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Spin Physics in Semiconductors pp 477–524Cite as

Diluted Magnetic Semiconductors: Basic Physics and Optical Properties

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Part of the book series: Springer Series in Solid-State Sciences ((SSSOL,volume 157))

Abstract

Diluted Magnetic Semiconductors (DMS) a class of magnetic materials, which fill the gap between ferromagnets and semiconductors (Galazka, Inst Phys Conf Ser 43, 133, 1979, [1]). In the early literature these DMS were often named semimagnetic semiconductors , because they are midway between non magnetic and magnetic materials. DMS are semiconductor compounds (A\(_{1-x}\)M\(_x\)B) in which a fraction x of the cations is substituted by magnetic impurities , thereby making the host semiconductor (AB) magnetic.

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Notes

  1. 1.

    This term can be misleading. Since Landé factors have different signs for s and d electrons in (Cd,Mn)Te, parallel spins means antiparallel magnetic moments. A consequence is the field-dependance of Larmor frequency of conduction electrons in a sample with a low Mn content, see Fig. 14.17.

  2. 2.

    A negative \(\beta \) means also an “antiferromagnetic” coupling between the Mn spins and a gas of holes occupying the upper levels of the valence band (both spin and energy are reversed by switching from electrons to holes). But the magnetic moments of antiparallel spins can be parallel if the effective Landé factors have opposite signs, as in (Cd,Mn)Te. Note that the exchange parameter \(\beta \) applies to the spin of the hole, not its total angular momentum.

  3. 3.

    Note that in literature the definition of \(J_{ij}\) may vary (sign and factor 2 which comes from a double summation over i and j on an ensemble of sites).

  4. 4.

    The modified Curie-Weiss law and the magnetization steps described in this section are caused by the interactions between two Mn spins having isotropic g-factor. Very similar deviations from the Brillouin function are also observed for isolated spins with anisotropic g-factor (non \(d^5\) ions), which exhibit level crossings already for a single spin.

  5. 5.

    The exchange interactions with distant Mn atoms are weak, but nevertheless they slightly shift the magnetization steps and must be taken into account for a precise determination of \(J_1\).

  6. 6.

    An equivalent expression of the free energy is given in [42].

  7. 7.

    Magnetic impurities with a non-\(d^{5}\) configuration have a non-zero orbital momentum. This usually gives rise to a more complex, anisotropic behavior of the magnetization, and also to a complex, but different, behavior of the giant Zeeman effect [65, 66]. The Jahn-Teller effect can significantly alter this anisotropy [50].

  8. 8.

    These large values of the giant spin splitting are often cast into an appealing but sometimes misleading form, that of an effective Landé factor which can be up to several hundreds. One must keep in mind that this g-factor is strongly temperature dependent, and that the giant Zeeman effect is highly non-linear in field.

  9. 9.

    This can be shown by introducing the three inequivalent directions in the expression of the free energy given by (14.14).

  10. 10.

    Assuming an electron gas confined in a quantum well with infinite barriers.

  11. 11.

    This is a classical effect, which can be described from the Bloch equations of the coupled hole and Mn spins. One can also start from the expression of the free energy (14.14) for Mn coupled to heavy-holes

    $$\begin{aligned} F(\mathbf {M},\mathbf {m})= \frac{M^2}{2\chi _{Mn}}-\mathbf {M}\cdot \mathbf {B}+\frac{m_x^2}{2\chi _h}-\frac{IM_{x}m_x}{(g\mu _B)(g_e\mu _B)},{(14.26)} \end{aligned}$$

    and calculating \(m_x\) so as to minimize the free energy (i.e., assuming a collective Mn-carrier behavior). For small tilt angle \(\theta \) the energy of the Mn system can then be expressed quantum mechanically using the creation and annihilation operators \(\hat{A}^\dag = [(1-\zeta )^{1/2}\hat{X}+i(1-\zeta )^{-1/2}\hat{P}]/\sqrt{2}\) and \(\hat{A}= [(1-\zeta )^{1/2}\hat{X}-i(1-\zeta )^{-1/2}\hat{P}]/\sqrt{2}\), with \(\zeta =I^2\chi _{Mn}\chi _h/(g\mu _B)^2(g_h\mu _B)^2\) the key parameter which goes to unity at the ferromagnetic transition. One finds the familiar expression for the quantum harmonic oscillator \(H=g\mu _BB(1-\zeta )^{1/2}(\hat{A}^\dag \hat{A}+1/2)\), the frequency being renormalized by \((1-\zeta )^{1/2}\).

  12. 12.

    This is a direct indication that the positive sign of the exchange integral \(\alpha \) results into an antiparallel configuration of the s and d magnetic moments in CdMnTe. At even larger field, the Larmor frequency of the conduction electrons vanishes: this is not a soft mode, but it provides a good configuration for a study of skyrmions.

  13. 13.

    The free energy reads

    $$\begin{aligned} F(\mathbf {M},\mathbf {m})= -(\mathbf {M}+\mathbf {m})\cdot \mathbf {B}-\frac{I\mathbf {M}\cdot \mathbf {m}}{(g\mu _B)(g_e\mu _B)}.{(14.27)} \end{aligned}$$

    For small tilt angles of \(\mathbf {M}\) and \(\mathbf {m}\), the energy can be expanded up to quadratic terms using the conjugated operators introduced before \(H=\frac{1}{2}(g_e\mu _BB+\varDelta )(\hat{x}^2+\hat{p}^2)+\frac{1}{2}(g\mu _BB+K)(\hat{X}^2+\hat{P}^2)-(K\varDelta )^{1/2}(\hat{X}\hat{x}+\hat{P}\hat{p})\) with the notations \(\varDelta =IM/g\mu _B\) and \(K=Im/g_e\mu _B\) for the Overhauser and (EPR) Knight shifts respectively. In terms of annihilation and creation operators this becomes \(H=\hbar \omega _e(\hat{a}\hat{a}^\dag +1/2)+\hbar \omega _{Mn}(\hat{A}\hat{A}^\dag +1/2)-\sqrt{K\varDelta }(\hat{A}\hat{a}^\dag +\hat{A}^\dag \hat{a})+...\) with \(\hbar \omega _e=g_e\mu _BB+\varDelta \) and \(\hbar \omega _{Mn}=g\mu _BB+K\). This expression of two coupled harmonic oscillators shows that the electron and Mn spin-flip excitations are coupled and the anticrossing energy is given by \(2\sqrt{K\varDelta }\) [162]. Note that, as for the soft mode, a classical description in terms of Bloch equations give the same result.

  14. 14.

    Care must be taken in the interpretation of Kerr or Faraday signal in magnetic materials, due complications introduced by dichroic bleaching. For instance the Faraday rotation angle is \(\theta _F \propto QM\), where Q is a magneto-optical coefficient. After excitation by a laser pulse the total variation of Faraday rotation contains two terms \(\delta \theta _F \propto M\delta Q+Q\delta M\), where the first term corresponds to dichroic bleaching, i.e. a change of magneto-optical properties due to photoexcited carriers, and the second term contains the information on magnetization dynamics [168,169,170].

  15. 15.

    If the effective field changes slowly with respect to the Larmor period, the magnetization follows adiabatically the total field direction and does not precess.

  16. 16.

    In magnetic quantum wells the coherent rotation is mainly induced by photocreated holes. The hole spin is generally locked along the growth axis due to confinement and strain, while the electron spin precesses rapidly. It is therefore possible to cancel the electron spin polarization using a second pump pulse delayed with respect to the first one by a half of the electron Larmor period. The amplitude of the tilted magnetization appears to be insensitive to the delay between the two pump pulses, indicating that the coherent rotation is induced mainly by the hole spin polarization [179].

  17. 17.

    In itinerant ferromagnets the distinction between carriers and spins is rather artificial as itinerant electrons contribute both to transport and magnetism (see [186]).

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Authors and Affiliations

  1. Universite Grenoble Alpes, 38000, Grenoble, France

    Joël Cibert

  2. CNRS, Institut Néel, 38000, Grenoble, France

    Joël Cibert

  3. Laboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Université de Montpellier, Montpellier, 34095, France

    Denis Scalbert

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  1. Laboratoire Charles Coulomb, Université Montpellier, Montpellier, France

    Mikhail I. Dyakonov

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Cibert, J., Scalbert, D. (2017). Diluted Magnetic Semiconductors: Basic Physics and Optical Properties. In: Dyakonov, M. (eds) Spin Physics in Semiconductors. Springer Series in Solid-State Sciences, vol 157. Springer, Cham. https://doi.org/10.1007/978-3-319-65436-2_14

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