- Email: [email protected]
Temperature dependence of 1.55 μm VCSELs
15 August 1998
Optics Communications 154 Ž1998. 43–46
Temperature dependence of 1.55 mm VCSELs J. Masum ) , N. Balkan, M.J. Adams Department of Physics, UniÕersity of Essex, Colchester CO4 3SQ, UK Received 26 February 1998; revised 14 May 1998; accepted 28 May 1998
Abstract The temperature for minimum threshold carrier concentration in 1.55 mm VCSELs can be significantly lower than that at which the peak gain matches the cavity resonance. A simple model is implemented to investigate the magnitude of this temperature difference and to aid the design of VCSELs for room temperature operation. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The development of 1.3 mm and 1.55 mm VCSELs has been slower than for shorter wavelength VCSELs, due to technological challenges posed by the InGaAsPrInP material system. Nevertheless there has been significant progress with 1.55 mm VCSELs exhibiting cw operation at room temperature using wafer fusion technology w1x and up to y258C with epitaxial InGaAsPrInP mirrors w2x. As the development of these devices continues, the need for theoretical modelling and simulation becomes important to optimise laser performance for specific applications. There already exist a number of sophisticated numerical models for VCSELs w3–5x. In the present contribution we use a simple model for optical gain to focus attention on the temperature sensitivity of the threshold current, with the objective of deriving design rules for 1.55 mm VCSELs. A plot of threshold current as function of temperature for VCSELs shows a roughly parabolic dependence with a minimum, which is ascribed to an optimum matching between the photon energy of the gain peak, EP , and the cavity resonance, E M w6,7x. However, modelling w8–10x reveals that the temperature at which the minimum threshold current is achieved can differ significantly from that at which the peak gain matches the cavity resonance. To investigate the intrinsic causes of this difference, attention is focused here on the effect of temperature on the threshold carrier density, n th , the energy gap, EG , and the resonant mode photon energy, E M .
2. Theory Two simple models for gain with no k-selection rule were implemented, one for VCSELs with bulk active region, and the other for quantum well devices. The threshold condition in bulk active region VCSELs is given using the approximation
)
E-mail: [email protected]
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 2 9 5 - 8
J. Masum et al.r Optics Communications 154 (1998) 43–46
44
Table 1 List of the parameters used for device modelling Symbol
Definition
Value
a B E Lz mc mv M N
Total loss Bimolecular recombination constant Photon energy
100 cmy1 1 = 10y1 0 cm3 sy1 0.8 eV
Quantum well width Effective mass of electron Effective mass of hole Number of wells Refractive index
˚ 100 A 0.047 0.63 9 3.4
due to Marinelli w11x as follows: Bc 2 2
4N p
ž
4p 2 m c m v h
2
3r2
DE
/ ž / E
2
sinh w Ž Fc q Fv y D E . r2 kT x cosh w Ž Fc q Fv y D E . r2 kT x q cosh w Ž Fc q Fv . r2 kT x
sa ,
Ž1.
where a is the total loss, i.e. the sum of all internal and end losses, Fc , Fv are the electron and hole quasi-Fermi levels, respectively, measured positively into the bands from the band edges, c is the speed of light, h is Planck’s constant, k is Boltzmann’s constant, T is the temperature, D E is the difference between E M and EG , and B is the bimolecular recombination constant which is assumed to be independent of temperature Ža weak temperature dependence has been reported w12x which does not affect the present results significantly.. The values of the parameters used are given in Table 1.
Fig. 1. Plot of Ža. threshold carrier density and Žb. bandgap, resonant mode and gain peak as functions of temperature for 1.55 mm VCSEL with bulk active region.
J. Masum et al.r Optics Communications 154 (1998) 43–46
45
In order to calculate the quasi-Fermi levels and hence the threshold carrier concentration, n th , Eq. Ž1. was used in conjunction with the charge neutrality condition using Joyce–Dixon approximation w13x, n
žž /
Fc s kT ln
1
n
'8
Nc
q
Nc
/
,
n
žž /
Fv s kT ln
1
n
'8
Nv
q
Nv
/
,
Ž2.
where n is the electron carrier density; Nc and Nv are the conduction and valence band effective densities of states respectively. Fig. 1 illustrates the calculated variation of n th , EG , E M and E P , with temperature for a bulk active region VCSEL for a s 100 cmy1. The temperature dependence of EG is taken to be the same as that of the InP bandgap w14x, and the experimental temperature variation of E M of 0.04 meVrK w15x is assumed. The relative offset between E M and EG is taken as a parameter in the model. If required, this can be related to the experimentally measurable offset between E M and the photoluminescence peak energy, as done for specific lasers in Ref. w10x. The value of Ž E M y EG . is defined in practice by the cavity length, and the results in Fig. 1 are for the case where this offset is zero at 220 K. It can be seen for this example that the minimum n th is achieved at Tmin of 290 K, while E M and E P match at Tmatch of 320 K. Similarly, the threshold condition in MQW VCSELs assuming recombination between the first sub-bands w8x is given by 2p BMc 2 m c m v kT 2
N E
2
hL2z
ln
w 1 q exp Ž FvrkT . xw 1 q exp ŽyFcrkT . x 1 q exp Ž Ž Fv y D E . rkT . 1 q exp Ž Ž D E y Fc . rkT .
sa ,
Ž3.
where M is the number of wells and Lz is the QW width. To calculate the threshold carrier density, n th , Eq. Ž3. was used in conjunction with charge neutrality condition, Eq. Ž4., including the first and second sub-bands which are separated by energies D En and D Ep w8x, 1 q exp
Fc
ž / kT
1 q exp
ž
Fc y D En kT
/
s 1 q exp
Fv
ž / kT
z
1 q exp
ž
Fv y D Ep kT
z
/
,
Ž4.
where z is the ratio of hole to electron mass Ž m vrm c .. Note that for the MQW case D E is the difference between E M and the effective bandgap, EGeff , allowing for the energy displacement of the first sub-bands.
3. Design rules A comparison between the results calculated for QW and bulk active region VCSELs is shown in Fig. 2; the QW devices have either 1 well or 9 wells, each 10 nm in width. Fig. 2 plots the temperature corresponding to minimum electron concentration, Tmin , against the temperature Tmatch at which E P matches E M . Each plot is calculated for the same value of loss a s 100 cmy1, B s 1 = 10y10 cm3 sy1 and is obtained by varying the relative offset between EM and EG Žbulk. or EGeff ŽMQW.. The results show the differences and similarities between the behaviour of VCSELs with bulk and MQW active regions. In each case the variation of Tmin with Tmatch is linear, but for the MQW case the values of these parameters are much closer Ž5–10 K. than is the case for bulk material where the difference can be as much as 30–40 K. This fundamental difference
Fig. 2. Plot of Tmin as function of Tmatch for bulk and QW VCSELs.
46
J. Masum et al.r Optics Communications 154 (1998) 43–46
between bulk and MQW VCSELs is a direct consequence of the step-like density of states for the latter, which causes the peak gain to occur at photon energies close to the bandgap. Hence, to design a device with minimum n th of 300 K, the cavity length should be adjusted to give Tmatch around 330 K and 310 K for bulk and MQW active regions respectively as shown in Fig. 2. The difference between Tmin and Tmatch increases with increasing loss a , and decreases with increasing number of quantum wells in the MQW case. Note that we assume here that the loss term, a , is independent of carrier concentration, i.e. we are deliberately omitting the effects of free-carrier absorption and inter-valence band absorption ŽIVBA.. In addition, the effects of band-gap renormalisation are also ignored. Also n th has not been converted into threshold current Ith . To accomplish the latter, it would be necessary to include in the model the effects of Auger and other non-radiative recombination, of carrier capture into the quantum wells and of leakage out of the wells. In addition, for cw threshold calculations it is necessary to take into account the heating effects, which in turn demands a structure-specific model. Although all these effects may contribute to the temperature dependence of threshold, they are neglected here in order to focus attention on the intrinsic causes of temperature sensitivity. Of the effects listed here we expect the dominant ones to be IVBA, which tends to increase Tmatch , and Auger recombination which tends to reduce Tmin . When these effects are taken into account, the superiority of MQWs over bulk VCSELs is further enhanced. Further details on these topics will be the subject of a future publication.
4. Conclusion A study has been made of the difference between the temperature for minimum threshold electron concentration and that at which the peak gain matches the cavity resonance in 1.55 mm VCSELs. Attention has been focused on the intrinsic causes of this difference, and underlying trends leading to design rules have been identified. The superiority of MQW over bulk active regions has been established, although more work remains to quantify the differences in terms of detailed modelling for both systems.
References w1x D.I. Babic, K. Streubel, R.P. Mirin, N.M. Margalit, J.E. Bowers, E.L. Hu, D.E. Mars, L. Yang, K. Carey, IEEE Photon. Technol. Lett. 7 Ž1995. 1225. w2x K. Streubel, S. Rapp, J. Andre, J. Wallin, IEEE Photon. Technol. Lett. 8 Ž1996. 1121. w3x G.R. Hadley, K.L. Lear, M.E. Warren, K.D. Choquette, J.W. Scott, S.W. Corzine, IEEE J. Quantum Electron. 32 Ž1996. 607. w4x J. Piprek, D.I. Babic, J.E. Bowers, J. Appl. Phys. 81 Ž1997. 3382. w5x J.W. Scott, D.B. Young, B.J. Thibeault, M.G. Peters, L.A. Coldren, IEEE J. Sel. Topics Quantum Electron. 1 Ž1995. 638. w6x B. Tell, K.F. Brown-Goebeler, R.E. Leibenguth, F.M. Baez, Y.H. Lee, Appl. Phys. Lett. 60 Ž1992. 683. w7x G. Goncher, B. Lu, W.-L. Luo, J. Cheng, S. Hersee, S.Z. Sun, R.P. Schneider, J.C. Zolper, IEEE Photon. Technol. Lett. 8 Ž1996. 316. w8x G.W. Taylor, P.A. Evaldsson, IEEE J. Quantum Electron. 30 Ž1994. 2262. w9x J. Piprek, SPIE 3003 Ž1997. 182. w10x S. Rapp, J. Piprek, K. Streubel, J. Andre, J. Wallin, J. Quantum Electron. 33 Ž1997. 1839. w11x F. Marinelli, Solid State Electron. 8 Ž1965. 939. w12x A. Haug, Appl. Phys. B 44 Ž1987. 151. w13x W.B. Joyce, R.W. Dixon, Appl. Phys. Lett. 31 Ž1977. 354. w14x Y.P. Varshni, Physica 34 Ž1967. 149. w15x M.A. Fisher, Y.-Z. Huang, A.J. Dann, D.J. Elton, M.J. Harlow, S.D. Perrin, J. Reed, I. Reid, M.J. Adams, IEEE Photon. Technol. Lett. 7 Ž1995. 608.
Optics Communications 154 Ž1998. 43–46
Temperature dependence of 1.55 mm VCSELs J. Masum ) , N. Balkan, M.J. Adams Department of Physics, UniÕersity of Essex, Colchester CO4 3SQ, UK Received 26 February 1998; revised 14 May 1998; accepted 28 May 1998
Abstract The temperature for minimum threshold carrier concentration in 1.55 mm VCSELs can be significantly lower than that at which the peak gain matches the cavity resonance. A simple model is implemented to investigate the magnitude of this temperature difference and to aid the design of VCSELs for room temperature operation. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The development of 1.3 mm and 1.55 mm VCSELs has been slower than for shorter wavelength VCSELs, due to technological challenges posed by the InGaAsPrInP material system. Nevertheless there has been significant progress with 1.55 mm VCSELs exhibiting cw operation at room temperature using wafer fusion technology w1x and up to y258C with epitaxial InGaAsPrInP mirrors w2x. As the development of these devices continues, the need for theoretical modelling and simulation becomes important to optimise laser performance for specific applications. There already exist a number of sophisticated numerical models for VCSELs w3–5x. In the present contribution we use a simple model for optical gain to focus attention on the temperature sensitivity of the threshold current, with the objective of deriving design rules for 1.55 mm VCSELs. A plot of threshold current as function of temperature for VCSELs shows a roughly parabolic dependence with a minimum, which is ascribed to an optimum matching between the photon energy of the gain peak, EP , and the cavity resonance, E M w6,7x. However, modelling w8–10x reveals that the temperature at which the minimum threshold current is achieved can differ significantly from that at which the peak gain matches the cavity resonance. To investigate the intrinsic causes of this difference, attention is focused here on the effect of temperature on the threshold carrier density, n th , the energy gap, EG , and the resonant mode photon energy, E M .
2. Theory Two simple models for gain with no k-selection rule were implemented, one for VCSELs with bulk active region, and the other for quantum well devices. The threshold condition in bulk active region VCSELs is given using the approximation
)
E-mail: [email protected]
0030-4018r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 2 9 5 - 8
J. Masum et al.r Optics Communications 154 (1998) 43–46
44
Table 1 List of the parameters used for device modelling Symbol
Definition
Value
a B E Lz mc mv M N
Total loss Bimolecular recombination constant Photon energy
100 cmy1 1 = 10y1 0 cm3 sy1 0.8 eV
Quantum well width Effective mass of electron Effective mass of hole Number of wells Refractive index
˚ 100 A 0.047 0.63 9 3.4
due to Marinelli w11x as follows: Bc 2 2
4N p
ž
4p 2 m c m v h
2
3r2
DE
/ ž / E
2
sinh w Ž Fc q Fv y D E . r2 kT x cosh w Ž Fc q Fv y D E . r2 kT x q cosh w Ž Fc q Fv . r2 kT x
sa ,
Ž1.
where a is the total loss, i.e. the sum of all internal and end losses, Fc , Fv are the electron and hole quasi-Fermi levels, respectively, measured positively into the bands from the band edges, c is the speed of light, h is Planck’s constant, k is Boltzmann’s constant, T is the temperature, D E is the difference between E M and EG , and B is the bimolecular recombination constant which is assumed to be independent of temperature Ža weak temperature dependence has been reported w12x which does not affect the present results significantly.. The values of the parameters used are given in Table 1.
Fig. 1. Plot of Ža. threshold carrier density and Žb. bandgap, resonant mode and gain peak as functions of temperature for 1.55 mm VCSEL with bulk active region.
J. Masum et al.r Optics Communications 154 (1998) 43–46
45
In order to calculate the quasi-Fermi levels and hence the threshold carrier concentration, n th , Eq. Ž1. was used in conjunction with the charge neutrality condition using Joyce–Dixon approximation w13x, n
žž /
Fc s kT ln
1
n
'8
Nc
q
Nc
/
,
n
žž /
Fv s kT ln
1
n
'8
Nv
q
Nv
/
,
Ž2.
where n is the electron carrier density; Nc and Nv are the conduction and valence band effective densities of states respectively. Fig. 1 illustrates the calculated variation of n th , EG , E M and E P , with temperature for a bulk active region VCSEL for a s 100 cmy1. The temperature dependence of EG is taken to be the same as that of the InP bandgap w14x, and the experimental temperature variation of E M of 0.04 meVrK w15x is assumed. The relative offset between E M and EG is taken as a parameter in the model. If required, this can be related to the experimentally measurable offset between E M and the photoluminescence peak energy, as done for specific lasers in Ref. w10x. The value of Ž E M y EG . is defined in practice by the cavity length, and the results in Fig. 1 are for the case where this offset is zero at 220 K. It can be seen for this example that the minimum n th is achieved at Tmin of 290 K, while E M and E P match at Tmatch of 320 K. Similarly, the threshold condition in MQW VCSELs assuming recombination between the first sub-bands w8x is given by 2p BMc 2 m c m v kT 2
N E
2
hL2z
ln
w 1 q exp Ž FvrkT . xw 1 q exp ŽyFcrkT . x 1 q exp Ž Ž Fv y D E . rkT . 1 q exp Ž Ž D E y Fc . rkT .
sa ,
Ž3.
where M is the number of wells and Lz is the QW width. To calculate the threshold carrier density, n th , Eq. Ž3. was used in conjunction with charge neutrality condition, Eq. Ž4., including the first and second sub-bands which are separated by energies D En and D Ep w8x, 1 q exp
Fc
ž / kT
1 q exp
ž
Fc y D En kT
/
s 1 q exp
Fv
ž / kT
z
1 q exp
ž
Fv y D Ep kT
z
/
,
Ž4.
where z is the ratio of hole to electron mass Ž m vrm c .. Note that for the MQW case D E is the difference between E M and the effective bandgap, EGeff , allowing for the energy displacement of the first sub-bands.
3. Design rules A comparison between the results calculated for QW and bulk active region VCSELs is shown in Fig. 2; the QW devices have either 1 well or 9 wells, each 10 nm in width. Fig. 2 plots the temperature corresponding to minimum electron concentration, Tmin , against the temperature Tmatch at which E P matches E M . Each plot is calculated for the same value of loss a s 100 cmy1, B s 1 = 10y10 cm3 sy1 and is obtained by varying the relative offset between EM and EG Žbulk. or EGeff ŽMQW.. The results show the differences and similarities between the behaviour of VCSELs with bulk and MQW active regions. In each case the variation of Tmin with Tmatch is linear, but for the MQW case the values of these parameters are much closer Ž5–10 K. than is the case for bulk material where the difference can be as much as 30–40 K. This fundamental difference
Fig. 2. Plot of Tmin as function of Tmatch for bulk and QW VCSELs.
46
J. Masum et al.r Optics Communications 154 (1998) 43–46
between bulk and MQW VCSELs is a direct consequence of the step-like density of states for the latter, which causes the peak gain to occur at photon energies close to the bandgap. Hence, to design a device with minimum n th of 300 K, the cavity length should be adjusted to give Tmatch around 330 K and 310 K for bulk and MQW active regions respectively as shown in Fig. 2. The difference between Tmin and Tmatch increases with increasing loss a , and decreases with increasing number of quantum wells in the MQW case. Note that we assume here that the loss term, a , is independent of carrier concentration, i.e. we are deliberately omitting the effects of free-carrier absorption and inter-valence band absorption ŽIVBA.. In addition, the effects of band-gap renormalisation are also ignored. Also n th has not been converted into threshold current Ith . To accomplish the latter, it would be necessary to include in the model the effects of Auger and other non-radiative recombination, of carrier capture into the quantum wells and of leakage out of the wells. In addition, for cw threshold calculations it is necessary to take into account the heating effects, which in turn demands a structure-specific model. Although all these effects may contribute to the temperature dependence of threshold, they are neglected here in order to focus attention on the intrinsic causes of temperature sensitivity. Of the effects listed here we expect the dominant ones to be IVBA, which tends to increase Tmatch , and Auger recombination which tends to reduce Tmin . When these effects are taken into account, the superiority of MQWs over bulk VCSELs is further enhanced. Further details on these topics will be the subject of a future publication.
4. Conclusion A study has been made of the difference between the temperature for minimum threshold electron concentration and that at which the peak gain matches the cavity resonance in 1.55 mm VCSELs. Attention has been focused on the intrinsic causes of this difference, and underlying trends leading to design rules have been identified. The superiority of MQW over bulk active regions has been established, although more work remains to quantify the differences in terms of detailed modelling for both systems.
References w1x D.I. Babic, K. Streubel, R.P. Mirin, N.M. Margalit, J.E. Bowers, E.L. Hu, D.E. Mars, L. Yang, K. Carey, IEEE Photon. Technol. Lett. 7 Ž1995. 1225. w2x K. Streubel, S. Rapp, J. Andre, J. Wallin, IEEE Photon. Technol. Lett. 8 Ž1996. 1121. w3x G.R. Hadley, K.L. Lear, M.E. Warren, K.D. Choquette, J.W. Scott, S.W. Corzine, IEEE J. Quantum Electron. 32 Ž1996. 607. w4x J. Piprek, D.I. Babic, J.E. Bowers, J. Appl. Phys. 81 Ž1997. 3382. w5x J.W. Scott, D.B. Young, B.J. Thibeault, M.G. Peters, L.A. Coldren, IEEE J. Sel. Topics Quantum Electron. 1 Ž1995. 638. w6x B. Tell, K.F. Brown-Goebeler, R.E. Leibenguth, F.M. Baez, Y.H. Lee, Appl. Phys. Lett. 60 Ž1992. 683. w7x G. Goncher, B. Lu, W.-L. Luo, J. Cheng, S. Hersee, S.Z. Sun, R.P. Schneider, J.C. Zolper, IEEE Photon. Technol. Lett. 8 Ž1996. 316. w8x G.W. Taylor, P.A. Evaldsson, IEEE J. Quantum Electron. 30 Ž1994. 2262. w9x J. Piprek, SPIE 3003 Ž1997. 182. w10x S. Rapp, J. Piprek, K. Streubel, J. Andre, J. Wallin, J. Quantum Electron. 33 Ž1997. 1839. w11x F. Marinelli, Solid State Electron. 8 Ž1965. 939. w12x A. Haug, Appl. Phys. B 44 Ž1987. 151. w13x W.B. Joyce, R.W. Dixon, Appl. Phys. Lett. 31 Ž1977. 354. w14x Y.P. Varshni, Physica 34 Ž1967. 149. w15x M.A. Fisher, Y.-Z. Huang, A.J. Dann, D.J. Elton, M.J. Harlow, S.D. Perrin, J. Reed, I. Reid, M.J. Adams, IEEE Photon. Technol. Lett. 7 Ž1995. 608.