VOLUME 83, NUMBER 19 PHYSICAL REVIEW LETTERS 8NOVEMBER 1999 Evolution of a Surface-Roughness Spectrum Caused by Stress in Nanometer-Scale Chemical Etching K.-S. Kim, J. A. Hurtado, and H. Tan Division of Engineering, Brown University, Providence, Rhode Island 02912 (Received 10 June 1999) It is reported that a flat free surface of a stressed solid is configurationally unstable under chemical etching and the surface roughness grows with different rates for different spatial frequencies. The theory described in this Letter predicts that with a shallow chemical etching the roughness with spatial frequency below a critical value grows while the roughness of higher frequency decays. The theory was verified via an atomic force microscope experiment with aluminum. This study provides a simple experimental method to measure stress in metals and ceramics. PACS numbers: 68.35.Ct, 46.80. + j, 68.45. – v, 81.65.Cf The phenomenon of stress-induced roughening of solid surfaces has recently gained great attention [1], especially because of its relevance in the morphological instability of flat surfaces and island formation during heteroepitaxial growth of thin films. There are two distinctive atomic- level mechanisms which allow variation of the surface configuration. One is the case of direct addition and re- moval of atoms on the surface from and to the surround- ings of the surface. Chemical etching belongs to this case. The other is the case in which atoms move along the sur- face itself. In the latter case a gradient in the chemi- cal potential results in diffusive mass transport along the surface [1]. This phenomenon is particularly relevant for processes characterized by high stress, high temperature, and small-size scales, such as during growth of heteroepi- taxial thin films. The general observation is that a flat surface under stress is unstable against diffusional pertur- bations with a sufficiently large wavelength l (or small wave number v  2p l); there exists a critical wave number v cr below which diffusional perturbations grow and above which they decay. A similar behavior is ex- pected in chemical etching processes. The stress-induced surface roughening in a shallow chemical etching is ana- lyzed, in this Letter, for isotropic homogeneous solids. The analysis shows that the stress roughening in this case also has the same critical wave number v cr ; however, the frequency-spectrum dependence of the growth (or decay) rate of the roughness is different for the two cases. This analysis provides a simple stress dependent function of the growth rate of the surface roughness for general two- dimensional frequency spectrum. An atomic force mi- croscope (AFM) has been used for direct measurements of the surface topography of a stressed aluminum sample between two etching steps. The surface topographies are then processed to reveal the frequency dependence of stress-induced roughening in chemical etching. This Let- ter provides, for the first time, direct experimental verifi- cation of this frequency dependence. (a) Etching kinetics and surface evolution. —The ge- ometry of chemical etching is depicted in Fig. 1. An elastic solid subject to a uniform biaxial stress field s is covered by an etching solution. The solid is considered to be isotropic and homogeneous, with shear modulus m and Poisson’s ratio n, and always in mechanical equilibrium. The tendency of the interface to change the shape of its reference configuration is represented by the chemical po- tential [2,3] along the surface x s  Vg 0 1 gks 1 ws, where s indicates a point on the surface, V is the atomic volume, g 0 is the electrochemical potential of the etching reaction, g is the interface energy density of the solid-etchant interface, k is the curvature (positive in sign for a concave surface), and w is the strain energy den- sity along the solid surface. As the surface configuration is varied, the positive interface energy tends to flatten the sur- face while the positive strain energy is inclined to roughen the surface, changing the chemical potential in opposite directions. This competition depends on the spatial fre- quency of the roughness. The speed of etching is assumed to be directly proportional to x s, thus ≠H≠t  Mg 0 1 gk 1 w q 1 1 j=Hj 2 , (1) where H is the height of the interface as shown in Fig. 1, M is the linear kinetic constant of etching, and = denotes the two-dimensional gradient operator. The projected average FIG. 1. Schematic of chemical etching of a solid surface under stress s . The height and roughness of the surface, at time t and position x, are represented as Hx, t  and hx, t , respectively. 3872 0031-9007 99  83(19)  3872(4)$15.00 © 1999 The American Physical Society