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Modelling of temperature effects on the characteristics of mid-infrared quantum cascade lasers
Optics Communications 281 (2008) 5385–5388
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Modelling of temperature effects on the characteristics of mid-infrared quantum cascade lasers A. Hamadou a,*, J.-L. Thobel b, S. Lamari c a
Département de Génie Mécanique, Faculté des Sciences et Sciences de l’ingénieur, Université Abdelhamid Ibn Badis de Mostaganem, BP 227 Mostaganem, 27000 Algérie, Algeria Institut d’Electronique, de Microélectronique et de Nanotechnologie (IEMN), UMR 8520, Université des Sciences et Technologies de Lille, Avenue Poincaré, BP 60069, 59652 Villeneuve d’Ascq Cédex, France c Département de Physique, Faculté des Sciences, Université Ferhat Abbas de Sétif, Cité Mabouda, 19000 Algérie, Algeria b
a r t i c l e
i n f o
Article history: Received 19 March 2008 Received in revised form 27 June 2008 Accepted 19 July 2008
Keywords: Quantum cascade laser Temperature dependence Thermionic emission Unsaturated modal gain Threshold current density Output power
a b s t r a c t The effect of temperature on the characteristics of GaAs/AlGaAs quantum cascade lasers operating in the mid-infrared range is theoretically investigated and compared with reported experimental results. We have included the dependence of the phonon scattering rate, linewidth variations and thermionic lifetime on temperature. It is found that the characteristics depend strongly on temperature. Our model yields threshold current densities in good agreement with the experiments. The effect of injection efficiency on the unsaturated modal gain is also considered. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Mid-infrared quantum cascade lasers (QCLs) have gained a great deal of attention in the last several years, because they offer a variety of advantages over conventional lasers. Since their first demonstration in InGaAs/AlInAs [1] and AlGaAs/GaAs [2] systems, several publications studying experimentally [3-10] and theoretically [3,10–12] the effect of temperature on the performance of QCLs have appeared. These studies are being used to improve the design of devices with a reduced threshold current density and an increased output power. The electron escape time into the continuum levels plays an important role in determining the performance of a QCL. At high operating temperatures, the escape of electrons affects the injection efficiency and thus the threshold current and output power of a QCL [13,14]. Other important parameters which affect the performance of QCLs at high temperatures are the broadening of the electroluminescence spectrum and non-radiative scattering times. The full width at half maximum (FWHM) of the electroluminescence spectrum in a QCL is governed by different scattering mechanisms, but in the mid-infrared range, the electron–longitudinal optical phonon interaction plays a dominant role. Therefore, we propose a simple compact model predicting the effect of tempera* Corresponding author. Tel.: +213 76937575; fax: +213 45211018. E-mail address: [email protected] (A. Hamadou). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.07.036
ture on the unsaturated modal gain, accounting for the dependence of the FWHM and scattering times on the Bose–Einstein factor. The analytical expressions for temperature-dependent threshold current and output power are also derived from the proposed model.
2. Rate equations The system of rate equations for electron numbers N3 and N2 in levels 3 and 2, and the photon number N ph in the resonator cavity can be written in the following form:
dN3 J N3 c0 r32 C ¼ gWL ðN3 N2 ÞNph ; e s3 dt V
ð1aÞ
dN2 J N3 N2 c0 r32 þC ¼ ð1 gÞWL þ ðN3 N2 ÞNph ; e s32 s21 dt V
ð1bÞ
dNph Nph c0 r32 ; ¼ NC ðN3 N2 ÞNph dt V sp
ð1cÞ
where spontaneous emission has been neglected and we have assumed for simplicity that all gain stages are identical. In the above equations, g is the injection efficiency into the upper laser state, e is the electronic charge, J is the electron current density tunnelling into level 3, W and L are the width and length of the cavity respectively, V ¼ NWLLp is the volume of the cavity (Lp is
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the length of a single stage of the cascade laser structure, N is the number of stages), c0 ¼ c=neff is the velocity of light in medium (neff is the effective refractive index). C and r32 are the mode confinement factor and the stimulated emission cross section, respectively. The temperature dependence of the parameter r32 ðTÞ is given by [15]
r32 ðTÞ ¼
4pe2 z232 ; neff e0 kð2c32 ðTÞÞ
ð2Þ
1
where ez32 is the dipole matrix element, k is the emission wavelength, e0 is the vacuum permittivity, T is the temperature and 2c32 is the FWHM which is mainly due to LO-phonon scattering. In Eq. (2), we assume that r32 depends on temperature only through FWHM. Therefore the temperature dependence of 2c32 can be calculated from [16]
c32 ðTÞ 2nq ðTÞ þ 1 ¼ ; c32 ð77Þ 2nq ð77Þ þ 1
ð3Þ
where nq ðTÞ is the phonon occupation number given by the Bose– Einstein factor
nq ðTÞ ¼
1 expðhxkTLO Þ 1
ð4Þ
;
with hxLO being the LO-phonon energy and k being the Boltzmann constant. sij (i = 3, 2 and j = 2, 1) is the phonon scattering time between levels i and j and s3 is the total lifetime of electrons in level 3. In general, s3 is a sum over all possible final states and all scattering mechanisms and, in principle, it could be calculated from the knowledge of wavefunctions and scattering potentials. However, such a calculation would be too heavy to be included in a simple compact model like the one described here and we prefer to use convenient approximations. In that spirit, we assume that s3 can be given by 1 1 1 s1 3 ¼ s32 þ s31 þ s3c , where s3c represents the escape time from the upper laser state towards continuum states. In the general case, transfers towards satellite valleys should also be considered and X valleys states should be treated in a similar way as C valley states. However, extensive investigations by Gao et al. [17-19] indicate that these contributions are small for AlGaAs/GaAs QCLs with Alrich barriers, and therefore we neglect them for the sake of simplicity. Moreover, we also neglect leakage due to tunneling [20] and we approximate s3c by a simple thermionic formula [21]:
sth ¼
2pm L2z kT
!12 exp
DEact ; kT
ð5Þ
DNðTÞ ¼
ðTÞ ðTÞ ð1 ss21 Þg ð1 gÞ ss213 ðTÞ 32 ðTÞ
1þN
s3 ðTÞWL eJ
V e0 neff k 4pe2 z232 c0 C
ab ¼ C em ij ðnq ðTÞ þ 1Þ þ C ij nq :
C em ij
ð8Þ
C ab ij ,
The prefactors and which refer to emission and absorption respectively, are in general different. However, for the kind of QCL considered in this article, we may use further approximations. Levels 2 and 1 are separated by just one phonon energy and thus the emission process dominates. On the contrary, state 3 is separated from the lower states by much more than a phonon energy, so that the prefactors for emission and absorption are comparable: ab C em 3j C 3j [22]. With these approximations one gets [22,23]
1
s21 ðTÞ 1
s3 ðTÞ
¼
¼
1
sc21 1
sth
ð1 þ nq ðTÞÞ;
þ
1
sc3
ð9Þ
ð1 þ 2nq ðTÞÞ:
ð10Þ
Here scij and sc3 ¼ ð1=sc31 þ 1=sc32 Þ1 are the scattering times between level i and level j, and the total lifetime of level 3 at cryogenic temperature, respectively.
3. Unsaturated modal gain The temperature dependence of modal gain GM ðTÞ for all gain stages is proportional to the temperature dependence of both population inversion DNðTÞ and stimulated emission cross section r32 ðTÞ, and can be expressed as follows:
GM ðTÞ ¼
C V
r32 ðTÞDNðTÞ:
ð11Þ
Replacing Eqs. (2) and (6) in Eq. (11), and using Eq. (3), the expression for the unsaturated modal gain at temperature T, normalized to that at cryogenic temperature, can be written as
c sc 2nq ðTÞþ1 s sc 2n ðTÞþ1 1 1 s21 g ð1 gÞ s21th nq ðTÞþ1 þ s21c nqqðTÞþ1 c nq ðTÞþ1 GM ðTÞ 32 3 : ¼ sc sc sc GcM ð2nq ðTÞ þ 1Þ 1 þ 2nq ðTÞ þ s 3 g ð1 gÞ s21c 1 s21 c th
32
3
Nph
;
ð6Þ
ph;sat ðTÞ
2c32 ðTÞ ; ðTÞ s3 ðTÞ 1 þ ss2131 ðTÞ þ ss21ðTÞ ðTÞ th
Thus, the relative unsaturated modal gain is a function of the injection efficiency, scattering times and phonon population.
4. Output power The temperature dependence of output power, calculated by replacing Eq. (6) into Eq. (1c) in conjunction with Eq. (7) and taking into account the relationship between the output power and the photon number P out ¼ g0 hxN ph =sp ; is given by
Pout ðTÞ ¼ g0 gr ðTÞ
where Nph;sat is the saturation photon number, given by
Nph;sat ðTÞ ¼
sij ðTÞ
ð12Þ
where m is the effective mass for electrons in level 3, DEact is the activation energy between the excited state of the laser and the bottom of the continuum states, and Lz is the width of active region. The thermionic emission time given by Eq. (5) is very sensitive to activation energy and temperature. The photon lifetime sp inside the cavity can be expressed as 0 s1 p ¼ c ðaw þ am Þ, where aw is the waveguide loss of the cavity and am is the mirror loss, given by am ¼ lnðR1 R2 Þ=ð2LÞ where R1 and R2 are the reflectivities of facet 1 and 2 respectively. Under steady state conditions, the population inversion DNðTÞ ¼ N 3 N 2 obeys the following relation:
and sij ðTÞ and s3 ðTÞ are the scattering times and the total lifetime of level 3 at temperature T. The lifetimes of levels are mostly determined by the non-radiative scattering mechanisms such as LO-phonon, acoustic phonon, electron–electron, interface roughness and impurity scattering processes. In our case, as long as the energy separation of the states exceeds the phonon energy, the scattering times are mainly controlled by LO-phonon emission and absorption, and thus, in general, their temperature dependence is given by
ð7Þ
x h NðI Ith ðTÞÞ: e
ð13Þ
In the above equation, h x is the energy of emitted photon. The power output coupling efficiency g0 [24] and the radiative efficiency gr ðTÞ are
g0 ¼
pffiffiffiffiffi ð1 R1 Þ R2 am pffiffiffiffiffi pffiffiffiffiffi ; ð1 R1 Þ R2 þ ð1 R2 Þ R1 am þ aw
ð14Þ
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gr ðTÞ ¼
ðTÞ ðTÞ ð1 ss21 Þg ð1 gÞ ss213 ðTÞ 32 ðTÞ
20
ð15Þ
ðTÞ ðTÞ ss21 1 þ ss213 ðTÞ 32 ðTÞ
: Our model
16
and Ith ðTÞ ¼ WLJth ðTÞ is the threshold current, with J th ðTÞ being the temperature-dependent threshold current density. An expression for Jth ðTÞ, obtained by equating the gain to the sum of the waveguide and mirror losses, is
J th ðTÞ ¼
18
Jth(kA/cm2)
Symbol : Ref [10]
14 12 10 8
e0 neff kLp aw þ am 2c32 ðTÞ : ðTÞ C 4pez232 s3 ðTÞðð1 ss2132 ðTÞ Þg ð1 gÞ ss213 ðTÞ Þ ðTÞ
ð16Þ
6 4 2 0
5. Numerical results and discussion
1.2 1 Without thermionic emission
0.8 c
G M (T)/ G M
100
150
200
250
300
Temperature (K)
With thermionic emission
Fig. 2. Calculated threshold current density (solid line) and experimental data (symbol) from [10] of QCL as a function of temperature with unit injection efficiency.
8
η=1
7
77 K
6
Output power (W)
In the following discussion, we study the temperature dependence of unsaturated modal gain, threshold current density and output power for the structure described in Ref. [5]. We use in our calculation the parameters taken from Refs. [3,5,25]: z32 ¼ 1:7 nm, s32 ¼ 2:1 ps, s3 ¼ 1:4 ps, s21 ¼ 0:3 ps, neff ¼ 3:27, 2c32 ð77Þ ¼ 12 meV, aw ¼ 20 cm1, DEact ¼ 58 meV, N ¼ 48, R1 ¼ R2 ¼ 0:29, C ¼ 0:32, Lp ¼ 45 nm, Lz ¼ 10 nm, L ¼ 1 mm, W ¼ 34 lm and m ¼ 0:067m0 (m0 is the free electron mass). In Fig. 1, the relative unsaturated modal gain (RUMG) GM ðTÞ=GcM is plotted as a function of temperature with and without thermionic emission effects for three injection efficiencies g ¼ 1, 0.75, and 0.5. The data have been normalized to cryogenic temperature, to simplify the comparison. As shown in this figure, the RUMG is very sensitive at high temperature to the injection efficiency. For higher values of the thermionic emission time (sth ! 1) at low temperatures (T 6 77 K), the escape process becomes practically negligible and therefore the RUMG without thermionic emission is given by GM ðTÞ=GcM ¼ 1=ð1 þ 2nq ðTÞÞ2 . Indeed, as the temperature is increased, the thermionic emission time in the active region decreases, reducing the total lifetime s3 of electrons in the upper laser state, and therefore directly reducing the RUMG. For example at room temperature, the RUMG is lowered by 5 times for g ¼ 1 and 10 times for g ¼ 0:5; compared to the cryogenic temperature conditions. Fig. 2 shows the temperature dependence of the threshold current density J th obtained from the theoretical calculations and the experimental work presented in [10]. The solid line is for the theoretical results obtained using Eq. (16) with unit injection efficiency. The increase of the threshold current density with temperature is attributed firstly to the reduction in the total lifetime of electrons in level 3, due to thermionic emission process, and secondly to the increase in the phonon number with temper-
0.6
50
200 K
5 4
240 K
3 280 K
2 300 K
1 0
1
2
3
4
5
6
7
8
9
Current (A) Fig. 3. Calculated output power per facet versus injection current for a QCL of section 1 mm 34 lm at several temperatures, where g ¼ 1.
ature. Our calculation shows good agreement with the experimental data. Similar results were obtained theoretically by Ortiz et al. at different Al concentrations with another model [3]. As compared to the experimental results, our model underestimates J th by 12% at T = 290 K. This underestimation is probably due to the non-unity injection efficiency into the upper laser level. Nevertheless the agreement between theoretical and experimental data supports the validity of the proposed model. We conclude our analysis with Fig. 3, where we show the influence of temperature on the calculated output power PðTÞ versus injection current for unity injection efficiency. The output power of the QCL varies linearly with the current injection and depends strongly on temperature. Moreover, a considerable increase in the threshold current occurs when increasing the values of temperature, as shown in this figure. Finally, the slope efficiency decreases when temperature is increased.
0.4
6. Conclusion 0.2 0 0
50
100
150
200
250
300
Temperature (K) Fig. 1. Comparison of relative unsaturated modal gain GM ðTÞ=GcM with and without thermionic emission. The injections efficiency are given respectively by g ¼ 1 (solid line), 0.75 (dashed line), and 0.5 (dotted line).
In summary, we have proposed a simple compact model in order to predict the effect of the temperature on the performance of a mid-infrared QCL. Our model is based on the assumption that the temperature dependence of unsaturated modal gain, threshold current density and output power is only due to the phonon scattering rate, linewidth variations and thermionic lifetime. The good agreement obtained between our model and experiments supports
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the validity of these assumptions. However, at high temperatures, one should consider the fact that the injection efficiency is less than unity. Acknowledgments The authors wish to acknowledge the help of F. Dessenne and T. Sadi, from IEMN. References [1] J. Faist, F. Capasso, D.L. Sivco, C. Sirtori, A.L. Hutchinson, A.Y. Cho, Science 264 (1994) 553. [2] C. Sirtori, P. Kruck, S. Barbieri, P. Collot, J. Nagle, M. Beck, J. Faist, U. Oesterle, Appl. Phys. Lett. 73 (1998) 3486. [3] V. Ortiz, C. Becker, H. Page, C. Sirtori, J. Cryst. Growth 251 (2003) 701. [4] H. Page, S. Dhillon, M. Calligaro, C. Becker, V. Ortiz, C. Sirtori, IEEE J. Quantum Electron. 40 (2004) 665. [5] H. Page, C. Becker, A. Robertson, G. Glastre, V. Ortiz, C. Sirtori, Appl. Phys. Lett. 78 (2001) 3529. [6] C. Pflügl, W. Schrenk, S. Anders, G. Strasser, C. Becker, C. Sirtori, Y. Bonetti, A. Muller, Appl. Phys. Lett. 83 (2003) 4698. [7] R.P. Green, A. Krysa, J.S. Roberts, D.G. Revin, L.R. Wilson, E.A. Zibik, W.H. Ng, J.W. Cockburn, Appl. Phys. Lett. 83 (2003) 1921. [8] L. Diehl, D. Bour, S. Corzine, J. Zhu, G. Höfler, B.G. Lee, C.Y. Wang, M. Troccoli, F. Capasso, Appl. Phys. Lett. 88 (2006) 041102.
[9] A. Wittmann, T. Gresch, E. Gini, L. Hvozdara, N. Hoyler, M. Giovannini, J. Faist, IEEE J. Quantum Electron. 44 (2008) 36. [10] S. Höfling, V.D. Jovanovic´, D. Indjin, J.P. Reithmaier, A. Forchel, Z. Ikonic´, N. Vukmirovic´, P. Harrison, A. Mircˇetic´, V. Milanovic´, Appl. Phys. Lett. 88 (2006) 251109. [11] J.M. Tavish, D. Indjin, P. Harrison, J. Appl. Phys. 99 (2006) 114505. [12] C.A. Evans, V.D. Jovanovic´, D. Indjin, Z. Ikonic´, P. Harrison, IEEE J. Quantum Electron. 42 (2006) 859. [13] H. Page, P. Kruck, S. Barbieri, C. Sirtori, M. Stellmacher, J. Nagle, Electron. Lett. 35 (1999) 1848. [14] P. Kruck, H. Page, C. Sirtori, S. Barbieri, M. Stellmacher, J. Nagle, Appl. Phys. Lett. 76 (2000) 3340. [15] J.T. Verdeyen, Laser Electronics, Prentice-Hall, Englewood Cliffs, NJ, 1995. [16] C.F. Hsu, O. Jeong-Seok, P. Zory, D. Botez, IEEE J. Sel. Top. Quantum Electron. 6 (2000) 491. [17] X. Gao, D. Botez, I. Knezevic, Appl. Phys. Lett. 89 (2006) 191119. [18] X. Gao, D. Botez, I. Knezevic, J. Appl. Phys. 101 (2007) 063101. [19] X. Gao, D. Botez, I. Knezevic, J. Comput. Electron. 6 (2007) 305. [20] F. Capasso, C. Sirtori, J. Faist, D.L. Sivco, S.N.G. Chu, A.Y. Cho, Nature 358 (1992) 565. [21] H. Schneider, K.V. Klitzing, Phys. Rev. B 38 (1988) 6160. [22] C. Sirtori, H. Page, C. Becker, V. Ortiz, IEEE J. Quantum Electron. 38 (2002) 547. [23] C. Becker, C. Sirtori, H. Page, A. Robertson, V. Ortiz, X. Marcadet, Phys. Rev. B 65 (2002) 085305. [24] L.A. Coldren, S.W. Corzine, Diode Lasers and Photonics Integrated Circuits, Wiley, New York, 1995. [25] S. Höfling, R. Kallweit, J. Seufert, J. Koeth, J.P. Reithmaier, A. Forchel, J. Cryst. Growth 278 (2005) 775.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Modelling of temperature effects on the characteristics of mid-infrared quantum cascade lasers A. Hamadou a,*, J.-L. Thobel b, S. Lamari c a
Département de Génie Mécanique, Faculté des Sciences et Sciences de l’ingénieur, Université Abdelhamid Ibn Badis de Mostaganem, BP 227 Mostaganem, 27000 Algérie, Algeria Institut d’Electronique, de Microélectronique et de Nanotechnologie (IEMN), UMR 8520, Université des Sciences et Technologies de Lille, Avenue Poincaré, BP 60069, 59652 Villeneuve d’Ascq Cédex, France c Département de Physique, Faculté des Sciences, Université Ferhat Abbas de Sétif, Cité Mabouda, 19000 Algérie, Algeria b
a r t i c l e
i n f o
Article history: Received 19 March 2008 Received in revised form 27 June 2008 Accepted 19 July 2008
Keywords: Quantum cascade laser Temperature dependence Thermionic emission Unsaturated modal gain Threshold current density Output power
a b s t r a c t The effect of temperature on the characteristics of GaAs/AlGaAs quantum cascade lasers operating in the mid-infrared range is theoretically investigated and compared with reported experimental results. We have included the dependence of the phonon scattering rate, linewidth variations and thermionic lifetime on temperature. It is found that the characteristics depend strongly on temperature. Our model yields threshold current densities in good agreement with the experiments. The effect of injection efficiency on the unsaturated modal gain is also considered. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Mid-infrared quantum cascade lasers (QCLs) have gained a great deal of attention in the last several years, because they offer a variety of advantages over conventional lasers. Since their first demonstration in InGaAs/AlInAs [1] and AlGaAs/GaAs [2] systems, several publications studying experimentally [3-10] and theoretically [3,10–12] the effect of temperature on the performance of QCLs have appeared. These studies are being used to improve the design of devices with a reduced threshold current density and an increased output power. The electron escape time into the continuum levels plays an important role in determining the performance of a QCL. At high operating temperatures, the escape of electrons affects the injection efficiency and thus the threshold current and output power of a QCL [13,14]. Other important parameters which affect the performance of QCLs at high temperatures are the broadening of the electroluminescence spectrum and non-radiative scattering times. The full width at half maximum (FWHM) of the electroluminescence spectrum in a QCL is governed by different scattering mechanisms, but in the mid-infrared range, the electron–longitudinal optical phonon interaction plays a dominant role. Therefore, we propose a simple compact model predicting the effect of tempera* Corresponding author. Tel.: +213 76937575; fax: +213 45211018. E-mail address: [email protected] (A. Hamadou). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.07.036
ture on the unsaturated modal gain, accounting for the dependence of the FWHM and scattering times on the Bose–Einstein factor. The analytical expressions for temperature-dependent threshold current and output power are also derived from the proposed model.
2. Rate equations The system of rate equations for electron numbers N3 and N2 in levels 3 and 2, and the photon number N ph in the resonator cavity can be written in the following form:
dN3 J N3 c0 r32 C ¼ gWL ðN3 N2 ÞNph ; e s3 dt V
ð1aÞ
dN2 J N3 N2 c0 r32 þC ¼ ð1 gÞWL þ ðN3 N2 ÞNph ; e s32 s21 dt V
ð1bÞ
dNph Nph c0 r32 ; ¼ NC ðN3 N2 ÞNph dt V sp
ð1cÞ
where spontaneous emission has been neglected and we have assumed for simplicity that all gain stages are identical. In the above equations, g is the injection efficiency into the upper laser state, e is the electronic charge, J is the electron current density tunnelling into level 3, W and L are the width and length of the cavity respectively, V ¼ NWLLp is the volume of the cavity (Lp is
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the length of a single stage of the cascade laser structure, N is the number of stages), c0 ¼ c=neff is the velocity of light in medium (neff is the effective refractive index). C and r32 are the mode confinement factor and the stimulated emission cross section, respectively. The temperature dependence of the parameter r32 ðTÞ is given by [15]
r32 ðTÞ ¼
4pe2 z232 ; neff e0 kð2c32 ðTÞÞ
ð2Þ
1
where ez32 is the dipole matrix element, k is the emission wavelength, e0 is the vacuum permittivity, T is the temperature and 2c32 is the FWHM which is mainly due to LO-phonon scattering. In Eq. (2), we assume that r32 depends on temperature only through FWHM. Therefore the temperature dependence of 2c32 can be calculated from [16]
c32 ðTÞ 2nq ðTÞ þ 1 ¼ ; c32 ð77Þ 2nq ð77Þ þ 1
ð3Þ
where nq ðTÞ is the phonon occupation number given by the Bose– Einstein factor
nq ðTÞ ¼
1 expðhxkTLO Þ 1
ð4Þ
;
with hxLO being the LO-phonon energy and k being the Boltzmann constant. sij (i = 3, 2 and j = 2, 1) is the phonon scattering time between levels i and j and s3 is the total lifetime of electrons in level 3. In general, s3 is a sum over all possible final states and all scattering mechanisms and, in principle, it could be calculated from the knowledge of wavefunctions and scattering potentials. However, such a calculation would be too heavy to be included in a simple compact model like the one described here and we prefer to use convenient approximations. In that spirit, we assume that s3 can be given by 1 1 1 s1 3 ¼ s32 þ s31 þ s3c , where s3c represents the escape time from the upper laser state towards continuum states. In the general case, transfers towards satellite valleys should also be considered and X valleys states should be treated in a similar way as C valley states. However, extensive investigations by Gao et al. [17-19] indicate that these contributions are small for AlGaAs/GaAs QCLs with Alrich barriers, and therefore we neglect them for the sake of simplicity. Moreover, we also neglect leakage due to tunneling [20] and we approximate s3c by a simple thermionic formula [21]:
sth ¼
2pm L2z kT
!12 exp
DEact ; kT
ð5Þ
DNðTÞ ¼
ðTÞ ðTÞ ð1 ss21 Þg ð1 gÞ ss213 ðTÞ 32 ðTÞ
1þN
s3 ðTÞWL eJ
V e0 neff k 4pe2 z232 c0 C
ab ¼ C em ij ðnq ðTÞ þ 1Þ þ C ij nq :
C em ij
ð8Þ
C ab ij ,
The prefactors and which refer to emission and absorption respectively, are in general different. However, for the kind of QCL considered in this article, we may use further approximations. Levels 2 and 1 are separated by just one phonon energy and thus the emission process dominates. On the contrary, state 3 is separated from the lower states by much more than a phonon energy, so that the prefactors for emission and absorption are comparable: ab C em 3j C 3j [22]. With these approximations one gets [22,23]
1
s21 ðTÞ 1
s3 ðTÞ
¼
¼
1
sc21 1
sth
ð1 þ nq ðTÞÞ;
þ
1
sc3
ð9Þ
ð1 þ 2nq ðTÞÞ:
ð10Þ
Here scij and sc3 ¼ ð1=sc31 þ 1=sc32 Þ1 are the scattering times between level i and level j, and the total lifetime of level 3 at cryogenic temperature, respectively.
3. Unsaturated modal gain The temperature dependence of modal gain GM ðTÞ for all gain stages is proportional to the temperature dependence of both population inversion DNðTÞ and stimulated emission cross section r32 ðTÞ, and can be expressed as follows:
GM ðTÞ ¼
C V
r32 ðTÞDNðTÞ:
ð11Þ
Replacing Eqs. (2) and (6) in Eq. (11), and using Eq. (3), the expression for the unsaturated modal gain at temperature T, normalized to that at cryogenic temperature, can be written as
c sc 2nq ðTÞþ1 s sc 2n ðTÞþ1 1 1 s21 g ð1 gÞ s21th nq ðTÞþ1 þ s21c nqqðTÞþ1 c nq ðTÞþ1 GM ðTÞ 32 3 : ¼ sc sc sc GcM ð2nq ðTÞ þ 1Þ 1 þ 2nq ðTÞ þ s 3 g ð1 gÞ s21c 1 s21 c th
32
3
Nph
;
ð6Þ
ph;sat ðTÞ
2c32 ðTÞ ; ðTÞ s3 ðTÞ 1 þ ss2131 ðTÞ þ ss21ðTÞ ðTÞ th
Thus, the relative unsaturated modal gain is a function of the injection efficiency, scattering times and phonon population.
4. Output power The temperature dependence of output power, calculated by replacing Eq. (6) into Eq. (1c) in conjunction with Eq. (7) and taking into account the relationship between the output power and the photon number P out ¼ g0 hxN ph =sp ; is given by
Pout ðTÞ ¼ g0 gr ðTÞ
where Nph;sat is the saturation photon number, given by
Nph;sat ðTÞ ¼
sij ðTÞ
ð12Þ
where m is the effective mass for electrons in level 3, DEact is the activation energy between the excited state of the laser and the bottom of the continuum states, and Lz is the width of active region. The thermionic emission time given by Eq. (5) is very sensitive to activation energy and temperature. The photon lifetime sp inside the cavity can be expressed as 0 s1 p ¼ c ðaw þ am Þ, where aw is the waveguide loss of the cavity and am is the mirror loss, given by am ¼ lnðR1 R2 Þ=ð2LÞ where R1 and R2 are the reflectivities of facet 1 and 2 respectively. Under steady state conditions, the population inversion DNðTÞ ¼ N 3 N 2 obeys the following relation:
and sij ðTÞ and s3 ðTÞ are the scattering times and the total lifetime of level 3 at temperature T. The lifetimes of levels are mostly determined by the non-radiative scattering mechanisms such as LO-phonon, acoustic phonon, electron–electron, interface roughness and impurity scattering processes. In our case, as long as the energy separation of the states exceeds the phonon energy, the scattering times are mainly controlled by LO-phonon emission and absorption, and thus, in general, their temperature dependence is given by
ð7Þ
x h NðI Ith ðTÞÞ: e
ð13Þ
In the above equation, h x is the energy of emitted photon. The power output coupling efficiency g0 [24] and the radiative efficiency gr ðTÞ are
g0 ¼
pffiffiffiffiffi ð1 R1 Þ R2 am pffiffiffiffiffi pffiffiffiffiffi ; ð1 R1 Þ R2 þ ð1 R2 Þ R1 am þ aw
ð14Þ
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gr ðTÞ ¼
ðTÞ ðTÞ ð1 ss21 Þg ð1 gÞ ss213 ðTÞ 32 ðTÞ
20
ð15Þ
ðTÞ ðTÞ ss21 1 þ ss213 ðTÞ 32 ðTÞ
: Our model
16
and Ith ðTÞ ¼ WLJth ðTÞ is the threshold current, with J th ðTÞ being the temperature-dependent threshold current density. An expression for Jth ðTÞ, obtained by equating the gain to the sum of the waveguide and mirror losses, is
J th ðTÞ ¼
18
Jth(kA/cm2)
Symbol : Ref [10]
14 12 10 8
e0 neff kLp aw þ am 2c32 ðTÞ : ðTÞ C 4pez232 s3 ðTÞðð1 ss2132 ðTÞ Þg ð1 gÞ ss213 ðTÞ Þ ðTÞ
ð16Þ
6 4 2 0
5. Numerical results and discussion
1.2 1 Without thermionic emission
0.8 c
G M (T)/ G M
100
150
200
250
300
Temperature (K)
With thermionic emission
Fig. 2. Calculated threshold current density (solid line) and experimental data (symbol) from [10] of QCL as a function of temperature with unit injection efficiency.
8
η=1
7
77 K
6
Output power (W)
In the following discussion, we study the temperature dependence of unsaturated modal gain, threshold current density and output power for the structure described in Ref. [5]. We use in our calculation the parameters taken from Refs. [3,5,25]: z32 ¼ 1:7 nm, s32 ¼ 2:1 ps, s3 ¼ 1:4 ps, s21 ¼ 0:3 ps, neff ¼ 3:27, 2c32 ð77Þ ¼ 12 meV, aw ¼ 20 cm1, DEact ¼ 58 meV, N ¼ 48, R1 ¼ R2 ¼ 0:29, C ¼ 0:32, Lp ¼ 45 nm, Lz ¼ 10 nm, L ¼ 1 mm, W ¼ 34 lm and m ¼ 0:067m0 (m0 is the free electron mass). In Fig. 1, the relative unsaturated modal gain (RUMG) GM ðTÞ=GcM is plotted as a function of temperature with and without thermionic emission effects for three injection efficiencies g ¼ 1, 0.75, and 0.5. The data have been normalized to cryogenic temperature, to simplify the comparison. As shown in this figure, the RUMG is very sensitive at high temperature to the injection efficiency. For higher values of the thermionic emission time (sth ! 1) at low temperatures (T 6 77 K), the escape process becomes practically negligible and therefore the RUMG without thermionic emission is given by GM ðTÞ=GcM ¼ 1=ð1 þ 2nq ðTÞÞ2 . Indeed, as the temperature is increased, the thermionic emission time in the active region decreases, reducing the total lifetime s3 of electrons in the upper laser state, and therefore directly reducing the RUMG. For example at room temperature, the RUMG is lowered by 5 times for g ¼ 1 and 10 times for g ¼ 0:5; compared to the cryogenic temperature conditions. Fig. 2 shows the temperature dependence of the threshold current density J th obtained from the theoretical calculations and the experimental work presented in [10]. The solid line is for the theoretical results obtained using Eq. (16) with unit injection efficiency. The increase of the threshold current density with temperature is attributed firstly to the reduction in the total lifetime of electrons in level 3, due to thermionic emission process, and secondly to the increase in the phonon number with temper-
0.6
50
200 K
5 4
240 K
3 280 K
2 300 K
1 0
1
2
3
4
5
6
7
8
9
Current (A) Fig. 3. Calculated output power per facet versus injection current for a QCL of section 1 mm 34 lm at several temperatures, where g ¼ 1.
ature. Our calculation shows good agreement with the experimental data. Similar results were obtained theoretically by Ortiz et al. at different Al concentrations with another model [3]. As compared to the experimental results, our model underestimates J th by 12% at T = 290 K. This underestimation is probably due to the non-unity injection efficiency into the upper laser level. Nevertheless the agreement between theoretical and experimental data supports the validity of the proposed model. We conclude our analysis with Fig. 3, where we show the influence of temperature on the calculated output power PðTÞ versus injection current for unity injection efficiency. The output power of the QCL varies linearly with the current injection and depends strongly on temperature. Moreover, a considerable increase in the threshold current occurs when increasing the values of temperature, as shown in this figure. Finally, the slope efficiency decreases when temperature is increased.
0.4
6. Conclusion 0.2 0 0
50
100
150
200
250
300
Temperature (K) Fig. 1. Comparison of relative unsaturated modal gain GM ðTÞ=GcM with and without thermionic emission. The injections efficiency are given respectively by g ¼ 1 (solid line), 0.75 (dashed line), and 0.5 (dotted line).
In summary, we have proposed a simple compact model in order to predict the effect of the temperature on the performance of a mid-infrared QCL. Our model is based on the assumption that the temperature dependence of unsaturated modal gain, threshold current density and output power is only due to the phonon scattering rate, linewidth variations and thermionic lifetime. The good agreement obtained between our model and experiments supports
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