Interface phonon modes in truncated conical self-assembled quantum dots

Interface phonon modes in truncated conicalself-assembled quantum dots

Cl�eement Kanyinda-Malu, Rosa Mar�ııa de la Cruz *

Departamento de F�ıısica, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Legan�ees (Madrid), Spain

Received 4 November 2002; accepted for publication 14 February 2003

Abstract

Within the framework of the standard dielectric continuum model (DCM), we have performed a theoretical study of

interface (IF) phonon modes in III–V semiconductor self-assembled quantum dots (SAQDs). We model the SAQD as a

truncated cone whose growth-axis is along the polar axis of the cone. Small and arbitrary polar angle approximations

are used to resolve the angular partial differential equation within the variables separation scheme of the Laplace�sequation. Our theory allows to obtain new features in the IF modes behavior as the geometry parameters change

(aspect ratio and angle of the conical dot). In fact, IF eigenfrequencies present an abrupt change at determined angles,

related with the change of sign in the IF dispersion law.

� 2003 Elsevier Science B.V. All rights reserved.

Keywords: Self-assembly; Quantum effects; Dielectric phenomena; Interface states; Phonons

1. Introduction

A new trend of current semiconductor physics is

to achieve self-assembled structures which are

homogeneous in their sizes and shapes. In this

challenge, strained and unstrained quantum dots(QDs) have been fabricated, yielding several

shapes and sizes, being pronounced pyramids and

lens domes the most reported in the literature [1].

Concerning optical properties of self-assembled

quantum dots (SAQDs), replica of phonons in PL,

PLE and Raman measurements were reported [2–

5]. Large broadening of FWHM in photolumi-

nescence was attributed to fluctuation of size,

shape and chemical composition which disguises

the expected rich excitonic fine structure [6]. At

theoretical level, to account for phonon modes,

many works treated the SAQDs as spherical, cylin-

drical or real pyramidal systems in the frameworkof dielectric continuum model (DCM) [7–10], hy-

drodynamic continuum model [11] or valence force

band scheme [12]. Within the framework of DCM,

theoretical calculations in spherical and cylindrical

QDs have demonstrated that the shape is very

important to determine the types of interface as

well as their behaviors when physical parameters

like the size of the dot are modified. Knipp andReinecke [9] reported that the interface phonon

frequencies of spherical dots are not affected by the

size, while on the contrary, the cylindrical quan-

tum dots probe two types of surface phonon

*Corresponding author. Tel.: +34-91-624-8733; fax: +34-91-

624-8749.

E-mail address: [email protected] (R.M. de la Cruz).

0039-6028/03/$ - see front matter � 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0039-6028(03)00335-2

Surface Science 529 (2003) 503–514

www.elsevier.com/locate/susc

mail to: [email protected]

modes, one of them being size-dependent [10].

Therefore, we believe that the pyramidal shapes of

strained III–V or II–VI semiconductor QDs will

affect deeply their phononmodes, and consequently

influence their optical properties. In particular,

localization of phonons can be expected at the topand vertex of the pyramids. Thus, it is necessary to

understand the phonon vibrations behavior at the

interface of SAQDs. Despite the number of works,

these pyramidal SAQDs have not been yet com-

pletely investigated respect to their quantized vib-

rational modes or phonons. To our knowledge, the

more detailed study on quantum dots having less

symmetrical shapes was performed by Knipp andReinecke [9] using DCM approach. For dots with

cusps, these authors calculated IF modes by means

of an integral equation, which seems to account

for the surface charge densities. The application to

the dots with cusps is restricted to the discussion of

the dispersion eigenfrequencies. Aside its impor-

tance, the behavior of IF frequencies is not clearly

discussed in their paper.In the present work, we model pyramidal dots

as conical structures and study their collectively

quantized vibrational modes. Throughout this

work, we pay great attention to experimentally

reported physical parameters like the aspect ratio

and the angle of the conical dot. This paper is

organized as follows. Section 2 describes our

truncated cone parameters and presents relevantexpressions needed to calculate the IF-modes.

Section 3 presents numerical results and discussion

for two lowest phonon modes. The small and ar-

bitrary polar angle approximations are used to

account for the effect of mathematical approaches

on the IF behavior. In Section 4, relevant conclu-

sions of the study are given.

2. Theoretical model

The geometry of the truncated cone (also called

conical horn) structure is shown in the Fig. 1. R1and R2 represent the top and bottom radii of

the bases respectively, h0 is the height-size of theSAQD and h (h > h0) is the total height of theexternally surrounding cone. Because of its axial

symmetry, it is possible to describe the cone by

means of spherical coordinates. Then, we choose

the origin of the spherical coordinates at the apex

of the cone with the polar axis directed along the

axis of the cone. The surface of the cone is defined

by h ¼ h0 and we define the region outside the coneby values of polar angle h0 < h < p. A similar

description of a conical structure by means of

spherical coordinates was done in Ref. [13]. Weassume that the dot-island semiconductor of di-

electric constant e1 is surrounded by another polarsemiconductor with dielectric constant e2. Forsimplicity in the present work, we shall neglect the

image potential as well as the mechanical bound-

ary conditions, where the transverse optical and

longitudinal optical modes can mix into TO–LO

coupled mode. In fact, the image potential wasseen to affect only very narrow nanostructures,

while its contribution is weak in large nanostruc-

tures. Therefore, according to macroscopic elec-

trodynamics, and in the absence of free charges

inside the dot and barrier materials, we have

ejðxÞr2Ujð~rrÞ ¼ 0 ðj ¼ 1; 2Þ; ð1Þwhere ejðxÞ and Uj are the dielectric function and

scalar potential in material j, respectively. In bothsemiconductors, the dispersionless dielectric func-

tions are supposed to be frequency-dependent, and

is described by

ejðxÞ ¼ ej;1x2j;LO � x2

x2j;TO � x2ðj ¼ 1; 2Þ; ð2Þ

where ej;1, is the high-frequency optical dielectricconstant, xj;TO and xj;LO are, respectively, the

Brillouin zone center frequencies of transverse-

θθ0

r

h0

R2

R1

h

φ

Fig. 1. Truncated cone model for the pyramid-like SAQDs.

504 C. Kanyinda-Malu, R.M. de la Cruz / Surface Science 529 (2003) 503–514

optical and longitudinal-optical modes in the ma-

terial j. Besides the trivial solution UjðrÞ ¼ 0, thereare two solutions to Eq. (1), i.e., either ejðxÞ ¼ 0(the confined bulk-like solution) or ejðxÞ 6¼ 0(which gives rise the interface mode solution). Inorder to resolve analytically the Laplace�s equationin separable variables, we assume that the elec-

trostatic potential is along the / axis and dependsonly on the coordinates r and h (see Fig. 1). Asimilar assumption for the dependence of the

magnetic field in the coordinates r and h was donefor a conical structure in order to resolve the wave

equation for the magnetic field by separation ofvariables [13]. Then, in separable variables scheme,

the trial Laplace solution Uðr; h;uÞ ¼ f ðrÞgðhÞ�expðimuÞ leads to a set of one-dimensional differ-ential equations

r2d2fdr2

þ 2r dfdr

þ a

�þ 14

�f ¼ 0 ð3Þ

and

d2g

dh2þ cotðhÞ dg

dh� a

�"þ 14

�þ m2

sin2 ðhÞ

#g ¼ 0:

ð4ÞHere, a is the separation constant to be determinedfrom the electrostatic boundary conditions. Knipp

and Reinecke [9] have used a similar separable

variables scheme to resolve the Laplace�s equationin quantum dots with cusps. It is understood that

boundary conditions determines the existence of

the interface modes. Due to the nature of the

truncated system, one expects two surface modes.

By analogy with cylindrical QDs, we will termbase-surface optical (BSO) modes those related to

the radial solution and the side-surface optical

(SSO) modes as a consequence of boundary con-

ditions applied on the angular solution of the cone.

Now, we shall find the BSO and SSO eigenfre-

quencies.

2.1. Radial solution and BSO modes in the truncated

cone

In Eq. (3), r ¼ 0 and r ¼ 1 are singular points.

One can choose a point called turning point [14] to

force the analyticity of f ðrÞ in a bounded region of

the domain. If we choose the turning point at r0,the solution of the radial differential equation can

be written using the series asymptotic expansion,

i.e.

f ðrÞ ¼ 1

r1=2exp i

Z r

r0

gr0dr0

� �ð5Þ

or in explicit form

f ðrÞ ¼ r�1=2 exp ig lnrr0

� �; ð6Þ

r0 can be interpreted as the distance from the topbase to the apex of the cone, i.e., r0 ¼ h� h0 (seeFig. 1). Notice that the complex conjugate of f ðrÞis also solution of the one-dimensional radial

equation. Due to the particular nature of the

conical geometry, standard electromagnetic bound-

ary conditions do not apply in the radial solution.Therefore, we adopt the boundary integral equa-

tion on the top and bottom bases of the cone to

analyze the BSO modes in this geometry. First, let

us assume that the electrostatic potential is a real

function of r. Upon this assumption, the imagi-nary part of f ðrÞ vanishes everywhere. We obtainthen

sin g lnh� h0

h

� �� ¼ 0 ð7Þ

at the bottom base of the cone. The above relationdefines uniquely the values of g for given h and h0which reads as

g ¼ �p lnh� h0

h

� �: ð8Þ

To find the phonon-modes on the base-edges of

the horn, let us assume that the electrostatic po-

tential on both top and bottom bases of the

truncated cone is identical. This assumption is

motivated by two facts: (i) the QD is surroundedby the same material and (ii) within the framework

of the DCM, microscopic aspects like roughness

effect in the interfaces can not be appropriately

treated; therefore, all heterointerfaces behave as

rigid walls. Then, the Green theorem permits to

write a local potential for any point in a selected

domain (here truncated cone) in terms of the po-

tential and its normal derivative for any point lo-cated on the boundary of this domain. Sareni et al.

C. Kanyinda-Malu, R.M. de la Cruz / Surface Science 529 (2003) 503–514 505

[15] have used this theorem to analyze the effective

dielectric constant in random composites, using

the balance of the flux of the electric displacement

vector. Following this model, with the assumption

of identical potential on the bases, we can write

e1ðxÞZS1

ofondS þ e2ðxÞ

ZS2

ofondS ¼ 0; ð9Þ

where S1 and S2 refer to the top and bottom sur-faces, respectively and n is the normal unit vector

to the surface. From this condition, we find that

e1ðxÞ h� h0h

� ��3=2

¼ e2ðxÞ cos g lnh� h0

h

� �� :

ð10ÞSubstituting Eq. (8) into Eq. (10) we obtain

e1ðxÞe2ðxÞ ¼ b ð11Þ

with

b ¼ � h� h0h

� �3=2: ð12Þ

Eq. (11) together with the relation of the dielectric

function expressed in Eq. (2), constitute the tran-

scendental equation for the IF phonons at the

basis of the horn.

2.2. Angular solution and SSO modes in the

truncated cone

2.2.1. Small polar angle approximation (h �)

If a is positive, one can set a þ 1=4 � g2 and thesolution of the surface phonon modes is obtained

for positive values of g2 with the electric field givenby E ¼ �rU. For very small polar angles (h �)and assuming x ¼ gh, Eq. (4) can be written as

x2d2g

dh2þ xdgdh

� ðx2 þ m2Þg ¼ 0; ð13Þ

which is a characteristic differential equation for

modified Bessel functions of the first and thirdkind, and m defines the order of the function. It isalso emphasized that the solution of Eq. (13),

which takes into account the geometry of the dot,

must vanish inside, while it decreases exponentially

outside of the region, so that the surface phonons

can be localized near the apex of the cone.

Therefore, we express this solution in terms of

modified Bessel functions, i.e.

gðhÞ ¼ AmImðghÞ ð06 h6 h0Þ; ð14Þ

gðhÞ ¼ BmKmðghÞ ðh06 h6 pÞ: ð15ÞThe existence of interface modes implies that gðhÞmust be finite for all values of h. Therefore, oneexpects that the continuity condition on the nor-

mal component of the electric displacement field

will apply also at the interface of the cone withsurrounding medium, i.e. at h ¼ h0. Using theelectric displacement field and dispersion relation

of the dielectric functions (see Eq. (2)) of respective

materials, we obtain the transcendental equation

which leads to the surface phonon modes,

e1ðxÞe2ðxÞ ¼

oox ln½KmðxÞ�½ �oox ln½ImðxÞ�½ � ; x ¼ gh0: ð16Þ

In order to have a compact formula, we set

e1ðxÞe2ðxÞ ¼ bm: ð17Þ

For m ¼ 0, and m ¼ 1, we have

b0 ¼ �K1ðgh0ÞI0ðgh0ÞK0ðgh0ÞI1ðgh0Þ

ð18Þ

and

b1 ¼ � I1ðgh0Þ gh0K0ðgh0Þ þ K1ðgh0Þ½ �K1ðgh0Þ gh0I0ðgh0Þ � I1ðgh0Þ½ � ; ð19Þ

respectively. For the reasons which will be dis-

cussed below, the dispersion relation can be ex-

pressed in terms of non-material dependent

parameter km,

km ¼ bm þ 1bm � 1 : ð20Þ

2.2.2. Arbitrary polar angle approximation

Following Mehler–Fock transformation [16]applied to boundary valued problems inside a

cone, the angular solution to Eq. (4) is

gðhÞ ¼ Cgm

Pm�1=2þigðcos hÞPm

�1=2þigð� cos h0Þ;06 h6 h0;

Pm�1=2þigðcos h0ÞPm

�1=2þigð� cos hÞ;h06 h6 p;

8>><>>:

ð21Þ


where Pm�1=2þigðcos hÞ is called Mehler or conical

function and is a pure real quantity [14,16] with

g2 ¼ a. We analyze the case a > 0, being a < 0 thewell-known case of the Legendre associated poly-

nomials, solutions of Laplace�s equation in sphe-roidal structures. Accordingly, gðhÞ must be finitefor all values of h, as in the small polar angle ap-proximation. Therefore, applying the continuity of

the normal component of the electric displacement

field at h ¼ h0 and using dispersion relation of thedielectric functions of respective materials, we

obtain the transcendental equation which leads to

the surface phonon modes, eigenfrequencies of theconical structure. Following Knipp and Reinecke

[9], we define the dispersion relation of phonon

eigenfrequencies as

kgm � e1ðxgmÞ þ e2ðxgmÞe1ðxgmÞ � e2ðxgmÞ

¼oox ln Pm

�1=2þigð�xÞPm�1=2þigðxÞ

h in ooox0 ln Pm

�1=2þigð�x0Þ=Pm�1=2þigðx0Þ

h in o��x¼x0¼cosh0

:

ð22Þ

In general, in Eq. (21) g varies continuously from 0to 1. Consequently, k ranges between kmax and 0,

where kmax is a function of both m and h. In thefollowing, we will use only values of g which matchwith the boundary conditions in radial solution.

The logarithmic derivatives in the above equation

is the non-material dependent expression of SSOinterface-mode frequencies. The transcendental

equation can be expressed in terms of series ex-

pansion of sinðh=2Þ and cosðh=2Þ. In fact, form ¼ 0, P�1=2þigðcos hÞ is expressed in serie of

sinðh=2Þ, while P�1=2þigð� cos hÞ is a complex ex-pression that involves indefined integral [14]. To

overcome the lengthy calculation, let us rewrite

P�1=2þigðe�ip cos hÞ as P�1=2þigðcos h0Þ, with h0 ¼ h�p. By simple substitution of h0 in the serie expan-

sion of P�1=2þigðcos hÞ and using trigonometrichints, we obtain

Pm�1=2þigð� cos hÞ ¼ 1þ

4g2 þ 122

cos2h2

� �

þ ð4g2 þ 1Þð4g2 þ 32Þ2242

� cos4 h2

� �þ � � � ð23Þ

By changing variables while derivating the conical

functions, it follows that Eq. (22) becomes for

m ¼ 0

and for m ¼ 1, we have

kg0 ¼ �1þ 4g

2 þ 322

9þ 2ð4g2 þ 1Þð4g2 þ 32Þ8

� � �cos h

4 2þ 3ð4g2 þ 1Þ8

þ ð4g2 þ 1Þð4g2 þ 32Þ43

1þ 1þ ð4g2 þ 32Þ2 � 42

� �sin2 h

� � � ð24Þ

kg1 ¼ 5 cos h

�þ 4g

2 þ 328

cos4h2

� �� 134sin2 h þ 22 sin2 h

2

� �

þ 3

4sin2

h2

� �2

��þ cos2 h

2

� ��þ sin h cos4

h2

� �cos h

�� 2 sin2 h

2

� �ð4g2 þ 9Þ2

�

� 7 cos h

(þ 4g

2 þ 98

24 sin2h2

� �� 2 sin2 h � cos h cos2 h

2

� �

þ 3ð4g2 þ 9Þ2

43sin2

h2

� �2

��þ cos2 h

2

� �� sin h cos4

h2

� �cos h

�� 2 sin2 h

2

� ��)�1

: ð25Þ


Similarly, we write the compact expression of the

DCM eigenfrequencies, with bgm given by

bgm ¼ 1þ kgm

1� kgm: ð26Þ

Higher order branches of interface modes can di-

rectly derived from Eq. (22) by means of the re-

currence relations between P�1=2þigðcos hÞ and

Pm�1=2þigðcos hÞ.

3. Numerical results and discussion

In the previous section, we have derived the

analytical expressions of the interface-phonon

frequencies for truncated conical quantum dots. In

order to illustrate the IF-behavior, we performed

numerical calculations in typical III–V semicon-

ductor SAQDs such as InAs/GaAs and InAs/InP.These QDs follow a Stranski–Krastanov growth

pattern with a nearly uniform distribution of

shapes and sizes. While the InAs/GaAs QDs are

widely documented, the precise experimental de-

scription of the dot parameters (shape, strain,

composition, . . .) for InAs/InP is scarcely reported.The interest in InAs/InP QDs is motivated by its

sensitivity to substitute the InAs/GaAs QDs insome optical devices. In fact, InAs/InP based-

lasers can emit at non-dispersive wavelength of 1.55

lm [17]. Also, InPAs alloys are usually used to

accommodate the higher lattice mismatch of InAs

on the GaAs substrate. On the other hand, de-

pending on the growth direction, growth of InAs

on InP can produce quantum wires or quantum

dots. The material parameters used in our calcula-tions are taken from Refs. [18,19] and are summa-

rized in Table 1. We will restrict the presentation

of our results to special cases where the InAs/

GaAs and InAs/InP QDs differ in phonon fea-

tures.

Fig. 2 shows the dependence of the BSO pho-

non frequencies with the height in InAs/InP

quantum dots for two ranges of h0: (a) 8–15 nmand (b) 8–18 nm. For these two investigated ran-ges, the barrier-like frequencies decrease as the

height increases, while the behavior is reverse for

the dot-like frequencies. Both interface frequencies

for (a) and (b) cases reach a constant value when

the height tends to reach the total height h of theexternal cone, in this case h ¼ 15 and 18 nm, re-spectively. The dependence of the BSO frequencies

with the height in InAs/GaAs quantum dots is alsoshown in Fig. 3. In both systems, higher values of

h imply high values of barrier modes (or low valuesof dot modes), independently with the h0 start-ing value. For InAs/GaAs QDs, the barrier-like

frequencies (dot-like) lie between barrier (dot)

reststrahl region, i.e. xj;TO < xj;IF < xj;LO (j ¼ 1; 2)(see values in Table 1). However, in InAs/InP

QDs, IF-modes are not restricted to the reststrahlregion of respective material compounds. A ten-

Table 1

Material parameters used in the calculations

Material e0 e1 xLO(cm�1)

xTO(cm�1)

InAs 15.15 11.7 241.0 218.0

GaAs 13.18 10.89 292.0 268.7

InP 12.61 9.61 345.0 303.7

8.0 10.0 12.0 14.0 16.0 18.0

240

250

260

270

280

290

300

310

InAs

InP

ω (

cm-1)

Height (nm)

Fig. 2. Dependence of the BSO frequencies with the height for

InAs/InP quantum dots.


tative explanation of this feature is given as

follows. Raman measurements and theoretical

calculations on InP(1 1 0) semiconductor have

demonstrated that upmost atomic layers of InP

present surface phonon frequencies located into

the phonon band-gap of this material. Middle

frequencies of such surface phonons are close tothe LO frequencies of the InAs semiconductor and

are predicted to be active in IR experiments [20].

In general, for a given material, the vibrational

frequencies of their surface phonons are different

from those of the bulk, since the upper bonding

partners are missing in the upmost atomic layers

which constitute the surface border. Besides, the

surface and bulk frequencies are best discriminatedfor materials with high band gap and the differ-

ences in surface and bulk phonons are enchanced

when a surface is terminated with atoms not pre-

sent in the bulk such as occurs in heterointerfaces

[20]. On the other hand, as it is well established in

the literature, the interface modes in low-dimen-

sional heterostructes like quantum wells, -wires

and -dots are localized near the heterointerface

decaying very fast outside this region. Therefore, it

is possible that vibrational characteristics of in-

terface and surface phonons present similarities. Infact, we may expect IF-modes to take values in the

range of surface modes. Thus, the barrier-like

mode of InAs/InP QDs have values ranging from

the InP reststrahl region up to the surface modes

reported in InP semiconductor [20]. Meanwhile,

the dot-like modes grow from its reststrahl re-

gion––for smaller values of h0 not shown herein,the dot-like frequencies lie in the reststrahl––up tohigher values which are close to the aforemen-

tioned InP surface phonons.

The other difference between InAs/GaAs and

InAs/InP QDs is reflected in the small rate of the

slowing (or increasing) of IF modes in the InAs/

GaAs QDs. Notice that in the InAs/GaAs QDs,

both the dot and the barrier have the same cation

(As) while in InAs/InP QDs, the anion (In) is equalin both materials. Also, the phonon band-gap

between the dot and barrier materials is higher in

InAs/InP than in InAs/GaAs. This high phonon

band-gap in InAs/InP QDs, together with the

small difference in optical dielectric constants be-

tween InAs and GaAs could probably explain

different behavior observed in BSO phonon modes

for InAs/GaAs QDs with respect to the InAs/InPas the dot height is changed. This study can be

improved with the inclusion of the strain effect on

the interface phonon modes in different lattice-

mismatched QDs.

Fig. 4 shows the dependence of the SSO phonon

frequencies of the modes m ¼ 0 and m ¼ 1 with theparameter gh for InAs/InP quantum dots within

the small polar angle approximation. We observethat the frequency values satisfy the well known

interface conditions: i.e. xj;TO < xj;IF < xj;LO

(j ¼ 1; 2) (see values in Table 1). On the otherhand, for the barrier-type modes, the frequencies

of the fundamental mode (m ¼ 0) are higher thanthat of the first excited mode (m ¼ 1) in the in-vestigated range of the parameter gh, while thisbehavior is reverse for the dot-type modes. Thisfeature was also observed in spheroidal QDs,

where phonon modes of higher indexes present

8.0 10.0 12.0 14.0 16.0 18.0

228

280

InAs

GaAs

ω(c

m-1)

Height (nm)

Fig. 3. Dependence of the BSO frequencies with the height for

InAs/GaAs quantum dots.


lower values of frequencies [21]. For increasing

values of gh, it seems that the modes m ¼ 0 andm ¼ 1 of barrier-type (dot-type) tends to convergeto a single-interface mode close to ðxj;LOþxj;TOÞ=2 (j ¼ 1; 2). This effect was also reported inIII–V semiconductor asymmetric quantum wells

where different branches of IF phonon modes

converge to asymptotic values [22]. The separation

of phonon-frequencies between m ¼ 0 and m ¼ 1 ispronounced for smaller values of gh. The barrier-like and dot-like modes in this approximation act

as dual branches of interface optical phonons re-ported in periodic multilayers or in k-componentFibonnaci multilayers [23]; i.e., the increase in the

barrier mode is accompanied by the decrease of

the dot-like mode and vice-versa. A similar de-

pendence for the SSO phonon frequencies of the

modes m ¼ 0 and m ¼ 1 with the parameter gh isfound for InAs/GaAs quantum dots. Why InAs/

GaAs and InAs/InP QDs do not exhibit differentbehavior as observed in BSO modes can be at-

tributed to the rotational symmetry that forbids

the modification of lateral morphology of the dot.

Another relevant applied parameter is the as-

pect ratio, defined as r ¼ h0=2R2. Typically re-ported experimental values for r span from 0.1 to0.3 [2,3,17]. Grassi et al. [24] reported aspect ratios

ranging from 0.1 to 0.6. In order to investigate the

influence of this parameter in the interface modes,we calculated the SSO frequencies as a function of

the polar angle for aspect ratios ranging from 0.1

to 0.7. The results obtained for r ¼ 0:1 and r ¼ 0:3are shown in Fig. 5. However, our discussion is

focused in the results for m ¼ 0 and m ¼ 1 inwhole range of aspect ratios investigated. For the

fundamental mode (m ¼ 0), the dependence ofdot-like and barrier-like frequencies with the angleis similar for r6 0:4. The barrier-like frequenciesincrease for increasing polar angle, while the op-

posite behavior is obtained for the dot-like fre-

quencies. For aspect ratios between 0.5 and 0.7, an

abrupt change occurs between these branches;

leading the dot-like frequencies to switch to the

barrier-like ones and vice-versa at specific values

of the polar angle. This abrupt change occurs atpolar angles which are dependent of the aspect

ratio. For instance, for r ¼ 0:6, the switching oc-curs at h � 32�, while for r ¼ 0:7 the change isaround h � 26�.Attending to the first excited mode (m ¼ 1),

interchange between dot and barrier branches is

not obtained for any aspect ratio investigated.

However, the dependence of these branches withthe polar angle change for different aspect ratios.

In fact, for r ¼ 0:1, the barrier-like frequenciesdecrease with increasing angles, while the dot-like

frequencies increase for increasing angles. When

r ¼ 0:2, the dot-like and barrier-like frequenciesare almost constant and for small angles, they

reach the values of the mode m ¼ 0. For rP 0:3,the barrier-like frequencies are increasing func-tions of the angle, while the dot-like frequencies

are decreasing functions. In the light of these re-

sults, we deduce that the fundamental and first

excited modes are critically sensitive with the as-

pect ratio. A possible origin of the different fea-

tures between these modes can be explained in

terms of a distinct dispersion relation for m ¼ 0and m ¼ 1 in the arbitrary polar angle approxi-mation (see Eqs. (24) and (25)). More details about

the dispersion relations will be given below.

220

240

320

340

m = 0

m = 1

m = 1

m = 0

InP

InAs

ω(c

m-1)

ηθ0.0 0.2 0.4 0.6 0.8 1.0

Fig. 4. Dot-like and barrier-like SSO frequencies as a function

of the parameter gh for InAs/InP quantum dots (small polar

angle approximation).


To illustrate the interchange between dot and

barrier branches of the fundamental mode, we

show in Fig. 6, the SSO frequencies for r ¼ 0:5 andseveral starting values of h0. We find that theabrupt change (i.e., the dot-like modes go to the

barrier-like ones and vice-versa) occurs aroundh � 38�, independently on h0. In fact, the startinginput value of h0 defines the beginning value of thepolar angle in our calculations. Before and after

this change, the barrier-like frequencies increase

for increasing angles, while the dot-like frequencies

decrease with the angle. Unlike in the flattened

conical QDs (r6 0:3), the first-excited barrier-like(dot-like) modes present frequencies located above(below) the fundamental frequencies values for

h0 ¼ 2 nm. This effect is observed up to an inter-cross level around 25�, where first excited barrier-like (dot-like) modes recover values smaller

(higher) than the fundamental ones. This effect,

together with the abrupt change seems to be in

agreement with the oscillating character reported

by Knipp and Reinecke [9] in dots with cusps. Inthe light of these results, it can be deduced that for

quantum dots of complex geometries, the IF-

modes show anomalous behaviors such as the

abrupt interchange of dot-like and barrier-like

branches. Notice that this phenomenon occurs for

aspect ratios in the range 0.5–0.7, values which

represent shapes of QDs less smoothed. Other kind

of anomalies are extensively reported elsewhere inthese low symmetrical geometries. For instance,

the IF modes localized at the infinitely sharp tips

reveal a continuous vibrational spectrum, while

those of the smoothed tips are characterized by a

discrete spectrum [9]. Similar phenomena are ob-

served in InAs/InP QDs.

The dispersion relations are shown in Fig. 7,

where the adimensional parameter k is plotted as afunction of the quantum dot height. For small

polar angle approximation, k is negative in all theinvestigated region. In arbitrary angle approxi-

mation, the fundamental mode dispersion relation

changes sign with the height, while the first-excited

mode dispersion is a growing function of the

height. Thus, whether k is positive or negativewould affect strongly the behavior of the IF fre-quencies. The sign change of k, reveals the im-portance of the geometry in the phonon modes.

73 74 75 76 77 78 79220

230

280

290

(a)

m = 0

m = 1

m = 1

m = 0

In As

GaAs

ω (

cm-1

)

Polar angle θ48 50 52 54 56 58 60

220

230

280

290

(b)

m = 0

m = 1

m = 1

m = 0

InAs

GaAs

ω (c

m-1

)

Polar angle θ

Fig. 5. Dependence of the SSO frequencies with the polar angle (arbitrary polar angle approximation) for InAs/GaAs quantum dots

with h ¼ 15 nm, and (a) r ¼ 0:1, (b) r ¼ 0:3.


This sign change could be interpreted in terms of

the possibility of modes localization in regions of

maximum curvature. In fact, Bennett et al. [25]

reported that the phonon dispersion law changesfrom positive to negative as the quantum wire base

changes from circular to elliptical shape. To in-

terprete this behavior, they compare the phonon

propagation wave with hole-like Schr€oodinger mo-tion and conclude that the distorsion from circular

shape induces the IF phonon modes to concentrate

in the region of highest curvature. Also, studies of

wave propagation in conical systems have dem-onstrated that the electromagnetic fields increase

anomalously when the tip of the cone is ap-

proached [13] giving rise to the concentration of

the electromagnetic energies at the apex or near

singular points of the domain.

The comparison with the dispersion relations

for spherical and cylindrical geometries (not re-

ported here) implies that the conical case is morecomplex. For example, for spherical quantum dots,

b ¼ �1� 1=l and gives rise to completely positivek, where dependence on orbital quantum numbermdoes not appear. In the case of cylindrical QDs, due

to the existence of symmetric and antisymmetric

modes, the dispersion relationship can be positive

or negative. Aside, while in spherical quantum dots

there is only one type of interface that yields con-

sequently one type of IF modes with two branches(dot-like and barrier-like), for cylindrical and

conical quantum dots, we have two interfaces

yielding TSO or BSO and SSO phonon modes with

the dot-like and barrier-like branches. However,

for conical QDs, the BSO and SSO modes do not

show symmetric and antisymmetric branches as

they do in cylindrical QDs. This can be explained in

terms of a loss of symmetry in the conical geome-try. The increasing of phonon modes number and

their trend changes with the parameter m can en-hance the number of active modes in less symmet-

rical geometries. Of course, a comparison with

existing theoretical works suggest that phonon

modes are size-dependent in non-regular geomet-

rical shapes. This indicates that polarization-de-

pending optical measurements can be used to probethe shapes and sizes of semiconductor QDs. Fi-

nally, within the framework of separable variables

scheme of our model, one can expect that incident

light parallel to the radial parameter of the cone

only excites the BSO vibrations, whereas the light

10 20 30 40 50

220

240

260

280

300m = 0

m = 1

m = 1

m = 0

InAs

GaAs

(a)

ω (c

m-1

)

Polar angle θ30 35 40 45

220

240

260

280

300

m = 0

m = 1

m = 1

m = 0

InAs

GaAs

(b) Polar angle θ

ω (c

m-1

)

Fig. 6. Dependence of the SSO frequencies with the polar angle (arbitrary polar angle approximation) for InAs/GaAs quantum dots

with r ¼ 0:5, h ¼ 15 nm, and (a) h0 ¼ 2 nm, (b) h0 ¼ 8 nm.


incident perpendicular to this axis mainly excites

the SSO modes.

4. Conclusion

The IF phonon modes are investigated in

truncated conical SAQDs as a function of several

geometrical parameters such as aspect ratio, polar

angle and height. Two semiconductor systems areinvestigated: InAs/GaAs and InAs/InP in the

DCM framework. To resolve the Laplace�s equa-tion, the separation of variables is used taking into

account the small and arbitrary angle approaches.

In the light of the obtained results, it can be con-

clude that the IF-frequencies of conical SAQDs

are size-dependent with complex dispersion rela-

tion. However, for heights close to the height of

external cone, the BSO frequencies are size-inde-

pendent. In this case, optical absorption experi-

ments could not distinguish QDs with heightsranging in the above interval. To the change of

sign in the dispersion relation (arbitrary-angle

approximation) corresponds abrupt changes in the

SSO modes, switching from barrier-like to dot-like

modes and vice-versa at specific polar angle for rranging from 0.5 to 0.7. On the other hand, this

geometrical extrapolation between pyramid and

truncated cone can be extended to other quantumdots shapes like lens or domes.

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2.0 4.0 6.0 8.0 10.0 12.0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

Dis

pers

ion

rela

tion

(a.u

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h (nm)

Fig. 7. Dispersion relation of interface phonons as a function

of the height of conical quantum dots. The symbol (�) is form ¼ 0; (}) is for m ¼ 1 within the arbitrary angle approxima-tion; and (�) is for m ¼ 0 within the small angle approxima-tion.


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Interface phonon modes in truncated conical self-assembled quantum dots

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