Nonlinear Optics: Fundamentals, Materials and Devices

The field of nonlinear optics developed gradually with the invention of lasers. After the discovery of second-harmonic generation in quartz, many other interesting nonlinear optical processes were rapidly discovered. Simultaneously theoretical programmes for the understanding of nonlinear optical phenomena were stimulated in accordance to develop structure-property relationships. In the beginning, research advances were made on inorganic ferroelectric materials followed by semiconductors. In the 1970's, the importance of organic materials was realised because of their nonlinear optical responses, fast optical response, high laser damage thresholds, architectural flexibility, and ease of fabrication. At present materials can be classified into three categories - inorganic ferroelectrics, semiconductors, and organic materials. Advances have also been made in quantum chemistry approaches to investigate nonlinear optical susceptibilities and in the development of novel nonlinear optical devices. Generally, inorganic and organic nonlinear optical materials and their related optical processes are reported in separate meetings. This book collects for the first time papers covering the recent developments and areas of present research in the field of nonlinear optical materials.

• Language: English
• Publisher: Elsevier Science
• Release date: Dec 2, 2012
• ISBN: 9780444596741

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Interaction"

ORGANIZING COMMITTEE of the Fifth TOYOTA CONFERENCE

Honorary Chairman

Professor. MITSURU NAGASAWA,     Department of Mechanical Systems Engineering, Faculty of Engineering, Toyota Technological Institute, 2-12 Hisakata, Tenpaku-ku, Nagoya 468, Japan

Chairman

Professor. SEIZO MIYATA,     Department of Material Systems Engineering, Faculty of Technology, Tokyo University of Agriculture & Technology, 2-24-16, Naka-machi, Koganei, Tokyo 184, Japan

Members

Professor. EIICHI HANAMURA,     Department of Applied Physics, Factulty of Engineering, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113, Japan

Professor. HACHIRO NAKANISHI,     Institute for Chemical Reaction Science, Tohoku University, Katahira, Aoba-ku, Sendai 980, Japan

Professor. HIROSHI NISHIHARA,     Department of Electronics, Faculty of Engineering, Osaka University, 2-1, Yamada-oka, Suita-shi, Osaka 565, Japan

Dr. HIROYUKI SASABE,     Department of Biopolymer Physics, Frontier Research Program, Institute of Physical & Chemical Research (RIKEN), 2-1, Hirosawa, Wako-shi, Saitama, 351-01, Japan

Auditor

Dr. OSAMI KAMIGAITO,     Executive Vice President, Toyota Central R&D Labs., Inc., 41-1, Yokomichi, Nagakute, Aichi 480-11, Japan

Secretary General

Dr. MASUHIKO KAWAMURA,     Associate Division Manager, Technical Information and Patent Division, Toyota Central R&D Labs., Inc., 41-1, Yokomichi, Nagakute, Aichi 480-11, Japan

TOYOTA CONFERENCES

The First TOYOTA CONFERENCE

Molecular Conformation and Dynamics of Macromolecules in Condensed Systems, 28 September – 1 October 1987 (Aichi, Japan), edited by M. Nagasawa

The Second TOYOTA CONFERENCE

Organization of Engineering Knowledge for Product Modelling in Computer Integrated Manufacturing, 2 – 5 October 1988 (Aichi, Japan), edited by T. Sata

The Third TOYOTA CONFERENCE

Integrated Micro-Motion Systems - Micromachining, Control and Applications, 22 – 25 October 1989 (Aichi, Japan), edited by F. Harashima

The Fourth TOYOTA CONFERENCE

Automation in Biotechnology, 21 – 24 October 1990 (Aichi, Japan), edited by I. Karube

The Fifth TOYOTA CONFERENCE

Nonlinear Optical Materials, 6 – 9 October 1991 (Aichi, Japan), edited by S. Miyata

The Sixth TOYOTA CONFERENCE

Turbulence and Molecular Processes in Combustion, 11 – 14 October 1992 (Japan), edited by T. Takeno (to be published)

Fundamentals

Outline

Chapter 1: Excitonic and Surface-State Enhancement of Optical Nonlinearity in Semiconductor Microcrystallites

Chapter 2: Coulomb effects in laser-excited semiconductor microstructures

Chapter 3: MOLECULAR ENGINEERING IMPLICATIONS OF ROTATIONAL INVARIANCE IN NONLINEAR OPTICS: OCTUPOLAR SYSTEMS FOR QUADRATIC PROCESSES

Chapter 4: Theoretical calculations on nonlinear susceptibilities of organic materials

Chapter 5: Local field effects in enhancing the nonlinear susceptibility of optical materials

Chapter 6: Frequency Dependence of Third Harmonic Generation in Cis- and Trans-Polyacetylene: Importance of the Degenerate Ground State to Nonlinear Optical Response

Chapter 7: Resonant interaction of photons with randomly distributed quantum dots

Chapter 8: Enhanced Excitonic Optical Nonlinearity of Quantum Dot Lattice

Excitonic and Surface-State Enhancement of Optical Nonlinearity in Semiconductor Microcrystallites

Eiichi Hanamura,     Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan

Abstract

Semiconductor-doped glasses, e.g., doped with CuCl, CdS or CdSe, show enhanced third-order-optical susceptibility χ(3) (Ω; Ω, Ω, Ω). In this paper, two origins of this χ(3)-enhancement are theoretically discussed: (1) excitonic enhancement, e.g., in CuCl microcrystallites in glasses or insulators and (2) surface-state enhancement of Ξ(3, e.g., in CdS-CdSe microcrystallites in glasses or polymers. In the first case, the superradiative decay of the exciton was also expected. These theoretical predictions and experimental results are presented and discussed.As to the surface state of semiconductor microcrystallites, we have not yet obtained its definite identification. However some characteristic features of nearly degenerate four-wave mixing due to the surface state can be described in terms of a few relaxation constants relevant to this state. Inversely, the analysis of this signal will clarify to some extent this surface state of the semiconductor microcrystallites embedded in glasses, insulators or polymers.

1 INTRODUCTION

Since Ekimov et al. [1] observed three-dimensional quantization of electronic levels in semiconductor microcrystallites (later abbreviated as SMC or MC) embedded in glasses, these systems have been studied extensively. Especially Jain and Lind [2] observed strong generation of phase-conjugated wave by CdS-Se doped glasses, and the large χ(3) was found for these materials [3].

The lowest elementary excitation in the crystal is a Wannier exciton in semiconductors. This Wannier exciton is characterized by exciton Bohr radius aB and binding energy Ebexc. The former (aB) describes an average separation between an electron in the conduction band and a hole in the valence band, while the latter (Ebexc) is stabilization energy of the exciton below free electron-hole pair. These microcrystallites are considered spherical in the shape. At least CuCl microcystallites in NaCl were confirmed to be almost spheres with a radius ranging from 10 Å to 100 Å [4,5]. We should distinguish two quantization schemes. First, when the exciton Bohr radius aB is much larger than a radius R of the SMC as in GaAs MC with R much smaller than its exciton Bohr radius aπ)²/2meRπ)²/2mhR. Second, oppositely when the exciton Bohr radius aB is much smaller than the SMC size R, e.g., in CuCl MC (aB = 6.7Å), the exciton becomes a good concept so that the center-of-mass motion is quantized in the MC. This quantization energy in the CuCl MC with R = is 200 meV. This second exciton in the SMC was theoretically shown to decay superradiatively and have the enhanced |χ(3) (Ω; Ω, Ω, Ω)| under resonant pumping of this exciton and a few experimental conditions [6,7]. This was experimentally confirmed [8–11], and is discussed in Section 2.

On the other hand, CdS and CdSe MCs are just on the border between the two schemes but large value of |χ(3)| was observed in some cases [2,3,12]. In Section 3, we will discuss how |χ(3) (Ω; Ω, Ω, Ω)| will be enhanced by the presence of the surface-trapped state. Several models of the trapped state or the surface state have been proposed to explain red-shifted luminescence spectrum: (1) donor-acceptor pair inside the CdS-Se MCs, (2) S or Se deficiencies at the MC surface, both of which trap the exciton and play the role of red-shifted radiative centers, (3) cations, e.g., Na+ ions in glasses close to the CdS-Se MC attract the exciton, the electron composing the exciton penerates out to the glass and the bound exciton is formed by deforming the exciton in the CdS-Se MC into Na+ ions, and (4) the image forces induced at the MC-matrix interface attract both the electron and the hole to just outside the SMC if the dielectric constant of the matrix is smaller than that of the SMC, so that the bound exciton will be formed around the interface. There is also a possibility that two of these combine to form the trapped state, e.g., (3) and (4) effects will work cooperatively. Once we assume the existence of these trapped states, we can evaluate χ(3) (Ω; Ω, Ω, Ω) in terms of the relaxation constants relevant to these trapped states. Then we find out large enhancement of χ(3) for some cases at the sacrifice of (slow) switching time in this case. It is, however, noted that the excitonic and surface-state enhancements coexist. We will also discuss theoretically in Section 4, nearly degenerate four-wave mixing signals characteristic of the trapped states at the surface, which will be effective in clarifying the nature of the surface state of the glass and other matricies.

2 EXCITONIC ENHANCEMENT OF |χ(3)| AND SUPERRADIANCE

Figure of merit for nonlinear optical responses is defined by |χ(3)|/ατ, where τ is switching time and α linear absorption coefficient at the relevant frequency. Unfortunately there exsits an empirical law of a constant figure of merit, almost independent of materials and pump-frequencies [13]. That is, we cannot avoid the trade-off relation between |χ(3)| and speed of switching (1/τ). On the other hand, resonant pumping of coherent excitons was proposed to show rapid radiative decay [7] and enhanced χ(3) [6], satisfying both requirements of large χ(3) and rapid switching at the same time as long as the switching is mainly determined by the superradiative decay [14,15].

R ≤ λ as

(1)

Here Eg is an energy gap between the valence and conduction bands, Ebexc the exciton binding energy in the lowest (1s) electron-hole relative motion, and n the principal quantum number for the center-of-mass motion with the mass M within the MC with the radius R. The transition dipolemoment to this exciton state from the crystal ground state [6]

(2)

in comparison to the band-to-band transition μcv. It is noted that the oscillator strength is almost concentrated on the lowest state n=1.

2.1 Superradiative decay

The first effect of mesoscopic transition dipolemoment is superradiative decay of the lowest exciton (Is, n=1). We consider such a dilute microcrystallite system as (1) excitations cannot transfer among MCs and (2) their radius R is larger than the microscopic size aB but much smaller than the macroscopic size λ of the wavelength of the exciting radiation field. Therefore this system is called mesoscopic. Under the condition (2), e.g., R ≤ 140Å for CuCl MC, the Mie scattering is almost negligible [15]. Let us consider the cases in which broadening of the lowest exciton level due to its phonon scattering becomes of the same order of magnitude as or larger than the energy quantization E2 - E1 = ²π²/(2MR²), or in which the phonon scattering rate of the lowest exciton into the higher levels becomes larger than the superradiative decay rate. In these cases, the lowest state exciton is hybridized with the higher levels with lower oscillator strength so that this superradiative decay will become relatively slow. Therefore the observation of the superradiative decay will be limited to the low temperature and/or to the microcrystallites of the intermediate size. Then the radiative decay rate is calculated in terms of Fermi’s golden rule [7] as

(3)

Note here the mesoscopic enhancement 64π(R/a. This exciton has the maximum cooperation number in the sense that this exciton state is made by a coherent superposition of all atomic excitations composing a SMC. For example, the exciton in CuCl MC with

R = .

Itoh et al. [8] succeeded in making almost spherical CuCl MCs in NaCl matrix well controling their sizes ranging from 17 Å to 100 Å. The absolute values as well as the size-dependence of the radiative decay time (Eq.(3)) were observed in good agreement with the theoretical predictions [7]. On the other hand, the size-dependence of T1 was observed to be R−².¹ deviating from R−³ dependence for CuCl MCs embedded in glasses and the effects of the surface states become evident for the MCs with the radius smaller than 30 Å [10]. These two effects may come from the fact that excitations can penertrate out into the glasses because of the smaller energy barriers in the glasses than those of NaCl matrix.

2.2 Enhancement of χ(3)

The second effect of the mesoscopic transition dipolemoment (Eq.(2)) is mesoscopic enhancement of χ(3) (Ω; Ω, Ω, Ω). These mesoscopic dipolemoment P1 are multiplied four-times in the expression of the third-order optical polarization so that the mesoscopic enhancement of χ(3 is also expected under nearly resonant pumping of the exciton. As far as the exciton is considered as a harmonic oscillator or an ideal boson, it cannot contribute to any nonlinear optical responses. Here, however, three factors make the excitons in MC deviate from the harmonic oscillators: (1) two excitons with the same spin structure in a MC interact with each other as

(4)

in the first Born approximation, (2) the longitudinal decay rate of the exciton 2γ, and (3) the transverse relaxation rate γ ≡ γ + γ’ with the pure dephasing constant γ‘. As a result, under such nearly resonant pumping the lowest exciton Eω1 as |Ω - ω1| ωint in SMC with a radius R such as aB < R < λ, χ(3) (Ω; Ω, Ω, Ω) is evaluated as follows [6]:

(a) ωint > |Ω - ω1| > Γ:

(5)

where Nc ≡ 3r/(4πR³) is number density of the SMCs with r a constant volume fraction. (b) ωint > Γ > |Ω - ω1|:

(6)

For the case (a), χ(3) is almost real and increases as R³ because the fourth power of P1 gives R⁶-dependence and overcomes R−3-dependence of Nc. For the case (b), χ(3) becomes almost imaginary and increases as R³ if the longitudinal decay rate 2γ is determined not by the superradiance but by size-independent processes. Oppositely when the superradiative decay 2γ is dominant over the pure dephasing γ, |χ(3)| decreases as R−³ as R increases. For example, a system of CuCl MCs with R = 80Å and the volume ratio r γ = 0.03 meV.

Masumoto et al.[9] observed the absorption saturation effects by the pump-probe method, which depend on the size of the CuCl MCs. They obtained -Imχ(3) ∝ R².⁶ and Im χ(3) = −10−3 esu for 0.12% CuCl MCs with the radius 100 Å, nearly in agreement with the theory [6,15]. When the longitudinal relaxation 2γ is determined by the superradiative decay but the dephasing γ’ is much larger than γ, - Imχ(3) ∝ R, the lowest exciton level is hybridized with higher levels with the smaller transition dipolemoments. As the radius R (w2 ωΓ increases so that the mesoscopic enhancement of |χ(3)| is reduced by this hybridizing effect as the size R increases beyond the critical size. Then we can expect the optimum size for the largest |χ(3)|, which depends on detuning |ω1 − Ω| and the lattice temperature.

3 SURFACE-STATE ENHANCEMENT OF χ(3)

We will evaluate the third-order optical susceptibility under nearly resonant pumping of an exciton which relaxes into a lower bound state such as a surface state of the SMCs embedded in glasses. The model in Fig. 1 show the 5-level system which consists of the three levels (a crystal ground state g, an exciton state n, and two-exciton state ra) and the trapped state T and higher excited states M which are connected by one-photon transition from the T state. We describe effects of reservoirs in terms of the longitudinal decay rates Γn→g(= Γm→n), Γn→T and ΓT→g, and the transverse relaxation constants Γng, Γmnωn. The density matrix ρ(t) of the electronic system in Fig. 1 obeys the following equation:

Figure 1 Energy diagram of the electronic system in a semiconductor microcrystallite. g:ground state, n:one-exciton state, m:two-exciton state, T:one-exciton trapped state, and M:state connected from T-state by one-photon transition. Γng, Γmn :transverse relaxation constants and Γn→g, Γn→T, ΓT→g: longitudinal ones.

(7)

where H0 denotes the Hamiltonian of electronic system, and H’ = − P· E describes the interaction between the external field E = 2E1 cos Ω1t + 2E2 cos Ω2t and the electric dipolemoment P = −e Σri of the electronic system H0.

(1) To the first-order process in H’, the stationary solution is obtained as follows:

(8a)

(8b)

(2) The 2nd-order stationary solution of (2Ω1) and three diagonal ones as follows:

(9a)

(9b)

(9c)

(9d)

(3) The third-order stationary polarization with the frequency component 2Ω1 - Ω2 is expressed in terms of the following three third-order-density matrices as follows:

(10)

3.1 χ(3) of CuCl microcrystallites in NaCl matrix

In this system which was discussed in the last section, we can neglect the trapped states of excitons so that the third-order polarization 〈P(3)(Ω)〈 = χ(3) (Ω; Ω, Ω, Ω)|E|² Ee−iΩt is described by the first two terms of Eq.(10). Taking account of the results in Section 2 under nearly resonant pumping the lowest exciton (1s, n = 1), we have

and

Inserting these expressions in the first two terms of eq.(10), we have the third-order optical susceptibility as

(11)

We have the same expressions Eqs.(5) and (6) for the relevant case ωint |Ω - ω1|, Γ in which the mesoscopic enhancement of χ(3) is realized as in ref. [16], but the full expression (11) is a little different from those in ref. [6] and [16]. The present treatment fully takes into account (1) all possible contributions under rotating-wave and nearly resonant approximations and (2) the normalization of the density matrix in the closed system correctly.

3.2 Effects of the trapped states on χ(3)

Exciton emisson line is observed at the same position as the absorption peak in the CuCl MCs embedded in NaCl matrix while the much stronger emission lines of CdS-Se MCs show red-shifts of an order of 1eV below the absorption peaks than the non-shifted emission line. These red-shifted emission lines come from trapped state

of the SMCs. Several microscopic models of these bound states have been proposed but these states have not been clearly identified yet. These trapped states may be localized at the MC surfaces so that these are called sometime surface states. In this paper, first we evaluate contribution of those trapped states to χ(3) (Ω; Ω, Ω, Ω) in terms of characteristic constants of these states. Second, in Section 4, we derive the signal spectrum of nearly degenerate four-wave mixing under nearly resonant pumping of the exciton. The spectrum of this signal intensity as a function of the difference frequency Ω1 - Ω2 of two incident beams gives fruitful information of the bound states. This four-wave mixing spectra give not only colorful phenomena but also will be useful in clarifying the nature of the bound states.

We will discuss first three effects of the trapped states on χ(3) (Ω; Ω, Ω, Ω).

(1) In many cases, the trapped state has long decay time. Then the electron in the ground state can not contribute to the polarization with frequency Ω while it is in the trapped state. This gives new contribution χ(3) and is represented by the third term of Eq.(10), i.e.,

(12)

Note that this contribution is the product of the dominant term in Eq.(11) and a factor Γn→T/ΓT→g, i.e., the factor measuring how long the excitation persists in the trapped state. It is also noted that the lowest excited-level in CdS-Se MC may be different from that of CuCl because the former is on the border between the weak and strong confinements. However the enhancement factor Γn→T/ΓT→g works independent of the detailed electronic state. Trapping time (Γn→T)−1 is considered to be an order of ns or ps while the decay time of the luminescence from the trapped state is estimated to be an order of µs. If this estimation is correct, χ(3) (Ω; Ω, Ω, Ω) is enhanced by a factor of Γn→T/ΓT→g ∼ 10³ to 10⁶ in comparison to the SMCs without the trapped state. However, the trapped state with the longer decay time is the easier saturated so that the higher order optical processes become non-negligible.

(2) The second effect of the trapped state on χ(3) is presented by the last term in is expressed as follows:

(13)

where

(14)

(15)

Here PTM denotes the transition dipolemoment between the trapped state T and the higher excited state M and ΓMT as one parameter. This contribution of Eq.(13) can be considered as new polarization process once the trapped state is populated. We do not know the details about higher excited states M. When the nearly resonant states Ω - ωMT has the same enhancement factor Γn→T/ΓT→g

(3) The third effect comes from the modification of the first two terms of Eq.(10) because Γn = Γn→g + Γn→T increases by a term Γn→T due to the presence of the trapped state. If the decay Γn→g is determined by the radiative process, it is of an order of ns. Therefore Γn increases by a factor of 10³ due to the presence of the trapped state. This reduces by this factor Γn→T/Γn→g the non-trapped state contribution described by the first two terms of Eq.(10).

It is noted that the excitonic enhancement of χ(3) is accompanied with the fast-switching due to the superradiative decay, resulting in the enhancement of the figure-of-merit, but that the trapped-state enhancement of χ(3) is accomplished with sacrifice of long switching time, which is determined by the longest time constant (ΓT→g)−1 ∼ 10−6 s.

4 NEARLY DEGENERATE FOUR-WAVE MIXING

Under application of two beams (Ω1, k1) and (Ω2, k2), both of which are nearly resonant to the exciton, the signal at (2Ω1 - Ω2, 2k1 - k2) is observed, originating from the third-order polarization 〈P(3)(2Ω1 - Ω2)〉. The last two terms of Eq.(10) come from reflection of the third wave (Ω1, k(Ω1 - Ω2, k1 - k2) on the trapped states created by the two incident beams Ω1 and Ω2. These contributions are obtained in the following form:

(16)

where

(17)

(18)

As Eqs. (9b) and (9c) show, the third-order polarization shows the singular behavior as a function of Ω1 - Ω2 around the origin Ω1 - Ω2 = 0 because Γn and ΓT→g are very small in comparison to Γng. Here Γn = Γn→g + Γn→T and the decay time of the exciton luminescence (Γn)−1 is observed to be of an order of ns, while the decay time of the trapped excitation (ΓT→g)−1 to be of an order of µs, in the case of exciton in GaAs quantum well systems. On the other hand, the transerve relaxation constant Γng of the exciton is of an order of ps. The signal of the degenerate four-wave mixing (DFWM) is proportional to the square of the absolute value of χ(3) (2Ω1 - Ω2). As a result, the spectrum of the DFWM shows the following hierarchical structure:(1) the very sharp Lorentzian structure (like a spike) with the halfwidth ΓT→g ∼ 10⁶S−1 around Ω1 - Ω2 = 0, (2) the rather sharp Lorentzian with the halfwidth Γn ∼ 10⁹s−1 around Ω1 - Ω2 = 0 and (3) the broad structure with the width Γng ∼ 10¹²s−1 around Ω1 ∼ ωng. These DFWM spectra have been observed by Steel and his coworkers for the exciton in GaAs quantum well [QW] systems [18]. Both real and imaginary parts of the first two terms and the third term of Eq.(10) have the same sign in almost all parts of the spectrum region while the last term of Eq.(10), i.e., the contribution of Eq.(18) to ΔPin Eq.(16), the sharp dip is expected in the broad background in the DFWM spectrum. Both the sharp spike and dip structures were observed in χ(3) (2Ω1 Ω2 : Ω1, Ω2, Ω1) spectrum of CuCl bulk exciton depending on the pump frequency Ω1 relative to the exciton peak ω1 [19]. The hierarchical sturcture has been observed in the DFWM spectrum under nearly resonant pumping of the exciton in GaAs quantum well systems [20]. Analysis of these observation has just begun along the theoretical line presented here. This kind of spectroscopy will work also to clarify the nature of the trapped state of the SMC embedded in glass, and this understanding will be help us in designing the nonlinear optical glasses.

Acknowledgements

This work is supported by the Grand-in-Aid for Scientific Researches on Priority Areas, Electronic Properties of Mesoscopic Particles and Electron Wave Interference Effects in Mesoscopic Structures, and Japan-US research cooperation on Dynamical and Nonlinear Optical Response under Resonant Pumping of Excitons in Semiconductor from the Ministry of Education, Science and Cultre of Japan.

5. REFERENCES

1. Ekimov, A.I., Efros, Al.L., Onuschchenko, A.A. Solid State Commun.. 1985; 56:921.

2. Jain, R.K., Rind, R.C. J. Opt. Soc. Am.. 1983; 75:647.

3. Roussignol, P., Ricard, D., Lukasik, J., Flytzanis, C. J. Opt. Soc. Am.. 1987; B4:5.

4. Itoh, T., Iwabuchi, Y., Kataoka, M. Phys. stat. sol. (b). 1988; 145:567.

5. Itoh, T., Iwabuchi, Y., Kirihara, T. Phys. stat. sol. (b). 1988; 146:531.

6. Hanamura, E. Phys. Rev.. 1988; B37:1273.

7. Hanamura, E. Phys. Rev.. 1988; B38:1228.

8. Itoh, T., Furumiya, M., Ikehara, T. Solid State Commun.. 1990; 73:271.

9. Masumoto, Y., Yamazaki, M., Sugawara, H. Appl. Phys. Lett.. 1988; 53:1527.

10. Nakamura, A., Yamada, H., Tokizaki, T. Phys. Rev. 1989; B40:8585

11. A. Nakamura et al., private commun.

12. Yumoto, J., Fukushima, S., Kubodera, K. Opt. Lett.. 1987; 12:832.

13. Auston, D.H., et al. Appl. Optics. 1987; 26:211.

14. Hanamura, E.Haug H., Banyai L., eds. Optical Switching in Low-Dimensional Systems. Plenum:, 1989:203.

15. E. Hanamura, in SPIE (Society of Photo Optical Instrumentation Engineers) 1268 (Application of Ultrashort Laser Pulses in Science and Technology) (1990) 96.

16. Hanamura, E., Gonokami, M., Ezaki, H. Solid State Commun.. 1990; 73:551.

17. Y. Masumoto, private commun.

18. Remillard, J.T., Wang, H., Steel, D.G., Oh, J., Pamulapati, J., Bhattacharya, P.K. Phys. Rev. Lett.. 1989; 62:2861.

19. M. Gonokami, private commun.

20. D. G. Steel, private commun.

Coulomb effects in laser-excited semiconductor microstructures

Hartmut Haug,     Institut für Theoretische Physik, J. W. Goethe Universität Frankfurt, Robert-Mayer-Strasse 8, D-6000 Frankfurt am Main, Germany

Abstract

The Coulomb interaction between the photo-excited free carriers gives rise to nonlinear optical properties of semiconductor microstructures by changing spectral properties and by determining the ultrashort-time kinetics. As examples for both aspects, the results of two recent studies will be reviewed: a) Calculations of the plasma-density dependence of the spectra of absorption and dispersion of a quasi-onedimensional quantum wire; b) Study of the Coulomb Boltzmann kinetics for a quasi-two-dimensional electron gas of a quantum well by means of Monte Carlo simulations and eigenfunction expansions.

Keywords

Quantum wires

quantum wells

optical nonlinearity

Coulomb kinetics

Boltzmann equation

Monte Carlo simulation

eigenfunction expansion

1 INTRODUCTION

The demand to increase information rates in communication systems stimulates the development of optical and electro-optical switching elements which are fast, small and reliable. In the spectral vicinty of their band edge, semicondutors have been shown to possess the necessary optical nonlinear properties which are needed for active or passive switching elements [for books on these topics see e.g. 1, 2, 3]. These nonlinear optical properties can be taylored and optimized [4] by confining the optically active electrons to spatial areas which are still large on the scale of the lattice constant but small enough that the electrons behave no longer as free waves, but as the lowest (or few lowest) mode(s) of a set of standing waves. Depending on the numbers of directions in which the electrons are confined, one speaks about quantum wells, quantum wires and quantum boxes (or dots). In these mesoscopic structures the optically active electrons form a quasi-two-, one-, or zero-dimensional Fermion gas, respectively. The word quasi indicates that these low-dimensional gases have only certain properties of a low-dimensional system, but remain three-dimensional in other respects. E.g. the Bloch part of the electron wavefunction remains the same as in bulk semiconductors, only the envelop part is changed as discussed above. Furthermore, the Coulomb interaction remains three-dimensional V(r) = e²/ε0r, because the field lines also close in the material surrounding the microstructure. This barrier material has often a dielectric constant which is close to ε0 of the material of the microstructure.

The interaction with laser light of appropriate frequency excites typically many electrons across the intrinsic gap between the conduction and valence band of a semiconductor. These optically excited, nonequilibrium electrons in the conduction band and holes in the valence band all interact with each other via the long-range Coulomb potential. In general, the theory of the nonlinear optical properties of semiconductors is a complicated nonequilibrium many-body problem. Not only the changes or renormalizations of the energies have to be calculated as in an equilibrium system, but also the kinetics in these newly defined states has to be determined possibly in a self-consistent way.

Fortunately, the necessary formalism exists in terms of the contour-time-ordered nonequilibrium Greens functions [5,6]. This formalism offers a systematic diagram technique to treat these difficult problems. In quasi-equilibrium situations the rapid Coulomb collisions between the photo-excited carriers established already Fermi distributions of the electrons and holes. The parameters of the Fermi distributions are determined by the exciting laser light. Under these conditions the theory of the nonlinear optical properties of bulk semiconductors and quantum well structures has been developed rather sucessfully [4].

Recent progress in microstructuring techniques has opened up the production of semiconductor quantum well wires (QWW) and the study of their optical properties [7–11]. Therefore, one can try to extend former calculations [12] of the interband polarization for 3d and 2d systems to quasi-one-dimensional quantum wires [13]. It will be shown that the obtained large optical nonlinearities in the vicinity of the exciton resonance make quantum wires interesting candidates for switching devices.

For ultrafast switching devices, the quasi-equilibrium assumption holds no longer, one has to treat the kinetics of the carriers in detail. In an electron-hole plasma the Coulomb scattering gives rise to a relaxation on the time scale of 10 to several 100 femtoseconds. The treatment of this ultrafast kinetics in optical devices is still in its childhood. Here, some results of a detailed study [14] of the Coulomb kinetics of a quasi-two-dimensional electron gas will be presented. Ensemble Monte Carlo simulations will be used to study the relaxation kinetics of the Coulomb Boltzmann equation. This method allows to follow the evolution of an ensemble in the two-dimensional momentum space directly on the computer screen and to obtain a very direct insight in the relaxation kinetics. For near-equilibrium situations also eigenfunction expansions for the linearized collision operator are used. In this regime perfect agreement with corresponding Monte Carlo simulations is obtained. The eigenfunction expansion allows to understand in detail how the relaxation depends e.g. on the position and width of the initial deviation from the equilibrium distribution, proving clearly that conventional relaxation approximations are not valid.

2 PLASMA-INDUCED NONLINEARITIES IN QUANTUM WIRES [13]

Present-day quantum wires are usually made from GaAs/GaAlAs quantum wells by adding an additional lateral confinement for the carriers. The wires available up to now are not really quasi-one-dimensional, because the energy separation between the subbands is not large enough (typically 1-5 meV). However, there is rapid improvement in the fabrication of these wires. Here, calculations of the dependence of the spectra of absorption and dispersion of GaAs/GaAlAs QWW’s with only one subband on the density of an optically generated electron-hole plasma will be reported. The calculations [13] are an extension of corresponding calculations for bulk and quantum well semiconductors [3,12,15], which are generally in good agreement with corresponding measurements.

Ω. In the direction of the wire axis the electrons and holes can move freely. The effective-mass approximation is used and only the lowest 1d subband is taken into account. The ground state wave functions of electrons and holes are supposed to have the same spatial extension which leads to spatial charge neutrality in the plane perpendicular to the wire axis. Dielectric effects due to differences in the static dielectric constants ε0 of the well and the barrier material are neglected. The Fourier transformation in the z-direction of the 2d-Coulomb interaction averaged with the ground state wave function of the confining oscillator potential [10]