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: \[\boldsymbol{J} = A(n-n^{eq}) V \exp \left( -B \sqrt{\phi} \right)\]

Tunnelling organic–organic

At organic heterojunctions, transport is often dominated not by pure quantum tunnelling but by trap-assisted transfer of carriers across the interface. In this case, carriers can drift into localized states at the boundary and hop across. The OghmaNano model treats this process phenomenologically: it vanishes at equilibrium and increases linearly with the carrier imbalance, much like surface recombination velocities.

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 rate constants describing the ease of carrier transfer. Unlike direct tunnelling, which is governed by an exponential dependence on barrier thickness and height, the organic–organic model captures hopping-like transfer through disordered states at the interface. It is therefore best understood as an effective transport channel layered on top of the usual drift–diffusion picture, enabling carriers to cross energetic “uphill” barriers that would otherwise be strongly suppressed.

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. \[\boldsymbol{J} = A(n-n^{eq}) V \exp \left( -\frac{q\phi}{kT} \right)\]