Optical Properties of Excited Silicon an

Optical Properties of Excited Silicon and Germanium at Low Temperatures - CaltechTHESIS
CaltechTHESIS
A Caltech Library Service
About
Browse
Deposit an Item
Instructions for Students
Optical Properties of Excited Silicon and Germanium at Low Temperatures
Citation
Lyon, Stephen Aplin
(1979)
Optical Properties of Excited Silicon and Germanium at Low Temperatures.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/8gcb-7j35.
Abstract
Part I of this thesis deals with 3 topics concerning the
luminescence from bound multi-exciton complexes in Si. Part II presents a model for the decay of electron-hole droplets in pure and
doped Ge.
Part I.
We present high resolution photoluminescence data for Si doped With Al, Ga, and In. We observe emission lines due to recombination of electron-hole pairs in bound excitons and satellite lines which have been interpreted in terms of complexes of several excitons bound to an impurity. The bound exciton luminescence in Si:Ga and Si:Al consists of three emission lines due to transitions from the ground
state and two low lying excited states. In Si:Ga, we observe a second triplet of emission lines which precisely mirror the triplet due to the bound exciton. This second triplet is interpreted as due to decay of a two exciton complex into the bound exciton. The observation of the second complete triplet in Si:Ga conclusively demonstrates that more than one exciton will bind to an impurity.
Similar results are found for Si:Al. The energy of the lines show that the second exciton is less tightly bound than the first in Si:Ga. Other lines are observed at lower energies. The assumption of ground state
to ground-state transitions for the lower energy lines is shown to produce a complicated dependence of binding energy of the last exciton on the number of excitons in a complex. No line attributable to the decay of a two exciton complex is observed in Si:In.
We present measurements of the bound exciton lifetimes for the four common acceptors in Si and for the first two bound multi-exciton complexes in Si:Ga and Si:Al. These results are shown to be in agreement with a calculation by Osbourn and Smith of Auger transition rates for acceptor bound excitons in Si. Kinetics determine the relative populations of complexes of various sizes and work functions, at temperatures which do not allow them to thermalize with respect to one another. It is shown that kinetic limitations may make it impossible to form two-exciton complexes in Si:In from a gas of free excitons.
We present direct thermodynamic measurements of the work functions of bound multi-exciton complexes in Al, B, P and Li doped Si. We find that in general the work functions are smaller than previously believed. These data remove one obstacle to the bound multi-exciton complex picture which has been the need to explain the very large apparent work functions for the larger complexes obtained by assuming
that some of the observed lines are ground-state to ground-state transitions. None of the measured work functions exceed that of the electron-hole liquid.
Part II.
A new model for the decay of electron-hole-droplets in Ge is presented. The model is based on the existence of a cloud of droplets within the crystal and incorporates exciton flow among the drops in the cloud and the diffusion of excitons away from the cloud. It is able to fit the experimental luminescence decays for pure Ge at different
temperatures and pump powers while retaining physically
reasonable parameters for the drops. It predicts the shrinkage of the cloud at higher temperatures which has been verified by spatially and temporally resolved infrared absorption experiments. The model also accounts for the nearly exponential decay of electron-hole-droplets
in lightly doped Ge at higher temperatures.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
(Applied Physics) ; Optical properties, silicon, germanium
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
McGill, Thomas C. (advisor)
Smith, Darryl L. (advisor)
Thesis Committee:
Unknown, Unknown
Defense Date:
24 July 1978
Record Number:
CaltechTHESIS:07182014-141717487
Persistent URL:
DOI:
10.7907/8gcb-7j35
Default Usage Policy:
No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
8568
Collection:
CaltechTHESIS
Deposited By:
Dan Anguka
Deposited On:
18 Jul 2014 21:54
Last Modified:
26 Nov 2024 23:07
Thesis Files
Preview
PDF
- Final Version
See Usage Policy.
15MB
Repository Staff Only:
item control page
CaltechTHESIS is powered by
EPrints 3.3
which is developed by the
School of Electronics and Computer Science
at the University of Southampton.
More information and software credits
OPTICAL PROPERTIES OF EXCITED SIL ICON AND GER~~NIUM
AT LOW TEMPERATURES

Thes i s by
Stephen Aplin Lyo n

In Partia l Fulfill ment of the Requirements
for t he Degree of
Doctor of Philosophy

California In stitute of Tec hnol ogy
Pasadena, California
1979
(S ubmitted July 24, 1978)

TO tW FATHER

iii
ACKNOWLEDGEMENTS
I would like to thank Drs. T. C. McGill and D. L. Smith for
their assistance during the course of this work.

Their guidance and

encouragement has been of great value.
I am also indebted to Drs. J. W. Mayer and R. B. Hammond for
convincing me to become involved in research early in my graduate
career and introducing me to the area in which I have worked.

Dr.

Hammond also helped me get started in the laboratory on the early
part of my work.

I had numerous helpful discussions with Dr. M.

Chen, K. R. Elliott, A. Hunter, G. Mitchard, G. Osbourn, Dr. D. S.
Pan, and J. N. Schulman.

I owe special thanks to G. Osbourn for his

calculation of bound exciton Auger rates used in Chapter 3, K. R.
Elliott for obtaining the solution to the rate equations for bound
multi-exciton complexes used in Chapter 4, and to Dr. M. Chen for
the experimental data quoted in Chapter 5 as well as his valuable
input in the formulation of the model presented there.
I would also like to thank the Extrinsic Silicon group at Hughes
Researc h Laboratories for supplying and characterizing many of the
samples I have used.

Without th eir generous assistance, many of these

experiments would have been impos s ible.

I would like to extend my

gratitude to V. Snell for typing this thes i s and for her excellent
and cheerful secretarial help.
I would like to thank Dr. J. M. Warlock for allowing me to use
hi s equipment and work in hi s l aboratory during a visit to Bell Labs.

iv
For financial support I owe thanks to the California Institute
of Technology, the Gilbert Fitzhugh Foundation, the ARCS Foundation,
the Office of Naval Research, and the Advanced Research Projects
Agency.
Finally, ar.d most importantly, I would like to thank my wife,
Gail, for her encouragement and support during my years as a graduate
student.

ABSTRACT
Part I of this thesis deals with 3 topics concerning the
lum inescence from bound multi-exciton complexes in Si.

Part II

presents a model for the decay of electron-hole droplets in pure and
doped Ge.
Part I.
We present high resolution photoluminescence data for Si doped
withAl, Ga, and In.

We observe emission lines due to recombination

of electron-hole pairs in bound excitons and satel l ite lines which
have been interpreted in terms of comple xes of several excitons bound
to an impurity.

The bound exc iton luminescence in Si:Ga and Si:Al

consists of three emiss ion lines due to transitions from the ground
state and two low lying excited states.

In Si:Ga, we obse rve a

second triplet of emission lines which precisely mirror the tripl et
due to the bound exciton.

Thi s second triplet is interpreted as due

to decay of a two exciton complex into the bound exciton.

The ob-

servation of the second compl ete triplet in Si:Ga conclusively
demon strates th at more than one exc iton will bind to an impurity.
Similar re s ults are found for Si:Al.

The energy of the lines show

that the second exciton i s less ti ghtly bound than the fir s t in Si :Ga .
Other lines are observed at lower energ i es.

The assumptio n of ground-

state to ground-state tra nsitions for the lower energy lines i s sr.own
to produce a complicated dependence of binding energy of the la st
exciton on tre number of excitons in a complex.

No line attributable

to the decay of a two exciton complex i s observed in Si :In.

vi
We present measurements of the bound exciton lifetimes for the
four common acceptors in Si and for the first two bound multi-exciton
complexes in Si:Ga and Si:Al .

These results are shown to be in agree-

ment with a ca l culation by Osbourn and Smith of Auger transition
rates for acceptor bound excitons in Si.

Kinetics determine the

relative populations of comp l exes of various s i zes and work function s ,
at temperatures whi ch do not allow them to thermalize with respect to
one another.

It is shown that kinetic li mitations may make it impos-

sible to form two- exciton comp l exes in Si:In from a gas of free
excitons.
We present direct thermodynamic measureme nts of the work functions of bound multi-exciton compl exes in Al, B, P and Li doped Si.
We find that in genera l the work functions are sma ller than previously
beli eved.

These data remove one obstacle to the bound multi-exciton

complex picture which ha s been the need to expl ain the very large
apparent work functions for the l arger compl exes obtained by assuming
that some of the observed lines are ground-state to ground-state
tran s itions .

None of the measured work functions exceed that of the

electron-hol e l iquid.

Part II
A new model for the decay of el ectron-hole-dropl ets in Ge i s
prese nted.

The mod el i s based on the ex i ste nce of a cl oud of dropl ets

within the crystal and incorporates exciton flow among the drops in
the cloud and th e diffu s ion of excitons away from the cloud.

It is

vii
able to fit the experimental luminescence decays for pure Ge at different temperatures and pump powers while retaining physically
reasonable parameters for the drop s .

It predicts the s hrinkage of

the cloud at higher temperatures which has

been verified by spa tially

and temporally resolved infrared absorption experiments.

The model

also accounts for the nearly exponential decay of electron-hole-droplets
in lightly doped Ge at higher temperatures.

viii
Parts of this thesis have been or will be published under the
following titles:
Chapter 2:

Edqe Luminescence Spectra of Acceptors in Si; Implications
for Multiexciton Complexes, S. A. Lyon, D. L. Smith and
T. C. McGill, Phys. Rev . B}I, 2620 (1978).

Chapter 3:

Bound Exciton Lifetimes for Acceptors in Si, S. A. Lyon,
G. C. Osbourn, D. L. Smith and T. C. McGill, Solid State
Commun. 23, 425 (1977).

Chapter 4 :

Thermodynamic Determination of Work Functions of Bound
Multiexciton Compl exes> S. A. Lyon, D. L. Smith, and T. C.
McGill, Phys. Rev. Lett. il• 56 (1978).
Temperature Dependence and Work Functi ons of Bound
Mu l tiexciton Complexes in Si , S. A. Lyon, D. L. Smith
and T. C. McGill, (in preparation).

Chapter 5:

Trans i ents of the Photoluminesce nce from EHD in Doped and
Undoped Ge, M. Chen, S. A. Lyon, K. R. Elli ott, D. L. Smith
and T. C. McGil l , Il Nuovo Cimento 398 #2 , 622 (1 977).
Transients of t he Photoluminescence Intensities of the
Electron-Hole-Droplets in Pure and Do ped Ge, M. Chen,
S. A. Lyon, D. L. Smith and T. C. ~lcGill, Phys . Rev. 817,
(1 978) .

ix
Publications not included in this thesis are :
Temperature Dependence of Relative Emission Intensities Via
Symmetry Allowed Phonon Processes in Si and Ge, D. L. Smith,
R. B. Hammond, M. Chen, S. A. Lyon and T. C. McGill, Proceedings
of the Thirteenth International Conference on the Physics of
Semiconductors, Rome , 1976, p. 1077.
Transient Decay of Satellite Lines of Bound Excitons in Si:P,
A. Hunter, S. A. Lyon, D. L. Smith, and T. C. McGill {in
preparation).
Fine Structure in the Photoluminescence Spectra of Lithium
Doped Silico~, S. A. Lyon, D. L. Smith and T. C. McGill
(in preparation).

TABLE OF CONTENTS
AC KNOVJL EDGH1EriTS

i i;

ABSTRACT
CHAPTER l :

INTRODUCTION

Background

Outline of Thes i s

III

Optical Processes in Indirect Semiconductors

REFEREfKES

PART I
CHAPTER 2:

FINE STRUCTURE OF ACCEPTOR BE AND BMEC
Itl Si

Introduction

J-J Splitting of Acceptor BE

III

Experiment

Experimental Results

Discussion and Conclusions

REFERENCES
CHAPTER 3:

LIFETIMES OF BE AND BMEC BOUND TO ACCEPTORS
IN Si

Introduction

Experimental Results

III

Auger Calculation of BE Lifet imes

Conclusions

REFERENCES
CHAPTER 4:

TEMP ERATURE DEPENDENCE AND WORK FUNCTIONS
OF BOUND MULTIEXCITON COMPLEXES IN Si

Introduct ion

Anal ys i s of the Temperature Data

III

Experimental r1et hods

Experimental Re s ults

Discussion and Conclusions

116

REFERENCES

120

PART II
CHAPTER 5:

CLOUD MODE L FOR THE DECAY OF EHD IN
PURE AND DOPED Ge

122

In t r odu ct ion

123

Desc riotion of the Model

127

III

Detailed Mathematical Formulation
of r1od el

136

Results of Calcul ation and Comparison
with Expe riments

146

Summary and Conclus ion

156

REFERENCES

158

CHAPTER 1
INT RODUCTION

Background
One of the important aspects in attempting to understand a

material is determining its el ectronic properties.

In general this

means gaining some knowledge of the possible electronic excitations
of the system. Since a compl ete description of 10 23 interacting
atoms is impossible, the first approach is to determine the s ingl e
particle excitations in a sol id .

Thi s approach corresponds to a

determination of the band struct ure.

Thi s one electro n picture is

very powerful and is the basis for much of our understanding and
most of the technology associated with semiconductors.

However,

treating the electronic excitations of a semiconductor as being independent of one another is an approximation which becomes invalid
at low temperatures and high excitation densities.

Under these

conditions it is possib l e to produce electron ic excited states of
the solid which are qualitatively different from those described in
the single particle, or Hartree-Fock, picture.

This thesis wi ll be

co ncerned with properties of excitations of this type in crystals
of s ilicon and ~ermanium.
One of the simplest excitations of a semiconductor not contained in the usual Hartree-Fock picture is the free exciton (FE)(l).
The FE consists of an electron and hole bound together by their
mutual coulomb attraction.

Typical ionization energies for excitons

in semiconductors range from about 1 to 20 meV (see Table 1.1) .

low temperatures where kT i s smal l compared to the exciton ioni zation
energy, free electrons and hol es in a crystal will bind to form

Table l.l
Free Excitons

* 0
Bohr radiu s , a (A)

Silicon

Germanium

43 (a)

114 (a)

14.7 (b)

4.15 (c)

Dissociation Energy
E (meV)

Effective ma ss , mex

.335m

Lifetime, Tex (~s)
EHD
Work function,~EHD (meV)

8.2

(d)

1.8

(c)

Pair density, n0 (em -3 )

3.3xlo18 (d)

2.4xlo 17 (e)

Pairs per droplet, v

6.6xl0 6

10 6-1o8

(f)

(g)

Total Fermi energy, EF
(EF

+EF )

(meV)

Lifetime, T ( ~s )

22 . 2

(d)

6.43

- .2

- 28

6.5

2-10

(e)

Critical Temperature

Radiu s , R0 ( ~m)
a)
b)
c)
d)
e)
f)
g)
h)
i)

(h)

( i)

obtained using E0 = e2 /2 E0 a *
K. L. Shakl ee and B. Nahory, Phys. Rev. Lett. 24, 942 (19 70) .
G. A. Thoma s , A. Frova , J. C. Hen sel, R. E. Miller, and P. A. Lee,
Phys. Rev. 813 , 1692 (1976) .
Ref. 28.
Ref. 16.
M. Capizzi, M. Voos , C. Benoit a l a Guillaume and J. C. McGroddy,
Solid State Commun. 16 , 709 (1975).
T. K. Lo, B. J. Feldman, and C. D. Jeffries , Phys. Rev. Lett. }1,
221 (1973).
Ref. 12.
v. S. Bagaev, et al, rFiz. Tverd. Tel a . 15, 3269 (1973)~ [ So v. Phys .
Solid State ~:-2179 (1974)].

excitons.

The ratio of the density of free carriers to the den s ity

of excitons i s given by mass action,
nenh

(1.1)

--a::

ex
where ne is the density of free electrons, nh the den s ity of free
holes, ne x the density of free excitons, and E0 the dissociation
energy of an exciton.
In order to understand the optical properties cf an exciton,
we nezd an approximation to its wave-function.

In a two band model,

and negl ec ting spin, the wave-function can be written ( 2 ) in terms
of the Hartree-Fock basis as,
( 1 . 2)

tVFE =

i s the one-particle wave-function for an e l ectro n in the
eke
conduction band with wavevecto r k , and x k is for a hole in the

where~

v h

valence band with wave-vector kh.

The sum on ke and kh range over

the Brillouin zone for each band.

The Fourier transform of

A(ke,kh)' F(r 1 ,r 2 ), is the "envelope functi on ", or slowly varyinq
part of the wave-function, giving its exte nt in space, and ~ and x
contain the rapidly varying periodic parts of th e Bloch fun ctions.
Putting this form for ~FE into Schrodinger's equation with the
potential e 2/ £ r, with £
the static dielectric constant, and

assuming isotropic effective masses, we obtain an equatio n for
F(r1 ,r 2 ) whose form is the same as that for a hydrogen atom.

Thus

* and
the exciton acts as a free particle with an effective mass mex'
associated center-of-mass energy, and has internal degrees of
freedom described by hydrogenic wave-functions.

The size of the

wave-function is characterized by a Bohr radius and the energy
scale i s set by an excitonic Rydberg.

Values for these parameters

for silicon and germanium are given in Table 1 .1,
Free excitons are excited s tates in silicon and germanium and
therefor e have a finite lifetime.

One decay route i s for the

electron and hole to recombine, emitting a photon .

In the indirect

materials such as Si and Ge a momentum-conserving phonon is also
emitted, and thus the energy of the photon is given by,
hv = E

ex - 1il1
· phonon

(1. 3)

where Eex , the energy of the excito n, is made up of a part due to
the center-of-mass kinetic energy and a part due to the creation
from the gro und state of the electron-hole pair in the exciton.
This l ast part is just the energy of an exciton with zero kinetic
energy, E0 = Egap - E0 , and s i nee Egap » E0 and Ega p » fln phonon'
the emitted photon has an energy just slightly less than the band
gap.

We expect the photons emitted in the decay of free excitons

to have an energy di st ribution sta rting at Egap - E0 - 1in phonon
and extending to s li ght ly high er energ i es due to their center-of-mass
motion.

At low densities the exc iton s ex hibit. a Boltzmann di stribution of kin eti c energ i es which 9ives a lumines cence lineshape ( 3 )

( 1 . 4)

where IFE(hv) i s the intensity of the luminesce nce emitted at v.
The spectrum in Fig. 1.1 was taken on high purity Si, and the electrans and holes were generated by exc iting the crystal with above
band gap light.

The line at high est energy i s due to the decay of

free excitons.

From the inset it can be seen that the theoretical

curve from Eq. 1.4 (convolved with the instrumental response function)
describes the lineshape extremely well.
At an energy lower than the free exciton in Fig. 1.1, we see a
broad line l abe l ed El ectron-Hol e Li qu id . Thi s luminescence line
was first observed in 1966 by Haynes ( 4 ) and interpreted as the
decay of one exciton in a bound pair of exc itons, a biexciton.
In 1968 Keldysh (S) suggested that at high excito n densities, the
excitons could condense in to a highly correl ated, Fermi-degenerate
pl asma of el ectrons a nd holes. Further work ( 6 · 7 ) has s hown that
the line originally ascribed to the decay of biexcitons actually
arises from the recombination of pairs within a droplet of this
plasma (EHD).

Observation of this metallic liquid within a crysta l

has been reported for several semiconductors, including Si, Ge,
GaP , ( 8 ) SiC ( 9 ), and CdS (lO). Theoretica l calculat ions show that
the liquid phase sho uld be more stable than the free exc iton in
many semiconductors (ll) _ There is now a large body of data which
s hows conclusively that the co nd ensate is a degenerate Fermi-liquid
in thermal equilibrium with the surrounding gas of free excitons (l Z)

Figure 1. 1
Photoluminescence spect rum of high purity Si at
l ow temperatures. The in set shows the FE at hi gher
resolution and t he so lid dot s are a fit us ing the
theoretica l lines hape discussed in the text. (from
R. B. Hammond, et ~. Phys. Rev. 813, 3566 (1 976). )

SILICON
LO - TO PHONON ASSISTED
RECOMBINATION RAD IATION
T= 2. 1°K , NA -N 0 = 7x I011 cm- 3

-1rTHEORY t_O 8 TO • • • •
COMPOSITE
EXPERIMENT

1--(f)

FREE
EXCITON

uoo

1.095

ENERGY (eV)

1---

E g, - hwpTOh
1.070

1.075

1.080

1.085

1.090

ENERGY (eV)

Figure l . l

1.095

1.100

At low excitation intensities (low exciton densiti es) no liquid i s
formed.

If the sample is below the liquid critical temperature and

the excitation inten sity is increa sed , a threshold for production
of droplets is reached.

Further increa ses in excitation increa se

the fraction of the volume occupied by EHD without increasing the
exciton density.

From meas ureme nts of the excitation threshold and

liquid den s ity as a function of temperature, approximate phase
diagrams have been determined for the exciton-EHD system (l 2 )
The EHD ha s been s tudied by a nu~ber of techni ques, one of the
most important being the observation of its luminescence as in Fiq.
1. 1.

An above band-gap light source was used to create the electron-

hole pairs for Fig. 1.1, but other methods of excitation such as
high energy electron beams (l 3 ) or electrical injection ( 14 · 15 ) of
carriers have also been used.

A s impl e model for the lineshape

shows that its width is determined by the el ectron and hole Fermi
energies and the temperature (l 6 )_ From the Fermi energies and effect ive masses of the carriers, the density of pairs within the
liquid can be determined.

Also, the energy difference between the

free exciton edg e and the high energy edge of the EHD gives the
work fun ction for a pair in the liquid, ¢EYD (see Fig. 1.1) .

From

careful fitting of luminescence as a function of temperature, these
parameters ha ve been meas ured as well as their var iation with
temperature for Si and Ge (see Table l .1).
Another fruitful approach to the study of EHD in Ge has been
the scattering and absorption of infrared light (l ? )

Typicall y a

10
wavelength of 3.39 ~m is used due to the large absorption coefficient
in EHD, and the availability of a laser (He-Ne) at thi s wavelength.
The presence of the condensed phase within a crystal will locally
change the real part of the index of refraction.

The unexcited Ge

crystal is transparent at these wavelengths, but the local index
changes will Rayleigh scatter a probe beam .

From the angular depen-

dence of this scattering the radii of the droplets of liquid in
unstrained Ge are found to range from about 2 to 10 ~m depending on
temperature.

Measurements of the spatial dependence of the scattering

and absorption show that the droplets form a cloud within the crystal
with a pump-power dependent radiu s as l ar ge as several mm .

From

measurements of the absolute absorption, it has been determined that
the fill factor (fract ion of the crystal volume occupi ed by liquid)
within the cloud is typically about 1%.

Recent doppl er~s hifted light

scattering experiments (lB) show that the droplets are pushed into
the crystal, probably through their interaction with phonons.
A third experimental method has been to look at the break-up of
EHD in a large electric fie ld(lg).
in one part of the crystal.

A device is made with a pn junction

EHD are created in another area and al-

lowed to drift into the junction where they are pul l ed apart by the
high fields present.

This produces a current s pike in the external cir-

cuit, and the number of carriers in the droplet i s approx imately equal
to the total charge within the spike. From the luminescence and light
scattering we know the density and s ize of a droplet and thus the
total numb er of carriers .

The value obtained from the junction noi se

experiment is in good agreement with the value calculated from the
other two experiments.
Until now we have been dis cussing experiments performed on
crystals with very low concentrations of electrically active impurities (most important aspect (at l east technologically) of semiconductors
is our ability to change th eir electrical properties through the introduction of small amounts of impurities. If a sma ll amount of a shallow
13
impurity (- lo ;cm for Si or about 1 part per billion) i s added to
a crystal, it becomes much easier to generate EHD ( 20 ) . The excitation threshold for production of the liquid is reduced although these
impurity concentrations are insufficient to cause s i gnificant changes
in such macroscopic properties of the droplets as

their work function,

density, or radius .

It is thought that the impurities are acting as
nucleation sites for the droplets ( 20) and reduce the supersaturation

of the excitonic gas needed to produce the li quid.

If in fact excitons

are building up on impurities to form EHD, then we s hould be able to
detect "embryonic droplets" co nsisting of only a few excitons bound
to an impurity .

In Fig. 1.1 we do see some lines la beled "Bound

Excitons" which are due to the binding of excitons to impurities.

know that the pair in the initial state i s immobile, for otherwise
we would expect a characteristic broadening towards higher energies
at increa sed temperatures due to increased kinetic energy as seen for
the FE.

The "Bound Exciton" 1 ines of Fig. 1. 1 show no significant

broadening with temperature, however.

12
The binding of a single exciton to an impurity is a common
phenomenon in semiconductors.

These bound excitons (BE) occur in

both direct and indirect materials .

It has been found experimental l y

that a neutral shallow impurity will bind an exciton with about onetenth the ionization energy of the impurity (Zl)

In Si ionization

energies of common shallow impurities range from about 30 meV to
over 100 meV, and thus from this phenomenological "Haynes rule''
we expect excitons to bind with from about 3 to over 10 meV as
observed (see Table l .2).

The BE lines in Si are typically less

than .5 meV in width and it is possible to resolve the BE due to
each of the shallow impurities in a Si crystal.

Impurity ionization

energies in Ge are about 10 meV or les s, which i mp lies that the RF
work fun ction s are all about 1 meV or les s and since the widths are
comparable to those in Si, the lines are difficult to separate .

The

line pos i t ion s for luminesce nce from excitons bound to various impurities in Si are kn own (see Table 1.2). We find that the highest energy
line of the three "Bound Excitons" in Fig. l.l i s due to the decay of a
single exciton bound to residual boron impurities. The two lower energy
lines in Fig. 1.1 do not correspond to BE on any known impurities, and
studies have s hown that they are associated with boron( 22 ). It is
thought that the se lines arise from decays within "embryonic droplets"
consisting of 2 excitons bound to a boron for the hi gher lying line and
3 exciton s for the lower lying line.
A series of luminescence lines associated with phosp horou s
i s shown in Fig . l .2.

Again the hi ghest energy l ine is due to the

decay of a single exciton bound to the impurity, whil e the lower l ying

13
Table 1.2
Shall ow Impurities in Si
Donors

Impurity Ionization Energy, EI'
(meV)(a)

BE work function
BE' (meV)

Li
Sb

33(c)

3.4(d)
4.64(e)

As
Bi
Acceptors
Al
Ga
In
Tl
a)
b)
c)
d)
e)
f)

42.7
45.5
53.7
69 (c)
44.5
68 . 5
72

155
260 (c)

BE(NP) line oosi tion, hvBE' (~ev)(b)

4.69
5.51(e)
7.7l(e)

1151.21 ±.02
1149. 97±. 11
1149. 92 ±. 05
1149.1 ±.11
1146.9 ±. 11

3.94
5.08
5.66
13.68
44.2 (f)

1150.67 ±.05
1149.53 ±.05
1148.95±.05
1140. 93 ±. 05
1110.4 ±. 1 (f)

F. Bas sa ni, G. Iadon i si, and B. Preziosi, Rep. Prog. Phys. lZ_, 1099
(1974).
Obtained assuming FE(NP) threshold = 1154 .61 meV and ¢BE from (e)
for Sb, As, Bi. For Li used BE (TO) from (d) and assumea Ero=58 meV.
Sze, Physics of Semiconductor Devices, (Wiley-Interscience, New
York, 1969) p. 30.
K. Ko sa i and M. Gershenzon, Phys . Rev. 89, 723 (1974).
E. C. Lightowlers, M. 0. Henry, and 1~. A. Vouk, J. Phys. C. lQ, L713
(1977).
K. R. Elliott, D. L. Smith, and T. C. McGill (to be published).

Figure 1. 2
Photoluminesce nce spectrum of the no-phonon replicas
of the BE and the first four BMEC in Si:P. The T (from R.
Sauer, W. Schmid, and J. Weber, Solid State CommuW. 24,
507 (1977)) are the mea sured lifetimes of the lines .--The
splittings between the FE threshold and the phosp horous
related lines are shown in the lower part of th e figure.

c....,

(f)

"'T1

-'•

r4= 105
r3=IZOns

(m = 3)

f-- 8.3 meV

r 2 =155 ns

---------1

------ 1
~ 4.7meV ·I

FE
(No-Phonon)
Threshold

1.140

ENERGY (eV)

1.145

1.150

1.155

11 .0 meV- - -- - - - - - - - - - - - l
13.7 m e V - - - - - - - - - - -- - - 1
~ 14.4 meV - - - - - - - - - - - - - - - - - - l
~----15 . 5 m e V - - - - - - - - - - - - -- - -

b5
{m=6lm =5)

(m = 4)

(m =2)

r 1 =270ns

Tbath = 4.2oK

- {f-

( m = I)

Np =4x1o' 4cm3
GaAs Laser "'25W-cm-2(peak)
lf-Lsec pulse, lOOkhz

Si: P

_.
c.n

lines arise from decays within groups of excitons bound to a singl e
impurity, a Bound Multi-Exciton Compl ex (BMEC).

Luminescence l ines
from the decay of a BMEC were first observed by Pokrovskii ( 23 ) in
connection with the nu cl eat i on of EHD in s ili con, as mentioned
earlier, and have subsequentl y been studied extens i vely ( 24 ). To
date t hey have been r eported in Si associated with the donor impurities: Li , Sb, P, As, and the acceptors B, Al, Ga. Analogous
lines have also been reported in Ge ( 25 ), cubic SiC, ( 26 )and
GaP ( 2?).

If we ass ume that the lines in Fig. 1.2 are in fact due to

these multi-exciton complexes, then we can label each line with an index, m, whi ch tells how many excitons are bound to the i mpurity in
the ini tia l state.

Thus the luminescence we observe is due to the

recombination of one electron-hol e pair , out of m pairs, leaving us
with m-1 pairs on the site.

One of the f irst questions tha t arises

is that of how tightl y can a complex bind an exci t on .

This work

f un ction for 1 complex with m-exc itons , ~m ' is just the difference in
energy between the state consisting of an exciton with zero momentum
and an m-1 exciton complex in its ground state, and the st ate consisting of an m exciton complex.
If we assume that the rad iative decay of an m- complex l eave s
the final (m-1) compl ex in its ground state, then we can det ermine
the work function spectroscopicall y.

The work function und er thi s

assumption is the spectroscopic difference, 6 , between th e complex's
luminescence l ine and the free exc iton edge. Thes e spectros copic dif ~
ferences are diagrammed in t he bottom half of Fi g. 1.2.

For the BE

lm=l) we know that the final state of the transition is the ground
state of the neutral impurity s ince the luminescence and absorption
lines occur at exactly the same energy.

Thu s for the bound exciton,

the spectroscopic difference. o1 , and the work function, ¢ 1 , are the
same.

For the BMEC the o increase monotonically with m.

the transitions are, in fact, ground state to ground state, then
this says that each exciton binds to the complex more tight ly than
the last exciton.

Also, since we know that the work function of the
EHD is about 8.2 meV ( 28 ), these data would seem to show that excitons

will bind more tightly to a complex than to an EHD.

This is somewhat

disconcerting since it would be expected that a l arge complex would
look very much like an EHD to an incoming exciton; the other carriers
would have screened out the impurity potentia l.

Furthermore, if these

complexes do bind excitons more tightly than the EHD, then under certain
conditions drop l ets should be unstable wit h respect to breaking up into
BMEC.

However, EHD luminescence is seen wit h doped Si .
Another problem with the BMEC model has been the observed split-

tings of th e lines with the crystal under stress or in a magneti c
field ( 29 , 30 )
It is found that each line splits into the same
number of components and that the components hav e the same energy
separations for every complex.

The only differ ences between compl exes

are the relative intensities of the components and the variation of
these ratio s with tempera ture.

I t is difficult to understand why a

complex with several excitons should show essentially the same simpl e
splittin gs as those for a singl e exciton bound to the impurity .

These

work function and field splitting arguments have convinced one of the
early proponents of th e BMEC model to abandon the concept ( 29 )
although no new model for the lines has been proposed.
Despite these problems, there are some reasons one would like
to retain the BMEC model.

First , as shown in Fig. 1.2 each of the

lines ha s a different lifetime, Tm.

The luminescence decays are found

to be exponential over at l east two decades.

Since the T are all

different, we know that each line arises from a different initial
state, and further the fact that T 1>T seems consistent with a multi~m
pair picture. Another piece of evidence for the BMEC picture is
that absorption lines correspondi ng to the luminescence lines for
m f l are not observed.

In Si :B , for exampl e , the intensity of the

luminescence for m=l and m=2 is comparable, as are the decay times,
which indicates that the two processes have similar oscillator
strengths.
seen.

However, in absorption only the bound exc iton (m=l) is

This i s to be expected in the BMEC mode l since to produce a

2-exc iton comple x with the absorptio n beam there must be bound excitons
in the crystal to start with.
The third and most important piece of data in support of the
BMEC picture concerns the variation of intensity of the lines with
excitation intensity.

It is found that the intensities are proportional

to some power of the excitation, and the exponent, a , is larger for
the bigger complexes.

A log~log plot of luminescence intensity versus

excitation level for Si:Al is shown in Fig . 1.3.

The upper curve is

for the BE and we see that its intensity vari es approximately linearly
with the excitat ion .

The curves labeled b1 (J =2 ) and b2 are for the

Figure 1.3
Data on th e pump power dependence of the BE and fir st
two BMEC in Si:A1. The straight hines were fit to the data
ass uming Intensity a (Exc itati on) . The values obtai ned for
a are s hown.

•-BE(J•O)
A-b 1 (J•2)

Si:AI
NA 1 = 51l10

D -bz

GoAs Loser 4..us pulse

104

Tboth"4. 5o K
100%-JOW em-2

en
z 10

a.= 2.70

10%

100%

EXCITATION

INTEN SITY

Figure 1. 3

m=2 and m=3 complexes, respectively.

l~ e

see that these lines have a

faster than l inear dependence on excitation.

The rate equations

based on the BMEC model predict this superlinear

behavior.

However,

in that picture the exponent for the m-exciton compl ex shoul d be m,
the number of excitons on the site.

The experimenta l numbers do not

exactly reproduce this 1, 2, 3 sequence, however, the excitati on
is only at the surface, not homogeneous, so that some deviation could
be expected.

Furthermore, the rate equations predict t hat the inten-

s iti es should saturate at high excitations as all the impurities are
occupied and this behavior is seen in Fig. 1 , 3 for the b1 and b2 lines.
The superlinear intensity dependence on excitation does suggest that
more than one exciton i s invo l ved for the processes in which m >
and also establishes an ordering of the lines according to their
exponent ,a .

This i s the origin of the assi9nment of m values to t he

lines in Fig. 2 up through m=4 .

II.

Outline of Thes is
This thesis is divided into two parts:

Part I deals with

some experimental investigations of bound multi-exciton compl exes
in Si.

Part II is a discu ss ion of a new model for the decay of

electron-hole-droplets in pure and li ght l y doped Ge.

Part I.
It has been suggested that acceptors in Si offer a test of
the BMEC concept. ( 26 ) Some of the acceptor BE have low lying excited states.

If a 2-exciton complex decays. it cou ld l eave the BE

in one of the excited states as well as the gro und state .

Chapter 2 we present experimenta l evidence that this does occur for
Si :Ga and Si:Al.

The data co nclusively demonstrate that more than

one exciton will bind to an im purity.

However. the apparent work

function s do not show a simple depend ence upon the number of excitons
on the site .

No BMEC are observed in Si:In.

Measurements of the l ifetimes of BE and BMEC on acceptors are
presented in Chapter 3.

We find that the BE lifetimes vary by almost

three orders of magnitude depending on the impurity and that this
variation can be accoun ted for by assuming that the dominant decay
mechanism i s a phononles s Auger process .

The measured lifetimes

have important implications for the kinetics of BMEC formation .

is shown that it may be impossible to form 2-exciton complexes in
Si:In from a gas of FE due to kinetic limitations.
In Chapter 4 we present direct thermodynamic measurements of th e

work functions of BMEC in Si doped with B~ Al, Li, and P.

We find

that the second exciton in Si:P and the third in Si :Bare the most
tightly bound and thereafter each exciton bind s with less energy than
the la st .

This i s in contrast to the monotonically increas ing work

functions for the larger complexes one obtains by assuming that the
observed lines are due to ground-state to ground-state transitions.
None of the measured work function s exceeds that of the EHD.

The

data eliminate the problem of accounting for unphysica lly large
binding energies of BMEC.

Part II.
Measurements of the decay of EHD in pure Ge exhibit an excitation
dependence which is not explained by the "avera9e drop" r.10de l.
Attempts to use this model produce unphysica l results.

In Chapter 5

a new model i s presented which takes into account exciton diffusion
and the fact that the EHD form a "cloud" within the crystal.

The

model i s able to fit the experimental results while retaining
physically reasonable parameters, and the exc itati on dependence is a
natural consequence.

In lightly doped Ge EHD decay much more sl owly

than in pure Ge under the same conditions.

The model presented here

i s also able to fit these data by assuming the impurities reduce the
exciton diffusion l ength .

24
III.

Optical Pro cesses in Indirect Semicondu ctors
In order to study aggregates of nonequilibrium carriers within a

semiconductor we need some way of generating them.

One of the

simpl est methods, and the one used for the exper iments discussed in
this thesis, is to illuminate the crystal with above band-gap
radiation.

In Fig. 1. 4 the optica l absorption curves for Si, Ge,

and GaAs are shown .

At a photo n energy below Eg for a particul ar

semiconductor, the absorption coefficient i s very sma ll.

For GaAs ,

a direct gap materia l, the absorpt ion coefficient i s l ar ge for photons
with energies on l y sl ightly abov e Eg.

The absorpt ion turns on more

s lowly for Si and Ge s ince they have indirect gaps.

The difference

between the 77°K and 300°K curv es i s du e to the change in band- gap
with temperature.

For th e experiments to be discussed the excitation

was either a GaAs l aser (1 .46 eV) or an Ar

l aser (2.41 eV) .

Since

the gaps change only sli ghtly with temperature below 77°K we can use
t hose curves to estimate the penetration dept h of our exc itin9 li ght.
Thus for Si the absorpt i on l ength for the GaAs la se r is - 50 ~m whi l e
for the Ar+ l aser i t i s - 2 ~m.

For Ge both l asers em i t photons with

an energy above the direct gap and thus are absorbed wi t hin l ess
than 1 ~m.
When the l aser photo ns are absorbed they create electron -hole pai rs
with energies cons iderabl y larger than the band gap.

This pl asma

proba bl y re l axes t o the l att i ce temperature in fract i ons of a nanosecond ( 3l), or at l ongest , a few nanoseconds ( 32 )
The fre e carriers

:- ~
' e 10

.....

....

...~... 10~

' ;y

"'0v

"'
C( 10

V>'
vI

I /,

~ l'I

300"K

- - - - - 77"1<

(,'

~~--

Go As
- - r- i-t- - 1- ~~

10°

0.6

0 .8

A:, ~" ~

~~~

0::

~ ~ .,.

: !

910

lw (e V)

Fi gure 1. 4
Meas urements of near-gap optical absorption at 77°K
a nd 3QQOK i n Si, Ge, and GaAs (from Sze, Phys i cs of Semiconductor Devices, (Wiley-Intersc i ence , New York, 1969)
p. 54.)

bind into excitons on these same time scal es .

These thermalization

times are in general short compared to typical decay times for EHD,
excitons, and BMEC, often making it possible to use equilibrium
thermodynamics in the analysis.
In radiative re combi nation the indirect gaps of Si and Ge play
an essential role

(2)

A photon with an energy of about 1 eV (com-

parable to the bandgaps for Si or Ge) has a wave-vector of about
l0- 3A-l
This is to be compared with the wave -vector at the edge

of the Brillouin Zone which is usually - lA-l

The crystal momentum

of the photon is negligible with respect to typical carrier momenta.
This leads to the conclusion that the dipole matrix el ement for
recombina-tion of free carriers is~

( 1 . 5)

where E is the pol arization of the photon and p = -i~V .

This matrix

el ement is zero unless k = k2 . The band structure for Si is shown
in Fig. 1 .5. We see that the valence band maximum occurs at k = 0;
however, the conduction band minimum occurs at k - (.85,0,0)~n.

Thus

the matrix element of Eq. 1.5 is zero for an electron-hole pair at
the band edges .

It is necessary to go to higher order to make this

transition allowed.

Thi s accounts for the l ong radiative lifetimes

and corresponding l y small radiative efficiencies of Si and Ge.

recombine radiatively, the pair emits both a phonon and a photon;

Figure 1.5
A calcul ation of the band structu r e of Si, neglectin g
spin-orbit effects. The top of the val ence ba nd i s l abeled
r25 ' (E=O). The conduction ba nd mi ni mum i s near the point
l abeled x1 . The minimum occu r s approximate l y at k=(.85,0,0)
2n/a. The band gap at T=O i s 1.169 eV. (from D. J. Chadi,
Phys. Rev. BJQ, 3572 (1 977)).

Figure 1.5

the photon carr ying off most of the energy and the phonon carrying off
the crystal momentum.

(The process involving phonon capture i s only

important at high er temperatures) .

The rate for this process looks

like ,

(l. 6)

where He-p i s the electron-phonon interact ion, and k ,k , l a bel the
1 1
intermediate states. Phonon ass i s ted recombination i s illustrated
schematica ll y in Fig . 1.6a.

This process requ ires the emissio n of a

phonon with wave-vector (.85,0,0) for Si, and from the disper s ion
curves of Fig. 1 .6b we see that there are severa l phonons whi ch ca n
participate.

The possibilities are a TA pho non (19 meV), an LA

(- 41 meV), an LO (56 meV), and a TO (58 meV).

Luminesce nc e i s seen

associated with all of these phonons except t he LA.

(The absence of

LA phonon assisted l ines is not well und erstood. )
The r eq uirement in Eq. 1.5 for co nservat i on of k is onl y strictly
true for free particles.

For bound states the crysta l momentum of a

singl e electron or hol e is no longer a conserved quantity.

These

states mu st be made up of a linear combination of Bloch function s
with different k.

Thus the wave function s exhibit a spread ing in

Figure 1 .6
a) Schematic illus tration of phonon assisted radiative r ecombination in an indirect semi conductor.
b) Phonon dispersion curves for Si in th e (l ,0,0)direction (from G. Dolling, Inelastic Scatterin g of
Neutrons in Solids and Li uids {International Atomic
Energy Age ncy , Vi enna 1962 , Vol . II, p. 37 . )

KPh:---- ~

Fig ure 1.6

k-space, and the extent of this spreading depends upon the nature of
the potential which is causing the binding .

In particu l ar, part of

the potential of an impurity is very short range (central cell part)
and consequentl y mixes states f r om over much of the zone into the
wave fu nct i ons of the particles bound to it.

If , for exampl e, we

have an exci ton bound to a neutral donor, the el ectrons will have a
finite amplitude at the zone ce nter.

Therefore, it is possible to

have the exc i ton recombine radiatively without emitting a phonon .
The strength of this "no-phonon'' (NP) process depends upon the
strength of the short range part of t he i mpurity potential and thus
on the part i cular impurity involved.
In the luminescence spectra of Si or Ge we will find replicas
of each feature due to the partic ipati on of different phonons.

Us ing

the case of a BE on a donor in Si as an example, we expect four
repl i cas.

At the highest energy we see a no-phonon replica .

Nineteen meV l ower in energy we observe l uminescence fro~ the BE a~ain
but now through a TA-phonon assisted process .

At 56 and 58 meV

below the NP line we find the LO and TO assisted reolicas, re spectivel y.
Other r epli cas involving 2 pho nons are also seen, but they are much
weaker than the single phonon processes.

References
1.

R. S. Knox, Theory of Excitons, (Academic Press, New York, 1963).

F. Bassani a nd G. Pastori Parravi c ini, El ectronic States and
Of2tical Transitions in Solids_, (Per9amon Press, Oxford, 1975).

R. J. Elliott, Phys. Rev. l 08, 1384 (1957).

J . R. Haynes, Phys. Rev. Lett . lZ_, 860 ( 1966).

v. Ke l dysh, in Proc . 9t h Int. Conf. Phys. of Semi cond.,

Moscow, 1968.
6.

(Nauka, Leningrad, 1968), p. 1303.

Y. E. Pok rovsk ii and K. I. Svistunova, Fiz. Tek. Polup. ±•
691 ( 1970) [Sov. Phys. - Semi cond.

±• 409 ( 1970)].

C. D. Jeffries, Science 189, 955 (1975), and references therein.

Ja gdeep Shah, R. F. Leheny, W. R. Hard in g, and D. R. Wi ght,
Phys . Rev. Lett . 38, 11 64 (1977) .

D. Bimberg, M. S. Sko lnick, and W. J , Choyke, Phys . Rev. Lett .
40, 56 (1978).

10.

R. F. Leheny and Jagdeep Shah, Phys . Rev. Lett . 38 511 (1 977) .

11.

G. Beni and T. M. Rice, Phys. Rev . Lett . 37, 874 (19 76), and
references therein .

12.

G. A. Thomas, T. M. Rice and J. C. Hensel, Phys. Rev. Lett. ?3,
219 (1974) .

13.

M. A. Vouk and E. C. Lightowlers, J. Phys.

14 .

V. Marrell o, T. F. Lee, R. N. Si lver, T. C. McGill and J. W.

c~ ,

3695 (1975) .

Mayer, Phys. Rev. Lett. lL· 593 (1973} .
15.

R. B. Hammond, V. Marrello, R. N. Si l ver, T. C. McG ill and J. W.
Mayer, Soli d State Comm . .l_i, 251 (1974).

34
16.

G. A. Thomas , T. G. Phillips , T. M. Rice and J. C. Hensel,
Phys . Rev. Le tt. ll· 386 (1 973).

17.

J . C. V. Mattos, K. L. Sha kl ee, M. Voos, T. C. Damen and J . M.
Warl ock, Phys. Rev. 813, 5603 (1976) , and references therein.

18 .

J . Doehler, J. C. V. Mattos, and J . M. Warlock, Phys. Rev. Lett.
38 ' 7 26 ( 1977) .

19.

J. M. Hvam and 0. Chri stenson, Solid State Comm. ~' 929 (1974),
and references therein.

20.

V. S. Bagaev, T. I. Ga lkina, and 0. V. Gogolin, in Proc. lOth
Int. Co nf . Phys. of Semicond. , Cambridge, 1970.

(USAEC, Oak

Ridge, 1970) .
21.

J. R. Haynes, Phys . Rev. Lett._!, 361 (1960).

22.

R. Sauer, Phys. Rev . Lett. 11, 376 (1 973).

23 .

A. S. Kaminskii, Y. E. Pokrov ski i, and N. V. Alkeev, Zh. Eksp.
Tear. Fiz. ~. 1937 (1970) [Sov. Phys.- JETP. E_, 1048 (1 971) ].

24.

S. A. Lyon. D. L. Smith and T. C. McGi ll, Phys. Rev. 817, 2620
(1978), and references therein .

25.

R. W. ~lartin, Solid State Comm. li• 369 (1 974).

26.

P. J. Dean, D. C. Herbert, D. Bi mberg, a nd~/ . J. Choyke, Phys.
Rev. Lett. ll· 1635 (1976).

27.

Private communication from M. Pilkuhn.

28.

R. B. Hammond, T. C. McGi ll and J. t3566 (1976) .

29.

R. Sauer and J. t~eber 7 Phys. Rev. Lett. ~. 48 (1 976 ).

30.

R. Sauer and J. Weber, Phys. Rev. Lett. 12_, 770 (1977).

31.

R.F. Leheny (Pri vate Communication).

32 .

J. C. Hensel and K. Suzuki , Phys. Rev. 89, 4219 (1974) .

PART I
CHAPTER 2
FINE STRUCTURE OF ACCEPTOR
BE AND BMEC IN Si

Introduction
As wa s expl ained in Chapter 1, the luminescence spectra of

silicon doped with shallow impurities contain several emission lines
at slightly l ower energy than the BE (l- 3 )
One explanation of the
origin of these satellites of the BE is that they result from electronhole recombination within bound multiple exciton complexes of different sizes.

As discussed in Chapter 1, this interpretation is

strongly supported by the excitation dependence of the lines and is
consistent with the lifetime and absorp tion measurements.

However,

if the lines are interpreted as ground-state to ground-state transitions, then the BMEC appear to have work functions whi ch increase
with the size of the complex and can become over twice the EHD work
function (l- 3 )
Furthermore, the stress and Zeeman splittin9s seem
much too simple for an entity as complicated as a BMEC {4 ,S).

Thus,

there are data which argue in a circumstantial way both for and
against the BMEC picture, but none of the se data definitively address
the question of whether more than one exciton ca n bind to an impurity.
The acceptors in Si offer an ideal test of the BMEC model, as
suggested by Dean and coworkers ( 6 , 7 ). The BE in Ga and Al doped Si
give a luminescence spectrum which cons ists of three closely spaced
lines corresponding to emiss ion from the BE gro und state and two low
lying excited states (g-ll)

(In Si :In, two BE emission lines are

obse rved and a third i s see n in absorption experiments .)

The three

states for the acceptor BE are due to a coupling of the two spin 3/2
holes in the BE {ll)

The ground state corresponds to the J =O state ,

and the two exc ited states correspo nd to the J ~2 state which is
further s plit by the crys ta l fi eld .

If the BMEC picture i s co rrect ,

an optical tran s ition from the two exciton complex to a BE shou ld
mirror the BE luminescenc e spectrum, because th ere ar e three avai labl e
final states for the t ransition .
In Si:Ga we see three s uch lines in high-resolu tion photoluminescence spect r a .

The energy positions of the lines are we ll

within the experimenta l re solution of th e positions expected for the
BMEC model .

The observation of these three emission lines conclu-

s ive l y demonstrates that, at least in Si: Ga, two excitons will bind
to the impuri ty.

The data for Si:Al are similar to those for Si:Ga,

however, only one 2- exci ton compl ex line i s observed.

For Si: In a

line at somewhat lower energies than the BE i s observed, but it i s
probably not assoc i ated with the decay of a two-exciton compl ex .

II .

J-J Splitting of Acceptor BE
The top of the valence band in silicon is made up of p-type

atomic wave-funct i ons on the Si cores.

When the el ectronic spin i s

included with this orbital part, the total angular mome ntum of the
electrons i s either J =3/2 or J =l/2 .

The J=l/2 states are moved to

lower energy by the spi n-orbi t interaction, mak i ng the top of the
valence band consist of J~ 3/2 states.

(Actua l ly, s ince the symmetry

i s cubic in the crystal, the states shou ld not be label ed by J .
With cubic symmetry J =3/2 becomes a r 8 state and J=l/2 is a r

state .
This distinction i s not important at the moment, and we will continue
to use J as a l abel.)

For an effective mass acceptor in Si, the

wave-function for a hole bound to the impurity is a l inear combination
of states near the top of the valence band.

Thus , this hole will act

li ke a J =3/2 particle .
When an exciton binds to an acceptor, there are two holes and one
el ectron bound to the i mpurity.

It i s thought that the two holes are

highly localized near th e impurity atom due to its strong attractive
potential, while the electron is spread out over a larger region (l 2 ).
The holes will interact with one another through the ir mutual coulomb
repuls i on, and thus we expect their angular momenta to be coupled (l 3 ).
Two J=3/2 particles can co upl e into J=3,2 ,1 ,0 states, but only the
J=2,0 states are allowed for the hol es by the Pauli exclusion principl e .
Thus, we expect the ground state of th e BE to be split into two states
corresponding to the tota l ang ul ar momentum of the holes being either

J=2 or J=Q.

A more careful treatment including the cubic crystal

environment shows that the J=2 state is split into two, giving us
three states (r 11 r 3 and r 5 ) for our BE.
An analogous splitting has been observed experimentally for acceptors in InP (l 4 ) and GaAs (l 5 ). Careful measurements of the
behavior of the lines with stress has established the J-J coupling
interpretation for the splittings.

For shallow acceptors in GaAs 1

the J =O line lies at higher energies as one would expect from re sults
in atomic physics (l 3 )
However, for the deeper acceptors the J=O
line moves to lower energies and becomes the ground state (l 5 )
t1ea s urements of the oscillator strengths for the three BE lines in
Si:Al, Si:Ga, and Si:In (The splitting i s small in Si:B.) are in aqreement with the relative strengths predicted by assuming the splitting
i s due to the J-J coupling and that the ground state is J =O (l 6 ,ll).
The ordering of the states i s the same for all three acceptors in
Si, and like the deeper acceptors in GaAs this ordering is different
from what one sees in atomic physics.
A sc hematic illustration of the photoluminescence spectrum we
expect to see is shown in Fig. 2.1.

On the left is an energy level

diagram (not to scale).

The states are labeled by the total angular

momentum of the holes ,

Thu s , at the bottom we have the ground state

of the acceptor (A0 ) with J =3/2 .

Approximately one band gap in energy

above the A0 is the BE which consists of the three states di sc ussed
above.

When the exciton decays the three different initi al states lead

to three luminescence lines as shown on the right.

The lowest lying

Figure 2. 1
Illu stration of the J-J coup ling of the two holes in
th e acceptor BE and its effect on the luminescence spectra .
In the energy l evel diagram on the l eft, J l abe l s the total
angu lar momentum of th e hol es. The expected luminesce nce
from these l evel s i s shown schematically on the right.

-...,

LLJ

J :l-

6II

-...,

-.,

LLJ

J :l-

<.9

J-J--

_.,

_j
Q_
(f)

..."
.,..

...
-.,.

...

C\J

II
-.,

(!)

Fig ur e 2. 1

......
f()

II
-.,

line i s the ground- state to ground-state transition (J =O). and the
hi gher l yin g lines arise from excited-state to ground-state t r ans ition s (J =2).

The three lines are s hown with different he ights to

indicate that the ratio of th e line intensities i s temperature dependent, since the splitting i s in the initial state.
At the top of Fig. 2.1, on the l eft, i s a state l abeled b1 ,
which i s our hypothetical 2-exc iton compl ex bound to th e impurity.
A J-value ha s not been assigned to thi s s tate s ince we do not know i ts
structure.

As discussed in the last sec tion, we expect to see lines

du e to the decay of one exciton in this compl ex leaving a BE.

Since

the BE has two low ~ l y in g exc ited states the decay of b1 s hould give
three lines for the three fi na l states.

(In principl e one or mo r e

of these lines could be forbidden though probably the complex cont ain s a sufficient mi x of states to make all t he tran s it i ons al l owed.
In the s imple pi ct ure of add ing a t hird hole to the J =3/2 s hel l , b
would be a J =3/2 state , and all the trans itions would be allowed.) A
schematic of the expected spectrum i s shown on the ri ght of Fi g. 2. 1 .
(The lines ar e la beled by the J of the final BE sta t e. )

Aga i n there

are three lines , but s in ce the spl itting is in the fin al state we
should see a mirror of the BE spectrum.

There will be two gro un d-state

to exc ited- s tate transi ti ons at lower energy, and the ground- state to
ground-state tran s ition produces t he hi gher l ying line .

Since the

splitting is due entirely to the BE , th e energy differences between
the lines should be exact l y th e same as the energy differ ences in t he
BE luminesce nce spectrum.

Also, the splitting i s in th e f inal sta t e

44
making the relative intensities of the lines independent of temperature.

45
III.

Experiment
The experiments were performed using samples of s ingl e crysta l

si li con.

The crystals were grown by the Float-Zone method, and the

dopants were introduced during growth.

The impurity concentrations

were determined by Hall measurements.

All samples, except the Si :Al.

were discs about l em in diameter and 3 mm thick.
was a wafer approximately 450 ~m thick.

The Si:Al crystal

The samples were lapped and

etched vd th HN0 3 : HF (7: l) to remove surface damage.

The crystals \

cleaned with methanol before being mounted in the dewar for an
experiment .
A dia gram of the experimental apparatus is shown in Fig. 2.2.
The crysta l s were mounted on a copper sampl e block and placed in a
Janis variable-temperature dewar.

Temperatures above 4.2°K were

obtained by heating the sampl e block in the He-vapor and regulated
with a commercia l temperature regulator.

For the experiments di s-

cussed in this section, a calibrated Ge temperature sensor held in
mechanical contact with the sampl e was used to read the temperature ,
as opposed to the Si diode shown in the figure.

Experi ence has s hown

that the Ge sensor used in this confiquration reads the sampl e
temperature to within 2°K at the l aser powers employed.

The res ults

in thi s chapter are in sensitive to this magnitude of uncertainty in
the temperat ure .
For most of the experiments the excitation so urce wa s an Ar-ion

l aser operati ng at 5145 A.

As shown in Fig. 2,2 the beam was filter ed

to remove any infrared lines in the region of interes t and brought to

"0
"0

(0•

::I

ro ro

(")

VlC:

roc.o

::I -'·

-'•,.,

0__,

::r

-o

READOUT

TEMPER~.TURE

1.8-300°K

TEf>lPERATURE
REGULATOR

LJ
'-1.1

SI DIODE
TEMPERAl

~LENS

AR~~E~ON I --~------~

FILTER

SEI·lSOR

LENS

.,._.

------.... [

COOLED
S-1 PMT

-- - ..

-------- ~·---

.,._.,. ... ____

DEWAR

I -

ANP
DISC.

GATED COUNTER
AND
1 - - - - 1 RECORDER
RATEMETER

EXPERI MENTAL SETUP

SPECTROMETER

GRATING

0'1

47
By moving the lens, the size of the laser

the sample through a lens.

spot on the crystal could be changed.

For some experiments a GaAs

la ser diode was used for excitation.

The diode was mounted on the

sample holder a couple mm from the Si crystal.

The spot size was

approximately 1 mm as determined by an infrared image converter.
The GaAs laser was operated in a pulsed mode, typically with 2 ~s
pul ses and a 4% duty factor.
calibrated Si photodiode.

The laser power was measured usi n9 a

The powers quoted for both the Ar-ion

and GaAs have not been corrected for reflection losses at the surface
of the sample.
The luminescence was collected from the edge of the crystal as
shown in Fig. 2.2.

The li ght exists predominantly through the edge

du e to total-internal refl ect ion within the Si.

Two lenses giving

a magnification of 3 coll ected the luminescence a nd focused it on
the entrance s lit of the spectrometer.

For most of the experiments a

double pass Spex 1400-I I spectrometer was used although for some
experiments a s ingl e pass Spex 1269 was used.

The output of the

spectrometer wa s detected by a liquid nitrogen cooled S- 1 photomultipli er tube (RCA 7102).

A fa st amplifier-discriminator at the

output of the photomultiplier enabled us to use sing l e-photon-counting
techniques.

The discriminator pulses were processed by a gated rate-

meter gi ving an analog output proportional to the intensity which was
recorded on a s trip chart r ecorder.

The dark co unt with the photo-

multiplier cold was about 4 counts per second, which made synchronous
det ection unneces sary.

Consequently, the experiments usi ng the Ar-ion

la ser were run CW l eqding to i mproved temperature stabi lity.

IV.

Experimental Results
The luminescence spectrum for Si:Ga is presented in Fig. 2.3 .

this spectrum, we obser ve transitions from the ground state of the BE,
labeled BE(J=O)~ and from the two excited states of the BE (label ed
BE (J= 2)).

The splitting between the BE(J=O) line and the lowest

line of the BE(J=2) doublet is 1.47 meVt and the splitting between
the BE(J=2) doublet is 0.33 meV.

More important to our di scuss ion

here are the three lines labeled b , which we will interpret as due
to decay of a two exciton complex to a BE, The two lines l abe led b
(J =2 ) correspond to transitions with the BE left in an excited state;
the line l abel ed b1 (J =O) corresponds to a transition in which the
BE is left in its ground state.

The splitti ngs between the b1 emis-

sion lines are equal to (within the resolution of the experiment,
0.05 meV) the corresponding splittings for the BE emissio n lines.

The

interpretation of the b1 lines as due to recombination of a two
exciton complex i s further supported by meas urements of the pump power
dependence of the b1 and BE line intensities.

These measurements show

that the intens ity ratios of the b1 lines are independent of pump
power; whereas, the ratio of the b1 line intensities to the BE line
intensities increa ses with pump power.

The observation of the com-

plete b1 triplet conclusively demonstrates that, at least in Si:Ga,
a complex of two excitons bound to a single shallow impurity does
occur.

In Fig. 2.3, we also see an emission line, l abe l ed b , which
could be interpreted as decay of a three exciton complex. The

emission line labeled Pis due to an exciton bound to a phosphorus

Figure 2.3
The photoluminescence spectra of Ga doped Si in the
energy range for no-ph onon ass i sted transitions. The lines
labeled BE, b1, and b2 are assoc ia ted with the Ga i mp ur it i e s .
The line l abeled P is associated with phosphorous impurities
in the Si.

-s
ro

1.0

(f)

..,.,
.....

5145A

"'0, 3 mm spot

0.7W

Tbath = 4.2 oK

NGa = 2 X lo'5 cm3

Si: Go

BE(J=O)

ENERGY (eV)

BE(J=2)

I \ J \ 40

1.150

-1r-

1.1505

\ ! I

p1 f-

1.1500

bl (JA 2) )

-H

1.1495

I ..

b (J=O)

1.1510

(J1
_.

52
impurity ( 2 , 3 )

Another line at even lower energies not shown in

Fig. 2.3 has been observed in the Si:Ga.

This line occurs at 1. 1441

eV.
The luminescence data for Si:Al are presented in Fig. 2.4.
data are similar to those reported above for Si:Ga.
shows lines due to BE(J=O) and BE(J=2 ) states.

The

The spectrum

The splitting between

the two components of the BE(J=2) line is about 0.22 meV.

At lower

energies a broad b1 line i s observed; it is probably made up of the
two components of the b1 (J =2 ) doubl et .

The b1 (J=O) line is not

observed, but s hould occur at an energy indicated by the arrow l abeled
b1 (J=O); that is, it is und er the much stronger BE (J =O ) line and
hence difficult to observe. The two other lines, l abe l ed b2 and b3 ,
obse rved at lower energy are associated with the Al impurity .
sion l ines labeled Pare due to pho sphorus impurities ( 2 , 3 ).
The data for Si:In are presented in Fig. 2.5 .

The data show a

simi lar structure to that observed in Si:Ga and Si:Al.

The BE(J=O)

line is spli t from a s ingle BE(J=2) line by about 3.1 meV .
BE(J=2 ) has been seen in absorption at higher energies.)
satel lite line is observed 4.0 meV below the BE(J=O) line.
satellite ha s a line s hape
line .

Emis-

(A second
This

simi l ar to that of the BE(J=2) emission

It would be tempting to interpret thi s line as due to the

decay of a 2-exciton complex into the BE(J=2 ); however, other factors
rule out this interpretation.

Most importantly, the line does not

exhibit the pump power dependence expected for a 2-exciton complex.
It shows the same dependence on excitation as the BE(J=O) and i s even

Figure 2.4
The photoluminescence s pectra of Al doped Si in the
energy range for no-phonon assisted transition s. The lines
label ed BE, b1 , b2, and b3 are associated with Al impurities. The lines l abeled Pare associated with phosphorous
impurities in the Si .

"'T1

....

"'S

(f)

f- I

b3
1.145

-0.3mm spot

0.4W 5145.8.

Tbath = 4.2°K

NAr = 5x lo'4cm3

Si :AI

I I

XIO I I

--- r

1.1505 1.1510

-n-

II XI

I\ 10

BE (J=O) BE (J=2)

l b (J=O)

b, (J =2)

ENERGY (eV)

-H-

1. 1480 1.1485

. J~i

-H-

(.11

Figure 2.5
The photoluminescence spectra of In doped Si in the
energy range of the no-phonon ass i sted tran s ition s. All
three lines are associated with In impurities .

I-

Z l

'"'S

([)

c::

(f)

_,,

..,..,

1.135

XIOO}

1.140

BE (J=O)

~t-

---5mm spot

ENERGY ( eV· )

\..

cm3

0.5 w 5145.8.

Tbath =7oK

N1n = 3.2 x I0

S i : In

1.145

t I x!OO

BE(J=2)

U1
0\

57
seen at low pump powers in heavily doped Si.
origin of this line i s unknown.

At the moment the

Discussion and Conclusions
We believe that our experimental data along with that previously

reported (g,lo) argues strongly for the existence of two-exciton
complexes bound to Si:Ga and Si:Al.

However, the results are not

as conclusive in the case of the b2 , b3 lines since there is no structure indicating as clearly that the decay is from a three-exciton
complex into a two-exciton complex or four-exciton complex into a
three-exciton compl ex.

If we take the additional lines to be due to

ground state transitions involving a change of one in exciton number,
then the energy separations between the free exciton, no-phonon
thre s hold and the satellite line energies give the binding energies
of an exciton to a BMEC consisting of zero to three excitons, as
discussed in Chapter l.

These line rositions alonq with a line

indicating the free exciton threshold and the energy per pair in the
electron -hole liquid (lS) are given in Fig. 2.6.

The data show that

for Si:B the binding energy of an additiona l exciton increases monotonical l y with comple x number.

For Si:Al, the first two excitons bind

with approximately the same energy while the third and fourth bind
with successively larger energy.
qualitatively different.

For Si:Ga, the situation is

The second exciton binds with a smaller

binding energy than the first.

Th e binding energy of the second

exciton in both Si:Ga and Si:Al i s l ess than in Si:B.

Further, it

i s interesting to note that the b lines for Si: B, Si:Al and Si :Ga
all occur at approximately the same energy. The b lines are inverted
in order in compari son to the ordering for the b1 lines. This

Figure 2.6
The position of the no-phonon assisted lines in photolu minescence for various lines associated with B, Al, Ga,
and In acceptors in Si. The line labeled FE was obtained
by shifting the position of the TO-phonon assisted line (Ref.
18) due to the FE by the TO-phonon energy of 58 meV (Ref. 3).
The line labeled ~FHD is the chemical potential of the
electron-hole dropTet (Ref. 18).

1.155--------------FE (Shifted to NP)

Si: 8

o Si: AI
...--...

• Si: Go

-- 1.150

• Si: In

(f)

J-L EHD (Shifted to NP)

itJ

~ 1.145

1.140

BE(J =O)

b 1(J =O)

LINE NUMBER

Figure 2.6

suggests that binding energy as a function of exciton number i s not
simply a monotonic function of the strength of the central cell
correction.
These data also show clearly that the binding of the excitons for
three and four exciton complexes forB, Al, and Ga are stron ge r than
that for an exciton in the EHD in spite of the fact that the first
exciton binds less strong ly than a pair in the EHD.

The fact that

binding energy obtained by interpreting the lines as due to groundstate to gr ound- state transitions dips s ubsta nti a ll y below that for
the EHD suggests that deeper lines b2 and b3 may in fact r es ult from
r ecombination in mu l t i exciton complexes where the complex i s left
in an excited state.

While lines that coul d be in terpreted as ground-

state to ground-state

for these compl exes are not observed , it shou ld

be noted that the correspondin g b1 (J =O) line i s not seen in Si:Al
because it is buried under the much larger BE .

Furthermore, the

ground-state to ground-state transition for the 2-exciton complex in
Si:Ga i s approx imate l y three times l ess intense than the assoc i ated
ground-state to excited-state tran s itions.
In summary, we have investi gated th e photoluminescence spectra
of Si doped with the acceptors Al, Ga, In .
BE are observed for th e Si:Ga and Si:Al.

Two excited states of the
In the Si: Ga we al so ob-

se rve three l ines associated with the decay of a two-exciton compl ex.
The f act that t hese lines mirror the structure of the BE i s conclusive
evidence that at l east two excitons will bind to a Ga impurity in Si.

The data for Si:Al are similar to those of Si:Ga and indicate th at
two excitons will al so bind to an Al impurity.

Howeve r , no lines

attributable to BMEC are observed in luminescence spectra of Si:In.
We have al so observed two lower energy satellites in Si:Ga and Si:Al
presumably due to the decay of larger BMEC.

Assuming the lines arise

from ground-state to ground-state tran s itions we find that the work
function s of the complexes do not show a simple dependence on either
exciton number or the strength of the central cell correction.

63
References
1.

A. S. Kaminsk ii , Y. E. Pokrovskii, and N. V. Al keev, Zh. Eksp. Tear.
Fix. 59, 1937 (1970) [Sov. Phys. JETP. 32, 1048 (1971)].
Lett.~.

R. Sauer, Phys. Rev.

K. Kosai and ~1. Gershenzon, Phys . Rev. 89, 223 (1974).

R. Sauer and J.

R. Sauer and J. Weber, Phys . Rev. Lett. 12_, 770 (1977).

P. J . Dean, D. C. Herbert, D. Bimberg, and W. J. Choyke, Phys . Rev.

~1eber,

376 (1973).

Phys. Rev. Lett., 36, 48 (1976).

Lett. lL, 1635 (1 976).
7.

P. J. Dean, D. C. Herbert, D. Bimberg, and W. J. Choyke, in Proceedin gs of the 13th International Conference on the Physics of Semiconductors, Rome, 1976. edited by F. G. Fumi (Tipografia Marves,
Rome, 1977) p. 1098 .

M. A. Vouk and E. C. Lightowl ers, in Proceedinqs of the 13th
International Conference on the Physics of Semiconductors, Rome,
1976, edited by F. G. Fumi (Tipografia r~a rves' Rome, 1977) p. 1098.

w. Thewa l t, Phys. Rev. Lett. 38' 521 (1977) .

M. L.

10.

11.

p. J. Dean, W. F. Fl ood and G. Kam insky, Phys. Rev. 163' 721 (1967).

12.

D. s. Pan, D. L. Smi th, and T. C. McGill, Solid State Commun.
~.

13 .

c. LightO'II1ers and M. 0. Henry, J. Phys. ClO, L247 ( 1977) .

1557 ( 1976).

See, for exampl e , E. U. Condon a nd G. H. Short1ey, Theory of
Atomic Spectra (Un iversity Press, Cambridge, 1970) p. 287.

14 .

A.M. YJhite, P. J. Dean and B. Day, J. Phys. Q, 5002 (1975).

15.

f'1. Schmidt, T. N. Morgan, and vJ. Schairer, Phys. Rev . .ED_!_, 5002

(1975).
16 .

K. R. Elliott, G. C. Osbourn, D. L. Smith and T. C. McGill,
Phys. Rev. Bl7, 1808 (1978) .

17.

G. Ki rczenow, Can. J. Phys. 55, 1787 ( 1977).

18.

R. B. Hammond, T. C. McGill and J. W. t·1 ayer, Phys. Rev. B13, 3566
(1976).

CHAPTER 3
LIFETIMES OF BE AND BMEC
BOUND TO ACCEPTORS IN Si

Introduction
In the la s t chapter we found that it i s possible to bind at

least two excitons to an impurity.
concerning the BMEC.

There are still other questions

Even at high excitation den sities we were un-

able to produce a 2-exciton complex in Si:In.

Furthermore, we find

that higher excitation densities are needed to produce BMEC for
deeper impurities.

For example, at 4.2°K in Si:B, the b line can be
as large or larger than the BE, with only modest pump power. For
Si:Ga, however , the b1 is always mu ch small er than the BE(J =O).

Also,

it is found that the BE saturates at a lower excitation inten sity
for boron than for gallium.

Aluminum lies in between the other two.

Another observation i s that the overall radiative efficiencies
of the bound excitons decrease for the deeper impurities.

The inten-

sity of the TO-phonon assisted BE replica is sig nificantly lower
for a sample of Si:Ga than for one of Si:B under the same excitation conditions.

The oscillator strengths of the phonon assisted

processes do not change much for the different impurities, however (l)
The variation in radiative efficiencies leads us to conclude that the
BE can have very different lifetimes with the deepe st impurities
having the fa stest de cay rate.
In Si, BE lifetimes have been measured for the donors Li ( 2 ),
P ( 3 ) and As ( 4 ); among acceptors, only the lifetime forB ( 3 ) has
been reported.

In thi s chapter, I will present measurements of the

BE lifetimes for the four common acceptors in Si (B, Al, Ga and In)
and lifetimes of the first two BMEC in Si:Al and Si:Ga.

(For Si:In,

we are only able to set an upper limit on the BE lifetime).

find very different lifetimes for the BE as sociated with the different acceptors; the lifetimes decrease rapidly (by about three
orders of magnitude) as the binding energy of the acceptor increases.
The lifetime of the acceptor BE in Si is most likely limited
by Auger transitions in which an electron recombines with one of
the holes in the BE and the energy is carried off by the second
hole.

We will compare our measurements with the results of a cal-

culation (S) of transition rates for this process.

The calculation

accounts for the strong dependence of the BE lifetime on acceptor
binding energy and i s in approximate quantitative agr eement with the
measured lifetimes .

II.

Experimental Re sults
The experimental se tup is essentially the same as that diagrammed

in Fig. 2.2.

A GaAs diode la ser was used for excitation in stead

of the argon-ion la ser.

The laser diode wa s mounted in the helium

dewar, a few mm from the sample.
diameter.

The la se r spot was about 1 mm in

The Si crysta l s were prepared by lapping and etching as

di scussed in Chapter 2.

The boron, ga llium and indium doped sampl es

were several mm thick while the Si:Al crystal wa s about 450 ~ m thick.
Bath temperature was measured with a Ge sensor in the sampl e block.
The spec trometer slits were adjusted to select a narrow wav el ength

band (- 3A) centered on the peak of eac h line.

For Si:B the TO

assisted lines were measured while for th e other crysta l s the NP
luminesce nce was studied.

The gated photon cou nter (see Fig. 2.2)

has an adjustable length gate which can be sca nned in t ime relative
to a tri gger pul se whi ch is derived from the diode pul s ing c ircuitry.
The experiment consists of adju stin g the spec trometer to the peak of
a line and then sca nnin g the gate to obtain a plot of the luminescence
intens ity as a f un ction of time after the end of the excitation pulse.
The minimum system response time was tested by measuring the
fall time of the l aser pulse; it was ap pro ximatel y 5 nsec .

measureme nts on Si:B , a 200 nsec gate was used; for Si :Al, Si :Ga
and Si: In, the 5 nsec gate was used.

The l aser exc itation power

was sel ec t ed so th at th e BE luminesce nce dominated the s pectrum.
Thus, the r es ult s s hould not be compli cated by exci ton captu r e and

BMEC decay.

The decay of the BE luminescence was exponential for

over an order of magnitude drop in the intensity.
The measurements were made at 4.2°K and l0°K.

There was

a small increase in the observed lifetime as the temperature was
raised to 10°K.

In the case of Si:B this increa se i s probably due

to the evaporation of excitons off the B impurities and the subsequent "feeding" of the BE by the longer 1 ived FE.

(There was a

reasonably large free exciton emission signal at l0°K, but this
emission signal was very weak compared to the BE emission at 4.2°K.)
In the case of Si:Ga, the BE has an excited s tate 1.47 meV above
the ground state.

At 10°K this excited state i s populated with a

probability comparable to that for the ground state.

If the Auger

rate for the excited state is slower than for the ground state,
population of the excited BE state in Si:Ga (as well as exciton
evaporation and recapture) can incr ease the observed lifetime as
the temperature is raised.
The measured BE lifetimes for the four acceptors are li sted
in Table 3.1.

We have also listed the decay time for the fir st

two BMEC in Si:Al and Si:Ga.

The result for Si:B at 4.2°K i s
identical to that previously reported ( 3 ). (We did not measure the

BMEC decay for Si:B because they are already available in the
literature ( 3 ).) The BE lifetime in Si:In was s horter than our
system re spon se time , and we can only set an upper bound on it.

<5 nsec

35 nsec a

58 nsec
39 nsec

46 nsec

1. 0 ~s ec
80 nsec
59 nsec

T(4.2°K)

67 nsec

1. 15 }..!Sec

T( 10°K)

155

meV

44.5 meV
68.5 meV

350 nsec at 0. 5 watt
300 nsec at 0.5 watt
300 nsec at 0.5 watt
200 nsec at 0.5 watt

2·1 015cm- 3
2·1016

350 nsec at 0.5 watt

6 }..!Sec at 0. 1 watt
350 nsec at 0.2 wat t
350 nsec at 0.5 watt

Excitation Conditions

5·10 14 cm - 3
3·10 16cm- 3
2·1 015 cm- 3

NA
2· 1015 em- 3
5 ·1 014 em - 3
5·10 14cm- 3

Measured lifetimes for bound excitons (BE) and bound multi ple exciton complexes for acceptors
in Si; b1 and b2 label the first and second BMEC, respectively; EA is the acceptor binding
energy and NA is the acceptor concentration.

a) The b2 line for Si:Ga was very weak and there was a background which decayed with a 47 nsec lifetime
beneath it. The presence of the background may cause t he measured decay ti me to be somewhat lar9er
than the actual lifetime .

Si :In(BE)

Si: Ga(BE)
bl(J=2)

Si:B(BE)
Si :Al (BE)
bl(J=2)

Table 3.1

'-J

III.

Auger Calculation of BE Lifetimes
From Table 3. 1, we see that the BE lifetime decreases rapidly

as the acceptor binding energy increases.

This behavior is similar
to that observed for acceptor BE lifetimes in GaP ( 6 ). This effect

can be understood as due to an increased spreading in k-space of
the BE wavefunction as the acceptor binding energy increases ( 6 ).
This idea has been quantified and calculations of Auger rates for
acceptor BE in Si have been carried out ( 5 )

Here I will give only

the physical picture and results of the detailed calculation .
From time dependent perturbation theory, the BE Auger transition rate is given by

FIVI I

Here IF

(3. 1)

is the final state which consists of a free hole, I I ·

is the initial state which co nsists of two holes and an electron
bound to the charged acceptor, and the interaction which leads to
the transition, V, is the Coulomb interaction between the carriers.
The final hol e state is an eigenstate of wavevector.
sible final states are restricted by energy conservation.

The posThe

initial BE state is not an eigenstate of wavevector becau se the
carriers are localized in space by the impurity potential.

order for the Auger transition to occur , the initial BE state must
have an amplitude to contain wavevectors which are acces s ible to
the final state hole.

Because the wavefunction for the holes in

the BE ar e peaked in k-space at k=O while the electron is peaked at
the conduction band minimum, the total wavevector for the BE is peaked
at the conduction band minimum.

This point is rather far, in k- space,

from the constant energy surface accessible to the final state hole.
Thus, spreading of the BE wavefunction in k-space is essential for the
Auger transition to occur.

For the acceptor BE, the holes are more

highly localized in space than the el ectron and the spreading of the
BE wavefunction in k-space is due primarily to the holes.

The extent

of this spreading depends on the impurity , it increases as the binding
energy of the acceptor increases because the holes are more strongly
localized in space for the more tightly bound acceptors .
In principle, it is possible that the Auger tran s itions are
phonon assisted.

If this were the case, the phonon would supply

the wavevector necessary for the transition to occur, and the
transition rate would not depend on the k-space spreading of the
BE

wavefunction.

Thus, one would expect the phonon as sisted Auger

rates to be insensitive to the acceptor type.

Since the observed

l ifetimes are, in fact, very sensitive to the acceptor type, we
believe the Auger transitions occur without phonon assistance.
The Auger transit i on rates for the acceptor BE were computed
using wavefunction s obtained from a variational calculation (S)
with a simpl ified model of the BE.
are shown in Fig. 3. 1.

The results of the cal culation

The calculation is seen to describe the

observed dependence of the lifetime on acceptor binding energy
reasonably well.

The computed l ifetimes are l arger than the

Figure 3. 1
Bound excito n lifetimes vs impurity binding ener gy
for the four common acceptors in Si. The so li d circles
are mea s ured values and the holl ow squares are calcul ati ons of the Auge r lifetime. For Si:In, the lifetime
was shorter than our system r esponse time, and we can
only set an upper li mit on it.

Bound Exciton
Auger Lifetime
Acceptors in Si

-Q)

Al 0

10- 7

( .)

10- 8

""0

:;)

10- 9

e Experiment

Theory

10- IO
40

100

120

140

Acceptor Binding Energy (meV)

Figure 3.1

160

experimental values by about a factor of three for all the
impurities.

Considering the sensitivity of the calculated lifetimes

on the BE wavefunction, we consider this agreement to be reasonable.

IV.

Conclusions
In conclusion, we see that these measurements an swer some

of the que stion s concernin g the systematics of BMEC as a function
of impurity.

The deeper impurities give rise to more rapid Auger

rate s for BE and BMEC and correspondingly l ower radiative efficiencies.

Also, the re s ults expl ain the higher pump powers needed

to make BMEC or saturate the BE in Si with deeper impurities.

saturate the BE we must supply excitons faster than they decay on
the impurities, which means , of course, that it is more diffi cult
to sa turate the BE if they decay rapidly.

Simil arly, to make an

m-exc iton complex, we need the (m-1)-exc iton to live l ong enough
to capture the next exciton.

For the more rapidly decaying com-

plexes thi s means that a high density of excitons is necessary to
r educe this capture time.

This explains the hi gh pump powers

needed to make sig nificant numbers of BMEC in Si:Ga.

It also

may explain the absence of 2-exc iton complexes in Si:In.

The capture

rate per impurity is given by

(3. 2)

where nex is the exciton density, vth the exciton thermal velocity
and a

is the cross section for a s ingle m-compl ex to capture an
exciton making it an (m+l)-complex. For Si:In, o 0 (capture by
a neutral impurity) has been measured (?),and it ha s a value of
-lo- 12 cm 2 at 2°K, decreasing rapidly at higher temperatures.

77
If we assume that the capture cross section for a second exciton
is about the same as for the first, and take a thermal velocity of
106 em/sec, then we can estimate the formation rate of 2-exciton
complexes in Si:In for various exciton densities.

In particular in

order to form 2-exciton complexes this capt ure rate must be larger
than the BE decay rate.

As suming a 5 nsec BE decay rate and the

values quoted above for the other parameters, we find that "ex must be
at least 2xlo 14 cm- 3 . However , we cannot reach this exciton density
at 2°K because EHD will grow and maintain a lower den s ity.

Therefore,

even if two excitons will bind to an indium impurity in si li con,
we cannot expect a BMEC to form out of a gas of free excitons .
In summa~y, we have measured the luminescence lifetimes of BE
on the four common acceptors in Si and lifetimes of the first two BMEC
in Si:Ga and Si:Al.

The BE decay rate s increa se rapidly as the bind-

ing energy of the acceptor increases, there being about a three order
of magnitude difference between the rates for B and In.

These results
are in approximate quantitative agreement with a calculation ( 5 )
which assumes that the lifetimes are limited by Auger transitions.
Knowledge of the decay rates is important to the understanding of the
kinetics of the growth and decay of BMEC.

We find that the BE life-

time in Si:In is so short (<5 ns) that it is probably impossible to
form 2-exc i ton complexes on In impuriti es out of a gas of fr ee exciton s.

References
1.

P. J. Dean, ~J. F. Flood and G. Kaminsky, Phys. Rev. 163, 721
(1967).
Rev.~.

K. Kosai, r1 . Gershenzon, Phys.

R. Sauer, Proc. Twelfth Int. Conf. Physics of Semiconductors

723 (1974).

Stuttgart, 1974), p. 42.
4.

D. F.

Nelson, J. D. Cuthbert, P. J. Dean, and G. D. Thomas,

Phys. Rev. Lett. lZ· 1262 (1966).
5.

G. C. Osbourn, and D. L. Smith, Phys. Rev. B16, 5426 (1977).

P. J. Dean, R. A. Faulkner, S. Kimura, and t·1. Ilegems, Phys.

Rev. 84,1926 (1971) .
7.

K. R. Elliott, D. L. Smith, and T. C. McGill, Solid State Commun.
24, 461 (1977).

CHAPTER 4
TEMPERATURE DEPENDENCE AND WORK FUNCTIONS
OF BOUND MULTIEXCITON COMPLEXES IN Si

Introduction
In Chapter 2 we saw that the Bt1EC interpretation has been firml y

established for two excitons bound to the acceptors aluminum and
gallium (l- 3 ). However, questions arise in extending the picture to
the donors and to the 1O'-'ler energy 1 i nes for the acceptors,

One

question involves the binding energies of excitons to the complexes .
In this chapter we will discuss thermodynamic measurements of the work
functions, ¢m' of the BMEC (m labels the number of pairs in the
complex).

At temperatures such that the thermalization rate of

excitons on and off of a complex is large comoared to its decay rate
we assume that the complex is in ther~al equilibrium with the surrounding gas of FE.

This assumption enables us to obta in the work

functions from our measurements of the line intensities as a function of temperature.
In the usual interpretation of the lines as correspondin9 to
ground-state to ground-state (G-G) transitions , the binding energy
of the last exciton is given by the energy separation between the
given BMEC line and the free exciton (FE).

However, there is no

evidence to support the assumption of G-G transitions.

In fact for

the 2-exciton complex in Si :Ga t he G-G tran s ition is at least three
times weak er than the transitions which leave the BE in an excited
state ( 3 )
Furthermore, with the usual interpretati on for the
larger complexes, each additional exciton is bound more tightly than
the last. In the context of particu l ar models ( 4 • 5 ) it has been
proposed that at least some of the lines are not G-G transitions.

The thermodynamic measurements give us a way of determining the
work functions of the BMEC while avoiding questions concerning the
final states of the optical transitions and without using a specific
model of the structure of the complexes in the interpretation of the
data.

In contrast to the large "binding enerqies" obtained by taking

spectroscopic differences we find from the thermal data, for the
dopants studied here, that the second or third exciton is the most
tightly bound and subsequent excitons bind with decreasing energy.
Furthermore, none of the measured work functions exceeds that of the
EHD ( 6 )
The e xperiments have been performed on crystals of Si doped with
Al, B, Li and P.

The position of the G-G transition is known for
them= 2 BMEC (l- 3 ) in Si:Al . For Si:Al our results aqree with the
previous work for the binding energy of the BE and m=2 BMEC.

How-

ever, our data show that the m=3 BMEC in Si:Al is bound less tightly
than its line position would indicate.

Similarly for Si:B the BE

and m=2 BMEC have work functions agreeing with their spectroscopic
differences.

However, the m=3 and m=4 BMEC are bound less strongly

than the G-G intepretation of the spectroscopy would indicate and
thus correspond to decays into excited states.

The thermodynamic

value for the BE work function in Si:Li agrees with the spectroscopic
difference.

Our measurement of the work function of the 2-exciton

complex indicates that it decays into an excited state of the BE.
We have observed this BE excited state as well as another one in the

Li doped Si .

For Si:P we obtain the correct binding energy for the

BE and our data show that all of the first four BMEC lines arise
from transitions to excited states and are thus l ess strongly bound
than previous l y believed.
The remainder of thi s chapter will be organized as follows:

Section I I we wi 11 clescri be th e method by \'Jhi ch work functions are
obtained from the temperature data.
the experimental methods.

In Sect ion III we will describe

Section IV will be a presentation of the

experimental work fun ctions f or BMEC in Si:Al , Si :B, Si :Li and Si:P.
Section V wil l be a discussion of these results and our conc lusions.

II.

Analys i s of the Temperat ure Data
At suffici ent l y high t emperat ures th e evapora tion rate of exc itons

off a complex will exceed the r ecombination rate for pairs within the
compl ex .

Under these condition s the complex i s in approximate t her ma l

eq uilibrium with the s urroundin g gas of free excitons.

Thus we can use

equ ilibrium stat i sti cs to r elate th e populations of m-exciton complexes, (m- 1) -complexes, an d free exc itons (FE).

An (m- 1)-complex

plus a FE with wave vector£ i s then vi ewed as a type of exc ited state
of an m-comp l ex and as such has a probability of being occupi ed
given by

P (m- 1 , £ )

( Ll . l )

wh er e gm i s the degeneracy of an m- compl ex , gex i s the d e~e nera cy of a
FE, m i s the vmrk funct i on of an m-complex,

ck is the kin eti c

energy of a FE wit h wave vector I, T i s the temp erature, and P(m)
i s the probability of find i ng a compl ex wit h m-exc itons .

Since we are

only interested in the density of FE and not their spec ific £ ' s we
ca n average over £.

In integrat ing over t he wave vecto r of the

exc i to ns, we will assume t hat th e FE di spersion curves are de scribed
s impl y by an effecti ve mass, m* .

Act uall y the FE ground s tate in Si

i s split by about .3 meV(7- g).

However, unlike Ge, the spl itti ng

do es not ge nerate a significa nt ma ss r eversa l(lO) and the . 3 meV
i s l ess than the un certai nti es in the present exper i ments .

Wit h

thi s ass umpt ion of a s in gl e exc iton band and r epl acing the proba bil i ties wi th densities, we obta in

ex m- 1

*kT ~.3/2

0m e

- cpm/kT

(4. 2)

9m-lgex
, nex i s the dens i ty of excitons and nm i s t he
gm
dens ity of m-comp l exes . Eq. (4.2 ) may be rearrang ed, yi eld ina,

\-Jhere Om _

' 2nm k

(4.3)

where now the ratio on the l eft i s access ibl e to experiment.

If we

assume that the lumine scence intensi t y of a given line is prorortional
to t he dens i ty of the assoc iated compl ex , then we have

( T3/2 I

R-n . I

i a; cpm/kT

(4.4 )

ex m-1 ,

where I

i s the int ens i ty of the lu mi nescenc e line due to the decay
of an m-complex and I
i s the in teqrated FE l umin esce nce in te ns ity.
ex
Ther efor e, a graph of the quant i ty on t he l ef t s i de of Eq . (4.4) versus
1/kT wil l produce a strai ght line with slope cpm,

I n the actu al experi-

ments, s urface exc i tat i on was used and t he den s iti es in Eq. (4 . 3) are
functions of pos ition within t he crysta l .

At every poi nt in the

crystal Eq. (4.3) holds, hm·Jever the

si9nal we obsel'Ve is the inter~ral

of the luminescence over the crystal volume.

As lonq as the

carrier profiles do not chan9.e v1ith temperature, then using spatially
integrated luminescence intensities in Eq. ( 4. 4) vii 11 not chanqe the
analysis.

As we lower the temperature, at some point the recombina-

tion rate of the m-exciton complex v1ill

beco~e

comparable to its

evaporation rate and thermal equilibrium will no longer exist.

understa nd this region, it is necessary to investigate the rate
equations which 9overn the BMEC . A system of rate eq uations has been
proposed ( 11 ) to describe the Bt1EC which assumes that each camp 1ex
obeys the relation
dn

-n
nm+l
m _ __!!l+ - - +n
n V a
-nn V a
~- Tm
Tm+l
m-1 ex th m-1
m ex th m

- n

Revap + n
Revap
m+ 1 m+ 1

m m

(4.5)

where Tm is the recombinative lifetime for an m-complex, Vth is the
exciton thermal veloci ty, am is th e exciton capture cross-section of
an m-complex and Revap
i s the evaporation rate of exc itons off an
m-complex.

The terms on the righthand s ide of Eq. (4 . 5) correspond

to r espectively: pair recombination on an m-complex, pair recombination on an (m+l)-complex (making an m-comp l ex ), exciton capture on an

(m-1)- complex (making an m-complex), exc iton capture on an m-complex,
evaporation of an exciton off an m-comp l ex~ and evaporation of an
exciton off an (m+l)-compl ex (making an m- complex).

These rate

equations assume that when a pair in a complex recombines , th e
only effect is to r educe the number of pairs by one .

However, the

dominant decay mec hanism for a compl ex i s probabl y an Auger process (l 2 )
which deposits about 1 eV into the BMEC.

Since the work funct i ons

f or the se compl exes are of orde r of a f ew meV, it may be more appropriate to assume that a non - radiat ive r ecombi nat i on str ips all th e
excitons off the impurity .

Thi s would change Eq . (4.5) by eliminatin9

the second term on the ri0hthand side .
If we ass ume that the rate equat i ons take the form of Eq.(4.5)
then the system can be solved for n

the next sma ll e~ comp l ex .

in terms of quantities inv o lvin ~

If we wri te the rate equation fo r

(~+1)-

compl exes the 2nd, 4th and 6t h t erms in Eq . (4.5) will appear but with
the opposite s i gn, so that add ing th e two equat i ons ca uses th ese terms
to cancel out.

Extrapo l ating this idea we find:

_ n Revap +
Revap
mm
n ~+ l ~+1

(4.6)

dnk
In steady state~= 0 for all k , and we can l et ~ be larqe enough so
that n.Q. = 0, th en so lvin g Eq. (4 .6 ) y i eld s

( 4 . 7)

m-1 ex

As we mentioned above , it may be appropria te to assume th at a
non-radiative r ecombinat ion destroys th e compl ex.

Thi s cha nges Eo .

( 4 . 7) to:

(4.8 )

This extra term in the denominator on th e righth and side i s usually
nm+l
negligible since nm+l << nm so that 1/ T >>
Only under near
nmTm+1
saturation conditions, where nm+l-nm' will this term be i mportan t.
In either case, (Eq. 4.7 or 4. 8) if th e temp eratu re i s hiqh enou~ h
to ·make the evaporation rate l arge compa r ed to the recombination t erms
then we can neol ect decay leaving
Revap

( 4 . 9)

Ass uming deta il ed ba l ance for the capture and release of exc itons,

Figure 4.1
Graphs of a ca l culati on of the expected temperature
dependence of BMEC from Eq. (4.7) for three forms of cr (T).
The graphs cover a temperature range in which modifications
to thermal equilibrium du e to finite lifetimes become important.

-6.0

-7.0

Decay Induced
Modifications
to Thermal
Equilibrium

f/kT/

/1 j

..----:--- -8.0
f-->c::

......

1.__..

-9.0

...........

-8

!J /

CJ"

1.__..

lc ocT

-eI

cp = 5me V
Tm = 7°K

- 10.0

2 CJ" ( T ) e- 1 k Tm
I IT = Tm

-11.0

_____ L_ _ _ __ __ L_ _ _ _ _ _
2 .0
2 .5
3.0

- 1 2 . 0L-----~~------J__L

0.5

1.0

1. 5

I /kT(meV- 1)

Figure 4.1

Revap = 2Tim kT
( h2

)3/2 0 V a

m th m-1

-cp /kT
e m

(4.10)

and Eq. (4.9) can be reduced to the thernal equilibriu~ result,E~ . (4.2) .
However , we s hould expect deviations from the eq ui librium result
when Revap~ l /T , as descr i bed by Eq. (4.7).

At l ower tempera tures the

curve we obtain from the Arrhen ius pl ot wi ll bend down away from the
stra i ght line of Eq . (4.4) an d become relatively constant. Thi s rounding
will app ear at about t he temperat ure , Tm, wh er e the evapora tion r ate
equa l s the decay rate.

(4. 11 )

The valu e of T~ is linea r in cpm whil e dependina only loqarith~cally
on the other terms , and , hence i s sensitive to cpm and in sensitive to
the precise values of the other parameters .
In Fig. 4.1 we show how we expect the te~perature data t o l ook for
three possible f orms for the capture cr oss section.

The curves are a

graph of the natural logarithm of T312 t i mes the righthand side of

Eq.(4 . 7) vers us 1/kT . This funct i on , t n
T 0 ~T)
-cp/~T • i s
_4 1/T + T a(T) e
pl otted for a(T) = consta nt, (T) a T , and a(T) a T-8. A measurement of a (T) has been made for the BE in Si :In ( l 3 ), and a was found to
increase rapidly wi th decreasina temperature .

A value of 5 meV was

used for $ and T wa s chosen to be 7°K.

fixes T .

Together with a(T) thi s

The dashed line in the figure corresponds toy = $/kT

and i s approached asymptotically by the calculated curves as T-~ .
We see that curve assuming a~ T- 8 shows essentially no rounding at
Tm.

The experimental data do exhibit breaks, indicatin9 that the
cross sec tion does not vary as rapidly as T- 8 . The graphs assuming
a = constant and a ~ T- 4 both round over near T

III.

Experimental Methods
The B, Al, and P doperl sampl es were cut from float zone s il icon

crystals in which the dopant s were introduced during growth.

The Si:B

and Si :P samples were 3 mm thick whi l e the Si:Al was 450 ~m thick .
Si :Li samples were prepared by diffu s ing Li onto Si crystals .

The

Lithium

doped si li con sample No . 1 was prepared from p-ty pe si li con with an
initi al room temperature resistivity of 15 kQ- cm.

A suspension of

Li in minera l oil was painted on both sides of the 1 mm thick crysta l
and diffu sed into the si li con at 370°C for one minute.

The crystal

was then r emo ved fro~ the furnace and li ght ly l apped to remove excess
Li.

A drive-in diffusion at 650°C for 1 1/2 hours was performed

to improve the uniformity of th e Li distribution.

The other Si:Li

sample (No . 2) was prepared from n-type s ili con with an initial
room temperature resistivity of 80 Q-cm.
for one minute at 400°C.

The pre-depos ition was

The rest of the preparation was the same

as for the first Li-doped sampl e.

The impurity concentration in

each of the samples was rletermined from room te~perature r es i stivity
measurements .

The crystals were all about one square centimeter in

area.

The samples were l apped and etched with HN0 :HF (7:1) to
remove surface damage. Immediately before mount ing in the dewa r

the crysta l s were washed with methano l.

The 5145 A line from an argon- ion l aser was used for excitation.
The excitation was continuous, not chopped, to make accurate meas ure -

ments of sampl e temperature possible .

The exciting beam was fir st

passed through an infrared filter to remove any extraneous emission
lines in the wavelen gt h r eg ion of interest.

Spot sizes us ed on the

sample ranged between 3 and 8 mm in diameter.

The luminescence wa s

collected with a l ens and after passing through a filter to remov e
the visible l aser light was focu sed on the entrance slit of a
grating spectrometer.

The ligh t was detected v1ith a liquid nitrogen

cooled S-1 photomultiplier and processed with photon counting equipment.
The sample temperature wa s measured with a calibrated silicon
diode thermometer solder~d directly to th e crystal .

The FE lines hape

was fit for th e higher pump powers and the temperature obtained agreed
with the sensor r eadi ng to within 15% .

For the Si :Al sample the

intensity ratio between the qround state and excited states of the BE
wa s also measured.

The agreement between this measure of th e sampl e

temperature and the sensor reading was again to within 15%.

Another

check on the se nsor s ability to measure tru e sampl e temperature was
obtained by observing the sensor reading as liquid helium wa s a llowed
to touch the bottom of the crystal.

With the sensor attached at the

top of the sample. as soo n as any portion was im~ersed in liquid
the reading dropped to within . 2°K of 4.2°K with l aser powers
typical of tho se used for these experiments.

Thus, the whole

crystal was at essentia ll y one temperature and that temperature
was accurately measured with th e Si sensor.

It was found that the

measured work function s depended somewhat on excitation intensity,
decreasing 10-20% at high pump densities.

This i s probably due to

inhomogeneous sa turation, leading to a t emperature dependent carrier profile as di sc ussed in the l ast sect ion .

To reduce this ef-

feet, we used as low excitation dens ities as was f eas ible for any
given line.
Spectra were taken at each temperature and any background was
subtracted to obtain the true inten s ity of the l ines.

The slits were

opened so that the peak inten si ty was proportional to the integrated intens i ty of the BMEC lines.

For the FE the integrated intensity is proportional toT times the peak intensity (l 4 ) . For
the BE we hav e ass umed that th e dens ity of occupied impurities i s

small compared to the total i mpurity density.

Therefore we avoided

saturating centers when measuring BE work functions.

95
IV.

Experimental Results

Si :B and Si :Al
The temperature dep endence of the ratios along values of ~

(±.5 meV form = l-3, ±l meV form= 4) for the BE and m = 2-4
BMEC for Si:B are shown in Fig.4.2.

At high temperatures the data

points fall on a straight line while the curves bend away from the
straight line at l ower temperatures,.

The rounding of the curves

is seen to occur somewhere between 7 and 8°K for m = 2,3 and at
a lower temperature for the m = 4 B~~C.

These values for T are in

a ran ge expected from Eq. (4. 11).
The uncertainti es \'Jere assigned by observinq that in Fi9. 4.1 for
a= constant or a~ T- a st raight line fit to the high temperature
portion of th e curve will mi s-estimate ~ by up to about .5 meV.

For

the larger complexes, (m = 4,5) we hav e data over a limited temperature range and uncertainties of ~ 1 meV are assumed.
The rounding is again apparent in Fig.4.3 which shows the data
and the valu es of ~m for the BE and m = 2,3 BMEC in Si;Al.

Using

Eq.(4.ll) we can estimate the temperature, Tm, at which the curves
should bend away from the straight line. Ass uming Tl = Bxlo- 8 sec ., (l 2 )
Vth = 2xl0 6 em/sec, m* = 0.6 m0 , a 0 = lo- 13cm 2 (l 3 ), and~~ = 5.1 meV,
we can estimate Tm for the BE in Si:Al and obtain T1 = 6.5 0 K, in
agreement with the data.
Fig. 4.4 is a graph of our r es ults for \-Jork functions of B~1EC

Figure 4.2
The temperature dependence of m = 1-4 BMEC lines in
Si:B. Here ¢ i s the work function obtained from the
stra i ght linemfit to the data.

T EMPERATURE (°K)
7.0
6 .0
5.5
5.0
4.5
I 0.0 9.0 8.0
10. 0 ~~~~-.~--~--~-----.----·~--~~---.~--~

Si:B
N8 = 8 x

9.0

to' 3 cm- 3
¢ 1 = 3 .6 meV

514 5 A Excitat ion

0.2W

8.0

~4 mm

Spot

¢ 2 = 6 .0 meV

7.0

• 0.2W

~4 mm

Spo t

--J 6.0
><
a>

ri f: 5.o

1-H

• • •

4 .0

cj>3 = 6. I meV
0.2W ~4mm Spot

3 .0

¢ 4 = 3 .0 meV
0. 1W ~2mm Spot

2.0

1.0

2.0

1.5

1/kT

(meV- 1)

Fi gure 4.2

2.5

Figure 4.3
The temperat ure depend ence of m = 1-3 BMEC lines in
Si:Al. Here ~m i s the wo r k function obta ined from the
stra i ght line fit to the data .

TEMP ERATURE (°K)
8.0

10.0

7.0

6 .0

Si: AI
9.0
8.0

NAI = 5 X 10 14
5145A Exc itati on
~4 mm Spot

cp 1 = 4.4 meV
O.IW

7.0

...---E

6.0

c/>2 = 4.6 meV
0.4W

f.H
.....___......

E. 5.o
4.0

c/> 3 = 6.8 meV
0 .4W

3.0
2.0
1.0

1.0

1.5

1/kT (mev- 1)

Figure 4. 3

100

in Si:Al and Si:B, as well as the values one obtains by taking
spectroscopic differenc es ( o

- hv
- hv ) as a function of the
ex
m'
number of pairs on the impurity (m). We know that the BE lines

arise from G-G transitions and, therefore, th e work functions,
¢ , and the spectroscopic differences,

o1 , are equal. We see in

the figure that for both boron and aluminum ¢ 1 equals o1 to within
the experimental un certainties.

(The unc ertainty in om is ±. 3 meV,

arising main ly from the uncertain ty in the FE edge.)

The position

of the G-G transition for a 2-exciton complex in Si:Al has been
established independe~tly( l- 3 ).

Thus we can determine ¢ 2 spectro-

scopically and the thermodynamic value is in a~reement with the
spectroscopic result.

We also see that ¢ 2 = 6 2 for Si:B, which

indi cates that them= 2 BMEC line i s a G-G transition.

For both

Si :Al and Si :B ¢

< 6 , and the difference is larqer than the
experimental un certainties, indi cati ng that them = 3 BMEC lines

arise from a transition l eavi ng the 2-exc i ton complex in an excited
state.

The value for ¢ 4 in Si:B i s considerably less than o4 , again

indicating an exc ited state.

One possible explana tion for the

substantia l decrease in work function for the fourth exciton in
Si:B is that thi s last pair puts a fifth hole in to the complex but
all four r

hole states are already occupied. The envelope function
will be forced to cha nge in order to accommodate the fifth hole,
possibly l ead ing to weaker binding .

l 01

Figure 4.4
The thermodynamic work functions and no-phonon spectroscopic differences as a function of the number of pairs on
the site for BMEC in Si:B and Si:Al. The spectroscopic differences were obtained assuming a no-phonon FE threshold of
1154.6~.3 meV.
For Si:Al th e m=2 G-G transition is used for
the spectro scopic difference. The energy of this transition
i s inferred from the position of the transitions l eavi ng the
BE in an excited state (Ref. l -3). The dashed line indi cates
the work function of the el ectron- hol e droplet (Ref. 6).
The experi menta l uncertainties in the thermodynamic numbers
are +.5 meV for m~ l -3 and ±l meV for m=4.

102

cpm -Thermodynamic

o 8m - NP Spectroscopic
difference

o,o

Si:B

~t&

Si : AI

..--..

-- ---- --- - --B--- -----

-.....

¢ EHD

<.9
0::

NUMBER OF PAIRS (m)

Figure 4.4

103

For Si :Al and Si:B we hav e found that wh e reve r ~m i s known
independ ently (the two BE and them= 2 gMEC in Si :Al) the thermodynam i c determination is in agreement with the previous r es ult .
However, for m = 3,4 we find th at the work function s are l ess than
the corr esponding spectrosco pic differenc es.

Thi s implies th at the

luminescence lines are due to transitions which l eave the fin al complex in an exc ited sta t e.

The energ i es of these excited states ar e

Si:Li
The temperat ure data and values o f ~

for the BE and m = 2 BMEC
in Si :Li are shown in Fig.4.5. As for Si :B and Si :Al we find that

~l = o1 in Si: Li (o1 = 3.4 me V (lS)). Again this was to be expected
si nce the BE i s known to be a G- G transition.

However , in co ntra st

to the two acceptors, we find that ¢ 2 < o2 (o = 6 . 1 meV ( 15) ). The
data f or the m = 3 and l arger compl exes did not show a break until
about 9°K or high er.

By these t empe rat ures t he lines were too weak

to obtain r eliabl e measurements of ¢ , but the valu es forT

ind icate

that the l arger complexes may have work fun ction s that are at least
as l ar ge as ¢ 2 .
We see that Si :Li i s qualitat ively different from Si:B and
Si:Al in t hat ¢ 2 < o2 .

The m=2 BMEC line corresponds to a trans i -

tion whi ch l ea ves the BE in an exc ited stat e , with an energy

o2 - ¢2 = 2.5 meV above the ground state.

Spectra of Si: Li are

104

Figure 4.5
The temperature dependence of ~=1 , 2 BMEC l ines in Si:Li .
Here ¢m i s the work functio n obtained from the straight li ne
fit to the data . T~5 data for m=l were obta ined fr om sampl e
No. 2 (Nli = 1. 2x10 cm-3) with 100 mW of exc ita tion and a
2 mm spot s ize . The data for m=2 were obtained from two diff ere nt expe riments. The data for 9°K and hi gher in temperature
were taken on samol e No . 2 with 600 mH of excitation and a 3
mm spot size. The data fo r goK and lower were taken on sample
No. 1 (N l i = l xlo13cm-3) with 200 mW of excitation and a 5
mm spot s1ze. The two sets of data were shifted vertically
to ~atch the 9°K data points from eac h experiment .

105

TEMPERATURE (° K)

Si: Li

5145 A Excitation
5.0

4.0

C'J

......_

E 3.0
r01-

2.0

1.0

1.5

1/kT

(m ev-1)

Figu re 4.5

106

Fi gure 4.6
Phot oluminescence spec tra at three tempera t ures of Li doped Si in the ener gy ran ge f or TO and LO phonon ass i s t ed
trans iti ons . The lines l abel ed BE are associ at ed with th e
Li i mpuriti es . The thermal dat a predi ct the exi s t ence of an
excited s t ate of the BE at th e position indi cated by the
arrow. The line l abel ed P i s associ ated with phos phorous
i mpuriti es and the line label ed FE i s the fr ee exciton
luminesce nce .

107

Si: Li
Li -2

NL,= 1.2 X 10

400mW 5145

- 3mm spo l

BEro
FE

-u-

~BE;

1.090

1.095

ENERGY (eV}

Figure 4. 6

108
shown in Fig.4.6 and we see in the upper spectrum (10.5°K) a
luminescence line, labell ed BE *2 , correspondin~ to an excited state
of the BE with the expected energy (indicated by the arrow).
know

that this line at 1. 09589 eV is an excited state of the BE

because it is not present in the two lower spectra taken at 4.2

and 2.1 K, in accordance with a Boltzmann factor. In addition to the
l . 09589 eV line and the BETO and BELO (the line labelled p i s due to
pho sphorous impurities) we see a line at 1.09451 eV, la bel l ed BE~.
Thi s line is large at 10.5°K, sma ll er (relative to

the BE) at

4.2°K, and entirely absent in the 2.1°K spectra.

This dependence

on temperature indi cates that it is also an excited state of the BE,
0.94 meV above the gro und state .

In Si : Li we have observed two

excited states of the BE, one of which is positioned correctly to
be the final state in th e radiative decay of the 2-exciton complex.
The G-G transition for this complex would li e 3.6 meV below the FE
and be buried under the BE line.
C.

Si: P
The data and values of ¢m for Si:P are given in Fi g. 4.7.

The

work functions ¢m and spectroscopic differences, 8m, as well as the
spectroscopic differenc es, to the B-lines, o8 , are plotted in Fig.4.8
as a fun ction of the number of pairs on the impurity. The B- lines
are interpreted as G-G transitions in the shell model ( 5 ), making
the o8 (o8 = hv - hv
, oB ~ 01 ), the work functions of the com1
8m-l
m m
ex
plexes in this model. We find that ¢ 1 = o1 = 8~ as expected. For

109

Figure 4.7
The temperature dependence of m=l -5 BMEC lines in Si:P .
Here ¢m is the work function obtained from the straight line
fit to the data.

TEMPERATU RE (°K)
8.0

10.0

7.0

6.0

Si :P

5145A Exci tati on

-3

• Np = 4x lo' cm• Np = 6 x 10

• cp 1 =4.9meV
O.IW
-Smm Spot

-6mm Spot

____....._.

cp3 = 4.5meV

E w

t-t 1-1

0.4W

-4mm Spot

f-t-1
..._____...,..

-4mm Spot
o ~~--~--~--~~--~--~--~~--~--~--~~

1.0

1.5

1/kT (meV- 1)

Figure 4.7

2.0

Figure 4.8
The thermodynamic work functions, no-phonon spectroscopic
differences, and spectro scopic differences to the B-lines (Ref.
17) as a function of the number of pairs on the s ite for BMEC
in Si:P . The no-phonon spectroscopic differences were obtained
by assuming a no- phonon FE threshold of 11 54.6±.3 meV and the
B-series spectroscopic differences were obtained from the data
in Ref. 1. The dashed line indicates the work function of the
e l ectron-hol e droplet (Ref . 6). The experimental uncertainties
in the thermodynamic numbers are ±.5 meV for m=l - 3 and ±l meV
for m=4,5.

11 2

Si : P
~ 8~- f3 series Spectroscopic
difference (shell model)

8m - NP Spectroscopic

difference

¢m- Thermodynamic

>(].)

->E

-------~---- - -----------

<.9
0::::

c/>EHD

NUMBER OF PAIRS (m)
Fi gure 4.8

113

the 2-exciton complex, ¢ 2 < 02 as we found for Si:Li.

However, the

graph shows that ¢ 2 > o~ which indi cates that the s1 line does not
arise from the decay of a 2-exciton complex bound to a phosphorous
impurity.

For m > 2 the ¢ m decrease as the number of pairs on the

impurity increases.

Thi s is in contrast to the beha vior of om and

0 ~ which both increase for increasing m.
¢m

None of the measured

exceed ¢EHD while both om and o~ are qreater than ¢EHD for suffi-

ciently l arge m.
Since 6 2 > ¢ 2 the m-2 BMEC trans iti on we observe in Si:P must
leave the BE in an excited state as in Si:Li.

There are two excited

states of the phosphorous BE observed in absorption and hi gh temperature luminescence spectra.
energy.

Neither of these lines have the correct

However, we expect to see the G-G transition at hv

-¢ .
The

Fig.4.9 is a spectrum taken on Si:P in the TO phonon region.
sample was prepared from ultra-high purity (NA-ND = 2xlo 11 ) Si by
tran smutation doping (l 6 )

The lines are labelled after Ref. 17 .

m is them-exciton complex line) , The arrow indicates the expected

position of the G-G transition for the decay of a 2-exc iton complex.
There i s a smal l line with energy 1.09058 eV at this position.

The

m=2 BMEC on boron lies at about this energy; however, the boron content of this sampl e is very low.

vie do not know whether the observed

line is due to P or whether the P-related transition is being masked
by the m=2 boron line.

114

Figure 4.9
Photoluminescence spectrum of P-doped Si in the energy
range for TO and LO phonon assisted transitions. The lines
l abel ed with a ' s and 8 ' s (Ref . 17) are associated wi th the P
impurities. The therma l data predict the ex i stence of the G-G
transition for the m=2 BMEC decay at the position indi cated by
the arrow .

115

Si: P
Np= 1.5X 1014 cm-3
400 mW

N 8 "'2XI0 11 cm3

5145 A "'3mm spot
Tbath = 4.2oK

._>(f)

t-i

1.086

1.090

ENERGY (eV)
Figure 4.9

116

Discussion and Conclusions
We have measured the temperature dependence of BMEC lines in Si

doped with B, Al, Li and P.

At the higher te~peratures we assume

that the complexes come into thermal equ ilibrium with the surrounding
exciton gas.

This assumption all ows us to extract values for the

work functions of the BMEC from the thermal data.

We in terpret the

lower temperature data in terms of a system of rate equations and
we see the expected transition to thermal equi libr ium as the
tempera ture is raised.
In all cases where the work functions are known independentl y,
(m=2 for Si :Al and the four BE)~ the values of ¢ obtained from the
thermal data are in agreement with the previous r es ults. He also
find that for the m=2 BMEC in Si:B ¢ 2 = 6 2 , indicati ng that the
line is a G-G transition.
that ¢m < 6m.

For the other lines measured, we find

This suggests t hat ground ~state to excited-state

transition s are be ing observed.

For Si :Li the temperature depen-

dence of th e m=2 BMEC predicts that the BE has an excited state
about 2.5 meV above the ground state.

Thi s ha s been confirmed by

our observation of an excited state of the BE at the expected
pos ition in high temperature luminescence spectra .

We also see a

line in spectra of Si:P which is positioned correctly to be the G-G

117

transition of the m=2 BMEC, though we have not yet made this identification with certainty.

Its sma ll size and the general lack of

lines due to G-G transitions for the B~EC are consistent with data
for Si:Ga in which the G-G transition for the m=2 B~EC i s at l east
a factor of three small er than the transitions leaving the BE in
an exc ited state.
We find that the dependence of the measured work functions on
the number of excitons in the compl ex is very different from that
of the corresponding spectroscopic differences .

In genera l the

spectroscopic differences increase with the size of the complex while
the values of ¢m show that for Si:Al and Si:B, the binding increases
form going from 1 to 3 and then decreases for m=4 in Si: B.
and Si:Li, ¢m increases form going from
going from 3 to 5 in Si ;P.

For Si:P

to 2, then decreases for m

The decrease in ¢

m for m=4 in Si;B may

be due to the necessity of changing the envelope function when the
fifth hole is added .
Our data answer some of the questions involving the work functions of BMEC but raise some interesting new questions.

All of the

measured work functions are l ess than ¢EHD"

In the limit of l arge

m, however , we expect ¢m to approach ¢EHD"

We have not been abl e

to measure t he work functions for l arge enough complexes to observe
this limi ting behavior.

It may be possible to understand the approach

to ¢EHD in terms of a surface energy correction to th e dropl et work

118

function for very small drops.

We also do not have a clear under-

standing of the decrease in ¢m form ranging from 2-5 in Si:P .

The

difficulty in minimiz ing the electron-el ectron in teractio n when the
fourth electron is added to the complex may contribute to the decrea se in work function go ing from m=2 to m=3 in Si:P.
As well as the dependence of the work functi ons on the size
of the BMEC, another question these data raise concerns the nature
of the final states of the optical transitions.

We fi nd that in

general the observed lines involve trans itions which leave t he
complex in an excited state.

In only a few cases (as ide from the

BE) are G-G transitions observed.

However, a back9round i s often

observed beneath the BMEC lines (l 5 ), and it could arise from a
superposition of G-G and other tran s ition s which are too weak to
appear as di screte lines.

We are still l eft with t he question of

why the complexes seem to decay predominantly into one particu l ar
excited sta te when the final compl ex probably has many excited
states.

In some cases the remaining complex i s l eft with enough

energy to ki ck off one or more excitons .

We know that the excited

st ate s mus t hav e li fe times of at leas t a picosecond to 9i ve th e observed linewidths.
In conclusion we have used a thermodynamic method to measure the
binding energ i es of the last exc iton for BMEC in Si :Al, Si:B,
Si:Li and Si;P.

For th e larger complexes the wor k functions we

11 9

meas ure are considerabl y smaller than the corresponding spec troscop i c differenc es.

Int erpret in~

the BMEC lines as beina due to

G-G transitions l eads to work function s which increase monotoni ca ll y
for the large r compl exes; whereas, we find that the second exc iton
in Si:P and third in Si:B are the most tightly bound and eac h
succeeding exc i to n is bound with l ess energy than the l ast.

find that, in general, the optica l transitions l eave the remaining
compl exes in exc ited st ates.

120
References
1.

t~. L. t.J.

Thewalt, Phys. Rev. Lett. 38, 521 (1977); and Can. J.

Phys. ~. 1463 (1977).
2.
3.

c. Lightowlers and M. 0 . Henry, J. Phys . ClQ, L247 ( 1977).
s. A. Lyon, D. L. Smith, and T. c. McGi 11 , Phys. Rev. B.!Z_, 2620
E.

( 1978).
4.

T. N. Morgan, in Proceedinqs of the Thirteenth International
Conference on the Physics of Semiconductors, Rome, 1976, edited
by F. G. Fumi (Tipografia t~arves, Rome, 1977). p. 825.

G. Kirczenow, Solid State Commun.

R. B. Hammond, T. C.

~1cGi 11 ,

£L, 713 (1977).

and J.

~1.

f~ayer,

Phys . Rev. BJ_l,

3566 (1976).
7.

R. B. Hammond, D. L. Smith and T. C. McGill, Phy s. Rev. Le tt.
35,1535 (1975).

M. Capizzi, J. C. Merle, P. Fiorini, and A. Frova, Solid State
Commun. 24, 451 (1977).

R. B. Hammond and R. N. Silver, Solid State Commun. (to be
publi s hed).

BJ2_, 4898 (1977).

10.

t~. Altarelli and N. 0. Lipari, Phys. Rev.

11.

R. N. Silver, Phys . Rev. Bll_, 1569 (1975).

12.

S . A. Lyon, G. C. Osbourn, D. L. Smith and T . C. McGill, Solid
State Cornmun. ~. 425 (1977).

13.

K. R. Elliott, D. L. Smith, and T. C. McGill, Solid State
Commun. 24, 461 (1 977 ).

121
14 .

The integrgted intensity of the FE i s given by
I · J (hv ) 1/ 2 e-hv/kT dv , or I
~ I (kT) 312 . The
ex
ex
intens ity at the peak of the FE lumi nesce nce line i s pro1 2
port ional to I 0 (kT) 1 . Therefo r e Iex ~ T x !peak'

15.

K. Ko sa i and M. Gershenzon, Phys. Rev.

16.

A. Hunter , et ~~ (unpublished).

17.

M.L.W. Thewalt , So lid St ate Commun. £1, 937 (1 977 ).

B~,

723 (1974).

122

PART II
CHAPTER 5

CLOUD MODEL FOR THE DECAY
OF EHD IN PURE AND DOPED Ge

l 23
I.

Introduction
As discussed in Chapter 1, the exi stence of electron-hole-droplets

in semi conductor s i s now well establi s hed.

Thei r eq uilibrium proo-

erti es have been stud i ed exte ns ively , es pecially in Ge.

However ,

th ere are still que s tion s concerning the transient behav ior of the
drop s in Ge.

Numerous inves tigations hcve been made of the growth

and decay of EHD (l-lO) .

The kinetics have been studied with both

surface ( B) and volume exc itat ion ( 9 ).

In the volume-excitat ion

experiments ( 9 ) th e EHD ar e observed to grow from a s uper- saturated
FE gas in agreement with the predi ction s of nu cleat ion theory (ll •12 )
The surface exc i ta ti on experiments are l ess well understood.

They

are compl icated by the transport of carrier s into the crystal.

The

EHD appe ar to be formed in a dense pla sma at the surface under the
l aser spot and then move into the sampl e forming a "cloud" of dropl ets
with a di ame ter of 1 mm or more.
Simil arly, at high excitation den s iti es the decay of EHD i s
complex.

The drop s are in a gas of fr ee excitons and their decays

are coupled.

The general system of equations which must be solved

i s,
d\!.
-\).
= 1
dt

aT \! i

2/3

-EHD/kT+bn

ex

(r . ) \!~/ 3

(5. l a )

fo r eac h dropl et, l abe ll ed by i, and

OV n

ex

(r) -

n (r)
- EHD/kT
2/3)
ex
+ aT 2\! 21.1 3 e
-bn (r) \). o(r-r.)
Tex
ex
(5.lb)

124
for the FE, where v. i s the number of electron-hole pairs in the

ith EHD located at ri, T and T

ex are respectively the EHD and FE
recombination lifetimes, a is the Richardson-Dushma n constant (l 4 )
T i s the temperature, nex(r) is the den s ity of FE at the point r,
and 0 i s the exciton diffusivity. The constant b i s equal to
nvth(4nn /3) -2/3 where vth is the average thermal velocity of the

excitons, and n0 i s the density of pairs in an EHD.

The three terms

on the righthand side of Eq. sJa are the recombination rate for pairs
in the EHD, the FE evaporation rate off EHD, and the exciton capture
rate on EIID, respectively.
Even in principle the system of equations 5.la,b i s in soluble
si nce we have no way of determining the pos ition of each EHO.

Some

simplifying assumptions are necessary to obtain tractable equations.
Pokrovsk ii (l 3 ) and Hensel ( 3 ) have proposed a model which has been
used to describe the decay of the EHD in the surface-excitation
experiments.

Thi s average or "single drop" picture i s based on the

assumption of a uniform number den s ity of identi ca l drop s , N, throughout the crystal and a uniform exciton den s ity nex. Then the sys tem of
equations 5.la,b reduces to two average equations.

For droplets we

have,
dv
-v
T2 2/3 -¢EHD/kT + bn v 213
-a
ex
dt

(5.2a)

and for FE,
bn Nv213
ex

(5.2b)

125
where v is the average number of e l ectron-hol e pairs in an EHO.
Usually the further approximation i s made that nex is zero, i.e., the
EHD are evaporating in a "vacuum", and thus the exciton capture rate
is zero.

This decouples Eq. 5.2a from the excitons and allows an

analytic solution,

v(t) = v(o)

(t -t)/3-r

- 1

(5.3)

e t/3-r - 1
with
v ( o) l/ 3
2 -¢EHD/kT
aT -re

(5.4)

At low temperatures the evaporation rate is very sma ll making
tc >> 3-r, and th e decay i s nearly exponential wi th a lifetime, -r .
At higher temperatures, the decay is faster than this exponential
and i s characterized by the cut-off time, t .

By adjusting t

it

is poss ible to fit a wide range of experimental data with this mode l.
However, tc is found to be a function of excitation intens ity. At
relatively high excitation ( 5 , 6 ) the values of t needed to fit the
113
data lead to values for v(o)
and consequently initial drop radii
whi ch are much l arge r than those deduced from other experiments (lS)
From fits to the decay tran s ients drop radii of se veral hundred microns
are obtained, whil e li ght-scattering experiments show that the drop s
are less than ten microns in s i ze ( 2 • 10 • 16 ). Furthermore, we know
from infrared absorption measurements that at all but the lowest

126

excitation intens ities the EHD form a cloud with interdrop distances
small compared to the FE diffu s ion length (l 6 )
We expect that the
FE dens ity in the body of the cloud to be nea r the thermodynamic
equilibrium value , not zero.

The re sulting r ecapt ure of exciton s

should r ed uce the net evaporation and its effect on th e decays at
high temper atures .
In this chapter a new model will be introduced which overcomes
some of the defi cienci es in the singl e drop picture.

Inst ead of

assuming the EHD are spread throughout the crystal, the new model
ass umes that there i s a uniform density of drops within some r egion
(the cloud) and non e outside.

The FE dens ity ins ide the cloud i s

determined by so lving the diffu s ion equation around an "averag e drop".
The diffu s ion equation i s again solved to determine the exciton profile
outside the cloud.

By incorporating the expe ri menta l fact that t he

EHD form a cloud, and including the diffu sion of excitons, it i s
possibl e to fit the data at all pump powers and t emperatures without
invoking the unphysica ll y l arg e drops.

From a fit to th e EHD decay

it accurately predi ct s th e FE luminescence decay .

Furthermore, it i s

possibl e to include the effect s of dopin g on the EHD luminescence
transients by dec r eas ing the FE diffus ion length.

127
II.

Desc ription of th e Model
Fig. 5.1 is a schematic illustration of the basic ideas under-

lying the cloud model.
dots excitons.

The smal l circles repre se nt droplets and th e

The cloud is assumed to be a hard-ed ged sphere in

that th ere i s a uniform density of EI-ID 1 s within some radius R and none
outs ide . We know that thi s is somewhat idea lized in that th e cl oud 1 s
shape i s much more complicated ( 16 •17 ) and the EHD may be in motion( 20).
However , the model r eta in s the essential idea thnt there i s a bounded
r egion in whi ch droplets are relatively close together , and the
exciton density is consequently non-negli gib l e.

Within the cloud it

is assumed that th ese exc itons have an average density (n ) which
ex
is determined by conditions in the body of the cl oud. Since the EHD 1 s
are t ake n to be occupying a region within an infinite medium, there
i s an exc iton diffu s ion tail extending out from th e s urface of the
cloud, the density being pinned to nex at th e surfa ce and going to
zero at infinity.

In the figure this is shown as a decreas ing

dens ity of dot s (e xc iton s ) moving out from th e cloud.

This diffu s ion

of excitons away fro~ the cloud provides one means of decay, since
it is assumed that they are s uppli ed by the evaporation of dropl ets
at th e surface, making th e cloud s hrink.
The in set in Fig. 5. ldepicts the situation in the body of th e
c l oud and i s th e bas i s for th e ca lculation of th e average exc iton
dens ity.

The cl ose proximity of the droplets, interdro p di stances

being muc h l ess than an exciton diffu s ion l e n~th for pure material,
impli es that eac h drop on the averag e only suppli es excitons to its

128

Figure 5.1
Schematic i llu stration of the cl oud of el ectron- hol edroplets and fre e-excitons. The radiu s of t he cl oud i s Rc·
Inset shows an enl arged vi ew of an el ectron-hol e dropl et surrounded by neighboring el ectro n- hole droplets . The centra l
el ect ron-hol e-droplet needs to suppl y exc i to ns only in to the
volume bou nded by Rs (das hed line ).

129

SCHEMATIC OF CLOUD
OF
ELECTRON - HOLE DROPLETS

..
o - ELECTRON - HOLE DROPLET
· - EXC ITON
Figure 5.1

130

immediate vicinity.

Each droplet is assi8ned a spherical volume of

radiu s Rs as shown in the figure by the dashed line, and the condition i s imposed that the net flux of excitons across these boundaries
is zero.

This is in contrast to a single drop picture in which the

exciton s outside the drop would have a boundary condition at
infinity.

Solving the diffusion equation within the region bounded

by Rs and averaging the exciton density over this volume gives the
average exciton density within the cloud.

Each droplet in the body

of the cloud now shrinks as pairs within it and the excitons inside
its region recombine.

Thus the number of pairs bound in droplets

and the associated luminescence signal decrease both through the
shrinking of individual droplets and of the cl oud as a whole.

This

approach allows us to treat both evaporation and backflow for an EHD
as well as the escape of excitons from the cloud; two important
aspects of the problem which are neglected in the single-drop model.
The differential equation describing the decay of the cloud is
determined by equating the rate of change of the total number of
pairs in the cloud to the sum of the recombination rate for pairs
bound in droplets, the recombination rate for excitons within the
cloud, and the diffusion current of excitons away from the cloud
at its surface .

-V c N0

- JD R

(5.5)

where Vc is the volume of the cloud, N0 the density of pairs bound

131

in EHD's (EHD density x fill factor), and J 0 the diffusion current of
free excitons, here evaluated at the radius of th e cloud, R .

Dis-

tributing the time derivative we obtain two terms one desc ribing
the change in Vc and the other change in total density of pairs .

If

we assume that recombination only affects the density of pairs in
the cloud without changing its size while exciton diffusion causes
the cloud to shrink but only perturbs the pair density slightly near
the surface, then Eq. (5.5) separates giving
dR
(No+ nex) dtc = IDvnexiR

(5.6)

and

= _ (No

+ nex)
Tex

(5. 7)

where D i s the exciton diffusivity, and thu s ~Dvn

ex R

i s th e excito n

flux evaluated at the surface of the cloud.

Thus, there are essen-

tially two problems which must be solved.

One is the shrinking of the

cloud due to the diffu s ion of excitons away from its surface (Eq. 5.6).
The other co ncerns the calculation of the exc iton profi le around eac h
dropl et.

From thi s part we obtain nex' which i s used in the solu tion

of Eq. {5.7) for EHD within the cl oud as we ll as in Eq. (5.6) for
the c loud's s ur face .
To obtain the exc iton diffus ion current on the righthand s ide of

132

Eq. (5.6) we need to solve the diffusion equation governing the FE's
outside the EHD cloud,

an

ex - ov2n

~ -

ex

- n

ex

/T

ex

( 5. 8)

where "ex is a function of both t and the distance r from the center
of the cloud.

The boundary conditions to be satisfied are

nex = nex
at r = Rc(t)

(5 . 9)

and
n =0
ex
at r =

00

(5.10)

Equation (5.8) is coupled to Eq. (5.6) for R through boundary condic
dR
tion Eq. (5.9). These eq uations ca n be solved for dtc in terms of
Rc (t) and nex (t).
We can rearrange Eq. (5.7 ) as

(5.11)

where the term in parentheses on the right is the average exciton

133
generation rate, which is just the net evapora tion rate per uni t
volume after the end of the excitation pulse.

Therefore, to deter-

mine the decay of EHD and FE inside the cloud, it is necessary to
solve the exciton diffusion Eq. (5.8) for the region surroundin g
a given droplet with two boundary conditions.

First, at the surface

of the drop, r = R0 (t), diffusion current away from the droplet
equals net evaporation rate, or

= aT

2 ~¢EHD/kT 2/3

(5.1 2)

Second, no exciton diffusion current flows across R , that i s ,
9n

ex = 0
(5.1 3)

The only time dependent part of N0 is the droplet radius, R0 ,
which is related to the diffusion current away from droplets by
Eq. (5.11).

The exciton diffusion equation can be solved to obtain

this current which when subst ituted into Eq. (5.11) yields dR 0/d t
as a function of time.
From the two parts of the problem, Eq. (5.6) and (5.7), two
equations are obtained

(5.1 4)

134

and

The functions f and g will be derived in the next secti on.

straightforward numerical integration of f and g yields R0 (t) and
Rc(t) .

From R0 (t) and Rc(t) we obtain

(5 .1 6)
and

4 3IFE(t ) - -3 nRc nex (t) + Mex (t)

( 5.17)

wher e M i s th e number of excitons outs ide the cloud and F is t he
ex
fill fa cto r. To determine Mex we numerically integr ate the source sink eq uation with r espect to time

- Me x ) d t • + t"' ( )
'lex o
ex

(5.1 8)

Wher e Mex (o), the number of exciton s outs ide the cloud at t

0, i s

calculated from the initial conditions.
Durin g the l aser pulse, a gener at ion term must be added to the
l eft hand side of Eq . (5.1 2).

Thi s genera t ion term drives the exciton

gradient at eac h dropl et surface pos itive so that exc i tons can flow
into drop l ets during the time the l ase r is on.

I t i s obvious that

si nce we hav e not introduced the gener ation term, we cannot start

135
out with the correct initiql condition.

Jn fact, our solutio n as-

s umes an initial condition suc h that the exciton gradient is negative
at the surface of each droplet.

However , si nce FE decay is fast, we

expect the exciton density to r elax from that at the end of the
laser pul se to the one we assume in about one exciton lifetime.
r elaxation can be observed in the FE transient in Fig. 5.3.

Thi s

Thus,

our solu tion s hould be accurate to describe the decay transients
after about one exciton lifetime from the end of the laser pulse.

136
III.

Detailed Mathematical Formulation of Model
In this section the forms of f(R 0 ) and g(R 0 ,Rc) in Eq. (5.14) and

(5.15) will be derived.

First we will solve Eq. (5.11) for the

interior of the cloud in order to obtain

nex (t) and No(t). As

explained in the last section 7 we assume that each droplet has a
volume associated with it and that there is no net flow of excitons
across the surface of this volume.

We take this region to be a

sphere of radius R defined by

_ ( R~(t=O)) l/
Rs F(t=O)

(5.19)

The decay of a single drop is governed by
-n o VD
- JD
dt (noVO) = T

(5.20)

Ro
with n the density of pairs in an EHD, v0 the average volume of a

drop, and J 0 the exciton diffusion current~ here evaluated at the
surface of the drop.

Since n is independent of time, we have

Fnl_

no

( 5. 21 )

where F is the flux of free excitons away from the surface. To find
Fo we must solve the diffusion equation for the exciton s with boundary
conditions at R0 and Rs

137

At r = Rs,

ex

(5.22)

anex
-ar= 0.
At the s urface of the drop, evaporation must

balance backflow plus diffu s ion current away from the drop.

In

equilibrium, however, the backfl ow ju s t balances the evaporation.
So at r = R0 ,
(5 .23 )

where vth i s th e exciton thermal velocity and n~x(T) is the measured
density of exci ton s in eq uilibrium with droplets as a function of
tempe r at ure.
Defining q = rne x and us ing the assumed spherical symmetry, the
diffu s ion eq uation becomes
(5 .24)

Since we are in the interior of the cloud (and the fill f ac tor i s large
enough for the cloud concept to be mea ningful) q vari es s lowl y with
time, and we can make an adiabatic ass umption that ~ = 0.

Thi s

it still has
remove s the ex plicit time dependence of nex (r) thouqh
an impli cit ti me de pendence through the boundary condition at R0 .
The diffu s i on eq uation , Eq. (5.24), ha s solutions of the form

138

ex

- a
(r) = q/r = - 0

no (a e
l ex

-(r-R )/t

ex + e

{r-R 0 )/t

ex)

(5.25)

where t ex is the exciton diffusion length, t ex = ~
ex

, and a 1 ,a 2

are determined by the boundary conditions at R0 and Rs.

If we let

which is just the ratio of drop radius to exciton mean free path,
and
(5.27)

then
a2 = y

R -t
s ex
R H

(5.28)

ex

and

(5.29)

This gives
Fo!
no

RD

no
ex ( a2+1 + a2-~)
= oa
1 no
ex

(5.30)

Subs tituting Eq. {5.30) into Eq . (5.21) we obtain

( 5. 31 )

139
with
a 2+1
( R

which is Eq. (5.14) in the last section.

~)

(5.32)

ex

This equation is integrated

numerically since a 1 and a2 are complicated functions of R .

From Eq.

(5.25) we obtain

[ (a 2-l)(Rs -~ ex ) + 2a 2 ~ ex ]. (5.33)

ex ·~

-o=
nex

and
(5.34)
Now with nex and N0 we can go back to the problem of the cloud
as a whole.

We need to determine the exciton diffusion profile out-

side the cloud to find F0 1R , the flux of excitons at its surface.
We have

anex

ar-- ov2nex - _Q
ex

(r > R(t))

(5.35)

and
anex = 2
Dil n
at
ex

~ +

Tex

3vth • F
(noex - nex)
RD

(r ~

R (t))

(5.36)

140

The first of these equations holds for excitons completely outside the
region in which there are droplets (r>R ).

The second equation i s

for excitons just inside the edge of the cloud with the last term
corresponding to the net evaporation of excitons per unit volume.
Thi s term leads to an average effective diffusion l ength for excitons
within the cloud which i s much smaller than £ ex and thus the perturbation in exciton concentration due to the cloud's surface extends
in only a small di stance.
In solving these equations, we must match the two solutions at
r = Rc(t) with boundary cond itions nex = 0 as r + oo and nex + nex as
r + o.

Deep inside the cloud nex satisfies the equation
anex = -nex + 3Fvth
at
Tex
R0

Defining w

(no
ex

n ) .

(5.37)

ex

nex - nex' and an effective lifetime due to capture by

droplets T c ~ RD/(3 F vth)' then s ubtracti ng Eq. (5.36) from Eq.
(5.37) yields
(5.38)
For reasonable parameters (see Tabl e5 .1) Tex / Tc - 100, and the effective diffusion l engt h,
£ef =

DT

ex c
( T +T )
ex c

(5.39)

141

is given approximately by

and i s quite short,~ ex I~ e f~lo.

Thus , the perturbation in excito n

density due to the cloud' s surface extends in only a short distance.
Furthermore, Tc i s so short that the exciton profile within the cloud
can react very rapidly to the motion of the surface as the cloud
shrinks.

This allows us to make an adiabatic approximation, setting

aw; at = 0 in the frame of reference of the cl oud ' s surface andreducing the equation to steady state .

Using sp herical symmetry the

resulting equation is readily solved, giving:
w(r) = S w(R )

( r/~ef

le

R / 9.. f

(e c

-r/~ef )

-e

_ e

-R / ~ ~

( r ~ R ( t) ) ,

ej

( 5. 40)

where the second term in the numerator keeps th e so lution finite at
the origin and w(R ) must be determined by matching to the outside

solu tion.
The solution to the equation for r > Rc(t) is complicated by
the fact that the exciton lifetime i s too long to allow us to make
the simpl e ad i abatic assumption in thi s region we used in the
r < Rc case.

Here the exciton profile doe s not simpl y move with

the cloud ' s su rface but it changes depending upon the velocity of
the surface.
rnex
Ass uming spherical symmetry and defining, u - fCIT) and

142
-aR
v = ate , we can reduce the equation for r > Rc to a one-dimensional
form,

au - D .Ll:!. - u (-1 - - vIR )
at- ar 2
'ex

(5.41)

Now we transform to the frame stationary with re spect to the cloud's
surface by defining

Defining

= r-R c ( t)

(5.42)

, , = RRc T- Tex v , the resulting equation is
c ex

.£Q = D a2 u - v ~ - u/ • '
at
a/
ax

(x > 0)

(5.43)

Thus, to find F0 1Rc, we must solve thi s equation subject to the
boundary conditions that the ins ide and outside solutions match at
Rc and that the density of exc itons is zero at infinity.

nex - w(Rc ) = u(x=O)

~~~ R = a~ (R c ~x)j x=O

(5.44)
(5.45)

and
u(oo ) = 0

(5.46)

The transformat ion ha s taken some of the time dependen ce out of u.
The parts still left are

143
an ex (R c )
at

av
at

a-r '
at

and

If we assume that these terms are sma ll. then we can make the
adiabatic assumption and say au/ at = 0.

This assumption allows an

analytic solution to Eq . (5.43).
Defining:
(5.47)

the solution is

ex

(r) = R (t)n

ex

(R )

(R -r )/ 9.. '

(5.48)

Matching the inside soluti on, Eq. (5 .40) , to the outside soluti on
and using the f act that Rc >> 9.- ef' gives

9,_ I

ex (R c ) = nex £ '+£

(5.49)

ef

Balancing the dens ity of carriers inside with the flux out gives

(5.50)
where a9v/ at has been ignored since it i s sma ll when the adiabatic
approximation is valid.

Defining
( 5. 51)

144

we get the solution (ignoring the slight dependence on v of n (R ))
ex c

(5.52)

with

~ex (~Recx +
Tex

-2
(R R)
g 0' c = a+2

a+2 - 2 ( ~ ex I Rc )

This is the equation given in the last section (5.15).

(5.53)

N0 and nex

are functions of R0 and thu s a is a function of R0 (t).
Thu s we have reduced the problem to two consecutive numerical
integrations .

First Eqs. (5.31) and (5.32) are integrated to obtain

R0 (t), and from it nex(t) (Eq. (5.33)) and N0 (t) (Eq. (5.34)) may be
found. These functions are then used in the integration of Eqs .
(5.52) and (5.53) to find Rc(t).

It is mos t convenient to calculate

the exciton lumine scence by breaking it up into the contribution
from excitons within the cloud and the contribution from those
outside (Eq. 5. 17).

The source-sink equation (Eq. (5.18)) i s used

to find the term arising from excitons outside the cloud.

This avoids

the problem of a careful calculation of the exciton profile which
would be needed for a spatial integration to determine IFE.

The

effective diffu s ion length,~·, i s a good characterization of the
diffusion profile near the cloud's s urface s ince it arises from the
surface moving past some excitons and thus giving less slope to the
profile than just~ex·

However, far from the cloud edge the exciton

profile is not a simple exponential due to the acceleration of the

145

surface of the cloud.
The theoreti cal curves were ca l culated with initi al conditions
chosen so as to avoid transients at the begi nning.

Thus, it was

assumed that the cloud and droplet edges have a finite vel ocity at

= 0.

Thi s was found by determining the diffu s i on profiles within

the cloud to ge t nex• and then solving self-cons i stentl y for aRc/ at .
The self- consistent approach is necessary to correctly include the
dependence of nex(Rc) on v.

The number of exciton s outside the

cloud at the beginning of the calcul ation, Mex (o), i s determined by
s pati ally integ rating the solutions for nex (r) at t = 0 .

146
IV.

Results of Calculation and Comparison with Experiments
In this section, the calculated decay transients of the EHD

and FE intensity will be presented and compared to experimental
data.
The theoretical curves presented in Fig. 5.2 were calculated
for three different initial cloud radii.

Values of the parameters

used in the calculation are li sted in Table 5.1 along with the values
of the same parameters reported in the literature.

Figure 5.2(a)

shows the calculated EHD luminescence decay transient, normalized to
unity at the start of the decay .

The corresponding curves for the

At 4.2°K, an increase of R (o) causes
the decay to be longer, because for l arger Rc (o) the initial surface

FE are shown in Figure 5.2(b).

to volume ratio is sma ll er, and this lessens the importance of
exciton diffusion away from the cloud . Spati ally-resolved optical
absorption experiments at 4.2°K ( 2l) s how that different initial
R (o) can be created by using different excitation powers. It can
be seen in Figure 5.2(b) that there are clear differences between
the three FE curves making them a sensitive, independent check on
the model.

In particular, these results are very different from the

solutions to the average rate equations in which there is no boundary
to the region occupied by EHD's.

Experimental results of the EHD and

FE decay transients after long (100 ~sec) excitation pulses for two
different pump powers are al so shown in Figure 5.2.

It can be seen

that excel lent agreement is obtained for both the EHD and the FE
curves when reasonable values for the parameters are used in the

147

calculation.

There are several points that should be noted however.

First, parameters can offset each other to some extent, e.g., reducing
the exciton diffusion length slows down the decays, but this may be
offset by increasing the equilibrium density of excitons.

Whenever

possible, values for the parameters determined from independent
measurements have been used to reduce the number of fitting parameters.

Second, optical absorption experiments have shown that ex-

citon recombination current at the Ge surface is non-negligible at
Furthermore, recent pictures of the cloud in Ge have
shown that it has a complicated shape which is only roughly hemispherical (l 7 )
Thus, our mode l is qualitatively correct in that the
essential physics has been retained, but it is difficult to get
accurate quantitative results from the fits.

Since the cloud is

not simple in shape, R is an average cloud radius characterizing
the volume to surface ratio. Figure 5.2(c) shows plots of Rc(t)
corresponding to the cases shown in (a) and {b).

At 4.2°K, it is

seen that the shrinking of the cl oud of EHD's is as important as
the shrinking of individual droplets in causing the decay in the
droplet luminescence intensity.

This collapse of the EHD cloud has

been obs erved in temporally and spatially resolved absorption ex.

per1ments

(21)

Our model also correctl y describes the EHD decay at 2°K.
The cloud radius, Rc, does not change noticeably with time because
nex is so small that FE recombination current cannot significantly
affect EHD decay.

148

Figure 5.2
Results of ca l cul ations of the model for 4.2°K. The
parameters used are appropriate for pure Ge and are given
in Table 5. 1. The initial radii of the cl oud for curves 1, 2
and 3 are 1.0, 1.4 and 1. 8 mm, respectively. Experimental
results for high-purity Ge are shown for comparison; pumppowers used are (!) 0.14 W and (•) 0.09W. (Experimental
results from M. Chen.)

149

>-

1(/)

1-

:::;

0:

0.4 -

>-

1-

Vi

1-

0 .8

LL

:::;

:E
0:

0.4

1.0
(/)

:2
0:

0.8

:::>

:::;

0.6

::::E
0:

0 .4

10
20
TIM E (p sec)

Figure 5,2

Table 5.1.

150
Values of parameters used in the model

Parameter
EHD lifetime, T
FE lifetime, Tex
Equilibrium den s ity of FE, n°
ex
2°K
4.2°K
Pair density in the EHD, n0
2°K
4. 2°K
Initi al EHD radius, R0 {0)
2°K
4. 2°K
FE diffusion l ength, ~
ex
Fill factor, F
Initial cloud r adius, R {0)
a.
b.
c.
d.
e.
f.
g.

h.
i.
j.
k.

Values used in
calculations
for Fig. 7
37 )JSec
7 l-!Sec

Values found in
the literature
36-45 JJ Sec(a-c)
6-8 )J Sec (c-d)

7xl 011 em - 3
3xlo14 cm- 3
2xlo 17 cm- 3
2x1o 17 cm- 3

2.lxl0 17 em -3 (e)
2.4xlo 17 cm- 3 (e)

2 )Jm
l 0 JJm
0.8 mm
2%
1.4- 1.8 mm

2 )Jm (f)
- 10 )Jm (g)
- 1 mm ( h ' i )
1-2% ( f' j)
l -2 mm {k,j)

Ref. 3
Ref. 4
Ref. 5
V. Marella, T. C. McGill and J. W. Mayer, Phys. Rev. Bl3, 1607 (1976).
G. A. Thomas, A. Frova, J. C. Hensel, R. E. Mill er and P. A. Lee,
Phys. Rev. Bl3, 1692 (1 976 ) .
J. M. Warlock, T. C. Damen, K. L. Shaklee and J. P. Gordon, Phys.
Rev. Lett. 33, 771 (1974).
Ref. 10
Ref. 13
Ref. 20
Ref. 19
Ref. 16

151
Under the same excitation conditions, the initial cloud
radius, initial fill factor, and exciton diffusion length can be
different in pure Ge and Ge doped with about 1o15 cm- 3 of impurities.
Spatial luminescence intensity scans show that the cloud of EHD's
does not penetrate as deeply in lightly doped Ge as in pure Ge( 20 • 21 ),
so Rc (o) should be smaller in lightly doped Ge compared to pure Ge.
This fact, coupled with the observation that the total luminescence
intensities from pure and doped Ge do not differ significantly,
impli es a larger fill factor in doped Ge.

Neutral impurity scatteri ng

of excitons can significantly alter the exciton diffusion length
at the 1015 cm- 3 dopin g level. Figure 5.3 s hows calculations for
4.2°K for various initial cloud radii, fill factors, and exciton
diffusion l engths.

For curve 1, an initial radiu s of 0.5 mm was

assumed, and the fill factor has been increa sed to 10% in keeping
with the greater confinement of the EHD's.
length, ~ex, is still 0.8 mm.

The exciton diffusion

The decay is seen to be slower in this

case compared to pure Ge {Figure 5.2(a)) but is still far from being
exponential.

Curve 2 assumes Rc (o) = 0.5 mm, fill factor = 2% and

ex 0.016 mm. Comparison of curves l and 2 shows the much greater
importance of diffu sion length over fill factor in slowing down the

~ _ =

decay.

Curve 3 incorporates all the expected changes for doped Ge

with R (o) = 0.5 mm, ~ = 0.016 mm, and a fill factor of 10%. As
ex
can be seen from the figure, altering the parameters to those for
doped Ge dramatically changes the decay curves, causing the luminescence transients to become nearly exponential for temperatures

152

as high as 4.2°K.

Excellent agreement with the experimental data,

shown as dots in Figure 5.3(a), is obtained .

It should be noted

that the value for exciton diffusion length in the doped material
was determined through fitting the luminescence decays and has not
been determined independently.

Figure 5.3(b) shows the calculated

decay curves for FE corresponding to the cases in Figure 5.3(a).
The decay time of the FE is controll ed by the evaporation of excitons
from droplets and is, therefore, directly related to the EHD lifetime.

For the case corresponding to lightly doped Ge, curve 3,

the FE decay is expected to be very slow compared to pure Ge.
Figure 5.3(c) shows the cal culated Rc(t) using the same parameters
as those used for the corresponding curves in Figures 5.3(a) and (b).
These curves illustrate clearly that reducing the diffusion length
stops the escape of exciton s from the cloud and the reduction of
R with time. Merely increasing the fill factor as for curve 1 does
not accomplish this, and consequently the calculated decay is much
too fast compared to experiment.

The net evaporation of excitons

from droplets must be shut off in order to account for the l ong
decays observed for li ghtly doped Ge.
The reduction of free exciton diffusion length in doped Ge
implies a short exciton diffus i on tai l and, therefore, fewer excitons around each drop.

Thus, it is expected t hat the rel ative EHD

to FE intensity should be small while EHD's are decaying.

This

reduction of FE intensity for doped Ge is observed experimentally.

153

Figure 5.3
Res ults of ca lculations of the Model for 4.2°K.
Curve 1: FE diffu s ion l ength- 0. 8 mm, f ill factor = 10%.
Curve 2: FE diffusion l engt h - 0.016 mm, fill factor = 2%.
Curve 3: FE diffu s ion length = 0.016 mm, fill factor = 10%.
Initial cloud radiu s for all three curves i s 0.5 mm, all
other parameters are the same as tho se for pure Ge given in
the text. Experi me ntal res ults of the luminescence intensity
decay of th e EHD in a Ge sample with 4xlol5cm3 are shown as
the dots. (Expe rimental re s ults from M. Chen.)

154

>-

1.0

Ge:A s
15
4XI0 As/cm3
T =4.2°K

t:
(/)

..__

0.8

:X:

0 .6

_I

0:

>..__

0.4

1.0

(/)

..__

0 .8

lJ....

0.6

_I

0:

0.4

2,3
(c)

(/)

:::>

0:

:::>

_I

_I

0:

20
40
TIME !,..sec}

Figure 5.3

155

Our model predicts, as observed experimentally, thEt the
decay of the EHD in li ghtl y doped Ge at 2°K should be determined by
the recombinative lifetime of the EHD.

156

V.

Summary and Conclu s ion
It has become apparent that the s ingle-drop model i s unabl e to

account for the EHD decay transients observed in many experiments.
For high excitation intensities and temperatures, unphys ical
.eters are needed to fit the data.

para~-

We have developed a new model

for the EHD luminescence intensity decay which takes into account
both the existence of a cloud of droplet s and exciton emission and
capture by droplets.

This model simultaneously gives excellent

fits to both the EHD and FE decay transients.

In the case of pure

Ge at 4.2 0 K, evaporation from the droplet s in the cl oud keeps the
exciton density inside th e cl oud at approximately the equi l ibrium
density after the excitation is turned off.

This exciton density

implies that backflow i s large within the cloud.

This high backflow

rate causes the observed EHD decay times at 4.2°K in pure Ge to be
longer than expected from the independent droplet model.

The decay

of the luminescence is due to a combi nation of the shrinking of
individual droplet s and the s hr inking of the cloud as a whole.
Pump-power dependence of the decay time i s a consequence of the
differ ent initial cloud radii generated by different pump conditions.
For the lightly doped Ge the FE exciton diffusion lengt h is
reduced from that in pure Ge.

This change in diffusion length can

produce a l arge reduction in the rate of FE evaporation from the
droplets.

At s uffici ently small diffu s ion lengths the dropl ets in

our model act independently.

The cl oud does not shrink, s ince

157

excitons are not supplied to a substantial region outside the cloud
or for that matter in between droplets .

Furthermore, the evaporation

of excitons from a given drop i s nearly canc.elled by the recapture
of those excitons by the same drop before they can diffuse away .
This reduction in net evaporation accounts for the nearly exponential behavior of the EHD decay transients in lightly doped Ge at
high temperatures.

158
References
1.

C. D. Jeffries, Science 189, 955 (1975), and references contained
therein.

2.

K. L. Shaklee, in Proceedings of Third International Conference
on Light Scattering in Solids, Campinas, Brazil, 1975, edited by
M. Balkanshi, R. C. C. Lute, and S. P. S. Porto (F1a~~arion
Sciences Press, Paris, 1976) p. 160.

3.

J. C. Hensel, T. G. Phillips and T. M. Rice, Phys. Rev. Lett .
30, 227 (1973).

4.

C. Benoit a la Guillaume, M. Capizzi, B. Etienne and M. Voos,
Solid State Commun. ]_§_, 1031 {1974).

5.

R. M. Westervelt, T. K. Lo, J. L. Staehli and C. D. Jeffries ,
Phys . Rev. Lett. lf_, 1051 (1974).

6.

B. M. As hkinad ze and I. M. Fishman, Fiz. Tekh. Po1uprovodn.
}l, 408 (1977) [Sov. Phys. Semicond. Jl, 235 (1977)].

7.

K. Fujii and E. Otsuka, Solid State Commun. }!, 763 (1974) .

8.

J. Shah, A. H. Dayem, M. Voos and R. N. Silver, Solid State
Commun. ]1_, 603, ( 1976).

9.

J. L. Staehli, Phys. Status Solidi B75, 451 (1976).

10.

V. S. Bagaev, N. A. Penin, N. N. Sibe1'din and V. A. Tsvetkov ,
Fiz. Tverd. Tela. ]_§_, 3259 (1973) [Sov. Phys. Solid State }i, 2179
(1974)].

11.

R. N. Silver, Phys. Rev. 811, 1569 and B11_, 5689 (1975).

12.

R. M. Weste rvelt, Phys. Status Solidi b74, 727 and b76, 31
(1976).

159
13.

Y. E. Pokrovskii, Phys . Status Solidi all, 385 (1972).

14 .

To apply to the excitons, the Richardson-Dushman Constant for
metals is corrected for exciton mass, binding energy and degeneracy. In addition, a factor (36) 113 n 213 i s incorporated in

which converts v
15.

213

to the droplet surface area.

J. M. Hvam and 0. Christensen, Solid State Commun. }2, 929
(1974).

16 .

J.C.V. Mattos, K. L. Shaklee, M. Voos, T. C. Damen and J. M.
Warlock, Phys. Rev. Bl 3, 5603 (1976), and references contained
therein.

17.

J. P. Wolfe, M. Greenstei n, and J. Arzegian, Bull. A.P . S. 23,
422 (1978).

18.

J. Doehler, J . C.V. Mattos and J. M. Warlock, Phys. Rev . Lett. 38,
726 (1977).

19.

K. R. Elliott, D. L. Smith and T. C. McGill (to be publi s hed).

20.

R. W. Martin, Phys. Status Solidi B61, 223 (1 974).

21.

M. Chen (unpublished).