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ИНФОРМАТИКА, ВЫЧИСЛИТЕЛЬНАЯ ТЕХНИКА И УПРАВЛЕНИЕ INFORMATION TECHNOLOGY, COMPUTER SCIENCE, AND MANAGEMENT
УДК539.2
DOI 10.12737/19692
Band structure and magnetic properties of cubic crystals InxGa1-xN: Ab initio calculations*
V. V. Ilyasov1, I. V. Ershov2, T. P. Zhdanova3, I. Ya. Nikiforov
i,2,3,4 Don state Technical University, Rostov-on-Don, Russian Federation
Зонная структура и магнитные свойства кубических кристаллов InxGa1-xN: Ab initio расчёты*
В. В. Илясов1, Т. П. Жданова2, И. В. Ершов3, И. Я. Никифоров4
1,2,з,4донской государственный технический университет, г. Ростов-на-Дону, Российская Федерация
Band structure of solid solutions InxGa1-xN with sphalerite structure and with considerable percentage of indium (x = 0,25; 0,5; 0,6; 0,7; 0,75; 0,9; 0,95; 0,97; 0,99; 1,0 ) is calculated using the density-functional theory (DFT) and the cluster version of the local coherent potential method within the frame of the multiple scattering theory. The electron structure of the ternary solutions of InxGa1-xN in sphalerite crystallographic modification is compared; the interpretation of their features is given. The concentration dependence on the energy gap for the entire variation range of the indium content in the solution is obtained. The spin polarization effect of the states of In, Ga, and N for the alloys with the considerable percentage of In, as well as the transition of the ternary solutions of In0.75Ga0.25N into the of magnetic semiconductor state is found out. The magnetic moments of In, Ga, and N atoms, and the saturation magnetization in InxGa1-xN semiconductor system are determined.
Keywords: band structure, magnetic moment, valence band, band gap, density of electronic states.
Зонная структура твёрдых растворов 1пхОа1-х№о сфалерит-ной структурой и большим содержанием индия (х = 0,25; 0,5; 0,6; 0,7; 0,75; 0,9; 0,95; 0,97; 0,99; 1,0) рассчитана методами теории функционала плотности (БП) и кластерной версии локального когерентного потенциала, в рамках теории многократного рассеяния. Проведено сравнение электронной структуры тройных растворов 1пхОа1-х№фалеритной кристаллографической модификации, дана интерпретация их особенностей. Получена концентрационная зависимость ширины запрещённой полосы для всего диапазона изменения содержания индия в растворе. Обнаружен эффект спиновой поляризации состояний 1п, Оа и N для растворов с большим молярным содержанием 1п и переход тройных систем 1п0.75Оао.25№ состояние магнитного полупроводника. Определены магнитные моменты на атомах 1п, Оа и N и намагниченность насыщения полупроводниковой системы ¡пхОа^К
Ключевые слова: зонная структура, магнитный момент, валентная зона, ширина запрещённой полосы, плотность электронных состояний.
Introduction. The semiconductor solid solutions InxGai_xN are the most promising materials for optoelectronics to obtain the blue and green light-emitting diodes which are used for high-density optical storage of information and high-power devices, for blue lasers for instance. The extraordinary property of these materials is the ability to operate over a wide temperature range and the insensitivity to X-ray irradiation [1, 2]. By varying of the indium percentage one can manage the value of the energy gap in the range from 1.92 eV (c-InN) [3] upto 3.2 eV (c-GaN) [3] that gives the opportunity to use these materials in various parts of spectrum [4, 5]. For the far ultraviolet region (240-300 nM) the semiconductor materials based on InxGa1-xN (0,5< x < 1) crystals are of high interest.
The alloys InxGa1-xN with the high percentage of indium are studied not sufficiently in comparison with other nitrides of III group elements because of the difficulties connected with the growing of such crystals [6]. On the top of c-GaN buffer lay it is possible to grow up a film of alloy In-Ga-N of high quality with cubic structure [3, 7] using the method of gas-source molecular beam epitaxy. The theoretical investigation of these alloys InxGa1-xN (0 < x < 0,75) were performed using the densi-
Работа выполнена в рамках инициативной НИР. " E-mail:viily@mail.ru,thijd@mail.ru, zhdanovatp@gmail.com "* The research is done within the frame of the independent R&D.
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ty-functional theory (DFT) in the local density approximation (LDA) [2, 4]. The structural, termodynamical properties, and the composition dependence of bowing parameter of the calculated band gap have been determined. It was shown that this bowing parameter was due to structure effect that is to the structure induced disordering of the bond lengths, and depends on the fraction of indium. Nevertheless the mechanism of the influence of indium atoms on the energy gap in InxGai_xN alloy has not been elucidated, for 0.75 <x< 1 especially.
The heightened interest to the search of new so called magnetic semiconductors (MS) is due to their key role for creation of new devices of spintronics [8, 9]. The most traditional way is to admix magnetic d-atoms, Mn-atoms for example, into the semiconductor matrix GaN. It leads to the rise of special magnetic properties particular for ferromagnetic materials with the high Curie temperature [10]. The admixture of Ga-atoms into the epitaxial films of GaN as was shown by magnetometry [11] results in the origin of the giant magnetic momentum at gadolinium atom. The alloyed layers of GaN:Gd demonstrate the fer-romagnetism with the temperature of transition into magnetic order state higher than room temperature. Because of this the system GaN:Me looks promising for spin electronics.
Quite other way of creation of dilute magnetic materials is possible using effect of spin polarization by admixture of non-magnetic atoms into the non-magnetic matrix. For instance for BeO: (B, C, N) system it was shown [12], that by the partial substitution oxygen atoms in BeO by atoms boron, carbon or nitrogen the spin polarization of (2p T - 2p^ )-states occurs, and the system BeO : X becomes to be magnetic semiconductor (BeO : B) or magnetic semimetal (BeO : C, N). The ternary systems InxGa1-xN and the binary basic systems GaN and InN are usually considered as non-magnetic semiconductors. Because of this there are no publications with spin-polarized calculations of zone structure of such materials.
In the present work the zone structure, forbidden energy band bowing and magnetic properties of wide band semiconductors InxGa1-xN with the great percentage of indium using the methods of functional density theory (spin-polarized version) and the cluster version of local coherent (spin-restricted version) were investigated.
Models and methods. The crystallographic structure of the ternary system InxGa1-xN is of cubic sphalerite-type, its space group is F43m. The coordination polyhedral (CP) of the atoms In and Ga are tetrahedral [MeN4 ]. Interatomic distances
are supposed to be: Ga - N = In - N = a -v/3/4 , where a is the lattice parameter, which was determined according to Vegard's law.
Zone calculations of electron structure of nitrides InxGa1-xN ( x = 0; 0,25; 0,5; 0,6; 0,7; 0,75; 0,9; 0,95; 0,97; 0,99; 1,0 ) were performed with the spin-polarized full-potential method of pseudopotential (program package Quantum-Espresso [13]) using the theory of the functional density (DFT) developed by Kohn and Sham [14] . In this theory the total energy of the unite cell is the sum of the one-electron energy, Hartree energy, exchange-correlation energy, and electrostatic Ewald energy, and the non-local exchange-correlation functional in the form Perdew-Burke-Ernzerhof (PBE) [15].
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The following electron configurations of atoms have been used: In — [Kr]5s 5p 4d , N — [He]2s 2p , Ga — [Ar]4524p:3d10. For atoms In, Ga and N the electrons in the states 5s25p14d10, 4s24p13d10, and 2s22p3 were attributed to valence ones correspondingly, whereas the electrons of completely occupied shells [Kr], [Ar], and [He] were considered as core-electrons. The influence of core-electrons was taken into account by using ultrasoft pseudopotentials for indium atom (code Dal Corso [13]) whereas for gallium and nitrogen atoms the code Vanderbilt (versions 735 u 734) [15] was used.
The plane waves set was "cut off' at the energy 20 Ry that provided the convergence of the total energy of the cell in 2 our self-consistent calculations not worse than 0,001 Ry. The density of electron states was calculated using the rectangular -g grid of points of dimensions 4x4x4, spaced evenly throughout the Brillouin zone (Monkhhorst-Pack grid).
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The electron densities p(r) in the real space and the partial densities of electron states (DOS) in valence and conduction bands were calculated separately for every spin projection similar way as in [12]. It gave us the possibility to determine
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connected with spin magnetic moment at every atom. ^ To investigate the electron structure of the system Ga-In-N with the great fraction of indium the local coherent poten-
tial ab initio method in the frame of multiple scattering theory was used. This method was developed in our group and applied for many systems, for Ti-Al-C system for instance [16]. The approximation of virtual crystal is quite effective approach for the system under investigation with the percentage of an element of any possible value. The muffin-tin approximation for crystal 112 potential was used. The latter was the sum of Coulomb, exchange (Slater's version) and Madelung contributions. The local
partial densities for every component of alloys was calculated for cluster of 250 atoms of the system In-Ga-N according to expression
In.E. ^[E^' j e2 ImTrTmX
n 1(E) =- J {rRl (r,E} dr-
% 0
Imt 0
where l is the orbital quantum number, Rl(r,E) are the radial wave functions, T is a matrix element of the scattering operator, and index 0 shows the centre at which the atom In is placed.
Results and discussion.
Fig. 1 - Total densities of states for InxGa1-xN (x = 0,25; 0,5; 0,75).
Band structure and magnetic properties. The band structure of cubic InxGa1-xN system changes significantly as the molar fraction x of indium increases. It concerns both occupied valence and unoccupied conduction bands. In Fig. 1 the total
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electron densities of states for both spin orientations (TDOS) for solid solutions InxGai_xN (x = 0.25; 0.5; 0.75) are shown. One can see the features in comparison to TDOS of the c-GaN (Fig.2).
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Fig. 2 - Total densities of states for InN and GaN (x = 0,25; 0,5; 0,75).
The most significant changes of electron structure happen for In025Ga075N and Ino.75Ga025N. These changes are connected with the interaction In4d- and Ga3d-obitals with 2p-obitals of nitrogen which results in the changes of the electronic and magnetic properties of the system under consideration.
The valence band (VB) is formed by states essentially of 2p-electrons of nitrogen atoms with the admixture of s- and /»-states of Ga and In and distinguishes the high energy localization of In5s- (3,8 eV), In5p- and Ga4p- (6,6 eV and 7,7 eV), and 2p-states of nitrogen (8.7 eV). The corresponding maxima of TDOS for both spin orientations can one see for all alloys. At the same time for In025Ga0.75N there are some band gaps which can be seen on the curves of local densities of electron states for both spin subsystems NT(E) u N^(E) of indium, gallium and nitrogen, as well as appearance of unoccupied states of In5s-and Ga4s-electrons at 0,6 eV from Fermi level (Fig.3) which can be treated as acceptor level.
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Fig. 3. Partial atomic densities of states for In0.25Ga0 75N (a) and In0.75Ga0.25N (b)
At significant molar fraction of indium (for example x = 0,75) the bands of occupied and free electron states shift to the greater energies by more than 1,0 eV in comparison to In0 25Ga075N. The energy gap Eg of In075Ga025N decreases by 0.5 eV and is equal to 1,9 eV.
By integration of electron local partial densities for every spin projection for the systems InxGai-xN (x = 0; 0,25; 0,5; 0,75; 1,0) provides us the local partial charges of atoms and the possibility to determine the spin magnetic moments at every kind of atoms (Table1). It was ascertained that the ternary system under investigation displayed rather small magnetic moment at atoms Ga, In, and N, which value increases in the row Ino.25Gao.75N ^ Ino.5Gao.5N ^ Ino.75Ga025N. The greatest spin splitting takes place for 4p-shell of gallium atom, that causes the appearance of magnetic moment at Ga atom 0,009^B, whereas the magnetic moments at atoms of matrix indium and nitrogen in In075Ga025N are equal to 0,008 (xB. In this system, when x = 0.75 the Ga atoms can be considered as admixture atoms.
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Table 1.
The structural parameters, magnetic moments at atoms and absolute magnetization Ms (at T = 0K)
in the systems InxGa1-xN
Lattice parameter, Â Cell volume, (a.u.)3 Magnetic moments, Ms, Цв/cell
In Ga N
c-GaN 4,500 153,73 - 0,009 - 0,009 0,05
In0.25Ga0.75N 4,655 680,69 0,006 0,007 0,007 0,17
In0.5Ga0.5N 4,740 718,67 0,007 0,008 0,008 0,18
In0.75Ga0.25N 4,770 732,40 0,009 0,009 - 0,008 0,19
c-InN 4,980 208,36 0,010 - 0,010 0,06
It is worth to mention that in binary compounds GaN and InN there are magnetic moments. For instance for c-GaN are there magnetic moments at atoms Ga (0,009 (xB) and N (- 0,009 (xB), but it does not lead to the ferromagnetism of this compound because two sublattices are magnetized antiparallel ways. These results agree with the molecular Weiss' theory, according to which the magnetization of every sublattice MA,B is determined by the sum of average statistical magnetization of sepa-
A B A B ^N A B A B
rate ions << | ' >> by analogy with [17] M • = — << | ' >> , where N - the number of atoms and << | ' >> - a quantum-mechanical average magnetic moment of the n-level.
Thus the resulting magnetization in binary and ternary systems InxGa1-xN (x = 0; 0,25; 0,5; 0,75; 1,0) is the sum of the magnetization of magnetic sublattices and is determined by the set of levels En in common nearly the ground state. Our ab initio calculations of the electron structure and the total and the absolute magnetization of binary and ternary systems gave for the total magnetization the zero estimation M = 0 but the absolute magnetizations Ms > 0 is different from zero (Table 1). The magnetization of saturation Ms (for T = 0 K) of the semiconductor materials under investigation increases monotonous way with x and achieves 0,19 ^B/cell for x = 0,75, being small for all x . It is possible to suppose that the systems GaN, InN, and InxGai-xN are in paramagnetic state. The appearance of ferromagnetic properties of GaN layers by admixture of Gd atoms was noted earlier [11], and was connected with the giant magnetic moment at the atom of gadolinium. Unfortunately the measurements of the magnetic moments at atoms of Ga and N have not been carried out.
The creation of magnetic moments at atoms of nitrogen (0,42 (xB) was observed in BeO : N [12] by the partial substitution of oxygen atoms by nitrogen atoms. The nature of the creation of magnetic properties of nitrogen atoms in the system BeO: N the authors of paper [12] explain by the mutual energy position of the valence O2p-band of matrix and 2p-band of nitrogen atoms which is determined by the difference of orbital energies Ae (O2p - N2/) = -1,749 eV, that leads to the spontaneous polarization of N(2pT - 2p^)-states and converts the system BeO : N into the state of magnetic semimetal (MSM). It was shown [12] that by diminishing of the difference of the orbital energies Ae in the system BeO : (B, C, N) the magnetic moments of admixturing atoms decrease noticeably (from 1,16 (xB for BeO : B to 0,42 (xB for BeO : N), magnetic moments at atoms of beryllium and oxygen as well as the total magnetization of such system decrease also.
However for the semiconductor system InxGa1-xN the results of the work [12] about the role of the mutual positions of p-orbital energies of admixtural atoms and the atoms of non-metallic lattice cannot be used because alloying in the system InxGa1-xN is performed into metallic sublattice and secondly, the energy of 2p-orbital of nitrogen (-11,4852 eV) [18] is situated lower than the energies of p-orbital In (- 4,6947 eV) [18] and Ga (- 4,9218 eV) [18] The differences of energies Ae of p-orbitals in one case are equal: for nitrogen Ae (Ga4p - N2p) = 6,563 eV and for indium Ae (In5p - Ga4p) = 0.227 eV. So for 2 instance in the system In0 75Ga0 25N the magnetic moment at the atom nitrogen is more than 4 times less than in system BeO:N, that can be connected with the influence of the element composition of matrix, crystallographic and electron zone structures.
The analysis of the partial charges of both spin subsystems NT(E) an N|(E) in the semiconductor alloy In0.75Ga0.25N (Table 2) allows us to state that spontaneous spin polarization of the electron states Ga (4pT - 4p^)-, In(5pT - 5pj), and N(2pT -J3 2p I) give main contributions into magnetic moment when there are no external magnetic fields. The contributions into magnetization of lattice ions of the electrons of Ga4s-, In5s-, and N2s-orbitals turn out to be 2 times less than of their p-electrons. It is ^ while to note that Ga3d- and In4d -states of the valence band electrons make no noticeable contribution into the magnetic mo-£ ments at atoms of the system InxGa1-xN.
Table 2.
Partial charge distribution of valence electrons at atoms for the spin subsystems N|(E) u Nj(E) in In0.75Ga0 25N
In Ga N
s P d s P d s P
n| 0,3819 0,6840 4,9855 0,4342 0,7205 4,9967 0,7473 2,1651
nj 0,3797 0,6790 4,9856 0,4313 0,7143 4,9967 0,7480 2,1721
(nî- nj) 0,0022 0,0050 -0,0001 0,0029 0,0062 0,0000 -0,0007 -0,0070
I 0,7616 1,3631 9,9711 0,8655 1,4347 9,9934 1,4953 4,3372
The dependence Eg (x) .By increasing of the molar fraction of indium in the system InxGa1-xN in spite of significant
change of the electron energy structure both in occupied and unoccupied bands, it is possible to state that the width of the valence band conserves its value, the VB and Fermi level shift by more than 1 eV to the higher energies, and the width of forbidden energies Eg decreases from 2,4 eV (x = 0,25) to 1,9 eV (x = 0,75). The calculated composition dependence Eg (x) in c-
InxGa1-xN(x = 0; 0,25; 0,5; 0,75; 1) appears to be nonlinear one and is shown in Fig. 4, where it is compared with the experimental data [2]. One can see the good agreement of received theoretically in the present work and experimental data.
3.50
3.00
2.50
2.00
1.50
0.00
0.40
0.80
Fig. 4. Composition dependence curve Eg (x) for alloys c-InxGa1-xN (x = 0;0,25;0,5;0,75; 1)
The form of the composition dependence curve Eg (x) for alloys c-InxGa1-xN (x = 0; 0,25; 0,.5; 0,75; 1) is analogous
of one found earlier [19] for the cubic alloys c-AlxGa1-xN and does not contradict usual notions of physics. The value of bowing parameter at indium fraction x = 0,5is the greatest one and equals to 0.25 eV.
Interatomic interactions. Some particularities of the interatomic interaction in cubic alloysc-InxGa1-xN on can observe on the maps of total electron density (ED) in the real space. For example in Fig. 5 the maps of ED are demonstrated for In0 25Ga0.75N in the planes (110) and (001). One can see, that along the lines Ga-N and In-N the covalent type bonds prevail. The localization of ED for In-N bond is expressed explicitly that is the evidence of creation of hybridized orbitals In4d-N2p and Ga3d-N2p. The interaction of atoms of metallic sublattice is metallic one, whereas the bonds Ga-In are of directing type and are formed by In2p- and Ga2p-states in the energy range from 6.5 to 11 eV.
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Fig. 5 - The total density maps in Ga-N and In-N (001) and (110) planes according to InxGa1-xN (x = 0,25; 0,5; 0,75)
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The analysis of electron energy spectra (Fig. 3a and 3b) which have the resonance character shows that the creation of localized hybridized orbitals In5s-N2p and Ga4s-N2p in the energy interval from 4 to 10 eV is possible.
Conclusion. With application of ab initio zone quantum-mechanical method using the density functional theory and with the cluster version of the local coherent potential approach the systematic calculations of electronic and magnetic properties of cubic alloys InxGa1-xN for various molar fractions 0 < x < 1 of indium were carried out.
In the row GaN^- In0.25Ga0.75N^ In0.5Ga0.5N^ In0.75Ga0.25N we have observed: the shift of VB to the higher energies by more than 1.0 eV, the decreasing of energy gap Eg , which equals for In0 99Ga001N ~ 1,7 eV, the appearance of acceptor
levels above Fermi energy due to In and Ga atoms and resonance electron states of s- and p-symmetry in unoccupied band.
The results of our calculations of partial local electron densities of states for spin subsystems NT(E) and Nj(E) permit us to assert the existence of the spin polarization effect of atoms In, Ga, and N in the system InxGa1-xN, that transmits it into the state of magnetic semiconductor. The magnetic moments of atoms In, Ga, and N has been calculated as well as saturation magnetization of the system InxGa1-xN, which can be treated as the evidence of the paramagnetic state of this material. Thus, in cubic crystals InxGa1-xN with the great fraction of indium the effect of spin splitting of zone Ga4(s-, p-), In 5(s-, p-), and N2p was observed which leads to creation of magnetic moments about 0.01^B at atoms of both sublattices. At the same time these atomic magnetic moments In (0,009^B), Ga (0,009^B), and N (-0,008^B) does not lead to ferromagnetism because the magnetic moments of atoms of two sublattices are antiparallel, which is in accordance with molecular theory of Weiss.
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Поступила в редакцию 07.12.2015 Сдана в редакцию 07.12.2015 Запланирована в номер 23.03.2016
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