A generalised thermal LED-model and its applications

Electric characteristics

The I–U-characteristic of an idealized diode was found by William Shockley of Bell Telephone Laboratories in 1949, [5, 10]. It is an exponential dependency of forward current I _F on forward voltage U _F,

Herein, n is the emission coefficient and U _T the thermal voltage. The details are depending on the certain compound of the semiconductor materials forming the junction and not of importance for the further discussion because the characteristic is usually provided by the LED manufacturers, see upper left of Figure 4 for a red low power LED. These characteristics are usually obtained by measurements done by LED manufacturers.

The thermal voltage U _T = k _B T _j/q relates the kinetic energy of the charges to the potential energy of the total charge q of the P–N-junction. This temperature dependence of the forward voltage is also provided by the manufacturers; see upper right of Figure 4 for the same red LED. The voltage difference, ΔU _F = U _F − U _F, typ, is based on a typical voltage, U _F, typ, which is explained Section 3.

Luminous characteristics

Prerequesite for efficient light emission is a high probability of radiating recombinations within the P–N-junction, [11]. This is usually the case in semiconductor materials with a direct band gap, e.g. GaAs. The wave length λ of the emitted radiation depends on the energy difference, E _g, between valence band and conduction band

It forms a distribution, as can be seen as red curve in Figure 1 for the red LED. The half width and the peak wave length of the spectrum depend, among others, on the temperature. The band gap and thus the energy of emitted photons sink with rising temperature. According to [4] this behaviour was found to be approximately linear for junction temperatures above 290 K according to Varshni formula

This leads to larger wave lengths and thus to a kind of redshift of the colour of the direct radiation as seen in Figure 5.

Extrapolations

Regression can only be done within the interval of data. If temperature or current outside this interval are used or reached there still shall exist a physically appropriate solution. Thus an extrapolation will be necessary. Let Ω be the domain of definition for the functions of interest according to equations (7) to (10).

Around the rectangular area Ω eight regions appear as can be seen in Figure 9. First we take the vertical and horizontal parts Ω _T −, Ω _T +, Ω _I − and Ω _I +.

Extrapolation of voltage

We take equation (7) and calculate the derivative with respect to T _j and I _F

We use the following physically motivated functions.

1. In Ω _T− we perform a linear ansatz m ⋅ x + a and for every (T, I) ∈ Ω _T− we find

1. In Ω _T+ we use an exponential ansatz a ⋅ e ^b/x and find for every (T _j, I _F) ∈ Ω _T+

with

1. In Ω _I− we do exponential extrapolation a ⋅ e ^b/x until zero I _F = 0 and find for all (T _j, I _F) ∈ Ω _I−

with

and

1. In Ω _I+ we do exponential extrapolation a ⋅ e ^b/x and find for all (T _j, I _F) ∈ Ω _I+

with

and

The remaining areas Ω_{−−/−+/+−/++} comprise two bounding areas which can be used for extrapolation directly.

As an example, we derive a value U at (T, I) ∈ Ω₊₊. From the extrapolated areas above we can observe that U _maxmax = U(T _j,max, I _F,max), U _I max = U(T, I _F,max), and U _T max = U(T _j,max, I) from Ω _I + resp. Ω _T + and the desired value via

Extrapolation light flux

We take equation (8) and calculate the derivative with respect to T _j and I _F

We use the following extrapolations motivated by physics.

1. In Ω _T− we do linear extrapolation m ⋅ x + b and find for every (T _j, I _F) ∈ Ω _T−

1. In Ω _T+ we apply an exponential extrapolation a ⋅ e ^b/x and get for every (T _j, I _F)∈ Ω _T+

with

and

1. In Ω _I− we perform an exponential extrapolation a ⋅ e ^b/x up to I _F = 0, and get for every (T _j, I _F)∈ Ω _I−

with

and

1. In Ω _I+ we perform an extrapolation with the approach A ( x ) + b and we find for every (T, I)∈Ω _I+

The other domains Ω_{−−/−+/+−/++} are treated the same as in the previous part on the voltage U.

The extrapolation of the colours like shown in Figure 8 is treated only with linear extrapolation. Otherwise, they are derived in the same way as the values above.