Radiations maximum at the first two sidebands and FNBW for the fundamental patterns of different values of SLL.
\r\n\tMain emphasis should be on its applications. In every field MOFs can be used due to its greater stability and high surface area, but the focus should be on applications.
",isbn:null,printIsbn:null,pdfIsbn:null,doi:null,price:0,priceEur:null,priceUsd:null,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"507abe0040ce1d146a7ed603648a1bb6",bookSignature:"Dr. Shobha Waghmode",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/7388.jpg",keywords:"MOFs, COFs, Nanomaterials, PANI, Agnps",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"September 10th 2018",dateEndSecondStepPublish:"October 1st 2018",dateEndThirdStepPublish:"November 30th 2018",dateEndFourthStepPublish:"February 18th 2019",dateEndFifthStepPublish:"April 19th 2019",remainingDaysToSecondStep:"a year",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,editors:[{id:"197394",title:"Dr.",name:"Shobha",middleName:null,surname:"Waghmode",slug:"shobha-waghmode",fullName:"Shobha Waghmode",profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:"Savitribai Phule Pune University",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"0",totalChapterViews:"0",totalEditedBooks:"0",institution:{name:"University of Pune",institutionURL:null,country:{name:"India"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"8",title:"Chemistry",slug:"chemistry"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"270935",firstName:"Rozmari",lastName:"Marijan",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/270935/images/7974_n.png",email:"rozmari@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. 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Venkateswarlu",coverURL:"https://cdn.intechopen.com/books/images_new/371.jpg",editedByType:"Edited by",editors:[{id:"58592",title:"Dr.",name:"Arun",surname:"Shanker",slug:"arun-shanker",fullName:"Arun Shanker"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"50344",title:"Advanced Scanning Tunneling Microscopy for Nanoscale Analysis of Semiconductor Devices",doi:"10.5772/62552",slug:"advanced-scanning-tunneling-microscopy-for-nanoscale-analysis-of-semiconductor-devices",body:'\nSince invention of solid-state electric junctions, charge carrier distribution has become the primary requirement of electronic device design to achieve desirable device performance. Typically, a spatial distribution of charge carriers in semiconductor devices is created by introduction of electronic impurity atoms with particular electron configuration allowing to donate a free electron to the host semiconductor (donor impurity) or to trap a valence electron (acceptor impurity) from the host material. Thus, the host semiconductor with donor impurity atoms has become a negative-charge (electrons) conductor and is called n-type. The host semiconductor with acceptor impurity atoms has become a positive-charge (holes) conductor and is called p-type. Typical semiconductor devices have concentration of impurity atoms in a range of 1015–1021/cm3, which is less than 1 % of total number of atoms. Defects and atom vacancy often behave like impurity atoms.
\nEarly days, charge carrier distribution was derived from spatial distributions of impurity atoms in semiconductor materials. Secondary ion mass spectrometry (SIMS) has been used to obtain a depth distribution profile of impurity atoms in semiconductor materials by sputtering with high-energy ions. As modern high-performance Si devices such as complementary metal-oxide-semiconductor (CMOS) transistors are less than 100 nm in size, and have complex material structures, the 1D SIMS profiling becomes inadequate. Figure 1 shows a typical structure of a metal-oxide-semiconductor field effect transistor (MOSFET) consisting of gate, channel, and source/drain regions with high impurity concentrations.
\n(a) A sketch of a MOSFET device in a cross-section. (b) 3D view of a charge distribution in a MOSFET measured by the STM gap modulation method. Charge concentration is emphasized by color: blue color for low charge concentration in p-Si channel and red color for high charge concentration in source (S), drain (D), and gate (G).
Recently, a new technique of tree-dimensional (3D) atom mapping, which is called atom probe tomography, was introduced based on counting of atom ions ejected from a needle-like device specimen [1–6]. Aside from complexity of the sample preparation and 3D data reconstruction of the atom probe technique, the charge carrier distribution is assumed to be equal to that of impurity atoms. However, the carrier distribution deviates significantly from the impurity atom distribution as a result of internal electric field at material interfaces, trapped charges in oxide, and fractional activation of impurity atoms in areas of high impurity concentration. Therefore, techniques allowing to measure local distribution of charge carriers within the electronic device interior have been a focus of attention from scientific and practical points of view.
\nSignificant attention has been addressed to high-spatial resolution analysis of modern sub-100-nm electronic devices, nanowire devices which meet miniaturization to less than 10 nm in order to achieve new functions and energy-efficient operation. Last decade, various techniques have been developed for charge carrier mapping. A common high-resolution imaging technique, scanning electron microscopy (SEM), has been upgraded with an energy-filtering option, allowing us to obtain the image contrast as a function of the surface electrostatic potential [7–10].
\nScanning probe techniques are an important tool for local probing of electric properties and have played important roles in scientific research on electronic materials and in evaluations of device structures in fabrication processes. Scanning probe microscopy (SPM) techniques are based on the ability to position a sharp probe electrode in very close proximity with high precision to the sample surface under investigation [11]. Different physical quantities can be measured by the probe including electric tunneling current, atomic and electrostatic forces, or other types of probe-sample interactions. By moving the probe laterally over the sample surface and performing measurements at different locations, two-dimensional distributions of surface atomic structure, electric current, electrostatic potential, or other properties can be obtained.
\nSPM techniques employed in local electrical measurements are atomic force microscopy with a conductive probe (c-AFM) [12], scanning spreading resistance microscopy (SSRM) [13], scanning Kelvin probe microscopy (SKPM) [14], and scanning tunneling microscopy (STM) [15]. These scanning probe techniques create two-dimensional (2D) maps of variations in the surface electric potential or electric current density along a cross-section of a semiconductor device, when the surface states, defects, adsorbates, and foreign particles on the cross-sectional surface do not affect the initial charge carrier distribution. In majority of cases, certain surface treatments of the cross-sectional surface are applied prior to measurements to eliminate undesirable surface effects. Quantitative impurity profiles by SSRM and SKPM have been demonstrated for high impurity concentrations, where a spatial resolution on the order of the probe tip radius (~5 nm) was obtained under optimum conditions [16–19].
\nSTM has been used for impurity distribution measurements in Si devices by analyzing current-voltage spectra [20–23]. To derive quantitatively variation in the charge carrier distribution from STM measurements, one must analyze complex dependence of the tunneling current on the bias voltage, the tunneling gap, and the band-bending potential beneath the STM probe tip on a semiconductor surface. Thus, simulations of STM operation are an essential part of the data analysis.
\nIn this chapter, we focus on advanced STM-based spectroscopy techniques as nanoscale methods for two-dimensional (2D) charge carrier analysis. It represents original development of scanning probe microscopy methods for Si device metrology with ultimate spatial resolution. We describe the principles of the advanced STM methods and give representative examples of applications to nanoscale analysis of Si CMOS devices and nanowires. Advantages, difficulties, and limitations of the advanced STM modes will be discussed in comparison with other techniques used in a field of device metrology.
\nThe chapter begins with description of device cross-section preparation methods and essential features of STM measurements on a semiconductor surface. Measurement principles of original STM-based techniques and application examples will be given. Current development in STM simulations will be outlined. Prospects toward research in new 2D materials will be elaborated.
\n\nFigure 2 shows a common way for making solid crystal cross-sections. The process includes a number of steps. (1) Cleavage and/or dicing of a thin crystal wafer are used to define a desired location of the cross-sectional plane. (2) Chemical-mechanical planarization-polishing (CMP) and focus ion beam (FIB) techniques are applied to tune location of the cross-sectional plane with a sub-micrometer accuracy. (3) Chemical and electric passivation of the cross-sectional Si surface by hydrogenation or thin oxide is carried out to prevent distortion of original charge carrier distributions by surface states and contamination.
\n(a) Fabrication of a device cross-section for STM measurements. (b)–(c) STM images (a set-point: 300 pA, 2.0 V) of an oxide-passivated Si surface before (b) and after (c) 1 ML C60 film formation. Color scale is 0.8 nm (b) and 2.5 nm (c). Insert shows an image of 10 C60 molecules.
Chemical and electric passivation of solid surfaces is the subject of extended research in catalysis to control on charge transfer process and chemical reactions in solid-liquid and solid-gas interfaces [24]. Moreover, chemical and electric passivation of semiconductor surfaces are a basic process in fabrication of modern Si devices, enabling to reduce off-state leakage current and photocarrier losses in solar cells [25]. Without passivation treatment, silicon surfaces have pronounced bands of surface states, which dominate the contrast of the STM images, so that it becomes difficult to characterize the underlying electrical interfaces. Therefore, passivation of Si surfaces by hydrogenation or oxidation has been employed in order to reproducibly prepare uniform surfaces of device cross-sections and to obtain very low density of surface states.
\nHydrogenation of Si surfaces is achieved by etching in fluoric acid solutions. Etching removes the native Si oxide and terminates the Si dangling bonds with hydrogen atoms making a stable, passivated surface with very low density of surface states in the Si band gap [26]. A number of investigations have confirmed that tunneling spectra of such stabilized Si surfaces show variation with dopant type and concentration due to passivation of dangling bond states and the suppression of surface states [27–32].
\n\nSi(111) surfaces can be atomically flattened by wet treatment in NH4F aqueous solutions [33]. In the procedure, the samples were dipped in a 5% HF solution to remove the residual oxide layer, then immersed in a 40% NH4F solution at room temperature, and rinsed in ultrapure water for 1 min. This treatment renders the Si surface mono-hydride, well suited for STM analysis. In this treatment, hydrogen also reacts with near-surface impurity atoms forming electrically inactive complexes, thus, changing the initial charge distribution. To reactivate the impurity atoms, heating of the samples around 200–250°C is necessary [33, 34].
\nTo prepare atomically flat Si(001) surfaces, a combined process is adopted, which consists of wet treatment using a fluoric acid solution and subsequent annealing in H2 atmosphere at ~600°C and a pressure of ~2 × 103 Pa [35]. The authors showed the formation of an atomically flat Si(001) surface that have well-ordered step-terrace structures in the active device area. The flattening was attributed to the enhanced migration of Si atoms when anisotropic etching was suppressed.
\nHydrogenation of Si surfaces may not always be compatible with processing steps in a particular application, as Si surface etching usually introduces topographic contrast due to etching rate dependence on doping concentration, crystal orientation, and material composition. An alternative way to passivate Si surface is oxidation. The passivation of Si surfaces by controlled growth of ultra-thin oxide layer relies on the layer-by-layer oxidation kinetics at low oxygen pressure [36–38]. We adopted the preparation of cross-sectional surfaces of Si devices as follows [39, 40]. First, dicing and ultra-fine polishing are used to expose either (100) or (110) surfaces of the device. The polished surfaces are cleaned by few cycles of etching in dilute fluoric acid solution and wet-oxidation in H2SO4:H2O2 (3:1) solution to remove a damage layer. Finally, ultra-thin (~0.3 nm) oxide layer is grown at ~600°C under an O2 pressure of 3 × 10−3 Pa following etch-cleaning in HF:HCl (1:19). This procedure left a flat surface without any ordered structure as seen in Figure 2(b), where the atomically flat terraces are separated by atomic steps of 0.24–0.27 nm in height. The oxide thickness was 0.32–0.35 nm as determined by x-ray photoelectron spectroscopy, and by scanning reflection electron microscopy (SREM). The low-pressure oxidation process results in a residual density of surface state traps of ~1012 cm−2 for Si(100) surfaces [24, 41, 42], which is suitable for STM spectroscopy analysis.
\nWhen a well-defined mono-molecular layer is prepared on a passivated surface, its molecular level can be utilized to quantitatively analyze the electrical properties of the underlying substrate. We call this method as a molecule-assisted spectroscopy. For this purpose, monomolecular thick films of C60 (fullerene) were formed by vapor sublimation of C60 to the oxidized Si surfaces to a thickness of 3–5 molecular layers. The excess of C60 layers was removed by sample heating at 170–190°C for 10 min. Because electrostatic interaction between the molecule and the underlying Si is stronger than the Van der Waals interaction between molecules within the film, a C60 molecules adjacent to the Si surface remain at high coverage (~80%) as seen in Figure 2(c) [41].
\nThe STM operation principle is based on quantum mechanical phenomenon—electron tunneling through a potential barrier formed by a gap between the outermost atoms on the metal tip and the sample. When the gap is about 1 nm or less, electrons from the STM tip can penetrate into the sample with certain probability owing to the wave nature of the quantum particle.
\nUnder external electric field, electron tunneling creates a measurable electric current, the tunneling current. In the single particle approximation, the tunneling current density is given by a difference in the particle flow across the gap from the STM tip and that from the semiconductor and is expressed as an integral over particle’s energy
where T(Z, E, V), the transmission factor, is a function of gap width (Z), electron energy (E), and external gap voltage (Vgap). ρtip(E) and ρsample(E) are the density of electron states at the surface of the STM tip and the sample, respectively. f(E) is the Fermi function describing which energy states are occupied with electrons.
\nHere, we outline the important features of the STM technique essential for analysis of charge carrier distribution in semiconductors. They are\n
tunneling barrier shape,
sharing of applied voltage between the tunnel gap and a surface band-bending region, and
surface charge density in the semiconductor beneath the STM probe electrode.
The principle of scanning tunneling microscopy of a semiconductor. (a) An STM setup, (b) an energy band diagram of a tunnel junction, and (c) a charge balance diagram.
The tunneling barrier shape determines the electron transmission factor and the value of the tunneling current. Figure 3 shows an STM measurements setup and an energy band diagram of an ideal STM junction for n-type Si. Rectangular shape of the tunneling barrier is used in simple STM models. The actual potential barrier profile is different because of image potential lowering (Δϕ) owing to strong Coulomb interaction between charge and image charge in conductive materials [43–45]. Also, the tunneling gap may include an insulating layer such as ultrathin oxide and a molecular film with different dielectric properties. Therefore, the tunneling electrons experience an effective potential barrier of a barrier height (BH) given by
where ΦM is the work function of the metal tip and EF is the Fermi energy of the semiconductor, q is the elementary charge. For electron energy smaller than BH, the transmission factor is approximated by [46].
The tunneling constant α = 10.2 when the gap width is in units of nanometer and BH – in eV.
\n\nBecause electric charge density in semiconductors is lower than that in metals, applied electric field penetrates deep beneath the semiconductor surface. To maintain the charge neutrality, a band-bending region is created beneath the STM probe. The applied voltage VS is shared between the gap and the band-bending region and is given by
where last term ϕMS = (ΦM − EF) is an electrostatic potential difference between the work function of the STM tip and the semiconductor Fermi energy, and χ is the electron affinity of the semiconductor. In thermal equilibrium and VS = 0 , the charge neutrality is conserved, and the electric charge in the STM tip (QM) is equal to the local electric charge at the semiconductor surface beneath the STM tip. At VS = 0, the band-bending region is created owing to the electrostatic potential difference ϕMS. Figure 3 illustrates the case when an electron depletion region is formed for n-type Si under an external positive bias voltage VS > 0 to the sample. For n-Si, the surface charge (QSS) includes positive charge of impurity atoms (QN) and mobile carriers (holes) (Q+):
According to the Gauss law [43, 47, 48], the voltage across the gap is given by
where ∈0, and ∈Si are the permittivity of the vacuum gap and Si, respectively. The depth of the band-bending region (w) depends on the electric field screening by the electric charge in the semiconductor and is given by
It is straightforward that the tunneling current strongly depends on the local electric charge at the semiconductor surface. When there were surface states and interface traps, these trapped charges would alter the initial charge carrier distribution, and great care must be taken to prepare clean, well-defined cross-sectional surfaces. In fact, conventional furnace oxidation produces a gap-state density of about 1010 cm−2 for Si(100) and less than 1012 cm−2 for Si(111) surfaces [47]. Low-pressure oxidation below 600°C results in a density of ~1012 cm−2 for Si(100) surfaces [26, 41, 42]. The surface oxidation effectively reduces density of surface states on Si surfaces, making that the current behavior becomes dependent on charge carrier concentration in the Si bulk beneath the STM probe.
\nTopographic STM images of a sample surface are formed when the STM probe is moved along the surface while keeping pre-determined tunneling current value (Itun) at an applied voltage (VS) by adjusting the gap width with a piezoelectric scanning unit. The STM technique offers ultimate spatial resolution down to a sub-nanometer range because tunneling current is strongly localized around the outermost atom of the STM tip owing to exponential current decay with the tip-sample distance. Three advanced STM-based modes discussed below rely on measurements and analysis of the tunneling current and, thus, offer high spatial resolution. Details of the SPM system construction and operation have been reviewed in original papers and textbooks [11].
\nTo study charge carrier distribution in semiconductor devices, we describe three STM-based techniques: a vacuum gap modulation method, a molecule-assisted probing method, and a dual-imaging method.
\nA vibrating electrode technique was used to measure the surface potential on solid surfaces by using the Kelvin method [49]. Present-day noncontact atomic force microscopy (nc-AFM) uses vibrating probes for detecting atomic, electrostatic and magnetic forces [50]. In metals, mechanical modulation of the tunnel barrier has been applied as a method to evaluate local work function of the sample [46, 51–54]. In semiconductors, a model of STM junction considering both transparency of the tunnel barrier and the band-bending potential was elaborated [22, 23].
\nWhen the STM probe vibrates normal to the sample surface, the gap width changes as
where ω = 2π ∙ f is the angular frequency, dz is an amplitude of the vibration. For small vibration amplitude, dz ≪ Z0, the transmission factor periodically changes with the time-dependent change of both the gap width and the gap voltage. When the STM probe approaches toward the surface, Vgap is reduced while increasing the surface potential (Vbb). A change of the gap voltage Vgap is related to the mean charge QSS at the surface by the Gauss law [43, 47, 48] and is expressed as
where dψ is a change of the band-bending potential.
\nTo determine the tunneling current response (dI) to a time-dependent variation of the gap width, the tunneling current is expressed as
where I0 is the mean tunneling current. In the linear approximation [46], the current response is dominated by variation of the mean transparency of the vacuum gap. Thus, in-phase amplitude of the tunneling current response is given as
In our experiments, the mean tunneling current I0 is held constant; thus, the quantity (dI1/dz) is proportional to the local charge density at the surface beneath the STM tip under the bias voltage. There is a 90°-phase-shifted current component representing a displacement current owing to change in the STM junction capacitance as discussed in details in Reference [55]. We used the capacitive signal for fine-tuning of the signal phase in the measurements of in-phase current by a lock-in technique.
\nIn the model above, terms due to the shape of the tunnel barrier and capacitance effects associated with modulation of the band-bending region beneath the STM probe are neglected, albeit the effects are essential at high frequency and low impurity concentration [55].
\nWhen the modulation of band-bending region is taken into account, the tunneling current response is given by two terms (Appendix A)
The first term represents the contribution of the gap width modulation, and the second term accounts for variations of Vgap and Vbb .
\nIt is constructive to take a look at origin of charge QSS for n-type and p-type Si under positive bias voltage. In n-Si in Figure 3, the electric field from the STM probe repels mobile electrons deep into the bulk creating a surface depletion region, and QSS = QN + Q+ ≈ QN > 0. The larger the bias voltage, the larger the amount of positive charge accumulated beneath the STM probe. As a consequence, the amplitude of the current response (dI) depends predominantly on density of accumulated positive charge. On the contrary, in p-type Si under the same polarity bias, the electric field attracts mobile majority carriers (holes) to the surface reducing amount of negative charge of acceptor impurities (QP) beneath the STM probe. As a consequence, the amplitude of the current response (dI) depends predominantly on small amount of accumulated positive charge, and QSS = QP + Q+ ≈ Q+. At the position of electrical p-n junction, the balance of positive and negative charges exists, and QSS ≈ 0. Thus, we are able to derive position of electrical p-n junction through analysis of the (dI/dz) profiles. In addition, detection of charge centres near the Si surface at a depth of ~1 nm has been reported for epitaxial Si layers [56].
\nExperimentally, differential tunneling current (dI/dZ) maps were obtained by vibrating the STM probe normal to the sample surface. The STM probe-sample gap was vibrated at a frequency of 12–50 kHz and an amplitude of 20–50 pm while keeping the vacuum gap at constant mean tunneling current I0 (the constant current mode). In-phase current response dI was measured with a lock-in amplifier at each point in the topographical image. The vibration frequency was selected sufficiently larger than the feedback circuit bandwidth (~10 kHz) and away from the electromechanical resonances of the STM measurement system.
\nThe ability of specific molecules to selective reactions on the surface is well known in catalysis. Recently, functionalization of SPM probes by attaching functional groups to achieve the chemical selectivity in recognition of DNA sequences and biological molecules has been performed, for example, see [57–59].
\nThe method described here is different. A molecule-assisted probing method makes use of a discrete energy level of an adsorbed molecule as a marker of the local Fermi energy. It takes advantage of resonant electron tunneling (RET) to monitor the energy level of the marker molecule, such as fullerene C60, introduced into a tunneling barrier between the STM probe and the oxidized Si surface. The fact that the C60-derived conductance peaks shift in energy depending on dopant concentration in the underlying substrate makes this technique usable as a probing method of the charge carrier profiling on semiconductors [39, 41, 60]. The C60 molecule was selected as it satisfies the selection criteria: small size, chemical stability, and an energy position of molecular orbital outside of the Si energy band gap.
\nA model of a double-barrier junction (DBJ) was elaborated based on the theory of planar resonant tunnel diodes [61] and alignment of molecular states [62]. Figure 4(a) and Figure 4(b) show the experimental setup and an energy band diagram of an ideal DBJ consisting of the vacuum gap (B1), the C60 layer and the thin oxide (B2) under a resonant injection bias VRET. EA is the electron affinity of the C60 layer, and Ei is the Fermi energy for intrinsic Si. At the resonance condition, the Fermi energy of the STM tip aligns with the lowest unoccupied molecular orbital (LUMO), and thus, the strength of electric field in the vacuum gap is given by F = (ΦM − EA)/Z0. For an ideal oxide and neutrality of C60, continuity of the electric displacement is preserved across the DBJ, and the RET voltage is given by
where d60 and dox are the thickness of C60 molecule and the oxide, respectively. ∈C60 and ∈ox are the permittivity of C60 and oxide, respectively. Vbb voltage is obtained as a function of the electric field F at the Si surface by solving the 3D Poisson equation at quasi-equilibrium.
\nTo measure the RET voltage, mono-molecular fullerene films were prepared by vapor sublimation of C60 to the oxidized Si surfaces at room temperature followed by re-evaporation of excess molecules as described in Section 2.3. Differential conductance (dI/dV) − V spectra in Figure 4(c) were obtained at a constant probe-sample gap by using a lock-in technique where a small ac voltage (20 mVpp, 50 kHz) was superimposed on the sample bias voltage. The initial tunneling conditions were set with a tunneling current of 200 pA at a set-point voltage of 2.5 V. Each (dI/dV) − V spectrum was fitted to Lorentzian function to determine a voltage of the C60-derived conductance peak, the RET voltage [41, 64]. For high conductance of the tunnel gap, the STM tip is close to the molecule layer, and another transport mechanism, the single electron tunneling [66], becomes apparent and hinders the RET voltage detection. Thus, optimization of the gap width is required.
\nMolecule-assisted probing method. (a) A setup. (b) An energy band diagram of a double-barrier junction under the resonance conditions. B1 is the tunneling gap, and B2 is thin oxide. (c) (dI/dV) spectra of C60 on p-type Si substrates with a boron concentration of 8 × 1014 cm− 3 (curve 1), 4 × 1015 cm− 3(curve 2), 3 × 1018 cm− 3(curve 3), and without C60. (d) RET voltage as a function of the Si Fermi energy (EF − Ei) from measurements (symbols) and 3D numerical calculations for oxide thickness of 0.3 nm (broken line) and 0.7 nm (solid line) according to Eq. (13) and Reference [41].
The measured RET voltage obtained for uniformly doped Si wafers with different dopant concentrations is shown in Figure 4(d). The data are well reproduced by the numerical calculations according to Eq. (13) where STM probe emitter was modeled as a cone with a hemispherical end and a radius of curvature of 10 nm, and Z0 = 1 nm, dC60 = 1 nm , and dox = 0.3 nm, ΦM = 4.5 eV for W(111) probes and EA = 2.6 eV. The good agreement between the calculated RET voltage and the experimental data for uniform-doped wafers verifies the calibration relationship for Si [41, 63].
\nThe spatial resolution of the method is restricted to the size of the marker molecule and to the electric field penetration length. It has been demonstrated by the (dI/dV) mapping that the RET peaks are localized within the C60 core (~1 nm) due to their origin in resonant tunneling mediated by one lowest unoccupied molecular orbital (LUMO+1) of C60 [41]. Since the LUMO+1 was localized at the pentagonal rings [65] and C60 molecule rotates at room temperature, the observed peak intensity represents the orientation-averaged orbital conductance of C60. The estimate of the penetration depth is a Debye length of ~1.5 nm for p-Si under large positive bias, though the length depends on the dopant concentration for n-Si [41, 63].
\nSTM technique is limited to conductive surfaces and is inapplicable to the imaging of novel device structures, including insulator surfaces such as silicon-on-insulator (SOI) devices. Strong interest to such measurements is stimulated by the fact that discrete dopant distribution enables attractive applications such as quantum computing [67] and single-electron devices [68]. Therefore, a dual-imaging method was developed to enable simultaneous measurements of electric current and interaction force acting on the scanning probe. It was achieved by attaching an STM metal tip to a special force sensor [67–76].
\n\nFigure 5 shows the experimental setup for the simultaneous measurement of tunneling current (Itun) and force between the metal probe tip and the Si surface. In our technique, the interaction force gradient between the metal probe tip and the surface was detected as a shift in the resonance frequency (Δf) of a quartz length extension resonator (qLER) which vibrated at ~1 MHz (Q factor ~50,000) with an amplitude of 0.05–0.3 nm [67–70]. The probe tips were made of a tungsten wire with a diameter of 10 μm. The wire was attached to the quartz resonator and sharpened by the focused ion beam technique (FIB). Typically, the probe tips had a diameter of Ø30 nm and the aspect ratio of more than 10, resulting in small stray capacitance. Detection of the frequency shift by electric means makes such sensors suitable for measurements in ultra-high vacuum environment and at different temperature, which are often required in nanomaterial and nanoscale device research.
\nThe advantages of our multimode scanning probe microscopy (MSPM) system are\n
tunneling current and forces acting on the probe tip are measured simultaneously at a mean probe-sample gap of about 1 nm in constant current (CC) or constant force (CF) operation modes;
small vibration amplitude (0.1–0.2 nm) enables us to drastically reduce the probe-sample gap, leading to better spatial resolution;
the sensitivity to electrostatic forces is increased at an optimal gap;
the force detection is performed in a noncontact manner, which is suitable for measurements of solid crystals and thin films.
In the CC mode, a force gradient map is measured while the mean gap (Z0) maintains a set-point tunneling current. Typically, the measurement condition corresponds to a gap of approximately 1 nm, as estimated from the distance dependence of the tunneling current [72]. The spatial variation of the frequency shift (Δf) reflects variations in the interaction force caused by charge carriers, impurity charges, and surface imperfections as illustrated in Figure 5(b). When a donor is present in proximity to the STM tip, the attractive force acting on the tip increases owing to Coulomb interaction between the donor charge and the image charge induced in the STM tip, leading to measurable change in the Δf value [75, 76]. The interaction strength depends on the depth of the donor location and the electrostatic screening by mobile carriers. Experimentally, lateral extent of 5–10 nm and a detection depth of ~1 nm have been reported for phosphorus and boron atoms in Si [32, 33, 76]. Change in the interaction force on grains with different work function was employed for recognizing crystal orientation of sub-10-nm-size grains in nano-crystalline TiN films [77].
\nDual-imaging method. (a) A measurement setup. (b) A sketch of interaction force acting on a vibrating STM probe. (c) (Itun-Z) and (Δf-Z) spectra showing ranges of repulsive interaction (1–2) and attractive Coulomb interaction (2–3) for an oxide-passivated Si(111) surface (a set-point: 30 pA, 2.0 V). (d) A measured (Δf-VS) spectrum at position 3(blue curve), and a result of fitting to Eq. (14) (red curve).
In the CF mode, a tunneling current (Itun) map is measured while the mean gap (Z0) is maintained at a constant frequency shift. There are two ranges in distance dependences of Itun and Δf as indicated in Figure 5(c) for an oxide-passivated Si(111) surface. At short distances (range 1–2), repulsive interaction dominates, and current exponentially grows when the STM tip approaches the surface. At longer distances (range 2–3), the electrostatic Coulomb interaction dominates. There is an optimal distance indicated as position 2 in Figure 5(c) where the sensitivity to electrostatic force is maximum [72]. At this distance, the (Δf − VS) spectrum has the largest curvature.
\n\nUnder the applied voltage VS, the electrostatic force gradient between the probe tip and the sample is expressed according to the theory in References [73, 78] for small vibration amplitude
where C is the effective tip-sample capacitance. CPD, the contact potential difference, refers to the difference between the work function of the metal probe (ΦM) and the Fermi energy of the underlying Si (EF), and is given by
where q is the elementary charge. A local value of the CPD voltage, which is determined by local charge concentration in the underlying Si, can be obtained by fitting of the spectrum to Eq. (14). In the example in Figure 5(d), a CPD voltage of +0.8 V was obtained for an oxidized p-Si(111) surface. The CPD voltage mapping was employed in 2D analysis of the built-in potential in small Si MOSFET devices [79] and p-n junctions [72] showing the attainable spatial resolution better than 3 nm. Particular applications of the CF mode also include analysis of impurity distribution profiles from Itun maps measured at different bias voltage [80], non-uniform distribution of photocarrier in Si stripes [81], and nanoscale conductance switching in phase-change GeSbTe thin films [82].
\nFor STM measurements, cross-sections of Si MOSFETs were prepared by ultra-fine polishing to expose (110) surfaces and were passivated by ultra-thin oxide layer as described in Section 2.2. Si n-type MOSFET with nominal gate lengths (LG) in the range of 20–150 nm were fabricated according to a process described in Reference [83]. The measurements were done with W(111) crystal probes in an ultrahigh vacuum (~4 × 10−9 Pa) at room temperature.
\n(a) Topographic image of a cross-section containing two small Si MOSFET devices. (b) A (dI/dZ) map of a device with a gate length of 31 nm (a set-point: 230 pA, 3.4 V, dz = 20 pm). (c)–(d) Line profiles measured at 12 nm depth beneath the gate electrodes showing the electric channel length (LS-D). (e) Profiles calculated by Eq. (12) for expected impurity distribution. (f) Measured electric channel length (symbols) as a function of gate length. Line is the calculation result.
Topographic image of two small MOSFET is shown in Figure 6(a), where the gate electrodes are surrounded by two black cavities produced by sidewall oxide etching during the surface preparation. The source/drain (S/D) extensions on the left- and right-hand sides of the gate electrode are seen as bright stripes in the (dI/dZ) map in Figure 6(b). Depletion regions separate the S/D extensions from the p-type channel beneath the gate electrode and the Si bulk. The extension depth is ~18 nm as measured from the gate oxide. The electric channel length (LS − D) was determined as the distance between 2 minima in (dI/dZ) line profiles measured at a depth of 12 nm beneath the gate oxide as indicated in Figure 6(c, d). Calculated profiles of the K3 factor in Figure 6(e) reproduce the measured (dI/dZ) profiles, confirming that each minimum in (dI/dZ) signal represents the position of the electric p-n junction. LG was determined from STM topographs. Results summarized in Figure 6(f) give an overlap value of 6 ± 1 nm, which is in excellent agreement with a transverse straggle of 7 nm for an implanted ion energy of 25 keV. An accuracy of the channel measurements was about 1 nm at 3.4 V, while the measurements were affected by random positions of individual ionized dopant atoms in the extension regions.
\n\nThe C60-assisted probing technique has been actually applied to quantitative analysis of charge carrier profiles on cross-sections of power MOSFET, where the precise control over the doping profile is essential to obtain low ON-state resistance and high breakdown voltage [39, 40]. Figure 7(a) depicts a schematic structure of a super-junction power MOSFET. Two p-type islands were formed by multiple boron ion implantations into the low-doped n-type epitaxial layer with a carrier density of ~1 × 1016 cm3. In Figure 7(b), we clearly see that two p-type islands are separately formed with the same peak concentrations, confirming the anticipated dopant concentration. Moreover, the experimental data revealed an extension of island 1 beyond the expected depth, which is attributed to a scatter-less travel of boron ions through Si crystal at high implantation energy, the ion channeling effect[84].
\n(a) Schematic structure of a super-junction device showing two p-Si islands made by boron ion implantation. (b) Depth profiles of the RET voltage along center of the device: measured data (symbols) taken with 20-nm steps. Profiles (lines) were calculated for the two boron density profiles shown in (c). Reproduced with the permission from Reference [39].
The ability of the dual-imaging method for characterization of modern silicon-on-insulator (SOI) devices is illustrated by analysis of the structure and electric conductance of SOI nanowires (NW) with different surface passivation. Note that the NW is the promising structure for sub-10-nm MOSFETs and for such functional devices as chemical sensors. Figure 8 shows high-resolution measurements of a Si NW with a cross-section area of 20 × 20 nm2 acquired at a set point of Δf = 0.6 Hz, dz = 95 pm, VS = − 1.5 V. We see in Figure 8(c) the current gradually decreases in the NW interior with the distance from the Si pad owing to the dependence of the NW resistivity on its length. We note that an apparent NW width in the current map is about 2-fold of that in the topograph. As the NW is protruded above the buried oxide (BOX) by 20 nm, a side surface of the sharp tip touches the NW as illustrated in the insert of Figure 8(c), and this results in a so-called “sidewall” current outside the Si NW body. The current value and fluctuations were reduced for the NW passivated with an ultrathin oxide layer compared to the hydrogen passivation. The tunneling current decreased within a distance of ~300 nm from the Si pad electrode for both types of surface termination. At the negative voltage, the tunneling current is defined by electrons traveling from large Si pad through the SOI nanowire, and the current value is determined by resistivity of the NW volume and the surface conduction. The macroscopic conduction model including the conductance contributions of the nanowire volume and the surface states confirmed the length-dependent conductance of thin Si nanowires [85].
\n(a) An experiment setup. (b) Topographic image of silicon-on-insulator nanowire with a cross-section of 20 × 20 nm2, and (c) corresponding current map acquired at -1.5 V and Δf = 0.6 Hz, dz = 95 pm. (d) Current profiles along A-A line for Si nanowires after hydrogen-passivation (curve 1), oxide passivation (curve 2), and along B-B line (curve 3). Adopted from Reference [85] (Copyright 2013 Trans. Mat. Res. Soc. Japan).
Photo-carrier generation in semiconductors is a fundamental process utilized in solar cells and photo-detectors. For reduced size of modern detectors, the role of structural elements in carrier accumulation and transport has been increasing [86]. In particular, photocarrier distribution on textured surfaces of Si can be a factor to improve the efficiency of solar cells. Analysis of spatial distribution of photocurrent (PC) in strained Si stripes under tilted illumination gives an insight into photocarrier behavior near the stripe edges with an effective spatial resolution of ~10 nm [81].
\n\nFigure 9 shows the sample structure and the measurement setup, where inhomogeneous light intensity profile was created under tilted (50° off-normal) illumination and different light wavelength (λ). Strained Si stripes of 50–1000 nm in width and 300 nm in height were fabricated on Si(001) wafer, and separated by SiO2. The stripe surface was passivated by an ultrathin oxide as described in Section 2.2. The light intensity was mechanically modulated at frequency of ~3 kHz, and the PC signal was measured by a lock-in unit. Topographs and PC maps were measured by the dual-imaging method where the tip-sample gap was set by a set-point of Δf = 1.2 Hz, dz = 130 pm, and VS = − 0.8 V, using the CF mode.
\n\nTopographic image in Figure 9(b) shows uniform surface of the Si stripe. The PC signal was not uniform, and large at a distance of ~50 nm from the stripe edge on the light illumination side, when stripes were illuminated with laser light and an intensity of 12 mW/cm2 as seen in Figure 9(c). Large PC signal at stripe edges was observed irrespective of the scanning directions, when light with λ = 405 and 364 nm was used as seen in line profiles in Figure 9(d, e). In contrast, illumination with red light (λ = 675 nm) produced uniform PC distribution. As the absorption depth in Si is ~11 nm for λ = 364 nm, ~130 nm for λ = 405 nm, and ~4000 nm for λ = 675 nm [87], the respective illumination produces different light intensity profiles. Calculated PC profiles in Figure 9(f) reproduced the observed PC distributions when a rectangular bar geometry, non-coherent light, and a photocarrier diffusion length of 100 nm were used [81].
\n(a) Photocurrent (PC) measurement setup. (b) A topograph and (c) corresponding PC map of a Si stripe under illumination with λ = 405 nm. (d)–(e) Measured line profiles of height (black lines) and PC (dotted lines) across the stripe edge under tilted illumination for two wavelengths (λ). (a set-point: Δf = 1.2 Hz, dZ = 130 pm, VS = − 0.8 V). (f) PC line profiles calculated for a rectangular bar exposed to light at top and side surfaces. Adopted from Reference [81] (Copyright 2012 The Japan Society of Applied Physics).
We note that the relative intensity of a PC peak at a position of ~30 nm for λ = 364 nm is ~3.2-fold the signal in the stripe interior. Enhancement of light intensity by ~3.5-fold at strained Si stripe edges has been reported for λ = 364 nm [88, 89]. The enhancement mechanism may be related to increased photocarrier generation owing to interference of coherent laser light [81], narrowing of the Si energy gap under stress [90] or increase in the tunneling probability through electromagnetic field coupling to the sharp STM tip [91].
\nSTM has the capability to 2D impurity profiling by employing advanced STM methods as shown above. Although, accurate analysis of charge carrier distributions in actual 2D and 3D device structures has been a substantial challenge. STM tunneling current is a complex function of structural, material, and electronic parameters of the system consisting of a 3D probe tip and a semiconductor. On the basis of fundamental theory, there have been theoretical discussions of 1D and 2D treatments for the STM junction geometry. A 3D numerical simulator has been reported that solves the 3D potential distribution of the sample STM probe system and calculates the tunneling current, so-called the potential-based model [23, 92, 93]. However, to describe the precise physics of STM measurements, the charge carrier flow in the sample must be included, as evidenced by the NW measurements in Figure 8. Recently, new model evolves solving the charge carrier transport between a probe tip and a sample consistently with the current continuity equation, so-called the current-continuity model. The current-continuity model accounts for charge carrier transport between states in an STM probe and the conduction and the valence band of Si and was implemented on the basis of a technology computer-aided design (TCAD) semiconductor device simulator code [94]. It is a significant advancement in the field.
\nAn analysis based on the current-continuity model has been applied successfully to extracting impurity distribution profiles in a MOSFET from experimental current maps measured by the dual-imaging method [80], and for evaluating photocarrier dynamics in Si nanowires with a cross-section of 10 × 10 nm2 [95].
\nThe remaining challenge is to include the effect of single impurity scattering on charge carrier transport in nanoscale devices. The impurity scattering for a thin semiconductor wire has been solved using the 3D Green function approach and the numerical Monte-Carlo method [96]. An atomistic view into an impurity atom appearance in STM images has been elaborated within the framework of a self-consistent-charge density functional tight-binding method (SCCDFTB), for example, see [97, 98].
\nAdvanced STM-based methods for 2D analysis of charge carrier distributions in semiconductor devices with high spatial resolution represent the substantial development of scanning probe microscopy. The described methods rely on detection and analysis of tunneling current which is strongly localized within an atomic dimension. This leads to significant improvement in the sensitivity and spatial resolution for measuring local electric characteristics of Si devices and nanowires, when effects of surface states are suppressed by adequate surface treatment.
\nThe gap modulation method can attain an ultimate spatial resolution comparable to that of STM topographic images in p-n junction regions, and can detect individual charged impurity atoms along the surface at a depth of few nanometers. Quantitative evaluation of charge distributions can be derived by comparing experimental data and simulations of the underlying charge concentration. The accuracy relies on the ability of the simulation to account for quantum phenomena, and further development of simulations based on the current-continuity model will be essential.
\nThe capability of the molecule-assisted probing method has been demonstrated with the use of C60 molecules. A spatial resolution of ~1 nm is determined by the size of the molecule. However, the C60 film on oxidized Si surfaces leaves ~20% uncovered areas. The coverage can be increased by the use of chemically modified C60 or other small molecules those formed a monomolecular-thick film on SiO2 surface. For high conductance of the tunnel gap, another transport mechanism, the single electron tunneling [66], becomes dominant and obscures the RET voltage measurements. Thus, optimization of the gap width is required.
\nThe presented methods can be used for measuring on rough surfaces, but careful data analysis should be performed to discard “artifacts.” In the gap modulation method, the tip vibration amplitude (dz) varies with tilt angle of the underlying surface, causing changes in the (dI/dZ) signal. In the dual-imaging method, large “sidewall” current such as shown in Figure 8 must be considered in data analysis. Also, atomically ordered surfaces can be obtained by cleavage, yet, to attain ultimate spatial resolution, STM measurements in well-controlled environment such as in an ultrahigh vacuum are necessary, where we can avoid undesirable effects caused by absorption of charged particles and molecules from air.
\nTo summarize, specific features of the presented 2D STM-based methods are (a) noncontact, stress-free measurements allowing analysis of delicate sample structures; (b) high spatial sensitivity to electrostatic field, which is substantial advancement in comparison with scanning Kelvin probe microscopy; (c) the ability to study nanoscale structures with a lateral size of 20 nm and below, which are inaccessible by other techniques.
\nFurther applications of the advanced STM methods will contribute to high-spatial resolution analysis of modern sub-100-nm electronic devices, functional nanowire devices, and novel devices incorporating two-dimensional materials such as graphene and topological superlattices. It will advance our understanding of charge carrier transport at nanoscale and encourage inventing novel energy-efficient devices.
\nThe tunneling current is described as a periodic function as
\nThe mean tunneling current is given in terms of the thermionic emission approximation including the vacuum tunneling term according to Reference [99] as
\nFor dz ≪ Z0, the factor K3 is derived considering only linear terms of dz and dψ, and is given by
\nThe area charge concentration at the Si surface (QSS) is obtained by solving the Poisson equation. An analytic solution for a 1D abrupt junction is given by [47]
\nΛ is the extrinsic Debye length, and volume densities of positive (p0) and negative (n) charge are in the Si bulk. The factor β = 1/kBT, and kB is the Boltzman constant, T is temperature.
\nFor 3D structures, a charge concentration at the semiconductor surface (QSS) is obtained by numerically solving the Poisson equation.
\nThe authors would like to thank colleagues of Nanoelectronics Research Institute (AIST, Japan) for valuable discussions and constructive comments motivating the research works.
\nIn any wireless communication system, the antenna is an essential component to transmit or receive a message signal. In many applications such as satellite communication, point-to-point communication, military communication, surveillance, radar, sonar, aircraft, etc., the antenna gain and directivity should be sufficiently high so as to direct most of the antenna-radiated power along a particular direction by reducing the power level (side lobe power) at other directions. A single radiator may not meet such requirements due to its omnidirectional power pattern and high side lobe level (SLL) in the far-field region. Moreover, radiation of huge amount of transmitter power from a single antenna element needs high-power amplification in the feed network. The high-power amplifier is not easy to design and safe to handle. Therefore, a number of antenna elements are arranged along a line, called linear antenna array (LAA), or in a plane called planer antenna array (PAA). The use of multiple antenna elements in the transmission and reception systems simplifies the power amplifier design problem by reducing the power level per transmitting antenna elements of the arrays. Some other advantages of using antenna arrays are to improve signal fading resistance or deliberately exploit the signal fading; mitigate the interfering signal coming from other directions, adaptive beam forming, and null steering at both transmitter and receiver; and increase system capacity. Due to its high gain and narrow beamwidth, the large antenna arrays also find applications in weather forecast, astronomy, image processing, and biomedical imaging.
\nAlthough the antenna array with uniform excitation amplitude and equally spaced antenna elements is the simplest one for practical implementation and also can be used to synthesize different patterns, due to the high value of peak SLL, it is impractical to use in such applications. In conventional antenna array (CAA) system, the low side lobe pattern is obtained by tapering the static excitation amplitudes. The well-known analytical techniques to taper amplitude distributions in nonuniformly excited antenna arrays are Dolph-Chebyshev (DC) and Taylor series [1]. However, the high dynamic range ratio (DRR) and complex excitation of the antenna elements are the major drawbacks of such CAA synthesis method with nonuniform excitation, because the complex excitation is practically difficult to realize and designing the practical antenna with high DRR of static amplitude tapering provides various errors such as systematic errors and random errors.
\nConversely, the ultralow SLL pattern in the far-field of the antenna array can be realized even in uniform amplitude antenna arrays by exploiting “time” as a fourth dimension [2, 3]. The introduction of the additional dimension “time,” into the antenna array system, results in time-modulated antenna array (TMAA). By using the fourth degree of freedom, “time” in antenna array system, various errors in realizing the low SLL pattern can be drastically reduced, and error tolerance levels become equivalent to those obtained in conventional antenna array system for the patterns of ordinary SLLs [4, 5]. Yet, the main disadvantage in TMAA is the generation of sideband signals which appeared due to the time modulation of the antenna signals by periodically commutating the antenna elements with the specified modulation frequency. Therefore, time modulation involves with the radiation or reception of electromagnetic energy at different harmonics of the modulation frequency that are termed as sidebands. In some applications where the antenna array is synthesized at center (operating) frequency, sideband signals are not useful. In such cases, sideband signals and associated power losses are suppressed to improve the radiation efficiency at the operating frequency of the antenna array [5, 6]. Presently, it is investigated that sideband signals are also effective in synthesizing multiple patterns and researchers are interested to exploit the same in some specific applications of the modern-day communication systems like harmonic beam forming [7], generation of multibeam radiation pattern [8], beam steering [9, 10], direction finding [11], wireless power transmission [12], etc. The interested readers may refer to Reference [13] for the state-of-the-art overview, applications, and present research trend on time-modulation theory and techniques.
\nThis chapter explains about the fundamental theory and techniques of different time-modulation strategies and such antenna array synthesis methods using optimization algorithms. The parameters involved with the use of optimization techniques and TMAA synthesis problem have also been presented.
\nLet us consider a linear antenna array of N number of mutually uncoupled isotropic radiators with inter-element spacing d0. The antenna elements are placed along the x-axis with the first element at the origin of the geometrical coordinate system as shown in Figure 1. In the XZ plane (one of the vertical principle plane), the array factor expression of CAAs can be obtained as in Eq. (1) [1]:
\nBasic antenna array of N element with inter-element spacing of d0.
where ω0 = 2πf0 = 2π/T0 is the angular frequency in rad/sec for the operating signal of frequency f0 in Hz; T0 is the time period of the operating signal; β = 2π/λ is the wave number with λ being the wavelength; p = 1, ……, N represents the element number of the antenna array; Ap and Фp\n
In order to control the antenna pattern by using the additional degree of freedom, namely, “time,” periodically the static excitation amplitudes of the antenna element are time-modulated. The commonly used and simplest way of doing that is to insert high-speed radio-frequency (RF) switches in the feed network, just prior to radiating sources as shown in Figure 2. Each array element is assumed to be connected to the RF switches with individually controlled switching circuits. The switches are periodically “on” and “off” according to a predetermined on-time sequence \n
Time-modulated linear antenna array (TMLAA) geometry.
Let us further assume that all the switches corresponding to the antenna elements in Figure 2 are on (short circuited) at the same instant of time, say at the beginning of each period “η*Tm” with “η” being the time period number 0, 1, 2,…, by using rectangular pulses of amplitude unity. Hence, the switches which are on for the whole time period Tm as shown in Figure 3(a) can be directly connected to the signal as time modulation is not required for such cases. On the other hand, the switches remained short circuited for their specific on-time duration and open circuited after their corresponding on-time duration (\n
The periodic pulse sequence of the TMLAA. (a) Unit pulse of periodicity TP. (b) On–off time duration of each antenna elements for one time-modulation period TP, and it is repeated at every TP time interval.
After the switching operation, the array factor expression of Eq. (1) can be written as in Eq. (3) [2]:
\nwhere \n
where \n
where \n
Putting Eq. (4) in Eq. (3), the array factor expression of Eq. (3) is obtained as
\nThus, Eq. (6) expresses that the signal is not only radiated at the operating frequency, ω0 for k = 0, but also the signals are radiated at different harmonics of the modulating frequency, kωm, with ω0 as the center frequency. The signal radiation at different harmonics is termed as sideband radiation (SBR). For such a TMLAA, the array factor expression at kth harmonic of the modulation frequency is readily obtained by combining Eqs. (5) and (6) as
\nTherefore, the array factor at the fundamental frequency, i.e., at operating frequency (for \n
From Eq. (8), it can be observed that τp’s\n
It can be observed from Eqs. (7)–(10) that, due to time modulation, the sideband signals inherently appeared around the center frequency spaced in multiples of the modulation frequency. In this section, the characteristics of harmonic signal radiated by an arbitrary time-modulated element are observed by varying the normalized switch-on time for its complete range from 0 to 1. Then by defining relative and normalized sideband power, the effects of reducing SLL on the first null beamwidth (FNBW) and maximum sideband power level are observed.
\nFrom Eq. (7), we can see that the array factor at different sidebands is the superposition of the harmonic signal radiated from the individual antenna element. Hence, sideband power pattern and total sideband power can be obtained from the harmonic characteristics of the time-modulated elements as expressed in Eq. (5). The normalized harmonic radiation of the individual time-modulated antenna element is given as [15]
\nwhere hpk is the normalized/relative harmonic radiation corresponding to the pth element. The variation of normalized harmonic power of the first three harmonics (k = 1, 2, and 3) with normalized switch-on time, τp, over its complete range (0, 1) is shown in Figure 4. As can be seen, at the lower value of τp, all hpkmax are almost the same, and for τp → 0, all hpkmax are exactly equal to 0 (zero) dB as it is expected from the Fourier series of unit impulse function. However, at the other extremes of τp, when τp → 1, all hpkmax → −∞, which is the predicted result as can be seen in Eqs. (5) to (10), with k = 1, 2, and 3. Again there is no radiation at hp2 for τp = 0.5 and at hp 3 for τp = 0.3 and 0.66 which can also be verified from Eq. (5) with k = 1, 2, and 3. Thus, Figure 4 indicates that the contribution of the harmonic component from a particular element to produce the sideband pattern depends on the on-time duration of the corresponding element. Therefore, the desired sideband power pattern can be synthesized in TMAAs by judiciously controlling the on-time sequence of the time-modulated antenna elements.
\nVariation of the first three harmonic powers from an antenna element with normalized switch-on time, τp.
Usually in TMAA, the radiation pattern is synthesized at center frequency by suppressing the sideband radiation level to sufficiently low value. Thus, the maximum of the power radiated at f0 is used to normalize the corresponding power pattern at center frequency. On the other hand, the sideband power is divided by the maximum power at f0 to measure the relative power level at different sidebands with respect to that of the radiation at center frequency. In this regard, the relative signal power radiated at different harmonics (k ≠ 0) is measured as in Eq. (12):
\nwhere “SBLk” represents the relative value of sideband level at kth harmonic (k = 1, 2, …), i.e., relative value of the array factor AFk in dB, and “max (AF0 (θ, t))” is the maximum value of the array factor at operating frequency ω0, i.e., the maximum radiation level at k = 0. Thus, with k = 0, Eq. (11) gives the normalized power pattern for the center frequency pattern, whereas, for the sideband radiations (with k ≠ 0), it is the relative power with respect to the maximum of the center frequency pattern.
\nIt is understood that in addition to the desired operating frequency (center frequency), TMAAs also radiate signals at the infinite number of different harmonics of the modulation frequency. When the desired power pattern is synthesized at the center frequency, the sideband power is wasted. In this section, the influences on the first null beamwidth (FNBW) and sideband radiation by reducing SLL of the center frequency pattern are observed. The SLL of the power pattern at f0 is reduced by using the conventional amplitude tapering technique, namely, Dolph-Chebyshev (DC) [1], and a heuristic search global optimization method, namely, genetic algorithm (GA) [16].
\nThe conventional antenna array synthesis technique such as Dolph-Chebyshev (DC) method [1] can be directly used to realize power pattern of the desired value of SLL at the center frequency. For a 30-element uniformly excited (UE) TMAA, the equivalent excitation coefficient of the DC pattern of desired SLL is made equal to the normalized on-time duration of the array elements. Following the DC method, the power pattern of different values of SLL is obtained at the center frequency.
\nIn order to reduce the SLL at the center frequency pattern using optimization technique, a cost function is required. A well-defined cost function of any optimization problem is important to obtain satisfactory performance. The cost function measures the distances between the desired and obtained values of the radiation parameters which are to be controlled. During the optimization process, the algorithms compare the obtained values of the radiation parameters with those of their respective desired values. Without considering sideband radiation and FNBW, the cost function to realize the patterns of desired SLLs at f0 is defined as
\nwhere SLLmax is the actual value of the SLL as obtained during each trial of the optimization process and SLLd is its desired value. Any heuristic search global optimization method can be employed to reduce the SLL of the power pattern at f0. Here, one of the useful stochastic search global optimization methods, namely, genetic algorithm (GA), is used to synthesize the power pattern of different values of SLL of the array under consideration [17].
\nIt can be seen from Eqs. (5)–(10) that the Fourier coefficients and hence amplitudes of the harmonic signals are decreasing gradually with increasing harmonic order. Thus, the radiation energy at the first few harmonics (called sidebands) is most significant. So, the influence on the maximum radiation at the first two harmonics of TMAA is observed by reducing the SLL of the center frequency pattern. Firstly, the SLL of the power pattern at f0 is reduced by using the Dolph-Chebyshev (DC) method [1]. Then a global optimization method is used to synthesize the same pattern as obtained via DC. In order to observe the effects of reducing SLL on SBL and FNBW, these values are noted for different power patterns. Table 1 shows the simulation results of the maximum sideband level (SBLmax) at the first and second harmonics for the fundamental pattern with different values of maximum SLL (SLLmax) ranging from −15 dB to −55 dB. The radiation pattern at f0 as obtained by GA and DC with SLL of −55 dB is shown in Figure 5. The first null beamwidth (FNBW) for different values of SLLmax of the main beam radiation pattern has been noted and is plotted in Figure 6. The maximum two harmonics are normalized with respect to the maximum value of the radiation at f0. For the different values of SLLs, the change in SBLmax at the first and second harmonics is shown Figure 7. Since, the Dolph-Chebyshev (DC) method gives the optimum pattern, i.e., the pattern with minimum FNBW for a specific value of SLL or vice versa. For the DC patterns of different SLLs, the corresponding FNBW, SBL1(max), and SBL2(max) are also given in Table 1. The plot SLL vs. FNBW is shown in Figure 6, and that for SLL vs. SBLmax is shown in Figure 7. Figure 6 depicts that for the DC method, FNBW is linearly increased when |SLLmax| is enhanced, whereas Figure 7 shows that SBL1max initially decreases from −3.35 dB and obtained its minimum value of −13.36 dB at −25 dB SLL pattern. Thereafter, it gradually increases and becomes almost steady at −12.5 dB after −30 dB SLL. From Figures 6 and 7, it can be seen that for the GA-based patterns of different SLLs, SBLmax and FNBW vary randomly as in the cost function, only SLL is considered without controlling FNBW and SBL.
\nThe patterns at f0 by DC | \nThe patterns at f0 by GA | \n||||||
---|---|---|---|---|---|---|---|
SLLmax (dB) | \nFNBW (deg) | \nSBL1(max) (dB) | \nSBL2(max) dB) | \nSLLmax (dB) | \nFNBW (deg) | \nSBL1(max) (dB) | \nSBL2(max) (dB) | \n
−15 | \n7.2 | \n−3.35 | \n−8.30 | \n−15.06 | \n8.4 | \n−10.01 | \n−17.91 | \n
−20 | \n8.4 | \n−7.2 | \n−17.83 | \n−20.03 | \n9.6 | \n−9.95 | \n−17.06 | \n
−25 | \n9.8 | \n−13.36 | \n−20.25 | \n−25.78 | \n10.4 | \n−8.51 | \n−14.27 | \n
−30 | \n11.2 | \n−12.28 | \n−19.31 | \n−30.28 | \n12.0 | \n−9.98 | \n−14.75 | \n
−35 | \n12.4 | \n−12.42 | \n−17.45 | \n−35.66 | \n17.2 | \n−12.19 | \n−17.39 | \n
−40 | \n13.8 | \n−12.42 | \n−17.45 | \n−40.04 | \n19.0 | \n−6.058 | \n−13.13 | \n
−45 | \n15.2 | \n−12.59 | \n−17.40 | \n−43.76 | \n20.2 | \n−10.12 | \n−14.572 | \n
−50 | \n16.6 | \n−12.58 | \n−17.40 | \n−50.52 | \n22.4 | \n−7.301 | \n−12.7870 | \n
−55 | \n18.0 | \n−12.55 | \n−17.37 | \n55.6 | \n23.6 | \n−7.3 | \n−12.9 | \n
Radiations maximum at the first two sidebands and FNBW for the fundamental patterns of different values of SLL.
GA- and Dolph-Chebyshev-based pattern of SLL of −55.6 dB at f0.
FNBW for different values of SLLs of the GA and Dolph-Chebyshev patterns at f0.
The plot of SBR1(max) and SBR2(max) for the different values of SLLs of the patterns at f0.
Different time-modulation strategies have been reported for synthesizing antenna arrays. These can be classified as (1) variable aperture size (VAS); (2) pulse shifting; (3) binary optimized time sequence (BOTS); (4) subsectional optimized time steps (SOTS); (5) variable aperture size (VAS) with quantized on-time (VAS-QOT) or quantized aperture size (QAS); and (6) nonuniform period modulation (NPM). From the array factor expression as given in Eq. (6), it can be observed that for TMAA, the array factor at different harmonic can be obtained if the Fourier coefficients of different time-modulated elements are known. Therefore, in the following sections, along with the brief description of different time-modulation approaches, the Fourier coefficients of time switching elements under the respective time-modulation scheme are presented.
\nThis is the first type of time-modulation strategy as reported in [2] where the aperture size of the antenna array is varied with time. The time-modulation principle as discussed in Section 2 falls under this category.
\nIn VAS time-modulation scheme, only the switch-“on” time duration is considered for deriving the array factor expression. However, when the RF switches are used to commutate the antenna elements in TMAAs, the radiation patterns at center frequency as well as at different harmonics depend not only on the switch-on time duration but also on the switch-“on” and switch-“off” time instants of the array elements [18, 19]. Thus along with the switch-on time durations as considered in VAS scheme, switch-on and switch-off time instants are also taken as another degree of freedom to control the power pattern in TMAA. For the pulse shifting strategy, periodic switching instants of the pth element over the modulation period are shown in Figure 8. In this case, both on-time instant \n
Switching instants defining pulse shifting strategy under two cases.
Hence, the normalized switch-on time duration, τp, is given as \n
Another possible situation may appear as shown in Figure 8(b) where \n
The complex Fourier coefficient for the pulse shifting strategy at kth harmonic due to the pth element under the two cases can be obtained, respectively, as [14].
\nBy taking into account the additional degree of freedom, namely, on-time instants of the antenna elements, improved array patterns can be observed. For example, more sideband reduction as compared to VAS approach is obtained when the same array pattern is synthesized at the center frequency [18, 19], and electronic beam steering [9] and harmonic beam patterns of different shapes [7, 8] can be realized without phase shifters.
\nIn binary optimized time sequence (BOTS), the switch-on time duration of an arbitrary pth element is divided into Q number of minimal time steps of equal length over a modulation time period Tm [20] as shown in Figure 9. The minimal time step, t0, is given by
\nSwitching function defining binary optimized time sequence (BOTS) strategy.
The periodic on–off sequence of the set of time steps corresponding to the pth element is represented by the switching function Up(t). If the on–off status of qth time step for the pth element is symbolized with a binary bit, \n
The complex Fourier coefficient of pth element at kth harmonic with the BOTS switching scheme can be obtained as [20]
\nwhere \n
In SOTS-based switching strategy, the time-modulation period (Tm) is divided into a number of subsections with variable lengths [21]. Let us assume that Tm is divided into Q number of time steps as shown in Figure 10 for the switching strategy of pth element of the array. For the qth time step, the on and off time instants of the switch are denoted by \n
The schematic of the periodic pulse sequence for SOTS switching strategy of the pth element of TMLAA.
The Fourier coefficient at the kth harmonics for the pth element can be written as
\nwhere \n
It can be observed that, if the number of subsections Q is 1, then SOTS is transformed into pulse shifting-based strategy. On the other scenario, if the on-time duration at each step, i.e., the separation between the on and off time instants, becomes multiples of \n
In Section 3, the different patterns of desired values of SLL at f0 are obtained by making the on-time sequence equal to the Dolph-Chebyshev coefficient of the corresponding patterns. Thus the appropriate set of on-time sequence is required to generate the desired pattern even in uniformly excited TMAAs.
\nIn this section, to generate different patterns in time-modulated antenna arrays (TMAAs) instead of considering continuous value of on-time duration [22], the modulation period is divided into a number of equal steps as in BOTS. However, in BOTS, multiple switching of on–off over the modulation period is considered. Such multiple changes of switching states over the modulation period need fast and complex switching circuit. Unlike BOTS, in this modulation scheme, the on–off states of the switches are assumed to change once over the complete modulation period like VAS. However, the on–off states of the switches are rounded off to the nearest quantization step to obtain quantized on-times (QOTs) of the corresponding elements as shown in Figure 11. In this time-modulation scheme, the time-modulation period, Tm, is quantized into “Q” number of discrete levels. At qth quantization level, the value of tq is given by q*(Tm/Q), where q = 1, 2… Q. The allowable on-time \n
The proposed time-modulation approach for the quantized on-time of the switches.
In all of the abovementioned switching strategies, all antenna elements are modulated with the same modulation frequency, ωm, and such time modulation is termed as uniform period modulation (UPM). Time-modulated antenna array (TMAA) based on UPM is commonly known as uniform TMAA (UTMAA). On the other hand, if the antenna elements of the array are time-modulated with different modulation frequencies as shown in Figure 12, it is defined as time modulation with nonuniform period modulation (NPM), and the corresponding array is defined as nonuniform TMAA (NTMAA) [23, 24]. Let us consider that the antenna elements are modulated with different modulation periods \n
Time-modulated array architecture with NPM switching strategy where f1 ≠ f2 ≠ … ≠ fN.
where \n
And finally, Fourier coefficient at the kth harmonics for the pth element is obtained as [24].
\nLet \n
where \n
The first summation indicates that the signals radiated at the center frequency \n
But in the case of NTMAA, the modulation frequencies are selected in such a way that f1 ≠ f2 ≠ … ≠ fN. So, due to different modulation frequencies of different antenna elements, the signals radiated from different harmonics appeared at different frequencies, and the term kfp in the second summation of [25] becomes different for different elements. That means the kth-order harmonics of different elements appear at different frequencies and the scenario is the same for all the other order harmonics. So, unlike UTMAA, the harmonic signals appeared at different frequencies and are distributed in space, which in turn decreases the resultant SBL [23]. Recently, some research works have reported the calculation of the sideband power of NTMAA [24, 25], and also the reduction of the sideband power losses using NTMAA is investigated [26].
\nIn Section 3.3, it is observed that, though the conventional amplitude tapering methods such as Dolph-Chebyshev and Taylor series can be used to obtain the power pattern of the desired SLL with minimum beamwidth at the operating frequency of time-modulated antenna arrays, these methods are not useful to control the undesired power radiated at different sidebands. Similarly, it is also observed that application of the stochastic computational technique, such as GA, for suppressing side lobe level of the center frequency pattern without taking into account the sideband radiation, cannot reduce sideband signal power. Also, the beamwidth of such patterns is unpredictable. The power pattern with low SLL and suppressed sideband is preferred for the different communication systems.
\nTherefore, the parameters to be considered to synthesize pencil beam pattern in TMAAs as shown in Figure 5 are SLL, FNBW, and SBL. However for the shaped beam pattern such as flattop and cosec squared, in addition to these three parameters, ripple level in the desired shaped region is another parameter to be taken into account. Further, it can be observed that while SLL is reduced, FNBW is increased and SBL is significantly large. In this regard, SLL, SBL, and FNBW for pencil beam pattern and SLL, SBL, FNBW, and ripple level for synthesizing shaped beam patterns are the conflicting parameters.
\nIn Eq. (13), the cost function is defined to synthesize the power pattern with a single objective that is to achieve the desired value of SLL in the synthesized power pattern. Conversely, the synthesized pencil beam patterns at the operating frequency should have reduced SLL along with sufficiently suppressed SBL and narrow beamwidth. Thus, TMAA synthesis problems are multi-objective optimization problems where the multiple objectives are low SLL and narrow beamwidth (BW) of the main beam at operating frequency and low value of maximum sideband level (SBLmax) for synthesizing pencil beam pattern while one more objective is low ripple level for synthesizing shaped beam patterns.
\nTMAA synthesis problem is non-convex and nonlinear in nature. A number of numerical techniques as already mentioned—Dolph-Chebyshev and Taylor series [1]—are available to synthesize pencil beam power pattern in conventional antenna arrays (CAAs). Also, some analytical methods are reported to generate shaped beam patterns and phase-only controlled multiple power patterns in CAAs [27, 28, 29]. Durr et al. described a modified Woodward-Lawson technique to design phase-differentiated multiple pattern antenna arrays with prefixed amplitude distributions [27]. The analytical technique reported in [28] is used to determine the nonlinear phase distribution of linear arrays. A method based on projection approach [29] is proposed to synthesize reconfigurable array antennas of a cosecant2 beam and a flattop beam (FTB) by using a common amplitude with phase-only control of analog phase shifters. Though these numerical and analytical techniques can also be applied to determine the nonlinear distributions of dynamic excitation coefficient and phase to synthesize power pattern at operating frequency of TMAAs, such methods have no control on sideband power level. Therefore, the powerful global stochastic optimization tools such as genetic algorithm (GA) [30], differential evolution (DE) [4, 5, 31, 32], particle swarm optimization (PSO) [7], simulated annealing (SA) [6, 33], and artificial bee colony (ABC) [22, 34] are essentially required to solve such multi-objective TMAA synthesis problems.
\nMost of the TMAA synthesis problems are solved by applying single-objective optimization method where all the objectives are added with different weighting factors to form a single cost function and the cost function is minimized by employing heuristic evolutionary algorithms. The different stochastic optimization techniques are used with the objective to synthesize desired patterns at the operating frequency by reducing SLL and SBL. One of the commonly used techniques to define the cost function of such conflicting multi-objective TMAA synthesis problem is as expressed in Eq. (26):
\nwhere \n
In Eq. (26), all the objectives are added with different weighting factors to form a single cost function. In such techniques, it is tedious and difficult to select proper weighting factor for the optimal solution. Improper set of weighting factors strongly effect on achieving the final values of the desired synthesizing parameters and hence on the performance of the optimization algorithm. Generally, some selected best results are presented without mentioning such difficulties. However, these values of the weighting factors are obtained by trial and error method [4]. Though multi-objective evolutionary algorithm (MOEA) [35, 36] can be used to solve such problems, the researchers are not comfortable with it as it has been used rarely as compared to single-objective optimization approaches.
\nIt is already discussed that time-modulated antenna array synthesis problems are non-convex as well as nonlinear. Therefore, stochastic, global computational techniques are required to solve such problems. In this regard, different population-based global searching techniques such as DE, SA, GA, PSO, ABC, and multi-objective evolutionary algorithm (MOEA) have been applied successfully to synthesize the desired pattern at the center frequency by suppressing sideband radiation to satisfactorily low levels. However, here the working principle of ABC and its implementation have been presented, and a novel approach to synthesize TMAA is discussed.
\nIn Section 4.5, the quantized aperture size (QAS) time modulation or variable aperture size with quantize on-time duration has been explained. In this section first to realize such time-modulation approach, a time modulator, namely, quantized time modulator (QTM), is presented. Then it is shown that though the quantized on-time duration has been used, however, by selecting a suitable number of quantization levels, the effect of quantization errors on the synthesized patterns can be reduced. In order to select the best possible set of quantized on-time values, the potentiality of artificial bee colony algorithm (ABC) has been exploited as the global searching algorithm. Thus, for the desired patterns, ABC finds the optimum set of unknown parameter values from the discrete search space of QOT. The synthesized results as obtained by using this quantized on-time are compared with that achieved by using continuous search space of on-time [6, 33]. Finally, considering the discrete search space of QOT, a low side lobe level (SLL) flattop pattern with low dynamic range ratio (DRR) is synthesized by utilizing a fully digitally controlled QTM. The major advantage of this approach is that by implementing the “time modulator” either as a discrete component on a printed circuit board or in an integrated circuit (IC), it can generate different patterns in the TMAA system.
\nFor appropriate switching operation at pth element, a current pulse with a pulse width of \n
The proposed quantized time modulator (QTM).
The wave form of the input and output pulses of different pulse widths that can be obtained at the outputs Oq\n\n∀\nq\n∈\n\n1\n…\nQ\n\n\n in Figure 13.
One of the most important features in TMAAs is to reconfigure different antenna patterns just by changing the on-time sequence across each element. Such a feature can easily be obtained in the proposed QTM employing PWS. The PWS consists of N number of (Q × 1) multiplexers and their outputs that are used to modulate antenna element using the quantized values of \n
Karaboga [37] introduced the artificial bee colony (ABC) algorithm to simulate intelligent food foraging behavior of the honeybee swarm. The ABC algorithm shows excellent performance for optimizing multivariable functions as compared to other similar algorithms like genetic algorithm (GA), differential evolution (DE), and particle swarm optimization (PSO). ABC is a robust search and optimization algorithm with relatively fewer control parameters [38]. Although GA is extensively used due to its efficiency to solve the optimization problems with binary/discrete variables, it requires high computational time as well as high memory consumption to store unnecessary binary data during the conversion of a real number to binary and vice versa. The decoding method as applied in ABC algorithm requires one-line MATLAB code which directly quantizes continuous values of the variables by rounding off them. The food foraging behavior of real bees and the implementation of the algorithm have been briefly discussed in the following section.
\nThe constituents of the food foraging systems are the unemployed bees (UBs) and the employed bees (EBs) in a beehive and food sources (FSs) in their surroundings. Initially, all the bees are unemployed, and after they find a rich food source, they become employed. UBs are categorized into scout bees (SBs) and onlooker bees (OBs). The food foraging process is initiated when the SBs start to explore the rich food source randomly from any location by moving toward any direction of the search space. When SBs find a rich food source, it becomes an EB and returns to the hive to attract other bees by performing a special dance known as the waggle dance. Depending on the quality of the food source, the EBs recruit some bees to extract nectar from the source. The EBs abandon the current food source when the nectar of the source is finished and becomes scout bees (SBs). However, in the dancing area, OBs examine the quality and quantity of the food sources with the information provided by the EBs, and after examinations EBs select a food source. Thus during the food foraging process, exploration is carried out by SBs, and exploitation is carried out by EBs and OBs. Due to the presence of both exploration and exploitation, ABC becomes a robust search and optimization algorithm. It is to be noted that the objective of the bees in ABC is to find out the location of the best possible food sources within the search space. Hence, the possible locations of the food sources are the possible solutions to this process. But in other swarm intelligence algorithms, e.g., particle swarm optimization (PSO), the locations of the individual agents are the possible solution within the search space. It is assumed that the number of employed bees (NE) and number of onlooker bees are equal in the colony and also these are equal to the number food sources (FN).
\nIn the following steps, the real bee colony behavior into the problem space is implemented:
Specifying objective: The objective is to synthesize far-field patterns at f0 by simultaneously minimizing SLL, SBLmax, and first null beamwidth (FNBW) or ripple (R).
Parameters to be optimized: Depending on the requirement in an array synthesis problem, suitable independent parameters are chosen as the optimization parameter vector χ. The number of parameters in χ represents the dimension (D) of the specific optimization problem.
Defining the cost function: According to the design parameters discussed above and multiple objectives of the synthesis problem, the cost function is defined as
where δh with h = 0, 1, and 2 are the instantaneous values of different parameters of the desired patterns, while δhd is the desired values of the specific parameters. For all examples as considered in Section 8, δ0 is the maximum SLL (SLLmax) of the pattern at f0 and δ1 is the value of SBLmax among the first five sidebands. But, for the first two examples, δ2 represents FNBW, and, for the third case, it is the ripple level of the flattop pattern for which the positions of δhd and δh are interchanged in the Heaviside step function \n
Initialization: The possible solution, χi, where i = 1, 2… FN, of an arbitrary number of food sources is generated randomly within the search space. With FN possible locations, each with D dimension is expressed in terms of a [FN × D] matrix.
Evaluating the quality of the food source: For all the possible solutions, the values of ψ and the corresponding fitness values, μi, are evaluated.
Employed bees’ stage: The greedy nature of the employed bees (EBs) is incorporated, and the new sources (si) surrounding its neighborhood are generated as follows:
where j\n
Onlooker bees’ stage: The quality of the food source is represented by the fitness value, μi, of the cost function, and onlooker bees select the new source by means of the probability, \n
where μmax is the maximum fitness value among the current possible solutions. Like employed bees (EBs), the greedy selection is also applicable to onlooker bees(OBs).
Scout bees’ stage: In this stage, the abandonment of a food source by the employed bees is simulated. If the fitness value of the cost function is not improved during a specified number of steps called “limit = FN*D” [25], it is ignored, and the parameter, \n
\n
Remembering the best solution: The overall new best solution as mentioned in the steps “e–h” replaces the previous best, and the value is then stored.
Stopping criterion: Steps “(e)” to “(i)” are repeated until the cost function converges to the desired value or a predetermined value of maximizing the number of cycles (MNC).
The VAS-based synthesis problems that have been reported in [6, 33] are considered at first, and the QAS-based time-modulation approach is applied to realize the patterns. Here, the modulation period Tm is quantized in 10 equal discrete levels, i.e., Q = 10. Hence, the discrete search space for the optimization problem (τp) becomes {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1}.
\nExample 1: A 30-element UE TMLAA is placed along the x-axis with one element at the origin, and a uniform inter-element spacing of 0.7λ is considered. It is desirable in practice for such an array to feed with {Ap} = 1 and {\n
ABC optimized power pattern obtained by using the discrete value of τp of Table 2.
Element numbers (p) | \nτp | \n
---|---|
1 | \n1 | \n
2 | \n0.30 | \n
3 | \n0.10 | \n
4–22 | \n1 | \n
23 | \n0.90 | \n
24 | \n0.90 | \n
25 | \n0.10 | \n
26 | \n0.10 | \n
27 | \n0.10 | \n
28 | \n0.90 | \n
29 | \n0.10 | \n
Optimum discrete values of \n
Example 2: In the second example, the synthesis problem as discussed in [33] is considered. From the list of static and dynamic excitations of one-half of the linear arrays as presented in Table 3, Ref. [33], it was found that out of the five edge elements, only three are time-modulated to synthesize the sum pattern, whereas, for the difference pattern, time modulation is applied only on four center elements. In this work, to synthesize the sum and difference pattern, the proposed method is applied in the following way. For the UE TMLAA, the sum pattern is synthesized by taking the discrete τp values of five edge elements (in one-half of the array) as “χ.” In order to compare the ABC optimized results with those of SA, during optimization, the three lower values of τp are rounded off to their nearest quantization levels, whereas the higher two τp values are kept to 1 so that the ABC optimized pattern is obtained by time modulating the same number of (i.e., three) elements as observed in SA. However, to synthesize the difference pattern, perturbation of discrete τp values of four center elements are considered. In Eq. (27), the same values of δhd’s as used in Example 1 are set. Figures 16 and 17 show the ABC optimized sum and difference patterns, respectively. For optimizing the sum and difference pattern with NE = 30 and limit = 450, the ABC takes only 23 and 5 iterations, respectively (refer to Figure 18). The corresponding optimum discrete values of τp are shown in Table 3. It can be observed that the sum and difference pattern is obtained by time modulating the same number of elements as found in [33]. As compared to [33], SLLmax and SBLmax of the sum pattern are improved by 2.03 and 1.5 dB, respectively. In case of difference pattern, the SBLmax is reduced by 2.37 dB with only 0.37 dB rise in SLL. Also, for both the sum and difference patterns, the amount of sideband power is found to be 3.35% and 4.69% of the total power which are 4.30% and 5.45% in the respective patterns of [33]. The FNBW of ABC optimized sum pattern and difference pattern was found as 6.12 and 4.56°, respectively, which are quite comparable to 5.88 and 4.59° as for the patterns in [33].
\nElement numbers | \n1 & 30 | \n2 & 29 | \n3 & 28 | \n4 & 27 | \n5 & 26 | \n6–11 & 25–20 | \n12 & 19 | \n13 & 18 | \n14 & 17 | \n15 & 16 | \n|
---|---|---|---|---|---|---|---|---|---|---|---|
τp | \nSum pattern | \n1 | \n1 | \n0.2 | \n0.9 | \n0.1 | \n1 | \n1 | \n1 | \n1 | \n1 | \n
Difference pattern | \n1 | \n1 | \n1 | \n1 | \n1 | \n1 | \n0.1 | \n0.9 | \n0.3 | \n0.1 | \n
Optimum discrete values of τp of ABC optimized sum and difference pattern, as shown in Figures 12 and 13.
ABC optimized sum pattern as obtained by time modulating the same percentage (20%) of elements as in [33]. SLL and SBLmax of the pattern are obtained as −17.87 and −31.44 dB, respectively.
ABC optimized difference pattern as obtained by time modulating the same percentage (26.7%) of elements as in [33]. SLL and SBLmax of the pattern are obtained as −16.05 and −31.44 dB, respectively.
Convergence characteristics of ABC for the synthesized sum and difference patterns of Figures 5 and 6.
Figure 19 shows SBLs of the first 30 sidebands for the synthesized patterns as considered in Example 1 and Example 2. It can be observed that at the higher sidebands also, the SBLs are below SBLmax. Further observation shows that the no radiation is produced at 10th, 20th, and 30th sideband with quantized values of τp as at these harmonics the array factor expression becomes zero for all elements.
\nSideband levels of the first 30 sidebands for the different patterns in examples 1 and 2.
Example 3: In this example, it is shown that the same time modulator can also be used to synthesize a flattop pattern. Accordingly, a symmetrical TMLA with element number N = 20 and inter-element spacing d0 = 0.5λ is considered. Here, the objective is to synthesize a flattop pattern in the broadside direction with digitally controlled static excitation amplitudes and phases by using five digital attenuators and phase shifters. A flattop pattern with a beamwidth of 30°, maximum ripple level (Rmax) at the flat region of less than 1 dB, and transition width of 8° is selected as the target pattern. Although such pattern with more stringent design specification is reported in [6], analog attenuators and phase shifters are required. Due to symmetry, the dimension of the parameter vector χ = {Ap, \n
ABC optimized space pattern at f0 and the first 30 sidebands. At f0, the flattop pattern is obtained with SLL, SBLmax, and Rmax of −29.31, −29.9, and 1.22 dB, respectively.
Element numbers (p) | \nNormalized on-time, τp | \nDiscrete values of excitation | \n|
---|---|---|---|
Amplitude, Ap | \nPhase, \n | \n||
1 & 20 | \n0.65 | \n0.200 | \n−33.75 | \n
2 & 19 | \n0.95 | \n0.325 | \n−22.50 | \n
3 & 18 | \n0.95 | \n0.475 | \n0 | \n
4 & 17 | \n1 | \n0.525 | \n33.75 | \n
5 & 16 | \n1 | \n0.675 | \n67.50 | \n
6 & 15 | \n1 | \n0.975 | \n90 | \n
7 & 14 | \n1 | \n1 | \n112.50 | \n
8 & 13 | \n1 | \n0.800 | \n135 | \n
9 & 12 | \n1 | \n0.600 | \n−180 | \n
10 & 11 | \n1 | \n0.700 | \n−146.25 | \n
Optimum discrete values of Ap, \n
In the continuous search space of VAS time-modulation method [2, 3, 4, 5, 6], the on-time duration of array elements can be of any value between 0 and Tm. In [2], for each time-modulated elements, the current pulse required with pulse width over the range of (0.1Tm < \n
Introduction of the additional degree of freedom “time” provides flexibility in synthesizing antenna array patterns and overcomes the shortfalls of realizing the patterns through conventional array synthesis methods. Among the different time-modulation strategies, QAS can be realized through a simple digital circuit consisting of a pulse generator, simple tapped delay line with equal delay at each tap output, flip-flops, and multiplexers. This circuit can be implemented in either an integrated circuit (IC) form or in a printed circuit board and can be used as a discrete component to generate different patterns. However, as far as the nonuniform period modulation is concerned, the function of the quantized time modulator (QTM) circuit needs to be investigated, specifically to time modulate the elements with multiple frequencies which need accommodation of multiple PLLs in the circuit for the multiple frequencies. Regarding other time-modulation approaches, complexity in the switching circuit increases as per the sequence, VAS, pulse shifting, BOTS, SOTS, and NPM, respectively, while their performance in synthesizing low SLL power patterns with suppressed SBL follows the reverse order. Thus, for a time-modulation approach, the improved performance in terms of the capability of synthesizing low side lobe power patterns by suppressing harmonic signal level is obtained at the cost of complex switching mechanism. However, due to the advancement in the semiconductor technology, availability of high-speed semiconductor switches makes it possible to realize such complex switching mechanism by writing simple program code in complex programmable logic devices (CPLDs).
\nIn all the time-modulation approaches except NPM, for the desired power pattern, optimization algorithm is required to determine the proper set of on-time sequence. The construction of suitable cost function with multiple objectives such as narrow beamwidth; low values of SLL and SBL, etc.; and the selection of corresponding weighting factors plays an important role to achieve the best possible power patterns. This chapter gives a brief fundamental insight toward all this issues.
\nThis work is financially supported by the Ministry of Electronics and Information Technology (MeitY), Govt. of India, under Visvesvaraya Young Faculty Fellowship of Visvesvaraya Ph.D. scheme (Grant No. PhD-MLA-4(29)/2015-2016) and DST-SERB project ref. file number EEQ/2016/00836, dated January 17, 2017.
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