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 ...
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