648
23 Electricity-to-Light Conversion
At first it seems reasonable to assume that always η w < 1. However, it has been reported that at
small current and at elevated temperature the wall-plug efficiency is found larger than 100% due to the
electrical work pumping heat from the lattice to the photon field in a GaSb-based diode[1941]. The
groundwork for such effect was laid in [1942], essentially predicting that ω > e V is possible.
Internal Quantum Efficiency
The internal quantum efficiency is the number of photons generated (inside the semiconductor) per
injected electron–hole pair. High material quality, low defect density and low trap concentration are
important for a large value of η int . The recombination current in the pn-diode is given in (21.136).
Light Extraction Efficiency
The light extraction efficiency of the LED chip is ratio of the number of photons leaving the device and
the total number of generated photons.
12 The geometry of the LED is of prime importance to optimize
χ ex . Due to the large index of refraction of the semiconductors (n s ∼ 2.5–3.5), light can leave the
semiconductor only under a small angle θ c from the surface normal due to total reflection (cf. (9.11)
and see right part of Fig. 9.4). Against air (n 1 ≈ 1) the critical angle is
θ c = sin
−1
(1/n s ) .
(23.7)
The critical angle for total reflection is 16
◦ for GaAs and 17
◦ for GaP. Additionally, a portion of the
photons that do not suffer total reflection is reflected back from the surface with the reflectance R given
by (cf. (9.15))
R =
n s − 1
n s + 1
2
.
(23.8)
We note that the above formula is valid strictly for vertical incidence. For the GaAs/air interface, the
surface reflectance (for normal incidence) is about 30%. Thus, the light extraction efficiency for a LED
is given by (1–R) and the critical angle by
χ ex ∼ =
4 n 1 n s
(n 1 + n s ) 2 (1 − cos θ c ) ≈
4 n s
(n s + 1) 2 (1 − cos θ c ) .
(23.9)
The latter approximation is valid when the outer medium is air. For GaAs, the light extraction efficiency
is 0.7 × 4% ≈ 2.7%. Thus, for this simple geometry, only a small fraction of generated photons can
leave the device and contribute to the LED emission.
23.3.4 Device Design
In the following subsections the strategies that have allowed significant improvement of the extraction
efficiency (Fig. 23.5) and thus LED performance are briefly discussed. The record by 2014 for light
extraction efficiency is 89% [1943], however not for a mass-produced device.
Nonplanar Surfaces
With curved surfaces, the problem of total reflection can be (partially) circumvented (Fig. 23.6). Spherically polished chips are feasible, but, very expensive. The epoxy seal of the standard LED case
(Fig.23.7a) and its shape play a similar role, however, with a smaller index of refraction than the
semiconductor, and are important for the beam shape.
12 Note that the light extraction efficiency is also important for solar cells, cf. Sect. 22.4.3.
23 Electricity-to-Light Conversion
At first it seems reasonable to assume that always η w < 1. However, it has been reported that at
small current and at elevated temperature the wall-plug efficiency is found larger than 100% due to the
electrical work pumping heat from the lattice to the photon field in a GaSb-based diode[1941]. The
groundwork for such effect was laid in [1942], essentially predicting that ω > e V is possible.
Internal Quantum Efficiency
The internal quantum efficiency is the number of photons generated (inside the semiconductor) per
injected electron–hole pair. High material quality, low defect density and low trap concentration are
important for a large value of η int . The recombination current in the pn-diode is given in (21.136).
Light Extraction Efficiency
The light extraction efficiency of the LED chip is ratio of the number of photons leaving the device and
the total number of generated photons.
12 The geometry of the LED is of prime importance to optimize
χ ex . Due to the large index of refraction of the semiconductors (n s ∼ 2.5–3.5), light can leave the
semiconductor only under a small angle θ c from the surface normal due to total reflection (cf. (9.11)
and see right part of Fig. 9.4). Against air (n 1 ≈ 1) the critical angle is
θ c = sin
−1
(1/n s ) .
(23.7)
The critical angle for total reflection is 16
◦ for GaAs and 17
◦ for GaP. Additionally, a portion of the
photons that do not suffer total reflection is reflected back from the surface with the reflectance R given
by (cf. (9.15))
R =
n s − 1
n s + 1
2
.
(23.8)
We note that the above formula is valid strictly for vertical incidence. For the GaAs/air interface, the
surface reflectance (for normal incidence) is about 30%. Thus, the light extraction efficiency for a LED
is given by (1–R) and the critical angle by
χ ex ∼ =
4 n 1 n s
(n 1 + n s ) 2 (1 − cos θ c ) ≈
4 n s
(n s + 1) 2 (1 − cos θ c ) .
(23.9)
The latter approximation is valid when the outer medium is air. For GaAs, the light extraction efficiency
is 0.7 × 4% ≈ 2.7%. Thus, for this simple geometry, only a small fraction of generated photons can
leave the device and contribute to the LED emission.
23.3.4 Device Design
In the following subsections the strategies that have allowed significant improvement of the extraction
efficiency (Fig. 23.5) and thus LED performance are briefly discussed. The record by 2014 for light
extraction efficiency is 89% [1943], however not for a mass-produced device.
Nonplanar Surfaces
With curved surfaces, the problem of total reflection can be (partially) circumvented (Fig. 23.6). Spherically polished chips are feasible, but, very expensive. The epoxy seal of the standard LED case
(Fig.23.7a) and its shape play a similar role, however, with a smaller index of refraction than the
semiconductor, and are important for the beam shape.
12 Note that the light extraction efficiency is also important for solar cells, cf. Sect. 22.4.3.