# 2.9 Mobility - Resistivity - Sheet Resistance

Table of Contents -
Glossary -
Study Aids -
¬
®

In this section:
- Bulk mobility
- Temperature dependence of the mobility

- Resistivity
- Sheet resistance

## 2.9.1 Bulk mobility

The mobility of electrons and holes in bulk silicon is
shown in the figure below.

resistiv.xls - mobility.gif
**Fig.2.9.1** *Electron and hole mobility versus
doping density for silicon*

This is an active figure which can be used to find the
bulk mobility for specific doping concentrations as well as the
related resisitivity and sheet resistance.
Note that the mobility is linked to the **total** number of
ionized impurities or the sum of the donor and acceptor rather than the free carrier density which
is to first order related to the difference between the donor
and acceptor concentration.

The minority carrier mobility also depends on the total impurity
density, using the curve which corresponds to the minority carrier
type.
The curves are calculated from
the empiric expression:

(mob10)

where *m*_{min},
*m*_{max},
*a* and *N*_{r} are fit parameters. These
parameters for Arsenic, Phosphorous and Boron doped
silicon are
provided in the table below:

tmob1.gif

Example 006

### 2.9.1.1 Temperature dependence of the
mobility

## 2.9.2 Resistivity

The conductivity of a material is defined to be the
current density divided by the applied electric field. Since
the current density equals the product of the charge of the
mobile carriers, their density and velocity it can be
expressed as a function of the electric field using the
mobility. To include the contribution of electrons as well
as holes to the conductivity, we add the current density
due to holes to that of the electrons, or:

(mob8)

The conductivity due to electrons and holes is
then obtained from:
(mob9)

The resistivity is defined as the inverse of the
conductivity, namely:
(mob5)

The resulting resistivity as calculated with the expression
above is shown in the figure below:

resistiv.xls - resistiv.gif

**Fig.2.9.2** *Resistivity of n-type (red curve) and p-type
(blue curve) silicon versus
doping density*

Example 003 -
Example 004

## 2.9.3 Sheet resistivity of a 14 mil thick wafer

The concept of sheet resistance is used the characterize both wafers as thin
doped layers, since it is typically easier to measure the sheet resistance
rather than the resistivity of the material. The sheet resistance of a
layer with resisitivity, *r*, and
thickness, *t*, is given by their ratio:

(mob7)

While strictly speaking the units of the sheet resistance is Ohms,
one refers to it as being in Ohms per square. This nomenclature
comes in handy when the resistance of a rectangular piece of material with
length, *L*, and width *W* must be obtained. It
equals the product of the
sheet resistance and the number of squares or:
(mob6)

where the number of squares equals the length divided by the width.
The figure below shows the sheet resistance of a 14 mil thick silicon wafer
which is n-type (blue curve) or p-type (red curve)

resistiv.xls - sheetres.gif
**Fig.2.9.3** *Sheet resistivity of a 14 mil thick
n-type (red curve) and p-type
(blue curve) silicon wafer
doping density. This active figure can be modified to
accomodate any layer thickness.*

2.8
¬
® 2.10

© Bart J. Van Zeghbroeck, 1996, 1997