# Carrier States in Ferromagnetic Semiconductors and Diluted Magnetic Semiconductors—Coherent Potential Approach—

## Abstract

**:**

## 1. Introduction

## 2. Conduction-electron States in a Ferromagnetic Semiconductor (FMS)

#### 2.1. Coherent Potential Approach to the s-f Model

#### 2.2. t-Matrix Elements and Their Thermal Average

**S**,

#### 2.3. Results for the Conduction-electron States in an FMS

**Figure 1.**The DOS shown as a function of ω/Δ for $IS$/Δ = 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0: (a) paramagnetic states ($\langle {S}_{z}\rangle $ = 0; left panel) and (b) completely ferromagnetic states ($\langle {S}_{z}\rangle $ = S; right panel). The solid curves show up-spin states and the dotted curves show down-spin states. The points on the energy axes indicate $-IS$ and $+I(S+1)$, and the straight dotted lines indicate $-(\Delta +IS$) and $+(\Delta +IS)$ for reference.

**Figure 2.**The DOS shown for magnetizations $\langle {S}_{z}\rangle /S$ = 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0: (a) $IS/\Delta =0.2$ (left panel) and (b) $IS/\Delta =0.8$ (right panel). The solid curves show up-spin states and the dotted curves show down-spin states.

**Figure 3.**(a) The shift of the bottom of the band in paramagnetic states normalized by the exchange energy, $({\omega}_{b}+\Delta )/IS$, shown as a function of $IS/\Delta $. The arrows indicate the experimentally obtained magnetic redshifts of EuO (0.27 eV), EuS (0.18 eV), and EuSe (0.13 eV) with $IS=0.35$ eV. (b) The energy of the bottom of the band depicted as a function of $T/{T}_{c}$ for $IS/\Delta =0.2,0.4,0.6,$ and $1.0$. The value of $-\langle {S}_{z}/S\rangle $, which corresponds to the result of the weak exchange interaction limit, is also shown.

## 3. Carrier States in Diluted Magnetic Semiconductors (DMSs)

#### 3.1. Model Hamiltonian for a Carrier in a DMS and the Application of the Dynamical CPA

#### 3.2. General Consideration for the Carrier States in a DMS

#### 3.3. The Case of Strong Exchange Interaction

**Figure 4.**Left panel: DOS as a function of $\omega /\Delta $ for various values of $\langle {S}_{z}\rangle /S$. The solid lines represent the down-spin carrier and dotted lines represent the up-spin carrier. The impurity levels ${E}_{a}=\pm 1.25\Delta $ are indicated by dots on the abscissa. Right panel: (a) A-site component of the DOS, $(1-x){D}_{\downarrow}^{A}\left(\omega \right)\Delta $ and $-(1-x){D}_{\uparrow}^{A}\left(\omega \right)\Delta $. (b) M-site component of the DOS, $x{D}_{\downarrow}^{M}\left(\omega \right)\Delta $ and $-x{D}_{\uparrow}^{M}\left(\omega \right)\Delta $. The thick, thin, and dotted lines represent the cases of $\langle {S}_{z}\rangle /S=1.0$, $0.5$, and $0.0$, respectively. Note the difference in the scale of the vertical axes of (a) and (b). From Takahashi and Kubo [42].

#### 3.4. The Case of Moderate Exchange Strength (II-VI-based DMSs)

**Figure 5.**The results for II-VI-based DMSs. Left panel: DOS as a function of $\omega /\Delta $ for various values of $\langle {S}_{z}\rangle /S$. The solid lines represent the down-spin carrier and the dotted lines represent the up-spin carrier. The arrow indicates the Fermi level ${\epsilon}_{F}/\Delta $ for $n=x\phantom{\rule{4pt}{0ex}}(=0.05)$. Right panel: (a) A-site component of the DOS, $(1-x){D}_{\downarrow}^{A}\left(\omega \right)\Delta $ and $-(1-x){D}_{\uparrow}^{A}\left(\omega \right)\Delta $. (b) M-site component of the DOS, $x{D}_{\downarrow}^{M}\left(\omega \right)\Delta $ and $-x{D}_{\uparrow}^{M}\left(\omega \right)\Delta $. The thick, thin, and dotted lines represent the cases of $\langle {S}_{z}\rangle /S=1.0$, $0.5$, and $0.0$, respectively. From Takahashi and Kubo [42].

**Figure 6.**The results for II-VI-based DMSs. (a) Low-energy part of the DOS $D\left(\omega \right)$. (b) Optical absorption spectrum $A\left(\omega \right)$ in arbitrary units (arb. units). The solid lines represent the down-spin carrier and the dotted lines represent the up-spin carrier. The values of the band-edge energy, ${\omega}_{b}/\Delta $, obtained by a simple approximation are plotted as dots on the lines of $\langle {S}_{z}\rangle =S$ and $\langle {S}_{z}\rangle =0$ (see text). Note that the energy of the bottom of the model band is $\omega =-\Delta $. From Takahashi [43].

**Figure 7.**Left panel: Absorption peak energy ${\omega}_{p}/\Delta $ as a function of $\langle {S}_{z}\rangle /S$. The solid line represents the down-spin carrier and the dotted line represents the up-spin carrier. The error bar represents the half-peak width. The values of ${\omega}_{b}/\Delta $ obtained by a simple approximation are plotted as dots on the lines of $\langle {S}_{z}\rangle =S$ and $\langle {S}_{z}\rangle =0$ (see text). Right panel: Exchange splitting $[{\omega}_{p}\left(\mathrm{up}\right)-{\omega}_{p}\left(\mathrm{down}\right)]/\Delta $ as a function of $x\langle {S}_{z}\rangle /S$ for various values of x. The straight lines are adjusted to obtain the best fit with each set of x data. From Takahashi [43].

#### 3.5. The Case of Strong Attractive Potential

**Figure 8.**The results for the DMSs with strong attractive potential. Left panel: DOS as a function of $\omega /\Delta $ for various values of $\langle {S}_{z}\rangle /S$. The solid lines represent the down-spin carrier and the dotted lines represent the up-spin carrier. The impurity level ${E}_{a}=-1.25\Delta $ is indicated by the dot on the abscissa. Right panel: (a) A-site component of the DOS, $(1-x){D}_{\downarrow}^{A}\left(\omega \right)\Delta $ and $-(1-x){D}_{\uparrow}^{A}\left(\omega \right)\Delta $, (b) M-site component of the DOS, $x{D}_{\downarrow}^{M}\left(\omega \right)\Delta $ and $-x{D}_{\uparrow}^{M}\left(\omega \right)\Delta $. The thick, thin, and dotted lines represent the cases of $\langle {S}_{z}\rangle /S=1.0$, $0.5$, and $0.0$, respectively. From Takahashi and Kubo [42].

#### 3.6. Mechanism of Carrier-induced Ferromagnetism in DMSs

**Figure 9.**The result for Curie temperature ${T}_{c}/\Delta $ as a function of carrier density n for various values of $IS/\Delta $ with $x=0.05$ and ${E}_{M}=0$. The result based on the assumption that an impurity band has the same shape as the model DOS is drawn as ‘LIMIT’ (see text). From Takahashi and Kubo [42].

**Figure 10.**Phase diagram for ${E}_{M}=0$ and $x=0.05$. The critical value ${n}_{c}$ (solid line; left scale) and the maximum ${T}_{c}$ (dotted line; right scale) are presented as functions of $\left|IS\right|/\Delta $. The carrier density ${n}_{x}$ at which ${T}_{c}$ reaches the maximum is included (solid line; left scale). From Takahashi and Kubo [42].

**Figure 11.**The DOS of the impurity band in the case of $IS/\Delta =-1.0$ and ${E}_{M}=0$. The thick, thin, and dotted lines represent the cases of $\langle {S}_{z}\rangle /S=1.0$, $0.5$, and $0.0$, respectively. Dot A indicates the impurity level for $x\to 0$. From Takahashi and Kubo [41].

**Figure 12.**The result for ${T}_{c}/\Delta $ as a function of n for various values of ${E}_{M}/\Delta $ with $x=0.05$ and $IS=-0.4\Delta $. The result denoted by `LIMIT’ is included (see text). From Takahashi and Kubo [42].

#### 3.7. Specific Results for (Ga,Mn)As

**Figure 13.**The lower-energy part of the DOS of Ga${}_{1-x}$Mn${}_{x}$As for various values of x: (a) $\langle {S}_{z}\rangle =0$ and (b) $\langle {S}_{z}\rangle =0$. The solid lines represent the down-spin carrier and the dotted lines represent the up-spin carrier. The impurity level ${E}_{a}=-1.057\Delta $ is indicated by a dot on the line $x=0$. From Takahashi and Kubo [42].

**Figure 14.**The results for Ga${}_{1-x}$Mn${}_{x}$As with $x=0.05$. Left panel: Low-energy part of the DOS shown for various values of $\langle {S}_{z}\rangle /S$. The solid lines represent the down-spin carrier and the dotted lines represent the up-spin carrier. The arrow indicates the Fermi level for $n=x(=0.05)$. The impurity level ${E}_{a}=-1.057\Delta $ is indicated by a dot on the horizontal line $\langle {S}_{z}\rangle =0$. Right panel: (a) Ga-site component of the DOS, $(1-x){D}_{\downarrow}^{A}\left(\omega \right)\Delta $ and $-(1-x){D}_{\uparrow}^{A}\left(\omega \right)\Delta $ and (b) Mn-site component of the DOS, $x{D}_{\downarrow}^{M}\left(\omega \right)\Delta $ and $-x{D}_{\uparrow}^{M}\left(\omega \right)\Delta $. The thick, thin, and dotted lines represent the cases of $\langle {S}_{z}\rangle /S=1.0,0.5$, and $0.0$, respectively. From Takahashi and Kubo [42].

**Figure 16.**The result for Ga${}_{1-x}$Mn${}_{x}$As with $x=0.005$: (a) low-energy part of the DOS $D\left(\omega \right)$, (b) optical absorption band $A\left(\omega \right)$ in arbitrary units (arb. units). The solid lines represent the down-spin carrier and the dotted lines represent the up-spin carrier. Note that the energy of the bottom of the model band is $\omega =-\Delta $. Along the upper horizontal axis of (a), energies for $\Delta =$2 eV are graduated in eV, where the origin of the energy, 0.0 eV, is taken at $\omega =-\Delta $. The impurity level ${E}_{a}=-1.057\Delta $ (or $-0.113$ eV) is indicated by a dot on the line $\langle {S}_{z}\rangle =0$. From Takahashi [43].

**Figure 17.**An illustration of the difference between the characters of a hole in a II-VI-based DMS and in a III-VI-based DMS.

## 4. Summary and Concluding Comments

## Acknowledgement

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## Appendix

## Dynamical CPA–Locator Formalism

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Takahashi, M. Carrier States in Ferromagnetic Semiconductors and Diluted Magnetic Semiconductors—Coherent Potential Approach—. *Materials* **2010**, *3*, 3740-3776.
https://doi.org/10.3390/ma3063740

**AMA Style**

Takahashi M. Carrier States in Ferromagnetic Semiconductors and Diluted Magnetic Semiconductors—Coherent Potential Approach—. *Materials*. 2010; 3(6):3740-3776.
https://doi.org/10.3390/ma3063740

**Chicago/Turabian Style**

Takahashi, Masao. 2010. "Carrier States in Ferromagnetic Semiconductors and Diluted Magnetic Semiconductors—Coherent Potential Approach—" *Materials* 3, no. 6: 3740-3776.
https://doi.org/10.3390/ma3063740