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Excitons at High Density
Edited by H. NEKAH and .S ENITIKIN
stnetnoC
I. Introduction
Survey. H. NEKAH and .S NIKmNE . . . . . . . . . . . . . . 2
Introduction to Exciton Spectroscopy. .S ENITIK1N ....... 5
II. Biexcitons
Properties of Biexcitons. S. NIKmNE . . . . . . . . . . . . . 81
Biexcitons- Bose Condensation and Optical Response. E. HAN~ 34
HI. Electron-Hole Droplets
Properties of Electron-Hole Drops in Germanium Crystals.
V. .S V~O~GAB . . . . . . . . . . . . . . . . . . . . . . . 27
Theory of Electron-Hole Drops in Germanium and Silicon.
T. M. RICE . . . . . . . . . . . . . . . . . . . . . . . . 19
IV. Biexcitons and Droplets
Spectroscopic Study of Exciton-Exciton Interaction (Biexcitons,
Drops) in Semiconducting Crystals. .B V. VOKIVON ...... 601
Exciton Condensation in Germanium. A. A. VEHChGOR ..... 721
V. Special Optical Properties of Excitons at High Density
Gigantic Oscillator Strength Inherent in Exciton Complexes.
E. I. ABHSAR . . . . . . . . . . . . . . . . . . . . . . . 051
Interaction between Excitons at High Concentration. R. ,YVEL
A. ,SAVIB J. B. ,NURG and .S NmmNE . . . . . . . . . . . . 171
VI. Laser Action of Excitons
Theory of Stimulated Emission by Excitons. H. NEKAH and
S. ENITIKIN . . . . . . . . . . . . . . . . . . . . . . . . 291
Experimental Investigation on the Competition of Stimulated
Emissions Involving Excitons. R. ,YVEL J. B. ,NURG and .S Nn<mNE 112
Experimental Studies of Excitons at High Densities. K. L. EELKAHS 221
IV stnetnoC
VII. Excitonic Polaritons at Higher Densities
Tests of Validity of Spatial Dispersion Theories on Lead Iodide
Crystals Spectra. M. ,NNAMSORG J. ,NNAMLLEIB and S. NIKITINE 242
Medium and High Polariton Densities. H. MAHR . . . . . . . . 265
Polaritons at High Intensities and in Bose Condensed Exciton
Systems. H. HAKEN, J. GOLL, and A. SCHENZLE 285
. . . . . . . .
Subjeet Index .... . . . . . . . . . . . . . . . . . . . 297
Classified Index of Authors and Titles . . . . . . . . . . . . . 299
yevruS
H. NEKAH and .S ENITIKIN
Excitons at high density and polaritons offer a new field of research within
the framework of semiconductor physics. The pioneering work of
Nikitine, Gross and others has established that hydrogen-like excitons
may be created in semiconductors. These excitons are made up of an
electron of the conduction band and a hole of the valence band, usually
described by effective masses, which are coupled together by the Coulomb
interaction, possibly modified by polarization effects. At this stage a
great variety of Rydberg constants and further generalizations beyond
the hydrogen atom are found, because both the effective masses and the
effective Coulomb interaction depend on the different crystals involved.
Furthermore, in certain crystals tensor masses must be taken into ac-
count, and excitons may belong to different valleys of the conduction
and valence band. Finally, the effective moments of the electron and
the hole may differ from that of free electrons.
Detailed investigations of the different bound states of excitons are
still in progress, but a new field has recently emerged thanks to the
possibility of creating high concentrations of excitons. The usual way
of producing such high concentrations is by shining laser light on the
insulating crystals. In large classes of crystals excitons may form ex-
citonic molecules ("biexcitons") in analogy to the hydrogen molecule.
On the other hand, the effective interaction between excitons may also
be repulsive, hence bose condensation of excitons may be possible. Bose
condensation of biexcitons has also been considered and there are some
experimental indications that such an effect exists.
Another field currently eliciting great interest si the formation of
electron-hole droplets. Here the excitonic binding has broken up and
some sort of metallic drop has been formed, bound together mainly by
the Coulomb exchange interaction between the electrons and holes.
A further possibility that has been discussed is the formation of poly-
excitons. It is even suggested that excitons may form a crystal, pre-
sumably of the type of a quantum crystal, within the crystal. Thus -xe
citons would form a new type of matter within matter. We are convinced
that the various possibilities are by no means exhausted, because the
mass ratio between the exciton and hole may vary over a wide range.
yevruS 3
Many of the properties of excitons are investigated by optical means,
so that the detailed study of the interactions between light and excitons
at high intensity is of great importance. We have therefore included in
this volume several articles dealing with the polariton concept. Polaritons
consist of an exciton and a photon. At high concentrations of excitons
or polaritons new effects of light propagation appear. Thus, one might
expect the dispersion curve of the polariton to be changed and perhaps
even to find self-induced transparency of excitons. Moreover, inter-
actions between polaritons can give rise to a number of different scat-
tering processes.
Excitons at high density may also produce stimulated emission of
light; this process is of great practical interest because of the extremely
high gain of such lasers. The present authors think that this field is
worthy of particularly intensive development because many different
processes may be competing with each other, and both temporal and
spatial transient states may play an important role. It is hoped that the
present book will stimulate further research work in this extremely
fascinating field. The articles in this book (except those by Novikov
and Rashba) were presented at an international symposium at Tonbach,
Germany. We wish to thank the Deutsche Forschungsgemeinschaft and
the Land Baden-Wtlrttemberg for financial support and Mrs. Funke
for her efficiency in organizing this meeting.
Introduction to Exciton Spectroscopy*
S. NIKITINE
Contents
.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
.2 Early Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
.3 Classification of Exciton Transitions . . . . . . . . . . . . . . . . . . . . 6
a) First-Class Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 7
b) Second-Class Transitions . . . . . . . . . . . . . . . . . . . . . . . 8
)c Indirect Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . 9
d) High-Energy Spectra . . . . . . . . . . . . . . . . . . . . . . . . . 9
.4 The Spectrum of Cu20 . . . . . . . . . . . . . . . . . . . . . . . . . 9
.5 Solid-State Spectroscopy and Band Structure ................ 21
Recent Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
1. Introduction
This short paper should not be considered as a review of exciton spec-
troscopy but only as an introduction. For detailed information, the
reader should consult review papers -1-4. Some of these papers are,
however, rather out-of-date. Exciton spectroscopy has developed very
rapidly and, to the author's knowledge, no attempt has been made to
summarize its present state. This introduction is not intended to fill this
gap but to facilitate the approach to the reports that follow. It is limited
to classic results: It is hoped that the non-specialist reader will find
these remarks helpful.
2. Early Observations
Though some work was carried out on solid-state spectroscopy in the
early twenties 5, 6, modern work was started some 22 years ago by
Hayashi Mazakuzu 7, by Gross 8 and his group, and by the Stras-
bourg group 9. Other groups joined this research later 10.
; This work was carried out within the frame of the agreement on French-German
scientific cooperation between C.N.R.S., Paris, Universit6 Louis Pasteur, Strasbourg, and
Universitiit Stuttgart.
6 S. Nikitine
Semiconductors and insulating crystals, ionic or covalent, exhibit
two types of spectra in the low-energy absorption edge:
a) This edge may exhibit one or several sharp absorption peaks.
In some cases, they form series converging to a continuum. The series
are sometimes hydrogen-like. The peaks have been shown to be due to
the optical formation of excitons, an excited state of the crystal in which
an electron and a hole are bound in a state resembling that of the
hydrogen atom.
b) The edge may exhibit absorption steps. These steps have been
recognized to be due to band-to-band transitions or to transitions to
exciton states with the cooperation of phonons. They are known as
indirect transitions.
The formation of excitons is essential for an understanding of this
part of the spectrum. In high-energy absorption rather broad peaks are
also observed. They might be ascribed to exciton transitions of higher
energies, or to band-to-band transitions at points in the Brillouin zone
where the joint density of states has a pronounced maximum.
Equidistant peaks apparently forming vibrational satellites of exciton
peaks have been reported 11. The separation of these peaks agrees
well with the LO phonon of the crystal.
It has been shown by the Strasbourg group that the reflection spectra
sometimes exhibit drastic anomalies 12. Most of these anomalies cor-
respond to the first and second exciton peaks of a series. Some exciton
spectra are, however, weak and do not give rise to anomalies. The anoma-
lies consist of a pronounced maximum and a sharp minimum of reflection,
called excitonic residual ray and missing ray, respectively. Maxima of
reflection can also be observed in the high-energy spectrum and can be
ascribed to high-energy exciton transitions or to transitions at par-
ticular points in the Brillouin zone.
Some weak lines on the low-energy side of exciton peaks depend on
the quality of the crystal and have been named "sensitive lines" by the
Strasbourg group 13. They have been ascribed to complexes formed
by excitons captured by defects or impurities.
Exciton luminescence was first observed by Grillot 14 independ-
ently by the Strasbourg group -15, and by Archangelskaja et al. 16.
This luminescence exhibits narrow and broad lines and sometimes
vibrational satellites.
3. Classification of Exciton Transitions
These observations have stimulated theoretical work and it has been
shown by Elliott 17 and Haken 18 that several classes of exciton
spectra can be described by the theory 3.
Introduction to Exciton Spectroscopy 7
a) First-Class Spectra
This class corresponds to transitions according to the selection rules
k = 0, A k = 0, where k is the wave vector of excitons. In this class ex-
citons in a spectroscopic S state are formed. A series of lines is expected:
v = o~v - R~x/n 2 with n=1,2,3..,
xeR is an excitonic Rydberg constant, xeR = (R/e z) (#/mo), where e is a
dielectric constant, 0m the mass of the electron in the vacuum at rest,
# the effective mass of the exciton, and R the atomic Rydberg constant.
Haken's correction 18 has to be used for the calculation of .5
The oscillator strength per unit cell is of the form f = const/e 3 n 3 and
typical values are about 10-2 to 01 -3. For v > ~ov the theory predicts
a continuous absorption, which corresponds to a band-to-band tran-
sition taking into account electron-hole interaction.
Some observed spectra are tentatively ascribed to this class. Though
the quantitative agreement is in many cases not very good, the quali-
tative description and the orders of magnitude are well predicted by the
theory. It has to be remembered that the theory is based on assumptions
that are only roughly realistic in many cases. CuI, CuC1 and CuBr are
quoted as examples of such transitions 19; one of these spectra is
shown in Fig. .1 Only this class of spectra shows excitonic residual
rays 12: Fig. 2 shows one of the residual rays reported by the Stras-
bourg group 20.
c.
31
d:l
& t.
t-
"O
o ,.
3550 36'00 36'50 37'00 37.50 38'00 3850 %(A~
Fig. .1 Spectrum of CuC1 (thin film) at 4.2 K showing a number of peaks. The spectrum
is ascribed to the first-class type of exciton transition. The spectrum has a somewhat com-
plicated character on account of the overlap of two series (the sharp and diffuse series)
separated by the spin-orbit splitting of the valence band.Vibrational satellites can be seen
on the high-energy side
8 S. Nikitine
%(R
100
80
O6
4O
20"
(~,1
x
a6oo 0 ' 3700 ' 38'00 ' a9bo , 4obo
Fig. .2 Reflection spectrum of CuC1 at 4.2 K showing strong anomalies, excitonic residual
rays and missing rays for the first line of both series
b) Second-Class Transitions
Elliott -17 has shown that, when the matrix element of the transition
moment is zero on account of the valence-band (V.B.) and the con-
duction-band (C.B.) wave functions having the same parity at the F
point, another class of transitions is possible in the electric dipole ap-
proximation. The selection rules are A k = 0, and k + 0 but small. In this
case excitons in a spectroscopic P state are created. The series of lines
is still given by
v = v~ - Rex/n 2 but n = 2, 3, 4 ....
The transition to the n = 1 state is forbidden in the dipole approximation
but can take place in the quadrupole or magnetic dipole approximations.
The oscillator strength is
f = const (n z - 1)/~32 n s .
Again Haken's corrections must be used for .e f is of the order of mag-
gnitude of 10-6 For the quadrupole transition the order of magnitude
is of about 10-9 For v > oov the absorption is again a continuum.
First- and second-class transitions are not expected to be possible for
the same transition unless the crystal has no center of symmetry.
It has been shown by the Strasbourg group that the yellow and green
series of Cu20 are well described by the theory of second-class transi-
tions. Indications in favor of such transitions have been reported for
SnO2, but transitions of this class are exceptional 211.
