ivchenko_bookreg

the pseudospin precession around Ω leads to a depolarization of the initial spin component perpendicular to Ω while the component parallel to Ω remains unchanged. This means that the steady-state value of S is obtained by projecting S0 onto the Ω direction, i.e., S = Ω(Ω ·S0)/|Ω|2. Therefore, the relation between the secondary and primary polarizations can be presented in the following form

ˆ

with the matrix Λ given by

One can see that, for nonzero anisotropic exchange splitting δ2, the longitudinal magnetic field restores the PL circular polarization and induces the linear polarization under circularly polarized excitation. It also follows that, under linearly polarized excitation, the magnetic field gives rise to two additional e ects: suppression of the alignment and polarization conversion with an appearance of circular polarization in the PL.

In [5.85], instead of measuring the PL polarization degrees Pl, Pl and Pc under a fixed position of the polarizer, the modulation technique is applied where the analyzer is in a fixed position and the sample is pumped by the incident light changing its polarization from circular or linear to orthogonal at a certain frequency. The measured values are then the e ective polarization degrees

where Iβα designates the PL intensity in the configuration (α, β) of the polarizer and analyzer, and α, β are linear polarizations along the axes [100],

¯

[010], [110], [110] or circular polarizations σ+, σ−. The theory shows that, under resonant excitation conditions and neglecting the level anticrossing effects, the set (5.160) is equivalent to the values of Plα, Plα and Pcα, where α indicates the polarizer position.

Figure 5.13 displays the dependencies ρcα(Bz ) and ρlα(Bz ) measured on one of the type-II GaAs/AlAs SL samples at the PL spectral maximum (λ =

urates from 2.5% to 5% in weak magnetic fields B ≈ 20 G and then gradually increases up to the level of 20% at Bz = 2.5 kG. Figure 5.13b clearly demonstrates the field-induced orientation-to-alignment conversion: ρc110(Bz )

272 5 Photoluminescence Spectroscopy

Fig. 5.13. E ect of the longitudinal magnetic field on optical orientation and alignment of localized excitons in a type-II GaAs/AlAs SL: (a)ρcσ+ , (b) ρc110 , (c) ρl110 , (d) ρlσ+ . Experimental data (T = 4.2 K) are shown as points. Solid curves are theoretical fits. From [5.85].

reaches a maximum value of 5% at Bz ≈ 0.7 kG and reverses its sign under the field inversion. Moreover, ρc110(Bz ) di ers in sign from the measured

dependence ρc110¯ (Bz ). The e ect of longitudinal magnetic field upon the optical alignment is illustrated in Fig. 5.13c. Note that the main variation of

ρl110 takes place at the same magnetic fields Bz ≈ 0.7 kG as for the function ρcσ+ (Bz ) in Fig. 5.13a. Figure 5.13d shows that the orientation-to-alignment

conversion e ect is reversible: the experimental dependencies ρc110(Bz ) and ρlσ+ (Bz ) are close to each other.

The solid curves in Fig. 5.13 are theoretical fits by using the theory taking into account the exchange splitting with Ω1 = 0, Ω2 = 0 in accordance with the mechanism of anisotropic exchange interaction in GaAs/AlAs shortperiod type-II SLs discussed in Sect. 5.5.1. The fast low-field increase of ρcσ+ in Fig. 5.13a is attributed to a spatially separated electron-hole pairs characterized by small values of exchange splitting. In the analysis this contribution is taken into account by adding a constant value of 5% to the theoretical curve ρcσ+ (Bz ). Except for this narrow region the experimental data can be described taking additionally into consideration the exciton spin relaxation between the radiative and nonradiative states and possible losses in the orientation and alignment during the quasi-resonant photoexcitation process [5.85]. Since in type-II GaAs/AlAs SLs there are two kinds of localized excitons with Ω1 di ering in sign, the calculated circular-to-linear and linear-to-circular conversion terms have to be multiplied by the imbalance

factor

N (+) − N (−) f = N (+) + N (−) ,

where N (±) denotes the concentration of excitons localized at the AlAs-on- GaAs and GaAs-on-AlAs interfaces, respectively. Thus, the magnetic-field- induced conversion between the circular and linear polarizations suggests an e ective method to measure an important structural parameter, namely, the imbalance factor f . The curves in Fig. 5.13 are calculated for δ2 = 1.8 µeV and f = 0.9. Since in type-II GaAs/AlAs SLs the frequency Ω2 is zero and Ω1τ 1, the theory in full agreement with experiment predicts no optical alignment in the cw excitation regime for the incident light polarized along the [100] or [010] axis, Λl l = 0, and no l -c or c-l polarization conversion,

Λcl = Λl c = 0.

5.5.4 Photoluminescence of Neutral and Charged Quantum Dots

Figure 5.14 displays the e ective polarization degrees of photoluminescence ρlα, ρlα, ρcα measured as a function of the longitudinal magnetic field (the Faraday geometry) on InAlAs QDs in an AlGaAs matrix [5.104]. The PL

˚

was recorded at the wavelength 6890 A under quasiresonant excitation with

˚

λ = 6764 A. The main experimental findings are observations of (a) the optical alignment of excitons for any in-plane direction of linear polarization

274 5 Photoluminescence Spectroscopy

of the exciting light, and (b) magnetic-field-induced conversion of the 110 alignment to orientation but no similar conversion of the 100 alignment. The MBE-grown InGaAs/GaAs-like QDs have a shape of pyramids with the height parallel to the growth direction z [001] and the base oriented along the 100 directions [5.89]. In case of the square base, QDs are characterized by the C2v point symmetry for which Ω1 = δ2 cos Φ2 = 0 and Ω2 = δ2 sin Φ2 = 0. In the general case of a rectangular base the symmetry is reduced to C2 and Ω1, Ω2 are both nonzero. The experimental results on InAlAs QDs can be explained by assuming that the positive and negative values of Ω2 are equally probable, the average value of Ω1 is nonzero, and the mean-square values of Ω12 and Ω22 are comparable. It follows then that the components Λij odd in Ω2 vanish after averaging over the ensemble of

QDs. As a result the secondary and primary polarizations are related by the matrix Λ(+)ij where

and the angle brackets mean averaging over the distribution of Ω1 and Ω2. While calculating the theoretical curves in Fig. 5.14 the Gaussian distribution

was assumed, allowance for nonzero average value of Ω1 was made and the

˜ ˜

dispersion parameters Ω1, Ω2 were found from the fitting procedure.

The studies of spin dynamics in self-organized InAs/GaAs QDs under pulsed photoexcitation supplement the continuous wave QD spectroscopy experiments. The best fit of the experimental data on time-resolved optical orientation, optical alignment and linear-to-circular polarization conversion leads to the anisotropic exchange splitting δ2 = 135 eV in InAs/GaAs QDs obtained after a nominal deposition of 2.2 InAs monolayers [5.105]. These experiments also evidence a spin relaxation quenching in semiconductor QDs at low temperature compared to bulk or 2D structures and bring experimental support to proposals using electron spins in QDs for quantum information encoding and processing in a solid-state system.

Cortez et al. [5.106] have demonstrated a possibility to manipulate the spin of the resident electron in an n-doped QD using nonresonant optical excitation. The time dependence of the luminescence and circular polarization of the ground state emission under excitation in the wetting layer is shown in Fig. 5.15. To interpret the set of obtained experimental results the following scenario can be proposed providing a simplified description of the three-particle complex (or QD trion) dynamics. Let us consider that the dots contain ideally a single resident electron and that at most a single electron-

276 5 Photoluminescence Spectroscopy

Fig. 5.15. Top: time-resolved photoluminescence detected at 1.15 eV under circularly polarized excitation at 1.44 eV of the n-doped QD sample. The inset shows a blowup of the fast dynamics at short times. Bottom: the corresponding circular polarization is shown (dark line), as well as the polarization in the presence of a counterpolarized pump pulse 10 ns earlier (grey line). From [5.106].

will further thermalize to the ground state s↓c . This explains the observation of copolarized PL at the very beginning of the recombination process (see inset in Fig. 5.15). The anisotropic electron-hole exchange interaction causes a simultaneous spin reversal of the hole and one electron. This flip-flop is followed by an irreversible relaxation of the electron subsystem to the S0 singlet state. The radiative recombination of the resulting frozen trion produces then counterpolarized light, due to the hole spin state. This path is much faster than the direct recombination. It is this mechanism that is assigned to

the reversal of PL polarization after a few tens of picoseconds in Fig. 5.15. In the case of the dark trion T−1 with the hole s↓h the most likely process is the single particle spin flip of an electron, maybe due to the e ect of spin-orbit interaction and dot shape anisotropy. This electron spin-flip relaxation is due to the slow increase of counter polarization during the recombination lifetime. Finally one can conclude that the spin of the electron remaining in the dot after recombination of the photoinjected electron-hole pair is polarized, with an increase of the spin ↓ population (for σ+ excitation). This allows to write and read the spin state of the resident electron in n-doped QDs.

5.6 Interface-Induced Linear Polarization of

Photoluminescence

In Sect. 3.3.3 we considered the in-plane anisotropy of type-I heterostructures where both electrons and holes are confined within the same layers. Now we will concentrate on type-II heterostructures with no-common atoms, like InAs/AlSb [5.107, 5.108], ZnSe/BeTe [5.109, 5.110], CdS/ZnSe [5.111], where electrons and holes are confined in adjacent layers and the interband optical transitions are indirect in the real space. The PL measurements in type-II structures are preferential because, for the indirect transitions, the oscillator strength is very small and the transmission experiments are much less sensitive to the possible anisotropy.

Figure 5.16 demonstrates a giant in-plane optical anisotropy of BeTe/ZnSe heterostructures. In order to detect the anisotropy signal induced by a single interface the PL from high-quality ZnSe/BeTe double barrier structure was studied. The layer sequence is shown in Fig. 5.16c together with current-voltage characteristics. The samples are as symmetrical as possible and comprise lattice matched BeZnSe contact layers, undoped ZnSe spacer layers. They act as electron emitters under applied bias voltage and BeTe/Zn1−xMnxSe/BeTe double-barrier structure with x = 0 or x = 0.1. All the normal and inverted interfaces are grown under Zn and Te termination and contain the Zn-Te and Te-Zn chemical bonds. PL is excited by an Ar-ion laser. The heterostructures demonstrate pronounced resonant-tunneling features which become less pronounced under illumination. Figures 5.16a and 5.16b present the PL spectra measured for two studied samples and detected

¯

in the linear polarization, Pl, along the [110] and [110] axes. The emission lines EL and ER originate from the spatially indirect transitions involving electrons from the ZnSe emitters and photoholes from the BeTe layers. The lines WL, WR observed only in the second sample are due to the quantumwell electrons. Splitting between the EL and ER lines can be attributed to small charge-asymmetry of the emitters.

One can see that the lines EL and ER are strongly polarized and, moreover, polarized orthogonally. The interpretation of these lines is confirmed by their behavior in an electric field under applied voltage. Positive electric

fields push o electrons from the rightmost interface and form a triangular potential near the leftmost interface pressing electrons towards this interface. As a result the relative intensity of the EL line increases and that for the ER line vanishes. A field reversal leads to an exchange of the role played by the interfaces, in this case it is the line ER that survives at negative biases. The surviving line is as strongly polarized as it was at zero bias. Thus one concludes that the radiation contributed by indirect optical transitions at a particular interface between BeTe and ZnSe has very high linear polarization and this polarization is almost insensitive to the electric field. The fact of nonzero lateral polarization is a natural consequence of the C2v point symmetry of a single interface in a (001)-grown heterostructure. However, giant values of this polarization need a theoretical explanation.

The temperatureand incident-power dependencies of the PL polarization in ZnSe/BeTe MQWs was studied in [5.110]. Complete sets of data on time-resolved and time-integrated spectra of the PL intensity and linear polarization Pl were obtained. The aim was to clarify whether the observed polarization is induced by localization of carriers at anisotropic defects and interface imperfections or it is an intrinsic property of the heterostructure. The crucial point is the stability of polarization against changes in various external and internal conditions, namely, against an increase in the excitation power by more than 7 orders of magnitude, against the temperature increase from 1.7 K to 300 K, and, in addition, there are no remarkable changes in the linear polarization with applying an external magnetic field up to 7 T. Under such conditions the PL spectrum exhibits tremendous modifications, its intensity changes by many orders of magnitude, the spectral maximum moves upwards or downwards by 200÷400 meV and the PL band halfwidth varies within broad limits as well. In contrast, the PL linear polarization degree remains stable varying between the limits of 50% and 75% and is almost constant within the halfwidth of the spectral band. All this means that any theoretical interpretation of the available experimental results should be based on intrinsic microproperties of a single (001) heterojunction between two semiconductors with a zinc-blende-like lattice.

In type-II ZnSe/BeTe systems the conductionand valence-band o sets are very large. The penetration depth for an electron into the BeTe layer and for a hole into the ZnSe layer has the order of the lattice constant. Therefore in such a system the wave functions of an electron and a hole participating in the spatially indirect radiative recombination overlap remarkably only over few atomic planes. For calculations of optical matrix elements in this case the conventional envelope-function approximation is invalid. One must instead use microscopic pseudopotential or tight-binding models.

In the following we outline a tight-binding theory suitable for the calculation of interband optical transitions on a type-II heterojunction [5.112]. Let us consider a periodic CA/C A heterostructure grown along the axis [001] and consisting of alternating layers of binary compounds CA and C A with

280 5 Photoluminescence Spectroscopy

di erent cations and anions. The electron wave function in the tight-binding method is written in the form

n,α

Here φnα(r) are the planar orbitals, n is the number of the atomic plane, and α is the orbital state index. For clarity, the coe cients in expansion (5.161) are supplied with an additional superscript b = a for the anion (n = 2l, where l is an integer) and b = c for the cation (n = 2l + 1). For the states with a zero lateral wave vector, i.e. for the states with kx = ky = 0, the planar orbitals are related to the atomic orbitals Φbα by

φnα = Φbα(r − an − n1o1 − n2o2) ,

n1,n2

where n1, n2 are arbitrary integers, o1 = (a0/2)(1, 1, 0), o2 = (a0/2)(1, −1, 0), a0 is the lattice constant of the face-centered cubic lattice, an is the position of any atom on the n-th atomic plane. We remind that the distance between neighboring cation and anion planes equals a0/4.

In the tight-binding method the wave equation for an electron with the energy E transforms into the system of linear equations for the coe cients

In the sp3 tight-binding model, the atomic s- and p-orbitals are taken into account only. Hence, the pair of superscripts α, b runs through eight values sa, sc, px a, py a, px c, py c, pz a and pz c. Taking into account the symmetry

¯

oriented along [110] and [110] are used instead of the orbitals px [100], py [010]. Neglecting the spin-orbit interaction, the tight-binding Hamiltonian in a homogeneous semiconductor crystal is described by nine parameters

Esa, Eca, Esp, Epa, Vss, Vxx, Vxy , Vsa,pc = Vpc,sa and Vsc,pa = Vpa,sc.

For the electron states with zero lateral wave vector, eight coupled linear equations for the coe cients Cnαb are decoupled into three independent sets of equations: four for s and pz orbitals, two for px orbitals and two for py orbitals.

In calculating the matrix elements for optical transitions we apply the relation between the velocity and coordinate operators