LETTERS PUBLISHED ONLINE: 23 NOVEMBER 2015 | DOI: 10.1038/NNANO.2015.258
Revealing giant internal magnetic fields due to spin fluctuations in magnetically doped colloidal nanocrystals William D. Rice1†, Wenyong Liu2, Thomas A. Baker2, Nikolai A. Sinitsyn3, Victor I. Klimov2* and Scott A. Crooker1* a 1
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Strong quantum confinement in semiconductors can compress the wavefunctions of band electrons and holes to nanometrescale volumes, significantly enhancing interactions between themselves and individual dopants. In magnetically doped semiconductors, where paramagnetic dopants (such as Mn2+, Co2+ and so on) couple to band carriers via strong sp–d spin exchange1,2, giant magneto-optical effects can therefore be realized in confined geometries using few3–7 or even single8,9 impurity spins. Importantly, however, thermodynamic spin fluctuations become increasingly relevant in this few-spin limit10. √ In nanoscale volumes, the statistical N fluctuations of N spins are expected to generate giant effective magnetic fields Beff, which should dramatically impact carrier spin dynamics, even in the absence of any applied field. Here we directly and unambiguously reveal the large Beff that exist in Mn2+-doped CdSe colloidal nanocrystals using ultrafast optical spectroscopy. At zero applied magnetic field, extremely rapid (300–600 GHz) spin precession of photoinjected electrons is observed, indicating Beff ∼ 15 −30 T for electrons. Precession frequencies exceed 2 THz in applied magnetic fields. These signals arise from electron precession about the random fields due to statistically incomplete cancellation of the embedded Mn2+ moments, thereby revealing the initial coherent dynamics of magnetic polaron formation, and highlighting the importance of magnetization fluctuations on carrier spin dynamics in nanomaterials. The past decade has witnessed a resurgent interest in magnetically doped semiconductors, motivated in part by the ability to tailor, via quantum confinement and wavefunction engineering, the spin interactions between embedded magnetic atoms and band carriers (electrons and holes)2,9,11,12. Materials for spin-based electronic and photonic applications include magnetically doped nanoribbons13,14, nanowires15, epitaxially grown quantum dots7–9,16–19, and colloidal nanocrystals3–6,12,20. However, in these evershrinking volumes, magnetization fluctuations necessarily play an increasingly essential role7,10. To illustrate, consider an electron with spin s and spatially uniform wavefunction ψ e (r) confined to a volume V of Mn2+ -doped material, so that |ψ e (r)|2 ∼ 1/V. The doping density is xMn , so that N = xMn V is the number of embedded Mn2+ , each with spin SMn . The s–d exchange energy between the electron and the Mn2+ is therefore Esd ∝ Ni |ψ e (ri )|2 s · SMn i . In applied magnetic fields sufficient to align the paramagnetic Mn2+ (a few with a total spin tesla at 4 K), the electron interacts ∝ N, so that Esd ∝ N|ψ e (r)|2 ∼ xMn . In typical diluted ST = SMn i
−10 40 K 10 K 5K
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Figure 1 | Quantifying the large sp–d spin interactions in Mn2+-doped CdSe nanocrystals using MCD spectroscopy. a, The measured MCD (and absorption) spectra from Mn:CdSe/CdS NCs and also from non-magnetic CdSe control NCs at applied magnetic field B = 7 T in the Faraday geometry ˆ Insets show the CdSe/CdS core/shell NCs, and the black arrows (B∥ˆz∥k). depict the embedded Mn2+ spins. The control NCs show small and temperature-independent MCD, as expected. The Mn:CdSe/CdSe NCs show large, inverted, temperature-dependent MCD, confirming strong sp–d spin coupling. b, The Zeeman splitting of the 1s exciton, EZ , versus B at different temperatures. EZ from the non-magnetic control NCs is small and increases linearly with B, with exciton g-factor gex = 1.3. In contrast, EZ from the Mn: CdSe/CdSe NCs is large, temperature-dependent, and follows a Brillouin function (black lines). At 5 K, |EZ | saturates at ∼27 meV, and can be characterized by an enhanced effective exciton g-factor of ∼140 in these NCs.
1
National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA. 2 Chemistry Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA. 3 Theory Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA; †Present address: Department of Physics and Astronomy, University of Wyoming, Laramie, Wyoming 82071, USA. * e-mail: [email protected]; [email protected] NATURE NANOTECHNOLOGY | ADVANCE ONLINE PUBLICATION | www.nature.com/naturenanotechnology
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Figure 2 | Ultrafast electron spin dynamics and precession at B = 0 in Mn2+-doped CdSe nanocrystals. a, A schematic of the time-resolved Faraday rotation (TRFR) experiment. The green triangles represent ultrafast optical pulses separated by time, Δt. Spin-polarized electrons with s∥ +ˆz are injected by circularly polarized pump pulses at t = 0. The net spin projection in the ensemble, 〈sz (t)〉 is detected via the Faraday rotation θF imparted to time-delayed, linearly polarized probe pulses. b, TRFR signals from non-magnetic CdSe control NCs show only the long-lived and monotonically decaying spin relaxation of the photoinjected electrons, in agreement with past work23,24. c, In contrast, TRFR from Mn2+ -doped CdSe/CdS NCs at B = 0 (solid trace) exhibits a rapid and strongly damped half-cycle oscillation occurring within a few picoseconds. This oscillatory signal indicates the very rapid precession of the (initially) spin-polarized electrons around the large effective internal magnetic field Beff in each NC that is due to Mn2+ spin fluctuations. Beff in each NC is different in both magnitude and direction, leading to rapid electron spin dephasing. The dashed trace shows TRFR from the more heavily doped Mn:CdSe/CdSe NCs. The extremely fast variation at t ∼ 0 is a coherent artifact caused by the temporally overlapping and degenerate pump and probe pulses. d, The corresponding power spectra of the time-domain TRFR signals shows that the zero-field electron spin precession frequencies in these lightly (heavily) doped NC ensembles are peaked at ∼290 GHz (∼600 GHz), indicating that electrons “see” 〈B2eff 〉 ∼ 15T (∼30 T) in these NCs.
magnetic semiconductors, this exchange energy equates to a giant effective magnetic field Beff “seen” by the electron, of order 100 T when xMn is a few per cent1,2. In contrast, at zero applied field the Mn2+ are unpolarized, and on (rms) average 〈ST 〉 = 〈Beff 〉 = 0. However, the root-mean-square isnot zero. effective field due √ tostochastic spin fluctuations, 〈B2eff 〉, √ √ 2 Rather, 〈S2T 〉 ∼ N , so√that Esd ∝ N |ψ e (r)| ∼ xMn / N , which is reduced by a factor of N from the high-field case. In bulk or 2D systems, where band electrons are delocalized and N is large, these fluctuations are comparatively insignificant and Beff ≈ 0 at zero applied field. However, in quantum-confined nanocrystals where V is small and N is of order 10, 〈B2eff 〉 can attain a sizable fraction of its high-field value, even in the absence of any applied magnetic field. Directly revealing these giant internal fluctuating fields in nanomaterial ensembles—and their expected dramatic impact on carrier spin dynamics—is highly desired but remains a significant challenge. Early Raman studies of donor-bound magnetic polarons in bulk CdMnSe established the relevance of spin fluctuations in 10,21 . More recently, micro-PL measurements small effective volumes showed that 〈B2eff 〉 influences the emission spectra from Mn2+ -doped epitaxial quantum dots, provided that individual dots were studied to avoid ensemble broadening7,8,16–18. Separately, static spectroscopies such as magnetic circular dichroism (MCD), commonly used to measure sp–d exchange in magnetically doped nanocrystals3–6, are sensitive to time- (and ensemble-) average magnetization 〈Beff 〉, but do not reveal fluctuating fields. 2
Here, we show that ultrafast optical studies can unambiguously reveal 〈B2eff 〉 in Mn2+ -doped CdSe nanocrystals via the rapid electron spin precession and relaxation that Beff induces even at zero applied field. Precession frequencies from 300–600 GHz are observed, indicating that electrons “see” 〈B2eff 〉 = 15−30 T. Because the signals occur within just a few picoseconds, these experiments directly reveal the initial coherent formation dynamics of the interesting “fluctuation magnetic polaron” that is well-known in magnetic semiconductor physics22; that is, when Beff is due to thermodynamic spin fluctuations alone, and well before any collective polaron can form due to additional Mn2+ alignment19,20. Several samples of 4–5 nm diameter Mn2+ -doped CdSe nanocrystal (NC) cores were synthesized, and then overcoated with CdS or CdSe shells to improve optical quality (see Methods). The 1s exciton absorption in these large NCs lies below the ∼2.15 eV 4 T1 6 A1 internal Mn2+ transition, suppressing excitation of the Mn2+ ions20. In this Letter we focus on two samples: Mn:CdSe/CdS NCs with 〈N〉 ≈ 10, and more heavily doped Mn: CdSe/CdSe NCs with 〈N〉 ≈ 20, where 〈N〉 is the average number of Mn2+ per NC. Control samples of non-magnetic CdSe NCs were also studied. Using conventional MCD spectroscopy (see Methods), we first verify strong sp–d exchange interactions in these NCs, by confirming an enhanced Zeeman splitting EZ of the 1s exciton, which tracks the average Mn2+ magnetization (Fig. 1). In general, EZ = gex μB B + ΔEspd 〈SMn z 〉, where the first term is the small intrinsic Zeeman splitting of the host semiconductor (gex ∼ 1), while the
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DOI: 10.1038/NNANO.2015.258 Precession about Beff ∝ ∑SiMn|Ψe(ri)|2 x
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Figure 3 | Modeling magnetization fluctuations and ultrafast electron spin dynamics in Mn2+-doped NC ensembles at B = 0. a, The incomplete cancellation of the N randomly oriented Mn2+ spins (SMn = 5/2) in any given NC generates an effective internal magnetic field, Beff, about which an electron spin s, initially polarized with s0 ∥ +ˆz at t = 0, will precess. Beff is different, both in orientation and magnitude, in every NC. Three such NCs having different Beff are depicted, along with the corresponding time evolution sz (t). b, Averaging sz (t) over all possible Beff in the NC ensemble shows that 〈sz (t)〉 exhibits a fast half-cycle precession with a minimum value of sz ∼ 0 and a recovery to a steady-state value of 1/3 (black trace). Including an additional empirical 7 ps spin relaxation time yields the orange curve, which matches the measured TRFR data very well (dashed blue trace). c, Solid curves show the expected per unit volume, 〈M2 〉 /V, of N unit paramagnetic spins ina fixed volume as a function of applied field at 2 K. At zero field, rms total moment √ 2 〈M2 〉 increases as N , while at high field it increases as N. The dashed curves show 〈M 〉 for N = 10 at other temperatures (1 K and 4 K). d, Curves show 〈M2 〉 /V (∝Beff ), for the case of a fixed density of moments xMn , for√ different volumes (using N = 10 when V = V0 ). As V drops (increasing quantum confinement), the zero-field rms magnetization increases in proportion to 1/ V .
second describes additional splitting due to sp–d exchange with the Mn2+ spins, which have average spin projection 〈SMn z 〉 along the applied field B (∥ zˆ )1–6. Since isolated Mn2+ ions are spin-5/2 paramagnets, 〈SMn z 〉 follows a temperature- and field-dependent Brillouin function B5/2 (5gMn μB B/2kB T), where the Mn2+ g-factor gMn ≃ 2.0, μB is the Bohr magneton, and kB is the Boltzmann constant. The overall magnitude of the sp–d exchange, parameterized by ΔEspd , depends on exchange constants and the overlap of the electron and hole wavefunctions with the Mn2+ dopants1,2,6. The non-magnetic CdSe NCs exhibit a small temperature-independent MCD (as expected), with gex ≃ 1.3. In contrast, the Mn: CdSe/CdS NCs show a large, inverted, and strongly temperaturedependent MCD, with EZ tracking 〈SMn z 〉 and saturating at −27 meV. At 5 K, EZ can be characterized by an enhanced exciton g-factor of ∼140; that is, as though the magnetic field in the NCs is vastly larger than the applied field, consistent with past work4–6. The highly doped Mn:CdSe/CdSe NCs exhibit even larger sp–d interactions, with |EZ | ∼ 50 meV (Supplementary Fig. 3). Crucially, however, static MCD spectroscopy reveals only how electrons and holes respond to the time- and ensemble-average effective field, 〈Beff 〉. At B = 0, 〈Beff 〉 = 0 and therefore MCD signals disappear, although Beff within any particular NC may be large (and fluctuating). However, a key result of this work is to demonstrate
that ultrafast spectroscopy can circumvent this limitation, to directly reveal 〈B2eff 〉 in NC ensembles, even at zero field. Ultrafast spin dynamics are measured by time-resolved Faraday rotation (TRFR; see Fig. 2a). Consistent with the angular momentum selection rules in these materials1,2, circularly polarized pump pulses inject electrons (and holes) at the 1s exciton resonance that are initially spin-polarized along +ˆz at t = 023–27. The projection of this spin polarization along zˆ is then detected via the Faraday rotation imparted on time-delayed, linearly polarized probe pulses (see Methods). TRFR techniques are extremely powerful probes of electron spin dynamics in non-magnetic NCs23,24 and epitaxial quantum dots25, and of coupled carrier–ion spin dynamics in Mn2+ -doped quantum wells and sol-gel films26,27. Here we apply TRFR to magnetic NCs, where carrier–Mn2+ interactions are enhanced by strong quantum confinement. At B = 0, TRFR signals from non-magnetic CdSe NCs (Fig. 2b and Supplementary Fig. 4) reveal long-lived monotonic spin relaxation of the injected electrons (the hole spins dephase quickly23), consistent with prior work23,24. In stark contrast, TRFR signals from both the lightly and more heavily Mn2+ -doped NCs are radically different (Fig. 2c): the injected spin polarization rapidly drops within 1–2 ps and then partially recovers before decaying away completely, exhibiting a strongly damped half-cycle
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Figure 4 | Ultrafast electron spin dynamics in Mn2+-doped NCs in applied transverse magnetic fields Bx . a, TRFR of Mn:CdSe/CdS NCs from Bx = 0−7 T at 40 K for short delay times. The half-cycle spin precession at 0 T evolves into a faster and more oscillatory precession signal at 7 T, as the individual Mn2+ spins align along Bx . b, Including Bx into the model of electron spin precession reproduces these trends. c, The power spectra corresponding to the raw TRFR data show a clear shift of the peak electron spin precession frequency, νe . d, νe versus Bx at 1.8, 5, and 40 K. Note that νe is independent of temperature at of νe in the more heavily doped Mn:CdSe/CdSe NCs at Bx = 0, since 〈B2eff 〉 is due to Mn2+ spin fluctuations alone at zero field. e, Temperature dependence
7 T. νe exceeds 2.5 THz at low temperatures. In these NCs, νe ∼ 600 GHz at zero field, indicating
oscillation. In general, oscillatory TRFR indicates electron spin precession about applied and/or effective magnetic fields23–27; these data therefore directly reveal an ultrafast electron spin precession at B = 0, due to 〈B2eff 〉 in these NC ensembles. These signals provide, for the first time, (i) quantitative and unambiguous evidence of the large effective magnetic fields that are always present in Mn2+ -doped NCs due to spin fluctuations, and (ii) direct observation of the initial coherent dynamics of magnetic polaron formation, when photoinjected carriers precess about Beff due to fluctuations alone, but well before any additional Mn2+ alignment occurs22. Figure 2d shows that these TRFR signals contain frequency components peaked at 290 and 600 GHz, indicating that electrons “see” 〈B2eff 〉 ∼ 15 T and ∼30 T in the lightly and more heavily doped NCs (using the bare electron g-factor ge = 1.4; see Supplementary Fig. 4). We confirm that these fast TRFR signals are not due to exciton recombination; transient absorption studies reveal nanosecond exciton lifetimes. Note that holes feel effective fields 5–6 times larger due to their larger p–d exchange interaction with Mn2+ 1,2. The damped half-cycle precession signal at B = 0 can be understood within a simple model that reproduces the data and captures the physics (Fig. 3). In any given NC, Beff is proportional to Nessential 2 Mn 2+ i Si |ψ e (ri )| , the weighted vector sum of the embedded Mn spins10,16. At B = 0, Beff is oriented randomly in each NC. The disNC ensemble is centred at 〈Beff 〉 = 0, tribution of Beff across the with characteristic width B2eff that depends on NC size and on 〈N〉. Supplementary Figs 6 and 7 show simulated distributions for different |ψ e (ri )|2 . Photoinjected electrons, all initially polarized with s∥ + zˆ at t = 0, subsequently precess about the Beff in each NC. TRFR detects 〈sz (t)〉, the average electron spin projection along zˆ . Three 4
〈B2eff 〉 ∼ 30T. Inset: 7 T TRFR signals at 5 and 30 K.
example NCs are depicted in Fig. 3a; each has different Beff and therefore different dynamics sz (t). Clearly, individual sz (t) traces add coherently only near t = 0. Figure 3b shows the result of numerically averaging over all possible Beff (see Methods): 〈sz (t)〉 drops − /(μ g 〈B2 〉), then recovers rapidly to nearly zero on a timescale πh eff B e to a third of its initial value, exhibiting a damped half-cycle oscillation even when each electron is assigned infinite spin lifetime. Imposing an additional (and realistic26) spin relaxation of 7 ps, probably due to incoherent electron-Mn2+ spin-flip scattering, gives simulated dynamics closely matching the TRFR data. This model is analogous to early treatments of electron spin dephasing in epitaxial quantum dots due to hyperfine coupling with nuclear spins28. It assumes “frozen” Mn2+ fluctuations, because electrons precess rapidly (∼300 GHz) while any Mn2+ dynamics are much slower—for example, Mn2+ spin precession (see below), or formation of collective magnetic polarons due to Mn2+ alignment (which requires ∼100 ps)19,20,22. We emphasize that ultrafast electron spin precession at zero applied field has never been observed in bulk or 2D diluted magnetic semiconductors, where carriers are more likely to overlap with many Mn2+ spins26. Rather, it uniquely arises from strong nanocrystal quantum confinement, with two inextricably linked consequences: giant magneto-optical effects from just a few Mn2+ dopants are possible, however, large (and potentially deleterious) magnetization fluctuations exist at zero or small applied fields. The qualitative dependence of 〈B2eff 〉 on applied magnetic fields B is readily anticipated. Figure 3c shows the expected rms total moment, 〈M2 〉, of N classical spins versus B. At B = 0, 〈M2 〉 is due to fluctuations alone√and is therefore independent of temperature and increases as N , while at large B (aligned spins) it increases as N. Crucially, the ratio of 〈M2 〉 at zero and large
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Figure 5 | Pump-induced coherent transfer of angular momentum to the Mn2+ spins. a, When Bx > 0, TRFR data of Mn:CdSe/CdSe NCs reveal an additional slow precession that persists for hundreds of picoseconds (long after the electron spin dynamics have ceased). Data at different Bx are intentionally offset for clarity. b, This slow precession frequency increases linearly with Bx at 28.1 GHz/T (gMn = 2.0), is independent of temperature (not shown), and appears only in Mn2+ -doped NCs, indicating a collective and coherent precession of the embedded Mn2+ spins themselves.
field therefore allows to directly determine N. Separately, Fig. 3d shows 〈M2 〉 /V (∝ Beff ) for material with fixed moment density xMn , but different volume. As V shrinks (increasing confinement), √ the zero-field rms magnetization grows in proportion to 1/ V . We confirmed these trends by applying transverse fields Bx . Figure 4a shows that the half-cycle spin precession observed at zero field evolves into an even faster oscillation with a deeper minimum as Bx increases. (The slower oscillation that develops after ∼2 ps is due to Mn2+ precession, discussed below.) This is expected, as Beff in each NC increases and orients along xˆ as Mn2+ aligns. Including Bx into our model reproduces the TRFR data nicely (Fig. 4b). The measured power spectra (Fig. 4c) clearly reveal the increase of the peak electron precession frequency ne with Bx . Figure 4d shows ne (Bx ) at 1.8, 5, and 40 K. Confirming predictions, ne (∼300 GHz) is independent of temperature at zero field (we note that this indicates that charged NCs in the ensemble, if any are present, do not influence these results; see Supplementary Fig. 5). ne increases with Bx at different temperature-dependent rates in accord with the Brillouin-like polarization of the Mn2+ spins (Fig. 3c), saturating at ∼1,000 GHz at 1.8 K, or approximately three times its zero-field value. As discussed above, this ratio suggests 〈N〉 ≃ 10 in these Mn:CdSe/CdS NCs. Moreover, we note that ne saturates at an energy (4 meV = 1,000 GHz) that is about 7 times less than the maximum exciton (electron + hole) splitting measured by MCD (27 meV; see Fig. 1), consistent with the known s–d and p–d exchange constants in bulk CdMnSe (ref. 2) and confirming that the TRFR oscillations arise from electrons alone. Figure 4e shows ne measured in the more heavily doped Mn: CdSe/CdSe NCs versus temperature. At zero field, ne ∼ 600 GHz and is temperature-independent, indicating very large effective fields 〈B2eff 〉 ∼ 30T due to spin fluctuations. When Bx = 7 T, ne exceeds 2.5 THz at low temperatures (Beff > 125 T). This larger ratio indicates that 〈N〉 is approximately twice that of the Mn:CdSe/CdS NCs shown in Fig. 4d, a conclusion independently supported by static MCD studies (Supplementary Fig. 3).
LETTERS
Finally, Fig. 5 shows that strong sp–d spin interactions also influence the Mn2+ spins themselves. When Bx > 0, a much slower precession signal appears that persists for hundreds of picoseconds. The frequency increases linearly with Bx at 28.1 GHz/T (gMn = 2.0), independent of temperature. This phenomenon, known from early studies of Mn2+ -doped quantum wells26, arises from the collective and coherent spin precession of the Mn2+ spins, due to ultrafast ‘tipping’ of the average Mn2+ magnetization (∥ˆx) by the transient exchange field of the initially polarized holes (∥ zˆ ). Once perturbed away from xˆ , the average magnetization precesses about Bx and is measurable with TRFR, enabling timedomain paramagnetic spin resonance studies of Mn2+ . In summary, studies of giant internal magnetic fields due to thermodynamic spin fluctuations, revealed here with ultrafast optical spectroscopy, represent an important but previously unexplored aspect of nanoscale magnetism in colloidal quantum structures. Together with increasingly precise control over the placement of dopants29,30, as well as the possibility for magnetic coupling between the dopants themselves (e.g. in ferromagnetic semiconductors), these experiments pave the way for fundamental studies of magnetically doped nanomaterials based on fluctuations alone.
Methods Methods and any associated references are available in the online version of the paper. Received 15 July 2015; accepted 6 October 2015; published online 23 November 2015
References 1. Furdyna, J. K. Diluted magnetic semiconductors. J. Appl. Phys. 64, R29–R64 (1988). 2. Introduction to the Physics of Diluted Magnetic Semiconductors (eds. Kossut, J. & Gaj, J.A.) (Springer, 2010). 3. Hoffman, D. M. et al. Giant internal magnetic fields in Mn doped nanocrystal quantum dots. Solid State Commun. 114, 547–550 (2000). 4. Norris, D. J., Yao, N., Charnock, F. T. & Kennedy, T. A. High-quality manganese-doped ZnSe nanocrystals. Nano Lett. 1, 3–7 (2001). 5. Archer, P. I., Santangelo, S. A. & Gamelin, D. R. Direct observation of sp-d exchange interactions in colloidal Mn2+ - and Co2+ -doped CdSe quantum dots. Nano Lett. 7, 1037–1043 (2007). 6. Bussian, D. A. et al. Tunable magnetic exchange interactions in manganesedoped inverted core-shell ZnSe-CdSe nanocrystals. Nature Mater. 8, 35–40 (2009). 7. Bacher, G. et al. Monitoring statistical magnetic fluctuations on the nanometer scale. Phys. Rev. Lett. 89, 127201 (2002). 8. Besombes, L., Léger, Y., Maingault, L., Ferrand, D. & Mariette, H. Probing the spin state of a single magnetic ion in an individual quantum dot. Phys. Rev. Lett. 93, 207403 (2004). 9. Kobak, J. et al. Designing quantum dots for solotronics. Nature Commun. 5, 3191 (2014). 10. Dietl, T. & Spałek, J. Effect of fluctuations of magnetization on the bound magnetic polaron: Comparison with experiment. Phys. Rev. Lett. 48, 355–358 (1982). 11. Abolfath, R. M., Hawrylak, P. & Zŭtić, I. Tailoring magnetism in quantum dots. Phys. Rev. Lett. 98, 207203 (2007). 12. Beaulac, R., Archer, P. I., Ochsenbein, S. T. & Gamelin, D. R. Mn2+ -doped CdSe quantum dots New inorganic materials for spin-electronics and spin-photonics. Adv. Funct. Mater. 18, 3873–3891 (2008). 13. Yu, J. H. et al. Giant Zeeman splitting in nucleation-controlled doped CdSe:Mn2+ quantum nanoribbons. Nature Mater. 9, 47–53 (2010). 14. Fainblat, R. et al. Quantum confinement-controlled exchange coupling in manganese (II)-doped CdSe two-dimensional quantum well nanoribbons. Nano Lett. 12, 5311–5317 (2012). 15. Wojnar, P. et al. Giant spin splitting in optically active ZnMnTe/ZnMgTe core/ shell nanowires. Nano Lett. 12, 3404–3409 (2012). 16. Wojnar, P. et al. Microluminescence from Cd1−x Mnx Te magnetic quantum dots containing only a few Mn ions. Phys. Rev. B 75, 155301 (2007). 17. Hundt, A., Puls, J. & Henneberger, F. Spin properties of self-organized diluted magnetic Cd1−x Mnx Se quantum dots. Phys. Rev. B 69, 121309(R) (2004). 18. Dorozhkin, P. S. et al. Longitudinal and transverse fluctuations of magnetization of the excitonic magnetic polaron in a semimagnetic single quantum dot. Phys. Rev. B 68, 195313 (2003). 19. Seufert, J. et al. Dynamical spin response in semimagnetic quantum dots. Phys. Rev. Lett. 88, 027402 (2002).
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20. Beaulac, R., Schneider, L., Archer, P. I., Bacher, G. & Gamelin, D. R. Lightinduced spontaneous magnetization in doped colloidal quantum dots. Science 325, 973–976 (2009). 21. Nawrocki, M., Planel, R., Fishman, G. & Galazka, R. Exchange-induced spin-flip Raman scattering in a semimagnetic semiconductor. Phys. Rev. Lett. 46, 735–738 (1981). 22. Yakovlev, D. R. & Ossau, W. in Introduction to the Physics of Diluted Magnetic Semiconductors (eds. Kossut, J. & Gaj, J. A.) Ch. 7 (Springer, 2010). 23. Gupta, J. A., Awschalom, D. D., Efros, A. L. & Rodina, A. V. Spin dynamics in semiconductor nanocrystals. Phys. Rev. B 66, 125307 (2002). 24. Fumani, A. K. & Berezovsky, J. Magnetic-field-dependent spin decoherence and dephasing in room-temperature CdSe nanocrystal quantum dots. Phys. Rev. B 88, 155316 (2013). 25. Greilich, A. et al. Mode locking of electron spin coherences in singly charged quantum dots. Science 313, 341–345 (2006). 26. Crooker, S. A., Awschalom, D. D., Baumberg, J. J., Flack, F. & Samarth, N. Optical spin resonance and transverse spin relaxation in magnetic semiconductor quantum wells. Phys. Rev. B 56, 7574 (1997). 27. Whitaker, K. M. et al. Spin-on spintronics: Ultrafast electron spin dynamics in ZnO and Zn1−x Cox O sol-gel films. Nano Lett. 11, 3355–3360 (2011). 28. Merkulov, I. A., Efros, A. L. & Rosen, M. Electron spin relaxation by nuclei in semiconductor quantum dots. Phys. Rev. B 65, 205309 (2002). 29. Ithurria, S., Guyot-Sionnest, P., Mahler, B. & Dubertret, B. Mn2+ as a radial pressure gauge in colloidal core/shell nanocrystals. Phys. Rev. Lett. 99, 265501 (2007). 30. Grumbach, N., Rubin-Brusilovski, A., Maikov, G. I., Tilchin, E. & Lifshitz, E. Manipulation of carrier-Mn2+ exchange interaction in CdTe/CdSe colloidal quantum dots by controlled positioning of Mn2+ impurities. J. Phys. Chem. C 117, 21021–21027 (2013).
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31. Vlaskin, V. A., Barrows, C. J., Erickson, C. S. & Gamelin, D. R. Nanocrystal diffusion doping. J. Am. Chem. Soc. 135, 14380–14389 (2013). 32. Ithurria, S. & Talapin, D. V. Colloidal atomic layer deposition (c-ALD) using self-limiting reactions at nanocrystal surface coupled to phase transfer between polar and nonpolar media. J. Am. Chem. Soc. 134, 18585–18590 (2012).
Acknowledgements We gratefully thank D. R. Yakovlev and D. L. Smith for helpful discussions and insight. W.D.R. acknowledges support from the Los Alamos LDRD programme. W.L., T.A.B., and V.I.K. are supported by the Office of Chemical Sciences, Biosciences, and Geosciences of the Department of Energy Office of Basic Energy Sciences. All optical measurements were performed at the National High Magnetic Field Laboratory, which is supported by NSF DMR-1157490.
Author contributions S.A.C. and V.I.K. conceived and directed the experiments. W.D.R. built and performed the optical experiments, and designed the numerical simulations. W.L. synthesized the nanocrystals, and T.A.B. made the films. N.S. provided theoretical insights. S.A.C. and W.D.R. analyzed the data and wrote the paper in close consultation with all authors.
Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to V.I.K. and S.A.C.
Competing financial interests
The authors declare no competing financial interests.
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Methods
Colloidal nanocrystal synthesis. Wurtzite CdSe NCs were initially prepared with 4–5 nm diameters. The 1s exciton energy in these NCs lies below the 4 T1 6 A1 internal Mn2+ transition at ∼2.15 eV, suppressing Auger excitation of the Mn2+ ions. Mn doping followed published procedures31 with necessary modifications. To eliminate Mn atoms and traps on the NC surfaces, an inorganic epitaxial shell was grown. For the lower doped Mn:CdSe/CdS NCs (〈N〉 ≃ 10), solution-based colloidal atomic layer deposition of CdS under ambient conditions was utilized32. The CdS shell is 1.2 nm thick (4 layers). For the more highly doped Mn:CdSe/CdSe NCs (〈N〉 ≃ 20), a CdSe shell was grown at elevated temperatures. Room-temperature emission quantum yields were thereby improved to ∼10–20%. Details of the colloidal NC synthesis, Mn2+ doping procedure, film preparation, and initial optical characterization are shown in Supplementary Figs 1 and 2 and the accompanying discussion. MCD measurements. MCD is based on measuring the normalized transmission difference between left- and right-circularly polarized light through the NC film, ˆ These circular (T L − T R )/(T L + T R ), in the Faraday geometry (applied field B∥ˆz ∥k). polarizations couple to the ‘spin-up’ and ‘spin-down’ (total angular momentum projection Jz = ±1) states of the nanocrystals’ 1s exciton respectively, which are Zeeman-split in B by the small intrinsic exciton g-factor gex of the host semiconductor (present in all materials), and also by the large sp–d exchange coupling between any local Mn2+ moments and the electron/hole spins of the NC. Thus, MCD spectra typically track the derivative of the 1s absorption. Probe light of tunable wavelength was derived from a xenon lamp directed through a spectrometer. The light was mechanically chopped, then polarization-modulated at 50 kHz using a photoelastic modulator, focused through the sample, and detected by an avalanche photodiode6. TRFR measurements. Time-resolved Faraday rotation (TRFR) studies using degenerate pump and probe pulses were performed in a 8 T superconducting magnet with direct optical access. A 76 MHz, 100 fs Ti:sapphire pulsed laser pumped an optical parametric oscillator, which produced ∼250 fs optical pulses that
were tuned in wavelength to the 1s exciton absorption peak (λexc = 570 nm; 2.175 eV). The pulse train was pulse-picked down to a 0.76 MHz repetition rate to avoid buildup of any long-lived magnetic effects. Circularly polarized pump light was generated using a photo-elastic modulator, with an estimated laser fluence at the sample of 10–20 µJ cm−2 (in the weak excitation regime, 0, the spin polarization along xˆ was calculated via the partition function. We then computed the time evolution of an electron spin s (initially oriented with s0 = +ˆz at t = 0) as it precesses about Beff 28: s(t) = (s0 · n)n + {s0 − (s0 · n)n} cos ωt + [{s0 − (s0 · n)n} × n] sin ωt
(1)
− . 〈s (t)〉 was obtained by where n = Beff /|Beff | and ω = 2πne = ge μB |Beff |/h z computing and averaging over many (104 − 105 ) simulated NCs. The linear contribution to ne from the intrinsic Zeeman splitting of the host semiconductor was also added. To model experimental data, an empirical (but realistic26) exponential spin relaxation time τ s ∼ 7 ps was also applied. The calculation (in Fig. 3) of the B, T dependence of the total rms moment, 〈M2 〉, of a collection of N moments (each with moment m0 ), was computed assuming classical spins via 〈M2 〉 = N 2 m20 {1/N + [1 − 1/N][ coth (b) − 1/b]2 }, where b = m0 gμB B/kB T.
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Revealing giant internal magnetic fields due to spin fluctuations in magnetically doped colloidal nanocrystals William D. Rice1†, Wenyong Liu2, Thomas A. Baker2, Nikolai A. Sinitsyn3, Victor I. Klimov2* and Scott A. Crooker1* a 1
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Strong quantum confinement in semiconductors can compress the wavefunctions of band electrons and holes to nanometrescale volumes, significantly enhancing interactions between themselves and individual dopants. In magnetically doped semiconductors, where paramagnetic dopants (such as Mn2+, Co2+ and so on) couple to band carriers via strong sp–d spin exchange1,2, giant magneto-optical effects can therefore be realized in confined geometries using few3–7 or even single8,9 impurity spins. Importantly, however, thermodynamic spin fluctuations become increasingly relevant in this few-spin limit10. √ In nanoscale volumes, the statistical N fluctuations of N spins are expected to generate giant effective magnetic fields Beff, which should dramatically impact carrier spin dynamics, even in the absence of any applied field. Here we directly and unambiguously reveal the large Beff that exist in Mn2+-doped CdSe colloidal nanocrystals using ultrafast optical spectroscopy. At zero applied magnetic field, extremely rapid (300–600 GHz) spin precession of photoinjected electrons is observed, indicating Beff ∼ 15 −30 T for electrons. Precession frequencies exceed 2 THz in applied magnetic fields. These signals arise from electron precession about the random fields due to statistically incomplete cancellation of the embedded Mn2+ moments, thereby revealing the initial coherent dynamics of magnetic polaron formation, and highlighting the importance of magnetization fluctuations on carrier spin dynamics in nanomaterials. The past decade has witnessed a resurgent interest in magnetically doped semiconductors, motivated in part by the ability to tailor, via quantum confinement and wavefunction engineering, the spin interactions between embedded magnetic atoms and band carriers (electrons and holes)2,9,11,12. Materials for spin-based electronic and photonic applications include magnetically doped nanoribbons13,14, nanowires15, epitaxially grown quantum dots7–9,16–19, and colloidal nanocrystals3–6,12,20. However, in these evershrinking volumes, magnetization fluctuations necessarily play an increasingly essential role7,10. To illustrate, consider an electron with spin s and spatially uniform wavefunction ψ e (r) confined to a volume V of Mn2+ -doped material, so that |ψ e (r)|2 ∼ 1/V. The doping density is xMn , so that N = xMn V is the number of embedded Mn2+ , each with spin SMn . The s–d exchange energy between the electron and the Mn2+ is therefore Esd ∝ Ni |ψ e (ri )|2 s · SMn i . In applied magnetic fields sufficient to align the paramagnetic Mn2+ (a few with a total spin tesla at 4 K), the electron interacts ∝ N, so that Esd ∝ N|ψ e (r)|2 ∼ xMn . In typical diluted ST = SMn i
−10 40 K 10 K 5K
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Figure 1 | Quantifying the large sp–d spin interactions in Mn2+-doped CdSe nanocrystals using MCD spectroscopy. a, The measured MCD (and absorption) spectra from Mn:CdSe/CdS NCs and also from non-magnetic CdSe control NCs at applied magnetic field B = 7 T in the Faraday geometry ˆ Insets show the CdSe/CdS core/shell NCs, and the black arrows (B∥ˆz∥k). depict the embedded Mn2+ spins. The control NCs show small and temperature-independent MCD, as expected. The Mn:CdSe/CdSe NCs show large, inverted, temperature-dependent MCD, confirming strong sp–d spin coupling. b, The Zeeman splitting of the 1s exciton, EZ , versus B at different temperatures. EZ from the non-magnetic control NCs is small and increases linearly with B, with exciton g-factor gex = 1.3. In contrast, EZ from the Mn: CdSe/CdSe NCs is large, temperature-dependent, and follows a Brillouin function (black lines). At 5 K, |EZ | saturates at ∼27 meV, and can be characterized by an enhanced effective exciton g-factor of ∼140 in these NCs.
1
National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA. 2 Chemistry Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA. 3 Theory Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA; †Present address: Department of Physics and Astronomy, University of Wyoming, Laramie, Wyoming 82071, USA. * e-mail: [email protected]; [email protected] NATURE NANOTECHNOLOGY | ADVANCE ONLINE PUBLICATION | www.nature.com/naturenanotechnology
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Figure 2 | Ultrafast electron spin dynamics and precession at B = 0 in Mn2+-doped CdSe nanocrystals. a, A schematic of the time-resolved Faraday rotation (TRFR) experiment. The green triangles represent ultrafast optical pulses separated by time, Δt. Spin-polarized electrons with s∥ +ˆz are injected by circularly polarized pump pulses at t = 0. The net spin projection in the ensemble, 〈sz (t)〉 is detected via the Faraday rotation θF imparted to time-delayed, linearly polarized probe pulses. b, TRFR signals from non-magnetic CdSe control NCs show only the long-lived and monotonically decaying spin relaxation of the photoinjected electrons, in agreement with past work23,24. c, In contrast, TRFR from Mn2+ -doped CdSe/CdS NCs at B = 0 (solid trace) exhibits a rapid and strongly damped half-cycle oscillation occurring within a few picoseconds. This oscillatory signal indicates the very rapid precession of the (initially) spin-polarized electrons around the large effective internal magnetic field Beff in each NC that is due to Mn2+ spin fluctuations. Beff in each NC is different in both magnitude and direction, leading to rapid electron spin dephasing. The dashed trace shows TRFR from the more heavily doped Mn:CdSe/CdSe NCs. The extremely fast variation at t ∼ 0 is a coherent artifact caused by the temporally overlapping and degenerate pump and probe pulses. d, The corresponding power spectra of the time-domain TRFR signals shows that the zero-field electron spin precession frequencies in these lightly (heavily) doped NC ensembles are peaked at ∼290 GHz (∼600 GHz), indicating that electrons “see” 〈B2eff 〉 ∼ 15T (∼30 T) in these NCs.
magnetic semiconductors, this exchange energy equates to a giant effective magnetic field Beff “seen” by the electron, of order 100 T when xMn is a few per cent1,2. In contrast, at zero applied field the Mn2+ are unpolarized, and on (rms) average 〈ST 〉 = 〈Beff 〉 = 0. However, the root-mean-square isnot zero. effective field due √ tostochastic spin fluctuations, 〈B2eff 〉, √ √ 2 Rather, 〈S2T 〉 ∼ N , so√that Esd ∝ N |ψ e (r)| ∼ xMn / N , which is reduced by a factor of N from the high-field case. In bulk or 2D systems, where band electrons are delocalized and N is large, these fluctuations are comparatively insignificant and Beff ≈ 0 at zero applied field. However, in quantum-confined nanocrystals where V is small and N is of order 10, 〈B2eff 〉 can attain a sizable fraction of its high-field value, even in the absence of any applied magnetic field. Directly revealing these giant internal fluctuating fields in nanomaterial ensembles—and their expected dramatic impact on carrier spin dynamics—is highly desired but remains a significant challenge. Early Raman studies of donor-bound magnetic polarons in bulk CdMnSe established the relevance of spin fluctuations in 10,21 . More recently, micro-PL measurements small effective volumes showed that 〈B2eff 〉 influences the emission spectra from Mn2+ -doped epitaxial quantum dots, provided that individual dots were studied to avoid ensemble broadening7,8,16–18. Separately, static spectroscopies such as magnetic circular dichroism (MCD), commonly used to measure sp–d exchange in magnetically doped nanocrystals3–6, are sensitive to time- (and ensemble-) average magnetization 〈Beff 〉, but do not reveal fluctuating fields. 2
Here, we show that ultrafast optical studies can unambiguously reveal 〈B2eff 〉 in Mn2+ -doped CdSe nanocrystals via the rapid electron spin precession and relaxation that Beff induces even at zero applied field. Precession frequencies from 300–600 GHz are observed, indicating that electrons “see” 〈B2eff 〉 = 15−30 T. Because the signals occur within just a few picoseconds, these experiments directly reveal the initial coherent formation dynamics of the interesting “fluctuation magnetic polaron” that is well-known in magnetic semiconductor physics22; that is, when Beff is due to thermodynamic spin fluctuations alone, and well before any collective polaron can form due to additional Mn2+ alignment19,20. Several samples of 4–5 nm diameter Mn2+ -doped CdSe nanocrystal (NC) cores were synthesized, and then overcoated with CdS or CdSe shells to improve optical quality (see Methods). The 1s exciton absorption in these large NCs lies below the ∼2.15 eV 4 T1 6 A1 internal Mn2+ transition, suppressing excitation of the Mn2+ ions20. In this Letter we focus on two samples: Mn:CdSe/CdS NCs with 〈N〉 ≈ 10, and more heavily doped Mn: CdSe/CdSe NCs with 〈N〉 ≈ 20, where 〈N〉 is the average number of Mn2+ per NC. Control samples of non-magnetic CdSe NCs were also studied. Using conventional MCD spectroscopy (see Methods), we first verify strong sp–d exchange interactions in these NCs, by confirming an enhanced Zeeman splitting EZ of the 1s exciton, which tracks the average Mn2+ magnetization (Fig. 1). In general, EZ = gex μB B + ΔEspd 〈SMn z 〉, where the first term is the small intrinsic Zeeman splitting of the host semiconductor (gex ∼ 1), while the
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DOI: 10.1038/NNANO.2015.258 Precession about Beff ∝ ∑SiMn|Ψe(ri)|2 x
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Figure 3 | Modeling magnetization fluctuations and ultrafast electron spin dynamics in Mn2+-doped NC ensembles at B = 0. a, The incomplete cancellation of the N randomly oriented Mn2+ spins (SMn = 5/2) in any given NC generates an effective internal magnetic field, Beff, about which an electron spin s, initially polarized with s0 ∥ +ˆz at t = 0, will precess. Beff is different, both in orientation and magnitude, in every NC. Three such NCs having different Beff are depicted, along with the corresponding time evolution sz (t). b, Averaging sz (t) over all possible Beff in the NC ensemble shows that 〈sz (t)〉 exhibits a fast half-cycle precession with a minimum value of sz ∼ 0 and a recovery to a steady-state value of 1/3 (black trace). Including an additional empirical 7 ps spin relaxation time yields the orange curve, which matches the measured TRFR data very well (dashed blue trace). c, Solid curves show the expected per unit volume, 〈M2 〉 /V, of N unit paramagnetic spins ina fixed volume as a function of applied field at 2 K. At zero field, rms total moment √ 2 〈M2 〉 increases as N , while at high field it increases as N. The dashed curves show 〈M 〉 for N = 10 at other temperatures (1 K and 4 K). d, Curves show 〈M2 〉 /V (∝Beff ), for the case of a fixed density of moments xMn , for√ different volumes (using N = 10 when V = V0 ). As V drops (increasing quantum confinement), the zero-field rms magnetization increases in proportion to 1/ V .
second describes additional splitting due to sp–d exchange with the Mn2+ spins, which have average spin projection 〈SMn z 〉 along the applied field B (∥ zˆ )1–6. Since isolated Mn2+ ions are spin-5/2 paramagnets, 〈SMn z 〉 follows a temperature- and field-dependent Brillouin function B5/2 (5gMn μB B/2kB T), where the Mn2+ g-factor gMn ≃ 2.0, μB is the Bohr magneton, and kB is the Boltzmann constant. The overall magnitude of the sp–d exchange, parameterized by ΔEspd , depends on exchange constants and the overlap of the electron and hole wavefunctions with the Mn2+ dopants1,2,6. The non-magnetic CdSe NCs exhibit a small temperature-independent MCD (as expected), with gex ≃ 1.3. In contrast, the Mn: CdSe/CdS NCs show a large, inverted, and strongly temperaturedependent MCD, with EZ tracking 〈SMn z 〉 and saturating at −27 meV. At 5 K, EZ can be characterized by an enhanced exciton g-factor of ∼140; that is, as though the magnetic field in the NCs is vastly larger than the applied field, consistent with past work4–6. The highly doped Mn:CdSe/CdSe NCs exhibit even larger sp–d interactions, with |EZ | ∼ 50 meV (Supplementary Fig. 3). Crucially, however, static MCD spectroscopy reveals only how electrons and holes respond to the time- and ensemble-average effective field, 〈Beff 〉. At B = 0, 〈Beff 〉 = 0 and therefore MCD signals disappear, although Beff within any particular NC may be large (and fluctuating). However, a key result of this work is to demonstrate
that ultrafast spectroscopy can circumvent this limitation, to directly reveal 〈B2eff 〉 in NC ensembles, even at zero field. Ultrafast spin dynamics are measured by time-resolved Faraday rotation (TRFR; see Fig. 2a). Consistent with the angular momentum selection rules in these materials1,2, circularly polarized pump pulses inject electrons (and holes) at the 1s exciton resonance that are initially spin-polarized along +ˆz at t = 023–27. The projection of this spin polarization along zˆ is then detected via the Faraday rotation imparted on time-delayed, linearly polarized probe pulses (see Methods). TRFR techniques are extremely powerful probes of electron spin dynamics in non-magnetic NCs23,24 and epitaxial quantum dots25, and of coupled carrier–ion spin dynamics in Mn2+ -doped quantum wells and sol-gel films26,27. Here we apply TRFR to magnetic NCs, where carrier–Mn2+ interactions are enhanced by strong quantum confinement. At B = 0, TRFR signals from non-magnetic CdSe NCs (Fig. 2b and Supplementary Fig. 4) reveal long-lived monotonic spin relaxation of the injected electrons (the hole spins dephase quickly23), consistent with prior work23,24. In stark contrast, TRFR signals from both the lightly and more heavily Mn2+ -doped NCs are radically different (Fig. 2c): the injected spin polarization rapidly drops within 1–2 ps and then partially recovers before decaying away completely, exhibiting a strongly damped half-cycle
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Figure 4 | Ultrafast electron spin dynamics in Mn2+-doped NCs in applied transverse magnetic fields Bx . a, TRFR of Mn:CdSe/CdS NCs from Bx = 0−7 T at 40 K for short delay times. The half-cycle spin precession at 0 T evolves into a faster and more oscillatory precession signal at 7 T, as the individual Mn2+ spins align along Bx . b, Including Bx into the model of electron spin precession reproduces these trends. c, The power spectra corresponding to the raw TRFR data show a clear shift of the peak electron spin precession frequency, νe . d, νe versus Bx at 1.8, 5, and 40 K. Note that νe is independent of temperature at of νe in the more heavily doped Mn:CdSe/CdSe NCs at Bx = 0, since 〈B2eff 〉 is due to Mn2+ spin fluctuations alone at zero field. e, Temperature dependence
7 T. νe exceeds 2.5 THz at low temperatures. In these NCs, νe ∼ 600 GHz at zero field, indicating
oscillation. In general, oscillatory TRFR indicates electron spin precession about applied and/or effective magnetic fields23–27; these data therefore directly reveal an ultrafast electron spin precession at B = 0, due to 〈B2eff 〉 in these NC ensembles. These signals provide, for the first time, (i) quantitative and unambiguous evidence of the large effective magnetic fields that are always present in Mn2+ -doped NCs due to spin fluctuations, and (ii) direct observation of the initial coherent dynamics of magnetic polaron formation, when photoinjected carriers precess about Beff due to fluctuations alone, but well before any additional Mn2+ alignment occurs22. Figure 2d shows that these TRFR signals contain frequency components peaked at 290 and 600 GHz, indicating that electrons “see” 〈B2eff 〉 ∼ 15 T and ∼30 T in the lightly and more heavily doped NCs (using the bare electron g-factor ge = 1.4; see Supplementary Fig. 4). We confirm that these fast TRFR signals are not due to exciton recombination; transient absorption studies reveal nanosecond exciton lifetimes. Note that holes feel effective fields 5–6 times larger due to their larger p–d exchange interaction with Mn2+ 1,2. The damped half-cycle precession signal at B = 0 can be understood within a simple model that reproduces the data and captures the physics (Fig. 3). In any given NC, Beff is proportional to Nessential 2 Mn 2+ i Si |ψ e (ri )| , the weighted vector sum of the embedded Mn spins10,16. At B = 0, Beff is oriented randomly in each NC. The disNC ensemble is centred at 〈Beff 〉 = 0, tribution of Beff across the with characteristic width B2eff that depends on NC size and on 〈N〉. Supplementary Figs 6 and 7 show simulated distributions for different |ψ e (ri )|2 . Photoinjected electrons, all initially polarized with s∥ + zˆ at t = 0, subsequently precess about the Beff in each NC. TRFR detects 〈sz (t)〉, the average electron spin projection along zˆ . Three 4
〈B2eff 〉 ∼ 30T. Inset: 7 T TRFR signals at 5 and 30 K.
example NCs are depicted in Fig. 3a; each has different Beff and therefore different dynamics sz (t). Clearly, individual sz (t) traces add coherently only near t = 0. Figure 3b shows the result of numerically averaging over all possible Beff (see Methods): 〈sz (t)〉 drops − /(μ g 〈B2 〉), then recovers rapidly to nearly zero on a timescale πh eff B e to a third of its initial value, exhibiting a damped half-cycle oscillation even when each electron is assigned infinite spin lifetime. Imposing an additional (and realistic26) spin relaxation of 7 ps, probably due to incoherent electron-Mn2+ spin-flip scattering, gives simulated dynamics closely matching the TRFR data. This model is analogous to early treatments of electron spin dephasing in epitaxial quantum dots due to hyperfine coupling with nuclear spins28. It assumes “frozen” Mn2+ fluctuations, because electrons precess rapidly (∼300 GHz) while any Mn2+ dynamics are much slower—for example, Mn2+ spin precession (see below), or formation of collective magnetic polarons due to Mn2+ alignment (which requires ∼100 ps)19,20,22. We emphasize that ultrafast electron spin precession at zero applied field has never been observed in bulk or 2D diluted magnetic semiconductors, where carriers are more likely to overlap with many Mn2+ spins26. Rather, it uniquely arises from strong nanocrystal quantum confinement, with two inextricably linked consequences: giant magneto-optical effects from just a few Mn2+ dopants are possible, however, large (and potentially deleterious) magnetization fluctuations exist at zero or small applied fields. The qualitative dependence of 〈B2eff 〉 on applied magnetic fields B is readily anticipated. Figure 3c shows the expected rms total moment, 〈M2 〉, of N classical spins versus B. At B = 0, 〈M2 〉 is due to fluctuations alone√and is therefore independent of temperature and increases as N , while at large B (aligned spins) it increases as N. Crucially, the ratio of 〈M2 〉 at zero and large
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Figure 5 | Pump-induced coherent transfer of angular momentum to the Mn2+ spins. a, When Bx > 0, TRFR data of Mn:CdSe/CdSe NCs reveal an additional slow precession that persists for hundreds of picoseconds (long after the electron spin dynamics have ceased). Data at different Bx are intentionally offset for clarity. b, This slow precession frequency increases linearly with Bx at 28.1 GHz/T (gMn = 2.0), is independent of temperature (not shown), and appears only in Mn2+ -doped NCs, indicating a collective and coherent precession of the embedded Mn2+ spins themselves.
field therefore allows to directly determine N. Separately, Fig. 3d shows 〈M2 〉 /V (∝ Beff ) for material with fixed moment density xMn , but different volume. As V shrinks (increasing confinement), √ the zero-field rms magnetization grows in proportion to 1/ V . We confirmed these trends by applying transverse fields Bx . Figure 4a shows that the half-cycle spin precession observed at zero field evolves into an even faster oscillation with a deeper minimum as Bx increases. (The slower oscillation that develops after ∼2 ps is due to Mn2+ precession, discussed below.) This is expected, as Beff in each NC increases and orients along xˆ as Mn2+ aligns. Including Bx into our model reproduces the TRFR data nicely (Fig. 4b). The measured power spectra (Fig. 4c) clearly reveal the increase of the peak electron precession frequency ne with Bx . Figure 4d shows ne (Bx ) at 1.8, 5, and 40 K. Confirming predictions, ne (∼300 GHz) is independent of temperature at zero field (we note that this indicates that charged NCs in the ensemble, if any are present, do not influence these results; see Supplementary Fig. 5). ne increases with Bx at different temperature-dependent rates in accord with the Brillouin-like polarization of the Mn2+ spins (Fig. 3c), saturating at ∼1,000 GHz at 1.8 K, or approximately three times its zero-field value. As discussed above, this ratio suggests 〈N〉 ≃ 10 in these Mn:CdSe/CdS NCs. Moreover, we note that ne saturates at an energy (4 meV = 1,000 GHz) that is about 7 times less than the maximum exciton (electron + hole) splitting measured by MCD (27 meV; see Fig. 1), consistent with the known s–d and p–d exchange constants in bulk CdMnSe (ref. 2) and confirming that the TRFR oscillations arise from electrons alone. Figure 4e shows ne measured in the more heavily doped Mn: CdSe/CdSe NCs versus temperature. At zero field, ne ∼ 600 GHz and is temperature-independent, indicating very large effective fields 〈B2eff 〉 ∼ 30T due to spin fluctuations. When Bx = 7 T, ne exceeds 2.5 THz at low temperatures (Beff > 125 T). This larger ratio indicates that 〈N〉 is approximately twice that of the Mn:CdSe/CdS NCs shown in Fig. 4d, a conclusion independently supported by static MCD studies (Supplementary Fig. 3).
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Finally, Fig. 5 shows that strong sp–d spin interactions also influence the Mn2+ spins themselves. When Bx > 0, a much slower precession signal appears that persists for hundreds of picoseconds. The frequency increases linearly with Bx at 28.1 GHz/T (gMn = 2.0), independent of temperature. This phenomenon, known from early studies of Mn2+ -doped quantum wells26, arises from the collective and coherent spin precession of the Mn2+ spins, due to ultrafast ‘tipping’ of the average Mn2+ magnetization (∥ˆx) by the transient exchange field of the initially polarized holes (∥ zˆ ). Once perturbed away from xˆ , the average magnetization precesses about Bx and is measurable with TRFR, enabling timedomain paramagnetic spin resonance studies of Mn2+ . In summary, studies of giant internal magnetic fields due to thermodynamic spin fluctuations, revealed here with ultrafast optical spectroscopy, represent an important but previously unexplored aspect of nanoscale magnetism in colloidal quantum structures. Together with increasingly precise control over the placement of dopants29,30, as well as the possibility for magnetic coupling between the dopants themselves (e.g. in ferromagnetic semiconductors), these experiments pave the way for fundamental studies of magnetically doped nanomaterials based on fluctuations alone.
Methods Methods and any associated references are available in the online version of the paper. Received 15 July 2015; accepted 6 October 2015; published online 23 November 2015
References 1. Furdyna, J. K. Diluted magnetic semiconductors. J. Appl. Phys. 64, R29–R64 (1988). 2. Introduction to the Physics of Diluted Magnetic Semiconductors (eds. Kossut, J. & Gaj, J.A.) (Springer, 2010). 3. Hoffman, D. M. et al. Giant internal magnetic fields in Mn doped nanocrystal quantum dots. Solid State Commun. 114, 547–550 (2000). 4. Norris, D. J., Yao, N., Charnock, F. T. & Kennedy, T. A. High-quality manganese-doped ZnSe nanocrystals. Nano Lett. 1, 3–7 (2001). 5. Archer, P. I., Santangelo, S. A. & Gamelin, D. R. Direct observation of sp-d exchange interactions in colloidal Mn2+ - and Co2+ -doped CdSe quantum dots. Nano Lett. 7, 1037–1043 (2007). 6. Bussian, D. A. et al. Tunable magnetic exchange interactions in manganesedoped inverted core-shell ZnSe-CdSe nanocrystals. Nature Mater. 8, 35–40 (2009). 7. Bacher, G. et al. Monitoring statistical magnetic fluctuations on the nanometer scale. Phys. Rev. Lett. 89, 127201 (2002). 8. Besombes, L., Léger, Y., Maingault, L., Ferrand, D. & Mariette, H. Probing the spin state of a single magnetic ion in an individual quantum dot. Phys. Rev. Lett. 93, 207403 (2004). 9. Kobak, J. et al. Designing quantum dots for solotronics. Nature Commun. 5, 3191 (2014). 10. Dietl, T. & Spałek, J. Effect of fluctuations of magnetization on the bound magnetic polaron: Comparison with experiment. Phys. Rev. Lett. 48, 355–358 (1982). 11. Abolfath, R. M., Hawrylak, P. & Zŭtić, I. Tailoring magnetism in quantum dots. Phys. Rev. Lett. 98, 207203 (2007). 12. Beaulac, R., Archer, P. I., Ochsenbein, S. T. & Gamelin, D. R. Mn2+ -doped CdSe quantum dots New inorganic materials for spin-electronics and spin-photonics. Adv. Funct. Mater. 18, 3873–3891 (2008). 13. Yu, J. H. et al. Giant Zeeman splitting in nucleation-controlled doped CdSe:Mn2+ quantum nanoribbons. Nature Mater. 9, 47–53 (2010). 14. Fainblat, R. et al. Quantum confinement-controlled exchange coupling in manganese (II)-doped CdSe two-dimensional quantum well nanoribbons. Nano Lett. 12, 5311–5317 (2012). 15. Wojnar, P. et al. Giant spin splitting in optically active ZnMnTe/ZnMgTe core/ shell nanowires. Nano Lett. 12, 3404–3409 (2012). 16. Wojnar, P. et al. Microluminescence from Cd1−x Mnx Te magnetic quantum dots containing only a few Mn ions. Phys. Rev. B 75, 155301 (2007). 17. Hundt, A., Puls, J. & Henneberger, F. Spin properties of self-organized diluted magnetic Cd1−x Mnx Se quantum dots. Phys. Rev. B 69, 121309(R) (2004). 18. Dorozhkin, P. S. et al. Longitudinal and transverse fluctuations of magnetization of the excitonic magnetic polaron in a semimagnetic single quantum dot. Phys. Rev. B 68, 195313 (2003). 19. Seufert, J. et al. Dynamical spin response in semimagnetic quantum dots. Phys. Rev. Lett. 88, 027402 (2002).
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20. Beaulac, R., Schneider, L., Archer, P. I., Bacher, G. & Gamelin, D. R. Lightinduced spontaneous magnetization in doped colloidal quantum dots. Science 325, 973–976 (2009). 21. Nawrocki, M., Planel, R., Fishman, G. & Galazka, R. Exchange-induced spin-flip Raman scattering in a semimagnetic semiconductor. Phys. Rev. Lett. 46, 735–738 (1981). 22. Yakovlev, D. R. & Ossau, W. in Introduction to the Physics of Diluted Magnetic Semiconductors (eds. Kossut, J. & Gaj, J. A.) Ch. 7 (Springer, 2010). 23. Gupta, J. A., Awschalom, D. D., Efros, A. L. & Rodina, A. V. Spin dynamics in semiconductor nanocrystals. Phys. Rev. B 66, 125307 (2002). 24. Fumani, A. K. & Berezovsky, J. Magnetic-field-dependent spin decoherence and dephasing in room-temperature CdSe nanocrystal quantum dots. Phys. Rev. B 88, 155316 (2013). 25. Greilich, A. et al. Mode locking of electron spin coherences in singly charged quantum dots. Science 313, 341–345 (2006). 26. Crooker, S. A., Awschalom, D. D., Baumberg, J. J., Flack, F. & Samarth, N. Optical spin resonance and transverse spin relaxation in magnetic semiconductor quantum wells. Phys. Rev. B 56, 7574 (1997). 27. Whitaker, K. M. et al. Spin-on spintronics: Ultrafast electron spin dynamics in ZnO and Zn1−x Cox O sol-gel films. Nano Lett. 11, 3355–3360 (2011). 28. Merkulov, I. A., Efros, A. L. & Rosen, M. Electron spin relaxation by nuclei in semiconductor quantum dots. Phys. Rev. B 65, 205309 (2002). 29. Ithurria, S., Guyot-Sionnest, P., Mahler, B. & Dubertret, B. Mn2+ as a radial pressure gauge in colloidal core/shell nanocrystals. Phys. Rev. Lett. 99, 265501 (2007). 30. Grumbach, N., Rubin-Brusilovski, A., Maikov, G. I., Tilchin, E. & Lifshitz, E. Manipulation of carrier-Mn2+ exchange interaction in CdTe/CdSe colloidal quantum dots by controlled positioning of Mn2+ impurities. J. Phys. Chem. C 117, 21021–21027 (2013).
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31. Vlaskin, V. A., Barrows, C. J., Erickson, C. S. & Gamelin, D. R. Nanocrystal diffusion doping. J. Am. Chem. Soc. 135, 14380–14389 (2013). 32. Ithurria, S. & Talapin, D. V. Colloidal atomic layer deposition (c-ALD) using self-limiting reactions at nanocrystal surface coupled to phase transfer between polar and nonpolar media. J. Am. Chem. Soc. 134, 18585–18590 (2012).
Acknowledgements We gratefully thank D. R. Yakovlev and D. L. Smith for helpful discussions and insight. W.D.R. acknowledges support from the Los Alamos LDRD programme. W.L., T.A.B., and V.I.K. are supported by the Office of Chemical Sciences, Biosciences, and Geosciences of the Department of Energy Office of Basic Energy Sciences. All optical measurements were performed at the National High Magnetic Field Laboratory, which is supported by NSF DMR-1157490.
Author contributions S.A.C. and V.I.K. conceived and directed the experiments. W.D.R. built and performed the optical experiments, and designed the numerical simulations. W.L. synthesized the nanocrystals, and T.A.B. made the films. N.S. provided theoretical insights. S.A.C. and W.D.R. analyzed the data and wrote the paper in close consultation with all authors.
Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to V.I.K. and S.A.C.
Competing financial interests
The authors declare no competing financial interests.
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DOI: 10.1038/NNANO.2015.258
Methods
Colloidal nanocrystal synthesis. Wurtzite CdSe NCs were initially prepared with 4–5 nm diameters. The 1s exciton energy in these NCs lies below the 4 T1 6 A1 internal Mn2+ transition at ∼2.15 eV, suppressing Auger excitation of the Mn2+ ions. Mn doping followed published procedures31 with necessary modifications. To eliminate Mn atoms and traps on the NC surfaces, an inorganic epitaxial shell was grown. For the lower doped Mn:CdSe/CdS NCs (〈N〉 ≃ 10), solution-based colloidal atomic layer deposition of CdS under ambient conditions was utilized32. The CdS shell is 1.2 nm thick (4 layers). For the more highly doped Mn:CdSe/CdSe NCs (〈N〉 ≃ 20), a CdSe shell was grown at elevated temperatures. Room-temperature emission quantum yields were thereby improved to ∼10–20%. Details of the colloidal NC synthesis, Mn2+ doping procedure, film preparation, and initial optical characterization are shown in Supplementary Figs 1 and 2 and the accompanying discussion. MCD measurements. MCD is based on measuring the normalized transmission difference between left- and right-circularly polarized light through the NC film, ˆ These circular (T L − T R )/(T L + T R ), in the Faraday geometry (applied field B∥ˆz ∥k). polarizations couple to the ‘spin-up’ and ‘spin-down’ (total angular momentum projection Jz = ±1) states of the nanocrystals’ 1s exciton respectively, which are Zeeman-split in B by the small intrinsic exciton g-factor gex of the host semiconductor (present in all materials), and also by the large sp–d exchange coupling between any local Mn2+ moments and the electron/hole spins of the NC. Thus, MCD spectra typically track the derivative of the 1s absorption. Probe light of tunable wavelength was derived from a xenon lamp directed through a spectrometer. The light was mechanically chopped, then polarization-modulated at 50 kHz using a photoelastic modulator, focused through the sample, and detected by an avalanche photodiode6. TRFR measurements. Time-resolved Faraday rotation (TRFR) studies using degenerate pump and probe pulses were performed in a 8 T superconducting magnet with direct optical access. A 76 MHz, 100 fs Ti:sapphire pulsed laser pumped an optical parametric oscillator, which produced ∼250 fs optical pulses that
were tuned in wavelength to the 1s exciton absorption peak (λexc = 570 nm; 2.175 eV). The pulse train was pulse-picked down to a 0.76 MHz repetition rate to avoid buildup of any long-lived magnetic effects. Circularly polarized pump light was generated using a photo-elastic modulator, with an estimated laser fluence at the sample of 10–20 µJ cm−2 (in the weak excitation regime, 0, the spin polarization along xˆ was calculated via the partition function. We then computed the time evolution of an electron spin s (initially oriented with s0 = +ˆz at t = 0) as it precesses about Beff 28: s(t) = (s0 · n)n + {s0 − (s0 · n)n} cos ωt + [{s0 − (s0 · n)n} × n] sin ωt
(1)
− . 〈s (t)〉 was obtained by where n = Beff /|Beff | and ω = 2πne = ge μB |Beff |/h z computing and averaging over many (104 − 105 ) simulated NCs. The linear contribution to ne from the intrinsic Zeeman splitting of the host semiconductor was also added. To model experimental data, an empirical (but realistic26) exponential spin relaxation time τ s ∼ 7 ps was also applied. The calculation (in Fig. 3) of the B, T dependence of the total rms moment, 〈M2 〉, of a collection of N moments (each with moment m0 ), was computed assuming classical spins via 〈M2 〉 = N 2 m20 {1/N + [1 − 1/N][ coth (b) − 1/b]2 }, where b = m0 gμB B/kB T.
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