## Abstract

Magnetic properties are introduced into solids by paramagnetic ions. These are transition-metal ions of the iron series with a partially filled electronic 3*d* shell or rare-earth ions of the lanthanide series with an incomplete 4*f* shell. In *magnetic semiconductors*, they represent a cation *component* of the crystal, while in *diluted magnetic semiconductors*, they are a substitutional *alloy* component on the cation sublattice. The magnetic moments of the paramagnetic ions are coupled by different kinds of exchange interactions. Superexchange mediated by *p* states of anion ligands favors antiferromagnetism with antiparallel alignment of the magnetic moments, while double exchange and *p–d* exchange favor ferromagnetism with parallel alignment. Magnetic ordering is disturbed if the thermal energy exceeds the exchange energy; critical Curie and Néel temperatures exist for the transition from the paramagnetic high-temperature range to magnetically ordered respective ferromagnetic and antiferromagnetic regimes at lower temperature.

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## Notes

- 1.
There is also a small diamagnetic contribution arising from filled electronic shells with zero spin and orbital angular momentum known as Larmor (or Langevin) diamagnetic susceptibility; this contribution does not depend on temperature and is generally counteracting paramagnetism in solids.

- 2.
- 3.
The Brillouin function is defined by \( {B}_J(x)=\frac{2J+1}{2J}\coth \left(\frac{2J+1}{2J}x\right)-\frac{1}{2J}\coth \left(\frac{x}{2J}\right) \) and varies from 0 to 1 for

*x*= 0 to ∞. For high temperature or small splitting,*x*≪ 1 applies, and the hyperbolic cotangent can be approximated by coth*y*= 1/*y*+*y*/3 –*y*^{3}/45 + …. - 4.
- 5.
The additional paramagnetism of nuclear spins is negligible compared to electronic contributions (a fraction below 10

^{−3}). - 6.
Besides ferro- and antiferromagnetic coupling, the magnetic moments can be ordered in a

*ferri*magnetic coupling with a not canceling antiparallel alignment for the moments of neighboring (not identical) magnetic ions. - 7.
Occasionally also incorporation on an interstitial site of the host crystal is found. Such incorporation in parallel to substitutional alloying may be detrimental for the intended magnetic properties as, e.g., pointed out for Ga

_{1-x }Mn_{ x }As in Sect. 2.2. - 8.
A slight departure from the Curie law found for Cd

_{1-x }Mn_{ x }Te and Cd_{1-x }Mn_{ x }Se at*T*< 0.1K was attributed to the small crystal-field splitting of the^{6}S ground state enabled by minute contributions of excited states and the additional effect of the nuclear spin. - 9.
Also a Curie–Weiss law according to Eq. 16 with a characteristic temperature |

| ≪ 1 was widely applied, indicating some minor residual coupling effects among the magnetic ions.*Θ* - 10.
The mass magnetization is given here in the conventionally used units of emu/g; the conversion factor to the SI unit is 1 emu/g = 10

^{−3}A × m^{2}= 10^{−3}J/T. - 11.
The sublattice of the cations is fcc in (cubic) zincblende and hcp in (hexagonal) wurtzite lattices.

- 12.
In some cases an observed ferromagnetism did solely originate from

*precipitates*of magnetic ions alloyed into nonmagnetic II–VI host crystals (Saito et al. 2003). - 13.
Molecular-beam epitaxy is performed at low deposition temperatures down to 180 °C to allow for high nonequilibrium alloying levels; due to the limited kinetics at such low temperature, point defects are easily created.

- 14.
At highest Mn composition, a deviation from expected values even for annealed samples indicates some onset of compensation.

- 15.
In the chalcopyrite structure (Fig. 13 in chapter “The Structure of Semiconductors”), the 1nn and 3nn neighbors are located at \( \left(\frac{a}{2},0,\frac{c}{4}\right) \)and \( \left(\frac{a}{2},\frac{a}{2},\frac{c}{2}\right) \), respectively.

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Böer, K.W., Pohl, U.W. (2018). Magnetic Semiconductors. In: Semiconductor Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-69150-3_9

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