Notes for Solid State Theory FFF051/FYST25

1 Notes for Solid State TheoryFYST25/EXTP90/NAFY017 Andreas WackerMatematisk FysikLunds UniversitetV artermin 2019iiA. Wacker, Lund University: Solid State Theory , March 22, 2019 These Notes give a summary of the lecture and present additional material, which may be lessaccessible by standard text books. They should be studied together with standard text booksof Solid State physics, such as Snoke (2008), Hofmann (2008), Ibach and L uth (2003) or Kittel(1996), to which is frequently State Theory is a large field and thus a point course must restrict the material. ,important issues such as calculation schemes for the electronic structure or a detailed accountof crystal symmetries is not contained in this marked with a present additional material on an advanced level, which may betreated very briefly or even skipped. They will not be relevant for the exam.

2 The same holdsfor footnotes which shall point towards more sophisticated that there are two different usages for the symbole: In these notee >0 denotes theelementary charge, which consistent with most textbooks (including Snoke (2008),Ibach andL uth (2003), and Kittel (1996)). In contrast sometimese <0 denotes thecharge 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 othersuggestion for want the thank all former students for helping in improving the text. Any further suggestionsas well as reports of misprints are welcome! Special thanks to Rikard Nelander for criticalreading and preparing several W. Snoke, Solid State Physics: Essential Concepts(Addison-Wesley, 2008).

3 P. Hofmann, Solid State Physics(Viley-VCH, Weinheim, 2008).H. Ibach and H. L uth, 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. Schrieffer, 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).

4 J. D. Jackson,Classical Electrodynamics(John Wiley & Sons, New York, 1998), 3rd W. Chow and S. W. Koch,Semiconductor-Laser Fundamentals(Springer, Berlin, 1999).iiiivA. Wacker, Lund University: Solid State Theory , March 22, 2019 List of symbolssymbolmeaningpageA(r,t)magnetic vector potential11aiprimitive lattice vector1B(r,t)magnetic field11D(E)density of states7 EFFermi energy7En(k)Energy of Bloch State with band indexnand Bloch vectork2eelementary charge (positive!)F(r,t)electric field11f(k)occupation probability16geLand e factor of the electron30giprimitive vector of reciprocal lattice1 Greciprocal lattice vector1 Hmagnetizing field29 Iradiation intensityEq. ( )MMagnetization29meelectron massmneffective massmeffof band n10 NNumber of unit cells in normalization volume3nelectron density (or spin density) with unit 1/Volume7nrefractive index41Pm,n(k) momentum matrix element10 Rlattice vector1unk(r)lattice periodic function of Bloch State (n,k)1 VNormalization volume3 Vcvolume of unit cell1vn(k)velocity of Bloch State with band indexnand Bloch vectork10 absorption coefficient41 (r,t)electrical potential11 magnetic dipole moment29 chemical potential16 0vacuum permeability29 BBohr magneton30 kelectric dipole moment44 mobility17 number of nearest neighbor sites in the lattice36 magnetic/electric susceptibility29/39 Contents1 Band Bloch s theorem.

5 Of Bloch s theorem by lattice symmetry .. K arm an boundary conditions .. Examples of band structures .. wave expansion for a weak potential .. of localized orbits for bound electrons .. Density of states and Fermi level .. and isotropic bands .. scheme to determine the Fermi level .. Properties of the band structure and Bloch functions .. degeneracy .. and effective mass .. Envelope functions .. effective Hamiltonian .. of Eq. ( ) .. 132 Semiclassical equation of motion .. General aspects of electron transport .. Phonon scattering .. Probability .. Boltzmann Equation .. conductivity .. in inhomogeneous systems .. and chemical potential .. 22vviA. Wacker, Lund University: Solid State Theory , March 22, effects .. Details for Phonon quantization and scattering.

6 Phonon spectrum .. potential interaction with longitudinal acoustic phonons .. interaction with longitudinal optical phonons .. 263 Classical magnetic moments .. Magnetic susceptibilities from independent electrons .. Diamagnetism .. by thermal orientation of spins .. paramagnetism .. Ferromagnetism by interaction .. Schr odinger equation .. band model for ferromagnetism .. and Triplet states .. model .. waves .. 374 Introduction to dielectric function and semiconductor The dielectric function .. relation .. to oscillating fields .. Interaction with lattice vibrations .. Interaction with free carriers.. Optical transitions .. The semiconductor laser .. description of gain .. current .. 485 Quantum kinetics of many-particle Occupation number formalism.

7 Rules .. operators .. Temporal evolution of expectation values .. Density operator .. Semiconductor Bloch equation .. Free carrier gain spectrum .. gain spectrum .. hole burning .. 586 Electron-Electron Coulomb effects for interband transitions .. Hamiltonian .. Bloch equations in HF approximation .. The Hartree-Fock approximation .. to the Coulomb interaction .. The free electron gas .. brief glimpse of density functional Theory .. The Lindhard-Formula for the dielectric function .. screening .. 707 Phenomenology .. BCS Theory .. Cooper pair .. BCS ground State .. from the BCS State .. transport in the BCS State .. of attractive interaction .. 79viiiA. Wacker, Lund University: Solid State Theory , March 22, 2019 Chapter 1 Band Bloch s theoremMost Solid materials (a famous exception is glass) show a crystalline structure1which exhibitsa translation symmetry.

8 The crystal is invariant under translations by alllattice vectorsRl=l1a1+l2a2+l3a3( )whereli Z. The set of points associated with the end points of these vectors is called theBravais lattice. Theprimitive vectorsaispan the Bravais lattice and can be determined byX-ray spectroscopy for each material. The volume of the unit cell isVc=a1 (a2 a3). Inorder to characterize the energy eigenstates of such a crystal, the following theorem is of utmostimportance:Bloch s Theorem:The eigenstates of a lattice-periodic Hamiltonian satisfying H(r) = H(r+Rl) for allli Zcan be written asBloch functionsin the form n,k(r) = eik run,k(r)( )wherekis theBloch vectorandun,k(r) is a lattice-periodic equivalent defining relation for the Bloch functions is n,k(r+Rl) = eik Rl n,k(r) for alllattice vectorsRl(sometimes called Bloch condition).For each Bravais lattice one can construct the corresponding primitive vectors of thereciprocallatticegiby the relationsgi aj= 2 ij.

9 ( )In analogy to the real lattice, they span the reciprocal lattice with vectorsGm=m1g1+m2g2+m3g3. More details on the real and reciprocal lattice are found in your define thefirst Brillouin zoneby the set of vectorsk, satisfying|k| |k Gm|for allGn, they are closer to the origin than to any other vector of the reciprocal lattice. Thus thefirst Brillouin zone is confined by the planesk Gm=|Gm|2/2. Then we can write each vectorkask= k+Gm, where kis within the first Brillouin zone andGmis a vector of the reciprocallattice. (This decomposition is unique unless kis on the boundary of the first Brillouin zone.)Then we have n,k(r) = ei k reiGn run,k(r) = ei k ru n, k(r) = n, k(r)1 Another rare sort of Solid materials with high symmetry are quasi-crystals, which do not have an underlyingBravais lattice. Their discovery in 1984 was awarded with the Nobel price in Chemistry 2011 Wacker, Lund University: Solid State Theory , March 22, 2019asu n, k(r) is also a lattice-periodic function.

10 Therefore we can restrict our Bloch vectors to thefirst Brillouin zone without loss of structure:For eachkbelonging to the first Brillouin zone, we have set of eigenstates ofthe Hamiltonian H n,k(r) =En(k) n,k(r)( )whereEn(k) is a continuous function inkfor eachband s theorem can be derived by examining the plane wave expansion of arbitrary wavefunctions and usingun,k(r) = {mi}a(n,k)meiGm r,see, chapter of Ibach and L uth (2003) or chapter 7 of Kittel (1996). In the subsequentsection an alternate proof is given on the basis of the crystal symmetry. The treatment followsessentially chapter 8 of Ashcroft and Mermin (1979) and chapter of Snoke (2008). Derivation of Bloch s theorem by lattice symmetryWe define thetranslation operator TRby its action on arbitrary wave functions (r) by TR (r) = (r+R)whereRis an arbitrary lattice vector.