MENDOZAALVAREZ, JG

NUNES, FD

PATEL, NB

Journal of Applied Physics

Repositorio da Producao Cientifica e Intelectual da Unicamp

Refractive index dependence on free carriers for GaAs 
J. G. Mendoza-Alvarez, F. D. Nunes a), and N. B. Patel 
Instituto de Ffsica "Gleb Wataghin ", Universidade Estadual de Campinas, Campinas, S.P. Brasil 
(Received 4 October 1979; accepted for publication 2 April 1980) 
This work reports on the influence of the injected free carriers on GaAs refractive index Nat 
297 oK. The variation of N caused by injected free carriers was theoretically calculated in a more 
complete way than has been performed earlier. New results were obtained rather than those of a 
reducing effect on N in a linearlike dependence on the injected free carrier concentration n. They 
are: (1) linearlike dependence of Non n occurs only beyond a certain value n I and (2) an increasing 
effect on N is caused by the injected free carriers for concentrations in the range between n = 0 and 
a certain value of n = nc (nc < n I)' In the range nc < n < n I a nonlinear decreasing effect was 
obtained. Effect (2) has not been noticed up to now, so far as the authors know. 
PACS numbers: 78.20.Dj, 42.55.Px, 42.1O.Ke 
The influence of free carriers on the refractive index of 
GaAs has been invoked to explain some phenomena ob-
served in GaAs lasers. 1-4 One important phenomena, for 
practical applications, is the laser waveguide formation3,4 in 
the direction parallel to the junction plane. 
In this case, injected free carrier perturbation on the 
refractive index dominates processes.4 Up to now the refrac-
tive index variation with injected carriers AN (n) has been 
taken as directly proportional to the injected carrier density 
n. The proportionality constant5 ,6 includes the plasma effect 
and the Burnstein shift of the absorption edge. Some experi-
mental data7,8 have shown agreement with this relation be-
tween AN and n. This relation was obtained by Thompsons 
with an approximate calculation. Another iridirect approxi-
mate calculation of N as a function of injection current was 
made by Cross and Adams9 by using an analytic fit to Marp-
les's refractive index data and considering the carrier ex-
change interaction and the injection only through an "effec-
tive band-gap energy". 
THEORETICAL FRAMEWORK 
In this work we present the results of a detailed calcula-
tion of the refractive index variation with free carriers, using 
a modified Kramers-Kronig analysis of a theoretical absorp-
tion coefficient calculation using Stem's treatment. 10 This 
calculation takes into account the band gap shrinkage 
caused by free carriers, II and the influence of the injected 
carriers on the bandtails and on the matrix element devel-
oped by Stem in Ref. 12. This matrix element describes the 
change from selection rule transitions for high energies, 
when parabolic bands take place, to nonselection rule transi-
tions that occur for low-energy transitions, when bandtails 
are involved. Besides the band-to-band absorption, our cal-
culation of the refractive index includes lattice absorption 
and intra and interband absorption. These effects were cal-
culated using the following expressions: 10 
ANlv~ -9X 1O- 4 INE 2, (1) 
(2) 
a)Present Address: Optoelectronic Devices Group-CPqD-TELEBRAS, 
Campinas, S.P, Brasil 
ANintra(electrons)(E)~ -9.6X 10- 21 niNE 2, (3) 
ANinter(holes)(E)~ - 6.3 X 1O-22pIE 2• (4) 
RESULTS 
In this section, results of our calculations will be shown. 
In Fig. 1 is shown the spectral absorption coefficient corre-
sponding to band-to-band transitions using bandtail densi-
ties of states, including the negative absorption part of the 
curve (or gain coefficient) for different injected carriers den-
sities n. 
In Fig. 2, the refractive index as a function of the photon 
energy, for different injected carrier densities, is shown. The 
refractive index shown in this figure was calculated by using 
the data of Fig. 1, including the effects described by Eqs. (1)-
(4). 
IcY 
Ie! 
10 
"'i 
E I 
~ _I 
./ 
/ 
I 
I 
'1 
I 
I 
i 
/ 
i 
I 
I 
i 
i 
i 
I 
·8 
I 
Go As 297 OK 
18 '1 
NAI..5xIO No'3. 10 
BAND TAILS 
n (em-'S) 
1_0 
2_2.5xld1 
3..5x10t1 
4_lx10'8 
5..15 X 10'8 
6_ 2 X 1018 
7_3.10'8 
8_4xld8 
_104~-::::~=-:-::-.L..J.::-'--...L.JL.J..~.L....l---'--L-L..J....L.-'----'---'-...L.JL.....L...L..I 
l34 136 1.38 140 1.42 144 l46 148 1.50 152 154 1.56 1.58 lSO 1.62 
E(eV) 
FIG. 1. Absorption coefficient vs photon energy as calculated by using a 
bandtail model for different injected carrier densities in the range n:O--
4X 10'8 cm -3, 
4365 J, Appl. Phys, 51(8), August 1980 0021-8979/80/084365-03$01.10 @ 1980 American Institute of Physics 4365 
FIG. 2. Resulting refractive index from the modified Kramers-Kronig anal-
ysis of the absorption for different injected carrier densities. 
The refractive index as a function of the injected carrier 
density is shown in Fig. 3, where we present curves for differ-
3.55 ---- -----------l 
z 
x 
w 
o 
~ 
o 
Go As 297"K 
n (em-' ) 
1405 
140 
1395 
139 
1.385 
138 
1375 
137 
1.365 
1.36 
1.355 
1.35 
FIG. 3. Refractive index as a function of the injected carrier density for 
different photon energies. . 
4366 J. Appl. Phys., Vol. 51, No.8, August 1980 
4 
3.5 
3 ~ 
..!I 
z 
<I 
2.5 , 
z 
Q 
~ 
2 IE 
~ 
x 
w 
1.5 0 
~ 
w 
> 
i= 
~ a: 
w 
O!i Q:: 
o 
-O!i 
r:. 
r:. + 
Go As 297 OK 
NA= 15x1d-
8 
No= 3x10
17 
Band Tails 
• -- 1.380oV 
• ---- 1385 eV 
+ ---.- 1 3900V 
r:. - .. - .. - 1.3950V 
o:-------+------~2!,-1 ------+3 -----T 
n (xld" em") 
FIG. 4. Refractive index variation -I1N as a function of the carrier injec-
tion n for different photon energies. Note that for n < 7 X 1017 cm " I1N is 
positive. 
ent energies, mainly those close to typical lasing photon en-
ergies. The data shown in this figure were obtained from Fig. 
2 by taking one energy and the refractive index correspond-
ing to different carrier densities. We note in this figure that 
up to around 6 X 1017 cm -3, the refractive index first in-
creases with n reaching a maximum value which is photon 
energy dependent. Beyond the carrier-concentration value 
no corresponding to this maximum value, the refractive in-
dex decreases with n crossing the value corresponding to the 
case of no injection (n = 0). For carrier concentration be-
yond this crossing point, the refractive index is a decreasing 
function of n: however, a linear behavior occurs only for 
values of n greater than a certain n l which is photon energy 
dependent. This can be seen clearly in Fig. 4 where the nega-
tive variation of the refractive index - ilN is shown as a 
function ofthe injected carrier density n. In the case of the 
linear behavior of ilN with n, the relation between them will 
be 
ilN= -An +B(hv). (5) 
Here we must point out that this expression should be fol-
lowed by the condition n > n p where n I is the value of n 
beyond which ilN varies linearly with n; therefore, (B / A ) 
here does not indicate the value of n for which ilN equals 
zero. In some cases, depending on photon energy (and also 
depending on doping level), Eq. (5) approaches the behavior 
of ilN with n between n 1 and no, and it may be used as a good 
approximation. 
Figure 5 shows the two parameters A and B as functions 
of the photon energy. It can be observed that A is almost a 
constant parameter while B is not. 
Mendoza-Alvarez, Nunes, and Patel 4366 
25~------------------------, 
Go As 297·K 
II 17 
N '1.5.10 N '3.10 
20 - Band Tolls 
15 f- 10 
I 
1 I 
OJ 101-------------
: I 
'" Q 5 
K 
., 
Go As 297-1( 
1 I. 1--------,------ ----" 
NA:; 1.5 x 1018 Ncr 31( tcf 
Band Toils 
FIG. 5. Constants A and B used for the 
straight lines fitting of the points (.:IN vs 
n) shown in Fig. 4 as a function of the 
photon energy. 
o~----------------
_ 5 '--__ ---'-____ LI -----~----'------" 
1.38 1.39 1.40 1.41 1.42 143 1.38 1.39 
E(oV) 
CONCLUSION 
We have obtained with our calculations the refractive 
index perturbation caused by the injected free carriers. The 
results indicate that free carriers do not only reduce the re-
fractive index in a linear way as was obtained by Thompson. 
For low injection levels free carriers increase the refractive 
index, and the decreasing effect only happens beyond a cer-
tain injection level. Also we observed that the decrease of 
refractive index with free-carrier concentration only occurs 
in an approximately linear way for carrier densities beyond a 
certain n I' This indicates that in GaAs lasers, the refractive 
index variation from the center to the edges of the waveguide 
along the junction plane does not vary monotonically. This 
fact may influence the guided mode the narrower the stripe 
width is. The increasing effect of injected free carriers has 
not been noticed up to now, as far as we know. However, the 
refractive index calculated by Zoroofchi et al., 13 from experi-
mental absorption coefficient data for different n- or p-doped 
GaAs samples, shows an increasing effect with doping con-
centration similar to that which we have calculated in the 
energy range close to the gap. Also, experimental measure-
ments of double-beam reflectance for n-type samples of 
GaAs made by Sell et al. 14 whose results in refractive index 
versus energy curves showed that in a certain energy range, 
the refractive index increases with n for low doping, and 
decreases as it is doped even more. 
These data also suggest that the injected free-carrier 
effect that we calculated could be doping level dependent. 
The higher the doping level, the smaller the injected free-
carrier effect. Calculations for different doping levels will be 
4367 J. Appl. Phys., Vol. 51, No.8, August 1980 
1.40 1.41 1.42 1.43 
E(oV) 
carried out. The differences observed among Thompson's 
results and our data can be related to the inclusion in the 
calculation of the band-gap shrinkage and tail length vari-
ation with free-carrier injection level not considered by him. 
ACKNOWLEDGMENTS 
We acknowledge the financial support of TELEBRAS 
and an author (J.G.M-A) acknowledges the support of 
CONACyT-CIEA/IPN of Mexico and CNPq of Brasil. 
IF. D. Nunes, N. B. Patel, and J. E. Ripper, IEEE J. Quantum Electron. 
QE-13, 675 (1977). 
2J. M. Vilela, F. D. Nunes, and N. B. Patel, IEEE J. Quantum Electron. 
QE-1S, 801 (1979). 
'Thomas L. Paoli, J. App!. Phys. 48, 1361 (1977). 
4J. Buus, Opt. Quantum Electron. 10,459 (1978). 
5G. H. B. Thompson, Opto-Electron. 4, 257 (1972). 
b A. K. Jonscher and M. H. Boyle, Proc. IPPS Symposium on GaAs, 78 
(1966). 
7p. R. Selway, G. H. B. Thompson, G. D. Henshall, and J. E. A. Whi-
teaway, Electron. Lett. 10,453 (1974). 
8M. R. Mathews, R. B. Dyot, and W. P. Carling, Electron. Lett. 8, 570 
(1970). 
9M. Cross and M. J. Adams, Opto-Electron. 6,199 (1974). 
IOFrank Stem, Phys. Rev. A 133,1653 (1964). 
lie. J. Hwang, Phys. Rev. B 2, 4117 (1970). 
12Frank Stem, Phys. Rev. B 3, 2636 (1971). 
I3J. Zoorofchi and J. K. Butler, J. App!. Phys. 44,3697 (1973). 
I4D. D. Sell, H. e. Casey, Jr., and K. W. Wecht, J. App!. Phys. 45, 2650 
(1974). 
Mendoza-Alvarez. Nunes, and Patel 4367 


REFRACTIVE-INDEX DEPENDENCE ON FREE-CARRIERS FOR GAAS

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REFRACTIVE-INDEX DEPENDENCE ON FREE-CARRIERS FOR GAAS

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