
PSCAN UNITS

Introduction and Definition

The definition of dimensionless variables and units used in PSCAN and
many other Josephson junction circuit simulators stems from the basic
equations of Josephson junction dynamics - see, e.g., Ref.  1.  In
dimensional units the equations describing the time evolution of
Josephson phases even in simplest circuits are rather bulky.  For
example, for a single Josephson junction (described within the RSJ
model [1]) closed with a superconductor loop we have:

C(phi/2pi)d2p/dt2 + (1/R)(phi/2pi)dp/dt + Ic*sin(p) + (1/L)(phi/2pi)p = 0

phi = h/2e = 2.07 mV-pS

Here C is the junction capacitance, R its normal resistance (including
external shunt, if any), Ic critical current, while L is the loop
inductance.  Introducing the dimensionless current, voltage, time,
resistance, inductance, and capacitance as follows:

i = I/Iu               ic = Ic/Iu
v = V/Vu               Vu = IcR
t = T/Tu               Tu = (phi/2pi)(1/Vu)
r = R/Ru               Ru = Vu/Iu
l = L/Lu               Lu = RuTu = (phi/2pi)(1/Iu)
c = C/Cu               Cu = Tu/Ru = (pji/2pi)(1/Vu)(1/Ru)

we can reduce Eq. (1a) to a much simpler form:

    cd2p/dp2 + dp/dt + icsin(p) + p/l = 0

PSCAN uses these dimensionless variables (i, v, t, r, l, and c). 
Notice that according to Eq.  (2a) the choice of the critical current
unit (Iu) remains arbitrary.  (For a single junction, Iu = Ic is an
evident choice [1], but a typical RSFQ circuit uses many junctions
with a variety of areas and hence Ic values.) PSCAN accepts a value of
0.125 mA which is equal to the typical critical current of the
smallest junctions in RSFQ circuits operating at liquid helium
temperatures.

Typical Values of Units


Variable     Notation of the Unit
                    Unit Values for Various Levels of Nb-trilayer Technology 
                        3.5 um          0.8 um          0.4 um 
 Voltage        Vu      0.3     mV      1.1     mV      2.0     mV 
 Current        Iu      0.125   mA      0.125   mA      0.125   mA 
 Time   tu      1.1     ps      0.30    ps      0.165   ps 
 Resistance     Ru      2.4     Ohm     8.8     Ohm     16      Ohm
 Inductance     Lu      2.64    pH      2.64    pH      2.64    pH
 Capacitance    Cu      455     fF      37      fF      10      fF

Comments:

* At proper shunting (or no shunting) the IcR product (and hence Vu)
  does not depend on the junction area.

* For RSFQ applications, larger (>0.5 um) Josephson junctions should
  be externally shunted to avoid hysteretic behavior [1].  The
  standard choice of the shunt resistance is that resulting in
  critical damping:  c = 1, i.e. Cu = AminC0, where Amin is the
  smallest junction area and C0 is its specific capacitance.  For
  Nb-trilayer junctions, C0 slowly increases at logarithm of the
  critical current density jc:  from 5 uF/cm2 at jc = 1 kA/cm2 to
  approximately 8 uF/cm2 at jc = 100 kA/cm2 [2, 3].

* For deep-submicron, self-shunted Nb-based junctions, the theoretical
  maximum of Vu = IcR is 2.2 mV; the first experimental data [4] are
  surprisingly close (~2.0 mV).  Theoretically, Vu should not depend
  on the junction area, but may depend on the details of the
  particular fabrication technology, so that the table above lists
  only approximate values of this parameter.

* The unit of current (Iu) depends solely on convention (see above).

* As soon as Vu has been determined (generally, from experiment) and
  Iu chosen, all other units are defined exactly by Eqs.  (2).

* The time unit defined by Eq.  (2c) is natural, but a little bit
  small for Josephson junction dynamics description.  For example,
  typical FWHM duration of the SFQ pulse is about 3 tauu.  This is why
  the canonical RSFQ text [5] uses the units of tau0 = (pi)x(tu). 
  (tau0 in picoseconds is conveniently close to the linear size of a
  0.125 mA junction in microns.) When comparing various results, it is
  necessary to mind this duality. 

