Holistic electronic response underlying the development of magnetism in co-doped diluted magnetic semiconductors

Codoping, i.e., the simultaneous doping by two kind of dopants or, equivalently, defects, has been proven to be a suitable approach in enhancing the magnetism of diluted magnetic semiconductors (DMSs) and doped transition metal oxides (TMOs) [1–15]. Despite the plethora of theoretical and experimental reports on codoped DMSs and TMOs (especially those based on ZnO [16–31], GaN [32], GaAs [33, 34], TiO₂ [35, 36], etc), our understanding of the origin of their magnetism is still incomplete. Classic model approaches i.e., double exchange, superexchange, p-d exchange, RKKY, etc, (see, for example, [37]) are usually employed to justify the magnetic features of these materials.

Recently, we proposed an additional process [14, 15] whose applicability is more pronounced in the codoped systems. This puts emphasis on the spin-polarization induced by the magnetic dopants on their neighboring anions. It was demonstrated that our proposed successive spin polarization (SSP) model could be competitive with the other classic processes thus leading to enhanced ferromagnetism in codoped DMSs and related materials. (In the following, we will use the symbol Ga(A,B)N to indicate a host material (GaN in this example) codoped with the codopants A and B).

The locality of the proposed SSP approach [14, 15, 38] made us put emphasis on the local features that are underlying the developing of magnetism in DMSs and TMOs. Thus, it was demonstrated that ferromagnetism is correlated well with those system features which characterize the neighborhood of the magnetic dopants. We identified some universal features underlying the development of magnetism in these materials. These included the strength and direction of the spin polarization induced on the anions neighboring the magnetic dopants, the charge transfer processes from and towards and magnetic dopants as well as the charge redistribution within them. Interestingly, no clear dependence on bond lengths developed around the magnetic dopants was found. However, we found dependencies on the type of host DMS or TMO system as well as whether the magnetic dopants belong to the early or the late 3d-series of the transition metals (TMs) [39]. Further investigation of the dependence of DMS and TMO magnetism on the host material [40] led us to the identification of the dependencies of the magnetic moment ${\mu }_{A-{dop}}^{X,{host}}$ of the A-dopant on the host material in the presence of the X-codopant. We demonstrated that ${\mu }_{A-{dop}}^{X,{host}}$ correlates well with the corresponding d-band center, ${d}_{c,A-{dop}}^{X,{host}}$ of the A-dopant as well as with the p-band center, ${p}_{c,N-{host}}^{A,X}$, of the N-anions of the host in the presence of the A and X codopants. (We consider parts of d−and p − bands below the Fermi energy, E_F).

In the present work we place our focus on a particular 3d-TM dopant, A, and look for those factors, including the host and a codopant type, X, which optimize the magnetic features of the codoped DMS or TMO material. Our search has revealed that the magnetism in these materials is the outcome of a holistic response of the host system in the presence of the codopants. We show that the codopants by affecting the band structure of the system shift the p-band center of the anions of the host which, in turn, pull the d-band center of the d-bands of the codopants relative to E_F thus dictating the value of the magnetic moment of the dopants.

We used ab initio calculations within the density functional theory (DFT) and the spin-polarized generalized gradient approximation (SGGA) at the Perdew-Burke-Ernzerhof (PBE) [41] level of approximation augmented by including a Hubbard-U correction (DFT/SGGA+U formalism) based on Dudarev's approach [42] as implemented in the Vienna Ab-initio Simulation Package (VASP) [43–45]. We used U-values which can produce the band gaps of the free hosts as close as possible to the experimental results (see [39]) in order to have better description of their dopant (impurity) states/bands. Better results are obtained by fitting U-values to experimental results than by fitting to theoretical results obtained either using the HSE06 or the B3LYP functionals [46]. Changing the Hubbard-U parameter, U_d,TM for the (co)dopants we found quantitative changes in the results for the spin-polarization induced by the (co)dopants on their 1nn anions. However, they do not invalidate the proposed SSP model. Thus, as we are looking mainly for trends, we kept U_d,TM the same for all (co)dopants and avoided the use of any other hybrid functional (HSE06, B3LYP, mBJ [47, 48]) and their computational demands. For more details, please see [40].

We carried out a systematic analysis of the host materials GaN, GaP and CdS, doped with one of the V, Mn, Co and Cu dopants and codoped with the transition metals (TMs) of the 3d-series. We calculated the magnetic moments, ${\mu }_{A-{dop}}^{X,{host}}$, A = V, Mn, Co, Cu; X = 3d-TM in everyone of the above mentioned hosts as well as the d-band center, ${d}_{c,A-{dop}}^{X,{host}}$, of the total and spin-resolved d-bands of A and X.

3.1. Mn-dopants in codoped Ga(Mn,X)N, X = 3d-TM

The left panel of figure 1 shows the dependence of the d-band center, ${d}_{c,{Mn}-{dop}}^{X,{GaN}}$, of Mn-dopants in GaN codoped with 3d-TM-dopants, X, indicated on the horizontal axis (black circles). For comparison, in the same panel we also present the d-band center, ${d}_{c,X-{dop}}^{{GaN}}$ of X-monodopants in GaN (green triangles). In the right panel of figure 1, we show the variation of the d-orbital energies (with respect to vacuum) of the free 3d-TMs atoms as given by Harrison [49].

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The correlation between the magnetic moment, ${\mu }_{{Mn}-{dop}}^{X,{GaN}}$, of the Mn-dopants in codoped Ga(Mn,X)N, X = 3d-TM (X indicated on the horizontal axis) and their corresponding d-band center, ${d}_{c,{Mn}-{dop}}^{X,{GaN}}$, is shown in the left panel of figure 2, while the dependence of ${\mu }_{{Mn}-{dop}}^{X,{GaN}}$ on the codopant X is shown in the right panel of figure 2.

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As is apparent from figure 1, the variation of ${d}_{c,X-{dop}}^{{GaN}}$ (X being a monodopant in Ga(X)N) follows, as expected, that of the d-orbital energies of the free TM-atoms. The interesting thing here is that this variation is also observed for the ${d}_{c,{Mn}-{dop}}^{X,{GaN}}$. Ie., the X-codopant dictates the energy position of the d-band of the Mn-atom the latter adjusted close to that of the X-codopant.

3.2. Mn- and Co-dopants in codoped Ga(A,X)P, A = Mn,Co and X = 3d-TM

Figure 3 shows the dependence of ${\mu }_{{Mn}-{dop}}^{X,{GaP}}$ (left panel) and ${d}_{c,{Mn}-{dop}}^{X,{GaP}}$ (right panel) on the X-codopant in Ga(Mn,X)P, X = 3d-TM. Correspondingly, figure 4 shows the dependence of ${\mu }_{{Co}-{dop}}^{X,{GaP}}$ (left panel) and ${d}_{c,{Co}-{dop}}^{X,{GaP}}$ (right panel) on the X-codopant in Ga(Co,X)P, X = 3d-TM.

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It is worth noting that the correlations (${\mu }_{{Mn}-{dop}}^{X,{GaP}},{d}_{c,{Mn}-{dop}}^{X,{GaP}}$) and (${\mu }_{{Co}-{dop}}^{X,{GaP}},{d}_{c,{Co}-{dop}}^{X,{GaP}}$) for the Mn and Co dopants, respectively, in Ga(Mn,X)P and Ga(Co,X), X = 3d-TM, show opposite trends. That is, while ${\mu }_{{Mn}-{dop}}^{X,{GaP}}$ increases as X moves from the left to the right side of the 3d-series, ${\mu }_{{Co}-{dop}}^{X,{GaP}}$ decreases, going deeper and away from E_F (left panels of figures 3 and 4). Opposite trends are also seen for ${d}_{c,{Mn}-{dop}}^{X,{GaP}}$ and ${d}_{c,{Co}-{dop}}^{X,{GaP}}$ (right panels of figures 3 and 4). Ie., an increase of ${\mu }_{{Mn}-{dop}}^{X,{GaP}}$ is associated with a shift in the corresponding ${d}_{c,{Mn}-{dop}}^{X,{GaP}}$ deeper and far away from E_F. The opposite happens for Co, i.e., a shift in ${d}_{c,{Co}-{dop}}^{X,{GaP}}$ closer to E_F is associated with an increase of the corresponding ${\mu }_{{Co}-{dop}}^{X,{GaP}}$.

3.3. V-, Co- and Cu-dopants in codoped Ga(A,X)N, A = V, Co, Cu and X = 3d-TM

In figures 5–7 we present the variation with the codopant of the magnetic moment and their corresponding d-band center for three dopants of different d-band filling, namely V, Co and Cu used as dopants in Ga(A,X)N, A = V, Co, Cu, X = 3d-TM.

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Figure 5: the ${\mu }_{V-{dop}}^{X,{GaN}}$ varies along two branches. The one is limited in the group of the early 3d-TM and shows almost no variation for ${\mu }_{V-{dop}}^{X,{GaN}}$. The other covers the late 3d-TM group and shows mostly a decreasing tendency of ${\mu }_{V-{dop}}^{X,{GaN}}$ as X tends towards the end of the 3d-series. The trend seen for ${d}_{c,V-{dop}}^{X,{GaN}}$ is that it decreases as X goes from the early to the late 3d-series with noticeable exemptions for X = Ti,V. It is worth noting that the variation of the d-band center of the X-codopant, ${d}_{c,X-{dop}}^{X,{GaN}}$, follows that of ${d}_{c,V-{dop}}^{X,{GaN}}$.

Observing the results of figure 5 it can be seen that ${\mu }_{V-{dop}}^{X,{GaN}}$ shows a complicated variation with the X-codopant although the corresponding ${d}_{c,V-{dop}}^{X,{GaN}}$ variation shows a rather linear decrease in gong from the left to the right of the 3d-series.

Figure 6: ${\mu }_{{Co}-{dop}}^{X,{GaN}}$ decreases along two different branches one along the early and the other along the late 3d-series as in the case of V (figure 5). Correspondingly, there is also an increase in ${d}_{c,{Co}-{dop}}^{X,{GaN}}$ along two branches dictating the variation of ${d}_{c,X-{dop}}^{X,{GaN}}$. There appears to be no major differences in the results obtained for Ga(Co,X₂)N and Ga(Co₂,X)N.

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Figure 7: ${\mu }_{{Cu}-{dop}}^{X,{GaN}}$ is seen to increase as X moves from the early to the late 3d series. Correspondingly, ${d}_{c,{Cu}-{dop}}^{X,{GaN}}$ shows a slight decrease, while ${d}_{c,X-{dop}}^{X,{GaN}}$ shows a rather larger decrease.

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Figures 5–7 show the trend in the correlation (${\mu }_{A-{dop}}^{X,{GaN}}$, ${d}_{c,A-{dop}}^{X,{GaN}}$), A = V, Co, Cu showing a noticeable dependence on the d-band filling.

3.4. Cu-dopants in codoped Cd(Cu,X)S, X = 3d-TM

In figure 8 we present the magnetic moment, ${\mu }_{{Cu}-{dop}}^{X,{CdS}}$, and the d-band center, ${d}_{c,{Cu}-{dop}}^{X,{CdS}}$, of Cu-dopants in Cd(Cu,X)S, X = 3d-TM. Additionally, in the left panel we include the variation of the corresponding energy gaps for spin-up and spin-down electrons (indicated as low/high gaps, ${E}_{{gap},{low}/{high}}^{X,{CdS}}$), while in the right panel we include the corresponding variation of the p-band center, ${p}_{c,S-{CdS}}^{X,{Cu}}$, of the S-anions in the presence of the X and Cu codopants.

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In this figure we see a similar variation with respect to the codopant X (i.e., the atomic number of the codopant) for ${\mu }_{{Cu}-{dop}}^{X,{CdS}}$, ${E}_{{gap},{low}}^{X,{CdS}}$ and ${p}_{c,S-{CdS}}^{X,{Cu}}$. We also observe similar features found for Mn and Co dopants in GaP (figures 3 and 4). Ie., an increase of ${\mu }_{{Cu}-{dop}}^{X,{CdS}}$ is associated with a shift in ${d}_{c,{Cu}-{dop}}^{X,{CdS}}$ moving it deeper and far away from E_F.

The conclusions derived from figures 1–8, for the codoped systems Ga(A,X)N, Ga(A,X)P and Cd(A,X)S, A = V, Mn, Co, Cu; X = 3d-TM, can be used to identify the following gross features:

• ${d}_{c,A-{dop}}^{X,{host}}$ of a dopant A = V, Mn, Co, Cu, in a specific host seen as a function of the codopant X = 3d-TM follows, in general, the trend of the energies of the 3d-TM free atoms, i.e., ${d}_{c,A-{dop}}^{X,{host}}$ moves away from E_F as A moves along the 3d-series from left to right with the only exception found for Co (see figures 4 and 6).
• ${d}_{c,A-{dop}}^{X,{host}}$ of the dopant A = V, Mn, Co, Cu, in a specific host also dictates that of ${d}_{c,X-{dop}}^{X,{host}}$, X = 3d-TM, i.e., that of its X-codopant (and vice versa).
• As ${d}_{c,A-{dop}}^{X,{host}}$, A = V, Mn, Co, Cu, in a specific host moves away from E_F, its corresponding ${\mu }_{A-{dop}}^{X,{host}}$ increases with a possible exception found for V shown in the results of figure 5.
• The implicit relation $[{\mu }_{A-{dop}}^{X,{host}}$, ${d}_{c,A-{dop}}^{X,{host}}{]}_{X=3d-{TM}}$, A = V, Co, Cu⁴ shows a noticeable dependence on the d-band filling.
• The variation with respect to the X-codopant is found to be similar for ${\mu }_{{Cu}-{dop}}^{X,{CdS}}$, ${E}_{{gap},{low}}$ and ${p}_{c,S-{CdS}}^{X,{Cu}}$ (see figure 8).

Most of the above listed observed features can be understood as resulting from the development of bonding and antibonding orbitals in terms of the d-orbitals, d_A, A = V, Mn, Co, Cu, of the dopants and the orbitals, d_X, of the X-codopants. In fact, the bonding energy, ${E}_{{AX}}^{{bond}}$, resulting from the hybridization of d_A and d_X coupled through the interaction V_AX is given by the following equation [49]:

$$\begin{eqnarray}&&{E}_{{AX}}^{{bond}}=\displaystyle \frac{{\epsilon }_{{d}_{A}}+{\epsilon }_{{d}_{X}}}{2}-{\left\{{\left(\displaystyle \frac{{\epsilon }_{{d}_{A}}-{\epsilon }_{{d}_{X}}}{2}\right)}^{2}+{V}_{{AX}}^{2}\right\}}^{1/2}.\end{eqnarray} \tag{ 1 }$$

In the case of $| {V}_{{AX}}| \ll 1$, equation (1) takes the following form:

$$\begin{eqnarray}{E}_{{AX}}^{{bond}}=\left\{\begin{array}{cc}{\epsilon }_{{d}_{X}}-\displaystyle \frac{{V}_{{AX}}^{2}}{{\epsilon }_{{d}_{A}}-{\epsilon }_{{d}_{X}}} & {\epsilon }_{{d}_{A}}\gt {\epsilon }_{{d}_{X}}\\ {\epsilon }_{{d}_{A}}-\displaystyle \frac{{V}_{{AX}}^{2}}{{\epsilon }_{{d}_{X}}-{\epsilon }_{{d}_{A}}} & {\epsilon }_{{d}_{A}}\lt {\epsilon }_{{d}_{X}}\end{array}\right\}\end{eqnarray} \tag{ 2 }$$

In view of equation (2), it is expected that in a particular host, ${d}_{c,{Cu}-{dop}}^{X,{host}}\,\approx {E}_{{CuX}}^{{bond}}$ to be a decreasing function of X = 3d-TM, because for Cu we have: ${\epsilon }_{{d}_{{Cu}}}\gt {\epsilon }_{{d}_{X}}\,\,\forall X\in [3d-{TM}]$. This is, in fact, what we find, as shown in figure 8. In agreement with equation (2) are the results for ${d}_{c,{Mn}-{dop}}^{X,{GaN}}$ (figure 1), those of ${d}_{c,{Mn}-{dop}}^{X,{GaP}}$ (figure 3), and those for ${d}_{c,V-{dop}}^{X,{GaN}}$ (figure 5) in which we can clearly see an increasing and a decreasing part joined at ${\epsilon }_{{d}_{X}}={\epsilon }_{{d}_{{Mn}}}$ and ${\epsilon }_{{d}_{X}}={\epsilon }_{{d}_{V}}$ correspondingly. However the results of ${d}_{c,{Co}-{dop}}^{X,{GaP}}$ (figure 4) and those of ${d}_{c,{Co}-{dop}}^{X,{GaN}}$ (figure 6) show opposite behavior to that expected from equation (2), while the results of ${d}_{c,{Cu}-{dop}}^{X,{GaN}}$ (figure 7) are in partial agreement with equation (2).

In order to look for the causes of this discrepancy, we recall that these may be due to d-band filling effects and the fact that in the presented figures we are considering the band centers of the combined spin-up and spin-down d-bands and not independently those of each spin-band themself. In order to check these points, we calculated, as an example case, the d-band centers of the spin-up and spin-down bands for the set of Ga(Co,X)N, X = 3d-TM systems and plotted the calculated d-band averages for the spin-up, spin-down (${d}_{c,{spin}-{up}}^{{aver}},\,{d}_{c,{spin}-{dn}}^{{aver}}$ respectively) and the total d-band, ${d}_{c,{Co}-{dop}}^{{aver}}$ (= ${d}_{c,{spin}-{up}}^{{aver}}+{d}_{c,{spin}-{dn}}^{{aver}}$), in figure 9. The averages are taken over systems having as one of its codopants a specific 3d-TM shown in the legends.

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As it becomes apparent from figure 9, although the ${d}_{c,{spin}-{up}}^{{aver}}$ for Co is almost the larger (in absolute value), however the corresponding ${d}_{c,{spin}-{dn}}^{{aver}}$ is almost the smaller. This means that the arguments based on equations (1) and (2) should be applied separately for spins-up and separately for spin-down bands. Additionally, it should be noted that the bonding ${t}_{2g,{up}}^{{bond}}$ state that is developed from the Co's and codopant's spin-up ${t}_{2g,{up}}$ bands will be populated by the five spin-up electrons of Co. As a result, the spin-up electrons of the codopant's ${t}_{2g,{up}}$ state will populate the antibonding ${t}_{2g,{up}}^{{antibond}}$ state. This means that for codopants with higher ${d}_{c,{spin}-{up}}^{{aver}}$ than that of Co, the contribution of the spin-up bands to the combined ${d}_{c,{Co}-{dop}}^{{aver}}$ band will follow the increasing order of the ${d}_{c,{spin}-{up}}^{{aver}}$ of the codopants. On the other hand, the contribution of the spin-down bands to ${d}_{c,{Co}-{dop}}^{{aver}}$ appears not to be significantly dependent on the codopant as the ${d}_{c,{spin}-{down}}^{{aver}}$ of the codopant bands span a narrow energy window as shown in figure 9. It is apparent that the relative strength of the spin-up and spin-down contributions to the ${d}_{c,A-{dop}}^{X,{host}}$, A = V, Mn, Co, Cu in a particular host, will depend on both their d-band filling and the d-band filling of their codopant. This is because these two numbers specify the populations of the bonding and antibonding states that they develop. Apparently the basic role of both the crystal field splittings and the exchange spin-splittings in the population of the hybridized orbitals are implicitly incorporated in our analysis.

It can be claimed that arguments based on equations (1) and (2) and their spin-resolution can capture the major features of our results for the dependence of ${d}_{c,A-{dop}}^{X,{host}}$, A = V, Mn, Co, Cu on their X-codopant in the considered hosts. This is quite significant because the knowledge of ${d}_{c,A-{dop}}^{X,{host}}$ allows us to predict the behavior of the magnetic moment, ${\mu }_{c,A-{dop}}^{X,{host}}$, in a particular host in the presence of 3d-TM X-codopants. In fact, the dependence of ${\mu }_{c,A-{dop}}^{X,{host}}$ on the atomic number of the X-codopant of the dopant A shows an almost linearly increasing (decreasing) variation when ${d}_{c,A-{dop}}^{X,{host}}$ decreases (increases) as a function of the X-codopant's atomic number.

The implicit correlation ${[{\mu }_{c,A-{dop}}^{X,{host}},{d}_{c,A-{dop}}^{X,{host}}]}_{X}$, X = 3d-TM, can be understood as a tendency indicating that as ${d}_{c,A-{dop}}^{X,{host}}$ is getting far away from E_F, the d_A orbitals become more localized and the dopant tends to obtain its free-atomic characteristics. The opposite is true in the case that ${d}_{c,A-{dop}}^{X,{host}}$ approaches E_F. Additional characteristics of the host and the codopants which affect the charge transfer processes may contribute to this variation. Thus, one could encounter dependencies on the relative atomic electronegativities and band fillings of host and codopant atoms.

The development of the trends shown in figures 1–9 and those found in [40] indicate that the development of magnetism in DMSs and TMOs is not limited to a hybridization process of the d-orbitals isolated from the rest of the material. Instead, it appears to be a synergistic process that is undertaken by the whole system in responding to the presence of the codopants. Ie., the introduction of the dopant/codopant bands brings significant changes in the band gap of the system. This leads to the relocation of the valence band maximum (VBM) which, in turn, pulls the p-band center of the anions of the host material into a new position. This, in turn, pulls the d-band centers of the hybridized orbitals of the codopants into their final position. In figure 8, we show this parallel variation of ${\mu }_{{Cu}-{dop}}^{X,{CdS}}$ and the gaps ${E}_{{gap},{low}/{high}}$ in the left panel and that of ${d}_{c,{Cu}-{dop}}^{X,{CdS}}$ and ${p}_{c,S-{CdS}}^{X,{Cu}}$ in the right panel for the systems Cd(Cu,X)S, X = 3d-TM.

The present results thus suggest a completely new approach in the investigation and understanding of the origin of the defect induced magnetism. This is strongly dictated by the way the defects (codopants) affect the electronic band structure of the host material, the hybridization and population of its bands.

Moreover, our results justify previous reports indicating that the location of the Fermi level, E_F, within the impurity band plays a crucial role in determining T_C through an estimation of the degree of localization of the impurity band holes and suggest that codoping with donor ions, or modulation doping can be used to engineer the location of E_F within the impurity band in an efficient manner [27, 50, 51].

The present results shed a new light on the electronic processes which underlie the development of magnetism in DMSs, TMOs and related materials. Dopants and codopants are found to elicit a holistic response from their hosting material to their presence. The codopant's additional contribution is to induce a rehybridization of dopant's d-spin-orbitals which become electron populated in accordance with the band fillings of both dopant and codopant. These processes dictate the absolute value of the magnetic moments of the codopants and therefore the magnetic features of the system. The type of the magnetic coupling among the dopants/codopants (FM or AFM or of Feri) could induce small quantitative changes related to the local aspects of the doping (intra-dopant and inter-dopant charge transfers). However it will not affect the conclusions of the present work which emphasize the holistic response of a host system upon its doping.

The results presented reveal the optimal choice for the codopant in order to optimize the magnetic moments of the dopants. On the other hand, the development of the magnetic coupling among the magnetic dopants is shown to be the outcome of competing processes among which the one of the successive spin polarization [14, 15, 38] seems to have a potential role.