Free excitons and polaritons

The optical properties of a semiconductor are connected with both intrinsic and extrinsic effects. For a start, the intrinsic excitonic features in the 3.376–3.450 eV range are discussed. The wurtzite ZnO conduction band is mainly constructed from the s-like state having ( ) symmetry, whereas the valence band is a p-like state, which is split into three bands due to the influence of crystal field and spin-orbit interactions. The ordering of the crystal-field and spin-orbit coupling split states of the valence-band maximum in wurtzite ZnO was calculated by our group using tight-binding theory. The obtained electronic band structure of wurtzite ZnO is shown in Fig. 2-5. The valence-band symmetry ordering

(A-7

Γc

Γ , B-9 , and C- ) in ZnO is the same as that observed in low temperature PL and magnetoluminescence measurements by other researchers.[15-17]

Γ7 Γ₇

Group theoretical arguments and the direct product of the group representations of the band symmetries (Γ₇ for the conduction band, Γ for the A valence band, ₉ upper Γ₇ for the B valence band, and lower Γ for the C valence band) will result ₇ in the following intrinsic exciton ground state symmetries:

7 9 5 6 , 7 7 5 1

Γ ×Γ → Γ + Γ Γ ×Γ → Γ + Γ + Γ₂

The and exciton ground states are both doubly degenerate, whereas and are both singly degenerate.

Γ5 Γ₆ Γ₁

Γ2 Γ₅ and Γ₁ are allowed transitions with E⊥c and E║c, respectively, but the Γ₆ and Γ are forbidden. ₂

Fig. 2-5 The electronic band structure of wurtzite ZnO. [unpublished]

Figure 2-6 displays the PL spectrum in the range of fundamental excitonic region measured at 10 K in the E⊥c polarization geometry for a high quality ZnO crystal.

The A-free exciton and its first excited state emission are observed at FXⁿ_A⁼¹= 3.3771 eV (3.3757 eV for Γ₆) and FX_Aⁿ⁼²= 3.4220 eV for Γ₅ (3.4202 eV for ) band symmetry, respectively. Although, at k = 0,

Γ6

Γ exciton is forbidden in the current 6

measurement mode of polarization, it is still observable, due to the fact that the photon has finite momentum. Geometrical effects such as not having the sample orientation exactly perpendicular to the electric field may be also a reason for observing transition. Using the energy separation of ground state and excited state peak positions, the exciton binding energy and band gap energy can be predicted.

The energy difference of about 45 meV gives an A-free exciton binding energy of 60 meV and a corresponding band gap energy of 3.4371 eV at 10 K. Based on the reported energy separation of the A- and B-free excitons (in the range of 9–15 meV),

Γ6

we assigned the weak emission centered at 3.3898 eV, which is about 12.7 meV apart from the A exciton, to the B exciton transition.

Fig. 2-6 Free excitonic fine structure region of the 10 K PL spectrum for the ZnO single crystal. [14]

Additional fine structure of exciton lines was also observed in low temperature PL spectra. In strongly polar materials like ZnO transverse Γ excitons couple ₅ with photons to form polaritons. Therefore, as indicated in Fig. 2-6, the FX_Aⁿ⁼¹( ) exciton line has two components. The higher energy component at 3.3810 eV, which is 3.6 meV apart from the A exciton, can be assigned to the so-called longitudinal exciton (upper polariton branch—UPB

Γ5

A). The lower energy component at 3.3742 eV, which is about 2.9 meV apart from the A exciton, corresponds to the recombination from the “bottleneck” region, in which the photon and free-exciton dispersion curves cross (lower polariton branch—LPBA). As a result, their dispersion curves are illustrated in Figure 2-7, and we will explain the energy diagram of a system consisting of a crystal and radiation below. Since Γ ₆

excitons do not have transverse character, they do not interact with light to form polaritons, and thus have only normal free-exciton dispersion curves as seen in the PL spectra.

The incident radiation in the exciton band region is converted to the exciton polariton inside the crystal and it has two modes, the upper and lower branches. The energy of the polariton is shown in Fig. 2-7 as a function of wave vector k. The broken line represents the relation between the energy and wave vector of the photon in vacuum. The absorption of a photon creates an upper-branch polariton at a, which is then scattered to state b of the lower-branch polariton. The polariton thus formed is thermalized from b to c through phonon and point-defect scattering. The thermalized polariton can be annihilated as a photon at d (direct radiative annihilation) or move in the crystal to be ionized or annihilated nonradiatively at imperfections, since the thermalized state has a longer lifetime. If the incident radiation energy is below the bottom of the upper-branch polariton, it generates the lower polariton directly.

Here two assumptions are necessary. First, most inelastic scattering is caused by the LO phonon. Second, the probability for a thermalized exciton to be ionized or annihilated at imperfections is larger than that for it to be annihilated as a photon at d after being scattered from b to c. Under these assumptions if the energy difference between the upper polariton state a and the lower polariton state d is an integral multiple of the LO phonon energy, the cascade scattering of the polariton into state d occurs very efficiently and it is annihilated as a photon. If this energy difference is not an integral multiple of the phonon energy, the polariton is scattered and thermalized in the state near c in the lower-branch polariton and then it is ionized or trapped at imperfections before reaching state d. Thus the LO phonon structure is

expected in the excitation spectrum of the free-exciton emission.

a

b c

d

Fig. 2-7 Dispersion curves of exciton and exciton polaritons.