Observation of Anomalous Negative Differential Resistance in Diode Breakdown Simulation Using Carrier Temperature Dependent Impact Ionization

When carrier temperatures are used to model impact ionization with self-consistent cooling effects, anomalous negative differential resistance (NDR) was found in diode breakdown simulation. Possible mechanisms responsible for the NDR are analyzed.

In recent years, the drift-diffusion (DD) model has been extended to the energy transport (ET) model and the hydrodynamic (HD) model to account for non- equilibrium effects from the elevated average carrier energy at high fields.Inclusion of electron and hole temperatures as state variables allows transport coef- ficients such as mobilities and impact ionization rates as functions of the carrier temperatures instead of the local electric field to remove the thermal equilibrium approximation [1].However, this new parametriza- tion of the transport coefficients has received contro- versial criticisms.Although much success has been achieved in fitting specific experiment measurements, especially in MOSFET substrate current cases (e.g., [2]), it has also been established that the distribution function, in particular the tail part that determines the impact ionization and MOSFET gate current, is not well characterized by the carrier concentration and average energy alone [3, 4].Nevertheless, if the impact ionization rates are chosen as functions of the carrier temperatures, we report here for the first time an anomalous negative differential resistance (NDR) in a simple pn junction breakdown simulation.Although this NDR is very small in magnitude and only happens at certain choice of parametrization of transport coefficients in the ET and HD models, iden- tification of such numerical behaviors may prevent erroneous interpretations when complex device struc- tures are simulated.Possible mechanisms that can cause this NDR are briefly summarized.
We have chosen a test pn junction diode with n + 1019cm -3 and p 5 x 1017cm -3 to avoid confus- ing interpretation when significant amount of minor- ity carriers reaches metallurgical contacts.For simplicity, no band-to-band tunneling model is used.The vicinity of the breakdown region in the reverse- bias IV curves, using field and energy dependencies for the mobility Ix and the impact ionization rate et, is shown in Fig. 1.The simulation is performed by PiS-CES-2ET with modified curve tracing techniques [5] to capture NDR.The DD model with Ix(F) and c(F) does not have any NDR region and the breakdown Present address. 404Phillips Hall, School of Electrical Engineering, Cornell University, Ithaca, NY 14853.Tel: (607)255-3998.
Fax: (607) 254-4777.1019cm -3 and p 5 x 1017cm-3.The drift-diffusion (DD) and the energy transport (ET) models with different mobility models are used FIGURE 2 Carrier temperature profiles when electrical break- down happens.Notice that the temperatures are the same at the onset (low current) and deep (high current) breakdown regions.
The profiles are those of the ET g(Tc) model in Fig.
voltage is close to the measured data.At the onset of breakdown, the ET model with offTc) and g(Tc) shows a small NDR.However, the ET model with (X(Tc) and g(F),although it is not based on physical reasoning and is only used here as a numerical comparison, does not show any NDR region.There are two possible mechanisms for the observed NDR, and further inves- tigation is necessary to identify which mechanism is more dominant.The first possible mechanism is due to the cooling effect feedback from impact ionization [6].In the electron (similar for hole) energy balance equation, the energy exchange rate Uwn related to impact ionization can be included self-consistently as 3 3 kBTp) gp,ii kBTn (1) t,t w.
g n E g -+--" where gn, ii and gp,ii are the impact ionization rates ini- tiated by electrons and holes, Eg is the band gap, and T n and Tp are the electron and hole temperatures.This cooling gives a positive feedback for enhancement of g(Tc), since asymptotically l.t o Tc -1 for velocity satu- ration.This small NDR will mostly disappear if It(F) is used or the cooling mechanism is not accounted for in the carrier energy balance equation.The second possible mechanism is bipolar multiplication.When diode breakdown happens, both the electric field and carrier temperatures stop increasing Distance (microns) FIGURE 3 Carrier concentration profiles at the onset (low) and deep (high) breakdown regions (shown in Fig. 2) and the carrier concentrations (shown in Fig. 3) and impact ionization generation rates (shown in Fig. 4) start building up within the original depletion region.If cx(F) is used, the peak generatio n rates for electrons and holes are always aligned.However, as can be seen in Figs. 2 and 4, the peaks of Cn(Tn) and Op(Tp), as well as T n and Tp, are  x 1017) under 6V bias tor M defined by the ratio of outgoing and incoming electron (or hole) currents in the depletion region can be expressed as [7]" /--%exp (n-p)dx dx (2) If n(X) =-p(X), the breakdown condition as M is simply .[0w a 1 and no NDR is possible as long as the depletion region remains the same width and a has only one peak of F within it.However, if c n (x) p(X), the reverse breakdown current is affected by the convolution of o n and Cp, and NDR can exist for slightly perturbed T n and Tp profiles.
Besides, it can be observed that the breakdown voltage predicted by c(Tc) is much larger than that by c(F) [9].This can be understood from two aspects.First, when only one carrier is considered (as in nin cases of Fig. 5), the generation rate calculated by (Tc) is smaller than (F) due to the latent effect from dF/dx (note that in the bulk case, o(Tc) o(F) by definition).Second, if the peaks of c n and (xp are dislocated, the integral in (2) will be more damped than when n and otp are aligned.

Biographies Edwin C. Kan is a research associate in Stanford
University.His research interests include semiconduc- tor device and process modeling, VLSI circuits, and numerical methods for partial differential equations.
Gyo-young Jin is a post-doctoral research affiliate with Stanford University.His research interest includes device physics and modeling, especially Monte Carlo simulation and microscopic modeling of the transport phenomena in the semiconductor devices.
Zhiping Yu is a senior research associate in Stan- ford University.His research interests include IC process, device and circuit simulation, and in particu- lar, the numerical techniques and modeling of devices with heterostructures.
Robert W. Dutton is professor of electrical engi- neering at Stanford University.His research interests focus on integrated circuit process, device and circuit

offset by a small distance [ 6 ]FIGURE 4
FIGURE 4  Profiles of generation rates initiated by electron and hole impact ionization

FIGURE 5
FIGURE 5 The generation rate in 0.4 gm nin device (with n 5