Download Inhomogeneous And Multiple-Band Superconductivity

Inhomogeneous And Multiple-Band
Superconductivity
Jason Sadowski, Kaori Tanaka
University of Saskatchewan
Motivation
Tight-Binding Model and Mean-Field Theory
Results
In a recent experiment on NbSe2 by Kiss et al [1], they have
shown that a Charge Density Wave (CDW) can enhance superconductivity (SC). This is a surprising results as it has
been long believed that CDW causes insulating behaviour
and should compete with SC. It is suspected that for this to
occur there must be multiple portions of the Fermi surface
interacting. The goal of this research is to to understand and
explain:
We model the motion of valence electrons by the tight-binding model, in which the electrons hop from one atom site j to the
next i on a discrete lattice with probability amplitude tij . Site-dependent impurity potentials are represented by i and energies
are measured from the chemical potential µ. The Uij represents the electron-phonon interaction between sites i and j.
X
X †
(i − µ) n̂iσ
tij ciσ cjσ +
H0 =
Single-particle Hamiltonian
In Fig. 5 we present the on-site pairing potential for a
nanoscale two-dimensional s-wave superconductor. Quantum interference effects due to surfaces result in nonuniform
superconducting order. Figures 6 and 7 show the tunnelling
conductance for such small superconductors with s-wave and
d -wave coupling, respectively, in the bulk (red curves) and
at the edge of the sample (green curves). The CDW density modulations coexisting with an s-wave superconducting
pairing potential are illustrated in Fig. 8.
1XX
H = H0 +
Uii n̂i↑n̂i↓ +
Uij n̂iσ n̂jσ0
Electron-phonon interaction
2
i
hiji σσ 0
X (H)
1 X (H)
1 X (F ) †
Heff = H0 +
Vii n̂iσ +
Vij n̂iσ −
Vij ciσ cjσ
2
2
iσ
hijiσ
hijiσ
X
1X
† †
† †
+
∆ii ci↑ci↓ +
∆ij ci↑cj↓ + H.c.
Mean-field Hamiltonian
2
i
X
ˆ The recent discovery in NbSe2: coexistence of
inhomogeneity (e.g., electron density modulations) and
superconductivity
ˆ Interference effects due to surfaces/interfaces (nanoscale
devices)
ˆ Recently discovered superconductors (MgB2, iron-based
superconductors): Effects of multiple Fermi surfaces
(bands)
iσ
hijiσ
Figure 2: Lattice for the
Tight-Binding Method
hiji
where n̂iσ =
†
ciσ ciσ
and
†
ciσ
Bogoliubov-de Gennes Equations
(ciσ ) is the electron creation (annihilation) operator with spin σ.
Self Consistent Fields
[1] T.Kiss, et al., Nat. Phys. 3 720 (2007)
BCS Superconductivity
In the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity the electrons condense into pairs called Cooper
pairs. We can understand this phenomenon with the analogy
shown in Figure 1.
Electron-Phonon Interaction
ˆ Negative electron causes the positive lattice of ions to
deform towards it.
ˆ Original electron moves away leaving a region of net
positive charge
ˆ Second electron now is attracted to the region of positive
charge due to the 1st electron
ˆ Cooper pairs form a macroscopic quantum state
Figure 1: Electron-Phonon Interaction
The net result is that two electrons which normally would
repel each other, now have a net attractive interaction via
the lattice.
s-Wave and d -Wave Superconductivity
In conventional electron-phonon superconductors, two electrons with opposite spins (singlet-pairing) are bound to form
a Cooper pair with zero relative angular momentum (swave). Some unconventional superconductors such as hightemperature cuprates have singlet Cooper pairs with relative
angular momentum two (d -wave).
After performing the Bogoliubov transformation to the
mean-field Hamiltonian we arrive at an eigenvalue problem
in terms of the particle and hole amplitudes un and vn with
energy eigenvalue En :
T̂ + V̂
(H)
ˆ?
∆
+ V̂
ˆ
∆
(F )
− T̂ + V̂
(H)
!
+ V̂
(F )
un
vn
= En
un
vn
Fermi-Dirac distribution at fundamental temperature τ :
1
,
fn = f (En ) = E /τ
e n −1
Hartree-Fock, Pairing Potentials, and Density distribution
h
i
X
hn
i
j
(H)
2
2
= Uij
Vij =Uij
|un (j)| fn + |vn (j)| (1 − fn )
2
n
1 X ?
(F )
[(un (i)un (j) + un (i)un?(j)) fn
Vij = Uij
2
n
Figure 5: s-wave Order Parameter
+ (vn?(i)vn (j) + vn (i)vn?(j)) (1 − fn )]
1 X
[un (i)vn?(j) + un (j)vn?(i)] (1 − 2fn )
∆ij = − Uij
2
n
Figure 6: Density of States: s-wave
Parallel Computation
The calculation is done on a large-scale “Beowulf”-class PC cluster, iglu, here at the U of S. Iglu has 128 Intel Xeon processors
clocked at 3.06 GHz, each with 2 GB RAM. Matrix diagonalization is performed with the Scalable Linear Algebra Package
(ScaLAPACK) designed for use on distributed memory parallel computers. ScaLAPACK organizes the available processors into
a PR × PC process grid.
For example if there are 8 processors available and we want to map them
onto a 2 × 4 grid, then we have a grid as shown in Figure 3. Once the
process grid has been initialized we can begin distributing the matrix
elements out to various processors. In order to ensure a good load balance
and communication time between the processors we use a 2-D block-cyclic Figure 3: Example of a PR = 2 and PC = 4
data distribution. The matrix is first divided into blocks and then handed Process Grid
out in a cyclic manner to the processes in the grid (similar to dealing out
a hand of cards). Figure 4 shows an example of a 9 × 9 matrix being
distributed amongst 4 processors in the grid. In this case we have divided
the matrix into 2 × 2 blocks.
Advantages
ˆ Each processor holds only a fraction of the global matrix
ˆ Balanced distribution ensures high efficiency from each processor
ˆ Minimal communication time between processors
ˆ We can solve problems that are much larger than what is feasible with a
Figure 4: A 9 × 9 matrix distributed across 4
single processor!
processors
Department of Physics and Engineering Physics, University of Saskatchewan
Figure 7: Density of States: d -wave
Figure 8: Charge Density Wave and Superconductivity
Future Work
We plan to generalize our calculation to systems with multiple bands (Fermi-Surfaces).
Nov 4th, 2009