InGaAs quantum cascade lasers

Solid-State Electronics 52 (2008) 1669–1673

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Tuning nonlinear susceptibility in strained AlGaAs/InGaAs quantum cascade lasers Denzil Roberts *, Gregory Triplett Department of Electrical and Computer Engineering, University of Missouri-Columbia, Columbia, MO 65211, United States

a r t i c l e

i n f o

Article history: Received 15 May 2008 Accepted 7 June 2008 Available online 27 July 2008 Review of this manuscript was arranged by A. Iliadis, C. Richter, and A. Zaslavsky Keywords: Quantum cascade laser Nonlinear susceptibility

a b s t r a c t This paper explores enhancement of nonlinear susceptibility in strained quantum cascade lasers (QCLs) that lead to unique intracavity characteristics that include frequency mixing producing both fundamental and higher order modes that propagate freely within a GaAs matrix. Doing so provides a QCL cavity design with emission near the 3.5 lm range, a region suitable for applications within the 3–5 lm atmospheric transmission window. In this study, a self-consistent Schrodinger–Poisson solver was employed to analyze the effects of strain within an AlGaAs/InGaAs active region between AlGaAs/GaAs injectors on a [1 1 1] GaAs matrix for the purpose of enhancing nonlinear susceptibility. Strain relief available through the use of [1 1 1] GaAs allows increased indium composition in the active region and results in observation of second-harmonics below the 5 lm range with tunable optical dipole moments and oscillator strengths. Results demonstrate the feasibility of strained AlGaAs/InGaAs devices on GaAs for producing higher order harmonics that lay below the 4 lm spectral limit. Ó 2008 Elsevier Ltd. All rights reserved.

1. Introduction Following the demonstration of the first intersubband quantum cascade (QC) laser [1] in 1994, significant progress and advancement has followed to include a wide range of wavelengths on GaAs and InP based material [2]. Specifically, the operating range includes mid- to far-infrared with peak power levels in the watt range and above room temperature pulsed operation for wavelengths from 4.5 to 16 lm [3]. Considering the progress made thus far, there is still much work to be done in producing QC lasers that emit efficiently in the atmospheric window (3–5 lm wavelength range), which is important for free-space communications and gas sensing applications [4–8]. Two common approaches in achieving QC lasers with shorter wavelength emission are through material selection, i.e. wider bandgap materials and type-II structures. In this study, an alternative method that includes nonlinear optical processes is explored for producing second-harmonic generation through enhanced nonlinear susceptibility. Within the QC laser cavity, second-harmonic generation can be employed, if both the fundamental and second-harmonic waves can propagate freely without being reabsorbed by the semiconductor. The production of second-harmonics essentially depends on the susceptibility (v), which is a material dependant parameter and is thought to be either optically linear or nonlinear. Linearity depends on whether the polarization of the material is proportional to the electric field that aligns the nucleus–electron dipole at an atomic site. In this work, the AlGaAs/GaAs structure is ex-

* Corresponding author. Tel.: +1 302 423 2069; fax: +1 573 882 0397. E-mail address: [email protected] (D. Roberts). 0038-1101/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.sse.2008.06.040

plored because of its lattice-matched characteristics and technological maturity and the fact that indium in the quantum well in the active region increases strain and conduction band offsets that is required for shorter wavelength emission. Conveniently, InGaAs is also optically nonlinear due to its non-centro-symmetric atomic arrangement. Therefore, the polarization (and thus the nonlinear susceptibility) of InGaAs can be further enhanced by inducing strain in this material. Appropriately, this study involves strained active regions (InGaAs) in AlGaAs/GaAs QC lasers with the goal of enhancing nonlinear susceptibility and photon generation in the 3–5 lm wavelength range. The photon energy generated in this approach by the fundamental mode and second-harmonic generation is well within the transparency region of the substrate material. To enhance nonlinear susceptibility, v(2), this approach aims to maximize the product of the dipole matrix elements of the intersubband transitions and minimize the energy denominators using resonant effects, which can be approximated as follows [9]:

vð2Þ ð2xÞ 

e3 hziii ihziiiii ihziiii i N ðhx  DEiii  i  Ciii Þð2hx  DEiiii  i  Ciiii Þ e0 ð1Þ

where e is the electron charge, e0 the permittivity of the vacuum, N is the electron density in the active region, DEij is the separation between subbands, Cij and hzij i are the half width at half maximum and the matrix element of the i ? j intersubband transition. To date, some work has explored the optical nonlinear susceptibility of AlInAs/InGaAs QC laser structures grown by molecular beam epitaxy (MBE) on InP [10–13]. The results from those studies show much promise in tuning the nonlinearity based on current

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density and applied bias. Still, very little attention has been given to GaAs-based systems in this regard. In a recent study of GaAs and InAs piezoelectric properties by Bester et al. [14], it was reported that first and second-order contributions are of comparable magnitude. In essence, the second-order piezoelectric tensor contributes significantly to the piezoelectric effects in GaAs and InAs. In fact, data extracted from experimental work reflect equally strong first- and second-order contributions. Based on this study, adding indium to GaAs quantum well not only increases DEc, but also enhances nonlinear susceptibility. Using density functional theory, Bester, et al. calculated linear and quadratic piezoelectric coefficients for GaAs, InAs, and InGaAs quantum wells on GaAs along the [1 1 1] direction. The magnitude of the coefficients is provided in Table 1, where e14 is the first-order tensor and Bujk are the quadratic coefficients [14]. As apparent from the data in Table 1, the magnitudes of the first-order and nonlinear bulk piezoelectric tensors are comparable. Based on this data, the investigation of AlGaAs/InGaAs QCL structures is appropriate for exploring second-harmonic generation, since the addition of indium to the GaAs quantum well not only increases DEc, but also enhances nonlinear susceptibility. It is important to note that the piezoelectric properties in the [1 1 1] direction are especially promising for this study. In fact, when a strained InGaAs layer is embedded between (Al)GaAs particularly along the [1 1 1] direction, the induced strain results in stronger dielectric polarization. Therefore, the growth of AlGaAs/ InGaAs QCLs on (1 1 1) GaAs surfaces potentially provides greater flexibility in the nonlinear characteristics of these structures. InGaAs/GaAs quantum well structures and QC lasers are mostly grown on the (0 0 1). However, growth on (1 1 1) substrates also has a significant advantage in the form of critical layer thickness (CLT). According to Anan et al. [15–17], the CLT for the (1 1 1) orientation is around three times greater than that for a (0 0 1) and two times greater than that for (1 0 0) orientations. Growth on the (1 1 1) surfaces enhances strain and piezoelectric effects and provides an additional degree of freedom for QC lasers device designers. Interest in the optical properties of films grown on misoriented substrates is increasing and a greater understanding of the piezoelectric effect in QC lasers could potentially lead to QC lasers emission below 4 lm range. 2. Modeling approach 2.1. Device structure Since most III–V and II–VI compounds (including GaAs and InP) are non-centro-symmetric materials, these materials can give rise to nonlinear susceptibility resulting in second-harmonic generation as long as the orientation of the susceptibility is nonzero with electric field. Therefore, laser structures under study consist of AlGaAs/GaAs superlattices and AlGaAs/InGaAs active regions with varying indium composition. The following is the injector/active region structure (Fig. 1) in nanometers starting from the injection barrier: 4.6/1.9/1.1/5.4/1.1/4.8/2.8/3.4/1.7/3.0/1.8/2.8/2.0/3.0/2.6/ 3.0. The bold fonts represent the three wells for the active region. The structures were assumed to be pseudomorphic and simulated

Table 1 Linear and quadratic piezoelectric coefficients (C/m2) along the [1 1 1] direction as calculated from density functional theory

GaAs InAs

e14

B114

B124

B156

0.230 0.115

0.439 0.531

3.765 4.076

0.492 0.120

GaAs (3.0nm) Al.45Ga.55As (2.6nm) GaAs (3.0nm) Al.45Ga.55As (2.0nm) GaAs (2.8nm) Al.45Ga.55As (1.8nm) GaAs (3.0nm) Al.45Ga.55As (1.7nm) GaAs (3.4nm) Al.45Ga.55As (2.8nm) InxGa1-xAs (4.8nm) Al.45Ga.55As (1.1nm) active region

InxGa1-xAs (5.4nm) Al.45Ga.55As (1.1nm) InxGa1-xAs (1.9nm) Al.45Ga.55As (4.6nm) (111) GaAs substrate

Fig. 1. Simulated QC laser device structure.

along different crystal planes (1 0 0) and (1 1 1). The simulated structures include a three-quantum well active region, but with indium composition that varies from 0% to 6% in the InGaAs matrix and 45% Al-content in the AlGaAs barriers. It is estimated that up to 0.4% of strain is induced with the addition of indium to the quantum wells in the active region. 2.2. Device simulation In this study, a self-consistent Schrodinger–Poisson solver is employed to examine a strained three-quantum well QC laser design, which potentially emits at short wavelength via second-harmonic and sum-frequency generation. The design parameters selected for these simulations include indium composition in the GaAs quantum well, electric field, and crystal orientation. The importance of these factors involves the ability to enhance nonlinear susceptibility, increase intersubband spacing, and explore tunability via applied electric field [11,18–20]. The device structures were computed using an effective mass approximation of the envelope function with plots of the relevant moduli squared of the wavefunctions and Dirichlet boundary. Calculations of direct and indirect energy gaps, their temperature dependences, and conduction and valence band deformation potentials that account for strain effects in pseudomorphic thin layers [17] are also incorporated in this simulation. The three-quantum well QC laser design is similar to [21], except device structures in this study are simulated on GaAs along (1 1 1) orientation. The key parameters in this study include crystal orientation, indium composition in the GaAs quantum well of the active region, optical dipole moment (l), oscillator strength (f), fundamental mode (E32 = E3 ? E2), secondharmonic (E42 = E4 ? E2), and applied electric field. Critical design

Table 2 Critical design parameters for the simulated AlGaAs/InGaAs structures Design parameters

Range

Units

Crystal orientation Indium composition Electric field

(1 0 0), (1 1 1) 0–6 40–53

% kV/cm

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1.5 1.4

4

1.2 3

1.1

2

1.0 0.9

1

0.8 0.7

3. Modeling results 0.6

T = 300 K, electric field: 52 kV/cm In0.05Ga0.95As/Al0.45Ga0.55As on (100) GaAs substrate λ = 7.7 μm SHG (λ = 4.0μm)

-20

-10

0 10 20 distance (nm)

30

40

Fig. 3. Simulated AlGaAs/GaAs injector with an AlGaAs/InxGa1xAs (x = 0.04) active region on a (1 0 0) GaAs substrate.

1.5 1.4 In0.04Ga0.96As Al0.45Ga0.55As

1.3

energy (eV)

Shown in Figs. 2–6 are the conduction-band-energy diagrams of QC lasers with plots of the moduli squared of the relevant wave functions. In Figs. 2–5, an electric field of F = 52 kV/cm was applied. In Fig. 6 42 kV/cm was applied. In order to achieve a larger conduction band offset at a fixed aluminum content of 45%, indium was added to three GaAs quantum wells in the active region, where optical transitions occur. No indium was added to the injector region to restrict strain to the active region only. In Fig. 2, simulation of an AlGaAs/GaAs injector with an AlGaAs/ GaAs active region on (1 1 1) GaAs is shown. In comparison to structures on (1 0 0), there are notable changes in the slope of the conduction band for these structures, and the E3 level is further lowered into the conduction band. The change in slope is associated with more dominant piezoelectric properties stemming from the (1 1 1) orientation. Using the (1 1 1) growth plane, the optical transition, E32, increases in energy from 9 to 8.3 lm. In Fig. 3, a structure with 4% indium in the active region on (1 0 0) is shown. Here, the E32 transition increases in energy from 9 to 7.8 lm, which is attributed to increases in intersubband spacing in the InGaAs well and the conduction band offset consistent with a smaller InAs energy gap material. In Figs. 4–6 for various indium compositions (4%–6%), the E32 transition is further increased to at least 7.2 lm due to the addition of indium in the active region and (1 1 1) GaAs substrate. This is expected as increasing mole fractions of indium in the InGaAs matrix will lower the quantum well energy gap and increase the conduction band offset, DEc. Along with an increase in the E32 transition, a resonance between E2 and an upper level, E4, is observed. This resonance corresponds with a frequency nearly twice that of the fun-

In 0.04Ga 0.96As Al0.45Ga 0.55 As

1.3

energy (eV)

parameters are summarized in Table 2. Crystal orientation include (1 0 0) and (1 1 1) GaAs, indium composition in the active region varied from null to 6%, and the device structures were simulated using a range of applied electric fields. The sum-frequency generated is attributed to the nonlinearities in the active regions, which can be associated with pumping of the electrons from the E3 ? E4 state, the use of indium in the active wells, and choice of orientation. The fundamental and second-harmonic modes were calculated according to the wavefunctions simulated and evaluated based on the optical dipole moment oscillator strength and intersubband energy separation. Corresponding nonlinear susceptibility v(2) (2x) is extracted from data using Eq. (1).

1.2

4

1.1

3 2

1.0 0.9 1

0.8 0.7 0.6 0.5

T = 300 K, electric field: 52 kV/cm In0.04Ga0.96As/Al0.45Ga0.55As on (111) GaAs substrate λ = 7.5 μm SHG (λ = 4 μm)

-20

-10

0

10

20

30

40

distance (nm) Fig. 4. Simulated AlGaAs/GaAs injector with an AlGaAs/InxGa1xAs (x = 0.04) active region on a (1 1 1) GaAs substrate.

1.5 1.3

1.4 GaAs Al Ga As 0.45 0.55

1.3

1.1

4

1.2

1.0

3

0.9

energy (eV)

energy (eV)

1.2

2

0.8

1

0.5

Al0.45Ga0.55As

4 3

1.1

2

1.0 0.9

1

0.8

0.7 0.6

In0.05Ga0.95As

0.7

T= 300K, electric field = 52 kV/cm λ = 8.3 μm GaAs/Al 0.45 Ga0.55 As on (111) GaAs substrate

T = 300 K, electric field: 52 kV/cm

0.6 In0.05Ga0.95As/Al0.45Ga0.55As on (111) GaAs substrate λ = 7.6μm SHG (λ = 4.0μm)

0.5 -20

-10

0

10

20

30

40

distance (nm) Fig. 2. Simulated AlGaAs/GaAs injector with an AlGaAs/GaAs active region (without indium) on (1 1 1) GaAs substrate.

-20

-10

0

10 20 distance (nm)

30

40

Fig. 5. Simulated AlGaAs/GaAs injector with an AlGaAs/InxGa1xAs (x = 0.05) active region on a (1 1 1) GaAs substrate.

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19.0

1.4

18.5 In0.06Ga0.94As Al0.45Ga0.55As

1.3

energy (eV)

intersubband dipole moment Å

1.5

1.2

4

1.1

3

1.0

2

0.9 1

0.8 0.7 0.6

T = 300 K, electric field: 42 kV/cm In0.06Ga0.96As/Al0.45Ga0.55As on (111) GaAs substrate λ = 7.38μm SHG (λ = 3.8μm)

-20

-10

0

10

20

30

18.0 17.5 17.0 16.5 16.0 15.5 15.0 14.5 14.0

4% indium on (111) 4% indium on (100) 0% indium on (100)

13.5 13.0

40

38

40

42

distance (nm)

44

46

48

50

52

54

electric field (kV/cm)

Fig. 6. Simulated AlGaAs/GaAs injector with an AlGaAs/InxGa1xAs (x = 0.06) active region on a (1 1 1) GaAs substrate.

Fig. 8. Influence of electric field on the intersubband dipole moment for 0% and 4% on both (1 0 0) and (1 1 1).

damental mode, E32. The wavelengths for the E42 transitions are found to be in the 4 lm wavelength range although the E4 state remains in the quasi continuum state, as shown in Figs. 4–6. The influence of electric field on the dipole moment and oscillator strength from the fundamental mode, E32, were also explored and are illustrated in Figs. 7–9. As anticipated, there exists a general trend that shows increasing E32 energy transitions with higher electric field and indium composition. Moreover, we are able to achieve much higher transition energy when moving from a structure that contains 4% indium on (1 0 0) to one with the same composition, but on the (1 1 1). This is significant because not only are we able to increase the fundamental transition energy but by using the (1 1 1) plane, we are also able to accommodate strained layers of InGaAs without compromising the transition energy. The increase in intersubband transition energy is observable for structures that include various indium composition in the active region on both (1 0 0) and (1 1 1) surfaces (Fig. 7). Comparison of the influence of electric field on structures with 0% and 4% indium on (1 1 1) shows comparable maximum dipole moment and oscillator strength near 1.8 nm and 0.9, respectively (Figs. 8 and 9). For structures that contain 4% indium in the active region on (1 0 0), there seems to be a reduction in its dipole moment and oscillatory strength compared to those without

indium in the active region on (1 0 0). However, when the same structure is simulated on (1 1 1) with 4% indium in the active region, its oscillator strength and dipole moment are increased above those data obtained for structures with 0% and 4% indium on (1 0 0). This effect can be attributed to the piezoelectric field, which is strongest when directed along (1 1 1) plane. Overall, structures simulated on (1 1 1) GaAs with indium ranging from 0%–6% in the active region include a variation in dipole moment and oscillator strength as a function of the applied electric fields. However, there may exist additional limitations regarding indium composition and applied electric field. In Fig. 6, which contains 6% indium in the active region, the simulated structures were limited to electric fields no higher than 45 kV/cm. For applied fields beyond this value, the injector and upper transition are not aligned and the device becomes non-operational. The higher electric fields enhance a stark-shift in the conduction band, which is induced by the piezoelectric field due to the (1 1 1) plane. Although, these strain-induced built-in fields provide an extra degree of freedom for bandgap engineering, they can also result in misalignment of the injector and upper laser level, E3 [22]. A similar scenario is observed in structures containing 4% and 5 % indium at higher fields. Moreover, these structures exhibit local maximum values of the dipole moment and oscillator strength at various electric fields,

4% indium on (111) 4% indium on (100) 0% indium on (100)

0.165

0.95 0.90 0.85

oscillator strength

energy (eV)

0.160

0.155

0.150

0.145

0.80 0.75 0.70 0.65 0.60

4% indium on (111) 4% indium on (100) 0% indium on (100)

0.55

0.140

0.50

38

40

42

44

46

48

50

52

54

electric field (kV/cm) Fig. 7. Influence of electric field on the energy for 0% and 4% on both (1 0 0) and (1 1 1).

38

40

42

44

46

48

50

52

54

electric field (kV/cm) Fig. 9. Influence of electric field on the oscillator strength for 0% and 4% on both (1 0 0) and (1 1 1).

D. Roberts, G. Triplett / Solid-State Electronics 52 (2008) 1669–1673

while their respective E32 energy transitions consistently increases with increasing field. This data suggests that these structures can be tuned via applied electric field with improvements to dipole moment and oscillator strength. The susceptibility values were also explored in this study. Susceptibility values were extracted from matrix elements of the i ? j intersubband transitions and with the use of half width at half maximum data [23] at electric field, F = 52 kV/cm. Assuming Ciii  Ciiii ¼ C where hC = 7 meV, N = 8  1017 cm3 and the corresponding optical dipole moments, hziii i ¼ 1:7 nm;hziiiii i ¼ 0:31 nm and hziii i ¼ 0:18 nm for AlGaAs/InxGa1xAs (x = 0.06), the nonlinear susceptibility is v(2) = 1.36  10(8) m/V. By comparison, the value estimated in [23] is greater by one order of magnitude. However, the values obtained in this study are approximately two orders above the nonlinear susceptibility of bulk GaAs (v(2)  3.8  108 m/V) [23]. The estimated nonlinear susceptibility values for AlGaAs/InxGa1xAs, where x = 0.05 becomes v(2)  1.31  10(8) m/V. At x = 0.04, the nonlinear susceptibility reduces to v(2)  4.85  10(9) m/V. These values are comparable to those obtained for InP-based SHG devices [24] for various applied electric fields. The nonlinear susceptibility data for those devices range from 0.5 to 5.8  108 m/V. Due to misalignment of injector and upper transitions bands for 6% indium composition in these structures, there appears to be a local maximum for v that lays between 0% and 5% indium composition for devices with applied field, F = 52 kV/cm. Above 5% indium composition, the stark-shift becomes more dominant. The range from 0% to 6% will be further explored through adjustment of the well thickness and other QCL designs. 4. Conclusion This paper explores enhancement of nonlinear susceptibility in quantum cascade lasers (QCLs) that lead to unique intracavity characteristics that include frequency mixing producing both fundamental and higher order modes. The QCL design includes an AlGaAs/InGaAs active region between AlGaAs/GaAs injectors on a [1 1 1] GaAs matrix for the purpose of enhancing nonlinear susceptibility. Results demonstrate that second-order harmonics can be observed with an AlGaAs/InGaAs design on the (1 1 1) crystal growth plane. Future work should include appropriate phase matching around the cavity that will lead to higher optical power levels. References [1] Faist J, Capasso F, Sivco D, Sirtori C, Hutchinson A, Cho A. Quantum cascade laser. Science 1994;264:53.

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