ivchenko_bookreg

A rigorous theoretical study [5.75], taking into account the k -governed heavy-light mixing, slightly modifies the dependence of the linear polarization P on the electron energy Ee as compared with the above simple consideration.

The width of the zero-LO-phonon hot-PL peak in bulk and QW samples is caused by the warping of the hole subbands as well as by inhomogeneous broadening of the acceptor levels. The lowand high-energy edges of the peak correspond to electrons excited with the lateral wave vector k {100} and k {110}, respectively. If excitation of hot PL is performed in the configuration e [110] the electrons with k {100} are not aligned and the low frequency edge should be unpolarized. Conversely, electrons with k {110} contribute to the high energy edge of the zero-phonon peak and their recombination should be polarized. Therefore, the polarization is expected to grow from the low to high frequency edge, in agreement with experimental observations on MQWs and SLs [5.73, 5.76].

5.5 Polarized Photoluminescence of Excitons

5.5.1 Fine Structure of Exciton Levels in Nanostructures

If allowance is made for free-carrier spin, then the ground state n = 1 of an exciton is degenerate, even in the case of simple bands. For direct-gap Γ -point excitons, the ground-state degeneracy is equal to the product of the conductionand valence-band degeneracies at k = 0. The wave functions of the ground state transform according to the representation Dexc = Γe × Γh, where the representations Γe and Γh characterize the symmetry of electron and hole states at the extremum point. The representation Dexc is reducible and may be decomposed into irreducible representations of the point group of the semiconductor system, bulk material or nanostructure. The electronhole exchange interaction partially removes the degeneracy and splits the 1s-exciton level into states transforming according to the irreducible representations. In the present subsection we touch the main aspects of the electron-hole interaction in semiconductors and briefly recall the fine structure of excitonic levels in QWs, QWRs and QDs.

252 5 Photoluminescence Spectroscopy

A consistent theory of electron-hole exchange interaction in semiconductors has been developed by Pikus and Bir [5.77] and Denisov and Makarov [5.78]. In the e ective-mass approximation, the Coulomb-interaction operator between an electron and a hole in a semiconductor crystal includes three

contributions, describing, respectively, the direct, or intraband, Coulomb interaction (UC ) and the long-range (Uexchlong) and short-range (Uexchshort) exchange interaction. We introduce the two-particle excited states of the crystal

|m, ke; n, kh , where ke,h is the wave vector of the electron or hole, indices m and n number the degenerate states of the electron in the conduction band and the hole in the valence band at ke,h = 0 (for definiteness, a direct-gap

semiconductor with cubic symmetry and an extremum at the Γ point is considered). Then the matrix elements of the UC and Uexchlong operators between

these states can be written as

14πe2

=−V æ0|ke − ke|2 δm mδn n δke +kh ,ke +kh ,

Here K is the total wave vector ke + kh = ke + kh; Eg is the band gap; pmn¯ is the matrix element of the momentum operator, calculated between the electronic Bloch functions |m, k = 0 and |n,¯ k = 0 (the hole state n, k and the electron state n,¯ −k are coupled to each other by the timeinversion operator); æ0 and æb are the dielectric constants, low-frequency and high-frequency (at the electron-hole-pair excitation frequency), and V is the volume of the crystal. The interaction represented by (5.115) can be interpreted as a result of the virtual recombination and generation of an electron-hole pair. This equation can be derived as well by one more method taking into account the macroscopic electric field generated by the electronhole pair and the self-consistent e ect of this field on the energy of the pair. It is worth to note that in (5.115) the retarding electron-photon interaction is neglected. The generalized form of this equation taking into account the successive emission and absorption of a transverse photon by an electron-hole pair can be found in [5.79].

The Fourier components of the Coulomb potential with the wave vectors b + ke −ke, where b represents a nonzero reciprocal-lattice vector, contribute to the short-range interaction. When ke and kh are small enough to satisfy the criterion of the applicability of the e ective-mass method, operator Uexchshort has the character of a contact interaction and can be represented in the form

where a0 is the lattice constant and the factor a30 is separated out in order to have the dimension of energy for the coe cients ∆m n ,mn. The dependence of these coe cients on the band indices is found from symmetry considerations, while their absolute magnitudes are determined by comparing theory with experiment on studies of the fine structure of exciton levels. The number of linearly independent coe cients coincides with the number of irreducible representations contained in the direct product Γe × Γh. For illustration, let us consider the pair of bands Γ6 and Γ7 in GaAs-type semiconductors: Γ6 ×Γ7 = Γ2 +Γ15. It is convenient to go to a basis of electron-hole excitations in which the three states |ν, ke, kh (ν = x, y, z) are optically active in the polarization e ν, while the optical transition to the fourth state |Γ2, ke, kh is forbidden. The long-range exchange interaction given by (5.114) involves only the states |ν, ke, kh and has in the new basis the form

where p0 is the interband matrix element of the momentum operator for the optical transition to the state |ν , see also (2.204).

The short-range interaction splits the exciton quartet Γ6 × Γ7(1s) into a dipole-allowed triplet Γ15 and an optically-inactive singlet Γ2. The long-range exchange interaction splits the 1s exciton states longitudinal and transverse with respect to the exciton wave vector K. The longitudinal-transverse splitting is expressed via the microscopic parameter p0 as

where aB is the 3D-exciton Bohr radius.

The ground state e1-hh1(1s) of a heavy-hole exciton in zinc-blende-based QWs is also fourfold degenerate but, in contrast to the previous case of the Γ6 × Γ7(1s) exciton, the exciton sublevels are characterized by the angular momentum component M = s + j = ±1, ±2, where s = ±1/2 and j =

±3/2 are the electron and hole spin components. In accordance with the multiplication law Γ6 × Γ6 = Γ1 + Γ2 + Γ5 for a QW structure of the D2d symmetry, the short-range exchange interaction splits the state e1-hh1(1s) into a radiative doublet Γ5 which is dipole-active in the e z polarization and corresponds to M = ±1 and two close-lying dipole-forbidden singlet levels Γ1 and Γ2 which are the symmetrized and antisymmetrized linear combinations of the exciton states with M = ±2. The long-range exchange interaction splits the longitudinal and transverse exciton states lifting the energy of the longitudinal exciton by

254 5 Photoluminescence Spectroscopy

where K = |K|, K is the in-plane wave vector of the 2D exciton, f (ρ) is the envelope function describing the relative electron-hole motion,

P (K) = dz dz ϕe1(z )ϕhh1(z ) ϕe1(z)ϕhh1(z) e−K|z−z |

and ϕe1(z), ϕhh1(z) are the single-particle size-quantized functions. For small values of K satisfying the condition Ka 1, the function P (K) or the co- e cient C is constant. Hence the long-range exchange interaction in a QW vanishes linearly when K goes to zero. This also means that no longitudinaltransverse splitting is expected for the exciton excitation with light propagating along the growth axis of the well. For Ka 1, the longitudinal-transverse splitting and the coe cient C can be expressed in terms of the exciton radiative damping Γ0 defined by (3.18) as

where k = nb(ω0/c).

In the following we will use the two pairs of basis states for the e1-hh1(1s) exciton quartet. The first set |M, K is characterized by a certain exciton angular momentum component M = ±1, ±2. The matrix element of the

where M0 is independent of the light polarization e. In the second basis the optically-active states are replaced by the excitonic states Γ5 polarized along the fixed axes α = x, y. It is worth to mention that in the basis |αK the e ective Hamiltonian for the states Γ5 can be written as a 2 × 2 matrix

The spin-dependent part of this matrix can be conveniently presented in the form

where Φ is the angle between K and x.

Now we consider the fine structure of 0D excitons and focus on the anisotropic splitting of the radiative doublet in semiconductor nanostructures, namely, heavy-hole excitons that are localized at a particular wellwidth fluctuations in type-I QWs, localized at a particular interface in typeII heterostructures, or confined by asymmetrical QDs. Despite the obvious di erences between these three kinds of excitonic states their exchange splitting can be described by a common exchange-interaction matrix which in the basis |M taken in the order |1 , | − 1 , |2 , | − 2 may be written as follows

00 eiΦ1 δ1 −δ0

Here the constants δ0, δ1, δ2 refer, respectively, to the quartet splitting into

the doublets | ± 1 and | ± 2 , nonradiative-doublet splitting into the states

√

(|2 ± eiΦ1 | − 2 )/ 2, and radiative-doublet splitting into the states

√

(|1 ±eiΦ2 |−1 )/ 2 which are dipole-active along the rectangular axes rotated

around the structure principal axis z by the angle Φ2/2 with respect to the fixed axes x and y. Note that the splitting δ0 retains even in the uniaxialsymmetry approximation, a nonzero value of δ1 appears taking into account the tetragonal symmetry D2d of an ideal QW, and the δ2-governed splitting is completely related to the low symmetry of an exciton-localizing potential. We will also use the frequencies Ωj (j = 0, 1, 2) defined by Ω0 = δ0, Ω1 = δ2 cos Φ2, Ω2 = δ2 sin Φ2. Then the matrix (5.123) for the two states | ± 1

We first discuss the energy and exchange splitting of an exciton localized at a fluctuation in the width of a type-I QW. Beyond the limits of the considered localization island, the QW is assumed to contain N monomolec-

ular layers (in GaAs, the width of a monolayer is a0 ˚

/2 = 2.8 A). One of the interfaces is flat while the other is shifted in the region of the island by a monolayer increasing the QW width here to (N + 1)a0/2. If the linear dimensions of the island exceed the exciton 2D Bohr radius, the envelope of the exciton wave function can be approximated by (2.188). If the shape of the island is anisotropic, the function F (X, Y ) in (2.188) is as well anisotropic which results in a nonzero value of δ2 in (5.123). Introducing the 2D Fourier transform

where the retardation is taken into account, as a consequence K is replaced

√

by K2 − k2 and the summation is performed over K > k.

Goupalov et al. [5.79] calculated the splitting δ2 for a rectangular island of the dimensions Lx, Ly , see the inset in Fig. 5.10. The function F (X, Y )

256 5 Photoluminescence Spectroscopy

Fig. 5.10. (a)Energy levels Enn of the localized exciton. (b) The splitting between localized states |n, n , x and |n, n , y as a function of Ly for a fixed value of Lx =

The parts of curves corresponding to negative and positive values of δ2 are shown by dashed and solid lines, respectively. [5.79]

where V is the energy di erence between the 1s-exciton levels in the two perfect QWs di ering in the width by one monomolecular layer, ε is the localization energy referred to the free-exciton energy in the thinner QW. While calculating δ2 the approximation of the factorized envelope function was used and F (X, Y ) was taken in the form of a product Fx(X)Fy (Y ), where Fx(X), Fy (Y ) are found self-consistently from (5.126). The localized states |nn are labelled by a pair of integer quantum numbers nn , e.g., 11, 21, 22 etc., describing the exciton in-plane confinement in the x and y directions.

Figure 5.10 illustrates the dependence of the energy and splitting of the ground and excited states on the island dimension. Note that Enn = V − εnn is the localized-exciton energy referred to the free exciton energy in the wider QW. Due to the long-range exchange interaction the level Enn is split into two sublevels Enn ,x, Enn ,y polarized along the x and y directions, respectively. According to (5.125), the sign of the splitting δ2 ≡ Enn ,x − Enn ,y is mainly determined by the sign of the di erence of the mean squaresKx2 and Ky2 . When n = n , the splitting of the Enn level is evidently negative if Ly < Lx and positive if Ly > Lx, and this agrees with curves 11 and 22 in Fig. 5.10. As the quantum number n (or n ) increases, the dispersionKx2 (or Ky2 ) increases. Therefore, for close-lying Lx and Ly , the signs of the di erences Enn ,x −Enn ,y and n−n coincide. If the sides of a rectangular island are rotated with respect to the fixed axes x, y by the angle Φ then, in the spin Hamiltonian (5.123), the angle Φ2 is nonzero and equal to 2Φ.

Gammon et al. [5.10, 5.11] studied the PL of GaAs/AlGaAs QWs in the optical near-field regime and measured the PL spectrum of a single QD formed by a large monolayer high island. They report a fine structure splitting of 20-50 µeV and linear polarization of the split sublevels for both the ground and excited states of the localized exciton. The sequence of signs of this di erence observed for the ground and four excited states of the localized

˚

exciton [5.11] is reproduced in Fig. 5.10 for values Ly lying between 420 A

˚

and 480 A.

In type-II GaAs/AlAs(001) SLs, the radiative level of localized heavy-hole excitons e1-hh1(1s) was found to be split into the sublevels polarized along

¯

the [110] and [110] directions in the interface plane [5.80–5.85]. Moreover, it was established that in the same sample there exist simultaneously two classes of excitons with equal absolute values but opposite signs of the energy

¯

x [110] and y [110], i.e., δ2 is nonzero and Φ2 = 0 or π if the reference axes are x and y. This anisotropic exchange splitting of excitonic levels cannot be explained in terms of the long-range exchange interaction in excitons localized by laterally-anisotropic islands because the exciton oscillator strength in the type-II heterostructures is too small. The exchange-interaction anisotropy was related in [5.86–5.88] to the heavy-light hole mixing under normal hole

258 5 Photoluminescence Spectroscopy

incidence on the (001) interface. Then the pair of wave functions at the bottom of the lowest heavy-hole subband hh1 can be written as

(hh1)

ψ±3/2 = F (z) |Γ8, ±3/2 ± iG(z) |Γ8, 1/2 + iGso(z) |Γ7, 1/2 . (5.127)

As compared with (3.116) an admixture of the spin-orbit-split valence band Γ7 is included into ψ(hh1). The real envelope functions F (z) and G(z), Gso(z) are even and odd with respect to the reflection z → −z for the origin z = 0 taken in the center of a GaAs well. The functions G(z), Gso(z) are proportional to the heavy-light hole mixing constant tl-h in the boundary conditions for the hole envelope function, see (3.115). Excitons contributing to the lowtemperature PL of undoped type-II SLs are bound two-particle excitations localized by the structure imperfections in the plane of interfaces with an X- electron and a Γ -hole confined inside two neighboring AlAs and GaAs layers. This presupposes the existence of localized excitons with a leftand right- hand-side electron labelled in the following as the XL and XR states. Let the localization length exceeds the exciton Bohr radius a˜, describing the inplane relative motion of the electron-hole pair. Then the anisotropic exchange splitting of the XL and XR levels is given by [5.88]

where uL,R(z) and vL,R(z) are the X1 and X3 envelopes of an X-electron confined, respectively, in the leftand right-hand AlAs layer. ε0 determines the bulk electron-hole short-range exchange interaction taken in the form

where σeα, σhα are the electron and hole spin Pauli matrices. Taking into ac-

count that u2L(z) = u2R(−z), vL2 (z) = vR2 (−z), F (z) is even and G(z), Gso(z) are odd, one immediately obtains opposite signs of the splitting for XL and

XR states. This explains the two classes of excitons observed experimentally. Figure 5.11 shows the calculated dependence of the anisotropic exchange splitting of the radiative doublet in GaAs/AlAs SLs as a function the SL period d for a fixed ratio of the GaAs layer thickness, a, and the AlAs layer thickness, b. Experimental values of δ2 measured by optically detected magnetic resonance and quantum beats technique are presented as well.

The Molecular-Beam-Epitaxy (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, Ω2 = δ2 sin Φ2 = 0. In the general case of a rectangular base the symmetry is reduced to C2 and both Ω1 and Ω2 are nonzero.

Period (Å)

Fig. 5.11. Anisotropic exchange splitting of the radiative doublet as a function of the period of type-II GaAs/AlAs SL. Triangles [5.80] and rhombes [5.83] show experimental results, solid and dashed curves are calculated by using slightly di erent models. According to [5.87].

For free heavy-hole excitons in ideal QWs of the symmetry D2d the spin Hamiltonian contains the term in (5.122) proportional to δ0 and representing the long-range exchange interaction as well as the terms in (5.123) proportional to δ0, δ1 and representing the short-range exchange interaction.

In an external magnetic field the exciton spin-Hamiltonian includes, in addition to (5.123), the Zeeman contribution

where sz = (1/2)σ and Jz are the z-components of the electron spin s = 1/2 and hole angular momentum J = 3/2. We neglect the Zeeman splitting of heavy-hole states in an in-plane magnetic field. In the Faraday configuration,

Bz, the split quartet levels are given by

With increasing the field, the level 2 goes downward, the level 3 goes upward and, at a particular value of Bz , they cross each other. Near the crossing point the states 2, 3 can be strongly mixed even by a weak symmetry-breaking potential that couples radiative and nonradiative states. Owing to the larger population of the nonradiative state the intensity of emitted light increases. This e ect known as the level-anticrossing can lead to a remarkable nonthermal exciton spin polarization or alignment of exciton oscillating dipole moment under unpolarized excitation resulting in a appearance of circular or linear PL polarization near the crossing point. Depending on the exciton e ective g factors the level 3 can also cross the level 1 at a higher value of the magnetic field where the field-induced PL polarization is opposite to that related with the 2-3 anticrossing [5.90, 5.91].

If the anisotropic-exchange constants δ1, δ2 are zero then at zero magnetic field the radiative and nonradiative levels are doubly degenerate. The transverse magnetic field (Voigt configuration) mixes the excitonic state m = ±1 with the state m = ±2 resulting in the shift of the levels

and retaining the double degeneracy. The degeneracy is removed in some extent only if δ1 or/and δ2 are nonzero.

We turn now to the fine structure of the ground-state exciton level in cubic and hexagonal NanoCrystals (NCs) of the spherical form. Due to the confinement of electrons and holes in all three directions, the exchange splitting of excitonic levels increases as R−3 with decreasing the dot radius R and is strongly enhanced as compared with that in bulk semiconductors. Large exchange splittings were observed in CdSe NCs embedded in glassy materials and polymer films [5.92–5.94].

In order to make transparent the comparison between the shortand long-range exchange contributions to the splitting of excitonic levels we first consider optical transitions between the simple conduction band Γ6 and the simple valence band Γ7 in semiconductors of the crystal class Td. In this case the 1s-exciton level is four-fold degenerate taking into account the electron and hole spin degeneracy. As well as in the case of bulk semiconductor, the direct product Γ6 × Γ7 is decomposed into the Γ15 and Γ2 irreducible representations. It is also convenient to use the spin basis |Γ15, ν (ν = x, y, z) and