Hall measurement using the van der Pauw technique is a common characterization approach that does not require patterning of contacts. Measurements of the Hall voltage and electrical resistivity lead to the product of carrier mobility and carrier concentration (Hall coefficient) which can be decoupled through transport models. Based on the van der Paw method, we have developed an automated setup for Hall measurements from room temperature to ∼500 °C of semiconducting thin films of a wide resistivity range. The resistivity of the film and Hall coefficient is obtained from multiple current-voltage (I-V) measurements performed using a semiconductor parameter analyzer under applied constant “up,” zero, and “down” magnetic field generated with two neodymium permanent magnets. The use of slopes obtained from multiple I-Vs for the three magnetic field conditions offer improved accuracy. Samples are preferred in square shape geometry and can range from 2 mm to 25 mm side length. Example measurements of single-crystal silicon with known doping concentration show the accuracy and reliability of the measurement.

## I. INTRODUCTION

High temperature material properties are important for a variety of applications including high-temperature electronics and thermoelectric devices. In semiconducting materials, besides the overall electrical resistivity, carrier mobility (*μ*) and carrier concentration (*n*) are crucial parameters for devices and can be obtained from Hall measurements. The Hall effect, discovered in 1879 by Edwin Hall,^{1} refers to the perpendicular voltage developed under a magnetic field and electrical current, while the van der Pauw method,^{2} a widely used technique, provides a way to obtain the Hall coefficient for an approximately square, flat sample by applying a magnetic field normal to the surface of the sample and a current between diagonally opposite corners. The voltage across the two other corners of the film with and without magnetic field application leads to the Hall coefficient which is the product of carrier mobility *μ* and concentration *n*. Hall setups have been developed to perform measurements under different conditions such as high pressure,^{3,4} different gas environments for gas sensing purposes,^{5} and on few tens of micrometer sized samples.^{6} Commercial Hall measurement setups use bulky electromagnets with powerful current source and cooling systems to generate the large magnetic field which required ∼1 T. While electromagnets have the advantage of providing a variable magnetic field, they also result in non zero offsets when reversing the magnetic field which requires additional calibrations.^{7} Magnetic field uniformity requires large electromagnets (12 in. coil diameter for 60 mm side length square samples^{8}). The maximum temperature commercial setups such as the MMR Variable Temperature Hall System can reach is ∼460 °C and the sample holders are designed for particular sample dimensions, typically less than 1 cm^{2} (MMR Technologies, Inc.). As an alternative to electromagnets, rare earth permanent magnets are good elements that we considered and use in our setup with the magnetic field perpendicular to the surface of the film. This direct trihedral configuration for magnetic field, current, and Hall voltage was proposed in early works for semiconductor surface Hall measurements^{9} and has been applied to bulk,^{3} thin film,^{6} and bilayer structures.^{10}

## II. HALL MEASUREMENTS

Figure 1 shows the schematics of the Hall effect, the transverse potential difference developed in a current carrying conductor immersed in a magnetic field B perpendicular to the direction of current flow.^{11} This phenomenon is used to characterize charge carrier mobility and concentration in semiconductor materials. An electron with velocity *v* in the conductor travels in the opposite direction of the current *I* and experiences the Lorentz force

where *q* is the elementary charge, *ν* is the carrier velocity, and *B* is the magnetic field.

Charge carriers will accumulate in one side of the sample establishing an electric field that opposes the Lorentz field. Charges will continue to migrate until equilibrium is reached when the two fields cancel each other, $E\u2192=\u2212v\u2192\xd7B\u2192$ or *E* = *vB* for a magnetic field normal to the current direction. The potential difference between the two sides of the conductor, Hall voltage, is hence given by

where *w* is the width of the conductor. For a uniform cross section sample, the current in turn is given by

where (*nv*) is the product of the effective carrier concentration and velocity, and *A* is the sample cross-section area (*A* = *wt* where *t* is the thickness of the conductor). The Hall voltage *V _{H}* is then given by

From the measured hall voltage *V _{H}* the effective free carrier concentration in the material can be obtained as

Depending on the type of charge carriers and their relative contributions, and assuming a given transport model, the carrier mobility can then be obtained. For example, for electronic-only conduction (and single electron mass), the electron mobility can be found by

In general, however, certain transport assumptions and simplifications are required to decouple *n* and *μ* in order to obtain approximate carrier concentrations and mobilities.

## III. SETUP DESCRIPTION

The setup consists of three main parts: The sample holder with heating elements, the movable stage with the magnetic frame, and the electronic control and data acquisition components.

The sample holder is a brass chuck of 6 cm × 1 cm × 22 cm lying on a glass-ceramic base of 10 cm × 0.4 cm × 28 cm (Fig. 1). Two cartridge heaters of 300 W each are inserted in one end of the chuck while the other end is designed to support samples. Four electrical probes are attached to the top surface of the ceramic base and connected to coaxial cables from the bottom of the base. The contacts are articulated in the middle so they can be positioned to measure square samples between 1 mm and 25 mm side-length. Two thermocouples are clamped to the chuck surface, one between the heaters for temperature control and the other close to the sample (1–2 mm) to record the sample temperature. When the temperature near the heaters is stable around the target, the system waits for 10 min during which the temperature of the sample asymptotically approaches a stable value and the measurement is then started. The thermocouple near the sample is not used for temperature regulation because of the long time response of the system and to avoid rippling of the sample temperature. This is important for characterization of materials that undergo irreversible changes with temperature, such as phase-change compounds.

When the measurement at a given temperature is complete, the target temperature is increased to the next step. The system repeats the stabilization tasks with measurements until the final temperature cycle. The chuck is enclosed by an aluminum cover, sealed on top of the ceramic base, to prevent light straying and to form a chamber that is filled with nitrogen to reduce oxidation of the samples and setup components at high temperatures. An opening along the center of the chuck drives nitrogen from the heaters side to the sample side inside the chamber. The movable magnetic frame is controlled with two stepper motors to apply an “up,” “down,” or no field to the samples (Fig. 2). Two N42 grade neodymium magnets, 4 in. × 2 in. × 1 in. thick rare earth NdFeB blocks that can provide a maximum field of 1.32 T each, with an operating temperature up to 80 °C, are fixed on the horizontal walls of an iron rectangular frame that measures 9 cm × 15 cm inside, 10.5 cm × 16.5 cm outside, and 6.3 cm in depth. The gap between the two magnets is about 3 cm so the magnetic frame can be freely moved to surround the sample holder. The magnetic frame is fixed on an aluminum movable stage from the 2 vertical sides through an axes in two bearings connected to a stepper motor to flip the magnetic field direction between “up” and “down,” normal to the surface of the sample. The magnetic frame is initially positioned at a 90° angle with the sample holder then 100 steps of the stepping motor (with 1.8° /step) will apply the magnetic field in the opposite direction. For the electronic control of the system, the setup uses two Arduino Mega 2560 cards^{12} connected through USB to a computer. One card is used for analog to digital conversion to measure temperature from the two thermocouples as described before, and to generate switching pulses for the relay card for temperature control and stabilization. The second card is used to drive the stepper motors to apply or remove the magnetic field. The two cards, as well as an HP 4145B semiconductor parameter analyzer, are simultaneously controlled with the measurement computer through a LabVIEW interface. Once the temperature is stabilized within a given range, the stepper motors move the magnetic frame to the sample and the parameter analyzer performs an I-V and transfers the data to the computer.

To avoid the influence of minority-carrier injection on the measurements, the current passing through the sample is maintained low enough to get linear I-Vs as discussed in Ref. 13. Bipolar thermomagnetic effect, or Ettingshausen effect,^{14} may also introduce an error in Hall coefficient measurements for the case of intrinsic semiconductors or semimetals. This effect refers to the potential that is created in addition to the Hall potential if there is a temperature difference between the voltage measurement points (Δ*V* = − *S* ⋅ Δ*T*, where *S* is the Seebeck coefficient). It is usually very small and considered a minor effect^{15,16} especially in cases of expected uniform thermal profile as in our case where the sample sits on a large metal chuck. The flow of nitrogen through the chuck into the sample chamber helps create a uniform temperature environment (Fig. 3). Additional thermocouples may be installed around the sample to account for this contribution in materials or setups where this Seebeck voltage may be significant.

The LabVIEW code analyzes and plots the data in real time while controlling the system. As an example of the acquired data, the temperature on the surface of the chuck as a function of the target temperature and time is shown in Figure 4.

The temperature between the heaters follows the target closely (with rippling of ±4 °C at 700 °C) while the temperature close to the sample is significantly lower (∼315 °C and 510 °C at chuck temperatures of 400 °C and 700 °C). Because of the thermal inertia of the large chuck, the temperature of the sample asymptotically approaches a lower value without rippling as mentioned above. The temperature difference between the surface of the sample and the surface of the chuck is expected to be insignificant based on finite element simulations.^{17} After some stabilization time, ∼10 min, a new temperature target is set and regulation starts. The increase in sample temperature during the time it takes to complete the measurement is ∼1 °C at 400 °C and less than 2 at 500 °C which is acceptable since it is in the order of the tolerance of the thermocouples used.

## IV. MEASUREMENTS

Electrical resistivity can be measured on any arbitrary shape sample using the four points van der Pauw method.^{2,18} The sheet resistance of the sample, and hence the resistivity, are calculated using the voltage measured between two adjacent contacts while passing a current between the two opposite adjacent contacts. Results are more precise as the sample shape is closer to square. In our measurement, we use the configuration illustrated in Figure 5.

The resistivity of the sample is given by^{19}

where *R _{S}* is the sheet resistance of the film,

*t*is the thickness,

*R*

_{12,34}=

*V*

_{34}/

*I*

_{12},

*R*

_{14,32}=

*V*

_{32}/

*I*

_{14}, and

*F*is a geometry correction factor that is given by

with *R _{r}* =

*R*

_{12,34}/

*R*

_{14,32}a measure of squareness of the sample. If

*R*is less than one, its reciprocal should be used instead. The variation of

_{r}*F*with the ratio

*R*is represented in Figure 6. If the sample is square,

_{r}*R*= 1, and

_{r}*F*= 1 and Eq. (7) can be reduced to

The use of semiconductor parameter analyzer allows us to keep the same contacts probed for both resistivity and Hall measurements by switching variables between the SMUs. To determine the geometry correction factor F, we sweep the voltage in the contact 1 and use 2 as a ground to measure *R*_{12,34}, then use 4 as ground to measure *R*_{14,32}. The parameters *R*_{12,34} and *R*_{14,32} correspond to the slopes of the curves *V*_{3}–*V*_{4} versus *I*_{1} and *V*_{3}–*V*_{2} versus *I*_{1}, respectively.

Carrier concentration and Hall mobility are calculated from another set of measurements involving application of magnetic field perpendicular to the surface of the sample as pictured in Figure 7.

The magnetic field intensity *B* is measured on the surface of a sample using F. W. Bell Gauss/Tesla meter model 4048. The accuracy of the Gauss-meter probe is 0.01 mT. *V _{H}*/

*I*is measured using the configuration represented in Figure 7 as half of the difference between the slopes of I-V curves taken with the

*B*oriented in the two opposite directions perpendicular to the sample,

with *R*_{13,24} = *V*_{24}/*I*_{13} and the sign +/ − for *B* determines whether the magnetic field is oriented toward the sample or in the opposite direction. Applying the magnetic field in the two opposite directions perpendicular to the sample allows to verify the influence of *B* on the current flowing in the sample and eliminate any systematic non-zero offset from the voltage measurement. The sign of *V _{H}*/

*I*according to the orientation of the magnetic field will determine whether the majority carrier is holes or electrons (p-type or n-type) as schemed in Figure 1.

^{13}The main parameters

*R*

_{12,34},

*R*

_{14,32}, $R13,24B+$, and $R13,24B\u2212$ used in the calculation are obtained from the slopes of the I-V curves performed with the parameter analyzer.

Carrier concentration is calculated using Equation (5). Since mobility and carrier concentration are related by

the mobility of carriers within the film is given by

and carrier concentration is

Both Equations (12) and (13) are involving the term *V _{H}*/

*BI*which can be calculated from the slopes of

*R*

_{13,24}vs.

*B*for more accuracy.

*R*=

_{H}*V*/

_{H}t*BI*in cm

^{3}/Coulomb is called the Hall coefficient.

Figure 8 summarizes the measurement procedure and control program. The initial temperature, the temperature step, and the number of tests performed at each temperature step, n, are set initially in the program. After the stabilization period (10 min in our case, enough to reach a stable state with a maximum increase in temperature of less than 2 °C during the measurements) the current-voltage measurements with and without magnetic field are performed using the parameter analyzer. In the measurements presented here, 3 I-Vs for each configuration and magnetic field condition were performed at each temperature step (total of 15 I-Vs at each temperature to obtain the different *R* parameters). In our measurements the total time to complete one temperature step, which includes moving and rotating the magnetic frame, was ∼15 min. After the measurements are complete, the next target temperature is set and the cycle is repeated. Figure 9 shows the variation of the coefficient *V _{H}*/

*BI*with temperature measured in multiple heating cycles on an

*n*-type (phosphorous) low doped single crystal silicon sample, approximately shaped as a square with 15 mm side length.

The *ρ*-T measurement results are shown in Fig. 10. Standard error on the slopes obtained from the linear fit of the I-V measurements and the average of the n measurements at each target temperature are used as weights to calculate the errors on *ρ* and *V _{H}*/

*BI*by the method described by York

*et al.*

^{20}This method allows accounting for measurement errors on

*B*in the x-axes besides the measurement errors on

*V*/

_{H}*I*in the y-axes (inset in Fig. 9) which is not available in data analysis software like Origin Lab (OriginLab Corp., Northampton, MA). The drop in the electrical resistivity around 300 °C is due to the onset of significant thermal generation of minority carriers (holes) resulting in bipolar transport that must be taken into account when solving for carrier concentration and mobility.

^{21–23}With significant minority carrier contribution, and assuming one effective mass for each carrier, the electrical resistivity is given by

where *μ _{e}* and

*μ*are electron and hole mobility.

_{h}If the electron mobility is approximately proportional to the hole mobility which is the case for silicon,^{19} then

The hole coefficient can be written as^{21}

where *c* is the ratio of electron mobility to hole mobility. To solve the equations for *μ*, *n*, *and**p*, it is necessary to add the charge neutrality condition, *p* − *n* + *N _{d}* = 0 where

*N*is the donor concentration. For low-doped single-crystal silicon,

_{d}*c*= 3.0 at high temperatures, and

*n*,

*p*, and

*μ*can be found from the resistivity and Hall coefficient by solving Equation (16).

The mobility and concentration of electrons and holes as a function of temperature, calculated using *V _{H}*/

*BI*and

*ρ*, are shown in Figure 11. Measurement results obtained for our sample at room temperature (

*n*∼ 8.5 × 10

^{15}cm

^{−3},

*μ*∼ 1100 cm

^{2}/V-s, and

*ρ*∼ 0.688 Ω cm) were in concordance with empirical relation between

*μ*and n at room temperature proposed by Guido

*et al.*

^{24}for phosphorus doped silicon (

*n*∼ 8.5 × 10

^{15}cm

^{−3},

*μ*∼ 1205 cm

^{2}/V-s) and with the experimental data gathered by Irvin

^{25}for the relation between

*ρ*and

*n*on bulk silicon (

*n*∼ 8.5 × 10

^{15}cm

^{−3},

*ρ*∼ 0.674 Ω cm). Standard error on the slope from the linear fit of the I-V curves is used for error calculation instead of the error from instrument (parameter analyzer) which is very small in comparison (∼0.5% instrument error compared to ∼2% standard error on the slope of I-Vs). Standard error on the slopes is used along with 0.02 T systematic error on the magnetic field to calculate the final error on μ and n as discussed previously.

In summary, we have presented a high temperature setup for Hall effect measurements on thin film samples. The electrical resistivity and Hall coefficient are simultaneously extracted from multiple current-voltage measurements performed by a semiconductor parameter analyzer. The fully automated setup uses rare earth permanent magnets for constant magnetic field generation and can reach up to ∼500 °C sample temperature, limited by the power of the heaters, heat conduction along the chuck, and oxidation of the electrical contacts. Articulated electrical contacts allow measurements on different size samples, from ∼2 mm to 25 mm side length. Multiple measurements on single crystal low doped silicon sample show good agreement with published data for room temperature.

## Acknowledgments

This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No. DE-FG02-10ER46774. Lhacene Adnane was also supported by the U.S. Department of Education through a GAANN Fellowship. The authors thank Sadid Muneer for valuable discussions and insight related to data interpretation and analysis.

## REFERENCES

_{3}Ceramics