Superconductivity in Group III-V Semiconductor AlN Under High Pressure

The electronic properties of cubic zinc blende type group III-V semiconductor AlN under pressure is studied using full potential linear muffin-tin orbital (FP-LMTO) method. At normal pressure, AlN is an indirect bandgap semiconductor with band gap value 4.56 eV. When the pressure is increased, there is enhanced overlapping between the wave functions of the neighboring atoms. As a result the widths of the valence and empty conduction bands increase. These changes lead to the narrowing and indirect closing of the band gaps in AlN (metallization). On further increase of pressure, AlN becomes a superconductor and AlN comes under the class of electron-phonon-mediated high pressure superconductors. The superconducting transition temperatures (Tc) of AlN are obtained as a function of pressure for the CsCl structure. It is also confirmed that the metallization, structural phase transition and onset of superconductivity do not occur simultaneously in this compound.


INTRODUCTION
Group III-V semiconductors have been extensively studied because it is considered as an important technological material in electronic and opto-electronic applications [1].Generally aluminum compounds (AlN, AlAs) crystallize in zinc blende (ZnS) structure [2].The effect of pressure on the electronic properties of group III-V compounds can be investigated in many ways [3,4].The technological applications of AlN compound require significant progress in the fundamental understanding of their behavior at normal and high pressures [5].Subjecting AlN to high pressure leads to pressure-induced metallization, structural phase transition, and superconducting transition [2].Wang et al. [6] presented the mechanical and electrical properties of twelve III-V semiconductors, under pressure, using Plane wave Pseudopotential method.Edgar [7] presented the properties of group III nitrides using experimental observations.Wagner et al. [8] reported the electronic and phonon deformation potentials of GaN and AlN using Ab initio calculations versus experimental values.There are no high pressureinvolving studies related to metallization and superconductivity in AlN.This motivated us to take up the present investigation.In this work, self-consistent full potential linear muffin tin orbital method (FP-LMTO) is employed to study the effect of pressure [9].We have analyzed the phenomena of metallization (NaCl structure) and superconductivity for high pressure (CsCl) structure of AlN [1,2].It is hoped that this analysis will enable us to make some general statement regarding the path to high Tc superconductivity in covalent compounds.

Calculative procedure
The electronic band structure and density of states calculations were performed for AlN corresponding to different reduced volumes in ZnS, NaCl and CsCl structures, by the first-principle FP-LMTO method with in generalized gradient approximation (GGA) .The electronic configurations of Al and N are [Ne] 3s 2 3p 1 3d 10 (Z = 13) and [He] 2s 2 2p 3 (Z = 7), respectively.The valence electronic configurations chosen in our calculations are 3s 2 3p 1 for Al, and 2s 2 2p 3 for N.There are 8 valence electrons

Full Paper
Orbital: Electron.J. Chem.7 (3): 226-230, 2015 contributing to the valence bands.The final energy convergence is within 10 -5 Ry.The calculated total energies were fit to Murnaghan's equation of state (EOS), to determine the phase-transition pressure and other ground-state properties [1,2].Murnaghan's equation of state is given by the following formula to obtain the equilibrium lattice constant and other ground state properties.
In our calculations we have chosen the ZnS structure for AlN at ambient pressure.The phase stability of the ZnS and NaCl structures of AlN is analyzed using the enthalpic calculation [1].The enthalpy H(P) is defined by: and the transition pressure corresponding to the phase transition from ZnS to NaCl is obtained from the relation: where HZnS and HNaCl are the enthalpies of the ZnS and NaCl phases, respectively.

The Band structure of AlN under pressure
The band structures of AlN were computed for various reduced volumes ranging from V/Vo=1.0 to 0.3 in steps of 0.05.Even though we have obtained the band structure for V/Vo values from 1.0 to 0.3, we have presented here the band structures of AlN along the symmetry directions -X-W-L--K and the corresponding density of states of AlN (Figs. 1 to 4).The volume compressions corresponding to V/Vo=1.0 and V/Vo=0.756for AlN is given.A single band nearer to the bottom arises from 2s 2 electrons of N (Fig. 1).The three bands appearing below the Fermi level are due to the 3s 2 , 3p 1 electrons of Al and 2p 3 electrons of N (Fig. 1).The empty conduction bands above the Fermi level are due to 3p, 3d states of Al and 3d, 2p states of N (Fig. 1).At normal pressure, the band gap of AlN is indirect with valence band maximum at  point and conduction band minimum at X point with band gap value 4.56 eV.The calculated energy gaps are in agreement with the experimental value of 4.8 eV (Table 1) [7].As pressure increases the width of the valence band and the empty conduction band get widened.These changes lead to the narrowing of the band gap under pressure (Fig. 3).The density of states (DOS) (states/Ry.)calculations for all the reduced volumes have been carried out.The density of states (DOS) histogram of AlN corresponding to normal pressure is shown in Fig. 2. At normal pressure, the levels arising from 3s 2 electrons of Al give the long spike near the origin.The short spikes near the Fermi energy are due to 2s 2 , 2p 3 electrons of N and 3p 1 electron of Al.The short peaks above the Fermi energy EF are due to the 3p, 3d states of Al and 3d, 2p states of N. The general features of the band structure and density of states (figures 1-4) are similar to that of the other group III-V compounds [1,2].

Ground state properties
The ground state properties and structural phase transitions are studied from the total energies obtained from our calculation.The total energy is calculated as a function of reduced volume (V/Vo) for ZnS, NaCl, and CsCl phases of AlN [1].Here, Vo is the experimental equilibrium volume corresponding to the experimental equilibrium lattice constant.In Table 1, the equilibrium lattice constant (ao), band gap (Eg), bulk modulus (Bo) and its pressure derivative (Bo 1 ) values are compared with experimental [7] and previous theoretical works [6].The values of reduced volume, pressure, and lattice constant are given in Table 2.The calculated total energies were fit to Murnaghan's equation of state to obtain the equilibrium lattice constant and other ground state properties.

Structural phase transition
In our calculation we have chosen the ZnS structure as the ground state structure for AlN.The phase stability of the B3 (ZnS), B1 (NaCl), and B2 (CsCl) structures of AlN is analyzed using the enthalpic calculation [1].The phase transition pressure (PT) and the corresponding reduced volume (V/Vo)T estimated in our calculation are given in Table 3.For AlN, our calculated phase transition pressure is in good agreement with the experimental and previous theoretical results [4,5].The mechanism for the phase transition is a geometric effect involving a change in the coordination number from 4 in the ZnS phase to 6 in the NaCl phase and to 8 in the CsCl phase under pressure [1,2].

Metallization
At normal pressure, AlN is a semiconductor.
With the increase of the pressure, the band gap decreases and at a particular pressure, there is a closing of the band gap.The band structure and density of states corresponding to the metallization of AlN are shown in figures 3 and 4, respectively.In AlN, the metallization occurs through indirect closing of the band gap between valence band maximum at  point and conduction band minimum at X point.The metallization volume of AlN is V/Vo=0.756(NaCl structure), which corresponds to the pressure PM = 1 Mbar.At the metallization pressure, the values for density of states at Fermi energy N(EF) are very small (pseudo gap), which indicate that metallization has just set in AlN (Fig. 4).Thereafter N(EF) increases slowly with pressure and becomes fairly large at a particular value of V/Vo.The values of EF and N(EF) corresponding to different V/Vo are used in studying the pressure variation of superconducting transition temperature.However, there are no experimental nor theoretical studies available for comparison at these pressures [1,2].

Superconductivity in AlN under pressure
The promotion of an s electron to the d shell in solids is one of the factors, which will induce superconductivity.Under very high pressures, aluminum compounds are not only metals but also superconductors.The theory of Gaspari and Gyorffy in conjunction with McMillan's formula is used to calculate Tc [1].
The electron -phonon mass enhancement factor,  is: where M is the atomic mass,  2  is an average of the phonon frequency square and I 2  is an average (over the Fermi energy) of the electronphonon matrix element square.
I 2  (in Rydbergs) can be written as: where l,l+1 = -l l+1 [(Dl(EF)-1)(Dl+1(EF)+l+2) + (EF-V(S))S 2 ] and in this, l is the radial wave function at the muffin-tin sphere radius corresponding to the Fermi energy, Dl is the logarithmic derivative of the radial wave function at the sphere boundary, V(S) is the muffin-tin potential at the sphere boundary, and S is the radius of the muffin-tin sphere.
The above quantities are taken from the band structure results.
The average of the phonon frequency square is: The onset of superconductivity occurs at V/Vo=0.5, with Tc= 0.024K.On further increase of pressure, Tc begins to increase.Our results indicate that AlN is a one-band rather than a one-electron material.The electron-phonon interaction parameter () and its variation under pressure shows that AlN is an electron-phonon mediated superconductor.In this material, the Tc increases with pressure.It is found that the Tc(P) follow the variation in N(EF) with pressure.The increase of Tc(P) depends mainly on the rate of increase of s  p electron numbers with pressure for AlN.At low pressures the rate of increase of these electron numbers are increasing with pressure resulting in the increase of Tc.The calculated values at high pressure (CsCl) structure are given in Table 4 for AlN.As pressure increases our computed value of Tc increases and reaches a maximum value.In our calculation, the highest Tc obtained in AlN is 8.864K at 7.0 Mbar (Table 4).This reflects the fact that the structural and band gap configurations play an important role in the superconducting (high Tc-max) behavior of these compounds under high pressure [1,2].

CONCLUSION
In the present investigation, the pressure dependent band structures and density of states of AlN is computed and the results are used to study the metallization and superconductivity under high pressure for the first time.When the pressure is increased there is enhanced overlapping between the wave functions of the adjacent atoms.As a result the widths of the valence and empty conduction bands increase.These changes lead to the narrowing and closing of band gaps (metallization).On further increase of pressure, AlN becomes a superconductor, and this material comes under the class of electronphonon-mediated high pressure superconductor.It is also confirmed that the metallization, structural phase transition, and onset of superconductivity do not occur simultaneously in aluminum compounds [2].
gives a good estimate of the Tc value.Here  * is the electron-electron interaction parameter which is estimated using the relation, EF) is the density of levels per atom per eV at EF.With the results obtained from the selfconsistent calculation, we have computed  D , ,  * and Tc as a function of pressure using Eqs.(4 -9)  .

Table 1 .
Equilibrium lattice constant (ao), bulk modulus (Bo) and its pressure derivative (Bo 1 ) of AlN in ZnS structure

Table 2 .
Reduced volumes, Lattice constant and Presssure values of AlN.

Table 3 .
Structural phase transition pressure for AlN.

Table 4 .
Variation of Tc as a function of pressure for AlN in CsCl structure.