Van der Pauw method
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The van der Pauw Method is a commonly ud technique to measure the sheet resistance of a material. The van der Pauw method is often ud to measure the Hall effect, which characterizes a sample of miconductor material and can be successfully completed with a current source, voltmeter, and a magnet. Products are available to automatically perform this procedure; however they typically offer the ability to do so at a range of temperatures. This means that they need accurate cooling and heating systems, and thus are expensive.
From the measurements made, the following properties of the material can be calculated: The sheet resistance, from which the resistivity can be inferred for a sample of a given
thickness.
The doping type (i.e. if it is a P-type or N-type) material.
The sheet carrier density of the majority carrier (the number of majority carriers per unit area).
From this, the density of the miconductor, often known as the doping level, can be found for
a sample with a given thickness.
The mobility of the majority carrier.
The method was first propounded by Leo J. van der Pauw in 1958 .[1]
Sample preparation
In order to u 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 is symmetrical. There must also be no isolated holes within the sample.
The measurements require that four ohmic contacts be placed on
the sample. Certain conditions for their placement need to be
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met:
They must be on the boundary of the sample (or as clo to it as possible).
They must be infinitely small. Practically, they must be as
small as possible; any errors given by their non-zero size
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of the contact and L is the distance between the contacts.In addition to this, any leads from the contacts should be
constructed from the same batch of wire to minimi thermoelectric 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-clockwi order, beginning at the top-left contact.
The current I 12 is a positive DC current injected into contact 1 and taken out of contact 2, and is measured in amperes (A).
The voltage V 34 is a DC voltage measured between contacts 3 and 4 with no externally applied magnetic field, measured in volts (V).
The sheet resistance R S is measured in ohms (Ω).
Resistivity measurements
Basic measurements
To make a measurement, a current is caud to flow along one edge of the sample (for instance, I 12)不见天
and the voltage across the opposite edge (in this ca, V 34) is measured. From the two values, a
resistance (for this example, R 12,34
) can be found using Ohm's law:
In his paper, van der Pauw discovered that the sheet resistance of samples with arbitrary shape can
be determined from two of the resistances - one measured along a vertical edge, such as R 12,34,
and a corresponding one measured along a horizontal edge, such as R 23,41. The actual sheet
resistance is related to the resistances by the van der Pauw formula
Reciprocal measurements
The reciprocity theorem [1] (www.du.edu/~jcalvert/tech/reciproc.htm) tells us that R AB,CD = R CD,AB
Therefore, it is possible to obtain a more preci value for the resistances R12,34 and R23,41 by making two additional measurements of their reciprocal values R34,12 and R41,23 and averaging the results.
洗我的衣服用英语怎么说We define
and
Then, the van der Pauw formula becomes
Reverd 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 reverd polarity measurements. The benefit of doing this is that any offt voltages, such as thermoelectric potentials due to the Seebeck effect, will be cancelled out.
Combining the 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 ction.
Measurement accuracy
Both of the above procedures check the repeatability of the measurements. If any of the reverd
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 tup, which should be investigated before continuing. The same principle applies to the reciprocal measurements—they should agree to a sufficient degree before they are ud 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
In most other scenarios, an iterative method is ud to solve the van der Pauw formula numerically for R S. Unfortunately, the formula doesn't fulfill the preconditions for the Banach fixed point theorem, thus methods bad on it don't work. Instead, nested intervals converge slowly but steadily.
Hall measurements
Background
Main article: Hall effect
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 ca the force
where Q is the charge on the particle in coulombs, v the velocity it is traveling at (centimeters per cond), and B the strength of the magnetic field (Wb/cm²). Note that centimeters are often ud to measure length in the miconductor industry, which is why they are ud here instead of the SI units of meters.
When a current is applied to a piece of
miconducting 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 (e electric current):
where n is the electron density, A is the cross-
ctional area of the material and q the elementary
charge (1.602×10-19 coulombs).
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If an external magnetic field is then applied
perpendicular to the direction of current flow, then the
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resulting Lorentz force will cau the electrons to
accumulate at one edge of the sample (e part (c) of
the figure). Combining the above two equations, and
noting that q 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 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 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 ε is q ε, 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 d is the depth of the material. Since the sheet density n s is defined as the density of electrons multiplied by the depth of the material, we can define the Hall voltage in terms of the sheet density:
Making the measurements
草莓种子怎么种Two ts 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, V13, P) and tho recorded with a negative field will have a subscript N (such as V13, 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 V13, P is recorded; note that the voltages can be positive or negative. This is then repeated for I13 and
V42, P.
As before, we can take advantage of the reciprocity theorem to provide a check on the accuracy of the measurements. If we rever the direction of the currents (i.e. apply the current I42 and measure V31, P, and repeat for I31 and V24, P), then V13, P should be the same as V31, P to within a suitably small degree of error. Similarly, V42, P and V24, P should agree.
Having completed the measurements, a negative magnetic field is applied in place of the positive on
e, and the above procedure is repeated to obtain the voltage measurements V13, N, V42, N, V31, N and V24, N.
Calculations
First of all, the difference of the voltages for positive and negative magnetic fields needs to be worked out:
V13 = V13, P − V13, N
V24 = V24, P − V24, N
V31 = V31, P − V31, N
V42 = V42, P − V42, N
蔡雅健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, t
he 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
