10
1 An Overview of Spintronics
Such observations were interpreted as being caused by spin angular momentum S,
having two values ±è/2 of its z components. Hence, the matrix operator S z should
be (i) 2 × 2 matrix and (ii) eigenvalues must be ±è/2. We understand that a 2 × 2
matrix having eigenvalues of ±è/2 would be the matrix of the form:
M 2×2 =
2
1 0
0 −1
.
(1.5)
In addition, Pauli also defined the first three dimensionless matrices σ x , σ y and σ z
such that
S x =
2
σ x ; S y =
2
σ y ; S z =
2
σ z .
(1.6)
Since, S x , S y and S z must have eigenvalues of ±è/2, σ matrices have eigenvalues
of ±1. Furthermore, Eq. (1.4) mandates that
σ x σ y − σ y σ x = 2iσ z
σ y σ z − σ z σ y = 2iσ x
σ z σ x − σ x σ z = 2iσ y .
(1.7)
According to Eqs. (1.5) and (1.6)
σ z =
1 0
0 −1
.
(1.8)
Hence, the other two matrices, which have eigenvalues of ±1 and obey Eq. (1.7),
are
σ x =
0 a
a
∗ 0
=
0 1
1 0
σ y =
0 b
b
∗ 0
=
0 −i
i 0
.
(1.9)
These are the famous Pauli spin matrices, which according to Eq. (1.6) act as
operators for the corresponding spin components. Additionally, square of each of
the Pauli spin matrices is the 2 × 2 unit matrix [I]. Thus,
|S|
2
= |S x |
2
+
S y
2 + |S z |
2
|S|
2
=
2
2
[I ] +
2
2
[I ] +
2
2
[I ]
|S|
2
= 3
2
2
[I ] + ¯
s(¯ s + 1)
2 [I ]