Control of hole localization in magnetic semiconductors by axial strain

Mn and Fe-doped GaN are widely studied prototype systems for hole-mediated magnetic semiconductors. The nature of the hole states around the Mn and Fe impurities, however, remains under debate. Our self-interaction corrected density-functional calculations show that the charge neutral Mn 0 and positively charged Fe + impurities have symmetry-broken d 5 + h ground states, in which the hole is trapped by one of the surrounding N atoms in a small polaron state. We further show that both systems also have a variety of other d 5 + h conﬁgurations, including symmetric, delocalized states, which may be stabilized by axial strain. This ﬁnding opens a pathway to promote long-range hole-mediated magnetic interactions by strain engineering and clariﬁes why highly strained thin-ﬁlms samples often exhibit anomalous magnetic properties.


I. INTRODUCTION
While hopes for room temperature ferromagnetism of Mn or Fe-doped III-V semiconductors have largely vanished [1], they emerge as model systems to study hole-mediated ferromagnetism and spintronics [2,3]. Hole localization governs the strength, nature, and range of magnetic interactions: extended (host-band-like) states are associated with long-range ferromagnetic interactions, whereas localized (impurity-bandlike) states usually lead to strong short-range interactions. For dilute systems where long-range interactions are crucial, hole localization or trapping usually is detrimental to long-range hole-mediated ferromagnetic interactions [4][5][6]. To assess whether or not the impurity simply captures carrier holes or electrons by changing its hybridization, one needs to identify the d n configuration or oxidation state [7] of the transition metal impurity. If the impurity itself does not capture carriers, we still need to consider the possibility of carrier trapping at other sites [8].
The electronic configuration of Mn in GaN has been studied extensively. In n-type GaN, Mn is in its singly negative charge state (Mn − , or Mn +II oxidation state) and has the d 5 configuration. Fe has the same d 5 configuration in its charge neutral state (Fe 0 , or Fe +III oxidation state). Under p-type doping, however, Mn is in the charge neutral state (Mn 0 ), and Fe in the singly positive charge state (Fe + ), but their oxidation states or d n configurations remain debated. Are these Mn 0 and Fe + states best described by d 4 or d 5 + h configurations? For Mn 0 , earlier theories suggest either a Jahn-Teller distorted d 4 configuration [9,10] or a d 5 + h configuration with the hole tightly bound in a state similar to the Zhang-Rice polaron [11][12][13][14][15], but experiments remain inconclusive on the nature of the Mn 0 electronic configuration [2, 16,17]. For Fe in GaN, even * hannes@ynu.ac.jp the existence of the Fe + charge state remains unclear [14,[18][19][20].
We show in this paper that around Mn or Fe impurities in GaN, hole carriers exhibit a variety of both trapped and delocalized states. We further show that by applying axial strain, the relative stability of the different hole states can be engineered, viz., carrier holes in unstrained GaN tend to be trapped by Mn or Fe impurities, but under tensile strain along the wurtzite c axis, delocalized hole states may be stabilized. Thus, the electronic as well as the ensuing magnetic properties of Mn or Fe-doped GaN can be engineered by axial strain due to, e.g., interfacial strain, mechanical compression, or the piezoelectric effect. This discovery sheds light on the possible origin of starkly different magnetic properties reported in Mn-doped GaN thin films [21,22] and deepens our understanding of the recently observed piezoelectromagnetic effect in Mn-doped GaN [23].

II. METHODS
So far, the theoretical study of the localization of carrier states in magnetic semiconductors has suffered from systematic failures of the different approximations to the density-functional theory [1]. The three basic failures are (i) misplacement of transition metal d levels with respect to band edges, (ii) band gap error and spurious mixing of d levels with host states, and (iii) incapability to localize holes or electrons in polaronic states. For transition metal impurities in semiconductors or insulators, not even hybrid functionals [24] or the GW approximation [25,26] offer systematic corrections to all three failures without additional empirical parameters. In the present paper, we employ a +U correction to accurately predict the transition metal d levels with respect to the GaN valence states [10,27] together with a self-interaction correction (SIC) on the nitrogen p levels, introduced to satisfy the generalized Koopmans condition [28,29].
We calculate the electronic structure of Mn and Fe-doped GaN in wurtzite supercells of 72 and 96 atoms. We replace one Ga cation by either a Mn or Fe impurity, the doping concentration thus being 2.1% and 2.8%, respectively. We use the projector augmented-wave method [30] implemented in the VASP code [31] with the GGA-PBE functional, on-site +U corrections of 3.9 and 3.5 eV for Mn and Fe d levels, respectively, and J fixed at 1.0 eV [32]. These +U parameters assure a correct description of the thermochemical stability of Mn and Fe in their different oxidation states [32] and agree well with values widely used in the literature [33]. The calculations are carried out with a plane-wave cutoff energy of 300 eV and a 4 × 4 × 4 k mesh for Brillouin-zone integrations. Structural relaxations are carried out such that forces acting on individual nuclei are <0.02 eV/Å. We then apply a hole-state potential on the N p levels such that the generalized Koopmans condition is fulfilled [28,29], yielding λ parameters of 5.1 and 3.9 eV for the Mn and Fe-doped systems, respectively. Notice that these λ are larger than 3.4 eV obtained for Mg doped GaN [29], likely due to the p-d hybridization in GaN:Mn and GaN:Fe; even though the λ parameters are system specific [34], the physical properties (charge density distribution, structural relaxation, magnetic moment) hardly vary with λ once it exceeds a critical value [28]. This theory has also, for the first time, correctly described the transition from extended to localized carrier states underlying a critical metal-insulator transition in a doped semiconductor [8]. For the present systems, this method accurately reproduces experimentally observed electronic and optical transitions (see Appendix), which further ensures the reliability of the present method for various total energy differences.

III. GROUND STATE OF THE ISOLATED DOPANT
We first investigate an isolated impurity, calculated in a supercell of 96 atoms, corresponding to the doping concentration of ∼2%. Our calculation shows that the ground state of both Mn 0 and Fe + in p-type GaN is a small polaron in d 5 + h configuration, with the hole tightly bound to one of the nitrogen atoms next to the impurity, i.e., instead of Mn 0 assuming the +III oxidation state, or Fe + the +IV state, one of the nitrogen atoms is oxidized to the +II state. Such small polaron ground states have previously been observed around full or empty d shell impurities, e.g., Mg (d 0 ), Be (d 0 ), or Zn (d 10 ) impurities in the same host [35][36][37][38], but never before around a half-filled d 5 shell.
The density of states for Mn 0 and Fe + in 96 atom supercells are given in Figs. 1(a)-1(d). The gap states show p-d hybridization, as described in Ref. [7]. The main difference between Mn 0 and Fe + is that Mn 0 has a clear crystal field resonance (CFR) state in the gap with a strong d character, whereas for Fe + , the gap states are mostly contributed by N p states, since the occupied Fe d levels are well below the valence band maximum E v . The Mn 0 CFR states were initially described as a partially occupied triply degenerate state of t representation, typical from the host symmetry [39][40][41], but here we find a doubly occupied e level clearly separated from an empty a level, as shown in Figs. 1(a) and 1(b). This splitting is associated with a broken-symmetry geometry, already described in Refs. [9,10]. Despite their correct description of the geometry, these recent works characterize the a level as mostly Mn d like. This a  level, however, has a strong p character due to one of the surrounding N atoms holding a small polaron, as revealed by the density of states shown in Figs. 1(a) and 1(b) and the real-space charge density distribution given in Figs. 2(a) and 2(b). This result points to a suitable explanation for Mn dopants in GaN based on d 5 plus a localized polaron. For Fe + , the lowest unoccupied state resembles that of Mn 0 -it is nondegenerate, strongly localized on one of the surrounding N atoms, and accompanied by a symmetry-breaking structural relaxation, thus best described as a small polaron state [see Figs. 1(c), 1(d), 2(c), and 2(d)]. Hence, both Mn 0 and Fe + exhibit d 5 + h ground states with the N p hole tightly bound to the transition metal impurity, regardless of whether the occupied transition metal d levels are in the gap or below E v .
Notice that for Mn the ground state polaron is along the [0001] direction (wurtzite c axis), but for Fe + the ground state polaron is along one of the 1101 directions (bond axis with one of the N atoms in the ab plane). These two differently oriented hole polarons are henceforth denoted by h c and h ab , respectively. Using this notation, the Fe + ground state configuration is d 5 + h ab and the Mn ground state configuration is d 5 + h c . However, for both impurities, both small polaron states are available: Fe + has a d 5 + h c configuration 25 meV above the ground state and Mn a d 5 + h ab configuration 14 meV above the ground state. This finding suggests that there may be other stable configurations with different hole states.

IV. MULTIFARIOUS HOLE STATES
For Fe + , in addition to the polaron ground state, we find a delocalized partially occupied dangling-bond hybrid (DBH) like state (Fe + ) * , akin to the one suggested earlier as the ground state [39][40][41]. The system exhibits a bistability because this (Fe + ) * configuration is only 37 meV less stable than the polaronic ground state. This delocalized quasi-1D state (h * 1D ) spreads over one of the six equivalent 1120 directions [see Figs. 2(e) and 2(f)], chosen randomly.
For Mn 0 , such a h * 1D state appears by increasing the impurity concentration to ∼2.8%, i.e., calculating the system in a 72  Table I. The fact that the impurity-to-impurity distance in the [1120] direction is exactly the same in both the 96 and 72 atom supercells suggests that this extended state is not simply a supercell effect [42], or a plurality effect [43] triggered by the presence of inactive spectator images, but indicates an insulator-metal transition when the Mn concentration is further increased [8]. Fe + calculated in the 72 atom supercell exhibits even more rich behavior. In addition to the h ab , h c , and (h * 1D ) states, we find another delocalized state [h * 2D , shown in Figs. 3(e)-3(f)], which is extended in the wurtzite ab plane. The h ab small polaron state is more stable, but the h c and h * 1D states (nearly degenerate in energy) are 18 and 19 meV less stable than the ground state (relative energies summarized in Table I). The emergence of delocalized states also for Fe + upon increasing impurity concentration evidences that Mn and Fe doped magnetic semiconductor systems are in the vicinity of an insulator-to-metal transition. However, as concentration further increases, clustering effects are expected to trap carrier holes [4][5][6], and/or, the system may phase separate [44][45][46][47]  and h * 2D are extended states. Uniaxial strain (+ for tensile, − for compressive) along the c axis is given by δ c , and the associated perpendicular lattice distortion by γ ab . The ground state configuration for each strain condition is set as E = 0, and the relative energies of higher energy configurations are given in meV. All data is calculated in 72 atom supercells. before exhibiting a Mott transition like Mn-doped GaAs [8]. Therefore it is desirable to find other means to stabilize the extended/delocalized hole states.

V. STRAIN CONTROL OF HOLE LOCALIZATION
GaN is a piezoelectric, so axial strain causes electric polarization P s , as illustrated in Fig. 4. For example, [0001] tensile strain displaces negative charge along the positive c axis, leading to polarization P s c. The small polaron d 5 + h systems imply a displacement of positive charge from three anions to the fourth neighbor of the d 5 center, which for the h c polaron implies a polarization P p c. Thus, an h c polaron in a GaN host under [0001] tensile strain exhibits opposite polarization than the surroundings, i.e., P s P p , which may lead to some destabilization as, in general, polarized systems have the tendency to align their polarizations.
We test this by applying such tensile strain for the defective systems (shown in Table I  the ground state. Thus, the relative stability of these two competing phases may be controlled by axial strain, due to, e.g., interfaces [48] or substrates [49]. The strain control of hole states being associated to electric polarization suggests that the hole localization/delocalization in these systems can equally well be controlled by electric gating. In fact, such intertwining of the piezoelectric effect and magnetic interactions has already been observed [23].

VI. DISCUSSION
We have shown that Mn 0 and Fe + dopants in GaN have polaronic d 5 + h ground states with holes trapped by one of the neighboring N atoms. However, we also find extended hole states slightly less stable than the polaronic ground states, and thus, the holes around the magnetic d 5 impurities exhibit multifarious behavior. Extended and trapped carrier states are associated with completely different magnetic interactions, so this multifarious behavior must be included in any attempt to describe the elusive, yet strong magnetic interactions observed in magnetic semiconductor materials.
The multifarious hole states can be engineered via axial strain, i.e., the relative stability of the different hole states can be controlled by varying strain conditions by interface [48] or substrate strain [49]. Compressive uniaxial strain along the [0001] direction stabilizes polarons oriented along the same [0001] direction, whereas tensile [0001] strain stabilizes polarons within the (0001) plane for Mn-doped GaN, and, extended hole states in the (0001) plane for Fe-doped GaN. The axial strain and different hole states are intertwined by the piezoelectric effect, such that strain causes electric polarization of GaN, which in turn affects the preferable direction of the polaron dipole moment, or destabilizes such a polaron state altogether. By the same token, the multifarious hole states can be controlled by external gating (inverse piezoelectric effect), as indeed recently was observed experimentally [23].
For Mn-doped GaN, the unstrained polaron ground state is unlikely to foster long-range ferromagnetic interactions, as evidenced by cryogenic Curie temperatures of around 10 K [17,22,23]. However, Fe-doped GaN has extended hole states under tensile uniaxial strain conditions similar as may be expected on GaN samples deposited on sapphire substrates [49]. This strain stabilization of the extended hole states happens at high impurity concentrations in the vicinity of a Mott transition [8]. Thus, suitable strain conditions and a sufficiently high impurity concentration may lead to much higher Curie temperatures even for Mn-doped GaN, which puts in new light the extraordinary high Curie temperature of 940 K observed in highly strained thin-film samples of Mn-doped GaN [21]. Because the strain induced transition from localized to delocalized states mostly involves host states, similar transitions may also occur for Mg (d 0 ), Be (d 0 ) or Zn (d 10 ) impurities in the same host [35][36][37][38], or, (d 0 ) or (d 10 ) impurities in oxides. This strain stabilization of extended states may even offer an explanation to the phantom ferromagnetism in thin-film oxides [50]. In the respective bulk materials, carriers tend to be trapped in polaron states [35][36][37][38] but may well become delocalized to facilitate ferromagnetic interactions due to substrate strain in thin-film samples.

APPENDIX: ELECTRONIC TRANSITIONS
Both Mn 0 and Fe + are assumed to be stable under p type doping, i.e., with the Fermi level in the lower part of the band gap. Thus we focus on the electronic transitions close to the valence band and calculate the thermal transition energies ε T (q/q − 1) = E(q − 1) − E(q) (Table II). For Mn, we also evaluate the optical transition energies in the Franck-Condon approximation [51][52][53], ε FC (q/q − 1) = E(q − 1; q) − E(q), where E(q) is the total energy for charge state q, and E(q − 1; q) refers to charge state q − 1 in the fixed geometry of q. Supercell size effects in total energies are treated as described in Refs. [54][55][56], including a rigid shift of the valence band by 0.27 eV [57] for transitions involving a CFR state. For Fe, the (+/0) transition lies slightly below the valence band edge, which means it requires degenerate hole doping of 10 21 cm −3 to be stabilized, which is larger than typically achieved in p-type GaN.

Comparison of GaN:Mn with optical experiment
The optical properties of GaN:Mn are well known-there are two absorption features at around 1.5 and 1.8 eV associated with Mn acceptor transitions. The band at 1.8 eV is  [58][59][60].
The other feature, which in fact has multiple lines around 1.5 eV, starting from a peak at 1.42 eV, was also initially attributed to the (0/−) acceptor transition [61,62] and later reinterpreted to be due to a Mn internal transition, to occur between either e and t levels [58], or the symmetry broken Mn CFR levels [9,10]. Typically, experimental absorption features are compared with optical transitions calculated in the Franck-Condon approximation, and indeed, ε FC (0/−) = 1.69 eV for Mn, as given in Table II, is well in agreement with the experimental feature observed at 1.8 eV. The metastable delocalized state found for Mn in the 72 atom supercell (Mn concentration of 2.8%) has a Franck-Condon transition only slightly lower in energy, ε FC (0/−) = 1.63 eV, which cannot be linked to the broad absorption peak at 1.5 eV. The ε FC (+/0) = 1.29 eV transition [63] may offer an interpretation for the 1.5 eV peak. However, it is questionable whether or not this (+/0) transition could yield that high intensities and the temperature activated changes observed in photocurrents [58]. Another interpretation for the 1.5 eV peak comes from the fact that the Franck-Condon approximation breaks down for polaron states [64,65]. As a consequence, in addition to the Franck-Condon spectrum of minimal kinetic energy [51][52][53], a sharp absorption peak may appear at the thermodynamic transition ε T (maximal kinetic energy [66]), here calculated as ε T (0/−) = 1.4 eV (see Table II), in good agreement with the experimental 1.5 eV peak. The obtained difference between ε F C and ε T , 0.29 eV, is the relaxation energy, or the thermal barrier required to activate the photocurrent at the ε T peak. It approaches the experimental value of 0.3 eV and also is comparable to the Stokes shift calculated in other Mn-doped systems such as nanocrystals [67].