Josephson Junction Model

**Type Name:** `jj`

The default Josephson junction model is an extended version of the RSJ
model as used by Jewett[11]. For level=3, a microscopic
tunnel junction ``Werthamer'' model is provided. This is based on the
open-source MitMoJCo project on `https://github.com/drgulevich/mitmojco`.

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.

JJ Model Parametersnameareaparameterunitsdefaultlevelmodel type - 1 coeffsetcoefficient set name - pijjpi junction - 0 rtypeQuasiparticle branch model - 1 cctCritical current model - 1 iconCritical current first zero A1.0e-2 icrit* Reference junction critical current A1.0e-3 vgorvgapGap voltage V2.6e-3 delvGap voltage spread V80.0e-6 cap* Reference junction capacitance F0.7e-12 cpicReference junction capacitance per critical current F/A0.7e-9 cmuCapacitance area/edge scaling 0.0 vmReference icrit*rsub V16.5e-3 rsuborr0* Reference subgap resistance vm/icrit icrnReference icrit*rnorm V1.65e-3 rnormorrn* Reference normal state resistance icrn/icrit gmuConductance area/edge scaling 0.0 icfctoricfactRatio of critical to gap currents - /4 forceDon't check conductances 0 vshuntVoltage to specify fixed shunt resistance V0.0 lsh0Shunt resistor series inductance, constant part H0.0 lsh1Shunt resistor series inductance, proportional part H/0.0 tsfactorTime step phase change limit /2 tsacceltime step accelerator 1.0 tcsuperconducting transition temperature K9.26 tnommodel reference temperature K4.2 deftempoperating temperature Ktnomtcfcttemperature dependence fitting parameter 1.74 vdbpakdropback voltage (read only)

Detailed information about these parameters is presented below.
Unless stated otherwise, this information also applies to the
Verilog-A Josephson junction model provided with * WRspice* in the
Verilog-A examples, and the microscopic model.

`level`

This specifies the model to use. There are three choices provided in. Level 1 (the default) is the internal RSJ model, and level 2 indicates the Verilog-A example RSJ model, which is available if it has been loaded. The third choice is level 3, which is an internal microscopic tunnel junction model, based on the open-source MitMoJCo project on*WRspice*`https://github.com/drgulevich/mitmojco`.`coeffset`**Level 3 only.**

This provides the name of the table of tunnel current amplitudes to use in the model. These are provided as files as produced from the*mmjco*utility provided with, and can also be produced with the Python program supplied with MitMoJCo. 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.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.

`pijj`**Levels 1 and 2 only.**

If this flag is given a nonzero integer value, the junctions will be modeled as a ``pi junction'' meaning that the zero-current phase is rather than zero. Such devices have been fabricated using ferromagnetic barrier materials. Although these devices have some interesting behavior, they are not at this point available or used with any frequency.`rtype`**Levels 1 and 2 only.**

The`rtype`parameter determines the type of quasiparticle branch modeling employed. Legal values for level 1 are listed below, only 0 and 1 are supported in level 2.**0**The junction is completely unshunted, all shunt conductances set to zero. **1**Standard piecewise-linear model. **2**Analytic exponentially-derived approximation. **3**Fifth order polynomial expansion model. **4**``Temperature'' variation, allow modulation of the gap parameter. Values for

`rtype`larger than 1 are not currently supported in the Verilog-A model supplied within the Verilog-A examples.*WRspice*The default is

`rtype=1`. Setting`rtype=0`will disable modeling of the quasiparticle current, effectively setting the RSJ shunt resistance to infinity. Conditions with`rtype=1`and`2`are as described by Jewett, however it is not assumed that the normal resistance projects through the origin. The`icfact`parameter can be set to a value lower than the default BCS theoretical value to reflect the behavior of most real junctions. The quasiparticle resistance is approximated with a fifth order polynomial if`rtype=3`, which seems to give good results for the modeling of some NbN junctions (which tend to have gently sloping quasiparticle curves).`Rtype=4`uses a piecewise-linear quasiparticle characteristic identical to`rtype=1`, however the gap voltage and critical current are now proportional to the absolute value of the control current set with a`control=`*src_name*entry in the device line. This is to facilitate modeling of temperature changes or nonequilibrium effects. For control current of 1 (Amp) or greater, the full gap and critical current are used, otherwise they decrease linearly to zero. If no device control source is specified, the algorithm reverts to`rtype=1`. It is expected that a nonlinear transfer function will be implemented with a controlled source, which will in turn provide the controlling current to the junction in this mode. For example, the controlling current can be translated from a circuit voltage representing temperature with an external nonlinear source. The functional dependence is in general a complicated function, but a reasonable approximation is 1 - (*T*/*T*_{c})^{4}. See the examples (A.3) for an example input file (`ex10.cir`) which illustrates`rtype=4`.It is currently not possible to use other than the piecewise linear model with temperature variation. If

`rtype=4`, then legal values for the critical current parameter are`cct=0`(no critical current) and`cct=1`(fixed critical current). If another value is specified for`cct`,`cct`reverts to 0. Thus, magnetic coupling and quasiparticle injection are not simultaneously available.`cct`**Levels 1 and 2 only.**

The`cct`parameter can take one of the following values in level 1, only 0 and 1 are supported in level 2.**0**No critical current. **1**Fixed critical current. **2**Sin(x)/x modulated supercurrent. **3**Symmetric linear reduction modulation. **4**Asymmetric linear reduction modulation. Values for

`cct`larger than 1 are not currently supported in the Verilog-A model supplied within the Verilog-A examples.*WRspice*The

`control`instance parameter should be used with devices using`cct`2,3, or 4. With`cct=2`, the first zero is equal to the value of the model parameter`icon`. For`cct=3`, the maximum critical current is at control current zero, and it reduces linearly to zero at control current =*icon*. Junctions with`cct=4`have maximum critical current at control current = -`icon`, and linear reduction to zero at control current = +`icon`. If`cct`is specified as 2, 3, or 4, the area parameter, if given, is set to unity. Otherwise, the model parameters are scaled appropriately by the area before use.`icon`**range:**1e-4 - 1.0**Level 1 only.**

This parameter applies when the`cct`parameter is set to one of the choices larger the 1, where critical current modulation is modeled. The value of`icon`is the first value for (assumed) full suppression of critical current.The parameter is not currently recognized by the Verilog-A Josephson junction model provided with

, as that model does not currently support values of*WRspice*`cct`larger than 1.`icrit`**range:**1nA - 0.1A

This is the critical current of the reference junction, which defaults to 1.0mA if not given. This parameter is not used if`cct`is 0. the superconducting current through a Josepjson junctions is*I*=*I*_{c}*sin*()*I*_{c}is the critical current. and the junction ``phase'' is= (2/)

The*V*(*t*)*dt*.*V(t)*is the junction voltage, and is the magnetic flux quantum.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`.`vgap`or`vg`**range:**0.1mV - 10.0mV

This parameter specifies the gap voltage, which in a hysteretic Josephson junction is a voltage at which there is a large and abrupt increase in conductivity. This parameter is material dependent, the default 2.6mV is appropriate for niobium junctions.`delv`**range:**0.001 - 0.2**Levels 1 and 2 only.**

This specifies the assumed width, in voltage, of the quasiparticle step region, or gap. In this region, current increases sharply with increasing voltage. The default value of 80uV is reasonable for high-quality niobium/aluminum oxide Josephson junctions independent of foundry.`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

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.`rsub`or`r0`**range:**8mV/`icrit`- 100mV/`icrit`

The reference junction subgap resistance can be given directly with this parameter, and a given value will override the`vm`value if also given.`icrn`**range:**1.5mV - 1.9mV**Levels 1 and 2 only.**

This is the product of the reference junction ``normal state'' resistance and the critical current, where the normal state resistance is the differential resistance measured well above the gap. The default value is that provided for the MIT Lincoln Laboratory SFQ5EE process[16] which is 1.65mV. This too is a commonly given parameter from Josephson foundries for characterizing junctions. If not specified explicitly, this provides the reference junction normal state resistance from the critical current.`rnorm`or`rn`**range:**1.5mV/`icrit`- 1.9mV/`icrit`**Levels 1 and 2 only.**

The reference junction normal state resistance can be given explicitly with this parameter, which will override`icrn`if this is also given.`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 and above gap 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 - 2.0mV

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.

`tc`**range:**0.1K - 280K

This is the transition temperature of the material used in the Josephson junction. We assume that both junction electrodes use the same material. The default value is 9.26K, the transition temperature of niobium.`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.`temp`**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 instance parameter of the same name. The default is the`tnom`value.`tcfct`**range:**1.5 - 2.5

The model utilizes a BCS interpolation formula to represent temperature variation of the gap parameter of the superconductor, with the current restriction that the same superconductor (and gap parameter) appears on either side of the barrier.=

Reference:*tanh*(*tcfct**tc*/*T*- 1))`https://physics.stackexchange.com/questions/192416/`

interpolation-formula-for-bcs-superconducting-gapHere, is the gap parameter, with the subscript indicating the value at temperature T=0. The

`tcfct`parameter is semi-empirical, commonly cited as 1.74, which is the default value. It is taken that the junction gap voltage and critical current are directly proportional to the gap parameter. The quasiparticle resistance is assumed invariant with temperature, which should be true for the low temperatures of interest.In order to apply temperature correction one takes the assumption that parameters for the reference junction are measured at nominal temperature

`tnom`, and we therefor have=

*tanh*(*tcfct**tc*/*tnom*- 1))The correction factor, which will multiply device critical current and gap voltage, is the ratio / . This is applied, in addition to area scaling, when extrapolating device parameters from reference junction parameters.

`vdpbak`(read only)

This parameter returns the computed value of the dropback voltage, which is the voltage at which the return trace of a hysteretic Josephson junction i-v curve snaps back to the zero-voltage state. If is also the voltage equivalent of the plasma resonance, and the product of critical current and shunt resistance for critical damping.