getelec.band_structure

Abstract base class representing a material's energy band structure.

Parameters

energy_resolution : float, default 0.01 The step size or resolution for the energy grid discretization, in electron-volts (eV)

Attributes

energy_resolution : float The step size or resolution for the energy grid discretization.

Represents a basic uniform metallic band structure grid.

Parameters

lower_energy_limit : float, default 0.001 The minimum boundary of the uniform energy grid. It represents the bottom of the conduction band and it is the zero for our energy reference system. upper_energy_limit : float, default 30.0 The maximum boundary of the uniform energy grid. It represents the upper limit of the energy grid and it can be above the top of the potential barrier. energy_resolution : float, default 0.01 The discretization step size for the grid.

Attributes

lower_energy_lim : float The minimum boundary of the uniform energy grid. upper_energy_lim : float The maximum boundary of the uniform energy grid. energy_resolution : float The discretization step size for the grid.

Represents a semiconductor band structure with conduction and valence bands.

Parameters

lower_energy_limit : float, default 0.001 The absolute lowest energy limit for the valence band generation. It represents the bottom of the valence band and it is the zero for our energy reference system. top_valence : float, default 12.5 The energy level corresponding to the top of the valence band. fermi_level : float, default 13.0 The Fermi energy level of the semiconductor system. band_gap : float, default 1.12 The energy band gap separating the valence and conduction bands. electron_eff_mass : float, default 1.64 The effective mass of electrons in the conduction band. It is a float that represents the relative effective mass of the electron when compared with the free electron mass. hole_eff_mass : float, default 0.68 The effective mass of holes in the valence band. It is a float that represents the relative effective mass of the hole when compared with the free electron mass. upper_energy_limit : float, default 30.0 The absolute maximum energy limit for the conduction band generation. energy_resolution : float, default 0.002 The discretization step size for the base grids. Finer than a metal's default because each band's grid ends at its band edge, where the TED has a finite slope: the trapezoid error in the current is then ~(h / k_B T)^2 / 12, about 1.1% at 0.01 eV and 300 K.

Attributes

lower_energy_lim : float The absolute lowest energy limit. upper_energy_lim : float The absolute maximum energy limit. energy_resolution : float The discretization step size. top_valence : float The top of the valence band. fermi_level : float The Fermi energy level. band_gap : float The semiconductor energy band gap. electron_eff_mass : float The effective mass of conduction electrons. hole_eff_mass : float The effective mass of valence holes.

Represents an adaptive metal grid system optimizing energy boundaries dynamically.

Notes

Use only with SN barrier. The rest of the barriers need their own optimisation protocol. If not SN barrier, use Metal or Custom.

The defaults are built for the currents an emitter actually produces. They hold the current density and the Nottingham heat within 1% of a far wider grid wherever the current density reaches 1e-12 A/cm^2, which is 1 pA from 1 cm^2 -- checked for the planar, sharp-tip and triangular barriers, work functions of 2.5 to 6 eV, fields of 0.2 to 12 V/nm and 300 to 3000 K. Below 1 pA/cm^2 the cuts start to show, and the smaller the current the more it costs to reach: the same 1% at arbitrarily small currents would need a supply_threshold around 1e-40 and the energies to match. If those currents are what you are after, pass a smaller supply_threshold and a larger barrier_width, or set the limits yourself with Metal or CustomMetal.

Parameters

barrier_width : float, default 3.0 The spatial width of the potential barrier, beyond which there is effectivively no tunneling. supply_threshold : float, default 5e-24 The numerical threshold constraint used to cap maximum grid energies, beyond which the emitted electrons do not substatially contribute to the total emitted current. The cut sits k_B T * ln(1 / supply_threshold) above the Fermi level, 53.7 k_B T at the default. Thermionic emission comes from the top of the barrier, so the grid has to reach it: at 32.2 k_B T (a threshold of 1e-14) the cut falls below the top once the top lies more than about 25 k_B T above the Fermi level, and the current is then underestimated -- by 7% for a 4.5 eV work function at 0.5 V/nm and 1500 K. energy_resolution : float, default 0.01 The structural resolution step size of the resulting mesh.

Attributes

barrier_width : float The spatial width of the potential barrier. supply_threshold : float The numerical threshold constraint.

Semiconductor band grids trimmed with the two criteria of SmartMetal.

The grids of Semiconductor, cut in each band:

• bottom: where the Schottky-Nordheim barrier becomes wider than barrier_width. As in SmartMetal, a band's grid never starts higher than 1 eV below the top of its occupied states, min(E_F, band top): field emission comes from within a few decay widths of that energy, and at low field the barrier criterion alone would start above it. For the conduction band that is 1 eV below E_F; for the valence band 1 eV below min(E_F, E_V), so a valence band lying more than 1 eV below the Fermi level keeps the energies just below E_V it emits from.
• top: where the supply ln(1 + exp(-(E - E_F)/k_B T)) falls below supply_threshold.

What survives is clipped to each band: the conduction grid never starts below E_C, and the valence grid never runs past the top of the Semiconductor valence grid. A band that lies entirely outside the window comes back empty and contributes nothing -- the conduction band of a semiconductor at 0 K with the Fermi level in the gap, for example. The points kept are exactly those of the Semiconductor grids with the same parameters, so the two agree wherever neither cut applies.

Parameters

lower_energy_limit : float, default 0.001 The bottom of the valence band, and the zero of the energy scale, in eV. top_valence : float, default 12.5 The top of the valence band, in eV. fermi_level : float, default 13.0 The Fermi level, in eV. Used when generate_band_structure is not given one; an emitter passes its barrier's. band_gap : float, default 1.12 The band gap, in eV. electron_eff_mass : float, default 1.64 Conduction band effective mass, relative to the free electron mass. hole_eff_mass : float, default 0.68 Valence band effective mass, relative to the free electron mass. upper_energy_limit : float, default 30.0 The top of the conduction band grid before trimming, in eV. energy_resolution : float, default 0.002 The grid step, in eV. See Semiconductor for why it is finer than a metal's. barrier_width : float, default 3.0 The barrier width, in nm, below whose energy the transmission is taken as negligible. supply_threshold : float, default 5e-24 The supply below which the electrons are taken as negligible. The cut sits k_B T * ln(1 / supply_threshold) above the Fermi level, 53.7 k_B T at the default, which keeps the barrier top inside the grid wherever the current reaches 1e-12 A/cm^2. See SmartMetal.

Attributes

lower_energy_lim, upper_energy_lim, energy_resolution : float As for Semiconductor. top_valence, fermi_level, band_gap : float As for Semiconductor. electron_eff_mass, hole_eff_mass : float As for Semiconductor. barrier_width, supply_threshold : float As for SmartMetal.

Notes

The barrier criterion uses the Schottky-Nordheim barrier, as SmartMetal does, whatever barrier the emitter holds.

As for SmartMetal, the defaults are set for currents down to 1e-12 A/cm^2 (1 pA from 1 cm^2). For currents far below that, pass a smaller supply_threshold and a larger barrier_width, or give the grids yourself with CustomSemiconductor.

Represents a static user-provided pre-defined metal band structure system. It can be used to set any arbitrary metallic band structure - the user should be ware of checking for consistency with the other modules

Parameters

custom_energy_array : numpy.ndarray The pre-computed custom structured array representation.

Attributes

custom_energy_array : numpy.ndarray The pre-computed custom structured array representation.

A semiconductor band structure on energy grids the user supplies, one per band.

The counterpart of CustomMetal: the energies are returned as given, for a range, a resolution or a spacing that Semiconductor does not offer. What a semiconductor needs beyond the energies -- the lower limit of each energy's E_z window, Eq. (8) of the paper -- follows from the band edges and effective masses, exactly as in Semiconductor, so the physics does not depend on which of the two built the grid.

The grids are checked when the band structure is generated, because points outside a band have no states and would make the distributions inconsistent: each must be a finite, strictly increasing 1-D array, the conduction energies must not lie below E_C = top_valence + band_gap, and the valence energies must not lie above top_valence. Either band may be empty, and then contributes nothing.

Parameters

custom_conduction_array : array_like Conduction band energies, in eV. custom_valence_array : array_like Valence band energies, in eV. top_valence : float, default 12.5 The top of the valence band, in eV. band_gap : float, default 1.12 The band gap, in eV. electron_eff_mass : float, default 1.64 Conduction band effective mass, relative to the free electron mass. hole_eff_mass : float, default 0.68 Valence band effective mass, relative to the free electron mass.

Attributes

custom_conduction_array, custom_valence_array : numpy.ndarray The energies of each band. top_valence, band_gap, electron_eff_mass, hole_eff_mass : float As for Semiconductor.

Notes

The distributions are verified on uniform grids, which is what the other band structures produce. A non-uniform grid is integrated correctly, but the transmission over each window is sampled at the grid's first step.

A metal whose energy grid and state count both come from a tabulated density of states.

The other metal band structures assume a free electron gas, whose density of states goes as sqrt(E) above the bottom of the conduction band. This one takes g(E) from a table -- a DFT calculation, say -- and weights every total energy by how many states it actually holds relative to that free electron reference,

R(E) = [g(E) / sqrt(E)] / [g(E_F) / sqrt(E_F)] ,

normalised so that R(E_F) = 1. The emitter multiplies the total energy distribution by R and rebuilds the normal energy distribution from the same double integral, so the current density, the Nottingham heat and both distributions stay consistent with each other.

The energies are the table's own, so the table sets both the range and the resolution of the grid. Energies at or below zero are dropped: zero is the bottom of the conduction band and the reference for every energy in this package, and 1 / sqrt(E) has no meaning below it.

Parameters

dos_energies : array_like Energies of the tabulated density of states, in eV, on the same scale as fermi_level -- that is, measured from the bottom of the conduction band, not from the Fermi level. Must be finite and strictly increasing. dos_values : array_like The density of states at those energies. Any units: only the shape survives the normalisation, which is why absolute currents remain comparable to the free electron result even for a projected density of states with no absolute normalisation of its own. fermi_level : float The Fermi level, in eV, measured from the bottom of the conduction band. Must lie inside the tabulated range, where the table must have states. work_function : float The work function, in eV. Used only to check that the grid reaches over the top of the barrier; the barrier itself holds its own copy.

Attributes

dos_energies, dos_values : numpy.ndarray The table, cropped to positive energies. fermi_level, work_function : float As above. energy_resolution : float The table's own step, taken as the median spacing of its energies.

Notes

What the weight is, and what it is not. In the supply integral the group velocity cancels the Jacobian, v_z dk_z = dE / hbar, so the quantity that belongs here is the number of forward-propagating channels at energy E -- the area of the constant energy surface projected on the surface plane, counted with multiplicity -- and not the density of states, which weights states by one over the same velocity. Writing R = g / sqrt(E) assumes the two are related as they are for free electrons. Where flat bands dominate g they have small normal velocity and small projected area, so their contribution is overestimated; for a d band metal that is a real effect, not a rounding one. Treat this as a model of the band structure's influence on emission rather than as a first-principles calculation of it. If the number of channels N(E) is known instead, it belongs in the same slot against its own free electron reference, which goes as E and not as sqrt(E): get_state_weights() is the only place the ratio is formed.

Range, and where this stops being physics. A table ends where the calculation that produced it ran out of bands, and its density of states falls to zero at that edge for that reason and not a physical one. Emission from the top of a table is therefore an artefact of the band count: the model reports no states where a real metal has a free electron-like continuum, and the current comes out too low.

Field emission is untouched by this, because it draws from within a fraction of an electronvolt of the Fermi level -- on a tabulated d band metal the share of the current density coming from the table's top 1 eV is around 1e-30 at 300 K, at every field from 1.5 to 7 V/nm. Thermionic emission is another matter: it draws from the top of the barrier, which sits close to the end of the table, and the same share reaches 0.99 at 1.5 V/nm and 2000 K, where the tabulated current density comes out 23 times below the free electron one for no physical reason.

So generate_band_structure warns when the top of the table clears the higher of the barrier maximum and the Fermi level by less than 8 k_B T. The barrier maximum is the Schottky-Nordheim one, E_F + phi - 2 sqrt(k_e F / 4), as in SmartMetal. Take the warning seriously: what it flags is not a discretisation error but a result set by where the band structure calculation stopped.

Examples

>>> band = DensityOfStatesMetal.from_file(          # doctest: +SKIP
...     "dos.txt", fermi_level=10.268, work_function=4.67)
>>> energies = band.generate_band_structure()       # doctest: +SKIP
>>> weights = band.get_state_weights(energies)      # doctest: +SKIP