- Email: [email protected]
Gain characteristics of ideal dilute nitride quantum well lasers
Physica E 13 (2002) 1102 – 1105
www.elsevier.com/locate/physe
Gain characteristics of ideal dilute nitride quantum well lasers S. Tomi$c ∗ , E.P. O’Reilly1 Department of Physics, University of Surrey, Guildford, Surrey GU2 7XH, UK
Abstract We use a realistic Hamiltonian to compare the gain characteristics of an ideal InGaAsN=GaAs quantum well laser structure emitting at 1:3 m with an equivalent N-free InGaAs=GaAs structure. The energy gap of InGaAs is reduced by the addition of N, due to a repulsive interaction between an N resonant band and the conduction band edge. This interaction increases the conduction band edge e1ective mass and decreases the value of the dipole matrix element linking the conduction and valence band edges. We 4nd that the addition of N reduces the peak gain and di1erential gain at 4xed carrier density, although the gain saturation value and the peak gain as a function of radiative current density are largely unchanged due to the incorporation of N. ? 2002 Elsevier Science B.V. All rights reserved. Keywords: Long-wavelength lasers; InGaAsN
Recently, dilute nitride III–V compounds have been proposed as a new class of material for the realisation of semiconductor quantum well (QW) laser diodes emitting at the 1:3 m optical window. It has been found that replacing a small amount of the group V element by nitrogen in a III–V compound reduces the energy gap and dramatically changes the electronic structure [1–3], thus o1ering a new route to band structure engineering and improved optoelectronic properties. In particular, the growth of strained InGaAsN=GaAs QW structures allows the bene4ts of compressive strain and of growth on a GaAs substrate. Such quantum wells can be incorporated into vertical cavity surface emitting laser (VCSEL) structures, using AlGaAs=GaAs Bragg mirrors with high refractive index contrast to achieve electrically pumped monolithic VCSELs at telecommunication wavelengths ∗
Corresponding author. Tel.: +44-1485-689402; fax: +441485-686781. E-mail address: [email protected] (S. Tomi$c). 1 Now at: NMRC, Lee Maltings, Prospect Row, Cork, Ireland.
[4 – 6]. In addition, the inclusion of N in the InGaAsN layers increases the conduction band o1set, leading to improved electron con4nement and decreased electron spill out at room temperature and above, when compared with conventional InGaAsP 1:3 m lasers. Although edge-emitting and VCSEL laser structures with impressive characteristics have already been reported [4 –7] the inFuence of N on the electronic structure and gain characteristics has not yet been fully elucidated. We present here a theoretical analysis of the gain as a function of carrier density and radiative current density in ideal InGaAsN=GaAs QW structures. We highlight how the N-induced changes in band structure modify the gain characteristics by comparing a 7 nm wide In0:36 Ga0:64 As0:98 N0:02 =GaAs QW structure, with a model calculation of a 7 nm “Inx Ga1−x As”=GaAs QW in which we retain the same valence band o1set and built-in strain, but arti4cially shift the conduction band edge down in energy to maintain the same transition energy between the lowest conduction (e1) and highest heavy-hole (hh1) valence state.
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 2 ) 0 0 3 1 3 - 2
S. Tomi$c, E.P. O’Reilly / Physica E 13 (2002) 1102 – 1105
a higher-lying nitrogen resonant band [1]. We have shown that it is necessary to introduce a modi4ed 10-band k · P Hamiltonian to describe the band dispersion in these compounds, adding two spin-degenerate nitrogen states to the conventional 8-band k · P Hamiltonian to account for the band-gap reduction and modi4ed conduction band dispersion [10]. The valence band o1set was calculated as follows: we 4rst used model solid theory [11] to determine the relative band alignment, Evo1set , of unstrained GaAs and In0:36 Ga0:64 As. The e1ect of replacing a fraction x of As by N was calculated following a similar procedure to Klar et al. [12], with the unstrained alloy conduction and valence band edges, Ecw and Evw assumed to vary with x as
1.60 1.50
e3 1.40
Energy [eV]
1.30
e2
1.20
e1
1.10 0.20
hh1
0.15
hh2
0.10
lh1
0.05 0.00 -0.05 0.00
1103
lh2
hh3
Ecw = Ecb − (Egb − Egh − Evo1set ) − x + x;
(1)
Evw = Evb + Evo1set + x
(2)
and the N resonant state energy, ENw(b) set at 0.02
0.04
0.06
0.08
0.10
k|| [1/Å] Fig. 1. The band structure of a 7 nm wide In0:36 Ga0:64 As0:98 N0:02 = GaAs QW (solid line), and the equivalent N-free structure (dashed line).
Both structures were assumed to have the same built-in compressive strain, with the lattice-mismatch xx ≈ 2:1%. Fig. 1 shows that the valence band dispersion is virtually identical in the two structures considered, so that any changes in the calculated gain characteristics are then due almost entirely to the inFuence of N on the conduction band structure. Previous gain calculations have compared an InGaAsN=GaAs 1:3 m laser with a lattice-matched InGaAsP=InP laser structure [8], showing comparable variation of the peak material gain as a function of carrier density in the two structures considered. These calculations con4rmed the potential bene4ts of dilute nitride alloys but, because many factors varied between the two structures considered, did not identify the speci4c inFuence of N on the gain characteristics. The energy gap initially decreases rapidly when a small fraction of group V atoms are replaced by nitrogen [9]. This reduction (of about 0:1 eV=% of N for x ¡ ∼0:03) occurs because of an anti-crossing interaction between the conduction band edge and
ENw(b) = Ecw(b) + 0:485[eV] − x + x;
(3)
where the superscript w(b) refers to the well (barrier) layers, respectively. Finally we shifted the well conduction and valence band edge energies using interpolated alloy deformation potentials to include the e1ects of the built-in hydrostatic and axial strain in the quantum well. The N-related band o1set parameters, , and were chosen as = 1:5 eV, = 3:5 eV, and = 3:5 eV [12], while the coupling parameter, VNc between the N resonant level and the conduction √ band edge was assumed to vary as VNc = 1:675 x eV. This value was chosen by using a weighted average of the theoretically calculated coupling parameters for all di1erent possible group III nearest neighbour environments [2]. The resulting heavy-hole valence band o1set used in the calculations was then 0:213 eV while the conduction band o1set assumed in the “InGaAs”=GaAs model structure was 0:345 eV. We note that the parameters used here reproduce very well the ground and excited state transition energies deduced from photomodulated reFectance spectroscopy on a series of InGaAsN=GaAs QW structures [13]. The coupling between the N level and the conduction band edge modi4es the conduction band edge wave function and reduces the interband transition matrix element |MT |2 compared to an N-free conventional
1104
S. Tomi$c, E.P. O’Reilly / Physica E 13 (2002) 1102 – 1105 7000
0.9 0.8
Peak Material Gain [1/cm]
Transition Matrix Elements in |M0|
2
1.0
TE: e1-hh1
0.7 0.6 0.5 0.4 0.3 0.2
TM: e1-hh1
5000 4000 3000 2000
200 [ K]
1000
300 [ K]
0
400 [ K]
-1000 0.0
0.1 0.0 0.00
6000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
18
18
18
18
19
19
2.0x10 4.0x10 6.0x10 8.0x10 1.0x10 1.2x10 3
N [1/cm ]
(a)
k|| [1/Å]
Peak Material Gain [1/cm]
7000
Fig. 2. The transition matrix elements for an In0:36 Ga0:64 As0:98 N0:02 =GaAs QW (solid line), and an equivalent N-free structure (dashed line). The matrix elements are given in units of |M0 |2 = m0 Ep =2, and Ep = 26:3 eV is the Kane energy.
6000 5000 4000 200 [ K]
3000
300 [ K]
III–V alloy. Fig. 2 shows the calculated variation as a function of in-plane wave vector k for the TE and TM interband matrix elements linking the 4rst con4ned electron (e1) and heavy-hole (hh1) states. The band edge, zone-centre, TE matrix element is calculated to decrease by ∼30% due to the incorporation of N. The nitrogen–conduction band coupling therefore leads to an increased conduction band edge e1ective mass. Using the density of states e1ective mass formula: 1 1 dEi (k ) = (4) m∗ ˝2 k d k
Fig. 3. (a) The temperature dependence of the peak material TE gain in a 7 nm In0:36 Ga0:64 As0:98 N0:02 =GaAs QW as a function of carrier density (solid line). Dotted lines are the same dependence for an equivalent nitrogen-free structure. (b) The temperature dependence of the peak material TE gain as a function of radiative current density. Solid lines represent a 2% nitrogen and dotted lines a nitrogen-free structure.
we calculate from Fig. 1 that the band edge e1ective mass in the 4rst conduction subband increases to 0:060m0 in the structure with 2% N compared to a value of 0:046m0 in the N-free case. This increase of about ∼30% con4rms that the product m∗e × |MT |2 stays approximately constant, as would be expected from k · P theory: the dominant contribution to the bulk conduction band inverse e1ective mass, m∗−1 , is c directly proportional to |MT |2 . The band structure and matrix elements presented in Figs. 1 and 2 were used to calculate the variation of material gain with temperature T and as a function of carrier density N and of radiative current density Jrad in both structures. We 4rst determined the electron and hole quasi-Fermi energies, EFc and EFv , as a function of N and T . The increase in the conduction
band e1ective mass leads to an increase in the carrier concentration at transparency for the InGaAsN structure and a decrease in the quasi-Fermi energy separation EFc − EFv for a 4xed carrier concentration. As a consequence the peak gain decreases at a 4xed carrier density when compared to an equivalent conventional III–V structure, Fig. 3(a). The gain and spontaneous emission spectra were calculated using density matrix theory [14], including Lorentzian broadening, with the intraband relaxation time in assumed to have a constant value, in = 0:1 ps. The variation of the radiative current density Jrad with N was then found by integrating over the spontaneous emission spectrum at each value of N . Fig. 3(a) shows the calculated peak gain as
i
k =0
2000
400 [ K]
1000 0 -1000 0
100
200
300
400
2
(b)
Jrad [A/cm ]
S. Tomi$c, E.P. O’Reilly / Physica E 13 (2002) 1102 – 1105
a function of carrier density and temperature. It is clear that the transparency carrier density increases and the peak gain decreases at 4xed carrier density in the InGaAsN structure as compared to the equivalent conventional “InGaAs” structure, because of the increased conduction band edge e1ective mass in the InGaAsN structure. There is however a much weaker variation in the joint optical density of states and hence in the calculated absorption spectrum (E) for low carrier density (the case of a 4lled valence band and empty conduction band). This weak variation reFects the fact that (Eg ) ˙ m∗r × |MT |2 ≈ const:, where m∗r is the reduced band edge e1ective mass: 1=m∗r = 1=m∗c + 1=m∗v . Likewise, because of the reduced interband matrix element, there is also little di1erence in the calculated gain as a function of radiative current density for the two structures considered, and the gain saturates at a similar level in the N-based and equivalent conventional QW structures considered (Fig. 3(b)). The increased conduction band non-parabolicity has little inFuence on the temperature dependence of the transparency carrier density for the dilute nitride structure considered, compared to the equivalent conventional structure. In summary, the addition of N reduces the peak gain and di1erential gain at 4xed carrier density in an ideal dilute nitride QW structure when compared to an equivalent conventional structure. However, the gain saturation value and the peak gain as a function of radiative current density are largely unchanged due to the incorporation of N, opening the possibility of optoelectronic devices based on GaAs at the technologically important wavelengths of 1.3 and 1:55 m.
1105
References [1] W. Shan, W. Walukiewicz, J.W. Ager III, E.E. Haller, J.F. Geisz, D.J. Friedman, J.M. Olson, S.R. Kurtz, Phys. Rev. Lett. 82 (1999) 1221. [2] A. Lindsay, E.P. O’Reilly, in: N. Miura, T. Ando (Eds.), Proceedings of 25th ICPS, Proceedings in Physics, Vol. 87, Springer, Berlin, 2001, p. 63. [3] B.N. Murdin, M. Kamal-Saadi, A. Lindsay, E.P. O’Reilly, A.R. Adams, G.J. Natt, J.G. Crowder, C.R. Pidgeon, I.V. Bradley, J.P.R. Wells, T. Burke, A.D. Johnson, T. Ashley, Appl. Phys. Lett. 78 (2001) 1568. [4] K.D. Choquette, J.F. Klem, A.J. Fischer, O. Blum, A.A. Allerman, I.J. Fritz, S.R. Kurtz, W.G. Breiland, R. Sieg, K.M. Geib, J.W. Scott, R.L. Naone, Electron. Lett. 36 (2000) 1388. [5] G. Steinle, H. Riechert, A.Yu. Egorov, Electron. Lett. 37 (2001) 93. [6] T. Kageyama, T. Miyamoto, S. Makino, Y. Ikenaga, N. Nishiyama, A. Matsutani, F. Koyama, K. Iga, Electron. Lett. 37 (2001) 225. [7] H. Riechert, A.Yu. Egorov, D. Livshits, B. Borchert, S. Illek, Nanotechnology 11 (2000) 201. [8] J. Hader, S.W. Koch, J.V. Moloney, E.P. O’Reilly, Appl. Phys. Lett. 77 (2000) 630. [9] M. Kondow, K. Uomi, A. Niwa, T. Kitatani, S. Watahiki, Y. Yazawa, Jpn. J. Appl. Phys. 35 (1996) 1273. [10] E.P. O’Reilly, A. Lindsay, Phys. Stat. Sol. (b) 216 (1999) 131. [11] C.G. Van de Walle, Phys. Rev. B 39 (1989) 1871. [12] P.J. Klar, H. Gruning, W. Heimbrodt, J. Koch, W. Stolz, P.M.A. Vicente, A.M. Kamal-Saadi, A. Lindsay, E.P. O’Reilly, Phys. Stat. Sol. (b) 223 (2001) 163. [13] S.A. Choulis, B.A. Weinstein, T.J.C. Hosea, M. Kamal-Saadi, E.P. O’Reilly, A.R. Adams, W. Stolz, Phys. Stat. Sol. (b) 223 (2001) 151. [14] D. Ahn, S.L. Chuang, IEEE J. Quantum Electron. 26 (1990) 13.
www.elsevier.com/locate/physe
Gain characteristics of ideal dilute nitride quantum well lasers S. Tomi$c ∗ , E.P. O’Reilly1 Department of Physics, University of Surrey, Guildford, Surrey GU2 7XH, UK
Abstract We use a realistic Hamiltonian to compare the gain characteristics of an ideal InGaAsN=GaAs quantum well laser structure emitting at 1:3 m with an equivalent N-free InGaAs=GaAs structure. The energy gap of InGaAs is reduced by the addition of N, due to a repulsive interaction between an N resonant band and the conduction band edge. This interaction increases the conduction band edge e1ective mass and decreases the value of the dipole matrix element linking the conduction and valence band edges. We 4nd that the addition of N reduces the peak gain and di1erential gain at 4xed carrier density, although the gain saturation value and the peak gain as a function of radiative current density are largely unchanged due to the incorporation of N. ? 2002 Elsevier Science B.V. All rights reserved. Keywords: Long-wavelength lasers; InGaAsN
Recently, dilute nitride III–V compounds have been proposed as a new class of material for the realisation of semiconductor quantum well (QW) laser diodes emitting at the 1:3 m optical window. It has been found that replacing a small amount of the group V element by nitrogen in a III–V compound reduces the energy gap and dramatically changes the electronic structure [1–3], thus o1ering a new route to band structure engineering and improved optoelectronic properties. In particular, the growth of strained InGaAsN=GaAs QW structures allows the bene4ts of compressive strain and of growth on a GaAs substrate. Such quantum wells can be incorporated into vertical cavity surface emitting laser (VCSEL) structures, using AlGaAs=GaAs Bragg mirrors with high refractive index contrast to achieve electrically pumped monolithic VCSELs at telecommunication wavelengths ∗
Corresponding author. Tel.: +44-1485-689402; fax: +441485-686781. E-mail address: [email protected] (S. Tomi$c). 1 Now at: NMRC, Lee Maltings, Prospect Row, Cork, Ireland.
[4 – 6]. In addition, the inclusion of N in the InGaAsN layers increases the conduction band o1set, leading to improved electron con4nement and decreased electron spill out at room temperature and above, when compared with conventional InGaAsP 1:3 m lasers. Although edge-emitting and VCSEL laser structures with impressive characteristics have already been reported [4 –7] the inFuence of N on the electronic structure and gain characteristics has not yet been fully elucidated. We present here a theoretical analysis of the gain as a function of carrier density and radiative current density in ideal InGaAsN=GaAs QW structures. We highlight how the N-induced changes in band structure modify the gain characteristics by comparing a 7 nm wide In0:36 Ga0:64 As0:98 N0:02 =GaAs QW structure, with a model calculation of a 7 nm “Inx Ga1−x As”=GaAs QW in which we retain the same valence band o1set and built-in strain, but arti4cially shift the conduction band edge down in energy to maintain the same transition energy between the lowest conduction (e1) and highest heavy-hole (hh1) valence state.
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 2 ) 0 0 3 1 3 - 2
S. Tomi$c, E.P. O’Reilly / Physica E 13 (2002) 1102 – 1105
a higher-lying nitrogen resonant band [1]. We have shown that it is necessary to introduce a modi4ed 10-band k · P Hamiltonian to describe the band dispersion in these compounds, adding two spin-degenerate nitrogen states to the conventional 8-band k · P Hamiltonian to account for the band-gap reduction and modi4ed conduction band dispersion [10]. The valence band o1set was calculated as follows: we 4rst used model solid theory [11] to determine the relative band alignment, Evo1set , of unstrained GaAs and In0:36 Ga0:64 As. The e1ect of replacing a fraction x of As by N was calculated following a similar procedure to Klar et al. [12], with the unstrained alloy conduction and valence band edges, Ecw and Evw assumed to vary with x as
1.60 1.50
e3 1.40
Energy [eV]
1.30
e2
1.20
e1
1.10 0.20
hh1
0.15
hh2
0.10
lh1
0.05 0.00 -0.05 0.00
1103
lh2
hh3
Ecw = Ecb − (Egb − Egh − Evo1set ) − x + x;
(1)
Evw = Evb + Evo1set + x
(2)
and the N resonant state energy, ENw(b) set at 0.02
0.04
0.06
0.08
0.10
k|| [1/Å] Fig. 1. The band structure of a 7 nm wide In0:36 Ga0:64 As0:98 N0:02 = GaAs QW (solid line), and the equivalent N-free structure (dashed line).
Both structures were assumed to have the same built-in compressive strain, with the lattice-mismatch xx ≈ 2:1%. Fig. 1 shows that the valence band dispersion is virtually identical in the two structures considered, so that any changes in the calculated gain characteristics are then due almost entirely to the inFuence of N on the conduction band structure. Previous gain calculations have compared an InGaAsN=GaAs 1:3 m laser with a lattice-matched InGaAsP=InP laser structure [8], showing comparable variation of the peak material gain as a function of carrier density in the two structures considered. These calculations con4rmed the potential bene4ts of dilute nitride alloys but, because many factors varied between the two structures considered, did not identify the speci4c inFuence of N on the gain characteristics. The energy gap initially decreases rapidly when a small fraction of group V atoms are replaced by nitrogen [9]. This reduction (of about 0:1 eV=% of N for x ¡ ∼0:03) occurs because of an anti-crossing interaction between the conduction band edge and
ENw(b) = Ecw(b) + 0:485[eV] − x + x;
(3)
where the superscript w(b) refers to the well (barrier) layers, respectively. Finally we shifted the well conduction and valence band edge energies using interpolated alloy deformation potentials to include the e1ects of the built-in hydrostatic and axial strain in the quantum well. The N-related band o1set parameters, , and were chosen as = 1:5 eV, = 3:5 eV, and = 3:5 eV [12], while the coupling parameter, VNc between the N resonant level and the conduction √ band edge was assumed to vary as VNc = 1:675 x eV. This value was chosen by using a weighted average of the theoretically calculated coupling parameters for all di1erent possible group III nearest neighbour environments [2]. The resulting heavy-hole valence band o1set used in the calculations was then 0:213 eV while the conduction band o1set assumed in the “InGaAs”=GaAs model structure was 0:345 eV. We note that the parameters used here reproduce very well the ground and excited state transition energies deduced from photomodulated reFectance spectroscopy on a series of InGaAsN=GaAs QW structures [13]. The coupling between the N level and the conduction band edge modi4es the conduction band edge wave function and reduces the interband transition matrix element |MT |2 compared to an N-free conventional
1104
S. Tomi$c, E.P. O’Reilly / Physica E 13 (2002) 1102 – 1105 7000
0.9 0.8
Peak Material Gain [1/cm]
Transition Matrix Elements in |M0|
2
1.0
TE: e1-hh1
0.7 0.6 0.5 0.4 0.3 0.2
TM: e1-hh1
5000 4000 3000 2000
200 [ K]
1000
300 [ K]
0
400 [ K]
-1000 0.0
0.1 0.0 0.00
6000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
18
18
18
18
19
19
2.0x10 4.0x10 6.0x10 8.0x10 1.0x10 1.2x10 3
N [1/cm ]
(a)
k|| [1/Å]
Peak Material Gain [1/cm]
7000
Fig. 2. The transition matrix elements for an In0:36 Ga0:64 As0:98 N0:02 =GaAs QW (solid line), and an equivalent N-free structure (dashed line). The matrix elements are given in units of |M0 |2 = m0 Ep =2, and Ep = 26:3 eV is the Kane energy.
6000 5000 4000 200 [ K]
3000
300 [ K]
III–V alloy. Fig. 2 shows the calculated variation as a function of in-plane wave vector k for the TE and TM interband matrix elements linking the 4rst con4ned electron (e1) and heavy-hole (hh1) states. The band edge, zone-centre, TE matrix element is calculated to decrease by ∼30% due to the incorporation of N. The nitrogen–conduction band coupling therefore leads to an increased conduction band edge e1ective mass. Using the density of states e1ective mass formula: 1 1 dEi (k ) = (4) m∗ ˝2 k d k
Fig. 3. (a) The temperature dependence of the peak material TE gain in a 7 nm In0:36 Ga0:64 As0:98 N0:02 =GaAs QW as a function of carrier density (solid line). Dotted lines are the same dependence for an equivalent nitrogen-free structure. (b) The temperature dependence of the peak material TE gain as a function of radiative current density. Solid lines represent a 2% nitrogen and dotted lines a nitrogen-free structure.
we calculate from Fig. 1 that the band edge e1ective mass in the 4rst conduction subband increases to 0:060m0 in the structure with 2% N compared to a value of 0:046m0 in the N-free case. This increase of about ∼30% con4rms that the product m∗e × |MT |2 stays approximately constant, as would be expected from k · P theory: the dominant contribution to the bulk conduction band inverse e1ective mass, m∗−1 , is c directly proportional to |MT |2 . The band structure and matrix elements presented in Figs. 1 and 2 were used to calculate the variation of material gain with temperature T and as a function of carrier density N and of radiative current density Jrad in both structures. We 4rst determined the electron and hole quasi-Fermi energies, EFc and EFv , as a function of N and T . The increase in the conduction
band e1ective mass leads to an increase in the carrier concentration at transparency for the InGaAsN structure and a decrease in the quasi-Fermi energy separation EFc − EFv for a 4xed carrier concentration. As a consequence the peak gain decreases at a 4xed carrier density when compared to an equivalent conventional III–V structure, Fig. 3(a). The gain and spontaneous emission spectra were calculated using density matrix theory [14], including Lorentzian broadening, with the intraband relaxation time in assumed to have a constant value, in = 0:1 ps. The variation of the radiative current density Jrad with N was then found by integrating over the spontaneous emission spectrum at each value of N . Fig. 3(a) shows the calculated peak gain as
i
k =0
2000
400 [ K]
1000 0 -1000 0
100
200
300
400
2
(b)
Jrad [A/cm ]
S. Tomi$c, E.P. O’Reilly / Physica E 13 (2002) 1102 – 1105
a function of carrier density and temperature. It is clear that the transparency carrier density increases and the peak gain decreases at 4xed carrier density in the InGaAsN structure as compared to the equivalent conventional “InGaAs” structure, because of the increased conduction band edge e1ective mass in the InGaAsN structure. There is however a much weaker variation in the joint optical density of states and hence in the calculated absorption spectrum (E) for low carrier density (the case of a 4lled valence band and empty conduction band). This weak variation reFects the fact that (Eg ) ˙ m∗r × |MT |2 ≈ const:, where m∗r is the reduced band edge e1ective mass: 1=m∗r = 1=m∗c + 1=m∗v . Likewise, because of the reduced interband matrix element, there is also little di1erence in the calculated gain as a function of radiative current density for the two structures considered, and the gain saturates at a similar level in the N-based and equivalent conventional QW structures considered (Fig. 3(b)). The increased conduction band non-parabolicity has little inFuence on the temperature dependence of the transparency carrier density for the dilute nitride structure considered, compared to the equivalent conventional structure. In summary, the addition of N reduces the peak gain and di1erential gain at 4xed carrier density in an ideal dilute nitride QW structure when compared to an equivalent conventional structure. However, the gain saturation value and the peak gain as a function of radiative current density are largely unchanged due to the incorporation of N, opening the possibility of optoelectronic devices based on GaAs at the technologically important wavelengths of 1.3 and 1:55 m.
1105
References [1] W. Shan, W. Walukiewicz, J.W. Ager III, E.E. Haller, J.F. Geisz, D.J. Friedman, J.M. Olson, S.R. Kurtz, Phys. Rev. Lett. 82 (1999) 1221. [2] A. Lindsay, E.P. O’Reilly, in: N. Miura, T. Ando (Eds.), Proceedings of 25th ICPS, Proceedings in Physics, Vol. 87, Springer, Berlin, 2001, p. 63. [3] B.N. Murdin, M. Kamal-Saadi, A. Lindsay, E.P. O’Reilly, A.R. Adams, G.J. Natt, J.G. Crowder, C.R. Pidgeon, I.V. Bradley, J.P.R. Wells, T. Burke, A.D. Johnson, T. Ashley, Appl. Phys. Lett. 78 (2001) 1568. [4] K.D. Choquette, J.F. Klem, A.J. Fischer, O. Blum, A.A. Allerman, I.J. Fritz, S.R. Kurtz, W.G. Breiland, R. Sieg, K.M. Geib, J.W. Scott, R.L. Naone, Electron. Lett. 36 (2000) 1388. [5] G. Steinle, H. Riechert, A.Yu. Egorov, Electron. Lett. 37 (2001) 93. [6] T. Kageyama, T. Miyamoto, S. Makino, Y. Ikenaga, N. Nishiyama, A. Matsutani, F. Koyama, K. Iga, Electron. Lett. 37 (2001) 225. [7] H. Riechert, A.Yu. Egorov, D. Livshits, B. Borchert, S. Illek, Nanotechnology 11 (2000) 201. [8] J. Hader, S.W. Koch, J.V. Moloney, E.P. O’Reilly, Appl. Phys. Lett. 77 (2000) 630. [9] M. Kondow, K. Uomi, A. Niwa, T. Kitatani, S. Watahiki, Y. Yazawa, Jpn. J. Appl. Phys. 35 (1996) 1273. [10] E.P. O’Reilly, A. Lindsay, Phys. Stat. Sol. (b) 216 (1999) 131. [11] C.G. Van de Walle, Phys. Rev. B 39 (1989) 1871. [12] P.J. Klar, H. Gruning, W. Heimbrodt, J. Koch, W. Stolz, P.M.A. Vicente, A.M. Kamal-Saadi, A. Lindsay, E.P. O’Reilly, Phys. Stat. Sol. (b) 223 (2001) 163. [13] S.A. Choulis, B.A. Weinstein, T.J.C. Hosea, M. Kamal-Saadi, E.P. O’Reilly, A.R. Adams, W. Stolz, Phys. Stat. Sol. (b) 223 (2001) 151. [14] D. Ahn, S.L. Chuang, IEEE J. Quantum Electron. 26 (1990) 13.