Influence of Bi and N related impurity states on the band structure and band offsets of GaSbBiN alloys

Current Applied Physics 16 (2016) 1687e1694

Contents lists available at ScienceDirect

Current Applied Physics journal homepage: www.elsevier.com/locate/cap

Influence of Bi and N related impurity states on the band structure and band offsets of GaSbBiN alloys D.P. Samajdar a, *, Utsa Das b, A.S. Sharma b, Subhasis Das b, S. Dhar b a b

Department of Electronics and Communication Engineering, Heritage Institute of Technology, Chowbaga Road, Anandapur, Kolkata, 700107, India Department of Electronic Science, University of Calcutta, 92 A.P.C. Road, Kolkata, 700009, India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 6 July 2016 Received in revised form 10 October 2016 Accepted 12 October 2016 Available online 14 October 2016

The perturbed band structure of a proposed material GaSbBiN, formed by the incorporation of N and Bi in GaSb, is calculated using a 16 band k·p Hamiltonian. The changes in band gap (Eg), spin-orbit splitting energy (DSO), conduction band offset (DEc) and valence band offset (DEv) are investigated as functions of N and Bi mole fractions. In the low temperature regime, the addition of Bi and N to GaSb causes substantial reduction in the band gap and enhances the spin-orbit splitting energy, thereby making Eg < DSO which is expected to improve the thermal stability and high-temperature efficiency of photonic devices by the suppression of different loss mechanisms. The values of DEc and DEv for GaSbBiN alloys, calculated with reference to the host GaSb lattice, increase with the increase in Bi and N concentrations. Calculations indicate that DEc > DEv which is required for electron confinement in order to get improved temperature-insensitive characteristics of optoelectronic devices. © 2016 Elsevier B.V. All rights reserved.

1. Introduction GaSb has emerged as a technologically important III-V semiconductor in the recent years for potential applications in high efficiency thermo-photovoltaics, mid-infrared lasers, photodetectors, high speed electronic devices and non-linear optics [1e3]. Such applications stem from several attractive properties of the material, such as, high hole mobility (850e10800 cm2/Vs), low carrier effective masses and relatively small direct band gap value [4,5]. Incorporation of 1% N in GaSb has been reported to reduce the material band gap by 230 meV [6]. According to the Band Anticrossing (BAC) model, the incorporation of nitrogen in the host III-V lattice results in the formation of an energy level extended in k space and resonant with the host conduction band. The interaction of the host conduction band with the nitrogen resonant state causes splitting of the conduction band into two non-parabolic subbands Eþ and E [7]. Nrelated defect levels occur both above and below the conduction band edge (CBE) in GaSb and the large number of lower-energy N states near the CBE results in the formation of strongly disordered band structure in GaSbN. Moreover, the formation of NeN pairs in the GaSb supercell has significant effects on the band structure of

* Corresponding author. E-mail address: [email protected] (D.P. Samajdar). http://dx.doi.org/10.1016/j.cap.2016.10.010 1567-1739/© 2016 Elsevier B.V. All rights reserved.

GaSbN due to the formation of two defect levels with odd and even symmetries [3]. The electronegativity mismatch between antimony and nitrogen is more than any other combination of commonly used Group V elements which is expected to enhance the bandgap reduction in dilute antimonide nitrides in comparison to dilute arsenide nitrides [8]. This has been demonstrated from the optical absorption spectra of molecular beam epitaxy (MBE) grown GaSb1xNx samples in which the incorporation of 1.52% N resulted in a band gap reduction of ~320 meV which is substantially larger than the band gap reduction of ~220 meV GaAsN containing 1.5% N [8]. Similar results were also obtained from photoluminescence (PL) and optical absorption measurements of dilute GaSbN bulk materials, with up to 1.4% N, grown by MBE on GaSb substrates with a band gap reduction of 300 meV at room temperatures [9]. The incorporation of Bi in III-V alloys causes similar reduction in materials band gap along with an enhancement in the spin-orbit splitting [10]. This effect is attributed to the formation of Bi related impurity levels EBi and EBi-so below the valence band maximum of the host III-V alloy which interacts with the extended states of the host valence band matrix. Studies on GaSbBi alloys gained significance in the past decade due to their potential optoelectronic applications in the important 2e5 mm region. Optical absorption, photoreflectance and photoluminescence measurements on different epitaxially grown GaSbBi layers indicate a band gap reduction in the range 30e40 meV/%Bi [11e14]. These experimental findings have been

1688

D.P. Samajdar et al. / Current Applied Physics 16 (2016) 1687e1694

supported by the well-established theoretical models based on density functional theory (DFT) [15], tight binding calculations [16] and kp modelling [17,18]. The temperature induced shift of the direct E0 transition between 15 and 290 K has been found to be ~70 meV which is comparable to the band gap shift in the GaSb host material over the same temperature range but this value is somewhat higher than other Bi containing III-V alloys such as GaAsBi, GaInSbBi and AlGaSbBi [19]. Structurally, small diameter N atoms in the III-V host produces lattice compression whereas, large diameter Bi atoms dilates the lattice. These opposite effects of N and Bi insertion have been recently utilized to grow GaAsBiN alloys containing small amounts of both Bi and N whose suitable combinations resulted in lattice matching with GaAs substrates [20,21]. The variation of band parameters such as band gap, spin-orbit splitting energy, band offsets and strain of GaAsBiN/GaAs heterostructures as functions of Bi and N mole fractions has been studied theoretically by Sweeney et al. [22]. Habchi et al. [23] reported the calculation of the band structure of GaAsBiN lattice matched to GaAs using k·p method and investigated the effective masses of the carriers near the zone centre with varying Bi and N concentrations. A large band gap reduction of 198 meV/%Bi and an increase of spin orbit splitting energy by about 56 meV/%Bi causes a crossover between Dso and Eg for GaN.039As.893Bi.068 which may be effective in suppressing the Auger recombination and Intervalence Band Absorption (IVBA) mechanisms in GaAsBiN based lasers and photodetectors [23]. Ashwin et al. investigated the properties of MBE grown quaternary alloy Ga1-yIn yNxSb1x which can improve the efficiency of GaSb based thermophotovoltaics in the 2e4 mm region and act as an alternative to HgCdTe for infrared sources and detectors operating in the 8e14 mm atmospheric transmission window. Addition of both In and N to GaSb improves the possibility of lattice matching of GaInNSb to GaSb provided In:N ratio of 4.14:1 is maintained following Vegard's law [24]. Similar advantages can be obtained by co-doping GaSb with Bi and N including lattice match of the resultant alloy material with GaSb substrates since addition of Bi causes a lattice dilation [11] whereas N in GaSb compresses the lattice [6]. However, to our knowledge, there is no report of the growth of GaSbBiN material till date. In this paper, we present a theoretical investigation of epitaxial GaSbBiN with Bi and N compositions corresponding to lattice match with GaSb substrates using a 16 band k·p model involving computer codes in order to get

2 6 6 6 6 6 6 0 6 6 6 6 1 6 pffiffiffiP * 6 þ 2 6 6 rffiffiffi 6 2 6 6 Pz 6 3 6 ¼6 6 1 6 pffiffiffiP * 6 6 6  6 6 6 6 0 6 6 6 6 1 6 pffiffiffiP z 6 6 3 6 6 4 1 * pffiffiffiP 3 C

mod H88

0

1 pffiffiffiPþ 2

C

0

0 1 * pffiffiffiPþ 6 rffiffiffi 2 Pz 3

rffiffiffi 2 Pz 3

the details of the band structure of proposed material. The paper is organized as follows: In the first part, we have given the introduction of III-V-Bi-N alloy systems followed by a description of the mathematical models we have used for the calculation of band structure, band offsets and strain. Section 3 summarizes the results and discussions obtained through our calculations. The final section concludes the contribution of our work. 2. Mathematical modelling 2.1. Calculation of band structure The band structure of GaSb1xyBixNy is calculated by using a 16  16 k·p Hamiltonian. This Hamiltonian is obtained from a basic 8  8 matrix H0 [25] which combines the effect of the Valence Band Anticrossing (VBAC) interaction in the valence band with the Band Anticrossing (BAC) in the conduction band. The matrix H0 is modified by the addition of the band offset terms DECBM, DEVBM and DESO to the diagonal elements corresponding to the G6, G7 and G8 mod . bands respectively [23] resulting in a modified Hamiltonian H88 The addition of N and Bi atoms to the host GaSb lattice results in the formation of defect levels EN, EBi and EBi-so. N-related defect level EN is located above the CBM of GaSb whereas Bi-related impurity level EBi and corresponding spin-orbit split-off level are below the VBM. The interaction between the nitrogen resonant level with the host conduction band matrix causes the generation of ECþ and EC conduction sub bands in GaSbN due to the large difference in electronegativity between Sb and N [26]. In a similar way, the anticrossing interactions between the VBM and Bi levels causes the splitting of the valence sub bands-light hole (LH), heavy hole (HH) and spin-orbit split-off (SO) leading to the band gap bowing effect through an upward shift of the VBM in GaSbBi [15,27]. Coupling of the Bi impurity states, the extended valence band states and the N localized states with the host conduction band states is taken care of by the inclusion of the coupling terms VBi(x) ¼ CBix and VN(y) ¼ CNy. The resultant 16  16 Hamiltonian can be written as,

H1616 ¼

N;Bi ðy; xÞ V88

N;Bi V88 ðy; xÞ

!

N;Bi E88

The matrix elements are as defined below,

0

1 pffiffiffiPz 3

1 pffiffiffiPþ 6

1 pffiffiffiP 6 rffiffiffi 2 Pz 3

1 pffiffiffiP 2

1 pffiffiffiPþ 3

H

a

b

0

ia pffiffiffi 2

a*

L

0

b

*

mod H88

iD pffiffiffi 2 rffiffiffi * 3 ia 2

b

0

L

a

1 * pffiffiffiP 2

0

b*

a*

H

pffiffiffi i 2b*

1 * pffiffiffiPþ 3

ia* pffiffiffi 2

rffiffiffi 3 ia 2

pffiffiffi i 2b

S

1 pffiffiffiPz 3

pffiffiffi i 2b*

iD* pffiffiffi 2 rffiffiffi * 3 ia 2

iD pffiffiffi 2

ia pffiffiffi 2

0

3 1 pffiffiffiP 7 7 3 7 7 7 1 pffiffiffiPz 7 3 7 7 7 pffiffiffi 7 i 2b 7 7 7 rffiffiffi 7 7 3 7 7 ia 2 7 7 7 iD 7 pffiffiffi 7 7 2 7 7 7 ia* 7 pffiffiffi 7 2 7 7 7 7 0 7 7 7 7 7 5 S

(1)

D.P. Samajdar et al. / Current Applied Physics 16 (2016) 1687e1694

0 B B B B B N;Bi V88 ðy; xÞ ¼ B B B B B @

0

N;Bi E88

EN B 0 B B 0 B B 0 ¼B B 0 B B 0 B @ 0 0

VN ðyÞ 0 0 0 0 0 0 0

0 EN 0 0 0 0 0 0

0 0 EBi 0 0 0 0 0

0 VN ðyÞ 0 0 0 0 0 0

0 0 0 EBi 0 0 0 0

0 0 VBi ðxÞ 0 0 0 0 0

0 0 0 0 EBi 0 0 0

0 0 0 0 0 EBi 0 0

0 0 0 VBi ðxÞ 0 0 0 0

0 0 0 0 VBi ðxÞ 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0

EBiso

0 0 0 0 0 VBi ðxÞ 0 0

0 0 0 0 0 0 VBi ðxÞ

1 0 0 C C 0 C C 0 C C 0 C C 0 C C 0 A VBi ðxÞ

(2)

1 C C C C C C C C C C A

Ep ¼ (3)

EBiso

where,

C ¼ Eg þ

1689

    Z2 1 Ep 2 1 2 2 2 þ DECBM ðx;yÞ k   þk þk x y z 2m0 m*e 3 Eg Eg þ Dso

 i Z2 h 2 kx þk2y ðg1 þ g2 Þþk2z ðg1 2g2 Þ þ DEVBM ðx;yÞ 2m0  i Z2 h 2 kx þk2y ðg1  g2 Þþk2z ðg1 þ2g2 Þ þ DEVBM ðx;yÞ L¼ 2m0

H¼

1 S ¼ ðLþHÞ Dso  DESO ðx;yÞ 2

P± ¼ P kx ±iky

2m0 Z2

P2

(6)

The individual parameters in Eq. (4) are the same as that of the parent 8  8 Hamiltonian as defined in Refs. [17] and [25], modified by the addition of the terms DECBM(x,y), DEVBM(x,y) and DESO(x,y) to the diagonal matrix elements. These terms take into account the linear change in the CBM, VBM and spin-orbit split-off band energies between the ternary alloys GaSbN and GaSbBi due to Virtual Crystal Approximation (VCA) [17]. A list of the VBAC and BAC parameters required for the calculation of the band structure of GaSb1xyBixNy is given in Table 1. At the G point (kx ¼ ky ¼ kz ¼ 0), all the off-diagonal elements of the modified 8  8 matrix becomes zero and the solution of the 16  16 Hamiltonian gives 4 set of equations corresponding to the Eþ and E levels of the conduction band (CB), heavy hole (HH), light hole (LH) and spin-orbit split-off (SO) bands. The reduction in band gap of GaSb1-x-yBixNy occurs due to the upward movement of the LH/HH Eþ sub band and the lowering of the conduction sub band CB E. The band gap for the quaternary alloy GaSbBiN as a function of Bi and N mole fraction is given by,

Pz ¼ Pkz

Z2 pffiffiffi 2 3 kz kx iky g3 2m0  i Z2 pffiffiffih 2 b¼ 3 kx k2y g2 2ikx ky g3 2m0

a¼

Eg ðx; yÞ ¼

D ¼ LH

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 GaSb Eg þ DECBM ðx; yÞ þ EN  ðC  EN Þ2 þ 4C 2N y 2  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1  DEVBM ðx; yÞ þ EBi  ðC  EBi Þ2 þ 4C 2Bi x 2

(4)

(7)

Eg is the unstrained band gap, Dso is the spin-orbit splitting energy and g1, g2 andg3 are the Luttinger parameters for GaSb. The valence band parameters (g1, g2, g3) used in the 8  8 Hamiltonian VB VB are not identical to the Luttinger parameters (gVB 1 ,g2 ,g3 ) used in the 6  6 Hamiltonian proposed by Alberi et al. [18] due to the inclusion of the conduction band in the 8  8 Hamiltonian [13]. The relation between the modified Luttinger parameters (g1, g2, g3) and VB VB the Luttinger parameters used in VBAC (gVB 1 ,g2 ,g3 ) is [25,28,29],

At the G point (k ¼ 0), the 16 band Hamiltonian reduces to a 6  6 Hamiltonian as given below:

g1 ¼

gVB 1

Ep  3Eg

g2 ¼ gVB 2  g3 ¼

gVB 3

Ep 6Eg

(5)

Ep  6Eg

P is the Kane matrix element which is defined in terms of energy units as [25,30],

Table 1 List of BAC parameters and their values used in this work. Parameters

Values used in the calculations (eV) structure

EBi EN EBi-so CBi CN

1.17 [27] 0.89 2.67 [27] 1.01 [27] 2.83 0.12

GaSbN DECBM GaSbBi DECBM GaSbN DEVBM GaSbBi DEVBM GaSbN DESO GaSbBi DESO

2.78 [27] 2.61 0.38 [27] 1.87 1.06

1690

D.P. Samajdar et al. / Current Applied Physics 16 (2016) 1687e1694

2

H66

0

C 6 0 6 6 0 ¼6 6 6 VN ðyÞ 4 0 0

0 . 0 H L 0 0 VBi ðxÞ 0 0

0 0 S 0 0 VBi ðxÞ

0

VN ðyÞ 0 0 EN 0 0 0

0 VBi ðxÞ 0 0 EBi 0

3 0 0 7 7 7 VBi ðxÞ 7 7 0 7 5 0 EBiso

hy

ECB ðεÞ ¼ ECB ðε ¼ 0Þ þ DECB (8)

0

where C ¼ EgGaSb þ DECBM ðx; yÞ and H ¼ L ¼ DEVBM ðx; yÞ. The solution of the above matrix yields six distinct eigen values given by the following three sets of equations:

EHH=LH± ¼

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 H0 þ EBi ± H 02  2H 0 EBi þ E2Bi þ 4VBi ðxÞ2 2

(9a)

ECB± ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    1 C 02  2C 0 EN þ E2N þ 4VN ðyÞ2 C 0 þ EN ± 2

(9b)

ESO± ¼

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 S þ EBiso ± S2  2SEBiso þ E2Biso þ 4VBi ðxÞ2 2

(9c)

1 hy EVBHH ðεÞ ¼ EVB ðε ¼ 0Þ þ DEVB  DES 2 1 hy EVBLH ðεÞ ¼ EVB ðε ¼ 0Þ þ DEVB þ DES 2

(13)





hy hy where DECB ¼ 2ac 1  cc12 ε and DEVB ¼ 2av 1  cc12 ε account for the 11 11 influence of hydrostatic component of strain on the band structure

and DES ¼ 2b 1 þ cc12 ε. The parameters ac and av are the hydrostatic 11 deformation potentials of the conduction and the valence bands, b is the axial deformation potential and ε denotes the relative lattice mismatch between GaSbBiN and GaSb. The parameters of the binary compound used in our calculations are taken from published results [35]. All the material parameters for GaSb1-x-yBixNy are obtained from the linear interpolation of the corresponding parameters of the parent binary component according to the equation,

2.2. Calculation of the band offsets The lattice constant of GaSb1xyBixNy can be obtained using Vegard's law as a linear interpolation of the lattice constants of the endpoint compounds [23,25].

aGaSb1xy Bix Ny ¼ ð1  x  yÞaGaSb þ xaGaBi þ yaGaN

(10)

Using the relationship (8), it has been calculated that GaSb1xy BixNy can be lattice-matched to GaSb for a Bi to N concentration ratio x/y of 0.146. Values of aGaBi, aGaN and aGaSb are taken in our calculations as 6.324 Å [31], 4.50 Å [30] and 6.095 Å [30], respectively. Using the VCA method, the conduction and the valence band edges for GaSb1-x-yBixNy can be calculated as [32],

EC ðx; yÞ ¼ EC ðyÞ þ DEC ðx; yÞ EV ðx; yÞ ¼ EV ðxÞ þ DEV ðx; yÞ

(11)

EC(y) and EV(x) respectively denote the CBM and the VBM levels due to the incorporation of N and Bi calculated using Conduction Band Anticrossing (CBAC) model [34] and VBAC model [18,34]. The band gap of GaSb1xyBixNy can be calculated using a combination of VBAC and CBAC model using the relation [32],

Eg ðx; yÞ ¼ EC ðx; yÞ  EV ðx; yÞ

(12)

and the conduction band and valence band offset ratios may respectively be obtained from the relations [32],QC ¼ (DEC/DEg) and QV ¼ (DEV/DEg) where DEg ¼ jEg(x,y)Eg(GaSb)j

2.3. Calculation of strain The strain induced shift of the conduction band minimum and the valence band maximum in strained GaSbBiN on GaSb was also taken into account in our calculation. Both the conduction band and the valence band get shifted by the hydrostatic component (DEhy) of strain. The shear component (DES) of strain gives separate EVB-HH and EVB-LH by removing the valence band degeneracy [33],

Fig. 1. Electronic band structure of (a) GaSb with the relative energy positions of CB, HH, LH and SO bands and the N and Bi related energy levels EN and EBi and EBi-so respectively (b) GaSbBiN lattice matched to GaSb substrate showing the splitting of CB, LH/HH and SO bands, calculated using a 16 band k$p Hamiltonian in the k directions near the G point.

D.P. Samajdar et al. / Current Applied Physics 16 (2016) 1687e1694

QGaSb1xy Bix Ny ¼ ð1  x  yÞQGaSb þ xQGaBi þ yQGaN

1691

(14)

3. Results and discussions We have calculated the band structure of GaSbBiN alloy, lattice matched to GaSb substrates, using the 16  16 kp Hamiltonian discussed earlier. The solution of the Hamiltonian yields eight doubly degenerate eigen values corresponding to the EHH±, ELH±, ESO± and ECB± energy levels. Fig. 1(a) shows the band structure of GaSb GaSb0.954Bi0.04N0.006 in the D and L directions near the G point along with the relative positions of the impurity levels EN, EBi and EBi-so. The interaction of the impurity level with ECB splits into two distinct energy bands ECBþ and ECB- to accommodate the electrons as occurring in a crystal following Pauli's Exclusion Principle. Similar splitting of the HH, LH and SO bands occur due to the interaction of the host valence band matrix with the impurity levels EBi and EBi-so. Fig. 1(b) shows the band structure of GaSb0.954Bi0.04N0.006 in the D and L directions showing the splitting of the energy bands of the host into Eþ and E sub bands. Clearly, the band diagram reveals the reduction in band gap of GaSbBiN alloy due to the downward movement of the ECB- sub band and the upward movement of the EHH/LHþ sub band with respect to the CB and HH/LH bands of GaSb. The decrease in the ECB- energy level by 0.256 eV and the increase in the VBM by 0.032 eV leads to a large band gap reduction of 0.288 eV. The spin orbit splitting energy DSOþ increases by about 0.071 eV. Increase in the value of DSOþ by about 17.5 meV/at%Bi is close to the value of 21 meV/%Bi obtained for GaSbBi [27]. The band structure for GaSbBiN/GaSb alloy is calculated at a temperature of 10 K considering the band gap value of GaSb as 0.812 eV and Bi and N concentrations of 4.0 and 0.6 at% respectively. The band gap reduction by 0.288 eV corresponds to a band gap of 0.524 eV for GaSb0.954Bi0.04N0.006. Fig. 2 represents the relative energy positions of the CB, HH/LH and SO sub bands as a function of Bi mole fraction under latticematched condition at the G point. The values of the energy levels ECB±, EHH/LH± and ESO± are calculated using Eq. (9a)e(9c). The upward and the downward movements of the corresponding Eþ and E- sub bands may be interpreted from a careful inspection of the deduced set of equations. As is observed from the Fig. 2, there is a repulsive interaction between the Eþ and E energy levels, thereby pushing the coupled bands away from each other. Band gap reduction of about 104 meV/%Bi is observed in GaSbBiN due to the upward and downward movement of EHH/LHþ and ECB- by about 8.4 meV and 95.4 meV/%Bi, respectively. The decrease in the calculated value of ECB- is much greater in comparison to the increase in the value of EHH/LHþ and it contributes to ~92% of band gap reduction in GaSbBiN alloys. This is due to the large conduction band offset of 2.78 eV for GaSbBi formed due to Bi alloying in GaSb unlike the valence band offset value of only 0.38 eV as can be observed from Table 1. Out of the total reduction in ECB- by 95.4 meV/%Bi, conventional alloying due to Bi 6s states located below the CBM causes a CBM reduction of 27.8 meV/%Bi. Almost similar value was obtained by Polak et al. [36] for GaSbBi. The balance 67.6 meV/%Bi reduction in ECB- is made up by the band anticrossing (BAC) interaction of the N related impurity states with the host conduction band matrix. The increase in the values of VN(y) and VBi(x) with the increase in Bi and N mole fraction increases the downward movement of ECB- and the upward movement of EHH/LHþ and hence enhances the band gap reduction. While the location of the EHH/LH- and ESO- energy levels is dependent on the impurity levels EBi and EBi-so, the location of ECB- level is dependent on EN.

Fig. 2. Relative positions of the Eþ and E energy levels corresponding to the CB, HH/ LH and SO bands for GaSb1xyBixNy.

Fig. 3 presents the variation of band gap energy with Bi mole fraction using two different theoretical models for GaSb11.146xBixN0.146x. The values of Eg using the 16 band kp model are calculated with the help of Eq. (7) whereas Eq. (12) is used for the calculation of band gap, combining the effects predicted by the VBAC and the CBAC models. For lower values of Bi and N concentrations, the two values of Eg are almost equal to each other but deviates for higher amounts of Bi and N in the material. This is due to the fact that, to some extent, both Bi and N respectively, perturb the conduction and the valence bands, the effect of which is not taken into consideration in the VBAC or CBAC models, taken alone. Hence, for a Bi concentration of 0.5 at% (N ¼ 0.07 at%), a higher value of band gap reduction of 407.4 meV is obtained using the 16 band k$p model in comparison to the value 338.3 meV calculated using a combination of the VBAC and the CBAC models. Moreover,

Fig. 3. Variation of band gap energy and spin-orbit splitting energy as a function of Bi mole fraction for GaSbBiN, lattice matched to GaSb. Inset shows the variation of strain with percentage Bi and N incorporation.

1692

D.P. Samajdar et al. / Current Applied Physics 16 (2016) 1687e1694

Fig. 4. Variation of (a) conduction band offset (CBO), (b) valence band offset (VBO), (c) conduction band offset ratio and (d) valence band offset ratio for GaSbBiN as function of Bi mole fraction (x) for five different values of N mole fraction in the material. Dashed lines are for the lattice matched condition (x/y ¼ 0.146).

an enhancement in spin-orbit splitting energy is also observed in GaSbBiN similar to that in GaSbBi [27,36] and in GaAsBi [17,37] and it may be observed that for a Bi concentration of 0.4 at%, DSO becomes greater than Eg which might manifest into a suppression of the Auger recombination and the Inter Valence Band Absorption (IVBA) processes in antimonide based semiconductor lasers and photodetectors. The inset in Fig. 3 shows the amount of strain generated due to the incorporation of N and Bi in GaSb. The small amount of strain in GaSbBiN alloys grown on GaSb is due to the strain compensation between GaSbN and GaSbBi. Unlike the smaller N atoms which give rise to tensile strain, the larger Bi atoms in GaSb lead to compressive strain, thereby compensating strain due to the incorporation of both Bi and N atoms [22].

Calculation of the band parameters, such as, CBO, VBO, CBO and VBO ratios, band gap energy and spin-orbit splitting-energy is essential from the practical aspects of device design. Fig. 4(a)e(d) show the plots of the band offsets and their ratios as functions of Bi mole fractions, drawn for five different values of N contents. Both the VBO and the CBO for GaSbBiN increase with the increase in Bi and N mole fraction. The CBO is found to be almost ten times greater than the VBO which could provide better electron confinement in optoelectronic devices fabricated using GaSbBiN alloys [32]. It can be seen from the plots that the Bi concentration is almost ten times greater as compared to the N concentration due to the fact that a Bi:N concentration ratio of at least 6.85:1 is to be maintained for lattice-matched condition. From Table 1, it may be

Fig. 5. Predicted band gap values of GaSb1-x-yBixNy as functions of Bi and N mole fractions obtained using (a) CBAC þ VBAC Model (b) 16 band k$p model.

D.P. Samajdar et al. / Current Applied Physics 16 (2016) 1687e1694

observed that the CBO for GaSbBi is much greater than that for GaSbN which coupled with the higher value of Bi concentration produces a large CBO for GaSbBiN, unlike the VBO which is much lower compared to the CBO due to the lower value of VBO for GaSbBi. Further, in contrast with the linear increase in both CBO and VBO with Bi mole fraction, there is a non-linear decrease and increase of the CBO and VBO ratios QC and QV respectively. Indeed, this occurs due to the non-linear dependence of Eg on Bi mole fraction as is evident from Fig. 3. Such opposite characteristics of QC and QV may be attributed to the greater contribution of the CBAC calculated conduction band lowering to the total reduction in band gap. The calculated values of the band gap of GaSb1-x-yBixNy with variations in both Bi and N mole fractions are shown in Fig. 5. Fig. 5(a) indicates the N mole fractions required to maintain a band gap in the range 0.7e0.1 eV with increments of 0.1 eV. The values of N mole fraction for a particular band gap and variable Bi mole fraction is calculated using a combination of VBAC and CBAC models. It can be observed from the figure that to maintain a band gap value of 0.1 eV for GaSbBiN, y ¼ 0.032 and x ¼ 0.05 or y ¼ 0.046 and x ¼ 0 is required. Fig. 5(b) shows almost a similar plot as that in Fig. 5(a) with the exception that the values of N mole fractions are obtained utilizing Eq. (7) for a constant value of Eg and Bi mole fraction varying in the range 0.01e0.05. In this case, Eg ¼ 0.1 eV can be obtained for y ¼ 0.051 and x ¼ 0 or y ¼ 0.03 and x ¼ 0.05. So, a lower percentage of nitrogen is required to obtain the same band gap value in the presence of Bi as compared to the former (CBAC þ VBAC) model indicating the influence of Bi on both CB and VB as is predicted by the 16 band k$p model. The predicted band offsets for conduction band (DEC), light hole (DEvlh) and heavy hole (DEvhh) valence bands of GaSbBiN, are shown in Fig. 6 as functions of Bi content for some discrete values of N compositions. Large band offsets are required to trap electrons and holes and prevent their leakage [22]. Large CBO of the order of 0.4 eV can be achieved for Bi and N compositions of 5 at% and 3.4 at % respectively. With N and Bi compositions of 5 at%, it can be observed from Fig. 7 that DEC is greater than DEvlh and DEvhh. The maximum values of DEC, DEvlh and DEvhh can reach 0.33 eV, 0.22 eV and 0.12 eV respectively at xBi ¼ yN ¼ 0.05. Thus the incorporation of N and Bi in GaSb helps to achieve a precise control over the band offsets. Larger values of band offsets provide better carrier

1693

Fig. 7. Predicted values of band offsets for conduction band (DEC), light hole (DEvlh) and heavy hole (DEvhh) valence bands of GaSbBiN on GaSb substrate for various Bi and N compositions.

confinement by suppressing carrier leakage which improves the high-temperature efficiency of light emitting devices [22]. 4. Conclusions We have calculated the band structure of GaSb1-x-yBixNy quaternary alloy, lattice matched to GaSb substrates, using a 16 band kp Hamiltonian. The anticrossing interactions of the conduction and valence bands with the N and Bi related impurity levels cause a large band gap reduction by about 288 meV for GaSb0.954Bi0.04N0.006. The spin orbit splitting energy increases by about 71 meV and a Bi concentration of 0.5 at% results in the formation of DSO > Eg regime. The band gap values for GaSbBiN are calculated and compared using two theoretical models of 16 band k$p Hamiltonian and a combination of conduction band anticrossing CBAC) and valence band anticrossing (VBAC) models. A higher value of band gap reduction is predicted using kp method as compared to that obtained using a combination of VBAC and CBAC models. This difference is supposed to be due to the slight perturbation of the valence and the conduction bands by N and Bi respectively which is not taken into consideration in the latter method. Conduction band and valence band offsets are also calculated as functions of Bi and N mole fractions. CBO as large as 0.4 eV can be achieved for Bi and N composition of 5 at% and 3.4 at% respectively which would help in getting improved carrier confinements. Calculated values of DEC is found to be greater than the light hole and heavy hole related valence band offsets DEvlh and DEvhh. References

Fig. 6. Predicted values of band offsets for conduction band (DEC), light hole (DEvlh) and heavy hole (DEvhh) valence bands for GaSbBiN as functions of both Bi and N mole fractions.

[1] A. Chandola, H.J. Kim, P.S. Dutta, S. Guha, L. Gonzalez, V. Kumar, J. Appl. Phys. 98 (2005) 093103. [2] A. Chandola, R. Pino, P.S. Dutta, Semicond. Sci. Technol. 20 (2005) 886. [3] J.J. Mudd, N.J. Kybert, W.M. Linhart, L. Buckle, T. Ashley, P.D.C. King, T.S. Jones, M.J. Ashwin, T.D. Veal, Appl. Phys. Lett. 103 (2013) 042110. [4] F. Ning, D. Wang, L-M. Tang, Y. Zhang, K.-Q. Chen, J. Appl. Phys. 116 (2014) 094308. [5] F. Ning, L-M. Tang, Y. Zhang, K.-Q. Chen, Sci. Rep. 5 (10813) (2015) 1, http:// dx.doi.org/10.1038/srep10813. [6] A. Mondal, T.D. Das, N. Halder, S. Dhar, J. Kumar, J. Cryst. Growth 297 (2006) 4. [7] P.H. Jeffereson, et al., Appl. Phys. Lett. 89 (111921) (2006). [8] T.D. Veal, L.F.J. Piper, S. Jollands, B.R. Bennett, P.H. Jefferson, P.A. Thomas, C.F. McConville, B.N. Murdin, L. Buckle, G.W. Smith, T. Ashley, Appl. Phys. Lett. 87 (2005) 132101.

1694

D.P. Samajdar et al. / Current Applied Physics 16 (2016) 1687e1694

[9] D. Wang, S.P. Svensson, L. Shterengas, G. Belenky, C.S. Kim, I. Vurgaftman, J.R. Meyer, J. Appl. Phys. 105 (2009) 014904. [10] J. Kopaczek, M.K. Rajpalke, W.M. Linhart, T.S. Jones, M.J. Ashwin, R. Kudrawiec, T.D. Veal, Appl. Phys. Lett. 105 (2014) 112102. [11] S.K. Das, T.D. Das, S. Dhar, M. de la Mare, A. Krier, Infrared Phys. Technol. 55 (2012) 156. [12] S.K. Das, T.D. Das, S. Dhar, Semicond. Sci. Technol. 29 (2014) 015003. [13] M.K. Rajpalke, W.M. Linhart, M. Birkett, K.M. Yu, D.O. Scanlon, J. Buckeridge, T.S. Jones, M.J. Ashwin, T.D. Veal, Appl. Phys. Lett. 103 (2013) 142106. [14] J. Kopaczek, R. Kudrawiec, W. Linhart, M. Rajpalke, T. Jones, M. Ashwin, T. Veal, Appl. Phys. Exp. 7 (2014) 111202. [15] M.P. Polak, P. Scharoch, R. Kudrawiec, Semicond. Sci. Technol. 30 (2015) 094001. [16] M. Usman, C.A. Broderick, A. Lindsay, E.P. O'Reilly, Phys. Rev. B 84 (2011) 245202. [17] K. Alberi, O.D. Dubon, W. Walukiez, K.M. Yu, K. Bertulis, A. Kroktus, Appl. Phys. Lett. 91 (2007) 051909. [18] K. Alberi, J. Wu, W. Walukiewicz, K.M. Yu, O.D. Dubon, S.P. Watkins, C.X. Wang, X. Liu, Y.J. Cho, J. Furdyna, Phys. Rev. B 75 (2007) 045203. [19] J. Kopaczek, R. Kudrawiec, W.M. Linhart, M.K. Rajpalke, K.M. Yu, T.S. Jones, M.J. Ashwin, J. Misiewicz, T.D. Veal, Appl. Phys. Lett. 103 (2013) 261907. [20] W. Huang, K. Oe, G. Feng, M. Yoshimoto, J. Appl. Phys. 98 (2005) 053505. [21] S. Tixier, S.E. Webster, E.C. Young, T. Tiedje, S. Francoeur, A. Mascarenhas, P. Wei, F. Schiettekatte, Appl. Phys. Lett. 86 (2005) 112113.

[22] S.J. Sweeney, S.R. Jin, J. Appl. Phys. 113 (2013) 043110. [23] M.M. Habchi, A. Ben Nasr, A. Rebey, B. El Jani, Infrared Phys. Technol. 61 (2013) 88. [24] M.J. Ashwin, D. Walker, P.A. Thomas, T.S. Jones, T.D. Veal, J. Appl. Phys. 113 (2013) 033502. [25] S.T. Ng, W.J. Fan, Y.X. Dang, S.F. Yoon, Phys. Rev. B 72 (2005) 115341. [26] W. Shan, W. Walukiewicz, J.W. Ager, E.E. Haller, J.F. Geisz, D.J. Friedman, J.M. Olson, S.R. KurtzPhys, Rev. Lett. 82 (1999) 1221. [27] D.P. Samajdar, T.D. Das, S. Dhar, Mater. Sci. Semicond. Process 40 (2015) 539. [28] M. Gladysiewicz, R. Kudrawiec, M.S. Wartak, J. Appl. Phys. 118 (2015) 055702. [29] C.R. Pidgeon, R.N. Brown, Phys. Rev. 146 (1966) 575. [30] I. Vurgaftman, J.R. Mohan, L.R. Ram-Mohan, J. Appl. Phys. 89 (2001) 5815. [31] S. Francoeur, M.J. Seong, A. Mascarenhas, S. Tixier, M. Adamcyk, T. Tiedje, Appl. Phys. Lett. 82 (2003) 3874. [32] S. Nacer, A. Aissat, K. Ferdjani, Opt. Quant. Electron. 40 (2008) 677. [33] F. Bousbih, S.B. Bouzid, R. Chtourou, F.F. Charfi, J.C. Harmand, G. Ungaro, Mater. Sci. Eng. C 21 (2002) 251. [34] D.P. Samajdar, S. Dhar, Sci. World. J. 2014 (2014) 704830. [35] D.P. Samajdar, S. Dhar, Superlattices Microstruct. 89 (2016) 112. [36] M.P. Polak, P. Scharoch, R. Kudrawiec, J. Kopaczek, M.J. Winiarski, W.M. Linhart, M.K. Rajpalke, K.M. Yu, T.S. Jones, M.J. Ashwin, T.D. Veal, J. Phys. D Appl. Phys. 47 (2014) 355107. [37] L. Bhusal, A. J Ptak, R. France, A. Mascarenhas, Semicond. Sci. Technol. 24 (2009) 035018.