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 , the magnetic-field and excitation-power dependencies of the electron and nuclear spin polarizations are calculated for the particular value of . 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.
pacs:71.70.Jp, 72.20.Jv, 72.25.Fe, 78.20.Bh
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 GaAsN, GaInAsN and semiconductor quantum wells Ga(In)AsN/GaAs at room temperature JETP1 (); JETP2 (); apl2009 (); current (); Int1 (); QW2010 (), see also Condensed () and references therein. The centers occupied by single spin-polarized electrons act as a spin filter Weisbuch (); Paget (); Condensed () 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 band-to-band photoluminescence (PL) and the photoconductivity, as compared to the linearly polarized photoexcitation JETP1 (); apl2009 (); current (); PRBcond ().
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 DP1972 (). 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 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. naturemat (); KalevichPRB (); JetpLett2012 (); APL2013 (); Buyanova2013 (); NatureComm (); Chen2014 (); Toulouse2014 (). 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 KalevichPRB (); JetpLett2012 (); NatureComm ().
The theory of spin-dependent Shockley–Read–Hall recombination derived in Ref. JETP2 (), see for more detais Condensed (), 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 , where the axis is parallel to the exciting light beam and coincides with the normal to the sample surface. The model of Ref. JETP2 () is unable to interpret the experimental data obtained in the longitudinal magnetic field . The initial way out KalevichPRB () was to assume the spin-relaxation time of bound electrons to depend on the magnetic field . 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 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. NatureComm (). 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. Toulouse2014 () 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. JetpLett2012 (). Nevertheless, the role of spin relaxation of nuclei in the system under consideration still remains open. In Ref. Toulouse2014 () 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 . This means nothing more than that the formulation of Ref. Toulouse2014 () 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 and 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 and . 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 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 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 of the electron-nuclear system, where and () are the bound-electron and nuclear spin projections upon the fixed axis , hereafter the normal to the sample surface, and is the angular momentum of a nucleus. In the first, general, part of the paper we will take to be arbitrary and then shift to the particular case of 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 GaAsN, the momentum is 3/2. A detailed analysis for this value of will be performed elsewhere.
In addition to , we also use the notation for the state with the electron spin and the total component of the angular momentum . In the following we take into account the hyperfine interaction of the electron and nuclear spins given by the Fermi contact Hamiltonian
where and () are the electron and nuclear spin operators. Moreover, we consider the normal incidence of the polarized exciting light in the external magnetic field (Faraday geometry), take into account the Zeeman Hamiltonian for the bound electrons and neglect the interaction between the magnetic field and the magnetic moments of the nuclei or conduction-band electrons. Here the bound-electron Landé factor and is the Bohr magneton.
The occupation of the defect with one bound electron is described by a spin-density matrix . In the Faraday geometry, the components with unequal total angular-momentum components and vanish. Therefore, it is enough to consider the components
which are normalized on the density of single-electron defects
The matrix with or contains only one non-zero component and can be presented as
It is worth to note that the electron spin-density matrix (22 matrix)
is diagonal whereas the matrices with contain off-diagonal components. In the geometry under consideration, the spin-density matrix of the defect singlet with two bound electrons is diagonal, its diagonal components are normalized on the density of double-electron defects, . The sum of and gives the density of deep defects, .
Thus, for a nucleus with , instead of 144 equations declared in Ref. Toulouse2014 () there are only 21 nonzero quantities to be found: 2 components and , 12 components with and , four components with , the densities of electrons in the conduction band with the spin and the unpolarized free-hole density .
Iii Kinetic equations for the spin-density matrix
The two kinetic equations
have the same form as those in the model of Ref. Condensed () where the hyperfine coupling was ignored. Here and are the densities of single-electron defects with the electron spin , their sum being , and are the generation rates of the spin-up and spin-down photoelectrons, and is the proportionality constant in the electron trapping rate by deep centers. We remind that, due to the relations
among the four densities and only two are linearly independent.
The steady-state kinetics of paired defects is described by the equations
The first term
describes generation of the defect states with two electrons due to the capture of a conduction-band electron onto a single-electron defect. The second term
describes the recombination of a free unpolarized photohole with one of the singlet-state electrons, is the proportionality constant. The final term describes the nuclear spin relaxation. For the nuclei with it has a simple unambiguous form
In case of the nucleus , 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 is characterized by one time parameter as follows DP1972 ()
The kinetic equations for the spin-density matrices () can be written in the compact form as
In Eq. (10) the first and second terms
describe the capture and loss of the second electron by a defect. The term on the right-hand side represents the hyperfine and Zeeman interactions with a 22 -dependent spin Hamiltonian
where , , and are the spin Pauli matrices. The bound-electron spin relaxation is phenomenologically described by the standard term
which is equivalent to
or, see Eq. (2),
We remind that, for nonzero density-matrix components, the sum coincides with which means that if . The set of equations (10) represents scalar equations, particularly, 6 equations for and 14 equations for .
The off-diagonal components of the spin-density matrix can be expressed via the diagonal components
Excluding the off-diagonal components we obtain for the diagonal components of the commutator in Eq. (10)
The factor is an even function of the longitudinal magnetic field whereas the factors with are asymmetric functions of because
Under circularly-polarized photoexcitation the electron-nuclear states with and () can be differently involved in the kinetics which is the main reason for the asymmetry of dependence observed experimentally.
The expression (17) describing the effect of hyperfine interaction can be rewritten in the form
allowing the interpretation in the spirit of Fermi’s golden rule for the probability rate
of the transition from the quantum state to the state , where and are the energies of these states, and are their average occupations, is the matrix element of the perturbation operator. In Eq. (21), the role of ideal -function is played by the smoothed -function with the damping
iii.1 The model neglecting nuclear spin relaxation
If the nuclear spin relaxation is neglected then the set of kinetic equations reads
We remind that, for , the value of vanish and, thus, is nonzero only for .
Surprisingly, the set (23) has a simple magnetic-field-independent solution
where is the total optical generation rate of photoelectrons into the conduction band (or, equivalently, photoholes into the valence band), is the initial degree of photoelectron spin polarization,
and the time is defined by Eq. (19). The factor is given by
and equals to 1/4 for and to for .
One can see that, for the steady-state solution (24), the values and coincide. This means that, on the first hand, the diagonal components of the commutator in Eqs. (15), (III) and (17) are switched off as if the hyperfine interaction were absent and, on the other hand, the nuclei are spin-polarized and their spin polarizations in the single- and double-electron defect states coincide
Since in the steady state the term in Eq. (23) vanishes the densities of conduction-band electrons, , and double-electron defects, , satisfy equations independent of the hyperfine constant and the magnetic field:
where , , and we use the dimensionless variables
iii.2 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
in Eq. (23) and, similarly to DP1972 (), 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 the components and reduce to a much simpler form
Moreover, the unphysical states with and should be excluded from Eq. (27).
Using the identity
we derive for the densities , and the polarization degree the following expressions
Replacing by their expressions (28) we find
The densities and satisfy Eqs. (25) where
should be replaced by
The spin relaxation of bound electrons can easily be incorporated into the balance equations if . For this purpose the sum in Eqs. (15) and (III) can be approximated by the sum calculated in the limit and given by . As a result the problem is reduced to solving a set of four equations, namely, the two equations (3), (4) and two additional equations
One can see from Eqs. (31) and (33) that, for the fast spin relaxation of double-electron defect states, the hyperfine interaction effectively leads to a decrease of the electron spin relaxation time governed by the parameter . Since is an even function of , see Eq. (29), in the approximation under consideration the point of minimum in the dependence lies at .
iii.3 Approximation of unpolarized nuclei
At low excitation powers when the lifetime of single-electron defect state is long compared with and that of two-electron states is longer than 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 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 by the sum .
Iv Hyperfine interaction for a nucleus with
In this case Eq. (7) reduces to two scalar equations for and which can be transformed to the equations for and :
Equations (10) for read
where, see Eq. (17),
Two additional equations for and have the form
From Eqs. (35) and (36) we conclude that the set of equations for the diagonal components of the spin-density matrix and occupations are dependent on the magnetic field through the square . This clearly demonstrates that, for , the electron spin polarization is a symmetric function of and has a minimum at the point .