1.
Introduction

Owing to their unique band structure, the valence electrons in III–V semiconductors can be excited directly into the conduction band; the choice of elemental combinations changes the band gap and lattice constant^{[2, 3]}. Thus, III–V semiconductor can be designed for emission/absorption devices of various photon wavelengths. Industrial applications are, such as, laser diodes, light-emitting diodes, and solar panel devices^[2–4]; a considerable effort is being devoted to their development. Nanotechnologies are opening new possibilities for electronic and photo devices^[5–7]. Semiconductor-based quantum systems are considered as possible quantum-computing devices^[8], in which quantum states, such as electron spin, photon polarity, and magnetic polarization, are manipulated. Ferromagnetic Mn-doped III–V semiconductors can supply polarized electrons and control electron spin states^{[9, 10]}, and small coils can be used to control the electron-nuclear-spin system^[11]. Influence of hyperfine interactions (HFI) between the lattice nuclei and electrons are to be considered for controlling electron-nuclear-spin systems.



The electron-nuclear-spin system of III–V semiconductors undergoes a ferromagnetic phase transition below a critical temperatureT_c, which occurs in a relaxation of artificially excited conduction electrons exhibiting thermally non-equilibrium spin distribution^{[1, 12–15]}. Contact HFI (the Overhauser effect^[16]) enables the conduction electrons to exchange spins with the lattice nuclei during relaxation, thus inducing a change in the nuclear polarization.



The Overhauser effect is usually observed in an applied external magnetic field, wherein the nuclear magnetization induced by the polarization of lattice nuclei is assumed to be negligible compared to the external magnetic field. However, internal nuclear magnetic field significantly affects the electron-nuclear-spin system in III–V semiconductors. In particular, the strength of the effective magnetic field acting on the conduction electrons with fully polarized lattice nuclei is estimated to be ?1.37, ?1.17, and ?2.76 T, respectively, for the nuclei⁶⁹Ga,⁷¹Ga, and ⁷⁵As in GaAs^{[14, 17]}.



Owing to the Overhauser spin-flip mechanism acting via the contact HFI, both the electron level structures and the nuclear magnetization (polarization) act on each other. This effect further enhances the nuclear magnetization and induces a positive feedback, thereby polarizing the lattice nuclei below T_c, which appears to be a second-order phase transition. In dynamic nuclear self-polarization (DYNASP), the large nuclear polarization is maintained because of the Overhauser effect in the large effective nuclear magnetic field that is produced by the magnetization of the lattice nuclei.



The DYNASP produced by linearly polarized light showed a stable, high-degree of nuclear polarization below the T_c, even in the presence of non-polarized conduction electrons, because the phenomenon is characterized by the relaxation of the conduction electrons exhibiting non-thermal distribution in the initial stage of excitation^[12–14]. However, electrons excited to the conduction band is generally achieved by elliptically polarized light; therefore, the DYNASP induced by elliptically polarized light was studied, which revealed some of its interesting features^{[1, 15]}. Therein, α of Eq. (9) in Ref. [1] denoted the effect of the conduction-electron polarization in the equation of nuclear polarization.



In our previous study^{[1, 15]}, however, the effect of the external magnetic field was not considered. This study therefore considers this effect, and uses α’ to denote the combined effect of α and the external magnetic field in the equations of nuclear polarization^{[1, 15]}.



The effect of conduction electrons on the nuclear polarization is overviewed herein. We then extend the DYNASP theory by introducing the effect of an external magnetic field. The newly obtained equations are applied to study the influence of the external magnetic field on the DYNASP phenomenon in an InP semiconductor.

2.
Theory of DYNASP

2.1
Spin polarization of optically excited electrons

The electron-nuclear-spin system in a semiconductor that is relevant to relaxation of the optically excitatied conduction electrons in DYNASP is overviewed herein. The left panel of Fig. 1 schematizes the band structure near the Γ-point of a III–V semiconductor. The arrows indicate an excitation-relaxation-and-recombination process. When the valence electrons are excited to the conduction band, they move around in the semiconductor. The Bohr radius of an electron trapped in a shallow donor potential can be as large as ~10 nm, which enables the electron to interact with ~10⁵ nuclei. The electrons therefore experience many magnetic fields produced each nuclei, which justifies the applicability of averaged magnetic field (i.e., the mean-field concept). In contrast, large electron densities are observed near the lattice nuclei because they attract electrons traveling through the semiconductor, thereby enhancing the effect of the nuclear magnetization on the electrons. The order of the effective magnetic field produced by fully polarized nuclei and that acts on the conduction electrons can be as large as 1 T^{[14, 17]}.






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Figure1.
Schematic band structure near the Γ-point of a III–V semiconductor. The abscissa is the wave vector k, and the ordinate is the excitation energy. The arrows indicate an excitation-relaxation-and-recombination process for an electron. The bands denoted by c, hh, lh, and sh are the conduction, heavy-hole, light-hole, and split-off bands, respectively, which have the corresponding angular momenta j = 0, 1, 0, and 0^[14]. (right) The relative interband transition probabilities between the magnetic sub-states of the valence band ($j=frac{3}{2}$) and the conduction band ($j=frac{1}{2}$).




The right-hand side of Fig. 1 shows the interband transition probabilities between the highest valence and lowest conduction bands. Using circularly polarized light of $sigma+ (sigma-)$ polarities, electrons are excited via transitions characterized by $Delta m_{{j}}=+1 (-1)$, where m_j is a magnetic sub-state of an electron state with an angular momentum j. If the lower states are populated uniformly, the number of electrons excited to the $+frac{1}{2} (-frac{1}{2})$ state by circularly polarized light of $sigma- (sigma+)$ is three times larger compared with those to thee other $-frac{1}{2} (+frac{1}{2})$ state, (i.e., only the hh and lh bands are considered). By varying the polarization of the light used for excitation, the spin polarization of the electrons can be controlled from ?0.5 to 0.5, where the polarization is defined by $S=frac{n_{
m{l}+}-n_{
m{l}-}}{n_{
m{l}}}$, where $n_{
m{l}+}$ and $n_{
m{l}-}$ are electron population of $-frac{1}{2}$ and $+frac{1}{2}$ states in the conduction band, and $n_{
m{l}}=n_{
m{l}+}+n_{
m{l}-}$. Because the lifetime of the conduction electron before it drops back into the valence band is much shorter than the spin-relaxation via week HFI, most electrons preserve their initial spin states.

2.2
DYNASP of lattice nuclei

As shown in Fig. 2, the energy levels of the conduction electrons in a magnetic field split into states with $m_{j}=pm1/2$. The interaction between a nucleus and an electron induces further hyperfine splitting. During thermal relaxation, the transitions caused by the contact HFI change the nuclear-spin states $Delta m_{
m{z}}=pm1$ by a transition of $Delta m_{{j}}=mp1$ (i.e., $Delta m_{
m{F}}=0$). The population ratio of the magnetic sub-states between $(+frac{1}{2}, ;m_{
m{z}})$ and $(-frac{1}{2}, ;m_{
m{z}}+1)$ tends to follow the Boltzmann distribution, thereby causing nuclear polarization.






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Figure2.
Energy levels of electrons in the conduction band. The ordinate is the excitation energy, and the abscissa is the magnetic field. As the magnetic field B increases, the energy gap $varepsilon$ between the magnetic sub-states $m_{j}=pmfrac{1}{2}$ increases. The hyperfine structure is shown on the right, where $m_{
m{F}}$ is the quantum number of the hyperfine magnetic sub-states. The dashed lines are transitions caused by the contact HFI. The nuclear spin $I=frac{3}{2}$ is assumed.




As the magnetic field resulting from the nuclear polarization increases, the energy splitting $varepsilon$ expands. This in turn enhances the nuclear polarization, thereby increasing the nuclear magnetization. The positive feedback in his thermal-relaxation induces a large nuclear polarization below the T_c.



To explain the DYNASP phenomenon a balance equation was introduced^{[1, 12–15]}. The population of conduction-electron spin states is assumed to follow the Boltzmann distribution during thermal relaxation. Solving the balance equation yields the following relation for the nuclear polarization P:

$$P = frac{{{I_{{ m{av}}}}}}{I} = {B_{ m{I}}}left[{Iln left( {frac{{1-{S_{ m{T}}}}}{{1 + {S_{ m{T}}}}}frac{{1 + S}}{{1-S}}} ight)} ight], $$

(1)

where $B_{
m{I}}(x)$ is the Brillouin function, I is the nuclear spin, $I_{
m{av}}$ is the projection of the nuclear spin along the magnetization axis, and S is the electron polarization. The electron polarization of the thermally equivalent condition $S_{
m{T}}$ is given by

$$frac{{1 - {S_{ m{T}}}}}{{1 + {S_{ m{T}}}}} = exp left( {frac{varepsilon }{{kT}}} ight), $$

(2)

where k is the Boltzmann constant (1.38 × 10^?23 J/K) and T is the temperature. The energy splitting of the conduction electron states $varepsilon$ is given by

$$varepsilon = AIP + {varepsilon _{{B_{ m{ex}}}}}, $$

(3)

where AIP is the contribution of nuclear polarization, and $varepsilon_{{B_{{
m{ex}}}}}$ is the contribution of external magnetic field $B_{
m{ex}}$: $varepsilon_{{B_{{
m{ex}}}}}=g_{
m{e}}mu_{
m{B}} B_{
m{ex}}$. The hyperfine splitting factor is A = $frac{2pi}{3I}mu_{0}g_{
m{e}}mu_{
m{B}}mu_{
m{I}}|phi(0)|^{2}$, where $mu_{0}$ is the permeability of free space, $g_{
m{e}}sim2.0$ is the g-factor of a free electron, $mu_{
m{B}}=$ 9.274 × 10^–24 J/T is the Bohr magneton, $mu_{
m{I}}$ is the magnetic moment of a nucleus, and $phi(0)$ is the amplitude of the electron wave function at the lattice position.



Introducing the parameters of the T_c of DYNASP defined by $T_{
m{c}}=I(I+1)A/(3k)$ and the electron polarization $alpha=frac{1}{2}{
m {ln}}left(frac{1+S}{1-S}
ight)$, we obtain

$$P = {B_{ m{I}}}left( {frac{{3I}}{{I + 1}}frac{{{T_{ m{c}}}}}{T}P + 2Ialpha + frac{{{varepsilon _{{B_{{ m{ex}}}}}}}}{{kT}}I} ight).$$

(4)

The first term $left(frac{3I}{I+1}frac{T_{
m{c}}}{T}P
ight)$ in the Brillouin function corresponds to the magnetic field induced by nuclear polarization, the second term $left(2Ialpha
ight)$ corresponds to the polarization of the conduction electrons, and the third term $frac{varepsilon_{{B_{{
m{ex}}}}}}{kT}I$ corresponds to the external magnetic field. The second and third terms in the Brillouin function can be combined as $2Ileft(alpha+varepsilon_{{B_{{
m{ex}}}}}/(2kT)
ight)=$ 2Iα′ Thus, we obtain

$$P = {B_{ m{I}}}left( {frac{{3I}}{{I + 1}}frac{{{T_{ m{c}}}}}{T}P + 2Ialpha '} ight).$$

(5)

The equations given above can be generalized to a lattice composed of several types of nuclei. Following the method proposed by Dyakonov et al.^[12] the average T_c is defined by

$$overline {{T_{ m{c}}}} = sumlimits_i {{a_{ m{i}}}} {T_{{ m{ci}}}} = sumlimits_i {{a_{ m{i}}}} frac{{{A_{ m{i}}}{I_{ m{i}}}({I_{ m{i}}}{ m{ + }}1)}}{{3k}}, $$

(6)

where $a_{
m{i}}$ is the density of the component nuclei in a unit cell of the lattice, $T_{
m{ci}}$ is the partial T_c, and i denotes the composing nuclei. Using the energy splitting of the conduction electrons $varepsilon=sum_{
m{i}}:a_{
m{i}}A_{
m{i}}I_{
m{i}}P_{
m{i}}+varepsilon_{{B_{{
m{ex}}}}}$, the nuclear polarization of a lattice nucleus denoted by j is obtained:

$${P_{ m{j}}} = {B_{{{ m{I}}_{ m{j}}}}}left( {frac{{3{I_{ m{j}}}}}{T}frac{{sumlimits_i : {a_{ m{i}}}{A_{ m{i}}}{I_{ m{i}}}{P_{ m{i}}}}}{{3k}} + 2{I_{ m{j}}}alpha '} ight).$$

(7)

3.
The DYNASP phenomenon

Section 2.2 theoretically reviews the DYNASP. The relation between nuclear polarization P (or $P_{
m{j}}$), conduction-electron spin polarization α, and external magnetic field $B_{
m{ex}}$ is achieved. The resulting Eqs. (5) and (7) are similar to those reported previously^{[1, 15]}; however, α is substituted by α′ ($=alpha+varepsilon_{
m{ex}}/(2kT)$); this indicates that the DYNASP functions similarly under both conditions. As an example, let us consider the polarization of ¹¹⁵In in an InP semiconductor.



We determined the ¹¹⁵In polarization by iterating Eq. (7) according to the Ref. [1], varying the parameters α′ and T. In this calculation, the values of the hyperfine constants and the quantity $eta$s from Ref. [1, 15, 18–20] were considered, and we neglected the low abundance of ¹¹³In (4%).



Fig. 3 shows the temperature dependence of the nuclear polarization of ¹¹⁵In. When $alpha'=0$, large nuclear polarization is achieved below T_c, and it vanishes above T_c . In contrast, when $alpha'>0$, nuclear polarization is achieved even above T_c. It should be noted that both ¹¹⁵In and ³¹P show almost identical T_c of approximately 2.3 K, although $T_{
m{ci}}$ of ³¹P and ¹¹⁵In are 0.11 K, and 4.4 K, respectively. The obtained T_c agrees with average critical temperature $overline{T_{
m{c}}}$ = 2.3 K deduced using Eq. (6). This is owning to the mean-field treatment, with the average magnetic field produced by a large number of lattice nuclei.






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Figure3.
Temperature dependence of the nuclear polarization of ¹¹⁵In in a InP semiconductor. The numbers on the curves are the values of α′.




Fig. 4 shows the nuclear polarization of ¹¹⁵In by varying α’. The hysteresis phenomenon was observed below 2 K, which is less than the T_c. Above the T_c, the nuclear polarization shows S-shaped curves. Below the T_c, the hysteresis loop becomes larger as the temperature decreases. It should be noted that the shapes of the hysteresis loops of ³¹P resemble those of ¹¹⁵In, whereas the nuclear polarization of ³¹P is smaller than that of ¹¹⁵In. This is owning to the smaller values of the nuclear spin of ³¹P (I = 1/2).






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Figure4.
The dependence of α′ on the nuclear polarization of ¹¹⁵In in a InP semiconductor. The arrows show the direction of the change in polarization of the hysteresis curves.




α′ is determined by the electron-spin polarization α and the energy splitting of conduction band state $varepsilon_{{B_{{
m{ex}}}}}$ induced by the external magnetic field. α takes the value of $-0.55 < alpha < 0.55$ for electron-spin polarization of $-0.5 < S < 0.5$. For example, the parameter $varepsilon_{B_{
m{ex}}}/(2kT) sim 0.5$ is achieved for an external magnetic field $B_{
m{ex}}=1$ T, and temperature T = 2 K. The contribution of DYNASP of an external magnetic field < 1 T at 2 K is therefore the same order of magnitude as that of electron-spin polarization α.

4.
Summary

The theory of DYNASP phenomenon in III–V semiconductors was reviewed herein, and the effect of an external magnetic field was introduced. We expect the same behavior to be observed in DYNASP, as discussed in Refs. [1, 15] by replacing the electron-spin polarization α with the α′ defined by $alpha'=alpha+varepsilon_{
m{ex}}/(2kT)$. The DYNASP phenomena discussed above are summarized as below:



(1) When $T < T_{
m{c}}$, the nuclear polarization increases steeply with decreasing temperature. The T_c agree with the averaged critical temperature $overline{T_{
m{c}}}$.



(2) When $alpha'=0$, no nuclear polarization is achieved above the T_c. However, the nuclear polarization is enhanced near and above the T_c with increase in α′.



(3) The nuclear polarization can be controlled by α′. Below the T_c, the nuclear polarization shows a hysteresis loop as a function of α′. Above the T_c, the nuclear polarization is simply determined by α′.



The DYNASP phenomenon enables us to achieve a high degree of nuclear polarization and to control the amount of nuclear polarization. By combining the NMR and DYNASP, the direction of the nuclear polarization can be manuipulated, thus widening the range of applicability of electron-nuclear-spin systems in semiconductors to other areas of research, such as NMR imaging^[21], quantum computing^[22], and magnetic-moment measurements of unstable nuclei^{[23, 24]}.