Resistivity and resistance measurements

Resistivity and resistance measurements silicon resistivity silicon electron and hole mobility

1.E+04 p-resistivity

1.E+03

mobility (cm^2/V·s)

1.E+02 resistivity

1600

n-resistivity

1.E+01 1.E+00 1.E-01 1.E-02 1.E-03

µp

1400

µn

1200 1000 800 600 400 200 0

1.E-04

1.0E+13

1.E-05 1.00E+13

1.00E+15

1.00E+17 doping

resistivity: ⇢ =

1

1.00E+19

1.00E+21

1.0E+15

1.0E+17

1.0E+19

1.0E+21

doping (cm^-3)

1 = 1µN

Reference text: “Semiconductor material and device characterization 2/e” by Dieter Schroder, John Wiley, 1998.) EE 432/532

Resistance – 1

To measure the doping level, we can make use of simple resistance measurements. EE 201 approach: Make a text-book resistor from a chunk of n-type material that has a uniform doping concentration, ND. (The exact same approach applies to p-type material, as well.) The dimensions of the sample are W x L x t. The current will be carried by electrons, which have a concentration of n = ND.

t

L

+

– V

W

I

V ⇢ L ⇢ R= = = · gr I tW t

There may be problems with contact resistance. EE 432/532

Resistance – 2

Four-point probe method Put 4 probes in a collinear arrangement onto the sample surface. The probes are equally spaced. The wafer thickness is t.

I

V

Pass a current between the outer two probes. Measure the voltage between the inner two probes.

s t

The use of separate current and voltage contacts gets away from problems with contact resistance. The ratio of the voltage to the current will give a quantity like a resistance. Not surprisingly, the resistance is the product of the sample resistivity and the a geometrical factor. EE 432/532

V ⇢ R= = · g4 I t

Resistance – 3

⇢ R = · g4 t



1 sinh (t /s ) g4 = ln ⇡ sinh (t /2s )

1 x sinh (x) = e 2

e

x

The geometric factor accounts for how much the current is “squished” in the layer as it flows between two outer probes. Depends on the ratio t/s. Thick wafer t >> 1 s

t g4 ! 2⇡s

⇢ R= 2⇡s

Thin wafer (or diffused layer at the top surface of a wafer) t