How to compute a capacitance-voltage curve

[JunwenDiao]

I was simulating the Capacitance vs Voltage curve for a HPGe detector with depletion_handling = true for calculate_electric_potential! and observed a strange behavior of the plot: it is being basically constant before depletion and starting to increase after the depletion voltage, while keeping the variation very small.

The problem is probably due to the ϵ_r for the undepleted region not being set to the scaling_factor_for_permittivity_in_undepleted_region but the same as the depleted ϵ_r (which is 16 for HPGe I believe)

the ϵ_r is not changing with the depletion:
0V:

Partially depleted:

Fully depleted:

So that above the depletion voltage the depleted region goes into the contact, turning the ϵ_r from 1.0 to 16.0, thus increasing the value of the capacitance in the calculation:

[fhagemann]

So, we do not scale up ϵ_r in the plot but in code, so the plots you are showing make perfect sense.

See for example these lines, where this is applied in-code:

SolidStateDetectors.jl/src/PotentialCalculation/PotentialCalculationSetup/PotentialCalculationSetupCartesian3D.jl

Lines 183 to 197 in bb477d9

	if depletion_handling
	for iz in axes(ϵ, 3)
	for iy in axes(ϵ, 2)
	for ix in axes(ϵ, 1)
	pt::CartesianPoint{T} = CartesianPoint{T}(mpx[ix], mpy[iy], mpz[iz])
	ig = find_closest_gridpoint(pt, point_types.grid)
	if is_undepleted_point_type(point_types.data[ig...])
	ϵ[ix, iy, iz] *= scaling_factor_for_permittivity_in_undepleted_region(det.semiconductor) * (1 - imp_scale[ig...])
	elseif is_fixed_point_type(point_types.data[ig...])
	ϵ[ix, iy, iz] *= scaling_factor_for_permittivity_in_undepleted_region(det.semiconductor)
	end
	end
	end
	end
	end

The formula you are quoting depend on the WEIGHITNG POTENTIALS, not the electric potential. In order to calculate the capacitance, you would then also need to calculate the weighting potential with depletion_handling = true. Is that what you are doing right now?

[fhagemann]

The variations you are seeing right now are just based on the different grid spacings resulting from simulations at different bias voltages.

[JunwenDiao]

Hi,
thank you very much for your reply!
I think that is real source of the problem. I made a stupid mistake when testing depletion_handling = true for calculate_weighting_potential which made we think that I can not do that! Now it is giving me the right plot

[fhagemann]

Great to hear that! I renamed the issue and moved it to the discussion Q&A section, so that others can find the answer when bumping into a similar problem!

[jasondet]

@fhagemann there's three things here that don't make sense to me still. I thought that the relative permittivity inside of a conducting material should be very large, and in depleted Ge it's somewhat large but not super high (@JunwenDiao tells me SSD uses a value of ~16). If this is correct:

1. It looks from @JunwenDiao's plots that SSD is setting that value to 1 on the contacts, i.e. the vacuum permittivity. Shouldn't it be a very large value consistent with the fact that the contacts are conducting?

2. If you put the detector at some voltage where it is partially depleted (with depletion_handling=true) and the user asks for the dielectric constant at some point where the detector is undepleted, you should get a very high value because the Ge is effectively conductive there. But @JunwenDiao's plots show that SSD returns the value 16 throughout the bulk regardless of whether the detector is depleted at that point or not.

3. @JunwenDiao's plots appear to show that region that is defined as a contact ends up getting smaller with increasing voltage. The contacts should stay fixed as the detector voltage is changed, no?

Have I misunderstood something?

[fhagemann]

Hi @jasondet,

you are correct in assuming that the relative permittivity should be very large in conducting (and undepleted) material.

The values shown in the ϵ_r plot are the ones calculated for the setup prior to determining its depletion, just from whatever is defined in MaterialProperties.jl.
Once we have computed the point types (and know which grid points are undepleted or part of a contact at a fixed potential), we scale up the relative permittivity in code (using the function scaling_factor_for_permittivity_in_undepleted_region), when running with depletion_handling = true. However, this is not reflected in the ϵ_r plot.

SolidStateDetectors.jl/src/SolidStateDetector/Semiconductor.jl

Lines 142 to 144 in 929daa7

	function scaling_factor_for_permittivity_in_undepleted_region(sc::Semiconductor{T})::T where {T}
	100000
	end

SolidStateDetectors.jl/src/PotentialCalculation/PotentialCalculationSetup/PotentialCalculationSetupCylindrical.jl

Lines 217 to 221 in 70fc59c

	if is_undepleted_point_type(point_types.data[ig...])
	ϵ[ir, iφ, iz] *= scaling_factor_for_permittivity_in_undepleted_region(det.semiconductor) * (1 - imp_scale[ig...])
	elseif is_fixed_point_type(point_types.data[ig...])
	ϵ[ir, iφ, iz] *= scaling_factor_for_permittivity_in_undepleted_region(det.semiconductor)
	end

Also note that ϵ_r is computed on a grid corresponding to the (extended) midpoints of the electric potential grid (which is why the ϵ_r grid dimensions are +1 higher in each dimension that the electric potential grid). Therefore, the plot might look like the contact's relative permittivity is set to 1, even though it's just showing the value for outside. This plotting artifact might get more enhanced if the contact thicknesses are small/zero, and if the contact is close to the boundary of the world that you defined in the config file.

The fact that the contact dimensions seem to change with the bias voltage is probably due to the adaptive grid.
At 0V, SSD will just use the initial grid of the simulation (because they are no steep gradients that would trigger adding additional grid points). The higher the voltage, the more steep gradients are to be expect, and therefore the final grid might have more or fewer ticks close to the contact region, depending on the bias voltage applied. One way to avoid this, would be to define a grid and pass it to SSD instead of relying purely on the default grid refinement.

using SolidStateDetectors, Unitful

# create a Simulation object
sim = Simulation(SSD_examples[:InvertedCoax])

# define a grid from the geometries in the simulation, assure that grid points are maximum 0.1mm apart
grid = Grid(sim, max_tick_distance = 0.1u"mm")

# pass the grid to calculate_electric_potential! and turn of automatic refinement
# --> this will ensure that the potentials are always calculated on the same grid
calculate_electric_potential!(sim, grid = grid, refinement_limits = missing)

This might also get rid of the fluctuations of the capacitance values shown in the plot in the original post in this thread.
However, the fineness of the grid will then also influence how long it will take for the simulations to run.

[jasondet]

Thank you for the fast response @fhagemann. A couple of follow-up questions:

1. How does one obtain a plot of the actual relative permittivity of a detector at a specified voltage?

2. Does the grid spacing just produce plotting artifacts or could this be affecting the calculations too? Because aren't the plotting routines just interpolating the grid the same way the calculations are? For example, it's worrying to me that at 0 V there are points with "point type = 0" at just under z = 5 cm, while once a voltage is applied that region all changes to depleted material. How does one draw a plot at 0 V that correctly identifies which regions are contacts and which regions are bulk?

[fhagemann]

1. This is not saved in sim, but it can be determined using the same algorithm as in PotentialCalculationSetup, e.g.

   using SolidStateDetectors
   using SolidStateDetectors: is_undepleted_point_type, is_fixed_point_type, scaling_factor_for_permittivity_in_undepleted_region
   using Plots, Unitful
   
   sim = Simulation(SSD_examples[:InvertedCoaxInCryostat])
   sim.detector = SolidStateDetector(sim.detector, contact_id = 2, contact_potential = 500u"V")
   calculate_electric_potential!(sim, grid = Grid(sim,max_tick_distance = 0.5u"mm"), depletion_handling = true)
   
   ϵr = let _ϵr = fill(SolidStateDetectors.get_precision_type(sim)(sim.medium.ϵ_r), axes(sim.electric_potential.data))
       for ig in CartesianIndices(sim.electric_potential.data)
           pt = GridPoint(sim.electric_potential.grid, Tuple(ig))
           # determine "raw" ϵr from semiconductor and passives
           if pt in sim.detector.semiconductor
               _ϵr[ig] = SolidStateDetectors.get_ρimp_ϵ_ρfix(CartesianPoint(sim.electric_potential.grid[ig]), sim.detector.semiconductor)[2]
           end
           if !ismissing(sim.detector.passives)
               for passive in sim.detector.passives
                   if pt in passive
                       _ϵr[ig] = max(_ϵr[ig], SolidStateDetectors.get_ρimp_ϵ_ρfix(CartesianPoint(sim.electric_potential.grid[ig]), passive)[2])
                   end
               end
           end
           # scale up ϵr for fixed (conducting) volumes and undepleted volumes
           if is_undepleted_point_type(sim.point_types.data[ig]) 
               _ϵr[ig] *= scaling_factor_for_permittivity_in_undepleted_region(sim.detector.semiconductor) * (1 - sim.imp_scale.data[ig]) 
           elseif is_fixed_point_type(sim.point_types.data[ig]) 
               _ϵr[ig] *= scaling_factor_for_permittivity_in_undepleted_region(sim.detector.semiconductor) 
           end
       end
       DielectricDistribution(_ϵr, sim.electric_potential.grid)
   end
   plot(ϵr, xunit = u"mm", yunit = u"mm", colorbar_scale = :log10)

   

2. The grid spacing will also affect the calculations. You can specify the grid tick spacing when running simulations:

   sim = Simulation(SSD_examples[:InvertedCoaxInCryostat])
   apply_initial_state!(sim, ElectricPotential)
   # or calculate_electric_potential!(sim, refinement_limits = missing, depletion_handling = true)
   plot(sim.point_types)

   

   sim = Simulation(SSD_examples[:InvertedCoaxInCryostat])
   apply_initial_state!(sim, ElectricPotential, Grid(sim,max_tick_distance = 0.1u"mm"))
   # or calculate_electric_potential!(sim, grid = Grid(sim,max_tick_distance = 0.1u"mm"), refinement_limits = missing, depletion_handling = true)
   plot(sim.point_types)