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Quasiparticle dynamics in ballistic weak links under weak voltage
bias: an elementary treatment Available online at http://www.idealibrary.com on
Article No. spmi.1999.0704
Superlattices and Microstructures, Vol. 25, No. 5/6, 1999
Quasiparticle dynamics in ballistic weak links under weak voltage
bias: an elementary treatment
H
ERBERT
K
ROEMER Department of Electrical and Computer Engineering, and QUEST, Center for Quantized Electronic
Structures, University of California, Santa Barbara, CA 93106, U.S.A.
(Received 22 March 1999)
A simple one-dimensional model for SNS weak links in the ballistic limit is presented.
In the presence of a bias voltage, the quasiparticle state at any given instant of time is
described as a superposition of that particular set of phase-dependent Andreev bound
states that belongs to the specic phase difference present at that instant between
the superconducting banks. The treatmentbasically a form of adiabatic perturbation
theoryhas a strong formal similarity to the treatment of the k</i>-space dynamics of an
electron in a periodic potential under perturbation by an external electric eld, sufciently
strong to cause transitions across the energy gaps between bands (Zener tunneling). It is
shown that the quasiparticle wavefunction retains its phase information during analogous
transitions between Andreev bands. The experimental observation of Shapiro steps at
one-half the canonical voltage follows naturally from the model, along with some of
the experimental properties of these steps, especially their much weaker temperature
dependence, compared to the canonical steps.
c 1999 Academic Press
Key words:
Andreev reections, ballistic, Shapiro steps, weak links.
1. Introduction: the problem
In a 1985 paper [1], K¨ummel and Senftinger (KS) studied the time evolution of a quasiparticle (QP)
wavepacket in an idealized SNS weak link, under the inuence of a weak external voltage bias, making the
idealizing assumptions of perfectly ballistic transport, perfectly transparent SN interfaces, and zero Fermi
velocity mismatch. The authors showed that, as a result of multiple Andreev reections (ARs), the QP would
pick up kinetic energy from the applied bias, until it was ejected from the Andreev well, into the downstream
superconducting bank. However, the paper did not address questions of the ac Josephson effect under voltage
bias, in the presence of this energy pickup.
Experimentally, a pronouncedand highly anomalousac Josephson effect in ballistic SNS weak links
has recently been reported by Drexler et al. [2] and Lehnert et al. [3, 4] (DL). In the present paper, we
re-examine the KS treatment and attempt to reconcile itat least qualitativelywith these experimental
observations, and especially with the very pronounced anomalies found by DL. E-mail: kroemer@ece.ucsb.edu
07496036/99/050877 + 13
$30.00/0
c 1999 Academic Press 878
Superlattices and Microstructures, Vol. 25, No. 5/6, 1999
In DL, the authors investigated the Shapiro steps induced, by high-frequency irradiation, in the dc current
voltage characteristics (CVCs) of superconducting weak links, which were based on InAs quantum wells as
a coupling medium between Nb electrodes. The two investigations differ in the details of the device structure
as well as the measurement technique, but both studies revealed a common behavior quite different from that
in more conventional Josephson devices:
(a) In addition to the canonical Shapiro steps at the voltage V =
/2<i>e, where is the irradiation
frequency, the devices also showed strong steps at one-half that voltage,
V
1
/2
= /4<i>e,
(1)
indicating the presence of a strong component in the ac Josephson current at the frequency 4 eV
/ , twice the
canonical Josephson frequency J
= 2 eV / .
(b) With increasing temperature, both kinds of steps decreased, but the half-integer steps did so much more
slowly, persisting to higher temperatures than the integer steps, into a temperature range close to the critical
temperature T
c
of the superconducting Nb banks, where indications of both the dc Josephson effect and the
integer step had all but disappeared.
(c) By varying the drive frequency over a wide range, Lehnert found that the half-integer steps became
more pronounced with increasing frequency.
Perhaps the most surprising of these observations is the temperature dependence; as pointed out by
Lehnert et al. it rules out many potential explanations one might otherwise offer.
It was shown by Argaman [5, 6] that the observations can be explained in terms of a certain nonequilibrium
model: in systems with long energy relaxation times, the voltages necessary to reach the Shapiro steps
are sufciently large to drive the quasiparticle (QP) energy distribution out of equilibrium, leading to a
distribution function that contains itself a component oscillating with the canonical Josephson frequency.
This ultimately causes the current to contain a component oscillating with twice the Josephson frequency.
The higher the drive frequency, the larger the Shapiro step voltage, and hence the larger the nonequilibrium
component, thus immediately explaining observation (c) above. Beyond that, Argamans theory makes
several quantitative predictions, essentially all of which were conrmed experimentally by Lehnert [3, 4].
For details of the theory, we must refer to the original papers [5, 6], which also give extensive references to
related theoretical work by others. Of those other theoretical papers the ones most relevant to the present
work are those by Averin and Bardas [7, 8], who consider related problems for superconducting quantum
point contacts, drawing on a signicantly different formalism.
The work by Argaman draws on the well-developed theoretical formalism for diffusive weak links, into
which he incorporates nonequilibrium effects via a dis-equilibrated distribution function of the QPs over
a quasi-continuum of Andreev bound states. However, the devices investigated were actually closer to
the ballistic limit. Argaman arguescorrectlythat the underlying physics should carry over to ballistic
devices. In fact, the nonequilibrium effects should be more pronounced in the ballistic limit, where the
perturbation by random processes is much weaker. It might therefore be useful to approach the same
nonequilibrium physics from the opposite end, the purely-ballistic limit. Here, the only scattering processes
considered are normal scattering and Andreev scattering at the super/normal interfaces, with scattering
processes inside the normal material being neglected, or at best treated as a weak perturbation. This is of
course again an over-simplication, albeit one in the opposite direction from the diffusive limit.
When a small external bias is applied to a ballistic weak link, two processes take place, one time-periodic,
and the other time-monotonic (nonperiodic):
(a) The time-periodic process is the conventional ac Josephson current, just as in Josephson tunnel diodes.
Its most obvious ngerprint is the occurrence of Shapiro steps in the dc CVC under irradiation with a
high-frequency signal (see, for example, Tinkham [9])
(b) In ballistic structures in which multiple Andreev reections (ARs) can occur, the additional process
studied by KS can occur, where the quasiparticles may pick up energy from the bias eld, in a way that does Superlattices and Microstructures, Vol. 25, No. 5/6, 1999
879
not oscillatory in time. The most obvious ngerprint of this phenomenon is the sub-harmonic gap structure
often observed in the currentvoltage characteristics (CVCs) of ballistic weak links (see, for example,
Klapwijk et al. [10]).
Each phenomenon by itself has been discussed extensively in the literature, at various levels of
sophistication. Our objective here is to give a simple unied treatment that treats both phenomena on a
common basis, but on a more elementary level than what appears to be available in the literature.
Our treatment differs from that of KS in two ways: (a) We drop the restriction to perfectly transparent
SN interfaces and zero Fermi velocity mismatch. (b) Rather than explicitly following the time evolution of
a localized QP wavepacket, we treat the problem in the spirit of adiabatic perturbation theory, in which the
time evolution of an extended state is viewed as that of a linear superposition of states from a time-dependent
set of Andreev bound states. Our treatment implicitly assumes a Bogoliubovde Gennes (BdG) Hamiltonian
as in KS, in which the dc voltage bias has been included, not through a conventional (time-independent)
scalar potential, but through a time-dependent vector potential. The resulting BdG Hamiltonian depends on
time parametrically, making the problem readily tractable as an adiabatic perturbation problem. However, we
shall not nd it necessary to invoke the BdG equations explicitly.
Instead, we draw on a very close formal similarity to the dynamics of an ordinary electron in a periodic
potential, under the inuence of an applied electric eld that is sufciently strong to cause interband transi-
tions (in semiconductor physics commonly referred to as Zener tunneling). Such a treatment leads to a very
simple theoretical description of the physics of the basic phenomena, including the anomalies listed earlier.
The experimental examples coming closest to our ballistic limit are probably those SNS weak links in
which the normal conductor is the high-mobility two-dimensional electron gas in a semiconductor quantum
well, like the InAs-based quantum wells studied at UCSB and elsewhere (for complete references, see
Thomas et a