5 Experiment Procedure

5.1 The experimental setup

The apparatus uses a Ge sample, cut from a P-doped ingot, placed inside a isothermal aluminum case. It may be positioned in the gap between the two poles of a permanent magnet, made of two Nd-Fe-B discs and a U shaped soft-steel core, acting like a torus.

The sample has 7 wires soldered in the positions shown in Fig. 5.1 and Fig. 5.2 as follows:
Simplified schematic of the sample circuitry

Figure 5.1: Simplified schematic of the sample circuitry

  • Contacts 1 and 4 are used to feed the bias current from 0 to 20mA \(I_b\) produced by high-stability constant current generator (Fig. 5.1).

  • Contacts 7 and 5 are used to measure (through the use of differential amplifiers, DA for short) the voltage across the sample, in a 4-wire (often called “Kelvin”) resistance measurement.

  • Contacts 2-3 and 6 are the output of the Hall voltage and fed to two DA.

    Contact 6 is the reference point for the Hall voltage and contacts 2 and 3 are used to set the balancing potentiometers P after having removed the sample from the magnetic field (the Hall voltage should be zero in absence of applied magnetic field). The potentiometer P is not directly connected to the sample as it would lower the impedance, thus affecting the measurement.

    Three contacts are needed for the Hall voltage because two contacts cannot be precisely aligned.

    This measurement method is often called “5 wire hall voltage measurement” and is to date the most accurate way to conduct this experiment.

Printed circuit board with germanium sample

Figure 5.2: Printed circuit board with germanium sample

The differential amplifiers DA gain can be varied from a value of 0 (disabled) to 4096 in increments by the power of two. \(G_{DA}=[0,1,2,4,8,16,32,64,128,256,512,1024,2048, 4096]\). For the Hall voltage, the gain of the first stage of the amplification circuit can be varied indipendently from the second stage, to allow an easy adjustment of the balancing potentiometer.

The gain of the two stages of the Hall voltage must of course be multipled together.

The DA input circuitry is floating with a maximun \(+5V\) input range referred to ground and is then referenced to a \(2.5V\) middle point.

The numbering of the contact on the sample corresponds to the number of the pins in the rj45 connector of the sample assembly.

The output voltages measured can be distorted in case of DA saturation, and setting the gains conservately is recommended for a successful experiment run.

The best value for the bias current is a compromise between the need to obtain a large \(V_H\) to make measurements precise reducing the SNR (Signal to Noise Ratio) and a low self-heating of the element due to the Joule effect: \(V_R\) and \(V_H\) signal are proportional to \(I\) while the Joule self-heating is \(P=RI^2\). A value of \(10mA\) is an excellent starting point. \(V_R = I*10 \Omega * 25\)

Additionally, a lock-in input is provided. When a \(5V\) voltage is present on this input, the bias current will be set to \(0mA\). This is not required during normal operation but can be used for especially difficult measurements such as finding earth magnetic north pole using the sample. The input is compliant with standard 5V TTL voltage level.

The instrument outputs are conveniently provided over two DE-15 connectors that can be equipped either with BNC cables or with custom cables. The left connector is used for standard operation, the right connector for advanced operation and diagnostics.

Table 5.1: Output cable pinout
Connector Cable Description
\(Left\) Red Hall Voltage, \(G=G_{STAGE 1}* G_{STAGE 2}\) , \(V_{offset}=2.5V\)
\(Left\) Green Voltage between contacts 5 and 7, amplified, \(V_{offset}=2.5V\)
\(Left\) Blue Amplified thermocouple voltage, see chapter on the temperature measurement
\(Left\) White/Grey Bias current, voltage output, \(V_I = I*10 \Omega * 25\), \(0V=0mA, 5V=20mA\).
\(Left\) Black Gaussmeter output, aplified, \(V_{offset}=2.5V\)
\(Right\) Red 2.5V offset, can be used to correct for errors
\(Right\) Green Voltage between contacts 2 and 6, amplified by stage 1 of Vh
\(Right\) Blue Voltage between contacts 3 and 6, amplified by stage 1 of Vh
\(Right\) White/Grey Lock-in input. A 5V signal sets bias current to 0mA. Max 100Hz square wave.
\(Right\) Black 5V output, 100mA max

5.2 Hall voltage and resistance measurements at room temperature

With a finite value of magnetic field B orthogonal to the large face of the sample, we should measure identical values for \(V_H\) (but with opposite sign) when rotating of \(180\,^{\circ}\) the sample. This behavior must be tested before proceeding to further measurements: if reversing the \(B\) direction (i.e. rotating the sample of \(180\,^{\circ}\) degrees) different values are measured, the offsets should be better adjusted using potentiometers P in Fig. 5.1. If adjusting the offset does not lead to an improvement, there might be a saturated DA in the amplifier circuit: reduce the gain and repeat the test.

The absolute value of \(B\) may be varied by changing the width of the gap between the magnetic poles (see Fig. 5.3). One of the two permanent-magnets mounted on the soft-steel structure may be moved horizontally by turning the screw: increasing the gap, while keeping the sample in the middle between the two magnets, decreases the value of \(B\).

The screw device for changing the effective magnetic field

Figure 5.3: The screw device for changing the effective magnetic field

A calibration of the magnetic field \(B\) as a function of the gap \(d\) may be made using a gauss-meter probe placed between the poles inside the positioning jig. (see Fig. 5.14)

The ltk-hall-ge apparatus is provided with an included gauss-meter probe with excellent linearity.

5.3 The set-up for changing and measuring the sample temperature

The stainless-steel dewar can be half-filled of liquid nitrogen or a mixture of acetone and dry-ice (solid carbon dioxyde). The cold finger (the aluminum bar screwed into the base of the sample) is surrounded by the liquid nitrogen, allowing the sample to cool down.

The slope (the Seebeck coefficient) of the thermocouple output signal strongly depends on temperature (not linear).

Figure 5.4: The slope (the Seebeck coefficient) of the thermocouple output signal strongly depends on temperature (not linear).

–> Warning: Care must be taken to avoid contact between the dewar black plastic (ABS) and the Acetone. The white plastic (PTFE) is acetone-resistant.

The temperature is measured by a type K (Chromel-Alumel) thermocouple thermally coupled to the sample. The small voltage generated by the thermocouple is amplified by an AD84955 integrated circuit. The output is roughly proportional to the temperature with a sensitivity of \(\approx 5\frac { mV }{ \,^{\circ}\mathrm{K} }\) is shifted to obtain \(V_{out \, T}=2.5V\) at \(273.15\,^{\circ}\mathrm{K}=0\,^{\circ}\mathrm{C}\). While the K type thermocouple is fairly linear in a small range near room temperature, it is not linear in the whole temperature range covered by the apparatus, as can be seen in Fig. 5.4.

In order to get a correct measurement it is necessary to compensate for the non-linearity (see Fig. 5.4) of the thermocouple using the following polynomial:

\[\begin{equation} t_{calc}=d_{ 0 }+d_{ 1 }E+d_{ 2 }E^{ 2 }+...+d_{ n }E^{ n } \tag{5.1} \end{equation}\]

where \(E\) is the output voltage of the thermocouple in \(mV\).

A fitting polynomial (5.1) of the fifth order is more than sufficient, given the precision of our equipment.

The table 5.2 shows the polynomial coefficients obtained from a best fit of the NIST6 data tables.

Table 5.2: Polynomial coefficients obtained from NIST K thermocouple tables (\(-200< t \, [^{\circ}\mathrm{C}] <200\)).
Coefficient Value
\(d_0\) -0.3713966
\(d_1\) 25.2201378
\(d_2\) -0.2833052
\(d_3\) 0.0718439
\(d_4\) -0.0139566
\(d_5\) 0.0010494

Fig. 5.5 shows the NIST \(t(E)\) data for K thermocouple compared with the results obtained using eq. 24 and the coefficient of table 1, and the residual errors in the range (\(-200< t \, [^{\circ}\mathrm{C}] <200\))

Residual errors of the polynomial fit

Figure 5.5: Residual errors of the polynomial fit

The voltage \(E\) at the thermocouple junction can be obtained7 from the following equation:

\[\begin{equation} E=\frac { V_{ outT }-{ V }_{ Ref }-{ V }_{ Offset } }{ Gain } \tag{5.2} \end{equation}\]

where \(V_{outT}\) is the output of the instrument (on the front panel), \(V_{Ref}=2.5V\) the voltage that indicates a temperature \(T=0\,^{\circ}\mathrm{C}\), \(V_{offset}\) is the error voltage at \(0\,^{\circ}\mathrm{C}\) to achieve 125 mV at \(25\,^{\circ}\mathrm{C}\) and \(Gain\) is the internal gain of the AD8495 amplifier.

Using the fitting polynomial (5.1) allows us to finally obtain the temperature in Celsius:

\[\begin{equation} t={ f }_{ comp }\left( E \right) \tag{5.3} \end{equation}\]

\[\begin{equation} t={ f }_{ comp } \left(\frac { V_{ out }-2.5-1.25\cdot 10^{ -3 } }{ 122.4 } \right) \tag{5.4} \end{equation}\]

Two electronically controlled resistive elements (heaters) are wound around the base of the sample, allowing it to heath up after reaching room temperature. Three leds start blinking when \(t \ge 60\, \,^{\circ}\mathrm{C}\) to indicate that the sample is no longer safe to touch with naked hands. The apparatus shuts down the heater if \(t \ge 150\, \,^{\circ}\mathrm{C}\) as a safety measure. Heating is allowed again after the sample has cooled down to \(t \le 100\, \,^{\circ}\mathrm{C}\).

Depending on environmental condition such as airflow, ambient temperature, and liquid nitrogen present in the dewar, it may not be possible for the apparatus to reach the desired temperature. In this case it is recommended to insulate the sample with an adeguate heath-resistant material, such as a fiberglass cloth or polyamide/Aluminium film.

5.4 Suggested experimental procedure

5.4.1 Standard experiment run

To obtain accurate measurements the best procedure is the following:

  1. Connect the left output connector to your datalogger
  2. Verify the grounding of the equipment. The equipment should be connected to ground through the dedicated grounding plug.
    • WARNING: In case the ground is not properly connected, interferance may be coupled causing the apparatus to misbehave.
  3. Place the sample in the middle of the gap. Choose a proper value for the current \(I_b\) within a suggested 5-15mA range
  4. Optionally, adjust the gains for \(V_H\) (firstly, leave the second stage at \(G=1\) and adjust the first stage, then adjust the second stage), for \(V_R\) and for \(I_B\).
    • Note : the resistance at higher temperature may exceed the value at room temperature by a factor 2, and also the \(V_H\) signal increases with temperature. To avoid saturation it is recommended the the output voltage stays between \(1V<V_{out}<4V\).
  5. Check that the \(V_H\) value changes sign when rotating the sample of 180° around vertical axis. Choose the orientation that gives positive \(V_H\).
  6. Fill about half of the dewar with liquid nitrogen and wait until the liquid surface is quiet.
  7. Insert the cold finger into the dewar (the PTFE dewar-cover should seat stable onto the dewar mouth, and the PTFE heater cover should be set with the hole hosting the pin protruding from the dewar-cover). Adjust the sample in the mid of the magnet-gap and start the data acquisition.
  8. Verify your datalogger setup (see datalogger chapter) and start data acquisition.
  9. When the plot temperature vs time shows a slope close to zero, pause the data acquisition.
  10. Empty the dewar (e.g. transferring the residual liquid nitrogen into another dewar), reposition the sample in the middle of the magnets-gap.
  11. Resume data acquisition: the temperature will start increasing.
  12. When the temperature vs. time slope start approaching zero, switch ON the heater to half-power. Once the lights start flashing the apparatus will have reached 60°C, do not touch the sample in order to avoid burns.

To obtain precise measurements, at least one and a half hour is required for the whole temperature sweep.

Note: it is not suggested to keep liquid nitrogen inside the dewar while heating-up the sample: the temperature would rise more slowly and more humidity would condense onto the outer surface of the aluminum probe envelope.

It is also useful to blow-off the frost in order to prevent water entering the probe envelope (this might affect the thermocouple’s weak signal).

Typically one hour is required to complete the full temperature span, and the recommended data acquisition rate is \(0.2 sample/s\)

5.4.2 VH Center balance trimming

In case the \(V_h\) presents an undesided offset while no magnetic field is present, it is recommended to perform the following trimming procedure

  1. Set bias current to \(I=15mA\)
  2. Set the gain of the second stage of \(V_h\) to 0, take note of the voltage, it should be \(V_h\simeq2.5V\)
  3. Set the gain of the second stage of \(V_h\) to 1.
  4. Place the sample far from the magnetic field and trim the center balance potentiometer (using the provided isolated screwdriver) until you get the previously measured \(V_h\simeq2.5V\).
  5. You may set the gain of the second stage of \(V_h\) to a value greater than 1 if your equipment is not sensitive enough.
  6. In case the Vh gain is still incorrect, you may use the left and right potentiometers to adjust beyond the capabilities of the center potentiometer.

5.5 Data acquisition setup with Vernier software

The following chapter will explain the usage of the LoggerPro or LoggerLite software for Labpro/LabQuest2 Dataloggers in conjunction with ltk-hall-ge. LabPro, LoggerPro, LoggerLite, Labquest, Labquest2 are products and trademarks of Vernier.

5.5.1 Standard experiment run

In order to acquire the data correctly, you may execute the following procedure when indicated:

  1. Connect the 4 cables to the 4 Analog Inputs of the DataLogger (LabPro or LabQuest2) as follows: red cable (\(V_h\)) to CH1, blue cable (\(V_T\)) to CH2, green cable (\(V_R\)) to CH3, gray cable (\(V_I\)) to CH4.
  2. Connect the DataLogger to the PC (and connect the power supply to AC main).
  3. Start the LoggerPro application on the PC.
  4. The 4 signals of the Ltk-Hall-Ge device must be configured as standard “Raw Voltage 0-5V” sensors (see figure 5.6 and 5.7). Depending on the type of DataLogger in use you may notice different windows. In both cases you can click onto the field of the selected channel and a pop-down list will appear (see fig. 5.7), where you should choose the Raw voltage (0-5V) sensor.
  5. Before starting the experiment you may build new columns for the data collection spreadsheet that will be filled with data calculated from the set of the measured raw-data : Hall voltage, Temperature, Magnetic field, Current, Resistance must be calculated from the raw data values as explained hereafter.
    • For example the temperature \(t\) may be calculated from the raw data generated by the thermocouple circuitry in two steps as follows: first we calculate the junction-efm (E) 5.8, then from the E values we calculate the temperature in Celsius as shown in figure 5.9.
  6. Select the proper values of the gain for the variious channels. These gain values must be used to calculate the actual values from the raw data (taking not of compensating for the 2.5V level shift introduced by the apparatus).
    • Eg. with \(V_h\) channel gain \(G_{V_h}=16\) and \(V_R\) channel gain \(G_{V_r}=2\) the column can be calcolated as shown in figures 5.10 and 5.11
    • Eg. With the bias current signal, knowing that the gain for this channel is fixed to \(G_{I}=25\) and that the “current sensor” is a resistor of value \(R=10\Omega\pm0.1%\) in series to the sample, the actual current (measured in A) should be calculated as shown in figure 5.12. Finally, the resistance may be calculated as ratio \(R=\frac{V}{I}\) as shown in figure 5.13
  7. Configure the data acquisition mode as “Time based data collection”.

Both during the experiment run and at completion, graphs may be created from the measured data.

Setting-up sensors with LabQuest (left) and LabPro

Figure 5.6: Setting-up sensors with LabQuest (left) and LabPro

Choosing Raw Voltage sensor

Figure 5.7: Choosing Raw Voltage sensor

Calculation of $E$ from the thermocouple voltage

Figure 5.8: Calculation of \(E\) from the thermocouple voltage

Calculation of $t$ with polynomial fit

Figure 5.9: Calculation of \(t\) with polynomial fit

Calculation of $V_rsh$ with an example gain $G=16$

Figure 5.10: Calculation of \(V_rsh\) with an example gain \(G=16\)

Calculation of $V_R$ with an example gain $G=2$

Figure 5.11: Calculation of \(V_R\) with an example gain \(G=2\)

Calculation of $I$

Figure 5.12: Calculation of \(I\)

Calculation of $R$

Figure 5.13: Calculation of \(R\)

5.5.2 Calibration of the magnetic field

To perform a complete calibration of the magnetic field \(B\) versus the gap between the magnets you may use the mode “Events with Entry”. After starting such acquisition mode, you will be able to get the measurements from the gaussmeter probe automatically, as well as manually entering the gap between the magnets when prompted.

5.5.3 Additional resources

Additional resources on Vernier’s software and hardware products may be found on the manufacturer website. A quick reference manual is also available, together with a great number of screencasts.

5.6 Typical results

The sample shown in figure 3 (and used to obtain the data in the following examples) has thickness \(t=0.5mm\), width \(w=10mm\) and lenght \(l=15mm\).

A calibration of the magnetic field intensity \(B\) vs. gap \(d\) between magnets is shown in Fig. 5.14.

Measured $B$ values vs gap width $d$

Figure 5.14: Measured \(B\) values vs gap width \(d\)

An example of the measured \(V_H\) vs. magnetic field \(B\) at room temperature is shown in Fig. 5.15.

Hall voltage versus magnetic field intensity $B$

Figure 5.15: Hall voltage versus magnetic field intensity \(B\)

Fig. 5.16 shows an example of the measured values \(V_h\), \(R\) and \(t\) vs time obtained with a constant bias current.

Measured $B$ values vs gap width $d$

Figure 5.16: Measured \(B\) values vs gap width \(d\)

In order to evaluate the Ge energy gap \(E_g\), a plot of \(ln(R)\) vs. \(\frac{1}{2kT}\) was built, after calculating from the Celsius temperature \(Tc\) the absolute temperature \(T\) (\(k\) is the Boltzmann constant \(k = 8.617 \cdot 10^{-5} \frac{eV}{K}\).

From the slope in the intrinsic region (high temperature region, see Fig. 5.18 ) we get the value of the energy gap \(E_g\), extrapolated linearly to \(T=0^{\circ}\mathrm{K}\), that can be compared to the known value for germanium (\(E_g^o=0.78\), see Appendix 3)

$V_h$ versus $t$, $R$ versus $t$ and $t$ versus time

Figure 5.17: \(V_h\) versus \(t\), \(R\) versus \(t\) and \(t\) versus time

Example of linear best fit in the intrinsic region (high temperature)

Figure 5.18: Example of linear best fit in the intrinsic region (high temperature)

It’s possible to find example experimental data at the following link