The van der Pauw Method is a technique commonly used to measure the
resistivity
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
and the
Hall coefficient
The Hall effect is the production of a voltage difference (the Hall voltage) across an electrical conductor that is transverse to an electric current in the conductor and to an applied magnetic field perpendicular to the current. It was discov ...
of a sample. Its power lies in its ability to accurately measure the properties of a sample of any arbitrary shape, as long as the sample is approximately two-dimensional (i.e. it is much thinner than it is wide), solid (no holes), and the
electrode
An electrode is an electrical conductor used to make contact with a nonmetallic part of a circuit (e.g. a semiconductor, an electrolyte, a vacuum or air). Electrodes are essential parts of batteries that can consist of a variety of materials de ...
s are placed on its
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pract ...
. The van der Pauw method employs a four-point probe placed around the perimeter of the sample, in contrast to the
linear four point probe: this allows the van der Pauw method to provide an average resistivity of the sample, whereas a linear array provides the resistivity in the sensing direction. This difference becomes important for anisotropic materials, which can be properly measured using the
Montgomery Method
Montgomery refers to:
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For people with the name Montgomery, see Montgomery (name)
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* Montgomery Square, Brussels
* Montgomery metro station, Brussels
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* Montgomery (town), British India, former name of Sahiwal, Punjab ...
, an extension of the van der Pauw Method (see, for instance, reference).
From the measurements made, the following properties of the material can be calculated:
* The
resistivity
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
of the material
* The
doping type (i.e. whether it is a
P-type or
N-type material)
* The sheet carrier density of the
majority carrier
In physics, a charge carrier is a particle or quasiparticle that is free to move, carrying an electric charge, especially the particles that carry electric charges in electrical conductors. Examples are electrons, ions and electron hole, holes. Th ...
(the number of majority carriers per unit area). From this the charge density and doping level can be found
* The
mobility
Mobility may refer to:
Social sciences and humanities
* Economic mobility, ability of individuals or families to improve their economic status
* Geographic mobility, the measure of how populations and goods move over time
* Mobilities, a contemp ...
of the majority carrier
The method was first propounded by Leo J. van der Pauw in 1958.
Conditions
There are five conditions that must be satisfied to use this technique:
1. The sample must have a flat shape of uniform thickness
2. The sample must not have any isolated holes
3. The sample must be
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
and
isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
4. All four contacts must be located at the edges of the sample
5. The area of contact of any individual contact should be at least an
order of magnitude
An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic dis ...
smaller than the area of the entire sample.
The second condition can be weakened. The van der Pauw technique can also be applied to samples with one hole.
Sample preparation
In order to use the van der Pauw method, the sample thickness must be much less than the width and length of the sample. In order to reduce errors in the calculations, it is preferable that the sample be symmetrical. There must also be no isolated holes within the sample.
The measurements require that four
ohmic contact
An ohmic contact is a non-rectifying electrical junction: a junction between two conductors that has a linear current–voltage (I–V) curve as with Ohm's law. Low-resistance ohmic contacts are used to allow charge to flow easily in both directi ...
s be placed on the sample. Certain conditions for their placement need to be met:
* They must be as small as possible; any errors given by their non-zero size will be of the order ''D/L'', where ''D'' is the average diameter of the contact and ''L'' is the distance between the contacts.
* They must be as close as possible to the boundary of the sample.
In addition to this, any leads from the contacts should be constructed from the same batch of wire to minimise
thermoelectric
The thermoelectric effect is the direct conversion of temperature differences to electric voltage and vice versa via a thermocouple. A thermoelectric device creates a voltage when there is a different temperature on each side. Conversely, when ...
effects. For the same reason, all four contacts should be of the same material.
Measurement definitions
* The contacts are numbered from 1 to 4 in a counter-clockwise order, beginning at the top-left contact.
* The
current
Currents, Current or The Current may refer to:
Science and technology
* Current (fluid), the flow of a liquid or a gas
** Air current, a flow of air
** Ocean current, a current in the ocean
*** Rip current, a kind of water current
** Current (stre ...
''I''
12 is a positive DC current injected into contact ''1'' and taken out of contact ''2'', and is measured in
ampere
The ampere (, ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to elect ...
s (A).
* The
voltage
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
''V''
34 is a DC voltage measured between contacts ''3'' and ''4'' (''i.e.'' ''V''
4 - ''V''
3) with no externally applied magnetic field, measured in
volt
The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827).
Defi ...
s (V).
* The
resistivity
Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
''ρ'' is measured in
ohms⋅
metre
The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
s (Ω⋅m).
* The thickness of the sample ''t'' is measured in
metre
The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
s (m).
* The
sheet resistance
Sheet resistance, is a measure of resistance of thin films that are uniform in thickness. It is commonly used to characterize materials made by semiconductor doping, metal deposition, resistive paste printing, and glass coating. Examples of thes ...
''R
S'' is measured in
ohms per square (Ω/sq or
).
Resistivity measurements
The average resistivity of a sample is given by ''ρ = R
S⋅t'', where the sheet resistance ''R
S'' is determined as follows. For an anisotropic material, the individual resistivity components, e.g. ''ρ
x'' or ''ρ
y'', can be calculated using the
Montgomery method
Montgomery refers to:
People
For people with the name Montgomery, see Montgomery (name)
Places Belgium
* Montgomery Square, Brussels
* Montgomery metro station, Brussels
Pakistan
* Montgomery (town), British India, former name of Sahiwal, Punjab ...
.
Basic measurements
To make a measurement, a current is caused to flow along one edge of the sample (for instance, ''I
12'') and the voltage across the opposite edge (in this case, ''V''
34) is measured. From these two values, a resistance (for this example,
) can be found using
Ohm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
:
:
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
, and a corresponding one measured along a horizontal edge, such as
. The actual sheet resistance is related to these resistances by the van der Pauw formula
:
Reciprocal measurements
The
reciprocity theore
tells us that
:
Therefore, it is possible to obtain a more precise value for the resistances
and
by making two additional measurements of their reciprocal values
and
and averaging the results.
We define
:
and
:
Then, the van der Pauw formula becomes
:
Reversed polarity measurements
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
The thermoelectric effect is the direct conversion of temperature differences to electric voltage and vice versa via a thermocouple. A thermoelectric device creates a voltage when there is a different temperature on each side. Conversely, when ...
, will be cancelled out.
Combining these methods with the reciprocal measurements from above leads to the formulas for the resistances being
:
and
:
The van der Pauw formula takes the same form as in the previous section.
Measurement accuracy
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.
Calculating sheet resistance
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
:
The quotient
is known as the van der Pauw constant and has approximate value 4.53236. In most other scenarios, an
iterative method
In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived fr ...
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
In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certa ...
, so methods based on it do not work. Instead,
nested intervals
In mathematics, a sequence of nested intervals can be intuitively understood as an ordered collection of intervals I_n on the real number line with natural numbers n=1,2,3,\dots as an index. In order for a sequence of intervals to be considered ne ...
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.
Alternatively, a Newton-Raphson method converges relatively fast. To reduce the complexity of the notation, the following variables are introduced:
:
:
:
Then the next approximation
is calculated by
:
Hall measurements
Background
When a charged particle—such as an electron—is placed in a
magnetic field
A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
, it experiences a
Lorentz force
In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
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
:
where
is the charge on the particle in
coulomb
The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI).
In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ...
s,
the velocity it is traveling at (centimeters per
second
The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds ...
), and
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
The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
of meters.
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
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The moving pa ...
):
:
where
is the electron density,
is the cross-sectional area of the material and
the
elementary charge
The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 . This elementary charge is a fundame ...
(1.602×10
−19 coulomb
The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI).
In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ...
s).
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
is the charge on an electron, results in a formula for the Lorentz force experienced by the electrons:
:
This accumulation will create an
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
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
Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
across the material, known as the Hall voltage
. 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
is
, we can say that the strength of the electric field is therefore
:
Finally, the magnitude of the Hall voltage is simply the strength of the electric field multiplied by the width of the material; that is,
:
where
is the thickness of the material. Since the sheet density
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:
:
Making the measurements
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 ''I
24'' 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 ''I
13'' 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 ''I
42'' and measure ''V
31, P'', and repeat for ''I
31'' 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''.
Calculations
First of all, the difference of the voltages for positives and negative magnetic fields needs to be worked out:
''V''
13 = ''V''
13, P − ''V''
13, N
''V''
24 = ''V''
24, P − ''V''
24, N
''V''
31 = ''V''
31, P − ''V''
31, N
''V''
42 = ''V''
42, P − ''V''
42, N
The overall Hall voltage is then
:
.
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
:
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.
Other calculations
Mobility
The resistivity of a semiconductor material can be shown to be
:
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 many orders-of-magnitude difference between the two concentrations, and so this equation can be simplified to
:
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
:
This can then be rearranged to give the majority carrier mobility in terms of the previously calculated sheet resistance and sheet density:
:
Footnotes
References
*
*
*
Measuring Electrical Conductivity and Resistivity with the van der Pauw Technique
{{DEFAULTSORT:Van Der Pauw Method
Electrical engineering
Hall effect