Physics Letters A 373 (2008) 165–168
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Physics Letters A www.elsevier.com/locate/pla
p-type doping of GaInNAs quaternary alloys Hongliang Shi a,∗ , Yifeng Duan b a b
State Key Laboratory for Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, PO Box 912, Beijing 100083, People’s Republic of China Department of Physics, School of Sciences, China University of Mining and Technology, Xuzhou 221008, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 25 June 2008 Received in revised form 4 November 2008 Accepted 11 November 2008 Communicated by R. Wu PACS: 61.72.Bb 71.20.Nr
a b s t r a c t Using the first-principles band-structure method, we investigate the p-type doping properties and band structural parameters of the random Ga1−x Inx N1− y As y quaternary alloys. We show that the MgGa substitution is a better choice than ZnGa to realize the p-type doping because of the lower transition energy level and lower formation energy. The natural valence band alignment of GaAs and GaInNAs alloys is also calculated, and we find that the valence band maximum becomes higher with the increasing In composition. Therefore, we can tailor the band offset as desired which is helpful to confine the electrons effectively in optoelectronic devices. © 2008 Published by Elsevier B.V.
Keywords: First-principles Alloy Doping Formation energy Transition energy Band offset
1. Introduction The band gaps and lattice constants of conventional III–V alloys A1−x Bx C can be individually tuned by changing the composition x. The band gap can be described as E g (x) = (1 − x) E g ( AC ) + xE g ( BC ) − bx(1 − x),
(1)
where b is the bowing parameter and is composition independent. GaAs1−x Nx and Ga1−x Inx N1− y As y alloys have simulated great interests because of their unique reduction of band gap and giant composition dependent bowing parameter [1–5]. Wei et al. suggested that the band gap variation as a function of x in GaAs1−x Nx can be divided into two parts: (i) a bandlike part where the bowing coefficient is relatively small and nearly constant; (ii) an impurity like one where the bowing coefficient is considerably large and composition dependent. These results reflect the large differences between N and As in the atomic orbital energies and sizes [2]. The band anti-crossing model has also been proposed to explain the band gap reduction for GaInNAs alloys, which is consistent with the experimental results [3]. The Ga1−x Inx N1− y As y alloys have the great potential applications for long-wavelength (1.3–1.88 μm) optoelectronic devices,
*
Corresponding author. Tel.: +86 10 82305133. E-mail address:
[email protected] (H. Shi).
0375-9601/$ – see front matter © 2008 Published by Elsevier B.V. doi:10.1016/j.physleta.2008.11.010
which have been studied extensively [6–8]. The quaternary alloys have been successfully synthesized by incorporating nitrogen to GaInAs alloys. However, the increase of nitrogen will lead to poor crystal quality due to the lattice constant mismatch. As a result, the more indium composition is requested, which can also reduces the band gap and tensile strains [7]. The 1.3 μm GaInNAs/GaAs quantum wells (QWs) with good structural and optical properties have been successfully synthesized by introducing a small amount of N into InGaAs/GaAs QWs by molecular-beam epitaxy [7]. In addition, the 1.59 μm GaInNAsSb/GaNAs single-quantum-well laser diodes have been obtained experimentally by incorporating a small amount of Sb to improve the optical quality of QWs positively [8]. GaInNAs/GaAs QWs have the advantage of a larger conduction band offset compared with GaInAsP/InP QWs, which can lead to a higher characteristic temperature (T 0 ) [6]. The quantum efficiency of multijunction solar cell is found strongly correlated with the carrier concentration in GaInNAs doped with Si, Se, or Zn [9]. Therefore, it is of great interests to investigate the p-type doping properties of the GaInNAs quaternary alloys. In this work, we systematically calculate the formation energies and transition energies of Mg and Zn doped in GaInNAs alloys, and the natural band offset of GaAs and GaInNAs alloys with different In compositions. The Letter is organized as follows. In Section 2, we describes the calculational methods and details used in this work. In Section 3, we discuss the defect formation energies and transi-
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tion energy levels of Mg and Zn dopants. Then, in Section 4, we calculate the natural valence band alignment of GaN, InN, GaAs, InAs, and GaInNAs. A brief summary of this work is given in Section 5.
The optimized results are shown in Table 2 and the values in parentheses are obtained by the Vegard law. The Vegard law is well obeyed. The band gap bowing parameters b for GaInAs alloys are described as
2. Calculational details and methods
E g (x) = (1 − x) E g (GaAs) + xE g (InAs) − bx(1 − x).
Our first-principles band-structure and total-energy calculations are based on the density functional theory (DFT) and the Vienna ab initio simulation package [10] using the local density approximation (LDA) [11] for the exchange correlation potential and the projector augmented wave (PAW) method [12]. The Ga 3d and In 4d electrons are chosen as valence electrons. The electron wave function is expanded in plane waves up to a cutoff energy of 400 eV and the Brillouin zone integration is performed with 2 × 2 × 2 k-meshpoints in the zinc-blende Brillouin zone. All the geometries are optimized until the quantum mechanical forces acting on the atoms are less than 0.01 eV/Å. In order to calculate the band structural parameters of the alloys, we adopt the more efficient special quasirandom structures (SQS) approach [13,14], in which a small unit cell is used to mimic the random alloy A1−x Bx C by considering the physically most relevant structure correction functions and the mixed-atom sites are occupied based on this correction functions. In our calculation, we use the 2a × 2a × 2a supercell 64 atoms SQS at x = 0.25 and 0.75 for Ga1−x Inx As, and substitute one As atom with one N atom randomly to mimic the GaInNAs quaternary alloys. 3. Results 3.1. Related binary, ternary and quaternary alloys In order to investigate the electronic and structural properties of GaInNAs quaternary alloys, it is essential to calculate the lattice constants and band gap parameters of GaAs, GaN, InAs, and InN. Note that the calculations are performed with cubic zinc-blende (ZB) structure. Although the ground-state structure of nitrides is wurtzite (WZ), the results of ZB nitrides are applicable to the case of WZ, since the band structures are very similar near the band edge at Γ point [2]. Table 1 shows our calculated equilibrium lattice constant a, bulk modulus B, and band gaps at Γ point of GaAs, GaN, InAs, and InN with the ZB structure, which are compared with the corresponding experimental values (in parentheses) [15,16]. Our calculated lattice parameters aGaAs , aGaN , aInAs , and aInN are 5.609, 4.461, 6.032, and 4.945 Å, respectively, which are obtained by fitting the Murnaghan equation of state [17]. The calculated lattice parameters and bulk modulus B are in good agreement with the experimental values. Since GaInNAs alloys are experimentally grown by incorporation of a small amount of nitrogen into GaInAs, we only calculate the structural parameters of related ternary Ga1−x Inx As alloys. We use the SQS 64 atoms at x = 0.25, 0.5 and 0.75. We first calculate the lattice constant by the Vegard law [18], and then obtain the optimized values by fitting the Murnaghan equation of state. The lattice constant of Ga1−x Inx As alloys can be described as a(x) = (1 − x)aGaAs + xaInAs .
(2)
Table 1 Calculated lattice constant a, bulk modulus B, and band gap E g for binary alloys GaAs, GaN, InAs, and InN. The experimental values suggested in Refs. [15] and [16] are also listed in parentheses.
a (Å) B (Mbar) E g (eV)
GaAs
GaN
InAs
InN
5.609(5.653) 0.75(0.75) 0.51(1.52)
4.461(4.531) 1.99(2.08) 1.94(3.17)
6.032(6.058) 0.60(0.58) −0.36(0.417)
4.945(4.98) 1.45(1.46) −0.34(0.7)
(3)
The calculated bowing parameters b are listed in Table 2. Our results for bGa0.25 In0.75 As , bGa0.5 In0.5 As and bGa0.75 In0.25 As are 1.247, 1.108, and 0.922 eV, respectively, which are slightly composition dependent and larger than the value of 0.477 eV recommended by Vurgaftman and Meyer [15]. We also find that for Ga1−x Inx As alloys with the increasing Ga composition, the lattice constant decreases while the bulk modulus increases, which can be attributed to the fact that the electronegativity difference between Ga and As is larger than that between In and As, the ionic bonds become stronger with increasing Ga composition in the Ga1−x Inx As alloys. The lattice constants a(x, y ) of Ga1−x Inx N1− y As y quaternary alloys can be described by Vegard law as a(x, y ) = (1 − x)(1 − y )aGaN + x(1 − y )aInN
+ (1 − x) yaGaAs + xyaInAs .
(4)
We mimic the Ga1−x Inx N1− y As y quaternary alloys by substituting one As atom with one N atom in the Ga1−x Inx As 64 atoms SQS at x = 0.25 and 0.5. Therefore, the N concentration is 3% approximately. The calculated lattice constants and bulk modulus of Ga0.5 In0.5 N0.03 As0.97 and Ga0.75 In0.25 N0.03 As0.97 are shown in Table 3, which are compared with the values listed in parentheses obtained by the Vegard law. Note that the deviation of lattice constants is less than 0.7%, and that the value of Ga0.5 In0.5 N0.03 As0.97 is set to 5.821 Å during the calculations. 3.2. The formation energies and transition energy levels of Mg and Zn in GaAs and GaInNAs quaternary alloys The ionization energy of an accepter (q < 0) with respect to the VBM in the impurity limit is calculated by the following procedure described in Ref. [19],
(0/q) = E (α , q) − E (α , 0) − q kD (0) /(−q) Γ + DΓ (0) − VBM (host) ,
(5)
where E (α , q) and E (α , 0) are the total energies of the supercell at charge state q and neutral, respectively, for defect α , kD (0) and DΓ (0) are the defect levels at the special k points (averaged) and Γ (host) is the VBM energy of at the Γ point, respectively, and VBM the host supercell at the Γ point and is aligned using core electron levels away from the defect. The first term on the right-hand Table 2 Calculated lattice constant a, bulk modulus B, and band gap optical bowing parameter b for ternary alloys Ga1−x Inx As. The lattice constants listed in the parentheses are obtained by the Vegard law.
a (Å) B (Mbar) b (eV)
Ga0.25 In0.75 As
Ga0.5 In0.5 As
Ga0.75 In0.25 As
5.945(5.927) 4.91 1.247
5.847(5.821) 5.22 1.108
5.738(5.715) 5.40 0.922
Table 3 Calculated lattice constant a, bulk modulus B for quaternary alloys Ga1−x Inx N1− y As y . The lattice constants listed in the parentheses are obtained by the Vegard law.
a (Å) B (Mbar)
Ga0.5 In0.5 N0.03 As0.97
Ga0.75 In0.25 N0.03 As0.97
5.821(5.786) 5.33
5.718(5.680) 5.55
H. Shi, Y. Duan / Physics Letters A 373 (2008) 165–168
Fig. 1. Crystal structure of the random Ga0.5 In0.5 N0.03 As0.97 quaternary alloys doped with MgGa or ZnGa . The brick red, green, yellow, blue, and magenta balls represent Ga, In, N, As, Mg(Zn) atoms, respectively. Table 4 Calculated formation energy H f (α , 0) (eV) and transition energy level (0/−) for GaInNAs:Mg and GaAs:Mg are listed, and the H f (α , 0) (eV) for interstitial Mg is also calculated.
3NN 5NN GaAs Mgi
H f (α , 0)
(0/−)
1.988 2.195 2.508 1.080
0.095 0.108 0.111
Table 5 Calculated formation energy H f (α , 0) (eV) and transition energy level (0/−) (eV) for GaInNAs:Zn and GaAs:Zn are listed, and the H f (α , 0) for interstitial Zn is also calculated.
3NN 5NN GaAs Zni
H f (α , 0)
(0/−)
2.602 2.668 2.748 1.832
0.117 0.148 0.119
side of Eq. (5) determines the U energy parameter (including both the Coulomb contribution and atomic relaxation contribution) of the charged defects calculated at the special k points, which is the extra cost of energy after moving (−q) charge from the VBM of the host to the neutral defect level. The second term gives the singleelectron defect level at the Γ point. For charged defects, a uniform background charge is added to retain the global charge neutrality of the periodic unit-cell. The defect formation energy of a neutral defect is defined as
H (α , 0) = E (α , 0) − E (host) + nGa μGa + nα μα ,
(6)
where E(host) is the total energy of the host supercell without defect α ; μGa and μα are the chemical potentials of constituents Ga and α relative to the element solids energy. The nGa (nGa > 0) and nα (nα < 0) are the numbers of Ga and extrinsic defects α . We calculate the chemical potentials for Ga(μGa ), Mg(μMg ), and Zn(μZn ) relative to α -Ga, fcc Mg and hcp Zn. Tables 4 and 5 show our calculated MgGa (ZnGa ) acceptor formation energies and (0/−) transition energy levels for GaAs and random Ga0.5 In0.5 N0.03 As0.97 quaternary alloys. We select the Ga0.5 In0.5 N0.03 As0.97 alloy because of the good performance of the lasers with In content as high as 42.5% experimentally [7,8]. We calculate MgGa (ZnGa ) at different sites for Ga0.5 In0.5 N0.03 As0.97 quaternary alloy. The nNN in Tables 4 and 5 denotes that there are n In atoms in the fcc nearest neighbor (NN) sites centered around the Mg atom. Fig. 1 shows the crystal structure of the
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random Ga0.5 In0.5 N0.03 As0.97 quaternary alloys doped with MgGa in the 3NN site. We note that the formation energy of neutral MgGa (ZnGa ) in 5NN sites is slightly larger than that in 3NN sites. This is because the strain introduced by MgGa (ZnGa ) substitution is smaller in the Ga rich region than in the In rich region. The formation energy of MgGa (ZnGa ) in Ga0.5 In0.5 N0.03 As0.97 alloy is about 0.313–0.520 (0.08–0.146) eV lower than that in GaAs, indicating that it is easier to dope in GaInNAs with MgGa (ZnGa ). For MgGa and ZnGa in GaAs, our calculated (0/−) acceptor levels locate at 0.111 and 0.119 eV, respectively, larger than 0.029 and 0.031 eV obtained by Wang et al. using the k · p theory and an empirical impurity potential model [20]. Note that the chemical trends of the acceptor energy level are of the same although the values are different quantitatively. For MgGa in Ga0.5 In0.5 N0.03 As0.97 quaternary alloy, the (0/−) acceptor levels increase from 0.095 eV when Mg has three fcc nearest neighboring In atoms to 0.108 eV with five In atoms as fcc nearest neighbors. This is caused by the increasing p–d coupling which pushes the acceptor levels higher relative to the VBM of the quaternary alloy, which can be explained by the fact that the defect levels consist of the impurity valence p orbitals, and the In 4d orbitals are slightly higher than Ga 3d orbitals. As a result, the p–d coupling repulsion increases as the number of the fcc nearest neighboring In atoms increases, and at the same time, the accept levels become higher. This conclusion is also applicable to the case of ZnGa acceptor levels. The acceptor defect levels of MgGa substitution are lower than those of ZnGa substitution in GaAs and GaInNAs alloy. This is because Zn has the occupied d orbitals while Mg has not, and the defect levels with p character couple with the Zn 3d levels pushing the defect levels higher. The MgGa (0/−) acceptor level in the quaternary alloy is also shallower due to the higher valence band maximum edge than GaAs (see below). Note that our calculated formation energies of interstitial Mg and Zn in the Ga0.5 In0.5 N0.03 As0.97 quaternary alloy are 1.080 and 1.832 eV, respectively. According to above analysis, we conclude that the MgGa substitution is a better choice than ZnGa to realize the p-type doping due to the lower acceptor level and smaller formation energy. 4. The natural valence band alignment The offset E v ( E c ) between the VBM (CBM) of two semiconductor alloys is an important parameter for the applications of the heterostructures and quantum wells in optoelectronic devices. Compared with GaInAsP/InP QWs laser, GaInNAs/GaAs QWs laser has a larger CBM offset E c to prevent the electrons overflow leading to a higher characteristic temperature. The conduction band offset between Ga0.93 In0.07 N0.02 As0.98 (lattice matched to GaAs) and GaAs is 300–400 meV [21]. Geisz et al. [9] also studied the photocurrent of Ga0.92 In0.08 N0.03 As0.97 (band gap 0.95 eV) multijunction solar cell. In the following, we will investigate the important band offset parameter of GaInNAs/GaAs. To calculate the valence band offset E v (AX/BY) between two semiconductors AX and BY, we adopt the Wei’s procedure [22], where the band offset is given by AX/BY
AX E v (AX/BY) = E BY v ,C − E v ,C + E C ,C .
(7)
Here, AX AX E AX v ,C = E v − E C
(8)
) are the core level (C ) to valence band (and similarly for E BY v ,C maximum energy separations for pure AX (and similarly for pure BY), while AX/BY
AX E C ,C = E BY C − EC
(9)
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calculated MgGa acceptor energy level is lower than that of ZnGa in the Ga0.5 In0.5 N0.03 As0.97 quaternary alloy, suggesting that MgGa substitution is a better choice for the p-type doping. We also show that the valence band maximum offset of Ga1−x Inx N1− y As y /GaAs becomes larger as the In composition increases. Our calculated results are expected to be helpful for enhancing the performance of the optoelectronic devices. Acknowledgements This work was supported by the National Basic Research Program of China (973 Program) grant No. G2009CB929300 and the National Natural Science Foundation of China under Grant Nos. 60521001 and 60776061. Fig. 2. The calculated natural valence band alignment of GaN, InN, GaAs, InAs, Ga0.75 In0.25 N0.03 As0.97 and Ga0.5 In0.5 N0.03 As0.97 alloys.
is the difference in core level binding energy between AX and BY BY at the AX/BY superlattice. We calculate the E AX v ,C and E v ,C at the equilibrium zinc-blende lattice constants for AX and BY, respectively. We use the average lattice constant a¯ (between AX and AX/BY BY) to calculate the E C ,C in the AX/BY superlattice. A more detailed description of the procedure can be found in Ref. [22]. Using the above procedure we investigate the natural valence band alignment of the related GaN, InN, GaAs, InAs, Ga0.75 In0.25 N0.03 As0.97 and Ga0.5 In0.5 N0.03 As0.97 alloys. We estimate the error for these calculations is less than 0.02 eV by taking different atoms as reference to define the core levels. Fig. 2 shows the natural chemical trends. We find that the E v between Ga0.75 In0.25 N0.03 As0.97 and GaAs is about 122 meV and that the VBM of the GaInNAs quaternary alloy becomes higher with increasing In composition. Therefore, we can tailor the E v and E c (combining with the band gap of GaInNAs quaternary alloy) as desired in the GaInNAs/GaAs QWs to enhance the performance of the optoelectronic devices. 5. Summary In summary, we have investigated the p-type doping properties and band structure of the random Ga1−x Inx N1− y As y quaternary alloys using the first-principles methods and the SQS approach. The
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