4. Experimental Methods

4.1 Atomic Force Microscope (AFM)

Atomic Force Microscopy technique provides three-dimensional surface topographies at nm resolution of a wide range of solid materials (conducting, insulating, magnetic). The basic principle is simple: a sharp tip scans the sample surface detecting the topography. Typical $\textrm{Si}_{3}\textrm{N}_{4}$ tips have a curvature radius of 10-15 nm and opening angle of about $20^\circ$. When the tip is brought very near to the surface to be analyzed, it undergoes attractive or repulsive forces. The cantilever, on which the tip is located, is deflected. An optical amplification system measures the tip movements. Figure 4.1 shows the system working principle: a laser beam points on the cantilever and four photodiodes in a cross configuration detect the reflected beam. While the sample is scanned, the tip-sample interaction is kept constant by feedback.

Figure 4.1: Schematic description of the operation principle of the AFM. The feedback loop (a compensation network or a computer program) monitors the cantilever deflection and keeps it constant by adjusting the vertical position of the cantilever.

Atomic Force Microscope has gone through many modifications for specific application requirements. There are several AFM operating modes: contact mode, tapping mode, phase imaging mode...

The first and foremost mode of operation, the contact mode, is very similar to a profilometer: the tip-sample forces are maintained at a constant level and with piezoelectric motion, the surface is scanned by the tip.
The tapping mode, belongs to a family of AC modes, which refers to the use of an oscillating cantilever. The tip intermittently touches or taps the surface. The natural resonance frequency is shifted by the tip-sample force. The shift is proportional to the second derivative of the corresponding potential. This information is then converted in a topographical image of the surface. In order to filter out the thermal noise, a lock-in amplification is introduced and a more stable detection is allowed.
In phase imaging mode, the phase shift of the oscillating cantilever relative to the driving signal is measured. This phase shift can be correlated with specific material properties, which affect the tip/sample interaction.

The forces acting between the tip and the surface are of different nature and depend on the tip-sample distance. The first interaction is the electrostatic force. It begins at $0.1-1 \thickspace {\mu m}$ and may be either attractive or repulsive depending on the material. At $10-100 \thickspace nm$, surface tension effects result from the presence of condensed water vapor at the surface. The tip is pulled down toward the sample surface with attractive force up to $200 \thickspace pN$. Van Der Waals forces appear at the angstrom level above the surface: the atoms in the tip and sample undergo a weak attraction. Coming even closer, electron shells from atoms on both tip and sample repulse one another, preventing further intrusion by one material into the other (Coulomb forces, contact mode). Pressure exerted beyond this level leads to mechanical distortion and the tip or the sample may be damaged.

The actual sharpness of a tip influences directly its ability to resolve surface features. Moreover, certain tip damages (e.g. double-pointed and cracked tips) occur very often. In Fig. 4.2, the tip-sample geometry and interactions are considered. One obvious surface limitation is caused by deep grooves (Fig. 4.2(a)): the tip is not long enough, or thin enough, to reach the bottom of a recess. Furthermore, the tip cannot detect walls of the sample with an angle steeper than itself.

Figure 4.2: Errors deriving from surface geometry (a) and tip geometry (b) during AFM characterization.

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More errors are introduced by the tip geometries: consider a $\textrm{Si}_{3}\textrm{N}_{4}$ tip with a radius $R$ measuring a series of parallel objects of radius $r$ (Fig. 4.2(b)). As $R$ increases, it decreases the ability of the tip to identify the radius $r$ and to resolve the separation of the objects.

4.2 Scanning Electron Microscope (SEM)

Conventional light microscopes use a series of glass lenses to bend light waves and create a magnified image. The Scanning Electron Microscope handles electrons instead of light waves. The resolution of an optical microscope is limited by the wavelength of the used light. For an electron microscope, the wavelength $\lambda$ from deBroglie is:

$$\lambda=\frac{h}{\sqrt{2mW}}=\frac{h}{\sqrt{2meV}} \thickspace ,$$ (4.1)

where $h$ is the plank constant, $m$ the electron mass and $W=eV$ is the kinetic energy of the electrons. For typical voltages between 5 and $20 \thickspace kV$, the wavelength is between $0.16$ and $0.08 \thickspace$   Å$$. These resolutions are very optimistic: the actual resolving power of a SEM is mainly depending on the apparatus. During this work, the scanning electron microscope Leo 982 has been used. It is equipped with the GEMINI field emission column providing typical resolutions of 4nm at 1kV, up to 1nm at 30kV.

After the air is pumped out from the column, an electron gun (at the top) emits a beam of high energy electrons. Four different sources can be used [Was03]: tungsten filament, $LaB_6$ crystal tip, Schottky emitter and the cold field emitter. In ultra-high vacuum conditions ($<10^{-9}$mbar), the best beam properties are supplied by cold field emitter. Schottky emitter is a good compromise between performance and costs.

The electron beam travels downward through a series of magnetic lenses designed to focus the electrons to a very fine spot. Near the bottom, a set of scanning coils deflect the focused beam back and forth across the specimen, row by row. The SEM images are normally rendered black and white.

Samples have to be prepared carefully to withstand the vacuum inside the microscope. Biological specimens are dried in a special way, which prevents them from shrivelling. Because the SEM illuminates them with electrons, they also have to be made to conduct electricity. Often SEM samples are coated with a very thin layer of gold by sputtering deposition.

4.3 Hall measurements

The Hall effect measurement is a widely-used technique to determine electrical properties of semiconductors like majority carrier type, concentration and mobility. The Hall effect is based on the deflection by a magnetic field of moving charged particles.

Figure 4.3: Sign convention and terminology for a rectangular Hall sample.

In Figure 4.3, a rectangular sample has been considered. A voltage $\textrm{V}_x$ and a magnetic field $\textrm{B}_z$ are applied. Electrons and holes flowing in the semiconductor will experience a force, bending their trajectories, and they will build up on one side of the sample, creating a potential $\textrm{V}_H$ across it (the so-called Hall voltage). Assuming that all the conduction electrons have the same drift velocity $\overrightarrow{v_{n}}$ and the same relaxation time $\tau_{c}$, the resulting Lorentz force acting on any electron is given by:

$$\overrightarrow{F} = -q(\overrightarrow{\mathcal{E}} + \overrightarrow{v_{n}}\times \overrightarrow{B})$$ (4.2)

$$\overrightarrow{F} = m_{e}\frac{d \overrightarrow{v_{n}}}{dt} + m_{e}\frac{\overrightarrow{v_{n}}}{\tau_{c}} \thickspace ,$$ (4.3)

where $m_{e}\overrightarrow{v_{n}}/\tau_{c}$ is a damping term accounting for the scattering. In the steady state, the derivative of $\overrightarrow{v_{n}}$ in equation 4.3 is zero. Combining equations 4.2 and 4.3 and using the relation $\overrightarrow{J}=-qn_{0}\overrightarrow{v_{n}}$ for electrons, we get:

$$\overrightarrow{\mathcal{E}}=\frac{\overrightarrow{J}}{q\mu_{n}n_{0}} + \frac{\overrightarrow{J}}{qn_{0}}\times \overrightarrow{B} \thickspace ,$$ (4.4)

where $\mu_{n} = q\tau_{c}/m_{e}$ is the mobility. Considering $B_{x}=B_{y}=0$ and $J_{y}=J_{z}=0$, the vector equation 4.4 can be written as two scalar equations:

$$\mathcal{E}_{x} = \frac{J_{x}}{\sigma}$$ (4.5)

$$\mathcal{E}_{y} = -\frac{J_{x}B_{z}}{qn_{0}}=-\mu_{n}B_{z}\mathcal{E}_{x} \thickspace ,$$ (4.6)

where $\sigma= q\mu_{n}n_{0}$ is the conductivity and relation 4.5 is just the Ohm's law. Equation 4.6 expresses the fact that, along the $y$ direction, the force on one electron due to the magnetic field ( $q\mu_{n}B_{z}\mathcal{E}_{x}$) is balanced by a force ( $-q\mathcal{E}_{y}$) due to the Hall field. (Eq. (4.6)) is generally written as

$$\mathcal{E}_{y}=R_{H}J_{x}B_{z} \thickspace,$$ (4.7)

where $R_{H}= -\frac{1}{qn_{0}}$ is called the Hall constant. For a p-type semiconductor, it can be shown that

$$\mathcal{E}_{y}=R_{H}J_{x}B_{z}=\mu_{p}B_{z}\mathcal{E}_{x}$$ (4.8)

with $R_{H}= \frac{1}{qp_{0}}$, where $p_{0}$ is the equilibrium concentration of holes in the sample. It is thus seen that $R_{H}$ has a negative sign for electrons and a positive sign for holes.

From 4.7, for a rectangular bar like in Fig. 4.3, the Hall constant and the Hall mobility can be expressed by:

$$R_{H} = \frac{V_{H}d}{I_{x}B}$$ (4.9)

$$\mu_{H} = \frac{L}{W} \frac{V_{H}}{V_{x}B} \thickspace ,$$ (4.10)

where $V_{H}$ is the measured Hall voltage, $V_{x}$ is the applied voltage along the length $L$, and $I_{x}$ is the current. For a semiconductor with comparable concentrations of electrons and holes, $R_{H}$ is given by

$$R_{H}=\frac{(\mu_{p}^{2}p_{0}-\mu_{n}^{2}n_{0})}{q(\mu_{p}p_{0}+\mu_{n}n_{0})^{2}}$$ (4.11)

and the interpretation of $R_{H}$ and ${\mu}_{H}$ is difficult. But if the material is strongly extrinsic, the relation (4.11) for a highly $n$-doped material reduces to

$$R_{H}=-\frac{1}{qn_{0}} \qquad \textrm{and} \qquad \mu_{H}=\mu_{n}$$ (4.12)

and for a highly $p$-doped one to

$$R_{H}=\frac{1}{qp_{0}} \qquad \textrm{and} \qquad \mu_{H}=\mu_{p}$$ (4.13)

In these two cases, the measurement of $R_{H}$ and ${\mu}_{H}$ determines the majority carrier concentration and the mobility, respectively.

The simplest arrangement for measuring the Hall voltage is the rectangular geometry shown in Fig. 4.3, but a number of spurious voltages is included in this measurement. All these spurious voltages are eliminated if four readings are taken by reversing the direction of the bias current $I_{x}$ and the magnetic flux $B$. The true Hall voltage is then obtained by taking the average of the four readings. The contacts used for measuring the Hall voltage in Fig. 4.3 should be infinitesimally small, so that they do not distort the current flow. The bridge shaped samples shown in Fig. 4.4(a) are often used to reduce the distortion of the current. The ears on the pattern allow a large area to be used for contacting the sample without a severe distortion of current flowing through the specimen. The Hall voltage is measured between contacts 1 and 2, and then between 3 and 4. The average is taken as a final value.

Figure 4.4: Hall measurement contact configurations and layouts

[A bridge-shaped Hall sample with two pairs of ears contacts.]

[Van der Pauw arrangement for an arbitrary shaped sample.]

In some cases, it may not be convenient to cut the specimen in the form of a rectangular bar. Van der Pauw [vdP58] suggested an interesting method, using an arbitrary shaped thin-flat sample. The above sample (simply connected, i.e., no holes or nonconducting islands or inclusions), contains four very small ohmic contacts placed on the sample periphery (preferably in the corners). In presence of a normal magnetic field to the sample surface, a current $I_{ac}$ is established between two opposite contacts (a and c), and the voltage is measured between the other ones (b and d), as shown in Figure 4.4(b).

The resistance, defined as:

$$R_{ac,bd}=\frac{V_{bd}}{I_{ac}}$$ (4.14)

is first measured with the magnetic field $B$, $R_{ac,bd}(+B)$ and then with the opposite field $-B$, $R_{ac,bd}(-B)$. If all the possibilities concerning the current and magnetic field polarity are considered, then the Hall constant is given by:

$$R_{H} =\frac{d}{8 B}( R_{ac,bd}(+B)-R_{ac,bd}(-B)+R_{bd,ac}(+B)-R_{bd,ac}(-B)$$

$$+R_{ca,db}(+B)-R_{ca,db}(-B)+R_{db,ca}(+B)-R_{db,ca}(-B))$$ (4.15)

4.4 Capacitance-Voltage measurements

A different way to determine the background concentration $N_{d}$ of carriers in a semiconductor, is the capacitance-voltage (C-V) measurement. The C-V method relies on the fact that the width of a reverse biased space charge region of a junction depends on the applied voltage. The space charge $Q_{sc}$ per unit area of the semiconductor junction under an applied bias V is:

$$\centering {Q_{sc}=qN_d W= \sqrt{{2q\epsilon_{s}N_d \left(V_{bi}-V-\frac{kT}{q}\right)}}} \thickspace ,$$ (4.16)

where $W$ is the depletion layer width defined in Eq. (2.51) and $V_{bi}$ is the built in potential defined in Eq. (2.49) (section 2.2.1). The depletion layer capacitance per unit area is given by:

$$\centering {C=\frac{\vert\partial Q\vert}{\partial V}= \sqrt{\f... ...\epsilon_s N_d}{(V_{bi}-V-\frac{kT}{q})}}=\frac{\epsilon_s}{W}} \thickspace .$$ (4.17)

Equation 4.17 can be written in the form

$$\frac{1}{C^2} = \frac{2(V_{bi}-V-\frac{kT}{q})}{q \epsilon_s N_d}$$ (4.18)

$$-\frac{d ({1/C^2})}{dV} = \frac{2}{q \epsilon_s N_d}$$ (4.19)

$$N_d = -\frac{2}{q\epsilon_s \frac{d ({1/C^2})}{dV}}$$ (4.20)

$$W = \frac{\epsilon_s}{C} \thickspace .$$ (4.21)

In the case of uniform doping concentration, $N_d$ is constant within the depletion region and a strait line should be obtained by plotting $1/C^2$ versus V. The intercept on the X-axis of the line represents $V_{bi}$, the built-in potential^4.1. If $N_D$ is not constant, the doping profile can be determined from the slope of the curve and equation 4.20.

4.5 TLM and CLM

The ohmic contacts are of great importance in relation with the quality and the reliability of the microelectronic devices. The resistance of an ohmic contact can be evaluated by the transmission line method (TLM). The TLM structure is a series of identical contact pads, of width $w$ and length $L_C$, spaced at varying intervals $d_1$, $d_2$, $d_3$, ... (Fig. 4.5A). The TLM structure requires an additional mesa etching, in order to restrict the current flow only between the contact pads, along the x direction (with the convention of Fig. 4.5A). A similar technique, the circular transmission line method (CLM) does not require additional mesa etching: the current confinement in the direction perpendicular to the contact is obtained with concentric circular contacts (Fig. 4.5B).

Figure 4.5: Contact resistance test patterns. (a) TLM and (b) CLM configuration.

In both cases, to avoid the eventual influence of external series resistances, a four point measurement (Kelvin method) is suggested: two probes carry the current and two probes sense the voltage. The expression of the measured resistance between two contacts separated by the distance $d_i$ is:

$$R_{meas}=R_C \cdot X + R_S \cdot Y \thickspace ,$$ (4.22)

where $R _C \medspace[\Omega \medspace mm]$ is the contact resistance, $R_S \medspace[\Omega / \Box]$ is the sheet resistance of the semiconductor, and X and Y depend on the geometry of the contact. For a TLM structure, we have:

$$X = \frac{2}{w}$$ (4.23)

$$Y = \frac{d_i}{w}$$ (4.24)

and for a CLM structure, it results:

$$X = \frac{1}{2\pi}ln \left(\frac{1}{r}+\frac{r}{r-d_i}\right)$$ (4.25)

$$Y = \frac{1}{2\pi}ln\left(\frac{r}{r-d_i}\right) \thickspace .$$ (4.26)

In Fig. 4.6, the measured resistances are plotted as a function of the distance $d_i$. Fitting Eq. (4.22), the values of $R_S$ and $R_C$ can be obtained.

Figure 4.6: Evaluation of the contact and sheet resistances for TLM measurements.

Then, the specific contact resistance, $\rho _c \medspace[\Omega \medspace cm^2]$ is given by:

$$\rho_C=\frac{R_C^2}{R_S} \thickspace ,$$ (4.27)

and the transfer length $L_T$ (also known as effective contact length) is:

$$L_T=\frac{R_C}{R_S} \thickspace .$$ (4.28)

It can be noticed that the intercept with the x-axis of the fitted line in Fig. 4.6 is twice $L_T$. For the validity of the above analysis, one assumption has been made: the transfer length $L_T$ must be much smaller than the contact length $L_C$ [MD82].

4.6 Short pulse DC measurements

Mainly the DC measurements have been carried out using a Curve Tracer Sony-Tektronix 370A. Additionally to this, a driver transistor has been included in the circuit, so that $80\thickspace \mu s$ pulse measurements could be possible. The measurement setup is shown in Fig. 4.7(top). The circuit of the setup operates as following: the base output of the Curve Tracer operates in pulse mode and drives the base of an external control transistor with current pulses of 1mA. The Collector output (voltage supply) is connected to the Gunn diode and then the circuit is closed through the control transistor. To avoid the eventual influence of external series resistance, a four point measurement (Kelvin method) is used. The Gunn diode chip is connected to the measurement circuit, using micro-manipulators with Au needles. The measurement table has a heater, which permits I-V characteristics measurements in a temperature range from room temperature till $200^\circ$C.

Figure 4.7: Measurement setup for continuous and pulse ( $80\thickspace \mu s$) DC measurements with heating possibilities from room temperature till about 200 $\thinspace$  $^\circ$C (top). Measurement setup for very short pulse ( $100 \thickspace ns$) DC measurements (bottom).

The study of the breakthrough voltage without damaging the Gunn diode requires DC measurements with very short pulses ( $\ll 80 \thickspace \mu s$). Moreover, with different pulse length, the influence of the diode selfheating on the I-V characteristics can be determined. For these two reasons, a $100 \thickspace ns$ measurement setup has been designed and realized [Pro04]. The principle scheme of the measurement set-up is presented in Fig. 4.7(bottom).

100 ns pulses are generated from a low-power pulse generator and are amplified by a custom-designed printed circuit board (PCB), powered by a computer controlled DC power supply. The pulses are forwarded to the terminal of the diodes. The short pulse length requires a special attention in the terminal connection, in order to avoid line reflections. Therefore, on-wafer measurements are performed with HF picoprobes. For the packaged diodes, a further custom adaptor is connected directly to the PCB. The voltage drop on the diode and on a known serial resistance within the PCB are measured with a digital oscilloscope. The serial resistance voltage drop is proportional with the current flowing through the diode. The sweep range of the voltage pulse can be tuned from $0 \thickspace V$ up to $20\thickspace V$ and the current is limited to $3 \thickspace A$. A HP-VEE program running on a PC controls the measurements through a IEEE488 interface bus and computes the diode DC characteristics from the oscilloscope raw data.

4.7 S-parameters measurements

4.7.1 Network Analyzer Measurement systems

Network analyzers characterize accurately components by measuring their effects on the amplitude and phase of swept-frequency and swept-power test signals. Network analysis satisfies the engineering need to characterize the behavior of passive and active components quickly and accurately over broad frequency ranges.

Figure 4.8: Vector network analyzer HP8510C (8510XF)

6.2cm

Most of the devices used to collect data for this work, have been measured from 500MHz up to 110GHz, with 201 frequency steps. The S-parameters measurements need to be performed over appropriate bias range. The bias voltage should be varied from negative to positive values, linearly increasing up to 5 or $8\thickspace V$ (depending on the breakdown voltage of the device).

Temperature dependent measurements have been more complicated and time consuming: they have required to build a compact custom-made temperature controller based on a peltier element and a PT100 thermosensor. The temperature is function of the resistance of the PT100, which is embedded in a flat copper block. The peltier element cools or heats (depending on the terminal polarity) the copper block, transferring the heat to or from a big metal reservoir, kept at room temperature. In the considered setup, the sample is fixed with vacuum on the copper block and the temperature can be tuned from 2 $\thinspace$  $^\circ$C up to $80^\circ$C. Lower temperatures can not be reached because of ice formation and higher ones could damage the HF picoprobes.

When making on-wafer measurements, an ongoing concern is how accurate and repeatable are the collected data. The hardware has to be as good as practically possible, balancing performance and costs; error corrections can be an useful tool to improve measurement accuracy. The basic source of measurement errors are:

Systematic errors:

due to imperfection in the analyzer and test setup. They are repeatable (and therefore predictable) and are assumed to be time invariant. Systematic errors are determined during the calibration process and they are used for corrections during the measurements.

Random errors:

unpredictable, since they vary in a random fashion. Therefore they cannot be removed by calibration. The main contributors to random errors are source phase noise, sampler noise and IF noise.

Drift errors:

are due to the instrument or test-system performance changing after the calibration has been done. The main cause is temperature variation and can be removed by further calibrations and/or providing a stable ambient temperature.

Before a network analyzer can make error-corrected measurements, the network analyzer systematic errors must be measured and removed. Calibration is the process of quantifying these errors by measuring known precision standards. Three techniques are commonly used to calibrate network analyzers and wafer probing stations:

Short Open Load Through (SOLT)

is a two-port calibration, typical for coaxial transmission medium. It uses a short, an open and a load at each port as impedance standard to measure reflection errors. A through standard measures the frequency response tracking. A load match and an open signal path measure the isolation between ports [Gol91]. With non-coaxial transmission medium, the calibration standards of the SOLT is difficult to fabricate.

Through Reflect Line (TRL)

is a two-port calibration, which requires a through connection, a reflection standard, and a reference transmission line. It uses calibration standards, which are easily fabricated and characterized; for this reason, it is preferred for non-coaxial environment.

Line Reflect Match (LRM)

is a two-port calibration similar to TRL. Instead of TRL, it is based on fixed loads and not transmission line(s), as reference impedance. LRM is a particularly convenient broadband calibration for non-coaxial environments with accuracy as good as TRL.

The LRRM^4.2, a variation of LRM, was used for most of the measurements with the HP8510C-XF vector network analyzer in Figure 4.8; it was usually software assisted by WinCal, a program developed by Cascade Microtech, which supplied also the picoprobes and the probing station.

4.7.2 Displaying data: Smith Chart

During HF measurements and after data has been collected, it is essential to have an overview over the produced results. The fastest way is to display data with some kind of graphical interfaces (normally integrated in the measurement system or in a software package for computer controlled measurement systems). The most popular diagram for each parameter are:

• Magnitude and Phase amplitude Chart,
• Polar Chart,
• Smith Chart^4.3 (Fig. 4.9).

Smith chart represents the standard for $S_{11}$ and $S_{22}$, while polar chart is more used for $S_{12}$ and $S_{21}$. Magnitude and Phase amplitude chart can be used for all S-parameters. Assuming that the concepts of magnitude and phase amplitude chart and of polar chart are clear to the reader, Smith chart is defined in the followings. To simplify the analysis two assumptions are made:

1. the transmission line is essentially lossless^4.4, meaning that the characteristic impedance is entirely real;
2. the load impedance $Z_L$ is normalized to the characteristic impedance $Z_0$^4.5.

The Smith Chart is derived from the relationship between complex reflection coefficient $\Gamma$ and complex impedance Z.

$$\Gamma=\dfrac{Z-1}{Z+1}$$ (4.29)

Writing the impedance and the reflection coefficient as function of the real and imaginary components, $Z=r+jx$ and $\Gamma=\Gamma_r+j\Gamma_i$, equation 4.29 is equivalent with:

$$r = \dfrac{1-\Gamma_r^2-\Gamma_i^2}{(1-\Gamma_r)^2+\Gamma_i^2}$$ (4.30)

$$x = \dfrac{2\Gamma_i}{(1-\Gamma_r)^2+\Gamma_i^2} \thickspace .$$ (4.31)

To actually see the form of the equations for deriving the Smith Chart, we rearrange the terms to obtain the equations for a set of circles:

$$\left(\Gamma_r-\dfrac{r}{1+r}\right)^2+\Gamma_i^2=\left(\dfrac{r}{1+r}\right)^2$$ (4.32)

$$(\Gamma_r-1)^2+\left(\Gamma_i-\dfrac{1}{x}\right)^2=\left(\dfrac{1}{x}\right)^2 \thickspace .$$ (4.33)

From equations 4.32 and 4.33 the Smith Chart can be easily drawn. Some key features of this chart (Fig. 4.9) are listed:

• The horizontal axis through the origin represents all purely resistive impedance.
• The outermost rim represents all purely reactive impedances.
• We can see a family of circles: each of them is a constant resistance set of points.
• We can see a family of arcs: each of them is a constant reactance set of points.

Figure 4.9: Smith Chart