Chapter 2 Experimental Methods
2.3 Hall and Resistivity Measurements
These properties are measured along the in-plane direction of a sheet sample in dynamic vacuum.
The homebuilt setup has a boron nitride (BN) ceramic sample holder placed in a slit vacuum chamber set in between two poles of a 2T iron-core electromagnet.
The BN sample holder (Figure 2.2) has four rectangular holes on its two sides. Each holds a rectangular cartridge heater that is special designed with heating zone only in the front half. About one third of each heater is exposed to keep the temperature at the leads cool enough (overheating this part leads to insulation failure and corrosion of exposed heater wire that leads to frequent heater failure way below its power rating). Each heater is rated 120V 200W (60 Ω), but are connected in parallel to a DC current source of 60V, 10A maximum output. The sample temperature is measured by two type C thermocouples that are placed within the sample holder underneath each sample.
When operating the holder is placed in a 0.8 inch (2 cm) wide stainless-steel slit with its wall attached to the water-cooled magnetic pole, therefore a significant heat loss through radiation can be expected. This would cause excessive power consumption, which means shorter heater life.
Besides, it would make the real temperature of the sample surface (or the average across its thickness) much lower than the value recorded by the thermocouple. To reduce the radiation loss, the sample holder is wrapped with two layers of woven glass fiber as radiation shield (the part on top of samples are removable). With radiation shield the temperature of samples are very close to (<10 °C at 600 °C) that around the thermocouples. The system works at only about 90 W to get to 600 °C (the maximum power output is around 250 W). At higher temperature (>700 °C) the screws
and poles used to press leads against the sample begin to fuse together and thermal expansion causes lost of the pressure for good electric contact. 700 °C should be regarded as the maximum operating temperature.
Figure 2.2. Sample holder of the Hall effect and resistivity measurement setup, a) schematic of sample and measurement geometry, b) c) picture of sample holder without and with insulation.
Measurements are based on (DC) Van der Pauw method, and each measurement is carried out in a quasi-steady-state manner: The temperature continue to ramp with a set rate once measurement is started till the set maximum then cool with the same rate, meanwhile the resistivity and Hall effect test are carried out repeatedly. Before performing each test, the contact resistance is checked, by grounding 1 probe and applying same positive voltage to the other three and measure the resistance.
This is done for all 8 probes (when there are two samples). Next, the temperature is measured, then a DC current (set by user, 100 mA by default) is passed through probe 1 and 2, while voltage is measured between 3 and 4, the resistance R12-34 is the average of 8 measurements. Similarly R23-41 is also obtained then the sheet resistance R is determined by solving numerically the equation:
e!!R12!34/R+e!!R23!41/R =1 Equation 2.1
The resistivity ρ is obtained by ρ = Rd, d being the thickness of the sample. After ρ is determined for both samples, the temperature is measured again and the average is recorded as the temperature for resistivity. Measuring ρ for two samples takes about 2 minutes. For the Hall effect test, the DC current is pass through probe 1 and 3 with the presence of magnetic field of 2T and measures the voltage between 2 and 4, then analogously for 2-4 and 1-3, after 8 measurements for each the magnetic field flips its direction and the same measurement is repeated. Two averaged resistance are calculated with:
RH,1=(V24(B)!V24(!B))d/I132B Equation 2.2
RH,2=(V13(B)!V13(!B))d/I242B Equation 2.3
21 The Hall coefficient is the average RH = (RH,1 + RH,2)/2. Measuring RH for two samples takes about 16 minutes. The current program will not record temperature for the Hall effect test; but instead uses the temperature for the just finished resistivity measurements. Within the 16 minute span the temperature would have increased 25 °C. Nonetheless, for normal semiconductors RH has only weak temperature dependence so the error due to temperature drift would well be within uncertainty of measurement itself.
Good Van der Pauw measurement requires the sample being regular shaped (not necessarily round or squre). Each probe should be placed on the edge (not anywhere inside) of the sample with contact area as small as possible, and be equally separated from its neighbors. Preferably the 4 probes should form a perfect cross. If the sample is not a full-disk, the probes should be placed on the (sharp) corners rather than edges. Good Ohmic contact (linear I-V curve) marked by low contact resistance is essential for reliable results. This value depends on the specific material system (how their chemical potential align with that of molybdenum) and varies from 1 to 2 for heavily doped samples to high tens for undoped samples. Readings from 4 probes should be comparable and an abnormally large contact resistance usually means loss contact, inhomogeneity or cracked sample.
The following should also be kept in mind to minimize error: first, the sample should be as thin as possible, given it is mechanically robust. Thinner samples produce larger Hall voltage, which means better signal-to-noise ratio. Second, the thickness should be as uniform as possible, otherwise taking a good average for thickness is helpful. Even for a sample with unparalleled surfaces (~ 10%
difference) as long as the right averaged thickness is used, could give very close result compared with the same sample when well polished (< 2% difference). Third, the size of the sheet should be much larger than its thickness. Also the ratio of contact area over sample thickness should also be small. Last, avoid using alumina spacer underneath thin samples for better pressure from the leads as it increases the interfacial thermal resistance that in turn lead to a big temperature difference between sample and thermocouple.
A BN spray is available for high temperature measurements, which suppresses the evaporation of elements under dynamic vacuum from the free surface. A BN coated sample after measurement would have both surfaces remaining gray and metallic look, whereas a free surface after the same run becomes dusty black.