by John Clark, C. Eng, M.I.Mech.E., F.B.H.I., Managing Director of Jandel Engineering Ltd.

General

Before attempting measurement one needs to know something of the sample or the wafer - is it silicon? (Germanium is easier to contact and measure). Metallic and other layers are also deposited on semiconductor, sapphire or ceramic wafers.

First, is the sample clean and fresh?

If the sample is old it may be etched, washed and dried which will remove oxide which can impede ohmic contact.

Secondly, is the sample homogenous i.e. is it uniformly doped or does it have a layer on its surface e.g. by epitaxy, diffusion, ion-implantation, or sputtering etc?

If the sample has a layer it must be of the opposite conductivity type to the substrate i.e. electrically insulated from the substrate. A layer of the same conductivity type cannot be measured by the four point method because the substrate offers an easier path for the current, and the measured resistivity is effectively that of the substrate.

If the layer is thin, meaning sub-micron, one must avoid puncturing the layer by excessive needle loading, by sharp or rough needle tips, or too rapid descent velocity of the probes, excessive current can also inject minority carriers.

All these effects cause some leakage into the substrate, so that the measuring current in the layer is reduced, and the resistivity measured is too low.

Limits of Measurement Capability

1. The material must be capable of being probed, i.e. the probes must be able to make ohmic contact with the material e.g. Germanium, Silicon and metals. Materials such as Gallium Arsenide cannot normally be probed unless it is doped and measured with special measuring techniques such as that in the Four Dimensions Inc. GaAs probe.


2. Very low resistivity material e.g. aluminium, gold, platinum may require the maximum current from the current source to achieve a readng on the digital voltage display.


3. High sheet resistivity material e.g. ion-implanted silicon wafers, silicon on sapphire, can be measured using very low currents (say 1 microamp or less) and trying to avoid a greater voltage indication than 100mV.


4. Low level measurements are beset with problems e.g. electrical noise due to poor conatct conditions, thermally induced voltages, actinic effects, offset voltages produced by devices in the current source, and general leakage in plugs, lead etc.

Calculation of Resistivity

A selection of correction factors are published by various authorities, covering the modifications to be made according to the specimen size and shape being measured, we show two examples for measurement of circular samples in the centre with a linear probe of spacing 's'.

Basically, bulk resistivity (for a semi-infinite volume) = 2 x pi x s x (V/I) ohm.cm where s is the spacing of the probe in cm, I the test current, and V the measured voltage.

Sheet resistance for wafers and films RS = 4.532 x V / I  ohms per square.

Bulk resistivity for wafers and films q = RS x t = 4.532 x V x t / I where t is the thickness in cm.

General Comments

1. Most wafers and films approximate to 'infinite sheets' at the present time, but if the thickness is greater than 5 times the probe spacing (normally 1.00mm) i.e. 5mm then the semi-infinite solid formula is within less than 1%.


2. From the other point of view a reasonable sized wafer may be measured with a four point probe using the above sheet resistance formula. Provided the wafer thickness does not exceed 0.625 of the probe spacing the calculation is within 1%.

   Please see table below.

   FPP Correction Factors for Sample Thickness t		FPP Correction Factors for Sample Diameter d
   t/s	C1(t/s)	d/s	C2(d/s)
   0.3	1.0000	10	4.1712
   0.4	0.9995	20	4.4364
   0.5	0.9974	30	4.4892
   0.6	0.9919	40	4.5080
   0.7	0.9816	50	4.5167
   0.8	0.9662	60	4.5215
   0.9	0.9459	70	4.5244
   1.0	0.9215	80	4.5262
   1.2	0.8643	90	4.5275
   1.4	0.8026	100	4.5284
   1.6	0.7419	200	4.5314
   1.8	0.6852	º	4.5320
   2.0	0.6337



3. Remember that other geometrical effects affect the result if the wafer is not measured at the centre because the number of possible current paths is limited.

More information regarding correction factors can be see at: http://www.four-point-probes.com/haldor.html (These are LARGE graphic files).

We recommend study of the following original papers:

a) Linear Array Probes

Circular wafers at centre:

1. D. E. Vaughan, Br.J. Appl. Phys., 12, 414 (1961)


2. M. A. Logan, Bell Sys. Tech. J., 40, 885 (1961)
   
   Off centre but on radius:


3. L. J. Swartzendruber, National Bureau of Standards Technical Note 199 (1964)
   
   Perpendicular to radius:


4. M. P. Albert and J. F. Combs, IEEE Trans. Electron Devices, ED-11, 148 (1964)


5. L. J. Swartzendruber, Solid State Electronics, 7, 413 (1964)
   
   Rectangular sample at centre and off centre:


6. M. A. Logan, Bell Sys. Tech. J., 46, 2277 (1967)
   
   Half cylinder:


7. E. B. Hansen, Appl. Sci. Res., 8B, 93 (1960)
   
   Circular rod:


8. H. H. Gegenwarth, Solid State Electronics, 11, 787 (1968)
   
   Rectangular bar:


9. A. Marcus and J. J. Oberly, IEEE Trans. Electron. Devices, ED-3, 161 (1956)
   
   Note: All the foregoing is based on measurement using a four point linear probe, the current being passed between the outer probes and the voltage measured across the inner two probes.

Four-Point-Probes is a division of Bridge Technology. To request further information please call Bridge Technology at (480) 988-2256 or send e-mail to Larry Bridge at: sales@bridgetec.com