Dynamics of the electron-nuclear and heteronuclear polarization transfers in optically-oriented semi-insulating InP:Fe
Dynamics of the electron-nuclear and heteronuclear polarization-transfer processes in the optically oriented semi-insulating iron-doped indium phosphide were investigated through the characteristic time scales of the processes. (1) We find in the optical nuclear orientation process that the buildup times for P and In nuclear polarizations are of the same order. From the analysis of the cross-relaxation process between photo-excited electrons and nuclei, it is concluded that electron-nuclear dipolar-couplings are responsible for the polarization transfer in this case. This example shows that the nuclear-site dependence of the buildup time can be utilized to identify hyperfine couplings responsible for the process. (2) We find in the heteronuclear polarization transfer (cross polarization) process between optically oriented P and In that the cross-relaxation time is rather short; it is an order of magnitude shorter than that expected for nuclear dipolar couplings when a magnetic field is applied parallel to the crystalline  axis. From the cross polarization spectral density analysis, it is concluded that a large -coupling of the order of 2 kHz exists between these nuclei and that its angular dependence is not of a simple pseudo-dipolar type.
pacs:78.30.Fs 32.80.Bx 76.70.Fz
Dynamic nuclear polarization (DNP) is a method of creating hyperpolarized nuclear spins in solids, liquids or gases in a non-equilibrium fashion. For decades, it has been utilized to enhance sensitivity in the nuclear magnetic resonance (NMR) method. Recently, nuclear-spin-related phenomena in semiconductors have attracted much attention, which has added renewed interests to DNP. Examples include electron spin dynamics in semiconducting nanostructuresschliemann03 and solid-state NMR quantum computers;shimizu02 the hyperpolarized nuclei are utilized as a means to control electron spin states in the former, while they are expected to serve as initialized states for quantum computation in the latter. With the emergence of these new applications, it has become increasingly important to control positions, degrees of polarization and the nuclear species to be polarized, i.e., to manipulate hyperpolarization.
To achieve this purpose, one needs to create, transfer and localize nuclear spin polarizations in desired positions efficiently, which may require a variety of techniques to be integrated.tycko98a Nuclear spin polarizer (NSP) is one of such schemes.goto03a ; goto04a In this scheme, hyperpolarization is created in compound semiconductors such as InP and GaAs by the optical orientation (optical pumping) method,meier84 ; tycko96 where nuclear spins are hyperpolarized by polarized electrons photo-excited by circularly polarized light with the band gap energies of the semiconductors. The hyperpolarization thus created is manipulated by means of various polarization transfer techniques in solids, such as the cross polarization, polarization transfer and spin diffusion, which enable us to transfer hyperpolarization between different nuclear species, different portions in the semiconductor, or even different materials through the interfaces. In the last case, the polarization can be localized at the interfaces. tycko98a ; goto03a ; goehring03 ; goto07a
The polarization transfers in this scheme rely on hyperfine- and heteronuclear-couplings in semiconductors so that it is essential to understand their characteristics. Our understanding on them is still insufficient, however. The characteristics of the hyperfine couplings responsible for the polarization transfer from photo-excited electrons to nuclei in the optical orientation process are still open to debate. patel99 ; paravastu04 ; paravastu05 On the other hand, the strengths and angular dependences of the heteronuclear couplings in InP have been measured in the thermal equilibrium,tomaselli98 ; iijima03 but those between optically oriented nuclei still remain undetermined.
In this paper, we address these issues from the viewpoint of the characteristic time scales of the polarization transfer processes i.e., the cross-relaxation time between hyperpolarized nuclei, and the buildup time for nuclear polarization by photo-excited electrons, . The former () provides us with information on heteronuclear couplings between hyperpolarized nuclei. In §III, we show in the case of the semi-insulating iron-doped InP (InP:Fe) that the polarization transfer is predominantly mediated by nearest-neighbor indirect -couplings, and that their angular dependence is not of a simple pseudo-dipolar type. The latter (), on the other hand, provides us with information on the hyperfine couplings responsible for the nuclear spin orientation, which are closely related to the states of the photo-excited electrons. In §IV, we show that the nuclear-site dependence of is a good measure to determine the types of hyperfine couplings and that, in the case of InP:Fe, the polarization transfer from photo-excited electrons to nuclei is primarily brought about by the electron-nuclear dipolar couplings.
Ii Experimental Methods
The optical-nuclear-orientation and cross-polarization experiments were performed at 10 K with the optical pumping double resonance system.goto06 The system includes a two-channel (XY) NMR spectrometer (Apollo, Tecmag Inc.), a Ti:Sapphire tunable laser (3900S) pumped by a diode-pumped Nd:YVO cw green laser (Millennia Vs, Spectra-Physics Inc.), and a home-built top-loading XY double-resonance probe with an optical fiber attachment. The probe is installed in a dynamic gas-flow cryostat (Spectrostat 86/62, Oxford Instruments Inc.), which is mounted on a 270 MHz (6.346 T) wide-bore superconducting magnet (Oxford Instruments Inc.). The Ti:Sapphire laser provides linearly polarized light with the wavelength ranging between 600 and 1000 nm, which is transmitted to the sample space at the probe end by a polarization maintaining optical fiber (PANDA, Fujikura Co. Ltd.), then converted to circularly polarized light with a quarter waveplate before being applied to a sample. The sample used in this study was a wafer of the semi-insulating iron-doped InP with the crystal orientation of (100) and the carrier density at room temperature of cm (Showa Denko, lot 20044202). It was set inside a pickup coil at the probe end with the surface normal to the magnetic field and the light beam.
The pulse sequence for (primary nuclei, P), (secondary nuclei, In) and IR (infrared light with the photon energy of and the helicity of ) used in the optical-orientation-cross-polarization experiments is schematically shown in Fig. 1. It consists of the following four processes, i.e., (1) saturation, (2) optical pumping, (3) cross polarization and (4) detection. (1) At the beginning, saturation pulses consisting of eight pulses are applied to both the nuclei, which extinguish the initial polarizations at thermal equilibrium. (2) The sample is irradiated only with the infrared light for the duration of , which creates polarizations of both the nuclei inside the illuminated region of the sample. The polarization in the bulk (outside of the illuminated region) can also grow toward the equilibrium state for this duration. The -polarization, however, does not recover because the spin-lattice relaxation time at P is much longer than .goto04b . The -polarization, on the other hand, is extinguished again by the saturation pulses at the end of the duration. Consequently, only the optically oriented -polarization in the illuminated region remains at the end of the duration. (3) The cross-polarization is applied between the - and - spins, which transfers the -polarization to in the illuminated region, but not in the bulk because of the lack of -polarization there. (4) The signal from only the illuminated region is detected as a free induction decay. In our experiments, the effective duration time was fixed at 120 s.
Iii Cross-relaxation in hyperpolarized nuclear spins
In general, the contact time () dependence of the -spin magnetization in the cross polarization process is given by,
which results from the competition between the polarization transfer process from the -spins with the characteristic time , and the decay process of the -spins in the rotating flame characterized by .
The dependence of the In magnetization under the IR irradiation is shown in Fig. 2. One may find that the decay process is negligible (). In fact, (P) was reported to be as long as 80 ms,tomaselli98 which is much longer than . Setting , Eq. (1) can be reduced to,
It is intriguing to see whether or not is affected by the difference in the photon energy . We measured at two photon energies, = 1.386 and 1.407 eV, at both of which the P polarization is strongly enhanced by the optical orientation effect. The result is summarized in Table 1, which shows that is independent of within the experimental error. One of the possible explanations for this result may be relaxation of the photo-excited electrons, which occurs with a time scale much faster than , so that the electrons excited with different would result in the same meta-stable state. We also measured at two different IR-irradiation times = 60 s and 240 s, which is intended to investigate the effect of nuclear spin diffusion process. It is expected that for greater , nuclear polarizations may spread out farther from the positions where polarizations are originally created. Provided the spin diffusion constant cm/sec,abragam61 the expected diffusion lengths = 24 and 49 nm for = 60 s and 240 s, respectively.goto04a The result is shown in Table 1, which indicates that is almost independent of . i.e., is not very sensitive to the distance from the photo-excited electrons, at least, up to about 50 nm.
The cross relaxation rate can be calculated provided that all the internuclear couplings are given. In the following, we calculate using the formalism by Demco et al.demco75 ; mehring83 and compare the results with the experimental values. We show that the contribution of nuclear dipolar couplings to is too small to account for the experimental values, and that indirect -couplings are inevitable to account for it.
The cross polarization process with spin-locking can be expressed in terms of the -component of the cross polarization spectrum density . Assuming a Gaussian function for , is approximated by,demco75 ; mehring83
Here, is the correlation time for given by,
where is the second Legendre polynomial, and three values are given by the following lattice-sums;S2
and are the coefficients of the secular terms in the homo () and hetero () nuclear couplings, respectively. If only the nuclear dipolar couplings are responsible for them, they are given by,
Here, and are the respective gyromagnetic ratios. and are the vectors corresponding to the and bondings, and and are the angles between and the corresponding vectors, respectively (see Fig. 3). is the second moment for the heteronuclear couplings, which is given in the case of the dipolar couplings by,
The off-resonance factors , and are schematically shown in Fig. 4. They are defined as,
where and are the pulse intensities () in units of angular frequency, and and () are the offsets and the effective fields defined by,
where and are the resonance and the transmitter angular frequencies, respectively. If both and are on-resonances () and the Hartmann-Hahn condition is fulfilled (),
so that Eq. (3) is reduced to,
Now, let us calculate the dipolar contributions to in our case. The off-resonance effect is negligible in our experiments because,
so that, , and , i.e., the conditions in Eq. (10) are nearly fulfilled.
The estimation of requires the calculation of lattice sums in Eq. (5). In the present case, all the nearest neighbor spins are at the magic angle positions () as seen in Fig. 3, so that the summations in Eq. (5) start with the second nearest neighbor sites. Using the lattice constant madelung04 the -values in Eq. (5) are calculated as,
This value is by one order of magnitude smaller than the experimental values shown in Table 1. This difference is presumably due to the -couplings with the nearest-neighbor P nuclei, .
Here, we neglect the small next-nearest-neighbor homonuclear -couplings P-P) of the order of 10 Hz.iijima03 On the other hand, is given by,
where is the contribution from the -couplings. Note that no cross-terms between the dipolar and -couplings exist in Eq. (16) because of the absence of the dipolar couplings between the nearest-neighbor sites.tomaselli98 The value of is determined so that the observed s in Table 1 is reproduced. As a result, we found that it falls within the range kHz. We assume rather large error in taking into account the fact that is approximated by Gaussian for simplicity. demco75 ; mehring83
The presence of is consistent with the line width obtained experimentally. Assuming the Gaussian form of the spectrum, the line width is given with the second moment by,mehring83
which yields = 2.4 kHz. In reality, it is found that the In spectrum is rather close to Lorentzian with the full width at half of the maximum intensity (FWHM) of kHz.goto07a The observed FWHM is 1.7 times as large as . Assuming that kHz, is calculated as,
which yields = 4.1 kHz, reproducing the experimentally observed FWHM. The assumed =1.4 kHz is rather small compared to that estimated from , kHz. This is probably due to the ambiguities in the estimation of as well as the determination of the FWHM in the spectrum whose shape is not a Gaussian.
Tomaselli et al. discussed the -couplings in InP in the cross polarization/magic angle spinning (CP/MAS) experiments.tomaselli98 Assuming of the pseudo-dipolar type,
with being the angle between the nearest-neighbor P-In bond and the magnetic field, they determined the isotropic and anisotropic parts of the -coupling as Hz and ) or () Hz. A similar conclusion has been reported by Iijima et al.iijima03 In the present case where , the anisotropic part of Eq. (21) is zero, so that , whereas the value kHz obtained in the present study is much greater than kHz.
One possible explanation for the large at is that the angular dependence of is not of a simple pseudo-dipolar type, but of the anisotropic pseudo-exchange type,bloembergen55
which is a generic form of the angular dependent -coupling including the pseudo-dipolar one as a special case. In this case, , which yields non-zero value except for corresponding to the pseudo-dipolar case. The present data are still insufficient to determine the angular dependence of the anisotropic . Nevertheless, they show that the -coupling is not of a simple pseudo-dipolar type. Since the dipolar type angular dependence is averaged out in the MAS experiments, the determination of the angular dependence of may require measurements of cross relaxation times in static cross polarization experiments.
Iv Buildup time in the optical pumping process
In the previous section, we have shown that the characteristic time for the cross polarization process provides us with information on the heteronuclear couplings responsible for the polarization transfer. A similar argument is possible for the characteristic time in the nuclear polarization process by the optical pumping, i.e., the buildup time . It provides us with information on the hyperfine couplings responsible for the nuclear spin orientation.
There are two types of possible hyperfine interactions in semiconductors, i.e., Fermi contact and dipolar interactions. In the former case, photo-excited electrons are captured at shallow donor levels whose wave functions have the diameter of the order of 100 Å. The nuclear spins inside the wave functions are directly polarized through the flip-flop terms in the Fermi contact interaction ().S In the latter case, on the one hand, photo-excited electrons are localized at donor sites such as deep centers. Since little Fermi interactions exist with the surrounding nuclei, the nuclear spins near the donor sites are polarized through the non-secular terms of the dipole interaction (). That is, the type of hyperfine coupling is closely related to the state of the polarized electrons, so that its elucidation is essential to understand the mechanism of the optical nuclear orientation.
Many authors have addressed this issues so far. In the 1970’s, Bagraev et al. examined the buildup time of Si in the presence of deep centers in compensated silicon and argued the types of hyperfine couplings responsible for the optical orientation in this material.bagraev77 More recently, Patel et al. addressed this issue in InP.patel99 He proposed that the two mechanisms can be distinguished from the difference in the sign of the nuclear polarization relative to that at thermal equilibrium, and concluded that it is the dipolar coupling that causes nuclear polarization in undoped n-type InP. Paravastu et al. suggested in the case of semi-insulating GaAs, that the photo-excited electrons localized at donor sites cannot be solely responsible for macroscopic nuclear polarization.paravastu04 A factor that brings complications into the arguments is the presence of nuclear spin diffusion, which is believed to convey polarization farther away from the photo-excited electrons to achieve bulk nuclear polarization. Goehring et al. pointed out in InP nanoparticles that the nuclear spin diffusion process is rather slow,goehring03 suggesting that the spin diffusion might be rather insufficient to convey nuclear polarization in bulk materials.
Here, we show that the nuclear-site dependence of provides a clue to identify hyperfine couplings responsible for the buildup. We previously reported the nuclear-site dependence of in InP:Fe,goto04b which is summarized in Table 2, together with that of the spin-lattice relaxation rate at 300 K. Ep-dependence At first sight, it seems rather peculiar that the values of at In and P are of the same order, although those of are different by four orders of magnitude between them. If the Fermi contact were responsible for both and , at In would be forty five times as long as that at P.goto04b This result indicates that and are subject to different mechanisms from each other. In fact, we show in the following that is primarily caused by the Fermi contact interaction with conduction electrons, while is mainly caused by the dipolar interactions with localized electrons. The nuclear-site dependence of stems from the difference in the probability of electrons at the nuclear sites, while that of stems from the difference in the distance from the localized electrons.
For the Fermi contact interaction with thermally excited electrons, is given by,abragam61
with being the probability of electrons/holes at the nuclear site, and being the carrier density. Hence, the large difference in between P and In originates from that in . At indium sites, the conduction band consists mainly of s-orbitals and has large , while it is small at phosphor sites where the wave function mainly consists of p-orbitals of the valence band.
The Fermi contact interaction, however, is less effective for non-degenerated trapped electrons because of the following reason. To conserve energy in the flip-flop process (), the electrons should be excited to the state with the small excitation energy of corresponding to the nuclear Zeeman energy, while no such excited states are available at Fermi level in the non-degenerated electrons. The dipolar interaction, on the other hand, contains non-secular terms such as , which flip nuclear spins without flipping electron spins . In this case, is given by,abragam61 ; patel99
which contains no . Hence, contrary to the case of the Fermi contact interactions where is the origin of the nuclear-site dependence, the nuclear-site dependence in this case stems from the differences in the gyromagnetic ratio () and the lattice-averaged , i.e., .
The nuclear-site dependence of allows us to estimate the ratio of between P and In. Taking into account the fact that , Eq. (24) yields,
Using the relation and listed in Table 2, one obtains,
which means that indium nuclei are closer to the polarized electrons in average. This result is consistent with the values estimated from the lattice sums of as shown below. The calculation of the lattice sums of can be performed with the same calculation scheme as that used in Eq. (13) in §III. Assuming that the polarized electrons are localized at phosphor sites, the calculation yields,
To summarize, the experimental finding that and in InP:Fe are of the same order indicates that the dipolar coupling is mainly responsible for in this material. This is consistent with the conclusion by Patel et al. deduced from the sign of the nuclear polarizations.patel99 The present data also indicate that the photo-excited electrons may be located at phosphor sites, which may be related to antisites or iron trapping centers. This example shows that the nuclear-site dependence of provides us with information on the types of hyperfine couplings responsible for the nuclear spin orientation in the optical pumping process.
In conclusion, we have investigated the characteristics of heteronuclear and hyperfine couplings in optically oriented semiconductors using the semi-insulating InP:Fe. We have focused on the time scales in the polarization transfer processes, i.e., the cross-relaxation time in the cross polarization and the buildup time in the optical nuclear orientation. We find that is greater than that expected from the nuclear dipolar couplings by one order of magnitude. This discrepancy can be accounted for by assuming the -couplings of the order of 2 kHz between nearest-neighbor In-P. The angular dependence of the -coupling is inconsistent with that of a simple pseudo-dipolar type previously assumed, suggesting different anisotropy of the -coupling. On the other hand, we show that the nuclear-site dependence of provides a clue to identify the hyperfine coupling responsible for the optical nuclear orientation. We find in the case of InP:Fe that and are of the same order, indicating that the electron-nuclear dipolar coupling is primarily responsible for the nuclear spin orientation. It is also suggested that the photo-excited electrons are likely to be located at phosphor sites.
This work was partially supported by Special Coordination Funds for Promoting Science and Technology of the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, Industrial Technology Research Grant Program from New Energy and Industrial Technology Development Organization (NEDO) of Japan, and Grant-in-Aid for Basic Research from Japan Society for Promotion of Science (JSPS).
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