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Spin-dependent recombination and hyperfine interaction at the deep defects

We present a theoretical study of optical electron-spin orientation and spin-dependent Shockley-Read-Hall recombination taking into account the hyperfine coupling between the bound-electron spin and the nuclear spin of a deep paramagnetic center. We show that the number of master rate equations for the components of the electron-nuclear spin-density matrix is considerably reduced due to the restrictions imposed by the axial symmetry of the system under consideration. The rate equations describe the Zeeman splitting of the electron spin sublevels in the longitudinal magnetic field, the spin relaxation of free and bound electrons, and the nuclear spin relaxation in the two 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 particular value of I = 1/2. The role of the nuclear spin relaxation in each of the both defect states is analyzed. The circular polarization and intensity of the edge photoluminescence as well as the dynamic nuclear spin polarization as functions of the excitation power are shown to have bell-shaped forms


I. INTRODUCTION
Spin-dependent recombination (SDR) via deep paramagnetic centers has recently attracted increased interest and proved to be an effective tool for obtaining an abnormally high spin polarization of free and bound electrons in nonmagnetic semiconductor alloys GaAs 1−x N x , Ga 1−y In y As 1−x N x and semiconductor quantum wells Ga(In)AsN/GaAs at room temperature [1][2][3][4][5][6], see also [7] and references therein. The centers occupied by single spinpolarized electrons act as a spin filter [7][8][9] and block the free electrons of the same spin polarization from escaping from the conduction band. As a result the spin polarization of free photoelectrons generated by circularly polarized optical excitation (as well as that of bound electrons) can be enhanced up to ∼100%. The amplification of spin polarization is accompanied by an increase in the concentration of photoelectrons, the intensity of bandto-band photoluminescence (PL) and the photoconductivity, as compared to the linearly polarized photoexcitation [1,3,4,10].
The hyperfine interaction between the localized electron and the nucleus of the deep center mixes their spin states resulting in (i) a reduction of the initial electron spin polarization and (ii) dynamic nuclear polarization of the defect atoms [11]. In the absence of an external magnetic field, the localized-electron spin polarization can be reduced down to 1/2 and 3/8 for the nuclear spin I = 1/2 and 3/2, respectively. The longitudinal magnetic field suppresses the hyperfine coupling and restores the electronic polarization as soon as the electron Zeeman energy exceeds the hyperfine interaction: The expected increase in the intensity and circular polarization of the edge PL in the longitudinal magnetic field has been confirmed experimentally. In addition, strong nuclear polarization effects, due to a combination of the spin-dependent recombination and hyperfine coupling, have been reported and discussed in Refs. [12][13][14][15][16][17][18][19]. Particularly, the dynamically polarized nuclei create an effective magnetic field (the Overhauser field) acting on the spins of localized electrons; this field is added to the external magnetic field and shifts the 'electron polarization vs. field' curve, with the shift changing the sign under reversal of the circular polarization of the exciting light [13,14,17].
The theory of spin-dependent Shockley-Read-Hall recombination derived in Ref. [2], see for more detais [7], ignores the nuclear effects. It has been successfully applied to describe the main features of optical spin orientation of conduction-band and deep-level electrons in GaAsN at zero and transverse magnetic field B ⊥ z, where the axis z is parallel to the exciting light beam and coincides with the normal to the sample surface. The model of Ref. [2] is unable to interpret the experimental data obtained in the longitudinal magnetic field B z. The initial way out [13] was to assume the spin-relaxation time τ sc of bound electrons to depend on the magnetic field B z . This assumption could explain the polarization recovery with increasing the field but faced with the pressing need to find a mechanism of the field dependence of τ sc which looked unresolvable. Moreover, the modified model cannot provide a reasonable interpretation of the observed shift of the polarization-field curve changing the sign under the reversal of circular polarization of the incident light.
The first attempt to give a theoretical description of the studied nuclear polarization processes has been performed by Puttisong et al., see Supplementary Methods for Ref. [17].
In that work, the hyperfine interaction is taken into account approximately by introducing magnetic-field-independent flip-flop processes in the electron-nuclear system and including an additional phenomenological parameter, the flip-flop spin relaxation time. This approximation obviously provides physical insight into the role of the nuclei but its validity for a quantitative description is not obvious. A kinetic theory of the spin-dependent recombination incorporating the hyperfine interaction of electronic and nuclear spins has been proposed recently by Sandoval-Santana et al. [19] The master equation approach for the spin-density matrix of the electron-nuclear system includes 144 equations which are solved numerically.
The numerical calculation reproduces the main experimental findings of Ref. [14]. Nevertheless, the role of spin relaxation of nuclei in the system under consideration still remains open.
In Ref. [19] the nuclear spins are polarized only in the deep-center states with single bound electrons. The nuclei with two bound electrons are characterized just by their steady state average concentration N 2 . This means nothing more than that the formulation of Ref. [19] is based on the assumption of very fast nuclear spin relaxation in the defect state with a pair of electrons. As far as we know, at present there are no grounds to take this assumption for granted. In general the spin relaxation times τ (1) n and τ (2) n for defect states with one and two bound electrons can be of the same order and even longer than the lifetimes of these states. In this work we develop a theory of the spin-dependent recombination and hyperfine coupling for the arbitrary values of τ (1) n and τ (2) n . The paper is organized as follows. In Sec. II we introduce the electron-nuclear spin-density matrix of the defect state with a single bound electron and the spin-density matrix of the defect with two bound electrons (in the singlet state) and discuss the restrictions imposed on the nonzero components of these matrices by the axial symmetry of the system in the longitudinal magnetic field. In Sec. III, we derive the rate equations for the spin density matrices taking into account both the hyperfine coupling for a nucleus with the angular momentum I and the electron and nuclear spin relaxation.
The particular limiting cases are analyzed in Secs. III A, B and C. The simplifications in the case of a nucleus with I = 1/2 are considered in Sec. IV. The results of numerical calculation and their discussion are presented in Sec. V. Section VI contains the concluding remarks.

II. ELECTRON-NUCLEAR SPIN-DENSITY MATRIX
We use the basic states |s, m of the electron-nuclear system, where s = ±1/2 and m (−I ≤ m ≤ I) are the bound-electron and nuclear spin projections upon the fixed axis z, hereafter the normal to the sample surface, and I is the angular momentum of a nucleus.
In the first, general, part of the paper we will take I to be arbitrary and then shift to the particular case of I = 1/2 which allows simplification of the kinetic equations for the densities and spin polarizations of the free and bound electrons. For the deep defect responsible for the spin-dependent recombination in GaAs 1−x N x , the momentum I is 3/2. A detailed analysis for this value of I will be performed elsewhere.
In addition to |s, m , we also use the notation |s, M − s for the state with the electron spin s and the total component of the angular momentum M = s + m. In the following we take into account the hyperfine interaction of the electron and nuclear spins given by the

Fermi contact Hamiltonian
where s α and I α (α = x, y, z) are the electron and nuclear spin operators. Moreover, we consider the normal incidence of the polarized exciting light in the external magnetic field It is worth to note that the electron spin-density matrix (2×2 matrix) is diagonal whereas the matrices ρ (M ) ss with |M | < I + 1/2 contain off-diagonal components.
In the geometry under consideration, the spin-density matrix of the defect singlet with two bound electrons is diagonal, its 2I + 1 diagonal components N 2,m are normalized on the density of double-electron defects, N 2 . The sum of N 1 and N 2 gives the density of deep defects, N c .

III. KINETIC EQUATIONS FOR THE SPIN-DENSITY MATRIX
The two kinetic equations have the same form as those in the model of Ref. [7] where the hyperfine coupling was ignored. Here N + = ρ e 1/2,1/2 and N − = ρ e −1/2,−1/2 are the densities of single-electron defects with the electron spin ±1/2, their sum N + + N − being N 1 , G + and G − are the generation rates of the spin-up and spin-down photoelectrons, and c n is the proportionality constant in the electron trapping rate by deep centers. We remind that, due to the relations among the four densities n, N 1 , N 2 and p only two are linearly independent.
The steady-state kinetics of paired defects is described by the 2I + 1 equations The first term In case of the nucleus I = 3/2, the spin-relaxation term is ambiguous. However, if the perturbation leading to the inter-sublevel mixing is nonselective then, similarly to Eq. (8), the relaxation for I = 3/2 is characterized by one time parameter as follows [11] The kinetic equations for the spin-density matrices ρ In Eq. (10) the first and second terms describe the capture and loss of the second electron by a defect. The term on the righthand side represents the hyperfine and Zeeman interactions with a 2×2 M -dependent spin Hamiltonian where Ω = A/h, β = gµ B B z /h, s α = σ α /2 and σ α are the spin Pauli matrices. The bound-electron spin relaxation is phenomenologically described by the standard term Similarly to Eqs. (8) and (9), the nuclear spin relaxation can simply be described by or, see Eq. (2), We remind that, for nonzero density-matrix components, the sum s + m coincides with s + m which means that s = s if m = m . The set of equations (10) represents 8I + 2 scalar equations, particularly, 6 equations for I = 1/2 and 14 equations for I = 3/2.
The summation of the terms in Eq. (10) over M yields the equations for the densities N ± of single-electron defects, see Eqs. (3) and (4), The off-diagonal components of the spin-density matrixρ (M ) can be expressed via the diagonal components Excluding the off-diagonal components we obtain for the diagonal components of the com- where and The factor U 0 is an even function of the longitudinal magnetic field whereas the factors U M with M = 0 are asymmetric functions of B z because Under circularly-polarized photoexcitation the electron-nuclear states with M and −M (M = 0) can be differently involved in the kinetics which is the main reason for the asymmetry of dependence P e (B z ) observed experimentally.
The expression (17) describing the effect of hyperfine interaction can be rewritten in the allowing the interpretation in the spirit of Fermi's golden rule for the probability rate of the transition from the quantum state |1 to the state |2 , where E 2 and E 1 are the energies of these states, f 2 and f 1 are their average occupations, V 21 is the matrix element of the perturbation operator. In Eq. (21), the role of ideal δ-function is played by the smoothed δ-function with the damping 8 A. The model neglecting nuclear spin relaxation If the nuclear spin relaxation is neglected then the set of kinetic equations reads We remind that, for M = ±(I + 1/2), the value of u M vanish and, thus, U M is nonzero only for |M | < I + 1/2.
Surprisingly, the set (23) has a simple magnetic-field-independent solution where G = G + +G − is the total optical generation rate of photoelectrons into the conduction band (or, equivalently, photoholes into the valence band), the initial degree of photoelectron spin polarization, and the time τ c is defined by Eq. (19). The factor C I is given by and equals to 1/4 for I = 1/2 and to [8(1 + P 2 c )] −1 for I = 3/2. One can see that, for the steady-state solution (24), the values ρ , a = c p /c n and we use the dimensionless variables In these notations the hole density p is given by (Y + Z)N c . Equations (25) are identical to Eqs. (20) in Ref. [7] derived neglecting electron-nuclear hyperfine interaction.
B. The model assuming fast nuclear spin relaxation in the paired singlet As an alternative limiting case, we assume the nuclear spin relaxation in the defect state with two electrons to be quite short and set N 2,m = N 2 2I + 1 in Eq. (23) and, similarly to [11], ignore the nuclear spin relaxation in the single-electron defects. For convenience, we will first ignore the spin relaxation of bound electrons and then will extend the obtained result to allow for this relaxation. The solution for the spin-density matrix can be presented in the form Note that since v ±(I+1/2) = 0 the components ρ Using the identity we derive for the densities N + , N − and the polarization degree P c the following expressions where Replacing N ± by their expressions (28) we find The densities n and N 2 satisfy Eqs. (25) where The spin relaxation of bound electrons can easily be incorporated into the balance equations if τ sc τ c . For this purpose the sum in Eqs. (15) and (16) can be approximated by the sum calculated in the limit τ sc → ∞ and given by ζGP e /2. As a result the problem is reduced to solving a set of four equations, namely, the two equations (3), (4) and two additional equations It follows then that Eqs. (30) and (31) are valid as well ifτ −1 s is replaced by One can see from Eqs.

C. Approximation of unpolarized nuclei
At low excitation powers when the lifetime of single-electron defect state τ c = (c n n) −1 is long compared with τ (1) n and that of two-electron states τ c2 = (c p p) −1 is longer than τ (2) n one can take the nuclei to be unpolarized and set It follows then that the third terms describing in Eqs. (15) the hyperfine interaction can be replaced by where τ −1 scn is the bound-electron spin relaxation rate induced by the nucleus and defined by Therefore, in this approximation the influence of nuclei is accounted for by replacing τ −1 sc by the sum τ −1 sc + τ −1 scn .
The substitution of (6), (39) into Eqs. (3), (4) gives us two equations connecting three unknown quantities n + , n − and N 2 . We replace them by their sum and difference and obtain Eq. (6) arriving at where a = c p /c n and hereafter, instead of n ± , we use the variables n = n + + n − and P e = (n + − n − )/(n + + n − ).
From Eq. (43) one has where L, M are independent of Z. Therefore the substitution of (44) into Eq. 1(c). In experiment this is observed, under circularly polarized interband optical excitation, via the magnetic-field induced increase in the PL circular polarization and intensity [13,14,16,17,19]. The set of parameters unrelated to the nuclei and hyperfine coupling is the same as used in the previous analysis [13]: τ * = 2 ps, τ * h = 30 ps, τ s = 140 ps, τ sc = 700 ps, P i = 0.13, N c = 3×10 15 cm −3 . For the deep centers in GaAsN the average hyperfine n . The nuclear spin polarization is characterized by the two polarization degrees Their dependence on the excitation power W calculated in the absence of magnetic field is depicted in Fig. 2. In the case (a) the values P N 1 and P N 2 are different but comparable in magnitude whereas in the case (b), as expected, the polarization P N 2 is small and the polarization P N 1 is of the same order as the polarizations in Fig. 2(a). As seen in Fig. 1(d), the average nuclear spin is the highest at B = 0 and exhibits depolarization with the increasing magnetic field since the field decouples the electron and nuclear spins. It is also worth to note that the zero-field nuclear spin polarization is a nonmonotonic function of the excitation power and reaches a maximum for the intermediate power ∼75 mW. This can be understood as follows: in the low-power limit, the system is slightly driven out of the equilibrium and the nuclear polarization is still weak; in the high-power limit, the lifetime of bound electrons τ c is very short, the hyperfine-coupling factor U M in Eqs. (17), (18) decreases and the dynamic nuclear polarization by the polarized electrons is strongly weakened.
The spin-filtering effect is demonstrated in Fig. 3(a). At very low excitation intensity this effect is not switched on, the degree P e is independent of the magnetic field and the two curves in the figure calculated at zero (solid) and infinitely high (dashed) magnetic field merge as W → 0. At very high pumping the curves again merge since the lifetimeτ c , see Eq. (22), becomes very short and the uncertainty caused by this reduction decouples the hyperfine interaction. Figure 3(b) shows the power dependence of the ratio of electron polarizations at the strong and zero magnetic fields (solid) and similar ratio of the PL intensities (dotted). The peaks of the two curves are shifted with respect to each other in agreement with the experiment, Fig. 4(b) in Ref. [13]. increases sublinearly and is more sensitive to the variation of A as compared to P e (0).

VI. CONCLUSION
Thus, due the axial symmetry