Fractals and device performance variability: The key role of roughness in micro and nanofabrication V. Constantoudis a,⇑ , G.P. Patsis a,b , E. Gogolides a a Institute of Microelectronics (IMEL), NCSR ‘‘Demokritos’’, P.O. Box 60228, Aghia Paraskevi, Attiki 15310, Greece b Department of Electronics, Technological Educational Institution (TEI) of Athens, 12210 Aegaleo, Greece article info Article history: Available online 24 April 2011 Keywords: Fractal geometry Roughness Device variability Line edge roughness (LER) Micro and nanofabrication MOSFET abstract Nano and micro-fabrication processes generate patterns with rough surfaces. This roughness becomes a significant fraction of pattern dimensions as feature size scales down and starts to affect surface proper- ties and device performance. It has been shown that in most times surface roughness exhibits scale- limited fractal behavior which is characterized by the fractal dimension. Up to now, fractal dimension has been related to the physicochemical properties of open surfaces. Here, we examine its effects on the performance variability of devices incorporating features with rough surfaces. It is shown that when the feature dimension which is parallel to rough surface becomes lower than about 10 times the surface correlation length then the decrease of rms roughness is followed by an increase in the variability of fea- ture dimension perpendicular to rough surface. Both dependencies are determined by the value of the fractal dimension, which means that fractal dimension controls the impact of roughness on the variability of device dimensions and hence performance. We demonstrate the role of fractal dimension in device variability, by examining the impact of the gate sidewall roughness on the leakage current of planar MOS- FET devices. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction Micro and nano-fabrication processes (lithography, etching, deposition, epitaxy, etc.), usually induce roughness on the surfaces of patterned features. With continuing scaling down of device dimensions, roughness becomes an important fraction of feature size and affects the physicochemical properties of open surfaces as well as the performance of devices with rough embedded sur- faces. Examples include the modification of the wetting or optical behavior of open surfaces due to changes in their roughness [1,2], and the variability in the performance of planar transistors caused by the sidewall surface roughness of their gates [3–5]. The understanding and control of these roughness effects demands mathematical and statistical tools for its measurement and characterization. The traditional approach to roughness characterization, devel- oped mainly by engineers to address tribology issues, has been re- lied on the estimation of a large number of parameters describing various aspects of roughness [6]. This approach was straightfor- ward, but overlooked hidden symmetries and more elaborate mod- eling of surface morphology. Fractal geometry, invented for the characterization of objects with irregular shapes, provided a more scientific insight on surface characteristics by suggesting that most surfaces obey, at least in some scale regime, the fractal self-affine symmetry, i.e. they remain statistically invariant under anisotropic scale transformations [7]. The parameter quantifying this symme- try is the fractal dimension, which indicates the relative contribu- tion of high frequency surface fluctuations to total roughness. The assumption of fractal symmetry has been verified in a large number of experimental and modeling works examining the sur- face roughness induced by micro and nano-fabrication processes [8–10]. Furthermore, several works have dealt with the relevance of fractal dimension to open surface functionality and physico- chemical properties [11–14]. However, to the best of our knowl- edge, the impact of fractality of surface roughness on the variability of device performance has not been discussed yet. The aim of this work is to fill this gap and examine the implica- tions of the fractal dimension of rough surfaces in the statistical variability of device performance. To this end, first we review the implications of fractal theory in roughness characterization by illustrating the importance of fractal dimension on the roughness metrics of an open surface (Section 2). Then, we clarify the impact of roughness on device variability and propose a methodology for the assessment of fractal dimension impact on devices (Section 3). In Section 4, we demonstrate this role in the specific case of the effects of the roughness in gate sidewalls (usually called Gate Length Roughness) on the variability of MOSFET performance. Finally, Section 5 summarizes the main findings of the work and discusses future prospects. 0167-9317/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2011.04.054 ⇑ Corresponding author. Tel.: +30 210 6503116. E-mail address: vconst@imel.demokritos.gr (V. Constantoudis). Microelectronic Engineering 90 (2012) 121–125 Contents lists available at ScienceDirect Microelectronic Engineering journal homepage: www.elsevier.com/locate/mee