ivchenko_bookreg

(Γ8, ±3/2) and light-hole (Γ8, ±1/2) subbands [6.58]. In this case the resonant intermediate states n, n in (6.71) are the excitons formed by an electron and a heavy or light hole, respectively. In resonant magneto-Raman scattering, a double resonance occurs if the separation of the valence Landau levels in nonmagnetic semiconductors or Zeeman splitting between the states |Γ8, 3/2 , |Γ8, −3/2 in semimagnetic semiconductors equals the energy of an optical phonon [6.59, 6.60].

Semiconductor nanostructures open new possibilities for realization of the double resonance conditions because resonant states for incoming and outgoing beams can be excitons attached to di erent size-quantized subbands in QWs or QWRs and quantum-confined levels in QDs. For example, Kleinman et al. [6.61, 6.62] observed the doubly resonant LO-phonon Raman scattering in GaAs/AlxGa1−xAs QWs due to the following process

ω1 → Xe1-hh3 → Xe1-hh1 + LO → ω2 + LO ,

where Xeν -hhν denotes the exciton involving an eν electron and hhν hole.

˚

In the sample containing QWs of the thickness a = 140 A separated by

˚

193-A x = 0.33 alloy barriers, the excitation-energy interval between the Xe1-hh3 and Xe1-hh1 excitons is close to ΩLO. As the incident and scattered photons go into resonance with these excitons, a very strong single-LO peak is seen in emission, its intensity being comparable in amplitude to the main photoluminescence broad peak. The scattering peak intensity as a function of the excitation photon energy ω1 shows a sharp maximum with a width of ≈ 4.2 meV, in agreement with FWHM = (4 ±1) meV seen for the Xe1-hh1 exciton in the photoluminescence excitation spectrum.

The double-resonance conditions can be adjusted by external fields. Calle et al. [6.63, 6.64] observed the e ect setting the outgoing and incoming channels on the light-hole excitons Xe1-lh3, Xe1-lh1 and varying the strength of the applied magnetic field. Agull´o-Rueda et al. [6.65] could induce a double resonance by applying a variable electric field to a GaAs/Al0.35Ga0.65As SL with

˚ ˚

a = 30 A and b = 35 A and forming the Stark ladder (Sect. 3.3.2). The incoming resonance was set on the intrawell interband transition lh1, n0 → e1, n0 and outgoing resonance was tuned to the transition lh1, n0 → e1, n0−1, where n0 is the index of the Wannier-Stark localized state. According to (3.206) the corresponding value of the electric field is determined by |e|F d = ΩLO which gives F ≈ 56 kV/cm in very good agrement with the value F ≈ 60 kV/cm at which a maximum of the Raman intensity is observed.

In the cited studies resonant states for incoming and outgoing photons were 1s-excitons attached to di erent size-quantized subbands or to di erent Landau levels. Here we discuss one more double optical resonance. It is observed on the states e1-hh1(2s) and e1-hh1(1s) belonging to the same excitonic series and related to the lowest electron and hole subbands e1 and hh1 [6.66]. GaAs-based heteropairs does not suit for an observation of the 2s-1s resonance, because in the corresponding heterostructures the exciton binding energy and, hence, the 2s-1s energy separation, ω21, is several

322 6 Light Scattering

times smaller than the LO phonon energy ≈ 37 meV. A suitable object is a CdTe-based QW structure because (i) in CdTe, the LO-phonon energy is relatively low, ΩLO = 21 meV , (ii) in QWs CdTe/Cd1−xMnxTe (or CdTe/Cd1−xMgxTe) the energy spacing ω21 remarkably increases as a result of the quantum confinement e ect and di ers from ΩLO just by a few meV, (iii) in the magnetic field oriented along the growth axis z, the 2s- exciton level undergoes a strong diamagnetic blue shift and the condition ω21 = ΩLO is fulfilled in a moderate field B < 10 T . The double 2s-1s resonance was observed in the backscattering Faraday geometry for circularly polarized analyzer and polarizer. In both photoluminescence and photoluminescence excitation spectra the intensity of the sharp 1LO-replica was found to increase rapidly with tuning to the double resonance conditions (Fig. 6.7). An important fact is that a strong 1LO-line has been observed not only in parallel circular polarizations but also in the crossed z(σ+, σ−)¯z or z(σ−, σ+)¯z configuration.

Fig. 6.7. The magnetic field dependence (circles) of the ratio between intensities of the 1LOand 2LOlines under double 2s-1s resonance observed in the

˚ ˚

CdTe/Cd0.86Mn0.14Te 85 A/95 A MQW structure. Squares show the 2s-1s energy spacing as a function of the magnetic field. [6.66]

Let us consider two possible mechanisms of the doubly-resonant 2s-1s secondary emission. For the direct doubly-resonant one-phonon Raman scattering,

ω1 → Xe1-hh1(2s, K = 0) → Xe1-hh1(1s, K = 0) + LO → ω2 + LO ,

optical excitation of the 2s exciton is followed by the LO-phonon-assisted transition to the bottom of the 1s-exciton subband and then by emission of a photon from the 1s state. However, the analysis shows that the vertical transition 2s → 1s + LO is ine ective. Really, in a bulk semiconductor, such a transition is forbidden in the dipole approximation, i.e. the corresponding matrix element V1LOs,2s qz , where qz is the scattered wave vector (the di erence between wave vectors of the incident and secondary photons). Remind that by similar reason in resonant spectra of secondary emission of perfect crystals the 1LO-line is much weaker than the 2LO line. On the other hand, in the 2D approximation the exciton envelope function Ψns is factorized so that

with the same single electron and hole envelopes ϕe1, ϕh1 for 1s and 2s states. Here fns(ρ) is the envelope function describing the relative electron-hole motion in the (x, y) plane and ρ = [(xe − xh)2 + (ye − yh)2]1/2. The condition of orthogonality between f1s(ρ) and f2s(ρ) immediately makes zero the matrix element V1LOs,2s since, for backscattering of normally-incident light, the exciton-

phonon interaction operator is ρ-independent. Thus, the value of V LO can

1s,2s

be nonvanishing only due to the Coulomb-potential e ect upon the electron or hole confinement along the z direction. Furthermore, for the direct process the exciton polarization cannot undergo a remarkable change which contradicts to the presence of the strong 1LO-line in the spectra observed in the crossed circular polarizations.

The more realistic mechanism

ω1 → Xe1-hh1(2s, K = 0) → Xe1-hh1(1s, K = 0)

→ Xe1-hh1(1s, K = 0) + LO → ω2 + LO

can be described as follows: the 2s-exciton is first scattered from the bottom of the 2s branch to the “hot” 1s states characterized by large in-plane center-of-mass wave vectors, K, then it is multi-scattered by elastic random potential of the heterostructure and only afterwards emits a LO-phonon. Obviously the second mechanism has more grounds to be interpreted as a luminescence rather than Raman scattering. It should be mentioned that similar static scattering processes with a large momentum transfer were suggested by Kleinman et al. [6.62] and Gubarev et al. [6.60] in order to interpret their results on doubly resonant LO-phonon-assisted secondary emission respectively in GaAs/AlxGa1−xAs QWs and bulk semimagnetic semiconductor Cd0.95Mn0.05Te.

324 6 Light Scattering

The concept of double resonance in Raman scattering is very productive. The double-resonance spectroscopy has been used to study the light confinement in semiconductor planar microcavities with MQWs as an active layer [6.67]. Kipp et al. [6.68] have reported on Raman scattering by electron chargeand spin-density excitations of a modulation-doped QW embedded inside a microcavity. Under conditions of optical double resonance, when both the exciting laser and the scattered light are in resonance with the cavity modes which is achieved by a variation of incidence and scattering angles, electronic excitations can be selectively enhanced. The enhancement factor between the single-resonant case, when only the incoming light is in resonance with the cavity, and the double-resonant case, when the outgoing light also is in resonance with the cavity, amounts up to three orders of magnitude. Linear and nonlinear optical properties of quantum microcavities is considered in the next chapter.

7 Nonlinear Optics

Two are better than one,

because they have a good return for their work;

If one falls down,

his friend can help him up...

Though one may be overpowered, two can defend themselves.

Ecclesiastes 4: 9-12

Many nonlinear e ects can be phenomenologically described by expanding the electric polarization P in powers of the light-wave electric field

+χ(2)αβγ (ω1, ω2)Eβ (ω1, k1)Eγ (ω2, k2) exp [i(k1 + k2) · r − i(ω1 + ω2)t]

+χ(3)αβγδ (ω1, ω2, ω3)Eβ (ω1, k1)Eγ (ω2, k2)Eδ (ω3, k3)

× exp [i(k1 + k2 + k3) · r − i(ω1 + ω2 + ω3)t] + ...

Here ωj is the frequency, kj the wave vector, E(ωj , kj ) the amplitude of the j- th constituent of the electromagnetic field. For single crystals or short-period SLs, r is the 3D radius-vector while, in QW structures, r is a vector with the components x, y, zl, where zl is the position of the l-th well, for instance, the coordinate of its center (the choice of point zl within a well is arbitrary if its width a is small compared with the light wavelength). Similarly, in a QWR directed along z, the coordinates of r in (7.1) are (xl, yl, z), where xl, yl give the position of the wire in the (x, y) plane. The frequencies in (7.1) may assume both positive and negative values, the corresponding amplitudes being related through

E (ω, k) = E(−ω, −k) .

For the sake of simplicity, we neglect in (7.1) the spatial dispersion of the susceptibility, i.e., the dependence of χ(n) on k, k1, k2...

The first-order susceptibility χ(1) describes the conventional linear response. The second-order susceptibility χ(2) describes the generation of optical harmonics at the sum or di erence frequency (three-wave mixing), in particular, at ω1 = ω2, second-harmonic generation. The special case ω1 = −ω2 is considered in the next chapter devoted to photogalvanic e ects. The third-order susceptibility χ(3) describes a variety of phenomena, namely, third-harmonic generation (at ω1 = ω2 = ω3), two-photon absorption of a monochromatic light wave ( two of the three values of ωj coincide, the third one di ering from them in sign, e.g., ω2 = ω3 = −ω1) or two monochromatic

326 7 Nonlinear Optics

waves (two of the three frequencies di er in sign and do not coincide with the third, e.g., ω1 = −ω2 = ±ω3), a photoinduced change in the absorption or reflection of a probe beam linear in pump-beam intensity (photoabsorption or photoreflection), photoinduced gyrotropy or birefringence, four-wave mixing of three light waves ωj , kj (j = 1, 2, 3) accompanied by generation of a new harmonic at the frequency ω1 + ω2 −ω3 and the wave vector k1 + k2 −k3. Degenerate four-wave mixing is a particular case of the latter where two beams with the coinciding frequencies and di erent propagation directions, k1 = k2, interact and induce a new spatial harmonic 2k1 − k2 or 2k2 − k1.

In this chapter we discuss, one after another, the two-photon absorption (Sect. 7.1), biexciton optical spectroscopy (Sect. 7.2), resonant four-wave mixing (Sect. 7.3) and second-harmonic generation (Sect. 7.4) in nanostructures. Linear and nonlinear optical properties of quantum microcavities are considered in Sect. 7.5.

7.1 Two-Photon Absorption

Two-photon absorption is an elementary excitation process in which two photons simultaneously give their energies to the medium. The two quanta are chosen such that the system under study cannot absorb either quantum separately. Conservation of energy then demands that the sum of the energies of the two photons be the energy of the electronic transition.

Two-photon spectroscopy has properties that distinguish it from onephoton spectroscopy, and therefore it represents an alternative spectroscopic technique that provides valuable complementary information on the band structure of solids. Indeed, because the selection rules for two-photon transitions are di erent from those ruling linear absorption, the excited states which are not accessible in conventional linear optics can be observed. Also, the added opportunity in varying the relative polarization and the frequencies of the absorbed photons increases the flexibility of the two-photon technique.

For a monochromatic light wave A = e−iωtAe+ c.c. propagating in a bulk intrinsic semiconductor, the second-order perturbation theory yields the twophoton transition rate in the form

Here, A, e, ω are the scalar vector-potential amplitude, polarization unit vector and frequency of the light wave, respectively, c and v are the conduction and valence band indices for which the argument of the δ-function can vanish. The second-order (composite) matrix element of the two-photon transition is defined as

prk,vk is the matrix element of the momentum operator taken between the Bloch states (r, k) and (v, k), and the index r stands for all possible intermediate states, both filled and unoccupied. Note that we take into consideration only dipole-allowed transitions and assume the photon wave vector to be zero.

In the two-quantum process described by (7.2, 7.3), the first photon is absorbed creating a virtual electron-hole pair. The absorption of the second photon then leads to the virtual state being taken into the final pair state. Here, for simplicity, we ignore the Coulomb interaction modifying the freepair excited states into the discrete and continuum excitonic states.

The two-photon absorption coe cient K(2) is related to W (2) through

and with the nonlinear susceptibility, through the expression

, '

K(2) Im χ(3)αβγδ (−ω, ω, ω)eαeβ eγ eδ I .

Dual-beam two-photon spectroscopy employs two light sources of high and low power densities [7.1]. Usually, the pump beam is fixed and the other, the probe, is allowed to vary its frequency. The frequencies ω1, ω2 of the pump and probe beams are chosen to satisfy the conditions

ω1 < Eg /2 , ω2 < Eg < ω1 + ω2 .

The first condition prevents the absorption of two photons from the intense beam, i.e., W (2)(e1, ω1) = 0. Due to low intensity of the second beam, its two-photon absorption can be ignored and, therefore, the main contribution to the interband optical transitions comes from simultaneous absorption of two photons from di erent beams. For the biharmonic vector potential

A = e−iω1tA1e1 + e−iω2tA2e2 + c.c. ,

the two-photon probability rate is given by

& 2

×&&Mcv(2)(e1, ω1; e2, ω2; k)&& δ(Eck − Evk − ω1 − ω2) ,&

kcv

It is worth to mention that, if the light wave is quasimonochromatic, the probability rate (7.2) derived for a single laser source should be multiplied by

328 7 Nonlinear Optics

the correlation factor g2 = I2 /I2, where the angular brackets denote averaging over the intensity distribution (with I ≡ I). For multimode radiation with a random phase distribution, g2 = 2, and for a biharmonic wave with close frequencies, ω1 ≈ ω2, one has

g2 = 1 + 2I1I2/I2 ,

where I1 and I2 are intensities of the biharmonic components, the total intensity being I = I1 + I2.

Several methods are used to study two-photon absorption including measurements of the light transmission through the sample, the two-photon- absorption-induced photoluminescence and the change in the conductivity under two-photon excitation.

Transitions v → c which occur in real semiconductors can reach the final state by using any path through every intermediate state r which is available. In typical direct-gap semiconductors with electric-dipole-allowed interband optical transitions at the extremum point Γ , the two-photon absorption in the frequency region near the fundamental edge is mainly contributed by the intermediate states r in the initial and final bands, v and c, respectively. Therefore, in (7.3), both interband and intraband matrix elements of the momentum operator appear. At small values of the electron wave vector k, the former matrix elements are constant whereas the latter are proportional to k. In other words, for allowed one-photon interband transitions, the twophoton transitions are “allowed-forbidden”. For interband transitions near the fundamental edge Eg , the frequency dependence of the oneand twophoton transitions rates (or absorption coe cients K(1), K(2)) is given by

It is worth to mention that, in noncentrosymmetric crystals, the allowance for linear-k spin-dependent terms in the intraband electron Hamiltonians as well as the intermediate states r in the third bands di erent from v and c leads to an “allowed-allowed” contribution to Mcv(2). However, this contribution practically plays no essential role and can be neglected. Now, if the Coulomb interaction is taken into account and the final excited state is an exciton, then s-excitons are observed in one-photon spectra while the two-photon spectra reveal p-exciton states not accessible in linear optics.

Equations (7.2, 7.3) or (7.5, 7.6) can be used to describe two-photon absorption in nanostructures as well, while bearing in mind that the indices c, r and v should include the numbering of minibands in SLs, subbands in QWs [7.2–7.5] or QWRs [7.6, 7.7] and discrete levels in QDs [7.8, 7.9]. A unified theory for two-photon absorption spectra of Wannier–Mott excitons in a low-dimensional semiconductor of an arbitrary dimension d = 0, 1, 2, 3 is presented in [7.10].

In the case of a single QW, the summation in (7.2) is performed over the 2D electron wave vector k . Note that the two-photon absorption rate,

WSQW(2) , in a QW is the number of 2D electrons created per unit area per unit time. In MQWs, similarly to (4.6), the two-photon absorption coe cient is

connected with WSQW(2) by

K(2)(ω) = 2 ω WSQW(2) (ω) .

I(a + b)

If the quantum-confinement energy is small compared to the band gap Eg ,

(2)

then the calculation of the composite matrix element Meν s,vνj (k ) for a QW does not require a special summation over virtual states. For instance, for

two-photon transitions between simple bands v, Γ7 and c, Γ6 in a GaAs-based QW, one has

entering (7.2). Optical transitions between the valence and conduction sub-

bands with coinciding index ν belong to allowed-forbidden type and, at small

k , Meνs,vνj(2) (k ) k . The optical transitions with ∆ν ≡ ν − ν = ±1 are forbidden for one-photon absorption and allowed-allowed for two-photon ab-

sorption. Bearing this in mind, we come to conclusion that, in a 2D nanostructure, the exponents 1/2 and 3/2 in the frequency dependence of the transition rates (7.7) are changed, namely

for transitions with ∆ν = ±1. Here the abbreviations “a”, “f”, “a-f” and “a-a” stand for “allowed”, “forbidden”, “allowed-forbidden” and “allowedallowed”, respectively.

Figure 7.1 presents one-beam two-photon excitation spectra of the photoluminescence due to recombination of the exciton e1-hh1(1s) in GaAs/AlGaAs MQWs for two polarizations e z (a), and e z (b). At the same time, one of the curves in Fig. 7.1a shows the frequency dependence of one-photon photocurrent with a clearly pronounced feature at the resonance frequency of the exciton e1-hh1(1s), or H1-C1(1s) in notation of [7.11]. In accordance with the selection rules for allowed-forbidden two-photon excitation of excitons, the strongly one-photon-allowed 1s state of the e1-hh1 exciton is absent in the two-photon spectrum. The onset of two-photon absorption occurs at the 2p-exciton feature shown as the shaded area in Fig. 7.1a. The 2p assignment is confirmed by the weak 2s-exciton feature seen in the linear spectrum.

330 7 Nonlinear Optics

Fig. 7.1. (a) Single-photon (right scale) and two-photon (left scale) light absorption spectra in GaAs/Al0.35Ga0.65As MQW structure. Single-photon spectrum was measured at normal incidence (e z). The dotted line below shows that for the

˚

sample with a = 40 A there is no two-photon absorption in e z polarization within the wavelength range studied. Vertical lines indicate the predicted positions of exciton resonances. The vertical arrow indicates the threshold of hh1 → e1 intersubband transitions. (b) Two-photon light absorption spectra for a similar struc-

˚

ture but with wider QWs (a = 110 A). The solid and dotted lines were measured in e z and e z polarizations, respectively. The solid and dotted vertical lines indicate predicted positions of exciton resonances which are active, accordingly, in two-photon allowed-forbidden and one-photon allowed transitions. From [7.11].

The linear increase of two-photon absorption at higher energy as well as the plateau seen in linear absorption is in agreement with (7.9) for d = 2.

In the polarization e z, other properties specific to the quantum confinement become apparent. In this case the two-photon transitions between subbands with the same index ν are not allowed, see (4.3). The 1s-hh2 transition is also suppressed because of the selection rules for optical transitions from heavy-hole-subbands (Table 2.2). Therefore, for e z, the onset of twophoton absorption occurs at the e1-lh2 transition. The next allowed transition is from lh1 to e2. The excitonic peaks e1-lh2(1s) and e2-lh1(1s) are followed by plateaus due to allowed-allowed two-photon transitions into the corresponding continuum states, in contrast to the linear frequency dependence for the e1-hh1 allowed-forbidden transitions.

The above selection rules are valid for symmetrical QWs. Shimizu [7.3] discussed what changes are expected in the two-photon spectra if one applies a static electric field F normal to the interface plane. In addition to the red shift due to the quantum-confined Stark e ect (Sect. 3.4.1), selection rules for ν − ν are removed, since F causes parity mixing in subband envelope functions. An experimental study of electric-field-induced changes in the twophoton absorption spectra is performed by Fujii et al. [7.12]. Particularly, in agreement with theory, they observed growth and redshift of the exciton peak 1e-lh1(1s) which vanishes at zero bias.