Direct determination of zero-field splitting for single Co²⁺ ion
embedded in a CdTe/ZnTe quantum dot

Fig. 1 presents photoluminescence spectrum of a bright neutral exciton in a CdTe/ZnTe QD with a single Co²⁺ ion in zero magnetic field. Emission is split due to $s,p$-$d$ exchange interaction between exciton and magnetic ion. Four observed lines result from four possible spin projection of Co²⁺ ion with spin $3/2$ onto the quantization axis given by heavy-hole exciton in QD.Kobak et al. (2014) More precisely, in excited state formed by exciton and Co²⁺ ion, for each projection of excitonic spin ($\pm 1$, corresponding to $\sigma^{+}$, $\sigma^{-}$ circular polarisations) there are four energy levels and in the final state after exciton recombination there are four quantum states (two doubly degenerate levels at zero field) of Co²⁺. In each circular polarization there are 16 possible transitions but only 4 are optically allowed due to requirement of Co²⁺ spin conservation for electric dipole transition (solid arrows on Fig. 2). In zero magnetic field the 4 lines for $\sigma^{+}$ polarization coincide with those for $\sigma^{-}$.

Intensity of two inner and two outer lines reflects differences in occupancy of various Co²⁺ states. Fig. 1 shows a typical case, when cobalt has a fundamental state with spin projection $\pm 3/2$ and consequently outer lines related to such a spin states are more pronounced than lines related to $\pm 1/2$ states (inner lines). However we argue that apart from Co²⁺ level ordering, such four main lines do not carry sufficient information about absolute value of Co²⁺ zero-field splitting. It is because line separation energy is determined by $s,p$-$d$ exchange interaction not by splitting of Co²⁺ states. Moreover, occupancy of various states of Co²⁺ under optical excitation is not simply governed by temperature and Boltzmann distribution. One could try to describe it by an effective temperature, but there is no good method of determination of such a parameter, which could be order of magnitude larger than actual experimental temperature (e.g. 10 or 30 K instead of 1.7 K in Supplementary Information of Ref. Kobak et al., 2014). Moreover, Fig. 1 shows that two outer lines of a QD with a single Co²⁺ ion exhibit not-equal intensity which is a fingerprint of efficient relaxation of Co²⁺ - exciton system during lifetime of a single exciton introduced to the QD. This makes measurement of occupancy distribution of Co²⁺ states in empty dot even more complex. Therefore we conclude that precise determination of Co²⁺ zero-field splitting from analysis of four main photoemission lines of a QD with a single Co²⁺ is not feasible.

In previous section we discussed four main optical transitions with conserved spin of Co²⁺ ion. In this section we report about additional bright neutral exciton transitions, which can be observed only if in–plane strain mixes Co²⁺ spin states. Since such mixing is influenced by $s$,$p$-$d$ exchange interaction between carriers and magnetic ion, Co²⁺ spin mixing is slightly different for a QD with and without exciton. When spin wavefunction of Co²⁺ is not identical in initial and final state of optical transition, part of oscillator strength is distributed between transitions where main spin component is different in initial and final state, but both states are not fully orthogonal. This effect opens possibility of studying partially allowed optical transitions shown by blue and green dashed arrows in Fig. 2. The intensity of such emission lines is strongly sensitive on parameters describing Co²⁺ ion. For values of parameters determined in this work, intensity ratio between main emission lines (red colour in Fig. 2) and partially allowed lines (blue and green color in Fig. 2) is 3 orders of magnitude. Measurements of such weak PL lines is experimentally challenging but it is worth of effort because partially allowed lines carry information about parameters describing spin structure of Co²⁺ ion. In particular, energy difference between the strongest emission lines in spectra (red outer lines) and weak, partially allowed lines at lower energy (blue lines) are equal exactly to cobalt spin state splitting ($\Delta_{Co}$) since such transitions have the same initial states ($\pm 3/2$ with admixture of $\mp 1/2$) but different final states ($\pm 3/2$ with admixture of $\mp 1/2$ and $\pm 1/2$ with admixture of $\mp 3/2$, respectively). The same energy distance ($\Delta_{Co}$) is between inner main lines (red color) and partially allowed emission lines at higher energy (green color). The energy splitting of low energy weak emission lines (blue) is related to $s$,$p$-$d$ exchange interaction in initial state, so it is the same as splitting of outer main emission lines (red). Similarly, high energy weak emission lines (green) exhibit splitting equal to splitting of inner main lines (red).

Experimental observation of partially allowed transitions is shown in Fig. 3(a) which presents typical low temperature magnetophotoluminescence of bright exciton in a QD with a single Co²⁺ ion in Faraday configuration, measured in two circular polarizations of detection. Line identification on the spectra is in agreement with simulation shown in Fig. 3(b). In both figures (a) and (b) we observe strong main emission lines (red lines on panel (b) or scheme of Fig. 2) corresponding to transitions with conserved spin of Co²⁺. Additionally we observe also three weak emission lines. The first one is associated with the dark exciton recombination (black line on panel (b)) which involves change of Co²⁺ spin from mainly +3/2 to mainly -3/2, so it exhibit the largest g-factor and it is a fingerprint of Co²⁺ spin states mixing, but it does not carry information about zero-field splitting of Co²⁺. Two more weak lines are related to bright exciton transitions partially allowed by cobalt $\pm 3/2$ and $\mp 1/2$ spin states mixing (blue lines on panel (b) and scheme of Fig. 2). We determine zero-field splitting of cobalt ($\Delta_{Co}=3.13$ meV for Co²⁺ shown in Fig. 3) by subtraction of emission energies of lines related to the the transitions for which initial state is the same (mainly $\pm 3/2$) but final state is different (mainly $\pm 3/2$ and $\mp 1/2$ for weak and strong lines respectively). Note that values of $\Delta_{Co}$ parameter estimated from two pairs of emission lines are the same.

Simulation presented in Fig. 3 (b) is based on simple model of exciton - Co²⁺ - ion system described in details in Methods of Ref. Kobak et al., 2014. The most important part of our theoretical description is given by the standard Hamiltonian of Co²⁺ ion, which is also the Hamiltonian of the final state of the system after the exciton recombination:

where the first term represents the Zeeman effect described by magnetic ion $g$-factor, Bohr magneton $\mu_{B}$, magnetic field $\overrightarrow{\mathbf{B}}$ and total magnetic momentum $\overrightarrow{\mathbf{J}}$ ($J=3/2$). The second and the third term describe magnetic anisotropy of a magnetic impurity with $D$ and $E$ representing axial and in-plane components of crystal field. Both parameters $D$ and $E$ can be determined basing on anticrossings discussed in detail in the next section.

Since g-factors are different for main and partially allowed transitions (see Fig. 3), approaching of various lines in magnetic field is inevitable. Series of anticrossigs between main and partially allowed transitions are observed in magnetophotoluminescence experiment shown in Fig. 4(a). Additionally, Fig. 4(b) presents corresponding simulation of the optical spectra and Fig. 4(c) presents simulated energy levels of neutral exciton in a CdTe QD with Co²⁺ (initial states of photoluminescence) and empty dot with single cobalt (final states). All panels of Fig. 4 show important spectral features closely related to the spin structure of Co²⁺ ion. Anticrossing between $+3/2$ and $-1/2$ cobalt spin states in the absence of exciton appears four times in excitonic spectra (due to final state of recombination), all appearances are at the same magnetic field (about 7$.5$ T for QD from Fig. 4a). They are denoted on panel (b) and (c) by black circles. Red and blue squares on panel (b) and (c) represent anticrossings of cobalt spin states modified by the presence of carriers forming neutral exciton, which acts on Co²⁺ as effective magnetic field of about 3 T for QD from Fig. 4a. As a consequence such anticrossings are shifted and are visible at about $4.5$ T and $10.5$ T for $\sigma^{+}$ and $\sigma^{-}$ polarization of detection respectively. Observed shift is analogical to the shift of the magnetic ion levels anticrossing from zero field to field of about 2 T which has been already reported for Mn²⁺+h complex in InAs/GaAs QDs,Kudelski et al. (2007); Krebs et al. (2009); Baudin et al. (2011) and for Fe²⁺ in CdSe/ZnSe QDsSmolenski et al. (2016) where $s$,$p$-$d$ exchange interaction with exciton was also acting on magnetic ion as an effective magnetic field.

Particularly important for this work are anticrossings in the final state which are marked in Fig. 4 by black circles. They give additional information about Co²⁺ properties in absence of carriers. In such an anticrossing separation energy of spectral lines corresponds directly to separation energy between Co²⁺ states close to the anticrossing of spin states +3/2 and -1/2. In Fig. 5 we plot the dependence of such Co²⁺ spin states splitting $\Delta_{Co}(B)$ as a function of the magnetic field for QDs shown in Fig. 3 (violet points), Fig. 4 (green points), and for three other QDs with a single Co²⁺ ion.

Solving of Hamiltonian (1) leads us to analytical formula describing magnetic field evolution of energy distance between mixed $\pm 3/2$ and $\mp 1/2$ spin states of Co²⁺:

where parameters $D$ and $E$ (axial and in-plane components of crystal field) can be extracted directly from the experimental data. Parameter $E$ can be determined from the value of $\Delta_{Co}$ for magnetic field at which anticrossing between $+3/2$ and $-1/2$ spin states appears ($\Delta_{Co}(B_{min\Delta})=2\sqrt{3}E$). Knowing $E$ parameter one can obtain $D$ parameter from the zero-field spin states splitting given by formula: $\Delta_{Co}(0)=2\sqrt{3E^{2}+D^{2}}$. Finally $g$-factor is equal to $-D/(\mu_{B}B_{min\Delta})$. Parameters $D$, $E$, and the $g$-factor of Co²⁺ ion can be also determined by fit of Eq. 2 to experimental data, as shown by solid lines in Fig. 5.

The values of the parameters describing Co²⁺ ions in all studied QDs are summarized in Table 1. The data are sorted in increasing order with respect to the zero-field splitting $\Delta_{Co}$, which varies from about $2.5$ meV to almost $8$ meV. The value of parameter $D$ is in the range between $-1.22$ meV and $-3.77$ meV. For all discussed QDs parameter $D$ is negative which favours states of spin equal $\pm 3/2$. Parameter $E$, which causes the mixing of Co²⁺ spin states, varies from $0.12$ meV to $0.55$ meV. The values of Co²⁺ $g$-factor strongly differ from each other starting from $1.82$ and ending with the value $2.91$. Analysis of the Table 1 shows that there is an anti-correlation between $\Delta_{Co}$ and $g_{Co}$. The smaller Co²⁺ zero-field splitting the larger value of Co²⁺ $g$-factor. Surprisingly, for the declining value of Co²⁺ zero-field splitting we do not observe the convergence of Landé factor of Co²⁺ to the limit defined by unstrained CdTe crystal,Ham et al. (1960) which is equal $2.31$. This shows that there are other important factors affecting the structure of the energy states of cobalt ion. One of possible explanations of g-facotor lower than 2.31 is that larger g-factor would be observed for other axis than growth axis tested in our experiment.

We note that distribution of measured values of zero-field splitting is probably not governed by distribution in the studied samples, but it is rather affected by preselection of studied Co²⁺ ions caused by our experimental limitations. At zero field or at magnetic field in range of a few Tesla, only relatively small zero-field splitting of Co²⁺ can be determined. For large zero-field splitting partially allowed transitions are too weak to be observed. Therefore we expect that real distribution Co²⁺ zero-field splitting in the samples is even wider than observed and that typical value is larger than about 3 meV observed a few times in our first experiments. This expectation is supported by high magnetic field experiment (black data of Fig. 5), where anticrossing of +3/2 and -1/2 is expected at about 35 T for Co²⁺ ion with $\Delta_{Co}(0)$ equal to almost 8 meV and parameter $D=-3.77$ meV.

Absolute values of parameter $D$ obtained in this work are significantly larger than values reported for semiconductor systems without strain. For Co²⁺ in zinc blende semiconductors like CdTe, ZnTe or ZnSe parameter $D$ is neglectedHam et al. (1960); Wu and Dong (2004); Grzybowski et al. (2015). In wurtzite structure diluted magnetic semicondutors with Co²⁺ parameter $D$ was found to be positive with absolute value much smaller than in our experminets on QDs: +0.06 meV for CdSe,Lewicki et al. (1991); Isber et al. (1995) +0.08 meV for CdS,Lewicki et al. (1991); Bindilatti et al. (1994) and +0.34 meV for ZnO.Estle and De Wit (1961); Macfarlane (1970); Koidl (1977); Jedrecy et al. (2004); Ferrand et al. (2005); Sati et al. (2006); Pacuski et al. (2006)

In strained semiconductor systems, parameter $D$ obtained in this work for Co²⁺ can be compared to values obtained for transition metal ions. For Mn²⁺ QDs CdTe/ZnTe, parameter $D\approx 0.007$ meV was determined from depolarization efficiency of optically oriented ionLe Gall et al. (2009) and from variation of coherent oscillation amplitude,Goryca et al. (2014) which were also used for determination of parameter $E=1.8$ $\mu$eV.Lafuente-Sampietro et al. (2016a) Much larger values, in range of meV, are expected from magneto-photoluminescence of main emission lines in QDs with transition metal ions exhibiting non-zero orbital momentum, $D>0.8$ meVSmolenski et al. (2016) for Fe²⁺, $D>2.25$ meVLafuente-Sampietro et al. (2016a) for Cr²⁺. Finally, in our first report on main emission lines of a QD CdTe/ZnTe with a single Co²⁺, using multi parameter fit we obtained value $D_{z}\approx-1.4$ meV,Kobak et al. (2014) in the same range as directly determined values presented in this work.