6. Experimental results and discussion

The electrical properties of the fabricated Gunn diodes and their temperature dependence are analyzed in this chapter from several points of view. Considering the numerous aspects to be treated, in order to avoid confusion, the discussion has been divided into three main topics.
The first two deal with the behaviour of the Gunn diodes, as single active devices, at low and high frequency, pointing out the electrical influence of each layer of the device structure. Special attention is given to the role of the two hot electron injectors: the graded gap barrier and the resonant tunneling double barrier.
The third part presents two different Gunn oscillator approaches: a conventional one based on a cavity resonator is compared with a novel quasi-planar monolithic integrated circuit (MMIC).

6.1 Gunn diode direct current behavior

6.1.1 Contacts

The parasitic connection resistances have a negative effect on the DC, high frequency and noise performance of an electronic device. Those resistances are basically the result of two components: the connection lines and the contact resistances. Because of the particular geometry of the Gunn diodes, the connection lines can be neglected, and the crucial parameter becomes the ohmic contact.

The available ohmic contact technology (section 5.2 [Lep97,Jav03]) did not require any improvement or optimization. However, TLM and CLM measurements (see section 4.5) have been systematically performed as quality verification to detect the process reproducibility or irregularity. The contact pads of the TLM structures are $150 \times 50 \thickspace \mu m^2$ squares and are separated by increasing distances of 5, 10, 15, 20, 40, 80 and 120 $\thickspace \mu m$. The six circular shaped CLM structures consist of outer circles with constant diameter of 100 $\thickspace \mu m^2$ and concentric inner circles with a decreasing diameters of 96, 92, 88, 84, 80 and 68 $\thickspace \mu m$.

Figure 6.1: Graphic analysis of the transmission line measurement. On the left, I-V characteristics for the different distances between contacts. On the right, resistance versus pad distance.

A measurement example for a TLM structure is shown in figure 6.1. The different linear I-V characteristics (Fig. 6.1 left side) are depending on the distance between pads. The interpolation of the resistance dependance on the contact distance (Fig. 6.1 right side) provides the contact resistance $R _C \medspace[\Omega \medspace mm]$, the effective contact length $L_T \medspace[ \mu m]$, the specific contact resistance $\rho _c \medspace[\Omega \medspace cm^2]$ and the sheet resistance $R_S \medspace[\Omega / \Box]$.
All these parameters are summarized in table 6.1 for different wafers.

Table 6.1: Ohmic contacts for different processed wafers

W-number		$R_C$	$L_T$	$\rho _c$	$R_S$
		$[\Omega \medspace mm]$	$[ \mu m]$	$[\Omega \medspace cm^2]$	$[\Omega / \Box]$
17008	GaAs	0.047	2.7	$1.2 \cdot 10^{-6}$	17.6
18006	GaAs	0.040	2.3	$9.2 \cdot 10^{-7}$	17.4
18038	GaAs	0.038	2.2	$8.2 \cdot 10^{-7}$	17.5
19032	GaAs	0.038	2.1	$8.0 \cdot 10^{-7}$	17.9
G695E2t	GaN	0.165	0.7	$1.1 \cdot 10^{-6}$	238
G695E2b	GaN	0.097	2.0	$1.9 \cdot 10^{-6}$	48

Concerning GaAs, the specific contact resistance $\rho_C$ remains under $\sim 1\cdot 10^{-6}$, which defines the upper limit for good ohmic contacts. The differences between the GaAs results are due to the process reproducibility and on the slightly different doping concentration of the considered wafers.
The values for GaN, instead, refer to the same wafer G695 and to the same sample E2. They have been prepared and annealed in the same time. The only difference is that G695E2t was processed on the top GaN contact layer and that G695E2b was processed on the bottom contact layer. Even if the doping nominal level is the same for the two layers, the top one is thinner than the bottom one ( $200\thickspace nm$ vs. $800\thickspace nm$); this could explain the lower values of G695E2b sheet resistance. Another aspect that should be taken into account, is the influence of the dry etching and the related crystal deterioration: the bottom ohmic contact G695E2b lays on a surface which was subjected to an aggressive chlorine based plasma process. An improved plasma etching process could reduce the reported inconsistency between top and bottom contacts.

6.1.2 I-V characteristics of graded gap injector GaAs Gunn diodes

Figure 6.2 shows the DC characteristics of two GaAs Gunn diodes: one with the graded gap injector (GGI) and the other one without. In our case, the current flows in the forward direction when a negative voltage is applied on top of the device. As expected, the I-V characteristics of the diode without injector is symmetric. The diode with the graded gap injector presents an asymmetric I-V curve with a well pronounced Schottky-like behavior. In addition, the maximum current in the reverse direction (positive voltages) is about 10% higher than in the forward one. This is an indication for different electron occupations of the L-valley, causing different drift velocities for the two current directions.

Figure 6.2: I-V characteristics for two Gunn diodes: one with the graded gap injector (solid curve), the other one without (broken curve). Negative voltages correspond to forward currents, positive voltages to reverse currents.

Figure 6.3: Low voltage I-V characteristics of diodes with different Al maximum concentration in the graded barrier.

In order to find the most effective injector, devices with different maximum aluminium concentrations have been compared (28%Al up to 34%Al). The DC characteristics of the diodes for low voltages are shown in Fig. 6.3: especially in the reverse direction, the Schottky-like behavior becomes more pronounced with increasing Al concentration, i.e. higher barriers.

In the steep increasing part of the I-V curves, the slopes in the two directions are nearly identical. These slopes reflect the quality of the ohmic contacts and no relevant change has been observed for different Al concentrations in the barriers.

After the linear region, the current peak and the DC negative differential resistance should be interpreted as a pure self heating effect. Considering the poor heat properties of GaAs, self heating is playing an important role (see section 6.1.6 and section 6.2.1).

In the reverse current direction, electrons enter the active-region directly without passing any hot electron injector. In this case, no enhanced L-valley occupation can be expected. This results in a current peak nearly independent of the Al-concentration.

In the forward direction, different injectors lead to different occupations of the L-valley and therefore to different peak drift velocities and peak currents. The forward current decreases with increasing Al concentration. The ratios between the peak current densities $J_p$ in both directions are presented in Table 6.2.

Table 6.2: Peak current densities in the forward and reverse direction for different Al concentrations in the injector.

	28% Al	30% Al	32% Al	34% Al
$J_p^{fw}$	<#17373#>	30.40	29.60	28.80	28.40
$J_p^{re}$			31.05	31.20	30.60
$J_p^{fw}/J_p^{re}$		0.98	0.95	0.94	0.92

The current maxima in the forward and in the reverse direction can be used as a measure of the injector effectiveness. The lower the ratio $J_p^{fw}/J_p^{re}$, the more efficient is the electron injection in the L-valley (Table 6.2). Nevertheless, only a high frequency investigation allows an analysis of the Gunn diode injector in the whole operating bias range ( $3\hspace{1mm} V\hspace{1mm} -\hspace{1mm} 6\hspace{1mm} V$), obtaining a quantitative estimation of the related L-valley occupations (see section 6.2.4).

6.1.3 Temperature dependant DC modelling of graded gap injector GaAs Gunn diodes

The graded gap injector of the Gunn diodes can be analyzed using temperature dependent DC measurements. An example of the DC temperature behavior of a GaAs Gunn diode is shown in Fig. 6.4 for an injector with 32% maximum Al concentration. As expected, the peak current is strongly temperature dependent. This effect is directly connected with an increase of the active region resistance with temperature.
The primary explanation comes from the relative occupations of the $\Gamma$ and L-valley: higher the temperature, higher the L-valley occupation and lower the average drift velocity and current. The effect is stronger in the reverse direction, because of the higher number of electrons remaining in the $\Gamma$ valley.
In the I-V characteristics region where the slope is positive, an important factor contributing to the change of the diode resistance is the increase of scattering processes with temperatures. Scattering is not connected with the injector properties nor with the current direction.

Figure 6.4: Forward (left diagram) and reverse (right diagram) I-V characteristics of a graded gap injector Gunn diode (32% Al) for different temperatures.

For low voltages, we assist to the opposite phenomena: the current increases rising the temperature. The low voltage behavior is extremely influenced by the injector: the thermionic emission of the electrons over the barrier increases dramatically with increasing temperatures.

6.1.3.1 Temperature dependent simulations of graded gap injector Gunn diodes

In chapter 2.2.2, the equivalent circuit for a graded gap injector Gunn diode has been described and Eq. (2.60) has been proposed for analytical modelling. A simulation package for an automatic evaluation of temperature dependent I-V characteristics in connection with Eq. (2.60) has been written. A complete description of the methods and the software code can be found in [FML03].

Using Eq. (2.60) and taking as fitting parameters $c_1$, $\mu$ and $I_s$, the measured I-V curves have been fitted at different temperatures (left diagram of Fig. 6.5). For each temperature, a set of $c_1$, $\mu$ and $I_s$ has been obtained.

Figure 6.5: Graphical procedure to obtain the barrier height $\phi$, the effective Richardson constant $A^*$ and the saturation current Is.

Assuming that the temperature dependance of the injector is described by a Richardson law (see section 2.2.1 and Eq. (2.53)) , $I_s$ is given by:

$$I_s = AJ_s=A A^* T^2 e^{-\frac{q\phi}{kT}}$$ (6.1)

where $\phi$ is the barrier height of the graded gap emitter and A the diode cross-section area. As seen from Eq. (6.1), the representation of $ln(I_s/T^2)$ versus 1/T is a straight line (right diagram of Fig. 6.5). The slope of the fitted line gives the barrier height $\phi$ and the intersection with the axis $1/T=0$ gives $A \cdot A^*$.

The described procedure has been applied to several diodes with different injectors. For a better visualization of the results, only four cases are considered, focusing on the influence of the maximum Al concentration in the graded barrier. The diode layer structures have been grown in a sequence and processed simultaneously to minimize inaccuracy or inhomogeneity. In Fig. 6.6, the barrier height $\phi$ is represented as a function of the saturation current $I_s$ at $50^\circ$C.

Figure 6.6: The graded gap barrier height as function of the saturation current $Is$ at $50^\circ$C for different maximum Al concentration (28%, 30%, 32% and 34%) in the graded gap injector. The processing and the measurements of the diodes considered in this diagram belongs to the cooperation project [Pro04].

The general tendency in Fig. 6.6 follows the expectations. Increasing the maximum Al concentration in the graded barrier, the saturation current decreases and the barrier height increases.

Once the layer-structure is decided and the parameters $c_1$, $\mu$ and $I_s$ are determined, the described procedure is suggested as a production benchmark. In a global environment, where costs and quality issues are strongly connected, minimum reliability targets lower then 2ppm^6.1 are standard requirements. Each production step is a potential thread for the fulfillment of these severe targets and needs a constant control. The proposed benchmark gives a full overview on the layer structure and processing quality with a small effort (DC measurements).

6.1.4 I-V characteristics of resonant tunnelling injector GaAs Gunn diodes

The typical I-V characteristics of a Gunn diode with resonant tunneling injector (RTI) is illustrated in Fig. 6.7. For low voltages, the I-V curve of the RTI Gunn diode in the forward and in the reverse direction is symmetric. This symmetry confirms the good quality of the double barrier structure and of the ohmic contacts. The low voltage asymmetry observed in the case of GGI Gunn diode (Fig. 6.3) was due to the Schottky-like behavior of the graded barrier. The Schottky-like characteristic is a distinctive feature of the graded gap injector and causes unwanted voltage drop.

Applying higher voltages, the current shows a peak and a negative slope witnesses the self heating of the device. As for the GGI Gunn diode, the forward current peak is much lower than the reverse one. This is a first hint of the resonant tunneling injector effectiveness. The already defined ratio between the peak current density in the forward ( $24.5 \thickspace kA/cm^2$) and in the reverse direction( $35 \thickspace kA/cm^2$), $J_p^{fw}/J_p^{re}$ is equal to 0.7. In comparison with the graded gap injector (0.9), this is a much better result. Moreover, the negative slope of the RTI diode in the forward direction is flatter than the one in the reverse: this gives the impression of a less influence of the forward current by self-heating effects.

Figure 6.7: I-V characteristics of a typical resonant tunneling injector Gunn diode.

One final remark for the RTD specialists concerns the resonance of the double barrier and the peak to valley ratio. Figure 6.7 does not show any steep negative differential resistance (typical in RTD structures) because of the particular design of the device. The peak current of the isolated double barrier structure has been estimated at current densities higher than $60kA/cm^2$. The Gunn diode active region prevents the flowing of such a high current and avoids the undesirable bistability. For thick AlAs double barriers ($\geq$6 monolayers), such a phenomenon is expected in the following conditions: the measurement has to be carried out at lower temperatures, with very short pulses, and in the reverse direction, where the Gunn diode current densities are higher.

6.1.5 I-V characteristics of GaN Gunn diodes

Figure 6.8 shows representative I-V characteristics of GaN Gunn diodes. The considered structure does not present an injector and no distinction between forward and reverse polarity is made because of the nearly symmetrical results. The measurements have been carried out in three different conditions: $80\thickspace \mu s$, $300\thickspace \mu s$ and no pulses. There is practically no difference between the three situations up to $7 \thickspace V$. In all three cases, the curves are linear, confirming the good quality of the ohmic contacts. After $7 \thickspace V$, the slope of the solid curve (no pulses) starts to decrease. The current saturates and does not show any maximum like in GaAs Gunn diodes without injector. The current starts to saturate at higher biases, 8.5 and $10 \thickspace V$ for 300 and $80\thickspace \mu s$ pulses, respectively. This effect is clearly connected with self-heating: even if GaN is a good thermal conductor, the Gunn diode active region is not in direct contact with an efficient heat-sink, but with a thick sapphire substrate. As reference, the thermal conductivity at room temperature of GaN is $130 \thickspace W m^{-1} K^{-1}$ but the one of sapphire is only $30 \thickspace W m^{-1} K^{-1}$. Moreover, the power density, to be dissipated, reaches $1 \thickspace MW/cm^2$.

Which is the real origin of the current saturation shown in Fig. 6.8? Is the intervalley transfer playing a role in this phenomenon? Concerning this aspect, it is difficult to make a real comparison of the DC properties of the presented GaN Gunn diode with data from the literature. Considering the threshold field $E_T$ as the electric field at witch the slope of I-V is zero, values of about $87.5 \thickspace kV/cm$ can be estimated ( $80\thickspace \mu s$ long pulses). This value is from 30% up to 40%, lower than the simulated values reported in the literature [KOB⁺95,AWR⁺98]. The discrepancy could be explained as misinterpretation of the device length (the active region has been considered $1.6 \mu m$ long in the previous estimation).

The maximal current density levels are around $90 \thickspace kA/cm^2$. Taking as reference value typical GaAs Gunn diodes without injector, the maximal current densities range from 30 to $40 \thickspace kA/cm^2$. The current density differences are due to the different doping concentrations ( $10^{17} \thickspace cm^{-3}$ for GaN and $10^{16} \thickspace cm^{-3}$ for GaAs) and to the respective maximal drift velocities, which were already described in chapter 2.1.2 and in Fig. 2.4.

Figure 6.8: I-V characteristics of a typical GaN Gunn diode with 80$\mu s$ and 300$\mu s$ long pulses.

6.1.6 $100 \thickspace ns$ pulse measurements: heat effects evidences

In section 6.1.3, it has been shown how sensitive are Gunn diodes to the environment temperature shifts. In section 2.4, it has been demonstrated that the inside temperature of the Gunn diode is much higher than the environment one, because of the self-heating. If we want to measure I-V characteristics without self-heating effects, very short pulses with a low duty cycle have to be used. For this purpose, very short pulse measurements have been carried out. The description of the measurement setup can be found in chapter 4.6.

Figure 6.9: Forward (left diagram) and reverse (right diagram) I-V characteristics of a graded gap injector Gunn diode (34% Al) with 100ns and 300$\mu s$ pulses.

In Fig. 6.9, the I-V characteristics of a graded gap injector Gunn diode with very short pulses (100ns) are illustrated. For comparison, measurement carried out with longer pulses (80$\mu s$) are also presented. It can be noticed, that with short pulses, the I-V curves do not show a negative slope. 80$\mu s$ pulses, instead, are enough long to heat up the diode. This confirms, beyond doubt, that self heating is the origin of the negative slope in the I-V characteristics of the Gunn diodes. A better estimation of the negative slope dependance on the heating time (pulse length) is given in section 6.2.1.
In the forward current direction, up to the current peak, there is practically no difference between long and short pulse I-V curves; after the current peak, the difference remains marginal. For the reverse bias, however, there is a large difference between the two cases. The current increase is dramatic when the short pulses are applied. The above evidences demonstrate the effectiveness of the injector. In the reverse direction, the electron occupation of the $\Gamma$ valley is high. A long pulse length causes a stronger self-heating of the active region leading to a higher inter-valley transfer, a lower $\Gamma$ valley occupation and a lower mean drift velocity. In the forward direction, the $\Gamma$ valley occupation is low because of the injector; a further decrease due to the heating has only marginal effects on the current.
The ratio between the peak currents in the two directions is now free from self-heating effects and can give a clear estimation of the injector effectiveness.

The short pulse setup was also used to determine the breakthrough voltage for diodes with different injectors and with different passivation [Pro04]. The breakthrough voltage is indicated by a sudden increase in the current behavior for high biases. Measuring the breakthrough voltage with long pulses, it damages irreversibly the Gunn diodes, due to the current stress. With short pulses, the influence of a $Si_3N_4$ passivation has been examined. The average value of the measured breakthrough voltages is about 9V. Negligible differences have been found between passivated and unpassivated diodes. As expected, the breakthrough voltage is slightly higher in the forward direction because of the lower current density.

6.2 Gunn diode high frequency behaviour

6.2.1 Impedance measurements up to 50MHz

In section 6.1.6, it was demonstrated that the I-V characteristics of a GaAs Gunn diode, measured with very short pulses ( $100 \thickspace ns$), does not have a negative slope after the threshold voltage. The negative differential resistance appearing for longer pulses ($80\mu s$) was explained as consequence of the heating of the diode. Due to restrictions of the pulse setup, it was not possible to determine the exact pulse length, which represents the transition from a negative slope behaviour to a positive one. An interesting method for this estimation is proposed in [Sto03]: the impedance-frequency characteristics of the Gunn diodes are measured, in order to study the transition frequency at which the resistance from negative becomes positive. The measurement is performed with an Agilent 4294 Precision Impedance Analyzer in a frequency range from $5\thickspace kHz$ up to $50 \thickspace MHz$.

Figure: Differential resistance vs. frequency in the forward direction. Diode parameters: no injector, area= $256\hspace{1mm} \mu m^2$, active region length $=1.6 \hspace{1mm} \mu m$. [Sto03]

In Fig. 6.10, the real part of the measured impedance is presented as function of the frequency for a GaAs Gunn diode without injector. The absolute value of the differential resistance increases for increasing bias voltages. For all the biases, the transition from negative to positive differential resistances appears at a frequency between 100 and $200 \thickspace kHz$. This transition can be better understood introducing an equivalent circuit for the temperature changes inside the diode. The thermal resistance and capacity of the Gunn diode mesa are defined as:

$$R_{TH} = \frac{h}{\lambda A} \thickspace ,$$ (6.2)

$$C_{TH} = c_v m \thickspace,$$ (6.3)

where $c_v$ is the specific heat capacity, m the mass, h the thickness, A the area, and $\lambda$ the thermal conductivity. It is now possible to express a time constant $\tau_{TH}$ and the related frequency for the defined RC system:

$$\tau_{TH} = R_{TH}C_{TH}=\frac{c_v m h}{\lambda A} \thickspace ,$$ (6.4)

$$f_{TH} = \frac{1}{2 \pi \tau_{TH}}=\frac{\lambda A}{2 \pi c_v m h} \thickspace .$$ (6.5)

Assuming an uniform material, the mass $m$ can be expressed as

$$m=\rho V=\rho A h \thickspace ,$$ (6.6)

where $\rho$ is the density and V the volume of the mesa. The time constant and the frequency can be written as:

$$\tau_{TH} = R_{TH}C_{TH}=\frac{c_v \rho h^2}{\lambda } \thickspace ,$$ (6.7)

$$f_{TH} = \frac{1}{2 \pi R_{TH}C_{TH} }=\frac{\lambda}{2 \pi c_v \rho h^2} \thickspace .$$ (6.8)

Equation 6.7 and 6.8 are now independent on the mass and area of the mesa. This property has been confirmed experimentally in [Sto03], where the transition frequency from negative differential resistance to positive one was constant with different diode areas.

For our GaAs mesa $c_v$ is $350 \thickspace J/(kgK)$, $\lambda$ is $46 \thickspace W/(mK)$, $\rho$ is $5500 \thickspace kg/m^3$ and h is $3 \thickspace \mu m$. $f_{TH}$ can be estimated as:

$$f_{TH}\approx 422\thickspace kHz \thickspace .$$ (6.9)

The computed $f_{TH}$ is two times higher than the transition frequency of the measured diodes in Fig. 6.10. The RC model, actually, is taking into account only the volume of the mesa. In reality, the effective volume should consider also a part of the semiconductor under the diode. Increasing the parameter h with $2 \thickspace \mu m$, $f_{TH}$ results:

$$f_{TH}\approx 152\thickspace kHz \thickspace .$$ (6.10)

The corrected value of $f_{TH}$ matches quite well the behaviour of the measured GaAs Gunn diodes without injector.

Impedance measurements have been also performed on graded gap injector GaAs Gunn diodes. A Gunn diode with a hot electron injector does not follow anymore the described model. The heating of the device is not uniform. Furthermore, the dissipated power density distribution is depending on the voltage. In Fig. 6.11, it can be noticed that the transition frequency changes with voltage. At $4\thickspace V$ the transition frequency is between 100 and $200 \thickspace kHz$, like for the diodes without injector. For lower voltages, the transition frequency shifts to much higher frequencies.

Figure: Differential resistance vs. frequency in the forward direction. Diode parameters: graded-gap-injector (32% Al), area= $196\hspace{1mm} \mu m^2$, active region length $=1.8 \hspace{1mm} \mu m$.

To understand Fig. 6.11 and the injector influence, let us consider a diode bias range from 2 up to $6 \thickspace V$. If we assume the current to be constant after the threshold voltage, the power dissipated in the injector is also constant. In the same bias range, the diode dissipated power increases linearly with the terminal voltage. From these considerations, at higher voltages, the influence of the injector can be almost neglected, explaining the similarity found at $4\thickspace V$ between diodes with and without injector.

6.2.2 High frequencies investigations of GaAs Gunn diodes

Figure 6.12: Conductance and susceptance, solid and broken curve, vs. frequency for different applied positive DC bias. Diode parameters: no injector, area= $196\hspace{1mm} \mu m^2$, active region length $=1.6 \hspace{1mm} \mu m$.

Gunn diodes are normally mounted in a cavity and the resulting oscillators are measured and used as microwave generators. The literature of the last 30 years deals extensively with such data. Despite the possibility to de-embed the package, the cavity and the outside circuit from the measured data, it is very hard to achieve a good characterization of the device. The resulting information does not help to explain completely the behavior of the diode itself. Therefore, S-parameter measurements up to 110 GHz have been performed, in order to evaluate the small signal behavior of Gunn diodes with and without hot electron injector, at very high frequencies.

A complete description of the S-parameters and of the network analyzer setup can be found respectively in chapter 2.3.1 and 4.7.1.

The conductance and the susceptance of a Gunn diode without injector for different positive bias voltages are shown in Fig. 6.12. The negative conductance presents a voltage dependent minimum. The corresponding frequencies are 96 GHz at $2\hspace{1mm} V$, 92 GHz at $3\hspace{1mm} V$ and 87 GHz at $4 \hspace{1mm} V$. In all the measured samples, the negative conductance minima frequency decreases with increasing voltage.

Figure 6.13: Conductance and susceptance, solid and broken curve, vs. frequency in the forward and the reverse direction. Diode parameters: graded-gap-injector (32% Al), area= $196\hspace{1mm} \mu m^2$, active region length $=1.8 \hspace{1mm} \mu m$.

Figure 6.14 shows the admittance of a Gunn diode with the graded gap injector for negative (forward direction) and positive (reverse direction) voltages. The reverse direction looks similar to the one of the diodes without injector. It is not surprising, considering that the injector is designed to work only in the forward direction. For negative voltage, in fact, the influence of the graded gap injector leads to a sharp peak-like appearance of the negative conductance minimum near 60 GHz. The sharp negative conductance minimum is a direct consequence of the dead zone reduction caused by the injector. Additionally, a shift to lower frequencies can be observed and the second harmonic minimum appears.

Figure 6.14: Conductance and susceptance, solid and broken curve, vs. frequency in the forward and the reverse direction. Diode parameters: resonant tunneling injector (6 AlAs monolayers per barrier), area= $196\hspace{1mm} \mu m^2$, active region length $=1.6 \hspace{1mm} \mu m$.

Figure 6.14 shows the admittance of a Gunn diode with the resonant tunneling injector for negative and positive voltages. Again, the reverse direction looks similar to the one of the diodes with the graded gap injector and without injector. In the forward direction, also the resonant tunneling injector presents a peak-like resonance of the negative conductance minimum. The minimum is sharper than for the graded gap injector and the shift to lower frequency is larger. These results represent a further evidence of the significant potentials of the resonant tunneling injector.

6.2.3 Drift velocity computation and operation mode classification

On the basis of the Gunn diode small signal admittance (see section 2.1.4), after McCumber and Heime [MC66,Hei71]

$$Y(\omega,v_d,\mu_0) = {A \epsilon \epsilon_0 v_{d}} \frac{\gamma^2}{e^{-\gamma L} + \gamma L - 1} \thickspace ,$$ (6.11)

$$\gamma = \frac{e n_0 \mu_0}{\epsilon \epsilon_0 v_{d}} + i \frac{\omega}{v_{d}} \thickspace ,$$ (6.12)

the electron drift velocity can be derived. In Eq. (6.11) and 6.12 $v_d$ is the drift velocity, $\mu_0$ the mobility, $\omega$ the angular frequency, A the device area, $\epsilon$ the relative permittivity, $\epsilon_0$ the vacuum permittivity, $e$ the electron charge, L the device active region length and $n_0$ the total carrier concentration. It could be shown that at certain frequencies ($f_{NCM}$), minima in the negative conductance appear. These frequencies are directly correlated with the drift velocity $v_d$ and the sample length L:

$$\frac{L \omega}{v_d} = 2 \pi m \hspace{1cm}(m=1,2,3..),$$ (6.13)

$$f_{NCM} = \frac{v_d}{L} m,$$ (6.14)

where the number m accounts for higher harmonics, which can be observed also in Fig. 6.13 in the forward direction, near 110 GHz.

Equation 6.14 can be applied to the measured diodes, in order to estimate the drift velocity dependance on the electric field. In Table 6.3, samples with different active region lengths ( $L_1=1.8 \hspace{1mm} \mu m$ and $L_2=1.5 \hspace{1mm} \mu m$) and different areas are compared at $4 \hspace{1mm} V$. The experimental values prove the predictions of Eq. (6.14): there is an inverse proportionality between the frequencies of the negative conductance minimum ( $f_{NCM}\approx 1/$transit time) and the active region length. The value $f_{NCM}\cdot L$ can be interpreted as the average group velocity of the electrons in the active region ( $\sim 1.06\cdot10^7 \hspace{1mm} cm/s$ for diodes with injector).

Table 6.3: Frequencies corresponding to the negative conductance minima ($f_{NCM}$) for diodes with different active region lengths (L) and areas at $4 \hspace{1mm} V$; as expected the product $f_{NCM}\cdot L$ remains constant.

	$L_1=1.8\mu m$		$L_2=1.5\mu m$
Area	$f_{NCM1}$	$f_{NCM1} \cdot L_{1}$	$f_{NCM2}$	$f_{NCM2} \cdot L_2$
396 $\mu m^2$	59.4 GHz	1.06E5 m/s	69.8 GHz	1.05E5 m/s
196 $\mu m^2$	59.0 GHz	1.06E5 m/s	70.8 GHz	1.06E5 m/s
144 $\mu m^2$	62.0 GHz	1.11E5 m/s	70.4 GHz	1.06E5 m/s

In Gunn diodes the drift velocity depends on three parameters: device temperature, electric field and injector barrier.

The shift with bias to lower frequencies of the negative conductance minima (Fig. 6.12) can be understood thinking on the electric field-drift velocity characteristic for GaAs: the increase of the applied voltage leads to a quasi-proportional change of the electric field, at the same time the drift velocity should decrease (we are considering electric fields higher than the critical electric field).

In Fig. 6.15 the measured drift velocity ( $f_{NCM}\cdot L$) is shown as a function of the supplied voltage. The diode without injector behaves symmetrically applying positive and negative biases. For voltages lower than $1.5 \hspace{1mm} V$, there is no Gunn effect; from $1.5 \hspace{1mm} V$ up to $2\hspace{1mm} V$, the drift velocity increases and reaches a maximum; for voltages higher than $2.2 \hspace{1mm} V$, the drift velocity decreases rapidly.

The drift velocity for diodes with injector is asymmetric (Fig. 6.16). In the reverse direction, it can be assumed that there is no injector, since in this configuration the graded gap barrier is situated at the end of the active region. Therefore, the voltage dependance of the drift velocity is very similar to the one without injector. In the forward direction, because of the injector barrier, the electrons are already hot when they enter the active region and the resulting drift velocity is smaller. A clear tendency can be observed as a function of the Al-content in the injector layer: the higher the Al-content, the lower the drift velocity and the flatter the slope of the curve. This property is directly connected with the diode frequency stability. Diodes with flatter drift velocity-voltage dependance should be less affected by temperature or voltage changes, because the occupation of the L-valley is mainly dominated by the hot electron injection rather than the electrical field or temperature activated process.

Figure 6.15: Drift velocity vs. voltage. Diode parameters: no injector, area= $196\hspace{1mm} \mu m^2$, active region length $=1.6 \hspace{1mm} \mu m$.

Figure 6.16: Drift velocity vs. bias voltage. Diode parameters: area= $144\hspace{1mm} \mu m^2$, active region length= $1.6\hspace{1mm} \mu m$, graded gap injector with maximum Al concentration from 28% to 34%.

A direct, voltage depending comparison between the negative conductance minima of the planar diodes and the oscillating frequencies of the packaged diodes cannot be made. When diodes are mounted inside a resonator, their behavior will be additionally affected by the resonator properties. However, with the help of the planar structures, some important trends in the high frequency behaviors have been identified, leading to a better layer structure and to an optimization of the packaged diodes.

By placing a transferred electron device in a cavity or resonant circuit, we can distinguish three main operational modes (see chapter 2.3.3):

The Transit Time mode.

The oscillation frequency $f_r$ is determined by the space charge or domain transit time $\tau_d$:

$$f_r=\frac{1}{\tau_d}=\frac{v_{drift}}{L} \thickspace .$$ (6.15)

The Delayed Domain mode.

The resonant period of the circuit $T_r$ is longer than $\tau_d$:

$$\frac{1}{2 \cdot \tau_d}=\frac{v_{drift}}{2 \cdot L}<f_r<\frac{1}{\tau_d}=\frac{v_{drift}}{L} \thickspace .$$ (6.16)

The Quenched Domain mode.

The resonant period of the circuit $T_r$ is shorter than $\tau_d$, but longer than the domain nucleation time $\tau_s$:

$$\frac{1}{\tau_d}=\frac{v_{drift}}{L}<f_r<\frac{1}{\tau_s} \thickspace .$$ (6.17)

Normally, a Gunn diode cavity oscillator has to be mechanically tuned, in order to get the largest power output at the corresponding desired frequency; comparing this target oscillating frequency with the drift velocity, the oscillating mode of the diode can be determined even before packaging. Different oscillation modes exhibit different powers, efficiencies and thermal stability properties [Hob72,Mak79].

In our case, a frequency of about 77 GHz is required, the active region is $1.6\hspace{1mm} \mu m$ long and a transit time mode is expected for drift velocities around $v_0=1.23\times 10^7 cm/s$. Drift velocities lower than $v_0$ correspond to the Quenched Domain mode, drift velocities higher than $v_0$ correspond to the Delayed Domain mode. Sample with high Al content in the injector (32% and 34%) can oscillate at 77 GHz only in the Quenched Domain mode.

In the case of a second harmonic oscillator, the required frequency is 38.5 GHz and the only operating mode is the Delayed Domain for all the considered structures.

This qualitative analysis, summarized in Fig. 6.15 and 6.16, does not need the complete exact circuit conditions but neglects one important parameter: the domain formation. In fact, the domain formation time and the corresponding dead-zone length can not be easily estimated from small signal measurements. The dead-zone, that is also responsible for temperature instabilities and high noise levels, is dramatically reduced with an optimized hot electron injector.

6.2.4 Estimation of the $\Gamma$ and L-valley occupation

From the determined drift velocities, quantitative values of the occupation of the $\Gamma$- and L-valley can be obtained. The drift velocity should be considered as the average of the high electron velocity in the $\Gamma$-valley and the low electron velocity in the $L$-valley. Defining $n_{\Gamma}$, $n_{L}$, $\mu_{\Gamma}$, $\mu_{L}$ as the carrier concentrations and the mobility of electrons in the two valleys, $v_d$ and $f_{NCM}$ can be rewritten as:

$$v_d = \frac{\mu_{\Gamma}n_{\Gamma}+\mu_{L}n_{L}}{n_{\Gamma}+n_{L}}{E},$$ (6.18)

$$f_{NCM} = \frac{\mu_{\Gamma}n_{\Gamma}+\mu_{L}n_{L}}{n_{\Gamma}+n_{L}} \frac{E}{L}.$$ (6.19)

As seen in Eq. (6.19), the change in $f_{NCM}$ is caused by a change in the ratio between $n_{\Gamma}$ and $n_{L}$. From the I-V characteristics (low voltages), we expect that the voltage drop on the injector is not symmetric in the forward and reverse directions. In order to have a reliable value of the electric field E in the active region, an accurate computation of the voltage drop on the injector has to be performed. For this purpose, results from Silvaco [Int] commercial package and manual fitting with a load-line model have been compared and taken into account. From Hall measurements of calibration samples, the $\Gamma$-valley electron mobility has been determined to be $\mu_{\Gamma}=5000 \hspace{1mm} \frac{cm^2}{Vs}$. According to McCumber ( $\mu_L=100 \hspace{1mm} \frac{cm^2}{Vs}$ [MC66]) and to Hakki ( $\mu_L=200 \hspace{1mm} \frac{cm^2}{Vs}$ [Hak67]) the $L$-valley mobility has been assumed 40 times lower than the $\Gamma$-valley one.

Figure 6.17: $L$-valley occupation for different electric fields; forward and reverse current directions (solid and dotted curve); injector with 32% Al concentration and active region $1.6\hspace{1mm} \mu m$ long.

Considering $n_0=n_{\Gamma}+n_L$ as the carrier concentration in the active region ( $n_0=1.1\cdot 10^{16}\hspace{1mm} cm^{-3}$), $n_{L}/n_0$ can be expressed by:

$$\frac{n_L}{n_0}=\frac{\mu_{\Gamma}-\frac{L\cdot f_{NCM}}{E}}{\mu_{\Gamma} -\mu_L} \thickspace.$$ (6.20)

The experimentally determined ratio of the carrier concentration in the $L$-valley ($n_{L}/n_0$) versus the electrical field E is presented in Fig. 6.17 for the structure with 32% aluminium content. It can be easily recognized the effectiveness of the injector, which is responsible for the occupation difference between the two current directions: this occupation difference decreases with E as the $L$-valley carrier concentration saturates at high electric fields.

6.2.5 Temperature dependance of the drift velocity

The Gunn diode temperature stability is a well known problem and has already been the topic of many publications ([Hob72,Mak79]) in the 1970's. Starting from a traditional approach, it is possible to identify five parameters, which concurrently contribute to the temperature dependance of the diode susceptance: low field mobility, domain transit time, domain formation, domain extinction and dielectric constant.

Figure 6.18: Drift velocity vs. bias voltage for different temperatures. Diode parameters: graded gap injector with maximum Al concentration 32%, area= $144\hspace{1mm} \mu m^2$, active region length= $1.6\hspace{1mm} \mu m$.

In this section, temperature dependant S-parameters are analyzed, showing how the Gunn diode hot electron injector influences the domain transit time and the domain creation. In order to change the active layer temperature of the diode, a custom designed heating arrangement was used: the measurements were performed at room temperature with no need to recalibrate the network analyzer. The substrate was in contact with a copper block cooled or heated by a temperature controlled peltier element (for more details see chapter 4.7.1).

From the measured S-parameters, the temperature dependant drift velocity has been found as a function of the bias voltage (see section 6.2.3). As expected, the drift velocity decreases with increasing temperatures: for the same bias voltage, at 70 $\thinspace$  $^\circ$C the drift velocity is lower than at 0 $\thinspace$  $^\circ$C. At 70 $\thinspace$  $^\circ$C more electrons occupy the L-valley, decreasing the average drift velocity. Figure 6.18 shows the described effect for a Gunn diode with a graded gap injector. Yellow boxes delimit the change of the drift velocity with the temperature at $+4\thickspace V$ and $-4\thickspace V$. It can be easily noticed, that the box in the forward current direction is much smaller than the box in the reverse direction. In other words, the considered diode is more temperature stable in the forward direction, where the graded gap injector is designed to work. Moreover, the drift velocity change with temperature of a Gunn diode with a graded gap injector is three times less than the one of a Gunn diode without a hot electron injector.

6.3 Gunn diode based oscillators

In the previous two sections, a detailed description of the main aspects concerning the electrical behaviour of single Gunn diodes has been presented. Further on, two examples of microwave voltage controlled oscillators^6.2 are discussed: a cavity waveguide Gunn oscillator and a MMIC Gunn oscillator. Both of them are based on a graded gap injector GaAs Gunn diode. The first VCO represents the classical solution, it is commercially available and combines good performances for a reasonable price. The second VCO is an attempt to integrate a planar resonating circuit with a Gunn diode, directly on the GaAs substrate used to grow the diode structure. The design of the coplanar wave guide (CPW) low-pass filter and of the complete planar oscillator is based on comparison of simulations with measurements.

6.3.1 The Gunn diode cavity oscillator

After the packaging, the Gunn diode chip is mounted in a resonant cavity. The current waveform of the Gunn diode is non-sinusoidal, so that higher harmonics of the fundamental frequency can be achieved. Typical cavity oscillators allow a fundamental or a second harmonic frequency generation. In the latter case, the Gunn diode is embedded in a resonator and the second harmonic is coupled out through a waveguide, which is below cutoff for the fundamental frequency. The resonator may consist in a part of the cavity itself [Bar81], or in a combination of a disk and a post in a coaxial configuration [Ond79].

Figure 6.19: Cross sectional view of a typical Gunn diode cavity oscillator.

The basic design of a cavity Gunn oscillator is shown in Fig. 6.19. The Gunn diode chip is pressed and fixed on the bottom of the cavity. In this way, the diode is automatically electrically grounded. On the top of the diode, the anode contact is connected to a disk and a post. Changing the diameter of the disk and the length of the post, it is possible to tune respectively a parallel capacitance and a series inductance. The post is combined with a bias choke from which the DC power is provided. The bias choke, which is carefully isolated from the waveguide cavity, is designed as an efficient low-pass filter, in order not to loose any HF power in the DC supply connection. In Fig. 6.19, one can see also a back-short: it is required to mechanically tune the oscillator response. Another adjustable side-short can be moved perpendicular to the cross section plane. The back-short influences strongly the oscillator power and the side-short allows a fine tuning of the frequency.

Figure 6.20: Frequency and output power, solid and broken curve, vs. voltages for a graded gap injector Gunn diode second harmonic cavity oscillator [Pro04].

The typical behavior of a second harmonic cavity oscillator is presented in Fig. 6.20. The oscillator contains a graded gap injector Gunn diode with a maximum Al concentration of 32% and a $1.6 \thickspace \mu m$ long GaAs active region. The frequency and power characteristics versus the bias voltage have been measured in CW conditions (no pulses). The Gunn diode starts to oscillate at $1.4 \thickspace V$, slightly after the threshold voltage. The turn on voltage, $V_{ON}$ (the voltage above the threshold at which coherent RF power is obtained) can be found around $3\thickspace V$. Between the threshold and the turn on voltage, the oscillations are incoherent. This incoherent regime is not usable; it is similar to the situation existing in a laser diode before the threshold current. From 3 to $6 \thickspace V$ the frequency increases monotonically with the bias voltage.

6.3.2 The planar low pass filter

The miniature low-pass filter, used in the MMIC Gunn oscillator, consists in a slow-wave periodic structure proposed by Sor et al. [SQI01]. The main goal is to increase the effective capacitance and inductance along the CPW line. The inductance is enhanced reducing the CPW center conductor width and further capacitances to the ground are created branching out the center conductor and the two grounds. The form of the cell with the equivalent lumped element circuit is illustrated in Fig. 6.21.

Figure 6.21: Schematic view and equivalent circuit of an unit cell of the periodic low pass filter.

Two cell and three cell configurations with $1 \thickspace mm$ and $0.8 \thickspace mm$ length per cell respectively, have been considered. For both of them, the scattering parameters have been simulated and measured. A detailed description of the measurement equipment can be found in section 4.7.1. The low-pass filter has been designed and simulated using Sonnet Design Suite V.9 [V.903]

Figure 6.22 shows the response of a periodic low-pass filter with three $0.8 \thickspace mm$ long cells. A good match between the simulated and experimental S-parameters can be noticed. The excellent periodic low-pass filter capabilities demonstrated in [SQI01] have been confirmed. With $1 \thickspace mm$ long cells, the cutoff frequency decreases. This can be understood considering that, with the same effective dielectric constant, the cut-off frequency is inverse proportional to the cell length.

Figure 6.22: Simulated and measured response of the three cell, periodic low-pass filter. The unit cell length is 0.8mm.

The influence of the cell number is demonstrated in Fig. 6.23, where two and three cell low-pass filter are presented. A sharper roll-off can be accomplished simply by inserting more cells. On the other hand, adding cells can be seen as a disadvantage, if the integrated circuit size is a priority. The two cell low-pass filter with $1 \thickspace mm$ long units resulted in the best tradeoff between performance and size.

Figure 6.23: Effect of additional cells on the low-pass filter roll-off. Simulated insertion loss of two and three cell. The unit cell length is 1 mm.

In conclusion, the periodic structure has demonstrated a compact size, low insertion losses at low frequencies (pass-band region), high attenuation levels at high frequencies (band-stop region) and a simple filter synthesis and fabrication.

6.3.3 The planar resonant circuit

A schematic view of the voltage controlled oscillator (VCO) is shown in Fig 6.24. Even if the Gunn diode is a two terminal device, it has been implemented in a two port configuration. The first port is connected to the resonator and the second one leads to the coupler and to the low-pass filter for the DC supply. All the passive elements have been simulated with Sonnet Design Suite V.9 [V.903].

Figure 6.24: Typical Gunn diode MMIC voltage controlled oscillator layout.

The coupler consists in an interdigitated capacitor with 5 fingers. Two kinds of resonators have been examined: the first one combines a coplanar line and two hairpins, the second one combines a slightly shorter coplanar line with a double T structure. Both of them are variations of the basic $\lambda/2$ resonator.

The oscillator design started from DC and S-parameters measurement data of the planar graded gap injector Gunn diode, as described in section 6.2.2.

The oscillating frequency is given by the generalized oscillating condition

$$Y_d + Y_L =0 \thickspace ,$$ (6.21)

where $Y_d$ is the admittance matrix of the diode and $Y_L$ is the admittance matrix for the passive elements. The diode conductance was negative over a wide frequency range, allowing us a simple design of the passive elements, using a short signal analysis. The simulation of the whole resonant circuit $Y_L$ is much time consuming: it requires a 3 port computation (DC-in, HF-out and the diode input) and a high meshing precision of the passive elements (resonator, coupler an low-pass filter). For this reason, only the interval 25 to $60\thickspace GHz$ has been considered, assuming no oscillation could take place out of this range. For frequencies lower than $25\thickspace GHz$, the low-pass filter influences $Y_L$ and strong resonances could appear. These resonances, on the other hand, can not lead to oscillations, because the real part of diode admittance, Re($Y_d$) is positive under $25\thickspace GHz$.

Figure: Conductance and susceptance, solid and broken curve, vs. frequency for the $36\thickspace GHz$ resonant circuit.

Figure: Conductance and susceptance, solid and broken curve, vs. frequency for the $44.5\thickspace GHz$ resonant circuit.

The simulated response of the two considered circuits (at the diode port) is presented in Fig. 6.25 and in Fig. 6.26. The first one shows a resonance at $36\thickspace GHz$ and the second one at $44.5\thickspace GHz$. The maximum intensity for the conductance is respectively 53 and $68 \thickspace mS$. Concerning the peak width, the $36\thickspace GHz$ resonance seems slightly sharper than the other one.

No direct measurement could confirm the computations: scattering parameters can not be measured in the considered configuration and the required layout change would influence the wave propagation, probably creating artifacts. However, an indirect verification is provided in the next section, where the oscillation frequency for two oscillators based on the described resonant circuits is presented.

6.3.4 The monolithic integrated voltage-controlled Gunn oscillator

The implemented VCO was characterized using wafer probing and a measurement setup composed from a 40-GHz HP8564E spectrum analyzer, two Agilent 11970 (A and U) harmonic mixers and a DPM-2A power meter. The HP-R281A coaxial to waveguide transition connected the picoprobes to the waveguide inputs of the mixers and power meter. As explained in section 6.3.3, two different oscillating circuits have been considered.

In comparison with a cavity Gunn oscillator, the planar one does not require any mechanical tuning: no back-short or side-short are available. Frequency and power characteristics are fixed by the lithographic patterns. If this constitutes a big advantage in a mass production environment, in a research or early development stage, it reduces flexibility and restricts the tolerances for the impedance matching.

The frequency and the HF output power versus the Gunn diode tuning voltage of the first oscillator is shown in Fig. 6.27. After reaching the threshold voltage, the Gunn diode starts to oscillate. The frequency is not stable and more peaks can be seen in the spectrum analyzer. At $3.2 \thickspace V$, we have the so called turn-on voltage $V_{ON}$. After $V_{ON}$, the frequency starts to be stable and increases monotonously with voltage. The frequency varies from $36.65 \thickspace GHz$ at $3.2 \thickspace V$ to $37.55 \thickspace GHz$ at $5.5 \thickspace V$. A typical behavior for a graded gap injector Gunn diode can be noticed: the turn-on voltage ( $3.2 \thickspace V$) is very close to the threshold ( $2.5\thickspace V$). For this reason, a graded gap injector Gunn diode allows coherent oscillations over a wider voltage range compared with a standard Gunn diode [NDS⁺89]. The frequency characteristics in Fig. 6.27 looks very similar to the one of the cavity oscillator (Fig. 6.20). The HF power can be scaled with the diode diameter and the efficiency is analogous.

Figure 6.27: Frequency and output power, solid and broken curve, vs. bias voltage for a graded gap injector Gunn diode monolithic oscillator (resonant circuit I).

The second planar oscillator with a different resonant circuit is presented in Fig. 6.28. At $4.5\thickspace V$, it generates $46.5\thickspace GHz$ with a peak power of $3.77\thickspace mW$. In this second oscillator, both the power and the frequency are higher than in the first one, but the voltage tuning of the frequency is inferior. Better tuning capabilities could be achieved adding to the second circuit a Schottky varactor.

A question remains. Why is the tuning range of the first oscillator about $1 \thickspace GHz$ and the range of the second less than $30 \thickspace MHz$? Are the two tuning ranges related only to the quality factors of the respective resonant circuits? Actually, an operating mode switch could explain different pushing^6.3 behaviors. In section 6.2.3, a classification of the operating modes was proposed in connection to the drift velocity. 37 and $46\thickspace GHz$ would correspond to a velocity of $6\times 10^6 cm/s$ and $7.4\times 10^6 cm/s$, respectively. Remembering that all the measured graded gap injector Gunn diodes had a drift velocity higher than $9\times 10^6 cm/s$, it can be concluded that both the planar oscillators are operated in the delayed domain mode.

Figure 6.28: Frequency and output power, solid and broken curve, vs. bias voltage for a graded gap injector Gunn diode monolithic oscillator (resonant circuit II).

A last consideration: the displayed power levels do not take into account the losses of the picoprobes and of the HF coaxial to waveguide transition. Estimating these losses, we expect that the total output power is with 2dB higher.