Analytical Models of Interband Tunneling


Gallium Nitride, Band-to-band Tunneling, PN diode


Given the difficulty of creating good ohmic contacts to \(n\)-doped GaN, the nitrides community would benefit greatly from alternative structures which allow large currents to flow from \(p\)-contacts into \(n\)-GaN. One such possibility is “spiked” \(pn\) junctions in GaN, where the regrowth of the junction interface produces a built-in sheet charge. This sheet charge notches the band diagram, forcing the high electric fields necessary for large tunneling currents. Recent work has derived analytical approximations to the band diagrams of these devices, yet, due to the highly non-uniform fields, simple expressions of the tunneling currents in these devices remain an open problem. Relating the design parameters, eg geometry and doping, of these interfaces to the tunneling current will be vital for rapid modeling if these structures are to find application in LEDs and more.

The initial theoretical studies of interband tunneling began with Keldysh (Keldysh 1958) and Kane (Kane 1960). In particular, the two-band model of Kane has proven an effective approximation for the study of many direct-gap semiconductors. A major intuitive simplification was introduced by Sze (Sze 1981), who approximated the problem as a classic tunneling wavefunction through a one-dimensional barrier, neglecting the details of the matrix element and band-structure. Both methods assumed, however, a uniform electric field. Within Sze’s picture, this assumption can be relaxed rather simply. Many authors (eg (Takayanagi 1991), (Tanaka 1994)) have since expanded Kane’s model as well to non-uniform field, though at the cost of greater mathematical complexity. Given the trade-offs of both Kane’s and Sze’s approaches, it will be useful to study the application of each model to the non-uniform field profile of the spiked \(pn\) junction.

Sze’s WKB method

Given the mathematical ease of Sze’s method, it is the more natural place to begin. In a semi-classical, effective mass/envelope function picture of electron transport in a crystal, the state is imagined to be a wavepacket with simultaneously well-defined and smoothly evolving position \(x\) and crystal momentum \(\hbar k\). The energy \(E\) of the tunneling electron is constant, but the bands \(E_C(x)\), \(E_V(x)\) bend throughout the device. At any position \(x\), one could then determine the electron’s energy relative to changing the band edges, and then, from the crystal dispersion relation, compute \(k(x)\) for every position.

As the electron enters the classically forbidden region, the \(x\)-component of its wavevector passes through zero onto the imaginary axis and the wavepacket decays in amplitude. As the electron exits the region, its wavevector passes back through zero once more and returns to the real axis. Once \(k(x)\) is known, the WKB approximation provides a simple expression for the transmission coefficient through a tunneling barrier: \[T(E)=\exp\left\{-2\int_{x_v}^{x_c}dx|k(x)|\right\} \label{WKB}\] where \(x_v\) and \(x_c\) are the valence and conduction-side boundaries of the tunnel region.

Reduction to a scattering problem

To evaluate this integral, \(k(x)\) should, in principle, come from the (complex) dispersion relation of the crystal, or some model thereof (eg (Guan 2011)). In our approach, we will properly evaluate \(k(x)\); however, it’s worth noting the clever trick which Sze leverages to ignore the details of the dispersion and map the solution onto an introductory-level scattering problem. In explaining this point, we will consider only electrons with no transverse momentum, and we briefly restore the assumption of uniform electric field, \(\xi(x)=\xi=E_G/q(x_c-x_v)\).

With the assumption of parabolic bands (both of effective mass \(m^*\)) for small \(k\), one can analytically continue the dispersions \(E(k)=E_v(x)-\hbar^2k^2/2m^*\), \(E(k)=E_c(x)+\hbar^2k^2/2m^*\) onto imaginary \(k\) to find valid solutions in the gap. (Note that this only holds near to the respective band edges because of the parabolicity assumption.) So, near \(x_v\), we have \(E=E_V(x)+\hbar^2|k|^2/2m^*\). Since, at the start of the tunnel region, \(E_V(x_v)=E\), and the slope of \(E_V\) is given by \(\xi\), \(E_V(x)=E-q\xi(x-x_V)\), and \[ik(x)=\sqrt{2m^*q\xi(x-x_v)/\hbar^2}\] The above is formally the same as the classic dispersion of an particle of energy \(E\) in a barrier \(U(x)\), where \(U(x)-E=q\xi(x-x_v)\). Playing the same game near \(x_c\), we find an effective barrier \(U(x)-E=q\xi(x_c-x)\). When the electron is deep into the tunneling region, the dispersion is of course more complicated. Nonetheless, Sze interpolates the simplest algebraic functional form for a barrier \(U(x)\) which fits to the above limits. That is the quadratic: \[U(x)-E=\frac{(E_G/2)^2-(q\xi x)^2}{E_G}\] where the zero of \(x\) has been implicitly set to the mean of \(x_c\) and \(x_v\). In Sze’s approximation, the transmission coefficient for interband tunneling is just given by the transmission coefficient of a particle with a familiar parabolic dispersion (and mass \(m^*\)) through the above parabolic potential barrier. Conveniently, this coefficient does not even depend on the energy of the tunneling electron (within the WKB approximation, for uniform fields, and ignoring transverse momentum). Note that, if the conduction and valence bands have different effective masses, \(m^*\) should be a reduced effective mass; in Sze’s convention, \(m^*\rightarrow m_r^*=2/(\frac{1}{m_e^*}+\frac{1}{m_h^*})\).

Allowing for transverse momentum can be shown to simply raise the parabolic barrier by \(E_\perp=\hbar^2k_\perp^2/2m^*\). Including this effect, and evaluating the WKB integral, we find \[T(E,E_\perp)=\exp\left\{-\frac{\pi\sqrt{m^*}(E_G+2E_\perp)^{3/2}}{2\sqrt{2}q\hbar\xi}\right\}\] Despite the roughness of the above derivation, Tanaka (Tanaka 1994) finds more rigorously that for a two-band model in a uniform field, the exponential dependance of the transmission coefficient is precisely Sze’s expression. Given \(T\), one can then sum over the group velocities of all the contributing \(k\)-states to find the current.