Tunneling times of acoustic phonon packets through a distributed Bragg reflector
© Lazcano et al.; licensee Springer. 2014
Received: 10 May 2014
Accepted: 12 August 2014
Published: 29 August 2014
The longwave phenomenological model is used to make simple and precise calculations of various physical quantities such as the vibrational energy density, the vibrational energy, the relative mechanical displacement, and the one-dimensional stress tensor of a porous silicon distributed Bragg reflector. From general principles such as invariance under time reversal, invariance under space reflection, and conservation of energy density flux, the equivalence of the tunneling times for both transmission and reflection is demonstrated. Here, we study the tunneling times of acoustic phonon packets through a distributed Bragg reflector in porous silicon multilayer structures, and we report the possibility that a phenomenon called Hartman effect appears in these structures.
KeywordsDistributed Bragg reflector Tunneling time Hartman effect
Phonons, the quanta of lattice vibrations, manifest themselves practically in all electrical, thermal, and optical phenomena in semiconductors and other material systems. The reduction of the size of electronic devices below the acoustic phonon mean free path creates a new situation for phonon propagation and interaction, opening up an exciting opportunity for engineering phonon spectrum in nanostructured materials . Since the early work of Narayanamurti et al. , important progress has lately emerged in the development of nanowave phononic devices including, e.g., mirrors, cavities, and monochromatic sources.
How long does it take for a particle to tunnel through a potential barrier? This is a question that has occupied physicists for decades and one for which there is still no definitive answer . The Hartman effect (HE) states that the tunneling time becomes independent of the barrier length . The independence of tunneling time on barrier length would imply arbitrarily large and indeed superluminal velocities for tunneling wave packets, if tunneling was in fact a propagation phenomenon.
Phonon tunneling studies have also revealed phenomena related to the HE. Recent experiments on tunneling acoustic waves have reported the breaking sound barrier [5, 6]. Yang et al. found that inside a phononic band, the group velocity increases linearly with the sample thickness, a rather remarkable effect that is a signature of tunneling in quantum mechanics [6, 7]. Villegas et al. have discussed the physical conditions under which the tunneling time for long-wavelength phonons through semiconductor heterostructures is independent on the system’s size, i.e., the effect equivalent to the HE for electrons . Experimental studies of HE in the context of the nanophononics have been carried out [9, 10]. In these works, very short transit times in the stop bands have been measured, one acoustic equivalent of HE of electron tunneling through potential barriers.
During the last decade, interest in achieving all-silicon-based opto- and microelectronics was highly stimulated by the discovery of the unique optical properties of porous silicon . Porous silicon is known as a versatile material with applications in light emission, sensing, and photonic crystal devices. It is well known that the introduction of artificial spatial periodicity in the elastic properties of a system results in Brillouin zone folding. Such folding is often accompanied by the appearance of bandgaps in the phonon frequency spectrum [12, 13]. In the last few years, this interest has been translated to porous silicon-based phononic systems [14–17]. Here, we study the tunneling times of acoustic phonon packets through a distributed Bragg reflector in porous silicon multilayer structures.
The paper is organized as follows: The ‘Methods’ section provides some fundamentals in both the long-wavelength model and the transfer matrix method. The main theoretical findings are presented in the ‘Results and discussion’ section. In particular in this section, from general principles such as invariance under time reversal, invariance under space reflection, and conservation of energy density flux, the equivalence of the tunneling times for both transmission and reflection is demonstrated. At the end of the paper, the main conclusions are given.
where Z j =ρ j v j is the acoustic impedance, ρ j the mass density, v j the velocity of sound, and d j the layer width.
Results and discussion
Equivalence of the transmission and reflection times in the tunneling of long-wavelength phonons
Invariance under time reversal
provided only that β(z) and ω Γ (z) are real functions. Observe that this equation has the same form as (3). Therefore, if u(z,t) is a solution of (3), then u∗(z,−t) is also a solution. u∗(z,−t) is often referred as the time-reversed solution. The behavior of the wave equation exhibited by (9) is called invariance under time reversal. For the stationary state, invariance under time reversal implies that if u(z) is a stationary-state wave function, then u∗(z) is also one.
Conservation of energy density flux
Invariance under space reflection
Falck et al. have obtained a similar result for symmetric scattering potentials .
Tunneling of acoustic phonons through a distributed Bragg reflector
The phonon modes with energies within the first minigap appear marked by the black solid line in Figure 3b,c. We observe that at the frequency f C ≅0.70 GHz, the amplitude of the wave shows an abrupt decay along the axis of the DBR. This qualitative behavior is very similar for the other physical quantities.
being vf the phase velocity.
In this paper, we studied tunneling times of acoustic phonon packets through a distributed Bragg reflector made of porous silicon layers. Under the assumption that the long-wavelength approximation is valid, and from general principles of symmetry and conservation, we report an explicit demonstration of the equivalence of the transmission and reflection times in the tunneling of long-wavelength phonons. Calculations of the vibrational energy density and the vibrational energy stored within the structure allows a better visualization of the physical phenomena occurring in this system. The description of the stress and strain fields complements the energetic description. We report the possibility that a phenomenon called Hartman effect appears in porous silicon multilayer structures, an acoustic equivalent of Hartman effect of electrons tunneling through potential barriers. The results of this study could be useful for the design of acoustic devices.
DV and RPA acknowledge the hospitality of Benemérita Universidad Autónoma de Puebla, México. DV acknowledges the support of S.R.E. (México). JA acknowledges Conacyt, México for the partial support under grant no. 167939.
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