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A hydrodynamic model calculation of linear and nonlinear optical response in an asymmetric parabolic quantum well
~
Solid State Communications, Vol. 88, No. i, PP. 5-7, 1993. Printed in Great Britain.
0038-1098/93 $6.00+.00 Pergamon Press Ltd
A hydrodynamic model calculation of linear and nonlinear optical response in a n a s y m m e t r i c p a r a b o l i c q u a n t u m well
W.L. Schaich
Department of Physics Indiana University Bloomington, IN ~7,~05 (Received 15 July 1993, acceptedfor publication 3 August 1993 by .4. H. MacDonald) We develop model calculations of the optical response of asymmetric parabolic quantum wells using a hydrodynamic description of the confined electrons' dynamics. Both first and second order responses are found, and the latter yield susceptibilities for second harmonic generation and optical rectification. The spectral dependence of each response coefficient shows a rich structure due to various plasmon standing wave resonances.
There has been a lot of interest recently in the nonlinear optical response of electrons trapped in quantum wells [1-8]. The usual theoretical model that has been used neglects most many body effects to concentrate on the coupling between a single subband below the Fermi level, EF, and several subbands above EF. By manipulating the well shape or by applying a static electric field, one can enhance matrix elements and make energy denominators (nearly) vanish, leading to very strong linear responses. In this note we examine the optical response for quantum wells with several partially occupied subbands. To avoid the complications of following in detail the interactions of the many possible single particle excitations, we use a hydrodynamic model which allows a reasonable treatment of the system's collective response. Resonant responses are still possible but now arise from standing plasma waves. This alternate physical picture has been found to represent well the infrared absorption of wide parabolic quantum wells (PQW) with a nearby grating coupler [9-11]. We apply it here to P Q W ' s in isolation, but which have an asymmetric profile (APQW). The asymmetry is needed to produce a nonzero second order response. Our theoretical treatment is adapted from earlier work for the surface of a semi-infinite metal [12,13]. Although the results found there have been shown to be quantitatively unreliable compared to treatments that allow for an extended tail of electron density into vacuum [14-16], the model should do better in P Q W systems since the confinement of electrons in non-neutral wells is significantly more stringent than at neutral metal (or well) surfaces [17]. W i t h both this rationale and the considerable simplifications of a hydrodynamic model over a full microscope evaluation in mind, we present
results here for the d-parameters that characterize the linear optical response [18,19] and the second order susceptibilities that describe second harmonic generation and optical rectification. Since the (bare) confining potential of the well is parabolic we can, via Poisson's equation, specify the A P Q W by a stepped profile of effective positive charge. In the simplest model, the equilibrium electron density will neutralize the effective positive charge over a bounded range. Assuming only one discontinuous j u m p in the effective positive charge, we take for the ground state electron density profile,
no(x) =
n n2
0 < x < w1 Wl < x < w
0
x
(1)
where w = wl + w2 and x is along the direction normal to the plane of the well. The specific choice for n l , n2, Wl and w2 as well as the values of the effective mass m*, host dielectric constant co, and incoherent scattering time 7- are taken from the characterization of an A P Q W that has been constructed by molecular beam epitaxy in a AlxGal_xAS system [10,20]. The equilibrium system described by Eq. (1) is subjected to a spatially constant electric field E(°ut), direction along 2. The superscript means t h a t E(°ut) is the field just outside the APQW, but inside the host semiconductor. It varies in time at the (first harmonic) frequency w. By solving the hydrodynamic equation of motion along with Poisson's equation and the equation of continuity in the separate density steps and matching solutions across boundaries, we determine the induced polarization through the well to first and second order in E (°ut). The boundary conditions on the polarization
6
ASYMMETRIC PARABOLIC QUANTUM WELL 200
.''"1 ....
....
Vol. 88 No. 1 I ....
I ....
I
'"1''"-
2
0
\i
0
1
-t
-800 v
-2
i!
-400
,, I .... -
0
20
40
60
80
....
-4
I00
0
I ....
20
v ( o m -1)
of the free charges are the same as used before [12,13] and the algebraic reduction is straightforward but too involved to be written out here. We only comment on the technical point of extracting second harmonic or static responses from second order quantities. The second order response is driven by the product of first order quantities. If one writes time variations in the form
Am(t) = Ame -im~t + A~neimwt
(2)
to describe the mth harmonic, then a product of AI(t)B1 (t) produces both a second harmonic response C2(t) with complex amplitude C2 = A1B1 and a static response Co(t) with complex amplitude Co = AIB~. Turning now to the numerical results [20], we begin in Fig.[1] with a plot of the only nontrivial d-parameter. Since the host dielectric function e0 has been taken to be real, the imaginary part of d describes absorption by the A P Q W [18,19]. The two strongest structures are slightly above the "bulk" plasma frequencies associated with nl and n2 at 51.5 cm -1 and 36.4 cm -1
i,,,,,,,,,
.... I' " 1 ' " l
I Q v
]
,, 0
2O
80
100
3. Plot of the susceptibility for second harmonic generation versus the frequency of the applied field. The solid (dashed) curve gives the real (imaginary) part of X~2) respectively. This upshift is due to spatial dispersion, which imposes an energy cost for making a standing plasma wave with a finite wavelength. In the nonlinear response one should expect structures near the frequencies of strong absorption and (for second harmonic generation) at half these frequencies. This expectation is borne out in Fig.[2], which plots the efficiency of optical rectification. The susceptibility is defined by X(2) = 2 Re
[I0
w P°(x)/lE~ °ut)]2
]
(3)
where Po(x) is the complex amplitude for the component of the static polarization in second order along and Re means "real part of". Even relatively weak structures in the imaginary part of d seem to lead to strong structures in X~ z).'^ Finally we show in Fig.[3] the complex susceptibility for second harmonic generation defined by
X(2)----fo~ -~" P2(x)/(E~l ut) 2
(4)
with P2(x) the second harmonic analogue of Po(x). There is a rich structure in the frequency dependence due to the interplay of resonances at w and 2w. The magnitude of X~ z)'^ shows a more modest variation but its phase grows by 151r across the frequency range of the figure.
m
0
I
I ....
60
I./(cm -1)
1. Plot of the d-parameter for the linear response to a fieM along the normal to the A P Q W versus the frequency of the applied field. The solid (dashed) curve gives the real (imaginary) part of d.
lO
I ....
40
40
60
80
100
(ore -I)
2. Plot of the susceptibility for optical rectification versus the frequency of the applied field.
The results in Figs. [1-3] are of course specific to our choice of parameters and one can wonder how much further the strong nonlinear effects can be enhanced. Since we do not have a transparent, analytic formula for the X'S it is not possible to systematically maximize them over the range of our parameters. We can say that the results shown here give a sense of the possible frequency variations that should be expected. By modest variations in the n's and w's we can produce peaks with an order of magnitude larger strength. Also the peak heights associated with first order resonances (i.e., peaks in the imaginary part of d) show a quadratic de-
Vol. 88, No. 1
ASYMMETRIC PARABOLIC QUANTUM WELL
pendence on 7, so more perfectly grown samples should yield large enhancements. On the theoretical side it would be interesting to extend the model to allow for a nearby grating coupler, which would allow the 2D plasmon that propagates in the plane of the PQW to contribute. The grating may also remove the need for the well to be asymmetric.
ACKNOWLEDGMENTS We thank Beth Gwinn for stimulating discussions. This work was supported in part by the National Science Foundation through grant DMR 89-03851. We also thank members of the Physics Department at Montana State University for their hospitality during a sabbatical visit.
REFERENCES 1. M.K. Gurnick and T.A. Detemple, J. Quantum Electron. 19, 791 (1983).
11. E.L. Yuh, E.G. Gwinn, and W.L. Schaich, unpublished.
2. J. Khurgin, Appl. Phys. Lett. 51, 2100 (1987).
12. M. Corvi and W.L. Schaich, 3688 (1986).
3. L. Tsang, D. Ahn, and S.L. Chuang, AppL Phys. Lett. 52, 697 (1988). 4. Z. lkonic, V. Milanovic, and D. Tjapkin, J. Quantum Electron. 25, 54 (1989).
Phys. Rev. B 33,
13. W.L. Schaich and A. Liebsch, Phys. Rev. B 37, 6187 (1987). 14. M. Weber and A. Liebsch, Phys. Rev. B 35, 4711 (1987); B36, 6411 (1987); B37, 1019(E) (1987).
5. M.M. Fejer, S.J.B. Yoo, R. L. Byer, A. Harwit, and S.J. Harris Jr., Phys. Rev. Lett. 62, 1041 (1989).
15. A. Chizmeshya and E. Zaremba, Phys. Rev. B 37, 2805 (1987).
6. P.F. Yuh and K.L. Wang, J. Appl. Phys. 65, 4377 (1989).
16. A. Liebsch and W.L. Sctmich, Phys. Rev. B 40, 5401 (1989).
7. E. Rosencher and Ph. Bois, 11315 (1991).
Phys. Rev. B 44,
8. C. Sirtori, F. Capasso, D.L. Sivco, A.L. Hutchinson, and A.Y. Cho, Appl. Phys. Left. 60, 151 (1992). 9. P.R. Pinsukanjana et al., Phys. Rev. B 46, 7284 (1992). 10. E.L. Yuh, E.G. Gwinn, P.F. Hopkins, and A.C. Gossard, unpublished.
17. J.F. Dobson, Phys. Rev. B 46, 10163 (1992). 18. W.L. Schaich and W. Chen, 10714 (1989).
Phys. Rev. B 39,
19. J. Zhang, S.E. Ulloa, and W.L. Schaich, Phys. Rev. B 41, 5467 (1990); B43, 9865 (1991). 20. We use nl = 2.70x 1016/cm 3, n 2 -- 1.35x 1016/cm 3, Wl = 416/~, w2 = 689A, m*/m = 0.071, e0 = 12.86, and r -- 1 ps.
Solid State Communications, Vol. 88, No. i, PP. 5-7, 1993. Printed in Great Britain.
0038-1098/93 $6.00+.00 Pergamon Press Ltd
A hydrodynamic model calculation of linear and nonlinear optical response in a n a s y m m e t r i c p a r a b o l i c q u a n t u m well
W.L. Schaich
Department of Physics Indiana University Bloomington, IN ~7,~05 (Received 15 July 1993, acceptedfor publication 3 August 1993 by .4. H. MacDonald) We develop model calculations of the optical response of asymmetric parabolic quantum wells using a hydrodynamic description of the confined electrons' dynamics. Both first and second order responses are found, and the latter yield susceptibilities for second harmonic generation and optical rectification. The spectral dependence of each response coefficient shows a rich structure due to various plasmon standing wave resonances.
There has been a lot of interest recently in the nonlinear optical response of electrons trapped in quantum wells [1-8]. The usual theoretical model that has been used neglects most many body effects to concentrate on the coupling between a single subband below the Fermi level, EF, and several subbands above EF. By manipulating the well shape or by applying a static electric field, one can enhance matrix elements and make energy denominators (nearly) vanish, leading to very strong linear responses. In this note we examine the optical response for quantum wells with several partially occupied subbands. To avoid the complications of following in detail the interactions of the many possible single particle excitations, we use a hydrodynamic model which allows a reasonable treatment of the system's collective response. Resonant responses are still possible but now arise from standing plasma waves. This alternate physical picture has been found to represent well the infrared absorption of wide parabolic quantum wells (PQW) with a nearby grating coupler [9-11]. We apply it here to P Q W ' s in isolation, but which have an asymmetric profile (APQW). The asymmetry is needed to produce a nonzero second order response. Our theoretical treatment is adapted from earlier work for the surface of a semi-infinite metal [12,13]. Although the results found there have been shown to be quantitatively unreliable compared to treatments that allow for an extended tail of electron density into vacuum [14-16], the model should do better in P Q W systems since the confinement of electrons in non-neutral wells is significantly more stringent than at neutral metal (or well) surfaces [17]. W i t h both this rationale and the considerable simplifications of a hydrodynamic model over a full microscope evaluation in mind, we present
results here for the d-parameters that characterize the linear optical response [18,19] and the second order susceptibilities that describe second harmonic generation and optical rectification. Since the (bare) confining potential of the well is parabolic we can, via Poisson's equation, specify the A P Q W by a stepped profile of effective positive charge. In the simplest model, the equilibrium electron density will neutralize the effective positive charge over a bounded range. Assuming only one discontinuous j u m p in the effective positive charge, we take for the ground state electron density profile,
no(x) =
n n2
0 < x < w1 Wl < x < w
0
x
(1)
where w = wl + w2 and x is along the direction normal to the plane of the well. The specific choice for n l , n2, Wl and w2 as well as the values of the effective mass m*, host dielectric constant co, and incoherent scattering time 7- are taken from the characterization of an A P Q W that has been constructed by molecular beam epitaxy in a AlxGal_xAS system [10,20]. The equilibrium system described by Eq. (1) is subjected to a spatially constant electric field E(°ut), direction along 2. The superscript means t h a t E(°ut) is the field just outside the APQW, but inside the host semiconductor. It varies in time at the (first harmonic) frequency w. By solving the hydrodynamic equation of motion along with Poisson's equation and the equation of continuity in the separate density steps and matching solutions across boundaries, we determine the induced polarization through the well to first and second order in E (°ut). The boundary conditions on the polarization
6
ASYMMETRIC PARABOLIC QUANTUM WELL 200
.''"1 ....
....
Vol. 88 No. 1 I ....
I ....
I
'"1''"-
2
0
\i
0
1
-t
-800 v
-2
i!
-400
,, I .... -
0
20
40
60
80
....
-4
I00
0
I ....
20
v ( o m -1)
of the free charges are the same as used before [12,13] and the algebraic reduction is straightforward but too involved to be written out here. We only comment on the technical point of extracting second harmonic or static responses from second order quantities. The second order response is driven by the product of first order quantities. If one writes time variations in the form
Am(t) = Ame -im~t + A~neimwt
(2)
to describe the mth harmonic, then a product of AI(t)B1 (t) produces both a second harmonic response C2(t) with complex amplitude C2 = A1B1 and a static response Co(t) with complex amplitude Co = AIB~. Turning now to the numerical results [20], we begin in Fig.[1] with a plot of the only nontrivial d-parameter. Since the host dielectric function e0 has been taken to be real, the imaginary part of d describes absorption by the A P Q W [18,19]. The two strongest structures are slightly above the "bulk" plasma frequencies associated with nl and n2 at 51.5 cm -1 and 36.4 cm -1
i,,,,,,,,,
.... I' " 1 ' " l
I Q v
]
,, 0
2O
80
100
3. Plot of the susceptibility for second harmonic generation versus the frequency of the applied field. The solid (dashed) curve gives the real (imaginary) part of X~2) respectively. This upshift is due to spatial dispersion, which imposes an energy cost for making a standing plasma wave with a finite wavelength. In the nonlinear response one should expect structures near the frequencies of strong absorption and (for second harmonic generation) at half these frequencies. This expectation is borne out in Fig.[2], which plots the efficiency of optical rectification. The susceptibility is defined by X(2) = 2 Re
[I0
w P°(x)/lE~ °ut)]2
]
(3)
where Po(x) is the complex amplitude for the component of the static polarization in second order along and Re means "real part of". Even relatively weak structures in the imaginary part of d seem to lead to strong structures in X~ z).'^ Finally we show in Fig.[3] the complex susceptibility for second harmonic generation defined by
X(2)----fo~ -~" P2(x)/(E~l ut) 2
(4)
with P2(x) the second harmonic analogue of Po(x). There is a rich structure in the frequency dependence due to the interplay of resonances at w and 2w. The magnitude of X~ z)'^ shows a more modest variation but its phase grows by 151r across the frequency range of the figure.
m
0
I
I ....
60
I./(cm -1)
1. Plot of the d-parameter for the linear response to a fieM along the normal to the A P Q W versus the frequency of the applied field. The solid (dashed) curve gives the real (imaginary) part of d.
lO
I ....
40
40
60
80
100
(ore -I)
2. Plot of the susceptibility for optical rectification versus the frequency of the applied field.
The results in Figs. [1-3] are of course specific to our choice of parameters and one can wonder how much further the strong nonlinear effects can be enhanced. Since we do not have a transparent, analytic formula for the X'S it is not possible to systematically maximize them over the range of our parameters. We can say that the results shown here give a sense of the possible frequency variations that should be expected. By modest variations in the n's and w's we can produce peaks with an order of magnitude larger strength. Also the peak heights associated with first order resonances (i.e., peaks in the imaginary part of d) show a quadratic de-
Vol. 88, No. 1
ASYMMETRIC PARABOLIC QUANTUM WELL
pendence on 7, so more perfectly grown samples should yield large enhancements. On the theoretical side it would be interesting to extend the model to allow for a nearby grating coupler, which would allow the 2D plasmon that propagates in the plane of the PQW to contribute. The grating may also remove the need for the well to be asymmetric.
ACKNOWLEDGMENTS We thank Beth Gwinn for stimulating discussions. This work was supported in part by the National Science Foundation through grant DMR 89-03851. We also thank members of the Physics Department at Montana State University for their hospitality during a sabbatical visit.
REFERENCES 1. M.K. Gurnick and T.A. Detemple, J. Quantum Electron. 19, 791 (1983).
11. E.L. Yuh, E.G. Gwinn, and W.L. Schaich, unpublished.
2. J. Khurgin, Appl. Phys. Lett. 51, 2100 (1987).
12. M. Corvi and W.L. Schaich, 3688 (1986).
3. L. Tsang, D. Ahn, and S.L. Chuang, AppL Phys. Lett. 52, 697 (1988). 4. Z. lkonic, V. Milanovic, and D. Tjapkin, J. Quantum Electron. 25, 54 (1989).
Phys. Rev. B 33,
13. W.L. Schaich and A. Liebsch, Phys. Rev. B 37, 6187 (1987). 14. M. Weber and A. Liebsch, Phys. Rev. B 35, 4711 (1987); B36, 6411 (1987); B37, 1019(E) (1987).
5. M.M. Fejer, S.J.B. Yoo, R. L. Byer, A. Harwit, and S.J. Harris Jr., Phys. Rev. Lett. 62, 1041 (1989).
15. A. Chizmeshya and E. Zaremba, Phys. Rev. B 37, 2805 (1987).
6. P.F. Yuh and K.L. Wang, J. Appl. Phys. 65, 4377 (1989).
16. A. Liebsch and W.L. Sctmich, Phys. Rev. B 40, 5401 (1989).
7. E. Rosencher and Ph. Bois, 11315 (1991).
Phys. Rev. B 44,
8. C. Sirtori, F. Capasso, D.L. Sivco, A.L. Hutchinson, and A.Y. Cho, Appl. Phys. Left. 60, 151 (1992). 9. P.R. Pinsukanjana et al., Phys. Rev. B 46, 7284 (1992). 10. E.L. Yuh, E.G. Gwinn, P.F. Hopkins, and A.C. Gossard, unpublished.
17. J.F. Dobson, Phys. Rev. B 46, 10163 (1992). 18. W.L. Schaich and W. Chen, 10714 (1989).
Phys. Rev. B 39,
19. J. Zhang, S.E. Ulloa, and W.L. Schaich, Phys. Rev. B 41, 5467 (1990); B43, 9865 (1991). 20. We use nl = 2.70x 1016/cm 3, n 2 -- 1.35x 1016/cm 3, Wl = 416/~, w2 = 689A, m*/m = 0.071, e0 = 12.86, and r -- 1 ps.