Absence of halfmetallicity in defect-free Cr, Mn-delta-doped
Digital Magnetic Heterostructures
We present results of a combined density functional and many-body calculations for the electronic and magnetic properties of the defect-free digital ferromagnetic heterostructures obtained by doping GaAs with Cr and Mn. While local density approximation/(+U) predicts half-metallicity in these defect-free delta-doped heterostructures, we demonstrate that local many-body correlations captured by Dynamical Mean Field Theory induce within the minority spin channel non-quasiparticle states just above . As a consequence of the existence of these many-body states the half-metallic gap is closed and the carriers spin polarization is significantly reduced. Below the Fermi level the minority spin highest valence states are found to localize more on the GaAs layers being independent of the type of electronic correlations considered. Thus, our results confirm the confinement of carriers in these delta-doped heterostructures, having a spin-polarization that follow a different temperature dependence than magnetization. We suggest that polarized hot-electron photoluminescence experiments might bring evidence for the existence of many-body states within the minority spin channel and their finite temperature behavior.
Digital Magnetic Heterostructures (DMH) are semiconductor heterostructures in which magnetic mono-layers are incorporated using the digital-alloy technique [1, 2]. This technique was developed in order to combine the conventional semiconductors with magnetic materials, with the desire to realize simultaneous band gap and magnetic engineering. Within the DMH technique, the quality of magnetic mono-layers is highly sensitive to growth conditions similarly to the case of diluted magnetic semiconductors, were low temperature thermal annealing [3, 4] and codoping [5, 6] has been extensively developed. The existing technological difficulties increase therefore the significance of first-principles investigation, which can predict many characteristics of DMH and indicate trends in their properties.
For DMH materials, most first-principles investigations were done for GaAs/Mn [7, 8]. The exchange interactions inside and between Mn mono-layers were analyzed, in particular the spacer dependent magnetic properties were recently discussed . Independent of the spacer thickness the electronic structure was predicted to be half-metallic and using the technique of injection of free holes  intra and interlayer exchange coupling mechanisms were discussed . In addition, previous studies discussed transport properties, optimal concentration of Mn, possible effect of anti-site defects, etc., [7, 8] . A number of calculations were also performed for Mn doped Si, Ge, and Ga [11, 12, 13].
Properties of these heterostructures are expected to deviate from the physical behavior presented by family of diluted magnetic semiconductors . The most common position occupied by Mn within the (III,V) host GaAs diluted magnetic semiconductor is the Ga site. The substitutional Mn contributes with its two 4s electrons in the crystal bonding similarly as the two Ga-4s electrons. Less favorable energetically is the interstitial Mn position and therefore is less common . While in Mn doped GaAs semiconductors the system is described as “bulk”, in the fabrication of GaAs/Mn-digital magnetic heterostructures, MnAs mono-layers were embedded into GaAs superlattices , so that the quasi-bidimensional geometry of the MnAs layer is preserved. Thus, the random substitution of positions (Ga occupied or interstitial) by the Mn ions in the case of diluted magnetic semiconductors shows up as an ordered replacement of a GaAs layers by MnAs layers in the case of DMH, with evident consequences upon the critical temperature. DMH exhibits an unexpected Curie temperature dependence on concentration of Mn ions. At a fixed value of Mn concentration (fixed number of MnAs layers), the Curie temperature decrease with the thickness of the GaAs, but saturates at large thicknesses ( GaAs mono-layers). Electronic structure calculations were also reported in the literature for the case of -doping with Mn of GaAs/AlAs superlatices . In the defect free cases a half-metallic heterostructure was obtained. It was also demonstrated that in the presence of As-anti-sites, the half-metallic state was destroyed and a p-type metallic conductance was evidenced because of majority spin electrons. In addition the computed magnetic coupling between the Mn ions becomes larger in the presence of As anti-sites . To our best knowledge no results which include electronic correlations were reported for the case of Cr or Mn even in the defect-free cases.
In this paper we present results of the first-principles electronic structure calculations within Local Spin Density Approximation (LSDA), LSDA+U and LSDA+Dynamical Mean Field Theory (DMFT) for the different defect free (MAs)/(GaAs) digital heterostructures, containing transition metal-arsenide mono-layers (M=Cr,Mn) introduced in the GaAs superstructure. The most important correlation effect in half-metals, the formation of many-body states within the gap can not be captured by the mean-field LSDA/+U-type calculations. Dynamical mean-field theory was demonstrated to be able to describe such effects , therefore is essential to compare directly the mean-field LSDA/+U and the DMFT results. We demonstrate that the half-metallic character is lost in -doped GaAs heterostrcutures containing Cr or Mn at finite temperatures and in the presence of dynamic correlations. Correlation effects soften the magnetic properties of the ferromagnetic layers, induced states above with tails that cross the Fermi level and reduce the carrier spin polarization. Within the minority spin channel below the Fermi level the character of states forming the band-edge remains not affected by electronic correlations and the states are mostly localized on GaAs layers confirming the experimentally observed confinement of carriers in these heterostructures [15, 17].
Ii Crystal structure and computational method
To compute the electronic structure for the digital magnetic heterostructures we used the standard representation of the tetragonal symmetry, space group . The tetragonal super cell was obtained from the FCC unit cell by a rotation in the basal plane and an integer translation along the (001) direction. The new translation vectors are: , , , where is the bulk GaAs lattice parameter, and , and they correspond to a (GaAs) super cell. Within this super cell a GaAs mono-layer is replaced by MAs, such that the obtained DMH has the unit cell formula (MAs)/(GaAs). For GaAs which has an open structure, a close packed structure is obtained by including empty-spheres. Such empty potential wells were used also in the present heterostructure geometry. We used identical muffin-tin spheres for all atoms in the unit cell having the average Wigner-Seitz radius corresponding to the experimental GaAs lattice parameter () . The basis used for the self-consistent calculations contains the spd-partial waves for transition metals, -partial waves for Ga, As and -partial waves for the empty spheres E. means that the -partial waves are downfolded within the self consistent calculations. Selfconsistency was performed for 105 irreducible -points and total energy convergence was achieved with an accuracy of Ry. Results for the density of states, magnetic moments and spin polarizations are presented in the next section.
Correlation effects in the valence orbitals are included via an on-site electron-electron interaction in the form . The interaction is treated in the framework of DMFT [18, 19, 20], with a spin-polarized T-matrix Fluctuation Exchange (SPTF) type of impurity solver  implemented within the EMTO basis set [23, 22]. The Coulomb matrix elements are expressed in the usual way  in terms of three Kanamori parameters , and . Here, destroys/creates an electron with spin on orbital on lattice site . The SPTF approximation is a multiband spin-polarized generalization of the fluctuation exchange approximation (FLEX) [25, 26], but with a different treatment of particle-hole (PH) and particle-particle (PP) channels. The particle-particle (PP) channel is described by a -matrix approach [27, 28] giving a renormalization of the effective interaction. This effective interaction is used explicitly in the particle-hole channel. Justifications, further developments and details of this scheme can be found in Ref. 21.
For the case of half-metallic ferromagnets it was demonstrated  by model as well as realistic electronic structure calculations that many-body effects are crucial for half-metals: they produce states with tails that cross the Fermi level so that the gap is closed and half-metallicity is lost [29, 30, 31, 32, 33]. The origin of these many-body non-quasiparticle (NQP) states is connected with “spin-polaron” processes: the spin-down low-energy electron excitations, which are forbidden for the HMF in the one-particle picture, turn out to be possible as superpositions of spin-up electron excitations and virtual magnons [34, 16]. Spin-polaron processes are described within the SPTF approach by the fluctuation potential matrix with , defined in a similar way as in the spin-polarized FLEX approximation :
The essential feature here is that potential (1) is a complex energy dependent matrix in spin pace with off-diagonal elements:
where represents the bare vertex matrix corresponding to the transverse magnetic channel, is an effective transverse susceptibility matrix and is the bare transverse susceptibility . are fermionic Matsubara frequencies and corresponds to the magnetic interaction channel [25, 26]. The local Green functions as well as the electronic self-energies are spin diagonal for collinear magnetic configurations. In this approximation the electronic self-energy is calculated in terms of the effective interactions in various channels. The particle-particle contribution to the self-energy was combined with the Hartree-Fock and the second order contributions . To ensure a physical transparent description the combined particle-particle self-energy is presented by its a Hartree and Fock types of contributions: where the particle-hole contribution reads:
Since the static contribution from correlations is already included in the local spin-density approximation (LSDA), so-called “double counted” terms must be subtracted. In other words, those parts of the DFT expression for the total energy that correspond to the interaction included in the Hubbard Hamiltonian has to be subtracted. Several double-counting schemes has been proposed and used for specific materials applications [35, 36]. It has been recognized that the results of the LSDA+U calculations may depend crucially on the choice of the scheme used . In particular for the class of moderately correlated metals as one may consider the half-metallic ferromagnets in the LSDA+U calculations performed here we used the “around mean-field”  double-counting scheme. Similarly to the LSDA+U in the DMFT approach the Hubbard Hamiltonian represents the underlying physics, while in contrary to the mean-field LSDA+U solution the time dependent dynamics are considered, while spatial fluctuations are neglected. Obviously double counting correction has to be considered also. To achieve this, we replace with  in all equations of the DMFT procedure . Physically, this is related to the fact that DMFT only adds dynamical correlations to the LSDA result. For this reason, it is believed that this kind of double-counting subtraction “” is more appropriate for a DMFT treatment of metals than the alternative static “Hartree-Fock” (HF) subtraction .
Iii Results and discussion
When the mono-layer of MAs, with M=Cr, Mn is introduced in the GaAs super cell, the symmetry lowering from cubic to tetragonal take place and the states split further into a double degenerate and situated at lower energies and the non-degenerate .
At higher energies the and are situated [39, 24]. Magnetic exchange interaction split the spin channels and contribute additionally to the crystal field in opening the minority spin gap. In Fig.1 we present spin resolved, total density of states (DOS), for the DMH structures (MAs)/(GaAs), M=Cr, Mn. In both minority and majority spin channels at energy of about (not shown in Figs.1) the As- and Ga- are present. In the energy range to As and Ga states are dominant characters in the bonding orbitals, while in the vicinity of the Fermi level in the energy range of to the dominant transition metal states are present (see also orbtial projected DOS of Fig.2). At the LSDA level the occupied minority spin channel for both compounds is similar, because in this energy range the transition metal -states are weakly present within the bonding orbital dominated by As and Ga- states. The Fermi level is situates for minority spin electrons within the gap, closer to the bottom of the conduction band. The values of the energy gaps are and . The difference between the Fermi level and the bottom of the conduction band is somewhat larger for the DMH with Cr, in comparison with the value obtained in the Mn-based DMH. Given the fact that the top of minority-spin valence band is situated at similar energies in these heterostructures, the magnitude of the gap is therefore determined by the energy distance between the Fermi level and the bottom of conduction band.
A more detailed analysis of the density of states can be made on the base of the orbital projected densities of states Fig. 2. Here the LSDA and the LSDA+U results for the correlated transition metal atoms are compared.
LSDA+U calculations were performed for different values of the average Coulomb interaction and taking the same value for the exchange parameter . In Fig. 2 we show the results for U=3eV and J=0.9eV, typical values for transition metals [24, 40]. The left column represents the results for the compound with Cr, while the right column shows the densities for the compound containing Mn. In each pannel the upper/lower part contains the results for the majority/minority spins. Although the similarities of the orbital projected DOS are visible, we note the general tendency of the LSDA+U namely to enlarge the minority spin gap and . An interesting feature is that most of the spectral weight at the Fermi level is provided by the orbital for both cases of Cr and Mn, however the larger value is obtained with the Cr substitution. An almost negligible spectral weight is obtained for -orbitals. As expected from this approach transition metal d-states are further pushed apart from the Fermi level. In particular for the Cr heterostructure within the majority spin channel the LSDA peaks (see Figs. 1,2) located at about -1eV are shifted to lower energies at -1.5eV. The occupied majority spin Mn states are within the bonding band at -3eV are also shifted to lower energies at -4eV. Within the minority spin channel for all compounds the group of states situated around 1eV above are shifted accordingly to higher energies (see Fig. 1, 2). For the other values of U and J a similar tendency of DOS was obtained for both heterostructures.
In the followings we plot the total As and Ga spin resolved density of states for (MAs)/(GaAs), M=Cr,Mn. Fig.3 presents the results for Cr- doping. The left/right columns show the As/Ga-DOS for the different layers. As one can see in the majority spin channel the most important contribution around the Fermi level comes from the arsenic and gallium atoms situated at and . These are in fact the positions in the immediate vicinity of the transition metal CrAs-layer. The similar trend is visible for the Mn- doped heterostructure in Fig.4. In addition we note that as expected at the Fermi level the dominant states are As-p states.
The general picture resulting from the extended analysis (by inspecting the atom and orbital resolved densities of states -Figs.2, 3 and 4 ) of the LSDA and LSDA+U results is that the -doped heterostructure presents a half-metallic behavior with a minority spin gap.
LSDA+DMFT calculations were performed for the same values of and and temperatures . In Fig. 5 we present the DMFT results for the atom resolved total density of states in a small energy window around . Further we show the 3d-orbitals contribution into the main peaks of Cr/Mn transition metals. The insets of Fig. 5 shows the results of non-magnetic calculations, from where we can roughly estimate the bandwidth which is of about 6eV. Based on the analysis of the values of the bandwidth, orbital occupations and the magnitude of U, one expects for the heterostructures in discussion, a behavior typical for a medium correlated metal.
As one can see the dynamic correlations captured by DMFT gave a completely different picture for the orbital distributions around the Fermi level. In the case of Cr significant majority spin density of states is obtained for the -orbitals in contradiction with the LSDA/+U results were these orbitals are shown to give a negligible contribution see Fig. 2. Above , in the unoccupied part the majority spin 3d electrons have a significant and at higher energies (not shown) character. In the minority spin channel from no significant spectral weight is present, while above the many body induced states have a predominant character. At higher energies a predominant character is present.
In the case of Mn- doped structure, in the majority spin channel As-p is dominant at the Fermi level, the Mn- contribution is seen around 0.3eV below the Fermi level. It is interesting to note that the position of this orbitals are not changed with respect to the LSDA picture. Contrary to the LSDA minority spin Mn- have a larger spectral weight below , while above -states are seen to cross the Fermi level. The general picture obtained by including dynamic correlations is that the gap within the minority spin channel is closed and the half-metallicity disappears. For both Cr and Mn compounds closure of the gap is due to the presence of many-body states with character only seen in DMFT.
The analysis of the orbital resolved density of states can be correlated with the behavior of the self-energy seen in Fig.6,
where the atom and orbital resolved imaginary part of self-energy is shown. The self-energies for -orbital were excluded since they are identical due to the cell symmetry with the ones. We observe that the imaginary part of the self-energy has a rather symmetric energy dependence around the Fermi level, with a normal Fermi-liquid-type behavior . The minority spin shows a significant increase just above the Fermi level which is more pronounced for the orbitals.
In Fig. 7 we plot the real part of the orbital resolved majority and minority self-energies. The left/right columns represent the results for Cr/Mn doped compounds. The figure also displays a line with a unitary negative slope as a guide for the analysis of the energy dependence of the real part of the self-energies around . As one can see for both spin directions, the self-energy has a negative slope at the Fermi energy, which confirms that the quasiparticle weight is reduced by correlations. For both Cr and Mn for majority spins is clearly less than unity. Contrary for the minority Cr/Mn- spins is larger than unity above the Fermi level (within our approximation, we cannot determine with sufficient accuracy), suggesting the nonquasiparticle nature of the minority-spin states within the gap. We have checked that the above behavior of the real and imaginary parts of the self-energy does not change significantly in the range of investigated temperatures. For smaller U values the slope is reduced slightly although, the same peculiar behavior is seen for the real and imaginary part of -orbital self-energy. Therefore we can conclude that the many-body induced NQP states within the minority spin gap situated just above the Fermi level, are mainly determined by the transition metal orbitals. As the closure of the minority spin gap happens by the extension of NQP state tails from above, towards the Fermi level and in order to have a complete picture it is important to understand the effect of correlations upon the minority spin states situated below . This can be studied by looking at the band edges within the minority spin channel. In particular we investigate the character of the electronic states below and above with or without including correlations. As discussed previously, above the the Fermi level transition metal 3 states provide the main character that developed many-body NQP states who’s tail cross . Below the dominant states have As-p character. Therefore, we looked at contributions of As states originating from different As layers to the top of minority spin valence band. The As layers are grouped into four classes, each having the distance , where with respect to the transition metal atom located in origin. The results of this analysis is presented in table 1.
As the distance between the transition metal mono-layer and As mono-layer increases, the As-(, ) character continue to increase. In the same time, the difference between values from different mono-layers decreases. These results suggest that the As mono-layer situated closer to the transition metal has its electrons confined by the hybridization. At larger distances, hybridization decreases, and more -character is available to form the top of the valence band. One can note that and contributions are different as a consequence of symmetry lowering present in the tetragonal geometry. These results are in agreement with previous calculations obtained using the macroscopic average of the total DFT potential [42, 7, 41].
In the followings we comment the comparison of the results obtained within the three different approaches to the electronic structure calculations within the DFT. A well known problem of the LSDA and its relative GGA is the underestimation of the band gaps (for many semiconductors) in addition to the missing electronic correlation effects. For the heterostructures in discussion we present results obtained with different methods that consider correlations effects: a simplified mean field LSDA+U and a more complex LSDA+DMFT approach. The LSDA+U scheme seeks to correct the problems of LSDA by adding a repulsive potential +U, obtained from a mean field decoupling of the local Coulomb interaction. Its consequence is that the occupied/unoccupied 3d levels are shifted to lower/higher energies as seen in Figs. 2,3,4 The ground state picture resulting from both this methods is a half-metallic ferromagnetic state.
On contrary to the mean-field methods, DMFT captures correctly the dynamics at low as well as high energies. This approach describes properly the quasiparticle formation around Fermi level simultaneously with the lower and upper Hubbard bands at higher energies [18, 19]. Dynamical correlations captured by DMFT change completely the physical picture of these compounds. For the majority spin states a slight spectral weight redistribution takes place, however this does not affect the value at the Fermi level which is similar to that obtained within LSDA (see Fig. 1). For minority spins, although the position of the top of the minority spin valence band is slightly changed including correlations, the As-p contributions from different As layers are similar in values to those obtained within the LSDA calculations, see Tab. 1. This demonstrates that the As-p character distribution within the top of the minority spin valence band is not sensitive to correlation effects.
The conclusion therefore is that the many-body interactions affects the half-metallic band structure predominantly within the minority spin channel, above , by introducing NQP states. The presence of these many-body states reduce significantly the ideal carrier spin polarization predicted by LSDA(+U). In order to discuss quantitatively the conduction electrons spin polarization we use the common definition [43, 44, 45]: , where and are the density of states and the velocity at . The values of determine what type of experimental measurements that can access the polarization. The corresponds to the spin-resolved photoemission experiments, while the higher orders corresponds to spin polarizations measured with point contact Andreev reflection in the ballistic (n=1) or diffusive regimes (n=2) [43, 44]. In Fig. 8 we show the results for the polarization obtained using the simplified , where the velocities for spin up and down are droped. One can see a clear distinction between the different temperature behavior of magnetization and polarization, similar to many potential half-metallic ferromagnets [29, 47, 46, 33, 41].
For the Mn- doped GaAs the temperature dependence of remnant magnetization of samples that consist of 100 period of GaAs(10 mono-layers)/MnAs(0.5 mono-layer) showed a quasi-linear dependence and a critical temperature of about . More recent measurements on the Mn--doped DFH include polarized neutron reflectometry and magnetometry performed on samples having 20, 50 and 100 mono-layer GaAs spacers in the temperature range from 30 to 40K . This analysis allowed to identify two types of contribution to the magnetization of Mn-doped DMH: one type that has the superlattice periodicity and the second one resulting from random clustering effects suggested by the discontinuous nature of the quasi-two-dimensional MnAs . A direct comparison with these experimental results are not possible for the present theoretical calculations, due to the fact that the supercells considered here contain a considerably smaller number of layers. Nevertheless certain qualitative agreements are obvious. As one can see in Fig. 8 the present LSDA+DMFT calculation captures a qualitative quasi-linear temperature dependence of the reduced magnetization also for the smaller numbers of mono-layers in discussion, a behaviour that is seen in the experiment. In addition the quasi-two-dimensional nature of the problem is reflected in the present calculations by the separation of electrons and holes among different layers (see Tab. 1 and the discussion below). A major difference is that for the current supercell geometry our estimate for the Curie temperature is around four times larger.
Let us further comment upon the recent polarization measurements of spin carriers in Mn- doped GaAs compounds using hot electron photoluminescence (HPL) . The hot-electron photoluminescent spectra were interpreted based on the previous analysis performed in diluted magnetic semiconductors . According to this analysis in Mn doped GaAs in a wide range of Mn content the valence band holes predominantly occupy the Mn acceptor impurity band. It was suggested that the polarization of the valence band holes is proportional to the sample magnetization [49, 17]. To interpret the data on digital Mn/GaAs heterostructures, a combined contribution from the -doped and the unintentionally doped interlayer region was considered as a superposition of both contributions. It was shown that carriers experience a strong exchange interaction with the ferromagnetic interfaces, which rapidly decease with increasing distance with respect to the ferromagnetic Mn- layer .
As one can see the electronic correlations play an important role in the properties of the defect-free DMH presented above. In the presence of dynamical many-body correlations the top of the valence band does not change significantly, however, NQP states appear in the minority spin channel, Fig.1 and 5, just above the Fermi level. At finite temperatures the tails of the NQP state cross the Fermi level and the half-metallicity is lost. In addition we have shown that carriers spin polarization and magnetization follow a different temperature behavior. We expect that optically excited radiative recombination of electrons and holes within the minority spin channel would help to demonstrate the presence of the many-body effects in particular the existence of non-quasiparticle states in these materials. Hot-electron photoluminescence spectroscopy was recently used to discuss hole spin polarization in diluted magnetic semiconductors . The HPL circular polarization under circularly polarized excitation provides detailed information on spin-relaxation mechanism [49, 51], and allows a finite temperature characterization of the HPL polarization . We believe that such experiments would demonstrate a different finite temperature behavior of magnetization and polarization as qualitatively demonstrated in Fig. 8.
The technology based on digital alloy technique is expected to produce heterostructures with an enhanced because of locally confined high transition metal concentration, and a better control of the ferromagnetic properties by controlling the GaAs layer thickness. The expected enhancement of the Curie temperature has not yet been realized, instead, it was found that magnetic properties depend strongly on interlayer thicknesses [15, 48]. Our results of the electronic structure calculations based on LSDA predict that the defect free delta doped digital magnetic heterostructures are half-metals in agreement with previous calculations . The same conclusion is given by computations that are using a mean field LSDA+U approach. On contrary including many-body correlations captured by DMFT the half-metallicity is lost. The computed finite temperature magnetization follows an almost linear temperature dependence similarly to the experimental measurements performed however for larger GaAs(10 mono-layers)/MnAs(0.5 mono-layer) heterostructures [15, 48]. In comparing the critical temperatures, our results overestimate by a factor of four the measured values. Certainly additional work is necessary in this direction in particular the extension of the theoretical approach to include dynamical correlations beyond locality. A particular importance in applications of DMH in spintronic devices is the effect of the ferromagnetic layers on the spin polarization of the carriers. Our results demonstrate that magnetization and polarization follows a different temperature dependence. We suggest that such effect might be captured by hot-electron photoluminescence spectroscopy. The quantitative analysis of the intensity of recombination radiation for transition between the electrons and holes within the minority spin channel in the presence of electronic correlations is in progress.
L.C. and E.-V.M., acknowledge the support from the PN II ID_672/2008 grant and the FWF projects no. P19630-N16, P18551-N16, P21289 and by the cooperation project NAWI Graz Grant No. F-NW-515-GASS. L.V. acknowledges the financial support from the Swedish Research Council and the Hungarian Scientific Research Fund (research projects OTKA 84078).
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