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Increasing Absorption Bistability in Coupled Band Quantum Wells
Physica A 283 (2000) 277–280
www.elsevier.com/locate/physa
Increasing Absorption Bistability in Coupled Band Quantum Wells M.F. Pereira Jr. ∗
à Instituto de Fisica, Universidade Federal da Bahia, 40210-340, Salvador, BA, Brazil
Abstract Increasing absorption bistability is studied under realistic conditions with a many-body approach for nonlinear optical absorption in semiconductor quantum wells, combining Coulomb and band structure engineering e ects. The theory predicts carrier induced all-optical increasing c 2000 Elsevier Science B.V. All rights reserved. absorption bistability for low temperatures. PACS: 71.10.−w; 42.65.Pc; 42.65.−k; 78.66.−w; 73.20.Dx Keywords: Increasing absorption bistability; Many-body e ects; Semiconductor quantum wells; Optical computing
1. Introduction The optical properties of semiconductor media have a strongly non-linear dependence on the, usually high, density of charged carriers generated by the interaction with laser beams, that may lead to multi-stability/instabilities [1]. Unstable and multi-stable systems are of great interest for basic studies of both equilibrium and non-equilibrium statistical mechanics, and among them, bistable devices have the further advantage that they can be used to create optical analogs of digital logic gates, which are important for optical processing or computing. In an optically bistable device two (meta-) stable values of output light intensity exist for a given value of input intensity, and in order to observe bistability some non-linear optical mechanism must be combined with additional feedback. Usually, the feedback is provided by semi-re ecting mirrors in an external cavity. However, under certain conditions, semiconductor media are capable of providing the feedback without the need of an external cavity. All optical and cavity-less bistability of the increasing absorption type (IAOB) has been measured
Supported by the Conselho Nacional de Pesquisas (CNPq) of Brazil. Fax: + 55-71-235-55-92. E-mail address: mauro@ s.ufba.br (M.F. Pereira Jr.)
∗
c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 1 6 7 - 9
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M.F. Pereira Jr. / Physica A 283 (2000) 277–280
and consistently explained for di erent mechanisms, e.g., broadening of bound-exciton lines, or by bandgap shrinkage which can be induced either by thermal phonons, or an electron–hole plasma [2]. The large room temperature non-linearities displayed by multiple semiconductor quantum wells (MQWs), led to an early theoretical investigation on the possibility of room temperature carrier-induced IAOB [3]. A more detailed theory for non-linear optical absorption of quantum wells was not available then, and a step-function approximation was used to describe the absorption at a given frequency as a function of carrier density. Although quite an achievement at the time, the approach can be misleading and a more accurate model is required for a realistic description of the phenomenon. In this paper steady-state IAOB in multiple quantum well systems is analyzed with a technique that combines the major many-body e ects within the relevant carrier density range with realistic band structure engineering methods.
2. The carrier-induced non-linearities The carrier-induced optical non-linearities in semiconductor quantum wells can only be consistently described by means of many-particle techniques, leading, e.g. to extended T -Matrix or semiconductor Bloch equations [4 –9]. In order to describe the evolution of the absorption, (!; N ), for arbitrary carrier densities, N , we must take into account that the near-resonant interaction of a laser eld with semiconductors is characterized by the creation (annihilation) of an electron–hole pair for each absorbed (emitted) photon. Under quasi-equilibrium conditions, the non-linear optical absorption at a given energy ! in an excited medium characterized by an inversion of population factor, fev (k) = fe (k) + fv (k) − 1 is obtained by numerically solving the integral equation for the inter band polarization, or equivalently, the equation for the optical P ∗ ev (k; !), [8] 1 susceptibility, , i.e., (!; N ) = (4 !)=( nb c ˝) kev kev X (! − Ee (k) − Ev (k)) ev (k; !) = fev (k) ev (k) + Wk−q ev (q; !): q6=0
(1) The focus here is on thin MQW samples, where di usion e ects can be neglected, and the density of electron–hole pairs generated by a light eld may be described by a rate equation, dN=dt = −N= + (!; N ) I=!. The steady-state solution for the light transmitted through the sample in the absence of di raction is given by Beer’s law, I (z) = I (0) exp(− (!; N ) z). 1
Here E and f are the renormalized energy and occupation factor which characterizes the sub band , nb denotes the background refractive index, is the sample volume, c is the velocity of light in vacuum, and the indices e; v, e = e1 ; e2 ; : : : ; v = v1 ; v2 ; : : : ; label the multiple electron and hole sub bands and incorporate the total angular momentum z-projection for electrons and a block diagonalization index which characterizes the coupled heavy- and light-hole bands. For more details see Ref. [8].
M.F. Pereira Jr. / Physica A 283 (2000) 277–280
279
Fig. 1. Absorption spectra of a 5 nm GaAs − Al0:23 Ga0:77As quantum well (a) solid: 200 K, detuning, = −1:1; (b) dashed: 300 K, = −1:6; dot-dashed: step function approximation. (b) comparison of the curves in (a) with data extracted from Ref. [9] (symbols). (c) Carrier Density (N × 1018 carriers=cm3 and (d) Output Power (Iout × 1 kW=cm2 ) as a function of Input Power (Iin × 1 kW=cm2 ) for a 5 nm GaAs Quantum Well. solid: 200 K, detuning, = −1:1; dashed: 300 K, = −1:6:
3. Numerical results and discussion In what follows, the approach brie y summarized in the previous section is applied to the description IAOB in coupled-band MQWs. The steady-state regime is discussed to simplify the analysis and highlight the contrast between the di erent cases studied. A total sample length of 1 and an input eld pulse width of 10 ns are assumed. The mechanism for IAOB can be summarized as follows. The system is excited below the bandgap, in a region of small absorption, where a strong laser beam can create a nonzero electron–hole density that gives rise to bandgap shrinkage. For increasing carrier densities the absorption edge is gradually shifted towards the laser frequency. The absorption then increases, reaches a maximum and eventually decreases again, as illustrated in Fig. 1a. Bistability then occurs for the carrier density and the output intensity as a function of input intensity as depicted respectively by Fig. 1c and d. The detuning from band gap is chosen in each case to maximize the bistability contrast. The more realistic model used here with parameters which lead to a good comparison with experimental absorption spectra, predict no IAOB at room temperature for the MQWs considered, in contrast to the conclusions of the simpli ed approach of Ref. [3]. The step-function approximation is interesting since it provides a simple analytical expression that allows for a qualitative discussion of the phenomenon. However,
280
M.F. Pereira Jr. / Physica A 283 (2000) 277–280
it can only be used to describe low-temperature experiments. The results presented here show that bandgap renormalization is a necessary but not sucient condition for IAOB. A small absorption at low carrier density and a good contrast between high- and low-carrier absorption are also needed, and are best achieved with sharp low-energy band tails in the absorption curves, which however depend strongly on temperature and sample quality. In summary, in this paper, a theory combining many-body and band structure e ects methods for non-linear quantum wells describes increasing absorption optical bistability. The approach predicts the e ect in III–V quantum wells only at low temperatures, in contrast with results found in the literature. Smart structures conceived to optimize the carriers con nement within the wells, with consequent increase in electron–hole overlap and e ective Coulomb interaction, with small barrier penetration, thus reducing the sensitivity to sample defects which broaden the absorption spectra and reduce the bistability contrast, should give rise to optimal conditions to observe the phenomenon and for the performance of possible all-optical logical devices. References [1] R. Zimmermann, Many-Particle Theory of Highly Excited Semiconductors, Teubner Texte zur Physik, Vol. 18, Leipzig, (1987). [2] K. Bohnert, H. Kalt, C. Klingshirin, Appl. Phys. Lett. 43 (1983) 1088. [3] S. Schmitt-Rink, C. Ell, S.W. Koch, H.E. Schmidt, H. Haug, Solid State Commun. 52 (1984) 123. [4] M.F. Pereira Jr., R. Binder, S.W. Koch, Appl. Phys. Lett. 64 (1994) 279. [5] A. Girndt, F. Jahnke, A. Knorr, S.W. Koch, W.W. Chow, Phys. Stat. Sol. B 202 (1997) 725. [6] M.F. Pereira Jr., K. Henneberger, Phys. Rev. B 58 (1998) 2064. [7] G. Manzke, Q.Y. Peng, K. Henneberger, U. Neukirch, K. Hauke, K. Wundke, J. Gutowkski, D. Hommel, Phys. Rev. Lett. 80 (1998) 4943. [8] M.F. Pereira Jr., Koch, Chow, Appl. Phys. Lett. 59 (1991) 2941. [9] R. Jin, K. Okada, G. Khitrova, H.M. Gibbs, M. Pereira, .W. Koch, N. Peyghambarian, Appl. Phys. Lett. 61 (1992) 1745.
www.elsevier.com/locate/physa
Increasing Absorption Bistability in Coupled Band Quantum Wells M.F. Pereira Jr. ∗
à Instituto de Fisica, Universidade Federal da Bahia, 40210-340, Salvador, BA, Brazil
Abstract Increasing absorption bistability is studied under realistic conditions with a many-body approach for nonlinear optical absorption in semiconductor quantum wells, combining Coulomb and band structure engineering e ects. The theory predicts carrier induced all-optical increasing c 2000 Elsevier Science B.V. All rights reserved. absorption bistability for low temperatures. PACS: 71.10.−w; 42.65.Pc; 42.65.−k; 78.66.−w; 73.20.Dx Keywords: Increasing absorption bistability; Many-body e ects; Semiconductor quantum wells; Optical computing
1. Introduction The optical properties of semiconductor media have a strongly non-linear dependence on the, usually high, density of charged carriers generated by the interaction with laser beams, that may lead to multi-stability/instabilities [1]. Unstable and multi-stable systems are of great interest for basic studies of both equilibrium and non-equilibrium statistical mechanics, and among them, bistable devices have the further advantage that they can be used to create optical analogs of digital logic gates, which are important for optical processing or computing. In an optically bistable device two (meta-) stable values of output light intensity exist for a given value of input intensity, and in order to observe bistability some non-linear optical mechanism must be combined with additional feedback. Usually, the feedback is provided by semi-re ecting mirrors in an external cavity. However, under certain conditions, semiconductor media are capable of providing the feedback without the need of an external cavity. All optical and cavity-less bistability of the increasing absorption type (IAOB) has been measured
Supported by the Conselho Nacional de Pesquisas (CNPq) of Brazil. Fax: + 55-71-235-55-92. E-mail address: mauro@ s.ufba.br (M.F. Pereira Jr.)
∗
c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 1 6 7 - 9
278
M.F. Pereira Jr. / Physica A 283 (2000) 277–280
and consistently explained for di erent mechanisms, e.g., broadening of bound-exciton lines, or by bandgap shrinkage which can be induced either by thermal phonons, or an electron–hole plasma [2]. The large room temperature non-linearities displayed by multiple semiconductor quantum wells (MQWs), led to an early theoretical investigation on the possibility of room temperature carrier-induced IAOB [3]. A more detailed theory for non-linear optical absorption of quantum wells was not available then, and a step-function approximation was used to describe the absorption at a given frequency as a function of carrier density. Although quite an achievement at the time, the approach can be misleading and a more accurate model is required for a realistic description of the phenomenon. In this paper steady-state IAOB in multiple quantum well systems is analyzed with a technique that combines the major many-body e ects within the relevant carrier density range with realistic band structure engineering methods.
2. The carrier-induced non-linearities The carrier-induced optical non-linearities in semiconductor quantum wells can only be consistently described by means of many-particle techniques, leading, e.g. to extended T -Matrix or semiconductor Bloch equations [4 –9]. In order to describe the evolution of the absorption, (!; N ), for arbitrary carrier densities, N , we must take into account that the near-resonant interaction of a laser eld with semiconductors is characterized by the creation (annihilation) of an electron–hole pair for each absorbed (emitted) photon. Under quasi-equilibrium conditions, the non-linear optical absorption at a given energy ! in an excited medium characterized by an inversion of population factor, fev (k) = fe (k) + fv (k) − 1 is obtained by numerically solving the integral equation for the inter band polarization, or equivalently, the equation for the optical P ∗ ev (k; !), [8] 1 susceptibility, , i.e., (!; N ) = (4 !)=( nb c ˝) kev kev X (! − Ee (k) − Ev (k)) ev (k; !) = fev (k) ev (k) + Wk−q ev (q; !): q6=0
(1) The focus here is on thin MQW samples, where di usion e ects can be neglected, and the density of electron–hole pairs generated by a light eld may be described by a rate equation, dN=dt = −N= + (!; N ) I=!. The steady-state solution for the light transmitted through the sample in the absence of di raction is given by Beer’s law, I (z) = I (0) exp(− (!; N ) z). 1
Here E and f are the renormalized energy and occupation factor which characterizes the sub band , nb denotes the background refractive index, is the sample volume, c is the velocity of light in vacuum, and the indices e; v, e = e1 ; e2 ; : : : ; v = v1 ; v2 ; : : : ; label the multiple electron and hole sub bands and incorporate the total angular momentum z-projection for electrons and a block diagonalization index which characterizes the coupled heavy- and light-hole bands. For more details see Ref. [8].
M.F. Pereira Jr. / Physica A 283 (2000) 277–280
279
Fig. 1. Absorption spectra of a 5 nm GaAs − Al0:23 Ga0:77As quantum well (a) solid: 200 K, detuning, = −1:1; (b) dashed: 300 K, = −1:6; dot-dashed: step function approximation. (b) comparison of the curves in (a) with data extracted from Ref. [9] (symbols). (c) Carrier Density (N × 1018 carriers=cm3 and (d) Output Power (Iout × 1 kW=cm2 ) as a function of Input Power (Iin × 1 kW=cm2 ) for a 5 nm GaAs Quantum Well. solid: 200 K, detuning, = −1:1; dashed: 300 K, = −1:6:
3. Numerical results and discussion In what follows, the approach brie y summarized in the previous section is applied to the description IAOB in coupled-band MQWs. The steady-state regime is discussed to simplify the analysis and highlight the contrast between the di erent cases studied. A total sample length of 1 and an input eld pulse width of 10 ns are assumed. The mechanism for IAOB can be summarized as follows. The system is excited below the bandgap, in a region of small absorption, where a strong laser beam can create a nonzero electron–hole density that gives rise to bandgap shrinkage. For increasing carrier densities the absorption edge is gradually shifted towards the laser frequency. The absorption then increases, reaches a maximum and eventually decreases again, as illustrated in Fig. 1a. Bistability then occurs for the carrier density and the output intensity as a function of input intensity as depicted respectively by Fig. 1c and d. The detuning from band gap is chosen in each case to maximize the bistability contrast. The more realistic model used here with parameters which lead to a good comparison with experimental absorption spectra, predict no IAOB at room temperature for the MQWs considered, in contrast to the conclusions of the simpli ed approach of Ref. [3]. The step-function approximation is interesting since it provides a simple analytical expression that allows for a qualitative discussion of the phenomenon. However,
280
M.F. Pereira Jr. / Physica A 283 (2000) 277–280
it can only be used to describe low-temperature experiments. The results presented here show that bandgap renormalization is a necessary but not sucient condition for IAOB. A small absorption at low carrier density and a good contrast between high- and low-carrier absorption are also needed, and are best achieved with sharp low-energy band tails in the absorption curves, which however depend strongly on temperature and sample quality. In summary, in this paper, a theory combining many-body and band structure e ects methods for non-linear quantum wells describes increasing absorption optical bistability. The approach predicts the e ect in III–V quantum wells only at low temperatures, in contrast with results found in the literature. Smart structures conceived to optimize the carriers con nement within the wells, with consequent increase in electron–hole overlap and e ective Coulomb interaction, with small barrier penetration, thus reducing the sensitivity to sample defects which broaden the absorption spectra and reduce the bistability contrast, should give rise to optimal conditions to observe the phenomenon and for the performance of possible all-optical logical devices. References [1] R. Zimmermann, Many-Particle Theory of Highly Excited Semiconductors, Teubner Texte zur Physik, Vol. 18, Leipzig, (1987). [2] K. Bohnert, H. Kalt, C. Klingshirin, Appl. Phys. Lett. 43 (1983) 1088. [3] S. Schmitt-Rink, C. Ell, S.W. Koch, H.E. Schmidt, H. Haug, Solid State Commun. 52 (1984) 123. [4] M.F. Pereira Jr., R. Binder, S.W. Koch, Appl. Phys. Lett. 64 (1994) 279. [5] A. Girndt, F. Jahnke, A. Knorr, S.W. Koch, W.W. Chow, Phys. Stat. Sol. B 202 (1997) 725. [6] M.F. Pereira Jr., K. Henneberger, Phys. Rev. B 58 (1998) 2064. [7] G. Manzke, Q.Y. Peng, K. Henneberger, U. Neukirch, K. Hauke, K. Wundke, J. Gutowkski, D. Hommel, Phys. Rev. Lett. 80 (1998) 4943. [8] M.F. Pereira Jr., Koch, Chow, Appl. Phys. Lett. 59 (1991) 2941. [9] R. Jin, K. Okada, G. Khitrova, H.M. Gibbs, M. Pereira, .W. Koch, N. Peyghambarian, Appl. Phys. Lett. 61 (1992) 1745.