# 2.9 Mobility - Resistivity - Sheet Resistance

In this section:

## 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 mmin, mmax, a and Nr are fit parameters. These parameters for Arsenic, Phosphorous and Boron doped silicon are provided in the table below:

tmob1.gif

Example 006

## 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