Teory and impementation beind: Universa surface creation - smaest unitce Bjare Brin Buus, Jaob Howat & Tomas Bigaard September 15, 218 1 Construction of surface sabs Te aim for tis part of te project is to create a possibe surfaces wit a given Mier Indice and te conventiona bu structure. In tis section, te teory beind te construction of surface sabs wi be outined and from tis fundament an impementation as been deveoped in Pyton. Tis impementation, wi be abe to create any surface for any type of structure incuding foowing common bu structures - te simpe cubic unit ce, te body centered cubic unit ce, te face centered cubic unit ce and te exagona cose paced unit ce. 1.1 Teory By introducing bot te rea and te reciproca attice spaces most pieces of te puzze of creating any surface is derived. In addition some integer matmatics wi be used. 1.1.1 Rea attice space First, we wi start by defining te system in rea space. We ave tree basis vectors tat span te crysta attice in te conventiona unit ce ( a 1, a 2, a 3 ). Tese tree vectors do not ave to be ortogona and te procedure wi terefore aso wor for cp structures. Additionay, te engts of te vectors wi not in a cases be te same, so te teoretica approac to tis probem, wi invove tree independent engts. For most bu structures tere wi ony be one or two different attice constants determining te unit ce due to symmetry, for instance te L1 and L12 aoys as mentioned in section XXX metod XXX. Te unit ce can be seen in drawing 1 wit te engts and directions. 1
a 3 a 3 a 1 a 2 a a 1 2 Drawing 1: Tis drawing sows te basis vectors and sizes for te system. A surface is defined by its Mier Indice (,,), were, and a are integers, wic in rea space can be described by te crysta panes tat are parae to te pane tat intersects te basis vectors ( a 1, a 2, a 3 ) at 1 a 1, 1 a 2, 1 a 3. Te Mier Indices are used for a four types of structures sc, bcc, fcc and cp. In case of one or more of te Mier Indice (,,) is zero, te pane does not intersect wit te corresponding axis. For instance, if is equa to zero, te norma vector to te pane, defined by (,,), wi be ortogona to a 2. A (,,) Mier Indice is unpysica and ence wi not be incuded in te teory section, owever a part of te impementation code wi notify te user tat te cosen surface is not possibe to create. 1.1.2 Reciproca attice space For te fu understanding of te attice construction, it is very usefu to introduce te reciproca vector space. Te basis vectors in te reciproca attice space are given by, b1 = a 2 a 3 a 1 ( a 2 a 3 ), b 2 = a 3 a 1 a 2 ( a 3 a 1 ), b 3 = a 1 a 2 a 3 ( a 1 a 2 ) (1) a i b j = δ ij (2) Wen introducing te reciproca attice vectors, te norma vector for a given surface pane wit te Mier Indices () is given by n = b 1 + b 2 + b 3 (3) 2
t 2 t 3 t 1 Drawing 2: Tis drawing sows a surface wit Mier Indices (2,1,1). Te repetition of a attice point is sown, and te vectors spanning tis surface can be found. Furtermore, a desired surface wit a norma vector n, drawing 2 sows tat for a set of non-zero Mier indices, tree vectors, wi be noted t 1,2,3, in te pane can easiy be found. Two vectors, ineary independent ofcourse, created by a inear combination of te vectors t 1,2,3, can span te desired surface. Tese vectors are given by a 3 a 2 a 1 a t 1 = 1 a 1 + 1 a 2, t 2 = 1 a 2 1 a 3, t 3 = 1 a 1 1 a 3 but it soud be noted tat te anti-parae versions of t 1,2,3 aso can be used. Since, and a are integers, we can mutipy wit te product and divide wit te indice, wic wi be common for eac of te attice vectors, a i, wit respect to bot of te attice vectors a i, so a new set of vectors end up being, t 1 = a 1 a 2, t 2 = a 1 a 3, t 3 = a 2 a 3 (4) For te specia cases of two of te Mier Indices being zero, it is a very straigtforward, to see tat te appropriate vectors to span te norma vector, wi be te corresponding basis vectors in rea space. If for instance, and bot are zero, it wi resut in a coice of v 1 = a 1 and v 2 = a 2. 1.1.3 Determination of te two surface vectors Having introduced te rea space and te reciproca space, most of te teory is avaiabe and ence te determination of te two vectors tat span te surface wit respect to a given Mier Indice is possibe. 3
Te simpe attice points r i,j,m are paced at r i,j,m = i a 1 + j a 2 + a 3 were i, j, m a are integers. Because of te arrangement of te attice points, not a surface panes wi go troug tese points. Te dot product between te norma vector n and te attice points r i,j,m gives, i + j + m = d and since a constants are integers, d must aso be an integer and terefore te vaues of d as been quantized. Tis equation is a Linear Diopantine Equation and te smaest d for wic tere exist an non-zero soution (i, j, m) wen (,, ) are non-zero is, accordingy to Bezouts Identity, wen d is te smaest common divisor of,, and. If one or two of te Mier Indices is zero, te identity is true, but ony wen coosing te argest common divisor for te non-zero parts of (,, ). If a Mier indice as a common divisor e > 1, te non-zero components of te Mier Indice can ten be reduced wit 1 e (,, ), and sti define te same surface. A soution wi terefore exist for i + j + m = d (5) and because of te reduction of te norma vector, te vaue for d wi be ±1. Te two surface vectors, must obey te fact tat tey are ortogona wit respect to te norma vector n, v 1,2 n =. (6) Because of tis, te cross product between te two surface vectors v 1 and v 2 must give a constant times te norma vector n. Te constant must be as sma as possibe, sti non zero, because te area spanned by v 1 and v 2 is equa to te engt of te cross product. And since te new norma vector is te smaest possibe, te constant must be ±1. Consider a coice of te two surface vectors, t 1 and t 3. Tey wi bot fufi equation??, owever te crossproduct between t 1 and t 3 wi be, t 1 t 3 = = 2 =. Uness te size of is ±1, te size of te surface area spanned by t 1 and t 3 wi be too big. It is terefore crucia to introduce a competey new inear combination of te tree vectors t 1, t 2 and t 3. A inear combination of t 1 4
and t 2 fufi te same requirements wen p and q are integers in p + q (p + q) = Tis eaves a more simpe equation to sove, (7) (p + q) = 1 (8) Te soution to tis equation can be found using te Extended Eucidean Agoritm to determine te unnowns integers, p and q. Te two new vectors, wic span te surface are described a 1 a 1 v 1 = p a 2 + q, v 2 = a 2 (9) a 3 a 3 However, tere are infinite possibe soutions for p and q but some of te soutions are better tan oters, in reation to visuaizing te surface. Terefore anoter criteria is impemented. Te coser to being ortogona te surface vectors are, te easier it becomes to appy adsorbates onto te te surface. Te procedure for tis wi be expained in section??, but te teory wi be expained ere. Te soution for p and q can be cosen to accommodate tis wit respect to an integer, c, by v 1 = (p + c) a 1 a 2 + (q c) a 1 a 3 (1) Tis cange of vector v 1, does not cange te crossproduct between v 1 and v 2, as sown in equation??, because te cross product between te canges and v 2 is zero. Tis is sown beow c a 1 a 2 c a 1 a 3 c c a 2 a 3 = = (11) Tis cange of v 1 resuts in an agoritm, wic wi be presented ater, to determine te most appropriate coice of vectors. 5
1.1.4 Finding te 3 rd vector for te new unit ce After determining te two vectors spanning te surface ( v 1, v 2 ), te tird basis vector( v 3 ) of te surface sab can be found. Tis vector does not need to be ortogona to te two surface vectors. Te vector wi go from one attice point to its repeated attice point anoter pace in te structure. Tis means tat te same contraints appy to tis vector as for v 1 and v 2, but some additiona constraints wi be added. Te vector wi ave to be an integer inear combination of te tree origina attice vectors ( a 1, a 2, a 3 ) and ave te coordinates (i 3 a 1, j 3 a 2, m 3 a 3 ). In addition v 3 cannot be ortogona to te surface norma, so v 3 n. To find te integers i 3, j 3 and m 3 by cacuating te dot product using normavector of te surface from equation??, and te definitions of te reciproca vectors ( b 1, b 2, b 3 ): n v 3 = ( b 1 + b 2 + b 3 ) (i 3 a 1 + j 3 a 2 + m 3 a 3 ) = i 3 + j 3 + m 3 = d, (12) were d must be a non-zero integer because a of,,, i 3, j 3 and m 3 are integers. It wi now be sown tat d = 1. Defining te voume of te conventiona unit ce to be V spanned by te conventiona basis a 1, a 2 and a 3. V = ( a 1 a 2 ) a 3 (13) and rewriting of te tree reciproca vectors to te foowing form V b 1 = a 2 a 3, V b 2 = a 3 a 1, V b 3 = a 1 a 2. (14) wi ease te cacuations, and wit a te pieces set, a determination of te vaue of d is possibe. Te voume of te ce spanned by ( v 1, v 2, v 3 ) is presented beow, were te constants (i,j,m) 1,2 refer to te constants defined by te formaism used for v 1, equation??, and for v 2, equation??. V = v 3 ( v 1 v 2 ) = v 3 {(i 1 a 1 + j 1 a 2 + m 1 a 3 ) (i 2 a 1 + j 2 a 2 + m 2 a 3 )} = v 3 {i 1 j 2 a 1 a 2 + i 1 m 2 a 1 a 3 + j 1 i 2 a 2 a 1 + j 1 m 2 a 2 a 3 } + v 3 {m 1 i 2 a 3 a 1 + m 1 j 2 a 3 a 2 } } = V v 3 {i 1 j 2 b3 i 1 m 2 b2 j 1 i 2 b3 + j 1 m 2 b1 + m 1 i 2 b2 m 1 j 2 b1 i 1 i 2 = V v 3 j 1 j 2 m 1 m 2 6 b1 b2 b3 (15)
Te cross product between (i 1, j 1, m 1 ) and (i 2, j 2, m 2 ) as been found previousy in equation?? using te Extended Eucidian Agoritm as (,, ). Inserting tis into te equation V = V v 3 ( b 1 + b 2 + b 3 ) = V d. (16) d is terefore equa to 1. Te nowedge of d in equation?? eads to a new equation to sove, to determine te tird vector v 3. n v 3 = ( b 1 + b 2 + b 3 ) (i 3 a 1, j 3 a 2, m 3 a 3 ) = i 3 + j 3 + m 3 = 1 (17) Tis equation can be soved using te Extended Eucidean Agoritm for tree variabes and terefore te tird vector v 3 is determined. Wit tese tree vectors, ( v 1, v 2, v 3 ), a basis for te new unit ce is created. Te impementation of tis wi be described in te foowing section, aong wit some ways to get around te numerica issues in Pyton. 1.2 Impementation in Pyton Based on te teory derived above an arbitrary surface can be created using te procedure found on te DTU nifeim custer. To create a surface using te procedure described in tis section a conventiona bu ce of te surface materia is needed aong wit te Mier indices and te dept of te sab. Te impementation in Pyton using ASE to setup te atoms consists of tree parts. First, a new basis is derived from te Mier indices wit two of te basis vectors ying in te surface pane. Secondy, te atoms in te conventiona bu ce are expressed in te terms of te new basis in a sab wit te seected dept. Finay, te unit ce of te sab is modified so te tird ce vector points perpendicuar to te surface and a atoms are moved into te unit ce. 1.2.1 Surface basis For any surface type described by a Mier indice (,, ) te surface basis ( v 1, v 2, v 3 ) is found reative to te conventiona bu unit ce. v 1 and v 2 are cosen to be in te surface pane. In te specia case were ony one of te Mier indices is non-zero v 1 and v 2 are simpy te unit vectors in te directions were te Mier indices are zero, respectivey and v 3 is te direction were te Mier indice is non-zero. 7
For a oter situations v 1 and v 2 are found by soving te inear equation?? using te Extended Eucidean Agoritm - in te script defined as ext_gcd(). Tis yieds an infinite set of soutions a of wic can be used. However, te optima structure is found wen te ange between te two base vectors are as cose to 9 o as possibe, as te structure wi be as compact as possibe and specific sites are easier to identify. Tis soution is found by minimizing te scaar product of te two base vectors by c Z. (p + c) a 1 a 2 + (q c) a 1 a 3 a 2 a 3 min(c) Tis can be expressed as 1 + c 2 min(c) and te soution is found wen c is equa to te fraction 1 2 rounded to te nearest integer. Because of numerica errors a toerance is used. In pyton tis is expressed as foows. p,q = ext_gcd(,) 1 = dot( p*(*a1-*a2)+q*(*a1-*a3), *a2-*a3) 2 = dot( *(*a1-*a2)-*(*a1-*a3), *a2-*a3) if abs(2)>to: c = -int(round(1/2)) p,q = p+c*, q-c* v1 = p*array((,-,))+q*array((,,-)) v2 = reduce(array((,,-))) a,b = ext_gcd(p*+q*,) v3 = array((b,a*p,a*q)) Te ast four ines define te base vectors for te surface using te Extended Eucidean Agoritm for two variabes to find v 1 and v 2 and tree variabes to find v 3. 1.2.2 Atom positions Wen te basis ave been found te atoms in te conventiona ce are base-canged to te new basis using for i in range(en(bu)): newpos = inag.sove(basis.t,bu.get_scaed_positions()[i]) scaed += [newpos-foor(newpos+to)] and ten moved so te scaed positions in te new basis are witin te box spanned by te basis. Te toerance is needed so atoms positioned exacty 8
on te boundary are treated consistenty despite numerica errors. Te ce in te new basis is ten repeated in te v 3 direction to create te required sap dept. For many appications it is usefu to ave te z-direction pointing perpendicuar to te surface to enabe eectrostatic decouping and to mae vacuum eigt and adsorbate distance we defined. Te next step in te procedure is terefore to aign te z-direction wit te cross product of v 1 and v 2 wit a engt so te ce voume is preserved. Te fina step before te sab is created is ten to move te atoms so te scaed coordinates are between and 1 in te v 1 and v 2 directions maing it obvious ow te atoms are ocated reative to eac oter wen te structure is visuaized. 9