ivchenko_bookreg

where H is the Hamiltonian. Taking the latter in the form Hαb bα(R , R) and introducing the matrix elements of the coordinate rαα (R, R ) we can find the tight-binding matrix elements of the velocity vαb bα(R , R). Here R = a + τb is the position of the atom specified by the location of an elementary cell, a, and the location τb of the atom of sort b within the cell. Usually only intrasite matrix elements

are taken into account with rα α describing the inter-orbital transitions within a single atomic site. In the tight-binding model of Lew Zan Voon and RamMohan [5.113] the inter-orbital matrix elements rα α are ignored. Then the velocity matrix elements contain only inter-atomic contributions and can be unambiguously expressed in terms of the tight-binding parameters as

It is seen that according to this theory the intra-atomic terms with R = R are indeed equal to zero, and the inter-atomic terms are directed along the vector R − R , i.e., along the chemical bond between the atoms R and R . In this case, the inter-atomic transitions between the planes 2l, 2l − 1 and 2l, 2l + 1 cause the emission of photons polarized in the direction of the axes

¯

x [110] and y [110], respectively.

The optical matrix elements corresponding to the emitted photons polarized along the axes x and y are written as

Here, Mj is the interband matrix element of the velocity operator, Vlx is the contribution to Mx from the inter-atomic transitions between the anion plane 2l and the cation plane 2l − 1, Vly is the similar contribution to My from transitions between the planes 2l and 2l+1, Cnsb, Cnpz b are the coe cients of the s- and pz -orbitals in expansion (5.161) for the electron states in the lowest conduction band Γ1, Cnpj b is the pj -orbital coe cient for the pj hole states in the valence band.

It is important to stress the principal di erence between calculations of optical matrix elements in type-I and type-II structures. The interband optical matrix element is proportional to the overlap integrals between the

282 5 Photoluminescence Spectroscopy

electron and hole wave functions. In type-I structures, the overlap integral is contributed by the whole QW layer, one needs to know the coe cients of admixture of heavyand light-hole states in the hole wave function at the lowest hole subband hh1. In this case the detailed information about behavior of the wave function near interfaces is not needed and the coe cients of admixture can be found from the e ective boundary conditions imposed on the hole envelope functions, as it was realized in Sect. 3.3.3. In type-II structures, the behavior of the hole wave function inside the layer, where a hole is confined, can be calculated as well by imposing the boundary conditions on the envelopes. However, in this case the calculation of interband matrix elements requires the knowledge of microscopic behavior of the wave functions at the interfaces and the transition oscillator strength is strongly a ected by an anisotropic orientation of interface chemical bonds. Thus one has to go forward beyond the generalized envelope function approaches.

The results of calculation of the e1-hh1 PL linear polarization

I ¯ − I110

Plin = 110

I ¯ + I110

110

are presented in Fig. 5.17. The tight-binding parameters for the ZnSe and BeTe layers can be chosen from the data on band structures of the corresponding bulk semiconductors. Note that the constants of lattices ZnSe and BeTe are close to each other, i.e., ZnSe/BeTe constitute a heteropair with matched lattice constants. However, they are di erent from those of the volume semiconductor ZnTe or BeSe. For this reason, the tight-binding coe cients for interface atoms can be considered as independent parameters of the theory. The three upper curves have been calculated for the ZnTe-like interface, while the three lower curves correspond to the BeSe-like interface. Each curve corresponds to a particular value of diagonal energy assumed for the p-like atomic orbital at the interface Zn atoms either at the interface Be atoms. From this figure we conclude that the theory allows high degrees of the PL linear polarization in type-II heterostructures. Moreover, as a rule the polarization follows the orientation of interface chemical bonds. Allowance for intra-atomic transitions modifies the PL polarization but leaves unchanged a possibility for the polarization to be very high.

The above-discussed lateral anisotropy is related to the tetrahedral orientation of the chemical bonds along the 111 directions and can be described by the reduced point-group symmetry C2v . An ideal QW structure with two equivalent interfaces has the higher symmetry D2d. It is uniaxially isotropic, because the chemical bonds at the opposite interfaces lie in mutually orthog-

¯

onal planes, (110) and (110), and their contributions to the anisotropy cancel. Thus, in ideal QW structures the in-plane anisotropy of single interfaces is hidden. However, under special conditions it can reveal itself through a linear polarization of vertically emitted radiation. Such is the case in the presence of an electric field applied normal to the plane of the well (Fig. 5.16) or in asymmetric QWs with di erent profiles of nonabrupt normal and inverted

Fig. 5.17. Linear polarization of the photoluminescence in the ZnSe/BeTe heterostructure as a function of the tight-binding parameter Vxy at the Zn-Te interface (curves 1, 2 and 3) or Be-Se interface (curves 4, 5 and 6). Curves 1, 2 and 3 are calculated, respectively, for the diagonal energy Epc = 5, 6, 7 eV for the interface Zn atom and curves 4, 5 and 6 for Epc = 3, 4, 5 for the interface Be atom. For the other interface parameters, mean values, for example, Esa(ZnTe) = [Esa(ZnSe) + Esa(BeTe)]/2, etc, were used. [5.112]

interfaces, and/or for heteropairs with no common atoms and with di erent kinds of chemical bonds at the abrupt interfaces (Sect. 3.3.3).

In [5.114,5.115] a method was proposed to uncover the hidden anisotropy of ideal QWs without breaking their invariance to the mirror-rotation operation S4 and reducing the uniaxial symmetry. The method is based on the analysis of the magnetic-field induced elliptical polarization of the photoexcited system consisting of optically anisotropic subsystems. Indeed, the indirect radiative recombination at a type-II (001) interface between two zinc-blende-lattice semiconductors, like BeTe and ZnSe, can be modelled by light-emitting 2D dipole oscillators. It su ces to assume that each oscillator can equally oscillate along the planar axes x and y and the coupling of x dipoles with the x -polarized light di ers from the y -y coupling by a factor of Λ. The circular oscillations x ± iy e ectively describe the electronhole states | − 1/2, 3/2 and |1/2, −3/2 and are characterized by the optical matrix elements for emission

where M0 is a constant and e is the polarization unit vector of an emitted photon. In fact, the model uses only one (real) parameter Λ which is the ratio, pcv,y /pcv,x , of the interband matrix elements of the momentum operator. According to (5.167), a photon emitted by the oscillation x + iy or x − iy is elliptically polarized with the principal axes of the ellipse parallel to x , y and the degrees of linear and circular polarization are given by

Here the sign ± coincides with the circularity of the oscillation. For each oscillation the emitted photon is completely elliptically polarized and the total degree of polarization P satisfies the condition

At zero magnetic field, the states | − 1/2, 3/2 and |1/2, −3/2 are equally populated. Due to that the emission is linearly polarized with Pl = Pl0 but lacks circular polarization, Pc = 0. In the presence of a longitudinal magnetic field the electron and hole spin states are split and, therefore, thermally populated. At high magnetic fields the carriers are completely polarized. It follows from (5.169) that the circular polarization under saturation di ers from ±100% and amounts to

Pcsat = ± 1 − (Pl0)2 ≡ Pc0. (5.170)

If, due to the structure asymmetry, the contributions, Ji( ω) and Jn( ω), to the spectral intensity from the normal (n) and inverted (i) interfaces are di erent then the linear polarization is given by

where η(E) = Ji(E)/Jn(E). The degree of circular polarization at saturating magnetic fields is determined by

which is identical with (5.170). Due to the function η( ω) the linear polarization Pl has a spectral dependence. In contrast, Pcsat is spectrally independent and determined by only one parameter, Pl0, of an individual interface in agreement with the experiment results presented in Fig. 5.18.

˚ ˚

Fig. 5.18. (a) Linearly polarized PL spectra of 100 A/50 A ZnSe/BeTe MQWs. B = 0 (magnetic field of 8.5 T does not modify the spectra). (b) Spectral variation of the PL linear and circular polarizations measured at B = 8.5 T. Pl(E) is identical at B = 0 and 8.5 T. Insets: Polarization of the indirect PL detected at 1.95 eV vs. excitation density (c) and magnetic field at W = 40 W/cm2 (d). Lines are the guide to the eye. T = 1.6 K. From [5.115].

6 Light Scattering

He loads the clouds with moisture;

he scatters his lightning through them.

Job 37: 11

Under scattering of light one understands the appearance, in the medium illuminated by an external source, of new electromagnetic waves with frequencies and/or propagation directions di erent from those of the initial wave. Note that neither specular reflection nor refraction on a smooth macroscopical boundary between two media are attributed to scattering processes.

In bulk semiconductors the light can be scattered (i) by free carriers, namely, by charge-density fluctuations (single-particle excitations and plasmons) and spin-density fluctuations (spin-flip transitions), (ii) by phonons, optical (Raman scattering) or acoustic (Brillouin or Mandelshtam-Brillouin scattering), and (iii) by static imperfections and inhomogeneities inside the medium (Rayleigh scattering), see Sect. 6.2. In QW and QWR structures, contributions to the scattering (i) can come not only from intrasubband transitions but also from intersubband transitions, i.e. from intersubband chargeand spin-density excitations (Sect. 6.3). The Raman e ect is enriched with scattering by folded acoustic phonons (Sect. 6.4) as well as by confined and interface optical phonons (Sect. 6.5). The spin-flip and double-resonance Raman spectroscopy is considered in Sects. 6.6 and 6.7, respectively.

6.1 The Physics of Light Scattering

In order to elucidate elastic and inelastic scattering of light by using the simplest possible model we consider a localized electric dipole with the moment P satisfying the equation of motion for a classical oscillator

where ω0 is the dipole eigenfrequency, ω and E0 are the frequency and electricfield amplitude of the incident monochromatic light wave, the coe cient q characterizes the interaction between the dipole and the electric field. Let us assume that, for any reason, this coe cient varies periodically in time and it can be presented as a sum of constant and alternating contributions

288 6 Light Scattering

where q0, q1 are constant coe cients and Ω is the modulation frequency, in the following we are interested in the limiting case Ω ω0. The solution of (6.1) with q chosen in the form (6.2) consists of three harmonics

Thus, time variation of the dipole moment is a triple-harmonic oscillation at the initial frequency ω and two other, combinational, frequencies ω ±Ω. The forced oscillations of the dipole moment give birth to secondary light waves at the same frequency ω (Rayleigh scattering) and two new frequencies ω + Ω and ω −Ω (Raman scattering). In a real system, the modulation frequency Ω can be any phonon frequency or the energy spacing between two eigenstates of the system related to .

It is instructive to consider a little more complicated model containing two dipoles with the moments P (t), P (t) that satisfy the set of two equations

d2P (t) + ω2P (t) = q0E0 cos ωt , dt2 0

(6.4)

d2P 2(t) + ω02P (t) = q (t)P (t) . dt

In the case under consideration the incident light directly interacts only with the first dipole characterized by the resonance frequency ω0 and the fixed coe cient q = q0. Oscillation of the second dipole appears because of the linear coupling between the dipoles with the coupling coe cient being time modulated, namely q (t) = q0 + q1 cos Ωt. Under stationary excitation the first dipole oscillates with the frequency ω of the primary light wave whereas the oscillation of the second dipole is a superposition of three harmonics

It follows then that the oscillation amplitudes P+1, P−1 and, hence, amplitudes of the scattered wave at the corresponding frequencies ω ± Ω should increase enormously as the condition of the double optical resonance ω = ω0 and ω + Ω = ω0 (or ω − Ω = ω0) is being approached. A real example of the double optical resonance is presented in Sect. 6.7.

In what follows we will use the notations ω1, q1, e1 and ω2, q2, e2 for the frequency, wave vector and polarization unit vector of the initial (primary) and scattered (secondary) electromagnetic waves, respectively. If a particle, or quasiparticle, involved in the act of scattering collides with a photon and changes its energy and wave vector from E1, k1 to E2, k2, the conservation laws read

If the scattering of a photon is accompanied by emission or absorption of an excitation characterized by the frequency Ω and wave vector Q the conservation laws take the form

In the process a (or b) the photon frequency decreases (increases), it is the so-called scattering in the Stokes (or anti-Stokes) spectral region. One can also introduce the transferred wave vector and frequency

In the presence of absorption in the medium at the frequency ω1 or ω2 the expression for q will contain the real parts of the wave vectors. For the collision (6.6), one has q = k2 − k1, ω = (E2 − E1)/ , and, for the scattering described by (6.7a) or (6.7b), q = ±Q, ω = ±Ω.

In terms of the Damen, Porto and Tell notation [6.1], scattering configurations are usually described by four symbols, two inside a parenthesis and two outside, for example, x(yz)y, z(xx)¯z or z(σ+, σ−)¯z. The symbols inside are, left to right, the polarization of the incident and of the scattered light, while ones to the left and right of the parenthesis are the propagation directions of the incident and scattered light, respectively. Thus, the configuration x(yz)y means that q1 x, e1 y, q2 y, e2 z, i.e., it corresponds to incoming light along the x direction with linear polarization y and the scattered light collected on the y direction with linear polarization z. The symbol z¯ means the axis reversed with respect to z so that the configuration z(σ+, σ−)¯z corresponds to the σ+ circularly-polarized incident light with the wave vector q1 z backscattered and detected in the σ− polarization.

QW structures lack the translational symmetry along the growth axis z. Therefore, the requirement of conservation of the wave-vector z-components should be excluded from (6.6, 6.7). Similarly, the restrictions imposed on the wave vectors by these equations are lifted in two cross-sectional directions in the case of a quantum wire and removed at all in a QD structure.

Phenomenologically, the light scattering can be described by adding to the material relation between the dielectric polarization and electric field, Pα = χ0αβ Eβ , a contribution

δχαβ (r, t) Eβ

290 6 Light Scattering

caused by fluctuations of the dielectric susceptibility. If δχ exp ( iΩt ± iQr), then the wave equation for the total electric field E contains an inhomogeneous term proportional to

exp (−iω1t iΩt) exp (iq1r ± iQr) ,

which serves as a source for secondary light waves. In the phenomenological description the spectral intensity of the scattered light is proportional to the squared fluctuation of the susceptibility,

where E0 is the amplitude of the initial wave, δχ(q, ω) is the space and time Fourier-component of the fluctuation δχ(r, t), and the angular brackets denote the averaging over the energy and wave-vector distribution of quasiparticles involved in the scattering.

We start the microscopical description from the simplest resonant light scattering which is the resonant fluorescence of two-level quantum systems (atoms, impurity centers in the crystalline matrix, localized excitons, QDs etc.). The spectral intensity of the scattered light is given by

with the scattering matrix element being

Here E0 is the amplitude of the initial wave, ω0 is the energy spacing between the ground, |i , and excited, |f , states; df i is the dipole-moment matrix element for the optical transition |i + ω1 → |f , the damping rate Γ is equal to (2τf )−1 with τf being the lifetime of the excited state |f (the ground state lifetime is assumed to be infinite). Note that for the process under consideration the frequencies ω2 and ω1 coincide.

The expression for I can be rewritten in the equivalent form

Such a secondary-emission process can be considered as light scattering because the spectrum of the secondary radiation is tied to the initial light frequency ω1 and shifts with shifting this frequency. On the other hand, the same process has features of the photoluminescence because it can be described in terms of a two-step process with a real intermediate state. Thus, the optical