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In 1986 it was reported , that a single beam ofpicosecond, A.=0.5 μm pulses at a high repetition rate of82MHz with intensity just bellow melting threshold can induce permanent periodic surface structures(relief gratings) on silicon with periods as small as an eighth of the light wavelength (~.06 μm). The grating rulings on (11 l)surtace were oriented along one of [110] axes. Later this phenomenon was more thoroughly investigated in Ref. {2]. The structures were produced on silicon and germanium and were studied as functions of laser wavelength, polariz.ati.on, repetition rate, pulse width and crystallographic orientation. Contrary to Ref.[1] it was fotmd that the surface relief gratings were oriented with wave vectors parallel to the polariz.ation vector of the linear polarized laser beam and the structure period in Si was observed to increase Vith laser pulse width 7: P and to decrease with increase ofthe pulse repetitionrateR (for 150 kHz-150 MHz). Besides atthe same conditions tht: structures with pulse independent periods were recorded in Si. In Ge the structures produced with 75 ps pulses had lesser period t.lian those produced with pulses of considerably less duration (5 ps) but with higher repetition rate. The effect of abrupt transition of structure period from 0.325μm to 0.096μm along the scan line was observed in Si. To our knowledge no model was able to explain the above features of ultrashort period laser-induced periodic surface structures formation. In this work we propose and develop such a model, the physical essence of which is as follows. Intense laser pulse excites electrons across the band gap and creates in a subsurface layer with thickness of order of absorption lengt.11. a. -l a hig..11. concentration of electron-hole plasma. In Si at temperatures T ~ Tm ( T,,. is the melting temperature) a -l - 4*10-<i cm [3J. During the action of a train of laser puls..."S with duration 't P and repetition rate R the subsurface plasma-enriched layer of thickness h < cx-1 is formed. We consider this thin plasma enriched layer as a "film" of thickness h tightly bonded to the underlying crystal ("substrate") (Fig. 1). The initial state with tmbended film and laterally wllfonn plasma concentration becomes unstable at a certain critical value of surface plasma concentration nc and due to the surface plasma-strain (SPS) instability [ 4] the state with periodically bended film and periodic piling up of plasma concentration is formed(Fig.1). The period of th.is SPS structure is shown to be d = 2h with the value of h being determined by interplay of three processes of plasma transport nozmal to the! surface: the canier diffusion, the carrier drift. induced by self-consistent strain and plasma renonnaliz.ation of forbidden band gap, and the canier drift induced by nonwllform heating. Auger recombination plays essential role in restricting the value of h. The competition of diffusion and drifts leads to establishing at times ts of quasistati.onary regime. At low repetition rat·~ ( R-1 < 7:0 , 7:0 is the linear recombination time) ts< 'tp• i.e. steady state is established before the end of a pulse and disappears before· the beginning of the next pulse ("'single pulse" regime of plasma layer formation). No surface plasma accumulation from pulse to pulse in a train occms in this regime and h is independent ofR .. At high repetition rate (R-1 < 7: 0) surface plasma accumulation from pulse to pulse takes place and the time of establishing of steady state is t1 < 7: 0 . In limiting cases of regimes controlled by either diffusion and strain, plasma-induced drift (DSPD) or by diffusion and temperature-induced drift (DTD) the obtained general solution of nonlinear plasma dynamics equation yields explicit formulas for has a function of laser intensity and parameters of the medium. The dep...'"Ildency of d = 2.h onR and 7: P in DSPD regime is shown to describe the corresponding experimental dependencies of structure periods in Si and Ge, recorded in Ret:[2]. The abrupt transition from DSPD to DTD regime is the cause of the occurrence of sudden change of structure period to ultrashortvalue(-0.095 μm ), observed in Si (2]. The orientations of SPS gratings are determined by the interplay of two factors reflecting the dual nature of plasma-strain instability: the crystallographic anisotropy of surface shear elastic modulus and strong linear polarized laser beaminduced anisotropy of lateral ambipolar diffusion coefficient Thus depending on irradiation conditions the SPS grating wave vector may be either crystallographically c·rieated or be directed along the el.tric ,,-ect~r E cf the i:Kident laser field. Tiris feature of the SPS instability model reconciles controversial experimental findings of Refs. [ 1] and ( 2 J. The problem of describing the phenomenon of ultrashort period swface structure formation thus cofuists of two parts. The first is the consideration of plasma transport in direction normal to the surface and calculation of the thickness of subsurface plasma layer, and second is the consideration oflateral plasma transport in self-consistent periodic strain field and calculation of the surface structure periods. In this work we formulate and consider both problems. The work is organized as follows. In Sec.2 plasma tra.'1Sport normal to the surface is treated on the base of nonlinear plasma transport equation and the thickness of subsurface plasma layer his found. In Sec.3 the "film on substrate" model of SPS instability is formulated and in Sec.4 a general set of nonlinear equations in mode representation, describing SPS instability, is obtained. In Sec.5 we perform linear stability analysis ofSPS-instability equations and find grmvth rates of surface periodic SPS structures. In Sec.6, using the obtained expressions for the growth rates, we calculate the characteristics of SPS structures and compare the theoretical results with experimental ones ofRefs.[l,2J.
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