Chapter 4 Magnetic response of single InAs QD
4.2 Magneto-photoluminescence results
4.2 Magneto-photoluminescence results
Sequentially, the cryostat was inserted into the bore of superconducting magnet, which provided the sample for an external magnetic field along the grown direction (Faraday geometry). We executed the magneto-µ-PL measurement on the total of seven QDs mentioned in last section. Representative spectra selected from particular dot (QD1) are shown in Fig. 4.6. With increasing the magnetic field, each peak splits into the Zeeman double corresponding to two spin states, the emission energy shift are related to two mechanisms. The first one is the Zeeman spin splitting that arises from the interaction between magnetic field and spin angular momentums of carriers and increases linearly with magnetic field. Another one is the well-known diamagnetic shift that usually increases quadratically with magnetic field for neutral exciton in QD.
Figure 4.6: (a) The magneto-PL spectra for QD1 under a magnetic field B=0–6 T. (b) The corresponding peak energies of different excitonic complexes as a function of B for QD2, where σ + and σ − in each form a Zeeman doublet. The dashed line is the average energy of σ + and σ −.
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Fig. 4.6 (b) plots the energies of two split peaks as a function of B for all excitonic complexes of QD1, which are denoted by the symbols 𝜎+ and 𝜎−. The Zeeman spin splitting and diamagnetic shift can be extracted by the energy difference 𝜎−− 𝜎+ and the average value of 𝜎+ and 𝜎−, as the following two relations,
Zeeman spin splitting = σ+− σ− (4.5)
Diamagnetic shift = σ++ σ−
2 (4.6) In Fig. 4.7 (a), the measured Zeeman spin splitting for four excitonic emission lines of four investigated QDs ( QD2, QD3, QD4 and QD5) are plotted as a functions of 𝐵. Generally, the Zeeman spin splitting is expressed as Eq. 4.7,
Zeeman splitting = 𝑔𝜇𝐵𝐵 (4.7)
where 𝜇𝐵 is the Bohr magneton equal to 5.7883818066 × 10−5 eV/T. The factor of 𝑔 indicates the magnitude of the splitting and is determined by the material parameters of band gap, spin-orbit splitting, and so on. It can be expressed as,
𝑔 = 𝑔0−4 3
𝑚0𝑃2 ℏ2
∆𝑠𝑜
𝐸𝑔(𝐸𝑔+ ∆𝑠𝑜) (4.8)
where 𝑚0 and 𝑔0 ≈ 2 are the free electron mass and the Lande factor, 𝐸𝑔 is the band gap, ∆𝑠𝑜 is the spin-orbit splitting of valence band, and 𝑃 = 𝑖(ℏ/𝑚0)⟨𝑆|𝑃𝑧|𝑍 is the Kane momentum matrix element formed between the s-antibonding conduction (𝑆) and p-bonding valence-band states (𝑍) [44]. Besides the material parameters, the quantum confinement of the quantum structure also acts another crucial factor for its 𝑔 factor. For example, the shapes and sizes of QDs are reported to connect to the values of the 𝑔 factor significantly in Ref. 45, 46, 47
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We fitted the data in Fig. 4.7 (a) by the Eq. 4.7 to get their 𝑔 factors. Clearly, the deduced 𝑔 factors for all four excitonic emission lines (𝑋−, 𝑋, 𝑋𝑋, and 𝑋+) of the same QD are almost identical, which gives a proof that the four excitonic emission lines are indeed belong to the same QD. On the other hand, the deduced 𝑔 factors for the four investigated QDs distribute over a range from 3.05 to 3.27, the slight difference in 𝑔 factors maybe attributes to the variation of QD’s shape and size.
In addition, we also plot the measured diamagnetic shifts for four excitonic emission lines of the four investigated QDs as a function of 𝐵2 in Fig. 4.7 (b).
Obviously, for 𝑋, 𝑋𝑋, and 𝑋+ in the four QDs, the measured diamagnetic shifts all display a quadratic dependence ∆E = γ𝐵2, from which we can get the diamagnetic coefficient denoted γ. Actually, the quadratic dependence is still hold for other investigated QDs not shown in Fig. 4.7. In very strong contrast, the diamagnetic shift for the 𝑋− does not always obey the quadratic dependence, and has quite large variations among individual QDs. For QD2 and QD5, the 𝑋− diamagnetic shifts still maintain the quadratic dependence, but for QD3, that is more close to a quartic dependence. Interestingly, QD4 exhibits an unexpected-negative diamagnetic shift for 𝑋−, i.e., a special paramagnetic behavior. However we still use a quadratic dependence to fit the anomalous 𝑋− diamagnetic shift to get 𝛾𝑋−, the deduced diamagnetic coefficients for four excitonic complexes of all investigated QDs are plotted as a function of 𝑋 emission energy as shown in Fig. 4.8. A clear trend of 𝛾𝑋 > 𝛾𝑋𝑋 ≅ 𝛾𝑋+ is observed for all QDs, and the diamagnetic coefficients 𝛾𝑋, 𝛾𝑋𝑋 and 𝛾𝑋+ become larger with the raise of 𝑋 emission energy. On the other hand, the diamagnetic coefficient 𝛾𝑋− was found to be the smallest one among the four excitonic species, and it seems not to have obvious dependence on 𝑋 emission energy. The difference of diamagnetic coefficients among the four excitonic complexes can be attributed to interparticle Coulomb interactions. For this trend of 𝛾𝑋 > 𝛾𝑋𝑋 ≅ 𝛾𝑋+, M. F. Tsai et al.
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have discuss its cause in Ref. 23. In their theory model, a 2D-disk with a parabolic-confined potential is considered, and the interparticle Coulomb energies are regarded as perturbation terms. They suggested that because of the wider lateral extent of electron than that of hole in small InAs QDs, the increasing rates of 𝑉𝑒𝑒 and 𝑉𝑒ℎ with magnetic field are more rapid than that of 𝑉ℎℎ, i.e., ∆𝑉𝑒𝑒(𝐵) ≅ ∆𝑉𝑒ℎ(𝐵) >
∆𝑉ℎℎ(𝐵), thus resulting in the trend 𝛾𝑋 > 𝛾𝑋𝑋 ≅ 𝛾𝑋+. However, the anomalous 𝑋− diamagnetic shifts are not yet explained in Ref. 23. We think that the supposition of regarding the interparticle Coulomb energies as perturbation terms is no more suitable for 𝑋− diamagnetic shifts, i.e., the carrier’s wavefunctions would be obviously changed by Coulomb interactions. Therefore, we would develop another numerical simulation to help us explain these diamagnetic shifts of all excitonic complexes, including the anomalous 𝑋− behavior in latter section.
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(a) (b)
QD5 QD5 QD5
QD4 QD4
QD4 QD2
QD3 QD3 QD3
QD2
Figure 4.7: The Zeeman splitting versus 𝐵 (a) and the diamagnetic shifts versus 𝐵2 (b) for 𝑋−, 𝑋, 𝑋𝑋, and 𝑋+ of four QDs (QD2, QD3, QD4, QD5). Points are the measured data, and the lines are guide for the eye.
QD2
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