2. Device Physics / Theoretical basis

In this chapter an overview of the basic principles behind a Gunn oscillator is presented. The phenomenon of the bulk negative conductivity concerning semiconductor with a particular band structure is explained. Relevance is given to the conditions under which oscillations, small signal amplification or only pure ohmic behaviour of the Gunn devices are achieved. A detailed description of different Gunn injectors will point out the possible device improvements. Finally the heat-sink issue is explained and a finite elemente thermal analysis illustrates the best geometric/material configurations for cooling Gunn devices.

2.1 Theory of the Gunn effect

2.1.1 History

The Gunn diode, also known as Transferred Electron Device (TED), is an active two-terminal solid-state device. It is unique in the sense that its voltage controlled negative differential resistance is only depending on bulk material properties rather than a junction or an interface.

The fundamental mechanism, the transferred-electron effect, was theoretically described by B. K. Ridley and T.B. Watkins in 1961 [RW61]. In 1962, Hilsum predicted the possibility of transfer-electron amplifiers and oscillators [Hil62]. In spite of Ridley-Watkins-Hilsum work, the transferred electron effect was named after an IBM researcher interested in the response of III-V semiconductors on pulsed voltages, J. B. Gunn.

Figure 2.1: J. B. Gunn's first results applying $16\thickspace V$, $10\thickspace ns$ voltage pulses. The device length was $\mathrm{2.5 \cdot 10^{-3}cm}$ and the corresponding oscillation frequency was $4.5 \thickspace GHz$ [Gun63].

In 1962, independently Gunn observed a "noisy" resistance, measured as a function of the voltage, applying $704 \thickspace V$ on a GaAs sample. "Why did the reflected signal from a 50$\Omega$ transmission line, terminated in GaAs sample, produce several ampere of noise?"
With better equipment Gunn detected regular current oscillations at about 5GHz and applied for a patent (1963). This spontaneous discovery founded the development of active-semiconductor devices to replace microwave vacuum tubes. After publishing his results [Gun63,Gun64,Dun65], many researchers started studying Gunn diodes.
H. Krömer was the first to link Gunn oscillations with the transferred-electron effect in 1964 ([Krö64]). A convincing evidence of such a correlation was delivered by A. R. Hutson, A. Jayaraman and A. G. Chynoweth from Bell Labs in 1965. They showed how hydrostatic pressure could first decrease the threshold field and then suppress the current oscillations, demonstrating that the Gunn oscillations are based on the electron transfer from the low- to the high-energy valley ($\Gamma$ and L).

2.1.2 The transferred-electron effect and the domain formation

Figure 2.2: Crystallographic directions in a zinc-blende crystal.

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Figure 2.3: Simplified band structures of zinc-blende (cubic) GaAs, InP and GaN. [Sze98,Mor99]

The transferred-electron effect arises from the particular form of the band structure of some III/V compound semiconductors like GaAs, InP and GaN. As shown in Fig. 2.3, these materials are direct band-gap semiconductors, having the conduction band main minimum in the $\Gamma$-point. Two other satellite valleys^2.1 L and X are in the directions [111] and [100], respectively.
At low electric fields, the conduction band electrons occupy the bottom of the central valley. Applying an electric field, the electrons accelerate until they collide with imperfections of the crystal lattice. Via collisions, the electrons loose a component of their momentum, which is directed along the electric field, and some kinetic energy which raises the lattice temperature (Joule heating). As the electric field is incremented further, the mean electron energy becomes higher, and higher energy states can be occupied in the conduction band. When the electron kinetic energy reaches the intervalley transfer energy $\Delta E=E_L-E_{\Gamma}$ (for GaAs $0.32\thickspace eV$), electrons have the additional choice of occupying one of the satellite valleys (for GaAs the L-valley), as long as a suitable momentum transfer is also involved. The electron effective mass $m^*$ is depending on the curvature of the band structure E(k) [IL03]:

$$\left(\frac{1}{m^*}\right)_{ij}= \frac{1}{\hbar ^2} \frac{\partial ^2 E(k)} {\partial k_i \partial k_j} \thickspace ,$$ (2.1)

where $m_{ij}^*$ is effective mass tensor, $\hbar$ is the Plank constant and k is the wave vector. Assuming that the effective mass tensor components (in the direction of the principal axes) are equal to $m^*$, equation 2.1 can be simplified as:

$$m^*=\frac{\hbar ^2}{d^2E(k)/d^2 k} \thickspace .$$ (2.2)

In the satellite valleys, the curvature is higher and the effective mass of the electrons is up to 6 times the $\Gamma$-valley effective mass^2.2. Electrons with sufficient energy have the choice of occupying either valley. For these electrons there is a higher probability of occupying the satellite valleys which provide a relatively high density of states. In the satellite valleys, the electrons not only posses a higher effective mass, but also undergo strong scattering processes [BHT72]. The combination of these two effects explains why the mobility in the side valley $\mu_{L}$ is up to 70 times lower compared to that in the central valley $\mu_{\Gamma}$. If $n_{\Gamma}$ and $n_{L}$ are the electron density in the central and satellite valleys, respectively, the mean drift velocity $v_d(\mathcal{E})$ is:

$$v_d(\mathcal{E})=\frac{\mu_{\Gamma}n_{\Gamma}+\mu_{L}n_{L}}{n_{\Gamma}+n_{L}} \thickspace \mathcal{E} \thickspace .$$ (2.3)

For higher electric fields, more electrons have the sufficient energy and the satellite valley occupation $n_{L}$ will increase. Although an increasing electric field should lead to a higher electron drift velocity in each valley, the intervalley electron transfer can compensate this effect and result in a negative differential mobility. Defining $\eta$ as the relative L-valley occupation

$$\eta=\frac{n_{L}}{n_{\Gamma}+n_{L}},$$ (2.4)

and assuming that the mobility does not depend on the electric field, Eq. (2.3) leads to:

$$\frac{d v_d(\mathcal{E})}{d\mathcal{E}}=\mu_{\Gamma}-(\mu_{\Gamma}-\mu_{L})\left(\eta+\frac{d\eta}{d\mathcal{E}}\mathcal{E}\right) \thickspace .$$ (2.5)

In order to have a negative differential mobility, the following condition has to be fulfilled:

$$\frac{d v_d(\mathcal{E})}{d\mathcal{E}}<0 \hspace{1cm} \Rightarro... ...rac{\frac{\mu_{\Gamma}}{\mu_{\Gamma}-\mu_{L}}-\eta}{\mathcal{E}} \thickspace .$$ (2.6)

Taking into account that $\frac{\mu_{\Gamma}}{\mu_{\Gamma}-\mu_{L}}$ is higher than 1 and $\eta$ lower than 1, $\frac{d\eta}{d\mathcal{E}}$ must always be positive, which means that the L-valley relative occupation ($\eta$) has to increase with the electric field. Equation 2.6 confirms that the intervalley electron transfer can cause a negative differential mobility.

Figure 2.4: Schematic view of the average electron drift velocity $v_{d}$ at 300K as a function of the electric field E for GaAs, InP and GaN [IL03,AWR⁺98].

Figure 2.4 shows the average electron drift velocity as a function of the electric field for GaAs, InP and GaN. The drift velocity raises with the electric field up to the threshold field $\mathcal{E}_T$, hence a negative differential mobility appears and the drift velocity starts to decrease. $\mathcal{E}_T$ is extremely dependant of the material: in GaAs $\mathcal{E}_T$ is about $\sim \thickspace 3.5 \thickspace KV/cm$, in InP it is $\sim \thickspace 10 \thickspace KV/cm$ and in GaN values between $140 \thickspace KV/cm$ and $170 \thickspace KV/cm$ are reported [KOB⁺95,AWR⁺98]

A frequency independent $v-\mathcal{E}$ characteristics can be used to describe electron transport in the presence of a time-varying electric field as long as the frequency of operation is significantly lower than the relaxation frequency $f^*$ defined as [AP00]:

$$f^*=\frac{1}{\tau_{ER}+\tau_{ET}} \thickspace ,$$ (2.7)

where $\tau_{ER}$ is the energy relaxation time and $\tau_{ET}$ is the intervalley relaxation time. For GaAs and GaN, $\tau_{ER}$ is 1.5 and $0.15 \thickspace ps$, respectively and $\tau_{ET}$ is 7.7 and $1.2\thickspace ps$ [BHT72,KOB⁺95]. Based on these values, the relaxation frequency $f^*$ of GaAs is found to be $109 \thickspace GHz$. The frequency capability of GaN is superior, as indicated by the $f^*$ of $740\thickspace GHz$.

In summary, the material requirements for transferred-electron negative differential mobility are [Hob74]:
$\parbox{16cm}{ \begin{itemize} \item The energy gap from the satel... ... the valleys must occur over a small range of electric fields. \end{itemize}}$

2.1.2.1 Internal instabilities and domain formation

The negative differential mobility seen in Fig. 2.4 can be exploited directly for AC power generation, if the Gunn diode is connected to an adequate circuit. Stringent conditions are placed on its realization and further description will be given in section 2.3.3 (Low Space-charge Accumulation, LSA operating mode). One difficulty encountered when utilizing directly Gunn diodes as AC negative differential resistances, arises from internal instabilities.

Figure 2.5: Schematic view of the growth of space-charge fluctuations to a stable dipole domain [Hob74]. C identifies the cathode (emitter) and A the anode (collector).

To explain how a high-field-domain builds up in a semiconductor with a negative differential mobility, a simplified example from Hobson [Hob74] is presented. Let us consider an uniformly-doped device with an electron concentration $n_0$. A noise process or a defect in the doping uniformity causes a fluctuation in the electron density $n$. The fluctuation is an electric dipole, consisting of a depletion region and an accumulation region (Fig. 2.5(a)). The electric field relation to the non-uniformity in the space charge is given by the Poisson equation:

$$\epsilon_0 \epsilon_r \frac{\partial \mathcal{E}}{\partial x} = e \left( n - n_0 \right) \thickspace ,$$ (2.8)

where $\epsilon_r$ is the relative permittivity and $\epsilon_0$ is the vacuum permittivity.

If the mean electric field is below the threshold field $\mathcal{E}_T$, electrons with a higher electric field move faster than electrons elsewhere. The space charge accumulation fills in the depletion region and the fluctuation is damped by dielectric relaxation.

If the mean electric field is above $\mathcal{E}_T$, the drift velocity of the electrons in the region of higher field is reduced. The space-charge region swells (Fig. 2.5(b)) and, as a consequence of equation 2.8, the electric field raises in the region of the domain. In the rest of the device, the electric field sinks because the total voltage drop must remain constant.

In figure 2.5(c), the domain has reached a stable condition. The level of the electric field outside the domain $\mathcal{E}_R$ is under the threshold and the electric field in the domain reaches $\mathcal{E}_P$. $\mathcal{E}_P$ and $\mathcal{E}_R$ correspond to the same drift velocity. Inside the domain, electrons travel as fast as outside the domain and the space-charge region stops growing. When a domain is stable, no other domain can build up while $\mathcal{E}_R$ is below the threshold.

2.1.3 Domain Dynamics

Figure: Illustration of the equal area relationship between $\mathcal{E}_R$ and $\mathcal{E}_p$ [But65,BFH66].

In this section, the behaviour of stable domains will be examined. The stable space charge profile, or domain, drifts at a constant velocity through the device. The electric field in the different parts of this nonlinear entity is strictly related to the domain growth and decay and to the corresponding stability. The analysis of the domain form requires non linear solutions of the Poisson equation and the current continuity equation taking the $v-\mathcal{E}$ characteristics into account. An analytical solution of the problem comes from Butcher et al. [But65,BFH66]. Their most important results can be summarized in three items:

• If the diffusion parameter D is assumed constant and independent of the electric field, the domains travel with the same velocity as the electrons outside the domains ($v_R$).
• The peak electric field, $\mathcal{E}_p$ and $v_R$ are related with the so-called dynamic characteristic (Fig. 2.6).
• The dynamic characteristic can be constructed geometrically from the $v-\mathcal{E}$ characteristics ( $F_{v-\mathcal{E}}$) as the locus of points ( $\mathcal{E}_p$,$v_R$), which satisfies the condition:

  $$Area\left(\mathcal{A}(\mathcal{E}_p,v_R)\right)=Area\left(\mathcal{B}(\mathcal{E}_p,v_R)\right) \thickspace ,$$ (2.9)

  
  
  where the region $\mathcal{A}$ is defined from
  $$\mathcal{A}(\mathcal{E}_p,v_R)=\{(\mathcal{E},v)\thickspace : \th... ...\thickspace \cap \thickspace v<F_{v-\mathcal{E}}(\mathcal{E})\} \thickspace ,$$
  and the region $\mathcal{B}$ is defined from
  $$\mathcal{B}(\mathcal{E}_p,v_R)=\{(\mathcal{E},v)\thickspace : \th... ...{E}_p \thickspace \cap \thickspace \mathcal{E}>\mathcal{E}_T \} \thickspace .$$

The conserved^2.3 current density $j$ is given by:

$$j = e n v_d(\mathcal{E}) \medspace + \medspace \epsilon_0 \epsil... ...artial t} \medspace - \medspace eD \frac{\partial n}{\partial x} \thickspace .$$ (2.10)

where the three terms represent, respectively, the drift, displacement and diffusion component. Equation (2.10) is simplified outside the domain, where $n$ and $\mathcal{E}$ are independent of the position. The current is carried entirely as drift current. Inside the domain, all three terms of Eq. (2.10) play a role. The electric field gradients move and determine the displacement current to flow. At the peak field of the domain, there is no displacement current because ${\partial E}/{\partial x}=0$, but the electron density gradient results in a diffusion current. Figure 2.6 illustrates the relative values between diffusion and drift currents. For the depletion region of the domain, the current is predominantly carried by a displacement current, while in the accumulation region there is a large drift current opposed by displacement and diffusion currents.

Figure 2.7: Domain profile in the limit condition of zero diffusion.

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Further on, the form of a stable high field domain will be examined, neglecting the diffusion and it will be shown that the domains travel at the same velocity ( $v_{domain}$) as the electrons outside the domain($v_R$). If the diffusion coefficient D is assumed to be zero, the electron density of the domain must be zero (depletion region) or $\infty$ (accumulation region) [Hob74]. In this simple case, the electric field in the domain is triangular, as shown in Fig. 2.7. Outside the domain, nothing changes and the carrier concentration remains $n_0$. The current density outside the domain is

$$j=n_0 e v_R \thickspace .$$ (2.11)

Inside the domain, in the case of a fully depleted region, the current density is written:

$$j^*=\epsilon_0 \epsilon_r \frac{\partial \mathcal{E}}{\partial t}... ..._0 \epsilon_r \frac{\partial \mathcal{E}}{\partial x} v_{domain} \thickspace .$$ (2.12)

The Poisson equation requires:

$$\epsilon_0 \epsilon_r \frac{\partial \mathcal{E}}{\partial x}=-n_0 e \thickspace ,$$ (2.13)

therefore

$$j^*=n_0 e v_{domain} \thickspace .$$ (2.14)

By current continuity ($j^*=j$), Eq. (2.11) and Eq. (2.14) require that

$$v_R=v_{domain} \thickspace .$$ (2.15)

In the case of a fully depleted domain, the result is not influenced by the electric field dependence of the diffusion constant D.

The next problem consists in finding the stable value of the outside field $\mathcal{E}_R$ in connection with the applied terminal bias $U_B$. If $L$ is the device length, $w$ the width of the domain and $U_D$ the domain voltage, $U_B$ is given by:

$$U_B=U_D+\mathcal{E}_R(L-w) \thickspace .$$ (2.16)

The domain length can be neglected, because it is small in comparison with the device length ($w<<L$), therefore:

$$\colorbox{grau}{$\displaystyle{U_B=U_D+\mathcal{E}_R L \thickspace .}$}$$ (2.17)

$U_D$, in the case of the triangular domain (with the diffusion coefficient D=0) is given by:

$$U_D=\frac{w (\mathcal{E}_p-\mathcal{E}_R)}{2} \thickspace.$$ (2.18)

From the Poisson equation

$$w=\frac{\epsilon_0 \epsilon_r (\mathcal{E}_p-\mathcal{E}_R)}{n_0e} \thickspace,$$ (2.19)

resulting for $U_D$ the expression:

$$\colorbox{grau}{$\displaystyle{U_D=\frac{\epsilon_0 \epsilon_r (\mathcal{E}_p-\mathcal{E}_R)^2}{2n_0e} \thickspace.}$}$$ (2.20)

Equation (2.20), the electrical boundary condition (Eq. (2.17)) and the equal area rule between $\mathcal{E}_p$ and $\mathcal{E}_R$ determine the domain behaviour for a given terminal bias $U_B$.

Figure 2.8: Diagram of the relationship between the domain voltage $U_D$ and the electric field outside the domain $\mathcal{E}_R$ imposed by the space charge dynamics (solid-line). At the same time, the device must satisfy the boundary conditions contained in the load-line (dashed line). Three possible configurations for three external biases $U_B$ are considered.

Figure 2.8 outlines the relationship between the domain voltage and the electric field outside the domain. In the diagram, the three load lines correspond to three different terminal biases $U_B$. The intercepts are the possible solutions for a given doping concentration $n_o$, device length $L$. The electrical field $\mathcal{E}_R$ is displayed versus the domain voltage $U_D$. Three configurations are possible:

Load line 1.

The average electric field $U_B/L$ (the intersection of the load-line with the $\mathcal{E}_R$-axis) is higher than threshold field $\mathcal{E}_T$. Under this condition, only one solution is possible and the solution is stable.

Load line 2.

The average electric field $U_B/L$ is equal to $\mathcal{E}_S=V_S/L$. The value $V_S$ is called sustaining voltage. Under this characteristic voltage the domain will be extinguished in flight. At the sustaining voltage, the load-line is tangent to the $U_D-\mathcal{E}_R$ curve. $V_S$ is extremely important concerning the analysis of the oscillation modes.

Load line 3.

The average electric field $U_B/L$ is lower than $E_T$ and higher than $\mathcal{E}_S=V_S/L$. If the value $U_B/L$ has been higher than $\mathcal{E}_T$ during the domain nucleation, two solutions are possible. The higher intercept is stable, the lower one unstable.
Keeping in mind that the load line condition (Eq. (2.17)) must be satisfied all the time and that the $U_D-\mathcal{E}_R$ curve refers only to a steady state domain, the instability of the lower intercept can be explained with a simple example. Let us consider a small noise fluctuation: a small increase of $U_D$ causes $\mathcal{E}_R$ to decrease (Eq. (2.17)). The new state is not anymore on the $U_D-\mathcal{E}_R$ curve, but higher. In the new state, the electrons outside the domains move faster than the domain. This means that both the accumulation and depletion regions of the domain will grow causing a further increase of $U_D$ (Poisson's equation). In the next state, the working point will move again higher on the load-line and the readjustment will continue up to the higher stable intercept.

2.1.4 The small signal behaviour of a transferred-electron device

In this section, the stability with respect to small current or voltage perturbations of the steady-state solutions will be considered [MC66,Hei71]. The analysis starts from the Poisson's and current equations:

$$\frac{\partial \mathcal{E}(x)}{\partial x} = \frac{e}{\epsilon_r \epsilon_0}\left( n(x) - n_0(x) \right) \thickspace ,$$ (2.21)

$$j = e \medspace n(x) \medspace v_d(\mathcal{E}) \medspace + \medspace \epsilon_r \epsilon_0 \frac{\partial \mathcal{E}}{\partial t} \thickspace ,$$ (2.22)

and from two assumptions about the fluctuations:

• the fluctuations are periodic in time;
• the fluctuations are small in comparison to the corresponding equilibrium values $\mathcal{E}_0$, $j_0$ and $v_0$.

The so defined fluctuations are governed by:

$$\mathcal{ E}(x,t) = E_0 \medspace + \medspace \Delta \mathcal{E}(x) \thickspace e^{i \omega t} \thickspace ,$$ (2.23)

$$j(t) = j_0 \thickspace \medspace + \medspace \Delta j \thickspace e ^{i \omega t} \thickspace ,$$ (2.24)

$$v(x,t) = v_0 \thickspace \medspace + \medspace \mu_0 \Delta \mathcal{E}(x) \thickspace e^{i \omega t} \thickspace .$$ (2.25)

From Eq. (2.21) and Eq. (2.23) it follows:

$$\frac{d \Delta \mathcal{E}}{d x} \thickspace e^{i \omega t} \meds... ...\medspace \frac{e}{\epsilon_r \epsilon_0} \left( n - n_0 \right) \thickspace ,$$ (2.26)

and

$$n \medspace = \medspace n_0 \medspace + \medspace \frac{\epsilon... ...pace \frac{d \Delta \mathcal{E}}{d x} \thickspace e^{i \omega t} \thickspace .$$ (2.27)

Equations (2.23), (2.24) and (2.25) can be substituted in the current equation (2.22) and $n$ can be eliminated with the help of Eq. (2.27), obtaining:

$$j_0 + \Delta j \thickspace e^{i \omega t} = e \left( n_0 + \frac... ...r \epsilon_0 \Delta \mathcal{E} \thickspace \omega \thickspace e^{i \omega t}.$$ (2.28)

After some simplifications:

$$\Delta j = \left( i \omega \epsilon_r \epsilon_0 + e \mu_0 n_0 \... ...al{E} \thickspace \frac{d \Delta \mathcal{E}}{d x} \thickspace e^{i \omega t}.$$ (2.29)

The last term in Eq. (2.29) can be neglected, because it is of the same order as $\cal O$ $((\Delta \mathcal{E})^2$) and $\Delta \mathcal{E}$ was assumed to be small.

$$\Delta j = \left( i \omega \epsilon_r \epsilon_0 + e \mu_0 n_0 \... ...al{E} + v_0 \epsilon_r \epsilon_0 \frac{d \Delta \mathcal{E}}{d x}\thickspace.$$ (2.30)

If the doping concentration in the active region is uniform, $n_0$ is not depending on $x$ and Eq. (2.30) results in a linear differential equation with constant coefficients. Choosing $\Delta \mathcal{E}(x=0)=0$ as boundary condition, the general solution is:

$$\colorbox{grau}{$\displaystyle{\Delta \mathcal{E}(x) = \frac{\Del... ...ilon_r \epsilon_0 v_0 \gamma} \left( 1 - e^{-\gamma x} \right)\thickspace ,}$}$$ (2.31)

where

$$\colorbox{grau}{$\displaystyle{\gamma = \frac{e n_0 \mu_0}{\epsilon_r \epsilon_0 v_{0}} + i \frac{\omega}{v_{0}}\thickspace .}$}$$ (2.32)

Finally, from Eq. (2.23) and Eq. (2.31), we have:

$$\mathcal{E}(x,t) =\mathcal{E}_0+ \frac{\Delta j}{\epsilon_r \epsi... ...{\epsilon_r \epsilon_0 v_0 \gamma} e ^{-\gamma x} e^{i \omega t} \thickspace .$$ (2.33)

The second term on the right side of Eq. (2.33) describes an uniform field oscillation, independent of $x$; this governs in the LSA operating mode (see section 2.3.3). The last term represents a wave propagating in the $x$ direction with the same phase and group velocity $v_0$, because using (2.32) we obtain:

$$v_{phase}\equiv \frac{\omega}{Im[\gamma]}=v_0 \thickspace ,\thic... ...v_{group}\equiv \frac{\partial \omega}{\partial(Im[\gamma])}=v_0 \thickspace .$$ (2.34)

This variable was defined as the drift velocity, at which the electrons move through the crystal under the influence of the external electrical field. The modulation $\Delta \mathcal{E} \medspace e^{i \omega t}$ of the electron density creates a space charge wave, which propagates with the electron drift velocity $v_0$.
If $\mu_0>0$, then $Re[\gamma] > 0$ and the wave is damped; if $\mu_0<0$, then $Re[\gamma] < 0$ and the wave amplitude increases. The sign of $\mu_0$ depends first of all on the electric field in relation to the threshold field $\mathcal{E}_T$, as already explained in section 2.1.2 and shown in Fig. 2.4.
The device is stable with respect to small perturbations if the real part of the impedance $Z$ is positive. $Z$ is defined as:

$$Z = \frac{\Delta U}{A \thickspace \Delta j}=\frac{\underset{0}{\overset{L}{\int}} \Delta E(x) \thickspace dx}{Ae \thickspace \Delta j}\thickspace,$$ (2.35)

where $A$ is the cross sectional area of the Gunn device. With Eq. (2.31) it follows:

$$\colorbox{grau}{$\displaystyle{Z(\omega) \thickspace = \thickspac... ..._0} \thickspace \frac{e^{-\gamma L} + \gamma L - 1} {\gamma^2}\thickspace .}$}$$ (2.36)

As demonstrated by McCumber and Chynoweth [MC66], Eq. (2.36) is well behaved for all finite $\omega$ and $Z(\omega)$ has no singularities. This means that the device described by Eq. (2.36) is always stable when it operates under constant current conditions.
In order to determine the stability under constant voltage conditions, the poles of the admittance $Y(\omega)=1/Z(\omega)$ or the zeros of $Z(\omega)$ must be investigated. Setting $s=\gamma L$, the zeros of Eq. (2.36) correspond to the zeros $s_n=\phi_n+i\thickspace \eta_n$ (n=0,1,2..) of

$$f(s)=e^{-s}+s-1=0 \thickspace .$$ (2.37)

It is necessary to divide f(s) into the real and imaginary part:

$$f(s_n) = e^{-\phi_n}(\cos(\eta_n) + i \sin(\eta_n))+\phi_n+i\thickspace \eta_n -1\thickspace ,$$ (2.38)

$$Re[f(s_n)] = e^{-\phi_n}\cos(\eta_n)+\phi_n-1\thickspace ,$$ (2.39)

$$Im[f(s_n)] = e^{-\phi_n}\sin(\eta_n)+ \eta_n \thickspace .$$ (2.40)

• The first zero is $s_0=0$: this means $\gamma L=0$. Either the length of the device is zero (L=0) or the frequency and the mobilities are zero ($\gamma=0$).
  $s_0=0$ is not a physically relevant zero of $Z(\omega)$.
• The other zeros $s_n$ (n=1,2,3..) satisfy the conditions:
  

  $$\eta_n = \frac{(4n+1)\pi}{2}-\arcsin\left[(1-\phi_n)e^{\phi_n}\right]\thickspace,$$ (2.41)

  $$\phi_n = -\frac{1}{2}\thickspace \ln\left[\eta_n^2+(1-\phi_n^2)\right]\thickspace,$$ (2.42)

  $$\lim_{n \to \infty}(\eta_n) = \frac{(4n+1)\pi}{2}\thickspace,$$ (2.43)

  $$\lim_{n \to \infty}(\phi_n) = -ln\left[\frac{(4n+1)\pi}{2}\right]\thickspace .$$ (2.44)

  
  The second zero is for $\phi_1=-2.09$.

If $\vert\phi_1\vert<2.09$, then the Re[Z] is positive and the device is stable. From Eq. (2.32), it results:

$$n_0L\thickspace<\thickspace2.09\thickspace\frac{\epsilon_0 \epsilon_r v_0}{e \vert\mu_0\vert}\thickspace.$$ (2.45)

Choosing the parameters for GaAs ( $\epsilon_r=12.9$, $v_0=10^7\thickspace cm^2/s$ and $\mu_0=2500\thickspace cm^2/Vs$), we have:

$$n_0L\thickspace<\thickspace6 \thickspace 10^{11} \thickspace cm^{-2}\thickspace.$$ (2.46)

Considering that the mobility depends on the doping level, the consistent stability condition for GaAs is given by:

$$n_0L\thickspace<\thickspace 10^{12} \thickspace cm^{-2}\thickspace.$$ (2.47)

It can be concluded that, at room temperature, GaAs devices will always be stable with steady state properties, if the product $n_0L$ is under the critical value. This result applies only for a small signal approach; nevertheless, with appropriate intensity of the electric field (i.e. $\gg \mathcal{E}_T$), samples with $n_0L$ lower than the critical value provide amplification for microwave signals (i.e. $Re[\gamma] < 0$).

2.2 Gunn diode with hot electron injectors

Hot electron injection is the process of raising the electron energy to the level of the L band before it enters the drift zone. Without a hot electron injector, the electrons will remain in the $\Gamma$-valley until they gain enough energy to transfer to the upper valleys. The distance required to gain such an energy depends on the electric field level and, on a minor scale, on the device temperature. The region near the emitter, where most electrons reside in the $\Gamma$-valley, is called dead-zone. Gunn domains can build only outside the dead-zone. The dead-zone not only narrows the active region (up to 17% in a $1.5 \thickspace \mu m$ GaAs Gunn device [NDS⁺89]), but introduces an undesirable positive serial resistance reducing the r.f. power and the efficiency of the device. Moreover, the dead zone is not constant and should be considered as an aleatory process having very bad influences on the device noise properties.

The solution to the described problems is to embed a hot electron injector in the emitter region just before the Gunn diode active region. An efficient injector leads to the following advantages:

• improved temperature stability of power and frequency,
• improved oscillator turn on characteristic, especially at low temperatures,
• increased efficiency (lower voltage drop on the injector),
• low phase noise.

The Schottky, graded gap and resonant tunneling injectors represent different approaches to fulfill these objectives.

2.2.1 Metal semiconductor Schottky contact injector

To understand how the Schottky contact injector works, it is useful to introduce briefly the physics behind a metal-semiconductor contact.

Figure 2.9: Energy-band diagrams for Schottky contact on n-semiconductor: (a) before contacting, (b) after contacting in equilibrium, (c) under reverse bias and (d) direct bias.

There are two kinds of metal-semiconductor contacts: ohmic and Schottky. An ohmic contact is a metal or silicide contact to a semiconductor with a small interfacial resistance and a linear I-V characteristics. In this case, the work function of the semiconductor must be greater than the work function of the metal ( $\phi_{s}>\phi_{m}$) [Nea97]. The described condition is valid only for ideal ohmic contacts (e.g. indium based compounds). Typically, semiconductor engineers consider ohmic contacts also Schottky contacts with a very thin potential barrier and a linear I-V characteristic.
In a Schottky contact, the work function of the semiconductor must be smaller than the work function of the metal ( $\mathrm{\phi_{s}< \phi_{m}}$). The ideal energy-band diagram for a particular metal and n-type semiconductor before making the contact is shown in Fig. 2.9(a), where $\mathrm{\phi_{m}}$ is the metal work function, $\mathrm{\phi_{s}}$ is the semiconductor work function and $\mathrm{\chi}$ is the semiconductor electron affinity. The vacuum level is used as a reference level. Before contacting, the Fermi level in the metal is above the Fermi level in the semiconductor. After contacting, in thermal equilibrium, the Fermi level has to be constant through the whole system and electrons from the semiconductor flow into the lower energy states in the metal. Positively charged donor atoms remain in the semiconductor, creating a space charge region (Fig. 2.9(b)). The potential barrier seen by electrons in the metal trying to move into the semiconductor is known as ``Schottky barrier''. Assuming an ideal interface, without Fermi pinning, the barrier height $\mathrm{\phi_{bn}}$ is given by:

$$\centering { \phi_{bn}=(\phi_{m}-\chi)}\thickspace .$$ (2.48)

On the semiconductor side, $\mathrm{V_{bi}}$ is the built-in potential barrier seen by electrons in the conduction band trying to move into the metal and is given by:

$$\centering { V_{bi}=(\phi_{bn}-\phi_{n})} \thickspace ,$$ (2.49)

where $\phi_{n}$ is the potential difference between the minimum of the conduction band in the semiconductor ( $\mathrm{E_{c}}$) and the Fermi level ( $\mathrm{E_{F}}$):

$$\centering \phi_{n}\ =\ \frac{E_{c}-E_{F}}{e} \thickspace ,$$ (2.50)

where $e$ is the electron charge.
In the reverse bias condition, where a positive voltage $\mathrm{V}$ is applied to the semiconductor with respect to the metal (Fig. 2.9(c)), the semiconductor-to-metal barrier increases, while $\mathrm{\phi_{bn}}$ remains constant in the idealized case. In the forward bias condition, a negative voltage is applied to the semiconductor with respect to the metal (Fig. 2.9(d)) and the semiconductor-to-metal barrier decreases, while $\mathrm{\phi_{bn}}$ still remains constant. In this case, electrons can move easily from the semiconductor into the metal.

Solving the Poisson's equation, it is possible to calculate the electric field present in the space charge region in the metal. The depletion layer width $W$ of region is given by:

$$\centering {W= \sqrt{\frac{2\epsilon_{s}(V_{bi}+V-\frac{kT}{e})}{eN_{d}}}} \thickspace ,$$ (2.51)

where $N_{d}$ is the donor concentration, $\epsilon_{s}$ is the dielectric constant of the semiconductor, k is the Boltzmann constant and T is the temperature in K. The term kT/e arises from the contribution of the majority-carrier distribution tail [Sze81].

Three basic transport processes through the Schottky barrier can be identified:

• thermionic emission of electrons over the potential barrier (dominant process for moderate doped semiconductor $N_d<10^{-17}cm^{-3}$),
• tunneling of electrons through the barrier (quantum-mechanical process important for heavily doped semiconductors and responsible for most ohmic contacts),
• recombination in the depletion region.

The ideal expression of the J-V characteristics taking into account the thermionic emission and the diffusion is [Sze81]:

$$J = J_s(e^{\frac{qV}{kT}}-1) \thickspace .$$ (2.52)

The saturation current density, $J_s$ for the thermionic emission case is:

$$J_s = A^*T^2 e^{-\frac{q\phi_{bn}}{kT}} \thickspace.$$ (2.53)

where $A^*$ is the effective Richardson constant. Tunneling, recombination and injection cause deviations from the ideal behaviour. Introducing the serial resistance^2.4 $R_s$ and the ideality factor $n$ leads to

$$\colorbox{grau}{$\displaystyle{J = J_s(e^{\frac{qV-R_s J}{nkT}}-1) \thickspace .}$}$$ (2.54)

Looking back at the first Gunn diodes, a Schottky barrier was always present. The quality of the ohmic contacts was extremely poor, leading to low uniformity, low reproducibility and high parasitic voltage drops.
Even if, in the last 30 years, amazing improvements have been achieved in metal-semiconductor contacts, an unsolvable problem remains for Schottky hot electron injectors: Fermi level pinning. The Schottky barrier on GaAs has a height of about $0.65 \thickspace eV$, not depending of the metal because of the Fermi level pinning. The barrier cannot be engineered properly and $0.65 \thickspace eV$ is not the optimum barrier height for an efficient electron transfer from $\Gamma$ to L valley. In a similar way, Fermi level pinning is present also in other III/V semiconductors, limiting the applications of the Schottky barrier injector Gunn diodes drastically.

2.2.2 Graded gap injector

The idea of the graded gap AlGaAs barrier comes from the need of a potential barrier which can be optimized, changing its height, width and shape [CBK⁺88,NDS⁺89,GWCE88]. As seen in section 2.2.1, these parameters are difficult to control in the case of the metal-semiconductor Schottky barrier, especially concerning the reproducibility.

Figure 2.10: Principle schema of a Gunn diode with a graded AlGaAs injector. C identifies the cathode (emitter) and A the anode (collector).

7.5cm

The epitaxial process offers an extremely high accuracy and tuning possibilities for the injector fabrication. The material system $Al_xGa_{1-x}As$ maintain nearly the same lattice constants, with the change of the Al concentration. Grading the Al concentration, it is possible to obtain a potential barrier. The AlGaAs barrier has to be nominally undoped and the Al concentration starts at 0% (on the cathode/emitter side) and increases linearly up to the maximal concentration (in the anode/collector direction). The height of the barrier has to be designed considering the energy needed by electrons for the $\Gamma$ to $L$ transfer.
Establishing the width of the graded AlGaAs barrier, two factors have to be considered. The barrier has to be wide enough to prevent electrons to tunnel at lower energy levels; only the electrons passing over the barrier have enough energy for the intervalley transfer. On the other hand, a wide undoped AlGaAs barrier results in a too high serial resistance. The second part of the injector is a thin $n^+$ doped layer of GaAs, connecting the AlGaAs barrier to the Gunn active region. A complete view of the graded gap injector is illustrated in Fig. 2.10.

In order to correctly understand the behaviour of the graded gap injector Gunn diodes, complex simulations are required. Monte Carlo computations have demonstrated that the graded AlGaAs barrier, followed by a thin highly doped layer, increases the intervalley electron transfer and reduces the dead-zone thereby improving noise performance, temperature stability and power conversion efficiency [LR90]. A much simpler approach for the simulation of the DC behaviour considers the Gunn diode composed of two elements in series: the graded gap injector and the diode active region plus the contact resistance. For low bias (voltages much lower than the threshold), the Gunn active region plus the contact resistance have a linear characteristics and the graded gap injector can be modelled like a Schottky barrier. Under these assumptions, a load-line model is defined like in Fig. 2.11 where the voltage drop on the diode $U_T$ is the sum of the single voltage drops, on the graded gap injector ($U_I$) and on the ohmic active region ($U_R$).

Figure 2.11: DC electrical model of a graded gap injector Gunn diode.

The following function describes the graded gap injector with a simple exponential law. Two important parameters are introduced: the saturation current $I_s$ (Eq. (2.53)) and the constant $c_1$, which is a measure of the tunneling current.

$$I_I=I_se^{c_1U_I} \thickspace .$$ (2.55)

Then, considering the device area $A$ and the active region length $l$, the current $I_R$ through the active region can be written as:

$$I_R=-\frac{(U_T-U_I)A}{l\rho} \thickspace .$$ (2.56)

The resistivity $\rho$ can be expressed as:

$$\rho=\frac{1}{ne\mu} \thickspace ,$$ (2.57)

where $n$ is the electron concentration, $\mu$ is the mobility and $e$ is the electron charge. Introducing the Lambert transcendental function W(x) and defining the diode current as $I_T=I_I(U_I)=I_R(U_I)$, it follows:

$$U_I = \frac{- W(\frac{c_1I_sle^{c_1 U_T}}{n\mu e A })+{c_1U_T}}{c_1} \thickspace ,$$ (2.58)

$$W(x) = \frac{x}{e^{W(x)}} \thickspace .$$ (2.59)

Substituting $U_I$ in Eq. (2.55), the diode current $I_T$ is obtained as function of the diode total bias $U_T$

$$\colorbox{grau}{$\displaystyle{I_T=I_s e^{{- W(\frac{c_1I_sle^{c_1 U_T}}{n\mu e A })+{c_1 U_T}}}. }$}$$ (2.60)

As shown in section 6.1.3, Eq. (2.60) matches the I-V behaviour of graded gap Gunn diode very well. The proposed model is fitted to the measured I-V characteristic in the reverse current direction for different temperatures. Moreover, taking into account the temperature dependence of $I_s$, the effective barrier height can be estimated.

2.2.3 Resonant tunneling injector

In section 2.2.2, the state of the art for hot electron injection in Gunn diodes has been discussed . This raises the question if a graded gap injector can be further improved. The solution might be in using the resonant tunneling effect, one of the most exciting quantum mechanical phenomena in semiconductor physics [CET74,TE73].

Figure 2.12: Bias-dependent band diagrams and current voltage characteristics for an RTD at 4 bias points: A, B, C, D.

Resonant tunneling diodes (RTDs) are based on a double potential barrier structure like AlAs/GaAs/AlAs. The structure is designed such that resonant bound states are present in the quantum well. Electrons can tunnel through the double-barrier if their transversal energy is equal to the energy of one of the quasi bound states in the quantum well. The I-V curve of a double barrier structure can be in principle understood with the help of Fig. 2.12. Close to zero electrical bias, a small fraction of the electrons has an energy equal with the energy of the first quasi bound state (Fig. 2.12A). As the voltage increases, the resonant states are shifted down towards the Fermi level on the emitter side and a greater number of electrons can tunnel, contributing to the current (Fig. 2.12B). At a certain voltage, the conduction band level of the emitter side is aligned to the quantum well resonant level and a maximum appears in the current (Fig. 2.12C). Beyond this voltage, the resonant level drops below the emitter conduction band edge, resulting in a sudden drop of the current (Fig. 2.12D). For higher voltages, the current rises again, because of the combination of two effects: the next resonant level lowers and a thermionic emission occurs over the barrier.

The negative differential resistance of the RTD (Fig. 2.12D) has been exploited for microwave analog devices like oscillators [BPBP93] and mixers [MMP⁺91] and for ultrafast digital devices like monostable-bistable transition logic elements (MOBILE) [MM93,SMI⁺01,Was03].

Figure 2.13: Band diagram of a typical GaAs/AlGas heterostructure (A) with the corresponding transmission probability T (B). The electron distribution before and after the double barrier points out the excellent energy filter capabilities of the device.

For RTD serving as a hot electron injector, no negative differential resistance is needed and only the small bias voltage region of the I-V curve should be considered. As a means to better understand the resonant tunneling injection, the different transport processes through a double barrier are examined [SHMS98]. In Fig. 2.13(A), the different transport paths are illustrated. The electrons with exactly the resonant energy $E_0$ tunnel through the double barrier with the probability of $\sim 1$ (process 1). In the emitter part, the electron accumulation in front of the first barrier bends the conduction band and causes the formation of a triangular potential well with discrete two-dimensional energy levels. Electrons can tunnel from the potential well through the double barrier with the help of a phonon (process 2). In a similar way, electrons undergoing the process 3, whose energy is a little higher than the resonant energy $E_0$, can tunnel due to an interaction with a phonon. In process 4, like in process 1, electrons have exactly the energy of the resonant level $E_1$. In process 5-6, high energy electrons cross the device, tunneling through only one barrier or overcoming the whole heterostructure (thermionic emission). Because of the required high energy, electrons involved in process 5 and 6 are relatively few.

The electron transmission probability T is a result of the different transport processes. In a first approximation, T can be expressed as a Lorentzian function of energy E with a half width $\Gamma_n$ [Dav98]:

$$T(E) = T_{max} \frac{\frac{1}{4}\Gamma_n^2}{\frac{1}{4}\Gamma_n^2+(E-E_n)^2} \thickspace ,$$ (2.61)

$$T_{max} = \frac{4 T_1 T_2}{(T_1 + T_2)^2} \medspace ,$$ (2.62)

$$\Gamma_n = \frac{\hbar}{\tau_n} \thickspace ,$$ (2.63)

where $T_1$ and $T_2$ represent the transmission probability of the single barriers, $\Gamma_n$ the full energetic width at half maximum of the transmission probability T and $\tau_n$ is the electron lifetime in the energy state $E_n$.
The simulated T versus the electron energy is plotted in Fig. 2.13(B). It can be noticed a tunneling probability of one for the quantized energy states. The width of the resonance is exponentially dependent on the thickness of the barrier. Direct effects of the resonance can be seen in the electron distribution behind the double barrier. This confirms that this device can be used for building a very precise energy filter.

2.3 Gunn diode based oscillators

2.3.1 Theory of two-port networks

All electromagnetic processes can ultimately be explained by Maxwell's four basic equations:

$$\nabla\cdot\vec{D} = \rho_{\mathrm{free}} \thickspace ,$$ (2.64)

$$\nabla\cdot\vec{B} = 0 \thickspace ,$$ (2.65)

$$\nabla\times\vec{E} = -\dfrac{\partial\vec{B}}{\partial t} \thickspace ,$$ (2.66)

$$\nabla\times\vec{H} = \vec{J}_{\mathrm{free}}+\dfrac{\partial\vec{D}}{\partial t} \thickspace .%$$ (2.67)

However, it is not always possible or convenient to directly use these equations. Solving them can be quite difficult. Efficient design requires the use of approximations such as lumped and distributed models.

Linear or nonlinear networks, operating with signals small enough for the networks to respond in a linear manner, can be completely characterized by parameters measured at the network terminals (ports), no matter of the network contents. Once the parameters of a network are determined, its behaviour in any external environment can be predicted, again without regard to the network contents. Although a network may have any number of ports, network parameters can be explained most easily by considering a network with only two ports, an input port and an output port, like in Fig. 2.14. To characterize a network, any of the several parameter sets can be chosen; each of these offers certain advantages [Pac96]. Each parameter set is related to a group of four variables associated with the two-port model. Two variables represent the excitation of the network (independent variables), and the other two represent the response of the network to the excitation (dependent variables).

Figure 2.14: General two-port network

2.3.1.1 Y-parameters

If the network is excited by voltage sources $v_{1}$ and $v_{2}$, the network currents $i_{1}$ and $i_{2}$ will be related to them by the following equations:

$$i_{1}=y_{11}v_{1}+y_{12}v_{2} \thickspace ,$$ (2.68)

$$i_{2}=y_{21}v_{1}+y_{22}v_{2} \thickspace ,$$ (2.69)

where $Y= \begin{pmatrix} y_{11} & y_{12}\\ y_{21} & y_{22} \end{pmatrix}$ is called the admittance matrix.
In the absence of additional information, four measurements are required to determine the four parameters $y_{11}$, $y_{12}$, $y_{21}$, $y_{22}$. Each measurement is carried out with one port of the network excited by a voltage source, while the other port is short circuited, as better listed in the follows:

$$y_{11}=\left.{\dfrac{i_{1}}{v_{1}}}\right\vert _{\parbox{6.5cm}{$v_2=0$, output short circuited,}}$$ (2.70)

$$y_{12}=\left.{\dfrac{i_{1}}{v_{2}}}\right\vert _{\parbox{6.5cm}{$v_1=0$, input short circuited,}}$$ (2.71)

$$y_{21}=\left.{\dfrac{i_{2}}{v_{1}}}\right\vert _{\parbox{6.5cm}{$v_2=0$, output short circuited,}}$$ (2.72)

$$y_{22}=\left.{\dfrac{i_{2}}{v_{2}}}\right\vert _{\parbox{6.5cm}{$v_1=0$, input short circuited.} }%$$ (2.73)

2.3.1.2 Z-parameters

Very similar to the admittance matrix is the impedance matrix. If the network is excited by current sources $i_{1}$ and $i_{2}$, the network voltages $v_{1}$ and $v_{2}$ will be related to them by the following equations:

$$v_{1}=z_{11}i_{1}+z_{12}i_{2} \thickspace ,$$ (2.74)

$$v_{2}=z_{21}i_{1}+z_{22}i_{2} \thickspace ,$$ (2.75)

where $Z= \begin{pmatrix} z_{11} & z_{12}\\ z_{21} & z_{22} \end{pmatrix}$ is called the impedance matrix.
In the absence of additional information, four measurements are required to determine the four parameters $z_{11}$, $z_{12}$, $z_{21}$, $z_{22}$. Each measurement is carried out with one port of the network excited by a current source while the other port is open circuited, as better listed in the follows:

$$z_{11}=\left.{\dfrac{v_{1}}{i_{1}}}\right\vert _{\parbox{6cm}{$i_2=0$, output open,}}$$ (2.76)

$$z_{12}=\left.{\dfrac{v_{1}}{i_{2}}}\right\vert _{\parbox{6cm}{$i_1=0$, input open,}}$$ (2.77)

$$z_{21}=\left.{\dfrac{v_{2}}{i_{1}}}\right\vert _{\parbox{6cm}{$i_2=0$, output open,}}$$ (2.78)

$$z_{22}=\left.{\dfrac{v_{2}}{i_{2}}}\right\vert _{\parbox{6cm}{$i_1=0$, input open.}}%$$ (2.79)

It is easy to see how impedance and admittance matrix are related:

$$Z=Y^{-1} \thickspace .$$ (2.80)

2.3.1.3 h-parameters

Slightly different to the admittance and impedance matrices (but not less important as it will be shown in the next sections) is the hybrid matrix. If the network is excited by input current source $i_{1}$ and output voltage source $v_{2}$, the network output current $i_2$ and the network input voltages $v_{1}$ will be related to them by the following equations:

$$v_{1}=h_{11}i_{1}+h_{12}v_{2} \thickspace ,$$ (2.81)

$$i_{2}=h_{21}i_{1}+h_{22}v_{2} \thickspace ,$$ (2.82)

where $h= \begin{pmatrix} h_{11} & h_{12}\\ h_{21} & h_{22} \end{pmatrix}$ is called the hybrid matrix.
In the absence of additional information, four measurements are required to determine the four parameters $h_{11}$, $h_{12}$, $h_{21}$, $h_{22}$, as better listed in the following:

$$h_{11}=\left.{\dfrac{v_{1}}{i_{1}}}\right\vert _{\parbox{6cm}{$v_2=0$, output short,}}$$ (2.83)

$$h_{12}=\left.{\dfrac{v_{1}}{v_{2}}}\right\vert _{\parbox{6cm}{$i_1=0$, input open, }}$$ (2.84)

$$h_{21}=\left.{\dfrac{i_{2}}{i_{1}}}\right\vert _{\parbox{6cm}{$v_2=0$, output short, }}$$ (2.85)

$$h_{22}=\left.{\dfrac{i_{2}}{v_{2}}}\right\vert _{\parbox{6cm}{$i_1=0$, input open.}}%$$ (2.86)

Table 2.1: Parameter conversion table.

S to Z $Z'=\frac{Z}{Z_0}$	Z to S $Z=Z' Z_0$
$$\begin{array}{l} \\ z_{11}'=\dfrac{(1+s_{11})(1-s_{22})+s_... ...s_{11})(1-s_{22})-s_{12} s_{21}} \nonumber\\ \\ \end{array}$$	$$\begin{array}{l} \\ s_{11}=\dfrac{(z_{11}' -1)(1+z_{22}')-... ...1}')(1+z_{22}')-z_{12}' z_{21}'} \nonumber\\ \\ \end{array}$$
S to Y $Y_{ }=Y_{ }' Z_0$	Y to S $Y_{ }'=\frac{Y_{ }}{Z_0}$
$$\begin{array}{l} \\ y_{11}'=\dfrac{(1-s_{11})(1+s_{22})+s_... ...s_{11})(1+s_{22})-s_{12} s_{21}} \nonumber\\ \\ \end{array}$$	$$\begin{array}{l} \\ s_{11}=\dfrac{(1-y_{11}')(1+y_{22}')+y... ...1}')(1+y_{22}')-y_{12}' y_{21}'} \nonumber\\ \\ \end{array}$$
S to h $h_{11}=h_{11}' Z_0$	h to S $h_{22}=\frac{h_{22}'}{Z_0}$
$$\begin{array}{l} \\ h_{11}'=\dfrac{(1+s_{11})(1+s_{22})-s_... ...s_{11})(1+s_{22})+s_{12} s_{21}} \nonumber\\ \\ \end{array}$$	$$\begin{array}{l} \\ s_{11}=\dfrac{(h_{11}'-1)(1+h_{22}')-h... ...{11}')(1+h_{22}')-h_{12} h_{21}} \nonumber\\ \\ \end{array}$$

2.3.1.4 S-parameters

Even if all parameter sets contain the same information about a network, and it is always possible to obtain any set from another set (Table 2.1), scattering parameters are important in microwave design, because they are easier to measure at higher frequencies than the others. They are conceptually simple, dimensionless, analytically convenient, and capable of providing a great insight into a measurement or design problem.

Scattering parameters are commonly called S-parameters; they relate to travelling waves, which are scattered or reflected when a n-port network is inserted into a transmission line. Their definition, given by Kurokawa [Kur65], starts with the normalized complex voltage waves:

$$a_{i}=\dfrac{V_{1}+Z_i I_{1}}{2 \sqrt{\vert Re(Z_i)\vert}} \thickspace ,$$ (2.87)

$$b_{i}=\dfrac{V_{1}-Z^*_i I_{1}}{2 \sqrt{\vert Re(Z_i)\vert}} \thickspace . %$$ (2.88)

In the available measurement systems, there is only a 2-port network (i=1,2) and the reference impedances $Z_i$ are real and equal to $Z_0$ ($50\Omega$ as described in the calibration section).

$$a_{1}=\dfrac{V_{1}+Z_0 I_{1}}{2 \sqrt{Z_0}}=\frac{\textrm{voltage wave incident on PORT1}}{\sqrt{Z_0}} \thickspace ,\\$$ (2.89)

$$a_{2}=\dfrac{V_{2}+Z_0 I_{2}}{2 \sqrt{Z_0}}=\frac{\textrm{voltage wave incident on PORT2}}{\sqrt{Z_0}} \thickspace ,\\$$ (2.90)

$$b_{1}=\dfrac{V_{1}-Z_0 I_{1}}{2 \sqrt{Z_0}}=\frac{\textrm{voltage wave reflected from PORT1}}{\sqrt{Z_0}} \thickspace ,\\$$ (2.91)

$$b_{2}=\dfrac{V_{2}-Z_0 I_{2}}{2 \sqrt{Z_0}}=\frac{\textrm{voltage wave reflected from PORT2}}{\sqrt{Z_0}}\thickspace .%$$ (2.92)

The linear equations describing the 2-port network are:

$$b_{1}=s_{11}a_{1}+s_{12}a_{2} \thickspace ,$$ (2.93)

$$b_{2}=s_{21}a_{1}+s_{22}a_{2} \thickspace ,$$ (2.94)

where $S= \begin{pmatrix} s_{11} & s_{12}\\ s_{21} & s_{22} \end{pmatrix}$ is the matrix of the following S-parameter:

$$s_{11}=\left.{\dfrac{b_{1}}{a_{1}}}\right\vert _{a_2=0}\hookright... ...the output port terminated by a matched load. $Z_L=Z_0 \Rightarrow a_2=0$,} \\$$ (2.95)

$$s_{12}=\left.{\dfrac{b_{1}}{a_{2}}}\right\vert _{a_1=0}\hookright... ...h the input port terminated by a matched load. $Z_S=Z_0 \Rightarrow a_1=0$,}\\$$ (2.96)

$$s_{21}=\left.{\dfrac{b_{2}}{a_{1}}}\right\vert _{a_2=0}\hookright... ... the output port terminated by a matched load. $Z_L=Z_0 \Rightarrow a_2=0$,}\\$$ (2.97)

$$s_{22}=\left.{\dfrac{b_{2}}{a_{2}}}\right\vert _{a_1=0}\hookright... ...the input port terminated by a matched load. $Z_L=Z_0 \Rightarrow a_2=0$.}. %$$ (2.98)

2.3.2 Negative differential conductance oscillators

An oscillator can be considered as the combination of an active multiport and a passive multiport (the embedding network). For the active device, we have:

$$[V]=[Z][I] \thickspace ,$$ (2.99)

and for the embedding network:

$$[V']=[Z'][I'] \thickspace .$$ (2.100)

In our case, $Z$ and $Z'$ are the impedance matrixes of the diode and of the passive elements.
By connecting the active and the embedding network:

$$[I]=-[I'] \thickspace ,$$ (2.101)

$$[V]=[V'] \thickspace ,$$ (2.102)

and from Eq. (2.99) and Eq. (2.100) results:

$$\{[Z]+[Z']\}[I]=0 \thickspace .$$ (2.103)

Now since $[I]\neq0$, the matrix [Z]+[Z'] is singular or

$$det([Z]+[Z'])=0 \thickspace .$$ (2.104)

Equation 2.104 is the generalized oscillation condition. Since Z is a function of the frequency and of the voltage, the oscillating frequency can be determined for a given bias point. It can be demonstrated that an equivalent condition of Eq. (2.104) is [KO81]:

$$det([S][S'] -1) =0$$ (2.105)

or in a similar way

$$det([Y] + [Y'])=0 \thickspace ,$$ (2.106)

where $S$ and $Y$ are the scattering and admittance matrices of the diode and $S'$ and $Y'$ are the scattering and admittance matrices for the passive elements, respectively. If maximum power output and efficiency are not of prime importance, a satisfactory oscillator design can be achieved using small-signal scattering matrix parameters. The major shortcoming of small signal oscillator design is that it does not provide any way of predicting the steady-state oscillating signal level. Oscillator design based on large-signal scattering matrix (i.e. non-linear) parameters is much more complicated because of the difficulty of obtaining large-signal parameters.

2.3.3 Gunn oscillation modes

In order to generate microwave power, a transferred electron device has to be placed in a cavity or in a resonant circuit. Several modes of operation can be distinguished. Each of them provides particular advantages or disadvantages from the point of view of tunability, stability and efficiency. The following four main operation modes will be discussed [Hei71,Hob74]:

• the Transit Time mode,
• the Delayed Domain mode,
• the Quenched Domain mode,
• the LSA mode.

2.3.3.1 The Transit Time mode.

Figure 2.15: Current and voltage time evolution of the Gunn oscillator operating in the Transit Time mode.

In a purely resistive circuit, the oscillation frequency $f_r$ is determined by the space charge or domain transit time $\tau_d$:

$$f_r=\frac{1}{\tau_d}=\frac{v_{drift}}{L} \thickspace .$$ (2.107)

This operating mode is commonly called Transit Time mode. The current flows with a sequence of short current pulses with the period $\tau_d$. The current spikes occur when a domain enters the anode and the next one is originating from the cathode. The voltage is sinusoidal with the same period (Fig. 2.15). If the circuit is heavily loaded, the R.F. voltage is small enough not to oscillate below $U_T$.

The efficiency of the Transit Time mode is not particularly high because of the narrowness of the current pulse and the small r.f. voltage amplitude. Moreover, the frequency range is fixed to the natural domain transit frequency. This disadvantage influences also the frequency stability, because the domain transit time is strongly temperature dependant.

2.3.3.2 The Delayed Domain mode.

In this case, the resonant period of the circuit $T_r=1/f_r$ is longer than $\tau_d$:

$$\frac{1}{2 \cdot \tau_d}=\frac{v_{drift}}{2 \cdot L}<f_r<\frac{1}{\tau_d}=\frac{v_{drift}}{L} \thickspace .$$ (2.108)

Figure 2.16: Current and voltage time evolution of the Gunn oscillator operating in the Delayed Domain mode.

The amplitude of the voltage sinusoidal waveform (Fig. 2.16) has to be large enough to cause the voltage to fall below threshold $U_T$ over a portion of the cycle. The domain reaches the anode and disappears during the second half of the voltage oscillation, while the voltage is below the threshold. Before the next domain can be nucleated, the voltage has to rise again over the threshold. The efficiency of the Delayed Domain mode is higher than for the Transit Time and, thanks to the larger current pulses, can reach up to 7.2% [War66]. The frequency is controlled by the resonant circuit, whose tunability can be mechanical or electrical (parallel Schottky varactors). A good temperature stability is achieved, generally, for aluminium cavities or DRO (Dielectric Resonator Oscillators).

2.3.3.3 The Quenched Domain mode.

Figure 2.17: Current and voltage time evolution of the Gunn oscillator operated in the Quenched Domain mode.

For this mode, the resonant period of the circuit $T_r=1/f_r$ is shorter than $\tau_d$, but longer than the domain nucleation and extinction time $\tau_s$:

$$\frac{1}{\tau_d}=\frac{v_{drift}}{L}<f_r<\frac{1}{\tau_s} \thickspace .$$ (2.109)

The circuit loading is further reduced and the voltage falls below the domain sustaining voltage $U_S$ for a portion of the cycle. The domain is generated in the cathode and, before reaching the anode, is quenched in flight, because the voltage falls below $U_S$. The next domain can not be nucleated until the terminal voltage rises above the threshold $U_T$. The main advantage of this mode consists in the generation of frequencies higher than the transit-time frequency. Considering the area under the current-pulse, it is evident that the efficiency of the Quenched Domain mode (5% [Hob74]) is higher than the one of the Transit Time mode, but lower than the one of the Delayed Domain mode.

2.3.3.4 The LSA mode.

It has been already anticipated, that a domain can be quenched before reaching the anode, if the device bias falls below $U_S$. The lifetime of the domain is connected with the resonance frequency of the circuit. If the frequency is high enough, the domain will not have time to fully nucleate and the diode operates in the LSA mode. LSA stands for Low Space-charge Accumulation. In this context, the I-V characteristics of the device should follow the v-E, which exhibits a region of negative differential mobility.
A first advantage of this mode consists in the high frequencies, which are achievable: the device length is not related to the oscillating frequency, which can be many times the transit time frequency (overlength Gunn oscillator). The second advantage concerns the R.F. output power: in the LSA mode, higher terminal voltages may be applied without causing impact ionizations [Hob74] and outstanding efficiencies up to 18.5% are reported [Cop67b].
The performances of the LSA mode are limited by four practical constraints to prevent the domain formation:

• The doping concentration divided by the frequency $n_0/f$ must remain in the interval $[2\cdot10^{10},2\cdot10^{11}] \thickspace s m^{-3}$ (GaAs case) [Cop67b].
• The homogeneity of the doping concentration and of the related low field conductivity must be better than 10% [Cop67b,Hob72].
• Not to distort the uniform-field conditions with the unavoidable accumulation layer, the device must be much longer than the space-charge transit length per cycle.
• The oscillating circuit must be designed in a way to prevent a long delay before starting the oscillation. During this delay, a domain could nucleate and cause device breakdown by impact ionization.

2.3.3.5 Operating modes overview for a transferred electron device.

Figure 2.18: Representation of the operating modes of a Gunn diode [Cop67a]. The allocation of the different categories depends on three parameters: $n_0L$, $fL$ and $n_0/f$.

Figure 2.18 summarizes the operating modes of a GaAs transferred electron device as a function of the doping concentration, frequency and device length. Three different situations can be distinguished: stable amplification, domain oscillation and LSA oscillation.
For $n_0L<10^{16} \thickspace m^{-2}$, no domain formation appears and the device can be used as an amplifier in frequency ranges around the transit frequency. For $n_0L>10^{16} \thickspace m^{-2}$ three different domain oscillation modes are possible depending on the resonator frequency: Transit Time, Delayed Domain and Quenched Domain mode. For higher frequencies and for $n_0/f \in [2\cdot10^{10}, 2\cdot10^{11}] \thickspace sm^{-3}$, we have the LSA mode.

The presented boundaries should not be regarded as absolute. The device behaviour next to the boundaries is also depending on the bias voltage, device temperature and circuit loading.

2.4 The thermal behaviour of a Gunn diode

In today's environment of high performance semiconductor devices, the common complaints that it is possible to "cook an egg" or "warm your coffee" on an electronic appliance like a typical modern computer are not that far fetched. Many semiconductor manufacturers have grappled with the difficulty of combining highly evolved thermal dissipation techniques with the dual requirement of packaging simplicity and reliability. In this context, the problem of the Gunn diode cooling is crucial.

As illustrated in the previous chapters, the efficiency of a Gunn diode is not very high. In order to achieve the required R.F. output levels, D.C. power densities greater than $140\thickspace kW/cm^2$ are reached^2.5. Therefore, in addition to good electric contacts, the Gunn diode requires good thermal contacts to the environment to avoid its destruction by overheating.

The standard packaging consists in the removal of the heat occurring from one end of the device with an integrated gold heat sink. The heat-sink is electroplated on the semiconductor and then bounded to a copper pedestal ultrasonically or by thermocompression. Only for research purposes, sometimes copper is replaced with diamond, for which the thermal conductivity at room temperature is 30 times higher than copper.
In this work, an original quasi-planar double-sided heat-sinking is presented: from the bottom side, the heat flows through the semiconductor substrate and from the top side, thick gold airbridges for electrical connections are exploited also as heat-sink.

In this section after an analytical description of the thermal problem, finite elemente simulations of the standard and the double-sided heat-sink are compared. The efficiency of the top contact heat-sink is presented here theoretically and in chapter 6 experimentally.

2.4.1 Analytical solution of the simplified static heat transfer problem

For most geometries, the detailed solution of the heat-flow problem through a small active device in a massive heat-sink is complicated.

Figure 2.19: Approximation of the device geometry to simplify the heat transfer problem [Hob74].

In the case of simple conduction^2.6, the heat transfer equation is given by:

$$-\nabla \cdot (\lambda \nabla T)=q \thickspace ,$$ (2.110)

where $q$ is the power density of the heat source $[Wm^{-3}]$ and $\lambda$ is the thermal conductivity^2.7.

In order to achieve simple analytical solutions, some assumptions are required. If interfacial and contact electrical resistances are negligible, all the heat is dissipated in the active region of the device. The composite geometry of the heat flow can be simplified as shown in Fig. 2.19. The Gunn device is considered as a series connection of a one dimensional active device region and a heat-sink with spherical symmetry heat flow.

The solution of of the first part of the heat flow problem for the one dimensional active region (dark-gray Fig. 2.19) with uniform heating is well-known [Hob74]:

$$T_{MAX}-T_0=\frac{Q l}{2\pi R^2 \lambda_{AR}} \thickspace ,$$ (2.111)

where $T_{MAX}$ is the maximum temperature, $T_0$ is the temperature at the border, $L$ is the diode length, $R$ is the diode radius, $\lambda_{AR}$ is the thermal conductivity of the active region and $Q = q \cdot (l\pi R^2)$ is the dissipated power. The solution of the second part of the problem (heat flow in the heat-sink, light-gray Fig. 2.19)) ,considering a point source and spherical symmetry, is given by [Hob74]:

$$T_{0}-T_{\infty}=\frac{Q}{\sqrt{2}\pi R \lambda_{HS}} \thickspace ,$$ (2.112)

where $\lambda_{HS}$ is the thermal conductivity of the heat-sink.

Combining together Eq. (2.111) and Eq. (2.114), the complete heat flow equation is

$$T_{MAX}-T_{\infty}=\frac{Q L}{2\pi R^2 \lambda_{AR}}+\frac{Q}{\sqrt{2}\pi R \lambda_{HS}}.$$ (2.113)

The thermal resistance $Z_{TH}$ of the whole system Gunn diode and heat-sink is:

$$Z_{TH}=\frac{T_{MAX}-T_{\infty}}{Q}=\frac{L}{2\pi R^2 \lambda_{AR}}+\frac{1}{\sqrt{2}\pi R \lambda_{HS}}.$$ (2.114)

Equation 2.113 confirms that, for a given device length, it is preferable to have the radius R as large as possible. At the same time, in order to keep the electrical resistance constant, the device resistivity must increase. The resistivity is depending on the electron concentration. However, too low doping concentrations have to be considered with caution, in order to avoid low rf efficiencies.

Another important factor for Eq. (2.113) is the thermal conductivity: it depends extremely of material. The active region thermal conductivity for GaAs, InP and GaN are: $\lambda_{GaAs}=45.5 \thickspace Wm^{-1}K^{-1}$, $\lambda_{InP}=68 \thickspace Wm^{-1}K^{-1}$ and $\lambda_{GaN}=130 \thickspace Wm^{-1}K^{-1}$. The excellent value of GaN are compensated by the power density, which this material requires before having transferred electron effects. Crucial for reducing the thermal resistance is also the material choice of the heat-sink. Materials, which are often used, are: copper ( $\lambda_{Cu}=400 \thickspace Wm^{-1}K^{-1}$), gold ( $\lambda_{Au}=317 \thickspace Wm^{-1}K^{-1}$) and diamond ( $\lambda_{diamond}=2500 \thickspace Wm^{-1}K^{-1}$).

As an example, the temperature within the GaAs is computed for the case of an integrated gold heat-sink. A diode length $L$ of $1.6 \thickspace \mu m$ and a radius R of $35 \thickspace \mu m$ have been considered with a dissipated power of Q = $5 \thickspace W$. For an environment temperature of $T_{\infty}$ = $300^\circ$ K, the resulting maximal temperature within the device is

$$T_{MAX} \approx 426^\circ K.$$ (2.115)

Replacing the gold heat-sink with a GaAs substrate, it follows

$$T_{MAX} \approx 1030^\circ K.$$ (2.116)

Without heat-sink, the temperature is more than the double and the diode would be destroyed immediately.

2.4.2 Finite elemente simulations of the temperature distributions in a Gunn diode

The problem of the heat flow in Gunn diodes has been solved numerically with the commercial software Finite Element Method Laboratory. FEMLAB is a modelling package for the simulation of physical processes, which can be described via partial differential equations. The modelling activity is a sequence of four steps:

• Create the diode geometry with a 3D CAD environment.
• Define the physics of the heat flow including the necessary material parameters.
• Generate the finite elemente mesh.
• Compute and analyze the solution.

The simulation has been performed on two structures: a conventional Gunn diode chip and a quasi-planar Gunn diode with air-bridges.

The Gunn diode chip is composed of a cylindrical top gold contact with a diameter of $70\thickspace \mu m$, a GaAs mesa with the same area and a conventional bottom heat-sink. The heat-sink consists of a cylindrical gold contact with a diameter of $350 \thickspace \mu m$ and a height of $40 \thickspace \mu m$, laying on top of a copper block with a diameter of $800 \thickspace \mu m$ and a height of $100 \thickspace \mu m$. As a boundary condition, the outer faces of the whole structure are kept thermally isolated, except for the copper bottom face, which is kept at a constant temperature of $300^\circ K$.

The quasi-planar Gunn diode is composed of a GaAs mesa on top of a $350 \thickspace \mu m$ thick GaAs substrate. The top face of the mesa ( $18 \times 20 \thickspace \mu m^2$) is connected with $3 \thickspace \mu m$ thick gold air-bridges to the substrate at a distance of $10 \thickspace \mu m$. The same boundary conditions have been chosen: the outer faces of the whole structure are thermally isolated, except for the GaAs substrate bottom face, which is kept at a constant temperature of $300^\circ K$.
In both cases, the GaAs mesa is divided into a $1.6 \thickspace \mu m$ active layer with a constant uniform power dissipation ( $5\thickspace V \times 26000 \thickspace kA/cm^2$) and two contact layers of $700 \thickspace nm$. No irradiation process has been considered.

The results of the simulations are presented in Fig. 2.20 - 2.23. In Fig. 2.20 and in Fig. 2.21, the temperature distribution is shown for the Gunn diode chip and for the quasi-planar Gunn diode, respectively. A 3D and a cross-sectional view of the two structures visualises the different temperature regions by means of different color levels. The maximal temperature reached by the Gunn diode chip is $471^\circ K$. This result agrees well with similar computations ( $475^\circ K$) reported by Cords and Förster [CF02]. In the case of the quasi-planar Gunn diode, a higher temperature is expected, considering the poor thermal conductivity of the GaAs substrate. Surprisingly, the maximal temperature is only nine degree higher ( $480^\circ K$): the gold air-bridges spread the heat from the top increasing the interface area between the diode and the GaAs substrate. The thicker the gold air-bridges, the better the top-side cooling.
In Fig. 2.22 and in Fig. 2.23, the heat flux profile is presented: the background color and the size of the cones are proportional with the heat flux intensity. For the Gunn diode chip, the maximal heat flux is close to the border of the active region at the interface between GaAs and gold. The maximal flux intensity of $192\thickspace kW/cm^2$ again agrees well with Cords' and Förster's result of $189\thickspace kW/cm^2$.
For the quasi-planar Gunn diode, a high heat-flux is located at the border of the active region at the interface with the GaAs substrates. This region seams to be material independent, since a similar zone can be seen in the Gunn diode chip. Anyway, the highest heat flux values can be found in the arms of the gold air-bridges, confirming the important role, which they have in planar Gunn diodes.

Figure 2.20: Heat-sink temperature profile for a standard Gunn diode chip.

Figure 2.21: Double-sided heat-sink temperature profile for a quasi-planar Gunn diode.

Figure 2.22: Heat flux profile for a standard Gunn diode chip.

Figure 2.23: Heat-flux profile for a quasi-planar Gunn diode.