vol. gain and intervalence band absorption quantum-well...

IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-20, NO. 7, JULY 1984 745

Gain and Intervalence Band Absorption in Quantum-Well Lasers

MASAHIRO ASADA, ATSUSHI KAMEYAMA, AND YASUHARU SUEMATSU, FELLOW, IEEE

Abstract-The linear gain and the intervalence band absorption are analyzed for quantum-well lasers.

First, we analyze the electronic dipole moment in quantum-well structures. The dipole moment for the TE mode in quantum-well structures is found to be about 1.5 times larger at the subbartd edges than that of conventional double heterostructures. Also obtained is the difference of the dipole moment between TE and TM modes, which results in the gain difference between these modes. Then we derive the linear gain taking into account the intraband relaxation, As an example, we applied this analysis to GaInAs/InP quantum-well lasers. It is shown that the effects of the intraband relaxation are 1) shift of the gain peak toward shorter wavelength with increasing injected carrier density even in quantum-well structures, 2) increase of the gain-spectrum width due to the softening of the profile, and 3) reduction in the maximum gain by 30-40 percent;

The intervalence band absorption analyzed for quantum-well lasers is nearly in the same order as that for conventional structures. However, its effect on the threshold is smaller because the gain is larger for quantum wells than conventional ones. The characteristic temperature TO of the threshold current of GaInAs/InP multiquantum-well lasers is calculated to be about 90 K at 300 K for well width and well number of 100 A and 10, respectively.

Q I. INTRODUCTION

UANTUM-WELL lasers with ultrathin active layers [ I ] , [2] have been found to have superior characteristics, such as ultralow threshold current [3] - [ 6 ] , less tempera-

ture dependence [7], narrow gain spectrum [8], etc., compared to conventional double heterostructure lasers for GaAs crystals. Although few have reported on long-wavelength GaInAsP/InP and GaInAs/AlInAs/InP quantum-well lasers [9] - [I I ] , it is expected that these lasers also have some of these superior characteristics.

In these conditions, it is important to discuss characteristics of quantum-well lasers theoretically. Until now, the theory of laser gain in quantum-well structures has been reported with simplified models [ 121 - [ 141 . On the other hand, a precise theory for conventional double heterostructure lasers [IS] , E161 , which explains many observed properties well, has been developed taking into account the electronic intraband relaxation. Such a theory is required for quantum-well lasers as well. The intraband relaxation is particularly important for quantum- well lasers with narrow gain spectrum since the gain spectrum is broadened due to this intraband relaxation.

Theoretical discussions are also required for temperature

Manuscript received November 8, 1983; revised March 6, 1984. This work was supported by a scientific grant-in-aid from the Ministry of Education, Science, and Culture, Japan.

The authors are with the Department of Physical Electronics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152, Japan.

Y (perpendicular to weti)

Fig. 1. Coordinate system used in the analysis where the y-direction is perpendicular to the quantum well, the 2-direction is parallel to the laser cavity, and w is the well width.

dependence of the threshold current of long-wavelength GaInAsP/InP quantum-well lasers (including GaInAs/InP as GaxInl-,AsyPl-y wi thy = 1, where the bandgap wavelength becomes maximum). For GaInAsP/InP conventional double heterostructure lasers, this problem has been discussed mainly with 1) the Auger effect [I71 -[20], 2) the intervalence band absorption [21], [22] (in particular, for 1.5-1.6 pm wavelength), and 3) carrier leakage over the heterobarrier [23] , [24]. For quantum-well lasers, the Auger effect has been reported theoretically [13], [25], [26], and it is interesting to discuss the intervalence band absorption, i.e., the optical transition of electrons from split-off to heavy-hole valence bands, in quantum-well structures.

In this paper, we analyze the linear gain of quantum-well lasers, taking into account the effects of the intraband relaxation. First, we obtain the dipole moment in quantum-well structures, and then derive the linear gain: As a numerical example, we calculate the linear gain of GaInAs/InP quantum- well lasers, and show the effects of the intraband relaxation.

We also analyze the intervalence band absorption in quantum- well lasers theoretically, and compare the results to those for conventional lasers. Using these results, we discuss the threshold current and its temperature dependence of GaInAs/InP quantum-well lasers.

11. LINEAR GAIN A. Dipole Moment

The probability of optical transition of electrons is propor- tional to the absolute square of the matrix element of the momentum or dipole moment. Here we analyze the dipole moment in quantum-well structures.

The coordinate system used in this paper is shown in Fig. 1. Supposing that the well width is much larger than the lattice period, the wavefunction of an electron confined in the well can be written as [27]

%kc,, = BCn (Y>u, (r> exp ( j h . r11) (1)

0018-9197/84/0700-0745$01.00 0 1984 IEEE

746 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-20, NO. 7 , JULY 1984

where the subscripts c (or h j and n = 0 , 1, 2 , . . denote the conduction band (or heavy-hole valence band) and the number of the quantized level (or the subband, in other words) in the well, respectively, u, (r j is the periodic part of the Bloch function of bulk crystals, kil and ril are the wave vector and the position vector both parallel to the interface of the heterojunction, i.e., within the x-z plane, and @,,(yj is the envelope function varying very slowly compared to u, (r), which is given in the Appendix. u, ( r ) and a,, ( y j are assumed to be normal- ized so that the absolute squares integrated within a unit cell and over all regions of y are equal to unity, respectively.

Using (I) , the matrix element of the dipole moment formed by an electron in subband n and a hole in subband m, R,, can be written as

where c

R = J uzeruhdv unit cell

The factor in the square brackets in ( 2 ) is approximated to be 6,, since the potential wells in the conduction and valence bands have the same symmetry with each other. Although a difference in effective mass and well depth (barrier height) between conduction and valence bands causes a slight devia- tion from 6,,, this is a good approximation for GaInAsP/InP systems with an error of about 0.01 both for m = n and for m f n .

We obtain Rch = 8k,khR for conventional lasers, since the functions a,, and ah,,, are the plane waves in this case.

The vector R , in ( 2 ) , of quantum-well lasers has the same style as that of conventional lasers. However, it differs from conventional ones in the average with respect to direction. We discuss this point below.

First, we discuss the vector R given by ( 3 ) for conventional lasers. Using the k . p method [ 2 8 ] , it is found that the vector R formed by a conduction electron and a heavy hole is rotating within the plane perpendicular to the wave vector k =kc = kh, as shown in Fig. 2. Representing the direction of k by the angles 8 and @ shown in Fig. 2 , the components of R are written as [ 2 8 ]

R (cos 0 sin @ + j cos @j for the x-direction

-R sin 8 for the y-direction (4)

R(cos 0 cos 6 - j sin 4) for the z-direction

where the length of the vector R is written as

R = (S[exIP,)/&

= (eh/'&J?ch) (Sla/axlpx)

(eh/2Ech)IEg(Eg + AO)/(Eg + 2A0/3)/nz,*I (5 1 where S and P, are the wave functions with the symmetries of atomic S-like and P,-like functions, respectively, e is the electron charge, Eg and A, are the bandgap and the spin-orbit splitting, respectively, Ech is the transition energy between the

I

Average is made along this circle.

k

(TE mode)

:ig. 2. Relation between directions of wavevector k and dipole moment R,h. The dipole moment is rotating in the plane perpendicular to the wavevector k . The averaged contribution of R,h to the electromagnetic wave in the quantum well is calculated by the integral of the square of the component of Rch parallel to the electric field, with moving k along the circle shown m the figure.

electron and the hole, and m: and mo are the effective mass in the conduction band and the free-electron mass, respectively.

The component of R parallel to the electric field E contrib- utes to electromagnetic waves as the gain or loss by the square of the scalar product [R . El2 . Since the wavevector k of electrons in a conventional laser distributes over all directions, the average of the square of the component of R parallel to any fixed E is obtained to be (5) R 2 by integrating IR . El2 over all directions.

Next, we consider the vector R of ( 3 ) for quantum-well structures. The component of the wave vector k parallel to the y-direction ky is discrete corresponding to the quantization of the energy levels in contrast with the conventional lasers. Therefore, the average of IR . El2 over all possible directions i s made by fixing the component k, for one subband, as shown in Fig. 2.

Here we suppose the TE-mode electromagnetic waves in the coordinate system shown in Fig. 1. In this case, the electric field E is along the x-direction. Using (4), the square of the component of R parallel to E is averaged for one subband as

27l

(R2) , = R 2 1 (cos2 8 sin2 @ + cos2 @)d$/271

= R 2 (1 + cos2 0 ) / 2

= R 2 (1 + Ecn/ecn)/2 (6 a>

R 2 (at subband edge) (6b)

where E,, and E,, are the quantized level and the total energy of subband n, which are shown in the Appendix. The origin of these energy levels is at the bottom of the potential well.

Using ( 2 ) , (6), and the result on the conventional lasers obtained above, the dipole moment for one subband averaged over all possible directions in a quantum-well structure, (R,?h)n, is expressed in terms of that in a conventional laser, (&'h)conv> as

( K k ) n E (#)6rnn6kc1l khil(1 +Ecn/ecn) ( ~ % ) c o n v (7a)

(7b)

e ($!)6mn6kcll khll (Rzh)conv (at subband edge)

for TE modes. In (7)> we have omitted the factor 6kckh i n (Rzh))conv-

ASADA et al. : QUANTUM-WELL LASERS 141

It is concluded from (7b) that the transition probability at the subband edges in quantum-well lasers is about 1.5 times larger than that in conventional lasers. If the number of subbands is not only one, the total average of the dipole moment is obtained by summing (Rzh)n given by (7a) over all subbands. This summation is shown below in (8), after multiplying the electronic distribution function. Since the number of the subbands increases with increasing well width, the total average of the dipole moment approaches the results of conventional lasers for the limit of large well width.

For TM modes, the electric field has two components along the y- and z-directions. Since the refractive index difference between the cladding and active layers is small in a semiconductor laser, the z-component of the electric field is much smaller than the y-component for TM modes. Therefore, the average ( R 2 ) , for TM modes is obtained by exchanging (1 t cos2 0)/2 and (1 + Ecn/ecn)/2 in ( 6 ) for sin2 f3 and (1 - Ecn/ecn), respectively. This result brings about the gain difference between TE and TM modes, as shown in Section II-C-(4).

B. Linear Gain Due to the intraband relaxation process such as the electron-

electron scattering, electron-phonon scattering, etc., broadening occurs in the gain spectrum of semiconductor lasers. Taking into account the intraband relaxation in the same way as that developed in conventional lasers [ 151 , [16] , the linear gain a(') in quantum-well lasers is written as

a( ' ) (0) = 0- m ~ m ~ / ( m ~ t mz)/nA w

where the origin of the energy levels is at the bottom of the conduction-band potential well except for the quantized levels for holes (Ehn) which are measured from the top of the valence band (bottom of the valence-band well) down to the inside of the valence band, w is the angular frequency of light, p is the permeability, e is the dielectric constant, mz is the effective mass of the heavy-hole band, Ech is the transition energy given by Ech = E,, - eh,,, w is the well width, M is the number of the quantized levels smaller of those of conduction and valence bands, T~~ is the intraband relaxation time [ 151 , [ 161 , and f , and f , are the Fermi functions given by

f, = + exP((%n - Efc)/kT)I-' >

(9) f , = t exp {(ehn - E')/kT)I-'.

In (8) and (9), we have assumed that electrons and holes in the wells are in equilibrium determined by the quasi-Fermi levels Ef, and Efi, respectively.

EfC and Efu are related to the densities of electrons and holes injected into the wells as

= (m,*kT/nfi2w)zln[1 + exp{(Efc - E,,)/kT}] n

P N N (10)

(mz kT/nA w ) z h [ 1 + exp - Ehn - Eg)/kT)] . n

In @)-(lo), we have neglected the light-hole band, since the density of states of this band is much smaller than that of the heavy-hole band.

C. Analysis of Linear Gain in GaInAslInP Quantum Well As a numerical example, we have calculated the linear gain

in Gao.47 In,. 53As/InP quantum-well structures. We have assumed that the effective masses rn: and rn; and the dielectric constant e are the same as those of conventional lasers [30] - [32]. We have assumed also that the intraband relaxation time q n is the same as that of conventional lasers 1221, i.e., T~~ = 1 X s, except in the following section where other values are also used to discuss the effects of the intraband relaxation. The estimation of T~~ by experiments (for example, see [22] and [41]) is a future problem for quantum-well lasers. The well structure and the quantized energy levels have been given in the Appendix. I) Effects of Intraband Relaxation: Calculation results of

the gain spectra of Gao.47 Inoas3 As/InP quantum-well structures for TE modes are shown in Fig. 3 for well width w = 100 A with the intraband relaxation time in = 1 X s and 03. As seen in Fig. 3, gain peaks exist corresponding to the quantized levels. Fig. 4 shows the maximum gain as a function of injected carrier density for various values of T ~ .

The following three effects of the intraband relaxation are obtained from Figs. 3 and 4. First, as seen in Fig. 3, the gain peaks are not fixed at the wavelengths determined by the quantized levels, but are shifting toward shorter wavelength with increasing injected carrier density for finite value of T ~ .

Second, the gain spectra around the peaks broaden due to the intraband relaxation. This effect is caused by the equiva- lent broadening of the quantized levels by an amount of&/Th included in the integral in (8). This effect of broadening is significant when M / T ~ is comparable to the difference between the levels E,, and E,, . For example, the quantum size effect may be hardly observed when well width is larger than 350 A for Gao. 47 Ino. 53 As/InP.

Finally, the third effect of the intraband relaxation is reduction in the maximum gain by about 30-40 percent, as seen in Fig. 4. This effect has also been obtained for conventional lasers [16], [22].

2) Well Width Dependence of Linear Gain: Fig. 5 shows the maximum gain as a function of injected carrier density for well width w = 50,100, and 200 A, and for conventional lasers [33] .

For small well widths, the gain is small in the region of low injection, and the gradients of the curves are steep, as seen in Fig. 5. This is because the density of states is larger for smaller well widths.

The break point observed in the curve of w = 200 is caused by the change of the maximum point of the gain spectrum from the 0th-0th transition to the 1st-lst, where the 0th and 1 st mean the quantized levels. The laser threshold must be de- signed avoiding such a break point to obtain stable single-mode oscillation.

3) Temperature Dependence o f Linear Gain: The temperature dependence of the linear gain is shown in Fig. 6(a) and (b). Here we have included the shift of the bandgap energy of InP and GaInAs with temperature as AEg/AT= -4 X (eV/K) [34]. As seen in Fig. 6, the decrease of the gain with temperature is less for quantum wells than conventional ones,

748 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-20, NO. 7, JULY 1984

20001 I I I I I I

CARRIER DENSITY N (10’8cma)

(a)

Wavelength ( p m

Fig. 3. Linear gain spectra of Ga0.~~Ino.53As/InP quantum-well lasers calculated for well width w = 100 A with the intraband relaxation time rin = 1 X s (solid curves and T~~ = m (dashed curves). The result for a conventional double heterostructure is also shown by a long-dashed curve.

- GalnAsllnP

2 50- W=100A T ~ 3 0 0 K

-

2000( I I I

9 100 GalnAsllnP T E mode W=100A

2o t I I I 1 0 1 2 3 L 5

CARRIER DENSITY N ClO*m”)

Fig. 4. Maximum gain as a function of injected carrier density calculated for Gao+471no.53As/InP quantum-well lasers with the intraband relaxation time 7in = 0.5 X s, 1 X s, and -.

Level 0-0 -

-

- I GalnAsllnP

TE mode

20 - i

0 1 2 3 4 , * -3 3” = 1 xl6’:ec - I

T=300K

10 I I I I

CARRIER DENSITY N ( 1 O c m

Fig. 5. Maximum gain as a function of injected carrier density calculated for Gao.471no.s3As/InP quantum-well lasers with the well width w = SO, 100, and 200 A and for conventional lasers.

since the broadening of carrier distribution with temperature is narrower for quantum wells. This advantage of quantum wells has been pointed out [7], [I21 , neglecting the effects of the intraband relaxation.

4 ) Gain Difference Between TE and TM Modes: Fig. 7 shows the difference of the linear gain spectra between TE and TM modes for a well width of 100 8. As seen, the gain for the

2o 1 Conventlonal

0 100 200 300 LO0

Temperature T ( K l

(b 1 Fig. 6. Temperature dependence of maximum gain calculated for

Gao.471no.s3As/InP quantum-well lasers. (a) Maximum gain as a function of carrier density for various temperatures, and (b) maximum gain as a function of temperature a t fixed carrier densities. For comparison, results for a conventional laser are shown by dashed curves.

2000 I I I I I

GalnAsAnP

t W=lOO& T =300K __ TE mode T i ,= l~ lO ‘~sec --- TM mode .. .

1000

Wavelength (pm)

Fig. 7. Gain spectra for TE and TM modes in Gao.47In0.53As/InP quantum-well lasers with the well width w = 100 A.

TE mode is much larger than that for the TM mode. This ten- dency is in agreement with the experimental result [29] where the measured gain for the TE mode was larger than that for the TM mode in multiquantum-well lasers.

111. INTERVALENCE BAND ABSORPTION

The absorption loss due to the transition of electrons between split-off and heavy-hole valence bands [21], [22] occurs in all subbands for quantum-well lasers. The intervalence


Valence-Band Top

- Fig, 8. Intervalence band absorption occuring in one subband of a

quantum-well laser.

band absorption occurring in one subband is shown sche- matically in Fig. 8. Summing this intervalence band absorption over all subbands, the total intervalence band absorption in a quantum-well structure is written as

ash (QW N &F (Rzh )n n

- ( f , - fh)m:m:/(d - m:)/(ga2w) (1 1)

where Rsh is the dipole moment, m: is the effective mass of the split-off band, f , and f h are the values of the Fermi function at the energy levels of the split-off and heavy-hole bands where the transition is occurring. In (1 l), we have neglected the intraband relaxation in order to make comparison easy between conventional and quantum-well lasers. f, is nearly equal to unity and m: is large enough to approxi-

mate 1 - fh by the Boltzmann distribution. Therefore, (1 1) can be approximated to be

ff,h(QW) = n o m mE/(m: - m:) ' (R;h)n exp [- {Ehn - Ei&z(n))/kTl

n

,/ exp (-Ehn / kT) *

= K O (QW)P (1 2)

where Et%, (n) has been shown in Fig. 8, which is dependent on the subband number n , P is the hole density, and KO (QW) is the constant of proportionality of ash to P in quantum-well structures.

The intervalence band absorption in conventional lasers, aSh(conv), can be calculated by the same approximation mentioned above [22]. The ratio between aSh(QW) and ctsh(conv) for TE modes with the same hole density can be written as

a,h(QW)/~,h(conv) = KO (QW)/Ko (conv)

N [(mi - m ~ ) n k T / m ~ / ( A w - A,)] ' I 2

. int [ w {2mi Eiba (conv) } 1/2 /nA ]

where int [ ] means discard of decimal fraction in the brackets. For the limit of large well width, the ratio in (13) converges to unity.

The exact calculation of the intervalence band absorption including the absolute value is possible if the valence band

parameters of the quaternary, in particular, the anisotropy of the valence band, are used [35]. However, it is rather compli- cated, and therefore, here we use the experimental results for the parameters of conventional lasers after calculating the ratio given by (13) to obtain the absolute value of the absorption in a quantum-well laser. K O (conv) -4 X lo-'? cm2 and &bU (conv) = 0.15 eV have been obtained from the measure- ment of the differentia quantum efficiency of 1.5-1.6 pm wavelength GaInAsP/InP conventional double heterostructure lasers [21] , [22] .

The calculation results of (13) are shown in Fig. 9 for Gao.47 In,.,,As/InP quantum-well structures at a photon energy of 0.8 eV. As seen in Fig. 9, the intervalence band absorption in quantum-well lasers is nearly in the same order as that in conventional lasers around room temperature.

IV. THRESHOLD CURRENT DENSITY OF GaInAs/InP QUANTUM -WELL LASERS

The threshold gain gth is calculated with the following equation:

gth = (1 - $)%x/$ ln(l/R)/(&) (1 4)

where $ is the optical confinement factor, a,, is the loss coefficient in the active region which is mainly due to the intervalence band absorption mentioned in the previous section, orex is the loss coefficient outside the active region, L is the cavity length, and R is the reflectivity of the end mirrors. The threshold carrier density Nth is obtained using (14) and Figs. 5, 6, and 9.

The threshold current density J th is obtained using Nth and the carrier lifetime 7,. For GaInAsP/InP and GaInAs/InP quantum-well lasers, T, includes the nonradiative Auger effect and the carrier leakage over the heterobarrier, as in conventional lasers. Although systematic studies including both of these nonradiative components have not yet been established, significant change is not expected in 7, for quantum-well lasers compared to conventional lasers. Here we assume the same carrier density dependence of 7, as that obtained experimentally for conventional lasers [37].

The threshold current density Jth is written as

Ah = ewNwNth/Ts

= ewN, BeffN;2h (15)

where e is the electron charge, B e f f is the effective recombination coefficient [37] , and N, is the number of wells.

We calculate two multiquantum-well (MQW) structures shown in Fig. lO(a) and (b), and compare the threshold current densities of these structures. We neglect the coupling of the wave- functions among the wells which causes the broadening of the quantized levels and thus the reduction of the gain. Using the Kronig-Penney model, this energy-level broadening has been obtained to be less than 1 meV for the lowest level of conduction band, when w 2 80 A and 120 A for MQWs shown in Fig. lO(a) and (b), respectively. This broadening is much narrower than that due to the intraband relaxation (h/rjn = 7 meV for T& = 1 X s). For the well widths smaller than the values mentioned above, the threshold current density is considered to be slightly larger than that obtained with the present model.

Calculation results of the optical confinement factor $ and the threshold current density Jth are shown in Fig. 11 (a) and

750 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-20, NO. 7, JULY 1984

I I

at hw:0.8eV -

-

8 3 I I g z a a IO0

0.5 200 300 400

a m Temperature T ( K )

Fig. 9. Ratio of intervalence band absorption between quantum-well and conventional lasers with equal hole densities for Ga0.47In0.53As/ InP at a photon energy of 0.8 eV.

Fig. 10. Multiquantum-well structures for which threshold current density is calculated in the text.

(b), respectively, as a function of the number of wells N,. The values of the refractive indexes in 1321 and the method in [42] have been used for the calculation of ,$, and a, and f? in (14) have been assumed to be the same as those of conventional lasers [36]. Discrete values obtained at each N, have been connected smoothly in these figures. As can be seen, there exists a minimum in each curve of &, in Fig. 11 (b).

The optical confinement factor [ is larger for the structure of Fig. 10(b), as shown in Fig. 11 (a). However, Jth is larger for this structure compared to that of Fig. lO(a) for large values of N,, as shown in Fig. 1 1 (b). This is because the gain in Fig. lO(a) is larger than that in Fig. IO(b) since the quantized levels in the latter are closer to one another than in the former due to the shallowness of the potential well. (See Table I in the Appendix.)

The intervalence band absorption increases proportionally to carrier density N(--P), as shown in (12). Thus, the intervalence band absorption suppresses the gain increase with N . This results in the increase of the threshold carrier density Nth and the threshold current density &, .

In quantum-well lasers, the gain increases with N more steeply than for conventional lasers. Thus, the effect of the intervalence band absorption is smaller for quantum-well lasers.

The gain increases more steeply for narrower quantum wells, as shown in Fig. 5. However, Nth is larger for narrower quantum wells since the optical confinement factor is small. This results in the larger magnitude of aSh(QW) in (12). Thus, the effect of the intervalence band absorption is not expected to be reduced by reducing the well width.

For GaAs/GaAlAs systems, ultralow threshold current den-

I I I

0.01 0 u 10 20

Well Number Nw

(a) I I I 1

- GaInAs/inP --- GalnAs/GalnAsP/lnP L =300um

0 1 1 I I J

(b) Fig. 11. Calculated results of (a) optical confinement factor 5, and

(b) threshold current density &, as a function of the number of wells N , for multiquantum-well structures shown in Fig. 10. The discrete value at each N , has been connected smoothly in these figures.

0 10 20 Welt Number Nw

Temperature T(K)

Fig. 12. Temperature dependence of the threshold current density calculated for the multiquantum-well structure shown in Fig. 10(a) with well width w = 100 A and well number N , = IO and for a conventional laser. The characteristic temperature To is found to be 90 K at 300 K for the multiquantum well, and 60 K for the conventional one.

sity has been obtained for a single quantum-well separated confinement structure [3] - [7] . In these single quantum-well structures, Nth is larger than for MQW structures since i: is small. Thus, by the same reason as mentioned above, single quantum-well structures appear to be unsuitable for GaInAsP/ InP and GaInAs/InP systems.

Fig. 12 shows the temperature dependence of the threshold current density calculated for MQW in Fig. lO(a) withN, = 10 and w = 100 8, compared to that calculated for a conventional laser. The characteristic temperature To is obtained to be 90 K at 300 K for the MQW laser, which is larger than that ob-


tained for the conventional one, 60 K. However, a steep increase of Jth is also occurring above about room temperature.

V. CONCLUSIONS

In this paper, we analyzed the linear gain and the intervalence band absorption for quantum-well lasers.

First, we analyzed the electronic dipole moment in quantum- well structures. The dipole moment for TE modes in quantum-well structures was obtained to be about 1.5 times larger at the subband edges than that in conventional ones. Also obtained was the difference of the dipole moment between TE and TM modes, which results in the gain difference between these modes. Then we derived the linear gain taking into account the intraband relaxation. As a numerical example, we applied this analysis to the GaInAs/InP quantum-well lasers. It was shown that the effects of the intraband relaxation were 1) a shift of the gain peak toward shorter wavelength with increasing injected carrier density even in quantum-well structures, 2) increase of the spectral width of the gain due to the softening of the profile, and 3) reduction of the maximum gain by 30-40 percent.

We also analyzed the intervalence band absorption in quantum-well lasers, which has been considered to be one of the factors causing the strong temperature dependence of the threshold current of 1.5-1.6 pm wavelength CaInAsPjInP conventional double heterostructure lasers. Although this absorption in quantum-well lasers was calculated to be nearly in the same order as that in conventional lasers, its effect on the threshold is smaller because the gain is larger for quantum- wells than conventional ones.

Finally, we calculated the threshold current density of GaInAsIInP multiquantum-well lasers. The characteristic temperature To was obtained to be about 90 K at 300 K for well width and well number of 100 A and 10, respectively. This value of To is larger than 60 K of conventional lasers.

APPENDIX QUANTIZED ENERGY LEVELS

First, we calculated the discontinuities of the band edges of conduction and valence bands at the heterojunction, AE, and AE,, based on the relative position of the top of the valence band E, obtained from the method of linear combination of atomic orbitals [38]. The value of E, for the quaternary Ga,In, - ,AsyPl - ,, was obtained from the interpolation of those for the related binaries, i.e., Gap, GaAs, InAs, and InP.

The values of AE, and AE, obtained with this method for GaAs/Gao.8Alo.zAs and Gao~131no~87As0~29P0,71/InP agrees well with the experimental results [ l] , [ 3 9 ] . The values of AEc and AE, calculated for GaInAsP/InP systems discussed in the text are listed in Table I.

Next, we calculated the quantized energy levels. The envelope function aCn (y) included in (1) is obtained from the following equation [27] :

[- ( h 2 / 2 m 3 ( a 2 / a ~ 2 ) + ~ ( ~ 1 1 ~ c n ( ~ ) = E c n c ~ , , ( ~ )

(A- 1)

TABLE I CALCULATION RESULTS OF BAND EDGE DISCONTINUITIES. THE ACTIVE

LAYER IS Gao47In0.53As (Eg = 0.75 eV [40])

Cladding Material (Eg) AE, (eV) AE, (eV)

InP (1.35eV) 0 . 3 3 0 . 2 7

Ga0.281n0.72As0.6P0.4 (0.95eV) 0.092 0.11

where the effective mass mz is distinguished between inside and outside of the well as mzl and mZ2, respectively, V ( y ) is the form of the potential well, and E,, is the quantized energy level to be obtained here. The origin of the energies is at the bottom of the conduction-band well. QCfl(y) is obtained from (A-1) as

acn = A { [ ~ ~ G J z - Y I ~ I { ) n; even

sin n ; odd

(lul >w/2) (A-2)

where A and B are constants, and w is the well width. Since Gcn ( y ) and its derivative are continuous at the hetero-

interface, E,, is obtained from the following equation as discrete values:

= { tan ) [ w . \ / m 2 & ] { } . (A-3) n; even

-cot n; odd

The well width at which the quantized level of number n (n=0 ,1 ,2 ; . * ) i s cu to f f i sg ivenby

(A-4)

Since electrons are free for the directions parallel to the heterojunction, the total energy of an electron ecn is given by

where a subband is formed accompanied by the quantized level Ecn as the bottom.

These calculations are also applied to the heavy-hole and split-off valence bands.

The radiative transitions occur between the subbands in conduction and valence bands with the same number. The wavelength corresponding to the transition between a conduction electron and a heavy hole at each subband edge is given by

A, = 1.24/ [Eg + Em + Ehn ] (pm) ('4-6)

where the unit of the energy levels is electron volt, and the quantized energy levels of heavy holes Ehn are measured down- ward from the top of the valence band. Calculation results of this wavelength for Gao. 47 Ino. 53 AsjInP are shown in Fig. 13.

7 5 2 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-20, NO. 7 , JULY 1984

I I I I I I

Well Thickness w ( b ) Fig. 13. Wavelengths corresponding to transitions between the quan-

tized levels of conduction electrons and heavy holes as a function of well width calculated for the Ga0.47In0.~3As/lnP quantum-well structure.

ACKNOWLEDGMENT The authors would like to thank Associate Professors K. Iga

and K. Furuya of the Tokyo Institute of Technology, and M. Yamada of Kanazawa University, for many stimulated discussions, and Y. Miyamoto of the Tokyo Institute of Technology for assistance with calculations of the optical confinement factor.

RFE‘EKENCES R. Dingle, “Confined carrier quantum states in ultrathin semiconductor heterostructures,” in Fesfkorperprobleme X V , H. J. Queisser, Ed. New York: Pergamon, 1975, pp. 21-48. N. Holonyak, Jr., R. M. Kolbas, R. D. Dupuis, and P. D. Dapkus, “Quantum-well heterostructure lasers,’’ IEEE J. Quantum Elec- tron., vol. QE-16, pp. 170-186, Feb. 1980. W . T. Tsang, “Extremely low threshold (A1Ga)As graded-index waveguide separate-confinement heterostructure lasers grown by molecular beam epitaxy,” Appl. Phys. Lett., vol. 40, pp. 21 7-219, Feb. 1982. S. D. Harsee, M. Baldy, P. Assenat, B. de Cremoux, and J. P. Duchemin, “Very low threshold GRIN-SCH GaAs/GaAlAs laser structure grown by OM-WE,” Electron. Lett., vol. 18, pp. 870- 871, Sept. 1982. R. D. Burnham, W. Streifer, D. R. Scifres, C. Lindstrom, T. L. Paoli, and N. Holonyak, Jr., “Low-threshold single quantum well (608) GaAlAs lasers grown by MO-CVD with Mg as p-type dopant,” Electron. Lett., vol. 18, pp. 1095-1097, Dec. 1982. T. Fujii, S. Yamakoshi, 0. Wada, T. Sakurai, and S. Hiyamizu, “GaAs-GaAlAs GRIN-SCH laser grown by MBE” (in Japanese), inNat. Conv. Rec. Appl . Phys. Soc. Japan, Sept. 1983. R. Chin, N. Holonyak, Jr., B. A. Vojak, K. Hess, R. D. Dupuis, and P. D. Dapkus, “Temperature dependence of threshold current for ,quantum-well A1,Gal.,As-GaAs heterostructure laser diodes,”Appl. Phys. Lett., vol. 36, pp. 19-21, Jan. 1980. H. Iwamura, T. Saku, T. Ishibashi, K. Otsuka, and Y. Horikoshi, “Dynamic behaviour of a GaAs-CaAlAs MQW laser diode,” Electron. Lett., vol. 19, pp. 180-181, Mar. 1983.

dependence of threshold current for coupled multiple quantum-

Phys., vol. 51, pp. 2402-2405, May 1980. well Inl-xGaxP1.,As,-InP heterostructure laser diodes,” J. Appl .

] H. Temkin, K. Alavi, W. R. Wagner, T. P. Pearsall, and A. Y. Cho, “1.5-1.6pm G a ~ . ~ 7 1 n ~ . s 3 A s / A l o ~ ~ ~ I n ~ ~ ~ ~ A s multiquantum well lasers grown by molecular beam epitaxy,” Appl . Phys. Lett., vol. 42, pp. 845-847, May 1983. [33

1 E. A. Rezek, N. Holonyak, Jr., and B. K. Fuller, “Temperature [31

[32

[ 11 ] T. Yanase, Y. Kato, I. Mito, M. Yamaguchi, K. Nishi, K. Koba- yashi, and R. Lang, “1.3 pm InGaAsP/InP multiquantum-well lasers grown by vapour-phase epitaxy,” Electron. Lett., vol. 19,

[12] N. K. Dutta, “Calculated threshold current of GaAs quantum well lasers,”J. Appl. Phys., vol. 53, pp. 7211-7214, Nov. 1982.

[13] A. Sugimura, “Auger recombination effect on threshold current of InGaAsP quantum well lasers,” IEEE J. Quantum Electron., vol. QE-19, pp. 932-941, June 1983.

[14] D. Kasemset, C. S. Hong, N. B. Patel, and P. D. Dapkus, “Graded barrier single quantum well lasers-Theory and experiment,” IEEE J. Quantum Electron., vol. QE-19, pp. 1025-1030, June 1983.

[15] Y. Nishimura and Y. Nishimura, “Spectral hole-burning and non- linear-gain decrease in a band-to-level transition semiconductor laser,” IEEE J. Quantum Electron., vol. QE-9, pp. 1011-1019, Oct. 1973.

[ 161 M. Yamada and Y. Suematsu, “Analysis of gain supprcssion in undoped injection lasers,” J. Appl . Phys., vol. 52, pp. 2653-2664, Apr. 1981.

[ 171 G.H.B. Thompson and G. D. Henshall, “Nonradiative carrier loss and temperature sensitivity of threshold current in 1.27 pm (GaIn)(AsP)/InP D. H. lasers,” Electron. Lett., vol. 16, pp. 42- 44, Jan. 1980.

[18] A. Sugimura, “Band-to-band Auger recombination effect on InGaAsP laser threshold,” IEEE J. Quantum Electron., vol. QE-17, pp. 627635 , May 1981.

[19] N. K. Dutta and R. J. Nelson, “Temperature dependence of threshold of InGaAsP/lnP double-heterostructure lasers and Auger recombination,’’ Appl. Phys. Lett., vol. 38, pp. 407409 , Mar. 1981.

[20] T. Uji, K. Iwamoto, and R. Lang, “Nonradiative recombination in InGaAsP/InP light sources causing light emitting diode saturation and strong laser-threshold-current temperature sensitivity,” Appl . Phys. Lett., vol. 38,pp. 193-195, Feb. 1981.

1211 A. R. Adams, M. Asada, Y. Suematsu, and S. Arai, “The temperature dependence of the efficiency and threshold current of Inl-xGaxAsyP1-y lasers related to intervalence band absorption,”

Japan. J. Appl. Phys., vol. 19,, pp. L621-L624, Oct. 1980. [22] M. Asada, A. R. Adams, K. E. Stubkjaer, Y. Suematsu, Y. Itaya,

and S. Arai, “The temperature dependence of the threshold current of GaInAsP/InP DH lasers,”lEEE J. Quantum Electron., vol. QE-17, pp. 611619 , May 1981.

[ 231 M. Yano, H. Nishi, and M. Takusagawa, “Temperature characteristics of threshold current in InGaAsP/InP double-heterostructure lasers,” J. Appl . Phys., vol. 51, pp. 40224028, Aug. 1980.

[24] S. Yamakoshi, T. Sanada, 0. Wada, I. Umebu, and T. Sakurai, “Direct observation of electron leakage in InGaAsP/InP double heterostructure,” Appl. Phys. Lett., vol. 40, pp. 144-146, Jan. 1982.

[25] N. K. Dutta, “Calculation of Auger rates in a quantum well structure and its application to InCaAsP quantum well lasers,” J. Appl. Phj’s., vol. 54, pp. 1236-1245, Mar. 1983.

[ 261 L. C. Chiu and A. Yariv, “Auger recombination in quantum-well InCaAsP heterostructure lasers,’’ IEEE J. Quantum Electron.,

[27] See, for example, C. Kittel, Quantum Theory of Solids. New York: Wiley, 1963, ch. 14, pp. 286-290.

[28] E. 0. Kane, “Band structure of indium antimonide,” J. Phys. Chem. Solids, vol. 1, pp. 249-261, 1957.

[ 291 H. Kobayashi, H. Iwamura, T. Saku, and K. Otsuka, “Polarisation- dependent gain-current relationship in GaAs-AIGaAs MQW laser diodes,”Electron Lett., vol. 19, pp. 166-168, Mar. 1983.

1301 R. J. Nicholas, J. C. Portal, C. Houlbert, P. Perrier, and T. P.

pp. 700-701, A u ~ . 1983.

VOI. QE-18, pp. 1406-1409, Oct. 1982.

Pearsall, “An experimental determination of the effective masses

vol. 34, pp. 492494 , Apr. 1979. for GaxInl-,AsyPl-y alloys grown on InP,” Appl . Phys. Lett.,

N. K. Dutta, “Calculated absorption, emission, and gain in Ino~~2Gao .z8As~ .6Po .~ , ” J. Appl . Phys., vol. 51, pp. 60956100, Dec. 1981. K. Utaka, Y. Suematsu, K. Kobayashi, and H. Kawanishi, “Ga- InAsP/InP integrated twin-guide lasers with first-order distributed Bragg reflectors at 1.3 pm wavelength,” Japan. J. Appl . Phys., vol. 19, pp. L137-L140, Feb. 1980. In the calculations of the linear gain for conventional lasers, we I

ASADA et aL: QUANTUM-WELL LASERS 7 5 3

Solids-The Physics of the Chemical Bond, San €&cisco,- CA: Freeman, 1980, ch. 10, pp. 252-256. R. Chin, N. Holonyak, Jr., S. W. Kirchoefer, R. M. Kolbas, and E. A. Rezek, “Determination of the valence-band discontinuity of InP-Inl~,Ga,P1.,As, (x - 0.13, z - 0.29) by quantum-well luminescence,” AppL Phys. Left., vol. 34, pp. 862-863, June 1979. R. E. Nahory, M. A. Pollack, W. D. Johnston, Jr., and R. L. Barns, “Band gap versus composition and demonstration of Vegard’s law for Inl-xGaxAsyP1- lattice matched to InP,” Appl. Phys. Lett., vol. 33, pp. 6 5 9 6 l 1 , Oct. 1978. M. Yamada, H. Ishiguro, and H. Nagato, “Estimation of the intraband relaxation time in undoped AlGaAs injection laser,” Japan. J . Appl. Phys., vol. 19, pp. 135-142, Jan. 1980. Y. Suematsu and K. Furuya, “Propagation mode and scattering loss of two dimensional dielectric waveguide with gradual distribution of refractive index,”IEEE Trans. Microwave Theory Tech., VOI. MTT-20, pp. 524-53 1, Aug. 1972.

Yasuharu Suernatsu (M162-SM’75-F’80), for a photograph and biogra- phy, see p. 139 of the February 1984 issue of this JOURNAL.

vol. gain and intervalence band absorption quantum-well...

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