**8. Modeling and simulation approaches for nanostructures: quantum effects, light-matter interaction, carrier transport**

Modeling approaches in nanostructures can be roughly divided into three categories of techniques, dealing with the different particles that come into picture in the detector physics. We broadly divide the discussion into bosonic and fermionic particles, as their physics are very different and therefore their modeling approaches have differed in their developments. It is impossible to cover all these myriad approaches in a single volume monolog, much less in a section of a chapter, therefore we provide only a very brief overview of these approaches along with some textbook references that can serve as starting point for deeper education and understanding.

For "bosonic" particles such as photons, plasmons (and various associated polaritons), phonons etc. it is customary to use a wavelike approach i.e. by solving a wave equation in a classical physics sense [32], and it is rarely necessary to approach them with quantum field theoretic approaches, i.e. second quantization methods. Photons are indeed extremely well modeled using a classical Lagrangian written by

formalizing the Maxwell's equation in terms of a 4-vector gauge theory [33]. Similar approaches work very well for plasmons, which are the collective oscillations of electrons, or phonons which are mechanical oscillations of the material lattice. Both plasmons and phonons show a dispersion relation, i.e. an energy-momentum relation with similar structure (optical and acoustic branches) and therefore are amenable to a unified approach to modeling and simulations.

This above fact is exploited by the many available computational electromagnetics software that convert the wave equation, which is a vector PDE boundary value problem into a linear algebra problem using the technique of Finite Element Analysis (FEA). This approach converts a physical domain into a collection of packed small triangular (most widely used shape in FEA) regions over which the solution is considered to be constant. In one iteration the solution over the "finite elements" are updated based on the previous step values, and this iteration continues till the error value over the whole grid reaches a minimum threshold, set as per the specifications of the problem. See [34] for a classic text on FEA. Other similar approaches include finite difference method, which instead of creating a mesh of triangles that can cover any possible physical geometry, uses a Cartesian grid on which the derivatives are discretized over "grid points" instead of triangular regions as used in FEA, and the problem is converted into a linear algebra problem. An associated method called the finite volume method (FVM) generalizes FEA and FEM to 3D problems. However these approaches are general enough that they can be extended to any dimensions, with associated approaches to discretization. The relative "mechanistic" approach of these methods that depend on linear algebraic methods have allowed development of extremely power simulators, including Ansys HFSS, COMSOL Multi-Physics, MATLAB etc. which can now utilize GPUs to accelerate the solution of very large problems.

For "fermionic" particles such as electrons and holes, the approaches differ in the sets of equations which are solved, depending on the application. Traditionally Boltzmann transport equation (BTE) has been a popular method in solving electron transport problems. However powerful many-body quantum statistical mechanics approaches have been developed as well, the central one being the Non-Equilibrium Green's Function (NEGF) method that has shown tremendous success in solving the problem of electron transport in complex material stacks such as superlattices and nanostructures. A classic exposition that covers both these approaches in an extremely approachable way is [35]. We next describe the fundamental principles underlying these methods and their application spaces.

The central entity of the Boltzman transport equation is the carrier distribution function *f t*ð Þ , *p*, *q* which is a function of time *t*, generalized position *p*, associated momentum *q*. The equation can be written as:

$$
\dot{f} + \dot{p}\nabla\_{\mathfrak{P}}f + \dot{q}\nabla\_{\mathfrak{P}}f = \mathbb{S}f \tag{13}
$$

Where ∇*f* denotes the gradient of *f* with respect to the subscripted variable, *S* is a scattering operator that mixes the distribution function in various *p* and *q* states, depending on the particular problem being addressed. In general solving this problem for even one particle involves keeping track of the distribution function in a 7D phase-space (6D in a steady state problem), which can quickly become intractable for even moderate sized problems, especially interacting systems. Therefore, the most commonly used approach to solve this PDE is to use Monte Carlo methods which allows us to construct the full solution by evaluating only certain trajectories through the phase-space, instead of calculating the full phase-space. Careful application of importance sampling can lead to fast tractable solutions to *f* from which it is possible to then calculate current-voltage relations (first moment), noise currents *Physics of Nanostructure Design for Infrared Detectors DOI: http://dx.doi.org/10.5772/intechopen.101196*

(second moment), and even higher order moments as necessary. It is possible to also calculate heat flows by energy weighted first moment of *f*.

NEGF method can be seen as a quantum analog of Boltzman transport equation. The central quantity of the interest in this formalism are the retarded Green's function *G<sup>r</sup>* , which the "causal" propagator of the carrier in the system, and the correlation function *G*<sup>&</sup>lt; which is the quantum density matrix. The NEGF method evolves the *G*<sup>&</sup>lt; under the influence of various scattering phenomena (Büttiker probes are a good example) under the action of the *G<sup>r</sup>* . The two equations are written as:

$$\mathbf{G}^r = \left[\mathbf{E} - H - \Sigma\right]^{-1} \tag{14}$$

$$-iG^{<} = G^{r} \Sigma^{in} G^{a} \tag{15}$$

Where *E* is the energy, *H* is the system Hamiltonian, Σ is the total "self energy" for all the scattering mechanisms applicable to the system, including physical contacts, momentum scattering, dephasing, inelastic scattering etc. Σ*in* is the corresponding sum of the "inflows" from these mechanisms into the system. We leave the details of how these may be written to the reference mentioned above and other follow on works from the author, a recognized authority on the subject.

The NEGF approach is well suited to handle any arbitrary combinations of material stacks, i.e. an arbitrary *H*, not necessarily confined to materials with parabolic bands. This method also forgoes any equilibrium or near equilibrium assumption that is commonly applied in BTE. Transport phenomena of lower dimensional, and indeed any quantum system is automatically incorporated in the NEGF approach. This makes NEGF a very powerful tool for modern detector structures which are increasingly based on superlattices. However, incorporating appropriate self-energy and inflow terms in the NEGF equation requires a nontrivial intellectual effort. As the field develops further we expect phenomena such as Auger recombination, Impact ionization, SRH recombination, multi-electron generation etc. will be included systematically in the toolset of NEGF methods, which will then yield a truly bottom-up quantum mechanical simulation approach for photonic devices.

#### **9. Metasurfaces for LWIR devices**

In this section, we now explore the possibility of exploiting the fast-developing area of metasurfaces—more generally known as flat optics—to add functionality to the existing IR sensors without affecting the performance or increasing SWaP. In addition, metasurfaces offer a possibility of increasing the temperature of operation. In reference to **Figure 1b**, the metasurface-coupled detector is an emerging technology to enhance optical absorption in a thin detector architecture. This becomes even more important where absorber layer is naturally thin—such as quantum dots, 2D materials— or require to be thin bulk absorber to minimize dark current and overall detector materials noise.

As the electromagnetic radiation passes through a resonant element, both its amplitude and phase undergo a large change. Interestingly the phase can be varied from 0 to π either for the frequencies in the vicinity of the resonant frequency as shown in **Figure 4** or equivalently for element size in the vicinity of size that is resonant at a given frequency. Particularly the latter observation along with little or no variation in amplitude can be exploited to introduce phase variation in the XYplane (for the light propagating in Z direction) of the optoelectronic device. With

**Figure 4.**

appropriate phase gradient on a dielectric surface—called metasurface—and the generalized Fresnel's equations [36, 37], we can change the propagation wave front to arbitrary shape and direction, leading to a number of interesting applications [36, 37].

Interestingly, the functions such as focusing, defocusing, total reflection, total transmission, total absorption, frequency filtering, non-specular reflection and/or transmission, polarization filtering, which often requires bulky optical elements, can be achieved with sub-wavelength-thick metasurface [38–44]. The metaelements include sub-wavelength and appropriately shaped metal—known as plasmonics—and all-dielectric, size-dependent Mie resonant scatterer. The advantage of metal plasmonics is that the size can be exceedingly small (often 1/10th of the wavelength) and field concentration/redirection can be huge. However, ohmic loss in metals often affects the optical absorption required in the optical devices. On the other hand, Mie resonant scatterer with a transparent dielectric can provide required phase change without any absorption but requires high index contrast for efficient light modulation. Since the Mie element size is λ/*n* (where *λ* is the design wavelength and *n* is refractive index of the dielectric) for resonance, the metasurface unit cell can contain only a few elements and hence a broadband design is often difficult. For illustration, we consider three applications—near-perfect reflection [42, 43] polarization filter [44] and broadband absorption [45, 46] for possible high temperature operation.

Our work in metasurface area is motivated by adding functionalities and operability to the existing IR devices. For example, a metasurface for near-perfect narrow band reflection. Highly reflective surfaces are of great interest for several applications, including sensor and eye protection under laser illumination, hardware hardening against laser irradiation, and optical elements in high-energy laser systems. In these applications, the surfaces require high reflectivity under intense illumination with little or no absorption or transmission. To protect the IR device from the high intensity radiation at a wavelength in the absorbing band, an ultrathin metasurface can be fabricated on the optical devices and the elements of the metasurface can be chosen to fully reflect the high intensity wavelength. We have used shapedependent Mie scattering to design ultra-thin metasurfaces and demonstrated [43] near-perfect reflection (>99.5%) in a narrow band (around 1550 nm) as shown in **Figure 5**. Note that 500 nm-tall Si nanopillars (with index 3.5) on low index SiO2 (index 1.45) are sufficient to achieve the high reflection without any metal component. The design principle is applicable to all reflection windows with appropriate changes to the element size and materials set.

*Physics of Nanostructure Design for Infrared Detectors DOI: http://dx.doi.org/10.5772/intechopen.101196*

**Figure 5.** *(a) Fabricated Si nanopillars on SiO2 and (b) the simulated (green) and measured (black) reflectance [43].*

The constituent materials for short-wavelength infrared to the long-wavelength infrared, can be, for example Si, InP, or GaAs for the high index and ZnS, BaF2, CaF2, or MgF2 for the low index materials that have high transparency with little or no absorption to 12 μm.

The polarization of the radiation can be exploited to improve imaging quality, particularly when the thermal contrast between a target and its background is insufficient for polarization-insensitive sensors [47]. Two kinds of polarizers absorptive polarizers and polarization beam splitters—are currently used. Absorptive polarizers such as wire grids, dichroic materials, or nanoparticle composites provide high degrees of polarization, but both polarizations cannot be analyzed simultaneously as needed in applications such as imaging and quantum information. The polarization beam splitters such as birefringent cubes or Brewster angle reflection in multilayer dielectric films preserve the rejected polarization by reflection or diffraction. However, they are bulky for chip-scale photonic and optoelectronic devices. We have slightly modified the design of **Figure 5** to introduce lattice asymmetry as shown in **Figure 6a** so that resonance frequency for S- and Ppolarizations are separated. Consequently, one polarization is reflected nearperfectly while the other is transmitted near totally as demonstrated (**Figure 6b**). The small difference in the transmitted component between simulations (dashed) and measured (solid) are because of over-etching of SiO2 around the pillars which can be corrected. Notice that mere 500 nm-thick metasurface, which can be monolithically integrated with the sensor can provide near perfect polarization splitting to add polarimetric functionality to the existing IR devices. The concept can be extended to other wavelength bands with appropriate choice of transparent materials. As can be noted, the reflectivity changes in a narrow band and the structures are useful in applications where the large bandwidth is not a requirement. However, several options including the optimization in height to diameter ratio, periodicity of the pillars, multiple element with differing size in the unit cell are available to increase the bandwidth.

Metasurfaces for broadband absorption and transmission will be more useful in IR sensors in increasing the operating temperature. We discuss the issues limiting the temperature of operations and then some metasurface designs that offer the possibility of addressing these issues.

Sensing technology in the infrared (IR) spectrum enables substantial advances in several applications including security, surveillance, industrial monitoring, and

**Figure 6.**

*(a) Fabricated Si nanopillars on SiO2 and (b) the simulated (dashed) and measured (solid) polarizationdependent reflectance [44].*

autonomous vehicle navigation. The reduction in SWaP of these sensors without sacrificing performance is particularly important. Among various IR technologies, long wavelength infrared (LWIR) imaging systems are crucial for target-acquisition tasks because of their advantages in adverse environments. Current detection and imaging technologies that cover the LWIR spectral regions typically operate at cryogenic temperatures of 77 K or lower. This requires expensive and bulky cooling systems that increase the overall SWaP of imagers. Uncooled LWIR detector would greatly alleviate the SWaP constraint and enhance the imaging system's portability, enabling broad applications in surveillance and reconnaissance under adverse conditions.

The increase in operating temperatures often requires unfavorable tradeoff between performance, cost, and power consumption. In photon detectors, the dark current increases exponentially with temperature as depicted in Eq. (6), resulting in more power consumption, high noise, deeper cooling requirements and thus poorer performance. Uncooled microbolometers for imaging have a considerably lower performance as compared to cooled photodetectors. Novel and nanocrystalline materials such as colloidal quantum dots (CQDs) show promise, but the synthesis of large sized particles for absorption in the LWIR with fewer surface states needs to

#### *Physics of Nanostructure Design for Infrared Detectors DOI: http://dx.doi.org/10.5772/intechopen.101196*

be mastered. There is a large number of publications [48, 49] demonstrating LWIR sensing at room temperature with 2D materials—often exploiting high absorption in thin layers and the Schottky barrier between them and Si—and also by measuring the changing properties resulting from high absorption in structured graphene [50], for example. While these results are encouraging, considerable improvements are needed to increase the QE—because only carriers within a few nm from the interface participate in absorption—and the path forward for integrating with ROICs for imaging is not clear.

For many years, there has been a continuing effort to increase the operating temperatures (*Td*) of IR detectors and arrays with conventional materials such as HgCdTe and type II superlattices (T2SL). The efforts include high-quality materials to reduce defect-mediated dark currents, low n-doping to achieve high depletion and suppress Auger and radiative-mediated dark currents, strained layer superlattices to reduce the Auger mediated dark currents, novel device architectures such as nBn and CBIRD, and others to reduce the diffusion currents, as well as optical immersion lenses, photon trapping structures, and plasmonic structures to reduce the collection volume and thus dark currents. However, the epitaxial layers of LWIR photodetectors are still very thick, requiring long growth times and creating sensitivity to chamber and source maintenance, inevitably promoting the numbers of SRH centers. Beyond some critical photodetector thickness, dark current begins to dominate and the signal-to-noise ratio of a photodetector decreases.

Current LWIR sensors employing p-n junctions require absorbers with a thickness equal to or greater than the targeted cut off wavelength (12 μm-thick for LWIR) for 80% absorption of the incident radiation [51]. However, as mentioned before, the thick absorption region results in a large dark current, arising from generation mediated by the Auger, radiative and Shockley-Read-Hall (SRH) mechanisms [52]. Efforts to reduce the dark current via higher bias (for depletion) increases the band-to-band and/or trap-assisted tunneling and thus do not offer a robust solution [53]. For example, higher depletion by very low doping of LWIR detectors did not increase the operating temperatures beyond 100 K [54]. Various prior efforts to pattern the entry surface—known as photon trapping or PT structures— succeeded in minimizing Fresnel reflection but achieved only a moderate decrease in absorption region thickness and a slight increase in operating temperature of larger band gap mid-wavelength infrared (MWIR) devices to near 200 K [55–57]. Since the absorption in the structured surface did not increase the photocurrent, the overall QE actually becomes poorer.

Recently, LWIR and MWIR photodetectors based on T2SLs, especially Ga-free ones, have shown promising performance [58]. However, the LWIR operation temperature is still limited to 77 K [59]. GaSb-based T2SLs have a higher operation temperature, but the temperature increase (up to 110 K) is still modest for a 10-μm cut-off [60]. In addition, epitaxially grown quantum dots (QDs), which are sometimes coupled with quantum wells, have been used to increase the operation temperature up to 200 K [61]. The challenge of epitaxial QDs lies in their low QE. CQDs have shown promise in the MWIR with a high QE. Still, the dark current and noise are high owing to their poor transport properties [62]. The requirements for high temperature of operation are illustrated [63] in **Figure 7**. The assumptions in this illustrative calculation include—8–10 μm spectral band, *f*/1 optics, 10 μm detector cut off, 70% in-band QE and 19 Me-integrated charge (30 Me-well). We note that when the absorber thickness *t* is 10 μm, the lowest noise equivalent differential temperature (NEdT) of 30 mK (red) is above the preferred value of 20 mK. When the collection volume is reduced *without loss* of QE (blue), we can increase *Td* by about 20–40 K. However, when this reduction in volume is combined with Auger suppression by a factor of 50, for example expected in superlattices because of flat

**Figure 7.** *Effect of collection volume and auger on detector performance [63].*

bands or in a lightly depleted absorber, the *Td* can be increased (dotted) to 270 K. More specifically, the designs for high temperature operation should meet the following criteria:


Having examined the aforementioned efforts of improving the operation temperature of LWIR photodetectors, we conclude that any approach to increase the operating temperature will require a substantial reduction in absorption thickness as we know thinner absorber reduces the collection volume, is easier to fully deplete with small voltages, and result in few defect centers.

The resonant pixel approach [64] developed by Choi et al. attempts to reduce the collection volume by allowing the light to enter through thin metallic layer and force it to go through multiple reflection by sandwiching the absorber between two metal layers. This effort has been successful, for example, in reducing the absorber thickness to 0.7–1.2 μm and still achieve quantum efficiency of 50–70% with about 2 μm bandwidth in LWIR region [63, 65]. The use of metal, required for resonant cavity, causes ohmic loss and further increase in bandwidth and quantum efficiency have not yet been demonstrated.

Several metasurfaces are proposed or demonstrated to achieve broadband absorption. However most of them use metallic elements, limiting the absorption in the dielectric absorber layer which cannot increase the photocurrent, and are not considered here. We will discuss two promising all-dielectric designs [45, 46]. To enable high absorption for solar application with thin MoS2 layers, the authors [45] proposed a structure shown in **Figure 8a** with corresponding absorption measurements shown in **Figure 8b**. Although this structure resembles that of Fabry-Perot cavity structure, they designed a photonic band gap (PBG) in standoff oxide layer to increase the absorption in MoS2 through increasing the interaction with guided

*Physics of Nanostructure Design for Infrared Detectors DOI: http://dx.doi.org/10.5772/intechopen.101196*

**Figure 8.**

*(a) Modified Fabry-Perot structure to include a photonic band gap design (periodic holes) in the oxide layer and (b) the predicted absorption spectrum (blue) in the monolayer MoS2 [45].*

resonance of the PBG. With a single layer MoS2 (of thickness 0.6 nm) and an absorption coefficient of <sup>2</sup> <sup>10</sup><sup>5</sup> cm<sup>1</sup> , a broadband average of 50% absorption in the visible is predicted. The scalable design can be extended to mid- and long-wave infrared region and even with 100 smaller absorption coefficient, the absorber thickness can be estimated to be 65 nm for this level of absorption. However, a detailed modeling absorption dispersion in the LWIR band is required to ensure that guided mode traverses through the entire absorber thickness required for even higher QE.

Interestingly, broadband all-dielectric transmitting metasurface—in the visible band—has been demonstrated [46]. Appropriately designed Si nanopillars on Si absorber (**Figure 9a**) has been fabricated (**Figure 9b**) and reflectance has been measured to be very low and broad band (**Figure 9c**, black). All the rest are transmitted and absorbed in Si. Notice that absorption is on the average over 95% broadband covering entire visible spectrum. The nanopillars behave like anantenna funneling the radiation over larger area, covering the nanopillar and the surrounding metal in the unit, through the pillars into the absorbing substrate. The funneling effect increases the electric field right below the nanopillars, resulting in absorption

#### **Figure 9.**

*(a) Cross section of the absorber (si) with metasurface (b) fabricated metasurface, and (c) measured reflectance without absorbing nanopillar (red), with absorbing nano pillar (blue) and with antireflection coated (black) [46].*

**Figure 10.** *Predicted absorption spectrum in 2 μm-thick absorber with a metasurface.*

enhancement in this layers. Note another important consequence of the funneling effect is that radiation does not impinge on the metal, allowing the metasurface to be transparent and conduction. This feature is particularly very important for all electrically operated IR devices.

Noting that above design is scalable, we have designed a similar structure with appropriate choice of materials for LWIR and the calculated absorption spectrum with 2 μm-thick absorber is shown in **Figure 10**. We see that about 80% broadband absorption is possible. Absorber without the metasurface will require to be at least 12 μm thick to achieve this level of absorption. In essence, the absorber volume can be reduced by a minimum of 6-fold reducing the dark current proportionately. Further, 2 μm low doped region is far easier to fully deplete, resulting in considerable reduction in Auger scattering, which further reduces the dark currents. Also, smaller bias is sufficient to achieve depletion, reducing the power consumption. The device development is in progress for possible 300 K operation.

#### **10. Summary**

In summary, the fast-developing field of metasurfaces show considerable promise for easier integration with the detector absorber layer to both, increase the SWAP performance and the operating temperature. This chapter discussed various approaches to using nanostructured metasurface designs to increase the photon capture by enhancing the apparent collection area *Aopt* while decreasing the electronic absorption volume, *Aex* . We discussed from ground-up the physics of the photodetection as an interplay of energy transfer from light to matter particles, which may take many routes, which all contribute to the large phase space of photodetector design. We discussed why there is a need for development of ultrathin and even lower dimensional materials for photodetection, to cut down on dark current and the corresponding decrease in the noise. This allows us to trade *D*\* enhancements with higher operating temperatures. We discussed the implications on the light absorption of ultra-thin and low dimensional materials, compared to their bulk counterparts. We also discussed some unique transport signatures that are enabled at low dimensions, including quantized conductance, reduction of flicker noise in ultra-clean 2D materials. We also mentioned issues of material integration and electrical controllability due to contact physics and quantum of capacitance. We briefly discussed the numerical approaches taken for simulation of photonic and electronic components. We provided a detailed exposition of the physics of metasurfaces to bring back the quantum efficiency in ultra-thin and low

#### *Physics of Nanostructure Design for Infrared Detectors DOI: http://dx.doi.org/10.5772/intechopen.101196*

dimensional detectors. We further discussed the increasing trend of using bandstructure engineering (T2SL) in building detectors, in particular nBn and complementary barrier detectors which engineer the electron transport properties of the detector, and how even such "novel" materials benefit from thinness of the detector material in achieving high operating temperature.

We hope that in this chapter we have provided a broad overview of the rich physics of photodetectors to the reader and provided further points to ponder over and references to follow for more in-depth education. The future prospects of infrared detectors are clearly in the direction of nanostructure designs to develop, not only systems that eliminates bulky cryogenic coolers, but also looks into thin sensor architectures that can be conformal. In addition, there will be a significant advancement in the area of focal plane array and readout electronic designs with subsequent system designs that will push intelligence near the sensor. The emergence of Artificial Intelligence, Machine Learning and neuromorphic approaches will also drive more and more functionalities in the sensor that will reduce the data pipeline, post processing and lag associated with current technologies.
