1 Solid State Physics Semiclassical motion in a magnetic ¯eld 16 Lecture notes by Quantization of the cyclotron orbit: Landau levels 16 Michael Hilke Magneto-oscillations 17 McGill University (v. 10/25/2006) Phonons: lattice vibrations 17 Mono-atomic phonon dispersion in 1D 17 Optical branch 18 Experimental determination of the phonon Contents dispersion 18 Origin of the elastic constant 19 Introduction 2 Quantum case 19 The Theory of Everything 3 Transport (Boltzmann theory) 21 Relaxation time approximation 21 H2O - An example 3 Case 1: F~ = ¡eE~ 21 Di®usion model of transport (Drude) 22 Binding 3 Case 2: Thermal inequilibrium 22 Van der Waals attraction 3 Physical quantities 23 Derivation of Van der Waals 3 Repulsion 3 Semiconductors 24 Crystals 3 Band Structure 24 Ionic crystals 4 Electron and hole densities in intrinsic (undoped) Quantum mechanics as a bonder 4 semiconductors 25 Hydrogen-like bonding 4 Doped Semiconductors 26 Covalent bonding 5 Carrier Densities in Doped semiconductor 27 Metals 5 Metal-Insulator transition 27 Binding summary 5 In practice 28 p-n junction 28 Structure 6 Illustrations 6 One dimensional conductance 29 Summary 6 More than one channel, the quantum point Scattering 6 contact 29 Scattering theory of everything 7 1D scattering pattern 7 Quantum Hall e®ect 30 Point-like scatterers on a Bravais lattice in 3D 7 General case of a Bravais lattice with basis 8 superconductivity 30 Example: the structure factor of a BCC lattice 8 BCS theory 31 Bragg's law 9 Summary of scattering 9 Properties of Solids and liquids 10 single electron approximation 10 Properties of the free electron model 10 Periodic potentials 11 Kronig-Penney model 11 Tight binging approximation 12 Combining Bloch's theorem with the tight binding approximation 13 Weak potential approximation 14 Localization 14 Electronic properties due to periodic potential 15 Density of states 15 Average velocity 15 Response to an external ¯eld and existence of holes and electrons 15 Bloch oscillations 16 2 INTRODUCTION derived based on a periodic lattice. However it appeared that most of the properties are very similar independent What is Solid State Physics? of the presence or not of a lattice. But there are many Typically properties related to crystals, i.e., period- exceptions, for example, localization due to disorder or icity. the disappearance of the periodicity. What is Condensed Matter Physics? Properties related to solids and liquids including crys- Even in crystals, liquid-like properties can arise, such tals. For example: liquids, polymers, carbon nanotubes, as a Fermi liquid, which is an interacting electron system. rubber,... The same is also true the other way around since liquids Historically, SSP was considers as he basis for the un- can form liquid crystals. derstanding of solids since many of the properties were Hence, the study of SSP and CMP are strongly interre- lated and can not be separated. These inter-correlations are illustrated below. The Big Bang of Condensed Matter Physics Crystal Crystal Supraconductivity Opto-effects Magnetism Metals Insulators Semiconductors properties 3 THE THEORY OF EVERYTHING See ppt notes H2O - AN EXAMPLE See ppt notes BINDING What holds atoms together and also keeps them from collapsing? FIG. 1: Graphical representation of a H2 molecule We will start with the simplest molecule: H2 to ask what holds it together. The total Hamiltonian can then be written as the sum between the non interacting H atoms and the cross terms Van der Waals attraction due to Coulomb interactions. Hence, The Van der Waals force arises simply from the change H = (H0) + (Hint) (1) in energy due to the cross Coulomb interactions between atom a and b, which is simply a dipole-dipole interaction. with µ 2 2 2 ¶ µ ¶ ~ ¡ 2 2¢ e e 2 1 1 1 1 H = ¡ r1 + r2 ¡ ¡ + e + ¡ ¡ (2) 2m r1a r2b rab r12 r1b r2a Using standard perturbation theory it is then possible which leads to a total potential of the form to evaluate the gain in energy due to Hint. This is left µ ¶ σ6 σ12 as an exercise. The result is Á(r) = ¡4² ¡ ; (4) r6 r12 ¡®a®b ¢E ' ; (3) which is usually referred to as the Lennard-Jones po- r6 ab tential. The choice of the repulsive term is somewhat arbitrary but it reflects the short range nature of the where ® are the atomic polarizabilities. x interaction and represents a good approximation to the full problem. The parameters ² and σ depend on the molecule. Derivation of Van der Waals Crystals Problem 1 What about crystals? Let's ¯rst think about what kind of energy scales are involved in the problem. If we assume Repulsion that the typical distance between atoms is of the order of 1Aº we have We just saw that there is an attractive potential of e2=1Aº ' 14:4eV for the Coulomb energy and (5) the form ¡1=r6. If there were only a Coulomb repul- sion of strength 1=r this would lead to the collapse of our µ ¶2 molecule. In fact there is a very strong repulsion, which 1 ~2 ' 3:8eV for the potential in a 1Aº quantum box comes from the Pauli principle. For the general pur- 1Aº pose this repulsive potential is often taken to be » 1=r12, (6) 4 In comparison to room temperature (300K' 25meV) the ground state energy is then given by E0=-13.6 eV. these energies are huge. Hence ionic Crystals like NaCl What happens if we add one proton or H+ to the system are extremely stable, with binding energies of the order which is R away. The potential energy for the electron is of 1eV. then e2 e2 U(r) = ¡ ¡ (10) Ionic crystals r jr ¡ Rj The lowest eigenfunction with eigenvalue -13.6eV is Some solids or crystals are mainly held together by the µ ¶ 1 1 3=2 electrostatic potential and they include the alkali-halides »(r) = p e¡r=a0 ; (11) + ¡ like (NaCl ¡! Na Cl ). ¼ a0 where a0 is the Bohr radius. But now we have two protons. If the protons were in¯nitely apart then the d general solution to potential 10 is a linear superposi- tion ,i.e., Ã(r) = ®»(r) + ¯»(r ¡ R) with a degener- ate lowest eigenvalue of E0 = E0=-13.6 eV. When R is not in¯nite, the two eigenfunctions corresponding to the lowest eigenvalues Eb and Ea can be approximated by Ãb = »(r) + »(r ¡ R) and Ãa = »(r) ¡ »(r ¡ R). See ¯gure 3. - + Ψ (r) b FIG. 2: A simple ionic crystal such as NaCl Ψ (r) V(r) a 0 The energy per ion pair is Energy e2 C 14:4eV C = ¡® + n = ¡® + n with 6 < n · 12; R Nionpair d d [d=Aº] d (7) + + where ® is the Madelung constant and can be calcu- 0 r lated from the crystal structure. C can be extracted ex- perimentally from the minimum in the potential energy and typically n = 12 is often used to model the e®ect of the Pauli principle. Hence, from the derivative of the FIG. 3: The potential for two protons with the bonding and potential we obtain: anti-binding wave function of the electron µ ¶ 12C 1=11 d = (8) 0 e2® The average energy (or expectation value of Eb) is ¤ ¤ so that Eb = hÃb HÃbi=hÃb Ãbi (12) 0 A + B Energy 11®e2 = E ¡ with (13) = (9) Z 1 + ¢ Nionpair 12d0 A = e2 dr»2(r)=jr ¡ Rj (14) How good is this model? See table below: Z B = e2 dr»(r)»(jr ¡ Rj)=r (15) Quantum mechanics as a bonder Z ¢ = dr»(r)»(jr ¡ Rj); (16) Hydrogen-like bonding R where h¢i ´ ¢dr and similarly, Let's start with one H atom. We ¯x the proton at A ¡ B E = E0 ¡ (17) r = o then we know form basic quantum mechanics that a 1 ¡ ¢ 5 Hence, the total energy for state Ãb is now This ¯gure illustrates why they are called bonding and total 2 Eb = Eb + e =R (18) anti-bonding, since in the bonding case the energy is low- ered when the distance between protons is reduced as When plugging in the numbers Etotal has a minimum at b long as R > R . 1.5Aº. See ¯gure 4 0 -8 Covalent bonding Covalent bonds are very similar to Hydrogen bonds, E tot a only that we have to extend the problem to a linear com- Antibonding -12 bination of atomic orbitals for every atom. N atoms E would lead to N levels, in which the ground state has a a E [eV] E bonding wave-function. tot -13.6 eV E b -16 E b R Bonding 0 Metals R [distance between protons] In metals bonding is a combination of the e®ects dis- FIG. 4: The energies as a function of the distance between cussed above. The idea is to consider a cloud of electrons them for the bonding and anti-binding wave functions of the only weakly bound to the atomic lattice. The total elec- electron trostatic energy can then be written as Z X 2 X 2 Z 2 e e e n(r1)n(r2) Eel = ¡ drn(r) + 0 + 1=2 ¢ dr1dr2 (19) r ¡ R R ¡ R jr1 ¡ r2j R R>R0 which corresponds to an ionic contribution of the form This last expressions leads to a minimum at rs=a0 = 1:6. We can now compare this with experimental values and 2 µ ¶1=3 ®e 3 the result is o® by a factor between 2 and 6. What went Eel = ¡ where rs = (20) 2rs 4¼n wrong. Well, we treated the problem on a semiclassi- cal level, without incorporating all the electron-electron and ® is the Madelung constant.