UNIVERSITY OF CALIFORNIA Santa Barbara. Thermal Oxidation-induced Strain In Silicon Nanobeams

Transcription

1 UNIVERSITY OF CALIFORNIA Santa Barbara Thermal Oxidation-indued Strain In Silion Nanobeams A Dissertation submitted in partial satisfation of the requirements for the degree Dotor of Philosophy in Materials by Adam Mihael Pyzyna Committee in harge: Professor David R. Clarke, Chair Professor Noel C. MaDonald Professor Frank W. Zok Professor Kimberly L. Turner Professor Glenn E. Beltz Marh 2005

2 The dissertation of Adam Mihael Pyzyna is approved. Noel C. MaDonald Frank W. Zok Kimberly L. Turner Glenn E. Beltz David R. Clarke, Chair January 2005

3 Thermal Oxidation-indued Strain In Silion Nanobeams Copyright 2005 by Adam Mihael Pyzyna iii

4 This work is dediated to Sarah Cline, who made onsiderable sarifie, and endured great hardship in hanging the path of her life so that I might finish what I set out to do. Had she asked me to hoose between herself and a dotoral degree, this volume would never have been written. And to Slim, who stayed by Sarah's side, and spent her final days stomping the mean streets of Brooklyn. And to my family, beause, in the words of L.L. Cool J., "You know, taking are of your hildren, taking are of, you know, your responsibility... you have to equip them with the tools, but they've got to limb the mountain themselves." iv

5 VITA OF ADAM MICHAEL PYZYNA January 2005 EDUCATION Bahelor of Siene in Mehanial Engineering, Northwestern University, June 2000 Dotor of Philosophy in Materials, University of California, Santa Barbara, Marh 2005 (expeted) PROFESSIONAL EMPLOYMENT : Co-op Engineering Eduation Program, Argonne National Laboratory : Lab Assistant, Department of Materials Siene and Engineering, Northwestern University PUBLICATIONS A. M. Pyzyna, D. R. Clarke, N. C. MaDonald, Thermal oxidation-indued strain in silion nanobeams, 17th IEEE International Conferene on Miro Eletro Mehanial Systems, MEMS 2004 Tehnial Digest, pp FIELDS OF STUDY Major Fields: Materials Siene and Engineering, Mehanial Engineering Studies in Materials and Miroeletromehanial systems with Professors David R. Clarke and Noel C. MaDonald Studies in Atom-probe field ion mirosopy (APFIM) with Professor David N. Seidman and Dr. Dieter Isheim v

6 ABSTRACT Thermal Oxidation-indued Strain In Silion Nanobeams Adam Mihael Pyzyna The injetion of self-interstitial atoms into bulk silion during thermal oxidation has long been assoiated with atomi sale phenomena, suh as oxidationenhaned diffusion (OED) and the growth of oxidation-indued staking faults (OISF). The urrent work suggests that interstitial injetion may also be impliated in a nanosale mehanial phenomenon. Previous researhers have found that the use sarifiial oxidation to fabriate suspended silion nanowires results in an unexplained aumulation of permanent strain, whih is manifested as bukling of the wires. We make use of suspended single-rystal silion beams of sub-miron thikness to quantify this oxidation-indued strain (OIS) and its dependene on oxidation onditions. Dry oxidation followed by removal of the oxide via ething in HF redues the thikness of the nanobeams, whih are fixed at both ends. As a result of the oxidation, the nanobeams undergo an inrease in length, whih is quantified by measuring the profiles of the bukled beams. The mirosale lateral dimensions of these beams failitate their measurement by optial interferometry, allowing us to take advantage of the innate amplifiation of displaement that aompanies bukling. This allows us to measure very small strains, in the range of mirostrain. A model for the oxidation-indued strain is developed based on the volumetri strain due to injetion of Si atoms, and transition state theory is employed to express strain rate in terms of thermally ativated proesses. The influene of stress on these proesses is also explored. vi

7 Contents List of seleted symbols used... 1 Introdution... 3 Chapter 1 Phenomenology of oxidation-indued strain The appearane of oxidation-indued strain in Si Oxidation-indued strain in Si nanobeams... 6 Chapter 2 Thermal Oxidation of Silion The linear-paraboli model The stress that aompanies oxidation Stress effets on oxidation kinetis Interstitial injetion Chapter 3 Oxidation of nanobeams Biaxial stress and strain in nanobeams Shape evolution at the nanobeam edge Chapter 4 Quantifying the oxidation-indued strain Volume fration of silion onsumed Measuring nanobeam displaement Determination of the oxidation-indued strain Bending equation for a bukled beam The end displaement effet Oxidation-indued strain data Chapter 5 Modeling oxidation-indued strain The interstitial injetion model Simple volumetri strain model Thermally ativated model Thermally ativated model with stress dependene Other mehanisms for oxidation-indued strain Creep Dopant segregation Summary of modeling as ompared with experiment Chapter 6 Methods for reduing oxidation-indued strain Inremental oxidation and ething Oxidation in the presene of hlorine Chapter 7 Conluding remarks and reommendations Disussion Summary and onlusions Future Work Appendix A Fabriation of Si nanobeams Appendix B The silion self-interstitial B.1 Struture B.2 Formation energy vii

8 B.3 Self-interstitial diffusion B.4 Misellaneous kinetis information Referenes viii

9 List of seleted symbols used (in order of appearane) σ g ε ε i g o h Si h h h Si ox growth stress in the oxide on a wafer during oxidation (ompressive) intrinsi strain in the oxide, determined by molar volume mismath growth strain in the oxide, what remains of ε i after visous flow (negative) initial thikness of the Si thikness of the Si after oxidation thikness of the oxide thikness of Si onsumed during oxidation f volume fration of Si onsumed during oxidation V onsumed volume of Si V o initial volume of Si ε oxidation-indued strain OI Si ε elast elasti strain in Si ox ε elast elasti strain in oxide ε oxide strain due to visous flow R vf Si = E E ratio of biaxial moduli of oxide and Si bi ox bi Si σ elasti stress in Si nanobeam during oxidation σ ox elasti stress in oxide during oxidation T oxidation temperature l length of a nanobeam before oxidation-indued strain l l o 1 ε 2 o Ω V i V θ n i n ν ν ν i o ν U D length of a nanobeam after oxidation-indued strain if ends were free length of a nanobeam after oxidation-indued strain when bukled initial strain present in Si over-layer of SIMOX wafer atomi volume of Si volume inrease due to injeted atoms volume of Si remaining after oxidation interstitial injetion ratio: the number ratio of Si atoms injeted to onsumed net number of Si atoms injeted during oxidation number of Si atoms onsumed during oxidation rate per unit area at whih Si atoms are injeted rate per unit area at whih Si atoms are onsumed to form oxide attempt frequeny per unit area for mehanism of interstitial aumulation attempt frequeny per unit area at Si/SiO 2 interfae using Debye frequeny ativation energy for mehanism of interstitial aumulation 1

10 U o V a A n Q ativation energy for mehanism of interstitial aumulation in absene of applied stress ativation volume for mehanism of interstitial aumulation pre-exponential onstant for power-law reep exponential onstant for power-law reep ativation energy for power-law reep α fration of σ Si that is effetive at driving power-law reep 2

11 Introdution In reent years, there has been onsiderable interest in the use of thermal oxidation to redue the dimensions of silion nanostrutures. Thermal oxidation an be used to make pre-defined silion features smaller by onsuming silion at the oxidation front, sine the formation of eah SiO 2 moleule requires one Si atom. For example, oxidation has been used to reate single eletron transistors from silion wires of a smaller diameter than is ahievable by nanolithography tehniques alone [1-3]. Furthermore, feature size an be reliably ontrolled through the use of selflimiting oxidation, whih is observed during the oxidation of ylindrial silion strutures. By taking advantage of this phenomenon, nanowires have been fabriated with diameters as low as 5 nm [4, 5]. The thermally grown oxide an also be used in a sarifiial manner, sine it is easily ethed away with hydrofluori aid, whih is highly seletive to SiO 2 over Si. In this way, thermal oxidation has been used to fabriate mehanial nanostrutures suh as suspended silion filaments [6]. Strutures of this type, whih have been used as speimens for mehanial testing [7], may also be useful for sensor appliations beause of their high surfae-to-volume ratio. Furthermore, silion is an attrative material for sensors beause of its ompatibility with existing IC proesses [8]. The thermal oxidation of silion is an extremely useful, but ompliated proess and long known (yet not fully understood) phenomena are playing an inreasing role in devie fabriation. For instane, the injetion of self-interstitial silion atoms from the oxidation front is problemati for eletrial devies, sine it leads to the formation of extrinsi staking faults that are responsible for leakage urrents. As suggested by the work ontained in this dissertation, though, interstitial injetion may also be impliated in a mehanial phenomenon. As the dimensions of silion strutures that are subjet to oxidation approah the nanometer sale, unantiipated results an our, suh as an unexplained soure of permanent strain without evidene of plasti flow. This oxidation-indued strain (OIS) leads to the bukling of suspended nanowires fabriated from bulk single-rystal silion (SCS) by means of sarifiial oxidation, as desribed above [9]. In this dissertation, we present evidene that thermal oxidation an ause a permanent aumulation of strain in SCS nanobeams as well as nanowires. The purpose of our study is to first use nanobeams to quantify oxidation-indued strain. These suspended silion beams, whih bukle due to the aretion of axial strain, are easily measured by optial interferometry. We then propose a model for OIS based on the injetion of self-interstitials, and finally investigate methods of reduing its effets. 3

12 Chapter 1 Phenomenology of oxidation-indued strain Thermal oxidation-indued strain (OIS) in silion was first observed by previous researhers who made use of sarifiial oxidation to reate suspended nanowires [9]. In the first setion of this hapter, we desribe the fabriation of these nanowires and the resulting strain as revealed by their bukling. In the following setion, we desribe our approah to reproduing and quantifying OIS using SCS nanobeams fabriated from silion-on-insulator material. 1.1 The appearane of oxidation-indued strain in Si Single-rystal silion nanowires were previously fabriated from bulk silion wafers by means of sarifiial oxidation [9]. As shown in Fig. 1.1, ommon miromahining proesses inluding thermal oxidation, photolithography, reative ion ething (RIE), plasma-enhaned hemial vapor deposition (PECVD), and KOH wet ething were first used to define a monolithi array of mirosale suspended beams. These beams were then thermally oxidized, thereby onsuming a onsiderable portion of the Si to form SiO 2. After removal of the oxide with buffered hydrofluori aid (BHF), the remaining silion wires ranged from 200 to 20 nm in diameter. Sine the wires were reated by sarifiial oxidation, the oxidation-indued strain was imparted during their fabriation. Figure 1.1 Shemati illustration, 1999 by B. W. Reed [9], of the fabriation of suspended silion nanowires from bulk single-rystal silion, as viewed parallel to the nanowire axis. In steps (a) through (f), suspended beams are reated using thermal oxidation, photolithography, RIE, PECVD and KOH ething. In step (g), the beams are thermally oxidized. When the width of the lower portion of the beam is suffiiently small relative to the upper portion, only a narrow silion ore remains. In step (h), the oxide is removed with BHF. 4

13 While the ross-setional dimensions of the beams were greatly redued by the proess desribed above, their length was disovered to have inreased. As shown in Fig. 1.2, this axial strain was revealed by the bukling of the nanowires, as observed by sanning eletron mirosopy (SEM). Sine both ends the wires were rigidly fixed, they ould only elongate by out-of-plane deformation. Figure 1.2 SEM image, 1999 by B. W. Reed [9], of bukled nanowires viewed at low angle to the surfae of the wafer. The nanowires span trenhes ethed deep into the silion wafer, whih are separated by vertial walls. The vertial lines are ridges on the wall that result from the oxidation step shown in Fig. 1g. 5

14 1.2 Oxidation-indued strain in Si nanobeams Although the bukling of nanowires is easily observed using SEM, it would be diffiult or impossible and extremely time onsuming to measure strain from images like that shown in Fig 1.2 with suffiient auray. Sine quantifying oxidation-indued strain is the first step to understanding it, though, we require a struture that an be measured easily, aurately and quikly. In this setion, we desribe the use of silion nanobeams to fulfill the requirements just mentioned. The fabriation of these strutures is made possible through the use of silion-on-insulator (SOI) wafers manufatured by means of separation by implantation of oxygen (SIMOX). The fabriation of SIMOX wafers onsists of two basi steps and results in the layered struture shown in Fig First, a silion wafer is implanted with a dose of high-energy oxygen ions, whih penetrate to a harateristi depth where they reat to form a buried layer of oxide preipitates. Then the wafer is annealed at high temperature (typially 4 to 6 hours at 1350 o C) in an argon rih environment. This auses the preipitates to oalese into a dense and uniform buried oxide (BOX) layer. The anneal step also heals damage that the Si suffered during implantation, thereby produing an eletroni-devie-grade, single-rystal over-layer of uniform thikness. The thiknesses of the silion over-layer and BOX layer are ontrolled by the implantation onditions. The thikness of the BOX layer depends on the implantation dose, whih typially ranges from O m. For example, a dose of O m produes a BOX layer that is about 400 nm thik. The thikness of the Si over-layer depends on the implantation energy as well as the dose. An implantation energy of 180 kev typially produes an over-layer ranging from 250 to 350 nanometers thik. The SIMOX wafers we use have a 217 nm thik over-layer and a 385 nm thik BOX layer. They are (100) oriented and lightly doped with boron to a resistivity of 10 to 20 Ω m. Figure 1.3 Cross-setional SEM image of a leaved SIMOX wafer beside a shemati illustration. 6

15 As shown shematially in Fig. 1.4, the SIMOX material lends itself quite naturally to the fabriation of nanobeams. The initial thikness of the nanobeams is determined by the thikness of the silion over-layer of the SIMOX wafer. The lateral dimensions, whih range up to 10 µ m wide and 200 µ m long, are ontrolled by photolithography and pattern transfer via deposition and ething steps similar to those used in the fabriation of nanowires, as desribed in Setion 1.1. A detailed proess flow and proessing onditions are provided in Appendix A. Figure 1.4 The Si over-layer of a SIMOX wafer is utilized to fabriate suspended nanobeams that span a trenh ethed into the silion substrate. Left: Shemati illustration with greatly exaggerated proportions. Right: SEM image of 6 µm wide by 49 µm long beams. Note: Although not depited in the shemati, the walls of the trenh are angled due to the anisotropi nature of the KOH release eth (see Appendix A). As with nanowires, the aretion of OIS an be demonstrated by the bukling of SCS nanobeams, whih have nanosale thikness. The larger lateral dimensions, however, enable us to take advantage of the innate amplifiation of displaement that aompanies bukling. Beause the oxidation-indued strain is so small, the axial elongation of a nanobeam is far too small to detet using mirosopy. However, sine the ends of the beam are kept fixed, the elongation is aommodated by bukling out of plane with amplitude muh larger than the inrease in length. In the ase of our nanobeams, an elongation on the order of 10 nm will produe a vertial displaement on the order of 1 µ m at the enter of the beam. Sine the lateral dimensions of the beams are large enough to resolve with optial mirosopy, this vertial displaement an be measured easily, aurately and quikly by optial interferometry. An example of a measurement obtained using a Wyko NT1100 Optial Profiling System is shown in Fig This tool is desribed later in Setion 4.2 7

16 Figure 1.5 A surfae profile of two beams aquired using an optial interferometry-based system. Beams akin to these measured 198 µm long by 11 µm wide by 58 nm thik and typially bukled to an amplitude of about 1.8 µm. The vertial sale is exaggerated. 8

17 The advantages of the nanobeam geometry made possible through the use of SIMOX material are summarized as follows: 1) The nanosale thikness allows us to onsume a signifiant fration of the beam s thikness in a reasonable oxidation time, thereby imparting measurable OIS. 2) The thikness of the silion over-layer is highly uniform, ensuring onsistent results on the wafer sale. 3) The high length-to-thikness ratio ensures that the beams will bukle easily, sine the ritial load for bukling is given by σ r 2 π h = E 3 l o 2 (1.1) where E is the elasti modulus, h is the beam thikness and l o is the length of the beam. 4) The high width-to-thikness ratio allows us equate the volume fration of silion onsumed during oxidation to the fration of thikness onsumed. This will be useful later when we quantify OIS in terms of the fration onsumed. 5) The mirosale length and width allow us use optial interferometry to measure the profiles of the bukled beams with high vertial resolution. 9

18 Chapter 2 Thermal Oxidation of Silion Silion has a high affinity for oxygen, and an amorphous native oxide film rapidly forms on Si upon exposure to an oxidizing ambient. Thermal oxidation may be performed with dry oxygen as the oxidant, whih we will refer to as dry oxidation, or water vapor, whih will be referred to as wet oxidation. Dry oxidation was used in our study aording to the following reation: Si + O 2 SiO2 (2.1) The growth of an oxide film on a silion wafer is illustrated in Fig Sine the molar volume of SiO 2 is larger than that of silion, oxidation of a wafer produes an oxide film of thikness hox 2.27h where h is the silion thikness onsumed to form the oxide [10]. This fator is in agreement with the molar volume ratio of SiO 2 to Si [11]. Figure 2.1 Oxide growth on a silion wafer. The Si-SiO 2 interfae moves into the silion as it is onsumed at the interfae to form oxide. Beause the volume oupied by a SiO 2 moleule is larger than that of a Si atom, the surfae of the oxide must move in the opposite diretion. 10

19 Figure 2.2 Some of the mehanisms present during the thermal oxidation of silion are illustrated. A) The oxidant is transported from the oxidizing atmosphere, reating a surfae onentration in aordane with Henry s law. B) The oxidant diffuses through the oxide. C) The oxidant reats with Si at the Si/SiO2 interfae. Due to a shortage of free volume for the reation, Si atoms are emitted from the interfae region. These atoms may D) be injeted into the silion lattie as self-interstitials or E) diffuse into the oxide. The transition from Si to SiO 2 is not abrupt, and the oxidation reation most likely ours over a reation zone, as indiated by the dotted line. The first setion of this hapter is dediated to the model proposed by Deal and Grove in 1965, whih has beome the standard model for oxidation of silion [12]. As will be desribed, Deal and Grove [12] arrived at their linear-paraboli model by onsidering the kinetis of oxidant transport and reation with silion at the Si/SiO 2 interfae. These kinetis are still highly relevant, but, as shown in Fig. 2.2, the importane of other mehanisms has also ome to light. We disuss some of these additional aspets of thermal oxidation in the setions following 2.1, sine they will be important in developing our model for oxidation-indued strain. 2.1 The linear-paraboli model The Deal-Grove model [12] is based on the assumption that oxidation progresses by the inward transport of the oxidant from the ambient to the Si/SiO 2 interfae rather than the outward transport of silion. In fat, studies using sequential oxidation in different oxygen isotopes show that the oxidation reation takes plae within 3 nm of the interfae. The flux of the oxidant is onsidered to proeed through three separate stages. First, the oxidizing speies are transported from the oxidizing ambient to the surfae of the existing oxide with flux F 1. Next, the oxidant diffuses through the existing oxide toward the Si-SiO 2 interfae with flux F 2. Finally, upon 11

20 arriving at the interfae, the oxidant reats with the silion to form silion dioxide aording to Eq. (2.1). The rate at whih the oxidant is onsumed by the reation at the interfae is expressed as flux F 3. For steady-state onditions, the total flux is F = F1 = F2 = F3. The flux for the first stage is assumed to be proportional to the differene between the oxidant onentration in the bulk gas and the oxidant onentration at the surfae of the oxide. In the absene of dissoiation, the equilibrium bulk * onentration of the oxidant in the oxide, C, is given by Henry s law. Thus, if the speies diffusing through the oxide is moleular oxygen, we have * ( ) F = hc C (2.2) 1 o where h is the gas-phase mass transfer oeffiient and C o is the onentration of the oxidant at the surfae of the oxide. Fik s law gives the flux of the oxidant aross the existing oxide. For steady-state onditions, F 2 is onstant and the onentration gradient must therefore be linear: dc C F D D C o i 2 = = (2.3) dx hox D is the diffusion oeffiient and C i is the onentration of the oxidant in the oxide near the Si-SiO 2 interfae. The flux orresponding to the oxidation reation is related to C through the reation rate onstant, k: i F = kc (2.4) 3 i For steady-state onditions, F = F and we have 2 3 C i and C o an be eliminated by setting F1 F2 = and * kc F = F1 = F2 = F = (2.5) k/ h + kh / D ox The rate of oxide growth an be determined by dividing the flux by the number of oxidant moleules inorporated into a unit volume of the oxide: * dhox kc = dt N k h kh D ( 1 + / + / ) ox (2.6) 12

21 Before integrating Eq. (2.6), it is useful to speify an initial film thikness, h i, assumed to exist at t = 0. Not only does this allow onsideration of multiple oxidation steps, but it an be used to aount for oxide growth before the assumptions of the model are valid. (Indeed, as mentioned above, the Deal-Grove model does not hold for the initial stages of growth. Instead, for oxidation in dry O 2, an initial rate enhanement is observed for films thinner than approximately 30 nm.) The solution to Eq. (2.6) then is h + Ah = Bt+ h + Ah (2.7) 2 2 ox ox i i If the initial thikness is aounted for by a shift, τ, in the time oordinate, we have where 2 ox ox ( ) h + Ah = B t+ τ (2.8) 1 1 A 2D = + k h (2.9) * 2DC B = (2.10) N τ h + Ah B 2 = i i (2.11) Eq. (2.8) an be solved for the oxide thikness as a funtion of time, giving hox t +τ /2 2 /4 = 1+ 1 (2.12) A A B The origin of the linear and paraboli omponents of Deal-Grove model are made apparent by onsidering the 2 limiting forms of Eq. (2.12). For relatively long oxidation times, suh that t A 2 /4B and t τ, Eq. (2.12) is mostly paraboli: 2 hox Bt (2.13) In this ase, the oxide is relatively thik and growth is therefore diffusion-ontrolled. The parameter B is referred to as the paraboli rate onstant. For very short oxidation times, suh that t A 2 /4B, Eq. (2.12) is mostly linear: 13

22 h ox B ( t+τ ) (2.14) A In this ase, the oxidant diffuses rapidly though the relatively thin oxide and growth is reation-ontrolled. The parameter BA is referred to as the linear rate onstant. 2.2 The stress that aompanies oxidation In general, a ompressive stress arises in a growing oxide film when its molar volume exeeds that of the material being oxidized [13]. As mentioned above, a large inrease in molar volume aompanies the onversion of Si to SiO 2. While the 3 3 volume of a SiO 2 moleule is 45 Å, the volume of a silion atom is only 20 Å. Sine expansion is restrited by the underlying bulk Si, the need for free volume leads to a large ompressive stress in a SiO 2 film grown on a Si wafer. This has been observed in situ via wafer urvature measurements by EerNisse [14] during wet oxidation and later by Kobeda, et al. [15] during dry oxidation. The origin of this stress, whih will be referred to as the growth stress, σ g, an be thought of in terms of an intrinsi strain, ε i, that aompanies the oxidation reation. Consider an unonstrained Si ube that is onverted to SiO 2. Sine the molar volume of SiO 2 is 127 % larger than that of Si, the length of eah side of the ube would inrease by approximately ε i = 31 %. However, a ube of material on the surfae of a silion wafer will not be allowed to hange its dimensions so freely. Beause the thikness of the wafer is muh greater than that of the oxide film, the silion wafer will remain virtually free of strain during oxidation. As a result, the growing oxide is laterally onstrained by the wafer and the 31 % length inrease is not allowed for diretions in the plane of the wafer. This onstraint would lead to an enormous in-plane stress if elasti ompression was required to limit the molar volume inrease. However, at elevated temperatures, silia and siliate glasses are not purely elasti solids and an deform in a visoelasti manner [16]. Consequently, the intrinsi strain is partly aommodated by visous deformation, allowing the volume inrease to be manifested hiefly as an inrease in thikness. The remaining elasti strain, whih will be referred to as the growth strain, ε, gives rise to the observed growth stress. The magnitude of the growth stress will therefore depend on the visous properties of the oxide and the time sale of the oxidation, as well as the thikness of the silion. Beause these fators depend on the oxidation temperature, reported values of σ also display a temperature dependene. Sine the visosity of g glasses dereases with inreasing temperature, the growth stress dereases as well. Referenes to a so-alled visous flow point abound in the literature, based on the observation by EerNisse that the oxide appears unable to support stresses above approximately 960 o C [17]. In EerNisse s experiments, wafers were oxidized in both wet and dry ambients. After ooling to room temperature, the wafers were g 14

23 urved due to thermal expansion mismath stress in addition to the above mentioned growth stress. They were then reheated while their urvature was monitored using refleted laser beams. The urvature was found to derease with temperature until the wafers beame flat at approximately 960 o C. Sine the oxides were grown at temperatures well above 960 o C, it an be inferred that visous flow ours above this temperature even if thermal expansion mismath was the only soure of stress. However, beause the visous relaxation of stress is time dependent and visosity a smooth funtion of temperature, we should not assume that this means there is no flow below 960 o C or that oxide grows free of stress above this temperature. In a later study, EerNisse reported large growth stresses measured in situ at 950 o C and below, but no stress at 975 o C and above [14]. However, these measurements were for wet grown oxide only. Wet grown oxide is known to have lower visosity than dry beause it is more porous and the silia network is weakened by the formation of non-bridging hydroxyl groups. For example, the bulk visosity of silia at 800 o C is larger for dry oxidation than wet oxidation by a fator of about 1000 [18]. It is therefore not surprising that other researhers report sizable growth stresses for films grown in dry oxygen at temperatures as high as 1150 o C. In Fig. 2.3, we show growth stress data from the literature that is obtained from a variety of wafer urvature measurements at room temperature [19-21]. The thermal expansion mismath omponent is subtrated from the measured stress and the remainder is reported as the growth stress. The validity of this method is demonstrated by omparison with in situ stress measurements during oxidation [15]. The empirial fit to the data shown in Fig. 2.3 is used to determine the values in Table 2.1, whih will be used later in analyzing our nanobeam data. 15

24 Figure 2.3 The reported growth stress as a funtion of dry oxidation temperature is approximated by an empirial fit to the data found in Ref. [19-21]. The stress does not display a thikness dependene for films thiker than approximately 50 nm [20, 22]. The stress varies within 50 nm of the interfae [22], though, and wafer urvature measurements therefore orrespond to the average stress through the thikness of the oxide. Oxidation Temperature ( o C) Growth Stress (MPa) Table 2.1 Growth stress determined from empirial fit shown in Fig The measurements made by Kobeda and Irene [15, 22] reveal a stress distribution through the thikness of the growing oxide. For thik oxides, the stress is virtually uniform exept for within a region extending approximately 50 nm from the interfae. Here, the stress begins to rise, approahing a ommon extrapolated value of about 450 MPa at the interfae, regardless of oxidation temperature in the range from 700 to 1000 o C. This observation is orroborated by infrared absorbane measurements, whih extrapolate to 460 MPa at the interfae [23]. The observation of a single ommon value for the maximum interfae stress implies that it results diretly from the onstant molar volume hange. 16

25 The shortage of free volume at the Si/SiO 2 interfae and the resulting growth stress have important reperussions for oxidation in general and the oxidation of nanobeams in partiular. In the following setion, we ll explain how stress auses deviations from the oxidation kinetis desribed in Setion 2.1. This will later be applied to the oxidation of nanobeams in Chapter 3. The shortage of free volume is also thought to be an important fator in the injetion of self-interstitials, whih we address in Setion Stress effets on oxidation kinetis In steady-state, the kinetis of silion oxidation desribed by the linearparaboli model in Setion 2.1 are determined by three physial parameters: the * equilibrium onentration of oxidant in the oxide, C, the diffusivity of the oxidant in the oxide, D, and the oxidation reation rate oeffiient, k. The values of these parameters, and therefore the oxidation rate, are expeted be influened by the stress state of the oxidizing system. This has been proven true by various observations, suh as enhaned growth rate subsequent to relaxation of the growth stress by high temperature annealing [24]. The relative ontribution from hange in eah parameter is still debated, though. For example, there is disagreement in the literature as to whether oxidation is primarily affeted by stress influene on the transport of the oxidant or the interfae reation. Also, while isotope oxidation experiments have revealed that stress has a onsiderable effet on transport, it is still not lear whether this is dominated by hange in solubility or diffusivity of the oxidant [24]. It may be that the stress dependene of eah of the parameters mentioned above plays a signifiant role. The influene of growth stress on oxidation kinetis was first made apparent by oxidation of two-dimensional strutures. Retarded oxide growth at both onvex and onave orners was first observed by Marus and Sheng [25] and more systematially explored by Kao et al. [26, 27]. In subsequent modeling of twodimensional oxidation, researhers have pointed to the effet of stress on C, D, k, * and the oxide visosity, η [27, 28]. As disussed in greater detail in Setion 3.2, Kao et al. [26, 27] were able to * explain their observations by onsidering the stress dependene of C, D, k, and η in addition to geometri effets. Aording to their model, oxidation of twodimensional strutures gives rise to stress normal to the Si-SiO 2 interfae, whih affets the reation rate by making the oxidation reation less energetially favorable. It follows then that the reation rate onstant will have an Arrhenius dependene with a stress-dependent barrier: σ rrv k k = ko exp kt B (2.15) 17

26 where k o is the stress-free rate onstant, σ rr is the stress normal to the interfae and V k is an ativation volume. The solubility and diffusivity are dereased in a similar manner, sine a positive pressure will redue the volume of the voids in the silia network that are oupied by the diffusing oxidant speies. Likewise, it is expeted that negative pressure will enhane solubility and diffusion. C PV = C * * C o exp kt B (2.16) PV D D= Do exp kt B (2.17) where * C o and the pressure and V C and D D o are the stress-free equilibrium onentration and diffusivity, P is V are ativation volumes. A pressure dependene was also proposed for visosity based on an empirial relationship that desribed the behavior of ertain SiO 2 glasses under pressure. As pointed our later by Sutardja and Oldham [28], though, the shear stress is more important in determining visosity. The following dependene was derived from the theory of rate proesses for fluids under high stress. τvη 2kT B η = ηo (2.18) sinh 2 ( τvη kt B ) where η o is the stress-free visosity, τ is the shear stress and V η is an ativation volume. The two-dimensional model by Kao et al. [26, 27] was quantitatively verified by one-dimensional numerial simulations that inorporated stress dependent values * for C, k, and η, but rigorous two-dimensional simulations showed that it ould not aount for the full range of experimental data. The model was later improved by Sutardja and Oldham [28], who inluded the stress dependene of diffusivity rather than solubility and replaed the pressure dependene of visosity with the shear stress dependene of Eq. (2.18). The effets of stress in the oxide are also important in planar oxidation, and have been used to explain observed deviations from the linear-paraboli kinetis. For example, non-arrhenius behavior has been reported for both the paraboli and linear rate onstants. The ativation energy for the paraboli onstant has been reported to rise from 0.9 ev at 1150 o C to 2.6 ev at 780 o C. This is as we would expet, sine the paraboli rate onstant haraterizes transport of the oxidant and the larger growth stress present at low temperatures would derease transport by dereasing solubility, diffusivity or both. 18

27 Stress/strain in the silion an also have an effet on oxidation rate through the interfae reation. Although many studies agree that the appliation of ompressive stress retards oxidation, there is disagreement on the effet of tensile stress. While some studies indiate that tensile stress enhanes the oxidation rate [29-31], others suggest the opposite [32]. 2.3 Interstitial injetion It is urrently well established that during thermal oxidation of silion, a small proportion of silion atoms are injeted into the silion lattie as interstitials. This idea was first suggested by Dobson [33, 34] and later emphasized by Hu [35] as an explanation for the simultaneous ourrene of two phenomena. The first is the formation of oxidation-indued staking faults (OISF). They are extrinsi in nature, onsisting of additional (111) planes of atoms, and they grow by absorbing interstitials. This requires either a loal supersaturation of interstitials or an undersaturation of vaanies. The seond phenomenon involves the diffusion of dopant atoms. Oxidation-enhaned diffusion (OED) is observed for dopants that diffuse predominantly by an interstitialy mehanism. It is also observed that oxidation-retarded diffusion (ORD) ours for Sb, whih diffuses predominantly by a vaany mehanism. Therefore, supersaturation of interstitials and undersaturation of vaanies must our simultaneously. Hu [35] argued that the oxidation proess is inomplete to a small degree, suggesting that perhaps 0.1% of the silion is not oxidized, but beomes free atoms that enter the lattie as interstitials. Several researhers have suggested that this is a result of the lak of free volume available to aommodate the large molar volume inrease upon onverting Si to SiO 2 [36-41]. Although muh of the neessary free volume is supplied by visous flow of the oxide, additional volume would be supplied by the flow of Si atoms away from the interfae. It is likely that most of these atoms diffuse into the oxide where they reat with the inoming oxidant and only a small fration enters the Si lattie. In fat, the presene of exess Si in the oxide near the interfae is established by muh experimental evidene of a transitional sub-oxide layer with stoihiometry near SiO. Various tehniques inluding photoeletron spetrosopy and pulsed laser atom probe mirosopy indiate that this layer is about 1.5 to 2 nm thik [42] and the reativity of this layer is demonstrated by studies using sequential oxidation in different oxygen isotopes. Tiller [41] addresses the shortage of free volume by proposing that the proess of silion oxidation involves the formation of an intermediate layer of a-ristobalite between the silion and the bulk amorphous oxide film. This transitional region between silion and vitreous SiO 2 is only a few moleular distanes wide. Tiller [41] points out that a-ristobalite an be formed at the (100) Si surfae with relatively low lattie mismath if the 4 atoms at the ¼ and ¾ positions of the Si unit ell are removed. These atoms are then assumed to leave the ell as interstitial defets. Sine the enthalpy of formation of an interstitial defet in Si is expeted to be muh higher 19

28 than that in SiO 2, the migration of an interstitial into the bulk Si requires an ativation step with an energy barrier equal to the differene in enthalpies. Beause the defet onentrations are not initially at their equilibrium values, there is also a differene in hemial potential of the defets on either side of the interfae. Tiller [41] takes this into aount when deriving a segregation oeffiient as well as an Arrhenius expression for the flux of interstitials into the silion. Lin, et al. [40], propose an alternative two-step oxidation reation followed by a segregation of exess silion at the interfae. Aording to his model, the free volume onstraint auses only about half of the lattie silion at the interfae to be immediately oxidized. As a result, a onentration of exess silion at the interfae builds rapidly. Most of these atoms flow into the bulk oxide, and quikly reat with inoming oxygen, but some esape into the silion bulk as interstitials. Tan and Gosele [37] question the approah of Lin et al. [40], asserting that the flow of Si atoms into the oxide would not supply the neessary free volume beause they still must reat to form SiO 2. They therefore onluded that that the free volume required for oxide growth is essentially supplied by visoelasti flow. However, they still point to the growth stress as the driving fore for injetion. They reason that the generation of interstitials is a onsequene of the oxide s inability to relax the growth stress instantaneously and examine the problem in terms of visous flow. Dunham and Plummer [38] aknowledge that the oxidation of interstitials in the oxide requires free volume, but points out that this expansion an be aommodated by visous flow beause the reatants are not bonded to the silion lattie. This implies that although the free volume is ultimately supplied by visous flow, there may be a signifiant redution in strain energy at the interfae if visous flow is failitated by the emission of Si into the oxide. Like Tiller [41] and Lin et al. [40], Dunham and Plummer envisage a onentration of interstitials in the oxide at the interfae that is determined by a balane between the fration of the interfae reations resulting in the reation of interstitials and the rate at whih they diffuse into the oxide. They plae an upper bound for this fration at 0.2, as determined by the exess volume reated by oxidation. The onentration of interstitials on the silion side of the interfae would then determined by some undetermined segregation oeffiient. Dunham supports his model by referring to evidene from OED experiments that the interstitial onentration at the interfae is not affeted by additional soures or sinks [43]. This indiates that the oxidizing interfae fixes the onentration of interstitials in the silion at the interfae rather than their flux into silion. Further support was reently provided by diffusion experiments using strong sinks, whih showed that the interfae onentration remained onstant while the injetion rates ranged over 4 orders of magnitude [44]. In an attempt to quantify the proportion of silion atoms injeted as interstitials, some researhers have measured the growth of extrinsi disloation loops produed by prior Si ion implantation [45, 46]. They estimated that at least 1/1000 of the Si atoms onsumed during oxidation are injeted as interstitials for 20

29 temperatures ranging from 850 to 950 o C in dry oxygen [45] and at 900 o C in pyrogeni steam [46]. The measurements are made by using TEM to monitor the growth of these extrinsi defets as they apture the injeted interstitials during oxidation. Similar experiments performed on SIMOX wafers, though, yield values lower by about an order of magnitude [47] for oxidation at temperatures above 1000 o C. However, these researhers did not implant with Si ions to form a pre-existing population of staking faults. Rather, other methods were used to inrease the density of nuleation sites for OISF, the growth of whih was monitored. A lower value was also determined earlier by Dunham and Plummer [38], who made a alulation based on the magnitude of OED. Using an interstitial supersaturation value obtained from OED experiments at 1000 o C, and taking the total self-diffusion oeffiient as an upper limit on the interstitial ontribution to the self-diffusion oeffiient, they onluded that, at most, one atom in 10 4 beomes an interstitial. Estimations of the average supersaturation of interstitials at the silion surfae have been made using staking faults as well as boron and phosphorous diffusion experiments [44, 45, 48]. For temperatures ranging between 750 to 950 o C, the supersaturation ranged between 5 and 25, the highest values being at the lowest temperatures. These results are in good agreement with the model proposed by Dunham and Plummer [38, 39]. Beause the struture and properties of the self-interstitial are important to this work, a detailed summary is given by Appendix B. 21

30 Chapter 3 Oxidation of nanobeams The purpose of this hapter is to desribe the oxidation of nanobeams in partiular. When silion wafers are oxidized, the thikness of the silion is muh thiker than the growing film. During the oxidation of nanobeams, though, the silion thikness is omparable to that of the oxide. As will be shown in Setion 3.1, this has important onsequenes due to the stress generated in the growing oxide. Before addressing this problem, though, we first desribe the oxidation experiments and establish some important notation that will be used throughout the remainder of this text. The majority of our studies on the oxidation of nanobeams were performed in dry oxygen over a temperature range from 900 to 1100 o C, whih spans the glass transition temperature of the silia formed. Eah nanobeam speimen was aompanied by a plain silion wafer piee of the same orientation and approximate doping as the SIMOX material. The protetive oxide remaining from the fabriation proess (see Appendix A) was removed with BHF immediately prior to oxidation. Speimens were held in dry nitrogen as the furnae was ramped up to oxidation temperature, at whih point the gas flow was swithed from nitrogen to dry oxygen. At the end of the oxidation, the gas flow was swithed bak to nitrogen and the temperature was ramped down at rate of 4 to 5 o C/min. At 700 o C, the furnae was slowly opened so that ooling of the speimens from 700 o C to room temperature took 10 to 12 minutes. The oxide was removed with BHF immediately prior to measurement. During the oxidation, the thikness of the nanobeams is redued as silion is onsumed to form oxide. To desribe the hanges in thikness that will be used in our treatment of stress and strain, we introdue the notation shown in the diagram in Fig

31 Figure 3.1 Thikness notation used to desribe the ross-setion of a nanobeam as it is onsumed during oxidation. During growth of an oxide layer of thikness h, a Si layer of thikness h is onsumed. As a result, the half-thikness of the beam is redued from the initial value of final value h. Si ox o h to the Si In order to make our findings appliable to any geometry, we hoose to quantify OIS in terms the volume fration of Si onsumed during oxidation: f V V (3.1) o where V in the volume of Si onsumed to form the oxide and V o is the volume of a silion struture before oxidation. Sine the thikness of a nanobeam is so muh smaller that its lateral dimensions, we an write f V h h = = (3.2) V o Si Si o o hsi where h Si is the half-thikness of the nanobeam and o h Si is the half-thikness before oxidation, as defined in Fig Fig. 3.2 shows the evolution of a nanobeam s profile as a thermal oxide is grown and subsequently removed. In this way, the beams thikness was redued to as little as 24 nm. The faeted walls of the trenh result from the anisotropi eth 23

32 used to release the beams. We note that the nanobeams are slightly bukled as fabriated. This results from displaement of the ends due to thermal expansion mismath between the BOX and the substrate, whih will be disussed in Chapter 4. Figure 3.2 A) Profile of a 217 nm thik nanobeam as fabriated from a silion-on-insulator wafer. (Proportions are greatly exaggerated.) The beam is slightly bukled initially, due to end displaements aused by the stress in the buried oxide. B) Silion is onsumed as thermal oxide is grown. The ombination of OIS and the oxide growth strain auses the bukled amplitude to inrease. C) When the beams are ooled to room temperature, the bukled amplitude further inreases due to the addition of thermal expansion mismath stress. D) When the oxide is removed, the elasti strain in the Si imposed by thermal expansion mismath and the growth stress is reovered, but some permanent strain remains in the now thinner beam. The oxide growth stress desribed in Setion 2.2 plays a role in the oxidation of nanobeams in two respets. In Setion 3.1, we employ 1-dimensional modeling to examine the effet of the growth stress on the stress-strain state of the nanobeam during oxidation. This will be important to the OIS model introdued in Chapter 5. In Setion 3.2, we also all attention to the effet of stress during two-dimensional oxidation, whih is the ase at the edge of a nanobeam. 3.1 Biaxial stress and strain in nanobeams As disussed in Setion 2.2, the inrease in molar volume that aompanies oxidation auses a large intrinsi lateral growth strain, ε i, in the oxide film. Beause of the amorphous nature of SiO 2 at elevated temperatures, muh of this strain is relaxed by visous flow, and the remaining elasti growth strain, ε g, gives rise to the 24

33 reported growth stress, σ g. In this setion, we all attention to an important differene between the oxidation of a wafer and the oxidation of a nanobeam. Taking this differene into aount, we then model the stresses in the oxide and silion during oxidation in terms of the reported growth stress. Beause a wafer is muh thiker than the growing oxide film, it does not undergo any appreiable strain during oxidation and the oxide is onstrained in the plane of the wafer. As a result, the molar volume inrease is aommodated by a ombination of visous flow and biaxial ompression of the oxide. The thikness of our nanobeams, however, is on the same size sale as that of the oxide, and signifiant strain an therefore our in the silion due to the load supplied by the oxide. Consequently, the elasti ompression in the oxide is somewhat redued as the silion is loaded in tension until mehanial equilibrium is ahieved. For the purpose of our derivation, we will desribe the mehanis of the nanobeam during oxidation in terms of strain as illustrated in Fig Thus, the elasti tensile stress in the nanobeam will have a orresponding positive elasti strain, Si ε elast. We also inlude the positive oxidation-indued strain in the nanobeam, ε OI. In order to speify a negative elasti strain that orresponds to the elasti ompressive stress in the oxide, we must onsider the orresponding hange in length, whih we denote as x. Sine the final length of the nanobeam is L, the initial length used to ompute the elasti strain is then ( L+ x), and we have ε ox elast L ( L+ x) ( L+ x). The negative strain due to visous flow of the oxide is defined as εvf Lvf Lo, where Lvf is the hange in length due to visous flow and L o is the initial length of the nanobeam. 25

34 Figure 3.3 Stain ompatibility during oxidation of a nanobeam. A) If the newly formed oxide was not onstrained, molar volume expansion would ause it s length to inrease by about 30% ompared to the initial length of the Si. B) Sine the length of the Si and SiO 2 are required to remain equal, the oxide undergoes a negative strain while the Si experienes a positive strain. In this illustration, the inrease in length of the nanobeam is exaggerated by an order of magnitude and the thiknesses are exaggerated by a few orders of magnitude. We begin our derivation by requiring the ompatibility of equal biaxial strains at the oxidation temperature. As illustrated in Fig. 3.3, we have Si ox ( ) ( ) ε + ε L = ε L + ε L+ x + ε L (3.3) elast OI o i o elast vf o The above expression an be simplified by the following approximations. If the ox initial value of ε elast is taken as ε g, we will have x L sine the reported values for growth stress orrespond to strain on the order of Given that the stress in the silion will be on the same order of magnitude and that the biaxial modulus of Si is over twie that of SiO 2, we an also expet that L Lo. Applying these assumptions, Eq. (3.3) beomes ε + ε = ε + ε + ε (3.4) Si ox elast OI i elast vf If we neglet the axial fores supported by the fixed ends of the bukled beam, fore balane at the oxidation temperature requires ε Eox Si ESi h = ε 1 ν 1 ν ox elast ox elast Si ox Si h (3.5) 26

35 where the thiknesses are as defined in Fig To simplify notation, we define R as the ratio of the room temperature bi-axial moduli of SiO 2 and Si [49, 50]. ( 1 Si ) ( 1 ν ) bi Eox E ν ox R = = 0.47 E E bi Si Si ox (3.6) In fat, the Young s Moduli are a funtion of temperature. For example, at 1000 o C, the modulus of Si in [100] diretion is about 8 % lower than the room temperature value [51]. However, in the following, we assume that the modulus of eah material is redued by a omparable fator, and the ratio will therefore remain approximately the same. We an then rewrite Eq. (3.5) as ε ox elast 1 hsi Si = εelast (3.7) Rh ox Combining Eqs. (3.4) and (3.7), we have ε Si elast εi + εvf εoi = 1 h Si 1+ R hox (3.8) When the Si is very thik ompared to the oxide, as is the ase during oxidation of a wafer, the elasti strain in the Si is negligible. For this ondition, Eq. (3.4) beomes ε = ε + ε + ε (3.9) ox OI i elast vf Sine wafer urvature measurements are sensitive to the elasti stress in the SiO 2, we an express some of our unknown strains in terms of the reported growth stress. Rearranging Eq. (3.9) and writing ε in terms of the growth stress, we have ox elast ( T) ε + ε ε = ε = σ E (3.10) ox bi i vf OI elast g ox whih is inserted into the numerator of Eq. (3.8), giving ε Si elast ( T) σ g E = 1 h Si 1+ R hox bi ox (3.11) 27

36 Using Eq. (3.11), we an now write an expression for the elasti stress in the Si in terms of the growth stress, ε Si elast σ = = Si elast bi ESi ( T ) σ ( T) g E bi ox 1 h Si 1+ R hox (3.12) and employing definition of R we have σ Si elast ( T ) = σ g ( T ) R+ h h Si ox (3.13) As mentioned in Chapter 2, relaxation of the intrinsi stress by visous flow produes an oxide film that is thiker than the onsumed silion layer by a fator of 2.27 for oxide growth on a wafer. This will not neessarily be true for all nanobeams, though. For reasons disussed above, the nanobeam will not maintain a rigid onstraint on the oxide s lateral dimensions. As a result, less expansion in the thikness diretion will be required. For example, SEM measurements of the beam shown in Fig. 3.7B indiate that the oxide is about 12.5 % thinner than the molar volume ratio would predit for 0.78 f = at 1000 o C. Nevertheless, we use the molar volume ratio in the present treatment so that the stresses may be expressed in terms of the fration of silion onsumed: o o h 2.27( hsi hsi ) ox h Si = = hsi hsi hsi (3.14) Using the definition of f, the above equation beomes h h ox Si f = f (3.15) Inserting the inverse of the above result into Eq. (3.13), we have σ Si elast ( T, f ) ( ) ( ) ( ) ( ) σ g T σ g T f = = R f f Rf f (3.16) The elasti stress in the oxide is determined from the fore balane as 28

37 σ ox elast ( T, f ) ( T)( f) ( f ) 0.44σ g 1 = Rf (3.17) Eqs. (3.16) and (3.17) are plotted in Fig. 3.4 using the values of growth stress found in table 2.1. Figure 3.4 Biaxial elasti stress in a nanobeam as a funtion of the volume fration of silion onsumed during oxidation at different temperatures. 3.2 Shape evolution at the nanobeam edge The observed thikness of silion nanobeams after oxidation is fairly uniform over almost all of their area. However, when the edge of a nanobeam is viewed in ross-setion after fraturing, as shown in Fig. 3.5, we observe a pinhed appearane, with the edge being thiker than the rest of the beam and the region near the edge being thinner. As shown in Fig. 3.6, the effet beomes more pronouned as more silion is onsumed. 29

38 Figure 3.5 Cross-setional SEM image of a nanobeam oxidized for approximately 27 hours at 900 o C. The oxide has been removed by ething in hydrofluori aid and the beam fratured for examination. Figure 3.6 A) Cross-setional SEM image of a nanobeam oxidized for approximately 4 hours at 1000 o C. B) Nanobeam oxidized for approximately 6 hours at 1000 o C. The oxide has been removed by ething in hydrofluori aid and the beam fratured for examination. When nanobeams are viewed before removal of the oxide, as in Fig. 3.7, it is apparent that the pinhing is the result of retarded oxidation at the orners and enhaned oxidation over a region near the edge. This an be explained by the effet of stress on the both oxidant transport and the oxidation reation. 30

39 Figure 3.7 A) Cross-setional SEM image of a nanobeam oxidized for approximately 14 hours at 900 o C. B) Nanobeam oxidized for approximately 6 hours at 1000 o C. The beams were fratured for examination. As reounted earlier in Setion 2.2.1, Marus and Sheng [25] found that during the oxidation of trenhes ethed into silion, retarded growth was observed for both onave and onvex orners. Later, Kao, et al. arried out omprehensive experiments to haraterize the oxidation of silion ylinders, rings and ylindrial holes [26]. In a ompanion paper, they proposed a model that explained the observed retardation on both onave and onvex strutures as well as the fat that oxide grown on onave surfaes is thinner than that grown on onvex surfaes [27]. If we only onsider the effets of geometry on the supply of oxidant to the silion surfae, we an reason that oxidation will be slower for onave surfaes than planar surfaes due to the dereased exposure to the ambient. When oxide is grown on a planar surfae, the area of the oxide/gas interfae is the same as that of the oxide/silion interfae. When oxide is grown on a onave surfae, though, the area of the oxide/gas interfae is less than that of the oxide/silion interfae. This is due to the differene in the radii of urvature and therefore ar length of the surfaes, whih have a ommon enter of urvature. As a result, the flux determined by Henry s law at the surfae will ultimately be distributed over a larger reating interfae, thereby reduing the growth rate. By the same reasoning, a onvex surfae would oxidize faster than a planar surfae, whih our images indiate is not the ase. Kao, et al. [27] point out another geometri effet that works against the oxidant supply effet, whih is based on volume onservation during visous flow of 31

40 the oxide. Sine the newly forming oxide has a larger molar volume than silion, the assoiated expansion is onstantly pushing out the old oxide, whih must rearrange itself by visous flow. Consider a ring of oxide formed on the surfae of a ylinder, suh as the one shown in Fig As new oxide is formed underneath it, the ring is pushed outwards to a larger radius, and is therefore strethed in the tangential diretion. If the oxide is onsidered inompressible, onservation of volume would then ditate that it beome thinner than a layer of oxide grown on a planar surfae, whose lateral dimensions are unhanged. However, onsidering this effet in addition to that of oxidant supply, Kao, et al. [27] still find that the oxide grown on onvex surfae would be slightly thinner than that grown on a planar surfae. In order to explain the retardation for onvex surfaes, Kao, et al. [27] showed that the effets of the growth stress on visosity, oxidant transport and the oxidation reation must all be onsidered. Central to their explanation is the presene of a hoop stress in the oxide growing on a onvex surfae. Let us return to the ring of oxide mentioned above. Beause the visosity of silia is very high, it will be able to support a signifiant tensile stress in the tangential diretion as it is onstantly strethed to make room for new oxide. As illustrated in Fig. 3.8, This leads to ompressive stress normal to the oxidation front, whih retards the reation rate by modifying the ativation energy for the oxidation reation. Sine the volume expansion must do work against the normal stress, the reation beomes less energetially favorable. This reasoning has also been used to explain self-limiting oxidation of silion nanowires [5]. Here we apply it to the retarded oxidation at nanobeam edge as well. Figure 3.8 Stress during oxidation of a Si ylinder, shown in ross setion. A tensile hoop stress is reated the existing oxide is fored outward by the volume expansion assoiated with the formation of new oxide at the interfae. This results in a normal fore ating on the interfae. 32

41 For relatively thik nanobeams, suh as the one shown in Fig. 3.7A, we notie that the oxide is thinner at the orners. This is onsistent with the above argument if we onsider the orners as two onvex regions separated by a planar region. For very thin nanobeams, though, suh as the one shown in Fig 3.7B, we find that the oxide growth between the orners is notieably retarded as well. This an be explained by onsidering the edge of the beam as a single onvex struture. The edge will approah this ondition as the separation between the orners dereases relative to the length sale of the oxide. In order to explain the enhaned oxidation on the planar surfaes near the beam edge, we must onsider the effet of stress on transport of oxidant to the silion. Transport, and hene the oxidation rate, will be inreased by tensile stress via two possible mehanisms. Tensile stress is expeted to inrease the volume of the voids in the silia network that are oupied by the diffusing speies. As a result, both the solubility and diffusivity of the oxidant should be inreased. Sine the oxide far from the edge is in ompression and the oxide at the edge is in tension, we expet a transition region in-between. The oxidation rate will be enhaned by the inreased transport in this region, but it will not be retarded as it is at the edge beause there is no urvature and therefore no normal stress. The importane of the stress effet on the oxidation rate an be demonstrated using the proess simulation program SSUPREM4. When the VISCOUS model is used, this software simulates two-dimensional oxidation based on the work of Chin, et al. [52, 53]. This model ombines a two-dimensional treatment of the Deal-Grove model with inompressible visous flow of the oxide. Sine the visosity of the oxide is high and the rate of deformation is slow, the hydrodynami flow equation is redued to a reeping flow equation, whih balanes the visous fore against the pressure gradient. Solution of this equation gives the veloity field of the flowing oxide elements. While this method produes a somewhat realisti shape for the bird s beak ommonly observed during loal oxidation of silion (LOCOS), it does not predit the observed retardation for onvex surfaes. This is demonstrated in Fig. 3.9 by omparison of a SSUPREM4 simulation to the nanobeam shown in Fig.3.7B. In fat, the simulation predits an enhanement of oxidation rate at the edge, whih is onsistent with the reasoning based only on oxidant supply. 33

42 Figure 3.9 A) Nanobeam pitured in Fig 3.7B. B) Result of SSUPREM4 simulation of dry oxidation for 6 hours at 1000 o C using the VISCOUS model without inluding the dependene of diffusivity, visosity and the reation rate oeffiient on stress. Considerable improvement is made to the simulation by setting the parameter STRESS.DEP=T in the OXIDE statement. When this is done, the stress dependene of diffusivity, visosity and the reation rate oeffiient is added to the model in aordane with the equations found in Setion A seond term is added to Eq. (2.15), though, in order to inlude the hange in the rate onstant due to stress parallel to the interfae. As shown in Fig. 3.10, the resulting shape is muh more realisti, with the oxide being onsiderably thinner at the edge. We note, however, that this model does not predit any oxidation enhanement near the edge. This may indiate the importane of the stress effet on the equilibrium onentration of the oxidant, whih the SSUPREM4 model does not inlude. 34

43 Figure 3.10 A) Nanobeam pitured in Fig. 3.7B. B) Result of SSUPREM4 simulation of dry oxidation for 6 hours at 1000 o C using the VISCOUS model with stress dependent diffusivity, visosity and reation rate oeffiient. 35

44 Chapter 4 Quantifying the oxidation-indued strain In this hapter, we desribe the measurement tehniques used to quantify OIS in terms of the volume fration of silion onsumed during oxidation at different temperatures. We ll first desribe how SEM imaging of nanobeam ross-setions revealed greater than expeted values for f. We then desribe the use of optial interferometry to measure the bukled profiles of nanobeams, whih are used to determine the oxidation-indued strain. Finally, OIS data is presented in terms of the fration onsumed. 4.1 Volume fration of silion onsumed Initially, f was determined from thikness measurements of SiO 2 grown on plain Si wafer piees, whih aompanied the nanobeam speimens during oxidation. These witness piees were of the same doping and orientation as the SIMOX material. As disussed Setion 2.2, a growing oxide is laterally onstrained by the Si wafer and undergoes visous flow so that the molar volume inrease is manifested predominantly as an inrease in oxide thikness. Thus, it is observed that an oxide film is about 2.27 times thiker than the Si onsumed during oxidation [10]. If the oxide grows on a nanobeam at the same rate as it does on a Si wafer, we an dedue the fration onsumed from the thikness of the oxide grown on the wafer piees as measured by refletane spetrosopy. o h = h h = 0.44h f Si Si ox h h = = 0.44 h h ox o o Si Si (4.1) However, measurements of SEM frature ross-setions suh as the one shown in Fig 4.1 reveal that the nanobeams are onsumed at a faster rate than is predited by the above assumption. For this reason, the thikness onsumed was re-evaluated for eah nanobeam speimen. The results are shown in Fig. 4.2, whih ompares the thikness onsumed during oxidation of a nanobeam with that inferred from the witness piees. The error bars for h represent two soures of error for the data from the nanobeams. The total error is the sum of the standard deviation of our SEM measurements and the unertainty introdued by the viewing angle of the beam rosssetion. When the beams are fratured for measurement, they are presumed to leave on {111} planes, produing a leaved surfae that forms an angle of o with the nanobeam surfae. Sine the beams are viewed at an angle of 5 o out of the plane of the beam, this angle will ause the beam to appear either thiker or thinner than it atually is, depending on the orientation of the fratured surfae. The error bars for the wafer measurements represent the standard deviation of many measurements of the oxide thikness made at different loations. 36

45 Figure 4.1 Cross-setional SEM image of a 39 nm thik nanobeam viewed at an low angle. Thiknesses were determined by averaging 25 to 50 measurements for eah beam. Oxide thikness measurements predited that this beam should have been 51 nm thik. 37

46 Figure 4.2 The thikness onsumed on a nanobeam is ompared the thikness onsumed on a wafer for eah oxidation time. As quantified in this plot, silion is onsumed faster when oxidizing a nanobeam than when a wafer is oxidized. The differene between h for a wafer and a nanobeam oxidized under the same onditions implies that nanobeams are oxidized at a faster rate than bulk Si. We believe this larger oxidation rate is likely due to a relaxation of the ompressive growth stress that arises in the oxide, as desribed earlier in Setion 3.1. The effet of stress relaxation on oxidation rate may be twofold. First, there is evidene that stress redution in the oxide inreases the oxidation rate onsiderably by ausing an inrease in oxygen transport though the already existing oxide [24]. This aeleration may be due to an inrease in either the solubility or diffusivity of oxygen. Seond, the oxidation reation rate may be enhaned by the tensile stress in the Si [29-31]. The trends shown in Fig. 4.3 support the above argument in two ways. In this figure, we show the differene between the thikness of Si onsumed on a nanobeam, nb w h, and the thikness onsumed on a wafer, h. Although there is onsiderable satter in the data, the linear urve fits indiate that the oxidation rate enhanement inreases with f at most temperatures. This is onsistent with our expetations, sine the stress magnitude dereases in the oxide and inreases in the Si as the ratio of oxide thikness to Si thikness inreases (see Setion 3.1). 38

47 Figure 4.3 The differene between thikness onsumed on a nanobeam and a wafer is plotted as a funtion of the fration onsumed. For most temperatures, the linear urve fits indiate a more pronouned effet as the fration onsumed inreases. We also note that the enhanement dereases with inreasing temperature. This is more learly shown in Fig. 4.4 by taking the average differene in thikness onsumed over all oxidation times for eah oxidation temperature. As disussed in Setion 2.2, the oxide growth stress is a dereasing funtion of oxidation temperature due to dereasing visosity. Sine the oxide growth stress is already relatively low at high temperatures, we expet a lesser hange in stress and, therefore, a less pronouned effet on oxidation rate. 39

48 Figure 4.4 The average differene in thikness onsumed over all oxidation times is evaluated for eah oxidation temperature. 4.2 Measuring nanobeam displaement In order to make a alulation of the strain arued in the nanobeams, we must first quantify their elongation. This an be obtained from measurements of vertial displaement of the bukled beams as funtion of length after removal of the oxide. Suh measurements are performed quikly and aurately with nanometer sale vertial resolution using the Wyko NT1100 optial profiling system manufatured by Veeo. The Wyko NT1100 is very muh like a standard optial mirosope in both appearane and use. As shown in Fig. 4.5, though, there are a few important differenes. These inlude a CCD detetor array, motorized vertial translation of the optial assembly and a Mirau interferometer. As shown in Fig. 4.6, the Mirau interferometer is similar to a Mihaelson interferometer in priniple, but not design. 40

49 Figure 4.5 Shemati diagram of the Wyko optial profiling system. Figure 4.6 Shemati illustration of a Mirau interferometer. Light travels down the opti axis, through the objetive, and is inident on a beam-splitter. Half of the light is refleted towards a referene mirror, then bak to the beam-splitter, then bak into the objetive. The other half of the light ontinues through the beam-splitter to the sample, where it is refleted bak through the beamsplitter and re-ombines with the referene beam before entering the objetive. Any differene in path length leads to interferene. When white light is used, a fringe envelope is produed, whose width is ditated by the oherene length of the light soure. 41

50 The fringe envelope pitured in Fig. 4.6 provides the basis for vertial sanning interferometry (VSI). First let us onsider a flat surfae being imaged by the mirosope. The interferometer is set up so that when the surfae is in fous, the path length differene is zero and the deteted intensity is therefore maximum. If the sample is moved from fous by vertial translation, the intensity will go through a series of bright and dark osillations, whose amplitude will quikly derease to zero. Alternatively, we an imagine tilting the surfae to give a simultaneous sampling of heights. This will result in a band of fringes viewed through the mirosope, as illustrated in Fig Figure 4.7 Diagram illustrating the intensity modulation as a funtion of height on a tilted surfae. The dots indiate the positions of the intensity maxima and the orresponding fringe pattern is shown above. Next we onsider a surfae of arbitrary topography. When the Wyko is used in VSI mode, the intensity of eah CCD pixel is monitored individually as the optial assembly (rather that the sample) is vertially translated. Sine the displaement is known through a motor alibration, the relative height of eah point on the surfae an be determined from the displaement at whih eah pixel s intensity is maximum. The lateral resolution is therefore limited by the area sampled per pixel. 42

51 In VSI mode, the deteted intensity is modulated by hanging the path length of the light that is refleted from the sample. The Wyko has another measurement mode, though, that operates by hanging the path length traveled by the light that is sent to the referene mirror. When phase-shifting interferometry (PSI) is used, the sampled surfae is kept at fous while the referene mirror is translated slightly by a piezoeletri transduer in order to ause a known phase shift between the sample and referene beams. Additional phase shift is then introdued by the variation in height of the sample. Phase data is obtained by integrating the intensity data olleted at many different relative phase shifts and the relative surfae height is alulated as λ hxy (, ) = φ, 4π ( xy) where λ is the wavelength of the soure and ( xy, ) φ is the phase information. A red filter is used to greatly broaden the fringe envelope so that high ontrast fringes are present through a greater depth of fous. Both VSI and PSI modes were investigated for nanobeam measurement and it was disovered that VSI measurements were affeted by an important limitation assoiated with transparent materials. Care must be taken when attempting to measure transparent strutures if there exists a refletive interfae below surfae of interest. As an example, we onsider the ase of transparent films, suh as photoresist and SiO 2, deposited on a refletive substrate, suh as silion. If this is the ase, and the thikness of the strutures is less than the oherene length of the light soure, the fringe envelope will span both refletive surfaes, thereby produing an intensity flutuation from both simultaneously. As a result, the net intensity will not be true to the position of the top surfae. If the sample is tilted as above, the fringe envelopes will overlap, produing and asymmetri pattern. If the thikness is large enough, the fringe patterns will be separated. Either ase will present a problem during measurement sine the software will detet two suessive intensity maxima while only expeting one. Both of these ases are illustrated in Fig

52 Figure 4.8 Multiple fringe envelopes our for transparent films on refletive substrates. On the left, a relatively thin SiO 2 film produes two overlapping fringe patterns one from the SiO 2 surfae and one from the Si/SiO 2 interfae. On the right, the muh thiker photoresist layer separates the fringe patterns. The are now three refletions. The patterns from the Si/SiO 2 interfae and the photoresist/sio 2 interfae no longer overlap due to the index of refration mismath at the photoresist/sio 2 interfae. 44

53 Although our nanobeams are made of silion, they are transparent beause their thikness is less than the absorption distane. As a result, the Wyko detets a refletion from both the top and bottom air/si interfaes when using white light in VSI mode and the resulting measurements are inaurate. This is illustrated in Fig. 4.9 by tilting a relatively flat antilever nanobeam, so as to produe overlapping fringe envelopes like those shown in Fig When red light is used in PSI mode, however, refletion from the bottom surfae of the beam does not seem to influene measurements. There are several possible reasons for this, but it may be related to the fat that the refletivity of silion drops dramatially with inreasing wavelength aross the visible spetrum. Due to the apparent limitations of VSI mode, all measurements reported here were made using PSI mode. An example of a Wyko profile is shown in Fig Figure 4.9 The fringe pattern refleted from a nanobeam is investigated by viewing a relatively flat antilever when tilted. A) When using white light in VSI mode, overlapping fringe envelopes are produed. As shown at right, this auses an overestimate of the overall hange in height as well as a highly exaggerated roughness. B) When PSI mode is used with a red filter, the fringes are evenly spaed with uniform ontrast. The resulting measurement is greatly improved and the roughness is onsistent with SEM observations. 45

54 Figure 4.10 Wyko optial profile of two upwardly bukled nanobeams that measure approximately 10 µm wide by 150 µm long. In a previous proess flow, we used isotropi reative ion ething (RIE) as a dry release step. The underutting due to this eth produed a ledge that subsequently bukled due to OIS, giving the ruffled appearane. Sine onsiderable distortion of the end onditions often resulted, the ledge formation was later eliminated by swithing to an anisotropi KOH release eth. 4.3 Determination of the oxidation-indued strain The axial strain responsible for the bukling of a nanobeam an be alulated from a two-dimensional line profile along length of the beam. Suh profiles are easily obtained from 3D profiles using the WykoVision32 software. In Setion 4.3.1, we disuss the use of beam theory to alulate an apparent strain from the measured profile of a bukled beam. In Setion 4.3.2, we onsider the effet of end displaements, whih have been disovered to ontribute a beam-length dependent omponent to strain as alulated by the method in This ompliation requires that urve fitting be used to extrat the strain of interest. The results are shown given in Setion Bending equation for a bukled beam As illustrated in parts A and B of Fig. 4.11, oxidation-indued strain would ause the length of a beam to inrease from it s initial value, l o, to a final value, l 1, if it were not onstrained in the axial dimension. Beause the oxidation-indued strain is so small, this elongation is far too small to detet using mirosopy. However, by keeping both ends of the beam fixed, we take advantage of a innate way to amplify displaement. If the beam is not permitted to lengthen in plane, it may still lengthen by bukling out of plane with an amplitude muh larger than the inrease in length. In the ase of our nanobeams, an elongation on the order of 10 nm will produe a bukled amplitude on the order of 1 µ m. 46

55 Figure 4.11 Shemati illustration of a OIS manifested in the bukling of a nanobeam. A) Before oxidation, the beam has initial length, l o, as defined during fabriation. B) After oxidation and removal of the oxide, OIS would ause a nanobeam to inrease to length l 1 if one end was permitted to translate. C) Both ends are fixed, though, and OIS will therefore give rise to axial ompression, whih is equivalent to the appliation of an axial load, P 1. D) The beam bukles sine P 1 exeeds the ritial load for bukling. The axial ompression is largely relaxed as the beam s length inreases to the ar length, s. As a result, the load dereases to P 2. The driving fore for bukling is the axial ompression that would arise were the beam not allowed to elongate by any means. The sum of the oxidation-indued strain, ε OI, and any strain initially present in the Si over-layer, ε o, would lead to a ompressive stress given by Hooke s law: 47

56 l l σ = ε + ε = (4.2) ( ) 1 o E OI o E l o As depited in Fig. 4.11C, this stress state is equivalent to the appliation of an axial load, P 1. One the beam bukles, a large portion of this load is relaxed sine the beam is able to inrease its length, thereby reduing the strain energy stored within it. In general, the ompression will not be ompletely relaxed beause bukling is also aompanied by an inrease in strain energy due to bending. Instead, fore and moment equilibrium will require that the axis of the beam be governed by the following differential equation [54]. 4 2 d z d z EI + P 4 2 = 0 (4.3) 2 dx dx where E is the modulus of elastiity, I is the moment of inertia and P 2 is the load shown in Fig. 4.11D. The load P 2 is positive by definition when the beam in ompression. The solution to Eq. (4.3) for a beam with lamped end onditions has the form z z = max 2π 1 os 2 lo x (4.4) When Eq. (4.4) is inserted into Eq. (4.3), the osine terms onveniently anel eah other and we are left with the following expression. 2 4EIπ + P2 = 0 (4.5) l 2 o The axial load P 2 an be dedued from the hange in length (and therefore elasti ompression) of the beam upon bukling: 2 o P2 = P1 EA l 1 l l l l l l l l = EA = EA l1 l1 l1 1 o 2 o 1 2 (4.6) After ombining Eqs. (4.6) and (4.5), the ross-setional area, A, and the moment of inertia are expressed in terms of the beams half-thikness, h Si. We then have 48

57 4π h l l 2 2 Si = 0 (4.7) 2 3lo l1 whih an be solved for l 1 : l = l π hsi 1 2 3lo (4.8) We measure l o diretly and l 2 is found by omputing the ar length of the beam s measured profile as l1 2 dz l2 = s = 1+ dx dx (4.9) 0 Finally, l 1 is used to alulate the total measured strain as ε ε ε l l 1 o m = OI + o = (4.10) lo In some ases, we find that nanobeam profiles, suh as the one shown in Fig. 4.12, are in exellent agreement with the ideal shape desribed by Eq. (4.4). However, for various reasons, we find that profiles are often not well suited to a osine fit. For example, the slope at the ends sometimes does not go to zero. Also, the profiles are sometimes asymmetri, as shown in Fig Deviation from the ideal osine shape is notieable at large f. 49

58 Figure 4.12 Eq. (4.4) is fit to the profile of a nanobeam that was annealed for 220 minutes at 1050 o C. Only one in 20 data points are shown. Figure 4.13 A) The profile of a 31 nm thik beam departs from the osine shape desribed by Eq. (4.4). B) A very good fit is ahieved by a 6 th order polynomial. Only one in 20 data points are shown. 50

59 The end onditions of the nanobeams are fixed by the interfae of the Si overlayer with the buried oxide layer. Not surprisingly, there are several possible reasons for distortion to our at the free edge of the buried oxide. For example, at room temperature, the buried oxide layer is in a state of ompressive stress due to the thermal expansion mismath between the oxide and the silion substrate. This is due to the fat that the BOX is formed during a high temperature anneal as the final proessing step in produing SIMOX material. When material is removed by ething to define our nanobeams, a free surfae is reated normal to the axis of the beam and some distortion ours as the stress is loally relaxed. This issue is disussed further in the following setion. Another soure of imperfetion in the end ondition is the BHF eth used to remove the oxide before and after oxidation. As we disovered during the ourse of this work, the buried oxide layer is underut non-uniformly in the region near the beam-ends. Distortions may also our at the beam-ends during oxidation due to the growth stress disussed in Chapters 2 and 3. In order to obtain an aurate measure of ar length even when the beams deviate from their ideal shape, the profiles are fit with a 6 th order polynomial rather than a sinusoidal funtion. As shown in Fig. 4.13B, very good fits are ahieved in this manner. 51

60 4.3.2 The end displaement effet When the measured strain is alulated using to the bending equation as desribed in Setion 4.3.1, we find that it has a strong dependene on the length of the beam measured (see Fig. 4.15). This dependene an be explained if we onsider the effet of a slight displaement of the beam-ends during fabriation. In fat, evidene for suh a displaement is given by the initial bukling of the beams as fabriated, before any oxidation treatment. Presumably, this displaement is due largely to the relaxation of thermal expansion mismath stress in the buried oxide layer. When material is removed during the fabriation of nanobeams, an oxide surfae is reated as illustrated in Fig Sine the free surfae annot support a load, displaement ours as the stress is relaxed in the region near the edge. Figure 4.14 Shemati illustration of the end displaement,d, due to the relaxation of ompressive stress in the oxide. A) The stress due to thermal expansion mismath is initially uniform in the lateral diretion. B) After material is removed, thereby reating a free surfae, stress is loally relaxed as the new surfae is displaed. The dashed line indiates the position of the surfae prior to relaxation. Intuitively, a given displaement is expeted to have a more signifiant effet on the bukled amplitude of a short beam than a long one. The relationship is derived as follows. The total measured strain was given by Eq. (4.10) as l ε = ε + ε = m OI o l l 1 o o in whih we assumed that l o would be provided by diret measurement. However, due to the end displaement, we do not measure l o. Instead, the measured length that the beam spans is lm = lo 2d. A more general form of Eq. (4.10) is therefore given by εm = εoi + εo + εd (4.11) where ε d is the strain due to the end displaements alone. This strain is given by 52

61 ε d lo lm 2d = = l l + 2d o m (4.12) and Eq. (4.11) an be restated as εm = εoi + εo + l m 2d + 2d Eq. (4.13) is then fit to the measured strain data, allowing us to extrat εoi fitting parameter. The results are presented in the following setion. (4.13) + ε as a o Oxidation-indued strain data In order to separate the strain of interest from that due to the end displaement, strain measurements for beams of different length are fit by Eq. (4.13). The results follow in Figs

62 Figure 4.15 The plotted strain is determined from nanobeam profiles as desribed in Setion for beams of different length. Eah data point gives the average value for 6 measured beams with the error bar representing the standard deviation. The strain at f = 0 was determined from speimens annealed in nitrogen for times omparable to the longest oxidation. The strain of interest is extrated by fitting Eq. (4.13) to the data. 54

63 Figure 4.16 The fits to the data in Fig yield the two omponents of strain that ause bukling. The strain due to the sum of OIS and the initial strain has no beam length-dependene. Sine the strain due to displaement of the ends is dependent on beam length, it is represented here by the end displaement. Error bars for f represent the same soures of error in h, as disussed in Setion 4.1. Error bars for strain and end displaement are given by the standard error alulated for the fits in Fig

64 Figure 4.17 The oxidation-indued strain is obtained by subtrating the initial strain from the strain plotted in Fig Sine the spread of strain values measured for annealed speimens (f = 0) is omparable to the satter in the data in general, we take their average as the initial strain. 56

65 Chapter 5 Modeling oxidation-indued strain Having quantified oxidation-indued strain as a funtion of the fration of silion onsumed in the previous hapter, we are now ready to investigate possible auses through omparison of our data with physial models. Beause the magnitude of OIS is quite small and we have not found any evidene for plasti deformation, we first onsider the possibility that the strain is due to the aumulation of selfinterstitial silion atoms, whih are known to be injeted from the Si/SiO 2 interfae during thermal oxidation. A model based on this phenomenon is developed in the three parts of Setion 5.1 and ompared with our experimental data. In Setion 5.2, we examine two other possibilities, whih are judged to be less likely andidates for the ause of OIS. We first show that the results of transmission eletron mirosopy (TEM), Nomarski mirosopy, and SEM provide no evidene of disloation plastiity. Sine it annot be ruled out, though, our data is ompared with the preditions of power-law reep, whih has been haraterized in the literature for single-rystal silion. We then address the redistribution of boron, whih is present as a dopant, but find that it annot aount for the magnitude of strain we observe. 5.1 The interstitial injetion model In this setion, we develop a model based on the assumption that the injetion of interstitials desribed in Chapter 2 is responsible for OIS. This is judged to be a plausible explanation given that the magnitude of the strain we measure is so small. We first develop a simple model that onsiders the volumetri strain ontributed by the injeted atoms and takes into aount the hanging geometry of the beam as its thikness is onsumed. From this model, we are able to obtain an empirial estimate of the number of atoms injeted relative to the number onsumed. We then expand our treatment by modeling the injetion as a thermally ativated proess, and ultimately suggest that the ativation is assisted by the biaxial stress desribed in Setion Simple volumetri strain model The following model is founded on onsideration of the inrease in volume of a Si rystal due to injetion self-interstitial atoms. Experiments have shown that ontinuum elastiity is appropriate for modeling the volumetri strain that results from bond relaxation assoiated with point defets in rystals [55]. In the spirit of Vegard s law [56], a linear relationship is found between the onentration of point defets and the resulting hange in lattie parameter. When point defets are reated in a rystal through a onservative proess, suh as interation with radiation, the hange in volume of the rystal is expressed in terms of the formation volume of the defet, V. The formation volume is the net hange in volume that would result from removing an atom from the surfae of the f 57

66 rystal and plaing it at an interstitial site within the rystal. Removing the atom from the surfae redues the volume of the rystal by approximately one atomi volume, Ω. When the interstitial is reated, a volume expansion ours due to relaxation of the loal bond struture. This is referred to as the relaxation volume, V r, and we an now express the differene as V f = V Ω r We note that V f and, therefore, the overall hange in volume, will be negative if V r <Ω. Indeed, alulations based on an orthogonal tight-binding model yield a value of V = 0.90Ω for the Si self-interstitial, resulting in a orresponding formation r volume V f = 0.10Ω [57]. If a non-onservative proess reates the interstitials, though, we only onsider the ontribution of V r and the overall hange in volume will be positive. Suh is the ase for interstitial injetion if we onsider the oxidation of a nanobeam as a system with a moving boundary orresponding to the Si-SiO 2 interfae. In this ase, the volume, V, is defined as the volume of Si that would remain after oxidation if no interstitials were injeted. Figure 5.1 A silion nanobeam element of unit length and width, L, is shown before and after an inrement in oxidation time. For small strain, an inremental inrease in OIS along the axis of the beam an be taken as one third of the volumetri strain that ours from the injetion of self-interstitial atoms during an inrement of oxidation time. As shown in Fig. 5.1, the oxidation time inrement has a orresponding inrement in the amount of silion 58

67 onsumed, whih is expressed as a thikness hange, introdued by Fig. 5.1, we have dh Si. Using the notation 1 dvi 1 dvi d ε = 2 3 V = 3 h L (5.1) Si where dv i is the inrease in volume due to injetion of interstitials. It is useful to define the net interstitial injetion ratio,θ, suh that dn i = θ dn (5.2) where dni is the hange in the net number of Si atoms injeted per unit area and is the hange in the number of Si atoms onsumed per unit area to form oxide. We define n i as the net number of interstitials injeted sine, in general, the net flux of interstitials aross the Si/SiO 2 interfae will have omponents in both diretions. The hange in injeted volume is then obtained by multiplying Eq. (5.2) by the relaxation volume, whih is taken as the atomi volume of silion, Ω. i i dn dv =Ω dn =Ω θ dn (5.3) Here, we assume that eah interstitial ontributes to the strain individually. However, self-interstitials are known to agglomerate into a variety of extrinsi defets. If most of the added volume is ontained in suh defets, the volume ontributed per injeted atom may be slightly lower than the relaxation volume of the individual atom. The hange in the number of atoms onsumed is written in terms of the hange in thikness of the Si as it is onsumed to form the oxide dn ( ) dv L h h + dh Ldh = = = Ω Ω Ω 2 2 Si Si Si Si (5.4) and Eq. (5.3) beomes dv i = θ (5.5) 2 LdhSi Inserting Eq. (5.5) into Eq.(5.1), we have dε θ 1 = dh 3 hsi Si (5.6) 59

68 And using the definition of f, f h o Si h h o Si Si o h = h (1 f ) Si Si dh df dh Si = h o Si = hdf o Si Si we an rewrite Eq. (5.6) as dε θ 1 = df 3 1 f (5.7) At this point, we an obtain an empirial estimate of θ from our data. Rearranging Eq. (5.7), we have dε θ = 31 ( f ) (5.8) df First, we find a good empirial fit to the strain data, as shown in Fig Sine the strain observed at different oxidation temperatures seems indistinguishable, we also inlude a fit for the strain at all temperatures onsidered as one set of data. 60

69 Figure 5.2 A) Empirial fit to OIS at eah oxidation temperature. B) Empirial fit to OIS at all oxidation temperatures when onsidered as a single set of data. The urves shown in Fig. 5.2 are differentiated with respet to f and the result is used in Eq. (5.8). The ensuing dependene of the injetion ratio on the fration onsumed is plotted in Fig 5.3. Figure 5.3 A) Empirial determination of the injetion ratio at eah temperature as a funtion of the fration onsumed using the urve fits shown in Fig. 5.2A. B) Empirial determination of the injetion ratio when OIS at all temperatures is onsidered as a single set of data, as in Fig. 5.2B. 61

70 Sine the injetion ratio apparently has a strong dependene on the fration onsumed, integration of Eq. (5.7) with θ as a onstant would not yield an aurate expression for OIS. We must therefore onsider the dependene of θ on f, whih is evident upon loser inspetion of the definition of θ. We ll find that a dependene on oxidation temperature is neessarily inluded as well. The injetion ratio is defined in terms of inremental hanges in number of atoms injeted and onsumed. If these inremental hanges are expressed as the produt of the instantaneous rates of injetion and onsumption with the time inrement, we have θ dn dn ν dt ν dt ν ν = i = i = i (5.9) where ν i is the rate per unit area at whih interstitials are injeted and ν is the rate per unit area at whih Si atoms are onsumed to form SiO 2. The rate of onsumption is diretly related to f as follows. We begin by expressing the number of atoms onsumed per unit area in terms of the thikness onsumed: n N ha 1 h = = = A Ω A Ω (5.10) where N is the total number of atoms onsumed and A is a unit area of the Si/SiO 2 interfae. Using the definition of f, Eq. (5.10) beomes h h f n = = Ω Ω o Si (5.11) and the rate of onsumption is o ν = dn hsi df dt = Ω dt (5.12) The quantity df / dt may be determined from our experimental measurements of f as a funtion of oxidation time and temperature. Sine f is linearly related to the f Tt, by fitting the following linear-paraboli oxide thikness, we an obtain ( ) expression to our data, as shown in Fig ( ) ( ) ( ) f Tt, = at 1+ bt t 1 (5.13) 62

71 The fitting parameters at ( ) and ( ) bt are analogous to the linear and paraboli rate onstants of the Deal-Grove model disussed in Setion 2.1. Their values are given in Table 5.1. Figure 5.4 A linear-paraboli fit to the experimental data it used to obtain the fration of silion onsumed as a funtion of oxidation time. The fitting parameters are given in Table 5.1. Oxidation Temperature ( o C) a(t) b(t) (1/s) Table 5.1 Values of the parameters obtained from the fits shown in Fig

72 We an now find df / dt by taking the time derivative of Eq. (5.13): d f dt ( Tt, ) ( ) ( ) + ( ) at bt = 2 1 btt The time dependene is then replaed with a dependene on (5.13) as (5.14) f by rearranging Eq. 2 f 2 f t = 2 ab + ab (5.15) and inserting it into Eq. (5.14), whih beomes 2 df ab = dt 2 f a ( + ) (5.16) We note that the time and temperature dependenies of f, a and b are not written, but will be implied from now on. The dependene of the onsumption rate on appears when Eq. (5.16) is ombined with Eq. (5.12) to give 2 o abhsi ν = 2 (5.17) Ω + ( f a) and this, in turn, produes a dependene of θ on f : The strain inrement is then θ ( f ) ( f a) 2Ω + 2 o abhsi = ν (5.18) i f dε 2Ω ν i a+ f = df 2 o 3abhSi 1 f (5.19) We an make an empirial estimate of the injetion rate (as we did above for the injetion ratio) by rearranging Eq. (5.19), 64

73 ν i 2 o 3abh Si 1 f dε = 2Ω a+ f df (5.20) whih is plotted in Fig Figure 5.5 A) The injetion rates are determined using the empirial fits at eah temperature separately, whih are shown in Fig. 5.2A. B) The injetion rates are determined using the empirial fit for all temperatures onsidered together, whih is shown in Fig. 5.2B. There is a urve for eah temperature beause of the dependene on temperature of the Si onsumption rate. The urves shown in Fig. 5.5 give us useful information about the dependene of the injetion rate on both f and oxidation temperature. The dependene on temperature is most obvious. It is quite lear from Fig. 5.5 that the injetion rate inreases with oxidation temperature. This behavior is expeted sine interstitial injetion is likely a thermally ativated proess. Suh a temperature dependene is therefore added to our model in The nature of the dependene of ν i on f is not so obvious. We notie that at most temperatures, the injetion rates seem to derease rapidly from a high initial value at f = 0. It is worth noting, though, that this derease takes plae over the range of f for whih we have the least amount of data. In fat, the urves shown in Fig. 5.5B derease over the range 0< f < 0.3, for whih we have no data points. Furthermore, the sensitivity of the injetion rate to the slope of the empirial fit inreases greatly as f 0. At f = 0, the injetion rate given by Eq. (5.20) is diretly proportional to the slope of the fit. It would therefore seem that the auray of our empirial method at low f is questionable. If we ignore the rapid initial 65

74 derease, it appears that the injetion rate as a funtion of fration onsumed is either lose to onstant or inreases slightly. Both of these possibilities will be onsidered in the following setions. But first, we ll assume that ν i is independent of f so that we may integrate Eq. (5.19) to obtain an expression for OIS from our simple model. In Setion 5.1.2, we onsider only the temperature dependene of ν i by introduing an Arrhenius expression for the thermally ativated injetion of interstitials. In Setion 5.1.3, we show that onsideration of the stress state desribed in Setion 3.1 adds a dependene on f to the injetion rate as well. Assuming that ν i is independent of f, integration of Eq. (5.19) over the appropriate limits yields the following result. ε εoi 2 f ν i dε 2 o abh (5.21) 0 Si 0 Ω a+ f = df 3 1 f 2Ων = f ( 1 + a) ln( 1 f ) (5.22) i OI 2 o 3abhSi Sine the injetion rate is expeted to depend on temperature, it is allowed to vary as a parameter when fitting Eq. (5.22) to our data, as shown in Fig The optimized values of ν i obtained from the fits are plotted in Fig We note that ν i appears to have an exponential dependene on temperature, as would be expeted for a thermally ativated proess. To be sure, we find that ν i is well fit by the following Arrhenius expression, ν i U = νoexp kt B (5.23) where ν o is a onstant, whose dimensions are frequeny per unit area of the oxidation front, and U is an ativation energy. Taking the natural logarithm of eah side of Eq. (5.23), we have lnν i U = lnνo (5.24) kt B In Fig. 5.7B, we plot lnν i as a funtion of 1 kt, B allowing us to determine ν o and U from a linear fit. 66

75 Figure 5.6 Curve fits to OIS data using Eq. (5.22). The injetion rate is allowed to vary as a fitting parameter. Figure 5.7 A) The fitting parameter values used to produe the urves in Fig. 5.6 are fit with Eq. (5.23). The error bars represent the standard error omputed for the fits in Fig B) Eq. (5.24) is used to obtain the apparent ativation energy and pre-exponential onstant. 67

76 5.1.2 Thermally ativated model In this setion, we adopt the Arrhenius dependene of the injetion rate that was introdued by Eq. (5.23) in Setion Using Eq. (5.23) in Eq. (5.22), we arrive at the following expression for oxidation-indued strain. ε 2Ων U = exp f ( 1+ a) ln( 1 f ) o OI 2 o 3abhSi kt B (5.25) By inorporating thermal ativation into our model, we replae one temperature dependent unknown with two unknowns that are expeted to be onstant with temperature. We may therefore use a nonlinear least squares method to searh for a single value for eah of these parameters, whih is optimized for all temperatures at one. In fat, it is neessary to do so beause the terms that inlude ν o and U have no dependene on f. In hoosing initial values for the optimization, we must make more speifi assumptions about the nature of the injetion proess. We ll onsider two possible senarios, whih predit very different values of ν o and U for a given injetion rate. The first will assume that the oxidizing interfae determines the flux of interstitials into the silion, while the seond assumes that thermal oxidation instead fixes the onentration of interstitials at the interfae. Although the speifi mehanism for injetion of a silion atom is not presently known, there must be an assoiated energy barrier, whih we assume is omparable to the energy of formation of self-interstitial. Let us assume that the interfae produes a onstant net flux of interstitials into the silion, irrespetive of the self-interstitial onentration. The strain rate would then be expeted to have an ativation energy omparable to the energy barrier for injetion of an interstitial. Although a wide range of alulated values is reported for the formation energy of the self-interstitial, we will take the most likely value of ~ 4 ev [58] as our initial guess. More referenes for formation energy alulations an be found in Appendix B. The pre-exponential fator will be regarded as the jump attempt frequeny per unit area of the interfae. This an be estimated from the areal density of silion atoms in the (100) plane and the Debye frequeny, Γ D. The attempt frequeny for an individual atom may be estimated as k θ h B D Γ D = (5.26) where θ D is the Debye temperature of Si. Using the value θ D = 645 K [59], we have 13 1 Γ D = s. If every Si atom at the Si/SiO 2 interfae is a andidate for injetion, we an use the areal density of atoms in the (100) plane to approximate the 68

77 2 frequeny per unit area, whih we will denote as ν D. There are 2 a atoms per unit area on the (100) fae of the diamond ubi unit ell. Using the lattie parameter a = nm for Si, we have ν D 2Γ 1 = = a s nm D (5.27) When the initial values are set as U = 4 ev and νo = νd = 9 10 s nm, we find that the optimization has onsiderable diffiulty in onverging on the optimum values. By varying our initial guesses in the range of 3 U 5 ev and 10 ν o 10 s nm and iteratively using the optimized values as initial guesses for subsequent runs, we find that there are many loal minima in the optimization funtion, whih produe poor fits. As an example, the urves shown in Fig. 5.8 use the values U = 4.26 ev and ν = s nm. o Figure 5.8 Example of poor urve fits that result from high initial guesses for the fitting parameters. Next, we onsider the possibility that the interstitial supersaturation in the silion at the interfae is pinned rather than the flux. In fat, this notion is strongly supported by experimental evidene, as desribed in Setion 2.3, and an be explained in terms of interstitial segregation aross the interfae to satisfy a hemial 69

78 potential balane. Sine the thikness of our nanobeams is very small ompared to the interstitial diffusion length (see Appendix B), we an onlude that the interstitial onentration within our nanobeams will be virtually uniform. If there are no sinks, we would expet the onentration to be everywhere equal to the onentration determined by the interfae, and the net flux of interstitials into the silion would therefore go to 0. As a result, our model would not predit that the strain varies with the volume fration of silion onsumed. However, as mentioned earlier, interstitials are known to form a variety of extrinsi defets, ranging from small lusters to staking faults with intermediate strutures suh as one-dimensional hains. There is also evidene for the formation of boron-interstitial omplexes. Sine all these strutures grow by absorbing interstitials, they would provide the sink neessary to maintain a flux if the onentration of interstitials remains onstant. We therefore postulate that the majority of the volume ontributed by the interstitials is ontained in one or more of the possible types of extended defet. It then follows that our model should be limited by the growth kinetis of the defets, rather than the injetion of interstitials at the interfae, sine even very strong sinks seem unable to perturb the interstitial onentration at the interfae [44]. In this ase, we expet a muh lower ativation energy, sine the formation energy per atom is quite low for many of the possible extended defets. The formation energies per atom of interstitial lusters are found to vary between 0.5 and 1.7 ev, depending on their size, and the formation energy of {113} defets approahes 0.7 ev asymptotially [60, 61]. Sine the value obtained from the empirial fit shown in Fig. 5.7B falls within this range, we will use it as the initial guess in the subsequent optimization. We also note that Hu [62] reported an ativation energy of 2.3 ev for OISF growth. Estimation of the pre-exponential fator, is muh more ompliated, but it is expeted to be muh lower than that the initial guess we used above for two reasons. First, the pre-exponential must reflet the effetive apture radius, sine we have a distribution of defets with finite apture area in a diffusion field. Seond, the effetive attempt frequeny per unit area of the interfae will now depend on the onentration of interstitials already in the silion, whih is expeted to be many orders of magnitude lower than the onentration of atoms on the (100) surfae. We also note that our model may be ompliated by a temperature dependene of the interstitial onentration. Although the supersaturation of * interstitials, CI C I, during oxidation is known to be a dereasing funtion of temperature [24, 39, 44, 48], this may be due to the fat that the equilibrium onentration, that C I * I C, rises with temperature. The data from Refs. [24, 39, 48] indiate * I C is smaller at 950 o C than at 900 o C by a fator of about 2. Aording to * the expression found in Appendix B.3, C I inreases by fator of about 6 for the same temperature range. However, this is only a rough estimate sine it is based on the ontroversial assumption that the equilibrium onentration of interstitials is higher than that of vaanies at the melting temperature. We will therefore make the 70

79 simplifying assumption that the interstitial onentration, and therefore ν o, are approximately independent of oxidation temperature. As shown in Fig. 5.9, the optimization was found to onverge with a good fit when the values obtained from the empirial fit in Fig. 5.7B are used as initial guesses. We also tried initial guesses signifiantly higher and lower these values followed by iteration as desribed above. In all ases, the optimum values onverged -1-2 to ν = 15 s nm and U = 1.5 ev. o -1-2 Figure 5.9 Optimized values for the fitting parameters onverged to ν = 15 snm and U = 1.5 ev with good fits when relatively low initial guesses were used. o Thermally ativated model with stress dependene In Setion 5.1.1, we determined from an empirial fit to our data that the interstitial injetion rate, ν i, has a lear dependene on temperature and an unertain dependene on the fration of silion onsumed. In Setion 5.1.2, we assumed for simpliity that ν i was independent of f and foused on the temperature dependene as ditated by transition state theory. We now propose a theory that adds an dependene through the influene of the biaxial stress state during oxidation, whih was previously investigated in Setion 3.1. f 71

80 We expet the tensile stress in the silion during oxidation to affet our thermally ativated strain model through a modifiation of the ativation energy, muh like the stress in the oxide affets the oxidation kinetis, as desribed in Setion In the previous setion, we onsidered two possible senarios for the aretion of OIS. In the first, we assumed that the strain rate was determined by a fixed flux of interstitials from the interfae. In the seond, we assumed that the strain rate was determined by the kinetis of defet growth in a fixed onentration. Here, we present a general formulation for the influene of stress, whih an be applied to either situation. In either ase, the applied stress is expeted to enhane the strain rate by adding a term to the ativation energy: 2 U = Uo σ SiVa (5.28) 3 where U o is the stress-free ativation energy, whih orresponds to that used in Eq. (5.25), σ Si is the elasti biaxial stress omputed in Setion 3.1 and V a is an ativation volume, whih is expeted to be omparable to the atomi volume. The fator of 2/3 is used to aount for the biaxial nature of the distortion. In the ase of fixed flux, the injetion rate will be enhaned by the effet of stress on the formation energy of an interstitial. In the lassial view, inserting an atom into a self-interstitial site requires work beause there is an assoiated volume inrease against the restoring fore supplied by the bonding of the lattie. Sine, the tensile stress auses a positive strain, the initial volume of the unoupied interstitial site will be inreased. Thus, the tensile stress ontributes work, whih subtrats from the work that must be done to insert an interstitial. The work done by the tensile stress will affet the fixed-onentration ase in muh the same way. If we assume that the onentration of free interstitials is presribed, we must turn our attention to the ativation energy for defet growth. The addition of an atom to a defet has an assoiated energy barrier sine the strain energy of the defet must inrease as it grows. However, the total hange in energy, and therefore the formation energy per atom, will be lowered by the superposed tensile strain. Upon inserting Eq. (5.28) into Eq. (5.19) we have, dε 2 2Ω ν Uo σ o exp 3 SiV a a+ f = df 2 o 3abhSi kt B 1 f (5.29) The dependene of σ Si on f ompliates integration of Eq. (5.29) onsiderably. However, as shown in Fig 5.10, we find that σ Si is very well approximated by a linear fit: 72

81 σ (, ) ( T) f ( f ) σ g T f = T ( ) f Rf Si (5.30) Figure 5.10 The elasti biaxial stress in the silion that was derived in Setion 3.1 (broken lines) is well approximated by a linear fit fored through the origin (solid lines). The slopes are given in Table 5.2 Oxidation Temperature ( o C) (T) (x10 8 Pa) Table 5.2 Slopes of the linear fits shown in Fig

82 When Eq. (5.30) is used to approximate the stress in the silion due to the oxide growth stress, Eq. (5.29) beomes dε 2Ων o Uo 2Vaf 3 a+ f = exp df 2 o 3abhSi kt B 1 f (5.31) whih is integrated to give ε OI Ων o U o = exp 2 o 3abhSi kt B 3kT B 2Va f exp 1 + Va 3kT B 2V 2 2Va( f 1 a V a ) 2( a+ 1exp ) Ei Ei 3 kt B 3 kt B 3 kt B (5.32) in whih Ei(x) is the exponential integral funtion, defined as y e Ei( x) = dy (5.33) x y We now have a new fitting parameter, V a, whih will be taken as the atomi volume, 29 3 Ω= m, as an initial guess. We find that the fitting is again poor when guessing Uo = 4 ev and νo = νd for the fixed flux ondition, as we did before. When using these initial values, the optimization returned the result shown in Fig with optimized values ν = 0.509ν, U = 5.25 ev and V = 23.5Ω. o D o a 74

83 Figure 5.11 Curve fitting yielded poor results when high values were used as initial guesses. When the optimized parameters were used as initial values for suessive runs, their values steadily dereased, with the quality of the fit improving. For low initial -1-2 guesses, the optimized values onverged to ν = 53 s nm, U = 1.7 ev and V a = 1.3Ω from both lower and higher values. The resulting urves are shown in Fig o o 75

84 Figure 5.12 Optimized values for the fitting parameters onverged to and V = 1.3Ω with good fits when low initial guesses were used. a -1-2 ν = 53 s nm, U = 1.7 ev o o The dependene of the injetion rate and injetion ratio on temperature and fration onsumed for the stress-dependent model are as follows: θ ν ( T f ) Uo 2Va f 3, = ν exp kt B ( T f ) i o, = ν U exp ( f a) 2V f 3 2Ω + o a o 2 o kt B abhsi (5.34) (5.35) The injetion rate and injetion ratio obtained using the optimized fitting parameters are plotted in Figs and

85 Figure 5.13 The interstitial injetion rates that orrespond to the urve fits shown in Fig Figure 5.14 The injetion ratios that orrespond to the urve fits shown in Fig

86 5.2 Other mehanisms for oxidation-indued strain Creep Plasti deformation of Si by onservative disloation motion has been observed at elevated temperatures with ritial resolved shear stresses on the order of a few Pa [63, 64]. As argued in Setion 3.1, tensile stresses of several hundreds of MPa are expeted to develop in the silion nanobeams during oxidation. Beause the stress applied by the oxide layers is biaxial and uniform, there are no in-plane shear stresses. Out-of-plane shear stresses an be resolved, though, for two slip diretions in eah of four nonparallel sets of planes that belong to a 2{111} < 110 >, whih is the primary slip system for the diamond ubi struture. We must therefore onsider the possibility that the nanobeams are deformed by disloation reep. However, as illustrated in Fig. 5.15, TEM observations have not revealed any evidene of disloation generation in a beam oxidized to f = 0.82 at 950 o C. We also found no disloations in a nanobeam oxidized to approximately f = 0.55 at 1000 o C. Figure 5.15 A) Dark field TEM image of a 40 nm thik beam oxidized at 950 o C. B) Close up of the noth seen near the top of A. Stress onentration would make the orner a likely site for disloations to nuleate. Inset: Diffration pattern with inident beam along the 001 zone axis. Yet, it is important to note that TEM observations are not onlusive, sine it is energetially favorable for disloations to slip ompletely out of suffiiently small 78

87 rystals. This fat is well known in TEM of disloation strutures. However, evidene of disloation motion should remain in the form of slip steps on the surfae of the nanobeam. Slip lines have not been observed, though, by either SEM (see Fig. 4.1) or Nomarski mirosopy. Nomarski interferene mirosopy, whih is also referred to as differential interferene ontrast (DIC) mirosopy, is a longestablished method for observation of slip lines in rystalline materials. When using refleted light DIC, ontrast is produed by path length differenes due to hanges in the slope of the speimen s surfae. The osine shape of a bukled nanobeam is thus made apparent by hanges in intensity shown in Fig Figure 5.16 DIC image of 10 µm wide by 100 µm long nanobeams viewed at 500 X magnifiation. The beams were redued to a thikness of 39 nm (f =0.82) by oxidation at 1050 o C. Observations were made at higher magnifiations, but there was not suffiient light intensity to apture a good image. Although slip steps have not been observed, we annot onlude that they did not exist during oxidation. For instane, prior to bukling measurements, the thermal oxide is removed with buffered hydrofluori aid. This eth is highly seletive to SiO 2 over Si with an eth rate ratio of SiO 2 :Si > 1000:1. However, slip steps may be damaged given that the atoms at the onvex edge of the step are expeted to be more vulnerable to attak. Sine the lak of evidene for disloation generation and motion is inonlusive, we must also onsider a reep model to explain OIS. Here we assume power-law reep, governed by an equation of the following form. dε dt n Q = Aσ exp kt B (5.36) 79

88 where A and n are onstants and Q is an ativation energy. For single-rystal 6 silion, the values A = and n = 5 are reported for tensile loading [65] and Q 2.3 ev [66, 67]. In Setion 5.1.3, we found that the tensile stress in the silion is well approximated as σ T ( ) f where ( ) Si T ranges from MPa for the temperature regime of interest. Steady-state single-rystal reep data gives strain rates greater than 10-6 /s for the orresponding shear stresses [68], whih are alulated using a Shmid fator of for stress applied in the [110] axial diretion. At this rate, the strain we observe would only take on the order of 100 s to aumulate. In ontrast, our oxidation times range from about 1 to 3 orders of magnitude longer, so we would expet muh larger strains to result from the alulated stresses if power-law reep were the dominant mehanism. However, for the following reason, it is not lear that the shear stress in the silion will be effetive at produing slip. Our beams are not subjet to a simple ase of tension supplied by end loading. Instead, the loading is supplied by the oxide ladding that surrounds the beam. As a result, it appears that the very soure of stress that would drive slip also opposes it. This is illustrated in Fig Figure 5.17 Axial loading supplied by the ompressive growth stress in the oxide and the resolved shear stresses on the {111} slip planes. Beause the stress in the oxide is ompressive while the stress in the silion is tensile, we find that the shear stress resolved on a given plane in the silion is opposed by the shear stress on the same plane in the oxide. In fat, if stati equilibrium is maintained, the shearing fores will be exatly balaned and no slip will be expeted. However, the situation is not stati sine the oxide is onstantly deforming by visous flow. We therefore onsider the possibility that slip is aused by transient or loal fore imbalanes. While we annot estimate the magnitude of the effetive stress that results, we will maintain the temperature dependene by assuming it is some fration, α, of the stress derived in Setion 3.1: 80

89 Using this stress in Eq. (5.36), we have ( ) σ = ασ = αt f (5.37) eff Si Si n Q dε = A( αf ) exp dt kt B The time dependene of strain an be replaed with an demonstrated earlier. Rearranging Eq. (5.16), we have (5.38) f dependene as ( f + a) 2 2 whih is substituted into Eq. (5.38) to give dt = df (5.39) ab ( f + a) n Q 2 dε = A( αf) exp df 2 kt B ab (5.40) The OIS is then obtained by integrating the above expression over f : ( α ) ε n OI (5.41) 2 0 B 0 f 2A Q n dε = exp ( f + a) f df ab kt ε OI n ( α ) Q a n+ 1 2A f = exp 2 + f ab kt B n+ 1 n+ 2 (5.42) If we use the reported values for A, n and Q, Eq. (5.42) an be fit to our data with 10 α as the only free parameter. The optimized value of α = produed the urves shown in Fig

90 Figure 5.18 The power-law reep model is fit to OIS data with the effetive stress allowed to vary as a fitting parameter Dopant segregation When silion is thermally oxidized, dopants initially present in the silion are known to redistribute at the interfae until their hemial potential is the same on eah side [11]. The SIMOX material used in our experiments is lightly doped with boron. Sine the equilibrium onentration of boron at the interfae is larger in the oxide than the silion, segregation aross the Si/SiO 2 interfae during oxidation redues the onentration of boron in the silion. Beause boron is a substitutional defet with a smaller atomi radius than Si, we must onsider the possibility that its depletion will be a soure of strain. However, we an show by a simple alulation that its ontribution will be negligible. The strain that will result from the replaement of boron with silion is ε B ( Ω V ) V nb B = = (5.43) 3V 3 where n B is the number of boron atoms removed per unit volume, V B is the volume oupied by a boron atom in the silion lattie and Ω is the atomi volume of silion. 82

91 For simpliity, let us assume that all of the boron is removed so that n B an be replaed with the initial onentration, C B. Furthermore, we will take the volume oupied by a boron atom to be zero in order to establish an upper limit on the strain, given by ε B ΩCB (5.44) For our material, CB 10 m and the maximum strain is therefore ε B 10, whih is about four orders of magnitude lower than the strain we observe. We an therefore onfidently onlude that segregation of boron is not responsible for the OIS we observe. 5.3 Summary of modeling as ompared with experiment At the start of this hapter, we introdued a model for OIS based on the hange in volume assoiated with the injetion of silion atoms from the oxidizing interfae. By assuming that the volume ontributed by eah injeted atom was equal to the atomi volume, we were able to express the volumetri strain in terms of the hanging geometry of the nanobeam and the injetion ratio, θ. Empirial fitting of our data yielded a measure of θ as well as the injetion rate, ν i, assuming that OIS is indeed aused by interstitial injetion. We found that θ ( f ) does not seem to depend strongly on oxidation temperature, but inreases onsiderably with the fration of silion onsumed. Presumably, this inrease is largely due to the dereasing value of ν with oxidation rate as the oxide gets thiker. It may also inlude a rise of the injetion rate with f, though. Indeed, a rise in ν i with f was later predited by the addition of stress-dependene to the model in Setion The empirially determined injetion rate was found to depend strongly on temperature, with an apparent Arrhenius behavior. This motivated the expansion of our model in Setion to inlude thermal ativation. The empirial dependene of ν i on f, however, is not as lear. The fits displayed a sudden drop from a large value at f = 0. This seemingly non-physial behavior may be an artifat of our lak of data for small f, but is more likely due to the inreasing sensitivity of the injetion rate to the slope of the empirial fit as f 0. Our stress-dependent physial model, whih fits the data quite well, does not predit the sudden drop in injetion rate, rather a steady rise with the fration onsumed (see Fig. 5.13). In Setion 5.1.3, we theorized that this rise results from the tensile stress applied to the silion during oxidation, whih is an inreasing funtion of f. It was reasoned that 83

92 the resulting strain would lower the ativation energy in our Arrhenius expression for ν i. In general, our theory onsiders injeted interstitial atoms as a soure of strain, but does not speify what form they may take one in the bulk of the silion. However, sine our model invokes transition state theory, we must be speifi about the rate limiting mehanism when hoosing initial values for urve fit optimization. If the oxidizing interfae reates a fixed interstitial flux, we expet this flux to be determined by ativation over a large energy barrier with a high attempt frequeny. Curve fits are poor when using the expeted values as initial guesses. If the oxidation front instead fixes the interstitial onentration near the interfae, we expet the flux to be limited by the growth of extrinsi defets. In this ase, a muh lower ativation energy and effetive attempt frequeny are expeted. This is in agreement with the results of urve fitting when the injetion rate is assumed onstant, but allowed to vary as a fitting parameter for eah temperature separately. Also, for the thermally ativated and stress-dependent thermally ativated models, optimization produes good urve fits that onverge on low values for the pre-exponential onstant and ativation energy when low initial guesses are used. In Setion 5.2, we onsidered alternative soures of strain. No evidene of disloation reep was found by TEM or Nomarski mirosopy, but our data was ompared with a power-law reep model nonetheless. Values from the literature were used for the pre-exponential and exponential onstants as well as the ativation energy. The effetive stress driving reep is expeted to be smaller than that derived in Setion 3.1, but should maintain its temperature dependene. The alulated stress was therefore multiplied by saling fator, whih was allowed to vary as a fitting parameter. The optimized value of the saling fator was indeed small, but the urve fits were poor. In Setion 5.3, we also onsidered the strain ontribution due to boron segregation, but found it to be negligible. 84

93 Chapter 6 Methods for reduing oxidation-indued strain In this hapter, methods for reduing the oxidation-indued strain are investigated. First, we onsider the effet of methods direted toward stress redution, whih may influene the interstitial injetion rate aording to the stressdependent model developed in Setion The stress levels experiened by a nanobeam oxidized to a given fration onsumed an be redued by an inremental proessing sequene in whih removal of the oxide is followed by re-oxidation. This is desribed in Setion 6.1. In Setion 6.2, we desribe an alternative method based on reports that oxidation in a hlorine-ontaining atmosphere redues OED and the growth of OISF [69-72]. The presene of a hlorine-ontaining ompound has been suggested to redue the interstitial supersaturation. The effet of hlorine may, in fat, be two-fold, sine it is also known to redue the growth stress during oxidation [20]. 85

94 6.1 Inremental oxidation and ething Sine the stress applied to the silion inreases with the thikness of the oxide, it an be effetively redued through a proess of inremental oxidation in whih the nanobeam is oxidized and the oxide is ethed away before re-oxidizing. Aording to the model presented in Setion 3.1, the stress in the nanobeam is ramped up as the oxide thikness, and therefore the fration onsumed, is inreased to a given value. However, as shown in Fig. 6.1, the overall stress experiened during oxidation an be redued if the given fration onsumed is ahieved in two steps with the oxide removed in-between. The stress history is shown shematially in terms of f for omparison with the interstitial injetion model. When onsidering the impliations of two-step oxidation with regard to reep, it is also helpful to express the stress history in time. We note that not only is the magnitude of the stress redued, but so is the total time over whih the stress is applied. This is a diret onsequene of the fat that the oxidation rate is muh faster when the film is thin. Figure 6.1 The approximate stress history is shown for speimens represented by points A and B in Fig. 6.2, whih were re-oxidized to give the data points A and B. The stress inreases as the oxidation proeeds to 0.6, when it is then redued to zero by removal of the oxide. The stress then begins inreasing again as the speimen is re-oxidized to 0.8. The stress for a beam oxidized to 0.8 in a single step is shown for omparison. Two of the nanobeams speimens originally oxidized at 950 o C were reoxidized as desribed above. As shown in Fig. 6.2, the resulting strain appears to be onsistently slightly lower than that whih results from a single oxidation for the same fration onsumed. However, the differene is omparable to the satter in our data. 86

95 Figure 6.2 After oxide removal and measurement, the speimens orresponding to data points A and B were re-oxidized to produe data points A and B. Optimization of the stress-dependent model of Setion is shown for omparison. 6.2 Oxidation in the presene of hlorine The addition of Cl ontaining ompounds suh as HCl, C 2 HCl 3 and C 2 H 2 Cl 2 and to dry O 2 is reported to redue the supersaturation of interstitials generated during oxidation. In partiular, the use of HCl has been shown to retard the growth of OISF s at oxidation temperatures greater than 1100 o C [69, 70] and to ause a redution of OED [71, 72] at temperatures greater than 1000 o C. If OIS is indeed a result of interstitial injetion and the effet of hlorine is to redue the interstitial supersaturation, we would therefore expet to measure lower strains when HCl is added to the oxidizing ambient. To test this idea, two speimens were oxidized with HCl at 950 o C using a liquid soure material known as Trans-LC, whose hemial formula is C 2 H 2 Cl 2. Trans-LC is delivered to the oxidizing ambient using dry nitrogen as a arrier gas, whih is bubbled through the liquid. Upon entering the furnae, Trans-LC is oxidized to produe HCl aording to the following reation. CHCl +2O 2HCl+2CO

96 Oxidation furnae restritions limited the available gas flow rates to 3000 sm of O 2 and 150 sm of N 2 arrier, resulting in an ambient that was approximately 4.8 % HCl by volume. The resulting strain data is shown in Fig Figure 6.3 Oxidation-indued strain arued during oxidation in dry O 2 is ompared with that when approximately 4.8 % HCl by volume is added to the ambient. As with the inremental oxidation treatment, the use of HCl appears to redue OIS slightly, but the differene is omparable to the spread in the data. In retrospet, the experiment should have been performed for a higher temperature, sine the effet of HCl beomes muh stronger as the oxidation temperature is inreased. For example, 0.3 % HCl by volume is suffiient to ause OISF retrogrowth at 1150 o C, while as muh as 1.0 % is required to do so at 1100 o C [69]. Moreover, OISF growth an be ompletely suppressed at 1150 o C, by the addition of 1.0 % HCl, while 3 to 6 % is required at 1100 o C. Therefore, sine the effet of HCl at 950 o C is expeted to be muh less than at higher temperatures, our HCl oxidation experiments are not as onlusive as they were intended to be. 88

97 Chapter 7 Conluding remarks and reommendations In Chapter 5, we onsidered several possible mehanisms for the origin of the oxidation-indued strain. In this hapter, we present an argument that an aumulation of injeted interstitial silion atoms is the most likely ause of OIS. In partiular, the net rate of injetion for nanobeams is thought to be limited by the growth of extrinsi defets, suh as interstitial lusters, hains and extrinsi partial disloations. This theory is in aordane with a number of experimental observations [43, 44] and interstitial injetion models [38, 40, 41], whih indiate that the supersaturation of interstitials at the oxidizing interfae is pinned by a hemial potential balane aross the interfae. Beause the nanobeams are very thin ompared to the diffusion length of self-interstitials, we assume that the oxidizing interfaes reate a onstant supersaturation throughout the beam. With no onentration gradient, there would be no net flux in the absene of defet growth. However, a flux ours if defets are present and serve as interstitial sinks. Sine an explanation for OIS based on dopant segregation was already ruled out in setion 5.2.2, we will first argue against reep before pointing to the strengths of the interstitial injetion models. 7.1 Disussion In Setion 5.2.1, we stated that no evidene for plasti deformation was found in the form of disloations or slip steps on the surfae of nanobeams. Nevertheless, we developed a model for OIS by power-law reep for omparison with our experiment. It was found, using reported values for parameters that desribe powerlaw reep in single-rystal silion, that urve fitting produed poor results. It should be noted that the values used for the pre-exponential and exponential onstants govern steady-state reep, whih may not be ahieved at the small strain we measure. However, Stage I reep rates are normally faster than the steady-state reep rate of Stage II. This would predit a sharper rise in OIS, making the urve fits even worse. Not only did the optimized urves deviate signifiantly from the data, but they also displayed a strong temperature rank-ordering, whih is not evident from our experimental results. Sine the predited strains derease greatly with inreasing temperature, it would appear that the reep rate is influened more strongly by the redution of stress at higher temperatures than the thermal ativation term. If this is the ase, we would expet a more signifiant redution of strain in the re-oxidation experiments of Setion 6.1. In fat, the inremental oxidation should have a two-fold effet on reep. As shown in Fig. 6.1, the time required to reah a given fration onsumed is signifiantly redued by inremental proessing as well. Therefore, not only would the strain rate be dereased by the lower applied stress, but it would also be integrated over a shorter time. On the basis of these arguments, it would appear that the experimental evidene negates disloation reep as the soure of strain, but we also offer some 89

98 reasoning for why it may not play a part. We have already argued that the effetive driving fore for slip of the silion will be ounterated by the shearing fores in the oxide layers. However, even if large net fores exist loally or transiently, the absene of plasti deformation in our nanobeams may be explained by the long reognized effet of sample size on yield strength. Enormous inreases in the yield strength of single-rystal metals has been observed for miron sale samples loaded in both tension [73, 74] and ompression [75]. Inreased yield strengths have also been observed in small silion strutures, even at elevated temperatures. At 300 o C, single-rystal silion wires with ross setion dimensions measuring 200 by 255 nm exhibited a ritial resolved shear stress of 6 GPa, whih is larger than that of millimeter sale speimens by a fator of 10 [76]. Sine our samples are nanosale in the relevant dimension, we should expet a onsiderable inrease in strength as well. The size effet desribed above may be related to the nuleation of disloations. Although disloations are relatively easy to move at high temperatures, they are not so easily reated. With threading disloation densities on the order of one or less per m 2, the SIMOX material we use an be taken as being initially free of disloations. We therefore expet plasti deformation of nanobeams to be limited by disloation nuleation. Sine the nuleation of disloations is likely aided by stress onentrations, there are two reasons that nanobeams may be so resilient to deformation. First, small strutures will inherently have small flaws and therefore small stress onentrations. Seond, existing surfae flaws will be removed as silion is onsumed to form oxide. Without signifiant stress onentration due to flaws, nuleation of disloations, and hene any disloation plastiity, may be unlikely. Sine the absene of reep an be reasoned and is supported by experiment, we turn now to the models based in interstitial injetion. At the beginning of Chapter 5, we introdued the interstitial injetion modeling with the assumption that the flux of silion atoms into the bulk would ause an inrease in volume measurable as a volumetri strain. The magnitude of this volume hange is expeted to depend to some extent on the onfigurations assumed by the extra atoms, should they aggregate into extended defets of some sort. However, the assumption that one atomi volume is added per injeted atom is judged to be a suffiiently good approximation for the evaluation of our model. We did not make any assumptions about whether suh defets are formed or, if so, whether the oxidation-indued strain rate is limited by their growth or the injetion mehanism. For either ase, though, we arrive at an expression of the same general form. The oxidation-indued strain as a funtion of the temperature and volume fration of silion onsumed ontains a thermally ativated term assoiated with the net flux of atoms into the bulk and an f dependene related to the hanging geometry of the nanobeam. The thermal ativation term in the oxidation-indued strain formulation is supported by empirial fitting of our data. Without making any assumptions about the net rate of interstitial injetion, we found that it must be an inreasing funtion of temperature if injetion is indeed responsible for the observed strain. Furthermore, when the injetion rate was assumed to be independent of the fration onsumed, we found that it was well desribed by an Arrhenius expression. We later expanded the 90

99 model to address the possible dependene of injetion rate on the fration onsumed by onsidering the effet of stress applied by the growing oxide. Sine this was implemented by a modifiation of the energy barrier in the Arrhenius term, though, it remains onsistent with a thermally ativated proess. It would appear that, within the resolution of our measurements, the magnitude of OIS is independent of oxidation temperature. This was initially surprising in the ontext of a thermally ativated model, sine the injetion rate is predited to inrease with temperature. However, it is important to note that when expressing strain in terms of f, it is neessary to inorporate the rate at whih atoms are onsumed, whih is also an inreasing funtion of temperature. Consequently, the rates of injetion and silion onsumption by oxidation produe ompeting dependenies on temperature when OIS is plotted against f. If oxidation-indued strain is instead plotted as a funtion of oxidation time, as shown in Fig. 7.1, we find a lear temperature ordering. Figure 7.1 Oxidation-indued strain data is plotted as a funtion of oxidation time. Before proeeding with urve fitting to evaluate the appliability of the injetion-based model, we found it neessary to be more speifi about the origin of the Arrhenius dependene. In general, we assume that OIS results from the injetion of self-interstitial atoms at the oxidizing interfae, regardless of whether or not they aggregate to form extrinsi defets one in the silion bulk, and that the strain rate 91

100 will be haraterized by the net flux of these point defets. This flux appears to be thermally ativated, but the flux-determining mehanism is not lear. When onsidering the following two possibilities, we find that the expeted ativation energies and pre-exponential onstants will differ greatly. If the flux is determined by the oxidizing interfae, we expet a large ativation energy assoiated with the formation energy of a self-interstitial, and a large attempt frequeny due to the high density of silion atoms at the interfae. In this ase, initial guesses for these values an be made with some degree of onfidene. Although a wide range of values for the self-interstitial formation energy have been reported over the years, the most reent state-of-the-art alulations point to a most likely value of 4 ev, and the attempt frequeny an be estimated from the mean phonon frequeny. These values differ greatly from those obtained by the empirial fitting, though, and they produe poor fits when used in both the thermally ativated and stress-dependent thermally ativated models. We also note that urve fits using the stress-dependent model require ativation volumes more than an order of magnitude higher than might be expeted. If we instead assume that the onentration is fixed by the oxidizing interfae, then the flux will be determined by the growth of sinks, suh as extrinsi defets. Beause the thikness of the nanobeams is so small and interstitials diffuse so quikly, we would expet the onentration over the entire volume of the nanobeam to be held onstant at the interfae value if no interstitial sinks existed. In this ase, the net flux would be zero and no strain would arue. However, if the presene of extrinsi defets were to provide a sink as they grow by olleting interstitials, a flux would be required to maintain the onentration. In this ase, the net flux would be determined by the growth kinetis of the defets. The ativation energy and effetive attempt frequeny are expeted to be signifiantly lower than for the fixed flux senario, but the estimation of these values is not so straightforward. For instane, both of these parameters will be influened by diffusion of the interstitials as well as the kinetis of their interation with the defets. The ativation energy for interstitial diffusion is likely quite low, but unertain, with most reported values ranging from about 0.4 to 1.1 ev (see Appendix B.3). The attempt frequeny for diffusion will also be low, but equally unertain beause it will depend on the onentration of interstitials. Experiments an give us information about the supersaturation of interstitials during oxidation, but their equilibrium onentration, C, and, therefore, their absolute onentration is * I * neessarily as vague as their diffusivity. Even so, reported estimations of C I and supersaturation values indiate that the onentration of interstitials per unit area of the interfae will be many orders of magnitude lower than onentration of atoms on the (100) surfae. The kinetis of interstitial apture is also diffiult to estimate. Sine the formation energy per atom of extended defets is lower than that of a self-interstitial, energy will be liberated by the apture of an atom. However this does not rule out an 92

101 unknown, though likely small, energy barrier for the apture mehanism. Also, the pre-exponential fator will depend on the size and distribution of the defets. Although the inlusion of a defet growth term greatly ompliates our model, this piture is supported by several of our observations: 1) Very good urve fits are produed with relatively low optimized values for ativation energy and the pre-exponential onstant. 2) The values of the optimum parameters obtained from fitting of our thermally ativated and stress-dependent models are omparable with those obtained from the empirial fitting, whih makes no assumption about the mehanisms that govern interstitial flux. 3) When using the stress-dependent model, we find that the optimized value for the ativation volume is quite lose to the atomi volume, as expeted. This is in ontrast to the high ativation volumes that aompany high attempt frequenies and ativation energies. 4) Sine the quality of the fits is not greatly affeted by the addition of stress dependene to the model, it seems that stress has a relatively small effet on the OIS. This may explain why our inremental oxidation experiments redued strain only slightly. 5) Our empirial determination of θ is at the low end of the range of reported values, as shown in Table 7.1. The reported values of θ appear to inrease with the strength of the interstitial sinks present during oxidation. This would be onsistent with the impliation that interstitial flux, and therefore θ, depends on the strength of the interstitial sinks present. Method Average θ Implantation SF growth [45, 46] 1-6 x 10-3 OISF growth [47] 2 x 10-4 This work ~1 x 10-4 OED [38] < 10-4 inreasing strength of interstitial sinks reated Table 7.1 The average value of the injetion ratio obtained from the stress-dependent thermally ativated model (Fig. 5.14) is given in omparison to values obtained by other experiments. The highest reported values of θ are determined from experiments involving a dense layer of staking faults indued by Si ion implantation prior to oxidation [45, 46]. This is expeted to be a very strong sink. Lower values are obtained when Si ion implantation is not used, but the density of nuleation sites for OISF is inreased [47]. The lowest estimates of θ are made by our work and OED experiments [38], in whih no pretreatment is performed to introdue sinks. 93

102 Sine the fixed-onentration approah is also onsistent with experimental measures of dopant diffusion phenomena [43, 44], we onlude that the most likely ause of OIS is extrinsi defet growth due to the injetion of self-interstitials. In our judgment, the smallest defets, suh as interstitial lusters, hains and staking fault nulei are the most likely andidates for three reasons. First, these types of defets would have been diffiult or impossible to detet by TEM, whereas larger defets suh as OISFs are easily imaged. We found neither. Seond, the large size and low number density of OISFs observed during oxidation under onditions similar to ours [77, 78] would be at odds with the uniformity of the strain we measure. With a 4 2 number density 10 m, only 1 in 10 nanobeams measuring 10 µ m wide by 100 µ m long would have a staking fault. However, as indiated by the error bars in Fig. 4.15, the standard deviation of our measurements from beam to beam is quite small. Third, the ativation energy of 2.3 ev reported for the growth of OISFs [62] is signifiantly larger than the values we find, whih range from 1.5 to 1.7 ev. 7.2 Summary and onlusions The purpose of this work was to investigate an unexplained soure of permanent strain that ours on the nanosale in single-rystal silion strutures when subjet to thermal oxidation. It was found that axial strain results when oxidation and subsequent oxide removal is used to redue the thikness of single-rystal silion nanobeams. This oxidation-indued strain, whih is manifested in the bukling of our nanobeams, was quantified by vertial-sanning optial interferometry measurements. Beause SiO 2 grows in ompression, a tensile stress is applied to the silion nanobeams during oxidation. Plasti deformation by disloation reep therefore beame an immediate suspet for the soure of OIS, sine the yield stress of silion is dereased signifiantly at elevated temperatures. However, for reasons argued above, reep is not believed to be an ative mehanism for OIS. Beause the magnitude of OIS is so small, we turned our suspiions to a long known phenomenon that ours during the thermal oxidation of silion the injetion of self-interstitial silion from the oxidizing interfae. We proposed that the measured OIS is a omponent of the volumetri strain that aompanies the injetion of silion atoms from the interfae. If OIS indeed results from the injetion of self-interstitials, the strain rate will be determined by the flux of atoms into the silion bulk. However, experimental evidene [43, 44] and modeling [38, 40, 41] indiate that the interfae establishes the onentration of interstitials at the interfae, rather than the flux. What's more, sine our beams are muh thinner than the diffusion length, we an expet that the onentration of interstitials throughout the beam is pratially onstant. With no onentration gradient, there would be no net flux and therefore no inrease in strain. However, we an expet a flux if extrinsi defets, whih an at as interstitial sinks, are able to form and grow. In fat, a variety of self-interstitial agglomerates are known to form in silion, ranging from pairs to small lusters, hains, and extrinsi disloation loops. When fit to our experimental data, our interstitial injetion based 94

103 models produe realisti parameters for OIS aumulation as limited by the growth kinetis of extrinsi defets with an ativation energy in the range of 1.5 to 1.7 ev. In onlusion, this work has taken a new perspetive on the injetion of selfinterstitial silion from the oxidizing interfae. In the past, evidene for this phenomenon and lues about its nature have been given by the formation of oxidation-indued staking faults and the enhanement/retardation of dopant diffusion. In this work, we investigate interstitial injetion via its mehanial effets on nanosale strutures. We find that a model for oxidation-indued strain based on self-interstitial injetion is onsistent with experimental observations, but only if we onsider the growth of extrinsi defets as a limit on the net flux of atoms into the silion. We find that for nanobeams, the interstitial injetion ratio is an inreasing funtion of the fration of silion onsumed aording to our model, and the average value is on the order of θ In the ontext of a fixed interstitial onentration at the avg interfae, we propose that θ depends on the strength of the interstitial sinks present, and that this aounts for the wide range of values reported in the literature. 7.3 Future Work Although our interstitial injetion model produes good fitting of the oxidation-indued strain data, our experiments do not diretly orrelate interstitial injetion with the buildup of strain. Nor do we diretly observe the extrinsi defets that are so essential in interpreting the values of our optimized fitting parameters. We annot diretly observe the injetion of atoms during oxidation, but we may be able to indiretly orrelate the phenomenon with strain by further oxidation experiments in a horine ontaining ambient. As disussed in setion 6.2, the presene of Cl is believed to redue the supersaturation of interstitials present during oxidation, and should therefore redue OIS. Our preliminary experiments seem to indiate a slight redution in OIS with the use of Cl. However, oxidation with Cl should be performed at higher temperatures if we expet to observe a definite effet. A redution in OIS would support the interstitial injetion piture, but we must note that it may also be onsistent with a reep model, sine Cl is known to redue the visosity of SiO 2, and therefore the growth stress as well [20]. There is another experiment, though, that may be able to orrelate interstitials with OIS while ruling out plasti deformation. It has been shown that SiO 2 an be removed by long-term, high temperature annealing in vauum [79-81] or argon [43] by redution of SiO 2 and Si to form SiO, whih desorbs at the gas/oxide interfae. Sine this requires a flow of Si into the oxide, an interstitial undersaturation or vaany supersaturation is established at the interfae, as evidened by staking fault shrinkage and the reversal of OED/ORD effets during annealing of thin oxide films in argon ambients [82]. These effets beome appreiable for films less than about 50 nm thik. Sine the oxide film ats as a sink rather than a soure of Si atoms under non-oxidizing onditions, annealing 95

104 may be able to reverse OIS as the extrinsi defets dissoiate to maintain the equilibrium onentration of self-interstitials in the silion. Furthermore, if this effet were observed, we ould argue against disloation reep sine the experiment is performed after the strain is imparted. Sine the stress in a non-oxidizing SiO 2 film will be quikly relaxed at high temperature, there would be no apparent mehanism for reversal of plasti strain. Further evidene against disloation reep may be provided by atomi fore mirosopy (AFM) measurements to searh for slip steps on the surfae of nanobeams. As noted in Setion 5.2.1, slip steps were not observed by Nomarski mirosopy or SEM. However, these methods may lak the spatial resolution neessary to disern nanosale slip. The experiments desribed above are aimed at orrelation of OIS with interstitial injetion. In order to support our model, though, it is also desirable to establish the presene of small extrinsi defets. This may be ahieved by more extensive TEM work, but the smallest defet strutures would not be detetable by this method. Instead, we may find evidene of defets via eletrial measurements. For example, the presene of defets should have a signifiant effet on arrier mobility. Also, interstitial lusters display both photoluminesene (PL) and deeplevel transient spetrosopy (DLTS) signatures [83-85]. The native oxide that forms on silion is a high quality dieletri with exellent mehanial properties. It was therefore the natural hoie made deades ago for the gate material used in silion transistors, whih remain at the heart of the miroproessor industry today. Beause the thermal oxidation of silion is so important, researhers have dediated a great deal time and effort to understanding this proess over the years. However, even after nearly half of entury, there are still many mysteries to be solved, and this dissertation has only alled attention to a few. 96

105 Appendix A Fabriation of Si nanobeams Figure A.1 Shemati illustration of proessing steps for fabriation of nanobeams using SIMOX wafers as viewed parallel to the beam axis. Proessing onditions are detailed on the following page. 97