Superconductivity, including high-temperature superconductivity
LOCALIZATION OF THE COHERENT WAVE IN A
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3. LOCALIZATION OF THE COHERENT WAVE IN A NONLINEAR LAYERED MEDIUM Let us turn to an analysis of a nonlinear layered medium containing a period structure of widely spaced parallel planes with properties different from those of the surrounding me- dium, i.e., Eq. ͑1͒ with an infinite number of ␦ functions on the right-hand side. In the case of weak dynamic coupling between defect planes we can use the results of Sec. 2 to reduce the problem to an effective system for an infinite chain of coupled anharmonic oscillators, which in the case of a coherent stationary wave are described by the following system of difference equations: Ϫi
ϩ͑ l Ϫ2 0 ͒U n ϪU n 3 ϩ 0 ͑2U n ϪU n ϩ1 ϪU n Ϫ1 ͒ϭ0. ͑48͒ Such equations are ordinarily used for interpreting the results of experiments on the localization of optical fluxes in layered nonlinear media, 10,11,18 but for this the parameters of the effective chain of oscillators are not specified. The math- ematical problem of the localization of excitations in discrete nonlinear systems of the type ͑48͒ and of the existence of discrete envelope solitons in them is now the subject of a large number of papers. 31,32 We restrict discussion to the simplest case, when the localization region of the nonlinear wave in the layered medium is much larger than the period of this structure. This condition imposes an additional re- striction on the wave amplitude: instead of the previous in- equality U
Ӷ 2 (U n Ӷ), we now have U n Ӷexp(Ϫa/2). Here the domain of admissible wave frequencies narrows, and inequality ͑24͒ goes over to
Ϫ
2 exp(
Ϫa). With the indicated stipulations, Eq. ͑48͒ can be replaced by the nonlinear Schro¨dinger differential energy for the function U ϭU(Z,t): Ϫi ץ
ץ
Ϫ4a 2
ץ 2
ץ
2 ϩ͑ l Ϫ2 0 ͒UϪU 3 ϭ0
with the well-known soliton solution U n Ϸ ͱ 2 ͱ l Ϫ2 0 Ϫ cosh ͓ ͱ ͑
Ϫ2
Ϫ ͒/ 0
͔
Ϫi t . ͑50͒ This solution describes analytically a nonlinear wave lo- calized in the transverse direction and propagating along a layered structure, as was observed, in particular, in the ex- periments of Refs. 10,11, and 18. This study was supported by the project INTAS-99 ͑Grant No. 167͒ and the program MNOP ͑Grant USU082087 ͒. 592 Low Temp. Phys. 26 (8), August 2000 I. V. Gerasimchuk and A. S. Kovalev * E-mail: igbox@iname.com ** E-mail: kovalev@ilt.kharkov.ua 1 Conference Digest of 14th International Colloquium on Magnetic Films and Surfaces, ⑀
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2 Abstract Book of 2nd International Symposium on Metallic Multilayers (MML ϭ
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550, 1212 ͑1988͒ ͓Sov. J. Low Temp. Phys. 14, 304, 669 ͑1988͔͒. 10 H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Phys. Rev. Lett. 81, 3383 ͑1998͒.
11 U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg, and Y. Sil- berberg, Appl. Phys. Lett. 75, 1348 ͑1999͒.
12 A. A. Stepanov and D. A. Yablonski , Fiz. Nizk. Temp. 15, 215 ͑1989͒ ͓Sov. J. Low Temp. Phys. 15, 122 ͑1989͔͒. 13 M. M. Bogdan, M. I. Kobets, and E. N. Khats’ko, Fiz. Nizk. Temp. 25, 263 ͑1999͒ ͓Low Temp. Phys. 25, 192 ͑1999͔͒. 14 D. Hennig, H. Gabriel, G. P. Tsironis, and M. Molina, Appl. Phys. Lett. 64, 2934 ͑1994͒.
15 Qiming Li, C. T. Chan, K. M. Ho, and C. M. Soukoulis, Phys. Rev. B 53, 15577 ͑1996͒.
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3153 ͑1993͒.
18 A. B. Aceves, C. De Angelis, T. Peschel, R. Mtuschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 ͑1996͒.
19 Yu. B. Gaididei, P. L. Christiansen, K. O. Rasmussen, and M. Johansson, Phys. Rev. B 55, R13365 ͑1997͒.
20 R. L. Stamps, R. E. Camley, R. J. Hicken, Phys. Rev. B 54, 4159 ͑1996͒. 21
͑1975͒ ͓Sov. J. Low Temp. Phys. 1, 742 ͑1975͔͒. 22 Yu. S. Kivshar and B. Malomed, J. Phys. A 21, 1553 ͑1988͒. 23 M. M. Bogdan, I. V. Gerasimchuk, and A. S. Kovalev, Fiz. Nizk. Temp. 23, 197 ͑1997͒ ͓Low Temp. Phys. 23, 145 ͑1997͔͒. 24 A. M. Kosevich and A. S. Kovalev, Introduction to Nonlinear Physical Mechanics ͓in Russian͔, Naukova Dumka, Kiev ͑1989͒. 25 S. M. Jensen, IEEE J. Quantum Electron. QE-18, 1580 ͑1982͒. 26 L. Thylen, E. M. Wright, G. I. Stegeman, C. T. Seaton, and J. V. Moloney, Opt. Lett. 11, 739 ͑1986͒.
27 S. Wabnitz, E. M. Wright, C. T. Seaton, and G. I. Stegeman, Appl. Phys. Lett. 49, 838 ͑1986͒.
28 A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Nonlinear Magnetiza- tion Waves. Dynamical and Topological Solitons ͓in Russian͔, Naukova Dumka, Kiev ͑1983͒.
29 A. S. Kovalev and E. S. Syrkin, Zh. E ´ ksp. Teor Fiz. 102, 522 ͑1992͒ ͓JETP 75, 277 ͑1992͔͒. 30 L. D Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed., rev. and enl., with L. P. Pitaevskii, Pergamon Press, Oxford ͑1984͒; Nauka, Moscow ͑1982͒. 31 S. Flach and C. R. Willis, Phys. Rep. 295, 181 ͑1998͒. 32 S. Flach, Discrete Breathers, Habilitationsschrift, Max-Planck-Institut fu¨r Physik komlexer Systeme, Dresden, Germany ͑1997͒.
Translated by Steve Torstveit 593
Low Temp. Phys. 26 (8), August 2000 I. V. Gerasimchuk and A. S. Kovalev On the quantum magnetic size oscillation effects in organic conductors M. Ya. Azbel’ School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel O. V. Kirichenko and V. G. Peschanski
͑Submitted April 5, 2000͒ Fiz. Nizk. Temp. 26, 810–814 ͑August 2000͒ The quantum magnetic size oscillations ͑QMSOs͒ of the thermodynamic quantities in layered organic conductors with a quasi-two-dimensional electron energy spectrum of arbitrary form are investigated theoretically. It is shown that the modulation of the QMSOs contains detailed information about the dispersion relation of the charge carriers. © 2000 American Institute
͓S1063-777X͑00͒00908-7͔ The Shubnikov–de Haas and de Haas–van Alfen quan- tum oscillation effects 1–3 are manifested most clearly in con- ductors of organic origin. This is due to the low-dimensional character of the energy spectrum of the charge carriers in organic conductors, which, as a rule, have a layered or fila- mentary structure with sharp anisotropy of the electrical con- ductivity. The electron energy spectrum of layered conduc- tors is quasi-two-dimensional, and the dispersion relation (p) for the charge carriers in filamentary conductors with high conductivity only along the filament obviously has a quasi-one-dimensional character. The high conduction of or- ganic conductors, if only in one direction ͑e.g., along the y axis
͒, attests to the large number of charge carriers in them, and these conductors have a metallic type of conductivity, at least in that one direction. The Fermi surface (p)ϭ
of quasi-one-dimensional conductors can be written in the form of slightly corrugated planes in momentum space. In layered conductors placed in a magnetic field H ϭ(0, 0, H) applied along the layers, a large fraction of the charge carriers moves along open trajec- tories in momentum space and, of course, do not take part in the formation of quantum oscillation effects in massive samples having thicknesses L much larger than the mean free path l of the charge carriers. 4 However, in thin conductors (L рl) with surfaces smooth enough to reflection the con- duction electrons in a nearly specular manner, the areas
,p x , p y ) of the open ͑cut off by specular reflections of the charge carriers on the boundary of the sample ͒ sections of the isoenergy surface (p)ϭ
by a plane p z ϭconst can take on only discrete values which differ by a multiple of 2 បeH/c, where e is the charge of the electron, ប is Planck’s constant, and c is the speed of light in vacuum. As a result, the conduction electrons on the open sections of the Fermi surface create a sort of oscillatory effect 5–7 wherein
the magnetosize quantum oscillations of the magnetization and magnetoresistance are showed by modulation of the am- plitude. In layered organic conductors a considerably larger number of charge carriers is involved in the formation of the quantum magnetosize oscillations than in ordinary quasi- isotropic metals on account of the weak dependence of S( p x , p y ) on the momentum projection p x in these materi- als ͑the x axis is directed along the normal to the layers͒. In quasi-one-dimensional conductors one expects that the quan- tum magnetosize effect will be still more clearly manifested, since for them the areas of the sections of the Fermi surface which are cut off by specular reflections depend weakly on p z as well.
8 As a result, averaging over these variables does not lead to a substantial decrease in the amplitude of the oscillations in comparison with the case of quasi-isotropic metals. Let us consider the oscillatory quantum magnetosize ef- fects in organic conductors with an arbitrary form of the electron energy spectrum ͑p͒ϭ ͚
ϭ0 ϱ
n ͑p y , p z ͒cos
ͫ an p x ប ϩ
␣ n ͑p y , p z ͒ ͬ . ͑1͒
The coefficients of the cosines in ͑1͒, as a rule, fall off rapidly with increasing number n, and the maximum value of the function 1
y ,y z ) on the Fermi surface is equal to
Ӷ F , where
is the quasi-two-dimensionality param- eter of the electron energy spectrum of the layered conduc- tor; a is the distance between layers, and ␣
( p y , p z ) ϭϪ ␣ n ( Ϫp y , Ϫp z ). In quasi-one-dimensional conductors the functions n ( p y , p z ) ϭ n ( Ϫp y , Ϫp z ), including 0 ( p y , p z ), depend weakly on p z . In a magnetic field parallel to the surface of a thin slab with sufficiently smooth faces y ϭ0,L, the quantization of the areas takes the form
͑,p x , p z ͒ϭ ͵ p x p x ϩeHL/c 2 p
͑,p x , p z ͒dp x ϭ2 ប eH c ͑nϩ ␥ ͒,
where Ϫ1Ͻ
␥ р0, and nϭ1,2,3, . . . , i.e., n is a positive integer. We assume, solely for the sake of brevity in the analysis of oscillatory effects, that the open sections of the Fermi surface are symmetric, p
( p x , p z ) ϭϪp y ( Ϫp x , p z ). In magnetic fields that are not too high, so that a is not only much smaller than L but also much smaller than the LOW TEMPERATURE PHYSICS VOLUME 26, NUMBER 8 AUGUST 2000 594 1063-777X/2000/26(8)/4/$20.00 © 2000 American Institute of Physics characteristic quantum radius ϭ(cប/eH) 1/2 , one can find the quantized energy spectrum of the conduction electrons from relation ͑2͒ with the use of their quasiclassical trajec- tories in the magnetic field. Solving the equation (p)ϭ, where
(p) is given by ͑1͒, for p y , we obtain the following expression for the average value of the momentum projec- tion p y :
¯ y ͑,p x , p z ͒ϭ
eHL ͵
x p x ϩeHL/c p y ͑,p x , p z ͒dp x ϭ ប L n. ͑3͒
If the quantum radius is comparable to the distance a between layers, as is the case for nanostructures and super- lattices, then the energy spectrum of the conduction electrons can be determined by solving the Schro¨dinger equation
ϭ . ͑4͒ The vector potential A ϭ(Hy, 0, 0). In the Landau gauge the vector potential of the Hamiltonian H
Ϫ(eH/c)y, pˆ y , p z ) is independent of x, z, and the generalized momen- tum component P
; the p z are good quantum numbers. For Ӷ1 the Hamiltonian depends weakly on the kinematic mo- mentum p x ϭ(P x ϪeHy/c) and, consequently, on y. In the limit
y will also be a good quantum number characterizing the state of a conduction electron, and the action of the operator pˆ y on the wave function in the
case of nonzero but small is written as pˆ y ϭp y 0 ϩ ␦
y , ͑5͒ where
␦ pˆ y goes to zero together with .
the solution of equation ͑4͒ can be written as ͑x,y,z͒ϭu͑y͒exp ͫ i ប ͑
x p x ϩyp y 0 ϩzp z ͒ ͬ . ͑6͒
In the linear approximation in the small parameter the equation for the function u(y ) has the form ͫ 0 ͑p y 0 , p z ͒ϩ
1 ͑p y 0 , p z ͒cos
ͩ a p x h Ϫ
c ប
ϩ ␣
͑p y 0 , p z ͒ ͪͬ ϫu͑y͒Ϫiបv y 0 ץ u ͑y͒ ץ
ϭu͑y͒, ͑7͒ where
v y 0 ϭ ץ 0 ( p y 0 , p z )/ ץ p y 0 . The solution of equation ͑7͒ must satisfy the boundary condition u(0) ϭu(L)ϭ0, which is what determines the quantized energy levels of the charge carriers. This boundary condition can be satisfied by a standing wave with nodes at y ϭ0 and L. After constructing the standing wave using so- lutions of equation ͑7͒, one can easily obtain the quantized energy spectrum of the charge carrier. In the leading approxi- mation in the parameter it is
n 0 ͑p z ͒ϭ
0 ͩ បn L , p z ͪ , ͑8͒ and the dependence of the energy levels on the magnetic field appears in small corrections to this quantity in the pa- rameter
. For determining the quantum magnetosize oscillations of the magnetization and the other thermodynamic quantities it is sufficient to calculate the thermodynamic potential ⍀ of a sample enclosed in a volume V: ⍀ϭϪ⌰ ͚ ͚ n ϭ0 ϱ 2V L ͑2 ប͒ 2 ϫ ͵ d p x ͵
z ln ͩ 1 ϩexp
Ϫϩ ⌰ ͪ , ͑9͒ where
⌰ is the temperature multiplied by Boltzmann’s con- stant,
ϭ Ϯ H, is the chemical potential, and is the
Bohr magneton. Using Poisson’s formula, we write the os- cillatory part of the potential as ⍀ ˜ ϭRe
͚ k ϭ1 ϱ ͚
k , ͑10͒ where I k ϭϪ⌰
2V L ͑2 ប͒ 2 ͵ Ϫ ␥ ϱ dn ͵
x ͵
z exp
͑2
͒ ϫln
ͩ 1 ϩexp Ϫϩ ⌰ ͪ . ͑11͒ Making a change of the variable of integration from n to the more convenient , we obtain I k ϭϪ⌰
2V L ͑2 ប͒ 2 ͵ 0 ϱ d ͵ d p x ͵
z ϫln
ͩ 1 ϩexp Ϫ ⌰ ͪ ץ n ץ exp ͑2 ikn ͒. ͑12͒ It should be noted that for ⌰Ӷ the main contribution to I k comes from the neighborhood of the point ϭ
. The limits of integration over p z , generally speaking, are deter- mined from the condition S Ͼ0. However, the oscillatory part of the magnetization of the conductor is formed by charge carriers with extremal values of S. For any dispersion relation of the conduction electrons S has an extremum on the central section of the Fermi surface by plane p z ϭ0, and
there can also be several more extrema if the spectrum of the charge carriers has a sufficiently complicated form. Let us consider the simplest model for the dispersion relation of a quasi-two-dimensional conductor, when S has one extremum at P
ϭ0, viz.
ϭ p y 2 ϩp z 2 2m ϩ
a p x ប , ͑13͒ where the constant A has the same value as
. A dispersion relation of this form allows one to obtain not only the quasiclassical but also the exact solution of equation
͑4͒. In calculating the oscillations of the magnetiza- tion it is sufficient to take into account only a small neigh- borhood of the point p
ϭ0 within which p z 2 рp 0 2 Ӷ2m. Then, after determining n from the quantization condi- tion
͑2͒, one can write it in the form of two terms, viz., a 595
Low Temp. Phys. 26 (8), August 2000 Azbel’
et al. main term which is independent of H, and a small correction which is proportional to and depends on the magnetic field: n ϭϪ ␥ ϩ L ͱ 2m 2 ប ͱ Ϫp z 2 /2m ͫ 1 Ϫ Ar 2L ͑Ϫp
2 /2m ͒ ϫsin
L 2r cos ͩ
x ប ϩ
L 2r ͪ ͬ
͑14͒ where r ϭcប/eHa. Let us assume that
Ӷ1. ͑15͒ Changing the order of integration in Eq. ͑12͒, we first inte- grate by parts over ; keeping only the rapidly oscillating terms, we obtain I k ϭϪ
L ͑2 ប͒ 2 ik ͵ Ϫ ប/a ប/a d p x ͵ Ϫp 0 p 0
z 1 2 ik ϫ ͵ 0 ϱ d f ͩ Ϫ
⌰ ͪ exp ͑2 ikn ͒, ͑16͒ where f (x) ϭ(1ϩexpx) Ϫ1 is the Fermi distribution function. Then we substitute expression ͑14͒ for n into ͑16͒: I k ϭ
L2 2 បa exp
ͩ Ϫ2 ik ␥ Ϫ i 2 ͪ ϫ ͵ 0 ϱ
f ͩ Ϫ ⌰ ͪ ͵ Ϫp 0 p 0
z exp
ͩ ikL ͱ 2m Ϫp z 2 ប ͪ ϫJ 0 ͩ
krA2m ប͑2mϪp z 2 ͒ 1/2 sin
L 2r ͪ ,
where J 0 is the Bessel function. In integrating over p z we use the method of stationary phase. Taking inequality ͑15͒ into account, it is easy to see that the most rapidly varying function in the integrand is exp(ikL ͱ 2m Ϫp z 2 / ប), which has a ??stationary point at p z ϭ0.
As a result of straightforward calculations we arrive at the expression I k ϭϪ 2 1/4 V ប 1/2 3/4
3/2
ak 5/2
L 5/2
m 1/4
⌿͑k⌫͒exp ͩ Ϫ2 ik ␥ Ϫi 4 ϩi kL ͱ 2m ប ͪ J 0 ͩ krA ͱ 2m ប ͱ sin
L 2r ͪ ,
where ⌿(z)ϭz/sinh z, ⌳ϭ( ⌰L/2ប ) ͱ 2m . In the
smoothly varying functions one can replace by , since H Ӷ . Using formulas ͑10͒ and ͑18͒, we can write the oscilla- tory part of the thermodynamic potential in the form ⍀ ˜ ϭ⍀
0 ͚
⌿͑k⌫͒
5/2
͚ cos ͩ Ϫ2 k ␥ Ϫ 4 ϩ kL ប ͱ 2m ͪ ϫJ 0 ͩ
krA ͱ 2m ប ͱ sin
L 2r ͪ ,
where ⍀ϭ2
5/4 V ប 1/2 3/4
/( 3/2 aL 5/2
). From here the calculation of the quantum oscillations of the thermodynamic quantities is done by elementary differ- entiation of expression ͑19͒. Let us determine the oscillatory part M ˜ of the magnetic momentum in the direction of the magnetic field, M ˜ ϭϪ ץ ⍀ ˜ ץ
. ͑20͒
Keeping only the leading terms in the parameter
,
M ˜ ϭ 2 ⍀ 0 H ͚
⌿͑k⌳͒
5/2
cos ͩ Ϫ2 k ␥ Ϫ 4 ϩ kL ប ͱ 2m ͪ R ϫ͑Ϫsin RϩR cos R͒J 1 ͩ
R sin R ͪ ,
where R ϭL/2r, ϭ
ប ͱ 2m ). The argument of the Bessel function J 0 in expression ͑21͒ goes to zero when the thickness L of the sample is a multiple of the period of the open electron trajectory 2
As the magnetic field is varied, the condition that L is a multiple of 2
This leads to modulation of the magnetosize oscillations ͑Fig. 1͒. It is easy to obtain the oscillatory dependence of the magnetization in the case of a quasi,two-dimensional elec- tron energy spectrum of arbitrary form. Keeping only the first two terms in relation ͑1͒, we obtain an expression for M
that is analogous to ͑21͒ but with
ͱ 2m replaced by 1
v 0 , where v 0 ϭ ץ (p y ,0)/
ץ p y . FIG. 1. Dependence of M /H on R ϭLeHa/2cប in relative units; ϭ100 ͑a͒ and 30
͑b͒. 596
Low Temp. Phys. 26 (8), August 2000 Azbel’
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