We find that the electronic dispersion in graphite gives rise to double resonant Raman scattering for excitation energies up to 5 eV. As we show, the curious excitation-energy dependence of the graphite D mode is due to this double resonant process resolving a long-standing problem in the literature and invalidating recent attempts to explain this phenomenon. Our calculation for the D-mode frequency shift (60 cm 21 ͞eV) agrees well with the experimental value. 81.05.Tp Single incoming or

more »

... g resonances are widely known in Raman spectroscopy and frequently used to study the electronic and vibrational properties of crystals or molecules. They occur if the energy of the incoming or the scattered photon matches the transition energy of an allowed electronic transition leading to a large enhancement of the Raman cross section [1] . Closely related is the idea of double resonant Raman scattering, where, in addition to the incoming or outgoing resonances, the elementary excitation makes a real transition. Double resonances are much stronger than single resonances. They were, however, only observed under very specific experimental conditions: The energetic difference between two electronic bands was adjusted to the phonon energy by applying electric or magnetic fields, uniaxial stress, or by a proper choice of the parameters of semiconductor quantum wells [2] [3] [4] [5] . Double resonant conditions were thereby realized for distinct excitation energies. In this Letter we study the double resonant Raman process for linearly dispersive bands as in semimetals. We show that double resonances are responsible for the observation of the defect induced D mode in graphite and its peculiar dependence on excitation energy. The double resonance considered has a much stronger enhancement than simple incoming or outgoing resonances explaining why the defect mode (and its second order peak) is so strong compared to the graphite G point vibration. The first order Raman spectra of graphite show besides the G point modes an additional defect induced peak, the so-called D mode [6-9]. The D mode is related to the finite crystallite size and disappears for perfect crystals [6,10]. Its frequency was found to shift with excitation energy at a rate of 40 50 cm 21 ͞eV over a wide excitation energy range [7,11-13], a phenomenon which has not been understood for almost 20 years. Tan et al. found that there is a curious discrepancy between the Stokes and anti-Stokes frequencies of the D mode, which they were unable to explain [14] . Various groups have recently attempted to explain the unusual excitation-energy dependence which was found also for the second order spectra, where the mode shifts at approximately twice the rate and is not defect induced. Sood et al. proposed a disorder in-duced double resonance above a gap D ഠ 1 eV in the band structure leading to a dependence of the phonon wave vector q and hence the phonon frequency on the energy of the incoming light E 1 as q ϳ ͑E 1 2 D͒ 1͞2 [12]. There is, however, no such gap in the electronic structure of graphite; it is a semimetal with valence and conduction band crossing the Fermi level at the K point of the Brillouin zone. Pócsik et al. introduced a new Raman mechanism for which the wave vector of the electron which is excited by the incoming resonant photon supposedly defines the wave vector of the scattered phonon [13] . This ad hoc k q quasiselection rule was applied to particular branches of the phonon band structure by Matthews et al. and Ferrari and Robertson [15, 16] . However, these explanations required a mysterious coupling of the optical branches to a transverse acoustic branch in the phonon band structure or did not yield the correct shift of the D mode. Single resonances have not been identified in the graphite Raman spectra because the linear dispersion of the electronic bands allows these resonances to occur at all energies E 1 and for the entire D band independently of q. There is no reason why the quasimomentum where the electronic transitions occur is transferred selectively to the phonon seen in the Raman spectra, and it is impossible to understand the difference between Stokes and anti-Stokes frequencies. In other words, the "quasiselection rule" invoked by Refs. [13, 15, 16] has no physical basis and cannot explain the experimental observations. To study the resonant Raman effect in a semimetal like graphite we first consider a one-dimensional example as depicted in Fig. 1 , where we have shown two linear bands with different Fermi velocities which cross at the Fermi level. The peculiarity of this electronic dispersion is that, in addition to single resonances, a double resonant transition is possible for a wide variety of excitation energies. The first step of the double resonance for a particular incident laser energy E 1 is to create an electron-hole pair at the k point matching the energy difference between the conduction and the valence band (i ! a). It is obvious that for a monotonically increasing phonon dispersion v ph ͑q͒ there exists a (v ph , q) combination which can scatter the electron to a state on the second band (a ! b). This 5214 0031-9007͞00͞85(24)͞5214(4)$15.00