Quasiparticle excitations and ballistic transport in the mixed state of mesoscopic superconductors
A. S. Mel’nikov
1,2
and V. M. Vinokur
2
1
Institute for Physics of Microstructures, Russian Academy of Sciences, 603950, Nizhny Novgorod, GSP-105, Russia
2
Argonne National Laboratory, Argonne, Illinois 60439
Received 4 January 2002; published 5 June 2002
As the size of the superconducting sample with a few fluxoids is less than the dephasing length new physics
comes into play. The quasiparticle excitations in vortices form coherent quantum-mechanical states providing
thus a possibility to control the phase-coherent transport through the sample by changing the number of
fluxoids and their configuration. Thus, mesoscopic samples with a few vortices realize a new type of magneti-
cally tunable Andreev waveguides. The sample conductance measured in the direction of the applied magnetic
field is determined by the transparency of different multivortex configurations giant multiquanta vortices and
vortex molecules which form a set of quantum channels. The transmission coefficient for each channel is
controlled by multiple Andreev reflections within the vortex cores and at the sample edge. These interference
processes result in a stepwise and/or oscillating behavior of the conductance as a function of the applied
magnetic field. This is a vortex-based switch with the magnetic field playing the role of the gate voltage.
DOI: 10.1103/PhysRevB.65.224514 PACS numbers: 74.25.Ha, 74.60.Ec
Modern microfabrication techniques opened a route for
studies of small superconducting structures of the size of
several coherence lengths. The pioneering works
1,2
revealed
a rich variety of different phases within given fluxoid states.
Magnetic field can penetrate the sample in the form of a
poligonlike vortex molecule or individual vortices can merge
forming multiquanta giant vortex. This transformation occurs
via the second-order phase transition. First-order izomeric
transitions between the different configurations of vortex
molecule seen as branching of the magnetization curves can
also take place. Numerical Ginzburg-Landau calculations
see, e.g., Ref. 3 confirmed that indeed vortices either merge
into a single giant vortex with a certain winding number m or
arrange in stable moleculalike configurations
4
with vortex
spacing a. The appealing question now is what are the result-
ing electronic states associated with different fluxoid struc-
tures and how do structural transitions in the vortex state of
a mesoscopic superconductor affect its electronic properties.
The low lying quasiparticle QP states bound at the isolated
vortex core carrying the flux quantum
0
= c / | e | were
found first by Caroli, de Gennes, and Matricon
5
and can be
viewed as the formation of standing quasiparticle waves due
to Andreev reflection of quasiparticles from the supercon-
ducting gap profile ( r
) confining the vortex core. The
quantitative theory of the quasiparticle states is based on the
Bogolubov–de Gennes BdG equations, and in s supercon-
ductors the QP spectrum for small values of the angular mo-
mentum quantum number is
5
= /( k
r
), where is
the gap value far from the vortex axis, is the coherence
length, k
r
=| k
| , k
is the wave vector in the plane perpen-
dicular to the vortex, and is half an odd integer. This is the
so-called anomalous branch of the QP energy, which, as
function of , varies from - to crossing zero as the
impact parameter b = / | k
| of the particle in the core varies
from - to + .
In this paper we report our findings on peculiarities of the
electronic structure of the QP Andreev states in a few-fluxoid
superconductor FFS. We also analyze the phase-coherent
transport through these states and demonstrate that conduc-
tance due to Andreev states in FFS’s reveals a variety of
oscillating behaviors. In particular, we find that local ballistic
conductance can alternate between the finite and the near-
zero values as a function of magnetic field. In this regime the
mesoscopic superconductor thus realizes a quantum vortex
switch where the external magnetic field plays the role of
gate voltage.
6
The bound states in the core can barely feel the presence
of the neighboring vortices as long as the intervortex dis-
tance a is much larger than the coherence length , i.e., as
long as H H
c 2
( a ). The formation of multiquanta vorti-
ces in infinite samples is not energetically favorable, which
can be understood as a result of the strong repulsion forces
between the singly quantized vortices. On the contrary, in
small enough mesoscopic samples multiquanta structures
may become stable for H H
c 2
due to compression forces
from shielding Meissner currents pushing vortices to the cen-
ter of the sample. As the distances between vortices compare
to the coherence length a , wave functions overlap, inter-
ference effects come into play, and fundamentally new fea-
tures of the QP spectrum, controlled by the geometry of both
the vortex molecule and the sample, appear as a result of
confinement. In small samples with radius R comparable to
the coherence length the behavior of the vortex states will
also be strongly affected by the edge electronic states. The
finite magnetic field suppresses order parameter near the disk
edge creating a potential well for quasiparticles. Bound qua-
siparticle states form due to both normal quasiparticle reflec-
tion at the disk edge and Andreev reflection from the bound-
ary of the classically unpenetratable region. The local density
of states DOS in such mesoscopic disc measured, e.g., by
the STM technique should exhibit strong oscillations as a
function of magnetic field. At the disk edge the period of
these oscillations with an increase in magnetic field should
correspond to the flux quantum, while at the disk center one
can observe two-quanta periodic behavior which is caused,
in fact, by the Aharonov-Bohm effect.
PHYSICAL REVIEW B, VOLUME 65, 224514
0163-1829/2002/6522/22451411/$20.00 ©2002 The American Physical Society 65 224514-1