Title Page
 Table of Contents
 Formation of porous silicon
 Photoluminescence enhancement and...
 Energy bands of silicon quantum...
 Carrier statistics and the current-voltage...
 Visible light emission from a p-n...
 Electrical band gap determination...
 Summary and conclusions
 Biographical sketch

Title: Fabrication and characterization of porous silicon light emitting diodes
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Permanent Link: http://ufdc.ufl.edu/UF00082368/00001
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Title: Fabrication and characterization of porous silicon light emitting diodes
Series Title: Fabrication and characterization of porous silicon light emitting diodes
Physical Description: Book
Creator: Chen, Zhiliang,
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Bibliographic ID: UF00082368
Volume ID: VID00001
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Table of Contents
    Title Page
        Page i
        Page ii
        Page iii
    Table of Contents
        Page iv
        Page v
        Page vi
        Page vii
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
    Formation of porous silicon
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
    Photoluminescence enhancement and saturation resulting from high temperature treatments of porous silicon
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
    Energy bands of silicon quantum wires
        Page 64
        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
    Carrier statistics and the current-voltage characteristics of silicon quantum wire pn junctions
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
    Visible light emission from a p-n porous silicon junction
        Page 99
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
    Electrical band gap determination of porous silicon using current-voltage measurements
        Page 115
        Page 116
        Page 117
        Page 118
        Page 119
        Page 120
        Page 121
        Page 122
        Page 123
        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
        Page 129
        Page 130
        Page 131
        Page 132
        Page 133
        Page 134
        Page 135
        Page 136
    Summary and conclusions
        Page 137
        Page 138
        Page 139
        Page 140
        Page 141
        Page 142
        Page 143
        Page 144
        Page 145
        Page 146
        Page 147
        Page 148
        Page 149
        Page 150
    Biographical sketch
        Page 151
        Page 152
Full Text








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.




ACKNOWLEDGEMENTS ....................................... ii

ABSTRACT ................................................. vi


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

POROUS SILICON .....................................47

3.1 Introduction ................................... 47
3.2 Experiments .....................................50
3.3 Discussion ......................................51
3.4 Conclusions .....................................57


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

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 Current-Voltage Characteristic of a pn
Junction Diode ...............................
5.5 The Current-Voltage Characteristic of Silicon
Quantum Wire pn Junction Diodes ..............
5.6 Conclusions ..................................




JUNCTION ............................................99

6.1 Introduction .................... .....
6.2 N-P Porous Layer and Device Fabrication
6.3 Measurements ...........................
6.4 Conclusions ............................


7.1 Introduction ...........................
7.2 Current Voltage Measurements ...........
7.3 Experiments ............................
7.4 Conclusions ............................




8 SUMMARY AND CONCLUSIONS ..........................


QUANTUM WIRES .............................140




REFERENCES .............................................145

BIOGRAPHICAL SKETCH .......................... ..


....... 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



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 I-V

measurements is strong evidence for the existence of a wide,

direct band gap in porous silicon.



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 silicon-on-insulator 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


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 X-ray 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 SiH-passivated sample can be

increased gradually by immersing the sample in ever-

increasing concentrations of HF solutions. Fourier-transform

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 red-shifted 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 a-Si:H. They noted a

similarity between optical band gap shrinkage of a-Si:H due

to loss of hydrogen and PL peak red-shifting 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 Si-O-H 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


Based on the above arguments and experimental results, they

conclude that the luminescence in anodically oxidized

silicon is due to Si-O-H 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 siloxene-like


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 blue-shifting 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

band-gap 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 free-standing 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 blue-shift 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 over-consumption 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

as-prepared porous Si which suggests that the chemical

composition is almost the same as that of the as-prepared

samples. From the experiments, they also noted that the

relative number of the Si-H2 surface bonds decreased during

oxidation whereas the measured PL intensity strongly

increased. This is contrary to suggestion of C. Tsai's


T. van Buuren et al. [Buu92, Buu93] measured the x-ray

absorption in the vicinity of the silicon L edge in porous

silicon. They found that the absorption threshold of porous

silicon blue-shifted by 0.3-0.4 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 blue-shift cannot be

explained by the presence of amorphous silicon (a-Si) on the

surface of the porous Si, since the blue-shift of a-Si

relative to crystalline Silicon (c-Si) 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

second-neighbor empirical tight-binding Koster-Slater 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 X-like conduction band minimum and an F-like 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


A. High temperature treatments of porous silicon using rapid

thermal oxidation (RTO) [Pet92, Bat93]. In this experiment,

the as-prepared 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 room-temperature

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


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 current-voltage 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 (I-V) 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 current-voltage

characteristics of porous silicon pn junction diodes is

reported in chapter 7. Finally, in chapter 8, summary and

conclusions are presented.





j 1940


r 1410

~ 1145
L 880


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

100 0

S-e- 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.


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
single-crystal 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].












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].




L = 31 A
S 0to -

C 0.5



-1.0 -0.5 0.0 0.5 1.0
Wave Vector k (Tr/a)

Figure 1.7. E-k diagram of a silicon quantum wire with
size L = 31 Angstrom [San92].



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].


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 anti-acid

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

set-up, 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, p-type (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

E-beam 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 Hewlett-Packard 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 F-terminated silicon complexes are unstable

in a HF solution. The polarization induced by Si-F bonds

causes HF molecules to attack the Si-Si weakened back bonds.

This is easy to understand from an inspection of the

following chemical formula

F F F F F"
HX s
+ 2HF
Si Si Sl SI
/ \ / \ / \ / \


\ /H \ / S\
SSl Sl +
/ \ / \ F/ \F


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 F-Si back bond. Then the second

layer of silicon dangling bonds having two electrons, which

used to form Si-Si covalent bonds with the first layer of

silicon, form a H-Si covalent bond with H+ ions. These bond

transformations are shown by the arrows in equation (2.1).

This reaction results in an H-terminated 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 p-type 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 n-type 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.



/ \ / \



Si Si
/ \ / \


+F+h -





" F F


Si Si
/ \ / \


As discussed previously, after a silicon wafer is immersed

in an aqueous HF solution, a surface terminated by H

results. This H-terminated surface is virtually inert

against further attack by F ions in the absence of holes.



/ \ / \


For a P-type 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, Si-H bonds can be attacked by fluoride ions

after a hole reaches the surface which makes reaction

centers accessible for F~ ions, and a Si-F bond is formed

with the simultaneous release of a hydrogen atom as shown in

equation (2.2a). The strong polarization of the Si-F 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

H-Si covalent bond, are re-allocated. After being attacked

by F-, one electron forms a new F-Si 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 F-terminated silicon surface with the

generation of an H2 molecule and the injection of one

electron into the bulk of the p-type wafer. This unstable F-

terminated silicon complex will be further attacked by HF

resulting in the H-terminated 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


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


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 well-known fact that porous

silicon formation is favored at high HF concentrations and

low-current densities, while electropolishing is favored at

low HF concentrations and high-current 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

column-like 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 over-potential, 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

Es-Es = EH-EH (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

VH = 2qNAe (Vs k)
CH q9 (2.10)

where CH is the Helmholtz capacity per unit area equal to


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( *I--ln(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 cm-3, 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 p-type 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


For a wafer with a constant doping density (NA fixed

with position), measurements show that the I-V 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


Figure 2.6 is a typical plot of voltage V across the

cell as a function of etching time T for P-type 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


A. Post-etching the non-luminescent porous silicon samples.

In the post-etching process the porous silicon samples are

subjected to a slow etch in a diluted HF solution for a

period of time. The effect of post-etching is to shrink down

the structure size of non-luminescent porous silicon samples

and to increase the porosity. An experiment on post-etching

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


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
(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

Figure 2.1. Experimental set up for porous silicon

HF Concentration

Figure 2.2. Topological distribution map for the
different regions of silicon dissolution as
a function of current density and HF




Region A
Pore Formation

,lishing c.o



Figure 2.3. Electric field distribution near a pore in
anodic silicon.

Figure 2.4. Cross section of porous silicon layer.

Va (Applied Potential)

0 C ........

Silicon Solution ": '
Silicon1 _
(d -4
. 0
U *


Ohmic SCR
contact layer





- qvH







Figure 2.5. Principal scheme of a P-type
silicon/electrolyte anodization system.
(a) the circuit; (b) the related energy band
diagram; (c) the potential distribution with
applied bias.






t r--


> :



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.


PS sample

-& --l

Figure 2.7. Experimental set up for PL and EL


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.


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


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 as-grown thermal nitride films have a

high structure density, which limits the diffusion of

nitrogen atoms through the silicon nitride, leading to a

"self-limiting" 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 oxy-nitride 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 open-opened furnace. It is found

that this surface film also possesses the self-limiting and

oxidation resisting properties of the nitride films [Rai75,

Mur79]. For example, an oxidation-resistant surface film of

less than 10 nm was found when an etched silicon substrate

was annealed at 980 oC in an open-ended 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 self-limiting 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


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 p-type, (100) silicon wafers

with a resistivity of 6-18 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 as-prepared Sample C and Sample B were at

roughly the same position. PL spectra were measured using a

30 mW Ar-ion laser. No PL was observed from as-prepared

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 as-prepared

samples were cleaved into small sub-samples and the PL was

measured on these sub-samples 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 procedure-I. PL measurements were carried

out on one set of sub-samples 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 procedure-II. The PL was measured as a

function of annealing time for the same physical sub-sample.

In this procedure, a sub-sample 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 as-prepared 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 as-prepared samples and

on the 30 min annealed samples. The infrared vibrational

spectra showed that the Six-Hy 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 procedure-I

(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 procedure-II (figure 3.3 and 3.4). In this

procedure, the PL curve was obtained on the same sub-sample

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 as-prepared

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. As-prepared 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 as-prepared Sample A into the luminescent

nanostructure range. Thus, as-prepared 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 as-prepared Sample C has roughly the same

PL peak as as-prepared Sample B (figure 3.4), as-prepared

Sample C should have roughly the same nanostructure feature

sizes as as-prepared 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 rapid-thermal-oxidized 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

2 hours in
post chemical etching No HF:Ethanol No

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.











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
* ''w. 'S

- I ,* 1\

* \

\ \N'



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.


9 1.75


| 1.70

1.65 Sample A-800 C
S----- Sample A-900 C
S-+- Sample B-800 C
-x- Sample B-900 C
-/- Sample C-800 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.


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 K-P 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(-iV-ko) 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


1. Neglect k3 or higher powers of k in the dispersion


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 non-degenerate 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 spin-orbit 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 spin-orbit 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 pseudo-Bloch 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


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 k-space 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(fk-kox)2 h2(ky-k0y)2 h2(kz-koz)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 z-direction is

b(k ) = 2(k + k) h2(k-koz)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(kz-koz)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 kz-koz 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 z-direction


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 X-like 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 two-fold

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 x-direction, the bulk dispersion relation is given as

h2(k2 + k) h2(kx-kox)2
e(k-k- 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 band-gap 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 four-fold 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 four-fold 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 off-diagonal 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)

Y ?(x,y) = sin ix sin jNy
L L L (4.24)

?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

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


EgW(L) = Egbulk + EWX1,1(L) X1,1,(L) (4.29)


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 non-periodic

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

tight-binding method, the pseudopotenial method, and a

first-principle 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


Brillouin zone, resulting in a direct band gap for the

silicon quantum wires.


Figure 4.1. Conduction band ellipsoids of constant
energy of bulk silicon.


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 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.






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

current-voltage 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 z-direction.

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 F2-m* i
Dn,m(E) = 2 / (E Ec nm) (5
22tV f2 (5.4)

Therefore, the total density of states per unit length


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)

F(E)= 1
exp(EEf) + 1
kT (5.7)

is the Fermi-Dirac 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


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 F-1/2 () Finally, we obtain

n =BcVkT F. /2 (1n,m)
n,m (5.9)

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 cm-3

and wire sizes L less than 3 nm, which is the general wire


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 F-1/2(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


5.4 The Current-Voltage Characteristic of a pn Junction Diode

In order to analyze the current-voltage characteristic

of a quantum wire pn junction diode in which the pn junction

is not passivated, we first consider the current-voltage

characteristic of a three dimensional (3D) bulk diode with a

cross-section 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 recombination-generation current in

the depletion region, surface recombination-generation

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-

of-art 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)


Ibds ~ ni2 (5.14)

and other symbols have their usual meaning.

5.4.2 Bulk Recombination-Generation Current Originating in
The Depletion Region

The bulk recombination-generation 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)


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 Recombination-Generation Current

This current comes from electron-hole 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


surface states provide additional generation-recombination

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 recombination-generation rate given by [Mul86]

Us = Nstvsta (psns n?)
ps + ns + 2ni cosh(E-T ) (
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 recombination-generation current, we obtain

Isr = Isrs exp[qV/(2kT)] (5.19)


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 freshly-prepared, 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 n-type 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 recombination-generation current flowing


in the x-direction and stemming from the channel-bulk space

charge region. Or

dI(y)/dy = WILsqUc (5.22)

where Uc is the recombination-generation rate described by

the Shockley-Read-Hall 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. 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)

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


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