<?xml version="1.0" encoding="ISO-8859-1"?><article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<front>
<journal-meta>
<journal-id>1409-2433</journal-id>
<journal-title><![CDATA[Revista de Matemática Teoría y Aplicaciones]]></journal-title>
<abbrev-journal-title><![CDATA[Rev. Mat]]></abbrev-journal-title>
<issn>1409-2433</issn>
<publisher>
<publisher-name><![CDATA[Centro de Investigaciones en Matemática Pura y Aplicada (CIMPA) y Escuela de Matemática, San José, Costa Rica.]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S1409-24332011000200003</article-id>
<title-group>
<article-title xml:lang="en"><![CDATA[Parallelization of a quantum-classic hybrid model for nanoscale semiconductor devices]]></article-title>
<article-title xml:lang="es"><![CDATA[Paralelización del modelo híbrido clásico-cuántico para un dispositivo semiconductor Mosfet nanométrico]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Salas]]></surname>
<given-names><![CDATA[Oscar]]></given-names>
</name>
<xref ref-type="aff" rid="A01"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Lanucara]]></surname>
<given-names><![CDATA[Piero]]></given-names>
</name>
<xref ref-type="aff" rid="A02"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Pietra]]></surname>
<given-names><![CDATA[Paola]]></given-names>
</name>
<xref ref-type="aff" rid="A03"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Rovida]]></surname>
<given-names><![CDATA[Sergio]]></given-names>
</name>
<xref ref-type="aff" rid="A04"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Sacchi]]></surname>
<given-names><![CDATA[Giovanni]]></given-names>
</name>
<xref ref-type="aff" rid="A05"/>
</contrib>
</contrib-group>
<aff id="A01">
<institution><![CDATA[,Universidad Nacional Department of Mathematics ]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
</aff>
<aff id="A02">
<institution><![CDATA[,Universities and Research (CASPUR)  ]]></institution>
<addr-line><![CDATA[ Roma]]></addr-line>
<country>Italy</country>
</aff>
<aff id="A03">
<institution><![CDATA[,Institute of Applied Mathematics and Information Technologies  ]]></institution>
<addr-line><![CDATA[Pavia ]]></addr-line>
<country>Italy</country>
</aff>
<aff id="A04">
<institution><![CDATA[,Institute of Applied Mathematics and Information Technologies  ]]></institution>
<addr-line><![CDATA[Pavia ]]></addr-line>
<country>Italy</country>
</aff>
<aff id="A05">
<institution><![CDATA[,Institute of Applied Mathematics and Information Technologies  ]]></institution>
<addr-line><![CDATA[Pavia ]]></addr-line>
<country>Italy</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>12</month>
<year>2011</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>12</month>
<year>2011</year>
</pub-date>
<volume>18</volume>
<numero>2</numero>
<fpage>231</fpage>
<lpage>248</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.sa.cr/scielo.php?script=sci_arttext&amp;pid=S1409-24332011000200003&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.sa.cr/scielo.php?script=sci_abstract&amp;pid=S1409-24332011000200003&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.sa.cr/scielo.php?script=sci_pdf&amp;pid=S1409-24332011000200003&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="en"><p><![CDATA[The expensive reengineering of the sequential software and the difficult parallel programming are two of the many technical and economic obstacles to the wide use of HPC. We investigate the chance to improve in a rapid way the performance of a numerical serial code for the simulation of the transport of a charged carriers in a Double-Gate MOSFET. We introduce the Drift-Diffusion-Schrödinger-Poisson (DDSP) model and we study a rapid parallelization strategy of the numerical procedure on shared memory architectures.]]></p></abstract>
<abstract abstract-type="short" xml:lang="es"><p><![CDATA[El transformar un software secuencial en uno paralelo, es costoso y difícil, lo cual constituye solo dos de los muchos obstáculos técnicos y económicos que se tienen que enfrentar cuando se desea hacer uso de sistemas HPC. En este trabajo investigamos la posibilidad de mejorar de forma rápida y eficiente el desempeño de un código numérico secuencial que se encarga de realizar la simulación del comportamiento y transporte de un flujo de electrones en un dispositivo semiconductor MOSFET doble puerta y de escala nanométrico. Se introduce el modelo Drift-Diffusion-Schrödinger-Poisson (DDSP) y se estudia una estrategia de paralelización rápida del procedimiento numérico, óptimo específicamente para arquitecturas a memoria compartida.]]></p></abstract>
<kwd-group>
<kwd lng="en"><![CDATA[Parallelization]]></kwd>
<kwd lng="en"><![CDATA[Shared memory paradigm]]></kwd>
<kwd lng="en"><![CDATA[Schrödinger equation]]></kwd>
<kwd lng="en"><![CDATA[Drift-Difusion system]]></kwd>
<kwd lng="en"><![CDATA[Subband model]]></kwd>
<kwd lng="en"><![CDATA[Nanotransistor]]></kwd>
<kwd lng="es"><![CDATA[Paralelezación]]></kwd>
<kwd lng="es"><![CDATA[Paradigma de Memoria Compartida]]></kwd>
<kwd lng="es"><![CDATA[Ecuación de Schrödinger]]></kwd>
<kwd lng="es"><![CDATA[Sistemas Drift-Difusion]]></kwd>
<kwd lng="es"><![CDATA[Modelo Subband]]></kwd>
<kwd lng="es"><![CDATA[Nanotubos]]></kwd>
</kwd-group>
</article-meta>
</front><body><![CDATA[ <div style="text-align: center;"><font size="4"> <span style="font-family: verdana; font-weight: bold;">Parallelization of a quantum-classic hybrid model for nanoscale semiconductor devices</span></font>    <br>     <br> <font size="4"><span style="font-family: verdana; font-weight: bold;">Paralelizaci&oacute;n del modelo h&iacute;brido cl&aacute;sico-cu&aacute;ntico para un dispositivo semiconductor Mosfet nanom&eacute;trico</span></font>    <br> </div> <font size="2">    <br> </font>     <div style="text-align: justify;"><font size="2"><span  style="font-family: verdana;">Oscar Salas<a href="#autor1"><sup><a>*</a> </sup></a></span></font>    <br> <font size="2"><span style="font-family: verdana;">Piero Lanucara<a  href="#autor2"><sup>&#8224;</sup></a></span></font>    <br> <font size="2"><span style="font-family: verdana;">Paola Pietra<a  href="#autor3"><sup>&#8225;</sup></a> </span></font>    <br> <font size="2"><span style="font-family: verdana;">Sergio Rovida<a  href="#autor4"><sup>&sect;</sup></a> </span></font>    <br> <font size="2"><span style="font-family: verdana;">Giovanni Sacchi<a  href="#autor5"><sup>&para;</sup></a></span></font>    ]]></body>
<body><![CDATA[<br> </div> <font size="2">    <br> <span style="font-family: verdana;"><a name="autor1"></a>*Department of Mathematics, Universidad Nacional, Heredia, Costa Rica. E-Mail: <a href="mailto:oscar.salas@unipv.it">oscar.salas@unipv.it</a></span>    <br> <span style="font-family: verdana;"><a name="autor2"></a>&#8224;Consortium for the Applications of Super-Computing for Universities and Research (CASPUR), c/o Universit&agrave; La Sapienza, P. Also Moro 2, 00185 Roma, Italy. E-Mail: <a href="mailto:lanucara@caspur.it">lanucara@caspur.it</a></span>    <br> <span style="font-family: verdana;"><a name="autor3"></a>&#8225;Institute of Applied Mathematics and Information Technologies, C.N.R., via Ferrata 1, 27100 Pavia, Italy. E-Mail: <a  href="mailto:pietra@imati.cnr.it">pietra@imati.cnr.it</a></span>    <br> <span style="font-family: verdana;"><a name="autor4"></a>&sect;Institute of Applied Mathematics and Information Technologies &#8212; CNR, Pavia, Italy. EMail: <a href="mailto:rovida@imati.cnr.it">rovida@imati.cnr.it</a></span>    <br> <span style="font-family: verdana;"><a name="autor5"></a>&para;Institute of Applied Mathematics and Information Technologies &#8212; CNR, Pavia, Italy. EMail: <a href="mailto:gianni.sacchi@imati.cnr.it">gianni.sacchi@imati.cnr.it</a>    <br>     <br> <a href="#correspondencia">Direcci&oacute;n para correspondencia</a>    <br> </span>    <br> </font>     ]]></body>
<body><![CDATA[<div style="text-align: justify;"><font size="3"><span  style="font-family: verdana; font-weight: bold;"></span></font> <hr style="width: 100%; height: 2px;"><font size="3"><span  style="font-family: verdana; font-weight: bold;">Abstract</span></font>    <br>     <br> <font size="2"><span style="font-family: verdana;">The expensive reengineering of the sequential software and the difficult parallel programming are two of the many technical and economic obstacles to the wide use of HPC. We investigate the chance to improve in a rapid way the performance of a numerical serial code for the simulation of the transport of a charged carriers in a Double-Gate MOSFET. We introduce the Drift-Diffusion-Schr&ouml;dinger-Poisson (DDSP) model and we study a rapid parallelization strategy of the numerical procedure on shared memory architectures.</span></font>    <br>     <br> <font size="2"><span style="font-family: verdana;"><span  style="font-weight: bold;">Keywords:</span> Parallelization; Shared memory paradigm; Schr&ouml;dinger equation; Drift-Difusion system; Subband model; Nanotransistor.</span></font>    <br>     <br> <font size="3"><span style="font-family: verdana; font-weight: bold;">Resumen</span></font>    <br>     <br> <font size="2"><span style="font-family: verdana;">El transformar un software secuencial en uno paralelo, es costoso y dif&iacute;cil, lo cual constituye solo dos de los muchos obst&aacute;culos t&eacute;cnicos y econ&oacute;micos que se tienen que enfrentar cuando se desea hacer uso de sistemas HPC. En este trabajo investigamos la posibilidad de mejorar de forma r&aacute;pida y eficiente el desempe&ntilde;o de un c&oacute;digo num&eacute;rico secuencial que se encarga de realizar la simulaci&oacute;n del comportamiento y transporte de un flujo de electrones en un dispositivo semiconductor MOSFET doble puerta y de escala nanom&eacute;trico. Se introduce el modelo Drift-Diffusion-Schr&ouml;dinger-Poisson (DDSP) y se estudia una estrategia de paralelizaci&oacute;n r&aacute;pida del procedimiento num&eacute;rico, &oacute;ptimo espec&iacute;ficamente para arquitecturas a memoria compartida.</span></font>    <br>     ]]></body>
<body><![CDATA[<br> <font size="2"><span style="font-family: verdana;"><span  style="font-weight: bold;">Palabras clave:</span> Paralelezaci&oacute;n; Paradigma de Memoria Compartida; Ecuaci&oacute;n de Schr&ouml;dinger; Sistemas Drift-Difusion; Modelo Subband; Nanotubos.</span></font><br style="font-family: verdana;">     <br> <font size="2"><span style="font-family: verdana;"><span  style="font-weight: bold;">Mathematics Subject Classification:</span> 65N55.    <br>     <br> </span></font> <hr style="width: 100%; height: 2px;">    <br> <font size="2"><span style="font-family: verdana;">Ver contenido disponible en pdf</span></font><br style="font-family: verdana;"> <font size="2"><span style="font-family: verdana;"></span></font>    <br> <font size="3"><span style="font-family: verdana; font-weight: bold;"></span></font> <hr style="width: 100%; height: 2px;"><font size="3"><span  style="font-family: verdana; font-weight: bold;">References</span></font>    <br>     <!-- ref --><br> <font size="2"><span style="font-family: verdana;">[1] Ben Abdallah, N.; Mhats, F. (2004) &#8220;On a Vlasov&#8211;Schr&ouml;dinger&#8211;Poisson model&#8221;, Comm.Partial Differential Equations 29(1-2): 173&#8211;206.    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=1946732&pid=S1409-2433201100020000300001&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --></span></font><br  style="font-family: verdana;">     <!-- ref --><br> <font size="2"><span style="font-family: verdana;">[2] OpenMP Architecture Review Board. OpenMP Application Program Interface. Version 2.5, May 2005.    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=1946734&pid=S1409-2433201100020000300002&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --></span></font>    <br>     <br> <font size="2"><span style="font-family: verdana;">[3] Brezzi, F.; Marini, L.D.; Pietra, P. (1987) &#8220;M&eacute;thodes d&#8217;&eacute;l&eacute;ments finis mixtes et sch&eacute;ma de Scharfetter&#8211;Gummel&#8221;, C.R. Acad. Sci. Paris S&eacute;r. I 305: 599&#8211;604.</span></font>    <!-- ref --><br>     &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=1946738&pid=S1409-2433201100020000300003&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><br> <font size="2"><span style="font-family: verdana;">[4] Davies, J.H. (1998) The Physics of Low Dimensional Semiconductors. Cambridge University Press.    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=1946739&pid=S1409-2433201100020000300004&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --></span></font>    <br>     <!-- ref --><br> <font size="2"><span style="font-family: verdana;">[5] Degond, P.; Levermore, C.D.; Schmeiser, C. (2004) &#8220;A note on the Energy-Transport limit of the semiconductor Boltzmann equation, Transport in transition regimes&#8221;, IMA Vol. Math. Appl., Springer, 135: 137&#8211;153.    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=1946742&pid=S1409-2433201100020000300005&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --></span></font>    <br>     ]]></body>
<body><![CDATA[<!-- ref --><br> <font size="2"><span style="font-family: verdana;">[6] Golse, F.; Poupaud, F. (1992) &#8220;Limite fluide des &eacute;quations de Boltzmann des semiconducteurs pour une statistique de Fermi&#8211;Dirac&#8221;, Asymptotic Analysis 6: 135&#8211;169.    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=1946745&pid=S1409-2433201100020000300006&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --></span></font>    <br>     <!-- ref --><br> <font size="2"><span style="font-family: verdana;">[7] Greenbaum, A. (1986) &#8220;Routines for Solving Large Sparse Linear Systems (SLAP)&#8221;. Lawrence Livermore Nat. Laboratory, Livermore Computing Center, USA.    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=1946748&pid=S1409-2433201100020000300007&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --></span></font>    <br>     <!-- ref --><br> <font size="2"><span style="font-family: verdana;">[8] Gummel, H.K. (1964) &#8220;A self-consistent iterative scheme for onedimensional steady state transistor calculations", IEEE Trans. on Elec Dev. 11(10): 455&#8211;465.    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=1946751&pid=S1409-2433201100020000300008&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --></span></font>    <!-- ref --><br> <br style="font-family: verdana;"> <font size="2"><span style="font-family: verdana;">[9] Nier, F. (1990) &#8220;A stationary Schr&ouml;dinger&#8211;Poisson system arising from the modelling of electronic devices", Forum Math. 2(5):489&#8211;510.    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=1946753&pid=S1409-2433201100020000300009&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --> </span></font><br style="font-family: verdana;"> <br style="font-family: verdana;"> <font size="2"><span style="font-family: verdana;">[10] Pietra, P.; Vauchelet N. (2007) &#8220;Modeling and simulations of the diffusive transport in a nanoscale double-gate MOSFET&#8221;, Technical Report 10PV07/10/0, IMATI-CNR, Pavia.    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=1946754&pid=S1409-2433201100020000300010&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --></span></font>    <br>     <!-- ref --><br> <font size="2"><span style="font-family: verdana;">[11] Poupaud, F. (1981) &#8220;Diffusion approximation of the linear semiconductor Boltzmann equation: analysis of boundary layers&#8221;, Asymptotic Analysis 4: 293&#8211;317.    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=1946757&pid=S1409-2433201100020000300011&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --></span></font>    <br>     <!-- ref --><br> <font size="2"><span style="font-family: verdana;">[12] Vauchelet, N. (2006) Mod&eacute;lisation Math&eacute;matique du Transport Diffusif de Charges Partiellement Quantiques. Ph.D. thesis, Universit&eacute; Paul Sabatier, Toulouse.    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=1946760&pid=S1409-2433201100020000300012&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --></span></font>    <br>     <!-- ref --><br> <font size="2"><span style="font-family: verdana;">[13] Vinter, B.; Weisbuch, C. (1991) Quantum Semiconductor Structures. Academic Press.    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=1946763&pid=S1409-2433201100020000300013&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><br>     ]]></body>
<body><![CDATA[<br>     <br> <a name="correspondencia"></a>Correspondencia a: </span></font><font  size="2"><span style="font-family: verdana;">Oscar Salas. </span></font><font  size="2"><span style="font-family: verdana;">Department of Mathematics, Universidad Nacional, Heredia, Costa Rica. E-Mail: <a href="mailto:oscar.salas@unipv.it">oscar.salas@unipv.it</a>    <br> </span></font><font size="2"><span style="font-family: verdana;">Piero Lanucara. </span></font><font size="2"><span  style="font-family: verdana;">Consortium for the Applications of Super-Computing for Universities and Research (CASPUR), c/o Universit&agrave; La Sapienza, P. Also Moro 2, 00185 Roma, Italy. E-Mail: <a href="mailto:lanucara@caspur.it">lanucara@caspur.it</a>    <br> </span></font><font size="2"><span style="font-family: verdana;">Paola Pietra. </span></font><font size="2"><span  style="font-family: verdana;">Institute of Applied Mathematics and Information Technologies, C.N.R., via Ferrata 1, 27100 Pavia, Italy. E-Mail: <a  href="mailto:pietra@imati.cnr.it">pietra@imati.cnr.it</a>    <br> </span></font><font size="2"><span style="font-family: verdana;">Sergio Rovida. </span></font><font size="2"><span  style="font-family: verdana;">Institute of Applied Mathematics and Information Technologies &#8212; CNR, Pavia, Italy. EMail: <a href="mailto:rovida@imati.cnr.it">rovida@imati.cnr.it</a>    <br> </span></font><font size="2"><span style="font-family: verdana;">Giovanni Sacchi. </span></font><font size="2"><span  style="font-family: verdana;">Institute of Applied Mathematics and Information Technologies &#8212; CNR, Pavia, Italy. EMail: <a href="mailto:gianni.sacchi@imati.cnr.it">gianni.sacchi@imati.cnr.it</a>    <br> </span></font> </div> <font size="2">    <br> </font>     <div style="text-align: center;"><font size="2"><span  style="font-family: verdana;"></span></font> <hr style="width: 100%; height: 2px;"><font size="2"><span  style="font-family: verdana;">Received: 23 Feb 2010; Revised: 24 Nov 2010; Accepted: 10 Dec 2010</span></font>    <br> </div> <font size="2">    ]]></body>
<body><![CDATA[<br> </font>     <br>      ]]></body><back>
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