Magnetic flux in a superconductive loop is naturally quantized and is determined by fundamental physical constants [5]
Single magnetic flux quanta (SFQ's) represent digital bits in RSFQ circuits which consist almost entirely of interconnected superconducting loops, each including several overdamped Josephson junctions. The junctions turn the loops into quantum interferometers, limiting the number of their stable states (see, e.g., Chapters 6, 7 of [6]). The junctions also allow fast switching between these states, i.e. insertion or extraction of a single flux quantum to and from the loop. For the rest of this thesis I will consider only the case of overdamped externally shunted junctions, adequately described by the RSJ (resistively shunted junction) model (see, e.g. Chapter 2 of [6]). The main equation of this model (neglecting thermal fluctuations)
simply states that the total current through the junction is a sum of
displacement, normal and supercurrent. The I-V curve of an RSJ
junction (Fig. 
) is a single-valued (non-hysteretic) curve and
there is no voltage across the junction as long as the current is less
than the critical value of 
.
Figure: I-V curve of an overdamped Josephson junction
The dynamics of the junction becomes clearer if Eq. () is
rewritten in the standard damped oscillator form:
Here
is the plasma frequency of oscillations around the equilibrium and
is the characteristic frequency of the junction. It is immediately
clear that, since both capacitance C and critical current are
proportional to the area A of the junction ( 
, 
is
critical current density), 
is independent of it. For the
standard 
- 
- 
HYPRES'
Nb-trilayer process [7] with specific capacitance of 
the frequency of plasma oscillations is
For a given technology, this number (which scales as 
)
roughly sets an upper limit for any SFQ-operating device. In RSFQ
circuits shunt value 
(for simplicity, I assume that shunt
value is much smaller than the normal resistance of the unshunted
junction) is usually chosen so that
but this is an obvious design trade-off: more strongly damped
junctions move more slowly and have more time to decide which way to
turn so that errors are less probable but this may also reduce the
maximum achievable speed. Damping is usually characterized by the
McCumber parameter 
:
In - 1-kA HYPRES' technology [7]
unshunted junctions typically have 
.
Condition () is satisfied if
(for the 1-kA HYPRES' technology [7]), where A is the
area of the junction. `` 
product'' of a critically shunted
junction is independent of its size and is about 
. The upper
limit for the area A of the junction is roughly set by the square of
the Josephson penetration depth 
(see, e.g., Chapter 8 of
[6]) which is the size at which junctions can no longer be
described as lumped circuit elements but rather as distributed
structures. For the critical current density 
,
. Since 
, the
product 
(the ``maximum critical current''
of Nb junctions) is independent of and is determined by the
properties of the superconductor. For the standard - - HYPRES' Nb-trilayer process [7]
the area A of the junction typically falls in the range of 
with critical currents from around 
up to 
. To zeroth approximation, the lower limit of the junction area is
determined by the smallest feature size of the technological process
but in practice smaller junctions are avoided as being more sensitive
to technological defects such as filmshift, missing area, etc. Also,
the critical current of a junction should be large enough to
make sure that the Josephson coupling energy 
is much
bigger than temperature 
, so that the probability of errors
induced by thermal noise is low. Typically, a ratio of 
is chosen and for helium temperatures the resulting
condition is 
.
