Carrier Mobility semiconductors

The carrier mobility deserves further study since it is directly linked to the conductivity and resistivity of a semiconductor. First we examine the doping dependence of the mobility and the corresponding doping dependence of the conductivity and resistivity. The concept of the sheet resistance is introduced next and applied to the calculation of the resistance of a semiconductor.

2.7.2.1. Doping dependence of the carrier mobility

The mobility of electrons and holes in silicon at room temperature is shown in Figure 2.7.3.

Figure 2.7.3 :

Electron and hole mobility versus doping density for silicon.

The electron mobility and hole mobility have a similar doping dependence: For low doping concentrations, the mobility is almost constant and is primarily limited by phonon scattering. At higher doping concentrations, the mobility decreases due to ionized impurity scattering with the ionized doping atoms. The actual mobility also depends on the type of dopant. Figure 2.7.3 is for phosphorous and boron doped silicon.

Note that the mobility is linked to the total number of ionized impurities or the sum of the donor and acceptor densities. The free carrier density, as described in section 2.6.4.1 is to first order related to the differencebetween the donor and acceptor concentration.

The minority carrier mobility also depends on the total impurity density. The minority-carrier mobility can be approximated by the majority-carrier mobility in a material with the same number of impurities.

The mobility at a particular doping density is obtained from the following empiric expression:

(2.7.10)

where mmin, mmax, a and Nr are fit parameters. These parameters for arsenic, phosphorous and boron doped silicon are provided in Table 2.7.2.

Table 2.7.2 :

Parameters for calculation of the mobility as a function of the doping density

The resulting mobilities in units of cm2/V-s are listed for different doping densities in Table 2.7.3.

Table 2.7.3 :

Mobility in silicon for different dopants and doping densities

2.7.2.2. Conductivity and Resistivity

The conductivity of a material is defined as 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 as described by equations (2.7.2a) and (2.7.2b), 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:

(2.7.11)

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

(2.7.12)

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

(2.7.13)

The resulting resistivity as calculated with the expression above is shown in Figure 2.7.4.

Figure 2.7.4:

Resistivity of n-type and p-type silicon versus doping density.

2.7.2.3. Sheet resistance

The sheet resistance concept is used to characterize both wafers and 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 uniformly-doped layer with resistivity, r, and thickness, t, is given by their ratio:

(2.7.14)

While the unit of the sheet resistance is Ohms, one refers to it as 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:

(2.7.15)

where the number of squares equals the length divided by the width. Figure 2.7.5 provides, as an example, the sheet resistance of a 14 mil thick silicon wafer for both n-type and p-type silicon.

Figure 2.7.5:

Sheet resistance of a 14 mil thick n-type and p-type silicon wafer versus doping density.

Example 2.9	A piece of silicon doped with arsenic (Nd = 1017 cm-3) is 100 mm long, 10 mm wide and 1 mm thick. Calculate the resistance of this sample when contacted one each end.
Solution	The resistivity of the silicon equals: where the mobility was obtained from Table 2.7.3. The resistance then equals: An alternate approach is to first calculate the sheet resistance, Rs: From which one then obtains the resistance:

Next, we generalize the concept of the sheet resistance to a semiconductor layer with non-uniform doping as illustrated with Figure 2.7.6.

Figure 2.7.6:

Non-uniform doping distribution.

The doping profile is representative of a diffused and/or ion-implanted p-type layer in a uniformly doped n-type substrate. The metallurgical junction depth, xj, occurs when the two doping densities are equal. The sheet resistance of the section with width, dx, is given by:

(2.7.16)

Where both the mobility, mp(x), and hole density, p(x), are dependent on position. The sheet resistance of the whole layer is the parallel resistance of all slices with width dx between zero and xj, resulting in:

(2.7.17)

where  is the average mobility in the doped layer. Ider Guerrero EES Secc: 1