1. Linear Poisson equation solver

QTCAD contains two Poisson solvers, a linear Poisson solver (which solves the Poisson equation usually introduced in basic electricity classes), and a non-linear Poisson solver which self-consistently accounts for the carrier densities in classical reservoirs. While the non-linear Poisson solver is typically used to model spin qubits in semiconductors, a linear Poisson solver is available for both semiconductor and superconducting devices.

The linear Poisson solver solves the partial differential equation

(1.36)\[-\nabla\cdot\left(\varepsilon\nabla\varphi\right)=\rho,\]

where \(\varepsilon\) is the dielectric permittivity of the medium, \(\varphi\) is the electric potential, and \(\rho\) is the (free) charge density.

Relevant device attributes

Parameter	Symbol	QTCAD name	Unit	Default	Setter
Permittivity	\(\varepsilon\)	`eps`	F/m	11.8 \(\varepsilon_0\)	`set_permittivity`
Charge density	\(\rho\)	`rho`	C/\(\mathrm m^3\)	0	`set_charge_density`

Variable	Symbol	QTCAD name	Unit
Potential	\(\varphi\)	`phi`	V

See Parameter setters for instructions on how to employ device parameter setters.

Another way to set the device permittivity attribute is through materials. See qtcad.device.materials for a description of materials attributes, which include permittivity.

Note

The default dielectric permittivity value is that of silicon,

Relevant solver attributes

The solver attributes are set by instantiating a SolverParams object with the desired attribute values and inputting it in the solver_params keyword argument of the solver constructor.

The default values of the relevant linear poisson solver attributes are given in the SolverParams API reference.

We remark that the same workflow is used to configure the parameters of all other QTCAD solvers.

Boundary Conditions

Here, we describe the boundary conditions that may be used in the linear Poisson solver.

Dirichlet boundaries (`API reference`)

On Dirichlet boundaries, the value of the electric potential is simply set to be equal to the applied potential \(\varphi_\mathrm{bias}\), i.e.

(1.37)\[\varphi = \varphi_\mathrm{bias}.\]

Natural boundaries

All boundaries at which no specific boundary condition is specified by the user are called natural boundaries. For the linear Poisson equation, such natural boundary conditions correspond to

(1.38)\[\mathbf{\hat n}\cdot \left(\varepsilon \nabla \varphi\right) = 0,\]

where \(\mathbf{\hat n}\) is the unit vector normal to the surface, pointing outside the simulation domain. In other words, natural boundaries set the component of the displacement field which is normal to the surface to zero.