Optimización de dispositivos híbridos 2D-piezoeléctricos mediante simulaciones numéricas

/* vim: set shiftwidth=2 tabstop=2 autoindent cindent expandtab filetype=css: */ /*! * Base CSS for pdf2htmlEX * Copyright 2012,2013 Lu Wang * https://github.com/pdf2htmlEX/pdf2htmlEX/blob/master/share/LICENSE */ /* Part 1: Web Page Layout: Free to modify, except for a few of them which are required by pdf2htmlEX.js, see the comments */ #sidebar { /* Sidebar */ position:absolute; top:0; left:0; bottom:0; width:250px; padding:0; margin:0px; overflow:auto; } #page-container { /* PDF container */ position:absolute; /* required for calculating relative positions of pages in pdf2htmlEX.js */ top:0; left:0px; margin:0; padding:0; border:0; /* required for lazy page loading in pdf2htmlEX.js (page visibility test) */ } @media screen { /* for sidebar */ #sidebar.opened + #page-container { left:250px; } #page-container { /* `bottom' and `right' are required for lazy page loading in pdf2htmlEX.js (page visibility test) * alternatively you may set width and height */ bottom:0; right:0; overflow:auto; } .loading-indicator_c00000 { display:none; } .loading-indicator_c00000.active { display:block; position:absolute; width:64px; height:64px; top:50%; left:50%; margin-top:-32px; margin-left:-32px; } .loading-indicator_c00000 img { position:absolute; top:0; left:0; bottom:0; right:0; } } @media print { @page { margin:0; } html { margin:0; } body { margin:0; -webkit-print-color-adjust:exact; /* enable printing background images for WebKit */ } #sidebar { display:none; } #page-container { width:auto; height:auto; overflow:visible; background-color:transparent; } .d_c00000 { display:none; } } /* Part 2: Page Elements: Modify with caution * The followings are base classes, some of which are meant to be override by PDF specific classes * So do not increase the specificity (e.g. ".classname_c00000" -> "#page-container .classname_c00000") */ .pf_c00000 { /* page */ position:relative; background-color:white; overflow: hidden; margin:0; border:0; /* required by pdf2htmlEX.js for page visibility test */ } .pc_c00000 { /* content of a page */ position:absolute; border:0; padding:0; margin:0; top:0; left:0; width:100%; height:100%; overflow:hidden; display:block; /* set transform-origin for scaling */ transform-origin:0% 0%; -ms-transform-origin:0% 0%; -webkit-transform-origin:0% 0%; } .pc_c00000.opened { /* used by pdf2htmlEX.js, to show/hide pages */ display:block; } .bf_c00000 { /* images that occupies the whole page */ position:absolute; border:0; margin:0; top:0; bottom:0; width:100%; height:100%; -ms-user-select:none; -moz-user-select:none; -webkit-user-select:none; user-select:none; } .bi_c00000 { /* images that cover only a part of the page */ position:absolute; border:0; margin:0; -ms-user-select:none; -moz-user-select:none; -webkit-user-select:none; user-select:none; } @media print { .pf_c00000 { margin:0; box-shadow:none; page-break-after:always; page-break-inside:avoid; } @-moz-document url-prefix() { /* fix page truncation for FireFox */ .pf_c00000 { overflow:visible; border:1px solid #FFFFFF; } .pc_c00000 {overflow:visible;} } } .c_c00000 { /* clip box */ position:absolute; border:0; padding:0; margin:0; overflow:hidden; display:block; } .t_c00000 { /* text line */ position:absolute; white-space:pre; font-size:1px; transform-origin:0% 100%; -ms-transform-origin:0% 100%; -webkit-transform-origin:0% 100%; unicode-bidi:bidi-override;/* For rtl languages, e.g. Hebrew, we don't want the default Unicode behaviour */ -moz-font-feature-settings:"liga" 0;/* We don't want Firefox to recognize ligatures */ } .t_c00000:after { /* webkit #35443 */ content: ''; } .t_c00000:before { /* Workaround Blink(up to 41)/Webkit bug of word-spacing with leading spaces (chromium #404444 and pdf2htmlEX #412) */ content: ''; display: inline-block; } .t_c00000 span { /* text blocks within a line */ /* Blink(up to 41)/Webkit have bug with negative word-spacing and inline-block (pdf2htmlEX #416), so keep normal span inline. */ position:relative; unicode-bidi:bidi-override; /* For rtl languages, e.g. Hebrew, we don't want the default Unicode behaviour */ } .__c00000 { /* text shift */ /* Blink(up to 41)/Webkit have bug with inline element, continuous spaces and word-spacing. Workaround by inline-block. */ display: inline-block; color: transparent; z-index: -1; } /* selection background should not be opaque, for fallback mode */ ::selection{ background: rgba(127,255,255,0.4); } ::-moz-selection{ background: rgba(127,255,255,0.4); } .pi_c00000 { /* info for Javascript */ display:none; } .l_c00000 { /* annotation links */ } /* transparent color - WebKit */ .d_c00000 { /* css drawing */ position:absolute; transform-origin:0% 100%; -ms-transform-origin:0% 100%; -webkit-transform-origin:0% 100%; } /* for the forms */ .it_c00000 { border: none; background-color: rgba(255, 255, 255, 0.0); } .ir_c00000:hover { cursor: pointer; } /* Base CSS END */ /* vim: set shiftwidth=2 tabstop=2 autoindent cindent expandtab filetype=css: */ /*! * Fancy styles for pdf2htmlEX * Copyright 2012,2013 Lu Wang * https://github.com/pdf2htmlEX/pdf2htmlEX/blob/master/share/LICENSE */ @keyframes fadein { from { opacity:0;} to { opacity:1;} } @-webkit-keyframes fadein { from { opacity:0;} to { opacity:1;} } @keyframes swing { 0% { transform: rotate(0deg); } 10% { transform: rotate(0deg); } 90% { transform: rotate(720deg); } 100%{ transform: rotate(720deg); } } @-webkit-keyframes swing { 0% { -webkit-transform: rotate(0deg); } 10% { -webkit-transform: rotate(0deg); } 90% { -webkit-transform: rotate(720deg); } 100%{ -webkit-transform: rotate(720deg); } } @media screen { #sidebar { background-color:#2f3236; /* modified from http://philbit.com/svgpatterns/#crossstripes */ background-image:url("data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0IiBoZWlnaHQ9IjQiPgo8cmVjdCB3aWR0aD0iNCIgaGVpZ2h0PSI0IiBmaWxsPSIjNDAzYzNmIj48L3JlY3Q+CjxwYXRoIGQ9Ik0wIDBMNCA0Wk00IDBMMCA0WiIgc3Ryb2tlLXdpZHRoPSIxIiBzdHJva2U9IiMxZTI5MmQiPjwvcGF0aD4KPC9zdmc+"); } #outline { font-family:Georgia,Times,"Times New Roman",serif; font-size:13px; margin:2em 1em; } #outline ul { padding:0; } #outline li { list-style-type:none; margin:1em 0; } #outline li > ul { margin-left: 1em; } #outline a, #outline a:visited, #outline a:hover, #outline a:active { line-height:1.2; color:#e8e8e8; text-overflow:ellipsis; white-space:nowrap; text-decoration:none; display:block; overflow:hidden; outline:0; } #outline a:hover { color:rgb(0,204,255); } #page-container { background-color:#9e9e9e; /* http://philbit.com/svgpatterns/#thinstripes */ background-image: url("data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1IiBoZWlnaHQ9IjUiPgo8cmVjdCB3aWR0aD0iNSIgaGVpZ2h0PSI1IiBmaWxsPSIjOWU5ZTllIj48L3JlY3Q+CjxwYXRoIGQ9Ik0wIDVMNSAwWk02IDRMNCA2Wk0tMSAxTDEgLTFaIiBzdHJva2U9IiM4ODgiIHN0cm9rZS13aWR0aD0iMSI+PC9wYXRoPgo8L3N2Zz4="); -webkit-transition:left 500ms; transition:left 500ms; } .pf_c00000 { margin: 13px auto; box-shadow: 1px 1px 3px 1px #333; /* Needed by IE to make box-shadow works * https://developer.mozilla.org/en-US/docs/Web/CSS/box-shadow */ border-collapse: separate; } .pc_c00000.opened { /* used by pdf2htmlEX.js, to show/hide pages */ -webkit-animation: fadein 100ms; animation: fadein 100ms; } .loading-indicator_c00000.active { /* * use 0.01s instead of 0s, * since YUI Compressor will change 0s to 0, * which is not recognized by Firefox */ -webkit-animation: swing 1.5s ease-in-out 0.01s infinite alternate none; animation: swing 1.5s ease-in-out 0.01s infinite alternate none; } .checked_c00000 { background: no-repeat url(data:image/png;base64,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); } } /* Fancy CSS END */ .ff0_c00000{font-family:sans-serif;visibility:hidden;} 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te sim ulaciones n um ´ ericas T raba jo Fin de Grado Autor: Iv´ an P alomo Lla v ona T utores: Ja vier Mart ´ ın S´ anc hez y V ´ ıctor Man uel Garc ´ ıa Su´ arez Curso 2021-2022 ´ INDICE 1 ´ Indice 1. In tro ducci´ on 2 2. Consideraciones te´ oricas 3 2.1. T eor ´ ıa del con tin uo el´ astico . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 . 1 . 1 . E l t e n s o r d e C a u c h y ................................. 3 2.1.2. Camp os de deformaci´ on en el plano . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.3. La relaci´ on en tre el stress y el strain . . . . . . . . . . . . . . . . . . . . . . . . 5 2 . 1 . 4 . T e n s o r d e e l a s t i c i d a d ................................ 5 2.2. Propiedades de emisi´ on ´ optica y estructura de bandas del W S e 2 ............ 6 2.2.1. Estructura cristalina del Wse2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.2. Propiedades ´ opticas de materiales 2D y aplicaciones . . . . . . . . . . . . . . 9 2.3. C´ alculos de estructura electr´ onica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.1. La ecuaci´ on de Sc hr¨ odinger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.2. T eor ´ ıa del funcional de la densidad . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.3. Tip os de programas para resolv er la ecuaci´ on de Sc hr¨ odinger . . . . . . . . . . 12 2.4. F undamen tos sobre piezo electricidad . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3. Sim ulaciones median te el soft w are comercial COMSOL Mult yph ysics 16 3.1. Disp ositiv os h ´ ıbridos en un material piezo el ´ ectrico 2D . . . . . . . . . . . . . . . . . . 18 3.1.1. Principio de funcionamien to . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1.2. Caso del actuador piezo el ´ ectrico monol ´ ıtico . . . . . . . . . . . . . . . . . . . . 20 3.1.3. Caso del actuador piezo el ´ ectrico basado en 2 ( ( patillas ) ) microm ´ etricas . . . . . 21 3.1.4. Caso del actuador piezo el ´ ectrico basado en 6 ( ( patillas ) ) microm ´ etricas . . . . . 22 4. Resultados 24 4.1. Cam bios en la direcci´ on y magnitud en el camp o de deformaciones in tro ducidos en la mono capa de W S e 2 ..................................... 2 4 4.1.1. Caso del actuador piezo el ´ ectrico monol ´ ıtico . . . . . . . . . . . . . . . . . . . . 24 4.1.2. Caso del actuador piezo el ´ ectrico basado en 2 ( ( patillas ) ) microm ´ etricas . . . . . 25 4.1.3. Caso del actuador piezo el ´ ectrico basado en 6 ( ( patillas ) ) microm ´ etricas . . . . . 27 4.2. Efectos de los camp os de deformaci´ on en la banda de energ ´ ıas prohibidas . . . . . . . 32 4.2.1. Caso del actuador piezo el ´ ectrico monol ´ ıtico . . . . . . . . . . . . . . . . . . . . 33 4.2.2. Caso del actuador piezo el ´ ectrico basado en 2 ( ( patillas ) ) microm ´ etricas . . . . . 34 4.2.3. Caso del actuador piezo el ´ ectrico basado en 6 ( ( patillas ) ) microm ´ etricas . . . . . 36 4.3. Comparaci´ on en tre los 3 mo delos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5. Conclusiones 39 6. Bibliograf ´ ıa 40 1 INTR ODUCCI ´ ON 2 1. In tro ducci´ on El desarrollo de n uev as tecnolog ´ ıas ha venido acompa˜ nado del descubrimiento de n uev os materiales con propiedades f ´ ısicas in teresan tes. La qu ´ ımica fue un gran a v ance en este sen tido, y a que esta nos p ermite v ariar la estructura in terna de los materiales v ariando su comp osici´ on, median te las reacciones qu ´ ımicas, Las propiedades de los materiales dep enden en ´ ultima instancia de su estructura in terna es decir la colo caci´ on de los ´ atomos del material, p or ejemplo el caso del carb ono. Si esta en la forma de grafito es un material blando y f´ acilmen te laminable, sin em bargo si tiene la estructura in terna del diaman te pasa a ser uno de los materiales mas duros que existen. ´ Recien temen te, la in v estigaci´ on de los llamados materiales bidimensionales (2D) ha supuesto un gran a v ance para su p otencial aplicaci´ on en n uev as tecnolog ´ ıas flexibles y ultracompactas. Estos ma- teriales tienen unas dimensiones laterales de v arias de decenas de micras, mien tras que su esp esor se man tiene en el rango de unos p o cos nanom ´ etros. En la pr´ actica, estos se pueden obtener delaminando cristales masiv os en los que las mono capas est´ an unidas median te fuerzas de V an der W aals en tre ellas. En general, estos materiales son f´ acilmen te deformables el´ asticamen te hasta v alores tan altos como un 20 %. Dado que, en ´ ultima instancia, las propiedades f ´ ısicas de los materiales cristalinos vienen condicio- nadas p or la distancia in terat´ omica de sus ´ atomos constituy en tes, v ariando dic ha distancia median te la in tro ducci´ on de camp os de deformaci´ on el´ asticas, se p o dr ´ ıa v ariar de manera rev ersible al mismo tiemp o sus propiedades f ´ ısicas. En este traba jo se realiza un estudio del sintonizado del band-gap de mono capas de WSe2 (un material 2D semiconductor) median te ingenier ´ ıa de camp os de deformaci´ on el´ asticos. P ara ello, se v alorar´ an div ersos dise ˜ nos de disp ositiv os h ´ ıbridos piezo el ´ ectrico-material 2D con el ob jetiv o de in tro ducir camp os de deformaci´ on con anisotrop ´ ıa y magnitud con troladas. El ( ( band gap ) ) , o gap para resumir, es la distancia en energ ´ ıas en tre el m´ aximo de la banda de v alencia y el m ´ ınimo de la banda de conducci´ on de un material, es una propiedad imp ortante en materiales, y a que nos determina el comp ortamien to aislan te o semiconductor del material y propie- dades sobre la emisi´ on de fotones. El material que mo dificaremos es el W S e 2 en mono capas, y a que presen tan un gap directo lo cual tiene v en ta jas a la hora de la emisi´ on de ´ optica como una may or eficiencia e in tensidad de emisi´ on. 2 CONSIDERA CIONES TE ´ ORICAS 3 2. Consideraciones te´ oricas 2.1. T eor ´ ıa del con tin uo el´ astico Se traba jar´ a con un ob jeto s´ olido, estos se pueden tratar como con tin uos y los efectos de fuerzas externas se pueden tratar median te teor ´ ıa del strain es decir el desplazamien to de la p osici´ on de equilibrio de los elemen tos del material, el comp ortamien to el´ astico del s´ olido est´ a descrito en el tensor de Cauc h y . 2.1.1. El tensor de Cauc h y Cuando se deforma un s´ olido el´ asticamen te, este en consecuencia resp onde con fuerzas para v olv er al estado inicial, si el s´ olido esta en equilibrio el sumatorio de to das las fuerzas o torques sobre el material deb e ser 0. Considerando un elemen to cubico de v olumen, definimos ∆ a i como una de las caras del cub o. Si ejercemos una fuerza  F j sobre esta cara, el material estar´ a sometido a una presi´ on o stress  σ :  σ ij =  F j ∆ a i i, j = x, y , z (1) De manera que en forma de matriz se describ e σ , el tensor de Cauc h y o stress:   σ xx σ xy σ xz σ y x σ y y σ y z σ z x σ z y σ z z   (2) El stress en un elemen to particular de sup erficie de v ector normal  n se obtiene sumando sobre to dos los elemen tos. σ i n = X j σ ij ·  n (3) Este tensor de stress se puede descomp oner en dos comp onen tes, las normales (o directas) σ ij donde i = j y las de cizalladura τ ij , en las que i  = j . Las comp onen tes del torque se pueden obtener como: T i = ϵ ij k τ ij (4) Como el torque tiene que ser 0 las comp onen tes del torque tienen que ser an tisim ´ etricas ( τ ij  = τ j i ).Esto causa que los v alores propios sean n ´ umeros reales y se pueden obtener diagonalizando el tensor: σ p =   σ 1 0 0 0 σ 2 0 0 0 σ 3   con σ p = ˜ Rσ ˜ R T y ˜ R = (  n 1 ,  n 2 ,  n 3 ) (5) En esta expresi´ on  n 1 ,  n 2 ,  n 3 son los v ectores propios, siendo n ij la comp onen te j del v ector propio i, ˜ R se construir´ a como: ˜ R =   n 11 n 12 n 13 n 21 n 22 n 23 n 31 n 32 σ 33   (6) Los v ectores propios nos determinan los ejes principales del s´ olido, el de ma y or v alor propio se llama ( ( ma jor ) ) y el de menor ( ( minor ) ) . Con los vectores propios y los v alores propios queda totalmen te definida la situaci´ on del stress del s´ olido. De hec ho, siempre es p osible determinar el estado del s´ olido median te la suma de tres stresses cada uno sobre un eje. 2 CONSIDERA CIONES TE ´ ORICAS 4 2.1.2. Camp os de deformaci´ on en el plano En el caso 2D el sistema es m uc ho m´ as sencillo y a que pasamos a tener solo 3 v ariables, los 2 v alores propios σ 1 , σ 2 del stress (siendo σ 1 el de ma y or m´ odulo y el stress ma jor, y σ 2 el minor ) y el ´ angulo ϕ que forma σ 1 , con el eje X (una direcci´ on arbitraria en el s´ olido cristalino). Un v ector de stress, σ n para cualquier in tercara definida a partir del v ector  n se obtiene del tensor de stress diagonalizado: σ n = ˜ Rσ p ˜ R T  n σ p =  σ 1 0 0 σ 2  R =  cosϕ − sinϕ senϕ cosϕ  σ p =  cos sen  (7) De manera que en general p o demos expresar σ n para cualquier alfa: σ n =  σ 1 + σ 2 2 cosα + ( σ 1 + σ 2 ) cos ( α − 2 ϕ ) σ 1 + σ 2 2 sinα + ( σ 1 − σ 2 ) sin ( α − 2 ϕ )  (8) En general, el estado de deformaci´ on del s´ olido se p o dr ´ ıa representar median te la ilustraci´ on mos- trada en la Figura 1. En dic ha ilustraci´ on, se represen ta el stress ejercido sobre un s´ olido (represen tado median te una elipse) donde se puede apreciar que es claramen te ma y or a lo largo de la direcci´ on del eje del ( ( stress ma y or ) ) formando un ´ angulo ϕ resp ecto del eje X, que se toma como referencia. Este eje X ser´ a ev en tualmen te una direcci´ on cristalina arbitraria en el cristal. Figura 1: Se represen tan los ejes del ( ( ma jor stress ) ) y ( ( minor stress ) ) . 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.x6_c00001{left:400.458667pt;} .xc_c00001{left:427.405333pt;} .x8_c00001{left:430.446667pt;} .x1a_c00001{left:435.180000pt;} .x24_c00001{left:489.496000pt;} .x1b_c00001{left:500.206667pt;} .x1c_c00001{left:512.425333pt;} .x1d_c00001{left:526.262667pt;} .x1e_c00001{left:537.885333pt;} .x1f_c00001{left:541.716000pt;} .x20_c00001{left:562.520000pt;} .x21_c00001{left:656.700000pt;} .x25_c00001{left:668.730667pt;} } 2 CONSIDERA CIONES TE ´ ORICAS 5 2.1.3. La relaci´ on en tre el stress y el strain El stress esta causado p or el desplazamien to de los comp onen tes del material de su p osici´ on de equilibrio,se puede relacionar esta deformaci´ on con la fuerza median te la ley de Ho ok e. Es decir, que la respuesta del material es lineal fren te al est ´ ım ulo. P ara los materiales is´ otrop os se puede expresar. σ i i = Y ∆ l i l i σ i i = Y ϵ i (9) Donde Y es el mo dulo de Y oung, ϵ es el strain y σ es el stress. Al aplicar una fuerza en una direcci´ on no solo se crea un stress, sino que tam bi ´ en se mo difican las p osiciones de equilibrio de los ´ atomos causando una reacci´ on de signo opuesto en las direcciones p erp endiculares ϵ j j , expres´ andolo de forma matem´ atica. ϵ j j = − ν ϵ i i ϵ j j = ν Y σ i i (10) ν es el llamado co eficien te de P oisson y este es el efecto P oisson. Si ahora consideramos una deformaci´ on p or cizalladura (es decir, que es paralela a la sup erficie de aplicaci´ on), esta se describ e median te el ´ angulo de cizalladura Θ y un m´ odulo de cizalladura µ : σ i j = µ Θ = µ ϵ i j µ = Y 2(1 + ν ) (11) Una deformaci´ on de cizalladura se puede considerar como 2 deformaciones normales en un sistema de co ordenadas rotado,donde µ ,Y y ν est´ an relacionados y el Θ se puede expresar como un cambio relativ o en longitud 2 ϵ ij . El tensor del strain nos caracteriza totalmen te la deformaci´ on de un s´ olido y adem´ as es sim ´ etrico, es decir ϵ ij = ϵ j i :   ϵ xx ϵ xy ϵ xz ϵ y x ϵ y y ϵ y z ϵ z x ϵ z y ϵ z z   (12) 2.1.4. T ensor de elasticidad En un material is´ otrop o y homog ´ eneo, y a queda totalmen te determinada la respuesta del material a la deformaci´ on una v ez tenemos las constan tes Y y µ . Sin em bargo los materiales cristalinos no son is´ otrop os (En un material is´ otrop o to das las direcciones son equiv alen tes, lo que define una estruc- tura cristalina son las direcciones cristalinas) y las constan tes del material v ar ´ ıan para las diferen tes direcciones. De manera que para mostrar que las propiedades son direccionales, se deb en generalizar las las expresiones, para emp ezar la ley de Ho ok e: σ ij = X lm C ij l m ϵ l m (13) C ij l m es el tensor de elasticidad, es de rango 4, como σ ij y ϵ lm tienen cada uno 6 en tradas (3 de la diagonal y 3 de las com binaciones) ha y 36 p osibles co eficien tes para describir las propiedades el´ asticas del material. P ara represen tarlo se utiliza la llamada notaci´ on de V oigh t median te una matriz 6 x 6, se expresa de la forma: C ij l m = C αβ con αβ ( xx, y y , z z , y z , z x, xy ) (14) Esto causa una redefinici´ on de los tensores de stress y strain como v ectores: σ α = ( σ xx , σ y y , σ z z , σ xy , σ y z , σ z x ) ϵ β = ( ϵ xx , ϵ y y , ϵ z z , ϵ xy , ϵ yz , ϵ z x ) (15) 2 CONSIDERA CIONES TE ´ ORICAS 6 Como el tensor deb e ser sim ´ etrico solo v a a tener 21 entradas (6 de la diagonal y 15 de los elemen tos sup eriores o inferiores y a que es sim ´ etrico) pudi ´ endolo expresar como : σ ij = X lm C ij l m ϵ l m ;      σ 1 σ 2 . . . σ 6      =      C 11 C 12 · · · C 16 C 21 C 22 · · · C 26 . . . . . . . . . . . . C 61 C 62 · · · C 66           ϵ 1 ϵ 2 . . . ϵ 6      (16) Esta matriz dep endiendo del caso, puede simplificarse, p or ejemplo para el caso de un sistema simple c ´ ubico se reducen a 3 el n ´ umero de en tradas distin tas, y a que la respuesta al stress es equiv alen te en cualquier plano, creando strain solo en ese plano:         C 11 C 12 C 12 000 C 12 C 11 C 12 000 C 12 C 12 C 11 000 000 C 44 0 0 0 0 0 0 C 44 0 00000 C 44         (17) 2.2. Propiedades de emisi´ on ´ optica y estructura de bandas del W S e 2 Una de las propiedades m´ as imp ortan tes de cualquier material es el llamado gap, es decir la distancia en tre el m´ aximo de la banda de v alencia y el m ´ ınimo de la banda de conducci´ on. El gap determina en tre otras cosas el comp ortamien to met´ alico, aislan te o semiconductor del material y la emisi´ on ´ optica de dic ho material. Ha y 2 tip os de gap: el directo y el indirecto, que sea de un tip o u otro dep ende de la p osici´ on relativ a del m´ aximo y el m ´ ınimo en el espacio de k. 2 CONSIDERA CIONES TE ´ ORICAS 7 Figura 2: Represen taci´ on del gap directo, en la parte som breada tenemos electrones y en blanco los h uecos, al pasar un electr´ on de la banda de conducci´ on a la de v alencia emitiremos un fot´ on. Figura mo dificada de la referencia [5] El gap es directo si el m´ aximo de la banda de v alencia est´ a en el mismo punto de la zona de Brillouin que el m ´ ınimo de la banda de conducci´ on, la emisi´ on de fotones sucede cuando un electr´ on de la banda de conducci´ on pasa a la banda de v alencia emitiendo un fot´ on de energ ´ ıa igual a la energ ´ ıa del gap, la frecuencia se determina a partir de E g = ¯ hw y la longitud de onda a partir de E g = h c λ , el suceso in v erso tam bi ´ en sucede es p osible la absorci´ on de un fot´ on p or un electr´ on de la banda de v alencia y que este pase a la banda de conducci´ on. 2 CONSIDERA CIONES TE ´ ORICAS 8 Figura 3: Represen taci´ on del gap indirecto, en la parte som breada tenemos electrones y en blanco los h uecos, al pasar un electr´ on de la banda de conducci´ on a la de v alencia emitiremos un fot´ on y un fon´ on. Figura mo dificada de la referencia [5] El gap se denomina indirecto si el m ´ ınimo de la banda de conducci´ on y el m´ aximo de la de v alencia est´ an en distin tos pun tos de la primera zona de Brillouin. T an to en la absorci´ on como en la emisi´ on tenemos un fon´ on in v olucrado, esto hace que el suceso sea menos probable que las transiciones cuando el gap es directo, adem´ as de que se gasta energ ´ ıa en el fon´ on, esto hace que estos materiales sean considerados p eores emisores de luz. 2.2.1. Estructura cristalina del Wse2 Figura 4: Las direcciones cristalogr´ aficas utilizando 3 o 4 v ectores para la base, las direcciones de la celda no son los ejes X e Y del espacio El W S e 2 es un material cuy a celda unidad es hexagonal,el efecto que v an a tener los distin tos strain sobre la celda del material ser´ a: 2 CONSIDERA CIONES TE ´ ORICAS 9 Figura 5: Observ amos los efectos de los 4 tip os de strain la parte sup erior ( a) y b) ) los normales y en la parte inferior los de cizalladura ( c) y d) ). Se observ a que los strain normales a) y b) ( ϵ xx y ϵ y y es decir en la direcci´ on x/y y aplicando en la cara normal a x/y ), claramen te aumen tar´ an las dimensiones de la celda (en el caso p ositiv o, en el negativ o seria el efecto in v erso las dismin uir´ an) y a que aumen tan claramen te la base o la altura del rom b oide, sin em bargo en el caso de los de cizalladura b) y c) ( ϵ xy y ϵ y x es decir en la direcci´ on x/y sobre la cara p erp endicular a y /x ) observ amos que estas se mantendr´ an constantes. 2.2.2. Propiedades ´ opticas de materiales 2D y aplicaciones Desde el aislamien to del grafeno a partir del grafito (material de V an der W aals que se exfoli´ o p or primera v ez) en el a ˜ no 2004, ha habido un crecien te in ter ´ es en explorar las propiedades de otros materiales de V an der W aals con propiedades f ´ ısicas tan dispares como: magn ´ eticas, semiconductoras, aislan tes, electro´ opticas, electr´ onicas, etc. La v ariedad de este tip o de materiales p ermite so ˜ nar con una n uev a tecnolog ´ ıa flexible basada en este tip o de materiales. En particular, para propiedades ´ opticas, los materiales 2D semiconductores de la familia de los ”metales de transici´ on”presen tan un bandgap sin tonizable dep endiendo de su esp esor, siendo de bandgap directo para el l ´ ımite de la mono capa. Esto es m uy in teresan te para aplicaciones de emisi´ on ´ optica en LEDs o detectores, y a que p ermite la emisi´ on eficien te de fotones. P or otra parte, la fuerte in teracci´ on luz-materia, los hace m uy in teresan te para aplicaciones en detecci´ on ´ optica. P ara la ma y or ´ ıa de los disp ositiv os pr´ acticos, la mo dificaci´ on del bandgap del material abrir ´ ıa un sinf ´ ın de p osibilidades para la fabricaci´ on de disp ositiv os m ulti-funcionales. En este sen tido, optimizar y desarrollar disp ositiv os que p ermitan in tro ducir camp os de deformaci´ on rev ersibles y con trol en su magnitud, as ´ ı como anisotrop ´ ıa, resulta m uy in teresan te. Este asp ecto se ab ordar´ a en esta tesis, utilizando una v ariedad de disp ositiv os piezo el ´ ectricos in tegrando mono capas de W S e 2 (un material de V an der W aals semiconductor con bandgap directo). 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.x1f_c00002{left:340.770667pt;} .x2d_c00002{left:352.297333pt;} .xf_c00002{left:367.493333pt;} .x20_c00002{left:375.624000pt;} .x2e_c00002{left:378.680000pt;} .x1a_c00002{left:386.630667pt;} .x21_c00002{left:392.833333pt;} .x1b_c00002{left:400.317333pt;} .x8_c00002{left:416.458667pt;} .x18_c00002{left:430.652000pt;} .x22_c00002{left:435.266667pt;} .x9_c00002{left:438.408000pt;} .x10_c00002{left:457.562667pt;} .xa_c00002{left:469.213333pt;} .x11_c00002{left:488.666667pt;} .x1e_c00002{left:502.326667pt;} .x2f_c00002{left:504.610667pt;} .x30_c00002{left:516.234667pt;} .x31_c00002{left:520.065333pt;} .x12_c00002{left:524.569333pt;} .x32_c00002{left:527.585333pt;} .x33_c00002{left:543.636000pt;} .x13_c00002{left:574.589333pt;} .x14_c00002{left:587.865333pt;} .x15_c00002{left:656.700000pt;} } 2 CONSIDERA CIONES TE ´ ORICAS 10 hetero estructuras, unidas mediate fuerzas de V an der W aals, p ermiti ´ endonos crear estructuras con diferen tes propiedades. En este traba jo, se estudiar´ a c´ omo in tro ducir camp os de deformaci´ on el´ asticos anis´ otrop os y sus efectos en los cam bios del gap en dic ho material. 2 CONSIDERA CIONES TE ´ ORICAS 11 2.3. C´ alculos de estructura electr´ onica 2.3.1. La ecuaci´ on de Sc hr¨ odinger P ara calcular los efectos sobre el gap v amos a tener que resolv er la ecuaci´ on de Sc hr¨ odinger inde- p endien te del tiemp o: H Ψ(  R 1 ,  R 2 ....  R N ,  r 1 ,  r 2 , ....,  r N ) = E Ψ(  R 1 ,  R 2 ....  R N ,  r 1 ,  r 2 , ....,  r N ) (18) En esta ecuaci´ on se considera la p osici´ on de los electrones (Con letras min ´ usculas), de los n ´ ucleos (Con letras ma y ´ usculas) y las distin tas in teracciones en tre estos. El hec ho de que los electrones in teract ´ uen en tre s ´ ı tiene la consecuencia de que no p o demos sacar una soluci´ on separada para cada electr´ on y luego unirlas. Resolv er esta ecuaci´ on incluso solamen te para los electrones (los n ´ ucleos, debido a su gran masa, se pueden considerar quietos, lo que se cono ce como la apro ximaci´ on de Born-Opp enheimer) es bastan te complejo para casos con m uc has part ´ ıculas, de manera que se han desarrollado m uc hos m ´ eto dos para resolv erla de manera n um ´ erica, estos m ´ eto dos suelen consistir apro ximar el problema a un sistema de electrones electrones no in teractuan tes, sustituy endo la in teracci´ on en tre electrones p or un p oten- cial promedio con el que se tienen en cuen ta to dos los electrones. Uno de estos m ´ eto dos es la teor ´ ıa del funcional dela densidad (DFT, Densit y F unctonal Theory), seg ´ un la cual las propiedades f ´ ısicas del estado fundamen tal ser´ an las mismas mien tras la densidad electr´ onica sea la misma en am b os casos. 2.3.2. T eor ´ ıa del funcional de la densidad DFT se basa en los teoremas de Hohen b erg-Kohn y el desarrollo p osterior de Kohn y Sham y p ermite sustituir, como se comen t´ o an teriormen te, un sistema con m uc hos electrones in teractuan tes p or un sistema de electrones no in teractuan tes en un p otencial efectiv o. Los teoremas de Hohen b erg-Kohn son los siguien tes: Existe una relaci´ on en tre la densidad electr´ onica y el funcional externo de forma que a partir de la densidad se puede obtener el hamiltoniano del que se obtiene la funci´ on de onda, es decir cualquier magnitud f ´ ısica es funcional de la densidad Como el estado fundamen tal es el de menor energ ´ ıa su densidad electr´ onica sera aquella que minimice la energ ´ ıa. Si se separa la energ ´ ıa de in teracci´ on electr´ on-electr´ on en 2 t ´ erminos la energ ´ ıa electrost´ atica cl´ asica y el t ´ ermino de in tercam bio, en este es donde est´ an los efectos cu´ an ticos y se determina como el pro ducto de la densidad de energ ´ ıa de in tercam bio y la densidad de carga. E [ n ] = T [ n ] + E ext [ n ] + E ee [ n ] = T [ n ] + E ext [ n ] + E clas [ n ] + E int [ n ] = ⟨ Φ | − 1 2 ∇| Φ ⟩ + Z n ( r ) V ext  dr + Z ϵ int n (  r )  dr + 1 2 Z n ( r ) n ′ ( r ′ ) | r − r ′ |  dr  dr ′ (19) Si se minimiza esta energ ´ ıa con resp ecto a la densidad p o demos encon trar el estado fundamen tal. δ δ n ( E [ n ] − µ + Z n (  r )  dr ) = 0 (20) Siendo µ el p otencial qu ´ ımico, la densidad est´ a dada p or m ´ ultiples funciones de onda de manera que se puede elegir una que no in teract ´ ue, es decir una dada p or un ´ unico determinan te de Slater, p or 2 CONSIDERA CIONES TE ´ ORICAS 12 lo que se p o dr´ a expresar como suma de densidades en un p otencial efectiv o: n (  r ) = X i | ϕ (  r ) 2 | (21) Una v ez se cono ce la dep endencia de la energ ´ ıa con la densidad, se puede obtener el hamiltoniano y su deriv ada. Con este obtener la funci´ on de onda y v olv er a calcular la energ ´ ıa, repitiendo este pro ceso hasta que la diferencia en tre la en trada y salida sea m uy p eque ˜ na. Los estados se obtendr´ an diagonalizando el hamiltoniano, si hemos utilizado N funciones en la base tendremos un hamiltoniano N x N de manera que al diagonalizar obtendremos N estados distin tos, en cada uno se colo car´ a un electr´ on (2 si consideramos iguales los electrones con el mismo spin, dep ender´ a del hamiltoniano). 2.3.3. Tip os de programas para resolv er la ecuaci´ on de Sc hr¨ odinger Los m ´ eto dos para resolv erla se pueden agrupar en 2 tip os seg ´ un la base, es decir el tip o de funciones que sumemos para llegar a la soluci´ on: Las soluciones basadas en la suma de orbitales lo calizados, es decir en tendiendo que los electrones est´ an m uy ligados a los n ´ ucleos. Aquellas basadas en la suma de ondas planas, es decir sup oner que partimos de electrones libres y les aplicamos ligaduras. Figura 6: Represen taci´ on en 1D de la suma de orbitales y ondas planas Cada m ´ eto do tiene sus v en ta jas e incon v enien tes, en el caso de orbitales lo calizados las soluciones de la base se parecen m uc ho a las de los ´ atomos y mol ´ eculas desde un inicio, sin em bargo si con unas p o cas no es p osible resolv er el sistema con la suficien te precisi´ on, es dif ´ ıcil a ˜ nadir n uev as funciones a la base. P ara ondas planas es al rev ´ es, las funciones se parecen p o co a la soluci´ on, sin embargo a˜ nadir m´ as funciones a la base es sencillo. Los electrones m´ as in ternos (tam bi ´ en llamados de core), se sustituy en p or un pseudop otencial ha- ciendo que las funciones sean m´ as sua v es, lo que resulta en que requieran menos pun tos en el mallado a la hora de calcular las funciones de onda. Los electrones mas externos (los de v alencia) son los que consideraremos como tal para los c´ alculos, estos electrones son los que participan en el enlace qu ´ ımico y son m´ as relev an tes para el material. El programa Quan tum-ESPRESSO [4] es un programa DFT de soft w are libre que est´ a basado en el uso de ondas planas y pseudop otenciales para calcular estructuras electr´ onicas, en este caso ha si- do utilizado para obtener el Gap de 1.5341 eV, y tam bi ´ en para obtener la v ariaci´ on del gap con el strain. 2 CONSIDERA CIONES TE ´ ORICAS 13 Figura 7: Estructura de bandas del W S e 2 en la parte sup erior un multicapa en la inferior una mono- capa, el 0 de am b os ejes es el niv el de F ermi. Imagen obtenida de la bibliografia [6] En el caso de la m ulticapa, el gap es indirecto y en el caso de la mono capa un gap directo. Con el programa tam bi ´ en se puede calcular el efecto del strain en el gap del material, p o demos v er los efectos de los strain de cizalladura y normales . Figura 8: La dep endencia de la v ariaci´ on de la energ ´ ıa del gap con el strain. Imagen obtenida de bibliograf ´ ıa [6] P ara los strain de cizalladura, v emos que la v ariaci´ on en el gap es muy peque ˜ na sin em bargo para los strain normales se observ a una clara dep endencia lineal para ϵ xx y ϵ y y a justando esta dep endencia a una recta se obtiene: ∆ E Gap = − 0 , 0638364 ϵ xx − 0 , 0636273 ϵ y y (22) 2 CONSIDERA CIONES TE ´ ORICAS 14 Una manera de en tender esta tendencia es recordar el caso molecular donde tenemos dos estados enlazan te y an tienlazan te, al acercar los ´ atomos la in teracci´ on en tre estos aumen ta y las funciones de onda de am b os se solapan m´ as, esto causa que los estados se separen. En n uestro an´ alisis suceder´ a algo parecido, al aplicar un strain negativ o acercamos las part ´ ıculas aumen tando el gap, cuando es p ositiv o estamos separ´ andolas dismin uy endo el gap. 2.4. F undamen tos sobre piezo electricidad V amos a describir el efecto piezo el ´ ectrico y que materiales lo presen tan. El efecto piezo el ´ ectrico se describ e como el hec ho de que algunos materiales presen tan una deformaci´ on al aplicarles una diferencia de p otencial. Se observ a en materiales cuy a celda unidad no es del to do sim ´ etrica ba jo in v ersiones, esto causa que se pro duzca un momen to dip olar el ´ ectrico en cada celda unidad, los cuales se alinear´ an con un camp o externo ba jo una determinada presi´ on. Los piezo el ´ ectricos cer´ amicos son capaces de man tener el momen to dip olar una v ez ha y a cesado la presi´ on externa, sin em bargo los dip olos estar´ an colo cados de manera aleatoria, de manera que el camp o neto ser´ a 0, una manera de evitar esto es p olariz´ andolos: esto consiste en aplicarles un camp o el ´ ectrico, una v ez este sup ere un v alor m ´ ınimo estos dip olos se alinear´ an. Figura 9: En a) tenemos el material con los dip olos sin p olarizar apun tando en direcciones aleatorias, aplicando un camp o externo en b) los dip olos se alinean s en la misma direcci´ on y sen tido, una v ez quitamos el camp o en c) los dip olos se desalinean un p o co p ero man tienen la orien taci´ on en general Este efecto ser´ a m´ as o menos imp ortan te seg ´ un la direcci´ on y la in tensidad de la deformaci´ on, sup oniendo que dep ende de manera lineal p o demos definir un tensor ˜ d de orden tres que nos indica la p olarizabilidad en cada direcci´ on P i . P i = X j k d ij k σ j k (23) P seria el momen to dip olar p or unidad de v olumen, ˜ d es el tensor piezo electricidad y σ es el tensor elasticidad. esta ecuaci´ on describ e el llamado efecto piezo el ´ ectrico directo, donde el v olta je es prop orcional al strain y al stress. P ara calcular este tensor necesitamos a la ecuaci´ on de Ho ok e generalizada (13) de manera que la p olarizaci´ on total es:      P 1 P 2 . . . P 6      =   d 11 d 12 · · · d 16 d 21 d 22 · · · d 26 d 31 d 32 · · · d 36        σ 1 σ 2 . . . σ 6      =   e 11 e 12 · · · e 16 e 21 e 22 · · · e 26 e 31 e 32 · · · e 36        ϵ 1 ϵ 2 . . . ϵ 6      (24) [Document text truncated for crawler view.]