Appendix A
Tensors
A.1 Introduction
A physical quantity T i j...m with a total of k indices that is independent of translations of the coordinate
system and transforms with respect to all indices like a vector is called a tensor of rank k.
Often, Einstein’s sum convention is used; a sum is built over indices with the same symbol. For
example, x
i = D i j x j shall be read as x
i =
3
j=1 D i j x j .
A.2 Rotation of Coordinate System
A rotation of the coordinate system is a transformation x → x
that is written in components as
x
i = D i j x j .
(A.1)
D is called the rotation matrix. The inverse of the rotation matrix is D
−1 with
D
−1
kl = D lk ,
(A.2)
i.e. it is the transpose of matrix D. The inverse transformation is x j = D i j x
i . Thus,
D i j D k j = δ i j .
(A.3)
A simple example is the azimuthal rotation around the z-axis by an angle φ (in the mathematically
positive direction)
D =
⎛
⎝
cos φ − sin φ 0
sin φ cos φ 0
0
0
1
⎞
⎠ .
(A.4)
For the description of an arbitrary rotation (x, y, z) → (X, Y, Z ), generally three angles are necessary.
Typically, the Euler angles (φ, θ, ψ) are used (Fig. A.1). First, the system is rotated by φ around the
z-axis. The y-axis becomes the u-axis. Then, the system is tilted by θ around the u-axis and the z-axis
becomes the Z -axis. Finally, the system is rotated by ψ around the Z -axis.
© Springer Nature Switzerland AG 2021
M. Grundmann, The Physics of Semiconductors, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-51569-0
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