Spin-dependent recombination and hyperfine interaction at deep defects

E. L. Ivchenko, L. A. Bakaleinikov, and V. K. Kalevich

Phys. Rev. B 91, 205202 – Published 11 May 2015

Abstract

We present a theoretical study of optical electron-spin orientation and spin-dependent Shockley-Read-Hall recombination in the longitudinal magnetic field, taking into account the hyperfine coupling between the bound-electron spin and the nuclear spin of a deep paramagnetic center. The master rate equations for the coupled system are extended to describe the nuclear spin relaxation by using two distinct relaxation times, $τ_{n 1}$ and $τ_{n 2}$ , respectively, for defect states with one and two (singlet) bound electrons. The general theory is developed for an arbitrary value of the nuclear spin $I$ . The magnetic-field and excitation-power dependencies of the electron and nuclear spin polarizations are calculated for the value of $I = 1 / 2$ . In this particular case the nuclear effects can be taken into account by a simple replacement of the bound-electron spin relaxation time by an effective time dependent on free-electron and hole densities and free-electron spin polarization. The role of nuclear spin relaxation is visualized by isolines of the electron spin polarization on a two-dimensional graph with the axes $\log_{2} (τ_{n 1})$ and $\log_{2} (τ_{n 2})$ .

Received 14 December 2014
Revised 3 April 2015

DOI:https://doi.org/10.1103/PhysRevB.91.205202

Authors & Affiliations

E. L. Ivchenko, L. A. Bakaleinikov, and V. K. Kalevich

Ioffe Physical-Technical Institute, 194021 St. Petersburg, Russia

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Vol. 91, Iss. 20 — 15 May 2015

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Images

Figure 1
Spin polarization degree of the conduction-band electrons (a,b), the intensity of interband PL (c), and nuclear spin polarization $P_{N 1}$ (d) as a function of the longitudinal magnetic field calculated for different excitation powers $W$ of the circularly polarized light. The curves 1–5 correspond, respectively, to the following values of $W$ : 10, 25, 75, 150, and 200 mW. The following set of model parameters is used in the calculations: $τ^{*} = 2$ ps, $τ_{h}^{*} = 30$ ps, $τ_{s} = 140$ ps, $τ_{s c} = 700$ ps, $P_{i} = 0.13, N_{c} = 3 \times 10^{15} {cm}^{- 3}$ , and $A = 17 μ eV$ . The relation $G = 2 \times 10^{23} W$ between the generation rate and excitation power is derived from the experiment, the units for $G$ and $W$ are ${cm}^{- 3} s^{- 1}$ and mW, respectively. In panel (a), both the nuclear spin relaxation time of the one-electron defect state ( $τ_{n 1}$ ) and that of the two-electron state ( $τ_{n 2}$ ) are taken to equal 150 ps. In panels (b), (c), and (d), $τ_{n 1}$ = 1000 ps and $τ_{n 2}$ = 1 ps. Note that the sets of calculated curves in panels (a) and (b) s are qualitatively similar.
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Figure 2
Nuclear spin polarization degrees $P_{N 1}$ (solid curves) and $P_{N 2}$ (dashed) as functions of the excitation power calculated at zero magnetic field for (a) $τ_{n 1} = τ_{n 2} = 150$ ps and (b) $τ_{n 1}$ = 1000 ps and $τ_{n 2} = 1$ ps. In panel (b) the values of $P_{N 2}$ are multiplied by a factor of 25.
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Figure 3
Dependence of the spin polarization of the conduction-band electrons on the magnetic field calculated for the excitation power 75 mW and different values of the nuclear spin relaxation time $τ_{n 1}$ (a) assuming $τ_{n 1}$ and $τ_{n 2}$ coincide and being equal to $0.15$ ps (curve 1), 50 ps (2), 150 ps (3), 450 ps (4), and $150 000$ ps (5); (b) keeping $τ_{n 2}$ = 150 ps fixed and taking $τ_{n 1}$ to equal 150 ps (curve 1), 300 ps (2), 750 ps (3), 1500 ps (4), and $150 000$ ps (5); and (c) $τ_{n 2}$ = 1 ps and $τ_{n 1}$ = 1 ps (curve 1), 2 ps (2), 5 ps (3), 10 ps (4), and 1000 ps (5). Other parameters are the same as those in Fig. 1.
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Figure 4
(a) Spin polarization of the conduction-band electrons as a function of the excitation power calculated at zero magnetic field (solid curve) and infinitely high magnetic field (dashed curve). The parameters are the same as those in Fig. 1. Solid and open circles represent the values measured in the absence of the magnetic field and at $B = 6.5$ kG [13]. (b) Isolines of the constant ratio $P_{e} (B \to \infty) / P_{e} (0)$ considered as a function of two variables, $\log_{2} (τ_{n 1})$ and $\log_{2} (τ_{n 2})$ , and calculated at the excitation power $W = 75$ mW. The times $τ_{n 1}$ and $τ_{n 2}$ are in picoseconds.
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Figure 5
Effect of the hyperfine interaction on the dependence of the spin polarization of the conduction-band electrons on the magnetic field. Curves 1–6 are calculated for the excitation power 75 mW and the following values of the hyperfine constant: $A = 0$ (horizontal line 1), $A_{0} / 4$ (curve 2), $A_{0} / 2$ (3), $A_{0}$ (4), $2 A_{0}$ (5), and $4 A_{0}$ (6), where $A_{0} = 17 μ eV$ . In panel (a), $τ_{n 1} = τ_{n 2}$ = 150 ps. In panel (b), $τ_{n 1} = 1000$ ps and $τ_{n 2} = 1$ ps. The values of the other parameters are the same as those in Fig. 1.
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