Dislocations are responsible for crystalline solids' fundamental mechanical properties of ductility and strength. Continuum based techniques such as Dislocations Dynamics (DD) are adept at dealing with dislocation mobility but cannot handle dislocation nucleation, which, being an atomistic phenomenon, can be best approached by atomistic studies. To date, most of the atomistic level simulation work done on dislocation nucleation has been on heterogeneous systems. However, there are an infinite number of configurations of heterogeneous systems, each with its own peculiar complexities, making it difficult to organize the subject in a coherent manner. The transition of dislocation nucleation cannot be studied at a fundamental level while focusing on inhomogeneous systems because the results are subject to influence by complex effects originating from heterogeneities such as stress fields, surfaces, interfaces, and other defects, which distort the physical picture. Also, the activation energies reported in such studies are presented as functions of nominal stresses, whereas the actual stresses at the dislocation core are much higher than those reported because of the presence of heterogeneities. To observe the phenomenon of dislocation nucleation in a fundamental way, free from complex effects such as stress concentrations, surfaces, etc., analysis of homogeneous dislocation is necessary. Homogeneous dislocation nucleation has also assumed great importance in view of the recent focus on nanoindentation as a means of ascertaining the onset of plasticity in single crystals. From this point of view, there have been efforts of understanding dislocation nucleation using Molecular Dynamics (MD). A serious drawback of MD, however, is the limited time-scale at its disposal. Consequently, very high strain-rates become necessary, resulting in a wide separation from the experimental time-scale of dislocation nucleation, which, being a stress-mediated thermally activated transition, occurs at stress levels significantly lower than the athermal stress.

The Nudged Elastic Band (NEB) method is a reaction pathway sampling algorithm that makes it possible to directly determine activation parameters (activation energy and activation volume) by searching the minimum energy path (MEP) of the transition. The highest point on the MEP is called the saddle-point and the energy difference between the saddle-point and the initial configuration is the activation energy input required in the form of thermal fluctuation if the transition is to occur.

In this study, using the NEB method we focus on homogeneous crystals of various classes of materials and this allows us to examine the problem of dislocation nucleation at a fundamental level, where complex surface and stress concentration effects are not allowed to distort the physical picture. Once a coherent methodology is established for predicting the favorability conditions for dislocation nucleation, the nucleated dislocation can be handed over to DD for further analysis, in the field of semiconductor and MEMS design.

The first application of the NEB method in the domain of homogeneous dislocation nucleation was Boyer's work on perfect crystal Cu. After reproducing Boyer's work on Cu, reaction pathway analysis for homogeneous dislocation nucleation in Si and Mo is carried out. Activation energy, activation volume, and the mechanics of the dislocation core are examined for all cases with the aim of finding out the influence of lattice structure on the phenomenon of dislocation nucleation. Activation volume assumes great importance because of the fact that unlike activation energy, it can be determined experimentally. The second aim of this research is to compare the atomistic results obtained with results based on the Peierls-Nabarro (PN) model in order to ascertain the limitations of approximate solutions such as the PN framework. Thirdly, this research seeks to link homogeneous dislocation nucleation with heterogeneous dislocation nucleation by comparison with one kind of heterogeneous system, namely dislocation from a sharp corner. Finally, this research also aims to deepen insight into the old but recently reinvigorated glide-shuffle debate in Si.

In total, this thesis is composed of 7 chapters. The contents of each chapter are summarized as follows.

Chapter 1: Introduction

In Chapter 1, the background of dislocation nucleation in which the present research is set is introduced, the aims of the thesis are listed, and the structure of the thesis is explained.

Chapter 2: Method

In Chapter 2, the simulations tools used in this research are introduced. The major simulation algorithm used in this research is the NEB method and other extended schemes, which are explained. However, classical atomistic calculation tools such as molecular dynamics (MD) and Conjugate Gradient (CG) relaxation are also extensively employed for preparing input images for the NEB algorithm.

Chapter 2 also contains details of artificial loop insertion, the methodology of applying the resolved shear stress on the slip-plane, interatomic potentials employed, and the visualization technique used throughout the thesis to show the mechanics of atomistic saddle-point configurations.

Chapter 3: Models and results

Chapter 3 contains the simulation results along with the model details for dislocation nucleation in face-centered cubic (FCC) Cu, body-centered cubic (BCC) Mo, and nucleation of dislocation of the shuffle-set and glide-set in diamond-cubic (DC) Si. In this chapter, simulations results of all calculations are presented without analysis, while the analyses are carried out using these results in the subsequent chapters that deal with the different aims of this thesis.

Chapter 4: Comparison between Si, Cu and Mo

Chapter 4 documents the comparison of homogeneous dislocation nucleation in three representative materials corresponding to three common lattice structures: FCC Cu, BCC Mo, and DC Si. The comparison is presented in the form of the activation energies, the mechanics of the saddle-point configurations and the activation volumes.

Atomistic results show that approximate indicators thought to be closely related to dislocation nucleation, such as the product of the unstable stacking fault energy and square of the Burgers vector, can be misleading even in the qualitative sense because they give higher activation energy for dislocation nucleation for Si than Mo, which is contrary to the atomistic results. Atomistic calculations are therefore necessary for accurate description of the phenomenon. Also made clear by atomistic calculations is that dislocation nucleation is a two-plane phenomenon in Cu, whereas depending upon the stresses involved it is accompanied by considerable atomistic displacements in planes above and below the slip plane in Mo as well as Si. Therefore, while the structural definition, which consists of integrating the contribution of activation volume from particles confined to the slip-plane, gives accurate results for Cu, its results can be misleading for Mo and Si.

Chapter 4 also presents the comparison of homogeneous dislocation nucleation with heterogeneous dislocation nucleation with an aim to improve the understanding about the link between the two fundamental types of dislocation nucleation. Since there are an infinite number of scenarios of heterogeneous dislocation nucleation, a few representative ones are chosen from literature and compared with homogeneous systems in terms of activation volumes and mechanics of the dislocation cores. The activation volumes are of the same order of magnitude for the two types of dislocations, with those for homogeneous dislocation nucleation slightly higher than the heterogeneous case. The reasons behind this fact are the difference between the shapes and radiuses of dislocation loops, as well as the surface effects.

Chapter 5: Comparison with PN based models

In Chapter 5 the atomistic results from the last chapter are compared with results of Peierls-Nabarro (PN) based hybrid model for homogeneous dislocation nucleation. The limitations of the PN model are highlighted along with the information that can only be made available by fully atomistic approaches. The P-N model results overestimate the atomic displacements at the saddle-point, and overestimate the activation energy requirements by a factor of 2 for Si and 5 for Cu.

The P-N based models, not being fully atomistic in nature, cannot accurately provide discrete atomistic details such as shape of the dislocation core, the presence or otherwise of extra slip-plane displacements and the actual directions of atomistic displacements. Atomistic simulation is therefore irreplaceable in order to get insight into the phenomenon of dislocation nucleation.

Chapter 6: The shuffle-glide controversy in Si

Chapter 6 deals with the shuffle-glide controversy in Si in terms of the activation energy, activation volume, and the mechanics of the dislocation cores in homogeneous dislocation nucleation. The role of mobility is also touched upon, along with comparison with results for heterogeneous dislocation nucleation regarding the shuffle-glide debate. The presence of two nucleation regimes is confirmed, with the cross-over point at 6.5 GPa. Nucleation accounts for bulk of the reaction pathway in the shuffle-set. In glide-set on the other hand, nucleation accounts for minor portion of the reaction pathway, but the low mobility of glide-set results in undulations in the energy curve. The inadequacy of the structural definition of activation volume is again proved, even more in the case of shuffle-set than the glide-set, in this chapter.

Chapter 7: Conclusions and future directions

In this chapter, the conclusions drawn from the thesis are summarized along with possible directions of future research and applications of atomistic simulation in the field of dislocation nucleation.

転位の生成現象に関する研究は、パイエルス、ライスらによる転位論からのアプローチが古くから行われている。しかしながら、連続体的取り扱いでは、原子スケールの転位芯の取り扱いがあいまいになり精度に欠けるという問題があった。

一方、近年の原子シミュレーションの進歩は、転位の生成に関して、一定の成果を挙げてきた。しかしながら、現在、最も大きな課題は、時間スケールが長い転位の生成現象に対して、分子動力学などの原子振動を扱う手法は、適用できないということであった。しかしながら、近年Nudged Elastic Band法を始めとする時間スケールの問題を克服する手法が多く提案され、その問題が解決に向かいつつある。

本論文では、Nudged Elastic band法(NEB法)をベースとした転位の反応経路解析を使って、固体中の転位の均質核生成の活性化エネルギー及び活性化体積の応力依存性を調べる手法をFCC, BCC, ダイヤモンド構造など様々な結晶構造へ応用し、結晶構造による物理現象の違いを明らかにするものである。さらに、シリコンに対しては、異なる種類の転位であるシャフルセット転位とグライドセット転位の核生成の違いを明らかにした。また、従来の連続体ベースの転位論では扱えない原子スケール特有の現象について指摘し、本手法の有効性について提案した。

第一章では、研究の背景と目的について述べられている。

第二章では、用いたシミュレーション手法であるNEB法の詳細と、転位の生成過程を表現するために不可欠である人工的な転位のすべり面への挿入手法が述べられている。これらは、NEB法の転位形状のレプリカを作成するために必須な手法である。

第三章では、FCC構造としてCu, BCC構造としてMo, ダイヤモンド構造としてSiの均質核生成解析モデルについての詳細が述べられている。

第四章では、転位の均質核生成プロセスに対して、Cu, Mo, Siについて、活性化エネルギー、鞍点の転位の形状、活性化体積の比較を行っている。結果、活性化エネルギー及び体積はMo, Si, Cuの順に高くなることがわかった。転位の形状については、Cuはすべり面内に限定されたすべり変位が生じるが、Mo, Siはすべり面外の変位が大きく、転位生成の障壁に重要な役割を果たすことがわかった。さらに、Moでは、バーガースベクトルと異なる方向へのすべりが生じていることがわかった。

また、従来転位の生成の障壁の指標として用いられてきた不安定積層欠陥エネルギーが誤った評価をもたらすことを示した。具体的には、不安定積層欠陥エネルギーは、MoよりSiのほうが大きかったが、活性化エネルギーはMoのほうが大幅に大きいことがわかり、定性的にも不安定積層欠陥エネルギーの評価は問題があることがわかった。さらに、均質核生成と角部や表面からの不均質核生成との比較も行われ、形状・応力の不均質性に起因して、主に臨界核の形状が異なり、活性化体積が小さくなることがわかった。

第五章では、パイエルスーナバロウモデルとの比較を行い、パイエルスモデルが活性化エネルギーをSiで2倍、Moで5倍程度過大評価すること、原子レベルでは、すべり変位がバーガースベクトルより小さいという転位芯の構造を正しく表現しないことが分かり、本原子スケールモデルの有効性を示した。

第六章では、シリコンのグライドセット転位とシャフルセット転位の比較を行った。結果、6.5GPa以上の応力ではシャフルセット転位の活性化エネルギーがグライドセット転位より小さくなり、それ以下の応力では逆にグライドセット転位のほうが小さくなることがわかった。これは、高温低応力ではグライドセット転位が観察され、低温高応力でシャフルセット転位が観察される実験結果を定性的に説明していると考えられ、長年議論されてきたシャフルーグライド論争を均質核生成の観点から明らかにした。

第七章では、結論と今後の課題が述べられている。さらなる、材料の展開、不均質核生成との関係の一般化が課題であることが示された。

よって、本論文は博士(工学)の学位請求論文として合格と認められる。