LMU M¨ unchen WS 2014/2015 Lehrstuhl f¨ ur Theoretische Nanophysik PD Dr. F. Heidrich-Meisner Dr. M. Piraud, Questions: [email protected] 2. Exercise Sheet TA1: Theoretical Solid State Physics To be discussed on Thursday, October 23, 2014. Exercise 1: Hall Effect Let us suppose that we have a plane-rod shaped conductor as shown in the above figure. If we apply an electric field in the x-direction, a current will be induced. Within the Drude approximation, this current will be in the x-direction (j = σ0 E with σ0 = ne2 τ /m). We now apply a magnetic field perpendicular to the plane-rod (H = H zˆ). If the carriers are electrons (i.e. q = −e) a Lorentz force will dislocate some charges to one of the sides of the plane-rod. In the steady state, the dislocated charge will generate an electric field to compensate the Lorentz force. This is known as the Hall effect. a. Draw on the figure the direction of the Lorentz force F L = q/cv × H. b. On which side of the bar will the negative charge accumulate, due to the Lorentz force ? c. Draw the direction of the induced electric field that will compensate the Lorentz force (beware that the carriers are electrons). d. Write down the equation of motion for the average of the momentum of the electron (i.e. dp/dt =?). e. In the steady state, the current is independent of time, thus dp/dt = 0. Calculate the Hall coefficient defined as: Ey . RH = jx H f. Discuss the dependence of the Hall coefficient with the magnetic field. g. Calculate the magnetoresistance, defined as ρ(H) = Ex /jx . Note: We are assuming that the rod is a nonmagnetic material. There is then no difference between B and H. Exercise 2: Sommerfeld expansion Let us suppose that we have an integral of the form: Z∞ dǫ H(ǫ)f (ǫ), −∞ where the Fermi distribution is: f (ǫ) = 1 e(ǫ−µ)/kB T +1 , and where H(ǫ) is a generic function that vanishes as ǫ → −∞ and diverges no more rapidly than some power of ǫ as ǫ → ∞. Also we assume the H(ǫ) is smooth close to the chemical potential µ. We want to derive the Sommerfeld expansion: Z∞ Zµ dǫ H(ǫ)f (ǫ) = dǫ H(ǫ) + (kB T )2n an n=1 −∞ −∞ ∞ X d2n−1 H(ǫ)|ǫ=µ , dǫ2n−1 (1) where the an are dimensionless numbers. Proceed as follows: a. Taking K(E) = Rǫ −∞ dǫ ′ H(ǫ′ ), Z∞ show that dǫ H(ǫ)f (ǫ) = −∞ Z∞ ∂f dǫ K(ǫ) − ∂ǫ . −∞ b. Expand the function K(ǫ) in a Taylor series around ǫ = µ. Using that ∂f /∂ǫ is an even function of ǫ, show that Z∞ −∞ dǫ H(ǫ)f (ǫ) = Zµ −∞ dǫ H(ǫ) + ∞ Z∞ X n=1 −∞ (ǫ − µ)2n dǫ (2n)! ∂f − ∂ǫ d2n−1 H(ǫ)|ǫ=µ . dǫ2n−1 c. With the substitution x = (ǫ − µ)/kB T you should obtain the Sommerfeld expansion Eq. (1) and find an expression for the dimensionless coefficients an for all n. Exercise 3: Free gas in 2D We consider a gas of N free and independent electrons, in two dimension in a box of linear dimensions Lx and Ly . a. What is the relationship between n and kF in two dimension ? b. Prove that in two dimension, the free electron density of states g(ǫ) is a constant independent of ǫ for ǫ ≥ 0 and for ǫ < 0. Calculate the constants. c. Show that, as g(ǫ) is constant, every term in the Sommerfeld expansion for n vanishes, except the T = 0 term. Consider that this expansion is exact, and deduce that µ = ǫF at any temperature. R∞ d. Using that n = −∞ dǫ g(ǫ)f (ǫ), deduce that when g(ǫ) is constant as in (c) then: ǫF = µ + kB T ln 1 + E −µ/kB T . e. Using the last equation, estimate the amount by which µ differs from ǫF . Comment on the numerical significance of this ’failure’ of the Sommerfeld expansion and on the mathematical reason for the failure. Exercise 4: Bravais Lattice a. Prove that any Bravais lattice has inversion symmetry on a lattice point. Hint: Express the lattice translation as linear combinations of primitive vectors woth integral coefficients. b. Prove that the diamond structure is invariant under an inversion in the midpoint of any nearest neighbour bond.