23.4 Lasers
675
23.4.9 Temperature Dependence
The threshold of a laser typically increases with increasing temperature as shown in Fig. 23.50a. Empirically, in the vicinity of a temperature T 1 the threshold follows an exponential law (see Fig. 23.50b)
j thr (T ) = j thr (T 1 ) exp
T − T 1
T 0
∝ exp
T
T 0
,
(23.38)
with T 0 being the so-called characteristic temperature.
20 It is the inverse logarithmic slope, T
−1
0
=
d ln j thr /dT .
T 0 summarizes the temperature-dependent loss and the carrier redistribution in k-space due to the
change of the Fermi distribution with temperature. With increasing temperature, populated states below
the quasi-Fermi level become unpopulated and nonlasing states become populated. Therefore, the gain
decreases with increasing temperature. This redistribution must be compensated by an increase of the
quasi-Fermi energy, i.e. stronger pumping. This effect is present for (even ideal) bulk, quantum well
(a)
2
1
0
0
100
200
25°C
115°C
95°C
GaAs/
AlGaAs
(b)
(c)
InGaAs/
GaAs
5.5
5.0
4.5
4.0
3.5
3.0
50 100 150 200 250 300
-2
Fig. 23.50 a P–I characteristic (cw output power per mirror facet) of stripe-buried heterostructure laser at various
temperatures of the heat sink between 25 ◦ C and 115 ◦ C in steps of 10 K. b Threshold current (in logarithmic scale)
of this laser as a function of heat-sink temperature and exponential fit (dashed line) with T 0 = 110 K. Parts a and b
adapted from [1998]. c Temperature dependence of the threshold current density of a quantum dot laser (3 stacks of
InGaAs/GaAs QDs, λ = 1150 nm) with T 0 (solid lines are fits) given in the figure
20 Since T 0 has the dimension of a temperature difference, it can be expressed in ◦ C or K. For the sake of unambiguity it
should be given in K.
675
23.4.9 Temperature Dependence
The threshold of a laser typically increases with increasing temperature as shown in Fig. 23.50a. Empirically, in the vicinity of a temperature T 1 the threshold follows an exponential law (see Fig. 23.50b)
j thr (T ) = j thr (T 1 ) exp
T − T 1
T 0
∝ exp
T
T 0
,
(23.38)
with T 0 being the so-called characteristic temperature.
20 It is the inverse logarithmic slope, T
−1
0
=
d ln j thr /dT .
T 0 summarizes the temperature-dependent loss and the carrier redistribution in k-space due to the
change of the Fermi distribution with temperature. With increasing temperature, populated states below
the quasi-Fermi level become unpopulated and nonlasing states become populated. Therefore, the gain
decreases with increasing temperature. This redistribution must be compensated by an increase of the
quasi-Fermi energy, i.e. stronger pumping. This effect is present for (even ideal) bulk, quantum well
(a)
2
1
0
0
100
200
25°C
115°C
95°C
GaAs/
AlGaAs
(b)
(c)
InGaAs/
GaAs
5.5
5.0
4.5
4.0
3.5
3.0
50 100 150 200 250 300
-2
Fig. 23.50 a P–I characteristic (cw output power per mirror facet) of stripe-buried heterostructure laser at various
temperatures of the heat sink between 25 ◦ C and 115 ◦ C in steps of 10 K. b Threshold current (in logarithmic scale)
of this laser as a function of heat-sink temperature and exponential fit (dashed line) with T 0 = 110 K. Parts a and b
adapted from [1998]. c Temperature dependence of the threshold current density of a quantum dot laser (3 stacks of
InGaAs/GaAs QDs, λ = 1150 nm) with T 0 (solid lines are fits) given in the figure
20 Since T 0 has the dimension of a temperature difference, it can be expressed in ◦ C or K. For the sake of unambiguity it
should be given in K.