Van der Pauw measurement method

The Van der Pauw measurement method is used to determine the electrical sheet resistance and the Hall coefficient of thin, homogeneous layers of any shape. In the semiconductor industry, measurements of the specific resistance and the Hall coefficient play an important role, since these two quantities can be used to determine the charge carrier concentration and mobility.

The measurement method was first described by Leo J. van der Pauw in 1958.

Measurement of specific resistance and sheet resistance

Determination of the resistance ${\displaystyle R_{\mathrm {AB,CD} }}$

The measuring structure consists of an area of ​​any shape without holes, which is contacted on its edge via 4 points (A – D). At this structure is the resistance

${\displaystyle R_{\mathrm {AB,CD} }={{U_{\mathrm {CD} }} \over {I_{\mathrm {AB} }}}}$ (see picture)

measured by impressing a current between contacts A and B and measuring the voltage drop between contacts C and D. The resistance is measured in a corresponding manner after the contacts have been interchanged cyclically. ${\displaystyle R_{BC,DA}}$

${\displaystyle R_{\mathrm {BC,DA} }={{U_{\mathrm {DA} }} \over {I_{\mathrm {BC} }}}}$

Using methods of conformal mapping , van der Pauw was able to show that the sheet resistance can be calculated from these two resistance values ​​and that neither the special shape of the structure nor the position of the contacts is included in this calculation. The following dependency results:

Geometry factor f for Van der Pauw method

${\displaystyle \exp {\left(-{{\pi d} \over \rho }\cdot R_{\mathrm {AB,CD} }\right)}+\exp {\left(-{{\pi d} \over \rho }\cdot R_{\mathrm {BC,DA} }\right)}=1}$

with = specific resistance and = layer thickness. ${\displaystyle \rho }$ ${\displaystyle d}$

If both the layer thickness and the resistances R _{AB, CD} and R _{BC, DA are} known, an equation with the unknown is obtained . For a structure of any shape, however, the specific resistance cannot be expressed using a closed formula. Therefore one determines either iteratively by nesting intervals or approximately according to the following formula: ${\displaystyle \mathbf {\rho } }$ ${\displaystyle \mathbf {\rho } }$

${\displaystyle \rho ={{\pi d} \over {\ln {2}}}\cdot {\frac {R_{\mathrm {AB,CD} }+R_{\mathrm {BC,DA} }}{2}}\cdot f}$

The correction factor f is taken from the picture opposite. For more precise measurements, a test structure is used which has at least one axis of symmetry and whose contacts are also arranged according to this symmetry. It then applies and the formula results for the specific resistance ${\displaystyle R_{\mathrm {AB,CD} }=R_{\mathrm {BC,DA} }}$

Shamrock structure

${\displaystyle \rho ={{\pi d} \over {\ln {2}}}\cdot R_{\mathrm {AB,CD} }}$

this can be done

${\displaystyle \rho \simeq 4{,}532\cdot d\cdot R_{\mathrm {AB,CD} }}$

sum up.

The procedure is only exact for ideal point contacts. If a suitable form of the measurement structure is selected, the influence of the contact size is negligible in practice. In semiconductor production , structures in the form of a clover leaf or a cross are used to measure the specific sheet resistance of thin layers, e.g. B. polysilicon or diffusion areas to determine. ${\displaystyle \rho _{\Box }}$

The sheet resistance of the layer is obtained by dividing the specific resistance obtained by the layer thickness : ${\displaystyle R_{\Box }}$ ${\displaystyle \rho }$ ${\displaystyle d}$

${\displaystyle R_{\Box }={\frac {\rho }{d}}}$

Measurement of the Hall coefficients

The same requirements apply for measuring the Hall coefficient as for measuring the coating thickness. In contrast, the current is impressed through contacts A and C and the resistance is measured. A homogeneous magnetic field Β is then applied perpendicular to the pane. Because of this, changes by the value Δ . ${\displaystyle R_{\mathrm {AC,BD} }}$ ${\displaystyle R_{\mathrm {AC,BD} }}$ ${\displaystyle R_{\mathrm {AC,BD} }}$

The Hall coefficient results from: ${\displaystyle A_{\mathrm {H} }}$

${\displaystyle A_{\mathrm {H} }={{d} \over {\mathrm {B} }}\cdot \Delta R_{\mathrm {AC,BD} }}$

The magnetic field Β causes a Lorentz force to act on the charge carriers q perpendicular to the streamlines and perpendicular to the magnetic field.

${\displaystyle {\vec {F}}=q\cdot {\vec {v}}\times {\vec {B}}}$

The current density can be expressed by . A transformation after and subsequent insertion results in a field strength of: ${\displaystyle \mathbf {J} }$ ${\displaystyle J=nqv}$ ${\displaystyle v_{x}}$

${\displaystyle E_{\mathrm {H} }={\frac {1}{nq}}\,J_{\mathrm {x} }B_{\mathrm {z} }=A_{\mathrm {H} }\,J_{\mathrm {x} }B_{\mathrm {z} }}$

${\displaystyle E_{\mathrm {H} }}$is proportional to and Β. ${\displaystyle J}$

${\displaystyle {\frac {1}{nq}}\,=A_{\mathrm {H} }\,}$

is the proportionality constant or the Hall coefficient. As it is known, the concentration of the charge carriers in the test structure can now be determined. ${\displaystyle q}$ ${\displaystyle n}$

literature

• LJ van der Pauw: A method of measuring specific resistivity and Hall effect of discs of arbitrary shape . In: Philips Research Reports . tape 13 , no. 1 , 1958, p. 1-9 .
• LJ van der Pauw: A Method of Measuring the Resistivity and Hall Coefficient on Lamellae and Arbitrary Shape . In: Philips Technical Review . tape 20 (1958/59) , no. 8 , p. 220–224 ( electron.mit.edu [PDF]).
• LJ van der Pauw: Measurement of the spec. Resistance and the Hall coefficient on discs of any shape . In: Philips Technical Review . tape 20 , 1959, pp. 230 .

Web links

Commons : Van der Pauw measurement  - collection of images, videos and audio files

• electron.mit.edu (PDF; 2.04 MB)

Individual evidence

1. ^ LJ van der Pauw: A Method of Measuring the Resistivity and Hall Coefficient on Lamellae and Arbitrary Shape . In: Philips Technical Review . tape 20 (1958/59) , no. 8 , p. 220–224 ( electron.mit.edu [PDF]).