Van der Pauw method

To make a measurement, a current is caused to flow along one edge of the sample (for instance, I₁₂) and the voltage across the opposite edge (in this case, V₃₄) is measured. From these two values, a resistance (for this example, ${\displaystyle R_{12,34}}$) can be found using Ohm's law:

${\displaystyle R_{12,34}={\frac {V_{34}}{I_{12}}}}$

In his paper, van der Pauw showed that the sheet resistance of samples with arbitrary shapes can be determined from two of these resistances - one measured along a vertical edge, such as ${\displaystyle R_{12,34}}$, and a corresponding one measured along a horizontal edge, such as ${\displaystyle R_{23,41}}$. The actual sheet resistance is related to these resistances by the van der Pauw formula

${\displaystyle e^{-\pi R_{12,34}/R_{s}}+e^{-\pi R_{23,41}/R_{s}}=1}$

The reciprocity theorem tells us that

${\displaystyle R_{AB,CD}=R_{CD,AB}}$

Therefore, it is possible to obtain a more precise value for the resistances ${\displaystyle R_{12,34}}$ and ${\displaystyle R_{23,41}}$ by making two additional measurements of their reciprocal values ${\displaystyle R_{34,12}}$ and ${\displaystyle R_{41,23}}$ and averaging the results.

We define

${\displaystyle R_{\text{vertical}}={\frac {R_{12,34}+R_{34,12}}{2}}}$

and

${\displaystyle R_{\text{horizontal}}={\frac {R_{23,41}+R_{41,23}}{2}}}$

Then, the van der Pauw formula becomes

${\displaystyle e^{-\pi R_{\text{vertical}}/R_{S}}+e^{-\pi R_{\text{horizontal}}/R_{S}}=1}$

A further improvement in the accuracy of the resistance values can be obtained by repeating the resistance measurements after switching polarities of both the current source and the voltage meter. Since this is still measuring the same portion of the sample, just in the opposite direction, the values of R_vertical and R_horizontal can still be calculated as the averages of the standard and reversed polarity measurements. The benefit of doing this is that any offset voltages, such as thermoelectric potentials due to the Seebeck effect, will be cancelled out.

Combining these methods with the reciprocal measurements from above leads to the formulas for the resistances being

${\displaystyle R_{\text{vertical}}={\frac {R_{12,34}+R_{34,12}+R_{21,43}+R_{43,21}}{4}}}$

and

${\displaystyle R_{\text{horizontal}}={\frac {R_{23,41}+R_{41,23}+R_{32,14}+R_{14,32}}{4}}}$

The van der Pauw formula takes the same form as in the previous section.

Both of the above procedures check the repeatability of the measurements. If any of the reversed polarity measurements don't agree to a sufficient degree of accuracy (usually within 3%) with the corresponding standard polarity measurement, then there is probably a source of error somewhere in the setup, which should be investigated before continuing. The same principle applies to the reciprocal measurements—they should agree to a sufficient degree before they are used in any calculations.

In general, the van der Pauw formula cannot be rearranged to give the sheet resistance R_S in terms of known functions. The most notable exception to this is when R_vertical = R = R_horizontal; in this scenario the sheet resistance is given by

${\displaystyle R_{S}={\frac {\pi R}{\ln 2}}}$

The quotient ${\displaystyle \pi /\ln 2}$ is known as the van der Pauw constant and has approximate value 4.53236. In most other scenarios, an iterative method is used to solve the van der Pauw formula numerically for R_S. Typically a formula is considered to fail the preconditions for Banach Fixed Point Theorem, so methods based on it do not work. Instead, nested intervals converge slowly but steadily. Recently, however, it has been shown that an appropriate reformulation of the van der Pauw problem (e.g., by introducing a second van der Pauw formula) makes it fully solvable by the Banach fixed point method.[7]

Alternatively, a Newton-Raphson method converges relatively quickly. To reduce the complexity of the notation, the following variables are introduced:

${\displaystyle s=e^{-\pi /{R_{s}}}}$

${\displaystyle R_{v}=R_{vertical}}$

${\displaystyle R_{h}=R_{horizontal}}$

Then the next approximation ${\displaystyle R_{s}^{+}}$ is calculated by

${\displaystyle R_{s}^{+}=R_{s}+R_{s}^{2}{\frac {1-s^{R_{v}}-s^{R_{h}}}{\pi (R_{v}s^{R_{v}}+R_{h}s^{R_{h}})}}}$

When a charged particle—such as an electron—is placed in a magnetic field, it experiences a Lorentz force proportional to the strength of the field and the velocity at which it is traveling through it. This force is strongest when the direction of motion is perpendicular to the direction of the magnetic field; in this case the force

${\displaystyle F_{L}=qvB\,\!}$

where ${\displaystyle q}$ is the charge on the particle in coulombs, ${\displaystyle v}$ the velocity it is traveling at (centimeters per second), and ${\displaystyle B}$ the strength of the magnetic field (Wb/cm²). Note that centimeters are often used to measure length in the semiconductor industry, which is why they are used here instead of the SI units of meters.

The Hall effect as it is used for the van der Pauw method.
(a) - a current flowing through a piece of semiconductor material
(b) - the electrons flowing due to the current
(c) - the electrons accumulating at one edge due to the magnetic field
(d) - the resulting electric field and Hall voltage ${\displaystyle V_{H}}$

When a current is applied to a piece of semiconducting material, this results in a steady flow of electrons through the material (as shown in parts (a) and (b) of the accompanying figure). The velocity the electrons are traveling at is (see electric current):

${\displaystyle v={\frac {I}{nAq}}}$

where ${\displaystyle n}$ is the electron density, ${\displaystyle A}$ is the cross-sectional area of the material and ${\displaystyle q}$ the elementary charge (1.602×10⁻¹⁹ coulombs).

If an external magnetic field is then applied perpendicular to the direction of current flow, then the resulting Lorentz force will cause the electrons to accumulate at one edge of the sample (see part (c) of the figure). Combining the above two equations, and noting that ${\displaystyle q}$ is the charge on an electron, results in a formula for the Lorentz force experienced by the electrons:

${\displaystyle F_{L}={\frac {IB}{nA}}}$

This accumulation will create an electric field across the material due to the uneven distribution of charge, as shown in part (d) of the figure. This in turn leads to a potential difference across the material, known as the Hall voltage ${\displaystyle V_{H}}$. The current, however, continues to only flow along the material, which indicates that the force on the electrons due to the electric field balances the Lorentz force. Since the force on an electron from an electric field ${\displaystyle \epsilon }$ is ${\displaystyle q\epsilon }$, we can say that the strength of the electric field is therefore

${\displaystyle \epsilon ={\frac {IB}{qnA}}}$

Finally, the magnitude of the Hall voltage is simply the strength of the electric field multiplied by the width of the material; that is,

${\displaystyle {\begin{aligned}V_{H}&=w\epsilon \\&={\frac {wIB}{qnA}}\\&={\frac {IB}{qnt}}\end{aligned}}}$

where ${\displaystyle t}$ is the thickness of the material. Since the sheet density ${\displaystyle n_{s}}$ is defined as the density of electrons multiplied by the thickness of the material, we can define the Hall voltage in terms of the sheet density:

${\displaystyle V_{H}={\frac {IB}{qn_{s}}}}$

Two sets of measurements need to be made: one with a magnetic field in the positive z-direction as shown above, and one with it in the negative z-direction. From here on in, the voltages recorded with a positive field will have a subscript P (for example, V_{13, P} = V_{3, P} - V_{1, P}) and those recorded with a negative field will have a subscript N (such as V_{13, N} = V_{3, N} - V_{1, N}). For all of the measurements, the magnitude of the injected current should be kept the same; the magnitude of the magnetic field needs to be the same in both directions also.

First of all with a positive magnetic field, the current I₂₄ is applied to the sample and the voltage V_{13, P} is recorded; note that the voltages can be positive or negative. This is then repeated for I₁₃ and V_{42, P}.

As before, we can take advantage of the reciprocity theorem to provide a check on the accuracy of these measurements. If we reverse the direction of the currents (i.e. apply the current I₄₂ and measure V_{31, P}, and repeat for I₃₁ and V_{24, P}), then V_{13, P} should be the same as V_{31, P} to within a suitably small degree of error. Similarly, V_{42, P} and V_{24, P} should agree.

Having completed the measurements, a negative magnetic field is applied in place of the positive one, and the above procedure is repeated to obtain the voltage measurements V_{13, N}, V_{42, N}, V_{31, N} and V_{24, N}.

Initially, the difference of the voltages for positive and negative magnetic fields is calculated:

V₁₃ = V_{13, P} − V_{13, N}
V₂₄ = V_{24, P} − V_{24, N}
V₃₁ = V_{31, P} − V_{31, N}
V₄₂ = V_{42, P} − V_{42, N}

The overall Hall voltage is then

${\displaystyle V_{H}={\frac {V_{13}+V_{24}+V_{31}+V_{42}}{8}}}$.

The polarity of this Hall voltage indicates the type of material the sample is made of; if it is positive, the material is P-type, and if it is negative, the material is N-type.

The formula given in the background can then be rearranged to show that the sheet density

${\displaystyle n_{s}={\frac {IB}{q|V_{H}|}}}$

Note that the strength of the magnetic field B needs to be in units of Wb/cm² if n_s is in cm⁻². For instance, if the strength is given in the commonly used units of teslas, it can be converted by multiplying it by 10⁻⁴.

The resistivity of a semiconductor material can be shown to be[8]

${\displaystyle \rho ={\frac {1}{q(n\mu _{n}+p\mu _{p})}}}$

where n and p are the concentration of electrons and holes in the material respectively, and μ_n and μ_p are the mobility of the electrons and holes respectively.

Generally, the material is sufficiently doped so that there is a difference of many orders-of-magnitude between the two concentrations, allowing this equation to be simplified to

${\displaystyle \rho ={\frac {1}{qn_{m}\mu _{m}}}}$

where n_m and μ_m are the doping level and mobility of the majority carrier respectively.

If we then note that the sheet resistance R_S is the resistivity divided by the thickness of the sample, and that the sheet density n_S is the doping level multiplied by the thickness, we can divide the equation through by the thickness to get

${\displaystyle R_{s}={\frac {1}{qn_{s}\mu _{m}}}}$

This can then be rearranged to give the majority carrier mobility in terms of the previously calculated sheet resistance and sheet density:

${\displaystyle \mu _{m}={\frac {1}{qn_{s}R_{s}}}}$