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Josephson Tunnel Junction Model
For level=3, a microscopic tunnel junction "Werthamer" model,
also known as a tunnel junction model (TJM) is indicated. The model
is more physics-based that the empirical RSJ model. The formulation
follows the method of
A. A. Odintsov, V. K. Semenov and A. B. Zorin, IEEE Trans. Magn. 23,
as implemented in the the open-source MitMoJCo project on https://github.com/drgulevich/mitmojco. The actual model
computations make use of predefined fitting parameters that can be
produced with the mmjco program provided with WRspice
(B.1). The mmjco program integrates the tunneling
current expressions producing a tunnel current amplitude (TCA) table.
This is compressed into a smaller representation using the OSZ
approach in the reference, which in addition to saving memory allows
rapid evaluation of the model expressions, basically replacing a
required integration by a short series expansion. Thus model
evaluation can be relatively inexpensive, though it is not as fast as
the simple RSJ model.
The parameters marked with an asterisk in the area column scale
with the ics parameter given in the device line, not necessarily
linearly. The present model paradigm assumes that the model
parameters apply to a ``reference'' junction, which is a typical
mid-critical current device as produced by the fouhdry.
Instantiations derive from the reference device for a desired critical
current. Appropriate scaling, not necessarily linear, will be applied
when formulating instance capacitance and conductances.
Josephson Tunnel Junction Model (Level 3) Parameters
|JJ Model Parameters
||Coefficient set name
||Quasiparticle current enabled
||Critical current enabled
||Parameter measurement temperature
||Superconducting transition temperature
||Superconducting transition temp side 1
||Superconducting transition temp side 2
||Debye temperature side 1
||Debye temperature side 2
||Riedel smoothing factor
||Terms in fit table
||Points in TCA table
||Fitting threshold parameter
||Reference junction critical current
||Reference junction capacitance
||Capacitance per critical current
||Capacitance scaling parameter
||Reference junction icrit*rsub
|rsub or r0
||Reference junction subgap resistance
||Conductance scaling parameter
|icfct or icfact
||Ratio of critical to step currents
||no limits imposed on vm, rsub
||Voltage to specify external shunt resistance
||Shunt resistor inductance constant part
||Shunt resistor inductance per ohm
||Phase change max per time step per 2
||Ratio max time step to that at <tt>vdp</tt>
|del1 (read only)
||Energy gap side 1
|del2 (read only)
||Energy gap side 2
|vg or vgap (read only)
Detailed information about these parameters is presented below.
This specifies the model to use, which is 3 in the present case.
This provides the name of the table of compressed tunnel current
amplitudes to use in the model. These are provided as files as
produced from the mmjco utility provided with WRspice (see
B.1). Single temperature ``.fit'' files will overrule any
other temperatures provided to the model. If a tempature-swept
``.swp'' file is provided, the model is able to accommodate
temperatures within the range of the sweep. The files are searched
for along a path provided by setting the tjm_path variable, or
a path can be provided directly. The default is to search the current
directory and $HOME/.mmjco if that directory exists.
These files are produced automatically as needed according to the
given model parameters and cached in the users .mmjco directory.
Therefor it is not common to use this parameter to load a set by name,
except to supply a name for a sweep file that the user has prepared
with mmjco which would provide precomputed data for all
temperatures that might be of use, thereby avoiding on-the-fly table
creation which can take some time.
There are two built-in coefficient sets, ``tjm1'' (the default) and
``tjm2''. These are the MitMoJCo NbNb_4k2_008 and NbNb_4K2_001 parameter sets, respectively. Both assume niobium at
temperature 4.2K and differ in the level of smoothing applied to
mitigate the Riedel singularity.
For the tunnel junction model, rtype is a flag, set by default,
that enables inclusion of the quasiparticle current. If set to 0,
quasiparticle current will not be included in the model.
For the tunnel junction model, cct is a flag, set by default,
that enables inclusion of the pair current. If set to 0, pair current
will not be included in the model.
range: 0.0K - 0.95*tc
This is the temperature at which all model parameters are measured.
The default is 4.2K, the boiling point of liquid helium.
range: 0.0K - 0.95*tc
This is the default operating temperature of instances of the model,
which can be overridden on a per-instance basis by specifying the
temp_k instance parameter. The default is the tnom
- tc, tc1, tc2
range: 0.1K - 280K
This is the superconducting transition temperature of the material(s)
used in the Josephson junction. The default value is 9.26K, the
transition temperature of niobium. The transition temperature may be
set separately on side 1 and side 2 of the junction using the tc1 and tc2 keywords. The tc keyword sets both sides.
If ambiguous, the last real or implied setting has precedence.
- tdebye, tdebye1, tdebye2
range: 40K - 500K
This is the Debye temperature of the material(s) used in the Josephson
junction. The default is 276K corresponding to niobium. As for the
transition temperature, the two sides of the junction can be set
independently. The model support computes the superconducting energy
gap as a function of temperature, transition temperature, and Debye
temperature using a BCS expression.
range: 0.001 - 0.099
This is a smoothing factor used when constructing the tunnel current
amplitude tables, the use of which eliminates the Riedel singularity.
The default value is 0.008. Higher values have larger smoothing,
reducing the impact of the peaks at the gap.
range: 6 - 20
The computed tunnel current amplitude tables are compressed to tables
having this many terms. The more terms that are included, the more
accurate are the parameter sets. However, the time to prepare the fit
tables grows rapidly with the number of terms. The default number of
terms is 8, which seems to provide reasonable accuracy.
range: 100 - 9999
This sets the number of energy values over which the tunnel current
amplitude tables are computed. The default is 500. Points are
computed between zero and twice the junction gap energy. More points
may provide more accurate results.
range: 0.1 - 0.5
This is the ratio of absolute to relative tolerance used in the table
compression algorithm. The default value of 0.2 seems to give good
range: 1nA - 0.1A
This is the critical current of the reference junction at nominal
temperature, which defaults to 1.0mA if not given. This parameter is
not used if cct is 0. The icrit parameter should not be
confused with the ics instance parameter. The latter is
actually a scale factor which specifies the instantiated device
critical current as well as appropriately scaling conductances and
capacitance, from the model reference current which is icrit.
range: 0.0 - 1nF
This is the capacitance of the reference junction, in farads. This
will override the cpic parameter if given, setting a fixed value
for reference junction capacitance, invariant with icrit. If
not given, junction specific capacitance is set via the cpic
parameter, see below.
range: 0.0 - 1e-9
This supplies the default capacitance per critical current in F/A.
This defaults to the MIT Lincoln Laboratory SFQ5EE process <a
href="tolpygo">[Tolpygo]</a> value (0.7pF for 1.0 mA), and will set
the junction capacitance if <tt>cap</tt> is not given. With
<tt>cap</tt> not given, changing <tt>icrit</tt> will change the
assumed capacitance of the reference junction.
range: 0.0 - 1.0
This is a new parameter in the current model, which is intended to
account for nonlinearity in scaling of capacitance with area (or
critical current, we actually define ``area'' as the actual over the
reference critical current). It is anticipated that the actual
junction capacitance consists of two components: a physical area
dependent ``bulk'' term, and a perimeter-dependent fringing term. The
cmu is a real number between 0 and 1 where if 0 we assume no
perimeter dependence, and if 1 we assume that all variation scales
with the perimeter. The default value is 0. The capacitance of an
instantiated junctions is as follows:
C = cap(A(1 - cmu) + cmu)
is the ``area'' scaling factor, which is the ratio of the
junction critical current to the reference critical current.
range: 8mV - 100mV, or 0
This is the product of the reference subgap resistance and the
reference device critical current. This parameter is commonly
provided by foundries, and is a standard indicator for junction
quality (higher is better). Values tend to decrease with increasing
critical current density. This defaults to the value for the MIT
Lincoln Laboratory SFQ5EE process, which is 16.5mV, The
reference junction subgap resistance is obtained from the value of
this parameter and the critical current, unless given explicitly.
The intrinsic subgap conductivity will be subtracted if smaller than
the given vm implies. If vm is set to 0, then no
additional conductivity will be added and only the intrinsic
conductivity will be seen. Often, the intrinsic subgap conductivity
is much smaller than observed in real junctions.
- rsub or r0
range: 8mV/icrit - 100mV/icrit, or 0
The reference junction subgap resistance can be given directly
with this parameter, and a given value will override the
vm value if also given.
The subgap conductance will be reduced by the intrinsic condutance if
this is smaller. If vm is given as 0 and this parameter is not
given, the parameter value will be 0. If the value is 0, no
additional conductance will be added.
range: 0.0 - 1.0
This is analogous to cmu, and applies to the subgap and normal
conductances. The vm, in particular, may vary with junction
physical size, with small junctions having lower vm than larger
ones. This parameter should capture this effect. It is taken that a
significant part of the conductivity is due to defects or
imperfections around the periphery of the junction area, and the
contribution would therefor scale with the perimeter. The scaling for
conductivity is as follows:
Gx = Gx0(A(1 - gmu) + gmu)
refers to either the subgap or normal conductance,
is the same parameter for the reference junction. The A
the scaling parameter, that is, the ratio of instance to reference
critical currents. The default value is 0, meaning that scaling is
assumed purely linear, which will be the case until a number is
provided through additional data analysis. It may prove necessary to
have separate scaling parameters for subgap and above gap condutance,
at which time a new model parameter may be added.
- icfct or icfact
range: 0.5 - /4
This parameter sets the ratio of the critical current to the
quasiparticle step height. Theory provides the default value of
which is usually adequately close. Characterization of
fabricated junctions would provide an improved number.
range: 0 or 1
If this flag is set, then the only range test applied to subgap
resistance values is that they be larger than zero. This affects the
parameters that set the quasiparticle branch conductance values, any
input other than a short circuit is allowed.
range: 0.0 - nominal gap voltage
This parameter is unique in that it does not describe an as-fabricated
junction characteristic. Rather, it is for convenience in specifying
a shunt resistance to use globally in SFQ circuits, If given (in
volts) conductance will be added automatically so that the product of
the total subgap conductance and the critical current will equal vshunt. This avoids having to calculate the value of and add an
explicit resistor across each Josephson junction, as used for damping
in these circuits. The designer should choose a value consistent with
the process parameters and the amount of damping required. Higher
values will provide less damping, usually critical damping is desired.
This parameter defaults to 0, meaning that no additional demping is
supplied by default.
range: 0.0 - 2.0pH
range: 0.0 - 10.0pH/
These parameters specify series parasitic inductance in the external
shunt resistor. the vshunt parameter must be given a value such
that the added external conductance is positive, or these parameters
are ignored. The inductance consists of a constant part (lsh0)
assumed to come from resistor contacts, plus a value (lsh1)
proportional to the resistance in ohms, intended to capture the length
range: 0.001 - 1.0
This is mainly for compatibility with the Verilog-A Josephson junction
model provided with WRspice in the Verilog-A examples. This is
equivalent to the WRspice dphimax parameter, but is
normalized to 2
. If not given, it defaults to
in WRspice, or 0.1 in the Verilog-A model not used in WRspice.
This is the maximum phase change allowed between internal time points.
range: 1.0 - 100.0
Time step limiting is performed relative to the Josephson frequency of
the instantaneous absolute junction voltage or the dropback voltage,
whichever is larger. The phase change is limited by tsfactor,
thus corresponding to a maximum time step relative to the period of
the frequency corresponding to the voltage. Note that in SFQ
circuits, where the junctions are critically damped, the junction
voltage is unlikely to exceed the dropback voltage, which is
numerically equal to the critical current times the shunt resistance
(vshunt). This implies that the maximum time step is a fixed
value by default.
When simulating SFQ circuits, between SFQ pulses there is often
significant time where signals are quiescent and one could probably
take larger time steps, speeding simulation. This appears true to an
extent, however one can see signs of instability if steps are too
The tsaccel parameter is the ratio of the longest time step
allowed to that allowed at the dropback voltage. In computing the
time step, the low voltage threshold is reduced to the dropback
voltage divided by tsaccel, so time steps will be inversely
proportional to voltages above this value.
Experimentation suggests that a value of 2.5 is a good choice for RSFQ
circuits, your results may vary.
- del1, del2 (read only)
These two read-only parameters return the gap potential in the
electrodes. These are computed internally as a function of
- vg or vgap (read only)
The junction gap voltage, equal to del1 + del2.
Next: TJM Model Temperature Dependence
Up: Josephson Junction Model
Previous: RSJ Model Temperature Dependence
Stephen R. Whiteley