getelec.potential_barrier

Position below which the image term makes the potential unphysical, in nm.

-k_e/(4x) diverges at the surface, so the barrier has to be cut off somewhere. The right place is where the potential first crosses zero on the way up -- the inner root of E_F + phi - F x - k_e/(4x) = 0,

x_c = (W - sqrt(W^2 - k_e F)) / (2F),   W = E_F + phi .

Below it the expression is negative only because of the divergence; above it, any negative value is the real barrier falling away past its outer turning point and must be kept.

Parameters

fermi_level, work_function, electric_field : float eV, eV, V/nm.

Returns

float

Abstract base class for potential barriers.

Defines the shared structural interface for computing spatial potential energy profiles relative to an emitting surface boundary.

Every parameter a barrier's shape depends on, as a hashable tuple.

Read from the instance's own attributes rather than a fixed list, so a subclass that introduces a new parameter is distinguished automatically. A fixed list silently collided: two barriers differing only in a new attribute produced the same cache key, and the solver returned the first one's potential for both.

Barriers that compute their profile from something other than plain attributes should override parameters to include it.

Planar surface potential model with Schottky image-charge correction.

Models the classical combination of a material's inherent work function barrier, a uniform linear applied external electric field, and the image-charge reduction term felt by a departing electron outside a planar conductive surface.

Parameters

fermi_level : float, default 9.5 The interior Fermi energy level of the material system. work_function : float, default 4.5 The inherent work function barrier of the surface. electric_field : float, default 3.0 The applied external electric field magnitude acting on the interface.

Attributes

fermi_level : float Stored value for the material chemical potential. work_function : float Stored value for the inherent surface work function. electric_field : float Stored value for the active external electric field.

Curved surface potential model for localized nano-emitter geometries.

Accounts for field enhancement and modified image-force interactions experienced at the boundary tips of high-curvature nanostructures (such as field emission needles), incorporating a tip radius and geometric scaling parameter ($\gamma$).

Parameters

fermi_level : float, default 9.5 The interior Fermi energy level of the material system. work_function : float, default 4.5 The inherent work function barrier of the surface. electric_field : float, default 3.0 The macroscopic baseline applied external electric field. radius : float, default 20.0 The physical radius of curvature of the micro/nano-emitter tip, in nm. The barrier is valid for 20-1000 nm (RADIUS_VALIDITY); outside that range the class warns. gamma : float, default 100 Geometric field enhancement/correction multiplier coefficient.

Attributes

fermi_level : float Stored value for the material chemical potential. work_function : float Stored value for the inherent surface work function. electric_field : float Stored value for the active baseline electric field. radius : float Stored value for the apex radius of curvature. gamma : float Stored value for the geometric enhancement tracking constant.

Notes

Across its valid range the potential is close to the planar Schottky-Nordheim barrier of SchottkyPotential: at R = 20 nm it lies within 0.3 eV of it across the barrier at the Fermi level (under 7% of the work function) at 3-8 V/nm, for gamma = 100 and E_F + phi = 12 eV. The current depends exponentially on the barrier, so it is not close: at R = 20 nm it is 0.22, 0.57 and 0.79 times the planar current density at 3, 5 and 8 V/nm, and at 50 nm 0.56, 0.80 and 0.91 times. The difference falls as 1/R; at 1000 nm, the top of the range, it is 2.8%, 1.1% and 0.6% at 3, 5 and 7 V/nm.

On the axis the applied term is exact to second order in x/R: its curvature is the one Laplace's equation requires at an apex of radius R. The third-order term, c3 F x^3 / R^2, is set by the shape of the emitter away from its apex, which radius and gamma do not describe: c3 = 4/3 for an ellipsoidal tip of any aspect ratio, about 1.05 for a hemisphere on a cylindrical post at gamma = 100, and gamma / (gamma - 1) here. The current density is sensitive to it, most of all near the outer turning point. Against the exact electrostatics of the two shapes (gamma = 100, phi = 4.5 eV, 300 K) this barrier under-predicts the current density at R = 20 nm by 3.3%, 0.7% and 0.26% at 3, 5 and 7 V/nm for an ellipsoidal tip, and by at most 0.5% for a hemisphere on a post; at R = 10 nm the ellipsoidal figures are 16%, 3.1% and 1.1%. The difference grows roughly as phi^(7/2) / (F^3 R^2), which is why the class warns below 20 nm.

Planar triangular barrier: the vacuum level tilted by the field, with no image charge.

The barrier of Fowler and Nordheim (1928),

V(x) = E_F + phi - F x    for x > 0,     V(x) = 0 inside the metal.

Without the image term there is no Schottky lowering: the maximum is the full E_F + phi, at the surface, and the barrier is taller and wider than SchottkyPotential at the same field, so it emits less. Past the outer turning point the potential keeps falling below zero, as the applied field makes it; nothing is clamped.

Parameters

fermi_level : float, default 9.5 The interior Fermi energy level of the material system, in eV. work_function : float, default 4.5 The inherent work function barrier of the surface, in eV. electric_field : float, default 3.0 The applied external electric field, in V/nm.

Attributes

fermi_level : float Stored value for the material chemical potential. work_function : float Stored value for the inherent surface work function. electric_field : float Stored value for the active external electric field.

Notes

This is the one barrier with a closed-form transmission: ~getelec.transmission_solutions.AiryTriangular (method="airy") is exact for it, below and above the barrier top. The analytic ~getelec.transmission_solutions.WKB solver is the Schottky-Nordheim form and does not describe this barrier.

The potential jumps by E_F + phi at the surface, which a grid solver resolves only to first order in its step -- by an amount set by where the surface falls between two grid points, so a node landing on the surface costs 1.3% in current density at h = 1e-3 nm. The Noumerov grid puts the surface halfway between two nodes, the placement that cancels the first-order term: against the exact solution the current density is within 1e-5 at the default step and the transmission within 6e-5, measured for E_F + phi = 5 to 13.5 eV, 1-12 V/nm and 300-2000 K.

The keyword names a callable will accept, past its first argument.

Returns "any" for a callable taking **kwargs, None when the signature cannot be read (a C callable), and otherwise the set of names. The first positional parameter is left out: it is the position array.

Barrier built from a potential the user supplies.

Takes either a callable V(x) or a tabulated profile and presents it to the solvers exactly as the shipped barriers do, so a potential worked out elsewhere -- a screened or self-consistent one, a profile from another code, measured points -- can be run through the same transmission solvers, emitters and distributions without subclassing anything.

Parameters

potential : callable or array_like Either V(x), taking positions in nm and returning energies in eV pointwise (same shape in, same shape out), or an array of potential values to be interpolated, in which case positions is required. positions : array_like, optional Sample positions for a tabulated potential, nm, strictly increasing. Not accepted for a callable, which is evaluated wherever the solver asks. fermi_level, work_function, electric_field, temperature : float, optional eV, eV, V/nm and K. Not used to build the potential -- that is entirely the user's -- but passed to the callable if it takes parameters of those names, and the first three are what ~getelec.transmission_solver.Noumerov reads to size its integration domain (auto_domain). Without them the domain falls back to the solver's max_barrier_width, 3 nm by default, which is too short for a low field. These four are the parameters update_params sets on every component at once, so handing this barrier to ~getelec.metal_emitter() replaces them with that call's arguments -- which is what keeps a barrier that depends on the temperature from disagreeing with the supply function about it. ~getelec.semiconductor_emitter() sets none of them: give them here. zero_inside : bool, default True Hold the potential at 0 for x <= 0 and evaluate the user's potential only outside, which is the convention of every other barrier here: the energy zero is the bottom of the conduction band, the metal is flat, and the vacuum level at the surface is E_F + phi. Set it to False only if the potential describes the metal side itself; the solver still needs it flat over the last two nodes of the metal, where the plane-wave match is made. **extra_parameters Any further named parameters of the callable, kept on the barrier and passed at every evaluation. A name the callable will not accept is an error, not a value silently dropped.

Attributes

potential_function : callable or None The user's callable, or None for a tabulated potential. table_positions, table_potential : numpy.ndarray or None The tabulated profile, or None for a callable. fermi_level, work_function, electric_field, temperature : float or None Stored values, passed to the callable and used to size the domain. zero_inside : bool Whether the metal side is held at zero. extra_parameters : dict The further parameters passed to the callable.

Notes

Units are this package's throughout: x in nm, V in eV, measured from the same zero as the other barriers (0 inside the metal).

The potential must fall below the emitted energies before the end of the solver's grid. A barrier still above them at the last node is not solved slightly wrongly, it is meaningless -- that is what auto_domain protects against for the shipped barriers, and it needs the three parameters above.

Nothing is clamped near the surface. A potential that diverges there -- an unscreened image term -- has to be cut off by the caller; get_image_cutoff() is the position the shipped barriers use.

Only the grid solvers read the barrier. method="wkb" and method="airy" build their transmission from E_F, phi and F alone, so with a barrier of your own they return the Schottky-Nordheim and triangular answers, not yours. Use method="noumerov", the default.

The solvers cache the profile against the barrier's parameters, so a callable is evaluated once per grid rather than once per energy. Its result must therefore depend only on its arguments: a callable reading a global that changes between calls returns the first profile again.

Examples

>>> triangular = Customised(
...     lambda x, fermi_level, work_function, electric_field:
...     fermi_level + work_function - electric_field * x,
...     fermi_level=7.5, work_function=4.5, electric_field=5.0)
>>> triangular.get_potential(np.array([-0.5, 1.0, 3.0]))
array([ 0.,  7., -3.])