Linearized density of states (DOS)
\\n\\n
Dr. Pletser’s experience includes 30 years of working with the European Space Agency as a Senior Physicist/Engineer and coordinating their parabolic flight campaigns, and he is the Guinness World Record holder for the most number of aircraft flown (12) in parabolas, personally logging more than 7,300 parabolas.
\\n\\nSeeing the 5,000th book published makes us at the same time proud, happy, humble, and grateful. This is a great opportunity to stop and celebrate what we have done so far, but is also an opportunity to engage even more, grow, and succeed. It wouldn't be possible to get here without the synergy of team members’ hard work and authors and editors who devote time and their expertise into Open Access book publishing with us.
\\n\\nOver these years, we have gone from pioneering the scientific Open Access book publishing field to being the world’s largest Open Access book publisher. Nonetheless, our vision has remained the same: to meet the challenges of making relevant knowledge available to the worldwide community under the Open Access model.
\\n\\nWe are excited about the present, and we look forward to sharing many more successes in the future.
\\n\\nThank you all for being part of the journey. 5,000 times thank you!
\\n\\nNow with 5,000 titles available Open Access, which one will you read next?
\\n\\nRead, share and download for free: https://www.intechopen.com/books
\\n\\n\\n\\n
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'
Preparation of Space Experiments edited by international leading expert Dr. Vladimir Pletser, Director of Space Training Operations at Blue Abyss is the 5,000th Open Access book published by IntechOpen and our milestone publication!
\n\n"This book presents some of the current trends in space microgravity research. The eleven chapters introduce various facets of space research in physical sciences, human physiology and technology developed using the microgravity environment not only to improve our fundamental understanding in these domains but also to adapt this new knowledge for application on earth." says the editor. Listen what else Dr. Pletser has to say...
\n\n\n\nDr. Pletser’s experience includes 30 years of working with the European Space Agency as a Senior Physicist/Engineer and coordinating their parabolic flight campaigns, and he is the Guinness World Record holder for the most number of aircraft flown (12) in parabolas, personally logging more than 7,300 parabolas.
\n\nSeeing the 5,000th book published makes us at the same time proud, happy, humble, and grateful. This is a great opportunity to stop and celebrate what we have done so far, but is also an opportunity to engage even more, grow, and succeed. It wouldn't be possible to get here without the synergy of team members’ hard work and authors and editors who devote time and their expertise into Open Access book publishing with us.
\n\nOver these years, we have gone from pioneering the scientific Open Access book publishing field to being the world’s largest Open Access book publisher. Nonetheless, our vision has remained the same: to meet the challenges of making relevant knowledge available to the worldwide community under the Open Access model.
\n\nWe are excited about the present, and we look forward to sharing many more successes in the future.
\n\nThank you all for being part of the journey. 5,000 times thank you!
\n\nNow with 5,000 titles available Open Access, which one will you read next?
\n\nRead, share and download for free: https://www.intechopen.com/books
\n\n\n\n
\n'}],latestNews:[{slug:"intechopen-partners-with-ehs-for-digital-advertising-representation-20210416",title:"IntechOpen Partners with EHS for Digital Advertising Representation"},{slug:"intechopen-signs-new-contract-with-cepiec-china-for-distribution-of-open-access-books-20210319",title:"IntechOpen Signs New Contract with CEPIEC, China for Distribution of Open Access Books"},{slug:"150-million-downloads-and-counting-20210316",title:"150 Million Downloads and Counting"},{slug:"intechopen-secures-indefinite-content-preservation-with-clockss-20210309",title:"IntechOpen Secures Indefinite Content Preservation with CLOCKSS"},{slug:"intechopen-expands-to-all-global-amazon-channels-with-full-catalog-of-books-20210308",title:"IntechOpen Expands to All Global Amazon Channels with Full Catalog of Books"},{slug:"stanford-university-identifies-top-2-scientists-over-1-000-are-intechopen-authors-and-editors-20210122",title:"Stanford University Identifies Top 2% Scientists, Over 1,000 are IntechOpen Authors and Editors"},{slug:"intechopen-authors-included-in-the-highly-cited-researchers-list-for-2020-20210121",title:"IntechOpen Authors Included in the Highly Cited Researchers List for 2020"},{slug:"intechopen-maintains-position-as-the-world-s-largest-oa-book-publisher-20201218",title:"IntechOpen Maintains Position as the World’s Largest OA Book Publisher"}]},book:{item:{type:"book",id:"3093",leadTitle:null,fullTitle:"Wave Propagation Theories and Applications",title:"Wave Propagation",subtitle:"Theories and Applications",reviewType:"peer-reviewed",abstract:"A wave is one of the basic physics phenomena observed by mankind since ancient time. 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\r\n\tUbiquitous computing is often mentioned as an application domain with blurry or even undefined rules, boundaries, and examples. Fortunately, there are several concepts, developments, and pieces of research that have been put forward into this book (data integration, distributed systems, data enhancement via blockchain or machine learning, etc.) which explain what ubiquitous computing is and how it can be used to everyone´s advantage. All these contributions are based on research works and developments involving areas of knowledge such as the Internet of Things, Autonomous Systems, Cyber-Physical Systems, and Wireless Sensor Networks. Among other topics, heterogeneous hardware integration, low capability distributed systems, microservices, security deployments for autonomous devices, or data mining procedures are fully mentioned and described here. In this way, this book provides a holistic view onto the field of ubiquitous computing and all its features, both related to the software and hardware, while at the same time offering knowledge from cutting edge projects that provide the backbone of the contents that have been included.
",isbn:"978-1-83968-690-0",printIsbn:"978-1-83968-689-4",pdfIsbn:"978-1-83968-691-7",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"adf308a0840ede98439d031a21ba73a6",bookSignature:"Dr. Ishwar Singh and Dr. Zhen Gao",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/9977.jpg",keywords:"Distributed Systems, Data Formatting, Actuators, Communication Protocols, Hardware Integration, Machine Learning, Interfaces, Unmanned Vehicles, Digital Twins, UAVs, UGVs, AUVs",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:null,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"September 3rd 2020",dateEndSecondStepPublish:"October 1st 2020",dateEndThirdStepPublish:"November 30th 2020",dateEndFourthStepPublish:"February 18th 2021",dateEndFifthStepPublish:"April 19th 2021",remainingDaysToSecondStep:"7 months",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:"Dr.Ishwar Singh has an extensive history of leadership, program, and course development, at Mohawk College, in the McMaster-Mohawk Bachelor of Technology (B.Tech.) Partnership and in SEPT. At Mohawk College, he designed and developed the first-ever degree program in Process Automation, in addition, to jointly securing funding with two other colleagues to establish a Process Automation Applied Research Centre.",coeditorOneBiosketch:"Dr. Gao has published over 100 journal and conference papers, 1 book,\r\nand 4 book chapters in areas of advanced robotics, automation, and artificial Intelligence. \r\nHe serves as the Editor-in-Chief of the International Journal of Intelligent Machines and\r\nRobotics; he is also the Editor-in-Chief of the Journal of Robotic and Mechatronic Systems.",coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"333793",title:"Dr.",name:"Ishwar",middleName:null,surname:"Singh",slug:"ishwar-singh",fullName:"Ishwar Singh",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0033Y000031TBdSQAW/Profile_Picture_1616070554137",biography:null,institutionString:"McMaster University",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"0",totalChapterViews:"0",totalEditedBooks:"0",institution:{name:"McMaster University",institutionURL:null,country:{name:"Canada"}}}],coeditorOne:{id:"353808",title:"Dr.",name:"Zhen",middleName:null,surname:"Gao",slug:"zhen-gao",fullName:"Zhen Gao",profilePictureURL:"https://s3.us-east-1.amazonaws.com/intech-files/0033Y0000364SnZQAU/Profile_Picture_1615808028038",biography:"Zhen Gao is an assistant professor in the School of Engineering Practice and Technology\r\n(SEPT) at McMaster University which he joined in July 2014. Since September 2020, he was\r\nappointed as Program Lead for a Master’s program in Systems & Technology which was focused\r\non Cyber Physical System. He has published over 100 journal and conference papers, 1 book,\r\nand 4 book chapters in areas of advanced robotics, automation and artificial Intelligence. Dr. Gao\r\nis the Associate Editor for the Conference Editorial Board of the IEEE Robotics and Automation\r\nSociety, 2021 IEEE International Conference on Robotics and Automation. He was the sessional\r\nchair For 9th International Conference on Interactive, Collaborative, and Blended Learning, and\r\nfor International Conference on Interactive Mobile Communication, Technologies and Learning.\r\nDr. Gao served as the Program Committee Member for World Congress on Intelligent Control\r\nand Automation, International Conference on Information and Automation, IEEE Conference on\r\nRobotics and Biomimetics, IEEE International Conference on Real-time Computing and\r\nRobotics, IEEE International Conference on Automation and Logistics, and International\r\nConference on Intelligent Robotics and Applications, and IEEE Canadian Conference on\r\nElectrical and Computer Engineering. His current research interests include industrial\r\ncontrollers, advanced robotics and automation, artificial intelligence, neural network and pattern\r\nrecognition. He serves as the Editor-in-Chief of International Journal of Intelligent Machines and\r\nRobotics; he is also the Editor-in-Chief of Journal of Robotic and Mechatronic Systems.",institutionString:"McMaster University",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"0",totalChapterViews:"0",totalEditedBooks:"0",institution:{name:"McMaster University",institutionURL:null,country:{name:"Canada"}}},coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"9",title:"Computer and Information Science",slug:"computer-and-information-science"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"280415",firstName:"Josip",lastName:"Knapic",middleName:null,title:"Mr.",imageUrl:"https://mts.intechopen.com/storage/users/280415/images/8050_n.jpg",email:"josip@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. 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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"72",title:"Ionic Liquids",subtitle:"Theory, Properties, New Approaches",isOpenForSubmission:!1,hash:"d94ffa3cfa10505e3b1d676d46fcd3f5",slug:"ionic-liquids-theory-properties-new-approaches",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/72.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"314",title:"Regenerative Medicine and Tissue Engineering",subtitle:"Cells and Biomaterials",isOpenForSubmission:!1,hash:"bb67e80e480c86bb8315458012d65686",slug:"regenerative-medicine-and-tissue-engineering-cells-and-biomaterials",bookSignature:"Daniel Eberli",coverURL:"https://cdn.intechopen.com/books/images_new/314.jpg",editedByType:"Edited by",editors:[{id:"6495",title:"Dr.",name:"Daniel",surname:"Eberli",slug:"daniel-eberli",fullName:"Daniel Eberli"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"57",title:"Physics and Applications of Graphene",subtitle:"Experiments",isOpenForSubmission:!1,hash:"0e6622a71cf4f02f45bfdd5691e1189a",slug:"physics-and-applications-of-graphene-experiments",bookSignature:"Sergey Mikhailov",coverURL:"https://cdn.intechopen.com/books/images_new/57.jpg",editedByType:"Edited by",editors:[{id:"16042",title:"Dr.",name:"Sergey",surname:"Mikhailov",slug:"sergey-mikhailov",fullName:"Sergey Mikhailov"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1373",title:"Ionic Liquids",subtitle:"Applications and Perspectives",isOpenForSubmission:!1,hash:"5e9ae5ae9167cde4b344e499a792c41c",slug:"ionic-liquids-applications-and-perspectives",bookSignature:"Alexander Kokorin",coverURL:"https://cdn.intechopen.com/books/images_new/1373.jpg",editedByType:"Edited by",editors:[{id:"19816",title:"Prof.",name:"Alexander",surname:"Kokorin",slug:"alexander-kokorin",fullName:"Alexander Kokorin"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"2270",title:"Fourier Transform",subtitle:"Materials Analysis",isOpenForSubmission:!1,hash:"5e094b066da527193e878e160b4772af",slug:"fourier-transform-materials-analysis",bookSignature:"Salih Mohammed Salih",coverURL:"https://cdn.intechopen.com/books/images_new/2270.jpg",editedByType:"Edited by",editors:[{id:"111691",title:"Dr.Ing.",name:"Salih",surname:"Salih",slug:"salih-salih",fullName:"Salih Salih"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"872",title:"Organic Pollutants Ten Years After the Stockholm Convention",subtitle:"Environmental and Analytical Update",isOpenForSubmission:!1,hash:"f01dc7077e1d23f3d8f5454985cafa0a",slug:"organic-pollutants-ten-years-after-the-stockholm-convention-environmental-and-analytical-update",bookSignature:"Tomasz Puzyn and Aleksandra Mostrag-Szlichtyng",coverURL:"https://cdn.intechopen.com/books/images_new/872.jpg",editedByType:"Edited by",editors:[{id:"84887",title:"Dr.",name:"Tomasz",surname:"Puzyn",slug:"tomasz-puzyn",fullName:"Tomasz Puzyn"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"70894",title:"Thermal Collective Excitations in Novel Two-Dimensional Dirac-Cone Materials",doi:"10.5772/intechopen.90870",slug:"thermal-collective-excitations-in-novel-two-dimensional-dirac-cone-materials",body:'Graphene, a two-dimensional (2D) carbon layer with a hexagonal atomic structure [1, 2, 3], has recently attracted outstanding attention from both academic scientists doing fundamental researches and engineers working on its technical applications [4]. Now, the scientific community is actively investigating the innovative semiconductors beyond graphene, with intrinsic spin-orbit interaction and tunable bandgap [5].
A remarkable feature of graphene is the absence of the bandgap in its energy dispersions. In spite of the obvious advantage of such bandstructure for novel electronic devices, electrons in graphene could not be confined due to the well-known Klein paradox [6]. To resolve this issue, graphene may be replaced with a material with a buckled structure and substantial spin-orbit interaction, such as silicene and germanene.
A new quasi-two-dimensional structure which has recently gained popularity among device scientists, is molybdenum disulfide monolayer, a honeycomb lattice which consists of two different molybdenum and sulfur atoms. It reveals a large direct band gap, absence of inversion symmetry and a substantial spin-orbit coupling. A summary of all recently fabricated materials beyond graphene is given in Figure 1. The last relevant example is black phosphorous (phosphorene) with a strong anisotropy of its composition and electron energy dispersion. Even though we do not study plasmons in phosphorene in the present chapter, there have been some crucial publications on that subject [7, 8].
Recently discovered two-dimensional materials: graphene and beyond graphene.
Plasmons, or self-sustained collective excitations of interacting electrons in such low-dimensional materials, are especially important, since they serve as the basics for a number of novel devices and their applications [9, 10] in almost all fields of modern science, emerging nanofabrication and nanotechnology. Propagation and detection of plasmonic excitation in hybrid nanoscale devices can convert to or modify existing electromagnetic field or radiation [11, 12, 13, 14]. Localized surface plasmons are particularly of special interest considering their interactions with other plasmon modes in closely-located optoelectronic device as well as with imposed electromagnetic radiation [15].
Finite-temperature plasmons are of special interest for possible device applications. Among them is the possibility to increase the frequency (or energy) of a plasmon by an order of magnitude or even more, specifically, as a consequence of the raised temperature. As it was shown in Ref. [16], the dispersion of a thermal plasmon is given as
At the same time, the damping rate, or broadening of the frequency, of such thermal plasmons varies as
In this chapter, we will consider thermal behavior of plasmons, their dispersions and damping rates. By equipping with this information, it is possible to predict in advance the thermal properties of an electronic device designed for a particular temperature range. In spite of a number of reported theoretical studies on this subject [16, 17, 18], there is still a gap on demonstrating experimentally these unique thermal collective features of 2D materials. Therefore, our review can serve as an incentive to address this issue.
All the novel 2D materials considered here could be effectively assigned to an individual category based on their existing (or broken) symmetries and degeneracy in their low-energy band structure. We started with graphene having a bandgap
Silicene and germanene, which represent buckled honeycomb lattices, possess subbands depending on valley and spin indices, and therefore are only doubly-degenerate. The electron-hole symmetry is broken for molybdenum disulfide and other transition-metal dichalcogenides (TMDC’s). For these situations, even though there exists a single electron-hole index
Energy dispersions and density of states (DOS)
The energy dispersions of buckled honeycomb lattices, obtained from a Kane-Mele type Hamiltonian, appear as two inequivalent doubly-degenerate pairs of subbands with the same Fermi velocity
where
The wave function of silicene, corresponding to eigenvalue equation in Eq. (1), takes the form [25]
where
Germanene, another representative of buckled honeycomb lattices [26, 27, 28, 29, 30], demonstrates substantially higher Fermi velocities and an enhanced intrinsic bandgap
MoS2 is a typical representative of transition-metal dichalcogenide (TMDC) monolayers. TMDC’s are semiconductors with the composition of TC2 type, where T refers to a transition-metal atom, such as Mo or W, while C corresponds to a chalcogen atom (S, Se or Te).
MoS2 displays broken inversion symmetry and direct bandgaps. Its most crucial distinction from the discussed buckled honeycomb lattices is its broken symmetry between the electrons and holes so that the corresponding energy bands are no longer symmetric with respect to the Dirac point, but could still be classified by a single index
Specifically, the energy bands of MoS2 can be described by a
where
In practical, we will neglect the
Using Eq. (3), we can verify that the degeneracy of two hole subbands (
One of the most important features in connection with plasmons at zero and finite temperatures is its dispersion relations, i.e., dependence of the plasmon frequency
where
The dielectric function introduced in Eq. (4) is determined directly by the finite-temperature
where the integration variable
The zero-temperature polarizability, which is employed in Eq. (5), is quite similar for all 2D materials considered here. The only difference originates from the degeneracy level of the low-energy band structure, such as
where
Finally, the full polarization function at zero temperature is obtained as
If the dispersions of low-energy subbands do not depend on the valley or spin indices
Integral transformation in Eq. (5), which is used to obtain the finite-temperature polarization function from its zero-temperature counterparts with different Fermi energies, was first introduced in Ref. [34]. It could be derived in a straightforward way by noting that the only quantity which substantially depends on temperature in Eq. (6) is the Fermi-Dirac distribution function
For accessible temperatures, the energy dispersions
We first look at intrinsic plasmons with
Additionally, finite-
where we introduce the notation
From Eq. (9), we find explicitly that
Here, the
Furthermore, for gapless (
where we have used the high-
On the other hand, for gapped (
where the constant
As we have seen from Section 3, we need know
The density of states (DOS), which plays an important tool in calculating electron (or hole) Fermi energy
where
This result is equivalent to the DOS of graphene except that there are no states within the bandgap region, as demonstrated by two unit-step functions
Finally, the chemical potential
where
At
where we have assumed that both subbands are occupied for simplicity. The discussions of other cases can be found from Ref. [18]. Consequently, minimum electron density required to occupy the upper subband of silicene is
On the other hand, by applying Eq. (16), in combination with DOS in Eq. (15), for silicene, a transcendental (non-algebraic) equation [43, 44] could be obtained for
where
Interestingly, the right-hand side of Eq. (18) contains terms corresponding to both pristine and gapless graphene, using which we find from Ref. [17].
as well as a well-known analytical expression for
An advantage of Eq. (18) is that it could be solved even without taking an actual integration. In fact, one can either readily solve it numerically using some standard computational algorithms, or introduce an analytical approximation to the sought solution near specific temperature assigned.
Numerical results for
Temperature dependence of the chemical potential
Temperature dependence of the chemical potential
Eq. (18) could also be applied to a wide range of 2D materials if its DOS has a linear dependence on energy
Now, we turn to calculate
where the calculation is based on a parabolic-subband approximation, i.e.,
From Eq. (22), we further seek an explicit expression for DOS in the form a piecewise-linear function of energy
or numerically,
The calculated numerical results for DOS in all regions are listed in Table 1. All introduced coefficients
Range index | Energy range | ||||
---|---|---|---|---|---|
Linearized density of states (DOS)
The critical doping density which is required to populate the lower hole subband in MoS2 is found to be
Therefore, for most experimentally accessible densities
Next, we would evaluate both sides of Eq. (16) for MoS2. As an example, we consider electron doping with density
From Eq. (27), we can easily find the electron Fermi energy
In a similar way, for hole doping with density
where
The right-hand side of Eq. (16) for TMDC’s could be expressed as a combination of electron and hole contributions
so that
For convenience, we introduce another function
where
where two terms with
Using these self-defined functions and their notations, we finally arrive at the “
where
Here, both
By using the calculated
Particle-hole modes and plasmon branch for extrinsic (doped) silicene layer at a finite temperature. Panels (a) and (b) show two comparative graphs for
In addition to graphene and silicene, another type of Dirac-cone materials is the one with fermionic states in which multiple Dirac points evolve into a middle flat band. One of the first fabricated materials with such a flat band is a dice or a
The low-energy electronic states of
The
The low-energy Dirac-Weyl Hamiltonian for the
where
Three energy bands from Hamiltonian in Eq. (38) or Eq. (39) are
where
Here, the components of wave functions in Eqs. (40) and (41) depend on valley index
Now we turn to deriving plasmon branches and their damping rates at finite
For
Structurally, Eq. (42) looks quite similarly to Eq. (6) for buckled honeycomb lattices and TMDC’s. The most significant difference comes as the existence of an additional flat band with
Here, we would limit our consideration to the case of electron doping with
where
for an arbitrary value of
Density plots for Landau damping with
Particle-hole modes, determined by non-zero
In a correspondence to the damping of plasmons presented in Figure 6, we show in Figure 7 the density plots for plasmon dispersions at
Plasmon branches for an isolated
In the last part of THIS CHAPTER, WE WOULD LIKE TO FOCUS ON finite-
Our schematics for an open system is shown in Figure 8. The dynamical screening to the Coulomb interaction between electrons in a 2D layer and in metallic substrate is taken into account by a nonlocal and dynamical inverse dielectric function
Schematics for a silicene-based open system and numerical results for the two plasmon branches and their damping in this system with
and the resonances in
By using the Drude model for metallic substrate, the dielectric function can be written as
where
As a special example, let us consider a silicene 2D layer with two bandgaps
where
In the limit of
where for convenience we introduced a coefficient
We notice from Eq. (48) that
Furthermore, using the notation defined by Eq. (49), we get from Eqs. (46) and (47) that
which leads to a bi-quadratic equation
Eq. (51) gives rise to two solutions
where
In Eq. (53), both plasmon branches contain a linear
Numerical results for thermal plasmons in open system are presented in Figure 9. Similarly to what we have found for graphene and silicene, there are two plasmon branches, both of which depend linearly on
Nonlocal hybridized plasmon dispersions for
In conclusion, we have developed a general theory for finite-temperature polarization function, plasmon dispersions and their damping for all known innovation 2D Dirac-cone materials with various types of symmetries and bandgaps. We have also derived a set of explicit transcendental equations determining the chemical potential as a function of temperature, which serves as a key part in calculating finite-temperature polarization function through the so-called thermal convolution path. The selection of a particular path with a specific
Using the calculated finite-temperature polarization function, we have further found the dispersions of hybrid plasmon-modes in various types of open systems including a 2D material coupled to a conducting substrate. The obtained plasmon dispersions in these 2D-layer systems are crucial for measuring spin-orbit interaction strength and dynamical screening to Coulomb interaction between electrons in 2D materials, as well as for designing novel surface-plasmon based multi-functional near-field opto-electronic devices.
We have generalized our developed theory for 2D materials further to most recently proposed
A.I. thanks Liubov Zhemchuzhna for helpful and fruitful discussions, and Drs. Armando Howard, Leon Johnson and Ms. Beverly Tarver for proofreading the manuscript and providing very useful suggestions on the style and language. G.G. would like to acknowledge the financial support from the Air Force Research Laboratory (AFRL) through grant FA9453-18-1-0100 and award FA2386-18-1-0120. D.H. thanks the supports from the Laboratory University Collaboration Initiative (LUCI) program and from the Air Force Office of Scientific Research (AFOSR).
All the authors declare that they have no conflict of interest.
As one of the qualified candidates for hypersonic propulsion system, scramjet combustor has attracted a large amount of attention all over the world. As is now well known, realizing high efficiency and steady combustion in a supersonic combustor always remains as a critical issue for scramjet engines. Numerous researches have indicated that the additions of cavity in combustor and the transverse fuel injection upstream of the cavity can promote the stabilization of combustion and flame [1, 2, 3]. Whereas, it is difficult yet for the fuel to reach properly mixing within the supersonic flow by the mechanical methods [4, 5]. Moreover, it is a great challenge for matching the transverse fuel jet up with the cavity under off-design condition [4]. Furthermore, certain stagnation pressure loss will appear using the approaches. Taken all account, some new methods are imminently needed for keeping stable and highly efficient combustion in combustor with the least penalties adding to the flowfield.
It is rather promising to adopt a discharge plasma for supersonic flow and/or combustion control in aerospace field [6, 7, 8]. It has been widely considered that plasma-assisted combustion is one of the most promising approaches for enhancing ignition and combustion in the environment of scramjet combustion [4, 6, 9]. As a further promising method, there are three advantages, that is: rapid response, less inertia, and flexibility [10]. Former studies clearly show that the quasi-DC discharge plasma can availably modify supersonic flow in a controllable manner among the discharge plasma mentioned above [8, 11], whereas the DBD plasma is commonly used in low-speed flow environment [12, 13]. If we can combine the quasi-DC discharge plasma with cavity and transverse fuel jet together, some new phenomena will surely appear, which may help in the improvement of combustor performance.
There in three primary mechanisms of the plasma effect on flow and combustion can be summarized: (1) momentum transfer, (2) fast local ohmic heating of the medium, and (3) active particles [6, 7, 14]. Ignoring the magnetic field, the mechanism of quasi-DC discharge plasma affecting a supersonic flow is mainly its fast local heating rather than the electrostatic force (i.e., the momentum transfer mechanism) [10, 15, 16].
This chapter aims at investigating the changes of the fuel jet, cavity, and whole scramjet combustor led by the quasi-DC discharge plasma based on the dominant thermal blocking mechanism. Here, a short cavity downstream of a fuel jet orifice is considered in order to simulate the combustor flowfield more realistic. The plasma is set as a controllable heat source. The
By applying a high voltage across the anode and cathode, a bright plasma filament appears above the two electrodes. With the impacting of incoming flow, the plasma occurs not only between the two electrodes but also several centimeters downstream on the wall [10]. In consideration of the oscillation character of its electrical parameters and mean temperature in the discharge path, this type of discharge plasma is called “quasi-DC discharge plasma.” However, it is a type of low temperature arc plasma in fact. The main properties of quasi-DC discharge plasma are described in [10, 11, 12, 15].
As depicted in Figure 1, the domain of quasi-DC discharge plasma filament is simplified as a cuboid region presented in red dashed lines for simulation based on its appearance. In Figure 1, the symmetric plotted line indicates that the cathode and anode are symmetrical for the central line of combustor wall and so does the plasma filament.
Computed configuration of quasi-DC discharge plasma in case one.
Because the quasi-DC discharge plasma releases a large amount of heat concentrated in its discharge path, the discharge path goes into very hot. This high temperature domain (i.e., the discharge path) obstructs the supersonic inflow due to a thermal blocking occurred. Hereby, the quasi-DC discharge plasma behaves as a virtual blockage in the high speed flow of scramjet combustor, which is called as “the dominant thermal blocking mechanism” [17]. On the foundation of dominant thermal blocking mechanism, the individual plasma filament can be simulated as a volumetric heat source [10, 12].
Generally speaking, the plasma input power and the effective heat power that transfers into circumambient gas are quite different. Besides, the percentage of loss commonly varies with the power supply and environment, so it is not a suitable way to use the plasma input power as the heat source value of quasi-DC discharge plasma in simulation. In order to straightforwardly describe the plasma heat strength, the average temperature of the plasma zone is acquired by using the numerical simulation here. This way is feasible when the average temperature of the plasma zone is within a reasonable range [10, 16, 18, 19]. Hence, based on the thermal blocking mechanism, a certain temperature which denotes the actuating strength (i.e., the input power) is specified for the plasma filament domain when the actuator works.
The electrodes are flush mounted and do not have influence on the main flow themselves. The length and the section dimensions of individual plasma filament heat source are 20 mm length and 3 × 3 mm, respectively. The plasma filament locates 40 mm upstream of the fuel orifice center. Besides, it is generated near the wall in the center of the combustor, which means the symmetric plane of plasma filament is within the symmetric
The unsteady Reynolds averaged 3D Navier-Stokes equations (URANS) with the
here,
where the coefficients can be found in [21]. The
where
S (K) | |||
---|---|---|---|
H2 | 273.11 | 96.67 | 8.411 × 10−6 |
O2 | 273.11 | 138.9 | 1.919 × 10−5 |
H2O | 416.67 | 861.11 | 1.703 × 10−5 |
N2 | 273.11 | 106.67 | 1.663 × 10−5 |
Parameters in Sutherland law.
Since this study focuses on the qualitative effect of plasma on the fuel jet and the scramjet combustor, the finite rate chemical model with the single step H2/O2 combustion kinetics model is applied. The reaction rate constant is derived from the Arrhenius formula. Hence, the computational time can be saved much and the well combustion flowfield can be obtained, too. The relevant parameters of H2/O2 one-step chemical model are shown in Table 2.
Equation | Pre-exponential factor | Temperature index | Activity energy [J/kg·mol] |
---|---|---|---|
9.87E + 8 | 0 | 3.1E + 7 |
H2/O2 one-step chemical model.
In order to capture the shock waves and other complex fluid structures better, the convective fluxes are evaluated using the advection upstream splitting method (AUSM) [22] with the second-order upwind approach at same time. The viscous fluxes are evaluated by using the second-order central differential scheme. Because the transport process of multispecies and the reaction both exist in the flowfield, a modified LU-SGS implicit method [23] is adopted for temporal integration.
Figure 2 shows the entire computational domain which is the half of scramjet combustor together with a short cavity and a fuel jet orifice. The inlet of combustor is located at
Computational domain and grids of the combustor.
The grids number is largely reduced by setting the symmetry face (
The entire computational domain is divided into 32 subdomains, and all assignments are completed by parallel computation on HP senior workstation, which takes about 320 h to obtain a convergence result. The inflow conditions in computation are as follows:
Three representative times are selected during one plasma control cycle for comparison after computation converged. The end of the actuator duty time (
The temperature distribution on the symmetric
Contours of temperature on the symmetric
The distribution of wall pressure near the fuel orifice on the symmetric plane of the combustor is shown in Figure 4, which is normalized by the value of the static pressure of inflow at the inlet. Compared with no plasma case, the positions that pressure starts to rise at different times move upstream from 56.6 to 52 mm due to the effect of plasma, which indicates the separation shock wave induced by the fuel jet moves upstream. And the first pressure peak decreases from 2.0 to 1.7, because of the weakening of the fuel jet induced shock on the symmetric plane. The results above are similar to the previous studies on nonreaction flow combustor [19]. More details indicate that the first pressure peak of time A is a bit higher than time B and C, but time B equals time C, which is due to the plasma control cycle too short for flow response and the duty cycle ratio also comparatively large in the flow condition here. Hence, the influence of plasma on the shock will be observed a little latter due to an inertia influence. Nevertheless, the effect of plasma on the shock wave for three typical times is highly similar on the whole.
Wall pressure distribution on the symmetric plane.
Because the separation zone upstream of the fuel orifice is primarily controlled by the separation shock wave, the size of this separation zone can be regulated by changing the location of the separation shock wave. On the one side, this zone behaves as an main ignition zone which can provide a high temperature and low flow speed environment in the scramjet combustor. On the other side, it will bring in certain pressure loss to the combustor. Hence, we can make use of the separation zone upstream of the fuel orifice by means of using the plasma with proper control parameters.
For the sake of determining the fuel mixing quality, the five cross-sectional planes distributed in equivalent ratio along the flow direction are given, as shown in Figure 5. The fuel jet arises with the actuator working, resulting in fuel decreasing near the wall including the bottom wall of the cavity. Whereas, the distribution shape of fuel in the cross-sectional planes varies from a distribution narrow and long profile into a circular profile in the upper space, which shows that the process of fuel spreading into the main supersonic flow is enhanced resulted from the plasma. Furthermore, the fuel distributions at time A are nearly the same as at time C, which is also correlated to the inertial effect mentioned above.
Distribution of equivalent ratio in cross-sectional planes. (a) Plasma off. (b) Plasma on, time A. (c) Plasma on, time C.
In Figure 6, the distribution of product water is shown. In order to distinguish the extent of product water between time A and C easily, the iso-surfaces of both are plotted by combining the half parts of them together as shown in Figure 6(b). Similar to the changes in Figure 5, more water is generated in the upper space due to plasma. Contrasted with the case without plasma actuator, the iso-surface of water expands much in its center and shrinks near the combustor wall especially downstream of cavity. These above should be attributed to the change of fuel jet spread, as given in Figure 5. Compared time A with C, the only distinction is a little more water formed at time A, which indicates that the combustion becomes weaker in the free time of a plasma control cycle.
Iso-surfaces of product water,
It can be seen that the fuel entrainment into the cavity is decreased and the combustion in cavity becomes weaker due to the plasma, as given in Figures 5 and 6.Whereas, the two figures also show that more fuel penetrates into the supersonic air flow, so the mass exchange between the inside cavity and its outside is enhanced due to the plasma. Hence, the plasma improves the fuel mixing above the cavity prominently, which can also be realized as the calculation results of combustion efficiency in Section 2.2.4.
The stagnation pressure recovery coefficient can indicate the pressure loss in a combustor, so it is an important index. When the stagnation pressure recovery coefficient goes up, the capability of combustor outflow will be enhanced. So, it is defined as the ratio of combustor outlet stagnation pressure to inlet stagnation pressure [24]. In fact, calculating the mass flow weighted mean stagnation pressure can gain the stagnation pressure in a cross section.
where
where
The calculated data about the stagnation pressure are given in Table 3. In order to analyze the tendency of stagnation pressure loss coefficient further, the average stagnation pressure at the outlet is replaced by the average stagnation pressure at different positions, as shown in Figure 7. At the actuator working time or even at the free time, the stagnation pressure loss can both be increased due to the plasma, as given in Table 3. In (7), the relative change rate of
where
Index of performance | Case | |||
---|---|---|---|---|
Plasma off | Plasma on, time A | Plasma on, time B | Plasma on, time C | |
1,098,093 | ||||
916,532 | 882,270 | 888,143 | 885,268 | |
0.83466 | 0.80346 | 0.80880 | 0.80619 | |
0.16534 | 0.19654 | 0.19120 | 0.19381 |
The inlet mean stagnation pressure
Distribution of the stagnation pressure loss coefficient.
In Figure 7, the difference of stagnation pressure loss between plasma cases is shown where no plasma case enlarges along the flow direction (
The combustion efficiency
where
Figure 8 plots the distribution of combustion efficiency in
Distribution of combustion efficiency downstream of fuel orifice.
In order to define the cost to effectiveness of quasi-DC plasma, it is calculated in value
The main results in this section are as follows: (1) The distribution of relatively high temperature zone moves downstream prominently due to the heat release from the quasi-DC discharge plasma. The separation shock wave induced by the fuel jet is partly weakened and moves upstream due to the plasma, which can regulate the size of recirculation zone upstream of the fuel orifice. (2) The fuel jet moves upward integrally resulted from the plasma heating effect. The fuel spread wider along the spanwise and penetrates into the leading flow deeper, resulting in the cross-section shape of the fuel jet varying from a narrow and long profile to a circular profile. Because of the variation of fuel mixing, in the upper space, more water forms while less appears near the wall compared with the case without plasma. (3) The stagnation pressure loss of combustor increases a little as actuator works, but the combustion efficiency in the combustor rises obviously. These above can be summarized as the comprehensive effects of flow structure changes caused by the plasma, including waves induced and heat transfer. Since it is negligible for the relative change of stagnation pressure recovery coefficient in the actuator working cases, and the ratio of deposited plasma energy to the increased combustion heat release is very little, it can be obtained that the quasi-DC discharge plasma can make more benefits than penalties for the scramjet combustor, when proper adopting control parameters of the plasma actuator.
On account of the direction of the quasi-DC discharge path for the inflow, the discharge modes include two types: longitudinal mode and transversal mode [12]. In case one, the quasi-dc discharge operates under transversal mode, while in case two, a longitudinal mode is adopted for the configuration. As shown in Figure 9, there are five anodes set upstream of the backward wall of cavity, while in the bottom wall of cavity the five cathodes are set. The five pairs of identical electrodes are arranged parallel and symmetrically. Besides, the flow will not be disturbed directly resulted from all the electrodes flush mounted in the wall. A filament plasma forms when applied a high voltage (generally 150–1200 V) between anode and cathode, which acts as strongly oscillating and bright and looks like an inverted “L” crossing the backward wall of cavity. Since the major heat energy of the quasi-DC discharge plasma focuses on the bright filament domain [10, 25], the filament plasma domain can be established in such a simplified shape as given in Figure 9.
The schematic of plasma filaments in the cavity.
Just like the way given in case one, every quasi-DC plasma filament is dealt as a volumetric heat source. To represent the plasma heat strength reasonably, the mean temperature of the plasma domain
Three representative times are chosen from one plasma cycle, which consists of the later actuator free duration and the actuator working duration, for comparison after computations converged. In the simulation, the duration of one plasma cycle is
The Mach number distribution of local cavity flowfield on the symmetrical
Mach number distribution: (a) no plasma; (b) with plasma, time A; (c) with plasma, time B; (d) with plasma, time C.
Owing to the periodic heat release from the plasma filaments and the thermal blocking functions, the plasma filaments behave as five knives cut the cavity shear layer. Hence, the mass transportation process will be disturbed by this “cutting” effect, and the moving direction of original shear layer has to be changed also. And then, the turbulence intensity and vorticity magnitude both are increased, so the fuel and air existing around the original shear layer can exchange through the cavity mouth more easily. Furthermore, the combustion enhancement downstream or over the cavity causes the decrease of local flow velocity, which results in the rise of local static pressure and blocks the incoming flow.
Figure 11 presents the distributions of wall pressure near the rear edge of cavity on
Wall pressure distribution near the rear edge of cavity. (a)
As presented in Figure 11b, the wall pressure of cavity rises distinctly on
Because the distributions of combustion products may be impacted by the change of cavity shear layer and waves, the mass fraction iso-surface of water (
Iso-surface of product water,
The above phenomena can be attributed to the reasons as follows: (1) Plasma filaments existed upstream of the cavity front edge release a large amount of heat, so the local static temperature increases and then the movement of fuel jet is promoted toward upper space. Because of the larger velocity far from the bottom wall (i.e., flow moves faster in the middle height of the combustor), the original product water will surely move downstream quickly. Therefore, the little water is found around the orifice when plasma actuators work, and the symmetry structures are formed in higher space downstream. It can be verified from the “branch pipes” presented in Figure 12b–d. (2) As stated above, the periodic fluctuation of cavity shear layer is related to the plasma filaments “cutting” effect on it, and then the mass transportation process is blocked which leads to the obvious change of combustion over the cavity. Hence, it affects the distribution of product water. Meanwhile, notice that the “branch pipes” curve structures match with the typical spanwise reverse vortex pairs in shape. Generally speaking, the spanwise reverse vortex pairs can extend the contact area between air and fuel by the methods of entraining air into its fuel core, which makes the mixing between fuel and air better and also the local combustion efficiency arisen. Whereas, the periodic disturbance from the plasma filaments impels the air and fuel transporting in the
The drag and mass exchange rate of cavity are two important cavity performance parameters [26]. The combustor’s drag is mainly generated by the cavity as it has a nearly constant cross section. The cavity drag includes two types: pressure drag and friction drag. Pressure drag is defined as the difference value between the force on cavity front wall and rear wall. Usually, the friction drag is too small compared with pressure drag so it always can be ignored. So the pressure drag is regarded as the cavity drag here. The drag coefficient of flame holding cavity is defined as:
where
In Table 4, the calculation results of the cavity drag and its drag coefficient are listed. It can be observed that the drag coefficient is 0.060 at the no plasma case, which is smaller than time B and C but larger than time A. In view of the change of cavity shear layer in Figure 10, it should be due to the moving upward of the shear layer as actuator works. As the shear layer moves upward, it deflects to the main flow and no more impacts on the cavity rear wall so the drag decreases. However, at time B and C, because of the variation of the shear layer fluctuation, stronger impact may appear the rear wall of cavity, and the combustion may be boosted in the cavity rear part, which can both lead to the increase of back pressure in the cavity. Considering the drag coefficient values at the three typical times, the time weighted average cavity drag coefficient is calculated to be 0.0668 which means the cavity drag usually increases by the plasma.
Front wall | Rear wall | Drag | Drag coefficient | |
---|---|---|---|---|
No plasma | −19.8 | 34.3 | 14.5 | 0.060 |
Plasma, A | −19.6 | 33.1 | 13.5 | 0.056 |
Plasma, B | −19.6 | 37.6 | 18.0 | 0.075 |
Plasma, C | −18.1 | 33.5 | 15.4 | 0.064 |
Calculation results about the cavity drag, unit: N.
As we know, the cavity drag in reaction flow is closely related to the combustion heat release zone of a cavity. So only reducing the cavity drag may not bring benefit for the whole combustor. It can be seen that the further detail analysis about the combustion distribution in flowfield should be done to understand the effect of plasma on the cavity performance.
The mass exchange rate of cavity
The mass exchange rate of cavity is given in Table 5. Because the species transportation is achieved through the vortices in the cavity shear layer, it is very significant for enhancing the diffusion ability of cavity shear layer. Owing to the break on the original stable structure of the shear layer resulted from the plasma filaments “cutting” effect, the mass exchange rates at plasma existing cases at times A, B, and C are 9.2, 197.2, and 107.8 times, respectively, than the “no plasma” case. In addition, the time-weighted average mass exchange rate is 8.217 g/s, which is 122.6 times than the “no plasma” case. And the variation of the mass exchange rate magnitude from time A to C is in related to fluctuation degree of cavity shear layer. As a result, the quasi-DC plasma does obviously promote the species transportation between external and internal cavity.
No plasma | Pulse, A | Pulse, B | Pulse, C | |
---|---|---|---|---|
0.067 | 0.614 | 13.013 | 7.223 |
Calculation results of cavity mass exchange rate, unit: g/s.
In the research fields related to “sound cavity,” “embedded magazine” and “flame holding cavity,” the dynamic study on pressure oscillation in a cavity is always greatly valued around the world [27]. According to the cavity shear layer distribution, four monitor points are chosen to capture the dynamic pressure feature, as shown in Figure 9. F-1 is located on the front wall of cavity with
where
The SPLs of F-1, F-2, R-1, and R-2 are 168, 193, 188, and 191 dB, respectively. However, at the no plasma case, their SPL values are 120, 118, 124, and 106 dB, respectively. It indicates that the SPL of all the monitor points go up sharply, and the relationship between point location and magnitude of SPL changes prominently. The points are near the mouth of cavity and the effect of plasma filaments on the cavity edges is strongest, so the SPL magnitude increases here. Moreover, through the analysis on cavity drag in Table 4, the conclusion can be obtained that the pressure disturbance on the monitor points is controlled by combustion, and the shear layer strongly affects the change of local combustion.
In Figure 13, the frequency spectrum characteristics of R-1 and R-2 by FFT are depicted. The several dominant frequencies of pressure oscillation are marked with red circles here, which show that the first dominant frequency is almost the double of the plasma actuation frequency 5 kHz. Whereas, it is barely equal to the plasma actuation frequency under nonreaction condition [19]. Furthermore, nearly every dominant frequency is an integer multiple of the plasma actuation frequency. The results above indicate that the pressure oscillation in the cavity is controlled by both local combustion flowfield and the plasma actuation frequency.
The normalized pressure frequency spectrum, with plasma. (a) R-1. (b) R-2.
The combustor stagnation pressure recovery coefficient of time A, B, C, and the no plasma case is 0.75, 0.81, 0.65, and 0.83, respectively. And the calculated time weighted average stagnation pressure recovery coefficient is 0.73. It suggests that the overall stagnation pressure loss is raised due to the plasma filaments, especially for time A and C. The increase of the combustor stagnation pressure loss is caused by several factors, such as the enlargement of separation zones, the rise of back pressure induced by local combustion and the strengthening of shock waves, which is based on the analysis of cavity shear layer and wall pressure distribution.
Considering the interesting variation of product water distribution as shown in Figure 12, the distribution of combustion efficiency in the flow direction may also vary influenced by the plasma filaments. Hence, the combustion efficiency
Combustion efficiency along
In a word, the combustion efficiency increases downstream of the front wall of cavity in most of the time due to the plasma. It becomes lower than the no plasma case at several positions for few moments, but the decrease is comparatively small. Furthermore, during one plasma cycle, the combustion efficiency gradually increases from beginning to end, which indicates that the plasma performs better during actuator free period for the pulsed actuation mode.
The ratio of power consumption of the plasma actuator to the increased combustion heat release of the combustor is calculated in value
The main results are as follows: (1) Because of the “cutting” effect of quasi-DC plasma filaments, the cavity shear layer becomes fluctuating and the local combustion changes which also results in the variation of wall pressure distribution near the cavity rear edge. On the mass fraction iso-surface of product water several “branch pipes” curve structures form periodically, which can be put down to the influence of plasma on the local combustion. (2) The effect of plasma on the cavity drag is relatively perplexing and the drag coefficient value is observed to be unsteady. Nevertheless, the mass exchange rate goes up prominently due to plasma, and the magnitude change from time A to C is in agreement with the fluctuation degree of cavity shear layer. (3) The SPLs of four cavity monitor points rise, and the frequency spectrum of monitor points near the rear edge presents that the first dominant frequency is twice the plasma actuation frequency, under the effect of plasma. Hence, the pressure oscillation in the cavity is controlled by both the plasma and local combustion flowfield. (4) On the one side, the combustion efficiency is usually increased due to plasma, and on the other side, the certain pressure loss increases resulted from the changes of combustion and waves in the flowfield by the plasma. Because the ratio of deposited plasma energy to the increased combustion heat release is extraordinarily small, it is considered that if optimal actuation parameters of the actuator are chosen the quasi-DC plasma can bring more benefits than penalties for the cavity in scramjet combustor.
This work is sponsored by the National Natural Science Fund of China, No. 51876219, No. 91441123, and No. 51777214. The authors would like to acknowledge the help of Dr. W. Feng in numerical simulation.
The authors declared that they have no conflicts of interest to this work.
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\n\nNote: All data represented above was collected by IntechOpen from 2013 to 2017.
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