Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
van der Pauw’s Theorem
on Sheet Resistance
Mike Bolt
Calvin College
Annual Meeting of the Michigan MAA
Hope College
April 11, 2015
Mike Bolt Calvin College
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
Mike Bolt Calvin College
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
Mike Bolt Calvin College
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
Ohm’s law for an ideal resistor: V = I R
I
I
I
V = voltage (V )
I = current (A)
R = resistance (Ω)
Mike Bolt Calvin College
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
Generalization to 3-d current density: E = ρ J
I
I
I
J
E = electric field (V /m)
ρ = resistivity (Ω · m)
J = current density (A/m2 )
Mike Bolt Calvin College
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
Resistivity (3-d)
Sheet Resistance (2-d)
Let d = uniform thickness so E = ρ J =
I
I
I
ρ
(dJ)
d
E = electric field (V /m)
ρ = sheet resistance (Ω = Ω/sq.)
J = current density (A/m)
Mike Bolt Calvin College
van der Pauw’s Theorem
E = ρJ
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
Theorem (van der Pauw, 1958)
On a simply connected region, take points A, B, C , D oriented
positively on the boundary and measure
RAB,CD =
VD − VC
jAB
RBC ,DA =
VA − VD
.
jBC
Then ρ satisfies exp(−πRAB,CD /ρ) + exp(−πRBC ,DA /ρ) = 1.
Mike Bolt Calvin College
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
Suppose our sample is a halfplane. Drive in a current j = jAB
at point A.
A
At distance r the current density J and electric field E satisfy
|J| =
j
πr
Mike Bolt Calvin College
|E| =
jρ
.
πr
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
One computes the associated voltage drop using E = −∇V
a
A
B
b
C
c
D
Then
Z
D
VD − VC =
C
jρ
−E · dr =
π
Mike Bolt Calvin College
Z
C
D
dr
jρ
a+b
=
ln
.
r
π
a+b+c
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
Next consider point B between A and C . Pull out a current j
at point B.
A
a
B
b
C
c
The associated voltage drop is
VD − VC = −
Mike Bolt Calvin College
jρ
b
ln
.
π
b+c
van der Pauw’s Theorem
D
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
Adding these together (with j = jAB ) gives
def
RAB,CD =
ρ
a+b
ρ
b
VD − VC
= ln
− ln
jAB
π a+b+c
π b+c
ρ (a + b)(b + c)
= ln
π
b(a + b + c)
Mike Bolt Calvin College
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
If rather we drive in a current j = jBC at B and pull out the
current at C , then we measure
def
RBC ,DA =
VA − VD
ρ (b + c)(a + b)
= ln
jBC
π
ac
Mike Bolt Calvin College
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
Easy calculation shows
exp(−πRAB,CD /ρ) + exp(−πRBC ,DA /ρ)
b(a + b + c)
ac
=
+
(a + b)(b + c) (b + c)(a + b)
= ···
=1
So van der Pauw’s theorem holds true for a halfplane.
Mike Bolt Calvin College
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
For general region Ω, drive in current j at A and pull the
current out at B.
This sets up an electric field E and electric potential V in Ω.
Mike Bolt Calvin College
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
Let f : Ω → Ω0 be conformal from Ω to halfplane Ω0 such that
f (A) = A0 , f (B) = B 0 , f (C ) = C 0 , f (D) = D 0 .
A¢
B¢
Mike Bolt Calvin College
C¢
van der Pauw’s Theorem
D'
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
Similarly, drive in current j at A0 and pull the current out at B 0 .
This sets up an electric field E0 and electric potential V 0 in Ω0 .
In fact, V = V 0 ◦ f and
RAB,CD =
VD − VC
VD 0 − VC 0
=
= RA0 0 B 0 ,C 0 D 0
j
j
That is, RAB,CD is conformally invariant.
Mike Bolt Calvin College
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
First sample
Mike Bolt Calvin College
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
First sample – measurement 1
Mike Bolt Calvin College
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
First sample – measurement 2
Mike Bolt Calvin College
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
First sample – numerics
Mike Bolt Calvin College
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
Second sample
Mike Bolt Calvin College
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
Second sample – measurement 1
Mike Bolt Calvin College
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
Second sample – measurement 2
Mike Bolt Calvin College
van der Pauw’s Theorem
Sheet Resistance
van der Pauw’s Theorem
Proof
Demonstration
Second sample – numerics
Mike Bolt Calvin College
van der Pauw’s Theorem