Surface Science Reports 63 (2008) 391–399
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Particle cross transfer Leonard Dobrzyński ∗ , Abdellatif Akjouj Centre National de la Recherche Scientifique, Université des Sciences et Technologies de Lille, Institut d’Electronique, de Microélectronique et de Nanotechnologie, Unité de Physique, Bâtiment P5, 59655 Villeneuve d’Ascq Cédex, France
article
info
Article history: Accepted 24 April 2008 editor: W.H. Weinberg Keywords: Particle Electron Phonon Transmission System
a b s t r a c t Particle and wave-packet duality lies in the heart of modern physics. A particle is a fermion or a boson. The electron is considered in this tutorial as a generic example of a fermion. Phonons represents the boson particle family. A phonon is a vibration quantum. Because of particle-wave duality, each particle can be described by a wave and each wave by a particle. The particle and the wave descriptions are complementary and equivalent, at least within the contents of this paper. The particle cross transfer effect avoids collisions between two identical particles travelling one towards the other by transferring them simultaneously from one incident guide to an output guide. This transfer leaves the propagation of all other incident particles contained in a given wave-length or energy window unperturbed. The corresponding system can be conceived out of finite mono-mode wires. A mono-mode wire accommodates only one particle within its lateral dimensions. The particle cross effect opens perspectives for new, more compact and more efficient devices. This review displays the simplest possible examples of such a system. © 2008 Elsevier B.V. All rights reserved.
Contents 1. 2.
3. 4.
5.
Introduction........................................................................................................................................................................................................................391 The theoretical language ...................................................................................................................................................................................................392 2.1. The language for systems conceived out of discrete dots ...................................................................................................................................392 2.2. The language for systems conceived out of continuous wires............................................................................................................................392 2.3. The language for systems conceived out of quantum dots and continuous wires ............................................................................................393 2.4. Summary ................................................................................................................................................................................................................393 Electron cross transfer in a system conceived out of continuous quantum wires ........................................................................................................393 3.1. Summary ................................................................................................................................................................................................................395 Phonon cross transfer ........................................................................................................................................................................................................395 4.1. Phonon cross transfer in a system conceived out of two dots and two continuous wires ...............................................................................395 4.2. Summary ................................................................................................................................................................................................................396 4.3. Phonon cross transfer in a system conceived out of dots ...................................................................................................................................397 Perspectives ........................................................................................................................................................................................................................399 References...........................................................................................................................................................................................................................399
1. Introduction The particle cross transfer effect avoids a collision between two identical particles travelling one towards the other by transferring them simultaneously from one incident guide to an output one. It leaves the propagation of all the other incident particles contained in a given wave-length or energy window unperturbed. The corresponding particle cross system is mono-mode when it is built out of mono-mode wires. This means that only one particle
∗
Corresponding author. E-mail address:
[email protected] (L. Dobrzyński).
0167-5729/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.surfrep.2008.04.001
mode can propagate in each elementary wire. This can be achieved when the two lateral dimensions of the wire are small compared with the particle wavelength. This report tries to be a comprehensive one, as well for undergraduate, graduate students and researchers. It reviews the Dobrzynski et al. original papers which first reported the particle cross transfer effect [1–3]. In what follows, is given first a chapter on the general theoretical language used in this tutorial. The most efficient way to understand a language is practice and study of the simplest possible examples. So most of the readers of this short tutorial will gain to read first the three simple illustrative examples which
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follow the language chapter. If this is still too abstract, the reading and studying of Akjouj, Sylla and Dobrzynski lecture notes full of much simpler examples of particle propagation in lamellar materials may be helpful to those interested to redo the analytical work reviewed in the present report [4]. A system exhibiting the particle cross transfer effect can be used as a multiplexer. A multiplexer is a system enabling to transfer one particle from one guide to another one. It is selective when only one type of particle is transferred, leaving the propagation of the other particles in the input guide unperturbed. Such transfer processes are particularly important in wavelength multiplexing and in telecommunication routing devices, see for example the pioneering achievements by Eugster and del Alamo; Haus and Lai; Orlov, Yariv and van Essen; and Fan et al. [5–8]. Some perspectives of further research on particle cross transfer systems are sketched at the end of this report. 2. The theoretical language The theoretical language used in this review paper is the Dobrzynski’s interface response theory one [9–12]. In what follows, is presented a summary of its basic concepts and equations. 2.1. The language for systems conceived out of discrete dots Consider a composite system contained in its space of definition D and formed out of N different constituents situated in their domains Di (1 ≤ i ≤ N). In this paper the constituents are quantum dots or finite wires made out of interacting quantum dots. In order to analyze such a composite system in closed form, it is assumed in a first approximation that each dot is a point with one single eigen energy. Then each dot wire has only one space dimension. This approximation is valid when the propagating particle wavelength is much greater than the lateral dimensions of realistic dot wires. Each constituent is bounded by an interface Mi , interacting in general with J other constituents through sub-interface domains Mik , (1 ≤ k ≤ J). The ensemble of all these interface spaces Mi is called the interface space M of the composite system. Consider now an infinite homogenous dot wire i described by an infinite matrix and a linear hamiltonian
[(E + jε)I − Hi ].
(1)
E is√the energy, Hi a linear hamiltonian, I the identity matrix, j = −1 and ε an infinitesimally positive small number. The inverse of this matrix is called the corresponding Green’s function Gi and
[(E + jε)I − Hi ]Gi = I .
(2)
and in the interface space M
∆(MM ) = I (MM ) + A(MM ).
(5)
The elements of the Green’s function g (DD) of a composite material are the elements of the matrix g (DD). This matrix is calculated as follows g (DD) = G(DD) − G(DM )[∆(MM )]−1 A(MD).
(6)
The new interface states of the composite system can be calculated from det[∆(MM )] = 0.
(7)
So the knowledge of ∆(MM ) in the interface space M is sufficient for the discovery of new interface states in a novel composite system, if such states exist. Moreover, if U (D) represents an eigenvector of the reference system formed by all the infinite materials i, Eq. (6) enables one to calculate the eigenvectors u(D) of the composite material u(D) = U (D) − U (M )[∆(MM )]−1 A(MD).
(8)
In Eq. (8), U (D), U (M ) and u(D) are row-vectors. Eq. (8) enables one to also calculate all the waves reflected and transmitted by the interfaces as well as the reflection and the transmission coefficients of the composite system. For this purpose, U (D) is a bulk wave launched in one homogeneous constituent of the composite system. 2.2. The language for systems conceived out of continuous wires Consider now linear differential hamiltonian and constituents made out of continuous wires. The elements of the Green’s function g (DD) of such a composite system can now be obtained from g (DD) = G(DD) − G(DM )[G(MM )]−1 G(MD)
+ G(DM )[G(MM )]−1 g (MM )[G(MM )]−1 G(MD).
(9)
G(DD) is the block diagonal Green’s function of the reference system. The elements of g (MM ) are the interface elements of the Green’s function of the composite system. The inverse [g (MM )]−1 ofSg (MM ) is obtained for any points in the interface space M = Mi as a superposition of the different [gi (Mi , Mi )]−1 , being the inverse of the [gi (Mi , Mi )] for each constituent i of the composite system. The latter quantities are given by the equation
[gi (Mi , Mi )]−1 = ∆i (Mi , Mi )[Gi (Mi , Mi )]−1 ,
(10)
where
∆i (Mi , Mi ) = I (Mi , Mi ) + Ai (Mi , Mi ),
(11)
A finite dot wire is now cut out of the infinite one, with the help of a cleavage operator Vi . This finite dot wire is defined in its space Di and has two free surfaces. Vi is defined in its interface space Mi . Define Asi as the truncated part within Di of
with I being the unit matrix,
Ai = Vi Gi .
and x0 , x00 ∈ Di and x ∈ Mi . In Eq. (12), the cleavage operator V0i acts only in the surface domain Mi of Di and cuts the finite or semi-infinite size block out of the infinite homogeneous medium. Ai is called the surface response operator of block i. The new interface states can be calculated from
(3)
In the same manner, Gsi is constructed out of the truncated part of the Gi . Define then block diagonal matrices G and As by juxtaposition of respectively all the Gsi and Asi defined for N different homogenous constituents i. A composite system is then constructed by assembling such finite constituents with the help of a coupling operator VI defined in the whole interface space M. Define then in the whole space D of the composite the matrices A = As + VI G
(4)
Ai (x, x ) = 0
Z
V0i (x00 )Gi (x00 , x0 )dx00 ,
det[g (MM )]−1 = 0,
(12)
(13)
showing that, if one is interested in calculating the interface states of a composite, one only needs to know the inverse of the Green’s function of each individual block in the space of their respective surfaces.
L. Dobrzyński, A. Akjouj / Surface Science Reports 63 (2008) 391–399
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Moreover, if U (D) represents an eigenvector of the reference system, Eq. (9) enables one to calculate the eigenvectors u(D) of the composite material u(D) = U (D) − U (M )[G(MM )]−1 G(MD)
+ U (M )[G(MM )]−1 g (MM )[G(MM )]−1 G(MD).
(14)
In Eq. (14), U (D), U (M ) and u(D) are row-vectors. Eq. (14) also enables one to calculate all the waves reflected and transmitted by the interfaces as well as the reflection and transmission coefficients of the composite system. In this case, U (D) must be replaced by a bulk wave launched in one homogeneous piece of the composite material. 2.3. The language for systems conceived out of quantum dots and continuous wires How may composite systems build out of quantum dots and continuous wires be analyzed ? Atomic materials have atoms as elementary constituents. So they are discrete and they should be approached with a matrix formulation. However when the wavelength of the particle under consideration is much bigger than the interatomic distances, some continuous approaches can be used. Consider a mixed material build out of continuous quantum wires as defined just above and some quantum dots smaller than the wavelength of the propagating particle and bigger than one single atom. Then such a mixed composite system has to be considered first as an atomic discrete one. In order to avoid analyzing such a composite system with the whole complexity of atomic models, it is possible to replace its continuous quantum wires by equivalent discrete ones, made out of quantum dots. Then the above language conceived for a composite system made out of quantum dots can be applied. The understanding of the physical properties of such composite systems is still in its infancy. 2.4. Summary A general theoretical language is given here above. It is illustrated in what follows by three simple examples. 3. Electron cross transfer in a system conceived out of continuous quantum wires One of the challenges of modern technology is to control electronic transport at the nanometer scale. At this scale length, since the electron mean free path becomes large compared to the size of the device, the majority of the observed features is due to the wave nature of electrons, being responsible for a variety of quantum-interference effects (see, for example Datta’s book, [13]). A system conceived by Dobrzynski [1] avoids collisions between two incoming electrons, heading one towards the other in a monomode input quantum wire. The collision is avoided thanks to a composite quantum system. This system cross transfers the two input electrons into an output wire. Selective transmission of an electron, from one quantum wire to another one, attracts attention because of its fundamental interest and practical importance, see for example the del Alamo et al. [14], Eugster et al. [5], Dagli et al. [15], Miskovic et al. [16], Dobrzynski et al. [17],. . . papers. To be more specific, tunable directional transfer from one waveguide to another is particularly important for signal multiplexing in various routing devices and as a potential electron-spectroscopy tool. The growing interest in this kind of system can also be attributed to the advancement in modern semiconductor technology that makes it possible to
Fig. 1. Sketch of the considered electron system. Proportions between the distances within the device correspond to the particular system studied in Fig. 2, assuming all the wires are 10 nm wide.
fabricate nanometer structures with a sufficient degree of control of their geometry and chemical composition. Accordingly, a resonant-coupling structure, built of semiconductor nanometer wires, which—under certain conditions— realizes the directional transfer of an electron with good selectivity is of great interest. More precisely, such a structure lets the electrons of every-but-one energy (from a certain energy window) propagate without perturbation from one input gate to an output gate, while transmitting one electron of a pre-selected and welldefined energy to a different output gate. This review stresses that by linear superposition, two electrons, having the same energy, simultaneously entering through two distinct input gates, can be cross-transferred to two distinct output gates. Moreover the simple system reviewed here enables one to obtain, with the help of the above theoretical language, closedform expressions for its characteristic wire lengths. It is then easy to determine all the parameters necessary for its tuning. The system under consideration is sketched in Fig. 1. All the wires are mono-mode semiconductor quantum guides. For an illustrative example, Gallium Arsenide quantum wires are taken. For symmetry purposes, the distance d0 between nodes 1 and 2 is the same as that between nodes 3 and 4, while the rest of the device is built out of only two distinct constituents of the same semiconductor, namely, finite wires of length d1 and d2 . More precisely, four identical single-stub structures, each consisting of a wire of length d2 grafted in the middle of another wire of length 2d1 , are connected between nodes 1 and 5, 5 and 4, 2 and 6, and 6 and 3, respectively. Such side-branch structures are known to exhibit transmission zeros at the eigen energies of the grafted resonator, see for example Shao et al. paper [18]. Between nodes 5 and 6, another double-stub structure of length 3d1 is stuck, with two wires of length d2 branched at equidistant positions. Owing to such simple building constituents of the system, there are only three parameters, namely, wire lengths d0 , d1 , and d2 . An incident electron injected into the coupling structure, for example from input gate 1, may be, as a result of scattering processes, reflected at gate 1 or ejected out of the multiplexer through gates 2, 3 or 4 (see Fig. 1). The corresponding reflection (t11 ) and transmission (t1n , n = 2, 3, 4) amplitudes can be expressed in terms of the elements of the Green’s function g of the system via [17] t11 = 2iFg (1, 1) − 1
(15)
and t1n = 2iFg (1, n)
(n = 2, 3, 4).
(16)
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√ In Eqs. (15) and (16), i = −1 stands for the imaginary unit, while 2
F =
h¯
2m∗
α
(17)
which, via Eqs. (28)–(31) enables one to determine the characteristic distances within the coupling structure. Define now the incoming electron probabilities of presence Ij = |ΨIj (E )|2
is related to the electron eigenvector α defined as 1√ ∗ α= 2m E . h¯
(18)
E and m∗ being respectively the electron energy and its effective mass. The necessary Green-function elements may be obtained taking into account the symmetry of the system. Consequently, for an electron incoming through gate 1, expressions for the reflection and transmission wave function amplitudes can be conveniently written as t11 = z1 + z2 + z3 + z4 − 1,
(19)
t12 = z1 + z2 − z3 − z4 ,
(20)
t13 = z1 − z2 + z3 − z4 ,
(21)
(j = 1, 2),
(36)
where ΨIj (E ) is the wave function of the incoming electron. In the same manner, we define Oj = |ΨOj (E )|2
(j = 1, 2, 3, 4),
the transmission probabilities at the output gates. Now, for a single input at gate 1, define a quality factor Q characterizing the selectivity of the directional forward transfer from gate 1 to gate 3 at the energy E0 by Q =
E0 2(E 0 − E0 )
,
(38)
such that O3 (E 0 ) = 21 I1 (E 0 ). With good accuracy, the quantum wire lengths should satisfy
π α0 d0 = (1 + 4n0 ) ,
(39)
2
and t14 = z1 − z2 − z3 + z4 ,
(22)
where
(37)
π α0 d1 = (1 + 4n1 ± δ) ,
(40)
2
and (23)
π α0 d2 = (2n2 ∓ δ) ,
(24)
δ=
y2 = A0 + B0 + A5 ,
(25)
y3 = A0 − B0 + A5 ,
(26)
is supposed to be small compared to one. Note that the positive value of δ in the expression for d1 corresponds to the negative value of δ in the expression for d2 and vice versa. Thus, Eqs. (39)–(42) enable one to estimate the wire lengths for the desired device operation, once the directional-transfer energy E0 and the corresponding quality factor Q are chosen. Note that several solutions for d0 , d1 , and d2 are possible, due to the different available values of integers n0 , n1 , and n2 . The performance of the proposed device can be tested by considering a structure build of Gallium Arsenide wires. Choosing E0 = 0.05 electron volt and Q = 500, the distances within the coupling structure are determined from Eqs. (39)–(42)) and rounded up or down to full nanometers, which results in d0 = 90 nm, d1 = 47 nm, and d2 = 43 nm. These wire lengths are next used to compute the reflection and transmission probabilities as a function of energy through the exact Eqs. (19)–(22). Kucharczyk et al. paper [19] gives these results, see Fig. 2 where the resulting transmission probability curves are presented. The expected sharp peak in the directional forward-transferred signal O3 (E ) (solid line), as well as the corresponding dip in the directtransmission curve O2 (E ) (dashed line) can be clearly identified. Moreover, both the initially selected energy E0 of the resonantly transferred electron and the quality factor Q characterizing the linewidth of the O3 (E ) peak are well-reproduced in numerical calculations. Additionally, the plateau of high values of O2 (E ) around E0 indicates the above-discussed energy window of electrons propagating without perturbation along the input wire. Finally, the directional backward-transferred signal O4 (E ) (dasheddotted line) is negligible, while the reflected signal O1 (E ) (dotted line) remains small over the considered energy range. Now, when two electrons are simultaneously transferred, one from gate 1 to gate 3 and the other from gate 2 to gate 4 (see Fig. 1), the output transmission probabilities are
zn =
i
(n = 1, 2, 3, 4),
2(i + yn )
with y1 = y2 −
2B25 2A5 + A6 + B6
,
and y4 = y3 −
2B25 2A5 + A6 − B6
,
(27)
where A5 = A1 − B5 = −
B21 2A1 + A2 B21
2A1 + A2
A6 = A1 −
1 2
,
(28)
,
(29) B21
2A1 + A2 − B1
+
B21
2A1 + A2 + B1
,
(30)
and B6 =
1 2
B21
2A1 + A2 − B1
−
B21 2A1 + A2 + B1
,
(31)
while Aj = −[tan(α dj )]−1
(32)
and Bj = [sin(α dj )]−1
(j = 0, 1, 2).
(33)
For a transfer of one electron from gate 1 to gate 3, one must satisfy the conditions A5 = A6 = 0
(34)
and B25 = B0 B6 ,
(35)
2 where nj (j = 0, 1, 2) are integers, while
3(1 + 4n1 + 2n2 )
1/2
2π Q
O1 = O2 = |2(z1 + z2 ) − 1|2
,
(41)
(42)
(43)
L. Dobrzyński, A. Akjouj / Surface Science Reports 63 (2008) 391–399
Fig. 2. The transmission probabilities O3 (kd) (solid line), O2 (kd) (dashed line), O4 (kd) (dotted line) and O1 (kd) (dotted-dashed line) as a function of electron energy e for the structure of Fig. 1 for one single input I1 = 1.
395
Fig. 4. Sketch of one possible geometry of the considered simple phonon system. It consists of two wires and two dots of mass M. The two dots are bounded to two fixed points by two interactions K and are restricted by an appropriate groove to move only in the direction parallel with the wires. They are bound also between themselves by an interaction β2 and to points (1−4) of the wires by four interactions β1 . One input I1 or two inputs I1 = I2 of longitudinal acoustic waves and four outputs O1 , O2 , O3 and O4 are considered.
4.1. Phonon cross transfer in a system conceived out of two dots and two continuous wires
Fig. 3. Output-signal intensities O1 = O2 (dashed line) and O3 = O4 (solid line) as a function of energy for the structure of Fig. 1 with d0 = 90 nm, d1 = 47 nm, and d2 = 43 nm, when two input electrons are simultaneously present.
and O3 = O4 = |2(z1 − z2 )|2 .
(44)
In other words, two electrons of particular energy E0 are crosstransferred through the electron cross transfer system to gates 3 and 4, respectively. This effect is illustrated in Fig. 3. Moreover, the derived closed-form expressions for the characteristic wire lengths within the coupling structure prove useful in finding the optimal parameters for the desired system operation. 3.1. Summary The discovery of a simple electron cross transfer effect is reviewed here above. The corresponding system was conceived out of continuous quantum wires. In what follows, a similar phonon cross transfer effect is illustrated on two other examples. The first is conceived out of two clusters and several continuous wires. The second example is conceived out of clusters with dimensions of the order of one nanometer. 4. Phonon cross transfer The directional ejection of one phonon from one wave guide to another is considered by Dobrzynski et al. [20,21]. In what follows, two examples of two phonons cross transferred are reviewed.
A simple system enabling two acoustic phonons to cross transfer simultaneously is now reviewed [2]. Such a structure is constructed out of two mono-mode acoustic wires and two dots bound together and to the wires by harmonic interactions. This simple structure can selectively transfer and in one direction one acoustic phonon from one wire to the other, leaving the propagation of neighboring acoustic phonons unaffected. Closed form relations enable one to obtain the values of the relevant physical parameters for this phenomena to happen at a chosen phonon frequency. An application to a specific phonon cross transfer system illustrates this subsection. A system enabling a directional ejection of one elastic phonon of a given wavelength from one wire to another should leave the other neighboring phonons to travel without perturbation of the input wire. At the same time, this phonon of one selected and welldefined wavelength is expected to be transferred to the other wire with a phase shift as the only admitted distortion. To meet the above requirements as closely as possible an appropriate coupling geometry is designed. A simple system, which under certain conditions, makes the directional transfer of one acoustic phonon possible with very good selectivity is reviewed. The system is depicted in Fig. 4. For the sake of simplicity, only in-plane transverse phonons are injected in the input wire. The wires are characterized by their linear mass density ρl and the speed c of the transverse acoustic phonons propagating along them. Each of the two identical dots is coupled to the motionless support by an interaction K. Both dots are restricted by an appropriate groove to move only in the direction parallel with the wires. The two dots are also coupled together by an interaction β2 . The interactions between the dots and the wires is modelled by four interactions β1 (see Fig. 4). The two input gates are (1 and 2), the output gates for the cross transferred phonons are (3 and 4) and the dots are 5 and 6. The distances between points (1, 2), (3, 4) and (5, 6) are d. The system shows two perpendicular mirror symmetry planes. Moreover the simple system reviewed here can be solved in closed form, with the help of the corresponding theoretical language reviewed above. This enables one to determine all the parameters necessary for its tuning, in order to observe the phonon cross transfer effect.
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Many other systems and geometries of this type can be devised, which differ in some details but show generally the same general effect. The acoustic dispersion relation of the acoustic transverse phonons is
ω = ck,
(45)
where k = 2π /λ is the propagation vector of the acoustic phonon of wavelength λ. Any incident acoustic wave injected in the input gate 1, generates, as a result of scattering processes, a reflected wave at node 1, along with three transmitted waves at nodes 2, 3, and 4 (see Fig. 4). The corresponding reflection (t11 ) and transmission (t1n , n = 2, 3, 4) amplitudes can be expressed in terms of the elements of the Green function g of the system via [20] t11 = 2iFg (1, 1) − 1
(46)
and t1n = 2iFg (1, n)
(n = 2, 3, 4). (47) √ In Eqs. (46) and (47), i = −1 stands for the imaginary unit, while F = c ρl ω.
(48)
The necessary Green’s function elements can be readily obtained, within the above appropriate theoretical language. Consequently, for acoustic waves incoming through gate 1, the expressions for the reflection and transmission wave function amplitudes can be conveniently written via Eqs. (19)–(23), with the yn ’s determined by the particular resonant-coupling structure under consideration [20]. For the structure of Fig. 4 y1 = y2 −
y2 = tan
2β
2 1
F (M ω2 − K − 2β1 ) kd
−
2
y3 = − tan
kd
β1 F
(49)
, −
β1 F
,
(51)
2β12 F (M ω − K − 2β1 − 2β2 )
.
(52)
For an incoming acoustic wave intensity I1 (kd) = 1, the outgoing acoustic wave intensities Oj (kd) are given by j = 1, 2, 3, 4.
(53)
The total acoustic wave transfer from a single input at gate 1 to the output 3, i.e. I1 (kd) = 1, O1 (k0 d) = 0, O2 (k0 d) = 0, O3 (k0 d) = 1 and O4 (k0 d) = 0 can be realized exactly at the angular frequency ω0 = ck0 , when the following conditions are fulfilled
(k0 d)2 =
(d/c )2 M
cos(k0 d) = −
[K + 2β1 + β2 ],
β2 , 2β1
(54) (55)
and sin(k0 d)
(k0 d)
=
ρl c (c /d)β2 . β1 2
(56)
The transferred wave has some width in kd around k0 d. If one wishes the corresponding peak in O3 (kd) to be symmetric, then one obtains another condition [20,21], namely k0 d = (1 + 4n0 )
π 2
,
Q (k0 d) =
k0 d
∆(k0 d)
,
n0 = 0, 1, 2, . . .
(57)
(58)
where ∆(k0 d) is the width of this signal for O3 (kd) = 0.5. An approximate value of this quality factor is found to be Q (k0 d) = (1 + 4n0 )2
2
Oj (kd) = |t1j |2 ,
However this condition and the ones given just above, are only fulfilled for β1 = 0 and β2 = 0. So in what follows a small dissymmetry of the peak in O3 (kd) and a small imprecision on the condition given by Eq. (57) is accepted. Let us also define the quality factor associated with the line width of the transferred signal by
(50)
−1
2
,
and y4 = y3 −
Fig. 5. The transmission coefficients O3 (kd) (solid line), O2 (kd) (dashed line), O4 (kd) (dotted line) and O1 (kd) (dotted-dashed line) as a function of kd for the system of Fig. 4 for n0 = 0, M = 0.001 kg, c = 10 m/s, ρl = 0.07 kg/m, d = 0.02 m, K = 475 kg/s2 , β1 = 110 kg/s2 and β2 = 22 kg/s2 , for one single input I1 = 1.
π 2 M c2 . 4 β2 d2
(59)
To give an illustrative and at the same time realistic example complying with the above assumptions one considers n0 = 0, the mass of the dots M = 0.001 kg, c = 10 m/s, ρl = 0.07 kg/m, d = 0.02 m, K = 475 kg/s2 , β1 = 110 kg/s2 and β2 = 22 kg/s2 . This set of parameters corresponds to typical elastic wires able to transmit transverse acoustic waves. Fig. 5 presents the transmission coefficients O3 (kd) (solid line), O2 (kd) (dashed line), O4 (kd) (dotted line) and O1 (kd) (dotteddashed line) as a function of the reduced wave vector kd. One remarks that the dissymmetry with respect to kd = π /2 is negligible. The peak in the transmission coefficient O3 (kd) shows a width at half maximum of the order predicted by Eq. (59). In this figure O2 (kd) is basically constant and equal to 1 after kd = π /2.This result comes from the parameters used in this calculation, but remains for other possible parameter sets as long as the analytical conditions given above are satisfied with good precision and the chosen quality factor is not too low. Now with two inputs of intensity I1 (kd) = I2 (kd) = 1 at gates 1 and 2, the output transmission probabilities are given by Eqs. (43) and (44). In other words, two transverse acoustic phonons of particular propagation vector k0 are cross-transferred through the structure to gates 3 and 4, respectively. This phonon cross transfer effect is illustrated in Fig. 6. 4.2. Summary This simple system can realize transverse acoustic wave multiplexing and also cross-transfer of two acoustic phonons,
L. Dobrzyński, A. Akjouj / Surface Science Reports 63 (2008) 391–399
Fig. 6. Output-signal intensities O1 (kd) = O2 (kd) (dashed line) and O3 (kd) = O4 (kd) (solid line) as a function of kd for the system of Fig. 4 for the same parameters as in Fig. 5 when two inputs of intensity I1 (kd) = I2 (kd) = 1 at gates 1 and 2 are simultaneously present.
respectively from gate 1 to gate 3 and from gate 2 to gate 4. Moreover, the above derived closed-form expressions enable one to easily find the optimal parameters for the desired system, enabling to tune it at will for specific applications. This simple phonon cross transfer system conceived out of two dots and two continuous wires is expected to stimulate further research. In what follows, the phonon cross transfer effect in a system conceived out of dots is reviewed. 4.3. Phonon cross transfer in a system conceived out of dots A simple phonon cross transfer system can be conceived out of two discrete dot chains and two other dots situated in between these chains. Each dot creates a deformation of the substrate and then an elastic dipole. The corresponding interaction between two dots is those between two elastic dipoles. The dot mass density is assumed to be larger than that of the substrate and the interaction between the dots to be smaller than the corresponding interactions within the substrate. With these assumptions, an acoustic phonon branch exists below the surface Rayleigh wave branch. A system enabling a directional transfer of an elastic wave of a given wavelength from one wire to the other should leave the other neighbor wavelengths travel without perturbation in the input wire. At the same time this wave of one selected and welldefined wavelength is expected to be transferred to the other wire with a phase shift as the only admitted distortion. To meet the above requirements as closely as possible an appropriate coupling geometry should be designed. A discrete phonon cross transfer system is depicted in Fig. 7. This system uses dots [3] rather than atoms [21] or macroscopic bodies and springs [2]. Here the interactions between the clusters are those between the elastic dipoles created by the deformations of the substrate on which the clusters are adsorbed. Indeed it is known that when a body of mass m is deposited on a substrate, its weight causes a deformation of the surface of this substrate and then an elastic dipole force. Such types of substrate mediated interactions exist between any type of clusters adsorbed on any type of substrates, see for example [22]. So in the system taken now as example, a substrate’s existence is required, although similar systems without an substrate could be created with macroscopic bodies. Only a simple general solution
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Fig. 7. Sketch of the geometry of the considered system. It consists of two dot chains and two other dots of mass M bounded to the substrate by a harmonic force constant K 0 . They are bound also between themselves by an interaction β2 and to dots (1 − 4) by interactions β1 . These interactions are due to the elastic dipole interaction between the dots. The elastic dipoles are created by the deformations of the substrate. Consider one input I1 or two inputs I1 = I2 of out of plane acoustic waves and four outputs O1 , O2 , O3 and O4 .
for such a phonon cross transfer system is reviewed here, leaving the choice of specific materials to experimentalists interested in constructing and testing such a system. The structure consists of two dot chains conducting out of plane transverse waves. The chains go respectively through points (1, 2) and (3, 4) and two additional clusters of mass M are at points 5 and 6. The distances between points (1, 2) and (3, 4) are Ld, where L = 1, 2, 3 . . .. The chains are characterized by the cluster mass m and the nearest neighbour force constant β related to these transverse vibrations of the chain. This force constant can be obtained as derivatives of the central energy of interaction between the two elastic dipoles due to the deformation of the substrate. This interaction energy is known to be inversely proportional to the distance d3 between the two clusters. Two identical clusters of mass M are assumed to be coupled to the motionless support by a force constant K 0 . These clusters are also coupled with each other by a force constant β2 inversely proportional to d52 , where d2 is the distance between these two clusters. Coupling of the wave motion in the cluster chains with the motion of the clusters of mass M is assured by four force constants β1 , inversely proportional to d51 , where d1 is the distance between clusters (5, 6) and the clusters (1 − 4), see Fig. 7. The system shows two perpendicular mirror symmetry planes. So the main parameters of this model are the ratios β1 /β and β2 /β between the force constants or equivalently d1 /d and d2 /d between the cluster distances, together with the ratios K 0 /β and M /m. Moreover the simple system presented here can be solved in closed form. The dispersion relation of the localized transverse modes of the dot chains is assumed to be mω2 = 2β(1 − cos kd),
(60)
where ω is the angular frequency and k the propagation vector. The chain dots are only weakly bound to the substrate and the influence of this coupling can be neglected in the above dispersion relation, at least for kd smaller than π /2. This assumption is realistic for dots weakly bound to the substrate. In general, one can expect a phonon dispersion relation more complicated than the one given by Eq. (60). However in the long wave limit (kd small) the realistic dispersion relation can be matched to the one given by this equation by an appropriate choice of the nearest neighbour dot interaction β .
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An incident wave intensity I1 (kd) = 1 launched onto the coupling structure, from the input gate 1, generates, as a result of scattering processes, the outgoing acoustic wave intensities Oj (kd), j = 1, 2, 3, 4 (see Fig. 7). The corresponding analytical expressions are given by Dobrzynski et al. paper [21] together with an application for optical phonons. Here also for an acoustical phonon the total phonon transfer from the input 1 to the output 3, i.e. O1 = 0, O2 = 0, O3 = 1 and O4 = 0 can be realized exactly at the angular frequency ω0 . This frequency and the force constants β , β1 , and β2 then should fulfil the following conditions M ω02 = K 0 + 2β1 + β2 , cos(k0 Ld) = −
β2 2β1
,
(61) (62)
and sin(k0 Ld)
ββ2 = 2 , sin(k0 d) β1
(63)
Fig. 8. The transmission coefficients O3 (kd) (solid line), O2 (kd) (dashed line), O4 (kd) (dotted line) and O1 (kd) (dotted-dashed line) as a function of kd for n0 = 0, L = 1, β1 /β = 0.2, β2 /β = 0.04, m = M, K 0 /β = 1 with one single input I1 = 1.
where k0 is the value of k for ω = ω0 in Eq. (60). The transferred wave has some width in kd around k0 d. If one wishes the corresponding peak in O3 (kd) to be symmetric, then one obtains another Dobrzynski et al. condition [20], namely Lk0 d = (1 + 4n0 )
π 2
,
n0 = 0, 1, 2, . . .
(64)
However this condition and the ones given by Eqs. (62) and (63) are only fulfilled for β1 = 0 and β2 = 0. So in what follows we will tolerate a small dissymmetry of the peak in O3 (kd) and a small imprecision on the condition given by Eq. (64). This is easily managed by adding a small quantity ε to the righthand side of Eq. (64) and then calculating β2 /β , β1 /β and K 0 /β in function of this ε . Define the quality factor associated with the bandwidth of the transferred signal by Q (k0 d) =
k0 d
∆(k0 d)
,
(65)
where ∆(k0 d) is the width of this signal for O3 (kd) = 0.5. An approximate value of this quality factor is Q (k0 d) = (1 + 4n0 )
h πM β πi sin (1 + 4n0 ) . L m β2 2L
(66)
One can notice that for large values of d2 , β2 may become very small and then the quality factor Q (k0 d) very large. However in all physical systems some loss mechanisms exist which prevents the experimental detection of too high a quality peak. This has to be taken into account before trying and choosing precise parameters for a characterization. Stress also that such a system is expected to work in the whole range of acoustic waves. An illustrative and at the same time realistic example complying with the above assumptions may be obtained for n0 = 0, L = 1, β1 /β = 0.2, β2 /β = 0.04, m = M, K 0 /β = 1. When the influence, of the force constants binding the chain clusters to the substrate, on the dispersion relation of the chain phonon can be neglected, it is possible to take advantage of the 1/d5 proportionality between the force constants and the cluster separation distance d and obtain an estimate for the distances d1 /d = 1.38 and d2 /d = 1.90. This set of parameters is realistic and could be used for any type of material, any size of the dots, as long as the dots are weakly bound to the substrate. If that is not the case, one can recalculate, using the above equations, other parameters for an acoustic phonon transfer in a smaller frequency range, where ω is proportional to k.
Fig. 9. Output-signal intensities O1 (kd) = O2 (kd) (dashed line) and O3 (kd) = O4 (kd) (solid line) as a function of kd for the same parameters as in Fig. 8 when two inputs of intensity I1 (kd) = I2 (kd) = 1 at gates 1 and 2 are simultaneously present.
Fig. 8 presents the transmission coefficients O3 (kd) (solid line), O2 (kd) (dashed line), O4 (kd) (dotted line) and O1 (kd) (dotteddashed line) as a function of the reduced wave vector kd. One remarks that the dissymmetry with respect to kd = π /2 is negligible. The peak in the transmission coefficient O3 (kd) shows a width at half maximum of the order predicted by Eq. (66). In this figure O2 (kd) is basically constant and equal to 1 after the dip due to the transfer. This result comes from the parameters used in this calculation, but remains for other possible parameter sets as long as the analytical conditions given above are satisfied with good precision and the chosen quality factor is not too low. Now with two inputs of intensity I1 (kd) = I2 (kd) = 1 at gates 1 and 2, by linear superpositions of the amplitudes the output transmission probabilities can also be obtained. In other words, two transverse acoustic phonons of particular propagation vector k0 are cross-transferred through the structure to gates 3 and 4, respectively. This phonon crossing effect is illustrated in Fig. 9 for the same parameters as those used in Fig. 8. These results show that the simple structure reviewed here can realize transverse phonon cross transfer respectively from gate 1 to gate 3 and from gate 2 to gate 4. Moreover, the above
L. Dobrzyński, A. Akjouj / Surface Science Reports 63 (2008) 391–399
derived closed-form expressions enable one to find the optimal parameters for the desired system, enabling one to conceive it at will for specific applications. Although this system does not need to be of nanometer size in order to operate, it is particularly well adapted for technologies of this size. Such a device could be excited by surface waves techniques, as done currently in many telecommunication devices [23]. 5. Perspectives One simple electronic system, see Fig. 1, on which the two particle cross transfer was discovered, is reviewed in this paper. Two other systems, see Figs. 4 and 7 are also reviewed for phonon cross transfer. The corresponding three geometries can be easily transposed for other particle cross transfer. For example photon, plasmon and magnon cross transfer systems may be easily conceived by using the corresponding one particle multiplexer systems reviewed by Vasseur et al. [24]. Moreover these three generic geometries may work, with appropriate adjustments, for all the different types of existing particles. The analytic general approach, reviewed in this tutorial, makes the conception of novel systems accessible even to undergraduate students. Of course,simulation researchers, experimentalists, engineers, technicians and industry managers are then needed in order to realize a prototype and try and introduce it on the market. From the experimental point of view, condensed matter particle cross systems can be build on well defined surfaces by the most advanced techniques enabling the design of micro, nanometer and even atomic surface structures. Many investigations are in progress all around the world, in order to overcome the difficulties in conceiving, designing,
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simulating and building the elementary constituents enabling one to build such devices. References [1] L. Dobrzynski, Phys. Rev. B 70 (2004) 193307; Virtual J. Nanoscale Sci. Technol. 10 (22) (2004). [2] L. Dobrzynski, P. Zielinski, A. Akjouj, B. Sylla, Phys. Rev. B 71 (2005) 047601. [3] L. Dobrzynski, H. Al-Wahsh, A. Akjouj, G. Hernandez-Cocoletzi, J. Phys.: Condens. Matter 18 (2006) 3151. [4] A. Akjouj, B. Sylla, L. Dobrzynski, Ann. Phys. Fr. 18 (1993) 363. [5] C.C. Eugster, J.A. del Alamo, Phys. Rev.Lett. 67 (1991) 3586. [6] H.A. Haus, Y. Lai, J. Lightwave Technol. 10 (1992) 57. [7] S.S. Orlov, A. Yariv, S. Van Essen, Opt. Lett. 22 (1997) 688. [8] S. Fan, P.R. Villeneuve, J.D. Joannopoulos, H.A. Haus, Phys. Rev. Lett. 80 (1998) 960. [9] L. Dobrzynski, Surf. Sci. 175 (1986) 1. [10] L. Dobrzynski, Surf. Sci. Rep. 6 (1986) 119. [11] L. Dobrzynski, Surf. Sci. 180 (1987) 489. [12] L. Dobrzynski, Surf. Sci. Rep. 11 (1990) 139. [13] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, 1995. [14] J.A. del Alamo, C.C. Eugster, Appl. Phys. Lett. 56 (1990) 78. [15] N. Dagli, G. Snider, J. Waldman, E. Hu, J. Appl. Phys. 69 (1991) 1047. [16] Z.L. Mišković, R.A. English, S.G. Davison, F.O. Goodman, J. Phys.: Condens. Matter 9 (1997) 10749. [17] L. Dobrzynski, B. Djafari-Rouhani, A. Akjouj, J.O. Vasseur, M.L.H. Lahlaouti, J. Phys.: Condens. Matter 11 (1999) L247. [18] Z. Shao, W. Porod, C.S. Lent, Phys. Rev. B 49 (1994) 7453. [19] R. Kucharczyk, L. Dobrzynski, B. Djafari-Rouhani, A. Akjouj, Physica E 24 (2004) 355. [20] L. Dobrzynski, B. Djafari-Rouhani, A. Akjouj, J.O. Vasseur, J. Zemmouri, Europhys. Lett. 46 (1999) 467. [21] L. Dobrzynski, A. Akjouj, B. Djafari-Rouhani, P. Zielinski, H. Al-Wahsh, Europhys. Lett. 65 (2004) 791. [22] P. Muller, A. Saúl, Surf. Sci. Rep. 54 (2004) 201. [23] E. Dieulesaint, D. Royer, in: L. Dobrzynski (Ed.), Handbook of Surfaces and Interfaces, vol. 2, Garland STPM Press, 1978, p. 65. [24] J.O. Vasseur, A. Akjouj, L. Dobrzynski, B. Djafari-Rouhani, E.H. El Boudouti, Surf. Sci. Rep. 54 (2004) 1.
