Spin Transfer Torque Magnetoresistive Random Access Memory
79
Fig. 13 Schematic of the
CIPT model, illustrating in
blue the theoretical
breakdown of current
flowing through probe 1.
Modelled after reference
[188]. In blue, used to model
the Eq. (56) to conserve
voltage. In white, used to
model Eq. (58) to decompose
the components of current
flow. In green, used to model
Eq. (60) to conserve current
The theory behind sheet resistance R s measurements using four-point probe will be
briefly discussed in this section. It is defined as R s =
ρ
t
, where ρ is the resistivity of a
material and t is the thickness. However, the geometric correction factors, thickness
and spacing between the probes should also be represented. The current injected
through a probe can be considered to flow radially outwards through the material in
the form of a cylinder with radius r and thickness t. Therefore, the current density J
is defined as J =
I
2πrt
.
Furthermore, the current also induces an electric field E =
dV
dr
= −ρ J for an
ohmic material, in which Ohm’s law can be applied to obtain the voltage drop over
a concentric radial distance r. From Fig. 13, the potential drop from point 2 to point
3 as a result of a current injected from point 1 can be calculated from the integration
of the E over two probe distances, resulting in the following expression:
r 3
∫
r 2
Edr = V 3 − V 2 =
ρ
t
I
2π
ln
1
r 3
− ln
1
r 2
= R s
I
2π
ln
r 3
r 2
.
(55)
Figure 14 illustrates a four point probe measurement with equal probe spacing x.
The drop in voltage measured between probes 2 and 3 is due to the current injected
from probe 4, which can be seen as a negative current flowing in the opposite direction
as compared to probe 1. Therefore, the vector sum of these two contributions leads
to the overall potential drop V = R s
I
2π
ln
r 3
r 2
− R s
I
2π
ln
r 2
r 3
. In the case of equal
probe spacing x or R s =
π
ln2
V
I
, the potential drop from the viewpoint of probe 1
can be expressed as V = R s
I
2π (2ln2).
The concept of sheet resistance can also be applied to the MTJ model as described
by Worledge et al. [188], in which the current passing through the tunnel barrier is
assumed to be perpendicular to the infinitely extended film plane (CPP configuration).
79
Fig. 13 Schematic of the
CIPT model, illustrating in
blue the theoretical
breakdown of current
flowing through probe 1.
Modelled after reference
[188]. In blue, used to model
the Eq. (56) to conserve
voltage. In white, used to
model Eq. (58) to decompose
the components of current
flow. In green, used to model
Eq. (60) to conserve current
The theory behind sheet resistance R s measurements using four-point probe will be
briefly discussed in this section. It is defined as R s =
ρ
t
, where ρ is the resistivity of a
material and t is the thickness. However, the geometric correction factors, thickness
and spacing between the probes should also be represented. The current injected
through a probe can be considered to flow radially outwards through the material in
the form of a cylinder with radius r and thickness t. Therefore, the current density J
is defined as J =
I
2πrt
.
Furthermore, the current also induces an electric field E =
dV
dr
= −ρ J for an
ohmic material, in which Ohm’s law can be applied to obtain the voltage drop over
a concentric radial distance r. From Fig. 13, the potential drop from point 2 to point
3 as a result of a current injected from point 1 can be calculated from the integration
of the E over two probe distances, resulting in the following expression:
r 3
∫
r 2
Edr = V 3 − V 2 =
ρ
t
I
2π
ln
1
r 3
− ln
1
r 2
= R s
I
2π
ln
r 3
r 2
.
(55)
Figure 14 illustrates a four point probe measurement with equal probe spacing x.
The drop in voltage measured between probes 2 and 3 is due to the current injected
from probe 4, which can be seen as a negative current flowing in the opposite direction
as compared to probe 1. Therefore, the vector sum of these two contributions leads
to the overall potential drop V = R s
I
2π
ln
r 3
r 2
− R s
I
2π
ln
r 2
r 3
. In the case of equal
probe spacing x or R s =
π
ln2
V
I
, the potential drop from the viewpoint of probe 1
can be expressed as V = R s
I
2π (2ln2).
The concept of sheet resistance can also be applied to the MTJ model as described
by Worledge et al. [188], in which the current passing through the tunnel barrier is
assumed to be perpendicular to the infinitely extended film plane (CPP configuration).