Ohms Per Square
What!
By Gene Chase, ETS, Inc.
Is it ohms per square meter or ohms
per square inch? Which is it? Actually, it is none of these, but
"ohms per square anything." However, this confusing term
that has been used to describe the Surface Resistivity (r
) of a material. Is it here to stay forever? The ESD Association
Glossary, ESDADV 1.01994 (1), describes Surface Resistivity
in the following way: "For an electric current flowing across
a surface, the ratio of DC voltage drop per unit length to the surface
current per width. In effect, the surface resistivity is the resistance
between two opposite sides of a square and is independent of the
size of the square or its dimensional units. Surface resistivity
is expressed in ohms per square. When using a concentric ring
fixture, resistivity is calculated by using the following expression,
where D1 = outside diameter on an inner electrode, D2 = inside diameter
of the outer electrode and R = measured resistance in ohms: (from
EOS/ESDS11.11  1993)(2)."
Surface Resistivity (r
_{s}) = {2p /[ln(D1/D2)]}R
Some have asked, why use this allegedly
ambiguous term and measurement? Can't we just use ohms? Because
of the geometry of the EOS/ESDS11.1 concentric ring electrode,
the resistance is simply r _{s}/10 ohms. One
could further argue, why not just always use this resistance in
ohms result?
In order to answer these questions,
we need to examine the history of ohms per square. For a
number of years the surface resistivity was a pure number
with no dimensions. Valdes (3) in 1954, wrote about the fourpoint
probe method to make resistivity measurements on germanium transistors.
However, all this work, and later work by Uhlir, (1955)(4), assumed
a threedimensional structures with one infinite dimension. Their
work was expanded by Smits (5) in 1958 for twodimensional structures.
Smits defined a fourpoint probe method of measuring "sheet resistivities."
This work eventually became an industry standard for measuring the
resistivity of diffused layers in semiconductors. He developed correction
factors for measuring sheet resistivities on twodimensional and
circular samples using a fourpoint probe where the two outer probes
source current and the two inner probes measure voltage. He found
that this method was not only useful for measuring diffused surface
layers, but was useful in obtaining "body resistivities"
of thin samples. Yet in all this work sheet resistivity (r
_{s}) had no dimensions, but was a pure number.
Although Smits showed that body resistivity (r ) was equal
to sheet resistivity (r _{s}) times w, where w is the
thin sample thickness, he did not assign the dimensions, ohmcm,
to this resistivity. The term he called "body resistivity," we now
commonly call "volume resistivity" or "bulk resistivity." It is
interesting to note that in Smit's work that he never uses the term
"sheet resistance." He developed the relationship that:
Sheet Resistivity (r
_{s}) = V / I (p /ln 2) = V / I (4.5324)
In 1962, Irvin (6) developed curves
showing the resistivity in ohmcm, versus Impurity
concentration of various doping levels in silicon. Here he defined
the "bulk resistivity" as ohmcm. The resistivity
is again dimensioned as ohmcm. There is no mention in this
publication of sheet resistance or ohms per square.
In 1968 in a book by Berry, etal. (7),
the authors state that the resistance of a thinfilm resistor is
directly proportional to the resistivity, r , and inversely
proportional to the thickness, d. They introduce the term
"sheet resistance (Rs)" to define thin film resistor parameters.
They define it as:
Rs = r / d
The authors further explain that the
sheet resistance may be thought of as a material property
since the film is essentially twodimensional. Therefore, a simple
thin film resistor consisting of a simple rectangle of length l
(in the direction of the current) and the width w has a resistance
of:
R = (r / d) (l/w) or
R = Rs (l/w)
The authors claim that the term (l/w)
is sometimes called the number of squares in the resistor, since
it is equal to the number of squares of side w that can be
superimposed on the resistor without overlapping. They assert that
the term "squares" is a pure number, having no dimensions. The author's
state that the sheet resistance has the unit of ohms, but it is
convenient to refer to it as "ohms per square" since the
sheet resistance produces the resistance of the resistor when multiplied
by the number of squares. They go on to say that the concept can
be broadened to include any arbitrarily shaped resistor by calling
the quantity Rd/r the effective number of squares.
The authors expand on the use Smit's fourpoint probe technique
and introduce new correction factors for the size of their substrate.
It turns out that the fourpoint probe is a useful tool to check
the uniformity of thinfilm resistors.
The term "sheet resistance"
has not only shown up in defining materials to control ESD. It is
also used to define resistive seas and overcoats of all types including
the coatings on cathode ray tube (CRT) monitors to reduce the second
anode electric fields that could be coupled to a person touching
the screen. It is also used to describe the resistance of the semitransparent
layer that composes one terminal of a liquid crystal display (LCD).
The term continues to be used to define the resistance of both thick
and thinfilm resistors. In a notable book on the physics of semiconductors
by Sze (8) in 1981, the term sheet resistance is not found
to describe the characteristics of semiconductors. Only the term
resistivity is used.
So now you know where the dimension
"ohms per square" apparently originated. It appears that
we are stuck with this term unless the authors of the ESD Association
Glossary decide to redefine it and use only the dimensions ohms
and ohmcm for surface and body (volume or bulk) resistivity
respectively.
Therefore, it would seem reasonable
that surface resistivity should always be measured in ohms
and volume resistivity in ohmcm, as Jonassen (9)
has argued for a number of years.
Maybe we should leave the term sheet
resistance and ohms per square to the thick and thin
film resistors and hybrid integrated circuit people, where it makes
some sense to them and stick to using ohms.
References
 ESD ADV1.01994, ESD Association Advisory for
Electrostatic Discharge Terminology  Glossary
 ANSI EOS/ESD S1.111993, EOS/ESD Association
Standard for Protection of Electrostatic Discharge Susceptible
Items  Surface Resistance Measurement of Static Dissipative
Planar Materials.
 Valdes, L., Resistivity Measurements on Germanium
transistors, Proceedings I.R.E., 42, Feb.1954, p420.
 Uhlir, A., The Potentials of Infinite Systems
of Sources and Numerical Solutions of Problems in Semiconductors
Engineering, Bell System Technical Journal, Jan 1955, p105.
 Smits F.M., Measurement of Sheet Resistivities
with the FourPoint Probe, Bell System Technical Journal, May
1958, p711.
 Irvin, J.C., Resistivity of Bulk Silicon and
Diffused Layers in Silicon, Bell System Technical Journal, 41,p387,
(1962).
 Berry, R.W., Hall, P.M., Harris, M.T., "Thin
Film Technology", Van Nostrand Reinhold Company, New York, NY,
1968.
 Sze, S.M., "Physics of Semiconductor Devices",
John Wiley and Sons, New York, NY, 1981.
 Jonassen, N., "Electrostatics", Chapman and
Hall and International Thomson Publishing, New York, NY,1998.
