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Interface phonon modes in truncated conical self-assembled quantum dots
Surface Science 529 (2003) 503–514 www.elsevier.com/locate/susc
Interface phonon modes in truncated conical self-assembled quantum dots Clement Kanyinda-Malu, Rosa Marıa de la Cruz
*
Departamento de Fısica, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Legan es (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 LaplaceÕs equation. 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. Ó 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-
*
Corresponding author. Tel.: +34-91-624-8733; fax: +34-91624-8749. E-mail address: rmc@fis.uc3m.es (R.M. de la Cruz).
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, cylindrical or real pyramidal systems in the framework of dielectric continuum model (DCM) [7–10], hydrodynamic 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 and Reinecke [9] reported that the interface phonon frequencies of spherical dots are not affected by the size, while on the contrary, the cylindrical quantum dots probe two types of surface phonon
0039-6028/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0039-6028(03)00335-2
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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 phonon modes, and consequently influence their optical properties. In particular, localization of phonons can be expected at the top and 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 completely investigated respect to their quantized vibrational modes or phonons. To our knowledge, the more detailed study on quantum dots having less symmetrical shapes was performed by Knipp and Reinecke [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 importance, 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 relevant expressions needed to calculate the IF-modes. Section 3 presents numerical results and discussion for two lowest phonon modes. The small and arbitrary polar angle approximations are used to account for the effect of mathematical approaches on the IF behavior. In Section 4, relevant conclusions 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. R1 and R2 represent the top and bottom radii of the bases respectively, h0 is the height-size of the SAQD and h (h > h0 ) is the total height of the externally surrounding cone. Because of its axial symmetry, it is possible to describe the cone by
r
R1
θ
h
θ0
h0
φ
R2
Fig. 1. Truncated cone model for the pyramid-like SAQDs.
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 cone by values of polar angle h0 < h < p. A similar description of a conical structure by means of spherical coordinates was done in Ref. [13]. We assume that the dot-island semiconductor of dielectric constant e1 is surrounded by another polar semiconductor with dielectric constant e2 . For simplicity in the present work, we shall neglect the image potential as well as the mechanical boundary conditions, where the transverse optical and longitudinal optical modes can mix into TO–LO coupled mode. In fact, the image potential was seen to affect only very narrow nanostructures, while its contribution is weak in large nanostructures. Therefore, according to macroscopic electrodynamics, and in the absence of free charges inside the dot and barrier materials, we have ej ðxÞr2 Uj ð~ rÞ ¼ 0 ðj ¼ 1; 2Þ;
ð1Þ
where ej ðxÞ and Uj are the dielectric function and scalar potential in material j, respectively. In both semiconductors, the dispersionless dielectric functions are supposed to be frequency-dependent, and is described by ej ðxÞ ¼ ej;1
x2j;LO x2 x2j;TO x2
ðj ¼ 1; 2Þ;
ð2Þ
where ej;1 , is the high-frequency optical dielectric constant, xj;TO and xj;LO are, respectively, the Brillouin zone center frequencies of transverse-
C. Kanyinda-Malu, R.M. de la Cruz / Surface Science 529 (2003) 503–514
optical and longitudinal-optical modes in the material j. Besides the trivial solution Uj ðrÞ ¼ 0, there are two solutions to Eq. (1), i.e., either ej ðxÞ ¼ 0 (the confined bulk-like solution) or ej ðxÞ 6¼ 0 (which gives rise the interface mode solution). In order to resolve analytically the LaplaceÕs equation in separable variables, we assume that the electrostatic potential is along the / axis and depends only on the coordinates r and h (see Fig. 1). A similar assumption for the dependence of the magnetic field in the coordinates r and h was done for a conical structure in order to resolve the wave equation for the magnetic field by separation of variables [13]. Then, in separable variables scheme, the trial Laplace solution Uðr; h; uÞ ¼ f ðrÞgðhÞ expðimuÞ leads to a set of one-dimensional differential equations d2 f df 1 r2 2 þ 2r þ aþ f ¼0 ð3Þ dr dr 4 and d2 g dg þ cotðhÞ 2 dh dh
"
1 aþ 4
# m2 þ 2 g ¼ 0: sin ðhÞ ð4Þ
Here, a is the separation constant to be determined from the electrostatic boundary conditions. Knipp and Reinecke [9] have used a similar separable variables scheme to resolve the LaplaceÕs equation in 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 term base-surface optical (BSO) modes those related to the radial solution and the side-surface optical (SSO) modes as a consequence of boundary conditions applied on the angular solution of the cone. Now, we shall find the BSO and SSO eigenfrequencies. 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
505
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. Z r 1 g 0 f ðrÞ ¼ 1=2 exp i dr ð5Þ 0 r r0 r or in explicit form r f ðrÞ ¼ r1=2 exp ig ln ; r0
ð6Þ
r0 can be interpreted as the distance from the top base to the apex of the cone, i.e., r0 ¼ h h0 (see Fig. 1). Notice that the complex conjugate of f ðrÞ is also solution of the one-dimensional radial equation. Due to the particular nature of the conical geometry, standard electromagnetic boundary conditions do not apply in the radial solution. Therefore, we adopt the boundary integral equation 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 imaginary part of f ðrÞ vanishes everywhere. We obtain then h h0 sin g ln ¼0 ð7Þ h at the bottom base of the cone. The above relation defines uniquely the values of g for given h and h0 which reads as
h h0 g ¼ p ln : ð8Þ h To find the phonon-modes on the base-edges of the horn, let us assume that the electrostatic potential on both top and bottom bases of the truncated cone is identical. This assumption is motivated by two facts: (i) the QD is surrounded by 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 potential and its normal derivative for any point located on the boundary of this domain. Sareni et al.
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[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 Z Z of of dS þ e2 ðxÞ dS ¼ 0; ð9Þ e1 ðxÞ S1 on S2 on where S1 and S2 refer to the top and bottom surfaces, respectively and n is the normal unit vector to the surface. From this condition, we find that 3=2 h h0 h h0 ¼ e2 ðxÞ cos g ln e1 ðxÞ : h h ð10Þ Substituting Eq. (8) into Eq. (10) we obtain e1 ðxÞ ¼b e2 ðxÞ
ð11Þ
h h0 b¼ h
3=2 :
ð12Þ
Eq. (11) together with the relation of the dielectric function expressed in Eq. (2), constitute the transcendental 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 the solution of the surface phonon modes is obtained for positive values of g2 with the electric field given by E ¼ rU. For very small polar angles (h ) and assuming x ¼ gh, Eq. (4) can be written as x2
gðhÞ ¼ Am Im ðghÞ ð0 6 h 6 h0 Þ;
ð14Þ
gðhÞ ¼ Bm Km ðghÞ
ð15Þ
ðh0 6 h 6 pÞ:
The existence of interface modes implies that gðhÞ must be finite for all values of h. Therefore, one expects that the continuity condition on the normal component of the electric displacement field will apply also at the interface of the cone with surrounding medium, i.e. at h ¼ h0 . Using the electric 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, e1 ðxÞ oxo ½ ln½Km ðxÞ ¼ ; e2 ðxÞ oxo ½ ln½Im ðxÞ
x ¼ gh0 :
ð16Þ
In order to have a compact formula, we set
with
Therefore, we express this solution in terms of modified Bessel functions, i.e.
d2 g dg þ x ðx2 þ m2 Þg ¼ 0; dh dh2
ð13Þ
which is a characteristic differential equation for modified Bessel functions of the first and third kind, and m defines the order of the function. It is also 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.
e1 ðxÞ ¼ bm : e2 ðxÞ
ð17Þ
For m ¼ 0, and m ¼ 1, we have b0 ¼
K1 ðgh0 ÞI0 ðgh0 Þ K0 ðgh0 ÞI1 ðgh0 Þ
ð18Þ
I1 ðgh0 Þ½gh0 K0 ðgh0 Þ þ K1 ðgh0 Þ ; K1 ðgh0 Þ½gh0 I0 ðgh0 Þ I1 ðgh0 Þ
ð19Þ
and b1 ¼
respectively. For the reasons which will be discussed below, the dispersion relation can be expressed in terms of non-material dependent parameter km , km ¼
bm þ 1 : bm 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 8 m m P ðcos hÞP1=2þig ð cos h0 Þ; > > < 1=2þig 0 6 h 6 h0 ; gðhÞ ¼ Cgm m Pm ðcos h0 ÞP1=2þig ð cos hÞ; > > : 1=2þig h0 6 h 6 p; ð21Þ
C. Kanyinda-Malu, R.M. de la Cruz / Surface Science 529 (2003) 503–514 m where P1=2þig ðcos 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 the well-known case of the Legendre associated polynomials, solutions of LaplaceÕs equation in spheroidal structures. Accordingly, gðhÞ must be finite for all values of h, as in the small polar angle approximation. Therefore, applying the continuity of the normal component of the electric displacement field at h ¼ h0 and using dispersion relation of the dielectric functions of respective materials, we obtain the transcendental equation which leads to the surface phonon modes, eigenfrequencies of the conical structure. Following Knipp and Reinecke [9], we define the dispersion relation of phonon eigenfrequencies as
e1 ðxgm Þ þ e2 ðxgm Þ e1 ðxgm Þ e2 ðxgm Þ n h io o m m ln P1=2þig ðxÞP1=2þig ðxÞ ox io ¼ n h o m m ln P1=2þig ðx0 Þ=P1=2þig ðx0 Þ ox0
kgm
507
where kmax is a function of both m and h. In the following, we will use only values of g which match with the boundary conditions in radial solution. The logarithmic derivatives in the above equation is the non-material dependent expression of SSO interface-mode frequencies. The transcendental equation can be expressed in terms of series expansion of sinðh=2Þ and cosðh=2Þ. In fact, for m ¼ 0, P1=2þig ðcos hÞ is expressed in serie of sinðh=2Þ, while P1=2þig ð cos hÞ is a complex expression that involves indefined integral [14]. To overcome the lengthy calculation, let us rewrite P1=2þig ðeip cos hÞ as P1=2þig ðcos h0 Þ, with h0 ¼ h p. By simple substitution of h0 in the serie expansion of P1=2þig ðcos hÞ and using trigonometric hints, we obtain 4g2 þ 1 h m 2 cos P1=2þig ð cos hÞ ¼ 1 þ 2 2 2 ð4g2 þ 1Þð4g2 þ 32 Þ 22 42 h cos4 þ 2 þ
:
x¼x0 ¼cos h0
ð22Þ In general, in Eq. (21) g varies continuously from 0 to 1. Consequently, k ranges between kmax and 0,
ð23Þ
By changing variables while derivating the conical functions, it follows that Eq. (22) becomes for m¼0
4g2 þ 32 9 þ 2ð4g2 þ 1Þð4g2 þ 32 Þ cos h 8 2 kg0 ¼ 3ð4g2 þ 1Þ ð4g2 þ 1Þð4g2 þ 32 Þ ð4g2 þ 32 Þ 2 þ 1 þ 1 þ h 4 2þ sin 8 43 2 42 1þ
ð24Þ
and for m ¼ 1, we have
4g2 þ 32 h 13 h 2 2 4 sin h þ 22 sin cos kg1 ¼ 5 cos h þ 2 4 2 8 3 h h h h 2 þ sin2 2 þ cos2 þ sin h cos4 cos h 2 sin2 ð4g2 þ 9Þ 4 2 2 2 2 ( 2 4g þ 9 h h 24 sin2 7 cos h þ 2 sin2 h cos h cos2 8 2 2 )1 2 3ð4g2 þ 9Þ h h h h þ sin2 : 2 þ cos2 sin h cos4 cos h 2 sin2 2 2 2 2 43
ð25Þ
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C. Kanyinda-Malu, R.M. de la Cruz / Surface Science 529 (2003) 503–514
Similarly, we write the compact expression of the DCM eigenfrequencies, with bgm given by 1 þ kgm : 1 kgm
InP
ð26Þ
290
280
-1
Higher order branches of interface modes can directly derived from Eq. (22) by means of the recurrence relations between P1=2þig ðcos hÞ and m P1=2þig ðcos hÞ.
300
ω (cm )
bgm ¼
310
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 semiconductor 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 description 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 in some optical devices. In fact, InAs/InP basedlasers 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, depending on the growth direction, growth of InAs on InP can produce quantum wires or quantum dots. The material parameters used in our calculations are taken from Refs. [18,19] and are summarized in Table 1. We will restrict the presentation of our results to special cases where the InAs/
Table 1 Material parameters used in the calculations Material
e0
e1
xLO (cm1 )
xTO (cm1 )
InAs GaAs InP
15.15 13.18 12.61
11.7 10.89 9.61
241.0 292.0 345.0
218.0 268.7 303.7
270
260
250
InAs
240 8.0
10.0
12.0
14.0
16.0
18.0
Height (nm) Fig. 2. Dependence of the BSO frequencies with the height for InAs/InP quantum dots.
GaAs and InAs/InP QDs differ in phonon features. Fig. 2 shows the dependence of the BSO phonon frequencies with the height in InAs/InP quantum dots for two ranges of h0 : (a) 8–15 nm and (b) 8–18 nm. For these two investigated ranges, 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 the external cone, in this case h ¼ 15 and 18 nm, respectively. The dependence of the BSO frequencies with the height in InAs/GaAs quantum dots is also shown in Fig. 3. In both systems, higher values of h imply high values of barrier modes (or low values of dot modes), independently with the h0 starting 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 reststrahl region of respective material compounds. A ten-
C. Kanyinda-Malu, R.M. de la Cruz / Surface Science 529 (2003) 503–514
GaAs
-1
ω(cm )
280
InAs
228
8.0
10.0
12.0
14.0
16.0
18.0
Height (nm) Fig. 3. Dependence of the BSO frequencies with the height for InAs/GaAs 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 to the 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 discriminated for materials with high band gap and the differences in surface and bulk phonons are enchanced when a surface is terminated with atoms not present in the bulk such as occurs in heterointerfaces [20]. On the other hand, as it is well established in
509
the literature, the interface modes in low-dimensional 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 interface and surface phonons present similarities. In fact, 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 region––for smaller values of h0 not shown herein, the dot-like frequencies lie in the reststrahl––up to higher values which are close to the aforementioned 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 equal in 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 between InAs and GaAs could probably explain different behavior observed in BSO phonon modes for InAs/GaAs QDs with respect to the InAs/InP as the dot height is changed. This study can be improved with the inclusion of the strain effect on the interface phonon modes in different latticemismatched QDs. Fig. 4 shows the dependence of the SSO phonon frequencies of the modes m ¼ 0 and m ¼ 1 with the parameter gh for InAs/InP quantum dots within the small polar angle approximation. We observe that 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 other hand, for the barrier-type modes, the frequencies of the fundamental mode (m ¼ 0) are higher than that of the first excited mode (m ¼ 1) in the investigated range of the parameter gh, while this behavior is reverse for the dot-type modes. This feature was also observed in spheroidal QDs, where phonon modes of higher indexes present
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m=0
340
m=1
InP
-1
ω(cm )
320
240
m=1 InAs m=0 220
0.0
0.2
0.4
0.6
0.8
1.0
ηθ 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).
lower values of frequencies [21]. For increasing values of gh, it seems that the modes m ¼ 0 and m ¼ 1 of barrier-type (dot-type) tends to converge to a single-interface mode close to ðxj;LO þ xj;TO Þ=2 (j ¼ 1; 2). This effect was also reported in III–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 is pronounced for smaller values of gh. The barrierlike and dot-like modes in this approximation act as dual branches of interface optical phonons reported in periodic multilayers or in k-component Fibonnaci 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 dependence for the SSO phonon frequencies of the modes m ¼ 0 and m ¼ 1 with the parameter gh is found for InAs/GaAs quantum dots. Why InAs/ GaAs and InAs/InP QDs do not exhibit different behavior as observed in BSO modes can be attributed to the rotational symmetry that forbids the modification of lateral morphology of the dot.
Another relevant applied parameter is the aspect ratio, defined as r ¼ h0 =2R2 . Typically reported experimental values for r span from 0.1 to 0.3 [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 0:3 are shown in Fig. 5. However, our discussion is focused in the results for m ¼ 0 and m ¼ 1 in whole range of aspect ratios investigated. For the fundamental mode (m ¼ 0), the dependence of dot-like and barrier-like frequencies with the angle is similar for r 6 0:4. The barrier-like frequencies increase for increasing polar angle, while the opposite behavior is obtained for the dot-like frequencies. 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 at polar angles which are dependent of the aspect ratio. For instance, for r ¼ 0:6, the switching occurs at h 32°, while for r ¼ 0:7 the change is around 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 with the polar angle change for different aspect ratios. In fact, for r ¼ 0:1, the barrier-like frequencies decrease with increasing angles, while the dot-like frequencies increase for increasing angles. When r ¼ 0:2, the dot-like and barrier-like frequencies are almost constant and for small angles, they reach the values of the mode m ¼ 0. For r P 0:3, the barrier-like frequencies are increasing functions of the angle, while the dot-like frequencies are decreasing functions. In the light of these results, we deduce that the fundamental and first excited modes are critically sensitive with the aspect ratio. A possible origin of the different features between these modes can be explained in terms of a distinct dispersion relation for m ¼ 0 and m ¼ 1 in the arbitrary polar angle approximation (see Eqs. (24) and (25)). More details about the dispersion relations will be given below.
C. Kanyinda-Malu, R.M. de la Cruz / Surface Science 529 (2003) 503–514
511
290
m=0
280
290
m=0
m=1
m=1 280
GaAs
ω (c m-1)
-1
ω (cm )
GaAs
In As
m= 1 InAs
230 230
m=1
m=0
m=0 220
220 73
(a)
74
75
76
77
78
79
Polar angle θ
48
(b)
50
52
54
56
58
60
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.
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 and several starting values of h0 . We find that the abrupt change (i.e., the dot-like modes go to the barrier-like ones and vice-versa) occurs around h 38°, independently on h0 . In fact, the starting input value of h0 defines the beginning value of the polar 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 (r 6 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 intercross level around 25°, where first excited barrierlike (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. In the 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 in these 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 observed in InAs/InP QDs. The dispersion relations are shown in Fig. 7, where the adimensional parameter k is plotted as a function of the quantum dot height. For small polar angle approximation, k is negative in all the investigated region. In arbitrary angle approximation, 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 negative would affect strongly the behavior of the IF frequencies. The sign change of k, reveals the importance of the geometry in the phonon modes.
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m=0
300
300
m=0
280
280
m =1
m=1
GaAs
ω (cm-1)
ω (cm-1)
GaAs 260
In As
260
InAs
240
240
m=1
m=1 220
220
m=0 10
(a)
20
30
m =0 40
Polar angle θ
30
50
(b)
35
40
45
Polar angle θ
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.
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 changes from positive to negative as the quantum wire base changes from circular to elliptical shape. To interprete this behavior, they compare the phonon propagation wave with hole-like Schr€ odinger motion 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 demonstrated that the electromagnetic fields increase anomalously when the tip of the cone is approached [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 reported here) implies that the conical case is more complex. For example, for spherical quantum dots, b ¼ 1 1=l and gives rise to completely positive k, where dependence on orbital quantum number m does 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 consequently 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 geometry. The increasing of phonon modes number and their trend changes with the parameter m can enhance the number of active modes in less symmetrical geometries. Of course, a comparison with existing theoretical works suggest that phonon modes are size-dependent in non-regular geometrical shapes. This indicates that polarization-depending optical measurements can be used to probe the shapes and sizes of semiconductor QDs. Finally, 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
C. Kanyinda-Malu, R.M. de la Cruz / Surface Science 529 (2003) 503–514
clude that the IF-frequencies of conical SAQDs are size-dependent with complex dispersion relation. However, for heights close to the height of external cone, the BSO frequencies are size-independent. In this case, optical absorption experiments could not distinguish QDs with heights ranging 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 r ranging from 0.5 to 0.7. On the other hand, this geometrical extrapolation between pyramid and truncated cone can be extended to other quantum dots shapes like lens or domes.
0.4
0.2
0.0
Dispersion relation (a.u.)
513
-0.2
-0.4
-0.6
References -0.8
-1.0
2.0
4.0
6.0
8.0
10.0
12.0
h (nm) Fig. 7. Dispersion relation of interface phonons as a function of the height of conical quantum dots. The symbol ( ) is for m ¼ 0; (}) is for m ¼ 1 within the arbitrary angle approximation; and ( ) is for m ¼ 0 within the small angle approximation.
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 are investigated: InAs/GaAs and InAs/InP in the DCM framework. To resolve the LaplaceÕs equation, 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-
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[17] P. Miska, C. Paranthoen, J. Even, N. Bertru, A. Le Corre, O. Dehaese, J. Phys.: Condens. Matter 14 (2002) 12301. [18] P. Bhattacharya, Properties of Lattice-Matched and Strained Indium Gallium Arsenide, INSPEC, The Institution of Electrical Engineers, London, UK, 1993. [19] Landolt-B€ ornstein, Numerical Data and Functional Relationships in Science and Technology, Springer-Verlag, Berlin, Heidelberg, 1982. [20] N. Esser, K. Hinrichs, W. Richter, http://pcoss.org/ icorsxm/paper/richter.pdf.
[21] F. Comas, C. Trallero-Giner, N. Studart, G.E. Marques, Phys. Rev. B 65 (2002) 073303. [22] P. Kinsler, R.W. Kelsall, P. Harrison, Physica B 263–264 (1999) 507. [23] P.W. Peng, G.J. Jin, Mu. Wang, A. Hu, S.S. Jiang, D. Feng, Phys. Rev. B 59 (1999) 3599. [24] M. Grassi Alessi, M. Capizzi, A.S. Bhatti, A. Frova, F. Materlli, P. Frigeri, A. Broacchi, S. Franchi, Phys. Rev. B 59 (1999) 7620. [25] C.R. Bennett, N.C. Constantinou, M. Babiker, B.K. Ridley, J. Phys.: Condens. Matter 7 (1995) 9819.
Interface phonon modes in truncated conical self-assembled quantum dots Clement Kanyinda-Malu, Rosa Marıa de la Cruz
*
Departamento de Fısica, Universidad Carlos III de Madrid, Avda. de la Universidad 30, 28911 Legan es (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 LaplaceÕs equation. 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. Ó 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-
*
Corresponding author. Tel.: +34-91-624-8733; fax: +34-91624-8749. E-mail address: rmc@fis.uc3m.es (R.M. de la Cruz).
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, cylindrical or real pyramidal systems in the framework of dielectric continuum model (DCM) [7–10], hydrodynamic 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 and Reinecke [9] reported that the interface phonon frequencies of spherical dots are not affected by the size, while on the contrary, the cylindrical quantum dots probe two types of surface phonon
0039-6028/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0039-6028(03)00335-2
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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 phonon modes, and consequently influence their optical properties. In particular, localization of phonons can be expected at the top and 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 completely investigated respect to their quantized vibrational modes or phonons. To our knowledge, the more detailed study on quantum dots having less symmetrical shapes was performed by Knipp and Reinecke [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 importance, 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 relevant expressions needed to calculate the IF-modes. Section 3 presents numerical results and discussion for two lowest phonon modes. The small and arbitrary polar angle approximations are used to account for the effect of mathematical approaches on the IF behavior. In Section 4, relevant conclusions 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. R1 and R2 represent the top and bottom radii of the bases respectively, h0 is the height-size of the SAQD and h (h > h0 ) is the total height of the externally surrounding cone. Because of its axial symmetry, it is possible to describe the cone by
r
R1
θ
h
θ0
h0
φ
R2
Fig. 1. Truncated cone model for the pyramid-like SAQDs.
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 cone by values of polar angle h0 < h < p. A similar description of a conical structure by means of spherical coordinates was done in Ref. [13]. We assume that the dot-island semiconductor of dielectric constant e1 is surrounded by another polar semiconductor with dielectric constant e2 . For simplicity in the present work, we shall neglect the image potential as well as the mechanical boundary conditions, where the transverse optical and longitudinal optical modes can mix into TO–LO coupled mode. In fact, the image potential was seen to affect only very narrow nanostructures, while its contribution is weak in large nanostructures. Therefore, according to macroscopic electrodynamics, and in the absence of free charges inside the dot and barrier materials, we have ej ðxÞr2 Uj ð~ rÞ ¼ 0 ðj ¼ 1; 2Þ;
ð1Þ
where ej ðxÞ and Uj are the dielectric function and scalar potential in material j, respectively. In both semiconductors, the dispersionless dielectric functions are supposed to be frequency-dependent, and is described by ej ðxÞ ¼ ej;1
x2j;LO x2 x2j;TO x2
ðj ¼ 1; 2Þ;
ð2Þ
where ej;1 , is the high-frequency optical dielectric constant, xj;TO and xj;LO are, respectively, the Brillouin zone center frequencies of transverse-
C. Kanyinda-Malu, R.M. de la Cruz / Surface Science 529 (2003) 503–514
optical and longitudinal-optical modes in the material j. Besides the trivial solution Uj ðrÞ ¼ 0, there are two solutions to Eq. (1), i.e., either ej ðxÞ ¼ 0 (the confined bulk-like solution) or ej ðxÞ 6¼ 0 (which gives rise the interface mode solution). In order to resolve analytically the LaplaceÕs equation in separable variables, we assume that the electrostatic potential is along the / axis and depends only on the coordinates r and h (see Fig. 1). A similar assumption for the dependence of the magnetic field in the coordinates r and h was done for a conical structure in order to resolve the wave equation for the magnetic field by separation of variables [13]. Then, in separable variables scheme, the trial Laplace solution Uðr; h; uÞ ¼ f ðrÞgðhÞ expðimuÞ leads to a set of one-dimensional differential equations d2 f df 1 r2 2 þ 2r þ aþ f ¼0 ð3Þ dr dr 4 and d2 g dg þ cotðhÞ 2 dh dh
"
1 aþ 4
# m2 þ 2 g ¼ 0: sin ðhÞ ð4Þ
Here, a is the separation constant to be determined from the electrostatic boundary conditions. Knipp and Reinecke [9] have used a similar separable variables scheme to resolve the LaplaceÕs equation in 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 term base-surface optical (BSO) modes those related to the radial solution and the side-surface optical (SSO) modes as a consequence of boundary conditions applied on the angular solution of the cone. Now, we shall find the BSO and SSO eigenfrequencies. 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
505
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. Z r 1 g 0 f ðrÞ ¼ 1=2 exp i dr ð5Þ 0 r r0 r or in explicit form r f ðrÞ ¼ r1=2 exp ig ln ; r0
ð6Þ
r0 can be interpreted as the distance from the top base to the apex of the cone, i.e., r0 ¼ h h0 (see Fig. 1). Notice that the complex conjugate of f ðrÞ is also solution of the one-dimensional radial equation. Due to the particular nature of the conical geometry, standard electromagnetic boundary conditions do not apply in the radial solution. Therefore, we adopt the boundary integral equation 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 imaginary part of f ðrÞ vanishes everywhere. We obtain then h h0 sin g ln ¼0 ð7Þ h at the bottom base of the cone. The above relation defines uniquely the values of g for given h and h0 which reads as
h h0 g ¼ p ln : ð8Þ h To find the phonon-modes on the base-edges of the horn, let us assume that the electrostatic potential on both top and bottom bases of the truncated cone is identical. This assumption is motivated by two facts: (i) the QD is surrounded by 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 potential and its normal derivative for any point located on the boundary of this domain. Sareni et al.
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[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 Z Z of of dS þ e2 ðxÞ dS ¼ 0; ð9Þ e1 ðxÞ S1 on S2 on where S1 and S2 refer to the top and bottom surfaces, respectively and n is the normal unit vector to the surface. From this condition, we find that 3=2 h h0 h h0 ¼ e2 ðxÞ cos g ln e1 ðxÞ : h h ð10Þ Substituting Eq. (8) into Eq. (10) we obtain e1 ðxÞ ¼b e2 ðxÞ
ð11Þ
h h0 b¼ h
3=2 :
ð12Þ
Eq. (11) together with the relation of the dielectric function expressed in Eq. (2), constitute the transcendental 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 the solution of the surface phonon modes is obtained for positive values of g2 with the electric field given by E ¼ rU. For very small polar angles (h ) and assuming x ¼ gh, Eq. (4) can be written as x2
gðhÞ ¼ Am Im ðghÞ ð0 6 h 6 h0 Þ;
ð14Þ
gðhÞ ¼ Bm Km ðghÞ
ð15Þ
ðh0 6 h 6 pÞ:
The existence of interface modes implies that gðhÞ must be finite for all values of h. Therefore, one expects that the continuity condition on the normal component of the electric displacement field will apply also at the interface of the cone with surrounding medium, i.e. at h ¼ h0 . Using the electric 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, e1 ðxÞ oxo ½ ln½Km ðxÞ ¼ ; e2 ðxÞ oxo ½ ln½Im ðxÞ
x ¼ gh0 :
ð16Þ
In order to have a compact formula, we set
with
Therefore, we express this solution in terms of modified Bessel functions, i.e.
d2 g dg þ x ðx2 þ m2 Þg ¼ 0; dh dh2
ð13Þ
which is a characteristic differential equation for modified Bessel functions of the first and third kind, and m defines the order of the function. It is also 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.
e1 ðxÞ ¼ bm : e2 ðxÞ
ð17Þ
For m ¼ 0, and m ¼ 1, we have b0 ¼
K1 ðgh0 ÞI0 ðgh0 Þ K0 ðgh0 ÞI1 ðgh0 Þ
ð18Þ
I1 ðgh0 Þ½gh0 K0 ðgh0 Þ þ K1 ðgh0 Þ ; K1 ðgh0 Þ½gh0 I0 ðgh0 Þ I1 ðgh0 Þ
ð19Þ
and b1 ¼
respectively. For the reasons which will be discussed below, the dispersion relation can be expressed in terms of non-material dependent parameter km , km ¼
bm þ 1 : bm 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 8 m m P ðcos hÞP1=2þig ð cos h0 Þ; > > < 1=2þig 0 6 h 6 h0 ; gðhÞ ¼ Cgm m Pm ðcos h0 ÞP1=2þig ð cos hÞ; > > : 1=2þig h0 6 h 6 p; ð21Þ
C. Kanyinda-Malu, R.M. de la Cruz / Surface Science 529 (2003) 503–514 m where P1=2þig ðcos 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 the well-known case of the Legendre associated polynomials, solutions of LaplaceÕs equation in spheroidal structures. Accordingly, gðhÞ must be finite for all values of h, as in the small polar angle approximation. Therefore, applying the continuity of the normal component of the electric displacement field at h ¼ h0 and using dispersion relation of the dielectric functions of respective materials, we obtain the transcendental equation which leads to the surface phonon modes, eigenfrequencies of the conical structure. Following Knipp and Reinecke [9], we define the dispersion relation of phonon eigenfrequencies as
e1 ðxgm Þ þ e2 ðxgm Þ e1 ðxgm Þ e2 ðxgm Þ n h io o m m ln P1=2þig ðxÞP1=2þig ðxÞ ox io ¼ n h o m m ln P1=2þig ðx0 Þ=P1=2þig ðx0 Þ ox0
kgm
507
where kmax is a function of both m and h. In the following, we will use only values of g which match with the boundary conditions in radial solution. The logarithmic derivatives in the above equation is the non-material dependent expression of SSO interface-mode frequencies. The transcendental equation can be expressed in terms of series expansion of sinðh=2Þ and cosðh=2Þ. In fact, for m ¼ 0, P1=2þig ðcos hÞ is expressed in serie of sinðh=2Þ, while P1=2þig ð cos hÞ is a complex expression that involves indefined integral [14]. To overcome the lengthy calculation, let us rewrite P1=2þig ðeip cos hÞ as P1=2þig ðcos h0 Þ, with h0 ¼ h p. By simple substitution of h0 in the serie expansion of P1=2þig ðcos hÞ and using trigonometric hints, we obtain 4g2 þ 1 h m 2 cos P1=2þig ð cos hÞ ¼ 1 þ 2 2 2 ð4g2 þ 1Þð4g2 þ 32 Þ 22 42 h cos4 þ 2 þ
:
x¼x0 ¼cos h0
ð22Þ In general, in Eq. (21) g varies continuously from 0 to 1. Consequently, k ranges between kmax and 0,
ð23Þ
By changing variables while derivating the conical functions, it follows that Eq. (22) becomes for m¼0
4g2 þ 32 9 þ 2ð4g2 þ 1Þð4g2 þ 32 Þ cos h 8 2 kg0 ¼ 3ð4g2 þ 1Þ ð4g2 þ 1Þð4g2 þ 32 Þ ð4g2 þ 32 Þ 2 þ 1 þ 1 þ h 4 2þ sin 8 43 2 42 1þ
ð24Þ
and for m ¼ 1, we have
4g2 þ 32 h 13 h 2 2 4 sin h þ 22 sin cos kg1 ¼ 5 cos h þ 2 4 2 8 3 h h h h 2 þ sin2 2 þ cos2 þ sin h cos4 cos h 2 sin2 ð4g2 þ 9Þ 4 2 2 2 2 ( 2 4g þ 9 h h 24 sin2 7 cos h þ 2 sin2 h cos h cos2 8 2 2 )1 2 3ð4g2 þ 9Þ h h h h þ sin2 : 2 þ cos2 sin h cos4 cos h 2 sin2 2 2 2 2 43
ð25Þ
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C. Kanyinda-Malu, R.M. de la Cruz / Surface Science 529 (2003) 503–514
Similarly, we write the compact expression of the DCM eigenfrequencies, with bgm given by 1 þ kgm : 1 kgm
InP
ð26Þ
290
280
-1
Higher order branches of interface modes can directly derived from Eq. (22) by means of the recurrence relations between P1=2þig ðcos hÞ and m P1=2þig ðcos hÞ.
300
ω (cm )
bgm ¼
310
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 semiconductor 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 description 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 in some optical devices. In fact, InAs/InP basedlasers 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, depending on the growth direction, growth of InAs on InP can produce quantum wires or quantum dots. The material parameters used in our calculations are taken from Refs. [18,19] and are summarized in Table 1. We will restrict the presentation of our results to special cases where the InAs/
Table 1 Material parameters used in the calculations Material
e0
e1
xLO (cm1 )
xTO (cm1 )
InAs GaAs InP
15.15 13.18 12.61
11.7 10.89 9.61
241.0 292.0 345.0
218.0 268.7 303.7
270
260
250
InAs
240 8.0
10.0
12.0
14.0
16.0
18.0
Height (nm) Fig. 2. Dependence of the BSO frequencies with the height for InAs/InP quantum dots.
GaAs and InAs/InP QDs differ in phonon features. Fig. 2 shows the dependence of the BSO phonon frequencies with the height in InAs/InP quantum dots for two ranges of h0 : (a) 8–15 nm and (b) 8–18 nm. For these two investigated ranges, 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 the external cone, in this case h ¼ 15 and 18 nm, respectively. The dependence of the BSO frequencies with the height in InAs/GaAs quantum dots is also shown in Fig. 3. In both systems, higher values of h imply high values of barrier modes (or low values of dot modes), independently with the h0 starting 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 reststrahl region of respective material compounds. A ten-
C. Kanyinda-Malu, R.M. de la Cruz / Surface Science 529 (2003) 503–514
GaAs
-1
ω(cm )
280
InAs
228
8.0
10.0
12.0
14.0
16.0
18.0
Height (nm) Fig. 3. Dependence of the BSO frequencies with the height for InAs/GaAs 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 to the 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 discriminated for materials with high band gap and the differences in surface and bulk phonons are enchanced when a surface is terminated with atoms not present in the bulk such as occurs in heterointerfaces [20]. On the other hand, as it is well established in
509
the literature, the interface modes in low-dimensional 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 interface and surface phonons present similarities. In fact, 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 region––for smaller values of h0 not shown herein, the dot-like frequencies lie in the reststrahl––up to higher values which are close to the aforementioned 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 equal in 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 between InAs and GaAs could probably explain different behavior observed in BSO phonon modes for InAs/GaAs QDs with respect to the InAs/InP as the dot height is changed. This study can be improved with the inclusion of the strain effect on the interface phonon modes in different latticemismatched QDs. Fig. 4 shows the dependence of the SSO phonon frequencies of the modes m ¼ 0 and m ¼ 1 with the parameter gh for InAs/InP quantum dots within the small polar angle approximation. We observe that 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 other hand, for the barrier-type modes, the frequencies of the fundamental mode (m ¼ 0) are higher than that of the first excited mode (m ¼ 1) in the investigated range of the parameter gh, while this behavior is reverse for the dot-type modes. This feature was also observed in spheroidal QDs, where phonon modes of higher indexes present
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m=0
340
m=1
InP
-1
ω(cm )
320
240
m=1 InAs m=0 220
0.0
0.2
0.4
0.6
0.8
1.0
ηθ 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).
lower values of frequencies [21]. For increasing values of gh, it seems that the modes m ¼ 0 and m ¼ 1 of barrier-type (dot-type) tends to converge to a single-interface mode close to ðxj;LO þ xj;TO Þ=2 (j ¼ 1; 2). This effect was also reported in III–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 is pronounced for smaller values of gh. The barrierlike and dot-like modes in this approximation act as dual branches of interface optical phonons reported in periodic multilayers or in k-component Fibonnaci 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 dependence for the SSO phonon frequencies of the modes m ¼ 0 and m ¼ 1 with the parameter gh is found for InAs/GaAs quantum dots. Why InAs/ GaAs and InAs/InP QDs do not exhibit different behavior as observed in BSO modes can be attributed to the rotational symmetry that forbids the modification of lateral morphology of the dot.
Another relevant applied parameter is the aspect ratio, defined as r ¼ h0 =2R2 . Typically reported experimental values for r span from 0.1 to 0.3 [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 0:3 are shown in Fig. 5. However, our discussion is focused in the results for m ¼ 0 and m ¼ 1 in whole range of aspect ratios investigated. For the fundamental mode (m ¼ 0), the dependence of dot-like and barrier-like frequencies with the angle is similar for r 6 0:4. The barrier-like frequencies increase for increasing polar angle, while the opposite behavior is obtained for the dot-like frequencies. 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 at polar angles which are dependent of the aspect ratio. For instance, for r ¼ 0:6, the switching occurs at h 32°, while for r ¼ 0:7 the change is around 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 with the polar angle change for different aspect ratios. In fact, for r ¼ 0:1, the barrier-like frequencies decrease with increasing angles, while the dot-like frequencies increase for increasing angles. When r ¼ 0:2, the dot-like and barrier-like frequencies are almost constant and for small angles, they reach the values of the mode m ¼ 0. For r P 0:3, the barrier-like frequencies are increasing functions of the angle, while the dot-like frequencies are decreasing functions. In the light of these results, we deduce that the fundamental and first excited modes are critically sensitive with the aspect ratio. A possible origin of the different features between these modes can be explained in terms of a distinct dispersion relation for m ¼ 0 and m ¼ 1 in the arbitrary polar angle approximation (see Eqs. (24) and (25)). More details about the dispersion relations will be given below.
C. Kanyinda-Malu, R.M. de la Cruz / Surface Science 529 (2003) 503–514
511
290
m=0
280
290
m=0
m=1
m=1 280
GaAs
ω (c m-1)
-1
ω (cm )
GaAs
In As
m= 1 InAs
230 230
m=1
m=0
m=0 220
220 73
(a)
74
75
76
77
78
79
Polar angle θ
48
(b)
50
52
54
56
58
60
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.
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 and several starting values of h0 . We find that the abrupt change (i.e., the dot-like modes go to the barrier-like ones and vice-versa) occurs around h 38°, independently on h0 . In fact, the starting input value of h0 defines the beginning value of the polar 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 (r 6 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 intercross level around 25°, where first excited barrierlike (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. In the 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 in these 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 observed in InAs/InP QDs. The dispersion relations are shown in Fig. 7, where the adimensional parameter k is plotted as a function of the quantum dot height. For small polar angle approximation, k is negative in all the investigated region. In arbitrary angle approximation, 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 negative would affect strongly the behavior of the IF frequencies. The sign change of k, reveals the importance of the geometry in the phonon modes.
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m=0
300
300
m=0
280
280
m =1
m=1
GaAs
ω (cm-1)
ω (cm-1)
GaAs 260
In As
260
InAs
240
240
m=1
m=1 220
220
m=0 10
(a)
20
30
m =0 40
Polar angle θ
30
50
(b)
35
40
45
Polar angle θ
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.
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 changes from positive to negative as the quantum wire base changes from circular to elliptical shape. To interprete this behavior, they compare the phonon propagation wave with hole-like Schr€ odinger motion 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 demonstrated that the electromagnetic fields increase anomalously when the tip of the cone is approached [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 reported here) implies that the conical case is more complex. For example, for spherical quantum dots, b ¼ 1 1=l and gives rise to completely positive k, where dependence on orbital quantum number m does 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 consequently 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 geometry. The increasing of phonon modes number and their trend changes with the parameter m can enhance the number of active modes in less symmetrical geometries. Of course, a comparison with existing theoretical works suggest that phonon modes are size-dependent in non-regular geometrical shapes. This indicates that polarization-depending optical measurements can be used to probe the shapes and sizes of semiconductor QDs. Finally, 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
C. Kanyinda-Malu, R.M. de la Cruz / Surface Science 529 (2003) 503–514
clude that the IF-frequencies of conical SAQDs are size-dependent with complex dispersion relation. However, for heights close to the height of external cone, the BSO frequencies are size-independent. In this case, optical absorption experiments could not distinguish QDs with heights ranging 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 r ranging from 0.5 to 0.7. On the other hand, this geometrical extrapolation between pyramid and truncated cone can be extended to other quantum dots shapes like lens or domes.
0.4
0.2
0.0
Dispersion relation (a.u.)
513
-0.2
-0.4
-0.6
References -0.8
-1.0
2.0
4.0
6.0
8.0
10.0
12.0
h (nm) Fig. 7. Dispersion relation of interface phonons as a function of the height of conical quantum dots. The symbol ( ) is for m ¼ 0; (}) is for m ¼ 1 within the arbitrary angle approximation; and ( ) is for m ¼ 0 within the small angle approximation.
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 are investigated: InAs/GaAs and InAs/InP in the DCM framework. To resolve the LaplaceÕs equation, 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-
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