Physics Letters A 374 (2010) 3370–3372

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Physics Letters A

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Simultaneous propagation of two Dyakonov–Tamm wavesguided by the planar interface created in a chiral sculptured thin filmby a sudden change of vapor flux direction

Jun Gao a,b,∗, Akhlesh Lakhtakia a, Mingkai Lei b

a NanoMM—Nanoengineered Metamaterials Group, Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA 16802, USAb Surface Engineering Laboratory, School of Materials Science and Engineering, Dalian University of Technology, Dalian 116024, People’s Republic of China

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 April 2010Received in revised form 8 June 2010Accepted 11 June 2010Available online 19 June 2010Communicated by P.R. Holland

Keywords:Dyakonov–Tamm wavesStructurally chiral materialSculptured thin film

While growing a chiral sculptured thin film by physical vapor deposition, a sudden change in the direc-tion of the vapor flux creates a plane of discontinuity. In effect, this is a planar interface between twodifferent structurally chiral materials. Theoretical analysis shows that more than one Dyakonov–Tammwaves, with different phase speeds and degrees of localization to this interface, may propagate in differ-ent directions guided by the interface.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

A structurally chiral material, exemplified by cholesteric liquidcrystals (CLCs) and chiral sculptured thin films (STFs), has a pe-riodic nonhomogeneity which is born of a continuous rotation ofanisotropic dielectric properties at a uniform rate along a specificaxis. CLCs are formed by the self-ordering of aciculate molecules ina stack of parallel sheets with successive sheets progressively ro-tating about the axis of stacking [1]. A chiral STF is a solid thinfilm whose helical morphology is nanoengineered by obliquely di-recting a collimated vapor flux in high vacuum towards a planarsubstrate that rotates about a fixed axis at a constant rate [2]. Theangle χv between the average direction of the vapor flux and thesubstrate plane determines the dielectric properties of the chiralSTF [3,4]. A sudden change of χv during the fabrication of a chiralSTF will create a plane of discontinuity. As this is, in effect, a pla-nar interface between two different structurally chiral materials, itcan be expected to guide surface waves [5].

These surface waves will have to be classified as Dyakonov–Tamm waves because they combine the features of Dyakonov sur-face waves [6] and Tamm states [7]. Dyakonov surface waves prop-agate localized to the planar interface of two homogeneous non-

* Corresponding author at: NanoMM—Nanoengineered Metamaterials Group, De-partment of Engineering Science and Mechanics, Pennsylvania State University, Uni-versity Park, PA 16802, USA.

E-mail address: jxg42@psu.edu (J. Gao).

0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2010.06.042

conducting materials, at least one of which is anisotropic. Sincethese surface waves can exist only when very restrictive condi-tions are satisfied, their existence was experimentally confirmedonly very recently [8]. Tamm electronic states are localized to thesurface of a solid occupying a half space, and were experimentallyobserved in 1990 on the surfaces of superlattices [9]. Comparedto Dyakonov surface waves, Dyakonov–Tamm waves have a muchlarger range of the propagation direction in the interface plane[10,11].

Earlier [11–13], we studied the propagation of Dyakonov–Tammwaves guided by the planar interface of two chiral STFs that aredissimilar only either in orientation or in structural handedness.In this Letter, we present a theoretical investigation of Dyakonov–Tamm waves guided by the plane of discontinuity supposedly cre-ated during the fabrication of a chiral STF by a sudden changein χv , while the substrate rotation speed was concurrently alteredso that the periodicity was not affected. As shown in Fig. 1, χ±are the angles of inclination of the helical nanocolumns that thechiral STF comprises on the two sides of the plane of discontinuityz = 0, and 2Ω is the structural period. Without loss of generality,the Dyakonov–Tamm wave is taken to propagate along the x axisin the xy plane, and it must decay as z → ±∞.

Section 2 outlines the relevant boundary-value problem whichyields the dispersion equation for Dyakonov–Tamm waves. Sec-tion 3 contains numerical results for a chiral STF made of titaniumoxide. An exp(−iωt) time-dependence is implicit, with ω denotingthe angular frequency. The free-space wavenumber and the free-space wavelength are denoted by k0 = ω

√ε0μ0 and λ0 = 2π/k0,

J. Gao et al. / Physics Letters A 374 (2010) 3370–3372 3371

Fig. 1. (Color online.) Schematic of the plane of discontinuity created in a chiralsculptured thin film by a sudden change in the direction of the vapor flux.

respectively, with μ0 and ε0 being the permeability and permit-tivity of free space. Vectors are underlined once but dyadics areunderlined twice. Cartesian unit vectors are identified as u x , u y ,and u z .

2. Theory

The frequency-dependent permittivity dyadic of the chiral STFis given by [2]

ε(z,ω) = ε0 S z(z) · S y(χ±) · ε±

ref (ω)

· S Ty

(χ±) · S T

z(z), z ≶ 0, (1)

where the reference relative permittivity dyadics

ε±ref (ω) = ε±

a (ω)uzu z + ε±b (ω)u xux + ε±

c (ω)u yu y, (2)

containing the relative permittivity scalars ε±a,b,c(ω), indicate both

frequency dependence and local orthorhombic symmetry. With theassumption that the chiral STF is structurally right-handed, thedyadic function S z(z) is written for both half spaces as

S z(z) = (u xu x + u yu y) cos

(π z

Ω+ γ

)

+ (u yux − u xu y) sin

(π z

Ω+ γ

)+ uzu z, (3)

where the angle γ allows us to consider the direction of surface-wave propagation arbitrarily with respect to the morphology of thechiral STF in the plane z = 0. The tilt dyadic

S y(χ±) = (u xu x + u zu z) cosχ±

+ (u zu x − u xu z) sinχ± + u yu y (4)

involves the angles of inclination χ± . The superscript T denotesthe transpose. The quantities ε±

a,b,c(ω) and χ± depend on the val-ues χ±

v of the vapor flux angle χv for z ≷ 0.As γ ∈ [0,2π) is arbitrary, we took the Dyakonov–Tamm wave

to propagate parallel to the x axis in the xy plane with all fieldsindependent of the y coordinate. Accordingly, the fields were rep-resented as [11]

E(r) = e(z)exp(iκx)

H(r) = h(z)exp(iκx)

}∀z, (5)

where ω/κ is the phase speed along the x axis. Following the pro-cedure described by Gao et al. [11–13], we substituted Eqs. (5) inthe frequency-domain Maxwell curl postulates to obtain separate4×4 matrix ordinary differential equations (MODEs) for z > 0 andz < 0. The standard boundary conditions were enforced across theplane z = 0, and the additional conditions that the fields must de-cay as z → ±∞ were imposed on the solutions of the MODEs, inorder to obtain the dispersion equation for the Dyakonov–Tammwave.

Fig. 2. Variation of κ/k0 with γ for χ+v = 30◦ and χ−

v = 7.2◦ .

3. Numerical results and discussion

In order to illustrate the characteristics of the Dyakonov–Tammwave supported by the plane of discontinuity z = 0, we fixed λ0 =633 nm and Ω = 197 nm, and the chiral STF was chosen to bemade of titanium oxide with [3,4]

ε±a = [1.0443 + 2.7394(

χ±v

π/2 ) − 1.3697(χ±

vπ/2 )2]2

ε±b = [1.6756 + 2.5649(

χ±v

π/2 ) − 0.7825(χ±

vπ/2 )2]2

ε±c = [1.3586 + 2.1109(

χ±v

π/2 ) − 1.0554(χ±

vπ/2 )2]2

tanχ± = 2.8818 tanχ±v

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

, (6)

where the values χ±v of the vapor flux angle are in radian. The

angles γ and χ±v were left as variables. A sequential combination

of standard numerical methods—the search, the bisection, and theNewton–Raphson methods [14]—was employed to solve the dis-persion equation for κ � 3k0, γ ∈ [0◦,180◦] and χ±

v ∈ (0◦,90◦].Let us begin with χ+

v = 30◦ and χ−v = 7.2◦ so that χ+ ≈ 59◦

and χ− ≈ 20◦ . Fig. 2 depicts computed values of the relativewavenumber κ/k0 for possible propagation of Dyakonov–Tammwaves against γ in 10◦-increments. These solutions of the disper-sion equation are organized in two branches: in the first branch,κ/k0 decreases from 1.840 to 1.725 as γ increases from 0◦ to 170◦;in the second branch, κ/k0 decreases from 1.847 to 1.840 as γ in-creases from 150◦ to 180◦ . When γ lies between 150◦ to 170◦ ,two solutions exist, one from the first branch and the other fromthe second branch. The value of κ/k0 at γ = 0◦ is the same as thatat γ = 180◦ due to the symmetry.

Although the depicted values of κ/k0 are solutions of the dis-persion equation, they may not necessarily indicate the propaga-tion of surface waves. Therefore, the time-averaged Poynting vectorP (z) on both sides of the plane z = 0 was examined for severalrepresentative solutions. As an example, the x-directed componentPx(z), after suitable normalization [11], is plotted against z/Ω forγ = 30◦ in Fig. 3. We see that the magnitude of Px(z) decays witha periodic variation for z > 0 (χ+

v = 30◦) and becomes negligiblysmall for z > 4Ω . On the other side (χ−

v = 7.2◦) of the interface,Px(z) decays much faster and is essentially confined to z > −Ω;the periodic variations for z < 0 can only be appreciated on a mag-nified scale. The variations of P x(z) allow us to conclude that thesolution we obtained indicates the existence of a surface wave—the Dyakonov–Tamm wave. This conclusion holds also for all othersolutions in Fig. 2.

Although the fields decay very rapidly as z → −∞ almost in-dependently of γ , their decay as z → ∞ does depend significantlyon γ . We found that the degree of localization in the half spacez > 0 to the plane z = 0 is maximum when γ ≈ 70◦ .

3372 J. Gao et al. / Physics Letters A 374 (2010) 3370–3372

Fig. 3. The x-directed component of the normalized time-averaged Poynting vectoras a function of z for γ = 30◦ , χ+

v = 30◦ , and χ−v = 7.2◦ .

Fig. 4. Variation of κ/k0 with γ for χ+v = 50◦ and χ−

v = 10◦ .

Clearly, the interface cannot exist if χ−v = χ+

v . In order to exam-ine the effect of |χ+

v −χ−v | on the propagation of Dyakonov–Tamm

waves, we also calculated κ/k0 as a function of γ for χ+v = 50◦

and χ−v = 10◦ . As shown in Fig. 4, the solutions are again orga-

nized in two branches, one decreasing from 2.087 to 2.054 as γincreases from 0◦ to 140◦ , and the other decreasing from 2.094 to2.087 as γ increases from 130◦ to 180◦ . Thus, two solutions ex-ist when γ lies between 130◦ and 140◦ . Comparing Figs. 2 and 4,we found that, with the increase of |χ+

v −χ−v |, (i) the phase speed

ω/κ decreases, (ii) the variability of the phase speed with the di-rection of propagation in the interface plane reduces, and (iii) thevalues of γ which allow two Dyakonov–Tamm waves—with differ-ing phase speeds—decrease in both magnitude and range.

Figs. 2 and 4 suggest the existence of a finite range of γ thatsupports the propagation of two different Dyakonov–Tamm waves,depending on the difference between χ+

v and χ−v . For illustration,

we fixed χ−v = 10◦ and varied χ+

v ∈ (0◦,90◦) to ascertain the min-imum value γmin and the maximum value γmax in this finite rangeof γ , and we found such a range to exist only for χ+

v ∈ (25◦,70◦),as shown in Fig. 5. With the increase of χ+

v beyond 25◦ , the maxi-mum value γmax first increases to its maximum 189◦ at χ+

v = 30◦and then decreases, whereas the minimum value γmin first de-creases to its minimum 128◦ at χ+

v = 60◦ and then increases.Herein, the solution at γ = 189◦ is the same as γ = 9◦ becauseof symmetry. The maximum width of the range of γ for the exis-tence of double solutions, γmax − γmin, is 38◦ at χ+

v = 30◦ .

Fig. 5. (Color online.) Variations of the minimum value γmin and the maximumvalue γmax of γ for the existence of double solutions with χ+

v when χ−v = 10◦ .

4. Concluding remarks

Chiral sculptured thin films are structurally chiral materialsgrown by directing a vapor flux obliquely towards a rotating sub-strate in high vacuum. A sudden change in the direction of thevapor flux creates a plane of discontinuity, which may be regardedas a planar interface between two different structurally chiral ma-terials. By solving a boundary-value problem wherein we assumedthat the structural periods on both sides of the interface are iden-tical, we found that more than one Dyakonov–Tamm waves, withdifferent phase speeds and degrees of localization to this inter-face, may propagate in different directions guided by the interface.All Dyakonov–Tamm waves are essentially confined to within twostructural periods on both sides of the interface.

Acknowledgements

JG gratefully acknowledges the financial support of ChineseScholarship Council (CSC). AL thanks the Charles Godfrey BinderEndowment at Penn State for ongoing support of his research ac-tivities.

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