COL 11(4), 041409(2013) CHINESE OPTICS LETTERS April 10, 2013
Population inversion study of GaAs/AlGaAs
three-quantum-well quantum cascade structures
Jinlong Liu (
444
777
999
)
1∗
, Xiaoli Zhang (
ÙÙÙ
www
)
1
, Jian Wang (
èèè
)
1
,
Yanwu L¨u (
½½½
ÿÿÿ
ÎÎÎ
)
1
, and Suqing Duan (
ããã
)
2
1
Department of Physics, School of Science, Beijing Jiaotong University, Beijing 100044, China
2
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
∗
Corresponding author: Jef f liu24@163.com
Received November 2, 2012; accepted December 7, 2012; posted online March 28, 2013
A population inversion study of GaAs/Al
x
Ga
1−x
As three-quantum-well quantum cascade structures is
presented. We derive the population inversion condition ( P IC) of t he active region (AR) and discuss the
PICs on different structures by changing structural parameters such as the widths of quantum wells or
barriers in the AR. For some instances, the PIC can be simplified and is proportional to the spontaneous
emission lifetime between the second and the first excited states, whereas some other instances imply that
the PIC is proportional to the state lifetime of the second excited state.
OCIS codes: 140.5965, 140.3070.
doi: 10.3788/COL201311.041409.
The quantum casc ade las e r (QCL) has beco me an im-
portant so urce of coherent radiation in the mid-infrared
region, and its frequency rang e has bee n extended to the
terahertz reg ion
[1]
. The QCL currently exhibits e xcel-
lent performance
[2−7]
. Theoretical research is an effective
tool to facilitate laser fabrication
[8]
. In this letter, we
study the mechanisms by which the widths of qua ntum
wells (QWs) and barr ie rs in the active region (AR) can
affect the population inversion of QCL.
The time-independent Schrodinger equation of an elec-
tron with mass m in one-dimensional potential V (x) is
b
H(x)φ(x) = −
~
2
2
d
dx
1
m
dφ
dx
+ V (x)φ(x) = Eφ(x), (1)
where V (x) includes –eF x, the potential caused by the
applied electric field F , and e is the electr on charge. To
solve Eq. (1), the wave function and potential are dis-
cretized. N+1 points are selected to equally divide the
AR, and the position c oordinate of the ele c tron can be
expressed as x
j
= j×h
0
, where h
0
is the step length, j
= 0, 1, 2,
...
, N , and x
j+1
= x
j
+ h
0
. The total length
of the AR of the QCL is L
a
= Nh
0
. At each selected
point, the wave function and potential ar e φ
j
= φ(x
j
)
and V
j
= V (x
j
), respectively. The first-order deriva-
tive of the discrete wave function at two adjacent points
can be expressed in the finite difference approximation
as
1
m
dφ(x
j
)
dx
=
φ(x
j
)−φ(x
j−1
)
m
j
h
0
. To obtain the second-order
derivative of the discrete wave function, three adjacent
points must be considered, resulting in the following ex-
pression:
d
dx
1
m
dφ(x
j
)
dx
=
1
m
j+1
h
2
0
[φ(x
j+1
) − φ(x
j
)]
−
1
m
j
h
2
0
[φ(x
j
) − φ(x
j−1
)]. (2)
After taking E q. (2) into Eq. (1), we obtain the matrix
equations of the AR:
(H − E1)φ(x) =
N
X
j=0
h
− u
j
φ(x
j−1
)
+(d
j
− E)φ(x
j
) − u
j+1
φ(x
j+1
)
i
= 0, (3)
in which the Hamiltonian is a symmetric dia gonal ma-
trix, where in 1 is the unit matrix, the diagonal ma trix
element is d
j
=
~
2
2h
2
0
(
1
m
j+1
+
1
m
j
) + V
j
, and the adjacent
non-diagonal matr ix element is u
j
=
~
2
2m
j
h
2
0
. Setting x
0
and x
N
as the boundar y coordinates of AR, the boundary
conditions o f the region are
φ(x
0
) = φ(x
N
), (4)
1
m
0
dφ
dx
x=x
0
=
1
m
N
dφ
dx
x=x
N
. (5)
We also obtain the matrix equation of the AR:
(H − E1)φ =
(d
1
− E) −u
2
0 . . −u
1
−u
2
(d
2
− E) −u
3
. . .
0 −u
3
(d
3
− E) . . .
. . . . . .
. . . −u
N−1
(d
N−1
− E) −u
N
−u
1
. . . −u
N
(d
N
− E)
φ(x
1
)
φ(x
2
)
φ(x
3
)
.
φ(x
N−1
)
φ(x
N
)
= 0. (6)
1671-7694/2013/041409(5) 041409-1
c
2013 Chinese Optics Letters