Download 3.5.4. Contact resistance to a thin semiconductor layer

3.5.4. Contact resistance to a thin semiconductor layer
The contact between a metal contact and a thin conducting layer of semiconductor can be
described with the resistive network shown in Figure 3.5.1, which is obtained by slicing the
structure into small sections with length ∆x, so that the contact resistance, R1 , and the
semiconductor resistance, R2 , are given by:
R1 =
ρc
W∆x
(3.5.1)
∆x
W
(3.5.2)
and
R2 = Rs
ρc is the contact resistance of the metal-to-semiconductor interface per unit area with units of
Ωcm2 , Rs is the sheet resistance of the semiconductor layer with units of Ω/o and W is the width
of the contact.
R1
R
-L
Figure 3.5.1
2
R1
R
R
2
0
R1
2
R1
R
R
2
x x + ∆x
2
d
Distributed resistance model of a contact to a thin semiconductor layer.
Using Kirchoff's laws one obtains the following relations between the voltages and currents at x
and x + ∆x.
V ( x + ∆ x) − V ( x ) = I ( x ) R 2 = I ( x)
I ( x + ∆x ) − I ( x ) =
Rs
∆x
W
V ( x)
W
= V ( x ) ∆x
R1
ρc
(3.5.3)
(3.5.4)
By letting ∆x approach zero one finds the following differential equations for the current, I(x),
and voltage, V(x):
dV I ( x) Rs
=
dx
W
(3.5.5)
(3.5.6)
dI I ( x )W
=
dx
ρc
Which can be combined into:
R
d 2 I ( x)
I ( x)
= I ( x ) s = 2 withλ =
2
dx
ρc
λ
ρc
Rs
(3.5.7)
The parameter λ is the characteristic distance over which the current occurs under the metal
contact and is also referred to as the penetration length. The general solution for I(x) and V(x)
are:
I ( x) = I 0
d−x
λ
d
sinh
λ
(3.5.8)
sinh
λ Rs
V ( x) = I 0
W
d−x
λ
d
sinh
λ
(3.5.9)
cosh
4.00E-04
3.50E-04
3.00E-04
2.50E-04
2.00E-04
1.50E-04
1.00E-04
5.00E-05
0.00E+00
1.20E-03
1.00E-03
8.00E-04
6.00E-04
4.00E-04
Current (A)
Voltage (V)
Both are plotted in Figure 3.5.2:
2.00E-04
0.00E+00
0
2
4
6
Distance (micron)
Figure 3.5.2
Lateral current and voltage underneath a 5 µm long and 1 mm wide metal contact
with a contact resistivity of 10-5 Ω-cm2 on a thin semiconductor layer with a sheet
resistance of 100 Ω/o.
The total resistance of the contact is:
Rc =
V ( 0) λ R s
d
=
coth =
I (0)
W
λ
ρ c Rs
W
coth
(3.5.10)
d
λ
In the limit for an infinitely long contact (or d >> λ) the contact resistance is given by:
ρ c Rs
Rc =
W
(3.5.11)
, for d >> λ
A measurement of the resistance between a set of contacts with a variable distance L between the
contacts (also referred to as a transmission line structure) can therefore be fitted to the following
straight line:
R=2
ρ c Rs
W
+ Rs
(3.5.12)
L
W
so that the resistance per square, Rs, can be obtained from the slope, while the contact resistivity,
ρc, can be obtained from the intersection with the y-axis. The penetration depth, λ, can be
obtained from the intersection with the x-axis. This is illustrated with Figure 3.5.3.
R
T
o
o
o
o
2Rc
−λ
Figure 3.5.3
0
L
Resistance versus contact spacing, L, of a transmission line structure.
In the limit for a short contact (or d << λ) the contact resistance can be approximated by
expanding the hyperbolic cotangent1 :
Rc =
1
coth x =
λ Rs λ d
ρ
1
d
( +
+ ...) = c + Rs
, for d << λ
W d 3λ
Wd 3 W
1 x x3
+ +
+ ... for x << 1
x 3 45
(3.5.13)
The total resistance of a short contact therefore equals the resistance between the contact metal
and the semiconductor layer (i.e. the parallel connection of all the resistors, R1 , in Figure 3.5.1),
plus one third of the end-to-end resistance of the conducting layer underneath the contact metal
(i.e the series connection of all resistors, R2 , in Figure 3.5.1).