But below 228K the situation alters drastically, as the value of n for the device crosses the value 2 and gradually increases to 3.93 due to further lowering of temperature to 110 K, which seems to indicate a feature, distinctly different from the mechanisms mentioned above, of tunneling phenomenon. In fact, a relatively temperature insensitive characteristic energy, ET = nKT of the device (approximately only about twenty percent change in the value of ET for a temperature change of 240 K), n greater than 2 and an exponential bias dependent relationship given by Eq. (1) are regarded as typical signatures of defect- assisted tunneling process which controls transport in the device below 228 K for region I. The nature and origin of traps are explored in a subsequent section.

Although an analysis on n with temperature gives tentative assignment of different transport mechanisms, in absence of a suitable theoretical model to fit the experimental data and for the purpose of extrapolation, it is found that the variation of n with temperature for region I as shown in Fig. 3, may be approximated by a quadratic function as suggested in ref. (Zhang et al. 2018)

Eq. (3) gives good agreement with experiment up to 148 K, shown in Fig. 3 by continuous line. At cryogenic temperatures below 148 K, n increases significantly, indicating for an additional effect, e.g., freeze-out of holes, which start to contribute intensely to carrier transport in this device (Park et al. 2016). Further, as an alternative to account for the temperature dependence of n for this device we have also tried an empirical formula n = ?T?, with ? and ? as temperature independent free parameters. The empirical formula fits our data well throughout the entire temperature range with the values of ? and ? given as 220 and 0.86, respectively (shown in Fig. 3 by dash-dash line).

In order to gain further insight into the mechanism of conduction let us have recourse to the model developed by Hirsch and Barriere for similar devices (Hirsch and Barriere 2003). In this model from a study of energy band profile the authors have assumed the current conduction to be dominated by two distinctly different energetic domains and radiative recombination occurs in the quantum well (QW) nearest to p-GaN due to low mobility of holes. The following expression for the total current density is used for this purpose

where vz, hole velocity along the z direction and dp, hole concentration with corresponding velocities between v and v + dv. Also, in the above expression Pe denotes the probability that a hole reaches the nearest QW. Pe is a function of both thickness and height of the potential barrier while the thickness of the barrier is fixed for a device its height is a function of hole energy as well as bias and thermal stress. When the hole energy is high the hole transfer is considered to be a certainty and Pe is treated as a constant and may be approximated to unity (Pe ? 1). For lower energies Pe is less than unity and need to be estimated in terms of the parameters of the device. Based on these considerations Hirsch and Barriere (2003) have divided the total current density into two parts . where Pt, tunnel transfer probability. This division immediately suggests that at high temperature the hole energy is greater than the activation energy and Pe ? 1. In this case, J1 alone contributes to the total current density and is given by (Hirsch and Barriere 2003)

where ?0, barrier height without an electric ?eld and ?V, potential drop due to GaN barrier. The activation energy Ea (= ?0 ? ?V) may be evaluated from an analysis of the experimental data by Eq. (6). It is interesting to note that the expression is similar to that of thermoionic hole emission given by Sze and Ng (2007). The equation also clearly shows that for a fixed temperature logI depends linearly on V.

Similarly, at low temperature and hole energy is not enough for thermoionic emission since Pe 228 K) a linear region occurs for all current versus inverse temperature curves for different voltages. We evaluate the value of Ea from Eq. (6) which may be approximated by taking log of both sides and retaining only the dominant term we get