Extremal transmission at the Dirac point of a photonic band structure

Extremal transmission at the Dirac point of a photonic band structure

Sepkhanov, R.A.; Bazaliy, Y.B.; Beenakker, C.W.J.

Citation

Sepkhanov, R. A., Bazaliy, Y. B., & Beenakker, C. W. J. (2007). Extremal transmission at the

Dirac point of a photonic band structure. Physical Review A, 75(6), 063813.

doi:10.1103/PhysRevA.75.063813

Version: Not Applicable (or Unknown)

License: Leiden University Non-exclusive license

Downloaded from: https://hdl.handle.net/1887/61281

Note: To cite this publication please use the final published version (if applicable).

Extremal transmission at the Dirac point of a photonic band structure

R. A. Sepkhanov,¹Ya. B. Bazaliy,^1,2and C. W. J. Beenakker¹

1Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

2Department of Physics and Astronomy, University of South Carolina, Columbia, South Carolina 29208, USA 共Received 19 March 2007; published 13 June 2007兲

We calculate the effect of a Dirac point共a conical singularity in the band structure兲 on the transmission of monochromatic radiation through a photonic crystal. The transmission as a function of frequency has an extremum near the Dirac point, depending on the transparencies of the interfaces with free space. The extremal transmission T₀=⌫0W / L is inversely proportional to the longitudinal dimension L of the crystal共for L larger than the lattice constant and smaller than the transverse dimension W兲. The interface transparencies affect the proportionality constant⌫0, and they determine whether the extremum is a minimum or a maximum, but they do not affect the “pseudodiffusive” 1 / L dependence of T₀.

DOI:10.1103/PhysRevA.75.063813 PACS number共s兲: 42.25.Bs, 42.25.Gy, 42.70.Qs

I. INTRODUCTION

In a two-dimensional photonic crystal with inversion symmetry the band gap may become vanishingly small at corners of the Brillouin zone, where two bands touch as a pair of cones. Such a conical singularity is also referred to as a Dirac point, because the two-dimensional Dirac equation has the same conical dispersion relation. In a seminal work 关1兴, Raghu and Haldane investigated the effects of broken inversion symmetry and broken time reversal symmetry on the Dirac point of an infinite photonic crystal. Here we con- sider the transmission of radiation through an ideal but finite crystal, embedded in free space.

As we will show, the proximity to the Dirac point is as- sociated with an unusual scaling of the transmitted photon current I with the length L of the photonic crystal. We as- sume that L is large compared to the lattice constant a but small compared to the transverse dimension W of the crystal.

For a true band gap, I would be suppressed exponentially with increasing L when the frequency ␻ lies in the gap.

Instead, we find that near the Dirac point I⬀1/L. The 1/L scaling is reminiscent of diffusion through a disordered me- dium, but here it appears in the absence of any disorder inside the photonic crystal.

Such “pseudodiffusive” scaling was discovered in Refs.

关2,3兴 for electrical conduction through graphene 共a two- dimensional carbon lattice with a Dirac point in the spec- trum兲. Both the electronic and optical problems are governed by the same Dirac equation inside the medium, but the cou- pling to the outside space is different. In the electronic prob- lem, the coupling can become nearly ideal for electrical con- tacts made out of heavily doped graphene 关2,3兴, or by suitably matching the Fermi energy in metallic contacts 关4,5兴. An analogous freedom does not exist in the optical case.

The major part of our analysis is therefore devoted to the question how nonideal interfaces affect the dependence of I on␻and L. Our conclusion is that

I/I₀=⌫0W/L 共1.1兲

at the Dirac point, with I₀the incident current per mode and

⌫0 an effective interface transparency. The properties of the

interfaces determine the proportionality constant⌫0, and they also determine whether I as a function of␻has a minimum or a maximum near the Dirac point, but they leave the 1 / L scaling unaffected.

In Sec. II we formulate the wave equations inside and outside the medium. The Helmholtz equation in free space is matched to the Dirac equation inside the photonic crystal by means of an interface matrix in Sec. III. This matrix could be calculated numerically, for a specific model for the termina- tion of the crystal, but to arrive at general results we work with the general form of the interface matrix共constrained by the requirement of current conservation兲. The mode depen- dent transmission probability through the crystal is derived in Sec. IV. It depends on a pair of interface parameters for each of the two interfaces. In Sec. V we then show that the extremal transmission near the Dirac point scales ⬀1/L re- gardless of the values of these parameters. We conclude in Sec. VI with suggestions for experiments.

II. WAVE EQUATIONS

We consider a two-dimensional photonic crystal consist- ing of a triangular or honeycomb lattice in the x-y plane formed by cylindrical air-filled holes along the z axis in a dielectric medium 共see Fig. 1兲. The crystal has a width W along the y direction and a length L along the x direction, both dimensions being large compared to the lattice constant a. Monochromatic radiation共frequency␻兲 is incident on the plane x = 0, with the electric field E共x,y兲e^i␻tpolarized along the z axis.

In the free space outside of the photonic crystal共x⬍0 and x⬎L兲 the Maxwell equations reduce to the Helmholtz equa- tion

共⳵x 2+⳵y

2兲E共x,y兲 +␻²

c²E共x,y兲 = 0. 共2.1兲 The mean 共time averaged兲 photon number flux in the x di- rection is given by关6兴

jH= ␧0c²

4iប␻²

冉

^E*⳵⳵^{Ex − E}

⳵^E*

⳵^x

冊

^{. 共2.2兲}

Inside the photonic crystal共0⬍x⬍L兲 the Maxwell equa- tions reduce to the Dirac equation关1兴

冉

^{− iv}D共⳵⁰x+ i⳵y兲 ^{− ivD共⳵}0^{x− i⳵y兲}

冊冉

^⌿⌿¹2

冊

^{=共␻−␻D兲}

冉

^⌿⌿¹2

冊

^,

共2.3兲 for the amplitudes ⌿1, ⌿2 of a doublet of two degenerate Bloch states at one of the corners of the hexagonal first Bril- louin zone.

As explained by Raghu and Haldane关1,7兴, the modes at the six zone corners Kp, K_p⬘共p=1,2,3兲, which are degenerate for a homogeneous dielectric, are split by the periodic dielec- tric modulation into a pair of doublets at frequency␻Dand a pair of singlets at a different frequency. The first doublet and singlet have wave vectors at the first set of equivalent corners Kp, while the second doublet and singlet are at K⬘_p^{. Each} doublet mixes and splits linearly forming a Dirac point as the wave vector is shifted by␦k from a zone corner. The Dirac equation共2.3兲 gives the envelope field ⬀e^i␦k·rof one of these doublets.

The frequency␻Dand velocity vDin the Dirac equation depend on the strength of the periodic dielectric modulation, tending to␻D= c⬘^兩Kp兩=c⬘^兩K⬘p兩=4␲^c⬘^{/ 3a andvD= c}⬘^{/ 2 in the} limit of weak modulation.共The speed of light c⬘ ^{in the ho-} mogeneous dielectric is smaller than the free space value c.兲

Equation共2.3兲 may be written more compactly as

− iv_D共⵱ ·␴兲⌿ =␦␻⌿, ␦␻⬅␻⁻␻D, 共2.4兲 in terms of the spinor⌿=共⌿1,⌿2兲 and the vector of Pauli matrices␴⁼共␴x,␴y兲. In the same notation, the velocity op- erator for the Dirac equation isv_D␴. The mean photon num- ber flux jDin the x direction is therefore given by

jD=vD⌿^*␴x⌿ = vD共⌿₁^*⌿2+⌿₂^*⌿1兲. 共2.5兲 The termination of the photonic crystal in the y direction introduces boundary conditions at the edges y = 0 and y = W which depend on the details of the edges, for example, on edges being of zigzag, armchair, or other type. For a wide and short crystal, WⰇL, these details become irrelevant and we may use periodic boundary conditions 关⌿共x,0兲

=⌿共x,W兲兴 for simplicity.

III. WAVE MATCHING

The excitation of modes near a Dirac point has been dis- cussed by Notomi关8兴, in terms of a figure similar to Fig.2.

Because the y component of the wave vector is conserved across the boundary at x = 0, the doublet near K₁=共Kx, Ky兲 or K₂=共−Kx, K_y兲 can only be excited if the incident radiation has a wave vector k =共kx, k_y兲 with kynear K_y. The conserva- tion of k_yholds up to translation by a reciprocal lattice vec- tor. We will consider here the case of 兩k兩⬍兩Kp兩, where no coupling to K₃ is allowed. The actual radius of the equal frequency contour in the free space at␻⁼␻Dwill depend on a particular photonic crystal realization.

The incident plane waves E_incident= E₀e^ik·r in free space that excite Bloch waves at a frequency␦␻⁼␻⁻␻Dhave ky

= Ky关1+O共␦␻^/␻D兲兴 and kx= k₀关1+O共␦␻^/␻D兲兴 with

k₀=

冑

共␻D/c兲²− K_y². 共3.1兲 For␦␻Ⰶ␻Dwe may therefore write the incident wave in the form

x

y W

L a

FIG. 1. 共Color online兲 Photonic crystal formed by a dielectric medium perforated by parallel cylindrical holes on a triangular lat- tice共upper panel: front view; lower panel: top view兲. The dashed lines indicate the radiation incident on the armchair edge of the crystal, with the electric field polarized in the z direction.

k_x k_y

K

K’

1

1 K₂

K’₂ K’₃

K₃

ω > ω_D

k₀

ω

k

ω

ω_D

-K K k

FIG. 2. Right panels: Hexagonal first Brillouin zone of the pho- tonic crystal共top兲 and dispersion relation of the doublet near one of the zone corners共bottom兲. Filled and open dots distinguish the two sets of equivalent zone corners, centered at K_pand K_p⬘, respectively.

The small circles centered at the zone corners are the equal- frequency contours at a frequency␻ just above the frequency ␻Dof the Dirac point. Left panels: Equal-frequency contour in free space 共top兲 and corresponding dispersion relation 共bottom兲. A plane wave in free space with k_x close to k₀ 共arrows in the upper left panel兲 excites Bloch waves in the photonic crystal with k close to K₁and K₂共arrows in the upper right panel兲, as dictated by conservation of k_yand␻ 共dotted horizontal lines兲.

SEPKHANOV, BAZALIY, AND BEENAKKER PHYSICAL REVIEW A 75, 063813共2007兲

063813-2

E_incident共x,y兲 = E+共x,y兲e^ik0x+iKyy, 共3.2兲 with E₊ a slowly varying function. Similarly, the reflected wave will have ky⬇Kyand kx⬇−k0, so that we may write it as

E_reflected共x,y兲 = E−共x,y兲e^−ik0x+iKyy, 共3.3兲 with E₋slowly varying.

The orientation of the Brillouin zone shown in Fig. 2 corresponds to an armchair edge of the triangular lattice at x = 0. For this orientation only one of the two inequivalent doublets is excited for a given k_y.共The other doublet at K₁⬘^, K₂⬘ is excited for −ky.兲 A 90° rotation of the Brillouin zone would correspond to a zigzag edge. Then a linear combina- tion of the two inequivalent doublets is excited near ky= 0.

For simplicity, we will restrict ourselves here to the case shown in the figure of separately excitable doublets.

While the conservation of the wave vector component parallel to the boundary determines which modes in the pho- tonic crystal are excited, it does not determine with what strength. For that purpose we need to match the solutions of the Helmholtz and Dirac equations at x = 0. The matching should preserve the flux through the boundary, so it is con- venient to write the flux in the same form at both sides of the boundary.

The photon number flux共2.2兲 for the Helmholtz equation may be written in the same form as the flux 共2.5兲 for the Dirac equation, by

jH=vHE^*␴xE, 共3.4a兲

vH=␧0c²k₀

4ប␻^{2, E =}

冉

^EE+⁺− E^{+ E−}₋

冊

^{. 共3.4b兲}

共In the prefactor k0 we have neglected corrections of order

␦␻^/␻D.兲 Flux conservation then requires

v_HE^*␴xE =v_D⌿^*␴x⌿, at x = 0. 共3.5兲 The matching condition has the general form关9兴

⌿ = 共vH/vD兲^1/2ME, at x = 0. 共3.6兲 The flux conservation condition共3.5兲 implies that the trans- fer matrix M should satisfy a generalized unitarity condition, M⁻¹=␴xM^†␴x. 共3.7兲 Equation共3.7兲 restricts M to a three-parameter form

M = e^␥␴ze^␤␴ye^i␣␴x 共3.8兲 共ignoring an irrelevant scalar phase factor兲. The real param- eters␣^,␤^,␥depend on details of the boundary at the scale of the lattice constant—they cannot be determined from the Helmholtz or Dirac equations共the latter only holds on length scalesⰇa兲.

We now show that the value of␣becomes irrelevant close to the Dirac point. At the boundary the incident and reflected waves have the form

Eincident= E₀

冉

¹1

冊

^{, Ereflected= rE0}

冉

− 1¹

冊

^{, 共3.9兲}

with r the reflection coefficient and E₀⬅E+共0,y兲 a slowly varying function. Both “spinors” are eigenvectors of ␴x, hence the action of e^i␣␴xonE is simply a phase factor

MEincident= e^␥␴ze^␤␴ye^i␣Eincident,

MEreflected= e^␥␴ze^␤␴ye^−i␣Ereflected. 共3.10兲 There is no need to determine the phase factor e^±i␣, since it has no effect on the reflection probability兩r兩².

A similar reasoning applies at the boundary x = L, where the matching condition reads

⌿ = 共vH/vD兲^1/2M⬘E, at x = L. 共3.11兲 Flux conservation requires that M⬘^{= e␥⬘␴ze␤⬘␴yei␣⬘␴x, with} real parameters␣⬘^,␤⬘^,␥⬘. The value of␣⬘is again irrelevant close to the Dirac point, because the spinor of the transmitted wave

Etransmitted= tE₀

冉

¹¹

冊

^共3.12兲

共with t the transmission coefficient兲 is an eigenvector of␴x. So

M⬘^Etransmitted= e^␥⬘␴ze^␤⬘␴ye^i␣⬘Etransmitted, 共3.13兲 with a phase factor e^i␣⬘that has no effect on the transmission probability兩t兩².

IV. TRANSMISSION PROBABILITY

We consider the case WⰇL of a wide and short crystal, when we may use periodic boundary conditions at y = 0 , W for the Bloch waves ⌿⬀e^i␦k·r. The transverse wave vector

␦^kyis then discretized at␦^ky= 2␲^{n / W}⬅qn, with mode index n = 0 , ± 1 , ± 2 , ± 3 , . . .. We seek the transmission amplitude tn

of the nth mode.

We first determine the transfer matrix M_n共x,0兲 of the nth mode⌽n共x兲e^iqnythrough the photonic crystal, defined by

⌽n共x兲 = Mn共x,0兲⌽n共0兲. 共4.1兲 From the Dirac equation 共2.4兲 we obtain the differential equation

d

dxMn共x,0兲 =

冉

^iv␦␻D

␴x+ qn␴z

冊

^{Mn共x,0兲, 共4.2兲}

with solution

Mn共x,0兲 = cos knx +sin knx kn

冉

ⁱv^␦␻D

␴x+ qn␴z

冊

^{. 共4.3兲}

We have defined the longitudinal wave vector

kn=

冑

共␦␻^/vD兲²− q_n². 共4.4兲 The total transfer matrix through the photonic crystal, includ- ing the contributions 共3.6兲 and 共3.11兲 from the interfaces at x = 0 and x = L, is

M = M⬘^{−1Mn共L,0兲M. 共4.5兲} It determines the transmission amplitude by

M

冉

^{1 + r1 − rnn}

冊

⁼

冉

^ttnn

冊

^⇒

冉

^{1 − r1 + rnn}

冊

=M^†

冉

^ttnn

冊

^⇒t1n

=1 2

兺

i=1

2

兺

j=1 2

Mij

*, 共4.6兲

where we have used the current conservation relation M⁻¹

=␴xM^†␴x.

The general expression for the transmission probability Tn=兩tn兩²is rather lengthy, but it simplifies in the case that the two interfaces at x = 0 and x = L are related by a reflection symmetry. For a photonic crystal that has an axis of symme- try at x = L / 2 both⌽共x兲 and␴y⌽共L−x兲 are solutions at the same frequency. This implies for the transfer matrix the sym- metry relation

␴yM␴y=M⁻¹⇒␴yM⬘␴y= M⇒␤⬘⁼␤, ␥⬘^{= −}␥, 共4.7兲 and we obtain

1 Tn

=

冉

^{␦␻vsin k}Dkn^nL

cosh 2␤^{− cos k}nL sinh 2␤^{sinh 2}␥

−qnsin knL kn

sinh 2␤^{cosh 2}␥

冊

²⁺

冉

^{cos knL cosh 2␥}

+qnsin knL kn

sinh 2␥

冊

^{2. 共4.8兲}

For an ideal interface共when ␤^{= 0 =}␥兲 we recover the trans- mission probability of Ref.关3兴.

At the Dirac point, where␦␻= 0⇒kn= iqn, Eq. 共4.8兲 re- duces further to

1 Tn

= cosh²共qnL + 2␥兲 + sinh²2␤^sinh2共qnL + 2␥兲.

共4.9兲 More generally, for two arbitrary interfaces, the transmission probability at the Dirac point takes the form

1 Tn

= cosh²共␤⁻␤⬘^兲cosh2␰n+ sinh²共␤⁺␤⬘^兲sinh2␰n,

␰n= qnL +␥−␥⬘^. 共4.10兲 While the individual T_n’s depend on ␥ ^and␥⬘, this depen- dence drops out in the total transmission兺nTn.

V. PHOTON CURRENT

The transmission probabilities determine the time aver- aged photon current I at frequency␻D+␦␻through the pho- tonic crystal

I共␦␻兲 = I0

兺

n=−⬁

⬁

Tn共␦␻兲, 共5.1兲

where I₀ is the incident photon current per mode. The sum over n is effectively cut off at兩n兩 ⬃W/LⰇ1, because of the exponential decay of the T_n’s for larger兩n兩. This large num- ber of transverse modes excited in the photonic crystal close to the Dirac point corresponds in free space to a narrow range␦␾⯝a/LⰆ1 of angles of incidence. We may therefore assume that the incident radiation is isotropic over this range of angles␦␾, so that the incident current per mode I₀does not depend on n.

Since W / LⰇ1 the sum over modes may be replaced by an integration over wave vectors 兺_n=−^⬁ _⬁→共W/2␲兲兰₋^⬁_⬁dq_n. The resulting frequency dependence of the photon current around the Dirac frequency is plotted in Figs.3 and4, for several values of the interface parameters. As we will now discuss, the scaling with the separation L of the interfaces is

β = 1 β = 0.5 β = 0.3β = 0

(L/vD)δω (L/W)I/I0

10 5 0 -5 -10 2.5

2 1.5 1 0.5 0

γ = 1 γ = 0.5 γ = 0.3γ = 0

(L/vD)δω (L/W)I/I0

10 5 0 -5 -10 2.5

2 1.5 1 0.5 0

FIG. 3. Frequency dependence of the transmitted current, for interface parameters␤⬘^{=␤, ␥}⬘^{= −}␥. In the top panel we take ␥=0 and vary␤, while in the bottom panel we take ␤=0 and vary ␥. The solid curves共␤=␥=0兲 correspond to maximal coupling of the pho- tonic crystal to free space. The curves are calculated from Eqs.共4.8兲 and共5.1兲, in the regime W/LⰇ1 where the sum over modes may be replaced by an integration over transverse wave vectors.

β = γ = 1 β = γ = 0.5 β = γ = 0.3

(L/vD)δω (L/W)I/I0

10 5 0 -5 -10 2.5

2 1.5 1 0.5 0

FIG. 4. Same as Fig.3for␤ and ␥ both nonzero.

SEPKHANOV, BAZALIY, AND BEENAKKER PHYSICAL REVIEW A 75, 063813共2007兲

063813-4

fundamentally different close to the Dirac point than it is away from the Dirac point.

Substitution of Eq.共4.10兲 into Eq. 共5.1兲 gives the photon current at the Dirac point

I共␦␻= 0兲 = I0⌫0

W L,

⌫0=arctan关sinh共␤⁺␤⬘兲/cosh共␤⁻␤⬘兲兴

␲^sinh共␤⁺␤⬘兲cosh共␤⁻␤⬘兲 , 共5.2兲 independent of the parameters␥^,␥⬘. For two ideal interfaces we reach the limit

lim

␤,␤⬘^→0I共␦␻= 0兲/I0= 1

␲ W

L, 共5.3兲

in agreement with Refs. 关2,3兴. Equation 共5.2兲 shows that, regardless of the transparency of the interfaces at x = 0 and x = L, the photon current at the Dirac point is inversely pro- portional to the separation L of the interfaces共as long as a ⰆLⰆW兲.

As seen in Figs.3 and4, the photon current at the Dirac point has an extremum共minimum or maximum兲 when either

␥^or␤are equal to zero. If the interface parameters␤^,␥^are both nonzero, then the extremum is displaced from the Dirac point by a frequency shift␦␻c. The photon current I共␦␻c兲 at the extremum remains inversely proportional to L as in Eq.

共5.2兲, with a different proportionality constant ⌫0 共which now depends on both␤^and␥兲.

The 1 / L scaling of the photon current applies to a fre- quency interval兩␦␻兩ⱗvD/ L around the Dirac frequency␻D. For 兩␦␻兩ⰇvD/ L the photon current approaches the L-independent value

I_⬁= I₀⌫W␦␻

␲^vD

, 共5.4兲

with rapid oscillations around this limiting value. The effec- tive interface transmittance ⌫ is a rather complicated func- tion of the interface parameters␤^,␤⬘^,␥^,␥⬘. It is still some- what smaller than unity even for maximal coupling of the photonic crystal to free space共⌫=␲^{/ 4 for}␤=␥= 0兲.

VI. CONCLUSION

While several experiments 关10,11兴 have studied two- dimensional photonic crystals with a honeycomb or triangu- lar lattice, the emphasis has been on the frequency range where the band structure has a true gap, rather than on fre- quencies near the Dirac point. Recent experiments on elec- tronic conduction near the Dirac point of graphene have shown that this singularity in the band structure offers a qualitatively new transport regime 关12兴. Here we have ex- plored the simplest optical analog, the pseudodiffusive trans- mission extremum near the Dirac point of a photonic crystal.

We believe that photonic crystals offer a particularly clean and controlled way to test this prediction experimentally. The experimental test in the electronic case is severely hindered by the difficulty to maintain a homogeneous electron density throughout the system 关13兴. No such difficulty exists in a photonic crystal.

If this experimental test is successful, there are other un- usual effects at the Dirac point waiting to be observed. For example, disorder has been predicted to increase—rather than decrease—the transmission at the Dirac point关14–16兴.

Photonic crystals could provide an ideal testing ground for these theories.

ACKNOWLEDGMENTS

We have benefited from discussions with A. R. Akhmerov, Ya. M. Blanter, and M. de Dood. This research was sup- ported by the Dutch Science Foundation NWO/FOM.

关1兴 F. D. M. Haldane and S. Raghu, e-print arXiv:cond-mat/

0503588; S. Raghu and F. D. M. Haldane, e-print arXiv:cond- mat/0602501.

关2兴 M. I. Katsnelson, Eur. Phys. J. B 51, 157 共2006兲.

关3兴 J. Tworzydło, B. Trauzettel, M. Titov, A. Rycerz, and C. W. J.

Beenakker, Phys. Rev. Lett. 96, 246802共2006兲.

关4兴 H. Schomerus, e-print arXiv:cond-mat/0611209.

关5兴 Ya. M. Blanter and I. Martin, e-print arXiv:cond-mat/0612577.

关6兴 H. van Houten and C. W. J. Beenakker, in Analogies in Optics and Micro Electronics, edited by W. van Haeringen and D.

Lenstra共Kluwer, Dordrecht, 1990兲.

关7兴 Reference 关1兴 considers a photonic crystal formed by dielectric cylinders in air, while we consider the inverse geometry of cylindrical perforations of a dielectric medium. Both geom- etries have a Dirac point in the band structure, see M. Plihal

and A. A. Maradudin, Phys. Rev. B 44, 8565共1991兲.

关8兴 M. Notomi, Phys. Rev. B 62, 10696 共2000兲.

关9兴 T. Ando, S. Wakahara, and H. Akera, Phys. Rev. B 40, 11609 共1989兲.

关10兴 D. Cassagne, C. Jouanin, and D. Bertho, Appl. Phys. Lett. 70, 289共1997兲.

关11兴 J.-Y. Ye and S. Matsuo, J. Appl. Phys. 96, 6934 共2004兲.

关12兴 A. K. Geim and K. S. Novoselov, Nat. Mater. 6, 183 共2007兲.

关13兴 A. H. Castro Neto and E.-A. Kim, e-print arXiv:cond-mat/

0702562.

关14兴 M. Titov, e-print arXiv:cond-mat/0611029.

关15兴 A. Rycerz, J. Tworzydło, and C. W. J. Beenakker, e-print arXiv:cond-mat/0612446.

关16兴 P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, e-print arXiv:cond-mat/0702115.