- Email: [email protected]
Microcavity effect on the nonlinear optical intersubband absorption in semiconductor quantum wells
15 February 1998
Optics Communications 147 Ž1998. 279–284
Microcavity effect on the nonlinear optical intersubband absorption in semiconductor quantum wells Ansheng Liu
1
Institute of Physics, Aalborg UniÕersity, Pontoppidanstræde 103, DK-9220 Aalborg Øst, Denmark Received 26 March 1997; revised 4 August 1997; accepted 27 August 1997
Abstract Based on a semi-classical local-field formalism in which retardation effect in connection with the electromagnetic propagation and nonlocal effect in the optical response of confined electron systems are included, we calculated the nonlinear optical intersubband absorption coefficient of a quantum well ŽQW. inside a microcavity. We show that, in the case where the cavity resonance energy is close to the intersubband resonance energy of the QW, an inclusion of the cavity effect results in a significant enhancement of the optical intersubband nonlinearities of the QW system in the vicinity of the cavity resonance frequency. As a consequence, the saturation intensity of the QW structure is substantially lowered compared with the results in the absence of the microcavity effect. The cavity effect on the nonlinear absorption lineshape of the QW is also discussed. q 1998 Published by Elsevier Science B.V. PACS: 78.66.–w; 42.50.Hz; 42.50.Ne Keywords: Nonlinear optical absorption; Quantum well; Microcavity effect
1. Introduction In the past decade the linear and nonlinear optical properties of semiconductor quantum well ŽQW. structures associated with intersubband transitions have attracted a great deal of attention w1–12x because of their potential infrared optoelectronic device applications. Particularly, it has been shown both theoretically w1,2,4x and experimentally w3x that the optical intersubband absorption of a two-level quantum well system decreases as the intensity of the incident light is increased. This is known as the optical saturation of the intersubband absorption which stems from the light-induced population redistribution between the two electronic subbands. Recently it has also been theoretically demonstrated that the intersubband resonance energy is downwards shifted and the absorption line shape is distorted with an increase in the excitation inten-
1
E-mail: [email protected].
sity because of the so-called depolarization Žor local-field. effect w5,6x. Such a light-induced redshift of the intersubband infrared absorption peak has been experimentally observed in a single Al 0.3 Ga 0.7 AsrGaAs square QW w9x. Nevertheless, the previous work w1–6,9x has mainly focused on the nonlinear optical intersubband response of the free-standing QW structure. The influence of QW geometric structures such as multiple interfaces and surfaces on the nonlinear optical absorption spectrum has not been addressed. It is known that these factors affect significantly the linear optical absorption spectrum of a multiple quantum well structure through multiple reflections of light in the system w10–12x. In this paper, we present a theoretical investigation of the microcavity effect on the nonlinear intersubband absorption of a QW system. The starting point of our theory is the microscopic retarded Maxwell-Lorentz equation combined with the light-induced current densities associated with intersubband transitions in the QW system. The field-induced current density is derived on the basis of the
0030-4018r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 7 . 0 0 4 8 8 - 4
280
A. Liu r Optics Communications 147 (1998) 279–284
'
Fig. 1. Schematic diagram showing the reflection Ž Er . of a p-polarized incident electromagnetic wave Ž Ei . from a QW-embedded microcavity. The Cartesian xyz coordinate system used in describing the QW structure is also indicated.
light is q Is Ž vrc0 . ´ 1 sin u , c 0 being the light velocity in vacuum. If we limit ourselves to the case where the QW has only two bound subbands involved in optical intersubband transitions within the conduction band, it is reasonable to assume that the light-induced current densities associated with intersubband transitions are localized within an effective width d which is somewhat larger than the thickness of the well layer, because the wave functions of electrons are exponentially decaying functions of z outside the well layer. Along the z axis the effective well layer is positioned at z s a, where a denotes the thickness of the barrier layer between the DBR and the well layer. In the random-phase approximation ŽRPA., and taking properly into account the nonparabolic dispersion of the electronic subbands of the QW, we find that the dominant components of the current density w J Ž v , z .x induced by the local electric field is given, in the long-wavelength and low-temperature limits, by Jz Ž v , z . s
one-body density-matrix formalism in the rotating-wave approximation ŽRWA. w13x. Thus in our theory the lightinduced electronic population redistribution between the subbands is taken into account in a nonperturbative manner. Using a combined transfer-matrix and Green’s-function method w12x, we solve the nonlinear wave equation exactly. In turn we derive a rigorous expression for the nonlinear optical reflectionrabsorption coefficient of a QW-embedded microcavity. We show that an inclusion of the cavity effect results in a significant enhancement of the intersubband optical nonlinearities of the QW system and a notable reduction in the saturation intensity. Physically, it is attributed to the cavity-induced enhancement of the local field across the QW.
ie 2 " 2 2pv Ž m ) .
=
kF
H0
2
NF Ž z .
E21Ž k I . D Ž k I . k I d k I
w " Ž v q irt 2 . x 2 y w E21Ž k I . x 2
,
Ž1.
where D Ž k I. s 1 q
t 1e2 < N < 2 t 2 Ž m )v .
2 y1
"2 =
Ns
aq d
Ha
w E21Ž k I . y " v x 2 q Ž "rt 2 . 2
F Ž z . Ez Ž z . d z ,
,
Ž2. Ž3.
and 2. Theory We consider a n-type doped QW sandwiched between a semi-infinite medium of a dielectric constant ´ 3Ž v . and a distributed Bragg reflector ŽDBR. constructed from Nm periods of two alternating component media having dielectric constants of ´ 1Ž v . and ´4Ž v .. A schematic illustration of this QW-embedded microcavity structure is given in Fig. 1. The barrier layers are characterized by a dielectric constant of ´ 2 Ž v .. The DBR is limited by a semi-infinitely extended medium 1 serving as a prism. In a Cartesian xyz coordinate system, it is assumed that z axis points along the direction normal to the interfaces of the QW structure, and that the media 1r2 and media 2r3 boundaries are positioned at z s 0 and z s L, respectively, as shown in Fig. 1. Let us now assume that a strong p-polarized electromagnetic wave of angular frequency v is incident at an angle u from the prism onto the QW structure Žsee Fig. 1.. Thus the parallel Ž x . component of the wave vector of
F Ž z . s c 1Ž z .
dc2Ž z .
y c2Ž z .
d c 1Ž z .
. Ž4. dz dz In the equations above m ) is the effective mass of electrons, and t 1 and t 2 are phenomenological intrasubband and intersubband relaxation times, respectively. E21Ž k I . ' . Žj E2 Ž k I . y E1Ž k I ., where Ej Ž k I . s E j q " 2 k I2 rŽ2 m ) Ij s 1, 2. gives the subband energy dispersion of the QW, Ž j s 1, 2. being the so-called parallel effective elecm) Ij tron mass of the quantum well, which originates in the conduction band nonparabolicity effect. In the present paper we have adopted a simple energy-dependent effective-mass scheme proposed by Ekenberg w14x to take into account the band nonparabolicity. The upper limit of integration, k F , appearing in Eq. Ž1. is the Fermi wave vector of the QW in the absence of the light field, which is determined from k F s 2 m ) I1 Ž EF y E1 . r", EF being the Fermi energy. The single-particle envelope wave functions w c 1Ž z . and c 2 Ž z .x and the corresponding energy eigenvalues Ž E1 and E2 . are calculated selfconsistently by solving the coupled one-band effective-mass Schrodinger equation ¨
(
A. Liu r Optics Communications 147 (1998) 279–284
and Poisson equation w15x, with taking into account the exchange and correlation effects in the local density approximation w16x. Note that Ez Ž z . appearing in Eq. Ž3. denotes the z component of the local electric field across the effective well layer, which should be determined selfconsistently from the wave equation. Incorporating the field-induced current density given in Eq. Ž1. into the Maxwell equation, and using a combined transfer-matrix and Green’s-function method described in detail in my previous work w12x, we find the so far unknown quantity N is related to the x component of the amplitude of the incident wave, A 0 , by W A0 Ns , Ž5. 2 1 y K˜ a Ž v , < N < . T11 y T12 r p
a Ž v ,< N < 2 .
r p s r p0 q
1 y K˜ a Ž v , < N < 2 .
WX ,
r 12 q r 23 exp Ž 2 iq H 2 L .
r p0 s
,
1 q r12 r 23 exp Ž 2 iq H 2 L .
ri j s
qH i ´ j Ž v . y qH j ´i Ž v . qH i ´ j Ž v . q qH j ´i Ž v . 2
q H i s Ž vrc0 . ´ i Ž v . y q I2 2
a Ž v ,< N <
2
m0 e "
.s
1r2
i s 1,2; j s 2,3 , ,
Ž8.
i s 1,2,3,4 ,
Ž9.
2
E21Ž k I . D Ž k I . k I d k I
kF
H0
,
Ž7.
2
2p Ž m ) .
=
Ž6.
w " Ž v q irt 2 . x
2
y w E21Ž k I . x
,
2
Ž 10. K˜ s K y UF Ž 0 . y VF Ž L . , Ks
c0
2
1
ž / v
aq d
´2
q I2
q
Ha
wF Ž z . x 2 d z
aq d
2 iq H 2
Ž 11.
HHa
F Ž z . exp Ž iq H 2 < z y z < .
=F Ž zX . d z d zX ,
F Ž0. s
q I2
2
c0
ž / v
2 i ´ 2 qH 2
aq d
Ha
v
=
2 i ´ 2 qH 2
aq d
Ha
Us
q I2
2
c0
ž /
1 q r 12 r 23 exp Ž 2 iq H 2 L .
Wsy
qI
exp Ž iq H2 z . F Ž z . d z ,
exp Ž iq H 2 L .
exp Ž yiq H2 z . F Ž z . d z ,
Ž 14 .
r 12 1 q r12 r 23 exp Ž 2 iq H 2 L . = w Sqq r 23 exp Ž 2 iq H 2 L . Sy x ,
Ž 15.
w r12 Sqy Sy x ,
Ž 16.
1 y r 12
q H 2 1 q r 12 r 23 exp Ž 2 iq H 2 L .
= w Sqq r 23 exp Ž 2 iq H 2 L . Sy x , qH2
Xsy
Ž 17.
1 q r 12
q I 1 q r 12 r 23 exp Ž 2 iq H 2 L .
= w F Ž 0 . q r 23 exp Ž iq H 2 L . F Ž L . x , S "s
aq d
Ha
Ž 18.
F Ž z . exp Ž "iq H 2 z . d z .
Ž 19.
The amplitude reflection coefficient Ž r˜p . of the QW structure, which is the amplitude ratio between the reflected and incident waves inside the prism Žcf. Fig. 1., is given by w12x r˜p s
T22 r p y T21 T11 y T12 r p
.
Ž 20 .
In Eqs. Ž5. and Ž20., Ti j is the ij element of the 2 = 2 matrix T, which is related to the transfer matrix M for the DBR by T s M Nm .
Ž 21.
For a p-polarized light, the explicit expressions for the transfer-matrix elements can be found, for example, in Ref. w12x. From Eq. Ž5. it is evident that the quantity < N < 2 has to be solved from the equation W
1
1 y K˜ a Ž v , < N < 2 . T11 y T12 r p
2
2 cos 2u I0
'
´ 0 c0 ´ 1
,
Ž 22.
Ž 12.
Ž 13. F Ž L. s
r 23 exp Ž iq H 2 L .
Vs
< N <2s
X
281
where I0 denotes the optical intensity of the incident light. Thus for a given I0 one can determine < N < 2 from Eq. Ž22., in turn calculate the reflection coefficient Ž r˜p .. When the light is totally reflected at the media 2r3 interface, which is the case for the QW system considered in this paper, the optical absorption coefficient of the QW structure is simply given by w6,12x A p s 1 y < r˜p < 2 .
Ž 23.
In Section 3 we shall use the above equations to calculate the optical intersubband absorption spectra of a QW inside a cavity at different excitation intensities and for various values of the cavity length.
282
A. Liu r Optics Communications 147 (1998) 279–284
3. Numerical results and discussion For a Al 0.33 Ga 0.67 AsrGaAs QW placed between vacuum Ž ´ 3 s 1. and a GaAsrAlAs DBR followed by a GaAs prism, we calculated the nonlinear optical absorption coefficient, A p , as a function of the frequency. ŽSince in this paper we consider only the optical response of the QW system to the pump field, the light absorption defined in Eq. Ž23. is thus for the pump beam.. In our calculations the following parameters are employed: the barrier height is 256 meV, m ) s 0.067m 0 , ´ 1 f 10.9, ´ 2 f 10.0, and ´4 f 8.4 w12x. The static dielectric constant that enters the Poisson equation is ´ r s 13.0. For our QW structure, we assume that the well layer is uniformly n-type doped. Without special notification, an ionized donor concentration of 2 = 10 18 cmy3 is used in our calculations. The well ˚ Using these parameters, we find width is taken to be 80 A. the energy spacing between the two lowest-lying subbands of the QW is E2 y E1 s 116.2 meV. The angle of incidence is u s 558. The thicknesses of GaAs and AlAs layers forming the DBR are L1 s 1.289 and L4 s 2.342 mm, respectively. These values of the GaAs and AlAs thicknesses are so chosen that q H1 L1 s q H 4 L 4 s pr2 at the central frequency of 127 meV. For the intrasubband and intersubband relaxation times, we choose "rt 1 s "rt 2 s 10 meV, unless specified particularly. In addition, we assume that the QW is placed at the middle of the cavity. In Fig. 2 we show the optical absorption spectra of the QW system at a cavity length of L s 2.978 mm for different light intensities, i.e. I0 s 0 Žcurve 1., 50 Žcurve 2., 100 Žcurve 3., 200 Žcurve 4. and 300 kWrcm2 Žcurve 5.. In all calculations the period number of the DBR is Nm s 4. It should be noted that at the above value of the cavity length one of the cavity resonance energies exactly matches the local-field-shifted intersubband resonance of the free-standing QW. Therefore, in this case one can
Fig. 2. Optical absorption Ž A p . spectra of a QW inside a cavity at a fixed cavity length of L s 2.978 mm for different light intensities, i.e. I0 s 0 Žcurve 1., 50 Žcurve 2., 100 Žcurve 3., 200 Žcurve 4., and 300 kWrcm2 Žcurve 5..
Fig. 3. Optical absorption Ž A p . spectra of a QW in front of a light-total-reflection dielectric mirror for different light intensities, i.e. I0 s 0 Žcurve 1., 50 Žcurve 2., 100 Žcurve 3., 200 Žcurve 4., and 300 kWrcm2 Žcurve 5.. The distance between the center of the well layer and the mirror is 1.489 mm.
expect that the intersubband-excitation-cavity coupling in our system is strong w12x. For comparison, we also plot in Fig. 3 the optical absorption spectra of the QW for the same values of the light intensities used in Fig. 2 but in the absence of the microcavity effect. In calculating Fig. 3, we still used Eqs. Ž6. – Ž23. but reset ´ 1 s ´ 2 s 10.0, Nm s 0, and u s 58.7848 which is precisely the refraction angle when the light is incident at the angle of 558 from the prism into the barrier layer. Actually, this is just the case in which the QW is placed in front of the light-total-reflection dielectric mirror, with the distance between the center of the well layer and the mirror being Lr2 s 1.489 mm. It appears from Fig. 2 that, at zero intensity Žsee curve 1. the optical absorption peak that is essentially located at the coupled cavity-intersubband-transition resonance energy w12x is rather narrow Žwith a FWHM linewidth of ; 4 meV. although a large subband dephasing Ž10 meV. is used in our calculations and the contribution to the line broadening from the conduction band nonparabolicity is taken into account as well. ŽWe remark that because of the local-field Ždepolarization. effect the band nonparabolicity line broadening is rather small w17,18x.. Such narrowing of the linear absorption line is solely due to the cavity effect, since one can clearly see from curve 1 in Fig. 3 that without the cavity effect the absorption linewidth of the QW is much larger. In a comparison of curves labelled 1 in Figs. 2 and 3, we also note that the cavity effect enhances significantly the peak absorption coefficient Žnote the different scales used in Figs. 2 and 3.. Returning to Fig. 2, one sees that, when the excitation intensity is increased Žsee curves 2 and 3., the peak absorption coefficient is significantly reduced. When the intensity of the incident light is further increased Žsee curves 4 and 5., the
A. Liu r Optics Communications 147 (1998) 279–284
Fig. 4. Optical absorption Ž A p . spectra of a QW inside a cavity at a fixed cavity length of L s 2.978 mm for different values of Nm , viz. Nm s 2 Žcurve 1., 4 Žcurve 2., and 6 Žcurve 3.. The light intensity is I0 s 300 kWrcm2 .
FWHM linewidth of the absorption spectra becomes larger and the absorption peak is apparently asymmetric. In a comparison of Figs. 2 and 3, one also notices that the cavity effect lowers the saturation intensity Ž Is . of the QW significantly. This is due to the cavity-induced enhancement of the local field across the QW. By the definition of the saturation intensity Žfor resonance frequencies., i.e. A p Ž Is . Peak s A p Ž 0 . Peak ln 2 w3x, we find that in the case of Fig. 2 Is f 87 kWrcm2 , whereas Is f 520 kWrcm2 in the case of Fig. 3. The latter value of Is is comparable to the reported data w3x. Here we have to stress that the saturation intensity defined above is valid only for the infinitely-extended medium with standard saturable absorption coefficient wA Ž1 q I0rIs .y1 x. Also, in principle, there is no justification to define a saturation intensity in a usual way w4,5x in our theory in which the influence of the QW surroundings on the nonlinear absorption coefficient is included. Nevertheless, it may be viewed as a reference to the magnitude of the optical nonlinearity of the QW system. In Fig. 4 we compare the absorption spectra of the QW system for different values of the period number Nm , namely Nm s 2 Žcurve 1., 4 Žcurve 2., and 6 Žcurve 3.. In the calculation, a light intensity of I0 s 300 kWrcm2 is used. Because in the vicinity of the central frequency the optical reflection coefficient of the DBR increases with increasing Nm w12x, one may expect that the cavity effect becomes more important when Nm is larger. From Fig. 4 one sees that, when Nm is small Žsee curve 1., the absorption peak is comparatively wide and its peak value is small. However, when Nm becomes larger Žsee curves 2 and 3., the cavity effect becomes stronger, and consequently leads to a substantial narrowing of the absorption line and a significant increase in the peak absorbance.
283
Fig. 5. Optical absorption Ž A p . spectra of a QW inside a cavity for different cavity lengths, namely L s 2.383 Žcurve 1., 2.978 Žcurve 2., 3.574 Žcurve 3., and 4.170 mm Žcurve 4.. The light intensity is I0 s 300 kWrcm2 and the period number is Nm s 4.
In order to see how the cavity length influences the nonlinear absorption coefficient, we show in Fig. 5 the optical absorption spectra of our QW system for different cavity lengths at the optical intensity of I0 s 300 kWrcm2 and for Nm s 4. It appears from Fig. 5 that, as the cavity length is increased, the coupled cavity-intersubband-transition resonance energy is decreased, as one would expect w12x. One can also see from Fig. 5 that the cavity length has a strong influence on the nonlinear optical absorption lineshape as well. At this stage we would like to mention that all results presented above were obtained with an assumption that t 1 s t 2 . One would expect, however, that a variation of the relaxation times affects significantly the nonlinear absorption spectrum. Fig. 6 shows the nonlinear absorption spectra of our QW-embedded cavity at a light intensity of
Fig. 6. Optical absorption Ž A p . spectra of a QW inside a cavity at a fixed cavity length of L s 2.978 mm for different intersubband relaxation times, i.e. " rt 2 s 4.0 Žcurve 1., 6.0 Žcurve 2., and 8.0 meV Žcurve 3.. The incident intensity is I0 s150 kWrcm2 .
284
A. Liu r Optics Communications 147 (1998) 279–284
the spectral lineshape is somewhat dependent on the electron density because the many-body effect leads to a slight change in the intersubband separation as well as the parallel effective masses.
4. Conclusion
Fig. 7. Optical absorption Ž A p . spectra of a QW inside a cavity at a fixed cavity length of L s 2.978 mm for different sheet electron densities Žin units of 10 12 cmy2 ., i.e. 0.1 Žcurve 1., 0.4 Žcurve 2.,1.0 Žcurve 3., and 2.0 Žcurve 4.. The light intensity is I0 s 300 kWrcm2 .
I0 s 150 kWrcm2 with different intersubband relaxation times, i.e. "rt 2 s 4.0 Žcurve 1., 6.0 Žcurve 2., and 8.0 meV Žcurve 3.. The other parameters used in calculating Fig. 6 are given as follows: "rt 1 s 10.0 meV, Nm s 4, L s 2.978 mm, and the donor concentration is 1.5 = 10 18 cmy3. From Fig. 6 we see that varying the intersubband relaxation time leads to an obvious change in not only the magnitude of the absorption coefficient but also the spectral lineshape. Finally we would like to discuss the dependence of the nonlinear absorption spectrum of our QW structure on the electron density. In the linear response of a QW without the cavity, it is well known that the absorption peak is blueshifted as the electron density increases because of the depolarization or local-field effect Žsee e.g. Ref. w18x and references therein.. In the presence of a cavity, the QWcavity interaction would change the electron-density dependence of the nonlinear absorption spectrum. In Fig. 7 we show the absorption spectra of our QW structure at a light intensity of I0 s 300 kWrcm2 for different sheet electron densities Žin units of 10 12 cmy2 ., namely, 0.1 Žcurve 1., 0.4 Žcurve 2., 1.0 Žcurve 3., and 2.0 Žcurve 4.. The remaining parameters used in Fig. 7 are "rt 1 s "rt 2 s 10.0 meV, Nm s 4, and L s 2.978 mm. It is clear from Fig. 7 that the peak absorption increases with increasing sheet electron density as one would expect. However, the peak location is almost not changed, and very close to the cavity resonance energy. One also notes from Fig. 7 that
In a semi-classical nonlocal local-field approach, we have studied the microcavity effect on the nonlinear optical intersubband absorption of a QW system. We found that, when the cavity length is so chosen that the cavity resonance energy is close to the intersubband resonance energy, and the period number Nm of the DBR is large enough, the cavity effect results in a significant enhancement of the optical nonlinearities of the QW system because of the cavity-induced enhancement of the local field. As a result, the saturation intensity of the QW is lowered when the strong cavity effect is included. It is also demonstrated that the nonlinear optical absorption spectrum is strongly dependent on the cavity length and the cavity quality Ž Nm .. References w1x D. Ahn, S.L. Chuang, J. Appl. Phys. 62 Ž1987. 3052. w2x D. Ahn, S.L. Chuang, IEEE J. Quantum Electron. QE-23 Ž1987. 2196. w3x F.H. Julien, J.M. Lourtioz, N. Herschkorn, D. Delacourt, J.P. Pocholle, M. Papuchon, R. Planel, G. Le Roux, Appl. Phys. Lett. 53 Ž1988. 116. w4x E.J. Roan, S.L. Chuang, J. Appl. Phys. 69 Ž1991. 3249. w5x M. Zaluzny, Phys. Rev. B 47 Ž1993. 3995. w6x A. Liu, O. Keller, Phys. Scr. 52 Ž1995. 116. w7x A. Liu, O. Keller, Optics Comm. 120 Ž1995. 171. w8x G.-Z. Zhao, S.-H. Pan, Solid State Commun. 99 Ž1996. 595. w9x K. Craig, B. Galdrikian, J.N. Heyman, A.G. Markelz, J.B. Williams, M.S. Sherwin, K. Campman, P.F. Hopkins, A.C. Gossard, Phys. Rev. Lett. 76 Ž1996. 2382. w10x A. Liu, J. Appl. Phys. 80 Ž1996. 1928. w11x J.Y. Duboz, J. Appl. Phys. 80 Ž1996. 5432. w12x A. Liu, Phys. Rev. B 55 Ž1997. 7101. w13x Y.R. Shen, The Principles of Nonlinear Optics, Wiley, New York, 1984. w14x U. Ekenberg, Phys. Rev. B 40 Ž1989. 7714. w15x See, for example, M.E. Lazzouni, L.J. Sham, Phys. Rev. B 48 Ž1993. 8948. w16x O. Gunnarsson, B.I. Lundqvist, Phys. Rev. B 13 Ž1976. 4274. w17x M. Zaluzny, Phys. Rev. B 43 Ž1991. 4511. w18x A. Liu, Phys. Rev. B 55 Ž1997. 7796.
Optics Communications 147 Ž1998. 279–284
Microcavity effect on the nonlinear optical intersubband absorption in semiconductor quantum wells Ansheng Liu
1
Institute of Physics, Aalborg UniÕersity, Pontoppidanstræde 103, DK-9220 Aalborg Øst, Denmark Received 26 March 1997; revised 4 August 1997; accepted 27 August 1997
Abstract Based on a semi-classical local-field formalism in which retardation effect in connection with the electromagnetic propagation and nonlocal effect in the optical response of confined electron systems are included, we calculated the nonlinear optical intersubband absorption coefficient of a quantum well ŽQW. inside a microcavity. We show that, in the case where the cavity resonance energy is close to the intersubband resonance energy of the QW, an inclusion of the cavity effect results in a significant enhancement of the optical intersubband nonlinearities of the QW system in the vicinity of the cavity resonance frequency. As a consequence, the saturation intensity of the QW structure is substantially lowered compared with the results in the absence of the microcavity effect. The cavity effect on the nonlinear absorption lineshape of the QW is also discussed. q 1998 Published by Elsevier Science B.V. PACS: 78.66.–w; 42.50.Hz; 42.50.Ne Keywords: Nonlinear optical absorption; Quantum well; Microcavity effect
1. Introduction In the past decade the linear and nonlinear optical properties of semiconductor quantum well ŽQW. structures associated with intersubband transitions have attracted a great deal of attention w1–12x because of their potential infrared optoelectronic device applications. Particularly, it has been shown both theoretically w1,2,4x and experimentally w3x that the optical intersubband absorption of a two-level quantum well system decreases as the intensity of the incident light is increased. This is known as the optical saturation of the intersubband absorption which stems from the light-induced population redistribution between the two electronic subbands. Recently it has also been theoretically demonstrated that the intersubband resonance energy is downwards shifted and the absorption line shape is distorted with an increase in the excitation inten-
1
E-mail: [email protected].
sity because of the so-called depolarization Žor local-field. effect w5,6x. Such a light-induced redshift of the intersubband infrared absorption peak has been experimentally observed in a single Al 0.3 Ga 0.7 AsrGaAs square QW w9x. Nevertheless, the previous work w1–6,9x has mainly focused on the nonlinear optical intersubband response of the free-standing QW structure. The influence of QW geometric structures such as multiple interfaces and surfaces on the nonlinear optical absorption spectrum has not been addressed. It is known that these factors affect significantly the linear optical absorption spectrum of a multiple quantum well structure through multiple reflections of light in the system w10–12x. In this paper, we present a theoretical investigation of the microcavity effect on the nonlinear intersubband absorption of a QW system. The starting point of our theory is the microscopic retarded Maxwell-Lorentz equation combined with the light-induced current densities associated with intersubband transitions in the QW system. The field-induced current density is derived on the basis of the
0030-4018r98r$19.00 q 1998 Published by Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 7 . 0 0 4 8 8 - 4
280
A. Liu r Optics Communications 147 (1998) 279–284
'
Fig. 1. Schematic diagram showing the reflection Ž Er . of a p-polarized incident electromagnetic wave Ž Ei . from a QW-embedded microcavity. The Cartesian xyz coordinate system used in describing the QW structure is also indicated.
light is q Is Ž vrc0 . ´ 1 sin u , c 0 being the light velocity in vacuum. If we limit ourselves to the case where the QW has only two bound subbands involved in optical intersubband transitions within the conduction band, it is reasonable to assume that the light-induced current densities associated with intersubband transitions are localized within an effective width d which is somewhat larger than the thickness of the well layer, because the wave functions of electrons are exponentially decaying functions of z outside the well layer. Along the z axis the effective well layer is positioned at z s a, where a denotes the thickness of the barrier layer between the DBR and the well layer. In the random-phase approximation ŽRPA., and taking properly into account the nonparabolic dispersion of the electronic subbands of the QW, we find that the dominant components of the current density w J Ž v , z .x induced by the local electric field is given, in the long-wavelength and low-temperature limits, by Jz Ž v , z . s
one-body density-matrix formalism in the rotating-wave approximation ŽRWA. w13x. Thus in our theory the lightinduced electronic population redistribution between the subbands is taken into account in a nonperturbative manner. Using a combined transfer-matrix and Green’s-function method w12x, we solve the nonlinear wave equation exactly. In turn we derive a rigorous expression for the nonlinear optical reflectionrabsorption coefficient of a QW-embedded microcavity. We show that an inclusion of the cavity effect results in a significant enhancement of the intersubband optical nonlinearities of the QW system and a notable reduction in the saturation intensity. Physically, it is attributed to the cavity-induced enhancement of the local field across the QW.
ie 2 " 2 2pv Ž m ) .
=
kF
H0
2
NF Ž z .
E21Ž k I . D Ž k I . k I d k I
w " Ž v q irt 2 . x 2 y w E21Ž k I . x 2
,
Ž1.
where D Ž k I. s 1 q
t 1e2 < N < 2 t 2 Ž m )v .
2 y1
"2 =
Ns
aq d
Ha
w E21Ž k I . y " v x 2 q Ž "rt 2 . 2
F Ž z . Ez Ž z . d z ,
,
Ž2. Ž3.
and 2. Theory We consider a n-type doped QW sandwiched between a semi-infinite medium of a dielectric constant ´ 3Ž v . and a distributed Bragg reflector ŽDBR. constructed from Nm periods of two alternating component media having dielectric constants of ´ 1Ž v . and ´4Ž v .. A schematic illustration of this QW-embedded microcavity structure is given in Fig. 1. The barrier layers are characterized by a dielectric constant of ´ 2 Ž v .. The DBR is limited by a semi-infinitely extended medium 1 serving as a prism. In a Cartesian xyz coordinate system, it is assumed that z axis points along the direction normal to the interfaces of the QW structure, and that the media 1r2 and media 2r3 boundaries are positioned at z s 0 and z s L, respectively, as shown in Fig. 1. Let us now assume that a strong p-polarized electromagnetic wave of angular frequency v is incident at an angle u from the prism onto the QW structure Žsee Fig. 1.. Thus the parallel Ž x . component of the wave vector of
F Ž z . s c 1Ž z .
dc2Ž z .
y c2Ž z .
d c 1Ž z .
. Ž4. dz dz In the equations above m ) is the effective mass of electrons, and t 1 and t 2 are phenomenological intrasubband and intersubband relaxation times, respectively. E21Ž k I . ' . Žj E2 Ž k I . y E1Ž k I ., where Ej Ž k I . s E j q " 2 k I2 rŽ2 m ) Ij s 1, 2. gives the subband energy dispersion of the QW, Ž j s 1, 2. being the so-called parallel effective elecm) Ij tron mass of the quantum well, which originates in the conduction band nonparabolicity effect. In the present paper we have adopted a simple energy-dependent effective-mass scheme proposed by Ekenberg w14x to take into account the band nonparabolicity. The upper limit of integration, k F , appearing in Eq. Ž1. is the Fermi wave vector of the QW in the absence of the light field, which is determined from k F s 2 m ) I1 Ž EF y E1 . r", EF being the Fermi energy. The single-particle envelope wave functions w c 1Ž z . and c 2 Ž z .x and the corresponding energy eigenvalues Ž E1 and E2 . are calculated selfconsistently by solving the coupled one-band effective-mass Schrodinger equation ¨
(
A. Liu r Optics Communications 147 (1998) 279–284
and Poisson equation w15x, with taking into account the exchange and correlation effects in the local density approximation w16x. Note that Ez Ž z . appearing in Eq. Ž3. denotes the z component of the local electric field across the effective well layer, which should be determined selfconsistently from the wave equation. Incorporating the field-induced current density given in Eq. Ž1. into the Maxwell equation, and using a combined transfer-matrix and Green’s-function method described in detail in my previous work w12x, we find the so far unknown quantity N is related to the x component of the amplitude of the incident wave, A 0 , by W A0 Ns , Ž5. 2 1 y K˜ a Ž v , < N < . T11 y T12 r p
a Ž v ,< N < 2 .
r p s r p0 q
1 y K˜ a Ž v , < N < 2 .
WX ,
r 12 q r 23 exp Ž 2 iq H 2 L .
r p0 s
,
1 q r12 r 23 exp Ž 2 iq H 2 L .
ri j s
qH i ´ j Ž v . y qH j ´i Ž v . qH i ´ j Ž v . q qH j ´i Ž v . 2
q H i s Ž vrc0 . ´ i Ž v . y q I2 2
a Ž v ,< N <
2
m0 e "
.s
1r2
i s 1,2; j s 2,3 , ,
Ž8.
i s 1,2,3,4 ,
Ž9.
2
E21Ž k I . D Ž k I . k I d k I
kF
H0
,
Ž7.
2
2p Ž m ) .
=
Ž6.
w " Ž v q irt 2 . x
2
y w E21Ž k I . x
,
2
Ž 10. K˜ s K y UF Ž 0 . y VF Ž L . , Ks
c0
2
1
ž / v
aq d
´2
q I2
q
Ha
wF Ž z . x 2 d z
aq d
2 iq H 2
Ž 11.
HHa
F Ž z . exp Ž iq H 2 < z y z < .
=F Ž zX . d z d zX ,
F Ž0. s
q I2
2
c0
ž / v
2 i ´ 2 qH 2
aq d
Ha
v
=
2 i ´ 2 qH 2
aq d
Ha
Us
q I2
2
c0
ž /
1 q r 12 r 23 exp Ž 2 iq H 2 L .
Wsy
qI
exp Ž iq H2 z . F Ž z . d z ,
exp Ž iq H 2 L .
exp Ž yiq H2 z . F Ž z . d z ,
Ž 14 .
r 12 1 q r12 r 23 exp Ž 2 iq H 2 L . = w Sqq r 23 exp Ž 2 iq H 2 L . Sy x ,
Ž 15.
w r12 Sqy Sy x ,
Ž 16.
1 y r 12
q H 2 1 q r 12 r 23 exp Ž 2 iq H 2 L .
= w Sqq r 23 exp Ž 2 iq H 2 L . Sy x , qH2
Xsy
Ž 17.
1 q r 12
q I 1 q r 12 r 23 exp Ž 2 iq H 2 L .
= w F Ž 0 . q r 23 exp Ž iq H 2 L . F Ž L . x , S "s
aq d
Ha
Ž 18.
F Ž z . exp Ž "iq H 2 z . d z .
Ž 19.
The amplitude reflection coefficient Ž r˜p . of the QW structure, which is the amplitude ratio between the reflected and incident waves inside the prism Žcf. Fig. 1., is given by w12x r˜p s
T22 r p y T21 T11 y T12 r p
.
Ž 20 .
In Eqs. Ž5. and Ž20., Ti j is the ij element of the 2 = 2 matrix T, which is related to the transfer matrix M for the DBR by T s M Nm .
Ž 21.
For a p-polarized light, the explicit expressions for the transfer-matrix elements can be found, for example, in Ref. w12x. From Eq. Ž5. it is evident that the quantity < N < 2 has to be solved from the equation W
1
1 y K˜ a Ž v , < N < 2 . T11 y T12 r p
2
2 cos 2u I0
'
´ 0 c0 ´ 1
,
Ž 22.
Ž 12.
Ž 13. F Ž L. s
r 23 exp Ž iq H 2 L .
Vs
< N <2s
X
281
where I0 denotes the optical intensity of the incident light. Thus for a given I0 one can determine < N < 2 from Eq. Ž22., in turn calculate the reflection coefficient Ž r˜p .. When the light is totally reflected at the media 2r3 interface, which is the case for the QW system considered in this paper, the optical absorption coefficient of the QW structure is simply given by w6,12x A p s 1 y < r˜p < 2 .
Ž 23.
In Section 3 we shall use the above equations to calculate the optical intersubband absorption spectra of a QW inside a cavity at different excitation intensities and for various values of the cavity length.
282
A. Liu r Optics Communications 147 (1998) 279–284
3. Numerical results and discussion For a Al 0.33 Ga 0.67 AsrGaAs QW placed between vacuum Ž ´ 3 s 1. and a GaAsrAlAs DBR followed by a GaAs prism, we calculated the nonlinear optical absorption coefficient, A p , as a function of the frequency. ŽSince in this paper we consider only the optical response of the QW system to the pump field, the light absorption defined in Eq. Ž23. is thus for the pump beam.. In our calculations the following parameters are employed: the barrier height is 256 meV, m ) s 0.067m 0 , ´ 1 f 10.9, ´ 2 f 10.0, and ´4 f 8.4 w12x. The static dielectric constant that enters the Poisson equation is ´ r s 13.0. For our QW structure, we assume that the well layer is uniformly n-type doped. Without special notification, an ionized donor concentration of 2 = 10 18 cmy3 is used in our calculations. The well ˚ Using these parameters, we find width is taken to be 80 A. the energy spacing between the two lowest-lying subbands of the QW is E2 y E1 s 116.2 meV. The angle of incidence is u s 558. The thicknesses of GaAs and AlAs layers forming the DBR are L1 s 1.289 and L4 s 2.342 mm, respectively. These values of the GaAs and AlAs thicknesses are so chosen that q H1 L1 s q H 4 L 4 s pr2 at the central frequency of 127 meV. For the intrasubband and intersubband relaxation times, we choose "rt 1 s "rt 2 s 10 meV, unless specified particularly. In addition, we assume that the QW is placed at the middle of the cavity. In Fig. 2 we show the optical absorption spectra of the QW system at a cavity length of L s 2.978 mm for different light intensities, i.e. I0 s 0 Žcurve 1., 50 Žcurve 2., 100 Žcurve 3., 200 Žcurve 4. and 300 kWrcm2 Žcurve 5.. In all calculations the period number of the DBR is Nm s 4. It should be noted that at the above value of the cavity length one of the cavity resonance energies exactly matches the local-field-shifted intersubband resonance of the free-standing QW. Therefore, in this case one can
Fig. 2. Optical absorption Ž A p . spectra of a QW inside a cavity at a fixed cavity length of L s 2.978 mm for different light intensities, i.e. I0 s 0 Žcurve 1., 50 Žcurve 2., 100 Žcurve 3., 200 Žcurve 4., and 300 kWrcm2 Žcurve 5..
Fig. 3. Optical absorption Ž A p . spectra of a QW in front of a light-total-reflection dielectric mirror for different light intensities, i.e. I0 s 0 Žcurve 1., 50 Žcurve 2., 100 Žcurve 3., 200 Žcurve 4., and 300 kWrcm2 Žcurve 5.. The distance between the center of the well layer and the mirror is 1.489 mm.
expect that the intersubband-excitation-cavity coupling in our system is strong w12x. For comparison, we also plot in Fig. 3 the optical absorption spectra of the QW for the same values of the light intensities used in Fig. 2 but in the absence of the microcavity effect. In calculating Fig. 3, we still used Eqs. Ž6. – Ž23. but reset ´ 1 s ´ 2 s 10.0, Nm s 0, and u s 58.7848 which is precisely the refraction angle when the light is incident at the angle of 558 from the prism into the barrier layer. Actually, this is just the case in which the QW is placed in front of the light-total-reflection dielectric mirror, with the distance between the center of the well layer and the mirror being Lr2 s 1.489 mm. It appears from Fig. 2 that, at zero intensity Žsee curve 1. the optical absorption peak that is essentially located at the coupled cavity-intersubband-transition resonance energy w12x is rather narrow Žwith a FWHM linewidth of ; 4 meV. although a large subband dephasing Ž10 meV. is used in our calculations and the contribution to the line broadening from the conduction band nonparabolicity is taken into account as well. ŽWe remark that because of the local-field Ždepolarization. effect the band nonparabolicity line broadening is rather small w17,18x.. Such narrowing of the linear absorption line is solely due to the cavity effect, since one can clearly see from curve 1 in Fig. 3 that without the cavity effect the absorption linewidth of the QW is much larger. In a comparison of curves labelled 1 in Figs. 2 and 3, we also note that the cavity effect enhances significantly the peak absorption coefficient Žnote the different scales used in Figs. 2 and 3.. Returning to Fig. 2, one sees that, when the excitation intensity is increased Žsee curves 2 and 3., the peak absorption coefficient is significantly reduced. When the intensity of the incident light is further increased Žsee curves 4 and 5., the
A. Liu r Optics Communications 147 (1998) 279–284
Fig. 4. Optical absorption Ž A p . spectra of a QW inside a cavity at a fixed cavity length of L s 2.978 mm for different values of Nm , viz. Nm s 2 Žcurve 1., 4 Žcurve 2., and 6 Žcurve 3.. The light intensity is I0 s 300 kWrcm2 .
FWHM linewidth of the absorption spectra becomes larger and the absorption peak is apparently asymmetric. In a comparison of Figs. 2 and 3, one also notices that the cavity effect lowers the saturation intensity Ž Is . of the QW significantly. This is due to the cavity-induced enhancement of the local field across the QW. By the definition of the saturation intensity Žfor resonance frequencies., i.e. A p Ž Is . Peak s A p Ž 0 . Peak ln 2 w3x, we find that in the case of Fig. 2 Is f 87 kWrcm2 , whereas Is f 520 kWrcm2 in the case of Fig. 3. The latter value of Is is comparable to the reported data w3x. Here we have to stress that the saturation intensity defined above is valid only for the infinitely-extended medium with standard saturable absorption coefficient wA Ž1 q I0rIs .y1 x. Also, in principle, there is no justification to define a saturation intensity in a usual way w4,5x in our theory in which the influence of the QW surroundings on the nonlinear absorption coefficient is included. Nevertheless, it may be viewed as a reference to the magnitude of the optical nonlinearity of the QW system. In Fig. 4 we compare the absorption spectra of the QW system for different values of the period number Nm , namely Nm s 2 Žcurve 1., 4 Žcurve 2., and 6 Žcurve 3.. In the calculation, a light intensity of I0 s 300 kWrcm2 is used. Because in the vicinity of the central frequency the optical reflection coefficient of the DBR increases with increasing Nm w12x, one may expect that the cavity effect becomes more important when Nm is larger. From Fig. 4 one sees that, when Nm is small Žsee curve 1., the absorption peak is comparatively wide and its peak value is small. However, when Nm becomes larger Žsee curves 2 and 3., the cavity effect becomes stronger, and consequently leads to a substantial narrowing of the absorption line and a significant increase in the peak absorbance.
283
Fig. 5. Optical absorption Ž A p . spectra of a QW inside a cavity for different cavity lengths, namely L s 2.383 Žcurve 1., 2.978 Žcurve 2., 3.574 Žcurve 3., and 4.170 mm Žcurve 4.. The light intensity is I0 s 300 kWrcm2 and the period number is Nm s 4.
In order to see how the cavity length influences the nonlinear absorption coefficient, we show in Fig. 5 the optical absorption spectra of our QW system for different cavity lengths at the optical intensity of I0 s 300 kWrcm2 and for Nm s 4. It appears from Fig. 5 that, as the cavity length is increased, the coupled cavity-intersubband-transition resonance energy is decreased, as one would expect w12x. One can also see from Fig. 5 that the cavity length has a strong influence on the nonlinear optical absorption lineshape as well. At this stage we would like to mention that all results presented above were obtained with an assumption that t 1 s t 2 . One would expect, however, that a variation of the relaxation times affects significantly the nonlinear absorption spectrum. Fig. 6 shows the nonlinear absorption spectra of our QW-embedded cavity at a light intensity of
Fig. 6. Optical absorption Ž A p . spectra of a QW inside a cavity at a fixed cavity length of L s 2.978 mm for different intersubband relaxation times, i.e. " rt 2 s 4.0 Žcurve 1., 6.0 Žcurve 2., and 8.0 meV Žcurve 3.. The incident intensity is I0 s150 kWrcm2 .
284
A. Liu r Optics Communications 147 (1998) 279–284
the spectral lineshape is somewhat dependent on the electron density because the many-body effect leads to a slight change in the intersubband separation as well as the parallel effective masses.
4. Conclusion
Fig. 7. Optical absorption Ž A p . spectra of a QW inside a cavity at a fixed cavity length of L s 2.978 mm for different sheet electron densities Žin units of 10 12 cmy2 ., i.e. 0.1 Žcurve 1., 0.4 Žcurve 2.,1.0 Žcurve 3., and 2.0 Žcurve 4.. The light intensity is I0 s 300 kWrcm2 .
I0 s 150 kWrcm2 with different intersubband relaxation times, i.e. "rt 2 s 4.0 Žcurve 1., 6.0 Žcurve 2., and 8.0 meV Žcurve 3.. The other parameters used in calculating Fig. 6 are given as follows: "rt 1 s 10.0 meV, Nm s 4, L s 2.978 mm, and the donor concentration is 1.5 = 10 18 cmy3. From Fig. 6 we see that varying the intersubband relaxation time leads to an obvious change in not only the magnitude of the absorption coefficient but also the spectral lineshape. Finally we would like to discuss the dependence of the nonlinear absorption spectrum of our QW structure on the electron density. In the linear response of a QW without the cavity, it is well known that the absorption peak is blueshifted as the electron density increases because of the depolarization or local-field effect Žsee e.g. Ref. w18x and references therein.. In the presence of a cavity, the QWcavity interaction would change the electron-density dependence of the nonlinear absorption spectrum. In Fig. 7 we show the absorption spectra of our QW structure at a light intensity of I0 s 300 kWrcm2 for different sheet electron densities Žin units of 10 12 cmy2 ., namely, 0.1 Žcurve 1., 0.4 Žcurve 2., 1.0 Žcurve 3., and 2.0 Žcurve 4.. The remaining parameters used in Fig. 7 are "rt 1 s "rt 2 s 10.0 meV, Nm s 4, and L s 2.978 mm. It is clear from Fig. 7 that the peak absorption increases with increasing sheet electron density as one would expect. However, the peak location is almost not changed, and very close to the cavity resonance energy. One also notes from Fig. 7 that
In a semi-classical nonlocal local-field approach, we have studied the microcavity effect on the nonlinear optical intersubband absorption of a QW system. We found that, when the cavity length is so chosen that the cavity resonance energy is close to the intersubband resonance energy, and the period number Nm of the DBR is large enough, the cavity effect results in a significant enhancement of the optical nonlinearities of the QW system because of the cavity-induced enhancement of the local field. As a result, the saturation intensity of the QW is lowered when the strong cavity effect is included. It is also demonstrated that the nonlinear optical absorption spectrum is strongly dependent on the cavity length and the cavity quality Ž Nm .. References w1x D. Ahn, S.L. Chuang, J. Appl. Phys. 62 Ž1987. 3052. w2x D. Ahn, S.L. Chuang, IEEE J. Quantum Electron. QE-23 Ž1987. 2196. w3x F.H. Julien, J.M. Lourtioz, N. Herschkorn, D. Delacourt, J.P. Pocholle, M. Papuchon, R. Planel, G. Le Roux, Appl. Phys. Lett. 53 Ž1988. 116. w4x E.J. Roan, S.L. Chuang, J. Appl. Phys. 69 Ž1991. 3249. w5x M. Zaluzny, Phys. Rev. B 47 Ž1993. 3995. w6x A. Liu, O. Keller, Phys. Scr. 52 Ž1995. 116. w7x A. Liu, O. Keller, Optics Comm. 120 Ž1995. 171. w8x G.-Z. Zhao, S.-H. Pan, Solid State Commun. 99 Ž1996. 595. w9x K. Craig, B. Galdrikian, J.N. Heyman, A.G. Markelz, J.B. Williams, M.S. Sherwin, K. Campman, P.F. Hopkins, A.C. Gossard, Phys. Rev. Lett. 76 Ž1996. 2382. w10x A. Liu, J. Appl. Phys. 80 Ž1996. 1928. w11x J.Y. Duboz, J. Appl. Phys. 80 Ž1996. 5432. w12x A. Liu, Phys. Rev. B 55 Ž1997. 7101. w13x Y.R. Shen, The Principles of Nonlinear Optics, Wiley, New York, 1984. w14x U. Ekenberg, Phys. Rev. B 40 Ž1989. 7714. w15x See, for example, M.E. Lazzouni, L.J. Sham, Phys. Rev. B 48 Ž1993. 8948. w16x O. Gunnarsson, B.I. Lundqvist, Phys. Rev. B 13 Ž1976. 4274. w17x M. Zaluzny, Phys. Rev. B 43 Ž1991. 4511. w18x A. Liu, Phys. Rev. B 55 Ž1997. 7796.