SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

198 CHAPTER 4. JUNCTIONS IN SEMICONDUCTORS:P-NDIODES

EC

EFp

EV

EFn hv

Jparasitic

(recombination

in junction)

Jn parasitic injection

Jp parasitic injection

J = Jparasitic+ eRspon tQW

Figure 4.33: Current flow mechanisms in a LED.

ratio of the current generating photons of the desired wavelength to the total current. The current
calculated for the p-n junction in the earlier sections are the wasted currents in the LED as they
calculate currents in the bulk and due to non-radiative centers. What remains to be calculated is
the spontaneous recombination rateRspon.a
As discussed in section 3.8.1, the radiative process is “vertical,” i.e., thek-value of the elec-
tron and that of the hole are the same in the conduction and valence bands, respectively. From
figure 4.34 we see that the photon energy and the electron and hole energies are related by

ω−Eg=

^2 k^2

2

[

1

m∗e

+

1

m∗h

]

=

^2 k^2

2 m∗r

(4.9.4)

wherem∗ris the reduced mass for thee-hsystem.
Ifanelectronisavailableinastatekandaholeisalsoavailableinthestatek(i.e.,iftheFermi
functionsfortheelectronsandholessatisfyfe(k)=fh(k)=1), the radiative recombination rate
is found to be
Wem∼ 1. 5 × 109 ω[eV s−^1 ] (4.9.5)