Superconductivity, including high-temperature superconductivity
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- 2. THE MODEL AND THE BASIC EQUATIONS
- 3. THE TRANSFER-MATRIX APPROACH
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1. INTRODUCTION The discovery of the quantum Hall effect in 1980 1 has
triggered intensive studies of a two-dimensional electron gas ͑2DEG͒ in an external quantizing magnetic field. These stud- ies have since been extended to different types of artificially fabricated semiconducting and metallic superlattices ͑SLs͒, organic conductors, and high-T c layered superconductors. Numerous studies, in particular, have been devoted to the problem of collective plasma and electromagnetic waves in 2DEG and layered conductors as well as in SLs in a high magnetic field. Generally, a three different physical cases should be distinguished in this problem: the case of classical SLs, the case of quantum SLs, and the case of layered con- ductors. In the first case constituent slabs of the SL are as- sumed to be so thick that one can neglect the electron energy quantization. The electromagnetic wave propagation in such SLs is determined completely by Maxwell’s equations and the appropriate boundary conditions. Quantum SLs have small separations between conducting layers, and the elec- tron dispersion across the layers in this case is due to the tunneling between neighboring layers. By layered conductors we shall understand a stack of 2D conducting planes sepa- rated by dielectric layers which prevent electrons from hop- ping between the neighboring planes. Layered conductors are realized in nature in the form of layered crystals such as dichalcogenides of transition metals, organic superconduct- ors, and high-T c cuprates. The high anisotropy of Tl- and Bi-based high-T
cuprates, 2 organic salts of ͑TMTSF͒ 2 X, 3 and ET families 4 makes them, like dichalcogenides of tran- sition metals, 5 good layered conductors in the sense formu- lated above. It is evident that layered conductors can also be fabricated artificially in the form of highly anisotropic SLs. All these materials are well described by the model of con- ducting planes embedded in a dielectric matrix. This model has proved to be useful in studies of different types of plasma
6–10 and electromagnetic 11–20 waves in layered con- ductors, superconductors, and superlattices. A quasi-two- dimensional nature of the conductivity in layered conductors brings some specific features into calculations of the collec- tive electromagnetic modes in them, especially in the pres- ence of an external magnetic field. Some new types of col- lective electromagnetic excitations have been predicted theoretically in a purely 2DEG in high magnetic fields under the conditions of the quantum and conventional Hall effects. Among them are surface polaritons, 21,22
magnetoplasma oscillations, 23 and quantum waves. 24,25 The variety of waves becomes richer in layered conductors. It is known that a quantizing magnetic field applied perpendicular to the layers makes possible the propagation of the helicons across the layers in both the conventional 11–14 and quantum 14,26–28 Hall-effect regimes. Real layered crystals and superlattices contain different types of defects within the layers as well as imperfections in their stacking which may give rise to new collective electro- magnetic modes such as, for example, magnetoimpurity waves 13
9,10,16 The infinite crystal is yet another idealization of the theoretical treatment of the problem, since any sample in experiments has a surface LOW TEMPERATURE PHYSICS VOLUME 26, NUMBER 8 AUGUST 2000 569
1063-777X/2000/26(8)/8/$20.00 © 2000 American Institute of Physics which is known to be a ‘‘structural defect’’ that generates surface modes decreasing into the bulk of the sample. Sur- face plasma modes have been studied extensively in the model of a semi-infinite layered electron gas. 7,8 Surface elec- tromagnetic waves have also been described in layered superconductors. 15 The purpose of this paper is to study the surface electro- magnetic waves in layered conductors in a perpendicular quantizing magnetic field. The basic equations describing the electric field components on the layers, E ␣ (z n ) ϵE ␣ (n), were derived in our previous publication 14 and can be written as follows ͑see Appendix for details͒: E x ͑n͒ϭ 4
c 2 ͚ n Ј
q
͑n,n Ј ͓͒ xx E x ͑n Ј ͒ϩ
xy E y ͑n Ј ͔͒,
͑1͒ E y ͑n͒ϭϪ 4
2 ͚ n Ј
q
͑n,n Ј ͓͒ y y E y ͑n Ј ͒
y x E x ͑n Ј ͔͒
Ϫ1 ͑n͒ ͑z
is a discrete coordinate of a conducting plane along the z axis ͒.
q ␣ (n,n Ј ) ϵG q ␣ (z n ,z n Ј ) in Eq. ͑1͒ satisfy the following equations: ͩ ץ 2 ץ
2 Ϫq 2 ͑z͒ ͪ G q 2 ͑z,z Ј ͒ϭ ␦ ͑zϪz Ј ͒,
ͩ ץ 2 ץ z 2 ϩU͑q, ,z ͒ ץ
z Ϫq 2
ͪ G q
͑z,z Ј ͒ϭ ␦ ͑zϪz Ј ͒,
where U ͑q, ,z ͒ϭ ͩ
q ͑z͒ ͪ 2 Ϫ1 ͑z͒ ץ ͑z͒ ץ z , ͑4͒ Here (z) is the dielectric constant of the matter be- tween the layers, ␣ ϵ ␣ (q, ,H) is the two-dimensional high-frequency conductivity tensor in an external magnetic field H; q stands for the wave vector, and q (z) is defined by the equation q 2 ͑z͒ϭq 2 Ϫ 2
2 ͑z͒. ͑5͒ 2. THE MODEL AND THE BASIC EQUATIONS Consider a regular semi-infinite layered crystal in which conducting planes occupy positions at a discrete periodic set of points z n ϭna (nϭ0,1,2...) along the z axis of the half space z Ͼ0. We assume that the dielectric constants are dif- ferent in the half spaces: 0 at z Ͻ0 and between the layers. The function (z) can be written analytically with the help of the Heaviside step function: ͑z͒ϭ ͑z͒ϩ 0 ͑Ϫz͒. ͑6͒ It then follows from Eq. ͑5͒ that the quantity q 2 (z) takes two different values in the half spaces: q 2 ͑z͒ϭ ͭ
2
z Ͼ0 2 , z Ͻ0,
͑7͒ where q 2
2 Ϫ 2 /c 2 and
2 ϭq 2 Ϫ 0 2 /c 2 . The Green’s function G q
(z,z Ј ) can be found with the help of the known general expression G q
͑z,z Ј ͒ϭ 1 W ͑ , ͒ ͕ ͑zϪz Ј ͒
͑z͒ ͑z Ј ͒ ϩ ͑z Ј Ϫz͒ ͑z Ј ͒
͑z͒ ͖ , ͑8͒ where
(z) and (z) are two independent solutions of the differential operator in the left-hand side of Eq. ͑2͒, and
W( , ) ϭ Ј Ϫ Ј is the Wronskian determinant. Choosing (z) ϭexp(Ϫq
(z) ϭcosh(q
ϩ(
/ q )sinh(q z) for z Ͼ0, we have G q
͑z,z Ј ͒ϭϪ 1 2q ͑e Ϫq ͉z Ϫz Ј ͉ ϩ ␦ e Ϫq ͉z ϩz Ј ͉ ͒, ͑9͒
z,z Ј Ͼ0, where ␦ ϭ͑q Ϫ ͒/͑q ϩ ͒. ͑10͒ The Green’s function G q
(z,z Ј ) ϵGˆ(z,z Ј ) in our model satisfies the following equation: ͩ ץ 2 ץ
2 Ϫq 2 ͑z͒ ͪ G ˆ ͑z,z Ј ͒ϩ⌬
␦ ͑z͒ ץ ץ
G ˆ ͑z,z Ј ͒ϭ
͑zϪz Ј ͒. ͑11͒ The quantity ⌬
⌬ ϭ2 ͫ q q ¯ ͬ 2 Ϫ
0 0 ϩ , ͑12͒ where the
following notations are adopted:
q ¯ ϭq 2 Ϫ( 2 /c 2 )
0 ϩ). The solution of Eq. ͑11͒ is trivially expressed in terms of the Green’s function G(z,z Ј ) that satisfies the very same equation but with ⌬ ϭ0: G ˆ ͑zϪz Ј ͒ϭG͑z,z Ј ͒Ϫ ⌬ 1 ϩ⌬ G Ј ͑0,0͒ G ͑z,0͒G Ј ͑0,z Ј ͒, ͑13͒ where we have used the
notation G Ј (0,z Ј ) ϭ lim x →0 ץ G(x,z Ј )/ ץ x. Taking into account that G(z,z Ј )
q
(z,z Ј ) for z,z Ј Ͼ0, we obtain from Eqs. ͑9͒ and ͑13͒ an exact formula for the Green’s function G
(z,z Ј ) in the positive half space: G q
͑z,z Ј ͒ϭϪ 1 2q ͑e Ϫq ͉z Ϫz Ј ͉ ϩ⌬ˆ
Ϫq ͉z ϩz Ј ͉ ͒, z,z Ј Ͼ0, ͑14͒ We have introduced the notation ⌬ˆ ϭ ␦ ϩ ⌬ ͑1Ϫ ␦ 2 ͒ 2 ϩ ␦ ⌬ . ͑15͒ Substituting Eqs. ͑9͒ and ͑14͒ into Eq. ͑1͒, we have E ␣ ͑n͒ϭ ͚ ,n Ј ϭ0
ˆ ␣ ͑e Ϫq
͉n Ϫn Ј ͉
␣
Ϫq
͉n ϩn Ј ͉
 ͑n Ј ͒,
where 570
Low Temp. Phys. 26 (8), August 2000 V. M. Gvozdikov ˆ ␣ ϭϪ 2
2
͑q, ,H ͒V ␣ , ͑17͒ and V ␣ is a matrix with the components V 11 ϭV 12 ϭ1, V 21 ϭV 22 ϭϪc 2 q 2 / 2 . The quantity ⌬ˆ ␣ takes two values: ⌬ˆ
ϭ ␦ and
⌬ˆ
ϭ⌬ˆ . 3. THE TRANSFER-MATRIX APPROACH To solve Eqs. ͑16͒ it is convenient to introduce new quantities A ␣ ͑n͒ϭ ͚  ˆ ␣ ͩ ͚ n Ј рn e Ϫq
͑nϪn Ј ͒
 ͑n Ј ͒
␣ ͚ n Ј ϭ0 e Ϫq
͑nϩn Ј ͒
 ͑n Ј ͒
͑18͒ and
B ␣ ͑n͒ϭ ͚  ˆ ␣ ͩ ͚ n Ј Ͼn e Ϫq
͑nϪn Ј ͒
 ͑n Ј ͒
. ͑19͒
The sum of A ␣ (n) and B ␣ (n) is exactly the electric field at the nth layer:
͑n͒ϭA ␣ ͑n͒ϩB ␣ ͑n͒. ͑20͒ Using Eqs. ͑18͒–͑20͒, one can easily obtain the recurrence relations A ␣ ͑nϩ1͒ϭe Ϫq
A ␣ ͑n͒ϩ ͚  ˆ ␣ ͓A  ͑nϩ1͒ϩB  ͑nϩ1͔͒, ͑21͒ B ␣ ͑nϩ1͒ϭe q
B ␣ ͑n͒Ϫ ͚  ˆ ␣ ͓A  ͑nϩ1͒ϩB  ͑nϩ1͔͒. ͑22͒ These equations may be recast in the matrix form: ͩ
␣ ͑nϩ1͒ B ␣ ͑nϩ1͒ ͪ ϭ ͚  Tˆ ␣ ͩ A  ͑n͒ B  ͑n͒ ͪ , ͑23͒ where the transfer matrix Tˆ ␣ has been introduced by the definition Tˆ ␣ ϭ ͩ ͑ ␦ ␣ ϩ ˆ ␣ ͒e Ϫq
␣
q
Ϫ
␣ e Ϫq
͑ ␦ ␣ Ϫ ˆ ␣ ͒e q
ͪ .
The transfer matrix satisfies the relation det Tˆ ␣ ϭTˆ ␣ 11
␣ 22
␣ 12
␣ 21
␦ ␣ . ͑25͒ As compared to the case of a one-component plasma oscil- lations in layered structures, which were discussed in Refs. 8 and 9 in terms of the transfer matrix of dimension 2 ϫ2, the matrix Tˆ ␣ given by Eq. ͑24͒ has a higher dimensionality (4 ϫ4) because of the two-component nature of the electro- magnetic waves in the system under study. Putting n ϭ0 in Eqs. ͑18͒ and ͑19͒, we arrive at the surface condition A ␣ ͑0͒ϭ⌬ˆ ␣
␣ ͑0͒ϩ
͚  ˆ ␣ ͑1ϩ⌬ˆ ␣ ͓͒A  ͑0͒ϩB  ͑0͔͒.
͑26͒ Before turning to the surface-mode calculations it is instruc- tive to address first the simpler case of an infinite layered conductor. In this case one can find the solution of the matrix equation ͑23͒ in the form A ␣ ͑n͒ϭC ␣ e ikan ,
␣ ͑n͒ϭD ␣ e ikan . ͑27͒ After substitution of these relations into Eq. ͑23͒, we have Det ͑
␣ Iˆ ϪTˆ ␣
͒ϭ0.
͑28͒ The symbol Det here stands for the determinant of the (4 ϫ4) matrix, while Iˆ is the (2ϫ2) unit matrix. Taking into account the condition given by Eq. ͑25͒, one can rewrite Eq. ͑28͒ in the form det
ͩ ␦ ␣ cos ka Ϫ 1 2 Tr Tˆ ␣ ͪ
͑29͒ which, after the substitution of the transfer-matrix compo- nents, yields the dispersion relation det
͓ ␦ ␣ ϩ
␣
͑q,k, ͔͒ϭ0,
͑30͒ where the structural form factor is given by S ͑q,k, ͒ϭ
͑q
͒ cosh
͑q
Ϫcos͑ka͒ . ͑31͒ Different types of electromagnetic waves in infinite lay- ered conductors have been studied on the basis of Eq. ͑30͒ under the conditions of the conventional and quantum Hall effects, in particular, magnetoimpurity waves 13 and helicons and helicons–plasmons. 14 The surface breaks the transla- tional invariance of Eq. ͑16͒ due to the term containing ⌬ˆ ␣
Because of that, the surface mode has no dispersion across the layers, and its field components damp into the bulk of the layered conductor. We assume this damping to be exponen- tial with a decrement ␥ and will find it below, E  ͑nϩ1͒ϭe Ϫ ␥a E  ͑n͒ϭ...ϭe Ϫ ␥an E  ͑0͒. ͑32͒ This equation means that A ␣ ͑n͒ϭA ␣ ͑0͒e Ϫ ␥an , B ␣ ͑n͒ϭB ␣ ͑0͒e Ϫ ␥an . ͑33͒
The above relations have the very same exponential form as those in Eq. ͑27͒, so that we can find the dispersion relation for the surface mode immediately from Eq. ͑30͒ by the substitution k →i ␥ . This yields det ͑Sˆ Ϫ1 ␦
Ϫ
␣ ͒ϭ0,
͑34͒ where the form factor Sˆ(q, ␥ ,
) ϭS(q,i ␥ ,
) is given by Sˆ ͑q, ␥ ,
͒ϭ sinh
͑q
͒ cosh
͑q
͒Ϫcosh͑ ␥
͒ . ͑35͒ To obtain an equation for the function ␥ ϭ ␥ (q ,
), we proceed as follows. First, writing the condition E ␣ (n ϩ1) ϭe Ϫ ␥a E ␣ (n) with the help of the transfer matrix and then putting n ϭ0, we arrive at the equation ͚ 
␣ 11 ϩT ␣ 21 ͒A  ͑0͒ϩ͑T ␣ 22
␣ 12 ͒B  ͑0͒]
ϭ͑A ␣ ͑0͒ϩB ␣ ͑0͒͒e Ϫ ␥a . ͑36͒
571 Low Temp. Phys. 26 (8), August 2000 V. M. Gvozdikov
Now using Eq. ͑24͒ for the transfer-matrix components, we obtain from Eq. ͑36͒ a relation for the ratio A ␣ /B ␣ at the
surface: A ␣ ͑0͒ B ␣ ͑0͒ ϭ⌫͑q, , ␥ ͒ϭ
q
Ϫe Ϫ ␥a e Ϫ ␥a Ϫe Ϫq
. ͑37͒ Combining this equation with the surface condition given by Eq.
͑26͒, we arrive at a pair of linear equations for the quan- tities B x (0) and B y (0), which have a nonzero solution if det ͓P ␣ ␦ ␣ Ϫ
␣ ͔ϭ0,
͑38͒ where
P ␣ ͑q, , ␥ ͒ϭ 1 1 ϩ⌬ˆ ␣ Ϫ 1 1 ϩ⌫ Ϫ ˆ ␣␣ . ͑39͒ Equations ͑34͒ and ͑38͒ form a closed system of equa- tions for the surface mode. This system can, however, be recast into a simpler pair of equations. Indeed, comparing Eqs.
͑38͒ and ͑34͒, we see that P ␣ ϭS Ϫ1 . This condition gives an equation for ␥ ϭ ␥ (q ,
): ͑1ϩ⌬ˆ ␣
␥a ϭ⌬ˆ ␣
q
ϩe Ϫq
. ͑40͒ Using this equation, we can eliminate ␥ from the form factor Sˆ ͓q, ␥ ϭ
(q , ), ͔ϵS¯(q, ) in Eq. ͑35͒, which yields the dispersion relation for the surface mode
ϭ s (q) det ͑
␣ Ϫ ˆ ␣ ͑q, ͒S¯͑q, ͒͒ϭ0,
͑41͒ where
S ¯ ͑q, ͒ϭ
1 ϩ⌬ˆ ␣
⌬ˆ ␣ ͪ ⌬ˆ ␣
q
ϩe Ϫq
sinh
͑q
͒ .
The amplitudes of this surface mode decrease exponentially into
the bulk
of the
layered conductor, E ␣ (n) ϭe Ϫ ␥an E ␣ (0), with a decrement ␥ ϭ ␥ (q,
(q)) given by ␥ ␣ ͑q͒ϭ 1
ln ͩ
␣
q
ϩe Ϫq
1 ϩ⌬ˆ ␣ ͪ , ͑43͒
where ϭ s (q). Being a collective excitation of the finite layered con- ductor, the surface mode also decreases exponentially into the left half space z Ͻ0 with a decrement
Ͼ0. This means that the condition q 2 Ϫ(
2 /c 2 )
0 Ͼ0 should hold, as well as the inequality q 2 Ϫ( 2 /c 2 ) Ͼ0, which has been tacitly as- sumed in the course of all the above discussion. Therefore, these two constraints together with Eqs. ͑41͒–͑43͒ comprise a complete set of equations describing the surface electro- magnetic mode in a layered conductor in an external mag- netic field within our approach. It is worthy of note that these dispersion relations are still rather general, since the 2D con- ductivity tensor that appears in them is as yet an arbitrary quantity. In the next section we will consider a Drude-like model for the conductivity of the 2DEG, leaving more com- plex models of the conductivity for further studies.
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