Two lasing thresholds in semiconductor lasers with a quantum-confined
active region
Levon V. Asryan
a)
State University of New York at Stony Brook, Stony Brook, New York11794-2350
and Ioffe Physico-Technical Institute, St. Petersburg 194021, Russia
Serge Luryi
b)
State University of New York at Stony Brook, Stony Brook, New York11794-2350
Received 22 July 2003; accepted 30 October 2003
We show that the free-carrier-density dependence of internal optical loss gives rise, in general, to the
existence of a second lasing threshold above the conventional threshold. Above the second
threshold, the light-current characteristic is two-valued up to a maximum current at which the lasing
is quenched. © 2003 American Institute of Physics. DOI: 10.1063/1.1636245
Internal optical loss adversely affects operating charac-
teristics of semiconductor lasers. Because of the lower value
of the optical confinement factor, the effect of internal loss is
stronger for lasers with a reduced-dimensionality active re-
gion, such as quantum well QW, quantum wire QWR,
and quantum dot QD lasers, than for bulk lasers.
1
All different processes contributing to the internal loss
can be grouped into two categories: one, such as free-carrier
absorption in the optical confinement layer OCL, or simply
waveguide, dependent on the injection carrier density; the
other, such as interface scattering or absorption in the clad-
ding layers, insensitive to this density. Absorption in the ac-
tive region of QW and QWR lasers is relatively small com-
pared to absorption in the OCL, at least at high injection
currents j or high temperatures T see Refs. 2 and 3. The
analogous process in the active region of QD lasers—which
is carrier photoexcitation from the discrete levels to the
continuous-spectrum states—is also small.
4,5
Neglecting
these processes, we must be concerned only with the free-
carrier density n in the OCL. Therefore, we need a relation
between n and the occupancy of states in the quantum-
confined active region, involved in the lasing transition. At
sufficiently high temperatures and below the lasing thresh-
old, this relation is given by equilibrium statistics and is of
the form
4
n =n
1
f
n
1 - f
n
, 1
where n
1
=N
c
OCI
exp(-E
n
/T), N
c
OCL
=2( m
c
OCL
T /2
2
)
3/2
, E
n
is the carrier excitation energy from a reduced-
dimensionality active region, and the temperature T is mea-
sured in units of energy. The function f
n
is the Fermi–Dirac
distribution, describing the occupancy of the confined states.
For QW or QWR lasers, f
n
is the occupancy of the subband-
edge level, involved into the lasing transitions. For a QD
laser, f
n
is the occupancy of the discrete level.
Assuming equal electron and hole occupancies ( f
n
= f
p
), and writing the total net internal loss coefficient
int
the quantity we shall refer to simply as the internal loss as
the sum of a constant
0
and a component linear in n , the
lasing threshold condition is brought into the form
g
max
2 f
n
-1 = +
0
+
int
n
1
f
n
1 - f
n
, 2
where g
max
is the maximum saturation value of the modal
gain g ( f
n
) =g
max
(2 f
n
-1), is the mirror loss, and
int
=const(n) can be viewed as an effective cross section for all
absorption loss processes for the type of carrier that domi-
nates absorption.
The solutions of Eq. 2 are Fig. 1a
f
n 1,n 2
= f
n
crit
f
n
crit
2
- f
n 0
-
1
2
0
g
max
, 3
where
f
n
crit
=
1
2
1 + f
n 0
+
1
2
0
g
max
-
1
2
int
n
1
g
max
4
is the ‘‘critical’’ solution when the cavity length equals its
minimum tolerable value see Eq. 13, and
f
n 0
=
1
2
1 +
g
max
5
is the solution in the absence of internal loss.
Both solutions 3 are physically meaningful and de-
scribe two distinct lasing thresholds. The lower solution
( f
n 1
) is the conventional threshold, similar to f
n 0
but modi-
fied by
int
The second solution ( f
n 2
) appears purely as a
consequence of the carrier-density-dependent
int
in the
OCL.
In the absence of lasing, the injection current density has
the following relation to f
n
:
4,6
j = j
spon
active
f
n
+ebBn
1
2
f
n
2
1 - f
n
2
, 6
a
Electronic mail: asryan@ece.sunysb.edu; URL: http://www.ioffe.rssi.ru/
DepTM/asryan.html
b
Electronic mail: Serge.Luryi@sunysb.edu;
URL: http://www.ece.sunysb.edu/serge
APPLIED PHYSICS LETTERS VOLUME 83, NUMBER 26 29 DECEMBER 2003
5368 0003-6951/2003/83(26)/5368/3/$20.00 © 2003 American Institute of Physics