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THE GENERATION OF LARGE k-VECTOR PHONONS
M. Colles
To cite this version:
M. Colles. THE GENERATION OF LARGE k-VECTOR PHONONS. Journal de Physique Colloques, 1972, 33 (C4), pp.C4-41-C4-43. �10.1051/jphyscol:1972409�. �jpa-00215086�
JOURNAL DE PHYSIQUE Colloque C4, supplkment au no 10, Octobre 1972, page C4-41
THE GENERATION OF LARGE k-VECTOR PHONONS
M. J. COLLES
Department of Physics, Heriot-Watt university, Edinburgh
RBsumB. - Le processus de rupture des phonons est examine de nouveau. I1 est conclu que les densites des phonons optiques qui sont nkessaires pour observer I'effet sont prohibitivement grandes si elles sont produites par des pulsations picosecondes via des effets Raman stimul6s. Une technique alternative pour la genkration cohkrente des phonons avec un grand k-vecteur, c'est- a-dire des effets Raman stimulCs pour deux phonons, est proposee et des rQultats experimentaux preliminaires dans NaCl sont discutes.
Abstract. - The process of phonon breakdown is reexamined. It is concluded that the optical phonon densities required to observe the effect, when they are produced via stimulated Raman scattering from picosecond pulses, are prohibitively large. An alternative technique for the cohe- rent generation of large k-vector phonons, stimulated two-phonon Rarnan scattering, is proposed and preliminary results of an experiment in NaCl are discussed.
For some years now it has been possible to generate high frequency optical phonons under conditions of coherent excitation via stimulated Raman scatter- ing [I]. Recently, with the help of picosecond pulses from mode locked lasers it has become possible to study the transient excitation of these phonons [2], [3], [4] and to directly measure their lifetimes [5], 161. The phonons generated via stimulated Rarnan scattering have wavevectors which are extremely close to the zone centre. There is therefore some considerable interest in extending these techniques to produce, coherently, both optical and acoustic phonons at points in the Brillouin zone away from k = 0. Two such techniques will be examined here.
Due to lattice anharmonicity optical phonons at a frequency o0 and wavevector ko
-
0 may, if their level of excitation is sufficiently high, parametrically gene- rate acoustic phonons satisfying the cncrgy and momentum matching conditions o,,,+
a,,, = o0 and ka,,+
k,,, = ko -- 0. This process, termed pho- non breakdown, was proposed by Orbach [7] and has recently been envoked to explain the results of experi- ments in diamond [8] where the acoustic phonons generated would lie close to the Brillouin zone boun- dary.Additional insight into this process may be gained by reexamining and extending the analysis given in reference [7]. The starting point was an interaction Hamiltonian of the form :
+ +
H = [A(k) c, c, c-,
+
c. c.] (1) with the c's creation and annihilation opcrators for the acoustic and optical phonons. The anharmonic coeffi- cient could then be evaluated in terms of the opticalphonon lifetime t o by utilising an expression derived by Vredevoe [9],
and assuming that both the group velocity and coupl- ing terms are isotropic. [If these quantities are not isotropic as is most probably the case then the resulting expression for I A(k) 1' should be multiplied by a factor j?, thc equivalent of a branching ratio in space rather than in energy. fl is not amenable t o simple calculation and, in what follows fl = 1 will be assumed, however, it is almost certainly somewhat less than this.]
From eq. (2) :
where V,, represents the combined group velocity in eq. (2). Orbach considered a rate equation in the form :
with S,, = n,,
+
n-,, the occupation number density of the acoustic phonons So the optical phonon occupa- tion number density and rk, = h/z,, the energy width of the terminal phonons (z,, their lifetime). It is useful to recast this equation in a clearer way. Now the number of states per unit frequency in a volume V for phonons is N(o) = l/(2 7 ~ ) ~ (4 nkz/Vk) Vand interpret- ing V, as the combined group velocity the number of two phonon states can be written as :Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972409
C4-42 M. J. COLLES Thus eq. (4) becomes :
with 6wk1 = the linewidth of the terminal pho- nons. This is now in a form completely compatible with a general rate equation expressing the possibility of stimulated gain provided a spontaneous process exists.
The first term in the brackets is the product of the probability of optical phonon decay somewhere (l/to) divided by the number of states within the linewidth over which this decay is most probable. The use of 6mk, in eq. (6) as this linewidth is questionable since the transition linewidth will be governed by either the initial or final state linewidth whichever is greater. That is by 60, (- l/to) or h o ~ , , ( l / ~ ~ , ) and in general the former quantity will be larger and should be used in eq. (6). With coherent excitation, however, this may no longer be the case since, under these conditions, a large number of optical phonons exist in a small number of modes. The vibrational wave with amplitude q appear- ing in the equation for stimulated Raman scattering represents a driven damped oscillator. The general solution combines a wave decaying at a rate I/T, with a component driven at the beat frequency olascr - oSlokes. The initial decaying part is responsible for the cc linewidth >) l / t o and would result from impulse or incoherent excitation. The driven part represents a wave with a linewidth given at best by l/zp where tp is the duration of the laser and Stokes pulses. It is this latter value Sw, = l/t, which should be used to replace 6wk, in eq. (6) provided t, < z,,.
Thus eq. 16) becomes :
The threshold for breakdown may then be expressed as :
The threshold condition however is less useful than an estimate of the value So required for an appreciable build up of acoustic phonons. Eq. (7) may be solved in the form :
S,,(t) = Sk,(0) e" , [Sk,(0) = 1 at threshold] (9) with
Taking as an example phonon breakdown in dia- mond with 7,
-
5 x lo-'' s, T, = lo-* s and N(ok,)-
5 x lo9 s-I eq. (1 1) gives 5'; = 1016 ~ m - ~ , a readily attainable figure. The prediction obtained for breakdown with picosecond pulses is, however, less encouraging. With T, = 10-l1 s and the other para- meters remaining the same Sh -- ~ m - ~ , an cxtre- mely difficult if not impossible phonon density to achieve 11 01.An alternative technique, and one which may well take specific advantage of short intense pulses, would be via stimulated two-phonon Raman scattering. For stimulated scattering the following expression yields the rate of growth of Stokes photons [I]
where da/dQ is the integrated cross-section per stera- dian, N the number of molecules per cm3, NL and Ns the laser and Stokes photon densities and h ( j J the lineshape factor. The quantity
spontaneous cross-section per mode of the electro- magnetic field to which the material system can couple.
The expression is perfectly general and applies to any process for which a spontaneous cross-section exists.
The difficulties involved in determining the power density required to achieve stimulated two-phonon Raman scattering are in the evaluation of h ( f , ) and in thc lack of any absolute or relative measurements of the two-phonon cross-sections. Taking NaCl as an example the Raman spectrum [ l l ] extends from 0 to 350 cm-' which would enable a very approximate halfwidth of
170 cm-' to be assigned to h(f,). A rough estimate of the cross-section may be obtained from the experi- mental equipment necessary to measure the spec- trum [I21 at about one tenth that of the cross-section of the vibrational bands of liquid water. The harmonic of a mode-locked ruby laser was focussed with a 30 cm lens into an 8 cm cell of water. The power density was sufficient to produce strong 1st and 2nd stokes scattering from the vibrational band in water (- 3 450 cm-' shift). Replacing the cell with an 8 cm long crystal of NaCl yielded some evidence of stimu- lated two-phonon Raman scattering although diffi- culties associated with working so close to the exciting line havc not yet been fully resolved. The peak of the Stokes radiation was shifted by
-
240 cm-'. On the basis of Krauzmans [ l l ] assignments this would correspond to the coherent generation of transverse Requiring that yz, = 30 for the attainment of largevalues of Sk, and neglecting I/tk, yields : acoustic phonons at the X point.
Sb = 15 nN(w,,) 2 (I1) Acknowledgments. - I would like t o thank Dr.
B. S. Wherrett of this Department for several illuminat- and a number density proportional to 11~:. ing discussions.
THE GENERATION OF LARGE k-VECTOR PHONONS
References
[l] See review by B ~ E M B E R G E N (N.), Am. J. Phys., 1967, [7] ORBACH (R.), IEEE Trans. Series, Ulfrasonics, 1967
35, 989. 14, 140.
[2] SHAPIRO (S. L.) e f al., Phys. Rev. Lett., 1967, 19, [81 OLLES (M. J.) and (J' Phys, Rev.
1093. Lett., 1971, 27, 670.
[9] VREDEVOE (L. A.), Phys. Rev., 1965, 140, 930.
[3] COLLES (M. J.), Opt. Commun., 1969, 1 , 169. [lo] A similar result was deduced on the basis of lifetime
[4] CARMAN (R. L.) et ul., Phys. Rev., 1970, 2 , 60. measurements by LAUBEREAU et al. (ref. 161).
[5] ALFANO (R. R.) and SHAPIRO (S. L.), Phys. Rev. [Ill KRAUZMAN (M.), C . R. Acad. Sci., Paris, 1968,
Lett., 1971, 26, 1247. 266B, 186.
[12] DUNCAN (I.) and STEWART (J. H.), in Light Scatter- [6] LAUBEREAU (A.) et al., Phys. Rev. Lett., 1971, 27, ing in Solids, ed. M. Balkanski, Flammarion,
802. Paris, 1971, p. 308.
DISCUSSION N. SHIREN. - Would you comment on the use of
rate equations, which neglect coherence phenomena, particularly in the case of polariton pumping ?
M. J. COLLES. - The situation in the case of pola- riton pumping could well be different. For phonon pumping if one assumed, as I did, that the process is perfectly phase-matched this assumption takes care of coherence effects, provided that the pump field is not being depleted.
C. ELBAUM. - 1. Could you comment on the physics involved in the conclusion that the phonon decay process under discussion is governed by the pulse duration rather than the line-width of the final state ?
2. What criteria are used to identify the optical line shift shown as due to generation of acoustic phonons ?
M. J. COLLES. - 1. Since the pulse duration is less than the estimated value of the acoustic phonon lifetime, the process is transient. Under transient conditions it is generally true that the maximum value attained by some quantity (in this case, the acoustic phonon intensity) will be governed by the duration of the interaction rather than the lifetime of the quan- tity. For example the absorption of a narrow line (Ao) will be apparently reduced if 117, > A o ; it will become proportional to 7,.
2. The line shift has not been positively identified as yet. Comparison with the spontaneous Raman spectrum, however, indicates that the line shift corres- ponds to a peak in this spectrum which has been assi- gned to the combination of two TA phonons at the X point.
