Phys 446 Solid State Physics Lecture 8 (Ch. 4.9 – 4.10, 5.1-5.6)Last time: Discussed the free electron (Drude) model applied to electronic specific heat and electrical conductivity. Today: Finish with Free electron model. Thermal conductivity. Motion in magnetic field: cyclotron resonance and Hall effectStart new chapter: energy bands in solidsSummary Last Lecture Free electron model – simplest way to describe electronic properties of metals: the valence electrons of free atoms become conduction electrons in crystal and move freely throughout the crystal. Summary Last Lecture(continued) Fermi energy - energy of the highest occupied electronic level at T = 0 K; 3D case: Density of states of 3D free electron gas:  Heat capacity of free electron gas at low temperatures kBT << EF :322232⎟⎟⎠⎞⎜⎜⎝⎛=VNmEFπ=3123⎟⎟⎠⎞⎜⎜⎝⎛=VNkFπ3123⎟⎟⎠⎞⎜⎜⎝⎛=VNmvFπ=ENEmVdEdNED2322)(212322=⎟⎠⎞⎜⎝⎛===π*2mneτσ−=Electrical conductivity:Thermal conductivity:Wiedemann-Franz law TLKσ=223⎟⎠⎞⎜⎝⎛=ekLBπρ= ρi+ρph(T)Motion in a magnetic field: cyclotron resonanceApplied magnetic field →the Lorentz force: F = −e[E+(v × B)] Perfect metal, no electric field - the equation of motion is:Let the magnetic field to be along the z-direction. Thenycxvdtdvω−=xcyvdtdvω=where - cyclotron frequencymeBc=ωFor moderate magnetic fields (~ few kG), ωc~ few GHz.e.g. for B = 0.1 T, fc= ωc/2π = 2.8 GHz Cyclotron resonance – peak in absorption of electromagnetic waves at ωcUsed to measure the effective mass in metals and semiconductorsabsorptionSummary of free electron model  Free electron model – simplest way to describe electronic properties of metals: the valence electrons of free atoms become conduction electrons in crystal and move freely throughout the crystal.  Fermi energy - the energy of the highest occupied electronic level at T = 0 K; Density of states of 3D free electron gas:  Heat capacity of free electron gas at low temperatures kBT << EF :322232⎟⎟⎠⎞⎜⎜⎝⎛=VNmEFπ=3123⎟⎟⎠⎞⎜⎜⎝⎛=VNkFπ3123⎟⎟⎠⎞⎜⎜⎝⎛=VNmvFπ=ENEmVdEdNED2322)(212322=⎟⎠⎞⎜⎝⎛===π*2mneτσ−=Electrical conductivity:Thermal conductivity:Wiedemann-Franz law TLKσ=223⎟⎠⎞⎜⎝⎛=ekLBπρ= ρi+ρph(T)The free electron model gives a good insight into many properties of metals, such as the heat capacity, thermal conductivity and electrical conductivity. However, it fails to explain a number of importantproperties and experimental facts, for example:• the difference between metals, semiconductors and insulators•It does not explain the occurrence of positive values of the Hall coefficient. •Also the relation between conduction electrons in the metal and the number of valence electrons in free atoms is not always correct.Bivalent and trivalent metals are consistently less conductive than the monovalent metals (Cu, Ag, Au)⇒ need a more accurate theory, which would be able to answer thesequestions – the band theoryLimitations of free electron modelThe problem of electrons in a solid – a many-electron problemThe full Hamiltonian contains not only the one-electron potentials describing the interactions of the electrons with atomic nuclei, but also pair potentials describing the electron-electron interactions The many-electron problem is impossible to solve exactly ⇒simplified assumptions neededThe simplest approach we have already considered - a free electron model The next step is an independent electron approximation: assume that all the interactions are described by an effective potential.One of the most important properties of this potential - its periodicity: U(r) = U(r+ T)Bloch theoremWrite the Schrödinger equation the approximation of non-interacting electrons: ψ(r) – wave function for one electron.Independent electrons, which obey a one-electron Schrödinger equation a periodic potential U(r) = U(r+ T) - Bloch electronsBloch theorem: the solution has the form where uk(r)= uk(r+T) - a periodic function with the same period as the latticeBloch theorem introduces a wave vector k, which plays the same fundamental role in the motion in a periodic potential that the free electron wave vector k plays in the free-electron theory. ħk is known as the crystal momentum or quasi-momentumAnother conclusions following from the Bloch theorem: the wave vector k can always be confined to the first Brillouin zoneThis is because any k' not in the first Brillouin zone can be written as k' = k + G Then, if the Bloch form holds for k', it will also hold for kEnergy bandsSubstitute the solutions in the Bloch form into theSchrodinger equation, obtain: with periodic condition: uk(r)= uk(r+T) For any k, find an infinite number of solutions with discrete energies En(k), labeled with the band index nFor each n, the set of electronic levels specified by En(k) is called an energy band. The information contained in these functions for different n and k is referred to as the band structure of the solid.Number of states in a bandThe number of states in a band within the first Brillouin zone is equal to the number of primitive unit cells N in the crystal.Consider the one-dimensional case, periodic boundary conditions. Allowed values of k form a uniform mesh whose unit spacing is 2π/L ⇒ The number of states inside the first zone, whose length is 2π/a, is (2π/a)/(2π/L) = L/a = N, where N is the number of unit cellsA similar argument may be applied in 2- and 3-dimensional cases.Taking into account two spin orientations, conclude that there are 2Nindependent states (orbitals) in each energy band. Nearly free electron (weak binding) modelFirst step: empty-lattice model. When the potential is zero the solutions of the Schrödinger equation are plane waves:0222)(0mkkE==rkr⋅=ickeV1)(0ψwhere the wave function is normalized to the volume of unit cell VcNow, turn on a weak potential. Consider it as a weak periodic perturbation in Hamiltonian. From perturbation theory have: ∑−++=jkjikikiiikEkEkiUkjUkEkE,'0020,0,0)()(,',)()(ψψwhere index i refers to ithband; 0refers to empty-lattice model. The first term is the undisturbed free-electron value for the energy.The second term is the mean value of the potential in the state i, k – constant independent of k – can set to zeroThe third term – the 2ndorder correction – vanishes except k' = k + GThe