Optoelectronic Semiconductor Devices Principals and Characteristics

4.2.1 
POWER AND LIGHT EXTRACTION EFFICIENCY: SURFACE-EMITTING LED

We have to introduce two terms in order to be able to quantify optical power. (Figure 

29.
). 
H - irradiance - the optical power ∆P per unit area received by the receiver 

---

r
P
H
S
∆
=
∆
(32) 
Figure 29.: Geometry for defining irradiance, radiant intensity, and radiance. 
[1]
J - radiant intensity
2
2
1
r
r
P
P
J
r
S n
W
S n
r
∆
∆
∆
=
=
=
∆
∆
P
∆
(33) 
Where 
2
r
W
S n r
∆ = ∆
is the solid angle of the power impinging on the receiver surface element. 
We can interpret J as the power ∆P contained in a solid angle ∆W. 
The light output power is nearly proportional to the injected current in the low-current range. The light 
power emitted in the active layer is given by the product of the number of photons emitted and the photon 
energy, hν. The product of the injected carrier density, J/qd, and the internal quantum efficiency gives us 
the number of photons created by the spontaneous recombination processing a unit volume of the active 
layer. Now, we can see that the light power emitted from a unit volume of the active layer is given by
i
act
J
P
h
qd
η
ν
=
(34) 
where η
i
- the internal quantum efficiency corresponding to the ratio of emitted photons to injected 
electrons:
(35) 
(
0
0
i
sp n
B
p
n
n
η
τ
=
+
+
)
∆
If we substitute formula (35) into (34) and using equations (29) and (30),
0
0
act
sp n
n
J
J
P
h
B
p
n
qd
qd
ν
τ
τ


=
+
+




(36) 
where τ
n
- the injected carrier lifetime. 

---

Under low-injection conditions, p
0
and 
0
n
J
n
qd
τ

, and the formula (36) can be rewritten for the p-type 
active layers (p
0
> n
0
) as
0
act p
sp
e
J
P
h B p
qd
ν
τ
−
=
(37) 
and for the n-type layers as
0
act n
sp
h
J
P
h B n
qd
ν
τ
−
=
(38) 
Under high-injection conditions, for 
0
n
J
p
qd

τ
and n
0
, we get:
2
act
sp
n
J
P
h B
qd
ν
τ


=




(39) 
In the last three formulas the variables only τ
n
and J. The light output power of the active layer then is 
proportional to the injected current density in the low-excitation range and to the square of the injected 
current density in the high-excitation range.