Solid State Theory Lecture Notes

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Notes for Solid State Theory FFF051/FYST25 Andreas Wacker Matematisk Fysik Lunds Universitet V artermin 2015ii A. Wacker, Lund University: Solid State Theory, VT 2015 These notes give a summary of the lecture and present additional material, which may be less accessible by standard text books. They should be studied together with standard text books of solid state physics, such as Snoke (2008), Hofmann (2008), Ibach and Luth  (2003) or Kittel (1996), to which is frequently referred. Solid state theory is a large eld and thus a 7.5 point course must restrict the material. E.g., important issues such as calculation schemes for the electronic structure or a detailed account of crystal symmetries is not contained in this course.  Sections marked with a present additional material on an advanced level, which may be treated very brie y or even skipped. They will not be relevant for the exam. The same holds for footnotes which shall point towards more sophisticated problems. Note that there are two di erent usages for the symbol e: In these note e 0 denotes the elementary charge, which consitent with most textbooks (including Snoke (2008),Ibach and Luth  (2003), and Kittel (1996)). In contrast sometimes e 0 denotes the charge of the electron, which I also used in previous versions of these notes. Thus, there may still be some places, where I forgot to change. Please report these together with other misprints and any other suggestion for improvement. I want the thank all former students for helping in improving the text. Any further suggestions as well as reports of misprints are welcome Special thanks to Rikard Nelander for critical reading and preparing several gures.Bibliography D. W. Snoke, Solid State Physics: Essential Concepts (Addison-Wesley, 2008). P. Hofmann, Solid State Physics (Viley-VCH, Weinheim, 2008). H. Ibach and H. Luth,  Solid-state physics (Springer, Berlin, 2003). C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, New York, 1996). N. W. Ashcroft and N. D. Mermin, Solid State Physics (Thomson Learning, 1979). G. Czycholl, Festk orperphysik (Springer, Berlin, 2004). D. Ferry, Semiconductors (Macmillan Publishing Company, New York, 1991). E. Kaxiras, Atomic and Electronic Structure of Solids (Cambridge University Press, Cambridge, 2003). C. Kittel, Quantum Theory of Solids (John Wiley & Sons, New York, 1987). M. P. Marder, Condensed Matter Physics (John Wiley & Sons, New York, 2000). J. R. Schrie er, Theory of Superconductivity (Perseus, 1983). K. Seeger, Semiconductor Physics (Springer, Berlin, 1989). P. Y. Yu and M. Cardona, Fundamentals of Semiconductors (Springer, Berlin, 1999). C. Kittel and H. Kr omer, Thermal Physics (Freeman and Company, San Francisco, 1980). J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, New York, 1998), 3rd ed. W. W. Chow and S. W. Koch, Semiconductor-Laser Fundamentals (Springer, Berlin, 1999). iiiiv A. Wacker, Lund University: Solid State Theory, VT 2015 List of symbols symbol meaning page A(r;t) magnetic vector potential 11 a primitive lattice vector 1 i B(r;t) magnetic eld 11 D(E) density of states 7 E Fermi energy 7 F E (k) Energy of Bloch state with band index n and Bloch vector k 2 n e elementary charge (positive) F(r;t) electric eld 11 f(k) occupation probability 16 g Land e factor of the electron 30 e g primitive vector of reciprocal lattice 1 i G reciprocal lattice vector 1 H magnetizing eld 29 I radiation intensity Eq. (4.10) M Magnetization 29 m electron mass e m e ective mass m of band n 10 n e N Number of unit cells in normalization volume 3 n electron density (or spin density) with unit 1=Volume 7 n refractive index 41 P (k) momentum matrix element 10 m;n R lattice vector 1 u (r) lattice periodic function of Bloch state (n; k) 1 nk V Normalization volume 3 V volume of unit cell 1 c v (k) velocity of Bloch state with band index n and Bloch vector k 10 n absorption coecient 41 (r;t) electrical potential 11   magnetic dipole moment 29  chemical potential 16  vacuum permeability 29 0  Bohr magneton 30 B  electric dipole moment 44 k  mobility 17  number of nearest neighbor sites in the lattice 36  magnetic/electric susceptibility 29/39Contents 1 Band structure 1 1.1 Bloch's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Derivation of Bloch's theorem by lattice symmetry . . . . . . . . . . . . 2 1.1.2 Born-von Karm  an boundary conditions . . . . . . . . . . . . . . . . . . . 3 1.2 Examples of band structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Plane wave expansion for a weak potential . . . . . . . . . . . . . . . . . 4 1.2.2 Superposition of localized orbits for bound electrons . . . . . . . . . . . . 5 1.3 Density of states and Fermi level . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.1 Parabolic and isotropic bands . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.2 General scheme to determine the Fermi level . . . . . . . . . . . . . . . . 8 1.4 Properties of the band structure and Bloch functions . . . . . . . . . . . . . . . 9 1.4.1 Kramers degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4.2 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4.3 Velocity and e ective mass . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Envelope functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5.1 The e ective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 11  1.5.2 Motivation of Eq. (1.23) . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5.3 Heterostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Transport 15 2.1 Semiclassical equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 General aspects of electron transport . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Phonon scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.1 Scattering Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.2 Thermalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4.1 Electrical conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.2 Transport in inhomogeneous systems . . . . . . . . . . . . . . . . . . . . 21 2.4.3 Di usion and chemical potential . . . . . . . . . . . . . . . . . . . . . . . 22 vvi A. Wacker, Lund University: Solid State Theory, VT 2015  2.4.4 Thermoelectric e ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22  2.5 Details for Phonon quantization and scattering . . . . . . . . . . . . . . . . . . 24 2.5.1 Quantized phonon spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5.2 Deformation potential interaction with longitudinal acoustic phonons . . 26 2.5.3 Polar interaction with longitudinal optical phonons . . . . . . . . . . . . 26 3 Magnetism 29 3.1 Classical magnetic moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2 Magnetic susceptibilities from independent electrons . . . . . . . . . . . . . . . . 30 3.2.1 Larmor Diamagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2.2 Paramagnetism by thermal orientation of spins . . . . . . . . . . . . . . 32 3.2.3 Pauli paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3 Ferromagnetism by interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.1 Many-Particle Schr odinger equation . . . . . . . . . . . . . . . . . . . . . 33 3.3.2 The band model for ferromagnetism . . . . . . . . . . . . . . . . . . . . . 34 3.3.3 Singlet and Triplet states . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.4 Heisenberg model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36  3.3.5 Spin waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4 Introduction to dielectric function and semiconductor lasers 39 4.1 The dielectric function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.1.1 Kramers-Kronig relation . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.1.2 Connection to oscillating elds . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Interaction with lattice vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3 Interaction with free carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4 Optical transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.5 The semiconductor laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46  4.5.1 Phenomenological description of gain . . . . . . . . . . . . . . . . . . . 47  4.5.2 Threshold current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5 Quantum kinetics of many-particle systems 49 5.1 Occupation number formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.1.1 De nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.1.2 Anti-commutation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.1.3 Field operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.1.4 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.2 Temporal evolution of expectation values . . . . . . . . . . . . . . . . . . . . . . 53vii 5.3 Density operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.4 Semiconductor Bloch equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.5 Free carrier gain spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.5.1 Quasi-equilibrium gain spectrum . . . . . . . . . . . . . . . . . . . . . . 57 5.5.2 Spectral hole burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6 Electron-Electron interaction 59 6.1 Coulomb e ects for interband transitions . . . . . . . . . . . . . . . . . . . . . . 60 6.1.1 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 6.1.2 Semiconductor Bloch equations in HF approximation . . . . . . . . . . . 60  6.1.3 Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.2 The Hartree-Fock approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 62  6.2.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.2.2 Application to the Coulomb interaction . . . . . . . . . . . . . . . . . . . 64  6.3 The free electron gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.3.1 A brief glimpse of density functional theory . . . . . . . . . . . . . . . . 66 6.4 The Lindhard-Formula for the dielectric function . . . . . . . . . . . . . . . . . 67 6.4.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.4.2 Plasmons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.4.3 Static screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 7 Superconductivity 71 7.1 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7.2 BCS Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 7.2.1 The Cooper pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 7.2.2 The BCS ground state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.2.3 Excitations from the BCS state . . . . . . . . . . . . . . . . . . . . . . . 77 7.2.4 Electron transport in the BCS state . . . . . . . . . . . . . . . . . . . . . 78  7.2.5 Justi cation of attractive interaction . . . . . . . . . . . . . . . . . . . . 79viii A. Wacker, Lund University: Solid State Theory, VT 2015Chapter 1 Band structure 1.1 Bloch's theorem 1 Most solid materials (a famous exception is glass) show a crystalline structure which exhibits a translation symmetry. The crystal is invariant under translations by all lattice vectors R =l a +l a +l a (1.1) l 1 1 2 2 3 3 where l 2Z. The set of points associated with the end points of these vectors is called the i Bravais lattice. The primitive vectors a span the Bravais lattice and can be determined by i X-ray spectroscopy for each material. The volume of the unit cell is V = a  (a  a ). In c 1 2 3 order to characterize the energy eigenstates of such a crystal, the following theorem is of utmost importance: Bloch's Theorem: The eigenstates of a lattice-periodic Hamiltonian satisfying H(r) = H(r + R ) for all l 2Z can be written as Bloch functions in the form l i ikr (r) = e u (r) (1.2) n;k n;k where k is the Bloch vector and u (r) is a lattice-periodic function. n;k ikR l An equivalent de ning relation for the Bloch functions is (r + R ) = e (r) for all n;k l n;k lattice vectors R (sometimes called Bloch condition). l For each Bravais lattice one can construct the corresponding primitive vectors of the reciprocal lattice g by the relations i g  a = 2 : (1.3) i j ij In analogy to the real lattice, they span the reciprocal lattice with vectors G =m g +m g + m 1 1 2 2 m g . More details on the real and reciprocal lattice are found in your textbook. 3 3 We de ne the rst Brillouin zone by the set of vectors k, satisfyingjkjjk G j for all G , m n i.e. they are closer to the origin than to any other vector of the reciprocal lattice. Thus the 2 rst Brillouin zone is con ned by the planes k G =jG j =2. Then we can write each vector m m k as k = k+G , where k is within the rst Brillouin zone and G is a vector of the reciprocal m m lattice. (This decomposition is unique unless k is on the boundary of the rst Brillouin zone.) Then we have ikr iG r ikr n (r) = e e u (r) = e u (r) = (r) n;k n;k n; k n; k 1 Another rare sort of solid materials with high symmetry are quasi-crystals, which do not have an underlying Bravais lattice. Their discovery in 1984 was awarded with the Nobel price in Chemistry 2011 http://www. nobelprize.org/nobel_prizes/chemistry/laureates/2011/sciback_2011.pdf 12 A. Wacker, Lund University: Solid State Theory, VT 2015 asu (r) is also a lattice-periodic function. Therefore we can restrict our Bloch vectors to the n; k rst Brillouin zone without loss of generality. Band structure: For each k belonging to the rst Brillouin zone, we have set of eigenstates of the Hamiltonian H (r) =E (k) (r) (1.4) n;k n n;k where E (k) is a continuous function in k for each band index n. n Bloch's theorem can be derived by examining the plane wave expansion of arbitrary wave functions and using X (n;k) iG r m u (r) = a e ; n;k m fmg i see, e.g. chapter 7.1 of Ibach and Luth  (2003) or chapter 7 of Kittel (1996). In the subsequent section an alternate proof is given on the basis of the crystal symmetry. The treatment follows essentially chapter 8 of Ashcroft and Mermin (1979) and chapter 1.3 of Snoke (2008). 1.1.1 Derivation of Bloch's theorem by lattice symmetry We de ne the translation operator T by its action on arbitrary wave functions (r) by R T (r) = (r + R) R where R is an arbitrary lattice vector. We nd for arbitrary wave functions: 0 0 T T 0 (r) =T (r + R ) = (r + R + R ) =T 0 (r) (1.5) R R R R+R 0 0 As T =T we nd the commutation relation R+R R +R T ;T 0 = 0 (1.6) R R 0 for all pairs of lattice vectors R; R . Now we investigate the eigenfunctions (r) of the translation operator, satisfying T (r) =c (R) (r) R 2ix i Let us write without loss of generality c (a ) = e for the primitive lattice vectors a with i i x 2 C (it will be shown below that only x 2 R is of relevance for bulk crystals). From i i Eqs. (1.1,1.6) we nd n n n 2i(n x +n x +n x ) ik R 1 2 3 1 1 2 2 3 3 n T (r) =T T T (r) = e (r) = e (r) R n a a a 1 2 3 where k =x g +x g +x g and Eq. (1.3) is used. 1 1 2 2 3 3 ik r Now we de ne u (r) = e (r) and nd ik (rR ) ik r ik R n n =u (r) u (r R ) = e (r R ) = e e (r R ) n n n z = (r) Thus we nd: If (r) is eigenfunction to all translation-operators T of the lattice, it has the form R ik r (r) = e u (r) (1.7) Chapter 1: Band structure 3 where u (r) is a lattice periodic function. The vector k is called Bloch vector. For in nite crystals we have k 2R. This is proven by contradiction. Let, e.g., Imfk  ag = 1  0. Then we nd 2 2 2nik a 2 4n 2 1 j (na )j =jT (0)j =je (0)j = e j (0)j 1 na 1 and the wave function diverges for n1, i.e. in the direction opposite to a . Thus (r) 1 2 has its weight at the boundaries of the crystal but does not contribute in the bulk of the crystal. As the crystal lattice is invariant to translations by lattice vectors R, the Hamiltonian H(r) for the electrons in the crystal satis es H(r) =H(r + R) for all lattice vectors R. Thus we nd T H(r) (r) =H(r + R) (r + R) =H(r)T (r) R R or  T H(r)H(r)T (r) = 0 R R z =T ;H(r) R As this holds for arbitrary wave functions we nd the commutation relation T ;H(r) = 0. R ThusfH;T ;T 0;:::g are a set of pairwise commuting operators and quantum mechanics tells R R us, that there is a complete set of functions (r), which are eigenfunctions to each of these operators, i.e. H (r) =E (r) and T (r) =c (R) (r) R As Eq. (1.7) holds, we may replace the index by n; k, where k = k is the Bloch vector and n describes di erent energy states for a xed k. This provides us with Bloch's theorem. 1.1.2 Born-von K arm an boundary conditions In order to count the Bloch states and obtain normalizable wave functions one can use the following trick. We assume a nite crystal in the shape of a parallelepiped with N N N unit cells. Thus 1 2 3 the entire volume is V = N N N V . For the wave functions we assume periodic boundary 1 2 3 c conditions (r+N a ) = (r) for simplicity. For the Bloch functions this requires kN a = 2n i i i i i with n 2Z or i n n n 1 2 3 k = g + g + g 1 2 3 N N N 1 2 3 3 Restricting to the rst Brillouin zone gives N N N di erent values for k. Thus, 1 2 3 Each bandn has within the Brillouin zone as many states as there are unit cells in the crystal (twice as many for spin degeneracy). If all N become large, the values k become close to each other and we can replace a sum over i 3 k by an integral. As the volume of the Brillouin zone isVol(g ; g ; g ) = (2) =V we nd that 1 2 3 c 2 This does not hold close to a surface perpendicular to a . Therefor surface states can be described by a 1 complex Bloch vector k . 3 If the Brillouin zone is a parallelepiped, we can writeN =2n N =2. But typically the Brillouin zone i i i is more complicated.4 A. Wacker, Lund University: Solid State Theory, VT 2015 Γ X W L Γ K X 3 2 1 0 000 100 1½0 ½½½ 000 ¾¾0 100 k 2π/a Figure 1.1: Free electron band structure for an fcc crystal together with a sketch of the Brillouin 2 2 zone (modi ed le from Wikipedia). The energy scale is =2m (2=a) . For the lattice  constant a = 4:05A of Al, we obtain 9.17 eV or 0.67 Rydberg. the continuum limit is given by Z X X V N N N V c 1 2 3 3 f(k) = Vol(k ; k ; k )f(k) d kf(k) (1.8) 1 2 3 3 3 (2) (2) k k for arbitrary functions f(k). 1.2 Examples of band structures The calculation of the band structure for a crystal is an intricate task and many approximation schemes have been developed, see, e.g., chapter 10 of Marder (2000). Here we discuss two simple approximations in order to provide insight into the main features. 1.2.1 Plane wave expansion for a weak potential In case of a constant potential U the eigenstates of the Hamiltonian are free particle states, 0 ikr 2 2 i.e. plan waves e with energy k =2m +U . These can be written as Bloch states by 0 decomposing k = k + G , where k is within the rst Brillouin zone. Then we nd n 2 2 (k + G ) n ikr (r) = e u (r) and E (k) = +U n 0 nk nk 2m The corresponding band structure is shown in Fig. 1.1 for a fcc lattice. A weak periodic potential will split the degeneracies at crossings (in particular at zone boundaries and at k = 0), thus providing gaps (see exercise 2). In this way the band structure of many metals such as aluminum can be well understood, see Fig. 1.2. 2 2 E(k) h /(2m a ) eChapter 1: Band structure 5 -0.2 -0.4 E f -0.6 E f -0.8 -1 -1.2 V c V c Figure 1.2: Calculated band struc- -1.4 ture of aluminum. After E.C. Snow, Δ Z Q Λ Σ Phys. Rev. 158, 683 (1967) Γ X W L Γ K 1.2.2 Superposition of localized orbits for bound electrons Alternatively, one may start from a set of localized atomic wave functions  (r) satisfying j   2  +V (r)  (r) =E  (r) A j j j 2m for the atomic potential V (r) of a single unit cell. The total crystal potential is given by A P V (r R ) and we construct Bloch states as A l l X 1 (n;k) ikR l (r) =p e c  (r R ) nk j l j N l;j P De ning v(r) = V (r R ) we nd A h h6=0 X   1 (n;k) ikR l H (r) =p e c E  (r R ) +v(r R ) (r R ) =E (k) (r) nk j j l l j l n nk j N lj p R 3  Taking the scalar product by the operation N d r (r) we nd i Z Z X X   (n;k) (n;k) (n;k) 3  ikR 3  l Ec + d r (r)v(r) (r)c + e d r (r) E +v(r R )  (r R )c i j j l j l i i j i j j l6=0;j Z X (n;k) (n;k) ikR 3  l =E (k) c + e d r (r) (r R )c n j l i i j l=0 6 ;j (n;k) which provides a matrix equation for the coecients c . For a given energy E (k), the n i atomic levels E  E (k) dominate, and thus one can restrict oneself to a nite set of levels i n in the energy region of interest (e.g., the 3s and 3p levels for the conduction and valence band of Si). Restricting to a single atomic S-level and next-neighbor interactions in a simple cubic crystal with lattice constant a we nd A + 2B(cosk a + cosk a + cosk a) x y z E(k)E + S 1 + 2C(cosk a + cosk a + cosk a) x y z with Z Z Z 3  3  3  A = d r (r)v(r) (r); B = d r (r)v(rae ) (rae ); C = d r (r) (rae ) S x S x S x S S S Energy Ry6 A. Wacker, Lund University: Solid State Theory, VT 2015 0 - Cl 3p -1 - Cl 3s Figure 1.3: Upper lled bands of KCl -2 + K 3p as a function of the lattice constant in atomic unit (equal to the Bohr radius  a = 0:529A) After L.P. Howard, -3 B Phys. Rev. 109, 1927 (1958). One + can clearly see, how the atomic or- K 3s -4 a = 5.9007 au 0 bitals of the ions broaden to bands with decreasing distance between the 4 6 8 10 ions. Lattice constant a u E = 0 F -2 -4 -6 Figure 1.4: Band struc- ture of copper with ex- -8 perimental data. After Calculation with parameter fit Experimental data R. Courths and S. Hufner,  Physics Reports 112, 53 -10 (1984) L Λ Γ Δ X K Σ Γ Thus the band is essentially of cosine shape, where the band width depends on the overlap B between next-neighbor wave functions. An example is shown in Fig. 1.3 for Potassium chloride (KCl), where the narrow band can be well described by this approach. The outer shell of transition metals exhibits both s and d electrons. Here the wave function of the 3d-electrons does not reach out as far as the 4s- electrons (with approximately equal total energy), as a part of the total energy is contained in the angular momentum. Thus bands resulting from the d electrons have a much smaller band width compared to bands resulting from the s electrons, which have essentially the character of free electrons. Taking into account avoided crossings, this results in the band structure for copper (Cu) shown in Fig. 1.4. The band structure of Si and GaAs as well as similar materials is dominated by the outer s and p shells of the constituents. This results in four occupied bands (valence bands) and four empty bands (conduction bands) with a gap of the order of 1 eV between. See Fig. 1.5. Energy below E eV F Energy RydbergsChapter 1: Band structure 7 L 4,5 6 Γ 8 Γ 8 L 6 Γ 7 Γ 4 7 GaAs X 7 2 X L 6 6 Γ Γ 6 6 Γ Γ 8 8 0 L 4,5 Γ L 7 Γ 6 7 -2 X 7 X 6 -4 -6 X L 6 6 -8 -10 L 6 X 6 Γ Γ -12 6 6 L Λ Γ Δ X U,K Σ Γ Wave vector k Figure 1.5: Band structure of GaAs After J.R. Chelikowsky and M.L. Cohen, Phys. Rev B 14,556 (1976) and Si from Wikipedia Commons, after J.R. Chelikowsky and M.L. Cohen, Phys. Rev B 10,5095 (1974) 1.3 Density of states and Fermi level The density of states gives the number of states per volume and energy interval. From Eq. (1.8) we obtain For a single band n the density of states is de ned by Z 1 3 D (E) = d k(EE (k)) n n 3 (2) 1:Bz The total density of states is then the sum over all bands. The density of states is obviously zero in a band gap, where there are no states. On the other hand, it is particularly large, if the bands are at as there are plenty of k-states within a small energy interval. Thus, copper has a large density of states in the energy range of 4eVE 2eV, see Fig. 1.4. Bulk crystals cannot exhibit macroscopic space charges. Thus, the electron density n must equal the positive charge density of the ions. As double occupancy of levels is forbidden by the Pauli principle, the low lying energy levels with energies up to the Fermi energyE are occupied F at zero temperature. If the Fermi energyE is within a band the crystal is a metal exhibiting a F high electrical conductivity (see the next chapter). In contrast, if the Fermi energy is located in a band gap, we have a semiconductor (with moderate conductivity which is strongly increasing with temperature), or an insulator (with vanishingly small conductivity). This distinction is not well-de ned; semiconductors have typically band gaps of the order of 1 eV, while the band gap is much larger for insulators. Energy eV8 A. Wacker, Lund University: Solid State Theory, VT 2015 1.3.1 Parabolic and isotropic bands For a parabolic isotropic band (e.g. close to the point, or in good approximation for metals) 2 2 k 2 2 we have E (k) =E + . Setting E = k =(2m ), we nd the density of states n n k e 2m e Z 1 parabolic 3D 3 D (E) =2(for spin) d k (EE E ) n k n 3 (2) 1:Bz Z Z Z 2  k max 1 2 = d' d sin dkk  (EE E ) n k 3 4 0 0 0 (1.9) z z p R 4 E m 2m E kmax e e k dE k 0 3 p m 2m (EE ) e e n = (EE ) n 2 3  Here (x) is the Heaviside function with (x) = 1 for x 0 and 0 for x 0. In the same spirit we obtain m e parabolic 2D D (E) = 2(for spin) (EE ) (1.10) n n 2 2 in two dimensions see Sec. 12.7 of Ibach and Luth  (2003) or Sec. 2.7.1 of Snoke (2008). If the states up to the Fermi energy are occupied, we nd in the conventional three dimensional case the electron density Z 3=2 E F 2m (E E ) e F c parabolic 3D n = dED (E) = : (1.11) c n 2 3 3 E c This an be used to estimate the Fermi energy of metals. E.g., Aluminium has a nuclear charge of 5 protons and two electrons are tightly bound to the nucleus within the 1s shell. Thus charge 3 3  neutrality requires 3 free electron per unit cell of volume 16.6 A (a forth of the cubic cell a for the fcc lattice). Using the free electron mass, this provides   2=3 2 3 2 E E = 3 = 11:7eV F c 3 2m  e 16:6A in good agreement with more detailed calculations displayed in Fig. 1.2. 1.3.2 General scheme to determine the Fermi level For ionic crystals like KCl, it is good to start with the atomic orbitals of the isolated atoms/ions. + Here the 3s and 3p states are entirely occupied both for the Cl and the K ion. Combining these states to bands does not change the occupation. Thus, the resulting bands should all be occupied and the Fermi level is in the gap above the band dominated by the 3p states Cl , see Fig. 1.3. A more general argument is the counting rule derived in Sec. 1.1.2. Here one rst determines the number of electrons per unit cell required for the upper bands to achieve charge neutrality. Assuming spin degeneracy, this is twice the number of bands which need to be occupied. E.g., Aluminium requires 3 outer electrons per unit cell and thus 1.5 bands should be occupied in average. Indeed for any k-point, one or two bands lie below the Fermi energy in Fig. 1.2. The same argument applies for Cu, where 11 outer electrons per atom (in the 4s and 3d shell) require the occupation of 5.5 bands in average, see Fig. 1.4.Chapter 1: Band structure 9 If there is more than one atom per unit cell, all charges have to be considered together. E.g. silicon crystallizes in the diamond lattice with two atoms per unit cell. As each Si atom has four electrons in the outer 3s/p shell, we need to populate 8 states per unit cell, i.e. 4 bands. Fig. 1.5 shows that these are just the four bands below the gap (all eight bands displayed result from the 3s/p levels), and the Fermi level is in the gap. The same argument holds for GaAs. 1.4 Properties of the band structure and Bloch functions 4 Most of the following properties are given without proof. 1.4.1 Kramers degeneracy AsH is a hermitian operator and the eigenenergies are real, we nd fromH (r) =E (k) (r) n;k n n;k the relation   H (r) =E (k) (r) n n;k n;k  ikr  Thus (r) = e u (r) (r) is an eigenfunction of the Hamilton-operator with n;k n;k n;k Bloch vectork and E (k) =E (k). n n The band structure satis es the symmetry E (k) =E (k). n n If the band-structure depends on spin, the spin must be ipped as well. 1.4.2 Normalization The lattice-periodic functions can be chosen such that Z 3  d ru (r)u (r) =V  (1.12) n;k c m;n m;k V c Furthermore they form a complete set of lattice periodic functions. Then the Bloch functions can be normalized in two di erent ways:  For in nite systems we have a continuous spectrum of k values and set Z 1 ikr 3  0 ' (r) = e u (r) ) d r' (r)' (r) = (k k ) 0 n;k n;k n;k m;n m;k 3=2 (2)  For nite systems of volume V and Born-von K arm an boundary conditions we have a discrete set of k values and set Z 1 ikr 3  p 0 ' (r) = e u (r) ) d r' 0(r)' (r) =  n;k n;k n;k m;n k;k m;k V V 1.4.3 Velocity and e ective mass The stationary Schr odinger equation for the electron in a crystal reads in spatial representation   2  +V (r) (r) =E (k) (r) n;k n n;k 2m e 4 Details can be found in textbooks, such as Snoke (2008), Marder (2000), Czycholl (2004), or Kittel (1987).10 A. Wacker, Lund University: Solid State Theory, VT 2015 ikr Inserting the the Bloch functions (r) = e u (r) can be expressed in terms of the lattice n;k nk periodic functions u (r) as nk   2 2 2 k E (k)u (r) = + k r  +V (r) u (r) n nk nk 2m m i 2m e e e Now we start with a solution u (r) with energy E (k ) and investigate small changes k: nk n 0 0   2 2 2 E (k +k)u (r) = H + k k + k r + k u (r) n 0 nk +k 0 0 nk +k 0 0 m m i 2m e e e (1.13) 2 2 2 k 0 with H = + k  r  +V (r) 0 0 2m m i 2m e e e Using the Taylor expansion of E (k), we can write n 2 X E (k ) 1 n 0 3 E (k +k) =E (k ) + k + kk +O(k ) (1.14) n 0 n 0 i j k 2 m (k ) n 0 i;j i;j=x;y;z where we de ned the   2 1 1 E (k) n e ective mass tensor = (1.15) 2 m (k) kk n i j i;j Now we want to relate the expansion coe cients in Eq. (1.14) to physical terms by considering Eq. (1.13) in the spirit of perturbation theory. In rst order perturbation theory, the change in energy is given by the expectation value of the perturbation/k with the unperturbed state:   2 1 2 E (k +k) =E (k ) + u k + r u k +O(k ) n 0 n 0 nk 0 nk 0 0 V m m i c e e 2 =E (k ) + P (k )k +O(k ) n 0 n;n 0 m e with the momentum matrix element R 3  d r (r) r (r) n;k m;k V i c R P (k) = : (1.16) m;n 3 2 d rj (r)j n;k V c Comparing with Eq. (1.14) we can identify E (k) n = P n;n k m e On the other hand, the quantum-mechanical current density of a Bloch electron is   e  J(r) = Re (r) r (r) nk nk m i e and P (k)=m =hJi=ehni is just the average velocity in a unit cell. Thus we identify the n;n e 1E (k) n velocity of the Bloch state v (k) = (1.17) n kChapter 1: Band structure 11 2 Thek in Eq. (1.13) provides together with the second order perturbation theory for the terms /k   X 1 1 2 P (k)P (k) n;m;i m;n;j =  + : (1.18) i;j 2 m (k) m m E (k)E (k) n e n m e i;j m(m=6 n) This shows that the e ective mass deviates from the bare electron massm due to the presence e 5 of neighboring bands. We further see, that a small band gap of a semiconductor (e.g. InSb) is related to a small e ective mass. 1.5 Envelope functions 1.5.1 The e ective Hamiltonian Now we want to investigate crystals with the lattice potential V (r) and additional inhomo- geneities, e.g. additional electro-magnetic elds with scalar potential (r;t) and vector poten- tial A(r;t) , which relate to the electric eld F and the magnetic induction B via A(r;t) F(r;t) =r(r;t) B(r;t) =r A(r;t) (1.19) t Then the single particle Schr odinger equation reads, see e.g. http://www.teorfys.lu.se/ staff/Andreas.Wacker/Scripts/quantMagnetField.pdf   2 (p +eA(r;t)) i (r;t) = +V (r)e(r;t) (r;t) (1.20) t 2m e wheree is the negative charge of the electron. For vanishing elds (i.e. A = 0 and  = 0) Bloch's theorem provides the band structure E (k) and the eigenstates (r). As these n nk eigenstates fopr a complete set of states, any wave functions (r;t) can be expanded in terms of the Bloch functions. Now we assume that only the components of a single band with index n are of relevance, which is a good approximation if the energetical seperation between the bands is much larger than the terms in the Hamiltonian corresponding to the elds. Thus we can write Z 3 (r;t) = d kc(k;t) (r) (1.21) nk 6 With the expansion coecients c(k;t) we can construct an envelope function Z 1 3 ikr f(r;t) = d kc(k;t) e (1.22) 3=2 (2) which does not contain the (strongly oscillating) lattice periodic functions u (r). If A(r) and nk (r) are constant on the lattice scale (e.g. their Fourier componentsA(q);(q) are small unless q g ) the envelope functions f(r;t) satis es the equation (to be motivated below) i h   i e i f(r;t) = E ir + A(r;t) e(r;t) f(r;t) (1.23) n t  e where E ir A +e is the e ective Hamiltonian. Here one replaces the wavevector k n in the dispersion relation E (k) by an operator. n 5 This is used as a starting point for kp theory, see, e.g., Chow and Koch (1999); Yu and Cardona (1999). 6 This is the Wannier-Slater envelope function, see M.G. Burt, J. Phys.: Cond. Matter 11, R53 (1999) for a wider class of envelope functions.12 A. Wacker, Lund University: Solid State Theory, VT 2015 Close to an extremum in the band structure at k we nd with Eq. (1.15) 0 "      X 1 1 i f(r;t) = E (k ) + +eA +eA e f(r;t) n 0 i j t 2 i x m (k ) i x i n 0 j i;j ij (1.24) 7 which is called e ective mass approximation . For crystals with high symmetry the mass tensor is diagonal for k , and Eq. (1.24) has the form of a Schr odinger equation (1.20) with the electron 0 mass replaced by the e ective mass. It is interesting to note, that Eq. (1.24) can also be derived for a slightly di erent envelope function f (r;t), which is de ned via the wave function as (r;t) = f (r;t)u (r). Both n n nk 0 de nitions are equivalent close to the extremum of the band. The de nition of Eqs. (1.21,1.22) has the advantage, that it holds in the entire band. On the other hand f (r;t) allows for a n multiband description, which is used in k p theory.  1.5.2 Motivation of Eq. (1.23) 8 Eq. (1.23) is dicult to proof. Here we restrict us to A(r;t) = 0, i.e. without a magnetic eld. For the electric potential we use the Fourier decomposition Z 3 iqr (r) = d q(q)e  and insert Eq. (1.21) into Eq. (1.20). Multiplying by (r) and performing the integration kn R 3 d r provides us with the terms (omitting the band index): Z Z 3  3 0 0 d r (r)i d k c _(k;t) 0(r) =ic _(k;t) k k   Z Z 2 p 3  3 0 0 d r (r) d k c(k;t) +V (r) 0(r) =E (k)c(k;t) k n k 2m 00 0 Substituting k = k + q we the potential part reads Z Z Z 0 ikr e 3  3 iqr 3 0 0 d r (r)(e) d q(q)e d k c(k;t) u 0(r) k k 3=2 (2) Z Z Z 00 ik r e 3  3 00 3 00 00 = d r (r) d k u (r)(e) d q(q)c(k q;t) k q k 3=2 (2) Z (u 00 u 00)for smallq k q k 3  (e) d q(q)c(k q;t) providing us with Z 3 ic _(k;t)E (k)c(k;t)e d q(q)c(kq;t) (1.25) n 7 section 4.2.1 of Yu and Cardona (1999) 8 See, e.g., the original article by Luttinger, Physical Review 84, 814, (1951) using Wannier functions. A rigorous justi cation, as well as the range of validity is a subtle issue, see G. Nenciu, Reviews of Modern Physics 63, 91 (1991).

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