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Abstract
The demand for faster magnetization switching speeds and lower energy consumption has driven the field of spintronics in recent years. The magnetic tunnel junction is the most developed spintronic memory device in which the magnetization of the information storage layer is switched by spin-transfer-torque or spin-orbit torque interactions. Whereas these novel spin-torque interactions exemplify the potential of electron-spin-based devices and memory, the switching speed is limited to the ns regime by the precessional motion of the magnetization. All-optical magnetization switching, based on the inverse Faraday effect, has been shown to be an attractive method for achieving magnetization switching at sub-ps speeds. Successful magnetization reversal in thin films has been demonstrated by using circularly polarized light. However, a method for all-optical switching of on-chip nanomagnets in high density memory modules has not been described. In this work we propose to use plasmonics, with CMOS compatible plasmonic materials, to achieve on-chip magnetization reversal in nanomagnets. Plasmonics allows light to be confined in dimensions much smaller than the diffraction limit of light. This in turn yields higher localized electromagnetic field intensities. In this work, through simulations, we show that by using localized surface plasmon resonances, it is possible to couple light to nanomagnets and achieve significantly higher opto-magnetic field values in comparison to free space light excitation.
© 2017 Optical Society of America
1. Introduction
Following the experimental demonstration of ultrafast (few hundreds of fs) demagnetization in 1996 utilizing optical pulses [1], the magnetization switching dynamics in these unprecedented time scales has been extensively investigated both theoretically and experimentally [2]. We note that magnetization reversal speed induced by magnetic fields or spin polarized currents is limited by the precessional motion of the magnetization. This limitation hinders the implementation of current spintronic devices in beyond-CMOS electronics and in particular, for on-chip integration of memory and logic spintronic-based devices. In contrast to CMOS devices that switch in ps time scales, state of the art spin-transfer-torque magnetic tunnel junctions (STT-MTJ), typically switch in the ns regime. Furthermore, high current densities of the order of 107 – 108 A/cm2 are required, making them unattractive from an energy consumption perspective. In addition such high current densities can result in electrical breakdown of the MgO tunnel barrier, negatively impacting their reliability.
All-optical magnetization switching has been shown in recent years to overcome precession dynamics limits and 40-fs circularly polarized laser pulses have been employed to reversibly switch the magnetization by changing the light helicity in a wide range of magnetic materials [2–6]. Whereas the full physical understanding underlying the ultrafast magnetization reversal remains incomplete, one plausible mechanism invokes the fact that circularly polarized laser pulses provide an effective magnetic field via the inverse Faraday effect. This opto-magnetic field, HOM, is proportional to ( is the magnitude of light electric field). HOM is largely responsible for the ultrafast magnetization reversal [5,7]. The magnitude of this field depends on the intensity of the incident light as given by the following equation [7]:
where β is the magneto-optical susceptibility of the magnetic material, ε0 is the permittivity of free space, is the direction of wave propagation and is the electric-field vector of the electromagnetic wave inside the magnetic material (* represents complex conjugate). Due to the cross-product, it is clear that Eq. (1) will only give non-zero values for circularly polarized light which we emphasize through. is, depending on the helicity, for circularly polarized light andfor linearly polarized light. The magneto-optical susceptibility is calculated using the following equation:where θF is the Faraday rotation at the wavelength λ for a magnetic layer of thickness d with magnetization M0 and with a real part of refractive index n. Vahaplar, et. al. [5] estimate that a laser pulse fluence of 1mJ/cm2 at 800nm generates a large opto-magnetic field of 5.2 T. In [5], The authors utilize 20nm thick films of GdFeCo and show magnetic domain reversal in sub-picosecond timescales for fluences lower than 4.5mJ/cm2. For higher fluences, their results show the formation of multi-domain states as well as helicity-independent switching which implies that there is an upper bound to the maximum light intensity that can be used for all-optical magnetization reversal under their experimental conditions. Their experimental setup consisted of a pump pulse of photon energy 1.54 eV with 70μm diameter and pulse width of 100 fs and a probe of photon energy 1.94 eV with 300μm diameter and 100 fs pulse width.All experiments and theoretical discussions regarding all-optical magnetization switching published thus far, relate to the direct interaction of the electric field of the incident free-space propagating light with the magnetization of the material. The use of gold antennas placed on the surface of SiN overcoat layers on TbFeCo thin films for magnetization switching with linearly polarized light has been reported by Liu et al [8]. Plasmonic nanostructures have also been used to enhance magneto-optical properties such as Faraday rotation in magnetic materials [9]. The influence of propagating surface plasmons on the in-plane and out-of-plane components of HOM have also been reported [10,11]. In this work, we study the effect on the opto-magnetic field of surface plasmons generated with circularly polarized light at the interface between magnetic materials and plasmon resonators. We find that the magnitude of the plasmon-generated opto-magnetic field, HOM, is enhanced in comparison to that resulting from the direct photon-magnetic material coupling. In particular, we find very large enhancements (~10x) when the magnetic material is either a dielectric or is integrated within a dielectric structure rather than a conventional metallic thin film. The utilization of surface plasmons for all-optical switching proposed in this work, paves the way for on-chip integration of nanoscale photonic and spintronic devices for beyond-CMOS circuitry.
2. Structure and materials
Plasmonics deals with the study of the oscillations of free electrons in a metal coupled to an electromagnetic field [12]. These oscillations, known as surface plasmons [13], are associated with wave-vectors much larger than that of free-space electromagnetic waves. Therefore, plasmonics enables the localization of electromagnetic energy to dimensions much smaller than the diffraction limit of light. We study the role of plasmonic resonators coupled to nanomagnet structures in the generation of opto-magnetic fields under illumination with circularly polarized light. In particular, through numerical simulations, we demonstrate that under identical laser fluences, significantly larger opto-magnetic fields are generated in the coupled plasmonic/magnetic nanostructures when compared to direct light excitation of the nanomagnet. Whereas we focus in this work on the switching of nanomagnets with perpendicular magnetic anisotropy (PMA), as they are the magnetic materials of choice for ultra-high density memory and logic devices [14], and therefore, our discussion pertains mainly the z-component of the opto-magnetic field generated by surface plasmons. However, our findings are readily applicable to in-plane magnets. Moreover, we also propose designs that would be useful for switching magnets engineered to exhibit tilted magnetization orientation [15].
Figure 1(a) shows a schematic diagram of the coupled magneto-plasmonic structure. It consists of a nanodisk stack comprising a plasmonic resonator (yellow), a nanomagnet (purple) and a capping layer (green) fabricated on an optically transparent substrate (white). With respect to the choice of materials for plasmonics, the use of gold [16] and silver [17] has dominated the literature. While these materials show excellent plasmonic properties, they suffer from drawbacks such as poor mechanical and thermal stability as well as incompatibility with CMOS processing. Consequently, over the past few years significant research efforts have been dedicated in the search of low-loss CMOS compatible metals that can substitute gold and silver [18,19]. In this regard, refractory transition metal nitrides have emerged as suitable candidates for the visible and near-infrared (NIR) electromagnetic spectrum [20]. In particular, titanium nitride (TiN) exhibits optical properties comparable to those of gold, it is compatible with current nanofabrication technologies and can be epitaxially grown on silicon [21], c-sapphire, and magnesium oxide [18]. Hence, in our simulations, we have considered the resonator to be made of TiN and the substrate to be Magnesium Oxide (MgO). We have performed simulations with two well-known magnetic PMA materials Bi-substituted iron garnet (BIG) [22] and Gadolinium Iron Cobalt (GdFeCo) [23]. Whereas we recognize that the magnetic properties of current BIG materials need to be improved to meet the thermal stability requirements for PMA nanomagnets, our goal in this paper in using BIG is to emphasize the potential improvements that can be obtained by employing dielectric nanomagnets. We also note that the dominant mechanism of switching in GdFeCo is believed to be heat-assisted and therefore it is not clear how significant an enhancement in opto-magnetic field in GdFeCo would prove to be for magnetic switching. Nevertheless our goal in choosing the two different magnetic materials is to highlight the impact of the optical properties of the magnet on the opto-magnetic field enhancement due to surface plasmons as a general case.
Fig. 1 (a) Schematic of proposed design. The substrate used is MgO. Each nanodisk consists of a plasmonic antenna (yellow), with a thin magnetic layer (purple) with a capping layer (green) on top. (b) Schematic of only a nanomagnet with the capping layer. In both figures, the red circular arrow at the bottom indicates that the illumination is with circularly polarized light and the curly red arrow indicates the direction of incidence.
As capping layer we selected Si3N4. The stack diameter is 50nm with the thickness of the TiN and the Si3N4 being 30nm and 20nm respectively. The thickness of the magnetic layer is kept constant at 10nm for both types of magnetic materials, BIG and GdFeCo. The choice of the stack diameter is selected to illustrate the key findings from our work. Later in the paper, we study the effect of reducing the nanodisk diameter to values commensurate with modern day magnetic memories, on the magnitude of the opto-magnetic field HOM. The optical properties of TiN depend on the material thickness and a 30nm nanodisk exhibits good metallic properties in the visible and NIR [24,25]. The choice of dielectric Si3N4 as the capping layer material and its thickness (20nm) allows us to tune plasmonic excitations in TiN in the visible/NIR wavelength region of interest. In all simulations, we consider normal incidence illumination by circularly polarized light through the MgO substrate, as shown in Fig. 1(a). The excitation wavelength is chosen to match the TiN nanodisk plasmonic resonance. This is determined to be 710nm for the material dimensions and optical constants employed. The illumination intensity is chosen as 1mJ/cm2 in order to compare our opto-magnetic field magnitude estimates with those of [5]. In Fig. 1(b), we show a schematic of the same structure as Fig. 1(a) but without the plasmonic resonator. We refer to the stack in Fig. 1(a) containing the plasmonic resonator as magneto-plasmonic stack (MPS), whereas that without the resonator is identified as non-plasmonic stack (NPS). In the following sections we report an almost one order of magnitude increment in the magnitude of the opto-magnetic field when the interaction with the nanomagnet is via the plasmonic resonator in the MPS sample. This significant increment is attributed to the near-field enhancement induced by the plasmonic resonator exhibiting Surface Plasmon Resonances (SPR). This enhancement indicates efficient coupling of electromagnetic radiation with an MPS structure as compared to direct illumination of nanomagnets as it is done in experiments reported thus far on all-optical-magnetization switching.
Figures 2(a) and 2(b) show the optical properties of BIG and GdFeCo in the visible and near-infrared regions. Figure 2(c) presents the optical properties of plasmonic TiN derived from spectroscopic ellipsometry measurements of sputter-deposited 30nm thick TiN thin films on MgO. Comparison of Figs. 2(a) and 2(b) indicate that at 710nm, BIG behaves as a low loss dielectric (ε”~0.01) whereas GdFeCo is a lossy metal. This results in plasmonic resonances in the BIG-MPS sample at the TiN-BIG interface, in contrast to the GdFeCo-MPS. There both GdFeCo and TiN are metallic in nature, the former being a relatively weaker metal (lower value of negative ε'). In this case we observe weak plasmonic resonances in the GdFeCo MPS sample at the GdFeCo-Si3N4 interface. More details are discussed in the following section.
Fig. 2 (a) Permittivity of Bismuth Iron Garnet (BIG) obtained from ref [26]. (b) Permittivity of GdFeCo obtained from ref [27]. (c) Permittivity of plasmonic TiN on MgO experimentally measured in our laboratory from spectroscopic ellipsometry measurements.
3. Results and discussion
In our simulations, we used a finite-element frequency domain solver using COMSOL software [28]. For all simulations illumination was with circularly polarized light at normal incidence. First we considered illumination of individual BIG-TiN MPS and NPS nanodisks separately and studied the electric field distribution characteristic of the plasmon resonance at 710nm wavelength. Utilizing Eq. (1), we derived the opto-magnetic field HOM. The magnetic susceptibility is calculated using Eq. (2) with experimentally reported values of [29] and [30] for BIG. In Fig. 3(a), we compare the magnitude of the z component of the opto-magnetic field component along the x-axis in BIG-TiN and BIG-MgO interface for MPS and NPS configurations respectively. One can clearly see an almost 10-fold enhancement in the z-component of the opto-magnetic field in the MPS sample compared to the NPS sample. In Fig. 3(b), we show the variation of the opto-magnetic field in the BIG MPS sample (50nm diameter, 10 nm thick) as a function of wavelength. The data is taken at the center of the MPS stack at the BIG-TiN interface. The enhancement in HOM,z due to the plasmon resonance at 710nm wavelength is apparent. The inset shows a volume plot of the opto-magnetic field component HOM,z over the entire magnet. It is clearly seen that HOM,z is highest towards the center of the magnet. Figure 3(c) shows how HOM,z decreases along the height of the BIG nanomagnet at the center as we move away from the TiN-BIG interface. The average value of HOM,z over the entire volume of the nanomagnets is 2.39T. For additional details, the reader is referred to Fig. 7 in the Appendix where we show the electric field amplitude color plots for light as a function of wavelength across the plasmon resonance.
Fig. 3 (a) Comparison of the z-component of the opto-magnetic field intensity along the x-axis of BIG-TiN interface for a 10nm thick BIG layer in the MPS (nanomagnet with TiN resonator) and NPS (only nanomagnet) sample. Illumination is with circularly polarized light of intensity 1mJ/cm2 at 710nm wavelength under normal incidence. (b) Wavelength dependence of the z-component of the opto-magnetic field for the MPS sample (50 nm diameter) at the stack center at the TiN-BIG interface. Inset: Plot of HOM,z over the entire volume of the magnet. (c) Plot of HOM,z along the axis of BIG nanomagnet. (z = 0nm refers to the TiN-BIG interface).
Next we present detailed electromagnetic analysis of the BIG MPS and NPS stacks when illuminated with circularly polarized light at 710nm. It is well known that when a plasmonic nanodisk is illuminated with circularly polarized light it generates a rotating dipole moment that is associated with the plasmonic resonance [31]. Figures 4(a) and 4(c) show the electric field components along the x-axis of the BIG-TiN and BIG-MgO interface for the MPS and NPS stack respectively. Comparing Figs. 4(b) and 4(d) we can clearly see the near-field enhancement for the MPS sample as compared to the NPS sample due to the plasmon resonance. Such a field enhancement is responsible for the increase in the opto-magnetic field of the MPS sample as noted earlier. It is noted that the electric field magnitude in Fig. 4(b) exhibits a maximum value at the edge of the structure. Whereas Fig. 3(a) shows that the opto-magnetic field component HOM,z is a maximum at the center of the nanomagnet. This apparent contradiction is easily resolved when we realize that HOM,z is a function of the x,y components of the electric field, as per Eq. (1), and therefore is not simply proportional to the overall magnitude of the electric field of light. Another noticeable feature of the plasmon resonance is that it introduces an out-of-plane component of the electric field Ez whereas the incident light has no such z-component. This is a well-known feature of the plasmon resonance [12]. For calculating the opto-magnetic field we have used only the out-of-plane component of the magnetic susceptibility as reported in literature. Thus, the opto-magnetic field reported in the figures, corresponds only to the z-component of HOM. We note that due to the presence of additional components of the electric field, plasmon resonances can in principle generate in-plane components of the opto-magnetic field HOM,x/y. The amplitudes of said components depend on the magnitude of the electric field in the xy plane as well as on the z-component of the electric field. Such an in-plane component of HOM can lead to more efficient switching of magnets with canted magnetization.
Fig. 4 (a) Electric field components along the x-axis of BIG-TiN interface for a 10nm BIG layer in the MPS sample. (b) Electric field intensity plot along the xy and yz plane for the 10nm BIG-TiN MPS structure under illumination with circularly polarized light of intensity 1mJ/cm2 at 710nm wavelength. Top inset: Schematic of the vertical cross-section of BIG-TiN MPS sample (c) Electric field components along the x-axis of BIG-MgO interface for a 10nm BIG layer in the NPS sample. (d) Electric field intensity plot along the xy and yz plane for the 10nm BIG NPS structure under illumination with circularly polarized light of intensity 1mJ/cm2 at 710nm wavelength. Bottom inset: Schematic of vertical cross-section of BIG-MgO NPS structure.
In Fig. 5(a), we present results on the opto-magnetic field HOM obtained for the MPS and NPS samples under illumination of circularly polarized light at 710nm when replacing the BIG with a 10nm thick GdFeCo nanomagnet. The magnetic susceptibility is obtained from [5] as 1.7*10−6 m/A. Although the value of as quoted in [5] is obtained for 800nm, we have used the value for 710nm as the magnetic properties of GdFeCo do not change significantly in this wavelength range [32,33]. As noted earlier in Fig. 2(b), GdFeCo is metallic in the visible/near-infrared region of the electromagnetic spectrum. Therefore in this case no plasmonic field enhancement at the GdFeCo-TiN interface is observed as both layers are metallic. In contrast, a plasmonic near-field enhancement at the GdFeCo-Si3N4 interface is observed in the MPS sample, although said enhancement is weaker. This is clearly shown in Figs. 5(b) and 5(c) where we plot the electric field intensity for the GdFeCo MPS and NPS sample respectively. The reason for the weaker field enhancement is because compared to TiN at 710nm wavelength, GdFeCo is a more lossy (higher ) and weaker (lower negative ) metal. The plasmonic resonance is mostly dominated by GdFeCo, with the TiN contribution to the plasmon resonance being significantly reduced. Hence the MPS structure with GdFeCo does not exhibit large opto-magnetic field enhancement compared to the NPS sample. We note an opto-magnetic field enhancement of only 3.4 in the MPS sample compared to the NPS sample which is less than what we observed for BIG due to aforementioned reasons. Although GdFeCo exhibits PMA for thicknesses > 10nm, the use of thicker layers would result in reduction of the contributions of the TiN plasmonic resonator on the opto-magnetic field enhancement.
Fig. 5 (a) Comparison of the z-component of the opto-magnetic field intensity along the x-axis of GdFeCo-Si3N4 interface for a 10nm GdFeCo layer MPS (nanomagnet with TiN resonator) and NPS (only nanomagnet) sample. (b) Electric field intensity plots along the xy(GdFeCo-Si3N4) and yz plane for the MPS(nanomagnet with TiN resonator) sample. (c) Electric field intensity plots along the xy(GdFeCo-Si3N4) and yz plane for the NPS(only nanomagnet) sample. Illumination is with circularly polarized light of intensity 1mJ/cm2 at 710nm wavelength under normal incidence.
We next consider the dependence of the opto-magnetic field enhancement on the MPS diameter size. It is well-known that plasmonic resonances in metal nanodisks strongly depend on their size. A change in size can alter the strength of the plasmon resonance. Figure 6 presents the simulation results for an MPS and NPS sample with 10nm thick BIG and 20nm stack diameter. In Fig. 6(a), we plot the comparison of the opto-magnetic field z-component along the x-axis in the x-y plane of the BIG-TiN interface and BIG-MgO interface for the MPS and NPS sample respectively. A clear enhancement in HOM,z of ~4x is observed compared to free space excitation. However, a comparison of Fig. 3(a) and Fig. 6(a) indicates that a reduction of the stack diameter of the MPS sample reduces the plasmonic enhancement in HOM from ~10 to 4, inspite of the nanodisk stack being excited at the resonant wavelength in both cases. Also the nature of the opto-magnetic field along the diameter is different. We believe that this is due to the weaker nature of the plasmon resonance for the sample with reduced diameter. Comparing Fig. 4(a) and Fig. 6(c), one can clearly see that the y-component of the electric field is reduced upon reducing the diameter. This leads to a reduction in HOM,Z. Also the variation of the y-component of the electric field of light along the stack diameter is slightly different upon reducing the diameter of the stack. Consequently there is a change in the way HOM,z varies along the diameter when the stack diameter is reduced. Nevertheless, the plasmonic enhancement in HOM,z still persists. This illustrates that plasmon-mediated magnetic switching is compatible with dimensionality reduction of nanomagnets for ultra-high density storage and it points to opportunities to increase the effect by matching the excitation light source to the plasmon resonance.
Fig. 6 (a) Comparison of opto-magnetic field intensity along the x-axis of BIG-TiN interface for MPS (magnet-plasmon coupled) structure and BIG-MgO interface for NPS(only nanomagnet) structure with 20nm diameter. Inset: Schematic of structures. (b) Electric field intensity plot at TiN–BIG interface for MPS sample. (c) Electric field components along the x-axis of the TiN-BIG interface for MPS sample. (d) Electric Field intensity plot along the y-z plane of the MPS sample. Illumination is with circularly polarized light of intensity 1mJ/cm2 at the resonant wavelength of 710nm.
Table 1 presents a summary of the obtained simulation results. The Figure-of-Merit FOM is calculated using the following formula:
Table 1. Summary of Results
We note that that in our simulations we neglect the optical anisotropy of both GdFeCo and BIG magnetic layers. This is because the off-diagonal terms of the permittivity tensor for both materials, obtained from the product of and M0 are at least an order of magnitude lower than the diagonal terms as shown in Figs. 2(a) and 2(b).
4. Conclusion
We report on a numerical study of surface plasmon-induced opto-magnetic fields for manipulation of the magnetic orientation of nanomagnets in on-chip, CMOS-compatible structures. We have used plasmonic titanium nitride and two commonly used PMA materials BIG and GdFeCo as nanomagnets to provide a fully CMOS-compatible material platform for the realization of future on-chip hybrid nanoplasmonic-magnetic devices. Our results show that surface plasmon resonances can lead to significant near-field enhancement and thereby much higher values of the opto-magnetic field HOM compared to free space excitation. We further show that the optical properties of the magnetic material play a crucial role in determining the strength of the plasmon resonance and hence an optimal choice of material is necessary in order to achieve the highest possible HOM. We also note the possibility of having in-plane components of the opto-magnetic field HOM with plasmon resonance. Thus, the proposed approach could potentially enable future on-chip plasmonic-magnetic devices for spintronics and memory.
Appendix
In Fig. 7, we present further results on the 50nm diameter BIG-TiN MPS sample simulation. Figure 7(a) shows the opto-magnetic field HOM,z at the center of the stack at the BIG-TiN interface as a function of illumination wavelength. It is the same as Fig. 3(b) in the main paper and is repeated here for convenience. In Fig. 7(b), we present electric field amplitude color maps as a function of the wavelength of the incident circularly-polarized. The fluence of the incident field is 1mJ/cm2 for all wavelengths. It is clear from the color maps in Fig. 7(b), that as one approaches 710nm, the near field intensity reaches a maximum. This in turn leads to the highest opto-magnetic field HOM,z at 710nm wavelength as shown in Fig. 7(a) and Fig. 3(b). At longer wavelengths, the near field intensity begins to decrease. This evidences the presence of a surface plasmon resonance for this structure at 710nm wavelength under illumination with circularly-polarized light.
Fig. 7 (a) Wavelength dependence of the z-component of the opto-magnetic field for a 50nm diameter MPS sample at the stack center at the TiN-BIG interface. Inset: Schematic of the structure. Blue arrows correspond to the wavelengths for which the electric field amplitude of light is plotted in b. (b) Electric field intensity plot along the xy and yz plane for the 10nm BIG-TiN MPS structure under illumination with circularly polarized light of intensity 1mJ/cm2. The wavelength of excitation is shown at the bottom left corner (xy plane refers to the BIG-TiN interface).
In Fig. 8, we show a comparison of the z-component of the opto-magnetic field at the BIG interface in the MPS sample with the TiN plasmonic resonator and with the TiN replaced by Si3N4. The objective of replacing the plasmonic resonator with a dielectric material was to illustrate the role of plasmonic resonance in the same stack structure. Figure 8(a) shows the opto-magnetic field HOM,z along the x-axis for the BIG-TiN and lower BIG-Si3N4 interface respectively under illumination by circularly polarized light at 710nm wavelength of 1mJ/cm2 fluence. One can observe an enhancement of more than 20 times due to the plasmonic resonator. Figure 8(b) shows the electric field intensity color maps for the two structures at 710nm and clearly illustrates the near field enhancement due to the TiN plasmonic resonator.
Fig. 8 (a) Comparison of the z-component of the opto-magnetic field intensity for a 10nm BIG layer along the x-axis of BIG-TiN interface and lower BIG-Si3N4 interface respectively in the MPS(nanomagnet with TiN resonator) and all dielectric (TiN replaced by Si3N4) sample. Illumination is with circularly polarized light of fluence 1mJ/cm2 at 710nm under normal incidence. Inset: Schematic of the two structures. (b) Electric field intensity color maps for the two structures under illumination with 710 nm circularly polarized light of 1mJ/cm2 fluence (xy plane refers to the BIG-TiN and lower BIG-Si3N4 interface in the top and bottom plots respectively).
Funding
Office of Naval Research (N00014-16-1-3003). AVK was also supported by the DARPA Optics and Imaging (EXTREME) program.
Acknowledgements
The authors would like to acknowledge useful suggestions provided by Prof. Vladimir Belotelov and Prof. Harald Giessen. The authors also acknowledge Dr. Kuntal Roy for helpful comments.
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