transverse acoustic waves in piezoelectric zno/mgo and gan/aln fibonacci-periodic superlattices

Surface Science 624 (2014) 58–69

Contents lists available at ScienceDirect

Surface Science

j ourna l homepage: www.e lsev ie r .com/ locate /susc

Transverse acoustic waves in piezoelectric ZnO/MgO and GaN/AlNFibonacci-periodic superlattices

D. Martínez-Gutiérrez, V.R. Velasco ⁎Instituto de Ciencia de Materiales de Madrid (ICMM), Consejo Superior de Investigaciones Científicas (CSIC), c/ Sor Juana Inés de la Cruz 3, 28049 Madrid, Spain

⁎ Corresponding author. Tel.: +34 91 334 90 45; fax: +E-mail address: [email protected] (V.R. Velasco).

0039-6028/$ – see front matter © 2014 Elsevier B.V. All rihttp://dx.doi.org/10.1016/j.susc.2014.01.020

a b s t r a c t
a r t i c l e i n f o
Article history:Received 13 June 2013Accepted 17 January 2014Available online 6 February 2014

Keywords:SuperlatticesAcoustic wavesPhonons

This work studies the transverse acoustic waves, including the piezoelectric coupling, in Fibonacci superlatticesformed by wurtzite ZnO/MgO and GaN/AlN, respectively. We examine also other superlattice structures formedby combining different kinds of Fibonacci sequences and finite periodic systems. The possibility to use differentFibonacci sequences including layers with double length of one of the constituent materials produces importantmodifications in the dispersion curves. The effect is more important in the lower frequency range and affects thegaps appearing in this frequency range. It is also possible to find narrow and flat bands cutting the original gapsand producing narrower ones. There are modes at different frequency ranges having spatial confinement in oneof the constituent parts of the superlattice period.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

In recent years the propagation of elastic waves in systems with pe-riodic structures has been intensively studied. Superlattices are one ofthese systems and they are of interest for their applications in differentareas, such as acoustics. Detailed information on the properties ofacoustic waves in layered materials can be found in [1]. The effects ofthe compositional or positional disorder on the properties of periodicstructures have also been studied. One of the reasons for this interestis the localization or spatial confinement of the waves due to disorder.The disorder can be low, intermediate or high. The highest degree of dis-order is represented by the completely disordered system, where ran-dom perturbations are present [2]. The lowest degree of disorder isrepresented by the presence of a compositional or positional defect inthe periodic structure. In this case highly localized modes may appearas very narrow bands within the gaps of the periodic structure. Becauseof this, these systems could be employed as frequency filters [3].

In thewide range of disordered systems the aperiodic ones are an in-termediate case between the periodic and random systems. In thesestructures layers of, at least, two different materials can be arranged ac-cording to a given aperiodic sequence. The corresponding structures ex-hibit two different orders at different length scales. At the atomic levelthe crystalline order determined by the periodic disposition of atomsin each layer is present. On the other hand at longer scales the aperiodicorder dictated by the growth of the different layers according to a given

34 91 372 06 23.

ghts reserved.

aperiodic sequence can be found. This allows using the aperiodic orderas a tool to modify different physical phenomena, with their own phys-ical scales, by tuning the characteristic length scales. Many works havebeen done on the optical, electronic, vibrational and magnetic proper-ties of aperiodic systems based on different generating sequences [4–9].

The systems including aperiodic parts present interesting physicalproperties. This has been the case in the optical capabilities of aperiodicsystems concerning the second [10] and the third harmonic generation[11], as well as the localization of light in these systems [12,13]. Hybrid-order devices formed by periodic and Fibonacci (aperiodic) blocks havebeen found to exhibit complementary optical responses [14]. Perfectoptical transmission has been found in symmetric Fibonacci-class mul-tilayers [15,16]. Broad omnidirectional reflection bands have been pre-dicted when combining Fibonacci sequences and periodic 1D photoniccrystals [17]. Phonon confinement has been predicted in 1D periodic/Fibonacci structures [18,19], as well as in 1D Fibonacci systems withmirror symmetry [20–22].

Although the full character of the aperiodic systems would bereached for very high generation orders of the different sequences,many characteristics can be obtained with lower generation orders.We shall use this approach in which we shall study different Fibonaccisequences forming the period of a superlattice.

The vibrational spectrum of aperiodic systems presents a highlyfragmented character [23,24]. By means of combinations of Fibonacciand periodic layer structureswe canmodify the structure of the primaryand secondary gaps in the different frequency ranges. We have also anadditional freedom, because we can start the Fibonacci sequence witha block A formed by a slab of a given material followed by a block B

http://crossmark.crossref.org/dialog/?doi=10.1016/j.susc.2014.01.020&domain=pdf

http://dx.doi.org/10.1016/j.susc.2014.01.020

mailto:[email protected]

http://dx.doi.org/10.1016/j.susc.2014.01.020

http://www.sciencedirect.com/science/journal/00396028

59D. Martínez-Gutiérrez, V.R. Velasco / Surface Science 624 (2014) 58–69

formed by a slab of a different material, or vice versa, thus generatingdifferent layered systems.

As materials we shall consider wurtzite ZnO, MgO, GaN and AlN.Many properties of ZnO are quite similar to those of GaN. When com-pared to GaN, ZnO can be grown as single crystals [25] and they canbe used as substrates for the growth of thin film devices. In this wayhigh quality films can be grown by homoepitaxy and thus avoid theproblems associated with the dislocation formation due to epitaxialmismatch in GaN growth. Besides this the ZnO bandgap (3.37 eV) [26]is quite close to that of GaN (3.4 eV) [27]. It can also be varied in a sys-tematic way by alloyingwithMgO [28,29] in a similar way to the case ofGaN with AlN and InN. Although MgO does not appear naturally in thewurtzite structure, several experimental groups achieved the growthof ZnO/MgO multiple quantum wells [30–34]. A MgO layer can takewurtzite structure on a high-quality ZnO buffer layer if the thicknessof the MgO layer is less than 10 nm [35].

All these materials are piezoelectric. In the case of materials belong-ing to the 6mm class with the C-axis parallel to the x3 direction and thedirection x2 normal to the interface there is a decoupling between mo-tion along the x1 and x2 directions and themotion along the x3 direction,due to the symmetry [36]. We shall consider here only the transverseacoustic wave having the electric potential coupled to the elastic dis-placement u3. The reference system and the constituent materials areshown in schematic form in Fig. 1.

Although no experimental values for the electronic, elastic and piezo-electric data of MgO in the wurtzite structure are available, they canbe obtained from theoretical calculations [37]. The data for ZnO aretaken from [38,39], whereas those of GaN and AlN are taken from [40,41].

We shall study the transverse acoustic waves of Fibonacci super-lattices of the above materials. We shall consider also the cases ofmore complex superlattices including Fibonacci and periodic parts.

The theoretical model and the method of calculation are presentedin Section 2. The dispersion relations of the transverse acoustic wavesand the spatial behavior of some modes in the systems consideredhere are presented and discussed in Section 3. Conclusions are drawnin Section 4.

2. Model and method of calculation

We shall consider superlattices formed by the periodic repetition ofmaterial blocks according to the Fibonacci sequence. The A block is

Fig. 1. Sketch of the superlattice having as period a second Fibonacci generation (it coin-cides with the usual binary superlattice). The materials belong to the 6 mm class withthe C-axis parallel to the x3 direction and we choose the normal to the interfaces as thex2 direction. The different layers with the corresponding thicknesses and the nomencla-ture for the different interfaces employed in Section 2 are shown also. The case of morecomplex superlattice periods, including more interfaces, is a generalization of thispicture [43].

formed by layers of material 1, in our case: ZnO and GaN, respectively.The B block is formed by layers of material 2, in our case: MgO andAlN, respectively.

A finite Fibonacci generation is produced by recursive stacking withthese A and B blocks, mapping the mathematical rule in the Fibonaccisequence

S1 ¼ Af g ; S2 ¼ ABf g ; S3 ¼ ABAf g ;

S4 ¼ ABAABf g ; ⋯; Sn ¼ Sn−1Sn−2;

ð1Þ

or

S1 ¼ Bf g ; S2 ¼ BAf g ; S3 ¼ BABf g ;

S4 ¼ BABBAf g ; ⋯; Sn ¼ Sn−1Sn−2:

ð2Þ

We shall consider also more complex superlattice systems formedby combining different Fibonacci sequences and hybrid systems includ-ing a given Fibonacci generation together with a finite periodic repeti-tion of blocks AB (BA). All the superlattices considered here areobtained by stacking the material layers along the x2 axis. A schematicview of the systems is presented in Fig. 1. There we show the layers ofthe constituent materials together with the axes orientation and thick-nesses. The crystal axes of all the materials forming the superlatticesare aligned. Taking into account the transverse elastic isotropy of thehexagonal crystals we have to consider only the absolute value κ ofthe parallel wavevector κ in the interfaces.

To obtain the dispersion curves of the transverse acoustic waves inthe multilayer systems we shall employ the surface Green functionmatching (SGFM) method [42]. This method is specially well adaptedto study the properties ofmultilayer systems formed by anisotropicma-terials and including complex periodswithmany interfaces [43]. All theformal and practical details can be found in [42,43]. Thus we shall givehere only the detailed expressions for the Green's functions and normalderivatives particular to the case of the transverse acoustic waves, in-cluding the piezoelectric coupling, in hexagonal crystals. The essentialformal expressions for the study of the superlattices will be also given.

The bulk material Green's function is given in this case by [44]

G ¼ϵ11N

− e15N

e15N

C44k2−ρω2

Nk2

264375; ð3Þ

where

N ¼ Ck2−ϵ11ρω2

; C ¼ C44ϵ11 þ e215: ð4Þ

N = 0 gives the dispersion relation of the bulk piezoelectric trans-verse acoustic wave

ω2 ¼ v2t k2

; v2t ¼ Cρϵ11

: ð5Þ

Table 1Elastic, dielectric and piezoelectric constants, mass densities for ZnO [38,39], MgO [37],GaN [40,41] and AlN [40,41] and the velocities of the transverse acoustic waves obtainedfrom these data.

Material C44(GPa)

ϵ11(10−11 F m−1) e15(C m−2) ρ(103 kg m−3) vt(103 m s−1)

ZnO 42.5 7.38 −0.37 5.606 2.813MgO 59.0 8.766 −0.428 3.6 4.119GaN 105.0 8.58 −0.3 6.156 4.150AlN 116.0 7.52 −0.48 3.255 6.048

60 D. Martínez-Gutiérrez, V.R. Velasco / Surface Science 624 (2014) 58–69

The surface projected elements of the bulk Green function are ob-tained from

Gij κ;ω2� �

¼ limϵ→0

12π

Z ∞

−∞exp iϵkx2h i

Gij κ; kx2 ;ω2

� �dkx2 ; ð6Þ

κ being the parallel wavevector, kx2 being the perpendicularwavevector in the bulk and i being the imaginary unit, whereas the nor-mal derivatives are obtained from

′G�ij κ ;ω2� �

¼ limϵ→0

12π

Z ∞

−∞exp ∓iϵkx2h i

ikx2Gij κ; kx2 ;ω2

� �dkx2 ; ð7Þ

thus giving

G ¼ϵ112Cβ

− e152Cβ

e152Cβ

Cβ−e215κ2ϵ11Cκβ

26643775; ð8Þ

and

′G� ¼� ϵ112C

∓ e152C

� e152C

�C−e2152Cϵ11

26643775 ð9Þ

Fig. 2. Dispersion curves of the transverse acoustic waves of a ZnO/MgO superlattice having aFibonacci generation; and (d) a fifth Fibonacci generation.

where

β ¼ κ2− ϵ11ρω2

C

!12

: ð10Þ

We need also

Gij κ ; x2; x′2;ω

2� �

¼ 12π

Z ∞

−∞exp ikx2 x2−x′2

� �h iGij κ ; kx2 ;ω

2� �

dkx2 ; ð11Þ

and

′Gij κ; x2; x′2;ω

2� �

¼ ∂∂x2

Gij κ; x2; x′2;ω

2� �

: ð12Þ

They are given by

G11 κ ; x2; x′2;ω

2� �

¼ ϵ112Cβ

exp −β x2−x′2��

;

G12 κ ; x2; x′2;ω

2� �

¼ − e152Cβ


;

G21 κ ; x2; x′2;ω

2� �

¼ e152Cβ


;

G22 κ ; x2; x′2;ω

2� �

¼ 12ϵ11

1κexp −κ x2−x′2

�� − e215

Cβexp −β x2−x′2

�� !;

ð13Þ

s period: (a) a second Fibonacci generation; (b) a third Fibonacci generation; (c) a fourth

image of Fig.�2


and

0G11 κ ; x2;x′2;ω

2� �

¼ −sgn x2−x′2� � �11

2Cexp −β x2−x′2

�� ;

0G12 κ ; x2;x′2;ω

2� �

¼ sgn x2−x′2� � e15


�� ;

0G21 κ ; x2;x′2;ω

2� �

¼ −sgn x2−x′2� � e15


�� ;

0G22 κ ; x2;x′2;ω

2� �

¼ −sgn x2−x′2� � 1

2�11

× exp −κ x2−x′2��

− e215C

exp −β x2−x′2�� !

:

ð14Þ

The boundary conditions at the different interfaces (continuity of theu3 displacement, the electric potential, the normal stress and the electricdisplacement) expressed in terms of these projections give the interfaceprojection of the Green function of the superlattice system [42], whoseformal representation is

eG−1s ¼ − eA−1

1 � eG−11 −eA−1

2 � eG−12

� �: ð15Þ

The generalization for N different interfaces is presented in [43] andshall not be discussed here.

Fig. 3. Same as Fig. 2 for a Mg

The elements entering in the former expression are defined in thefollowing way:

eG1 ¼ G1 e−iqDG1 l;mð ÞeiqDG1 r; pð Þ G1

" #;

eA1 ¼ A−11 e−iqDA1 l;mð Þ

−eiqDA1 r;pð Þ −Aþ1

" #;

ð16Þ

eG2 ¼ G2 G2 l; rð ÞG2 r; lð Þ G2

� �; eA2 ¼ Aþ

2 A2 l; rð Þ−A2 r; lð Þ −A−

2

� �; ð17Þ

where l,m, p and r denote the x2 coordinates from the interfaces shownin Fig. 1, D is the superlattice period, q is the normal wavevector associ-atedwith the superlattice periodicity and the dependencies on κ andω2

must be understood.The normal components of the stress and the electric displacement

are represented by the eA through theA andA� operators. In the presentcase they are given by

A ¼C44

∂G11

∂x2þ e15

∂G21

∂x2C44

∂G12

∂x2þ e15

∂G22

∂x2−e15

∂G11

∂x2þ ϵ11

∂G21

∂x2−e15

∂G12

∂x2þ ϵ11

∂G22

∂x2

26643775; ð18Þ

O/ZnO (BA) superlattice.

image of Fig.�3


and

A� ¼ C440G�

11 þ e150G�

21 C440G�

12 þ e150G�

22

−e150G�

11 þ ϵ110G�

21 −e150G�

12 þ ϵ110G�

22

" #: ð19Þ

The dispersion curves are obtained from the peaks of the local den-sity of states (LDOS), which is obtained from the trace of the interfaceprojection of the system Green function eGS [42,43].

To obtain the dispersion curves we study the LDOS as a function ofκD. For each κD value we sample the LDOS for 100 qD values rangingfrom zero to π, D being the period of the structure being considered.The allowed transverse acoustic branches have non-zero LDOS values.The forbidden gaps have LDOS zero values. A small imaginary part(0.001) is added to the frequency to perform the numerical calculations.

3. Transverse acoustic waves in Fibonacci superlattices and othercomplex structures

In order to illustrate the properties of the transverse acoustic wavesin these systems, we shall assume that the thicknesses of the differentlayers are d1(ZnO) = d1(GaN) = 1.7 nm, and d2(MgO) = d2(AlN) =4.2 nm. We choose these values because they are equal to those of thefirst Fibonacci superlattices grown experimentally [45]. It is easy tosee that in this case d1 = 0.29 D and d2 = 0.71 D. It is clear that otherthickness values could be chosen.

Fig. 4. Same as Fig. 2 for a Ga

In Table 1 we give the mass densities, elastic, dielectric and piezo-electric coefficients, togetherwith the velocities of the transverse acous-tic waves for the materials considered here.

We shall consider now the dispersion relation of the transverseacoustic waves of Fibonacci superlattices and other more complexsuperlattices whose periods are formed by Fibonacci generations andperiodic parts. The period length of these systemswill be quite different,due to its dependence on the number of A and B blocks.

We shall work with the reduced parallel wavevector κD and the re-duced frequency ωD

vM, vM being an average velocity as that of the classical

binary superlattice [46], given by

vM¼ v1v2D

v1d2 þ v2d1; ð20Þ

which reflects the inner structure of the supercell modulation.This can be generalized in the case of more complicated structures

having N1 layers of material 1 and N2 layers of material 2, thus giving

vM¼ v1v2D

v1D2 þ v2D1; ð21Þ

being D1 = N1d1, D2 = N2d2 and D = D1 + D2.We shall present in the following results for the second, third, fourth

and fifth generation Fibonacci superlattice, in order to see the evolutionof the dispersion curves with increasing order generation. We mustnote that the second order Fibonacci generation, having AB as period,

N/AlN (AB) superlattice.

image of Fig.�4


is nothingmore than the classical superlattice. In the sameway the thirdorder Fibonacci generation, having as period AAB or BBA is also a normalsuperlattice having A or B layers with double thickness.

Fig. 2(a) presents the dispersion curves of the transverse acousticwaves of a ZnO/MgO second generation Fibonacci superlattice.Fig. 2(b), (c) and (d) gives the dispersion curves for superlattices havingas periods a third, fourth and fifth Fibonacci generation, respectively.The first gap opens at κD = 0 at ωD=vM∼3, but is very narrow and it isnot seen at the scale of the figure.We can see in all the cases a wide sec-ond gap quite wide at non-zero κD values. There are higher gaps of sim-ilar characteristics. It can be seen that for the second and thirdgenerations the second gap is wider at higher κD values than the firstone. On the other hand for periods formed by higher order generationsthe second gap is wider at lower κD values than the first one. The fifthgeneration exhibits all the features found in the dispersion curves ofhigher order generation periods.We have calculated also the dispersioncurves neglecting the piezoelectric coupling. We have found no impor-tant changes in the dispersion curves. This is also true for all other sys-tems studied in this work. Nevertheless, in the case of piezoelectricmaterials the inclusion of the piezoelectric coupling is mandatory toget the transverse surface waves [36,47] and the transverse interfacewaves [48].

Fig. 3(a) presents the dispersion curves of the transverse acousticwaves for a MgO/ZnO (BA) second generation Fibonacci superlattice.Fig. 3(b), (c) and (d) gives the dispersion curves for superlattices havingas periods a third, fourth and fifth Fibonacci generation, respectively.We see here changes due to the different ordering of the A and B blocks.Now we have BB pairs instead of AA ones. We see now that from the

Fig. 5. Same as Fig. 2 for an Al

third generation onwards there is a closed and narrow gap just abovethe lower and wide gap. We observe also that a new feature appearsfrom the fourth generation onwards. This is a closed and very narrowgap in the bands above the lower wide gap seen before. The highergaps are narrower than those found in Fig. 2 for the AB ordering. As be-fore the fifth generation exhibits all the features found in higher ordergeneration periods.

Fig. 4(a) presents the dispersion curves of the transverse acousticwaves for a GaN/AlN (AB) second generation superlattice. Fig. 4(b), (c)and (d) gives the dispersion curves for superlattices having as periodsa third, fourth and fifth Fibonacci generation respectively. In this casewe see that we have reasonablywide gaps opening at κD=0, for differ-ent frequency ranges. From the fourth generation onwards we see thatthe first gap starting at κD= 0 becomes narrower and is only visible atnon-zero κD values. The fifth generation exhibits all the features foundin higher order generation periods.

Fig. 5(a) presents the dispersion curves of the piezoelectric trans-verse acoustic waves for an AlN/GaN (BA) second generationsuperlattice. Fig. 5(b), (c) and (d) gives the dispersion curves forsuperlattices having as periods a third, fourth and fifth Fibonacci gener-ation respectively. In this casewe see a similar situation to that found inFig. 3 for the MgO/ZnO (BA) system. From the fourth generation on-wards there is a first closed gap opening at zero κD values above thefirst gap.

Due to the differences seen in the dispersion curves of the sequencesordered following the AB or BA stacking in the Fibonacci rule, it is rea-sonable to look for the changes introduced in the dispersion curveswhen combining Fibonacci sequences with both ordering schemes.

N/GaN (BA) superlattice.

image of Fig.�5


This is presented in Fig. 6 for the ZnO/MgO system. Fig. 6(a) shows thedispersion curves of the piezoelectric transverse acoustic waves forthe ABBA system (second generation). This corresponds to a normalsuperlattice having double period. Fig. 6(b), (c) and (d) gives the disper-sion curves for superlattices with periods formed by joining bothstackings of the fourth, fifth and sixth Fibonacci generations, respective-ly. The case corresponding to the third generation coincideswith the bi-nary superlattice stacking ABABAB. In all the cases we see several lowergaps being wide at non-zero κD values and narrow higher gaps withclosed form.

As it has been quoted before, the mixing of Fibonacci aperiodicblocks together with periodic blocks opens the possibility for new fea-tures in the dispersion relations of the acoustic waves in multilayersystems.

In Fig. 7 we present the dispersion curves of a superlattice having asconstituents a ZnO/MgO fourth Fibonacci generation and a single ZnO/MgO block, with different orderings. Fig. 7(a) corresponds to theABAAB/AB case. Fig. 7(b) illustrates the ABAAB/BA case. Fig. 7(c) corre-sponds to the BABBA/AB case and Fig. 7(d) to the BABBA/BA case. Wecan see that the region of the lower gaps is quite different in all thecases considered. We must compare with the dispersion curves ofFigs. 2(c) and 3(c) corresponding to the fourth Fibonacci generationABAAB and BABBA respectively, in order to see the changes introduced.Fig. 7(a) shows only small differences when compared with Fig. 2(c).This can be easily understood because the additional AB block doesnot introduce any substantive change to the structure. On the other

Fig. 6.Dispersion curves of the transverse acousticwaves of ZnO/MgO superlattices having as peeration; (c) a fifth Fibonacci generation; and (d) a sixth Fibonacci generation.

hand the BB block introduced in the superlattice corresponding toFig. 7(b) was not present in the original structure. The differences intro-duced in the dispersion curves for the lower frequencies are clear andimportant, showing a wider first gap than in Fig. 2(c). Fig. 7(c) whencompared with Fig. 3(c) shows important differences due to the pres-ence of the AA pair not present in the fourth Fibonacci generationBABBA. We see now that the first gap is substituted by three narrowergaps. Fig. 7(d) shows no important differences when compared toFig. 3(c), because no new pairs are introduced in the structure.

The same is true for the GaN/AlN systems represented in Fig. 8.Fig. 8(a) and (d) shows almost no differences with Figs. 4(c) and 5(c),respectively, for the reasons discussed above. In Fig. 8(b) we can see awider first gap starting at κD = 0, than the one present in Fig. 4(c). Itis clear there that the two lower gaps are different in both cases. Thisis due to the presence of the BB pair in the structure. In Fig. 8(c) wesee the influence of the AA pair in the narrow bands that appear nowin the region of the first gap seen in Fig. 5(c).

In Fig. 9 we present the case of a GaN/AlN superlattice whose periodis formed by blocks (ABAB) or (BABA) sandwiched between two fourthorder Fibonacci generations. In this way we have four material slabsin the periodic part and five material slabs in the aperiodic one.We can see now differences in the lower gap region of Fig. 9(b) and(c)when comparedwith Figs. 4(c) and 5(c) respectively, for the reasonsdiscussed above. In the same way there are no important differen-ces in Fig. 9(a) and (d) when compared to Figs. 4(c) and 5(c),respectively.

riods (AB…|BA…) blocks of: (a) a second Fibonacci generation; (b) a fourth Fibonacci gen-

image of Fig.�6

Fig. 7.Dispersion curves of the transverse acoustic waves of ZnO/MgO superlattices having as periods the following stacking of layers: (a) ABAAB|AB; (b) ABAAB|BA; (c) BABBA|AB; and (d)BABBA|BA.


Fig. 10 presents a complementary case inwhich the period is formedby the fourth Fibonacci generation sandwiched between two AB or BAblocks respectively. The results are analogous to those of Fig. 9.

In the different superlattices studied here very narrow bands, forfixed κD values, are present, thus indicating the existence of flat, or al-most flat, bands.We shall illustrate this in Fig. 11 for the case of the dis-persion curves of Fig. 6(c) at κD = 26.0. Fig. 11 gives the dispersioncurves as a function of the superlattice reduced normal wavevectorqD. We see some flat bands at different frequency ranges.

We shall look now at the spatial distribution of the spectral strength,represented by the local density of states (LDOS as a function of x2/D),of some of the modes represented in Fig. 11. This is obtained from thesuperlattice Green function GS(κ,q,x2,x2′,ω2)) [42,43]. Fig. 12 gives thisinformation for the followingmodes: (a) (ωDvM

¼ 25:02; qD¼0); (b) (ωDvM¼

25:02; qD ¼ π); (c) (ωDvM¼ 31:68; qD ¼ 0); (d) (ωDvM

¼ 40:02; qD ¼ π);

(e) (ωDvM¼ 36:42; qD ¼ 0); and (f) (ωDvM

¼ 35:76; qD ¼ π). We see that

the lower frequency modes at the center and border of the Brillouinzone are confined to the domain of the (AB⋯) block in the superlattice pe-riod.We can see also that they are essentially localized in theAA zone cor-responding to 0.125 ≤ x2/D ≤ 0.197. There are intermediate frequencymodes extended along the whole period. On the other hand we findalso intermediate frequency modes predominantly confined to the do-main of the (BA⋯) block in the superlattice period. In this case they aremainly localized in the BB zone corresponding to 0.57 ≤ x2/D ≤ 0.75.Thus we see that in some of the systems analyzed here there are flat or

very narrow bands and that for some modes there exists the spatial con-finement of the elastic displacement and the electric potential in the dif-ferent parts of the total superlattice period.

We have found that by using the two kinds (AB⋯) and (BA⋯) ofFibonacci sequences we can modify the structure of the lower gap re-gions for the acoustic waves of the superlattice structures andmaterialsconsidered here. Somemodes are spatially confined in different parts ofthe superlattice period.

4. Conclusions

Wehave studied the transverse acoustic waves, including the piezo-electric coupling, of wurtzite ZnO/MgO and GaN/AlN Fibonaccisuperlattices. We have considered also the case of more complexsuperlattices formed by combining different sequences of Fibonacciand finite periodic blocks. The possibility to use (AB⋯) and (BA⋯)Fibonacci sequences including AA and BB pairs allows for importantmodifications of the lower gap region. It is possible to introduce narrowand flat bands that divide the original gaps in narrower ones. We havefound modes at different frequency ranges having spatial confinementin one of the constituent parts of the superlattice period. The case oftransverse acoustic waves propagating along symmetry directions of(100) and (110) interfaces of piezoelectric materials belonging to thecubic system is governed by equations similar to those of the hexagonalcrystals studied here. Thus we can expect similar results to those pre-sented here for analogous structures of cubic crystals.

image of Fig.�7

Fig. 8. Same as Fig. 7 for a GaN/AlN system.


Acknowledgments

This work has been supported by the Ministerio de Economía yCompetitividad (Spain) through Grant MAT2009-14578-C03-03. D.M.-G. acknowledges support from the FPI program of the SpanishMinisterio de Economía y Competitividad.

References

[1] E.H. El Boudouti, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, Surf. Sci. Rep. 64(2009) 471.

[2] D. Berdekas, S. Ves, J. Phys. Condens. Matter 21 (2009) 275405.[3] M. Trigo, A. Bruchausen, A. Fainstein, B. Jusserand, V. Thierry-Mieg, Phys. Rev. Lett.

89 (2002) 227402.[4] E. Maciá, F. Domínguez-Adame, Electrons, Phonons and Excitons in Low Dimension-

al Aperiodic Systems, Editorial Complutense, Madrid, 2000.[5] E.L. Albuquerque, M.G. Cottam, Phys. Rep. 376 (2003) 225.[6] W. Steurer, D. Sutter-Widmer, J. Phys. D 40 (2007) R229.[7] EnriqueMaciá Barber, Aperiodic Structures in CondensedMatter, Fundamentals and

Applications, , CRC Press, Boca Raton, 2009.[8] J. Milton Pereira Jr., R.N. Costa Filho, Phys. Lett. A 344 (2005) 71.[9] S.V. Grishin, E.N. Beginin, Yu.P. Saraevskii, S.A. Nikitov, Appl. Phys. Lett. 103 (2013)

022408.[10] S.N. Zhu, Y.Y. Zhu, Y.Q. Qin, H.F. Wang, C.Z. Ge, N.B. Ming, Phys. Rev. Lett. 78 (1997)

2752.[11] Y.B. Chen, C. Zhang, Y.Y. Zhu, H.T. Wang, N.B. Ming, Appl. Phys. Lett. 78 (2001)

577.[12] W. Gellermann, M. Kohmoto, B. Sutherland, P.C. Taylor, Phys. Rev. Lett. 72 (1994)

633.[13] T. Hattori, N. Tsurumachi, S. Kawato, S. Nakatsuka, Phys. Rev. B 50 (1994) 4220.[14] E. Maciá, Phys. Rev. B 63 (2001) 205421.[15] H. Huang, Y. Wang, C. Gong, J. Phys. Condens. Matter 11 (1999) 7645.

[16] X.Q. Huang, S.S. Jiang, R.W. Peng, A. Hu, Phys. Rev. B 63 (2001) 245104.[17] Jian-Wen Dong, Pen Hang, Wang He-Zhou, Chin. Phys. Lett. 20 (2003) 1963.[18] A. Montalbán, V.R. Velasco, J. Tutor, F.J. Fernández-Velicia, Phys. Rev. B 70 (2004)

132301.[19] H. Aynaou, E.H. El Boudouti, B. Djafari-Rouhani, A. Akjouj, V.R. Velasco, J. Phys.

Condens. Matter 17 (2005) 4245.[20] A. Montalbán, V.R. Velasco, J. Tutor, F.J. Fernández-Velicia, Surf. Sci. 594 (2005)

174.[21] A. Montalbán, V.R. Velasco, J. Tutor, F.J. Fernández-Velicia, Surf. Sci. 603 (2009)

937.[22] A.C. Hladky-Hennion, J.O. Vasseur, S. Degraeve, C. Granger, M. de Billy, J. Appl. Phys.

113 (2013) 154901.[23] M. Kohmoto, L.P. Kadanoff, C. Tang, Phys. Rev. Lett. 50 (1983) 1870.[24] E. Maciá, F. Domínguez-Adame, Phys. Rev. Lett. 76 (1996) 2957.[25] D.P. Norton, Y.W. Heo, M.P. Ivill, K. Ip, S.J. Pearton, M.F. Chisholm, T. Steiner, Mater.

Today 7 (2004) 34.[26] Ü. Özgür, Ya.A. Alivov, C. Liu, A. Teke, M.A. Reshchikov, S. Doğan, V. Avrutin, S.-J. Cho,

H. Morkoç, J. Appl. Phys. 98 (2005) 041301.[27] S. Strite, H. Morkoç, J. Vac. Sci. Technol. B 10 (1992) 1237.[28] A. Ohtomo, M. Kawasaki, T. Koida, K. Masubuchi, H. Koinuma, Y. Sakurai, Y. Yoshida,

T. Yasuda, Y. Segawa, Appl. Phys. Lett. 72 (1998) 2466.[29] A. Ohtomo, M. Kawasaki, I. Ohkubo, H. Koinuma, T. Yasuda, Y. Segawa, Appl. Phys.

Lett. 75 (1999) 980.[30] T. Makino, C.H. Chia, N.T. Tuan, H.D. Sun, Y. Segawa, M. Kawasaki, A. Ohtomo, K.

Tamura, H. Koinuma, Appl. Phys. Lett. 77 (2000) 975.[31] T. Makino, K. Tamura, C.H. Chia, Y. Segawa, M. Kawasaki, A. Ohtomo, H. Koinuma,

Appl. Phys. Lett. 81 (2002) 2355.[32] Th. Gruber, C. Kirchner, R. Kling, F. Reuss, A. Waag, Appl. Phys. Lett. 84 (2004)

5359.[33] B.P. Zhang, N.T. Binh, K.Wakatsuki, C.Y. Liu, Y. Segawa, N. Usami, Appl. Phys. Lett. 86

(2005) 032105.[34] Sz. Fujita, H. Tanaka, Sg. Fujita, J. Cryst. Growth 278 (2005) 264.[35] Sz. Fujita, T. Takagi, H. Tanaka, Sg. Fujita, Phys. Status Solidi (B) 241 (2004) 599.[36] J.L. Bleustein, Appl. Phys. Lett. 13 (1968) 412.[37] Y. Duan, L. Qin, G. Tang, L. Shi, Eur. Phys. J. B 66 (2008) 201.

http://refhub.elsevier.com/S0039-6028(14)00031-4/rf0005


























































image of Fig.�8

Fig. 9. Dispersion curves of the transverse acoustic waves of GaN/AlN superlattices having as periods the following stacking of layers: (a) ABAAB|ABAB|ABAAB; (b) ABAAB|BABA|ABAAB;(c) BABBA|ABAB|BABBA; and (d) BABBA|BABA|BABBA.


[38] T.B. Bateman, J. Appl. Phys. 33 (1962) 3309.[39] I.B. Kobiakov, Solid State Commun. 35 (1980) 305.[40] I. Vurgaftman, J.R. Meyer, J. Appl. Phys. 94 (2003) 3675.[41] M.S. Shur, A.D. Bykhovski, P. Gaska, M.A. Khan, GaN based Pyroelectronics and

Piezoelectronics, in: Colin E.C. Wood (Ed.), Handbook of Thin Film Devices,Heterostructures for High Performance Devices, volume 1, Academic Press, SanDiego, 2000, p. 299.

[42] F. García-Moliner, V.R. Velasco, Theory of Single and Multiple Interfaces, World Sci-entific, Singapore, 1992.

[43] R. Pérez-Alvarez, F. García-Moliner, V.R. Velasco, J. Phys. Condens. Matter 7 (1995)2037.

[44] V.R. Velasco, Surf. Sci. 128 (1983) 117.[45] R. Merlin, K. Bajema, R. Clarke, F.-Y. Juang, P.K. Bhattacharya, Phys. Rev. Lett. 17

(1985) 1768.[46] B. Jusserand, M. Cardona, in: M. Cardona, G. Güntherodt (Eds.), Light Scattering in

Solids V, Springer, Berlin, 1989, p. 49.[47] Yu.V. Gulyaev, Sov. Phys. JETP Lett. 9 (1969) 37.[48] C. Maerfeld, P. Tournois, Appl. Phys. Lett. 19 (1971) 117.



















image of Fig.�9

Fig. 12. Local density of states along the superlattice period-length for several modes of those presented in Fig. 11: (a) (ωDvM

= 25.02, qD= 0); (b) (ωDvM= 25.02, qD= π); (c) (ωDvM

=

31.68, qD = 0); (d) (ωDvM= 40.02, qD = π); (e) (ωDvM

= 36.42, qD = 0); and (f) (ωDvM= 35.76, qD = π). The vertical line gives the position of the interface separating both fifth Fibonacci

generations (S5jS5) forming the superlattice period.


image of Fig.�12

transverse acoustic waves in piezoelectric zno/mgo and gan/aln fibonacci-periodic superlattices

Documents