ivchenko_bookreg

|Γ2 rather than |m, n (m, n = ±1/2). The state |Γ15, ν is optically active for the light polarized along the α direction while the state |Γ2 is inactive. In the chosen basis non-zero components of the shortand long-range terms can be expressed as

in terms of the bulk exciton Bohr radius aB, the singlet-triplet splitting ∆bulkST and the longitudinal-transverse splitting ωLT of the bulk exciton level. According to (5.134) for an exciton confined within a spherical QD of the radius R aB, the singlet-triplet splitting is given by [5.95]

While deriving (5.135), equation (2.89) for the single-particle envelope function quantum-confined in a spherical QD is used assuming the barrier to be infinitely high. Taking for reasonable estimations ∆bulkST 0.1 meV, ωLT 1 meV we obtain that, among the two terms in (5.135), the long-range contribu-

to ∆QDST can also be derived from equation (3.134) for the renormalized resonance frequency of the exciton assuming qR → 0 and replacing Φ(r) by

(2πRr2)−1 sin2 (πr/R) .

At finite values of qR one has to use a more general equation (3.134). Moreover, in this case the damping ΓSQD defined by (3.135) becomes nonzero, it determines the radiative decay of the 0D exciton and is connected by ΓSQD = (2τ0)−1 with the exciton radiative lifetime τ0. A large long-range contribution to the electron-hole exchange interaction in semiconductor QDs has been also considered by using a many-body approach based on atomistic pseudopotential wave functions [5.96].

Note that in case of di erent high-frequency dielectric constants æ(1)b and æ(2)b , respectively inside and outside the dot, the second term in (5.135) should be multiplied by the factor [5.95, 5.97]

262 5 Photoluminescence Spectroscopy

For Γ6 × Γ8 excitons the valence band Γ8 is described by the 4 × 4 Luttinger Hamiltonian containing the three band parameters γ1, γ2 and γ3. The ground state of a hole confined in a spherical QD has the symmetry Γ8 and is characterized by the angular momentum component n = ±3/2, ±1/2. In the spherical approximation, γ2 = γ3, the four ground-state wave functions can be written as

Jα (α = x, y, z) are the angular momentum matrices in the Γ8 basis, R0(r) and R2(r) are the radial functions introduced in [5.92] and reflecting the fact that the envelopes Ψn(n)(r) are formed by coupled S (L = 0) and D (L = 2) states.

The short-range exchange interaction has the form [5.92]

where pcv = S|pˆx|X is the interband matrix element of the momentum operator.

In hexagonal crystals one has to add the crystal splitting term

to the Luttinger Hamiltonian. If a value of the crystal splitting ∆cr is small as compared with the characteristic confinement energy the combined electronhole exchange interaction in QDs of the radius R aB is described by the Hamiltonian [5.97]

ωLT is given by (2.204), χ(β) and ζ(β) are coe cients dependent on the ratio β = mlh/mhh between the lightand heavy-hole e ective masses. For β = 0.3 one has χ(β) ≈ 0.77 and ζ(β) ≈ 0.62. Note that ωTF is the short-range exchange induced splitting between the bulk exciton states with n = ±1 and n = ±2, and ωLT is the longitudinal-transverse splitting for a free exciton propagating perpendicular to the hexagonal axis of the bulk wurtzite-lattice crystal. Inset in Fig. 5.12a presents exciton level diagrams described by the Hamiltonian

where ∆ = ∆cr v(β) with v(β) ≈ 0.94 for β = 0.3. The excitonic sublevels are labelled in accordance with the value of total angular-momentum component on the C6 axis. The splitting, ∆QDTF (¯η), between the M = ±2 and lower M = ±1 states of a 0D exciton is given by

If the long-range exchange contribution is neglected the parameter η¯ in

(5.140) is replaced by

η = aB 3 R

which is smaller than η¯ by 4. Note that this result has alternatively been derived in [5.100] from consideration of resonant Rayleigh scattering of light by a QD following the procedure described in Sect. 3.2.1. Figures 5.12a and 5.12b present the measured and calculated dependence of the splittings ∆QDTF and ∆1, ∆2 on the nanocrystal radius.

In [5.101] the electron-hole exchange interaction is analyzed in the framework of the empirical tight-binding method. It is demonstrated that interatomic and intra-atomic (or intrasite) contributions to the long-range interaction enter in an inequivalent way. In particular case of the Γ6 × Γ7 exciton in a spherical dot, the long-range singlet-triplet splitting is given by

where ωLT is the longitudinal-transverse splitting for the bulk exciton and ωLT(intra) is the contribution to ωLT caused by the intra-atomic transitions.

Thus as compared to the envelope function approach valid for ωLT(intra) = 0, the splitting (5.141) di ers by a factor

where Mcv(inter), Mcv(intra) are the contributions to the interband optical matrix element due to the interand intra-atomic transitions in the tight-binding

5.5.2 Optical Orientation and Alignment of Free Excitons in Quantum Wells

Similarly to free carriers, exciton spins (or angular momenta) can be optically oriented. In the absence of an external magnetic field spin-polarized excitons are generated under optical pumping with circularly polarized photons: due to spin-orbit coupling of electronic states the selection rules for optical interband transitions provide a conversion of photon polarization into exciton spin orientation. This immediately follows from (5.121) for the matrix elements,

M1 e− = 0 , M−1 e+ = 0 ,

whereas, for the σ− photons, M1 = 0, M−1 = 0, in agreement with the angular momentum conservation law mexc = mphot, where mexc and mphot are the z components of exciton and photon angular momenta. In accordance with the same selection rule, mphot = mexc, the radiative recombination of spin-polarized excitons results in the PL circular polarization thus making possible optical detection of the spin polarization. The optical orientation of excitonic spins is a particular case of the more general phenomenon, namely, the selective optical excitation of excitonic sublevels. Another example of the selective excitation is the so-called optical alignment of excitons by linearly polarized radiation: in contrast to the optical orientation which means just the photoinduced di erence in the populations of the exciton states |m with the

spin m = ±1, the linearly-polarized light can excite preferentially the exciton

√

state (|1 + eiΦ|−1 )/ 2 with a definite direction of oscillating electric-dipole moment (a value of the phase Φ is determined by the direction of the light polarization plane).

Optical orientation and alignment of excitons can be described by using the exciton spin-density matrix. The method is especially applicable to bound

where d0αβ is the polarization density matrix of the initial radiation. As in Chap. 3, see (3.228), the light polarization is characterized by the circular polarization degree, Pc, and the degrees of linear polarization, Pl and Pl , referred to the two pairs of rectangular axes labelled here as x, y and x , y .

In order to make transparent the description of polarized PL we give here a simplified description of the optical orientation and alignment of excitons. For resonant excitation conditions and in the absence of exciton spin relaxation between the pairs m = ±1 and m = ±2, the optically-inactive sublevels remain unpopulated, the only nonzero components of the density matrix ρmm are those with m, m = ±1 and the e1-hh1(1s) exciton acts as a two-level system. Recall that any two levels can be considered as two states of an e ective 3D pseudospin with S = 1/2. The 2×2 spin density matrix ρmm (m, m = ±1) is expressed in terms of the average pseudospin S as

where σ˜α(α = x, y, z) are the Pauli matrices and N is the steady-state exciton concentration. The pure exciton states |1 and | − 1 are equivalent to the pseudospin polarized parallel or antiparallel to the z axis, respectively. The exciton states

√ √

|X = (|1 + | − 1 )/ 2 , |Y = −i (|1 − | − 1 )/ 2 ,

dipole-active along the x or y axis are described by a pseudospin with Sx =

|Y = (|X − |Y )/ 2 polarized in the x and y directions correspond to a pseudospin with nonzero component Sy = 1/2 or Sy = −1/2. Thus, one can rewrite (5.142) as

The generation matrix Gmm for m, m = ±1 can be expanded similarly to (5.146)

Here G0 is the total generation rate of excitons, Pl0, Pl0 and Pc0 characterize the initial light polarization, S0 is the exciton pseudospin at the moment of resonant excitation.

If the exciton is formed by binding of free electrons and holes into pairs,

ˆ

then G is proportional to the product ρˆeρˆh of single-particle spin-density

268 5 Photoluminescence Spectroscopy

matrices at the moment of binding. In this case one may observe only exciton orientation.

In terms of the pseudospin Pauli matrices, the spin-dependent long-range exchange interaction is written as

where

Ωx = CK cos 2Φ , Ωy = CK sin 2Φ .

The longitudinal-transverse splitting of the exciton state produces a motional narrowing type of spin relaxation similar to the D’yakonov-Perel’ mechanism considered in Sect. 5.3.2. Indeed, the o -diagonal terms of the Hamiltonian (5.150) represent an e ective magnetic field in the (x, y) plane. If such a field is fixed then, in the absence of a real external magnetic field, the pseudospin will process about this field which results in a spin decoherence in the exciton ensemble. In the multi-scattering regime, the exciton wave vector K is changed by each scattering event, the e ective field is randomly oscillating and the exciton spin decoherence is slowed down. If K changes faster than the pseudospin procession, i.e. if CKτp(2) 1, then one has for the exciton spin relaxation times [5.102]

where ΩK = Ωx2 + Ωy2 = CK and τp(2) is the momentum scattering time for

the angular harmonics cos 2Φ, sin 2Φ of the free-exciton distribution func-

tion. It should be noted that the times Ts1 ≡ τzzs and Ts2 ≡ τxxs = τyys describe the relaxation of exciton optical orientation and alignment, respec-

tively. The factor of 2 is easily understood by using the fact that the in-plane pseudospin, say Sx, is only a ected by the y component of the depolarizing field, whereas the longitudinal polarization Sz is relaxed by both x and y components. Therefore, the theory predicts that the PL experiments with linearly polarized light should exhibit a longer spin-relaxation rate driven by exchange than those performed with circularly polarized light.

In terms of the pseudospin (5.143) reduces to

where τ is the exciton lifetime in the state m = ±1, the spin-relaxation term −(∂S/∂t)s.r. has the components

Tsi (i = 1, 2) are the longitudinal and transverse pseudospin relaxation times including those described by (5.151). In addition, the pseudospin precession in an external longitudinal magnetic field B z is taken into consideration with the exciton Larmor frequency Ω given by (0, 0, g µB Bz / ) and the e ective exciton g factor given by g = gh −ge . Note that in weak magnetic fields satisfying the condition Ω τp 1 one can neglect the field e ect on the pseudospin relaxation times Tsi [5.102].

At zero magnetic field, the polarizations of the secondary and primary

In a longitudinal magnetic field B z, the PL circular polarization remains una ected while Pl, Pl change to

where Pl(0), Pl (0) are zero-field polarizations given by (5.153). One can see that, under linearly-polarized excitation in a longitudinal magnetic filed, the plane of polarization of the exciton radiation is rotated around z by the angle θ = (1/2) arctan Ω T2, while the degree of linear polarization, Plin =

Pl2 + Pl2, decreases by the factor [1 + (Ω T2)2]1/2.

Physically, equation (5.154) can be derived by using the following intuitive approach. Optical pumping in the polarization e0 x excites the electricdipole moment d0 of the exciton oscillator directed along the x axis. In a longitudinal magnetic field this dipole will rotate around z axis with angular frequency Ω /2. Hence, for the exciton created at t = 0 the components of the vector d exhibit the quantum beats

Taking into account the finite exciton lifetime and spin-relaxation time the PL intensity Iα(t) in the polarization α shows damped oscillations

Iα(t) e−t/T2 d2α(t) .

On the other hand, the steady-state intensity Iα is related with the timeintegrated quantum beats Iα(t) by

270 5 Photoluminescence Spectroscopy

integrating over t we obtain the Stokes parameters (5.154) for Pl(0) = 1, Pl (0) = 0.

In the absence of an anisotropic exchange interaction (δ1,2 = 0) and spin relaxation (Tsi → ∞), the transverse magnetic field B z does not a ect the PL polarization but can change the PL intensity [5.103]. The latter e ect occurs because the e ective lifetime of resonantly-excited excitons increases with increasing the field in the presence of both radiative and nonradiative recombination channels.

In this subsection we concentrated on the polarized PL of free heavy-hole excitons. The secondary radiation of light-hole excitons can be considered in the same way with allowance for nonzero optical matrix elements in both polarizations e z and e z.

5.5.3 Optical Orientation and Alignment of Zero-

Dimensional Excitons

Neglecting, for simplicity, the exciton spin relaxation one can obtain for the PL circular polarization under resonant circularly-polarized excitation in the longitudinal magnetic field [5.82]

where the exchange-related frequencies Ω1,2 are introduced in (5.124) and τ is the 0D-exciton lifetime in the radiative states m = ±1. If the exciton lifetime τ is long enough so that max (Ω1,2τ ) 1 then, because of the exchange, the exciton optical orientation is quenched. With increasing the magnetic field the PL circular polarization is restored and reaches a value of Pc0 as Bz tends to ∞. Thus, the magnetic field suppresses the depolarizing e ect of anisotropic exchange interaction and permits to observe the optical orientation of excitons even if max (Ω1,2τ ) 1.

As in the previous subsection, the behavior of exciton optical orientation and alignment is very convenient to interpret physically in terms of the exciton pseudospin. In a longitudinal magnetic field, the pseudospin Hamiltonian is a sum of the exchange term (5.124) and the σ˜z -dependent Zeeman term

The pseudospin S rotates around the vector Ω = (Ω1, Ω2, Ω ) with the

e ective Larmor frequency Ω = Ω12 + Ω22 + Ω2. In the realistic case