Radiative properties of multi-carrier bound excitons in GaAs
Excitons in semiconductors can have multiple lifetimes due to spin dependent oscillator strengths and interference between different recombination pathways. In addition, strain and symmetry effects can further modify lifetimes via the removal of degeneracies. We present a convenient formalism for predicting the optical properties of excitons with an arbitrary number of charge carriers in different symmetry environments. Using this formalism, we predict three distinct lifetimes for the neutral acceptor bound exciton in GaAs, and confirm this prediction through polarization dependent and time-resolved photoluminescence experiments. We find the acceptor bound-exciton lifetimes to be where . Furthermore, we provide an estimate of the intra-level and inter-level exciton spin-relaxation rates.
The radiative properties of excitons in semiconductors are of fundamental interest in current semiconductor physics as well as of technological interest due to their impact on optoelectronic device performance Chuang (2009). While the optical selection rules for the recombination of a conduction electron and valence hole are well understood Meier and Zakharchenia (1984); Chuang (2009), the selection rules for excitonic complexes with more than two carriers are complicated due to the multiple spin and angular degrees of freedom. In high symmetry environments, exciton lifetimes can be modified by interference between different recombination pathways Efros et al. (1996). This effect in quantum dots, for example, is responsible for the dark exciton, a radiative bottleneck in applications requiring bright sources de Mello Donegá et al. (2006) or alternatively a possible long-lived storage state for quantum information applications Poem et al. (2010). Exciton lifetimes are also modified by reduced symmetry environments, which can remove the possibility of interference by energetically separating the excitonic states.
In this work, we provide a convenient and general framework for describing the optical properties of arbitrary excitonic complexes. We use the second quantization formalism Kira and Koch (2012) for calculating dipole matrix elements of an excitonic complex with an arbitrary number of electrons and holes in a III-V direct band gap semiconductor. Using a generalized Weisskopf-Wigner theory, we show how special spontaneous emission eigenstates and multiple radiative lifetimes may emerge. We predict three radiative lifetimes of the neutral acceptor bound-exciton (AX) in bulk GaAs. We confirm the theory by performing polarization dependent and time-resolved photoluminescence experiments on the AX system.
Exciton lifetimes in III-V direct band gap semiconductors can be derived from the dipole operator for band-to-band recombination between a conduction-band electron and a valence-band hole Chuang (2009); [pages23-24]opticalOrientation. In the second quantization formalism, the dipole operator is
where () is the annihilation operator for an electron (hole) in the angular momentum state , H.C. is the Hermitian conjugate, and is a spin-independent constant A0X (). We define the coordinate system to be oriented along the ,  and  crystallographic directions. The hole angular momentum state is labeled with the opposite sign of the corresponding unoccupied electron angular momentum state. This dipole operator can be conveniently used to calculate the dipole matrix element between exciton states with an arbitrary number of charge carriers. For example, the dipole matrix element corresponding to the recombination of a two-carrier exciton with electron spin and hole spin is
where is the semiconductor vacuum state.
We describe the radiative lifetimes of excitons using a generalized Weisskopf-Wigner theory. The spontaneous emission rates from a set of degenerate excited states to a set of degenerate ground states are the eigenvalues of , where A0X (). Here is a matrix of the vector dipole matrix elements between ground state and excited state , , the frequency of the transition, the permittivity of the material, the index of refraction, and the speed of light. The time dependence of the excited state probability amplitudes satisfy
corresponding to exponential decay. Physically, Eq. 2 implies that radiative lifetimes are modified by constructive or destructive interference between different recombination pathways. In addition, it highlights how exciton states organize into spontaneous emission eigenstates (eigenvectors of ) with decay rates given by the eigenvalues of .
Before applying this formalism to the three-carrier acceptor bound-exciton system, as an example we treat the simpler two-carrier light-hole exciton. Light-hole excitons, consisting of an valence hole and a conduction electron, split from heavy-hole excitons () in reduced symmetry environments such as quantum dots, quantum wells and strained GaAs. The formalism yields the expected four spontaneous emission (SE) recombination rates Efros et al. (1996):
The aligned spin and excitons decay independently because of the orthogonal polarizations ( and ) of the transitions. On the other hand, constructive and destructive interference of the recombination pathway leads to a bright and dark exciton. While it is experimentally challenging to observe the brightest light-hole exciton due its polarization, this exciton has recently been observed using magnetic-field measurements in strain-engineered quantum dots Huo et al. (2014). We note that identification could alternatively be made through lifetime measurements at zero magnetic field.
We now turn to the neutral acceptor-bound exciton (AX), consisting of two holes and one electron bound to a substitutional acceptor impurity Bogardus and Bebb (1968); Haynes (1960). By recombination of the electron with one of the holes, AX decays radiatively to a neutral acceptor (A, a hole bound to an acceptor). Effective mass theory can be used to show that A has hydrogenic levels , , etc. In high-purity p-type GaAs, AX to A 1s, 2s, etc. photoluminescence (PL) is readily observed and provides a useful probe for resonant excitation, as shown in Fig. 1. Remarkably, the ensemble transition linewidths of this solid-state ensemble system are less than 40 eV (spectral linewidths in Fig. 1 are limited by the instrument resolution) A0X ().
Though the origin of the AX fine structure was once a controversy, strain experiments support hole-hole and crystal field coupling as the dominant mechanisms for splitting the 12 fold degenerate AX Mathieu et al. (1984); Karasyuk et al. (1998). In this scheme, the two holes lie in antisymmetric spin states with total spin 0 and 2 Mathieu et al. (1984). Hole-hole coupling splits the states from states. In zinc-blende semiconductors which possess crystal fields with symmetry, the states further split into two manifolds: with multiplicity 3 and with multiplicity 2. The full specification of AX also includes the spin of the electron, denoted as or (Fig. 2).
Our theory and experiments show that AX has multiple radiative lifetimes. To find the AX radiative lifetimes, we first compute the dipole matrix elements for the different A and AX states A0X (). The eigenvalues of are the radiative recombination rates of AX. An energy splitting between excited states causes a fast oscillation in the Weisskopf-Wigner theory which destroys the coupling between non-degenerate states A0X (). As such, only degenerate excited states are included in the dipole matrix when calculating the eigenvalues of .
The AX spontaneous emission rates in spherical symmetry and no hole-hole spin coupling are proportional to
When hole-hole spin coupling is introduced, states split from states, but the spontaneous emission rates remain unchanged:
With the inclusion of the zinc-blende crystal fields (which cause a - splitting), we find that the spontaneous emission rates become proportional to
Whereas previous studies of AX report only one lifetime Hwang (1973); Finkman et al. (1986) ( ns), a full study including spin and symmetry shows that AX has multiple lifetimes differing by up to a factor of 4. We can experimentally test this theory by studying the polarization dependence of photoluminescence (PL). If the system starts in an incoherent mixture of the four ground A states, excitation light of polarization resonant with will create an excited state density matrix in the subspace proportional to
where , is the incident polarization and are the dipole matrix elements corresponding to A0X (). Eq. 16 is valid in the limit of low excited state population. The PL emission from the states in with polarization is proportional to
Eq. 4 can be used to compute the arbitrary polarization dependence of AX-A transitions. In the case of exciting with linear polarization at an angle in the - plane and collecting linearly polarized light at ( corresponds to polarization along ), the angle dependent PL intensity is given by
where is a constant. These functions are plotted in Fig. 3a for and . Here we note that this simple angular dependence of excitonic PL can be used to verify the relative inter-carrier and crystal field coupling for AX, once a subject of debate, without the need for applied strain or magnetic fields Mathieu et al. (1984); Karasyuk et al. (1998); A0X (). In the case where crystal fields have an observable effect, the excitonic PL can also be used to determine crystal orientation: e.g. emission will be strongest when exciting along  and collecting .
We measure the polarization dependence of the transition using resonant continuous-wave (CW) excitation. Experiments were performed on a p-type GaAs crystal (6 m GaAs grown by molecular beam epitaxy on a GaAs substrate, ). The sample was mounted without strain in pumped liquid He (1.9 K) and excited with a Ti:Sapphire laser. Fig. 3a shows the polarization dependence of emission under resonant excitation of with and polarized excitation. The polarization visibility observed is somewhat less than would be expected from the ideal theory. The difference can be explained by relaxation between AX spin states.
We investigate the effect of inter-level relaxation on the diminished polarization visibility with time- and polarization-resolved measurements. Either the or transitions were excited resonantly with 2 ps Ti:Sapphire pulses, spectrally filtered to obtain 16 ps pulses with 0.03 nm bandwidth. Photoluminescence to was collected and imaged using a combined spectrometer/streak camera setup with a timing resolution of 27 ps. Four excitation conditions were studied, resonant excitation of or with or 45 linearly polarized light. PL polarized parallel and perpendicular to the excitation polarization was collected. The complete time-resolved data set is shown in Fig. 3b.
We observe a strong initial polarization visibility at that later decays because of inter-level relaxation. The initial polarization visibility of for excitation is close to the ideal value 9 for no excited state relaxation. (The uncertainty here is due to the uncertainty in the time.) The decay of polarization visibility indicates the existence of spin flip processes on the same timescale as the radiative lifetime.
We use the time-resolved data to obtain estimates of the inter- and intra-level relaxation rates in the exciton system (Fig. 3b). The time resolved data were fit to a 12 state density matrix model including inter-state relaxation A0X (). In the model, an optical pulse of a given polarization coherently creates an excited state density matrix given by Eq. 16. The subsequent time evolution of the excited state density matrix satisfies
where is a diagonal matrix of the excited state energies, the second term describes radiative recombination and is the Linbladian operator describing phenomenological relaxation between excited states A0X (). From the solution we calculate the relative PL intensity emitted into different polarizations using Eq. 4.
This model gives a good fit to the observed time dependence of AX emission (Fig. 3b). The 16 curves in Fig. 3b are fit simultaneously using 6 fit parameters: overall spontaneous emission rate (1.48 ns), inter-level relaxation (0.89 ns), intralevel relaxation (3.6 ns), intralevel relaxation (1.8 ns), temperature (4.7 K) and overall intensity normalization (5100 counts). These relaxation rates are shown schematically in Fig 2. The resulting AX radiative lifetimes are where lies in the range 0.49 to 0.74 ns. A detailed error analysis found the main source of uncertainty in the spontaneous emission rate to be due to an ambiguity in the choice of the background level A0X (). Since hole spin flips are predicted to be much faster than electron spin flips Fu et al. (2006); Kroutvar et al. (2004), we do not include electron spin flip processes in the model (shown schematically in Fig. 2). Temperature was included as a fit parameter because the effective temperatures for bound excitons can be larger than the bath temperature Rühle and Klingenstein (1978).
Using the best fit model parameters from the time resolved experiment, we are now able to predict the polarization dependence of PL in resonant CW excitation in the presence of spin relaxation. These curves are shown in Fig. 3a as “theory,” and agree well with the experimental data.
In conclusion, we presented a convenient and general formalism for calculating the optical properties of excitons in III-V semiconductors with an arbitrary number of carriers. We used this formalism to derive a model of the optical properties of AX in strain-free bulk GaAs which predicts 3 distinct radiative lifetimes. The model was confirmed using polarization and time-resolved experiments. The results are in contrast to previous reports for this system and highlight the importance of a unified treatment of all recombination pathways when deriving the radiative properties of multi-carrier excitons.
This material is based upon work supported by the National Science Foundation under Grant No. 1150647, DGE-0718124 and DGE-1256082. We would like to thank T. Saku for growing the material in NTT. YH acknowledges support from SORST and ERATO programs by JST.
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Radiative properties of multi-carrier bound excitons in GaAs
Appendix A Dipole Operator
The vector dipole operator for transitions between the conduction band and the heavy-hole or light-hole band in a zinc-blende direct band gap semiconductor (e.g. GaAs, InP, etc.) can be derived from the electron and hole basis functions. We derive the dipole operator in second quantization, which is convenient for calculating recombination rates for excitons with more than two charge carriers.
The valence band angular momentum states arise from coupling between p-like orbital states and the electron spin Yu and Cardona (2010). These couple together to form total angular momentum and :
The are known as heavy holes, as light holes and as split-off holes. Spin-orbit interaction splits the from the states and typically the split-off holes can be ignored in experiments.
Using angular momentum addition rules, the heavy hole and light hole states are
where are electron orbital wave functions transforming as and is the spin of the electron Chuang (2009). The hole angular momentum state has the opposite sign of the corresponding electron angular momentum. The conduction band states are and where denotes a spherically symmetric periodic part of the Bloch wave function. In spherical symmetry and for a exciton, the coordinate system can be taken to lie in an arbitrary direction. However for an exciton with non-zero momentum , the coordinate system must be taken with the z axis in the direction, thus somewhat complicating further analysis Chuang (2009). In what follows, we will restrict our discussion to excitations.
These basis functions can be used to calculate matrix elements of the dipole operator . As an example, the dipole matrix element for recombination of a spin down electron with a heavy hole is
The ordering of the matrix element reflects the transition occurring, in this case an electron moving from the conduction band to the valence band. In a bulk cubic crystal, by symmetry the matrix elements
are all identical. Further simplifying, we find this transition results in the production of right handed circularly polarized light:
The same procedure can be used to find the other dipole matrix elements.
We can now introduce creation operators () Kira and Koch (2012); Knox (1963) for the creation of an electron in the angular momentum state and some particular but unspecified spatial state. Because of the anti-symmetrization requirement, the creation and annihilation operators satisfy anti-commutation relations
where the anti-commuator is defined as . We will also introduce creation operators for holes using ; i.e. the linear and angular momentum of the hole has the opposite sign of the unfilled electron state Kira and Koch (2012). Instead of labeling the band index, we restrict hole creation/annihilation operators to act in the valence band and electron operators in the conduction band. For example, an exciton state can be written as
where is the semiconductor vacuum state with a filled valence band and empty conduction band.
The dipole operator can be written in second quantization as
where we have restricted to be in the valence band and to be in the conduction band, and Kira and Koch (2012). The first term corresponds to exciton annihilation and the second to exciton creation. Using the matrix elements calculated above, the dipole operator for a conduction band electron recombining with a heavy-hole or light-hole is
Appendix B Basis states for AX
The AX consists of two holes and one electron. Hole-hole coupling dominates, while the crystal fields split the levels further Mathieu et al. (1984). From the two holes, there are four possible total spin states: Sakurai and Napolitano (2010). The two holes in AX lie in a symmetric spatial state. On account of the Pauli principle, the spin state must therefore be antisymmetric with respect to interchange, resulting in only total spin 2 and 0 being allowed.
We will use
as shorthand for the creation of an antisymmetric state of two holes Kira and Koch (2012). Note that the ordering of the creation operators matters, consistent with the commutation relations in Eq. 1. Using this notation, we can write the total angular momentum states for the coupling of the two holes as:
In zince-blende semiconductors, hole-hole coupling splits the and states (Fig. S2).
In the presence of the crystal field with symmetry, the states split into three different irreducible representations Mathieu et al. (1984); Dresselhaus et al. (2008); Francoeur and Marcet (2010) (Fig. S2):
In order to derive these basis states, it is necessary to choose a coordinate system in which to write the symmetry operations of the crystal. Since we chose to use axes aligned along ,  and , , and correspond to the three crystallographic directions.
Appendix C Dipole Matrix Elements of AX
We use the dipole operator (3) to calculate the dipole matrix element between AX and A. To illustrate the method, we will calculate the matrix element as an example. First, we expand the matrix element and the dipole operator
All terms with an electron annihilation operator go to zero because the electron in is spin up. Using the fact that , the expression becomes
Using the fact that , and the commutation relations in Eq. 1, the dipole matrix element is
Repeating this calculation for each matrix element produces the dipole matrix elements for the AX-A system, given in Table S1.
Appendix D Generalized Weisskopf-Wigner theory for spontaneous emission from multiple excited levels
The Wiesskopf-Wigner theory of spontaneous emission Scully and Zubairy (1997); Berman and Malinovsky (2011); Hindmarsh (1967) can be generalized to calculate the spontaneous emission rate from a set of excited states to a set of ground states. The excited and ground states are not necessarily degenerate.
The wavefunction of the system is
where is the state with the atom in ground state and a photon in mode , polarization (), and is the excited atomic state and no photon. The sum on contains an implicit sum over the two polarizations .
In the interaction picture and rotating wave approximation, the Hamiltonian governing the time evolution of the atom and field is
are dipole matrix elements, is the annhilation operator for a photon in mode , , is the polarization of the photon, is the transition frequency, , is the material permittivity, is the material dielectric constant, is the speed of light, and is the quantization volume Scully and Zubairy (1997).
The time evolution is given by the Schrödinger equation
This yields the coupled differential equations
By formally solving the second equation and plugging into the first, the time evolution of the excited state probability amplitude satisfies
where and is some choice of natural frequency for the system. Assuming the modes are closely spaced in frequency, the sum over may be converted to an integral:
By introducing the matrix
and changing variables to integrate on , the equation of motion (Eq. 8) becomes
Since the integral over is only appreciable when , may be replaced with in the integrand and the lower frequency limit may be replaced by Scully and Zubairy (1997); Yamamoto and Imamoglu (1999). Using the delta function identity,
we arrive at the differential equations in the desired form:
The matrix can be computed from in a simple way. Using the parameterization
for the polarization, the integrand (9) contains a sum of integrals of the form
where is a unit vector. Performing the angular integrals, this becomes
Thus we arrive at a convenient shorthand for computing given the dipole matrix:
where the dot product is evaluated using . This shows the angular integral can be replaced with a simple dot product.
In matrix language, the differential equation governing excited state probability amplitudes for degenerate excited states is
By choosing a basis for the excited states in which is diagonal, the decay of each state is uncoupled from the others. Thus we see that the the eigenstates of decay independently at spontaneous emission rates equal to the eigenvalues of .
This method can be used to compare spontaneous emission rates between different manifolds of excited states. If the states are split in energy by a large amount compared to the radiative lifetime, the fast oscillating term in Eq. 10 will result in non-degenerate excited states becoming uncoupled. As an example, we solved Eq. 10 for two excited states and one ground state, whose lifetimes can be modified if the excited levels are degenerate. The differential equations governing the excited state probability amplitudes are
The solution is plotted in Fig. S3 for ns and various detunings . If the transitions are well resolved, the time dependence of the system follows that of two independent subsystems (Fig. S3a). On the other hand, if there is significant overlap between the Lorentzian line shapes, interference effects modify the radiative lifetime. In this toy model, near-degeneracy results in the existence of a dark state and long lived excited state population (Fig. S3b).
Therefore, to find the spontaneous emission rates for different non-degenerate sets of states, it is only necessary to calculate the eigenvalues of within each degenerate subspace. When comparing eigenrates between non-degenerate manifolds, the existence of in the pre-factor modifies the lifetimes. For excited state splittings of 10s of GHz (as for AX) at optical frequencies, this leads to a correction of a part in , which can often be neglected.
Appendix E Density matrix model
The time evolution of the excited state density matrix is described by
where is a diagonal matrix of the energies of the excited state energies, is the dipole matrix with elements and is the Linbladian operator. The first term describes unitary evolution, the second term spontaneous emission and the third excited state relaxation and decoherence. In this section, we describe the construction of this model.
A population transfer out of the system is mathematically identical to the spontaneous emission process in the excited state subspace. Population reduction can be accomplished mathematically using an anti-commutator Scully and Zubairy (1997). The spontaneous emission process is characterized by the decay of diagonal density matrix terms in the basis of the spontaneous emission eigenstates:
where and are the spontaneous emission eigenvalues and eigenstates of . Using the fact that any operator is diagonal in the basis of it’s eigenvectors, this becomes
Numerically, it is an advantage to include spontaneous emission in this way as the ground states do not need to be included in the density matrix. For AX, the excited state density matrix has 56 differential equations (ignoring off diagonal terms between non-degenerate excited states), whereas a treatment using the full density matrix including ground states would have 120.
Phenomenological relaxation between the excited states is included as a population transfer and decoherence. The excited state relaxation rate from state to arises from coupling between the spins and their environment. This model includes an intralevel relaxation rate between states in a degenerate manifold, and an interlevel rate between different manifolds. For and in different manifolds, the rates are modulated by the energy difference between the initial and final states,
where is the interlevel relaxation rate. This correctly reproduces the fact that in equilibrium the ratio of populations in different states is given by a Boltzmann factor. Many of the rates from states in the same irreducible representation can be shown to be the same by symmetry. For the purposes of this model, we assumed that all states within a given irreducible representation have the same phenomenological relaxation rate.
Phenomenological relaxation affects both the on- and off-diagonal elements of the density matrix. The total rate of population leaving () or entering () state is
This population relaxation also causes a decay of the off-diagonal terms in the density matrix. This can be accomplished with the anti-commutator:
where the second term enforces conservation of excited state population.
Appendix F Fit of time resolved data to model
We numerically integrated the equation of motion Eq. 11 to find the excited state density matrix as a function of time . In the case that is excited with polarization , the initial density matrix in the subspace is
where and all other terms of the density matrix are zero. From the solution of the density matrix as a function of time , the PL emitted from at polarization as a function of time is
These PL curves predict the time dependence of AX emission under different excitation conditions.
The model was fit to the data using a weighted least-squares residual due to the Poisson distributed nature of photon counting data Turton et al. (2003). Temperature was included as a fit parameter because in individual fits with K, the best fit relaxation rate depended on the excitation state. This implies that the effective exciton temperature was higher than 1.9 K, consistent with experiments on free excitons in GaAs where the effective temperature of free excitons was found to be somewhat higher than that of the bath Rühle and Klingenstein (1978).
We tested modifying the relaxation rate matrix so that electron spin flips can occur. In this case, the model also fits the data with different rates of interstate relaxation: interlevel relaxation (0.45 ), intralevel relaxation (1.2 ) and intralevel relaxation (0.71 ). Because we obtain good fits with either electron spin flips allowed or disallowed, the experiment is not sensitive to the rate of electron spin flips. However, the best fit spontaneous emission rate constant was the same 1.48 regardless of whether electron spin flips are allowed.
In order to estimate the uncertainty in the the measured spontaneous emission rate, we characterized the uncertainty due to random Poisson noise in the measurement data as well as systematic error due to the uncertainty in model parameters (e.g. pulse arrival time, background level). The largest uncertainty in measured spontaneous emission rate arises from the uncertainty of background level (Tab. S2). The raw data shown in Fig. S4 shows that there is a long lived emission above the background. This long lived emission may indicate some long lived state, e.g. exciton hopping into a metastable state and subsequent slow repopulation. Due to the uncertainty of the true background value, the AX best fit parameters acquire some uncertainty. Choosing a higher background level results in a faster best fit spontaneous emission rate, as the effective curvature of the decay becomes greater (Fig. S4). We take the confidence interval of the background level to be 0 to 30, this produces an uncertainty in spontaneous emission rate of (Fig. S4). Another way to estimate this uncertainty would be to incorporate a metastable excitonic level in the model. Because this introduces the danger of overfitting the model with too many adjustable parameters, we used the background level as a proxy for the uncertainty introduced by possible metastable states.
|Effect||Uncert. () in Spon. Emission|
|Data Cut Off||0.15|
|5% Laser Power Fluctuations||0.0047|
Next, we investigated whether changing the maximum number of data points collected (1-2 ns) modified the best fit spontaneous emission rate. In fitting the data, there is a somewhat arbitrary choice of when additional data points at longer times no longer improve the fit. Within reasonable choices of the data time cut-off of 1-2 ns, we found that the best fit spontaneous emission rate changed from 1.36 to 1.66 . This level of uncertainty is lower than that present from the unknown background level.
Next, we used a Monte Carlo simulation to determine the uncertainty in spontaneous emission due to Poisson noise. The raw data was used as the mean for new Poisson-distributed datasets. The model was fit to the new random datasets using the same weighted least-squares algorithm. The standard deviation of the resultant spontaneous emission rates is 0.0082 .
Monte Carlo simulations were also employed to calculate the uncertainty due to laser power fluctuations between experimental runs. The photon counting data was modulated by random 5% power fluctuations and passed through the least squares algorithm. We found the standard deviation of best fit spontaneous emission rates to be 0.0047 due to power fluctuations of the laser. These simulations demonstrate that the measurement is robust against Poisson noise and laser power fluctuations.
In summary, we have found that the spontaneous emission rate for AX lies within the range 1.36 to 2.03 . This corresponds to a lifetime constant in the range of 0.49 to 0.74 ns.
Appendix G Polarization of PL to determine dominant coupling in AX
While the splitting of the AX states into three sets of states is now understood, it was at one point a subject of debate. Two theories, the coupling scheme (JJCS) and the crystal-field scheme (CFS) can be used to explain some of the optical properties of AX Karasyuk et al. (1998); Mathieu et al. (1984). In both schemes, hole-hole coupling first rearranges the two hole states into and manifolds. In the JJCS, electron-hole coupling further splits the AX states, resulting in (arising from ) and ,