43opm-4- Room203 DLTM.4

Analytic Expressions for Optical Gain and Differential Gain in 1.3 pm Strained Quantum-well Lasers

C.H. Lin, Y.H. Lo School of Electrical Engineering, Cornell University, Ithaca, NY 14853

People have studied the effect of strain and number of wells on optical gain and differential gain and proved that

the strain effect can improve the performance of semiconductor lasers because of modified valence band structures.

Although, the physical mechanisms are unveiled, there exists no simple formula for device design because of the

large number of parameters involved. In this paper, we propose the analytic expressions for optical gain and diffenmtial gain in 1.3 pn separate confinement heterostructure (SCH) strained Infial-xAs/InGuAsP quantum-well

lasers. Except the barriers are assumed to be 1.1 pm wavelength latticed-matched to Id' , all other parameters such as

well width, strain, and number of wells, etc. can be varied. Using our expressions, one can find the structure

producing the lowest threshold current density for a given threshold gain, and the differential gain as functions of

deviceparameters.

We fmt calculated the optical gain and differential gain for strained quantum wells using the multiband effective

mass method considering the valence band mixing and intraband relaxation, and then fitted these results using

empirical formulas. The lasing wavelength was kept at 1.3 pm by changing strain and well thickness simultaneously. Figure 1 shows the relation between In composition (x %) and the corresponding well width (Ld.

The approximated formula is given in Table I.

Figure 2 shows the optical gain (g) versus the current density (J) for different x. Considering the strain and

saturation effect, the optical gain for single well can be well approximated by the following expression:

g ( c m - l ) = 1OOO* f ( x ) * I n ( J / J b ) ( 1 ) where f(x) is a straindependent constant, and Jb is the transparency current density. Both are given in Table I.

Because the barrier states a significantly populated at transparency, Jtr becomes a sublinear function of Nw,

proportional to Nw3/4 instead of Nw. The confmement factor (r) can be written as yNwk, where y is about 2x10-

(A-l). The expressions of modal gain (gmd) for multiple well lasers and the threshold current density for any

gmod required are given in Table I.

From the fact that &/ax < 0, we conclude the threshold current density always decreases with the strain. In other word, the more strain, the lower threshold. Assuming aJ/aNw = 0, we obtained the number of wells producing the

lowest threshold for given x and g m d . For example, for x = 6096, Nw is 1 for 0 < gmod < 22,2 for 22 < gmod <

37, and 3 far 37 < gmod < 42. The expression for differential gain (8') can be written as g' ( cm2 ) = 6.63~10-~ * NW3I4 * Lz * f(x) / JID ( 2 )

The relations of g'-J and g'-gmod for x = 60% and Nw=l, 3.5 are shown in Figure 3 and 4, respectively. The extra

differential gain enhancement in multiple quantum-well lasers is due to their less serious band filling in barrier states

than single well lasers. In conclusion, well approximated expressions for optical gain and differential gain are proposed. They can be used

to optimize laser perfmance without going through very complicated fmt principle calculations. 38

Table 1 : Useful empirical formulas for 1.3 pm InxGal-xAs/InGaAsP strained QW lasers.

empirical formulas comments

well width (Lz) (A)

optimum number Nw = gmod / W.75 TLZ f(x) 1

threshold Jth (Ned) Jth = Jt, Exp[ gmod /(yLz Nw f(x)) I transparency J P W (Ned) Jtr = (0.166~2 - 2 3 . 7 ~ + 887 ) Nw3/4

modal gain gmod (l/cm) gmod =Y Lz Nw In ( J / Jtr 1 f(x) differential gain g' (cm2) g' = 6.63x10-9*Nw314 Lz f(x) / J

b ( x ) = 80-5.38(x-53)+ 0.191(x-53)2 h G a l - x A s ( x 96 )

have lowest threshold

for given g m d and Nw

for given x, Nw and J for given x, Nw and J

-0.0027(x-53)3

ofwells Nw f(x) = -O.oooO64x~+o.O14x~-O.86x+17 ( y = 2x10-4 A-1)

o : caseswecalculated - : from empirical formula - 70

:so

i s , 50

10

80 50 55 60 65 70 75 Indium composition x %

X

c4

I - d

W

00

- : first principle data

f/ x = 6 0 % I

/->-I x = 53

. I I "0 100 200 300 400 500 600 700 800

Current Density J ( A / cu? ) ~ i g 1 . The for each Indiun c o m p ~ ~ n 96 Fig 2. The material gain done well vs current density

for x = 53%,60 % and 70%.

x 10'; . 8

I : fvst principle data - : cmpir i i formula < Y '*'I Nw-f rZ5

1.2 I

01 0 200 400 600 80

Current Density J ( A / d ) Fig 3. The ditrerential gain g' vs current density for

Nw=1,3,5 and x=6096

A I U .I : first principle data

' A - : empirical formula

' '00

W ' "I

os ' a

1 I . . . L I 1 0 20 40 (0 80 100 120

Modal Cain g w ( l / c m ) Fig 4. The differential gah g' vs modd gain &WII for Nw=lJJ

<J=Jta) 39