Scaling of the Interface Roughness in Fe-Cr Superlattices: Self-Affine versus Non-Self-Affine J. Santamaria, 1, * M. E. Go ´mez, 1,† J. L. Vicent, 2 K M. Krishnan, 3 and Ivan K. Schuller 1 1 Department of Physics, University of California–San Diego, La Jolla, California 92093-0319 2 Departamento de Fı ´sica de Materiales, Universidad Complutense, 28040 Madrid, Spain 3 Department of Materials Sciences, University of Washington, Seattle, Washington 98195-2120 (Received 15 April 2002; published 16 October 2002) We have analyzed kinetic roughening in Fe-Cr superlattices by energy-filtered transmission electron microscopy. The direct access to individual interfaces provides both static and dynamic roughness exponents. We find an anomalous non-self-affine scaling of the interface roughness with a time dependent local roughness at short length scales. While the deposition conditions affect strongly the long-range dynamics, the anomalous short-range exponent remains unchanged. The different short- and long-range dynamics outline the importance of long-range interactions in kinetic roughening. DOI: 10.1103/PhysRevLett.89.190601 PACS numbers: 68.55.Jk, 05.40.-a, 81.10.Aj, 81.15.Cd The analysis of surface morphology of thin films is a subject of intense activity. Despite the complexity of the growth process, theoretical and experimental studies have shown that kinetic roughening frequently obeys relatively simple scaling laws (self-affine roughness) [1]. In super- lattices, with many (buried) interfaces, the evolution of roughness is considerably more complicated and remains unexplored. Unfortunately, determining quantitively the interface structure from diffuse x-ray scattering [2–5] requires assumptions on the nature of the roughness (as- sumed generally self-affine [6]) and the existence of a single lateral correlation length for the various interfaces. However, a direct proof of these assumptions is lacking. Here by imaging directly the interfaces we determine simultaneously static and dynamic critical exponents. We provide evidence for non-self-affine anomalous rough- ness in superlattices with a power law dependent local interfacial slope. A rough surface can be characterized by fluctuations of surface height around its mean value, zx; t hx; t hhx; ti, at a position x and a time t. The local surface width is defined as l; t  hz 2 x; ti 1=2 , where the brackets denote an average over a lateral length scale, l, in a system of size Ll<L. Alternatively, roughness correlations over a distance l can be obtained from the height-height correlation function gl; t  hhx  l; t hx; t 2 i 1=2 . Both pictures are equivalent since numerical simulations show gl; t/ l; t [1]. For many surfaces, the surface width scales according to the Family-Vicsek [7] scaling ansatz, l; t t  fl=t 1=z ; (1) where the scaling function fu behaves as u  if u  1 and is constant for u  1. The local surface width thus increases as l; t/ l  until it saturates at a value  s t, for l larger than an in-plane cutoff length, , the lateral roughness correlation length. The saturation surface width and lateral roughness correlation length grow with time (thickness) following power laws with different dynamic exponents,  s t/ t  and t/ t 1=z , and the relation z  = holds [1]. In self-affine surfaces, local [l; t/ l  ] and global [L; t L  ] interface width scale with the same roughness exponent  [1], and for short length scales l; t is independent of deposi- tion time. Recently, growth models [8–10] with different rough- ness exponents at short ( loc ) and long () length scales have been proposed which correspond to a non-self-affine scaling ansatz [11,12]. The local interface width is no longer time independent at short distances; it follows a power law with a new exponent  loc , while at long lengths it scales as t  . The lateral roughness correlation length still scales as t/ t 1=z , but the exponents obey  loc     loc z and  loc    loc z [11]. A nonstationary local interface width of the form l; t l   lnt p used in models incorporating surface diffusion [13,14] was observed in thin films with a linear diffusion limited growth [15,16]. However, the power law dependence of the local interface width at short length scales has been observed only very recently in Cu [17] and in polymer thin films [18]. For superlattices the interfacial width and in-plane correlation length may increase from layer to layer, although their evolution is not clear a priori, due to the presence of more than one constituent. In other words, the extension of scaling concepts from single surfaces to superlattices is not straightforward. So far, interface characterization used mainly diffuse x-ray reflectivity (XRR), which provides averaged information over all interfaces. Extracting quantitative information relies on assumptions concerning the nature of the roughness; i.e., (a) the individual interfaces are self-affine and (b) a single correlation length is assumed for all interfaces (correlated roughness). X-ray spectra are simulated using a single height-height correlation function assum- ing hh n x  l; t h n x; t 2 i 2 2 f1  expl= 2 g which implies a single (averaged) roughness cutoff VOLUME 89, NUMBER 19 PHYSICAL REVIEW LETTERS 4NOVEMBER 2002 190601-1 0031-9007= 02=89(19)=190601(4)$20.00  2002 The American Physical Society 190601-1