ivchenko_bookreg

bulk. Jiang et al. [7.34] investigated second-order nonlinear optical properties of Zn1−xCdxSe/ZnSe asymmetric coupled QWs by using the reflective second-harmonic generation technique. The photon energy, ω = 1.17 eV, of Nd:YAG (Yttrium Aluminum Garnet) laser was close to the half band gap of the studied structures and, therefore, conditions for the half-band- gap resonance were realized. Compared with the second-harmonic generation intensities in ZnSe bulk material, a significant enhancement of the signals (at least one order of magnitude) was observed due to the structure inversion asymmetry. Note also that the p-in/p-out intensity was two orders of magnitude larger than the s-in/s-out one.

The second-order susceptibility due to the intraband intersubband transitions can be greatly enhanced due to a possibility to realize the conditions of double resonance for three subbands 1, 2 and 3

Replacing i1, i2, i3 by 1, 3, 2, respectively, we obtain from (7.29) for the double-resonant contribution to the second-order susceptibility of a QW [7.36, 7.38, 7.39]

where the dephasing constants Γi i are introduced into the energy denominators and, for simplicity, we omit the summation over the spin indices of the electron (or hole) states |ik . The tuning to the double-resonance conditions can be achieved by varying parameters of an asymmetrically stepped QW or asymmetrical double QWs and/or applying an external electric field.

Enhancements as big as three orders of magnitude in χ(2) were observed in doubly resonant experiments on n-doped QW structures with equal e2-e1 and e3-e2 transition energies [7.40, 7.41]. In these experiments, the only element of the susceptibility tensor that exhibits a strong double-resonant enhancement is χ(2)zzz . Thus, the exciting and generated electric fields have to possess nonzero z-components which limits the usefulness of the element χ(2)zzz for device applications. In contrast to the conduction subbands, the hole energy dispersion is di erent for each valence subband, leading to a relatively low reduced density of states for intersubband transitions in p-doped QWs. Therefore, the double-resonance enhancements which can be obtained in the latter structures are expected to be lower. However, the presence of elements χ(2)αβγ

di erent from χ(2)zzz provides more functionality of the p-doped structures, since the incident and outgoing (second-harmonic) waves can have various polarizations. Bitz et al. [7.39] demonstrated second-harmonic generation in p-doped GaAs/AlxGa1−xAs asymmetrically stepped MQW structures using

342 7 Nonlinear Optics

the emission of a free-electron laser in the wavelength interval between 13 and 18 µm. In their samples the double-resonance conditions were realized for the transitions between the first heavy-hole, first light-hole and second heavy-hole subbands. The measured maximum enhancement by a factor of 16 in the second-harmonic susceptibility of the QWs with respect to that of bulk GaAs agrees with the theoretical expectations taking into account the quality of investigated samples.

For centrosymmetric materials, such as single Si or Ge crystals of the point group Oh, the second-harmonic generation is forbidden within the electricdipole approximation. However, QW structures grown from these materials can lack such a center and allow the second-harmonic generation. Indeed, depending on the properties of an (001)-interface between the alloy Si1−xGex and Si, its symmetry on average can be C2v or C4v [7.42]. The former point group describes the symmetry of an ideal heterointerface with the interfacial chemical bonds lying in the same plane. A nonideal interface containing monoatomic fluctuations has two kinds of flat areas with interfacial planes shifted with respect to each other by a quarter of the lattice constant a0. The local symmetry of each area is C2v as well. However if the both kinds are equally distributed, the interface overall symmetry increases up to C4v . The both symmetries, C2v and C4v , allow the second-harmonic generation at a single interface. The symmetry of a Si1−xGex/Si QW structure containing two interfaces is described by one of five point groups: D2d or D2h in case of two ideal interfaces with odd or even number of monolayers between them; C2v for a pair of ideal and rough interfaces; C4v or D4h for two non-ideal interfaces of the overall symmetry C4v each. For QWs of the symmetry D2d, C2v and C4v the contributions from the leftand right-hand-side interfaces to the second-harmonic generation do not compensate each other. Ghahramani et al. [7.32] performed a full-band-structure calculation of the components χ(2)x x z and χ(2)zx x for short-period SinGen SLs grown on the Si(001) substrates. They obtained that the bulk value of χ(2) for SLs with odd n = 1, 3, 5 (point-group D2d) is of the same order as that of bulk GaAs, one of the most commonly employed electro-optic semiconductor materials. Unfortunately, the presence of atomic height steps on the substrate and the incapability of precise control of layer thickness in SLs in the present molecular-beam epitaxy prevents up to now the realization of a macroscopic χ(2) comparable to that of GaAs [7.43]. In addition, weak second-harmonic generation from GeSi nanostructures can arise from inhomogeneous strain, miscutting of the substrate, bulk quadruple mechanism. Zhang et al. [7.44] systematically analyzed contributions from di erent sources to the second-order susceptibility χ(2) in two kinds of p-doped asymmetrical GeSi structures, namely, in Si0.75Ge0.25/Si0.57Ge0.43 step asymmetric QWs and electric-field biased Si5Ge5 SLs. The largest evaluated value of χ(2) was found in the biased SL, under the electric field F > 100 kV/cm the main source of the nonlinear susceptibility arose due to the electric-field-induced asymmetry. In addition to

the second-harmonic generation, the linear and circular photogalvanic e ects discussed in Chap. 8 can be used to question an existence of the inversion center in GeSe QW structures.

The further enhancement of χ(2) has been demonstrated, both theoretically and experimentally, for intraband transitions in n- and p-doped QD structures [7.45,7.46]. In the study [7.46], the n-doped sample consisted of 30

˚

layers of InAs/GaAs self-assembled QDs separated by 500-A thick GaAs barriers and grown on the GaAs substrate. Second-harmonic power as a function of the pump photon energy shows a narrow peak at ω = 61 meV superimposed on a background signal corresponding to the contribution of bulk GaAs. The peak is an enhancement due to the double resonance for the intersubband e100-e000 (or p-s) and e110-e000 (or d-p) transitions.

In [7.47], a quantum cascade laser is employed to demonstrate that intersubband optical transitions in stacked semiconductor QWs can function as a simultaneous source of fundamental pump and second-harmonic light.

7.5 Nonlinear Optical Phenomena in Quantum

Microcavities

A natural way to enhance the light-matter interaction is to exploit band-edge resonances by tuning the incident-light frequency to the exciton-resonant spectral region. The quantum confinement of excitons in semiconductor nanostructures leads to the further increase in the resonant optical response. The photon confinement in quantum microcavities, i.e., in microcavities with embedded QW’s shown schematically in Fig. 7.3, has opened a way for additional considerable enhancement of exciton-photon coupling [7.48]. Among reasons for the interest in these multi-layered structures, we mention the following three. First, the microcavities have potential applications in the development of low-threshold vertical-emitting lasers. Second, fundamental aspects of the interaction of confined photon modes (2D photons) with matter have opened a new field in quantum electrodynamics. And, finally, the quantum microcavities can be particularly interesting for nonlinear optics since a nonlinear response has a stronger dependence upon the coupling constant.

In the next subsection we introduce the two-oscillatory model of quantum microcavities and present the dispersion of 2D exciton-cavity polaritons. Other subsections are devoted to nonlinear optics of the microcavities.

7.5.1 Exciton Polaritons in a Quantum Microcavity

Semiconductor microcavity is a multilayer heterostructure consisting of an active layer B of the thickness Lb sandwiched between Nl pairs of the C2/C1 mirror, or Distributed Bragg Reflector (DBR), and Nr pairs of C1/C2 DBR grown on the substrate D. If one or few QWs (material A) are embedded

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Fig. 7.3. Schematic representation of a quantum macrocavity.

inside the active layer the structure is called Quantum Microcavity (Fig. 7.3). As a rule we will consider the quantum microcavity with a single embedded QW.

The thickness Lb and the thicknesses a1 and a2 of the C1 and C2 layers in DBR’s are assumed to satisfy the conditions

where Nb is an integer that expresses the cavity length in half-wavelengths, ni (i = b, 1, 2) is the refractive index of the corresponding material, and ω¯ is an arbitrarily chosen frequency which turns out to be the resonance frequency of the microcavity, see below. In quantum microcavities, it is usually taken to lie close to the QW exciton resonance frequency ω0.

Exciton-polariton modes in quantum microcavities can be modelled by a pair of two coupled classical oscillators, one of them representing an exciton in the QW and another representing the photon mode. Taking the averaged dielectric polarization P (t) defined according to (3.41) and the electric field, E(t), inside the QW as two variables for the two-oscillatory model we can write for them a standard set of equations

Here Γ , γ¯ are the nonradiative damping rate of a bare 2D exciton and the damping rate of the bare-photon mode. The solution of (7.33) is sought in the exponential form P (t) = P e−iωt, E(t) = Ee−iωt. If the damping constants

Γ, γ¯ and the di erence, ω0 − ω¯, of the bare resonance frequencies are small compared to these frequencies, (7.33) reduces to

where, instead of q1 and q2, we introduced other parameters γj = qj /(2ω¯). One of them is easily found from a comparison of (7.34a) with (3.44), namely,

The expressions for γ¯ and γ2 are given below, see (7.51, 7.52, 7.55). From (7.34) we obtain the following complex eigenfrequencies of the mixed modes

Let us analyze the structure with ω¯ tuned exactly to ω0. In the weakcoupling regime defined by the condition (Γ − γ¯)2 > 4γ1γ2, one has for the eigenfrequencies

i.e., their real parts coincide but the imaginary parts are di erent. In the strong-coupling regime where (Γ − γ¯)2 < 4γ1γ2, the frequencies ω± di er in the real part

− ˜

In the latter case the di erence ω+ ω− = 2Ω is called the Rabi splitting. It is worth to mention that, as follows from Chap. 3, the conventional

single-QW structures are open systems where the 2D electronic excitations (2D-excitons) interact with 3D photons, the renormalization of exciton resonance frequency, see (3.18), is small and the exciton-photon coupling leads mainly to the exciton radiative damping. In a quantum microcavity with thick DBR’s of high quality, both excitonic and photonic states are size-quantized in the growth direction. As a result, the bare resonance frequencies ω0, ω¯ can be strongly renormalized. In real semiconductor microcavities the Rabi splitting amounts few meV and in some cases it can even exceed 10 meV. Now we turn to establishing the relation between γ2, γ¯ and parameters of the quantum microcavity.

The optical properties of the DBR’s, or mirrors, are characterized by the amplitude reflection coe cients rmj , rmj from the left, j = l, or right, j = r, mirror for the light incident, respectively, from the active layer and from the

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external medium, vacuum or substrate, and the transmission coe cients tmj , tmj through the mirror j defined in the similar way. In the following we ignore the di erence between the refractive indices of the active layer and the QW. Then the amplitude reflection and transmission coe cients, rQW and tQW, of a QW are given by (3.21).

Assuming the monochromatic light to be incident from the vacuum side under the angle θ0, we can present the amplitude reflection and transmission coe cients for the whole quantum microcavity as

where the transmission tQM refers to the light field at the boundary between the second DBR and the substrate, the phase ψb is defined by

the value Lb includes the QW thickness, the angle θb and incidence angle in vacuum are connected by the Snellius relation sin θ0 = nb sin θb, and

Substituting (7.40) into (7.38) we can present rQM , tQM as fractions with the denominator

DM C = rmlrmr(t2QW − rQW2 ) exp (2iψb) + (rml + rmr)r exp (iψb) − 1 . (7.41)

The dispersion equation for the exciton polaritons in the structure of Fig. 7.3 is given by

where q is the exciton-polariton in-plane wave vector (ω/c) sin θ0. The dispersion equation for the photon mode in a semiconductor microcavity without embedded QWs (or empty cavity) is expressed by (7.41, 7.42) with rQW = 0, tQW = 1, i.e.,

the reflection coe cient under light incidence from the left upon the Bragg mirror

The eigenfrequencies of exciton polaritons with q = 0 can be similarly derived taking into account that, at oblique incidence, the reflection and transmission coe cients rQW, tQW are given by (3.51) and (3.53) for the s- and p-polarized light, respectively. On the other hand, the empty-cavity photon mode has the dispersion

In a quantum microcavity the exciton and photon modes are mixed to lead to the following dispersion of the upper and lower TE-polarized excitonpolariton branches

where the exciton and photon-mode decay rates are ignored. Moreover, the exciton e ective mass is set to infinity because the photon-mode mass m¯ is extremely light and usually amounts 10−5-10−4 m0. For a quantum microcav-

The dispersion of TM-polarized exciton polaritons in quantum microcavities was discussed in [7.53].

In the strong-coupling regime, the split frequencies ω± reveal themselves as dips in the reflection spectra and peaks in the transmission or absorption spectra. For illustration of the strong-coupling regime, Fig. 7.4 shows the measured normal-incidence reflectivity of a λ cavity with quarter-wave pairs of AlAs (n1 = 2.95) and GaAs (n2 = 3.61) and with a single QW In0.04Ga0.96As embedded in a GaAs active layer. The cavity-photon resonance was tuned to

350 7 Nonlinear Optics

ω

Fig. 7.4. Reflection spectrum from a quantum microcavity with a single QW embedded in the center of 1λ active layer. From [7.52].

the exciton resonance, ω¯ = ω0. One can see two well-resolved reflectivity dips with a high splitting-to-linewidth ratio. The dip positions determine the energies, ω±, of the 2D-exciton–2D-photon coupled modes. The di erence(ω+ − ω−) ≈ 3.5 meV gives the Rabi splitting. The coherent interaction between the single-mode optical field and excitons in a QW leading to the Rabi splitting ω+ −ω− can also be studied as quantum beats in the transient reflection or transmission spectra. It is worth to note that the Rabi splitting can be enhanced by inserting few QWs near the electric-field antinode of a λ-cavity or in di erent antinodes of a macrocavity with the thicker active layer [7.52, 7.53].

It is instructive to survey the existing variety of exciton-polariton quasiparticles focusing attention at the dimensionalities of bare exciton and photon states. In bulk semiconductors an exciton polariton is formed by a 3D exciton and a 3D photon. Heterostructures with QWs, quantum wires and quantum dots allow one to study coupling between 3D photons and 2D, 1D or 0D excitons, respectively. In a planar quantum microcavity, coupling of a 2D exciton in the embedded QW with a 2D photon confined in the cavity is realized. The 2D-1D and 2D-0D exciton-polariton states can be engineered in 2D microcavities with embedded quantum wires [7.54–7.57] and QDs or QD SLs [7.58, 7.59]. G´erard et al. [7.60] and Reithmaier et al. [7.61] have succeeded in further reducing the dimensionality of bare-photonic quantum states and fabricated laterally structured semiconductor microcavities