What is a Gunn diode 2

Manufacture and characterization of GaAs Gunn diodes for applications at 77 GHz

Transcript

1 Research Center Jülich in the Helmholtz Association ff Institute for Layers and Interfaces Institute 1: Semiconductor Layers and Components Production and characterization of GaAs Gunn diodes for applications at 77 GHz Jürgen Stock

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3 reports from Forschungszentrum Jülich 4069

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5 Manufacture and characterization of GaAs Gunn diodes for applications at 77 GHz Jürgen Stock

6 reports from Forschungszentrum Jülich; 4069 ISSN Institute for Layers and Interfaces Institute 1: Semiconductor Layers and Components Jül-4069 D82 (Diss., Aachen, RWTH, 2003) Available from: Forschungszentrum Jülich GmbH - Central Library D Jülich - Federal Republic of Germany 02461 / Fax: 02461 / e

7 Production and characterization of GaAs Gunn diodes for applications at 77 GHz In this work, the properties of GaAs Gunn diodes with a hot electron injector were investigated. be used in radar and satellite technology or in intelligent control systems in automotive electronics. Various measurement methods were used to characterize the diodes, which allow an analysis of the DC characteristics, the small-signal behavior and the domain operation in a harmonic oscillator. Furthermore, clean room processes have been developed that enable both the production of Gunn diode chips with an integrated heat sink for use in oscillators and the processing of coplanar Gunn diodes for small-signal measurements. The functional principle of the Gunn diode is based on the so-called Gunn effect, the origin of which is a special electron scattering process in the conduction band of GaAs. By using a hot electron injector consisting of a linearly graded AlGaAs barrier and a subsequent doping tip, this scattering process is intensified and the efficiency of the component is increased. It could be shown that an undoped GaAs spacer layer between the AlGaAs barrier and the doping tip prevents unwanted diffusion of the doping atoms in the direction of the AlGaAs barrier and thus improves the effectiveness of the injector. The homogeneity and reproducibility of the manufacturing process could be increased compared to existing processes by the introduction of an etch stop when the substrate is thinned back and by the use of a plasma etching process to define the Gunn diode mesa. Fabrication and Characterization of GaAs Gunn Diodes for Applications at 77 GHz In the present thesis the properties of GaAs Gunn diodes with a hot electron injector have been investigated which are used to generate microwave power e.g. in radar and satellite technology or in intelligent control systems in the field of automotive electronics. The diodes have been characterized using different measurement methods which allow the analysis of the DC curves, the small signal behavior and the second harmonie mode operation in an oscillator. Furthermore the process technology has been developed to fabricate Gunn diode chips with integrated heat sink used in oscillators and coplanar Gunn diodes for small signal measurements. The operation principle of a Gunn diode is based on the so-called Gunn effect which originates from a particular electron scattering process in the conduction band of GaAs. Using a hot electron injector consisting of a linearly graded AlGaAs barrier and an adjacent delta doped layer, the scattering process is reinforced and the efficiency of the device is improved. It has been shown that an undoped GaAs spacer layer between the AlGaAs barrier and the delta doped layer can prevent the unintentional diffusion of doping atoms towards the AlGaAs barrier and therefore enhances the effect of the injector. The homogeneity and reproducibility of the fabrication process has been increased compared to existing methods by introducing an etch stop layer for the substrate removal and by using a plasma etching process for the definition of the Gunn diode mesa.

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9 Contents 1 Introduction 1 2 Theoretical basics The Gunn effect Discovery by JB Gunn Physical causes of the Gunn effect Development of high-field domains and current oscillations Material-specific cut-off frequencies The theory of the Gunn effect Stationary current-voltage characteristic Small-signal behavior Dynamics of high-field domains Behavior in the oscillator The structure of a Gunn- Diode standard structure Injectors Doping tip at the injector Doping profiles in the drift zone Thermal considerations Solution of the thermal conduction equation Influence of the temperature dependence of the thermal conductivity Influence of the moving high field domains Simulation of the temperature distribution Technology MBE layer structure Process sequence for the production of Gunn diode chips Vapor deposition of the ohmic gold side contact Production of the integrated heat sink from Sample with nickel Removal of the GaAs substrate

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11 CONTENTS iii B Definition of S parameters 111 C Process parameters 113 C.1 Cleaning processes C.2 Lithography processes C.3 Wet chemical etching processes C.4 Plasma etching processes C.5 Metallization processes C.6 Electroplating processes D Mask layout 123 D.1 Mask set for Gunn- Diode chips D.2 Mask set for coplanar Gunn diodes List of Figures 129 Bibliography 133 Acknowledgments 139

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13 Chapter 1 Introduction In recent years, the focus of microwave applications has shifted from the military to the civilian sector. In addition to the classic areas of application in radar and satellite technology, commercial applications, for example in automotive electronics, are now gaining in importance. Radar beams are used here in systems for automatic speed control or as anti-collision wheels. Such systems require powerful microwave sources in the frequency range between 60 and 90 GHz. These requirements are met by the so-called Gunn diode, the physical functional principle of which was examined in more detail in this work. To generate microwaves with Gunn diodes, the so-called Gunn effect is used, which was first observed experimentally in 1963 by J. B. Gunn [Gun63]. A few years earlier, Ridley, Watkins and Hilsum described the underlying physical principle [Rid61, Hi162]. The Gunn effect is based on an electron scattering process that occurs above a certain threshold field strength in the conduction band of certain semiconductor compounds (for example GaAs or InP). In addition to a main minimum at the I'point, these materials also have at least one further, energetically higher secondary minimum with a larger effective electron mass in the conduction band. When a certain threshold field strength is reached, the electrons can gain enough energy to switch from the main minimum to the secondary minimum. As a result of this scattering process, the mean drift speed of the electrons decreases and an area with negative-differential mobility is created. Due to this negative differential mobility, small fluctuations in the electron density can grow into large charge dipoles and migrate through the semiconductor. This causes oscillations in the current in an external circuit, the frequency of which is determined by the drift speed of the charge dipoles and the length of the semiconductor sample. Gunn diodes compete with oscillators based on integrated transistor circuits, so-called MMICs. In the frequency range above 75 GHz, fast HEMT transistors (High Electron Mobility Transistor) are typically used to set up such circuits [Siw99, Rad01]. In this type of transistor, the conductive channel is formed by a two-dimensional electron gas (2DEG). Due to the high electron mobility in the 2DEG, the HEMT has very good high-frequency properties and, similar to the Gunn diode, a low noise level. The performance data of the MMIC oscillators are in

14 2 CHAPTER 1. INTRODUCTION this frequency range is still well below the values ​​that can be achieved with Gunn diodes, so that the Gunn diode plays the more important role in practical use. The power output of Gunn diodes has been continuously expanded in the past. In particular, the advances in the production of semiconductor layers, for example by means of molecular beam epitaxy, allow the precise production of complex semiconductor layer structures in order to increase the efficiency of the Gunn diode. A significant improvement can be achieved by using a hot electron injecton, which gives the electrons the energy required for the scattering process into the secondary minimum of the conduction band as soon as they enter the drift region of the Gunn diode. This eliminates the otherwise necessary acceleration path for the electrons, and the Gunn diode works much more effectively. As part of this work, the physical properties of such a Gunn diode with an injector were examined in more detail. The injector used consists of an AlGaAs barrier with a linearly increasing aluminum content and a thin, highly doped GaAs layer as a connection between the AlGaAs barrier and the drift zone of the Gunn diode. The influence of the various components of the injector on the DC characteristics and the HF behavior of the Gunn diode was determined using suitable measurement methods. For high-frequency characterization, measurements on a Gunn oscillator as well as scatter parameter measurements on specially prepared Gunn diodes were carried out. The heat dissipation during the operation of a Gunn diode in an oscillator is of particular importance, since the input power of around 5 W is converted to a large extent into heat due to the low efficiency. For this reason, a process sequence for the production of Gunn diode chips with an integrated heat sink is presented. A particular challenge of this method is the necessary double-sided processing of the wafer, which requires complete removal of the substrate and consequently an alternative stabilization of the sample. Scattering parameter measurements allow access to the intrinsic high frequency properties of the Gunn diode without the influence of an oscillator. For this purpose, planar Gunn diodes with lower peak currents and coplanar leads are required. For this purpose, the process sequence was modified accordingly and, in particular, the use of a heat sink was dispensed with. Overview In the next chapter, some physical fundamentals are first discussed, which are important for understanding the relationships in the further course. In particular, calculations are carried out for the small and large signal behavior of the Gunn diode and for heat dissipation. The process flows for the production of Gunn diode chips with an integrated heat sink and of coplanar Gunn diodes are then presented. The technological methods and process parameters used as well as the semiconductor layer systems used are discussed. Then the measurements carried out on the Gunn diode chips are described. The modifications of the semiconductor layer system are explained on the basis of the direct current characteristics, before the high-frequency behavior of the Gunn diodes in an oscillator as a function of operating voltage and temperature is discussed. The following chapter deals with the scattering parameter measurements on coplanar Gunn diodes, with which the frequency-dependent behavior of the reflection factor and impedance was determined. Finally, the results of this work are summarized.

15 Chapter 2 Theoretical Basics This chapter explains the theoretical basics that are necessary to understand the Gunn diode and the measurements carried out. For this purpose, after a brief historical analysis, a qualitative description of the Gunn effect and its physical causes is given. This is followed by a simple quantitative analysis of the small-signal behavior of Gunn diodes and the dynamics of stable high-field domains. Finally, the influence of a resonator on the formation and propagation of domains is described. In chap. 2.3 different approaches to the optimization of the material system of a Gunn diode are discussed, which enable an improvement of the power yield and efficiency. The last section deals specifically with the problem of thermal stress on the component. 2.1 The Gunn effect Discovery by JB Gunn The Gunn effect [Gun63, Gun64], first observed by JB Gunn in 1963 and named after him, describes the occurrence of high-frequency current oscillations in certain III-V semiconductor samples when the electrical field is applied exceeds a critical value. Gunn first investigated the effect on n-doped GaAs and InP samples, which were ohmically contacted by alloying a suitable metallization (Sn for GaAs, In for InP). The current-voltage characteristics of the samples prepared in this way were measured with voltage pulses of 10 ns duration. Above a critical voltage, which corresponded to an electric field strength of Vcm-1, current fluctuations occurred in the form of periodic oscillations, the frequency of which was in the microwave range (Fig. 2.1). Gunn showed that the period of the oscillations is proportional to the sample length and approximately equal to the electron transit time between the two electrodes. Through further experiments, an influence of the ohmic contacts could be excluded, so that it seemed to be solely a property of the semiconductor material. Measurements of the microwave power output resulted in up to 0.5 W at 1 GHz; this corresponded to about 1-2% of the input power. Gunn recognized that these values ​​were his discovery for a technological application inter-

16 4 CHAPTER 2. THEORETICAL PRINCIPLES Figure 2.1: Current oscillations after applying a voltage pulse with an amplitude of 16 V and a length of 10 ns. The sample length was cm, the frequency of the measured oscillations was 4.5 GHz [Gun63]. made essential. He suggested several possible interpretations for his observations, including: the electron transfer to higher conduction band minima, but could not unambiguously clarify all open questions. The interpretation of the Gunn effect finally succeeded Krömer [Kro64], who explained the observations made by Gunn with the help of a theory developed by Ridley, Watkins and Hilsum. Ridley and Watkins [Rid61] had already shown a few years earlier that a negative differential conductivity can occur in certain semiconductors if, at high field strengths, electrons pass into an energetically higher minimum of the conduction band with less mobility. Hilsum [Hi162] confirmed this with a quantitative investigation for GaAs. The main features of the theory of Ridley, Watkins and Hilsum will be explained in more detail in the next section Physical causes of the Gunn effect The cause of the occurrence of the Gunn effect lies in the special conduction band structure of some semiconductors such as GaAs or InP (Fig.2.2) . In these semiconductor connections, besides the main minimum at the I'point, there are other so-called secondary minima in the power band, which are one amount of energy AE higher. The minor minimum at the L point of GaAs is, for example, 0.36 ev above the band edge at the I'point. In the main minimum, the density of states and thus also the effective electron mass is small (mi = 0.07 me for GaAs). This results in a relatively high mobility pi of the electrons. In contrast, the density of states in the secondary minimum at the L point is significantly higher, i.e. the effective mass is large (m2 = 0.4 me for GaAs) and the mobility M2 is consequently low.

17 2. 1. THE GUNN EFFECT 5 GaAs T '= 3 (1 (1 K InP T = 300 K Nebenmum Nebensaimurn r AL, CV I, eitung hasad E - {). 54; 2 a- [1.aitun sband 0 Hauptin in miun Eg-1.43oV 1tauptinintmusn E 1.34 el '5'alenzband \ -alesszbaisd Y x L, [) "[1001 pli] 4, * [1001 wave ektor Figure 2.2: Simplified representation of the band structure of GaAs and InP [Sze98 ]. If the external electric field is very small, the electrons only occupy the lowest energy states in the main minimum of the conduction band up to an energy that is given by the thermal energy of the electrons (, ~ ev at 300 K) Electrons accelerate and exchange the energy obtained in this way through collisions with the crystal lattice. This increases the mean electron energy, so that higher energy states in the conduction band can now also be occupied. If the mean electron energy reaches the interval transfer energy of 0.36 ev, the electrons can also the minor minimum Occupy at the L point. There the effective mass of the electrons is about six times as large as in the main minimum, and the electronic density of states is therefore significantly greater. Electrons with sufficient energy will therefore primarily occupy the secondary minimum. Although a further increase in the electric field causes a separate increase in the drift speed in the major and minor minimum, a range with negative differential mobility can occur in the combination due to the electron transfer. This should be made clear by means of a simple estimate. If one denotes the electron density in the main minimum with nl and in the secondary minimum with n2, then the following applies to the drift velocity v (e) = (ni / ii + n2p2) E where n = nl + n2 is the total electron concentration. If one sets for the relative occupation (2.1)

18 6 CHAPTER 2. THEORETICAL PRINCIPLES p of the secondary minimum we get from Eq.(2.1) Electric field strength E (kvlcin) Figure 2.3: Schematic representation of the mean electron drift speed as a function of the electric field strength for GaAs and InP. n2 n2 P n ni + n2 '(2.2) v (e) = (/ Il (1- p) + f12p) E. (2.3) If one assumes that the mobility is independent of the electric field strength, then de - , ~ 1 - (PI -, 12) (P + de E) (2.4) This results in dv 0 dp> ul -P2 p (2.5) ie the differential mobility becomes negative if the change in the occupation of the secondary minimum with the electrical Field strength exceeds a certain threshold. In this case, the relationship between drift speed and electric field strength has the form shown in Fig. 2.3 for the corresponding semiconductor. After reaching the limit field strength ET, the drift speed decreases and the differential mobility becomes negative. For the occurrence of negative differential mobility through electron transfer, the following requirements must be placed on the semiconductor: 1. The energetic distance AE between the main minimum and the secondary minimum must be significantly greater than the thermal energy kt of the electrons in order to thermally occupy the secondary minimum for small ones to avoid electrical fields.

19 2.1. THE GUNN EFFECT 7 2. The band gap Eg must be larger than AE in order to prevent impact ionization of electrons across the band gap from occurring before an electron transfer into the secondary minimum. 3. The effective mass of the electrons in the secondary minimum must be noticeably greater than the effective mass in the main minimum, so that there is a high probability that the secondary minimum will be occupied by electrons with the appropriate energy due to the resulting high density of states. 4. The mobility of the electrons in the secondary minimum must be significantly smaller than the mobility in the main minimum. 5. The electron transfer between the conduction band minima must take place within a narrow range of the electric field strength. An important experimental confirmation of the theory of Ridley, Watkins and Hilsum was achieved by Hutson et al. [Hut65] through experiments with GaAs under hydrostatic pressure. By increasing the pressure, the energy difference between the conduction band minima can be reduced in a controlled manner. The associated shift in the critical field strength ET towards lower values ​​was shown by Hutson et al. proven. Above a pressure of 26 kbar, the current oscillations finally stopped completely. This pressure agrees very well with theoretical calculations of the pressure that is necessary to lower the secondary minimum to the energetic level of the main minimum. In this case, the secondary minimum can already be occupied at room temperature without an external electric field. A field-induced transfer of electrons cannot take place and consequently no negative-differential mobility occurs. Similar experiments were carried out by Shyam et al. [Shy66] carried out by targeted application of pressure along certain crystal directions. Development of high-field domains and current oscillations Already Ridley [Rid63] showed that in a semiconductor, for which the dependence between drift speed and electric field shown in Fig. 2.3 applies, a splitting of the electric field can occur along the sample in domains with different field strengths, the domains passing through the sample with the mean drift speed of the electrons. This was demonstrated experimentally by Gunn [Gun65]. Using a capacitively coupled probe that could be moved along the sample surface, he was able to determine the potential profile along the sample. The measurement showed that the current oscillations he observed are associated with the occurrence of areas of high field strength, which arise at the cathode and run through the sample to the anode. The emergence of high-field domains in a semiconductor with negative-differential mobility can be illustrated using a simple qualitative consideration. Let there be a spontaneous fluctuation in the electron density n, which can occur, for example, as a result of noise effects or a deviation from the otherwise uniform doping concentration no. If the electric field is below the limit field strength ET, the random charge fluctuation Qo according to the exponential law becomes Q (t) = Qo exp (- t) (2.6) Td

20 8 CHAPTER 2. THEORETICAL PRINCIPLES E C A C A C Figure 2.4: Schematic representation of the creation of a stable high-field domain from a charge fluctuation. C denotes the cathode and A the anode [Hob74]. dismantled again. Td is the so-called dielectric relaxation time. If the semiconductor sample is represented in a simplified manner by a parallel connection of a differential resistance L Rd = (2.7) enpda and a differential capacitance Cd ecoa = (2.8) L, the dielectric relaxation time is Td = RdCd -. ECO enyd (2.9) where A is the cross-sectional area of ​​the sample, L the sample length and pd the differential mobility. If the electric field strength is above ET, i.e. if the differential mobility Pd is negative, then Td G 0, and the charge fluctuation increases according to Eq. (2.6) exponentially. This can be explained clearly with the aid of Fig. 2.4 as follows: The charge fluctuation has the form of an electrical dipole consisting of a depletion zone and an accumulation layer (Fig. 2.4a). The electric field E connected to this space charge zone via the Poisson equation eeo ~ = e (n - no) (2.10) is slightly increased at the point of the space charge zone compared to the surroundings. This results in a reduction in the drift speed of the electrons in the depletion zone, so that they fall back in relation to the moving space charge zone. As a result, on the one hand the depletion zone is expanded and on the other hand the accumulation layer is enlarged. The growth of the space charge zone

21 2. 1. THE GUNN EFFECT 9 causes according to Eq. (2.10) again increases the electric field strength in the area of ​​the space charge zone (Fig.2.4b). Since the voltage drop is constant along the entire sample, an increase in the field strength in the area of ​​the domain means a decrease in the field in the rest of the sample. Inside the domain the field strength increases to ED, outside it decreases to EA (Fig. 2.4c). ED and EA adjust themselves in such a way that the electron velocities inside and outside the domain are the same. With such a stable domain, the electric field strength EA outside the domain is below the limit field strength ET. This guarantees at the same time that no further domain can arise as long as a stable high-field domain passes through the sample. In a Gunn diode, the domains usually develop near the cathode. If the domain is stable, the electric field outside the domain has reached the constant value EA G ET, and a constant current I thus flows in the external circuit. The domain passes through the sample in time TD approximately at the mean drift speed of the electrons outside the domain until it reaches the anode. There the domain potential is reduced, whereby the electric field strength outside the domain and thus also the current in the external circuit increases. When the field strength reaches ET, a new domain is formed and the current I drops back to the initial constant value. The current curve results from a constant basic component on which periodic current pulses are superimposed at a time interval of TD Material-specific cut-off frequencies The maximum frequencies at which a Gunn diode delivers a significant microwave power are influenced by a large number of parameters. These include the heating of the sample due to the high power dissipation, space charge and diffusion effects, the impedance matching between the diode and the resonance circuit, the skin effect and the influence of the packaging. The following chapters deal with individual parameters at the appropriate point, a detailed discussion of all influencing variables can be found in [Sa183]. In this section, however, the purely material-specific restrictions are specifically discussed [Hob74]. The scattering process between the main minimum and the secondary minimum takes a finite time, which limits the maximum possible frequency. This is because negative differential mobility cannot occur before an interval spread has taken place. To explain the relationships, it is assumed that the field strength changes sinusoidally with time. If the electric field is above the limit field strength, the electrons in the main minimum can acquire enough energy to switch to the secondary minimum. The actual change takes place through a scattering process with a suitable phonon, which enables the electron to change its wave vector necessary for the change to the secondary minimum. The mean time between two scattering processes for GaAs at 300 K is approximately 10-12S, and the upper limit frequency due to the interval scattering is approximately 150 GHz. An even more severe limitation of the maximum frequency results from the relaxation time of the electrons in the main minimum. When the sinusoidal electric field drops again, the electrons have to switch back from the secondary minimum to the main minimum. Due to the high density of states in the secondary minimum, however, only a correspondingly large

22 10 CHAPTER 2. THEORETICAL PRINCIPLES number of electrons go over to the main minimum, when the mean energy of the electrons has dropped so far that there are enough free states. The relaxation time for GaAs is approximately S, and the result is a cut-off frequency of a maximum of 100 GHz. The relaxation time for InP is shorter, so that frequencies of up to 200 GHz can be achieved with this material system. Another way to get to higher frequencies is to use harmonics instead of fundamental oscillation. By means of special resonators adapted to harmonic operation, it is also possible to achieve sufficiently high microwave powers in the so-called second harmonic mode. 2.2 Theory of the Gunn Effect After the qualitative considerations in the first section, a theoretical treatment of the Gunn effect based on the Poisson equation and the current equation will now be presented. Both the DC behavior and the small and large signal behavior are analyzed. The latter includes the actual Gunn effect, i.e. the formation and spread of high-field domains. Finally, the different operating modes of a Gunn diode in a resonance circuit are examined. 2.11) aax EEo j = en (x) v (e). (2.12) Here x is the spatial coordinate in the direction of the current flow from the cathode to the anode, p (x) the charge density, e the relative dielectric constant of the semiconductor material, Eo the dielectric constant of the vacuum and v (e) the field-dependent drift speed of the electrons. The charge density p (x) is composed of 1. the concentration n (x) of the negatively charged, freely moving electrons (the hole concentration p is neglected because of p GG n and fcp «/ In) and 2. the concentration no (x) the positively charged, stationary donors. Thus, and inserted in Eq. (2.11) we get p (x) = -e (n (x) - no (x)) (2.13) ae (x) - c (n (x) _ no (x ». (2.14) aax EEo) with the help of Eq. (2.12) the electron concentration n (x), we get aae (x) noc j ai EEo (enov (e) - 1) (2.15)

23 2.2. THEORY OF THE GUNN EFFECT Local coordinate x / x, Figure 2.5: Field strength E as a function of the local coordinate x at different current densities j. The field strength E is normalized to the limit field strength ET and the spatial coordinate x to the length xo, which assumes the value xo = 21.5 / im for a doping of no = cm-3. The current density of the individual curves is given in units of it, the current density for ET [Cum66]. If there is charge neutrality in the semiconductor, then n = no. Because j = e n v, the quotient in the brackets of Eq. (2.15) then equals one and it follows a (x) = 0 =:> E (x) = const., (2.16) i.e. Ohm's law applies. However, if n 7 ~ no, i.e. j 7 ~ enov, then Eq. (2.15) can only be solved numerically. Fig. 2.5 shows the result of a corresponding calculation by McCumber and Chinoweth [Cum66] for no = 1013 cm-3. It can be seen from the figure that the field strength E increases monotonically with the spatial coordinate x and the current density j. Because of U = JE (x) dx (2.17), the voltage U increases monotonically with the current density j. This means, however, that the steady-state solutions of the current-voltage characteristic show no negative-differential resistance. Because of j = e n v and the course of the v (e) characteristic curve known from Fig. 2.3, one would have expected a range with negative-differential resistance in the current-voltage characteristic curve. In this calculation, however, it must be taken into account that changes over time are neglected and the doping is assumed to be spatially constant. In the considerations in Chap. However, a doping fluctuation was identified as the cause of the domain formation

24 1 2 CHAPTER 2. THEORETICAL BASICS assumed. To investigate the influence of such a doping fluctuation, McCumber and Chinoweth modified the homogeneous doping in their calculations by slightly lowering the doping concentration in a small area near the cathode. The dimensions of the sample remained unchanged. It was found that for a doping of no = cm-3, which was reduced to cm-3 over a small area, stable characteristics always occurred. However, if the doping was increased overall by a factor of 10, the result was characteristic curves which became unstable above a certain field strength. In the following section it will be shown that stationary characteristics can only be observed if the product nol G is 1012 cm-2, where L denotes the length of the sample. Small signal behavior To determine the influence of temporal disturbances on the stability of the stationary solutions , one starts again from the Poisson equation and the current equation, which is expanded here by the displacement current [Cum66, Hei71]: Furthermore, it should be assumed that the disturbances are periodic and small compared to the respective equilibrium values ​​Eo, jo and vo. This results in the following approaches: From Eq. (2.18) follows with the help of the approach from Eq. (2.20) E (x, t) = Eo + AE (x) e2wt (2.20) j (t) = jo + Aj e iwt (2.21) v (x, t) = vo + hoae (x) e2wt. (2.22) Solved according to the electron concentration, this results in ae (x) - e (n (x) _ no (x », (2.18) OOx eeo j = en (x) v (e) + eeo ae. (2.19) d0e ezwt =? (n _ no). (2.23) dx eeo eeo d0e n = no Zwt + e. (2.24) dx If the perturbation approaches Eq. (2.20) to Eq. (2.22) are put into the current equation Eq. (2.19) and eliminates n using Eq. (2.24), one obtains: ZWt eeo d0e t jo + Aj e = e (no + ezwt) (vo + po AE e zw) + i eeoae we ZWt. e dx (2.25 ) After simplification, it follows Aj d0e d0e = (iweeo + epono) AE + voeeo, + poeeoae, ezwt (2.26)

25 2.2. THEORY OF THE GUNN EFFECT 13 The last term in Eq. (2.26) can be neglected because it is in the order of magnitude of C7 ((DE) 2). This leaves Aj = d E. (iweeo + epono) AE + v0eeo d (2.27) Assuming that the doping concentration no is independent of the position coordinate x, then Eq. (2.27) is an inhomogeneous, linear differential equation with constant coefficients. If one chooses AE (x = 0) = 0 as the boundary condition, the general solution of this differential equation is: with the abbreviations AE (x) = ~ j (1 - e-yx) (2.28) EEOVO'Y y = '3X + io, ßx = enoico - wx 'wx - enomo = 1' EEOVO VO EEO Td Vo It follows from Eq. (2.28) AE (x) ezwt = Oj ezwt EEOVO'y EEOVO'Y x eiwt (2.29) The first term on the right-hand side of Eq. (2.29) describes a homogeneous field oscillation, which is important for the development of the LSA mode (see Chapter). The second term is a wave advancing in the x-direction with the phase velocity w VPh = 7 = Vo (2.30) and the group velocity aw vgr = - = VO. (2.31) The electron density is modulated by the electric wave AEe2Wt and generates a space charge wave that propagates with the electron drift velocity vo. For 13x> 0 this wave is damped, for ßx <0 the amplitude of the wave increases. This condition is fulfilled if the mobility yo <0, i.e. - according to the relationship v (e) between drift speed and field strength in GaAs - if the electric field strength has exceeded the threshold value ET. The sample is stable against small disturbances if the real part of the impedance Z is positive. The following applies to the impedance Z: (2.32) where A is the cross-sectional area of ​​the sample. With L AU = IAE (x) dx (2.33) 0

26 14 CHAPTER 2. THEORETICAL PRINCIPLES and Eq. (2.28) then follows Z L / ~ -, yl A Oj J AE (x) dx 1. = AEoV0 e L_ ~ 2 0 (2.34) One can show [Cum66] that Eq. (2.34) has no singularities, i.e. with Aj = 0, DU = 0 is not also at the same time. A GaAs sample is therefore always stable with current control (Aj = 0). In order to determine the behavior with voltage control (DU = 0), the real part of the impedance Z must be examined. McCumber and Chynoweth [Cum66] could from an analysis of the expression in Eq. (2.34) deduce that the stability of the sample is determined by the product of the doping concentration no and the length L of the active layer. The nol product can also be derived from a simple qualitative consideration.In Chap. The dielectric relaxation time Td = EEO (2.35) eno / io was introduced as the time constant characteristic for the construction of domains. The domain transit time T must therefore be at least equal to or greater than the absolute value of Td so that Gunn oscillations can occur. From this it follows L VO EEO eno Ipo = ITd1 (2.36) or nol> EEOVO e I pol, (2.37) Since the negative mobility po changes with the electric field strength, the nol product also changes depending on the field strength. Approximately one obtains nol> cm-2. (2.38) If the nol product of a GaAs sample is below this limit, the real part of the impedance Z is positive, i.e. the sample is stable and there is no Gunn effect. With a suitable field strength, however, such samples show negative-differential mobility and can therefore be used to amplify microwave signals. Dynamics of high-field domains If doping and sample length are chosen so that the no L product is greater than cm-2, high-field domains form in the semiconductor and the actual Gunn effect can be observed. The formation of such high-field domains has already been explained in Chapter and with the aid of Fig. 2.4. In this section, the behavior of stable domains will be investigated, i.e. domains whose shape no longer changes during movement through the semiconductor.

27 2.2. THEORY OF THE GUNN EFFECT 15 The consideration is based again on the Poisson equation and the current equation, which now also contains the diffusion current [But65, But66]: cae aax e EEO (2.39) OOE aj = env (e) + EEo - e - (The). c ~ t We are looking for solutions with the following properties: (2.40) 1. The domain should be stable, i.e. it should move through the semiconductor at the speed VD without changing its shape. 2. Outside the domain, the electron concentration n should be constant and equal to the doping concentration no, i.e. n ~ no = const. For x ~ ± oo. 3. The electric field EA outside the domain should be constant, i.e. E ~ EA = const. For x ~ ± oo. Under these conditions, the electric field E and the electron concentration n can be written as functions of the variable r = x-vdt (2.41). This transformation brings about the transition to a coordinate system that moves with the domain velocity VD. As a result of the transformation, the time dependency in Eq. (2.40), which expresses the unchangeable form of the domains. Because of de Or de ae de ar de dr aax dr 'ât dr ât -VD dr', Eq. (2.39) and from Eq. (2.40) j = env - EEOVD ~ E - e- (Dn). (2.43) r The current density in the areas outside the domain, in which, according to the boundary conditions, the constant field strength EA and the constant doping no is present, is due to the lack of diffusion and displacement current With Eq. (2.44) and Eq. (2.42) follows from Eq. (2.43) (Dn) = dr (Dn) - = dr (Dn) de _ -_ e - (n - no) (2.42) dr EEO j = enov (ea) = enova. (2.44) enova = env - CVD (n - no) - c r (Dn) (2.45)

28 16 CHAPTER 2. THEORETICAL BASICS or dr (Dn) = n (v - VD) - no (va - VD) - (2.46) If this equation is divided by Eq. (2.42), one obtains d (Dn) -_ eeo n (v VD) no (va - VD). (2.47) de e n - no If one assumes for the sake of simplicity that the diffusion constant D is field-independent, dn _ Eco n (v - vd) - no (va - vd). de n - no then follows (2.45) If one integrates this equation starting with the conditions at infinity up to a point in the domain n no (E) dn f «v (e ') (VA non' Den o - vd) n EA - vd l de ', (2.49) results in no - In (no) _1 = E Deno.1 «v (e') - VD) EA (VA n - VD)) de '. (2.50) At the location of the maximum field strength within the domain E = ED, deldr = 0, i.e. according to the Poisson equation n = no, i.e. the left side of Eq. (2.50) is zero: ED o EA 0 Den, 1 ((v - VD) n (VA - VD) I del. (2.5l) The integration of EA to ED can be done over the area of ​​the accumulation layer (path 1 in Fig 2.6) as well as along the depletion zone (path 2 in Fig. 2.6). In any case, 0 = «V - VD) - n0 (VA - VD)) de '_ ((v - VD) - n ( va - VD)) de '. (2.52) EA Wegl EA Weg2 Since along route 1 n> _ no and along route 2 n <_ no applies and thus the second term in the integrals from Eq. (2.52) is different, the above equation can only be fulfilled if VA = VD. (2.53) The calculation shows that the domain velocity VD is equal to the drift velocity of the electrons especially outside the domain.

29 2.2. THEORY OF THE GUNN EFFECT 17 Path 1 Path 2 Figure 2.6: The integration in Eq. (2.52) can take place both along path 1 via the accumulation layer and along path 2 via the depletion zone. Since the ratio no / n in the integral depends on the respective path, there is a contradiction if va = vd does not apply. For E = ED, Eq. (2.51) on ED ED 1 (v - VD) de '= 1 (v - va) de' = 0. (2.54) EA EA This is the so-called area rule of Buteher, the clear interpretation of which is shown in Fig. 2.7. At a certain voltage U on the semiconductor sample, a field EA outside the domain is established. Ea also defines va = v (ea) = vd. The associated field strength ED is according to Eq. (2.54) is defined by the equality of the hatched areas in Fig. 2.7. For different value pairs EA, above all, the associated values ​​for ED, vd lie on the dashed curve, which is also called the dynamic characteristic and can be constructed with the aid of the area rule. Now the shape of a stable high-field domain in the case of neglected diffusion is to be investigated in more detail. For D = 0, using the area rule va = VD results from Eq. (2.43) and Eq. (2.44) en ova = env - CCOVAE. (2.55) If one eliminates n with the help of the Poisson equation Eq. (2.42), then -de - = eno (v VA) (v - VA). (2.56) dr Eeo The two possible solutions of this equation are v (e) = va = v (ea), i.e. E = EA (2.57) and de dr _ eno eeo (2.58)

30 1 8 CHAPTER 2. THEORETICAL PRINCIPLES ranmic characteristic equal area Ei electric field strength E E D Figure 2.7: Illustration of the relationship between EA and ED using Butcher's area rule [But65]. In the borderline case without diffusion, the domain therefore consists of a depletion zone completely free of mobile charge carriers, over which the field strength according to Eq. (2.58) decreases linearly, and an infinitely thin accumulation layer at which a jump in the electric field strength occurs. The domain thus has a triangular profile (Fig. 2.8). Finally, a relationship between the field strengths EA and ED and the voltage UB applied to the semiconductor sample is to be derived. The following applies: UB = UD + EA (L - b), (2.59) where L is the sample length, b is the width of the domain and UD is the domain voltage. b «L, one can write in a simplified way Is UB = UD + EAL. (2.60) UD is defined as UD = 1 (E - EA) dr (2.61) E> EA and can be viewed as a measure of the size of the domain. If diffusion is neglected, the domain has a triangular shape and the integral in Eq. (2.61) corresponds to the triangular area UD = (ED - EA) 2. (2.62) With the domain width b, the drop in field strength in a triangular domain can also be expressed in the form de _ ED - EA (2.63) dr b

31 2.2. THEORY OF THE GUNN EFFECT 19 n E, EA Figure 2.8: Profile of a domain neglecting diffusion. express. By comparison with Eq. (2.58) is obtained and inserted into Eq. (2.62) results in UD b = EEO (ED - EA) (2.64) eno 960 2eno (ED - EA) 2. (2.65) After ED and EA have been related with the help of the area rule and the dynamic characteristic, Eq. (2.65) shows a relationship between the domain voltage UD and the field strength EA outside the domain [Cop66, Hob74]. The additional boundary condition in Eq. (2.60). Fig. 2.9 shows the relationship between UD and EA for a given doping no as well as the boundary condition according to Eq. (2.60) is plotted as a load line for a fixed voltage UB and a given sample length L. If the mean field strength UBIL (point of intersection of the load line with the EA axis) is greater than the limit field strength ET (load line 1 in Fig.2.9), there is exactly one point of intersection that clearly corresponds to the values ​​UB, L and no Field strength EA determines. A stable domain can also exist if the mean field strength UB / L falls below ET (load line 2 in Fig. 2.9). The prerequisite is that the UB / L value was briefly above ET in order to generate the domain. Load Line 2 has two intersections with the UD-EA curve, of which the lower one is unstable in this case. If namely - starting from the unstable working point - the domain voltage increases due to a small disturbance, the field strength EA decreases according to Eq. (2.60), but remains larger than given by the UD-EA curve. As a result, the electrons move faster than the domain outside the domain, so that both the accumulation layer and the depletion zone grow. According to the Poisson equation, this leads to a further increase in UD, as a result of which the operating point continues to move upwards until it finally reaches the stable position. There is also a certain minimum voltage, the so-called holding voltage UH, which is set at a given

32 20 CHAPTER 2. THEORETICAL PRINCIPLES Field strength E., Figure 2.9: Representation of the relationship between the domain voltage UD and the electric field EA outside the domain. Furthermore, three load lines for different supply voltages UB are shown. The intersection of the load line and the UD-EA curve determines the working point of the Gunn element. Sample length L must be applied so that the domain is preserved. 3 in Fig. 2.9 is a tangent to the UD-EA curve. The load line behavior in the oscillator belonging to UH In order to use a Gunn diode to generate microwaves, the component must be built into a resonance circuit. This can be, for example, a waveguide resonator or a planar microstrip live resonator. In this section, the behavior of a Gunn diode in a resonance circuit should therefore be illustrated by means of simple considerations. In addition to the static current-voltage characteristic, you also need the dynamic current-voltage characteristic, i.e. the current-voltage behavior in the event that a domain crosses the Gunn diode. In the static case, i.e. without the formation of domains, the component behaves like an ohmic resistor. In simplified form, the characteristic curve is a straight line up to the limit voltage UT (Fig. 2.10). Domains form above UT. When the domain has reached its stable form, the field strength outside the domain has dropped to the value EA and consequently the current also drops by the amount AI. A further increase in the external voltage U essentially causes an increase in the domain voltage UD (see Fig. 2.9). The field strength EA, however, hardly changes, so that the current I remains almost constant. The external voltage U can drop to the holding voltage UH without the domain being extinguished during its migration. However, the domain shrinks a little if the voltage drops below UT, as a result of which the field EA outside the domain and thus also the current I increase slightly.