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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.
(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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(II) SUMMARY OF MAIN TOPICS TO BE COVERED BY INVITED SPEAKERS
(1) Molecular conductors, by Khaled Al-Hassanieh (1 week)
(2) Spin systems and MC techniques, by Roger Melko
(1 class)
(3) Numerical studies of Manganese oxides, by Gonzalo Alvarez (1 week)
(4) Density Matrix Renormalization Group and quantum
spin chains,
by Fabian Heidrich-Meisner
(1 week)
(5) Diluted Magnetic Semiconductors, by Yucel Yildirim (1 week)
Other possible speakers are Cengiz Sen (manganites),
Yucel Yildirim (DMS), Florentin
Popescu (DMFT),...
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(III) HOMEWORK DESCRIPTION AND
PROJECTS:
Below you can find the titles
of several possible projects. The presentation of the project
will consist of: (1) a brief (~5-10 pages) document
with the main ideas, followed by (2)
an oral presentation. The main goals of this ``homework''
are: (i) to learn the
most basic aspects of
each item, collecting the most relevant
references, and (ii) to provide
an opportunity to the student to practice lecturing
in a friendly environment.
The enfasis of both the written document and the oral seminar must be
on the
intuitive main issues.
Each document/seminar must contain a set of simple ideas, cartoons,
concepts that are easy to remember.
Complicated math or
complicated experimental details
are not of
interest, but ideas and concepts.
If you have preferences
on
what project to select, let me know and I will try to adapt
to your wishes. Otherwise I will assign at random.
The first set of
projects must
be ready
by mid February, the second set by mid/late March.
The presentation of the report must be in PDF or PS or DOC
format. The students
will deliver a talk on the subject of their research,
which will must last approx. 20-30 minutes.
The
best would be
if this talk is given using powerpoint tools.
Topics
for first presentation (and student in charge):
1. Bose-Einstein condensation in atoms (Davi
Ortega)
Presentation
Paper
2. Introduction to neutron scattering (Jason Haraldsen)
Presentation
Paper
3. Introduction to carbon nanotubes
(Ard)
Presentation
Paper
4. Introduction to spintronics
(Schaefferkoetter)
Presentation
Paper
5. Scanning tunneling microscopy (Biao
Hu)
Presentation
Paper
6. Magnetic nanoclusters, such as Mn12
(Krause)
Presentation
Paper
7. Excitons and relation with solar energy (Shuhua
Liang) Presentation
Paper
8. Superconducting cobaltites (Nick
Vence)
Presentation
Paper
Topics
for second presentation (and student in charge):
1. Molecular conductors
(Krause)
Presentation
Paper
2. Oxide heterostructures
(Liang)
Presentation
Paper
3. Stripes in high-Tc
(Vence)
Presentation
Paper
4. Transport in nanotubes
(Ard)
Presentation
Paper
5. Diluted magnetic semiconductors
(Schaef.)
Presentation
Paper
6. Molecular Magnets
(Haraldsen)
Presentation
Paper
7. Ruthenates
(Hu)
Presentation
Paper
8. Biophysics
(Ortega)
Presentation
Paper
When?
First round:
February 22, 27, and
March 1 (who speaks when is to be decided)
Second round: (topics to
be decided)
April 19, 24, and 26
Emails
Shaun
Ard
sard*utk.edu
Jason
Haraldsen
jharalds*utk.edu
Biao
Hu
bhu1*utk.edu
Adam
Krause
akrause*utk.edu
Shuhua
Liang
sliang*utk.edu
Davi
Ortega
dortega*utk.edu
Josh Schaefferkoetter jschaeff*utk.edu
Scott
Thornton
wthornto*utk.edu
Nick
Vence
nevence*utk.edu
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