What is Solid state Theory

what is the solid state according to kinetic molecular theory and solid state theory seminar, and solid state theory an introduction pdf free download
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Dr.LeonBurns,New Zealand,Researcher
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Solid State Theory Spring Semester 2014 Manfred Sigrist Institut fur  Theoretische Physik HIT K23.8 Tel.: 044-633-2584 Email: sigristitp.phys.ethz.ch Website: http://www.itp.phys.ethz.ch/research/condmat/strong/ Lecture Website: http://www.itp.phys.ethz.ch/education/fs14/sst Literature:  N.W. Ashcroft and N.D. Mermin: Solid State Physics, HRW International Editions, 1976.  C. Kittel: Einfuhrung  in die Festk orperphysik, R. Oldenburg Verlag, 1983.  C. Kittel: Quantentheorie der Festk orper, R. Oldenburg, 1970.  O. Madelung: Introduction to solid-state theory, Springer 1981; auch in Deutsch in drei B anden: Festk operphysik I-III, Springer.  J.M. Ziman: Principles of the Theory of Solids, Cambridge University Press, London, 1972.  M.P. Marder: Condensed Matter Physics, John Wiley & Sons, 2000.  G. Grosso & G.P. Parravicini: Solid State Physics, Academic Press, 2000.  G. Czychol: Theoretische Festk orperphysik, Springer 2004.  P.L. Taylor & O. Heinonen, A Quantum Approach to Condensed Matter Physics, Cam- bridge Press 2002.  G.D. Mahan, Condensed Matter in a Nutshell, Princeton University Press 2011.  numerous specialized books. 1Introduction Solid state physics (or condensed matter physics) is one of the most active and versatile branches of modern physics that have developed in the wake of the discovery of quantum mechanics. It deals with problems concerning the properties of materials and, more generally, systems with many degrees of freedom, ranging from fundamental questions to technological applications. This richness of topics has turned solid state physics into the largest sub eld of physics; furthermore, it has arguably contributed most to technological development in industrialized countries. Figure 1: Atom cores and the surrounding electrons. Condensed matter (solid bodies) consists of atomic nuclei (ions), usually arranged in a regular (elastic) lattice, and of electrons (see Figure 1). As the macroscopic behavior of a solid is determined by the dynamics of these constituents, the description of the system requires the use of quantum mechanics. Thus, we introduce the Hamiltonian describing nuclei and electrons, b b b b H =H +H +H ; (1) e n ne with 2 2 X X pb 1 e i b H = + ; e 0 2m 2 jr r j i i 0 i i6=i 2 b 2 X X P 1 Z Z 0e j j j b H = + ; (2) n 2M 2 jR R 0j j j j 0 j j=6 j 2 X Z e j b H = ; ne jr Rj i j i;j b b whereH (H ) describes the dynamics of the electrons (nuclei) and their mutual interaction and e n b H includes the interaction between ions and electrons. The parameters appearing are ne 31 m free electron mass 9:1094 10 kg 19 e elementary charge 1:6022 10 As 3 4 M mass of j-th nucleus  10 10m j Z atomic (charge) number of j-th nucleus j The characteristic scales known from atomic and molecular systems are 52 2 10 Length: Bohr radius a = =me  0:5 10 m B 2 4 2 2 2 Energy: Hartree e =a =me = =mc  27eV = 2Ry B 2 with the ne structure constant = e =c = 1=137. The energy scale of one Hartree is much less than the (relativistic) rest mass of an electron ( 0:5MeV), which in turn is considered small in particle physics. In fact, in high-energy physics even physics at the Planck scale is considered, at least theoretically. The Planck scale is an energy scale so large that even gravity is thought to be a ected by quantum e ects, as r r c G 2 19 35 E =c  10 GeV; l =  1:6 10 m; (3) Planck Planck 3 G c 11 3 1 2 whereG = 6:673 10 m kg s is the gravitational constant. This is the realm of the GUT (grand uni ed theory) and string theory. The goal is not to provide a better description of electrons or atomic cores, but to nd the most fundamental theory of physics. highe- nergy physics solid state physics astrophysics and cosmology 10 meV 10 eV 1 MeV metals electrons, cores phenomenological GUT semiconductors particle physics atom string theory magnets standard model superconductors Mt- heory ferroelectrics ...... effective most fundamental models theory known and established Figure 2: Energy scales in physics. In contrast, in solid state physics we are dealing with phenomena occurring at room temperature (T  300K) or below, i.e., at characteristic energies of about E k T  0:03eV = 30meV, B which is even much smaller than the energy scale of one Hartree. Correspondingly, the impor- tant length scales are given by the extension of the system or of the electronic wave functions. The focus is thus quite di erent from the one of high-energy physics. There, a highly successful phenomenological theory for low energies, the so-called standard model, exists, whereas the underlying theory for higher energies is unknown. In solid state physics, the situation is reversed. The Hamiltonian (1) describes the known 'high-energy' physics (on the energy scale of Hartree), and one aims at describing the low-energy properties using re- duced (e ective, phenomenological) theories. Both tasks are far from trivial. Among the various states of condensed matter that solid state theory seeks to describe are metals, semiconductors, and insulators. Furthermore, there are phenomena such as magnetism, superconductivity, ferroelectricity, charge ordering, and the quantum Hall e ect. All of these states share a common origin: Electrons interacting among themselves and with the ions through the Coulomb interaction. More often than not, the microscopic formulation in (1) is too compli- cated to allow an understanding of the low-energy behavior from rst principles. Consequently, the formulation of e ective (reduced) theories is an important step in condensed matter theory. On the one hand, characterizing the ground state of a system is an important goal in itself. How- ever, measurable quantities are determined by excited states, so that the concept of 'elementary excitations' takes on a central role. Some celebrated examples are Landau's quasiparticles for 6Fermi liquids, the phonons connected to lattice vibrations, and magnons in ferromagnets. The idea is to treat the ground state as an e ective vacuum in the sense of second quantization, with the elementary excitations as particles on that vacuum. Depending on the system, the vacuum may be the Fermi sea or some state with a broken symmetry, like a ferromagnet, a superconductor, or the crystal lattice itself. 1 According to P. W. Anderson, the description of the properties of materials rests on two princi- ples: The principle of adiabatic continuity and the principle of spontaneously broken symmetry. By adiabatic continuity we mean that complicated systems may be replaced by simpler systems that have the same essential properties in the sense that the two systems may be adiabatically deformed into each other without changing qualitative properties. Arguably the most impres- sive example is Landau's Fermi liquid theory mentioned above. The low-energy properties of strongly interacting electrons are the same as those of non-interacting fermions with renormal- ized parameters. On the other hand, phase transitions into states with qualitatively di erent properties can often be characterized by broken symmetries. In magnetically ordered states the rotational symmetry and the time-reversal invariance are broken, whereas in the superconduct- ing state the global gauge symmetry is. In many cases the violation of a symmetry is a guiding principle which helps to simplify the theoretical description considerably. Moreover, in recent years some systems have been recognized as having topological order which may be considered as a further principle to characterize low-energy states of matter. A famous example for this is found in the context of the Quantum Hall e ect. The goal of these lectures is to introduce these basic concepts on which virtually all more elab- orate methods are building up. In the course of this, we will cover a wide range of frequently encountered ground states, starting with the theory of metals and semiconductors, proceeding with magnets, Mott insulators, and nally superconductors. 1 P.W. Anderson: Basic Notions of Condensed Matter Physics, Frontiers in Physics Lecture Notes Series, Addison-Wesley (1984). 7Chapter 1 Electrons in the periodic crystal - band structure One of the characteristic features of many solids is the regular arrangement of their atoms forming a crystal. Electrons moving in such a crystal are subject to a periodic potential which originates from the lattice of ions and an averaged electron-electron interaction (like Hartree- Fock approximation). The spectrum of extended electronic states, i.e. delocalized eigenstates of the Schrodinger  equation, form bands of allowed energies and gaps of "forbidden" energies. There are two limiting starting points towards the understanding of the band formation: (1) the free electron gas whose continuous spectrum is broken up into bands under the in uence of a periodic potential (electrons undergo Bragg scattering); (2) independent atoms are brought together into a lattice until the outer-most electronic states overlap and lead to delocalized states turning a discrete set of states into continua of electronic energies - bands. In this chapter we will address the emergence of band structures from these two limiting cases. The band structure of electrons is essential for the basic classi cation of materials into metals and insulators (semiconductors). 1.1 Symmetries of crystals 1.1.1 Space groups of crystals Most solids consist of a regular lattice of atoms with perfectly repeating structures. The minimal repeating unit of such a lattice is the unit cell. The symmetries of a crystal are contained in the space groupR, a group of symmetry operations (translations, rotations, the inversion or combinations) under which the crystal is left invariant. In three dimensions, there are 230 1 di erent space groups (cf. Table 1.1). 1 All symmetry transformations form together a set which has the properties of a group. A groupG combined with a multipliation "" has the following properties:  the product of two elements ofG is also inG: a;b2G ) ab =c2G.  multiplications are associative: a (bc) = (ab)c.  a unit element e2G exists with: ea =ae =a for all a2G. 1 1 1  for every element a2G there is an inverse a 2G with a a =aa =e. A group with ab =ba for all pairs of element is called Abelian group, otherwise it is non-Abelian. A subset 0 G G is called a subgroup ofG, if it is a group as well. Guides to group theory in the context of condensed matter physics can be found in the textbooks - Mildred S. Dresselhaus, Gene Dresselhaus and Ado Jorio: Group Theory - Application to the Physics of Condensed Matter - Peter Y. Yu and Manuel Cardona: Fundamentals of Semiconductors, Springer. 8We consider here a crystal displayed as a point lattice, each point either symbolizing an atom or a whole unit cell (Fig.1.1 for a square lattice). Translations in the space group are represented by linear combinations of a basic set of translation vectorsfag (primitive lattice vectors) connecting i lattice points. Any translation that maps the lattice onto itself is a linear combination of the fag with integer coecients, i R =n a +n a +n a : (1.1) n 1 1 2 2 3 3 a 2 R a 1 Figure 1.1: Crystal point lattice in two dimensions: the vectors a anda form the basic set 1 2 of translations andR = 2a +a in the gure. The shaded area is the Wigner-Seitz cell which 1 2 is obtained by drawing perpendicular lines (planes in three dimensions) through the center of all lines connecting neighboring lattice points. The Wigner-Seitz cell also constitutes a unit cell of the lattice. General symmetry transformations including general elements of the space group may be written in the notation due to Wigner, 0 r =gr +a =fgjagr; (1.2) where g represents a rotation, re ection or inversion with respect to lattice points, axes or planes. The elements g form the generating point groupP. In three dimensions there are 32 point groups. We distinguish the following basic symmetry operations: basic translations fEjag, rotations, re ections, inversions fgj0g, screw axes, glide planes fgjag, whereE is the unit element (identity) ofP. A screw axis is a symmetry operation of a rotation followed by a translation along the rotation axis. A glide plane is a symmetry operation with re ection at a plane followed by a translation along the same plane. The symmetry operations fgjag, together with the associative multiplication 0 0 0 0 fgjagfgjag =fggjga +ag (1.3) form a group with unit elementfEj0g. In general, these groups are non-Abelian, i.e., the group elements do not commute with each other. However, there is always an Abelian subgroup of R, the group of translationsfEjag. The elements g2P do not necessarily form a subgroup, 9because some of these elements (e.g., screw axes or glide planes) leave the lattice invariant only in combination with a translation. Nevertheless, the relation 0 1 0 fgjagfEjagfgjag =fEjgag (1.4) 1 0 1 0 fgjag fEjagfgjag =fEjg ag (1.5) holds generally. IfP is a subgroup ofR, thenR is said to be symmorphic. In this case, the space group contains only primitive translationsfEjag and neither screw axes nor glide planes. 2 The 14 Bravais lattices are symmorphic. Among the 230 space groups 73 are symmorphic and 157 are non-symmorphic. crystal system point groups space group numbers ( point groups, space groups) Sch on ies symbols international tables triclinic (2,2) C ;C 1-2  1 1 monoclinic (3,13) C ;C ;C 3-15 2 s 2h orthorhombic (3,59) D ;C ;D 16-74 2 2v 2h tetragonal (7,68) C ;S ;C ;D ;C ;D ;D 75-142 4 4 4h 4 4v 2d 4h trigonal (5,25) C ;S ;D ;C ;D 143-167 3 6 3 3v 3d hexagonal (7,27) C ;C ;C ;D ;C ;D ;D 168-194 6 3h 6h 6 6v 3h 6h cubic (5, 36) T;T ;O;T ;O 195-230 h d h Table 1.1: List of the point and space groups for each crystal system in three dimensions. 1.1.2 Reciprocal lattice We de ne now the reciprocal lattice which is of importance for the electron band structure and x-ray di raction on a periodic lattice. The reciprocal lattice is also perfectly periodic with a translation symmetry with a basic setfbg de ning arbitrary reciprocal lattice vectors as i G =m b +m b +m b ; (1.6) m 1 1 2 2 3 3 where m are integers and i a b = 2 ; i;j = 1; 2; 3; (1.7) i j ij such that a a b b j k j k b = 2 and a = 2 : (1.8) i i a  (a a ) b  (b b ) i j i j k k The reciprocal lattice of a simple cubic lattice is simple cubic. However, a body centered cubic (bcc) lattice has a face centered cubic (fcc) reciprocal lattice and vice versa (see slides). It follows that any real space lattice vector R and any reciprocal lattice vector G have the n m property that G R = 2(m n +m n +m n ) = 2N (1.9) m n 1 1 2 2 3 3 2 Crystal systems, crystals, Bravais lattices are discussed in more detail in - Czycholl, Theoretische Festk orperphysik, Springer 10with N being an integer. This allows us to expand any function f(r) periodically in the real lattice as X iGr f(r) = f e (1.10) G G with the obvious property: f(r +R) =f(r). The coecients are given by Z 1 3 iGr f = d rf(r)e (1.11) G UC UC where the integral runs over the unit cell of the periodic lattice with the volume . Finally, we UC de ne the ( rst) Brillouin zone as the "Wigner-Seitz cell" constructed in the reciprocal lattice (see Fig.1.1 and 1.2). 1.2 Bloch's theorem and Bloch functions We consider a HamiltonianH of electrons invariant under a discrete set of lattice translations ffEjagg, a symmetry introduced by a periodic potential. This implies that the corresponding b translation operator T on the Hilbert space commutes with the HamiltonianH =H +H a e ie (purely electronic HamiltonianH , interaction between electrons and ionsH ), e ie b T ;H = 0: (1.12) a b b This translation operator is de ned throughT jri =jr +ai andhrjT =hraj. Neglecting the a a interactions among electrons, which would be contained inH , we are left with a single particle e problem 2 pb HH = +V (rb); (1.13) 0 2m whererb andpb are position and momentum operators, and X V (r) = V (rR ); (1.14) ion j j describes the potential landscape of the single particle in the ionic background. WithR being j the position of thej-th ion, the potentialV (r) is by construction periodic, withV (r+a) =V (r) b for all lattice vectorsa, and representsH . Therefore,H commutes with T . For a Hamilto- ie 0 a b nianH commuting with the translation operator T , the eigenstates ofH are simultaneously 0 a 0 b eigenstates of T . a b Bloch's theorem states that the eigenvalues of T lie on the unit circle of the complex plane, a 3 which ensures that these states are extended. This means l l b b b T (r) = (ra) = (r); T (r) = (rla) =T (r) = (r) (1.16) a a la a a withl an integer (positive or negative). In order to be bounded (renormalizable) and delocalized in the periodic potential the wave function satis es, 2 2 l 2 2 2l 2 j (r)j =j (rla)j =j jj (r)j =j j j (r)j ; (1.17) a a requiring i' a j j = 1 )  =e : (1.18) a a 3 Transformation of wave function: 0 0 0 b b j i =T j i ) (r) =hrj i =hrjT j i =hraj i = (ra): (1.15) a a 11ikr This condition is satis ed if we express the wave function as product of a plane wave e and a periodic Bloch function u (r) k 1 ikr p (r) = e u (r): (1.19) n;k n;k with b T u (r) =u (ra) =u (r); (1.20) a n;k n;k n;k ika b T (r) = (ra) =e (r); (1.21) a n;k n;k n;k H (r) = (r): (1.22) 0 n;k n;k n;k The integern is a quantum number called band index,k is the pseudo-momentum (wave vector) and represents the volume of the system. Note that the eigenvalue of (r) with respect n;k ika i(k+G)a ika b to T , e , implies periodicity in the reciprocal space, thek-space, because e =e a for all reciprocal lattice vectorsG. We may, therefore, restrictk to the rst Brillouin zone and  = . n;k+G n;k Bloch's theorem simpli es the initial problem to the so-called Bloch equation for the periodic function u , k   2 b (p +k) +V (rb) u (r) = u (r); (1.23) k k k 2m where we suppress the band index to simplify the notation. This equation follows from the relation ikr ikr pbe u (r) =e (pb +k)u (r); (1.24) k k which can be used for more complex forms of the Hamiltonian as well. There are various numerical methods which allow to compute rather eciently the band energies  for a given k HamiltonianH. 1.3 Nearly free electron approximation We start here from the limit of free electrons assuming the periodic potential V (r) is weak. Using Eqs.(1.10) and (1.11) we expand the periodic potential, X iGr V (r) = V e ; (1.25) G G Z 1 3 iGr V = d rV (r)e : (1.26) G UC UC The potential is real and we assume it also to be invariant under inversion (V (r) = V (r))  withr = 0 being an inversion center of the crystal lattice, leading to V =V =V . Note G G G that the uniform componentV corresponds to an irrelevant energy shift and may be set to zero. 0 Because of its periodicity, the Bloch function u (r) is expressed in the same way, k X iGr u (r) = c e ; (1.27) k G G where the coecients c =c (k) are functions ofk, in general. Inserting this Ansatz and the G G expansion (1.25) into the Bloch equation, (1.23), we obtain a linear eigenvalue problem for the band energies  , k   2 X 2 0 0 (kG)  c + V c = 0: (1.28) k G GG G 2m 0 G 12This represents an eigenvalue problem in in nite dimensions with eigenvectors c (k) and eigen- G values  as band energies. These  include corrections to the bare parabolic dispersion, k k (0) (0) 2 2  = k =2m, due to the potential V (r). Obviously, the dispersion  is naturally parabolic k k in absence of the potential V (r) whereby the eigenstates would be simply plane waves. As a lowest order approach we obtain the approximative energy spectrum within the rst Brillouin 2 2 zone, considering only all parabolic bands of the type  (G) = (kG) =2m centered around k the reciprocal wave vectorsG (see dashed line in Fig.1.2)). Example: 1-dimensional system We illustrate here the nearly free electron method using the case of a one-dimensional lattice. 2 2 Assuming that the periodic modulation of the potential is weak, sayjV j G =2m (takingG G as a characteristic wavevector in the periodic system), the problem (1.28) can be simpli ed. Let us start with the lowest energy values around the center of the rst Brillouin zone, i.e. k 0 (jkj=a). For the lowest energy eigenvalue we solve Eq.(1.28) by 8 1 for G = 0 c  (1.29) 2mV G G :  1 for G =6 0 2 2 2 f(kG) kg leading to the energy eigenvalue 2 2 2 2 2 X k jV j k G    +E (1.30) k 0 2  2 2 2m 2m (kG) k G6=0 2m with 2 X 2 2 E = j j G (1.31) 0 G 2m G6=0 and 8 9 = X 1 1 2 = 1 4 j j (1.32) G  m m: ; G=0 6 2 2 with =V =f G =2mg (j j 1). We observe that  is parabolic with a slightly modi ed G G G k  (e ective) mass,mm m. Note that this result resembles the lowest order corrections in the Rayleigh-Schr odinger perturbation theory for a non-degenerate state. This solution corresponds to the lowest branch of the band structure within this approach (see Fig.1.2). The parabolic  approximation of the band structure at a symmetry point with an e ective massm , is a standard way to approximate band tops or bottoms. It is calledkp-approximation ("k-dot-p"). We stay at the zone center and address the next eigenstates which are dominated by the parabola 2 2 originating from G =2=a =G which cross for k = 0 at a value G =2m. Restricting  ourselves to these two components we obtain the two-dimensional eigenvalue equation system, 0 10 1 2 2 c G (kG)  V k 2G A A = 0: (1.33) 2m 2 2 c V (k +G)  G k 2G 2m The eigenvalues are obtained through the secular equation, 2 3 2 2  (kG)  V k 4 5 det 2G = 0; (1.34) 2m 2 2 V (k +G)  k 2G 2m 13leading to 2 3 s   n o n o 2 2 2 1 2 2 2 2 2 4 5  = (k +G) + (kG)  (k +G) (kG) + 4jV j k 2G 2 2m 2m 2 2 2 2 = G jV j + k 2G  2m 2m  (1.35) with the e ective mass  1 1 1  1 2j j (1.36)  G m m    where m 0 and m 0 asj j 1. We observe a energy band gap separating two bands + G  with opposite curvature (see Fig.1.2). Note that the curvature diverges for V 0 asm 0.  2G The wavefunctions at k = 0 are given by  V () () () 2G c = c =c (1.37) G +G +G jV j 2G where we have chosen V to be real and positive. Thus, 2G 8 sinGx for  k=0;+ u (x) = (1.38) k=0 : cosGx for  k=0; one being even and the other odd under parity operation xx. A similar analysis can be done at the boundary of the rst Brillouin zone where two energy parabolas cross. For example at k ==a we nd the two dominant contributions originate fromG = +2=a and2=a, respectively, together withG = 0. Also here the energy eigenvalues show a band gap with parabolic bands centered at k ==a (boundary of the rst Brillouin zone in one-dimension) with positive and negative e ective mass (see Fig.1.2). Analogous as for the band center we can distinguish the wavefunction with even and odd parity for the two bands at the Brillouin zone boundary. Indeed every crossing energy parabola centered around di erent reciprocal lattice points contributes to a band gap. By construction we can extend the band structure beyond the rst Brillouin zone and nd a periodic energy spectrum with  = (1.39) k+G k whereG is a reciprocal lattice "vector". Moreover, we nd in Fig.1.2 that = due to parity k k as well as time reversal symmetry, like for free electrons. 1.4 Tight-binding approximation We consider now a regular lattice of atoms which are well separated such that their atomic orbitals have small overlaps only. Therefore, in a good approximation the electronic states are rather well represented by localized atomic orbitals,  (r). The discrete spectrum of the atoms n is obtained with the atomic Hamiltonian, H (R) (rR) =  (rR); (1.40) a n n n for an atom located at positionR, so that 2 p H (R) = +V (rR) (1.41) a a 2m 14E band gap 1st Brillouin zone 2/// 2/ − − 0 k a a a a Figure 1.2: Band structure obtained by the nearly free electron approximation for a regular one-dimensional lattice. withV (r) as the rotation symmetric atomic potential as shown in Fig.1.3 a). The indexn shall a include all necessary quantum numbers, besides the principal quantum number also angular momentum (l;m) and spin. The single-particle Hamiltonian combines all the potentials of the atoms on the regular lattice (see Fig.1.3 b)), 2 X p H = + V (rR ) =H (R ) + V (r) (1.42) a j a j R j 2m R j where we single out one atomic potential (the choice of R is arbitrary) and introduce the j correction X 0 V (r) = V (rR ): (1.43) a R j j R 6=R 0 j j a) b) V V extended states localized atomic orbitals Figure 1.3: Potential landscape: a) a single atomic Coulomb potential yields a discrete spectrum electronic states; b) atoms arranged in a regular lattice give rise to a periodic potential which close to the atom sites look much like the attractive Coulomb-like potential. Electron states of low energy can be considered as practically localized at the atom sites, as the extension of their wave functions is very small. The higher energy states, however, extend further and can delocalize to form itinerant electron states which form bands. 151.4.1 Linear combination of atomic orbitals - LCAO We use here a linear combination of atomic orbitals (LCAO) to approximate the extended Bloch states X 1 ikR j (r) =p e  (rR ); (1.44) n k n j N R j whereN denotes the number of lattice sites. This superposition has obviously the properties of ika 4 a Bloch function through (ra) =e (r) for all lattice vectorsa. Note that this is n k n k similar to the Hund-Mullikan ansatz for molecular orbitals. First we determine the norm of this Bloch function, Z Z X 1 3  3 ik(R R )  0 j j 0 0 0 0 h1i (k) = d r (r) (r) = d re  (rR ) (rR ) n n n k n k j n j n N R ;R j 0 j Z X 3 ikR  j = d re  (rR ) 0(r) j n n R j X ikR j 0 0 = + e (R ) n n n n j R6=0 j (1.46) where due to translational invariance in the lattice we may setR 0 = 0 eliminating the sum over j R 0 and dropping the factor 1=N. To estimate the energy we calculate, j Z X 1 3 ik(R0R )  j j hHi 0(k) = d re  (rR )fH (R 0) + V (r)g 0(rR 0) n n j a j R n j n 0 j N R ;R0 j j Z X 1 3 ik(R0R )  j j =E 0h1i 0(k) + d re  (rR )V (r) 0(rR 0) n n n j R n j n 0 j N R ;R0 j j X ikR j =E 0h1i 0(k) + E 0 + e 0(R ) n n n n n n n j R =0 6 j (1.47) where Z 3  0 0 E = d r (r)V (r) (r) (1.48) R =0 n n n 0 n j and Z 3  0(R ) = d r (rR )V (r) 0(r): (1.49) n n j j R =0 n n 0 j From this we can now calculate the band energies through the secular equation, det hHi 0(k) h1i 0(k) = 0: (1.50) n n k n n The merit of the approach is that the tightly bound atomic orbitals have only weak overlap such that both 0(R ) and 0(R ) fall o very quickly with growingR . Mostly it is sucient to n n j n n j j 4 We apply the translation operation to the wave function (1.44), X X 0 1 1 ikR ik(R a) 0 b j j T (r) = (ra) =p e  (raR ) =p e  (rR ) a n k n k n j n j z N N R R j j 0 R j (1.45) X 0 1 ika ikR 0 ika j =e p e  (rR ) =e (r): n n k j N 0 R j 16takeR connecting nearest-neighbor and sometimes next-nearest-neighbor lattice sites. This is j for example ne for bands derived from 3d-orbitals among the transition metals such as Mn, Fe or Co etc.. Also transition metal oxides are well represented in the tight-binding formulation. Alkali metals in the rst row of the periodic table, Li, Na, K etc. are not suitable because their outermost s-orbitals have generally a large overlap. Note that the construction of the Hamiltonian matrix ensures that k k +G does not change  , if G is a reciprocal lattice k vector. 1.4.2 Band structure of s-orbitals The most simple case of a non-degenerate atomic orbital is the s-orbital with vanishing angular momentum (` = 0). Since these orbitals have rotation symmetric wavefunctions, (r) = (jrj), s s the matrix elements only depend on the distance between sites,jRj. As an example we consider j a simple cubic lattice taking nearest-neighbor (R =(a; 0; 0); (0;a; 0) and(0; 0;a)) and j next-nearest-neighbor coupling (R = (a;a; 0); (a; 0;a); (0;a;a)) into account. For j simplicity we will neglect the overlap integrals (R ), as they are not important to describe ss j the essential feature of the band structure. 8 t R connects nearest neighbors j (R ) = (1.51) ss j : 0 t R connects next nearest neighbors j which leads immediately to n:n: n:n:n: X X ikR 0 ikR j j  =E + E t e t e k s s R R j j (1.52) =E + E 2tfcos(k a) + cos(k a) + cos(k a)g s s x y z 0 4t cos(k a) cos(k a) + cos(k a) cos(k a) + cos(k a) cos(k a)g x y y z z x 0 Note that V (r) 0 in most cases due to the attractive ionic potentials. Therefore t;t 0. R j There is a single band resulting from this s-orbital, as shown in Fig.1.4. We may also consider thekp-approximation atk = 0 which yields an e ective mass 2 0 2  =E + E + 6t + 12t + k + (1.53) k s s  2m with 1 2 0 = (t + 4t ): (1.54)  2 m 0 Note thatt andt shrink quickly, if with growing lattice constanta the overlap of atomic orbitals decreases. 1.4.3 Band structure of p-orbitals We turn to the case of degenerate orbitals. The most simple case is the p-orbital with angular momentuml = 1 which is three-fold degenerate, represented by the atomic orbital wavefunctions of the form,  (r) =x'(r);  (r) =y'(r);  (r) =z'(r); (1.55) x y z with'(r) being a rotation symmetric function. Note thatfx;y;zg can be represented by spher- ical harmonics Y . We assume again a simple cubic lattice such that these atomic orbitals 1;m remain degenerate. Analyzing the properties of the integrals by symmetry, we nd, E =E =E =E and E 0 = E  0 : (1.56) x y z p n n p n n 17s-orbitals p-orbitals Figure 1.4: Band structures derived from atomic orbitals with s- (one band, upper panel) and p-symmetry (three bands, lower panel) in a simple cubic lattice. Left side: First Brillouin zone of the simple cubic lattice. Dispersion given along thek-line connecting XRM. We 0 0 0 00 choose the parameters: t = 0:2t for the s-orbitals; t = 0:2t, t = 0:1t, t = 0:05t and t = 0:15t. For the band derived from atomic p-orbitals, the irreducible representations of the bands are given at the symmetry points: (d = 3);X (d = 1) ,X (d = 2);R (d = 3);M (d = 1), 15 2 5 15 2 M (d = 2) where d is the dimension of the representation showing the degeneracy. 5 The overlaps Eq.(1.49) for nearest neighbors, 8 t R = (a; 0; 0)kx ( bonding) j (R ) = (1.57) xx j : 0 t R = (0;a; 0); (0; 0;a)?x ( bonding) j 0 and analogous for and , while 0 = 0, ifn 6=n . For next-nearest neighbors by symmetry yy zz n n we obtain, 8 t R = (a;a; 0); (a; 0;a) j (R ) = (1.58) xx j : 0 t R = (0;a;a) j and analogous for and . Next-nearest neighbor coupling also allows for inter-orbital yy zz matrix elements, e.g. 00 (R ) = (R ) =t sign(R R ) (1.59) xy j yx j nx ny for R = (a;a; 0) and analogous for (R ) and (R ). The di erent con guration of j yz j zx j nearest- and next-nearest-neighbor coupling is shown in Fig.1.5. Now we may setup the coupling matrix, 0 1 00 00 E (k) 4t sin(k a) sin(k a) 4t sin(k a) sin(k a) x x y x z 00 00 A 0 hHi = 4t sin(k a) sin(k a) E (k) 4t sin(k a) sin(k a) (1.60) x y y y z n n 00 00 4t sin(k a) sin(k a) 4t sin(k a) sin(k a) E (k) x z y z z 18next- nearest neighbors nearest neighbors " - bonding - bonding no coupling Figure 1.5: The con gurations for nearest- and next-nearest-neighbor coupling between p- orbitals on di erent sites. The p-orbitals are depicted by the dumb-bell structured wavefunction with positve (blue) and negative (red) lobes. For nearest-neighbor couplings we distinguish here -bonding (full rotation symmetry around connecting axis) and -bonding (two-fold rotation symmetry around connecting axis). Generally the coupling is weaker for - than for-bonding. No coupling for symmetry reasons are obtained between orbitals in the lower panel. with 0 E (k) = E + E + 2t cos(k a) 2t (cos(k a) + cos(k a)) x p p x y z (1.61) 0 +4t cos(k a)(cos(k a) + cos(k a)) 4t cos(k a) cos(k a) x y z y z and analogous forE (k);E (k). The three bands derived from the atomic p-orbitals are obtained y z by solving the secular equation of the type Eq.(1.50) and shown in Fig.1.4. Also in this case we may consider akp-approximation around a symmetry point in the Brillouin zone. For the -point we nd the expansion aroundk = 0: 0 1 2 2 2 Ak +B(k +k ) Ck k Ck k x y x z x y z 2 2 2 A 0 hHi =E + Ck k Ak +B(k +k ) Ck k (1.62) x y y z n n y z x 2 2 2 Ck k Ck k Ak +B(k +k ) x z y z z x y 0 0 2 2 0 0 00 withE =E + E + 2t 4t + 4t 4t ,A =a (t + 4t),B =a (t 2t + 2t ) andC =4t . p p These band energies have to be determined through the secular equations and lead to three bands with anisotropic e ective masses. 1.4.4 Wannier functions An alternative approach to the tight-binding approximation is through Wannier functions. These are de ned as the Fourier transformation of the Bloch wave functions, X X 1 1 ikR ikR p p (r) = e w(rR) w(rR) = e (r) (1.63) k k N N R k 195 where the Wannier functionw(rR) is centered on the real-space lattice siteR. We consider here the situation of a non-degenerate band analogous to the atomic s-orbital case, such that there is only one Wannier function per site. Wannier functions obey the orthogonality relation Z Z X 1 0 0 3  0 ikRikR 3  d rw (rR )w(rR) = e d r (r) (r) 0 k k N 0 k;k (1.65) X 1 0 0 ikRikR = e  0 = 0: kk RR N 0 k;k 2 2 We consider the one-particle Hamiltonian to be of the formH = r =2m +V (r), with a periodic potential V (r). Then,  can be expressed through k Z Z X 1 0 3  ik(RR) 3  0  = d r (r)H (r) = e d rw (rR )Hw(rR) k k k N 0 R;R (1.66) Z X ikR 3  = e d rw (rR)Hw(r); R where we took translational invariance of the lattice into account. With the de nitions Z 3   = d rw (r)Hw(r); (1.67) 0 Z 3  t(R) = d rw (rR)Hw(r) for R =6 0 (1.68) the band energy can be written as X ikR  = + t(R)e : (1.69) k 0 R This is the same type of tight-binding band structure as we have derived above from the LCAO view point. We can extend the Wannier function to the case of several bands, like the p-orbital bands. Then we de ne X 1 ikR (r) =p e c (k)w (rR) (1.70) nk nn n N R;n where for allk, X  0 0 c (k)c (k) = : (1.71) nn nn nn n The matrix c (k) rotates the Wannier function from the band basis into the atomic orbital nn basis, i.e. for p-bands into the three Wannier function with symmetry likefx;y;zg. 5 Ambiguity of the Wannier functions: The Wannier function is not uniquely de ned, because there is a "gauge freedom" for the Bloch function which can be multiplied by a phase factor i(k) (r)e (r) (1.64) k k where(k) is an arbitrary real function. In particular, we nd di erent degrees of localization ofw(rR) around its centerR depending on the choice of (k). 20Again we can express the band energy in terms of a tight-binding Hamiltonian, Z 3   = d r (r)H (r) nk nk nk Z XX  ikR 3  = c 0 (k)c (k)e d rw 0(rR)Hw (r) nn n n n n (1.72) 0 R n; n XX  ikR 0 = c 0 (k)c (k)e t (R): nn n n n n 0 R n; n 1.4.5 Tight binding model in second quantization formulation The tight-binding formulation of band electrons can also be implemented very easily in second quantization language and provides a rather intuitive interpretation. For simplicity we restrict ourselves to the single-orbital case and de ne the following Fermionic operators, y c creates an electron of spin s on lattice siteR ; j j;s (1.73) c annihilates an electron of spin s on lattice siteR ; j;s j in the corresponding Wannier states. We introduce the following Hamiltonian, X X y y H =  c c + t c c (1.74) 0 j;s ij j;s j;s i;s j;s i;j y with t = t real. These coecients t are called "hopping matrix elements", since c c ij ji ij j;s i;s annihilates an electron on site R and creates one on site R , in this way an electron moves j i (hops) fromR toR . Thus, this Hamiltonian represents the "kinetic energy" of the electron. j i Let us now diagonalize this Hamiltonian by following Fourier transformation, equivalent to the transformation between Bloch and Wannier functions, X X 1 1 y y ikR ikR j j p p c = a e and c = a e (1.75) j;s ks j;s ks N N k k y where a (a ) creates (annihilates) an electron in the Bloch state with pseudo-momentum k ks ks and spin s. Inserting Eq.(1.75) into the Hamiltonian (1.74) leads to 8 9 = X X X X 1 0 1 0 y y i(kk )R ikRikR i j i H =  e + t e a a =  a a (1.76) 0 ij 0 ks k ks ks k s :N N ; 0 i i;j k;s k;k ;s y wherea a =n constitutes the number operator for electrons. The band energy is the same ks ks ks as obtained above from the tight-binding approach. The Hamiltonian (1.74) will be used later for the Hubbard model where a real-space formulation is helpful. The real-space formulation of the kinetic energy allows also for the introduction of disorder, non- periodicity which can be most straightforwardly implemented by site dependent potentials  0  and to spatially (bond) dependent hopping matrix elements t =t(R;R )6=t(R R ). 0i ij i j i j 1.5 Symmetry properties of the band structure The symmetry properties of crystals are a helpful tool for the analysis of their band structure. They emerge from the symmetry group (space and point group) of the crystal lattice. Consider 6 b the action S of an elementfgjag of the space group on a Bloch wave function (r) k fgjag 6 In Dirac notation we write for the Bloch state with pseudo-momentum k as (r) =hrj i: (1.77) k k 211 1 1 b S (r) = (fgjag r) = (g rg a): (1.81) fgjag k k k b Becausefgjag belongs to the space group of the crystal, we have S ;H = 0. Applying a 0 fgjag b b 0 pure translation T =S 0 to this new wave function and using Eq.(1.5) a fEjag 1 0 ik(g a ) b b b b b T 0S (r) =S T 1 0 (r) =S e (r) a fgjag k fgjag g a k fgjag k 0 i(gk)a b =S e (r) fgjag k 0 i(gk)a b =e S (r); (1.82) fgjag k 0 i(gk)a b the latter is found to be an eigenfunction of T 0 with eigenvalue e . Remember, that, a b according to the Bloch theorem, we chose a basisf g diagonalizing both T andH . Thus, k a 0 7 apart from a phase factor, the action of a symmetry transformationfgjag on the wave function corresponds to a rotation fromk to gk. b S (r) = (r); (1.84) fgjag k fgjag gk 2 withj j = 1, or fgjag b S jki = jgki: (1.85) fgjag fgjag in Dirac notation. Then it is easy to see that 1 b b  =hgkjHjgki =hkjS H S jki =hkjHjki = : (1.86) gk 0 0 0 k fgjag fgjag Consequently, there is a star-like structure of equivalent points gk with the same band energy ( degeneracy) for eachk in the Brillouin zone (cf. Fig. 1.6). For a general point k the number of equivalent points in the star equals the number of point b The action of the operator S on the statejri is given by fgjag 1 1 b b S jri =jgr +ai and hrjS =hg rg aj; (1.78) fgjag fgjag such that 1 1 b hrjS j i = (g rg a): (1.79) k k fgjag The same holds for pure translations. Note that this de nition has also implications on the sequential application of transformation operators such as 1 b b b b b S S (r) =hrjS S j i =hfgjag rjS j i fg ja g fg ja g k fg ja g fg ja g k 1 1 fg ja g k 1 1 2 2 1 1 2 2 2 2 (1.80)  1 1 1 1 =hfgjag fgjag rj i = fgjag fgjag r : 2 2 1 1 k k 2 2 1 1 This is important in the context of Eq.(1.82). 7 Symmetry behavior of the wave function. X X 1 1 1 1 1 1 ikr iGr ik g rg a iG g rg a ( ) ( ) b b S (r) =p S e c (k)e =p e c (k)e fgjag k fgjag G G G G X X 1 1 i(gk)a i(gk)r i(gG)r i(gk)a i(gk)r iGr =p e e c (k)e =e p e c 1 (k)e G g G (1.83) G G X 1 i(gk)a i(gk)r iGr =e p e c (gk)e = (r); G gk fgjag G b where we use the fact thatc =c (k) is a function ofk with the propertyc (k) =c (gk) i.e. S u (r) = G G 1 G g G fgjag k u (r). gk 22Figure 1.6: Star of k-points in the Brillouin zone with degenerate band energies: Left panel: Star ofk; Right panel: contour plot of a two dimensional band =2tfcos(k a)+cos(k a)g+ k x y 0 4t cos(k a) cos(k a). The dots correspond to the star ofk with degenerate energy values, demon- x y strating  = . nk n;gk group elements for thisk (without inversion). Ifk lies on points or lines of higher symmetry, it is left invariant under a subgroup of the point group. Consequently, the number of beams of the star is smaller. The subgroup of the point group leavingk unchanged is called little group ofk. If inversion is part of the point group,k is always contained in the star ofk. In summary, we have the simple relations  = ;  = ;  = : (1.87) nk n;gk nk n;k nk n;k+G We can also use symmetries to characterize Bloch states for given pseudo-momentumk. Let us take a set of degenerate Bloch states belonging to the bandn,fj ;n;kig satisfying the eigenvalue equation, Hj ;n;ki = j ;n;ki (1.88) nk For givenk we consider the little group operations. Operating an element g of the little group b on a statej ;n;ki we obtain again an eigenstate of the HamiltonianH, asS commutes with fg;0g H, X 0 0 b b S j ;nki = j ;nkih ;nkjS j ;nki : (1.89) fg;0g fg;0g z 0 =M (g) 0 ; We transform only within the subspace of degenerate statesj ;nki and the matrix M 0 (g) is ; a representation of the group element g on the vector space of eigenstatesfj ;nkig. If this rep- resentation is irreducible then its dimension corresponds to the degeneracy of the corresponding set of Bloch states. Looking back to the example of tight-binding bands derived from atomic p-orbitals (Fig.1.4). The symmetry at the -point (k = 0) is the full crystal point group O (simple cubic lattice). h The representation is three-dimensional corresponding to a basis setfx;y;zg (p-orbital). 15 At the X-point (symmetry point on the Brillouin zone boundary) the group is reduced to D 4h (tetragonal) and the representations appearing are X (one-dimensional corresponding to z) 2 andX (two-dimensional corresponding tofx;yg). Note that generally the little group ofk has 5 lower symmetry and leads to splitting of degeneracies as can be seen on the line M where the bands, degenerate at the -point split up into three and combine again at the M-point into two level of degeneracy one and two, respectively. The symmetry group of the M-point is D while for arbitrarykk 110 it is C containing only four elements leavingk invariant: 4h 2v C D O . 2v 4h h 23

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