Semiconductor Interface Models in OghmaNano: Drift–Diffusion, Tunnelling, and Charge Doping

The equations below are adapted from section 4.16.3.1 "Possible Conduction Mechanisms" in the chapter Electronic Properties of Alkanethiol Molecular Junctions: Conduction Mechanisms, Metal–Molecule Contacts, and Inelastic Transport from the book Comprehensive Nanoscience and Technology. They are referenced back to Sze SM (1981) Physics of Semiconductor Devices. By default, carriers in OghmaNano already drift and diffuse across interfaces according to the gradients of the conduction and valence bands. Uphill band offsets suppress carrier flow, while downhill alignments allow easy transfer. The additional interface models described here sit on top of that baseline drift–diffusion picture, providing extra transport channels (for example, tunnelling or hopping) that can assist carriers in overcoming energetic barriers.

Direct tunnelling

\[\boldsymbol{J} = A(n-n^{eq}) V \exp \left( -\frac{2d}{\hbar} \sqrt{2m q\phi} \right)\] Here, \(A\) is a constant, \(V\) is the applied bias, and \(\phi\) is the barrier height calculated from the band structure. \(m\) is the electron mass, and \(d\) is the thickness of the barrier. In OghmaNano this is implemented in a simplified form: \[\boldsymbol{J} = A(n-n^{eq}) V \exp \left( -B \sqrt{\phi} \right)\]

Interface-assisted transport (organic–organic)

At organic heterojunctions, carrier transfer across the interface is often mediated by localized states rather than by direct band transport. In this regime, the interface behaves as a finite conductance that enables carriers to transfer between the two materials via trap-assisted or hopping-like processes.

The OghmaNano model represents this behaviour phenomenologically as a linear response to the departure from equilibrium, ensuring that the transfer current vanishes at equilibrium and increases with carrier imbalance across the interface.

For holes: \[\boldsymbol{J_p} = q T_{h}\,\big((p_{1}-p_{1}^{eq})-(p_{0}-p_{0}^{eq})\big),\] and for electrons: \[\boldsymbol{J_n} = -q T_{e}\,\big((n_{1}-n_{1}^{eq})-(n_{0}-n_{0}^{eq})\big).\]

Here, \(T_{h}\) and \(T_{e}\) are phenomenological transfer coefficients that describe the strength of coupling across the interface. This formulation is equivalent to introducing a two-sided surface recombination velocity or an interface conductance, and can be interpreted as the linearised limit of trap-assisted or hopping-mediated transfer between localized states.

The model should therefore be viewed as an effective transport channel superimposed on the drift–diffusion equations, enabling current flow across energetic barriers that would otherwise strongly suppress carrier transport.

Fowler–Nordheim tunnelling

\[\boldsymbol{J} = A(n-n^{eq}) V^2 \exp \left( -\frac{q4d\sqrt{2m} \phi^{3/2}}{3q \hbar V} \right)\] Not yet implemented but could be on request. Here \(A\) is a constant, \(V\) is the applied bias, and \(\phi\) is the barrier height calculated from the band structure, \(m\) is the electron mass, and \(d\) is the thickness of the barrier. In the model this is implemented as: \[\boldsymbol{J} = A(n-n^{eq}) V^2 \exp \left( -\frac{B \phi^{3/2}}{V} \right)\]

Thermionic emission

\[\boldsymbol{J} = A(n-n^{eq}) T^2 \exp \left( -\frac{q\phi -q\sqrt{qV/ 4 \pi \epsilon d}}{kT} \right)\]

Not yet implemented but could be on request. Here \(A\) is a constant, \(V\) is the applied bias, and \(\phi\) is the barrier height calculated from the band structure, \(m\) is the electron mass, and \(d\) is the barrier thickness. In the model this is simplified to: \[\boldsymbol{J} = A(n-n^{eq}) T^2 \exp \left( -\frac{q\phi -B\sqrt{V}}{kT} \right)\]

Hopping conduction

Not yet implemented but could be on request. \[\boldsymbol{J} = A(n-n^{eq}) V \exp \left( -\frac{q\phi}{kT} \right)\]