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Spin fluctuations dynamics on the insulating antiferromagnetic stepped surface model
Surface Science 560 (2004) 158–166 www.elsevier.com/locate/susc
Spin fluctuations dynamics on the insulating antiferromagnetic stepped surface model M. Tamine
*
Laboratoire de Physique et Chimie Quantique, Institut de Physique, Universite de Tizi-Ouzou, BP 17 RP-15000 Tizi-Ouzou, Algeria Received 29 October 2003; accepted for publication 13 April 2004 Available online 6 May 2004
Abstract We present a numerical method to calculate the spin fluctuations dynamics on the stepped surface. The model discussed here consists in an extended antiferromagnetic surface steps on the surface boundary of an insulating antiferromagnetic substrate. The stepped surface is formed by two straight steps dropped randomly and the spins moments of the steps and the substrate are considered as local with no electronic effects. The calculations concern in particular the energies of localized spin-wave modes near the stepped surface and employ the matching procedure in the randomphase approximation and mean field approximation. Only nearest neighbors exchange interactions are considered between the spins in the model. The theoretical formalism presented here solves for the three dimensional evanescent crystal spin field in the bulk and the surface domains around the steps, that arises owing to the breakdown of magnetic translation symmetry of the system. The results are used to calculate the spin modes energies associated with the steps and surface terraces. We show the presence of acoustic and optical localized spin-wave modes propagating along the surface and the steps, their spin fluctuations fields are also described as evanescent in the plane normal to the stepped surface and depend on the nature of the exchange parameters near the steps. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Stepped single crystal surfaces; Magnetic phenomena (cyclotron resonance, phase transitions, etc.)
1. Introduction The study of film growth has been increasingly characterized by the application of surface science methods to understand growth at the atomic level. Work in this field has been motivated by the ever more stringent requirements on the quality of thin films needed for developing advanced microelectronic, optical, and magnetic devices, as well as the
*
Tel.: +213-26-21-48-48; fax: +213-26-22-37-66. E-mail address: [email protected] (M. Tamine).
thrust toward nanometer-scale structures. As device miniaturization reaches submicrometer and nanometer length regimes, atomic level control of the fabrication processes for both novel materials and new devices have become of great importance. The presence of nanostructures such as random steps, kinks and other defects on crystal surfaces is important to their equilibrium topography as well as to a number of other surface properties. For instance, it is known that the presence of steps can modify the growth modes of surfaces. Most earlier studies on surface magnetism are based on the assumption that magnetic surface is
0039-6028/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2004.04.027
M. Tamine / Surface Science 560 (2004) 158–166
morphologically perfectly smooth (ideal bulk termination). Real films, however, have a rough surface. The atomic heights of surface atoms can differ by a few atomic spacings because of the formation of a variety of surface defects including diffusional disorder, surface reconstruction, vacancies and step changes in the height of the film [1,2]. Such surface roughness are expected to affect magnetism. Let us note that these surface defects may lead to a numerous effects including changes in the thermal properties of the film and short lifetimes for spin-waves as evidenced by large linewidths in Brillouin scattering experiments. During the last decade, an increasing interest from the theoretical and experimental points of view has been devoted to study the dynamics of disordered surfaces [3–6]. The study of spin-waves in ultra thin films has proved to be very useful, in particular, for determining magnetic anisotropy constant [7,8] using Brillouin light scattering [9,10], that provides a tool to probe these magnetic excitations in ultrathin layers. Previous studies have shown that there is a strong reflection of spin-waves at monoatomic steps and that spin-wave modes localized to the step edge occur at a single step. The effects of localized imperfections on spin-wave propagation in thin ferromagnetic films have been examined, in the case where these imperfections are assumed to be materially confined to a few lattice sites causing local changes in anisotropy and exchange fields [11]. Another work calculates, in the framework of a quasi-one-dimensional model, the reflection and transmission coefficients for a spin-wave which suffers diffraction on a step like atomic discontinuity [12]. Few years ago, it was demonstrated the existence of spin-wave localized modes occurring at a single step in the ferromagnetic films models [13] and at the monolayer Fe films on stepped W surfaces [14]. Breaking of the translational symmetry in one direction may result in new localized modes as compared to the smooth film case. The case of an antiferromagnet isolated step, has been recently examined [15]. In contrast, and to our knowledge, the case of two isolated steps has not been treated yet. In this paper we investigate the full problem arising from the absence of translational symmetry in two
159
directions due to a presence of steps. We present a precursor model system with the aim to study the surface dynamics due to an extended magnetic surface steps on the surface boundary of an insulating magnetic substrate. Such study is important because surfaces are never perfectly flat in practice and atomic steps, for example, on vicinal surfaces, are known to occur [16].
2. Model stepped surface dynamics Fig. 1a shows the geometry used in this work and the problem is formulated in terms of a Heisenberg model with two sublattices A and B. We assume, for simplicity, a simple cubic structure with lattice parameter a. The microscopic Hamiltonian used is a sum of the exchange and Zeeman terms. The stepped surface is formed by two straight steps dropped randomly, the spins moments of the steps and the insulating substrate being considered as local with no electronic effects. The site position perpendicular to the step is indexed by n, whereas the layer number is labeled by m. The two-dimensional reference cross section of the magnetic steps is taken geometrically at the plane indexed by the integer s ¼ 0. A schematic representation of the model showing the different exchange parameters is illustrated in Fig. 1b. We consider only nearest neighbors exchange interactions with translational symmetry in y-direction. Three main regions all having the same twodimensional periodicity along the direction normal to the stepped surface are identified: (i) a bulk spin sites region (m P 2) relatively removed from the steps having three-dimensional (3D) periodicity where spin-wave dispersion curves are first worked out, (ii) the surface terrace sites (m ¼ 1) consisting of an arbitrary reconstructed or relaxed layers inside which translational symmetry is lost along that direction not contained in the surface plane, and (iii) the steps region (1 6 m < 1) corresponding to spin sites belonging strictly to the steps planes inside which translational invariance symmetry remains. To analyze the spin dynamics on a given site (n; s; m) located on A or B sublattice, consider the local spin-deviation creation operators aþ n;s;m on
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M. Tamine / Surface Science 560 (2004) 158–166
z
(a)
y x
0
(n,s,m)
Spin Down (sub-lattice B)
Spin Up (sub-lattice A)
(b) m = -1
J S//1
J S//1 J c⊥
J c⊥
m=0
J S//2
J c⊥ JS//
m=1
J c⊥
J S//2
J c⊥
J S//1
J c⊥
J S//2
J c⊥
J S//
J
J m=2
J S//1
J c⊥ J c⊥
J S//2
J S//
J c⊥
J J
J -3
J S//
J S//
J J
J
-2
-1
0
1
n Fig. 1. (a) Schematic view of an antiferromagnetic stepped surface. The integers n, s, m count the spin site along the x, y and zdirections, respectively. (b) Two-dimensional cross-section of the insulating antiferromagnet model, showing the different exchange parameters. The z-axis is normal to the surface substrate, whereas the x-axis is perpendicular to the steps.
sublattice A and bþ n;s;m on a sublattice B. Each of them being described using a static and dynamic parts, as follows: þ þ iwt aþ n;s;m1 ðtÞ ¼ an;s;m1 ð0Þ þ an;s;m1 wn;s;m1 e
þ þ iwt aþ n1;s;m ðtÞ ¼ an1;s;m ð0Þ þ an1;s;m wn1;s;m e
and þ þ iwt ; bþ n1;s;m ðtÞ ¼ bn1;s;m ð0Þ þ bn1;s;m wn1;s;m e
ð1bÞ
and þ þ iwt bþ ; n;s;m1 ðtÞ ¼ bn;s;m1 ð0Þ þ bn;s;m1 wn;s;m1 e
ð1aÞ
þ and bþ where aþ n;s;m1 ð0Þ ðan1;s;m ð0ÞÞ n;s;m1 ð0Þ þ ðbn1;s;m ð0ÞÞ are the time independent part in the direction of the magnetic order along the z (x)-axis, wn;s;m1 and wn1;s;m characterize the spin fluctuating
M. Tamine / Surface Science 560 (2004) 158–166
variable from the direction of magnetic order, giving rise to spin excitations [17–19]. Introducing the phase factor expðiky aÞ and using Bloch theorem in the y-direction, we may write for aþ n;s1;m and bþ their wave-like representation, so that n;s1;m þ þ þ iky a iwt an;s1;m ðtÞ ¼ an;s1;m ð0Þ þ an;s1;m e e and þ þ iky a iwt e ; bþ n;s1;m ðtÞ ¼ bn;s1;m ð0Þ þ bn;s1;m e
ð2Þ To give a complete description of the spin fluctuations field, we need to detail the equations of motion for any site (n; s; m) in the bulk domain removed from the steps as well as on the surface terrace. These can be described as: n 1 þ þ Ean;s;m ¼ ðcosðkx a þ cos ky aÞÞ bþ nþ1;s;m þ bn1;s;m 2 o þ þ þ bþ þ b a n;s1;m n;sþ1;m n;s;m n o þ þ þ þ bn;s;mþ1 þ bn;s;m1 an;s;m
Ebþ n;s;m
þ l H a aþ ; A A n;s;m n 1 þ ðcosðkx a þ cos ky aÞÞ aþ ¼ nþ1;s;m þ an1;s;m 2 o þ þ þ aþ þ a b n;s1;m n;sþ1;m n;s;m n o þ þ þ þ an;s;mþ1 þ an;s;m1 bn;s;m þ lB HBa bþ n;s;m ;
ð3Þ with E ¼ ðhw lAðBÞ Ho Þ=JS. Ho is the applied magnetic field (lAðBÞ are the gyromagnetic ratios for a ions on A or B-sites, respectively). HA;B characterize the uniaxial anisotropy acting on A and B sublattices, respectively. Inserting Eqs. (1) and (2) in Eq. (3) yields to the bulk secular equation in form: g¼5 X
ðg1Þ
ng wn;s;m ¼ 0:
ð4Þ
g¼1
As has been pointed out elsewhere [20,21] both wn;s;m ðfÞ and w1 n;s;m ðfÞ are solutions owing to the Hermitian nature of the bulk dynamics. The frequencies of the bulk spin modes are obtained from Eq. (4), with satisfying condition jwn;s;m ðfÞj ¼ 1. For arbitrary values of wn;s;m ðfÞ, however, Eq. (4)
161
does not provide on its own the required unique solutions for the stepped surface system. For obtaining these one also needs to investigate the spin dynamics on the stepped surface region. As regards the stepped surface dynamics, it is necessary to specify in the surface boundary substrate, five elementary domains due to the broken symmetry along the x and z-directions. The primary domain consists of the spin sites belonging strictly in the first step (m ¼ 1). The second corresponds to two domains for spin sites also relatively removed from the step belonging in the second step (m ¼ 0). The third characterizes surface terrace sites (m ¼ 1) which refers the two regions for spin sites located in the left of the surface plane (n < 0; s; m ¼ 1) and right of the second step (n > 0; s; m ¼ 1). Introducing the following k k k k k k parameters: e1 ¼ JS1 =J , e2 ¼ JS2 =J , eS ¼ JS =J , k k k ? ? ? ? ? ? ? e? 1k ¼ JS1 =Jc , e2k ¼ JS2 =Jc , eSk ¼ JS =Jc ec ¼ Jc =J , the equations of motion for spin sites located in the left (n < 0; s; m ¼ 1) and on the surface substrate (n > 0; s; m ¼ 1) are hence obtained from Eq. (3) and expressed, respectively, by h n k þ þ Eaþ n;s;1 ¼ eS ð1 cosky aÞ bnþ1;s;1 þ bn1;s;1 o i þ þ þ bþ þ b a n;s;1 n;s1;1 n;sþ1;1 hn o i k þ ? þ þ eSk bn;s;0 þ eS bn;s;2 aþ n;s;1 þ lA HAa aþ n;s;1 ; h n k þ þ Ebn;s;1 ¼ eS ð1 cosky aÞ aþ nþ1;s;1 þ an1;s;1 o i þ þ þ aþ þ a b n;s1;1 n;sþ1;1 n;s;1 hn o i k þ ? þ a þ þ eSk an;s;0 þ eS an;s;2 bþ n;s;1 þ lB HB bn;s;1 : ð5aÞ h n k þ þ Eaþ n;s;1 ¼ eS ð1 cos ky aÞ bnþ1;s;1 þ bn1;s;1 o i þ þ þ bþ n;s1;1 þ bn;sþ1;1 an;s;1 h i k þ a þ þ eS bþ n;s;2 an;s;1 þ lA HA an;s;1 ; h n k þ þ Ebþ n;s;1 ¼ eS ð1 cos ky aÞ anþ1;s;1 þ an1;s;1 o i þ þ þ aþ n;s1;1 þ an;sþ1;1 bn;s;1 h i k þ a þ þ e S aþ n;s;2 bn;s;1 þ lB HB bn;s;1 :
ð5bÞ
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M. Tamine / Surface Science 560 (2004) 158–166
For the first step layer (1 < n 6 1, s; m ¼ 1), this yields: h n k þ Eaþ ¼ e ð1 cos k aÞ bþ y 1 nþ1;s;1 þ bn1;s;1 n;s;1 o i þ þ þ bþ n;s1;1 þ bn;sþ1;1 an;s;1 h i þ þ a þ þ e? 1k bn;s;0 an;s;1 þ lA HA an;s;1 ; h n k þ þ Ebþ n;s;1 ¼ e1 ð1 cos ky aÞ anþ1;s;1 þ an1;s;1 o i þ þ þ aþ n;s1;1 þ an;sþ1;1 bn;s;1 h i þ þ a þ þ e? 1k an;s;0 bn;s;1 þ lB HB bn;s;1 : ð6Þ For the second step layer (m ¼ 0), we may write the equations of spin fluctuations variables corresponding to the quarter-infinite half spaces to the left (1 < n < 1) and to the right (1 6 n 6 0) of the second semi-infinite step. These are expressed in the following forms: for spin sites belonging on the domain (1 < n < 1; s; m ¼ 0): h n k þ þ Eaþ n;s;0 ¼ e2 ð1 cos ky aÞ bnþ1;s;0 þ bn1;s;0 o i þ þ þ bþ n;s1;0 þ bn;sþ1;0 an;s;0 h n o i þ þ þ þ e? b þ b a 2k n;s;0 n;s;1 n;s;1 Ebþ n;s;0
þ lA HAa aþ n;s;0 ; h n k þ ¼ e2 ð1 cos ky aÞ aþ nþ1;s;0 þ an1;s;0 o i þ þ þ aþ n;s1;0 þ an;sþ1;0 bn;s;0 h n o i þ þ þ þ e? 2k an;s;1 þ an;s;1 bn;s;0
ð7Þ
þ lB HBa bþ n;s;0 : whereas, for spin sites belonging strictly on the second step terrace (1 6 n 6 0, s; m ¼ 0): h n k þ Eaþ ¼ e ð1 cos k aÞ bþ y 2 n;s;0 nþ1;s;0 þ bn1;s;0 o i þ þ þ bþ n;s1;0 þ bn;sþ1;0 an;s;0 h i þ þ a þ þ e? 2k bn;s;1 an;s;0 þ lA HA an;s;0 ; h n ð8Þ k þ Ebþ n;s;0 ¼ e2 ð1 cos ky aÞ an1;s;0 o i þ þ þ aþ n;s1;0 þ an;sþ1;0 bn;s;0 h i þ þ a þ þ e? 2k an;s;1 bn;s;0 þ lB HB bn;s;0 :
To solve for the spin dynamics localized on the stepped surface, we need to match the propagating bulk modes to the corresponding evanescent field near the stepped surface region using the matching procedure [15,22,23]. This is done employing equations (5), (6), (7) and (8). For spin sites somewhat removed from the first step yet on the second to the right and left of the step, the fluctuations spin variables may be represented in terms of evanescent field as follows: for spins located on the domain (1 < n 6 1, s; m ¼ 0Þ, we may put: aþ 1
nX e þnb
wfð1
f¼1
bþ 1
¼
nX e þnb
ð9Þ wfð1
f¼1
For spins sites down from the step labeled by the indices (1 6 n 6 0; s; m ¼ 0), the Eq. (8) may be written as aþ 1 6 n 6 0;s;m¼0 ¼
nX e þnb
wfð1 6 n 6 0;s;0Þ ðE; k; fÞ;
f¼1
bþ 1 6 n 6 0;s;m¼0
¼
nX e þnb
ð10Þ wfð1 6 n 6 0;s;0Þ ðE; k; fÞ:
f¼1
whereas for embedded sites surrounding the first step, the matching equations of bulk spin fluctuations variables with the localized modes on steps may be expressed in the following forms: aþ n;s;m ¼
nX e þnb
Rrfðn<0;s;mÞ CðA; wfðn;s;m>1Þ Þ
f¼1 m1 wfðn;s;m>2Þ ðE; k; fÞ;
bþ n;s;m ¼
nX e þnb
ð11aÞ
Rlfðn>0;s;mÞ CðB; wfðn;s;m>1Þ Þ
f¼1 m1 wfðn;s;m>1Þ ðE; k; fÞ
ð11bÞ
The weighting coefficients Rrfðn<0;s;mÞ and Rlfðn>0;s;mÞ depict basis unit vectors for projecting the evanescent field in respectively the quarter-infinite half spaces to the left and to the right of the two magnetic steps. CðAðBÞ; wfðn;s;m P 2Þ Þ are the normalized corresponding polarization vectors of
M. Tamine / Surface Science 560 (2004) 158–166
spin-waves. The subscript f carried out over all 3D travelling (jwfðn;s;mÞ j ¼ 1 and jw1 fðn;s;mÞ j ¼ 1) and exponential-like (jwfðn;s;mÞ j < 1 and jw1 fðn;s;mÞ j > 1) Bloch waves. Using the above matching procedure in two dimensions, it is finally possible to recast the equations for the dynamics of the spin fluctuation variables in the steps domain employing Eqs. (4), (9)–(11). This yields a square matrix form described as: ½MS ðt; h; fwf gÞjV i ¼ j0i;
ð12Þ
þ þ where jV i ¼ ½aþ 1
3. Numerical analysis and discussion Salient numerical examples are presented to demonstrate the essential features of the stepped surface spin-wave modes and the influence of bulkstepped surface exchange parameters on the localized spin modes. It is important to note that in the present calculations, due to the lack of experimental data, we neglect in our numerical results the effective magnetic field. It is quite possible that other kinds of magnetic interactions play also a role in the behavior and frequencies of the spin-waves localized on the surface and steps in which case these interactions should be considered in the spin fluctuations field calculations. Furthermore, let us quote that for any thorough surface study, we must include surface relaxation or reconstruction effects which can influence exchange constants. Consequently, we could also consider variations in exchange constants from side to side, but we will not explore this further in this work considering the degree of complexity of problem examined here. Consequently, only the k k k ? ? case where e? c ¼ 1 and e1 ¼ e2 ¼ eS ¼ e1k ¼ e2k ¼ ? eSk is investigated. k When the surface exchange interactions (JS ) k differ from those in the bulk (J ), JS may be larger
163
or smaller than J , the behavior of the surface magnetization with changing roughness is much more complex and interesting. The reduced atomic coordination at a surface produces a narrower band width and hence a larger magnetic moments k [24,25], favoring JS > J . On the other hand, the surface lattice spacing can be larger than the bulk lattice spacing, leading to a weaker spin–spin k interaction and favoring JS < J . We therefore consider both possibilities and the corresponding dispersion curves with the propagating and evanescent modes versus the wave-vector ky are shown in Fig. 2. Let us mention that in Fig. 2a and b, the case where the exchange parameters for surfacesteps region are weaker than in the bulk domain is considered, whereas in Fig. 2d and e the opposite case is assumed. The Fig. 2c depicts the free stepped surface configuration. Let us consider, firstly, the case of weak bulkk stepped surface exchange parameter (e1 ¼ 0:5) as illustrated in Fig. 2a. Three acoustic branches will be truncated at some values of kc (kc1 ¼ 0:44, kc2 ¼ 0:52, kc3 ¼ 0:87) corresponding to EB ðkc Þ ¼ ES ðkc Þ. One optic mode (O1 ) occurs tangentially with the bulk continuum with energy range 662 K 6 ES 6 706 K. All branches are doubly degenerate. Two localized modes (A1 and A2 ) correspond to those occurring along the two steps edges, whereas the third one (A3 ) characterizes the surface terrace mode. When a weak enough bulkstepped surface exchange coupling is considered k (e1 ¼ 0:8), the two truncated acoustic modes (A1 and A2 ) occur with energies ESA1 ðky Þ ¼ ESA2 ðky Þ and the energy value for the third acoustic mode remains constant with ESA3 ðky Þ 355 K as illustrated in Fig. 2b. k For ‘‘free stepped surface’’ case (e1 ¼ 1), the surface spin-waves modes are seen to split into two branches for small enough wave vector and they become approximately degenerate at larger wave vector. Since the splitting of the acoustic spin-wave surface branches is most apparent for small wave vector, these surface effects in films can in principle be studied experimentally by light scattering techniques. Furthermore, the two acoustic modes localized on the two steps are practically superposed and occur in the energy range 215 K 6 ES 6 222 K as illustrated in Fig. 2c.
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Fig. 2. Numerical solutions for the dispersion relations for bulk and stepped surface spin modes for several typical choices of the k model parameters e1 . The plot is given against ky a. The region containing the bulk modes is shown shaded, whereas the set of curves (dotted lines) represent the surface and steps modes.
The strong bulk-stepped surface exchange k coupling corresponding to e1 ¼ 1:4 and 1.8 shows the presence of optical modes as described in Fig. 2d and e, respectively. No acoustic wave branches
k
exist for e1 P 1. In Fig. 2d, four modes appear, both occurring above the bulk spin-wave region so-called ‘‘optical modes’’. Hence, there is an energy range, approximately 188 K 6 ES 6 319 K for
M. Tamine / Surface Science 560 (2004) 158–166
which there are localized interface modes as well as surface and step modes. Two modes (O2 , O3 ) will 0 be truncated at some values of kc0 (kc2 ¼ 1:7 and 0 kc3 0:9). One complete mode (O4 ) occurs origin of the first Brillouin zone and become the magnetic cell with ES ðky ! 0Þ 242 K. Fig. 2e displays the surface and steps spin-waves modes. Again the shaded area corresponds to the region occupied by the bulk modes. The Fig. 2e is presented emphasizing the distribution of the frequencies of the localized modes and the effects of changes in the surface-bulk exchange parameters on the surface-steps spin-wave branches that can occur. For k sufficiently small values of e1 , the assumed antiferromagnetic ground state can become unstable, corresponding to a reorientation of spins near the surfaces. Such an effect has been discussed by Mills in certain anisotropic ferromagnets [26].
165
ities, the growth of magnetic substrate and surface optical properties. We have to remark that our conclusions are based on a Heisenberg Hamiltonian model, with no inclusion of any reconstruction, pinning, step fluctuations, step-step interactions or magnetic anisotropy [27–29]. Our calculation is also restricted to a well defined model surface and to a reasonable sampling of the Brillouin zone. We believe, however, that we have uncovered some important points and that we have determined qualitatively the salient features of the spin-wave spectrum of stepped magnetic surfaces. Finally, the analytic procedure developed here for calculating the energies and intensities of the bulk and localized spin-wave modes in the vicinity of two steps can be applied to other low dimensional structures and we intend to report on that elsewhere. Brillouin light scattering would provide a suitable probe of these effects.
4. Conclusion Matching procedure is used in this work to calculate the spin-waves modes localized on surface step for the insulating antiferromagnet model. We emphasize that the present model is simple insofar that it considers only magnetic exchange interactions between the ordered spins, yielding hence the exchange-dominated step localized spinwaves. Following the analytical procedure developed in this paper, it is possible to introduce other forms of magnetic interactions in the present model in a relatively direct manner. The theoretical formalism described for the present case can readily be generalized to other surface steps problems concerning in particular the imperfect antiferromagnets films such as NiO or the perovskite structure KNiF3 . In both of these materials the magnetic Ni ions lie on a simple cubic lattice, and the exchange interactions are very strong making the use of Heisenberg type Hamiltonian appropriate. In the other hand, the frequencies of these localized modes may provide information concerning the local magnetic anisotropy and exchange interactions in the neighborhood of such model, and will contribute to understand more fully the role that may play in surface phenomena such as surface- steps instabil-
Acknowledgements The author is very indebted to Dr. J.M. Greneche and Dr. L. Berger (UMR CNRS 6087––Universite du Maine (France)) for their hospitality, encouragement and stimulating discussions and acknowledges a very useful correspondence with the referees and their remarks.
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Spin fluctuations dynamics on the insulating antiferromagnetic stepped surface model M. Tamine
*
Laboratoire de Physique et Chimie Quantique, Institut de Physique, Universite de Tizi-Ouzou, BP 17 RP-15000 Tizi-Ouzou, Algeria Received 29 October 2003; accepted for publication 13 April 2004 Available online 6 May 2004
Abstract We present a numerical method to calculate the spin fluctuations dynamics on the stepped surface. The model discussed here consists in an extended antiferromagnetic surface steps on the surface boundary of an insulating antiferromagnetic substrate. The stepped surface is formed by two straight steps dropped randomly and the spins moments of the steps and the substrate are considered as local with no electronic effects. The calculations concern in particular the energies of localized spin-wave modes near the stepped surface and employ the matching procedure in the randomphase approximation and mean field approximation. Only nearest neighbors exchange interactions are considered between the spins in the model. The theoretical formalism presented here solves for the three dimensional evanescent crystal spin field in the bulk and the surface domains around the steps, that arises owing to the breakdown of magnetic translation symmetry of the system. The results are used to calculate the spin modes energies associated with the steps and surface terraces. We show the presence of acoustic and optical localized spin-wave modes propagating along the surface and the steps, their spin fluctuations fields are also described as evanescent in the plane normal to the stepped surface and depend on the nature of the exchange parameters near the steps. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Stepped single crystal surfaces; Magnetic phenomena (cyclotron resonance, phase transitions, etc.)
1. Introduction The study of film growth has been increasingly characterized by the application of surface science methods to understand growth at the atomic level. Work in this field has been motivated by the ever more stringent requirements on the quality of thin films needed for developing advanced microelectronic, optical, and magnetic devices, as well as the
*
Tel.: +213-26-21-48-48; fax: +213-26-22-37-66. E-mail address: [email protected] (M. Tamine).
thrust toward nanometer-scale structures. As device miniaturization reaches submicrometer and nanometer length regimes, atomic level control of the fabrication processes for both novel materials and new devices have become of great importance. The presence of nanostructures such as random steps, kinks and other defects on crystal surfaces is important to their equilibrium topography as well as to a number of other surface properties. For instance, it is known that the presence of steps can modify the growth modes of surfaces. Most earlier studies on surface magnetism are based on the assumption that magnetic surface is
0039-6028/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2004.04.027
M. Tamine / Surface Science 560 (2004) 158–166
morphologically perfectly smooth (ideal bulk termination). Real films, however, have a rough surface. The atomic heights of surface atoms can differ by a few atomic spacings because of the formation of a variety of surface defects including diffusional disorder, surface reconstruction, vacancies and step changes in the height of the film [1,2]. Such surface roughness are expected to affect magnetism. Let us note that these surface defects may lead to a numerous effects including changes in the thermal properties of the film and short lifetimes for spin-waves as evidenced by large linewidths in Brillouin scattering experiments. During the last decade, an increasing interest from the theoretical and experimental points of view has been devoted to study the dynamics of disordered surfaces [3–6]. The study of spin-waves in ultra thin films has proved to be very useful, in particular, for determining magnetic anisotropy constant [7,8] using Brillouin light scattering [9,10], that provides a tool to probe these magnetic excitations in ultrathin layers. Previous studies have shown that there is a strong reflection of spin-waves at monoatomic steps and that spin-wave modes localized to the step edge occur at a single step. The effects of localized imperfections on spin-wave propagation in thin ferromagnetic films have been examined, in the case where these imperfections are assumed to be materially confined to a few lattice sites causing local changes in anisotropy and exchange fields [11]. Another work calculates, in the framework of a quasi-one-dimensional model, the reflection and transmission coefficients for a spin-wave which suffers diffraction on a step like atomic discontinuity [12]. Few years ago, it was demonstrated the existence of spin-wave localized modes occurring at a single step in the ferromagnetic films models [13] and at the monolayer Fe films on stepped W surfaces [14]. Breaking of the translational symmetry in one direction may result in new localized modes as compared to the smooth film case. The case of an antiferromagnet isolated step, has been recently examined [15]. In contrast, and to our knowledge, the case of two isolated steps has not been treated yet. In this paper we investigate the full problem arising from the absence of translational symmetry in two
159
directions due to a presence of steps. We present a precursor model system with the aim to study the surface dynamics due to an extended magnetic surface steps on the surface boundary of an insulating magnetic substrate. Such study is important because surfaces are never perfectly flat in practice and atomic steps, for example, on vicinal surfaces, are known to occur [16].
2. Model stepped surface dynamics Fig. 1a shows the geometry used in this work and the problem is formulated in terms of a Heisenberg model with two sublattices A and B. We assume, for simplicity, a simple cubic structure with lattice parameter a. The microscopic Hamiltonian used is a sum of the exchange and Zeeman terms. The stepped surface is formed by two straight steps dropped randomly, the spins moments of the steps and the insulating substrate being considered as local with no electronic effects. The site position perpendicular to the step is indexed by n, whereas the layer number is labeled by m. The two-dimensional reference cross section of the magnetic steps is taken geometrically at the plane indexed by the integer s ¼ 0. A schematic representation of the model showing the different exchange parameters is illustrated in Fig. 1b. We consider only nearest neighbors exchange interactions with translational symmetry in y-direction. Three main regions all having the same twodimensional periodicity along the direction normal to the stepped surface are identified: (i) a bulk spin sites region (m P 2) relatively removed from the steps having three-dimensional (3D) periodicity where spin-wave dispersion curves are first worked out, (ii) the surface terrace sites (m ¼ 1) consisting of an arbitrary reconstructed or relaxed layers inside which translational symmetry is lost along that direction not contained in the surface plane, and (iii) the steps region (1 6 m < 1) corresponding to spin sites belonging strictly to the steps planes inside which translational invariance symmetry remains. To analyze the spin dynamics on a given site (n; s; m) located on A or B sublattice, consider the local spin-deviation creation operators aþ n;s;m on
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z
(a)
y x
0
(n,s,m)
Spin Down (sub-lattice B)
Spin Up (sub-lattice A)
(b) m = -1
J S//1
J S//1 J c⊥
J c⊥
m=0
J S//2
J c⊥ JS//
m=1
J c⊥
J S//2
J c⊥
J S//1
J c⊥
J S//2
J c⊥
J S//
J
J m=2
J S//1
J c⊥ J c⊥
J S//2
J S//
J c⊥
J J
J -3
J S//
J S//
J J
J
-2
-1
0
1
n Fig. 1. (a) Schematic view of an antiferromagnetic stepped surface. The integers n, s, m count the spin site along the x, y and zdirections, respectively. (b) Two-dimensional cross-section of the insulating antiferromagnet model, showing the different exchange parameters. The z-axis is normal to the surface substrate, whereas the x-axis is perpendicular to the steps.
sublattice A and bþ n;s;m on a sublattice B. Each of them being described using a static and dynamic parts, as follows: þ þ iwt aþ n;s;m1 ðtÞ ¼ an;s;m1 ð0Þ þ an;s;m1 wn;s;m1 e
þ þ iwt aþ n1;s;m ðtÞ ¼ an1;s;m ð0Þ þ an1;s;m wn1;s;m e
and þ þ iwt ; bþ n1;s;m ðtÞ ¼ bn1;s;m ð0Þ þ bn1;s;m wn1;s;m e
ð1bÞ
and þ þ iwt bþ ; n;s;m1 ðtÞ ¼ bn;s;m1 ð0Þ þ bn;s;m1 wn;s;m1 e
ð1aÞ
þ and bþ where aþ n;s;m1 ð0Þ ðan1;s;m ð0ÞÞ n;s;m1 ð0Þ þ ðbn1;s;m ð0ÞÞ are the time independent part in the direction of the magnetic order along the z (x)-axis, wn;s;m1 and wn1;s;m characterize the spin fluctuating
M. Tamine / Surface Science 560 (2004) 158–166
variable from the direction of magnetic order, giving rise to spin excitations [17–19]. Introducing the phase factor expðiky aÞ and using Bloch theorem in the y-direction, we may write for aþ n;s1;m and bþ their wave-like representation, so that n;s1;m þ þ þ iky a iwt an;s1;m ðtÞ ¼ an;s1;m ð0Þ þ an;s1;m e e and þ þ iky a iwt e ; bþ n;s1;m ðtÞ ¼ bn;s1;m ð0Þ þ bn;s1;m e
ð2Þ To give a complete description of the spin fluctuations field, we need to detail the equations of motion for any site (n; s; m) in the bulk domain removed from the steps as well as on the surface terrace. These can be described as: n 1 þ þ Ean;s;m ¼ ðcosðkx a þ cos ky aÞÞ bþ nþ1;s;m þ bn1;s;m 2 o þ þ þ bþ þ b a n;s1;m n;sþ1;m n;s;m n o þ þ þ þ bn;s;mþ1 þ bn;s;m1 an;s;m
Ebþ n;s;m
þ l H a aþ ; A A n;s;m n 1 þ ðcosðkx a þ cos ky aÞÞ aþ ¼ nþ1;s;m þ an1;s;m 2 o þ þ þ aþ þ a b n;s1;m n;sþ1;m n;s;m n o þ þ þ þ an;s;mþ1 þ an;s;m1 bn;s;m þ lB HBa bþ n;s;m ;
ð3Þ with E ¼ ðhw lAðBÞ Ho Þ=JS. Ho is the applied magnetic field (lAðBÞ are the gyromagnetic ratios for a ions on A or B-sites, respectively). HA;B characterize the uniaxial anisotropy acting on A and B sublattices, respectively. Inserting Eqs. (1) and (2) in Eq. (3) yields to the bulk secular equation in form: g¼5 X
ðg1Þ
ng wn;s;m ¼ 0:
ð4Þ
g¼1
As has been pointed out elsewhere [20,21] both wn;s;m ðfÞ and w1 n;s;m ðfÞ are solutions owing to the Hermitian nature of the bulk dynamics. The frequencies of the bulk spin modes are obtained from Eq. (4), with satisfying condition jwn;s;m ðfÞj ¼ 1. For arbitrary values of wn;s;m ðfÞ, however, Eq. (4)
161
does not provide on its own the required unique solutions for the stepped surface system. For obtaining these one also needs to investigate the spin dynamics on the stepped surface region. As regards the stepped surface dynamics, it is necessary to specify in the surface boundary substrate, five elementary domains due to the broken symmetry along the x and z-directions. The primary domain consists of the spin sites belonging strictly in the first step (m ¼ 1). The second corresponds to two domains for spin sites also relatively removed from the step belonging in the second step (m ¼ 0). The third characterizes surface terrace sites (m ¼ 1) which refers the two regions for spin sites located in the left of the surface plane (n < 0; s; m ¼ 1) and right of the second step (n > 0; s; m ¼ 1). Introducing the following k k k k k k parameters: e1 ¼ JS1 =J , e2 ¼ JS2 =J , eS ¼ JS =J , k k k ? ? ? ? ? ? ? e? 1k ¼ JS1 =Jc , e2k ¼ JS2 =Jc , eSk ¼ JS =Jc ec ¼ Jc =J , the equations of motion for spin sites located in the left (n < 0; s; m ¼ 1) and on the surface substrate (n > 0; s; m ¼ 1) are hence obtained from Eq. (3) and expressed, respectively, by h n k þ þ Eaþ n;s;1 ¼ eS ð1 cosky aÞ bnþ1;s;1 þ bn1;s;1 o i þ þ þ bþ þ b a n;s;1 n;s1;1 n;sþ1;1 hn o i k þ ? þ þ eSk bn;s;0 þ eS bn;s;2 aþ n;s;1 þ lA HAa aþ n;s;1 ; h n k þ þ Ebn;s;1 ¼ eS ð1 cosky aÞ aþ nþ1;s;1 þ an1;s;1 o i þ þ þ aþ þ a b n;s1;1 n;sþ1;1 n;s;1 hn o i k þ ? þ a þ þ eSk an;s;0 þ eS an;s;2 bþ n;s;1 þ lB HB bn;s;1 : ð5aÞ h n k þ þ Eaþ n;s;1 ¼ eS ð1 cos ky aÞ bnþ1;s;1 þ bn1;s;1 o i þ þ þ bþ n;s1;1 þ bn;sþ1;1 an;s;1 h i k þ a þ þ eS bþ n;s;2 an;s;1 þ lA HA an;s;1 ; h n k þ þ Ebþ n;s;1 ¼ eS ð1 cos ky aÞ anþ1;s;1 þ an1;s;1 o i þ þ þ aþ n;s1;1 þ an;sþ1;1 bn;s;1 h i k þ a þ þ e S aþ n;s;2 bn;s;1 þ lB HB bn;s;1 :
ð5bÞ
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M. Tamine / Surface Science 560 (2004) 158–166
For the first step layer (1 < n 6 1, s; m ¼ 1), this yields: h n k þ Eaþ ¼ e ð1 cos k aÞ bþ y 1 nþ1;s;1 þ bn1;s;1 n;s;1 o i þ þ þ bþ n;s1;1 þ bn;sþ1;1 an;s;1 h i þ þ a þ þ e? 1k bn;s;0 an;s;1 þ lA HA an;s;1 ; h n k þ þ Ebþ n;s;1 ¼ e1 ð1 cos ky aÞ anþ1;s;1 þ an1;s;1 o i þ þ þ aþ n;s1;1 þ an;sþ1;1 bn;s;1 h i þ þ a þ þ e? 1k an;s;0 bn;s;1 þ lB HB bn;s;1 : ð6Þ For the second step layer (m ¼ 0), we may write the equations of spin fluctuations variables corresponding to the quarter-infinite half spaces to the left (1 < n < 1) and to the right (1 6 n 6 0) of the second semi-infinite step. These are expressed in the following forms: for spin sites belonging on the domain (1 < n < 1; s; m ¼ 0): h n k þ þ Eaþ n;s;0 ¼ e2 ð1 cos ky aÞ bnþ1;s;0 þ bn1;s;0 o i þ þ þ bþ n;s1;0 þ bn;sþ1;0 an;s;0 h n o i þ þ þ þ e? b þ b a 2k n;s;0 n;s;1 n;s;1 Ebþ n;s;0
þ lA HAa aþ n;s;0 ; h n k þ ¼ e2 ð1 cos ky aÞ aþ nþ1;s;0 þ an1;s;0 o i þ þ þ aþ n;s1;0 þ an;sþ1;0 bn;s;0 h n o i þ þ þ þ e? 2k an;s;1 þ an;s;1 bn;s;0
ð7Þ
þ lB HBa bþ n;s;0 : whereas, for spin sites belonging strictly on the second step terrace (1 6 n 6 0, s; m ¼ 0): h n k þ Eaþ ¼ e ð1 cos k aÞ bþ y 2 n;s;0 nþ1;s;0 þ bn1;s;0 o i þ þ þ bþ n;s1;0 þ bn;sþ1;0 an;s;0 h i þ þ a þ þ e? 2k bn;s;1 an;s;0 þ lA HA an;s;0 ; h n ð8Þ k þ Ebþ n;s;0 ¼ e2 ð1 cos ky aÞ an1;s;0 o i þ þ þ aþ n;s1;0 þ an;sþ1;0 bn;s;0 h i þ þ a þ þ e? 2k an;s;1 bn;s;0 þ lB HB bn;s;0 :
To solve for the spin dynamics localized on the stepped surface, we need to match the propagating bulk modes to the corresponding evanescent field near the stepped surface region using the matching procedure [15,22,23]. This is done employing equations (5), (6), (7) and (8). For spin sites somewhat removed from the first step yet on the second to the right and left of the step, the fluctuations spin variables may be represented in terms of evanescent field as follows: for spins located on the domain (1 < n 6 1, s; m ¼ 0Þ, we may put: aþ 1
nX e þnb
wfð1
f¼1
bþ 1
¼
nX e þnb
ð9Þ wfð1
f¼1
For spins sites down from the step labeled by the indices (1 6 n 6 0; s; m ¼ 0), the Eq. (8) may be written as aþ 1 6 n 6 0;s;m¼0 ¼
nX e þnb
wfð1 6 n 6 0;s;0Þ ðE; k; fÞ;
f¼1
bþ 1 6 n 6 0;s;m¼0
¼
nX e þnb
ð10Þ wfð1 6 n 6 0;s;0Þ ðE; k; fÞ:
f¼1
whereas for embedded sites surrounding the first step, the matching equations of bulk spin fluctuations variables with the localized modes on steps may be expressed in the following forms: aþ n;s;m ¼
nX e þnb
Rrfðn<0;s;mÞ CðA; wfðn;s;m>1Þ Þ
f¼1 m1 wfðn;s;m>2Þ ðE; k; fÞ;
bþ n;s;m ¼
nX e þnb
ð11aÞ
Rlfðn>0;s;mÞ CðB; wfðn;s;m>1Þ Þ
f¼1 m1 wfðn;s;m>1Þ ðE; k; fÞ
ð11bÞ
The weighting coefficients Rrfðn<0;s;mÞ and Rlfðn>0;s;mÞ depict basis unit vectors for projecting the evanescent field in respectively the quarter-infinite half spaces to the left and to the right of the two magnetic steps. CðAðBÞ; wfðn;s;m P 2Þ Þ are the normalized corresponding polarization vectors of
M. Tamine / Surface Science 560 (2004) 158–166
spin-waves. The subscript f carried out over all 3D travelling (jwfðn;s;mÞ j ¼ 1 and jw1 fðn;s;mÞ j ¼ 1) and exponential-like (jwfðn;s;mÞ j < 1 and jw1 fðn;s;mÞ j > 1) Bloch waves. Using the above matching procedure in two dimensions, it is finally possible to recast the equations for the dynamics of the spin fluctuation variables in the steps domain employing Eqs. (4), (9)–(11). This yields a square matrix form described as: ½MS ðt; h; fwf gÞjV i ¼ j0i;
ð12Þ
þ þ where jV i ¼ ½aþ 1
3. Numerical analysis and discussion Salient numerical examples are presented to demonstrate the essential features of the stepped surface spin-wave modes and the influence of bulkstepped surface exchange parameters on the localized spin modes. It is important to note that in the present calculations, due to the lack of experimental data, we neglect in our numerical results the effective magnetic field. It is quite possible that other kinds of magnetic interactions play also a role in the behavior and frequencies of the spin-waves localized on the surface and steps in which case these interactions should be considered in the spin fluctuations field calculations. Furthermore, let us quote that for any thorough surface study, we must include surface relaxation or reconstruction effects which can influence exchange constants. Consequently, we could also consider variations in exchange constants from side to side, but we will not explore this further in this work considering the degree of complexity of problem examined here. Consequently, only the k k k ? ? case where e? c ¼ 1 and e1 ¼ e2 ¼ eS ¼ e1k ¼ e2k ¼ ? eSk is investigated. k When the surface exchange interactions (JS ) k differ from those in the bulk (J ), JS may be larger
163
or smaller than J , the behavior of the surface magnetization with changing roughness is much more complex and interesting. The reduced atomic coordination at a surface produces a narrower band width and hence a larger magnetic moments k [24,25], favoring JS > J . On the other hand, the surface lattice spacing can be larger than the bulk lattice spacing, leading to a weaker spin–spin k interaction and favoring JS < J . We therefore consider both possibilities and the corresponding dispersion curves with the propagating and evanescent modes versus the wave-vector ky are shown in Fig. 2. Let us mention that in Fig. 2a and b, the case where the exchange parameters for surfacesteps region are weaker than in the bulk domain is considered, whereas in Fig. 2d and e the opposite case is assumed. The Fig. 2c depicts the free stepped surface configuration. Let us consider, firstly, the case of weak bulkk stepped surface exchange parameter (e1 ¼ 0:5) as illustrated in Fig. 2a. Three acoustic branches will be truncated at some values of kc (kc1 ¼ 0:44, kc2 ¼ 0:52, kc3 ¼ 0:87) corresponding to EB ðkc Þ ¼ ES ðkc Þ. One optic mode (O1 ) occurs tangentially with the bulk continuum with energy range 662 K 6 ES 6 706 K. All branches are doubly degenerate. Two localized modes (A1 and A2 ) correspond to those occurring along the two steps edges, whereas the third one (A3 ) characterizes the surface terrace mode. When a weak enough bulkstepped surface exchange coupling is considered k (e1 ¼ 0:8), the two truncated acoustic modes (A1 and A2 ) occur with energies ESA1 ðky Þ ¼ ESA2 ðky Þ and the energy value for the third acoustic mode remains constant with ESA3 ðky Þ 355 K as illustrated in Fig. 2b. k For ‘‘free stepped surface’’ case (e1 ¼ 1), the surface spin-waves modes are seen to split into two branches for small enough wave vector and they become approximately degenerate at larger wave vector. Since the splitting of the acoustic spin-wave surface branches is most apparent for small wave vector, these surface effects in films can in principle be studied experimentally by light scattering techniques. Furthermore, the two acoustic modes localized on the two steps are practically superposed and occur in the energy range 215 K 6 ES 6 222 K as illustrated in Fig. 2c.
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Fig. 2. Numerical solutions for the dispersion relations for bulk and stepped surface spin modes for several typical choices of the k model parameters e1 . The plot is given against ky a. The region containing the bulk modes is shown shaded, whereas the set of curves (dotted lines) represent the surface and steps modes.
The strong bulk-stepped surface exchange k coupling corresponding to e1 ¼ 1:4 and 1.8 shows the presence of optical modes as described in Fig. 2d and e, respectively. No acoustic wave branches
k
exist for e1 P 1. In Fig. 2d, four modes appear, both occurring above the bulk spin-wave region so-called ‘‘optical modes’’. Hence, there is an energy range, approximately 188 K 6 ES 6 319 K for
M. Tamine / Surface Science 560 (2004) 158–166
which there are localized interface modes as well as surface and step modes. Two modes (O2 , O3 ) will 0 be truncated at some values of kc0 (kc2 ¼ 1:7 and 0 kc3 0:9). One complete mode (O4 ) occurs origin of the first Brillouin zone and become the magnetic cell with ES ðky ! 0Þ 242 K. Fig. 2e displays the surface and steps spin-waves modes. Again the shaded area corresponds to the region occupied by the bulk modes. The Fig. 2e is presented emphasizing the distribution of the frequencies of the localized modes and the effects of changes in the surface-bulk exchange parameters on the surface-steps spin-wave branches that can occur. For k sufficiently small values of e1 , the assumed antiferromagnetic ground state can become unstable, corresponding to a reorientation of spins near the surfaces. Such an effect has been discussed by Mills in certain anisotropic ferromagnets [26].
165
ities, the growth of magnetic substrate and surface optical properties. We have to remark that our conclusions are based on a Heisenberg Hamiltonian model, with no inclusion of any reconstruction, pinning, step fluctuations, step-step interactions or magnetic anisotropy [27–29]. Our calculation is also restricted to a well defined model surface and to a reasonable sampling of the Brillouin zone. We believe, however, that we have uncovered some important points and that we have determined qualitatively the salient features of the spin-wave spectrum of stepped magnetic surfaces. Finally, the analytic procedure developed here for calculating the energies and intensities of the bulk and localized spin-wave modes in the vicinity of two steps can be applied to other low dimensional structures and we intend to report on that elsewhere. Brillouin light scattering would provide a suitable probe of these effects.
4. Conclusion Matching procedure is used in this work to calculate the spin-waves modes localized on surface step for the insulating antiferromagnet model. We emphasize that the present model is simple insofar that it considers only magnetic exchange interactions between the ordered spins, yielding hence the exchange-dominated step localized spinwaves. Following the analytical procedure developed in this paper, it is possible to introduce other forms of magnetic interactions in the present model in a relatively direct manner. The theoretical formalism described for the present case can readily be generalized to other surface steps problems concerning in particular the imperfect antiferromagnets films such as NiO or the perovskite structure KNiF3 . In both of these materials the magnetic Ni ions lie on a simple cubic lattice, and the exchange interactions are very strong making the use of Heisenberg type Hamiltonian appropriate. In the other hand, the frequencies of these localized modes may provide information concerning the local magnetic anisotropy and exchange interactions in the neighborhood of such model, and will contribute to understand more fully the role that may play in surface phenomena such as surface- steps instabil-
Acknowledgements The author is very indebted to Dr. J.M. Greneche and Dr. L. Berger (UMR CNRS 6087––Universite du Maine (France)) for their hospitality, encouragement and stimulating discussions and acknowledges a very useful correspondence with the referees and their remarks.
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