# Physics 880.06: Condensed Matter Physics (Winter, 2002)

#### Introduction and General Format

Physics 880.06 is the second quarter of a three quarter sequence on Condensed Matter Physics. The text for this quarters will continue to be Ashcroft and Mermin, Solid State Physics'' (Saunders, 1976), though I will depart from the book from time to time. The third quarter will devoted to a single topic, superconductivity, and use a different topic.

The course will meet MWF from 10:30 to 11:18 in Rm. 215 of the Aviation Building (across 19th St. from Smith). The instructor is David Stroud.

Grading will be based on homework and a small term project.

#### Syllabus

Note: the syllabus below is subject to change; I doubt I will get through all of it, and the order may change. The items most likely to be cut back are nos. 4, 5, and 6.

1. Quantum theory of lattice vibrations; phonons.

2. Formalism of second quantization

3. Transport in solids

4. Electron-electron interactions (dielectric properties, Hartree-Fock, density-functional theory).

5. Optical properties of solids.

6. Defects in solids (impurities, non-periodic potential, possibly extended defects such as dislocations, glasses, methods for treating these)

7. Semiconductors, possibly semiconductor devices

8. Magnetic properties of solids (including paramagnetism, diamagnetism, Landau levels, integer and fractional Hall coefficients, ferro- and anti-ferromagnetism, spin waves (magnons); quantum models of magnetism; other cooperative phenomena.

#### Problem Set 1

Due Friday, January 18, 2002

1. Ashcroft and Mermin, Chapter 23, Problem 2.

2. Ashcroft and Mermin, Chapter 23, Problem 3. [Hold off on part (b) for the moment.]

3. Show, using the Debye approximation, that the mean-square displacement of an atom from its static lattice site diverges at finite temperature in two dimensions, and even at zero temperature in one dimension. Hence, there cannot exist periodic solids of the conventional kind in one dimension at any temperature and in two dimensions at finite temperatures.

#### Problem Set 2

Due Friday, February 1, 2002

1. Ashcroft and Mermin, Chapter 23, Problem 3(b).

2. Ashcroft and Mermin, Chapter 24, Problem 3(a) and (c).

3. Ashcroft and Mermin, Chapter 28, Problem 1.

4. Ashcroft and Mermin, Chapter 28, Problem 3.

#### Problem Set 3

Due Friday, February 22, 2002

1. Ashcroft and Mermin, Chapter 17, Problem 1.

2. Ashcroft and Mermin, Chapter 17, Problem 4.

3. Consider a harmonic oscillator with a quartic perturbation. The Hamiltonian is then

H = ku^2/2 + p^2/(2m) + Gu^4,

where k is the spring constant, m is the mass, u is the displacement, p is the momentum, and G is a constant which measures the strength of the quartic perturbation.

Calculate the shift in the nth harmonic oscillator level, to first order in G.

4. Consider an electron with an effective mass {\em tensor} with principal values m_x, m_y, and m_z, and principal axes parallel to the x, y, and z, axes, and suppose that it is placed in a magnetic field B parallel to the z axis. Calculate the eigenvalue spectrum and calculate the degeneracy of each Landau level.

#### Problem Set 4

Due Friday, March 8, 2002

1. Consider an electron moving in the xy plane. There is (i) a magnetic field B in the z direction, and (ii) a scalar potential of the form V(x) = +eEx. The term eEx can be thought of as the scalar potential of an electric field.

(a) Find the energy eigenvalues of the Hamiltonian which describes this system. What happens to the degeneracy of the Landau levels? Be sure to pick a suitable gauge in which to solve the Schrodinger equation.

(b). Find the y component of velocity for an electron in a given eigenstate.

2. In a system consisting of two spin-1/2 electrons, consider the state

(|\uparrow \downarrow > + |\downarrow\uparrow>)/\sqrt{2}.

Show that this state is an eigenstate of the operator S^2 with eigenvalue S(S+1) = 2, and of the operator S_z with eigenvalue 0, where S^2 is the operator representing the square of the total spin, and S_z is the z component of total spin.

3. It was shown in class that the spin wave modes in a Heisenberg ferromagnet behave like Bose excitations (like phonons). In this problem, assume that the dispersion relation for these modes is \omega = Dk^2, where D is the spin wave stiffness constant.''

(a). Show that, in a three-dimensional Heisenberg at sufficiently low temperatures, the specific heat should vary as T^{3/2}, and find the dependence of the coefficient on D.

Let us write the magnetization at temperature T as M(T). At low temperatures, it can be shown that the quantity M(0) - M(T) is proportional to the total number of magnons per unit volume which are thermally excited at temperature $T$.

(b). Show that M(0) - M(T) is proportional to T^{3/2}.

#### Supplement to P. S. 4

(Optional; not to be turned in)

4. Find the spin-wave spectrum for the anisotropic ferromagnetic spin-1/2 Heisenberg model:

H = -\sum_< ij > [(1/2)J_perp(S_{i+}S_{j-} + S_{i-}S_{j+}) + J_zS_{iz}S_{jz}].

Here J_perp and J_z are positive constants, and the sum runs over distinct pairs of nearest neighbor spins. If J_perp = J_z, one recovers the usual isotropic Heisenberg model.

Hint: There will generally be a gap in the spin wave spectrum.

5. Repeat problem 1 above, but for an isotropic ferromagnetic Heisenberg model in an external magnetic field parallel to the z axis. Hint: Once again, there will be a spin wave gap.

6. Consider the antiferromagnetic (but isotropic) Heisenberg model with nearest neighbor interactions. (In other words, the hamiltonian is

H = -J\sum_< ij >S_i . S_j,

with J < 0.)

Assume that the lattice consists of two Bravais sublattices (e. g., the body-centered-cubic lattice).

Show that the antiferromagnetic state (with spins pointing in the + or -z direction on the two sublattices) is not an eigenstate of H and, therefore, cannot be its ground state.

7. Consider the antiferromagnetic Ising model, as given in class. Calculate the low-field susceptibility in the mean-field approximation, for temperatures both above and below the mean-field Neel temperature, and show that this susceptibility has a cusp at T_N.

8. Consider the ferromagnetic spin-1/2 Heisenberg model with nearest-neighbor interactions. Show that, in two dimensions (d = 2) there are an infinite number of spin-wave excitations at any nonzero temperature. (Hence, the magnetization vanishes at any finite temperature.)

Note: Problem sets are due by 5PM Friday in either the mailbox of the grader, Wissam Al-Saidi (preferred) or my mailbox.