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Polar optical phonon states and dispersive spectra of wurtzite ZnO nanocrystals embedded in zinc-blende MgO matrix
Superlattices and Microstructures 50 (2011) 242–251
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Polar optical phonon states and dispersive spectra of wurtzite ZnO nanocrystals embedded in zinc-blende MgO matrix Li Zhang ⇑ Department of Mechanism and Electronics, Guangzhou Panyu Polytechnic, Guangzhou 511483, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 3 June 2011 Received in revised form 18 June 2011 Accepted 20 June 2011 Available online 30 June 2011 Keywords: ZnO/MgO quantum dots Interface optical and quasi-confine phonon modes Dispersive spectra
a b s t r a c t Based on the macroscopic dielectric continuum model and Loudon’s uniaxial crystal model, the polar optical phonon modes of a quasi-0dimensional (Q0D) wurtzite spherical nanocrystal embedded in zinc-blende dielectric matrix are derived and studied. It is found that there are two types of polar phonon modes, i.e. interface optical (IO) phonon modes and the quasi-confined (QC) phonon modes coexisting in Q0D wurtzite ZnO nanocrystal embedded in zincblende MgO matrix. Via solving Laplace equations under spheroidal and spherical coordinates, the unified and analytical phonon states and dispersive equations of IO and QC modes are derived. Numerical calculations on a wurtzite/zinc-blende ZnO/MgO nanocrystal are performed. The frequency ranges of the IO and QC phonon modes of the ZnO/MgO nanocrystals are analyzed and discussed. It is found that the IO modes only exist in one frequency range, while QC modes may appear in three frequency ranges. The dispersive frequencies of IO and QC modes are the discrete functions of orbital quantum numbers l and azimuthal quantum numbers m. Moreover, a pair of given l and m corresponds to one IO mode, but to more than one branches of QC. The analytical phonon states and dispersive equations obtained here are quite useful for further investigating Raman spectra of phonons and other relative properties of wurtzite/zinc-blende Q0D nanocrystal structures. Ó 2011 Published by Elsevier Ltd.
1. Introduction Currently, a great deal of interest has been focused on the group-II–VI ZnO-based semiconductors, such as ZnO, MgO, CdO and their ternary compounds, which can be used to make optical devices ⇑ Tel.: +86 20 84749738 (O); fax: +86 20 347377068 (H). E-mail address: [email protected] 0749-6036/$ - see front matter Ó 2011 Published by Elsevier Ltd. doi:10.1016/j.spmi.2011.06.006
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operating in the blue and ultraviolet wavelength regions due to their wide adjustable directed bandgaps [1–3]. In addition, semiconductor nanocrystals and quantum dots (QDs) are nanostructures where the three-dimensional confinement of carriers results in a discrete energy spectrum [4–6]. This can further improve the performance of optoelectronic devices, such as low-threshold lasers and light polarization insensitive detectors. With the technique advancement of the crystal-growth, such as metal-organic chemical vapor deposition, the molecular-beam epitaxy and hydride vapor phase epitaxy, the ZnO-based nanocrystals and QDs can be fabricated in experiments [6–12], and it provides explicit and important research object of ZnO-based nanostructures. Hence the investigation of various physical properties in quasi-0-dimensional (Q0D) ZnO-based quantum structures has become a hot topic during the last decade [1–3,6–15]. Among the experimental and theoretical investigations on the Q0D ZnO-based nanostructures, the ZnO nanocrystals embedded in the MgO matrix attracted substantial attention [6–12,16–18]. This is mainly due to the fact that the capping MgO shell outside of the ZnO core can not only reduce the surface-related defects and enhance the stability of nanocrystal, but also confine the charge carriers into the core region due to the wider band offset potential, which greatly improve the luminescent characteristics such as quantum efficiency and photostability [6–10]. Zeng et al. [6] grown ZnO/MgO QDs based on a metal-organic chemical vapor deposition method. And a remarkable enhancement of free exciton emission from the QDs after the growth of MgO layer was observed in the photoluminescence (PL) measurement. Lee and cooperators [7] found that, in contrast to the bare ZnO nanocrystal, the luminescent efficiency of green emission in core/shell structure of ZnO/MgO nanocrystals increases by 30%. Bera and coworkers’ experiment [8] reveals that the ZnO/MgO QDs, synthesized by a sol– gel process, exhibited a quantum yield of 13% compared with less than 5% for bare ZnO QDs. By a sequential preparative procedure and capped with carboxymethyl b-cyclodextrin cavities, Rakshit and Vasudevan [9] synthesized the core-shell ZnO/MgO nanocrystals. Their experiment shows that MgO on ZnO nanocrystals prevented the aggregation of the nanoparticles and formation of nonradiative recombination sites, which results in a more intense and stable green photoluminescence. The bound electronic states, donor and acceptor impurity states, and excitonic states in the Q0D ZnO-based core-shell structure nanocrystals and QDs have been widely investigated [6,9,16–18]. However, only limited data is available on the polar optical phonon states and their dispersive spectra in wurtzite systems [19–22]. The phenomena of phonon replicas in the emission spectra, the homogeneous broadening of excitonic line width and the relaxations of hot carriers to the fundamental band edge are directly relative to the lattice vibration of semiconductor materials [23,24]. Theories and experiments reveal that not only the carriers, but also the lattice vibrations (phonon modes) are influenced greatly by the heterostructures of low-dimensional quantum systems [25–27]. Moreover, phonon modes have quite important influence on the optoelectronics and electronics properties of wurtzite ZnO(GaN)-based low-dimensional quantum structures [28–34]. Hence it is necessary and important to study the phonon modes in ZnO-based nanocrystal structures. It is well known that ZnO material usually crystallizes in the hexagonal wurtzite structure, whose physical properties behavior anisotropic in space. However, according to recent reports [35], if doping concentration x in MgxZn1xO is over than 0.36, the MgZnO ternary compound will change from wurtzite structure to cubic zinc-blend structure. The experiment of Lee et al. [7] clearly reveals that the core ZnO behaviors as hexagonal wurtzite phase, while the capping MgO displays as zinc-blende phase. For simplicity, we will investigate a wurtzite ZnO nanocrystal embedded in zinc-blende MgO matrix. Even though, the phonon modes in wurtzite/zinc-blende ZnO/MgO nanocrystal are more complicated than those the pure zinc-blende GaAs-based confined structures [36–39] due to the wurtzite anisotropy of ZnO material. In fact, within the framework of the dielectric continuum model (DCM), the optical phonon vibrating modes in pure zinc-blende GaAs-based Q0D QDs [36–39] and pure wurtzite GaN- (ZnO-) based QDs [32–34] have been deduced and analyzed. Klein et al. [36] studied the size dependence of electron-phonon coupling in cubic CdSe nanospheres. Roca et al. [37] investigated the polar optical vibration modes in a GaAs/AlAs sphere QD. Li and Chen [38] derived the polar IO phonon modes of a GaAs/AlGaAs cylindrical QD, and two types of IO modes (i.e. the top IO modes and the surface IO modes) are found in the Q0D quantum structures. Fonoberov and Balandin [19–22] analyzed the polar optical phonon modes in pure wurtzite ZnO/MgZnO nanostructures. Recently, we extended the phonon mode works to the pure wurtzite Q0D GaN-based and ZnO-based cylindrical QD
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structures, and that four types of polar mixing optical phonon modes may coexist in the Q0D wurtzite quantum systems has been predicted [32–34]. However, to the best of our knowledge, the integrated optical phonon modes and their dispersive properties in wurtzite/zinc-blende ZnO/MgO Q0D nanocrystals and QDs have not been investigated. The present work is just to fill a gap of optical phonon mode theories in wurtzite/zinc-blende ZnO/MgO nanocrystals. We will study the fully phonon vibration modes and dispersive spectra in a Q0D sphere ZnO/MgO nanocrystal structures in the present paper. The main accomplishments and significance of this work can be summarized into the three points as follows. (i) Based on the solutions of the Laplace equation in wurtzite spherical ZnO nanocrystal embedded in zinc-blende MgO matrix, it is confirmed that there are two types of polar optical phonon modes, i.e. the IO modes and the QC modes coexisting in the wurtzite/zinc-blende ZnO/MgO Q0D quantum structure. And the frequency ranges of the two types of phonon modes are characterized. (ii) Based on the DCM and Loudon’s [40] uniaxial crystal model, the explicit phonon states and dispersive equations of the IO and QC modes in the wurtzite/zinc-blende spherical nanocrystals are given. (iii) Numerical calculations on the dispersive relations of IO and QC modes in the ZnO/MgO nanocrystals are performed. It is observed that the dispersive frequencies of IO and QC modes are the discrete functions of orbital quantum numbers l and azimuthal quantum numbers m. A pair of given l and m correspond to one IO mode, but to more than one branches of QC modes. A detailed comparison with pure Q0D zinc-blende spherical QD and pure wurtzite cylindrical QD systems are carried out, and relative reasons resulting in the difference are analyzed in depth from both viewpoints of physics and mathematics. The remaining sections of the paper is organized so: in Section 2, the uniform phonon states of IO and QC phonon modes and their dispersive equations in the structures are presented. In Section 3, the numerical calculations on the dispersive frequency of the two types of phonon modes are carried out and discussed; and finally, we summarize the mainly conclusions obtained in the paper in Section 4. 2. Theory Let us consider a wurtzite-core/zinc-blende-shell spherical nanocrystal with radius R (referring to Fig. 1). The z-axis is taken along the direction of the c-axis of the wurtzite-core material and denote the axial- (radial-) direction as z (t). Within the framework of the macroscopic DCM and Loudon’s uniaxial crystal model, the electrostatic potential U1(r) of the polar vibrating modes within the wurtzite spherical nanocrystal is determined by the second-order differential functions (1) under the prolate spheroidal coordinate, i.e.
1 R2 ðn2 g2 Þ
^n
@ @ @ @ ðn2 1Þ þ g^ ð1 g2 Þ U1 ðn; g; uÞ @n @n @g @g
@2 U1 ðn; g; uÞ ¼ 0: R2 ðn2 1Þð1 g2 Þ @ u2
t
ð1Þ
And the electrostatic potential U2(r) outside of the wurtzite spherical nanocrystal is given by the second-order differential functions (2) under the spherical coordinate, i.e.
d
) 1 @ 1 @ @ 1 @2 2 @ r 2 sin h þ U2 ðr; h; uÞ ¼ 0: 2 r 2 @r @r r sin h @h @h r2 sin h @ u2
ð2Þ
where ed(x) is the dielectric function of the zinc-blende dielectric matrix out side of the wurtzite core ^¼^ ^ ,t,z) in Eq. (1) is the dielectric function of the wurtzite core matematerial. The function p^ ðxÞ (p n; g ^-direction. The dielectric functions t(x) and z(x) along the t- and z-directions of the rial along the p wurtzite core material as well as the dielectric function of the zinc-blende matrix material are respectively given by
v ðxÞ ¼ 1 v
x2 x2v ;L ; v ¼ d; t; z: x2 x2v ;T
ð3Þ
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z /c-axial MgO
ZnO R o
y
x Fig. 1. Schematic view of the wurtzite ZnO spherical nanocrystal with radius R embedded in zinc-blende MgO matrix. The zaxis is taken along the c-axis direction of the wurtzite ZnO material.
Fig. 2. The function g(x) for ZnO material as a function of x. The characteristic frequencies of ZnO and MgO materials divide the frequency axis into seven ranges, which are labelled as range I, II,. . ., and VII from left to right. The subscripts ‘‘1’’ and ‘‘2’’ correspond to the ZnO and MgO, respectively.
In Eq. (3), xz,L, xz,T, xt,L and xt,T are the zone center characteristic frequencies of A1(LO), A 1(TO), E1(LO), and E1(TO) modes of wurtzite bulk material, respectively. In order to solve the Eq. (1) conveniently, we define a function g(x) as
gðxÞ ¼ z ðxÞ=t ðxÞ:
ð4Þ
The meanings of this function and its influence on the phonon potential properties of vibration or decaying will be discussed later in Fig. 2. The phonon potentials U(r) of the wurtzite-core/zincblende-shell nanocrystal, namely the solutions of Eqs. (1) and (2), which are finite everywhere inside the nanocrystal and vanishes far away from the nanocrystal can be found analytically as follow:
UðrÞ ¼
8
C
Pm ðnÞ m l P ð ðn0 Þ l Pm l
gÞeimu ;
lþ1
R riþ1
imu Pm ; l ðcos hÞe
within outside
ð5Þ
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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where n0 ¼ 1= 1 gðxÞ, P m l ðxÞ are associated Legendre polynomials of the first kind, and integers l (l P 0) and m (jmj 6 l) are the orbital and azimuthal quantum numbers of the phonon modes in Q0D nanocrystals, respectively. In terms of the relationships of spheroidal/spherical coordinates with the Cartesian coordinates, it is easy to get the relations between the coordinates (n, g) and the coordinates (r, h), i.e.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 ¼ r sin h ¼ R 1=gðxÞ 1 ðn2 1Þð1 g2 Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z ¼ r cos h ¼ R 1 gðxÞng:
ð6Þ
It is obvious that the interface of the nanocrystal is described by r = R in spherical coordinates. Based pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi on the Eq. (6), the nanocrystal interface also can be denoted as n0 ¼ 1= 1 gðxÞ and g0 = cosh in spheroidal coordinates. Via the boundary conditions (BC) in spheroidal/spherical coordinates, it is found that the continuum condition of phonon potential functions at the interface of wurtzite nanocrystal, namely U1 ðrÞn¼n0 ;g¼cos h ¼ U2 ðrÞr¼R , is automatically satisfied. Next we use the continuum condition of electric displacement vector D perpendicular to the interface at r ¼ R to obtain the dispersive equation of phonon modes in the spherical nanocrystal. In terms of the relation (6) and the gradient operator in prolate spheroidal coordinates:
O¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ ^ 1 1 @ u @ ^ 1 g2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n^ n2 1 þ g þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2 @u R @n @g 2 2 R n 1 1 g n g
ð7Þ
one can get
t ðxÞgðxÞ @ U1 ðrÞ D1 ðn; g; uÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 1 gðxÞ @n n¼n
ð8Þ
0 ;g¼cos h
and
D2 ðr; h; uÞ ¼ d ðxÞ
@ U2 ðrÞ : @r r¼R
ð9Þ
Substituting Eq. (5) into the above Eqs. (8) and (9), and letting D1 = D2, we can get the follow formula, i.e.
z ð xÞ
m m 1 ðl þ 1Þðl þ mÞP m l1 ðn0 Þ lðl m þ 1ÞP lþ1 ðn0 Þ d ðxÞð2l þ 1Þðl þ 1Þðn0 n0 ÞP l ðn0 Þ ¼ 0: ð10Þ
This Eq. (10) is just the dispersive equation of the polar optical phonon modes in a wurtzite-core/zincblende-shell spherical nanocrystal. Substituting the dielectric functions Eq. (3) into Eq. (10), the dispersive frequencies x of the phonon modes can be worked out as functions of the quantum numbers l and m. It is interesting to note that, as the anisotropy of the dielectric function in wurtzite core material is neglected, (i.e. z(x) = t(x) = 0(x), thus g(x) = 1 and n0 ? 1), the relation (11) can be obtained:
x
x
@Pm l l ðnÞ ¼ 0 ðxÞ: R x @n
t ð Þgð Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g!1;n¼n0 !1 RP m ðn Þ 1 gð Þ 0 l
lim
ð11Þ
Using this relation, the dispersive Eq. (10) will reduced to the form as
l0 ðxÞ þ ðl þ 1Þd ðxÞ ¼ 0;
ð12Þ
which is just the dispersive equation of IO phonon modes in isotropic spherical QD [36,37]. This also illustrates that the dispersive equation obtained here in wurtzite-core/zinc-blende-shell nanocrystal is more general than the previous ones in cubic or pure zinc-blende QDs.
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3. Numerical results and discussions In order to get a clear picture for the types of phonon modes and their dispersive properties in the wurtzite/zinc-blende Q0D nanocrystal system, a numerical calculation is performed on a wurtzite/ zinc-blende ZnO/MgO spherical nanocrystal. The physical parameters of the materials used in our calculation are listed in Table 1 [41–43]. Before analyzing the dispersive properties of phonon modes, let us first discuss the types of phonon modes and the frequency ranges of these phonon modes in ZnO/MgO nanocrystal system. Shi et al. [44] pointed out that five types of polar optical phonon modes including the IO modes, QC modes, propagating modes (PR), half-space (HS) modes and exactly confined (EC) modes may coexist in pure wurtzite low-dimensional quantum heterostructures. Apart from the EC modes, whose frequencies x = xtL, all the other phonon modes are dispersive. In terms of the data given in Table1, the function g(x) of ZnO material as a function of x is shown in Fig. 2. The characteristic frequencies of ZnO and MgO materials are labelled on the abscissa. From the figure, it is observed that the characteristics frequencies of ZnO and MgO divide the frequency axis into seven ranges. For convenience, these ranges are labelled as range I, II, . . ., and VII from left to right. In each frequency range, the signs for the dielectric functions of ZnO and MgO semiconductors and g(x) function are certain (referring to Table 2). In m terms of the feature of associated Legendre function Pm l ðnÞ of the first kind, it is known that P l ðnÞ is a vibrating function as 0 < n < 1 (i.e. g(x) < 0). But as n > 1 (i.e. 0 < g(x) < 1) or in > 0 (i.e. g(x) > 1), Pm l ðnÞ behaves as decaying function. Thus the dispersive Eq. (10) gives the unified forms for the full phonon modes in wurtzite-core/zinc-blende-shell spherical nanocrystals. This is obviously different from the situation of pure wurtzite cylindrical quantum wires and QDs [32–34]. From the Fig. 2 or Table 2, we observe that the function g(x) takes negative values in ranges II, III and V, and it takes positive values in the ranges I, IV, VI and VII. Moreover, the values of g(x) are lower (over) than 1 in ranges I and VI (IV and VII). Based on the feature analysis of P m l ðnÞ above, it is found Pm l ðnÞ behaves as decaying (vibrating) waves in frequency ranges II, III and V (I, IV, VI and VII). Based on the properties of IO phonon modes, namely behaving as decaying waves in the core material and the dielectric matrix, it is easy to find that the IO modes of the system may exist in the ranges I, IV, VI and VII. The QC modes behave as oscillating waves in the core material, and as decaying waves in the dielectric matrix, which are quite similar to the electronic wavefunctions in quantum wells with finite-high potential barrier [44,45]. Thus the QC modes may appear in these three ranges II, III and V. Due to the dielectric matrix being isotropic zinc-blende semiconductor, only the decaying waves and exact confined waves can appear in this region. Thus there are no PR and HS modes in ZnO/ MgO nanocrystals. But IO and QC modes can exist in this structures because both the modes demand
Table 1 Zone-center energies (in meV) of polar optical phonons, dielectric constants of wurtzite ZnO material and zinc-blende MgO dielectric matrix [41–43]. ZnO MgO
xtT
xtL
xzT
xzL
z1
t1
47.2
71.7
50.7
73.6
3.78
3.7
xLO
xTO
d1
d0
90
49.32
2.96
9.86
Table 2 Signs of the functions t(x), z(x), g(x) and MgO semiconductors.
ZnO
MgO
d(x) in seven frequency ranges divided by the characteristic frequencies of ZnO and
Ranges
I
II
III
IV
V
VI
VII
t(x) z(x) g(x)
+ + +
+
+
+
+
+ + +
+ + +
d(x)
+
+
+
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decaying waves in the dielectric matrix. Taking into account the condition of solution (the dielectric functions of ZnO and MgO cannot take positive or negative signs simultaneously), the dispersive Eq. (10) of IO and QC modes have no solution in the ranges I, IV and VII. Thus the QC phonon modes can appear in the three ranges: II, III and V, while the IO phonon modes can only exist in the frequency range VI. Moreover, the n values of P m l ðnÞ are between 0 and 1, namely 0 < n < 1 in frequency range VI for IO phonon modes in the system. Fig. 3 plots the dispersion frequencies ⁄x of the full phonon modes as functions of the orbital quantum numbers l and azimuthal quantum numbers m in the ZnO/MgO nanocrystals. From the figure, it is seen that the characteristic frequencies xtT1, xTO2, xzT1, xtL1 and xzL1 (dash dot lines in the figure) ascertain three frequency ranges, which are the frequency ranges of QC modes. The topmost range falls into [xzL1,xLO2], which is the frequency range of the IO phonon modes. These observations are completely consonant with the analysis of Fig. 2 and Table 2. All the dispersive frequencies ⁄x of IO and QC modes are the discrete functions of quantum numbers l and m. This is quite analogous to the electronic energy levels of Q0D QDs [45]. The discrete dispersive spectra of phonon modes in the Q0D nanocrystals also differ obviously from those in quasi-2-dimensional (Q2D) quantum wells and quasi-1-dimensional (Q1D) quantum wires structures [32,33,44]. The phonon dispersive frequencies in Q2D (Q1D) quantum well (wire) systems in general, are the continuum functions of free phonon wave-numbers in planes (axial directions). For a given orbital quantum number l, there are l + 1 branches of IO phonon modes. They corresponds to the IO phonon modes with azimuthal quantum numbers m = 0, 1, 2, . . ., l, respectively. For a certain m, the dispersive frequencies of IO modes are the monotonic and decreasing function of l (referring the dash lines with symbols, such with squares, circles and triangles). With the increase of l, the dispersions of these modes become weaker and weaker. Moreover, the frequency intervals
Fig. 3. Dispersion frequencies ⁄x of the IO and QC phonon modes as functions of the orbital quantum numbers l and azimuthal quantum numbers m in the ZnO/MgO nanocrystals.
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of two IO modes with adjacent azimuthal quantum numbers (i.e. m and m + 1, m = 0, 1, 2, . . .) decrease with the increase of l. These are the common features of phonon modes in quantum confined systems [27,32–34,42,44]. Though the dielectric function d(x) takes different signs (referring to Table 2) in the frequency subranges of [xtT1, xTO2] and [xTO2, xzT1], the two subranges constitute a integrated range for QC phonon modes. And some dash curves, such as one of the line with squares, and that with circles, span and extend into the two subranges. Just as the IO modes, all the curves of QC modes in the two subranges are the monotonic and decreasing function of l. It is observed that, for given l and m, there is one branch or more than two branches of QC modes in the two subranges. For example, there are two branches of QC modes with l = 2 and m = 1, and quantum numbers l = 4 and m = 1 correspond to three branches of QC modes. This is different from the situation of IO modes. For IO modes in the systems, a pair of given l and m correspond to one IO mode. In addition, we notice that there is no QC modes for orbital quantum number l = 1 in the frequency range of [xtL, xzL1]. The QC modes begin to appear in this frequency range as l is over than 1. This characteristic is similar to the case of reducing behavior of phonon modes in wurtzite GaN-based quantum structures [32,44]. And it originates from the anisotropy of the wurtzite crystal structures of ZnO core material. The dispersive dash-curves of QC modes in this range are the monotonic and increasing function of l, which distinguishes the cases of IO modes and QC modes in the integrated range of [xtT1, xzT1]. At last, it should be pointed out that the dispersive frequencies of IO and QC modes have no relation to the size R of the Q0D nanocrystal, which can be seen clearly from the dispersive Eq. (10). In fact, this result has been verified by the recent experimental measurement of Zhang and cooperators [46]. Since the theoretical approach of this work is similar to the one developed by Fonoberov and Balandin [19–21], it is necessary to compare the results of the present work with those of Fonoberv and Balandin. In contrast to the freestanding wurtzite ZnO nanocrystal [21], the introduce of the barrier MgO material has obvious influence on the dispersive spectra of phonon modes in the Q0D nanocrystal structures. The characteristic frequency xTO, MgO divided the frequency range of QC modes in freestanding ZnO nanocrystal into two subranges (namely ranges II and III in Fig. 2). The phonon range of IO modes [xtT,ZnO, xzL, ZnO] in freestanding ZnO nanocrystal is forbidden in ZnO/MgO nanocrystal. And a new frequency range [xzl, ZnO, xLO, MgO] of IO modes appears in the wurtzite ZnO nanocrystal embedded in zinc-blende MgO matrix. Moreover, relative to the Fig. 1 of Ref. [20], the Fig. 2 of this work explicitly gives not only the signs of the function g(x), but also the specific value of g(x). In particular, the magnitude relationship between g(x) and 1 is displayed clearly as a function of x, which is quite useful for analyzing the forms and the decaying/vibrating properties of Pm l ðnÞ functions. Fonoberov and Balandin’s works numerically discussed the influences of quantum structures (ratio of spheroidal semi-axes, a/c and c/a) [20] and nonpolar dielectric matrix (dielectric constant d of exterior matrix varies from 1 to 4) [19] on the dispersive spectra of phonon modes in freestanding wurtzite ZnO and ZnO/MgZnO nanocrystals. The present work mainly analyzed the dependance of dispersive spectra on the orbital quantum numbers and azimuthal quantum numbers in wurtzite ZnO-core embedded in zinc-blende MgO dielectric matrix. Hence the present paper can be looked as a complementarity and extension of previous works [19–21].
4. Conclusions In conclusion, we have studied and discussed the polar optical phonon modes in a wurtzite-core/ zinc-blende-shell Q0D nanocrystal structure in the present work. The analytical phonon states of phonon modes are obtained by means of the DCM and Loudon’s uniaxial crystal model. It is confirmed that, via solving the Laplace equations in wurtzite and zinc-blende crystals under the spheroidal and spherical coordinates, there are two types of polar phonon modes, namely the IO modes and the QC modes in wurtzite/zinc-blende cylindrical nanocrystal structures. The unified phonon states (Eq. (5)) and dispersive equation (Eq. (10)) for the IO and QC modes are also obtained. Numerical calculations on a wurtzite ZnO spherical nanocrystal embedded in zinc-blende MgO matrix are performed. The frequency ranges of the phonon modes in the ZnO/MgO nanocrystal are analyzed in detail. It is found that the IO modes of the structure only exists in one frequency range, and the QC
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modes may appear in three frequency ranges. The dispersive frequencies of the IO and QC modes as functions of the orbital quantum number l and azimuthal quantum number m are plotted, and detailed features for these curves are discussed. It is found that the dispersive frequencies are the discrete functions of quantum numbers l and m, which is analogous to electronic energy levels in Q0D QDs [45], and differs obviously from the dispersive spectra of Q2D quantum wells [42,44] and Q1D quantum wires [32,33]. A pair of given l and m correspond to one IO mode. But more than one branches of QC may appear for a given orbital and azimuthal quantum numbers. The present theoretical and numerical results can be used to analyze and discuss the dispersive properties, Raman spectra and other relative properties of wurtzite/zinc-blende Q0D nanocrystal systems. We hope that the present work will stimulate further theoretical and experimental investigations of lattice dynamical properties, as well as device applications based on the wurtzite/zinc-blende Q0D nanocrystal structures. Acknowledgments Project was jointly supported by the NNSYF of China (Grant No. 60906042), Yangcheng Scholar Project (Grant No. 10B010D) and STPAA (Grant No. 10B001) of Guangzhou City. The author would like to acknowledge the valuable guidance and discussion of Prof. J.J. Shi of Peking University. References [1] Ü. Özgür, I. Alivov, C. Liu, A. Teke, M.A. Reshchikov, S. Dogan, C. Avrutin, S.-J. Cho, H. Morkoç, J. Appl. Phys. 98 (2005) 041301. [2] Z.L. Wang, Material Today 10 (2007) 20. [3] A. Janotti, C.G.V. de Walle, Rep. Prog. Phys. 72 (2009) 126501. [4] A.D. Yoffe, Adv. Phys. 50 (2000) 208. [5] A. Nakamura, K. Okamatsu, T. Tawara, H. Gotoh, J. Temmyo, Y. Matsui, Jpn. J. Appl. Phys. 47 (2008) 3007. [6] Y.J. Zeng, Z.Z. Ye, F. Liu, D.Y. Li, Y.F. Lu, W. Jaeger, H.P. He, L.P. Zhu, J.Y. Huang, B.H. Zhao, Crystal Growth Design 9 (2009) 263. [7] J.J. Lee, J. Bang, H. Yang, J. Phys. D: Appl. Phys. 42 (2009) 025305. [8] D. Bera, L. Qian, P.H. Holloway, J. Phys. D: Appl. Phys. 41 (2008) 182002. [9] S. Rakshit, S. Vasudevan, Nano 2 (2008) 1473. [10] J.G. Ma, Y.C. Liu, C.L. Shao, J.Y. Zhang, Y.M. Lu, D.Z. Shen, X.W. Fan, Phys. Rev. B 71 (2005) 125430. [11] G.N. Panin, A.N. Baranov, Y.-J. Oh, T.W. Kang, Current Appl. Phys. 4 (2004) 647. [12] G.N. Panin, A.N. Baranov, I.A. Khotina, T.W. Kang, J. Kor. Phys. Soc. 53 (2008) 2943. [13] K.A. Alim, V.A. Fonoberov, M. Shamsa, A.A. Balandin, J. Appl. Phys. 97 (2005) 124313. [14] K.K. Zhuravlev, W.H.H. Oo, M.D. McCluskey, J. Huso, J.L. Morrison, L. Bergan, J. Appl. Phys. 106 (2009) 013511. [15] C.J. Pan, K.-F. Lin, W.T. Hsu, W.-F. Hsieh, J. Phys. Cond. Matter 19 (2007) 186201. [16] S.Y. Wei, Q. Chang, Z. Zeng, Current Appl. Phys. 11 (2011) 16. [17] X. Zhao, S.Y. Wei, C. Xia, G. Wei, J. Lumin. 131 (2011) 297. [18] J.A. Davis, C. Jagadish, Laser Photon. Rev. 3 (2009) 85. [19] V.A. Fonoberov, A.A. Balandin, Phys. Rev. B 70 (2004) 233205. [20] V.A. Fonoberov, A.A. Balandin, J. Phys. Cond. Matter 17 (2005) 1085. [21] V.A. Fonoberov, A.A. Balandin, Phys. Stat. Sol. C 1 (2004) 2650. [22] V.A. Fonoberov, A.A. Balandin, J. Nanoelectron. Optoelectron 1 (2006) 19. [23] A.A. Balandin, J. Nanosci. Nanotech. 5 (2005) 1015. [24] X.B. Zhang, T. Taliercio, S. Kolliakos, P. Lefebvre, J. Phys.: Cond. Matter 13 (2001) 7053. [25] K. Sood, J. Menendez, M. Cardona, K. Ploog, Phys. Rev. B 54 (1985) 2111–2115. [26] A. Tanaka, S. Onari, T. Arai, Phys. Rev. B 45 (1992) 6587. 47 (1993) 1237. [27] H.J. Xie, C.Y. Chen, B.K. Ma, Phys. Rev. B 61 (2000) 4827. [28] A. Ashrafi, J. Appl. Phys. 107 (2010) 123527. [29] V.I. Kushnirenko, I.V. Markevich, L.V. Borkovska, B.M. Bulakh, Phys. Stat. Sol. C 7 (2010) 1605. [30] J.M. Wesselinowaa, A.T. Apostolov, J. Appl. Phys. 108 (2010) 044316. [31] V.A. Fonoberov, A.A. Balandin, J. Vac. Sci. Technol. B 22 (2004) 2190. [32] L. Zhang, J. Shi, T.L. Tansley, Phys. Rev. B 71 (2005) 245324. [33] L. Zhang, J.J. Shi, S. Gao, Semicond. Sci. Technol. 23 (2008) 045014. [34] L. Zhang, Turk. J. Phys. 34 (2010) 123. [35] C.H. Ahn, S.K. Mohanta, B.H. Kong, H.K. Cho, J. Phys. D: Appl. Phys. 42 (2009) 115106. [36] M.C. Klein, F. Hache, D. Ricard, C. Flytzanis, Phys. Rev. B 42 (1990) 11123. [37] E. Roca, C. Trallero-Giner, M. Cardona, Phys. Rev. B 49 (1994) 13704. [38] W.S. Li, C.Y. Chen, Physica B 229 (1997) 375. [39] L. Zhang, H.J. Xie, C.Y. Chen, Phys. Rev. B 66 (2002) 205326. [40] R. Loudon, Adv. Phys. 13 (1964) 423. [41] F. Güell, J.O. Osso, A.R. Goni, A. Cornet, J.R. Morante, Superlatt. Microstruc. 45 (2009) 271.
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Polar optical phonon states and dispersive spectra of wurtzite ZnO nanocrystals embedded in zinc-blende MgO matrix Li Zhang ⇑ Department of Mechanism and Electronics, Guangzhou Panyu Polytechnic, Guangzhou 511483, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 3 June 2011 Received in revised form 18 June 2011 Accepted 20 June 2011 Available online 30 June 2011 Keywords: ZnO/MgO quantum dots Interface optical and quasi-confine phonon modes Dispersive spectra
a b s t r a c t Based on the macroscopic dielectric continuum model and Loudon’s uniaxial crystal model, the polar optical phonon modes of a quasi-0dimensional (Q0D) wurtzite spherical nanocrystal embedded in zinc-blende dielectric matrix are derived and studied. It is found that there are two types of polar phonon modes, i.e. interface optical (IO) phonon modes and the quasi-confined (QC) phonon modes coexisting in Q0D wurtzite ZnO nanocrystal embedded in zincblende MgO matrix. Via solving Laplace equations under spheroidal and spherical coordinates, the unified and analytical phonon states and dispersive equations of IO and QC modes are derived. Numerical calculations on a wurtzite/zinc-blende ZnO/MgO nanocrystal are performed. The frequency ranges of the IO and QC phonon modes of the ZnO/MgO nanocrystals are analyzed and discussed. It is found that the IO modes only exist in one frequency range, while QC modes may appear in three frequency ranges. The dispersive frequencies of IO and QC modes are the discrete functions of orbital quantum numbers l and azimuthal quantum numbers m. Moreover, a pair of given l and m corresponds to one IO mode, but to more than one branches of QC. The analytical phonon states and dispersive equations obtained here are quite useful for further investigating Raman spectra of phonons and other relative properties of wurtzite/zinc-blende Q0D nanocrystal structures. Ó 2011 Published by Elsevier Ltd.
1. Introduction Currently, a great deal of interest has been focused on the group-II–VI ZnO-based semiconductors, such as ZnO, MgO, CdO and their ternary compounds, which can be used to make optical devices ⇑ Tel.: +86 20 84749738 (O); fax: +86 20 347377068 (H). E-mail address: [email protected] 0749-6036/$ - see front matter Ó 2011 Published by Elsevier Ltd. doi:10.1016/j.spmi.2011.06.006
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operating in the blue and ultraviolet wavelength regions due to their wide adjustable directed bandgaps [1–3]. In addition, semiconductor nanocrystals and quantum dots (QDs) are nanostructures where the three-dimensional confinement of carriers results in a discrete energy spectrum [4–6]. This can further improve the performance of optoelectronic devices, such as low-threshold lasers and light polarization insensitive detectors. With the technique advancement of the crystal-growth, such as metal-organic chemical vapor deposition, the molecular-beam epitaxy and hydride vapor phase epitaxy, the ZnO-based nanocrystals and QDs can be fabricated in experiments [6–12], and it provides explicit and important research object of ZnO-based nanostructures. Hence the investigation of various physical properties in quasi-0-dimensional (Q0D) ZnO-based quantum structures has become a hot topic during the last decade [1–3,6–15]. Among the experimental and theoretical investigations on the Q0D ZnO-based nanostructures, the ZnO nanocrystals embedded in the MgO matrix attracted substantial attention [6–12,16–18]. This is mainly due to the fact that the capping MgO shell outside of the ZnO core can not only reduce the surface-related defects and enhance the stability of nanocrystal, but also confine the charge carriers into the core region due to the wider band offset potential, which greatly improve the luminescent characteristics such as quantum efficiency and photostability [6–10]. Zeng et al. [6] grown ZnO/MgO QDs based on a metal-organic chemical vapor deposition method. And a remarkable enhancement of free exciton emission from the QDs after the growth of MgO layer was observed in the photoluminescence (PL) measurement. Lee and cooperators [7] found that, in contrast to the bare ZnO nanocrystal, the luminescent efficiency of green emission in core/shell structure of ZnO/MgO nanocrystals increases by 30%. Bera and coworkers’ experiment [8] reveals that the ZnO/MgO QDs, synthesized by a sol– gel process, exhibited a quantum yield of 13% compared with less than 5% for bare ZnO QDs. By a sequential preparative procedure and capped with carboxymethyl b-cyclodextrin cavities, Rakshit and Vasudevan [9] synthesized the core-shell ZnO/MgO nanocrystals. Their experiment shows that MgO on ZnO nanocrystals prevented the aggregation of the nanoparticles and formation of nonradiative recombination sites, which results in a more intense and stable green photoluminescence. The bound electronic states, donor and acceptor impurity states, and excitonic states in the Q0D ZnO-based core-shell structure nanocrystals and QDs have been widely investigated [6,9,16–18]. However, only limited data is available on the polar optical phonon states and their dispersive spectra in wurtzite systems [19–22]. The phenomena of phonon replicas in the emission spectra, the homogeneous broadening of excitonic line width and the relaxations of hot carriers to the fundamental band edge are directly relative to the lattice vibration of semiconductor materials [23,24]. Theories and experiments reveal that not only the carriers, but also the lattice vibrations (phonon modes) are influenced greatly by the heterostructures of low-dimensional quantum systems [25–27]. Moreover, phonon modes have quite important influence on the optoelectronics and electronics properties of wurtzite ZnO(GaN)-based low-dimensional quantum structures [28–34]. Hence it is necessary and important to study the phonon modes in ZnO-based nanocrystal structures. It is well known that ZnO material usually crystallizes in the hexagonal wurtzite structure, whose physical properties behavior anisotropic in space. However, according to recent reports [35], if doping concentration x in MgxZn1xO is over than 0.36, the MgZnO ternary compound will change from wurtzite structure to cubic zinc-blend structure. The experiment of Lee et al. [7] clearly reveals that the core ZnO behaviors as hexagonal wurtzite phase, while the capping MgO displays as zinc-blende phase. For simplicity, we will investigate a wurtzite ZnO nanocrystal embedded in zinc-blende MgO matrix. Even though, the phonon modes in wurtzite/zinc-blende ZnO/MgO nanocrystal are more complicated than those the pure zinc-blende GaAs-based confined structures [36–39] due to the wurtzite anisotropy of ZnO material. In fact, within the framework of the dielectric continuum model (DCM), the optical phonon vibrating modes in pure zinc-blende GaAs-based Q0D QDs [36–39] and pure wurtzite GaN- (ZnO-) based QDs [32–34] have been deduced and analyzed. Klein et al. [36] studied the size dependence of electron-phonon coupling in cubic CdSe nanospheres. Roca et al. [37] investigated the polar optical vibration modes in a GaAs/AlAs sphere QD. Li and Chen [38] derived the polar IO phonon modes of a GaAs/AlGaAs cylindrical QD, and two types of IO modes (i.e. the top IO modes and the surface IO modes) are found in the Q0D quantum structures. Fonoberov and Balandin [19–22] analyzed the polar optical phonon modes in pure wurtzite ZnO/MgZnO nanostructures. Recently, we extended the phonon mode works to the pure wurtzite Q0D GaN-based and ZnO-based cylindrical QD
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structures, and that four types of polar mixing optical phonon modes may coexist in the Q0D wurtzite quantum systems has been predicted [32–34]. However, to the best of our knowledge, the integrated optical phonon modes and their dispersive properties in wurtzite/zinc-blende ZnO/MgO Q0D nanocrystals and QDs have not been investigated. The present work is just to fill a gap of optical phonon mode theories in wurtzite/zinc-blende ZnO/MgO nanocrystals. We will study the fully phonon vibration modes and dispersive spectra in a Q0D sphere ZnO/MgO nanocrystal structures in the present paper. The main accomplishments and significance of this work can be summarized into the three points as follows. (i) Based on the solutions of the Laplace equation in wurtzite spherical ZnO nanocrystal embedded in zinc-blende MgO matrix, it is confirmed that there are two types of polar optical phonon modes, i.e. the IO modes and the QC modes coexisting in the wurtzite/zinc-blende ZnO/MgO Q0D quantum structure. And the frequency ranges of the two types of phonon modes are characterized. (ii) Based on the DCM and Loudon’s [40] uniaxial crystal model, the explicit phonon states and dispersive equations of the IO and QC modes in the wurtzite/zinc-blende spherical nanocrystals are given. (iii) Numerical calculations on the dispersive relations of IO and QC modes in the ZnO/MgO nanocrystals are performed. It is observed that the dispersive frequencies of IO and QC modes are the discrete functions of orbital quantum numbers l and azimuthal quantum numbers m. A pair of given l and m correspond to one IO mode, but to more than one branches of QC modes. A detailed comparison with pure Q0D zinc-blende spherical QD and pure wurtzite cylindrical QD systems are carried out, and relative reasons resulting in the difference are analyzed in depth from both viewpoints of physics and mathematics. The remaining sections of the paper is organized so: in Section 2, the uniform phonon states of IO and QC phonon modes and their dispersive equations in the structures are presented. In Section 3, the numerical calculations on the dispersive frequency of the two types of phonon modes are carried out and discussed; and finally, we summarize the mainly conclusions obtained in the paper in Section 4. 2. Theory Let us consider a wurtzite-core/zinc-blende-shell spherical nanocrystal with radius R (referring to Fig. 1). The z-axis is taken along the direction of the c-axis of the wurtzite-core material and denote the axial- (radial-) direction as z (t). Within the framework of the macroscopic DCM and Loudon’s uniaxial crystal model, the electrostatic potential U1(r) of the polar vibrating modes within the wurtzite spherical nanocrystal is determined by the second-order differential functions (1) under the prolate spheroidal coordinate, i.e.
1 R2 ðn2 g2 Þ
^n
@ @ @ @ ðn2 1Þ þ g^ ð1 g2 Þ U1 ðn; g; uÞ @n @n @g @g
@2 U1 ðn; g; uÞ ¼ 0: R2 ðn2 1Þð1 g2 Þ @ u2
t
ð1Þ
And the electrostatic potential U2(r) outside of the wurtzite spherical nanocrystal is given by the second-order differential functions (2) under the spherical coordinate, i.e.
d
) 1 @ 1 @ @ 1 @2 2 @ r 2 sin h þ U2 ðr; h; uÞ ¼ 0: 2 r 2 @r @r r sin h @h @h r2 sin h @ u2
ð2Þ
where ed(x) is the dielectric function of the zinc-blende dielectric matrix out side of the wurtzite core ^¼^ ^ ,t,z) in Eq. (1) is the dielectric function of the wurtzite core matematerial. The function p^ ðxÞ (p n; g ^-direction. The dielectric functions t(x) and z(x) along the t- and z-directions of the rial along the p wurtzite core material as well as the dielectric function of the zinc-blende matrix material are respectively given by
v ðxÞ ¼ 1 v
x2 x2v ;L ; v ¼ d; t; z: x2 x2v ;T
ð3Þ
L. Zhang / Superlattices and Microstructures 50 (2011) 242–251
245
z /c-axial MgO
ZnO R o
y
x Fig. 1. Schematic view of the wurtzite ZnO spherical nanocrystal with radius R embedded in zinc-blende MgO matrix. The zaxis is taken along the c-axis direction of the wurtzite ZnO material.
Fig. 2. The function g(x) for ZnO material as a function of x. The characteristic frequencies of ZnO and MgO materials divide the frequency axis into seven ranges, which are labelled as range I, II,. . ., and VII from left to right. The subscripts ‘‘1’’ and ‘‘2’’ correspond to the ZnO and MgO, respectively.
In Eq. (3), xz,L, xz,T, xt,L and xt,T are the zone center characteristic frequencies of A1(LO), A 1(TO), E1(LO), and E1(TO) modes of wurtzite bulk material, respectively. In order to solve the Eq. (1) conveniently, we define a function g(x) as
gðxÞ ¼ z ðxÞ=t ðxÞ:
ð4Þ
The meanings of this function and its influence on the phonon potential properties of vibration or decaying will be discussed later in Fig. 2. The phonon potentials U(r) of the wurtzite-core/zincblende-shell nanocrystal, namely the solutions of Eqs. (1) and (2), which are finite everywhere inside the nanocrystal and vanishes far away from the nanocrystal can be found analytically as follow:
UðrÞ ¼
8
C
Pm ðnÞ m l P ð ðn0 Þ l Pm l
gÞeimu ;
lþ1
R riþ1
imu Pm ; l ðcos hÞe
within outside
ð5Þ
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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where n0 ¼ 1= 1 gðxÞ, P m l ðxÞ are associated Legendre polynomials of the first kind, and integers l (l P 0) and m (jmj 6 l) are the orbital and azimuthal quantum numbers of the phonon modes in Q0D nanocrystals, respectively. In terms of the relationships of spheroidal/spherical coordinates with the Cartesian coordinates, it is easy to get the relations between the coordinates (n, g) and the coordinates (r, h), i.e.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 þ y2 ¼ r sin h ¼ R 1=gðxÞ 1 ðn2 1Þð1 g2 Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z ¼ r cos h ¼ R 1 gðxÞng:
ð6Þ
It is obvious that the interface of the nanocrystal is described by r = R in spherical coordinates. Based pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi on the Eq. (6), the nanocrystal interface also can be denoted as n0 ¼ 1= 1 gðxÞ and g0 = cosh in spheroidal coordinates. Via the boundary conditions (BC) in spheroidal/spherical coordinates, it is found that the continuum condition of phonon potential functions at the interface of wurtzite nanocrystal, namely U1 ðrÞn¼n0 ;g¼cos h ¼ U2 ðrÞr¼R , is automatically satisfied. Next we use the continuum condition of electric displacement vector D perpendicular to the interface at r ¼ R to obtain the dispersive equation of phonon modes in the spherical nanocrystal. In terms of the relation (6) and the gradient operator in prolate spheroidal coordinates:
O¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ ^ 1 1 @ u @ ^ 1 g2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n^ n2 1 þ g þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2 @u R @n @g 2 2 R n 1 1 g n g
ð7Þ
one can get
t ðxÞgðxÞ @ U1 ðrÞ D1 ðn; g; uÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 1 gðxÞ @n n¼n
ð8Þ
0 ;g¼cos h
and
D2 ðr; h; uÞ ¼ d ðxÞ
@ U2 ðrÞ : @r r¼R
ð9Þ
Substituting Eq. (5) into the above Eqs. (8) and (9), and letting D1 = D2, we can get the follow formula, i.e.
z ð xÞ
m m 1 ðl þ 1Þðl þ mÞP m l1 ðn0 Þ lðl m þ 1ÞP lþ1 ðn0 Þ d ðxÞð2l þ 1Þðl þ 1Þðn0 n0 ÞP l ðn0 Þ ¼ 0: ð10Þ
This Eq. (10) is just the dispersive equation of the polar optical phonon modes in a wurtzite-core/zincblende-shell spherical nanocrystal. Substituting the dielectric functions Eq. (3) into Eq. (10), the dispersive frequencies x of the phonon modes can be worked out as functions of the quantum numbers l and m. It is interesting to note that, as the anisotropy of the dielectric function in wurtzite core material is neglected, (i.e. z(x) = t(x) = 0(x), thus g(x) = 1 and n0 ? 1), the relation (11) can be obtained:
x
x
@Pm l l ðnÞ ¼ 0 ðxÞ: R x @n
t ð Þgð Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g!1;n¼n0 !1 RP m ðn Þ 1 gð Þ 0 l
lim
ð11Þ
Using this relation, the dispersive Eq. (10) will reduced to the form as
l0 ðxÞ þ ðl þ 1Þd ðxÞ ¼ 0;
ð12Þ
which is just the dispersive equation of IO phonon modes in isotropic spherical QD [36,37]. This also illustrates that the dispersive equation obtained here in wurtzite-core/zinc-blende-shell nanocrystal is more general than the previous ones in cubic or pure zinc-blende QDs.
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3. Numerical results and discussions In order to get a clear picture for the types of phonon modes and their dispersive properties in the wurtzite/zinc-blende Q0D nanocrystal system, a numerical calculation is performed on a wurtzite/ zinc-blende ZnO/MgO spherical nanocrystal. The physical parameters of the materials used in our calculation are listed in Table 1 [41–43]. Before analyzing the dispersive properties of phonon modes, let us first discuss the types of phonon modes and the frequency ranges of these phonon modes in ZnO/MgO nanocrystal system. Shi et al. [44] pointed out that five types of polar optical phonon modes including the IO modes, QC modes, propagating modes (PR), half-space (HS) modes and exactly confined (EC) modes may coexist in pure wurtzite low-dimensional quantum heterostructures. Apart from the EC modes, whose frequencies x = xtL, all the other phonon modes are dispersive. In terms of the data given in Table1, the function g(x) of ZnO material as a function of x is shown in Fig. 2. The characteristic frequencies of ZnO and MgO materials are labelled on the abscissa. From the figure, it is observed that the characteristics frequencies of ZnO and MgO divide the frequency axis into seven ranges. For convenience, these ranges are labelled as range I, II, . . ., and VII from left to right. In each frequency range, the signs for the dielectric functions of ZnO and MgO semiconductors and g(x) function are certain (referring to Table 2). In m terms of the feature of associated Legendre function Pm l ðnÞ of the first kind, it is known that P l ðnÞ is a vibrating function as 0 < n < 1 (i.e. g(x) < 0). But as n > 1 (i.e. 0 < g(x) < 1) or in > 0 (i.e. g(x) > 1), Pm l ðnÞ behaves as decaying function. Thus the dispersive Eq. (10) gives the unified forms for the full phonon modes in wurtzite-core/zinc-blende-shell spherical nanocrystals. This is obviously different from the situation of pure wurtzite cylindrical quantum wires and QDs [32–34]. From the Fig. 2 or Table 2, we observe that the function g(x) takes negative values in ranges II, III and V, and it takes positive values in the ranges I, IV, VI and VII. Moreover, the values of g(x) are lower (over) than 1 in ranges I and VI (IV and VII). Based on the feature analysis of P m l ðnÞ above, it is found Pm l ðnÞ behaves as decaying (vibrating) waves in frequency ranges II, III and V (I, IV, VI and VII). Based on the properties of IO phonon modes, namely behaving as decaying waves in the core material and the dielectric matrix, it is easy to find that the IO modes of the system may exist in the ranges I, IV, VI and VII. The QC modes behave as oscillating waves in the core material, and as decaying waves in the dielectric matrix, which are quite similar to the electronic wavefunctions in quantum wells with finite-high potential barrier [44,45]. Thus the QC modes may appear in these three ranges II, III and V. Due to the dielectric matrix being isotropic zinc-blende semiconductor, only the decaying waves and exact confined waves can appear in this region. Thus there are no PR and HS modes in ZnO/ MgO nanocrystals. But IO and QC modes can exist in this structures because both the modes demand
Table 1 Zone-center energies (in meV) of polar optical phonons, dielectric constants of wurtzite ZnO material and zinc-blende MgO dielectric matrix [41–43]. ZnO MgO
xtT
xtL
xzT
xzL
z1
t1
47.2
71.7
50.7
73.6
3.78
3.7
xLO
xTO
d1
d0
90
49.32
2.96
9.86
Table 2 Signs of the functions t(x), z(x), g(x) and MgO semiconductors.
ZnO
MgO
d(x) in seven frequency ranges divided by the characteristic frequencies of ZnO and
Ranges
I
II
III
IV
V
VI
VII
t(x) z(x) g(x)
+ + +
+
+
+
+
+ + +
+ + +
d(x)
+
+
+
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decaying waves in the dielectric matrix. Taking into account the condition of solution (the dielectric functions of ZnO and MgO cannot take positive or negative signs simultaneously), the dispersive Eq. (10) of IO and QC modes have no solution in the ranges I, IV and VII. Thus the QC phonon modes can appear in the three ranges: II, III and V, while the IO phonon modes can only exist in the frequency range VI. Moreover, the n values of P m l ðnÞ are between 0 and 1, namely 0 < n < 1 in frequency range VI for IO phonon modes in the system. Fig. 3 plots the dispersion frequencies ⁄x of the full phonon modes as functions of the orbital quantum numbers l and azimuthal quantum numbers m in the ZnO/MgO nanocrystals. From the figure, it is seen that the characteristic frequencies xtT1, xTO2, xzT1, xtL1 and xzL1 (dash dot lines in the figure) ascertain three frequency ranges, which are the frequency ranges of QC modes. The topmost range falls into [xzL1,xLO2], which is the frequency range of the IO phonon modes. These observations are completely consonant with the analysis of Fig. 2 and Table 2. All the dispersive frequencies ⁄x of IO and QC modes are the discrete functions of quantum numbers l and m. This is quite analogous to the electronic energy levels of Q0D QDs [45]. The discrete dispersive spectra of phonon modes in the Q0D nanocrystals also differ obviously from those in quasi-2-dimensional (Q2D) quantum wells and quasi-1-dimensional (Q1D) quantum wires structures [32,33,44]. The phonon dispersive frequencies in Q2D (Q1D) quantum well (wire) systems in general, are the continuum functions of free phonon wave-numbers in planes (axial directions). For a given orbital quantum number l, there are l + 1 branches of IO phonon modes. They corresponds to the IO phonon modes with azimuthal quantum numbers m = 0, 1, 2, . . ., l, respectively. For a certain m, the dispersive frequencies of IO modes are the monotonic and decreasing function of l (referring the dash lines with symbols, such with squares, circles and triangles). With the increase of l, the dispersions of these modes become weaker and weaker. Moreover, the frequency intervals
Fig. 3. Dispersion frequencies ⁄x of the IO and QC phonon modes as functions of the orbital quantum numbers l and azimuthal quantum numbers m in the ZnO/MgO nanocrystals.
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of two IO modes with adjacent azimuthal quantum numbers (i.e. m and m + 1, m = 0, 1, 2, . . .) decrease with the increase of l. These are the common features of phonon modes in quantum confined systems [27,32–34,42,44]. Though the dielectric function d(x) takes different signs (referring to Table 2) in the frequency subranges of [xtT1, xTO2] and [xTO2, xzT1], the two subranges constitute a integrated range for QC phonon modes. And some dash curves, such as one of the line with squares, and that with circles, span and extend into the two subranges. Just as the IO modes, all the curves of QC modes in the two subranges are the monotonic and decreasing function of l. It is observed that, for given l and m, there is one branch or more than two branches of QC modes in the two subranges. For example, there are two branches of QC modes with l = 2 and m = 1, and quantum numbers l = 4 and m = 1 correspond to three branches of QC modes. This is different from the situation of IO modes. For IO modes in the systems, a pair of given l and m correspond to one IO mode. In addition, we notice that there is no QC modes for orbital quantum number l = 1 in the frequency range of [xtL, xzL1]. The QC modes begin to appear in this frequency range as l is over than 1. This characteristic is similar to the case of reducing behavior of phonon modes in wurtzite GaN-based quantum structures [32,44]. And it originates from the anisotropy of the wurtzite crystal structures of ZnO core material. The dispersive dash-curves of QC modes in this range are the monotonic and increasing function of l, which distinguishes the cases of IO modes and QC modes in the integrated range of [xtT1, xzT1]. At last, it should be pointed out that the dispersive frequencies of IO and QC modes have no relation to the size R of the Q0D nanocrystal, which can be seen clearly from the dispersive Eq. (10). In fact, this result has been verified by the recent experimental measurement of Zhang and cooperators [46]. Since the theoretical approach of this work is similar to the one developed by Fonoberov and Balandin [19–21], it is necessary to compare the results of the present work with those of Fonoberv and Balandin. In contrast to the freestanding wurtzite ZnO nanocrystal [21], the introduce of the barrier MgO material has obvious influence on the dispersive spectra of phonon modes in the Q0D nanocrystal structures. The characteristic frequency xTO, MgO divided the frequency range of QC modes in freestanding ZnO nanocrystal into two subranges (namely ranges II and III in Fig. 2). The phonon range of IO modes [xtT,ZnO, xzL, ZnO] in freestanding ZnO nanocrystal is forbidden in ZnO/MgO nanocrystal. And a new frequency range [xzl, ZnO, xLO, MgO] of IO modes appears in the wurtzite ZnO nanocrystal embedded in zinc-blende MgO matrix. Moreover, relative to the Fig. 1 of Ref. [20], the Fig. 2 of this work explicitly gives not only the signs of the function g(x), but also the specific value of g(x). In particular, the magnitude relationship between g(x) and 1 is displayed clearly as a function of x, which is quite useful for analyzing the forms and the decaying/vibrating properties of Pm l ðnÞ functions. Fonoberov and Balandin’s works numerically discussed the influences of quantum structures (ratio of spheroidal semi-axes, a/c and c/a) [20] and nonpolar dielectric matrix (dielectric constant d of exterior matrix varies from 1 to 4) [19] on the dispersive spectra of phonon modes in freestanding wurtzite ZnO and ZnO/MgZnO nanocrystals. The present work mainly analyzed the dependance of dispersive spectra on the orbital quantum numbers and azimuthal quantum numbers in wurtzite ZnO-core embedded in zinc-blende MgO dielectric matrix. Hence the present paper can be looked as a complementarity and extension of previous works [19–21].
4. Conclusions In conclusion, we have studied and discussed the polar optical phonon modes in a wurtzite-core/ zinc-blende-shell Q0D nanocrystal structure in the present work. The analytical phonon states of phonon modes are obtained by means of the DCM and Loudon’s uniaxial crystal model. It is confirmed that, via solving the Laplace equations in wurtzite and zinc-blende crystals under the spheroidal and spherical coordinates, there are two types of polar phonon modes, namely the IO modes and the QC modes in wurtzite/zinc-blende cylindrical nanocrystal structures. The unified phonon states (Eq. (5)) and dispersive equation (Eq. (10)) for the IO and QC modes are also obtained. Numerical calculations on a wurtzite ZnO spherical nanocrystal embedded in zinc-blende MgO matrix are performed. The frequency ranges of the phonon modes in the ZnO/MgO nanocrystal are analyzed in detail. It is found that the IO modes of the structure only exists in one frequency range, and the QC
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modes may appear in three frequency ranges. The dispersive frequencies of the IO and QC modes as functions of the orbital quantum number l and azimuthal quantum number m are plotted, and detailed features for these curves are discussed. It is found that the dispersive frequencies are the discrete functions of quantum numbers l and m, which is analogous to electronic energy levels in Q0D QDs [45], and differs obviously from the dispersive spectra of Q2D quantum wells [42,44] and Q1D quantum wires [32,33]. A pair of given l and m correspond to one IO mode. But more than one branches of QC may appear for a given orbital and azimuthal quantum numbers. The present theoretical and numerical results can be used to analyze and discuss the dispersive properties, Raman spectra and other relative properties of wurtzite/zinc-blende Q0D nanocrystal systems. We hope that the present work will stimulate further theoretical and experimental investigations of lattice dynamical properties, as well as device applications based on the wurtzite/zinc-blende Q0D nanocrystal structures. Acknowledgments Project was jointly supported by the NNSYF of China (Grant No. 60906042), Yangcheng Scholar Project (Grant No. 10B010D) and STPAA (Grant No. 10B001) of Guangzhou City. The author would like to acknowledge the valuable guidance and discussion of Prof. J.J. Shi of Peking University. References [1] Ü. Özgür, I. Alivov, C. Liu, A. Teke, M.A. Reshchikov, S. Dogan, C. Avrutin, S.-J. Cho, H. Morkoç, J. Appl. Phys. 98 (2005) 041301. [2] Z.L. Wang, Material Today 10 (2007) 20. [3] A. Janotti, C.G.V. de Walle, Rep. Prog. Phys. 72 (2009) 126501. [4] A.D. Yoffe, Adv. Phys. 50 (2000) 208. [5] A. Nakamura, K. Okamatsu, T. Tawara, H. Gotoh, J. Temmyo, Y. Matsui, Jpn. J. Appl. Phys. 47 (2008) 3007. [6] Y.J. Zeng, Z.Z. Ye, F. Liu, D.Y. Li, Y.F. Lu, W. Jaeger, H.P. He, L.P. Zhu, J.Y. Huang, B.H. Zhao, Crystal Growth Design 9 (2009) 263. [7] J.J. Lee, J. Bang, H. Yang, J. Phys. D: Appl. Phys. 42 (2009) 025305. [8] D. Bera, L. Qian, P.H. Holloway, J. Phys. D: Appl. Phys. 41 (2008) 182002. [9] S. Rakshit, S. Vasudevan, Nano 2 (2008) 1473. [10] J.G. Ma, Y.C. Liu, C.L. Shao, J.Y. Zhang, Y.M. Lu, D.Z. Shen, X.W. Fan, Phys. Rev. B 71 (2005) 125430. [11] G.N. Panin, A.N. Baranov, Y.-J. Oh, T.W. Kang, Current Appl. Phys. 4 (2004) 647. [12] G.N. Panin, A.N. Baranov, I.A. Khotina, T.W. Kang, J. Kor. Phys. Soc. 53 (2008) 2943. [13] K.A. Alim, V.A. Fonoberov, M. Shamsa, A.A. Balandin, J. Appl. Phys. 97 (2005) 124313. [14] K.K. Zhuravlev, W.H.H. Oo, M.D. McCluskey, J. Huso, J.L. Morrison, L. Bergan, J. Appl. Phys. 106 (2009) 013511. [15] C.J. Pan, K.-F. Lin, W.T. Hsu, W.-F. Hsieh, J. Phys. Cond. Matter 19 (2007) 186201. [16] S.Y. Wei, Q. Chang, Z. Zeng, Current Appl. Phys. 11 (2011) 16. [17] X. Zhao, S.Y. Wei, C. Xia, G. Wei, J. Lumin. 131 (2011) 297. [18] J.A. Davis, C. Jagadish, Laser Photon. Rev. 3 (2009) 85. [19] V.A. Fonoberov, A.A. Balandin, Phys. Rev. B 70 (2004) 233205. [20] V.A. Fonoberov, A.A. Balandin, J. Phys. Cond. Matter 17 (2005) 1085. [21] V.A. Fonoberov, A.A. Balandin, Phys. Stat. Sol. C 1 (2004) 2650. [22] V.A. Fonoberov, A.A. Balandin, J. Nanoelectron. Optoelectron 1 (2006) 19. [23] A.A. Balandin, J. Nanosci. Nanotech. 5 (2005) 1015. [24] X.B. Zhang, T. Taliercio, S. Kolliakos, P. Lefebvre, J. Phys.: Cond. Matter 13 (2001) 7053. [25] K. Sood, J. Menendez, M. Cardona, K. Ploog, Phys. Rev. B 54 (1985) 2111–2115. [26] A. Tanaka, S. Onari, T. Arai, Phys. Rev. B 45 (1992) 6587. 47 (1993) 1237. [27] H.J. Xie, C.Y. Chen, B.K. Ma, Phys. Rev. B 61 (2000) 4827. [28] A. Ashrafi, J. Appl. Phys. 107 (2010) 123527. [29] V.I. Kushnirenko, I.V. Markevich, L.V. Borkovska, B.M. Bulakh, Phys. Stat. Sol. C 7 (2010) 1605. [30] J.M. Wesselinowaa, A.T. Apostolov, J. Appl. Phys. 108 (2010) 044316. [31] V.A. Fonoberov, A.A. Balandin, J. Vac. Sci. Technol. B 22 (2004) 2190. [32] L. Zhang, J. Shi, T.L. Tansley, Phys. Rev. B 71 (2005) 245324. [33] L. Zhang, J.J. Shi, S. Gao, Semicond. Sci. Technol. 23 (2008) 045014. [34] L. Zhang, Turk. J. Phys. 34 (2010) 123. [35] C.H. Ahn, S.K. Mohanta, B.H. Kong, H.K. Cho, J. Phys. D: Appl. Phys. 42 (2009) 115106. [36] M.C. Klein, F. Hache, D. Ricard, C. Flytzanis, Phys. Rev. B 42 (1990) 11123. [37] E. Roca, C. Trallero-Giner, M. Cardona, Phys. Rev. B 49 (1994) 13704. [38] W.S. Li, C.Y. Chen, Physica B 229 (1997) 375. [39] L. Zhang, H.J. Xie, C.Y. Chen, Phys. Rev. B 66 (2002) 205326. [40] R. Loudon, Adv. Phys. 13 (1964) 423. [41] F. Güell, J.O. Osso, A.R. Goni, A. Cornet, J.R. Morante, Superlatt. Microstruc. 45 (2009) 271.
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