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Vol. 14, pp. 1339—1341, 1974.

Pergamon Press.

Printed in Great Britain

COLLECTIVE EXCITATIONS OF THE POLARON-GAS L.F. Lemmens* and J.T. Devreeset Universitaire Instelling Antwerpen, Universiteitsplein 1, 2610 Wilrijk, Belgium (Received 27 March 1974 by S. Lundqvist)

The dispersion of the collective excitations of a polaron-gas is calculated. It is found that the interaction between the L.O. phonons and the plasmons induces a relatively strong wave vector dependence in the cf, o.,modes. This dispersion is primarily a consequence of the interaction of the L.O. phonons with the resonant plasmon branch in the pair excitation spectrum.

IT IS WELL known that the interaction between a charge-carrier and the lattice polarisation leads to the formation of a quasi particle commonly named polaron.”2

longitudinal modes of the dielectric medium (the L.O. phonons) occurs. l’his interaction has been studied by Varga4 in the long wavelength limit and the phonon—plasmon modes are observed in several polar semiconductors.5

l’his concept is used to discuss electronic, transport and optical properties in polar materials, and has been successful to provide a physical insight into the magneto-optical behaviour of polar semiconductors.3

First recall that if the plasma of carriers does not interact with the dielectric medium, it is charact. erized by collective excitations with a relatively stable ‘branch’ (the plasmon)6 and a decaying resonance in the pair excitation region (of the ~—q plane).7 Both modes depend strongly on the wave-vector. The noninteracting dielectric medium has a longitudinal polarisation mode (the L.O. phonon mode). in the model, studied in the present letter both the L.O. phonon and the T.O. phonon are considered as dispersionless.

However, when the carrier density is increased (by doping or by raising the temperature), the one particle approximation for the polaron problem breaks down and one has to consider the many-body interactions of the carriers explicitly. This means that one has to take into account the statistics and the mutual Coulomb interactions of the carrier. It is a convenient approximation to consider the charge carriers as a plasma imbedded in a dielectric continuum characterized by a single polarisation mode (the L.O. phonon mode). For such a system a relatively strong interaction between the longitudinal modes of the carrier plasma (the plasmons) and the

*

t

Describing the electron—phonon interaction with the Fröhlich Hamiltonian’ and calculating the total dielectric function (for a test charge) of the interacting system in the Random Phase Approximation one obtains:

Aspirant van het Nationaal Fonds voor Wetenschappelijk Onderzoek (N.F.W.O.) Also at: R.U.C.A. Antwerpen en S.C.K./C.E.N. Mol

= K’~(q,w) + cxL(q,w) (I) The ~ (q,w) is the Lindhard polarisabiity8 and KD is the dielectric function of the dielectric medium.

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COLLECTIVE EXCITATIONS OF THE POLARON-GAS

/

//

//

/

/ /

The roots of this equation can be calculated analytically in the long wavelength limit (q -+ O)~and in the long wavevector limit (q -+ co). In the long wavelength

/ /

/

/

//

/ //

/

IIII~2IIIIIIIIIIII~

UTO

/

//

o

•

ULO

_________________________________________ 2kF (a)

the L.O. frequency in the limit (q

00).

t

In both cases the frequency w(q) increases witl~ increasing wave vector until a maximum is reached;

//

//

I

than it thedecreases frequency the optic cJ~(q) then plasmon in thealso pair and a of excitation maximum joins the region. acoustic inphonon. pair The branch excitation frequency of the region.reaches When increases further, itthedecreases and the qunperturbed L.O. phonon mode, reaches

//

—

a minimum typical phonon for mode. alland densities, This finally behaviour increases but is less oftowards the pronounced w~(q) themode L.O. foris lower densities.

/

o

(b)

2kF

FIG.!. The frequencies of the collective modes for a polaron gas are shown (full line) for two densities ~a, ~~b• The borders of the pair excitations spectrum are indicated by a thin line. The optical constants of GaAs are used with n 7cm3 (a) and 0 = The 1.4 frequencies 1 0’ = 1.8 lO’8cm3 (b). of the dielectric medium (O~Oand WTO) are indicated by two interspaced lines, 2

KD (q,w)

q -÷ 0 and only one if q -+00 lead us to solve equation (3) numerically. The frequencies obtained this way are shown in Fig. 1 for two different densities of the carriers. Fig.the la the unperturbed frequency is smallerInthan frequency of the plasma optic phonon, in Fig. lb the unperturbed plasma frequency is larger

intersects

~oTO~

limit we fInd three modes: the two modes discussed by Varga and a resonant mode in the pair excitation region which is acoustic in nature. In the long wave vector limit only one mode is found whose frequency increases with increasing wave vector and approaches

This peculiar fact that three solutions appear as

/ -

Vol. 14, No. 12

=

K~

—

2

2 WLO 2

(2)

Interesting features of these renormalised modes are: (1) that there is a strong interaction between the original L.O. phonon and the acoustic plasmon resonance. (2) that there is a gap between the J(q) mode and the T.O. phonon; this gap is equal to ~ WTO where ~it~ 1~is the lowest frequency of w~(q),which occurs in the pair excitation spectrum. —

In PbTe the wave vector-dependence of the L.O. modes is measured and discussed in reference 9, for highly degenerate samples. The calculation of the

W

modes, however relies on the zero frequency dielectric function for the plasma in the semiconductor.

where K,, is the high frequency dielectric constant. The frequencies of the collective modes in the polaron-gas are then obtained as the zeros of the real part of the dielectric function. Re Kt0t(q,~.,)

=

0

(3)

This approximation is not allowed in virtue of the acoustic resonant plasmon branch, which as is shown in the present letter repels the non-renormalised quantum-mechanics. modes in 0a of way analogous to the non-crossing theorem’

Vol. 14, No. 12

COLLECTIVE EXCITATIONS OF THE POLARON-GAS

Recently the optical absorption of CdO was measured by Finkenrath et al..11 These authors observe relative minima in the vicinity of both the LO. frequency and the cJ~(0)frequency. We suggest that these minima are the onsets of phonon and

1341

plasmon emission by the carriers. According to the previously developed theories,4’9 no minimum would occur in the vicinity of the L.O. phonon frequency, because of the absence of the gap between C~$~rnand C.I~TOin these theories.

REFERENCES 1. 2.

KUPER C.G. and WHITFIELD D.C., (editors) Polarons and Excitons, Oliver & Boyd, London (1967). DEVREESE J.T., (editors) Polarons in Ionic C,ystals and Polar Semiconductors, North Holland, Amsterdam (1972).

3.

LARSEN D.M., in reference 2.

4. 5.

VARGA B.B.,Phys. Rev. 137, A1896 (1965). MOORADIAN A. and WRIGHT GB.,Phys. Rev. Lett. 16,999 (1966).

6. 7.

PINES D. and NOZIERES P. in The Theory of Quantum Liquids, Benjamin, New York (1966). COHEN MARVIN L. in Superconductivity (edited by PARKS) Marcel Dekker, New York (1969).

8.

UNDHARD J., Kgl. Danske Vidensk.ab. Seiskab. Mat. Fys. Medd. 28,408 (1954).

9.

COWLEY R.A. and DOLLING G.J.,Phys. Rev. Lett. 14, 549 (1965).

10.

VAN NEUMANN J. and WIGNER E.P., Z. Phys. 30,467 (1929).

11.

FINKENRATH H., FRICKE W. and UHLE M.,Phys. Status Solidi 60, 341 (1974).

Les fr~quencesrenormalisées des modes collectives d’un gas de polarons sont calculées. La dispersion de ces fr~quencesest trIs forte. Cette dispersion est la consequence du fait que les phonons optiques longitudinaux interagissent avec la resonance des plasmons dans la region d’excitations des paires.

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