ivchenko_bookreg

Fr¨ohlich et al. [7.13] performed two-photon magnetoabsorption measurements in GaAs/AlGaAs QW structures using as an intense radiation source a CO2 laser with the photon energy ω1 = 0.117 eV that is much smaller than the band gap Eg . In these conditions, the transitions playing the dominant role in two-photon absorption are v + ω2 → c + ω1 → c in which case the photon ω2 is absorbed at the first step, and v + ω1 → v + ω2 → c, where the interband transition is also induced by the photon ω2, and the intermediate states lie in the valence band. As a result, the principal contribution

proportional to (e1 ·k )e2. Therefore, one has the following selection rules for the transitions hh1 → e1 for circularly polarized CO2 radiation: Nc −Nv = 1 for σ+ polarization, and Nc − Nv = −1 for σ− polarization, where Nc,v are the Landau-level indices in the conduction and valence subbands. The twophoton selection rules for inter-Landau-level transitions allowed Fr¨ohlich et al. to determine the electron and hole cyclotron frequencies separately and, therefore, the e ective masses me and mhh.

Cingolani et al. [7.6] studied two-photon absorption in GaAs/AlGaAs

˚× ˚

quantum well wires of the 100A 600A cross-section along the x and y confined directions. In the studied sample, the wire width, Ly , along y is roughly 4 times longer than the 2D-exciton Bohr radius. Therefore, 1s-excitons behave almost like 2D particles quantized as a whole along y while 2p-excitons, having a larger extent of the relative motion, exhibit almost 1D character. The experiment [7.6] provides evidence for the strongly anisotropic selection rules in the in-plane polarizations e y and e z y. In particular, in the e z geometry the matrix elements of the two-photon process (hh1, ny ) → (e1, ny ) do not vanish only for transitions with ny = ny , where ny is the index of lateral confinement, and are proportional to kz (allowed-forbidden transition). Conversely, for e y only allowed-allowed transitions with ∆ny ≡ ny −ny = 0 are possible. In terms of the e1-hh1(1s) and e1-hh1(2p) excitonic final states, this means that 1s-excitons are excited in the e y geometry for ∆ny = 0 two-photon transitions whereas 2p-excitons are accessible for ∆ny = 0 twophoton transitions in the polarization e z.

For an array of spherical QDs randomly distributed in the 3D medium, one can introduce the e ective absorption coe cient K(2) which is related to

Here N is the concentration of QDs, R is the dot radius and f (R) is the size distribution function. Analytical expressions for the absorpion coe cient K(2)(ω) are derived by Fedorov et al. [7.9] for the strong confinement regime and the Lifshits-Slezov distribution

6

if R < 3 , f (R) = 2

if R > 32 ,

332 7 Nonlinear Optics

where R = R/R0 and R0 is the average QD radius.

7.2 Biexcitonic Nonlinearity

In nonlinear optics, there are two fundamentally di erent mechanisms of biexciton photogeneration. The first mechanism that dominates in studies of biexcitons in long-time scale measurements, e.g., photoluminescence measurements, is the binding of two incoherent excitons from a population of excitons. It is described by using the kinetic theory and calculating the exciton distribution function and the generation rate of biexcitons. It should be mentioned that generation of two spatially separated uncorrelated excitons and subsequent biexcitonic binding is possible only in systems with at least one direction of free motion, in bulk semiconductors and QW or QWR structures, but not in QDs. In the other mechanism, biexcitons are generated through a coherent two-photon excitation. Here we consider the latter mechanism on the basis of a three-level model including the ground state |0 , the excitonic state |X and the biexciton state |XX [7.14, 7.15]. Taking into account the biexcitonic nonlinearity, the linear equation (3.44) for the exciton averaged polarization P ≡ PX in a QW is extended to

where Γ, ξ, Γ0 are the same parameters as in (3.44), B is the amplitude of the biexcitonic wave function, and γbi is proportional to the matrix element of the photon-induced exciton-biexciton transition. In addition to (7.11), an independent equation for B should be written. It has the form

where δbi ≡ εbi, Γbi are the biexciton binding energy and damping rate. Moreover, the relation (3.40) between E and PX must be generalized to

where PXX = (γbi/ξΓ0)BPX is the dielectric polarization due to the biexcitonexciton optical transition.

Assuming Γ0 Γ we can consider E as an external electric field. Then, in the lowest nonvanishing approximation, the biexciton amplitude under single-beam optical excitation is given by

and, therefore,

The same result can be obtained by using the equation for the rate of twophoton excitation of biexcitons

ments proportional to the electric field E.

Under normal incidence upon QWs of the D2d symmetry, the polarization dependence of the two-photon probability rate is characterized by three

variant of the D2d point group and, for e z, linearly-independent invari-

and is expressed is terms of the chosen set of invariants. We remind that onephoton absorption for e z is independent of the light polarization state. In the uniaxial approximation, the coe cient a3 vanish and WQW(2) is insensitive to the in-plane polarization of the linearly polarized light. However, even in this case two-photon absorption is di erent for the linear and circular polarizations

For allowed-forbidden two-photon transitions into the continuum electronhole pair states, the ratio Λ(2) ≡ WQW(2) (lin)/WQW(2) (circ) characterizing the linear-circular dichroism di ers from unity by few tens percent. The observed biexciton state is a singlet with the electron spins (as well as the hole angular momenta) being antiparallel. Hence, the two photons of the same circular polarization cannot excite biexcitons, the coe cient a1 in (7.15) vanish and

Λ(2) → ∞.

334 7 Nonlinear Optics

In 1970s, two-photon absorption measurements brought new experimental evidence for the existence of biexcitons, or excitonic molecules, in bulk semiconductors [7.16]. The two-photon PhotoLuminescence Excitation (PLE) spectrum reached a maximum at a frequency separation equal (ω0 − δbi/2) in agreement with the energy conservation law 2ω0 − δbi = 2ω. The similar e ect was observed on QW structures, ZnSe/ZnMnSe MQWs [7.17] and GaAs/AlGaAs SQWs [7.18, 7.19]. Using a micron-sized photoluminescence probe Brunner et al. [7.18] studied excitons and biexcitons bound to

˚

a single island-like defect, a 4 A deep protrusion of the GaAs QW into the

˚

Al0.35Ga0.65As barrier layer with the lateral size of about 400 A. The PLE spectrum detected at the biexciton emission line ω0 − δbi shows the resonant behavior at ω = ω0 −δbi/2. In the studied structure, the excitation energy of the e1-hh1 ground-state exciton equals ω0 = 1.6544 eV and the biexciton binding energy δbi = 4.2 meV. The full width at half maximum of the twophoton resonance at ω = 1.6565 eV is extremely narrow and even limited by the linewidth of the exciting laser beam, which is characteristic for the 0D excitations.

7.3 Degenerate Four-Wave Mixing

Time-resolved degenerate four-wave mixing has proven to be a powerful tool to provide much information on the exciton dynamics and loss of coherence as well as on nonlinear mechanisms of exciton-photon interaction in bulk semiconductors and QW structures. In a typical two-pulse self-di raction setup (see inset in Fig. 7.2), a sequence of two coherent pulses 1 and 2 with the wave vectors k1, k2 are tuned to the exciton resonance, interfere and produce an excitonic grating. One of the two pulses, say, the pulse 2, is di racted by this grating into the direction k3 = 2k2 −k1. The magnitude of the di racted signal is then recorded as a function of the time delay T = t2 − t1 between the pulses.

Let us consider the four-wave mixing in a single QW sandwiched between semiinfinite barrier layers. We present the electric field of the incident twopulse radiation in the form

˜(j)

where ω˜ is the current frequency of the coherent light pulses and E0 (t) (j = 1, 2) are the slowly varying amplitudes. For example, for Lorentzian pulses one can write the electric field inside the QW in the form

! "

E(x, t) = e−iωt˜ E0,1e−|t−t1|/τp eik1x x + E0,2e−|t−t2|/τp eik2x x + c.c. ,

where E0,j are time-independent vectors. We assume (x, z) to be the plane of incidence in which case kj = (kjx, 0, kjz ). Usually in four-wave mixing

Fig. 7.2. Degenerate four-wave mixing signals in GaAs/AlGaAs MQWs vs. time delay for di erent temperatures and low excitation intensity. Inset: experimental configuration. From [7.20].

experiments the k1 and k2 vectors make small angles with the sample normal z. Thus, one may ignore an e ect of the z component of the electric field and neglect angular dependence of the reflection and transmission. In the following we omit the exponential functions exp (ikj,xx) bearing in mind that

2k2 − k1 independently detected.

For simplicity, we first assume the configuration of collinear polarized

dynamics in the QW are written as

where T2 = (Γ + Γ0)−1 is the polarization dephasing time and δ = ω˜ − ω0 is the laser detuning. Substituting the solutions (7.21) into (7.20) we obtain

FN L(t) = (ξΓ0)2N1 N22θ(t − t1)θ(t − t2)e−iδT e−(2t−t1−t2)/T2

! "

× iβ1ξΓ0eiδ(t−t2)e−(t−t2)/T2 + β2δ(t − t2) ,

˜(3)

where β1 = β1 + (i/ξ)β2. We will not write the full solution for P (t) but present only its dependence on the time delay T

where the functions F± are independent of T . It follows then that the timeintegrated four-wave mixing signal

where S±(3) are coe cients proportional to |N1|2|N2|4. One can see that the slope of the function ln S(3)(T ) is independent of the detuning δ and, hence, the measured signal is controlled by the homogeneous dephasing time T2 but not by the inhomogeneous broadening of the excitonic transition.

Figure 7.2 shows the time-integrated di racted signals versus the time

˚

delay taken from a 170-A GaAs/AlGaAs MQW structure. The signals are asymmetric in time delay and consist of exponentially rising (T < 0) and decaying (T > 0) wings. In accordance with (7.24) the rise time, τR, is about a factor of 2 smaller than the decay time, τD. Particularly, at low temperatures τR ≈ 650 fs and τR ≈ 1150 fs. With increasing lattice temperature, both the decay and rise times get shorter keeping the ratio ≈ 2.

According to (3.48, 3.49), in a QW structure with a top layer of finite thickness, the dephasing rate T2−1 is modified to

338 7 Nonlinear Optics

where the notations are the same as in (3.49).

In Chap. 3 we have shown that the dynamics of optical excitations in MQWs di ers considerably from that in a nominally identical single QW and depends strongly on the number of QWs, N , and the interwell separation, d. A special situation arises in a resonant Bragg MQWs structure, where the period equals an integer multiple of half light wavelength at the exciton resonance frequency. In such a structure there exists a superradiant mode which is characterized by an N times enhanced radiative decay rate, see (3.92, 3.98). In Sect. 3.2.5 we discussed linear optical properties of the resonant Bragg structures. H¨ubner et al. [7.21] used femtosecond degenerate four-wave mixing to study light-induced collective e ects in MQWs and clearly demonstrated e cient radiative coupling of excitons in high-quality Bragg MQW samples. The single QW displays an exponential signal yielding the decay time τD = 4.3 ps. In agreement with the theoretical prediction, the MQWs exhibit an initial-stage decay which is remarkably faster as compared to the signal from the single QW. The experimentally observed two-stage decay of the Bragg structures indicates slight deviation of the periodicity from λ(ω0)/2. Note that the four-wave mixing in semiconductor microcavities is separately considered in Sect. 7.5.

Recently, Shackelette and Cundi [7.22] have reexamined di erent nonlinear contributions to the optical Bloch equations in the density-matrix approach, including the saturation, local-field e ect, excitation-induced shift of ω0 and excitation-induced dephasing. They ignored the biexciton non-

˚

linearity because, in the studied MQW structures containing 83-A-thick

˚

In0.14Ga0.86As wells and 80-A-thick GaAs0.71P0.29 barriers, the same spectrallyresolved four-wave mixing signals were obtained for collinearly and cocircularly polarized pulses. However in many other heterostructures there exists strong biexcitonic contribution to four-wave mixing in the direction 2k2 −k1 [7.23–7.26]. The first indication of this contribution in GaAs-based MQWs was discovered for negative delays by Feuerbacher et al. [7.23]. In their experiment the nonlinear interaction can be explained as a two-step process. In the first step, two photons of pulse 2 create a biexciton which then, in the second step, interacts coherently with pulse 1 and gives the measured four-wave mixing signal. Breunig et al. [7.26] have clearly demonstrated the coherent control of the biexcitonic polarization in measurements of the time-integrated spectrally-resolved four-wave mixing on a ZnSe/ZnSxSe1−x single QW. The maximum energy of the laser pulses was set to the biexcitonic resonance and the narrow spectral width of the pulses allowed for exclusive excitation of the heavy-hole exciton-biexciton system. In the spectrally-resolved four-wave mixing signal, both the excitonic and biexcitonic resonances are observed for cross-linearly polarized pulses while the biexcitonic resonance is strongly suppressed in the case of cocircular excitation in agreement with spin selection rules for the formation of a bound biexcitonic state.

In addition to four-wave mixing, one can observe six-wave mixing in the direction 3k2 − 2k1 or 3k1 − 2k2 due to the second-order di raction of the pulse pair [7.27, 7.28]. To describe this nonlinear signal one needs to start in the expansion (7.1) from the fifth-order susceptibility χ(5) and consider participation of six waves in the buildup of a polarization in the corresponding direction. A microscopic theory based on the dynamics controlled truncation scheme, see for details [7.27], is able to reproduce the experimental results.

7.4 Second-Harmonic Generation

The second-harmonic generation is described in (7.1) by the third-rank tensor χ(2)(ω1, ω2) with ω1 = ω2 ≡ ω. It is an elementary process of the annihilation of two light quanta ω and the creation of one new quantum with twice the energy. Obviously, the second-harmonic power is quadratic in the incidentpump power, as experimentally checked, e.g., by Seto et al. [7.29]. Neglecting

the spatial dispersion of χ(2)(ω, ω) we conclude that the components χ(2)αβγ are symmetrical with respect to interchange of the two last indices,

Therefore, neglecting the spatial dispersion, the second-harmonic generation occurs only in piezoelectric crystals and nanostructures of the point symmetry belonging to the piezoelectric classes. For bulk crystals of the tetrahedral point group Td, the symmetry reduces the number of linearly independent elements of χ(2)(ω, ω) to one and, in a frame of references with the axes x, y, z along the basic crystallographic directions [100], [010] and [001], one has

In a (001)-grown QW of the symmetry D2d, there are two independent components

The point-group C2v of an asymmetrical QW allows five linearly independent

components χ(2)x x z = χ(2)x zx , χ(2)y y z = χ(2)y zy , χ(2)zx x , χ(2)zy y

340 7 Nonlinear Optics

χ(2)zzz that is zero in bulk and originates from the superstructure potential asymmetry, other components are nonzero also in a bulk semiconductor, they originate from both the bulk and structure inversion asymmetry. The symmetry increases from C2v to C4v when one uses the parabolic Hamiltonian for electrons in the conduction band, the Luttinger Hamiltonian for the valence band neglecting odd-k terms and the Bastard boundary conditions (2.34). In this case χ(2)x x z and χ(2)y y z or χ(2)zx x and χ(2)zy y equalize and the constitutive equations take the form [7.30]

Here Vd is the volume of the d-dimensional sample, i1, i2, i3 denote quantum numbers labelling the electron states |i and the eigen energies Ei, Ei2i1 = Ei2 −Ei1 , fi1i3 = fi1 −fi3 , and fi is the Fermi-Dirac distribution, the curly brackets {...}s mean a symmetrized form, e.g., {Rβγ }s = (Rβγ +Rγβ )/2. In general, the existing theoretical and experimental studies of the secondharmonic generation in nanostructures can be divided into two groups, depending on the light frequency ω: those dealing with energies ω in the region of valence-to-conduction interband transitions (see, e.g., [7.30, 7.33, 7.34] and references therein) and the second group using the photon energy in the region of intraband intersubband transitions [7.35–7.37]. In all cases, an enhancement of the nonlinear susceptibility as compared to the susceptibility of bulk materials was reported.

Atanasov et al. [7.30] addressed the problem of second-harmonic generation in asymmetrical QWs in the region of interband transitions including exciton e ects. In the excitonic contribution to χ(2), the energy di erences Ei2i1 , etc., in the denominators of (7.29) are replaced by the excitation energies of excitons attached to a pair of the corresponding subbands. As shown in [7.30], for step-like and double asymmetrical QWs, the inclusion of

exciton states leads, particularly, to the sharp half-band-gap resonances atω = Eeν ,hν;1s/2 for χ(2)x x z (ω), χ(2)zzz (ω) and to the near-band-gap resonances at ω = Eeν ,hν;1s for χ(2)zx x (ω). Note that the resonant peaks of |χ(2)zzz (ω)| are related with light-hole excitons only because the interband transitions from

heavy-hole subbands are forbidden in the e z polarization. The theory predicts values of second-harmonic-generation susceptibility larger than in the