Multidimensional quantum well laser and temperature dependence of its threshold
current
Y. Arakawa and H. Sakaki
Citation: Applied Physics Letters 40, 939 (1982); doi: 10.1063/1.92959
View online: http://dx.doi.org/10.1063/1.92959
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Multidimensional quantum well laser and temperature dependence of its
threshold current
Y. Arakawa and H. Sakaki
Institute of Industrial Science, University of Tokyo, Minato-ku, Tokyo 106, Japan
(Received 19 January 1982; accepted for publication 23 March 1982)
A new type of semiconductor laser is studied, in which injected carriers in the active region are
quantum mechanically confined in two or three dimensions (2D or 3D). Effects of such
confinements on the lasing characteristics are analyzed. Most important, the threshold current of
such laser is predicted to be far less temperature sensitive than that of conventional lasers,
reflecting the reduced dimensionality of electronic state. In the case of 3D-QW laser, the
temperature dependence is virtually eliminated. An experiment on 2D quantum well lasers is
performed by placing a conventional laser in a strong magnetic field (30 T) and has demonstrated
the predicted increase of To value from 144 to 313 0c.
PACS numbers: 42.55.Px, 73.60.Fw, 78.45.
+ h, 78.20.Ls
The two-dimensional (2D) nature of electron motion in
the quantum well (QW) structure introduces several unique
features to semiconductor lasers. For instance, the threshold
current J lh ofQW lasers is found less temperature sensitive
than that of conventional double heterostructure (DH) lasers. 1.2 Such improved behavior of J lh is ascribed to the
change in the state density Pc (E) of electrons, which is
brought forth by the decreased dimensionality of the freeelectron motion from 3D to 2D. Consequently, further improvements are expected if one modifies the form ofPe(E). In
this letter, we propose and analyze a new type laser "the
multidimensional (2D or 3D) quantum well (MD-QW) laser" as an extension of the conventional QW laser, which we
call1D-QW laser, hereafter. The most remarkable feature to
be shown is thatJ lh ofMD-QW lasers is much less temperature sensitive than that of the 1D-QW laser. We show,
further, that a conventional DH laser placed in a strong magnetic field behaves as a 2D-QW laser and the observed temperature sensitivity indeed decreases in accordance with our
theoretical prediction.
Figure l(a) shows an illustration of the active layer in
conventional DH lasers, in which the z axis is taken normal
to the active layer. 1D-QW lasers are realized by reducing
the thickness Lz of the active layer to the order of the de
Broglie wavelengthAe of carriers, as shown in Fig. l(b). MDQW lasers are defined as lasers, in which not only the thickness Lz but also the length Ly, and/or the width Lx are
reduced down to the order of Ae , as shown in Figs. I(c) and
I (d). Although the fabrication of such structures at present is
still technically difficult even with the most advanced device
technology, 2D-QW or 3D-QW structures can be effectively
achieved if we place conventional DH lasers or ID-QW
structures in a strong magnetic field, in which the electron
motion is confined in two dimensions, as will be discussed
later. To achieve the efficient population inversion and also
the efficient optical confinement, a number of mutually isolated quantum wells should be stacked in practice, so that
the group of QW occupies the volume identical with the
active layer of the conventional DH laser.
As the dimension ofQW increases from ID to 2D or
3D, the degree offreedom in the free-electron motion de939
Appl. Phys. Lett. 40(11), 1 June 1982
creases, leading to a change inpe(E). For the (3-i)-dimensional electron gas in the i-dimensional QW,p~)(E) is expressed as
follows:
101
_
Pc (E) plll(E)
(2me/122f/2
(2rr)
=L
e
pI21(E) =
e
p~I(E) =
n
~
11
(1)
E,
me
H [E - Ez(n)],
(1Tfz2Lz)
(mJ2122)112 /(1TLyLz)
(3)
,
'f.t [E-Ey(l)-Ez(n)]l12
L 1 o[E - Ex(k) n./,k
(2)
(LzLyLx)
Ey(l) - Ez(n)),
(4)
where me is the electron effective mass, E is the energy measured from the conduction-band edge Ec, 12 is Planck's constant, H (E) is a unit step function with H (E>O) = I and
H (E < 0) = 0, and 0 (E) is the delta function. Ez(n), Ey(1 ), and
Ex(k ) denote the quantized energy levels with the quantum
numbers n, I, and k, respectively, over which summation
should be carried out. In case the potential barrier is sufficiently high, the quantum levels are given by
€Z(n) = (122rr12me)(n/Lzf, Ey(l) = (122rr/2m)(I/Ly)2, and
Ex(k) = (Ij2rr/2m e)(k / LX)2. Notethatp~11(E),p~21(E), andp~I(E)
are very different from the parabolic state density p~OI(E). Similar behaviors are also expected for the state density p~I(E)
I
lei
Idl
FIG. 1. Illustration of various active layers for the conventional laser (a),
and the multidimensional QW lasers. (b), (c), and (d) correspond to lD-, 2D-,
and 3D-QW structures.
0003-6951/82/110939-03$01.00
© 1982 American Institute of Physics
939
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of holes.
The analysis of J th and its temperature dependence of
the conventional laser has been done by Lasher and Stem 3
(referred to as LS, hereafter) and by Adams. 4 The LS theory
is recently extended to the analysis ofa ID-QW laser,5 and
the steplike nature ofp~II(E) is shown to be mainly responsible
for the reduced temperature dependence of J th' In the following, we extend the LS theory further to evaluate the behavior of 2D-QW lasers as well as 3D-QW lasers. If we use
"no k-selection rule", 3 the gain coefficient g(E) of the i-dimensional QW laser for the photon energy E can be formally
expressed as
I~ 1.5
u
(5
Idl _ _ _---="""~=---
£
Ie)
~ 1.0
u
Ibl
Cl>
la,
N
~
lal
Ibl
L..
gUI(E) = (
C
= qdRspl17,
(6)
where q is the electronic charge, d is the active layer thickness, and 17 is the quantum efficiency. The threshold current
J th thus formulated can be calculated only numerically in
general, but analytical expressions can be obtained for IDQW and 3D-QW lasers as given by
q:
;~z poE (IlkTln(1 + Q),
_~~
2
poE III kTln
=
17 7Tfl Lz
(mu kT )
2'
(7)
Po7Tfl Lz
(at high temperature)
J
~( a V
l31
I31 _
th -
17
A
+ ~
(3)
- Po) poE 131,
(8)
= ("rc 2 fz3 In;E~)B iii,
Q = I [(yIC)D + (D - 1)112(1 + C)]/(1 + C - CD W,
D = exp[ a(II(~LzfIA olkTmcm v ]'
C = 11[ exp(po~LzlmvkT) - I], and mv is the hole effective mass. Equations (7) and (8) indicate that J th of a IDQW laser is proportional to Tln(T Iconst) near room temperature, whereas J th of a 3D-QW laser is independent of T.
We have also calculated numerically J th for GaAslAIGaAs
where V = LxLyLz, A
iii
2D-QW lasers as well as for conventional DH lasers. We
940
To ~ 104
To :::285
To "'481
To
·c
.(
·c
:::= ·c
(5)
in which nr is the refractive index, c is the light velocity, Eg is
the energy gap, and B ('1 is a constant representing the probability of dipole transitions. Consequently, gli)(E) can be calculated, once the distribution functionsf, (E) andfv (E) of electrons and holes are fixed. To determinefv(E) we follow LS
and assume the active layer to be heavily doped with acceptors such that the hole concentration Po is constant. To determinef,(E) we adjust the quasi-Fermi level in such a way that
the maximum gain gli)(Emax) satisfies the threshold condition, i.e., glil(Emax ) is equal to the total optical loss in the laser
cavity which we assume to be independent of the
temperature.
The rate R ~b of the total spontaneous emission is then
calculated by the energy integral of the spontaneous emission rate r.~(E) which can be uniquely determined whenpe'
Pu,f" andfu are given [see Eq. (6a) of Ref. 3]. By usingR ~b at
just below the threshold, J th can then be expressed as
J~~ =
Idl
E)
X [fe(E') -fv(E' -E)]dE'
J~~
lei
o
~t;~ )B 'i'iE -Egp~I(E')p~I(E' -
___
Appl. Phys. Lett., Vol. 40, No. 11, 1 June 1982
-40
-20
0
20
40
60
temperature ( • C )
FIG. 2. Numerical example of threshold current J lh calculated by extending the model of Lasher and Stem for conventional lasers (a) and quantum
well lasers for (b) 10-, (c) 20-, and (d) 3D-QW structures. J Ih is normalized
by J lh at O·c.
assume here that electrons populate only in the ground subband, which is valid when Lx, Ly, and Lz are chosen sufficiently small. The results for Tnear room temperature are
summarized in Fig. 2, and clearly show that the temperature
dependence of J th depends drastically on the dimension of
quantum well. If we express the result in terms of
J th = J o exp(T ITo), To for ID, 2D, and 3D are equal to
285 ·C, 481 ·C, and infinity ( 00 ), respectively, and exceeds by
for J th (104·C) of conventional DH lasers.
To understand the reason for such dramatic increase of
To, we like to remind here that the temperature dependence
of J th of the conventional GaAs-GaAIAs laser is ascribed to
the thermal spreading of the injected carriers over a wider
energy range of states, which leads to decreases of the maximum gain g(Emax) at a given injection level. Consequently, in
ID-QW lasers, wherep~l)(E) andp~l)(E) are steplike, the effect of such thermal spreading is expected to be smaller. 5 In
case of2D-QW lasers, one expects further suppression of the
temperature effect because p~21(E ) has a peaked structure and
is a decreasing function of E. In 3D-QW lasers, the thermal
spreading of carriers should vanish because the state density
is delta-functionlike. Hence, the temperature dependence of
J th will totally disappear, as long as the electron population
in higher subbands remains negligibly small.
We consider next the possibility of demonstrating this
unique feature of the MD-QW laser experimentally. In view
of the technical difficulties offabricating such structures, we
have investigated the conventional GaAs-GaAIAs DH laser
placed in a strong magnetic field. As we have used a channelled substrate planar (CSP) laser with the nondoped active
layer having relatively high carrier mobility, and applied the
high (pulsed) magnetic field Bz up to 30 T perpendicularly
(liz) to the active layer, fLB ( = WeT) is much greater than 1.
Hence carriers in the active layer are expected to complete
their cyclotron orbit. 6 The motion of such electrons is known
Y. Arakawa and H. Sakaki
940
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1·5 , . . . - - - - - - - - - - - - - - - - - - - ,
•
~O
.J:
1-;'
~:;
1·0
u
:g
o
0
.J:
If)
~
•
£;
al
.!::!
'iii
E
c
o
•
:sc
0·5
-40
-20
o
lal
0
B =24 T
( T. = 313·C )
[bl
•
B = 0 T
( T. = 144·C )
20
40
temperature ( • C )
FIG. 3. Temperature dependence of threshold currentJ th with and without
magnetic field B (24 T). J th is normalized by J th atO ·C, which is 52 rnA at
B = 0 and 54 rnA at B = 24 T.
to be quantized in the two transversal directions (x andy) and
forms a series of discrete Landau levels
[E = wc(n + 1/2) + (fi-r/2mc)kz; OJc = IilqlB fmc]'
Hence, the conventional laser placed in a strong magnetic
field can be regarded approximately as a 2D-QW laser,
which is indeed demonstrated quite recently by the independent work of Bluyssen.7 We, therefore, have measured the
temperature dependence of J th with and without the magnetic field B and the results are shown in Fig. 3. While To is
found to be 144·C for B = 0, To for B = 24 T is shown to
increase especially at lower temperature. When averaged (in
the sense of least-mean-square fitting) over the temperature
range (230-300 K) To for B = 24 T is determined to be
941
Appl. Phys. Lett., Vol. 40, No. 11, 1 June 1982
313 ·C, which is in fair agreement with our prediction. A
nonlinear behavior ofln J th with respect to T is possibly due
to the contribution from higher Landau levels; we leave the
detailed description of the experiment and its interpretation
as the subject of a separate paper and simply note here that
the mean time between collision is much larger than 1/OJc
(since OJ c 1'> 1), and that w c term is greater than k T when B
exceeds 20 T. Hence, the two-dimensional confinement of
carriers by the field Bz is effective, and introduces the peaked
structure in the state density although there are some contributions from higher subbands .
In summary, we have proposed a new type multidimensional QW laser and have shown theoretically that the dramatic increase of T is expected. An experimental proof of
such prediction has been successfully done for a 2D-QW
laser by placing a conventional laser in a strong magnetic
field.
We wish to express our sincere gratitudes to Professor
N. Miura and Dr. G. Kido for allowing the use of pulse
magnets, Dr. M. Nakamura and Dr. K. Aiki of Hitachi Ltd.
for supplying the laser diodes, Professor J. Hamasaki and
Professor Y. Fujii for their support and encouragement, and
Mr. M. Nishioka for his excellent technical assistance. The
work is supported by the Ministry of Education, Science,
and Culture.
I R. Chin, N. Holonyak, J r., and B. A. Vojak, Appl. Phys. Lett. 36, 19 (1980).
2N. Holonyak, Jr., R. M. Kolbas, R. D. Dupuis, and P. D. Dapkus, IEEE J.
Quantum Electron. QE-16, 170 (1980).
3G. Lasher and F. Stern, Phys. Rev. 133, A553 (1964).
4M. J. Adams, Solid-State Electron. 23, 585 (1980).
5K. Hess, B. A. Vojak, N. Holonyak, Jr., R. Chin, and P. D. Dapkus, SolidState Electron. 23, 585 (1980).
6R. B. Dingle, Proc. R. Soc. London A 211, 517 (1952).
7H. J. A. Bluyssen and L. J. van Ruien, IEEEJ. Quantum Electron. QE-17,
880(1981).
'Y. Arakawa, H. Sakaki, M. Nishioka, G. Kido, and N. Miura
(unpublished).
Y. Arakawa and H. Sakaki
941
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