Controlled electromigration on vicinal surfaces

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on Vicinal Surfaces

^ Cormac 6 Coilcain

Department of Physics

University of Dublin

A thesis submitted for the degree of

Doctor of Philosophy

Ar scath a cheile a rnhaireann na daoine

The work of a PhD. can be lonely and in the words of an ex-teacher of mine, “Caithfaidh tii do ghort fcin a treabhadh”. However this endeavour could not have been possible without the contributions, inspiration and support of so many and it would be an egregious transgression to fail to acknowledge their input.

Firstly I need to thank and acknowledge my supervisor Prof. Igor Shvets who, through a chance encounter some five years ago while walking across campus to­ gether, ultimately provided me with the opportunity and resources to do this re­ search. My first first encounter with Igor however was as a secondary school transi­ tion year student, more than a decade ago, when he accepted me as a lab assistant and allowed me to experience the research environment within his group, an expe­ rience and an eduction whose culmination is this thesis.

cluttered desk, Roseanne for the many conversations over cuppas, Michele for the office teddy-bear, James for his persistent but futile attempts to best me, Sankar for the cups of tea, Nuala, Hugo for being a graceful rugby player, Nina for her basement empathy, Diego, Ehsan, Hye Young and Kay. And the many other nerds that inhabit these buildings. Plamen for advice and this template. Kaan and Gen for a hand. To the staff of CRANN and the Science gallery, in particular Des.

1 am immensely grateful for the support physics department technical staff, in par­ ticular John Kelly and Jemmer who have been present, facilitating and supportive through my whole time at Trinity College and workshop, Dave, Dave, Mick, Gordon and Pat, who have done their best to facilitate my design demands.

To my fellow Aikido club members who provided a stress outlet, in particular Aaron, Keith, Kevin and Kayoko, who also know the trouble of postgraduate study.

To my friends, whose support who cannot be quantified. Without favouritism, dear comrade Giorgos, my long time college friends Gerard, Marita, Ellen, Lennon, Tom (and their respective better halves) all who have been inordinately kind and patient and provided much needed companionship.

Agus nf feidir Horn cuir sfos no thuairisc no ceartas a dheanamh chuig ce chomh riachtanach is a bhf tacafocht rno chlann, mo thuismitheoirf. Art, Aidan, Orna agus Diarmaid, go leir, bhf sibh chomh foighneach liom, leis an “tionscadal” seo.

2^his thesis is submitted by the undersigned for examination for the degree of Doc­

tor of Philosophy at the University of Dublin. It has not been submitted as an excercise for a degree at any other Univesity.

Apart from the advice, assistance and joint effort mentioned in the under signed and in the text, is entirely my own work.

Critical field behavior and antiband instability under controlled surface

electromigration on Si(lll) - C. O Coileain, V. Usov, I. V. Shvets, and

S. Stoyanov.

Phys. Rev. B

84(7), August 9 2011.

Experimental quantitative study into the effects of electromigration

field moderation on step bunching instability development on Si(lll).

- V. Usov, C. 6 Coileain, and I. V. Shvets.

Phys. Rev. B,

83(15), April

25 2011.

Influence of electromigration field on the step bunching process on

Si(lll). - V. Usov, C. 6 Coileain, and I. V. Shvets.

Phys. Rev. B,

82(15), October 6 2010.

Planar nanowire arrays formed by atomic-terrace low-angle shadow­

ing. - F. Cuccureddu, V. Usov, S! Murphy, C. O C6ileain,''a,nd I. V.

Shvets.

Rev. Sci. Instrum.,

79(5), May 2008.

The influence of an electric field on the dynamics of steps, on vicinal surfaces, dur­ ing high temperature annealing was experimentally studied. An experimental setup was designed, constructed and employed to explicitly isolate, for the first time, the electronhgration and thermal effects in the dynamics of the step-bunching process on the vicinal Si(lll) surface. Unlike conventional experiments, the setup’s config­ uration allowed combined but independent control direct-current joule-heating and irradiative heating of samples. This was achieved by placing electrically contacted samples into an alumina heating crucible in ultra high vacuum (UHV). Reduction of the applied electric field results in a corresponding reduction in joule heating, to compensate irradiative heating was increased to maintain a constant temperature.

Atomic force microscopy was used to characterise and analyse the morphologies of the surfaces. The morphologies of Si(lll) annealed at 1130 °C and 1270 °C surfaces are presented. A distinct difference was observed in how the (1x1) to (7x7) phase transition manifests itself on vicinal Si(lll) surfaces off-cut in the [112] and [112] directions. Reduction of the electric field results in a significant expansion of step- bunch width and an elongation of the crossing steps running along the terraces. The theoretically predicted systematic increase in the number of crossing steps with reduced electromigration force has been experimentally observed.

°C). This result was compared with the predictions of the transparent step model, which correctly predicts the formation of the step bunching instability by step-up adatom electromigration. The scaling relation obtained by experiment was found to be different than that deduced from the model.

For the first time values of the critical electric field {Ecr) were determined, i.e. the minimum field required to induce the stepr-bunching instability. The dependence of critical field on the mean initial inter-step distance is investigated and discussed. From experimental results a scaling relationship of approximately Ecr oc j was found for Regime III. This suggests that for crystal sublimation in this Regime the movement of surface adatoms is diffusion limited, i.e. relatively slow surface diffu­ sion and fast kinetics at the atomic steps. A weaker relationship was observed for Regime II, Ecr oc While this scaling relationship is yet to be understood theo­ retically it indicates that different mechanisms are responsible for the .step bunching in each regime.

The anti-band surface instability, which arises on Si(lll) after extended annealing times, was investigated. This was the first experimental study of the effects of a controlled electromigration field on the onset of antibands. The initial or onset stage of antiband formation on step-bunched surfaces was examined for a constant temperature of 1270 °C whilst systematically varying the applied electromigration field. The relationship between the electromigration field and minimum terrace width required to initiate the antiband formation was established. From this it was possible to estimate the effective charge (^eff) of tho surface atoms. The effect of a reduced initial inter-step distance on the onset of the antiband instability is also discussed and experimentally examined.

Declaration v

Publications vi

Summary vii

List of figures xii

List of tables xiii

Nomenclature xvii

1 Introduction 1

2 Vicinal Morphology 7

2.1 Vicinal Surfaces and Crystalline Materials... 7

2.1.1 Sapphire... 7

2.1.2 Silicon... 9

2.2 The step>-bunching mechanisms on Si(lll)... 11

2.2.1 Electromigration... 12

2.2.2 Generalised BCF Theory ... 13

2.2.3 Transparent Step Model... 18

2.2.4 Experimentally Derived Scaling Relationships... 22

2.3 Critical Electromigration Field on Si(lll)...23

2.4 Antibands... 26

2.5 Summary Table of Characteristics ... 32

3 Experimental Methodology 33 3.1 Instruments... 33

3.1.1.1 Contact Mode... 34

3.1.1.2 Non-contact Mode... 34

3.1.1.3 Tapping Mode... 35

3.1.1.4 Phase Imaging... 35

3.1.2 Ultra High Vacuum (UHV)system... 35

3.2 System Calibrations... 37

3.2.0.1 Temperature Calibrations... 37

3.2.1 Resistance Calibrations...38

3.2.2 Surface Cleanliness Verification...39

3.3 Annealing Procedure...40

3.4 Sapphire Substrate Preparation...42

4 Results 45 4.1 Vicinal Si(lll) Surface Characteristics...45

4.1.1 Step-Bunch Morphology...45

4.1.1.1 Regime II (1130°C)...46

4.1.1.2 Regime III (1270°C)...48

4.1.2 Crossing Step Density...50

4.1.3 The Effect of Temperature on terraces... 51

4.1.4 A comparison of the [112] and [112] crystallographic off-cuts... 52

4.2 Step-Bunch Analysis...54

4.2.1 Regime II (1130°C) ...54

4.2.2 Regime III (1270°C)...57

4.3 Ecr the Critical Electric Field...61

4.4 Antibands... 63

4.5 Vicinal a — AI2O3 annealing...67

4.6 Summary Table of Results...71

5 Conclusions 73

A ATLAS 79

B Experimental Analysis 83

C Thesis Key 87

1.1 Vicinal & Bunched Surfaces... 2

1.2

E

- Field Direction with respect to Steps ... 3

2.1 Sapphire or a — ^4^203 structure... 8

2.2 Silicon Structure... 10

2.3 Adatom motion and concentration... 15

2.4 Model Steps ... 16

2.5 Transparent step-bunch adatom concentration... 21

2.6

...27

2.7

... 28

2.8 Antiband formation ...29

3.1 Chamber photograph and sample stage schematic... 36

3.2 AES spectra demonstrating clean Si(lll) surface... 39

4.1 1130°C: F-field effect on Morphology... 47

4.2 1270°C: F-field effect on Morphology... 49

4.3 AFM: Adatoms on Terraces... 51

4.4 Miscut Dependent Surface Reconstruction on Si(lll) ...53

4.5

ym vs. h

in Regime II... 55

4.6 Slope Cross Sections for moderated

E

in Regime II... 56

4.7

ym vs. E

in Regime II... 57

4.8

ym vs. h

in Regime III... 58

4.9 Slope Cross Sections for moderated

E

in Regime III... 59

4.10

ym vs. E

in Regime III ...60

4.12

Influence of inter-step (/) distance on

E^r

...

4.13

Influence of £’-field on steady state crossing steps...

4.14

£’-field

vs.

Antiband onset

{Lm)

...

4.15

.E-fleld

vs.

Antiband onset (L^) for decreased

I...67

4.16

2° a-Al203 143 V/cm

1500 °C...

4.17 J5'-fleld moderation on annealed sapphire...69

4.18

ym vs. h

for Q;-Al203 71 V/cm... 70

A.l

Shallow angle deposition...

A.2

ATLAS

A. 3

ATLAS

B. l

Contaminated Step-Bunches on Si(lll)...

B.2

Analysis of AFM images ...

B.3

Raw maximum slope data set... 86

2.1 Summary of information Si(lll) step-bunching... 32

3.1 Crucible Temperature Calibration ... 38

3.2 Summary of Annealing Conditions... 43

4.1 Summary of Experimental Results... 71

Roman Symbols

A Entropic and stress mediated repulsion strength a Width of an atomic site

b Length of an atomic site -Ds Coefficient of surface diffusion dg Characteristic length

E Electric field Ecr Critical E - field F Force

g Step repulsion coefficient

Gr Adatom concentration gradient across a terrace h Step height

lb Step width within a bunch

Lm Minimum antiband onset terrace width

Imin Minimum step width within a bunch N Number of steps within a bunch

rig Equilibrium adatom concentration ris Adatom concentration

^eff Adatom effective charge

T Temperature

t time

U Interaction energy per unit length

V Step velocity

VcT Adatom drift velocity at E'er

Vdrift Adatom drift velocity y Step-bunch slope

ym Maximum slope within a step-bunch Greek Symbols

P Step kinetic coefficient

P Step stiffness K Rate parameter

As Mean diffusion distance

Step chemical potential

ft Atomic site area on crystal

Acronyms

AES Auger Electron Spectroscopy

AFM Atomic force microscopy

ATLAS Atomic Terrace Low Angle Shadowing

BCF Burton Cabrera Frank theory

MBE Molecular Beam Epitaxy

Introduction

Die Gedanken sind frei

in the fabrication of nano-scale devices that has driven recent investigations of steps on vicinal crystalline surfaces (see appendix page 79). Particularly as bottom-up solutions are sought for surface ordering rather than resource intensive top down fabrication processes. Thus controlling the evolution (or “self-organisation”) of surfaces is advantageous, and this was the objective of this study. To achieve this understanding the intricacies of these surfaces and particularly their steps’ motion is necessary, and has posed a long-standing problem of great scientific interest.

In this thesis I primarily discuss the results of numerous experiments carried out to study the evolution of surfaces off-cut from the silicon (111) face heated to high temperatures in an ultra high vacuum (UHV) environment. When a crystal surface is imperfectly aligned with a low index crystal plane it creates a stepped or vicinal surface as each successive atomic layer or parallel low index plane is intersected, these two configurations are shown in figure 1.1. The process of step-bunching on these surfaces, in which the initial train of equally spaced steps is transformed into large surface-structures consisting of demsely packed steps and wider flat terrace regions, has attracted significant attention [1]. Step-bunching can be induced by a number of factors including surface stress [2], chemical etching [3], deposition [4], the absorp­ tion of a foreign material [5-8] (considered in this study to be tittle more than a nuisance) and electromigration[4, 9]. Electromigration in particular, induced by annealing a material with a

Figure 1.1: Vicinal & Bunched Surfaces

direct current, has long been known to produce macro-scale steps on metal surfaces [10, 11], though the exact nature of these features has yet to be properly examined. This step-bunching process can be used to make the vicinal surfaces more functional as templates [12, 13] and importantly beyond the prospective technological applications, stepnbunches can be used to reveal the underlying physics of atomic step interactions [14, 15]. The characteristics of these step-arrangements can be tuned in size ranging from a few nm up to many pm [16]. For techno­ logical applications, of particular interest are the surfaces semiconducting materials and indeed one of the most fascinating examples of step-bunching is on slightly miscut silicon surfaces. However despite this phenomenon being well documented experimentally and discussed theo­ retically there are many unresolved questions as to its nature and notably experimental progress has lagged behind theoretical models.

The electrically induced step-bunching behaviour on the Si(lll) surface was first observed by Latyshev et al [9]. It was found that step-bunching occurred under the influence of direct current annealing, with the field oriented perpendicular to the step direction. However the most remarkable aspect of the discovery was that the direction of the current required to induce the step-bunching instability switches multiple times with increasing temperature. The schematic in figure 1.2 shows the two basics conhgurations, step-up and step-down electric fields, which are provided by electrodes in direct contact with the vicinal surface. Four separate distinct temper­ ature regimes were demonstrated. They found in Temperature Regimes I (~ 850 — 950°C ) and III (~ 1200 — 1300°C) current orientation must be down relative to the vicinal step direction, while in Regimes II (~ 1040 — I190°C) and IV (> 1300°C) the opposite step-up configuration is required to initiate step-bunch instability growth. It is generally accepted that the basic

Step-Up Current

IIS

E

Step-Down + Current

Figure 1.2:

E

An edge-on schematic showing the electric field configuration with respect to the vicinal steps. The left side of the image shows the current (j5-field) in the up-step direction, the right shows the current

(B-field) in the down-step configuration. The electrodes are in direct contact with the surface supplying

principle responsible for the movement and evolution of the steps is the electromigration of surface adatoms, which is to say that the atoms acquire a small “effective” charge and thus drift parallel to the applied current under the influence of the electric field [17-20]. The effec­ tive charge can generally divided into two parts, an electrostatic force part and a wind force part due to momentum transfer from the carriers. However exactly which of these mechanism is responsible for the movement of adatoms, (i.e. responsible for electromigration), has not been conclusively determined. This is different from classical studies of surface shape evolu­ tion, which is driven by the reduction of the surface free energy without an external driving force [21-24]. Not only does this give rise to step-bunches but depending on the electric field strength, direction [25] and annealing duration electromigration gives rise to features such as step-wandering [26-29], step density waves [30] and anti-bands [26, 31]. With the appropriate electric field at high temperature, as the annealing time is extended, the surface morphology steadily progresses through successive stages; from single atomic steps to step density waves (undulations in step density with double and triple steps), then to step-bunches followed by antibands (twisting of steps crossing wide terraces forming opposite inclined step bands) and finally step-wandering (periodic wandering of parallel steps) [1].

A number of theories have been put forward to explain and describe the step-bunching be­ haviour on Si(lll) [32-35]. A model proposed by Stoyanov was the first to account for the drift of surface adatoms under the influence of an external force, using an adapted form of existing crystal growth theory [14, 36, 37]. While Stoyanov did propose a separate theory to explain a reversed applied electric field [38], models have been unable to fully address the experimentally observed reversals in the bunching current with temperature, though it is suggested that the switching may be due to changes in the “transparency” of the atomic step to crossing atoms [39-42]. The morphology of stei>bunches in all annealing regimes share similar characteristics and their analysis can be used used to unravel the surface dynamics [43-45]. The inter-step distances within the ste]>bunches are determined by a balance between the step repulsion and electromigration’s propensity to induced bunching. It is the inter-step repulsion that prevents the steps coalescing to form macrosteps. Thus a reduction of the electric field should result in an expansion of stej>-bunch width.

that the relationship between field strength for a specific temperature regime cannot be ex­

amined. This thesis demonstrates how for the first time the effects of electromigration in the

dynamics of the step bunching process on the vicinal Si(lll) surface were isolated experimen­

tally. Unlike in conventional experiments, the annealing of Si(lll) was conducted in a specially

engineered setup enabling independent temperature and in-plane electric field tunability by

providing independent control of the radiative and direct current heating. This enabled study

of the step-bunching process in a manner that could not be probed in conventional experiments,

specifically changing electric field while keeping the sample temperature constant. Thus allow­

ing investigation of the effect of the electroinigration force F on the step bunching process in the temperature regimes II (1130 °C) and III (1270 °C). This control, opened the possibility

for a host of theoretically predicted dependencies on E to be tested.

This thesis seeks to experimentally examine the effects of controlling the applied force on

mobile surface atoms by moderating the electric field the surface is subjected to during an­

nealing. This thesis addresses this experimental challenge for the first time and describes the

experimental set-up and procedure devised to accomplish controlled electromigration. This

was done to assess the existing theoretically derived models and their validity was determined

comparison to the predicted scaling dependencies. Specifically the relationships lmin{E) and lmin{Ef) are examined for Regimes II and III, obtained from the maximum slope of the step bunches, where Imin is the minimum inter-step distance within a bunch and N is the number of steps in the bunch. The scaling exponents for these relationships are extracted and compared

to the predictions of theoretical models. Prom this process it was also possible to find the

minimum electric field required to sustain the step-bunching instability and its dependence on

the initial step spacing. Also presented are the step bunch morphologies, obtained on Si(lll) at

fixed temperatures but for different applied electric fields. Other factors, including the effect of

off-cut orientation on surface reconstruction, are also discussed. The effect of a moderated elec­

tromigration force on the evolution of steps on a surface subjected to prolonged annealing was

also examined. The conclusions drawn from these experiments are important in determining

the nature of the surface kinetics, allowing the refinement of theoretical models and answering

questions such as whether step)-bunching on Si(lll) is driven by diffusion or step-kinetic limited

sublimation.

This thesis reports on the the first observation of step-bunching induced by electromigra­

Vicinal Morphology

2.1 Vicinal Surfaces and Crystalline Materials

Before the intricacies of surface evolution can be discussed, familiarisation with fundamentals is imperative. When a crystal surfaee is imperfectly aligned with a high index crystal plane it creates a stepped or vicinal surface as it intersects each successive crystal plane. As cutting a perfectly aligned surface is infeasible, in practice vicinal surfaces always occur. A schematic of an untreated initial surface is shown in Figure 1.1 (a). If the initial step height ho is determined by the thickness of a single atomic layer, the average initial terrace width I can be determined from the simple expression tan0 = ho/lo, where 6 is the degree of miscut. If the surface is subjected to suitable annealing conditions these steps can coalesce to form bunches separated by wide atomically flat terraces, as sketched in Figure 1.1 (b). It should also be noted that annealing generally preserves the degree of miscut.

2.1.1 Sapphire

Sapphire or q — AI2O3 has a hexagonal crystal structure with a rhombohedral primitive cell consisting of 10 atoms. It is an insulator, chemically non-reactive and has a high melting point (~ 2050°C'). It has many technological uses, in particular its use as an insulating substrate for the growth of epitaxial thin films [46-48]. While it is not the primary material investigated in this study, some fascinating new surface behaviour was observed when vicinal substrates were annealed at high temperature in vacuum under the influence of an electric field.

Figure 2.1 (a), is not equivalent to A which is normal to the basal plane. For substrate applications, the most commonly encountered crystallographic planes in a — AI2O3 are the (0001) C-, (1102) R- and (1120) A-planes (Figure 2.1 (a)).

a)

A-plane

R-plane

Figure 2.1: Sapphire or o; —

AI2O3

structure

a) Unit cell of C-plane a — AI2O3, with the other primary planes highlighted b) The atomic structure of Oxygen (Red) and Aluminium (Blue) as viewed the C-plane

C-plane (0001) sapphire, shown in Figure 2.1 (b), substrates are the most widely used and it was chosen for study because it is used as template for the growth of nanowires due to the well-ordered steps which can be formed on its surface. The (0001) surface can terminate at three chemically different surface terminations. It can be either an oxygen surface layer, a surface with two aluminium layers or a surface ending between the two aluminium layers. The last of these has been predicted to have the lowest surface energy from theoretical calculations while the oxygen-terminated surface has the highest surface energy [49]. Several studies have been carried out in order to understand the surface evolution at high temperature.

steps or hill and valley structures have been attributed to faceting of the surface [51]. That is,

the surface is allowed to reach a thermodynamic equilibrium state at high temperature, where

flat low-energy facets alternate.

Investigations have concentrated on simple thermal annealing in air as it is a convenient way

to create a uniform stepped surface. However, this process is ultimately limited by the equilib­

rium structure for a specific temperature, which is reached after a sufficiently long annealing

time. Studies of annealing in UHV produced similar structures and steps [53, 56], however

annealing at higher temperatures (> 1650 °C) produced dendritic structures [57]. It is noted

that a (\/3T x ± 9° reconstruction forms above 1400 °C [53, 58], which can be reversed to the (1x1) structure during cooling to 900 °C. This is significant, as the (\/3T x \/3T)R± 9°

reconstruction, which is metallic-like (or more covalent), is produced above 1400 °C as opposed

to the (1x1) configuration which is insulating [59, 60].

htirthermore there is another significant difference between air and UHV annealed samples.

The A1 surface termination is generally the most stable when annealing in air, while the O

terminated surfaces becomes more favourable only if hydrogen present [61]. Thus the preferen­

tial 2n.c/6 step height (where n in an integer) observed after annealing air [51, 54] cannot be

assumed to be valid for annealing in vacuum. Instead a 3n ± 0.5.c/6 height distribution is ob­

served [53], suggesting the co-existence of adjacent terraces with different chemical terminations

[62-64]. This raises the question of surface stability and step polarity [65]. Under sublimation

conditions, high temperature in vacuum, it is believed that the surface decomposes into O and

AlO gas phases [66], which may be susceptible to a biased drift under the influence of an elec­

tric field. To conclude, while the surface structures and conductivity of sapphire have been

examined at high temperature [67, 68], the effect of an applied electric field on the morphology

and behaviour of the steps has not been examined and warrants study.

2.1.2 Silicon

Crystalline bulk silicon has a diamond cubic crystal structure with a lattice spacing of 5.43lA

and a bond length of 2.352A. A ball and stick schematic of the unit cell is shown in Figure 2.2

(a). Typically the most favoured wafer crystal orientation for industrial uses is (100), however

the (111) is also widely used. The 3-fold symmetry of the bulk terminated Si(lll) shown

Figure 2.2 (b), where the larger circles represent the atoms on the top surface layer with a

single dangling bond pointing up into vacuum, while the atoms of second layer are denoted by

adopt the more stable (7x7) Takayangi type reconstruction above 400 °C [69]. However above

« 850°C, which is required to induce step-bunching on Si(lll), this becomes a disordered (1x1) phase [5, 70-73]. It should also be noted that a variety of reconstructions related the (7x7)

phase have been observed on Si(lll) and that the incidence at which any phase is observed is influenced by the rate at which the surface is cooled through (7x7) to (1x1) transition [74].

b)

tool]

Figure 2.2: Silicon Structure

a) The unit cell of crystalline Si has a lattice spacing of 5.431 A. b) The unreconstructed (111) surface {i.e. 1x1). The atoms on the top layer have a single dangling bond pointing up into the vacuum (larger balls), the smaller balls are the atoms of the second layer, c) Difference between the inequivalent vicinal (111) surfaces cut in the (112) (left) and (112) (right) directions. The unreconstructed (112) off-cut has twice the number of dangling bonds at the step edge (atoms highlighted in red).

The (7x7) reconstruction and its industrial uses for bi-polar devices has made Si(lll) one of the most highly studied surfaces in surface science [13, 15], however the report of curious step-bunching behaviour on slightly off-cut Si(lll) by Latyshev et al. added to its scientific

interest [9]. In this study we use vicinal Si(lll), intentionally cut in either the [112] or the [Ii2]

by 1 - 4°, which gives a step height of ho ~ 3.14A aa the lattice spacing between Si(lll) layers. It should be highlighted that while these two off-cuts are quite similar they are not entirely equivalent [75, 76]. On unreconstructed surface they have an inequivalent number of dangling

While this thesis does not deal directly with atomically resolved features such as reconstruc­

tion, as all surfaces were examined ex-situ where a native oxide builds up on the surfaces once

they are exposed to ambient conditions and Atomic Force Microscopy (AFM) measurements

did not resolve individual atoms, understanding the atomic nature of the surface is vital if

the morphology is to be correctly interpreted and analysed. (It should be noted that despite

oxidation macroscopic features are preserved and can be resolved [77-79].)

2.2 The step-bunching mechanisms on Si(lll)

Beyond their use as mere templates step-bunches are a convenient way of probing the dynamics

of atomic steps on vicinal crystal surfaces. As has been previous alluded to, since the first

observation of step-bunching on vicinal Si(lll) by Latyshev et al. [9], no sufficient explanation

has been found for the behaviour. This is despite the surface being the focus of significant

scientific interest [14, 43, 80, 81].

In Latyshev’s original experiment direct current was passed through the samples perpendic­

ular to the orientation of the steps. This has a twofold effect, it heats the sample and applies

an electric fields (Fl-field). While some studies deal with electric fields parallel to the steps

[29, 75, 82], we will be primarily concerned with electric fields oriented in either in the step-up

or step-down direction relative to the vicinal orientation of the steps. This layout is summarised

in Figure 1.2.

The observed behaviour of Si(lll) can be summarised in a few sentences, as follows. Heating

vicinal Si(lll) with an step-down Fl-field leads to step-bunching in either of two temperatures

intervals. Regime I ( ~ 850 — 950°C) and Regime III ( ~ 1200 — 1300°C). Step-bunching

can also be induced with a step-up E-field with the appropriate temperatures, Regimes II

(~ 1040 — 1190°C) and IV (> 1300°C). The limits of the temperature intervals for each regime

vaxy at most by 50 °C in literature [27]. If the electric field orientation is opposite to those

described in the above regimes, it results in a regular step train with smaller steps and the

surfaces are stable against bunching.

It is important to define the basic the terminology and mechanisms used to describe the

movement of surface atoms under these experimental conditions. This can be achieved using

a simplistic but informative 3D model where each atom is represented in a cubic configuration

with six possible bonding sites. The lowest possible energy state for an atom is when all the

surfaces, so atoms will be linked to either five other atoms if they are incorporated into a

terrace or four if they are part of a continuous atomic step. If sufficient energy is supplied to

an atom its bonds can be broken, this will either leave a vacancy hole in a terrace or produce a

kink along a step. The addition of energy to the system is typically in the form of heat leading

to the diffusion of surface atoms. If a solitary atom on the surface, known as an adatom, does

not desorb from the surface it can move relatively freely by hopping between atomic sites until

such time as it encounters a lower energy site such as a step edge or more preferably a kink

position.

2.2.1 Electromigration

On the surface of a solid the atoms are less tightly bound than they are in the bulk, if the surface

is thermally heated these adatoms can move around by random hopping from site to site, i.e.

surface diffusion. Eloctromigration is a mass transport phenomenon driven by an externally

applied electrical force, i.e. biased diffusion. In a sense electromigration is the transport of a

part of conductor itself. The maximum current density is only limited by the temperature rise

due to Joule heating, which is ultimately below the melting point.

The driving force F acting on the adatoms in the electromigration is expressed as

F = (2.1)

where E is the electric field and is the effective adatom charge [83, 84]. The effective

charge can be written as the sum of two component charge terms, gjf = e{Zei + Z^), where e

is the elementary charge, Z^i considers the force due to the electrostatic interactions between

the atom and the electric field, while Z.u, is the so called “wind” term charge is associated

with the scattering of atoms produced by the electron current. Depending on screening due to

current carriers and on the position of the atom, Z^t lies in a range from zero to the atomic

valence. Z^ is a product of the number of current carriers, their mean free path and the cross

section of the adatom. It is the competition of these two factors that determines the direction

in which the adatoms drift. The value for the effective charge of an adatom on a heterogeneous

surfaces is easily determined from the rate of drift, this contrast is unfortunately unavailable

in homogenous systems such as silicon adatoms on a silicon substrate and thus the effective

2.2.2 Generalised BCF Theory

The first model to explicate the step)-bunching instability specifically on Si(lll), incorporating a

directional drift caused by the applied £'-field, was proposed by Stoyanov [36]. Prior models did

not include or allow for fj-field induced adatom drift. This theory modified the BCF (Burton

Cabrera Prank) [37] theory of crystal growth to account for adatoms with an average drift

velocity of — DsF/ksT [14, 86], i.e a directional drift. Where Dg is the coefficient of

surface diffusion, /cb is the Boltzmann constant and T is temperature. F, the force experienced

by the adatoms, is a result of the adatoms’ acquired effective charge ^effi thus F = QefffJ.

While this model is intrinsically ID, it is however a good approximation for the steps on vicinal

Si(lll) as the step-bunches, barring any major defects, are generally straight and parallel.

It should be noted however that this is not the first model to adapt the BCF theory to

include an anisotropy. The first was an asymmetry in the attachment-detachment processes at

the steps proposed by Schwoebel [87]. However, in contrast to the Ehrlich-Schwoebel effect,

the influence of an electromigration force can be controlled by manipulating the electric field,

accompanied by the electric current flowing through the crystal. It is this that provides the

basis for the experimental studies on the step dynamics.

A qualitative example can help demonstrate how the drift of the adatoms effects the move­

ment of the steps on the surfaces [36]. Beginning with the reasonable premise that steps

have only a limited capability to release and accommodate atoms, a concentration gradient

of adatoms will be created on a terrace due to the biased drift of atoms. This is explained

with the aid of Figure 2.3 (a). The step at site A cannot immediately supply new adatoms to

compensate for those that quickly depart, correspondingly the step at site B cannot instantly

accept all the atoms arriving to it. As result of this higher concentration of adatoms close to B

there is a diffusion flux towards A, thus reducing the net flux from A to B. With wider terraces

the adatom gradient is weaker, thus lessening the back diffusion of adatoms and resulting in

an increase in the net flux. However at high temperatures in vacuum there is significant des­

orption of adatoms from the surface, meaning that the concentration is lower than equilibrium

across the whole step. Figure 2.3 (b). The inclusion of desorption should have a profound effect

on whether or not the surface is stable against stei>bunching. Despite the desorption some

adatoms will cross the terraces to the lower step. This means that the adatoms released at

A will retard the motion of B with respect to A, since it will result in a net reduction of the

adatoms lost by the step. This retardation is stronger the shorter the width of AB, since fewer

of the step spacing, which is verified by experiment [88]. The consequence of this process is that, if terrace between A and B is shorter than the neighbouring terraces (this is a somewhat

realistic scenario as experimentally there will always be steps that are perturbed from their

equilibrium position), it will grow until reaches the average width of the vicinal surface. Under these conditions the step train is stable against stei>bunching but if the force on the adatoms

is reversed the opposite behaviour is expected. Increased adatom concentration in the vicinity of A will retard its motion with respect to B and narrow terraces will shrink further. It should

noted however that while the terraces can be compressed their width cannot become zero.

The surface transport can be described by the equation [36, 39, 42]

d^n,

dx^ ksT dx Ts- — = 0 (2.2)

Where Hs is the adatom concentration and is the average time for desorption of an adatom from a flat surface. There are three primary terms in this equation. The first term on the left considers the movement of adatoms by diffusion within the system, i.e. the diffusion of adatoms away from high to low concentration areas. For the model steps in figure 2.3 it would

be the back diffusion flux of adatoms from B to A and from the adatom sources. The second term considers the biased movement of adatoms induced by the by presence of a force on the adatoms. The final term considers the lifetime of adtoms on the surface before they are lost to

desorption. It should be noted that an additional term can be added to this equation if there is a deposition flux onto the surface [45], however this consideration is beyond the scope of this

study.

The boundary conditions which relate the surface flux and adatom concentration for a

general step i in the position Xi on a surface with positive slope, as shown in Figure 2.4 (a),

are given by the expressions [42]

Pu Usi - n^^{xi-i)

I

~~dx 1^) ~ ah n^{xi-i)Pd risi - n%{xi) ah

for x = Xi-i

for X = Xi

(2.3)

n^ixi)

where nsi{x) denotes the concentration of adatoms on the terrace between the (i-1 )-ih. and the i-th step (i.e., on the terrace between Xi^i and Xi) and n® being the concentration for

crystal-vapour equilibrium. The terms Pu and Pd (or just P) are the step kinetics coefficients^,

b)

n

Figure 2.3:

Adatom motion and concentration

(Reproduced variation of schematic in Ref. [36]) a) Schematic of adatom movement on a vicinal surface. Detaching from a step edge (at A or B), the adatoms favour “diffusion” in the direction of the applied force F. Adatoms may be lost to evaporation before they attach to a step, b) Adatom concentration under substantial desorption of adatoms from the surface, the adatom concentration is lower than equilibrium nj over the whole terrace. The adatoms supplied by at A quickly depart in the direction of the force F, reducing the concentration in the vicinity of the step, however some of these adatoms will successfully cross the terrace, attaching to the next step. Thus slowing the motion of the step at B. However if there terrace is narrower than neigbouring terraces this will result in a sharper adatom concentration gradient, resulting in greater back diffusion towards A from B, thus narrower terraces will contract further under these conditions.

describing the atom exchange between the crystal phase and the 2D gas of adatoms, for the up­

per and and lower terraces respectively. They have the dimensionality of velocity and determine

the potential or power of the step as a stock of atoms during crystallisation, thus describing

the atom exchange between the crystal phase and the 2D gas of adatoms on the terraces. /3

depends on the step structure, interatomic interactions and the temperature. The area ab (or

3)

J

n

s,i-l s.i

i-2 i-1 i+1

Figure 2.4: Model Steps

a) A generalised vicinal surface with a positive slope. The coordinate of the ith step is Xi, which has an adatom concentration function of na,i (x), an adatom flux Ji(x) and a terrace width U = Xi — Xi-i. b) The grey steps show an infinitely large crystal surface with a finite IN number of elementary steps of height h. In black the one-dimensional continuum equivalent centred at the origin. The width of the bunch is Lb, the dashed line the approximated assumed shape of the bunch.

As described above, for this model the equilibrium concentration n® depends on the distances to the neighbouring steps. This has the effect that the steps repel each other. The equilibrium concentration varies according to the expression [39, 42]

Tf,{xi) = <e

^0

^t+1

n+1

^0

— 1 n+1

(2.4)

where n® here is the equilibrium concentration for steps of equal width and Iq is a charac­ teristic length related to the step-step repulsion of the form

f nabA \

'

\Nsf)

repulsion between atomic steps, is related to the interaction energy per unit length of step by

the expression

u

= r

and I is the inter-step distance. The value of n = 2 is generally accepted as a match for

experimental data.

By solving the diffusion equation (2.2) with the boundary conditions (2.3), an expression for

the adatom concentration ni{x) on the terrace between i-1 th and i th steps can be obtained.

From this an expression for the step velocity ij as a function of its neighbouring terraces can

be found by substituting solutions for ni{x) and ni+i{x).

^ ns,i+i{xi) - _ ns,i{xi) - n^(xi

dt nlixi) n-{xi) (2.7)

While this provides some insight into the time evolution of the steps, the question arises

as to how these equations are relevant to an experimental study, particularly as many of these

quantities would be difhcult to measiire directly and moreover what can be newly investigated.

The most accessible quantities for measurement are the dimensions of the steps, height and

width. Numerical integration in the theoretical paper by Stoyanov and Tonchev [42] shows

that ste]>bunching does occur under these conditions for a stepndown electromigration force,

while a step-up force resulted in a regular distribution of steps. Furthermore their simulations

suggested a host of dependencies which could be experimentally tested. Of note in these was the

relationships between the step width within a bunch It and the strength of the applied electro­

migration force F, and Zb’s dependence on the number of steps within the step-bunch N. The

relationship between minimum step width and the number of steps in a bunch has previously

been experimentally measured and has been found to coincide well with the results obtained

from the numerically integrated model. From these numerical models it was determined that

(himi.) oc /v^-o-68 ^ where Imin this the minimum step width within a bunch. However the

dependence of Imin/1 on the electromigration inducing force showed two distinct branches de­

pending on the strength of the applied force for Fl/2kBT > 10“^ => {Imin/l) cx ~ F~3

while for 10-® < {Fl/2kBT) < => (Imin/l) oc F-°-^^ r^F-^.

As the electromigration force is directly dependent on the applied electric field, the re­

lationship between F and the step-step repulsion can be probed using measurements of the

inter-step distance within a step-bunch lb for different applied fields. To extract a relationship

numerical simulation, a continuum model rather than the purely discrete sharp step model is

more applicable. Considering a function z{x, t) that describes the crystal surface shape in the

moment t, for the general surface in Figure 2.4 (a) it can be written

dz ^ dxi/dt

dt Xi - Xi-x (2.8)

where h is the single step height and _Xi—i)—n is the local step density. By substituting

equation 2.7 into 2.8, Stoyanov derived the relationship [42]

1

(2.9) 2 f 18aA\

where R is a constant, numerically determined D — 0.63. While this is in good agreement

with the numerically derived results, it was suggested that using the numerically derived Imin

scaling relationship was more reliable [42].

2.2.3 Transparent Step Model

While the generalised BCF theory comfortably describes the step-bunching behaviour for step-

down electromigration, as for Regimes I and II, it predicts stable surfaces for step-up £'-fields

and thus cannot be used to explain the instability observed in Regimes II and IV. Different at­

tempts have been made to explain the £-field reversals required to maintain the step-bunching

instability with increasing temperature, with varying degrees of success [20, 38, 44, 89, 90]. It

should be noted at this stage that while it may be tempting to attribute the change in the

required electric field orientation to a change in sign of the the effective charge of the adatoms

this idea, while this has been theoretically investigated [20], it has not be experimentally sub­

stantiated [17]. Stoyanov suggests separate models to explain step-bunching for £'-fields in the

step-up and the step-down directions [14, 38]. However the proposal that the instability should

be dependent on net surface sublimation[90] has been refuted [91]. The model above, which is

used to explain the surface behaviour in Regimes I and III, considers the steps as impermeable

or non-transparent, i.e. the kink density is so high that adatoms hopping onto a step must par­

ticipate in the exchange between 2D gas of adatoms and the crystal phase and attach to the step

edge. This simplifies calculations as all interactions are restricted to between nearest-neighbour

steps. Thus the diffusion problem on the crystal surface can be reduced to a diffusion problem

for a generalised single terrace since the surface transport is effectively interr\ipted at each

step by the high rate exchange of atoms with the crystal phase. A separate continuum model

which a number of equivalent scaling relationships are derived between the step-width within

a bunch, bunch height and electric field, the core discussion will be outlined here. Within this

model, unlike the generalised BCF model, the steps are assumed to be (partially) transparent or

permeable, which means there is a direct exchange of adatoms between neighbouring terraces.

This could occur if the step had a sufficiently low density of attachment sites or kinks resulting

in the steps being bypassed by drifting adatoms. In which case the use of boundary conditions

as given in equation (2.3) are not justified. This can be expressed as ^ where /3/6 is

the frequency of atom attachment to a kink and Dgnl is the frequency of atom attachment to a

atomic site at the step edge. In his transparent step model Stoyanov argued that a step-bunch

can be consider a continuum source of adatoms during sublimation with a generating power

proportional to the step density and the local under-saturation. Thus expressing the velocity

of step motion during evaporation in terms of adatom concentration

V = —/3ris -

n.

(2.11) the number of adatoms generated in a unit time per unit length of step is

_

P

n%

The shape of the crystal surface (see Figure 2.4 (b)) is described by the function z{x,t) an

infinitely large crystal surface with a finite number of N steps, each with an individual height

h.

An equation, similar to that of equation 2.2, can be written to describe the concentration

of adatoms on the surface [38]

dus d^Us DgF dus I dz /' P ng — n'. dt * dx^ ksT dx ^ h dx \ fl nf

0

-^

(2.12)

the difference being the additional term to include the product of the density of steps

and the number of adatoms these will provide. It should be noted that this equation was

somewhat simplified, as diffusion on a surface with transparent steps is characterised by an

effective coefficient that is less than the coefficient of diffusion on a terrace, which is used in

this equation. This is because the average time an adatom spends at the more energetically

favourable step edge adsorption site is greater than the time spent in an adsorption site on a

terrace. Stoyanov makes a number of other simplifications and assumptions to find the steady

state of a stepHbunch and calculate the rate at which it travels along the surface. It is assumed

X = —^ and at X = ^ for equation 2.12, while dzjdx = hN/Lhbetween these two intervals. So

that the real shape of the crystal shown in Figure 2.4 b) surface is replaced by a simplified array

of N equidistantly distributed steps between these two points. A steady adatom concentration

field with time is assumed, i.e. dns/dt = 0. The mean diffusion distance = \/D^ is much

greater than width of the bunches L^. When integrating equation 2.12 the desorption term

^ needs to be taken into account only where it dominates, where there are no steps outside

-^ < x < and the change in height {dz/dx) is zero.

The .sohition of the diffusion equation (2.12), for ^ ^ with an adatom concentration

that tends to zero at the limits of x (also noting ns{x) and its first derivative are assumed

to be continuous functions at x = ±Lfc/2), provides an expression for the ratio of relative

adatom saturation at position x. The relative adatom saturation appears as a component in

the expression for the velocity of a step (equation 2.10). The results are split by two separate

and distinct evaporation kinetics situations; near-to- and far-from equilibrium states. For the

former, as the name would suggest, the concentration of adatoms in the middle of the bunch is

close to equilibrium, that is « n®, while for the latter Us For the near-to-equilibrium

expression.

ns(,x) = 1

-2 D^n^Sl Fxs ,.2 1

NX /3 2k bT XgLb for <x< ^ (2.13)

is obtained, which describes the concentration field for a solitary bunch, consisting of N steps

and a simplified inter-step distance of ^ (the far-from-equilibrium situation will be considered

later). This expression is produced by neglecting all but the linear terms of the electromigraton

force. Thus the expression describes the adatom concentration along the surface, at a constant

temperature, in terms the the equilibrium concentration, the step’s potential to produce addi­

tional adatoms, the force and diffusion form its maximum at the centre of the bunch. Without

the presence of an electromigration force this gives a symmetric adatom distribution across the

step bunch. It is noted though that the presence of a positive electromigration field (F > 0)

causes the adatom concentration field to be spread asymmetrically across the the step-bunch,

with an increased concentration at the leading edge of the bunch at i = L(,/2 . This change is

illustrated in Figure 2.5. This increased adatom concentration may be sufficient to compensate

for the Gibbs-Thomson effect on the surface geometry. The concentration of adatoms at Lb/2,

the leading edge of the bunch is then

Figure 2.5: Transparent step-bunch adatom concentration

The concentration ns(x) of adatoms on a crystal surface containing a bunch of steps, situated between X = (a) In the absence of electromigration (F = 0) the adatom concentration is symmetric around the middle of the bunch where it displays a maximum, (b) In the presence of eletromigration (F > 0) the adatom concentration field becomes shifted and the high values of ns(x) gX x < could compensate the Gibbs-Thomson effect.

For step-bunches to be stable the condition > 1 needs to be satisfied, to compensate

for adatom desorption, as below this the step)-bunch will dissipate. The local chemical potential

on a stepr-bunch is dictated by the local curvature, its relationship with the inter-step repulsion,

described by the equation U = A/l^ with n = 2 as before, was used to derive an expression for

the steady state shape of a bunch.

- ^

(snAV-VnWf)

Where It = It should be noted that this very obviously suggests a different scaling

relationship than that for bunches with non-transparent steps (Equation 2.9).

1. The same methodology can be used for the opposite limit of Nli,P/4Dsfln^ 3> 1 for large

bunches, which produces the expression;

(2.16)

Prom this it can be seen that the size scaling relationship in equation 2.15 is only valid for

bunches containing relatively few steps.

A further scaling relationship expression can be derived by using n = 1 in equation 2.6

governing the step-step repulsion energy. Ultimately giving

= (2.17)

As stated previously these scaling relationships were derived considering only the near-to-

equilibrium evaporation kinetics. The far-from-equilibrium equivalent of equation 2.13 is give

by

^ PX,N Fx Lb

ns{x) = nl 1 - (2.18)

XsLb) DaTigCl 2kBT 2As

Stoyanov concludes that the scaling expressions for the average inter-step distance lb becomes

AX,Q\ 5

and

2 fGAXsO.

V kF

(2.19)

(2.20)

for n = 2 and n = 1 in the stepnstep repulsion law U = AfF, respectively.

These theories predict a rather broad range of possible behaviours, however further experi­

mental and theoretical studies have refined the understanding of this surface [43, 78, 80, 91, 92].

2.2.4 Experimentally Derived Scaling Relationships

A general form of the scaling expression for both transparent and non-transparent steps can be

written in the form

' a \ / . oc /v~“ ( —

f'mxn LA i V ^ (2.21)

where a and q are positive scaling exponents. The scaling exponents a and q depend on the

The constant A, as before from equation 2.6 is the strength of the entropic and stress mediated

step repulsion. They can be determined experimentally, by directly measuring the lmin{N) and

lmin{F) dependencies and fitting them to a power law. This is what makes electromigration

induced step-bunching a valuable tool for studying the fundamental mechanisms of adatom

diffusion and the distance dependence of the repulsive interaction between atomic steps.

Previous experimental studies examined the scaling relationship between the height of the

step-bunch {h) and its maximum slope (?/„) [43, 78]. These quantities are directly related to

the number of steps within the bunch {N) and the minimum terrace width (Imin) respectively,

through the expressions h = Nho and Imin = ho/vm where ho is the height of an atomic step.

These prior experimental observations suggested that {Imin /I) OC N 0-68±0.03 {Imin/l) «

^-0.70±0.03 j-Qj. sm-faces annealed with a stejr-down current, which is close the predicted values

determined by numerical and analytical methods. For steps annealed with a step-up current

the scaling relationship (Imin/l) tx was found. Again this was found to be in good

agreement with theoretical predictions for transparent steps with the adatom concentration far

from equilibrium.

While these studies did address some of the scaling relationships and even went so far as to

estimate the effective charge of the adatoms [43], they did not examine the relationship between

the electroniigration force and Imin- This however is due to the limitations imposed by the

conventional experimental method which couples the applied electric field and the temperature.

The experimental sections in the next chapter of this thesis discusses how this problem was

addressed for the first time [30, 92].

It should also be noted at this stage that it was originally theoretically predicted that in

the presence of net deposition a reversal of current direction from step-up to step-down would

be required to induce the stepnbunching instability [38, 39]. While this was in agreement with

earlier experimental studies [90], it was not observed in subsequent experiments [91].

2.3 Critical Electromigration Field on Si(lll)

When applied electric field is reduced below critical value, (Ecr)i the electromigration force is

no longer sufficient to initiate the coarsening step-bunching process, as characterised by the

gradual growth of step-bunches heights and terraces widths with the annealing time. Taken

experimentally we can define Ecr as maximum filed for which the step bunching instability is not

because it is related to the fundamental thermodynamic quantity g{T), which is associated with

the contribution of step-step repulsion to the surface free energy f{p) of vicinal crystal surfaces,

in the classical treatment of steps to minimise surface engergy, given by equation [19, 24, 93]

f{p) = /(O) + kp + gp^ (2.22)

where p is a density of steps. It should also be intuitive, that the presence a repulsive inter-step

interaction would necessitate a stronger critical field to induce the step-bunching process on

surfaces with reduced initial inter-step distance I [88, 94], which is determined by the surfaces

overall miscut angle (0) from a principal index high symmetry surface. Theoretical models

provide different relationships between Ec.r and I, depending on the sublimation conditions on

the surface [14].

The temperature dependent step repulsion coefficient g{T) can be expressed in terms of the

step stiffness P via the equation [45],

2

9{T) =

(nkeTY

24P 1 + t 1 +

4Ap{T)

{kaTf

which also provides its relationship with the related quantity A. For crystal vapour equilibrium

A{T) w g(T) [81]. For this equation g is a function of temperature, however for these studies

only constant temperature settings were considered.

Generally, crystal sublimation involves two groups of processes. The first group includes

surface diffusion accompanied by adatom desorption on terraces separated by atomic steps.

The second group includes the interaction of adatoms with the atomic steps, i.e. attachment to

the step edges, migration along the edges to the kink sites, detachment from the kink positions

and detachment from the steps. Depending on the relative rates of these processes, there

are two distinguishable sublimation states in the Regime III. Firstly, the attachment-limited

state, which is characterised by relatively slow adatom attachment-detachment kinetics and the

dominating factor is the fast diffusion on terraces. Secondly the diffusion limited state, which

has the opposite attributes of relatively slow surface adatom diffusion and fast kinetics at the

steps. Sublimation proceeds differently in these regimes, therefore they should be considered

and analysed independently and results compared with the experiment.

For both sublimation categories, stability of a vicinal surface with respect to the unavoidable

fluctuations in the step density is determined by the sign of the parameter

where q is the wave number of the Fourier mode [19, 95, 96]. This expression is derived using

linear stability analysis. Using convenient dimensionless quantities T]j = ‘-f to describe the step

length where Ij is the width of the jth step, small fluctuations of the uniform terrace distribution

rjj = I + Arij, where Arfj <C 1, can be considered. The time evolution of these changes Ar]j, is

approximately governed by linear equations having a solution in the form Arjj = where

i in this case is the imaginary unit. As understood the vicinal surface will be unstable {i.e.

the fluctuations will grow with the evaporation time) when the real part of the parameter s is

positive. Thus the expression (2.24) approximates the real part of s of this for small values of

the wave number q < 1. Hence as seen, the sign of s is determined by the sign of D2 and the

surface is unstable for B2 > 0, i.e. the fluctuations in the distribution of the inter-step distances

grow with the sublimation time t. While for B2 < 0, the amplitude of the fluctuations decreases,

indicating that the surface is stable. Thus the critical onset of this behaviour is determined by

B2 = 0.

In the case of attachment-detachment limited sublimation (fast diffusion on terraces), the

linear stability analysis gives [14, 80, 96, 97] :

Z?2 =

2ds ^ drift+ UcPTs_ (2.25)

Where dg = Dgj fi is the characteristic length, defined by the adatom surface diffusion coefficient

[Df) and the step kinetic coefficient (/9). The adatom drift velocity as before is given by the

equation = DsqeRE/ksT. Ver = has units of velocity and characterises the

step)-step repulsion. The surface becomes unstable {B2 > 0) when the drift velocity is negative

{i.e. adatom electromigration is in the down-step direction) and its absolute value is larger

than the second term in the square brackets in equation 2.25. Thus, the onset of step-bunching

instability is determined by the condition:

drift ® (2.26)

expanding this and solving for the electric field E results in the equation for critical electri-

cromigration field

Ecr'-This relationship is derived for the limit where dg ^ 1. However, in the opposite limit of

linear stability analysis of these equations gives [14];

Bo =

-;21

^ drift+ (2.28)

where r? = yi+(2|^y

before resulting in:

1. An equation for Ecr can be obtained in the same way as

I2flg

(2.29)

Thus investigating the Ecr{l) dependency in comparison to theoretical predictions would

provide valuable information in determining which sublimation mechanism is responsible for

the development the of step-bunching instability on Si(lll). However prior to the experimental

investigations of this thesis any data on Ecr was unavailable and its dependence on the initial

inter-step distance and sublimation temperature was unknown.

2.4 Antibands

Studies of vicinal Si(lll) are often primarily concerned with the earlier stages of the step-

bunching instability, which is induced by the maximum attainable electric field {dc joule heat­

ing) applied along the miscut direction [43, 78]. Consequently, despite extensive experimental

and theoretical work, few advances have been made in understanding the phenomena that arise

on surfaces subjected to extended annealing. Subjecting the surface to prolonged annealing,

with the dc current driven along the miscut direction, allows the surface morphology to fur­

ther develop, giving rise to new patterns [31]. Specifically, electromigration of adatoms causes

steps crossing the terraces to twist until they acquire a reversed alignment and form bands

with opposite inclination, as compared to the primary step-bunches, close to the terrace edges

(antibands). To help illustrate what this looks like a schematic of this morphology is shown

in Figure 2.6, to represent what is seen on the AFM as shown in Figure 2.7 (a). Two mecha­

nisms responsible for the formation of these antibands were identified in previotis experimental

studies: sublimation spirals and the gradual evolution of the atomic steps crossing the terraces.

Figure 2.7(b) demonstrates neighbouring terraces where the antibands have developed by these

different mechanisms. The dislocations responsible for the sublimation spirals are relatively

rare on the surface and act primarily as a localised source of crossing steps, which eventually

develop into antibands. However it is the evolution of the crossing steps towards antiband

Terrace

Figure 2.6: Schematic of Antibands

Schematic of antibands on a step-bunched surface. Antibands are a bunch of steps inclined in the opposite direction to the of the primary step-bunch bands, which divide the terraces between the primary stei>step bunches.

of antibands has received relatively little attention as compared to other aspects of the devel­

opment of annealed Si(lll) [98, 99]. Although, there have been theoretical studies on the step

bending effects produced by electric currents driven parallel to the atomic steps [29].

As discussed previously, an adatom concentration gradient is created across terrace as a

result of adatom electromigration in the down-step direction and the limited rate of adatom

attachment and detachment at the step edges. In the case of step kinetics limited sublimation,

adatoms arriving at the lower edge cannot be instantly accepted by atomic steps thus the local

adatom concentration is comparatively elevated. The concentration gradient is responsible for

the atomic crossing steps to reced along terraces in an uneven manner, i.e. the step velocity

near the upper step-bunch edge of the terrace is greater t