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dielectric nanocomposite multilayer films
Surface and Coatings Technology 130 Ž2000. 100᎐109
Microscopic fundamentals of electromagnetic functional metalrdielectric nanocomposite multilayer films U
L.S. Wena, , R.F. Huangb, D.C. Liuc , X.T. Huc , X.D. Baia , C. Suna , J. Gonga a
Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110015, PR China b City Uni¨ ersity of Hong Kong, Hong Kong, PR China c Beijing Institute for Special Electrical Machine, Beijing, PR China
Abstract To understand the physical mechanism of electromagnetic response and potential energy conversion mode in nanostructured materials, especially in metalrdielectric nanocomposite multilayers, a theoretical framework has been suggested. It involves the prerequisite of response, transmission effect, mode of response, plasma model, efficiency of response mode, and size effect of constitutive characteristics. Experimental results have proved that the size effect exists for electrical conductivity, optical constants, permittivity, permeability and carrier density in ultrathin Al, Ti and Fe films as the thickness of the film decreases to a nanometer size range. Principles for designing nanostructures are discussed. 䊚 2000 Elsevier Science S.A. All rights reserved. Keywords: Metalrdielectric multilayer; Nanostructure; Electromagnetic response
1. Introduction Electromagnetic response ŽEMR. of matter is the result of complex mutual interaction between the electron system of matter and the electromagnetic wave field. Its mechanism and regularity could be understood according to the quantum field theory. In this sense, microwave quantum physics reveals a series of effects of atomic᎐molecular processes, resulting in the microwave spectra of matter w1,2x. However, even more abundant microwave spectra could be issued from nanostructured materials providing great perspectives for new electromagnetic functions. The reason for this could be the intimacy between the electron energy sublevel spacing of nanostructures and the energy quantum of electromagnetic waves from radio frequency band to microwave. Therefore, there appears to U
Corresponding author. Tel.: q86-24-23843531; fax: q86-2423891320. E-mail address: [email protected] ŽL.S. Wen..
be continuous activity for exploration in microscopic theory of EMR of nanostructured materials w3᎐6x, besides the results of microwave quantum physics. Considering the complexity and difficulty of the question, it would be better to study the question both theoretically and experimentally at the same time. Metalrdielectric nanocomposite multilayer films represent a system of potential electromagnetic functional materials and devices at the present time. Its structural periodicity and possibility of precise control of its characteristic size Žthickness of the single layer d or composition modulation wavelength ⌳ in it. provides favorable conditions for simplified theoretical treatment of the system as well as experimental identification of the results of the theoretical study. This paper reviews our recent results on microscopic fundamentals for metalrdielectric nanomultilayer film systems. A theoretical framework has been suggested as the result of a theoretical analysis based on solid state physics, plasma physics, electronics and the theory of electromagnetic fields and waves w7x. However, due
0257-8972r00r$ - see front matter 䊚 2000 Elsevier Science S.A. All rights reserved. PII: S 0 2 5 7 - 8 9 7 2 Ž 0 0 . 0 0 6 8 9 - 7
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to the complexity and difficulty of the system, it is certainly not a comprehensive theoretical treatment in its present status, but only a thesis of the future theory. However, it could be a necessary beginning for a further long-term exploration. At the same time, experimental work has been done w7᎐15x. It is concentrated now on revealing the size effects of constitutive characteristics of low-dimensional materials and its physical mechanisms.
2. Framework of microscopic fundamentals The framework of microscopic fundamentals has been obtained as the recent result of a theoretical study of general EMR behavior for metalrdielectric nanomultilayer films. It includes the following main points. 2.1. The prerequisite for EMR of matter As the starting point, we noticed that the prerequisite for electromagnetic response of matter to the electromagnetic wave field, is the entrance of the electromagnetic wave into the matter. Otherwise, the response will be too weak and could be neglected. 2.2. Transmission effect of ultrathin conducting film For conducting media, the electric field intensity EX for the one-dimensional case along the direction of wave propagation, is given by the plane wave solution of Maxwell’s equations w16x EX s E0 eya z eyj z
Ž1.
Where E0 is a constant vector, j is an imaginary unit, a is an attenuation constant,  is a phase constant and z is the ordinate along the direction of wave propagation. For conducting media, ␣ and  are approximately equal and could be represented as follows w16x af  f Ž ¨ . y1 r2
Ž2.
Fig. 1. Classical transmission effect of ultrathin conducting film.
between air and the film. The transmittance t of the film will be given by the ratio of ErE0 , t s ErE0 s eya z
Ž4.
Eq. Ž4. gives the transmission effect of conducting thin films deduced from the classical theory of electromagnetic field and waves. The effect could be, therefore, called the classical transmission effect in order to differ it from the quantum transmission effect due to tunneling. Fig. 1 gives the transmittance curve for an aluminium thin film. Its electric conductivity s 3.54 = 107 Srm; relative permeability r s 1.000021 and permeability of free space 0 s 4 = 10y7 have been used for calculation of the transmittance curve. It is evident from Fig. 1, that at the passed distance of 0.1 m from the interface between air and the film, we already have t ) 85%. For conducting films, the transmission behavior of electromagnetic waves is usually characterized by its penetration depth Žor skin depth. ␦ which is the depth at which the amplitude of E attenuates to 1re of its value at the airrfilm interface. Substituting t s 1re and z s ␦ into Eq. Ž4., the following formula for ␦ could be deduced: ␦ s 1ras Ž ¨ . 1r2
Ž5.
where ¨ is the frequency of the electromagnetic wave, and are the permeability and electrical conductivity of the matter. Eq. Ž1. indicates that for highly conductive metallic films, the electric field intensity E attenuates during propagation and the amplitude of it could be represented by
For example, at a frequency of 3 GHz, the value of ␦ for Al, Au, Cu, Ag are 1.6, 1.4, 1.2, and 1.2 m, respectively w17x. Substituting Eq. Ž5. into Eq. Ž4. and d for z, we obtain the formula of transmission for conducting films in dependence with thickness d:
Es E0 eya z
t s eyŽ d r ␦.
Ž3.
Eq. Ž3. gives the penetrating electric field intensity E in the conducting film at position z from the interface
Ž6.
Eq. Ž6. indicates the size effect for the classical transmission effect. The transmission effect will be
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negligible, if the characteristic size Žthickness d . of the conducting films is in the macroscopic range Ž d4 ␦ .. In that case, the conducting films are in a highly reflective state. However, if the thickness of the film is in the low-dimensional Žor nanometer sized. range Ž d< ␦ ., its t tends towards 1. That means, for the low-dimensional films, transmission becomes the main effect and the conducting films become highly transparent for electromagnetic waves. For intermediate range thickness of the conducting film, electromagnetic percolation Žor leakage. through the film could be observed. 2.3. Modes of EMR in the metalr dielectric nanomultilayer film system Now, we consider the possible modes of EMR in the metalrdielectric nanomultilayer film. The thickness of the metal layers d Žcharacteristic size of ultrathin film., can be variated in the range of several through several hundred nanometers. Metal layers could be magnetic or non-magnetic. However, the thickness of the dielectric layers was kept with the same value as at present. There are four types of EMR of metalrdielectric multilayers which we considered: 2.3.1. Plasma resonance In metallic layers of multilayers, the EMR could be due to the collective oscillations of free electrons in the layer. In that case, metal layers play an active role in formation of EMR in the form of plasma resonance Žplasmon., while the dielectric layers play only a role as an isolation spacer. In its nature, a plasma wave is a special form of electromagnetic wave. However, it differs from the conventional electromagnetic wave in that it is an electrical charge density wave and has a form of a longitudinal wave w18x. The condition of its resonance with an external field is characterized by the plasma frequency P , and given by w18,19x P s Ž Ne2r0 m . 1r2
Ž7.
where N is the electron density in unit volume, e is the charge of the electron, 0 is permittivity of vacuum, and m is the mass of the electron. The resonance of the plasma wave is called a volume plasmon or simply a plasmon w20x. The fact that s 0 at s P corresponds to a pole of the Green function, confirms that P is an excitation frequency of the system w19x. 2.3.2. Interface plasmon Under special conditions of interface structure, such as a two-dimensional periodic array of tips or attenuated total reflection coupler, surface plasmons can be
excited by an external electromagnetic wave field w21x. Previously, Ritchie has shown first w22x the coherent fluctuation performed by electron charges at a metal interface and called it a surface plasma oscillation. Its existence has been demonstrated in electron energy-loss experiments by Powell and Swan w23x. In the case of metalrdielectric multilayers, it will be excited at the metalrdielectric interfaces. Therefore, it could be called an interface plasmon. The angular frequency of an interface plasmon is related to that of a volume plasmon by w21x: Ž I . s Ž P .r'2
Ž8.
where Ž I . and Ž P . are the angular frequencies for the interface plasmon and the volume plasmon, respectively. The field equation of an interface plasma wave is w21x w Ž .x EI s E" 0 exp qi kX x" kz zy t
Ž9.
where EI , is the electrical field vector of an interface plasma wave, kX and kz are wave vectors in directions x and z, respectively. Amplitude E0" is positive for z) 0, negative for z - 0. The wave vector kX s 2 rP
Ž 10.
lies parallel to the x direction, where P , is the plasma resonance wavelength. In the direction perpendicular to the surface Žalong the z-axis., the surface plasmon attenuates quickly as an exponential function of the penetrated distance ŽFig. 2. w21x. The distance at which a surface plasmon attenuates to 1re, is the transverse skin depth of the surface plasmon. Here, the word transverse means that the skin depth is not along the direction of surface wave propagation, but perpendicular to it in the z direction. For a metal surface, it is limited within the Thomas᎐Fermi screening length of ˚ indicating the high localization of approximately 1 A, the surface plasmon. However, for a metalrdielectric interface the value of transverse skin depth of the T interface plasmon ␦IP becomes quite different for different interface materials. For metal layers with permittivity 1 T Ž . Ž X1 q 2 . r1’2 ␦IP 1 s r2
1r2
Ž 11.
where is the wavelength of electromagnetic waves. For a dielectric layer with permittivity 2 X ␦TIP 2 s Ž r2 . Ž 1 q 2 . r22
1r2
Ž 12.
L.S. Wen et al. r Surface and Coatings Technology 130 (2000) 100᎐109
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Fig. 2. Localization of surface plasmon.
where permittivity of the metal layer 1 has a complex form X
Y
1 s Ž 1 q 2 .
Ž 13.
˚ one obtains for silver ␦TIP1 s 240 A, ˚ For s 6000 A, T ˚ w21x. That means that the transverse skin ␦IP 2 s 3900 A depth of the interface plasmon is one order smaller than the wavelength and in the same order as the wavelength for the metal and dielectric layers, respectively. For the nanomultilayer system, the result means the propagation of interface plasmon in full thickness of the metal layer. An interface plasmon at the interface in a metalrdielectric nanomultilayer system is an evanescent wave propagating with a phase velocity 1r2 ¨ s rkX s cr Ž 0 . sin0
Ž 14.
with Ž 0 .1r2 sin0 ) 1 and ¨ F c. The resonance condition for an interface plasmon 1r2 rkX s c wŽ q 1 . r x 1r2 s cr Ž 0 . sin0
Ž 15.
of electrons in matter. When the metal layer is ferromagnetic, there could be the following types of ferromagnetic resonance in metalrdielectric multilayers: ferromagnetic resonance, anti-ferromagnetic resonance, ferrimagnetic resonance, and Faraday rotation. 2.3.3.1. Ferromagnetic resonance. We discuss at first ferromagnetic resonance. The spins of electrons in ferromagnetic materials are bound together by exchange interaction and make precession synchronously around the external permanent magnetic field vector H0 . The expression of the precession frequency of electron spin 0 w26,27x 0 s y␥H0
is usually called the equation for ferromagnetic resonance where ␥ is the gyromagnetic ratio and H0 , is the external permanent magnetic field. The resonance absorption is usually characterized by its half-width ⌬ H and frequency R . The half-width ⌬ H is proportional to the damping coefficient and can be calculated by the following formula:
could thus be fulfilled, particularly because the character of both waves is the same w21x. The intensity of an interface plasmon propagating along a smooth interface decreases with the penetrated distance x according to:
⌬ H s y2R D r␥
EX s E0 exp Ž y2 kX ⬙ .
R s y␥H0 R X
s E0 exp y2 Ž rc .wŽ 1 2 .
½
Ž X1 q 2 .x 3r2 Y1r2 Ž X1 . 2 x
5
Ž 16.
The passed distance L at which the intensity of interface plasmon decreases to 1re, is then given by X
X
L s Ž 2 kx . y1 s 2 Ž rc .wŽ 1 2 . r Ž 1 q 2 .x Y
½
X
Y1r2 Ž 1 . 2
5
y1
3r2
Ž 17.
The absorbed energy heats the film, and can be measured with a photoacoustic cell w24,25x. 2.3.3. Ferromagnetic resonance Ferromagnetism is formed due to the uncoupled spin
Ž 18 .
Ž 19 .
The frequency of ferromagnetic resonance R , is determined by Ž 20 .
where H0 R is the external permanent magnetic field under the resonance condition. For ferromagnetic resonance, R is influenced by the demagnetizing field Hd and magnetic anisotropy. The latter will be evident only for a single crystal. Considering the influence of Hd the expression of R for the ellipsoidal sample should be modified as follows w2x: R s y␥ w H0 R q 4 MS Ž NY y NZ .x w H0 R q 4 MS Ž NX y NZ .x4 1r2
Ž 21 .
where MS is the saturation magnetization, NX , NY , NZ are demagnetizing factors along three principal axes of the ellipsoidal sample. For the slice sample, when the magnetic field is parallel to the slice plane, NX s 1,
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NY s NZ s 0. The expression for R becomes w2x: R s y␥ H0 R w H0 R q 4 MS x4 When plane, comes
1r2
Ž 22 .
the magnetic field is perpendicular to the slice NX s NY s 0, NZ s 1. The formula for R bew2x:
R s y␥ Ž H0 R y 4 MS .
Ž 23.
2.3.3.2. Anti-ferromagnetic resonance. There are two characteristic modes of anti-ferromagnetic resonance as follows w2x: "s y␥ H0 " w HA Ž HA q 2 Hex .x
½
1r2
5
Ž 24 .
where q, corresponds to right-hand precession of the magnetic moment around an external permanent magnetic field, y corresponds to left-hand precession, H0 is the external field, HA is the anisotropic effective field of the magnetic crystal, Hex is the exchange field and directly proportional to the exchange integral J. 2.3.3.3. Ferrimagnetic resonance. For ferrimagnetic resonance, there are also two modes: the ferromagnetic resonance branch and exchange resonance branch. The former is the same as the resonance in ferromagnet, for the latter, the resonance frequency of left-hand precession around H0 is w2x: ex f ␥ Ž H0 y Hex q HA . f y␥Hex
Ž 25.
2.3.3.4. Faraday rotation. When an electromagnetic wave propagates in an anisotropic magnetic medium, its polarization plane rotates as it moves. This is the Faraday rotation effect. The angle of Faraday rotation relates to the phase of electromagnetic wave by: s y Ž qy y . 1r2
Ž 26.
The relation of the rotation angle is w26x: s Ž 1 Ž 0 .
1r2
1r2 r2 . w 1 y Ž ␥MS r Ž 0 y ..x
½
y w 1 y Ž ␥MS r Ž 0 q ..x 1r2
5
Ž 27.
For 4 0 , and 4 y␥ MS , the approximate relation for the Faraday rotation angle could be obtained by w28x: f ␥MS 1 Ž 0 .
1r2
r2
Ž 28 .
2.3.4. Spin wa¨ e resonance A spin wave is the elementary excitation of a spin system in a wave-like form. In magnetically ordered crystals, the spins of electrons are coupled by an ex-
change interaction. The latter plays a role as the restoring force against the perturbation of the external fields. Its quantum Žmagnon. is a quasi-particle and subordinates to Bose᎐Einstein statistics w2x: ² nk : s 1r w exp Ž បkrkT . y 1 x
Ž 29 .
where បk is the energy of the spin wave with a wave vector k., ² nk : is the number of spin waves with a wave vector k at temperature T. Such an excitation can propagate in volume of a magnetic media as a spin wave Žmagnon. or at the interface as an interface spin wave Žinterface magnon., so as for a plasmon and an interface plasmon, respectively. The frequency spectrum of a spin wave is given by w2x: k s ␥
Ž H0 y 4 MNZ q Hex ␣ 2 k2 .Ž H0 y 4 MNZ
qHex ␣ 2 k2 q 4 Msin2 k .
1r2
Ž 30 .
where k is the angle between the propagation direction and the external magnetic field H0 . 2.4. Plasma model All the above-mentioned four types of EMR could be described by using the corresponding equations of plasma physics. First of all, a plasma wave in a metal layer is a volume propagation of a longitudinal plasma electron oscillation Žplasma wave.. Similarly, an interface plasma wave is a propagation of the longitudinal plasma electron oscillation at the metalrdielectric interface. Their resonance Žplasmon and interface plasmon. has been already described by using the plasma equations w18,19,21x. Then, ferromagnetic resonance of precession, Faraday rotation and spin waves could be described using the same basic equations as for an electron cyclotron resonance in a magnetized plasma w27x. In addition, a tensor permeability for a ferrimagnetic and a tensor permittivity of plasma have similar matrix expressions w27x. The main advantage of the plasma model lies in providing a general mathematical means for description of the different EMR mechanisms and the possibility to find out the factors effecting the location of the resonance peaks in the electromagnetic wave spectra of nanocomposite multilayers. 2.5. Efficiency of electromagnetic response model For the question of efficiency of EMR, a theory of the slow wave system will be useful. Theoretical analysis based on electronic physics, has shown in general that for high energy transfer efficiency between an electron system and a travelling wave field, good synchronism between the electron movement and the travelling wave, low dispersion, high strength of the effec-
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tive internal field and high coupling impedance are favorable w27,28x. These requirements are undoubtedly very instructive for efficiency analysis of EMR and designing of metalrdielectric multilayer nanostructures. Among the above-mentioned requirements, a good synchronism between the electron movement and the travelling wave is of primary importance. Usually, to accelerate an electron to a velocity comparable with the velocity of light is extremely difficult. It would be preferable to retard the velocity of the electromagnetic wave to that which could be attained by the natural velocity of electrons in matter or by acceleration of the electron through an electromagnetic field. That is the goal of the slow-wave system Žor delay line. designing. There are two categories of slow-wave systems. The first is using an appropriate material with r r ) 1 for wave travelling. Another category uses periodic structures. The experiences of both categories of slow-wave systems are very useful for the topic of this paper and could serve even as a part of the fundamentals of the topic. However, for metalrdielectric multilayer nanostructures, the characteristic size of the structural component could be much smaller than that for an usual slow-wave system and up to as small as a nanometer size range Žor low-dimensional.. In that case, different physical mechanisms and size effects intrinsic for nanocomposite systems should be taken into account during nanostructure efficiency designing. 2.6. Size effect of constituti¨ e characteristics for lowdimensional materials Besides energy conversion mechanisms, the constitutive characteristics of materials play an important role in nanostructure designing as well. It is known for a long time that electrical conductivity of ultrathin metal film deviates from its bulk value. This phenomenon has been described as the classical size effect due to the surface scattering and grain boundary scattering of electrons when the film thickness is in the same order of magnitude with that of the mean free path of the electron w29,30x. When the film thickness is in the same order of magnitude with that of the De-Brogie wavelength of electrons in the matter, the phenomenon has been described as the quantum size effect w31x. On the other hand, the results of a recent theoretical analysis has shown w13x that of a nanowire depends on the local position when the radius of the nanowire decreases to a level comparable with the mean free path of an electron in it. In that case, the r⬘ ratio increases steeply with the decreasing radius of the nanowire.
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3. Experimental results on the size effect of constitutive characteristics of low-dimensional materials 3.1. Experimental details All the above-mentioned results published in journals on the size-effect, involve only electric conductivity of ultrathin aluminium film. To identify the generality of the size effect of constitutive characteristics of lowdimensional materials, more types of characteristics have been involved, such as electrical conductivity , optical constants n, k, permittivity and permeability . The waveband of measurement has been greatly extended, including visible light, infrared and microwave. Aluminium, Ti and Fe were used as the metal layer materials, while AlN was used as the spacer. The dc electrical conductivity of the films has been measured in-situ in the plating chamber of the magnetron sputtering apparatus by using a dc Wheatstone bridge from outside with a lead from the film sample to the instrument through a vacuum seal. Sheet resistances of Al, Ti, Fe film samples have been measured at first. Then, data of dc electrical conductivity were calculated using the data of sheet resistance and thickness measurement for Al, Ti, Fe films, respectively w8,10x. Optical transmittance, reflectance and absorptance in the visible light region for Al, Ti, Fe films have been measured by using a WFZ900-D4 UVrVIS spectrophotometer w7,12,15x. Similarly, transmittance, reflectance and absorptance in the infrared ŽIR. region for Al, Ti, Fe films have been measured by using a WQF-300 Fourier IR spectroscope w7,12,15x. Their optical functions n and k in the visible light and the infrared ŽIR. regions are obtained by the Newton᎐Simpson recurrent method w7,12,15x. The real and imaginary parts of permittivity ⬘ and Y in the visible light and the infrared ŽIR. regions have been calculated using the following relations w7,15x: s n2 y k2
Ž 31 .
s 2 nk
Ž 32 .
Measurement of ⬘, Y , ⬘, Y for both Al and Fe films at a frequency of 10 GHz have been completed by a Wiltron 560A scalar network analyzer in combination with a Wiltron 6647A sweep generator in microwave bands in the College of Microwave Engineering of the Northwest China University of Technology w16x. The dependence of carrier density in ultrathin Al films on the thickness of the film were obtained through Hall-effect measurement in the Chinese University of Hong Kong w20x. The room temperature measurement was completed by using a Bio-Rad HL5500 PC Hall-
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L.S. Wen et al. r Surface and Coatings Technology 130 (2000) 100᎐109
Fig. 3. Dependence of dc electrical conductivity on the thickness of ultrathin Al, Ti and Fe films.
Effect Measurement System. The cryogenic measurement has been made with a self-assembled PCcontrolled measurement system consisting of a CTL Cryostat, Lake Shore 330 temperature controller and electromagnet.
Fig. 4. Dependence of optical constant n and k in the visible region on the thickness of ultrathin Al, Ti and Fe films.
3.4. Size effect of optical functions and permitti¨ ity in the infrared region
3.2. Size effect of dc electrical conducti¨ ity Results of dc electrical conductivity ŽFig. 3. w8᎐10x show the common trend as indicated in references w28,29x that for ultrathin Al, Ti, Fe films, decreases with decreasing d. However, the character of the curves is quite different from each from other. The curve of Al is typical for a metal with high . It begins to deviate evidently from that of bulk Ti at a thickness of the film of approximately 100 nm, and drops steeply at approximately 10 nm. The curve of Ti is typical for a metal with moderate . It deviates evidently from that of bulk Al at a thickness of the film in the micrometer range and drops quickly from approximately 25 nm. The curve of Fe deviates evidently from the bulk value at approximately 150 nm and drops quickly from approximately 20 nm.
Results of optical functions and permittivity in the infrared region for Al, Ti, Fe films ŽFigs. 6 and 7. show also a complex size effect w7,12,15x. Among them, k ŽTi., k ŽFe. show a monotonic decrease in the nanometer size region, while nŽAl., nŽTi., nŽFe., k ŽAl., ⬘ ŽAl., ⬘
3.3. Size effect of optical functions and permitti¨ ity in the
¨ isible light region
The results of optical functions and permittivity in the visible light region for Al, Ti, Fe films ŽFigs. 4 and 5. show a complex size effect w7,12,15x. Among them, k ŽTi., k ŽFe. show a monotonic decrease in the nanometer size region, while nŽAl., nŽTi., nŽFe., k ŽAl., ⬘ ŽAl., ⬘ ŽTi. , ⬘ ŽFe., Y ŽAl., Y ŽTi., Y ŽFe. give a maximum in the nanometer size region.
Y
Fig. 5. Dependence of permittivity ⬘ and in the visible region on the thickness of ultrathin Al, Ti and Fe films.
L.S. Wen et al. r Surface and Coatings Technology 130 (2000) 100᎐109
Fig. 6. Dependence of optical constant n and k in the IR region on the thickness of the film for Al, Ti and Fe ultrathin films.
ŽTi. , ⬘ ŽFe., Y ŽAl., Y ŽTi., Y ŽFe. give a maximum in the nanometer size region. 3.5. Size effect of permitti¨ ity and permeability in the microwa¨ e band Results of the measurement of permittivity and permeability in the microwave band at a frequency of 10 GHz for both Al and Fe films show a monotonically increasing ⬘ and Y with decreasing d, while the
107
Y
Fig. 8. Dependence of permittivity ⬘ and in the microwave band Ž10 GHz. on the thickness of the film for ultrathin Al, Ti and Fe films.
curves of ⬘ and Y for Ti films have a more complicated size effect dependence d with a maximum at thickness approximately 25 nm ŽFig. 8. w7,15x. The dependence curves of ⬘ and Y of Fe on d are even more complicated with oscillations consisting of a minimum at approximately 18᎐20 nm and a maximum at approximately 31᎐33 nm ŽFig. 9. w7,15x. 3.6. Size-effect of carrier density and plasma frequency Dependence of carrier density in ultrathin Al films
Y
Fig. 7. Dependence of permittivity ⬘ and in the IR region on the thickness of the film for Al, Ti and Fe ultrathin films.
Y
Fig. 9. Dependence of permeability ⬘ and in the microwave band Ž10 GHz. on the thickness of the film for ultrathin Fe films.
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L.S. Wen et al. r Surface and Coatings Technology 130 (2000) 100᎐109
Fig. 10. Dependence of carrier density on the thickness of the film for ultrathin Al films.
on d reveals a strong oscillation of carrier density With d in the nanometer size region with minimums at film thicknesses of approximately 15 and 26 nm and maxima at 18.5 and 37 nm ŽFig. 10. w7,15x. Based on the data of carrier density, dependence of plasma frequency P on d has been calculated using plasma frequency formula Ž7.. The results demonstrate also a strong oscillation of P in a film thickness range of 15᎐40 nm ŽFig. 11. w15x.
EMR, transmission effect of low-dimensional conducting film, modes of EMR, plasma model, efficiency of the EMR mode, and size effect of constitutive characteristics of low-dimensional materials, are explained. Four potential EMR modes are discussed. They are plasmon, interface plasmon, ferromagnetic resonance, and spin wave resonance. Quantitative investigation of the modes of EMR and size effect of constitutive characteristics would be of primary importance for clarification of the principles of nanostructure designing and data collection at the present time. In the range of traditional materials science, the constitutive characteristics should not depend upon the size of the sample. That is true for bulk materials when its characteristic size is in the macroscopic range. However, size effect of constitutive characteristics will become natural, evident and could not be ignored for low-dimensional and nanostructured materials when their characteristic size decreases to the size range of the effective zone of local field in matter, for example, diameter of the Lorentz circle. The size effect of constitutive characteristics gives a challenge for traditional materials science. But it brings a vast possibility of new materials designing principles for nanostructured materials at the same time. The unique feature and advantage of nanostructured materials consist, therefore, in their size effect. The quantitative size effect relationship of constitutive characteristics could become in future a crucial part of the basis for nanostructured materials designing.
4. Discussion A theoretical framework has been suggested based on the recent results of a study on microscopic fundamentals of metalrdielectric nanomultilayers. The principles of its main points including the prerequisite for
5. Conclusion For metalrdielectric nanocomposite multilayers, the following points of view on their microscopic fundamentals could be concluded: 1. A framework of microscopic fundamentals for electromagnetic functional metalrdielectric nanocomposite multilayer films has been suggested. 2. A mechanism of plasmon, interface plasmon, ferromagnetic resonance and spin wave resonance has been discussed as for potential EMR modes with application perspectives. 3. It has been determined experimentally that the size dependence of constitutive characteristics is one of the unique features of low-dimensional materials. 4. Different size effect behaviors of constitutive characteristics provide new possibilities for nanostructured materials designing. References
Fig. 11. Calculated dependence of plasma frequency P on the thickness of the film for ultrathin Al films.
w1x Yu.A. Il’inskii, L.V. Keldysh, Electromagnetic Response of Material Media, Plenum Press, New York, 1994.
L.S. Wen et al. r Surface and Coatings Technology 130 (2000) 100᎐109 w2x Z.G. Ren, Microwave Quantum Physics, Academic Press, Beijing, 1980. w3x S. Uto, E. Tazoh, M. Ozaki, K. Yoshino, J. Appl. Phys. 82 Ž1997. 2791᎐2794. w4x R.E. Camley, D.L. Mills, J. Appl. Phys. 82 Ž1997. 3058᎐3067. w5x S. Rakhmanova, D.L. Mills, Phys. Rev., B 57 Ž1998. 476᎐484. w6x R.L. Stamps, R.E. Camley, J. Magn. Magn. Mater. 177-181 Ž1998. 813᎐814. w7x L.S. Wen, R.F. Huang, Frontiers of Thin Film Technology, accepted by Chinese Academy of Engineering and Chinese Materials Research Society, for publication by Shangdong Publishers Co. w8x Z.L. Tang, R.F. Huang, L.S. Wen, Acta Metall. Sinica 32 Ž3. Ž1996. 308᎐312. w9x X.H. Cao, Z.L. Tang, R.F. Huang, L.S. Wen, Acta Metall. Sinica 32 Ž1996. 4᎐405. w10x Z.L. Tang, R.F. Huang, L.S. Wen, Chin. J. Mater. Res. 11 Ž1997. 438. w11x X.D. Bai, R.F. Huang, L.S. Wen, Acta Metall. Sinica 34 Ž9. Ž1998. 1005᎐1008. w12x X.D. Bai, R.F. Huang, L.S. Wen, Vac. Sci. Technol. 19 Ž2. Ž1999. 108᎐110. w13x W.R. Zhao, D.Y. Lui, R.F. Huang, L.S. Wen, Acta Metall. Sinica 34 Ž1998. 819᎐823. w14x X.D. Bai, R.F. Huang, L.S. Wen, accepted and to be published in J. Mater. Sci. Technol. 16 Ž2000. 5. w15x X.D. Bai, Doctor Thesis in Institute of Metal Research, Chinese Academy of Sciences, June, 1999. w16x D. Jun Zheng, Field and Wave Electromagnetics, Translated from English edition, April, 1983 to Chinese, Jiaotong University Press, Shanghai, 1994.
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w17x P. Lorain, D.R. Corson, Electromagnetic Fields and Waves, Second Edition, Chinese translation, 1980, from Table 11-1, Press of People’s Education, Beijing, 1970, p. 334. w18x A. Halger, The optical constants of bulk materials and films, in: E.R. Pike, W.T. Welford ŽEds.., The Adam Halger Series on Optics and Optoelectronics, IOP Publishing Ltd, Bristol and Philadelphia, 1988. w19x M.G. Cottam, D.R. Tilley, Introduction to Surface and Superlattice Excitation, Press Syndicate of the University of Cambridge, 1989. w20x R.A. Ferrell, E.A. Sterne, Am. J. Phys. 30 Ž1962. 810. w21x H. Raether, Surface plasmons on smooth and rough surfaces and on gratings, Springers Tracts in Modern Physics, V11, Springer-Verlag, 1988. w22x R.H. Ritchie, Phys. Rev. 106 Ž1957. 874. w23x C.J. Powell, J.B. Swan, Phys. Rev. 118 Ž1960. 640. w24x R. Inagaki, K. Kagami, E.T. Arakawa, Phys. Rev. B24 Ž1981. 3644. w25x B. Rothenhauser, J. Rabe, P. Korpium, W. Knoll, Surf. Sci. 1 ¨ Ž1984. 373. w26x P.Z. Ning, D.F. Ming, Microwave Information Transmission Technology, Shanghai Science and Technology Press, Shanghai, 1985. w27x Q.G. Shui, Microwave Technology, Defense Industry Press, Beijing, 1986. w28x R. Chatterjee, Elements of Microwave Engineering, Ellis Horwood and John Wiley & Sons, Chichester, 1986. w29x E.H. Sondheimer, Adv. Phys. 1 Ž1952. 1. w30x A.F Mayadas, M. Shazkes, Phys. Rev. B1 Ž1970. 1382. w31x T. Ando, A.B. Fowler, F. Stern, Rev. Mod. Phys. 54 Ž1982. 437.
Microscopic fundamentals of electromagnetic functional metalrdielectric nanocomposite multilayer films U
L.S. Wena, , R.F. Huangb, D.C. Liuc , X.T. Huc , X.D. Baia , C. Suna , J. Gonga a
Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110015, PR China b City Uni¨ ersity of Hong Kong, Hong Kong, PR China c Beijing Institute for Special Electrical Machine, Beijing, PR China
Abstract To understand the physical mechanism of electromagnetic response and potential energy conversion mode in nanostructured materials, especially in metalrdielectric nanocomposite multilayers, a theoretical framework has been suggested. It involves the prerequisite of response, transmission effect, mode of response, plasma model, efficiency of response mode, and size effect of constitutive characteristics. Experimental results have proved that the size effect exists for electrical conductivity, optical constants, permittivity, permeability and carrier density in ultrathin Al, Ti and Fe films as the thickness of the film decreases to a nanometer size range. Principles for designing nanostructures are discussed. 䊚 2000 Elsevier Science S.A. All rights reserved. Keywords: Metalrdielectric multilayer; Nanostructure; Electromagnetic response
1. Introduction Electromagnetic response ŽEMR. of matter is the result of complex mutual interaction between the electron system of matter and the electromagnetic wave field. Its mechanism and regularity could be understood according to the quantum field theory. In this sense, microwave quantum physics reveals a series of effects of atomic᎐molecular processes, resulting in the microwave spectra of matter w1,2x. However, even more abundant microwave spectra could be issued from nanostructured materials providing great perspectives for new electromagnetic functions. The reason for this could be the intimacy between the electron energy sublevel spacing of nanostructures and the energy quantum of electromagnetic waves from radio frequency band to microwave. Therefore, there appears to U
Corresponding author. Tel.: q86-24-23843531; fax: q86-2423891320. E-mail address: [email protected] ŽL.S. Wen..
be continuous activity for exploration in microscopic theory of EMR of nanostructured materials w3᎐6x, besides the results of microwave quantum physics. Considering the complexity and difficulty of the question, it would be better to study the question both theoretically and experimentally at the same time. Metalrdielectric nanocomposite multilayer films represent a system of potential electromagnetic functional materials and devices at the present time. Its structural periodicity and possibility of precise control of its characteristic size Žthickness of the single layer d or composition modulation wavelength ⌳ in it. provides favorable conditions for simplified theoretical treatment of the system as well as experimental identification of the results of the theoretical study. This paper reviews our recent results on microscopic fundamentals for metalrdielectric nanomultilayer film systems. A theoretical framework has been suggested as the result of a theoretical analysis based on solid state physics, plasma physics, electronics and the theory of electromagnetic fields and waves w7x. However, due
0257-8972r00r$ - see front matter 䊚 2000 Elsevier Science S.A. All rights reserved. PII: S 0 2 5 7 - 8 9 7 2 Ž 0 0 . 0 0 6 8 9 - 7
L.S. Wen et al. r Surface and Coatings Technology 130 (2000) 100᎐109
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to the complexity and difficulty of the system, it is certainly not a comprehensive theoretical treatment in its present status, but only a thesis of the future theory. However, it could be a necessary beginning for a further long-term exploration. At the same time, experimental work has been done w7᎐15x. It is concentrated now on revealing the size effects of constitutive characteristics of low-dimensional materials and its physical mechanisms.
2. Framework of microscopic fundamentals The framework of microscopic fundamentals has been obtained as the recent result of a theoretical study of general EMR behavior for metalrdielectric nanomultilayer films. It includes the following main points. 2.1. The prerequisite for EMR of matter As the starting point, we noticed that the prerequisite for electromagnetic response of matter to the electromagnetic wave field, is the entrance of the electromagnetic wave into the matter. Otherwise, the response will be too weak and could be neglected. 2.2. Transmission effect of ultrathin conducting film For conducting media, the electric field intensity EX for the one-dimensional case along the direction of wave propagation, is given by the plane wave solution of Maxwell’s equations w16x EX s E0 eya z eyj z
Ž1.
Where E0 is a constant vector, j is an imaginary unit, a is an attenuation constant,  is a phase constant and z is the ordinate along the direction of wave propagation. For conducting media, ␣ and  are approximately equal and could be represented as follows w16x af  f Ž ¨ . y1 r2
Ž2.
Fig. 1. Classical transmission effect of ultrathin conducting film.
between air and the film. The transmittance t of the film will be given by the ratio of ErE0 , t s ErE0 s eya z
Ž4.
Eq. Ž4. gives the transmission effect of conducting thin films deduced from the classical theory of electromagnetic field and waves. The effect could be, therefore, called the classical transmission effect in order to differ it from the quantum transmission effect due to tunneling. Fig. 1 gives the transmittance curve for an aluminium thin film. Its electric conductivity s 3.54 = 107 Srm; relative permeability r s 1.000021 and permeability of free space 0 s 4 = 10y7 have been used for calculation of the transmittance curve. It is evident from Fig. 1, that at the passed distance of 0.1 m from the interface between air and the film, we already have t ) 85%. For conducting films, the transmission behavior of electromagnetic waves is usually characterized by its penetration depth Žor skin depth. ␦ which is the depth at which the amplitude of E attenuates to 1re of its value at the airrfilm interface. Substituting t s 1re and z s ␦ into Eq. Ž4., the following formula for ␦ could be deduced: ␦ s 1ras Ž ¨ . 1r2
Ž5.
where ¨ is the frequency of the electromagnetic wave, and are the permeability and electrical conductivity of the matter. Eq. Ž1. indicates that for highly conductive metallic films, the electric field intensity E attenuates during propagation and the amplitude of it could be represented by
For example, at a frequency of 3 GHz, the value of ␦ for Al, Au, Cu, Ag are 1.6, 1.4, 1.2, and 1.2 m, respectively w17x. Substituting Eq. Ž5. into Eq. Ž4. and d for z, we obtain the formula of transmission for conducting films in dependence with thickness d:
Es E0 eya z
t s eyŽ d r ␦.
Ž3.
Eq. Ž3. gives the penetrating electric field intensity E in the conducting film at position z from the interface
Ž6.
Eq. Ž6. indicates the size effect for the classical transmission effect. The transmission effect will be
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negligible, if the characteristic size Žthickness d . of the conducting films is in the macroscopic range Ž d4 ␦ .. In that case, the conducting films are in a highly reflective state. However, if the thickness of the film is in the low-dimensional Žor nanometer sized. range Ž d< ␦ ., its t tends towards 1. That means, for the low-dimensional films, transmission becomes the main effect and the conducting films become highly transparent for electromagnetic waves. For intermediate range thickness of the conducting film, electromagnetic percolation Žor leakage. through the film could be observed. 2.3. Modes of EMR in the metalr dielectric nanomultilayer film system Now, we consider the possible modes of EMR in the metalrdielectric nanomultilayer film. The thickness of the metal layers d Žcharacteristic size of ultrathin film., can be variated in the range of several through several hundred nanometers. Metal layers could be magnetic or non-magnetic. However, the thickness of the dielectric layers was kept with the same value as at present. There are four types of EMR of metalrdielectric multilayers which we considered: 2.3.1. Plasma resonance In metallic layers of multilayers, the EMR could be due to the collective oscillations of free electrons in the layer. In that case, metal layers play an active role in formation of EMR in the form of plasma resonance Žplasmon., while the dielectric layers play only a role as an isolation spacer. In its nature, a plasma wave is a special form of electromagnetic wave. However, it differs from the conventional electromagnetic wave in that it is an electrical charge density wave and has a form of a longitudinal wave w18x. The condition of its resonance with an external field is characterized by the plasma frequency P , and given by w18,19x P s Ž Ne2r0 m . 1r2
Ž7.
where N is the electron density in unit volume, e is the charge of the electron, 0 is permittivity of vacuum, and m is the mass of the electron. The resonance of the plasma wave is called a volume plasmon or simply a plasmon w20x. The fact that s 0 at s P corresponds to a pole of the Green function, confirms that P is an excitation frequency of the system w19x. 2.3.2. Interface plasmon Under special conditions of interface structure, such as a two-dimensional periodic array of tips or attenuated total reflection coupler, surface plasmons can be
excited by an external electromagnetic wave field w21x. Previously, Ritchie has shown first w22x the coherent fluctuation performed by electron charges at a metal interface and called it a surface plasma oscillation. Its existence has been demonstrated in electron energy-loss experiments by Powell and Swan w23x. In the case of metalrdielectric multilayers, it will be excited at the metalrdielectric interfaces. Therefore, it could be called an interface plasmon. The angular frequency of an interface plasmon is related to that of a volume plasmon by w21x: Ž I . s Ž P .r'2
Ž8.
where Ž I . and Ž P . are the angular frequencies for the interface plasmon and the volume plasmon, respectively. The field equation of an interface plasma wave is w21x w Ž .x EI s E" 0 exp qi kX x" kz zy t
Ž9.
where EI , is the electrical field vector of an interface plasma wave, kX and kz are wave vectors in directions x and z, respectively. Amplitude E0" is positive for z) 0, negative for z - 0. The wave vector kX s 2 rP
Ž 10.
lies parallel to the x direction, where P , is the plasma resonance wavelength. In the direction perpendicular to the surface Žalong the z-axis., the surface plasmon attenuates quickly as an exponential function of the penetrated distance ŽFig. 2. w21x. The distance at which a surface plasmon attenuates to 1re, is the transverse skin depth of the surface plasmon. Here, the word transverse means that the skin depth is not along the direction of surface wave propagation, but perpendicular to it in the z direction. For a metal surface, it is limited within the Thomas᎐Fermi screening length of ˚ indicating the high localization of approximately 1 A, the surface plasmon. However, for a metalrdielectric interface the value of transverse skin depth of the T interface plasmon ␦IP becomes quite different for different interface materials. For metal layers with permittivity 1 T Ž . Ž X1 q 2 . r1’2 ␦IP 1 s r2
1r2
Ž 11.
where is the wavelength of electromagnetic waves. For a dielectric layer with permittivity 2 X ␦TIP 2 s Ž r2 . Ž 1 q 2 . r22
1r2
Ž 12.
L.S. Wen et al. r Surface and Coatings Technology 130 (2000) 100᎐109
103
Fig. 2. Localization of surface plasmon.
where permittivity of the metal layer 1 has a complex form X
Y
1 s Ž 1 q 2 .
Ž 13.
˚ one obtains for silver ␦TIP1 s 240 A, ˚ For s 6000 A, T ˚ w21x. That means that the transverse skin ␦IP 2 s 3900 A depth of the interface plasmon is one order smaller than the wavelength and in the same order as the wavelength for the metal and dielectric layers, respectively. For the nanomultilayer system, the result means the propagation of interface plasmon in full thickness of the metal layer. An interface plasmon at the interface in a metalrdielectric nanomultilayer system is an evanescent wave propagating with a phase velocity 1r2 ¨ s rkX s cr Ž 0 . sin0
Ž 14.
with Ž 0 .1r2 sin0 ) 1 and ¨ F c. The resonance condition for an interface plasmon 1r2 rkX s c wŽ q 1 . r x 1r2 s cr Ž 0 . sin0
Ž 15.
of electrons in matter. When the metal layer is ferromagnetic, there could be the following types of ferromagnetic resonance in metalrdielectric multilayers: ferromagnetic resonance, anti-ferromagnetic resonance, ferrimagnetic resonance, and Faraday rotation. 2.3.3.1. Ferromagnetic resonance. We discuss at first ferromagnetic resonance. The spins of electrons in ferromagnetic materials are bound together by exchange interaction and make precession synchronously around the external permanent magnetic field vector H0 . The expression of the precession frequency of electron spin 0 w26,27x 0 s y␥H0
is usually called the equation for ferromagnetic resonance where ␥ is the gyromagnetic ratio and H0 , is the external permanent magnetic field. The resonance absorption is usually characterized by its half-width ⌬ H and frequency R . The half-width ⌬ H is proportional to the damping coefficient and can be calculated by the following formula:
could thus be fulfilled, particularly because the character of both waves is the same w21x. The intensity of an interface plasmon propagating along a smooth interface decreases with the penetrated distance x according to:
⌬ H s y2R D r␥
EX s E0 exp Ž y2 kX ⬙ .
R s y␥H0 R X
s E0 exp y2 Ž rc .wŽ 1 2 .
½
Ž X1 q 2 .x 3r2 Y1r2 Ž X1 . 2 x
5
Ž 16.
The passed distance L at which the intensity of interface plasmon decreases to 1re, is then given by X
X
L s Ž 2 kx . y1 s 2 Ž rc .wŽ 1 2 . r Ž 1 q 2 .x Y
½
X
Y1r2 Ž 1 . 2
5
y1
3r2
Ž 17.
The absorbed energy heats the film, and can be measured with a photoacoustic cell w24,25x. 2.3.3. Ferromagnetic resonance Ferromagnetism is formed due to the uncoupled spin
Ž 18 .
Ž 19 .
The frequency of ferromagnetic resonance R , is determined by Ž 20 .
where H0 R is the external permanent magnetic field under the resonance condition. For ferromagnetic resonance, R is influenced by the demagnetizing field Hd and magnetic anisotropy. The latter will be evident only for a single crystal. Considering the influence of Hd the expression of R for the ellipsoidal sample should be modified as follows w2x: R s y␥ w H0 R q 4 MS Ž NY y NZ .x w H0 R q 4 MS Ž NX y NZ .x4 1r2
Ž 21 .
where MS is the saturation magnetization, NX , NY , NZ are demagnetizing factors along three principal axes of the ellipsoidal sample. For the slice sample, when the magnetic field is parallel to the slice plane, NX s 1,
L.S. Wen et al. r Surface and Coatings Technology 130 (2000) 100᎐109
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NY s NZ s 0. The expression for R becomes w2x: R s y␥ H0 R w H0 R q 4 MS x4 When plane, comes
1r2
Ž 22 .
the magnetic field is perpendicular to the slice NX s NY s 0, NZ s 1. The formula for R bew2x:
R s y␥ Ž H0 R y 4 MS .
Ž 23.
2.3.3.2. Anti-ferromagnetic resonance. There are two characteristic modes of anti-ferromagnetic resonance as follows w2x: "s y␥ H0 " w HA Ž HA q 2 Hex .x
½
1r2
5
Ž 24 .
where q, corresponds to right-hand precession of the magnetic moment around an external permanent magnetic field, y corresponds to left-hand precession, H0 is the external field, HA is the anisotropic effective field of the magnetic crystal, Hex is the exchange field and directly proportional to the exchange integral J. 2.3.3.3. Ferrimagnetic resonance. For ferrimagnetic resonance, there are also two modes: the ferromagnetic resonance branch and exchange resonance branch. The former is the same as the resonance in ferromagnet, for the latter, the resonance frequency of left-hand precession around H0 is w2x: ex f ␥ Ž H0 y Hex q HA . f y␥Hex
Ž 25.
2.3.3.4. Faraday rotation. When an electromagnetic wave propagates in an anisotropic magnetic medium, its polarization plane rotates as it moves. This is the Faraday rotation effect. The angle of Faraday rotation relates to the phase of electromagnetic wave by: s y Ž qy y . 1r2
Ž 26.
The relation of the rotation angle is w26x: s Ž 1 Ž 0 .
1r2
1r2 r2 . w 1 y Ž ␥MS r Ž 0 y ..x
½
y w 1 y Ž ␥MS r Ž 0 q ..x 1r2
5
Ž 27.
For 4 0 , and 4 y␥ MS , the approximate relation for the Faraday rotation angle could be obtained by w28x: f ␥MS 1 Ž 0 .
1r2
r2
Ž 28 .
2.3.4. Spin wa¨ e resonance A spin wave is the elementary excitation of a spin system in a wave-like form. In magnetically ordered crystals, the spins of electrons are coupled by an ex-
change interaction. The latter plays a role as the restoring force against the perturbation of the external fields. Its quantum Žmagnon. is a quasi-particle and subordinates to Bose᎐Einstein statistics w2x: ² nk : s 1r w exp Ž បkrkT . y 1 x
Ž 29 .
where បk is the energy of the spin wave with a wave vector k., ² nk : is the number of spin waves with a wave vector k at temperature T. Such an excitation can propagate in volume of a magnetic media as a spin wave Žmagnon. or at the interface as an interface spin wave Žinterface magnon., so as for a plasmon and an interface plasmon, respectively. The frequency spectrum of a spin wave is given by w2x: k s ␥
Ž H0 y 4 MNZ q Hex ␣ 2 k2 .Ž H0 y 4 MNZ
qHex ␣ 2 k2 q 4 Msin2 k .
1r2
Ž 30 .
where k is the angle between the propagation direction and the external magnetic field H0 . 2.4. Plasma model All the above-mentioned four types of EMR could be described by using the corresponding equations of plasma physics. First of all, a plasma wave in a metal layer is a volume propagation of a longitudinal plasma electron oscillation Žplasma wave.. Similarly, an interface plasma wave is a propagation of the longitudinal plasma electron oscillation at the metalrdielectric interface. Their resonance Žplasmon and interface plasmon. has been already described by using the plasma equations w18,19,21x. Then, ferromagnetic resonance of precession, Faraday rotation and spin waves could be described using the same basic equations as for an electron cyclotron resonance in a magnetized plasma w27x. In addition, a tensor permeability for a ferrimagnetic and a tensor permittivity of plasma have similar matrix expressions w27x. The main advantage of the plasma model lies in providing a general mathematical means for description of the different EMR mechanisms and the possibility to find out the factors effecting the location of the resonance peaks in the electromagnetic wave spectra of nanocomposite multilayers. 2.5. Efficiency of electromagnetic response model For the question of efficiency of EMR, a theory of the slow wave system will be useful. Theoretical analysis based on electronic physics, has shown in general that for high energy transfer efficiency between an electron system and a travelling wave field, good synchronism between the electron movement and the travelling wave, low dispersion, high strength of the effec-
L.S. Wen et al. r Surface and Coatings Technology 130 (2000) 100᎐109
tive internal field and high coupling impedance are favorable w27,28x. These requirements are undoubtedly very instructive for efficiency analysis of EMR and designing of metalrdielectric multilayer nanostructures. Among the above-mentioned requirements, a good synchronism between the electron movement and the travelling wave is of primary importance. Usually, to accelerate an electron to a velocity comparable with the velocity of light is extremely difficult. It would be preferable to retard the velocity of the electromagnetic wave to that which could be attained by the natural velocity of electrons in matter or by acceleration of the electron through an electromagnetic field. That is the goal of the slow-wave system Žor delay line. designing. There are two categories of slow-wave systems. The first is using an appropriate material with r r ) 1 for wave travelling. Another category uses periodic structures. The experiences of both categories of slow-wave systems are very useful for the topic of this paper and could serve even as a part of the fundamentals of the topic. However, for metalrdielectric multilayer nanostructures, the characteristic size of the structural component could be much smaller than that for an usual slow-wave system and up to as small as a nanometer size range Žor low-dimensional.. In that case, different physical mechanisms and size effects intrinsic for nanocomposite systems should be taken into account during nanostructure efficiency designing. 2.6. Size effect of constituti¨ e characteristics for lowdimensional materials Besides energy conversion mechanisms, the constitutive characteristics of materials play an important role in nanostructure designing as well. It is known for a long time that electrical conductivity of ultrathin metal film deviates from its bulk value. This phenomenon has been described as the classical size effect due to the surface scattering and grain boundary scattering of electrons when the film thickness is in the same order of magnitude with that of the mean free path of the electron w29,30x. When the film thickness is in the same order of magnitude with that of the De-Brogie wavelength of electrons in the matter, the phenomenon has been described as the quantum size effect w31x. On the other hand, the results of a recent theoretical analysis has shown w13x that of a nanowire depends on the local position when the radius of the nanowire decreases to a level comparable with the mean free path of an electron in it. In that case, the r⬘ ratio increases steeply with the decreasing radius of the nanowire.
105
3. Experimental results on the size effect of constitutive characteristics of low-dimensional materials 3.1. Experimental details All the above-mentioned results published in journals on the size-effect, involve only electric conductivity of ultrathin aluminium film. To identify the generality of the size effect of constitutive characteristics of lowdimensional materials, more types of characteristics have been involved, such as electrical conductivity , optical constants n, k, permittivity and permeability . The waveband of measurement has been greatly extended, including visible light, infrared and microwave. Aluminium, Ti and Fe were used as the metal layer materials, while AlN was used as the spacer. The dc electrical conductivity of the films has been measured in-situ in the plating chamber of the magnetron sputtering apparatus by using a dc Wheatstone bridge from outside with a lead from the film sample to the instrument through a vacuum seal. Sheet resistances of Al, Ti, Fe film samples have been measured at first. Then, data of dc electrical conductivity were calculated using the data of sheet resistance and thickness measurement for Al, Ti, Fe films, respectively w8,10x. Optical transmittance, reflectance and absorptance in the visible light region for Al, Ti, Fe films have been measured by using a WFZ900-D4 UVrVIS spectrophotometer w7,12,15x. Similarly, transmittance, reflectance and absorptance in the infrared ŽIR. region for Al, Ti, Fe films have been measured by using a WQF-300 Fourier IR spectroscope w7,12,15x. Their optical functions n and k in the visible light and the infrared ŽIR. regions are obtained by the Newton᎐Simpson recurrent method w7,12,15x. The real and imaginary parts of permittivity ⬘ and Y in the visible light and the infrared ŽIR. regions have been calculated using the following relations w7,15x: s n2 y k2
Ž 31 .
s 2 nk
Ž 32 .
Measurement of ⬘, Y , ⬘, Y for both Al and Fe films at a frequency of 10 GHz have been completed by a Wiltron 560A scalar network analyzer in combination with a Wiltron 6647A sweep generator in microwave bands in the College of Microwave Engineering of the Northwest China University of Technology w16x. The dependence of carrier density in ultrathin Al films on the thickness of the film were obtained through Hall-effect measurement in the Chinese University of Hong Kong w20x. The room temperature measurement was completed by using a Bio-Rad HL5500 PC Hall-
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L.S. Wen et al. r Surface and Coatings Technology 130 (2000) 100᎐109
Fig. 3. Dependence of dc electrical conductivity on the thickness of ultrathin Al, Ti and Fe films.
Effect Measurement System. The cryogenic measurement has been made with a self-assembled PCcontrolled measurement system consisting of a CTL Cryostat, Lake Shore 330 temperature controller and electromagnet.
Fig. 4. Dependence of optical constant n and k in the visible region on the thickness of ultrathin Al, Ti and Fe films.
3.4. Size effect of optical functions and permitti¨ ity in the infrared region
3.2. Size effect of dc electrical conducti¨ ity Results of dc electrical conductivity ŽFig. 3. w8᎐10x show the common trend as indicated in references w28,29x that for ultrathin Al, Ti, Fe films, decreases with decreasing d. However, the character of the curves is quite different from each from other. The curve of Al is typical for a metal with high . It begins to deviate evidently from that of bulk Ti at a thickness of the film of approximately 100 nm, and drops steeply at approximately 10 nm. The curve of Ti is typical for a metal with moderate . It deviates evidently from that of bulk Al at a thickness of the film in the micrometer range and drops quickly from approximately 25 nm. The curve of Fe deviates evidently from the bulk value at approximately 150 nm and drops quickly from approximately 20 nm.
Results of optical functions and permittivity in the infrared region for Al, Ti, Fe films ŽFigs. 6 and 7. show also a complex size effect w7,12,15x. Among them, k ŽTi., k ŽFe. show a monotonic decrease in the nanometer size region, while nŽAl., nŽTi., nŽFe., k ŽAl., ⬘ ŽAl., ⬘
3.3. Size effect of optical functions and permitti¨ ity in the
¨ isible light region
The results of optical functions and permittivity in the visible light region for Al, Ti, Fe films ŽFigs. 4 and 5. show a complex size effect w7,12,15x. Among them, k ŽTi., k ŽFe. show a monotonic decrease in the nanometer size region, while nŽAl., nŽTi., nŽFe., k ŽAl., ⬘ ŽAl., ⬘ ŽTi. , ⬘ ŽFe., Y ŽAl., Y ŽTi., Y ŽFe. give a maximum in the nanometer size region.
Y
Fig. 5. Dependence of permittivity ⬘ and in the visible region on the thickness of ultrathin Al, Ti and Fe films.
L.S. Wen et al. r Surface and Coatings Technology 130 (2000) 100᎐109
Fig. 6. Dependence of optical constant n and k in the IR region on the thickness of the film for Al, Ti and Fe ultrathin films.
ŽTi. , ⬘ ŽFe., Y ŽAl., Y ŽTi., Y ŽFe. give a maximum in the nanometer size region. 3.5. Size effect of permitti¨ ity and permeability in the microwa¨ e band Results of the measurement of permittivity and permeability in the microwave band at a frequency of 10 GHz for both Al and Fe films show a monotonically increasing ⬘ and Y with decreasing d, while the
107
Y
Fig. 8. Dependence of permittivity ⬘ and in the microwave band Ž10 GHz. on the thickness of the film for ultrathin Al, Ti and Fe films.
curves of ⬘ and Y for Ti films have a more complicated size effect dependence d with a maximum at thickness approximately 25 nm ŽFig. 8. w7,15x. The dependence curves of ⬘ and Y of Fe on d are even more complicated with oscillations consisting of a minimum at approximately 18᎐20 nm and a maximum at approximately 31᎐33 nm ŽFig. 9. w7,15x. 3.6. Size-effect of carrier density and plasma frequency Dependence of carrier density in ultrathin Al films
Y
Fig. 7. Dependence of permittivity ⬘ and in the IR region on the thickness of the film for Al, Ti and Fe ultrathin films.
Y
Fig. 9. Dependence of permeability ⬘ and in the microwave band Ž10 GHz. on the thickness of the film for ultrathin Fe films.
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Fig. 10. Dependence of carrier density on the thickness of the film for ultrathin Al films.
on d reveals a strong oscillation of carrier density With d in the nanometer size region with minimums at film thicknesses of approximately 15 and 26 nm and maxima at 18.5 and 37 nm ŽFig. 10. w7,15x. Based on the data of carrier density, dependence of plasma frequency P on d has been calculated using plasma frequency formula Ž7.. The results demonstrate also a strong oscillation of P in a film thickness range of 15᎐40 nm ŽFig. 11. w15x.
EMR, transmission effect of low-dimensional conducting film, modes of EMR, plasma model, efficiency of the EMR mode, and size effect of constitutive characteristics of low-dimensional materials, are explained. Four potential EMR modes are discussed. They are plasmon, interface plasmon, ferromagnetic resonance, and spin wave resonance. Quantitative investigation of the modes of EMR and size effect of constitutive characteristics would be of primary importance for clarification of the principles of nanostructure designing and data collection at the present time. In the range of traditional materials science, the constitutive characteristics should not depend upon the size of the sample. That is true for bulk materials when its characteristic size is in the macroscopic range. However, size effect of constitutive characteristics will become natural, evident and could not be ignored for low-dimensional and nanostructured materials when their characteristic size decreases to the size range of the effective zone of local field in matter, for example, diameter of the Lorentz circle. The size effect of constitutive characteristics gives a challenge for traditional materials science. But it brings a vast possibility of new materials designing principles for nanostructured materials at the same time. The unique feature and advantage of nanostructured materials consist, therefore, in their size effect. The quantitative size effect relationship of constitutive characteristics could become in future a crucial part of the basis for nanostructured materials designing.
4. Discussion A theoretical framework has been suggested based on the recent results of a study on microscopic fundamentals of metalrdielectric nanomultilayers. The principles of its main points including the prerequisite for
5. Conclusion For metalrdielectric nanocomposite multilayers, the following points of view on their microscopic fundamentals could be concluded: 1. A framework of microscopic fundamentals for electromagnetic functional metalrdielectric nanocomposite multilayer films has been suggested. 2. A mechanism of plasmon, interface plasmon, ferromagnetic resonance and spin wave resonance has been discussed as for potential EMR modes with application perspectives. 3. It has been determined experimentally that the size dependence of constitutive characteristics is one of the unique features of low-dimensional materials. 4. Different size effect behaviors of constitutive characteristics provide new possibilities for nanostructured materials designing. References
Fig. 11. Calculated dependence of plasma frequency P on the thickness of the film for ultrathin Al films.
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