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Contents lists available at ScienceDirect
Physics Letters A
www.elsevier.com/locate/pla
Shape deformation induced enhancement of ferromagnetism
in δ-(Ga, Mn)As
Y. Zhu a, C.L. Ma b, D.N. Shi a, K.C. Zhang c
a College of Science, Nanjing University of Aeronautics and Astronautics, Nanjing 211100, Chinab Department of Applied Physics, University of Science and Technology of Suzhou, Suzhou 215011, Chinac Department of Physics, Bohai University, Jinzhou 121013, China
a r t i c l e i n f o a b s t r a c t
Article history:Received 18 January 2014Received in revised form 2 May 2014Accepted 8 May 2014Available online xxxxCommunicated by R. Wu
Using density functional calculations, the mechanism of enhancement of exchange energy of δ-(Ga, Mn)As with shape deformation was elucidated. The initial Tc was 250 K with exchange energy 395 meV, thereafter Tc could be obtained as 308 K with a higher exchange energy when deformation was considered. Lattice constant variation in the z-direction (perpendicular to the Mn layer) had a significant effect on the exchange energy, while that in the x- or y-direction would not. Compared with pure GaAs, the anti-ferromagnetic (AFM) energy of δ-(Ga, Mn)As changed abnormally with shape deformation. Enhancement of this exchange energy could mainly be attributed to the high, narrow AFM t2g peak on the Fermi surface with shape deformation. These findings shed new light on the physics underpinning the design of dilute magnetic semiconductors with shape deformation for spintronic applications.
© 2014 Published by Elsevier B.V.
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1. Introduction
Recently, diluted magnetic semiconductors (DMS) have at-tracted much attention, especially with regard to the doping of Mn in GaAs [1], which has inspired much interest in the new field of spintronics [2–5]. According to the Zener model, the Curie tem-perature of GaMnAs increases with the concentration of Mn [6,7]. Unfortunately, the solubility limit of Mn in GaAs is rather small [8]. However the Mn concentration in the GaAs/MnAs superlattice can increase to 50% in a zincblende structure [9]. Based on this idea, Sanvito and Hill [10] investigate the half-metallic (HF) properties of a digital ferromagnetic heterostructure composed of a layer of substitutional Mn on Ga sites in GaAs (referred to as δ-GaMnAs) from first principles. They find that the hole carriers of Mn at the Fermi energy are strongly confined within a few monolayers around the two-dimensional (2-d) MnAs plane. The exchange in-teraction is much stronger than any other random alloy with the same Mn concentration and HF properties [10]. Indeed, it has been reported that the Curie temperature of δ-GaMnAs samples has reached 250 K [11]. However this temperature is still below room temperature, which means that the sample is unsuitable for many applications. Nevertheless, 250 K is not far from the freezing point of water at approximately 273 K and room temperature (approxi-mately 300 K). Some physical methods can be explored by which
E-mail addresses: [email protected] (Y. Zhu), [email protected] (D.N. Shi).
http://dx.doi.org/10.1016/j.physleta.2014.05.0060375-9601/© 2014 Published by Elsevier B.V.
the exchange energy may be increased. Meanwhile the HF nature of δ-GaMnAs remains unchanged. Since shape deformation (strain) has been used to improve doping solubility in semiconductor sys-tems [12], and the MnAs in δ-GaMnAs is in the form of a 2-d layer, shape deformation should be one of the effective ways by which the Curie temperature may be increased. As the conclusions can be applied directly to these δ-DMS, such as substitutional δ-SiMn [13,14], interstitial δ-SiMn [14,15], SiFe and GeFe in their digital forms [16], δ-ZnCrS [17], and δ-ZnCrS [18], it is important to ob-tain reliable theoretical guidance about the shape deformation in δ-DMS. Although, in principle, exchange energy can be attributed to the confinement of hole carriers, the details of shape deforma-tion and the fundamental mechanism behind the contributions of shape deformation toward the magnetic ordering of δ-DMS is far from clearly understood.
The authors concentrated on the exploration of the shape de-formation induced enhancement of ferromagnetism by means of systematic density functional calculations in δ-GaMnAs. Shape de-formation in the z-direction (perpendicular to the Mn layer) will change the exchange energy almost linearly, but it is not obvious when it deforms in the x- or y-direction. Compared with the en-ergy of pure GaAs in the same supercell, the anti-ferromagnetic energy of δ-(Ga, Mn)As is abnormal in the presence of shape de-formation. These findings and their fundamental origin should be important for the successful fabrication of improved δ-GaMnAs DMS for spintronic applications at room temperature. Detailed
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Fig. 1. (Color online.) The supercell for MnAs1/GaAs15 configurations in the ball-stick format. Big purple and light blue balls represent As and Ga atoms respectively. Small grey balls are for Mn atoms. The labeled atom As is one of the atoms connected to two Mn atoms. x and z are axis of the supercell, y axis is perpendicular to the xz plane and the same as x axis (not shown). a and c are lengths of one crystal cell at the x and z direction respectively. h1, h2 and h3 are the height of layers and b1, b2 and b3 present the bond lengths.
discussion is provided in Section 3, following a brief description of the method and computational aspects in Section 2.
2. Method and computational details
The present calculations were carried out using the ab initiopseudopotential plane wave method, as implemented in the VASP package [19]. We adopted the generalized gradient approximation (GGA) [20] for the modelling of the exchange-correlation potential. An energy cut-off of 300 eV was chosen for the expansion of the wave functions. Testing the calculations with Ecut = 360 eV and 400 eV indicated that the strength of the Mn–Mn exchange inter-action, measured as the difference between the total energy of the AFM and FM states, �EAFM-FM = EAFM − EFM, converged to within less than 2 meV. The magnetic moments of two Mn atoms are par-allel in their FM state, and anti-parallel in AFM states. 4 × 4 × 2k-point grids were used for Brillouin zone sampling. Both the lat-tice sizes and atomic positions are fully relaxed with a criterion that requires the calculated forces to be less than 10−2 eV/Å, and 8 × 8 × 4 k-point grids were used for static calculations.
The crystal constant is chosen as 5.74 Å in the GGA calculations. Four GaAs cubic cells (eight atoms in a tetrahedral crystal cell) are aligned along the z-direction (see Fig. 1). One of the Ga planes is substituted by Mn and periodic boundary conditions are applied. This leads to an infinite superlattice MnAs1/GaAs7 referred to as δ-(Ga, Mn)As below, where an MnAs zincblende monolayer is sep-arated by a 5.74 ×4 Å GaAs layers, as shown in Fig. 1. This cell con-tains 32 atoms, and the substitution of two Mn atoms corresponds to an Mn concentration of 12.5%, which is less than the extreme experimental measurement 50% for δ-type (Ga, Mn)As [9]. The two Mn atoms couple through As atoms. This supercell is large enough to allow the consideration of all kinds of shape deformation [10].
Fig. 2. (Color online.) Exchange energy (a) and ferromagnetic energy (b) of MnAs1/GaAs15 as a function of a value and aligned with different c value. Simi-lar to (a), (c) presents exchange energy as a function of c value and aligned with different a value. The lines from L1 to L4 in (a) represent different situations, which have different kinds of shape deformation. L1 and L2 are lines in which c and a val-ues are lattice constant respectively. In L3, the volume keeps constant. In L4 all axis change linearly.
The δ-(Ga, Mn)As cell changes with two parameters: c and a, as shown in Fig. 1. For example, when lattice constants c and achange, respectively, the length of the supercell in the z-direction will become 4c, while that in the x- and y-directions will both be a.
3. Results and discussion
How does shape deformation affect the exchange energy? Su-percells with different c and a values are considered. The c and avalues are set around the GaAs crystal constant of 5.74 Å, by in-creasing and decreasing in increments of 0.1 Å. The magnetic cou-pling energy (exchange energy) for each configuration is obtained from the difference of FM and AFM total energies, EAFM-FM. The re-sult is shown in Fig. 2(a). The solid lines show the exchange energy with a fixed value of c, while each dotted line shows the progress of shape deformation. Each solid color line represents the concept that the exchange energy is a function of a with one fixed value
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of c. The olive line (L1) corresponds to a c value of 5.74 Å, the crys-tal constant of GaAs, while the lines with other colors correspond to different c values with each increment of 0.1 Å from 5.74 Å. The a value is fixed at 5.74 Å for the dotted line (L2), while cchanges in increments of 0.1 Å on each side of the central value of 5.74 Å. The dotted line (L3) shows a special process, in which the volume of the supercell is constant during the shape deforma-tion process. In L4, a- and c-axis values change in similar incre-ments. The intersection of the dotted lines and L1 corresponds to the case without shape deformation, i.e., EAFM-FM = 395 meV/cell, which is consistent with other results [10].
All configurations in Fig. 2(a) are strong FM cases. It was found that almost all kinds of shape deformation could make the ex-change energy bigger than 395 meV when c > 5.74 Å. For example, when a is kept at 5.74 Å and c is changed to 5.84 Å, (referred to as point (a5.74, c5.84)), EAFM-FM becomes 456 meV/cell. The largest EAFM-FM is 501 meV/cell at (a5.94, c6.04). The shape deformation does enhance the exchange energy of δ-(Ga, Mn)As. The Tc of the supercell can be simply considered as linearly varying with the ex-change energy. The initial Tc is 250 K with an exchange energy of 395 meV [11], then for other shape deformation configurations:
Tc = 250EAFM-FM/395 (1)
So Tc = 289 K is obtained for (a5.74, c5.84), which is higher than the freezing point of water. For (a5.94, c6.04), Tc will be 317 K, which is higher than room temperature. Certainly, when EAFM-FMis higher than 395 meV, such a large exchange energy enhanced the Curie temperature of δ-(Ga, Mn)As.
The possibility of experimentally realizing shape deformation should be considered. As shown in Fig. 2(b), the lowest FM en-ergy corresponds to the initial supercell and is set as zero. Other FM energy accompanying shape deformation is subtracted from the initial energy, and all the data are then positive. If the FM energy is too large, the shape deformation process will be diffi-cult to realize in experiments. So only those structures with an FM energy of less than 0.5 eV were considered, hence the four particular lines: L1 to L4 in Fig. 2(a), are analyzed. Below 0.5 eV, there are only a few points at which the exchange energies are larger than 395 meV. In L1, it was found that the exchange energy decreases further when only the a-axis deviates from the crys-tal constant. Hence, shape deformation along the direction of the a-axis does not enhance the Curie temperature of δ-(Ga, Mn)As. Maybe L3 represents an easier experimental route to its realiza-tion, as the system retains FM energy in the low region shown in Fig. 2(b) at a constant volume. However, the exchange energy only makes Tc slightly larger than the freezing temperature of water. Following the route afforded by plot L2, in which only the c-axis increases offers a better alternative. Clearly, the exchange energy of (a5.74, c5.84) represented by the pink line and (a5.74, c5.94) in dark yellow are higher than 395 meV as shown in Fig. 2(a), and the FM energy of these two points is also below 0.5 eV as shown in Fig. 2(b). Based on Eq. (1), the highest Curie tempera-ture in L2 will be 308 K, which is higher than room temperature. Considering all of the points on L2, the exchange energy increases with increasing c except for (a5.74, c6.04) on the navy-blue line. On the whole, all of the data in Fig. 2(a) are separated by the cvalue, especially in the low exchange energy region. So the c value is the most important parameter during shape deformation. The way in which the supercell volume increases with crystal constant as shown in L4, could enhance the exchange energy, albeit at the expense of a much larger FM energy.
With regard to how the variation of the value of c affects the exchange energy, it was found that, when c is below 5.94 Å, the exchange energy will be almost linear with c. Especially, when awas set to 6.04 Å, a straight line (olive-green) is seen (Fig. 2(c)).
Fig. 3. (Color online.) FM energy (black square), AFM energy (red circle) with shape deformation in different line for MnAs1/GaAs15. The lowest FM energy is set as zero. The green triangle line is the energy with shape deformation for GaAs16, and the energy without shape deformation is set as zero.
The a value will also affect the result. As shown in Figs. 2(c) and (a), with the increase in a, the exchange energy increases and reaches its maximum, then decreases thereafter. As a func-tion of the c value, the highest exchange energy becomes larger with a larger fixed a value. Therefore, in a real experimental envi-ronment, the shape deformation should be considered to increase the c value first. At a certain c value, the a value is ideally sug-gested to be larger than 5.74 Å.
Between the FM and AFM states, one may wonder which states vary considerably with shape deformation. The FM and AFM en-ergy of δ-(Ga, Mn)As in L1 to L3 is plotted in Fig. 3. The energy of pure GaAs with shape deformation is also shown. In the top panel of Fig. 3, we can see that the FM and AFM energy lines do not keep up with pure GaAs when the a value decreases and c is kept at 5.74 Å. When a increases, the FM energy matches that of pure GaAs, while the AFM energy did not. In all, for c = 5.74 Å, the largest difference between the FM and AFM energies is at the point (a5.74, c5.74), and any change in a value militates against the enhancement of the exchange energy. In L2, where a is fixed at 5.74 Å, the trend in the variation of the FM energy is almost the same as that exhibited by pure GaAs. It means that the FM interaction will not vary with shape deformation. As compared to pure GaAs, the AFM energy deviating from pure GaAs means that the AFM interaction changes significantly in δ-(Ga, Mn). When the c value decreases from 5.74 Å, the AFM state will become more stable, and the FM state keeps up with the pure GaAs, so the exchange energy will decrease. When the value of c increases, the AFM state will become increasingly unstable, leading to an in-creased exchange energy. So the instability of the AFM state with increasing c value is the main reason behind the enhancement of the exchange energy. In plot L3, the same trend was seen as in L2 (see Fig. 3). The abrupt change in AFM can be described by the change of layer height and bond length as shown in Fig. 4.
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Fig. 4. (Color online.) The change of the height h1–h3 and bond length b1–b3 as shown in Fig. 1 as a function of c axis value in mode L2. Black and red lines rep-resent FM and AFM states respectively. Thick and solid lines are for h1 and b1, thin and solid lines are for h2 and b2, and thin and dotted lines are for h3 and b3.
Each layer height in the FM state varied linearly with c, and the GaAs heights labeled as h2 and h3 in the AFM state were also able to be plotted as a straight line. The MnAs height h3, denoted by the thick red line, in the AFM state looks different from other lines, although h3 can be claimed to be quasi-linear. In the AFM state, the MnAs layer is thin with a small c value, and thick when c is large. It means that the AFM state is sensitive to changes in the cvalue. This is also shown in the MnAs bond length. All other bonds follow the same trend except for the MnAs bonds in the AFM state. The different phenomenon shown along L1 is described in Fig. 5 in which only the value of a changes. With increasing a value, the height of the GaAs or MnAs layers remains practically constant, while the MnAs layer in its AFM state undergoes a quasi-linear decrease in height. All the bonds in L1 increase linearly with the avalue. Overall, the abnormal AFM change leads to a larger exchange energy for reasons that may be revealed by an examination of its electronic structures.
As shown in Fig. 6, the partial density of states (PDOS) of three kinds of structures in L1 and total density of states (TDOS) at a = 5.74 Å are chosen. Compared with PDOS and TDOS at a = 5.74 Å, the states on the Fermi level, which are essential for the nature of the crystal, stem mainly from Mn atoms: the PDOS of the Mn atom is analyzed below. In the spin-up channel, the e.g. states of Mn atom are connected to As atoms, and all lie well be-low the Fermi level both in FM and AFM couples. The t2g states are across the Fermi energy band, where the differences between FM and AFM are dominant. In the FM panel, with the magnetic moments of two Mn atoms being arranged parallel to each other, the double exchange is shown in the wide density of states (DOS) distribution across the Fermi energy band. While in the AFM panel, the t2g states do not interact with each other and then a high peak is seen across the Fermi level. A high peak Fermi energy will re-sult in the higher energy of AFM pairs. So in L1, all the structures are in an FM state as shown in Fig. 2(a). In AFM states, the differ-
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Fig. 5. (Color online.) The height and bond length change in mode 1.
Fig. 6. (Color online.) The FM DOS (left panel) and AFM DOS (right panel) of Mn atom of L1 with three kinds of cases in top three layers. And the bottom layer is TDOS for FM and AFM a5.74 structure. Zero energy indicates the position of the Fermi level. Positive and negative values of DOS are for the spin-up and spin-down channels, respectively.
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Fig. 7. (Color online.) The FM DOS (left panel) and AFM DOS (right panel) of L2 with four kinds of cases. Zero energy indicates the position of the Fermi level. Positive and negative values of DOS are for the spin-up and spin-down channels, respec-tively.
ences between three kinds of examples are complex, but we can still find something of interest therein. Both of the spin-up DOS at a = 5.74 Å and a = 5.44 Å are split into two parts and the Fermi energy lies in the valley between two peaks. The peak in the spin-down channel is just across the Fermi energy at a = 5.74 Å, and at a = 5.44 Å, the spin-down peak around the Fermi energy is split, therefore its AFM state is more stable. At a = 6.04 Å the spin-up and spin-down peaks are all below Fermi energy, with only the edge of the peaks crossing it. So the AFM interaction there will also be more stable than at a = 5.74 Å. With the most unstable AFM pairs, in contrast, a = 5.74 Å has the most stable exchange energy as shown in L1.
In L2, because of the high peak around the Fermi energy in the AFM panel, all the structures with shape deformation are also ferromagnetic. The differences between them are shown in the shapes of their peaks (Fig. 7). At c = 5.44 Å, Fermi energy is across the edge of the high peak in the t2g states. The conditions at c = 5.74 Å are the same as at a = 5.74 Å in Fig. 6. The peak in the spin-down channel at c = 5.94 Å is higher than the Fermi energy, but one of the split peaks in the spin-up channel is very narrow and just crosses the Fermi energy threshold. Differing from that at c = 5.94 Å, the peak at c = 6.04 Å is wider. So the structure at c = 5.94 Å has the most unstable AFM pairs and therefore also has
the highest exchange energy. So shape deformation will result in a change to the AFM t2g peak around the Fermi level. When the peak is just across the Fermi level, the higher exchange energy be-tween FM and AFM emerges. This is the main reason why shape deformation can enhance the Curie temperature of δ-(Ga, Mn)As.
4. Conclusions
In summary, especially with an increasing c value, shape de-formation can enhance the exchange energy of δ-(Ga, Mn)As. With shape deformation, the abnormal changes occur in the AFM state, which are both shown in the Mn–As bond and the MnAs layer height. The reason is shown in AFM DOS, in which the high peak around the Fermi energy varies with shape deformation. When the high peak becomes narrower and just crosses the Fermi energy threshold, the AFM couple will become more unstable, and then the exchange energy will be larger. Considering the experimental environment, shape deformation in the direction of the c-axis is the best way to enhance the exchange energy experimentally.
Acknowledgements
This work was financially supported by the NSFC (Grant Nos.: 11204131, 11374159, and 11247428). Y. Zhu also thanks the CPSF for their financial support (Grant No. 20110491420), and the NUAA Fundamental Research Fund (Grant Nos. 56XZA11035 and 56XNA12089). K.C. Zhang acknowledges the financial support from The Natural Scientific Foundation of Liaoning Province (Grant No. 20121078).
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