## Abstract

We calculate the loss and confinement factors of modes in terahertz quantum cascade laser structures at frequencies of 1–4 THz. The determination of the total loss splits naturally into the calculation of free-carrier losses in the active region and the waveguide losses. For both, we employ the Drude model. In the case of waveguide losses, we incorporate it into the formalism of the optical scattering matrix and trace the net threshold gain for laser operation in the waveguide for various frequencies as a function of thickness and doping of the buried contact layer. The results indicate that at lower frequencies and high doping, the preferred mode switches character from extended to tightly confined. This may have consequences for the creation of simplified longer-wavelength devices.

©2004 Optical Society of America

## 1. Introduction

Since the demonstration of the first mid-infrared quantum cascade laser (QCL) by Faist *et al*. [1], a race has ensued to extend this operational concept to ever-longer wavelengths where compact sources are rare. With increasing wavelengths, new types of waveguides with single-[2] and double-plasmon [3] confinement had to be developed to ensure sufficiently good overlap of the laser mode with the gain region. The first successful QCL in the terahertz (THz) frequency range, i.e., operating at photon energies below the optical-phonon energy of the host material, was implemented by Köhler *et al*. [4]. It was based on the single-plasmon waveguide design involving a thin doped contact layer above an intrinsic substrate in order to minimize free-carrier absorption. Rochat *et al*. [5] developed a QCL at 4.5 THz using a double-plasmon waveguide, where the THz mode is confined between a metal layer and a highly doped substrate, but at the cost of higher losses. Another approach was demonstrated by Williams *et al*. [6], who thinned the THz QCL device down to the active region alone and contacted it on both sides with thick gold layers, thus achieving high mode confinement and low losses. Motivated by these designs, we set out to explore such waveguide structures with respect to minimizing the threshold gain in the active layer. We find that lower frequencies require smaller net threshold gains in order to overcome waveguide losses, and that increased doping and contact layer thicknesses favor a second mode [7,8] which shows higher confinement and in the limit of infinitely thick doped substrates becomes the double-plasmon mode. For both modes, free-carrier absorption in the active region appears to be the ultimately limiting factor to QCL operation at lower frequencies.

## 2. Theoretical approach

#### 2.1 Layer structure

The structure which we analyzed is shown in Fig. 1. We assumed an undoped active region of varying thickness, a contact region with varying thickness and doping, and an undoped substrate region. The entire structure was enclosed between semi-infinite gold layers to ensure confined modes in the semiconductor layers.

#### 2.2 Scattering matrix

We chose the scattering matrix formalism introduced by Ko and Inkson [9] for electrons in heterostructures and applied by Ko and Sambles [10] and Ager and Hughes [11] to optical transmission through multilayer systems. Its strong advantage over the transfer matrix treatment [12] is numerical stability in the presence of evanescent waves [10], a fact which allows the identification of the strongly confined mode discussed below. The formalism starts with the standard transfer matrix treatment, in which tangential *E* and *H* fields are matched across boundaries. With each new layer, a scattering matrix is updated which relates the incoming field components to the outgoing ones for the whole layer subsystem. In the case of confined modes, the determinant of the scattering matrix diverges, further simplifying the identification of propagation constants relative to the transfer matrix treatment.

#### 2.3 Drude model and material parameters

The scattering matrix algorithm was implemented in MATLAB, and the positions of the poles in the determinant of the scattering matrix were tracked as a function of the various device parameters. The dielectric functions were given within the Drude model as *ε* = *ε*
_{0}
*ε*_{b}
(1-${\omega}_{\mathit{\text{pl}}}^{2}$/(*ω*
^{2} + *iω*/*τ*)) , where *ω* is the angular frequency of the radiation, *ε*_{b}
(*ε*
_{0}) the relative (free-space) dielectric constant, *ω*_{pl}
=(*ne*
^{2}/(*ε*
_{0}
*ε*_{b}
*m*))^{1/2} the plasma frequency, *e* the elementary charge, *n* the carrier concentration, *m* (*m*
_{0}) the effective (free) electron mass, and *τ* the momentum relaxation time which is related to the DC mobility over *μ* = *e*
^{2}
*τ*/*m* . The parameters for gold were taken as *n* = 5.9×10^{22} cm^{-3}, *τ* = 21 fs, *ε*_{b}
=8, and *m* = *m*
_{0} [13]. For the semiconductor, the values of GaAs were taken, with *ε*_{b}
= 12.4 , *m* = 0.067*m*
_{0}, and *τ* = 100 fs [14].

#### 2.4 Bulk and modal quantities

With the complex propagation constant *β* ≡ *k* + *iα* obtained for each mode (*k* being the wave number and *α* the field attenuation constant), the field in the layer structure was reconstructed. Next, the intensity profile of the z-polarized electric field (in growth direction) was determined and its confinement factor Γ was calculated, defined as the fractional overlap of |*E*_{z}
|^{2} with the active region.

With respect to the total attenuation, we treated losses by free-carrier absorption in the active layer and waveguide losses separately. The waveguide loss *l*
_{w} was calculated by assuming a laser structure with vanishing carrier density in the active region. The modal free-carrier loss *l*
_{fc} arising from the doping in the active region was determined by *l*
_{fc} = (Γ*L*
_{fc}), with *L*
_{fc} =[2*ω*
^{2}(|*ε*|-Re(*ε*))/(*c*
^{2}
*ε*
_{0})^{1/2} being the Drude expression for the frequency-dependent material (bulk) free-carrier attenuation. This assumes that the Drude model holds in coupled quantum well layers and that the effective dimensionality of the carriers is three, an assumption which was demonstrated in a superlattice [15] and which is plausible in QCLs. For frequencies above the plasma frequency, free-carrier absorption displays the known *λ*
^{2} -behavior: assuming an average doping of 4×10^{15} cm^{-3} [4], the free carrier losses are 6, 23, and 103 cm^{-1} at 4, 2, and 1 THz, respectively. This increasing loss is perhaps the key limiting factor regarding the extension of the QCL concept to lower frequencies, often far outweighing the waveguide losses.

#### 2.5 Threshold gain

In order to achieve laser action, a minimal (threshold) modal gain *g*
^{th} = *l*
_{w} + *l*
_{fc} is required. The corresponding material threshold gain is *G*
_{th} = *g*
_{th} /Γ = 2 |*α*|/Γ. If only waveguide losses are considered, this reduces to *G*
_{w,th} = *l*
_{w} /Γ . This quantity is the inverse of the modal quality factor discussed in this connection [6–8].

For the waveguide modes investigated in the following, the assumption of a gain-independent Γ is adequate. For the highest-loss cases considered, explicit calculation showed that “gain guiding” can increase the modal overlap by ten percent, but as these modes are technologically unfeasible, this effect was neglected.

## 3. Results

#### 3.1 Two laser modes

We performed the above calculations on the two modes with the largest real parts of the propagation constant, both of which simultaneously have the highest confinement factors and require the lowest threshold gains. The character of the two modes in the case of a transparent active region is shown in Fig. 1. Mode 0, which is almost entirely confined between the buried contact layer and the top gold cladding, is similar to the double-plasmon mode existing in a subwavelength semiconductor layer between two semi-infinite conductors [6] or between a metallic top layer and a buried metallic layer [7,8]. Its attenuation in the contact layer is quite high but the mode confinement approaches 100%, thus enhancing its response to material gain. Almost no z-polarized electric field extends into the substrate, and this tight confinement implies a high divergence of the beam coupled out of the laser. The next mode, Mode 1, corresponds to that found in the first THz-QCL structure of Ref. [4]. Here the intensity of the z-component of the electric field strongly extends into the substrate. Due to the low modal overlap with the doped contact layer, the associated absorption losses are considerably lower than for Mode 0, making it the first mode to reach threshold in THz QCLs with buried-contact layer thicknesses below 1 μm. In addition, higher substrate modes exist as well but show little modal overlap with the active region and do not reach laser threshold.

#### 3.2 Variation of active-layer thickness

As a first analysis, we hold the thickness of the buried contact constant at 1 μm and investigate the influence of the active-layer thickness on the threshold gain. The relevant plots of *G*
_{w,th} = *l*
_{w} /Γ (i.e., considering the influence of the waveguide alone and neglecting
free-carrier absorption in the active region) are shown in Fig. 2 for a frequency of 2 THz. The data for 1 and 4 THz do not differ significantly from those shown in this graph. It is apparent that for Mode 1, increasing the active layer thickness *d*
_{active} brings advantages at all doping levels, with the threshold gain scaling as 1/*d*
_{active}. This is a direct consequence of the increasing mode overlap, which is approximately linear in the active-layer thickness. Mode 0, on the other hand, is insensitive to the active-layer thickness. This is a consequence of its nearly full confinement within the active layer, independent of its thickness. In fact, as the active layer thickness increases, a slight increase in the threshold gain may be observed at low doping values.

#### 3.3 Variation of contact doping and thickness

Next, results in the form of the threshold gain as a function of contact doping density and thickness are presented in Fig. 3. We assumed a constant active layer thickness of 10 μm, roughly equal to that found in the literature [4,6]. For reasons of numerical stability, the thickness of the intrinsic substrate layer was chosen as 20 μm for tightly confined modes and 200 μm for loosely confined modes. The salient result is that at larger contact-layer thicknesses, the preferred mode switches from Mode 1 over to Mode 0.

In the plot of *G*_{w,th}
= *l*
_{w} /Γ for the 4-THz modes, one sees that at a contact doping level of 2×10^{18} cm^{-3} and thickness of 0.8 μm as chosen by Köhler *et al*. [4], the lowest threshold is indeed displayed by Mode 1. In this point, our value for the modal attenuation coefficient *l*
_{w} is 13 cm^{-1} which yields a total attenuation *l*
_{w} + (Γ*L*
_{fc}) of 16 cm^{-1}(with Γ = 0.47) and matches the value of 16 cm^{-1} reported in Ref. [4]. At the chosen doping density, the threshold gain is close to a local minimum: reducing the doping density by a factor of 2.5 results in an increased threshold by a factor of 1.5, while doubling of the doping density results in an increased threshold by a factor of 2. Considering the inherent uncertainty involved in doping MBE-grown layers, as well as effects such as carrier depletion and freeze-out, this fact might explain the reported absence of lasing in nominally identical QCL structures. Should one impose the restraint of a constant areal density of carriers (*nd*) = (2.5×10^{18} cm^{-3}) (0.8×10^{-4} cm), the doubling of *d* requires a greater threshold by a factor of 25, while the halving *d* requires a smaller threshold by a factor of 5. For Mode 1, this means that for a given contact resistivity, thin, highly doped contact layers are at an advantage. Finally, it is apparent that above contact thicknesses of 2 μm, Mode 0 prevails with values of *G*
_{w,th} down to 35 cm^{-1}, yielding a threshold material gain *G*
_{w,th} + *L*
_{fc} of 41 cm^{-1} which is not much higher than the corresponding value of 34 cm^{-1} for Mode 1. This implies that a simpler realization of a THz QCL involving a fully doped substrate may be feasible. This approach would circumvent the need for wet etching to contact buried layers, or for mechanically thinning the substrate for high confinement. However, one should work with substrate doping levels exceeding 5×10^{18} cm^{-3} .

In the plot for a frequency of 2 THz, one observes a slight but general drop in the threshold of both modes at all doping levels. In the contact thickness interval between 0.5 and 1 μm, the most advantageous doping level for Mode 1 is around 1×10^{18} cm^{-3}. However, this represents a weak minimum, and the difference to neighboring doping levels is not marked. The operating point of Mode 1 corresponding to the layer parameters used in [4] requires a threshold gain *G*
_{w,th} of 13.7 cm^{-1}. Again, Mode 0 becomes favored when *d* is enlarged, exhibiting threshold gains down to 26 cm^{-1}. Compared to such values, bulk free-carrier loss with a value of 63 cm^{-1} is now becoming dominant. Nevertheless, the use of a QCL operating at 2.3 THz has recently been reported [16], where a magnetic field was used for carrier confinement.

At 1 THz, the overall threshold falls yet again. Mode 1 is again favored by small contact layer thicknesses. There, for fixed thickness, an optimum in the doping density no longer exists. Inserting the layer parameters of Ref. [4] gives a threshold gain *G*
_{w,th} of 7.4 cm^{-1}. At buried-contact thicknesses above 1.5 μm and at higher doping densities, Mode 0 prevails with a reasonably small threshold down to 22 cm^{-1}. One has to note, however, that the waveguide losses are now negligible compared to the bulk free-carrier loss in the active region of 103 cm^{-1}. To achieve laser action at 1 THz appears extremely challenging, requiring reduced dimensionality either by magnetic or size confinement.

Over the investigated parameter range and frequencies, the lowest absolute thresholds are exhibited by Mode 1. Above doping ranges of 1×10^{18} cm^{-3}, the threshold of Mode 1 generally scales with *n* , while the threshold of Mode 0 goes as 1/ *n* . The fact that counter to general assumption, waveguide losses do not vary inversely with the squared frequency is due to the fact that the operating frequencies lie far below the plasma frequencies prevailing in the contact layers and gold cladding. Such behavior in waveguide losses has been reported elsewhere [17]. In layers with sufficiently light doping, a loss component inversely proportional to squared frequency is observed [18], an expected behavior when free-carrier absorption in the active layer is included in the calculations.

## 4. Conclusion

Using the optical scattering matrix formalism we have calculated the attenuation coefficient and the mode confinement for the two modes with the lowest threshold gain in a heterostructure consisting of a thin intrinsic (“active”) layer, a doped thin buried (“contact”) layer, and an extended intrinsic substrate layer, all enclosed between two semi-infinite gold layers. These numbers were combined to yield the minimal gain in the active region in order to achieve laser operation. This quantity was tracked as a function of 1) thickness of the active region, 2) thickness of the contact region, 3) doping of the contact region, and 4) frequency. The central result of our analysis is the existence of a crossover in the threshold gain between an extended mode (Mode 1) and a fully confined mode (Mode 0), with the consequence that the operation of the fully confined mode might be possible in a simplified structure with an *n*
^{++} doped substrate.

## Acknowledgment

Funding by the DFG (contract RO 770/16) and ESA (project 16863/02/NL/PA “Optical Far-IR Wave Generation”) is gratefully acknowledged.

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