Solid State Communications, Vol. 81, No. 6, pp. 525-527, 1992. 003%1098/92 $5.00 + .OO Printed in Great Britain. Pergamon Press plc

EXCITONIC EFFECTS IN OPTICAL SPECTRA OF A QUASI-ONE-DIMENSIONAL ELECTRON GAS

Pawel Hawrylak

Institute for Microstructural Sciences, National Research Council of Canada, Ottawa, KlA OR6, Canada

(Received 5 September 1991 by J. Taut)

The radiative recombination of electrons in a quasi-one-dimensional gas with valence band holes localized by potential fluctuation or acceptors is investigated. For holes localized by potential fluctuations the emission spectrum contains the singularity at low energy due to the divergence in single particle density of states and a singularity at the Fermi level (Fermi Edge Singularity) due to the electron-hole pair excitations. For holes localized on acceptors only the Fermi Edge Singularity survives as the dominant structure in the spectrum.

1. INTRODUCTION

THERE IS currently a great deal of experimental interest in fabricating modulation doped quantum wires [ 1, 21. For very low carrier density such systems would provide an experimental realization of a one- dimensional electron gas. The response of an electron gas to a localized perturbation is characterized by infrared divergencies and manifests itself in the form of Fermi Edge Singularity (FES) [3]. Here we inves- tigate theoretically the photoluminescence spectrum of a quasi-one-dimensional electron gas using three basic assumptions: (a) electrons are treated as non- interacting (b) valence holes are localized, (c) screen- ing of electron-hole interaction is instantaneous. These assumptions have been used successfully in explaining [4] the emission spectrum in modulation doped quantum wells and heterojunctions [5-71.

2. THE MODEL

We consider a quantum wire filled with N elec- trons occupying a single conduction subband. After the illumination process a photo-excited electron relaxes to its lowest energy state and a hole in the valence band becomes localized by a defect. If the defect is positively charged it corresponds to an accep- tor [4, 51, if it is neutral it describes e.g. wire-width fluctuations [6]. In the emission process an electron makes a transition into the localized level annihilating a hole with the simultaneous emission of a photon. The single particle states and energies of (N + 1) conduction electrons in the presence of a valence hole trapped on a defect are denoted by Ik) and ek. The single particle states and energies of (N) conduction electrons and one electron trapped on a defect (no valence hole) are denoted by In) and e, and I/z) and

-w, respectively. This is our final basis and the normal state one would like to probe optically.

The emission spectrum E(o) has been derived directly [4] from Fermi’s Golden Rule and we only repeat the essential elements here. E(w) can be written as a Fourier transform of a time dependent emission function E(t):

E(o) = 2 Re 1 dt e-ico-wma,)‘E(t), 0

(1)

where o,,, is the maximum photon frequency given by the difference between the ground state energies of N + 1 particles before and after photon emission (we set h = 1). The time dependent spectra1 function E(t) is given in terms of vertex G and self-energy C func- tions as:

E(t) = e iET/o,

e-iC(‘) g, m, e+““‘G,,.(t)m,, (2)

Here ml are the single particle transition matrix elements m, = p”, (hlA.> corresponding to a transition from a conduction state IA) to a localized state (h). pUc are conduction to valence momentum matrix elements and @\A) is the overlap of the conduction and local- ized electron envelope wavefunctions. The time evol- ution of the vertex (G) and self-energy (C) functions is governed by a set of nonlinear differential equations:

i G,,.(t) = - ieAGi,x(t) + i c G,,.(t)erG,-,.(t), 1”

l C(t) = 2 C ej,GA,l(t). I

The filling of phase space of initial states enters via the

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526 EXCITONIC EFFECTS IN OPTICAL SPECTRA Vol. 81. No. 6

initial condition for the matrix G(0):

G,,,(O) = 1 <~lk>(kl~‘>. k<kF

The overlap matrix elements (kin) between the initial and final states are solutions of the Wannier equation:

(4)

The electron-valence hole interaction Qk. is the change in the one electron potential between initial and final bases as seen from the initial basis. In the case of an acceptor the initial basis corresponds to a charge neutral complex while the final basis corre- sponds to a repulsive ionized acceptor. For the hole trapped by a neutral defect the initial basis contains an attractive center. In the emission process the annihil- ation of the hole as seen from the initial basis corre- sponds to the switching on of the repulsive potential. Hence the emission process always involves switching on a repulsive potential in the initial basis.

3. RESULTS

We calculate the emission spectrum for two cases: (a) a quasi 1-D electron gas confined by a parabolic potential interacting with a statically screened point like hole potential, (b) an ideal 1-D system of size L with a finite number of electrons N interacting with a model repulsive potential of the acceptor.

We first consider a parabolic confinement. The electron gas in a parabolic potential has been inves- tigated in detail by e.g. Hu and O’Connell [8]. The parabolic potential leads to subband energy spacing w, and a corresponding length scale b = (l/ma,)“‘, where m is the electron effective mass. The screened attractive interaction produced by a localized photo- hole can be written as V,(q) = V’(q)/&(q) if only a single subband is occupied and a hole is treated as a point charge. The bare interaction is given by V’(q) = (e2/co) exp (q2b2/8)Ko(q2b2/8), where a0 is the back- ground dielectric constant and K. is a Bessel function. The dielectric function c(q) is given by:

c(q) = 1 + exp W)2/4)~oWY/4) In (W& + d*

Here a, is the effective Bohr radius, KF is the Fermi wavevector, and d describes the smearing of the screen- ing singularity at 2Kr. The energy is measured in effective Rydberg Ry. We solve the Wannier equation (equation (4)) by discretizing the continuous energy spectrum on a set of Gaussian levels and weights. The initial condition for matrix G is then constructed and propagated in time using the standard Runge-Kutta techniques. In Fig. 1 we show a typical calculated

-5 -4 -3 -2 -1

energy CRY)

Fig. 1. The emission spectrum (solid line) from a quasi-one-dimensional electron gas confined by a parabolic potential. The energy separation between subbands is w. = 2 Ry, Fermi energy EF = 1 Ry. The hole is treated as a statically screened attractive point charge. The joint single particle density of states is shown by the dashed line.

emission spectrum for a parabolic wire. An energy spacing w. = 2 Ry, and Fermi energy EF = 1 Ry have been assumed. The spectrum consists of two peaks. The low energy peak reflects the l/(E)“* diver- gence of the single particle density of states in the absence of the photo-excited valence hole. This is illustrated in the single particle density of states (dashed curve). The second peak at higher energy corresponds to the Fermi Edge Singularity and reflects the electron-hole pair excitation spectrum of a quasi- one-dimensional electron gas. We find that in most cases considered the peak at low energies is stronger than the peak in the vicinity of the Fermi energy.

To test some of our results based on discretization of the energy spectrum in equation (4) we consider a simple example of an ideal 1-D channel of length L. In the center of the channel we place a repulsive impurity (a crude model of the negatively charged acceptor) in the form of a barrier of width a and height V,. Upon illumination a hole in the valence band is localized by a negatively charged acceptor. We assume perfect screening i.e. the potential of the hole screens the potential of the impurity completely. Hence the initial basis corresponds to electronic states on a ideal I-D channel of length L. With such a simple single particle basis the energy spectrum, optical matrix elements, and overlap matrix elements (kin) between initial and final basis can be easily calculated. In Fig. 2 we show the emission spectrum of 15 electrons occupying 40 levels due to the recombination of one electron with the valence hole. We chose V, = 4 Ry, a = 0.5 uB and

Vol. 81,‘No. 6 EXCITONIC EFFECTS IN OPTICAL SPECTRA 527

the state of the valence hole. For holes localized by potential fluctuations the emission spectrum consists of the low energy peak due to the single particle den- sity of states and a Fermi Edge Singularity at higher energies. For the emission spectrum due to recom- bination on acceptors a single asymmetric peak associated with FES is expected.

-0.5 ~

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0

mm

Fig. 2. The emission spectrum for an ideal 1-D channel of length L = lOa, repulsive impurity (model accep- tor) with size a = 0.5 uB and potential strength J’, = 1 Ry. There are 15 electrons present and 40 energy levels were included in the calculation. The dashed line shows the joint single particle density of states. The energy is measured in units of Fermi energy EF.

L = lOa. The dashed line shows the single particle density of states which peaks at low energies simply because the overlap of the hole and electron wave- function is large for low energy states. However, the shake up effects of the Fermi sea in the emission process shift the oscillator strength toward higher energies resulting in a single asymmetric peak. Similar results have been found for a parabolic confining potential and a repulsive impurity.

In summary, we find that the emission spectrum of a quasi-one-dimensional electron gas depends on

1.

2.

3.

4. 5.

6.

7.

8.

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