FABRICATION AND CHARACTERIZATION OF POROUS SILICON
LIGHT EMITTING DIODES
By
ZHILIANG CHEN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE
UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1994
ACKNOWLEDGEMENTS
I wish to express my sincere appreciation and deep
gratitude to my advisor Professor Gijs Bosman for his
inspiration, encouragement, and support throughout the
course of this research. His suggestions for research
problems, his insight in carrying out the work, and his
careful comments on the written product are all appreciated.
I would especially like to thank Professors S. S. Li,
F. A. Lindholm, T. Nishida, and C. J. Stanton for their
help, and for being on my supervisory committee. My special
thanks also go to Professor A. Neugroschel for his valuable
suggestions and comments during this research.
I am grateful to Professor L. L. Hench for financial
support during this work. Gratitude is also extended to
Professors P. Zory, R. E. Hummel, and J. H. Simmons for
providing the facilities to carry out the optical
measurements, and to Dr. R. Ochoa, Mr. S. S. Chang, and Miss
Li Wang for performing the photoluminescence spectroscopy.
I also thank Mr. J. Chamblee, Mr. T. Vaught, Mr. A.
Herrlinger, Mr. K. Rambo, and Mr. S. Schein for their
technical assistance. Thanks are also extended to many of
my colleagues and friends, Mr. E. W. Deeters, Mr. Y. H.
Wang, Mr. Daniel Wang, Mr. T. Y. Lee, and Ms. G. Sbrocco for
their support and encouragement.
I am greatly indebted to my father, mother and brother
for their love, sacrifice and inspiration.
Last but by no means least, I owe a great debt to my
wife Rong for her patience, understanding and support. I
thank her most sincerely.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ....................................... ii
ABSTRACT ................................................. vi
CHAPTER
1 INTRODUCTION ...................................... 1
1.1 Introduction .................................... 1
1.2 Two Opposing Controversial Interpretations
of Porous Silicon Luminescence ..................2
1.3 Organization of the Dissertation ...............12
2 FORMATION OF POROUS SILICON ........................23
2.1 Introduction ................................... 23
2.2 Experimental Setup ............................23
2.3 Silicon Surface Dissolution Mechanisms .........24
2.4 Porous Silicon Formation Model .................30
2.5 Process Related Photoluminescence of
Porous Silicon .................................35
2.6 Conclusions .....................................36
3 PHOTOLUMINESCENCE ENHANCEMENT AND SATURATION
RESULTING FROM HIGH TEMPERATURE TREATMENTS OF
POROUS SILICON .....................................47
3.1 Introduction ................................... 47
3.2 Experiments .....................................50
3.3 Discussion ......................................51
3.4 Conclusions .....................................57
4 ENERGY BANDS OF SILICON QUANTUM WIRES ..............64
4.1 Introduction ................................... 64
4.2 Effective Mass Theory ..........................65
4.3 Conduction Band Confinement in Silicon
Quantum Wires .................................. 70
4.4 Valence Band Confinement in Silicon
Quantum Wires .................... .............. 75
4.5 The Band Gap of Silicon Quantum Wires ..........78
4.6 Conclusions .....................................79
5 CARRIER STATISTICS AND THE CURRENTVOLTAGE
CHARACTERISTICS OF SILICON QUANTUM WIRE
PN JUNCTIONS ........................................84
5.1 Introduction ...............................
5.2 Density of States in a One Dimensional System.
5.3 Electron Density in Quantum Wires ............
5.4 The CurrentVoltage Characteristic of a pn
Junction Diode ...............................
5.5 The CurrentVoltage Characteristic of Silicon
Quantum Wire pn Junction Diodes ..............
5.6 Conclusions ..................................
..84
..85
..86
..88
..94
..95
6 VISIBLE LIGHT EMISSION FROM A PN POROUS SILICON
JUNCTION ............................................99
6.1 Introduction .................... .....
6.2 NP Porous Layer and Device Fabrication
6.3 Measurements ...........................
6.4 Conclusions ............................
7 ELECTRICAL BAND GAP DETERMINATION OF POROUS
SILICON USING CURRENTVOLTAGE MEASUREMENTS
7.1 Introduction ...........................
7.2 Current Voltage Measurements ...........
7.3 Experiments ............................
7.4 Conclusions ............................
........99
.......100
.......103
.......105
.......115
.......115
.......116
.......119
.......127
8 SUMMARY AND CONCLUSIONS ..........................
APPENDIX A
DERIVATION OF THE CONFINEMENT ENERGY OF
THE SECOND VALENCE BAND IN SILICON
QUANTUM WIRES .............................140
APPENDIX B
ELECTRON DISTRIBUTIONS IN DOPED
SILICON QUANTUM WIRES ..............
.......142
REFERENCES .............................................145
BIOGRAPHICAL SKETCH .......................... ..
.137
....... 151
Abstract of Dissertation Presented to the Graduate School of
the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
FABRICATION AND CHARACTERIZATION OF POROUS SILICON
LIGHT EMITTING DIODES
By
Zhiliang Chen
April 1994
Chairman: Gijs Bosman
Major Department: Electrical Engineering
The recent discovery of intense photoluminescence from
porous silicon has generated considerable interest in this
kind of material as it holds potential for application in
optoelectronic devices and VLSI technologies. This
dissertation deals with the systemati study of fabrication,
characterization and modeling of visible Light Emitting
Diodes (LED) made from porous silicon.
A full picture of the porous silicon formation
mechanisms, the fabrication process of porous silicon, and
the relationship between fabrication process and
photoluminescence are discussed in detail in this work and
supported by experimental studies. High temperature
annealing experiments on porous silicon in nitrogen ambient
are carried out. The experimental data strongly support the
quantum confinement model which is one of the models
proposed for porous silicon luminescence.
The occurrence of a wide, direct band gap of porous
silicon can be well explained by a simple picture based on
the effective mass approximation and quantum confinement
theory. The charge carrier statistics and device
characteristics of silicon quantum wires are derived.
The first visible light emitting diodes with a peak
wavelength of 640 nm (1.94 eV) made from porous silicon
homojunction pn diodes were fabricated and characterized. A
dc electrical characterization reveals a 2.20 eV electrical
band gap for porous silicon. The agreement between the
porous silicon band gap extracted from electroluminescence
measurements, photoluminescence measurements, and from IV
measurements is strong evidence for the existence of a wide,
direct band gap in porous silicon.
vii
CHAPTER 1
INTRODUCTION
1.1 Introduction
Porous silicon, obtained by electrochemical etching
of silicon in diluted HF at moderate current density levels,
is quickly becoming an increasingly important and versatile
electronic material. Its reactive porous nature and nano
scale structure allow for the selective formation of unique
electronic components. The doping selectivity of the
anodizing process and the rapid oxidation rate of porous
silicon due to very large surface areas have been utilized
in silicononinsulator technology [Ima84, Yon87, Tsao89].
Recently, the discovery of intense photoluminescence from
porous silicon by L. T. Canham [Can90] has generated
considerable attention to this kind of material as it holds
potential for application in optoelectronic devices and VLSI
technologies.
The most interesting aspect of the intensive
photoluminescence (PL) of porous silicon is its generally
broad spectrum between 600 and 800 nm corresponding to 2.1 
1.2 eV photon energy (figure 1.1). Since this large photon
energy can not be explained by bulk silicon properties and
its 1.12 eV indirect band gap, many studies [Gar91, Can91,
Rob92, Pro92, Tis92, Tsa92, Buu93, Lav93, Beh93] have
focused on the physical origin of the light emission from
porous silicon. Canham attributed this emission to a quantum
confinement effect in which the indirect silicon band gap
changes from 1.12 eV to a large, direct band gap around 1.7
eV after formation of porous silicon.
The understanding of the phenomenon is crucial for
further development of porous silicon. Currently, most of
the research groups focus their studies on the exploration
of the origin of the luminescence by employing various
techniques such as Secondary Ion Mass Spectroscopy (SIMS)
[Can91], Electron Paramagnetic Resonance (EPR) [Bha92],
Raman Spectroscopy [Tsa92] and Xray absorption [Buu92,
Buu93] A few papers have been published on
electroluminescence (EL) from porous silicon [Ric91, Kos92,
Nam92, Bre92, Che93, Ste93] and one paper has reported on a
high sensitivity photodetector application based on porous
silicon [Zhe92], indicating the potential of porous silicon
for applications in optoelectronics.
1.2 Two Opposing Controversial Interpretations of Porous
Silicon Luminescence
Although a clear description for the luminescence
mechanism has not yet been given, there are presently two
accepted explanations for porous silicon light emission,
namely the chemical compound model and the quantum
confinement model. These two models are discussed in detail
in the following sections.
1.2.1 Chemical Compound Model
The idea that the luminescence of porous silicon
excited by an UV lamp or Argon laser might be due to a
chemical compound absorbed on the vast surface area of
porous silicon was first proposed by Z. Y. Xu et al. [Xu92].
This Australian group observed an irregular dependence of
the PL spectrum on temperature for different samples and for
different spots even on the same sample. C. Tsai et al.
[Tsa92], after observing that the PL intensity of porous
silicon significantly decreases following annealing at
temperatures between 300 and 400 OC, (which is the same
temperature range for changing the surface termination from
mainly dihydride to predominantly monohydride,) used a
remote H plasma to form a predominant monohydride
termination on the surface of porous silicon. They found
that the very weak PL of a SiHpassivated sample can be
increased gradually by immersing the sample in ever
increasing concentrations of HF solutions. Fouriertransform
infrared (FTIR) spectroscopy shows that the number of SiH2
bonds increases with increasing HF concentration. Observing
the correlation between the silicon hydride density and the
PL intensity of porous silicon as shown in figure 1.2, they
conclude that SiH2 plays a key role in the porous silicon
luminescence process.
Using micro PL, S. M. Prokes et al. [Pro92] examined
the PL spectra of porous Si as a function of distance from
the top surface and at different annealing temperatures
(from 20 690 OC) Their results show that the peak
wavelength of the PL spectra is insensitive to depth and
that the peak intensity decreases with increasing distance
from the top surface. In addition, they observed that the
peak of the PL spectra redshifted as the annealing
temperature increased both with an argon ambient and a
vacuum environment and that the intensity decreased with
increasing temperature and became too weak to be observed at
690 oC. They attributed luminescence of porous Si to the
presence of hydrogen complexes (SiH, SiH2, SiH3 or (SiH2)n)
which leads to new bonding states formed deep within the
silicon valence bands, as happens in aSi:H. They noted a
similarity between optical band gap shrinkage of aSi:H due
to loss of hydrogen and PL peak redshifting of porous
silicon in the same temperature range. The explanation for
their experimental results is that the peak position of the
PL is related to the type of hydride present, and that the
intensity is a function of the surface area (i.e. the number
of hydrides).
Work by M. S. Brandt et al. [Bra92] has shown that the
photoluminescence and vibrational spectra of porous silicon
can possibly be attributed to SiOH compounds derived from
siloxene (Si603H6). In experiments, they first obtained the
PL spectra produced by siloxene after annealing at 400 OC in
air and found that these spectra agree well with the PL
spectra obtained from porous silicon. And then, by comparing
the infrared (IR) vibrational spectra measured on porous
silicon at room temperature and on annealed siloxene at 400
OC, they found that the same chemical bonds are present in
both types of samples. In addition they argue that
a. Like the luminescence of porous silicon, the luminescence
of siloxene can be tuned by substituting other ligands such
as halogens, OH or alcohol groups for H.
b. The bright chemiluminescence of siloxene which occurs
during oxidation is similar to visible electroluminescence
in porous silicon during anodic etching.
c. Both porous silicon and siloxene exhibit a pronounced
luminescence fatigue.
d. The decay of the photoluminescence after pulsed
excitation in both porous silicon and siloxene is strongly
nonexponential.
Based on the above arguments and experimental results, they
conclude that the luminescence in anodically oxidized
silicon is due to SiOH compounds derived from siloxene.
Chemiluminescence in the visible region from porous
silicon treated with a nitric or persulfate solution was
reported by P. McCord et al. [McC92]. Similar to the
situation as it occurs with siloxene prepared from CaSi2, a
drop of concentrated HN03 on the surface of dry porous
silicon can result in a flash of light with an audible pop.
Based on these facts, they conclude that the luminescence of
porous Si is mainly due to the formation of siloxenelike
compounds.
1.2.2 Ouantum Confinement Model
The quantum confinement model was first proposed by
Canham [Can90] and Lehmann and Gosele [Leh91] independently.
Canham found efficient photoluminescence from porous Si in
the energy range of 1.4 to 1.8 eV, far higher than 1.12 eV,
the intrinsic crystalline silicon band gap. The peak
position of the PL spectra can be blue shifted when porous
silicon is etched slowly in HF. Canham attributed the
luminescence to a quantum confinement effect. The
electrochemical, anodic etching of silicon results in a free
standing silicon quantum wire network which makes up the
porous layer as shown in figure 1.3. Since the electrons and
holes are confined in these quantum wires, the band
structure of the silicon will change from an indirect band
gap to a larger, direct band gap. The blueshifting of the
peak position of the PL spectra is evidence of this quantum
confinement and can be explained in terms of wire shrinking
during slow HF etching. Independent from Canham, Lehmann and
Gosele measured the transmission of monochromatic light
through porous silicon samples and observed a drastic shift
of the fundamental absorption edge of the free standing
porous silicon layers to 1.76 eV at room temperature as
shown in figure 1.4. The data show that the increase in the
bandgap energy is in reasonable agreement with the
prediction of the quantum confinement model using the
quantum wire size as measured by TEM.
Instead of slow HF etching, a thermal oxidation of
porous silicon to reduce the size of the freestanding wires
was performed by S. Shih et al. [Shi92]. In their
experiment, the porous silicon samples were thermally
oxidized for different periods of time and the PL spectra
were measured immediately after removing the oxide by
dipping the sample in an HF solution. As shown in Fig 1.5,
three pronounced results were observed which can be
explained using the quantum confinement model.
a. The gradual blueshift of the peak of the PL spectrum
during the oxidation can be interpreted as resulting from
wire shrinking effects due to the oxidation which cause the
band gap to increase.
b. The intensity of the PL initially increases with
increasing oxidation time and then drops quickly for longer
times and eventually levels off. This result may be
understood as follows: initially, increasing the oxidation
time increases the total number of luminescent structures
and enhances the quantum efficiency by reducing the Si wire
sizes which were originally too large to efficiently confine
the carriers. As a consequence, the PL intensity increases.
This process will continue until the gain in the number of
luminescent structures and efficiency are totally
compensated by the loss of luminescent structures mainly due
to overconsumption during oxidation. At this point, the PL
intensity reaches a maximum. After this, oxidation will
reduce the PL intensity.
c. The spectral width decreases monotonically with
increasing oxidation time. Due to the stress, thin wires are
harder to be consumed by oxygen than thick wires. In other
words, thick wires oxidize with a higher consumption rate
along the radial direction than the thin ones. The higher
wire shrinking rates of the thick wires drive the original
broader spectrum into a more compact shape with increasing
oxidation time.
A. Nakajima et al. [Nak92] measured the changes in PL
spectra of porous Si samples by oxidizing chemically. In
their work, the oxidization was carried out by dipping the
samples into H202 or HNO3 solutions for 30 min. at room
temperature. The samples were deoxidized in an HF solution.
The PL spectra were measured after each process. Fourier
transform infrared (FTIR) transmission spectral measurements
were performed to examine the extent of the oxidation or the
deoxidation. Their PL results clearly show that the PL
shifts to shorter wavelength (or higher energy) and
increases in intensity (figure 1.6). From a comparison of
the IR data measured before and after chemical treatment,
they concluded that the PL spectral change is not caused by
a change in the chemical composition of the porous Si but
results from the shrinking of the wires, i.e., a quantum
size effect, since the IR spectral data clearly show that
the deoxidation spectrum is almost the same as that of the
asprepared porous Si which suggests that the chemical
composition is almost the same as that of the asprepared
samples. From the experiments, they also noted that the
relative number of the SiH2 surface bonds decreased during
oxidation whereas the measured PL intensity strongly
increased. This is contrary to suggestion of C. Tsai's
[Tsa92].
T. van Buuren et al. [Buu92, Buu93] measured the xray
absorption in the vicinity of the silicon L edge in porous
silicon. They found that the absorption threshold of porous
silicon blueshifted by 0.30.4 eV with respect to
crystalline silicon and that the shift in the absorption
edge in porous Si depended on the HF concentration in the
etching solution and increased with electrochemical etching
time. They point out that this blueshift cannot be
explained by the presence of amorphous silicon (aSi) on the
surface of the porous Si, since the blueshift of aSi
relative to crystalline Silicon (cSi) is almost zero, but
can be explained by the quantum confinement model in which
the energy of the bottom of conduction band is raised. The
absorption spectrum of porous Si can be fitted by a model in
which the absorption spectrum of crystalline silicon is
shifted up in energy to simulate the average quantum shift
and broadened by the distribution of quantum wire sizes.
Theoretical calculations of the energy band structure
of free standing Si quantum wires as formed in porous Si
were carried out by several research groups [San92, Rea92,
Bud92, Ohn92, Wan93]. Even though different models and
computing methods are employed in these calculations, their
results are in general agreement with each other. Figure 1.7
shows the results of the band gap calculation by G. D.
Sanders and Y. C. Chang [San92]. In the calculation, a
secondneighbor empirical tightbinding KosterSlater model
was used and the silicon dangling bonds at the surface of
the wire were assumed to be passivated by hydrogen derived
from the HF acid used during the fabrication. It is well
known that bulk Si has an indirect band gap; however, the
calculation shows that an Si wire has a direct band gap with
an Xlike conduction band minimum and an Flike valence band
maximum both occurring at the zone center as shown in figure
1.7. In addition, it is found that the band gap increases
with decreasing wire size (figure 1.8) when the Si wire size
L falls in the quantum size range.
1.2.3 Summary
Two main opposing models proposed for porous silicon
luminescence have been addressed above. Although no
conclusive argument for the porous silicon light emission
has been given yet, it is now agreed upon that the quantum
size effect in porous silicon certainly plays a key role in
its optical properties as suggested originally by Canham. No
experiments disproving the quantum confinement model have
been reported so far. However, several experimental studies
of the porous silicon light emission phenomenon point
against the chemical model. Here we briefly summarize these
studies.
A. High temperature treatments of porous silicon using rapid
thermal oxidation (RTO) [Pet92, Bat93]. In this experiment,
the asprepared porous silicon was rapidly thermally
oxidized at 900 OC for one or two minutes. The RTO porous
silicon samples have shown (a) PL intensity increased and
the PL peak blue shifted; (b) the PL stability increased
dramatically; (c) Infrared spectroscopy measurements showed
that the hydrogen concentration at the silicon surface was
below the detection limit of the experimental setup. The
observation of the increasing PL intensity with decresing
hydrogen concentration after RTO treatment points strongly
against the chemical model as discussed previously.
Observations of (a) and (b) can be explained well within the
framework of the quantum confinement model.
B. Quantitative analysis of experimental data has indicated
that (a) porous silicon PL has no correlation with surface
hydrogen species [Rob93] and (b) the siloxene is not
generally responsible for the observed roomtemperature
luminescence in porous silicon [Fri93].
C. The fact that porous silicon may produce red, orange, and
also blue [Hou93, Lee93] luminescence is hard to explain in
terms of just one type of siloxene compound.
D. Porous silicon samples made by "dry" spark erosion of
crystalline silicon in a nitrogen atmosphere [Hum92] show
the same PL spectra as those resulting from the wet etching
process. This experiment goes against the idea that chemical
contaminants which have been chemisorbed in the pores during
anodic etching are solely responsible for the observed PL
spectra.
1.3 Organization of The Dissertation.
Motivated by the discovery of intense
photoluminescence of porous silicon, this dissertation
describes the fabrication, characterization and modeling of
light emitting diodes made of porous silicon. The work is
primarily based on the framework of the quantum confinement
model and the understanding of porous silicon formation. The
goal of this study is twofold:
A. Investigate the possibility of making light emitting
diodes using porous silicon.
B. Systematically study the band gap of silicon quantum
wires, the ideal realization of porous silicon.
To that end, we studied the luminescence phenomenon in
porous silicon from both the experimental and the
theoretical point of view. High temperature annealing
experiments on porous silicon in nitrogen ambient were
carried out. The results strongly support the quantum
confinement model, which gave a fundamental foundation to
this study. The fact that photoluminescence and
electroluminescence of successfully fabricated porous
silicon pn junction diodes have the same peak wavelength
indicates that porous silicon has indeed a wide band gap.
Without involving extensive numerical calculations, the
direct band gap of a silicon quantum wire due to quantum
confinement can be well explained by a simple picture using
the effective mass approximation and quantum confinement. A
study of the currentvoltage characteristics of porous
silicon pn diodes indicates in addition that the band gap of
porous silicon changes from a 1.12 eV bulk band gap to a
wider band gap of around 2 eV. The results of band gap
experiments employing both optical methods (PL and EL) and
an electrical method (IV) agree well with each other. The
quantum confinement model has thus been proven to explain a
variety of experiments in both the electrical and the
optical domain.
Following this introduction, the fabrication process of
porous silicon as well as the relationship between
fabrication process and the photoluminescence are discussed
in chapter 2. In chapter 3, high temperature treatments of
porous silicon and its PL variations with treatment are
described. The results strongly support the quantum
confinement model. Based on the effective mass theory, a
simple, insightful picture of the quantum confinement model
of porous silicon is developed in chapter 4. In this
chapter, the band gap of porous silicon is also calculated
and compared with other, numerical results. The framework of
carrier statistics for a quantum wire and the current
voltage characteristics of a pn quantum wire diode have been
established in chapter 5. In chapter 6, electroluminescence
from np porous silicon devices is presented. A study of the
porous silicon band gap by measuring currentvoltage
characteristics of porous silicon pn junction diodes is
reported in chapter 7. Finally, in chapter 8, summary and
conclusions are presented.
3000
2735
2470
2205
j 1940
1675
r 1410
~ 1145
.J
L 880
615
350
600 625 650 675 700 725 750 775 800 825 850
Wavelength (nm)
Figure 1.1. PL spectrum of p type porous silicon.
1000 2.0
Cu
100 0
Se PL ntensity (AU)
H n r SH2 concentrai on
1 0.0
a 10 20 30
[HIF] wt %
Figure 1.2. Room temperature PL intensity and SiH2U)
recovery of porous silicon as a function of
HF concentrations [Tsa92].
(b) (c)
Figure 1.3. Idealized porous silicon layer. (a) Cross
sectional view; (b) Plan view of the layer;
(c) Plan view of a high porosity layer.
1.0
0.8 / 0"
/ 0 PSL onp
S0.6 / o PSLon p+
. o+ SI reference
0 +
0 0.4 .
o +
500 700 900 1100
Wavelength (nm)
Figure 1.4. Measured transmission for monchromatic light
of porous slicon samples grown on a p type
silicon substrate (squares), on a p+ type
silicon substrate (circles) and of a silicon
singlecrystal reference sample [Leh91].
100 200 300
Oxidation Time (sec)
0 100 200 300
Oxidation Time (sec)
Figure 1.5.
PL variations after porous silicon oxidation
at 7000C. (a) Measured room temperature PL
peak position and peak intensity as a
function of oxidation time; (b) PL spectral
width at FWHM as a function of oxidation
time [Shi92].
8000
7000
6000
5000
4000
A
B
u
t3
al
3
u
U
0.
>1
(n
0
Q1
0C
550
950
650 750 850
Wavelength (nm)
Figure 1.6. The change of porous silicon PL spectra
after oxidation and deoxidation. (a) as
prepared; (b) oxidation with aqueous HN03
solution; (c) deoxidation with aqueous HF
solution after oxidation [Nak92].
2.5
2.0
1.5
L = 31 A
S 0to 
I
C 0.5
0.0
0.5
1.'0
1.0 0.5 0.0 0.5 1.0
Wave Vector k (Tr/a)
Figure 1.7. Ek diagram of a silicon quantum wire with
size L = 31 Angstrom [San92].
I I
I I
5 10. 15 20 25
Wire Size, L (A )
30 35 40
Figure 1.8. Variation of band gap (solid line) and
exciton energy (dashed line) with different
wire size L [San92].
CHAPTER 2
FORMATION OF POROUS SILICON
2.1 Introduction
The formation of porous silicon, which was first
reported in 1956 [Uhl56], results from an anodization
process performed in an chemical cell, in which the silicon
wafer is used as the anode and a Pt (or any other antiacid
electrode) in the electrolyte is the counter electrode. This
anodization process is controlled by the electrochemical
activity in the porous structure which depends on several
conditions such as the concentration of the electrolyte, the
silicon wafer dopant type and doping density, and the anodic
current and anodic environment (dark or illumination).
In this chapter, we first discuss the experimental
setup, followed by description of the surface dissolution
chemistry and a porous silicon formation model.
2.2 Experimental Setup
The experimental set up for the fabrication of porous
silicon samples is shown in figure 2.1. The chemical cell
made of teflon components was built in our lab. In order to
get a uniform porous silicon layer, a good ohmic contact to
the back side of the sample is needed to establish an
uniform current distribution across the silicon wafer. In
our experiments, ptype (100) silicon wafers were implanted
with boron on the back side to provide a P+ layer for good
ohmic contact. Aluminum was evaporated onto the P+ layer by
Ebeam following annealing. A teflon coated wire was pasted
onto the back side of the wafer with silver epoxy for
connection to the external circuit. A good ohmic contact
resulted. The back and the edges of the wafer were covered
with wax for protection from the HF. The samples formed the
anode of the chemical cell filled with diluted HF and Pt was
used as cathode. A HewlettPackard 4145B semiconductor
parameter analyzer was programmed to provide a constant
current for a specific amount of time and was also used to
monitor the voltage V across the chemical cell during
electrochemical etching.
2.3 Silicon Surface Dissolution Mechanisms
Using an aqueous HF solution for cleaning silicon
wafers has proven to be an effective means to passivate
surface states on silicon [Hua92,Hig90]. The surface
passivation is achieved by H termination of silicon dangling
bonds during the HF etching. The reason for H termination on
silicon surfaces rather than F termination is that, although
the relative strength of SiF (6eV) is higher than that of
SiH (3eV), the Fterminated silicon complexes are unstable
in a HF solution. The polarization induced by SiF bonds
causes HF molecules to attack the SiSi weakened back bonds.
This is easy to understand from an inspection of the
following chemical formula
F F F F F"
HX s
SI HF SI H+
+ 2HF
Si Si Sl SI
/ \ / \ / \ / \
I S +SS
F F
\ /H \ / S\
SSl Sl +
/ \ / \ F/ \F
(2.1)
As shown in the above equation, we assume that the first
layer (or surface layer) of silicon dangling bonds is
terminated by F. The large electronegativity of F compared
to that of Si causes a strong polarization in which the F
side is negative and the Si side is positive. Thus F ions
can easily attack and break the first layer of silicon back
bonds and then form a new FSi back bond. Then the second
layer of silicon dangling bonds having two electrons, which
used to form SiSi covalent bonds with the first layer of
silicon, form a HSi covalent bond with H+ ions. These bond
transformations are shown by the arrows in equation (2.1).
This reaction results in an Hterminated surface after
releasing silicon fluorides into the solution. The H
terminated surface is virtually inert against further attack
by F ions because the electronegativity of H is about that
of Si and the induced polarization is low. Furthermore,
accurate quantum chemical calculations show that a
significant, high activation barrier prevents SiH bonds
[Hig90], formed according to the above formula, from attack
by HF.
It is found in experiments, under cathodic
polarizations for both n and ptype material, that silicon
is normally stable. Only under anodic polarizations does
silicon dissolution occur. It is believed that silicon
surface atom dissolution during the anodization process is
possible only in the presence of holes. This means that it
is difficult for ntype material to dissolve since holes are
normally absent, unless under illumination, high fields, or
in the presence of other hole generating mechanisms.
The dissolution of silicon under anodic polarization
leads to a porous silicon layer or to electropolishing
depending on the anodization conditions. The morphology of a
porous layer also strongly depends on the exact anodization
conditions, such as HF concentration, silicon type, dopant
concentration and the anodic potential. When the anodic
potential is higher than a critical value, the silicon
surface electropolishes and a smooth, planar morphology will
result. Current efficiency measurements have been carried
out [Bea85] and indicate that only two of the four available
silicon electrons or holes participate in a direct
interfacial charge transfer during pore formation and that
all four silicon electrons are electrochemically active
during electropolishing. Based on this charge transfer
observation, the dissolution mechanism of anodic silicon for
forming porous silicon can be formulated as in equation
(2.2) in which the reaction proceeds completely as an
oxidation process and the holes act as oxidizing agents for
surface bonds.
H..
H H
SI
SI SI
/ \ / \
F H
Sl
Si Si
/ \ / \
F H H
+F+h 
SI Si
F H F
+F SI I
SI Si
" F F
SI
Si Si
/ \ / \
(2.2b)
As discussed previously, after a silicon wafer is immersed
in an aqueous HF solution, a surface terminated by H
results. This Hterminated surface is virtually inert
against further attack by F ions in the absence of holes.
F
Sl
SI SI
/ \ / \
(2.2a)
For a Ptype silicon wafer under anodization, the holes can
overcome the surface barrier, formed between silicon and the
electrolyte (as discussed in the following section), to
reach the silicon surface. From the point of view of the
local chemical bond, an excess hole concentration at the
silicon surface can be translated into an electron being
released from the bonding valence states. As a consequence
the average bond strength of surface atoms is reduced and
they become therefore accessible for chemical attacks
[Ten86]. Thus, SiH bonds can be attacked by fluoride ions
after a hole reaches the surface which makes reaction
centers accessible for F~ ions, and a SiF bond is formed
with the simultaneous release of a hydrogen atom as shown in
equation (2.2a). The strong polarization of the SiF bond
allows another F ion to attack and bond. It should be noted
that three electrons, one from the F ion and two from the
HSi covalent bond, are reallocated. After being attacked
by F, one electron forms a new FSi bond, one constitutes
an H atom which forms an H2 molecule later with another H
atom, and the third one injects into the bulk as shown in
equation (2.2b). So the total reaction presented by equation
(2.2), initiated after a hole reaches the silicon surface,
results in a Fterminated silicon surface with the
generation of an H2 molecule and the injection of one
electron into the bulk of the ptype wafer. This unstable F
terminated silicon complex will be further attacked by HF
resulting in the Hterminated surface as depicted in
equation (2.1). If other holes are available, the anodic
silicon dissolution will continuously follow the cycle from
equation (2.1) to (2.2) and then back to (2.1), dissolving
silicon in the process. It should be noted that, as
indicated in equation (2.2), there are only two charges
participating in the charge transfer process for one Si atom
dissolved.
The dissolution mechanism of silicon in the
electropolish mode is almost the same as the one we
discussed above. The difference is that a large over
anodization potential leads to more holes at the silicon
surface so that the fluoride ions can attack all four
silicon bonds resulting in all silicon atoms to be
dissolved.
Figure 2.2 shows a topological distribution map for
the different regions of silicon dissolution as a function
of current density and HF concentration [Smi92]. This
graphically demonstrates the wellknown fact that porous
silicon formation is favored at high HF concentrations and
lowcurrent densities, while electropolishing is favored at
low HF concentrations and highcurrent densities. The
different regions A (pore formation), B (transition), and C
(electropolishing) are labeled as shown in the figure. A
porous silicon formation model which involves the
electrolyte concentration, the anodic current density, and
the wafer doping density will be discussed in detail in the
following section.
2.4 Porous Silicon Formation Model
A mediated charge transfer mechanism at the interface
of electrolyte and anodic silicon results in the localized
dissolution of silicon which is essential to form the porous
silicon layer. In the porous silicon layer, the dissolution
of silicon always occurs in those places where holes are
most easily available. In the very beginning, this
dissolution is usually triggered randomly for clean, smooth
surface planes or starts at locations where the electric
field is enhanced by for example surface defects or the
convex shape of rough surfaces, etc. (figure 2.3). After
silicon atoms are removed from these locations, the convex
shape becomes more pronounced. Since the dissolution prefers
locations with an enhanced field, the pores will grow in the
direction of the enhanced field. It can be shown that the
electric field is inversely proportional to the curvature
radius of a pore, r, and so the largest, enhanced field is
always found at the pore tip [Bea85,Lec90]. Thus, in
general, dissolution will most likely occur in the pore tip
which results in pore development toward bulk silicon in the
direction of the enhanced field or, in other words, in the
direction of the current. In the ideal case, the pores would
form a straight line network in which all pores are parallel
with each other (figure 1.2). In reality, however, straight
pore progression rarely happens since other conditions, such
as defects, a nonuniform current distribution inside the
wafer, etc., may lead to silicon dissolution in other
directions and thus alter the pore growth direction as
depicted in figure 2.4. A TEM picture of porous silicon
obtained by 0. Teschke et al. [Tes93] clearly shows the
columnlike network morphology of porous silicon.
The discussion of the dissolution mechanism of silicon
during anodization in section 2.3 indicates that the
anodization potential plays a key role in such dissolution.
For a large or overpotential, all four silicon electrons
participate in a direct interfacial charge transfer which
results in electropolish, leading to a smooth planar
morphology. But for a moderate potential, only two of the
four available silicon electrons participate in the charge
transfer and porous silicon formation results. Figure 2.5
shows the P type silicon/electrolyte anodization circuit,
the related energy band diagram, and the potential
distribution through the whole system with applied bias Va.
In the figure, Vp is the potential between the Fermi level
and the valence band, Vs is the surface potential of the
semiconductor, VH is the potential across the Helmholtz
layer formed by electrolyte ions in the vicinity of the
semiconductor and Vr is the reference potential. The voltage
drop in the bulk electrolyte region can be neglected due to
the high ionic concentration in this region. Thus we have
Va = Vr Vp Vs VH (2.3)
If the current flow through the Schottky barrier Vs is
dominated by majority carriers, the thermionic emission
theory [Sze81] will give
I = A* T2.exp[q(Vs+Vp)/kT] (2.4)
where A* is the effective Richardson constant, k is the
Boltzmann constant and T is temperature. If the bulk silicon
doping density is NA, then Vp can be expressed as
Vp = (kT/q)ln(Nv/NA) (2.5)
where Nv is the effective density of states of the valence
band equal to 1.04 x 1019/cm3 for silicon.
Substituting equation (2.5) into (2.4), Vs can be
expressed in terms of NA and I by
Vs = (kT/q) [ln(A*T2/I)+ln(NA/Nv)] (2.6)
For simplicity, if we assume a constant electric
field, EH, in the Helmholtz layer having thickness t, the
Helmholtz potential drop can be expressed as
VH = EH't (2.7)
and EH can be related to the electric field Es at the
depleted silicon surface by the boundary condition
EsEs = EHEH (2.8)
It is easy to show, that the surface electric field Es
depends on the surface potential Vs by [Sze81]
2qNA kT
Es =(Vs
EV q (2.9)
After combining (2.7),(2.8),(2.9), we have
VH= ~t 2qNA (V kT
EH Es q
or
VH = 2qNAe (Vs k)
CH q9 (2.10)
where CH is the Helmholtz capacity per unit area equal to
t/EH.
Finally, by substituting (2.6) and (2.10) into (2.3),
we obtain a relationship between Va, the silicon doping
density NA, and the anodization current I
(A) Ms.N/kTUn( *Iln(NA)I]
Va = Vr kT In(A ) 2qesNA kT [n(Ah ) InNA 11
q I CH I (2.11)
The second term in the RHS of the above equation stems from
the surface depletion layer potential of the semiconductor
whereas the third term stems from the potential drop in the
Helmholtz layer. The value of CH depends on the
concentration of HF and the semiconductor doping and is
generally around 10 20 9F/cm2 [Lec90, Bsi90, Gas89]. If we
assume CH is 10 9F/cm2 and NA is 1015 cm3, the value
of V2qEsNA/CH is 1.8 x 10 V1/2 which is so small that the
contribution of the third term can be neglected for the case
of low doping. Therefore for low doping, most voltage drops
across the semiconductor space charge region, which limits
the current flow and results in an exponential dependence of
current on the applied voltage as expressed by equation
(2.4). At high doping densities, the Helmholtz potential can
no longer be neglected and both the Schottky layer and the
Helmholtz layer play a role in the formation of the porous
silicon layer.
It should be noted that in the derivation of equation
(2.11), the thermionic model was employed for charge
transport in the Schottky barrier region. This model is
valid for doping densities up to 1019 /cm3 for ptype silicon
[Gas89, Ron91] It is invalid for more heavily doped
materials in which charge tunneling through the Schottky
barrier occurs. In such case, equation (2.11) has to be
modified.
For a wafer with a constant doping density (NA fixed
with position), measurements show that the IV curves don't
depend on the thickness of the formed porous layer [Gas89,
Ron91]. This result strongly demonstrates that the chemical
reaction only takes place at the pore tips which effectively
makes the active chemical layer dynamics independent of
position.
Figure 2.6 is a typical plot of voltage V across the
cell as a function of etching time T for Ptype porous
silicon formation under a constant current bias. After a
short transient (the duration depends on the magnitude of
the supply current), a constant value of voltage, which
corresponds to stable pore growth and progression, is
obtained during pore progression through the silicon wafer.
2.5 Process Related Photoluminescence of Porous Silicon
The experimental set up for both photoluminescence and
electroluminescence (will be discussed later) is shown in
figure 2.7. For photoluminescence, the porous silicon
samples were excited by an Argon laser (488 or 514 nm lines)
or UV laser and the computer controlled monochromator
automatically recorded the luminescence spectrum.
Not all the porous silicon samples luminesence. It is
known that photoluminescence only can be observed on those
porous silicon samples which have a high porosity and/or
fine structure [Cul91,Voo92]. A high porosity and fine
structure can be achieved via the following two fabrication
processes.
A. Postetching the nonluminescent porous silicon samples.
In the postetching process the porous silicon samples are
subjected to a slow etch in a diluted HF solution for a
period of time. The effect of postetching is to shrink down
the structure size of nonluminescent porous silicon samples
and to increase the porosity. An experiment on postetching
will be discussed in chapter 3.
B. Selectively use the anodic current density and HF
concentration. Referring to figure 2.2, the proper
conditions exist not only within pore formation region 'A'
but also close to transition region 'B'. This is because, as
discussed in section 2.3, the porous silicon samples made in
the regions close to transition region 'B' will have a high
porosity. High porosity will result in small feature sizes
of the crystalline silicon wires and thus in a large band
gap (which will be discussed in detail in chapter 4). The PL
spectrum, which reflects the band gap variation, will change
in accordance with the porosity change. To test this
hypothesis experimentally, we fabricated porous silicon
samples A, B, C, D, and E under the same anodic current with
different HF concentrations as indicated in Table 2.1. The
PL spectra are shown in figure 2.8. From the figure, it
becomes clear that the lowest HF concentration etching
condition (sample A) results in the shortest PL peak (around
650 nm) due to the highest porosity and thus the largest
band gap. The PL peaks shift to longer wavelength with
higher HF concentrations (samples B, C, and D). No PL could
be detected for sample E with our experimental system.
2.6 Conclusions
A model for porous silicon formation and the silicon
dissolution mechanism have been discussed in this chapter.
Porous silicon basically results from electrochemically
etching silicon wafers in HF electrolyte. The dissolution of
silicon under anodic etching can lead to either a porous
silicon layer or to electropolishing depending on the anodic
37
etching conditions. The photoluminescence peak wavelength of
porous silicon strongly depends on the anodic etching
condition. It is found that porous silicon samples with high
porosity will have a short peak wavelength PL.
Table 2.1. Sample fabrication conditions
Sample sets A B C D E
Anodic current
2 ^30 30 30 30 30
density (mA/cm) 30 30 30 30 30
Duration
(min) 10 10 10 10 10
HF concentration
(%) 20 40 50 60 80
PL peak position
(nm)PL peak position 650 700 750 770 No
(nm)
Figure 2.1. Experimental set up for porous silicon
fabrication.
HF Concentration
Figure 2.2. Topological distribution map for the
different regions of silicon dissolution as
a function of current density and HF
concentration.
Region
Electropc
.....7.
Region A
Pore Formation
,lishing c.o
x








~,\\\\\~\\\\\\\\\\\\\\\\\\\\\\\\\~,~'~
Figure 2.3. Electric field distribution near a pore in
anodic silicon.
Figure 2.4. Cross section of porous silicon layer.
I
Va (Applied Potential)
0 C ........
Silicon Solution ": '
Silicon1 _
(d 4
. 0
U *
C,)
Silicon
Ohmic SCR
contact layer
Ec
EF
Ev
Helmholtz,
layer
n7
Solution
 qvH
Constant
E
Reference
Silicon
4>
SVVp
SSolution
I
Figure 2.5. Principal scheme of a Ptype
silicon/electrolyte anodization system.
(a) the circuit; (b) the related energy band
diagram; (c) the potential distribution with
applied bias.
Metal
Metal
Va
!

I
t r
0.5
> :
0.4
CD
0
o>
0.3
0.2
0 100 200 300
Anodic etching time (s)
Figure 2.6. The voltage across the chemical cell
measured as a function of time during
electrochemical etching of a p type silicon
wafer in the dark. The cell was biased with
a current source.
I I
PS sample
& l
Figure 2.7. Experimental set up for PL and EL
measurements.
X
0.21 / i\
0.0 
600 700 800 900
Wavelength (nm)
Figure 2.8. Normalized PL spectra for porous silicon
samples fabricated under different
conditions as listed in Table 2.1.
CHAPTER 3
PHOTOLUMINESCENCE ENHANCEMENT AND SATURATION RESULTING FROM
HIGH TEMPERATURE TREATMENTS OF POROUS SILICON
3.1 Introduction
Wide band, efficient room temperature
photoluminescence from porous silicon has generated a great
interest in the field of semiconductor physics and
technology. The realization of visible electroluminescence
from porous silicon [Kos92, Nam92, Bre92] was a first step
towards optoelectronic device applications. There is still a
controversy, however, concerning the origin of the porous
silicon luminescence. The controversy stems mainly from the
fact that porous silicon has a vast surface area on which
many molecules may adsorb. As described in chapter 1, some
of these molecules such as siloxene, hydride complexes, and
Si:O:H compounds are thought to be responsible for the
porous silicon photoluminescence by some researchers. Others
claim that quantum confinement is responsible for the
observed phenomena.
It has been reported that the PL intensity
dramatically decreases with an increase of the annealing
temperature in the range from 300 700 oC in vacuum, N2, H2,
Ar, and air [Ook92, Rob92, Tis92, Tsa92, Pro93, Seo93]. The
PL degradation was attributed to the decomposition of the
silicon hydride species [Tsa92, Pro93], to surface structure
changes[Seo93], or to dangling bond formation [Rob92].
Enhanced PL intensities obtained after dry oxidation
of porous silicon in a high temperature range from 800 
1000 OC were reported recently. These results were obtained
by low pressure dry oxidation [Yam92] and by rapid thermal
oxidation (RTO) [Pet92, Tsa93]. Since the typical dimensions
of silicon nanostructures in luminescent porous silicon are
less than 10 nm, a slight over oxidation will consume all
the silicon resulting in a loss of nanostructures. An
oxidation time of less than a minute is required to observe
the PL enhancements in RTO [Pet92, Tsa93]. Rapid thermal
oxidation of porous silicon also dramatically improved the
PL stability [Bat93]. Both the PL enhancements and the PL
stability improvements are due to high quality SiO2 grown on
the porous silicon surface which passivates the silicon
surface.
Direct thermal nitridation of silicon in nitrogen and
amonia gas at 700 1400 oC has been reported in the
literature [Ito78, Mur79]. As in the silicon dioxide growth
process via thermal oxidation, silicon nitride grows at the
interface between silicon and silicon nitride with nitrogen
atoms diffusing through the silicon nitride layer. It is
well known that the asgrown thermal nitride films have a
high structure density, which limits the diffusion of
nitrogen atoms through the silicon nitride, leading to a
"selflimiting" growth process [Wu82]. This unique property
thus forms a major difference between the thermal
nitridation and the thermal oxidation of silicon. In the
case of oxidant impurity contamination in nitrogen or
ammonia ambient, a surface oxynitride film results. The
oxidant contamination may result either from initial, native
silicon dioxide grown on the etched silicon substrates
before annealing or from a backstream of oxidant impurities
from the atmosphere in an openopened furnace. It is found
that this surface film also possesses the selflimiting and
oxidation resisting properties of the nitride films [Rai75,
Mur79]. For example, an oxidationresistant surface film of
less than 10 nm was found when an etched silicon substrate
was annealed at 980 oC in an openended furnace in nitrogen
ambient but in the presence of oxidant contaminations from
atmospheric backstreaming for 10 days. A detailed study of
these surface films are beyond the scope of this
dissertation, but is reported in the literature [Rai75,
Mur79]. This slow and selflimiting growth process enables
us to track the PL changes in N2 ambient at elevated
temperatures without the risk of consuming and/or
significantly changing the size of the silicon
nanostructures.
In this chapter, we will present the results of our
studies on thermal annealing effects on the PL of porous
silicon. We annealed porous silicon samples at elevated
temperature in N2 ambient up to 3 hours and observed
remarkable PL intensity enhancements.
3.2 Experiments
The porous silicon samples used in this work were
formed by anodic etching of ptype, (100) silicon wafers
with a resistivity of 618 9 cm. Typical sample preparation
procedures were described in chapter 2. Three sets of
samples (A, B, and C) were made and their fabrication steps
are listed in Table 3.1. The anodic etching and the post
chemical etching conditions were chosen in such way that the
peaks of the PL of asprepared Sample C and Sample B were at
roughly the same position. PL spectra were measured using a
30 mW Arion laser. No PL was observed from asprepared
Sample A.
Thermal annealing was performed in a furnace at a
temperature of 800 OC and 900 oC, with a constant nitrogen
flow rate (4 liter/min). For comparison, the asprepared
samples were cleaved into small subsamples and the PL was
measured on these subsamples after annealing at different
temperatures and for different time intervals. Since some
native silicon dioxide is expected to grow on the samples
when they were transported in and out of the furnace, we
kept the transportation time constant for all samples. Two
different annealing procedures were used to unravel the
effects of this parasitic silicon dioxide growth on the
measured PL spectra.
Annealing procedureI. PL measurements were carried
out on one set of subsamples cleaved from Sample A. In this
procedure, each of these small samples was annealed just
once for a specific time interval and temperature. The PL
was measured on each small sample after annealing. Figure
3.1 shows the integrated PL intensity versus annealing time
curves resulting from this experiment. The PL spectra of the
samples annealed at 900 OC are shown in figure 3.2.
Annealing procedureII. The PL was measured as a
function of annealing time for the same physical subsample.
In this procedure, a subsample was annealed several times
at one temperature. The PL was measured on the same sub
sample after every annealing interval. Figure 3.3 shows the
integrated PL intensity versus annealing time curves
resulting from this experiment. The peak positions of the PL
spectra are shown in figure 3.4.
3.3 Discussion
It is found that high temperature annealing of porous
silicon sample in nitrogen ambient not only results in the
growth of a thin, self limiting surface film, but also
results in chemical compound decomposition at the surface.
In asprepared porous silicon samples, chemical impurities
such as hydrogen, fluorine and hydroxyl groups are present.
The maximum out effusion occurs at the following
temperatures [Har85]:
SiH2 at 300 OC
SiH3 at 300 C
SiH2F at 400 OC
CH3 at 500 C
H2 at 500 oC
H20 at 500 C
SiH at 500 C
SiHF2 at 500 OC
The thermal annealing temperatures (800 oC and 900 oC) used
in our experiment are higher than the above listed
temperatures. It is therefore expected that these chemical
compounds will discompose at the porous silicon surface
after high temperature annealing.
Fourier Transform Infrared (FTIR) spectral
measurements were carried out on the asprepared samples and
on the 30 min annealed samples. The infrared vibrational
spectra showed that the SixHy modes, which exist in the as
prepared samples, disappear after the high temperature
annealing step.
An enhancement of PL intensity was clearly observed
after annealing porous silicon in N2 ambient at 800 and
9000C, as shown in figure 3.1. A slight blue shift of the PL
spectra was found after 12 min annealing at 900 oC (figure
3.2). The PL enhancement and blue shift presented in figures
3.1 and 3.2 are similar to those resulting from slow wet
etching [Can90, Rob93] and from the RTO process.
Our experimental results are difficult to explain
within a silicon hydride model, a siloxene model, or a
Si:O:H compound model which have been proposed as a possible
explanation for porous silicon luminescence. According to
these models, the PL changes are caused by rearrangements of
the chemical bonds. In N2 thermal annealing, however, in the
absent of H, it is hard to understand with these models why
the enhancement and blue shifts of PL follow similar trends
as the ones observed from slow wet etching and from RTO. The
fact that no PL was observed in vacuum annealing at an
elevated temperature [Yam92] rules out the possibility that
the PL changes are due to surface chemical bond
rearrangement during high temperature annealing.
Our experimental results can be understood in terms of
a quantum confinement effect. Thermal nitridation of silicon
at 800 and 900 oC results in a very thin surface layer grown
on the silicon surface at a very slow rate due to its "self
limiting" property. Since we maintained the time needed for
transporting all our samples in and out of the furnace
constant, the PL differences in annealing procedureI
(figure 3.1 and 3.2) solely reflect N2 annealing effects on
the PL. In this annealing procedure, as depicted in figure
3.2, the slow growth rate of the thin layer leads to no
significant blue shift but the PL intensity increases
significantly during the first 8 minutes of 900 oC
annealing. The PL intensity increase is believed to be due
to the passivation of the silicon by the surface layer.
Since no significant layer thickness is achieved during this
short annealing time, no significant amount of silicon is
consumed and as a result the PL peaks remain at the same
energy position. After 8 minutes annealing, slight blue
shifts and peak intensity degradations were observed. We
attribute these phenomena to shrinkage of the silicon wire
size and hence loss of PL volume. For the samples annealed
at 800 oC, the surface layer growth rate is much slower than
at 900 oC, thus the PL intensity is expected to increase
slowly with annealing time (figure 3.1). Almost no peak
shift was observed in this experiment. Because of the "self
limiting" property of the surface layer, the thickness of
the layer will saturate for long annealing times [Wu82]. As
a result, the PL intensity and peak position should become
constant for long annealing times. This phenomenon is
confirmed by our experiment as is clearly shown in figure 1
and in the inset which shows an annealing time up to three
hours for T = 800 oC. This unique feature makes thermal
annealing porous silicon in nitrogen quite different from
thermally oxidizing porous silicon in oxygen. Silicon
dioxide has a high growth rate and its layer thickness never
saturates with time at 800 and 900 OC. The effects of
silicon dioxide on our samples are clearly shown in
annealing procedureII (figure 3.3 and 3.4). In this
procedure, the PL curve was obtained on the same subsample
which was annealed several times at the same temperature.
Since some native silicon dioxide will grow on the sample
surface not covered by the surface layer for every time the
sample is being taken in and out of the furnace, a lack of
PL intensity saturation and large blue shift caused by the
silicon oxidation effects are clearly shown in figure 3.3
and 3.4.
In the quantum confinement model, a thermally grown
layer has two effects on a PL spectrum. One is' passivation
of the surface and the other is the shrinkage of the silicon
nanostructures. The passivation role enhances the PL
efficiency by eliminating the nonradiative centers stemming
from unpassivated silicon dangling bonds. The shrinkage
role, however, can either enhance or degrade the PL
intensity depending on the feature size of the asprepared
porous silicon samples. In our experiment, we intended to
give Sample A bigger nanostructure sizes (smaller porosity)
than Sample C by choosing different anodic etching
conditions. The fact that we observed no PL signal from as
prepared Sample A is a result of such big nanostructures.
After annealing at 800 and 900 OC, the passivation and the
shrinkage of porous silicon by the surface layer first
enhance the PL of Sample A as shown in figure 3.3. The
continuous reduction of the nanostructure feature sizes by
annealing, however, degrades the PL intensity due to loss of
effective luminescence volume. Asprepared Sample B is a
result of anodic etching (identical to Sample A) plus two
hour post chemical etching. The chemical etching, like
thermal shrinkage, will reduce the large, nonluminescent
nanostructures of asprepared Sample A into the luminescent
nanostructure range. Thus, asprepared Sample B has a
detectable PL with a peak around 1.75 eV (figure 3.4) which
indicates that most of its nanostructure feature sizes are
of the order of 25 Angstrom (this will be discussed in
chapter 4). In this range, it can be expected that even a
few angstrom shrinkage of the nanostructures would lead to a
large loss of PL volume. Indeed, as shown in figure 3.3, the
PL intensity of Sample B does not increase but decreases
after annealing. In addition, the PL drops more rapidly at
900 OC than at 800 OC due to a higher rate of PL volume loss
at 900 oC. Since asprepared Sample C has roughly the same
PL peak as asprepared Sample B (figure 3.4), asprepared
Sample C should have roughly the same nanostructure feature
sizes as asprepared Sample B. The density of the
nanostructures of Sample C, however, is obviously higher
than that of Sample B since Sample C was subjected to anodic
etching only. Therefore, the PL intensity of Sample C is
stronger than that of Sample B as depicted in figure 3.3.
It is very interesting to note that we observe first a
red shift in the PL and then a blue shift with increasing
annealing time for Sample B and Sample C as shown in figure
3.4. No indication about a possible red shift is available
for Sample A because of no detectable PL emission of as
prepared sample A. A similar phenomenon was observed by C
Tsai [Tsa93] in rapidthermaloxidized porous Si. They
observed a red shift in the PL at first and then a blue
shift with increasing temperature, but the RTO process
failed to restore the PL to the original peak position. In
our case, annealing samples at 900 oC restores the peak
position and even shifts it to higher positions. The
mechanism of this red shift is not clear yet. It might be
due to the strain which is produced at the interface between
silicon and the surface layer. It is known that strain
induces bulk silicon band gap shrinkage [Smi78]. In porous
silicon, the stress produced by the passivation layer might
be very large and thus might induce a significant band gap
reduction. In that case, the initial red shift after
annealing is attributed to the strain caused by the
formation of the passivation layer. The blue shifts observed
upon further annealing would be the net result of the
competition between a red shift induced by strain and a blue
shift caused by the shrinkage of the nanostuctures.
3.4 Conclusions
Our results indicate that thermal annealing of porous
silicon in nitrogen ambient provides a good method to track
PL changes resulting from passivation and nanostructure
features shrinkage. PL enhancement and saturation as a
function of annealing time were clearly observed
experimentally. FTIR data showed that hydrogen disorbs from
the porous silicon surface after high temperature
treatments. Since, in the literature, the proposed hydride
model, siloxene model, or Si:O:H compound model involve
hydrogen, the FTIR results eliminate the possibility of
these models being responsible for the porous silicon
luminescence. Instead, our experimental results are in good
agreement with predictions based on the quantum confinement
effect. A detailed quantum confinement interpretation will
be given in following chapters.
Table 3.1. Fabrication conditions for sample
B, and C.
sets A,
Sample sets A B C
Anodic etch solution
(HF:Ethanol) 4:1 4:1 2:3
Anodic current density
(mA/cm2) 7 7 30
Duration 10 10 10
(min)
2 hours in
post chemical etching No HF:Ethanol No
(1:1)
0 2 4 6 8 10 12 14 16 18 20 22
Annealing time (min)
Figure 3.1.
Integrated PL intensity of Sample A versus
annealing time resulting from annealing
procedure I. Inset shows the integrated PL
intensity of the sample annealed at 800 C
versus annealing time up to three hours.
2500
2000
1500
1000
500
0t
1.4
1.6
1.8
2.0
Photon energy (eV)
Figure 3.2. PL spectra of Sample A after annealing at
900 OC for different time intervals using
annealing procedure I.
a I I I
,8 "A
I , 2min
I  min
,' ,' '1 4 min
\ 20  12min
j !/i 2 20min
'IV
S4
* ''w. 'S
 I ,* 1\
* \
\ \N'
'"'"""''"'
J
0 2 4 6 8 10 12 14 16
18 20 22
Annealing time (min)
Figure 3.3. Integrated PL intensity versus annealing
time for samples annealed at 800 and 900 OC
'using annealing procedure II.
1.80
9 1.75
C
 1.70
0.
.
1.65 Sample A800 C
S Sample A900 C
S+ Sample B800 C
x Sample B900 C
/ Sample C800 C
1.60 /  Sample C900 C
0 2 4 6 8 10 12 14 16 18 20
Annealing time (min)
Figure 3.4. PL peak position versus annealing time for
samples annealed at 800 and 900 oC using
annealing procedure II.
CHAPTER 4
ENERGY BANDS OF SILICON QUANTUM WIRES
4.1 Introduction
Although the physical origin of porous silicon
luminescence is not quite clear yet, the high temperature
treatments of porous silicon described in chapter 3 indicate
that the luminencesce is most likely due to quantum
confinement effects. The quantum confinement model predicts
a wide, direct band gap for porous silicon and thus highly
efficient porous silicon photoluminescence becomes possible.
This is supported by several numerical band calculations
based on the tight binding method [San92], the first
principles methods [Rea92, Bud92], and on the ab initio
pseudopotential method [Ohn92], etc.. These theoretical
calculations indicate that the energy band gap of silicon
quantum wires changes from the indirect band gap of bulk
silicon to a direct band gap. These results, however, are
obtained using extensive and complicated numerical
calculations which are in general are not easy to
understand. In this chapter, starting with a description of
the effective mass theory, we investigate the band structure
of quantum wires within the framework of the effective mass
theory in order to obtain a simple picture of the energy
band change due to quantum confinement. Our specific
interest here will be in the way quantum mechanical
confinement alters both the conduction and the valence bands
from their bulk silicon structure.
4.2 Effective Mass Theory
In semiconductors, the electron wavefunctions, Yo, in
the conduction and valence bands are found by solving the
Schroedinger equation which relates the system Hamiltonian,
H, of the crystal lattice to the energy, e(k), of the
electron. It can be written as
HYo(r) = [ 2V2 + UT(r)]Yo(r) = e(k)Yo(r)
2m, (4.1)
where mo is the free electron mass, UT is the system's total
potential energy which consists of the periodic lattice
potential UL plus the external potential U.
Directly solving the above equation is extremely
tedious due to the presence of lattice potential. This
equation, however, can be transformed to the so called
"effective mass equation" within the framework of the
effective mass theory. The effective mass theory has found
extensive use in the analysis of carrier transport in
semiconductors, especially in the analysis of
heterostructures and superlattices. A complete discussion
can be found in the literature [Lut55, Dre55, Dat89] and
will not be repeated here. Rather the main results of the
theory and its primary assumptions will be presented.
Because of the degeneracy of the valence bands, the
conduction bands and valence bands of semiconductors have to
be treated separately in the effective mass theory.
4.2.1 Conduction Band Effective Mass Equation
It is generally known that semiconductor conduction
bands are nondegenerate. In the case of nondegenerate energy
bands, the eigenfunctions Tnk of H are used as the basis for
the effective mass equation of a single band. Thus the
Schroedinger equation can be written as
H(r)Yn,k(r) = n(k)Tn,k(r) (4.2)
where En(k) are the eigenvalues, and n and k are indices
representing a particular band and crystal momentum
respectively. It has been shown most notably by Luttinger
and Kohn [Lut55] (using a KP formalism) that the above
Schroedinger equation can be rewritten as an effective mass
equation having the following form
[%n(iV ko) + U(r)]Y(r ) = n(k)Y(r) (4.3)
where En(iVko) is the bulk dispersion relation (with
respect to an extreme point ko) operator in band n. The
wavefunction '(r) appearing in the effective mass equation
is often called an "envelope function." The true
wavefunction Yo(r) is approximately equal to the product of
the envelope function P(r) and the periodic part of the
Bloch function.
It should be noted that two assumptions are made when
the K*P method is used to derive the above effective mass
equation:
1. Neglect k3 or higher powers of k in the dispersion
relation.
2. (a) the fractional change of U(r) over a unit cell is
small and
(b) U(r) must cause negligible band to band coupling.
In practice, assumption 1 is easily satisfied if we
confine our study to the vicinity of the edge of the bands.
Assumption 2(a) will not cause any problem generally due to
the macroscopic value of external potential U(r). Assumption
2(b) will hold in the case of nondegenerate energy bands in
which the interband coupling (or interaction) is weak enough
to be negligible.
The advantage of using the effective mass equation as
compared to equation (4.2) is that the Bloch functions have
been removed from the equation, and that the effect of the
periodic lattice potential is now accounted for by the
dispersion relation in which the effective mass, which can
be determined from cyclotron experiments, enters. Therefore,
using this approximation, the electron motion in quantum
wells and quantum wires truly becomes a "particle in a box"
problem with Y(r) as the wavefunction, and the material band
edges as the potential U(r).
4.2.2 Valence Band Effective Mass Equation
The effective mass equation (4.3) relies on the
assumption that the interband interaction is negligible. For
bands degenerate in energy, however, the assumption of weak
interaction is violated and the above outlined approach
cannot be used. The valence bands of most semiconductors,
unfortunately, are degenerate. These multiple valence bands
overlap in energy and even a weak static potential can
induce interband transitions. For the case of degenerate
bands, the effective mass equation (4.3) must be modified to
include the strong degenerate band interaction.
Without strain or spinorbit splitting the valence
band edge of silicon is a sixfold degenerate p multiple.
This sixfold multiple is comprised of three bands each
twofold degenerate due to spin. If the spinorbit coupling
interaction is taken into account, two of these three bands
("heavy hole" and "light hole" bands) are still degenerate
at the energy maximum at k = 0, and the third band ("split
off" band) obtains a maximum energy (at k = 0) at A = 44 meV
below the top the the valence band. In the following, we
will neglect the spin orbit coupling since the quantum wire
confinement energy is expected to be much greater than the
44 meV spin orbit separation.
In order to circumvent the use of assumption 2(b), we
can construct pseudoBloch functions from the original basis
functions [Dre55, Lut55, Dat89]; that is, the newly
constructed basis set eikrun,k=o(r) are eigenfuctions of the
crystal translation operator, but are not in general
eigenfunctions of the Hamiltonian. With the new basis set,
we still obtain an effective mass equation for each
degenerate band similar to the single band effective mass
equation (4.3). However, since the basis functions are not
eigenfunctions of H, a coupling term is introduced.
Therefore, a matrix representation is used to describe these
three degenerate bands. Due to coupling, the matrix
representation H is obviously no longer diagonal and can be
expressed as [Dre55, Lut55]
ik, + m(k0 + k2) akxky akxkz
H = nkky ^ + a(k + k2) ikykz
xkkz ikykz ik + a(k + k) ( 44 4)
where the Luttinger parameters 1, m, n are 6.8, 4.43, and
8.61, respectively [Mad82], and are expressed in units of
h2/2mo. To solve the effective mass equation in the valence
bands becomes a problem of calculating the eigenvalue, X, of
lkx + m(k + k) kxky kxkz
akxky ,2y + m(ki + kQ) & kykz = 0
kxkz nakykz Rik + m(ki + k) (4.5)
Based on the theoretical framework developed above, we
are now in a position to handle the conduction and valence
bands of silicon in quantum mechanical terms. As we will see
in subsequent sections, the effective mass theory is not
only is easily implemented in practice, but also is a
powerful tool to give us a simple physical picture of why
and how the band gap change in silicon quantum wires due to
confinement.
4.3 Conduction Band Confinement in Silicon Ouantum Wires
It is known that the conduction band edges of bulk
silicon have six minima located close to the X point in k
space whereas the valence band edges are located at the F
point (kx = 0, ky = 0, kz = 0) This implies that bulk
silicon has an indirect band gap. Figure 4.1 shows the
constant energy surfaces in kspace for an energy just above
the bottom of the conduction band. The dispersion relation
Eb(K) in the vicinity of the conduction band minima can be
expressed as:
b(kko) 2(fkkox)2 h2(kyk0y)2 h2(kzkoz)2
E (.K Ko)  +   +  
2mx 2my 2mz (4.6)
where the kx, k, kz axis are in the direction of <100>,
<010> and <001>, respectively. For the ellipsoids along the
x axis,
mx = mi, my = mz = mt, kox = 0.85(27t/a), k0y = koz = 0;
for the ellipsoids along the y axis,
my = ml, mx = mz = mt, k0y = 0.85(27K/a), k0x = koz = 0;
for the ellipsoids along the z axis,
mz = ml, mx = my = mt, koz = 0.85(2i7/a), kox = koy = 0,
where mi, mt are the longitudinal mass and the transverse
mass, respectively, and a is the silicon lattice constant.
The bulk dispersion relation for the conduction band
in the vicinity of the minimum in the zdirection is
b(k ) = 2(k + k) h2(kkoz)2
2mt 2ml (4.7)
upon substituting equation (4.7) into equation (4.3), the
effective mass equation becomes
aT + +)+ 1 2 (Eb U) = 0
mt ax2 ay2 MI a 2 (4.8)
Now let's consider a silicon quantum wire with width Lx and
Ly in the x and y direction and infinitely long in the z
direction. Therefore, four of the band minima ( kox, koy)
are in directions of confinement and the remaining two
minima ( koz) are in the unconfined directions. In this
confinement picture, the external potential
U(r)=0 < x
oo otherwise (4.9)
Let's first consider the bands in the z direction
(unconfined direction). Taking advantage of the effective
mass equation, we may try a solution of the form
Y(r) = Asin(nxx/Lx)sin(m7cy/Ly)ei(kzkoz)Z (4.10)
which satisfies the boundary conditions of the quantum wire.
The parameter A is a normalizing factor, n and m are quantum
integers. The quantity kzkoz represents the real momentum of
the electron. Substituting equation (4.9) and (4.10) into
equation (4.8), the dispersion relation (or confinement
energy) of a quantum wire in the unconfined zdirection
becomes
nim(k) = h2n272 + h2m2 I2 h2(kz kz)2
2L mt 2 mt 2ml (4.11)
For the case of Lx = Ly = L, equation (4.11) will be
ewzm() = 2 (n2 + m2) h2(kz koz)2
2L2 mt 2ml (4.12)
Obviously, this dispersion relation in the unconfined
direction indicates that the band is still Xlike in nature
and that the minimum is located at koz. Due to the
confinement in the x, y direction, the dispersion relation
is no longer a function of kx and ky, but rather depends on
the integer quantum numbers n and m. For the ground state n
= m = 1, we have
e =(k) = 2 + h2(kz koz)2
2L2 mt 2mi (4.13)
There is an identical set of subbands at koz in the
negative z direction. The ground state is thus twofold
degenerate. Compared with the bulk case, equation (4.13)
clearly shows that the band edge is shifted up by A272/(L2mt)
due to confinement.
The remaining four bands in the confined directions
are treated in a similar way. For example, for the band in
the xdirection, the bulk dispersion relation is given as
h2(k2 + k) h2(kxkox)2
e(kk k) = k+2
2mt 2mi (4.14)
Likewise we use a trial solution of the form
(r) = Asin (nx/Lx) sin (mny/Ly) eikzz (4.15)
The dispersion relation then becomes
exm(k) = (2n2 ( + ) 2k
2L2 ml mt 2mt (4.16)
This equation shows that, unlike bulk silicon, the minimum
of this set of subbands does not occur at ko, but rather at
zero on the kz axis. These bands form a direct bandgap with
the valence bands. This phenomenon is well explained in
terms of zone folding due to quantum confinement. The ground
state of these subbands is
wx (k) = h2a2 + 1) + 2kz
S 2L2 m mt 2mt (4.17)
This is a fourfold degenerate energy level because there
are three additional, identical subbands in the negative x
and y directions respectively.
After having derived equation (4.13) and (4.17), we
now can investigate the band gap characteristics of a
silicon quantum wire. The band edge of band Ewz(k), located
at kz = koz, has a minimum energy of h212/(L2mt). The band
edge of band Ewx, located at kz = 0, has a minimum energy of
(h2K2/2L2)/(1/ml+l/mt) The band edge difference of these two
bands is thus
Ae = eW = 1,1 h2nt2
2L2 ti mi (4.18)
For silicon with mt=0.1905 mo and mi = 0.9163 mo, As is
larger than zero. This means that the band edge of the
ground state of EWZ(k) is higher than that of the ground
state of EWX(k). It is obvious that the difference in the
silicon longitudinal and transverse effective mass leads to
the direct band gap of silicon quantum wires.
The E(k) dispersion relations as given by equations
(4.13) and (4.17) are illustrated in figure 4.2. Since bands
with a larger effective mass exhibit a smaller energy shift
due to carrier confinement, the fourfold degenerate EWX(k)
bands result in a direct band gap for the silicon quantum
wire as clearly shown in figure 4.2. This phenomenon becomes
more pronounced when the wire size becomes smaller due to
the fact that the difference in band edge energies is
inverse proportional to L2.
4.4 Valence Band Confinement in Silicon Ouantum Wires
The benefit of using the effective mass theory to
study the conduction band characteristics of silicon quantum
wire has been demonstrated above. In contrast to the
conduction bands, however, the valence bands in silicon are
degenerate and therefore can not be treated in the same way.
The Hamiltonian in matrix representation has been developed
for the case of multiple bands. We will use this matrix form
to study the valence bands of silicon quantum wires. The
price we have to pay is the added complexity of having to
solve a system of multiple, coupled differential equations
instead of a single differential equation. In general, the
matrix equation (4.5) is not easy to solve. Fortunately, the
maxima of the valence bands of silicon quantum wires all
occur at kz = 0. In the vicinity of kz = 0, most of the off
diagonal term in equation (4.5) is zero and thus the
equation can be simplified to
Ik/ + k X kxky 0
ykxky k k 0 = 0
k0 k0 m(kx + k2) X9
0 0 m(ki+k (4.19)
where kx = ix/L and ky = jlT/L (i, j = 1, 2, 3,...) for the
quantum wire case. As discussed before, finding the maxima
of the valence bands is equivalent to finding the eigenvalue
X. Obviously, the first set of the eigenvalues are given by
hl,i,j = m(kx2 + ky2) = '(i2 + j2)72/L2
with ground state 21,1,1 = 2mK72/L2. The other set of
eigenvalues are determined by the determinant
fk i + mky l ^nkxky
Ixkky + k (4.20)
In the following, we will use perturbation theory to solve
the above equation. The offdiagonal terms in the above
equation can be treated as a perturbation
a 2
H = nkxky = n
DxDy (4.21)
Thus, the first order correction for energy will be
2,i,j j H I (4.22)
the second order correction will be
+j: = I(Ti lH' I0 )12
p,qpij i,j (p,q (4.23)
where
Y ?(x,y) = sin ix sin jNy
L L L (4.24)
and
?2,i,j = (i2 + j2)
L2 (4.25)
For the ground state
2, I, = (i + m) 
L2 (4.26)
The first and the second order correction terms are (see
appendix A)
2,1,1 = 0
= ( 2 k4(4k2 15
(1+m) k=l (4.27)
Therefore, the second set of eigenvalues is given by (up to
second order of perturbation):
%2,1,1 = X2,1,1 + )2,1,1 + %2,1,1
or
2,1,1 = + ) ()2 l k4 (4k2 1)5
L2 ++m) 7 k=1
which can be written as
X2,1,1 = ( + m) 1 4 k4 (4k2 1)5)
L2 )2 k=l (4.28)
With the actual values of 1, m, and n for silicon, the value
of the second term inside the bracket of this equation
equals 0.1. Therefore, the confinement energy of the valence
bands (maximum of the valence bands) due to quantum
confinement will be determined by X1,1,1 since 1X2,1,11 =
A0. )2/ Xi = 272/L
0.9(I+m)g2/L2 > 1,1,11 = 2mR2/L2.
4.5 The Band Gap of Silicon Ouantum Wires
The band gap of silicon quantum wires is determined by
the minimum of conduction bands, which is located at the
zone center (kz = 0), and the maximum of the valence bands.
Since both the minimum of the conduction bands and the
maximum of the valence bands in silicon quantum wires are a
function of confinement, the quantum wire band gap is given
by
EgW(L) = Egbulk + EWX1,1(L) X1,1,(L) (4.29)
or
EgW(L) = 1.12 + 2.26/L2 + 3.33/L2
= 1.12 + 5.59/L2 (4.30)
which is a function of confinement L. In this equation L is
expressed in nanometers. Due to quantum confinement, the
conduction bands shift up and the valence bands shift down
with a decrease in wire size L. Thus the band gap increases
with L decreasing. Figure 4.3 shows the silicon quantum wire
band gap variation with wire width L. The results of the
other calculations mentioned earlier are also indicated in
this figure. In general, our work is in agreement with other
band gap calculations (especially for wire widths larger
than 2 nm) which generally involve extensive numerical
computations. Our overestimation of the band gap for wire
sizes less than 2 nm is probably due to the nonperiodic
nature (in the confined direction) of the latice potential
due to which the effective mass theory fails.
4.6 Conclusions
In this chapter, the silicon quantum wire band gap is
calculated within the framework of the effective mass
theory. For wire sizes larger than 2 nm, our work agrees
well with other computational calculations such as the
tightbinding method, the pseudopotenial method, and a
firstprinciple calculation, etc.. The overestimation of
band gap values for small wire sizes (less than 2 nm) in our
work is probably due to a limitation of the effective mass
theory. The advantage of employing the effective mass theory
for a silicon quantum wire band gap calculation is two fold:
1. The effective mass theory gives a fast and good
description of the electronic states for thick wires (sizes
larger than 2 nm) where numerical computation techniques
have a difficulty, or are sometimes unable, to calculate the
band gap due to the large number of atoms involved.
2. The effective mass theory gives a simple and sound
physical explanation of the direct band gap nature of
silicon quantum wires. According to the effective mass
theory, the difference in longitudinal and transverse
effective mass gives the order of the conduction band minima
and shift the conduction band minma to the center of the
80
Brillouin zone, resulting in a direct band gap for the
silicon quantum wires.
kx
Figure 4.1. Conduction band ellipsoids of constant
energy of bulk silicon.
0.8
0.0 
5 0 5 10 15
Kz (1/nm)
Figure 4.2. Quantum wire E(k) dispersion diagram of two
conduction band ground states for different
wire sizes. Zero energy refers to the bulk
conduction band edge.
This work
0o \ San91
+ Wan93
N \ Ohn92
+A E Ohn92
+A \ A Rea92
S + ,\ Pro92
+
S+0
o
S I .. . I . . I . . I . .
1 2 3 4
Wire width L (nm)
Figure 4.3.
The variation of silicon quantum wire band
gap with wire size. The solid line results
from the effective mass approximation as
discussed in the text. Results obtained by
others are also indicated in the figure.
0
C
.
c
1
CHAPTER 5
CARRIER STATISTICS AND THE CURRENTVOLTAGE CHARACTERISTICS OF
SILICON QUANTUM WIRE PN JUNCTIONS
5.1 Introduction
The discussion in previous chapters indicate that the
quantum confinement in quantum wires significantly alters
both the conduction and the valence band structures of bulk
silicon, altering almost every property of the material to
one degree or another. Therefore, the expressions which are
used to describe carrier statistics and transport in the bulk
material have to be modified for the case of quantum wires.
In order to correctly study and characterize porous silicon,
the theoretical framework for the silicon quantum wire needs
to be established first. In this chapter, starting with the
description of the density of states (DOS) in an one
dimensional (1D) system, we will derive an expression for the
pn product in quantum wires. Our results indicated that,
under specific conditions, the pn product of a quantum wire
will have a similar form as that of the bulk pn product. The
currentvoltage expressions for bulk silicon diodes apply to
the quantum wire case with some modifications.
5.2 Density of States in a One Dimensional System
In quantum wires, since the confinement is in two
directions (say, x and y directions), an electron possesses
only one degree of freedom along the unconfined zdirection.
Within the framework of the effective mass theory described
in chapter 4, the electron wavefunction in such an one
dimensional system can be described by
P(r) = Asin(nnx/L)sin(mn/L)eikzZ (5.1)
where we assume that the quantum wire has dimensions Lx, Ly,
and Lz, with Lx = Ly = L. The electron energy bands can then
be written as
A k2
En,m(kz) = Ec + En,m + z
2m* (5.2)
In an one dimensional system, the periodic boundary
condition in the unconfined z direction requires that the
wavevector kz must satisfy
kz = 27l/Lz (1 = 1, 2, ...)
The interval in kz space occupied by one eigenstate is
therefore 2K/Lz. The density of states (DOS) is defined as
the number of states between kz and kz + dkz or E and E + dE.
Accounting for the two spin orientations of each electron,
the subband density of states in k space is given by
Dn,m(kz) dkz = 2*dkz/(2C/Lz)
With total energies between E and E + dE, the subband density
of states is given by
Dn,m(E) dE = Dn,m(kz) dkz
= Dn,m(kz) (dkz/dE) dE
= (Lz/I) (dkz/dE) dE (5.3)
Substituting equation (5.2) in equation (5.3), we have
L z F2m* i
Dn,m(E) = 2 / (E Ec nm) (5
22tV f2 (5.4)
Therefore, the total density of states per unit length
becomes
D(E)= (E Ec nE) m5
n,m27c V ? (5.5)
5.3 Electron Density in Ouantum Wires
Using the density of states of the one dimensional
system derived above, we now can calculate the electron
density in quantum wire cases. Basically, the total line
density of electrons in the conduction band may be obtained
by multiplying the density of states by the Fermi function
and integrating over the conduction bands.
n = D(E) F(E) dE
1w1 (5.6)
where
F(E)= 1
exp(EEf) + 1
kT (5.7)
is the FermiDirac distribution function. By substituting
equation (5.5) and (5.7) into equation (5.6), we have
n= m1 2m* (E Ec nm) 1 dE
n,m 22n h 2 exp() +1
.kT
or
n= X' (B(E 6. dE
n,m 27 V h2 exp( ) + 1
iEc (5.8)
The integral in the above equation can be expressed in terms
of the Fermi integral F1/2 () Finally, we obtain
n =BcVkT F. /2 (1n,m)
n,m (5.9)
where
Bc 2 n,m 2 Ef E e=
B = 1 r2m _EfE_
27tV h2 T1nm kT
Using space charge neutrality and equation (5.9), The Fermi
level position with respect to the quantum wire conduction
band edge was calculated as a function of doping density. The
results for 2, 3, 4, and 5 nm wire sizes are depicted in
figure 5.1. In this plot, the summation in equation (5.9) was
carried out up to the third subband since, for example, for
the largest wire with wire size of 5 nm, the number of
electrons in subbands higher than the third is less than 1%
of the number of electrons in the ground state (appendix B).
It is not easy to evaluate the density of electrons in
the quantum wires due the summation in equation (5.9).
Fortunately, for the case of doping densities up to 1019 cm3
and wire sizes L less than 3 nm, which is the general wire
88
size range of the luminescence porous silicon, a detailed
calculation (appendix B) indicates that most of the electrons
occupy the first lowest subband (n=m=l) and that the Fermi
integral F1/2(Tn,m) can be approximated by \KCexp(Tn,m). Under
such conditions, we can neglect second and higher order terms
in the summation in equation (5.9) without causing a large
error. A simple form for the electron density in the
conduction band results:
n = Be exp (Ef Ec e1,1
x kT (5.10)
The valence bands can be treated in a similar way, and
the hole density in the valence band is given by
p=BvkTexp(Ef Ev +11,)
kT (5.11)
where %1,1,1 is the hole confinement energy described in a
previous chapter. The pn product becomes
pn = BeB kT exp ( )
kT (5.12)
where EgW is the quantum wire band gap given in a previous
chapter.
5.4 The CurrentVoltage Characteristic of a pn Junction Diode
In order to analyze the currentvoltage characteristic
of a quantum wire pn junction diode in which the pn junction
is not passivated, we first consider the currentvoltage
characteristic of a three dimensional (3D) bulk diode with a
crosssection as shown in figure 5.2a and 5.2b [McW54, Cut57,
Sah62]. The junction current in the diode may be divided into
four components according to the location of the
recombination and generation of carriers. They are bulk
diffusion current, bulk recombinationgeneration current in
the depletion region, surface recombinationgeneration
current, and surface channel current [Gov67, Sah57, Sah61,
and Sah62]. In most bulk pn junction diodes, the latter two
are usually insignificant and thus can be neglected due to
the good quality of the passivation layer made by the state
ofart technology. For the cases of no or low quality
passivation, the latter two can not be neglected. As a matter
of fact, they will dominate the junction current
characteristic [Gro67] especially under reverse bias
conditions. The first two current components have been
extensively documented in the literature [Mul86, Sze81] and
will not be further discussed here. Only their formula will
be presented. In order to correctly characterize pn junctions
in silicon quantum wires in which no good and controlled
passivation layer was formed, the latter two current
components will need to be taken into account. In the
following we follow closely Sah's treatment of pn junction
currents [Sah61, Sah62].
5.4.1 Bulk Diffusion Current
This current, usually called the diffusion current,
comes from carrier recombination and generation outside the
depletion region in regions labeled I and I' in Fig. 5.2a.
The current may be approximated by
Ibd = Ibds [exp(qV/kT) 1)] (5.13)
where
Ibds ~ ni2 (5.14)
and other symbols have their usual meaning.
5.4.2 Bulk RecombinationGeneration Current Originating in
The Depletion Region
The bulk recombinationgeneration current stemming from
depletion region II enclosed by abcda, excluding the surface
region ab in figure 5.2a equals
Ibr = Ibrs exp[qV/(2kT)] (5.15)
where
Ibrs ni (5.16)
In the reverse bias regime, Ibr = Ibrs niv1/2 for a step
junction and nivl/3 for a graded junction.
5.4.3 Surface RecombinationGeneration Current
This current comes from electronhole recombination in
the depletion region at the surface (ab in Fig. 5.2a).
Electron and hole recombination and generation at the surface
takes place due to the fact that a semiconductor surface has
an abundance of localized states having energies within the
forbidden gap. Even though the presence of a passivation
layer of silicon dioxide over the semiconductor surface
dramatically reduces the number of surface states, residual
91
surface states provide additional generationrecombination
centers over those present in the bulk. This region may
contribute a considerable amount of junction current if a
surface channel is not formed. The current is given by
Isr = q Ls Us dx
o (5.17)
where w is the depletion region width (segment ab in the
figure 5.2a), Ls is the junction circumference, and Us is the
surface recombinationgeneration rate given by [Mul86]
Us = Nstvsta (psns n?)
ps + ns + 2ni cosh(ET ) (
kT (5.18)
where Nst is the surface density of surface recombination
generation centers, and a is their capture cross section
(assumed to be the same for electrons and holes) The
subscript s denotes concentrations and conditions near the
surface and Est is the energy of the surface recombination
generation centers.
Following a calculation similar to the one used for the
bulk recombinationgeneration current, we obtain
Isr = Isrs exp[qV/(2kT)] (5.19)
where
Isrs niw (5.20)
In the reverse bias regime w v1/2 for an abrupt junction
(v1/3 for the graded junction). Therefore, Isr ~ niv1/2 in the
reverse bias regime.
5.4.4 Surface Channel Current
It has been found that a large excess reverse current
flows in a silicon and/or germanium pn junction diode having
a freshlyprepared, unoxidized surface [Mcw54, Cut57, Sah62].
This current is attributed to the fact that, without proper
oxide protection, at the interception of the junction and the
surface, surface charges or ion migration along the surface
forms a surface channel and thus produces a sizable leakage
current, which is called surface channel current.
The mathematical model for surface channel current is
based on the schematics of the surface channel as shown in
region abea in figure 5.2b [McW54, Cut57, Sah61]. The channel
is along the y direction and the channel depth is along the x
direction. It is assumed that the current is entirely carried
by electrons flowing into the channel from the ntype emitter
and that the channel electron density is independent of
lateral position (y direction). Thus, neglecting diffusion,
I (y) = qDnnWILs(q/kT)(dV/dy) (5.21)
where Dn is the diffusion constant of electrons in the
channel, n is the electron concentration in the channel, WI
is the channel width in the x direction and which may be a
function of y, Ls is the junction circumference, and V(y) is
the voltage drop in the channel. Current continuity requires
that the change in current flow along the channel in the y
direction equals the recombinationgeneration current flowing
93
in the xdirection and stemming from the channelbulk space
charge region. Or
dI(y)/dy = WILsqUc (5.22)
where Uc is the recombinationgeneration rate described by
the ShockleyReadHall recombination model given by
U = 1 (pn ng)
op + n + 2ni cosh(TE)
kT (5.23)
where to is the lifetime associated with the recombination of
excess carriers in a region with a density Nt of
recombination centers and Et is the recombination center
energy level.
5.4.4.1 Reverse bias
Under reverse bias, only those recombination centers
whose energy level Et is near the intrinsic Fermi level Ei
contribute significantly to the generation rate. Thus
Uc = ni/(2To) (5.24)
From equations (5.21), (5.22), and (5.24), we have
dI(y) WLsqni 9qDn 1
dV 2o kT I(y) (5.25)
or
2 W2Lq2Dnnni qV(y)
'To kT (5.26)
The boundary condition is that at y = , V(y) = 0, I(y) = 0;
and at y = 0, V(y) = V, the metallurgical junction voltage,
I(y) = Iscs. Thus the reverse bias channel current Iscs
becomes
