Microscopic States and the Verwey Transition of Magnetite

Subscriber access provided by UNIV OF DURHAM

Communication

Microscopic states and the Verwey transition of magnetite nanocrystals investigated by nuclear magnetic resonance Sumin Lim, Baeksoon Choi, Sang Young Lee, Soonchil Lee, Ho-Hyun Nahm, Yong-Hyun Kim, Taehun Kim, Je-Geun Park, Jisoo Lee, Jaeyoung Hong, Soon Gu Kwon, and Taeghwan Hyeon Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b04866 • Publication Date (Web): 20 Feb 2018 Downloaded from http://pubs.acs.org on February 21, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Nano Letters is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Microscopic states and the Verwey transition of magnetite nanocrystals investigated by nuclear magnetic resonance Sumin Lim1, Baeksoon Choi1, Sang Young Lee1, Soonchil Lee 1*, Ho-Hyun Nahm2, YongHyun Kim2, Taehun Kim3,4, Je-Geun Park3,4, Jisoo Lee5,6, Jaeyoung Hong5,6, Soon Gu Kwon5,6, and Taeghwan Hyeon5,6 Department of Physics, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141, Republic of Korea 2 Graduate School of Nanoscience and Technology, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141, Republic of Korea 3 Center for Correlated Electron Systems, Institute for Basic Science (IBS), Seoul 08826, Republic of Korea 4 Department of Physics and Astronomy, Seoul National University (SNU), Seoul 08826, Republic of Korea 5 Center for Nanoparticle Research, Institute for Basic Science(IBS), Seoul 08826, Republic of Korea 6 School of Chemical and Biological Engineering, and Institute of Chemical Processes, Seoul National University, Seoul, 08826, Korea

1

*

Corresponding Author: [email protected]

57

Fe Nuclear Magnetic Resonance (NMR) of magnetite nanocrystals ranging in size

from 7 nm to 7 μm is measured. The linewidth of the NMR spectra changes drastically around 120 K, showing microscopic evidence of the Verwey transition. In the region above the transition temperature, the linewidth of the spectrum increases and the spin-spin relaxation time decreases as the nanocrystal size decreases. The linewidth broadening indicates the significant deformation of magnetic structure and reduction of charge order compared to bulk crystals, even when the structural distortion is unobservable. The reduction of the spin-spin relaxation time is attributed to the suppressed polaron hopping conductivity in ferromagnetic metals, which is a consequence of the enhanced electron-phonon coupling in the quantum-confinement regime. Our results show that the magnetic distortion occurs in the entire nanocrystal, and does not comply with the simple model of the core-shell binary structure with a sharp boundary. Keywords: magnetite, nanocrystal, NMR, Verwey transition, charge order, hopping

1

ACS Paragon Plus Environment

Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nanoparticles of magnetic compounds have been actively studied for a rich variety of potential applications, such as magnetoresistive random access memory1 in nanoelectronics, contrast agents for magnetic resonance imaging2, and heat mediators3,4 in biomedicine. Technological advances in synthesis and characterization drove recent research development. Magnetite (Fe3O4) nanocrystals can be synthesized with a size deviation of several nanometers or less5,6, and nm-resolution images are available via transmission electron microscopy (TEM).

Magnetite, the oldest magnetic material known to mankind, still has many properties that are not fully understood, including the Verwey transition. The material has an inverse spinel structure in which the tetrahedral (A) sites are occupied by Fe3+ ions and the octahedral (B) sites are alternately occupied by Fe2+ and Fe3+ ions. Between the Curie temperature (TC = 858 K) and the Verwey transition temperature (TV = 123 K7), the material is ferrimagnetic and halfmetallic due to the double exchange interaction. As temperature passes through the Verwey transition temperature from above, the conductivity decreases by 2 orders of magnitude, the crystal symmetry changes from cubic to monoclinic8-10, and the magnetization suddenly decreases. A microscopic explanation of these behaviors is the electron hopping between Fe2+ and Fe3+ ions in B sites. Above TV, the easy electron hopping occurs, so that the average valence of all ions is 2.5+ and the crystal has cubic symmetry. Below TV, the strength of the double exchange interaction11, and subsequently, ferromagnetic order of the iron spin in the B sites are reduced12. This means that hopping rates drop significantly. Moreover, the Jahn-Teller distortion of Fe2+ ions with 3d6 configuration reduces the crystal symmetry. The charge and Fe2+ orbital order appear in a complicated way with localized electrons distributed over three Fe ions13. The observation of the Verwey transition of magnetite nanocrystals is of great importance because it is the hallmark of good crystallinity and bulk properties. Recently, the 2


Page 2 of 21

Page 3 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

size dependence of the Verwey transition was studied with stoichiometric magnetite nanocrystals by conductance, magnetization, and specific heat measurements14. The study discovered that TV decreases with decreasing the crystal size and the transition is gradually suppressed until it disappears at 6 nm. Similar measurements showed the transition occurs in octahedral magnetite nanoparticles of 6-14 nm in size but not in spherical nanoparticles of similar size15. A jump in the bandgap from the dI/dV measurement using scanning tunneling spectroscopy was also observed for a 10 nm nanocrystal16. Nuclear Magnetic Resonance (NMR) techniques have been invaluable for studying the microscopic states of magnetite near the Verwey transition temperature. For instance, NMR was used to study temperature dependence of magnetization17, and nuclear spin relaxation rates18. The charge and orbital ordering structures can be investigated when the detailed split peaks were obtained with high quality samples8,12,19,20. Comparison of the density functional theory calculations with the NMR resonance frequencies confirmed the electronic structure below the Verwey transition21,22. The first NMR study on the magnetite nanoparticle compared the Fe NMR spectrum to that of the maghemite (Fe2O3) nanoparticle23. In this work, we study the electronic and magnetic states of magnetite nanocrystals over a wide range of temperatures around the Verwey transition temperature using 57Fe NMR. Our results show that the NMR spectral linewidth change drastically near 120 K, providing a microscopic evidence of the Verwey transition in magnetite nanocrystals. The linewidth variation with the nanocrystal size indicates distortions in charge and spin orders. The crystal size dependence of the hopping conductivity is extracted from the spin-spin relaxation time (T2) above the transition temperature. The result is explained by the polaron hopping with the electron-phonon coupling enhanced in the quantum-confinement regime. We used several stoichiometric nanoparticles in various sizes (7, 16, 25, 36, 40, and 80 nm) (Fig. 1). Nanoparticles were synthesized by thermally decomposing iron acetylacetonate 3


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(Fe(acac)3) in benzyl ether and oleic acid under the reductive gas mixture of CO/CO2 4/96 in mass ratio. The detailed description on the synthesis and characterization can be found in ref. 14. In addition, a 7 μm sample was prepared by applying heat treatments on commercial Fe3O4 (Alfa Aesar). Because the 7 μm sample exhibits bulk-like characteristics in various measurements including NMR, it is used as the reference sample when comparing the nanoparticles to bulk. The 57Fe NMR spectrum and spin-spin relaxation time were measured at room temperature to liquid helium temperature with zero external magnetic field. Conventional Hahn-echo pulsed sequence with 0.5 μs and 1 μs pulse widths was used to obtain an echo signal in a home-built NMR spectrometer. Rf pulse power was carefully controlled to obtain signal only from the domains. Because of the linewidth broadening and the decrease in the spin-spin relaxation times, the signal-to-noise ratio rapidly decreases with the decrease of crystal size. Saturated magnetic moment of nanocrystals was measured by a Quantum Design SQUID magnetometer MPMS 5XL. More details of the experiment and supplementary data can be found in the supporting information file.

4


Page 4 of 21

Page 5 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Figure 1 - TEM images of the magnetite nanoparticles.

Figure 2 (a) The Fe NMR spectral intensity of the 25 nm sample obtained at temperatures around 𝑻𝑽 . NMR spectra of the nanocrystals and the bulk reference (7 μm) obtained (b) at 130 K (above TV) and (c) at 110 K (below TV). Short vertical lines at the bottom of (c) represent the positions of the split peaks observed in bulk crystals with good crystallinity19,20.

Typical

57

Fe NMR spectra of the magnetite nanocrystals obtained at various 5


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 21

temperatures between 100 K and 130 K are plotted in Fig. 2(a) for the 25 nm sample. The abrupt change of the spectrum near 118 K specifies the Verwey transition. The linewidth increases from about 0.5 MHz to 1 MHz as the temperature decreases below 118 K. Other nanocrystalline samples exhibit the same feature: the spectral shape and linewidth remain constant in the temperature range away from TV, but undergoes a rapid jump at the transition (Figure S1, Supporting Information). The spectra at the bottom of Fig. 2(b) and (c) are the Fe NMR spectra of the 7 μm sample obtained at 130 K and 110 K, above and below TV, respectively. The spectrum above TV shows a sharp single peak with the linewidth of about 3 kHz. The Fe NMR signals around 70 MHz come from the iron ions in A sites. The signals from B sites are much smaller than those from A sites above TV, and hardly observable below TV for the nanocrystalline samples. The resonance frequency of the NMR signal for magnetic materials in zero external field is proportional to the internal field experienced by the nucleus: 𝐴𝜇 + 𝐵𝑑𝑖𝑝 ,

(1)

where 𝐴 and 𝜇 represent the hyperfine coupling constant and the electronic magnetic moment in the same ion, respectively, and 𝐵𝑑𝑖𝑝 is the dipolar field generated by the electronic moments in the neighboring ions. The hyperfine coupling constant sensitively depends on the state of the spin and orbital of a magnetic ion. Therefore, the hyperfine field at A sites in magnetites reflects the oxygen environment surrounding the Fe3+ ion, while the dipolar field reflects the distribution of Fe2+ and Fe3+ ions in the neighboring B sites, that is, the state of the charge order. At temperatures above TV, where the bulk magnetite structure is cubic, all A sites are crystallographically equivalent and all iron nuclei experience the same hyperfine field. Every A site sees the same amount of the dipole field because the electron hopping averages the 6


Page 7 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

valence charge of the B sites. Therefore, a sharp single peak is observed in the NMR spectrum above TV. On the other hand, at temperatures below TV where the crystal symmetry of the bulk magnetite is reduced to monoclinic, eight A sites are crystallographically inequivalent. Accordingly, eight NMR peaks are expected and sharp split peaks are observed in high-quality bulk single crystals19,20. The positions of the split peaks are depicted as short vertical lines at the bottom of Fig. 2(c). In the spectrum of our 7 μm sample, each of the peaks are broadened and merged into three broad peaks as shown in the figure due to deteriorated crystallinity. This is consistent with previous observations in magnetite bulk samples with mediocre crystallinity17,18. The hyperfine and dipolar fields determine not only the resonance frequency but also the broadening in NMR for magnetic materials. Defects and impurities in deteriorated crystals may broaden the spectrum of each peak by varying the hyperfine and/or the dipolar fields. The hyperfine field is diversified when the symmetry-breaking distortion is induced in the oxygen environments in A sites. The broadening due to the diversified hyperfine field should be observed whether the structure is monoclinic (under TV) or cubic (above TV). This is not the case of our observation. The dipolar field is diversified when the charge order in B sites is reduced by the defects and impurities. The broadening due to the diversified dipolar field should be observed only under TV because the dipolar field is averaged out by hopping above TV. This is indeed what we observe in Fig. 2(b). Therefore, the broadening of the spectrum below TV is not due to the structural distortion in A sites but due to the disorder in the charge order in B sites. X-ray absorption spectroscopy (XAS) and X-ray powder diffraction (XRD) measurements on our samples also did not show appreciable difference in structure from bulk14. The results imply that the charge order in magnetites is very sensitive to the crystallinity. Thus when the imperfection in the crystallinity is too small to be observed from XAS or XRD, NMR measurements can be a good indicator of the crystal quality of magnetite. 7


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The spectra of the nanocrystalline samples obtained above and below TV are plotted together with those of the 7 μm sample in Fig. 2(b) and (c), respectively. We can see that the spectral shape of each nanocrystal below TV is approximately given by convoluting its spectrum above TV (Fig. 2(b)) with the 7 μm sample spectrum below TV (Fig. 2(c)). In other words, two factors determine the spectral linewidth of the nanocrystals. One is the broadening due to the reduced charge order in B sites as in the bulk sample, and the other which is irrelevant with the charge order and increases with decreasing size. Stoichiometry is believed to be unrelated to the linewidth broadening. The stoichiometry of the nanocrystals were estimated from the high resolution powder diffraction (HRPD) data in our previous X-ray diffraction study14. The offstoichiometry parameter δ, defined as Fe3(1-δ)O4, of the nanocrystals are independent of size and within error range of 6.5 % at maximum. Moreover, XAS and XRD measurements did not detect any considerable structural distortions, which should necessarily accompany offstoichiometry, as discussed in the previous paragraph.

8


Page 8 of 21

Page 9 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Figure 3 - (a) NMR frequency (f) and magnetization (M) obtained for various sample sizes near the Verwey transition temperature, normalized with respect to the 𝟕 𝛍𝐦

sample. The solid

lines are guides to the eye. (b) Full width at quarter maximum (FWQM) of the spectra versus temperature. Experimental errors are smaller than the symbol size. The solid lines are guide to the eye. The inset is the zoom-in near the transition temperature.

The most plausible reason for the spectral broadening of the nanocrystals with decreasing size is the distortion of the magnetic structure. The resonance frequency of the NMR for nanocrystals shifts to lower frequency with decreasing size as shown in Fig. 2(b) and (c). The amount of shift is approximately 1 % of the resonance frequency as the crystal size 9


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

decreases from 7 μm to 16 nm. Then the equation (1) predicts similar amount of decrease in the magnetic moment. However, magnetization measurements in Fig. 3(a) show that the saturated magnetic moment of the 16 nm sample decreases more than 30 % compared to that of the 7 μm sample, indicating spin structure distortion. The hyperfine field is not affected by the change of individual spin direction, because the d electron distribution of Fe3+ ions is spherical. However, the dipole field is heavily dependent on the direction of the neighboring spins in B sites. The increased broadening with decreasing size implies that the distortion of the spin structure in nanocrystals is strongly affected by the area of surface or shell, where spin interaction is different from bulk. It is worthwhile to note, however, that our NMR result does not comply with the simple core-shell model24 where two regions with different degrees of spin order are separated by an unphysical sharp boundary between them. For that binary magnetic structure, a superposition of broad and narrow spectra or a spectrum with two peaks are expected, contrary to our observation. Considering the fact that the band structure near the surface is generally different from that of bulk only within a couple of layers from the surface, it is rather unexpected that the entire spin structure of a nanocrystalline particle is significantly affected by surface or shell area. In Fig. 3(b), we plotted the full width at quarter maximum (FWQM) of the spectra to represent effectively the linewidth of the complicated spectra below TV. The transition temperature decreases as the crystal size decreases, from 123 K for the bulk sample to 115 K for the 16 nm sample. The linewidths are almost constant as a function of temperature except the transition temperature region. The size-dependent linewidth above the transition temperature can be taken as a measure of disorder in the magnetic structure.

10


Page 10 of 21

Page 11 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Figure 4 - (a) The spin-spin relaxation rate (1/T2) of nanocrystals versus temperature. The powder data is from ref. 18. Inset: The spin-spin relaxation rate of the 36 nm sample near 𝑻𝑽 . (b) The spin-spin relaxation time of nanocrystals versus crystal size obtained at 130 K. According to equation (2), this graph indicates that the hopping conductivity changes with crystal size. The solid line is the linear fit. The Inset shows the decay curves of spin echo amplitude.

To check the dynamics of electronic and magnetic disorders, the temperature dependence of the NMR spin-spin relaxation rate (1/T2) is studied for the nanocrystals. Fig. 4(a) shows the temperature behavior of 1/T2 compared to the data in ref. 18 obtained from powder samples. The relaxation rate at low temperatures is relatively independent of size except for the 11


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

7 nm sample and rapidly increases with temperature until they start to show size-dependent behavior around 25 ~ 30 K. At TV, the relaxation rates jump to higher values and then shows weak temperature dependence for T > TV. The relaxation rate monotonically increases with the decrease of crystal size in this temperature region. The dominant source of nuclear spin-spin relaxation in A sites is the fluctuation of the dipole field due to the polarized electron migration in B sites below TV, and above TV, the hyperfine field modulated through the A-B exchange coupling by electron hopping18. Below TV, the correlation time of the spin fluctuation increases while the amplitude of the fluctuation decreases as temperature goes down, yielding the maximum in 1/T2 in bulk samples. Here we focus on the analysis of the size dependence of the relaxation rate above TV. The sudden increase of the spin-spin relaxation rate at TV confirms that the hopping is the main source of the relaxation behavior above TV. When the hopping rate is faster than the observation frequency of NMR experiment, the relaxation rate is expected to be proportional to the hopping time25. In other words, the coherence time becomes longer when the hyperfine field generated by the hopping electrons fluctuates faster. It was shown experimentally that the relation holds in ferromagnetic metals based on the double exchange interaction such as magnetite18 and manganite26. The relaxation rate (hopping rate) of the magnetite nanocrystals shows a clear monotonic increase (decrease) with decreasing size above TV. This decrease in the hopping rate is consistent with the observation of the entropy decrease at the Verwey transition with decreasing size14. The electron hopping time should be related to the electrical conductivity. For a simple hopping model of noninteracting electrons, the Einstein relation predicts that conductivity, σ, should be inversely proportional to hopping time, τ27. Then the following qualitative relation is expected when both the internal field fluctuation and electrical conduction originates from the hopping: 12


Page 12 of 21

Page 13 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

(2)

T2 ~ 1⁄τ ~ 𝜎.

In other words, the NMR spin-spin relaxation time is proportional to the hopping conductivity of the magnetite nanocrystals, and the graph of T2 above TV versus crystal size plotted in Fig. 4(b) implies that the relative conductivity due to hopping changes with crystal size. The surface effect may lead to the decrease of the hopping rate with size reduction in some cases, but it is unlikely in our case. For the surface effect to be dominant, the relaxation rate at the surface should be quite different from that in bulk and the region of the surface where the effect occurs should be large. Then the NMR amplitude is expected to decay doubleexponentially (sum of two exponential decays) in time, but our curves fit very well to singleexponential decays (inset of Fig 4(b)). This observation is also against the model of the simple core-shell binary magnetic structure with a sharp boundary. Disorder in a ferromagnetic spin alignment may also reduce the hopping rate based on the double exchange interaction28,29, but magnetization reduction of 30 % is too small to fully explain the observed change in the relaxation rate. Here we suggest the enhanced electron-phonon (e-ph) coupling due to the reduced crystal size as a plausible scenario. Electrical conductivity in magnetite results from the superposition of the small polaron hopping conduction and the band conduction30. In the small polaron hopping, the weak temperature dependence of hopping is attributed to the zero-point 1

energy of phonon. For 𝑇 < 2 𝜃𝐷 where 𝜃𝐷 is the Debye temperature, the hopping 1

conductivity with activation energy, 𝐸𝑎 , is expected to be proportional to 𝑒𝑥𝑝(−𝐸𝑎 / 2 𝑘𝐵 𝜃𝐷 ) rather than the general relation, 𝑒𝑥𝑝(−𝐸𝑎 /𝑘𝐵 𝑇)31. The activation energy, which is determined by a half of the polaron binding energy, can be approximated as 𝐸𝑎 ~𝑔2 /2𝜔0 32, where 𝑔 is the e-ph coupling coefficient and 𝜔0 is the frequency of the longitudinal optical phonon. Raman and photoluminescence studies of nanostructures have suggested an increase in 𝑔 due 13


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 21

to quantum confinement (QC) effect33-35, while the typical energy of the longitudinal optical phonon is not significantly affected by QC and can be treated as a constant compared to acoustic phonons. Although there have been several theoretical works on the enhancement of the e-ph coupling under the QC condition36,37, its rigorous derivation is still unknown up to now. Based on the simple assumption that the hopping conductivity indefinitely decreases with decreasing size and saturates to the bulk value with increasing size, we adopt a 𝑑

phenomenological formula of 𝑔 ~ 𝑔𝑏𝑢𝑙𝑘 (1 + ( 𝑑𝑐 )𝛼 ) to represent the tendency of the activation energy as a function of the size d, where the critical size 𝑑𝑐 and the exponent 𝛼 are to be determined experimentally. Then for 𝑑 > 𝑑𝑐 , the nanocrystals should satisfy the following equation: ln 𝜎 ~ ln 𝑇2 ~ ln 𝑇2,𝑏𝑢𝑙𝑘 − (2

𝐸𝑃,𝑏𝑢𝑙𝑘 k B 𝜃𝐷

𝑑

)( 𝑑𝑐 )𝛼 , where 𝐸𝑃,𝑏𝑢𝑙𝑘 and T2,bulk are

the polaron binding energy and T2 in bulk, respectively. A linear fit of our data with the reported values (𝜃𝐷 ~ 500 K38, 𝐸𝑃 ~ 200 meV39, and T2,bulk ~ 30 ms18) gives 𝛼 ~ 0.54 and 𝑑𝑐 ~ 8.3 nm. The exponent 𝛼 decreases and the critical size 𝑑𝑐 increases as T2,bulk increases. T2,bulk of 50 ms gives 𝛼 ~ 0.48 and 𝑑𝑐 ~ 9.2 nm, for example. Though the spin hopping is believed to be the main source of the relaxation, there could be additional relaxation mechanisms. Therefore, the experimentally observed T2 for bulk is regarded as the lower limit of the relaxation time solely attributed to hopping and the rough estimates of 𝛼 and 𝑑𝑐 are about 0.5 and 8~9 nm, respectively. The resulting value of 𝑑𝑐 is close to the size of nanocrystals from which the Verwey transition starts to disappear (< 10 nm)14. It is also consistent with our experimental observation that 𝑇2 of the 7 nm sample is orders of magnitude different from those of other larger nanocrystals (Fig 4(a)). The QC analysis of phonons provides some hints on the size effect of magnetite nanocrystals on the Verwey transition. The relation, 𝑔 ~ 𝑑 −0.5 , may provide information for the development of a rigorous theory about the e-ph coupling in QC limits. The 14


Page 15 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

environmental screening effects in nanoparticles, which has also been observed in band gap dependency in quantum dots40,41, seem to make the exponent deviate from unity. In summary, the NMR spectrum and spin-spin relaxation time were obtained for magnetite nanocrystals over a wide temperature range around the Verwey transition. The spectral linewidth was approximately constant with temperature except near 120 K, where a drastic change clearly provides the microscopic evidence of the Verwey transition. The transition temperature decreases with the reduction of crystal size. The linewidth of the nanocrystals consists of two parts: one that is narrowed by the hopping mechanism while the other is not. The former indicates the reduction in the degree of the charge order due to the deteriorated crystallinity, and the latter indicates the distortion in the spin structure due to the surface effect. The spin-spin relaxation time monotonically decreases with the crystal size above TV. The DC hopping conductivity of the nanocrystals was estimated based on the small polaron hopping characterized by the enhanced electron-phonon coupling in the quantumconfinement regime. The analysis provides some valuable quantities such as the critical size and exponent of quantum-confined polaron hopping in nanoparticles.

Notes The authors declare no competing financial interest. Supporting Information Available: [NMR spectra of 16 nm, 40 nm, 80 nm, bulk (7 μm) samples. Experimental details.]

Reference 1) Waser, R.; Aono, M. Nat. Mater. 2007, 6, 833 - 840. 2) Laurent, S.; Forge,D.; Port, M.; Roch, A.; Robic, C.; Vander Elst, L.; Muller, R. N. Chem. Rev. 2008, 108, 2064 - 2110. 3) Lee, N.; Yoo, D.; Ling, D.; Cho, M. H.; Hyeon, T.; Cheon, J. Chem. Rev. 2015, 115, 10637 - 10689. 4) Stanley, S. A.; Gagner, J. E.; Damanpour, S.; Yoshida, M.; Dordick, J. S.; Friedman, J. M. Science. 2012, 336, 604 - 608. 5) Park, J.; Lee, E.; Hwang, N. M.; Kang, M.; Hwang, Y.; Park, J. G.; Noh, H. J.; Kim, 15


Nano Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

J. Y.; Park, J. H.; Hyeon, T. Angew. Chem. Int. Ed. 2005, 44, 2872 - 2877. Sun, S.; Zeng, H. J. Am. Chem. Soc. 2002, 124, 8204 - 8205. Verwey, E. J. Nature. 1939, 144, 327 - 328. Mizoguchi, M. J. Phys. Soc. Jpn. 1978, 44, 1512 - 1520. Iizumi, M.; Koetzle, T. F.; Shirane, G.; Chikazumi, S.; Matsui, M.; Todo, S. ActaCrystallogr. Sect. B: Struct. Crystallogr. Cryst. Chem. 1982, 38, 2121 - 2133. 10) Yoshida, J.; Iida, S. J. Phys. Soc. Jpn. 1977, 42, 230 - 237. 11) McQueency, R. J.; Yethiraj, M.; Chang, S.; Montfrooij, W.; Perring, T. G.; Honig, J. M.; Metcalf, P. Phys. Rev. Lett. 2007, 99, 246401. 12) Mizoguchi, M. J. Phys. Soc. Jpn. 1978, 44, 1501 - 1511. 13) Senn, M.; Wright, J.; Attfield, J. Nature. 2011, 481, 173 – 176. 14) Lee, J.; Kwon, S. G.; Park, J.; Hyeon, T. Nano Lett. 2015, 15, 4337 - 4342. 15) Mitra, A.; Mohapatra, J.; Meena, S.; Tomy., C.;Aslam, M. J. Phys. Chem. C. 2014, 118, 19356 - 19362. 16) Hevroni, A.; Bapna, M.; Piotrowski, S.; Majetich, S.; Markovich, G. J. Phys. Chem. Lett. 2016, 7, 1661 - 1666. 17) Boyl, E. L. Phys. Rev. 1963, 129, 1961 - 1964. 18) Mizoguchi, T.; Inoue, M.; J. Phys. Soc. Jpn. 1966, 21, 1310 - 1323. 19) Mizoguchi, M. J. Phys. Soc. Jpn. 2001, 70, 2333 -2344. 20) Novak, P.; Stepankova, H.; Englich, J.; Kohout, J.; Brabers, V. A. M. Phys. Rev. B 2000, 61, 1256 - 1260. 21) Patterson, C. Phys. Rev. B 2014, 90, 075134. 6) 7) 8) 9)

22) Reznicek, R.; Chlan, V.; Stepankova, H.; Novak, P. Phys. Rev. B 2015, 91, 125134. 23) Kristan, P.; Chilan, V.; Stepankova, H.; Reznicek, R.; Gornert, P.; Payer, P. Acta Phys. Pol. A 2014, 126, 138 - 139. 24) Morup, S; Brok, E.; Frandsen, C. J. Nanomater. 2013, 2013, 720629. 25) Moriya, T. Prog. Theor. Phys. 1956, 16, 641 - 657. 26) Savosta, M. M.; Borodin, V. A.; Novak, P. Phys. Rev. B 1999, 59, 8778 - 8783. 27) Kundig W.; Hargrove, R. S. Sol. Stat. Comm. 1969, 7, 223 - 227. 28) Viret, M.; Ranno, L.; Coey, J. M. D. Phys. Rev. B 1997, 55, 8067 - 8070. 29) Muroi, M.; Street, R.; McCormik, P. G. Phys. Rev. B 2001, 63, 052412. 30) Ihle, D.; Lorenz, B. J. Phys. C 1985, 18, L647 – L650. 31) Coey, J. M. D.; Viret, M.; Ranno, L.; Ounadjela, K. Phys. Rev. Lett. 1995, 75, 3910 - 3913. 32) Holstein, T. Ann. Phys. (N.Y.) 1959, 8, 325 - 342. 33) Zhao, H.; Wachter, S.; Kalt, H. Phys. Rev. B 2002, 66, 085337. 34) Cheng, H. M.; Lin, K. F.; Hsu, H. C.; Hsieh, W. F. Appl. Phys. Lett, 2006, 88, 261909. 35) Gong, K.; Kelley, D. F.; Kelley, A. M. J. Phys. Chem. C 2016, 120, 29533 - 29539. 36) Han, P.; Bester, G. Phys. Rev. B 2012, 85, 235422. 37) Melnikov, D. V.; Fowler, W. B. Phys. Rev. B 2001, 64, 245320. 38) Koenitzer, J. W.; Keesom, P. H.; Honig, J. M. Phys. Rev. B 1989, 39, 6231 - 6233. 39) Huang, H. Y.; Chen, Z. Y.; Wang, R. –P.; de Groot, F. M. F.; Wu, W. B.; Okamoto, J.; Chainani, A.; Singh, A.; Li, Z. –Y.; Zhou, J. –S.; Jeng, H. –T.; Guo, G. Y.; Park, J. G.; Tjeng, L. H.; Chen, C. T.; Huang, D. J. Nat. Comm. 2017, 8, 15929. 40) Franceschetti, A; Zunger, A. Phys. Rev. Lett, 1997, 78, 915 - 918. 41) Yavidov, B. Y. Eur. Phys. J. B. 2010, 75, 481 - 488. 16


Page 16 of 21

Page 17 of 21

16 nm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 100 nm 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 100 nm 36

40 nm

Nano Letters

25 nm

100 nm

80 nm


100 nm

(a)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(b)

Nano Letters

130 K (c)


Page 18 of 21

110 K

Page 19 of 21

(a)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 (b) 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

Nano Letters


Nano Letters

Page 20 of 21

(a)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

7

(b) 6

5

4

3

0.08

0.12

0.16 ACS Paragon Plus Environment

D-0.54(nm-0.54)

0.20

0.24

Page 21 of 21 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Nano Letters


Microscopic States and the Verwey Transition of Magnetite

Recommend Documents