Appendix E - Thermal Considerations

E.1 - Introduction

Thermal effects have been ignored in the preceding chapters. The next section justifies this approach, by showing that the temperature differential between the ends of the diode and its interior regions is always low, so that if good heat-sinking is used the entire device will remain cool. It is also shown that thermal effects occur on a much longer time scale than the electrical switching effects.

E.2 - Theory

The thermal resistance of a structure of length L and and cross-sectional A can be calculated in a manner analogous to electrical resistance, that is:

(E.1)

where κ is the thermal conductivity. The thermal capacity can be calculated by multiplying the specific heat capacity, C_p, by the mass of the structure:

(E.2)

where is the ρ density. The R_thC_th product represents the thermal time constant, and from (E.1) and (E.2) is given by:

(E.3)

Taking the fabricated WFSRD as an example (with L = 150 m), and the appropriate physical constants for silicon (C_p = 0.7 J/gK, ρ = 2.328 g/cm³, κ = 1.5 W/cmK [Sze81]) yields τ_th = 245 μs. Since this is 5 orders of magnitude higher than the electrical switching time, one can conclude that the temperature of the WFSRD does not vary appreciably during the switching transient. Similar results hold for the DSRD.

The effect of a heat flux φ on a one-dimensional slab in thermal equilibrium can be expressed as [Edwa89]:

(E.3)

where T is temperature. Consider again the WFSRD, and assume that the power dissipated by the diode is concentrated in the center of the structure (x = L/2) and that the temperature at the two ends of the diode (x = 0 and x = L) is fixed by good heating-sinking. Then the heat flux at any point (except at L/2) is given by

(E.4)

since half of the average dissipated power is absorbed by each end. Then, from (E.3), the temperature difference between the center (which is the hottest point) and either end (the coolest points) will be given by:

(E.5)

If one assumes that the WFSRD is operate at low duty cycles, such that the bulk of the power dissipation occurs during the forward biasing, then P_avg I_FV_F. If we take I_F 200 mA and V_F 1 V as typical worst-case values for a WFSRD with ideal ohmic contacts, and use the fabricated values of L = 150 μm and A = 1.35 mm² in (E.5), one obtains ΔT>> 0.04 K.

Power dissipation will be higher if the device is not operated at low duty cycles, since the current can be very large during the storage time. If we assume as a worst case that the reverse conduction storage time t_S represents 10% of the period, then P_avg>> 0.1 I_RV_F. For I_R>> 6 A and V_F>> 1 V, then ΔT>> 0.1 K.

The devices reported in Chapter 7 did not have perfect ohmic contacts, with the results the the forward voltage drop was on the order of 6 V, rather than 1 V. However even so, ΔT < 1 K. Thus for a device with effective heat-sinking, such that the temperature at the device ends is near room temperature, thermal effects should be negligible.

Similar results hold for the DSRD.