First-principle study on electronic structure property of GaN/AlN

IOP Conference Series: Materials Science and Engineering

First-principle study on electronic structure property of GaN/AlN

To cite this article: N Benayad et al 2012 IOP Conf. Ser.: Mater. Sci. Eng. 28 012006

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Zinc-blende AlN and GaN under pressure:

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-

First-principle study on electronic structure property of GaN/AlN

N. Benayad

, M. Dine El Hanani

, M. Djermouni

a

Laboratoire des Matériaux Magnétiques, Département de Physique, Faculté des Sciences, Université Djilali Liabes de Sidi Bel Abbés, Sidi Bel Abbés 22000, Algérie

b

Laboratoire de Modélisation et Simulation en Sciences des Matériaux, Faculté des Sciences, Université Djilali Liabes de Sidi Bel Abbés, Sidi Bel Abbés 22000, Algérie E-mail: [email protected]

Abstract. Ferromagnetism of GaN and AlN based magnetic semiconductors was investigated

by ab initio calculations based on the local density approximation. In a system of Mn atom doped GaN and AlN, the ferromagnetic ordering of Mn magnetic moments was induced by hole doping. It was also found that

ferromagnetic ordering of their magnetic moments in GaN and AlN without any additional carrier doping treatments. Appearance of the ferromagnetism in these systems suggests possibility for a fabrication of a transparent ferromagnetic which will have great impact on industrial applications in magneto optical devices.

1. Introduction

III-V nitrides are important in electronics, microelectronics and materials science. In particular GaN/AlN semiconductors have recently attracted extensive attention because of their importance to both science and technology for optoelectronic applications in the short-wave length range as well as for high-temperature, high-power and high-frequency electronic devices. This group III-V nitrides is considered as non-magnetic semiconductors. We can obtain by alloying them a new class: DMS (semiconductor magnetic diluted), with a sizable amount (a few percents or more) of magnetic elements, such as manganese Mn. The studies of DMS based on III-V and their heterosructures have offered a wide variety of materials and structures, making it possible to explore further the effect of the exchange interaction in semiconductors. Most remarkably Mn doping in GaN and AlN leads to ferromagnetism [1]. This behavior is generally called carrier-induced ferromagnetism because hole carriers introduced into the system mediate the ferromagnetic coupling between the Mn ions [2], although its microscopic mechanism has been controversial. The key to understanding the mechanism of the carrier-induced ferromagnetism is to elucidate the nature of the doped hole carriers as well as the exchange interaction between the holes in the host valence band and the localized d orbitals of magnetic ions, so-called p-d exchange interaction. Recently, there have been a few first principles electronic structure calculations of Mn doped GaN systems. Fong and al [3] performed electronic structure calculations of Fe and Mn doped GaN using the Tight-binding linearized muffin tin orbital (TB-LMTO) method. Kronik and al [4] considered (Mn, Ga) N in wurtzite structure and performed

1 To whom any correspondence should be addressed.

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electronic calculations using a plane-wave pseudo-potential method. Sanyal and al [5] studied the exchange interaction in (Mn, Ga) N and compared it with that in (Mn, Ga) As using the ab initio plane wave method within the local spin density approximation LSDA and (LSDA+U) calculations. The aim of this paper is to show that by using the full potential linear muffin tin orbital (FP-LMTO) within the LDA, the results of structural and electronic properties calculations are accurate compared to others works. The crystal is divided into two regions: non- overlapping muffin tin spheres surrounding every atom and the interstitial region between the spheres. Within the spheres, the potential is expanded in spherical harmonics, while the interstitial regions it is expanded in terms Fourier series. Provides the best accuracy at the price of increasing the computational time.

2. Results and discussion 2.1. Structural optimization

The calculations were first carried out assuming ideal zinc blend (ZB) and wurtzite (WZ) structures.

For wurtzite structures, the axis ratio c/a and the internal parameter u for both GaN and AlN are c/a = 1.636, 1.616, u = 0.375, 0.376 respectively were used. The calculated total energies and pressures for several lattice constants were fitted with the empirical Birch equation of state to obtain equiliberium lattice constants. For zincblende and wurtzite structures, the forces acting on atoms and the elements of the stress tensor have converged within 10

Ry/(a.u)

. Table I summarizes the results of our calculations for lattice parameters.

Table I. Calculated lattice constants a

and c

, bulk modulus, pressure derivative (B

) and internal parameter u compared to experiment and other works of GaN and AlN zinc blend (B

) and wurtzite (B

) phases.

Materials References a

(A

) c

(A

) B

(GPa) B

c/a u (A

) GaN (B

)

Present Experiment Other works

4.47 4.50

4.46

- - -

194.58 185

202

3.97 - 4.32

- - -

- - - GaN (B

)

Present Experiment Other works

3.158 3.192

3.18

5.166 5.183

5.151

194.2 188

-

4.1 3.2

-

1.636 1.624

1.62

0.375 0.375

-

AlN (B

)

Present Experiment Other works

4.34 4.37

4.349

- - -

201 215

211

4.002 - 3.90

- - -

- - -

AlN (B

)

Present Experiment Other works

3.07 3.11

3.11

4.961 4.979

4.979

205 185

-

3.996 5.7

-

1.616 1.601

1.601

0.376 0.385

0.382

a

Ref [6]

Ref [7]

Ref [8, 9, 10,11]

Ref [12]

For zinc blend (B

) and wurtzite (B

), the lattice constants from LDA calculations are slightly smaller than the experimental values. The equilibrium lattice constants to obtain from LDA calculations are in good agreement with those calculated by first principles method within different approximations. Our calculated bulks are slightly larger than experimental and other theoretical works except for AlN zinc blend (B

). This is essentially due to the effect of the LDA on the calculations. We note an anomaly in comparing our results of pressure derivative of the bulk modulus of AlN wurtzite (B

).

2.2. The relative stability and phase transition

In order to calculate the ground state properties of GaN and AlN, the total energies are calculated in GaN/AlN (NaCl B

, Zinc blend B

, Wurtzite B

) structures for different volumes around the

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equilibrium cell volume V

. Figure 1 show the fitted E

versus V curves of the three phases of GaN and AlN considered, calculated by using LDA calculation. The pressure is given by:

P =











  

 



 



 





1

' 0

' 0 0 0

B

V V B B

From this equation we deduce the transition volume:

V (P) =V

0

/ '

1

0 '

1

B

B P B





 



 

 



 

Where B

is the bulk modulus at the equilibrium V

and B

is its pressure derivative.

Figure 1. Calculated total energies as a function of volume for zinc blend, wurtzite and rocksalt GaN and AlN with LDA calculation .

Table II: transition pressures P

for GaN and AlN.

a

Ref [14]

Ref [15]

Ref [16]

Ref [17]

It’s clear from the E(V) curve, that the wurtzite (B

) structure is the most stable at room condition, which is consistent with experimental results and other theoretical works. The structure phase stability is determined by calculating Gibbs free energy for both phases (G = E

+PV-TS) since the theoretical calculations are performed at 0 K, Gibbs free energy becomes equal to enthalpy (H= E

+PV) [13].

The transition pressure (P

) of the zinc blend to rocksalt and wurtzite to rocksalt of GaN and AlN are determined from the constraint of equal static lattice enthalpy. The results for the pressure transition P

obtained are listed in table II, compared with other avaible theoretical results and experimental data.

The present results agree with the theoretical values and experimental data. Our results are similarity P

V

(B

) V

(B

) V

(B

)/V

V

(B

)

/V

 V%

GaN

Zinc blend - Rocksalt

GaN

Wurtzite - Rocksalt

Present Experiment Other Works Present Experiment Other Works

33.9 - 38.15

47.20 47-50

42.9

132.46 - - 127.39 - -

110.34 - - 107.43 - -

0.87 - - 0.84 - -

0.72 - - 0.71 - -

14.6 - - 13.2 - - AlN

Zinc blend - Rocksalt AlN

Wurtzite - Rocksalt

Present Experiment Other work Present Experiment Other Works

6.48 - 7.1

15.83 14-16.6

9.1

133.87 - - 128.311 - -

106.81 - - 103.689 - -

0.97 - - 0.75 - -

0.77 - - 0.93 - -

19.6 - - 17.9 - -

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between the zinc blend and wurtzite and show that the gradient corrections to the LDA have very small effects on P

of GaN/AlN.

2.3 Elastic constants

In the following, we study the GaN and AlN compounds in their metastable zinc blend phase because there are little experimental data and theorical works under strain and stress effect, behind this we will also study it in the wurtzite phase.

The elastic properties define the properties of material that undergoes stress, deforms and then recovers and returns to its original shape after stress ceases. These properties play an important part in providing valuable information about the binding characteristic between adjacent atomic planes, anisotropic character of binding and structural stability. Hence, to study the stability of these compounds in zinc blend (B

) and wurtzite (B

) structures, we have calculated the elastic constants at equilibrium lattice parameter. The elastic moduli require knowledge of the derivative of the energy as a function of the lattice strain. It is possible to choose this strain in such a way that the volume of the unit cell is preserved. In the case of cubic system, there are only three independent elastic constants namely C

, C

and C

. Thus for the calculation of them, we have used the Mehl method [18, 19], which state the use of the volume conserving orthorhombic and monoclinic tensors in determining the shear constants C

-C

and C

respectively. So set of three equations means to determinate all the constants, which means that three types of strain must be applied to the starting crystal.

The first type involves calculating the elastic modulus (C

+2C

), which are related to the bulk modulus B: B = 1/3 (C

+ 2C

).

The second type involves calculating the elastic modulus (C

-C

), which are related to the equilibrium volume: P/V

= C

-C

.

The third type involves calculating C

= 2P/V

.

Table III: Elastic constants of zinc blend GaN and AlN.

C

C

C

GaN Present 279 152 271

calcul

285 159 155

calcul

296 154 206

AlN Present 309 147 283

calcul

313 160 192

calcul

304 152 199

Ref [20]

Ref [21]

For zinc blend -GaN and zinc blend -AlN, there are no experimental data and there are several other calculations. Our calculations are all in good agreement with the other works.

In the case of wurtzite (B

) phase, there are five independent elastic constants namely C

, C

, C

and C

. These elastic constants are related to the bulk modulus B

and four shear constants (C

+C

, C

-C

, 1/2 C

and 2C

) [22]. Hence, a set of five equations is needed to determine all the constants.

The first equation involves calculating the bulk modulus B which related to the elastic constants by: 2/9(C

+C

+2 C

+1/2 C

)

The second and third ones calculating the shear constants (C

+C

) and C

by applying the uniform in plane strain tensor.





















 0 0

0 0

0

0

And the inter- layer strain tensor

















 0 0

0 0 0

0 0

0

Respectively.

Application of these strains changes the total energy from its unstrained values as following:

 E (V

) = V

(C

+C

) 

and  E (V

) = 1/2 V

C



. The fourth equation involves applying volume conserving monoclinic strain tensor

















 0 0 0

0 0

0 0





Which transform the energy to:  E (V

) = V

(C

-C

) 

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Finally, for the last type of deformation, we used the volume conserving triclinic strain tensor given by:















 0 0

0 0 0

0 0





This change the total energy to  E (V

) = 2V

C



The four shear constants are determined by a polynomial fit of the calculated total energy as a function of the shear strains curves. The calculated values of elastic constants in wurtzite-B

are listed in table IV. The positive values of the elastic constants, indicating that B

phase is elastically stable.

Table IV: Elastic constants of wurtzite GaN and AlN.

C

C

C

C

C

GaN Present 114 241 177 449 384 calcul

365 135 114 381 109

AlN Present 150 128 383 40 144 calcul

396 137 108 373 116

a

Ref [23]

Our results for both GaN and AlN are note in good agreement with others calculations obtained with ab initio calculations.

2.4 electronic properties

The electronic band structures of zinc blend and wurtzite in both GaN and AlN along symmetry lines are displayed in figures 2 and 3 for LDA calculation. The calculated band energies at normal symmetry points in the Brillouin zone are given in Table V and VI. The band structures are calculated at the appropriate theoretical equilibrium lattice constants for LDA.

In zinc blend-B

and wurtzite-B

for GaN, we find the valence band maximum (VBM) at  point and the conduction band minimum (CBM) at  point; predicting that this compound is a direct gap material (  -  ) for both phases 1.52, 2.3 (eV) respectively. In the absence of spin-orbit interaction, the valence band below the Fermi levelare triply and doubly degenerate at  point.

In zinc blend phase the calculated (  -  ) band gap value is in agreement with the others results [24, 25, 26] and the experiment results [27], but for others gaps (  -X), (  -L) we have either theoretical or experimental values to compare with them.

In wurtzite-phase the calculated (  -  ), (  -K) and (  -M) band gaps values are in agreement with the others calculations. For the zinc blend/ wurtzite AlN, the zero energy reference is the valence band maximum. It occurs at the  point, whereas the conduction band minimum occurs at the X point. Therefore, the band gap of zinc blend Al N is (  -X) indirect.

The calculated (  -  ), (  -X) and (  -L) band gaps values are in agreement with others predictions calculations [28], [29] (see the table V for comparison).

In wurtzite AlN, the band gap is direct at  point, this is in close agreement with the results of [30], [31].

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(a) Zinc blend phase (b) wurtzite phase

Figure 2. LDA band structures of GaN along the principle high-symmetry directions in the Brillouin zone in (a) zinc blend and (b) wurtzite phases.

(a) Zinc blend phase (b) wurtzite phase

Figure 3. LDA band structures of AlN along the principle high-symmetry directions in the Brillouin zone in (a) zinc blend and (b) wurtzite phases.

2.5. Density of state (a)

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(b)

Figure 4. Calculated total density of states in (a) GaN (zinc blend- wurtzite) and (b) AlN (zinc blend- wurtzite) respectively.

In order to check the accuracy of our band structures results. We have calculated the total density also the partial density of state (PDOS) is calculated but not shown. In this case we present the calculated total density of states (TDOS) of GaN/AlN in zinc blend and wurtzite structures.

For both zinc blend-GaN and wurtzite-GaN, the total DOS presents three regions, the lower part of the valence bands are dominated by N-2s states and the upper part by N-2p and Ga-4p states.

Figure 4 shows the total density of states (TDOS) for both zinc blend AlN and wurtzite AlN structures. While not much difference is expected between DOSs obtained for the tow structures. The total DOS presents three regions: the lower part of valence bands is dominated by N-2s sates and the upper part by N-2p and Al-3p states. The Al-3s states contribute to the lower valence bands.

2.6. Band gaps pressure coefficients

In order to investigate the pressure effect on zinc blend GaN and AlN energy gaps, we examine the band energies at the selected symmetry points as a function of the pressure. The results of our calculation for the direct and indirect band gaps E

(  ), E

(X) and E

(L) versus the pressure are shown in figures 5, table VI summarize the results of our calculated linear and sub-linear coefficients of band gaps compounds compared to the avaible theoretical works [32,33,34]. We notice that our results are in good agreement with the other prediction calculations. Our results for first pressure derivative coefficients (  ) of band gaps agree well with FP-LMTO calculations of Kim and al [35] and FP- LAPW calculations of Wei and Zunger [34]. Our results for second pressure derivative coefficients (  ) of band gaps are also in good agreement with LMTO calculations of Christensen and Gorczyca [32]. Since the linear pressure coefficients of E

(  ), E

(X) and E

(L) are positive these main band gaps are also increased under pressure. However the increase of E

(X) is less important than that of Eg (  ) and E

(L).

We also study the behavior of the energy band gaps versus the volume for GaN and AlN in zinc blend structure (see figures 5).

From figure 5, we notice that at equilibrium volume the GaN has an direct gap (  -  ), and the AlN has an indirect gap (  -X). The slope dE

/dV of the lines are given in table VII, the negative linear pressure coefficients indicate that the indirect band gaps (  -X) decreases with increase of the pressure.

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Figure 5. Direct and indirect band gap energies versus presure in zinc blend (a) GaN (b) AlN respectively .

Figure 6. Calculated volume coefficients of energy gap in zinc blend structure of GaN and AlN respectively

Table VI: pressure coefficients related to E

(P) = E(a

) +  P+  P

of the calculated values of the energy gaps for zinc blend GaN/AlN compared with other calculations respectively.

GaN  (meV/GPa)  (meV/GPa)

Present work Other calculations Present work Other calculations E ( 

- 

)

E ( 

- X

) E ( 

- L

)

37.4 2.16 35.6

31

, 32

, 40

1.7

32.1

, 42

-0.363 -0.024 -0.356

-0.38

-0.03

-0.38

AlN  (meV/GPa)  (meV/GPa)

Present work Other calculations Present work Other calculations E ( 

- 

)

E ( 

- X

) E ( 

- L

)

42.9 1.21 44.05

42

, 31

1.9

, 1.9

, 1.7

41.1

, 44

-0.59 -0.013 -0.067

-0.34

-0.03

-0.38

Table VII: calculated volume coefficients of energy gap in zinc blend structure of GaN and AlN

respectively.

GaN  (eV/A

)  (eV/A

)

Present work Other calc Present work Other calc E ( 

- 

)

E ( 

- X

) E ( 

- L

)

-0.092 -0.0025

-0.113

- - -

1.58 3.74 2.17

- - -

AlN  (eV/A

)  (eV/A

)

Present work Other calc Present work Other calc E ( 

- 

)

E ( 

- X

) E ( 

- L

)

-0.13 -0.16 -0.15

- - -

2.51 3.61 3.05

- - -

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2.7. Ternary alloys

The calculated spin-polarized band structures for Ga

Mn

N and Al

Mn

N alloys at predicted equilibrium lattice constants are depicted in figures 7 and 8, which are drawn along the symmetry directions in the first Brillouin zone. We define the majority-spin component to be the one that contains the largest number of electron. The states in the energy range from -8.3 to -3.1 eV for Ga

Mn

N (10.1 to 14.1 eV for Al

Mn

N) come mainly from N-p states with minor contributions from Mn-d Ga (Al)-s, p states.

The Fermi level is running through the impurity band, the Ga

Mn

N and Al

Mn

N alloys are half metallic. In this case, they possess a Fermi surface for the majority spin.

In this figures we see that both majority and minority spin components display a band gap. This indicates that the introduction of Mn does not destroy the semiconducting nature of these materials.

The calculated total of states (DOS) curves at the predicted equilibrium lattice constants for ferromagnetic GaN: Mn and AlN: Mn are shown in figure 9. In comparison with Ga and Al, Mn has extra four valence electrons, which fill spin-up e

and t

band. The doubly degenerated band is fully occupied, while the triply degenerated band is only two thirds filled so the Fermi level falls just into the latter 100% spin polarized band.

(a) (b)

Figure 7. Spin-polarized band structure of zinc blend Ga

Mn

N: (a) minority spin (b) majority spin.

(a) (b)

Figure 8. Spin-polarized band structure of zinc blend Al

Mn

N : (a) minority spin (b) majority spin.

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(a) zinc blende Ga

Mn

N (b) zinc blende Al

Mn

N

Figure 9. Total and partial spin polarized DOS in ferromagnetic states.

3. Conclusion

In this study we have presented a complete theoretical analysis of the structural, elastic and high pressure properties of GaN and AlN by using FP/LMTO method. The use of LDA for the exchange correlation potential permitted us to obtain good structural parameters. The results regarding the high pressure structural phase transition are in excellent agreement with the experimental data and agree with the other theoretical works. Plus we have performed a first principles FP/LMTO calculation of the electronic structures and magnetic properties of ordered ferromagnetic GaMnN and AlMnN with the zinc blend lattice. Optimal bulk geometries have been calculated by total energy minimization. The Ga-3d semi-core electrons have been explicitly treated as valence electrons. We found that both materials possess a band of well defined spin, which is primarily due to hybridization of Mn-3d and N- 2p orbitals.

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