On the generalized Hartman effect presumption in semiconductors and photonic structures
© Simanjuntak and Pereyra; licensee Springer. 2013
Received: 26 July 2012
Accepted: 7 March 2013
Published: 28 March 2013
We analyze different examples to show that the so-called generalized Hartman effect is an erroneous presumption. The results obtained for electron tunneling and transmission of electromagnetic waves through superlattices and Bragg gratings show clearly the resonant character of the phase time behavior so that a generalized Hartman effect is not expected to occur. A reinterpretation of the experimental results in double Bragg gratings is proposed.
with T2 and T the double- and single-barrier transmission coefficients, respectively, k the wave number, ω the frequency and A i , A r , F, and G simple functions of the potential parameters (P. Pereyra and H. P. Simanjuntak, unpublished work). Despite this clear dependence on L, involved and contradictory arguments lead to establish that τ becomes independent of L[8, 10, 11]. In the following we will consistently use a for the separation between barriers.
For the inference of a generalized Hartman effect to be meaningful for multi-barriers, double superlattices (SLs) or double Bragg gratings (BG), one would of course need to keep the physical parameters [like the energy (wavelength) of the particle (wave)] fixed as the length between barriers is increased. The tunneling and transmission times behavior should be taken with care when one tries to find a Hartman effect due to barrier separation in multi-barrier systems [8, 11] since, in general, the density of resonance energies grows rapidly as the separation increases. It is well known that the non-resonant gaps in the band structure of a SL or a BG become resonating when these systems are divided and separated; and the separation is increasingly varied. This was already recognized in  (for double SL) and in  (for double BG). On the other hand, it is well known that the tunneling time follows the resonant band structure [7, 16]. Thus, it is not possible to keep increasing the separation between barriers and superlattices without crossing resonances. For this reason, visualized here with specific examples for electrons and electromagnetic waves, the existence of a generalized Hartman effect is a rather questionable issue. For these examples we perform first principle calculations using the actual transmission coefficient of the system (such as that of double BG in the experiment in ) so that we can justify completely that the so-called generalized Hartman effect is erroneous.
To study the Hartman effect and to criticize the presumption of a generalized Hartman effect in superlattices, Bragg gratings, and multi-barrier systems, we will use the theory of finite periodic system that allows straightforward calculation of the phase time. For electron tunneling, we shall assume periodic and sectionally constant potentials with cells of length ℓ c =a+b and a barrier of width b and strength V o in the middle. For electromagnetic waves, each cell consisting of dielectrics 1 and 2 will contain a dielectric 2 of length b in the middle. In this case ϵ i , n i , and μ i (with i=1,2) are the corresponding permittivities, refractive indices, and permeabilities; the regions outside the SL are assumed to be air. For Bragg gratings, the refractive indices are periodic.
The tunneling time in Equation 2 is exact and general and valid for arbitrary number of cells, barrier width, and barrier separation. Thus, one can check the existence or not of a (generalized) Hartman effect at will. For concrete examples, we consider superlattices like (GaAs/Al0.3Ga0.7As) n /GaAs, with electron effective mass mA=0.067 m in GaAs layers, mB=0.1 m in Al0.3Ga0.7As layers (m is the bare electron mass) and Vo=0.23 eV, and Bragg gratings with periodic refractive index.
Results and discussion
with and . When m A , m B and Vo are taken as fixed parameters, we choose a=100 Å and b=30 Å.
This is the well-known Hartman effect. Since this quantity becomes also independent of the barrier separation [8, 11]a, it has been taken as the analytical evidence of a generalized Hartman effect. However, such an approximation that leads to the independence on a and n is obtained by taking the limit of large b first that is strictly speaking infinite, which makes the first barrier the only one that matters for the incoming wave to penetrate while the rest of the SL is immaterial. This was also pointed out by Winful . However, Winful  used an approximation: The transmission of the double square barrier potential to model the transmission through the double BG. Here, we present calculations using the actual transmission coefficient through the double BG. As mentioned before, for the generalized Hartman effect to be meaningful, it should not matter whatever limit we take first whether on a, b, or n. It turns out that a non-resonant energy region becomes resonant as the separation a increases (see the discussion on the double Bragg gratings in section ‘Hartman effect in two Bragg gratings systems’).
The Hartman effect and the electromagnetic waves
Hartman effect in two Bragg gratings systems
Using parameters of Longhi et al.  for n0,n1, kB, and L o , the non-resonant gap becomes resonant as the gratings separation increases. Though details are beyond the purpose of this paper, we plot in Figure 6 the PT as a function of the separation a for incident-field wavelength λ=1542 nm, and as a function of the frequency ω, for a=42 mm. Recall that in , λ≃1,550 nm was considered. While the PT appears completely in graph (c), in (b) it is plotted in a different range to compare with the experiment. The resonant behavior of the PT with a and the absence of any generalized Hartman effect are evident. Similar results are obtained when λ=2Π/k B .
We have shown that the presumption of generalized Hartman effect for tunneling of particles and transmission of electromagnetic waves is not correct.
The authors would like to thank Professor Norman H. March for comments and suggestions on the manuscript.
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