CONDENSED MATTER II
Physics Department, University of Tennessee
Instructor:   Prof. ELBIO DAGOTTO , Spring 2008

Web page of Elbio Dagotto             Web page of Dagotto/Moreo's group.

This course web page contains:

(I)  Summary of main topics to be covered in Dagotto's lectures

(II) Topics covered by invited speakers:

Speaker : Dr. Luis Dias - UT/ORNL
 Quantum Dots (Mar 27) Lecture 1 (PDF) Kondo Effect (Apr 1st) Lecture 2 (PDF) Kondo Effect in Nanostructures (Apr 3rd) Lecture 3 (PDF)

(III)  Homework projects: Project 1   Project 2

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(I) SUMMARY OF MAIN TOPICS TO BE COVERED IN DAGOTTO'S LECTURES:

(1) Aschroft+Mermin, chapter 17. Here we start with the analysis of screening in metals.
We will define epsilon(q) and chi(q), using the ``linear approximation'' or ``linear response''
assumption. Before this subject, details about the homework projects will be discussed in the first class.

(2) To evaluate chi(q), we will use the Thomas-Fermi  approx. where the potential is
assumed to change slowly with position. A formula will be derived in q-space and transformed
to real space, showing the presence of an exponential suppression of the original 1/r potential
of a charged impurity. The screening length associated with the cloud of electrons around
the impurity will be estimated for standard metals and it will be found to be ~0.5 - 1 Angstrom.
The Lindhard theory of screening is discussed next. The perturbative calculation will be sketched.
The limit q-->0 will be shown to give Thomas Fermi. The real space total potential has oscillations
(Friedel oscillations) related with singular behavior at q=2kF in q-space. The formula for
epsilon(q,omega) will also be briefly discussed.

(3) Special cases of Lindhard function. Kohn anomalies in phonons. Explanation  of singular
behavior at 2kF based on Fermi surface arguments.  RKKY interaction between spins. Plasma
oscillations in electrons. Exponential or oscillatory decay of electric fields inside  the sample.
Plasmon and its energy vs. k relation. Explanation in terms of long-range forces.
Plasmon frequency for positive ions without electrons. Inclusion of electrons and relation
between phononic and Fermi velocities.

(4) Dielectric function of a metal. Time-delayed interactions between electrons, mediated
by positive ions. Retardation.  Overscreening,  intuitive meaning. Attraction between carriers,
once retardation caused by heavy ions is considered. Switch from external charges to a pair of
electrons within the sample (q->k-k', omega -> Delta E/hbar).  Representation in terms of Feynman
diagramms, conservation of momentum and energy at every vertex. Discussion of Cooper's paper
(Phys. Rev. 104, 1189 (1956)), following Kittel's book. We will end with the writing of the
Schroedinger equation in terms of the energy e and the matrix element <e|V|e'> of the potential.

(5) Solution of the 2 electron Cooper problem in the presence of many other inert electrons
near the Fermi level.
Relevance of other electrons to existence of bound state for weak attraction V.
Formula for Delta
as a function of omega_Debye, density-of-states at EFermi, and attraction V. Size of pair.
Relevant Hamiltonian in the space of pairs. BCS wave function. Values of uk and vk, and gap Delta.

(6) We will start with high temperature superconductors following Rev. Mod. Phys. 66, 763 (1994).
Introduction on page 765: typical chemical compositions, Tc's, structures, ionic charge of
the many elements. Coppers are in a state of spin 1/2.
We will discuss single- and bi-layer
materials. Chemical doping such as La replaced by Sr. Phase diagram of LSCO.     Optimal,
underdoped and overdoped doping. Parent insulating antiferromagnetic compound (x=0).
Doping is through holes, but it can also be electrons (NdCeCuO). Linear resistivity, as opposed
to quadratic as in standard metals.

You can find a copy of the RMP66, 763 (1994) article clicking here.

(7) Then,  we move to the brief notes ``High-Tc Superconductors'' (copy to be provided),  which
describe the degrees of freedom of relevance in Cu++. The splitting of levels analysis leads to
a ``hole'' picture where at each Cu++ there is a S=1/2 spin.  The antiferromagnetic J is
estimated using a 4-steps process (perturbation in t_{pd}). J is found to be ~0.1eV,
much smaller than other energy scales, yet still 10 times the value of kBTc for superconductivity.
Heisenberg model for x=0 will be discussed. We will finish the brief set of notes on high-Tc,
addressing the t-J model. It will be discussed the Zhang-Rice singlet and the reduction from a
problem with Cu and O degrees of freedom to just Cu. We will also discuss the Hubbard model

and the large U/t limit.

(8) We will study the instructor's notes (hardcopy to be provided) on the mean-field solution of the
Hubbard model. Since the math is fairly clear in the notes, only the important concepts will be
addressed. We will discuss up to page 10 approx. The presence of a gap induced by the interaction
U is important (Hubbard gap , aka Mott gap; Hubbard-Mott insulator). Similarities with formulas for
BCS superconductors are clear. The mean-field Hamiltonian appears to not conserve momentum
since electrons with momentum p can be transformed into p+Q. Explanation will be provided based
on the spin-waves of momentum Q which can be easily created
. We will continue with the study
of the one-band Hubbard model, analyzing the ``quasiparticles'' gamma
which are a mixture of
electrons with momentum p and spin sigma, and electrons with  momentum p+Q and spin sigma.
We will discuss the similarities with the BCS formalism.

(9)  We will study spin-ladders as examples of materials that have a ``spin gap''. Basic ideas and
suggestion of superconductivity upon doping. Experimental realizations.  Reference:
Science 271, 618-623 (1996) .

(10)  We will learn what ``manganites'' are, namely the materials with the colossal magnetoresistance (CMR).
Hamiltonians for manganites will be presented and they will involve many active orbitals, and strong
Hund couplings. The experimental phase diagram will be discussed, as well as the magnetoresistance
effects. Monte Carlo results will suggest the main competing phases and also the relevance of phase
separation ideas. Orbital order will be explained. Experimental evidence of inhomogeneities will be
discussed. Numerical results suggesting that the inhomogeneities are crucial to understand the CMR
will be given.

We will follow E. Dagotto et al., Phys. Reports 1, 344 (2001). A copy can be found here.

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PROJECT 1 :

PROJECT 2 :