The Simulation Standard Page 10 April, May, June 2013

TCAD Simulation of Multiple Quantum Well Infrared Photodetector (QWIP)

AbstractWe demonstrate the capabilities of SILVACO TCAD tools for the design and simulation of intersubband GaAs/Al-GaAs multiple quantum well infrared photodetectors. We compute the photoconductive gain spectrum of the de-vice self consistently with the charge flow, non-radiative capture-escape, and intersubband transitions in the ac-tive region. The solid state calculations provide input to a light propagation modeling tool that allows for multiple reflections in the layered geometry of the active region. From this full calculation, we extract light absorption and quantum efficiency as a function of the angle of incidence and the wavelength of a mono-spectral beam of light. This TCAD simulation can be used as a starting point by de-sign engineers for obtaining guide lines to analyze and optimize quantum well infrared photodetectors.Keywords: Intersubband transitions, QWIP, Capture-Escape, k.p,

TCAD, Zincblende (z.b), Superlattice

IntroductionThe development of infrared photodetectors (IR) was started after the Second World War, and it has since been progressing at a rapid pace. Research in this field has al-ready made important technological impact in thermal imaging, night vision, and other IR applications. The first and second generation IR detectors exploited the intraband transitions, and were dominated by InSb and HgCdTe (also referred to as MCT). Both these materials could be used for detectors operating in the mid-wave-length regime (MWIR), with wavelength in the range of at 2-5 μm. The long-wavelength regime (LWIR) region is accessible only to MCT via its tailorable energy band gap over the range of 1–30 μm, large optical absorption for high quantum efficiency, and favorable intrinsic re-combination mechanisms. The MCT based IR technol-ogy is often considered matured. However uniformity and yield continue to be major limitations for MCT due to a weak Hg-Te bond that results in bulk, surface and interface instabilities. These limitations provide strong motivation to search for alternatives to MCT.

Multiple quantum well infrared detectors (QWIP) based on GaAs are an attractive alternative due to the well-de-veloped growth and processing technology. They have shown promising results, comparable with HgCdTe photodiodes at low temperatures in LWIR and very long wavelength regions (VLWIR). GaAs based QWIP tech-nology also provides the flexibility of monolithic inte-gration with GaAs electronics and into large focal plane arrays.

The principle underlying the operation of a QWIP is that of a quantum particle in a box. A set of quantum wells, with geometry and composition chosen to support only one bound state, collectively form a superlattice that supports a quasi-continuum of extended states carrying finite momentum. As a result, a bound state electron ex-cited into an extended state by a photon generates a finite photocurrent. Thus the bound to extended transitions form the fundamental absorption mechanism that deter-mines both the wavelength and the quantum efficiency of the detector. Perhaps the most attractive feature of a QWIP in this respect is the fact that absorption spectrum can be tuned by quantum well thickness and the barrier composition. On the other hand, the requirement of at least 30-50 wells in a single device is still a limiting fac-tor, which arises from the limited ability to control the growth over such thicknesses.

Technology Computer Aided Design by SILVACO pro-vides a flexible, cost-effective, and a rapid alternative for research and design optimization of QWIP. It allows re-searchers to gain insight into the complex physics inside a QWIP, and provides the ability to predict the material and geometrical parameters to create a device with the desired operating characteristics. This results in a signif-icantly more effective and efficient use of resources for device fabrication.

Simulation ConditionsThe structure of multiple quantum well we simulate consists of three GaAs quantum wells (n-type doped with concentration of 1.2×1018 cm-3), sandwiched between the wide band gap AlxGa1-xAs (x=0.25) layers, which act as barrier for the bound state carriers. The barriers should ideally be intrinsic, but we model them more realistically with n-type doping of 1011 cm-3. The thickness of quantum well is 4 nm, which we determined by lowering the thickness until only one bound state was supported. We chose the thickness of the barriers to be 50 nm to mini-mize the coupling of bound states in adjacent quantum wells, thereby making the wells virtually independent of each other. Inter-well coupling leads to inefficiencies due to increased radiative and non-radiative tunneling, and affects the spectrum in ways that are difficult to control as a function of electrical conditions. The structure of the active region and its electrostatic conditions may be directly compared with the middle of the 50 well region in the structure studied by Bethea et al [2].

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To bias the active region, we include 100 nm thick GaAs n-type contacts at the top and bottom of the active re-gion, and doped to the same concentration as the wells. The conduction band diagram with electron quasi-Fermi are shown in the Figure 1. In Figure 2, we show the pro-files of bound states acting as the initial states, and in Figure 3 we show the profile of a typical traveling state acting as the final state in optical absorption. The bound states decay sufficiently rapidly in the barriers that the well states can indeed be considered decoupled. In Figure 3 we note that the extended states are character-ized by highly oscillatory structure. Since the Schroding-er equation solves (see below) for the entire profile, these oscillations require a mesh with large spatial resolution.

We simulated 3 quantum wells instead of the full 50 well device in [2] because the large oscillations of quantum states require spatial resolution of less than 1 Angstrom, such that the simulation domain becomes limited by the

CPU, memory, and simulation time. However, this does not affect the conclusions reached in the simulation due to the fact that the quantum wells are independent to a very good approximation. Thus their optical response is additive. An additional difference arises from the fact that the quasi-continuum of extended states of the full device is replaced by a discrete spectrum in the smaller device modeled here. This is easily accounted for with a physically reasonable yet sufficiently large broadening of absorption line shape. Nonetheless, the users also have the ability to model absorption into a 1D continuum with its edge at the first extended state above each well.

We found from the simulation that the highly oscillatory nature of the extended states results in a very slow conver-gence of the magnitude of the optical transition matrix el-ements with respect to spatial resolution. This only affects the magnitude of the spectral response, while its shape converges rapidly at much coarser mesh spacing. Thus the meshes or nodes play a significant role for the accurate estimation of the range of detectable wavelengths, as well as the peak detection wavelength. The present simulation also provides an accurate estimate of quantum efficiency as a function of the incidence angle, taking into account both the fact that intersubband transitions are driven purely by TM modes (see below), as well as the effects of multiple reflections at material interfaces. Depending on the desired accuracy of magnitude of efficiency per well, users may need to further refine the mesh, resulting in longer simulation times.

The Schrodinger and Poisson equations may introduce non-linear corrections as the active region is lengthened. However, the electrostatic potential obtained from the self consistent solution provides us with a measure to gauge the importance of these corrections. This is so be-cause the potential essentially determines the location

Figure 2. Bound state wavefunctions in each of the three GaAs quantum wells. The wells support only one bound state.

Figure 3. The lowest travelling state wavefunction above the top of the leftmost quantum well. Note the abrupt phase shifts at well boundaries and the increasing oscillation frequency ap-proaching towards the collector. Both these require a high spa-tial resolution of the mesh.

Figure 1. Spatial profiles of conduction band edge and electron quasi Fermi level for a QWIP biased at 0.15 V. The structure consists of 3 GaAs quantum wells of thickness 4 nm separated by Al0.25GaAs0.75.

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of energy levels and the wavefunction profiles, which in turn completely determine the intersubband transitions and absorption spectrum. From Figure 1 we note that the quasi Fermi-level shows a linear drop of 0.15 V across the active region, which translates to about 3 V across the 50 well counterpart of this active region. Thus the electro-static potentials relative to the well depth, and thus the Hamiltonians governing quantum states in each well, are expected to be similar in both cases.

Calculation MethodologyThe quantum well modeling, including its optical re-sponse, is activated by specifying the QWELL model for the quantum well regions. The model discretizes the Schrödinger equation along the direction specified by SP.GEOM, which in this case is along the length of the active region. The discretized equation is solved via the method of finite differences. The spatial dependence of effective mass is taken into account via the current con-servation boundary conditions.

The SLATT model activates the calculation of extended states over the union of regions for which this model is specified. This model also solves a discretized Schro-dinger equation, but the solutions at each energy satisfy boundary conditions of a plane wave of the same energy escaping into the contact regions. The effective mass and band edge parameters are derived from the multiband k.p model. The bound wavefunction is assumed to van-ish outside this range. Users have the option to set the wave function penetration length in microns via the WELL.MARGIN parameter. In the present case, the pen-etration length into barriers is 0.01 μm.

The carriers in the states above the barrier band edge comprise a current flow that is essentially 3D and semi-classical in nature. The carriers from these states are captured by bound states via phonon emission, and they escape from these bound states via phonon absorption. This in- and out-flow of carriers is included in the self consistent drift-diffusion model for the entire device by selecting the WELL.CAPT model, and is specified by a phenomenological time constant (WELL.TAUN) for carrier capture lifetime. This introduces an addi-tional balance equation between the 3D semi-classical and 2D quantum mechanical bound state densities. The WELL.CAPT model modifies the conduction and va-lance bands in the barrier region to avoid bound states at edges. The resulting coupled drift-diffusion, Poisson, and Schrödinger equations are all solved until self con-sistency is reached amongst them at each bias point.

The QWELL model automatically searches for all bound states, and includes in the optical response only up to a maximum of WELL.CNBS many states of the conduction subbands. If the number of bound states, say B, support-ed (as dictated by the Hamiltonian) is less than WELL.

CNBS, then all B bound states are reported. In the pres-ent case, we use this feature to verify that only one bound state is supported by physical conditions of well thickness and material affinities. We set WELL.CNBS=2, and then verify below (see Results and Discussion) that only one state is indeed reported. The users can adjust the barrier height via the AFFINITY parameter on the MATERIAL statement and also verify that more bound states appear in the results as barrier heights increase.

The multiple quantum well structure is illuminated by a black body source at room temperature. Planck’s for-mula completely specifies the spectral distribution of a black body in terms of the wavelength λ (in μm) and the temperature T (in Kelvin). The absorption of this light occurs via intersubband transitions from the bound to extended states in each quantum well, and the presence of WELL.CAPT parameter specifies this connection be-tween the QWELL and SLATT models.

To calculate the spectrally resolved absorption of long wavelength infrared radiation, the models consider the electromagnetic field to be a plane wave of frequency ω = 2πc/ λ. The coupling between the electronic states and the electric field, E, is described by the interaction Hamiltonian,

(1)

where dnm are the vectorial matrix elements of the dipole operator between the electronic states n,m. In the present simulation both n and m represent the different sub-bands of the same conduction band, in other words, the matrix elements dnm represent intersubband transitions.

When differences between the effective masses of the con-duction bands in the wells and the barriers do not change the wavefunctions significantly, the intersubband transi-tions yield dnm = dnm s, where s is the direction normal to the surface and dnm is the magnitude of the matrix element. Since s.E yields the TM component of E, we obtain the fa-miliar experimental fact that only the TM component of the incident light contributes to intersubband transitions. The steady state TM gain coefficient is extracted using the Fermi Golden rule, and takes the form,

(2)

Note that the gain formula as reported in literature is dif-ferent from the one in Eq. (2) in that the gain is expressed in terms of the photon energy associated with incident wave, instead of its frequency. Here, nr is the real part of refractive index, and is the dipole matrix element be-tween initial and final states, and in the direction perpen-dicular to the plane of the quantum wells. is the density of transverse states in two dimensions (a constant), and

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Γ is the line width due to Lorentzian broadening arising from the scattering of the electrons with lattice vibra-tions, as well as collisions with interface roughness. Em,n is the difference between energies En and Em of the initial and final subbands of the same carrier type respectively. fn and fm are the Fermi-Dirac distribution functions for the initial and final intersubband states, and integrated over the entire transverse energy range Et.

Equation (2) gives the number of photons with TM polar-ization generated by inter-subband electronic transitions per unit length. For a beam of light incident at an angle with respect to the normal to the well, the dependence of s.E on the incidence angle with the TM component of the field yields the final gain coefficient, which is also input to the optical propagation module, as

(3)

The factor of ½ in this formula arises from taking the beam to be unpolarized [2]. The incidence angle specifies only the plane of the electric field, and for a polarized beam for which the angle of the field within this plane is φ, the inner product |s.E|2=sin2(θ)cos2(φ). An unpolarized beam contains a uni-form distribution of all φ, so that integrating the gain over all cos2(φ) over 0 to 360 degrees produces the factor ½.

Results and DiscussionWe begin with a discussion of the electronic transitions. Figure 4 shows the TM gain for the device. Note that absorption is simply the negative of the gain, and it is clear from the figure that the device is absorptive over the entire range of wavelengths shown. The spectral bandwidth for absorption is approximately 2 μm, while the peak absorption occurs at approximately 10 μm. The spectral broadening in this calculation is 10 meV, and ab-sorption above each well is limited to 5 traveling states above the local barrier height.

Due to thermal excitation, a very small fraction of car-riers is expected to populate the extended states at 300 K, and contribute to dark current. The downward tran-sitions from these states to the bound states are domi-nated by phonons with a time constant WELL.TAUN = 1 ps. However, there is also very small contribution from emission of photons representing the reverse process of photocurrent generation. This radiative capture rate is spectrally resolved in Figure 5 and its integral with re-spect to energy yields a total radiative time constant of 100 ps. Thus we also confirm that, as is normal in QWIP systems, the radiative capture is of negligible importance compared to non-radiative capture emission included in the self consistent model.

The absorption results can be clearly interpreted in terms of the energies of the bound states, which in turn can be directly related to quantum well widths and material composition of barriers (in particular the barrier height). We can thus tune the peak detection wavelength by sim-ply adjusting the well width.

Figure 4. TM gain versus wavelength. Note that negative gain simply means positive abosroption of the same magnitude.

Figure 5. Spectrally resolved radiative capture rate, which when integrated over the shown energy range yields an effective time constant of about 100 ps.

Figure 6. Total absorption as a function angle of incidence for an unpolarized mono spectral beam of wavelength 10 µm. Note that the ratio of photocurrents represents quantum efficiency. The vertical axis is dimensionless.

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We now show the results of total absorption of a mono spectral beam of light incident on this device at various angles and wavelengths. Figure 6 shows absorption and the ratio of available to source photocurrent as a func-tion of the angle of angle of incidence. The source pho-tocurrent is the incident beam power converted to units of electrical current, while the available photocurrent is the photocurrent generated by absorption. Thus in the absence of other loss mechanisms for IR photons, and the generated electrons, the ratio simply reproduces the absorption curve.

In Figure 6, the dependence of absorption on the angle of incidence θ shows the characteristic sin2(θ) form at small angles. We note that the maximum lies around 70 de-grees, and not at 90 degrees as would be expected from sin2(θ) alone. As the angle of incidence increases the beam propagates a greater distance along the surface, and thus a lower fraction interacts with the wells before being reflected back out of the device. The Luminous module in Atlas properly accounts for all the reflections to compute the actual angular dependence of the total absorption, and it clearly shows the expected drop in ab-sorption as compared sin2(θ) as incidence angles reach 60 degrees and beyond.

Figure 7 shows results of the calculation in which we vary the wavelength of the mono spectral beam while keeping its incidence angle at 30 degrees with respect to the normal. Here, as expected, we see that the total ab-sorption follows the gain curves since the intersubband transitions are the only source optical loss in the spectral range shown. The peak of the curve is again near 10 μm, and the curve follows the gain curve of the well nearest the optical source. Thus most of the total photocurrent is generated from this well.

We have assumed that all the photons absorbed in the quantum well excite the bound carriers and generate a photo current. The internal quantum efficiency of quantum well infrared detector can thus be computed by dividing the Source Photo Current to the Available Photo Current.

In the present simulation, only 3 wells are considered, and clearly, the maximum internal quantum efficiency η for these wells is very low. However, as pointed out above, addition of more wells would result in a linear increase in absorption with respect to the number of wells. This also explains why 50 or even more quantum wells are required to obtain a practical detector, as also reported in literature.

The spectral current responsivity of the quantum well infrared detector is the figure of merit and may be cal-culated from the gain and quantum efficiency data. The spectral current responsivity of QWIP is

(4)

Here, λ is the wavelength, η is the quantum efficiency, h is the Planck’s constant, c is the velocity of light, q is the electron charge, and G is the photoelectric current gain.

The dark current characteristics of a QWIP is shown in Figure 8 This can be performed by performing a series of simulations with the same bias voltage scan but different temperature specified on the models statement. The cur-rent plotted on a logarithmic scale is computed for a large bias range than in order to display its typical profile. Com-parison f the dark current curves for the present device at 77 K and 300 K shows the temperature dependency. The dark current is almost zero under zero bias condition, as biasing increases the current varies linearly. The curves clearly show the striking increase in dark current at high-er temperature. This results from the much higher ther-mal excitation at higher temperature. On the other hand, the differential dependence on the voltage remains very similar in both temperature regimes.

Figure 7. Total absorption as a function wavelength at the inci-dence angle of 30 degrees with respect to the normal. Note that the ratio of photocurrents represents quantum efficiency. The vertical axis is dimensionless.

Figure 8. Dark current as a function of bias voltage at low and high temperatures.

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The sensitivity of dark current to temperature also re-veals the further need for optimization with respect to the bound states. One the one hand, single bound state is necessary to ensure that all optical transitions generate a current, which increases efficiency. On the other hand, this also prevents us from having a deep bound state from which the thermal excitation can be suppressed even at high temperature. TCAD tools can be highly useful in finding an optimal balance between efficiency, detection wavelength, and dark current. Another prom-ising application of these tools is to search for alternative designs where these issues can be uncoupled and opti-mized independently.

Finally, we comment on various simplifications made in this simulation. We have neglected the effects tunneling assisted excitation, which further modify the response of the detector. The interface between the barriers and wells is taken to be ideal, i.e. without any interfacial de-fects or traps. The defect assisted tunneling is the major noise source in QWIP at low temperatures. We have not taken any noise into consideration in the above discus-sion. Users can incorporate the different types of noise associated with quantum wells using various noise models in the SILVACO model library. Finally, the resis-tance area product can be obtained by the differentiation of the dark current density with respect to voltage.

ConclusionWe demonstrated the use of SILVACO TCAD tools for design and optimization of quantum well infrared pho-todetectors based on GaAs in the long wavelength re-gion. We performed room temperature infrared absorp-tion, and dark current-voltage analysis. Although GaAs based QWIP technology has undergone aggressive de-velopment, improving the quantum efficiency remains a continuing challenge for researchers in this field. TCAD tools provide insight into the physical mechanisms plac-ing fundamental limitations on efficiency, and the ability to develop strategies to minimize their impact.

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