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, 763 (1987).as implemented in the the open-source MitMoJCo project on

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 ParametersnameareaparameterunitsdefaultlevelModel type - 3 coeffsetCoefficient set name - rtypeQuasiparticle current enabled - 1 cctCritical current enabled - 1 tnomParameter measurement temperature K4.2 deftempOperating temperature KtnomtcSuperconducting transition temperature K9.26 tc1Superconducting transition temp side 1 K9.26 tc2Superconducting transition temp side 2 K9.26 tdebyeDebye temperature K276 tdebye1Debye temperature side 1 K276 tdebye2Debye temperature side 2 K276 smfRiedel smoothing factor - 0.008 ntermsTerms in fit table - 8 nxptsPoints in TCA table - 500 thrFitting threshold parameter - 0,2 icrit* Reference junction critical current A1.0e-3 cap* Reference junction capacitance F0.7e-12 cpicCapacitance per critical current F/A0.7e-9 cmuCapacitance scaling parameter 0.0 vmReference junction icrit*rsubV16.5e-3 rsub orr0* Reference junction subgap resistance vm/icritgmuConductance scaling parameter 0.0 icfct oricfactRatio of critical to step currents - /4 forceno limits imposed on vm,rsub0 vshuntVoltage to specify external shunt resistance V0.0 lsh0Shunt resistor inductance constant part H0.0 lsh1Shunt resistor inductance per ohm H/0.0 tsfactorPhase change max per time step per 2 /2 tsaccelRatio max time step to that at <tt>vdp</tt> 1.0 del1(read only)Energy gap side 1 Vdel2(read only)Energy gap side 2 Vvgorvgap(read only)Gap voltage V

Detailed information about these parameters is presented below.

`level`

This specifies the model to use, which is 3 in the present case.`coeffset`

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(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*WRspice**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. </dl>`rtype`

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.`cct`

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.`tnom`**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.`deftemp`**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`value.`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.`smf`**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.`nterms`**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.`nxpts`**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.`thr`**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 results.`icrit`**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`.`cap`**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.`cpic`**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.`cmu`**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*)*A*is the ``area'' scaling factor, which is the ratio of the junction critical current to the reference critical current.`vm`**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[16], 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.`gmu`**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:*G*_{x}=*G*_{x0}(*A*(1 -*gmu*) +*gmu*)*G*_{x}refers to either the subgap or normal conductance,*G*_{x0}is the same parameter for the reference junction. The*A*is 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 /4 which is usually adequately close. Characterization of fabricated junctions would provide an improved number.`force`**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.`vshunt`**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.`lsh0`**range:**0.0 - 2.0pH`lsh1`**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 dependence.`tsfactor`**range:**0.001 - 1.0

This is mainly for compatibility with the Verilog-A Josephson junction model provided within the Verilog-A examples. This is equivalent to the*WRspice**WRspice*`dphimax`parameter, but is normalized to 2 . If not given, it defaults to /2 in, or 0.1 in the Verilog-A model not used in*WRspice*. This is the maximum phase change allowed between internal time points.*WRspice*`tsaccel`**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 large.

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 temperature.`vg`or`vgap`(read only)

The junction gap voltage, equal to`del1 + del2`.