Letter pubs.acs.org/NanoLett
Circularly Polarized Near-Field Optical Mapping of Spin-Resolved Quantum Hall Chiral Edge States Syuhei Mamyouda,† Hironori Ito,† Yusuke Shibata,† Satoshi Kashiwaya,‡ Masumi Yamaguchi,¶ Tatsushi Akazaki,¶ Hiroyuki Tamura,¶ Youiti Ootuka,† and Shintaro Nomura*,† †
Division of Physics, University of Tsukuba, Tennodai, Tsukuba 305-8571, Japan National Institute of Advanced Industrial Science and Technology (AIST), Umezono, Tsukuba 305-8568, Japan ¶ NTT Basic Research Laboratories, NTT Corporation, Morinosato-Wakamiya, Atsugi 243-0198, Japan ‡
S Supporting Information *
ABSTRACT: We have successfully developed a circularly polarized nearfield scanning optical microscope (NSOM) that enables us to irradiate circularly polarized light with spatial resolution below the diffraction limit. As a demonstration, we perform real-space mapping of the quantum Hall chiral edge states near the edge of a Hall-bar structure by injecting spin polarized electrons optically at low temperature. The obtained real-space mappings show that spin-polarized electrons are injected optically to the twodimensional electron layer. Our general method to locally inject spins using a circularly polarized NSOM should be broadly applicable to characterize a variety of nanomaterials and nanostructures. KEYWORDS: near-field optical microscope, quantum Hall effect, edge state, spin injection, chiral, nanophotonics, circular polarization, scanning probe microscopy
A
As a demonstration, we perform real-space mapping of quantum Hall chiral edge states near the edge of a Hall-bar structure of a GaAs/Al0.3Ga0.7As modulation-doped single heterojunction by injecting spin polarized electrons using a circularly polarized NSOM at low temperature. In the quantum Hall effect,18 one-dimensional current-carrying edge states are formed under a perpendicular magnetic field near the edge of a sample.19,20 The quantum Hall chiral edge states have a unique property that the edge currents are chiral, namely, backscatterings of the electrons by disorders are prohibited, and hence, the edge states are characterized by a long coherence length.21 The edge states have recently attracted much attention after discoveries of the quantum spin Hall effect16 and the quantum anomalous effect,17 where spin polarized edge current plays an essential role. The spin degree of freedom has been predicted to dramatically change the potential profiles near the quantum Hall chiral edge state.22−24 However, the difficulty in injecting spin-polarized electrons hinders a realspace mapping of the spin-resolved edge states. We utilize the optical approach that straightforwardly probes spin polarizations of the electrons in the edge states.25 Because the optical approach allows direct spatial mapping of the underlying electronic structures with minimum disturbance, this approach has been used to image the quantum Hall chiral edge states.10,26−29
photon with right (RCP) or left circular polarization (LCP) has a well-defined angular momentum component +1 or −1, respectively, along the direction of the propagation. Upon an optical excitation, the angular momentum of a photon is transferred to an electron and a hole following the optical selection rule. This method has been widely used in far-field spectroscopy.1 To study nanomaterials or nanostructures, however, we need to apply an optical imaging method with spatial resolution beyond the diffraction limit. One such method is a near-field scanning optical microscope (NSOM).2,3 NSOMs utilize the optical field that is confined to subdiffraction limited volume by a plasmon resonance at the apex of a sharp nanoprobe4−6 or by an opaque nanoprobe with an aperture typically 30−200 nm in size.2,3,7,8 NSOMs are widely used to obtain images by scanning a nanoprobe on the surface of an object.2−10 An NSOM, however, has not been used for illumination of circularly polarized light because a standard single mode optical fiber does not maintain optical polarization, and small tensions or torsions provoke straininduced birefringence at the NSOM probe, although circularly polarized illumination to subdiffraction limited volume is expected to clarify spin-related phenomena in a variety of nanomaterials and nanostructures.9,11−17 Here, we report on our method to illuminate sample surface with a circularly polarized light by using an NSOM tip with an aperture with good axial symmetry and compensating the residual retardation by controlling the polarization of the incident light such that the light emitted from an NSOM tip to be circularly polarized. © 2015 American Chemical Society
Received: December 11, 2014 Revised: February 15, 2015 Published: March 2, 2015 2417
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Nano Letters In this Letter, we show a first real-space mapping of spinresolved quantum Hall chiral edge states by injecting spinpolarized electrons using a circularly polarized NSOM. The obtained real-space mappings show the formation of spin-split incompressible strips induced by the exchange energy enhanced spin-gap, accordingly to a model calculation based on the local spin-density approximation. Our results demonstrate that our circularly polarized NSOM enables us to optically injected spinpolarized electrons to subdiffraction limited area and may find applications to investigate a variety of nanomaterials and nanostructures. Experimental Section. A double tapered NSOM probe was fabricated at the cleaved edge of an optical fiber by wet chemical etching7 and was coated with chromium or chromium/gold. An aperture of about 100 nm was fabricated by focused ion beam slicing of the apex of the tip. A single mode optical fiber was inserted from the top flange of a dilution refrigerator straight down to the NSOM probe to minimize the birefringence induced by torsion or bending. This also minimizes possible birefringence accompanied by the Faraday effect for a diamagnetic material, which is known to induce phase retardance of the two components of the electric field vectors at the presence of birefringence in an optical fiber subject to the external magnetic field.30,31 The phase retardance by the Faraday effect was estimated to be smaller than 0.005 rad. A laser light from a tunable semiconductor diode laser was led to a polarizer and a Berek compensator and then coupled to a single mode optical fiber as schematically shown in Figure 1a. The retardance and the orientation of the Berek compensator were set such that circularly polarized light was emitted from the NSOM tip inside the dilution refrigerator. Degree of circular polarization after compensation of the retardance
added by the optical fiber and the NSOM probe was better than 80%. The density of two-dimensional electron system (2DES) in the GaAs/Al0.3Ga0.7As modulation-doped single heterojuntion was 4.6 × 1011 cm−2 with a mobility of 1.8 × 106 cm2/(V s). We used a Hall-bar structure with the width and the length of 25 and 300 μm, respectively. The 2DES layer was located 90 nm from the surface. We denote that magnetic field parallel to the vector pointing from the rear to the front side of the sample as positive B. The sample on the scanning stage was illuminated with a laser light through an NSOM probe under a shear force feedback control in a dilution refrigerator with a base temperature of 70 mK. The sample temperature during measurements was below 250 mK. The scanning range of the tube piezoscanner was 2.1 μm. The optical excitation power was kept to be small to avoid any heating of carriers, and was 25 pW and 1.1 nW for the excitation photon energy of 1.5194 and 1.5140 eV, respectively. The former and the latter correspond to 11.1 and 5.7 meV above the onset of absorption of GaAs single heterojunction of 1.5083 eV,10 respectively. The change of the electron density due to local optical excitation was negligibly small because of the small excitation power. Photovoltage was amplified by a differential preamplifier and detected synchronously with a lock-in amplifier. Results and Discussion. Quantum Hall chiral edge states are formed near the edge of 2DES by the interplay of the confinement potential and the electrostatic screening in a magnetic field.20,32 Regions of constant electron density, called incompressible strips, are formed where the Fermi level is located in a finite gap due to the cyclotron motion of electrons. In between incompressible strips, regions of flat electrostatic potential and of smoothly varying electron density are formed and called compressible strips. Figure 1b shows a result of mappings of quantum Hall edge chiral states excited by unpolarized light at 1.5194 eV, which is above the onset of absorption of the bulk 2DES region and the exciton ground state energy in bulk GaAs buffer layer and below the onset of Al0.3Ga0.7As barrier layers. To better visualize the observed features, the first spatial derivative of photovoltage (∂V/∂y) is plotted with lateral position and B. Periodic photovoltage signals correlated with the electron filling factor ν are observed (Figure 1b, c). The laser light creates photoinduced carriers. The photoinduced holes move to the rear side of the sample and contribute to bulk diffusion current, whereas the photoinduced electrons near the edge of the sample with large excess energy relax rapidly to the Fermi level. Depending on ν and the position of the optical excitation, photoinduced electrons either diffuse to the compressible strips at the same side of the sample of the optical excitation or to the compressible strips at the opposite side of the sample. This changes the Hall voltage slightly to balance the diffusion current, resulting in positive or negative V for the former and the latter cases, respectively.10 At B < 1.3 T, photovoltage signals due to the spin unpolarized incompressible strips are observed. The position of incompressible strips is given by32 yk = (d0/(1 − (νL/ν)2)), where νL is a local filling factor, and d0 is a depletion layer thickness as estimated to be 134 nm (Figure 1b). With increase in B, the photovoltage signals split at B ≈ 1.5 T. At B ≥ 1.7 T, we observe photovoltage signals at around 1.7, 2.1, and 2.7 T in between the spin-unpolarized incompressible strips at yk. The photovoltage tends to zero at even ν at the condition for the quantum Hall effect, as shown in Figure 1c, in agreement with previous reports.10,28,29,33 Figure 1c also shows that the
Figure 1. (a) Schematics of the measurement setup and the sample structure. (b) Mappings of the first derivative of photovoltage near the edge of a Hall-bar structure. The mapping was obtained by exciting the sample by unpolarized light at 1.5194 eV, which is 11.1 meV above the onset of absorption of GaAs single heterojunction and below the onset of Al0.3Ga0.7As barrier layers, at 250 mK at magnetic fields between 1.00 and 3.50 at 0.02 T step. Black and white curves are the positions of the spin-unpolarized and polarized incompressible strips yk (see text). (c) Magnetic field dependence of photovoltage. The position of the illumination spot was 500 nm from the edge of the Hall-bar structure. 2418
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Figure 2. Circular polarization dependent mappings of the first spatial derivative of photovoltage as functions of lateral position and magnetic field for (a) right and (b) left circularly polarized light at B > 0 at the excitation energy of 1.5140 eV, which corresponds to 5.7 meV above the onset of absorption of GaAs single heterojunction. The black ovals denote the positions where the first spatial derivatives of photovoltage are stronger in LCP than in RCP. (c) Degree of polarization of the first spatial derivative of photovoltage P+ for B > 0 and (d) P− for B < 0. Black and white curves are the positions of the spin-unpolarized and polarized incompressible strips yk (see text).
photovoltage tends to zero at odd ν. The positions of the observed photovoltage signals partly deviate from the theoretically predicted positions of the incompressible strips yk. This indicates that the confinement potential in the sample is harder near the edge than the potential assumed in the calculation.32 Figure 2a and b show results of mapping of quantum Hall chiral edge states obtained by illuminating the sample locally with circularly polarized light at 1.5140 eV, which is 5.7 meV above the onset of absorption for RCP (Figure 2a) and LCP (Figure 2b) at B > 0, where ∂V/∂y is plotted. We can identify photovoltage signals that shift inward with an increase in |B|. We have found that the photovoltage signal at 3.5 T (ν = 5.4) is clearly stronger in LCP of −20 μV/μm than in RCP of −8.4 μV/μm as indicated in the ovals in Figure 2a and b. We have confirmed that the sign of the photovoltage is reversed by reversing the direction of B as shown in the line profiles of photovoltage in Figure 3a and b. Similar circular polarization dependence of the photovoltage was observed at 2.58 T (ν = 7.4) (Figures 2a, b and 3c). Figures 2c (B > 0) and 2d (B < 0) show degrees of polarization of the first spatial derivative of photovoltage Pη =
∂yV η↓ |∂yV η↓|
− +
∂yVη↑ |∂yVη↑|
Figure 3. Circular polarization dependent photovoltage as a function of lateral position. The sample was excited at 1.5140 eV at (a) 3.50 T (ν = 5.4), (b) −3.50 T (ν = 5.4), (c) 2.58 T (ν = 7.4), (d) 2.74 T (ν = 6.9), (e) 3.06 T (ν = 6.2), and (f) 3.34 T (ν = 5.7).
white dashed curves, indicating the opening of the exchange enhanced spin-gap34,35 up to at least 0.2 μm from the sample edge. By contrast, |P±| along the black curves is small, typically | P±| < 0.2. These results clearly show that spin-polarized electrons are injected optically to the 2DES layer as will be discussed in more details later. Figure 2c and d show scattered signals at ν ≈ integer because both denominator and numerator of the right-hand side of eq 1 are small at the condition for the quantum Hall effect. Figure 4 shows calculated subband structures near the edge of a 2DES as functions of X = −lB2 ky by a calculation in the Landau gauge based on the local spin-density approximation formalism,25 where lB and ky are the magnetic length and y component of the wave vector, respectively. Details of the
(1)
where ∂yVs↓ and ∂yVs↑ are the first spatial derivative of photovoltage at circular polarization for creating ↓- and ↑ -spin electrons, respectively, and η = ± is the sign of B. We find V↓+= VLCP and V↑+= VRCP for B > 0, and V↑−= VLCP and V↓−= VRCP for B < 0, where VLCP and VRCP are photovoltage at LCP and RCP excitations, respectively. Figure 2c shows blue strips at |B| ≈ 3.4, 2.5, and 2.0 T moving inward with increase in |B|. Remarkably, |P±| is close to unity at the tip position between 1.5 and 0.2 μm along the 2419
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hinders the diffusion of the down-spin electron inward, resulting in diffusion of more up-spin electrons than downspin electrons from the edge to the bulk, giving V+LCP > V+RCP as shown in Figure 3a, c, and f. Large |Ps| at 2m − 1 < ν < 2m in Figure 2c, d is in agreement with this argument. At 2m < ν < 2m + 1, a spin-unpolarized incompressible strip separates the bulk and the compressible strip of down-spin electron as shown in Figure 4b, c. The positive gradient of the subband energies of the up-spin electron on C↓ (Figure 4b, c) results in smaller diffusion of the up-spin electron to the bulk, LCP (Figure 3e), but this effect is less evident in giving VRCP + >V+ Figure 3e because the spin-unpolarized incompressible strip formed due to the cyclotron energy ℏωc is more robust to disorders and its width is wider than that of the spin-split incompressible strip formed due to the exchange-enhanced spin-gap. Consequently, the diffusion of both up- and downspin electrons across the incompressible strip to the bulk is small, and this explains the large positive photovoltage with small circular polarization dependence at ν = 6.2 in Figure 3e. At ν = 6.9 (Figure 3d), a narrow spin-unpolarized incompressible strip separates the bulk and edge. The absolute value of the observed photovoltage is small partly because the diffusion of photoinduced electrons to the sample edges is nearly balanced in this case, partly because ν is close to integer number. The contribution of the circular dichroism of optical absorption to the observed circular polarization dependence of photovoltage in Figure 2 and 3 is small because the photons are predominantly absorbed in the thick GaAs layer below the heterointerface at the optical excitation energy of 1.5140 eV, which is only 5.7 meV above the onset of absorption at 1.5083 eV.10 This also results in the small difference in the ionization rates of photogenerated excitons between LCP and RCP excitations because the ionizations are primarily due to the strong electric field perpendicular to the heterointerface that is not spin-dependent. The large widths of I and Is, typically several times larger than lB, give small probability of direct tunneling between the edge and the bulk at the Fermi-level. The electron transport between the edge and the bulk is due to slow processes, such as Frenkel-Poole transport36 near the Fermi-level, and/or the diffusion through the excited states ℏ ωc above the Fermi-level. This results in the lack of spinblockade due to the occupations of up- and down-spin levels in the bulk region. In fact, both the up- and the down-spin electrons diffuse across Is in the case of ν slightly smaller than even numbers as observed in Figure 3f. Although the Hall photovoltage is built up after optical absorption, ionization, and diffusion, the above results show that the nature of the innermost incompressible strip, whether it is spin-unpolarized or spin-split, is the primal factor for accounting the spinresolved Hall voltage mappings. In conclusion, we have successfully developed a circularly polarized NSOM that enables us to irradiate circularly polarized light from near-field optical probe tip with spatial resolution below the diffraction limit. We have shown that our newly developed circularly polarized NSOM enables us to locally inject spin-polarized electrons in the 2DES layer that is buried several tens of nanometers below the surface of a sample and is not accessed with scanning tunneling microscopy.37,38 As a demonstration, we investigate real-space mapping of the quantum Hall chiral edge state near the edge of a sample of GaAs/Al0.3Ga0.7As modulation-doped single heterojunction, and find strips of high-degrees of polarization |P±| at the tip
Figure 4. Calculated subband structures near the sample edge for (a) ν = 7.4, (b) ν = 6.9, (c) ν = 6.2, and (d) ν = 5.7. Spin-polarized incompressible strip (Is) separates the bulk and the edge states in (a) and (d), whereas spin-unpolarized incompressible strip (I) separates the bulk and the edge states in (b) and (c). wa, wb, wc, and wd denote the widths of the innermost incompressible strips.
calculation can be found in Supporting Information. The quantum Hall edge states without including spin are characterized by the formation of compressible strips with constant electrostatic potential by perfect screening and incompressible strips (I) with a finite energy gap at the Fermi level.20 Introduction of the spin-splitting does not change the concept of incompressible strips; however, it has been predicted23,24 that the compressible strips are suppressed and instead, narrow spin-polarized compressible strips (C↑, C↓) and spin-split incompressible strips (Is) in-between them are formed due to the exchange-enhanced spin gap as shown in Figure 4. This model explains the observations in Figures 2 and 3 as follows. The equilibration between the edge and the bulk of the sample explains the sign of the photovoltage.10,33 The diffusion between the bulk and the edge regions is governed by the width of the innermost incompressible strip. Hall photovoltage is established such that to balance the diffusion of photoinduced electrons as can be found in Supporting Information in more details. The innermost Is in Figure 4a at ν+ has the width wa = 176 nm, which is larger than the width of the innermost Is in Figure 4d of wd= 121 nm at ν−. Similarly, the innermost I in Figure 4c at ν+ has the width wc = 362 nm, which is larger than the innermost width of I in Figure 4b of wb = 236 nm at ν−. As a result, the diffusion of the photoinduced electrons from the edge to the bulk is small at ν+, resulting in the positive photovoltage as shown in Figures 3a, 3c, and 3e. By contrast, the negative photovoltage is observed in Figures 3d and 3f since the diffusion of the photoinduced electrons from the edge to the bulk is large due to smaller width of the innermost Is or I (Figures 4b, 4d) at ν− than the width wa or wc at ν+. The observed circular polarization dependence of the photovoltage is explained by whether the edge and the bulk are separated by the spin-unpolarized or the spin-split incompressible strips. At 2m − 1 < ν < 2m for m integer, a spin-split incompressible strip Is separates the bulk and the compressible strip of up-spin electron C↑ as shown in Figure 4a, d. On C↑, the subband energies of the up-spin electron are flat, whereas the subband energies of the down-spin electron increase with increase in the distance from the edge. This 2420
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Nano Letters position between 1.5 and 0.2 μm, indicating the formation of spin-split incompressible strips due to the opening of the exchange energy enhanced spin-gap. We find that the spinsplitting of the innermost incompressible strip governs the diffusion of optically created electrons and, hence, the optically induced Hall voltage in the quantum Hall chiral edge states. This result shows that our circularly polarized NSOM enables us to locally excite sample with circularly polarized light and to inject spin-polarized electrons. We expect the degree of circular polarization of the emission from an NSOM tip increases with decrease in size of the aperture based on the calculation of the electric field emitted from a small circular aperture within the Bethe−Bouwkamp model.39 Although our measurements were performed using near-infrared light, we expect that the spectral region may be extended to far-infrared regime as the first realspace mapping of the edge state was performed using farinfrared light.26 Our method to locally inject spins using a circularly polarized NSOM is generally applicable to a variety of nanomaterials and nanostructures, such as polarization dependent phenomena of surface plasmons in plasmonic nanostructures,9,11 the valley-spin coupling effect of transition-metal dichalcogenide thin films,12,13 chiral molecules,14,15 and the spin-selective mappings of the edge states of topological insulators.16,17
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(10) Ito, H.; Furuya, K.; Shibata, Y.; Kashiwaya, S.; Yamaguchi, M.; Akazaki, T.; Tamura, H.; Ootuka, Y.; Nomura, S. Phys. Rev. Lett. 2011, 107, 256803. (11) Day, J. K.; Neumann, O.; Grady, N. K.; Halas, N. J. ACS Nano 2010, 4, 7566−7572. (12) Xiao, D.; Liu, G.-B.; Feng, W.; Xu, X.; Yao, W. Phys. Rev. Lett. 2012, 108, 196802. (13) Wang, Q. H.; Kalantar-Zadeh, K.; Kis, A.; Coleman, J. N.; Strano, M. S. Nat. Nano. 2012, 7, 799. (14) Abdulrahman, N. A.; Fan, Z.; Tonooka, T.; Kelly, S. M.; Gadegaard, N.; Hendry, E.; Govorov, A. O.; Kadodwala, M. Nano Lett. 2012, 12, 977−983. (15) Ben Dor, O.; Morali, N.; Yochelis, S.; Baczewski, L. T.; Paltiel, Y. Nano Lett. 2014, 14, 6042−6049. (16) König, M.; Wiedmann, S.; Brüne, C.; Roth, A.; Buhmann, H.; Molenkamp, L. W.; Qi, X.-L.; Zhang, S.-C. Science 2007, 318, 766− 770. (17) Chang, C.-Z.; et al. Science 2013, 340, 167−170. (18) Klitzing, K. v.; Dorda, G.; Pepper, M. Phys. Rev. Lett. 1980, 45, 494−497. (19) Halperin, B. I. Phys. Rev. B 1982, 25, 2185−2190. (20) Chklovskii, D. B.; Shklovskii, B. I.; Glazman, L. I. Phys. Rev. B 1992, 46, 4026−4034. (21) Ji, Y.; Chung, Y.; Sprinzak, D.; Heiblum, M.; Mahalu, D.; Shtrikman, H. Nature 2003, 422, 415−418. (22) Dempsey, J.; Gelfand, B. Y.; Halperin, B. I. Phys. Rev. Lett. 1993, 70, 3639−3642. (23) Manolescu, A.; Gerhardts, R. R. Phys. Rev. B 1995, 51, 1703− 1713. (24) Ihnatsenka, S.; Zozoulenko, I. V. Phys. Rev. B 2006, 73, 155314. (25) Nomura, S.; Aoyagi, Y. Phys. Rev. Lett. 2004, 93, 096803. (26) Merz, R.; Keilmann, F.; Haug, R. J.; Ploog, K. Phys. Rev. Lett. 1993, 70, 651−653. (27) van Haren, R. J. F.; Blom, F. A. P.; Wolter, J. H. Phys. Rev. Lett. 1995, 74, 1198−1201. (28) Shashkin, A. A.; Kent, A. J.; Owers-Bradley, J. R.; Cross, A. J.; Hawker, P.; Henini, M. Phys. Rev. Lett. 1997, 79, 5114−5117. (29) van Zalinge, H.; Ö zyilmaz, B.; Böhm, A.; van der Heijden, R. W.; Wolter, J. H.; Wyder, P. Phys. Rev. B 2001, 64, 235303. (30) Smith, A. M. Appl. Opt. 1978, 17, 52−56. (31) Cruz, J.; M.V, A.; Hernandez, M. Appl. Opt. 1996, 35, 922−926. (32) Lier, K.; Gerhardts, R. Phys. Rev. B 1994, 50, 7757. (33) van Zalinge, H.; van der Heijden, R. W.; Wolter, J. H.; Ö zyilmaz, B.; Böhm, A.; Wyder, P. Semicond. Sci. Technol. 2004, 19, 1153. (34) Ando, T.; Uemura, Y. J. Phys. Soc. Jpn. 1974, 37, 1044−1052. (35) Nomura, S.; Tamura, H.; Yamaguchi, M.; Akazaki, T.; Hirayama, Y. Phys. Rev. B 2013, 87, 085318. (36) Frenkel, J. Phys. Rev. 1938, 54, 647−648. (37) Morgenstern, M.; Klijn, J.; Meyer, C.; Getzlaff, M.; Adelung, R.; Römer, R. A.; Rossnagel, K.; Kipp, L.; Skibowski, M.; Wiesendanger, R. Phys. Rev. Lett. 2002, 89, 136806. (38) Wiesendanger, R. Rev. Mod. Phys. 2009, 81, 1495−1550. (39) van Labeke, D.; Barchiesi, D.; Baida, F. J. Opt. Soc. Am. A 1995, 12, 695−703.
ASSOCIATED CONTENT
S Supporting Information *
Details of NSOM probe fabrication, polarization control procedure, circular polarization and direction of spin of optically created electrons, method of calculation of subband structure near the edge, and diffusion of electrons near the edge in magnetic field are described. This material is available free of charge via the Internet at http://pubs.acs.org/.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank M. Iitake at AIST for technical assistance. A part of this work was done in Nano-Processing Facility, AIST. This work was partly supported by Kakenhi Nos. 25103704, 26610079.
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REFERENCES
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