The case where the exchange integrals in the vicinity of the surfaces are modified has been dealt with by a couple of authors [9]. The 6 1.0 n=20 k ul b C T 0.5 / C T 0.0 0 5 10 15 20 n n=15 n=10 4 n=7 3 n=5 z2 S n=4 n=3 1 Surface Layer n=2 Centre Layer 0 n=1 0.0 0.1 0.2 0.3 0.4 T / eV Fig. 1. Layer-dependent magnetizations, hSzi, of sc(100) films as a function of α temperature for various thicknesses n.For all temperatures and film thicknesses the hSzi increase from the surface layer towards the centre of the films. Inset: Curie α temperature as a function of thickness of the sc(100) films. single-ion ansitropy which plays the mere role of keeping the magnetizations at finite temperatures was choosen D /J = 0.01. 0 Figs. 1 and 2 show the temperature and layer-dependent magnetizations of, respectively, simple cubic (sc) and face-centered cubic (fcc) films with the surfaces parallel to the (100)-planes. For the following Z means the coordination s number of the atoms in the surface layers and Z is the coordination num- b 7 1.0 n=20 k ul b C T 0.5 / C T 0.0 0 5 10 15 20 n n=15 n=10 4 n=7 3 n=5 z2 S n=4 n=3 1 Surface Layer n=2 Centre Layer 0 n=1 0.0 0.2 0.4 0.6 0.8 1.0 T / eV Fig. 2. Same as Fig. 1 for fcc(100) films. ber in the centre layers of the films. For the case of a monolayer, n = 1, the curves for the sc(100) and the fcc(100) ’film’ are identical, both having the same structure. With increasing film thickness the Curie temperatures of the films increase. For fcc(100) films the increase in T is steeper resulting in the C limit of thick films in a Curie temperature about twice the value of that of the according sc(100) films due to the higher coordination number of the fcc 3D-crystal (Z = 12) compared to the sc 3D-crystal (Z = 6). The larger b,fcc b,sc difference between surface and centre layer magnetization of the fcc(100) films compared to the sc(100) films can be explained by the lower ratio between Z s and Z (Z /Z = 8/12 and Z /Z = 5/6). b s,fcc(100) b,fcc s,sc(100) b,sc 8 Concluding, we have have shown that the presented approach is a useful and straigthforward method for calculating the layer-dependent magnetizations of filmsofvariousthicknesses andwitharbitraryspinS ofthelocalizedmoments. Acknowledgements We would like to thank P. J. Jensen for helpful discussions and for bringing Ref. [8] to our attention. One of the authors (R. S.) would like to acknowl- edge the support by the German National Merit Foundation. The support by the Sonderforschungsbereich 290 (”Metallische du¨nne Filme: Struktur, Mag- netismus und elektronische Eigenschaften“) is gratefully acknowledged. References [1] W. Haubenreisser,W. Brodkorb,A.Corciovei, andG.Costache, phys.stat. sol. (b) 53, 9 (1972). [2] Diep-The-Hung, J.C.S.Levy,and O.Nagai, phys.stat. sol. (b)93, 351 (1979). [3] Long-Pei Shi and Wei-Gang Yang, J. Phys.: Condens. Matter 4, 7997 (1992). [4] D. K. Morr, P. J. Jensen, and K.-H. Bennemann, Surf. Sci. 307-309, 1109 (1994). [5] D. A. Yablonskiy, Phys. Rev. B 44, 4467 (1991). [6] N. M. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966). [7] H. B. Callen, Phys. Rev. 130, 890 (1963). [8] M. E. Lines, Phys. Rev. 156, 534 (1967). [9] T. Wolfram and R. E. DeWames, Prog. Surf. Sci. 2, 233 (1972); D. L. Mills, In V. M. Agranovich and R. Loudon, editors, Surface Excitations. North Holland, Amsterdam (1984); M. G.Cottam andD. R.Tilley, Introduction to surface and superlattice excitations, Cambridge University Press (1989). 9