List of gas phase reactions and corresponding rate constants [24].
\\n\\n
Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\\n\\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'IntechOpen is proud to announce that 179 of our authors have made the Clarivate™ Highly Cited Researchers List for 2020, ranking them among the top 1% most-cited.
\n\nThroughout the years, the list has named a total of 252 IntechOpen authors as Highly Cited. Of those researchers, 69 have been featured on the list multiple times.
\n\n\n\nReleased this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\n\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
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Nonlinear optical phenomena such as self-focusing, self-phase modulation, soliton formation and propagation, higher harmonic generation, different types of stimulated light scattering, and four-wave mixing have attracted interest from the fundamental point of view of the investigation of light/matter interaction, and as a basis for applications in contemporary optical communications and optical signal processing. Nonlinear Optics - Novel Results in Theory and Applications contains novel results concerning the mathematical methods of nonlinear optical phenomena analysis, soliton formation and propagation in optical fibers, and peculiarities of nonlinear optical phenomena in micro- and nanostructures. The book may be interesting for researchers and engineers interested in nonlinear optics, lasers, and optical communications.",isbn:"978-1-78985-164-9",printIsbn:"978-1-78985-163-2",pdfIsbn:"978-1-83962-009-6",doi:"10.5772/intechopen.77311",price:119,priceEur:129,priceUsd:155,slug:"nonlinear-optics-novel-results-in-theory-and-applications",numberOfPages:154,isOpenForSubmission:!1,hash:"a3ad4a3553a3ec59f7992d4f6495ac07",bookSignature:"Boris I. Lembrikov",publishedDate:"February 6th 2019",coverURL:"https://cdn.intechopen.com/books/images_new/7582.jpg",keywords:null,numberOfDownloads:4367,numberOfWosCitations:0,numberOfCrossrefCitations:2,numberOfDimensionsCitations:2,numberOfTotalCitations:4,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"April 18th 2018",dateEndSecondStepPublish:"May 9th 2018",dateEndThirdStepPublish:"July 8th 2018",dateEndFourthStepPublish:"September 26th 2018",dateEndFifthStepPublish:"November 25th 2018",remainingDaysToSecondStep:"3 years",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:"Edited by",kuFlag:!1,biosketch:null,coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"2359",title:"Dr.",name:"Boris",middleName:"I.",surname:"Lembrikov",slug:"boris-lembrikov",fullName:"Boris Lembrikov",profilePictureURL:"https://mts.intechopen.com/storage/users/2359/images/system/2359.jpg",biography:"Boris I. Lembrikov is a senior lecturer at the Faculty of Electronics, Electrical and Communication Engineering of the Holon Institute of Technology (HIT), Holon, Israel. B. I. Lembrikov received his Ph.D. in Nonlinear Optics at the Technion – Israel Institute of Technology in 1996. Since then he was an invited researcher at the Haifa University, at the Max Planck Institute High Magnetic Field Laboratory at Grenoble, France, at the Technion, Haifa, Israel. Dr. B. I. Lembrikov is an author of the book \\Electrodynamics of Magnetoactive Media\\, a number of chapters in scientific books, a large number of papers in international peer reviewed journals and reports delivered at the international scientific conferences. He actively participated in a number of research projects concerning optics of nanoparticles, optical communications, UWB communications. The main research fields of interest of Dr. B. I. Lembrikov are nonlinear optics, optical and UWB communications, nanostructures, quantum dot lasers.",institutionString:"Holon Institute of Technology (HIT)",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"10",totalChapterViews:"0",totalEditedBooks:"3",institution:{name:"Holon Institute of Technology",institutionURL:null,country:{name:"Israel"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"1225",title:"Optical Physics",slug:"optics-and-lasers-optical-physics"}],chapters:[{id:"65062",title:"Introductory Chapter: Nonlinear Optical Phenomena",slug:"introductory-chapter-nonlinear-optical-phenomena",totalDownloads:683,totalCrossrefCites:0,authors:[{id:"2359",title:"Dr.",name:"Boris",surname:"Lembrikov",slug:"boris-lembrikov",fullName:"Boris Lembrikov"}]},{id:"64727",title:"Nonlinear Schrödinger Equation",slug:"nonlinear-schr-dinger-equation",totalDownloads:825,totalCrossrefCites:0,authors:[{id:"198550",title:"Ph.D.",name:"Jing",surname:"Huang",slug:"jing-huang",fullName:"Jing Huang"}]},{id:"63615",title:"Three Solutions to the Nonlinear Schrödinger Equation for a Constant Potential",slug:"three-solutions-to-the-nonlinear-schr-dinger-equation-for-a-constant-potential",totalDownloads:569,totalCrossrefCites:0,authors:[{id:"93519",title:"Dr.",name:"Gabino",surname:"Torres-Vega",slug:"gabino-torres-vega",fullName:"Gabino Torres-Vega"}]},{id:"63619",title:"Hydrodynamic Methods and Exact Solutions in Application to the Electromagnetic Field Theory in Medium",slug:"hydrodynamic-methods-and-exact-solutions-in-application-to-the-electromagnetic-field-theory-in-mediu",totalDownloads:338,totalCrossrefCites:1,authors:[null]},{id:"64097",title:"Polarization Properties of the Solitons Generated in the Process of Pulse Breakup in Twisted Fiber Pumped by ns Pulses",slug:"polarization-properties-of-the-solitons-generated-in-the-process-of-pulse-breakup-in-twisted-fiber-p",totalDownloads:462,totalCrossrefCites:0,authors:[null]},{id:"63461",title:"Towards Enhancing the Efficiency of Nonlinear Optical Generation",slug:"towards-enhancing-the-efficiency-of-nonlinear-optical-generation",totalDownloads:421,totalCrossrefCites:0,authors:[null]},{id:"63480",title:"Widely Tunable Quantum-Well Laser: OPO Diode Around 2 μm Based on a Coupled Waveguide Heterostructure",slug:"widely-tunable-quantum-well-laser-opo-diode-around-2-m-based-on-a-coupled-waveguide-heterostructure",totalDownloads:399,totalCrossrefCites:0,authors:[null]},{id:"63398",title:"Stimulated Raman Scattering in Micro- and Nanophotonics",slug:"stimulated-raman-scattering-in-micro-and-nanophotonics",totalDownloads:679,totalCrossrefCites:1,authors:[null]}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"185543",firstName:"Maja",lastName:"Bozicevic",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/185543/images/4748_n.jpeg",email:"maja.b@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. 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Injury has also been found when some sesquiterpenes are artificially deposited on leaves during tests of their ability to deter herbivores [2]. Therefore, the sequestration of terpenoid in specific compartments by sensitive metabolic processes may be essential to avoid adverse effects. The morphology of these structures varies according to the conditions of irrigation and also according to the toxicity of intracuticular contents [3].
These risks implicate the presence of specialized structures for the storage and the secretion of these compounds, those structures changed with the content and function. Largely named glandular trichomes which are multicellular epidermal hairs, found in some families such as Lamiaceae, Asteraceae, and Solanaceae, which secrete terpenes in an extracellular cavity located at the apex of the trichome [4, 5].
Previous studies have been able to observe these structures as well as its different constituents (Figure 1). The glandular trichome is composed of several cells performing different functions: the terpenes synthesized by the secretory cells pass into a subcuticular space located at the apex of the trichome for accumulation, and the basal cells ensure the attachment of the structure to the epidermis. These glands vary morphologically, biochemically, and secretionally.
Glandular hairs structure (Iriti et al. [12]).
The secretory glands of aromatic plants come in different shapes and sizes, in order to ensure a specific function. This function consists mainly in the protection of the different organs of the plant and the attraction of pollinators. These glands are subdivided into peltate and capitate hairs. Werker et al. [4] classified glands into short-term glands and long-term glands; short-term glands are glands that secrete rapidly to protect young organs. The long-term glands are glands in which the secretory substance accumulates gradually in the subcuticular space and play a role in the protection of mature organs such as the flower, as well as in pollination. According to this definition, he inferred that the capitate hairs are short-term glands, while the peltate hairs are long-term glands. The difference between these two types of glands consists of several aspects like structure, mode of secretion, and timing of secretion.
Capitate usually consists of a single or bicellular head and rarely more than two cells in some species with lipophilic content. This content is ready to be released just after its production via a porous cuticle [3].
The glands of secretions are localized in all the plant organs, leaves, and stems and even at the root. The location of these structures depends on the organ in which they are located and the function and nature of the substances stored and secreted by these glands.
Due to the enormous diversity of these structures, morphology, origin, size, location, microstructure of the head, secretory capacity, secretion mode, function, etc., their classification is made difficult. The use of one of these criteria renders the classification incomplete which requires a method based on different types of criteria to classify these glands.
The major classification divides the glands into two groups: glandular hair and nonglandular hair. The main characteristic of distinction between these two types is the morphology as well as the nature of the substances to be secreted.
Nonglandular trichomes differ in morphology, anatomy, and microstructure. Essentially, they are classified according to their morphology. They can be unicellular or multicellular and branched or unbranched. The unbranched multicellular trichomes can be uniseriate, biseriate, or multiseriate. They can be distinctly articulated between cells or transverse walls which make the distinction of these structures impossible on the surface. They may differ in length, size, and shape of cells and may be symmetrical or asymmetrical. This type of gland provides a protective function as a mechanical barrier and does not ensure secretion of any substance.
The term “glandular hair” refers to a wide variety of glands. They differ according to the chemical composition of the substances they secrete, accumulate, or absorb; according to their mode of production, structure, and location; and according to their functions. All of these differences serve a certain level, which overlaps the classification.
In many of the Lamiaceae, two main types of glandular trichomes are encountered, the capitate and the peltate. They differ according to the shape of their secretory head and according to the morphology as well as the nature of the substances to be secreted. The head of the capitate glands consists of 1–4 more or less rounded secretory cells, generally oriented horizontally; a stem, one to several long cells; and a basal cell. The head cell can sometimes be very large, as in some species of Salvia [5]. The peltate hairs’ head consists of 4–18 more flattened cells on a horizontal plane, a stem cell, and a basal cell. Thus, intermediate shapes can be encountered.
There is great variability in the secreted materials of glandular trichomes: polysaccharides, sugars, salts, lipids, essential oils, resins, proteins, etc. Uphof [6] proposed to classify glandular trichomes according to the nature of their secretory products. Fahn [7] classified secretory substances in plants in general into two groups:
Unmodified or slightly modified substances, such as salts secreted by certain glands, as well as nectar.
Substances synthesized by secretory cells. This can be hydrophilic (as in trichomes which secrete mucilage and glandular trichomes of digestive carnivorous plants) or lipophilic (such as in the glandular trichomes of Lamiaceae, Asteraceae, Geraniaceae, Solanaceae, and Cannabaceae).
The difficulty with this type of classification is that some glands secrete more than one type of substance. For example, the glands of Inula viscosa secrete lipophilic substances, polysaccharides, and proteins at different stages of life and by different organelles [8]. In some carnivorous plants, the same glandular hairs produce both seductive (nectar) and digestive substances (enzymes).
In Lamiaceae, in addition to lipophilic substances, relatively large amounts of polysaccharides, which vary in amount between species, are secreted by certain capitate glands [8]. Many of these secreted materials are phytotoxic.
The function of glandular trichomes varies according to their location, the substances it secretes, and the moment of secretion. Structurally, similar trichomes can produce different materials, but even when similar materials are produced, when the trichomes are located differently, they may have different functions. For example, the functions of seed mucilage trichomes differ considerably from those of leaves.
Different types of glandular trichomes may differ according to the stage of development of the organ. In Lamiaceae, from a functional viewpoint and according to their mode of secretion, the secretion materials of certain types of capitate glands are released outside just after production, when the leaves are still young and developing, often with a porous cuticle or mechanical breakage. The lipophilic fraction acts as a repellent on young leaves even before being touched. In the peltate glands, the secretory material is gradually secreted from the glandular cells in a space formed by elevation of the cuticle. The cuticle can break through a predator, releasing secreted material, which can then act as repellents. The glandular hairs can also be classified in “short term” corresponding with the first group of capitate glands and “long term” corresponding with the peltate glands [3].
Protection: the plant needs protection against various external factors such as herbivores and pathogens, extreme temperatures, excessive loss of water, allelopathy against competing plants, etc. When nonglandular hairs are densely distributed (sometimes forming a carpet thicker than the leaf itself, found in many plants growing in arid conditions), they can serve as a mechanical barrier against most of the factors externally. A correlation between trichome density and pest resistance has been demonstrated by Levin [9]. The glands, which secrete lipophilic substances, can be used in chemical protection against various types of herbivores and pathogens by repulsing or poisoning them. Secreted mucilage can trap insects by sticking.
Absorption of water: ensure a permanent function of water absorption and soil minerals. They are usually short-lived, unicellular, unbranched, and well elongated or with one or more swellings.
The epiphytic plants, which have no contact with the soil, as well as the plants that grow in xeric conditions, sometimes have other types of glands for the collection of fog water such as pineapple (Ananas comosus) [10].
Salt secretion: consist of cells that actively secrete a solution of different minerals.
Seductive trichomes: secretory glands of certain nectars, such as the unicellular glands of Lonicera and Tropaeolum and the multicellular glands of Abutilon. The most striking examples are carnivorous plants that have special requirements to catch their prey; they need to seduce and trap the various preys (sensory-triggering, digestion, and absorption devices). These functions are mainly fulfilled by trichomes [11].
The microscopic observations made on the Salvia plants (Figure 2) show the existence of two types of structures (secretion glands and protective hairs) which play various roles during the growth of the plants.
Scanning electron microscope observations of the central region of Salvia officinalis leaf (a), showing the presence of two types of glands: peltate glands (b, c) and capitate glands (d) with GP (peltate glands) and P (protective hairs).
Glandular hairs consist of a base cell, a short unicellular stem, and a secretion head generally composed of 12 cells (sometimes 16) arranged in the form of a shield of 4 central cells surrounded by 8 peripheral cells (Figure 2a,b). Moreover, Figure 2c shows a sessile gland with multiple cells of the head. These structures are called peltate glands.
The second category of glands or capitate glands have a variable morphology since one distinguishes two types of glands: glands consisting of a short stem with one or two cells (Figures 3a and 4a–c) and a head that can be unicellular or bicellular. In this type of gland, the cuticle is thin, with the presence or absence of a small subcuticular space. The second type consists of glands observed in old leaves at the end of flowering (Figure 5) and has a thin cuticle with the presence of a large subcuticular space directly to the epidermis. This structure is characterized by the presence of an accumulation pocket which is used to store the secretory material suggesting the presence of secretory cells below this pocket and whose synthesis products would arrive at the accumulation pocket through a structure (channel or pore) located under this pocket. The absence of this structure in young leaves and their appearance in old leaves suggests that these structures have a specific function related to the age of the leaf and are therefore specialized in the accumulation of a different substance.
Light microscopic observations of the different glands in Salvia officinalis. (a) Capitate gland, (b) detached peltate gland, (c) peltate gland in secretion phase, and (d) post-secretion peltate gland.
Observations in light and scanning electron microscopy of capitate glands in Salvia officinalis. (a & b) Glands with single-cell head; (c) Gland with two-cell head.
Observations with light microscopy and scanning electron microscopy of capitate glands in Salvia officinalis. (a) Distribution of the pocket; (b&d) empty pocket; (c&e) pocket fill with secretory material.
Capitate glands are usually characterized by a single-cell head (Figure 4a,b). However, the light microscopic observations of the Salvia officinalis leaves show the presence of capitate glands that can present a two-celled head (Figure 4c).
According to the classification proposed by Werker et al. [4], the capitate glands are short-term glands specialized in the synthesis of non-terpenic substances frequently found in young leaves to provide a protective function against predators. The peltate glands are called long-term glands that are present in the leaves throughout the growth phases and whose number increases at flowering and are specialized in the synthesis of terpene substances.
Observations by the LM as well as SEM of the peltate glands (Figure 3c,d) do not reveal the presence of pores through which the secretory material can exude, which makes the material not released until the break of the cuticle, whether due to mechanical events or at the end of gland life.
As in the case of sage, light microscopy observations made on Pelargonium sp. leaves show that this plant is characterized by the presence of two types of secretory glands, the peltate glands or long-term glands and the capitate glands or short-term glands, as well as protective hairs or non-secretory glands (Figure 6).
Scanning electron microscopic observations of the ventro-central part of a Pelargonium sp. leaf, showing the presence of two types of secretion glands, peltate. (a): observation of non glandular hairs and glandular hairs (c) and capitate (with red arrow).
Capitate glands (short-term glands) are divided into two categories according to their forms: capitate glands with a stem consisting of four cells (Figure 7b) and capitate glands with one head and unicellular stem (Figure 7c). At the level of these glands, the synthesized secretion material is ready to be released just after its production via a porous cuticle (Figure 7b).
Observations made by light microscopy on Pelargonium sp. leaf showing the two types of capitate glands. (a) Global vision; (b) capitate glands with a stem consisting of four cells; (c) capitate glands with one head and unicellular stem.
Moreover, we could only meet one type of peltate glands (long term), with a multicellular stem and a large head in which one finds the cells of synthesis as well as a storage pocket of essential oils. The appearance of these glands varies with their stage of maturity. Figure 8a shows a gland that has an empty aspect of any content and confirms the juvenile condition of the gland. These glands are gradually filled by the EO as they mature (Figure 8b,c), and their cuticle eventually breaks to release their contents at the end of their maturation (Figure 8c).
Observations in light microscopy of peltate glands, secreting essential oils at different stages of maturity in Pelargonium sp. (a) Beginning of maturation, (b, c) filling with the EO, and (d) bursting of the gland. C, cuticle; EO, essential oil (Gx 400).
These peltate glands have a thick and rigid cuticle, but along the maturity stages of these glands, the thickness of this cuticle decreases to facilitate bursting and release of contents toward the end of the maturity of the gland or by contact with some insects (Figure 8).
Environmental scanning electron microscopy observations of basil leaves show the presence of two types of glands (Figure 9), peltate gland and capitate gland. The peltate glands have a round shape and a fairly large diameter that can exceed 70 microns. The capitate glands are small not exceeding 40 microns and come in two forms, glands capitate with one head (CH) and two bilobed heads (C2H).
Observation by scanning electron microscopy of basil leaf showing the presence of pelt glands (P) and capitate glands (CH: capitate gland with one head and capitates gland with head bilobed C2H).
The observations made by the light microscopy of the capitate glands confirm those made by scanning electron microscopy, with the presence of capitate glands with a single head (Figure 10a) and capitate glands with two heads (Figure 11b). On the other hand, the glands with a single head can be unicellular (Figure 10a) or bicellular (Figure 10b).
Light microscopy observations of a capitate gland with a single head and a single cell (a) and with two cells (b). C, cuticle; E, epidermis; CG, glandular cell (GX400).
Observations made by light microscopy and scanning electron microscopy of a double-headed gland in basil plants.
In basil plants, the peltate glands show a variability in the number of secretory cells. There are large glands with four cells (Figure 12a). In addition, we have been able to highlight the presence of glands that have eight secretory cells (Figure 12b,c). A thick cuticle covers both types of glands. With the filling of the essential oil structure, this cuticle becomes thinner to facilitate the release of the contents.
Light microscopic observations of a cross section (central part of a basil leaf O. gratissimum) showing two types of peltate glands: glands with four secretory cells (a) and glands with eight secretory cells (b, c).
Figure 13 shows peltate glands of basil after bursting and release of their terpenic contents; according to this figure, we notice the existence of four secretory cells (Figure 13a) with an internal diameter (without the cuticle which encompasses the gland) that touches 62 μm. Furthermore, Figure 13b shows an immature gland, but burst under pressure (exerted by SEM), and shows a thick cuticle which confirms the state of non-maturity that has already been reported.
Observation by scanning electron microscopy of a pelted basil gland after natural bursting (a) and mechanics (b).
The microscopic observations made in this work show the presence of two major groups of secretory glands in the three plants (sage, pelargonium, and basil). The first group is peltate glands or long-term glands, and the second group is capitate glands or short-term glands. Within the same group, we can see a morphological variation from one plant to another, depending on the content, the role, and the phenological stage of the plant, which makes the classification difficult. But to ensure a credible classification, various authors are mainly based on the morphological criterion, the content, as well as the mode of secretion [4, 5, 6, 7, 8, 12].
In this context, scientists have given a definition to these two names; they have described a short-term gland as any gland whose secretion is rapid to protect young organs and a long-term gland as any gland in which the secretory substance accumulates gradually in the subcuticular space and serves for the protection of mature organs in the flower, as well as for pollination. The difference between the two types consists of several aspects, namely, the structure, the mode of secretion, and the moment of secretion.
The microscopic observations made on our plants go in the direction of this classification and show the two groups of glands, peltate and capitate. Moreover, two major types of peltate glands could be determined in this study, peltate glands with stem consisting of a single-cell and peltate glands glued directly to the epidermis. In the latter group, there is a major difference in the number of cells of each gland, since there are 4 and 8 cells in O. gratissimum and up to 12 in Salvia. These results are confirmed by the work of Tissier [13], Werker [8], and Gang et al. [14]. According to these authors, the peltate glands can have a large number of cells that can reach 16 cells.
These peltate glands are generally characterized by a large subcuticular space and a rigid cuticle which thins during the maturation of the gland which facilitates bursting at maturity or after mechanical contact with predators.
The captioned glands are subdivided into several categories according to their forms:
Capitate glands possessing a basal cell and a united or multicellular stem (1–3 cells) and a head
Capitate glands with a stem cell and a united or bicellular head
Capitate glands with a basal cell and a head with one or two cells
Capitate glands in the form of a pocket glued directly to the epidermis
These capitate glands are characterized by a thin cuticle with the synthesized secretory material which is ready to be released just after its production via a porous cuticle and usually consist of a united or bicellular head with lipophilic content.
Similar observations have been found in Salvia [15], since according to these authors the capitate glands are morphologically very variable and four types can be distinguished. Type I has a single or double bicellular stem and a united or bicellular head. The cuticle is thin and there is no subcuticular space. The secretory material is released slowly through the cuticle and can be released suddenly if the cuticle is broken. Type II is very small, possessing a unicellular stem and a small subcuticular space. The secretory material is probably secreted by a pore. Type III is a large gland with a long stem made up of several cells (about three or more), a neck cell, and a unicellular head, which probably releases the secretory material often collected as a drop on the head. Type IV is a large gland with a long thin stem, made up by four cells.
According to these authors, each type of gland is responsible for the synthesis of a distinct substance. Venkatachalam et al. [16] found a correlation between the increased number of peltate glands and increased camphor and thujane synthesis.
These observations are in agreement with those of Pedro et al. [17] who deduced the presence of three types of glands, a peltate type and two capitate types. According to these authors, the glands with a unicellular stem represent a distinct capitate gland with a specific content different from that of the peltate structure. According to the same work, these three types of glands ensure the synthesis of three types of different substances, terpenoids, phenols, and flavonoids.
In other works by Ko et al. [18], Pelargonium’s peltate and capitate glands all have a single basal cell. These two types of glands differ in the number of stem cells and in the diameter of the head depending on the stage of maturity [3]. According to these authors, the glands defined in our study as glands capitate to a head and a unicellular stem represent only one stage of the formation of the peltate glands. However, observations made in Salvia leaves show structures identical to this structure, possessing a small unicellular head and a single-celled stem. This structure has been classified as a capitate gland.
These kinds of peltate and capitate glands are found in several species (Mentha piperita, Origanum dictamnus, Monarda fistulosa, Pogostemon cablin, etc.) and have been described in ancient works [16]. In mint, the peltate glands have eight head cells, and the capitate glands have a single-stem cell and a head cell. In Origanum dictamnus, the head of the peltate glands has 12 cells, whereas in Pogostemon cablin, it has only 4 cells.
According to Werker [8], young basil leaves (O. basilicum) that are highly vulnerable to predators are heavily covered by capitate glands that are characterized by rapid secretion to repel herbivores. When the plants become older, the rate of the peltate glands increases; these terpenoid-rich glands will be used for defense against pathogens in case of injury and pollinator attraction.
The microscopic observations made on the three plants (basil, pelargonium, and sage) show the presence of two major types of secretory glands. The first type is peltate glands or long-term glands, and the second types are capitate glands or short-term glands. Within the same type of gland, we can see a morphological variation from one plant to another, depending on the content, the role, and the phenological stage of the plant.
In basil, we have demonstrated for the first time the existence of eight secretory cell glands in the peltate glands.
In Salvia, we noted the presence of two types of glands, capitate and peltate, the latter being constituted by 12–16 cells arranged in the form of a shield. On late flowering, we noted the presence of pocket secretions in older leaves. In this plant, we have mainly highlighted the presence of sessile glands, and this observation has never been confirmed in the literature.
The authors of this work (Zakaria Hazzoumi, Youssef Moustakime, Khalid Amrani Joutei) declare that they have no conflict of interest.
“The scientist is not a person who gives the right answers, he is one who asks the right questions. Claude Levi-Strauss’’.
In statistical physics only a few problems can be solved exactly. For complex problems, numerical methods can give exact results for problems that could only be solved in an approximate way. Numerical simulation can be a way to test the theory. The numerical results can be compared to the experimental results. The numerical simulation is placed between the fundamental and the experimental treatment; it has a quasi-experimental character (numerical experience). For problems of statistical physics, the most widely used simulation methods are the Monte Carlo method and the molecular dynamics method.
The first Monte Carlo simulation (MCS) was proposed by Metropolis et al. in 1953 [1]. The second Monte Carlo simulation was proposed by Wood and Parker in 1957 [2]. The obtained results were in good agreement with the experimental results of Bridgman [3] and those of Michels et al. [4]. In this method we attribute a series of initial positions chosen randomly to a system of N particles interacting through a defined potential. A sequence of particle configurations is generated by giving successive displacements to particles; we only retain configurations to ensure that the probability density is that of the chosen.
Molecular dynamics simulation (MDS) has been first introduced to simulate the behavior of fluids and solids at the molecular or atomic level. MDS was used for the first time by Alder and Wainwright in the late 1950s [5, 6] to study the interactions of hard spheres. The principle is the resolution of equations of motion for a hard sphere system in a simulation cell. The basic algorithm is Verlet’s algorithm [7].
In this chapter, we will present techniques of numerical simulations using the Monte Carlo method. We will present an application on the gas phase during plasma-enhanced chemical vapor deposition (PECVD) of thin films. The application concerns collisions between particles. Particles are in Brownian motion. Collisions, elastic or inelastic, are considered to be binary. Non-elastic collisions result in effective chemical reactions.
In Section 2, we cite some MCS and MDS works on PECVD processes. Section 3 presents general rules on numerical simulation methods. Section 4 presents how to simulate a physical problem using MCS? We present the Metropolis algorithm as a scheme to trait random configurations and different modules related to elaborate an MCS code. In Section 5, we apply the MCS on SiH4/H2 gas mixture during a PECVD process. Finally the conclusion summarizes the contents of the chapter.
The PECVD is the most widely used technique to produce hydrogenated amorphous silicon thin films (a-Si:H) for solar cells and for film transistors and electronic devices [8, 9]. Reactions during plasma deposition are complex and are not understood completely.
Gorbachev et al. [10, 11, 12] have developed a model that is based on chemical reactions and different processes in a PECVD reactor. The model takes into account the formation of SinHm oligomers (n ≤ 5). It presents a simulation of the growth of the films. Gorbachev et al. found that Si2H5 and Si3H7 strongly influence the growth of the film [11].
Valipa et al. [13] calculated the β reactivity of the SiH3 radical on a surface of a silicon lattice plane during the growth of a-Si:H using MDS. The mechanisms of physical and chemical interactions of low temperature plasmas with surfaces can be explored using MDS [14].
For a CH4/H2 mixture, Farouk et al. used the Monte Carlo method (PIC/MC); they calculated the ionization rate of the plasma and the deposition rate of the thin layer [15]. Rodgers et al. [16] have developed three-dimensional Monte Carlo simulations of diamond (100) surface CVD. Other works on MCS are in [17, 18, 19].
In our previous works [20, 21, 22, 23, 24], we were interested in the study of the gas phase and the interaction of plasmas with the surface, for SiH4/H2 and CH4/H2 gas mixtures during PECVD processes. The used numerical simulation techniques were MCS and MDS. To complete the studies, we used the fluid model [25].
The starting point of numerical simulation is a physical phenomenon; its purpose is to obtain useful physical results. Between these two points, several steps can be identified. These steps are general and they are applicable for MCS. The steps can be summarized as follows:
The physical phenomenon must be defined by the description of the dominant domain of physics. The main assumptions and simplifying approximations are necessary to understand the physical phenomenon and the design of the first model.
Mathematical model requires a mathematical formulation of the problem. It may be a problem of elements or discrete object or a problem of a continuous medium; it may be a spatiotemporal problem or frequency problem and may be a deterministic or probabilistic problem.
It would be interesting to know the mathematical equations that govern the phenomenon:
The forces between particles and elements
The potential interaction
The determination of a time scale
The determination of a length scale
Definition of constant magnitudes of motion and equilibrium magnitudes
Continuity equations, balance equations, transfer equations, etc.
The MCS technique has been chosen for this work; knowing its basic algorithm is necessary for elaborating the simulation. This step requires some actions:
Validation of the model on simple cases
Simulation calculation on complex phenomena
The MCS is based on a probabilistic process with a random choice of configurations and samples of the situation of the physical system. The two pedagogical examples most cited in the literature are the integration of a single variable function and Ising’s model of spin. In the following subsection, we define the integration of a single variable function. We introduce the Ising model at the end of Section 4.2.2.
Calculation of the definite integral for a function f(x) of a single variable x on domain {a, b} has been proposed (Figure 1):
The integral of a function f(x).
Let:
Let xi and yi be real random numbers (i = 1, 2,…, N), and let H be a real number greater than the f(x) for x belonging to the domain {a, b} (or x ∈ {a, b}).
Let r1 and r2 be two random numbers belonging to the domain {0, 1} according to a uniform distribution law. Generators (e.g., Ran, RANDOM, RANDUM, or other IMSL mathematical libraries) of random numbers can be used:
where xi and yi are random numbers (xi ∈ {a, b} and yi ∈ {0, H}).
The Monte Carlo (MC) method is based on a probabilistic process. Let N be the total number of cases chosen (possible cases). It is necessary to count the number of favorable cases (or the number of points below the curve y = f(x)); let yi ≤ f(xi)). The number of favorable cases is Nfav. When N➔∞, the value I of the integral is [26]:
An example [26] is the calculation of the value π by calculating the integral I on a quarter circle of unit radius (R = 1.0). The pairs of random numbers (xi, yi) satisfying the condition: xi2 + yi2 ≤ 1. The function f(x) is equal to
We take a = 0.0, b = 1.0, and H = 1.0.
For different values of N, we show that the numerical solution tends to π = 4I.
Although this integral is simple, it shows the strength and simplicity of the method. The technique can be generalized for the integration of multivariate functions.
We note that integration by the MC method is based on:
The choice of random configurations according to a uniform distribution law
Each configuration chosen is either favorable or unfavorable (the “or” is exclusive).
For statistical physics problems, the probabilistic choice of configurations is not always deterministic; the favorable and unfavorable cases are not exclusive. According to the Metropolis algorithm [26, 27], the steps of the simulation are:
Choice of a simulation cell of adequate shape to the studied phenomena. The size of the simulation cell is related to a scale of length characteristic of the forces and interaction potential of the studied phenomenon. This cell may contain Npc particles (and/or elements).
Choice of an initial configuration that responds to some physical and thermodynamic properties. The total or internal energy of the system is Ei.
Infinitesimal random displacement of a particle (or element of the system) and calculation of the new internal energy of the system Ef. This displacement is related to the physical magnitudes: time scale and length scale. The physical system tends toward a minimization of the internal energy of the system with some fluctuation. Let ΔE = Ef-Ei the fluctuation.
If ΔE ≤ 0; the new configuration is retained (favorable) and the different averages can be obtained; go to step (c).
If ΔE > 0; a random number ε is chosen such that 0 < ε < 1. Let the probability Pr equal to: Pr = exp. (−ΔE/kBT) (where kB is the Boltzmann constant and T is the temperature).
If ε < Pr, accept the move and in any case go back to step (c) for a new choice of an infinitesimal displacement (new configuration). Note that if such a trial move is rejected, the old configuration is again counted in the averaging with probability Pr.
Figure 2 shows how to choose between the selected configurations. Let ε be a random number following a uniform law; If ε1 ≤ Pr the configuration is retained, and if ε2 > Pr the configuration is rejected.
Configuration choice according to Metropolis scheme.
Numerical simulation using the MC method is a very important tool for the study of static properties. The basic algorithm is based on probability notions. Understanding of the distribution function and/or interaction potentials is the heart of the calculation.
In equilibrium statistical physics, the system has a certain probability that can be in any states. The probability of being in a state μ with energy H(μ) is given by the Boltzmann distribution P(μ):
where T is the absolute temperature and kB is called Boltzmann’s constant. It is conventional to denote the quantity (kBT)−1 by the symbol β. The normalizing factor Z, or partition function, is given by:
The average of a quantity Q fora system in equilibrium is:
The internal energy U, is given by:
which can be written in terms of a derivative of the partition function:
From thermodynamics we have expressions for the specific heat C, the entropy S, and the Helmholtz free energy F:
or
and
and
We can calculate other parameters affecting the system.
The Monte Carlo method is an excellent technique for estimating probabilities, and we can take advantage of this property in evaluating the results. The simplest and most popular model of a system of interacting variables in statistical physics is the Ising model. It consists of spins σi which are confined to the sites of a lattice and which may have only the values (+1) and (−1). These spins interact with their nearest neighbors on the lattice with interaction constant J; they can interact with an external magnetic field B coupling to the spins. The Hamiltonian H for this model is [26]:
The Ising model has been studied in one and two dimensions to obtain results of thermal properties, phase transition, and magnetic properties [26, 27, 28]. For chosen values of J and/or B, different steps may be taken for the calculations (simulation cell, initialization, configurations, boundary conditions, calculation algorithms). For any configuration, each spin takes the two possible directions. The detail of the calculation procedure is not the purpose of this chapter.
We give a system of N particles (atoms, molecules, ions or particles) placed in a cell of fixed volume, generally of cubic form. The initial positions may, depending on the case, be distributed randomly according to a certain law (uniform or otherwise) or have a given symmetry. In a fluid, a gas, or a plasma, the particles may have random positions in general; in a solid or surface, with a crystal structure, the particles take ordered positions. The choice of random initial positions allows great freedom on the choice of the number of particles in the cell.
At the first step, the particles are given velocities that are generally selected to have a zero total momentum. If the system is in thermodynamic equilibrium, the initial velocities will be randomly chosen according to a Maxwell-Boltzmann law. In the general case, the velocity distribution is according to the problem dealt with. All other phase properties can be initialized to the particles; the main thing is the conservation of the total quantities of the system.
The particles interact with each other according to chosen interaction potentials. Since the interaction potentials are specific for each “numerical experiment,” the main part of the work consists in calculating the interaction energies for each proposed configuration.
The choice of interaction potentials is directly related to the mathematical formulation of the problem according to the state of the medium: fluid, gas, plasma, or solid. It can be Lennard-Jones potential, Coulomb potential, Debye potential, Morse potential, Stillinger-Weber potential, Born-Mayer potential, Moliere potential, or others.
In general, two main boundary conditions are used: periodic boundary conditions (PBC) and minimum image convention (MIC) [29].
To minimize the surface effect, periodic boundary conditions (PBC) [30] are invariably imposed. The simulation cell is reproduced throughout the space to form an infinite mesh. We can simulate the properties of an infinite system. The particles that we follow are in the central cell; if a particle crosses a wall with a certain velocity, its image returns with the same velocity by the opposite wall. Under these conditions, the number of particles in the central cell, and consequently the density, is constant. These conditions also allow the conservation of the energy and the momentum of the system and do not introduce periodic effects (because of the interaction between particles).
According to the hypotheses and according to the geometry of the problem, other boundary conditions are proposed [26]. For example, in order to model thin films, the simulation cells are longitudinal and parallel to the film; one uses PBC in the directions parallel to the film. In the direction normal to the film, free edge boundary conditions can be used. In such cases, it may be appropriate to also include surface fields and surface interactions. In this way, one can study phenomena such as wetting, interface localization-delocalization transitions, surface-induced ordering and disordering, etc.
The core of the program includes calculating the potential energies of particle configuration and particle collisions. The interactions and collisions between particles can be elastic or inelastic; they can be binary or collective. For computation, the interaction energy of a particle with its neighbors is carried out by refocusing a base cell on the particle. This particle only interacts with particles in this region. This is called the “minimal image convention” (MIC) [1].
Generally, a RANDOM generator of real random numbers ri belonging to the domain {0, 1} (or ri ∈ {0, 1} is available. This distribution law is uniform.
To have a real random number xi belonging to the domain {a, b} (or xi
To have a real random number xi belonging to the domain {a, b} (or xi ∈ {a,b}) according to a formula (or law) of nonuniform distribution f(x), a histogram technique is used. Let Nm be the number of intervals. If the mesh is regular (Figure 3):
Random number selection according to f (x) distribution.
We define:
We define the sequence:
and the sequence:
Hence each real random number ri belongs to the domain {0, 1} (where ri ∈ {0, 1}) (according to the uniform law); this number belongs to the domain {rxj-1, rxj}. It corresponds to a random value xran of the domain {xj-1, xj}; this number satisfies the formula (or the law) of nonuniform distribution f(x).
This technique can be generalized for a nonuniform distribution law f(x) with an irregular mesh Δxi, or with tabular data f(xi) with i = 1,…, m.
The technique can be generalized, too, for a discrete distribution law f(i) with i = 1,…, m.
In the literature, the reader can find simple algorithms for the choice of random numbers of some simple functions (Gaussian, etc.).
It is necessary to find some parameters allowing the control of the smooth course of the evolution of the system. We must look for the constants of movement. For example for an isolated system, we have the conservation of the total energy and the quantity of matter.
By using the numerical simulation, it is possible to calculate many spatiotemporal quantities F(r,t). These quantities can be positions, speeds, kinetic moments, particle energies, concentrations, transport coefficients, etc. It would then be possible to calculate all other quantities related to F(r,t).
For the calculation of the averages, one can note the quantities on the space, on the time or on both. The histogram methods can be used. Static or dynamic distribution functions and spatial or temporal correlation functions can be calculated. It should be noted that the SMC is much more adequate for static properties because of the probabilistic choice of configurations.
Any calculated function or parameter F(r,t) can be used for another application in another calculation program.
In the MCS model discussed extensively in this chapter, it’s more about collisions between particles. It’s particle-particle MCS or PP-MCS. In many problems of physics, the general idea is the same, but the applications and proposed models are numerous.
Other MCS models, named particle-in-cell MCS (PIC-MCS), are based on particle-cell interactions. In these last models, we also use a probabilistic choice of configurations and small variations in the state of the system (following the Metropolis algorithm); the interaction is between the particle with a cell, a mesh, or a drop. The parameters and variables of the cell, although local and instantaneous, are macroscopic. These parameters and variables can be thermodynamic, fluid, or electromagnetic. An example of the model based on PIC-MCS is described by Mattei et al. [31] for simulation of electromagnetic particle-in-cell collision in inductively coupled plasmas. Several works can be found in the literature on this same line of work. Other MCS models using particles may be considered. [32].
For statistical physics problem solving (such as thin film deposition problems), MCS models use experimental, numerical, or theoretical data from other methods and models. Models can be improved to hybrid models. In the hybrid models, connections between two modules can be realized. The first module is MCS; the second module is fluid, electromagnetic, or other. An example of a three-module hybrid model is presented by Mao and Bogaerts [33] to study gas mixtures in PECVD system. The three modules are MCS, fluid, and electromagnetic. The first module EM calculates the electromagnetic fields by solving Maxwell equations. These fields are used as inputs in the module MCS, where the electron density, electron temperature, electron energy distribution function, and electron impact reaction rates can be computed with a Monte Carlo procedure. Subsequently, the module fluid calculates densities and fluxes of the various plasma species (i.e., heavy particles and electrons) with continuity equations and the electrostatic field with Poisson’s equation. This electrostatic field is used as input again in the EM. This cycle is iterated until convergence. The schematic of the hybrid model is given in Figure 4.
Schematic of a hybrid model of three modules used to study gas mixtures in the PECVD [33].
To solve statistical physics problems with evolutions as a function of time, kinetic models of MCS (kMCS) are used. Using kMCS, Battaile and Srolovitz [17] described kinetic phenomena of the diffusive motion of a single interstitial atom in a close-packed metal crystal. The motion of the interstitial atom is usually limited to two types: vibration of the atom around the center of the interstitial hole in which it resides and hops to nearest-neighbor interstitial sites. The atom can hop into any of the nearest-neighbor interstitial sites; it executes a random walk. In an MC simulation of this diffusion process, the new position of the interstitial atom is chosen at random from a list of the adjacent interstitial sites.
Other CVD and PECVD works on MCS are presented in Ref.s [15, 34, 35, 36, 37, 38]. They show how MCS methods can study properties of gas mixtures and properties of the growth of thin films.
In this section, we present an example of PP-MCS of collisions and reactions in gas phase of SiH4/H2 mixture used in PECVD process. Some paragraphs have been treated in previous works [21, 24].
We use a MCS to study collisions and chemical reactions in gas phase of SiH4/H2 mixture used in the PECVD process. In this phase, important reactions have been identified that contribute to the production and the consumption of hydrogen (H), silylene (SiH2), and silyl (SiH3). The hydrogen consumption reactions SiH4 + H → SiH3 + H2 and SiH3 + H → SiH2 + H2 are found to play a central role in deciding the distribution of hydrogen [39].The plasma chemistry indicates that H atoms and SiH3 radicals play an important role in the a-Si:H deposition process [40]. Experimentally, it is generally accepted that SiH3 radicals dominate a-Si:H and μc-Si film growth from SiH4 plasmas in the PECVD; it is the key precursor of a-Si:H deposition [41]. The proposed MCS allowed to get the ratio SiH2/SiH3 and mean value of densities of species. It provides information on SiH4 dissociation and on the production of SiH3, H, SiH2, and Si2H6 and other important parameters.
The plasma in the PECVD reactor is weakly ionized. For our study, the mixture gas contains 22% of SiH4 and 78% of H2; the pressure is 100 mtorr, the temperature of the gas ranges from 373 to 723 K, the electron temperature is about 2.5 eV, and the electron density is 3. 108 cm−3. The process is considered to be stationary. We take into account electrons and eight neutral species (SiH4, SiH3, SiH2, H, H2, Si2H6, Si2H5, SiH). Reactions taken into account include seven electron-neutral and 14 neutral-neutral reactions. Table 1 shows the 21 reactions and rate constants Kreac. At low temperature, the neutrals interact occasionally with each other and move under the effect of thermal agitation; their velocity distribution function is Maxwell-Boltzmann distribution. Electrons have the mean velocity with kinetic energy Te.
Symbol | Reactions | Kreac (cm3/s) |
---|---|---|
R1 | SiH4 + e→SiH3 + H+e | k1 = 3 × 10−11 [42] |
R2 | SiH4 + e→SiH2 + 2H + e | K2 = 1.5 × 10−10 [42] |
R3 | SiH4 + e→SiH + H + H2 + e | K3 = 9.34 × 10−12 [42] |
R4 | SiH4 + e→SiH2 + H2 + e | K4 = 7.19 × 10−12 [42] |
R5 | H2 + e→2H + e | K5 = 4.49 × 10−12 [42] |
R6 | Si2H6 + e→SiH3 + SiH2 + H + e | K6 = 3.72 × 10−10 [42] |
R7 | Si2H6 + e→SiH4 + SiH2 +e | K7 = 1.1 × 1010× (1.(1./(1. + (0.63 × P)))) [43] |
R8 | SiH4 + H→SiH3 + H2 | K8 = 2.8 × 10−11 × exp.(−1250/T) [44] |
R9 | SiH4 + SiH2→Si2H6 | K9 = 1.1 × 1010 × (1.−(1./(1. + (0.63 × P)))) [43] |
R10 | SiH3 + SiH3→SiH4 + SiH2 | K10 = 0.45 × 1.5 × 10−10 [44] |
R11 | SiH4 + Si2H5→SiH3 + Si2H6 | K11 = 5 × 10−13 [42] |
R12 | SiH3 + H→SiH2 + H2 | K12 = 2 × 10−11 [44] |
R13 | SiH3 + Si2H6→SiH4 + Si2H5 | K13 = 4 × 10−10 × exp. (−2500/T) [44] |
R14 | SiH2 + H→SiH + H2 | k14 = 2 × 10−11 [44] |
R15 | Si2H6 + H→Si2H5 + H2 | K15 = 0.66 × 2.4 × 10−10 × exp. (−1250/T) [43] |
R16 | Si2H6 + H→SiH4 + SiH3 | K16 = 0.34 × 2.4 × 10−10 × exp. (−1250/T) [44] |
R17 | SiH + H2→SiH3 | K17 = 2 × 10−12 [43] |
R18 | SiH2 + SiH3→Si2H5 | K18 = 3.77 × 10−13 [43] |
R19 | SiH2 + H2→SiH4 | K19 = 3 × 10−12 × (1. + (1./1. + (0.03 × P))) [43] |
R20 | 2SiH3→Si2H6 | K20 = 0.1 × 1.5 × 10−10 [43] |
R21 | SiH4 + SiH→Si2H5 | K21 = (1.−(1./(1. + (0.33 × P)))) × (6.9 × 10−10) [43] |
List of gas phase reactions and corresponding rate constants [24].
Let
And chemical reaction for the production of A is as:
Rate production and consumption for any species A are taken as:
The MCS is based on binary collisions at the microscopic level. Elastic collisions are between all particles, and inelastic collisions (or effective collisions) are those that result in a chemical reaction. A chemical reaction needs a collision involving at least two particles (atoms, ions, electrons, or molecules). According to kinetic theory, gases consist of particles in random motion. These particles are uniformly distributed in a cell which has a parallelepiped form of sizes Lx, Ly, and Lz (Figure 5). These particles move in a straight line until they collide with other particles or the walls of their container. Dimensions and volume of Monte Carlo cell must take into consideration the mean free path of species.
Form of the simulation cell.
Let ni be the density of neutral spice i (i = 1,…, 8). The first particle i is randomly chosen according to a probability of neutral species Prsp,I (nonuniform discrete distribution) given by:
The chosen particle takes randomly three components of space in cell ri(xi, yi, zi) according to the normal distribution (nonuniform distribution). It takes also randomly three components of velocity vi (vxi, vyi, vzi) according to Maxwell-Boltzmann distribution.
Let ni and nj be the densities of species i and j in the gas and Vij the relative velocity between the two species i and j.
According to the kinetic theory of gases, we have for an incident particle i on a target particle j the average collision frequency νij as:
where <sij> is the cross section of the particle j.
The mean free path <λι> of species i is:
The time between two collisions τij is then:
For chemical effective reactions (inelastic collisions) between two reactive species i and j giving products i’ and j’, the rate constant reaction verifies [45]:
General rules of collision theory are applied:
The new velocities of the colliding particles are calculated using conservation of energy and momentum for elastic collisions.
Conservation of total energy as isolated system.
Movement of the center of mass and relative motion around the center of mass.
The reader can refer to some fundamental physics books that deal with general notions of collisions and corresponding parameters [45, 46, 47, 48].
The plasma in the PECVD reactor is weakly ionized. At low temperature, particles interact occasionally with each other and move under the effect of thermal agitation. In reality, only a small fraction of collisions are effective (result in a chemical reaction) [21].
In our MCS, after traveling a random walk given by a Gaussian distribution, the first chosen particle collides with a second particle (molecule, atom, radical, or electron). The last particle j is randomly chosen according to a (i-j) collision probability Prcol,j (nonuniform discrete distribution) given by:
where
The activation energy is given by:
where the pre-exponential factor is assumed to be the collision frequency factor and Kreac is the rate constant of the gas phase reaction.
The two colliding particles (e.g., the electron and SiH4 molecule) can interact by several reactions (R1, R2, R3, and R4 in Table 1); we choose randomly one of gas phase reactions occurring according to a, nonuniform discrete distribution reaction probability Prreac (i,j):
where
All chemical systems go naturally toward states of minimum Gibbs free energy [21, 24]. A chemical reaction tends to occur in the direction of lower Gibbs free energy. To determine the direction of the reaction that is taking place, we use the old and new values of Kreac and the equilibrium constant with reactants and product concentrations. Each set of binary collisions can be related or converted into time. As cited in section (a), Table 1 gives gas phase reactions and corresponding rate constants used in this MCS.
To continue the simulation, after the elastic collision, particle i takes new values of components velocity and new mean free path; mean free path is taken from a normal (nonuniform) distribution (Gaussian distribution). If the collision is inelastic, we have to take a new particle.
From Metropolis algorithm, the scheme of this MCS is as follows:
Choices of particle of spice i with random position, velocity, and mean free path; periodic boundary conditions are used to keep particles in the elementary cell.
Choices of random collision with a spice j.
Study of collision type (elastic, inelastic). If the collision is elastic the particle i move with a new velocity and mean free path, and we return to step (b). If the collision is inelastic particles i and j give new particles i’ and j’, according to Metropolis scheme, and we return to step (a) or (b). Periodic boundary conditions are used to keep particles in the elementary cell.
At each step, we can note the different statistics.
Once the species are selected for the simulation model, an estimate of species densities should be made. Following the model of interaction and collisions between particles (binary, collective, etc.), a first choice of the minimum number Ni of particles of each species is made. A first estimate of the sizes (Lx, Ly, Lz) of the elementary cell is made.
The study of the types of interaction potentials and the calculation of the approximate values of the force ranges, the kinetic energies, the internal energies, and the energies of activation make it possible to correct the minimal numbers Ni of particles and the sizes (Lx, Ly, Lz) of the elementary cell.
Let kp be the number of a species, kp = 1,…, 9. The minimal numbers Qnp(kp) and the sizes (Lx, Ly, Lz) have to be discussed for statistical calculations.
For numerical programming, according to the programming language used and according to the size (or the computational capacity) of the computer, it is necessary to find a judicious choice of the tables of integer or real values and which values would be useful to save all during simulation. Let Ncol,m be the maximum number of elastic collisions per particle, and let Ncycle be the number of cycles to average the simulation calculations.
For this MCS, the numerical chosen values are in Table 2.
Cell dimensions and steps for collisions | Number of species Kp | Initial number of particles in cell | ||
---|---|---|---|---|
Lx (m) | 4.68 10−6 | 1 | Qnp(SiH4) | Qnp1 |
Ly (m) | 4.68 10−6 | 2 | Qnp(SiH3) | 10 |
Lz (m) | 20.0 10−3 | 3 | Qnp(SiH2) | 10 |
4 | Qnp(H) | 10 | ||
Ncol,m | 500 | 5 | Qnp(H2) | Qnp5 |
Ncycle internal cycle | 2000 | 6 | Qnp(Si2H6) | 10 |
Ncycle external cycle | 200,000 | 7 | Qnp(SiH) | 10 |
8 | Qnp(Si2H5) | 10 | ||
9 | Qnp(e) | Qnp9 |
Used quantities and parameters in calculations for the gas temperature Tg = 520 K.
For radicals (e.g., SiH3), particle numbers Qnp(k) are very small; we take Qnp(k) = 10. These numbers cannot take value 1 or 0, even if a species k is in trace form in the gas. The value 0 for a species k means that any other species k’ does not make a collision with the species k; and the value 1 means that we have no collisions between particles of the same species in the cell.
Qnp1, Qnp5, and Qnp9 are calculated from the volume of cell, the pressure, the temperature, and the total number of particles in the cell (Qnp1 = 0.81187824 * 109; Qnp5 = 0.20296956 * 109; Qnp9 = 131).
As we have chosen a stationary regime, we must reach the values and properties at equilibrium. The results of the simulation show this trend. In MCS, averaged values, distribution functions, autocorrelation functions, and correlation functions can be calculated. To ensure rapid convergence of calculations, it would be useful to look for statistically symmetric (or stationary or unsteady) parameters [26, 50].
As an example for our MCS calculation, we have:
The number of Si2H6, SiH, and Si2H5 particles reaching the surface is negligible.
Let Ns,i and Ns, H2 be the densities of a species i and H2 reaching the surface. The ratios Ns,i/Ns, H2 are too small (Table 3).
Let Ns,i be the density of a species i reaching the surface and Nv,i the density of same species i in volume. The ratios Ns,i/Nv,i are too small (Table 4); the surface effect is negligible.
The reactions begin with the dissociation (consumption) of H2 and SiH4 by R5, R1, and R2 reactions.
The production of SiH3 is done by R8, and then there is production of SiH2 by R12.
The reaction R2: SiH4 + e → SiH2 + 2H + e plays the central role in SiH4 dissociation by electron impact [24]. This result is compatible with [39].
The second important chemical reaction in the SiH4 dissociation is R1: SiH4 + e → SiH3 + H + e [24]. This result is compatible with that of Perkins et al. [51] and that of Doyle et al. [52].
Type | H2 | SiH4 | H | SiH3 | SiH2 |
---|---|---|---|---|---|
Ns,i/Ns, H2 | 1 | 0.23 | 1.67 10−4 | 8.60 10−5 | 9.86 10−6 |
Ratios Ns,i/Ns, H2 of particles reaching the surface compared to H2.
Type | SiH4 | SiH3 | SiH2 |
---|---|---|---|
v, j | 6.695 10−6 | 7.965 10−6 | 775 10−6 |
Ratios Ns,i/Nv,i of particles reaching the surface compared to volume.
MCS is a widely used method in statistical physics to study thermodynamic, structural, or phase properties. It is based on random and probabilistic processes. The purpose of this chapter is to present the technique for general use in physics for the study of thin film deposition problems. The technique can be generalized to other fields of science: biology, economics, transportation, and social sciences.
We started by presenting general rules for numerical simulation methods. Metropolis algorithm has been considered as the basic algorithm. After, we presented the different steps for the realization of a MCS code. We chose the particle-particle model MCS (PP-MCS) to explain the different steps and procedures to be applied in the deposition of thin layers by PECVD processes. We have shown that this technique can be generalized to the particle-in-cell MCS (PIC-MCS) case or kinetic MCS (kMCS), as it can be joined with other modules to give hybrid models. It is important to know how to choose random configurations from the laws or probability distributions in the system.
A numerical application is presented for collisions in a SiH4/H2 gas mixture in the PECVD process. A preliminary work of determination of the chemical reactions between molecules and radicals is made. A choice of the simulation cell is made, and the definition of the probabilities of the collisions between peers is made. The Metropolis algorithm makes it possible to follow the various elastic and inelastic collisions; it also makes it possible to make the statistics of the interactions with the surface. The results are compatible with [39, 51, 52].
Other questions may be asked to account for molecular ions, surface and volume kinetics, or thin film formation. The techniques and different models of the MCS (PP-MCS, MCS-PIC, kMCS) allow taking care of these questions.
The interconnection of the MCS with other models (MDS, hybrid model, fluid model, electromagnetic model, etc.) would allow answering more questions. The methods can be applied to other specialties than the physical sciences.
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