Patent Grant
10243660
Disclosed herein are methods of all-optical modulation of light. More particularly, the methods are directed to ultrafast all-optical modulation of the visible and infrared spectrum with nanorod arrays.
While active plasmonics in the ultraviolet to visible has been demonstrated, control in the near-infrared (NIR) to mid-infrared (MIR) spectral range has recently attracted significant attention for importance in telecommunications, thermal engineering, infrared sensing, light emission and imaging. Infrared plasmonics has been demonstrated with materials such as colloidal quantum dots, Si and InAs, and graphene. However, challenges include material instabilities and strong infrared absorption of solvents (quantum dots in solution), limited carrier densities (Si and InAs) and spectral range (graphene).
Noble metallic nanostructures possess large negative permittivity in the visible and near-infrared (NIR) range, and can therefore concentrate optical fields into subwavelength dimensions with enhanced nonlinear plasmonic response. However, the high electron concentration in noble metals limits the extent to which the electron distribution can be modified and with it the achievable permittivity modulation. In addition, strong interband transitions in the visible range (such as those from the d-band to the Fermi-surface in gold at an energy of ˜2.4 eV) give rise to a large dispersion of the permittivity modulation versus wavelength, which furthermore can overlap with their plasmonic resonances, thereby complicating the design of nonlinear optical devices.
As a result, there is a need for new materials and methods for all-optical modulation over a broad spectral range (from the visible to the infrared range) with ultrafast dynamics.
Disclosed herein is a method of optical modulation, the method comprising irradiating an optical switch with a control beam at a first control time and irradiating the optical switch with a signal beam at a signal time. The transmitted intensity of the signal beam in a direction depends on the delay time between the first control time and the signal time and the transmitted intensity of the signal beam in the direction is detectably different than a static signal. The optical switch comprises a nanorod array, the nanorod array comprising a plurality of nanorods extending outwardly from a substrate. In some embodiments, the nanorod array is arranged in a regular periodic pattern. In particular embodiments, the nanorod array has a periodicity of about 30 nm to about 5000 nm. In particular embodiments, the nanorod array comprises a plurality of nanorods having an average height of about 30 nm to about 5000 nm. In particular embodiments, the nanorod array comprises a plurality of nanorods having an average edge length of about 10 nm to about 500 nm.
The plurality of nanorods may comprise a transparent conducting oxide, a transparent conducting nitride, a transparent conducting carbide, or a transparent conducting silicide. In a particular embodiment, the plurality of nanorods comprise indium tin oxide. In some embodiments, plurality of nanorods comprise a plurality of film coated nanorods. In some embodiments, the substrate comprises indium tin oxide, yttria stabilized oxide, and/or aluminum oxide.
The method may further comprise irradiating the optical switch with a second control beam at a second control time, wherein the transmitted intensity of the signal beam in the direction depends on the delay time between the second control time and the signal time and the transmitted intensity of the signal beam in the direction is detectably different than a static signal. In particular embodiments, the first control time and the second control time are in controlled displacement.
The center wavelength of the control beam may be in the visible spectrum, the near infrared spectrum, mid infrared spectrum, or a combination thereof. The signal beam probe comprises wavelengths from the visible spectrum, near-infrared spectrum, mid-infrared spectrum, or a combination thereof. In particular embodiments, the signal beam is a broad band signal beam. In some embodiments, the control beam and/or the signal beam irradiate the nanorod array from an angle of incidence of 0° to 70°. In some embodiments, the control beam and/or the signal beam has a fluence less than 10 mJ/cm2.
The method may further comprise detecting the transmitted intensity of the signal beam in the direction. In some embodiments, the direction is substantially parallel with a forward propagation direction of the signal beam. In other embodiments, the direction is oblique with a forward propagation direction of the signal beam.
In some embodiments, the control beam excites a vibrational mode of the plurality of nanorod arrays and the transmitted intensity of the signal beam in the direction is modulated by the frequency of the vibrational mode. In particular embodiments, the vibrational mode is an extensional mode or a breathing mode.
Disclosed herein are all-optical switches and methods for all optical modulation. The all-optical switch is a device that permits controlling an optical signal with another optical signal. These devices have similarity to electrical transistors, the building blocks of computers, in which an electrical signal is controlled by another electrical signal. The ability to control signals allows for information processing functionalities for computing and communications. However, problems in electrical transistors include limited operating speed (limited to the gigahertz range). An alternative to signal processing is to use optical based techniques, where signals are all processed optically. For this purpose, photonic integrated circuits are proposed and actively pursued. This invention can be useful for performing functionalities in optical networks such as changing the amplitude of the beam (to realize the “on” and “off” states), changing the direction of the beam (to redirect signal transmission direction), and the polarization of the light. The light can be switched at hundreds of femto-seconds corresponding to a bandwidth in the terahertz regime.
The switching device is composed of periodic, vertically-aligned nanorod arrays. The switching device may be prepared by epitaxially growth of nanorods on a lattice matched substrate. Using an optical control beam to pump the nanorod array at its tunable plasmon resonance, one can switch a signal beam in the visible and infrared range with switching speed less than one picosecond, and absolute transmission modulation amplitudes of tens-of-percentage. In addition to turn-on and off functionalities, the nanorod array can also redistribute light intensities among different grating orders in the visible range in sub-picosecond time scales, and produce a periodic modulation of the signal beam with gigahertz frequency (corresponding to 10s of picosecond period), with differential transmission modulation amplitude. The switching in the infrared range primarily arises from a change of the plasma frequency of the nanorods, whereas the switching in the visible range primarily results from a change of the background permittivity of the nanorods. Both of change of the plasma frequency and background permittivity lead to a change of index of refraction of nanorods in respective ranges. The periodic modulation stems from the coherent acoustic vibrations of the nanorods following the optical pump.
The switching device allows for a number of different applications. The device allows for switching optical signals in the infrared. This can be used to switch telecommunication signals at 1550 nm, or switch (and control) infrared signals for infrared thermal imaging and engineering. In addition, the device may produce periodic oscillations of visible light at ˜20 gigahertz frequency. This is due to generation of acoustic vibrations of the nanorods. As the material's conductivity is geometry dependent, one can optically drive a periodically varying resistivity in the nanorods, which can be used for electrical signal manipulation. In addition, the vibrational frequency is highly environmentally sensitive, and thus can be explored for ultrasensitive mass sensors. Furthermore, the invention allows redistribution (or beam steering) of the visible spectrum in a sub-picosecond time scale. This opens doors for compact optical elements.
Optical modulation over a broad spectral range is accomplished by exploiting the non-parabolicity of the conduction band of nanorod arrays composed of transparent conducting materials. This non-parabolicity accounts for extraordinary, pump-induced carrier temperatures, resulting in subpicosecond modulation with up to 300% transmission change, and operation at telecom wavelengths and in the mid-infrared fingerprint region.
It is known that one of the most important optical properties of a material is the dielectric permittivity, ∈, which is a function of frequency. At high enough frequencies, ∈ can be solely determined by the plasma frequency ωp (if loss is negligible). The plasma frequency represents the natural resonant frequency of a collective oscillation, or plasmon, of a free-electron gas. For frequencies below ωp, the material can behave like a reflective metal (∈ is negative), whereas it acts like a transparent dielectric (∈ positive) for frequencies above ωp. Therefore, ωp represents an effective ‘knob’ that allows tuning of optical material properties and plasmon resonances. Many ways exist to change ωp. In its simplest form, ωp2=ne2/m (with electron charge e), ωp is insensitive to the electron temperature for a constant effective electron mass m and can be altered by tuning the electron density n. Whereas metals are characterized by a constant carrier density that fixes the plasma resonance frequency (typically visible or near-infrared frequencies), in semiconductors n is adjustable. This is most commonly realized by photoexcitation above the bandgap with intense laser pulses. But ωp can depend on temperature in cases where the parabolic band-structure approximation fails where the electrons in a solid no longer follow the parabolic energy-momentum relation observed at band extrema. This non-parabolicity results in a momentum dependence of the effective mass, m=m(k), which can significantly alter the plasma frequency even for a constant density of free electrons. This is the unique condition realized by the present technology.
In contrast to the noble metals with fixed carrier densities, metamaterials are characterized by a tunable carrier density and mobility that is achieved by doping or post-synthesis processing. Hence, plasmons in metamaterials can access the infrared fingerprint region for material identification and chemical sensing. In addition, metamaterials have a large bandgap. As a consequence, infrared or low-energy plasmons in metamaterials may experience much less damping than those in noble metals, where the interband transitions are close in energy to the plasmon resonances.
One may modify the LSPR on an ultrafast, subpicosecond timescale. The high-energy LSPR may be resonantly pumped with a control beam below the bandgap of a metamaterial. Owing to the absence of the interband excitation, the carrier density remains constant. Consequently, the change in plasma frequency observed in transmission at the LSPR with a signal beam does not stem from a carrier-density-induced modulation of plasma frequency as previously reported in other types of semiconductor. The pronounced pump probe signal is attributed to the conduction-band non-parabolicity in combination with a low carrier density and low heat capacity. These characteristics create a condition in which electrons in the conduction band are heated by the pump pulse to an astonishingly high carrier temperature. For comparison, gold, one of the most investigated plasmonic materials, exhibits far lower carrier temperatures of under similar pumping conditions. The exceptionally high carrier temperature ensures an electron distribution with a pronounced high-energy tail that is especially sensitive to the deviation of the conduction band from the parabolic form. The increase in effective mass at elevated energy states leads to a transient reduction in ωp. This is accompanied by relative transmission changes in the order of 300% for moderately high excitation intensities. The modulation depth is 1-2 orders of magnitude larger than in metals or semimetals (for example graphene) and comparable to other semiconductors. Furthermore, the timescale of the transient modulation is in the subpicosecond range, much faster than those observed in metals or semiconductors and on a par with single-layer graphene.
Spectral tuning of the plasmonic resonances in the near- and mid-infrared range can be achieved in different ways, for example by adjusting the carrier density, the pump fluence, or the photon energy by which a different LSPR mode can be excited. Additional flexibility is offered by shifting the resonances by means of the incidence angle and the geometry of the nanorod arrays. This allows one to target specific frequency windows with a broadband response. Electrical gating can, in principle, be included for adjusting the carrier density and hence ωp. The compatibility of the presented nanorod arrays with semiconductor processing technology is certainly an advantage over competing schemes, for instance graphene devices, and electrical gating would add only moderate complexity.
The switching device provides a number of advantages over existing technologies. First, the switch is ultra-broadband, which covers from the ultraviolet (about 355 nm) to the visible (710 nm), as well as the infrared range (1.5 micron to ˜10 microns). This outperforms other materials such as graphene, noble metals, or traditional semiconductors, where the spectral range of the modulation that can be achieved is very limited. The switching capability is usually enhanced at optical resonances, in this cases being multiple plasmon resonances in the infrared, and a number of interference-induced transmission dips in the ultraviolet to the visible. Second, the sub-picosecond switching speed is an order-of-magnitude faster than other nonlinear plasmonic materials (such as noble metal gold and silver) based all-optical switches, and is orders of magnitude faster than other type of optical switches. For example, mechanical switch has speed from milliseconds to microseconds, whereas semiconductor waveguide, and electric-optic switches have switching speed of nanoseconds). Third, the periodic modulation with tens of gigahertz frequency is faster than acousto-optic modulators, which have modulation frequencies up to 1 gigahertz. Forth, the absolute transmission intensity modulation, which is up to 35% in the visible and beyond 20% in the near-infrared and mid-infrared (corresponding to several hundred percent differential transmission modulation), is significantly larger than existing ultrafast all-optical switching devices (in gold based systems, the modulation amplitude is a few percent at most). Fifth, the sample is technologically easy to handle and fabricate. It does not degrade in air; it is stable under high optical pump powers up to tens of mJ/cm2 (while noble metal based structures melt at a few mJ/cm2); it can be grown over large areas by conventional chemical vapor deposition means combined with large-scale patterning techniques (such as nanoimprint or soft-lithography). Sixth, by simply tailoring the geometry of the nanorod array (including the height, periodicity, and edge lengths of the nanorods), or the carrier concentration of ITO by annealing in different oxygen environments, one can spectrally tune the normally-off or normally-on switching wavelengths. Seventh, no waveguide or fiber is required; both the control and signal beams can be directly coupled to the switch from free space. Moreover, the diffraction property of the nanorod arrays in the visible permit a dynamic intensity redistribution between the zero order and higher order diffraction modes.
One aspect of the invention is a method of optical modulation. The method comprises irradiating an optical switch comprising a nanorod array with a control beam at a control time and a signal beam at a signal time. The transmitted intensity of the signal beam in a particular direction depends on the delay time between the control time and the signal time. The transient behavior of the transmitted intensity of the signal beam is the result of a control beam and signal beam interacting with the nanorod array. This, in turn, is a key to all optical modulation and allows one to control the signal output over a broad range of frequencies and time scales suited to a number of different applications.
The controlled variation of the signal beam will enable a number of different functions. In one instance, the signal beam may be contain an optical bit. This may be accomplished by evaluating the transient signal beam relative to its static counterpart. When the intensity of the signal beam is greater than the threshold, that signal beam may be characterized as a “1” in an analogous manner to semiconductor transistor. Similarly, when the intensity of the signal beam is greater than the threshold, that signal may be characterized as a “0”. The switching between the “1” and “0” state is accomplished by controlling the interaction of the control beam and the signal beam with the nanorod arrays. This allows for the propagation of information and, by extension, logic gates and/or logic circuits.
Moreover, the propagation direction of the signal beam may be used to incorporate information. Because the signal beam is vectorial, intensity as well as directional information is contained in the transmitted signal.
The optical modulation is accomplished by optical switches comprising nanorod arrays. The nanorod arrays comprise a plurality of nanorods extending outwardly from a substrate. The plurality of nanorods may be composed of a metamateral. Metamaterials are artificial, engineered materials with rationally designed compositions and arrangements of nanostructured building blocks. These materials have an extraordinary response to electromagnetic, acoustic, and thermal waves that transcends the properties of natural materials. Examples of materials that may be used to prepare the nanorod arrays include transparent conducting oxides, transparent conducting nitrides, transparent conducting carbides, or transparent conducting silicides. Transparent conducting oxides include oxide semiconductors such as zinc oxide, cadmium oxide, tin oxide, and indium oxide that is doped to make them conducting. Examples of transparent conducting oxides include indium tin oxide (ITO), aluminum-doped zinc oxide (AZO), gallium-doped zinc oxide (GZO), indium-doped cadmium oxide (In:CdO), or fluorine-doped tin oxide (FTO). Examples of transparent conducting nitrides include TiAlN, TaN, ZrN, Zr3N4, YN, VN, NbN, Cu3N and WN. An example of a transparent conducting carbide includes SiC. Examples of transparent conducting silicides include silicides formed from metals such as Co, Cr, Fe, Hf, Ir, Nb, Ni, Os, Pt, Pd, Re, Rh, Ru, Ta, Ti, V, W, Zr, Ca, Mg and alkali metals. In particular embodiments, the nanorod array consists essentially of indium tin oxide.
The nanorod arrays, in some embodiments, may be film coated. The film coating may comprise any of the transparent conducting oxides, transparent conducting nitrides, transparent conducting carbides, or transparent conducting silicides described above.
The physical parameters of the nanorod array may affect the particular signal beam to be modulated as exemplified below. Physical parameters that may be varied include, but are not limited to, the arrangement of the nanorods on the substrate, the height of the nanorods, and the edge length of the nanorods. The nanorod array may comprise a plurality of nanorods arranged in any particular manner. The pattern may be a random pattern or a regular periodic pattern. The regular periodic pattern may be any pattern so long as it repeats at least 3 times along one or more directions. In particular embodiments the pattern may repeat at least 10 times, 25 times, 50 time, or 100 time along one or more directions. Examples of regular periodic patterns include the plurality of nanorods arranged in columns and rows, in a triangular pattern, in a hexagonal pattern, or a circular pattern. In in particular embodiments the periodic pattern is a regular square pattern of columns and rows. In certain embodiments the nanorod array has an average periodicity of about 30 nm to about 5000 nm in one or more directions that may be measured from the center of a nanorod to its nearest neighbor, including any interval therebetween. In particular embodiments, the nanorod array has a periodicity greater than 50 nm, 100 nm, 150 nm, 200 nm, 250 nm, 300 nm, 400 nm, 450 nm, 500 nm, 550 nm, 600 nm, 650 nm, 700 nm, 750 nm, 800 nm, 850 nm, 900 nm, 950 nm, 1000 nm and less than 5000 nm, 4500 nm, 4000 nm, 3500 nm, 3000 nm, 2500 nm, 2000 nm, 1500 nm, 1450 nm, 1400 nm, 1350 nm, 1300 nm, 1250 nm, 1200 nm, 1150 nm, 1100 nm, 1050 nm, 950 nm, 850 nm, 800 nm, 750 nm, 700 nm, 650 nm, 600 nm 550 nm, or 500 nm.
The plurality of nanorods may have an average height of about 30 nm to about 5000 nm measured from the substrate to the top of the nanorod, including any interval therebetween. In particular embodiments, the nanorod array has an average height greater than 50 nm, 100 nm, 150 nm, 200 nm, 250 nm, 300 nm, 400 nm, 450 nm, 500 nm, 550 nm, 600 nm, 650 nm, 700 nm, 750 nm, 800 nm, 850 nm, 900 nm, or 950 nm, 1050 nm, 1100 nm, 1150 nm, 1200 nm, 1250 nm, 1300 nm, 1400 nm, 1450 nm, 1500 nm, 1550 nm, 1600 nm, 1650 nm, 1700 nm, 1750 nm, 1800 nm, 1850 nm, 1900 nm, 1950 nm, 2050 nm, 2100 nm, 2150 nm, 2200 nm, 2250 nm, 2300 nm, 2400 nm, 2450 nm, 2500 nm, 2550 nm, 2600 nm, 2650 nm, 2700 nm, 2750 nm, 2800 nm, 2850 nm, 2900 nm, or 2950 nm and less than 5000 nm, 4500 nm, 4000 nm, 3500 nm, 3000 nm, 2500 nm, 2000 nm, 1500 nm, 1450 nm, 1400 nm, 1350 nm, 1300 nm, 1250 nm, 1200 nm, 1150 nm, 1100 nm, 1050 nm, 950 nm, 850 nm, 800 nm, 750 nm, 700 nm, 650 nm, 600 nm 550 nm, or 500 nm.
The plurality of nanorods may have an average edge length of about 10 nm to about 500 nm, including any interval therebetween. In particular embodiments, the nanorod array has an average edge length greater than 10 nm, 20 nm, 30 nm, 40 nm, 50 nm, 60 nm, 70 nm, 80 nm, 90 nm, 110 nm, 120 nm, 130 nm, 140 nm, 150 nm, 160 nm, 170 nm, 180 nm, 190 nm, or 200 nm and less than 500 nm, 450 nm, 400 nm, 350 nm, 300 nm, 250 nm, or 200 nm.
The nanorods extend outwardly from a substrate. The substrate may be any material that supports or facilitates the growth of the plurality of nanorods. In some embodiments, the substrate is at least partially transparent to light over specific or a wide range of wavelengths.
The substrate may comprise one or more materials. This may include a film of the material of the nanorods themselves and more or more additional materials. In some embodiments, the substrate is an oxide but other materials may be used as well. In some embodiments, the substrate comprises a film comprising indium, tin, and oxygen. In some embodiments, the substrate comprises yttria stabilized oxide or aluminum oxide (sapphire). In particular embodiments, the substrate comprises both a film comprising indium, tin, and oxygen as well as yttria stabilized oxide.
With the use of the nanorod arrays described above, one may optically modulate light over a broad range of wavelengths and over a broad range of time scales. The method comprises irradiating a nanorod array with a control beam at a control time and a signal beam at a signal time. In some embodiments, the method further comprises irradiating the nanorod array with a second control beam at a second control time. Where two control beams are used, controlling the displacement between the delay time between the first control time and the signal time and the delay time between the second control time may be equal or different may affect the intensity of the signal beam in the direction. Depending on the temporal proximity of a control time with a signal time, the signal beam may be detectably different than a static signal or not depending on the circumstances.
The use of two control beams may allow for optical processing in an analogous manner to semiconductor logic gates. This may be accomplished when the signal beam is detectably different than a static signal. The delay time between the first control time and the signal time and the delay time between the second control time and the signal time may be equal or different. For example, where a first and a second control signal are necessary to induce a change in the emitter signal above a threshold value over a static value, that may be analogized to an AND logic gate. As another example, where either a first or a second control signal may induce a change in the emitter signal above a threshold value over a static value, the may be analogize to an OR logic gate.
Depending on the number of optical switches used and arrangement, logic circuits may be prepared. The signal beam emanating from an optical switch may be directed onto another optical switch comprising a nanorod array. The signal beam in this instance may act as either a control signal or a probe. In some cases, the signal beam may be split and act as both a control signal and a probe.
The method may further comprise detecting the signal beam. Detectors and methods of detecting optical signals are known in the art, and no particular detector or method of detection is a necessary limitation of the present technology.
Over a period of time, a signal beam that is detectably different transiently than a static signal will revert to the static signal. As a result, the static signal beam is equivalent to a transient signal beam in the limit that the delay time between the control time and the probe time are infinite. This provides the baseline for the temporal response. Depending on the wavelengths of the control beam and signal beam, as well as the physical parameters of the nanorod array, the temporal response may be on the order of femtoseconds, picoseconds, nanoseconds, or microseconds. This may allow for the nanorod arrays described to be used in applications.
The center wavelength of a control beam may be any wavelength capable if inducing an electronic transition. In some embodiments, center wavelength of the first control beam is in the visible spectrum, the near infrared spectrum, mid infrared spectrum, or a combination thereof. In particular embodiments, the center wavelength may be tuned to induce an electronic transition associated with a localized surface plasmon resonance. Particular wavelengths capable of inducing that transition include control beams having a center wavelength about 1500 nm.
A control beam may irradiate the nanorod array at any angle of incidence capable of inducing an electronic transition. In some embodiments, the control beam irradiates the nanorod array from an angle of incidence of 0° to about 70°, including without limitations angles of incidence of between any range between any of 0°, 5°, 10°, 15°, 20°, 25°, 30°, 35°, 40°, 45°, 50°, 55°, 60°, 65°, and 70°.
The control beam may have a wide range of fluences. A surprising advantage of the present invention is that the nanorod arrays allow for high fluences. Particularly, the nanorod arrays allow for fluence high enough to irreversibly degrade other dielectric materials such as noble metals. The fluence may be high enough to allow for an emitter signal to be detectably different than a static signal. In other embodiments, the fluence may only be high enough to allow for an emitter signal to be detectably different than a static signal when a second control signal irradiates the nanorod array. In certain embodiments, the control signal has a fluence less than about 30 mJ/cm2. In particular embodiments, the control signal has a fluence less than 25 mJ/cm2, 20 mJ/cm2, 15 mJ/cm2, or 10 mJ/cm2.
Another surprising advantage of the present invention is that the nanorod arrays allow for a wide range of wavelengths to be modulated. The signal beam may be narrowly banded such that the full width half maximum bandwidth is tens or hundreds of nanometers. Alternatively, the signal beam may be a broad band probe that has a bandwidth greater than hundreds of nanometers. The center wavelength of the signal beam may be in the near ultraviolet spectrum, visible spectrum, the near infrared spectrum, mid infrared spectrum, or a combination thereof. In some embodiments, the signal beam is between 350 nm to 6000 nm. In a particular embodiment, the signal beam is between about 350 nm and 750 nm, 1500 nm and 6000 nm, or both. In certain embodiments, the signal beam has a fluence less than about 30 mJ/cm2. In particular embodiments, the control signal has a fluence less than 25 mJ/cm2, 20 mJ/cm2, 15 mJ/cm2, 10 mJ/cm2, 9 mJ/cm2, 8 mJ/cm2, 7 mJ/cm2, 6 mJ/cm2, 5 mJ/cm2, 4 mJ/cm2, 2 mJ/cm2, or 1 mJ/cm2.
The transient transmission signal may be a transient bleaching or induced absorption. As an alternative, the transient transmission signal may be a transient red-shifting or blue-shifting of a spectral peak. The kinetics of any of the bleaching, induced absorption, red-shifting, or blue-shifting may determine the temporal response. The absolute change in the intensity of the transmission may be greater than 5%, including changes greater than 10%, 15%, 20%, 25%, 30%, or 35%.
In some embodiments, the emitter signal is diffracted by the nanorod array acting as an optical grating. Any of the grating modes may be used as the signal beam. In some embodiments, the signal beam is detected at a zero-order grating mode. In other embodiments, the signal beam is detected at a non-zero-order grating mode. In particular embodiments, the signal beam is detected at both a zero-order grating mode and a non-zero-order grating mode.
Ultrafast plasmon modulation in the near-infrared (NIR) to mid-infrared (MIR) range by intraband pumping of nanorod arrays allows for the preparation of optical switches and methods for optical modulation. In contrast to noble metals, the lower electron density in nanorod arrays comprising indium, tin, and oxygen enables a remarkable change of electron distributions, yielding a significant plasma frequency modulation and concomitant large transient bleaches and induced-absorptions, which can be tuned spectrally by tailoring the nanorod array geometry. The low electron heat capacity explains the sub-picosecond kinetics that is much faster than noble metals.
Our work demonstrates a new scheme to control infrared plasmons for optical switching, telecommunications and sensing. A control signal with a center wavelength to the trans-LSPR wavelength may be used to maximize the sample absorption. A NIR probe that spans the trans-LSPR or a MIR probe that covers the long-LSPR may be used to prepare an emitted signal. The nanorod arrays support two LSPRs with collective electron oscillations along orthogonal directions. We denote the LSPR where electrons oscillate perpendicular, or parallel to the nanorod long axis as the transverse-LSPR (trans-LSPR), or longitudinal-LSPR (long-LSPR), respectively. The static transmission spectra of nanorod arrays measured using un-polarized light reveal a strong NIR absorption, which is associated with the trans-LSPR that can be excited at a range of incidence angles. The MIR transmission spectra under p-polarization shows strong transmission dips and a slight blueshift of the transmission dip under an increasing incidence angle, which are absent in the s-polarization analogue. The MIR transmission dips under p-polarization are attributed to the long-LSPR, which can only be excited by electric field component along the long axis. Due to weaker geometrical confinement, the long-LSPR occurs at a longer wavelength than the trans-LSPR.
The well-known sequence of events in plasmonic systems following pump excitation include electron dephasing, electron-electron scattering, electron-phonon coupling and lattice heat dissipation, which take place at different time scales. The temporal response in the infrared may include a sub-picosecond component is followed by a much weaker, slower-decaying tail that stays almost constant during the entire measured delay time up to tens of picoseconds. The sub-picosecond component of the emitter signal may be ascribed to electron-phonon coupling, whereas the slow-decaying, weaker component of the emitter signal results from the gradual cooling of the lattice.
In the examples below, we demonstrate both static LSPRs and their transient behaviors spanning the NIR to MIR range exhibited by nanorod under intraband excitations. Moreover we show that a high electron temperature achieved in nanorod arrays accounts for the sub-picosecond decay that is faster than that observed for noble metals. The low electron density of of the nanorod arrays enables a significant redistribution of electron energies under intraband pumping, which results in a remarkable change and thereby large differential and absolute transmission modulations. Furthermore, this spectral modulation can be tuned in the MIR through tailoring the sample geometry. Our results pave the way for robust manipulation of the infrared spectrum using heavily-doped, semiconductor-enabled material platforms.
Moreover, sub-picosecond optical nonlinearity of the nanorod arrays following intraband, on-plasmon-resonance optical pumping enables modulation of the full-visible spectrum with large absolute change of transmission, favorable spectral tunability and beam-steering capability. Furthermore, we observe a transient response in the microsecond regime associated with the slow lattice cooling, which arises from the large aspect-ratio and low thermal conductivity of the nanorod arrays. A number of transmission minima in the visible range, arising from collective light diffraction by the periodic dielectric nanorod array, give rise to a pump-induced transmission modulation with absolute amplitude up to ±20%. Our results demonstrate that all-optical control of the visible spectrum can be achieved by using wide-bandgap semiconductors in their transparent regime with speed faster than that of noble metals.
Moreover, the large scattering cross-section of the dielectric nanorod arrays (as opposed to the large absorption cross-section of noble metal nanostructures) allows for a dynamic redistribution of light intensities among different diffraction orders, and the spectral response of the nanorod arrays can be tuned by simply adjusting the incidence angle or tailoring the length of the nanorods. In the temporal domain, we found both a sub-picosecond response stemming from the electron-phonon coupling and a microsecond response arising from the lattice cooling in ITO.
The visible spectrum shown exhibits pronounced transmission minima. The transmission minima in the visible regime are not due to resonant absorption but are simply standing wave resonances. Each transmission minimum wavelength the waves reach an out-of-phase condition at the interface of the nanorod and substrate (which is at the bottom boundary of the nanorod). While the NIR LSPR is a localized phenomenon, the transmission minima in the visible range are due to coherent light diffraction by the nanorod arrays and therefore is attributed to an array effect. Effectively, the nanorod arrays acts as a two-dimensional diffraction grating that supports not only the forward propagating zero-order mode, but also non-zero-grating orders propagating in oblique directions. The dielectric nature of nanorod array in the visible range dictates that intensities of the non-zero grating orders should be complementary to that of the zero grating order.
Coupling light with acoustic vibrations in nanoscale optical resonators offers optical modulation capabilities with high bandwidth and small footprint. When using the nanorod arrays described herein as the operating media, optical modulation covering the visible spectral range with GHz bandwidth is achieved through the excitation of coherent acoustic vibrations. This broadband modulation results from the collective optical diffraction by the dielectric, and a high differential transmission modulation is achieved through efficient near-infrared, on-plasmon-resonance pumping.
The present disclosure is not limited to the specific details of construction, arrangement of components, or method steps set forth herein. The compositions and methods disclosed herein are capable of being made, practiced, used, carried out and/or formed in various ways that will be apparent to one of skill in the art in light of the disclosure that follows. The phraseology and terminology used herein is for the purpose of description only and should not be regarded as limiting to the scope of the claims. Ordinal indicators, such as first, second, and third, as used in the description and the claims to refer to various structures or method steps, are not meant to be construed to indicate any specific structures or steps, or any particular order or configuration to such structures or steps. All methods described herein can be performed in any suitable order unless otherwise indicated herein or otherwise clearly contradicted by context. The use of any and all examples, or exemplary language (e.g., “such as”) provided herein, is intended merely to facilitate the disclosure and does not imply any limitation on the scope of the disclosure unless otherwise claimed. No language in the specification, and no structures shown in the drawings, should be construed as indicating that any non-claimed element is essential to the practice of the disclosed subject matter. The use herein of the terms “including,” “comprising,” or “having,” and variations thereof, is meant to encompass the elements listed thereafter and equivalents thereof, as well as additional elements. Embodiments recited as “including,” “comprising,” or “having” certain elements are also contemplated as “consisting essentially of” and “consisting of” those certain elements.
Preferred aspects of this invention are described herein, including the best mode known to the inventors for carrying out the invention. Variations of those preferred aspects may become apparent to those of ordinary skill in the art upon reading the foregoing description. The inventors expect a person having ordinary skill in the art to employ such variations as appropriate, and the inventors intend for the invention to be practiced otherwise than as specifically described herein. Accordingly, this invention includes all modifications and equivalents of the subject matter recited in the claims appended hereto as permitted by applicable law. Moreover, any combination of the above-described elements in all possible variations thereof is encompassed by the invention unless otherwise indicated herein or otherwise clearly contradicted by context.
Recitation of ranges of values herein are merely intended to serve as a shorthand method of referring individually to each separate value falling within the range, unless otherwise indicated herein, and each separate value is incorporated into the specification as if it were individually recited herein. For example, if a concentration range is stated as 1% to 50%, it is intended that values such as 2% to 40%, 10% to 30%, or 1% to 3%, etc., are expressly enumerated in this specification. These are only examples of what is specifically intended, and all possible combinations of numerical values between and including the lowest value and the highest value enumerated are to be considered to be expressly stated in this disclosure. Use of the word “about” to describe a particular recited amount or range of amounts is meant to indicate that values very near to the recited amount are included in that amount, such as values that could or naturally would be accounted for due to manufacturing tolerances, instrument and human error in forming measurements, and the like.
No admission is made that any reference, including any non-patent or patent document cited in this specification, constitutes prior art. In particular, it will be understood that, unless otherwise stated, reference to any document herein does not constitute an admission that any of these documents forms part of the common general knowledge in the art in the United States or in any other country. Any discussion of the references states what their authors assert, and the applicant reserves the right to challenge the accuracy and pertinence of any of the documents cited herein. All references cited herein are fully incorporated by reference, unless explicitly indicated otherwise. The present disclosure shall control in the event there are any disparities between any definitions and/or description found in the cited references.
While not wishing to be bound by theory, the theoretical discussions are meant only to be illustrative and are not meant as limitations on the scope of the invention or of appended claims.
The following examples are meant only to be illustrative and are not meant as limitations on the scope of the invention or of the appended claims.
Here we describe the transient behavior of the localized surface plasmon resonances (LSPRs) of indium-tin-oxide nanorod arrays (ITO-NRAs) following intense, ultrafast laser excitation. On comparing the static and transient results we observe redshifts of LSPRs in sub-picosecond time scales under resonant, intraband optical pumping, which we attribute to a conduction band nonparabolicity-induced plasma frequency (ωp) reduction. We generalize the calculation of ωp to cover the case of a nonparabolic band and quantitatively determine the fluence dependent ωp shifts. We further show that the ultrafast, sub-picosecond response time stems from a high electron temperature, a direct result of the lower electron density of ITO in comparison to its noble metal counterparts. In addition, the LSPR modulation, based on modification of the collective-electron behavior of ITO-NRAs, enables differential transmission modulation beyond +100% and absolute transmission modulation beyond +20% with fluence <7 mJ/cm2 in both the NIR and MIR. The large bleaching and induced-absorption can enable both normally-off and normally-on optical switching functionalities, and can be further tuned spectrally by tailoring geometrical parameters of the ITO-NRAs.
Static LSPRs of the ITO-NRA
The uniform ITO-NRA with 1 μm periodicity, 2.6 μm height and 180 nm edge length shown in
The LSPRs were further confirmed by finite-element simulations with ITO modeled using the Drude formula
(see below), where ∈∞=3.95 is the background permittivity, γ=0.12 eV the damping factor, and ωp=2.02 eV the plasma frequency.
Transient Response of the Trans-LSPR and Long-LSPR
Knowing the static LSPR features we then performed transient absorption (TA) experiments to investigate the transient response of the LSPRs following ultrafast optical excitation. Two TA experiments were performed; in both cases we matched the pump center wavelength to the trans-LSPR wavelength of 1500 nm to maximize the sample absorption. We used a NIR probe that spans the trans-LSPR in the first experiment (denoted as NIR-probe-TA), then a MIR probe that covers the long-LSPR in the second (denoted as MIR-probe-TA). Schematic diagrams of the two TA experiments are shown in
The ΔOD spectral map shown in
Studies of noble metallic nanoparticles have shown the effect of intraband pumping on interband transitions by probing the system with photon energies comparable to the interband gap energy. There the intraband pump leads to a Fermi-surface smearing and change of the imaginary permittivity, Δ∈″(ω), for optical transitions involving the perturbed electronic states, which in turn gives rise to a change of the real permittivity, Δ∈′(ω), through the modification of background permittivity, Δ∈∞(ω). A plasmon redshift arose owing to a positive Δ∈∞(ω), which in the quasistatic limit can be understood as the LSPR frequency
(neglecting the damping term; ∈m is the permittivity of the surrounding medium) becomes smaller with an increasing ∈∞.
The long-LSPR redshift of the ITO-NRA we observed, however, cannot originate from Δ∈∞(ω). Revisiting the Drude permittivity
∈′(ω) at the trans-LSPR of 1500 nm is about −1.9, but at the long-LSPR wavelength of 4000 nm ∈′(ω) reaches −33. While ∈(ω) is sensitive mainly to Δωp but still partly to Δ∈∞(ω) in the NIR range, ∈(ω) behaves markedly different around the long-LSPR as it becomes much more sensitive to Δωp than to Δ∈∞(ω), owing to the one order of magnitude smaller ω (0.31 eV at the long-LSPR, 0.83 eV at the trans-LSPR) in comparison to the 2.02 eV ωp. Furthermore, Δ∈∞(ω) due to the modification of interband transition is strongly frequency dependent, reaching its maximum around the band gap energy and rapidly converging to a weak constant at longer wavelengths. Simulations with various Δ∈∞(ω) show that an unreasonably large constant Δ∈∞ of 1.0 in the long-LSPR range results in a peak ΔOD of only −0.06 (See
Theoretical Modeling of the Plasma Frequency Reduction
Without wishing to be bound to theory, theoretical modeling may provide insights into the present technology. Modulations of ωp for TCOs have been achieved by tuning the electron density (denoted as n) through electrical, electrochemical, and optical (interband pumping or charge injection) methods as these two quantities are related by
for a free electron gas, where n is the carrier density, e the elementary charge and m the effective mass. The intraband pumping used in our work, however, cannot change n, so the reduction of ωp must result from the detailed electronic structure of ITO which cannot be captured by the free electron gas model. In particular, we attribute the reduction of ωp to the conduction band nonparabolicity, which has been observed in several highly-doped TCOs whose electronic states are highly populated such that the chemical potential lies far above the conduction band minimum (CBM). Instead of being constant in a parabolic band, the effective mass in a nonparabolic band becomes k (the electron wave vector) dependent and increases for higher energy states. To quantitatively model the reduction of ωp under intraband pumping, we derived semi-classically the ωp for an electron gas in an isotropic, nonparabolic band. We adopted the formula
originally proposed by Kane and Cohen. (ℏ is the Planck constant, E is the electron energy referenced to CBM; 1/Eg denotes the nonparabolicity, where Eg is on the order of but does not represent the band gap). Note that the conduction band dispersion is fully determined by 1/Eg and m. Starting from the linearized collisionless Boltzmann equation, ωp is derived as
Similarly, the electron density n and electron energy density U become
where T is the electron temperature, μ the electron chemical potential and n a constant due to the conservation of electrons. Once the band structure parameters (1/Eg and m) and the static ωp are known, equations (1), (2) and (3) allow the complete determination of T dependent μ, ωp and U. The static ITO-NRA LSPRs correspond to an electron gas at T=300 K and following optical excitation the electron gas occupies a temperature T>300 K. Heat exchange between the electrons and lattice through electron-phonon coupling results in a time dependent T, μ, ωp and U. The electron configurations before and after the pump excitation are schematically illustrated in
In
In
The temperature changes shown in
valid for T<<TF, where kB is the Boltzmann constant, D(μ) is the electronic density of states at the Fermi level and TF is the Fermi temperature
Note that the Sommerfeld form well reproduces the numerically calculated C(T) at low temperatures for all nonparabolicities. In addition, the Sommerfeld form for ITO, although overestimating C(T) beyond low T range, is about one order of magnitude lower than the Sommerfeld C(T) for gold, confirming the much higher T that is achieved for ITO. In the two-temperature model for describing the dynamic energy exchange between the electrons and lattice, the electron-energy-loss-rate is proportional to the difference of the electron temperature rise, ΔT, and the lattice temperature rise, ΔTL. Based on the estimated ΔTL (
Spectral Tunability of the Long-LSPR in the MIR
To illustrate the spectral tunability of the long-LSPR, we plot in
The Generalized Plasma Frequency in Nonparabolic Bands
Assuming the collective oscillation of electrons produces an electron density of the form n(r,t)=n0+δn(r,t) with
the electric field and electric potential of the electron gas follow as E(r,t)=−∇φ(r,t) and ∇2φ(r,t)=−(−e)δn(r,t). From the linearized collisionless Boltzmann equation for electrons
where f0 is the Fermi function, and assuming a spatial and time dependence of the form φ(r,t)=φ(q)e−ω
and subsequently
Here,
E represents the electron energy referenced to the conduction band minimum, and E is the electric field. (Note that we have suppressed collisions, which is justified since ωp and γ obtained from finite element simulations are 2.02 eV and 0.12 eV, respectively). In the long wavelength limit where the plasmon wave vector |q| is small, expanding the denominator in equation (*) gives the equation
Terms that are odd in θ vanish on integrating over angle, hence keeping the second order term we obtain the expression for the plasma frequency
The isotropic, nonparabolic band structure is written as
in which m and 1/Eg fully determine the conduction band structure. (Note that consistent with the original notation in reference 1, Eg here is not the band gap in our notation but simply a parameter that characterizes the nonparabolicity of the conduction band). Taking derivative with respect to k of both sides of the nonparabolic equation gives the electron velocity
The plasma frequency is then written as
which can be further simplified as
which is equation (1) in the main text. Notably, when Eg→∞ eV (parabolic band condition), the usual expressions for plasma frequency
and electron density
are recovered. At arbitrary temperature T, the conservation of electron density under intraband pumping in our experiments fixes the chemical potential μ through
or equivalently (by substituting in k),
which is equation (2) in the main text. The energy density of electrons (in eV/cm3) referenced to the conduction band minimum can be calculated as
which is equation (3) in the main text.
The relative permittivity of ITO was modeled using the Drude formula
with ∈∞=3.95 (taken from Granqvist, C. G. et al). A good match between the experimental and simulated NIR experimental transmission spectra of the ITO-NRA (
We obtained the static (T=300 K) plasma frequency ωp(300 K)=2.02 eV from the finite element simulation fittings. Based on ωp(300 K)=2.02 eV we calculated μ(300 K) from equation (1), and with μ(300 K) we further determined n from equation (2). Under the intraband pumping in our study, conservation of n means that n is a temperature independent property. As a result, for electrons at temperature T>300 K, the constant n allows the determination of μ(T) from equation (2). Based on μ(T) we can further calculate ωp(T) and the energy density U(T) using equation (1) and equation (3), respectively. Using the reported 1/Eg=0.4191 eV−1 and m=0.263 m0 for ITO (m0 being free electron mass), we determined μ(300 K)=1.24 eV and n=1.59×1021 cm−3. The calculated electron energy density rise U(T)−U(300 K) (in eV/cm3) represents the energy required to raise the electron temperature from 300 K to T. Note that knowing [U(T)−U(300 K)] v.s. T one can calculate the heat capacity C(T) for the electron gas, which is a temperature dependent quantity.
The Electron Heat Capacity for ITO-NRA
In
In
where D(μ) is the density of states at the Fermi level. Calculations are performed for different 1/Eg with a fixed m=0.263 m0. The black dotted line represents a constant heat capacity of
(the Dulong-Petit limit), where kB is the Boltzmann constant.
In
Estimating the Experimental Excitation Energy Density
Transmission and reflection measurements showed that the ITO-NRA sample transmits ˜50% and reflects ˜5% of the pump power (relative to a gold film taken as 100% reflection) regardless of the pump fluence used in the TA experiments (up to 6.5 mJ/cm2). As a result, we estimated the actual absorbed pump fluence to be 45% of the pump fluence. We then converted the experimental pump fluence in mJ/cm2 to an excitation energy density in eV/cm3. Relevant parameters are: pump spot diameter at normal incidence, D (in cm); energy per pump pulse, F (in mJ); incidence angle (angle between the pump beam and the substrate normal), θinc (in degree); area of the sample being pumped,
pump fluence,
energy per pump pulse absorbed by the sample, 0.45×F (in mJ); nanorod array periodicity, height and edge length, a, H and L (in cm); total volume of the nanorods illuminated by the pump,
The experimental excitation energy density is then
or, equivalently,
As a result, the pump fluence
multiplied by a factor
gives the experimental excitation energy density (in eV/cm3). Using nanorod array periodicity a=1000×10−7 cm, height H=2600×10−7 cm and edge length L=180×10−7 cm, the factor for converting pump fluence in mJ/cm2 to excitation energy density in eV/cm3 is estimated as
which appears in the upper x-axis in
MIR-Probe-TA Under Various Incidence Angles, Polarizations, and Fluences
Since the pump and probe have the same polarization direction (as illustrated by the black arrow in
Estimation of the Lattice Temperature Rise
Due to the absence of a literature reported heat capacity for ITO, we estimated the heat capacity of ITO using the reported heat capacity for In2O3 at room temperature, which is 99.08 J mol−1 K−1. We used literature reported ITO lattice parameters (space group of Ia
Effects of the Underlying ITO Film on the Transmission Spectra of ITO-NRA
An epitaxial ITO film of about 10 nm was pre-sputtered before the ITO-NRA growth to facilitate electron beam lithography and immobilize the gold seeds during the nanorod growth. The cross-sectional transmission electron microscopy (TEM) image in
The ITO film thickness slightly increased after the vapor-liquid-solid nanorod growth, possibly due to a much slower but finite vapor-solid growth. This thin ITO film becomes less transparent and more reflecting at longer wavelengths in the MIR. To investigate further the effect of the underlying ITO film on the static transmission property of the ITO-NRA we performed static transmission measurements on both the ITO film and the ITO-NRA.
Scattering of Electrons at Elevated Temperatures
The electron-electron and electron-phonon scattering rates, which determine the damping term γ, depend on the electron and lattice temperature as well as the availability of states involved in the scattering processes. This is simplified in our analysis by the use of a constant damping rate γ in the finite element simulations performed for
shows a linear dependence of v on k, which in a classical picture suggests a larger damping rate γ due to more collisions when pumping the electrons to higher k states. In a nonparabolic band, however, the velocity becomes
This form suggests that rising of v due to the increasing
term is compensated by the increase of the denominator term (1+2E/Eg). This is consistent with the absence of spectral-broadenings of NIR and MIR LSPRs in both the NIR-probe and MIR-probe TA experiments that would otherwise lead to more pronounced induced-absorptions. Notably, in the limiting case of 1/Eg→∞ eV, the electron velocity
and becomes independent of k.
Discussion on Electron Thermalization
We note that comparison of
Sample Fabrication.
The ITO-NRAs were fabricated with a modified version of the procedure described earlier. Briefly, a 10 nm thick epitaxial ITO film was sputtered on YSZ (001) substrate at 600° C. under a 5 mTorr 20 sccm Ar gas flow. After spin-coating 70 nm GL-2000 electron beam resist (Gluon Labs), a large area (0.6 cm by 1 cm) array of 150 nm dots with a chosen periodicity was patterned with electron beam lithography (JEOL 9300). The substrate was then developed in Xylene at room temperature for 1 min, rinsed with IPA, followed by the deposition of 2 nm Cr and 15 nm Au, which were subsequently lifted off by immersing in Anisole at 75° C. for 1 hour. The ITO-NRAs were then grown in a three-zone furnace, in which the source (100 mg mixture of In and SnO with a molar ratio of 9:1) and substrate were kept at 900° C. and 840° C., respectively. The gas flow was a mixture of 6 sccm 5% O2 (balanced by N2) and 80 sccm pure Ar under a pressure of 130 mTorr.
Optical Measurement
Near to mid-infrared transmission spectra were measured with FTIR (Thermo Nicolet 6700). A ZnSe lens was used to focus the light to a 1 mm diameter spot. Transient absorption measurements were performed using a 35 fs amplified titanium: sapphire laser operating at 800 nm with a repetition rate of 2 kHz. Near-infrared pump pulses at 1500 nm were generated via a white light seeded optical parametric amplifier and were reduced in repetition rate to 1 kHz. Near-infrared probe pulses were generated by focusing a portion of the amplifier output into a 12 mm thick sapphire window. Mid-infrared probe pulses were produced from difference frequency mixing of the signal and idler beams produced from a second white-light seeded optical parametric amplifier. Pump-probe time delays were produced via a variable path delay stage and retroreflector. In MIR-probe-TA, p-polarization is achieved by rotating the sample around a rotation axis perpendicular to the propagation direction and polarization direction of the beam (as shown in
Simulations and Calculations.
FEM simulations of the periodic ITO-NRA were performed with the RF module of COMSOL Multiphysics. Codes for numerical calculations of the integrals were implemented with Matlab R2015a.
Here we demonstrate large optical nonlinearity of indium tin oxide nanorod arrays (ITO-NRAs) in the dielectric range from 360 nm to 710 nm (denoted as the visible range) when pumped at the localized surface plasmon resonance (LSPR) in the NIR. A number of transmission minima in the visible range, arising from collective light diffraction by the periodic dielectric nanorod array, give rise to a pump-induced transmission modulation with absolute amplitude up to ±20%. These transmission minima also act as sensitive “probes” for the quantification of permittivity change and thereby the optical nonlinearity of ITO. We show that a positive change of the real part of the permittivity is achieved throughout the visible range, which is attributed to a modification of the interband transitions in ITO. Moreover, the large scattering cross-section of the dielectric ITO-NRAs (as opposed to the large absorption cross-section of noble metal nanostructures) allows for a dynamic redistribution of light intensities among different diffraction orders, and the spectral response of the ITO-NRAs can be tuned by simply adjusting the incidence angle or tailoring the length of the nanorods. In the temporal domain, we found both a sub-picosecond response stemming from the electron-phonon coupling and a microsecond response arising from the lattice cooling in ITO.
Static Spectral Features of the ITO-NRA
While the NIR LSPR is a localized phenomenon, the transmission minima in the visible range are due to coherent light diffraction by the ITO-NRA and therefore is attributed to an array effect. Effectively, the ITO-NRA acts as a two-dimensional diffraction grating that supports not only the forward propagating (0, 0) order, but also the (1, 0) and (1, 1) grating orders propagating in oblique directions (See
Transient Absorption Experiments on the ITO-NRA
The transient spectral response of the ITO-NRA was investigated by pump-probe transient absorption (TA) experiments. To fully characterize the dynamics, we performed both nanosecond TA experiments (denoted as short-delay-TA experiments) and microseconds TA experiments (denoted as long-delay-TA experiments). In both experiments the center wavelength of the pump was tuned to the LSPR wavelength of 1500 nm, which permits large on-resonance absorption in the metallic region of the ITO-NRA. Pumping the sample at 800 nm (off-resonance) was found to give significantly weaker response in comparison to the on-resonance pumping (See
The Sub-Picosecond Component
The results of short-delay-TA experiments are summarized in
To estimate the fluence and wavelength dependent Δ∈′(ω) observed in our TA experiments, we carried out waveguide simulations, in which we arbitrarily introduced Δ∈′(ω) ranging from 0 to 1 on top of the static permittivity, and calculate the wavelength dependent mode index neff(ω) as a function of Δ∈′(ω). This allows for the calculation of the spectral locations of the five transmission minima associated with these values of Δ∈′(ω) using equation (1). The dependence of neff(ω) on Δ∈′(ω) and wavelength is color-coded in
Examination of
The Microsecond Component
We now discuss the slow component of ΔT(t)/T(0).
The microsecond decay time of the slow component (best illustrated by
Spectral Tunability and Beam-Steering Capability
In addition to the measurements on the (0, 0) order, we performed additional short-delay-TA experiments to analyse one of the four equivalent (1, 0) and (1, 1) diffraction spots. Examination of the spectral maps for the higher diffraction orders (
To further demonstrate the spectral tunability achievable by adjusting the geometric parameters, we performed static and short-delay-TA experiments on two additional ITO-NRAs with nanorod heights of 1.4 μm and 2.9 μm, respectively (the SEM images, NIR transmission spectra and visible ΔT(t)/T(0) spectral maps for these two ITO-NRA samples are presented in
Sample Fabrication
Briefly, an epitaxial ITO film of 10 nm thickness was deposited on YSZ (001) substrate using magnetron sputtering at 600° C., 5 mTorr under 20 sccm Ar gas flow. A 70 nm thick GL-2000 electron beam resist (Gluon Labs) was then spin coated on the substrate, followed by exposure of an array of 150 nm dots with designed pitch sizes (JEOL JBX-9300FS electron beam lithography system). The exposed sample was developed in Xylenes at room temperature for 60 seconds, and then rinsed by IPA. 2 nm Cr and 15 nm Au was thermally evaporated on the sample, which was subsequently lifted off in Anisole at 75° C. for 1 hour. The nanorod growth was performed at a customized tube furnace system.
Steady State Measurements
Transmission spectra in the near-infrared range were measured with FTIR (Thermo Nicolet 6700). A pair of ZnSe lenses were used to focus the light down to a 1-mm-diameter spot. Transmission spectra in the visible range were measured with an UV/Vis/NIR spectrophotometer (Perkin Elmer Lambda 1050).
Transient Absorption Measurements
Transient absorption experiments with delay times up to 1000 ps were performed using a 35 fs amplified titanium:sapphire laser operating at 800 nm at a 2 kHz repetition rate. Pump pulses at 1500 nm were generated via a white light seeded optical parametric amplifier and were reduced in repetition rate to 1 kHz. Broadband probe pulses were generated by focusing a portion of the amplifier output into a CaF2 window (2 mm thick). The probe pulses were mechanically time-delayed using a translation stage and retroreflector. The pump spot diameter on the sample was 396 μm. Full spectral maps for the (0,0) order appear in
Longer time-delay transient absorption measurements were performed with ˜100 ps time resolution using a 100 fs pump pulse and an electronically delayed white light probe pulse. The probe pulse is generated via self-phase modulation of a Nd:YAG laser in a photonic crystal fiber. Instabilities in the probe pulse were compensated by monitoring a beam-split portion of the pulse in a separate detector. Signal to noise ratios achieved with this system are notably lower than that those obtained for the higher time-resolution transient absorption system, primarily owing to the lower probe pulse-to-pulse stability. The pump spot diameter on the sample is 220 μm. Full spectral maps for the (0,0) order appear in
Finite-Element Simulations
The optical simulation and waveguide simulation were performed with the Wave Optics module of COMSOL Multiphysics. The optical simulation was full three-dimensional simulation in which periodic boundary conditions were applied along the in-plane directions; transmission and reflection of the ITO-NRA can be obtained. The waveguide simulation was a two-dimensional simulation, in which a eigenmode analysis was performed on the cross section of ITO nanorod for calculating the effective mode index. The heat-transfer simulation was enabled by the heat-transfer module of COMSOL Multiphysics. More details about optical, waveguide and heat-transfer simulations appear below.
Calculation of the Grating Order Intensities
The electromagnetic waves scattered by a periodic phased array can be decomposed into orthogonal eigenmodes, which are essentially the grating orders including both propagating and evanescent ones. Since the nanorod spacing of 1 μm is comparable to the wavelength in the visible range, higher order propagating modes (besides the zero order mode) can be produced. To extract intensities of these higher order modes from optical simulations, we decomposed the transmitted electric fields according to the procedures shown by J. Jin et al. Briefly, a two dimensional Fourier transform was performed on the electric field at the bottom boundary of the YSZ interface (the array being in the x-y plane and the bottom boundary is at z=z0),
where the coefficients
are the electric field intensities for the (n, m) order. S=a2 is the cross-sectional area of a unit cell, kxn=kx0−2πn/a and kym=ky0−2πm/a. Here kx0=k0·sin θ·cos φ, ky0=k0·sin θ·sin φ, k0=2π/λ is the incident wave vector, θ=0 is the incident angle and φ=0 is the azimuthal angle (the incident wave vector is normal to the substrate). The wave vector in the z direction is kznm=(k02−kxn2−kym2)1/2; a mode is propagating when kznm is real and evanescent when kznm is imaginary. Transmission of the (n, m) grating order is calculated as
where Einc is the electric field of the incident wave. The (1, 0) and (1, 1) orders are illustrated in the photograph of
To further verify that the wave propagating along the nanorod follows the fundamental HE11 mode, we plot in
Permittivity of ITO
The static transmission spectrum of the ITO-NRA from 360 nm to 710 nm was fitted using the Drude-Lorentz model, ∈(ω)=∈∞+AL/(ωL2−ω2−iγLω)−ωp2/(ω2+iγpω). Parameters that yield a good match between the simulated and the experimental transmission spectra are ∈∞=3.95, AL=(1.4 eV)2, ωL=3.8 eV, γL=0.01 eV, ωp=2.18 eV, and γp=0.12 eV. The Drude-Lorentz model was adopted simply to provide reasonable wavelength dependent permittivity for the subsequent waveguide simulations, from which the effective mode index neff(ω) can be obtained. The single Lorentz pole is not expected to accurately describe the permittivity of ITO in the ultraviolet range (below 360 nm). In addition, the value of ωp=2.18 eV obtained by fitting the visible spectrum is slightly larger than ωp=2.02 eV obtained by fitting the NIR spectrum using a pure Drude model described previously; in this work ωp=2.02 eV was used for the calculation of the fluence dependent electron distribution. The relative permittivity of ITO in the visible (with the Drude-Lorentz model) and near-infrared range (with the Drude model) are plotted in
The transmission spectrum of a bare YSZ substrate shown in
Transmission and Reflection of the ITO-NRAs Measured Using an Integrating Sphere
Theoretical Modelling of the Permittivity Change of the ITO-NRAs
We first generalize the calculation of the imaginary part of the relative permittivity to cover the case of direct interband optical transition in a semiconductor with a non-parabolic conduction band (CB) and a parabolic valence band (VB). The dispersion relations are ℏ2k2/2mv=Ev for holes in the VB, and ℏ2k2/2mc=Ec+CEc2 for electrons in the CB; here Ev is the hole energy referenced to the valence band maximum (VBM), Ec is the electron energy referenced to the conduction band minimum (CBM), mv is the hole effective mass, mc is the electron effective mass (at CBM), and c is the non-parabolicity of the CB. Both Ec and Ev are taken as positive.
ℏω=Eg+Ec+Ev (1).
If we let k2=2mc(Ec+CEc2)/ℏ2, equation (1) becomes ℏω=Eg+Ec+R(Ec+CEc2). This is a quadratic equation in Ec and can be rewritten as RCEc2+(R+1)Ec+(Eg−ℏω)=0, with the solution
The derivative of Ec with respective to the photon energy ℏω is given by
The density-of-states (DOS) for electrons at Ec is
or equivalently,
Using ρ(ℏω)d(ℏω)=ρ(Ec)d(Ec), where ρ(ℏω) is the joint-density-of-states (JDOS) for optical transition with photon energy ℏω, we get ρ(ℏω)=[d(Ec)/d(ℏω)]·ρ(Ec), which can be calculated numerically using equation (3) and (4). The absorption coefficient α(ω) arising from the considered transition can be written as
where n′(ω) is the real part of the refractive index and M is the electric dipole matrix element10.
For the highly doped materials considered here, f(Ev)=1, hence
Since α(ω)=2ωn″(ω)/c, we can write
where n″(ω) is the imaginary part of the refractive index. Combining equation (5) and (6) gives
which is a dimensionless quantity. Now the intraband optical pumping in our study gives rise to a redistribution of the electrons in the conduction band, whose temperature T can be calculated based on our earlier study. As the Fermi function term f(Ec) in equation (7) is electron temperature dependent, a temperature dependent ∈″(ω) can be calculated from
where Δ∈″(ω,T)=∈″(ω,T)−∈″(ω,T0) and Δf(Ec,T)=f(Ec,T)−f(Ec,T0) with T0=300K corresponding to the static case. Knowing Δ∈″(ω, T) we can further obtain Δ∈′(ω, T) using the Kramers-Kronig relation,
The procedure described above was used as a model to theoretically calculate the change of real part of the relative permittivity (shown in
Estimating the Electron and Lattice Temperatures
Measurements of the pump power showed that nearly 50% is transmitted and about 5% is reflected (reflection was referenced to a 200 nm thick gold film) by the ITO-NRA for all fluences used in the short-delay-TA experiments (up to 10.72 mJ·cm−2). We therefore conclude that 45% of the pump energy is absorbed by the ITO-NRA. Assuming a spatially uniform excitation profile, the energy (in mJ) deposited per unit volume of ITO nanorod (in cm3) per pump pulse can be calculated as
where p is the pump fluence in mJ·cm−2, L is the edge length in cm, H is the height in cm, and a is the periodicity in cm of the ITO-NRA. Note that in numerical calculations p was treated as a continuous variable.
The electron temperature at te,0 is denoted as Te,0. This was estimated using the procedure described earlier. To assess the lattice temperature (denoted as T1,0) achieved at t1,0, we used the heat capacity data from E. H. P. Cordfunke et al. for In2O3 measured for the range from 0 to 1000 K. To convert this data into the required units we used the In2O3 molecular weight of 277.64 g·mol−1 and a mass density of 7.16×103 kg·m−3 (calculated from the lattice constant of cubic ITO, 1.01 nm). For comparison purposes, Cordfunke's heat capacity is equivalent to 2.567×106J·m−3·K−1 at 298 K, which is to be compared with a value of 2.58×106J·m−3·K−1 adopted in the independent work by T. Yagi et al.
Details of the Heat-Transfer Simulations
The heat transfer equation is given by ρCp(∂TL/∂t)+∇·(−κ∇TL)=0 where the temperature TL is a function of both time and position, and κ is the thermal conductivity. This equation was solved using COMSOL Multiphysics in the time domain. A uniform temperature profile in the nanorod was used as the initial condition (with temperatures obtained from
The thermal conductivity κ of ITO was calculated from the equation κ=κel+κph, where κel and κph are thermal conductivities contributed by mobile electrons and phonons, respectively. According to T. Ashida et al, κph is almost constant (3.95 W·m−1·K−1 for ITO films with different electron concentrations), whereas κel is well described by the Wiedemann-Franz law of κel=LTσ, where L is the Lorentz number (2.45×10−8WΩ·K−2) and σ is the electrical conductivity. In our heat-transfer simulations we considered κel=LTσ as a temperature dependent quantity, as opposed to κph which was assumed to be temperature independent. To get a reasonable estimate for σ, we performed Hall measurement (Van der Pauw method, Ecopia HMS-5000) on an epitaxial ITO film grown on YSZ substrate, whose electron concentration and mobility were found to be ˜1.3×1021 cm−3 and 47 cm2·V−1·s−1, respectively, yielding a value of 9.4881×105 S·m−1 for σ. The thermal conductivity of ITO at 300 K is determined to be 10.9 W·m−1·K−1, which is more than an order of magnitude smaller than that of gold (314 W·m−1·K−1). The thermal conductivity and heat capacity of YSZ were taken to be 2.5 W·m−1·K−1 (from K. W. Schlichting et al17) and 60.4 J·mol−1·K−1 (from T. Tojo et al18), respectively. Both quantities were assumed to be temperature independent, since the temperature rise in the YSZ substrate is negligible in comparison to that of ITO. YSZ's molecular weight and mass density were 123.218 g·mol−1 and 6.0 g·cm−3, respectivelyl17. To further explore the geometrical dependence of the lattice heat dissipation rate, we performed extra heat-transfer simulations for ITO nanorods with different heights and edge lengths; the results are summarized in
Full ΔT(t)/T(0) Spectral Maps
Spectral Maps of ΔOD(t) and T(t)
In TA experiments
where I0 is the intensity of the beam transmitting through air (taken as the background in all measurements), and I(0) and I(t) are beam intensities transmitting through the sample before and at delay time t after the pump, respectively. Another commonly used quantity, ΔOD(t), is related to ΔT(t)/T(0) as, ΔOD(t)=−log10[1+ΔT(t)/T(0)]. The ΔOD(t) spectral map is plotted in
We combine the unique optical and mechanical properties of ITO to achieve strong modulation and steering of light via coherent acoustic vibrations in periodic indium-tin-oxide nanorod arrays (ITO-NRAs). Due to the low carrier concentration compared to noble metals, ITO-NRAs exhibit an LSPR in the near-infrared (NIR), as well as a number of transmission minima in the visible resulting from the collective light diffraction by the periodic array. By resonantly pumping the ITO-NRA in the NIR, we demonstrate coherent acoustic vibrations which modulate, and steer the probe signals in the visible range at ˜20 GHz frequency with a maximal differential transmission modulation amplitude up to ˜10%. In addition, two complementary transient absorption (TA) measurement techniques were employed to probe the delay time windows of 0-1 ns and 0-50 ns; together they permit a detailed investigation of both the breathing and extensional modes of the ITO-NRAs with a large aspect-ratio. By comparing the experimental vibrational frequencies with the finite-element simulation yielded counterparts, we for the first time report the anisotropic elastic tensor for single-crystalline ITO, which can shed light on the design and integration of mechanically robust, ITO-based electronic and optical devices, especially when their critical dimensions approach the tens of nanometer scale.
The ITO-NRA shown in
Having determined the static spectral response, we then studied the transient behaviors using pump-probe TA experiments. To efficiently excite the coherent acoustic vibrations, we tuned the center wavelength of the pump to the transverse-LSPR of the ITO-NRA at 1500 nm. A white light probe covering the visible range was used to study the vibration-induced intensity modulation. To ensure sampling the array instead of an individual nanorod, the pump beam was adjusted to have a diameter of 190 μm. Both the pump and probe beams were normal to the substrate (as indicated in
We first ran TA experiments with delay times ranging from 0 to 1 ns using probe pulses generated via a translational delay stage.
The coherent acoustic vibrations of the ITO-NRA are manifested by the temporal oscillations of the ΔT/T signals with a period of 50 ps shown in
Due to the dielectric nature of the ITO-NRA, the vibration-induced intensity modulation of the (0, 0) order is expected to be associated with complementary modulation of the (1, 0) and (1, 1) orders. This was investigated in extra TA experiments by collecting the (1, 0) and (1, 1) diffracted spots of the probe beam using an optical fiber. Only narrow spectral windows were measured due to the large spatial dispersion of the higher grating modes (
The frequency signatures of the ΔT/T spectral map for the (0, 0) mode (
Nanorods exhibit three types of vibration modes, namely breathing, extensional and bending modes; the first two correspond to expansion and compression along the radial and longitudinal directions, respectively. The frequencies of the breathing and extensional modes are inversely proportional to the radial and longitudinal dimensions, and are expected to differ by an order of magnitude in the present case (aspect ratio of ITO nanorod is ˜14). Since ˜100-nm radius nanorods exhibit breathing modes at a few to tens of GHz, we assign the 18.7 GHz and 22.1 GHz frequencies to the 1st and 2nd breathing modes, respectively. These two modes are consistent with observations in pentagonal gold nanowires, in which the displacement fields for the two modes were found to concentrate at the corners and edges of the cross-sectional plane, respectively, due to the break of the cylindrical symmetry.
To determine the elastic properties of ITO based on the vibrational frequencies, we performed finite-element simulations using COMSOL Multiphysics. Since simulations of a long aspect-ratio nanorod with sharp corners are computationally expensive, we first reduced the problem to a two-dimensional (2D) cross-section; this accurately captures the vibrational behaviors of the breathing modes which are dominated by lattice displacements within the cross-sectional plane. Justifications regarding the use of the 2D model appears below and from
We first attempted simulations by taking ITO as an isotropic material described by the Young's modulus, E, and Poisson's ratio, υ. As shown in
In order to determine C12, we further examined the extensional mode of the ITO-NRA using a different TA technique, in which a maximal delay time of ˜50 ns was realized via electronically delayed white light probe pulses. The measured ΔT/T spectral map and associated Fourier transform are shown in
With the three elastic constants in hand, we determine Young's modulus of ITO along any crystalline direction (See below). The Young's modulus of ITO along its three primary directions are determined to be E[100]=217.9 GPa, E[110]=110.5 GPa and E[111]=94.9 GPa, which suggests that ITO is elastically anisotropic, as further presented graphically by the orientation dependent Young's modulus diagram in
Static Optical Measurements.
The near-infrared transmission spectrum was measured using FTIR (Thermo Nicolet 6700). A ZnSe lens was used to focus the light into a 1 mm diameter spot. The visible transmission spectrum was measured with an UV/Vis/NIR spectrophotometer (Perkin Elmer Lambda 1050).
Transient Absorption Measurement.
Transient absorption measurements with 1 ns delay time were performed using a 35 fs amplified titanium:sapphire laser operating at 800 nm with a repetition rate of 2 kHz. The broadband probe pulses were generated by focusing a portion of the amplifier output into a 2 mm thick CaF2 window. The 1500 nm pump pulses were generated via a white light seeded optical parametric amplifier and were reduced in repetition rate to 1 kHz. The probe pulses were time-delayed using a mechanical translation stage and retroreflector. The pump spot diameter on the sample was 190 μm. Transient absorption measurements with 50 nm delay time were performed using a 100 fs pump pulse and an electronically delayed white light probe pulse with about 100 ps time resolution. The probe pulse is generated via self-phase modulation of a Nd:YAG laser in a photonic crystal fiber. Instabilities in the probe pulse were compensated by monitoring a beam-split portion of the pulse in a separate detector. The pump spot diameter on the sample for the latter measurements was 220 μm. The differential transmission change, ΔT/T, is defined as [T(t)−T0)]/T0, with T0 and T(t) denoting the static transmission and transmission at delay time t, respectively, both normalized to air.
Finite-Element Simulations.
Acoustic simulations were performed with the Structural Mechanics module of COMSOL Multiphysics in the frequency domain. A uniform strain in the entire simulation domain was applied and the vibrational response of the nanorod was subsequently analyzed.
Dependence of Vibration-Induced Oscillations on Wavelength and Fluence
Verification of the Two-Dimensional Cross-Sectional Simulations for the Breathing Modes
In linear elasticity theory the general form of Hooke's law can be written as {right arrow over (σ)}={right arrow over (∈)} (where
is the elasticity tensor), or alternatively {right arrow over (∈)}=
{right arrow over (σ)} (where
is the compliance tensor that is related to the elasticity tensor by
=
−1. Owing to its cubic symmetry, only three independent elastic constants, C11, C12 and C44, are required for ITO. The elasticity tensor can then be written as
In contrast, for isotropic materials only two independent elastic constants, C11 and C12, are required, with C44 given by C44=(C11−C12)/2. Under small deformations where linear elasticity applies, the volumetric change is ΔV/V=∈xx+∈yy+∈zz, where ΔV is the change in volume and ∈ii (ii=xx, yy or zz) are the principal strains.
In our simulations a uniform initial strain is applied to the entire volume and the mechanical response of the nanorod is calculated in the frequency domain. The volumetric strain is integrated over the entire nanorod volume and the vibrational modes are identified as the frequencies that give a peak in the total volumetric change. A full three-dimensional (3D) simulation of a sharp-cornered ITO nanorod confined on a substrate with large aspect-ratio is computationally expensive. This prevents varying the elastic constants, Cii, C12 and C44, in fine increments over large ranges; therefore, we performed two-dimensional (2D) simulations of the cross-sectional response, which are computationally inexpensive and permit the sweeping of C11, C12 and C44 independently over large ranges so as to obtain a best match between the simulated breathing mode frequencies and their experimental values. The use of a 2D model is supported by comparing it with full 3D simulations, in which we simulated an ITO nanorod with the experimental edge length of 180 nm (as used in 2D simulations) and heights varying from 600 nm to 2600 nm. Two limiting cases are considered: a free-standing nanorod (which represents no mechanical coupling between the ITO nanorod and the substrate) and a nanorod with its bottom boundary rigidly fixed (with zero displacements, as the substrate fully restricts the ITO nanorod bottom boundary). The 3D simulation results are presented in
We note that the bottom boundary does not influence the breathing mode frequencies when the nanorod height is greater than 600 nm as shown in
To study the impact of the nanorod sharpness on the breathing mode frequencies, we performed 2D cross-sectional simulations in which the cross-section is evolving from a perfect square to a square with 90°-arc corners.
Estimating the Mass Density of ITO
Based on the literature reported ITO lattice constant of 1.01 nm with space group of Ia
Finite-Element Simulations of an Elastically Isotropic ITO Nanorod
We performed 2D cross-sectional simulations by considering ITO as an elastically isotropic material with two independent elastic constants, Young's modulus E and Poisson's ratio υ.
Displacement Field Distributions for the Breathing Modes
From the Hooke's law for cubic crystals,
and considering that in 2D simulations no deformation exists in the z direction (indicating ∈zz, ∈yz, ∈zx=0), we obtain σxx=C11·∈xx+C12·∈yy, σyy=C12·∈xx+C11·∈yy, σzz=C12(∈xx+∈yy), σxy=C44·∈xy, and the volumetric strain ΔV/V=∈xx+∈yy+∈zz=∈xx+∈yy. Note that σzz reflects the transverse stress in the z direction caused by principle strain in the x-y plane. In
Calculation of the Weighted Average and Weighted Standard Deviation of the Extensional Mode Frequency
To calculate the weighted average and weighted standard deviation of the extensional mode frequency, we first denote the data shown in
The average frequency at the jth wavelength was calculated as of afj=Σi(xi,j·fi)/Σi(xi,j), which is shown in
The weighted average and the weighted standard deviation of the extensional mode frequency were finally calculated as A=(Σjwj·aλj)/(Σjwj) and
std=sqrt {Σj[wj·(aλj−A)2]/[(m−1)·(Σjwj)/m]}. Similar approach was also used for calculating the average and standard deviation for the breathing mode frequencies.
Calculation of the Orientation Dependent Young's Modulus for ITO
From =
−1 the components of the compliance are determined to be S11=4.59 TPa−1, S12=−1.28 TPa−1, and S44=29.59 TPa−1. The Young's modulus along any direction [a b c] in a cubic crystal can be calculated using (E[a b c])−1=S11−2(S11−S12−0.5·S44)(l2m2+m2n2+n2l2), or equivalently, (E[a b c])−1=(E[1 0 0])−1−3[(E[1 0 0])−1−(E[1 1 1])−1](l2m2+m2n2+n2l2) where l, m and n are the direction cosines defined as l=a/|k|, m=b/|k| and n=c/|k|, with k=ax+by+cz. Note that the Young's modulus in the (1 1 1) plane is orientation independent, which is a common result for cubic crystals. This is due to that directions in the (1 1 1) plane must satisfy (1 1 1)·[a b c]=0 and [a b c]·[a b c]=|k|2, which yields a constant l2m2+m2n2+n2l2=¼, and subsequently a constant Young's modulus for any directions lying on the (1 1 1) plane.
The Zener ratio, defined as Z=2C44/(C11−C12), is a measure of the elastic anisotropy of a cubic crystal. In our case Z is determined to be 0.396, which is close to that of single-crystalline YSZ used as the underlying substrate.
Bounds of the Young's Modulus for Polycrystalline ITO
The Voigt (assuming uniform strain) and Reuss (assuming uniform stress) moduli provide upper and lower bounds on the true Young's modulus for polycrystalline materials based on their single crystal elastic constants.
In the Voigt limit, it follows that 9KV=(C11+C22+C33)+2(C12+C23+C31), and 15GV=(C11+C22+C33)−(C12+C23+C31)+3(C44+C55+C66). Whereas in the Reuss limit, we have 1/KR=(S11+S22+S33)+2(S12+S23+S31), and 15/GR=4(S11+S22+S33)−4(S12+S23+S31)+3(S44+S55+S66), KV, GV, KR and GR are the Voigt bulk modulus, Voigt shear modulus, Reuss bulk modulus and Reuss shear modulus, respectively. Note that for cubic ITO C22=C33=C11, C23=C31=C12 and C55=C66=C44 (similar expressions hold for the compliance components). In polycrystalline materials, the Poisson's ratio and Young's modulus are related to the shear and bulk modulus as: υ=½·[1−3G/(3K+G)] and 1/E=1/(3G)+1/(9K).
Using C11=277.5 GPa, C12=107 GPa and C44=33.8 GPa, the bounds for the shear modulus, bulk modulus and Young's modulus for polycrystalline ITO are determined to be GR=44.6 GPa, GV=54.4 GPa, KR=KV=163.8 GPa, ER=122.6 GPa, EV=146.9 GPa. Note that KR=KV holds for all crystals with cubic symmetry. In addition, the Voigt and Reuss Poisson's ratios are calculated to be υV=0.35 and υR=0.38, respectively.
The Voigt and Reuss theorems predict that the Young's modulus for polycrystalline ITO (denoted as Epc) lies in the range of 122.6 GPa to 146.9 GPa. We note that literature-reported values of Epc (based on polycrystalline ITO films) are lack of consistency. For example, a 116 GPa was a deduced from X-ray diffraction measurements, whereas both a 190 GPa and a 100 GPa were obtained by nano-indentation analysis.
This application claims priority benefit from U.S. Application Ser. No. 62/290,908, filed 3 Feb. 2016, the entirety of which is incorporated herein by reference.
This invention was made with government support under DE-AC02-06CH11357 awarded by the U.S. Department of Energy and DMR1121262 awarded by the National Science Foundation. The government has certain rights in the invention.
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| 6782154 | Zhao et al. | Aug 2004 | B2 |
| 20070104417 | Tanaka | May 2007 | A1 |
| 20130240348 | Mi | Sep 2013 | A1 |
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| Number | Date | Country | |
|---|---|---|---|
| 20170222724 A1 | Aug 2017 | US |
| Number | Date | Country | |
|---|---|---|---|
| 62290908 | Feb 2016 | US |
