6 Symmetry Properties of Electronic and Photonic Band Structures W.Hergert,M.Dane,and D.Kodderitzsch Martin-Luther-University Halle-Wittenberg,Department of Physics, Von-Seckendorff-Platz 1,06120 Halle,Germany Abstract.Group theoretical investigations have a huge potential to simplify calcu- lations in solid state theory.We will discuss the application of group theory to elec- tronic and photonic band structures.The symmetry properties of the Schrodinger equation and Maxwell's equations as well will be investigated.We have developed methods to simplify group theoretical investigations based on the computer algebra system Mathematica. 6.1 Introduction The majority of physical systems exhibit intrinsic symmetries which can be used to simplify the solution of the equations governing these systems.Group theory as a mathematical tool plays an important role to classify the solu- tions within the context of the underlying symmetries.Extensive use of group theory has been made to simplify the study of electronic structure or vibra- tional modes of solids or molecules.There exists a number of excellent books illustrating the use of group theory.[6.1-6.4] Similarities between the solution of the Schrodinger equation,or the Kohn-Sham equations in the framework of density functional theory for a crystal,and the solution of Maxwell's equations have been already pointed out by Joannopoulos et al..6.5 Therefore,it is clear that the same group theoretical concepts like in the theory of electronic band structures including two-dimensional and three-dimensional structures,surfaces as well as defects should be applicable to photonic band structure calculations,if we take into account the vectorial nature of the electromagnetic field. There are several publications about group theoretical investigations of photonic crystals in the literature.Sakoda has extensively studied the sym- metry properties of two-dimensional and three-dimensional photonic crystals (cf.[6.6-6.9),starting from a plane wave representation of the electromag- netic fields.Group theoretical investigations are done also by Ohtaka and Tanabe.[6.10]They investigate the symmetry properties of photonic crystals represented by an array of dielectric spheres.In this case a series expansion in terms of vector spherical harmonics is used. The problem at the end for electronic and the photonic band structure calculations is:How to apply group theory in the actual research work,if one goes away from all the textbook examples.The aim of the paper is to show, W.Hergert,M.Dane,and D.Kodderitzsch,Symmetry Properties of Electronic and Photonic Band Structures,Lect.Notes Phys.642,103-125(2004) http://www.springerlink.com/ C Springer-Verlag Berlin Heidelberg 2004
6 Symmetry Properties of Electronic and Photonic Band Structures W. Hergert, M. D¨ane, and D. K¨odderitzsch Martin-Luther-University Halle-Wittenberg, Department of Physics, Von-Seckendorff-Platz 1, 06120 Halle, Germany Abstract. Group theoretical investigations have a huge potential to simplify calculations in solid state theory. We will discuss the application of group theory to electronic and photonic band structures. The symmetry properties of the Schr¨odinger equation and Maxwell’s equations as well will be investigated. We have developed methods to simplify group theoretical investigations based on the computer algebra system Mathematica. 6.1 Introduction The majority of physical systems exhibit intrinsic symmetries which can be used to simplify the solution of the equations governing these systems. Group theory as a mathematical tool plays an important role to classify the solutions within the context of the underlying symmetries. Extensive use of group theory has been made to simplify the study of electronic structure or vibrational modes of solids or molecules. There exists a number of excellent books illustrating the use of group theory. [6.1–6.4] Similarities between the solution of the Schr¨odinger equation, or the Kohn-Sham equations in the framework of density functional theory for a crystal, and the solution of Maxwell’s equations have been already pointed out by Joannopoulos et al.. [6.5] Therefore, it is clear that the same group theoretical concepts like in the theory of electronic band structures including two-dimensional and three-dimensional structures, surfaces as well as defects should be applicable to photonic band structure calculations, if we take into account the vectorial nature of the electromagnetic field. There are several publications about group theoretical investigations of photonic crystals in the literature. Sakoda has extensively studied the symmetry properties of two-dimensional and three-dimensional photonic crystals (cf. [6.6–6.9]), starting from a plane wave representation of the electromagnetic fields. Group theoretical investigations are done also by Ohtaka and Tanabe. [6.10] They investigate the symmetry properties of photonic crystals represented by an array of dielectric spheres. In this case a series expansion in terms of vector spherical harmonics is used. The problem at the end for electronic and the photonic band structure calculations is: How to apply group theory in the actual research work, if one goes away from all the textbook examples. The aim of the paper is to show, W. Hergert, M. D¨ane, and D. K¨odderitzsch, Symmetry Properties of Electronic and Photonic Band Structures, Lect. Notes Phys. 642, 103–125 (2004) http://www.springerlink.com/ c Springer-Verlag Berlin Heidelberg 2004
104 W.Hergert,M.Dane,and D.Kodderitzsch that computer algebra tools are appropriate to simplify group theoretical discussions connected with the calculation of electronic and photonic band structures. After a short discussion of the usefulness of computer algebra systems, we will introduce basic concepts of group theory.Representation theory will be discussed next,followed by the analysis of the symmetry properties of the Schrodinger equation and of Maxwell's equations.We want to solve the electronic and photonic problem for a lattice periodic situation.In the one case we have a periodic potential V(r)=V(r+R)in the other case a periodic dielectric constant e(r)=e(r+R).The consequences of that periodicity will be considered.A simplification of the solution of both kinds of problems is possible,if symmetry-adapted basis functions are used.This is discussed in more detail for the electronic problem.At the end we apply group theory to the calculation of photonic band structures. 6.2 Group Theory Packages for Computer Algebra Systems Computer algebra(CA)systems like Mathematica or Maple have been devel- oped to allow formal mathematical manipulations.Nowadays those systems are complete in such a sense,that formal manipulations,numerical calcula- tions,as well as graphical representations are possible in an easy and intuitive way with the same software.Apart from such general purpose CA systems, there exist also systems developed for special applications in mathematics. Group theoretical considerations,although conceptionally easy,lead very of- ten to time-consuming algebraic calculations,which are error-prone.There- fore group theory is an excellent field for the application of CA systems.Some systems for abstract group theory are available [6.11],but they are not very helpful for considerations in solid state theory.K.Shirai developed a Math- ematica package for group theory in solid state physics [6.12].We followed similar ideas but tried to make the package more easy to use. We have constructed a package for the CA system Mathematica which allows to do group theoretical manipulations which occur in solid state theory. 6.13]The software allows basic considerations with point groups,contains tight-binding theory for the electronic structure of solids,but also special applications to photonic crystals.The package is accompanied by an on line help,which is integrated in Mathematica's help system.All considerations discussed in this paper can be found in a Mathematica notebook which is part of our package.We will give references to the commands implemented in the package throughout the paper
104 W. Hergert, M. D¨ane, and D. K¨odderitzsch that computer algebra tools are appropriate to simplify group theoretical discussions connected with the calculation of electronic and photonic band structures. After a short discussion of the usefulness of computer algebra systems, we will introduce basic concepts of group theory. Representation theory will be discussed next, followed by the analysis of the symmetry properties of the Schr¨odinger equation and of Maxwell’s equations. We want to solve the electronic and photonic problem for a lattice periodic situation. In the one case we have a periodic potential V (r) = V (r+R) in the other case a periodic dielectric constant (r) = (r + R). The consequences of that periodicity will be considered. A simplification of the solution of both kinds of problems is possible, if symmetry-adapted basis functions are used. This is discussed in more detail for the electronic problem. At the end we apply group theory to the calculation of photonic band structures. 6.2 Group Theory Packages for Computer Algebra Systems Computer algebra (CA) systems like Mathematica or Maple have been developed to allow formal mathematical manipulations. Nowadays those systems are complete in such a sense, that formal manipulations, numerical calculations, as well as graphical representations are possible in an easy and intuitive way with the same software. Apart from such general purpose CA systems, there exist also systems developed for special applications in mathematics. Group theoretical considerations, although conceptionally easy, lead very often to time-consuming algebraic calculations, which are error-prone. Therefore group theory is an excellent field for the application of CA systems. Some systems for abstract group theory are available [6.11], but they are not very helpful for considerations in solid state theory. K. Shirai developed a Mathematica package for group theory in solid state physics [6.12]. We followed similar ideas but tried to make the package more easy to use. We have constructed a package for the CA system Mathematica which allows to do group theoretical manipulations which occur in solid state theory. [6.13] The software allows basic considerations with point groups, contains tight-binding theory for the electronic structure of solids, but also special applications to photonic crystals. The package is accompanied by an on line help, which is integrated in Mathematica’s help system. All considerations discussed in this paper can be found in a Mathematica notebook which is part of our package. We will give references to the commands implemented in the package throughout the paper
6 Symmetry Properties of Electronic and Photonic Band Structures 105 6.3 Basic Concepts in Group Theory In this section we will introduce the basic concepts of group theory.We will focus on definitions which will be necessary for the following discussions.To illustrate the concepts,let us discuss the symmetry group of a square in two dimensions,i.e.we are interested in all transformations which transform the square into itself.(cf.Fig.6.1)This example will be of later use for the discussion of photonic band structures. A set g ofelements A.B,C...is called a group if the following four axioms are fulfilled:i)There exists an operation,often called multiplication,which associates every pair of elements of g with another element of g:A Eg,BE g→A·B=C,C∈g,ii)The associative law is valid:A,B,C∈g→ (A·B).C=A·(B.C)=A·B.C,iii)in the set exists an identity element: A,E∈g→A·E=E·A=A,iv)for all A∈G exists an inverse element A-1∈G with A·A-1=A-1·A=E. The symmetry group of the square consists of rotations of /2 around the z-axis and mirror operations.The normal vectors of the mirror planes are the x and y-axis,Oa and Ob.A mirror symmetry may be expressed as a twofold rotation,followed by an inversion.All the symmetry operations of the group of the square,named C4v are: CAv=[E,Cz,CAz,Cz,ICx,IC2y:IC2a,IC2b} (6.1) If the group theory package is included in a Mathematica notebook by means of the command Needs ["GroupTheory'Master'"]1,the newly defined com- mand c4v=InstallGroup["C4v"]will install the group in terms of the ro- tation matrices of the elements.The matrices are stored in the list c4v.The command c4vs=GetSymbol[c4v]will transform the elements of the group into symbolic form (cf.(6.1)).The symbols will be stored in the list c4vs. Operations on the group can be done in both representations of the group el- ements.The group multiplication is implemented in our Mathematica package by a redefinition of the infix-operator (see 6.25). Fig.6.1.Symmetry of a square. 1 Mathematica commands are marked using this particular font
6 Symmetry Properties of Electronic and Photonic Band Structures 105 6.3 Basic Concepts in Group Theory In this section we will introduce the basic concepts of group theory. We will focus on definitions which will be necessary for the following discussions. To illustrate the concepts, let us discuss the symmetry group of a square in two dimensions, i.e. we are interested in all transformations which transform the square into itself. (cf. Fig. 6.1) This example will be of later use for the discussion of photonic band structures. A set G of elements A,B,C ... is called a group if the following four axioms are fulfilled: i) There exists an operation, often called multiplication, which associates every pair of elements of G with another element of G: A ∈ G, B ∈ G → A · B = C, C ∈ G, ii) The associative law is valid: A, B, C ∈G→ (A · B)· C = A ·(B · C) = A · B · C, iii) in the set exists an identity element: A, E ∈G→ A · E = E · A = A, iv) for all A ∈ G exists an inverse element A−1 ∈ G with A · A−1 = A−1 · A = E. The symmetry group of the square consists of rotations of π/2 around the z-axis and mirror operations. The normal vectors of the mirror planes are the x and y-axis, −→ Oa and −→ Ob. A mirror symmetry may be expressed as a twofold rotation, followed by an inversion. All the symmetry operations of the group of the square, named C4v are: C4v = E,C2z, C4z, C−1 4z ,IC2x,IC2y,IC2a,IC2b (6.1) If the group theory package is included in a Mathematica notebook by means of the command Needs["GroupTheory‘Master‘"]1, the newly defined command c4v=InstallGroup["C4v"] will install the group in terms of the rotation matrices of the elements. The matrices are stored in the list c4v. The command c4vs=GetSymbol[c4v] will transform the elements of the group into symbolic form (cf.(6.1)). The symbols will be stored in the list c4vs. Operations on the group can be done in both representations of the group elements. The group multiplication is implemented in our Mathematica package by a redefinition of the infix-operator ⊕ (see [6.25]). Fig. 6.1. Symmetry of a square. 1 Mathematica commands are marked using this particular font
106 W.Hergert,M.Dane,and D.Kodderitzsch For the graphical user interface of Mathematica we designed two addi- tional palettes.The palette SymmetryElements contains all the symmetry elements of the 32 point groups.The palette PointGroup contains all the operations defined in the package. The inspection of the multiplication table(MultiplicationTable[c4v]) of the group shows,that the table is not symmetric,indicating that the group multiplication is not commutative in this group.Therefore the group is not abelian (AbelianQ [c4vs]returns False.).All elements of the group can be also generated by successive multiplication of only two elements of the group, the so called generators.For our example we find the generators C4z,IC2x. (Generators[group_])2 Any set of elements of a group which obeys all the group postulates is called a subgroup of the group.Examples for subgroups are:[E,CCzC and [E,IC2y}(SubGroupQ[group_,subgroup_]performs a test). Another structure of the group is introduced by the definition of conjugate elements.An element B of the group g is said to be conjugate to A if their exists a group element X such that B=XAX-1.A class(classes [c4vs]) is a collection of mutually conjugate elements of a group.The classes of the group Cav are:(E),(Cz),(Cz,C),(ICx,ICy),(IC2a,ICb).It is easy to prove that:i)E always forms a class on its own,ii)every group element in g is a member of some class of g,iii)no element can be the member of two classes of g,iv)if g is an abelian group,every element forms a class on its own. We have already seen,that the group in the example can be represented by the abstract symbols or the rotation matrices.The relation between groups is formulated more rigorously in terms of homomorphism and isomorphism. Two groups g=[A,B,C,...}and C'=[A',B',C',...}are called homo- morphic,if i)to each element of g corresponds one and only one element of G'ii)for all elements holds A-A',B-B'AB-A'B'.The set of elements which are mapped to E'E g in a homomorphism is called ker- nel of the homomorphism Nk.In contrast to the homomorphism the map- ping in the isomorphism is one-to-one.Whereas the mapping of the symbols on the rotation matrices constitutes an isomorphism,the following mapping E,Cz,Caz,C+1 and IC2x;IC2y:IC2a:IC2b-1 constitutes a ho- momorphism. 6.4 Representation Theory 6.4.1 Matrix Representations of Groups Matrir representations of symmetry groups are the essential tools to inves- tigate symmetry properties of solutions of field equations as Schrodinger's equation or Maxwell's equations. 2 The formal arguments of the commands indicate,which type of information has to be plugged in.For more information use the help system or the example notebook
106 W. Hergert, M. D¨ane, and D. K¨odderitzsch For the graphical user interface of Mathematica we designed two additional palettes. The palette SymmetryElements contains all the symmetry elements of the 32 point groups. The palette PointGroup contains all the operations defined in the package. The inspection of the multiplication table (MultiplicationTable[c4v]) of the group shows, that the table is not symmetric, indicating that the group multiplication is not commutative in this group. Therefore the group is not abelian (AbelianQ[c4vs] returns False.). All elements of the group can be also generated by successive multiplication of only two elements of the group, the so called generators. For our example we find the generators C4z,IC2x. (Generators[group_]) 2 Any set of elements of a group which obeys all the group postulates is called a subgroup of the group. Examples for subgroups are: E,C2z, C4z, C−1 4z and {E,IC2y} (SubGroupQ[group_,subgroup_] performs a test). Another structure of the group is introduced by the definition of conjugate elements. An element B of the group G is said to be conjugate to A if their exists a group element X such that B = XAX−1. A class (Classes[c4vs]) is a collection of mutually conjugate elements of a group. The classes of the group C4v are: (E), (C2z), (C4z, C−1 4z ), (IC2x,IC2y), (IC2a,IC2b). It is easy to prove that: i) E always forms a class on its own, ii) every group element in G is a member of some class of G, iii) no element can be the member of two classes of G, iv) if G is an abelian group, every element forms a class on its own. We have already seen, that the group in the example can be represented by the abstract symbols or the rotation matrices. The relation between groups is formulated more rigorously in terms of homomorphism and isomorphism. Two groups G = {A, B, C, . . . } and G = {A , B , C ,... } are called homomorphic, if i) to each element of G corresponds one and only one element of G ii) for all elements holds A → A , B → B → AB → A B . The set of elements which are mapped to E ∈ G in a homomorphism is called kernel of the homomorphism NK. In contrast to the homomorphism the mapping in the isomorphism is one-to-one. Whereas the mapping of the symbols on the rotation matrices constitutes an isomorphism, the following mapping E,C2z, C4z, C−1 4z ⇒ +1 and IC2x,IC2y,IC2a,IC2b ⇒ −1 constitutes a homomorphism. 6.4 Representation Theory 6.4.1 Matrix Representations of Groups Matrix representations of symmetry groups are the essential tools to investigate symmetry properties of solutions of field equations as Schr¨odinger’s equation or Maxwell’s equations. 2 The formal arguments of the commands indicate, which type of information has to be plugged in. For more information use the help system or the example notebook.
6 Symmetry Properties of Electronic and Photonic Band Structures 107 A group of square matrices,with the matrix multiplication as relation between the elements,which is homomorphic to a group g is called a matrix representation of g.Each element Ag corresponds to a matrix I(A): T(A)T(B)=I(C)V A,B,CEg T(E)=E (identity matrix) (6.2) T(A-1)=T(A)-1 It is obvious that an I-dimensional representation consisting of a set of(Ix l)- matrices can be transformed in another representation by means of a similar- ity transformation I(A)=S-1I(A)S using a non-singular(I x I)-matrix S. The rotation matrices of a finite symmetry group form a three-dimensional matrix representation in that sense.The mapping of the group elements of Cav onto the numbers +1,-1 represents also,in this case a one-dimensional, matrix representation of the group. From two matrix representations T and T2 of g with the dimensions l and l2,a(l1+l2)-dimensional representation I can be built,forming the block matrices T(A (6.3) The representation I is called the direct sum of the representations T and T2:T=IT2.Representations which can be transformed into direct sums by similarity transformations are called reducible.If such a transformation is not possible,the representation is called irreducible.The rotation matrices of the group Cav are given by: 100 -100 010 T(E)= 010 (C2z) 0-10 I(Caz)= 100 001 001 001 0-10 -100 100 r(IC)= 10 0 (IC2x)= 010 -10 001 001 001 0-10 010 T(IC2a)= -100 T(IC2b)= 100 001 001 The rotation matrices are block matrices and therefore reducible into a two- dimensional and a one-dimensional matrix representation of Cav. An important quantity of a matrix representation which will not change under a similarity transformation is the trace of the rotation matrix.This trace is called the character x of the representation matrix
6 Symmetry Properties of Electronic and Photonic Band Structures 107 A group of square matrices, with the matrix multiplication as relation between the elements, which is homomorphic to a group G is called a matrix representation of G. Each element A ∈ G corresponds to a matrix Γ(A): Γ(A)Γ(B) = Γ(C) ∀ A, B, C ∈ G Γ(E) = E (identity matrix) (6.2) Γ(A−1) = Γ(A) −1 It is obvious that an l-dimensional representation consisting of a set of (l×l)- matrices can be transformed in another representation by means of a similarity transformation Γ (A) = S−1Γ(A)S using a non-singular (l × l)-matrix S. The rotation matrices of a finite symmetry group form a three-dimensional matrix representation in that sense. The mapping of the group elements of C4v onto the numbers +1, −1 represents also, in this case a one-dimensional, matrix representation of the group. From two matrix representations Γ1 and Γ2 of G with the dimensions l1 and l2,a(l1 + l2)-dimensional representation Γ can be built, forming the block matrices Γ(A) = Γ1(A) 0 0 Γ2(A) . (6.3) The representation Γ is called the direct sum of the representations Γ1 and Γ2 : Γ = Γ1⊕Γ2. Representations which can be transformed into direct sums by similarity transformations are called reducible. If such a transformation is not possible, the representation is called irreducible. The rotation matrices of the group C4v are given by: Γ(E) = 100 010 001 Γ(C2z) = −100 0 −1 0 0 01 Γ(C4z) = 0 10 −100 0 01 Γ(IC−1 4z ) = 0 −1 0 100 001 Γ(IC2x) = −100 0 10 0 01 Γ(IC2y) = 100 0 −1 0 001 Γ(IC2a) = 0 −1 0 −100 0 01 Γ(IC2b) = 010 100 001 The rotation matrices are block matrices and therefore reducible into a twodimensional and a one-dimensional matrix representation of C4v. An important quantity of a matrix representation which will not change under a similarity transformation is the trace of the rotation matrix. This trace is called the character χ of the representation matrix.
108 W.Hergert,M.Dane,and D.Kodderitzsch (6.4) i=1 The character system of a group is called the character table.The character table can be generated automatically using CharacterTable [c4vs].We will get: C1C22C32C42C5 1111 11 C1=(E) 3111-1-1 C2=(C2z) r411-1 1-1 C3=(C4,Cz) 211-1-11 C4=(IC2x,IC2x) 52-2100 Cs=(IC2a,IC2b) The character table represents a series of character theorems,which are im- portant for later use:i)the number of inequivalent irreducible representations of a group g is equal to the number of classes of g,ii)the characters of the group elements in the same class are equal,iii)the sum of the squares of the dimensions of all irreducible representations is equal to the order of the group,iv)two representations are equivalent if their character systems are equivalent.Especially important are the following theorems: -A representation I is irreducible,if: ∑1xT)P=9 (6.5) Teg (The sum of the squares of the characters of the rotation matrices of Cv exceeds the group order g =8,indicating that this representation cannot be irreducible. -The number n,how often an irreducible representation I,or a repre- sentation equivalent to I,is contained in the reduction of the reducible representation I,is given by: n=1∑xT)xT) (6.6) 9 Teg (g-order of the group,x(T),x'(T)-character of the group element in the representation I and I) Orthogonality theorems for characters ∑x(Ck)'x(Ck)Nk=gi (6.7) ∑X'(Ck)'X(C)N=g (6.8)
108 W. Hergert, M. D¨ane, and D. K¨odderitzsch χ(A) = l j=1 Γ(A)jj (6.4) The character system of a group is called the character table. The character table can be generated automatically using CharacterTable[c4vs]. We will get: C1 C2 2 C3 2C4 2C5 Γ1 11 1 1 1 Γ3 1 1 1 -1 -1 Γ4 1 1 -1 1 -1 Γ2 1 1 -1 -1 1 Γ5 2 -2 1 0 0 C1 = (E) C2 = (C2z) C3 = (C4z, C−1 4z ) C4 = (IC2x,IC2x) C5 = (IC2a,IC2b) The character table represents a series of character theorems, which are important for later use: i) the number of inequivalent irreducible representations of a group G is equal to the number of classes of G, ii) the characters of the group elements in the same class are equal, iii) the sum of the squares of the dimensions of all irreducible representations is equal to the order of the group, iv) two representations are equivalent if their character systems are equivalent. Especially important are the following theorems: – A representation Γ is irreducible, if: T∈G | χ(T) | 2= g . (6.5) (The sum of the squares of the characters of the rotation matrices of C4v exceeds the group order g = 8, indicating that this representation cannot be irreducible.) – The number n, how often an irreducible representation Γi , or a representation equivalent to Γi , is contained in the reduction of the reducible representation Γ, is given by: n = 1 g T∈G χ(T)χi (T) ∗ (6.6) (g- order of the group, χ(T), χi (T) - character of the group element in the representation Γ and Γi ) – Orthogonality theorems for characters k χi (Ck) ∗χj (Ck)Nk = g δij (6.7) i χi (Ck) ∗χi (Cl)Nk = g δkl (6.8)
6 Symmetry Properties of Electronic and Photonic Band Structures 109 (x(C),x(Ck)characters of elements in class Ck in irreducible represen- tations I and I,N&-number of elements in Ck.In the first relation the summation runs over all classes of the group.In the second relation it runs over all inequivalent irreducible representations of the group.) The character theorems can be easily checked for all the point groups using Mathematica. 6.4.2 Basis Functions of Irreducible Representations Our final goal is the investigation of the symmetry properties of the solu- tions of Maxwell's equation.Therefore we have to associate the symmetry expressed by the symmetry group,i.e.the point group Cav in our example, with the symmetry properties of scalar and vector fields. We define a transformation operator to a symmetry element T as P(T). For a scalar function it holds: P(T)f(r)=f(T-r) (6.9) The operators P(T)form a group of linear unitary operators.If we discuss Maxwell's equations,we have to deal with vector fields in general.Instead of(6.9)the following transformation has to be applied to the vector field F: P(T)F(r)=TF(T-r) (6.10) Now we define the basis functions of an irreducible representation (IR) and concentrate on scalar fields.If a set of l-dimensional matrices I(T)forms a representation of the group g and (r),...,o(r)is a set of linear inde- pendent functions such that P(T)pn(r)=∑T(T)mnm(r）n=l,2,,l (6.11) m=1 then functions on(r)are called partners in a set of basis functions of the rep- resentation T.on is said to transforms like the nth row of the representation. We can write any function o(r),which can be normalized,as a sum of basis functions of the irreducible representations I'P of the group. p (r)= (6.12) p n=l Note,that this is not an expansion like the Fourier series.We don't expand a function with respect to an orthonormal complete set of functions here.The set of functions o(r)depends on o(r)itself.The functions op(r)in (6.12) can be found by means of the so called projection operators
6 Symmetry Properties of Electronic and Photonic Band Structures 109 (χi (Ck), χj (Ck) characters of elements in class Ck in irreducible representations Γi and Γj , Nk - number of elements in Ck. In the first relation the summation runs over all classes of the group. In the second relation it runs over all inequivalent irreducible representations of the group.) The character theorems can be easily checked for all the point groups using Mathematica. 6.4.2 Basis Functions of Irreducible Representations Our final goal is the investigation of the symmetry properties of the solutions of Maxwell’s equation. Therefore we have to associate the symmetry expressed by the symmetry group, i.e. the point group C4v in our example, with the symmetry properties of scalar and vector fields. We define a transformation operator to a symmetry element T as Pˆ(T). For a scalar function it holds: Pˆ(T)f(r) = f(T −1r) (6.9) The operators Pˆ(T) form a group of linear unitary operators. If we discuss Maxwell’s equations, we have to deal with vector fields in general. Instead of (6.9) the following transformation has to be applied to the vector field F: Pˆ(T)F(r) = TF(T −1r) (6.10) Now we define the basis functions of an irreducible representation (IR) and concentrate on scalar fields. If a set of l-dimensional matrices Γ(T) forms a representation of the group G and φ1(r),...,φl(r) is a set of linear independent functions such that Pˆ(T)φn(r) = l m=1 Γ(T)mnφm(r) n = 1, 2,...,l (6.11) then functions φn(r) are called partners in a set of basis functions of the representation Γ. φn is said to transforms like the nth row of the representation. We can write any function φ(r), which can be normalized, as a sum of basis functions of the irreducible representations Γp of the group. φ(r) = p lp n=1 φp n(r) (6.12) Note, that this is not an expansion like the Fourier series. We don’t expand a function with respect to an orthonormal complete set of functions here. The set of functions φp n(r) depends on φ(r) itself. The functions φp n(r) in (6.12) can be found by means of the so called projection operators
110 W.Hergert,M.Dane,and D.Kodderitzsch >(T)mP(T),Pn9(r)=6p(r)(6.13) T∈G ∑x(T)*P(T) (6.14) TeG The projection operators are implemented in the package. (cf.CharacterProjectionOperator[classes_,chars_,func_].)As an ex- ample,we investigate the symmetry properties of a function o(r)=(ax+ by)g(r)(a and b are constants).We apply the projection operators connected to the group Cav,to the function and get the following result (r)=Pio(r)=az g(r)Pi2o(r)=bz g(r) (6.15) Pio(r)=ayg(r) (r)=P2(r）)=byg(r） It is impossible to project out parts of (r)transforming like other IRs of Cav.Therefore our trial function o(r)=(ax+by)g(r)is a sum of functions, transforming like the representation I'5 (E). If we consider three-dimensional photonic crystals we cannot resort to scalar fields anymore.We have to take into account the full vectorial nature of the fields also in symmetry considerations.A more detailed discussion of the subject can be found in 6.14-6.16]. 6.5 Symmetry Properties of Schrodinger's Equation and Maxwell's Equations Here we want to investigate the symmetry properties of scalar or vector fields, which we get as solutions of Schrodinger's equation or Maxwell's equations. The operators P(T)form a group of linear unitary operators.This group is isomorphic to the group of symmetry elements T.The Hamilton-Operator H(r)of the time-independent Schrodinger equation H(r)=E(r)is given by h202 H(r)=-2m8m2+V(r). (6.16) For an arbitrary transformation T the transformation behavior of the Hamil- tonian is given by (r)=P(T)(Tr)P(T)-1. (6.17) For transformations which leave产invariant,.i.e.a(Tr）=H(r）we get: [a,P(T1=0. (6.18)
110 W. Hergert, M. D¨ane, and D. K¨odderitzsch Pp mn = lp g T∈G Γp(T) ∗ mnPˆ(T), Pp mnφq i (r) = δpqδni φp m(r) (6.13) Pp = lp g T∈G χp(T) ∗Pˆ(T) (6.14) The projection operators are implemented in the package. (cf. CharacterProjectionOperator[classes_,chars_,func_].) As an example, we investigate the symmetry properties of a function φ(r)=(a x + b y)g(r) (a and b are constants). We apply the projection operators connected to the group C4v, to the function and get the following result φ5 1(r) = P5 11 φ(r) = axg(r) P5 12 φ(r) = bxg(r) P5 21 φ(r) = ay g(r) φ5 2(r) = P5 22 φ(r) = by g(r) (6.15) It is impossible to project out parts of φ(r) transforming like other IRs of C4v. Therefore our trial function φ(r)=(a x + b y)g(r) is a sum of functions, transforming like the representation Γ5 (E). If we consider three-dimensional photonic crystals we cannot resort to scalar fields anymore. We have to take into account the full vectorial nature of the fields also in symmetry considerations. A more detailed discussion of the subject can be found in [6.14–6.16]. 6.5 Symmetry Properties of Schr¨odinger’s Equation and Maxwell’s Equations Here we want to investigate the symmetry properties of scalar or vector fields, which we get as solutions of Schr¨odinger’s equation or Maxwell’s equations. The operators Pˆ(T) form a group of linear unitary operators. This group is isomorphic to the group of symmetry elements T. The Hamilton-Operator Hˆ (r) of the time-independent Schr¨odinger equation Hψˆ (r) = E ψ(r) is given by Hˆ (r) = − 2 2m ∂2 ∂r2 + V (r) . (6.16) For an arbitrary transformation T the transformation behavior of the Hamiltonian is given by Hˆ (r) = Pˆ(T)Hˆ (Tr)Pˆ(T) −1 . (6.17) For transformations which leave Hˆ invariant, i.e. Hˆ (Tr) = Hˆ (r) we get: [H, ˆ Pˆ(T)] = 0 . (6.18)
6 Symmetry Properties of Electronic and Photonic Band Structures 111 All transformations,which let H invariant,form a group.The corresponding operators P(T)form an isomorphic group,the group of the Schrodinger equa- tion.All elements of the group of the Schrodinger equation commute with H.Because the operator of the kinetic energy is invariant under all rotations forming the group O(3),the symmetry of the Hamiltonian is determined exclusively by the potential V(r). We want to consider photonic crystals.Therefore we have to extend the analysis to Maxwell's equations. 7D(r,t)=0 VxE(r,)=-是B(r,) (6.19) 7B(r,t)=0 V×H(r,)=景D(r,t) We assume that we have no free charges and currents.Using the materials equations D=eeoE,B=uoH and a harmonic time dependence for the fields,i.e.E(r,t)=E(r)exp(iwt)the basic equations are given by E)=aTxW×Er)=()'Eo例 (6.20) aH(r)=V× ×H=()H e(r) (6.21) If we assume a two-dimensional photonic crystal,i.e.the dielectric constant varies in the x-y-plane(rl)we can resort to the solution of scalar equations for the field components in z-direction. E:(rI)=-c(rn) 1 02 2+2 E(r=() Ez(rI） (6.22) 咒H(r)=- 010. 01 Oz e(rl)Ox ay e(r)ay H(ru) =()H.o 3 (6.23) Corresponding to (6.18)we have to find all the transformations T which leave the operators n for the three-dimensional case or,for the two-dimensional case invariant.It can be shown that the group of Maxwell's equations is formed by the space group of the photonic crystal,i.e.consists of all translations and rotations which transform e(r)into itself. 6.6 Consequences of Lattice Periodicity Here we want to illustrate the consequences of lattice periodicity to the so- lution of Schrodinger equation or Maxwell's equation.We consider crystals which have d-dimensional translational symmetry and can be represented as a set of points sitting on a Bravais-lattice (for convenience we do not take
6 Symmetry Properties of Electronic and Photonic Band Structures 111 All transformations, which let Hˆ invariant, form a group. The corresponding operators Pˆ(T) form an isomorphic group, the group of the Schr¨odinger equation. All elements of the group of the Schr¨odinger equation commute with Hˆ . Because the operator of the kinetic energy is invariant under all rotations forming the group O(3), the symmetry of the Hamiltonian is determined exclusively by the potential V (r). We want to consider photonic crystals. Therefore we have to extend the analysis to Maxwell’s equations. ∇D(r, t)=0 ∇ × E(r, t) = − ∂ ∂tB(r, t) ∇B(r, t)=0 ∇ × H(r, t) = ∂ ∂tD(r, t) (6.19) We assume that we have no free charges and currents. Using the materials equations D = 0E, B = µ0H and a harmonic time dependence for the fields, i.e. E(r, t) = E(r) exp(iωt) the basic equations are given by ΞˆEE(r) = 1 (r) ∇ × (∇ × E(r)) = ω c 2 E(r) (6.20) ΞˆHH(r) = ∇ × 1 (r) ∇ × H(r) = ω c 2 H(r) (6.21) If we assume a two-dimensional photonic crystal, i.e. the dielectric constant varies in the x-y-plane (r||) we can resort to the solution of scalar equations for the field components in z-direction. Ξˆ2D E Ez(r||) = − 1 (r||) ∂2 ∂x2 + ∂2 ∂y2 Ez(r||) = ω c 2 Ez(r||) (6.22) Ξˆ2D H Hz(r||) = − ∂ ∂x 1 (r||) ∂ ∂x + ∂ ∂y 1 (r||) ∂ ∂y Hz(r||) = ω c 2 Hz(r||) (6.23) Corresponding to (6.18) we have to find all the transformations T which leave the operators ΞˆE, ΞˆH for the three-dimensional case or Ξˆ2D E , Ξˆ2D H for the two-dimensional case invariant. It can be shown that the group of Maxwell’s equations is formed by the space group of the photonic crystal, i.e. consists of all translations and rotations which transform (r) into itself. 6.6 Consequences of Lattice Periodicity Here we want to illustrate the consequences of lattice periodicity to the solution of Schr¨odinger equation or Maxwell’s equation. We consider crystals which have d-dimensional translational symmetry and can be represented as a set of points sitting on a Bravais-lattice (for convenience we do not take
112 W.Hergert,M.Dane,and D.Kodderitzsch into account a possible arrangement of basis atoms around the lattice points here).We want to denote the translation vectors which form the Bravais lattice {T}as miai mi=0,±1，±2.. (6.24) i=1 with a;being a basic lattice vector.Associated with the set [T}is a set of symmetry transformations (translations),which leave the crystal invariant. It can easily established that this set forms a infinite,discrete group,denoted here as T. Associated with every lattice point might be a set of symmetry operations consisting of rotations,mirror reflections and so on,which leave the crystal and the point they are applied to also invariant.The group formed from this set is called a point group 9o. Let us introduce a new symbols to state the considerations made so far more clearly.The Seitz-operator [Rt takes some vector r to a new vector r'by first rotating and then translating it r'=Rt'r=Rr +t. (6.25) Here R denotes a rotation matrix and t is a vector.Then the translation group T can be characterised by the elements [ET}(E is the unity matrix) and the elements of the point group elements as {RO}.The space group g of a crystal is defined as a group of operations [Rt which contains as a subgroup the set of all pure primitive translations of a lattice,T,but which contains no other pure translations.Here we want to consider only symmorphic space groups whose symmetry operations consists of a rotation followed by a primitive translation T (this excludes the case of having for examples glide-planes and screw axis as symmetry elements of the crystal, where fractions of T,t=T/n,n EN are involved). In what follows we want first use the properties of T to arrive at the Bloch- theorem which makes a statement about the form of the wavefunction in translational invariant systems.As stated above T is an infinite group behind of which the construct of an infinite crystal lies.To proceed we form a finite group from T by imposing periodic boundary conditions in d dimensions. We consider a building block of the crystal containing Ni x...x Na=Nd primitive cells.This imposes the following condition on the wavefunction of the crystal 3 This is done for convenience,because we then can use all statements for finite groups. 4 This is an approximation which is not severe if we disregard surface effects. Note,that whereas imposing periodic boundary conditions in one dimension can be thought of forming a ring out of a chain of lattice points,in three dimensions this is topological not possible
112 W. Hergert, M. D¨ane, and D. K¨odderitzsch into account a possible arrangement of basis atoms around the lattice points here). We want to denote the translation vectors which form the Bravais lattice {T} as T = d i=1 miai, mi = 0, ±1, ±2 ... (6.24) with ai being a basic lattice vector. Associated with the set {T} is a set of symmetry transformations (translations), which leave the crystal invariant. It can easily established that this set forms a infinite, discrete group, denoted here as T . Associated with every lattice point might be a set of symmetry operations consisting of rotations, mirror reflections and so on, which leave the crystal and the point they are applied to also invariant. The group formed from this set is called a point group G0. Let us introduce a new symbols to state the considerations made so far more clearly. The Seitz-operator {R|t} takes some vector r to a new vector r by first rotating and then translating it r = {R|t}r = Rr + t. (6.25) Here R denotes a rotation matrix and t is a vector. Then the translation group T can be characterised by the elements {E|T} (E is the unity matrix) and the elements of the point group elements as {R|0}. The space group G of a crystal is defined as a group of operations {R|t} which contains as a subgroup the set of all pure primitive translations of a lattice, T , but which contains no other pure translations. Here we want to consider only symmorphic space groups whose symmetry operations consists of a rotation followed by a primitive translation T (this excludes the case of having for examples glide-planes and screw axis as symmetry elements of the crystal, where fractions of T, t = T/n, n ∈ N are involved). In what follows we want first use the properties of T to arrive at the Blochtheorem which makes a statement about the form of the wavefunction in translational invariant systems. As stated above T is an infinite group behind of which the construct of an infinite crystal lies. To proceed we form a finite group3 from T by imposing periodic boundary conditions in d dimensions. We consider a building block of the crystal containing N1 ×···× Nd = Nd primitive cells.4 This imposes the following condition on the wavefunction of the crystal 3 This is done for convenience, because we then can use all statements for finite groups. 4 This is an approximation which is not severe if we disregard surface effects. Note, that whereas imposing periodic boundary conditions in one dimension can be thought of forming a ring out of a chain of lattice points, in three dimensions this is topological not possible.