We study the phonon-assisted process of dissociation of a magnetoroton, in a
fractional quantum Hall liquid, into an unbound pair of quasiparticles. Whilst
the dissociation is forbidden to first order in the electron-phonon
interaction, it can occur as a two-phonon process. Depending on the value of
final separation between the quasiparticles, the dissociation is either a
single event involving absorption of one phonon and emission of another phonon
of similar energy, or a two-phonon diffusion of a quasiexciton in momentum
space. The dependence of the magnetoroton dissociation time on the filling
factor of the incompressible liquid is found.Comment: 4 pages, no figure

A. Griffin

A.F.G. Wyatt

C.J. Mellor

F. Schultze-Wischler

J.P. Wolfe

K.A. Benedict

M. E. Portnoi

M.A.H. Tucker

R.B. Laughlin

S.M. Girvin

T. Ando

U. Zeitler

V. M. Apalkov

V.E. Zhitomirskii

V.I. Fal’ko

Yu.A. Bychkov

Physical Review B

English

arXiv

Copyright © 2002 The American Physical SocietyWe study the phonon-assisted process of dissociation of a magnetoroton, in a fractional quantum Hall liquid, into an unbound pair of quasiparticles. Whilst the dissociation is forbidden to first order in the electron-phonon interaction, it can occur as a two-phonon process. Depending on the value of final separation between the quasiparticles, the dissociation is either a single event involving the absorption of one phonon and the emission of another phonon of similar energy, or a two-phonon diffusion of a quasiexciton in momentum space. The dependence of the magnetoroton dissociation time on the filling factor of the incompressible liquid is found

Apalkov, V.M.

Portnoi, M.E.

Open Research Exeter

RAPID COMMUNICATIONSPHYSICAL REVIEW B 66, 121303~R! ~2002!Two-phonon scattering of magnetorotons in fractional quantum Hall liquidsV. M. Apalkov* and M. E. Portnoi†School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom~Received 26 June 2002; published 10 September 2002!We study the phonon-assisted process of dissociation of a magnetoroton, in a fractional quantum Hall liquid,into an unbound pair of quasiparticles. Whilst the dissociation is forbidden to first order in the electron-phononinteraction, it can occur as a two-phonon process. Depending on the value of final separation between thequasiparticles, the dissociation is either a single event involving the absorption of one phonon and the emissionof another phonon of similar energy, or a two-phonon diffusion of a quasiexciton in momentum space. Thedependence of the magnetoroton dissociation time on the filling factor of the incompressible liquid is found.DOI: 10.1103/PhysRevB.66.121303 PACS number~s!: 73.43.2f, 73.43.FjRecent advances in phonon spectroscopy1 have yieldedfascinating results in several branches of condensed-matterphysics. These results range from the observation of the re-markable anisotropy of phonon propagation in crystallinesolids2 to the detection of R2 rotons in liquid helium.3 Ar-guably the most interesting results are obtained when ballis-tic nonequilibrium phonons interact with low-dimensionalmany-body systems. For example, recent experiments inphonon-induced quantum evaporation have provided the firstdirect evidence for a Bose-Einstein condensate on the surfaceof liquid 4He.4 A similar approach has been used to probequasi-two-dimensional ~2D! electron systems based on semi-conductor heterostructures. In this method, the nonequilib-rium acoustic phonons injected into the system are absorbedby the 2D electron gas, and this changes its transport5–8 andoptical9 properties at low temperatures. Such a technique hasrecently been used to obtain information about a magneti-cally quantized 2D electron gas at fractional filling factors.7The absorption of nonequilibrium phonons by a 2D in-compressible liquid, underlying the fractional quantum Halleffect, results in an increase of the effective electron tem-perature, which is detected by measuring the dissipative con-ductivity of the system. The phonon absorption process canbe schematically divided into two stages. During the firststage, the phonons are absorbed by the incompressible liq-uid, with the creation of nonequilibrium neutral excitations~magnetorotons! with wave vectors of the order of the in-verse magnetic length.10 During the second stage, the non-equilibrium electron system is thermalized, with the creationof charged quasiparticles which are free to participate in thedissipative conductivity. The process of thermalization canbe understood as the dissociation of magnetorotons intowell-separated quasielectrons and quasiholes. Initially, it wasproposed that such a process could be intensified by the ab-sorption of nonequilibrium phonons from the pulse.11 It waslater shown12 that due to the almost flat dispersion of theneutral excitations of an incompressible liquid at large wavevectors, the absorption of a phonon by a magnetoroton isallowed only for a very small momentum transfer, which isnot enough to dissociate the magnetoroton. However, due tothe electroneutrality of the magnetoroton, the detectedchanges in dissipative conductivity cannot be explainedwithout the dissociation of the magnetoroton into chargedquasiparticles. The main goal of this paper is to uncover the0163-1829/2002/66~12!/121303~4!/$20.00 66 1213mechanism of magnetoroton dissociation. Understanding thismechanism is essential for the interpretation of the phononabsorption spectroscopy data, and for the prediction of thepossible outcome of experiments at different filling factors.We show that the phonon-pulse-induced dissociation of themagnetoroton occurs as a second-order process in theelectron-phonon interaction. It is a common feature for allsystems containing dispersionless particles that two-phononprocesses provide the main contribution to the transportproperties of the system.13,14The neutral low-energy excitations of the incompressibleelectron liquid have a nonzero gap D(q) for all wave vectorsq, with a minimum value D* at q5q*;1/l0, where l0 is themagnetic length. The excitations close to q* are well de-scribed by the single-mode approximation,15 and are calledmagnetorotons by analogy with roton excitations of super-fluid helium. At large wave vectors the neutral excitation is aquasiexciton, which consists of a fractionally-charged quasi-electron and quasihole.15–18 In what follows, we considerquantum Hall liquids with a filling factor n5p/m , where p isan integer and m is an odd number. For this filling factor thecharge of a quasielectron ~quasihole! is 1/m (21/m) of theelectron charge. The effective magnetic length for these qua-siparticles is l0*5Aml0. The dispersion of neutral excitationsof the incompressible liquid at large values of momentumcan then be written in the form D(q)5D‘2«0 /m3q , where«05e2/kl0 is the Coulomb energy. We use the magneticlength l0 and the Coulomb energy «0 as the units of lengthand energy, respectively. To study the phonon-induced disso-ciation of a magnetoroton, we should consider phonon-assisted transitions of the magnetoroton with momentumclose to q* into a quasiexciton with momentum larger than acritical momentum value qcr . This critical momentum is de-fined by the conditionD‘2D~qcr!5Tcr , ~1!where Tcr is a characteristic temperature ~in units of «0 /kB).To find the two-phonon scattering rate for neutral excita-tions of an incompressible liquid, we assume that within thewhole range of our interest (q.q*), a magnetoroton can bedescribed as a quasiexciton, i.e., a bound state of a quasielec-tron and a quasihole. Although this is not a very good ap-proximation at q;q*, we use this assumption to illustrate©2002 The American Physical Society03-1RAPID COMMUNICATIONSV. M. APALKOV AND M. E. PORTNOI PHYSICAL REVIEW B 66, 121303~R! ~2002!the importance of two-phonon processes for the thermal re-laxation of incompressible liquids.For convenience we label the 3D vectors with capital let-ters, e.g., QW , and their projections with the correspondinglowercase letters: QW 5(qW ,qz). We neglect the effects relatedto phonon polarization and crystal anisotropy, and write thequasielectron-phonon and quasihole-phonon interactionHamiltonian in the formHint5(QW1mVe~Q !AVZ~qz!rˆ e~qW !@bˆ †~2QW !1bˆ ~QW !#2(QW1mVe~Q !AVZ~qz!rˆ h~qW !@bˆ †~2QW !1bˆ ~QW !# ,~2!where bˆ †(QW ) and bˆ (QW ) are creation and annihilation opera-tors of a phonon with momentum QW , V is a normalizationvolume, and rˆ e and rˆ h are the quasielectron and quasihole2D density operators, respectively. Because the quasiparti-cles are in the lowest subband of size quantization, the z partof the density operator is accumulated in the form factorZ(qz):Z~qz!51~11iqz /b!3, ~3!where the Fang-Howard approximation19 for the lowest sub-band wave function is used, and b is the parameter of theFang-Howard wave function.The response of a quasiparticle to the external perturba-tion is m times weaker than the corresponding response of anelectron. For the quasihole this response has the opposite12130sign. Therefore, in Eq. ~2! the quasiparticle-phonon interac-tion is characterized by the matrix elements 6(1/m)Ve(Q),where the upper sign is for quasielectrons and the lower signis for quasiholes. Here, Ve(Q) is the matrix element of theelectron-phonon interaction, which is given by theexpression20Ve~Q !5ABpQ 2iABdQ , ~4!where Bp and Bq are the constants of piezoelectric and de-formation potential couplings. We use material parameters ofGaAs in our numerical estimates.Using Eq. ~4!, one can rewrite the Hamiltonian ~2! in theformHint5(QW1mZ~qz!AVSABpQ 2iABdQ D cˆ ~qW !@bˆ †~2QW !1bˆ ~QW !# , ~5!where cˆ (qW )5rˆ e(qW )2rˆ h(qW ) is the quasiparticle charge-density operator.The matrix elements of the operator cˆ (qW ) between thequasiexciton states, which are characterized by the 2D mo-menta kW and kW 1, are given by the expression^kW 1ucˆ ~qW !ukW &522ie2mq2/4sinS m qxky2qykx2 D d~kW 12kW2qW !,~6!where the momentum is in units of 1/l0.The rate of the two-phonon scattering of the quasiexcitonto states with momentum greater than qcr isW~qcr!52p\«03l061m4E dwq*2p Eq f .qcrd2qW fE Ed3QW~2p!3d3QW 8~2p!3uZ~qz!u2uZ~qz8!u2uM ~qW *,qW f ,QW ,QW 8!u23N~Q !@N~Q8!11#dD~q*!2D~q f !2s~Q82Q !d~qW *1qW 2qW f2qW 8!, ~7!whereM ~qW *,qW f ,QW ,QW 8!5^qW *uHˆ ~QW !uqW *1qW &^qW *1qW uHˆ ~QW 8!uqW f&Da2sQ81^qW *uHˆ ~QW 8!uqW *1qW 8&^qW *1qW 8uHˆ ~QW !uqW f&Db1sQ8, ~8!Da5D(qW *1qW )2D(qW f), Db5D(qW *1qW 8)2D(qW *), andHˆ ~QW !5FABpl0Q 2iABdQl0 Gcˆ ~qW !, ~9!where s is the average speed of sound, which is s’0.03 in units of «0l0 /\ . In Eq. ~7!, the first integral is the average of thescattering rate over all possible directions of the initial momentum qW *. N(Q) is the nonequilibrium phonon distributionfunction created by the external phonon source. The two d-functions represent the conservation of energy and momentum.Due to the exponential dependence @Eq. ~6!# of the matrix elements of the charge-density operator cˆ (qW ) on the phononmomentum q, the main contribution to the integrals in Eq. ~7! comes from the region where the difference between q and q8is small, and the vectors qW * and qW f are parallel. In this case, substituting Eqs. ~6! and ~9! into Eq. ~8!, we obtain the expression3-2RAPID COMMUNICATIONSTWO-PHONON SCATTERING OF MAGNETOROTONS IN . . . PHYSICAL REVIEW B 66, 121303~R! ~2002!uM ~qW *,qW f ,QW ,QW 8!u2’16~Bpl021BdQ2!~Bpl021BdQ82!QQ8l02@D1~q f !1sQ8#2H F dDD1~q f !1sQ8G 2F sinmqq*sin f2 G43Fcosmqq8sin 2f2 G21F sinmqq8sin 2f2 sinmqq fsin 2c2 G2J , ~10!where D1(qcr)’Da’2Db’D(@qcr1q*#/2)2D(q*), dD5Da1Db’2D(qW *1qW )2D(qW f)2D(qW *), f is the angle be-tween vectors qW and qW *, and c is the angle between vectorsqW * and qW f . From Eq. ~10!, one can see that the amplitude oftwo-phonon scattering of the quasiexciton vanishes when thewave vectors of the phonons, qW and qW 8, have the same direc-tion. We assume that the angles f and c between the wavevectors are small. This approximation is good for large val-ues of q f . The first term in the curly brackets in Eq. ~10!contains the factor dD , which describes the nonlinearity ofthe excitation spectra and depends strongly on the shape ofthe dispersion curve D(q). There is no good approximationto describe the magnetoroton spectrum in the intermediateregion between the magnetoroton minimum and the begin-ning of the quasiexcitonlike dispersion. As an upper limit fordD we take the value dD50.005, which corresponds to aquarter of the magnetoroton binding energy; the term con-taining dD then gives only a small correction to the wholeexpression. In what follows, we shall disregard this term andonly consider the contribution from the second term in thecurly brackets.If the width of the wave function in the z direction ismuch larger than 2/dqcr , i.e., the Fang-Howard parameter bis much smaller than dqcr/2, then by substituting Eq. ~10!into Eq. ~7! and performing the integrations, we obtain theexpression for the rate of quasiexciton dissociation:W~qcr!’W01m432p21s3qcrAqcrAq*F b2sdqcrD0~qcr!G7/23N~dqcr/2!@N~dqcr/2!11#3F 11 Bddqcr24Bpl02 G2 1dqcr3S 113 qcr~q*1qcr!2D erfcSAm2 dqcrD , ~11!wheredqcr5qcr2q*,erfc~x !512erf~x !5~2/Ap!Ex‘dy exp~2y2!is the complementary error function, and the constant W0 isgiven by W05Bp2/(\«03l04).12130In the opposite case of a narrow quantum well, i.e., b@dqcr/2, W(qcr) takes the formW~qcr!’W01m416p31s3qcrAqcrAq*N~dqcr/2!@N~dqcr/2!11#31dqcr S 113 qcr~q*1qcr!2D erfcSAm2 dqcrD , ~12!where the main contribution comes from the piezoelectricinteraction.From Eqs. ~11! and ~12! one can see that the rate W(qcr)is exponentially small for large qcr . The two-phonon disso-ciation of the quasiexciton as a single event is allowed onlyfor small values of critical momentum. For example, forqcr’3, which corresponds to a characteristic temperaturelarger than Tcr;1/m3qcr;1.5 K @see Eq. ~1!#, we find fromEq. ~12! the rate of quasiexciton dissociation W(qcr)’33108 s21, or the characteristic time of dissociation ;331029 s. For this calculation the temperature of nonequilib-rium phonons is taken as Tph510 K. For a larger value ofthe critical momentum qcr , the rate of direct two-phonondissociation of a quasiexciton becomes very small. For ex-ample, at Tcr50.5 K we obtain qcr;1/m3Tcr;7, whichgives an exponentially small value of the erfc function:erfc@(A3/2)(qcr2q*)#;10216. In this case one should con-sider the series of subsequent two-phonon transitions, whichrepresent the diffusion of the quasiexciton in momentumspace.Following the derivation of Eqs. ~11! and ~12!, we esti-mate the average change in the momentum of the quasiexci-ton, dq0, during a two-phonon scattering event: dq0’2/Am , which is equal to 1.15 for n51/3.The time of the two-phonon scattering event can be esti-mated from Eq. ~7!, where the lower integration limit istaken as q* (q f.q*). For a narrow quantum well, this timeis given byt0’1W06p3m4s3N~dq0/2!@N~dq0/2!11#3Aq*~q*1dq0!3/2dq0S 113 qcr~q*1qcr!2D 21. ~13!The time of diffusion of the quasiexciton, from the magne-toroton state with momentum q* to the quasiexciton statewith critical momentum qcr , can then be estimated from theequationtcr’t0S qcrdq0D2’t01m6S 1Tcrdq0D2, ~14!3-3RAPID COMMUNICATIONSV. M. APALKOV AND M. E. PORTNOI PHYSICAL REVIEW B 66, 121303~R! ~2002!where Eq. ~1! has been used.For n51/3, s50.03, and for the nonequilibrium phonontemperature Tph510 K, our estimate gives t0’0.4310210 s. In this case the time of the magnetoroton disso-ciation is about tcr’231029 s for Tcr50.5 K.We assume that the phonon distribution is characterizedby some equilibrium temperature Tph of the phonon pulse:N(q)5(esq/Tph21)21. The typical phonon energy in Eq.~13! is about sdq0/25s/Am , which corresponds to a tem-perature Tph* ’s/Am . At n51/3 we have Tph* ’2 K. If thephonon nonequilibrium temperature is much larger thanTph* ,Tph@Tph* , then the dependence of t0 on the denomina-tor m of the filling factor has the power law t0}m7/2. FromEq. ~14!, the time of magnetoroton dissociation tcr is then12130proportional to m23/2.In conclusion, we have shown that a flux of nonequilib-rium phonons induces dissociation of magnetorotons of anincompressible fractional quantum Hall liquid into well-separated pairs of quasiparticles. This process is only pos-sible to second order in the quasiparticle-phonon interaction.Depending on the value of the critical quasiexciton momen-tum, the magnetoroton decay can be considered as a directtwo-phonon dissociation or as the result of the two-phonondiffusion of a quasiexciton in momentum space.We are grateful to K. A. Benedict, C. J. Mellor, and A. F.G. Wyatt for stimulating discussions and to D. G. W. Parfittfor valuable advice. This work was supported by the UKEPSRC.*Present address: Department of Physics, University of Utah, SaltLake City, Utah 84112.†Electronic address: m.e.portnoi@ex.ac.uk1 The state of the art in phonon spectroscopy is best represented inProceedings of the Tenth International Conference on PhononScattering in Condensed Matter, edited by M. N. Wybourneet al. @Physica B 316-317 ~2002!#.2 J.P. Wolfe, Phys. Today 48~9!, 34 ~1995!.3 M.A.H. Tucker and A.F.G. Wyatt, Science 283, 1150 ~1999!.4 A.F.G. Wyatt, Nature ~London! 391, 56 ~1998!; see also A. Grif-fin, ibid. 391, 25 ~1998!.5 L. J. Challis and A. J. Kent, in Aspects of Semiconductor Nano-structures, edited by G. Bauer, F. Kuchar, and H. Heinrich~Springer, Heidelberg, 1992!.6 C.J. Mellor, R.H. Eyles, J.E. Digby, A.J. Kent, K.A. Benedict,L.J. Challis, M. Henini, C.T. Foxon, and J.J. Harris, Phys. Rev.Lett. 74, 2339 ~1995!.7 U. Zeitler, A.M. Devitt, J.E. Digby, C.J. Mellor, A.J. Kent, K.A.Benedict, and T. Cheng, Phys. Rev. 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Girvin, The Quantum Hall Effect~Springer, Heidelberg, 1987!.18 T. Chakraborty and P. Pietilainen, The Fractional Quantum HallEffect, 2nd ed. ~Springer, New York, 1995!.19 T. Ando, A.B. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437~1982!.20 V. F. Gantmakher and Y. B. Levinson, Carrier Scattering in Met-als and Semiconductors ~North-Holland, Amsterdam, 1987!.3-4

Two-phonon scattering of magnetorotons in fractional quantum Hall liquids

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Two-phonon scattering of magnetorotons in fractional quantum Hall
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Two-phonon scattering of magnetorotons in fractional quantum Hall liquids

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