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## Schottky diode mixer in the sub-millimeter wavelength range for imaging applications

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1 Schottky diode mixer in the sub-millimeter wavelength range for imaging applications presented by Jan Steffen Schür from Erlangen Erlangen 2013 to the Technical Faculty of the Friedrich-Alexander-Universität Erlangen-Nürnberg for obtaining the doctoral degree of Doctoral Engineer (Dr.-Ing.)

2 Approved as a dissertation by the Technical Faculty of the Friedrich-Alexander-Universität Erlangen-Nürnberg. Oral examination day: May 6th, 2013 Chairman of the doctoral body: Prof. Dr.-Ing. habil. Marion Merklein Reviewer: Prof. Dr.-Ing. Lorenz-Peter Schmidt Prof. Dr. Closely. Dr.-Ing. h.c. mult. Hans Hartnagel

3 For my pioneers and companions

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5 Contents 1 Introduction The frequency range of the sub-millimeter wave technology Motivation and structure of the work Fundamentals of frequency conversion with resistive mixers Frequency conversion on non-linear characteristics Current and voltage relationship on diode characteristics Estimation of the minimum conversion loss Noise behavior of Schottky diodes Types and properties of available GaAs-THz- Diodes GaAs Schottky single diodes Diodes for flip-chip assembly Film diode Monolithic integrated diode Antiparallel GaAs diode pair Micromechanical structure of compact THz mixer Waveguide environment Waveguide production Planar circuit technology Micromechanical construction technology Simulation and modeling of THz mixers Fundamental wave mixer Circuit simulations

6 vi Field simulations Harmonic mixers Circuit simulations Field simulations Metrological verification of the implemented THz mixer Measurements on scaled models Diode characterization Measurements on the fundamental wave mixer Sensitivity Mixing loss, mixer noise temperature Measurements on the harmonic mixer Mixing loss, mixer noise temperature System concepts and further work Summary 121 Bibliography 125 Summary 129 Acknowledgments 133 Curriculum vitae 137

7 List of the most important formula symbols and abbreviations Natural constants Symbol Value Unit Meaning c 0 2, ms 1 Vacuum speed of light e 2, Euler's number ε 0 8, AsV 1 m 1 Dielectric constant k 1, J / K Boltzmann constant µ 0 4π 10-7 VsA 1 m 1 permeability constant q 1, As elementary charge Formula symbol Symbol Unit Description am Geometry size bm Geometry size bn cm 2 V 1 s 1 Electron mobility in the semiconductor CSF junction capacitance at the operating point C j0 F junction capacitance of the uncontrolled diode CPF parallel housing capacitance of the diode C tot F total capacitance of the diode d KS m geometry size EV / m vector of the electric field strength E g ev band gap in the semiconductor f Hz frequency

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10 x Abbreviation LO LSE LSM MWS PLL REM RWO TE TEM TUD WR x ZF Description local oscillator electrical longitudinal mode magnetic longitudinal mode Mircowave Studio (CST) phase-locked loop, phase-locked loop, scanning electron microscope, reverse wave oscillator, transversal, electrical, transversal, electro-magnetic frequency band, technical University of Darmstadt, rectangular hollow

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13 Chapter 1 Introduction 1.1 The frequency range of sub-millimeter wave technology From a high-frequency point of view, the electromagnetic spectrum has already been developed in wide areas through mature technology and a large number of applications. From an electrical engineering point of view, even power supply networks with a relatively low operating frequency of 50 Hz require an approach that is typical of high frequencies, in which wave propagation and transit time effects must be taken into account, especially with long line lengths. Classic radio and broadcasting processes are linked to low-frequency applications in the spectrum, extend to over 100 MHz in the spectrum and are generally considered to be the first pure high-frequency applications. Modern communication and radar systems are now located in frequency ranges up to over 100 GHz and here primarily use the advantage of a high system bandwidth and, at the same time, a low relative bandwidth. Imaging radar systems for distance ranges of up to around 200 m use increasingly higher absolute operating frequencies and the associated shorter wavelengths for improved lateral resolution. Exploratory research investigates, for example, the potential for improvement of high-resolution 150 GHz radar systems for use in the automotive sector [1]. Looking further at technical systems at higher frequencies, the first optical systems in the far, mid and near infrared are only found again in the electromagnetic application spectrum above approx. 30 THz, followed by systems in the visible spectral range. Without claiming to be exhaustive, the examples mentioned are intended to give a rough overview of the part of the electromagnetic spectrum in which classic high-frequency technology is located. For this work, the focus should be on the frequency range between the millimeter waves (in the frequency range up to approx. 300 GHz)

14 2 1 Set up the introduction and frequencies in the far infrared (from approx. 30 THz). The range of sub-millimeter waves between 0.3-3 THz is still relatively little developed technically and places some special demands on components in this frequency range. The greatest challenge for THz systems is the high atmospheric attenuation, which is essentially due to water vapor and which has a strong adverse effect on the propagation of THz signals in the atmosphere. In addition, the short wavelengths (λ THz 1 mm) for a large number of components require dimensions of a similar order of magnitude and thus require highly precise manufacturing technology. In addition to these two aspects, which make the sensible use of THz technology more difficult, it should also be emphasized that, in addition to thermal radiation, there are hardly any natural THz sources on earth in this frequency range. The sub-millimeter wave range is therefore a very "quiet" frequency range in the natural electromagnetic spectrum on earth. The lack of natural (and historically also technical) THz sources means that the first, and until a few years ago still strongest, scientific interest in this frequency range was in radiometric THz applications, which investigated the intrinsic thermal radiation of bodies has been. As early as 1895, Ernest Fox Nichols in Berlin investigated the reflectivity of metals in the far infrared and expanded his observation spectrum to include longer wavelengths that extended into the THz range. A few years later, Heinrich Rubens and Ferdinand Kurlbaum jointly researched the intrinsic thermal radiation of bodies at different temperatures over a very large frequency range. The results achieved here also covered spectral ranges in the THz spectrum and inspired Max Planck in 1900 to formulate his radiation law. These very early THz experiments were increasingly supplemented by broadband, spectroscopic experiments in the years that followed. The analog use of optical components such as wire polarizers and prisms in the longer-wave sub-millimeter wave range coined the term quasi-optics and for a long time represented the basis of THz circuit technology. From a physical point of view, the often postulated "THz gap" was therefore between the millimeter waves and the infrared spectral range already closed at the beginning of the 20th century [2]. The engineering-scientific development of the THz area did not take place until about 50 years later due to the progress of semiconductor technology. With the advent of very small, low-capacitance Schottky diodes, the first heterodyne receivers emerged, which for the first time enabled the measurement of both the amount and the phase of a THz signal. This advance in the

15 1.1 The frequency range of sub-millimeter-wave technology 3 Measuring ability and accuracy opened up new perspectives and questions for the sub-millimeter-wave range and led to an increased interest in this frequency range, especially in the field of astronomy and astrophysics. In addition to the improvements in Schottky technology suitable for room temperature, the discovery of the Josephson effect and the invention of the SIS mixer 1 also made a new, high-performance semiconductor component available for the heterodyne reception of THz signals. A major disadvantage of SIS technology, however, is the need for cryogenic cooling in SIS mixers, which have to be operated at temperatures of a few Kelvin. As further technological impulses for the metrological development of the THz range, there are also new, pure detector components, such as hot electron bolometers, which, however, also require complex cooling for high sensitivities. If you take a closer look at the scientific experiments in the THz range over the past 10 years, you (still) see a clear focus on the areas of astrophysics and astronomy. There is great interest in the formation and history of galaxies in these scientific disciplines. A multitude of astrophysical questions can be answered by knowing the presence or absence of certain basic molecules. The very strong emission line of ionized carbon (C +) at 158 µm (corresponding to 1.9 THz) can be cited as an example, which, however, is covered by the thermal emission of dust clouds in very young areas of the universe [3]. Other emission lines in the THz range that are interesting from the astronomical point of view are caused by water, oxygen, carbon monoxide and nitrogen, among other things, and can be clearly identified and quantified with a sufficiently high spectral resolution. The qualified measurement of these radiometric emission lines is made more difficult by the already described high attenuation by water vapor in the atmosphere. In order to still be able to record these weak THz signals on the ground, very sensitive THz receivers are required. In order to enable a clear spatial assignment in interstellar space, very large antenna apertures in the form of radio telescopes with high angular resolution are used. To minimize atmospheric attenuation, such radio telescopes are operated on high mountains, such as the 15 m James Clerk Maxwell Telescope on Manua Kea (Hawaii). Probably the best-known project of a very powerful sub-millimeter-wave radio telescope is the Atacama Lar- 1 SIS = Superconductor-Insulator-Superconductor, which is currently under construction

16 4 1 Introduction ge millimeter array ALMA in Chile, which on the one hand uses the extremely dry climate and, through the use of 66 parabolic antennas, offers an extremely high angular resolution for interferometric measurements. In addition to the earth-based radio telescopes, satellite-based THz experiments such as the ESA Herschel system are also used to answer these questions. The new availability of powerful THz receivers and the advances in the development of THz sources, which are not considered here, have led to a rapidly increasing interest in THz experiments and systems in the last few decades. The new scientific focus in this frequency range led to the fact that the term "terahertz" was added to the Oxford English Dictionary in 1970, the number of publications relating to the THz range increased sharply and is nowadays commercialized with the term T-Ray technology . The driving applications in the modern THz work area are almost exclusively close-range systems from the areas of non-destructive material testing (e.g. [4], [5]), short-pulse spectroscopy ([6]) and imaging systems with a focus on safety applications ([7], [ 8th]). 1.2 Motivation and structure of the work With the current short-range THz applications, the requirements for THz receivers have changed significantly. Compared to the astronomical applications in radio telescopes and satellites, in which minimum receiver noise temperatures and maximum sensitivity have to be obtained with considerable effort, more and more practical aspects are in the foreground with the newer THz applications. By dispensing with cryogenic cooling of the receiver, significantly more compact systems can be developed, which enable the systems to be used more easily at the expense of system performance. The lower sensitivity of the THz receiver and the increased noise power in the measurement signal have less of an effect on the relatively short measurement distances of a few meters. From a technological point of view, Schottky technology, which is suitable for room temperature, is particularly suitable for such applications, which has proven its suitability through the advances in recent years both for the development of powerful detectors and for heterodyne THz mixers. In particular, the work of the US company Virginia Diodes Inc. 1 has extremely powerful, planar Schottky diodes for the THz frequency range 1

17 1.2 Motivation and structure of thesis 5 produced. Another aspect in the development of compact THz systems is the circuit technology used. The far-reaching departure from the previously used quasi-optical circuit technology with polarization grids and focusing mirrors towards waveguide-integrated planar circuits enables significantly smaller and more robust circuits and THz systems. The core of the present work is the question of the extent to which powerful Schottky diode mixers can be implemented in miniaturized waveguide circuit technology that circumvent the existing problems and limitations of existing open mixer concepts and are particularly suitable for imaging THz systems. In order to answer this central question, suitable, waveguide-integrated circuit technologies must first be evaluated, which enable the use of low-capacitance, planar Schottky diodes. Furthermore, the required passive circuit elements, such as filter structures and line transitions, must be developed under the manufacturing-related boundary conditions. Based on these principles, the performance of uncooled planar Schottky diodes with regard to minimum receiver noise temperature and conversion loss is to be demonstrated through the design and construction of a 600 GHz fundamental mixer with a single diode, as well as its metrological characterization. The results are partly based on a DFG research project 1, the aim of which was the development and evaluation of planar, low-noise GaAs Schottky diodes. In this project, Schottky technology from the Technical University of Darmstadt was used and GaAs diodes were optimized for use in planar THz circuits through work at the Chair for High Frequency Technology at the Friedrich-Alexander University Erlangen-Nuremberg. The intensive research activities in the field of Schottky technology at the TUD have meanwhile led to the establishment of the company ACST GmbH 2, which manufactures high-performance GaAs Schottky diodes and sells them commercially. In addition, mixer concepts for highly flexible THz mixers are examined in this thesis, which through subharmonic pump concepts and wired local oscillator feeds have the potential to completely dispense with quasi-optical circuit elements in an imaging short-range THz system. The investigations are intended to take into account the changed boundary conditions for the local oscillator source at higher 1 DFG funding numbers SCHM 1535 / 1-2 and 1-3: Noise-optimized, uncooled Schottky diode mixers for the higher terahertz range; Nov Dec

18 6 1 Introduction Take into account subharmonic mixer concepts and show to what extent the performance of available pump sources can be used for modern, imaging THz systems. Also, depending on the local oscillator frequency, suitable cable designs for the local oscillator feed must be designed and analyzed. Suitable subharmonic mixer concepts can be compared with fundamental mixers with the same signal frequency and their suitability for imaging THz systems in the near range can be analyzed. Of particular interest in these investigations is the question of how frequency- and phase-stabilized pump sources for THz mixers can be used with such a concept and whether it is possible to supply the LO signal with the help of flexible, dielectric waveguides. A system structure in this way could illustrate the potential of this mixed concept to make THz receiving systems very compact, robust and mobile. The amplitude and phase sensitivity of THz receivers in close-range imaging processes also motivates the design of flexibly usable subharmonic mixed concepts in which the focus is not on the highest sensitivity, as in radiometric systems, but on a high integration potential for active imaging radar processes. The flexibility required for this with regard to the positioning of the receiver can potentially be achieved through the use of low-attenuation dielectric lines in the local oscillator feed. LO source mirror d x, critical diplexer mixer transmitter mirror measurement object polarization grating mirror mirror Figure 1.1: THz mapping concept in quasi-optical circuit technology with fundamental wave mixer. Critical distances are marked in red, degrees of freedom in the positioning of components in blue.Figures 1.1 and 1.2 illustrate the great advantage of subharmonic THz mixers in imaging systems when using a

19 1.2 Motivation and structure of the thesis 7 dielectric local oscillator feed compared to a system with fundamental wave mixer, in which the measurement signal and the local oscillator signal have to be superimposed with a quasi-optical diplexer. Due to the high accuracy with which the quasi-optical circuit elements have to be aligned, only the measurement object can be moved when using a fundamental wave mixer with a common signal path for the measurement signal and local oscillator signal in such systems (Fig. 1.1). In contrast to this, the system concept according to Fig. 1.2a offers, in addition to a significantly more compact structure, the possibility of moving the transmitter / receiver unit consisting of the lighting source and mixer. (a) (b) LO source LO source R min mixer & transmitter diel. Line R min transmitter diel. Line mixer, transmitter, measurement object, measurement object Figure 1.2: THz mapping concepts with subharmonic mixer and dielectric lines for local oscillator feed (a) and bistatic, thinned line array for SAR processes (b). Critical distances are marked in red, degrees of freedom in the positioning of components in blue. This receiver concept is very attractive in the extension according to Fig. 1.2b, in which a mobile mixer can be used to simulate a receiver line that spans a synthetic aperture. With additional stationary transmission elements, a 2-dimensional synthetic aperture radar can thus be generated, which allows a fully 3-dimensional, volumetric image with a further vertical movement of the measurement object and using tomographic imaging methods. The flexibility of the receiver positioning can still be used in this system to investigate thinning concepts for multi-channel receiver lines. The problems mentioned are to be considered in the context of the present work from a scientific point of view and evaluated using two demonstrators. For this purpose, the basics of frequency conversion on resistive nonlinearities are first explained in Chapter 2 and the resulting requirements and technological boundary conditions are worked out. Following the basics of resistive mixing, Chapter 3

20 8 1 Introduction the GaAs Schottky diodes used in this work are presented. For the realization of the planned mixer, a waveguide-integrated, planar circuit technology is used, which is clearly superior to the previously used open construction of whisker-contacted diodes in corner cube reflectors, both in terms of performance and in terms of mechanical robustness. The requirements for the micromechanical production of the miniaturized, passive circuit components are examined in more detail in Chapter 4. In this chapter, the influences of manufacturing tolerances are also considered and the alternative line routing for the local oscillator signal is presented. Based on the theoretical and technological boundary conditions that have been developed, the circuit simulations for the mixer design are presented in Chapter 5 and estimates of the expected performance of possible mixer concepts are evaluated using a diode equivalent circuit diagram. On the basis of the simulations, a fundamental mixer at 600 GHz with optimized efficiency and very low noise temperature is presented, which should allow an evaluation of the performance of the GaAs Schottky diodes used. Furthermore, subharmonic mixer concepts are investigated, which reduce the requirements for the local oscillator source and are suitable for use in imaging close-range applications. Based on the design and modeling of individual passive core elements such as line transitions from rectangular waveguides to planar line technologies and line filters, the real diode geometries are also taken into account and optimized in the further 3D field simulations. The metrological verification of the manufactured THz mixers takes place in Chapter 6. Here, the measurement methods used are first presented and then the determined performance data of the mixers are compared with the estimated simulation data. Following the metrological characterization of the implemented mixers, possible system concepts based on the developed THz receivers are presented in Chapter 7 and the results of the work are incorporated into further research activities of the chair. The knowledge gained and the results obtained from the work carried out are finally summarized and critically assessed in Chapter 8.

21 Chapter 2 Fundamentals of frequency conversion with resistive mixers The frequency conversion of signals is often one of the central functions in high-frequency circuits. A distinction is made here between upmixing, in which lower-frequency signals are converted to higher frequencies, if possible without loss of information, in order to be able to be transmitted via a radio interface, for example. The second case of frequency conversion to lower frequencies can reverse the upward mixing, for example in a radio receiver, in order to be able to evaluate the information-carrying signal. Another area of application for downmixing is the analysis of very high-frequency signals that cannot be processed directly in their original frequency range. Downmixing is an effective way of evaluating the amplitude and phase of the RF signal in a limited bandwidth without losing the original amplitude and phase information. For the frequency conversion, non-linear components are required regardless of the direction of the conversion. A controlled mixture is usually always possible if the non-linear behavior can be controlled in a targeted manner. Essentially, components with controllable capacitive behavior (e.g. Schottky diodes in reverse direction as varactors) or components with controllable resistances (e.g. Schottky diodes in forward direction) are used. This work now focuses on the downmixing on the resistive nonlinearity of planar Schottky diodes. In this chapter the basic properties of Schottky diodes are explained and the analytical relationships and limits of the resistive mixture are clarified. The real GaAs Schottky diodes used are the subject of consideration in the following chapter.

22 10 2 Fundamentals of frequency conversion with resistive mixers 2.1 Frequency conversion using non-linear characteristics In order to understand the mixing of non-linear components, it is initially useful to clarify the concept of linearity. The linear behavior of a component is understood to mean its property of reacting to changes in the modulation with a proportional output variable. A simple example is the ohmic resistance, in which the current through the component is directly proportional to the applied voltage. According to Ohm's law, the proportionality factor between voltage and current corresponds exactly to the resistance value R. In the current-voltage characteristic there is therefore a straight line with a gradient of 1 / R. The modulation-independent proportionality also means that the resulting current has exactly the frequency of the excitation, i.e. the applied voltage. Further spectral components cannot occur. The relationships between excitation, characteristic curve and effect using the example of an ohmic resistor are shown in Figure 2.1a. I 1 / R = const II 1 / R = f (u) IU ωt U 2Θ ωt UI (ω) UI (ω) a) ω b) ωt ωt ω Figure 2.1: Modulation of a linear (a) or nonlinear (b ) Characteristic curve with a monofrequency alternating voltage In contrast to this, non-linear components have a modulation-dependent behavior, ie the property of the component depends on the momentary amplitude of the excitation. Figure 2.1b illustrates this relationship using a resistive characteristic in the form of a kink straight line. The characteristic curve is linear in sections, but if the characteristic curve is controlled in such a way that the excitation traverses both areas of the characteristic curve, a non-linear behavior can be seen at

23 2.1 Frequency conversion on non-linear characteristics 11 in which the proportionality factor between voltage and current varies in sections between the value infinite and r. This can be clearly seen in the course of the current. If, in addition to the temporal course of the current, its spectral components are considered, one recognizes, in addition to a direct current component, the occurrence of higher frequency components that are harmonic to the fundamental frequency. This example of frequency multiplication or rectification on a non-linear characteristic curve illustrates the creation of harmonic frequencies in the stimulating signal, but does not yet describe the frequency conversion during the mixing process. For this purpose, the non-linear characteristic curve of the kinking line is now to be controlled with an additional small signal in addition to a large-signal alternating voltage, and the effect of this process is to be considered using the resulting current. For this and all other considerations, the cosine-shaped large-signal modulation is referred to as the local oscillator and the additional small-signal modulation is referred to as the HF signal. In addition, the following chapters only consider the case of downmixing, in which the resulting intermediate frequency is lower than the frequency of the local oscillator or the RF signal. When analyzing the mixing process, it is also useful to consider the large-signal modulation by the local oscillator signal and a DC voltage separately from the small-signal modulation by the RF signal. As can be seen in Figure 2.1b, the current local oscillator amplitude defines the conductance of the characteristic curve effective at the corresponding point in time; an additional bias voltage U V acts here additively to the local oscillator amplitude and shifts the mean value of the local oscillator signal on the kink straight line to a favorable operating point. Expressed mathematically, this means that a large-signal voltage U D0 is applied to the non-linear component, which can be expressed by the bias voltage U V and the local oscillator signal U LO as follows: U D0 = UV + ÛLO cos (ω LO t) (2.1) This voltage controls the bending line of the form {0 UID (UD) = D <0 G d UDUD 0 (2.2). For the relationship between current and voltage, it is therefore necessary to know the conductance G d at all times. With the definition of the current flow angle Θ according to Figure 2.1b, the time-controlled conductance G (ω LO t) according to [9] can be given as follows:

24 12 2 Fundamentals of frequency conversion with resistive mixers 0 π ω LO t <Θ G (ω LO t) = G d Θ ω LO t + Θ 0 + Θ <ω LO t + π (2.3) with (Θ = arccos U) V Û LO (2.4) For the analysis of the spectral components of the current I D, it is useful to break down the controlled conductance into a Fourier series G (ω LO t) = n = G ne jnωlot (2.5). The coefficients G n of the series representation for the case of the buckling line are calculated as G n = 1 π G (ω LO t) e jnωlot d (ω LO t) = (2.6) 2π π = G Θ de jnωlot d (ω LO t) = 2π Θ = sin (nθ) G d nπ To describe the frequency conversion of the RF signal U S to the signal U ZF, a small signal analysis is now carried out. With the small-signal voltage U D = ÛScos (ω S t) + ûzfcos (ω ZF t) (2.7) and the time-controlled conductance, the small-signal current i d can now be determined on the non-linear characteristic:

25 2.1 Frequency conversion using non-linear characteristics 13 id = G (ω LO t) UD (2.8) with G (ω LO t) = di D (UD) du D UD = U D0 In order to clarify the effect of the frequency conversion, the series expansion of the controlled conductance according to Eq. 2.5 terminated after the second term (fundamental wave mixing in a sideband). In Eq. 2.6 it can also be seen that the coefficients G n are purely real and therefore G n = G n is given. For the following calculation, the following applies to the series representation of the conductance G (ω LO t) = G 0 + G 1 e jωlot + G 1e jωlot (2.9) As can be seen in the following calculation, the multiplication of the individual cosine terms results the series representation of the conductance with the cosine term of the small-signal voltage cosine terms at new frequencies. It can easily be shown that the resulting combination frequencies ω k must satisfy the following equation: ω k = ± mω LO ± nω S, m, n N 0 (2.10) For the case of downmixing in a sideband on the fundamental wave of the local oscillator, which was first considered m = 1 can be set. From Eq it can be seen that there are two possible combinations with identical intermediate frequencies. These two cases will be considered separately in the following.

26 14 2 Fundamentals of frequency conversion with resistive mixers 1st case: Downward mixing in equal position (m = 1 and n = 1) In this case the signal frequency is above the local oscillator and ω IF = ω LO + ω S applies. With equations 2.8 and 2.9 results from the non-linearity for the small-signal current: id = (IS e jωst + IS e jωst + I ZF e jωzft + I ZF e jωzft) = (2.11) = G (ω LO t) UD = = (G 0 + G 1 e jωlot + G 1 e jωlot) (US e jωst + US e jωst + U ZF e jωzft + U ZF e jωzft) = = G 0 [(US e jωst + US e jωst + U ZF e jωzft + U ZF e jωzft)] + + G 1 US ej (ωlo + ωs) t + G 1 US} ej (ωlo ωs) t {{} + e jω ZF t + G 1 U ZF ej (ωlo + ωzf) t} {{} e jω S t + G 1 U ZF ej (ωlo ωzf) t + + G 1 US ej (ωlo + ωs) t} {{} e jω ZF t + G 1 US ej (ωlo ωs) t + + G 1 U ZFe j (ωlo + ωzf) t + G 1 U ZF ej (ωlo ωzf) t} {{} e jω S t If one only considers the interesting spectral components at ± ω S and ± ω ZF and observes the relationship between the ZF- Frequency and the frequencies of the local oscillator and the RF signal, Gl can be given as a so-called conversion equation in matrix form: (IS I ZF) () () G0 G = 1 US G 1 G 0 U ZF (2.12)

27 2.2 Current and voltage relationship on diode characteristics Case: downward mixing in the inverted position (m = 1 and n = 1) Analogous to the previous case, the small signal current can also be specified if the signal frequency is less than the local oscillator frequency (ω IF = ω LO ω S) . The conversion equation in this case reads: (IS I ZF) () G0 G = 1 G 1 G 0 () US UZF (2.13) If you look at the two cases more closely, it becomes clear that due to the symmetry of the cosine function in the Intermediate frequency ω ZF no more statement can be made as to whether the frequency of the RF signal was converted from the upper sideband (mixing in equal position) or the lower sideband (mixing in inverted position). In both cases, the second converted frequency in the IF signal, which cannot be distinguished from the HF signal, is referred to as the image frequency. The problem of the image frequency can be eliminated by filtering or suitable mixer design (single sideband mixer). If the series expansion is continued in equation 2.9, the mixing process at subharmonic pump frequencies can be described in an analogous manner. 2.2 Current and voltage relationship on diode characteristics The frequency conversion using the example of a non-linear characteristic in the form of a kink straight line illustrates the effect of the resistive mixture, but is more of an academic example due to the shape of the characteristic. However, the kinking line is a first approximation for the characteristic of real Schottky diodes. A more realistic description of the current-voltage relationship at a Schottky diode in the pass band is possible using an exponential function: () I D = I S e qu D n d kt 1 (2.14) The diode current is expressed here by an exponential relationship with the applied diode voltage. The exponential growth is typical for real diodes, although there is no general restriction to semiconductor diodes. With all these diodes, the strong increase in current with increasing voltage levels is the result of a threshold value for the electrical field strength. In vacuum diodes, for example, this threshold value essentially depends on the work function of the electrons

28 16 2 Fundamentals of frequency conversion with resistive mixers from the cathode material. In semiconductor diodes, the equivalent to this is the potential barrier at the semiconductor-semiconductor or semiconductor-metal junction. The saturation current strength I S and the ideality factor d as parameters of the diode design and the diode materials can be recognized as characteristic curve parameters in Gl. In addition to the elementary charge q and the Boltzmann constant k as natural constants, the temperature T also influences the exact course of the characteristic. For the operation of the diode at room temperature, however, the temperature can also be regarded as constant, and in this case it is advisable to express the constant values as the so-called temperature voltage U T: UT = kt q 26 mv (2.15) Figure 2.2a shows the exemplary course of a current-voltage characteristic of a Schottky diode with linear axis scaling 1. Due to the large range of values of the diode current, it is often advantageous to scale the ordinate axis logarithmically. This is shown in Figure 2.2b. 3.5 a) linear representation 10 5 b) semi-logarithmic representation 3 Diode current ID in A Diode current ID in A Diode voltage UD in V Diode voltage UD in V Figure 2.2: Characteristic curve of a Schottky diode with linear (a) and semi-logarithmic (b) axis scaling The approximation of the The characteristic curve of real Schottky diodes using an exponential function according to Eq is relatively accurate for small voltage levels. In the case of large modulations, however, the ohmic losses in the series resistance R S of the diode become increasingly effective.The influence of this 1 The following parameters were used to calculate the characteristic: n d = 1.2; T = 290 K; I S = 10 fa

29 2.2 Current and voltage relationship on diode characteristics 17 series resistance can be seen in Figure 2.3 using the example of a GaAs Schottky diode. For small diode voltages, the two characteristic curves are congruent, but from a diode voltage of approx. 0.8 V the curve of the diode with finite series resistance becomes increasingly flat diode current ID in ARS = 0 Ω RS = 10 Ω diode voltage U in VD Figure 2.3: Influence of the Series resistance on the characteristic of a Schottky diode In addition to the series resistance, real diodes have other parasitic properties which, due to their reactive behavior, are not noticeable in the DC characteristics. Essentially, these are inductively effective properties that can be represented by serial equivalent circuit diagram elements that are caused by very narrow lines or bond wire connections. Furthermore, undesired, capacitive effects can occur, which are largely caused by the structure of the housing of the diode. In addition to these static effects, in addition to the modulation-dependent conductance of the diode, the junction capacitance must be taken into account as a variable, parasitic property in the mixing process 1. To answer the question of the origin of the parasitic properties in real diodes, please refer to Chapter 3. In this chapter, the planar GaAs Schottky diodes used are presented and it is made clear by which geometries of the diode structure the parasitic effects are caused. The parasitic properties of real diodes shown can be used in a simple equivalent circuit diagram using a few components for the following considerations: 1 The level-dependent junction capacitance can in principle be used as a non-linearity for frequency conversion, analogous to the controlled conductance of the diode. For an efficient implementation, however, specially configured varactor diodes with a large capacitance change C max / c min are to be preferred.

30 18 2 Describe the fundamentals of frequency conversion with resistive mixers in sufficient detail [9]. Figure 2.4 shows the equivalent circuit diagram for a typical Schottky diode. CS (U) URSLSR d (u) CP Figure 2.4: Simple equivalent circuit diagram of the Schottky diode The two impedances for the level-dependent, resistive (R d (U)) and capacitive behavior (CS (U)) of the diode are connected in parallel here to recognize. The ohmic losses through supply lines and their inductance can be found in the series connection of S and S. The case capacitance C P is parallel across all other elements. It can be clearly seen that the capacities in particular play an important role for the suitability of diodes in the THz range, since these become increasingly low-resistance at high frequencies and short-circuit the differential resistance R d (u). The resulting idea of reducing the capacities by miniaturizing the diode, however, comes up against technological limits. Possible miniaturizations of the diodes also have an advantageous effect on the series inductance L S, since the line lengths and thus the inductive behavior decrease. The current state of diode technology [10] now allows low-capacitance Schottky diodes with junction capacitances in the order of magnitude of 1 ff and housing capacities <10 ff with an ideality factor of n d 1.25. The track resistances can usually be kept below 15Ω. The series inductances are below 1 ph and are therefore usually negligible. 2.3 Estimation of the minimal conversion loss The estimation of the minimal conversion loss of a resistive single diode mixer is to be sketched in the following chapter based on the general, theoretical derivations according to SALEH [11]. For this consideration, a large signal analysis is first necessary to determine the modulation of the diode by the local oscillator and the setting of the operating point

31 2.3 Estimation of the minimum conversion loss 19 due to a direct voltage must be taken into account. In addition, an ideal, exponential diode characteristic according to Eq. For the mixing process, only the fundamental waves of the local oscillator and the RF signal are considered; a purely reactive termination is assumed for all other harmonics. However, the image frequency of the RF signal is taken into account. In the following, the influence and the mode of action of the RF signal and the resulting intermediate frequency signal through a small signal modulation on the non-linear diode characteristic will now be considered in more detail. For the following description, a single diode mixer connected in series as shown in Figure 2.5 is used. With this mixer circuit it is initially irrelevant whether an up or down mixer is being examined. The down mixer case, which is interesting for this work, is given when the signal is fed into the local oscillator gate and the parallel resonant circuit on the input side is suitably dimensioned at the same time. In this case, the bandwidth of the parallel resonant circuit is to be selected so that the resonant circuit forms an idle both for the local oscillator signal and for the RF signal and its image frequency. First of all, the mixer circuit is only fed with the local oscillator signal and a DC voltage for the large-signal consideration, and the resulting values at the diode are considered. UDI LO IDI ZF U LO ω LO ω ZF U ZF UVIVCV Figure 2.5: Single diode mixer in series connection For a monofrequency, cosine-shaped local oscillator signal of frequency f LO, a diode voltage U D of the form UD (t) = UV + ÛLOcos (ω LO t) results . (2.16)

32 20 2 Fundamentals of frequency conversion with resistive mixers From this it follows with the exponential diode characteristic according to Eq. 2.14, the diode current I D (t) to [()] ID (t) = IS e 1 nd UUV + ÛLO cos (ωlot) T 1. (2.17) For the following small signal consideration it makes sense to measure the diode current ID (t) through to break down a Fourier series representation into its spectral components: ID (ω LO t) = IDI Dn cos (nω LO t) (2.18) n = 1 The Fourier coefficients I Dn of the diode current are calculated as I Dn = 1 π π π ID (ω LO t) cos (nω LO t) d (ω LO t), n N 0 (2.19) When representing the Fourier coefficients of the diode current according to Gl with the time function of the diode current according to Gl, terms of the form Ĩ n (A) = 1 π occur e Acos (x) cos (nx) dx (2.20) 2π π. These terms correspond to the modified Bessel functions of the first kind of order n. With the aid of the modified Bessel functions and the abbreviation α = 1 nd UT (2.21), the Fourier series representation according to Eq and the corresponding Fourier coefficients can also be rewritten as follows: ID (ω L0 t ) = IS e [Ĩ0 αuv (αûlo) + 2Ĩ1 (αÛLO) cos (ω LO t) +] + 2Ĩ2 (αÛLO) cos (2ω LO t) + ... IS (2.22) This representation allows negligible, opposing saturation current IS a direct indication of the large signals occurring. For the resulting direct current I V as a function of the operating point voltage U V, the

33 2.3 Estimation of the minimum conversion loss 21 The amplitude of the local oscillator voltageÛLO and the diode parametersI S and α follows from Eq. 2.22: I V = I S e αuv Ĩ0 (αÛLO) (2.23) The current at the local oscillator frequency can be specified in an analogous way. I LO = 2I S e αuv Ĩ1 (αÛLO) (2.24) With knowledge of the pump voltage ÛLO and the pump current I LO, the pump power P LO can now be calculated and the large-signal conductance GD (ω LO) can be approximated at the local oscillator frequency: P LO = IS α eαuv αûlo Ĩ1 (αÛLO) (2.25) GD (ω LO) 2αI S e αuv Ĩ1 (αÛLO) αûlo (2.26) Since the large signals according to Eq essentially depend on the argument αûlo of the modified Bessel functions, a limit value observation is for very large levels αûlo >> 1 interesting. With this assumption and the approximation of the diode characteristic by an exponential function according to Eq, the large signals can be approximated as follows: IV, max IS eα (ûlo + uv) 2πα Û LO (2.27) I LO, max 2I V, max (2.28) P LO , max IV, max ÛLO (2.29) GD, max (ω LO) 2 IV, max Û LO (2.30) The conversion equations should now be determined to estimate the conversion loss of the single diode mixer. Here, the considerations are again limited to those that are interesting for this work

34 22 2 Fundamentals of frequency conversion with resistive mixers Case of the down mixer with a signal frequency very close to the local oscillator frequency (in relation to the IF frequency). This constellation represents one of the three observation cases according to SALEH, and is referred to there as broadband input. Essentially, this case is characterized in that the signal and the associated image frequency have identical signal paths and see identical impedances at the resistive non-linearity and in the direction of the source. This condition is met very well in the case of the single diode mixer with signal and local oscillator frequencies around 600 GHz and an IF decoupling of less than 15 GHz. The single diode mixer is now viewed as a modulation-dependent conductance in series with the small-signal equivalent circuit diagram according to Figure 2.6. When viewed in the master value form, this topology is generally referred to as a Y mixer. IS gd (t) I ZF US f S f ZF U ZF Figure 2.6: Small signal equivalent circuit diagram for a mixer in series connection The choice of the network topology with the description of the diode as a non-linear conductance does not represent a restriction of the generality. The following considerations can be equivalent for mixer circuits in parallel topology (Z mixer). The considered Y-mixer can now be considered as a three-port for the interesting spectral components at the signal, image and intermediate frequency. 1 I S Z Q U S I ZF 3 I Sp Mixer U ZF Z L Z Q = Z Q U Sp Z ZF Figure 2.7: Mixer description as three-gate 2

35 2.3 Estimation of the minimum conversion loss 23 For the calculation of the conversion equation, the Fourier series of the modulation-dependent diode conductance according to Eq. 2.5 to cancel only after the third term. In this case, the series representation of the conductance results as follows: G (ω LO t) = G 0 + G 1 e jωlot + G 1e jωlot + G 2 e j2ωlot + G 2e j2ωlot (2.31) The diode current can now go directly through the description of the conductance and the small-signal voltages U S, u Sp and u ZF can be calculated. id = (IS e jωst + IS e jωst + I Sp e jωspt + I Spe jωspt + (2.32) + I ZF e jωzft + I ZFe jωzft) = = (G 0 + G 1 e jωlot + G 1e jωlot + G 2 e j2ωlot + G 2e j2ωlot) (US e jωst + US e jωst + U Sp e jωspt + USp e jωspt + + U ZF e jωzft + U ZF e jωzft) If this calculation takes into account the relationships between the frequencies and their combinations specify the conversion matrix: ω ZF = ω LO + ω S (2.33) ω ZF = ω LO ω Sp (2.34) ω S = 2 ω LO ω Sp (2.35) ω Sp = 2 ω LO ω S (2.36) ISG 0 G 1 G 2 USI ZF = G ISp 1 G 0 G 1 U ZF (2.37) G 2 G 1 G 0 USp

36 24 2 Fundamentals of frequency conversion with resistive mixers For the calculation of the conversion loss and the input and output impedances with the help of the conversion matrix, SALEH bases the conversion matrix in the form of Gl IS m 11 m 12 m 13 USI ZF = m 21 m 22 m ISp 21 U ZF (2.38) m 13 m 12 m 11 USp and introduces the following constants, which only depend on the mixer and not on the terminations of the three gates: ǫ 1 = m 12m 21 m 11 m 22 = G 1G 1 G 2 0 (1) (2.39) θ = m 13 m 11 = G 2 G 0 (θ 1) (2.40) ǫ 2 = 2ǫ 1 1 + θ (1) (2.41) ǫ 3 = ǫ 1 1 θ 1 ǫ 1 1 + θ (1) (2.42) (1 ǫ 2) = (1 ǫ 1) (1 ǫ 2) (2.43) With these abbreviations the following mixer sizes are calculated: [Z ZF = m 22 1 ǫ] 2 1 + x 2 (2.44) x 2 = ZQ / m 11 1 + θ (2.45) Z ZF, opt = m 22 1 ǫ2 (2.46)

37 2.3 Estimation of the minimum conversion loss 25 L ZSB = 2 m 12 m 21 (x 2 +1) (x 2 +1 ǫ 2) x 2 ǫ 2 (2.47) L ZSB, opt = 2 m ǫ2 m 21 1 (2.48) 1 ǫ 2 ZQ, opt = m 11 (1 + θ) 1 ǫ 2 (2.49) Similar to the approximations of the large-signal parameters in the equations, a limit value analysis for very large local oscillator modulations αûlo >> 1 can also be carried out for the double sideband conversion loss L ZSB become. In this case, the mixer constants in the equations are calculated as follows: () ǫ 1 = m12 m 21 = G 1 2 m 11 m 22 G 2 0 =) 2 (Ĩ1 (αûlo) 1 1 Ĩ 2 (αûlo) αûlo (2.50) θ = m 13 m 11 = G 2 G 0 = Ĩ2 (αÛLO) Ĩ 0 (αûlo) 1 2 αûlo + 1 (αûlo) 2 (2.51) ǫ 2 = 2ǫ 1 1 + θ (2.52) 2 (αÛLO) 2 ǫ 3 = ǫ 1 1 ǫ 1 θ 1 1 + θ αûlo (2.53) With the help of these constants the conversion loss for very large local oscillator modulation can be calculated as 2 L ZSB 2 (1+) (2.54) αûlo L db ZSB 3 + 4,343 2 αûlo

38 26 2 Fundamentals of frequency conversion with resistive mixers can be approximated. The borderline case of 1 / αÛLO 0 represents the minimum achievable double sideband conversion loss. This results from L ZSB, min = 2 or L db ZSB, min = 3 db. (2.55) The case of the single-sideband mixer, in which the image frequency is terminated purely reactively within the mixer (and therefore no real power is consumed at this frequency), can be treated in an analogous way and approximated for high levels. Figure 2.8 shows the course of the conversion losses for these two cases. 5 Conversion loss in db L ZSB L ESB αûlo Figure 2.8: Comparison of the single-side and double-sideband conversion loss of a single diode mixer Here, for the borderline case of 1 / αÛLO 0, the known relationship between the difference between single-side and double-sideband conversion loss L ZSB, min = 2 L ESB, min or (2.56) L db ZSB, min = LdB ESB, min +3 db can be recognized. The two limit values therefore each represent ideal mixers, on the one hand a mixer in which an RF signal with an unchanged level is mixed to an IF frequency (single sideband mixer with

39 2.4 Noise behavior of Schottky diodes 27 L ESB = P S / P ZF = 1). The second mixer case is an ideal mixer whose IF power is composed by lossless conversion of two RF signals from the upper (OSB) and lower (USB) sideband (symmetrical around the local oscillator frequency) (double sideband mixer with L ZSB = PS, OSB / P ZF = PS, USB / P ZF = 1/2). In reality, however, this value cannot be achieved due to the kinking characteristic curve (see Figure 2.3) due to a finite series resistance in real diodes. In practice, however, very good fundamental wave mixers achieve double sideband conversion losses of less than 4 db. Conversion loss limits for sub-harmonic pumped mixers are theoretically possible in a similar way. However, the assumptions made here of purely reactive terminating impedances of frequencies and combination frequencies not taken into account (in particular for the spectral components between the local oscillator and the RF frequency) represent a very unrealistic assumption. For this reason, the subharmonically pumped mixer in this work dispenses with such a theoretical estimate. In Chapter 5, however, realistic estimates for subharmonically pumped mixers are carried out in the form of circuit simulations, the results of which can be used in a much more practical way. 2.4 Noise behavior of Schottky diodes The noise behavior of Schottky diodes is a decisive quality criterion for many applications in the submillimeter wavelength range due to the low signal level. The production of low-noise diodes for the THz frequency range is therefore just as crucial for the performance of diode-based THz components as is a low-loss, adapted RF circuit design. For the modeling of the noise behavior for the optimization of the diode technology or for the design of the RF circuit, a large number of physical noise processes must be taken into account. The dominant noise contributions for the use of Schottky diodes in mixers are thermal noise and shot noise, as well as hot electron noise and trapping effects. For the use of Schottky diodes in detectors, the 1 / f noise is still of great importance. In [12], the essential noise processes were modeled in great detail and verified by measurement using whisker-contacted GaAs Schottky diodes from the TU Darmstadt. The thermal noise can be relatively easily identified by the resistive loss

40 28 2 Fundamentals of frequency conversion with resistive mixers and can therefore be taken into account in the circuit design by the series resistance in the equivalent circuit diagram of the diode according to Fig. 2.4. The models of the other noise processes are, however, much more complex and heavily dependent on parameters of the semiconductor technology, which cannot be measured directly. In addition to the material sizes and doping densities, these include e.g. also the geometry sizes of the diode. If these diode details are not available or cannot be determined by measurements, only rough estimates of the noise behavior can be carried out with the noise models. If all diode parameters are known, the models can be used to use the noise behavior to optimize the diode technology. For the RF design of a diode circuit, the implementation of these mostly non-linear noise models in a circuit simulator is often not expedient or too complex. Especially when designing a typical single diode fundamental wave mixer, e.g. is to be used as a low-noise receiver, the strong correlation between the conversion loss of the mixer and its mixer noise figure can be exploited.This relationship follows directly from the comparison of the consideration of the Schottky diode as an initially lossless, resistive non-linearity and a purely thermally noisy resistor [13]. Starting from the lossless diode characteristic as an approximation with an exponential curve according to Gl, the shot noise of the diode current I D () I D = I S e qu D n d kt 1 is considered here in more detail as the dominant noise effect. If the diode current ID is now interpreted as a superposition of a forward current ID, v running in the flow direction and a reverse current ID, r running in the opposite direction, the following must apply for the two currents: and ID, v = IS e qu D nd kt = (ID + IS)> 0 (2.57) ID, r = IS <0. (2.58) For both streams the noise power densities of the shot noise can be given as follows:

41 2.4 Noise behavior of Schottky diodes 29 W id, v (f) = 2q (ID + IS) and (2.59) W id, r (f) = 2qI S. (2.60) Since both currents are statistically independent, the two Noise power densities can simply be combined to form a noise power density of the entire diode: W id (f) = 2q (ID + 2I S). (2.61) The resistive conductance of the diode required for further consideration corresponds to the differential conductance on the characteristic curve at the operating point: G 0 = di D du D = IS qnd kt e qud nd kt = (2.62) = qnd kt [ID + IS ] (2.63) With the known noise power density of the diode current (Eq. 2.61) and the small-signal conductance (Eq. 2.63) of the diode at the operating point, the shot noise of the diode can be described by an equivalent conductance at an effective noise temperature T eff. In general, the noise power density of the shot noise of a current through a conductance G is the comparison of the noise power densities W ig (f) = 4kTG. (2.64) 2q (ID + 2I S) = 4kT eff G 0. (2.65) allows the effective noise temperature T eff of the diode to be specified taking into account the differential conductance according to Gl: T eff = 2q (ID + 2I S) 4kG 0. ( 2.66) If it is also taken into account here that the diode current ID at the operating point of the diode is much greater than the reverse saturation current IS, the effective noise temperature of the lossless Schottky diode can through

42 30 2 Fundamentals of frequency conversion with resistive mixers T eff n dt 2. (2.67) can be approximated. In the special case of the ideal diode with n d = 1, the effective noise temperature T eff is therefore only half as large as the physical temperature T of the diode. It is also evident from Eq that with ideality factors greater than 1 and constant amplitude of the modulation, the effective noise temperature increases. If ohmic losses in the form of a finite series resistance are taken into account, a further increase in the noise temperature can be observed. However, the noise temperatures of complete diode mixers (receiver noise temperatures) are still significantly higher than the noise temperatures of the individual diode. For the description of the noise behavior of a complete, uncooled diode mixer including its wiring, the shot noise of the diode current or the effective noise temperature of the diode derived from it according to Gl only plays a subordinate role. The losses of the frequency conversion in the mixing process as well as ohmic and adaptation losses within the mixer circuit come to the fore here. Although these loss mechanisms have no original physical similarities, they can be summarized for noise modeling as equivalent thermal noise, since all three mechanisms reduce the signal-to-noise ratio and have no frequency dependence. If the mixer is interpreted as a passive, self-reflection-free two-port whose input is fed at the signal frequency and at whose output the intermediate frequency signal is available, the conversion loss L of the mixer represents the transmission attenuation 1 / s 21 2 of the passive two-port Two-port this means for the effective noise temperature according to [14]: T eff = T 0 (1 s 21 2) s (2.68) Expressed by the noise figure F = (T eff / T 0) + 1 is the relationship between the conversion loss and the noise figure of the mixer under the assumptions and approximations made: F = 1 = L. (2.69) s 21 2

43 2.4 Noise behavior of Schottky diodes 31 The comparison of this approximation, which is very helpful for the design of uncooled diode mixers, with published measurement results can be seen in the following figure: Conversion loss L in db Mixer noise figure F in db Planar diodes 335 GHz in [15] 585 GHz in [16] 585 GHz in [17] 585 GHz in [18] 585 GHz in [18] 640 GHz in [19] 640 GHz in [18] 690 GHz in [17] Whisker Diodes 318 GHz in [16] 345 GHz in [16] 340 GHz in [16] 557 GHz in [18] L = F Figure 2.9: Published conversion losses and mixer noise figures for Schottky diodes fundamental wave mixers (double sideband sizes in each case) The graph shows the measured values conversion loss and mixer noise figure for diode mixers in Frequency range of GHz compared in two groups: on the one hand mixers with planar diodes in planar or waveguide-integrated mixer circuits (square markers) and on the other hand mixers with whisker-contacted diodes in open corner cube Superstructures. With one exception each ([15] and [16]), all mixers with whisker-contacted diodes are above the straight line L = F and thus have conversion losses that are numerically greater than the mixer noise figure. The relatively large losses in these cases are essentially due to the limited coupling efficiency of the HF and LO signals into the open corner cube reflector and the whisker antenna. Theoretically, this coupling of a Gaussian beam into the whisker antenna is already limited to below 70% [20] and thus represents a significant loss factor. The measured values of the mixers with planar diodes, on the other hand, collect below the straight line L = F the losses caused by z. B. finite coupling efficiencies are lower or are less important in comparison to the noise behavior. Rather, it stands to reason that

44 32 2 Fundamentals of frequency conversion with resistive mixers the increased current densities in the diodes due to the lower coupling losses the effective diode temperature due to a significantly increased physical temperature of the diode rise so much that they can no longer be neglected. Furthermore, the increased current densities increase noise effects that have not been taken into account so far (e.g. hot electron noise and hot electron trapping [12]).

45 Chapter 3 Structures and Properties of Available GaAs THz Diodes For resistive frequency conversion in high-frequency technology, components with non-linear current-voltage relationships are required. For the electrical description of the resistive non-linearity, a level-dependent, positive resistance or conductance is usually used. In contrast to linear, resistive components, an infinite number of output signals that are harmonic for the excitation are formed due to the non-linear relationship between the modulating variable and the resulting output signal. In addition to this effect of multiplying resistive non-linearities in the case of large-signal modulation, the emergence of combination frequencies of the two input frequencies can be observed with an additional small-signal modulation. In this case, we are talking about mixing on a non-linear characteristic. The focus of the present work is the frequency conversion to lower frequencies by down mixers in the THz frequency range. The mathematical basics have already been discussed in the previous chapter, in this chapter the gallium arsenide diodes used will now be presented. GaAs Schottky diodes are characterized by a very high charge carrier mobility (the electron mobility b n for GaAs is around cm 2 V 1 s 1), which is absolutely necessary for operation at frequencies greater than 100 GHz. As an alternative to GaAs, another III-V compound semiconductor called gallium nitride (GaN) for operation at millimeter waves can be mentioned. GaN has an electron mobility that is over a factor of 8 lower, but is characterized by a very large band gap (E g, gan = 3.2 ev in contrast to E g, gaas = 1.4 ev) and greater thermal conductivity, which means this compound semiconductor is very well qualified for circuits with large signal amplitudes and power. For

46 34 3 Structures and properties of available GaAs-THz diodes High-performance THz mixers, however, GaN has no significant significance; GaAs Schottky diodes are preferred here. Furthermore, due to the metal-semiconductor contact (Schottky contact), there are practically no capacitive effects due to charge carrier diffusion, so that the Schottky contact can be described as a pure resistive non-linearity in a first approximation. Parasitic properties of the diode can, as in the equivalent circuit diagram according to Figure 2.4, be modeled as being independent of modulation except for the junction capacitance. The GaAs Schottky diodes used in this work are the result of a DFG-funded research project with the Technical University of Darmstadt (TUD). The starting point for the subsidized optimization of planar, low-noise Schottky diodes were the first planar diodes that arose from the work on whisker-contactable honeycomb diodes at TUD. The cooperation within the project framework offered the possibility to optimize the diodes on the diode production side with regard to the semiconductor properties and additionally to improve the diode geometries in three-dimensional field simulations for the mixer operation within the scope of this work. The following are the initial diode designs and the improved diode designs. 3.1 GaAs Schottky single diodes For use in low-loss and low-noise fundamental wave mixers, GaAs single diodes are the method of choice. For the optimization of the diode geometries and the design of a high-performance mixer circuit, a single diode was initially available for flip-chip integration on a planar circuit substrate. In close cooperation with the research group at the Institute of Microwave Technology and Photonics at TUD, the initial design of the diode could be optimized with regard to the parasitic diode properties and two new diode designs for the hybrid integration of the diodes on the circuit substrate were evaluated. The interdisciplinary cooperation between Schottky technology at TUD and THz circuit technology at LHFT enabled a comprehensive optimization of the planar diodes for use in low-noise THz mixers. The diodes described below were optimized and evaluated with regard to semiconductor technology at TUD. The results from the simulations presented in Chapter 5 were able to flow in, which give indications of optimization potential for use in the sub-millimeter wavelength range.

47 3.1 GaAs Schottky single diodes 35 A detailed description of the Schottky technology used for the manufacture of the diodes can be found in [21] Diodes for flip-chip assembly The initial design for the diode optimization and the mixer design is a GaAs Schottky single diode . For mounting on a planar circuit substrate, in this design the diode is contacted upside down on the soldering pads with the line structures (flip-chip mounting). With this design, as well as with the following optimized geometries, the soldering pads are always higher than the air bridge for anode contact in order to prevent damage at this critical point. Figures 3.1 and 3.2 show a 3D simulation model of the diode structure and a sectional view through the diode. If this diode geometry is operated in the pass band, the current flow would be coming from the right via the soldering pad and the air bridge to the platinum anode in the center of the mesa. The anode diameter of this diode is 1 µm. The ohmic rear contact, which is embedded in the GaAs substrate, is located directly under the mesa. This has a conductive connection to the second soldering pad and thus completes the current path through the diode. Due to the vertical flow of current through the mesa, a quasi-vertical diode geometry is also used in this case, since the flow of current, as in the case of whisker-contacted diodes, occurs in a vertical direction, but macroscopically a horizontal flow of current through the entire diode can be observed. Ohmic rear contact air bridge Mesa solder pad 14 µm solder pad 70 µm 150 µm Figure 3.1: Output design of the Schottky single diode

48 36 3 Structures and properties of available GaAs THz diodes Solder pad Mesa air bridge Solder pad Ohmic back contact GaAs substrate Figure 3.2: Sectional view of the Schottky single diode Series resistance RS, a series inductance LS and a parallel capacitance CP are modeled. From the geometry of the flip-chip diode shown above, it can be seen that the parasitic variables, in particular the parallel capacitance in the equivalent circuit diagram, cannot be explained by individual structures of the diode geometry. Rather, due to the design, there are capacitive couplings between individual diode elements at several points, which are described in total by a parallel capacitance in the modeling. The question of which components of the diode structure primarily influence the total capacitance was investigated in the early course of the project using simulative estimates at the TU Darmstadt. Table 3.1 shows the relevant diode structures with their capacitance contribution to the total capacitance of approx. 10 ff of the flip-chip diode. For comparison, the junction capacitance of the Schottky contact of the unconnected diode is also listed in the table. From a high-frequency point of view, the parasitic capacitances are parallel capacitances to the actual Schottky contact and therefore represent low-impedance signal paths for the THz and local oscillator signals that prevent efficient frequency conversion at the Schottky contact. For an optimized, low-parasit diode design, it is useful to minimize the capacitance contributions of the diode structures. Here, however, compromises have to be found between the geometric minimization of the metallic structures and the distance between the metallic areas in order to

49 3.1 GaAs Schottky single diodes 37 Diode structure Capacitance contribution [ff] Junction layer capacitance (CS (0)) 1.8 Ohmic rear contact 4.2 Air bridge base at the anode contact 2.3 Air bridge 1.8 Air bridge base on the soldering pad 1.5 Table 3.1: Capacitance contributions of the diode structures reduce the total capacitance of the diode. By optimizing the diode design in a 3D field simulator, especially by optimizing the dimensions of the airlift, the capacity contributions could be reduced by approx. 1.5 ff. For the simulative optimization, the Schottky contact was replaced by a lumped element port in the field simulator and the power transmission between the soldering pads and the modeled Schottky contact was maximized. This procedure enables very simple parameter variations of the diode geometries in the field simulator, which, on the basis of the scattering parameters, allows a direct statement to be made about the achieved improvement in the signal coupling into the Schottky contact. The thickness of the ohmic rear contact is coupled to the thickness of the GaAs substrate due to the manufacturing technology. The minimization of the rear-side contact to reduce the capacitive behavior is therefore only possible to a very limited extent, since a reduction in the substrate thickness below 10 μm reduces the mechanical stability of the flip-chip diode too much. From the point of view of minimizing capacitance, a very thin rear-side contact would be advantageous here, since this could also reduce the volume of the GaAs substrate. The relatively high permittivity of GaAs (ε r 13) has a very negative effect on the capacitance contributions of the individual diode structures. Another reason that speaks against reducing the thickness of the rear side metallization too much is the thermal effect of the rear side contact, which is required as a heat sink for the very high current densities in the Schottky contact. In contrast to the thickness of the back contact, however, the width can be significantly reduced, which does not have too great an influence on the thermal effect, but only marginally reduces the capacity contribution. The parasitic, ohmic conduction losses, which have not been taken into account so far and which make a contribution to the series resistance R S in the equivalent circuit diagram, can only be influenced to a negligible extent by geometry optimizations without changes to the semiconductor doping and are therefore not discussed further. The inductive series impedance L S is elegant

50 38 3 Designs and properties of available GaAs THz diodes are given by the width and length of the airlift. In this case, however, minimizing the inductance by widening the airlift is contrary to optimizing the capacity contribution of the airlift [21]. The simulative investigations of the airlift geometry have shown a greater benefit here due to the narrowing of the airlift width. A shortening of the air bridge to minimize the inductance also stands in the way of an increase in the total capacitance, since the shorter air bridge also reduces the distance between the soldering pads. Here, however, a compromise could be found through simulations, which maximizes the power coupling from the soldering pad into the Schottky contact.The optimized geometries, especially in the area of the upper gold metallization, have already been able to effectively minimize the parasitic, capacitive effects, but the remaining total capacitance of the order of magnitude of 10 ff is still in an unsatisfactory range if the diodes are used for frequencies greater than 200 GHz should. For this reason, further, fundamental changes in the diode design were pursued in order to improve the suitability of the diodes for THz applications. In the following two sections, diode designs are presented in which the already determined optimizations of the air bridge and solder pad dimensions are used and in addition the influence of the GaAs substrate is minimized. Since very promising approaches were pursued here, the optimized flip-chip design of the diode was not manufactured. Film Diode In principle, two approaches are conceivable to minimize the parasitic diode capacitance. On the one hand, this is a minimization through optimized geometries of the metal volumes, as shown in the previous section. Another starting point are the insulating materials between the metallizations. Here, the GaAs with its relatively high permittivity is essentially decisive for the total capacitance of the diode. However, since the doped GaAs layers in the epitaxial layer are essential for the formation of the required Schottky and rear-side contacts, these GaAs volumes cannot be changed. However, the doped GaAs layers represent only a very thin layer of the entire GaAs substrate. For clarification, Table 3.2 shows the epitaxial layering of the GaAs wafers used. The original wafer thickness in the manufacturing process is 350 µm and is achieved using a combined wet-chemical and mechanical etching process

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