Magnetic KronigPenney model for Dirac electrons in single-layer graphene
Identifieur interne : 001690 ( Istex/Corpus ); précédent : 001689; suivant : 001691Magnetic KronigPenney model for Dirac electrons in single-layer graphene
Auteurs : M. Ramezani Masir ; P. Vasilopoulos ; F M PeetersSource :
- New Journal of Physics [ 1367-2630 ] ; 2009.
Abstract
The properties of Dirac electrons in a magnetic superlattice (SL) on graphene consisting of very high and thin (-function) barriers are investigated. We obtain the energy spectrum analytically and study the transmission through a finite number of barriers. The results are contrasted with those for electrons described by the Schrdinger equation. In addition, a collimation of an incident beam of electrons is obtained along the direction perpendicular to that of the SL. We also highlight an analogy with optical media in which the refractive index varies in space.
Url:
DOI: 10.1088/1367-2630/11/9/095009
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<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">Magnetic KronigPenney model for Dirac electrons in single-layer graphene</title><author><name sortKey="Ramezani Masir, M" sort="Ramezani Masir, M" uniqKey="Ramezani Masir M" first="M" last="Ramezani Masir">M. Ramezani Masir</name><affiliation><mods:affiliation>Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: mrmphys@gmail.com</mods:affiliation></affiliation></author><author><name sortKey="Vasilopoulos, P" sort="Vasilopoulos, P" uniqKey="Vasilopoulos P" first="P" last="Vasilopoulos">P. Vasilopoulos</name><affiliation><mods:affiliation>Department of Physics, Concordia University, 7141 Sherbrooke Street West Montreal, Quebec, H4B 1R6, Canada</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: takis@alcor.concordia.ca</mods:affiliation></affiliation></author><author><name sortKey="Peeters, F M" sort="Peeters, F M" uniqKey="Peeters F" first="F M" last="Peeters">F M Peeters</name><affiliation><mods:affiliation>Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium</mods:affiliation></affiliation><affiliation><mods:affiliation>Departamento de Fsica, Universidade Federal do Cear, Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, Cear, Brazil</mods:affiliation></affiliation><affiliation><mods:affiliation>Author to whom any correspondence should be addressed.</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: francois.peeters@ua.ac.be</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:A31C27D6643EBB9C403400C3EFE0356AA8E204AE</idno><date when="2009" year="2009">2009</date><idno type="doi">10.1088/1367-2630/11/9/095009</idno><idno type="url">https://api.istex.fr/document/A31C27D6643EBB9C403400C3EFE0356AA8E204AE/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">001690</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Magnetic KronigPenney model for Dirac electrons in single-layer graphene</title><author><name sortKey="Ramezani Masir, M" sort="Ramezani Masir, M" uniqKey="Ramezani Masir M" first="M" last="Ramezani Masir">M. Ramezani Masir</name><affiliation><mods:affiliation>Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: mrmphys@gmail.com</mods:affiliation></affiliation></author><author><name sortKey="Vasilopoulos, P" sort="Vasilopoulos, P" uniqKey="Vasilopoulos P" first="P" last="Vasilopoulos">P. Vasilopoulos</name><affiliation><mods:affiliation>Department of Physics, Concordia University, 7141 Sherbrooke Street West Montreal, Quebec, H4B 1R6, Canada</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: takis@alcor.concordia.ca</mods:affiliation></affiliation></author><author><name sortKey="Peeters, F M" sort="Peeters, F M" uniqKey="Peeters F" first="F M" last="Peeters">F M Peeters</name><affiliation><mods:affiliation>Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium</mods:affiliation></affiliation><affiliation><mods:affiliation>Departamento de Fsica, Universidade Federal do Cear, Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, Cear, Brazil</mods:affiliation></affiliation><affiliation><mods:affiliation>Author to whom any correspondence should be addressed.</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: francois.peeters@ua.ac.be</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">New Journal of Physics</title><title level="j" type="abbrev">New J. Phys.</title><idno type="ISSN">1367-2630</idno><idno type="eISSN">1367-2630</idno><imprint><publisher>IOP Publishing</publisher><date type="published" when="2009">2009</date><biblScope unit="volume">11</biblScope><biblScope unit="issue">9</biblScope><biblScope unit="page" from="1">1</biblScope><biblScope unit="page" to="21">21</biblScope></imprint><idno type="ISSN">1367-2630</idno></series><idno type="istex">A31C27D6643EBB9C403400C3EFE0356AA8E204AE</idno><idno type="DOI">10.1088/1367-2630/11/9/095009</idno><idno type="articleID">318400</idno><idno type="articleNumber">095009</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">1367-2630</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract">The properties of Dirac electrons in a magnetic superlattice (SL) on graphene consisting of very high and thin (-function) barriers are investigated. We obtain the energy spectrum analytically and study the transmission through a finite number of barriers. The results are contrasted with those for electrons described by the Schrdinger equation. In addition, a collimation of an incident beam of electrons is obtained along the direction perpendicular to that of the SL. We also highlight an analogy with optical media in which the refractive index varies in space.</div></front></TEI><istex><corpusName>iop</corpusName><author><json:item><name>M Ramezani Masir</name><affiliations><json:string>Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium</json:string><json:string>E-mail: mrmphys@gmail.com</json:string></affiliations></json:item><json:item><name>P Vasilopoulos</name><affiliations><json:string>Department of Physics, Concordia University, 7141 Sherbrooke Street West Montreal, Quebec, H4B 1R6, Canada</json:string><json:string>E-mail: takis@alcor.concordia.ca</json:string></affiliations></json:item><json:item><name>F M Peeters</name><affiliations><json:string>Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium</json:string><json:string>Departamento de Fsica, Universidade Federal do Cear, Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, Cear, Brazil</json:string><json:string>Author to whom any correspondence should be addressed.</json:string><json:string>E-mail: francois.peeters@ua.ac.be</json:string></affiliations></json:item></author><language><json:string>eng</json:string></language><originalGenre><json:string>paper</json:string></originalGenre><abstract>The properties of Dirac electrons in a magnetic superlattice (SL) on graphene consisting of very high and thin (-function) barriers are investigated. We obtain the energy spectrum analytically and study the transmission through a finite number of barriers. The results are contrasted with those for electrons described by the Schrdinger equation. In addition, a collimation of an incident beam of electrons is obtained along the direction perpendicular to that of the SL. We also highlight an analogy with optical media in which the refractive index varies in space.</abstract><qualityIndicators><score>6.556</score><pdfVersion>1.4</pdfVersion><pdfPageSize>595.276 x 841.89 pts (A4)</pdfPageSize><refBibsNative>true</refBibsNative><keywordCount>0</keywordCount><abstractCharCount>566</abstractCharCount><pdfWordCount>7199</pdfWordCount><pdfCharCount>31828</pdfCharCount><pdfPageCount>21</pdfPageCount><abstractWordCount>88</abstractWordCount></qualityIndicators><title>Magnetic KronigPenney model for Dirac electrons in single-layer graphene</title><genre><json:string>research-article</json:string></genre><host><volume>11</volume><publisherId><json:string>nj</json:string></publisherId><pages><total>21</total><last>21</last><first>1</first></pages><issn><json:string>1367-2630</json:string></issn><issue>9</issue><genre><json:string>journal</json:string></genre><language><json:string>unknown</json:string></language><eissn><json:string>1367-2630</json:string></eissn><title>New Journal of Physics</title></host><publicationDate>2009</publicationDate><copyrightDate>2009</copyrightDate><doi><json:string>10.1088/1367-2630/11/9/095009</json:string></doi><id>A31C27D6643EBB9C403400C3EFE0356AA8E204AE</id><score>0.21924792</score><fulltext><json:item><original>true</original><mimetype>application/pdf</mimetype><extension>pdf</extension><uri>https://api.istex.fr/document/A31C27D6643EBB9C403400C3EFE0356AA8E204AE/fulltext/pdf</uri></json:item><json:item><original>false</original><mimetype>application/zip</mimetype><extension>zip</extension><uri>https://api.istex.fr/document/A31C27D6643EBB9C403400C3EFE0356AA8E204AE/fulltext/zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/A31C27D6643EBB9C403400C3EFE0356AA8E204AE/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a" type="main" xml:lang="en">Magnetic KronigPenney model for Dirac electrons in single-layer graphene</title></titleStmt><publicationStmt><authority>ISTEX</authority><publisher>IOP Publishing</publisher><availability><p>IOP Publishing and Deutsche Physikalische Gesellschaft</p></availability><date>2009</date></publicationStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a" type="main" xml:lang="en">Magnetic KronigPenney model for Dirac electrons in single-layer graphene</title><author xml:id="author-1"><persName><forename type="first">M</forename><surname>Ramezani Masir</surname></persName><email>mrmphys@gmail.com</email><affiliation>Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium</affiliation></author><author xml:id="author-2"><persName><forename type="first">P</forename><surname>Vasilopoulos</surname></persName><email>takis@alcor.concordia.ca</email><affiliation>Department of Physics, Concordia University, 7141 Sherbrooke Street West Montreal, Quebec, H4B 1R6, Canada</affiliation></author><author xml:id="author-3"><persName><forename type="first">F M</forename><surname>Peeters</surname></persName><email>francois.peeters@ua.ac.be</email><affiliation>Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium</affiliation><affiliation>Departamento de Fsica, Universidade Federal do Cear, Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, Cear, Brazil</affiliation><affiliation>Author to whom any correspondence should be addressed.</affiliation></author></analytic><monogr><title level="j">New Journal of Physics</title><title level="j" type="abbrev">New J. Phys.</title><idno type="pISSN">1367-2630</idno><idno type="eISSN">1367-2630</idno><imprint><publisher>IOP Publishing</publisher><date type="published" when="2009"></date><biblScope unit="volume">11</biblScope><biblScope unit="issue">9</biblScope><biblScope unit="page" from="1">1</biblScope><biblScope unit="page" to="21">21</biblScope></imprint></monogr><idno type="istex">A31C27D6643EBB9C403400C3EFE0356AA8E204AE</idno><idno type="DOI">10.1088/1367-2630/11/9/095009</idno><idno type="articleID">318400</idno><idno type="articleNumber">095009</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>2009</date></creation><langUsage><language ident="en">en</language></langUsage><abstract><p>The properties of Dirac electrons in a magnetic superlattice (SL) on graphene consisting of very high and thin (-function) barriers are investigated. We obtain the energy spectrum analytically and study the transmission through a finite number of barriers. 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Phys.</jnl-abbreviation><jnl-shortname>NJP</jnl-shortname><jnl-issn>1367-2630</jnl-issn><jnl-coden>NJOPFM</jnl-coden><jnl-imprint>IOP Publishing</jnl-imprint><jnl-web-address>stacks.iop.org/NJP</jnl-web-address></jnl-data><volume-data><year-publication>2009</year-publication><volume-number>11</volume-number></volume-data><issue-data><issue-number>9</issue-number><coverdate>September 2009</coverdate></issue-data><article-data><article-type type="paper" sort="regular"></article-type><type-number type="paper" numbering="article" artnum="095009">095009</type-number><article-number>318400</article-number><first-page>1</first-page><last-page>21</last-page><length>21</length><pii></pii><doi>10.1088/1367-2630/11/9/095009</doi><copyright>IOP Publishing and Deutsche Physikalische Gesellschaft</copyright><ccc>1367-2630/09/095009+21$30.00</ccc><printed></printed></article-data></article-metadata><header><title-group><title>Magnetic Kronig–Penney model for Dirac electrons in single-layer graphene</title><short-title>Magnetic Kronig–Penney model for Dirac electrons in single-layer graphene</short-title><ej-title>Magnetic Kronig–Penney model for Dirac electrons in single-layer graphene</ej-title></title-group><author-group><author address="nj318400ad1" email="nj318400ea1"><first-names>M</first-names><second-name>Ramezani Masir</second-name></author><author address="nj318400ad2" email="nj318400ea2"><first-names>P</first-names><second-name>Vasilopoulos</second-name></author><author address="nj318400ad1" second-address="nj318400ad3" alt-address="nj318400aad4" email="nj318400ea3"><first-names>F M</first-names><second-name>Peeters</second-name></author><short-author-list>M Ramezani <italic>et al</italic></short-author-list></author-group><address-group><address id="nj318400ad1" showid="yes">Departement Fysica, <orgname>Universiteit Antwerpen</orgname>, Groenenborgerlaan 171, B-2020 Antwerpen, <country>Belgium</country></address><address id="nj318400ad2" showid="yes">Department of Physics, <orgname>Concordia University</orgname>, 7141 Sherbrooke Street West Montreal, Quebec, H4B 1R6, <country>Canada</country></address><address id="nj318400ad3" showid="yes">Departamento de Física, <orgname>Universidade Federal do Ceará</orgname>, Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, Ceará, <country>Brazil</country></address><address id="nj318400aad4" showid="yes" alt="yes">Author to whom any correspondence should be addressed.</address><e-address id="nj318400ea1"><email mailto="mrmphys@gmail.com">mrmphys@gmail.com</email></e-address><e-address id="nj318400ea2"><email mailto="takis@alcor.concordia.ca">takis@alcor.concordia.ca</email></e-address><e-address id="nj318400ea3"><email mailto="francois.peeters@ua.ac.be">francois.peeters@ua.ac.be</email></e-address></address-group><history received="7 May 2009" online="30 September 2009"></history><abstract-group><abstract><heading>Abstract</heading><p indent="no">The properties of Dirac electrons in a magnetic superlattice (SL) on graphene consisting of very high and thin (<inline-eqn><math-text>δ</math-text></inline-eqn>-function) barriers are investigated. We obtain the energy spectrum analytically and study the transmission through a finite number of barriers. The results are contrasted with those for electrons described by the Schrödinger equation. In addition, a collimation of an incident beam of electrons is obtained along the direction perpendicular to that of the SL. We also highlight an analogy with optical media in which the refractive index varies in space.</p></abstract></abstract-group></header><body refstyle="numeric"><sec-level1 id="nj318400s1" label="1"><heading>Introduction</heading><p indent="no">During the last five years single-layer graphene (a monolayer of carbon atoms) has become a very active field of research in nanophysics [<cite linkend="nj318400bib1">1</cite>, <cite linkend="nj318400bib2">2</cite>]. It is expected that this material will serve as a base for new electronic and opto-electric devices. The reason is that graphene's electronic properties are drastically different from those, say, of conventional semiconductors. Charge carriers in a wide single-layer graphene behave like ‘relativistic’, chiral and massless particles with a ‘light speed’ equal to the Fermi velocity and possess a <italic>gapless, linear</italic> spectrum close to the <inline-eqn><math-text><italic>K</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>K</italic>′</math-text></inline-eqn> points. One major consequence is the perfect transmission through arbitrarily high and wide barriers, referred to as Klein tunneling.</p><p>One of the most challenging tasks is to learn how to control the electron behavior using electric fields in graphene. This task is made complicated precisely by the Klein tunneling according to which Dirac electrons in graphene can tunnel through arbitrarily wide and high electric barriers [<cite linkend="nj318400bib3">3</cite>].</p><p>Alternatively, one can apply a magnetic field to control the electron motion. It has been shown in numerous papers that an inhomogeneous magnetic field can confine standard electrons described by the Schrödinger equation [<cite linkend="nj318400bib4">4</cite>]–[<cite linkend="nj318400bib6">6</cite>]. The question then arises whether it can confine Dirac electrons in graphene. Up to now semi-infinite magnetic structures, that are homogeneous in one direction, have been considered, making the task simpler by converting the problem into a one-dimensional (1D) one [<cite linkend="nj318400bib7">7</cite>]–[<cite linkend="nj318400bib15">15</cite>]. In particular, magnetic confinement of Dirac electrons in graphene has been reported in structures involving one [<cite linkend="nj318400bib7">7</cite>] or several magnetic barriers [<cite linkend="nj318400bib8">8</cite>, <cite linkend="nj318400bib9">9</cite>] as well as in superlattices (SLs), without magnetic field for some very special values of the parameters involved [<cite linkend="nj318400bib16">16</cite>]. In such structures standard electrons can remain close to the interface and move along so-called snake orbits [<cite linkend="nj318400bib5">5</cite>] or in pure quantum mechanical unidirectional states [<cite linkend="nj318400bib4">4</cite>].</p><p>Given the importance of graphene, it would be appropriate to study this magnetic confinement more systematically. We make such a study here by considering a <italic>magnetic</italic> Kronig–Penney (KP) model in graphene, i.e. a series of magnetic <inline-eqn><math-text>δ</math-text></inline-eqn>-function barriers that alternate in sign. This model can be realized experimentally in two different ways: <ordered-list id="list1" type="arabic" pattern="5"><list-item id="list1-i1" marker="1."><p>One can deposit ferromagnetic strips on the top of a graphene layer but in a way that there is no electrical contact between graphene and these strips. When one magnetizes the strips along the <inline-eqn><math-text><italic>x</italic></math-text></inline-eqn> direction, cf figure <figref linkend="nj318400fig1">1</figref>(a), by e.g. applying an in-plane magnetic field, the charge carriers in the graphene layer feel an inhomogeneous magnetic field profile. This profile can be well approximated [<cite linkend="nj318400bib22">22</cite>] by <inline-eqn><math-text>2<italic>B</italic><sub>0</sub><italic>z</italic><sub>0</sub><italic>h</italic>/<italic>d</italic>(<italic>x</italic><sup>2</sup>+<italic>z</italic><sub>0</sub><sup>2</sup>)</math-text></inline-eqn> on one edge of the strip and by <inline-eqn><math-text>−2<italic>B</italic><sub>0</sub><italic>z</italic><sub>0</sub><italic>h</italic>/(<italic>x</italic><sup>2</sup>+<italic>z</italic><sub>0</sub><sup>2</sup>)</math-text></inline-eqn> on the other, where <inline-eqn><math-text><italic>z</italic><sub>0</sub></math-text></inline-eqn> is the distance between the two-dimensional electron gas (2DEG) and the strip, and <inline-eqn><math-text><italic>d</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>h</italic></math-text></inline-eqn> the width and height of the strip (see figure <figref linkend="nj318400fig1">1</figref>(b)). The resulting magnetic field profile will be modeled by two magnetic <inline-eqn><math-text>δ</math-text></inline-eqn>-functions of height <inline-eqn><math-text>2π<italic>B</italic><sub>0</sub> <italic>h</italic></math-text></inline-eqn>. Such ferromagnetic strips were deposited on the top of a 2DEG in a semiconductor heterostructure in [<cite linkend="nj318400bib23">23</cite>].</p></list-item><list-item id="list1-i2" marker="2."><p>It was recently shown that local strain in graphene induces an effective inhomogeneous magnetic field [<cite linkend="nj318400bib24">24</cite>] (figure <figref linkend="nj318400fig1">1</figref>(c)). When one puts the graphene layer on a periodically structured substrate the graphene at the edges of the substrate becomes strained and the situation can be described by a magnetic <inline-eqn><math-text>δ</math-text></inline-eqn>-function profile such as that shown in figure <figref linkend="nj318400fig2">2</figref>.</p></list-item></ordered-list></p><figure id="nj318400fig1" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="26.0pc" printcolour="no" filename="images/nj318400fig1.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj318400fig1.jpg"></graphic-file></graphic><caption type="figure" id="nj318400fc1" label="Figure 1"><p indent="no">(a) Layout of the system: a ferromagnetic strip on the top of a bilayer graphene sheet separated by a thin oxide layer. (b) Magnetic field and corresponding vector potential at a distance <inline-eqn><math-text><italic>z</italic><sub>0</sub>=0.1</math-text></inline-eqn> under the strip parallel to it. (c) A graphene layer on the top of a periodic structured surface.</p></caption></figure><figure id="nj318400fig2" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="22.0pc" printcolour="no" filename="images/nj318400fig2.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj318400fig2.jpg"></graphic-file></graphic><caption type="figure" id="nj318400fc2" label="Figure 2"><p indent="no">(a) Two opposite magnetic <inline-eqn><math-text>δ</math-text></inline-eqn>-function barriers, indicated by arrows; the vector potential is shown by the shaded (green) area. (b) Angles related to the propagation of an electron through this system. (c) Schematics of a periodic vector potential (shaded areas) and corresponding magnetic field indicated by the black arrows. (d) Arrangement of magnetic <inline-eqn><math-text>δ</math-text></inline-eqn>-functions that leads to a periodic and alternating in sign vector potential.</p></caption></figure><p>In a quantum mechanical treatment of the above two systems, the vector potential <inline-eqn><math-text><bold>A</bold>(<italic>x</italic>)</math-text></inline-eqn> is the essential quantity and, within the Landau gauge, <inline-eqn><math-text><bold>A</bold>(<italic>x</italic>)</math-text></inline-eqn> is nothing else than a periodic array of step functions. The Hamiltonian describing this system is periodic and consequently we expect the energy spectrum of the charge carriers in graphene to exhibit a band structure. The advantage of this <italic>magnetic</italic> KP model is mainly its analytical simplicity that provides some insight and allows a contrast with the same model for standard electrons [<cite linkend="nj318400bib27">27</cite>]. To do that we adapt a method developed in optics, for a medium with a periodic-in-space refractive index. This optical method is clear and very well suited to the problem. Incidentally, there are many analogues of optical behavior in electronics, such as focusing [<cite linkend="nj318400bib17">17</cite>]–[<cite linkend="nj318400bib19">19</cite>], [<cite linkend="nj318400bib30">30</cite>], [<cite linkend="nj318400bib31">31</cite>], collimation or quasi-1D motion of electrons and photons [<cite linkend="nj318400bib4">4</cite>, <cite linkend="nj318400bib8">8</cite>, <cite linkend="nj318400bib16">16</cite>, <cite linkend="nj318400bib20">20</cite>], and interference [<cite linkend="nj318400bib21">21</cite>] in a 2DEG.</p><p>The paper is organized as follows. In section <secref linkend="nj318400s2">2</secref>, we present the method and evaluate the spectrum and electron transmission through two antiparallel, <inline-eqn><math-text>δ</math-text></inline-eqn>-function magnetic barriers. In section <secref linkend="nj318400s3">3</secref>, we consider SLs of such barriers and present numerical results for the energy spectrum. In section <secref linkend="nj318400s4">4</secref>, we consider a series of <inline-eqn><math-text>δ</math-text></inline-eqn>-function vector potentials and our concluding remarks are given in section <secref linkend="nj318400s5">5</secref>.</p></sec-level1><sec-level1 id="nj318400s2" label="2"><heading>Characteristic matrix for Dirac electrons</heading><p indent="no">An electron in a single-layer graphene, in the presence of a perpendicular magnetic field <inline-eqn><math-text><italic>B</italic>(<italic>x</italic>)</math-text></inline-eqn>, which depends on <inline-eqn><math-text><italic>x</italic></math-text></inline-eqn>, is adequately described by the Hamiltonian
<display-eqn id="nj318400eqn1" textype="equation" notation="LaTeX" eqnnum="1"></display-eqn>where <inline-eqn><math-text><bold>p</bold></math-text></inline-eqn> is the momentum operator, <inline-eqn><math-text><italic>v</italic><sub><upright>F</upright></sub></math-text></inline-eqn> the Fermi velocity, and <inline-eqn><math-text><bold>A</bold>(<italic>x</italic>)</math-text></inline-eqn> the vector potential. To simplify the notation, we introduce the dimensionless units: <inline-eqn></inline-eqn>, and <inline-eqn></inline-eqn> Here <inline-eqn><math-text>ℓ<sub><upright>B</upright></sub></math-text></inline-eqn> is the magnetic length and <inline-eqn><math-text><italic>t</italic></math-text></inline-eqn> the tunneling strength. In these units equation (<eqnref linkend="nj318400eqn1">1</eqnref>) takes the form
<display-eqn id="nj318400eqn2" textype="equation" notation="LaTeX" eqnnum="2"></display-eqn>Then the equation <inline-eqn><math-text><italic>H</italic>Ψ(<italic>x</italic>,<italic>y</italic>)=<italic>E</italic>Ψ(<italic>x</italic>,<italic>y</italic>)</math-text></inline-eqn> admits solutions of the form
<display-eqn id="nj318400eqn3" textype="equation" notation="LaTeX" eqnnum="3"></display-eqn>with <inline-eqn><math-text>ψ<sub><upright>I</upright></sub>(<italic>x</italic>,<italic>y</italic>), ψ<sub><upright>II</upright></sub>(<italic>x</italic>, <italic>y</italic>)</math-text></inline-eqn> obeying the coupled equations
<display-eqn id="nj318400eqn4" textype="equation" notation="LaTeX" eqnnum="4"></display-eqn><display-eqn id="nj318400eqn5" textype="equation" notation="LaTeX" eqnnum="5"></display-eqn></p><p>Due to the translational invariance along the <inline-eqn><math-text><italic>y</italic></math-text></inline-eqn> direction, we assume solutions of the form <inline-eqn><math-text>Ψ(<italic>x</italic>,<italic>y</italic>)=<upright>exp</upright> <upright>i</upright><italic>k</italic><sub><italic>y</italic></sub><italic>y</italic>(<italic>U</italic>(<italic><italic>x</italic></italic>), <italic>V</italic> (<italic><italic>x</italic></italic>))<sup><upright>T</upright></sup></math-text></inline-eqn>, with the superscript ‘T’ denoting the transpose of the row vector. For <inline-eqn><math-text><italic>B</italic>(<italic>x</italic>)∼δ(<italic>x</italic>)</math-text></inline-eqn>, the corresponding vector potential is a step function <inline-eqn><math-text><italic>A</italic>(<italic>x</italic>)∼Θ(<italic>x</italic>)</math-text></inline-eqn>. For <inline-eqn><math-text><italic>A</italic>(<italic>x</italic>)=<italic>P</italic></math-text></inline-eqn> constant, equations (<eqnref linkend="nj318400eqn4">4</eqnref>) and (<eqnref linkend="nj318400eqn5">5</eqnref>) take the form
<display-eqn id="nj318400eqn6" textype="equation" notation="LaTeX" eqnnum="6"></display-eqn><display-eqn id="nj318400eqn7" textype="equation" notation="LaTeX" eqnnum="7"></display-eqn>Equations (<eqnref linkend="nj318400eqn3">3</eqnref>)–(<eqnref linkend="nj318400eqn7">7</eqnref>) correspond to those for an electromagnetic wave propagating through a medium in which the refractive index varies periodically. The two components of <inline-eqn><math-text>Ψ(<italic>x</italic>,<italic>y</italic>)</math-text></inline-eqn> correspond to those of the electric (or magnetic) field of the wave [<cite linkend="nj318400bib28">28</cite>, <cite linkend="nj318400bib29">29</cite>].</p><p>Equations (<eqnref linkend="nj318400eqn6">6</eqnref>) and (<eqnref linkend="nj318400eqn7">7</eqnref>) can be readily decoupled by substitution. The result is
<display-eqn id="nj318400eqn8" textype="equation" notation="LaTeX" eqnnum="8"></display-eqn>where <inline-eqn><math-text><italic>Z</italic>=<italic>U</italic>, <italic>V</italic></math-text></inline-eqn>. If <inline-eqn><math-text><italic>E</italic><sup>2</sup>→<italic>E</italic>′</math-text></inline-eqn> and <inline-eqn><math-text>(<italic>k</italic><sub><italic>y</italic></sub>+<italic>P</italic>)<sup>2</sup>→<italic>V</italic><sub><upright>eff</upright></sub></math-text></inline-eqn>, equation (<eqnref linkend="nj318400eqn8">8</eqnref>) reduces to a Schrödinger equation for a standard electron where <inline-eqn><math-text><italic>V</italic><sub><upright>eff</upright></sub>(<italic>k</italic><sub><italic>y</italic></sub>,<italic>x</italic>)=(<italic>k</italic><sub><italic>y</italic></sub>+<italic>P</italic>)<sup>2</sup></math-text></inline-eqn> can be considered as an effective potential. Taking <inline-eqn><math-text>&thetas;<sub>0</sub></math-text></inline-eqn> as the angle of incidence, we have <inline-eqn><math-text><italic>k</italic><sub><italic>x</italic></sub>=<italic>E</italic> <upright>cos</upright> &thetas;<sub>0</sub>=[<italic>E</italic><sup>2</sup>−<italic>k</italic><sub><italic>y</italic></sub><sup>2</sup>]<sup>1/2</sup></math-text></inline-eqn> and <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>=<italic>E</italic> <upright>sin</upright> &thetas;<sub>0</sub></math-text></inline-eqn> as the wave vector components outside the medium and <inline-eqn><math-text><italic>k</italic>′<sub><italic>x</italic></sub>=<italic>E</italic> <upright>cos</upright> &thetas;=[<italic>E</italic><sup>2</sup>−(<italic>k</italic><sub><italic>y</italic></sub>+<italic>P</italic>)<sup>2</sup>]<sup>1/2</sup></math-text></inline-eqn> as the electron wave vector inside the medium; <inline-eqn><math-text>&thetas;=<upright>tan</upright><sup>−1</sup>(<italic>k</italic><sub><italic>y</italic></sub>/<italic>k</italic>′ <sub><italic>x</italic></sub>)</math-text></inline-eqn> is the refraction angle. This renders equation (<eqnref linkend="nj318400eqn8">8</eqnref>) simpler with acceptable solutions for <inline-eqn><math-text><italic>U</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>V</italic></math-text></inline-eqn><display-eqn id="nj318400eqn9" textype="equation" notation="LaTeX" eqnnum="9"></display-eqn><display-eqn id="nj318400eqn10" textype="equation" notation="LaTeX" eqnnum="10"></display-eqn>For future purposes, we write <inline-eqn><math-text><italic>U</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>V</italic></math-text></inline-eqn> as a linear combination of <inline-eqn><math-text><italic>U</italic><sub>1</sub>, <italic>U</italic><sub>2</sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>V</italic><sub>1</sub>, <italic>V</italic><sub>2</sub></math-text></inline-eqn>:
<display-eqn id="nj318400eqn11" textype="equation" notation="LaTeX" eqnnum="11"></display-eqn>We now multiply the equations of the first row by <inline-eqn><math-text><italic>U</italic><sub>2</sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>U</italic><sub>1</sub></math-text></inline-eqn>, respectively, and those of the second by <inline-eqn><math-text><italic>V</italic><sub>2</sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>V</italic><sub>1</sub></math-text></inline-eqn>. The resulting equations lead to
<display-eqn id="nj318400eqn12" textype="equation" notation="LaTeX" eqnnum="12"></display-eqn>where <inline-eqn><math-text><bold>D</bold>=<upright>det</upright> <italic>D</italic></math-text></inline-eqn> and
<display-eqn id="nj318400eqn13" textype="equation" notation="LaTeX" eqnnum="13"></display-eqn>Equation (<eqnref linkend="nj318400eqn12">12</eqnref>) shows that the determinant of the matrix (<eqnref linkend="nj318400eqn13">13</eqnref>) associated with any two arbitrary solutions of equation (<eqnref linkend="nj318400eqn8">8</eqnref>) is a constant, i.e. <inline-eqn><math-text><italic>D</italic></math-text></inline-eqn> is an invariant of the system of equations (<eqnref linkend="nj318400eqn11">11</eqnref>). This also follows from the well-known property of the Wronskian of second-order differential equations. For our purposes a convenient choice of particular solutions is
<display-eqn id="nj318400eqn14" textype="equation" notation="LaTeX" eqnnum="14"></display-eqn>such that
<display-eqn id="nj318400eqn15" textype="equation" notation="LaTeX" eqnnum="15"></display-eqn>Then the solution, with <inline-eqn><math-text><italic>U</italic>(0)=<italic>U</italic><sub>0</sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>V</italic>(0)=<italic>V</italic><sub>0</sub></math-text></inline-eqn>, can be expressed as
<display-eqn id="nj318400eqn16" textype="equation" notation="LaTeX" eqnnum="16"></display-eqn>or, in matrix notation, as
<display-eqn id="nj318400eqn17" textype="equation" notation="LaTeX" eqnnum="17"></display-eqn>Since <inline-eqn><math-text><italic>D</italic></math-text></inline-eqn> is constant, the determinant of the square matrix <inline-eqn><math-text><italic>N</italic></math-text></inline-eqn> is a constant; its value, found by setting <inline-eqn><math-text><italic>x</italic>=0</math-text></inline-eqn>, is det <inline-eqn><math-text><italic>N</italic>=<italic>Fg</italic>−<italic>f G</italic>=1</math-text></inline-eqn>. It is usually more convenient to express <inline-eqn><math-text><italic>U</italic><sub>0</sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>V</italic><sub>0</sub></math-text></inline-eqn> as a function of <inline-eqn><math-text><italic>U</italic>(<italic>x</italic>)</math-text></inline-eqn> and <inline-eqn><math-text><italic>V</italic> (<italic>x</italic>)</math-text></inline-eqn>. Solving for <inline-eqn><math-text><italic>U</italic><sub>0</sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>V</italic><sub>0</sub></math-text></inline-eqn>, we obtain <inline-eqn><math-text><bold>Q</bold><sub>0</sub>=<bold>M</bold><bold>Q</bold></math-text></inline-eqn>, where
<display-eqn id="nj318400eqn18" textype="equation" notation="LaTeX" eqnnum="18"></display-eqn>This matrix <inline-eqn><math-text><bold>M</bold></math-text></inline-eqn> is unimodular, <inline-eqn><math-text>|<bold>M</bold>|=1</math-text></inline-eqn>. Now we can find the characteristic matrix from equations (<eqnref linkend="nj318400eqn9">9</eqnref>) and (<eqnref linkend="nj318400eqn10">10</eqnref>) as
<display-eqn id="nj318400eqn19" textype="equation" notation="LaTeX" eqnnum="19"></display-eqn></p><sec-level2 id="nj318400s2.1" label="2.1"><heading>Bound states</heading><p indent="no">With regard to the average of vector potential we shall consider two different systems: one with zero average and the other with nonzero average along the <inline-eqn><math-text><italic>x</italic></math-text></inline-eqn> direction. First let us consider the magnetic field profile as shown in figure <figref linkend="nj318400fig2">2</figref>(a) for which the corresponding vector potential is
<display-eqn id="nj318400eqn20" textype="equation" notation="LaTeX" eqnnum="20"></display-eqn>where <inline-eqn><math-text>Θ(<italic>x</italic>)=0(<italic>x</italic><0)</math-text></inline-eqn>, <inline-eqn><math-text>1(<italic>x</italic>>0)</math-text></inline-eqn> is the theta function. This vector potential has a nonzero average, and the corresponding effective potential becomes (see figure <figref linkend="nj318400fig3">3</figref>(a)) as
<display-eqn id="nj318400eqn21" textype="equation" notation="LaTeX" eqnnum="21"></display-eqn>Here <inline-eqn><math-text><italic>L</italic></math-text></inline-eqn> is measured in units of the magnetic length <inline-eqn><math-text><italic>l</italic><sub><upright>B</upright></sub></math-text></inline-eqn>. There are two different cases that we have to consider.</p><figure id="nj318400fig3" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="31.9pc" printcolour="no" filename="images/nj318400fig3.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj318400fig3.jpg"></graphic-file></graphic><caption type="figure" id="nj318400fc3" label="Figure 3"><p indent="no">The effective potential for <inline-eqn><math-text><italic><italic>k</italic></italic><sub><italic><italic>y</italic></italic></sub><−<italic><italic>P</italic></italic>/2</math-text></inline-eqn> and <inline-eqn><math-text><italic><italic>k</italic></italic><sub><italic><italic>y</italic></italic></sub>><italic><italic>P</italic></italic>/2</math-text></inline-eqn> for two different cases, (a) vector potential with nonzero average corresponding to figure <figref linkend="nj318400fig2">2</figref>(a), and (b) vector potential with zero average corresponding to figure <figref linkend="nj318400fig2">2</figref>(c). (c) Minimum (blue full curve) and maximum (red dashed curve) of the effective potential versus <inline-eqn><math-text><italic><italic>k</italic></italic><sub><italic><italic>y</italic></italic></sub></math-text></inline-eqn> corresponding to the situation depicted in (a) for <inline-eqn><math-text><italic><italic>P</italic></italic>=0.1</math-text></inline-eqn>.</p></caption></figure><p><italic>Case</italic> 1 for <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub><−<italic>P</italic>/2</math-text></inline-eqn>: as shown in figure <figref linkend="nj318400fig3">3</figref>(a) by the full red curve, we have a 1D symmetric quantum well that, as is well-known, has at least one bound state (see also figure <figref linkend="nj318400fig3">3</figref>(c)). For <inline-eqn><math-text><italic>E</italic><sup>2</sup><<italic>k</italic><sub><italic>y</italic></sub><sup>2</sup></math-text></inline-eqn>, the particle will be bound, while for <inline-eqn><math-text><italic>E</italic><sup>2</sup>><italic>k</italic><sub><italic>y</italic></sub><sup>2</sup></math-text></inline-eqn>, we have scattered states, or equivalently the electron tunnels through the magnetic barriers.</p><p><italic>Case</italic> 2 for <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>> −<italic>P</italic>/2</math-text></inline-eqn>: as shown in figure <figref linkend="nj318400fig3">3</figref>(a) by the dotted blue curve, the effective potential is similar to that of a barrier. We have a pure tunneling problem. With reference to figure <figref linkend="nj318400fig2">2</figref>, <inline-eqn><math-text><italic>x</italic><sub>1</sub>=0</math-text></inline-eqn> and <inline-eqn><math-text><italic>x</italic><sub>2</sub>=<italic>L</italic></math-text></inline-eqn>, the solutions are as follows. For <inline-eqn><math-text><italic>x</italic><0</math-text></inline-eqn>, the wave function is
<display-eqn id="nj318400eqn22" textype="equation" notation="LaTeX" eqnnum="22"></display-eqn>where <inline-eqn><math-text>κ=<italic>E</italic><upright>cosh</upright> ξ</math-text></inline-eqn> and <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>=<italic>E</italic> <upright>sinh</upright> ξ</math-text></inline-eqn>, while for <inline-eqn><math-text><italic>x</italic>><italic>L</italic></math-text></inline-eqn> it is
<display-eqn id="nj318400eqn23" textype="equation" notation="LaTeX" eqnnum="23"></display-eqn>In the middle region, <inline-eqn><math-text>0<<italic>x</italic><<italic>L</italic></math-text></inline-eqn>, the wave function is given by
<display-eqn id="nj318400eqn24" textype="equation" notation="LaTeX" eqnnum="24"></display-eqn>with <inline-eqn><math-text><italic>k</italic>′ =[<italic>E</italic><sup>2</sup>−(<italic>k</italic><sub><italic>y</italic></sub>+<italic>P</italic>)<sup>2</sup>]<sup>1/2</sup>=<italic>E</italic><upright>cos</upright> &thetas;</math-text></inline-eqn>. Matching the wave functions at <inline-eqn><math-text><italic>x</italic>=0</math-text></inline-eqn> and <inline-eqn><math-text><italic>x</italic>=<italic>L</italic></math-text></inline-eqn> leads to a system of four equations relating the coefficients <inline-eqn><math-text><italic>C</italic></math-text></inline-eqn>, <inline-eqn><math-text><italic>D</italic></math-text></inline-eqn>, <inline-eqn><math-text><italic>F</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>Q</italic></math-text></inline-eqn>. Setting the determinant of these coefficients equal to zero, we obtain a transcendental equation whose solution gives the energy spectrum
<display-eqn id="nj318400eqn25" textype="equation" notation="LaTeX" eqnnum="25"></display-eqn>For the special value of <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>=−<italic>P</italic></math-text></inline-eqn>, we have <inline-eqn><math-text><upright>sin</upright> &thetas;=0</math-text></inline-eqn> and can rewrite equation (<eqnref linkend="nj318400eqn25">25</eqnref>) as <inline-eqn><math-text><upright>cos</upright> (<italic>EL</italic>)=0</math-text></inline-eqn> or equally <inline-eqn></inline-eqn>. The resulting bound states, as a function of <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn>, are shown by the red full curves in figure <figref linkend="nj318400fig4">4</figref>(a). The area of existence of bound states is delimited by the lines <inline-eqn><math-text><italic>E</italic>=−<italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>E</italic>=−(<italic>k</italic><sub><italic>y</italic></sub>+<italic>P</italic>)</math-text></inline-eqn>. The number of bound states increases with <inline-eqn><math-text>|<italic>k</italic><sub><italic>y</italic></sub>|</math-text></inline-eqn> which is also clear from the behavior of the minimum and maximum of the effective potential (see figure <figref linkend="nj318400fig3">3</figref>(c)). No bound states are found for <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>>−<italic>P</italic>/2</math-text></inline-eqn> as is also apparent from figure <figref linkend="nj318400fig3">3</figref>(c). For <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>→−<italic>P</italic>/2</math-text></inline-eqn>, the potential is shallow and only one bound state exists. The average velocity <inline-eqn><math-text><italic>v</italic><sub><italic>n</italic></sub>(<italic>k</italic><sub><italic>y</italic></sub>)</math-text></inline-eqn> along the <inline-eqn><math-text><italic>y</italic></math-text></inline-eqn> direction is given by
<display-eqn id="nj318400eqn26" textype="equation" notation="LaTeX" eqnnum="26"></display-eqn>where <inline-eqn><math-text><italic>j</italic><sub><italic>y</italic></sub>=−<upright>i</upright>(<italic>U</italic><sup>*</sup><italic>V</italic>−<italic>V</italic><sup>*</sup><italic>U</italic>)</math-text></inline-eqn>. From figure <figref linkend="nj318400fig4">4</figref>(a) it is clear that these bound states move along the <inline-eqn><math-text><italic>y</italic></math-text></inline-eqn> direction, i.e. along the magnetic barriers. Their velocity <inline-eqn><math-text><italic>v</italic><sub><italic>y</italic></sub>>−<italic>v</italic><sub><upright>F</upright></sub></math-text></inline-eqn> is negative for <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub><−<italic>P</italic></math-text></inline-eqn>, but as the electron is approaching <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>→−<italic>P</italic></math-text></inline-eqn> we have <inline-eqn><math-text><italic>v</italic><sub><italic>y</italic></sub>→0</math-text></inline-eqn>. For <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>>−<italic>P</italic></math-text></inline-eqn>, the velocity <inline-eqn><math-text><italic>v</italic><sub><italic>y</italic></sub>><italic>v</italic><sub><upright>F</upright></sub></math-text></inline-eqn> is positive. This can be understood from the maximum and minimum of the effective potential which is shown in figure <figref linkend="nj318400fig3">3</figref>(c). The energy bound states can only exist between these two lines. Note that the slope of min <inline-eqn><math-text><italic>V</italic><sub><upright>eff</upright></sub></math-text></inline-eqn> is negative for <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub><−<italic>P</italic></math-text></inline-eqn>, while it turns positive for <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>>−<italic>P</italic></math-text></inline-eqn>, which explains the <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn> dependence of the velocity. From figure <figref linkend="nj318400fig4">4</figref>(a) it is clear there are two different classes of bound states. The bound state that follows very closely the <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>=−<italic>P</italic></math-text></inline-eqn> curve and extends to the region <inline-eqn><math-text>−<italic>P</italic><<italic>k</italic><sub><italic>y</italic></sub><0</math-text></inline-eqn> with energy close to zero has a wave function that is concentrated around the position of the two magnetic delta-functions and decays exponentially in the region <inline-eqn><math-text>0<<italic>x</italic><<italic>L</italic></math-text></inline-eqn>. The wave functions of the other bound states are concentrated in a region between the two magnetic delta-functions (i.e. in a standing wave fashion) and decay exponentially outside this region.</p><figure id="nj318400fig4" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="31.9pc" printcolour="no" filename="images/nj318400fig4.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj318400fig4.jpg"></graphic-file></graphic><caption type="figure" id="nj318400fc4" label="Figure 4"><p indent="no">(a) Energy spectrum (red full curves on white background) and contour plot of the transmission (color background) through two magnetic, but opposite in direction <inline-eqn><math-text>δ</math-text></inline-eqn>-function barriers for <inline-eqn><math-text><italic>L</italic>=200</math-text></inline-eqn> and <inline-eqn><math-text><italic>P</italic>=0.1</math-text></inline-eqn>. (b) The same as (a) but now for the configuration shown in figure <figref linkend="nj318400fig2">2</figref>(b) for <inline-eqn><math-text><italic>a</italic>=<italic>b</italic>=<italic>c</italic>=200</math-text></inline-eqn> and <inline-eqn><math-text><italic>P</italic>=0.1</math-text></inline-eqn>. (c) Transmission versus energy through the magnetic <inline-eqn><math-text>δ</math-text></inline-eqn>-function barriers in (a) for <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>=−0.05</math-text></inline-eqn>. (d) Same as (c) but now for the configuration shown in figure <figref linkend="nj318400fig2">2</figref>(b) for <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>=0</math-text></inline-eqn>. (e) Transmission versus <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn> through the system described in (a) for fixed <inline-eqn><math-text><italic>E</italic>=0.1</math-text></inline-eqn>. The red arrow lines indicate the position of the bound states. (f) Same as (e) but now for the configuration shown in figure <figref linkend="nj318400fig2">2</figref>(b) for <inline-eqn><math-text><italic>E</italic>=0.15</math-text></inline-eqn>.
</p></caption></figure><p>Next, we consider a structure with zero-average vector potential as shown in figure <figref linkend="nj318400fig2">2</figref>(b), with corresponding effective potential shown in figure <figref linkend="nj318400fig3">3</figref>(b). The effective potential for <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub><−<italic>P</italic>/2</math-text></inline-eqn> and <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>><italic>P</italic>/2</math-text></inline-eqn> consists of a potential well and a potential barrier and therefore has at least one bound state. Thus, we expect bound states for all <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn> with energy between <inline-eqn><math-text><italic>E</italic>=−(+)(<italic>k</italic><sub><italic>y</italic></sub>+<italic>P</italic>)</math-text></inline-eqn> and <inline-eqn><math-text><italic>E</italic>=−(+)k<sub><italic>y</italic></sub></math-text></inline-eqn> when <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub><−<italic>P</italic>/2</math-text></inline-eqn> (<inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>><italic>P</italic>/2</math-text></inline-eqn>). The dispersion relation for those bound states results from the solution of
<display-eqn id="nj318400eqn27" textype="equation" notation="LaTeX" eqnnum="27"></display-eqn>where <inline-eqn><math-text><bold>M</bold></math-text></inline-eqn> is the transfer matrix for the unit shown in figure <figref linkend="nj318400fig2">2</figref>(d). These bound states are shown by the red full curves in figure <figref linkend="nj318400fig4">4</figref>(b). Because of the spatial inversion symmetry of the vector potential the spectrum has the symmetry <inline-eqn><math-text><italic>E</italic>(−<italic>k</italic><sub><italic>y</italic></sub>)=<italic>E</italic>(<italic>k</italic><sub><italic>y</italic></sub>)</math-text></inline-eqn>. Note that for <inline-eqn><math-text>−<italic>P</italic><<italic>k</italic><sub><italic>y</italic></sub><<italic>P</italic></math-text></inline-eqn> the lowest bound state has energy <inline-eqn><math-text><italic>E</italic>≈0</math-text></inline-eqn>. For <inline-eqn><math-text>−<italic>P</italic>/2<<italic>k</italic><sub><italic>y</italic></sub><<italic>P</italic>/2</math-text></inline-eqn>, we have two potential barriers and therefore no bound states.</p></sec-level2><sec-level2 id="nj318400s2.2" label="2.2"><heading>Reflection and transmission coefficients</heading><p indent="no">Consider a plane wave incident upon a system of two <inline-eqn><math-text>δ</math-text></inline-eqn>-function magnetic barriers, identical in height but opposite in direction, placed at <inline-eqn><math-text><italic>x</italic>=0</math-text></inline-eqn> and <inline-eqn><math-text><italic>x</italic>=<italic>L</italic></math-text></inline-eqn>, as shown schematically in figure <figref linkend="nj318400fig2">2</figref>(a). In this case, the vector potential is constant for <inline-eqn><math-text>0⩽<italic>x</italic>⩽<italic>L</italic></math-text></inline-eqn>, zero outside this region, and homogeneous in the <inline-eqn><math-text><italic>y</italic></math-text></inline-eqn> direction. Below we derive expressions for the amplitudes and intensities of the reflected and transmitted waves.</p><p>Let <inline-eqn><math-text><italic>A</italic></math-text></inline-eqn>, <inline-eqn><math-text><italic>R</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>T</italic></math-text></inline-eqn> denote the amplitudes of the incident, reflected and transmitted waves, respectively. Further, let <inline-eqn><math-text>&thetas;<sub>0</sub></math-text></inline-eqn> be the angle of incidence and exit as shown in figure <figref linkend="nj318400fig2">2</figref>(b). The boundary conditions give
<display-eqn id="nj318400eqn28" textype="equation" notation="LaTeX" eqnnum="28"></display-eqn>The four quantities <inline-eqn><math-text><italic>U</italic><sub>0</sub>, <italic>V</italic><sub>0</sub>, <italic>U</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>V</italic></math-text></inline-eqn> given by equations (<eqnref linkend="nj318400eqn28">28</eqnref>) are connected by the basic relation <inline-eqn><math-text><bold>Q</bold><sub>0</sub>=<bold>MQ</bold></math-text></inline-eqn>; hence, with <inline-eqn><math-text><italic>J</italic>=<italic>m</italic>′ <sub>11</sub>+<italic>m</italic>′<sub>12</sub><upright>e</upright><sup><upright>i</upright>&thetas;<sub>0</sub></sup></math-text></inline-eqn> and <inline-eqn><math-text><italic>K</italic>=<italic>m</italic>′ <sub>21</sub>+<italic>m</italic>′ <sub>22</sub><upright>e</upright><sup><upright>i</upright>&thetas;<sub>0</sub></sup></math-text></inline-eqn>, we have
<display-eqn id="nj318400eqn29" textype="equation" notation="LaTeX" eqnnum="29"></display-eqn>where <inline-eqn><math-text><italic>m</italic>′ <sub><italic>ij</italic></sub></math-text></inline-eqn> are the elements of the characteristic matrix of the medium, evaluated at <inline-eqn><math-text><italic>x</italic>=<italic>L</italic></math-text></inline-eqn>. From equation (<eqnref linkend="nj318400eqn29">29</eqnref>) we obtain the reflection and transmission amplitudes
<display-eqn id="nj318400eqn30" textype="equation" notation="LaTeX" eqnnum="30"></display-eqn>In terms of <inline-eqn><math-text><italic>r</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>t</italic></math-text></inline-eqn>, the <italic>reflectivity</italic>and <italic>transmissivity</italic> are
<display-eqn id="nj318400eqn31" textype="equation" notation="LaTeX" eqnnum="31"></display-eqn>The characteristic matrix for a homogeneous vector potential is given by equation (<eqnref linkend="nj318400eqn19">19</eqnref>). Labeling with subscripts 1, 2 and 3 quantities which refer to the regions, respectively, I, II and III of figure <figref linkend="nj318400fig2">2</figref>(a), and by <inline-eqn><math-text><italic>L</italic>=<italic>x</italic><sub>2</sub>−<italic>x</italic><sub>1</sub></math-text></inline-eqn> the distance between the magnetic <inline-eqn><math-text>δ</math-text></inline-eqn>-functions, we have (<inline-eqn><math-text>β=<italic>EL</italic> <upright>cos</upright> &thetas;<sub><italic>i</italic></sub></math-text></inline-eqn>)
<display-eqn id="nj318400eqn32" textype="equation" notation="LaTeX" eqnnum="32" lines="multiline"></display-eqn>The reflection and transmission amplitudes <inline-eqn><math-text><italic>r</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>t</italic></math-text></inline-eqn> are obtained by substituting these expressions in those for <inline-eqn><math-text><italic>J</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>K</italic></math-text></inline-eqn> that appear in equation (<eqnref linkend="nj318400eqn30">30</eqnref>). The resulting formula can be expressed in terms of the amplitudes <inline-eqn><math-text><italic>r</italic><sub>12</sub>, <italic>t</italic><sub>12</sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>r</italic><sub>23</sub>, <italic>t</italic><sub>23</sub></math-text></inline-eqn> associated with the reflection at and transmission through the first and second ‘interface’, respectively. We have
<display-eqn id="nj318400eqn33" textype="equation" notation="LaTeX" eqnnum="33"></display-eqn>and similar expressions for <inline-eqn><math-text><italic>r</italic><sub>23</sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>t</italic><sub>23</sub></math-text></inline-eqn>. In terms of these, expressions <inline-eqn><math-text><italic>r</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>t</italic></math-text></inline-eqn> become
<display-eqn id="nj318400eqn34" textype="equation" notation="LaTeX" eqnnum="34"></display-eqn>The amplitude <inline-eqn><math-text><italic>t</italic></math-text></inline-eqn> of the transmission through the system is given by [<cite linkend="nj318400bib8">8</cite>, <cite linkend="nj318400bib9">9</cite>, <cite linkend="nj318400bib12">12</cite>, <cite linkend="nj318400bib24">24</cite>, <cite linkend="nj318400bib25">25</cite>],
<display-eqn id="nj318400eqn35" textype="equation" notation="LaTeX" eqnnum="35"></display-eqn>where <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>=<italic>E</italic><upright>sin</upright> &thetas;<sub>0</sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>+<italic>P</italic>=<italic>E</italic> <upright>sin</upright> &thetas;<sub><italic>i</italic></sub></math-text></inline-eqn>. This equation remains invariant under the changes <inline-eqn><math-text><italic>E</italic>→−<italic>E</italic></math-text></inline-eqn>, <inline-eqn><math-text>&thetas;<sub>0</sub>→−&thetas;<sub>0</sub></math-text></inline-eqn> and <inline-eqn><math-text>&thetas;<sub><italic>i</italic></sub>→−&thetas;<sub><italic>i</italic></sub></math-text></inline-eqn>. A contour plot of the transmission is shown in figure <figref linkend="nj318400fig4">4</figref>(a) and slices for constant <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>E</italic></math-text></inline-eqn> are shown, respectively, in figures <figref linkend="nj318400fig4">4</figref>(c) and (d). By imposing the condition that the wave number <inline-eqn><math-text><italic>k</italic><sub><italic>x</italic></sub></math-text></inline-eqn> be real for incident and transmitted waves, we find that the angles <inline-eqn><math-text>&thetas;<sub>0</sub></math-text></inline-eqn> and <inline-eqn><math-text>&thetas;<sub><italic>i</italic></sub></math-text></inline-eqn> are related by
<display-eqn id="nj318400eqn36" textype="equation" notation="LaTeX" eqnnum="36"></display-eqn>Equation (<eqnref linkend="nj318400eqn36">36</eqnref>) expresses the angular confinement of the transmission elaborated in [<cite linkend="nj318400bib8">8</cite>, <cite linkend="nj318400bib9">9</cite>, <cite linkend="nj318400bib12">12</cite>, <cite linkend="nj318400bib25">25</cite>, <cite linkend="nj318400bib26">26</cite>]. Notice its formal similarity with Snell's law. Using equation (<eqnref linkend="nj318400eqn36">36</eqnref>) we obtain the range of the angles of incidence <inline-eqn><math-text>&thetas;<sub>0</sub></math-text></inline-eqn> for which transmission through the first magnetic barrier is possible
<display-eqn id="nj318400eqn37" textype="equation" notation="LaTeX" eqnnum="37"></display-eqn>For the special value of the energy <inline-eqn><math-text><italic>E</italic>=<italic>P</italic>/2</math-text></inline-eqn> and <inline-eqn><math-text>&thetas;<sub>0</sub></math-text></inline-eqn> in the range <inline-eqn><math-text>−π/2⩽&thetas;<sub>0</sub>⩽π/2</math-text></inline-eqn>, we have <inline-eqn><math-text>&thetas;<sub><italic>i</italic></sub>=π/2</math-text></inline-eqn>, while for <inline-eqn><math-text><italic>E</italic>=−<italic>P</italic>/2</math-text></inline-eqn> the result is <inline-eqn><math-text>&thetas;<sub><italic>i</italic></sub>=−π/2</math-text></inline-eqn>. Alternatively, we can put <inline-eqn><math-text>&thetas;<sub><italic>i</italic></sub>=±π/2</math-text></inline-eqn> in equation (<eqnref linkend="nj318400eqn36">36</eqnref>) and obtain, for <inline-eqn><math-text><italic>P</italic>>0</math-text></inline-eqn>, the result
<display-eqn id="nj318400eqn38" textype="equation" notation="LaTeX" eqnnum="38"></display-eqn>where the <inline-eqn><math-text>+(−)</math-text></inline-eqn> sign corresponds to <inline-eqn><math-text><italic>E</italic>>0</math-text></inline-eqn> (<inline-eqn><math-text><italic>E</italic><0</math-text></inline-eqn>). A contour plot of the transmission as function of <inline-eqn><math-text><italic>E</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn>, obtained from equation (<eqnref linkend="nj318400eqn35">35</eqnref>), is shown in figure <figref linkend="nj318400fig4">4</figref>(a). In figure <figref linkend="nj318400fig4">4</figref>(a), we distinguish three different regions. In the region between <inline-eqn><math-text><italic>E</italic>=−(<italic>k</italic><sub><italic>y</italic></sub>+<italic>P</italic>/2)</math-text></inline-eqn> and <inline-eqn><math-text><italic>E</italic>=−<italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn>, the wavevector of the incident wave is imaginary and they are evanescent waves. In this region, <inline-eqn><math-text><italic>k</italic>′</math-text></inline-eqn> is real and it is possible to find localized states. The <inline-eqn><math-text><italic>k</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>k</italic>′</math-text></inline-eqn> for the second region between <inline-eqn><math-text><italic>E</italic>=<italic>k</italic><sub><italic>y</italic></sub>+<italic>P</italic>/2</math-text></inline-eqn> and <inline-eqn><math-text><italic>E</italic>=−(<italic>k</italic><sub><italic>y</italic></sub>−<italic>P</italic>/2)</math-text></inline-eqn> are real and the electron can tunnel through the magnetic <inline-eqn><math-text>δ</math-text></inline-eqn>-barriers. In the blue shadow region between <inline-eqn><math-text><italic>E</italic>=<italic>k</italic><sub><italic>y</italic></sub>+<italic>P</italic>/2</math-text></inline-eqn> and <inline-eqn><math-text><italic>E</italic>=<italic>k</italic><sub><italic>y</italic></sub>−<italic>P</italic>/2</math-text></inline-eqn>, <inline-eqn><math-text><italic>k</italic></math-text></inline-eqn> is real but <inline-eqn><math-text><italic>k</italic>′</math-text></inline-eqn> is imaginary and solutions inside the barrier are evanescent and there is very little tunneling which very quickly becomes zero. The transmission probability <inline-eqn><math-text>|<italic>T</italic>|=<italic>t</italic>·<italic>t</italic><sup>*</sup></math-text></inline-eqn> is equal to <inline-eqn><math-text>1</math-text></inline-eqn> for <inline-eqn><math-text><upright>cos</upright> (2β)=1</math-text></inline-eqn>. In this case, the energy becomes
<display-eqn id="nj318400eqn39" textype="equation" notation="LaTeX" eqnnum="39"></display-eqn>The condition <inline-eqn><math-text><upright>cos</upright> (2β)=1</math-text></inline-eqn>, or equivalently <inline-eqn><math-text>β=<italic>n</italic>π=<italic>Eh</italic><upright>cos</upright> &thetas;<sub>2</sub></math-text></inline-eqn> with <inline-eqn><math-text><italic>n</italic></math-text></inline-eqn> an integer, should be combined with that for the transmission to occur in the region delimited by the curves <inline-eqn><math-text><italic>E</italic>=±(<italic>k</italic><sub><italic>y</italic></sub>+<italic>P</italic>)</math-text></inline-eqn> and <inline-eqn><math-text><italic>E</italic>=±<italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn>. For example, in figure <figref linkend="nj318400fig4">4</figref>(a) for <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>=−0.05</math-text></inline-eqn> and <inline-eqn><math-text>0<<italic>E</italic><0.2</math-text></inline-eqn> we have <inline-eqn><math-text>12</math-text></inline-eqn> maxima. It is readily seen that with these parameters in equation (<eqnref linkend="nj318400eqn39">39</eqnref>) we find <inline-eqn><math-text>12</math-text></inline-eqn> different energies as shown in figure <figref linkend="nj318400fig4">4</figref>(c).</p><p>Figure <figref linkend="nj318400fig4">4</figref>(b) shows a contour plot of the transmission for the structure shown in figure <figref linkend="nj318400fig2">2</figref>(d), which is symmetric around <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>=0</math-text></inline-eqn>. Note that the number of resonances has increased substantially as compared to the previous case which is due to the fact that we have twice as many magnetic barriers in our systems.</p></sec-level2></sec-level1><sec-level1 id="nj318400s3" label="3"><heading>A series of units with magnetic <inline-eqn><math-text>δ</math-text></inline-eqn>-function barriers</heading><sec-level2 id="nj318400s3.1" label="3.1"><heading><inline-eqn><math-text><italic>N</italic></math-text></inline-eqn> units</heading><p indent="no">We consider a system of <inline-eqn><math-text><italic>N</italic></math-text></inline-eqn> units, such as those shown in figures <figref linkend="nj318400fig2">2</figref>(a) and (d) with periods <inline-eqn><math-text><italic>L</italic>=<italic>a</italic>+<italic>b</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>L</italic>=<italic>a</italic>+<italic>b</italic>+<italic>c</italic>+<italic>d</italic></math-text></inline-eqn>, respectively. The corresponding periodic vector potential is <inline-eqn><math-text><bold>A</bold>(<italic>x</italic>)=<bold>A</bold>(<italic>x</italic>+<italic>nL</italic>)</math-text></inline-eqn> and the magnetic field <inline-eqn><math-text><bold>B</bold>=<bold>B</bold>(<italic>x</italic>+<italic>nL</italic>)</math-text></inline-eqn>, with <inline-eqn><math-text><italic>n</italic>=1, 2, …, <italic>N</italic></math-text></inline-eqn>. The characteristic matrix for one period <inline-eqn><math-text><bold>M</bold>(<italic>L</italic>)</math-text></inline-eqn> is denoted by
<display-eqn id="nj318400eqn40" textype="equation" notation="LaTeX" eqnnum="40"></display-eqn>On account of the periodicity we have
<display-eqn id="nj318400eqn41" textype="equation" notation="LaTeX" eqnnum="41"></display-eqn>To evaluate the elements of <inline-eqn><math-text><bold>M</bold>(<italic>NL</italic>)</math-text></inline-eqn> we use a result from the theory of matrices, according to which the <italic>N</italic>th power of a unimodular matrix <inline-eqn><math-text><bold>M</bold>(<italic>L</italic>)</math-text></inline-eqn> is (<inline-eqn><math-text><italic>u</italic><sub><italic>N</italic></sub>(χ)≡<italic>u</italic><sub><italic>N</italic></sub></math-text></inline-eqn>)
<display-eqn id="nj318400eqn42" textype="equation" notation="LaTeX" eqnnum="42"></display-eqn>with <inline-eqn></inline-eqn> and <inline-eqn><math-text><italic>u</italic><sub><italic>N</italic></sub></math-text></inline-eqn> the Chebyshev polynomials of the second kind:
<display-eqn id="nj318400eqn43" textype="equation" notation="LaTeX" eqnnum="43"></display-eqn>where
<display-eqn id="nj318400eqn44" textype="equation" notation="LaTeX" eqnnum="44"></display-eqn>Here <inline-eqn><math-text>ζ</math-text></inline-eqn> is the <italic>Bloch phase</italic> of the periodic system [<cite linkend="nj318400bib33">33</cite>], which is related to the eigenfunctions of <inline-eqn><math-text><bold>M</bold></math-text></inline-eqn>. In the limiting case of <inline-eqn><math-text><italic>N</italic>→∞</math-text></inline-eqn>, we have total reflection when <inline-eqn><math-text>ζ</math-text></inline-eqn> is outside the range <inline-eqn><math-text>(−1, 1)</math-text></inline-eqn>.</p></sec-level2><sec-level2 id="nj318400s3.2" label="3.2"><heading>SL</heading><p indent="no">Here we consider a finite number <inline-eqn><math-text><italic>N</italic></math-text></inline-eqn> of lattice units shown in figure <figref linkend="nj318400fig2">2</figref>(c). We set
<display-eqn id="nj318400eqn45" textype="equation" notation="LaTeX" eqnnum="45"></display-eqn>The characteristic matrix <inline-eqn><math-text><bold>M</bold><sub>2</sub>(<italic>L</italic>)</math-text></inline-eqn> for one period is readily obtained, in terms of these quantities, as in section <secref linkend="nj318400s2">2</secref>, and from that the characteristic matrix <inline-eqn><math-text><bold>M</bold><sub>2<italic>N</italic></sub>(<italic>NL</italic>)</math-text></inline-eqn> of the multilayer system according to equation (<eqnref linkend="nj318400eqn41">41</eqnref>). Its elements are
<display-eqn id="nj318400eqn46" textype="equation" notation="LaTeX" eqnnum="46" lines="multiline"></display-eqn>where <inline-eqn><math-text><italic>s</italic>=<italic>p</italic><sub>2</sub><italic>p</italic><sub>1</sub></math-text></inline-eqn>, <inline-eqn><math-text><italic>u</italic><sub><italic>N</italic></sub>≡<italic>u</italic><sub><italic>N</italic></sub>(χ)</math-text></inline-eqn>, and
<display-eqn id="nj318400eqn47" textype="equation" notation="LaTeX" eqnnum="47"></display-eqn>The reflection and transmission coefficients of the multi-unit system are immediately obtained by substituting these expressions into equation (<eqnref linkend="nj318400eqn30">30</eqnref>). The numerical results are shown in figures <figref linkend="nj318400fig5">5</figref> and <figref linkend="nj318400fig6">6</figref> for finite SL with <inline-eqn><math-text><italic>N</italic>=10</math-text></inline-eqn> units. Two different types of structures are considered as shown in the insets to figures <figref linkend="nj318400fig6">6</figref>. The transmission does not have <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>→−<italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn> symmetry for the periodic system with magnetic delta up-down as is apparent from figure <figref linkend="nj318400fig6">6</figref>(a). We contrast these results with the case in which we used an arrangement of magnetic delta-function as in the previous structure plus another unit with an opposite direction of magnetic delta-function. As is clearly shown in figure <figref linkend="nj318400fig6">6</figref>(b), we have <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>→−<italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn> symmetry for the transmission probability through this structure. The transmission resonances are more pronounced, i.e. the dips become deeper, when the number of barriers increases for both types of units. But the gaps occur when the wave is mostly reflected. The position of these gaps, which are especially pronounced as <inline-eqn><math-text><italic>N</italic></math-text></inline-eqn> increases, can also be found from the structure of the Bloch phase <inline-eqn><math-text>ζ</math-text></inline-eqn>, as shown in figures <figref linkend="nj318400fig5">5</figref>(c) and (d). In figure <figref linkend="nj318400fig5">5</figref>, the bound states are shown by the blue solid curves that are situated in the area <inline-eqn><math-text>−<italic>k</italic><sub><italic>y</italic></sub>−<italic>P</italic>/2<<italic>E</italic><−<italic>k</italic><sub><italic>y</italic></sub>+<italic>P</italic>/2</math-text></inline-eqn> in case (a) and in <inline-eqn><math-text>−<italic>k</italic><sub><italic>y</italic></sub>−<italic>P</italic>/2<<italic>E</italic><−<italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn> for case (b) plus an area located symmetric with respect to <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn>. Notice in figure <figref linkend="nj318400fig5">5</figref>(a) that several bound states merge into a resonant states at <inline-eqn><math-text><italic>E</italic>=−<italic>k</italic><sub><italic>y</italic></sub>+<italic>P</italic>/2</math-text></inline-eqn>. This is different from figure <figref linkend="nj318400fig4">4</figref>(a) where each bound state becomes a resonant state at <inline-eqn><math-text><italic>E</italic>=−<italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn>.</p><figure id="nj318400fig5" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="31.9pc" printcolour="no" filename="images/nj318400fig5.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj318400fig5.jpg"></graphic-file></graphic><caption type="figure" id="nj318400fc5" label="Figure 5"><p indent="no">Contour plot of the transmission (a) and Bloch phase (c) through <inline-eqn><math-text><italic>N</italic>=10</math-text></inline-eqn> magnetic <inline-eqn><math-text>δ</math-text></inline-eqn>-function barriers with <inline-eqn><math-text><italic>a</italic>=10</math-text></inline-eqn>, <inline-eqn><math-text><italic>b</italic>=10</math-text></inline-eqn> and <inline-eqn><math-text><italic>P</italic>=1</math-text></inline-eqn>. (b) and (d) The same as in (a) and (c) for <inline-eqn><math-text><italic>a</italic>=5</math-text></inline-eqn>, <inline-eqn><math-text><italic>b</italic>=5</math-text></inline-eqn>, <inline-eqn><math-text><italic>c</italic>=5</math-text></inline-eqn>, <inline-eqn><math-text><italic>d</italic>=5</math-text></inline-eqn> and <inline-eqn><math-text><italic>P</italic>=1</math-text></inline-eqn>, single unit.</p></caption></figure><figure id="nj318400fig6" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="37.6pc" printcolour="no" filename="images/nj318400fig6.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj318400fig6.jpg"></graphic-file></graphic><caption type="figure" id="nj318400fc6" label="Figure 6"><p indent="no">(a) and (b) Transmission versus energy through <inline-eqn><math-text><italic>N</italic>=1, 5, 10</math-text></inline-eqn> magnetic units of <inline-eqn><math-text>δ</math-text></inline-eqn>-function barriers shown on the left. The upper unit has <inline-eqn><math-text><italic>a</italic>=10</math-text></inline-eqn>, <inline-eqn><math-text><italic>b</italic>=10</math-text></inline-eqn>, <inline-eqn><math-text><italic>P</italic>=1</math-text></inline-eqn>, <inline-eqn><math-text><italic>E</italic>=1.5</math-text></inline-eqn> and the bottom one <inline-eqn><math-text><italic>a</italic>=<italic>b</italic>=<italic>c</italic>=<italic>d</italic>=5</math-text></inline-eqn>, and <inline-eqn><math-text><italic>P</italic>=1</math-text></inline-eqn> and <inline-eqn><math-text><italic>E</italic>=2.5</math-text></inline-eqn>.</p></caption></figure></sec-level2><sec-level2 id="nj318400s3.3" label="3.3"><heading>Spectrum of an SL</heading><p indent="no">Let us take <inline-eqn><math-text><italic>N</italic>→∞</math-text></inline-eqn>. We can find the energy-momentum relation from the previous standard calculation [<cite linkend="nj318400bib32">32</cite>, <cite linkend="nj318400bib33">33</cite>] by using
<display-eqn id="nj318400eqn48" textype="equation" notation="LaTeX" eqnnum="48"></display-eqn>where <inline-eqn><math-text><italic>M</italic></math-text></inline-eqn> is the characteristic matrix of one period. This results in
<display-eqn id="nj318400eqn49" textype="equation" notation="LaTeX" eqnnum="49"></display-eqn>With reference to the regions I and II shown in figure <figref linkend="nj318400fig2">2</figref>(a), we write <inline-eqn><math-text><italic>k</italic><sub>1</sub>=[<italic>E</italic><sup>2</sup>−<italic>k</italic><sub><italic>y</italic></sub><sup>2</sup>]<sup>1/2</sup></math-text></inline-eqn> and <inline-eqn><math-text><italic>k</italic><sub>2</sub>=[<italic>E</italic><sup>2</sup>−(<italic>k</italic><sub><italic>y</italic></sub>+<italic>P</italic>)<sup>2</sup>]<sup>1/2</sup></math-text></inline-eqn> and show the solution for <inline-eqn><math-text><italic>E</italic><sup>2</sup>>(<italic>k</italic><sub><italic>y</italic></sub>+<italic>P</italic>)<sup>2</sup></math-text></inline-eqn> in figure <figref linkend="nj318400fig9">9</figref>. Equation (<eqnref linkend="nj318400eqn49">49</eqnref>) differs from the corresponding result of [<cite linkend="nj318400bib27">27</cite>], equation (<eqnref linkend="nj318400eqn7">7</eqnref>), for the case of Schrödinger electrons, in the term <inline-eqn><math-text><italic>P</italic><sup>2</sup></math-text></inline-eqn> in the prefactor of the second term on the right-hand side and the linear <inline-eqn><math-text><italic>E</italic></math-text></inline-eqn> versus <inline-eqn><math-text><italic>k</italic></math-text></inline-eqn> spectrum instead of the quadratic one in [<cite linkend="nj318400bib27">27</cite>]. If <inline-eqn><math-text><italic>P</italic></math-text></inline-eqn> is large the differences become more pronounced. Our numerical results for the energy spectrum are shown in figures <figref linkend="nj318400fig7">7</figref>–<figref linkend="nj318400fig11">11</figref>. The results for standard and Dirac electrons show not only similarities but also important differences. The first band shows a qualitative difference near <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>≈0</math-text></inline-eqn>, see figure <figref linkend="nj318400fig9">9</figref>. As figures <figref linkend="nj318400fig7">7</figref>, <figref linkend="nj318400fig9">9</figref>(a) and (b) show, the band behavior in the <inline-eqn><math-text><italic>k</italic>=<italic>k</italic><sub><italic>x</italic></sub></math-text></inline-eqn> direction for fixed <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn> is constant and almost symmetric about <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>=0</math-text></inline-eqn>; the motion becomes nearly 1D for relatively large <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn>. From the contour plots of figures <figref linkend="nj318400fig9">9</figref>(b) and (d), as well as from figure <figref linkend="nj318400fig7">7</figref>(c), we infer a collimation along the <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn>-direction, i.e. <inline-eqn><math-text><italic>v</italic><sub><italic>y</italic></sub>∝∂<italic>E</italic>/∂<italic>k</italic><sub><italic>y</italic></sub>≈<italic>v</italic><sub><upright>F</upright></sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>v</italic><sub><italic>x</italic></sub>≈0</math-text></inline-eqn>, which is similar to that found for an SL of electric potential barriers [<cite linkend="nj318400bib16">16</cite>] for some specific values of the barrier heights. Also, there are no gaps for <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>≈0</math-text></inline-eqn> in figure <figref linkend="nj318400fig10">10</figref>(a) but there are for the case of Dirac electrons as seen in figure <figref linkend="nj318400fig10">10</figref>(b). This difference can be traced back to the presence of <inline-eqn><math-text><italic>P</italic><sup>2</sup></math-text></inline-eqn> in the dispersion relation equation (<eqnref linkend="nj318400eqn49">49</eqnref>) when compared to the same equation for the standard electron. The even-number energy bands in figure <figref linkend="nj318400fig11">11</figref>(b) are wider than those in figure <figref linkend="nj318400fig11">11</figref>(a) and, as a function of the period, the energy decreases faster for Dirac electrons. This behavior of the bands for Dirac electrons is very similar to that for the frequency <inline-eqn><math-text>ω</math-text></inline-eqn> versus <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn> or <inline-eqn><math-text><italic>L</italic></math-text></inline-eqn> in media with a periodically varying refractive index [<cite linkend="nj318400bib29">29</cite>]. This is clearly a consequence of the linear <inline-eqn><math-text><italic>E</italic>−<italic>k</italic></math-text></inline-eqn> relation. Notice the differences between the lowest bands shown in panels (a) and (b) in figure <figref linkend="nj318400fig12">12</figref> and in particular the difference between the corresponding drift velocities as functions of <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn>.</p><figure id="nj318400fig7" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="31.9pc" printcolour="no" filename="images/nj318400fig7.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj318400fig7.jpg"></graphic-file></graphic><caption type="figure" id="nj318400fc7" label="Figure 7"><p indent="no">(a) The first two energy bands and contour plot of second (b) and first (c) band for a magnetic SL of <inline-eqn><math-text>δ</math-text></inline-eqn>-function barriers with <inline-eqn><math-text><italic>a</italic>=4</math-text></inline-eqn>, <inline-eqn><math-text><italic>b</italic>=4</math-text></inline-eqn>, and <inline-eqn><math-text><italic>P</italic>=1</math-text></inline-eqn>.</p></caption></figure><figure id="nj318400fig8" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="31.9pc" printcolour="no" filename="images/nj318400fig8.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj318400fig8.jpg"></graphic-file></graphic><caption type="figure" id="nj318400fc8" label="Figure 8"><p indent="no">Dispersion relation (<inline-eqn><math-text><italic>E</italic></math-text></inline-eqn> versus <inline-eqn><math-text><italic>k</italic></math-text></inline-eqn>) for a standard electron in (a) and a Dirac electron in (b). The fixed values of <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn> are shown in the panels, <inline-eqn><math-text><italic>L</italic>=8</math-text></inline-eqn> and <inline-eqn><math-text><italic>P</italic>=1</math-text></inline-eqn> (the energy for the standard electron is measured in units of <inline-eqn></inline-eqn> with <inline-eqn></inline-eqn> and all distances in units of <inline-eqn></inline-eqn>).</p></caption></figure><figure id="nj318400fig9" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="26.3pc" printcolour="no" filename="images/nj318400fig9.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj318400fig9.jpg"></graphic-file></graphic><caption type="figure" id="nj318400fc9" label="Figure 9"><p indent="no">First energy band for (a) standard and (c) Dirac electron in SL of magnetic <inline-eqn><math-text>δ</math-text></inline-eqn>-function barriers with <inline-eqn><math-text><italic>a</italic>=100</math-text></inline-eqn>, <inline-eqn><math-text><italic>b</italic>=100</math-text></inline-eqn>, and <inline-eqn><math-text><italic>P</italic>=0.1</math-text></inline-eqn>. (b) and (d) Corresponding contour plots of (a) and (c) (the energy for the standard electron is measured in units of <inline-eqn></inline-eqn> with <inline-eqn></inline-eqn> and all distances in units of <inline-eqn></inline-eqn>).</p></caption></figure><figure id="nj318400fig10" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="31.6pc" printcolour="no" filename="images/nj318400fig10.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj318400fig10.jpg"></graphic-file></graphic><caption type="figure" id="nj318400fc10" label="Figure 10"><p indent="no">Dispersion relation for (a) a standard electron and (b) a Dirac electron. The period is <inline-eqn><math-text><italic>L</italic>=<italic>a</italic>+<italic>b</italic>=8</math-text></inline-eqn> and the shaded (in green) regions are the lowest six allowed bands. The solid curves in both panels, the dash-dotted curves in (a) and the dashed ones in (b) show bound states for a free electron (the energy for the standard electron is measured in units of <inline-eqn></inline-eqn> with <inline-eqn></inline-eqn> and all distances in units of <inline-eqn></inline-eqn>).</p></caption></figure><figure id="nj318400fig11" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="26.0pc" printcolour="no" filename="images/nj318400fig11.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj318400fig11.jpg"></graphic-file></graphic><caption type="figure" id="nj318400fc11" label="Figure 11"><p indent="no">Energy versus period <inline-eqn><math-text><italic>L</italic>=<italic>a</italic>+<italic>b</italic></math-text></inline-eqn> for (a) a standard electron and (b) a Dirac electron with fixed <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub>=4</math-text></inline-eqn> (the energy for the standard electron is measured in units of <inline-eqn></inline-eqn> with <inline-eqn></inline-eqn> and all distances in units of <inline-eqn></inline-eqn>).</p></caption></figure><figure id="nj318400fig12" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="31.9pc" printcolour="no" filename="images/nj318400fig12.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj318400fig12.jpg"></graphic-file></graphic><caption type="figure" id="nj318400fc12" label="Figure 12"><p indent="no">(a) and (b) Lowest-energy band for a standard electrons and drift velocity versus <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn> for three different values of <inline-eqn><math-text><italic>L</italic></math-text></inline-eqn>. (c) and (d) First (black curves) and second (red curves) band for Dirac electrons and the drift velocity versus <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn> for three different values of <inline-eqn><math-text><italic>L</italic></math-text></inline-eqn> (the energy for the standard electron is measured in units of <inline-eqn></inline-eqn> with <inline-eqn></inline-eqn> and all distances in units of <inline-eqn></inline-eqn>).</p></caption></figure></sec-level2></sec-level1><sec-level1 id="nj318400s4" label="4"><heading>A series of <inline-eqn><math-text>δ</math-text></inline-eqn>-function vector potentials</heading><p indent="no">In the limit that the distance between the opposite-directed magnetic barriers decreases to zero the vector potential approaches a <inline-eqn><math-text>δ</math-text></inline-eqn>-function [<cite linkend="nj318400bib24">24</cite>]. We consider a series of magnetic <inline-eqn><math-text>δ</math-text></inline-eqn>-function vector potentials <inline-eqn></inline-eqn> as shown in figure <figref linkend="nj318400fig11">11</figref>(a). First we consider a single such potential that is zero everywhere except at <inline-eqn><math-text><italic>x</italic>=0</math-text></inline-eqn>. We start with equations (<eqnref linkend="nj318400eqn6">6</eqnref>) and (<eqnref linkend="nj318400eqn7">7</eqnref>) which now become
<display-eqn id="nj318400eqn50" textype="equation" notation="LaTeX" eqnnum="50"></display-eqn><display-eqn id="nj318400eqn51" textype="equation" notation="LaTeX" eqnnum="51"></display-eqn>The solutions are readily obtained in the form
<display-eqn id="nj318400eqn52" textype="equation" notation="LaTeX" eqnnum="52"></display-eqn><display-eqn id="nj318400eqn53" textype="equation" notation="LaTeX" eqnnum="53"></display-eqn>Integrating equations (<eqnref linkend="nj318400eqn50">50</eqnref>) and (<eqnref linkend="nj318400eqn51">51</eqnref>) around <inline-eqn><math-text>0</math-text></inline-eqn> gives
<display-eqn id="nj318400eqn54" textype="equation" notation="LaTeX" eqnnum="54"></display-eqn><display-eqn id="nj318400eqn55" textype="equation" notation="LaTeX" eqnnum="55"></display-eqn>and
<display-eqn id="nj318400eqn56" textype="equation" notation="LaTeX" eqnnum="56"></display-eqn>We now consider the entire series of <inline-eqn><math-text>δ</math-text></inline-eqn>-function vector potentials shown in figure <figref linkend="nj318400fig13">13</figref>(a) and use equations (<eqnref linkend="nj318400eqn55">55</eqnref>) and the periodic boundary condition <inline-eqn><math-text>Ψ<sub><italic>I</italic></sub>(0)=<upright>e</upright><sup><upright>i</upright><italic>k</italic><sub><italic>x</italic></sub><italic>L</italic></sup>Ψ<sub><italic>II</italic></sub>(<italic>L</italic>)</math-text></inline-eqn>. The resulting dispersion relation for the SL is
<display-eqn id="nj318400eqn57" textype="equation" notation="LaTeX" eqnnum="57"></display-eqn>where <inline-eqn><math-text>η=1+ℓ<sub><upright>B</upright></sub><italic>A</italic><sub>0</sub></math-text></inline-eqn>. From equation (<eqnref linkend="nj318400eqn57">57</eqnref>) we find the energy spectrum as
<display-eqn id="nj318400eqn58" textype="equation" notation="LaTeX" eqnnum="58"></display-eqn>We can define
<display-eqn id="nj318400eqn59" textype="equation" notation="LaTeX" eqnnum="59"></display-eqn>and obtain
<display-eqn id="nj318400eqn60" textype="equation" notation="LaTeX" eqnnum="60"></display-eqn>The energy bands around the Dirac point are plotted in figure <figref linkend="nj318400fig12">12</figref>(b). Notice that: (i) there is an opening of a gap at the Dirac point, (ii) the motion is strongly 1D, i.e. along the <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn>-direction, and (iii) higher subbands have a smaller dispersion.</p><figure id="nj318400fig13" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="31.6pc" printcolour="no" filename="images/nj318400fig13.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj318400fig13.jpg"></graphic-file></graphic><caption type="figure" id="nj318400fc13" label="Figure 13"><p indent="no">(a) A series of <inline-eqn><math-text>δ</math-text></inline-eqn>-function vector potentials. (b) Dispersion relation for the system shown in (a).</p></caption></figure></sec-level1><sec-level1 id="nj318400s5" label="5"><heading>Concluding remarks</heading><p indent="no">We developed a <italic>magnetic</italic> KP model for Dirac electrons in graphene. The model is essentially a series of very high and very narrow <italic>magnetic</italic> <inline-eqn><math-text>δ</math-text></inline-eqn>-function barriers alternating in sign. The treatment of the transmission through such a series of barriers followed closely the one developed in optics for media in which the refractive index varies in space [<cite linkend="nj318400bib28">28</cite>, <cite linkend="nj318400bib29">29</cite>]. We contrasted a few of the results with those for standard electrons described by the Schrödinger equation [<cite linkend="nj318400bib27">27</cite>].</p><p>In several cases the energy spectrum or the dispersion relation was obtained analytically, cf equations (<eqnref linkend="nj318400eqn25">25</eqnref>), (<eqnref linkend="nj318400eqn39">39</eqnref>), (<eqnref linkend="nj318400eqn47">47</eqnref>), (<eqnref linkend="nj318400eqn48">48</eqnref>) and (<eqnref linkend="nj318400eqn57">57</eqnref>), largely due to the simplicity of the model and the adapted method from optics. For only two <italic>magnetic</italic> <inline-eqn><math-text>δ</math-text></inline-eqn>-function barriers, opposite in sign, we saw several bound states, whose number increases with <inline-eqn><math-text>|<italic>k</italic><sub><italic>y</italic></sub>|</math-text></inline-eqn>, and a reduction of the wavevector range for which tunneling is possible, cf figure <figref linkend="nj318400fig4">4</figref>(a). This is in line with that reported earlier for single [<cite linkend="nj318400bib7">7</cite>] and multiple [<cite linkend="nj318400bib9">9</cite>] barriers. The reduction becomes stronger as we increase the number of barriers, cf figure <figref linkend="nj318400">4</figref>(b). We also made contact with Snell's law in optics, cf equation (<eqnref linkend="nj318400eqn36">36</eqnref>): the term <inline-eqn><math-text><italic>P</italic>/<italic>E</italic></math-text></inline-eqn> represents the deviation from this law.</p><p>An important feature of the SL results is a collimation of an incident electron beam normal to the superlattice direction at least for large wavevectors. As easily seen from figures <figref linkend="nj318400">7</figref> and <figref linkend="nj318400fig8">8</figref>, for <inline-eqn><math-text>|<italic>k</italic><sub><italic>y</italic></sub>|⩾2</math-text></inline-eqn> we have <inline-eqn><math-text><italic>v</italic><sub><italic>x</italic></sub>∝∂<italic>E</italic>/∂<italic>k</italic><sub><italic>x</italic></sub>≈0</math-text></inline-eqn> for the first three minibands in the middle panels and nearly five minibands in the right panels. This occurs for both standard electrons and Dirac electrons, but notice the important difference for <inline-eqn><math-text>|<italic>k</italic><sub><italic>y</italic></sub>|≈0</math-text></inline-eqn> shown clearly in figure <figref linkend="nj318400fig9">9</figref>. This collimation is similar to that reported in [<cite linkend="nj318400bib16">16</cite>] for SLs involving only electric barriers but with somewhat unrealistic large barrier heights.</p><p>It is also worth emphasizing the differences and similarities in the first two minibands and the corresponding drift velocities as functions of <inline-eqn><math-text><italic>k</italic><sub><italic>y</italic></sub></math-text></inline-eqn> for different periods <inline-eqn><math-text><italic>L</italic></math-text></inline-eqn> and constant <inline-eqn><math-text><italic>k</italic><sub><italic>x</italic></sub></math-text></inline-eqn> as shown in figure <figref linkend="nj318400fig12">12</figref>. Notice, in particular, the resemblance between the drift velocities in the lowest miniband for standard electrons and the second miniband for Dirac electrons.</p><p>Given that ferromagnetic strips were successfully deposited on the top of a 2DEG in a semiconductor heterostructure [<cite linkend="nj318400bib23">23</cite>], we hope they will be deposited on graphene too and that the results of this paper will be tested in the near future.</p></sec-level1><acknowledgment><heading>Acknowledgments</heading><p indent="no">We thank Professor A Matulis for helpful discussions. 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Phys.</jnl-title><volume>69</volume><pages>137</pages><crossref><cr_doi>http://dx.doi.org/10.1119/1.1308266</cr_doi><cr_issn type="print">00029505</cr_issn></crossref></journal-ref></reference-list></references></back></article></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="eng"><title>Magnetic KronigPenney model for Dirac electrons in single-layer graphene</title></titleInfo><titleInfo type="abbreviated"><title>Magnetic KronigPenney model for Dirac electrons in single-layer graphene</title></titleInfo><titleInfo type="alternative" lang="eng"><title>Magnetic KronigPenney model for Dirac electrons in single-layer graphene</title></titleInfo><name type="personal"><namePart type="given">M</namePart><namePart type="family">Ramezani Masir</namePart><affiliation>Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium</affiliation><affiliation>E-mail: mrmphys@gmail.com</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">P</namePart><namePart type="family">Vasilopoulos</namePart><affiliation>Department of Physics, Concordia University, 7141 Sherbrooke Street West Montreal, Quebec, H4B 1R6, Canada</affiliation><affiliation>E-mail: takis@alcor.concordia.ca</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">F M</namePart><namePart type="family">Peeters</namePart><affiliation>Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, B-2020 Antwerpen, Belgium</affiliation><affiliation>Departamento de Fsica, Universidade Federal do Cear, Caixa Postal 6030, Campus do Pici, 60455-760 Fortaleza, Cear, Brazil</affiliation><affiliation>Author to whom any correspondence should be addressed.</affiliation><affiliation>E-mail: francois.peeters@ua.ac.be</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="paper">paper</genre><originInfo><publisher>IOP Publishing</publisher><dateIssued encoding="w3cdtf">2009</dateIssued><copyrightDate encoding="w3cdtf">2009</copyrightDate></originInfo><language><languageTerm type="code" authority="iso639-2b">eng</languageTerm><languageTerm type="code" authority="rfc3066">en</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType></physicalDescription><abstract>The properties of Dirac electrons in a magnetic superlattice (SL) on graphene consisting of very high and thin (-function) barriers are investigated. We obtain the energy spectrum analytically and study the transmission through a finite number of barriers. The results are contrasted with those for electrons described by the Schrdinger equation. In addition, a collimation of an incident beam of electrons is obtained along the direction perpendicular to that of the SL. We also highlight an analogy with optical media in which the refractive index varies in space.</abstract><relatedItem type="host"><titleInfo><title>New Journal of Physics</title></titleInfo><titleInfo type="abbreviated"><title>New J. Phys.</title></titleInfo><genre type="journal">journal</genre><identifier type="ISSN">1367-2630</identifier><identifier type="eISSN">1367-2630</identifier><identifier type="PublisherID">nj</identifier><identifier type="CODEN">NJOPFM</identifier><identifier type="URL">stacks.iop.org/NJP</identifier><part><date>2009</date><detail type="volume"><caption>vol.</caption><number>11</number></detail><detail type="issue"><caption>no.</caption><number>9</number></detail><extent unit="pages"><start>1</start><end>21</end><total>21</total></extent></part></relatedItem><identifier type="istex">A31C27D6643EBB9C403400C3EFE0356AA8E204AE</identifier><identifier type="DOI">10.1088/1367-2630/11/9/095009</identifier><identifier type="articleID">318400</identifier><identifier type="articleNumber">095009</identifier><accessCondition type="use and reproduction" contentType="copyright">IOP Publishing and Deutsche Physikalische Gesellschaft</accessCondition><recordInfo><recordContentSource>IOP</recordContentSource><recordOrigin>IOP Publishing and Deutsche Physikalische Gesellschaft</recordOrigin></recordInfo></mods></metadata><enrichments><json:item><type>multicat</type><uri>https://api.istex.fr/document/A31C27D6643EBB9C403400C3EFE0356AA8E204AE/enrichments/multicat</uri></json:item></enrichments><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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